NATIOrAl BUREAU OF

Qa ooTfansfer of Radiation

Radiative Transfer and the

Spectra of Celestial Bodies

Prepared in collaboration with

D. G. Hummer

from a draft translation by Eileen Weppner

Joint Institute for Laboratory Astrophysics

National Bureau of Standards

and

University of Colorado

Boulder, Colorado 80302

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

Issued November 1973

051.

>38 in

V. V. Ivanov

English language edition of
Library of Congress Catalog Number: 73-600273

National Bureau of Standards Special Publication 385

Nat. Bur. Stand. (U.S.), Spec. Publ. 385, 480 pages (Nov. 1973) CODEN: XNBSAV

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

(Order by SD Catalog No. C13.10 :385). Price $7.45. Stock Number 0303-01 188
ABSTRACT

This book is a revised and somewhat extended version of V. V. Ivanov's

Radiative Transfer and the Spectra of Celestial Bodies, published in Moscow

in 1969. The principal subject is the transfer of radiant energy through a gas composed ot atoms with two discrete levels. Although the emphasis of the

book is on analytical methods, extensive numerical and graphical results are presented

Key words: Astrophysics; energy loss; photons; radiative transfer;

spectral line profiles; stellar atmospheres.
FOREWORD TO THE ENGLISH EDITION

This book is a revised and somewhat extended version of the author's

Radiative Trans fer and the Spectra of Celestial Bodies published in Russian

in 1969. The book is devoted to a rather limited problem of radiative transfer theory, namely radiative transport of excitation in a tenuous gas composed

of two-level atoms. Although this problem is highly restricted, it is the

heart of the whole theory of transfer of line radiation.

Our approach may seem old-fashioned, since it is mainly analytical

rather than numerical. Although numerical methods play an essential and

constantly increasing role in transport theory, the analytical approach must

not be underestimated. In the final analysis, a computer solution of a problem without a prior investigation of the nature of its solution is only

a mathematical analog of a physical experiment. Analytical methods provide

the basis for radiative transfer theory, as they do for any theory.

We use the now-classical analytical methods of transport theory developed

mainly by E. Hopf, V. A. Ambar ts umi an , S. Chandrasekhar and V. V. Sobolev.

The use of these methods rather than Case's method of the normal mode expansion may also seem somewhat old-fashioned. The majority of the results presented in this book may be readily obtained by Case's method. It seems,

however, that the asymptotic solutions of line transfer problems are more directly obtained by the classical methods.

The manuscript of the original Russian edition of the book was completed

in June 1968. In preparing the English language edition I have revised the

whole manuscript and added more than 100 pages, mainly to account for results

that were published since 1968. Sections 3.6, 4.5, 6.9, 6.10, 8.10 and 8.11

are new, and sections 2.6, 3.8, 3.9, 4.4, 7.5, 8.5, 8.6 and 8.9 are substantially enlarged; minor additions have been made throughout the book.

I am deeply indebted to Dr. D. G. Hummer for his efforts to organize the

publication of the Engl ish- language version of this book and for his excellent

editorial work. Dr. Hummer also succeeded in fitting smoothly into the translated text of the original book all of the additions supplied by the author in

a rather non-colloquial form of the English language.

V. Ivanov

Leningrad

December 1971
FOREWORD TO THE RUSSIAN EDITION

With more than a dozen books on transfer theory now available, the need

for yet another might well be questioned. However, in recent years a new and

important area of transfer theory, the scattering of radiation within spectral

lines, has been developing rapidly. The discussions of this topic in the

existing monographs do not reflect its present state; yet this area will

undoubtedly continue to grow in importance, as it provides the theoretical

basis for the interpretation of line spectra of optically thick plasmas.

That radiative transfer theory for spectral lines has been developed

primarily by astrophysicists is not accidental, of course, but reflects the

important role that radiation transfer plays in astrophysics. In fact, much

of our understanding of celestial objects is based on the study of their line

spectra. Unfortunately specialists in transfer theory are not yet able to

solve the seemingly simple problems that have long been the concern of astrophysicists. However, there is now reason to hope that in the foreseeable

future we shall have the reliable methods so urgently needed by astrophysicists

for computing the intensities and profiles of spectral lines. These methods

will evolve from those currently in use, primitive as they may be in many

respects. The recent successful applications of transfer theory, especially

through numerical methods, to spectral lines provide an encouraging sign.

This book avoids concrete as trophys ical problems. However, most of the

model problems solved here arose in connection with the study of spectra of

j celestial objects. Although the models considered were usually too crude to

/ be used directly in interpreting the observations, they are valuable in helping us to understand the physics of the phenomenon and to develop a feeling

for the problem. Further, models of this kind allow one to estimate the

accuracy and range of validity of the approximations and numerical methods

H used in transfer theory.

I This book is addressed primarily to theoreticians, although certain of

i its topics will be of interest to observers as well. The sequence of thought,

in essence, is "from the physics of the problem, to mathematics, and back

again to physics." As to the level of mathematical development, two criticisms

i are possible: some readers will find the discussion insufficiently rigorous,

j while others will feel that the attention to purely mathematical questions

; has been excessive. Perhaps the best description of the point of view adopted

j here is to say that the book gives the solution of a number of problems of

;j applied mathematical physics at a physical level of rigor.
vi FOREWORD TO THE RUSSIAN EDITION

The selection o£ material was largely dictated by the author's interests.

Considerable space has been given to results obtained in recent years by a group of astrophysicists at the University of Leningrad, and an appreciable

amount of this material is being published here for the first time.

The list of references is not intended to be exhaustive. From the vast

literature we have referred only to those papers bearing directly on the

problems chosen for inclusion in the book. Decisions on questions of notation proved to be far from easy. The problems considered are intrinsically

quite cumbersome as their solutions involve many parameters. Lack of agreement on notation among various authors further complicated the situation.

For the reader's convenience, a table of the basic symbols is given at the

end of the book, including the number of the page on which each appears for

the first time.

Equations are given two numbers within a chapter. The first number is

that of the section of the chapter in which the equation appears; the second

reflects its order within that section. For example, (6.56) in Chapter II indicates equation 56 of section 2.6. In referring to an equation in another

chapter, the number of that chapter precedes the "internal" number. Thus,

equation (6.56) of Chapter II will be referred to in any other chapter as (2.6.56).

I wish to expressed my profound appreciation to Professor V. V. Sobolev,

who has played a crucial role in shaping my views on the problems of transfer

theory. He has read the book in manuscript and many of his suggestions have

been incorporated into the final draft.

For several years the author has worked closely with Dr. D. I. Nagirner,

who has contributed to the formulation and clarification of many important

problems. Dr. Nagirner has provided many of the results appearing in this

book, including the majority of previously unpublished numerical data. My

correspondence with Dr. D. G. Hummer (USA) has also played a significant role

and his permission to use a number of important unpublished results has been

most useful.

Mr. H. Domke of the Potsdam Observatory assumed the difficult task of

checking the formulae. Invaluable assistance in drawing the figures and

preparing the manuscript for the printer was rendered by T. M. Maksimova,

I. I. Lebedeva and L. F. Gromova.

To all of these people I extend my warmest thanks.

V. Ivanov
EDITOR'S PREFACE

This book had its inception in a letter written by Dr. Ivanov to me in

1967, asking permission to use some o£ my unpublished material in a book on

line formation that he was writing. In my reply authorizing the use o£ that

material, I also offered to arrange for the translation and publication of

an English- language version of his book. At that time the need for a wellwritten up-to-date book on the radiative transfer problems of spectral line

formation was acute. I felt confident from my knowledge of Dr. Ivanov's

published work and from our correspondence extending back to 1963 that his

book would meet that need admirably. In this way I could also abandon, with

a clear conscience, my own plans to write a book on this subject.

After the book appeared in 1969, I contacted a large number of publishers

but found little interest in the translation as a commercial venture. At the

encouragement of the then- Chairman of JILA, Dr. Peter Bender, and with the

I assistance of the Chief of the Laboratory Astrophysics Division of the

I National Bureau of Standards, Dr. Steven J. Smith, I arranged for the publicaJ tion of the book by the Government Printing Office. As I was by then too busy

lj to translate the book without assistance, we arranged for Eileen Weppner to

provide me with a draft translation and for Alice Levine to assist in the editorial work, to design the book and to oversee the typing and final assembly

of the camera copy. The draft translation was prepared and edited in the

spring and early summer of 1971. This was sent to Dr. Ivanov who made extensive corrections and additions. The final editing was begun in the spring of

1972 and was completed on New Year's Day, 1973. The last corrected proofs were received from Dr. Ivanov in July, 1973.

A word on the design of this book may be in order. Having at our disposal the newly-developed IBM Selectric II (a 10- and 12-pitch typewriter),

we endeavored to use its flexibility to produce a design that is both

attractive and readable. The large page format was selected to minimize

the effect of the lack of right- justification and to allow most of the

longer equations to be typed on one line. As a considerable amount of

thought and experimentation has been invested in this design, we hope that

it may be of value to other people wishing to publish technical books on

a very low budget.

A venture of this magnitude depends critically on the assistance of many

people. I am deeply indebted to Eileen Weppner for supplying a draft translation in a very short time, to Leslie Haas and Dorothy Harris for typing the

first draft, to Olivia Briggs and Harriet Ortiz for quickly and accurately

typing the camera copy and to Lorraine Volsky, head of the JILA editorial

service, for her advice and assistance in many matters. I would like to
vi i i EDITOR'S PREFACE

thank Drs . Bender and Smith for providing funds from the ARPA contracts under

their control to pay for the draft translation and the editorial help.

Although Dr. Smith allowed me to work on this project as an official duty of

my NBS position, it is probably not surprising that most of the editorial

work was done at home in the evenings and on weekends. I am grateful to my

wife, Janet, and son, Julius, for their patience in postponing many family

activities" until the book is finished." Dr. John Davis of the Naval Research

Laboratory has read the proof sheets and detected a number of errors. To

Alice Levine I owe a special debt of gratitude, for without her cheerful,

intelligent and professional assistance in all phases of this project -- editorial, design, preparation of camera copy and proofreading -- the job

simply would not have been finished. Finally I would like to say that my

satisfaction with the outcome of this project has been enormously enhanced

by the pleasure of working with Dr. Ivanov, for in addition to contributing

much new material and extensively re-writing the existing text in a way that

significantly increased the value of the book, he played a crucial role in

checking my editorial work and in carefully proofreading the camera copy in

a most efficient manner. Despite all of our efforts, it is inevitable that errors remain. If, as readers detect errors, they would communicate them to me , I will maintain an erratum list that may be obtained on request from the JILA publications

office.

D. G. Hummer

Boulder

July 1973
ACKNOWLEDGMENTS

We wish to thank the authors and publishers for permission to use the

i following tables and figures.

Table 2 from E.H. Avrett and D.G. Hummer, Non-coherent scattering II. M.W.R.A.S. 13_0 , 295 (1965).

Table 8 after D.W.N. Stibbs and R.E.Wier, On the H- functions for isotopic

scattering. M.N.R.A.S. 119 , 512 (1959).

Table 10 after M.A. Heaslet and R.F. Warming, Radiative source- function predictions for finite and semi-finite non- conservative atmospheres.

Astrophys. and Space Soi. 1^, 460 (1968).

Table 36 after D.G. Hummer, Non-coherent scattering III. The effect of

continuous absorption on the formation of spectral lines.

M.-N.R.A.S. Vh^, 73 (1968).

Tables 39, 40 after A.L. Crosbie and R. Viskanta, Effect of band or line

shape on the radiative transfer in a non-gray planar medium.

J.Q.R.S.T. 10, 487 (1970).

Figure 37 after G.D. Finn, Probabilistic radiative transfer

203 (1971).

J.Q.R.S.T. 11,

Figures 39, 40, 41 from E.H. Avrett and R. Loeser, Kernel representations in

in the solution of line- transfer problems. Smithsonian

Astrophys. Obs . Special Rept. No. 201 (1966).

Figures 43, 44, 45, 46 from D.G. Hummer, Non-coherent scattering III. The

effect of continuous absorption on the formation of

spectral lines. M.N.R.A.S. 1_38, 73 (1968).

Figure 53 after A.G. Hearn, Radiative transfer of Doppler broadened resonance

lines II. Proa. Phys. Soa. 84_, 11 (1964), Figure 2. Copyright by

The Institute of Physics.
X ACKNOWLEDGMENTS

Figures 54, 55, 56 after E.H. Avrett and D.G. Hummer, Non-coherent scattering

II. Line formation with a frequency independent source

function. M.N.R.A.S. 150 , 295 (1965).
CONTENTS

BASIC CONCEPTS 1

1.1 RADIATION 1 Intensity of radiation 1 Radiation density and flux 2 Thermodynamic equilibrium (TE) 3 Radiation temperature 4

1.2 MATTER 5 Preliminaries 5 Thermodynamic equilibrium and local thermodynamic equilibrium (TE and LTE) 6 Non- equil ibr ium gas 7

1.3 INTERACTION PROCESSES 9 Elastic collisions 10 Inelastic collisions 10 Discrete

radiative processes 13 Continuum radiative processes 15

1.4 EQUATIONS OF STATISTICAL EQUILIBRIUM 18

General case 18 Thermodynamic equilibrium (TE) 21 Local

thermodynamic equilibrium (LTE) 22 Partial LTE 23

Locally-controlled state 24

1.5 THE SCATTERING OF RADIATION IN SPECTRAL LINES 24

Line absorption coefficient 24 Frequency redistribution during

scattering 29 Approximation of complete frequency redistribution 35

1.6 RADIATION TRANSFER 37

Transfer equation for line frequencies 37 Transfer of continuum

radiation 40 Radiative transfer equation for spectral lines

superimposed on the continuum 43 The cases of TE and LTE 44

THE LINEAR APPROXIMATION 49

2.1 MONOCHROMATIC SCATTERING 50

Basic assumptions and definitions 50 Transfer equation 51

2.2 THE TRANSFER EQUATION FOR SPECTRAL LINES 53

Gas of two-level atoms 53 General case 55 Standard form of

the transfer equation 55

2.3 PLANE AND SPHERICAL GEOMETRIES 57

Plane geometry 57 Spherical geometry 61 Two scales of optical

distance 62
CONTENTS

2.4 INTEGRAL EQUATIONS OF STATISTICAL EQUILIBRIUM 63

Derivation o£ the basic equation 63 Homogeneous media 65

Plane geometry 66 Spherical geometry 69

2.5 THE REDUCTION OF NONLINEAR TWO- LEVEL PROBLEMS TO LINEAR ONES 71 Linear case 71 Basic equations in the nonlinear case 73

Reduction of the nonlinear problem to a linear one 74

2.6 KERNEL AND RELATED FUNCTIONS: GENERAL PROPERTIES 76

Definitions and basic relations 76 Series expansions 80

Asymptotic behavior at infinity 80 Integral relations 83

Characteristic exponent 83 Alternative form of the asymptotics 86 Concluding remarks 87

2.7 KERNEL AND RELATED FUNCTIONS: PARTICULAR CASES 8 7 Doppler profile 87 Voigt profile 91 Lorentz profile 94

III. mNOCHROmiC SCATTERING 97

3.1 THE GREEN'S FUNCTION FOR AN INFINITE HOMOGENEOUS MEDIUM 97

Basic equation and its transformation 97 Explicit expression

for the Green's function 99 Alternative representation of the

Green's function 101

3.2 THE DIFFUSION LENGTH AND RELATED QUANTITIES 103

3.3 THE GREEN'S FUNCTION FOR AN INFINITE MEDIUM: GENERAL ANALYSIS 105

Behavior at infinity 105 Integral properties 106 Domain of

applicability of the asymptotic theory 108

3.4 THE GREEN'S FUNCTION FOR AN INFINITE MEDIUM: DETAILED ANALYSIS 108

Behavior in the vicinity of the source 108 Detailed asymptotic

theory 109 Numerical data 111 Nearly conservative scattering 114 Accumulation effect 114

3.5 PLANE ISOTROPIC SOURCE 116

The source function 116 Analysis of the source function 117

Additional remarks 119

3.6 OTHER SOURCE DISTRIBUTIONS 120

Plane problems: General relations 120 Plane problems: Asymptotic solutions 121 Spherical problems 123

3.7 SEMI- INFINITE MEDIUM: GENERAL THEORY 125

Basic relations 125 The Ambartsumian H- function 128 Resolvent

function 133

3.8 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS 136

Uniformly distributed sources 136 The Milne prablem (conservative

case) 138 Diffuse reflection 141 The Milne problem (nonconservative case) 144

3.9 SEMI- INFINITE MEDIUM: DEEP LAYERS 145

The relation of asymptotic solutions for infinite and semi-infinite

media 145 The source function in deep layers 147
CONTENTS XI 11

INFINITE mm 149

4.1 THE GREEN'S FUNCTION FOR A HOMOGENEOUS MEDIUM 150

Basic equations 150 Explicit expression for the Green's function 152 Alternative representation of the Green's function 153

4.2 BEHAVIOR OF THE GREEN'S FUNCTION NEAR THE SOURCE 157

4.3 BEHAVIOR OF THE GREEN'S FUNCTION FAR FROM THE SOURCE 159

Basic formula 159 Nearly conservative scattering 160 Auxiliary

functions 162 Alternative derivation of the asymptotics 165

4.4 MODIFIED FORM OF THE GREEN'S FUNCTION. NUMERICAL DATA 166

Conservative scattering 166 Numerical data for the conservative

case 168 General case, 0 < X < 1 170 Numerical data for the non- conservative case 171

4.5 LONGEST FLIGHT APPROXIMATION 171

The basic expression 171 Physical interpretation 173

4.6 ACCUMULATION EFFECT. THERMAL I ZAT I ON LENGTH 174

Accumulation effect 174 Thermal ization length: leading considerations 177 Thermal i zat ion length: definition and limiting

cases 178 Concluding remarks 181

4.7 PLANE ISOTROPIC SOURCE 181

Basic formulae 181 Small-T behavior 182 Large-x behavior 183

Modified form of $^(t) . Longest flight approximation 186

Numerical data 188

SEFII-INFINITE FlEDIUFl: GENERAL THEORY 193

5.1 BASIC EQUATIONS 194

The transfer equation 194 The resolvent and the resolvent function 198 Auxiliary equation. H- function 200

5.2 PROBABILISTIC INTERPRETATION 204

Single scattering 204 Multiple scattering 205

5.3 THE GREEN'S FUNCTION FOR A HALF-SPACE 208

Derivation of the basic relation 208 Explicit expression for

the resolvent function 210

5.4 THE H- FUNCTION 212

Introduction 212 Explicit expressions for H(z) 213 Moments of

H(z) 216 Alternative integral equations for H(z) 218 Behavior

of H(z) for small z 218 Behavior of H(z) for large z (conservative case) 219 Behavior of H(z) for large z (non- conservative) 221 Qualitative behavior of H- functions 222

5.5 THE RESOLVENT FUNCTION AND ASSOCIATED QUANTITIES 223

Behavior for small t 224 Behavior for large x 224 Modified

form of the resolvent function and its asymptotics. Approximate

forms 225 Integral relations 228 Functions related to

$(x) 229

5.6 THE CASE OF THE DOPPLER PROFILE 2 31

Conservative scattering 231 Non- conservative scattering 234
CONTENTS

5.7 THE CASES OF VOIGT AND LORENTZ PROFILES 236

Asymptotic expansions of the basic functions 236 Numerical data

for the conservative case 240 Numerical data for the non- conservative case 244

VI. SEMI-INFINITE MEDIUM: STANDARD PROBLEMS 251

6.1 UNIFORMLY DISTRIBUTED SOURCES. GENERALIZED MILNE PROBLEM 251

A medium with uniformly distributed sources 251 Generalized

Milne problem: the source function in the deep layers 253 The

Milne problem as a constant flux problem 255

6.2 THICKNESS OF THE BOUNDARY LAYER 257

Definition and order- of -magnitude estimate 257 Limiting

cases 258

6.3 DEPARTURES FROM LTE IN THE BOUNDARY LAYERS OF AN ISOTHERMAL

GAS 259

Formulation of the problem 259 The exact and approximate solutions 260 Physical interpretation 261

6.4 DIFFUSE REFLECTION 266

Emergent intensity expressed in terms of the H- function 266

Analysis of the line profile 269 Albedo of the medium 272

6.5 EXPONENTIAL DISTRIBUTION OF SOURCES 274

Formulation of the problem 274 Analysis of the line profile 275

6.6 GAS ILLUMINATED BY CONTINUUM RADIATION 278

The line source function and emergent intensity 278 Physical

significance of the results. Similarity relations 279 Hot

gas illuminated from outside 281

6.7 SOURCES OF OTHER TYPES 283

Recurrence relation 283 Particular case. Alternative form of

the recurrence relation 284

6.8 MEAN NUMBER OF SCATTERINGS 2 85

Basic expression 285 N expressed in terms of the resolvent

function 286 Particular cases 288

6.9 INTEGRAL PROPERTIES OF THE ESCAPING RADIATION 289

Physical significance of the Hopf function 289 A counterpart of

the Hopf function for line transfer problems 290 Frequency

spread of the escaping radiation 293

6.10 THE EFFECT OF MULTIPLE SCATTERING ON THE ESCAPING RADIATION 294

Introductory remarks 294 Photon escape probability 295

Escape of photons from great depths 296

VII. CONTINUUM CONTRIBUTIONS TO LINE FORMATION 301

7.1 KERNELS OF THE INTEGRAL EQUATIONS AND THE ASSOCIATED FUNCTIONS 302

Basic formulae 302 The case of the weak continuum absorption 304

The functions U and V 307

7. 2 INFINITE MEDIUM 310

The Green's function 310 Integral relations 314
CONTENTS XV

7.3 THERMAL I ZAT I ON LENGTH 315

Definition of the thermali zation length 315 Mean optical path

per scattering 316 Thermalization length. Limiting cases 320

7.4 ANALYSIS OF THE INFINITE MEDIUM GREEN'S FUNCTION 322

General analysis 322 3-solution 323 X-solution 324 A more general case 324

7.5 SEMI- INFINITE MEDIUM 325

Transfer equation 325 The resolvent function and the H- function 32 7

7.6 LINE FORMATION IN AN ISOTHERMAL ATMOSPHERE 333

Formulation of the problem 333 Limiting cases 335 Boundary

layer 338 Line profiles 341 Concluding remarks 344

7.7 STRUCTURE OF THE BOUNDARY LAYER 345

PLANE LAYER OF THE FINITE THICKNESS 3^9

8.1 BASIC EQUATIONS 349

The resolvent and the resolvent function 349 Auxiliary equation.

X- and Y-functions 352 Resolvent function expressed in terms of

the X- and Y-functions 353 Alternative representation of the

solution 355 The transfer equation, emergent intensities and

flux relations 355

8.2 X- AND Y-FUNCTIONS 358

Preliminaries 358 Equations satisfied by the X- and Y-functions 358 Integral rel.ations 361 Series expansions of X

and Y 364 X(o°) and its estimates 366

8.3 ROLE OF DISSIPATIVE PROCESSES 370

Dissipation measure of the medium 370 Analysis of the limiting

cases 372

8.4 ASYMPTOTIC SOLUTIONS FOR MONOCHROMATIC SCATTERING 374

Resolvent function 374 X- and Y-functions 376 Limiting forms

of the asymptotic expressions 378

8.5 LINE TRANSFER PROBLEMS: ASYMPTOTIC SOLUTIONS 380

X- and Y-functions (conservative case) 380 Resolvent function.

Large-scale description (conservative case) 389 Non- conservative

case 393

8.6 THE SCHUSTER PROBLEM 397

Basic equations 397 Emergent intensities 399 Line source function 400 The Milne problem as a limiting case of the

Schuster problem 401 Curve of growth 402

8.7 DIFFUSE REFLECTION AND TRANSMISSION. A MEDIUM WITH UNIFORMLY

DISTRIBUTED SOURCES 405

Diffuse reflection and transmission 405 Uniformly distributed

sources 408

8.8 LINE FORMATION IN AN ISOTHERMAL ATMOSPHERE 413

Basic equations 413 The source term 415 Limiting cases 418

Numerical data 420
xvi CONTENTS

9 MEAN NUMBER OF SCATTERINGS 42 3 Basic relations 423 Uniformly distributed sources 427 Homogeneous sphere with uniformly distributed sources 431 Concluding

remarks 433

10 MEAN LENGTH OF A PHOTON PATH 4 34

Energy balance in a medium 434 Basic relations 434 Estimates

and asymptotic forms 436

11 APPROXIMATE SOLUTION OF THE BASIC EQUATION 438

Basic formula 438 Uniformly distributed sources 440

REFERENCES 4^15

NOTATION ^57
TRANSFER OF RADIATION

IN SPECTRAL LINES

CHAPTER I

BASIC CONCEPTS

The study of the equilibrium state o£ a gas in its own radiation field

is still in a preliminary stage in which the theory is being developed,

rather than one in which the principal results have already been obtained.

It will therefore be helpful to begin our discussion with a more or less

detailed analysis of some basic concepts. The specific assumptions that are

at present required for complete solution of the problems need not be introduced at the outset.

This chapter has two purposes: it provides an introduction to the subject and outlines the limits of applicability of the theory to be developed,

simultaneously sketching a more general approach to the problem.

1,1 RADIATION

INTENSITY OF RADIATION. In the theory of multiple scattering of light, radi

ation is considered to be an aggregate of photons that constitute a photon

gas. Photons are regarded as particles moving with the velocity of light c. Accordingly, each photon is characterized by (1) the energy E or the frequency V, related by E = hv (h being Planck's constant); (2) the three com- ponents of the radius vector r, which specifies the location of a photon in

space relative to some frame of reference; and (3) the two angular variables

that give the direction of motion of the photon, a direction that we shall

characterize by the unit vector co.

The set of six variab les -- frequency , three spatial coordinates, and two

angular variables (e.g., direction cosines of vector aj)--defines a six-dimen

sional phase space. Each photon is represented by a point in this space.

Transfer theory is a statistical theory in which the radiation field is

considered to be an ensemble of photons that are described by the parameters

enumerated above. A complete description of this statistical ensemble is

given by the corresponding distribution function f(v,r,cio,t), such that

f (v ,r ,a3,t)dvdVdco is the number of photons with frequencFes from v to v + dv

locateH" at time t in a volume dV near a point r and moving within a solid

angle dw around the direction ix>. If f does not depend on time t, the radiation field is said to be steady. We shall consider only such fields.
2 BASIC CONCEPTS

In radiative transfer theory one is commonly concerned not with the distribution function f itself, but with the associated quantity I (r,a)) defined

by the relation ^

Iv (^I'iii^ = chvf(v ,r,w) . (1.1)

The quantity 1^ is called the intensity of radiation and has the following

physical meaning. Let there be photons with frequencies from v to v+dv,

moving in the vicinity of a point r in directions lying within the solid

angle dco about ai. Then 1^ (r ,0)) dv dadw represents the energy these photons

transfer per unTt time througF the area da perpendicular to oj, located at

point r. The intensity I^ and the phase density f each compTetely describe

the raH'iation field provided the polarization effects may be neglected. The

description of a radiation field when polarization is considered is thoroughly discussed by S. Chandrasekhar (1950). At the present stage of development

in the areas of radiative transfer theory to which this book is devoted, the

consideration of polarization effects is not of immediate concern. RADIATION DENSITY AND FLUX. An important integral characterizing the radiation field at a given point is its radiation density P^(2l)> defined such that

p^dvdV is the radiant energy in the frequency interval from v to v+dv,

within the volume element dV. According to this definition

p^(r) = hvyi (v,r, )daj' , (1.2)

where the integration encompasses all directions. With (1.1) in mind, we can

also write

Pv (r) = -^JI^(r,a)')da3' , (1.3)

or % c "v ^-

where is the mean intens i ty

Another integral characterizing the radiation field at a given point is the

radi ation flux vector f^Fy (i;_) , defined by

^fy(l) = y*aj' I^ (r,(£' )doa' . (1.6)

The projection of the flux vector on a given direction is called the flux in

that direction. Having designated the flux in the direction u as TrFvca(l.) . we have

^Pva)(l) = y*Iv (r,aJ')cosYdaj' , (1.7)

where y is the angle between directions w and w' , so that cos Y = '^•'^ . From the last equation it follows that 'fF^Q)(r) is the energy flowing across

a unit area perpendicular to w per unit time in a unit frequency interval.

In specifying the direction co we have defined the orientation of the area perpendicular to this directTon as well. Consequently the quantity ttF (r)

also represents the flux through this area. ~
1.1 RADIATION 3

The radiation flux through an area is easily expressed in terms o£ the

fluxes in the directions of the coordinate axes. Let the direction cosines

of the vector w be cos9-j^, cose2, cosG^, so that

u = i_cosO-|^ + jcosO^ + kcosBj , (1-8)

where i^, j, k are unit vectors of the coordinate axes. Similarly we can write ~

oj' = i_cos0j^ + jcose; + kcosSj • (1-8')

Since

cosy = = cos0^ cos6-j^ + cosS^ cos92 + cos6^ cos9^ ,

we find from ri.71 that

F = F cos9, + F cos9^ + F cos9, , (1.9)

VO) vx 1 vy 2 vz 3 ' ^ '

where ttFvx» '"'^vy '"^vz are fluxes in the directions of the x- , y- and z- axes, respectively. The first of these is

'^^vx^^'^

= yi^(r,a)')cos9^dao' (1.10)

The quantities TTF^y(r) and 7TF^^(r)are defined analogously.

Thus the amount of radiative energy per unit volume is characterized

by the radiation density, whereas the direction and velocity of its flow at

a given point are characterized by the flux vector. However, these quantities alone do not, of course, completely specify the radiation field.

THERMODYNAMIC EQUILIBRIUM (TE) . In this book we shall address ourselves to

the study of the physical conditions in a gas that is not in a state of

thermodynamic equilibrium (TE) . As a rule, the intensity of radiation in

such a gas changes substantially from point to point. The radiation field

is non- isotropic and its spectral composition depends upon both coordinates

and direction. Thus in a non-equilibrium system the intensity I (r,to) depends, generally, upon all phase coordinates. As it is possible tEat for

certain values of the phase variables the conditions are quite different

from TE, while for others the conditions are more or less near equilibrium,

we might usefully review briefly the properties of a radiation field in TE

In TE the intensity of radiation does not depend upon coordiriates and

direction so that the radiation field is homogeneous and isotropic. Moreover,

the frequency dependence of the intensity is a universal function of the system's temperature T and does not depend upon the properties of the matter

with which the radiation interacts. The intensity of radiation in TE will,

as is customary in astrophysics, be denoted by Planck's function B (T)

This function is defined by

V

(T) = Ji^^e^v/kT _ ^ ^^^^^^

where k and h are Boltzmann's and Planck's constants. It is convenient to

introduce the special notation u (T) for the radiation density in TE. We

have, from (1.3) and (1.11), ^
4 BASIC CONCEPTS

The integrated radiation dens i ty

oo u(T) = J

u^(T)dv (1.13)

0 is seen from (1.12) to be proportional to the fourth power of the temperature

(the Stef an-Boltzmann law)

u(T) = al"^, (1.14)

where

^ •

• (1.15)

ISh^c^

The quantity a is related to Stefan's constant a by a = 4a/c.

In the short wavelength region, i.e. hv >> kT , Planck's function re- duces to Wien's function

2hv"^ -hv/kT

B^(T) = e , hv >> kT , (1.16)

and in the long wavelength region, where hv << kT , it reduces to the

Ray leigh- Jeans law:

B^(T) = v^ hv << kT. (1.17)

We note that the substitution of Wien's function for Planck's is equivalent

to the assumption that stimulated emission is negligible compared with spontaneous emission.

RADIATION TEMPERATURE. In the absence of TE , it is often convenient to

express the radiation intensity in terms of the Planckian intensity at some temperature T. This can be done in several ways. The simplest is to express

I^ as a fraction of B (T) , where T is the local temperature of the gas. However, the deviation of I^ from By)(T) is often quite large. It is therefore

more convenient to introduce the radiation temperature T-^. by means of the

definition

I^(r,a.) = ^e'^v/kT, _

' (.^^g^

or

I = B (T ) . (1.18')

V V r-^ Strictly speaking, (1.18) is not a definition of the radiation temperature.

Radiation is a photon gas and its temperature should be introduced according

to the general principles of statistical physics, i.e. in terms of entropy.
1 . 2 MATTER 5

In this way it may be shown that (1.18) is a consequence o£ the general concept o£ the temperature of a non- equi librium system, and not simply an ad hoc

definition of an interpolation parameter Tj. (see, e.g., J. Oxenius , 1966).

It is clear that the radiation temperature Tj- is a function of the same arguments as 1^; i.e., in general, T^. = Tj.(v,r,aj). The difference T^- - T

provides a measure of the deviation of 1^ from By(T), where T is the local

gas temperature. This description of the radiation field has the feature

that in the Wien region a large variation in the intensity corresponds to a relatively small change in the radiation temperature.

Just as the radiation temperature may be used to characterize the intensity of radiation with frequency v at a given point and propagating in a given direction, a temperature parameter may also be int^roduced to describe

the mean intensity or radiation density. The quantity T-^ defined by

J^(r) = B^(T^), (1.19)

or equivalently , by

p^(r) = u^(T^), (1.19-)

may be called the mean radiati on temperature for the frequency v at a given

point. It is clear that the_mean_radiation temperature is a function of both

frequency and coordinates: Tp = Tj.(v,r_).

From the theoretical point of view, this description of the radiation

field in terms of temperature parameters, provides nothing new. Moreover,

since these parameters are related to the intensity in a rather complex

manner, their use only leads to unnecessarily complicated expressions.

Therefore in transfer theory the various kinds of radiation temperatures are rarely encountered. However, because it is easy to visualize, the temperature description is rather widely used (primarily by oDservers and experimentalists). Unfortunately, it is not always precisely specified what any given

"temperature" describes. Yet it is absolutely necessary to do this, especially when it is recalled that other temperature parameters are used to describe the state of matter. The failure to distinguish different "temperatures" has more than once been the source of unfortunate misunderstandings.

1.2 MATTER

PRELIMINARIES. We shall now consider the description of the state of a gas

interacting with radiation. We shall regard as known the chemical composition of the gas, or more precisely, the relative number of atoms of various

elements, and also the density of the gas at each point. The properties of

the atoms comprising the gas, such as their energy levels, transition probabilities, collision cross sections, etc., are also regarded as given. The

problem, then, is to describe the state of the gas considered as an aggregate

of individual particles with known properties.

In each elementary volume, atoms of a particular element are present in

various stages of ionization. To describe the state of a gas, therefore,

one must determine the degree of ionization of the atoms. Furthermore, one must know the distribution of the ions over the discrete energy levels as well as the velocity distribution of ions in a given state of excitation and

ionization. The velocity distribution function of the free electrons formed
6 BASIC CONCEPTS

as the atoms are ionized must also be found. Once all of these parameters

are known, the state o£ the gas is completely described.

In practice, simplifications of different types arise (or, in any case,

are introduced during theoretical considerations), so that it is not necessary to discuss the question in such general terms. We shall now discuss

typical examples of situations that are encountered.

THERMODYNAMIC EQUILIBRIUM AND LOCAL THERMODYNAMIC EQUILIBRIUM (TE and LTE)

We start with the simplest case: that in which the gas may be considered to

be in a state of TE. The following conditions must be met:

(1) The velocities of each species of particle (let us say, r-times ionized

atoms of the 1-th element, in the i-th state) must have a Maxwellian

distribution, characterized by the same temperature T for all species.

Let ni be the total number of such particles per unit volume. Then the

number having velocities from v to v+dv is

dn. = „.4.

' ^„ e-*'/2" v^dv , (2.1)

(2TTMkT)-^/^

where M is the particle mass. The free electrons must also have a Maxwellian velocity distribution at the same temperature T.

(2) The distribution of atoms over the energy levels must be given by

Boltzmann's law

n. g. -hv ./kT

= ^ e , [2.2)

^1 ^1

where n^ is the population of level i, i.e. the number of atoms in this

level per unit volume, gi is the statistical weight of the level, and

hvj^^ is its excitation energy.

(3) Ionization must follow the Saha equation

^ 1/ _ g: 2(2.mkT)^/^ ^-hv^^/kT

'

^ ^1 §1 h3

where n^^ is the number of atoms in the ground state in some (e.g., the

r-th) stage of ionization per cm^, n"*" is the concentration of ions in

the ground state of the next (r+l-th) stage of ionization, ne is the

concentration of free electrons, g-^ and g"*" are the statistical weights

of the ground states of the "atom" and "ion," respectively, m is the

electron mass, and hv^,; is the energy required to ionize an r-times

ionized atom in its ground state.

(4) The radiation intensity must be given by Planck's function (1.11) with

the same temperature T appearing in expressions (2.1) - (2.3).

With sufficient rigor for our purposes, these four conditions can be

regarded as defining the state of TE. It is obvious that these conditions

are rather stringent and are met only in exceptional cases. However, it is often found that when one or more of these conditions is

violated the others are satisfied with sufficient accuracy. One example of

a state of this type is the so-called state of local thermodynamic equi - librium , or LTE. If conditions 1 to 3 are satisfied at each point in a gas

and the temperature T is permitted to change from point to point, the gas is
1 . 2 MATTER 7

said to be in LTE. It must be stressed that in LTE the radiation can have

arbitrary intensity. As long as conditions 1 to 3 are not violated, the

j intensity can deviate arbitrarily from the Planck intensity at the local

temperature T. When collisional processes dominate radiative processes in

j the ionization and excitation o£ atoms, the state of the gas is close to LTE

! (see Sec. 1.4 for further details).

Initially LTE was understood in astrophysics in a somewhat different

sense. Namely, the medium was said to be in LTE when conditions 1 to 3 were satisfied only if the intensity at every point was sufficiently close to the

: Planck intensity at the local gas temperature. In this more restricted

sense LTE exists if the variation of T is small over distances of the order

of the maximum mean free path of the photons that contribute substantially

to the level populations. In this book the expression LTE will be used in

the wider sense explained above, with no restrictions imposed on I^.

NON-EQUILIBRIUM GAS. In astrophysics, cases in which more than one of condi-

' tions 1 to 4 are violated are more frequently encountered than those described by LTE. If at least one species of particle has a Maxwellian velocity

I distribution, it is possible to introduce the concept of kinetic temperature.

We shall refer to such a gas as being in partial equilibrium . If the kinetic

i Maxwellian temperature cannot be introduced for even one species of particle,

the system is then said to be in a completely non-equilibrium state.

' We shall consider only systems in partial equilibrium. However, this

, restriction still leaves us most of the problems of practical interest. Actu-

! ally, even under such extreme conditions as those existing in interstellar

I'j space free electrons, as well as atoms and ions in the ground state, have a

j Maxwellian velocity distribution. Moreover, the electron and ion temperaII tures are practically identical. Perhaps still further from TE is the

: plasma of a gaseous discharge. In this case there are many situations in

which one may speak of electron and ion temperatures that are frequently

, very different.

An aggregate of particles of a given species with a Maxwellian velocity

! distribution can be called an equilibrium part of a system, or equi librium

j: subsystem . (The word "equilibrium" is used here to emphasize the fact that

' the distribution of the particles of the subsystem over the trans lational

j degrees of freedom is the same as in TE . ) There may be several such equi

j librium subsystems, exactly as many as there are species of particles having

j a Maxwellian velocity distribution. A system that as a whole is not in

equilibrium may be composed of a set of equilibrium subsystems. Obviously,

! this can only occur if the particles of each species have a Maxwellian

[i velocity distribution,

ii]

i| The kinetic temperatures of individual subsystems may, in general,

i'l differ from one another. However, even if they are equal, the system as a

I whole is not necessarily in TE ; equal kinetic temperatures merely indicate

i that at each point in the medium a single kinetic temperature exists for all

'! particles, i.e. that condition 1 (page 6) is met. We shall refer to such

I systems as "single temperature." That a s ingle - temperature gas is not

necessarily in equilibrium follows from the fact that individual subsystems

are not necessarily in equilibrium with each other: the degree of ionization and excitation in such a gas is not, generally speaking, described by

ill the Saha and Boltzmann laws. The interstellar gas is a good example of a

ji single- temperature system that is very far from TE. In those regions of the

interstellar medium in which hydrogen is strongly ionized (the so-called

I H II regions), the degree of ionization, for example, of hydrogen atoms (equal

.1 to 10-^-10^) is several orders of magnitude smaller than the equilibrium

|( value for a gas with a kinetic temperature T==104°K and a density n^^lQl-lO-^.
8 BASIC CONCEPTS

Particles whose velocity distribution differs from Maxwellian form non- equilibrium subsystems (say, a subsystem of r-times ionized atoms of the 1-th

species, in level i) . As we shall see later, deviations from a Maxwellian

distribution are more of a rule than an exception for excited atoms whose

concentration is controlled principally by radiative processes.

For the sake of simplicity we shall consider only single- temperature

gases. As a rule, the generalization of the results to the case of a gas

with different electron and ion temperatures is trivial. In accordance with

the above discussion, for a complete description of the state of a singletemperature gas at a given point one must know the kinetic temperature and

the concentrations of particles of each species, i.e. the level populations

and the state of ionization. However, other methods of description, theoretically equivalent to that just indicated, but more convenient in practice,

are often used.

In the first

gas temperature T

species of ion in

atoms of a given

TE concentrations

centration of r + question. Let n^ some element , and

in the next stage

(2.3) ,

of these methods each point has assigned to it a kinetic

, an electron density ng, and the concentration of each

its ground state. The concentrations of r-times ionized

element in excited levels are expressed as fractions of the

corresponding to these values of T and tiq and of the con- 1-times ionized atoms of this element at the point in

be the population of level i of an r-times ionized atom of

n* the concentration of ground state atoms of that element

of ionization. Then in TE , as follows from (2.2) and

n - = n n

1 e

hv. /kT

ic'

2(2TTmkT)-^/^

(2.4)

where hvic is the energy to ionize an r-times ionized atom in level i. For a single- temperature gas, n^^ may be expressed as

+ Si h^ic/^^

^i = "e^ — 77:^ ' ^2.5)

g 2(2TrmkT)3/2

Here the level population is characterized by the dimensionless parameter

bi — the so-called Menzel factor (D. Menzel, 1937). This means of expressing

the concentration of excited atoms is especially useful when the levels are populated primarily from above, by recombinations and transitions from

higher-lying states.

It should be noted that values of b-j^ greater than unity do not necessarily imply that the population of level i relative to the ground state is

higher than the equilibrium population corresponding to the temperature T. When the degree of ionization is less than that at equilibrium, the population of level i may be lower than the Boltzmann value even though bj^ > 1. Another means that is sometimes used to describe the state of a gas

differs from the one just discussed only in that the populations of the ex- cited levels, instead of being represented by the parameters b^^, are characterized by the numbers c^ , defined by the relation

n. g. -hv-,./kT ^ = — e c. . (2.6)

^1 §1 '
1.3 INTERACTION PROCESSES 9

The dimens ionless parameter is the factor by which the population of level

i of an r-times ionized atom differs from the Boltzmann population corresponding to local temperature T. With this description the population of level i is expressed in terms of the concentration of ground state ions of the same

(r-th) stage of ionization, whereas when the parameters are used, the population is referred to the concentration of (r + 1) -times ionized atoms. The

quantity Cj^ is sometimes called the reduced concentration of excited atoms.

From (2.5) and (2.6) it follows that

+ „+ , ,„s3/2 -hv, /kT c n § 2(2TrmkT) Ic i n — - — —^ e — • (2.7)

e n^ gi bi

Thus the ratio ci/b^ indicates the factor by which the degree of ionization

differs from that in TE for local temperature T and electron density n^.

Since the left side of (2.7) does not depend upon i, we conclude that the

quantities c^ and b^ differ only by a constant factor.

Another means of representing the state of a gas, which may be called

the thermal description, is as follows. The kinetic temperature T, electron

density ng , and concentration of ions in some i-th (usually the ground)

level are given. The populations of the other levels are characterized by

parameters T^]^ defined by the expression — = — e , (2.8)

. 1 ^1

where hv^j^ is the energy difference of the levels i and k. The quantity

T^]^ is called the excitation temperature of level i relative to level k. In

a similar way, the degree of ionization is described by the ionization

temperature T^^^, defined by the expression

^ n^ _ 2(2TrmkT)^/^ ^-^^Ic/^^lc ^2 9)

^ ^1 §1

In LTE the values of T^i^ and T]^^ for all levels of all atoms, in any stage

of ionization, are equal and identical to the local gas temperature T.

Values of Tik and/or Ti^; differing from the kinetic temperature T reflect

deviations from LTE and indicate how strongly the equilibrium between the

individual equilibrium subsystems making up the s ingle -temperature gas has

been disturbed.

In this section we have considered alternative descriptions of a non- equilibrium gas in the steady state. This steady non- equi libr ium state can

exist only if it is maintained by an external mechanism. Radiation incident

upon the gas from outside is one example; but it is by no means the only

possibility. In any case, a steady non-equilibrium gas can never be thought

of as an isolated system.

1.3 INTERACTION PROCESSES

The steady state of a gas is established through various kinds of

interactions. Let us discuss briefly those processes that play a decisive
10 BASIC CONCEPTS

role under the conditions encountered in astrophysics. First we will con- sider the interaction of particles (atoms, ions, and free electrons) not

related to the processes of light emission and absorption; then we will

discuss radiative phenomena, i.e. the interaction of radiation and matter.

ELASTIC COLLISIONS. Elastic collisions cause changes only in the velocity

of the particles and do not precipitate any internal reorganization in the

particles, i.e. ionization and transitions between levels. These processes

lead to the establishment of a Maxwellian velocity distribution. If elastic

collisions occur much more often than inelastic collisions and radiative pro- cesses for particles of a given species, then the velocities of these particles have a Maxwellian distribution. For atoms and ions in the ground state,

these conditions usually seem to be satisfied. As a rule electrons also

have a Maxwellian velocity distribution.

The situation for excited atoms is quite different. The lifetime of

excited states is sufficiently short that at low enough densities the atom

will usually complete a downward transition long before an elastic collision

has occurred. In order, therefore, to ascertain whether or not the velocity

distribution of the atoms in some excited level is Maxwellian, one must

closely examine the populating mechanisms. If radiative processes dominate

collisional processes and the intensity of radiation incident upon the atoms

varies strongly across the line in which excitation occurs, appreciable deviations from the Maxwellian distribution may occur. We shall consider this

question in greater detail in Sec. 1.5.

INELASTIC COLLISIONS. Inelastic collisions cause excitation and ionization

of atoms together with the inverse processes of de-excitation and recombination. Under astrophysical conditions the most important of the inelastic

collisions are those of atoms and ions with free electrons; we shall confine

our attention to such collisions.

Let us first consider collisional transitions between discrete levels.

A collision leading to the excitation of an atom at the expense of the incident electron's kinetic energy is called a collision of the first kind. The reverse process is known as a collision of the second kind, or de-activation.

Let ni and nj^ be the populations of the lower and upper levels, respectively,

and ne the electron concentration. The number of collisional transitions

occurring in a unit volume per unit time is:

for i ^- k collisional excitations: n.n C-, , 1 e IK

for k -> i de-activations (collisions of the second kind):

^i<;"-e^ki ' C3.1)

The quantity Cik is related to the excitation cross section qikCv) by

the relation

oo ^ik = /qik(v)vf(v)dv, (3.2

^ik

where v is the velocity of the incident electron and £(v) is the velocity

distribution function of the electrons. Here v . is the threshold velocity,

determined by the obvious condition

mv.j^

hv . 1 ik

(3.3)

where hv., is the difference between the energies of levels i and k.
1.3 INTERACTION PROCESSES 11

I£ the velocity distribution o£ the electrons is Maxwellian with a temperature T, the rate coefficients Cj^-y. and Cj,^ are related by the expression

^ki (3.4)

Actually, in TE each process is exactly balanced by its inverse, so that

n . C . 1 1 ik ^kSi (3.5)

I Moreover, in this case the ratio of the level populations is given by

Boltzmann's law (2.2). From (3.5) and (2.2) it immediately follows that in

TE the relation (3.4) is indeed valid. However, since Cik and C]^i are deterI mined only by atomic properties (cross sections) and by the velocity distribution of the electrons, (3.4) must hold in the absence of TE as well, provided that the velocity distribution of the electrons is the same as in TE

,

i.e. Maxwellian.

To obtain an order of magnitude estim.ate of the number of transitions

induced by electron impacts, one can proceed from the semi-classical ex- pression for the excitation cross section Rj^]^('^)'

lik (V)

,23 3e c

2 2 g.

\^ hv.^ mv2

;

(3.6)

where Aj^^ is the Einstein coefficient for the spontaneous transition k i and e is the electronic charge. From (3.6) and (3.2) it follows that for

electrons having a Maxwellian velocity distribution with temperature T

^ik

2 3 e c m ^ki gk -hVik/^T

^hv3 (2TMkT)-^ Si

ik

(3.7)

where

P(x)

3*^

277 [1 - xe^Ej^(x) ] , (3.8)

I and Ej^(x) is the first exponential integral:

-x dx (3.9)

I

A somewhat better estimate of C^v is obtained if the values of the

function P(hvi}^/kT) are not obtained from (3.8), but instead the values given

in Table 1 are used (after H. van Regemorter, 1962). In this particular case
12 BASIC CONCEPTS

TABLE 1

THE FUNCTION P(X)

Positive Positive

X Atoms Ions X Atoms Ions

<0 005

2 TT E^ (X)

2tt E^ (X) 0 . 4

1

0

0

209

100

0

0

290

214

0 01 1 . 160 1 . 160 2 0 063 0 201

0 02 0 . 956 0 . 977 4 0 040 0 200

0 04 0 . 758 0 . 788 10 0 023 0 200

0 1 0 . 493 0 . 554

0 2 0 . 331 0 . 403 >10 0 . 0 6 6x~ %0 200

it is convenient to calculate the values of Ei(x) appearing in the table byusing its series expansion:

E,(x) = - Inx -Y* + E (-1)^^^ . (3.10)

^ j=l 3-3:

where y* - 0.5772 is Euler's constant. The expression (3.7) is applicable

when the k -» i transition is optically allowed.

Electron impact transitions between the discrete states and the continuum are described in a similar manner. There are two such processes: electron impact ionization and three-body recombination, that is, recombination

involving the collision o£ three particles. During the latter process, which

is essentially a collision o£ the second kind, the energy released in the

capture o£ the electron is carried away by the third particle. We shall

assume that this third particle is also an electron, although this is not

always the case. The number o£ transitions between level i and the continuum

(c) occurring in a unit volume per unit time is:

£or electron impact ionization i c : n-n C. . ^ 1 e ic

£or three-body recombination c ^ i: n^n^C -. - (3.11)

The quantity C^c is expressed in terms o£ the impact ionization cross section

qic(v) by (3.2) and (3.3), in which the subscript k must be replaced by c throughout. A line o£ reasoning similar to that leading to (3.4) indicates

that £or electrons with a Maxwellian velocity distribution
1.3 INTERACTION PROCESSES 13

An order of magnitude estimate o£ the rate constant for electron impact

ionization of an atom is given by

, -Iiv. /kT

4 1 ic

C. =4T^2m^el 1 ^ ^ pf^^), (3.13)

3/3 hv.^ (2TTmkT)^ \ kT /

where n is the number of equivalent optical electrons, and the function P is

given by (3.8). This expression is obtained by using Thomson's formula for

the ionization cross section:

^ ' ^ - ^ (3.14)

^ic (V) = 2^^^ (-1 2_\ . 2 Ihv. 2 mv j \ i-C mv /

DISCRETE RADIATIVE PROCESSES. Let us now proceed to radiative processes.

We first consider those associated with transitions between discrete levels.

Let ni and nj^ be the populations of the lower and upper levels, respectively,

and the mean intensity of radiation:

Jv

For the present we shall assume that within the line the mean intensity does

not depend upon frequency and is equal to Jij^. Then per unit time per unit

volume the number of transitions is:

for k i spontaneous transitions: n. A, . , K K

for k -> i stimulated transitions:

^Jc^lci'^ik ' (3.16)

for i -> k photo-excitations: n-B., J-i . ^ 1 ik ik

The quantities A]^j^ , Bj^j^ , and Bj^]^ are. the Einstein coefficients for

spontaneous and stimulated emission and absorption, respectively (calculated

for mean radiation intensity rather than for radiation density) . They are related by the well-known expressions

2hv^,

I so that it is sufficient to have only one of these coefficients. Usually one

uses either the transition probability Aj^j^ or the oscillator strength fik,

I'which is related to k^i by the equation
14 BASIC CONCEPTS

0 2 2 2

A, . = — — . - (3.19)

ki gv „„3 ik

mc

There are numerous summaries o£ the values o£ fiki see, £or example,

C. W. Allen (1963); W. L. Wiese, M. W. Smith and B. M. Glennon (1966); W. L. Wiese, M. W. Smith and B. M. Miles (1969). Methods for the theoretical calculation and experimental determination o£ oscillator strengths, excitation

and ionization cross sections, and probabilities of other elementary processes are discussed in detail, e.g., by I. I. Sobel'man (1963), D. R. Bates

(1962) and J. B. Hasted (1964), to whom we refer the reader for further information. A detailed summary of the data on the cross sections for electron

impact ionization has recently been published by W. Lotz (1967).

The number of photo- excitations can be expressed in terms of the Einstein coefficient B-iv and the atomic absorption coefficient k-jv as follows.

Then ^

j^jj. and the atomic absorption coefficient kj^i^

Let I be the intensity of radiation radiation with frequency frequency v within tn(he line i -> k

{7- I k . T ( v) n . dvdco hv V ik^ ^ 1

is the number of i k photo- excitations per unit time per unit volume induced by radiation with frequencies v, v + dv , incident within the solid

angle dio around a certain direction. If the velocity distribution of atoms

in level i is isotropic, then kj^]^(v) does not depend on direction. The total

number of i -> k photo-excitations per sec per cm3 is obviously

1 /- 1 dv

f

I dw

V (3.20)

or

4TTn dv

. f k., (v) J — ij ik^ V Yxv (3.21)

The absorption coefficient kii^(v) has a sharp maximum at the center of the

line. Therefore the value of the integral in (3.21) is wholly determined by

the values of the integrand in a narrow band of frequencies near line center,

and the latter expression may be replaced by

4Tr n

00 (3.22)

where v^^^ is the line-center frequency. If the mean intensity depends

weakly upon frequency within the line, then it may be set equal to Ji^, so that instead of (3.22) we have
1.3 INTERACTION PROCESSES 15

4tt t

(3.23)

This is the desired expression for the number of photo-excit.ations in terms

of the absorption coefficient kj^]^(v)^ Equating it to n j^Bj^]^ J j^j^

, we arrive

at the relation

yk.^(v)dv = -Tf . (3.24)

0

If the intensity cannot be assumed to be independent of the frequency

within the line, then (3.22) must be used to find the number of photo-excitations. Using (3.24), it may be written in the form n.B.j^^J.j^, with

oo oo /k., (v)J dv /k.,(v)dv/l — if ik^ ^ V ^ ik^ ^ J v^TT

oo ~ oo y*k.j^(v)dv y-k.j^(v)dv ^ik = — = — • ^^-25)

0 Thus for radiation with an arbitrary frequency dependence incident upon a volume element, the number of photo-excitations_occurring in it is given by

! the second of the expressions (3.16), in which J^j^ is defined by (3.25).

Let us now consider stimulated transitions k -> i . We assume that the

photons emitted in spontaneous transitions k -> i have a frequency distribution proportional to the line absorption coefficient kj^v(v) and that they

i are emitted with equal probability in all directions. (These assumptions

ji are equivalent to the approximation of complete frequency redistribution

!i discussed in detail in Sec. 1.5.) Then the number of photons with frequencies from V to V + dv emitted per cm^ per sec within the solid angle doo is

n. A, . k . , (v) dv J ik^ ^ do)

k k i oo 4 TT j

k,,(v)dv

0

To find the number of stimulated transitions one has to multiply the number

of spontaneous transitions by (c2/2hv3) l^ (see Sec. 1.6). Integrating the

resulting express_ion over all frequencies and directions, we arrive at the

expression n^^Bj^^J^j^ postulated above, with J^]^ given by (3.25). As is clear

from the derivation, this expression does not always hold. However, since we shall assume complete frequency redistribution throughout the book, we ar_e justified in setting the number of stimulated transitions equal to ^]^Bj^^ J^-j^

.

CONTINUUM RADIATIVE PROCESSES. Let us now turn to transitions in which one

of the states belongs to the continuous spectrum and the other to the discrete. There are three such processes: spontaneous radiative recominations

,

stimulated radiative recombinations, and photo- ionizations . Let the free

electrons have a Maxwellian velocity distribution at temperature T. We shall

denote the number of such processes occurring per unit volume per unit

time as:

522-519 O - 74 - 3
16 BASIC CONCEPTS

for c i spontaneous radiative recombinations: n n'''A . . e ci '

for c i stimulated radiative recombinations: n n"'^B .J. . e ci ic'

for i c photo-ionizations : ^i^ic"^ic ' (3.26)

The coefficient A^-^ is a function of electron temperature, namely,

CO ^ci = / ei(v)vf(v)dv , (3.27)

0 where 6.(v) is the cross section for radiative recombination, and fX.v) is the

velocit^ distribution function for the electrons. The quantity ^dJiQ, which

gives the number of stimulated recombinations, is expressed in terms of the mean intensity and the capture cross section 3i(v) as follows

r

2 B .J. = / B-(v)vf(v) —— J dv, (3.28) ci ic J ""i^ ' ^

' 2hv3 ^

where

2 hv = hv. + — . (3.29) ic 2

Finally, the number of photo-ionizations from level i is

n.B. J. = 4^n. f k. (v)J — , (3.30)

1 ic ic 1 J ic^^

hv

V .ic

where k^Q(v) is the cross section for photo-ionization from level i.

The quantity 3j^(v) is expressed in terms of 5^-j^(^(^) by the Milne relation

3.(v) 1^-^ = Jl^L- ^k. (v) , (3.31) 222 + ic^^' c^m^v^ g

which is easily derived from the condition of detailed balance in TE

The expressions (3.27), (3.28), and (3.30), together with (3.31), show

that when the mean intensity is known, only one atomic property need be

known in order to calculate the number of radiative transitions from level i to the continuum and back: namely, the cross section kj^^(v) for photo-ionization from level i.

For hydrogenic ions with charge Z:

k. (V) =

Z TT e mz (.^^ (3.32)

ic^ ^ 6 5 3 ic

3/3ch i V

where gic^^^ -^^ Gaunt factor. In as trophys i cal calculations of the num- ber of radiative ionizations and photo- recombinations , the Gaunt factor is
1.3 INTERACTION PROCESSES 17

often set equal to unity, with errors o£ only a few percent. In the approximation gic(v) = 1 (Kramers' approximation) for a gas with kinetic temperature

T, the value of A(-i , as follows from (3.27), (3.31), and (3.32), is

1 &i hv. /kT /hv. . A.=K -^-^ ^ e eA-^\ (3.33)

where

1^ g 2(2TrmkT)-^/^ v

^9 5 10 „4

° 3/3c3h6

If the intensity of radiation is Planckian with temperature T^ = T, then

1 Si hVi^/kT /hv^^

i -J. =K -i- tl e Q* _i£ I (3.35)

° 2(2.mkT)^/2 n kT '

'

1. J. = K

1 c 1 c

1 I* (l!^) , (3.36)

o i5 \ kT / .

where

/

Q*(x) = / 1

""^^

, (3.37)

x(e^ - 1)

/

I*(x) = / — . (3.38)

x(e^ - 1)

We note that

I*(x) = E^(x) + Q*(x) (3.39)

and

*

CO R .j^2j 1

where y* = 0.5772 is Euler's constant, and are the Bernoulli numbers

(B2 = 1/6, B4 = -1/30,...). The series (3.4oj converges for all x > 0;

but it is useful for computations only for small x. For the derivation of

(3.40), see V. V. Sobolev and V. V. Ivanov (1962).
18 BASIC CONCEPTS

Free-free transitions, in which both states belong to the continuum,

i.e., spontaneous and stimulated bremsstrahlung and inverse bremsstrahlung

,

are described similarly. If the electrons have a Maxwellian velocity distribution with temperature T, then for hydrogenic ions with charge Z the freefree absorption coefficient, per ion and per electron, is

k^rv) = ^-^^ J_g^^(v) , (3.41)

3/3mch (2TTmkT)^ ^

where gcc(^^) "^^^ Gaunt factor, which in the optical region is on the

order of unity.

1.4 EQUATIONS OF STATISTICAL EQUILIBRIUM

GENERAL CASE. If the state of a gas, as characterized by the specification

of its kinetic temperature and the degree of ionization and excitation of

each atomic species, does not change with time, the gas is said to be in a steady state. Such states are described by the equations of statistical

equilibrium . They express the equilibrium between the various elementary

processes that lead to the establishment of that state.

A tremendous range of steady states, highly specialized examples of

which are the states of TE and LTE , is possible, depending on the particular

elementary processes that are dominant in the specific situation. We shall

be studying primarily those states that are maintained largely by photo-excitation by the radiation field of the gas itself. However, at the outset

we shall not limit ourselves to any specific case but, rather, discuss equations of statistical equilibrium in general, including all of the main types

of elementary processes.

The state of a single- temperature gas at a given point will be completely described if we know the distribution of atoms over energy levels and the

degrees of ionization as well as the temperature. We shall consider the

equations that determine these quantities. The concentration of atoms in

level i can be found by equating the number of transitions into this level to

the number of transitions out of it. The number of atoms of a given element

arriving in level i per cm-^ per sec for each elementary process of interest

is listed below:

(1) spontaneous and stimulated radiative transitions from higher discrete

levels

oo ^k^\i ^ \i^"ik^ k=i + l

(2) spontaneous and stimulated radiative recombination:

n n"^ (A . + B .J. ) ; e ^ ci ci ic-^

(3) transitions from above induced by collisions of the second kind:

oo n / n, C, . ; e X—/ k ki ' k=i + l
1.4 EQUATIONS OF STATISTICAL EQUILIBRIUM

(4) three-body recombination:

n^n^C . e ci

(5) photo- excitation from lower-lying levels

i-1

V n.B. . J. . ;

j = l

(6) collisional excitation:

n

e

i-1

En .C . . .

3 Thus, the total number o£ transitions into level i per sec per cm is

c i-1

n, (A, . + B, . J. 1 + n C, . ) + n . (B . . J . . + n C . . ) , (4 i—t ki ki ik e ki-^ < j 31 ji e ji-^ ' ^ k=i+l 3=1

where the first summation includes both discrete (k = i+1, i+2,...,)

and continuum (k = c) states, with n^ being understood as ngn"^.

We now list the number of transitions from level i by the various

processes : (1) radiative transitions (spontaneous and stimulated) into lower-lying

levels:

i-1

n. "V (A. . + B. . J. . ) ;

3 = 1

(2) downward transitions induced by collisions of the second kind

i-1

(3) photo-excitation into higher levels:

ik ik ' k=i + l
20 BASIC CONCEPTS

C4) upward transitions through electron impact:

00

^e^i E ^ik ' k=i + l

(5) photo- ionization

n.B. J. ; 1 ic ic

(6) ionization by electron impact:

n n.C e 1 ic

For the total number o£ transitions from level i we therefore have

i-1 c n. V (A.. + B..J.. + n C.) + n. (B.,J., + n^C.,) . (4.2)

1 ij ij ji e ij-^ 1

^ ik IK e ik-' j=l k=i+l

Equating the number of populating and de -populating transitions, we arrive at the following set of equations for the populations n^:

c i-1

En, (A, . + Bi -J-i + n C, . ) + V* n.(B..J.. + n^C..) = k^ ki ki ik e ki-^ ji ji e ji^

(4.3)

k=i+l j=l

i-1 c

= n. y (A.. + B-.J.. + n C.) + n. (B-^J-i, + n C,), i = l,2... . 1 ^ 1] ij 31 e ij-^ 1 ^ ik ik e ik^ ' j=l k=i+l

The system (4.3) can be used to find the state of a gas at a given point

in the medium only if certain additional information is available: (1) the

radiation intensity; (2) the temperature; and (3) the electron density ng and

ionic concentration n"^. If any of these factors are unknown, the problem is more complicated. Thus if the concentration of n+ ions is not known, the

system (4.3) must be solved simultaneously with an ionization equilibrium

equation expressing the equality of the numbers of ionizations and recom- binations in a unit volume:

00 00 y n.(B. J. + n C. ) ^ = n n^ V (A . + B .J. + n^C^.). (4.4)

3^ jc jc e jc^ e ^ ^ c] cj jc e cj^

j=l j=l

Here the total number of photo- ionizations and electron impact ionizations

from all levels is given on the left; the total number of recombinations

composed of spontaneous and stimulated radiative recombinations and

three-body recombinations into all levels appears on the right.
1.4 EQUATIONS OF STATISTICAL EQUILIBRIUM 21

If, in addition, the temperature is not known, the system (4,3) -(4.4)

must be coupled with an energy balance equation. This equation expresses

the fact that in a steady state the energy acquired by an elementary volume

must equal the energy lost. For a single-temperature gas, only the electron

temperature need be found. Energy is introduced into the electron gas, i.e.

it is heated through a number of processes: (1) electrons acquire the energy

of an absorbed photon by inverse bremsstrahlung ; (2) in a three-body re- combination the binding energy of the level into which the electron is captured is transformed into heat; (3) in a collision of the second kind, energy

equal to the difference of energies of the levels is produced; (4) in photoionization the electron gas gains energy equal to the difference between the

energy of the ionizing photon and the binding energy of the level from which

ionization occurred. The electron gas may be heated by other mechanisms as well. As for cooling, the electron gas loses energy through bremsstrahlung,

radiative recombination, and collisional excitation and ionization.

An energy balance equation allowing for all of these processes can be

formulated. However, since usually only a few of these processes are important, it hardly seems worthwhile to write out this general equation, particularly since we henceforth regard the temperature of the gas as known.

In order to solve the equations of statistical equilibrium (4.3)-(4.4),

the radiation field in the medium must be known. If the mean intensity J is

known for the relevant frequencies, calculating the steady state in principle

presents no difficulties as it involves only the solution of a set of simultaneous linear algebraic equations. However, when an important role is

played in the gas by transitions induced by its own radiation field, the

situation is very much more complicated. A knowledge of the radiation field

then depends on the solution of the statistical equilibrium equations, which

must be solved concurrently with the equations of radiative transfer. In

this case the condition of statistical equilibrium is expressed in terms of

integral equations. Physically this means that the conditions at each point

are determined by the state of the medium as a whole, because volume elements

far from each other interact effectively through the exchange of radiation.

A more detailed discussion of the statistical equilibrium equations for this

important case is given in the next chapter.

Let us now look in detail at several special steady states for a singletemperature gas.

THERMODYNA^lIC EQUILIBRIUM (TE) . In this state the velocity distribution of

all particles is Maxwellian, the degree of excitation and ionization is

given by Boltzmann's and Saha's laws, and the intensity of radiation is given

by Planck's function. The state of the system is completely determined by

one parameter — the temperature T.

In TE each process is exactly compensated for by its inverse (the so- called principle of detailed balance). In particular, the number of radiative

transitions from an upper level i to a lower level j then equals the number

of j

-> i photo - excitations

n + B. . u

13 4lT i = 2 , . . . ; (4.5)

where u is the Planckian radiation density of frequency v.., given by

^ji

(1.12). Similarly, the number of radiative ionizations from level i,
22 BASIC CONCEPTS

n.B. — u

1 ic V ic

is equal to the number of radiative recombinations into that level

nn''(A.+B.— u \ = n.B. , i = l,2, . . . . e V ci ci 4tt ^ic) ^ ^ (4.6)

For collisional transitions the detailed-balance relations are

n.C.. = n.C.., ij = l,2, (4.7)

n n'^C - e ci

= n.C.

1 ic i=l,2, (4.8)

LOCAL THERMODYNAMIC EQUILIBRIUM (LTE) . As has been said, LTE is that state

of a Maxwellian gas in which the level populations accord with the Boltzmann

distribution, ionization follows Saha's equation, while the radiation intensity is not, in general, given bv Planck's function. Contrary to the case of

TE , the temperature may change from point to point. In discussing the physical conditions required to establish such a state, we shall limit our consideration to a statement of sufficient conditions.

In (4_i_3) and (4.4) we shall temporarily set equal to zero both the mean intensity J and the A- coeff i cients . We obtain

c i-1 i-1 c L Vki * E "jC., = n. ^

C.. . n. Cik' i = (4.9)

k=i+l j=l j=l k=i+l

En.C.3 3 = n n"^ C . c e -t—' cj

3=1 3=1

(4.10)

If the levels are populated according to the Boltzmann distribution, and the

degree of ionization is given by the Saha equation, the detailed-balance re- lations (4.7) and (4.8) are satisfied. Using them, it is not difficult to

verify the fact that in this instance (4.9) and (4.10) are also satisfied.

Since these equations are linear in the quantities n^/n^n'^ , the equilibrium

values of the populations of the discrete levels and the continuum are in

fact the only solution. Moreover, the temperature entering the Boltzmann and

Saha equations may be a function of position. Thus a gas whose steady state

is described by equations (4.9) and (4.10) is in LTE.

The system (4.9)-(4.10) differs from (4.3)-(4.4) in the absence of terms

allowing for radiative transitions. Thus if these terms are negligible in

comparison with those describing the collisional transitions the gas may be

considered to be in LTE. In other words, a sufficient condition for the

assumption of LTE to be valid is that radiative transitions be negligible in

comparison with collisional transitions . Since the role of collisions
1.4 EQUATIONS OF STATISTICAL EQUILIBRIUM 23

increases with electron density, LTE should exist at sufficiently high electron densities. For numerical estimates, see D. Sampson (1969).

PARTIAL LTE. Under as trophys ical conditions, LTE in the strict sense is

hardly ever encountered since the densities in the atmospheres are not, as a rule, sufficiently high. More frequently a gas is in a state that might be

called partial LTE, by which we mean the following. Let us consider a cer- tain volume of gas in a given field of non- Planckian radiation and trace

the change in its state as the density is decreased. The initial density is

assumed to be sufficiently high for the gas to be in LTE. Then as the density decreases, at a certain point one will no longer be able to neglect

radiative transitions in comparison with collisional ones for all levels.

Deviations from LTE usually develop first in the lower levels, while the

populations of the higher levels still remain in equilibrium with respect to

the continuum. The number of levels whose populations depart noticeably

from the equilibrium values gradually increases as the density decreases.

There is a wide range of densities for which the higher levels have essentially the same populations with respect to the continuum as in LTE, while the

populations of the lower levels may be far from equilibrium. This situation

arises because one cannot neglect the effect on the lower levels of radiative

transitions compared to the collisional ones. This, then, is the state of

partial LTE.

Roughly speaking, the entire negative-energy spectrum of the atom can

in this case be broken down into two regions — equilibrium and non-equilibrium,

Let io be the number of the level dividing these regions. The relative populations of the levels in the equilibrium region, i.e. close enough to the

continuum, are the same as in LTE (bj^ - 1 for i ^ ig) • And the populations

of levels lying in the non-equilibrium energy region (i < ig) are now governed not only by local temperature and density, but also by the radiation field

at the given point, so that the coefficients b^ for these levels can be very

different from unity. It is obvious that the size of the non -equilibrium

energy "gap," which is characterized by the number Iq of the critical level,

is determined by the properties of the atoms making up the gas, by its

density, and by the radiation field at the point in question.

For conditions in stellar atmospheres D. Sampson (1969) has obtained the

following equation for ig for atoms of effective charge Z:

n^pfx. .^A - 2.S.10^^ (-^\^ ^^sSlhll (4.11)

where X^j^ = hv^-^/kT, hv]^^- is the ionization energy of ground-state hydrogen

and P(x) is the function given in Table 1 (p. 12). It is assumed that

Iq > 2. On levels where i > Iq deviations from LTE do not exceed 5 percent.

The gas in stellar atmospheres, in the upper layers of planetary atmospheres, and in nebulae and the interstellar medium is in a state of partial

LTE. Of course, the number of levels of, say, hydrogen atoms whose populations are not in equilibrium with the continuum will be substantially different in stellar atmospheres from the number in H II regions of the interstellar medium. However, there is no basis for the a priori assumption that

in stellar atmospheres the level populations of the atoms of any element have

equilibrium values. This assumption, the so-called LTE hypothes is , has been

employed in the theory of stellar atmospheres for some fifty years. The

applicability of this hypothesis, so attractive because of the tremendous
24 BASIC CONCEPTS

simplifications it introduces, must be justified in each specific case. A

consistent theoretical approach to the study of stellar atmospheres should

be based on a detailed examination of the elementary processes. Active

research in this field is under way; the results have been reviewed recently

by J. T. Jefferies (1968) and D. Mihalas (1970),

LOCALLY CONTROLLED STATE. If the state of a gas at a given position in a medium does not depend on conditions in other parts of the system, we may

speak of a locally-controlled state. LTE is an example, although not the

only one, of such a state.

Interaction between remote volume elements occurs via the radiation

field. Therefore, if the conditions at a given position are not to depend

on the state of the medium as a whole, radiative processes caused by the

gas' own radiation field should not be significant. If collisions dominate

all radiative processes, this condition is obviously met (in this case the

gas is in LTE). But if collisions do not dominate and radiative transitions

cannot be neglected, then in order for a locally-controlled state to exist

the gas must be practically transparent. The first situation exists, for

example, in a high-pressure laboratory gas discharge. The second case corresponds to the solar corona. The equations of statistical equilibrium

for this situation are algebraic, and their solution raises no theoretical

difficulties, apart from the fact that usually the probabilities of the

elementary processes are poorly known.

1.5 THE SCATTERING OF RADIATION IN SPECTRAL LINES

LINE ABSORPTION COEFFICIENT. Radiative processes associated with transitions

between discrete levels deserve closer scrutiny. In the first place, one must consider the frequency dependence of the line absorption coefficient.

According to the quantum theory of radiation, for an isolated, motionless

atom the absorption coefficient for a line corresponding to the i k

transition is (see W, Heitler, 1954)

2 gi Av„

k (V) = S !k R

_ (5.1)

2 2 g. ^ . ^ 2 . . ^ 2

where

i-1 k-1

j = l £=1

The quantity Avj^ is called the natural, or radiation line width. At a distance Avj^ from the line center, the absorption coefficient is half as large

as at the central frequency (at v = Vg) .

If an atom is not isolated, the effect of the surrounding particles is

to increase the line width (so-called pressure effects). As is well known

(see, e.g., I. I. Sobel'man, 1963), the shape of the line is often similar

to that in the preceding case and is described by (5.1), except that Avj^

must be replaced by
1.5 THE SCATTERING OF RADIATION IN SPECTRAL LINES 25

Av^ - AVj^ + AV(. , (5.3)

where Av^; is the collisional line width.

Atoms are, in fact, not at rest, but are always in thermal motion. Let

us consider radiation propagating along the z axis, and take the atomic

velocity component in that direction to be . Then, because o£ the Doppler

effect, the central frequency of the line is shifted and becomes

c

The absorption coefficient in this case for radiation of frequency v within

the line is obviously

c^ Sk

^ 8Tr2v^j^ g. Cv-v')2 + (Av J

2

In order to obtain the absorption coefficient per atom of a volume element of

gas, the expression (5.5) must be averaged over the distribution function of

the z-components of the velocities of the absorbing atoms. If atoms in the

lower level have a Maxwellian velocity distribution with temperature T, then

the fraction of the atoms having z-components of velocity from v to v + dv

is z z z

|7 exp I Idv^ , (5.6)

(ZTTMkT)^

where M is the atomic mass. Averaging (5.5) over the distribution (5.6),

and taking (5.4) into account, we find that

where

2 e dy U(a,x) = ^

j

^ '±1

. (5.8)

it3/2 / (x-y)2 + a2

In the two last expressions, the quantity

'D 'c' ( M

. !ii im.y (5.9)

is the Doppler line width, x is the dimensionless frequency measured from the

line center in Doppler widths
26 BASIC CONCEPTS

X

V - V,

Av,

(5.10)

and finally

a = Av.

Av

D

AVp^ -H Av^

Av

(5.11)

D

The function U(a,x) is known as the (normalized) Voigt function. It describes the frequency dependence of the line absorption coefficient, and

therefore plays an important role in all matters relating to the interpretation of the shape of spectral lines.

The expression (5.7) is applicable only when the absorbing atoms have a Maxwellian velocity distribution. In the case of lines of the resonance

series, i.e. arising in transitions from the ground state, this assumption

is not as a rule open to doubt. However, for lines of subordinate series,

as we shall soon see, the assumption of a Maxwellian velocity distribution

for the absorbing atoms may be rather crude. Nevertheless, for lack of anything better, this assumption is universally employed.

At the densities and temperatures of interest in astrophysics, the parameter a is small (a 10"^ - 10'4). In this case the frequency dependence

of the absorption coefficient in the central parts of the line differs substantially from that in the far wings. That is to say, there is a certain

critical distance from the line center (which we designate as |xq|), such

that for |x| significantly less than | xq | we have, with sufficient accuracy,

U(a,x) ^ ^-^e"^ (5.12)

For |x| significantly larger than | xg | , it can be assumed that in the first

approximation

U(a,x)

TTX''

The value of | xq | is the root of the equation

(5.13)

2 e-^0 aTT -k 1 (5.14)

As a decreases, | x„ | increases

a 10-2 10 10-"

2.67 3. 12 3.51
1.5 THE SCATTERING OF RADIATION IN SPECTRAL LINES 27

Along with the Voigt absorption coefficient (5.7) — (5.8), we shall

also make frequent use of its limiting forms, corresponding to a = 0 and

a = °°. In the first of these cases we get the so-called Doppler absorption

coefficient

c^ Sk \i -x^

In the second we have

k,j^(v) c

'ik

2 g,. A

^k \i 1 . (5.16)

87t2v?j^ ^i Av^ 1 + x^

In (5.16), as distinct from the preceding formulae of this section, the

dimensionless frequency x is measured not in Doppler widths Avj), but in

units of Av*:

X = V - Vq

The absorption coefficient of (5.16) is named after Lorentz.

Comparing these expressions with the approximate forms of U (a, x) for

|x| < I x^ I and |x| > I Xq | , we see that in the central parts of the line the

absorption coefficient is close to the Doppler form, while far from the line

center it can be considered Lorentzian. Accordingly, one often refers to

the Doppler core and the Lorentz wings of a line.

Let us look a little more closely at the Doppler absorption coefficient,

It corresponds to a = 0, i.e., the case in which Av^ is negligible in com- parison with AVj^. As Av^ ->• 0 we have

Av^

(v-Vq)2 + (AvJ2

7t6(v-Vq) ,

where (S(v-vq) is the delta function. Therefore, in this limiting case, the

absorption coefficient of an atom at rest reduces to

2 g, k.v(v) = —-— — A, .6(v-v.) . (5.17)

8Trv2^ Si ki ^ 0^

Taking into account the Doppler shift of the central frequency of the line

for a moving atom and averaging (5.17) over the Maxwellian distribution (5.6), we obtain (5.15). This derivation of (5.15) makes the physical significance

of the frequency dependence described by the Doppler absorption coefficient

quite obvious: it is a direct reflection of the Maxwellian velocity distribution of the absorbing atoms.
28 BASIC CONCEPTS

Of primary importance in line transfer problems is the behavior of the

absorption coefficient in line wings. The asymptotic properties of the solution of the line transfer equation are insensitive to the details of the behavior of the absorption coefficient in the central parts of the line, but

are very sensitive to its behavior in the wings. The exact meaning of this

statement will become clear later. We mention it now because if the absorption coefficient cannot be assumed to be represented by the Voigt function,

the theory of collisional broadening is usually more reliable for the line

wings than for the central parts of the line. In particular, under rather

mild restrictions it may be assumed that far from the line center kj^i^(v) is

proportional to Iv-vqI"^, where k is a constant, 1 < k < °°.

The frequency dependence of the line absorption coefficient is usually

described by the so-called profi le function a(x) , defined as the ratio of

the absorption coefficient for frequency v to its value at line center:

k.

a(x) = TT^r—T ' ^^'^^^ kik(vo)

so that a(0) = 1. Profiles corresponding to the three cases considered above

(a = 0, 0 < a < <», and a = ») will be referred to by the names Doppler,

Voigt, and Lorentz, respectively, and will be denoted by subscripts D, V,

and L

2 Doppler: a^M = e'^ , (5.19)

Voigt: a,,(x) = LLlavO ^ (5,20)

^ U(a,0)

Lorentz: a, (x) = . (5.21)

^ l+x2

The characteristic frequency interval used to define the dimensionless frequency X (equal to Avq in the D and V cases, and to Av^ in the L case), will

henceforth be designated simply as Av. The normalization constant A,

defined by the relation

00 A f a(x)dx = 1 , (5.22)

- 00

is equal, in the D, V, and L cases respectively, to

Ap = tt"'^; A^ = U(a,0); A^ = tt"^ . (5.23)

As we have already mentioned, profiles of the form

a(x) ^ Wlxl'"^, |x| ^ «=

, (5.24)
1.5 THE SCATTERING OF RADIATION IN SPECTRAL LINES 29

where W and k are constants, 1 < k < <», are also o£ substantial importance.

We shall also often make use of a (somewhat artificial) rectangular profile,

which will be denoted by the subscript M. This M may be thought of as an abbreviation of either the name Milne or the word "monochromatic":

Milne: a^^^Cx) =

j J; |x| > } ' = f

' ^^.25)

The rectangular (or Milne) profile is the particular case (corresponding

to p = 0) of the profiles

(l-|x|)P, Ixl < 1

0 , |x| > 1

where p is a parameter, -1 < p < «>. Obviously, in practical line transfer

problems such profiles are not encountered. However, in the theoretical

analysis of line transfer problems this family of profiles was found to be

very useful. In concluding the discussion of the absorption coefficient in

a spectral line, w3 note that if temperature and density vary from point to

point the absorption coefficient is a function not only of frequency, but

also of position. If the velocity distribution of absorbing atoms is non-

!! isotropic, it then also depends upon direction. However, the problems in

I which ki]<;(v) and a(x) depend on r and/or o) are outside the scope of this

book. It will be assumed throughout that ki]<;(v) and a(x) are given functions

I of frequency that are independent of both position and direction.

The strong frequency dependence of the absorption coefficient is characteristic of line transfer problems. An immediate consequence of this is

the very large difference between the mean free paths of core and wing pho-

' tons. Thus, if we have a Doppler profile (and there is no continuum absorpI tion) , a photon with x = 3 has a mean free path that is e^ - 10^ times

larger than the mean free path of a photon with x = 0. Therefore, within one

line we have photons with mean free paths differing by several orders of

I

magnitude. As it is scattered, the photon's frequency may change, causing a large change in its mean free path.

, FREQUENCY REDISTRIBUTION DURING SCATTERING. So far we have been concerned

J with the photo- excitation of atoms. We must now consider the emission processes involved in radiative transitions. As this is a much more complicated

i question, we will be forced to quote the majority of the results we use within out proving them.

First let us discuss terminology. An i k photo-excitation process

1 followed by a k i radiative transition will be called scattering of a

I! photon in the line. If all processes de -populating the upper level k can be

I neglected in comparison with the k i radiative transition, and there is no loss of line photons in flight (due to photo- ionizations , etc.), we shall say

I that pure, or conservative, scattering occurs. We shall lump together under

' the general name of (true) absorption of radiation in the line all processes

causing transitions from level k following i -> k photo- excitation , with the

Ij exception of the k i radiative transitions. Examples of true absorption

' processes are collisions of the second kind, ionization from the upper level,

spontaneous k -> j transitions (j ^ i) , etc. In the last-named process a photon with energy hv^j^ is transformed into a photon of energy hvj^.:; it

j| should be stressed that this process is also treated as a true absorption

I process for radiation in the line i ^ k. Further on we shall say that a

a(x) =
30 BASIC CONCEPTS

photon in a line corresponding to the transition between levels i and k comes from primary sources , i£ its emission during a radiative k i transition

does not occur directly after an i > k photo-excitation caused by the

medium's own line radiation. Thus when the chain. of radiative transitions

i-^k^j->k->-i occurs , it is assumed that one photon in the i -> k line was lost as a consequence of true absorption, and a new photon was then born,

coming from primary sources. However odd this terminology may look at first

sight, it turns out to be very convenient.

Let us consider the scattering processes in more detail, beginning with

a simple illustrative example. We shall trace the fate of photons in a resonance line corresponding to the 1^2 transition, assuming that pure

scattering occurs and that the gas is tenuous enough for pressure effects to

be neglected. Moreover, for the sake of simplicity, we shall neglect the

broadening of the upper level, i.e. we assume that the absorption coefficient

of a motionless atom is given by (5.17). We shall assume that the absorbing

atoms have a Maxwellian velocity distribution. Let us look at an elementary

volume of gas, illuminated by isotropic monochromatic radiation of frequency

\>l, and ask for the shape of the line emitted by this volume through the

resonance scattering of the incident radiation. The first conclusion that

comes to mind, "Doppler, of course," turns out to be incorrect. s This can be confirmed, for example, in the following way. With the

above assumptions, the profile of the emission line is a direct reflection of

the velocity distribution of the excited atoms, just as the Doppler absorption coefficient reflects the Maxwellian velocity distribution of the absorbing atoms. Now the velocities of the excited atoms will not, in the case at

hand, be distributed according to Maxwell's law. The frequency of the radiation absorbed by an atom is v = vq + vqCv^/c) , where v^ is the component of

the atomic velocity in the direction of propagation of the radiation.

Therefore, for the atom to be able to absorb a photon of frequency v^, its

total velocity must not be less than

^0

(5.26)

Consequently, among the excited atoms there will be none whose velocities are less than v^.

Analysis shows that in this case the velocity distribution function for

the excited atoms takes the form

dn2 = —M

exp

z IcT

M(v^-vJ)

2kT

V dv

V < V,

V > V,

(5.27)

where n2 is the population of the upper level, dn2 is the number of excited

atoms with velocities from v to v + dv, and T is the kinetic temperature of

ground state atoms. The velocity distribution (5.27) is completely unlike

the Maxwellian distribution.

In order to answer the question concerning the form of the line emitted

by the volume element, the atomic velocity distribution must first be con- verted to a distribution of the component along a line of sight that is con- veniently taken as the z-axis. The result is that the normalized distribution function of the z-component of the velocity has the form
1.5 THE SCATTERING OF RADIATION IN SPECTRAL LINES 31

where |v| = vi for IV2I < vi and |v| = Iv^l for jv^l ^ V]^ . It is obvious

that the intensity of radiation with frequency V2 emitted by a unit volume

through the resonance scattering of the radiation incident upon it is determined by the equation

hv^2

2 4'rr

where ~ "^0 ""^ "^0

'•^z''''"-^

' Consequently

where X2 is the dimensionless frequency corresponding to The population

of the upper level is determined by the equation of statistical equilibrium

2 21 hvi2 ^ V

or

2

4-rr n "^l

^2^21 = i;;;7j"i^i2(^o) '-1 ' ^'-''^

where I^-j^ is the intensity of the incident radiation, and x-^ is the dimensionless frequency corresponding to v^. From (5.29) and (5.30) we find,

taking (5.2 8) into account.

\ = ^1^^12(^0)Iv^^f^l'^2) ^ , (5.31)

where

~ 2 r(x-,^,X2) = / e'''^ dt (5.32)

and |x| is the larger of [x^l and | X2 | . Thus the shape of the line is

described by the function r(X]^,X2) (Fig. 1) and is quite unlike the Doppler

profile. In particular, it depends on the frequency of the incident radiation. The width of the flat portion of the profile increases with |x-j^|.

The expression (5.32) was found by W. Unno (1952a) and, independently, by

522-519 0-74-4
32 BASIC CONCEPTS

V. V. Sobolev (1955, 1956). The derivation given here is due to V. V. Ivanov

(1967)

— 0.4

CM 0.3 ^ 0.1

0 0.5 1.0 1.5 ^2

Fig. 1. Frequency redistribution function for Doppler broadening.

The function r{x\, X2) describes the conversion o£ photons of one frequency into photons of another within an elementary gas volume. In the simple

example that has just been considered, this transformation is caused by the

Doppler effect. In more complicated cases other mechanisms are responsible

as well for the frequency redistribution of photons within the line. However,

whenever the radiation incident onto the volume is isotropic, and the velocity

distribution function of atoms in the lower state is Maxwellian, the intensity

of the radiation scattered by the volume can be represented by an expression

of the form (5.31), although, of course, the function r(xj,X2) will not then,

in general, be given by (5.32).

The quantity r(x]^,X2) is called the redistribution function . In all cases mentioned below, as well as in the example just discussed, the redistribution functions are normalized so that

J

r(x-j^ ,X2)dx2 = a(x-|^) . (5.33)

The physical significance of r(x2,X2) follows from (5.31). The quantity

r(x]^,X2)dx2 is the atomic absorption coefficient for radiation of frequency

XI which is subsequently emitted as a photon with frequency in the interval

(X2, X2 + dx2) , expressed as a fraction of the absorption coefficient at

line center.

Let us turn now to a slight modification of our previous example. We now assume that the volume element is illuminated by unidirectional radiation

instead of by isotropic radiation and leave the other assumptions in force.

Let the incident radiation propagate in direction co^ . It is clear that the

velocity distribution of the excited atoms will now depend upon direction.

The spherical symmetry that existed for isotropic illumination gives place

to axial symmetry, with the axis of symmetry parallel to . The intensity

of the scattered radiation will depend upon both frequency and the angle with

the direction of incidence. Instead of (5.31) we have

'^v^^-l^ " n^k^2(^0^^v^'^-l^^*^^l'^2'^^ ~^ ' (5.34)

where y is the angle between and (£2, so that

cosy = 00, • co^ . (5 . 35)
1.5 THE SCATTERING OF RADIATION IN SPECTRAL LINES 33

The functions t(xi,X2; y) and r(xi,X2) are related by

r(x^,X2) = ZTT^rCx^ ;y) siny dy (5. 36)

0 The function r(xx,X2;Y) is the angle - dependent redistribution function. In

the particular case of zero natural line width, as was shown by R. N. Thomas

(1957) ,

r(x-|^,X2 ;y)

4tt -^/^siny

exp

2 2 x^+X2 - 2x^X2COSY

sm Y (5.37)

The function r(X]^,X2), corresponding to (5.37), is given by (5.32).

There is an additional effect that we have ignored. An atom can scatter

non- isotropically the radiation incident upon it. We shall call x(Y)d.u)/4Tr

the probability that radiation is scattered within the solid angle dco alDOUt

the direction forming an angle y with the initial beam. The function x(y) is

called the phase function , or scattering indicatrix. Until now atoms were

assumed to scatter incident radiation in all directions with the same probability (isotropic scattering), a situation described by the isotropic or spherical phase function

x(y)' = 1 (5. 38)

In reality we are concerned with dipole scattering, for which we have the

Rayleigh phase function:

xCy) = - (1 + cos^y)

4 (5 . 39)

Strictly speaking, one should take this circumstance into account. However,

the difference of phase functions from the spherical case is usually ignored

in the study of multiple scattering in spectral lines. This is by no means

the least accurate of the approximations employed in this theory.

If the assumption of a zero natural line width is not made, the redistribution functions become more complicated. If, as before, a resonance line

is being considered and the gas density is so low that pressure effects may

be ignored, then the photons are again redistributed in frequency by the

Doppler effect alone. It is found (V. G. Levich, 1940; L. G. Henyey , 1940;

W. Unno, 1952b; V. V. Sobolev, 1955, 1956) that

00 .(x.,x,) = 1 fe-^y'

^ ^ 7T3/2u(a,0) y

t)

2 r arctg + arctg j dy , (5.40)

r(x, ,x, ;y) = —— exp ( — t^csc"^ - lU(a sec-,s sez!^ \ , (5.41)

^ ^ 47T3/2u(a,0)sinY \ 2/ \ 2 2/
34 BASIC CONCEPTS

where

|x -x^l X +x

t = —i— s = ——- . (5.42)

Here a is the ratio of the natural to the Doppler width and U(a,x) is the

(normalized) Voigt function (5.8). When a = 0, (5.40) and (5.41) reduce to

(5.32) and (5.37), respectively.

If pressure effects cannot be ignored, the picture changes. Thus far the

photon frequencies in the atom's rest frame before and after scattering were assumed to be the same although in the observer's rest frame they were in fact

different. Now a new mechanism for frequency redistribution is added to the

Doppler effect. While an atom is in the excited state, the positions of sur- rounding particles that cause a displacement of the level may change completely. Thus, if the density is on the order of 10l2 cm"^, then the average distance between particles is "^10'^ cm. At a temperature of lO^^K the average

relative velocity of hydrogen atoms is '\^2'106 cm/sec, so that an atom will

travel '\^10"* cm in 5«10'11 sec. Since the lifetime of an atom in an excited

state is 'x^lO"^ sec, no correlation at all can be expected between the positions of the surrounding particles at the moments of absorption and emission.

Although this treatment is very crude, and the question is indeed much more complicated, the basic conclusion that correlation between the frequencies

of absorbed and emitted photons is absent appears to be correct. It follows

that in the atom's rest frame the probability that emission of a photon of

frequency V2 will follow absorption of a photon of frequency is independent

of and is proportional to the absorption coefficient for the frequency

The Doppler effect provides an additional frequency redistribution mechanism

whose action was discussed in sufficient detail above. The resulting redistribution function is very complicated; we shall not reproduce it here. The

properties of this function were studied by G. D. Finn (1967) . Special attention should be paid to the fact that if |x, | and |xp| are large, then

r(x, ,x,) ax^ . (5.43)

This result reflects the lack of correlation between the frequencies of the

absorbed and emitted photons, and is valid in situations in which the Doppler

effect plays a negligible part in the frequency redistribution.

Until now we have limited our discussion to resonance lines. For lines

of the subordinate series the situation is even more complicated. First, the

assumption of a Maxwellian velocity distribution for the excited atoms must

be used with caution. In this connection we recall that the above analysis

was based on the fact that the velocities of atoms in the lower level were supposed to have a Maxwellian distribution. Second, since the lifetime of an atom in either of the levels is finite, neither can be regarded as infinitely

narrow. Because of this the frequency of the photon may change even when it

is scattered from an isolated, motionless atom. This redistribution mechanism has been the topic of numerous studies. It is discussed in detail by

R. v.d. R. Woolley and D. W. N. Stibbs (1953), where references are given to

the earlier publications. [Ed. note: An important clarification of this

situation has been published by Omont, Smith and Cooper, Astroiphys . J. 175 , 185-199, 1972.]

In this book we shall be primarily concerned with resonance lines.

Therefore, we shall not discuss the question of redistribution functions for
1.5 THE SCATTERING OF RADIATION IN SPECTRAL LINES 35

lines o£ subordinate series in more detail. Moreover, the approximate

expressions that are always used for them involve the same approximations as for resonance lines (the so-called approximation of complete frequency redistribution; see below).

A comprehensive discussion of the redistribution function in various

cases, which includes the derivation of the formulae given above without proof

and an exhaustive list of references on this subject, has been given by

D. G. Hummer (1962, 1965a); a useful review is given by J. T. Jefferies (1968),

APPROXIMATION OF COMPLETE FREQUENCY REDISTRIBUTION. The extremely cumbersome

form of the redistribution functions makes imperative the introduction of an approximation that retains the basic properties of the process of frequency

redistribution while avoiding unimportant details. Such an approximation was

first published, so far as we know, by J. Houtgast (1942), and thereafter by

a number of authors. Known as the approximation of complete frequency redistribution, it involves the following two simplifications. First, it is

assumed that the dependence of redistribution functions on the scattering

angle may be disregarded, i.e. r(x]^,X2) is used throughout instead of

r(xi,X2;Y)- Second, the exact redistribution function r(xi,X2) is replaced

by an approximate one, setting

r(x, ,x,) Aa(x^) a(x2) (5.44)

where a(x) is the profile of the absorption coefficient in the line.

This approximation, which is now generally accepted, provides the basis

for all of our subsequent discussion. What considerations can be adduced in

favor of this approximation, and when does it become reasonably accurate?

The main argument for neglecting the angular dependence of the redistribution

functions rests on the fact that the radiation being scattered is usually

incident from all sides. With the exception of regions in immediate proximity

to the boundary of the region occupied by the gas, the radiation intensity

does not as a rule depend strongly on direction. The second approximation,

expressed by (5.44), is a much more serious matter. It is based on two facts.

First, if the absorption coefficient has a Voigt profile, then in the Lorentz

wings, as follows from (5.20) and (5.13),

V 7rU(a,0)

(5.45)

so that for sufficiently large |x-j^| and I X2 1 , according to (5.44),

2

r \ a - 2 - 2 r(x^ ,x,) ^ —— x^ .. ^ Tr2u(a,0) ^ (5.46)

When pressure effects are substantial, th'e result agrees with that for the

exact redistribution function (5.43). Second, in the line core, where frequency redistribution is caused almost entirely by the Doppler effect, a fraction of the photons will always be scattered with complete frequency re- distribution. This follows from (5.37), which shows that for photons scattered through a right angle (y = Tr/2) ,
36 BASIC CONCEPTS

- X. otpCx^) ap(x2) - . C5.47)

When pressure effects are substantial (Av(3>>Avj^) , these considerations

ensure that the assumption of complete frequency redistribution is reasonable

for all frequencies. But if pressure effects are unimportant, the approximation of complete redistribution can be used only for the central frequencies

of the line, and the atoms must be excited mainly by radiation flowing within

the Doppler core. Detailed calculations confirm this qualitative line of

reasoning.

Further information on the accuracy of the approximation of complete

frequency redistribution may be found in a monograph by V. V. Sobolev (1956)

and in papers by R. N. Thomas (1957), J. T. Jefferies and 0. R. White (I960),

A. G. Hearn (1964a), D. G. Hummer (1963, 1965a), J. Oxenius (1965), M.

Dobrowolny and F. Engelmann (1965), and the author (V. V. Ivanov, 1967).

Special attention is called to a paper by D. G. Hummer (1969) where one can find the most detailed study of this problem to date.

According to the basic idea of the approximation of complete frequency

redistribution, the fraction of the energy emitted within a volume element

in the frequency interval (v, v+dv) by spontaneous k ^ i transitions is

eil<;(v)dv; ^^^(v) does not depend on the mechanism by which the level k is

populated, and is proportional to the absorption coefficient The

normalization condition

oo /

e.j^(v)dv = 1 , (5.48)

together with the relation (3.24), makes it possible to determine the coefficient of proportionality. In this way we get

or

e., (v) = — a(x) . (5.49')

This expression for ei]^(v) will be used in the next section in deriving the

equation of radiative transfer in spectral lines.

An approximation that takes the opposite approach to that of complete

frequency redistribution was widely used in the as trophys ical literature for

almost half a century, and is still occasionally used. It postulates that

the frequency of a photon is unchanged during scattering so that

r(x-^,X2) = a(x-^)6 (x^-X2) , (5.50)
1.6 RADIATION TRANSFER 37

where 5(x) is the de Ita- function . For complete frequency redistribution, a photon completely forgets its initial frequency during scattering, while

according to (5.50) it remembers and preserves it.

There are no real physical grounds for approximation (5.50), and it

should be abandoned. Calculations using this approximation directly contradict the results of several experiments. On the other hand, experiments

especially designed to check the applicability of the assumption of complete

frequency redistribution have shown that it is in fact a good approximation

(A. V. Phelps, 1959; P. Walsh, 1959; A. V. Phelps and A. 0. McCobrey, 1960).

We shall conclude with some remarks on terminology. Scattering with

complete frequency redistribution is often called completely non- coherent

,

while the term "coherent scattering" is widely used to designate scattering

in which the frequency does not change. This terminology can hardly be

called apt as the point of interest is not, in fact, the phase relations between the incident and scattered waves, but rather the changes in the frequency of a photon during scattering. Instead of the expressions "coherent"

and "completely non- coherent" scattering, we shall use "monochromatic

scattering" and "scattering with complete frequency redistribution."

1.6 RADIATION TRANSFER

Before the equations of statistical equilibrium can be solved, the radiation field at every point in the medium must be known. The radiation field

is specified by the equation of radiative transfer.

TRANSFER EQUATION FOR LINE FREQUENCIES. Let us consider the radiation in a spectral line corresponding to the transition between the lower i-th and

upper k-th levels. We assume first for simplicity that the i k photoexcitations and the inverse k ^ i transitions are the only relevant elementary processes in the range of frequencies of interest. Let Iy(X.,iil) be the

intensity of radiation of frequency v at point r in direction w. Then, by

definition, I ^ ( r , oj) dvdadou is the energy flowing per unit time through an area

da perpendicular to w in the frequency interval (v , v+dv) within the solid

angle doj. The transfer equation describes the change of intensity along the

beam. At the point r + wds the intensity in direction w equals I (r+oj^ds , o)) . The difference v —

—

[I (r+tods,aj) — I (r , w) ] dvdadco (6.1)

then represents the change in the intensity along the path ds

This change occurs in two ways. On one hand, energy is expended in

exciting the atoms of the gas. If ki^i^(v) is the line absorption coefficient

per atom in state i, then the energy expended per unit time in atomic excitation is

^i''^ik ^— '— (6.2)

On the other hand, atoms in the upper level k emit photons during k -> i transitions, leading to an increase in the intensity. The energy emitted

in spontaneous k ^- i transitions by a volume dads per unit time within the

solid angle doj is (hvi]^/4Tr) Aj^ j^nj^dadsdio . In the approximation of complete
38 BASIC CONCEPTS

frequency redistribution, the fraction o£ this energy emitted in the

frequency interval (v,v+dv) is eii^(v)dv. Thus the contribution to the

original beam due to spontaneous emission is

hv .

,

— e^j,(v)nj^Aj^^dvdadsdto . (6.3)

Stimulated transitions must also be taken into account. The radiation

field is a boson gas. Therefore, according to the general principles of

quantum statistics, in order to allow for induced emission, the probability

of a spontaneous transition in a given cell of the phase space of coordinates

and momenta must be multiplied by 1 + N, where N is the number of particles

(in this case of photons) in this cell. If f is the phase density of photons

in the space (v, r, w) (the intensity ly is given by 1^ = chvf; see Sec. 1.1)

and N is the number of photons in a cell of the phase space of coordinates r and momenta p, then

fdvdVdoj = 2N iii-

, h

2 where dr is the volume element of the space (r, £_) , i.e. dr = p

dpdVdca.

Here p is the momentum of the photon: p

= hv7"c; the factor 2 allows for two

possible states of polarization. Hence

N = -— f , 2v2

or

XT 2 N = c2hv5

I, . (6.4)

Therefore, to account for stimulated emission, the probability of spontaneous

transition is multiplied by

2

1 + N = 1 + —— I , . 3 2hV^

Thus due to k -> i radiative transitions, both spontaneous and stimulated, the

energy in the original beam is increased by

e.j^(v)nj^Aj^. (1 + — I^(r,u))] dvdadsdw . (6.5)

Using (5.49), we can rewrite (6.5) as
1.6 RADIATION TRANSFER 39

2hv^ g. / 2

Combining (6.1), (6.2), and (6.6), we must have, to conserve energy,

[I^(r + tods,oj) - I^(r,ti)) ]dvdada) =

2hv?, g. / 2 - k.j^(v)n.I^(r,w) +

iii ^

^ik'^^'^^'k I ^ ^ —^ ^v^I'^^ 1

1 dvdadsdo).

Expanding I^(r + a)ds,w) in a Taylor series, we get

9I^(r,w) 31 '^-'-^ ^Iv'^I'-^

cos9, + cosB- + cos6, = ^ ay 9z

= - k., (v)n.I (r.aO ^ k., (v)n, (l - ^^L^lA > \

2hvjj^ /

where the quantities cosQ^ are the direction cosines o£ the vector w_:

(£ = i^cos6-j^ + jcos02 + kcosBj . (6.8)

Equation (6.7) may also be written in the form

w-VI^(r,a)) = - k-j^(v)n.I^(r,a3) +

2hv\ g. I 1 . ^

k., (v)n, I 1 + — I (r,w) 1 , c2 Sk il^' ' M 2hv?j^ ^ -'- '

or, regrouping the terms on the right.

(g- \ 2hv?, g. -

it "kj ^v^I'H) ' gt ^ik^^^ ^k • (6.9-)

(6.9)
40 BASIC CONCEPTS

This is the equation of radiative transfer in a spectral line (in the approximation of complete frequency redistribution).

The coefficient of -I^ on the right side of equation (6.9') is the

volume absorption coefficient in the line:

The second term on the right side of (6.9') is the energy spontaneously

emitted into the line per unit volume per unit frequency interval around frequency V per unit time per unit solid angle. This quantity is known as the

line em.ission coefficient ^j^j^C"^) •

^ik

2hv ?iik

(V) = _^-Jk.,(v) . (6.11)

The ratio of the line emission coefficient to the line absorption coefficient

is called the line source function S . It follows from (6.10) and (6.11) that

under the assumption of complete frequency redistribution, the line source function is independent of both frequency and direction and is therefore a function of position only. It is related to level populations by

-

c2 \Si "k /

Defining T^}^ as the excitation temperature of level k relative to level i, as in (2.8), we can rewrite (6.12) as

(^ik^I^) S.j,(r) = BvijT^^viL)] , (6.12')

where Bv(T) is Planck's function. By using (6.10) and (6.12), the transfer

equation (6.9') may be expressed in the form

a).V I rr,a3) = -a . , (v) I (r , o)) - S . , (r) . (6.13)

TRANSFER OF CONTINUUM RADIATION. We shall now obtain the equation of transfer for continuum radiation. The change in intensity along the path ds is

caused, on the one hand, by losses occurring in the photo- ionization of atoms

and in free-free absorption and, on the other, by contributions from spontaneous and stimulated radiative recombinations and free-free emission.

If radiation with frequencies (v,v + dv), propagating v;ithin the element

of solid angle dto around the direction w is incident upon a volume dads, the

energy lost per second in photo- ionizing atoms is
1.6 RADIATION TRANSFER 41

k^^(v)njI^(r,aj)dvdadsdco , (6.14)

j = i

where kj^(v) is the atomic absorption coefficient in the j-th continuum,

so that

k.^(v) = 0, V < v.^, j=l,2,... (6.15)

The free-free energy loss is

"e^^^cc*-^-* ^v'-i'-'''^^^'^'^^'^'^ ' (6.16)

where kpp(v) is the corresponding absorption coefficient. Combining (6.14)

and (6.16), we get the total energy loss:

^^kj ^ (v) n

j I^ (r_, to) dvdadsdoj , (6.17)

3 = 1

where, as in Sec. 1.4, the summation extends over both the discrete states

(j = l,2,...) and the continuum (j = c), with n^ =

^e^"""* In spontaneous radiative recombinations to level j the energy emitted

in an element of phase volume dvdVdo) about the phase point (v,r_,oj) is

(dV = dads)

, , ,2hv^ -hv/kl + Sj '^^jc^^'^ , , rA k. (v) e n n —4- t— e dvdadsdtu . (6.18)

c2 ^ 2(27TmkT)3/2

This expression is obtained by assuming a Maxwellian velocity distribution

for the electrons corresponding to temperature T. Its derivation, which

presents no difficulty, makes use of (3.31).

The energy produced by free-free emission is given by (6.18), with j 0 a

3/2

replaced by c, on the assumption that v^^ = 0 and

^ 3

h-^

In accordance with the general rule for including the effects of stimulated transitions, (6.18) must be multiplied by 1 + (c2/2hv3)i^. Summing

the resulting expressions over all j, including the continuum, we find,

finally, that the total energy emitted by radiative recombination and freefree emission is
42 BASIC CONCEPTS

]+ ^

g. hv . /kT

n n >^ k. (v) —^ e dvdadsdw

(6. 20)

2 L 2hv3 ^

^

j = l

From (6.1), (6.17), and (6.20) it follows that

aj-VI^(r,(£) = k^^(v)n^I^(r,(£)

j = l (6.21)

3 r _2 n . _1 g. hv. /kT

^ 2hv" -hv/kT + e c2

1 + I rr,co)l n n"" V k. (v)

^ e 2hv

Equation (6.21) is the radiative transfer equation iu the continuum for a Maxwellian gas.

It is sometimes useful to group the terms on the right side of (6.21) a little differently, and also to convert from the level populations nj to the

Menzel parameters bj , according to the expression (2.5). Equation (6.21) will

then assume the form

a:.VI^(r,a)) = _ n^n^ ^k . Jv) |l e^""^ ^^^^^b^ - e '^"^Z^^) I^ (r ,(£) + .

3 = 1 c (6.21')

^ 2hv^ -hv/kT

c2

^ ^

gj hv /kT

n n Y^k. (v) -1 e

j = l

with b = 1. c The quantity

a

- - e

j = l

is the volume absorption coefficient in the continuum. The function a (v)

exhibits discontinuities at the series limits (the famous "saw" whose diagram.matic representation appears in all textbooks on astrophysics) .

The emission coefficient e''(v) in the continuum is given by
1.6 RADIATION TRANSFER 43

2hv'' g-hv/kT hv.^AT

n n^Vk. (v)

^ e

^ (6.23)

c c The ratio e (v)/a (v) is the continuum source function:

S^(r) £''(v)

a''(v)

(6.24)

In terms o£ the quantities just introduced, we can write the transfer equation (6.21') in standard form:

VI^ (r , w) a''(v) I^(r,c£) - S^(r) (6.25)

The transfer equation is, of course, subject to boundary conditions.

Let the gas occupy a volume bounded by a convex surface, which does not

reflect radiation incident upon it. In this case the boundary condition is

oj-n < 0 (6. 26)

where r^ is the radius -vector of an arbitrary point on the boundary, n is the

unit vector along the outward normal to the boundary at point r_o > ^\)(~^0>!£) is a specified function representing the intensity of radiation of

frequency v incident from the outside onto the boundary of the medium at

point r_Q in direction co. If the boundary is capable of reflecting radiation,

or if the boundary surTace is not convex, the condition (6.26) must be

altered correspondingly.

RADIATIVE TRANSFER EQUATION FOR SPECTRAL LINES SUPERIMPOSED ON THE CONTINUUM.

Strictly speaking, we were not completely consistent in the derivation of

the above set of transfer equations, since we did not allow for the interaction of the continuous and line spectra. An example of a process involving

this interaction is the photo- ionization of excited atoms by line radiation

of the same element. Thus Lyman-alpha photons can ionize hydrogen atoms from

excited states. We have also tacitly assumed that the gas should have only

one component, i.e. be composed of atoms of only one element, and that this

element is present in only two stages of ionization, with the ions of the

higher stage all in the ground state. Obviously these assumptions are satisfied strictly only for pure hydrogen.

If these assumptions are not valid, then in the line transfer equations

one must take into account not only radiative processes caused by i i k

transitions, but also the loss of radiation in photo- ionizing the atoms

that provide the primary source of opacity in the continuum. Emission from

radiative recombinations in this element and free-free processes must also

be included. The line transfer equation then assumes the form
44 BASIC CONCEPTS

a)-VI^(r,a)) = - ^a.j^(v) + o^j^^ I^^I'^^ ^ ''ik '^''^ ^ik "^I^ ^ ^ik' l^^-^^^

c c where aj^]^ and e^-^ are the absorption and emission coefficients in the con- tinuum at frequency \>iy^. Within the line they can be assumed to be independent of frequency.

Let us introduce the total source function

S = — ^ . (6.28)

Clearly it can be represented as

^ik'^''^ ''ik^ c S = — S., + — S^ , (6.29)

aij^(v)+a.j^ a.j^(v).a.j^ ik

where S^y^ is the line source function and S^j^j^ is the continuum source function at the line frequency. With (6.29) in mind, we can rewrite the transfer

equation (6.27) in the form

aj.Vl^(r,(£) = -^a.j^(v)+a.j^^^ ^I^(r,(£)-S^(r)^ . (6.30)

THE CASES OF TE AND LTE . We now have the complete set of coupled transfer

and statistical equilibrium equations. From this set it is instructive to

find the relations describing TE . For simplicity we assume the gas to have

only a single component, with kinetic temperature T.

We assume first that the gas fills the whole space and that the radiation field is homogeneous and isotropic, i.e. I^ does not depend on position

and direction. Then the left side of the transfer equation (6.9') vanishes,

so that

3 2hv.-, g. - k., (v) (n. - — n, ) I + — — k., (v)n, = 0 . (6.31)

ft >)

Consequently the intensity of radiation within the line is independent of

frequency and

I = ('A^-L-y . (6.32,

In the case at hand Jik = Ivt,- The direct substitution of (6.32) then shows

that radiative transitions between the discrete levels satisfy the detailedbalance conditions:
1.6 RADIATION TRANSFER 45

^kf%i ^ ^ki ^ik^ = n. B.^ J.^, i=l,2,..; k>i . (6.33)

We emphasize that so far neither the level populations nor the intensity of

radiation are specified. The only restriction imposed is that they satisfy

the relation (6.32). Therefore, this relation is the condition for the

radiative i Z k transitions to be in detailed balance.

With (6.33) taken into account, the set of statistical equilibrium equations (6.3) reduces to

c i-1

n (A . + B . J. ) + n, n Ci • + "V^ n.n C-. = c^ ci ci ic^ / ^ k e ki / j ] e ji

k=i+l j=l

= n.> nC.+n.B. J. +n. "S ^n C . , , i = l , 2 ,

1 / > e ij 1 ic ic 1 / J e ik' ' '

j-1

j=l k=i+l

Let us seek its solution in the form

n. g. -hv^./kT

1 ^1 - li

(6.34)

^1

e , i=l,2,...;c, (6.35)

which is a compact way of writing both the Boltzmann (i=l,2,...) and Saha

(i=c) laws. In this case the collisional transitions satisfy the detailedbalance relations (4.7) and (4.8). Hence, in the statistical equilibrium

equations (6.34), terms that account for collisional transitions cancel. It remains to be shown that if the level populations satisfy (6.35) and the radiation field is homogeneous and isotropic, then

n (A . + B . J. ) = n.B. J. , i=l,2,... . (6.36)

C Cl Cl IC^ 1 IC ic' '

To prove this result let us turn to the radiative transfer equation in

the continuum. Since, according to the initial assumption, I^ does not depend

on r, (6.21) gives

2hV

V 1 r ^ hv/kT / k . (v) n . g e jc^ j^c

j = l ^

hv. /kT

> k. (v)n g. e Z—/ jc^ ^ c*j

j = l

-1

(6.37)
46 BASIC CONCEPTS

whence, taking (6.35) into account

i.e. the intensity o£ radiation is Planckian with T^, = T. Using (3. 27) - (3. 30)

it is easy to verify that the relation (6.36) is indeed satisfied, provided

the intensity of radiation is Planckian and the level populations satisfy

(6.35). With (6.35) and (6.36) in mind, we find that the ionization equilibrium equation (4.4) is also satisfied.

Hence, the assumption that the radiation field in a Maxwellian gas filling the whole space is homogeneous and isotropic leads to the inevitable

conclusion that this intensity is Planckian, and that the level populations

and the degree of ionization necessarily obey the Boltzmann and Saha laws.

In short, if the radiation field in a Maxwellian gas is homogeneous and isotropic, the gas is in TE

The assumption that the gas fills the whole space is not necessary. Let

us isolate in an infinite gas an arbitrary volume V. In the present model

the interaction of remote volumes is due only to the exchange of radiation.

The effect of the remainder of the gas is only to irradiate the volume V from

the outside with radiation of intensity (T) . Therefore, if the gas fills

only the volume V and is illuminated by radiation with the intensity B^(T)

the conditions in V will be exactly the same as in the infinite medium. This means that TE holds in an arbitrary cavity filled by gas and illuminated by

Planckian radiation.

In deriving the statistical equilibrium equations we used the relations

found from thermodynamic considerations. Hence, strictly speaking, this line

of reasoning does not constitute a proof. However, it does seem to be of

interest. This argument clearly shows how, in the specific model, the

kinetic equations lead to the equilibrium relations as a limiting case. Now a few words about LTE . In LTE the coupled statistical equilibrium

and radiative transfer equations are radically simplified. The solution of

the statistical equilibrium equations for the discrete levels and the con- tinuum is given by Boltzmann's and Saha's laws. Since the level populations

are known, the transfer equations become first-order differential equations.

We shall consider in a little more detail the limiting form they assume in

this case. Since the level populations obey Boltzmann's law, the volume absorption

coefficient in the line, as expressed by (6.10) is now

/ .hv /kT\

a.j^(v) = k.j^(v)n. h-e ). (6.39)

The line source function (6.12) reduces to Planck's function for the local

kinetic temperature:

(6. 40)
1.6 RADIATION TRANSFER 47

so that the transfer equation (6.13) becomes

a)-Vl^(r,aj) a-i^Cv) - (T(r)) (6.41)

Analogously, the absorption coefficient in the continuum given by (6.22) assumes the form

a^(v)

3 = 1 ^ (6.42)

and the continuum source function is found to be equal to Planck's function:

S^(r) = B^(T(r)) , (6.43)

so that instead of (6.25) we have

VI^(r ,co) = - a^(v) I (r,oj) - B y:T(r))] (6.44)

As follows from (6.41) and (6.44), when the temperature is a known function

of the coordinates, the calculation of the intensity involves a theoretically

simple, but sometimes cumbersome, quadrature. Since the temperature can change from point to point in LTE , the absorption coefficients vary with

position

Until now the kinetic temperature of the medium has been regarded as given. Yet in fact it is determined by an energy equilibrium equation which

must be solved simultaneously with the statistical equilibrium and radiative

transfer equations. If absorption and emission of radiation are the only

mechanisms for heating and cooling the gas, and its state does not change

with time,_ then such a gas is said to be in a state of radiative equilibrium . In this case the equation expressing the energy balance is called a radiative

equilibrium equation. For a single- component gas it obviously has the form

00 /

(i<k)

(v) (r) dv

CO

i,k

(i<k)

dv (6.45)

and in particular, in LTE

00 j

[^^(^) " 2 ^ikt^)][jvd) -^(nr))] dv

i,k

(i<k)

(6. 46)

Even in LTE the simultaneous solution of transfer and radiative equilibrium equations is rather difficult. This constitutes the central problem of

522-519 O - 74 - 5
48 BASIC CONCEPTS

the classical theory of model stellar atmospheres, a theory based on the

a priori assumption of the existence of LTE in stellar atmospheres (see, e.g.,

G. Miinch, 1960).

Henceforth we shall always regard the temperature as known. However, if

the gas density is low and LTE cannot be assumed, the situation is still very

complicated. In the statistical equilibrium equations terms appear depending

on the intensity at a given point in the medium. On the other hand, the intensity, which is found by solving the transfer equation, depends in turn

on the level populations. Therefore, if radiative processes play an important

role in populating the levels, the calculation of statistical equilibrium

states involves the simultaneous solution of the statistical equilibrium and

radiative transfer equations. This is exactly the problem to which this book

is addressed.
CHAPTER II

THE LINEAR APPROXIFIATION

The system of statistical equilibrium and radiative transfer equations

is nonlinear, as we have already seen. As the solution of this system, in

full generality, entails tremendous difficulties, which have yet to be

completely overcome, we are limited to rather schematic models. Consequently,

it seems logical to study in detail all of the specific cases for which

exact solutions can be obtained in closed form. Such cases are far from

plentiful; moreover, they all relate to highly idealized situations that are sometimes rather far removed from the concrete problems encountered in the

interpretation of observations. Nevertheless, such model problems are of

great interest

First of all, these simple models can illustrate specific features of

the general problem. The study of such model problems aids us in reaching

a clear understanding of the physical aspects of the problem, and provides

the orientation necessary, in the more complicated cases, to sift the

essential ideas from a mass of details. Second, because these model problems

have exact solutions, they serve as standards for estimating the accuracy and

limits of applicability of various approximate and numerical methods. Finally,

I these problems are, as a rule, of interest in their own right. The greater

part of this book, then, is devoted to a study of these standard problems.

It is natural that an exact solution is much more readily obtained in

those cases in which the set of statistical equilibrium and radiative trans-

. fer equations can be linearized. With this in mind, we shall consider the

linearized form of these equations in this chapter.

I The linear theory of radiative transfer in spectral lines is a generalij zation of the now classical theory of monochromatic scattering. We begin,

I therefore, with a brief survey of the fundamental concepts of the phenomenological theory of radiative transfer when frequency changes are neglected. A

detailed discussion then follows of the simplifications that arise when the

set of statistical equilibrium and radiative transfer equations is linearized.

It is shown that in certain instances intrinsically nonlinear problems can be

reduced to linear ones. Finally, at the end of the chapter we study the

j properties of a number of special functions that play an important role in

the study of a state of a gas in its own radiation field.
50 THE LINEAR APPROXIMATION

2.1 MOKOCHROMATIC SCATTERING

BASIC ASSUMPTIONS AND DEFINITIONS. Two basic assumptions are made in the

classical theory of multiple scattering of radiation. First, the radiation

is assumed to have no effect upon the medium in which it is propagated,

thereby guaranteeing the linearity of the transfer equation. Second, the

frequency of the photons is assumed not to change as the radiation interacts

with the matter. Although this assumption is definitely not valid for multiple scattering within a line (see Sec. 1.5), there are many cases in which it

does represent a very good approximation to reality. As an example, one might point to the multiple scattering of light by small particles. We may

assume the frequency to be constant if over the spectral interval that includes the possible initial and final frequencies of the photon, the optical

properties of the medium do not change substantially. The frequency dependence of the optical properties may then be neglected and the frequency of

the photon becomes immaterial. We may therefore assume that the frequencies

of all photons are equal; i.e. we may speak of the scattering of monochromatic radiation with a fixed frequency.

The theory of multiple scattering of light is very closely related to

neutron transport theory. When the energy of the neutrons changes only

slightly as they are scattered by nuclei (thermal neutrons), the dependence

of the scattering cross section upon energy may be disregarded. In neutron

transport theory this situation is referred to as the constant cross section

approximation, or the one -group approximation. In this approximation neutron

transport is mathematically identical to monochromatic scattering of radiation. Therefore, although we shall speak of photons and use the terminology

of the theory of light scattering, all of the results in this section (and

also in Chapter III) apply to neutrons as well.

In the theory of monochromatic scattering the optical properties of an elementary volume of the medium are characterized by three quantities —the

volume absorption coefficient o, the probability A that a photon survives

the act of scattering, and the phase function or scattering indicatrix xCy)*

These quantities are defined as follows. Let radiation of intensity I (the

subscript v may be omitted throughout in the theory of monochromatic scattering) be incident upon a unit volume within a unit solid angle about a certain

direction. Then ol is the radiant energy interacting with matter per unit

time. This interaction may be of two types. Part of the energy, say o

,

is absorbed, i.e. is converted to another form of energy (for example, heat).

The quantity is called the volume coefficient of true absorption. The

remainder of the radiation only changes its direction of propagation in the

interaction with matter, and conserves its initial frequency. We shall denote this so-called scattered energy by Ogl. The quantity Og is the volume

scattering coefficient. It is obvious that

0=0^^0^. (1.1)

The ratio

A = — = 5— ' (1.2)

a 0 + a

s a

is the probability that a photon survives the act of scattering. This quantity is known as the albedo for single scattering. (Other notations for the same quantity frequently encountered in the literature are: , cog, 1 — e.)

If X = 1, then the scattering is said to be pure, or conservative. As stated
2.1 MONOCHROMATIC SCATTERING 51

in Sec. 1.5, the phase function x(y) (often also designated as x(y), P(y).

etc.) is the probability density of scattering through an angle y from the

initial direction. Its normalization is

fx(Y)dco = 1. (1.3)

In general, the medium emits, as well as absorbs, radiant energy. The

energy emitted by a unit volume per unit of time within a unit solid angle

is called the emission coefficient e. For monochromatic scattering, the

emission coefficient e and the intensity I may refer to any spectral interval

over which the optical properties of the medium do not depend upon frequency,

and not simply a unit frequency interval.

The emission coefficient £(r,a}) depends, in general, on both position

and direction, and is composed oT two parts. First, a volume element scatters

a part of the radiation incident upon it. The contribution to the emission

coefficient e from scattering is obviously equal to

A£ yI(r,a)')x(Y)da3' ,

where y is the angle between co and w ' . Second, within the volume element

there can be internal sources of radiation whose strength is independent of

the intensity of radiation incident upon it, and is regarded as given. These

are known as primary or true sources. Denoting their contribution to the

emission coefficient as e*(r,a)), we have

£(r,a3) = ^ /'l(r,a)')x(Y)da3' + £*(r,a3) . (1.4) — — An J — — — —

The ratio of the emission coefficient to the absorption coefficient is the

so-called source function:

£ (r ,0)) S(r,a3) = 7

-

. (1.5)

a(r)

If the phase function is spherical, i.e. x(y) = 1, and the primary sources

are isotropic, so that e* = £*(r), then the source function does not depend

on direction and is equal to

S(r) = — f I(r,co')da)' + S*(r) , (1.6) — /\t\ J — — —

where S* is the primary source function or the source term:

S* = — . (1.7)

a

TRANSFER EQUATION. The radiative transfer equation is obtained, as always,

by considering the change in intensity along the beam. For monochromatic

scattering it is

co«VI = -al + e , (1.8)
52 THE LINEAR APPROXIMATION

or wVI = -a(I - S) (1.9)

This equation has the same form as (1.6.13) and (1.6.25). However, in (1.9)

the absorption coefficient a is regarded as given, whereas the quantities

aj^]^(v) and aC(v) depend on the level populations which, in general, are not

known at the outset.

For the spherical phase function and isotropic primary sources, (1.6)

and (1.9) give us

wVI (r ,Lo) = -o (r) [ I (r ,co) ~47T Jf

I (r ,oj' )dLo' - S*(r) ] (1.10)

We shall henceforth deal exclusively with (1.10) in order to avoid unneces- sary complications. The boundary condition for (1.10) is

I(rQ,w) = I^(rp|,w), a)*n < 0 ^0 , (1.11)

where r^ is an arbitrary point on the boundary surface, and n is the unit

vector normal to the boundary in the outward direction at the point r ^ . However, with no loss of generality, this condition may be changed to

I (r^ ,w) co*n < 0 (1.12)

for if I (r,t£) is the intensity of radiation at point r_ in direction co that

has come directly from external sources and has been attenuated by the medium,

then one may set

S*(r:

a -/4tt I°(r,to')doo' (1.13)

and solve the transfer equation (1.10) with the boundary condition (1.12).

By adding I*^(r,w) to I (r ,0)) the total intensity at a given point can be obtained. The quantities I^ and I are called the intensities of direct and

diffuse radiation, respectively.

If the absorption coefficient does not depend upon the coordinates

(a = const), the transfer equation (1.10), together with boundary condition

(1.12), is equivalent to the following integral equation for the source function:

S(r)

^^ J

exp (—0 I r — r '

I

)

S(r')dr' + S*(r) (1.14)

r — r

where the integration extends over the entire volume occupied by the medium,

Equation (1.14) is called the Peierls equation. (Its derivation may be

found, for example, in B. Davison's book (1958); see also Sec. 2.4.) When

the source function is found, the solution of the transfer equation reduces

to quadrature, as may be seen from (1.9).
2.2 THE TRANSFER EQUATION FOR SPECTRAL LINES 53

When the medium and the distribution o£ primary sources possess a particular symmetry, (1.10) and (1.14) are simplified. However, we shall postpone for a moment the analysis of these simplifications in order to clarify

the analogy between the problems of monochromatic scattering and the linear

problems of radiative transfer in spectral lines.

2.2 THE TRANSFER EQUATION FOR SPECTRAL LINES

GAS OF TWO-LEVEL ATOMS. The simplest problem in which one calculates the

steady state of a gas in its own radiation field is as follows. An isothermal gas, composed of atoms having only two discrete levels, occupies a given

region. The density of the gas and the electron concentration are known functions of the coordinates. The gas is not illuminated from the outside, and

has a sufficiently low temperature so that the average thermal energy of the

particles is much less than the excitation energy of the upper level:

kT << hvx2- We desire to find the degree of excitation of the gas and to

calculate the radiation field in the line.

Under these assumptions, the population 1x2 of the excited level will

everywhere be small compared to the concentration n^ of ground state atoms.

An isothermal medium of infinite extent will be in TE , with the level populations given by Boltzmann's law. If hvi2 kT, the degree of excitation

will be very low, even in an infinite medium, and if the gas occupies a finite region, then it will obviously be lower still. Therefore the total con- centration of atoms n = n^ + n2 is very nearly equal to the population ni of

the lower level. Furthermore, the line would have a Planckian intensity in

an infinite medium, and the factor 1 + (c2/2hv^)I^, allowing for the stimulated emission, would differ from unity only by terms of order exp (—hv 2/^^) . In a finite medium from which some radiation escapes, this quantity is even

closer to unity because the intensity is less than its equilibrium value.

Consequently stimulated emission may be neglected. The statistical equilibrium and radiative transfer equations therefore assume the form

"2^^21 ^ ^e^21^ = ^lf^l2^12 ^ ""e^lZ^ ' ^^"^^

2hv^,^ g,

- 03-^1^^^'^^ = -k]_2^^^^1^v^I'^^ * f^^^l2f^^^2 • ^2.2)

Using the familiar relations between the Einstein coefficients, expressing

C12 in terms of C21, and introducing the explicit expression for J12 given

by (1.3.25), we have instead of (2.1)

n 2 g / k 2(v')J ,dv' g2 -hv^^/kT — (A +n C ) = — — A.. - — + — e n C.^,

^^-21 "e 21'' ^ r. ^71 «> rr e 21

2h^l2 ^1 ^ / k,,(v')dv

0 12

We now define
54 THE LINEAR APPROXIMATION

2hv^. g, S = ~ — — . (2.5)

The quantity A is the probability that a downward radiative transition will

occur following the excitation o£ an atom. In other words, A is the probability that a photon survives the scattering process. The quantity S is

the line source function when stimulated emission is disregarded.

Using (2.4) and (2.5), we can write the condition of statistical equilibrium (2.3) in the form

/o^2'^v')dv' f

I^, Cr,a)')dcu'

S = — + S* , (2.6)

4 TT ~

whe re

2hv^-, -hv.^/kT

S* = e (1 - A) (2.7)

and 0]^2(^) is the volume line absorption coefficient:

0^2^^) = ^i2'^^^'^l • (2-8)

The transfer equation may be rewritten as

a3-VI^(r,(^) = -0^2^'^^ l^^v^^l'^^^ -S(r)) • (2.9)

Substituting (2.6) into (2.9), we finally arrive at the following linear

integro-dif ferential transfer equation:

w*VI^(r,w) =

/o^2^v')dv' / I^, (r,a)')do

-a,2Cv)

( I^(r,.) - S*(r)l. (2.10)

We stress that in the present case the line absorption coefficient is known,

since n-^ can be assumed to be equal to the total concentration of atoms,

which is regarded as given. The transfer equation (2.10) is subject to the

boundary condition

I^(rQ,aj) = 0, oj-n < 0 , (2.11)
2.2 THE TRANSFER EQUATION FOR SPECTRAL LINES 55

which expresses the absence of radiation incident on the medium from the outside. Once the radiation intensity is found, the problem is solved, since

from the statistical equilibrium equation the population of the upper level

may be calculated directly.

GENERAL CASE. The assumption that an atom has only two levels is not necessary for the linearization of the transfer equation. If stimulated emission

may be neglected and the population of the lower level is regarded as given,

the linearity of the transfer equation will be ensured. Let us consider the

problem of computing the radiation field in a line formed by transitions

between levels i and k. If we disregard stimulated emission, the line trans

fer equation (1.6.9) assumes the form

u)«VI^Jr,aj) = -k^j^Cv)n. I^, Cr ,co) +

1 V

2hv.T g. ik *i

c^ Sk— k.j^(v)n^ , (2.12)

and the condition of statistical equilibrium may be written as

\^ - ^^ik^ik * ^k (2.13)

where n^Dk is the number of transitions from level k per unit of time by all

possible means except k -> i radiative transitions, and E]^ is the number of

transitions into level k due to all processes except i k photo-excitations

by diffuse radiation. Repeating literally the arguments used in deriving

(2.10), we get the following linear transfer equation from (2.12) and (2.13)

w«VI^(r,(£) = -a(v)

where

I (r,oj) - —

CO /a(v')dv' / I^, (r,a)')dc

CO ^a(v')dv'

- S*(r) , (2.14)

X = Aki ^ Dk

(2.15)

2hvrik

c2 gk H^^ki ^ Dk)

(2.16)

Since the population of the lower level is regarded as known, the absorption

coefficient a(v) (v)ni is known. The situation is more complicated for

the quantities X and S*. Through Dk and Ek , X and S* depend, in general, on

the populations of all levels except the k-th, and also upon the radiation

fields in all lines and continua except that in the i i k line. In several

instances, however, these quantities can also be regarded as known. Examples

can be found in the work of V. V. Sobolev (1962) and V. P. Grinin (1969).

STANDARD FORM OF THE TRANSFER EQUATION. Throughout this book we shall assume

that the frequency dependence of the line absorption coefficient is identical

at all points in the medium. Then we can set
56 THE LINEAR APPROXIMATION

a(v) = a(vQ)a(x) = n^k^^ ) a (x) , (2.17)

the profile o£ the absorption coefficient a(x) being independent of position.

If the broadening of the line is caused by the Doppler effect, this assumption implies that the medium is regarded as isothermal (since temperature

variations lead to changes in the Doppler width in terms of which the dimensionless frequency x is defined). If, moreover, the line broadening is

caused by collisions (Lorentz profile), the density should also be constant.

If both line broadening mechanisms are significant simultaneously (Voigt profile) , then the assumption that the absorption coefficient is independent of

position also implies that both the temperature and the concentration of the

particles responsible for the pressure broadening are constant throughout the

medium

Substituting (2.17) into (2.14), we get the radiative transfer equation

in the standard form:

03* VI (r, CO, x) = — a (Vq) a (x) I (r , w ,x) +

00

+ oCVq) ^ Aa(x) J

a(x')dx'y I (r ,0) ' , x ' ) doj ' + a (Vq ) a (x) S* (r) , (2.18)

or

aj-VI(r,w,x) =-o(vQ)a(x) ^I(r,a3,x) -S(r)^ , (2.19)

where S(r) is the line source function:

oo S(l) = J

a(x')dx'y I (r ,0)' ,x' )doj' + S*(r) , (2.20) —OO

and A is the normalization constant:

00 Ay a(x)dx = 1 . (2.21) —oo Although we are now using x as the frequency variable, it is important to

stress that the normalization of the intensity is unchanged , so that

I(_r,a),x) dadtodv is the energy flowing per second through an area da perpendicuTar to the beam direction, within the solid angle dco in the frequency

interval (v,v+dv), where v = Vq + xAv.

For cases in which absorption and emission in the continuum must be

taken into account, the line transfer equation assumes the fo'rm (see (1.6.27))

w-VI(r,aj,x) = -a(vQ)a(x)^I(r,a),x) - S (r) j -a^ ^I (r , oj ,x) -S^(r)^ , (2.22)

where S (r) is the continuum source function:
2.3 PLANE AND SPHERICAL GEOMETRIES 57

(2.23)

and S(r) is, as before, given by (2.20). The functions S* and S describe

the distribution of primary sources of radiation in a given line. As has

already been mentioned in Sec. 1.5, whenever the k ^ i radiative transition

does not occur following an i -> k photo-excitation by the medium's own line

radiation, a photon is considered to have come from primary sources.

The transfer equation (1.10) describing isotropic monochromatic scattering is a special case of (2.18). Indeed, let us suppose that the profile of

the absorption coefficient is rectangular, so that

( 1 , |x| < 1 , a(x) = < (2.24)

( 0 , |x| < 1 .

From (2.18) it then follows that within the line (in the case at hand for

|x| < 1), the intensity does not depend on frequency, and equation (2.18)

reduces to (1.10). The physical explanation of this is as follows. If the

profile is rectangular, all of the line photons have the same mean free path,

which is unchanged by frequency redistrilDution . Hence all the photons may be

regarded as having the same frequency.

2.3 PLANE AND SPHERICAL GEOMETRIES

If the medium has any particular symmetry, the transfer equation becomes

simplified. We shall consider the cases of plane and spherical geometry.

PLANE GEOMETRY. When the absorption coefficient at line center a(vo), the

continuum absorption coefficient o'^ , the photon survival probability A, and

the functions S* and S*^, which describe the distribution of primary sources,

depend upon only one spatial coordinate (for example, z) , the system has

plane symmetry. The intensity of radiation I is then a function of z, the

frequency, and the angle between the positive z-axis and the direction of

propagation of the radiation. This angle, designated as 63 in the previous

chapter (in Sec. 1.1 and 1.6), will henceforth be denoted simply as 0 (Fig.

2). Instead of (2.22) we now have, taking (2.20) into account,

z

cos i?=yU.

Fig. 2. Notation used in the transfer equation for plane geometry.
58 THE LINEAR APPROXIMATION

+ 0

cose '^^'^^^^'^^ =-^a(vQ)a(x) + a^^ I(z,e,x) +

00 (Vq) ^ Aa(x) J

a(x')dx'y I ( z , 6 ' ,x ' ) doj ' + o (v^ ) a (x) S* ( z

) + a'^S^(z)

(3.1)

It is convenient to define

^0 ^0

T f

a(Vg)dz' = k^j^Cvg) f

n.(z')dz' (3.2)

and to refer to this quantity as the optical distance at line center from the

plane z = Zq. This distance is measured in mean free paths of a photon at

line center (for = O) . Setting y = cosG, we have, finally,

i ^^^"^;'^^ = (a(x) - I(t,m,x) -

1 J

- a(x')dx'y ^

Aa(x) I

a(x')dx'y I (t ,p ' ,x ' ) dy ' -a(x)S*(T) -6S^(t) , (3.3)

or

^dIXT^£_^ = a(x) (l(T,M,x) - S(T)^ + B ^I(T,y,x) - S^(t)^ , (3.4)

where

00 1 S(t) = aix')dx'

J' I(t,m' ,x')dy' + S*(t) (3.5)

_oo _1

and 3 is the ratio of the continuum to the line center absorption coefficients :

- - (3.6)

o(vo)

We now obtain the boundary conditions to be satisfied by the intensity.

If the integral

CO Tq = j

a(VQ)dz' (3.7) —oo

converges, the medium is said to have a finite optical thickness Tq. To

avoid dealing with negative optical distances, it is convenient to define the

T-coordinate system by the expression
2.3 PLANE AND SPHERICAL GEOMETRIES 59

00

= J

a(vQ)dz' , (3.8)

i.e. to set zq = °o in (3.2). The quantity t in this case is known as the

optical depth ; it varies from 0 to tq. The boundary conditions expressing

the absence o£ radiation incident upon the medium are

1(0, y,x) = 0 , M < 0 , I(TQ,y,x) = 0 , y > 0 . (3.9)

I£ the medium is illuminated by external radiation, the transfer equation may,

as before, be solved with the boundary conditions (3.9), after the function

describing the distribution of primary sources has been redefined and the

direct and diffuse radiation fields have been separated (see Sec. 2.1).

If we consider the line source function S(t) to be known, the solution

of (3.4) with the boundary conditions (3.9) can at once be written:

I(t,1j,x) =

T

0

a(x)S(T') + 6(t')S'-(t

T J

exp |-- (a(x)(T' -t) + 6(t)dt^ J

6(t)dt

T

dT

y < 0

I(T

^0 a(x)S(T' ) + B(t' )S''(t

(3.10)

exp i ^a(x) (t' - t) + y

B(t)dt^ dx ' y > 0 (3.11)

These expressions give only a formal solution of the transfer equation, since

the line source function S(t), as is seen from (3.5), itself depends on the

intensity

If the integral (3.7) diverges, and (3.8) converges at any z ^—°°, then

the medium is referred to as semi - infinite , or as a layer of infinitely large

optical thickness. In this case it is convenient to define the x-coordinate

according to (3.8). The first of the boundary conditions of (3.9) remains

valid and the second is replaced by the requirement that at xg = °° the integral in (3.11) converges for all x. Finally if at any z the integrals

J"

o(vQ)dz' and j' o(vQ)dz'

diverge simultaneously, the medium may be regarded as optically infinite.
60 THE LINEAR APPROXIMATION

Nearly all of the current research in the theory o£ radiative transfer

in spectral lines is concerned with the solution of (3.3) under additional

assumptions of one kind or another. Usually it is assumed that X and 6 do

not depend on optical depth. Moreover, since for strong lines B << 1, it is

often assumed that 6=0. The transfer equation then becomes

y —dl (t , y , x) r ^ T r ^

'-r-^—- = a(x)I(T,y,x)

dT

00 1 -

^

Aa(x) y* a(x' )dx' y I (x , y

' , x' ) dp ' -a(x)S*(T) , (3.12)

which has the formal solution

T a (x) . , . ^

- (t ' — T ) I(T,y,x) = - /* S(T')e ^ a(x)— , y < 0 , (3.13)

u

0 a(x)^^,

T

I(T,y,x) = / S(T')e ^ a(x)— , y > 0 . (3.14)

y

In particular, the intensity of radiation emerging through the boundary

T = 0 is

^0 a(x) ^,

1(0, y,x) = f S(T')e ^ a(x)— , y > 0 . (3.15) J y

0

The physical content of this result becomes quite obvious if S(t) is replaced

by the expression (2.5) for the line source function in terms of the level

populations and the variable of integration t' is replaced by the geometrical

depth z '

T rn ^ ik Aa (x) '

I (0 ,y ,x) =

- 4iT Av

A,,n^(z')e ^ ^. (3.16)

The transfer equation for isotropic monochromatic scattering in plane

geometry, as follows from (1.10), is

1 ^Lil^ = i(x,y)-i f I(T,y')dy' -S*(T) , (3.17)

dT 2 ^

-1

where t is the optical depth:
2.3 PLANE AND SPHERICAL GEOMETRIES 61

00 /

T = y

a(z')dz' , (3.18)

z

For a layer o£ optical thickness tq with no external illumination, the formal

solution of (3.17) is

T T

' — T I(t,m) = - I S(T')e ^ dx;.

, y < 0 , (3.19)

0 T^0 T ' — T I(T,y) = y

S(T')e ^ , y > 0 . (3.20)

T

Many of

applied to (3

to a study of

is discussed,

line transfer

Chapter IV it

is studied in

placed by the

the case of a given the stu

the methods developed for solving (3.17) may be successfully

.12) as well. The next chapter, therefore, is entirely devoted

monochromatic scattering; in particular, the solution of (3.17)

In subsequent chapters the same procedure is used to solve the

equation. Equation (3.12) is studied in Chapters IV - VI. In

is assumed that the medium is infinite; a semi - infinite medium

Chapters V and VI. In Chapter VII the condition 3 = 0 is re- less stringent requirement 6 = const. Finally, in Chapter VIII

finite value of tq is considered, particular attention being

dy of the radiation field in an optically thick layer (tq >> 1) • SPHERICAL GEOMETRY. Let the optical properties of a medium, i.e. the quantities A, o(vo) and a'^ , and the source strength depend only upon distance r from some point, which we shall take as the origin. From the symmetry of

this configuration, the intensity will depend only on the frequency, the

radius r, and the angle 9 between the radius -vector to a point, and the direction of radiation there. In this case we have

wvi = cosi I sine 31 - 9r r 3(

Defining y = cos 9, from (2.22) and (3.21) we find

(3.21)

. ^^^^>^>^^ + ^-^-H- = - (a(vja(x) ^ a^) I(r,u,x)

3 r r 3y

(3.22)

+ a(vQ)

Y

Aa(x) a(x')dx' f I ( r , y

' , x ' ) dy ' + o (v^ ) a (x) S* (r) + a^s'^(r) . _oo — 1

If cr(v„) and do not depend on r (homogeneous medium), then, introducing the optical distance at line center

T = a(vQ)r (3.23)
62 THE LINEAR APPROXIMATION

we obtain

>^I(t,ij,x) ^ 1 — 9I(t,ij,x) i r . n l t r . y —!^

'-^—^ + = —'-^—J- = _ict(x) + 3jI(T,y,x) + 3t t 9y y y

(3.24)

+ 2 Aa(x) y a(x')dx'

J'

I (t , y

' , x ' ) dy ' +a(x)S*(T) +3S^(t) . - — CO _ 1

With the same assumptions concerning the symmetry o£ the system, the transfer

equation for monochromatic scattering with 3=0 has the form

9iIijlH1 + 1 - 9i(T>y) ^ 8t T 9y

^a(x) + 3^

1 = - I(T,y) + ^ f

I(T,y')dy' + S*(t) . (3.25)

^1

Equation (3.24) for an infinite medium with 3 = 0 is studied in Chapter

IV. In Chapter VII the results are generalized to the case 3 = const > 0.

TWO SCALES OF OPTICAL DISTANCE. In conclusion, we comment on the choice of

the optical distance scale. We have introduced here optical distances t meas- ured at line center . Sometimes the optical distance x is used, which is

related to t by

T = A? . (3.26)

The quantity T has the merit of being independent of the shape of the profile.

When stimulated emission is neglected, t = kii^(vo)Ni, where N^ is the number

of atoms in the lower level along the line of sight and kik(vo) is the absorption coefficient per atom at line center. Using the well-known relation (see

Sec. 1.3)

oo / k,,(v)dv

hv . 1 ik

4tt ik

'0

and the fact that a(x) =

^^ij^ (^ ) /'^j^j^ (^0 ' write

k.j^(vQ) Ja(x)dv = — B. ,

0

whence
2.4 INTEGRAL EQUATIONS OF STATISTICAL EQUILIBRIUM 63

and therefore

k., (v„) = A —— B.,

- 1 ^^ik

T = AA = —— B.., N. . (3.27) 4uAv ik 1

^

^

Thus T, unlike t, depends only on the number of absorbing atoms and the

strength of the atomic transition (and also on the characteristic line width

Av) , but not on the shape of the profile. The use of T instead of x is

necessary when results obtained for various absorption coefficients must be

compared (say, for Voigt profiles with different values of a).

2.4 INTEGRAL EQUATIONS OF STATISTICAL EQUILIBRIUM

DERIVATION OF THE BASIC EQUATION. In the linear case the solution of an inte

gro-dif ferential line transfer equation is, as has already been noted, equivalent to a determination of the population n]^ of the upper level as a function of the coordinates. Once the radiation intensity is known, (2.20) gives

us the line source function S(r) or equivalently ni^(r). We shall now consider an integral equation that expresses directly the condition of statistical

equilibrium. The solution of this equation gives us the line source function

from which nj^ is readily found. It is then easy to calculate the radiation

field in the medium as well.

We shall start from the statistical equilibrium equation (2.13), which

for convenience is reproduced here:

^k^\i ^ ^k^ = ^^ik^ik ^ ^k • ^^-1)

We define as ny- the population of level k in the absence of i k photo-excitations by the medium's own radiation (both in line and continuum). We assume that_ the direct radiation from external sources is not included in

Jiy^, i.e. J]_-^ refers only to the diffuse radiation fi^ld, while photo -excitations by direct radiation are included in Ej^. Then nj^ is determined by the

equilibrium condition

We note that from (4.2) and (2.16) it follows that

3

* * ^^^ik Si "k

S = — — — . (4.3)

*

|| This expression makes clear the physical meaning of the function S . From

' (4.1) and (4.2) we have, taking (2.15) into account,

nj^(r) = An.(r) ^

J.j^(r) + nj(r) . (4.4)

Aki

522-519 O - 74 - 6
64 THE LINEAR APPROXIMATION

An explicit expression for the i ^ k photo-excitation can be obtained, although a certain amount of care is required.

Let dV be the volume element near point r, and dV the volume element

near x' (Fig- 3). Radiative i ^ k transitions in dV are induced by radiation

arriving there from all points of the medium. Let us calculate the contribution due to radiation from the volume element dV . In dV the energy emitted

Fig. 3. Derivation of the equation of statistical equilibrium in integral form.

per unit time in the frequency range (v , v + dv) within the k -> i line is

[hvikAkink(r' )A(a(x)/Av) + 47T£C ' J ] dvdV . We shall denote by da the projection of the volume element dV onto a surface perpendicular to the line

joining points r and x' • The volume dV is seen from dV to subtend a solid

angle da|r - r'|'2. Within this solid angle the energy emitted in dV per

second in frequencies from v to v + dv is

4tt r — r h^ik^i^k^l'^^ Av

+ 4TTe (r ] dodvdV (4.5)

The energy is partially absorbed along the path from r' to r, and partially

scattered in all directions by the medium, so that of~"this energy dV receives

only the fraction

exp {-a(x) T (r ,r ' ) -T'^(r,r')} , (4.6)

where T(r,r') is the line center optical distance between points r and r ' , i.e.

s T(I.I') = ^ik^^O^ / i^i(l")ds' , (4.7)

0 with s = |r — x'l- The integration is performed along the line joining r' and r. Analogously, T'^(r,r') is the optical distance between r and r' in the

continuum.

Of the energy emitted by the element dV in the frequency range (v , v + dv)

and incident on the volume element dV, the following fraction is absorbed by

the atoms responsible for the line:
2.4 INTEGRAL EQUATIONS OF STATISTICAL EQUILIBRIUM 65

k.^(vQ)n. (r)a(x)ds , (4.8)

where ds is the length o£ the element dV along the direction of propagation.

Multiplying (4.5), (4.6), and (4.8) together, integrating over frequency,

and taking into account that dods = dV, we get the total energy absorbed in

dV due to i k transitions from the radiation emitted in dV:

hv

n, (r)k,, (vj I M,(r,r')n, (r')A, — —- i^;-. , j^.vQ, -2^^'^ ^^'k^-^ ^^ki

4tt r - r' '

+ M, (r,r') -AlA^ £^(r') I dVdV ,

(4.9)

where

CO

^ ^ J

ct^(;^3exp{-a(x)T(r,r' ) - T^(r,r')}dx , n = 1,2 . (4.10) — oo

Finally, dividing this energy by hv^j^ and by dV and integrating over the en- tire volume of the medium, we obtain the total number of i ^ k photo-excitations per unit volume near the point r:

ni(r)B.,J.^(r) = n.(r)k.^(v,) M,(r.r-)dV' . - - (4.11)

1 /-Mid,!:') 4^^^

Substituting this expression into (4.4), we finally arrive at the following

statistical equilibrium equation for level k:

M ( r r ' ) = "iWku(-o' hj i' :';,|2 "k(i'"j^' *

, /.M (r ,r' ) . . *

+ n. (r)k., (vj

^

/ ^ ~ ~ , ^^^^ E:^(r')dV' + n, (r) . (4.12 1^-^ ik^ 0^ |r - r' |2 AAkihvik "

-

HOMOGENEOUS MEDIA. If the population of the lower level n^ and the continuum

opacity do not depend on position, then from (4.12) we obtain, after some algebra, the following equation for the line source function S(t_):
66 THE LINEAR APPROXIMATION

where

S, Ct) = B— / e II sC(T')dT' + S (t) . (4.13')

^ 4Try |T^_-r'|2 - "

Here t_ = nikik(vo)r is the optical radius- vector, di;' is the volume element

of _t' -space, 3 = a'^/nikij^ (vq ) is the ratio of the continuum to the line cen- ter absorption coefficients, and

oo

= A J

a^(x)e~°'^''^^dx , k = 1,2 . (4.14)

The physical significance of the individual terms on the right side of

(4.13) is as follows. The first term represents atomic transitions from the

lower level i into the upper level k by photo-excitation due to the radiation

field of the medium in the k -> i line; the term containing S*" (x) expresses

the photo-excitation by the medium's continuum radiation; and S*(x) accounts

for all other ways in which atoms can enter level k in addition to i ^ k

photo-excitations by the medium's radiation field. It is obvious that the

solution of (4.13) is equivalent to the solution of the integro-dif ferential

transfer equation (2.18) subject to the boundary condition expressing the

absence of external illumination.

Equation (4.13) (for the special case 3=0) was obtained, independently

and at about the same time, by L. M. Biberman (1947) and T. Holstein (1947),

and is sometimes called the Biberman-Holstein equation. It describes the

scattering of line radiation in homogeneous media, neglecting nonlinear

effects. The assumption of homogeneity provides a substantial simplification:

the kernel of the equation becomes a function of the variable |r — r' | , and

not of r and r' separately, as in the general case.

When the medium has a particular symmetry, the integral equation of

statistical equilibrium (4.13) assumes a simpler form. We shall consider

the cases of plane and spherical geometry.

PLANE GEOMETRY. This case is very important from the point of view of applications, primarily in astrophysics. The requirement of a homogeneous medium

would constitute a very considerable limitation, since the density usually

changes rather rapidly with depth (e.g., in stellar atmospheres). Fortunately, in the case of a plane geometry the integral equation of statistical

equilibrium reduces to an equation with a kernel depending on the difference

of the arguments (displacement kernel) even when the population of the lower

level depends arbitrarily upon depth. It is sufficient to require only that

the ratio of the continuum to the line-center absorption coefficients is

depth - independent (3 = const).

The following procedure can be used to obtain the equation for the line source function in the case of a plane layer of optical thickness tq . We

substitute the formal solution of transfer equation (3.10) - (3.11) into

(3.5), which expresses the line source function in terms of the intensity.

For 3 = const we have
2.4 INTEGRAL EQUATIONS OF STATISTICAL EQUILIBRIUM

0

67

S(t) =^kj a(x')dx'j-y ^y[aCx')S(T') + 3S^(T')]e —<» 1—10

a(x' )S(t ' ) + 6S' Ct')]

T —

(a(x')+3)^

dx '

dx' +

(4.15)

+ S (T)

Hence we find that

S(t) (|t - T' I ,e)S(T')dT' + S^(t) , (4.16)

where

r

sJ(t) = K^^(|t - t- I ,6)S^(T')dT' + S*(t) , (4.16-)

and

CO K^(t,6) = a y* ct^(x)E^((a(x) + 3)t) dx , (4.17)

CO :^^(T,3) = A J

a(x)E^((a(x) + 3)t) dx (4.18)

Here Ej^(t) is the exponential integral function

1 t E^(t) - f f 0 ^ (4.19)

If the line-center optical thickness is infinite (semi - infinite medium),

then tq = °° in equations (4.16) and (4.16'). And if the medium has infinite

optical thickness in both directions (infinite medium) , then the line source function is the solution of the equation

S(t)

oo

= \ f

K^dl - T- I ,3)S(T')dT' + S*(T) , (4.20)

in which
68 THE LINEAR APPROXIMATION

*

oo /

K^^(|t - T- I ,3)S^[T')dT' +

* S Ct) (4.20')

The continuum absorption coefficient is often much smaller than the linecenter absorption coefficient. If continuum absorption and emission are com- pletely neglected, (4.16) then reduces to

S(T) (|t - T' I )S(T')dT' + S (t) , (4.21)

whe re

K^(T) E K^(T,0)

oo /

(x)E^(a(x)T) dx (4.22)

and instead of (4.20) we have

S(T)

OO

T '

I )S(t • )dT ' + S (t) (4.23)

According to the discussion at the end of Sec. 2.2, one may obtain the

equations describing monochromatic scattering, by introducing the rectangular

profile. From (4.22) it follows that in this particular case K-j^(T) = Ei(t) . Thus for monochromatic scattering the source function in a plane layer is

the solution of the equation

S(T) = ^

2 E^(|t - t' i)S(T')dT' + S (t) (4.24)

whereas for an infinite medium the basic integral equation is

S(T)

CO

t' |)S(T')dT' + S (x) (4.25)

The properties

related to K]^(t) are equations (4.24) and

in detail in Chapter

(4.23). Equation (4

Chapter V the case o Chapter VI more spec

depth - dependence of

Tg = <=° is studied in

Chapter VIII

of the kernel function K2(t) as well as various functions

studied in Sec. 2.6 and 2.7. The solution of integral

(4.25), describing monochromatic scattering, is discussed

III. In Chapter IV results are obtained for equation

.21) is studied for tq = «> in Chapters V and VI; in

f an arbitrary function S*(t) is considered, and in

ific results are derived for certain special forms of the

the source strength. Equation (4.16) with 8^0 and

Chapter VII, and the case of finite tq is taken up in
2.4 INTEGRAL EQUATIONS OF STATISTICAL EQUILIBRIUM 69

SPHERICAL GEOMETRY. First we shall consider scattering in a homogeneous

sphere o£ optical radius tq . For simplicity we assume that there is no absorption and no emission in the continuum (3 = 0) . According to the assumption o£ spherical symmetry, the line source function S and the distribution

of primary sources S* depend only on distance from the center of symmetry,

which is, of course, taken as the origin. Equation (4.13) is now

/•M (|t - T- I) S(T) = / ^ — S(T')dT' + S (T) , (4.26)

where x = , and the integration is to be performed over the sphere of

radius Tq. Introducing the spherical coordinates t' , 9' ,cj) ' , we have

di' = T ' ^dx ' sine ' de ' d4> ' ,

I I 2 2 2 1^— ^'1 =x +x' — 2xx'cos0' .

Setting y' = cosG' and using (4.14), we find instead of (4.26)

^0 « 1 . S(T) = ^ / S(x')x'2dT'A /* a2(x')dx' f

exp{-a(x')/x2.x'2-2xx'y'} ^^,^s*^^^^

2 J J J t2 + t' 2 - 2xx'm'

0 -00 -1 (4.2 7)

Transforming to a new variable

t = a(x' ) (x^ + X ' ^ - 2xx 'y ' )'^

,

we obtain

T 00 a(x')(x + x')

S(x) = ly S(x')x'^dx'A Ja^(x')dx' / e'''^ ,~r * S*(x) , (4.28)

a (x ' ) I X—X

'

whence

^0 (4.29)

xS(x) = - /

- t'I) - K^(x + x')] x'S(x')dx' + xS*(x) ,

0 where K-|^(x) is, as before, given by (4.22). The solution of this integral

equation is equivalent to the solution of the integro-dif ferential transfer

equation (3.24) with 6=0 subject to the boundary condition I(xo,y,x) = 0 for y < 0.

It is worth noting that the problem of determining the source function

in a homogeneous sphere is essentially reduced to that for a plane layer.
70 THE LINEAR APPROXIMATION

it Indeed, if we define S (x) (and S(t)) for negative values of x by the relations

S*(-x) = S*(x), S(-x) = S(x) , (4.30)

then (4.29) may be rewritten in the form

It follows that

^0

J

K^(|x - x' |)x'S(x')dx' + xS*(x), _Xg < X < Xq . (4.31)

-^0

s(x^ + x)

S(t) = ^

, (4.32)

where s(x) is the solution of the equation

s(x) = I J

K^(ix - x'|)s(x')dx' + s*(t) , (4.33)

0 in which

s*(T) = (x - Xq)S*(x - Xg) . (4.34)

Hence, to determine S(x) for a homogeneous sphere of radius xq , one has to

find the source function in a layer of thickness 2xq with a fictitious source

s*(x) (which is negative for x < xq) . More complicated systems possessing spherical symmetry may be considered

similarly. Thus, for the source function in a homogeneous spherical shell,

which surrounds a perfectly black sphere and receives no external illumination, the following equation may be derived:

^2

xS(x) = y y I^K^(lx-x'l) - K^(/x2 - + /t ' 2 _ t ^

2 )J

^ ' S (x ' ) dx ' +

Here x = Oj^jj.(vQ)r is the optical distance from the center, X] is the inner

optical radius of the shell, and X2 — x^ is its optical thickness. The

equation is even more complicated for a hollow spherical shell. This situation was studied by T. A. Germogenova (1966). However, if the thickness of

the shell is small in comparison with its radius (say, the inner radius),

the equation simplifies greatly and assumes the form
2.5 THE REDUCTION OF NONLINEAR TWO-LEVEL PROBLEMS TO LINEAR ONES 71

S(T) = ^[K^(|t - t' I) + T + t' ) ]SCt' ]dT ' + S (t) (4.36)

0

The integral equation (4.29) has been considered by S. Cuperman, F. Engelmann and J. Oxenius (1963, 1964) and by V. V. Sobolev (1962), where re- sults of numerical solutions for several special cases are given. M. Weinstein

(1962) has obtained an equation describing scattering in a homogeneous sphere

with a partially reflecting boundary. T. A. Germogenova (1960) has studied

equation (4.35) with K]^(t) = E]^(t) (monochromatic scattering). Finally, V. V.

Sobolev (1959a, 1965) and D. I. Nagirner (1965) have considered equations

(4.29) and (4.36) for large Tq.

For application to the study of transfer of resonance radiation in gasdischarge tubes, a number of authors have also considered media with cylindrical symmetry (T. Holstein, 1951; M. Weinstein, 1962, and M. A. Heaslet and

R. F. Warming, 1966)

2.5 THE REDUCTION OF NONLINEAR TWO-LEVEL PROBLEMS TO LINEAR ONES

LINEAR CASE. Since radiation drives some of the atoms from the lower to the

upper level, the population of the lower level cannot, strictly speaking, be

regarded as given, and must be found from the solution of the statistical

equilibrium equations. If the intensity of line radiation causing redistribution of the atoms between levels is not too great, this effect may be neglected in the first approximation. So far we have done so. However, when

the radiation intensity becomes very large this situation will change. The

line opacity of the medium will decrease because an appreciable fraction of

the atoms are driven from the lower to the upper level. At the same time

stimulated emission begins to play a role.

These nonlinear effects are, in general, rather difficult to treat

exactly. There are, however, specific cases in which nonlinear problems can

be reduced to linear ones. This section will be devoted to the discussion

of one such case.

Let the medium have plane symmetry, so that all quantities depend on only one spatial coordinate — the geometrical depth z, measured from the

boundary of the medium. We shall assume that the gas is isothermal and con- sists of two-level atoms. The total concentration of atoms

and the electron density n^ are regarded as known constants. We wish to

know the steady state of such a gas.

If the temperature of the gas is low (kT << h.\>i2) and the intensity of

radiation incident upon it from outside is not too great, we have a special

case of the problem discussed at the beginning of Sec. 2.2. For a plane geometry, the transfer equation (2.9) assumes the form

n = n, (z) + n, (z) (5.1)

y dl (z ,y ,x) ^ dz

= -a(x)k, ^ (v„)n^ (I (z ,y ,x) -S(z)) (5.2)

or
72 THE LINEAR APPROXIMATION

^ dI(T,y,x) ^ a(x)(l(T,y,x) - S(t)) , (5.3)

where the optical depth x is defined by the relation

lo

= J k^^C^o^^l'^^ " ^12 '^'^O^^l '^^O ~ ' (5-4)

here zq is the geometrical thickness o£ the layer. In the linear case the

value of n^ is known and may be set equal to the total concentration of atoms

n, since n2 << ni. The line source function S(t) in (5.3) is related to the

level populations by (2.5), and according to (3.5) and (2.7) it may be ex- pressed in terms of the intensity of diffuse radiation as follows:

3 r r 2hv^^ -hv,^/kT ^ S(t) =|a / a(x')dx' / I(T,y' ,x')dy' + -Lie (1-X)+Sq(t) , _co _ 1 ^

(5.5)

where

I '^(^q)

S*(t)=^A f a(xQ)dxQy lQ(0,yQ,XQ)e ^^0 dy^. (5.6)

Here Ig(0,yQ,XQ) is the azimuth- averaged intensity of the radiation of frequency Xg incident at an angle arccos yg with the inward normal to the boundary (for the sake of simplicity we assume that only the boundary t = 0 is

illuminated). The last term on the right side of (5.5) represents photoexcitation by external radiation attenuated by the medium. The probability

A of photon survival is

X = ^ . (5.7)

A2I + neC2i

For n^ = const and T = const, A does not depend on position.

Substituting the formal solution of (5.3) into (5.5), we arrive at the

following equation for the line source function:

""Q 3

, r 2hv,^ -hv-i^/kT ^ S(t) = A

y

K^(|t-t' |)S(T')dT' + ^e (1-A) + Sq(t) , (S.8)

0 ^ .

in which K2(t) is given by (4.22), and the optical thickness of the layer is

"^0 = ^12^^0^ j ^i^^^dz ,
2.5 THE REDUCTION OF NONLINEAR TWO-LEVEL PROBLEMS TO LINEAR ONES 73

or, since << n^,

BASIC EQUATIONS IN THE NONLINEAR CASE. If the gas temperature is high

(kT ~ hvi2) if the radiation incident upon it is very intense, stimulated

transitions can no longer be ignored, and the problem becomes nonlinear. The

radiative transfer equation then takes the form (see Sec. 1.6)

2hvj3 Si ''-'^

+ a(x)k^2 (^0'' 7 "2 '

and the statistical equilibrium equation is written as

^2^^21 ^ ^21^12 ^ ^0^21^ = "lf^l2^12 ^ ^eCl2) ' ^^-^^^

where

1 Jl2 = \

^ f

a(x')dx' f (l(z,y'x') + 1° ( z , y

' , x

' )) dp ' . (5.11) — 00 _ 1

Here I and 1° are the intensities, respectively, of the diffuse radiation

and the azimuth-averaged direct radiation. We shall show that the combined

solution of the nonlinear equations (5.9) and (5.11) can be reduced to a set

of linear problems. Let

T = k^2(^o^

}^ I §1 "2^"'^\

The quantity x is now the line-center optical depth when stimulated

emission is regarded as negative absorption. In the transfer equation (5.9)

the geometrical distance z may be transformed into the optical depth t. This

transformation, it is true, is of a formal nature, because until the level

populations have been found as functions of z, the dependence of t on z re- mains unknown.

If the line source function

S-'^l'-l^'A-y (5.13)

c2 VSl "2 /

is introduced, then the transfer equation (5.9), written in terms of the

variable t, has the same form as in the linear case.
74 THE LINEAR APPROXIMATION

On the other hand, from the statistical equilibrium equation (5.10) the

ratio of level populations vii/ni, and consequently the source function, may

easily be expressed in terms of the intensity. After some algebra we find

/

1

^ (5.14)

S(t)=|a j

a(x')dx' y I(T,y' ,x')dy' + (1 - X)B^^2fT) + S*(t) ,

whe re

A

2 1 X = — — , (5.15)

At, + n C-,, 21 e 2

By^2(T) is Planck's function

A and Sq(t) is given by (5.6), with t defined according to (5.12). Finally, by

substituting into (5.14) the formal solution of the transfer equation (5.3),

given by (3.13) and (3.14), we obtain the following linear equation for S(t):

S(T ) = I y

K^(|t - T - |)S(T')dT' + (1 - A)B^^2f'^) ^ S*(T) . (5.17)

REDUCTION OF THE NONLINEAR PROBLEM TO A LINEAR ONE. At first glance the

nonlinear case hardly seems to differ from the linear one: the expression

(5.13) is used instead of (2.5) for the source function; the parameter X is

calculated according to (5.15) instead of (5.7); and, finally, in the integral equation for S(t), the coefficient of (1 — X) is Planck's function rather

than Wien's [cf. (5.8) and (5.17)]. However, this is not the only difference;

for the value of tq for which (5.17) is to be solved is also unknown. According to (5.12) the optical thickness tq of the whole layer is given by the

expression

^0 = ^12f^0) dz (5.18)

which, at first sight, can be evaluated only after the level populations have

been found as functions of z, i.e. only after the problem has been solved.

However, a procedure is available for obtaining tq without a preliminary determination of the z-dependence of level populations.

We define tq as the limiting value of optical thickness that the medium

would have if all atoms were in the lower level:

k^2(Vo)nzQ (n^(z') + n2(z'))dz' (5.19)
2.5 THE REDUCTION OF NONLINEAR TWO-LEVEL PROBLEMS TO LINEAR ONES 75

We shall call this quantity the limiting optical thickness o£ the medium. It

is obvious that the actual optical thickness tq is less than t§, since some

of the atoms are in the upper level. We also introduce the limiting optical

depth

z, *

T = k ^2(vQ)n(zQ - z) = k^2(^o^ / ("if^^'^ ^^z*^^'^) ' • (5-20)

If the value of Tq is known and in some way or other the source function

has been obtained as a function of the limiting optical depth t*, it is clear

that the problem is then completely solved.

For the remainder of this discussion it is important that we regard the source function defined by (5.17) to depend on tq as a parameter: S = S(t,

tq) . In order to obtain the "real" value of tq for which the solution of

(5.17) gives the dependence of the source function on optical depth t, we pro

ceed as follows. From (5.12) and (5.20) we have

* dT

dx

n + / g \ -1

I 1 - — — 1

, (5.21)

or

^"^ = 1 + ^1

+ S(T,Tj , (5.22)

whence

T„ = T

0

+ (l + —]-^ f S(T,T )dT . (5.23)

This equation may be used to find Tq. In fact, if (5.17) is solved for all

Tq less than tq, then the right side of (5.23) will be a known function of Xg

Having found the "real" value of tq given by the root of (5.23), we can then

select from all the functions S(x, xg) with tq ^ xg the one that corresponds

to this value of xq.

In this function the variable x must still be converted to the limiting

optical depth x*. This may be done by means of the relation (21 + 52_\_c^ f S(x',x.)dx' , (5.24)

which follows from (5.22). It must be stressed that in this equation, xq

is the real optical thickness of the medium, i.e. the root of Eq . (5.23).

Thus the solution of a nonlinear problem may be divided into three

stages: (1) solving (5.17) for all xq less than xq; (2) determining the real

optical thickness xg from (5.23); (3) transforming the argument of S(x, xg)

from X to the limiting optical depth x* via (5.24), and thereby to z. Thus

if a linear problem reduces to the solution of one integral equation for
76 THE LINEAR APPROXIMATION

S(t, tq) , in the nonlinear case a set of such equations with tq < tq must be

solved. This is the price paid for nonl ineari ty

There are, however, two important special cases in which the situation

becomes quite simple: those of infinite and semi - infinite media. Let us consider, for example, a semi - infinite medium. It is clear that even if

allowance is made for the decrease of the optical depth because of excitation

of atoms from the lower level to the upper one, it remains semi - infinite as before. Here, therefore, the real optical thickness tq need not be determined. The T-dependence of the source function is found by solving only one integral equation, (5.17), with tq = «>. Then it is simple, using (5.24) with

Tq = to convert from optical to geometrical depth. And if one is interested in the intensity of emergent radiation, rather than the source function

itself, one need only determine S(t) without considering its dependence on

z, since

00 a(x) ^

1(0, y,x) = / S(T')e ^ a(x)— .

' (5.25)

This last observation also applies to a layer of finite optical thickness.

The problem discussed in this section was first considered by E. A.

Milne (1930), who, however, assumed that the frequency of a phxiton remains

constant during scattering. The same problem, allowing for frequency redistribution, was studied by R. N. Thomas (1957). Both Milne and Thomas assumed

a semi - infinite medium, and did not discuss tlie conversion from optical to

geometrical depth. A method for determining tq and transforming from t to z has been suggested by V. A. Ambartsumian (1964, 1966); see also N. B. Yengibarian (1966), V. Yu . Terebizh (1967) and Yu . Yu. Abramov, A. M. Dykhne

and A. P. Napartovich (1967a).

2.6 KERNEL AND RELATED FUNCTIONS: GENERAL PROPERTIES

DEFINITIONS AND BASIC RELATIONS. The function Ki(t), which appears in the

kernel of the integral equation for the line source function, and several

functions related to it play an essential role in all of the problems of

radiative transfer in spectral lines. In this section we study the most frequently encountered of these special functions (for the case in which the absorption in the continuum may be ignored) . These results will be used con- stantly in subsequent chapters. This and the following section are based on

the work of D. I. Nagirner and V. V. Ivanov (1966).

Let us define

oo

Mj^(t) = a y a^(x)e~"'^^^'^dx, t > 0 ; k = 1,2, ... , (6.1)

and

oo

^nk^^^ = ^ f

a^(^)E^(«(x)T)dx , t > 0; k ,n = 1,2, ... , (6.2)

where E^(t) is the n-th exponential integral
2.6 KERNEL AND RELATED FUNCTIONS: GENERAL PROPERTIES 77

1 E^(t) = j e

2 dy . (6.3)

0

The most frequently encountered of the functions Kj^}^(t) are K-|^2(''^) ^'^d K2i(t). Special notations will be used for them:

The function K^Ct) determines the kernel of the integral equation for the

line source function in media with plane and spherical geometry (see Sec.

2.4), and will be referred to as the kerne 1 function , or sometimes simply

the kernel. The normalization of K]^(t1 is

CO (T)dT = 1

We note that

and, in particular.

T From (6.1) - (6.3) it follows that

00

T

oo K2(t) = y* K^(t) dt (6.6)

K„^(T) = ^ J

M^(t)

^ , k,n = 1,2,... . (6.7) nk^ ^ y k

T From (6.5) and (6.7) we find

(n-l)K^j^(T) = Mj^(T) - tK^_^^ k+1^^^ ' k = 1,2,...; n = 2,3,... . (6.8)

In particular,

K2(t) = M^(t) - tK^(t) . (6.9)

Substituting (6.3) into (6.2) and introducing z = y/a(x), we obtain

1

=° ct (x) _ 31 K^j^(t) = A f a^''''~^(x)dx J

e

^ z^"^dz . (6.10)

-co 0
78 THE LINEAR APPROXIMATION

Here and throughout the rest o£ the book we shall assume, unless the contrary

is stated, that a(x) is an even continuous function that decreases monotonically as |x| increases. Changing the order of the integrations in (6.10), we obtain

•o

where

G (z) = 2A

CO /m+ 1 a (x') dx' , z < 1 , (6.12)

00

x(z) /

G^(z) = 2A / a"^^^x')dx' , z > 1 , (6.13)

and x(z) is defined by the expression

a(x(z)) = -i- ; x(z) > 0 . (6.14)

We shall drop the subscript 1 in Gj^(z), so that

G^(z) E G(z) . (6.15)

Then, in particular,

T

CO 'J^ Ko(t) = / e

^ G(z) dz . (6.17)

Along with M]^(t) and Kj^]^(t), the functions

V(

oo u) - /"-^

*^ 1 + u 2,2

dz (6.18)

and

00 U(z) = z^ / -£IllL dz'

J z2_2>2

(6.19)
2.6 KERNEL AND RELATED FUNCTIONS: GENERAL PROPERTIES 79

also play an important role in line transfer problems. The first of these

differs only by a constant factor from the Fourier transform of K]^ ( | x | ) :

CO CO VCu) = \ f

( I T I )e^'^"dT = J

K^[t)costu dT , (6.20)

0 from which, incidentally, it follows that

00 K^(t) =^f V(u)cosTudu . (6.21)

The second is related to the one-sided Laplace transform of the kernel function

CO CO K^(s) E y* e-^^K^(T)dT = f

-^l^^dz- (6.22)

0 0 ^

by the expression

By substituting the explicit expression for G(z) from (6.12) - (6.13) into

(6.18) and (6.19) and integrating by parts, we obtain

OO V(u) = - A / a^(x) arctg dx , (6.24)

a(x)

U(z) = z - f a^(x) In ^""^^^ ^

^ dx . (6.25)

2 J za(x)—

CO

When z is real, the ratio (za (x) +1 ) / ( za (x) —1 ) in (6.25) should be replaced by

its modulus, and the integral (6.19) is to be interpreted as a principal value.

These functions reduce to the "classical" ones appropriate to monochromatic scattering if we assume that the profile is rectangular, i.e.

a(x) = aj^(x) = 1,

0,

< 1

> 1

This rectangular profile was discussed by E. A. Milne (1930) and will be

referred to as the Milne profile. It may be assumed that for the Milne profile G(z) = 1, z < 1, G(z) = 0, z > 1. In this case the kernel function

Ki(t) is Ei(t), the function K2(t) reduces to E2(t), and the functions V(u)

and U(z) are

Vm(u) arctg u

u (6.26)

522-519 O - 74 - 7
80 THE LINEAR APPROXIMATION

the quantity (z+l)/(z— 1) being replaced by its absolute value for — 1 < z < 1. As we have already mentioned in Sec. 1.5, the subscript M will be used to

denote functions referring to monochromatic scattering.

SERIES EXPANSIONS. Series expansions are readily obtained for -the functions

M]^(t) and Kj^]^(t). Let us define

CO

a^ = A / a-' "(x)dx , j = 0,1,2,... , (6.28)

DO /

a^(x) Jin a(x)dx . (6.29)

Expanding the exponent in the integrand of (6.1), we find

oo

j=0

Similarly, the well-known expansions of the exponential integrals (see, e.g.,

V. Kourganoff, 1952) may be substituted into (6.2) to obtain series representations for Kj^]^(t). In particular,

GO , _^ K^(x) = -a^£nT-a^Y*-a . T.^^-^)' -^.2 ^^l^i^^^.iy. '

'

OO * _ . ^, j +2 (6.32)

K (t) = 1 + a T£nT + (ay -a + a)T + 2^(-l)^ a

^

I 1 1 1 ^

j+z (j+i)(j+2):

3=0

* where y = 0.577216 is Euler's constant. The series (6.30) - (6.32) converge

for all T , 0 < T < °° ; however they are only useful when t is not large.

From (6.24) and (6.25) we have

oo

\jf ^ 1 ^ V* r ^21+2 —2(j+l) ^ T V ( u ) = — a T - + 7. ( -1 ) — u , u>l, (6.33)

2 1 u j=o 2j + 1

00 U(z) = 22 JalL z2j+2 ^ 0 < z < 1 . (6.34)

j=0 2j +

ASYMPTOTIC BEHAVIOR AT INFINITY. The asymptotic forms that will be derived

in this subsection are much more important than the series expansions just

given

Let us assume that the function x(z) defined by (6.14) is such that for

arbitrary y, 0 < y < «>,

T

• x ' (x/y) 2 6 ( -z^-\ lim —) ( \ = y > (6.35)

-j-->oo X (Xj
2.6 KERNEL AND RELATED FUNCTIONS: GENERAL PROPERTIES 81

with 6 > 0. The functions x(z) encountered in cases of practical interest

usually satisfy this condition. Let us consider the behavior of M]^(t) and

Kj^]^(t) at infinity, supposing that (6.35) holds. Substituting a(x) = 1/z

in (6.1) and transforming from integration over x to integration over z, we obtain

00 Mj^(t) = 2A j

Q-^'^ x'(z) ^ , (6.36)

1

^

or

T

\if r ^ T A /* — y k—2 x'(T/y) J x'(t)

M, (t) = 2A / e ^ y —\

' \' dy • \^ . ^ J X' (t) t-K—1

0 ^

Using (6.35), we finally find

M, (t) ~ 2Ar(k + 26-l) 2L1IiI , t ^ «> , (6.37)

^ T-k - 1

where r is the gamma function. This method of obtaining the leading term of

the asymptotic expansion of Mj^(t) is due to Yu. Yu. Abramov, A. M. Dykhne

,

and A. P. Napartovich (1967a, 1967b).

Substituting (6.37) into (6.7) and using the same trick, we get the leading term of the asymptotic expansion of Kj^k(T)

K^j^(x) ~ 2A '}^'\'7^}

^ , T ^ 00 . (6.38)

k+n + 26-2 T^~^

In particular.

K fT) ~ 2A ^^LLll , (6.39)

1 26+1 T

K.(t) - 2k ^Sl^ X' (t) . (6.40)

2 26 + 1

Let us now obtain an expression for ^^{'^) when z is large. The substitution a(x') = 1/z' reduces (6.13) to

G (z) = 2A / x' (z • )

' m J (z')"^*l

'

z

^

^ from which, using the device employed in the derivation of (6.37), we obtain

G (z) ~ , z - » • (6.41)

I" 26+m ziTi

and, in particular.
82 THE LINEAR APPROXIMATION

G(z) ~ -i^ . (6.42)

26 + 1 z

We note in passing the asymptotic relations that follow from (6.37), (6.40),

and (6.42):

M-, (t)

K^Ct) ~ r(26)TG(T) , T ~ . (6.43)

^ 26 + 1

To get the asymptotic form of V(u) for u ^ 0 , we first note that

G(z)dz = 1 , (6.44) /0 which may be proven by equating the right sides of (6.17) and (6.32) and setting T = 0. Using (6.44), we find from (6.14)

2 V(u) = 1 - /* ^ ^^^^ dz . (6.45) ^ 1 + u^z^

At this point the derivation depends on whether or not the second moment of

the kernel function exists. Let

00 00 0^ = I

T^K^(T)dT = 2 / Z^G(z)dz . (6.46)

2 If a < °° , then obviously for u 0

2

^ 1 - V(u) ~ ^ u^ . (6.47)

On2the other hand, if the second moment of the kernel does not exist

(a = «>) , then for small u the main contribution to the integral in (6.45) comes from large values of z, and for G(z) we can substitute its asymptotic

form. Using this fact, we may readily show that if 0 < 6 < 1, in the limit

as u 0

7t6 \ u / 1 - V(u) — x'[-l . (6.48)(26+l)sin ^ '

'

2 2 For 0 < 6 < 1 we always have a = oo

; for 6 > 1 necessarily a The case

of 6 = 1 is more complicated. Here one may have both < oo and = oo. in

the first of these cases (6.47) applies. The second case, i.e. 6 = 1 , = oo^ is exceptional, and neither (6.47) nor (6.48) are valid. However, (6.48)

shows that in this case, as u -> 0 , 1 — V(u) tends to approach zero more slowly than does :x'(l/u). An example of this exceptional case will be given

shortly

The following asymptotic forms of U(z) can be obtained from (6.19) in a similar way: if < oo

^ then
2.6 KERNEL AND RELATED FUNCTIONS: GENERAL PROPERTIES 83

U(z) - 1 - - ^^^^^g x'(z) , z - - . (6.49)

Z 0 + 1

2 and i£ a = «> and 6 < 1, then

2 The case o£ a = «> and 6 = 1 is again an exception,

(6 . 50)

It is evident that the asymptotic behavior of the kernel and the related

functions determines, in many respects, the behavior of the solution of the

integral equation for the source function. As we have just shown, these

asymptotics depend on the behavior of the function x(z) at large z, or, equivalently, on the behavior of the absorption coefficient in the range of x- values where a(x) is small, i.e. in the line wings. Therefore, one might ex- pect to obtain important information about the steady state of a gas in its own radiation field by knowing only the behavior of the absorption coefficient in the line wings. As we shall see in subsequent chapters, this expectation is fully justified.

INTEGRAL RELATIONS. If the second moment of the kernel diverges (a^ = 0°),

the following useful integral relations hold:

CO / U(z)G(z)dz = 1/2 , (6.51)

•o

dz TT^ 2 j

U(z)G(z) ^ = Y

. (6.53)

0

^

2 The derivation of these relations is omitted. If o < °°, these relations are invalid. Such is the case, for example, for monochromatic scattering. In

this instance some analogs of these relations do exist, but they are of no interest since Uj^^(z) and Vf^j(u) are elementary functions.

CHARACTERISTIC EXPONENT. Let Ki(t) be a given non-negative function normalized to unity on the interval (0 ,<») and let V(u) be its cosine- trans form

[essentially, the Fourier transform of K^^ ( | t | ) ] . We further assume that

1 - V(u) ~ 4)(u)u^'^ , u ^ 0 , u > 0 , (6.54)

where 0 < y < 1 and <^{u) is a function that varies slowly as u ^ 0, i.e. is

such that for arbitrary a > 0

u-^0 cj) (u)
84 THE LINEAR APPROXIMATION

In particular, (t)(u) may be a constant. The value of y will be referred to as the characteristic exponent of K-j^ (t ) . Comparison of (6.54) and (6.47) shows that if K]^(t) has a finite second

moment, then y = 1 and the function (l)(u) is simply a constant: ^ = o^/Z.

If 02 = oo and the condition (6.35) is satisfied, then for 6 < 1 (6.54) and

(6.48) give

l)sinTT6,..2Y ,'^rt^ 4'(u)u , (6.56)

ttA

Substituting this expression into (6.35) and taking into account that (l)(u) varies slowly, we find that y = 6 . It can be shown that in the "exceptional"

case of 0^ = oo and 6 = 1 we also have y = 6( = 1). The results may be summa- rized as follows

2 1. If a < then y=l, 6>l,(i) = — . (6.57)

00 2 If = oo and Y

= 1, then y = 1( = 6) and (j,(u) - f

" ^^^^^^ . ^^"^^^ ^ 1+u^z^

If 0^ = ~ and 6 < 1, then y = & and (j)(u) — u~^^x'(i). (6.59)

(2Y + l)sinTTY \u/

Let us consider as illustrations several important specific cases.

(1) Milne rectangular profile (monochromatic scattering):

( 0,

x| < 1 ,

x I > 1

- T

* 2 We have Kj(t) = Ei(t) ~ e /t for i ^ «>, so that a < oo ; hence we are con- cerned here with case 1. Further, we have V[vj(u) = arctg u/u = 1—u2/3 + so that

Milne: y = 1 , <t> = 1/3 . (6.60)

(2) Parabolic profile:

X , I X I < 1 , a(x) = i:. xl > 1 .

In this case x'(z) ~ l/(2z ) for z ^ °° , and (6.35) gives 6 = 1, whereas (6.39)

shows that K-^(t) ~ 1/(2x3) for x ^ °o , so that o2 = oo (case 2). It is easy

to show that in this particular case (j)(u) = — (£n u)/4.

(3) Doppler profile:

2 a^ix) = e

•
2.6 KERNEL AND RELATED FUNCTIONS: GENERAL PROPERTIES 85

We have xq(z) = (In z)^^^ and, according to (6.35), 6 = 1/2 (case 3). Using

(6 . 59) we get

^1/2 / i\ -1/2

Doppler: y = 1/2, <t> (u) = - Mn - | (6.61)

4 \ u)

(4) Lorentz and Voigt profiles: For the Lorentz profile a, (x) = (l+x2)-l we have xl(z) = (z-l)l/2, and ^

Lorentz: y = 1/4, (}) = . (6.62)

For the Voigt profile defined by (1.5.20) and (1.5.8),

1 / \^/^

Voigt: Y

= 1/4, (f> = I

l2TTaU(a,o)J , (6.63)

(5) Profiles that decrease as power laws in the wings:

a(x) ~W|x| , |x| ,

where W and < are constants, W>0,l<K<a>. Here we have

,1-2y K-1 , _ 1-2y ttAW^ 2< 1 + 2y sinTTY

(6.64)

The profiles used in these examples are of interest either because of

their importance in applications of the theory (e.g., the Doppler and Voigt

profiles) or for purely theoretical reasons (say, the parabolic profile).

To clarify many of the points of the theory set forth in the next chapters it is quite useful to consider a one -parameter family of profiles having

the form

a(x) = ^1-qx^y ^'^O(l-qx^) , (6.65)

where 0(s) is the unit step function: 9(s) = 0 for s < 0, 0(s) = 1 for

s > 0, and q is a parameter, q > —2 . For q < 0 the profiles have wings ex- tending to + infinity; for q > 0 the width of the line is finite. In the

particular case of q = —1 we have the Lorentz profile; in the limit as q ^ 0 we obtain the Doppler profile; q = 1 corresponds to the parabolic profile;

and in the limit as q ^ we obtain a 6-function profile (monochromatic

scattering)

Using the family of the profiles (6.65) as a working example, we can

trace the distinction between the constant 6 introduced according to (6.35),

and the characteristic exponent y defined by (6.54). Using (6.65), (6.14),

and (6.35) we find
86 THE LINEAR APPROXIMATION

q < 0

6 q > 0

SO that 6 can be arbitrarily large. For 6 not exceeding unity, y = 6; while

for 6 > 1 we have y = 1.

The profiles of this family with infinitely extended wings have 5 < 1/2,

while lines of finite width correspond to 6 > 1/2. This feature is of a general nature; that is, it can be shown that if a line has infinitely extended

wings, the characteristic exponent does not exceed 1/2 . The proof is based

on the consideration of the function x(z) defined by (6.14). From the definition one can infer that if the wings extend to infinity, then x(z) ^ <» for

z

-> °°. On the other hand, (6.56) gives

The integral on the right is divergent for z «> only if 6 < 1/2, which

proves our assertion.

Throughout this book it will be assumed that y < 1 if the contrary is

not explicitly stated or if it is not obvious from the context, as in the case

of monochromatic scattering. Since y = 6 for y < 1, the characteristic exponent y will be often identified with 6. The main attention will be given to

lines with infinitely extended wings, i.e. to values of y in the interval

0 < y < 1/2.

ALTERNATIVE FORM OF THE ASYMPTOTICS. The asymptotics of the kernel and re- lated functions found above can be rewritten in another form if (6.56) is

used. This alternative form will be used extensively in what follows. To

facilitate reference we reproduce the most important of the asymptotics in

this alternative form (t °°, z -> <») ;

z x(z)

M, (t) ~ - (2y + l)r(k + 2y-l)sin7Ty 0(1/t)

Tk+2y-l

(6.66)

K, (t) ~ I

r(2y+l)sinuy

^2y + l (6.67)

K2(t) ~ ^r(2y)sin^y (6.68)

G(z) 2 . — smiry ^(1/z)

,2y ^ 1 (6.69)

U(z) (6.70)
2.7 KERNEL AND RELATED FUNCTIONS: PARTICULAR CASES 87

CONCLUDING REMARKS. The functions Kx(t) encountered in radiative transfer

theory satisfy the condition (6.54). This condition enables one to develop,

in a sense, a closed theory for the asymptotic behavior of the solutions of

the transfer equation without appealing to detailed information about the

line absorption coefficient. Roughly speaking, instead of a function a(x),

one has to know only a number y, which determines the functional form of the

asymptotics (apart from a slowly varying function, which usually degenerates

into a constant close to unity). The physical reason for this is as follows.

The asymptotics are governed by photons having large mean free paths, i.e.

wing photons. We can ignore the details of the frequency dependence of the

absorption coefficient, for the results depend only on the rate at which

a(x) decreases as x and this information is completely specified by only

one number, y- The kernel function K]^(t) has an immediate probabilistic interpretation:

K]^(t) is the probability density for the direct radiative transfer of excitation over a distance t (see Sec. 5.2). The characteristic feature of problems of radiative transfer in spectlral lines with infinitely extended wings

is that the mean-square shift of excitation is infinite, i.e. = «> , This

characteristic distinguishes in a fundamental way these problems from the

whole class of monochromatic scattering problems, both isotropic and non-isotropic, for which is always finite.

In the language of probability theory, the fact that (6.54) holds means

that the functions K-^^t) dealt with in radiative transfer theory fall within

the domains of attraction of stable probability distributions (see, e.g.,

I. A. Ibragimov and Yu. V. Linnik, 1965; V. Feller, 1966). This observation

seems to be very important, and the relation of line transfer problems to the

theory of stable probability distributions deserves the most careful consideration. So far only the very first steps have been taken in this direction

(V. V. Ivanov and S. A. Sabashvili, 1972).

Once the form of the absorption coefficient is specified, more detailed

information can be obtained about the functions discussed in the preceding

section

DOPPLER PROFILE. In this case

2.7 KERNEL AND RELATED FUNCTIONS: PARTICULAR CASES

2 a^(x) = — X

e C7.1)

the normalization constant is Ap = tt ing to (6 . 12) - (6 .13)

,

'2 (1.5.23) , Xd(z) (£n z)^, and, accordz > 1

(7.2)
88 THE LINEAR APPROXIMATION

Using the well-known asymptotic expansion of the probability integral, we find from (7.2) :

Gn(z) ^~ . z - oc, (7.3)

^ 2^'2z2(£n z)^2 [in z)-J

whe re

g = (-i)j (7.4)

and it is assumed that (—1)11 = 1. The expansion (7.3) refines, for the case of the Doppler profile, the asymptotic form (6.42) of G(z) for large z found

in the preceding section, where only the leading term of (7.3) was obtained.

The coefficients aj and a^ in the power series expansions of the functions under consideration are, in this case,

a° = (j + 1)~'\ a° = -2-^/2 _ ^7^5)

The asymptotic series for Mv(t) and K^^kC^^) x ^ 0° are readily obtained

(V. V. Ivanov and V. T. Shcherbakov, 1965a, 1965b; E. H. Avrett and D. G. Hummer, 1965). It is found that

00 M^x)^—i ^ ill=mi T^^h^) . (7.6)

where r'--^-' is the j - th derivative of the gamma function. Specifically,

,,D, , 1 /, 0.28861 ^ 0.74179 ^ 1.7015 6. 4426 ^ \ .77. M (t) ; 1 + r + + . . . j

, (7.7)

^ 7T^2T(Jln t)'2 \ in T (£n t)^ (£n t)^ (£n t)^ /

-,D, , 1 /t . 0.21139 ^ 0. 30888 ^ 0 . 15296 ^ 0. 48726 ^ \ rv (t) ~ ( 1 + + + + T + . . . I . ( / . 8)

TT^T2(£n t)^ VJin t (£n t)^ [in t)3 [in t)4 /

For T = 10^ the expansion (7.7), as shown, gives M^(t) to three significant

figures, and (7.8) determines Mp(102) to within a few units in the fifth

digit. Tables of m5(t) and m5(t) for t < 10^ are given by V. V. Ivanov and

V. T. Shcherbakov ( 1965a). The function M^(t) has also been tabulated by

T. Tomatsu and T. Ogawa (1966).

The asymptotic expansion of Knk(T) for large t is

00 k\(t) 1 V a., m-i)j_:__^ (7 93

TT^^Tk(£n T)h (2j):: [in t)J
2.7 KERNEL AND RELATED FUNCTIONS: PARTICULAR CASES 89

where

j ^ikn = Z (-l)'cir^j-i^(k) (7.10

and the C^ are binomial coefficients. In particular,

^D. . 1 A 0.03861 ^ 0.33784 0.26933 \

K [Tj — II + + ...) , (7.11)

2TT'2T'^(£n t)'5 \ in T (£n x)^ (£n t)3 y

^D. . 1 /, 0.53861 ^ 1.14576 3.1337 ^ \

(Tj ~ — I 1 + + ...). (7.12)

2TT^T(£n t)^ \ in T (in t)^ (£n t)-^ /

The expansions (7.11) and (7.12) as shown give values of K^(t) and K2(t) at

T = 100 which are accurate to two units in the third significant figure. Tables of these functions are in papers by V. V. Ivanov and V. T. Shcherbakov

(1965b), T. Tomatsu and T, Ogawa (1966), and A. L. Crosbie and R. Viskanta

(1970a). Various approximate representations for K^(t) have been obtained

which are well adapted for rapid evaluation on computers (D. G. Hummer and

G. B. Rybicki, 1967, E. G. Avrett and R. Loeser, 1966).

The asymptotic expansion of Vp(u) for u 0 may be obtained as follows.

From (6.24) we have

00 A {u[l - V(u)]} = 2Au^ f

^Mdx _ ^7^^3^ ^ u2 + a2(x) _ 2 ^ For the Doppler profile we set e = ut , to obtain

_d_

du

SO that for small u

_d_

du

oo oo {u[i - vju)]} ~ , " T t^" - ^i'- f

t^" -r-^

Evaluating the integrals which appear here, we obtain

^ - 2n

^^-'4^

-5- {u[l - Vn(u)]} u (4n - 1) :

/iV if i i
90 THE LINEAR APPROXIMATION

where E2n ^^^s Euler's numbers (Eg = 1, E2 = —1, E4 = 5, E5 = —61, ...)• For

small values of u, we wish to express V]3(u) in the form

Vp(u) (7.15)

where the are unknown constants. Substituting (7.15) into (7.14), differentiating, and equating coefficients of identical powers of In 1/u, we obtain

the following recurrence relations for the numbers vj :

4n+l

2n + l 2n

n = 0,1,2, .. . ,

(4n+3)

2n+2 [4(n+l)

, , / \2n + 2

1 + (4n+5)(4n+l)

^2n+2l 16 2n

(7.16)

n = 0,1,2,

with VQ = 1. The first few coefficients v-: are vi = —0.25000; V2 = 1.1128;

V3 = -1.3910; V4 = 10.7577; V5 = -24.205. The expansion (7.15), including

terms through 3 = 5, gives 1 — V£)(u) for u < 10"~5 to four significant figures,

and for u < 10~^ to six significant figures.

The asymptotic series for Uj3(z) as z ^ <=° is obtained in a similar

manner. From (6.25) we have

_d_

dz

U(z) = A /

g (x)dx

1— z 2 (x)

(7.17)

Thus, for the Doppler profile, as z ^ °°

3/2

2n + 2 (4n+i) : : 1-2

z(£n z)3/2 (4n+2):: 2n+2

2n

2n + 2

1 (£n z)2n

(7.18)

where . are Bernoulli's numbers

2j

2 6' 4 " 30' ''6 " 42 ' •

We seek U^{z) for z <» in the form
2.7 KERNEL AND RELATED FUNCTIONS: PARTICULAR CASES 91

"d^^^ ~ 1 " 371 22 S- • (7.19)

16z(£n z)-^/^ (£n z)-'

From (7.18) and (7.19) we obtain the following recurrence relations for Uj

_ 4n + 5 ^2n + l "Zn ' (7.20)

^ (4n^5):: (22nM V tt^""'^ .g , , (4n.5)(4n.5)

n = 0 1 2 ^2n + 2 ^ (4n-^6):: ^ n + 2 r2n + 4| i6 ^2n ' ^•••

with = 1. In particular, Ui = -0.75000; u? = 4.0218; ut = -7.0381; ua = 63. 779 ; U5 = -175. 39.

VOIGT PROFILE. In this case

(x) = , (7.21) ^ U(a, 0) ' ^ ^

where U(a, x) is the normalized Voigt function defined by (1.5.8). Since the

absorption coefficient is given by a rather complicated non - e lementary function, simple expressions cannot be obtained even for Gy(z), not to mention

M]^(t) and The normalization constant A and the coefficient ai are

the only quantities that can be expressed in terms of tabulated functions,

name ly

,

A,, = U(a, 0) = - e^'

V TT

00 dx , (7.22)

^ 1 U (a/2,0)

1 /2 U(a,0)

(7.23)

For other quantities of interest, only partial results are available.

The function U(a, x) itself can be regarded as known. A detailed study

of its properties may be found in the review by B. H. Armstrong (1967), which

contains an exhaustive bibliography. There are also numerous tables of

U(a, x) , the most complete being the 8-figure table of D. G. Hummer (1965b)

(x = 0.00 (0.05) 5.00 (0.1) 10.0; 24 values of a from a = 10-4 to a = 0.5).

For large |x| the following asymptotic expansion for U(a, x) is valid:

U(a,x)

TTX j=0

(2j + l)'. x2J

(7.24)

where
92 THE LINEAR APPROXIMATION

2i

c.(a) = y; (-1)^ ^ ——- . J ^„ (2i + l): (j-i) : 22 (j-i)

i = 0

For a > 10 and x > 5 the expression

U(a,x) =

7TX' [•(t-1i-(—15 r 2 ^ 4 — 5a + a

4

) (7.25)

is accurate to four significant figures; this becomes six places for x > 10.

Using (7.25) it is readily shown that the function xy(z) for z ^ °° has

the expansion

Xy (z)

3 _ 2\ U(a,0) 1 ^ \^U(a,0) a z y L /

With this in mind, we obtain from (6.13)

(7.26)

/aU(a,0)\"- _1_ r _a/3 _^2\ ,u(a,0)J-. 0(z2)

'

\ TT y ^3/2 [ 10 \^2 j

^ ' az ^

'

, z ^ CO (7.27)

It can also be shown that for t ^ <»

M^d) = (aU(a,0))''^ ^ i _ 1 ^| _ a^^ ^U(a,0) j- + 0(t ^)

4(T) = I

(aU(a,0))

(7.28)

(7.29)

Asymptotic forms of M^ii) and K2(t) may now be obtained by differentiation,

s ince

M^d) = - ^

M^(t), K^(t) = - ^ K^d)

We therefore have

My(,3 = i^u(a,o)y^ _i_

3/2

.V. . _ (aU(a,0)) "^ 1 1 3 ^3/2

1-A(l_a^) .U(a,0)J-.O(x-^)]

l-A(|-_.2>),U(a,0)i-.O(.-2)]

(7.30)

(7.31)

(7.32)

A few words should be said about the region in which it is practical to

apply asymptotic expressions. Let us take mY(t) as an example. From (7.28)
2.7 KERNEL AND RELATED FUNCTIONS: PARTICULAR CASES 93

it is clear that this expansion may be used only when the second term in the

square brackets is small compared to unity. Since the value of a is small in

all cases o£ practical interest, the coefficient of l/ax in the square brackets is of order unity, and the condition just stated takes the form

T >> a"-"- . (7.33)

The expansion (7.28) is then valid only when (7.33) is satisfied; the same

is true for (7.29), (7.31), and (7.32). Thus the domain of applicability of

these expansions depends on the value of a, and increases with a. Values of the function AK-[^(At)/2 for Voigt profile with a = 0.001 and

0.01, and also for the Doppler (a = 0) and Lorentz (a = °°) profiles are shown in Table 2, which is reproduced from the paper of E. H. Avrett and

D. G. Hummer (1965). Here, as usual, A is the constant normalizing to unity

the integral of a(x) over all x. An approximate representation as a sum of

exponentials has also been obtained for kY(t) for several values of a, which

allows rather accurate evaluation of this function without recourse to numerical integration (E. H. Avrett and R. Loeser, 1966).

As for the functions Vy(u) and (Jy(z), from (6.18) and (6.19) it is readily shown, by using (7.27), that

TABLE 2

THE FUNCTION yK^(AT)

log: T a = 0 a = 0 . 001 a = 0.01 a = °°

4 1 ,, 8861 1 ,, 8833 1 ..8587 8 .0883 - 1

3 1 ,,4269 1 ,,4249 1 .. 4067 6 .2561 - 1

2 9 ,,6842 - 1 9 ,,6711 - 1 9 ..5549 - 1 4 .4255 - 1

1 5 ,,1728 - 1 5 ,,1668 - 1 5 ..1137 - 1 2 ,.6102 - 1

0 1 .,3071 - 1 1 ,, 3066 - 1 1 ,.3016 - 1 9 ,.3712 - 2

1 2 ,.1316 - 3 2 ,.1363 - 3 2 ,.1803 - 3 6 .4738 - 3

2 1 ,. 25 70 - 5 1 ,.2718 - 5 1 ,.4547 - 5 1 .6913 - 4

3 9 ..9495 - 8 1 ,. 1142 - 7 3 ,.2428 - 7 5 .2779 - 6

4 8 ,.5043 - 10 2 ,. 6470 - 9 1 ,.6138 - 8 1 .6669 - 7

5 7 .5518 -12 1 ,. 5907 - 10 5 .2591 - 10 5 .2705 - 9

6 6 . 8624 - 14 5 ,.2592 - 12 1 .6663 - 11 .6667 - 10

7 6 ,. 3334 - 16 1 ,. 6663 - 13 5 .2704 - 1 3 5 .2705 - 12

8 5 .9 108 - 18 5 ,.2705 - 15 1 .6667 - 14 1 .6667 - 13

9 5 . 5630 -20 1 .6667 - 16 5 ..2705 - 16 5 .2705 - 15

10 5 .2 703 -22 5 . 2 705 - 18 1 .6667 - 17 1 .6667 - 16
94 THE LINEAR APPROXIMATION

V^(u) = 1 - (2TraUCa,0)) '^ _ A

^|

_ .UU,0)^ + 0(u2)j , u -> 0 ,

(7.34)

(7.35)

LORENTZ PROFILE. In this case

a, (x) = ^- . (7.36)

In the limit as a «= the expansions found in the preceding subsection

give the corresponding expansions for the Lorentz profile, if we use the

fact, which follows from (7.22), that

lim aU(a,0) = 1/tt

However, for the Lorentz profile one can obtain substantially more detailed

information. The majority of the quantities of interest can be expressed in

terms of elementary or higher transcendental functions.

\, The normalization constant is = 1/tt. Further, Xj^(z) = (z—1) ^ and

Gl(z) =

a\ = 1/2 , z < 1 , (7. 37) if .1 z > 1 /?=r \ — larcsm — — I

,

For z > 1 the function Gj^(z) may be expanded in the series

00

^ z3/2

^

(2j+3)(2j):: J j=0

The constants a^ and are

J ^L ^ iZj-l) !

a^ = - - £n 2 . (7.39)

J (2j):: 2

The functions M2'(t) and M2(t) can be expressed in terms of the modified

Bessel functions ;

M[iT) = 3 2 Iq^I^ , (7.40)
2.7 KERNEL AND RELATED FUNCTIONS: PARTICULAR CASES 95

so that they may be regarded as known. Substituting the asymptotic expansions of the modified Bessel functions we obtain (t <»)

oo M^x) - -h S ^^^^^^ ^ , (7.42)

u-2T-^ 2J(2j):: tJ

^ ^5-2^3/2 2J+l(2j) : 1 t3

and, in general,

oo Mhx) - ^ E

(^j-i)::(2j.2k-3):: j_ _

From (7,44) and (6.7) asymptotic expansions of the functions K^, (t) are eas-

\ ily obtained for large t: ^

k\(t) 1— y; (2j-i)::(2j.2k-3):: _i_

^%Tk-^2 2j''k-2[2(j+k + n)-3] (2j) I : tJ

In particular,

Ki'(T) - (l + — - + — + — +...], (7.46)

135 J- . 175 -L . 224 t2 128 t3

27 J- . 25 J- . 221 t2 128 ,3

K^(t) ~ —r^TTT (l + — - + — + — +...). (7.47) 2' 3-^^2^1/2 \ 20 T 7 i?« _^ i

•

Tables of kWt) and K2(t) for 0 < t < 50 are given by A. L. Crosbie and R, Viskanta (1970a).

The functions Vj^(u) and (^^(2) are

V (u) = 1 48) i- + u)^ + — arctg r(2u)'''(u + X+u2")^0 , (7, ^ v/TH 2u L J

I 1 _ I /— - i /TZI . i an lll^ , 0 < z < 1 ,

I 2 2 2 l + /l_z

Ul(z) = (7.49)

( 1 _ i /— . ^ £n . z > 1 , 2 2 /J

and can be expanded in the series

522-519 0-74-8
96 THE LINEAR APPROXIMATION

j=0

'

' u^' , 0 < u < 1 , (7.50)

(2j+3)(2j)::

U^(z) = 1 /I 3=0

j (2j-i) :

j_

[2j+3)(2j):: zj

z > 1 , (7.51)

where [a] is the largest integer not exceeding a. Expressions (7.48) and (7.49) are most simply obtained by the use o£

(7.13) and (7.17). For the Lorentz profile, the integrals appearing on the

right sides of these equations are not too difficult to evaluate. We have

^ [uV (u) ] = 1 du ^

— z

,2 _d_

dz

(Z)

u(u+/l + u ) 2(l+u-)

1 1

2 /T+ z

2 /T+ z

1 < z <

^ 1

2

(7.52)

(7.53)

the second of the expressions (7.53) being valid for all complex z with

Re z > 0 , except for points of the real axis [1, <») . Integrating (7.52) and

(7.53) over u and z, respectively, we obtain (7.48) and (7.49).
CHAPTER III

MONOCHROMATIC SCATTERING

Problems o£ radiative transfer in spectral lines differ in several essential features from the classical problems of monochromatic scattering. Since

we shall refer continually to these differences, it seems advisable to preface the solution of radiative transfer problems in spectral lines with a special chapter devoted to monochromatic scattering, in order to make available for convenient reference the essentials of this theory. Further information may be found in the following monographs: E. Hopf (1934); S. Chandrasekhar (1950); V. Kourganoff (1952); K. M. Case, F. de Hoffmann and G. Placzek, (1953); V. V. Sobolev (1956); B. Davison (1958); I. W. Busbridge

(1960); G. M. Wing (1962); K. M. Case and P. F. Zweifel (1967); and in many

other publications, some of which will be mentioned below.

3.1 THE green's FUNCTION FOR AN INFINITE HOMOGENEOUS MEDIUM

BASIC EQUATION AND ITS TRANSFORMATION. We begin our study of the theory of

monochromatic scattering by determining the radiation field that arises from

an isotropic point source in an infinite, isotropically-scattering , homogeneous medium. We assume that the source lies at the origin of the coordinate

system. Let us denote by Sp the source function for this problem, i.e., the

ratio of the emission to the absorption coefficient. Since the Peierls equation is linear, Sp is essentially the Green's function for this equation in

the case of an infinite homogeneous medium. From considerations of symmetry

it is clear that the Green's function Sp depends only on the distance r from

the source, which is conveniently measured in optical units (mean free paths

of a photon). The function Sp(T) is determined from the Peierls integral

equation (see Sec. 2.1):

Here t = (Tx,Ty,Tz) is the optical radius vector, x = |t| is the optical

distance from the source, and X is the probability that a photon survives the

act of scattering (the ratio of the scattering to the total cross section).

The integration in (1.1) extends over all space. In writing this equation

the source strength (per unit frequency interval) is assumed to be 47t/o^,

where a is the absorption coefficient, or more precisely, the extinction

(1.1)
98 MONOCHROMATIC SCATTERING

coefficient (total macroscopic cross section). The subscript p on the Green's

function indicates that it describes the radiation field of a point source.

The physical significance of (1.1) is as follows. The emission of a vol- ume element located at a distance x from the source has two components: the

direct radiation from the source scattered by this element, and the. re-emitted

radiation arriving from all other points of the medium. This equation is a special case of (2.4.26), discussed in Sec. 2.4, which describes the transfer of radiation in a spherically symmetrical medium. In the present case

we have assumed a rectangular profile and set S*(t) = Aexp (— t) /4ttt2 .

It is useful to transform (1.1) before obtaining its solution. Let the

components of the optical radius -vector to the field point M be t^, Ty, x^.

By integrating the Green's function S*(x) over the plane t-^ = const, we obtain

the quantity

—CO CO

Transforming from rectangular coordinates x , x , x to cylindrical coordinates x^, 4), x^ (Fig. 4), we find x y

z

CO ^^(r^) = 2ii /Sp(x)x^dx . (1.3)

Taking into account that

we have finally

0

X = Xi + X

1 z

00 ^ Z

oo ./

(xj = 2tt / Sp(x)xdx . (1.4)

z

Fig. 4. Transformation of the equation for the Green's function.
I 3.1 THE GREEN'S FUNCTION FOR AN INFINITE HOMOGENEOUS MEDIUM 99

Similarly, we obtain

CO CO f f ^ dx^dXy = Ztt J

dx = 2t:E^(|tJ) (1.5)

— CO — CO

I X I

and, finally,

exp ( — I X—X

'

I

)

r r exp(.-| x-x' i J / / dx dx = 2ttE^(|x -x'

J J i x-x • 1 2 ^ y

1 ^

1 z z

) . (1.6)

Therefore, by integrating (1.1) over all Xx and Xy, we obtain

OO CO oo

^ CtJ = \ \ E,(|x -x'|)dx' / / S (x')dx'dx' + t Ei (I - oo>- 2 J

1 ' z z'^ z / / p

^ X y 2 1 z

00 — 00 tJ) . (1.7)

Using (1.2) once again, and writing x for the independent variable x^, we finally arrive at the following equation for $oo(t):

CO

^coC-^) = \ j E^(|x - X' l)*,„CT')dx' + I

E^(|x|) . (1.8) — oo

It is evident that $ (x) is an even function, so that only x > 0 need be

considered.

From the solution (1.8), we readily obtain the solution of (1.1), since

Sp(x) and ^>^(x) are related by

S (x) = ^ — $ (x) , (1.9)

P 2ux dx ^

'

which follows from (1.4).

EXPLICIT EXPRESSION FOR THE GREEN'S FUNCTION. Equation (1.8) is typical of

the class of equations readily solved by the use of Fourier or two-sided

Laplace transforms (see e_^g . E. C. Titchmarsch, 1937; P. Morse and H. Feshbach,

1953). Let us denote by f(s) the one-sided Laplace transform of the function

£(T) :

oo f(s) = y e~^'^f(x)dx . (1.10) ~~ 0

Applying the two-sided Laplace transform to (1.8) and using the convolution

theorem, we obtain

li,
100 MONOCHROMATIC SCATTERING

E (s) + E (-S)

2 i_|e (s) - |e (-S)

But

00 E^(s) = / e~^^E-^CT)dT = -£n(l + s) , (1.12)

so that the last equation may also be written in the form

A „ 1 + s rr—ln-; ? (S) + ? (-S) = , ,

. (1.13) " 1 - ^in^ 2 s 1-s

To obtain ^'^(t) for t > 0 one must invert the Laplace transform, i.e.

evaluate the contour integral

^ooC-^) = ^ /

^oo(s)e^''ds , (1.14)

where all singularities of $ (s) lie to the left of the line Re s = ag. We

require the solution of (l.sf that is bounded at infinity. The Laplace transform ^^(s) of such a solution is a function of the complex variable s regular

in the right half-plane. Therefore in (1.14) we can set oq = 0 , and take the

imaginary axis as the path of integration. Using (1.13) and taking into ac- count the regularity of $oo(—s) in the left half-plane, which is a consequence

of the regularity of ¥ (s) in the right half-plane, we can write, instead of

(1.14),

+ icO r- _1 "I

—100

The substitution of s = iu reduces this integral to the form

0 Finally, application of (1.9) gives

00

$ (T) = i /" Aarctgu ^^^^ _ (-1^16)

°° TT 7 u-Xarctg u

oo

S (T) = -J— f_AiI£lgi^usinTudu . (1.17)

P z-n^T J u-Aarctg u

0 This is the desired explicit expression for the Green's function for an infinite homogeneous medium.
3.1 THE GREEN'S FUNCTION FOR AN INFINITE HOMOGENEOUS MEDIUM 101

ALTERNATIVE REPRESENTATION OF THE GREEN'S FUNCTION. The representation (1.17)

is inconvenient for calculation because the integrand oscillates. However, a further transformation of the integral in (1.15) is possible, leading to an expression that does not have this disadvantage. This transformation is obtained by deforming the path of integration.

For Re s < 0 the integrand has the following singularities:

1. A pole on the real axis at s = — k where k is the positive root of the

so-called characteristic equation —X „ 1+k

In

2 k 1-k

(1.18)

2. A branch point at s =-1, where In (1 + s) is not s ingle - valued

We make a cut along the real axis from — to —1 and deform the path of

integration in (1.15) so that it encompasses all of the singularities of the

integrand in the left half-plane (Fig. 5). To evaluate the integral along

the cut, we note that above the cut s = x+iO, and

1 ^ n 1+s

1 £n2 s 1-s

1 - -^in

2x

x+1

x-1 2x

Below the cut s = x-iO, so that

1 ^ r, 1+S , X . 1 In =1 In

2s 1-s 2x

x+1

x-1

, Xtt . + 17^ •

Therefore the integral along the path £, encompassing the cut, is equal to

-1 r. .-1

In

2ui y \\ 2x x-1

lle^^ dx +

ZTii [V 2x x-1 2x / J

le^^dx ,

(s;

Fig. 5. Path of integration for the evaluation of the Green's function

for monochromatic scattering.
102 MONOCHROMATIC SCATTERING

or

1 \^ 2x x-iy \^ 2xy

In evaluating the integral along the path £ we have taken into account the

fact that the contribution from the small circle of radius r centered at

s = —1 tends to zero as r ^ 0.

Now let us calculate the contribution to the integral (1.15) from the

pole at s = —k. The residue at this pole is —k(l-k^) -kx

— e X-l+k2

Differentiating (1.18) with respect to \, we find that the coefficient of

e~kT is just X I dk/dA j . The pole term may therefore be written as

X I dk/dX I e""'^''^ . Substituting x = 1/y in (1.19), adding the contribution from

the pole, and allowing for the fact that $00(1^) is an even function, we finally

obtain

1 I T I $ (t) = 2kB^e""^l^l + i /* e ^ R(y)

^ , (1.20)

0 whe re

R(y) =

l-k^ X dk

2(k2+X-l) 2k dX

(1.21-)

Using the relation (1.9), we find from (1.20) that the radiation field

arising from an isotropic point source of strength 47t/o2 in an infinite homogeneous medium has the source function

S (T) = -A- f^k^ e-^-^ . ( e"^ R(p)

^ , (1.22)

P 4uT V X J ' ,,2 /

'

and, in particular, in the conservative case

1 T

S (t) P 4ttt

0 1-^3 e ^ R(M) dH.j

, X = 1 . (1.22-)

The derivation of this expression for the Green's function has been the

main object of this section. We shall analyze it in detail below. In most

of the standard texts on transfer theory, an analysis of this kind is either

omitted or given very briefly, leaving unanswered a number of questions of

special interest to us. An exception is the book by K. M. Case, F. de
3.2 THE DIFFUSION LENGTH AND RELATED QUANTITIES 103

Hoffmann and G. Placzek (1953), from which most of the results presented in

Sec. 3.2 — 3.6 are taken.

It is not completely accurate to refer to Sp(T) as the Green's function

for an infinite medium. It would be more nearly correct to refer to the

function

Goo^l'l') = <S(|t-t'|) + Sp(|T-T'|) , (1.23)

where 6 ( | ;r —t '

| ) is the delta- function , as the Green's function. We hope that

this minor Tncons istency will not cause confusion.

The solution of the Peierls equation for an infinite medium,

X / exp (I T—T

4tt /

I

'

I 2 S(t) = — / ""^^ I- -' I

S(T')dT' + S*(t) (1.24)

* with an arbitrary source term S (x), is expressed in terms of G as follows

S(T) = JS (l')G^(T,T')dT' , (1.25)

or, by virtue of (1.23) ,

S(t) = S*(t) + y*S*(T')Sp(|T-T' |)dT' . (1.26)

The T_' - integration in (1.23) — (1.26) extends over the whole space.

3.2 THE DIFFUSION LENGTH AND RELATED QUANTITIES

In monochromatic scattering the structure of the radiation field depends

in an essential way on the parameter k introduced above as the positive root

of the transcendental characteristic equation (1.18), for it determines the

position of the pole of the Laplace transform of the function $oo(t). In Sec.

3.3 it will be shown that far from the source the radiation field is entirely

determined by the value of k.

The root of the characteristic equation and a number of quantities re- lated to it also play an important role in other problems of monochromatic

scattering. We shall now examine the way in which these quantities depend on

X.

The quantity k is a monotonic function of X, varying from k = 1 at X=0tok=0atX=l (Fig. 6). The reciprocal of k is called the diffusion length , which we shall denote by t^. It is important to emphasize that

the diffusion length differs significantly from unity, i.e. from the mean

free path of a photon, only when absorption is relatively unimportant _ in

comparison with scattering. For example, if every second photon survives the

scattering process (X = 0,5), the diffusion length is approximately 5 percent

greater than the mean free path. At X = 0.9 the diffusion length becomes
104 MONOCHROMATIC SCATTERING

0 0.2 0.4 0.6 0.8 1.0 X

Fig. 6. The root k of the characteristic equation as a function of the

albedo for single scattering X. approximately twice as great, and only for very weak absorption (1 — X £ 0.01)

do the diffusion length and the mean free path differ by an order of magnitude . From (1.18) it is easy to obtain the following asymptotic expansions,

which are useful for calculations when X is small:

k 1 - 2e-2/^(l . . 2A-12X^x' ^-4/X ^ ^ ^2.1)

~ 1 - 4e-^/^fl . izile-^/^ . 24-20X.5X^ ^-4/X ^ \ ^ 2)

In the opposite case when X is very close to unity (1 — X << 1) , we have:

(3(1-X))'2

1,2 3(1-X)

2 1 2 2 2 = 1 - -(1-X) i^(l_X)^ £_(i_x)-^ + 5^ ' 175^ ' 125

1 - -(1-X) + ^(1_X)2 + _i-(l_X)^ +

5 175 175^ '

(2.3)

(2.4)

It is thus evident that in the extreme case of very weak absorption, the

diffusion length is approximately (3(1—X)) 2. From (1.18) it follows that

dk^ ^ 2k^ (1-k^) ^ 4k^ ^2

dX X (X-l+k2)

(2.5)

We have already encountered this quantity; it is the coefficient of the exponential term in (1.22). Differentiating (2.2) and (2.4), we find the following useful expansions
3.3 THE GREEN'S FUNCTION FOR AN INFINITE MEDIUM: GENERAL ANALYSIS 105

_ . . 8-6X^-2/X ^ 72-84xW ^ \

, , ^ ^ (2.6)

-f(l-X) . * 3^(l-A)' * ...) , 1 - X « 1 . (2.7)

The values of the quantities just mentioned and the quantity A encountered in

the next section are given in Table 3.

3.3 THE green's FUNCTION FOR AN INFINITE MEDIUM: GENERAL ANALYSIS

BEHAVIOR AT INFINITY. Let us now discuss the properties of the Green's function for an infinite homogeneous medium. It can be seen from (1.22) that the

function xSpCx) is the sum of two terms. One of them — the term not involving the integral — decreases with distance as e"^"^ , with k < 1. The second,

being a superposition of the functions ' with y < 1 , should decrease at

least as fast as e~^ (it tends to zero even slightly faster; see below). For

large enough distances from the source, therefore, the first term should be- come predominant, i.e. it describes the asymptotic behavior of the solution

for T 0°. The integral component, on the other hand, may be regarded as a deviation from this asymptotic solution which is significant only when t is

sufficiently small, i.e. near the source. Thus the Green's function (1,22)

TABLE 3

THE CHARACTERISTIC ROOT k AND RELATED QUANTITIES

X k

dk^

dX

A

0 . 0 1 0000 1 0000 0 0000 0 00000

0 . 3 0 9974 0 9948 0 1162 0 . 08176

0 . 4 0 9856 0 9715 0 3733 0 2305

0 . 5 0 9575 0 9168 0 7319 0 3992

0 . 6 0 9073 0 8233 1 1460 0 5568

0 . 7 0 8286 0 6866 1 5900 0 .6947

0 . 8 0 .7104 0 5047 2 05 11 0 .8128

0 . 9 0 .5254 0 .2761 2 5224 0 9 136

0 . 92 0 4740 0 .2247 2 6175 0 .9320

0 . 94 0 .4140 0 .1714 2 .7128 0 .9498

0 . 96 0 . 3408 0 .1162 2 8083 0 .9670

0 . 98 0 . 2430 0 .0590 2 .9041 0 .9838

0 . 99 0 .1725 0 .0298 2 .9520 0 .9919

100 0 .0000 0 .0000 3 . 0000 1 .0000
106 MONOCHROMATIC SCATTERING

may be represented in the form

where

Sp(T) = S^3(T) . S^^(T) , C3.1)

TTT

and S-(--j.Ct) describes the behavior o£ the Green's function in the near zone. The subscripts "as" and "tr" denote "asymptotic" and "transition," respective'

ly. The feasibility of this division into asymptotic and nonasymptotic parts

is the most important feature of the Green's function. As will become clear

later, this feature is not peculiar to the problem of a point source in an infinite homogeneous medium, but is characteristic of the entire class of

monochromatic scattering problems.

But in reality, how universal is the possibility of dividing the Green's

function into two parts in this way? At "large enough" distances from the

source, of course, the exponential term is dominant and the division becomes

valid. We must consider, however, the possibility that these distances are

so large that radiation from the source is unlikely to reach this asymptotic

region.- When the survival probability of a photon is significantly different

from unity, this is just what does occur. INTEGRAL PROPERTIES. In order to analyze this question properly, we shall

first consider the integral properties of the Green's function (1.22). We

wish to ascertain what fraction of the photons in the medium is described by

the asymptotic part of the solution. In other words, we wish to evaluate the

following expression:

S^3(T)dT

(3.3)

/Sp(T)dT

where the integration in both cases extends over all space.

The numerator may be calculated directly:

oo

S (T)dT = 4k^B^ / Te"^"^ dx = 4B^ . (3.4)

0 Evaluating the integral in the denominator is somewhat more complicated.

Using (1.9) and integrating by parts, we find that

oo /t^^ Sp(T)dT = 2(2n + 1) j T^''$^(T)dT , n = 0,1,2, ... . (3.5)

0

For small values of n the integral on the right can be calculated in the

following way. In the domain of regularity of the function ^^(s), i.e. for

Re s > — k, it can be expanded in a Taylor series:
3.3 THE GREEN'S FUNCTION FOR AN INFINITE MEDIUM: GENERAL ANALYSIS 10 7

00 00 00 CO

= f

^„(T)e-^^ dT = y* $^(T)dT - S J

T<f^(T)dT + ^ f

T^$^(T)dT - ... (3.6)

0 0 0 0

Consequently in the strip — k < Re s < k

GO OO

^cofs) +^00^-5) = 2 f

$^(T)dT + 5^ fT^$^(T)dT + ... (3.7)

0 0 Substituting this expansion into (1.13), we obtain

°° OO _^

2 f

$^(T)dT + s2 f ThjT)dt = (l - ^^ni^j - 1 . (3.8)

Expanding the right side in a power series and equating the coefficients of

equal powers of s, we find the values of the integrals in question. In particular ,

00 f $ (T)dT = , (3.9) J ^ 2(1-X) ' 0

OO / T2$^(T)dT = ^— . (3.10)

0 3(1-A)2

From (3.5) and (3.9), it follows that

fS (T)dT = — . (3.11) ^ P - 1-A

This equation has a simple physical significance. In order to interpret this

equation, we note that the ratio of the total number of photons emitted in the

medium per second to the number of photons created by the sources per unit

of time gives the mean number N of scatterings of a photon, i.e.

7

N = -W- , (3.12)

fs^dx

7i where S is the primary source function, and integration extends over the

entire volume of the medium. (For more detail, see Sec. 6.8.) We shall

apply this formula to the problem at hand. The free term of the original

integral equation (1.1) must be substituted for S*, and the solution Sp(T)

of this equation must replace S. We obtain

N = 1 /* S (T)dT , (3.13)
108 MONOCHROMATIC SCATTERING

which, combined with (3.11), gives us

N =

1 (3.14) 1-A

Thus (3.11) is a direct consequence of the physically obvious fact that in an infinite homogeneous medium the mean number of scatterings of a photon is

equal to (1 - X)-l.

DOMAIN OF APPLICABILITY OF THE ASYMPTOTIC THEORY. We now resume the calculation of the fraction of the photons described by the asymptotic term. Substituting (3.4) and (3.11) into (3.3), we obtain finally

Turning to the values of A in Table 3 (p. 105), we see that for strong

absorption (small A) the asymptotic term describes a rather small fraction

of all photons. Thus for A = 0.4 this fraction does not exceed 25 percent,

and for A = 0.3 it is less than 10 percent. Only for nearly pure scattering

(1 — A << 1) is there an overwhelming preponderance of photons in the zone where the asymptotic behavior, described by (3.2), is established. Consequently, the greater the role played by absorption, the more cautious one must be in drawing any conclusions about the nature of the radiation field if

information about only the asymptotic part of the solution is available. Such

conclusions are at best valid for only a small fraction of all photons. It can be shown that the range of parameter values for which the substitution of

the asymptotic for the exact solution provides reasonable accuracy is even narrower than might be thought on the basis of the figures just mentioned.

In (3.3) the main contribution to the integral containing S^5(t) comes from

those regions where StrC^) is still not negligible in comparison with S^s(t),

Therefore the fraction of the photons found in the asymptotic region, i.e.

where 335(1) >> StrC^), is substantially less than A.

3.4 THE green's FUNCTION FOR AN INFINITE MEDIUM: DETAILED ANALYSIS

BEHAVIOR IN THE VICINITY OF THE SOURCE. Having discussed in general terms

the properties of the Green's function, and having outlined the limits of

applicability of the asymptotic theory, we now turn to a detailed analysis of

the expression (1.22). This analysis is essentially a study of the "near"

zone, where the exact solution of (1.22) cannot be replaced by the asymptotic

form (3.2). The word "near" is in quotation marks here because in fact, as we have seen, the dimensions of this zone are very large when A is small. We

have a special reason for studying the behavior of the Green's function in the

"near" zone as thoroughly as possible: when we study radiation scattering in

spectral lines, we shall constantly be dealing with a situation that strongly

resembles that of the "near" zone.

A = i(i_A)B^ . (3.15)

In particular, when 1 — A is small,

A = 5^ 175^ ^ 175^ ^ (3.16)

First let us study the radiation field in the immediate vicinity of the

source, i.e. for small x. Multiplying both sides of the integral equation
3.4 THE GREEN'S FUNCTION FOR AN INFINITE MEDIUM: DETAILED ANALYSIS 109

(1.1) by T and taking the limit as t ^ 0 we obtain

2 > 2 X rexp(-|T-T' I) lim t^S^Ct) =-^+limT^-^/ ^s^Ci'^dT' . (4.1)

T->0 P 4^ ^^J t-t'|2 P

The limit on the right equals zero. Therefore for sufficiently small t

The contribution to Sp(T) from radiation that has arrived from the source without being scattered is

S*(T) = 2- £-1 , (4.3)

477 ^2 which coincides with (4.2) for x << 1. Such a result might have been expected.

The behavior of Sp(T) for small t may be studied in more detail. The

technique for such a study, based on the use of (1.17) and (1.22), is ex- plained in detail in Sec. 4.2. The result is

The intensity I clearly depends on two variables: the distance from the

source, and the angle 6 between the position vector and the direction of

propagation of the radiation. It can be shown (M. G. Smith, 1964) that for

y = cos 9 > 0 and x ^ 0

I(x,u) = ^— (t\ - arcsin(l-y^)^) i + O(Jlnx) , (l-y2 3% \ /

t (4.5)

2 J' rr ^ ^ ^ arcsin(l— y )^ ^ I (t ,-y) = — + 0 (£nx) , 4tt Q_y2-)% X

or, if we convert from the variable y to the angle 6,

I(x,e) = — - + 0(£nx) , 0 < 0 < X ^ 0 . (4.5')

4tt sin9 X

We note some errors in Smith's paper: a factor \/2 is omitted in the right

side of the expansion, and the second term is incorrect throughout. Instead

of using this term, we give in (4.5) and (4.5') only an estimate of its

growth rate for x ^- 0

l) DETAILED ASYMPTOTIC THEORY. Let us turn back to the Green's function. Comparison of (4.4) and (1.22) shows that for small x the main contribution to
110 MONOCHROMATIC SCATTERING

Sp(x) comes from the non- asymptotic part of the solution, i.e. from

(t) tr ^

' 4TrT ^

^2

(4.6)

which is conveniently represented in the form

X -T

4ttt' (4.7)

where ^r^^i^ allows for the deviation of S^.-p(T) from the expression (4.3) for

S*(t). ^Thus

£p(T) = xe^ /e-^/^R(y)dH (4.8)

It is advantageous to write S^j.(t) this way because at moderate values

of T (let us say, for t < 20) the function ?p(T) is of the order of unity.

It can be considered as a correction factor, and set equal to unity in estimates. And when t is large, Cp(T) simplifies greatly, and a rather simple

asymptotic representation can be obtained for it as follows. The substitution

1

1 + x reduces (4.8) to

?p(T) = T'/ • (-)e dx

(4.9)

(4.10)

For large t the main contribution to this integral comes from small values of

X. In this region the function R(l/l+x) can be approximated by

(4.11)

whe re

Xg = 2e

-2 /A

Substituting (4.11) for R in (4.10) and introducing y = tx, we find finally,

for large t,

^ (t) F(t) , (4.12)
3.4 THE GREEN'S FUNCTION FOR AN INFINITE MEDIUM: DETAILED ANALYSIS 111

where

oo I-l

; and

F(t) = y* + A. (£ny-t)^j e ^'dy (4.13)

t = InZx - - . (4.14)

The expression (4.12) exhibits the asymptotic behavior of ?p(T) for

large t. It is remarkable that the quantity \^E,^{t:), which depends on two

variables t and X, in the extreme case of large t reduces to a rather simple

function of the single variable t. Naturally, this is a great simplification. In later chapters we shall see that similar simplifications play a tremendously important role in the study of scattering in spectral lines.

NUMERICAL DATA. To obtain the Green's function it is necessary to evaluate

F(t) . For t2 < tt2 this function may be expanded in a Taylor series

2 3 F(t) = 0.88322 - 0.04684t - 0.11500^ + 0.01863— +

2 3!

t^ t^

+ 0.08280± 0.0202— ... , (4.15)

41 51

'

which, taking all of the terms appearing here, for |t| ^ 0.5 gives F(t) to

four significant figures. The following asymptotic expansion is valid for

t^ > tt2:

(4.16)

* where y is Euler's constant. When t does not satisfy one of the inequalities |t| << TT

, |t| >> IT, F(t) can be evaluated numerically from (4.13) (Table

4).

The expressions ( 4 . 12 ) -( 4 . 14) give ?p(T) for t >> 1. In the opposite

case when t << 1, we have

Cp(T) = 1 - ^hJ^-^\^y + ^ZX-Aidy^ni + 0(t2) . (4.17)

All we lack now for a full evaluation of the Green's function Sp(T) are values of ?p(T) in the intermediate region, which we can find from (4.8) by

numerical integration. K. M. Case, F. de Hoffmann and G. Placzek (1953) give

4-s.f. tables of ^p(T) for t < 20 and several values of X. These tables,

unfortunately, are inaccurate. We give a three- figure table of Cp(T) for

X = 1 obtained by rounding off the values of ?p(T) given by Case et al.

(Table 5). At x = 20 the asymptotic expression (4.12) is already quite accurate (Fi g. 7) . After a study of the non- asymptoti c part of the solution, it is helpful

to revert to the discussion of the region for which the asymptotic theory

522-519 O - 74 - 9
112 MONOCHROMATIC SCATTERING

TABLE 4

THE FUNCTION F(t)

t F(t) t F(t) t F(t3

-20 0 ,,0259 -2. 0 .. 7670 5 0 ,.2600

-18 0 ..0320 -1 0 ,.8723 8 0 ,.1236

-16 0 ..0407 -0 . 5 0 .. 8920 10 0 ,.0835

-14 0 ,.0535 0 0 ,.8832 12 0 ,.0601

-12 0 ,.0732 0 . 5 0 .. 8460 14 0 ,.0452

-10 0 .1058 1 0 ,.785 1 16 0 ,.0352

- 8 0 ,. 1640 2 0 ,.6256 18 0 ,.028 1

- 5 0 ,. 3624 3 0 ,. 4686 20 0 ,.0230

- 3 0 ,.6208

TABLE 5

THE FUNCTION ^ (t) FOR X = 1

T S

(t) T fp (t) T

0 . 0 1 , 000 0 . 8 0 . 791 4 . 0 0.544

0 . 1 0 . 956 0 . 9 0 . 775 4 . 5 0.525

0 . 2 0 . 922 10 0 . 761 5 . 0 0.508

0 . 3 0 . 893 1

5 0 . 702 6 . 0 0 .479

0 . 4 0 . 868 2 . 0 0 . 657 7 .0 0 .455

0 . 5 0 . 846 2 . 5 0 . 62 1 8 .0 0.435

0.6 0 . 826 3 . 0 0 . 591 9 . 0 0.417

0 . 7 0 . 808 3 . 5 0 . 565 10 .0 0 . 402
3.4 THE GREEN'S FUNCTION FOR AN INFINITE MEDIUM: DETAILED ANALYSIS 113

1.0

0.8 S 0.6

0.4

0.2

0

-2.0 0 2.0

t

Fig. 7. The accuracy of the asymptotic form of Cp(T) : the curve is the

function F(t); the points are values of (Att/2)25 {20) obtained from (4.8)

by numerical integration.

0 2 4 6 8 10 12 14 16 18 20

' T

j

Fig. 8. Fraction of photons described by the asymptotic term.

I

based on the neglect o£ S-j-jCx) in comparison with S^sC^) is applicable.

Using the results quoted above, it is easy to find the ratio Sas(T)/Sp(T) as

a function of t. This ratio appears in Fig. 8.

It is evident from Fig. 8 that for weak absorption the main contribution

to the Green's function comes from the asymptotic term even at distances as small as one mean free path from the source. In the conservative case

(X = 1), neglecting StrC^) in comparison with Sas(T) gives an error of less

than 2 percent for t > 2. For t > 3 the error does not exceed 0.5 percent.
114 MONOCHROMATIC SCATTERING

NEARLY CONSERVATIVE SCATTERING. As we have just seen, i£ 1 - A << 1, we need

to take S-t;j-(''^) into account only for small t. But in this region S^j.(t) for

1 — A << 1 can be assumed to be approximately equal to StrC^) foi" A = 1. Therefore when the scattering is nearly conservative, the Green's function

may be approximated by

^p"^^'^^ ~ ^as^^'^^ ^ S^^(T,1). 1 - A « 1 , (4.18)

where the second argument is added to emphasize the A-dependence of the functions involved. Furthermore, in the expression (3.2) for Sa^s(''^)> 'the factor

can be replaced without great loss of accuracy by the leading term of

its expansion for 1 — A << 1 and S^;f.(T) can be replaced by Cp(T) according to

(4.7). Then (4.18) assumes the form

S^(T) ^ (sxe-^^ + e-U^T)) , 1 - A << 1 ,

" (4.19)

where ^p(T) refers to A = 1. Hence it follows that for t << t, e 1/k the

Green's function for 1 — A << 1 is equal to the Green's function for A = 1 to within the accuracy of the leading term of the expansion in terms of

(1 — A)^. In other words, for nearly conservative scattering, absorption may

be ignored in the first approximation at distances from the source that are small compared with the diffusion length; the effects of absorption appear

only at a distance on the order of the diffusion length. In the next chapter

we shall see that a similar situation exists for radiation in spectral lines,

except that the dimension of the conservative region is governed by the so- called thermalization length rather than by the diffusion length.

ACCUMULATION EFFECT. We shall now turn to the cumulative effect of multiple

scattering. More precisely, we want to see how the value of the source function at a given distance from the source depends on the degree to which the

radiation interacts with the -medium.

2 Let us consider a point source of strength 4TT/a located in a vacuum,

with a test volume of matter a distance r from it. We assume that the properties of this matter (its absorption coefficient a, value of the parameter

A, etc.) are the same as those of the matter occupying the whole space in the

system we are discussing. Let the volume element have the shape of a cylinder with cross section 1 cm^ , length dr, and with its axis directed toward

the source. The flux of radiation incident upon this volum.e is

4tt 1 ^ J_

(j2 ^2

2 An amount of energy odr/x is absorbed from the incident flux per unit time,

of which a fraction A is emitted in all directions. The source function in

the test volume is therefore

Snd) = .

' (4.20)

If the source were in an infinite homogeneous medium, then at a distance t from the origin the source function is equal to the Green's function S (t).

Therefore the ratio
3.4 THE GREEN'S FUNCTION FOR AN INFINITE MEDIUM: DETAILED ANALYSIS 115

i(T) =

47tt' Sp(T) (4.21)

indicates the factor by which the source function within a test volume located at a distance r = i/a from the source changes under the influence of the

medium. The quantity i(T) can be called the medium- effect coefficient and is

shown for several values of X, in Fig. 9. (This figure is based on data given

by K. M. Case, F. de Hoffmann, and G. Placzek, 1953.) Our object is to discuss the general behavior of the function i(T).

The radiation is severely attenuated in a strongly absorbing medium

(small X) . In the extreme case A -> 0 the attenuation is exponential (e"""^) . The rate at which the medium-effect coefficient decreases with increasing t becomes smaller as X becomes larger. For X > 4/tt2 = 0.405 a region around

the source exists where the medium acts as a kind of amplifier, or, more precisely, as a storage element. However, as long as absorption plays an important role, the size of this region and the accumulation of photons are not very large. Thus, even for X = 0.9 the maximum value of the medium- effect

coefficient is only about 1.9, and the radius of the accumulation zone, where

i(T) > 1, does not exceed 4.8 mean free paths. Only for very weak absorption

(1 — X << 1) does the accumulation effect become significant.

It is clearly of interest to consider the case of nearly conservative

scattering in a little more detail. From (4.21) and (4.19) it follows that

i(T) ~ 3Te ^"^ + e ^^p(T) , 1 - X << 1 . (4.22)

For large i the second term on the right is negligible compared to the first;

so for large t

i(T) 3Te-kT (4.23)

X = 0.9

Fig. 9. Medium-effect coefficient i(T)
116 MONOCHROMATIC SCATTERING

It is evident from Fig. 9 that as the role of absorption decreases, the

maximum of i(T) shifts in the direction of large t. Therefore in estimating

the position of the maximum one can proceed from (4.23). Differentiating

(4.2 3) and equating the derivative to zero, we find that the maximum accumu- lation occurs at a distance t^^^x " from the source. In other words, for

nearly conservative scattering the maximum accumulation is observed at a distance of one diffusion length from the source. Since k ~ /3 ( 1—X )

, for nearly

conservative scattering we may use (4.23) to find the maximum value of the

medium-effect coefficient

1^ i(T ) /'—V , 1 - A << 1 . (4.24) ^ max-^ \1— A/

To determine the radius of the accumulation zone, i(x) must be set equal

to unity in (4.23):

3x^e~^^l = 1 . (4.25)

For reasonably small values of k the solution of this equation gives a value

for the radius of the accumulation zone that is several times the diffusion

length X(j = 1/k. For example, for X = 0.99 we find - 4.3X(j, and for

X = 0.9999 we have xi - 7.1x^.

What is the physical nature of the accumulation effect? In a vacuum the

photon trajectories are rays originating from the source. When the source

is surrounded by matter, the photons are scattered and follow, zigzag paths.

Consequently, the escape of photons from the vicinity of the source is inhibited, resulting in their accumulation there. The degree of accumulation of

photons in an infinite medium is limited by absorption.

We have deliberately discussed these obvious concepts in considerable

detail. In the next chapter we will consider the radiation field in a spectral line. One might expect a similar accumulation to occur in that case as well, but it is not quite the same. It turns out that the possibility of

frequency shifts during scattering greatly changes the picture, for in some

cases it sharply reduces the accumulation; while in others nullifies it com- pletely .

3.5 PLANE ISOTROPIC SOURCE

THE SOURCE FUNCTION. Having found the Green's function Sp (x) , we have in

principle solved all the problems of scattering in an infinite medium for

arbitrary distributions of the primary sources. Because of the linearity of

the integral transfer equation, evaluation of the source function requires

the integration of Sp(x) over the source distribution.

Let us consider one important special case. We imagine a uniform distribution of sources in a given plane, and assume them to emit isotropically

with a source strength per unit frequency per cm^ equal to 4tt. We choose the

frame of reference so that the origin lies in the source plane. The source function then will depend only on the_ optical distance x from this plane. We

denote the source function as Spi(x) (subscript "pi" for plane). Let x' be

the optical distance from a field point to an arbitrary point on the emitting

surface, and ar be the optical distance of the latter point from the base of

the perpendicular dropped from the field point onto the plane. The source

function Spi(x) is the superposition of the contributions of each of the
3.5 PLANE ISOTROPIC SOURCE 117

"point" sources distributed in the plane t = 0. Considering that Sp refers

to source strength 4tt/o2, we have

00 Sp^d) = 2^ J

a'-Sp(T')rdr (5.1)

But

T ,2

= T 2^22 + a r

There fore

Sp,(T) = 2./ Sp(T')T' dT' . (5.2)

After comparing (5.2) with (1.4), we conclude that

Sp^/T) = f„(T) (5.3)

Thus the function ^'^(t) has a double physical significance. On one hand, as was shown in Sec. 3.1, it is the integral of S for an isotropic source of

strength 47r/a2 over a plane at a distance t from the source. On the other

hand, as we have just seen, is equal to the source function at a distance

T from a plane isotropically emitting 47t units of energy per sec per cm^

This coincidence should be expected from physical considerations.

ANALYSIS OF THE SOURCE FUNCTION. We can now use many of the results of the

previous sections. The most important of these results is the explicit ex- pression (1.20) for the function $^(t)

the sum of two terms -- one asymptotic

Like Sv3(t), the function $ (t) is

as

(t) = 2kB^e~^'^' (5.4)

and the other non- asymptotic

0

'R(y)^ (5.5)

The first dominates for large values of |t|, and the second for small values

It is useful to write the non- asymptotic term as

tr (T) = A E^(|t|)C^(t) (5.6)

where ?^(t) is the factor by which the function $^^(t) differs from the value

corresponding to single scattering. Values of ^Ji'^) for x < 30 and X = 1 are given in Table 6 (the corresponding table given by K. M. Case, F. de

Hoffmann, and G. Placzek, 1953, is inaccurate; the values of ?„(t) have been

recalculated) .
118 MONOCHROMATIC SCATTERING

TABLE 6

THE FUNCTION ? (t) FOR A = 1

T So:oc (T) T

CO T

CO ^ T K oo

^

0 . 0 1

0000 1 . 2 0 .6733 6 . 0 0 . 4594 16 0 . 3352

0 . 1 0 . 8679 1 . 4 0 .6548 7 . 0 0 .4385 17 0 . 3283

0 . 2 0 . 8320 1 . 6 0 .6382 8 . 0 0 .4208 18 0 .3220

0 . 3 0 . 805 1 1 . 8 0 .6232 9 . 0 0 .4054 19 0 .3161

0 . 4 0 . 7830 2 . 0 0 . 6095 10 0 . 39 19 20 0 . 3105

0 . 5 0 . 7640 2 . 5 0 .5798 11 0 .3799 22 0 . 3005

0 . 6 0 . 7473 3 . 0 0 .5550 12 0 . 3692 24 0 . 2916

0 . 7 0 . 7322 3 . 5 0 .5338 13 0 . 3595 26 0 .2836

0 . 8 0 . 7185 4 . 0 0 .5153 14 0 . 3507 28 0 . 2764

0 . 9 0 . 7059 4 . 5 0 . 4990 15 0 . 3426 30 0 .2698

1 . 0 0 . 6943 5 . 0 0 .4844

For jxl << 1 one can obtain the following expansion for $ (t) :

+ (5.7)

* where y is Euler's constant and

)!i

= / (R(y) - 1) ^ . (5.8)

Values of p_, are given in Table 7. From (5.6) and (5.7) it follows that,

for I T

I << 1

,

A

?o.(^) - 1 - — ^ • ^^-^^

iln

I T I + Y To completely determine the function K^i."^) , we must establish its behavior

for large |t|. Since for r >> 1

E;l(t) ~ ^ , (5.19)
3.5 PLANE ISOTROPIC SOURCE 119

TABLE 7

THE VALUES OF p*

A p*j X p*

0 0 0 0000 0 6 0 1668

0 1 0 1478 0 7 0 0459

0 2 0 3083 0 8 -0 0715

0 3 0 3991 0 9 -0 1825

0 4 0 3744 1 0 -0 2861

0 5 0 2 830

we find from (5.6) that

1 5.(T) - Ixlel^l /

e-|^l/^R(y)^ . (5.11)

0

Let. us compare this expression with (4.8). For large |t| the main contribution to both integrals comes from values of y close to unity. Therefore the

difference of a factor y in the integrands is unimportant, and Coo(t) has the same asymptotic behavior as E, (t) , given by ( 4 . 12 ) - ( 4 . 14) . Th^ results shown

in Sec. 3.4 are also directly applicable here.

ADDITIONAL REMARKS. We could conclude our discussion of the plane isotropic

source at this point. However, a number of additional comments seem to be of some interest. The first of these relates to the behavior of the solution

for weak absorption. When 1 — A is small, the leading term in the expansion

of the quantity 2kB2, which appears as a factor in the pole term of *^(t), is

2kB^ ~ — . (5.12)

2(1-X)^ 2k

Consequently the function $^(t) diverges as X -> 1 . This indicates that a stationary radiation field cannot exist in an infinite medium with an isotropic plane source if the scattering is conservative. In reality, scattering is never strictly conservative, and the medium is never infinite, so that divergence is avoided.

However, the tendency of ^>„(t) to approach infinity as X ^ 1 significantly influences the behavior of the solution for nearly conservative scattering. Since the non- asymptoti c term $^j,(t) remains finite as X -> 1, the

divergence occurs only in the asymptotic term (5.4). Therefore, no matter

how small a value of t we take, a value of X exists such that for larger
120 MONOCHROMATIC SCATTERING

values o£ A, ^as'^''^) will exceed ^-t-^CT). Consequently the smaller the role o£

absorption, the smaller the interval over which the exact expression for

cannot be replaced by the asymptotic term ^asC"!^)* In the extreme case of

weak absorption (1 — X << 1), we have

^ooCt) ~ ^ e-^l"^' ^ i E^CItD^^Ct) , 1 - X « 1 , (5.13)

where ^^(t) refers to X = 1. This expression is a counterpart of (4.19).

The second observation is as follows. Until now the medium has been

considered to be homogeneous. This assumption was essential for the point

source problem. In the case of a plane source, however, it may be replaced

by the less stringent requirement that the absorption coefficient a depend

only upon the distance from the source plane (see Sec. 2.4). All results of

the present section remain valid and only the relation of optical to geometrical distance is changed. Instead of t = az we have

= y a(z')dz' , (5.14)

where z is the geometrical distance from the source plane.

3.6 OTHER SOURCE DISTRIBUTIONS

PLANE PROBLEMS: GENERAL RELATIONS. If the source distribution has plane

symmetry, then the determination of the source function involves just the

solution of the equation

00 S(T) = I y

E^(|t-t' |)S(T')dT' + S*(T) ,

' (6.1)

oo * where S (t) describes the primary source distribution. Let r^(T,T') be the

resolvent of (6.1), i.e. the solution of

00 r^(T,T') = I y* E-^(|T-t|)r^(t,T')dt + A e^(|t-t' I) . (6.2)

00

It follows that the resolvent depends not on t and t' separately, but only on

I T—T

'

I . There fore

,

r<„(T,T') = r^(T' ,T) (6.3)

and .

'

r^(T+T^, t'+T^) = r^(T,T') , (6.4)

where ti is an arbitrary constant. The last equation expresses the transla-
3.6 OTHER SOURCE DISTRIBUTIONS 121

tional invariance of the resolvent, i.e. its independence of the choice of

reference frame in which x is defined.

Sometimes the plane Green's function G (t,t') is used instead of the.

resolvent. It is defined as the solution o?

oo G^(t,t') = ^ J

E^(|T-t|)G^(t,T')dt + 6(T-T') , (6.5)

where 6(t—t') is the de Ita- function . Obviously,

G^Ct.t') = r^CT.T') + 6(t-t') . (6.6)

Comparing (6.2) and (1.8) and using the trans lational invariance of the resol'

vent, we get

r„(T,T') = r^(O.T-T') = *^(t-t') . (6.7)

We note in passing that the resolvent, being a function of the absolute difference of its arguments, satisfies the equation

ar ar oo CO + = 0 . (6.8)

Once the resolvent of (6.1) is known, its solution can at once be written :

CO S(t) = S*(T) + J

S*(T')r^(T,T')dT' , (6.9)

00

or

00 S(T) = y S*(T')G^(T,T')dT' . (6.9-)

00

With (6.7) in mind, we can write (6.9) in the form

00 S(t) = S*(t) + y S*(T')f^(T-T')dT' . (6.10)

00

This is the final expression. The function may be considered known (see

the preceding section)

PLANE PROBLEMS: ASYMPTOTIC SOLUTIONS. If S*(t) tends to zero rapidly enough

as |t| -> oo the behavior of the source function S(t) for large | t | turns out

to be universal; namely, S(t) decreases proportionally to e~k | x 1 . Indeed,
122 MONOCHROMATIC SCATTERING

from (6.10J and (5.4) it can be readily shown that i£

e" ' S (T')dT' < <»

, (6.11)

then

00 S^g(T) = 2kBM f e-^^'s*(T')dT' j e~^'^' , t ± ~ , (6.12)

^ oc. '

where the subscript "as" stands for "asymptotic."

A If S (t) > 0, the last expression can be rewritten in the form

5^3(1) = S^$^Jt + tJ , t ^ ±~ , (6.13)

whe re (y ®^^'^*<^^' J

® ^^'s*(t' )dT'j '

, (6.14)

) is given by (5.4), and is the root of the equation

e 2kT^ y* g k^'s*(T')dT' = y e^^'s*(T' )dT' . (6.15)

This representation enables one to make the following physical interpretation of the asymptotic form of S(t). If the sources are sufficiently con- centrated (in particular, if they are located entirely within a finite range

of T values), then at large enough distances their effect is the same as that

of a plane source. The value of determines the strength of this effective

source, and gives its position.

It is always possible to choose the t reference frame in such a way that

(30 /

* S (t' )sh kx' dT* = 0 . (6. 16)

From (6.15), we see that this choice leads to = 0, so that

S^^(t) = S^$^^(T) , T - ±00 , (6.17)

and, as follows from (6.14) and (6.15),
3.6 OTHER SOURCE DISTRIBUTIONS 123

S^ = y e ^^'s*(T')dT' . . (6.18)

00

We note also another particular case o£ (6.12). If

S*(-T) = -S*(T) , (6.19)

then the function S(t) is also odd, so that (6.1) for t > 0 can be written

as

oo S(t) = -J FE^dT-T'I) - E^(t + t') S(T')dT' + S*(t) . (6.20)

0 According to (6.12) and (6.19), if S (t) decreases steeply enough as t -> °°, the asymptotic form of the bounded solution of this equation is

S^^(t) = S^^kB^e-^^ , T - ~ (6.21)

where

00 S^ = y S*(t' )sh ki'dx' . (6.22)

0 Apart from the notation, this expression is identical to that found by T. A.

Germogenova (1960) by another route. (Germogenova' s expression contains a typographical error.) In the limit X -> 1 (6.21) reduces to Davison's expres

sion (see R. Marshak, 1947):

00 S(~) = ^ f

T'S*(T')dT' , X = 1 . (6.23)

0

SPHERICAL PROBLEMS. Let us now consider an infinite homogeneous medium with

spherically symmetric sources. The functions S* and S depend only on the op

tical distance t from the center of symmetry. We shall show that if S (t)

decreases rapidly enough at infinity, then the asymptotic form of S(t) is,

apart from a constant factor, the same as that of Sp(T).

In Sec. 2.4 we have shown that the equation for S in an infinite homogeneous spherically symmetric medium can be reduced to the form

00 tS(t) = - f ["E^dT-T'l) - E^(t + t')| T'S(T')dT' + tS*(t) . (6.24)

^ 0 ^

Comparing it with (6.20) and using (6.21) and (6.22), we find.
124 MONOCHROMATIC SCATTERING

S (T) = S^-^ 4k2B^e , T ^ 00 (6.25)

as /I'j^j

whe re

00

0

= — / t'S*(t' )sh kt'dx' . (6.26)

In particular, for k = 0 (conservative scattering)

= 4iT y T'^S*(T')dT' . (6.27)

The quantity is the strength of a point source at t = 0 which is asymptotically equivalent in its effect to the distributed source S*(t).

The foregoing derivation of the asymptotic form (6.25) is formal. We

shall also present a derivation based on more easily visualized considerations. This approach will elucidate the physical significance of expression

(6.26)

Let t', 6', (})' be the spherical coordinates as shown in Fig. 10. We

wish now to find the contribution to the source function S(t) from the volume

element located about the field point Ui with the coordinates (t', 6', ({)').

Since we are interested only in the asymptotic behavior of the source function S(t) for large x, we may assume that the rays directed to the point of

observation from all the points with t' e |t'| << t are parallel. Hence the

contribution of the volume element around Mi is the same as the contribution

of a point source with S^ equal to S* (t ' ) t

' 2 di ' s inB ' d6 ' dcj) ' located at the

distance t — x from the point of observation, with x = t'cosG'. In other

words, this contribution is

S*(T')T'^dT'sin0'de'dct>' J_ 4k^B^e~'^'^^~^'^°^®^ . 4TrT

The contribution to S(t) given by the volume element around M-, is, evidently.

Fig. 10. Derivation of the asymptotic solution for the spherically-symmetric

case.

I
3.7 SEMI- INFINITE MEDIUM: GENERAL THEORY 125

S*(t' ) t' ^di' sine' de' d(})' 1 4j^2g2 -k(T + T'cose')

4ttt

so that the total contribution o£ M, and M-, equals

2S (t' ) t' ch(kT' cose ')dT'sine' de' d(j)' 1 kx

4ttx

Therefore, the factor 2 ch (kx ' cos 6

' ) allows for absorption arising from the

differences in the lengths of the rays from Ui, M2 , and 0 to the point of

observation located at a distance x >> 1 in the direction 6' = 0. Integrating

the last expression over 4)' and 6', we get the contribution to S(x) from the sources located in a spherical shell of radius x' and thickness dx'. To obtain (6.25) one has only to integrate the resulting expression over all x'.

BASIC RELATIONS. Until now we have been discussing the scattering of radiation in an infinite medium. We now must consider the effect of boundaries

on the radiation field in the medium. The geometry of a scattering region

can vary greatly, and it is, of course, impossible to study boundary effects

for a general case. However, if the radius of curvature of the boundary surface is much greater than the photon mean free path, the boundary may be con- sidered to be approximately plane, and we obtain the classical problem of

radiation scattering in a semi - inf ini te medium. Such a semi- infini te medium,

bounded by a plane, can, in particular, be regarded as a very good idealization of a stellar atmosphere, which is precisely why this problem was first

studied. More recently similar problems were encountered in the study of

neutron transport.

A semi-infinite medium may be considered to be a good model only for

those regions of the medium where only one (the nearest) boundary significantly affects the radiation field. When "interference" of the boundaries must

be taken into consideration, the problem becomes much more complicated (see

Chapter VIII) . The radiative transfer equation for a plane-parallel medium is (see Sec.

where y is the cosine of the angle between the direction in which the radiation propagates and the outward normal to the boundary, x is the optical

depth, and S*(x) is the function describing the distribution of primary

sources. The solution of the transfer equation is subject to the boundary

condition

3.7 SEMI-INFINITE MEDIUM: GENERAL THEORY

2.3)

1 (7.1)

1(0, y) = 0 y < 0 (7.2)

expressing the absence of external illumination.

The equations (7.1) -(7. 2) define the same problem as the integral equation for the source function S(x) (see Sec. 2.4):
126 MONOCHROMATIC SCATTERING

00 S(t) = ^ J

E^(It-t' |)S(T')dT'

0

+ S (t) (7.3)

The source function is related to the intensity o£ radiation by

1 S(t) = ^ / I(T,y')dy' + S*(t) (7.4)

A huge body of literature is devoted to the study of these equations,

and a discussion of the methods available for solving them may be found in

any text on the theory of multiple scattering of light. We shall therefore

confine ourselves to an outline of the basic ideas, referring for details to

the monographs mentioned at the beginning of the chapter, and to the original

papers cited below. Although an exact solution of (7.3) can be found in ex- plicit form, it is extremely unwieldy for use in practical calculations.

Therefore, in addition to summarizing the formulae expressing this solution,

we will also present tables of auxiliary quantities to facilitate their use. Many of the equations given here are also valid for the general case of scattering with complete frequency redistribution and an arbitrary profile. Derivations of these general relations will be given in Chapter V.

A The solution of (7.3) with an arbitrary source term S (t) may be regarded

as a search for its resolvent r(T,T'), satisfying the equation

CO r(T,T') = I J

E^(lT-t|)r(t,T')dt + ^

e^(It-t'I) . (7.5)

0

when the resolvent r(T,T') is known, the source function is found by quadrature :

S(t) = S*(t) + JS*(t' )r(T,T' )dT' .

0

(7.6)

It may be shown (see Sec. 5.1) that to obtain the resolvent r(T,T') depending

on two arguments, x and t', one has to find only the function r(0,T) = 1(1,0)

of one variable, which we denote as $(t):

$(t) = r(0,T) (7.7)

The function $(t) will be referred to as the resolvent function . As follows

from (7.5) and (7.7), it is the solution (bounded at infinity) of the equation

<J)(t

00

) = J y E^(|t-t' |)$(T')dT' + ^

E^(T) (7.8)

The resolvent is expressed in terms of the resolvent function $(t) by
3.7 SEMI- INFINITE MEDIUM: GENERAL THEORY 127

r(T,T') = $(|t-t'|) + / $(T-t) $(T'-t)dt , (7.9)

where is the smaller o£ t and x' (B . Davison, 1958, Chapter VI; V. V. Sobolev, 1956, Chapter VI, see also Sec. 5.1).

The function 0(t) has the following physical interpretation. Consider

a plane isotropic source at optical depth x in a semi - infinite medium whose

strength is 4it per cm^ . Then the source function at the boundary of the

medium is *(x). It is also possible to regard $(x) as the source function

at a depth x in a medium, bounded by a plane isotropic source emitting 4tt units of energy per unit area. That the physical significance of the function

|J>(x) is open to a dual interpretation of this kind is a particular consequence

of the reciprocity principle.

The following explicit expression can be obtained for the function $(x)

(I. N. Minin, 1958; see also Sec. 5.3):

$(x) = 2kBerk(T-^Xe) X r^-x/yR(y) dy

2 J H(y) y (7.10)

where the notation is identical to that of (1.20), H(y) is the solution of

the nonlinear integral equation

H(y) = 1 + ^

yH(y) f

^^^^ dy'

2 J y+M'

(7.11)

and X is the so-called extrapolation length defined by the relation

e = B (7.12)

In particular, for conservative scattering

$(t) = ^ + 1 y

-T/y R(y) dy ^ ^ H(y) y

' (7.10')

The resolvent function $(x) and the function H(y) are the basic special functions associated with the problem of light scattering in a s emi - infinite medium.

We shall call the function H(y) the Ambartsumian function (V. A.

Ambartsumian , 1942, 1943; see also V. A. Ambartsumian, 1960). In all fairness

it must be mentioned that (7.11) was first obtained and solved by 0. Halpern,

R. K. Luneburg, and 0. Clark (1938); however, their study, unlike that of

Ambartsumian, did not attract the attention it deserved.

522-519 O - 74 - 10
128 MONOCHROMATIC SCATTERING

THE AMBARTSUMIAN H-FUNCTION. An explicit expression for H(y) can be obtained

in the form of a definite integral. The integral representation of H(y) (for

the special case of conservative scattering) was first found by E. Hopf

(1934). Later other representations were found for H(y); 0. Halpern, R. K. Luneburg and 0. Clark (1938), and, independently, V. A. Fock (1944) showed

that

00 (7.13)

From this expression it follows, in particular, that for 0 < y < 0° the

tion H(y) increases monotonically with y. Its limiting values are

funcH(0) = 1 ; H(«>) = (1-A)-

The function H(y) is usually considered only for values of the argument

y in the interval 0 < y < 1. As seen from (7.11), once H(y) is known in this

"basic" interval, its value for any other value of y can be found by simple

integration.

There are numerous tables of H(y) for 0 < y < 1. The tables given by

S. Chandrasekhar (1950) were obtained by solving numerically (7.11) using an iterative method. Most complete are the tables of D. W, N. Stibbs and R. E.

Wier (1959), obtained from an integral representation of H(y) by numerical

integration. Table 8 is based on the data of Stibbs and Wier. D. W. N.

Stibbs (1963) has also published a table of H(y) for X = 1 and y > 1.

Let us now consider the integral properties of H(y)

the i-th moment of H(y), i.e.

We denote by aj^

1 = y* y"^H(y)dy , i = 0,1,2, (7.14)

Letting y tend to infinity in (7.11), and using the fact that H(<») = (1—X) we find

a, = i (l - /1-x) . (7.15)

The other moments cannot be expressed in terms of elementary or special functions; they must be found by numerical integration, except for ai when

X = 1. It can be shown that for 1 — X << 1

= 4 [l -

^ /3 L /3q(oo) (1-X)''^ -H |-(1-A) + ..] (7.16)

We note also that for X

^ /3 ^ (7.17)
3.7 SEMI- INFINITE MEDIUM: GENERAL THEORY 129

TABLE 8

THE FUNCTION H(y)

X

0 . 5 0 . 7 0 . 8 0.9 0 950 0 975 0 995 1 000

0 00 u u u u

11 u u u u i n r\ r\ Ci u u u u i n n n n U u u u i n r\ n r\ U U u U i

fi n n n U U U U 1 n n n n U U U U 1 n r> n n U U U U

0 05 1 nU AH A^ Jz 1 \J 0 1 1 1 n sj 1 Q u o i y

11 n Q Q 7 u y y / i

1 1 1 1 i i i) i i i y 4 11 9 Q 7 i z y o 11 7 A ^ 1 O D D

0 10 1 01 7 A 11 1

1 11 o7 nu i

1 7 Q Q i J> 0 O i i / Z 1

1i i y D z i Z

9 111 1 1 1 1 9 7 1 A Z o 1 o 1 Z 4 / 4

0 1

I n Q /I 7 u y 4 / 1 1 b U o i i O O D i o 7 /I n Z J 4y 1 z 0 9 4 1 2 9 3 6 i

7 T r 3 Z 0 6 17 r n o 3 5 0 0

0 20 i i 1 i 0 1

1 o o r 1 0 Z 5 1 o o o 2 2 8 6 1 2 9 14 1 3 3 7 3 1 3 70 3 1 4146 1 4 5 0 4

0 25 1L

1 o n 7 i z y / 1 z 1 U 9 1 2 6 6 3 1 34 33 1 40 0 7 1 44 2 7 1 5001 1 5 4 7 3

0 30 1 14 3 9 1 2 364 1 30 06 1 3914 1 4605 1 5117 1 5 829 1 6 4 2 5

0 35 i

1 r ^ £ 15 6 6 1 2 59 5 1 3 3 2 0 1 4363 1 5171 1 5 7 7 8 1 6635 1 7 36 4

0 40 i 1 0 » U 1 2 8 0 6 1 3611 1 4 7 8 5 1 5 710 1 6 414 1 7 4 2 1 1 8 2 9 3

0 45 i 1

7 A A r\ 3 0 0 0 1 3 8 81 1 518 3 1 ^ o T r 6 2 z 5 1 7 0 2 7 1 819 1 1 n o 1 7 y Z 1 3

0 50 i

1 O "7 "7 loll 1 317 9 1 413 3 1 5 5 6 0 1 6 7 18 1 7 6 2 1 1 89 4 6 Z n 1 o o U 1 Z 0

0 55 1 1964 1 3346 1 4368 1 5918 1 7192 1 8196 1 9688 2 1037

0 60 1 2043 1 3501 1 4590 1 6259 1 7647 1 8753 2 0417 2 1941

0 65 1 2117 1 3646 1 4798 1 6583 1 8086 1 9295 2 1134 2 2842

0 70 1 2186 1 3781 1 4995 1 6893 1 8509 1 9822 2 1840 2 3740

0 75 1 2249 1 3909 1 5182 1 7190 1 8918 2 0334 2 2536 2 4635

0 80 1 2309 1 4029 1 5358 1 7474 1 9313 2 0834 2 3222 2 5527

0 85 1 2365 1 4142 1 5526 1 7746 1 9695 2 1320 2 3898 2 6417

0 90 1 2417 1 4250 1 5685 1 8008 2 0065 2 1785 2 4565 2 7306

0 95 1 2466 1 4351 1 5837 1 8259 2 0424 2 2258 2 5223 2 8193

1 00 1 2513 1 4447 1 5982 1 8501 2 077 1 2 2710 2 5873 2 9078

Here is the value at infinity of the Hopf function q(T) (see Sec. 3.8).

Later we shall encounter the quantity

1 (7.18)

which is the convergent part of the minus-first moment of the H-function. It can be shown (J. W. Chamberlain and M. B. McElroy, 1966) that
130 MONOCHROMATIC SCATTERING

a*-j^ = 2JlnH(l) . (7.19)

Values of , a^, a-^, and a2 are given in Table 9.

As well as satisfying the nonlinear equation (7.11), the function H(ij)

is also a solution of the linear equation (see, e.g. V. V. Sobolev, 1956,

Chapter IV)

H(y)fl - Aji.ni^) = 1 - /^dy' , (7.20)

\ 2 y-1/ 2 ^

y-y'

where (y+l)/(y-l) is to be replaced by its absolute value for 0 < y < 1, and

the integral on the right side is to be understood as the Cauchy principal

value. This equation holds for all complex y, except —1 < y <0. Substituting

y = 1/k in (7.20) and invoking (1.18), we find that

TABLE 9

MOMENTS OF THE FUNCTION H(y)

A a*

^ a

0 ^1 ^2

0 . 1 0 0723 1 0263 0 .5156 0 3444

0 . 2 0 15 14 1 0557 0 .5332 0 3568

0 . 3 0 2388 1 0889 0 .5531 0 3710

0 . 4 0 3368 1 1270 0 .5762 0 3875

0 . 5 0 4483 1 1716 0 .6035 0 4070

0 . 6 0 5785 1 225 1 0 .6366 0 4309

0 . 7 0 7359 1 2922 0 .6787 0 4614

0 . 8 0 9378 1 3820 0 .7358 0 5032

0 . 85 1 0671 1 4416 0 .7744 0 5316

0 .90 1 2305 1 5195 0 .8253 0 5694

0.925 1 3336 1 5700 0 . 8588 0 5944

0.950 1 4620 1 6345 0 .9019 0 6268

0.975 1 6404 1 7269 0 .9645 0 6741

0.985 1 7435 1 7818 1 .002 1 0 7027

0.995 1 9013 1 8679 1 .06 17 0 7485

1 .000 2 1348 2 0000 1 .1547 0 8204
3.7 SEMI-INFINITE MEDIUM: GENERAL THEORY 131

f "I

J 1-]

HClO dy = 1 (7.21)

We also note the relations

X XkH(l/k) f

HMIii^ dy = 1

n l-k2y2

(7.21')

1 XkB2 / _HHlHi_ dy = H(l/k)

Q (1-ky)

(7.21")

To prove the first of these relations we set y = 1/k in (7.11)

the resulting equation

H(l/k) 1 1 -

J

fl A / HiiilidyA

V

2 J l^kyI

= 1

To show that

is identical to (7.21') it is sufficient to replace unity in the brackets by

the left side of (7.21). The relation (7.21") may be derived as follows.

From (7.11) it follows that

dp VH(ii) I y=-i/k

y'H(y')

(l-ky')2

dy' ,

while (7.20) gives

_d_

dy

y=-l/k y 2 J l+ky' / dy \ 2 y-1 / y=-l/k

2B'

H(l/k)

Equating the right sides of these two equations we get (7.21").

We now reproduce the expressions characterizing the behavior of H(y) for

small and large y. From (7.11) it is easy to find that

H(y) = 1 -Ay£ny+ )rO._^\y + 0((£ny)^) , y 0 , (7.22)

so that the derivative of H(y) diverges logarithmically as y -> 0 . Combining

(7.13) and (7.22) it can be shown that as y

-> 0
132 MONOCHROMATIC SCATTERING

X \ * £n H(y) = --\iin y+ +

I

2v (7.22-)

+ ^UX - A^^jy^ + 0(y\n y) .

This expansion was obtained by C. Mark (1947) for the special case A = 1. Using (7.22') it is easy to obtain the coefficients of all terms in the expansion of H(y) — 1 which tend to zero more slowly than y^£ny as y 0 . For

y > 1 it can be found from (7.11) and (7.15) that

1 - (1-X)^- + + . (7.2 3)

H(y) 2 (v\ y ^2 ^3

Hence for y > ( X/ 2 ) ( 1-X)~ ''^a^

H(y) = (1-X) ^^l_A(i_x) ''^a^i + . .

. j ; (7.24)

whereas for 1 < y < (X/2) (1—X) 1 H(y)=--^(l + — -i(l-X)^^ Ji_ + . . .\ . (7.2 5)

X a-j^ y y X a-|^ /

In particular, for pure scattering (X=l) we find from the last expansion,

bearing in mind (7.16) and (7.17),

H(y) = /"y + /3q(-) - ^ 0

- (oo)^ i + 0(y-^) , X = 1 , y - » . (7.26)

It follows from (7.10) and (7.12) that to calculate $(t) one must have

values of H(y) for y = 1/k, or values of the extrapolation length Xg. Values

of these quantities appear in Table 10 (after M. A. Heaslet and R. F. Warming,

1968a; see also I. Ku^^^er, 1953). For small values of 1—X the following ex- pansion is useful:

For nearly conservative scattering (1—X<<1) the function H(y) can be

expressed in terms of the conservative H-function:

H(y,X) = H(y,l)j^l - /3y(l-X)^ + ^3y^-r (y )^

( 1-X)

+ /3y(|-3y^r(y)^(l-X)^/2 ^ 0^(1-X)^)J ,

(7.28)

whe re

J. r(y) = ^ ( R(y') (7.29)

2 J y+y'
3.7 SEMI- INFINITE MEDIUM: GENERAL THEORY 133

TABLE 10

i THE VALUES OF T

e AND H(i

k

X T

e

0 . 3 2 . 4947 1 . 1269

0 . 4 1 . 8249 1 . 1843

0 . 5 1 . 4408 1 . 2552

0 . 6 1 . 1923 1 . 3479

0 . 7 1 . 0181 1 . 4799

0 . 8 0 . 889 1 1 . 6955

0 . 9 0 . 7896 2 . 1710

0 .95 0 . 7479 2 . 8339

0 .96 0 . 7401 3 . 0998

0 .97 0 . 7324 3 . 4888

0 .98 0 . 7250 4 . 1401

0 . 99 0 . 7176 S . 6078

1 .00 0 . 7104 OO

and R(y') refers to X = 1. Values o£ the function r(y) for 0 < y < 1 are given in Table 11. For the proof of (7.28) see Sec. 5.4. The first two

terms of this expansion are easily obtained directly from (7.11) and (7.15).

From (7.28) it follows that this expansion is useful in practice for those

values of y for which ij(l-X)^ << 1. RESOLVENT FUNCTION. Now we are ready to discuss the properties of the resolvent function <J>(t). Using the same method as in Sec. 4.2, we can show that

for small x

X r * * X 2 ^'(t) = — —Jlnt—Y +a_^+—T£n t —

* * 1

^ ^^'^^^ - j(a*i+l-Y*)T£nT + 0(t) ,

* where y = 0.577216 is Euler's constant. The derivation of this expansion,

though it is simple in essence, requires a lengthy calculation.

In the opposite extreme case of large t we have from (7.10)

$(t) = 2kBe~^'^'^*^e) + o(e ^) (7.31)
134 MONOCHROMATIC SCATTERING

0 ,,00 0 ..0000

0 ..05 0 ,. 0695

0 .. 10 0 ,. 1077

0 ,. 15 0 ,. 1356

0 ,. 20 0 ,.1576

0 ..25 0 ,.1757

0 ,. 30 0 ,. 1909

TABLE 11

THE FUNCTION t

U r (y)

0 ,. 35 0 ,. 2040

0 .. 40 0 ,.2 154

0 ,.45 0 ,.2255

0 ..50 0 ,.2344

0 ..55 0 ,.2424

0 ..60 0 ,. 249 7

0 ..65 0 ..2562

U r (y)

0 ,. 70 0 ,.2622

0 . 75 0 ..2677

0 ,. 80 0 ..2728

0 ,. 85 0 ..2 774

0 ,.90 0 ,.2818

0 .95 0 ,.2 858

1.00 0 .. 2 896

For values of t for which neither of these expansions can be used, the function *(t) must be found numerically from its explicit expression (7.10). It

is useful to represent $(t) in a form similar to that used in Sec. 3.5 for

$^(t) , namely

,

$(t) = ZkBe'^'^^^^e) + A E^(t)^(t) , (7.32)

where only the correction factor ?(t) need be tabulated. This factor is of

order unity when t is not too large. For large values of t, C(t) has the

asymptotic form

?(t) (7.33)

where F(t) and t are given by (4.13) and (4.14), respectively. Thus, in the

asymptotic region, ^(t) differs from Cp(T) and C^(t) only by the constant

factor 1/H(1) . ^

For nearly conservative scattering the contribution from the second term

in (7.10) is substantial only for small t, where ?(t) for A < 1 differs little

from C(t) for X = 1. Using (7.27) and the expansions given in Sec. 3.2, we

find that, approximately,

$(t) ~ /"e"^^ + |e^(t)C(t) , 1 - X << 1 , (7.34)

or
3.7 SEMI-INFINITE MEDIUM: GENERAL THEORY 135

$(t,X) ~ /3(e ^^-l) + $(t,1) . (7.35)

In (7.34) ^(t) refers to X = 1.

In contrast to $ (t) , the exponential (pole) term in <I>(t) remains bounded

as X ^ 1 with a limiting value of /3; see (7.10'). This behavior results

from the escape of photons through the boundary which limits their accumulation in the medium.

The function $(t) satisfies several useful integral relations, which we

give for reference without proof.

oo (T)$(T)dT = , (7.36)

oo

1 $(T)dT = (l-X)"''^ - 1 , (7.38)

00 /

A "l T$(T)dT = - — , (7. 39)

2 1-X

oo J

$^(T)dT = I

^P*i-a*^^ ^ 2kB^ , (7.40)

oo ye^($(T)-2kBe-k^""^e)jdT = iM e'^-^e _ i . (7.41)

0

For X = 1 instead of ( 7 . 38) -

( 7 . 41) we have, respectively,

OO y* [$(t) - /3]dT = /3q(~) - 1 ,

^ (7.42)

0

CO y* T[$(T) - /IldT = ^ |-q^(~)] . (7.43)
136 MONOCHROMATIC SCATTERING

/ [$^(T)-3]dT = 2q(c») -1 , (7.44)

•o

^

00 / e^[$(T) - /3]dT = /I - 1 . (7.45)

*0

All of these relations can be obtained by elementary means although in a num- ber of cases the calculations are rather tedious (especially the proof of

(7.44)) .

It is also interesting to note that the resolvent function $(t) for

A = 1, which is by definition the solution, bounded as t <», of an integral

equation of the second kind,

$(t

00

) = E^(|t-t' |)$(T')dT' + I

E^(t) , (7.46)

is also the solution of two equations of the first kind:

f

E^dx-T' |)HT')dT = J- _ £3(1) (7.47)

0

and

00 y E2(t'-t)<I)(t' )dT' - f E2(T-T')$(T')dT' = £2(1) . (7.48)

T Q

To prove these assertions we differentiate (7.47) twice and (7.48) once. In both cases we obtain (7.46). Further, for x = 0 (7.47) and (7.48) are satisfied, since we obtain the relations that follow from (7.37). This completes

the proof.

3.8 SEMI-INFINITE MEDIUM: STANDARD PROBLEMS

The evaluation of the radiation field in a s emi - infini te medium with

some specified distribution of primary sources can be carried out by means of

the general expressions introduced in the preceding section. We shall therefore confine ourselves to a summary of the results relevant to several standard problems that will be considered in detail later in the context of scattering in spectral lines .

UNIFORMLY DISTRIBUTED SOURCES. The radiation field in a medium with uniformly distributed sources can be determined by solving the transfer equation

(7.1) or the integral equation (7.3) with S = const. The source function in

this case is (V. V. Sobolev, 1957a; see also Sec. 6.1):
3.8 SEMI-INFINITE MEDIUM: STANDARD PROBLEMS 137

S(T) = S*C1-X) , (8.1)

where

T ^Ct) =1 + 1 $(T')dT' (8.2)

* _ 1^ The source function increases monotonically from S(0) = S (1—A) ^ to

S(oo) = s*(l—X)~l (Fig. 11). The intensity of the emergent radiation is found

to be proportional to Ambartsumian' s function H(ij):

1(0, y) = S (1-X) ^H(y) (8.3)

As the angle of emergence increases, the intensity decreases with a rate that

increases with A (Fig. 12).

Let us consider the case of nearly conservative scattering (1 — X << 1)

in a little more detail. Using (7.35), we have approximately, instead of

(8.1),

S(t) _ * /3 S k(i-x)^^ L 1-e-k

''+kq(T)J , (8.4)

where q(T) is the Hopf function (see (8.10)). Since k ~ /3(1—X) for

3-2-1 0 I 2 3 4 5

Fig. 11. Depth dependence of the source function in a medium with uniformly

distributed sources.
138 MONOCHROMATIC SCATTERING

Fig. 12. Angular dependence of the intensity of radiation emerging from a semi-infinite medium with uniformly distributed sources.

1 — X << 1, we have

S(T)

* r ^

[lk2 L (8.5)

For the intensity o£ the emergent radiation, one can obtain from (8.3) and

(7.28) the first terms of its expansion in powers of k:

1(0, y) — * /3S H(y) (l-ky) , 1 - X << 1 , (8.6)

where H(y) is the conservative H-£unction.

From (8.5) it follows that far from the boundary

A S(t) - ^

d-e"^"") , T >> 1 , 1 - A << 1 ; (8.7)

k^

2 from which it is evident that in this region the product k S(t) depends not

upon T and \ separately, but only on the combination t e kx . For t < 1 the source function increases rather rapidly with depth. When t reaches the

order of unity, the source function begins to saturate and approaches its

limiting value S(<») = S*(l—X)—1. Thus the effect of the boundary is significant to depths corresponding to a value of t on the order of unity, i.e. for

T on the order of = 1/k. A similar picture is also observed for scattering

in spectral lines with the difference that now the thickness of the boundary

layer is given not by the diffusion length, but by some other quantity (see

Sec. 6.2).

THE MILNE PROBLEM (CONSERVATIVE CASE). .Let us consider one important special

case in a little more detail. We put S = /i — A and then go to the limit
3.8 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS 139

X 1. We then find from (8.1) that the solution of the homogeneous equation

00 S(T) = i f

E^(|t-t' |)S(T')dT' , (8.8)

normalized so that S(0) = 1, can be represented as

S(t) = /3[T+q(T)] , (8.9)

whe re

qCT) = 4i (l + f $(T')dT' I - T . (8.10)

{ I

According to (8.3) the emergent intensity in this case is equal to

1(0, p) = H(y) , (8.11)

and the corresponding flux is

1

ttF = 2u f l(0,y)udy = 2wa, = Az ^ , (8.12)

where a^^ is the first moment of the H- function. The functions $ and H in

(8.10) and (8.11) refer to A = 1

The solution of (8.8) and the determination of the corresponding emergent intensity is known as the (conservative) Milne problem, which has been

studied in dozens, if not hundreds, of papers (see, in particular, the book

by V. Kourganoff (1952), almost completely devoted to this problem). Equations (8.9)— (8.11) give the solution of the Milne problem.

The function q(T), which appears in the expression for S(t), is known

as the Hopf function. The physical significance of q(T) is elucidated in

Sec. 6.9. Substituting (7.10') into (8.10) we obtain the following explicit

expression for q(T), first found by C. Mark (1947):

q(T) = 4: - ^

H^-^~"^')¥^^^ ' fS.13)

/3 2/3 ;f \ /Hfy)

We immediately see that the Hopf function increases monotonically , with

q(0) = 1//3 = 0.577... . The value of q (<») , according to (8.13), is equal to
140 MONOCHROMATIC SCATTERING

1 q(~) = 4: - /

^

dy (8.14)

/3 2/3-^ H(y)

and was calculated by E. Hop£ (1934), who showed that

tt/2

f

0

q(«,) =- f I i 1 de = 0.710 ... . (8.15)

^ J \sin20 1-ectge

j

Many other integral representations can be obtained for q(°^) besides (8.14)

and (8.15); for example,

q(oo) = 1 _ r(l)

1 / * * \

where r(p) , a_i , and p_2 are given by (7.29), (7.18), and (5.8). G. Placzek

and W. Seidel (1947) rewrote (8.15) in the form

T7/2

q(oo) = A + i/' /J_ 1

) de (8.16)

^2 W y^i 1-ectge

0 j

and used this expression to calculate q(°°) to eight places. Su-shu Huang

(1952) and J. I. F. King, R. V. Sillars and R. H. Harrison (1965), using

different methods, obtained q(T) to ten places, which is, of course, far in excess of any practical needs. In a recent paper by H. C. van de Hulst

(1968), the value of q("») calculated to 12 places by K. Grossman is quoted:

q(«') = 0. 710 446 0 89 800 . . .

There are numerous tables of q(T), varying in accuracy and detail. For

a long time the record for accuracy was held by V. Kourganoff's tables (1952),

which gave q(T) to seven places. In 1965 the palm was yielded to J. I. F. King, R. V. Sillars and R. H. Harrison, (1965), who published an 8-figure table of Hopf's function. We give q(T) to five places (Table 12). We note that

for small t, as follows from (7.30) and (8.10),

q(T) = /3

(A A \ l+a_ -Y \

/3 j- j

T +

+ —(T£nT)^ , 16 ^ 16

A A 2a T + 3—2y

2 - T £nT + 0(1^) . (8.17)

C. Mark (1947) found this expansion by another route.

There are two reasons for this great interest in Hopf's function. One

of them has a practical, or more precisely, an applied, basis because the
3.8 SEMI-INFINITE MEDIUM: STANDARD PROBLEMS 141

TABLE 12

THE HOPE FUNCTION q(T)

T q(T) T q(T3 T q(T) T q(T)

0 . 00 0 57735 0 . 08 0 62185 0 . 70 0 690 11 ——

2 . 00 0 70792

0 . 01 0 58824 0 . 09 0 62499 0 . 80 0 69353 2 . 25 0 70867

0 . 02 0 59539 0 . 10 0 .62792 0 . 90 0 69629 2 . 50 0 709 19

0 . 03 0 60124 0 . 20 0 64955 1 . 00 0 69854 2.75 0 70955

0 . 04 0 60629 0 . 30 0 66337 1 . 25 0 70257 3 .00 0 70981

0 . 05 0 61076 0 . 40 0 67309 1 . 50 0 70513 3 . 25 0 70999

0 . 06 0 61479 0 . 50 0 .68029 1 . 75 0 70680 00 0 71045

0 . 07 0 61847 0 . 60 0 .68580

solution of the conservative Milne problem is related to the evaluation of

the distribution of temperature with depth in solar-type atmospheres (see,

e.g. A. Unsold, 1955). The second reason is of an entirely different nature

As is apparent from (8.10), the resolvent function $(t) for X = 1 is simply

related to the Hopf function. Therefore, in terms of q(T) one can express

not only the solution of the Milne problem itself, but also the solutions of

all other conservative scattering problems in a semi - infinite medium.

DIFFUSE REFLECTION. Let parallel rays be incident on a semi- infinite medium

at an angle arc cos yg to the inward normal, and let Iq be the net flux per

unit area normal to the rays. The problem is to find the intensity of the

diffusely reflected radiation. This problem goes back to A. S. Eddington,

who considered it in connection with the reflection effect in binary stars.

Its solution for the conservative case was given by E. Hopf (1934), and for

arbitrary X, 0 < X < 1, by 0. Halpern, R. K. Luneburg, and 0. Clark (1938),

who have shown that

This problem is also of great importance in the study of light scattering in

planetary atmospheres. The problem of diffuse reflection has been solved by

V. A. Ambartsumian (1943) (see also V. A. Ambartsumian , 1960) througha new approach known in transfer theory as the principle of invariance. This approach made it possible to obtain from simple physical considerations the

expression (8.18) and equation (7.11) for the function H(u) which now bears

his name. Ambartsumian ' s method, having proved extremely fruitful, was developed and widely used by S. Chandrasekhar (1950), V. V. Sobolev (1956), and

R. Bellman and his associates (see, in particular, the book by G. M. Wing,

1962, which contains numerous references to the early work of R. Bellman's

grdup on applications of the invariance principles to the theory of multiple

H(y)H(yQ)
142 MONOCHROMATIC SCATTERING

scattering and to other problems). We shall not extend the scope o£ this

book to cover these questions.

Along with the intensity of the diffusely reflected radiation, it is

also useful to know the corresponding source function. It is easy to see that

it must satisfy the equation

00 S(T,Po) = \J E^(|t-t' |)S(T' .yg)dT' + I, -T/y,

4tt (8.19)

in which we have explicitly shown the dependence of the source function S(t)

on the parameter yg. The solution of (8.19) has the form (see, e.g., V. V.

Sobolev, 1956, Chapter III, and also Sec. 6.4 of this book):

A / ~^/^n r -f^-T' ) /Pn ^ S(T,yQ) = lQ^H(yQ)(e J

^ %(T')dT') . (8.20)

In particular.

S(0,yQ) = Iq a H(yQ) (8.21)

and for large t

S(T,y.) ~ In :r- 2kB ^.^1^^ e-^^^^^e) . (8.22)

The last result is obtained from (8.20) by substituting the explicit expression (7. 10)for ^(t).

Substituting (8.22) into the transfer equation

y M = I _ S , (8.23)

dx

we find that for large t

I(T, y,y ) - Iq ^ 2kB

, e'^^^'^e) . ^8.24)u U 4^ (1-kyg) (1 + ky)

It follows that deep within the medium the directional distribution of intensity becomes more nearly isotropic as the role of absorption decreases.

Specifically, for A = 1 , we have k = 0, and (neglecting terms on the order of

e"""^ ) the intensity does not depend on direction; the flux is therefore equal

to zero. Thus the mean intensity

1 J(T) = J

I(T,y)^ = 1 JI(T,y)dy , (8.25)
3.8 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS 143

to the same accuracy, does not depend on depth, and is equal to

J(t) ~ Iq 0 ^O^^^O^ , X = 1 , T >> 1 . (8.26)

I£ the medium is illuminated by isotropic radiation o£ intensity I„, the

intensity o£ the reflected radiation and the source function may be obtained

from (8.18) and (8.20) by integrating over direction:

1 1(0, u) = 2u y I(0,y,yQ)dyQ , (8.27)

0

1 S(T) = 2tt J

S(T,yQ)dyg . (8.28)

0

On the basis of the relations found in Sec. 3.7, it is easy to show that

1(0, y) = Iq [1 - (l-A)^^H(y) ] , (8.29)

S(t) = IqLI - , (8.30)

where ^(t) is defined by the expression (8.2).

For weak absorption the second term in (8.30) approaches the order of

the first only for large x, where the asymptotic expression can be used for

Y(T) :

'^(t) ~ (l-A)~^(l-e~'^^) , T >> 1 , 1 - X << 1 . (8.31)

This result follows from (7.10) and (8.2) for 1 — X << 1. Therefore for nearly conservative scattering we have the approximation

S(t) - IqS"^^ . (8.32)

As is clear from the derivation, this expression is valid for large x. However, it follows directly from (8.30) and (8.31) that for small x and

1 — X << 1, the source function differs only slightly from Ig. The same re- sult is given by (8.32), which is therefore valid for all x.

Two conclusions follow from (8.32). The first is that although the

direct radiation falls off as e— , the diffuse radiation penetrates to great

depths; the smaller k the greater the depth of penetration. The external

radiation penetrates to depths on the order of the diffusion length xj = 1/k.

The second conclusion is as follows. Let there be two nearly conservative

media with values of X equ al to -^1 and ''^2. The corresponding values of k are

k^ ~ /3(1 - \-^) and k^ ~ /3(1 - X^) . Then the source function in the first medium at depth X]^ will be asymp522-519 O - 74 - 11
144 MONOCHROMATIC SCATTERING

totically (X ^ 1) equal to the source function in the second medium at a depth where and are related by the equation k-j^i-j^ = '^2^2

(1 - = (1 - X2JT2 . (8.33)

The relation (8.33) is a similarity principle. The closer and ^2 are to

unity, the more accurately (8.53) is fulfilled. Analogous similarity relations play an important role in the theory of transfer of line radiation

(see, in particular. Sec. 6.6).

THE MILNE PROBLEM (NON- CONSERVATIVE CASE). The Milne problem for an arbitrary

A, 0 < A < 1, involves the evaluation of the radiation field in a semi-infinite medium with no sources at finite depths. The radiation field is generated by a source (generally, of infinite strength) located infinitely deep in

the medium.

Let S(t) be the^source function of the Milne problem. According to the

preceding paragraph S(t) must be a solution of the homogeneous equation

S(t) = ^ / E^(|t-t' |)S(T')dT' . (8.34)

Without any loss of generality we may assume that

S(0) = 1 . (8.35)

The solution of (8.54) is readily expressed in terms of the resolvent

function $(t). In the problem of diffuse reflection the value of appearing

in (8.19) and (8.20) is the cosine of the angle of incidence of the external

radiation, so that 0 < pq < 1. However, we can consider \\q just as a parameter. By (7.20), H(ijq) has a pole at yg " — 1/k, where k is the positive root

of the characteristic equation (1.18). Hence, if we take Iq = ( 47t/ A) (H (y q ) )~

in (8.19) and then let = —1/k, this equation reduces to (8.54). From

(8.20) it then follows that

T S(t) = e^^ + j

e^^^~^' ^<^(t' )dT' , (8.56)

and (8.18) gives

1(0, y) = . (8.57)

1-ky

For conservative scattering (A = 1) one has k = 0, and these expressions re- duce to (8.9) and (8.11), respectively.

After some reductions, which are omitted for brevity, the following ex- plicit expression for S(t) can be obtained from (8.56) and (7.10):

1
3.9 SEMI- INFINITE MEDIUM: DEEP LAYERS 145

The asymptotic behavior o£ S(t) for large t is described by the first term

on the right. If this asymptotic term is extrapolated to negative x, it

vanishes at x = — Tg, which explains the term "extrapolation length."

Considering (7.12), we can rewrite (8.38) in the form

S(x) = H(l/k) [e^^ - D(x) ] , (8.39)

where

1

D(,) = e-^f^^2xe) , _K f

-x/y RM

2H(l/k) 0 (l+ky)H(y)

Tables and graphs of D(x) are given by M. A. Heaslet and R. P. Warming (1968a)

who denote by f(x) our D(x).

3.9 SEMI-INFINITE MEDIUM: DEEP LAYERS

THE RELATION OF ASYMPTOTIC SOLUTIONS FOR INFINITE AND SEMI - INFINITE MEDIA.

In the conservative case there is a simple relationship between the asymptotic

solutions for infinite and s emi - infini te media. As will be shown in Sec.

6.1, a similar relationship exists for scattering in spectral lines.

Let us define for x > 0

l'^(x) = 1 + 4tt Jx'^Sp(x')dx' . (9.1)

This function increases m.onotonically from *i'^(0) = 1 to

>!/ («>) = J_ . (9.2)

°° 1-X

The physical significance of ^^(t) is as follows. Let us consider monochromatic scattering as s cattering°°of radiation by two-level atoms whose absorption coefficient has a rectangular profile (see the end of Sec. 2.2). Then

( (1-X) /X) (^^(x)-l) is the probability that the excitation, which initially

occurs at x°°= 0 and then migrates into the medium will be destroyed (through

collisions of the second kind, etc.) at a point that is located not further

than X from the point of initial excitation.

From (9.1) and (1.9) it follows that

'i'^(x) = 1 - 2x$^(x) + 2 f

<I>^(x')dx' . (9.3)

0

Substituting here the explicit expression (1.20) for *<„(t), integrating, setting X = «>, and using (9.2), we obtain
146 MONOCHROMATIC SCATTERING

f R(y)dy = ^ - i (9.4)

0

In particular, for conservative scattering

1 R(u)du = y 5 R(y)dy 1 , X = 1 . (9.5)

0

It is convenient to introduce a special notation for 1' (t) when X = 1

S (t) E ^ (t) , X = 1 . (9.6)

Although the resolvent ^^(t) diverges as A ^ 1, the difference

T

^^oo(^) - / f^(T')dT'

remains finite, so that

S (t) = 1 - 21im T<5

CO ^ c A ^ 1 L

0

(9.7)

From (9.7) and (1.20) we find, taking (9.5) into account, that S^(t) can be

represented as

where

,^(.) = |_l/e-/^(l.l)R(y)dy

(9.8)

(9.9)

The function q^(T) is similar to Hopf's function: QooCf) is a monotonically

increasing function that varies within rather narrow limits:

For T >> 1, (8.9) and (9.8) assume the forms

(9.10)
3.9 SEMI- INFINITE MEDIUM: DEEP LAYERS 147

SCt) ~ /3[T+q(<») ] , C9.ll)

S^(T) ~ I

[T^+q^Coo)] . (9.12)

Consequently a simple asymptotic relation exists between the solutions for an infinite and a semi - in f ini te medium:

S(t) ~ /2[S^(t) , T - - . (9.13)

In other words, the asymptotic behavior of the solution of Milne's problem

is simply related to the asymptotic form of the Green's function for a con- servative infinite homogeneous medium.^ It is worth noting that the functional

form of the^ relation between SC^) and S^(t) — the proportionality of SCt)

and (S^(t))^ — is preserved in 1 ine - frequency scattering, and the coefficient

of proportionality in all the cases of practical interest is close to unity

(see Sec. 6 . 1) .

THE SOURCE FUNCTION IN DEEP LAYERS. If the source strength S*(t) decreases

rapidly enough as t , we can readily show that the source function in deep

layers decreases as e—ki.

We have

CO S(t) = S*(t) + / S*(t' )r(T ,t' )dT' . (9.14)

i

The resolvent r(T,T') is expressed in terms of the resolvent function $(t) in

the following manner (see Sec. 3.7 and 5.1):

r(T,T') = $(|t-t'|) + / $(T-t) $(t '-t)dt , (9.15)

0 where = min (t,t'), and

$(t) = 2kB e~^^'^^'^e) + o(e~^) , t ^ «> . (9.16)

An immediate consequence of (9.15) and (9.16) is that the asymptotic (t ^ <») form ofr(T,T')is

r^^(T,T') = 2kB e-k^^"^e3 ^e^,^ ' . f

^"^

' "^^ $ (t) dt^ . (9.17)

or, if use is made of (8.36),
148 MONOCHROMATIC SCATTERING

r fx,!') = 2kB e"^^^^'^e)s(T'

)

as (t)S(t') (9.17')

where S is the solution of the Milne problem and $as '^^e asymptotic form

of the resolvent function i^Ct). We note that in Sec. 3.5 and 3.6 the notation

$as was used for another function, namely the asymptotic term of ^^(t) . We

hope this inconsistency will not cause confusion. Upon substituting (9.17')

into (9.14) we finally obtain the following asymptotic form of S(t):

= S^2kB e-k(T + Te) (9.18)

or

(9.18')

whe re

00

S (T')S(T')dT' (9.19)

The condition for this expression to be valid follows from (9.19): S (x)

must decrease fast enough as x -> to ensure the convergence of the integral

(9.19). Since s(t) increases as e^T for x °o , it is necessary that S*(x) = o j^e""''^''^) for X ^ ». With this in mind, in deriving (9.18) we have neglected

S (x) in comparison with the term containing the integral. The asymptotic

form (9.18) was found by T. A. Germogenova (1960) in another way. In the

specific case X = 1 (9.18) reduces to Davison's formula (B. Davison, 1951):

S^^(x) = /3 S^ , A = 1 (9. 20)

Substituting (9.18) into the transfer equation

dx

we find that in the deep layers the intensity of radiation is asymptotically

equal to

(9.21)
I

1 CHAPTER IV

INFINITE MEDIUM

I

- . .

I In Chapter I we discussed the description of the steady states o£ a

j system consisting of gas and a radiation field in circumstances when thermoI dynamic equilibrium is not established. The basic processes leading to the

establishment of such states were also discussed, and the statistical equi-

; librium equations for macroscopic masses of gas were derived. These equations were considered in greater detail in Chapter II. Since problems of

radiative transfer in spectral lines and problems of monochromatic scattering

, are very closely related, Chapter III was devoted entirely to a study of the

: solutions of radiative transfer equations for monochromatic scattering.

! Now we proceed to carry out our basic task of solving the equation of

radiative transfer for spectral lines. The principal assumptions are:

ij scattering occurs with complete frequency redistribution, and X = const.

" For the sake of simplicity, we will assume in Chapters IV - VI that the

absorption coefficient in the continuum is negligible compared to the line

absorption coefficient. The more complex problems arising when both line

and continuum absorption must be considered simultaneously are discussed in

j Chapters VII and VIII.

I In this chapter we will discuss the radiation field in an infinite

I medium. In reality, of course, a medium can never be infinite. However,

quite often the dimensions of the region occupied by a gas are much greater

i than the mean free path of a photon with the line-center frequency, i.e. the

gas has a very large optical thickness in the line. The theory presented

': in this chapter is relevant to the inner parts of such systems. It was

; developed mainly by D. I. Nagirner and the author (D. I. Nagirner, 1964a,

, 1964b; D. I. Nagirner and V. V. Ivanov, 1966; V. V. Ivanov and D. I. Nagirner,

Ij 1966) , and is presented here in a somewhat extended form.

I There are two reasons for undertaking a detailed study of scattering in

I an infinite medium. First, as has already been mentioned, when certain conI ditions are met, a medium of finite dimensions may be considered to be infin-

|; ite. The second reason, which we would like to stress, is that knowledge of

the radiation field in an infinite medium greatly facilitates the study of

the complex effects that arise in the boundary regions of a gas occupying

a finite volume. In the following chapters we will analyze the radiation
150 INFINITE MEDIUM

field in these boundary regions where the escape of radiation must be con- sidered.

4.1 THE green's FUNCTION FOR A HOMOGENEOUS MEDIUM

BASIC EQUATIONS. We wish to calculate, in the linear approximation, the

spatial distribution of excited atoms in an infinite homogeneous medium with

an isotropic point source. The medium is considered to be composed of twolevel atoms. Just as for monochromatic scattering, the corresponding source function is essentially the Green's function for an infinite homogeneous

medium. If the source is taken to define the origin, the transfer equation

assumes the form (see Sec. 2.3)

9l(-^,Vi,x) ^ iV 3I(T,y,x) _c.(x)I(T,y,xl .

2 AaCx)

y

a(x')dx'y

(t , y

' ,x ' ) dy ' + a(x)S*(T), (1.1)

where S*(t) is the primary source function which represents radiation arriving directly from the source, and I(T,y,x) is the intensity of diffuse

radiation

To determine S*(t), we let Q be the total source strength, integrated

over all line frequencies and over all directions. We shall consider a unit volume in the shape of a cylinder with a base 1 cm^ and an axis directed

toward the source, located a geometrical distance r from the source. When

viewed from the origin this volume element subtends a solid angle r"2. Within this solid angle the source radiates energy Q/4TTr2 per unit time; the

energy falling in the frequency interval (v, v + dv) is

Aa(x)

^ . 4^r2 Av

As the radiation is attenuated along the path from the source to the volume,

the energy incident on the volume is

Aa(x) e-^«f^^

4TTr2 Av

where t is the optical distance of the volume from the source:

T = k22(vo)nir. A fraction a(x) k]^2 (^o)"l °f this energy is absorbed in ex- citing atoms in the volume. The total energy acquired by the volume per

unit time from the absorption of direct radiation is therefore

oo Q A 2 , . - Ta(x) J 1 r ^ ^ Ay a (x) e ''dx-k^2 C^o^^l ' 4^r

or ^2(1)

47tt2

Q [k;^2^^0^^l]^ '
4,1 THE GREEN'S FUNCTION FOR A HOMOGENEOUS MEDIUM 151

where M2 ( x) is given by (2.6.1). The fraction of this energy re-emitted in

the frequency interval (v, v+dv) is AAa (x) (dv/Av) . Therefore the primary

emission coefficient of the volume under consideration is

A ^h^^^ na r "12 = — |kl2f^n^^i

4tt t2 4TrAv L » 1

The corresponding prim.ary source function is

S*(t) = a(x)k-^2

(^O-'^l or

S'==(t) = ± ^ Ll£_2—y_ Q . (1.2)

4Tr T- 4TrAv

If, for the sake of simplicity, we assume that

4ttAv

^[^12(^0)^1]

^

we obtain the following transfer equation:

2 9ICT,y,x) ^ 1-y 3I(T,y,x) .

. r ^ y

^—2— + — —2-^-2

- = — a(x)I(T,y,x) + 8t t 9y

(1.3)

+ - Aa(x) / a(x')dx' / I (x ,y ' ,x' ) dy ' + -^L a(x) . (1.4)

As shown in Sec. 2.4, the transfer equation (1.4) is equivalent to the

integral equation for the line source function Sp(x) having the form

A r^oClT-x'!) . M (x)

^f^^ ^ T- / ~

7, S (x')dx' + -i- -i— , (1.5)

P 4tt / T- T

'

I 2 P 4tt T-2

where x = |x_|. The source function Sp(x) is the sum of the last two terms

on the right side of (1.4), divided by a(x) , and is related to the level

populations n-, and n^ by the expression
152 INFINITE MEDIUM

S (t) P

2hv

12 1 ^^2

.2 g2

(1.6)

The integration in (1.5) extends over the whole space. The function Sp(T) is,

except for the 6-function, the Green's function for an infinite homogeneous

medium. The physical significance of this equation has been discussed in

Chapter II, where the notation is also defined. The solution is obtained in

generally the same way as for monochromatic scattering. Referring to Sec.

3.1 for all the details (see also Sec. 7.2), we shall briefly outline only

the general trend of the argument. We shall discuss in more detail those

aspects that differentiate the two problems.

Integration of both sides of (1.5) over the plane = const reduces its

solution to that of the following equation for the function ^'^^(t) :

oo

^oc^T) = J

y*K^(|T-T' |)$^(T')dT' -.AK^dTl), (1.7)

where K]^(t) is defined by (2.4.22)

lated by the equation

The functions S (t) and

P

(t) are reSp(T) 1 _d_

2ttt dT

,(T) (1.8)

EXPLICIT EXPRESSION FOR THE GREEN'S FUNCTION. In order to obtain $co(t) , we take the double-sided Laplace transform of (1.7) to obtain the expression

. K. (s)+K (-S)

*^(s) + $^(-s) = - i ^— , (1.9)

where f(s) is the one-sided Laplace transform of f(T). Using (2.6.23), we can rewrite (1.9) as

*„(s) * *„(-s) = hU (I-IO)

From (1.10) it follows that the solution of (1.7), which tends to zero as

|t| ^ °° , can be represented by an integral of the form

+ 1°° ll\

(T) = ^

f

-A^si^ e^^ds , (1.11)

2^i J l-^^fj) - 1°°
4.1 THE GREEN'S FUNCTION FOR A HOMOGENEOUS MEDIUM 153

or

CO

^ / i-xvru~

^co(^) = r /

'

' cos du , (1.12)

-XV(u)

where V(u) is given by (2.6.24). From (1.12) and (1.8) we finally get the

desired explicit expression for the Green's function:

oo

S (t) = —— I

^^^^^

u sin Tu du . (1.13)

We note that the expressions (1.12) and (1.13) are of the same form as for monochromatic scattering. In that case V(u) is given by (2.6.26), and

(1.12) and (1.13) reduce, respectively, to (3.1.16) and (3.1.17).

ALTERNATIVE REPRESENTATION OF THE GREEN'S FUNCTION. By deforming the path of

integration in (1.11) we can obtain a representation of the Green's function

Sp(T) which is more suitable for calculation. We seek the solution of (1.7)

which tends to zero as t °°. The Laplace transform $oo(s) of such a solution

is regular in the right half-plane. From (1.10) it follows that for Re s < 0 the singularities of $oo(s) should be the same as those of the expression on the right side of (1.9). Let us therefore consider the function

= 1 -^K^(s) K^(-s) . (1.14)

The function K-|^(s) is a Cauchy-type integral:

CO K (3) =

^ /f 1+sz'

dz '

I, We shall assume from now on that the profile of the absorption coefficient

[ a(x) is the continuous monotonic function of |x| (this condition is obviously

not satisfied for the rectangular profile) . It is readily seen that under

these assumptions (which can easily be weakened) the function G(z) is posijl tive for all finite z, 0 < z < °°. From the general properties of Cauchy-type

I integrals it follows (see, e.g., N. I. Muskhelishvili , 1962 , F. D. Gakhov

,

j 1963), that under rather mild assumptions concerning G(z), which are satisfied

i in all the cases of practical interest (the Holder condition) , the function

' K^Cs) is regular on the whole plane of the complex variable s, except for

points of the negative real semi-axis, where it is non- s ingle -valued . Moreover, it can be shown that the fun£tion (1.14) does not vanish (see Sec. 7.2).

.j Therefore the only singularity of $oo(s) in the left half-plane is the branch

line (-°°,0). Consequently integration in (1.11) along the imaginary axis

j can be replaced by integration along the contour i, shown in Fig. 13. Let us suppose that the absorption coefficient is such that the second moment of the

kernel
154 INFINITE MEDIUM

s)

Fig. 13. Path of integration for the evaluation of the Green's function

in line transfer problems.

oo /

dT (1.15)

diverges. If the line has infinitely extended wings, divergence is assured

(see Sec. 2.6). It can be shown that if = °° , the integral along the

small circle of radius r centered on s = 0 tends to zero as r ^ 0.

Therefore we find that for t > 0

$ (t)

CO ^2tti X/ I

K^(-x-iO) - ^

K^(x)

[l -

I

K^(-x + iO)

In

e dx (1.16)

According to the Sokhotski i - Plemel j formulae (see, e.g., N. I. Muskhelishvili

,

1962 , F. D. Gakhov, 1963) ,

where

K^(-x±iO) = K^(-x) + ^icf^) \ , (1.17)

OO

-L J 1 -XZ '

with the integral being understood as the principal value. Substituting

(1.17) into (1.16), introducing z = 1/x, and taking into account that <J><^(t) is an even funct.i.on, we find
4.1 THE GREEN'S FUNCTION FOR A HOMOGENEOUS MEDIUM

where

(1.18)

(1.19)

and ti(z) is given by (2.6.19) or (2.6.25). From this, by means of (1.8), we obtain finally the following expression for the Green's function Sp(T) with

which we are concerned:

S„(T) = ^

/ e"^/" R(z) G(z) ^ . (1.20)

P 4TrT / z2

The functions Oco(t) and S (x) were first obtained in this form by D. I. Nagirner (1964a, 1964b). P

7 Now let the second moment of the kernel be finite (a < °°) . If we assume, as before, that G(z) > 0 for all finite z ^ 0, then in the non-con- servative case (A < 1) , expressions (1.18) and (1.20) still hold. However,

because of the non-uniform convergence, one cannot go to the limit A ^ 1 in

(1,18) and (1.20) in order to obtain the conservative Green's functions.

In the conservative case the expression for the Green's function contains an extra term. This term accounts for the contribution coming from the integral

along a small circle centered on s = 0:

S (t) = -

P 4

OO -(- * f

e'^'^ R(z) G(z) ^)

(1.21)

From (1.21) and (1.8) it follows that the function $ (x) in this case diver00 ges .

The Green's function will be thoroughly analyzed and tabulated in subsequent sections, but first we should like to make two observations. The

first of these relates to the derivation of (1.20) and (1.21). In the case

of monochromatic scattering the function T(l/s) is so simple that the

Sokhotskii-Plemel j formulae are not needed to evaluate the integral along

the branch cut. Line transfer problems are another matter; here the use of

these formulae is essential.

The second observation relates to the structure of the solution (1.20).

In contrast to the case of monochromatic scattering, there is no exponentially decreasing pole term. This fact has far-reaching consequences. It implies that in radiative transfer problems for spectral lines with infinitely

extended wings the diffusion approximation, which essentially neglects the

integral term in comparison with the pole term, is in principle invalid.

From other considerations, this conclusion had already been reached in the

pioneering work of L. M. Biberman (1947) and T. Holstein (1947); see also

S. I. Braginskii and G. I. Budker (1958).
156 INFINITE MEDIUM

I£ a(x) is identically zero outside a finite range in x, it may happen

that < oo (see Sec. 2.6). According to (1.21), in this case the asymptotic form of the conservative Green's function for large x is

S (t)

p , as ^

^ X = 1 (1.22)

2 This function satisfies the diffusion equation, so that if a is finite and

scattering is conservative, the diffusion approximation may be used. Thus,

contrary to widespread opinion, there do exist problems of radiative transfer in spectral lines with complete frequency redistribution for which the

diffusion approximation is valid. However, these problems are rather artificial (lines with finite wings).

In concluding this section we give three integral relations satisfied

by R(z)

00 /

R(z)G(z)dz

1-X

(1.23)

OO /

(R(z)-l)G(z) dz 2 f XV^(u)

^ / l-XV(u)

du , (1.24)

OO /

(R(z) -l)G(z)-y = — a^ (1.25)

where a-^ is given by (2.6.28). The conditions for the validity of these

relations are the same as those under which the Green's function S (t) is

given by (1 . 20) . ^

9 The first of these relations can be obtained as follows. It is clear

that the mean number of scatterings of a photon in an infinite homogeneous

medium equals (l-X)-l. Consequently it can be asserted (see Sec. 3.3) that

oo

S (t) T dx

p 1-X (1.26)

Substituting (1.20) into (1.26), we arrive at (1.23). Expressions (1.24)

and (1.25) will be proven in Sec. 4.7 and 4.2, respectively. We emphasize

that relations ( 1 . 2 3) -( 1 . 2 5 ) cannot be applied to monochromatic scattering,

although certain similar relations are known. Thus the analog of (1.23)

is relation (3.9.4).
4.2 BEHAVIOR OF THE GREEN'S FUNCTION NEAR THE SOURCE 157

4.2 BEHAVIOR OF THE GREEN S FUNCTION NEAR THE SOURCE

Let us turn to a study of the Green's function Sp(T). In the first

place we shall consider its behavior for small t, i.e. in the vicinity of

the source. We shall consider only the case in which Sp(T) is given by

CI. 20), which is the usual situation in line transfer problems.

Let us rewrite (1.20) in the form

S (t) = —

P 4TTT

00 /^(R(z)-l)G(z)^ +

oo /^G(z)i| (2.1)

and examine, as x -> 0, the behavior of the integrals appearing here. Differentiating (2.6.16) and (2.6.31), we find that

oo /^G(z)^

z

^ — - + 0(T)

Further

,

oo '^/'(R(z)-l)G(z)^

oo /

(R(z)-l)G(z)^ + z^

(2.2)

/^^l)(R(z)-l)G(z)^

z

^ + (2.3) y

(e"''/"-l)(R(z)-l)G(z)^ + /

0

^

For T < 1 we have the following estimate for the third term on the right:

oo y* (e"^/^-l)(R(z)-l)G(z)^ = 0(T) . (2.4)

Turning to the second term, we note that for z < 1 , as follows from (1.19),

(2.6.12) , and (2.6.34) ,

R(z)

.1

= ^2Xa2 - ^^y-a^^ + O(z^) , (2.5)

so that

1 /

(e"^/"-l)(R(z)-l)G(z)^ a-^^2Xa2-A2ya^^ (E2(t) -1) + 0(T) , (2.6)
158 INFINITE MEDIUM

or J

(e""/^-!) (R(z) -l)G(z)^ = a-, (zAa^-A^llafjTJlnT + 0(t)

0 Therefore we can rewrite (2.3) as

7z

L, ( ZAa^ -A^—a^ I' 2 4 ly

0

(R(z)-l)G(z)

00 dz_ ^ f

,2 J

dz (RCz)-l)G(z)^ + z- Q- z^

(2.7)

. a^^2Xa2-A2^a2^^ T£nT + 0(t) . (2.8)

Coinbining this result with (2.2), we find for the Green's function S (t)

for small t: P

S (T)

4itt'

dz (R(z)-l)G(z)^ - SiAT +

,2tt^ 2\ 2„

+ a^l2Aa2-A —a^^ji^ ^nr 0(1) (2.9)

The integral appearing in this expansion can be expressed simply in terms of

the constant In studying the behavior of the Green's function for t << 1, we could have started from its representation in the form of (1.13). Substitution of y = Tu would then have given

oo f

-1 y s m y

T dy (2.10)

For u > 1, using (2.6.33), we get

1 l-XV(u)

1 n ^ 1 - Xa, -

2 u

•IT./2 ,u 2\1 —A( —a, -X—a, I—

2 \u 2 2 (2.11)

From (2.10) and (2.11) we find that

Sp(T) = (2.12)

Equating (2.12) and (2.9), we obtain (1.25). Taking this relation into

account, we have finally, instead of (2.9),
4.3 BEHAVIOR OF THE GREEN'S FUNCTION FAR FROM THE SOURCE 159

S (t)

P 4ttt2

Jinx + 0(1) (2.13)

We note that the first term depends linearly on X, the second term conta'ms

A2 J and the third contains This indicates that as the distance from the source to an element of volume increases, so does, on the average, the number

of scatterings experienced by those photons that excite atoms in this volume,

This result might have been anticipated from physical considerations.

4.3 BEHAVIOR OF THE GREEN S FUNCTION FAR FROM THE SOURCE

BASIC FORMULA. Simplifications in the Green's function in the other extreme

case — that of large distances from the sour ce — depend on the behavior of the

absorption coefficient in the wings of the line.

We shall again consider only the case of Sp(T) given by (1.20). Let

x(z) be a non-negative function defined by the relation

a(x(z)) (3.1)

We assume that the limit

f(y) = lim

exists and is equal to

x'(t)

(3.2)

f(y) = y2^, 0 < Y < 1 (3.3)

Specifically, as was shown in Sec. 2.6, these conditions are satisfied both

by the Doppler profile (y = 1/2) and by absorption coefficients that de- crease in the wings as |x|"'^, where 1 < k < °o

. in the latter case

K-1

Y = , 2k

so that 0 < Y < 1/2. As was mentioned in Sec. 2.6, if the line has infinite

wings, then the characteristic exponent does not exceed 1/2. Hence, we shall be primarily concerned with the interval 0 < y - 1/2. We shall con- sider the large x behavior of Sp(x) assuming that (3.2) -(3.3) hold.

For X >> 1 the main contribution to the integral (1.20) comes from the

values of the integrand for large z. Therefore, one can substitute for U(z)

and G(z) their asymptotic forms found in Sec. 2.6. Using (2.6.50) and

(2.6.42), we find that for large z the function R(z) , defined by (1.19), can be approximated by

2Y + 1 J

ttA 2Y + 1

-1 2 X' (z)

522-519 O - 74 - 12
160 INFINITE MEDIUM

Introducing this expression into (1.20), letting t/z = y, and substituting

y^''''x'(T) for x'(T/y), which is possible by

T >> 1, we obtain, after minor reductions.

virtue of (3.2)-(3,3) since

47TT' r (2y.l) lAx^Ill V!L

T (1-X)2

(3.4)

where

CO

rr2Y+2) / (t+

y 1+2y j

y dy

F (t) =

, P ^(2y+2) ^

(t+y2'^ctgTTY)2+y^Y

(3.5)

t = (1-X) 2y + 1 XttAx ' (t)

(3.6)

and r(x) is the gamma- function . This is the desired asymptotic form of

Sp(T) for large x. Strictly speaking, (3.4) is the asymptotic form of Sp(T)

as T -> « and at the same time A ^ 1 in such a way that (1-A)/x'(t) = const.

A considerable simplification arises because, in the asymptotic domain, the

quantity

4ttt^ (1-A)

^ x'(t)

S (t)

P

^

instead of depending on the two variables x and A, becomes a function of the

single variable t, which is simply expressed in terms of x and X. We have

already encountered a similar simplification in Sec. 3.4 in our study of

monochromatic scattering.

NEARLY CONSERVATIVE SCATTERING. When the role of absorption is small

(1-X << 1) , three zones can be distinguished in the asymptotic region

(t >> 1).

1. Zone of nearly conservative scattering . In this zone absorption

has little effect. This is the region in which x >> 1, but t << 1. The

closer X is to unity, the greater the extent of this region, and in it, as follows from (3.4)-(3.6),

S ^ _i L (1-4y )tg UY 1 P 4ttt2 ttA r(2Y) XX ' (x)(3.7)

In particular, for the Doppler absorption coefficient we have

Y

= 1/2 and

(3.8)
4.3 BEHAVIOR OF THE GREEN'S FUNCTION FAR FROM THE SOURCE 161

Going to the limit y -> 1/2 in (3.7) and using (3.8), we obtain

S ( t) (£nT)^ (3.9)

For X = 1 the condition t << 1 does not bound values of t from above. There

fore for conservative scattering (3.9) gives an asymptotic form of the

Green's function which is valid for all x >> 1. If the few first terms are taken in the expansions of Uj^^z) and Gj)(z) for z >> 1, (2.7.3) and (2,7.19),

it is possible to obtain from (1.20) not only the first, but also the subsequent terms of the large x asymptotic expansion of S^(x) for A = 1:

47Tx2 ^3/2

(£nx)^ ^^ l + 2y* _J 2tt^ + 3y*^ + 3y* + 3

2.nx 24

+ 0^(ilnx) '^^

X = 1 , X ^ 0°

,

(3.10)

or

S (x)

Attt^ ^3/2

(£nx)^ 1 . 0-5386 _ 1.0613 , o(ilnT)-A

£nx (2.nx)2 \ /

(3.11)

In (3.10), Y* = 0.577216 is Euler's constant.

For the Voigt profile, y - 1/4. Considering (2.7.26), we find from

(3.7) that in that part of the nearly conservative zone where asymptotic

forms appropriate for the Voigt profile can be used.

SV(,) 3 P 47Tx2 2^(aU(a,0))^.

X

2 (3.12)

If we allow a to go to infinity here, we get the asymptotic result for the

Lorentz profile. As follows from (2.7.22),

aU(a,0) (3.13)

Therefore

S^(x)

P^

^ 4ttx2 27t^ (3.14)

For conservative scattering, a more refined asymptotic form can be derived,

similar to the expansion (3.10) in the Doppler case:

5!:(t) =

^ -k

4'n-x^ 2tt'2

1 +

9 1 20 X 1120 t2

1053 1 , 0^^-33 , X = 1, X ^ °°. (3.15)

II
162 INFINITE MEDIUM

This expansion is valid for all t >> 1.

2. Transition zone. In this zone absorption is more important, although it still does not play the dominant role. This region corresponds to

distances from the source for which the parameter t is on the order of unity.

The Green's function can be found from (3.4).

3. Strong absorption zone. This is the region in which t >> 1. The

Green's function is given by the following expression, resulting from (3.4)

and (3.5):

S^(T) r(2Y + l)

,

• (3.16)

^ 4^t2 (1-A)^ t

Since this zone always exists, even when A is not close to unity, we have

retained the factor X in this expression. Obviously, when 1-A << 1, A may

be replaced by unity.

AUXILIARY FUNCTIONS. The function Fp(t) must still be evaluated in order to

comnletely determine the Green's function for t >> 1. l^'e shall consider the cases of Doppler and Voigt profiles. For the Doppler profile we have

Y

= 1/2, and (3.5) assumes the form

y,2

FD(t) = ^ f ^:idy . (3.17)

^ 2 / t2.y2

This integral may be expressed in terms of known functions, namely:

2 3 F^(t)=-^-^(cit-sint-sit-cost) , (3.18)

where si t and ci t are the integral sine and cosine:

OO 00 si t = - fiilLil- dx, ci t = - f dx . (3.19)

t t

From (3.18) an expansion of Fp(t) can be obtained which converges rapidly

for small t. The first few terms of this expansion are:

FP(t) = li Tl - Zt - t^^nt + (1 - Y*)t2 + It^ + U\nt + 0(t4)l ,(3.20)

V 2 I 2 46 J

where y* = 0.577216 is Euler's constant. For large t the function Fp(t) has

the asymptotic expansion

F^t)

P a--

11

.2

6:

t6

(-1)^ (2n+2)

4. 2n

)

(3.21)
4.3 BEHAVIOR OF THE GREEN'S FUNCTION FAR FROM THE SOURCE 16 3

which is easily obtained from its integral representation (3.17). Values o£

F^(t) are given in Table 13.

For the Voigt profile, y = 1/4, and (3.5) gives

3/2,

,v

0 (t + /y)

(3.22)

For t << 1, Fp(t) can be evaluated from the expansion

F^t) = it^ - 2 t3 . . P 3 3 (3.23)

and for t >> 1, from the asymptotic series

TABLE 13

THE FUNCTION F (t)

P

t F^t)

P

t t F^t)

P

t P°Ct,

0 . 0 1 .00000 2 . 6 0 . 5034 6 . 4 0 .8122 13 . 6 0 .9434

0 . 1 0 .00435 2 . 8 0 .5 320 6 . 8 0 . 8275 14 . 0 0 .9462

0 . 2 0 .01545 3 . 0 0 .55 86 7 . 2 0 .8411 14 4 0 .9489

0 . 3 0 .03118 3 . 2 0 .5833 7 . 6 0 . 8533 14 8 0 .95 13

0 . 4 0 .05011 3 . 4 0 . 6062 8 . 0 0 .8642 15 2 0 .9535

0 . 5 0 .07122 3 . 6 0 .6275 8 . 4 0 .8740 15 6 0 .9 55 7

0 . 6 0 .09377 3 . 8 0 .6474 8 . 8 0 .8828 16 0 0 .9 577

0 . 7 0 .1172 4 . 0 0 . 6658 9 . 2 0 . 8907 16 4 0 .9596

0 . 8 0 .1411 4 . 2 0 .6830 9 . 6 0 . 8979 16 8 0 .9613

0 . 9 0 . 1652 4 . 4 0 .6991 10 . 0 0 .9045 17 2 0 .9629

1 . 0 0 .1893 4 .6 0 . 7140 10 . 4 0 .9 104 17 6 0 .9645

1 . 2 0 .2365 4 . 8 0 .7280 10 . 8 0 .9159 18 0 0 . 9659

1 . 4 0 .2820 5 .0 0 .7411 1

. 2 0 .9209 18 4 0 .9673

1.6 0 .3252 5 . 2 0 . 7533 11 . 6 0 .9254 18 8 0 .9686

1 . 8 0 . 3658 5 .4 0 . 7647 12 . 0 0 .9296 19 2 0 .9698

2 . 0 0 .4039 5 .6 0 . 7755 12 . 4 0 .9335 19 . 6 0 .9709

2 . 2 0 . 4395 5 . 8 0 . 7855 12 . 8 0 .9371 20 . 0 0 .9720

2 . 4 0 .4726 6 . 0 0 . 7950 13 . 2 0 .9404
164 INFINITE MEDIUM

3tt2 1^ = 0 \ 2 / tk

= 1 - 16 1 h t

+ 5 (3.24)

For intermediate values of t the function F^ft] must be found from (3.22) by

numerical integration (Table 14) . P

We emphasize that the form of the function Fp(t) does not depend on the

value of the Voigt parameter a. In particular, for a = » the Voigt profile

becomes the Lorentz profile (see Sec. 1.5). Thus FV(t) = FL(t). However,

TABLE 14

THE FUNCTION F (t)

P

t t F^t)

P

t V F;(t) t VF^(t)

0 . 0 0 .00000 2 . 8 0 . 3963 7 . 2 0 .6710 14 . 8 0 .8190

0 . 1 0 .00299 3 .0 0 .4173 7 . 6 0 . 6844 15 . 2 0 . 8232

0 . 2 0 .01078 3 . 2 0 .4372 8 . 0 0 . 6968 15 . 6 0 .8272

0 . 3 0 .02199 3 . 4 0 . 4558 8 . 4 0 . 7083 16 . 0 0 . 8311

0 . 4 0 .03562 3 . 6 0 .4734 8 . 8 0 .7189 16 . 4 0 .8347

0 . 5 0 .05091 3 . 8 0 .4899 9 . 2 0 . 7288 16 . 8 0 . 8383

0 . 6 0 . 06733 4 . 0 0 . 5055 9 . 6 0 .7381 17 . 2 0 .8416

0 . 7 0 .08445 4 . 2 0 .5203 10 . 0 0 . 7467 17 . 6 0 . 8448

0 . 8 0 .1020 4 . 4 0 .5342 10 . 4 0 . 7549 18 . 0 0 .8480

0 . 9 0 .1197 4 . 6 0 .5474 10 . 8 0 . 7625 18 . 4 0 . 8510

1 . 0 0 .1374 4 . 8 0 . 5600 11 . 2 0 . 7696 18 . 8 0 . 8539

1 . 2 0 .1722 5 . 0 0 .5719 1

. 6 0 .7763 19 . 2 0 . 8566

1 . 4 0 . 2060 5 . 2 0 .5832 12 . 0 0 . 7828 19 . 6 0 . 8593

1 . 6 0 .2382 5 . 4 0 . 5939 12 . 4 0 .7888 20 . 0 0 . 8619

1 . 8 0 . 2687 5 .6 0 . 6041 12 . 8 0 . 7945 30 . 0 0 .9052

2 . 0 0 . 2974 5 . 8 0 .6138 13 . 2 0 . 8000 50 . 0 0 .9418

2 . 2 0 . 3245 6 . 0 0 .6232 13 . 6 0 .8050 100 . 0 0 .9704

2 . 4 0 . 3499 6 . 4 0 .6405 14 . 0 0 . 8099

2 . 6 0 . 3738 6 . 8 0 .6564 14 . 4 0 .8145
4.3 BEHAVIOR OF THE GREEN'S FUNCTION FAR FROM THE SOURCE 165

the region for which the expression (3.4) with y ~ 1/4 is valid depends on a. The minimum value o£ t for which this expression provides a reasonable

approximation increases as a becomes smaller.

ALTERNATIVE DERIVATION OF THE ASYMPTOTICS. The asymptotic behavior of the

Green's function for large t and 1-X << 1 can also be obtained in another

way, which is of interest because it offers the possibility of considering

line scattering and monochromatic scattering in a unified form. In other

words, this method works for all values of the characteristic exponent Y,

0 < Y - 1- Since our purpose now is to illustrate the alternative method we shall study only the simplest case, A = 1. Our point of departure is the

representation of the Green's function in the form (1.13). For large x and

1-X << 1, the main contribution to the integral (1.13) comes from the

region of small u, in which V(u) in the numerator of the integrand of (1.13) can be set equal to unity, and l-V(u) in the denominator can be replaced by

(see Sec . 2.6)

where (|5(u) is a slowly varying function of u as u -> 0 (in particular, it can

be a constant). In Sec. 2.6 it was shown that for Y < 1, i.e. in the cases

so far considered in this section,

For monochromatic scattering Y

= 1 and <p = 1/3.

Substituting (3.26) into (1.13), we find that for A = 1 and large

enough x

,

l-V(u) 4'(u) u (3.25)

(3.26)

S (x)

P

du (3.27)

Introducing xu we have

y and recalling that (l)(u) varies slowly for small u,

oo

S (x) P siny dy (3.28)

The integral on the right is to be understood in a certain generalized

sense, for example, as the limit of the integral

oo

0

as c ^- 0 . Evaluating this integral, we finally find
166 INFINITE MEDIUM

1 1-2Y T^^"^

S (T) - ^-I — , T - CO, X = 1, 0 < Y ^ 1 . (3.29)

^ 4ttt^ r(2Y)cos Try ^(1

The expression (3.7), found earlier, is a special case o£ (3.29). It

is obtained from (3.29) by the use of (3.26). In the case of monochromatic

scattering V(1/t) ~ 1-1/3t^ for t ^ ~, and (3.29) gives

sJJ(T) ~ , (3.30)

which is the result already obtained in Chapter III by another route.

The case of nonzero 1-A can be treated similarly..

4.4 MODIFIED FORM OF THE GREEN's FUNCTION. NUMERICAL DATA

The above study of the asymptotic behavior of the Green's function

suggests a modified form of its representation for y < 1 . This modified

representation has important practical advantages over the representations

found in Sec. 4.1.

CONSERVATIVE SCATTERING. We begin with the simplest case of conservative

scattering (A = 1) and consider the asymptotic form (3.7) of the Green's

function. Using the asymptotics of the functions Mj^(t) (see Sec. 2.6; we recall that for y 1 one has 6 = y) ,

M, (t) ~ 2Ar(k + 2y-l) ^lill , r -> oo

, (4,1)

^ ^k

T

we can rewrite (3.7) in the form

P 4ttt2 [Mi(t)]2

where

C M^(t)

S^(t) ~ — , T ^ - , (4.2)

S

= ^^^^^ • (4.3)

The coefficient C„ decreases monotonically with y from Cp = 1 for y = 0 to

Cp = 0 for y = 1. It is essential that Cp is close to unity for those

values of y that are of primary importance for us, namely 0 < y ^ 1/2.

In particular, Cp = 3/Tr = 0.955... for y = 1/4 (Lorentz and Voigt profiles),

Cp = 8/tt2 = 0.811... for y = 1/2 (Doppler profile).

At first glance it may seem that the representation (4.2) is artificial

and has no advantages over the original form (3.7). However, this is not so.
4.4 MODIFIED FORM OF THE GREEN'S FUNCTION- NUMERICAL DATA 167

First, since M-^Ct) and M2(t) can be computed for all t > 0, one might try to use the asymptotic result (4.2) as an approximation for Sp(T) in the non- asymptotic domain as well, i.e. for small t. Second, the fact that Cp is

close to unity, combined with the simple physical interpretation of M]^(t) and

M2(t), suggests that the asymptotic results (apart from factors of the order

of unity) are amenable to direct physical interpretation. As we shall see shortly, both ideas are highly fruitful.

1 Accordingly, we represent Sp(T) for all t in the form

C M (T)

S(t) = —^ 5 (t) , (4.4)

[M-|^(t)]

where 5p(T) is the correction factor that transforms the asymptotic equality

of (4.2j into exact equality. The last equation is the definition of ?p(T).

The function Cp(T) may differ substantially from unity only for relatively

small T. Substituting into (4.4) the expansions of Sp(T) and M]^(t) for small t found in Sec. 4.2, (2.13), and Sec. 2.6, (2.6.30), respectively, we get

5p(T) = _1_ ca-^T +

f 2a, - ILl a^M Inx + Od^) (4.5)

Thus, unless y is close to unity (in particular, for y < 1/2), the function

5p(t) for small t is not very different from unity. For the Doppler and

Lorentz profiles it is also easy to estimate the rate at which 5p(T) tends

to unity as x -> °°. Equating the right sides of (4.4) and (3.10) and using

(2.7.6), we get

5^(T) = 1 - 1 -1- - ihlllll . 0((£nx)-^), X - - . (4.6)

P 4 5,nT 48 (£nx)2

In a similar way it can be shown that

P 5 X 140 j2

It is a striking characteristic of the representation (4.4) that for the

profiles of practical interest the function ?p(x) is close to unity for all x (see the numerical data in the next subsection). Therefore 5p(x) can be

regarded as a rather unimportant correction factor. If only an estimate of

the Green's function is needed, this factor can be replaced by unity. If we also disregard the difference of C from unity, we get the simple approximation P

S^ (T) = -J- _i _ , (4.8)

p,a

4ttx^ [M-^(x)]
168 INFINITE MEDIUM

where the subscript a emphasizes that it is an approximate expression.

Usually, the smaller the characteristic exponent y, the more accurate this

approximation is.

The representation o£ the solution o£ the integral equation for the line source function in a form similar to (4.4) is possible not only for an infinite medium, but also for other cases of conservative and non-conservative

scattering. In these representations the "main" part of the x -dependence

,

which is expressed in terms of kernel functions, is multiplied by a correc- tion factor, which is close to unity. These representations give rise to

approximate expressions of the type of (4.8) which will be shown in the next

section to have an immediate physical interpretation.

Expressions of the type of (4.4) play an important role in the theory

of transfer of line radiation. The source function was represented in this

form first for a semi - infinite medium (V. V. Ivanov, 1965), and later for an infinite medium (V. V. Ivanov and D. I. Nagirner, 1966) and a layer of

finite optical thickness (V. V. Ivanov, 1972).

NUMERICAL DATA FOR THE CONSERVATIVE CASE. The values of the Green's

functions for the conservative infinite medium are given in Table 15 for

Doppler and Lorentz profiles. They were obtained from (1.20) by numerical

integration. This table is extracted from a more detailed table given by

D. I. Nagirner and A. B. Schneeweis (1973).

Having obtained the Green's function Sp(T), one can easily find Cp(T)

from (4.4). In this calculation the values of M]^(t) and M2(t) are needed.

For the Doppler profile they are given by V. V. Ivanov and V. T. Shcherbakov

(1965a). For the Lorentz profile, Mi(t) and M2(t) can be expressed in terms

of the modified Bessel functions (see Sec. 2.7), and thus also may be con- sidered known.

Graphs of Cp(T) and 5p(T) are shown in Figs. 14 and 15. In both cases

5p(T) is rather close to unity for all t, so that Cp(T), indeed, may be con- sidered as a correction factor. Although in the Doppler case (y = 1/2) the

maximum deviation of ?p(T) from unity is larger than in the Lorentz case

(y = 1/4), it is still not too large.
4.4 MODIFIED FORM OF THE GREEN'S FUNCTION. NUMERICAL DATA 169

TABLE 15

THE GREEN'S FUNCTION S (t) FOR THE DOPPLER AND LORENTZ PROFILES WITH X = 1

T 5^(1}

P

T p Sp^(x)

0, 01 5 .679 + 2 3 .998 + 2 7 3 .627 -3 5 .410 -4

0 . 05 2 . 355 + 1 1 .627 + 1 8 2 . 856 -3 3 . 867 -4

0 . 1 6 . 146 0 4 . 144 0 9 2 , 310 -3 2 . 874 -4

0 . 2 1 . 662 0 1 . 068 0 10 1 .909 -3 2 . 204 -4

0 . 3 7 .923 -1 4 . 855 - 1 12 . 1 .371 -3 1 . 391 -4

0 , 4 4 . 744 -1 2 .778 - 1 14 1 .035 -3 9 .432 -5

0. 5 3 .212 -1 1 . 800 - 1 16 8 . 101 -4 6 . 737 -5

0 . 6 2 . 347 -1 1 . 261 -1 18 6 .523 -4 5 . 007 -5

0 . 7 1 . 805 -1 9 .316 -2 20 5 .372 -4 3 . 840 -5

0 . 8 1 .441 -1 7 . 155 -2 25 3 . 554 -4 2 . 190 -5

0 , 9 1 . 183 -1 5 . 659 -2 30 2 .532 -4 1 .385 -5

10 9 .929 -2 4 .579 -2 35 1 . 900 -4 9 . 405 -6

125 6 . 859 -2 2 .906 -2 40 1 .480 -4 6 .726 -6

1

50 5 .074 -2 1 .987 -2 45 1 . 187 -4 5 . 005 -6

1

75 3 .932 -2 1 . 431 -2 50 9 . 738 -5 3 . 842 -6

2 . 0 3 . 150 -2 1 .070 -2 60 6 .909 -5 2 .432 -6

2 . 5 2 . 168 -2 6 . 506 - 3 70 5 . 166 -5 1 .653 -6

3. 0 1 . 593 -2 4 .277 -3 80 4 .013 -5 1 . 183 -6

3. 5 1 .224 -2 2 .975 -3 90 3 .211 -5 8 . 807 -7

4 , 0 9 .720 -3 2 . 159 -3 100 2 .629 -5 6 . 764 -7

4. 5 7 .920 -3 1 .621 -3 200 7 .021 -6 1 . 193 -7

5 . 0 6 .585 -3 1 . 251 -3 500 1 .210 -6 1 . 206 -8

6. 0 4 .773 -3 7 .959 -4 1000 3 . 179 - 7 2 .131 -9
170 INFINITE MEDIUM

GENERAL CASE, 0 < A < 1. Now let us abandon the assumption of conservative

scattering. We now modify the definition of the quantity t from its original

form (3.6) to

t = nuiii^j^ . (4.9)

For large t the values of t given by the two expressions are asymptotically

equal. However, (4.9) is more convenient in that it also enables one to

calculate t for values of t that are not particularly large. Using (4.1), we can rewrite (3.4) as

S^(t) ~ F (t) , T -> X -> 1, t = const, (4.10)

P 4 (1-A) 2 P

where Fp('t) is, as before, given by (3.5).

Let us now represent the Green's function for all t and X in the form

S^(t) = ^

-i r- F (t) C^(T,A) , (4.11)

P 4ut2 (1-A)2 P P

where t is given by (4.9), and ?p(T.,A) is the correction factor to the asymptotic form (4.10). We emphasize that the equality in (4.11) is exact, and

not asymptotic.

For nearly conservative scattering (small 1-A), absorption becomes significant only at considerable distances from the source, while ^ (t,A) deviates significantly from unity only for relatively small x. Therefore the error caused by substituting Cp(T) = Cp(T,l) for Cp(T,A) decreases as A in- creases and hence for all t and small 1-A

1 M (t)

S^(t) ~ -±-

^

7

F ^n^^^^'

^< 1- ^4.12)

P 4^x2 (1-A)2 P P

This equation shows that in the limiting case of nearly conservative scattering, the Green's function, which depends on two variables, t and A, is

asymptotically expressed in terms of functions of a single variable.

Obviously, this is a substantial simplification.

In the zone of conservative scattering, i.e. for such x that t << 1,

from (4.12) we have

M (x)

S (X) - 2 (4_13)

P 4^x2 [M-^(x)]2 P

Here §p(x) differs substantially from unity only in that part of the zone relatively close to the source. For the transition zone and the strong
4.5 LONGEST FLIGHT APPROXIMATION 171

absorption zone, in which values of t are not much less than unity, (4.12)

may be replaced by

P 4^T^ (1-A)2 P

Specifically, for t >> 1 (strong absorption zone), we find from (3.5) F (t)~l,

so that (4.14) assumes the form ^

Sp(T) ~ ^ . (4.15)

P 4-ITT'^ (1-^)^

No matter how little X differs from unity, for sufficiently great distances

from the source the asymptotic behavior described by (4.15) will eventually

be reached.

NUMERICAL DATA FOR THE NON - CONSERVATIVE CASE. The values of the Green's

function for the Doppler and Lorentz profiles and A = 0.9, 0.99, and 0.999,

calculated from (1.20) by numerical integration are given in Table 16 (after

D. I. Nagirner and A. B. Schneeweis, 1973).

We note that for the Doppler and Lorentz profiles the expansion (2.13)

gives Sp(T) for t < 0.02 to four significant figures. The expansion (3.15),

with all the terms retained, also gives four significant figure accuracy for

T > 20. For the Lorentz profile the relative error of the representation

(3.4) does not exceed 1.5 percent for x s 30 and all A, 0 < A < 1; for

T ^ 100 the error is less than 0.5 percent. For the Doppler profile the

asymptotic form (3.4) gives the Green's function for x > 100 and arbitrary A with an error of not more than 2.5 percent.

We now conclude our formal study of the Green's function for an infinite

homogeneous medium. A physical interpretation of the results will be given

in the next section. However, before proceeding with this, we note that

B. A. Veklenko (1957, 1959) and Yu . Yu . Abramov and A. P. Napartovich (1968)

have studied the non-stationary radiation field arising from an "instantaneous" source located in an infinite homogeneous medium that scatters radiation with complete frequency redistribution. More precisely, the non- stationary Green's function for an infinite homogeneous medium has been

found, and its behavior has been studied in various extreme cases (large distances and considerable time lapses after the initial "flash")

4.5 LONGEST FLIGHT APPROXIMATION

In this section an approximate form is proposed for the infinite medium

Green's function. The approximation is suggested by the asymptotic results

found in the preceding section, and is valid for y < 1.

THE BASIC EXPRESSION. Comparison of (4.2) and (4.15) suggests an approximation to the Green's function Sp(x) of the form — T AM (t)

S (x) = ^ 2 (5.1)

P'^ 4ux2 [l-A + AM-^(x) ]2
172 INFINITE MEDIUM

TABLE 16

THE GREEN'S FUNCTION FOR THE DOPPLER AND LORENTZ PROFILES WITH X < 1

: 0 . 9 X = 0 . 99 X = 0 999

T S^

p (T) S

P

S

P S p

S P S^

P (T)

0 1 5 . 418 0 3 .683 0 6 .071 0 4 . 097 0 6 . 138 0 4. 140 0

0 2 1 . 431 0 9 . 369 - 1 1 .637 0 1 . 054 0 1 . 660 0 1 . 066 0

0 3 6 . 654 -

4 . 208 - 1 7 .779 -4 . 788 - 1 7 . 908 - 1 4 . 846 - 1

0 5 2 . 564 -

1 .524 - 1 3 . 132 -1 .771 - 1 3 . 203 -1 1 . 797 - 1

0 7 1 . 370 -

7 .713 -2 1 . 748 -

9 . 143 -2 1 . 799 - 1 9 . 299 -2

1 0 6 . 992 -2 3 .674 -2 9 . 508 -2 4 . 479 -2 9 .882 -2 4. 569 -2

1 5 3 . 166 -2 1 .518 -2 4 . 768 -2 1 .933 -2 5 .039 -2 1 . 9 81 -2

2 0 1 . 747 -2 7 .818 -3 2 .903 -2 1 .036 -2 3 . 120 -2 1 . 067 -2

3 0 7 . 039 -3 2 . 876 -3 1 .412 -2 4 . 101 -3 1 .570 -2 4 . 259 -3

5 0 1 . 899 - 3 7 .313 -4 5 . 391 -3 1 . 180 -3 6 . 424 -3 1 . 244 -3

7 0 7 . 072 -4 2 . 823 -4 2 . 743 -3 5 .031 -4 3 .500 -3 5 . 371 -4

10 2 . 205 -4 1 . 005 -4 1 .284 -3 2 .014 -4 1 .811 -3 2. 184 -4

15 5 . 073 -5 3 .051 -5 5 . 096 -4 7 .073 -5 8 . 404 -4 7. 834 -5

20 1 . 652 -5 1 . 297 -5 2 .520 -4 3 . 360 -5 4 . 810 -4 3 . 788 -5

30 3 . 156 -6 3 . 826 -6 8 .615 -5 1 .174 -5 2 . 142 -4 1 . 362 -5

50 3. 722 -7 8 .0 15 -7 1 . 903 -5 3 . 099 -6 7 .375 -5 3 . 757 -6

70 9 . 091 -8 2 .817 -7 6 . 330 -6 1 .282 -6 3 .522 -5 1 . 609 -6

100 2 . 055 - 8 9 . 168 -8 1 . 799 -6 5 . 000 - 7 1 .544 -5 6 . 548 -7

where the subscript a stands for "approximate." For a reason that will be- come clear in the next subsection, this approximation is referred to as the longest flight approximation to the Green's function.

Approximation (5.1) has the following remarkable properties: (1) It

gives the leading term of Sp(T) for x ^ 0. (2) In the non-conservative case

(X ^ 1) it correctly describes the asymptotic behavior of Sp(T) as t -> °°. (3) It gives the exact value of the mean number of scatterings of a photon,

i.e. it satisfies the equation

OO . -

'

-

At\ / S CT)T^dT = — •

0

^ ^
4.5 LONGEST FLIGHT APPROXIMATION 173

(4) It describes correctly the functional form of the leading term of the

asymptotic expansion of Sp(T) for those t >> 1 that satisfy the inequality

Mi(t) >> 1 — A, i.e. in the asymptotic part of the conservative zone (provided

it exists)

With all this in mind, one might expect (5.1) to be an excellent approximation for all T and X, with arbitrary profiles of practical interest. As a rule, the accuracy of (5.1) increases as the characteristic exponent y de- creases . PHYSICAL INTERPRETATION. G. B. Rybicki and D. G. Hummer (1969) have shown

that (5.1) expresses the fact that the net effect of all scattering processes

can be approximated by the single longest flight in the sequence of scatterings undergone by a photon. We reproduce their reasoning.

Let us derive the probability density of longest flights. If a photon

undergoes precisely n flights bef.ore destruction the probability that the

longest flight lies between t and t + dx is given by

n-1

nM2 (t) dT M2(T')dT' (5.2)

Indeed, the probability that the first flight ends between x and x + dx

is M2(x)dx and the probability that the remaining n-1 flights are shorter

than X is the bracketed expression raised to the power n-1. Since the

longest flight might also be the second, third, etc., flight we multiply by n, Since the probability that the photon experiences precisely n flights is

X^'l(l-X), the net probability density of longest flights is

oo (l-A)M2(x) n X

j n=0 1)

M2(T )dx

n-1

(l-X)M2(x)

.1-X J"^M2(x')dx'

(5.3)

or

(l-X)M^(x)

[l-X + XM-^(x) ]2

The series has been evaluated by recognizing it as the derivative of a geometrical series. We also used the fact that (see Sec. 2.6)

I

X M2 (x')dx' = 1 - M-^(x)
174 INFINITE MEDIUM

Let us now assume that an excited atom is born at t = 0 . The excitation

is then transferred by radiation and is eventually destroyed at some point.

The probability that the excitation energy will be converted into heat in a spherical shell of radius x and thickness dx is evidently (cf. Sec. 3.9)

According to (5.1), this probability is approximated by the distribution of

longest flights (5.4), i.e. the net result of multiple flights is approximated by the longest single flight.

We emphasize that the approximation (5.1) is valid only for y < 1, with

the conservative limit given by (4.8). This approximation was obtained from

(4.4) by setting Cp(T) and Cp equal to unity. Since Cp tends to zero as

Y

->- 1 , the approximation (5.1) breaks down for y = 1. In particular, it is

inapplicable if the second moment of M]^(x) exists, since then we always

have Y

= 1

There is a sharp physical distinction between cases in which the second

moment of M2(x) does, or does not, exist. Since the existence of this moment

implies that the distribution of free flights is not too heavily dominated by

long flights, some average flight can be chosen as representative. Since

the mean number of scatterings in an infinite medium is (1-A)~1, we may imagine that the photon undergoes a random walk of (1-A)"1 steps, each, when the

second moment exists, of roughly constant length. Because all directions

are equally likely, one expects the average distance travelled tOj^be proportional to the square root of the number of scatterings, or (l-A)-2^ in

agreement with the result (6.14) below. In this case (5.4) would be a poor

approximation to (5.5).

If the second moment of M2(x) does not exist, the excitation at large, x is dominated by the relatively infrequent long flights. For example,

the most likely situation in line problems is for a photon created at a point

in the line core to be imprisoned there until a relatively improbable scattering event deposits it in the wing. The photon then moves a very large

distance in comparison with the net effects of all previous scattering in the core. Finally, it remains imprisoned near this new point until its ultimate

destruction. The validity of this physical picture is verified by the

approximate agreement between Sp(x) and Sp a^^"^)- this situation the distance travelled by the photon increases wi^h the number of scatterings, not

because the net distance is increased by the addition of more individual

scattering steps, but because the probability of a scattering into the wing

region has increased, thereby increasing the longest scattering length.

ACCUMULATION EFFECT. In the preceding chapter (Sec. 3.4) it was shown that

in a weakly absorbing medium an accumulation region exists around the source

in the case of monochromatic scattering. In this region the source function

is greater than it would be in a test volume located in a vacuum at the same distance from the source. We shall now examine how frequency redistribution

affects the accumulation effect.

1-X 2 4iTx S (x)dx . (5.5)

4.6 ACCUMULATION EFFECT, THERMALIZATION LENGTH

For scattering of line radiation with complete frequency redistribution

the source function in a test volume located in a vacuum at a distance r from the source of strength
4.6 ACCUMULATION EFFECT. THERfiALIZATION LENGTH 175

Q = 4fTAv

equals

4u ^2

SqCt) = — , (6.13

where t = ki2(^o3^1^* '''h® factor a-]^ is the value of M2(t) at t = 0 . It appears because the number of photons with frequencies from x to x + dx emitted

by the source which afterward cause photo -excitation in the test volume is

proportional to a^Cx)dx. When this quantity is integrated over all frequencies, the factor ai appears. By definition, the medium-effect coefficient

is (cf. Sec. 3.4)

i(T) = S (t). (6.2)

Aai P

Let us first consider the case of the Doppler profile. For 1 — X << 1 and T >> 1 we find from (6.2), using (4.14),

oo

.3)

I og r

Fig. 16. The medium-effect coefficient for scattering with the Doppler profile.

Qualitatively, everything remains the same as for monochromatic scattering —

around the source an accumulation zone exists in which i^(T) > 1, and as

1 - A decreases, the size of this zone increases. However, from the quantitative viewpoint there is a tremendous difference. For the Doppler profile

the medium-effect coefficient in the accumulation zone is of the order of

unity, while with monochromatic scattering it can reach very large values.

For example, for 1 - A = 10'^ we have iD(Tniax) ~ 5-6. whereas i^(Tinax) ~ 0.6 • 103 (see (3.4.24)). This difference is caused by the frequency redistribution during scattering, which leads to progressive "pumping" ofphotons

from the core of the line into the wings. As the absorption coefficient in

the line wings is small, a photon emitted there traverses a relatively great

522-519 0 - 74 - 13
176 INFINITE MEDIUM

distance without scattering. Such a photon is much less likely to excite

atoms than a photon whose frequency is close to the line center.

Thus two phenomena with opposite effects occur during scattering. On one hand, the direction of the photon changes, leading to the "trapping" of

its trajectory and giving rise to the accumulation effect. On the other

hand, the frequency of the photon changes. At each scattering, a fraction of

radiation enters the line wings. As the length of the path traversed by

radiation in the medium increases, the number of scatterings experienced by

the photons also increases, and the radiation gradually loses the ability to

excite atoms. In the case of the Doppler profile, the first effect dominates

the second, although to a very small degree. This is why i^Ci) is of the

order of unity in the accumulation zone. For scattering with the Lorentz profile the redistribution effect dominates. Even in the conservative case, when the accumulation effect should

be greatest, it does not exist, since, as is seen from (6.2) and (3.15),

iL(^) = A 1 [i + A 1. _ x » 1 . (6.4)

In the immediate vicinity of the source, it is true, a small region exists

in which i^(T) > 1. This is seen, for example, from (6.2) and (2.13),

which lead to the result that, for small t

iL(T) = 1 +!V - + (6.5) (4 -

1)

However, the behavior of i (t) quickly changes from increasing to decreasing

(Fig. 17), and throughout the asymptotic region (t >> 1) the medium - ef feet

coefficient is much smaller than unity. In contrast, for monochromatic

scattering and in the case of the Doppler profile, i(T) > 1 at all t > 0.

Thus for the Lorentz profile the presence of even a pure scattering

medium decreases the concentration of excited atoms at great distances from

the source, compared with the value found in a test volume located at the same distance from the source, but in a vacuum. In this case the

0 12 3 4 5 T

Fig. 17. Medium-effect coefficient for conservative scattering with the

Lorentz profile.
4.6 ACCUMULATION EFFECT. THERMALI ZATION LENGTH 177

concentration of excited atoms in a pure scattering medium falls off as r"^/^, where r is the distance from the source (see (3.14)). At first glance

this result might appear strange. In a pure scattering medium the flux

I through a specified area decreases as r'^, and it would therefore seem that

the density of excited atoms could not decrease faster. In fact, this is not

; so. The flux, integrated over the line, does decrease as r"^, but the fre-

;• quency distribution of the radiation also changes with distance. Because

i of the strong frequency dependence of the line absorption coefficient,

this change in frequency should certainly be kept in mind. It can lead to an

j even more rapid decrease in the concentration of excited atoms. The decisive

factor is the behavior of the profile in the far line wings. The more slowly

the absorption coefficient decreases into the wings, the more rapidly the

concentration of excited atoms far from the source falls off in a pure scatj tering medium. From (3.29) and (6.2) it follows that in a conservative

medium i(T) behaves asymptotically as T2Y-l/(t)(l/T) for t ^ o°. Hence for

Y < 1/2 the accumulation effect is absent; for y > 1/2 it is always present;

and in the case of y = 1/2 one can have both i(T) > 1 (for example, the

Doppler profile) and i(T) < 1.

THERMALI ZATION LENGTH: LEADING CONSIDERATIONS. Line photons emitted by the source undergo multiple scatterings in the medium. Each scattering consists

of the excitation of an atom followed by a spontaneous downward transition.

The farther an excited atom is from the source, the more scatterings the

j average photon will have undergone before exciting this atom. At the same

t time, the probability that a non-radiative transition follows the excitation

of an atom and the photon "dies" is 1 — A. Therefore, no matter how small

the quantity 1 — X, far enough from the source the destruction of photons

j becomes significant. And near the source, where excitation is due primarily

I to photons having experienced a relatively small number of scatterings, the

radiation field should be nearly the same as if photons did not die at all.

I In accordance with the foregoing discussion, for 1 — A << 1 a region

I should exist around the source in which scattering can, to a good approximaI tion, be considered as conservative, i.e. one can assume that X = 1. It is

clear that as the role of absorption decreases, the size of this region of

conservative scattering should increase. This qualitative conclusion is

valid for all cases (for any line absorption profile, any phase function,

etc.). If one wishes to speak quantitatively, then everything is determined

I by the details of the processes occurring in the elementary act of scatter-

'f ing.

As we have seen in Chapter III (Sec. 3.4), for isotropic monochromatic

scattering the size of the conservative region is of the order of the diffusion length = 1/k. Provided that 1 - X << 1, from (3.2.3) we have

^

. (6.6)

^3(1-X)

In line - frequency scattering everything depends on the behavior of the

absorption coefficient in the line wings. From the results of the preceding

sections it is clear that absorption may be ignored, in the first approximation, for distances from the source corresponding to small values of the

parameter t defined by (4.9). The effects of absorption become significant

only when t approaches unity. An order of magnitude estimate of the dimensions of the conservative scattering region is therefore given by the value

of T corresponding to t = 1.
178 INFINITE MEDIUM

These considerations indicate that a certain characteristic length ought

to exist for any medium in which "true" absorption processes occur along with

scattering. This length characterizes the mean distance from the place where

a photon is born to the place where it dies. In other words, there is a cer- tain mean distance from the place where the atom receives its primary excitation to the place where the energy utilized in that excitation is converted

into heat. This basic length is called the thermalization length . Since we are interested primarily in an order of magnitude estimate for

the case of nearly conservative scattering, one might define the thermalization length as the root of the equation

M^(t^) = 1 - a . (6.7)

This equation is obtained from the condition t = 1 just mentioned if the

factor r(2Y+2)/XTTY is replaced by unity. The definition (6.7) is, however,

inapplicable in the case of monochromatic scattering. This is a serious

shortcoming. Taking into account this fact as well as some other considerations which we will not explore here, we find it more convenient to define

the thermalization length in a somewhat different manner. Evidently one must

require the values of defined in this way to be of the same order of magnitude as those given by (6,4). Such a definition is given below.

THERMALIZATION LENGTH: DEFINITION AND LIMITING CASES. We define, as in

Sec. 3.9,

T

-l'^(T) = 1 + 47r/ S (t)t^dt . (6.8)

0 The value of 'i'c»(T) is obviously the number of excited atoms within a sphere

of radius t around the source of unit strength. Clearly 1'oo(<=°^ = (1 — A)"-*- (see the end of Sec. 4.1). Furthermore, let S_^(t) be the function *i'^(T) for

the conservative medium:

1 ^/

?

o

II 1

X2 >x 1

0

1

]

log T

Fig. 18. Definition of the thermalization length (schematic curves).
4.6 ACCUMULATION EFFECT. THERMALI ZATION LENGTH 179

S (t) = ^ (t) , X = 1 (6.9)

We now define the thermalization length as the root of the equation

-1

(6.10)

We can see the physical significance of this definition from the following

argument. Let us consider a point source in a medium with a given X < 1,

and the same source in a conservative medium. The thermalization length is

then the radius of a sphere around the source in the conservative medium

which contains the same number of excited atoms as are in the whole non-con- servative medium. To aid the visualization of this definition, it is shown

graphically in Fig. 18.

Let us consider this definition for several specific cases. For X = .0 we have x-;- = 0, indicating that a non-radiative downward transition follows

each atomic excitation. The extreme opposite case, 1 — X << 1, is much

more interesting. From (4.2) and (6.8) -(6.9) it follows that, for y < 1,

S (t) M. (t)(6.11)

and (6.10) gives

M^(T^) ~ C (1-X) , 1 - X ^< 1 , (6.12)

which in the most important case of 0 < y < 1/2 agrees with the estimate

given by (6.7). Using the asymptotic form of Mj^d) given in Sec. 2.7, we find from (6.12), for the most important types of absorption coefficients,

Doppler

3/2

(1-A) Vln A

1-

(6.13a)

Voigt

Lorentz

— aU(a,0) (1-X)

1 (i-x)-2

9

(6 . 13b)

(6.13c)

We note that (6.13b) is applicable only for >>

ing in the wings as Ixl"!^, (6.12) gives

For profiles decreas

T, oc (1-A)-1/2y , (6.13d)
180 INFINITE MEDIUM

where y = (ic-1)/2k. Finally, for the rectangular profile (monochromatic

scattering), for 1-X << 1 we get from (6.10) and (3.9.12)

~ ~ , (6.14)

•^Sd-X)

where is the diffusion length.

For nearly conservative scattering we can obtain a general asymptotic

expression for the thermalization length which is valid for an arbitrary

value of the characteristic exponent y, 0 < y - 1. Using the asymptotic

form of the conservative Green's function as given by (3.29), we find from

(6.8) -(6.9) ,

2Yr(2Y)cos^Y <J'(1/T)

Substituting this expression into (6.10) and using the fact that (j) is a slowly varying function, we finally obtain

~ r-pr- • 1 - X << 1 , (6.16)

' (1-A)1/2y

where

f

= 2Yr(2Y)cos uy ^L^_^^l/2y\ ^/^Y

^^^^^^ 1-2y V /.

In practice f is usually of the order of unity. Equation (6.16) is valid

for both line - frequency and monochromatic scattering and gives, in these

cases, the results (6.12) and (6.14) respectively.

The general conclusion that can be drawn from these results can be

summarized in the following way. For nearly conservative scattering the

thermalization length is very sensitive to the behavior of the absorption

coefficient in the line wings. The more slowly the absorption coefficient

decreases in the wings, the larger the thermalization length. The extent of

the effect arising from differences in the form of the absorption coefficient is illustrated by the following example. For 1-A = 10"^ (in practice

much smaller values of 1-A are often encountered), the magnitude of the

thermalization length is 2-103 for the Doppler absorption coefficient,

3'107 for the Lorentz profile, and O.S-IO^ for monochromatic scattering.

Thus the difference amounts to several orders of magnitude. However, even today one still occasionally encounters an error that has been very widespread in recent years, namely, the identification of the thermalization

length with the diffusion length not only for monochromatic scattering

(which is quite acceptable), but also for 1 ine - frequency scattering for

every type of absorption coefficient, which is absolutely incorrect. We

have just seen the enormous errors that this can cause.
4.7 PLANE ISOTROPIC SOURCE 181

' CONCLUDING REMARKS. The concept o£ thermalization length is extremely useful

for estimating the nature of the solutions of most problems involving linefrequency radiative transfer. The value of defines the radius of the

I "sphere of influence" of a point source throughout which the primary excitation is "smeared." If sources whose strengths vary only slightly over a distance of the order of are distributed throughout a medium, the source function will have roughly the same depth dependence as the strength of the

primary sources. Conversely, if the strength of the primary sources varies

greatly over distances of the order of , "smearing" begins to exert a subi stantial effect, and there is no longer any simple relation between S(t_) and

S*(t_). Roughly speaking, the source function at a specific point is a cer- tain mean of S*(t_) over the surrounding region with dimensions of the order

of !(-. A detailed qualitative discussion of this question is given by R. N.

I Thomas (1965a, 1965b) .

j The ultimate goal of optical plasma diagnostics is to recover the values

' of the physical parameters of the medium— for example, the temperature and

k density distributions — from the fharacteristics of the observed radiation.

! It is evident that the radiation intensity directly reflects the distribution

of excited atoms, whereas temperatures and densities can be found through the

' primary source distribution for the radiation field. The "smearing" effect

of which we have just spoken therefore sets a theoretical limit on the information that can be obtained by optical diagnostics.

I The systematic use of the thermalization length in transfer theory began

!; with the paper by E. H. Avrett and D. G. Hummer (1965) which explained the

1 important role of this characteristic length. The original definition of

[ was not completely satisfactory, and efforts were made to improve it. The

I literature therefore contains various definitions of t^, differing only

,i slightly from one another (see G. B. Rybicki and D. G. Hummer, 1969; G. D.

j Finn and J. T. Jefferies, 1968; V. V. Ivanov, 1966; V. V. Ivanov and D. I.

\ Nagirner, 1966) . An improved definition of has recently been discussed at

length by a number of people working on problems of line transfer theory

(D. G. Hummer, G. B. Rybicki, J. C. Stewart, V. V. Ivanov). The definition

(6.10) expresses the point of view developed by the author as this discussion

proceeded. For another point of view, see G. B. Rybicki and D. G. Hummer

(1969).

BASIC FORMULAE. Until now we have been concerned with the point source. Now

let us turn our attention to the problem of a plane isotropic source in an infinite medium. We shall assume that its strength per unit area is 4itAv/A.

If T is the line center optical distance from the emitting surface and S ^(t^)

is the corresponding line source function, then, as in monochromatic scattering (cf . Sec. 3.5)

,

4.7 PLANE ISOTROPIC SOURCE

(7.1)

where $ (t) is the solution (bounded at ± infinity) of the equation

OO (7.2)

- oo

I
182 INFINITE MEDIUM

Explicit expressions for $ (t) were obtained in Sec. 4.1 and are given by

0 (t)

CO

= I f

^V(u)

^ / l-XV(u)

cos Tu du , (7.3)

$ (t) = y e-'"l/^R(z)G(z) ^ (7.4)

where V(u) and R(z) are given by (2.6.24) and (1.19) respectively. For these

expressions to be valid it is sufficient (though not necessary) to suppose

that ct(x) is a continuous monotonically decreasing function of |x|.

SMALL-T BEHAVIOR. Proceeding as in Sec. 4.2, from (7.4) we find that for

small ItI

0

T^£n| TI 1 + O(t^) ,

where

CO (R(z)-l)G(z) dz

(7.5)

(7.6)

On the other hand, (7.3) and (2.6.21) give

(t) = - K

' 2

00 ^ (It I) +1 r XY (u)

^ ^ / l-XV(u)

so that for small t

$ (t) a-j^ ^n ] T I + a-|^Y*-a +

CO . 2 f

XV^u)

^ 0^ l-XV(u)

o(|t|)

and thus, in addition to the representation (7.6), p*^ may also be written

in the form „ ,\ - I f

^V'(^ ) du

^/ l-AV(u)

(7.7)
4.7 PLANE ISOTROPIC SOURCE 183

Incidentally, we have proved the identity (1.24). However, what is now

of interest to us is not the identity itself, but an interesting consequence

of (7.7). If the characteristic exponent y is less than 1/2 (in particular,

if the absorption coefficient decreases in the line wings according to an arbitrary power law, i.e. is proportional to |x|"'^, 1 < k < °=>) , then the

quantity p*i remains finite for A ^ 1. For the Doppler profile (y = 1/2) the

picture is different. As is seen from (7.7) and (2.7.15), when \ tends to

unity, the value of p*-^ increases indefinitely. Thus in this case a steady

radiation field cannot exist in an infinite, pure scattering medium. In this

respect scattering with the Doppler profile is no different from monochromatic scattering (see Sec. 3.5). The physical reason for this similarity is

that the accumulation effect is present in both instances (see Sec. 3.4 and

4.6). The values of q*i for the Doppler and Lorentz profiles are listed in

Table 17. This and the other tables in this section are based on the calculations of D. I. Nagirner and A. B. Schneeweis (1973).

LARGE -T BEHAVIOR.

Y < 1 is

The asymptotic form of $oo(t) for large in the case of

(2y+1) |t|(1-A)2

(7.8)

where

(1-A) 2y + 1 XttAx' ( I t I ) (7.9)

F (t)

_ CO

r(2Y+i)Q^

e ^ y^^ dy

2y . 2 4y (t+y 'ctg ^y) +y

^ (7.10)

TABLE 17

THE VALUES OF p*

1_JV Doppler Lorentz

0 . 5 0 6630 0 3624

0 . 4 0 8940 0 4755

0 . 3 1 207 0 6172

0 . 2 1 679 0 8068

0 . 1 2 584 1 094

10-2 6 673 1 652

10-^ 12 60 1 799

10-4 20 . 25 1 825

10-^ 29 .41 1 . 829
184 INFINITE MEDIUM

The result (7.8) is derived directly from (7.4), by using the asymptotic

forms of G(z) and U(z) found in Sec. 2.6. The use of the asymptotic expression for Sp(T) obtained in Sec. 4.3, m conjunction with (1.8), also leads

to (7.8), but by a more indirect route.

The functions Foo(t) for the Doppler (y = 1/2) and the Voigt and Lorentz

profiles (y = 1/4) are tabulated in Tables 18 and 19, respectively. We

note that for t -> 0

F°(t) = t^(-£nt-Y* + - t + (7.11)

TABLE 18

THE FUNCTION F^(t)

F^t) F^t) CD ^ ^ F^t) on -*

0 0 0 00000 3 . 4 0 . 7504 9 .6 0 .9451

0 1 0 01865 3 . 6 0 . 7666 10 0 0 ,9489

0 2 0 05175 3 . 8 0 ,7814 10 4 0 9522

0 3 0 08965 4 . 0 0 . 7948 10 8 0 ,9553

0 4 0 1290 4 . 2 0 . 8072 11 2 0 .9581

0 5 0 1682 4 . 4 0 .8185 11 6 0 .9606

0 6 0 2064 4 . 6 0 . 8290 12 0 0 .9629

0 7 0 2431 4 . 8 0 . 8386 12 4 0 .9651

0 8 0 2783 5 . 0 0 . 8474 12 8 0 .9670

0 9 0 3117 5 . 2 0 . 8556 13 2 0 .9688

1 0 0 3434 5 . 4 0 . 8632 13 6 0 ,9705

1 2 0 4017 5 . 6 0 . 8702 14 0 0 .9720

1 4 0 4537 5 . 8 0 .8767 14 4 0 ,9734

1 6 0 5000 6 . 0 0 . 8828 14 8 0 9747

1 8 0 5413 6 . 4 0 .8937 15 2 0 ,9760

2 0 0 5782 6 . 8 0 .9032 15 6 0 9771

2 2 0 6112 7 . 2 0 .9116 16 0 0 9782

2 4 0 6408 7 . 6 0 .9189 16 8 0 9801

2 6 0 6674 8 . 0 0 .9254 17 6 0 9817

2 8 0 69 13 8 . 4 0 . 9312 18 4 0 9832

3 0 0 7130 8 . 8 0 .9364 19 2 0 9845

3 2 0 7326 9 . 2 0 .9410 20 0 0 9857
4,7 PLANE ISOTROPIC SOURCE 185

Vlit) = t\l + ^ t£nt ... ) , (7.12)

whereas for t ^ <»

k=0 .2k

m (k.i)lL]2fk^i^/2r(5sil) J_

4 J \ 2 / i-k

(7.13)

(7.14)

TABLE 19

THE FUNCTION F^(t)

t VF (t) t V

F

CO

(

^ t)' t V

F ( t)

0 . 0 0 . 00000 3 . 4 0 .5515 9 .6 0 . 7962

0 . 1 0 .00725 3 .6 0 .5678 10 .0 0 .8033

0 . 2 0 .02348 3 . 8 0 .5830 10 . 4 0 . 8098

0 . 3 0 .04436 4 .0 0 . 5972 10 . 8 0 . 8160

0 . 4 0 .06766 4 . 2 0 .6106 11 . 2 0 . 8217

0 . 5 0 . 09209 4 . 4 0 .6231 11 .6 0 .8271

0 . 6 0 . 1169 4 . 6 0 .6348 12 . 0 0 . 8322

0 . 7 0 . 1415 4 . 8 0 .6459 12 . 4 0 . 8370

0 . 8 0 . 1656 5 . 0 0 .6563 12 . 8 0 .8417

0 . 9 0 .1892 5 . 2 0 .6662 13 . 2 0 . 8459

1 .0 0 .2119 5 . 4 0 .6755 13 .6 0 . 8499

1 . 2 0 . 2550 5 . 6 0 .6843 14 . 0 0 . 8538

1 . 4 0 . 2947 5 . 8 0 . 6927 14 . 4 0 .8575

1 . 6 0 .3311 6 . 0 0 . 7006 14 . 8 0 . 8610

1 . 8 0 . 3645 6 . 4 0 .7154 15 , 2 0 . 8643

2 . 0 0 . 3951 6 . 8 0 . 7288 15 . 6 0 . 8675

2 . 2 0 .4232 7 . 2 0 . 7410 16 . 0 0 .8705

2 . 4 0 .4491 7 . 6 0 . 7521 16 . 8 0 . 8761

2 . 6 0 .4728 8 . 0 0 . 7624 17 . 6 0 .8813

2 . 8 0 .4948 8 , 4 0 . 7719 18 . 4 0 .8861

3 . 0 0 .5151 8 . 8 0 . 7806 19 . 2 0 . 8905

3 . 2 0 .5340 9 . 2 0 . 7887 20 . 0 0 . 8946
186 INFINITE MEDIUM

For conservative scattering (and for y < 1 > well as for 1 — X << 1 in

the asymptotic part of the conservative scattering zone), we have from (7.8)

$ ~ (2Y-^l)tg7rY 1 . 2ATTr(2Y) |t|x'(|t|)

In particular, for the Voigt profile y = 1/4 and the asymptotic form of the

function x(t) is given by (2.7.26). From (7.15) we find

$I(t) ~| ^ 1x1"^, |t1 » a'^ . (7.16)

2^ (aU(a,0))^

In the limit as a , we get the asymptotic form of the conservative

function Ooo(t) for the Lorentz profile:

^^(j) ^ _1_ |t| . (7.17)

°° 2 /if

MODIFIED FORM OF $oo(t). LONGEST FLIGHT APPROXIMATION. As in the case of a point source, the formation of homologous combinations of arguments can be

used to show that for y < 1/2 the asymptotic expression (7.8) will also give

good results in the non-asymptotic region. The source function for any x and

A can be represented in the form

A Ki(|x|)

*..(T) F^(t)5^(x,A) , (7.18)

2 2 (1-A)

where

^ 1-

t = ^ r(2y) (7.19)

TT ak,(|t|)

and Coo(x',^) is the correction factor to the asymptotic form. The equality in

(7.18) is exact, and not asymptotic.

For nearly conservative scattering (1-A << 1) and Y < 1/2 the factor

Coo(x,A) can be replaced by ?oo(x) = 5c»(x,l), and it can be set equal to unity

to obtain an estimate of the source function. For conservative scattering

(A = 1) we have from (7 . 18) - (7 . 19)

C ( I T I ) *oc(T) = — ^ooC^). ^ = 1 . (7.20)

^ k,(|t|)

where

= . (7.21)

Try
4.7 PLANE ISOTROPIC SOURCE 187

I£ X * 1 , then for | t | ->

$ (t)

2 Ki(|t|)

(7.22)

but for 1 — X << 1 the region in which this asymptotic representation can be

applied is limited to very large values of |t| (namely, those corresponding

to values of t much greater than unity)

We emphasize that, in contrast to the analogous formulae for the Green's

function, when t << 1 it is impossible to take the limit y 1/2 in (7.8) and

(7.18) in order to get the asymptotic forms appropriate to the Doppler

profile

Sec

The function

4.5 is

,(t) in the longest flight approximation introduced in

(T) = - 2 K^dxl)

1-X+XK2( I T I

)

(7.23)

TABLE 20

THE FUNCTION <I)^(t) FOR CONSERVATIVE SCATTERING

T $^(t) T ^''"(t) T $^(t) T ^^(t) 00 "^ oo^-^ oo^-^ m^-'

0 01 2 017 1 4 0 .7013 3 8 0 4438 9 . 0 0 .2 86 1

0 05 1 610 1 5 0 6819 4 0 0 4326 9 . 5 0 2783

0 1 1 431 1 6 0 6638 4 2 0 422 1 10.0 0 2711

0 2 1 248 1 7 0 6470 4 4 0 4123 11 0 2582

0 3 1 138 1 8 0 6312 4 6 0 403 1 12 0 2470

0 4 1 059 1 9 0 6165 4 8 0 3945 13 0 2372

0 5 0 9960 2 0 0 6026 5 0 0 3864 14 0 2 284

0 6 0 9442 2 2 0 5772 5 5 0 3681 15 0 2205

0 7 0 9001 2 4 0 5545 6 0 0 352 1 16 0 2134

0 8 0 8617 2 6 0 5340 6 5 0 3380 18 0 2010

0 9 0 82 78 2 8 0 5155 7 0 0 3254 20 0 1906

1 0 0 7975 3 0 0 4986 7 5 0 3 141 22 0 1816

1 1 0 7701 3 2 0 4832 8 0 0 3039 24 0 1738

1 2 0 7452 3 4 0 4690 8 5 0 2946 25 0 . 1702

1 3 0 722 3 3 6 0 4559
188 INFINITE MEDIUM

r The properties and the physical significance o£ this approximation are similar to those of the longest flight approximation of the Green's function

SpCx). However, it should be noted that C^o in (7.20) tends to infinity as

Y

-> 1/2. Hence in the conservative scattering zone (7.23) breaks down for

Y

> 1/2. In particular, for the Doppler profile (y = 1/2) it must be a poor

approximation in the conservative region, i.e. for such t that 1 -X<<K2 (

|

t | ) . NUMERICAL DATA. We shall now present some numerical results. First let

us consider the cases of the Lorentz and Doppler profiles. Table 20 gives

values of the funct ion $co (t) for conservative scattering with the Lorentz

profile, obtained from (7.4) by numerical integration. Using the values of

$^(t) shown in the table, ?^(t) was evaluated from (7.20) (prior to which

K^(t) and were tabulated) . It was found that the function C^(t) in- creases from tt/4 = 0.79 at t = 0 to 1.46 at t = 0.1, and then decreases

rather rapidly. For x > \, ^p^^-* deviates from unity by less than 8 percent.

Thus, for the Lorentz profile the representation (7.17) with Coo(t) = 1 gives

an accuracy of better than 50 percent for all t; for t > 1 the error does not

exceed 8 percent and for t > 5 it is less than 3 percent.

TABLE 21

THE FUNCTION $ (t )

T A = 0 . 9 A = 0 99 A = 0.

0 . 1 1 . 768 0 3 963 0 6 961

0 . 2 1 . 526 0 3 689 0 6 684

0 . 3 1 . 377 0 3 517 0 6 509

0 . 5 1 . 179 0 3 280 0 6 267

0 . 7 1 .040 0 3 107 0 6 089

1 . 0 8 . 862 - 1 2 905 0 5 880

1 . 5 7 .058 - 1 2 647 0 5 610

2 . 0 5 . 779 - 1 2 445 0 5 395

3.0 4 .070 - 1 2 135 0 5 056

5.0 2 .272 - 1 1 712 0 4 571

7.0 1 . 404 - 1 1 427 0 4 220

10 7 . 724 - 2 1 131 0 3 824

15 3 . 560 - 2 8 199 3 352

20 1 .962 -2 6 247 3 007

30 8 . 194 -3 3 969 2 . 5 15

50 2 . 691 -3 1 970 1 907

70 1 . 300 -3 1 146 1 . 529

100 6 .053 -4 6 055 -2 1 . 161
4.7 PLANE ISOTROPIC SOURCE 189

The values o£ $oo(t) for the Doppler and Lorentz profiles and for several

X's are shown in Tables 21 and 22. As we have already mentioned, $£(t)

diverges as X -> 1

The effect of the Voigt parameter a on the form of $oo(t) is graphically

illustrated by Figs. 19-22, made available through the courtesy of D. G.

Hummer. The values of •I'ooCt) used in the construction of these curves were

found by numerical solution of equation (7.2). Fig. 19, which is relevant

to the Doppler profile, gives a good illustration of the divergence of <I>B(t) as A. ^ 1 . When a = 10'^, the x-region in which $J^(t) can be considered equal

to the conservative function 'fXC'^) ^° reasonable accuracy exists only for

1 — X ;^ 10"^ 10"6. This is evident from Fig. 20. The region in which it

is practical to apply the asymptotic form (7.16) for a = 10"^ would therefore

appear to be rather small, although a comparison of Figs. 20-22 shows that

it expands as a increases. For the Lorentz profile (a = °°) a x-region in

which the asymptotic form (7.17) can be used exists for 1 — X = 10"^ and

rapidly increases in size as 1 — X decreases.

TABLE 22

THE FUNCTION $o^(T)

T X = 0.9 X = 0 99 X = 0.999

0 . 1 9 . 603 1 330 0 1 415 0

0 . 2 7 .985 1 149 0 1 232 0

0 . 3 7 .026 1 040 0 1 122 0

0 . 5 5 . 805 8 999 9 800 -1

0 . 7 5 .001 8 057 8 843 -1

10 4 . 165 7 052 7 819 -1

1

r 3 . 258 5 924 6 666 -1

2 . 0 2 . 666 5 154 5 875 -1

3. 0 1 .935 4 153 4 840 -1

5. 0 1 .227 3 084 3 723 -1

7. 0 8 .885 -2 2 513 3 116 -1

10 6 .228 -2 2 012 2 578 -1

15 4 .096 -2 1 555 2 077 -1

20 3 .014 -2 1 290 1 782 -1

30 1 .928 -2 9 859 -2 1 434 -1

50 1 .072 -2 6 941 -2 1 089 -1

70 7 . 184 -3 5 463 -2 9 066 -2

100 4 .642 -3 4 203 -2 7 453 -2
190 INFINITE MEDIUM

-2 -10 1 23456789 10

I og T

Fig. 19. The function $ (x) for the Doppler profile.
4.7 PLANE ISOTROPIC SOURCE 191

-2-1 0 1 2 3456789 10 II log r

Fig. 22. The function ^o^Ct) for the Lorentz profile.

522-519 O - 74 - 14

CHAPTER V

SEFII-INFINITE MEDIUM: GENERAL THEORY

In Chapter IV the general problem o£ radiative transfer in an infinite

medium was discussed. Consequently no attention was paid to boundary regions,

in which both absorption and the escape of radiation play important roles.

We shall now study the radiation field in these regions, which are particularly important as primary contributors to the radiation field emerging from, the

medium. It is natural to begin by considering an idealized medium, occupying

a half-space. We will devote Chapters V and VI to this subject.

The problem of radiation scattering in a semi - infinite medium is one of

the few problems in the theory of multiple scattering whose exact solution

can be obtained in closed form. Thus aside from our interest in this problem

as such, it serves as a touchstone for verifying the accuracy and limits of

applicability of the various approximate and numerical methods of transfer

theory. This justifies both the prominence given to this problem and our policy of quoting numerical results which, unfortunately, are. still far from

complete. It is also no accident that the study of the asymptotic properties

of the solution receives so much attention. As will be shown shortly, when

the scattering is nearly conservative, so that almost every photo-excitation

of an atom is followed by a downward radiative transition, the boundary layer

becomes very thick. The asymptotics that we have just mentioned reflect simplifications in the structure of this thick boundary layer at optical depths

much greater than unity. These simplifications are of a sufficiently general

character that they must pertain not only to a medium bounded by a plane, but

also to media having a more complicated geom.etry.

In this chapter a formal solution will be developed for evaluating the

radiation field in a half-space for a source strength that depends on depth

in an arbitrary way. The functions appearing in the solution are studied in

detail, and some of them are tabulated. In the next chapter these general

results are applied to standard model problems, and the physical aspects of

these problems are discussed.

The standard tool for the solution of half-space problems of multiple

scattering is the Wiener-Hopf technique fsee , e.g., E. Hopf, 1934 ; E. C. Titchmarsh, 1937; I. W. Busbridge, 1960). However, as we shall see in this

chapter, if the Green's function for an infinite medium is known, a more

direct and physical approach can be used. It enables one to solve half-space
SHMI- JMFTMITB MKDIUM: GEMHRAL THEORY

probleifis by eleTnentary me?jns and elucidate?-; the physical meaning of unany quantities and relations v/hich remain formal in the Wiener-IIopf approach.

5.1 BASIC EOIJATIOMS

'I'lli: 'i'RANSFER EQUATION. We shall assume that the strength of the radiation

sources depends on only one spatial coordinate — the distance from the boundary. 'I'he source stren^jth is constant on any surface parallel to the boundary

of the rnoflium. Tn vi ov/ of the symmetry of the problem, the intensity will depend only on doptfj, iho anfle with the normal, and frequency. Let x be the

usual dimonsionlcss frequency, c, the optical depth, i.e. the distance along

the normal from the boundary of the medium measured in mean free paths of a photon of frequency x = 0, and p, the cosine of the angle between the direction of propagation of radiation and the outward normal. 'fhen 1 = I(T,y,x).

fn tfio absence of continuum absorption the equation of transfer within a line

is f see Sec. 2.7))

d 1 r T , y , X ) f. . f d T

1 n . 1

)

-

I

A'/fxj j" 'jA/a' J

f fr ,x' ) dy ' -a(xjS*rTj

J. CO _ ]

and the l;oundary condition expressing absence of radiation incident from outs ide is writ ton as

If0,lj,xj = 0 , u

<- 0 . (1.2J

'I'lic solution of" ri.lj V'/itfi the l)oundary condition (1.2) and an arbitrary primary source rijnc:tion [','''({) is the subject of this chapter.

Introducing, the line source functi.on

1

•Sr-r) = jA y^y.fx'jdx' J

f fx , p

' , x' ) d)j ' + s'^Tt) , (1.3)

-co _ 1

we can rewrite the transfer equation in the form

dT(T,y,x) ^ K,,,,x)-S(T) , ri.4j

a(x) di

from which it is evident that in fact the intensity depends not on y and x separately, but only upon the combination

z = . (1.5)

a(x)

We shall again denote the intensity as a function of t and z by I, which

should not cause confusion. Its normalization is unchanged, so that

I (t , z) dado)dv is the energy flowing per unit of time within an element of
5.1 BASIC EQUATIONS 195

solid angle doj in the frequency interval (v , v + dv) through an area da located at optical depth t and oriented perpendicular to the direction of propagation of the radiation.

Using the notation (1.5), we have, instead of (1.4) and (1.2) respectiveZ = I(T,Z) _ S(T) , di

I (0 , z) = 0 , z < 0 .

(1.6)

(1.7)

The expression for the source function S(t) can also be reorganized by substituting z' = y'/a(x') in the second integral in (1.3):

l/a(x'

)

S(t) = / a"(x')dx' I

I(T,z')dz' + s'^(t)

-l/a(x'

)

00 (1.8)

changing the order of integration, we obtain

S(t)

oo 4/I(t ,z')G(z')dz' + S (t) , (1.9)

whe re

G(z)

00

= A a^(x ')dx' , |z| < 1

00 2A / a" (x' ) dx' , I z I > 1 , x(z)

(1.10)

and x(z) is defined by

a(x(z)) = -i- , x(z) > 0

We note that

G(z) = G(-z)

Further, (1.9) and (1.3) give the identity
196 SEMI- INFINITE MEDIUM: GENERAL THEORY

1 J

I(T,z')G(z')dz' = A J

aix')dx' f I f t , ^\

J dp '

Setting 1=1 and using the fact that G(z) is an even function, we find

00 y* G(z) dz = 1 . (1.11)

0

Explicit expressions for G(z) for the Doppler and Lorentz profiles were given

in Sec . 2.7.

For the intensity of radiation at depth t, we find from (1.6) and (1.7)

(see (2.3.13) and (2.3.14))

T I(t,z) = -y*S(T')e~^^'~'^^/^dT'/z , y < 0

0

00 I(t,z) = y S(T')e~'^^'~^^/^dT'/z, P > 0 .

(1.12)

T Specifically, the intensity of the emergent radiation is

CO 1(0, z) = / S(T')e ^'/^di'/'z . (1.12')

0

We give now for reference expressions for the three quantities of physical interest: the total radiation flux in the line, the density of radiation,

and the radiation pressure. Let ttF(t) be the total flux of radiation in the

line in the direction of the normal to the layers (it is assumed that the flux

is positive if the energy flows in the negative-i direction):

r

7tF(t) = 2u

I dv / I(T,y,x)ydiJ

This expression can be reduced to

TrF(T) = Av ^ y I(T,z)G(z)z dz . (1.13)

Substituting I(t,z) from (1.12), we get
5.1 BASIC EQUATIONS 197

00

ttF(t) = Av

-y y K2(|t-t' |)sgn(T'-T)S(T')dT' , (1.13')

0

whe re

OO 00 K^Ct) = J

e~^/^ G(z)dz = A y a(x)E2(a(x)T)dx .

0

The properties o£ K2(t) were considered in Sec. 2.6 and 2.7.

By definition, the total density of radiation in the line is

1

p(t) = 27T i y dv y I(T,y,x)dy , ^0 -1

from which

CO p[T) = Av 1^ f

ICT,z)GQ(z)dz , (1.14)

/\c ^00

or, if use is made of (1.12),

00 P(T) = Av |I y K]_^(|t-t' i)S(T')dT' , (1.14-)

where

00

= A y (x) Ca(x) t) dx .

OO

For the properties of K^^ and G-j^ , see Sec. 2.6.

Finally, the force of radiation pressure exerted on a unit volume at

depth T is _ <7(Vn) r r

P^(t) = 2tt — I

a(x)dv / I(T,y,x)ydy , ^

0 -1

where o(vq) is the line-center absorption coefficient. The quantity Pj,(t) is
198 SEMI- INFINITE MEDIUM: GENERAL THEORY

positive if the force is directed in the negative-i direction. It can easily

be shown that

P^(t) = Av

^ 0 ^ 2tt ^

y* I(T,z)G2(z)zdz , (1.15)

or, considering (1.12),

P^(T) Av ~r~ ^ J

K22(|t-t' |)sgn(T'-T)S(T')dT' . (1.15')

We can easily obtain an integi'al equation for S(t) whose solution is

equivalent to the solution of the transfer equation (1.1) with the boundary

condition (1.2). One need only substitute into (1.9) the expressions for the

intensity given in (1.12). This yields the following equation for the line source function S(t):

00 S(t) = K^(|t-t' |)S(T')dT' + S*(T) , (1.16)

whe re

op CO

= J

e~^/^'G(z' )dz' /z' " ^ y

a^(x)E^(a(x)T)dx . (1.17)

Equation (1.16) is the basic integral equation for the problem at hand.

In what follows we shall base our analysis on this equation rather than on the transfer equation in its differential form.

THE RESOLVENT AND THE RESOLVENT FUNCTION. To solve the integral equation

(1.16) we shall use the method developed by V. V. Sobolev (1956; 1958a; 19S9b ; 1967a) and K. M. Case (1957a, 1957b). In recent years this method has found

rather widespread application in the theory of radiative transfer.

The solution of (1.16) for an arbitrary source term S (t) is essentially

equivalent to the evaluation of its Green's function G(t,t'), defined by

00 G(T,T')=|y* K^(|T-tl)G(t,T')dt + 6(T-T') , (1.18) I

0

where 6 (x) is the de It a- funct ion . Equivalent ly , we could find the resolvent

r(T,T') that satisfies the equation

00 r(T,T') = K^(|T-t|)r(t,T')dt + ^

k^(It-t'I) . (1.19)
5.1 BASIC EQUATIONS 199

The Green's function and the resolvent are related by

G(t,t') = rCT,T') + 6(t-t') . (1.20)

When r(T,T') is known, the source function S(t) is obtained by integrating over the distribution of primary sources:

00 S(t) = S*(t) + J

r(T ,T')S*(T')dT' . (1.21)

Because of the symmetry of the kernel of equation (1.16), the Green's

function and the resolvent are symmetrical, i.e.

r(T,T') = r(T' ,t) . (1.22)

The resolvent, which depends on two variables, may be simply expressed in

terms of a function of one argument, $(t), which we shall call the resolvent

function . It is the value of r(T,T') for x' = 0:

$(t) e r(T,0) = r(0,T) . (1.23)

To show this we shall first of all prove that if a function f(T) is bounded

as T 0 , then

00 00 ^ f

K^(|T-t|)f (t)dt = J

K^(|T-t|)f' (t)dt + f(0)K^(T) . (1.24)

^0 0

Splitting the integral on the left into two parts — one from 0 to x and

the other from x to <» — and substituting y = x — t in the first, and y

=

t — X in the second, we obtain

00 f

00 j K^(lx-t|)f(t)dt = J

K^(y)f (x-y)dy + (y) f (x+y) dy , (1.25)

from which

_d_

dx

0

T

00 J

K^(|x-tl)f(t)dt = f(0)K^(x) +

J

K^(y)f' (x-y)dy + j (y) f + ' (x+y ) dy . (1.26)

"Q 0
200 SEMI- INFINITE MEDIUM: GENERAL THEORY

Rearranging the sum o£ the two last terms on the right, and using the identity

(1.25), we get (1.24).

Now, differentiating (1.19) with respect to t and then to t', we add the

equations thus obtained term by term. Considering (1.24) and (1.23), we find

00 K,(|x-tl, * ^) dt * H.n \

K,(T) . (1.27,

0

Introducing the notation (1.23), we have from equation (1.19)

00 $(t) = I J

K^(|T-t|)$(t)dt + I

K^(t) . (1.28)

Comparing (1.27) and (1.28), we conclude that

iL + Jil = <i>(t)$(t') . (1.29)

The solution of this equation satisfying the condition (1.23) can, for

t' > T, be written as

T r(T,T') = $(t'-t) + / $(T-t)$(T'-t)dt . (1.30)

Hence, because of the symmetry of the resolvent, it follows that for arbitrary

T and t'

r(T,T') = $(|t-t'|) + / <|)(T-t)$(T'-t)dt , (1.31)

0

whe re is the smaller of x, x' Expression (1.31) shows that, indeed, the evaluation of the resolvent

of equation (1.16) involves the determination of a function of one variable

$(x), defined by (1.28). This function plays a fundamental role in problems

of radiative transfer in a semi - in f ini te layer.

Equation (1.31) for the specific case K]^(x) = 'E-^i'i) was first derived

by G. Placzek in 1945 (see B. Davison, 1958, Chapter VI). The derivation

above was given by V. V. Sobolev (1958a). It is interesting that (1.30) can also be obtained directly from physical considerations (see the next section)

AUXILIARY EQUATION. H-FUNCTION. Turning now to the derivation of relations

that are used in the determination of $(x), we first consider the function

S(x,z), defined as the solution (bounded as x ») of the auxiliary equation
5.1 BASIC EQUATIONS 201

00 S(t,z) = A J

K^(|t-t' |)S(t' ,z)dT' — t/z

(1.32)

Let us compare this equation with (1.28). From (1.17) we see that the free

term in (1.28) is a superposition of the free terms in (1.32) for different

values of the parameter z. Because of the linearity of these equations, the

solution of (1.28) is then a superposition of the solutions of (1.32), namely

00 4/s(.,z,G(.)dz/z . (1.33,

0

Differentiating (1.32) with respect to t and using (1.24), we find

00 3S(T,z) __ X f K r|x-T'|) i^ill^ dT- ^

+ S(0,z) A K (t) - i e-^/" 2-1- z

From (1.34), (1.32), and (1.28), it follows that

From the last equation we have

S(t,z) = H(z)

(1.34)

^^^^>^^ = - i S(T,z) + S(0,z)$(t) . (1.35)

(e-"^/" . J

e-^"-^')/"$(T')dT') , (1.36)

where the notation

H(z) = S(0,z) (1.37)

has been introduced. Substituting (1.36) into (1.33), we obtain a Volterra

equation for 'I'(t) with a displacement kernel (convolution- type equation)

T $(t) = N(t) + y* N(t-t' )<I'(t' )dT' , (1.38)

where
202 SEMI- INFINITE MEDIUM: GENERAL THEORY

00 N(t) = ^ I

e H(z')G(z')dz'/z' . (1.39)

This result shows that the form of $(t) is determined entirely by the properties of H(z). We shall now obtain a nonlinear integral equation for the

latter function.

From (1.21) and (1.32) we have

00 S(T,z) = e""^/^ + y r(T,T')e~^'^^dT' , (1.40)

0

or, considering (1.20),

S(t,z) = / G(T,T')e ^'/^dx' . (1.41) /0

Thus the function S(t,z) is essentially the Laplace transform of the Green's

function with respect to one of the variables. For this reason S(t,z) will

play an important part in our analysis. Setting x = 0 in (1.40), we obtain

00 H(z) = 1 +y $(x' )e""^' /^dx' . (1.42)

0

By means of (1.38), we can express the integral appearing here in terms of

H(z), to obtain an equation for this function. To do this, both sides of

(1.38) must be multiplied by e""^/^ and integrated over x from 0 to 0°. After

minor manipulation, we obtain

oo H(z) = 1 + A zH(z) / iillll G(z')dz' . (1.43)

2 ^

z+z'

This is the desired equation for H(z), which occupies the central position in

the study of the transfer of line radiation in a semi - infini te medium. We

note also a useful relation, which will be used extensively later: /, H(z)H(Zn)

S(x,z)e~^/^0 dx = — zz„ . (1.44)

0 '^'0 Z + Zo 0

It is readily obtained from (1.36) if use is made of (1.42)
5.1 BASIC EQUATIONS 203

Expression (1.42) shows that H(l/s) — 1 is the Laplace transform o£

$(1). The determination of $(t) therefore involves the inversion of the

Laplace transform, that is, the evaluation of the integral

e^^ds

However

,

poss ib le

physical

for the evaluation

to express $(t) in cons iderat ions

of 0(t) we

terms of $. shall use an(t) and H(z)

approach which makes it

proceeding directly from

We note that

to being given by

the intensity of the

(1.12') , may also be

emergent radiation l(0,z)

written as in addition

00 /

1(0, z) = y S (t)S(t ,z)dT/z

0

(1.45)

In fact, from (1.12') and (1.21) it follows that

1(0, z)

00 00 /(s*(x.) S*(t') ./+ / S*(T)r(T' ,T)dTU Tie ^'/^dx'/z . (1.46)

0 V 0

Inverting the order of integration in the second term, we obtain

1(0, z)

00 y 00 — T r(T' ,T)dT' JdT/z (1.47)

Remembering the symmetry of the resolvent r(T,T'),

in brackets is just S(t,z), as follows from (1.40)

therefore identical to (1.45).

we find that the quantity

The last equation is

Let us summarize the main steps of this method for solving the basic

integral equation (1.16). In solving this equation we seek its resolvent

r(T,T'), which can be expressed as a function of one variable $(t) (1.31).

This function satisfies an integral equation (1.38) of the convolution type.

The solution of this equation involves the inversion of the Laplace transform

of the function H(l/s) - 1, where H(z) is the solution of the nonlinear integral equation (1.43). Thus the form of the resolvent r(T,T') is determined

essentially by the properties of H(z).

This method is directly applicable to the solution of equations having

the form (1.16) when the kernel function K]^_(t) can be represented as a superposition of exponentials. Equation (1.16) is treated with rather broad as- sumptions as to the form of Ki(t), by V. A. Pock (1944), K. M. Case (1957a),

and others. The most complete study of equations of this type is that of

M. G. Krein (1958)
204 SEMI- INFINITE MEDIUM: GENERAL THEORY

5.2 PROBABILISTIC INTERPRETATION

A simple probabilistic interpretation, proposed by V. V. Sobolev (1951,

1956), may be given to the linear problems of radiative transfer theory. It

does not in itself lead to any new methods for solving the problem; however,

it does enable us to see the problem from another point of view. Moreover,

this approach makes it possible to obtain many important relationships directly from probabilistic considerations, avoiding a long series of intermediate

trans formations.

SINGLE SCATTERING. Let us consider an excited atom located at a depth t. With probability X the atom will make a downward radiative transition, and a line photon will be emitted. Under the assumption of complete frequency

redistribution, the probability that this photon is emitted within an element

of solid angle doj in the dimensionless frequency interval from x to x + dx is

AAa(x)dx — . (2.1)

4Tr

If the photon is emitted in the direction corresponding to arccos y < 71/2,

then the probability that it will escape from the medium without any further

interaction with matter is

g-a(x) T/y

Thus

Aa(x)e-°'^^)^/^dc.dx

4tt (2.2)

is the probability that the excited atom, situated at depth t, will emit a photon v;ith a frequency from x to x + dx which will, without subsequent scatterings, escape from the medium at an angle arccos y to the normal within the

solid angle doj. Integrating (2.2) over all frequencies and angles of emergence, we get

I

A^ a(x)dx fe~"'^^^^/^dy = \

^ Ja(x) E2 (a(x) T)dx = |

K2(t) .

00 Q — 00 The function K2(t) is discussed in detail in Sec. 2.6. Thus the total probability that an excited atom located at optical depth t will emit a photon in

the line which will escape from the medium without scattering is equal to

(A/2)K2(t) .

In Sec. 2.6 it was shown that

K2(t) = y e ^'^^G(z)dz .

0

It is therefore clear that (A/2) G(z) is the probability that an excited atom

emits a photon w-ith such a frequency and in such a direction that the ratio

y/a(x) lies between z and z+dz. We note that the relation (1.11) thereby

gains an obvious probabilistic interpretation.
5.2 PROBABILISTIC INTERPRETATION 205

Now let us go on to consider two excited atoms, one of which is at depth

T, and the other at depth i+dx. The corresponding probabilities that photons will be emitted and escape directly from the medium are (X/2)K2(t) and

(X/2) K2 (T+dr) , respectively. It is obvious that the difference between them,

which, as follows from (2.6.5), is equal to ( A/ 2 ) K]^ (t ) di , gives the probability that a photon will be emitted and subsequently experience its first absorption in a plane layer of thickness di , located at a distance t from, the

emitting atom. From this discussion, the probabilistic meaning of the kernel

of the basic integral equation (1.16) is apparent.

MULTIPLE SCATTERING. Until now the discussion was confined to the probabilistic interpretation of quantities characterizing single scattering. Now let us turn to the problems of multiple scattering, restricting ourselves to a con- sideration of a semi - infinite medium. Let p (x ,y ,x) dwdx be the probability

that an excited atom at depth t will emit a photon that will escape through

the boundary (in general, after a number of scatterings) at an angle arccos y to the normal within the solid angle dto in the frequency interval from x to

x+dx. The probability that the photon will be emitted and interact with

matter for the first time somewhere in the layer between t' and x'+di', exciting an atom in that layer, is (X/2 ) K]^ ( | t-t '

| ) di ' . The probability that this

atom will in its turn emit a photon that will escape from the medium at an angle arccos y with frequency x is p (t ' , y , x) dcudx. Integrating the quantity

( X/2 ) K]^ ( I T— T '

I )p ( T ' , y , x) dojdx over all t', we obtain the probability that the

photon will escape from depth t after at least one scattering. Adding to it

the probability of direct escape (2.2), we should obtain p (x , y , x) dwdx . Thus we arrive at the following equation for p(x,y,x):

00 p(x,y,x) = AjK^(|x-x'|)p(T',y.x)dx' + Aa(x)^e-°'^''^^/^ . (2.3)

0

The function p(x,y,x) is called the escape probability of a photon . From

(2.3) it is seen that the function

S(T,z) = ill P^'^''-'>^^ (2.4)

X Aa(x)

does not depend on the quantities y and x themselves, but only on the combination z = y/a(x) , and satisfies the equation

S(x,z) = ^ K, (|x-x' ns(x' ,z)dx' + e-^/^ , (2.5)

which is the auxiliary equation (1.32) introduced in the preceding section.

A comparison of (2.4) and (2.1) shows that the escape probability of a photon with a given frequency and direction is changed by a factor of S(x,z)

through its interaction with the medium. This makes clear the physical signij ficance of the function S(x,z), which was introduced in Sec. 5.1 in a purely

I formal manner.

' From (1.37) it is apparent that the quantity ( X/2 ) G ( z) H ( z ) dz is the

probability that the appearance of an excited atom on the boundary of a semiinfinite medium will be followed by the escape of a photon for which y/a(x)
206 SEMI- INFINITE MEDIUM: GENERAL THEORY

lies between z and z+dz. Integrating this probability over all z, we get

the total probability that once an excited atom appears on the boundary of

the medium a photon will escape from it. This probability is equal to

(A/2)aQ, where

oo

= /

H(z)G(z)dz . (2.6)

•o The probability that the photon will be "trapped" in the medium, or, more accurately, that the energy of the original excitation will not be carried out

of the medium by radiation is then 1 — (A/2)ao. These comments will be of

use later.

Now let us consider the resolvent T C^i ,^2) . Considering the probabilistic interpretation of the function Ki{t) , we conclude from (1.19) that

r(T]^,T2)dT2 is proportional to the probability that the appearance of an excited atom at depth T]^ will sooner or later lead to the photo- excitation of an atom in the layer between 12 and T2+dT2. This is readily seen by expressing

r(T2,T2) as a Neumann series

00 r(T-^,T2) = ^n^l^l-^2l^ ' ^^-^^ n=l

whe re

00 ^n^'^l-^2l^ = I y K^(|t^-t' |)Vl^l^'-^2')'l^' ' ^2.8)

^^(It^-t^D = A K^(|T^-T2l) . (2.9)

It is clear that the n-th term of this series is the probability that the

excitation is transferred from depth ti to T2 with n—1 intermediate photoexcitations . The probabilistic interpretation of che resolvent given above makes it

easy to obtain the relation (1.30), which expresses the resolvent in terms

of the resolvent function $(t). Excitation migrates from depth t-^ to depth

T2 in the following manner. An excited atom at depth x^^ emits a photon.

During the subsequent random-walk process in the medium, it arrives at 12

where it excites an atom. The probability that the appearance of an excited

atom at depth will sooner or later entail the excitation of an atom in a layer of unit optical thickness lying at depth T2 will, for brevity, be called

Ti ->- T2 transition probability. We make the situation definite by assuming

that T2 > Ti. Let us divide all possible photon trajectories into two categories. The first contains those trajectories that never intersect the plane

T = T-j^, and the second, all the others (Fig. 23). Clearly the probability of

a T I ^ T 2 transition along all trajectories of the first type equals the

probability of the 0 ^ '^2"''^! transition, i.e. equals r(0, T2-T]^). Turning

to trajectories of the sec.ond type, let x = t be the depth closest to the

boundary x = 0 reached by a photon in its walks along a trajectory of the

second type (t < ti ) . It is obvious that the atom must be excited at this

depth, since the photon would otherwise continue to move in the direction of
5.2 PROBABILISTIC INTERPRETATION 207

TYPE I TYPE n

r = 0

Fig. 23. Derivation of the expression for r(T^, t^) in terms of 0(1)

decreasing x. The trajectory described by a photon exciting an atom anywhere

in the layer between t and t+dt does not cross the x = t level, and is otherwise arbitrary. The probability of the transition x ]^ x with photoexcitation

in a layer of thickness dt is therefore T (t i—t , 0)dt. Excitation from depth

t is transferred to depth X2, subject only to the limitation that the trajectory does not cross the plane x = t, with a transition probability r(0,

T2—t) . Thus the probability of the t-^ ^ x^ transition along those trajectories

of the second type for which the photo-excitation closest to the boundary

T = 0 occurs between t and t+dt, is

r(x-^-t,0)r(0,X2-t)dt

Integrating this expression over t from 0 to x^, and adding t'le contribution

from trajectories of the first type, we get the total probability of the

Hi X2 transition, equal to r(Ti,X2). In this way we arrive at the relation (x2 > :

r(x^,x2) = r(0,x2-T^) + y* r(x^-t, 0)1(0, x2-t)dt , (2.10)

0

which, in view of (1.23), is identical to (1.30).

It is also easy to show that

r(x^,x2) = r^(x^,x2) ~ y* <5CT^+t)$(x2+t)dt . (2.11)

0

This expression shows that in a semi - infinite medium, contrary to the situation in an infinite one, photon trajectories that intersect the plane x = 0 are "forbidden."

522-519 O - 74 - 15
208 SEMI INFINITE MEDIUM: GENERAL THEORY

Similar considerations also make possible probabilistic interpretations

of many other relations obtained in the preceding section.

5.3 THE green's FUNCTION FOR A HALF-SPACE

DERIVATION OF THE BASIC RELATION. In Chapter IV the radiation field arising

from a point source located in an infinite homogeneous medium was calculated,

i.e. the Green's function for the radiative transfer equation was obtained

for an unbounded region. Now we shall find the resolvent of the integral

equation (1.16), or equivalently , the Green's function for a half-space. Once we have the Green's function for an infinite medium, this can be done very

easily (V. V. Ivanov, 1964a). We shall show that the resolvent function 4>(t) can be expressed in terms of ^^(t) and the H- function. The resulting expression can be used to find the H-function explicitly (see Sec. 5.4) and hence

to completely solve the problem of determining the half-space Green's function.

The Green's function for a half-space is

G(t-^,T2) = ^(T^-T^) + r(T^,T2) , (3.1)

where r(T,,T-,) is the solution of the equation

r(T. 1 J

J^l^l"^!-^' :2)dT' . AK^(|T^-r2|) (3.2)

The most direct way of obtaining r(Ti,T2) is to solve this equation by iteration, yielding a representation of r(T-j^,T2) in the form of a Neumann series

(2.7) - (2.9). Physically, the solution of this equation by iteration cor- responds to the successive calculations of scatterings of different orders.

However, such a solution is of limited practical interest, since the series

converges slowly unless A is small, whereas the most interesting case is precisely the one in which X is close to unity. Here, clearly, some other approach is needed so that all scatterings can be considered at once, and

T^T-^fT^) can be obtained in closed form.

Such an approach might, for example, be based on probabilistic considerations of the kind used in the preceding section to derive equation (2.10).

Let us classify all trajectories leading to the ^ t2 transition in an infi

nite medium into two types. To the first type belong those trajectories that never intersect the plane t = 0, and to the second all that remain (it is

assumed that ti and t2 are non-negative). We calculate separately the probability of a Ti T2 transition in an infinite medium along the two types of

trajectories

Since trajectories of the first type do not cross the plane t = 0, and

are otherwise arbitrary, the probability of the transition along these trajectories is equal to the total probability of a ti -»• 12 transition in a semiinfinite medium, i.e. r(Ti,T2). Let us find the probability of this transi- | tion along trajectories of the second type. Photons describing such trajec- ' tories undergo at least one scattering in half-space t < 0, since otherwise,

having left the t > 0 region, they would not be able to return to depth

T2 > 0. Trajectories of the second type can therefore be classified according to the depth at which the first scattering occurs in the half-space
5.3 THE GREEN'S FUNCTION FOR A HALF- SPACE 209

T < 0. Let us obtain the T2 transition probability along those trajectories for which the first scattering in the region of negative t occurs in

the layer from -t to -t+dt, with t > 0. For a transition along such a trajectory, the photon should escape from depth out of the half-space t > 0,

continue without scattering into the half-space t < 0 to a depth —t , be absorbed between -t and -t+dt, and finally come from this depth to depth

T2 > 0 (generally, again after random walks in all the infinite space). It

is easy to see that the probability of this complex event is

2^ Jdxj dy'p(T^,y' ,x')e

" ' ^

' (-t , T2 ) a (x ' ) dt /y ' . (3.3)

or, if use is made of (2.4),

00 J

S(T^,z')G(z')e-^/^'r^(-t,T2)dt dz'/z' , (3.4)

0

where r^(T, x') is the Green's function for an infinite medium. The integration over z' in this expression allows for the escape of the photon from the

half-space t > 0 at any angle and with any frequency. By integrating this

expression over t from 0 to <», we allow for all trajectories of the second

type. Thus the total probability of the 12 transition along such trajectories is equal to

00 00

I y

S(T^,z')G(z')dz'y* e-^/^'r^(-t,T2)dt/z' . (3.5)

0 0

The Ti T2 transition in an infinite medium occurs along a trajectory

of either the first or the second type. Therefore the sum of r(Ti,T2) and

(3.5) should equal the total probability of the transition for an infinite

medium, i.e. r^(Ti,T2). Consequently,

00 00

^o.(^l>^2^ = r(T^,T2) " 7/ S(T^,z')G(z')dz' y* e-'^/"'r^(-t,T2)dt/z' . (3.6)

0 0

This expression provides an important relation between the Green's functions

for an infinite and a semi-infinite medium and allows the function $(t) to be

readily expressed in terms of ^^(t) and the H- function.

For this purpose, we set X]^ = 0 in (3.6) and recall that according to

(1.37) S(0,z) = H(z). We obtain

00 00 ^J^2^ = + Y /

r„(-t,X2)dt j e-^/^'H(z')G(z')dz'/z' . (3.7)
210 SEMI- INFINITE MEDIUM: GENERAL THEORY

Since it is obvious physically that in an infinite medium the probability of

the Ti T2 transition depends not on the quantities and t2 themselves,

but only on Iti-t2| Csee Sec. 3.6), we can assert that T^(-t, 12) = '^'„CT2 + t) . Therefore, replacing the independent variable t2 by t in°°(3.7), we find the

desired expression

$(t) = $^(T) -| y*$^(T+t)dt y e-^/^'H(z')G(z')dz'/z' . (3.8)

Since r(T2^,T2) is expressed in terms of <I>(t) , this result in essence expresses the Green's function for a half-space in terms of the Green's function

for an infinite medium and the H-function. Here we have obtained this relation on the basis of simple physical considerations, but it can, of course,

be obtained by purely formal means (see K. M. Case, 1957a, 1957b).

EXPLICIT EXPRESSION FOR THE RESOLVENT FUNCTION. Substituting into (3.8) the

explicit expression for $^(1), one obtains a useful integral representation

of 3>(t). Let us first consider the case in which the infinite medium Green's

function is given by (4.1.20) and, moreover, the function $^(t) exists (hence

e.g., the conservative Doppler case is excluded for a moment). In Sec. 4.1

it was shown that in this case, for t > 0,

$ (t)

00 /z R(z)G(z)dz/z , (3.9)

where R(z) is given by (4.1.19). By integrating over t and rearranging, we obtain

$(t)

00 00 . 0^0 ' G(z')dz' G(z)dz/z (3.10)

The quantity in brackets is 1/H(z), as follows from the basic equation (1.43)

for H(z). Therefore, finally.

$(t)

00

0

T/z R(zl G(z)dz/z

H(z)

(3.11)

In this case ^(t) differs from $^(t) only in the factor 1/H(z) in the

integrand. This similarity, of course, is not accidental; it arises from

the fact that all of the physics of the problem is already contained in

(|>^(t). The difference between '^'^oC^) and 3>(t) comes entirely from the difference in geometry, and the factor 1/H(z) allows for this difference.

An explicit expression is known for the H-function (see the next sec- tion). Therefore (3.11) explicitly expresses $(t) , and thereby r(Ti,T2)>

in terms of a(x) . The result (3.11) was first obtained by D. I. Nagirner

(1964a, 1964b) by another means.
5.3 THE GREEN'S FUNCTION FOR A HALF- SPACE 211

In Sec. 4.1 we showed that if the characteristic exponent y > 1/2, the

conservative function $^(t) does not exist. It also may not exist for

Y

= 1/2 (as in the case of the Doppler profile, for example). We now show,

however, that the resolvent function $(t) remains finite in th.e conservative

case for all y ^ 1 ^.nd is given by

oo

= o f 2 J

f(T) = / e ''/^R(z)G(z) , X = 1 , (3.12)

zH ( z )

2 where a is the second moment of the kernel function:

oo oo

= y T^K^(T)dT = 2 y z^G(z)dz . (3.13)

2 Usually, in line transfer problems a = «>, so that the first term on the

right in (3.12) is absent. In particular, this is always the case if the

characteristic exponent y is less than unity.

To prove (3.12) we use the explicit expression for the conservative

Green's function for an infinite homogeneous medium found in Sec. 4.1,

00 SpCT)=^^-^+ f

e-^/"R(z)G(z)dz/z2 I (3.14)

2 (if a = °°, the first term on the right automatically vanishes). We further

note that the function $'^(t), related to Sp(T) by (see Sec. 4.1)

$'^(T) = -2TTTSp(T) , (3.15)

remains finite in the conservative case for all y* Differentiating (3.8) we obtain ( A = 1) :

00 00

$' (t) = $'^(t) - - f

*'jT + t)dt y e~''^/^'H(z' )G(z' )dz' /z' . (3.16)

0 0

Now we substitute into (3.16) the explicit expression for 'J''oo(''^) which results

from (3.15) and (3.14), and make use of (1.43) and the fact that in the con- servative case

oc /

= / H(z)G(z)dz = 2 , X = 1 . (3.17)

0
212 SEMI- INFINITE MEDIUM: GENERAL THEORY

The result (3.17) follows from the probabilistic interpretation of ag men- tioned in the preceding section (cf . also the next section) . The resulting

expressionfor$'(T)is

$'(t)=-7 f e-''/^R(z)G(z) —^i— , (3.18)

^ / z^H(z)

whence

$(t) = <5(oo) + i / e ^/^R(z)G(z) --^ . (3.19)

2 J zH(z)

0

The proof of the fact that, in accordance with (3.12),

z^G(z)dz| , (3.20)

will be given in Sec. 5.5.

The expressions just found for $(t) are valid under the same conditions

for which Sp(T) is given by (4.1.20) and (4.1.21). In particular, these

expressions are valid if a(x) is an even monotonically decreasing continuous

function.

We also note that from (3.8) and (3.1.20) it is easy to obtain the explicit expression (3.7.10) for $(t) for monochromatic scattering, which was given in Sec, 3.7 without proof.

5.4 THE H-FUNCTION

INTRODUCTION. The function H(z) is the most important special function occur- ting in problems of radiative transfer in a semi - infinite medium. As we have

just seen, it enters into the explicit expression for the corresponding Green's

function. Moreover, as we shall see in the next chapter, -in many cases the

intensity of emergent radiation may be expressed directly in terms of H(z).

This function is also of importance in the study of scattering in an optically

thick layer (see Sec. 8.5).

The function H(z) is a generalization of the Ambartsumian function Hf^(y)

to the case of scattering with complete frequency redistribution and an ar^bitrary profile. The nonlinear integral equation which it satisfies,

H(z) = 1 + -zH(z) / liIlDG(z')dz' , (4.1)

2 J z + z 0

is a generalization of Ambartsumian' s equation
5.4 THE H- FUNCTION 213

0

(4.2)

As has been noted more than once, isotropic monochromatic scattering is a special case of scattering with complete frequency redistribution, corre- sponding to a rectangular profile, i.e.

1, |x| < 1 ,

0, lx| > 1 .

(4.3)

For z < 1 we then have z e y/a(x) = y, and the function Gf,j(z) is equal to

j unity for z < 1 and to 0 for z > 1, so that (4.1) in this particular case

ij reduces to (4.2). On the other hand, (4.1) is itself a special case of the

i more general equation obtained by V. V. Sobolev (1949, 1954) in a study of

I the scattering of line- frequency radiation with absorption in the continuum

taken into account (see Sec. 7.5). Problems of monochromatic scattering

I with non-spherical phase functions also lead to H- functions (S. Chandrasekhar

,

1950), which are defined by (4.1) with G(z) = 0 for z > 1. If the phase

function is a sum of a finite number of Legendre polynomials, then for z < 1

I the function G(z) is an even polynomial in z. The function (A/2)G(z) in this

case is known as Chandrasekhar ' s characteristic function. We may therefore

j regard (X/2)G(z) as a generalization of the characteristic function to prob-

" lems of scattering with frequency redistribution. For the physical significance of G(z) in these problems, see Sec. 5.2.

H-functions are the subject of a substantial literature in the theory of

ij monochromatic scattering (see, in particular, S. Chandrasekhar, 1950 , and

I. Busbridge, 1960). Some of the results obtained there are also applicable

1 to the case of line- frequency scattering. However, as frequency redistribution has a substantial effect on the behavior of the H-functions, there are

I fundamental differences between the results for monochromatic and line-frequency scattering.

j EXPLICIT EXPRESSIONS FOR H(z). A knowledge of the Green's function for an infinite medium makes it possible to obtain an explicit expression for H(z)

. by means of elementary manipulations, without appealing to the theory of

functions of a complex variable. Differentiating (1.28) with respect to X,

i we obtain

I a^T.X) __ A f K (|x-T'|) dT' ^ i nx,X) , (4.4)

I 3X2^ oX \

0

jj where we have explicitly shown the X-dependence of the resolvent function

I If the function $(t,X) on the right is regarded as known, this relation may

t; be considered as an integral equation for 9$/8X. According to (3.11), the

\ free term, of this equation, i.e. (1/X)$(t), is a superposition of functions

i of the form e~'^ / . Therefore the solution of (4.4) should be a superposition

1 of the solutions of (1.32); to be precise:

ii
214 SEMI- INFINITE MEDIUM: GENERAL THEORY

9X

00 ft^'l f

S(T.z.)Mlll G(z.,dzVz' . (4.5)

Multiplying both sides of this equation by e""""^/^, integrating over t from 0 to °° , and using (1.42) and (1.36), we find, after a simple rearrangement,

CO ^^""^^'^^ = i f R(z') ^iLia dz- . (4.6)

dX 2 J z + z'

^ 0

Taking (4.1.18) into account, we then have

X 5,nH(z) = J

A'^ dX'/A' . (4.7)

0 \^ /

where $ (s) is the one-sided Laplace transform of $ (t) :

CO

$ (s) = A / R(z') £Li_!l dz

2 y 1+sz'

It follows from (4.7) that

H(z,X) = H(z,l)exp

i - / ^oo(p^')d^'/^' (4.7')

It is easy to verify that this relation, as well as the representation

(4.7), is also valid for monochromatic scattering (whereas terms must be

added on the right sides of (4.5) and (4.6) to allow for the pole). Using

(3.1.20), we find that (4.7') gives us the following relation between the

Ambartsumian functions for X < 1 and for the conservative case:

i\i(y,A) = 1+ky exp

7 J y+y' J

R(y' ,X' )dX' (4.7")

From this relation the expansion (3.7.28) of H[vi(y,X) in powers of /I—X , given

in Sec. 3.7 without proof, can readily be obtained.

The integration over X in (4.6) can be performed directly. We define

X^ zG(z)

e(z) = arctg , , 0 < 0(z) < tt . (4.8)

i— A U Z J
5.4 THE H- FUNCTION 215

Direct calculation gives

9e(z)

3X I

zR(z)G(z)

From [4.6) we therefore find

H(z) = exp

CO e(z') dz'

z ' (z + z

' ) (4.9)

Keeping (4.1.23) in mind, for X < 1 , we can rewrite (4.6) as

9&nH(z , X) ^

3 A 2(1-

00 L__i

-A) 2 Jf

0

R(z')G(z') z'dz'

z + z'

' (4.6-)

from which we arrive at the following representation of H(z), wliich, unlike

(4.9), is not valid for X = 1:

H(z) = (1-X) ^ exp

OO e(z') dz'

z + z

(4.9')

From a comparison of (4.9) and (4.9'), incidentally, it follows that

exp

00 -1/ e (z) dz/z 1-X (4.10)

Similar integral representations can also be obtained for the Ambartsumian

function H^^da) . These representations differ from (4.9) and (4.9') in that

they have extra factors allowing for the contribution from the pole.

Another m.ethod of deriving the expression (4.7"), as well as other similar relations, and a detailed study of these results is given by T. W.

Mullikin (see, in particular, J. L. Carlstedt and T. W. Mullikin, 1966, which

gives a summ.ary of the results found in this way, and a useful list of references). The reasoning used here to obtain the explicit expression for H(z) seems to be the most elementary one as regards the mathematics. (Another

derivation of (4.9) has been given by R. F. Warming, 1970a.) In accordance

with the statement at the beginning of the chapter, knowledge of the Green's

function for an infinite medium suffices, without an appeal to the theory of

functions of complex variables, to find explicitly both the H-function and

the half-space Green's function. It is often believed that much more sophisticated mathematics is needed to solve the problem of multiple scattering in

a half-space. ; In addition to (4.9), a number of other integral representations are known for H(z). A form especially convenient for computing is
216 SEMI- INFINITE MEDIUM: GENERAL THEORY

H(z) exp

00

0

Jin [l-AV(u) ] du

1 + z^u2„2

(4.11)

where V(u) is given by (2.6.18) or (2.6.24). The derivation o£ (4.11) may be

found, for example, in the book by S. Chandrasekhar (1950). The relation

between the various representations of H(z) is discussed by D. I. Nagirner

(1968) . MOMENTS OF H(z). Let us consider the general features of the behavior of

H(z). From the basic equation for H(z), rewritten in the form

H(z)

oo

1 _ A-z f

^ilill G(z')dz'

2 J z+z'

^ 0

- 1 (4.12)

it is clear that H(z) is a monotonical ly increasing function of z, with

H(0) = 1. From the equation directly following (4.7) and from (4.1.23) we have

2(1-A)

Setting z = 00 in (4.7), we therefore find that

(4.13)

H(oo) = (1-A)- (4.14)

In the conservative case (X = 1) the H-function increases indefinitely as

z -> oo. The nature of the divergence of H(z) for z oo in the conservative

case, as well as the rate at which H(z) for X < 1 approaches its limiting

value H(oo) = (i—X)"'^^ is determined by the behavior of the absorption coefficient in the line wings. However, we shall delay the detailed discussion of

these important questions for the moment to consider some useful integral

relations

Let be the i-th moment of the H-function with weight G(z)

00

0

z^H(z)G(z)dz (4.15)

We emphasize that ai , from some value of i on, may diverge. Thus, if the

second moment of the kernel diverges, as is usually the case in line transfer

problems, then in the conservative case ai and all higher moments do not

exis t

Going to the limit z ^ oo in (4.1), we get

H(oo) = 1+1 H(oo)a^

It follows, on using (4.14), that
5 . 4 THE H- FUNCTION 217

aQ = i (1 - /1-A) . (4.16)

This relation is found to be most useful. For the physical significance of

ttg , see Se c , 5.2.

In the conservative case one can also find ai explicitly (if it exists),

Multiplying (4.1) by z2g(z) and integrating over z, we get

00 / 00 \ 00 y (h(z) -

J

zH(z) y ^^+^^ G(z')dz'J z^G(z)dz = z^G(z)dz .

0 \ 0 / 0

Using the identity

z = I _ z

z+z ' z + z

and the fact that in the conservative case, according to (4.16),

y* H(z)G(z)dz = 2 , X = 1 , 0

we get

00 oo oo

I J

H(z)z^G(z)dz y

^ '^j^'

G(z')dz' = y z^G(z)dz .

0 0 0

Applying the above identity once again, we find from this equation

00 00 ^

00 1 ct^ _ 1 / H(z)zG(z)dz / iUiiili G(z')dz' = / z^G(z)dz

2 ^ 2 J J z + z' J

0 0 0

Adding the last two equations, we obtain

00 i = 2 y z2G(z)dz ,

from which we finally obtain

= 2 f y z^G(z)dz| , X = 1 , (4.17)
218 SEMI- INFINITE MEDIUM: GENERAL THEORY

or

a-^ = /2 a , (4.17')

2 where a is the second moment of the kernel function K-^{t) . ALTERNATIVE INTEGRAL EQUATIONS FOR H(z). The equation for HCz) can be rear- ranged. Clearly we can rewrite (4.1) in the form

oo H(z) = 1-^1 H(z)ao -

I

H(z) f ^^^j^f^ G(z')dz' , 0

from which, on using (4.16), we finally obtain

H(z) ( /r=I + - /

^ G(z')dz' 1=1. (4.18)

\ 2 J z+z' / \ 0 /

This equation assumes an especially simple form in the conservative case:

00

J

H(z) y ^'^^J^V

'' G(z')dz' = 1 , X = 1 . (4.18')

0

It can be shown that H(z) also satisfies the linear equation

00 H(z) [l-XU(z) ] = 1 --Z / iilili G(z')dz' , (4.19)

2 J z-z'

0

which is a generalization of (3.7.20). This equation is valid for any complex

z, except those on the negative real semi-axis. When z is a positive real

number, the integral on the right and U(z) are to be understood as Cauchy

principal values.

BEHAVIOR OF H(z) FOR SMALL z. It can be shown that for z > 0

JlnH(z) = -y a^z^nz + |

^a*^ - a^ z +

(4.20)

whe re

^2a2 - X ^a^^z^ + 0(z^£nz)

OO a*^ = y* (H(z)-l)G(z)dz/z , (4.21)
5.4 THE H- FUNCTION 219

and the constants a^ and a are defined by (2.6.28) and (2.6.29), respectively.

From (4.20) it follows that the derivative of H(z) diverges logarithmically

as z ^ 0. The proof of (4.20) is omitted.

We note, again without proof, that

X

^ll = J f

P*^(X')dX' , (4.22)

0

where p_]_ is given by (4.7.6) or (4.7.7). Using these expressions for p—i, we

find for a^-^ the representations

00

^ll " y Kn[l-AV(u)] + XV(u)| du , (4.22')

OO

BEHAVIOR OF H(z) FOR LARGE z (CONSERVATIVE CASE). The asymptotic form of

H(z) in the conservative case is easily obtained directly from (4.18') (V. V.

Ivanov, 1968). Let us first assume that the characteristic exponent is less

than unity: 0 < y < !• Then the second moment of the kernel function K2(t)

is necessarily infinite (see Sec. 2.6). It can be shown that for this reason

the main contribution to the integral in (4.18') for z ^ » comes from the

values of the integrand for z' -> «>. Multiplying and dividing the integrand

by H(z)G(z), we have

oo -H^(z)G(z) /

H(z' )G(z' ) ^ ^ ^ (4.23)

2 J z+z' H(z)G(z)

0

We substitute z ' = zy and assume the existence of the (as yet unknown) limit

£ (y) = ^(.^y)G(^y') (4.24)

^ z-co H(z)G(z)

Then from (4.23) we find that as z

°°

H(z) ~ C^(zG(z)^ '

,

• (4.25)

where
220 SEMI- INFINITE MEDIUM: GENERAL THEORY

Substituting (4.25) into (4.24) and using (2.6.42), we discover that

fH(y)=i75) (4.27)

whe re

f(y) = lim ^'

[""/l^ , (4.28)

Z^ oo ^ I ^ J

and x(z) is a function such that a(x(z)) = 1/z, x(z) > 0. As in Sec. 2.6, we assume that

f(y) = y^^ , (4.29)

with 0 < Y < !• Specifically, for the Doppler profile, y = 1/2, and for a profile decreasing proportionally to Ix|"~'^ in the wings,

Y

= V- . (4.30)

2<

Substituting (4.27) into (4.26), and taking (4.29) into account, we

find

= sin^Y^ . (4.31)

The expression (4.25) with given by (4.31) is the desired asymptotic form

of H(z) as z ^ «>,

As we have seen in Sec. 2.6 (2.6.69), the leading term of the asymptotic

expansion of G(z) for y < 1 can be written as

nr 2 . (i)(l/z) G(z) ~ — smTTY —- . ^ z2Y+l

Substituting it into (4.25), we get

H(z) ~ ((})(l/z)^ ' z^, A = 1 , z ^ «> . (4.32)

This expression is also readily obtained from the integral representation

(4.11) of the H- function. For large z the main contribution to the integral

in (4.11) comes from small values of u, and hence the asymptotic form

1 — (t)(u)u2Y can be substituted for V(u) . Then u = t/z and using the fact

that <^(u) is a slowly varying function, we arrive at (4.32). Equation (4.32)

holds for all y» 0 < Y - 1» which is an advantage of this representation over

(4.25). (Equation (4.25) is valid only for y < !•)

For the most important profiles, we have

Milne: Hj^(z) = s'^'z + 0(1) , (4.33a)
5. 4 THE H- FUNCTION 221

Doppler: Hj.(z)

Voigt: H,,(z)

Lorentz : H, (z)

Error estimates in these equations do not follow from (4.32). For the Milne

rectangular profile the estimate is a consequence of (3.7.26); for the Doppler

profile it will be derived in Sec. 5.6, and for the Voigt and Lorentz profiles, in Sec. 5.7. The results (4. 33b) - (4. 33d) were first found (without error estimates) by the author (V. V. Ivanov, 1962a). An asymptotic expression for H(z) as z 0°, essentially the same as (4.25), was recently found by

Yu. Yu. Abramov, A. M. Dykhne , and A. P. Napartovich (1967b) by more subtle

arguments than those used here.

The asymptotic behavior of the conservative H- function for z -> <» was recently discussed also by R. F, Warming (1970b), but his results are less

complete than those given here.

BEHAVIOR OF H(z) FOR LARGE z (NON- CONSERVATIVE CASE). The behavior of the

H-function for large z in the non- conservative case is more complicated. The

simplest way of studying this situation is to start with a representation of

the H-function in the form of (4.11).

When z is large, the main contribution to the integral on the right side

of (4.11) comes from values of the integrand for u close to zero. Therefore

in (4.11),V(u) can be replaced by its asymptotic form 1 — ())(u)u2y (see Sec.

2.6). As a result we have

2.-1/4,1/2^,^,31/4 , o(zl/^£nz)-3/4) ^

( y^'zl/4 , o(z-^/4^nz) , V2TTaU(a,0)/

(4. 33b)

(4. 33c)

(4. 33d)

£nH(z)

00

Substituting zu = y and using the fact that (})(u) is a slowly varying function,

so that 4)(y/z) ~ (t)(l/z), z ->- «>, we find that with the same accuracy

00 £nH(z) ~ -7 y

iln^l-A + X(J)(l/z)y^^)-^ ,

or

H(z) ~ (1-X) ^h(q) , (4. 34)

whe re

00 £nh(q)=-i f inll^qy^A-^ (4.35)
222 SEMI- INFINITE MEDIUM: GENERAL THEORY

and

q = Xz~^Y(^(l/z) (1-A)~^ . (4.36)

Thus in the asymptotic region (z >> 1), the function (1—X)^H(z), depending on two variables — z and X — becomes a function of the single variable q, which is simply related to z and X. This is the main simplification. The

asymptotic representation of (4.34) is due to V. V. Ivanov and D. I. Nagirner

(1965). The function h(q) is essentially the asymptotic form of (1—X)^H(z)

for the case in which the limits z

o° and X ^ 1 are taken in such a way that

the quantity q, defined by (4.36), remains constant.

As for the function h(q), a knowledge of its values for 0 < q < 1 is

sufficient, since the following relation holds:

h^i^ = q''^h(q) . (4.37)

This result is a direct consequence of (4.35).

, From (4.32) and (4.36) it is easily seen that for 1 — X << 1 the quantity

q can be represented as g;,„

t n y T e rl t n' (1-X)h2(z,1)

(4.38)

where H(z,l) is the conservative H- function. With this in mind and considering (4.37), we find from (4.34) that

H(z,X) ~ H(z,l)h^i^ , l^X << 1 . (4.39)

In contrast to (4.34), this equation is valid for all z > 0, and not only for

z >> 1. The accuracy of (4.39) increases as X tends to unity.

QUALITATIVE BEHAVIOR OF H-FUNCTIONS. To better visualize the behavior of Hfunctions, in Fig. 24 we plot £gH(z) versus Jlgz for the Doppler profile using

the values of Hj)(z) given by V. V. Ivanov and D. I. Nagirner (1965). The

parameter of the curves is the value of 1 — X. Although these curves refer

to the Doppler profile, they reflect the general features of H-functions for

other profiles as well. ' 0" When X is not too close to unity, the amplitude of the variation of H(z)

is not large, and it approaches its asymptotic value H(°°) = (1— X)~^ at rather

small z. As X increases, so do the values of H(z). However, H(z) increases

non-uni formly . As X becomes close to unity, a region of z-values develops in

which H(z) becomes practically independent of X and can be approximated by

the conservative H- function. As 1— X decreases, the size of this conservative

region increases. However, no matter how closely X approaches unity, the curves of H(z) for X < 1 and for X = 1 eventually diverge, the first approaching the asymptote H(«') = (1— X)"~^, and the second increasing to infinity. Let

Zg be the value of z for which q = 1. According to (4.38), the value of Zs for 1 — X << 1 is the root of the eaua.tioji

c
5.5 THE RESOLVENT FUNCTION AND ASSOCIATED QUANTITIES 223

i Fig. 24. The H-functions for the Doppler profile,

i; H(z^,l) = (l-X)-'' .

The quantity Zs is the abscissa o£ the point at which the curve of H(z,l)

ii intersects the horizontal asymptote of H(z,A). Substituting H(z,l) from

1 (4.32) into the last equation, we find that asymptotically as A -> 1

z^ - Qd-X)-^^^^ , 1-A << 1 ,

where

is a slowly varying (in particular, constant) factor, usually close to unity.

For z << Zs one can assume that H(z,X) •= H(z,l) approximately. In the

part of this region for which z >> 1, the asymptotic form (4.32) can be used.

If the opposite inequality holds, i.e., z >> Zg , we have H(z,X) ~ (1—X)~'^.

In the intermediate region, where z is of the order of Zg , values of the Hfunction can be found from (4.34) or (4.39). Therefore, to obtain the Hfunction for all X's which are close enough to unity, one has to tabulate,

first, the conservative H-function for not too large z and, second, the function h(q) for 0 < q < 1.

5.5 THE RESOLVENT FUNCTION AND ASSOCIATED QUANTITIES

Having studied the properties of the H-functions, we can now examine the

behavior of the fundamental function $(t) , in terms of which the resolvent of

the basic integral equation for the line source function is expressed. As we have already laentioned in Sec. 5.1, we shall call $(t) the res olvent function .

522-519 O - 74 - 16
224 SEMI- INFINITE MEDIUM: GENERAL THEORY

BEHAVIOR FOR SMALL t. From the integral equation for $(t),

CX3

) = y

y* K^(|t-t' |)$CT')dT' + A k^(t) , (5.1)

it follows that

lim

T->0

00 Is(T')$(T')dT' (5.2)

But

00

(t' ) $(t' )dT'

0 0

oo oo J

G(z)dz/z j/ $(T')e ^'^^dT' (5.3)

With the aid of (1.42), we obtain

00 00 (T')$(T')dT' = J

^H(z)-ljG(z)dz/z = a_-^

0 0

(5.4)

We now find from (5.2), using the expansion of (x) for small t (see Sec,

2.6),

X \ ( * * ~\

$(t) = a, Uni + — (a a,Y —a I 2 1 2 \ -1 1 /

(5.5)

By means of essentially simple, but rather cumbersome, rearrangements of the

explicit expressions for $(t) given by (3.11) and (3.12), additional terms of

the expansion can also be obtained. It turns out that

X X / * * .-

+ V a^T (Unr)^ + ^ a^ ^a-^ (y *-l ) +a-a*^^ xJlnT + 0(t) , t 0 . (5.6)

4 1\ 1

In the last two equations y is Euler's constant.

BEHAVIOR FOR LARGE t. In the opposite limiting case, i.e. for large t, the

resolvent function can also be simplified. Let x(z) be defined by a(x(z)) = 1/z. If

lim , , . J-X30 ^ I T J
5.5 THE RESOLVENT FUNCTION AND ASSOCIATED QUANTITIES 225

and 0 < y < 1 , then for t >> 1

^ A

2Ar(y.l) xllll F(t)

2 2Y.1 T (i_x)3/2

where

00

'^'^ ~ r(2Y-l) J { 2y 2 4Y(h/ Y^^ \

"

Here the function h(q) is given by the expression (4.35), and t is related to

T and X in the following manner:

^ = (1-^) , ,

• (5.9)

XttAx' (t)

Expressions (5.7)-(5.8) describe the asymptotic behavior of $(t) as t ^ and X ^ 1 simultaneously in such a way that t = const. The result (5.7) is

obtained from the explicit expression for $(t). Its derivation does not differ from that used in Sec. 4.3 to study the behavior of the Green's function

Sp(T) for large t. Although the representation (5.7) is by no means simple,

it is at least much less complicated than the exact expression (3. 11) . According to (5.7), in the asymptotic region (x >> 1) the function /I—X t$(t)

does not depend on t and X separately, but only on their combination t. This

greatly simplifies matters.

MODIFIED FORM OF THE RESOLVENT FUNCTION AND ITS ASYMPTOTICS. APPROXIMATE

FORMS. Equation (5.7) is a natural point of departure in obtaining exact,

asymptotic and approximate expressions for the revolvent function in more con- venient forms.

Let

<f(T) = ^ ^— F(t)UT,X) ' ^^''^^^

2 (l-X)^/2

where F(t) is given by (5.8), and the quantity t by definition is related to

T and X by

t = ir(2Y) . (5.11)

IT XK2(t)

In (5.10), unlike (5.7), the equality is not asymptotic but exact. When t is

large enough the values of t given by (5.9) and (5.11) are asymptotically

equal, so that (5.10) reduces to (5.7), and ?(t,X) 1 as t ^ The factor

?(t,X) thus allows for the deviation from the asymptotic form when x is not

too large. When X is close to unity, we have ?(x,X) ~ ?(x,l) = C(t), so that

X K (x)

2 (l-X)3/2

Hence for nearly conservative scattering we have a substantial simplification:

the function which depends on x and X, can be expressed asymptotically as

X -> 1 in terms of functions of one variable for all x.
226 SEMI- INFINITE MEDIUM: GENERAL THEORY

In order to obtain an expression for the resolvent function in the case

of conservative scattering, we must let X tend to unity in (5.10). According

to (5.11)^ in this limit we have t 0 . Equations (4.37) and (4.35) show that

h(q) ~ q— ^ for q

-> °°. Keeping this in mind, it is easy to find from (5.8)

that as t -> 0

F(t) IT (s imry)

2 Yr(Y)r(2Y)

.3/2 (5.13)

From (5.10) and (5.13) we have

r

K (t)

^'Ct) = - —i—- ax) , X

2 [K,(t)]3/2

1 , (5.14)

whe re

C =

1 Yr(Y) \tt

r (2y) siniTY (5.15)

As Y increases, C decreases monotonically from C=latY=OtoC=Oat

Y

= 1. In the region of the y values of primary importance for us, that is

0 < Y ^ 1/2, the coefficient C is close to unity (for y = 1/2 we have

C = 23/2Tr~l = 0.900). The more slowly the absorption coefficient decreases

in the line wings, the closer C is to unity.

The representation of the conservative function $(t) in the form of

(5.14) was suggested by V. V. Ivanov (1965). It is convenient because in cases of practical importance the function 5(t) is close to unity for all t From a comparison of (5.14) and (5.6), we find

S(0) = 1/C . (5.16)

On the other hand,

C(t) (5.17)

If all that is needed is an estimate of the resolvent function for X = 1, the

differences of 5(t) and C from unity can be ignored. As a result we obtain

a very simple approximation (subscript a stands for "approximate")

1 (t)

^ 2 [K,(T)]3/2

(5.18)

valid for all t and for an arbitrary profile with y < !• As a rule, the

approximation improves as y decreases.

From (5.14) and (5.17) we find that

$(t)

K]^(t)

2 [K2(t)]3/2

, T °° , X = 1 (5.19)
5.5 THE RESOLVENT FUNCTION AND ASSOCIATED QUANTITIES 227

Equation (5.19) gives the leading term of the asymptotic expansion of

the conservative function ^C^) for y < 1 , If y = 1, one can deduce from

(5.19) only that <I>(t) = o (K^ (t) /K2^/ ^ (t) ) as t ^ °°. However, the asymptotic

expression for $(t) may be rewritten in a form that is valid for any value

of the characteristic exponent y, 0 < Y ^ !• Let us proceed from the equation

(see Sec. 5.1)

00 H(z) = 1 + y e~^/^$(T)dT .

0

Substituting the asymptotic form (4.32) of the conservative H- function from

the preceding section, we obtain

00 ($(l/z))~^z^ ~ y e~^/^$(T)dT , z ^ 0°

Since for large z the main contribution to the integral on the right comes from the region of large t values, one can obtain the asymptotic form of $(t).

It turns out that

^(t) 1 -1- , T -> 00

, X = 1 . (5 . 20 ) r(Y)(c})(l/T))-^ t1-y

In particular, if y = 1 and the second moment of the kernel exists, the function (J) degenerates into the constant (see Sec. 2.6)

00 00

= J y* T^K^(T)dT = y Z^G(z)dZ ,

and (5.20) gives

'(~) =

( f

z^G(z)dz| , A = 1 , Y

= 1 . (5.21)

.0

This result was already given (without proof) at the end of Sec. 5.3.

In particular, for the most commonly considered profiles (5.20) gives:

.M. ^ /T . ^/ e Milne: ^^\t) = /3 + o( —? -) , (5.22)

\T(£nT)2/

Doppler: ,\r) - 2." ^^^'^/^ (.nx) , 0 (.'^ ^ ^Znr)-', (5.23)
228 SEMI- INFINITE MEDIUM: GENERAL THEORY

Voigt: ^'^(t) = T + Ol^^\ , (5.24)

(2TTaU(a,0))^rCl/4) '(^)

Lorentz: <|)^(t) = t + o(^^) . [5.25)

2^r(l/4)

For the error estimates in (5.22), (5.23), and (5 . 24) - (5 . 25) , see Se<:. 3.7,

5.6, and 5.7, respectively. As the asymptot"" form (5.24) is valid only when

T >> a""-*- , the region o£ applicability increases with a. The results (5.23)-

(5.25) and their analogs for a one - dimens i onal semi- infinite medium in which

scattering occurs only strictly forward or backward were found by the author

(V. V. Ivanov, 1960, 1962a, 1962b). Let us now turn to hearly conservative

Let us now turn to nearly conservative scattering. We assume that

Y < 1.. As in the case of an infinite medium, when X is close to unity, three

zones exist. For those values of t for which the values of t calculated from

(5.11) are much less than unity, it may be assumed that $(t,A) ~ $(t,1). This

is the zone of nearly conservative scattering. In the asymptotic (t >> 1)

part of this zone, the asymptotic form (5.19) may be used. As t increases

further t becomes of the order of unity. Here the resolvent function must be

calculated according to (5.7), with F(t) found by numerical integration (transition zone). Finally, when t is so large that t >> 1, we are in the zone of

strong absorption, and the situation is quite simple. Since F(t) -> 1 as

t ^ °°, we find from (5.10)

*(t) ~ ^ i—— , t >> 1 . (5.26)

2 (l-A)3/2

For A 7^ 1 the zone of strong absorption inevitably exists, whereas the zone

of conservative scattering exists only for 1 — A << 1, and becomes smaller as

A decreases further from unity.

Comparison of (5.18) and (5.26) suggests the following simple approximation for the resolvent function $(t) :

<5a(T) = I

K^(t)(i-A + AK2(t)^ . (5.27)

V/hen 1 — A is sufficiently small, this approximation, for all t, must give

$(t) to within a factor of the order of unity. The properties of this approximation are similar to those of the longest flight approximation for the infinite medium Green's function given in Sec. 4.5. The simple probabilistic

meaning of all the quantities appearing in (5.27) suggests that this approximation is amenable to direct physical interpretation.

INTEGRAL RELATIONS. In addition to (5.4), the function $(t) satisfies three

other integral relations, namely:

00 /

2 K2(T)$(T)dT = y(l - /1-A) - 1 , (5.28)

y $(T)dT = (l-A)"'^ - 1 , (5.29)

0
5.5 THE RESOLVENT FUNCTION AND ASSOCIATED QUANTITIES 229

0

00 f <D^(T)dT = ^^p*^ - a*^^ , (5.30)

where is given by (4.7.6) or (4.7.7). The proof o£ the first of these

is similar to that of (5.4) and the second follows from (1.42) for z = ». The third is derived from (2.11) by setting T2 = 0 and then letting t-^ tend

to zero:

oo /

)^(T)dT = lim[$^(T) - $(t)] . (5.31)

Substituting on the right side the expansions of $ (x) and <I>(t) for t << 1 given by (4.7.5) and (5.6), we arrive at (5.30).

FUNCTIONS RELATED TO ^(t). In many standard problems the following function

is encountered:

T(t) = 1 + J <l>(T')dT' . (5.32)

0

Except for a constant factor, it is equal to the source function in a medium

with uniformly distributed primary sources (see Sec. 6.1). From (5.6) and

(5.32) we find that for small t

= 1 -

1

a^TJlni + J

^a*^ + a-|^(l-Y*)-ay

(* ~ ^^1 * 1 2 2

a-j^Y +a ^ ^—ij'^

^'^'^

?(t) = 1 - ^ a^TJlnT + ^ ( a*T+a-, (1-Y )-a)T +

(5. 33)

A 22^rt ^2 X I + — a^T (ilnx) + — a,la,Y "^a—

16 1 e

'

*

'

By representing H'(t) in the form

oc f(T) = (1-X) ' - / $(T')dT' , (5.34)

T

which follows from (5.32) and (5.29), using (5.26) we obtain the asymptotic

form of this function for x -> <» (which is valid for y < 1)

(5. 35)

This expansion is applicable in the strong absorption zone, i.e. t >> 1. When 1-X << 1, a transition zone (t ~ 1) and a zone of nearly conservative

scattering (t << 1) also exist.
2 30 SEMI- INFINITE MEDIUM: GENERAL TFIEORY

It can be shown that for small 1 — X and all t > 0, the function T(t)

has an asymptotic representation

(5 . 36)

Here t is given by (5,11), the function F(t) increases monotonically from zero

at t = 0 to unity at t = and for small t behaves as

F(t) /s imry

Yr(Y)

(5.37)

Finally C(t) is a correction factor of the order of unity, with the asymptotic value ^(t) ^ 1 for x -> °°. We shall not give the explicit expression for

F(t} because of its complexity. The simplification appearing in (5.36) is a familiar one: a quantity depending on two variables is expressed, in the

limit of small 1 — A, in terms of functions of one argument.

Substituting (5.27) into (5.32), we obtain an approximation

>i'^(T) = (1 - X + XK^d))" (5. 38)

This approximation gives the exact value of 'F(O) and properly describes the

leading term of (t) as t ^ 0 and possesses the rigorous asymptotic form

(5.35) as 1 ^ °= . Moreover, for values of t >> 1 'that satisfy the inequality

K2(t) >> 1 — X, it gives the correct functional form for the leading term of

the T-dependence of T(t), with a numerical coefficient which differs, for an arbitrary profile with infinitely extended wings, from its asymptotically

exact value by, at worst, 10 percent. With all this in mind, it is not sur- prising that in the nearly conservative case the approximation (5.38) is quite

accurate for all t.

The function H'(t) for conservative scattering is of special importance,

It gives the solution of the corresponding Milne problem (see Sec. 6.1).

Special notation will be used for this function:

s(T) = nx) , X = 1 (5.39)

If 1 — X << 1, in the conservative scattering zone, i.e

have ^(t) ~ s(t) . Substituting (5.19) into (5.32), we get

for t << 1, we

S(t) (5.40)

with C given by (5.15). This asymptotic form fails for y = 1- However, if

(2.6.68) is substituted for K2(t) in (5.40), we obtain the expression which

is valid for an arbitrary value of the characteristic exponent y , 0 < y < 1:
5.6 THE CASE OF THE DOPPLER PROFILE 231

S(T) ~ (<^(l/T)) '-^T^, !-><». (5.41)

In particular, for the profiles of immediate physical interest we have:

Milne: S^^) = /3t + 0(1) , (5.42a)

Doppler: S°(t) = 4tt~^/^t^^^ (£nT) ^^"^ + 0{t^ ^

^ {Inj)'^ ^ ^) , (5.42b)

Voigt: S^T) = i^^L- T^/^ . oIJ^ , (5.42c)

(2uaU(a,0))l/4r(l/4) V^'^^/

Lorentz: S (x) = x . 0^—j . (5.42d)

Error estimates in these expressions do not follow from (5.41). They are derived in Sec. 3.8, 5.6, and 5.7.

Letting X tend to unity in (5.36) and using (5.37), we obtain S(t) in a modified form:

S(t) = ^ at) . (5.43)

/K2(t)

Here 1(t) is a weakly changing correction factor to the asymptotic form (5.40),

which is usually close to unity for all x. Setting gii) and C equal to

unity, we get the approximate solution of the Milne problem:

S,(t) =

^

. (5.44)

^ v/K2(t)

This result also follows from (5.38) for X = 1.

5.6 THE CASE OF THE DOPPLER PROFILE

In this section a number of the results obtained above will be discussed

in more detail for the special case of the Doppler profile. Numerical data

will be presented.

CONSERVATIVE SCATTERING. We shall start by considering conservative scattering (X = 1). In Sec. 5.4, we gave in (4.33b) the leading term in the asymptotic form of H£)(z) for z >> 1. From (4.19) one can obtain a recurrence re- lation for the coefficients of an asymptotic series whose leading term is

given by (4.33b). The calculations, however, are very cumbersome and are not

reproduced here. It is found that

°° h

H„(z) ~ 2TT~'^z^(£nz)'^ V ^ , z - - , (6.1)

where the constants h, are defined by the recurrence relations
232 SEMI-INFINITE MEDIUM: GENERAL THEORY

with h

0

k-1

4k J j=0

k-i

1=0 m= j +Jl

, k = 1,2,

1. Here the coefficients g are given by (2.7.4),

i

(6.2)

f.. = 4-. B.,,

^ 2i~2(i+2):

(6.3)

where the Bj^ are the Bernoulli numbers, and the numbers are defined according to (2.7.20). From (6.2) we find, in particular,

h^ = 0.125000 , = -1^JL12. =-0.84138

1 2 128

The constants hi and h.2 were obtained earlier by different methods by

D. I. Nagirner (unpublished) and the author (V. V. Ivanov, 1968). These

methods, however, do not enable one to obtain the general recurrence relation

(6.2). For the proof of (6.2), see V. V. Ivanov (1970a). As we have already

mentioned, the derivation of (6.2) proceeds from the linear integral equation

(4.19) for H(z). An attempt to find the coefficients hk directly from the

explicit integral representations of H(z) turns out to be much more cumbersome, and only the first few of the constants hj^ can be found in this way.

Values of the conservative function Hd(z) are given in Table 23. They were obtained from (4.11) by numerical integration (V. V. Ivanov and D. I. Nagirner, 1965). When z is large, values of H£)(z) can be found by the expansion (6.1). For z = 100, the first term alone gives an accuracy of about 1 percent; if three terms of the expansion are retained, the error is reduced

to 0.05 percent.

TABLE 23

THE FUNCTION H (z) FOR THE CONSERVATIVE CASE

z (z) z (z) z (z) z (z)

0 .00 1 . 0000 8.0 5 . 3180 90 20 .556 1000 76 . 897

0 . 05 1 . 0887 8.5 5 . 4970 95 21 . 185 1100 80 .950

0 . 1 1

1566 9 . 0 5 . 6718 100 21 . 798 1200 84 . 832

0 . 2 12743 9 . 5 5 . 8427 110 22 .985 1300 88 . 563

0 . 3 1 . 379 1 10 6 . 0100 120 24 . 123 1400 92 . 161

0 . 4 1 . 4759 11 6 . 3349 130 25 .218 1500 95 .639

0 . 5 1 . 5671 12 6 . 6478 140 26 .575 1600 99 . 009

0.6 1 . 6538 13 6 . 9502 150 27 .298 1700 102 . 28

0 . 7 1 . 7369 14 7 . 2431 160 28 . 290 1800 105 . 46

0.8 1 . 8168 15 7 . 5273 170 29 . 254 1900 108 . 56

0.9 1

8941 16 7 . 8038 180 30 . 192 2000 111 .58

(Continued)
5.6 THE CASE OF THE DOPPLER PROFILE

TABLE 23 (Continued)

233

z «D

Cz) z z »D

z

1

0 1 . 969 1 17 8.0730 190 31 . 105 2200 117.42

12 2 . 1129 18 8 .3356 200 31 . 997 2400 123.01

1

4 2 . 249 1 19 8.5921 220 33 . 721 2600 128.38

16 2 . 3809 20 8 . 8429 240 35 374 2800 133.56

18 2 . 5070 22 9 . 3289 260 36 . 963 3000 138.57

2. 0 2 . 6288 24 9 . 7963 280 38 . 497 3200 143.42

2 . 2 2 . 7466 26 10 .247 300 39 . 980 3400 148.12

2 . 4 2 . 8610 28 10.683 320 41 . 418 3600 • 152.70

2 . 6 2 . 9 722 30 11.106 340 42 . 814 3800 157.16

2 . 8 3 . 0805 32 11.516 360 44 . 173 4000 161.50

3. 0 3 . 1863 .34 11.9 15 380 45 . 49 7 4200 165.75

3. 2 3 . 2896 36 12.304 400 46 . 788 4400 169 . 90

3. 4 3 . 3906 38 12.684 420 48 . 050 4600 173.96

3 . 6 3 . 4896 40 13.055 440 49 . 283 4800 177.94

3 . 8 3 . 5 866 42 13.418 460 50 . 49 1 5000 181.83

4 . 0 3 . 6818 44 13 . 773 480 5 1 . 674 5500 191.27

4 . 2 3 . 77S3 46 14.121 500 52 . 835 6000 200 . 30

4 . 4 3 . 8671 48 14.462 550 55 . 644 6500 208.98

4 . 6 3 . 9575 50 14 . 797 600 58 . 335 7000 217 . 35

4. 8 4 . 0464 55 15.6 10 650 60 . 9 23 7500 225.43

5 . 0 4 . 1339 60 16 . 389 700 63 . 419 8000 233.26

5 . 5 4. 3471 65 17.140 750 65 . 832 8500 240.86

6. 0 4 . 5531 70 17.866 800 68 . 170 9000 248 . 25

6. 5 4 . 75 26 75 18.568 850 70 . 440 9500 255 . 45

7. 0 4 . 9463 80 19 . 249 900 72 . 648 10000 262.47

7 . 5 5 . 1346 85 19.911 950 74 . 799

Let us now turn to functions S (t) and $ (t) . The asymptotic forms of

these functions for large x obtained in the preceding section can also be

refined. This is most easily done as follows. From (1.42), (5.39), and

(5.32) we have

00 zH(z) = J

e~^/^S(T)dT , X = 1 . (6.4)

0

For large t we will attempt to express S (t) in the form
234 SEMI- INFINITE MEDIUM: GENERAL THEORY

S^T) ~ 4.-3/Sl/2c.nT)^/4 V j . (6.5)

Substituting (6.5) and (6.1) into (6.4) and carrying out a series of rather

lengthy transformations, one can show that

J [2/ TT

1 22jj'. II.

j=l m=k-j

Since Sq = 1, we find from (6.6)

s = i [2 (Y*+2£n2)-3] = 0.11588 , 8

= "YY^

[4tt^ + 12 (Y* + 2£n2)^ - 36(Y* + 2£n2) + 81] = -0.75044 ,

where y = 0.577216 is Euler's constant. If necessary, values of Sj^ for

k > 2 can also be calculated.

Differentiating (6.5) and using the fact that in the conservative case

T S(t) = 1 +y$(T')dT' , (6.7)

we get

00 ^D, , ^ -3/4 -1/2, „ ,1/4 *3

$ (t) ~ 27T ' T (Jinx) / J —r , T ^ «=

, (0.8)

where

*o = ^' *k ^ " ^k-i' ^ = ^'2' ••• • f^-^^

Specifically, cf)^ = 0.61588, and ^2 =-0.92426.

NON- CONSERVATIVE SCATTERING. Now let us consider an absorbing medium. We

begin with the H-functions. The function ¥{y)(z) for several values of X is

given in Table 24. Much more detailed tables have been given in V. V. Ivanov

and D. I. Nagirner (1965). For small 1 — A and z >> 1 the asymptotic representation (4.34) can be used to compute the H- function. In the present case of the Doppler profile, we have
5.6 THE CASE OF THE DOPPLER PROFILE 235

TABLE 24 ^

THE FUNCTION H (z) FOR NON- CONSERVATIVE SCATTERING

log(I-A ] z -2 -3 -4 -5 -6

0.0 1 0000 1 0000 1 0000 1 0000 1 0000

0 . 1 1 1503 1 1556 1 1565 1 1566 1 1566

0 . 2 1 2611 1 272 1 1 2739 1 2742 1 2742

0 . 5 1 5301 1 5608 1 5662 1 5670 1 5671

1 . 0 1 8840 1 9539 1 9667 1 9687 1 9690

2 2 4274 2 5906 2 622 7 2 6279 2 6286

5 3 5045 4 0002 4 1118 4 1307 4 1335

10 4 5 76 5 5 6651 5 9500 6 0011 6 00 88

20 5 7702 7 9761 8 6807 8 8180 8 8394

50 7 2 79 7 12 069 14 212 14 702 14 784

100 8 2025 15 762 20 307 21 541 21 760

200 8 879 1 19 583 28 356 31 314 31 894

500 9 4389 24 095 41 909 50 452 52 449

1000 9 6796 26 718 53 602 70 994 75 876

2000 9 8211 28 589 65 283 97 557 108 93

5000 9 9192 30 110 78 623 141 19 172 82

10000 9 9563 30 759 86 179 177 89 240 55

£nh^(q) =-i f iln(l+qy) (6.10)

and

q

= . (6.11)

1-^ 4z(£nz)^

Differentiating (6.10) and evaluating the integral on the right, we find

± iinh^(q) = -i . 1

^ . (6.12)

from which

,

q S,nh^(q) =-i IniW) - ^

arctgq - ^ f ^^^^ dx . (6.13)

0
236 SEMI- INFINITE MEDIUM: GENERAL THEORY

For small q we have the expansion

inh^iq) = ^

qilnq - 1 - 9_ + . . . , q ^ 0 . (6.14)

For large q, (6.14) and (4.37) give us

£nhp(q) = -i anq - ^ - ^ + ... , q - 00 . (6.15)

Values o£ the function h£)(q) for 0 < q < 1 are given in Table 25. For q > 1 the function hD(q) is easily calculated from the tabulated values by means of

(4.37).

The accuracy of the asymptotic representation (4.34) for the Doppler profile can be indicated as follows. At z = 10 this expression gives values of

Hp(z) for X > 0.9 with a maximum error of around 3 percent. When z > 1000,

the accuracy is better than 1.7 percent and for z = 10,000 the error does not

exceed 1.1 percent.

Thus these asymptotic expressions are valid over a fairly wide region.

In combination with the available tables of 1\d(z) , they make it possible to

obtain values of the H- function for any X and z with an accuracy that is quite

sufficient for any application of the theory. For practical purposes it suffices to have the H-function to 2 or 3 significant figures. One may question

whether Hd(z) really need be tabulated with the great accuracy of Table 23.

It seems to us that this question must have'a positive answer. In practice,

the accuracy of most of the approximate methods used to solve the transfer

equation is difficult to predict beforehand. It therefore appears desirable

to tabulate very accurate numerical values of the exact solutions for several

standard problems.

As will be shown in the next chapter, in many important particular cases

the intensity of the emergent radiation can be expressed in terms of the Hfunction. In order to avoid interpolating between the tabulated values of

Hj)(z), it is convenient to calculate the intensity of the emergent radiation

for values of the frequency x that correspond to the tabular values of z. For

example, x = 1.52 corresponds to the value of z 5 ye^^ = 10 for y = 1, etc.

It may be expected that the tabulation of the functions $^(t) and ^^(t)

will be completed in the near future; see also Sec. 8.11 where the accuracy

of approximation (5.38) is considered.

5.7 THE CASES OF VOIGT AND LORENTZ PROFILES

ASYMPTOTIC EXPANSIONS OF THE BASIC FUNCTIONS. For conservative scattering

with Voigt and Lorentz profiles the asymptotic forms obtained in Sec. 5.4 and

5.5 can be refined.

We shall begin with the H-functions. From (4.11) we find that

± ,nH(z) =-1 f

^Xl(u)

dz ^ J l-XV(u'

udu

XV(u) i+z2u2

(7.1)
5.7 THE CASES OF VOIGT AND LORENTZ PROFILES 237

TABLE 25

THE FUNCTION h (q)

q h q h D(q) q h D(q) q h D(q) q h oCq)

0 . 00 1 .000 0 . 20 0 . 840 0 . 40 0 . 761 0 . 60 0 . 705 0 . 80 0 . 663

0.01 0 .982 0 .21 0 . 835 0 .41 0 . 758 0 .61 0 . 703 0 . 81 0 . 661

0.02 0 .969 0 .22 0 . 830 0 .42 0 . 755 0 .62 0 . 701 0 .82 0 .659

0 .03 0 .958 0 . 23 0 .826 0 . 43 0 . 751 0 .63 0 .698 0 .83 0 .657

0 .04 0 .947 0 . 24 0 . 821 0 . 44 0 . 748 0 .64 0 . 696 0 . 84 0 .655

0 .05 0 .938 0 . 25 0 .817 0 , 45 0 . 745 0 .65 0 .694 0 .85 0 .653

0 . 06 0 .^29 0 . 26 0 .813 0 . 46 0 . 742 0 .66 0 .692 0 . 86 0 .652

0 .07 0 .921 0 .27 0 . 808 0 .47 0 . 740 0 .67 0 .689 0 . 87 0 .650

0.08 0 .913 0 . 28 0 . 804 0 .48 0 . 737 0 .68 0 .687 0 . 88 • 0 . 648

0 , 09 0 .905 0 . 29 0 . 800 0 .49 0 . 734 0 . 69 0 .685 0 . 89 0 . 646

0 . 10 0 . 898 0 . 30 0 . 796 0 . 50 0 .731 0 . 70 0 .683 0 .90 0 .645

0.11 0 . 89 1 0 .31 0 . 792 0 .51 0 . 728 0 .71 0 .681 0 .91 0 .643

0.12 0 . 885 0 . 32 0 . 789 0 .52 0 . 726 0 .72 0 .679 0 .92 0 .641

0 . 13 0 . 879 0 . 33 0 . 785 0 .53 0 .723 0 .73 0 .677 0 .93 0 .640

0 . 14 0 . 873 0 . 34 0 .781 0 .54 0 .720 0 . 74 0 .675 0 .94 0 .638

0 . 15 0 . 867 0 . 35 0 . 778 0 .55 0 .718 0 . 75 0 .673 0 .95 0 .636

0 . 16 0 . 861 0 . 36 0 .774 0 . 56 0 .715 0 . 76 0 .671 0 .96 0 .635

0.17 0 . 856 0 . 37 0 .771 0 .57 0 .713 0 . 77 0 . 669 0 .97 0 .633

0.18 0 . 850 0 . 38 0 . 767 0 .58 0 . 710 0 . 78 0 . 667 0 .98 0 .631

0 . 19 0 . 845 0 . 39 0 . 764 0 . 59 0 . 708 0 . 79 0 . 665 0 .99 0 .630

1.00 0.628

For z > 1

The re fore

_d_

dz

00 f

V (u) udu ^ q/J_\

/ l-XV(u) i + z2u2 Vz7

1

JlnH(z) = -i f

^111}^ ^idu_ , Jl\ , , > 1 . (7.2)

In Sec. 2.7 it has been shown that as u 0
238 SEMI- INFINITE MEDIUM: GENERAL THEORY

r ^ 1 (2TTaU(a,0)) ' h V^(u) = 1 - ^

' u'

Substituting this expansion into (7.2), we find that for the Voigt profile

and A = 1

dz 4z 10 V2 /

U(a,0) £nz

The comparison of this equation with (4.33c) shows that in the conservative

case

Hy(z) = Qr 1 + 10 \2 j Si z \z (7.3)

where

Q

- 1-/^ \

Tih) y2^aU(a,0)

j

(7.4)

For small values of a (let us say, for a £ 0.1), the second term in the

braces is small compared to unity only for z >> a—1 . The z-values for which

the Voigt asymptotic forms are applicable for the H-function must satisfy

this inequality.

Letting a tend to infinity in (7.3) and recalling that aU(a,0) 1/tt as

a ^ 00 (see Sec. 2.7), we get the expansion of the conservative Lorentz Hfunction

1 -

lOlT

Z -> 00 (7.5)

Let us consider, further, the behavior of the conservative resolvent

function $V(t) as x ^ oo. Using (7.3) and the expansions of Uy(z) and Gy(z)

for z -> 00 given in Sec. 2.7, we can find from the explicit expression (3.11)

for $(t) that

.V. . „ -3/4

^ (t) = Qt 1 -

40

(7.6)

In the limit as a ^ oo we find

'(T) = 2l/4r

/3 -3/4

1 + 40tt t \t/

(7.7)

From (7.6) it follows that the Voigt asymptotic form of the resolvent

function is not applicable for all t >> 1, but only for t >> a~l (cf. Sec.

2.7).

The result (7.5) leads to an interesting relation satisfied by the con- servative Voigt $—function, namely.
5.7 THE CASES OF VOIGT AND LORENTZ PROFILES 239

oo J

_ $V^^^J = 1 . (7.8)

Actually, as was shown in Sec. 5.1

c» )dT = H(z) - 1 ,

from which we obtain

00 y^"^^^ Qt"-^/"^ - 3>^'(T)jdT = 1 - H^(z) + ^^^^ • ^''•^^

Letting z go to infinity here and using (7.3), we arrive at (7.8).

From (7.8) it is possible to obtain the second term in the asymptotic

expansion of the solution of the Milne problem for the Voigt profile. We have

T S(t) = 1 + J

$(t)dt , 0

so that for the Voigt profile

T

S^^(T) = 4Qxl/^ ^ 1 ^ / (^""(t) -Qt-3/4)dt

Using (7.8) we can rewrite this equation in the form

S^T) = 4Qt1/4 - /

(^\t) - Qt-2/^ jdt

Substituting '^'^(t) from (7.6) we finally obtain

S^^(T) = 4Qt1/4 ^1 . a

(I

_ a^)

^ . 0(1/t)^ . (7.10a)

In the limit as a ^ ~ we get

sL(x) = T^/^ fl - ^ ^ ^ Od/T)") . (7.10b)

r(l/4) \ 407T T J

522-519 O - 74 - 17
240 SEMI- INFINITE MEDIUM: GENERAL THEORY

NUMERICAL DATA FOR THE CONSERVATIVE CASE. Let us now consider the numerical

data relating to the Voigt and Lorentz H-functions. We shall first of all

examine the effect of the Voigt parameter a, i.e. the ratio of the collisional

and damping width to the Doppler width. Table 26 gives values of the conser- vative H-functions for several values of a. The Doppler H-function corre- sponds to a = 0, and the Lorentz to a = <». For z < 1 the table gives values

of H(z), and for z > 1 it gives values of the quantity H(l/a(x)) tabulated

according to the argument x. It is convenient to tabulate H(l/a(x)) as a function of x to facilitate the construction of line profiles.

TABLE 26

THE H-FUNCTIONS FOR THE VOIGT AND LORENTZ PROFILES (CONSERVATIVE SCATTERING)

H(z)

z a = 0.001 a = 0.01 a == 0.1 a = °°

0 .00 1 000 1 000 1 000 1 .000

0.02 1 0 41 1 041 1 0 3 8 1.027

0 . 06 1 103 1 102 1 094 1 .064

0 . 1 1 156 1 154 1 141 1 .096

0 . 2 1 273 1 269 1 243 1.161

0 . 3 1 378 1 370 1 332 1.216

0 . 4 1 474 1 464 1 411 1 . 264

0 . 5 1 565 1 551 1 .485 1 . 307

0 . 6 1 651 1 634 1 554 1 . 347

0 . 7 1 .733 1 713 1 618 1 . 383

0 . 8 1 813 1 789 1 680 1.418

0 . 9 1 889 1 862 1 738 1 . 450

1 . 0 1 963 1 933 1 793 1 . 480

\a(

X a = 0.001 a = 0.01 a := 0.1 a = oo

0.0 1 963 1 933 1 793 1 .480

0 . 2 1 993 1 960 1 813 1 . 492

0 . 4 2 087 2 048 1 874 1.525

0 . 6 2 263 2 211 1 983 1.577

0 . 8 2 552 2 476 2 151 1 .643

1 . 0 3 013 2 890 2 394 1 .718

1 . 1 3 336 3 173 2 549 1 . 759

1 . 2 3 742 3 523 2 729 1 . 800

1 . 3 4 252 3 951 2 935 1.842

1 . 4 4 896 4 475 3 168 1 .885

(Continued)
5.7 THE CASES OF VOIGT AND LORENTZ PROFILES

TABLE 26 (Continued)

241

Va(x)/

a=0.001 a = 0.01 a = 0.1 a = °°

1, 5 5 707 5 109 3 427 1 928

16 6 730 5 . 871 3 711 1 971

1. 7 8 021 6 775 4 016 2 015

1. 8 9 . 645 7 831 4 339 2 058

1. 9 11 . 68 9 040 4 674 2 101

2 . 0 14

• 22 10 39 5 014 2 144

2 . 1 17 34 11 87 5 353 2 187

2. 2 21, 14 13 43 5 685 2 230

2 . 3 25 . 64 15 05 6 004 2 272

2 4 30 85 16 67 6 305 2 .314

2 5 36 66 18 26 6 585 2 355

2 6 42 90 19 11 6 843 2 396

2 7 49 36 21 18 7 080 2 436

2 8 55 81 22 44 7 297 2 477

2 9 62 08 23 54 7 .497 2 516

3 0 68 01 24 47 7 682 2 556

3 1 73 44 25 26 7 856 2 594

3. 2 78 19 25 92 8 019 2 633

3 3 82 12 26 50 8 174 2 671

3 4 85 25 27 02 8 323 2 709

3. 5 87 70 27 48 8 466 2 746

3 75 92 00 28 54 8 806 2 837

4 00 95 17 29 5 1 9 126 2 926

4 25 98 00 30 43 9 430 3 0 13

4 50 100 7 31 31 9 721 3 098

4 75 103 3 32 16 10 002 3 180

5 00 105 8 32 99 10 27 3 261

5 5 110 7 34 57 10 79 3 418

6 0 115 3 36 08 11 28 3 568

6 5 119 7 37 52 11 75 3 712

7 0 124 0 38 90 12 20 3 85 1

7 5 128 2 40 24 12 63 3 986

8 0 132 2 41 .53 13 05 4 116

8 5 136 1 4 2 78 13 45 4 243

9 0 139 9 44 00 13 84 4 366

9 5 143 6 45 18 14 22 4 485

10 0 147 2 - 46 34 14 59 4 601
242 SEMI- INFINITE MEDIUM: GENERAL THEORY

Figure 25 shows the effect of the parameter a upon the H-function in the case of pure scattering. One of the conclusions that may be drawn from this

figure is that even for a = 10—3 the values of Hy(z) are close to H^iz) only

for rather small z. The practical region of applicability of the results

obtained for the Doppler profile is therefore rather small. This is especially true of the asymptotic results corresponding to z >> 1 (and t >> 1). Of

course, this does not mean that the asymptotic theory considered in detail in

the previous section is completely devoid of interest. From a purely theoretical point of view it is necessary for the development of the theory itself. However, even from the practical side it is useful, since the idealized

case of the Doppler profile is often used to test various approximate and

numerical methods for solving radiative transfer problems in spectral lines.

The second conclusion to be drawn from Fig. 25 is that, in agreement with

the result obtained above, for a 7^ 0 and with sufficiently large z, the function H(z) increases proportionally to z'a. This asymptotic result, so attractive in its simplicity, appears to be valid over a very wide region. We now consider the accuracy of this result.

The leading terms of all the asymptotic expansions associated with the

Voigt profile are obtained by replacing the normalized Voigt function U(a,x)

with its asymptotic form

U(a,x) ~ ^ , (7.11)

which is valid for sufficiently large |x| (see Sec. 1.5). Therefore

V(x) = "^^>;j ~ ^ L . (7.12) U(a,0) TTU(a,0) ^2

Substituting this expression into (7.3), we find that the leading term in the

log z

Fig. 25. H-functions for the conservative scattering.
5.7 THE CASES OF VOIGT AND LORENTZ PROFILES 243

expansion of the conservative Voigt H-£unction for sufficiently large |x| is

(7.13) "v(a(x)) ^ (2)

A comparison of the exact values of the H-functions given in Table 26 with

the values given by this result shows that when a = 10~3^ (7.13) provides an accuracy of 5 percent for all |x| ^ 3.2, and 1 percent for |x| ~ 10 . For a 10-2 the error does not exceed 5 percent for |xl Z 2.9, and 1 percent for

|x| ~ 6.7. When a = 0.1, the corresponding values of fx| are approximately

2.9 and 4.0.

For the Lorentz profile (a 0 > (7.5) gives us (^) ~ (jf

^

(7.14)

This expression is essentially the same as (7.13), since for the Lorentz profile the frequency is measured in units of the collisional width and not in

Doppler widths, i.e. x^ = x/a. When the Lorentz H-function is calculated

from the asymptotic expression (7.14), an accuracy of 5 percent is attained

at X ~ 1.8; for x > 3.3 the error does not exceed 1 percent. More detailed information about the accuracy of the approximations (7.13) and (7.14)

may be obtained from Fig. 26, which gives for several values of a curves of the

relative errors

A(x)

^ot(x.

(7.15)

X

Fig. 26. Accuracy of the asymptotic representations (7.13) and (7.14)
244 SEMI- INFINITE MEDIUM: GENERAL THEORY

as a function of x. In the last equation H^s is the asymptotic form of the

H-function, calculated from (7.13) for 0 < a < °° and from (7.14) for a = «>.

In summary, it can be said that for conservative scattering the asymptotic theory gives a very good description of the behavior of the H-functions

beyond the Doppler core of the line. The asymptotic form of H(l/a(x)), for

a given value of x, is approximately as accurate as (7.11) for the same value

of X. NUMERICAL DATA FOR THE NON- CONSERVATIVE CASE. Now let us turn to the numerical results and asymptotic forms for the non- conservative case. Tables 27,

28, and 29 present values of the H-functions for A < 1. They were obtained

from the representation of H(z) in the form (4.11) by numerical integration

(values of the H-functions for A = 1 were found in the same manner). The

tables we give ,in this section were compiled by D. I. Nagirner, who plans to

publish in the near future more detailed numerical data as well as a description of the method of calculation. R. F. Warming (1970a) has recently published detailed six-place tables of the Lorentz H-functions.

To compute H-functions beyond the Doppler core for any A < 1, one can use the asymptotic theory which, as for A = 1, provides completely adequate accu- racy here. As has been shown in Sec. 5.4, in the Voigt asymptotic region,

i.e. for z >> a~l

,

H,,(z) ~ (1-A) ^h,,(q) , (7.16)

where

00 £nh (q) =-i / £n(l+qvy) (7.17)

0 1+y^

and

- = (1-A) z^' . (7.18)

^ (2^aU(a,0))'^

Values of the function hv(q) for 0 < q < 1 are given in Table 30. The relation (4.37) must be used to find hY(q) for q > 1. We note that

h^(q) = 1 - _ i q^^nq + . .. , q - 0 , (7.19)

The function h(q) for the Lorentz profile is identical with hy(q)

For nearly conservative scattering it is more convenient to use another

asymptotic expression, also found in Sec. 5.4,

H(z,A) - H(z,l)hfi^ , 1 - A << 1 , (7.21)
5.7 THE CASES OF VOIGT AND LORENTZ PROFILES 245

where

q = [(1-X)h2(z,1)]-^

and H(z,l) is the conservative H-function. This expression is valid for all

z, and becomes more accurate as 1 — A decreases. We can obtain an idea o£

Tts accuracy from the following example. For 1 — X = 10""^ and a = 0.01,

values of H(z) calculated from (7.21) using the data of Tables 26 and 30 differ from the accurate ones given in Table 29 by only a fraction of a percent.

TABLE 27

THE H-FUNCTIONS FOR THE VOIGT AND LORENTZ PROFILES (X =0.99)

H(z)

z a = 0.001 a = 0.01 a == 0.1 a = CXI

0 .00 1 000 1 000 1 000 1 000

0.02 1 040 1 040 1 037 1 026

0 . 06 1 099 1 .098 1 .092 1 063

0 . 10 1 150 1 149 1 138 1 094

0 . 2 1 261 1 258 1 236 1 158

0.3 1 358 I 353 1 321 1 211

0.4 1 447 1 440 1 397 1 258

0.5 1 529 1 521 1 467 1 300

0.6 1 607 1 597 1 532 1 338

0.7 1 680 1 669 1 593 1 374

0.8 I 750 1 737 1 650 1 407

0.9 1 818 1 802 1 704 1 438

1.0 1 882 1 865 1 755 1 467

(a(x)^1 X a = 0.001 a = 0.01 a := 0.1 a = OO

0.0 1 882 1 865 1 755 1 467

0.2 1 908 1 889 1 774 1 478

0.4 1 988 1 967 1 830 1 510

0.6 2 136 2 108 1 930 1 560

0. 8 2 373 2 332 2 083 1 622

1 .00 2 733 2 670 2 300 1 694

1 . 25 3 425 3 308 2 677 1 790

1 . 50 4 461 4 240 3 17. 1 890

1 . 75 5 835 5 440 3 760 1 991

(Continued)
246 SEMI- INFINITE MEDIUM: GENERAL THEORY

TABLE 27 (Continued)

Va(x)

X a = 0.001 a = 0.01 a = 0.01 a = 00

2 . 00 7 326 6 713 4 363 2 091

2. 25 8 564 7 778 4 906 2 188

2 . 50 9 329 8 486 5 341 2 282

3.0 9 838 9 088 5 910 2 462

q 9 1 ? q 9L. o^ Ro D ^ u o i. , 4.0 9 926 9 346 6 534 2 785

4.5 9 934 9 411 6 755 2 . 931

5 . 0 9 941 9 464 6 944 3 . 068

6 . 0 9 950 9 543 7 254 3 . 319

7. 5 9 960 9 625 7 605 3 . 647

10 , 0 9 969 9 710 8 015 4 . 103

TABLE 28

THE H-FUNCTIONS FOR THE VOIGT AND LORENTZ PROFILES (X == 0.999)

z

H(z)

a = 0.001 a = 0.01 a = 0.1 a = CO

0. 00 1

000 1 000 1 000 1 000

0.02 1 . 041 1 041 1 .038 1 027

0 .06 1

102 1 101 1 093 1 064

0 . 10 1

156 1 153 1 141 1 096

0 . 2 1

272 1 267 1 242 1 161

0 . 3 1

375 1 368 1 330 1 215

0 . 4 1

470 1 461 1 410 1 263

0 . 5 1 . 559 1 547 1 483 1 306

0 , 6 1

644 1 629 1 55 1 1 346

0 . 7 1

725 1 707 1 616 1 . 382

0 . 8 1

803 1 782 1 675 1 . 417

0 . 9 1. 878 1 854 1 734 1 . 448

1 . 0 1 . 950 1 924 1 789 1 . 479

(Continued)
5.7 THE CASES OF VOIGT AND LORENTZ PROFILES

TABLE 28 (Continued)

247

X a = 0.001 a = 0.01 a = 0 . 1 a = 00

0 . 0 1. y 5 u 1 , n o ,1

, y 2 4 1

, 7 89 1 ,.4 79

0 . 2

^1 ,. y /y i , OCT , y 5 i 1 ., 80 8 1 ,.49 0

0 . 4 2 , A "7 1

. (J 7 1 2 ,,037 1 ,, 869 1 ,.5 24

(J . 0 I .,242 Z .. ly / 1 .,974 1 ,,575

0 . 8 2 . coo ,522 2 .

-1 r "7 .45 7 2 ,,14 4 1 ,,641

1.00 2 ,, y 5 4 2 , o r r\

, o 5 y 2 , 7 O y1 ,384 1 ,,716

1.25 3 , O "7 "7. 0 7 7

73 ,,56 4 2 ,.811 1 ,,818

1.50 5 , AAA .444 4 , o ^ r\, y oy 3 ,, 39 5 1

,924

1 . 75 QO ,, U D U 6 .. y 5

4 . 1 O 7 ,12 3 2 . A 7 O ,032

2.00 1 2 ,, 0 5 y ,,618 4 ,,928 2 .,139

o or 2.25 i 7 ,. Z 4 1 z ,, 0 /

rD , "7 1 C i . O /I yl

, z 4 4

2 . 5 U 2 2 , 7 O, 0 0 1 b ,. 3 is D ,, 4 U / Z , 7 /I "7 ,34/

2.75 2 5 .. 0 5 1 7 .. 4 7 6 ,, y b / z ,,44/

3.00 2 o ., U 4

L Q >o ,. o 1

•7/ ,, 4 U i z , C /I c

, D 4 0

3 . 2 0 2 0 . n "7 ly

. D i

"7 "7 "7 1 Z , ^ /I A , D 4 U

3 . 5 U 2y ,. 3d I U ,. 1 O

oO ,, u y 0 z , "7 7 7 3.75 29 ,. 5 4 2U ,. 5 7

o0 ,, 3y 4 z , Q O T

, 0 Z O 4.00 29 ,. 6 6 20 .. 9 4 8 ,,6 70 2 , A 1 A ,910

4.25 29 ., 76 2 1 ,, 2 8 8 ,,931 2 , A A r ,995

4.50 29 ,. 85 2 1 ,. 59 9 ,,17 8 3 ,,0 79

4.75 29 ,. 9 3 2 1 ,, 88 9 ,,414 3 ., 160

5.00 30 ,. 00 2 2 ,, 15 9 ,.640 3 ,, 2 39

5 . 5 30 ., 12 2 2 ,. 6 3 1 0 ,, 0 6 3 ,,39 2

6 . 0 30 . 23 23 ,.07 1 0 ,. 46 3

,538

6 . 5 30 ,. 3 2 2 3 ,. 46 1 0 ,. 84 3 . ^; "7 A

,6 79

7.0 30 . 40 23.. 81 11,. 19 3., 814

7.5 30 .47 24,. 13 11 ,.52 3 .,945

8.0 30 .53 24,.42 11 ,, 84 4 .,071

8.5 30 ,.58 24 ,,69 12 ,, 14 4 ,, 193

9.0 30 .63 24,.94 12 ., 42 4 ,,311

9 . 5 - 30.68 25 . 17 12 ,. 70 4 .,425

10 . 0 30 .72 25 ,.38 12,,95 4 ,,538
248 SEMI- INFINITE MEDIUM: GENERAL THEORY

TABLE 29

THE H-FUNCTIONS FOR THE VOIGT AND LORENTZ PROFILES (A == 0.9999)

H(z)

z a = 0.001 a = 0.01 a = 0.1 a = 00

0 ,00 1 , 000 1 . 000 1 .000 I .000

0.02 1 . 041 1 041 1 .038 I .027

0 . 06 1 . 103 1

101 1 .09 4 I .064

0 . 10 1 . 156 1 , 154 1 .141 I .096

0 . 2 1 . 2 73 1 . 268 1 .243 I .16 1 0 . 3 1

377 1 . 370 I .331 I t ?^ X^ 6\J 0 . 4 1 . 474 1 . 46 3 I .411 1 . 264

0.5 1 . 564 1 . 551 1 .485 1 . 307

0.6 1 . 650 1

634 1 .554 1 . 347

0.7 1 . 732 1 . 713 1 .618 1 383

0.8 1 . 811 1 . 788 1 .679 1 418

0.9 1 . O 0 o O O 0 1

861 1 .737 1 ,45 0

1 n 1 . r\ o y 6 2 1 . y 5 z 1 . 79 3 1 , 4 80

X a = 0.001 a = 0.01 a == 0.1 a = 00

0 . 0 196 2 1

,

932 I 79 3 I _ 480

0 . 2 99 1 1

9 59 1X I _ 49 2

0 . 4 2 . 0 85 2 . 047 I 8 7 3 I 5 25

0 . 6 2260 2 . 210 1 9 82 I , 5 7 7

0 . 8 2 . 5 4 8 2 . 474 2 151 1 , 6 4 3 1.00 3 . 00 7 2 . 887 2 39 3 1 . 718

1.25 3 . 969 3 . 719 2 82 7 1

,

8 21

1.50 5 . 6 7 2 5 . 094 3 4 2 3 I , 9 2 8 1.75 8 . 6 8 3 7 . 246 4 _ 1 70 2 , 0 36

2.00 1 3 . 88 10 . 30 5 005 2 . 144

2.25 22. 20 1 4 . 0 3 5 . 832 2 . 250

2 .50 33. 49 17. 86 6 . 565 2 . 354

2. 75 45 . 33 2119 7 . 165 2 . 456

3.00 54 . 65 23. 58 7. 652 2 . 554

3.25 61 . 03 25 . 14 8 . 06 1 2 . 651

3.50 64. 24 26 . 25 8. 425 2 . 744

(Continued)
5.7 THE CASES OF VOIGT AND LORENTZ PROFILES

TABLE 29 (Continued)

249

\a(x)

X a = 0 .00 1 a = 0 .01 a = 0 . 1 a = oo

0 . / D D D . 93 0 7 17 oo . 760 2 .8 36

4 . U U D / . 1

1 Q L O . 00 r\y

0 7 4 2 .925

D O • 12 O 0 78 y .

•7 "7 7 3 7 3 3 .011

A c n 4 . o U o y . 03 Iv . 53 y

659 3 .09 6

4 . / O Q oy . 88 7 n oU , 24 y

9 3 4 3 .17 3

o . u u / u . 68 z n oU , 9 2 1 U . Z U 3 .259

c c 3 . O

7 9 1 3X 0 J t . 22 I U . 7 1 / 1

•7 3

/lie .415

A n 0 . u

7 "Z 43 % 7 o o , 43 1 1 . 1 O

1 O 3 .565

o . o

7 A 70 7 A 5 7 i i . A A 0 4 3

7 A A

. 709

7.0 75 . 66 35 . 65 12 . 08 3 . 848

7.5 76 . 63 36 . 67 12 . 49 3 .982

8.0 77. 5 2 37 . 65 12 . 90 4 . 112

7 S 34 38. 58 1 O • ? Q A4 9 7

9 . 0 79 . 10 39 . 48 13 . 66 4 . 360

9 . 5 79 . 80 40 . 33 14 . 03 4 .479

10 . 0 80 . 46 41 . 16 14 . 38 4 . 595

TABLE 30

THE FUNCTION h(q) FOR THE VOIGT AN U LORENTZ PROFILES

q h(q) q h(q) q h(q) q n (qJ

0.00 1 . 0000 0 . 26 0 . 8690 0 . 51 0 .7882 0 . 76 0 .7282

0.01 0 .9931 0 . 27 0 .8652 0 . 52 0 . 7854 0 .77 0 .726 2 0.02 0 .9865 0 . 28 0 . 8614 0 . 53 0 . 7827 0 .78 0 .7241

0.03 0 .980 1 0 . 29 0 . 85 77 0 . 54 0 . 780 1 0 . 79 0 .7220

0.04 0 .9740 0 . 30 0 .854 1 0 . 55 0 .7775 0 . 80 0 . 7200

0 . 05 0 . 9680 0 . 31 0 . 8505 0 . 56 0 . 7749 0.81 0 .7180

0 . 06 0 .9621 0 . 32 0 . 8469 0 . 57 0 .7723 0 . 82 0 .7160

0.07 0 .9565 0 . 33 0 . 8435 0 . 58 0 . 7697 0.83 0 . 7140

0.08 0 .9510 0 . 34 0 . 8400 0 . 59 0 . 7672 0 . 84 0 .7120

0 . 09 0 . 9456 0 . 35 0 .8366 0 . 60 0 . 7647 0 . 85 0 .7101

0 . 10 0 .9403 0 . 36 0 .8333 0 . 61 0 .7623 0 . 86 0 . 7082

CContinued)
250 SEMI- INFINITE MEDIUM: GENERAL THEORY

TABLE 30 (Continued)

q h(q) q h(q) q h(q) h h(q)

0 . 11 0 ,.9351 0 . 37 0 ,,8300 0 , 62 0 .,759 8 0 . 87 0 ,.7062

0 . 12 0 ,,9301 0 . 38 0 .,8268 0 . 63 0 ,.75 74 0 . 88 0 ., 7044

0 . 13 0 ,.9252 0 . 39 0 .,8236 0 . 64 0 ,,7551 0 . 89 0 ,.7025

0 . 14 0 .9204 0 . 40 0 .,8204 0 . 65 0 ,,7527 0 . 90 0 ..7006

0 , 15 0 ..9156 0 . 41 0 .,8173 0 . 66 0 ..7504 0 . 9 1 0 ,.6988

0 . 16 0 ,.9110 0 . 42 0 .,8142 0 . 67 0 ,.7480 0 . 92 0 .69 69

0 . 17 0 ,.9064 0 . 43 0 .,8112 0 . 68 0 ,.7458 D . 93 0 ,.6951

0 . 18 0 ..9020 0 . 44 0 .,8082 0 . 69 0 ., 7435 0 . 94 0 ,. 6933

0 . 19 0 ,. 89 76 0 . 45 0 ,, 8052 0 . 70 0 ,, 7413 0 . 9 5 0 ,.6915

0 . 20 0 ,. 8933 0 . 46 0 ,, 8022 0 . 71 0 ,. 7390 0 . 96 0 ,.6898

0 . 21 0 ,.889 1 0 . 47 0 ,. 7994 0 . 72 0 ,. 7368 0 . 97 0 ,.6880

0 .22 0 ,. 8849 0 . 48 0 ., 7965 0 . 73 0 ,, 7347 0 . 98 0 ,.6863

0 .23 0 ,. 8808 0 . 49 0 ,, 79 37 0 . 74 0,.7325 0 . 99 0 ,.6845

0 .24 0 ,. 8768 0 . 50 0 ., 7909 0 . 75 0 ,. 7304 1

00 0 ,.6828

0.25 0.8729
CHAPTER VI

SEFII-INFINITE MEDIUM: STANDARD PROBLEMS

The results obtained in the preceding chapter enable us to write in ex- plicit form the solution of the integral equation describing the scattering

of line radiation in a semi - infini te layer with a primary source whose

strength depends in an arbitrary way upon depth. Unfortunately, this solution is rather cumbersome. However, if the depth dependence of the source strength has a reasonably simple form, such as, for example, a constant or decreasing exponential function, etc., the solution can be greatly simplified

and put into a more compact form. It is naturg.1 to consider the simplest standard cases in greater detail. Such standard problems enable one to form a clear physical picture of the phenomenon and to better understand the nature

of the problems of spectral- line radiative transfer arising from the change

in frequency of a photon during scattering.

In this chapter, we apply the general results obtained in Chapter V to

the solution of several standard problems. Our goal in this chapter is twofold. We will show how the general methods previously developed can be

applied to concrete cases. In addition, we anticipate that it will be useful

as a handbook of complete solutions of standard problems.

6.1 UNIFORMLY DISTRIBUTED SOURCES. GENERALIZED MILNE PROBLEM

A MEDIUM WITH UNIFORMLY DISTRIBUTED SOURCES. Let us first study the radiation

field in a medium with uniformly distributed sources, i.e., one with S*(t) = S* = const. In this case the basic integral equation for the line source function is

00

+ s (1.1)

0

It is easily shown that
252 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

SCt) = S(0) ^1+ y

$(T')dT'j (1.2)

0

Indeed, differentiating (1.1) and taking (5.1.24) into account, we have

CO S'(t) = ^ /

K^(|t-t' 1)S' (T')dT' + S(0) A k^(t) . (1.3)

0

From a comparison with equation (5.1.28) for the resolvent function $(t) we conclude that

S'(t) = S(O)nT) , (1.4)

from which (1.2) directly follows.

The value of S(0) may be obtained as follows. As x <» the function

S(t) should tend to the source function in an infinite medium wi th S"(t) = S"'. In fact, when the sources are uniformly distributed, sufficiently deep in a semi - infini te medium conditions differ little from those in a medium that is

infinite in all directions, because at great depths absorption processes dominate the escape of radiation through the boundary. In an infinite medium each

photon undergoes an average of (1 — ^)~-'- scatterings. The source function

for an infinite medium with S*(t) = S is therefore S*(l — X)~l. From what

has just been said, we have

* lim S(t) = , (1.5)

or

S1-X

(00 . 1+ y$(T')dT'j (1.6)

Using (5.5.29), we find

= S(0)

0

S(0) = S*(l-X) ^ , (1.7)

and, finally,

S(t) = S*(l-A) '"^ f

$(T')dT'j , (1.8)

or, in the notation of (5.5.32),

S(t) = S*(1-A)~'^T(t) . (1.9)
6.1 UNIFORMLY DISTRIBUTED SOURCES- GENERALIZED MILNE PROBLEM 253

0£ course, this expression may also be obtained from the general relations found in Sec. 5.1. For S*(t) = S* the expression (5.1.21) gives

0

It can be shown (see Sec. 6.8 below) that

(oo . 1+ y

r(T,T')dT'j . (1.10)

oo

1+J

r(T,T')dT' = (1-X) ^T(t) , (1.11)

0

which brings us back to (1.9).

If the source function S(t) is known, it is easy to find the intensity

of radiation at any point in the medium. We limit ourselves to determining

the intensity of the emergent radiation l(0,ij,x). This is usually the quantity of greatest interest, since it is most readily found from experiment.

In astrophysical problems l(0,y,x) is, as a rule, the only quantity that can

be determined by observation. As is well known,

00 1(0, y,x) = y S(T)e-^^^^^/^a(x)^ . (1.12)

0

Substituting (1.8) here, integrating by parts, and using (5.1.42), we get

1(0, y,x) = S*(l-X)"''^H(y/a(x)) (1.13)

Thus when the sources are uniformly distributed throughout the medium, the

angular and frequency distribution of the emergent radiation is proportional

to H(z) , with z = y/a(x).

GENERALIZED MILNE PROBLEM: THE SOURCE FUNCTION IN THE DEEP LAYERS. From

(1.9) it is apparent that (1.1) has no solution for pure scattering (X = 1).

However, we can set S* = /I — X in (1.1) and (1.9) and then allow X to approach unity. Then (1.1) reduces to the homogeneous equation

oo S(t) =1 y K^(|t-T' |)S(T')dT' , (1.14)

0

and, as follows from (1.9), its solution, normalized so that S(0) = 1, is

T S(t) = y$(T')dT' . (1.15)

0
254 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

According to (1.13), the emergent intensity is

1(0, y,x) = H(y/a(x)) . (1.16)

The functions $ and H in (1.15) and (1.16) refer to A = 1 . Equation (1.14) represents a generalization of the conservative Milne

problem (cf. Sec. 3.8). Since an explicit expression for $(t) is known (see

Sec. 5.3), the result (1.15) gives the solution of the generalized Milne problem in the form of an integral, i.e., the closed, albeit rather cumbersome

expression

Zl. . 1 . 1 JM _ ^-t/z, or.^ G(z) S(T) = I^T + 1 + i

J

(1 - e ^^^) R(z)

^

dz . (1.17)

0

2 We emphasize that if the second moment of the kernel is infinite (a = °°) , the term proportional to t vanishes. In Sec. 5.5-5.7 simple expressions

describing the behavior of S(t) for large and small values of i^were obtained.

In particular, the leading term of the asym.ptotic expansion of S(t) for large

lis

S(T) - ^^^^/^^^'"^

,y , T - 00 . (1.18)

Yr(Y)

It would be very useful to have detailed tables of S(t) for the absorption

coefficients most often encountered in practice. So far very little has been

done in this direction; only a modest three- figure tab'Te of sD(t) has been

published (E. H. Avrett and D. G. Hummer, 1965, Table 2).

An asymptotic relation exists between the solution of the generalized

Milne problem and the Green's function Sp(T) for an infinite, homogeneous,

conservative scattering medium.

We define

T S^(T) = 1 + 4^^ t2Sp(t)dt , A = 1 . (1.19)

0

In Sec. 4.3 it has been shown that in the conservative case

.2y-1

S (T) ~ ^

P 4^t2 r(2Y)cosTTY <J)(1/t)

'

Substituting this expression in (1.19) and utilizing the fact that ({)(u) is a slowly varying function we get

S (T) 1-2^-- -

2Yr(2Y)cos^Y 4>(1/t)

Comparing this equation with (1.18), we arrive at the desired relation:
6.1 UNIFORMLY DISTRIBUTED SOURCES. GENERALIZED MILNE PROBLEM 255

SCt) ~ c ( S^(^)j , T -> oo

, (1.20)

where

c

= (mill costtyY

'^

VYr2(Y) V

As Y varies from 0 to 1/2, the value of c increases monotonically from 1 to

/2 , _For Y

= 1 (in particular, for monochromatic scattering) we also have

c = /2 , which agrees with the result obtained in Sec. 3.9.

THE MILNE PROBLEM AS A CONSTANT FLUX PROBLEM. In the case of the generalized

Milne problem the transfer equation is

1 I(T^y,x)

^ a(x)I(T,y,x) - 1 Aa(x) y* a (x ' ) dx ' y* I ( t , y

' , x ' ) dy ' , (l'.21)

and the boundary condition is 1(0, y,x) = 0 for y < 0. The source function

S(t) is now just the radiation intensity, averaged over frequency (with the

weight Aa(x)) and over direction:

00 i S(t) = a(x')dx'y* I(T,y' ,x')dy'

-1

(1.22)

Integrating the transfer equation over y from -1 to +1 and over x from -<» to"

+00, we obtain

1

-1

I (t ,y ,x) ydy = 0 (1.23)

The quantity following the derivative operator is the total flux in the line

which, from (1.23), is clearly independent of depth. In this respect the

generalized Milne problem is no different from the classical one. However,

in the generalized Milne problem the flux in the line may be infinite (although its derivative according to (1.23) is equal to zero) while for mono- chromatic scattering it is finite.

The integral equation (1.14) for the function S(t) is a generalization

for an arbitrary profile of the (first) Milne equation

00 S^(T) = E^(|T-T'|)S^^T')dT' . (1.24)

0

As is well known, the function S^'^(t) , normalized so that S^'(O) = 1, satisfies

in addition to (1.24) the equation

522-519 O - 74 - 18
256 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

00 1 J

E2 (t'-t)S^(t' )dT' - J

E2(t-t' )S^(t' )dT' = 4= , (1.25)

T 0

'^^

which is sometimes called the second Milne equation. Using (3.7.37), one m.ay readily show that

00 /

E, (t' )S^Vt' )dT' = 4= . (1.26)

0 ^3

Therefore, (1.25) can also be rewritten in the form

/

[E.(t'-t) - E,(t') ]S"(T')dT' - ^2^^ -Lj - ^2

T -/

T Mr.. [E2(t-t') + E2 (t' ) ]S"(t' )dT' = 0 . . (1.27)

The generalization of this equation to the case of an arbitrary profile is

00 J

[K2(t'-t) - K2(T')]S(T')dT' -

T

- J

[K2(t-t') + K2(t') ]S(T')dT' = 0 . (1.28)

0

From (1.14), the generalized Milne problem is seen to correspond to the case in which there are no sources at any finite depth. The radiation field

may be thought of as arising from a source of, generally, infinite strength,

located infinitely deep in the medium. Physically, this means that the atomic

excitation in the surface layers is caused by radiation that is generated at

great depths.

We note in passing that the conservative resolvent function

$(T) = S(T) , dx

in addition to satisfying (5.1.28) with X = 1 , is also the solution of the

following two equations:

00 x J

K2 (t'-t) $(t' )dT' - JK2(T-T')0(T')dT' = K2(t) , (1.29)
6.2 THICKNESS OF THE BOUNDARY LAYER 257

00 y* [KjCt') - KjCIt-t' |)]*(T')dT' = K3CT) , (1.30)

where

T K3(t) = j K2(T')dT' . (1.31)

0

In the special case of the rectangular profile, i.e., for monochromatic scattering, (1.29) and (1.30) become, respectively, (3.7.48) and (3.7.47).

The generalized Milne problem has been formulated and investigated by

the author (V. V. Ivanov, 1962b, 1965, 1968); it has also been discussed by

D. G. Hummer and J. C. Stewart (1966). It can be regarded as the simplest

standard problem in the theory of line- frequency radiative transfer.

In the next two sections the discussion of a medium with uniformly distributed sources will be continued, with emphasis on the physical aspects of

the problem.

6.2 THICKNESS OF THE BOUNDARY LAYER

DEFINITION AND ORDER- OF- MAGNITUDE ESTIMATE. The effect of the boundary is to

allow radiation to leave the medium from those regions in its vicinity. Thus

the boundary plays the role of an absorbing wall. Obviously the escape of

radiation becomes more important as the boundary is approached. Therefore,

with a uniform distribution of primary sources, the mean intensity and the

photo- excitation rate near the boundary will decrease. This uncompensated

decrease causes the excitation to fall as the boundary is approached. Consequently, a boundary layer will exist in the gas.

It follows from (1.8) that

= (l-X)''« , (2.1)

S(«>)

i.e. within the boundary layer the source function decreases by a factor of

(1—X)~'2. Leaving for a moment the detailed behavior of S(t), we shall consider the important problem of estimating the thickness of the boundary layer.

An order- of-magnitude estimate can be obtained from simple physical considerations (V. V. Ivanov, 1965, 1966). The probability per scattering that a photon is destroyed by inelastic processes is 1 — X. The probability that an excited atom at a depth x will emit a photon that will immediately (without

scattering) escape is (X/2)K7(t) (see Sec. 5.2). Since, in line transfer problems the whole process of photon random walks can be approximated by a single

flight (see Sec. 4.5), it is clear that for those values of t for which

(X/2)K2(t) << 1 — X, the escape of radiation is unimportant in the first

approximation. On the other hand, for (X/2)K2(t) >> 1 - X the role of inelastic processes is small in comparison with the effect of the boundary. We

shall denote the optical thickness of the boundary layer as Xj^ (the subscript

b stands for boundary). Thus, when 1 — X << 1, an order of magnitude estimate

of ttq is given by the root of the equation
258 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

I

K^Ct^) = 1 - a . (2.2)

This equation could be taken as a definition of . However, for a number of reasons we prefer to define Xj^ a little differently, but in a way that does

not contradict the estimate given by (2.2).

We shall define the optical thickness of the boundary layer as the value

T = that satisfies the equation

S(T^) = (1-X) , (2.3)

where S(t) is the solution of the homogeneous equation (1.14) normalized so that S(0) = 1. LIMITING CASES. Let us consider several special cases. For A = 0 we have

S(t|)) = 1, so that = 0 . In the extreme case X 0 the role of radiative

transitions is negligibly small, and therefore the boundary does not affect

the degree of excitation. The boundary layer is absent. In the opposite

extreme case 1 — A << 1 we have from (2.3), using (1.18),

"b [Y^r^Y)Hi/T^)]^^^\i-x)"^/^^ ,

whence, using the fact that 4) is a slowly varying function,

~ D^(l-A)~^/2'^ . (2.4)

Here D|^ is the slowly varying function of X:

I^b ^2p2^^3^^^,_,3l/2y^- 1/2y

(2.5)

In cases of practical interest, is of the order of unity

In particular, for monochromatic scattering (y = 1 , 4* = 1/3), we get

from (2.4) and (2.5)

^ A (1-X)

~

i.e. for nearly conservative monochromatic scattering the thickness of the

boundary layer is asymptotically (X -> 1) equal to the diffusion length. Combining (2.3) and (5.5.40), we find that if the characteristic exponent y is

less than unity, then

K2(tj^) ~ C^d-X) (2.6)

whe re
6.3 DEPARTURES FROM LTE IN THE BOUNDARY LAYERS OF AN ISOTHERMAL GAS 259

C = —1— (l r(2Y)sin^Y) '

. (2.7)

Since the factor C is close to unity for values of y of primary interest,

namely, for 0 < y ^ 1/2, the values of x^^ given by (2.2) and (2.6) are in

close agreement.

For nearly conservative scattering a simple asymptotic relation exists

between the thickness of the boundary layer t-^ and the thermalization length

T+-, which follows from a comparison of (2.4) and (4.6.16):

CT, (2.8)

where

! c= /iliill cos^Y/2t ^2.9)

I \yT2iy) 1-2 Y ;

The coefficient c decreases monotonically as y increases from_0 to 1, with

; ciO) = e = 2.718; c(l/4) = 2.327; c(l/2) = 2.000 and c(l) = /2 = 1.414.

! Thus, for nearly conservative scattering the thermalization length

and the thickness of the boundary layer Xj^ are quantities of the same order

of magnitude. In making estimates, therefore, there is no need to distinguish

between them. In particular, all of the conclusions drawn in Sec. 4.6 about

the effect on the thermalization length of the behavior of the absorption

' coefficient in line wings can be applied as well to the thickness of the boundary layer. As the rate at which the absorption coefficient decreases in the

' line wings becomes smaller, the thickness of the boundary layer increases, and

the dependence of x^ on the line shape is very strong. For nearly conservaI tive scattering the thickness of the boundary layer is very large. It may be

I several orders of magnitude larger than the mean free path of the line center

photon.

6.3 DEPARTURES FROM LTE IN THE BOUNDARY LAYERS OF AN ISOTHERMAL GAS

' FORMULATION OF THE PROBLEM. Let us consider the simplest physical problem,

,j which involves the solution of the equation discussed in Sec. 6.1. This pro-

' blem is ideally suited to illustrate the physics of the line transfer problem,

jl and is undoubtedly of interest in its own right.

i'

Consider a homogeneous, isothermal gas filling a half-space. The gas is

composed of atoms with the populations of the ground and first excited levels

equal to nx and n2 , respectively. In addition to the atoms, there are free

electrons, whose concentration ng is everywhere the same. The electron temperature is equal to the atomic kinetic temperature, and is thus independent

I of depth.

We shall consider two processes ivhich populate the excited level: (1)

transitions from the ground level due to electron impact, and (2) photoexcitation; and two processes which depopulate it: (1) downward radiative

transitions (spontaneous and stimulated) , (2) collisions of the second kind.

The problem is to calculate the degree of excitation nz/ni as a function of

i| depth. We assume that no radiation is incident upon the gas from the outside.

II i
260 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

As was shown in Sec. 2.5, this problem involves the solution of the

following integral equation for S(tJ:

oo S(t) = ^ J

K^(|t-t' |)S(T')dT' H- Bvq(T)(1-A) . (3.1)

Here S(t) is the line source function:

2h ^ 1 S(T) = ( £i !ll - i) , (3.2)

c2 \gi ^2 I

^vg (T) is Planck's function at the line center frequency vq , and X is the

probability that a photon survives the scattering process, vhich in this case

is related to the probabilities of the elementary processes by the expression

X

= — . (3.3)

21 e 2 1

1 — exp ^—

Thus, we are dealing with the equation considered in Sec. 6.1, in which

S* = (I-X)B^q(T) . (3.4)

THE EXACT AND APPROXIMATE SOLUTIONS. Using (1.9), we find that the solution

of (3.1) is

S(T) = (l-X)''%^(T)nT) . (3.5)

As was shown in Sec. 5.5, for 1 — X << 1 the function T(t) can be represented

in the form

^(t) ~ (l-X)~''^F(t)C(T) , (3.6)

where

t = - r(2Y)

^ . (3.7)

K2(T)

The merit of such a representation of *i'(T) is that the function C(t)

varies within narrow limits. For rough estimates it can safely be replaced

by unity.

For t << 1, i.e. x << t^, , we get from (3.5) and (3.6), with the use of

(5.5. 37) ,

S(t) ~ (1-X)\ (T) —^ t(r) , T << T, , (3.8)

0 /kTTTT
6.3 DEPARTURES FROM LTE IN THE BOUNDARY LAYERS OF AN ISOTHERMAL GAS 261

where C is given by (2.7). This result can also be written in the form

S(T) ~ C1-X)'^Bv^(T)S(t) , t << , (3.9)

where_^S(T) is the solution of the homogeneous equation (1.14), normalized so that S(0) = 1. Thus, in the surface layers the source function is proportional to the solution of the generalized Milne problem. The values of t in- crease as T increases. When t becomes of the order of unity, i.e. when t is

of the order of the thickness of the boundary layer t]-, , the rate at which the source function increases becomes smaller, and for t >> 1 it practically

vanishes. From (3.5) and (5.5.35) we find that for values of t in this region

7 K (t)

SCt) = B^^(T) 11 - —^-^ ^ ... j

, T » . (3.10)

By substituting (5.5.38) into (3.5) we get an approximate solution of (3.1):

S a

This approximation is extremely useful. It properly describes all the important physical features of the solution. If one needs only an estimate of the

solution to obtain physical information, this approximation always suffices.

In the physical analysis given below it would be possible to restrict our- selves to this approximation rather than to the exact results.

D. G. Hummer and J. C. Stewart (1966) have proposed another approximation, namely

S (T) = B^ (T)

^^-^'^ 'g^^^^

^

. (3.11b)

^ 0 (1+(1-X)S2 (t))^

This approximation is usually more accurate than (3.11a). However, to use it

one has to tabulate in advance the solution of the generalized Milne problem.

We note in passing that (3.11b) reduces to (3.11a) if the approximate form

(5.5.44) is substituted for S(t).

In the case of the Doppler profile the approximate form (3.11b) is accurate to within a few percent.

The results just obtained are shown graphically in Fig. 27, in which

plots of lg(S(T)/S(0) ) as a function of lg(/TfT) are given for the Doppler

profile (according to D. G. Hummer and J. C. Stewart, 1966).

PHYSICAL INTERPRETATION. It is evident from the figure that as t ^ «> the source function tends to a constant value. This result has the following

interpretation. Since the gas is isothermal and is not illuminated by external radiation, the deviation from thermodynamic equilibrium must arise because

radiation escapes across the boundary. As we go further from the boundary,

this process becomes less important and the degree of excitation ultimately

approaches the equilibrium value.

Close to the boundary the departure from equilibrium increases rapidly

and can become very large. We shall assume, for the sake of simplicity, that
262 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

Fig. 27. Depth dependence of the line source function in a medium with

uniformly distributed sources. Doppler profile.

the temperature of the gas is sufficiently low so that the average thermal

energy of the particles is substantially smaller than the excitation energy

of the upper level (kT << hvg). We shall also assume that the gas density

is so low that spontaneous transitions occur much more often than collisions

of the second kind (A2 i >> ngC2i). Neither of these assumptions is essential

and they are only made so that our conclusions will be more clear.

The first of these assumptions enables us to disregard stimulated emission in comparison with spontaneous emission. Formally, this assumption

leads to the following simplifications. First, X is given by

A

X = 21

A2l"^eC21

(3.12)

instead of by (3.3).

function. Third, (3,

Second, Planck's function can be

2) must be replaced by

replaced by Wien's

S(T)

^^^0 §1 ^

c2 g2

(3.13)

Let us discuss the second assumption. To order of magnitude C21 ~ '^^21'

where v is the average thermal velocity of the electrons (on the order of

108cm/sec for T = 10?°), and ^2.1 average value of the cross section,

within an order of magnitude of the gas kinetic value ( 10""!^ cm'^ ) . For allowed

optical transitions, A21 ~ 108sec~l. Therefore for tem.pe ratures around 104°K

the inequality A2 1 >>neC2i is satisfied for electron densities much less than

10l6cm-3. In as trophys ical problems one often deals with significantly lower

densities. In laboratory plasmas the densities and temperatures are such that

the condition A2 1 >>ngC2i is again often fulfilled. In such cases it can be

assum.ed that
6.3 DEPARTURES FROM LTE IN THE BOUNDARY LAYERS OF AN ISOTHERMAL GAS 263

^21

1 - X = n (3.14) ^21

and consequently 1 — X << 1. Substituting (3.13) and (3.14) into (3.5) and replacing Planck's function

with Wien's function, we find

n- -hVn/kT/C

n^ e

Specifically, for those values of t corresponding to t << 1,

g^ -hv /kT/C.

n2 = ^ e ^ (-±^jS(T) . (3.16)

Thus, for low electron concentrations and temperatures that are not too high,

the population of the first excited level of the atoms near the boundary of

an isothermal, s emi - infinite layer of a gas is proportional to the square root

of the electron density. We stress that this conclusion is valid only when

the destruction of photons in flight may be disregarded (i.e. for

1 — A << 36(3), see Sec. 7.6), and when photo- ionizations from the excited

level are infrequent compared with quenching collisions, which is the only

process competing with spontaneous transitions.

As follows from (3.7) and (2.2), for t << T^Q the corresponding values

of t are much less than unity. Therefore the region in which the expression

(3.16) is valid is restricted to values of t much less than the thickness of

the boundary layer. As the electron density decreases, the quantity 1 — X decreases in proportion to ng, see (3.14). The thickness of the boundary

layer then increases in proportion to ng—^/^Y, where y is the characteristic

exponent.

The decrease in the excitation toward the boundary may be described in

terms of the excitation temperature Tgy, defined by the relation

n

or, equi valently

,

12 ^-hvQ/kTg^

^1 Si

S(T) = B^o(Tg^(T)) . (3.18)

In Fig. 28 a plot of Tgj./T as a function of Ig t appears as an illustration,

The curve refers to the Doppler profile, with 1 — X = 10~6 and hvg/kT = 5.

The deviation of the excited level_ population from the Boltzmann value

indicates that the radiation intensity will differ from the Planckian. According to (1.13) and (3.4),
264 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

.00

X 0.50

log T

Fig. 28. Depth dependence of the excitation temperature in a homogeneous, isothermal gas (Doppler profile, hv^/kl =5, 1 — X = 10"^).

1(0, p,x) = B^^(T) (l-X)'''^H(y/a(x)) . (3.19)

Line profiles calculated from this expression are shown for a number of values

of X in Fig. 29. The plots refer to the case of the Doppler profile, with the

radiation emerging along the normal to the boundary, so that y = 1. The values of the Doppler H- functions used to construct these curves were taken from

V. V. Ivanov and D. I. Nagirner (1965). It is evident from the figure that

the boundary of a homogeneous, isothermal p-lasma occupying a half-space emits

Planckian radiation, which forms a background on which the absorption line is seen. As y decreases, the line becomes somewhat deeper, and its width increases slightly (Fig. 30). We emphasize that this absorption line appears in the

spectrum of a gas that is isothermal as regards trans lational degrees of freedom because an inner degree of freedom, the excitation, is not isothermal

(see Fig. 2 8) .
6.3 DEPARTURES FROM LTE IN THE BOUNDARY LAYERS OF AN ISOTHERMAL GAS 265

Throughout this section numerical data referring to the Doppler profile

have been used to illustrate general statements. We shall also show some results for the Voigt and Lorentz profiles. Fig. 31 illustrates the effect

of the Voigt parameter a on the depth- dependence of the source function (according to E. H. Avrett and D. G. Hummer, 1965) . The corresponding line

profiles computed from thie tables of H- functions and their asymptotic forms

I I

I I 1 I 1 I I 1 1 L -3-2-1012345678

log t/A

Fig. 31. The line source function in a homogeneous isothermal gas with

1 — X = 10"*. The numbers on the curves are the values of the Voigt

parameter a.
266 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

given in Sec. 5.7, are shown in Fig. 32. From these figures it is evident

that as a increases, the thickness of the boundary layer increases rather

rapidly. The line has a clearly expressed Doppler core and wide damping

wings .

The problem considered in this section was first formulated and studied

by E. A. Milne (1930) who essentially assumed the rectangular line profile.

Under the assumptions of complete frequency redistribution and a Doppler profile the problem was first considered by L. M. Biberman (1949). Biberman

obtained (3.1) for the case of hvg >> kT and solved it by iteration for two

values of A. A. G. Hearn (1962, 1964b) re-examined the problem under the same assumptions as those used by Biberman. Using numerical methods, he

found the source function and calculated the line profiles for several values of X, some of them rather close to unity. For the more general case in

which both Doppler and collisional broadening determine the line shape, the

problem was first studied by E. H. Avrett and D. G. Humme.r (196 5). Equation

(3.1) was solved numerically not only for constant X, but also for A depending on T (E. H. Avrett, 1965).

6.4 DIFFUSE REFLECTION

EMERGENT INTENSITY EXPRESSED IN TERMS OF THE H- FUNCTION. Let us now consider

the radiation field in a medium that is illuminated from the outside. We

shall assume that the medium is illuminated by parallel beams of monochromatic radiation which is constant in intensity from point to point on the boundary. We shall Be particularly interested in determining the angular and

frequency distribution of the emergent radiation. This is the so-called problem of diffuse reflection.

Fig. 32. Intensity of normally emerging radiation for 1 — A = 10 Numbers on the curves are values of the Voigt parameter a.
6.4 DIFFUSE REFLECTION 267

A rigorous solution o£ this problem for conservative isotropic mono- chromatic scattering was obtained by E. Hop£ (1934). He expressed the intensity o£ the emergent radiation in terms of the corresponding H-function and

obtained an explicit integral representation of this function. Hopf's result

can be extended to the case of scattering with complete frequency redistribution with an arbitrary profile.

We shall consider incident radiation of unit irradiance on a surface

perpendicular to its direction of propagation, with dimensionless frequency

xq and the cosine of the angle of incidence yg. We shall denote by y the

cosine of the angle between the direction of propagation of radiation and

the external normal. The problem then reduces to the solution of the transfer equation

1

^ dI(T^,y.x)

^ a(;x)I(T,y,x) - A Aa(x) J

a(x')dx' JI (x , y

' , x ' ) dy ' -

-1

(4.1)

with the boundary condition

1(0, y,x) = 0 , y < 0 . (4.2)

The line source function corresponding to this case satisfies the equation

00 S(T,yQ,XQ) = J

y* K^(|t-t' |)S(T' ,yo,XQ)dT' +

(4.3)

A A , , -(a(Xo)/yQ)T

4iT Av ^

From this it is apparent that the quantity

depends only on t and on the ratio

Zr, = ^0

'0 a(xo)

and is determined by the equation

-t/z,

(4.5)

S(t,Zq) = A

J

k^(|t-t' 1)S(T' ,ZQ)dT' + e V (4-6)
268 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

i.e. it is identical with the auxiliary function introduced in Sec. 5.1.

The intensity of the emergent radiation 1(0, y,x; y^.x^) is expressed in

terms o£ the source function by

00 I(0,y,x;yQ,XQ) = J

S

(

t , y^

,

Xq ) e" )^ a (x) di /y . (4.7)

0

Hence, according to (4.4) it follows that 1(0, y,x; yg.xg) can be represented

as a product of a(xQ) times a function depending only on z s y/a(x) and zq.

For brevity we shall omit the arguments yg and Xg and write the emergent intensity simply as 1(0, z).

In order to find 1(0, z), the expression (5.1.45) can also be used instead

of (4.7), which in the case at hand can be written as:

I(0,z)=-A.A / S(T,z)e ^ '°a(x )dT/z . (4.8)

4iT Av y ^

Using (5.1. 44) , we get

, H(z)H(z ) 1(0, z) = ^ y , (4.9)

4ttAv ^'^0

with zg given by (4.5). Since the function H(z) can be regarded as known,

this result gives the solution of the problem of diffuse reflection.

We note that the line source function S(T,yg,xg) is given by the expression

(—

t/z f — (t—t')/z \

e

^ ^ J ^ °<i>(T')dT'j , (4.10)

0

'

which follows from (4.4) and (5.1.36). Of course, this expression could also

be obtained from the general result (5.1.21), if (5.1.31) were used. However,

for the simple case under consideration, this method would scarcely be the

most direct. The integral term in (4.10) can be rearranged by partial integration. We get as a result

'^"'"O'^O^ - ^

Aa(Xo)H(zQ)^T(T) -

\^ j

e %(T')dT'j . (4.11)

The advantage of this representation as compared to (4.10) is that >1'(t') re- mains finite as t' -> 0 while $(t') diverges. Hence (4.11) is more useful for

computati on.
6.4 DIFFUSE REFLECTION 269

It is clear that expressions (4.9) and (4.10) specify the intensity o£

emergent radiation and the source function not only in a medium illuminated

by monochromatic parallel rays, but also in all cases in which there are internal radiation sources in the medium whose strength decreases exponentially

with depth. An example of this will be discussed in detail in the next section.

ANALYSIS OF THE LINE PROFILE. Let us examine the results given by the ex- pressions we have obtained. The spectral distribution of the reflected radiation, i.e. the profile of the line, seems very distinctive. Let us rewrite

(4.9) by substituting z = ij/a(x) , in order to exhibit explicitly the dependence of l(0,z) on the frequency x and on the angular variable p. We obtain

XA

1(0, y,x) =

^ a(x)y . (4.12)

4TrAv v + z^aix) ^

The line profile r(y,x), normalized so that the normally emergent (y=l) intensity at line center (x=0) is set to unity, is given by the expression

Trn > (1+Zn)ci(x) h( . ) ^ I (0 ,y ,x) ^ 0^ ^ \a(x)/

1(0,1,0) y+ZQa(x) H(l)

If the angle of incidence of the illuminating radiation tends to 77/2,

then \iq and Zq approach zero. From (4.13) it is evident that for Zg = 0

"(a(x)) r(y,x) = _x±s±i/_ , (4.14)

y H(l) ^

In this case the external rays hardly penetrate into the medium, but only

irradiate its outermost layers. Here, therefore, we are faced with the problem of emission by a medium whose sources are concentrated at the boundary.

From (4.14) we see that in this case an emission line is formed. Actually,

although the function H(y/a(x)) increases with the distance from the line

center (i.e. as |x| increases), this increase is insufficient to counteract

the rapid decrease in the factor a(x). As a result, r(y,x) appears to be a monotonically decreasing function of |x|.

The opposite limiting case is obtained when radiation in the far wing of

the line (|xg|>>l) is incident on the medium with any angle arccos yg it/2.

Then a(xQ)<<l, and the parameter Zg is large. In the limit as z„ ^ <» we have

r(y,x) = (4.15)

As the distance from the center of the line increases, the quantity y/a(x)

,

and hence H(y/a(x)), increases. Therefore r(y,x) is now an increasing function of |x|, i.e. we have an absorption line. Comparing the last expression

with (1.13), we see that they are essentially identical. This is hardly sur- prising. The greater the quantity Zq , the deeper the external radiation

penetrates without significant attenuation. In the limit as zg 0° we get a

[ uniform distribution of sources through the medium, for which the result

[ (1.13) has already been derived.
SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

For any finite value of zq an emission line is formed, which increases

in width with zq (Fig. 33) . When zq is large enough a depression appears at

the center of the line, whose depth increases with zg . Simultaneously, the

separation of the intensity maxima increases. The limiting shape of this cen- tral self- reversal is given by (4.15). Conversely, in the far wing of the

line, where ij/a(x) >> Zq , from (4.13) we have

a(x) \a(x)/ r(vi,x) = (1 + z ) _1^|±2^ + ... . (4.16)

u y H(l)

Thus, in the line wings the frequency dependence appears to be the same as for Zg = 0: the intensity decreases in proportion to H (y/a (x) ) a (x) as |x|

increases. Let us consider this dependence in a little more detail.

If X is not very close to unity, then it may be assumed that H(y/a(x))

is approximately equal to (1 — A)~'^, since in the line wings y/a(x) >> 1. Therefore instead of (4.16) we get

r(y,x) = —— + . .. , (4.17)

H(l) y(l-X)'5

omitting terms that approach zero more rapidly than a(x) . Hence, the intensity in the far wing of the line is proportional to the absorption profile.

^ I 50 -

X

Fig. 33. Line profiles for exponential distributions of primary sources. Doppler profile. Curves are for the case X = 1, y = 1. Numbers on curves

are values of l/zg-
6.4 DIFFUSE REFLECTION 271

When X is close to unity, the situation is slightly more complicated.

Clearly (4.16) is still valid. However, it is now possible for a region of

frequencies |x| to exist for which the condition ij/a(x) >> zq for the validity

of (4.16) is satisfied, while H(y/a(x)) is still much less than its limiting

value (1 - A)~^. For such frequencies the function H(y/a(x)) increases rather

rapidly with |x|, and therefore r(y,x) decreases considerably less rapidly

than a(x). In this case the general asymptotic expressions obtained in Sec.

5.4 must be used for H(ij/a(x)). Frequencies |x| >> 1 may exist, for which

(4.16) is applicable, while the asymptotic form corresponding to pure scattering can still be used for H(y/a(x)). Such cases arise when 1 — X << 1 and the

value of Zq is much less than the thickness of the boundary layer . If this

situation occurs, then, for example, with the Doppler profile, we can use

(4.16) and (5.4.33b) to obtain the expression for the normally emergent intensity in this frequency region.

l+z„ 2 2 r(l,x) = :—^ e Hp(e^ ) + = 2ti'

l + z,

Hj3(l)

(4.18)

Thus the intensity in this case decreases with the distance from line cejiter

in proportion to /| x| exp (— x^/2) , i.e. considerably less rapidly than the

absorption coefficient exp(—x^).

Going further from the line center, z = y/a(x) becomes so large that the

rate at which H(z) increases becomes smaller, at first only slightly, and then

more and more strongly. Ultimately it nearly vanishes; H(z) approaches its

asymptotic value H(oo) = (1 — X) 2, and the intensity of the emergent radiation

becomes proportional to the absorption coefficient a(x)

The result (4.13) also makes it possible to trace the dependence of the

line profile on y, the angle of emergence. The . y- dependence describes the

change in the shape of a line going from the center of the disc of a star or planet (y = 1) to its limb (y = 0). The general nature of the changes occuring is clear from Fig. 34. As y increases the line becomes wider and the

central depression becomes deeper.

Fig. 34. Line profiles for various angles of emergence. Curves refer to

Doppler profile, X = 1 and 1/zq = 0.30.

522-519 O - 74 - 19
272 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

We are not going to make a special study o£ the i-dependence of the source function S(T,yQ,XQ), i.e. the variation in the degree of excitation

with depth, but instead we shall limit ourselves to some very brief remarks.

Vifhen Zq is small, the degree of excitation decreases mono toni cal ly with depth

In this case an emission line is formed (see Fig. 33, p. 270). As Zq increas

es the T-dependence of S(T,yo>^o) develops a maximum. In this case an emission line is formed with a central se 1 f - revers al , which reflects the drop in

the excitation temperature toward the boundary. As zq increases further,

becoming much larger than the thickness of the boundary layer, saturation

sets in. The ratio of the degree of excitation at the maximum to that at the

boundary increases much less slowly as zq continues to grow and the maximum

on the curve of S(T,yQ,XQ) becomes ever flatter. Corresponding to this," the

line profile also changes — the intensity maximum becomes less and less sharp

ALBEDO OF THE MEDIUM. In many instances it is necessary to know the total

energy reflected in the line in all directions per unit surface. We denote

this quantity as E.

The energy leaving the medium in a spectral line through a unit area per

unit time within the solid angle doj is equal to

CO

0

The total energy E reflected per unit time in all directions is

CO 1 (4.19)

0 0

Substituting (4.9) here, we get

(4.20)

Instead of y we introduce the variable z = y/a(x). Then

1 a(x)

(4.21)

Changing the order of integration, we get

00

0

(4.22)
6.4 DIFFUSE REFLECTION 273

From the equation for the H-function written in the form (5.4.18) it follows

that

00

Therefore, finally,

liilll G(z)dz = —i /1-X . (4.23)

^^^0 H(Zq)

E = PqCI-IUZq) /1-X ) . (4.24)

Since we are assuming in this section that the emission of the medium

arises from its illumination by parallel rays, the reflected energy is naturally expressed in terms of the incident energy. The latter quantity is

equal, per unit area, to Eg = yg.' We therefore find that of all the energy

incident on the boundary of the medium, the fraction

Ag = |- = 1 - H(Zq)/~ (4.25)

0 is reflected, while the fraction

1 - Aq = H(Zq) /1-X (4.26)

is absorbed in the medium. The quantity A^ is known as the albedo of the

me d i um

It follows from (4.26) that the role of absorption increases with Zq . This is related to the fact that as zg becomes larger the external radiation

penetrates deeper into the medium and the number of scatterings experienced

by a photon before escaping increases. Since for each scattering there is a certain probability that the photon will be destroyed, 1—Ag will increase

with Zg.

From (4.26) it is easy to obtain the mean number of scatterings IT ex- perienced by a photon. For this it suffices to divide the fraction of the

photons absorbed in the medium by the probability per scattering that a photon is destroyed, which is equal to 1 — A.. As a result we get

N = (1-A)"'2h(z.) . (4.27)

(See also Sec. 6.8, where this expression is found from other considerations.)

As has already been mentioned at the beginning of the section, many of

the relations found here are analogous to results obtained earlier for the

case of monochromatic scattering. The method we are using was developed by

V. V. Sobolev (1956). The present discussion is based on a paper of the

author (V. V. Ivanov, 1962b), but contains much more detail.
274 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

6.5 EXPONENTIAL DISTRIBUTION OF SOURCES

FORMULATION OF THE PROBLEM. In the preceding section we mentioned that the

results obtained there apply not only when parallel rays are incident on the

medium, but also when there are internal radiation sources within the medium

whose strength varies exponentially with depth. Let us discuss in detail an example in which the problem reduces approximately to this situation.

Earlier, in Sec. 6.3, the concentration o£ atoms in the first excited

level in a homogeneous, isothermal gas was calculated on the assumption that

the line broadening was caused by thermal motion of the atoms, i.e. the absorption coefficient was assumed to have a Doppler profile. The population

of the upper level was assumed to be established by the balance of radiative

and collisional excitations, on one hand, and downward radiative transitions

and quenching collisions, on the other. Let us now discard the assumption

that the gas is isothermal.

When the temperature is not constant, the rates of electron collisions

of the first and second kinds, C12 and C21, which depend on the electron temperature T, will now be functions of x, whose explicit form is determined by

the deptn dependence of the temperature. However, this is not the only way

that temperature variations affect the situation. The Doppler width of the

line, in units of which the frequencies are measured, also now becomes a function of t. Consequently the statistical equilibrium equation no longer

reduces to an integral equation with a kernel depending on the modulus of the

difference of two depths. Therefore an exact treatment of this problem seems

to be much more complicated than for those previously considered.

We shall disregard the dependence of the Doppler width Avp upon temperature, as our first additional approximation. This assumption will be used

throughout the rest of this section. In this approximation the equation of

statistical equilibrium, with stimulated emission neglected, has the form

A /* n^Ci) (A2^+ngC2P =-|i J

( | t-t '

| (t ' ) dx ' ^n^n^C^^ ' ^^'^^

0

The quantity Cji depends weakly upon the temperature, and in an approximate

treatment of the problem this dependence can also be neglected. However,

because of the presence of an exponential factor in the relation between C21

and C12 > the temperature dependence of C12 appears to be strong and cannot

be neglected. The probabilities of the elementary processes will be assumed

to depend on T only via this exponential factor. This will constitute the

second additional approximation. The value of C21 can, for example, be taken

to correspond to the temperature at the depth t = 0. Under these assumptions

(5.1) may be rewritten as

, / 2hv^ -hv./kT(T)

S(t) =- / K (|t-t' l)S(T')dT' + (i-X)-—^ e , (5.2)

0 ^ -

where S(t) is the usual line source function, related to the level populations

by (3.13), and the quantity X is expressed in terms of the probabilities of

the elementary processes by (3.12).
6.5 EXPONENTIAL DISTRIBUTION OF SOURCES 275

If the depth- dependence o£ the temperature has the form

T(0) T(T)

l+mi

(5.3)

where m is a constant, then (5.2) becomes

S(T)

oo

t' I )S(t' )dT' + Qe

-t/z.

(5.4)

where

e 21 2hvQ -hvQ/kT(0)

(5.5)

'0 kT(0) 1 hv

(5.6)

0

Thus in this case we are dealing with the radiation field of a medium with

an exponential distribution of sources, i.e. we have recovered the problem

analyzed in detail in the preceding section. Using the results obtained

there, we can at once write the expression for the intensity of the emergent

radiation

1(0, y,x) = Qz

0

H(z)H(Zq)

(5.7)

where z = ij/a(x). Our aim is now to analyze this expression, with the purpose of showing how the shape of the line emitted by the medium can be used

to estimate the variation of the temperature with depth. It is necessary to

understand clearly that this analysis is valid only for those extremely

specialized assumptions that we have just made. This analysis does not give

a ready-made prescription for practical use. On the contrary, this analysis

might serve as a warning to anyone attempting to draw conclusions about the

physical conditions in a gas from the form of self- reversed lines.

ANALYSIS OF THE LINE PROFILE. As is seen from (5.7), the profile of the line

emitted by the gas depends on the value of the parameter zq. According to

(5.6) this parameter, in turn, is related to the quantity m, i.e. it depends

on the depth distribution of the temperature. This fact can be used to obtain the depth dependence of T through the line profile.

The probability X that a photon survives the act of scattering tends to

unity as the electron density decreases, whereas the value of zq, according

to (5.6), is independent of density, and under the present assumptions is

determined only by the temperature distribution. The thickness of the boundary layer increases as the density decreases and for sufficiently low n^ it

becomes much larger than Zr,. In this limiting case, as has been shown in the

preceding section, the profile of the line ceases to depend on X (in (5.4)

the quantity X can be set equal to unity) and is instead determined entirely

by the value of Zq. Thus the line profile in the limiting case of low densities is determined only by the depth- dependence of the temperature.
SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

Figure 33 (p. 2701 shows line profiles for radiation emerging in the

direction of the normal to the boundary (y = 1), computed from (5.7) for the

Doppler absorption coefficient and X = 1. The value of l/zg = fhvQ/kT(0))m

is given as the parameter of the curves. It is evident from the figure that

as m becomes smaller, i.e. as the temperature decreases more slowly with

depth, the line becomes wider and the depth of the central self -re vers al in- creases . The first conclusion that follows from this figure is that in and of itself a distribution of kinetic temperature that increases toward the surface

does not guarantee the appearance of an unreversed emission line. It is only

when the temperature increase occurs rapidly enough (small m) that the line

will not have a depression in the center. What causes the appearance of this

intensity minimum? Because the temperature increases toward the boundary,

the number of collisional excitations increases as t becomes smaller. However, at the same time, the escape of radiation from the medium becomes more important and causes the number of photo-excitations to decrease. If the

temperature does not increase rapidly enough, the second factor dominates,

causing the concentration of excited atoms to decrease near the boundary. In

this way the appearance of a self- reversal in the center of the line may be

unde rs tood.

The second conclusion that may be drawn from Fig. 33 is that the depth

of the central minimum is significantly more sensitive to the temperature distribution than the width of the line as measured, for example, by the separation of the intensity maxima. Let us consider this point in a little more

detail. The frequency at which the intensity maximum occurs is defined by

the condition

^nO,V,^) . 0 , (5.8)

9x

which, by means of (5.7), may be expressed in the form

dH(z) ^ H(z)_ ^ - ^5^9-,

dz z+Zq

We denote the root of this equation by z^. Knowing this root, we can calculate the frequencies x^^ of the intensity maxima from the condition

a(x^) = -H- . (5.10)

^m

Specifically, for the Doppler profile

= -Un^l . (5.11)

For small values of m, when the temperature varies slowly with depth,

the line width depends only slightly on m. We shall verify this by finding

explicitly the dependence of |xj^| on m for small m. When m is small enough,

the value of zq , as is seen from (5.6), becomes large. In this case the root

of equation (5.9), i.e. z^,^, should also be large. Therefore in solving equation (5.9) we can use the asymptotic expression for H(z) (see Sec. 5.4),
6.5 EXPONENTIAL DISTRIBUTION OF SOURCES 277

(5.12)

noting that this expression leads to the following equation for z^:

(5.13)

whence z Therefore for sufficiently small m we have instead of (5.11)

1/2

(5.14)

-1 Since Zq is proportional to m , it follows from the last expression that for

small m the distance between maxima increases approximately as {In 1/m)-'-/^,

i.e. very slowly indeed.

It is also easy to estimate the depth of the central minimum. From

(5.7) we have

1(0, y,x ) y + z„ H(z ) 1(0, y,0) Zj„ + Zq H(y)

(5.15)

When m is small, as we have seen, z^ Zq >> 1 for the Doppler profile, and

instead of the last expression we have approximately

-1/4

1(0, y,0) Hp(y)

(ZpAnzQ) 1/2 (5.16)

Thus the maximum intensity, expressed as a fraction of the line center intensity, increases as m decreases, roughly speaking, in proportion to vZg , i.e.

as l//m. Although this dependence is not very strong, it can be used to • determine the value of m from the line profile. We can conclude generally

that for a Dopp ler-b roadene d line emitted by a hom.ogeneous non- is othe rmal gas,

with a temperature that decreases slowly with depth, the line parameter most

sensitive to the temperature distribution is the depth of the central selfreversal.

The situation is quite different if the line is collis ionally broadened,

so that the absorption coefficient can be assumed to be Lorentzian. We shall

restrict ourselves to the case in which the temperature varies slowly with

depth (small m) . Our goal will be to show how the width of the line and the

depth of the central depression depend on m. The line of reasoning used above

for the Doppler line can be followed here as well, except that in order to obtain the root of equation (5.9) for small m we must now use the asymptotic

expression not of the Doppler, but of the Lorentz H- function, which has the

form (see Sec. 5.4)

Hl(z) (I)

1/4 1/4 (5.17)

Substituting this expression into (5.9), we find that z^ is given by the
2 78 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

equation

(5.18)

whence z Zg » 1

' Zo/3, Thus from (5.10) we have for the Lorentz profile and

0 3p

(5.19)

where | x„ | is the distance of the intensity maxima from the center of the

line, measured in collisional widths

follows from (5.15) and (5.17) that

As to the height of the maxima, it

1(0, y,x^) ,5/4 1/4

1(0, y,0) 29/4Hj^(m)

(5.20)

The quantity Zq is proportional to 1/m. Therefore, according to (5.19), as m decreases the separation of the intensity maxima increases in proportion

to l/Zin, whereas the height of the maxima increases only as m~l/4; see (5.20)

Hence, for the Lorentz profile the line width is more sensitive than the

depth of the central depression to the temperature distribution. We recall

that for the Doppler profile the situation was exactly the opposite. This

example demonstrates how carefully one must analyze the profiles of emission

lines in order to obtain information about physical conditions in a medium.

The analysis in this section has been based on the assumption that the

effect of temperature variation on the Doppler width may be ignored. This

and other approximations discussed at the beginning of the section are very

rough. A more careful discussion of the problem of line formation in a medium with the temperature depending on depth has been given by D. G. Hummer

and G. B. Rybicki (1966) and G. B. Rybicki and D. G. Hummer (1967), who used

a numerical approach.

6.6 GAS ILLUMINATED BY CONTINUUM RADIATION

THE LINE SOURCE FUNCTION AND EMERGENT INTENSITY. From the results of Sec.

6.4, it is easy to find the intensity of the diffusely reflected radiation

and the degree of atomic excitation in several other simple cases

First we shall obtain the line source function for a gas illuminated by

isotropic radiation whose intensity is independent of frequency within the

line. Thus we may assume that continuum radiation is incident upon the

medium from all directions. This illuminating radiation is a superposition

of separate beams, each being characterized by its angle of incidence

arccos and frequency Xq . It is evident that the line source function must

then also be a superposition of the functions S(T,yQ,XQ) determined by (4.3).

Therefore, if the intensity of the incident radiation is Ig, the line source function for this problem is

S(T) = 1^2.

1 00 / /

S(T,yQ ,XQ)dvQ (6.1)
6.6 GAS ILLUMINATED BY CONTINUUM RADIATION 279

From (4.3) it now follows that the source function satisfies the equation

oo S(t) = ^ J

K^(|t-t' hS(T')dT' + IojK2(t) , (6.2)

where

00 1 oo K2(t) = A /*a(x)dx/'e~("^^^/^)^dy = j e'"^/^G(z)dz . (6.3)

-o. Q 0

The solution of this equation has the form

S(t) = Iq[1-(1-X)^/2>^(t)] , (6.4)

where, as usual,

T I'(t) = 1+

J

$(T')dT' . (6.5)

This result is readily verified by direct substitution of (6.4) into (6.2),

if the identity

00

I /

K^(1t-t' |)dT' = 1 - i K2(t) (6.6)

0

and the results obtained in Sec. 6.1 are used. The intensity of the emergent

radiation corresponding to the source function (6.4), as is easily seen, is

1(0, y,x) = Iq (6.7)

PHYSICAL SIGNIFICANCE OF THE RESULTS. SIMILARITY RELATIONS. Let us elucidate

the physical meaning of these results. First we express the intensity Ig of

the radiation incident on the boundary in terms of a radiation temperature Tjthrough the equation (see Sec. 1.1)

Iq = B^^(T^) , (6.8)

where By)q(T) is Planck's function. Also, instead of the line source function

S(t) we introduce the excitation temperature Tg-j^-, i.e. we define

S(T) = B (T_(T)) . (6.9)

^0

^
280 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

Then (6.4) assumes the form

^VQ^'^ex^ = B^^(T^) [l-(l-X)^/2nT) ] . (6.10)

We shall consider the case that is most interesting from the as trophys i cal

point of view, in which X is very close to unity. Then, as was shown in Sec.

5.5, with an accuracy which increases as X approaches unity,

(1-X)^/^Y(t) ~ F(t)aT) , (6.11)

whe re

t = 1 r(2Y) . . (6.12)

^ K2(t)

Replacing ^(t) with unity here, we obtain an expression that is significantly

in error only for small t. However for small t quantity (1—X)1/2t(t) may be

neglected in comparison with unity in (6.10). Therefore, substituting (6.11)

with ^(t) = 1 into (6.10), we get an asymptotic relation that is valid for

all T, with an accuracy that X approaches unity:

B,^(T^,) ~ B^^(Tp[l-F(t)] , 1 - X « 1 . (6.13)

At depths small in com.parison with the thickness of the boundary layer

, the parameter t is small; from the last equation it follows that

T ~ T , T << T, , (6.14) ex r ' b ' ^

-'

since F(t) 0 as t -> 0 (see (5.5.37)). Thus for t << the degree of excitation is nearly independent of depth, and is such that the excitation temperature of the upper level relative to the lower is equal to the temperature

of the incident radiation. Starting with depths of the order of 15, the degree of excitation begins to drop. Here the situation is the direct opposite

of the one we found in Sec. 6.3 in our study of a self-luminous isothermal

gas. There throughout the boundary layer of thickness the excitation temperature Tex substantially less than the gas temperature T. Now, on the

contrary, within a layer of the same thickness the radiation and the population of the upper level are found to be in equilibrium, in the sense that

'^ex ~

'^r* other words, T]-, represents the depth to which effective interaction occurs between the external continuum radiation and the gas.

There is one important consequence of (6.13): for 1 — X << 1 the degree

of excitation does not depend on t and X separately, but only on the combination (6.12). In other words, the following similarity relation holds: the

degree of excitation at optical depth t = ti in a medium with X = Xi , illuminated by continuum radiation, is the same as that in a medium with x = -^2 at a depth t = 12, where and t2 are related by

(1-X2)K2(t^) = (1-X^)K2(t2) • (6.15)

The closer Xi and X2 are to unity, the more accurate the similarity relation

becomes
6.6 GAS ILLUMINATED BY CONTINUUM RADIATION 281

When the temperature o£ the incident radiation is large, a significant

fraction of the atoms in the boundary region enter the excited state, so that

the optical depth at any point is substantially reduced. In the case of a semi - infini te medium this effect has no influence on the angular and frequency

distribution of escaping radiation; however, it must be taken into consideration in relating the optical to the geometric depths. The relevant technique

was presented in Sec. 2.5.

HOT GAS ILLUMINATED FROM OUTSIDE. In the case so far considered, the external radiation was assumed to be the only source of atomic excitation in the

medium. A more complicated case can be studied, in which there are internal

sources as well as the external radiation. We shall consider one example.

Let us assume that there is incident upon a homogeneous, isothermal gas

of kinetic temperature T, occupying a half-space, isotropic radiation whose

intensity does not depend on frequency within the line and is characterized

by the radiation temperature T^.. Two mechanisms for populating the upper

level are operating — electron impact and photo-excitation, and two mechanisms

for depopulating it — radiative transitions and collisions of the second kind.

In short, all the assumptions of Sec. 6.3 are retained except for one: we now assume that external radiation is incident upon the gas, whereas in Sec.

6.3 no such radiation was present.

It is clear that in this case the source function is determined by the

following equation:

00 S(t) = A f K (It-t- |)S(T')dT' + B (T )Ak (t) + B (T)(l-X) , (6.16)

^ ""O ^ 2 ^ ^0

where X, as before, is given by (3.3). Here the last term on the right allows

for collisional excitation, the second term for photo-excitation induced by

the external radiation and, finally, the integral term for radiative excitations by the gas' own radiation. The solution of (6.16) is a linear combination of the solutions of equations (6.2) and (3.1), and has the form

S(t) = B^, (T ) [l-(l-X)^/^>i'(T) ] + B^^ (T) (1-X)^/2h'(t) . (6.17)

Our task is to explain the nature of this solution. In order to make the

results easier to visualize physically, we shall transform from the source

function S(t) to the excitation temperature Tex> through the relation

S = BvQ(Tgx)' We shall denote T^^^ the excitation temperature of the upper

level m the same medium when no radiation is incident from outside, i.e. for

T^ = 0. It is obvious that T^^^ is defined by the relation

B^^ (T° ) = B (T)(1-X)^/^'I'(T) . (6.18) ex Vq

If the function I'(t) is known, there is little difficulty in computing the

excitation temperature as a function of x according to (6.17). We shall limit

ourselves to one illustrative example and use it to explain the general nature

of the results in various cases. The discussion will center on results obtained for the Doppler profile with the following parameter values: 1—X=10~6;

hvg/kT = 5; T-^/T = 2.0, 1.5, 1.0, 0.8, 0.5, 0.0. The results of a numerical

solution of the transfer equation, made available to the author by D. G.
282 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

Hummer (see also E. H. Avrett and D. G. Hummer, 1965), were used to find

^(t). From the calculated values of the excitation temperature Tg;j^(T) , the

plots of Tgj^(T)/T as a function of Ig x appearing in Fig. 35 were constructed.

This figure makes it possible to understand how analogous curves would behave

for other values of the parameters. Values of T^ fall naturally into three

regions, corresponding to the three types of curves in Fig. 35:

1 . The radiation temperature T^ is higher than the gas temperature T . Then for depths much less than the thickness of the boundary layer, the

excitation temperature Tg^j^ is practically constant, and is close to Tj-. Then

it begins to fall and ultimately it becomes equal to T (curves corresponding

to Tj./T = 2.0 and 1.5).

2 . The radiation temperature Tj. is less than T, but significantly higher

than T^x TO), i.e. the excitation temperature at the boundary of a medium not

illuminated from outside . In this case a region also exists near the boundary in which the excitation temperature is close to Tj. and varies only

slightly. However, the size of this region is now determined not by the

thickness of the boundary layer, but by the difference T — Tj., and increases

as this difference decreases. Further from the boundary the excitation temperature increases and approaches T (curves Tj-/T =0.8 and 0.5).

3. The radiation temperature T;^- is significantly lower than T^^^ (0 ) . In this case the external radiation is not significant in determining the

atomic excitation. At all depths, beginning with the boundary itself, Tgx(T)

differs little from tO^(t) (curve T^/T =0).

The profile of a line emitted by the gas is also easily expressed in

terms of the function H(y/a(x)) for the present problem. Without dwelling on this question in detail, we mention that if the temperature of the incident

radiation T is higher than the gas temperature T, an emission line is seen

Fig. 35. Depth dependence of the excitation temperature in an isothermal

gas of temperature T, illuminated by radiation of temperature T . Doppler

profile, hv /kT =5, 1 - X = 10-6. ^
6.7 SOURCES OF OTHER TYPES 283

against the background of the continuum, and the greater the di fference between

Tj. and T, the greater the intensity of this line. For Tj. = T we have a purely

continuous spectrum without any trace o£ a line, and for T^. < T an absorption

line is formed whose depth increases as T decreases.

6.7 SOURCES OF OTHER TYPES

The intensity of the emergent radiation can be expressed in terms of the

function H(z) for a number of situations in which the source strength depends

on depth in more complicated ways than those already discussed. In this sec- tion we shall consider the case in which the source strength is given by the

product of a polynomial in x and an exponential (V. V. Ivanov and D. I. Nagi rner , 196 5).

RECURRENCE RELATION. It is obviously sufficient to express in terms of H(z)

the emergent intensity corresponding to

S*(T) = S*(t) = T^e ^^^^

, n = 0,1,2, 0 < Zq < co . (7.1)

V/e denote the corresponding source function as Sn(T,zo), and the intensity of

escaping radiation as Ij^(0,z,zq). We have the following equation for Sj^(t,zo);

oo T / Z S^(T,Zg) = I J

K^C|t-T' |)Sj^(t' ,ZQ)dT' + I'^e

^ '°

. (7.2)

-t/z,

f K^(|t-T' |)S^(T' ,ZQ)dT' +

0

Differentiating with respect to , we get

00 3S (t,z.) r 9S(t',z) n+1 -t/z. ^=7 / K(1t-t'|)-^4 ^dT' V (7.3)

oZ„ L J oZn 7^

0 0 • " 0

Comparing these two equations, we conclude that

From the expression

9S (T,z )

"

0

1^(0, z,Zq) = y S^(T,ZQ)e ^/^dx/z (7.5)

it therefore follows that

81 (0,z,z )

From (7.2) for n = 0 we have So(t,zo) = S(x,Zg), where S(x,z) is the auxiliary
284 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

function introduced in Sec. 5.1. Therefore

/

S(T,ZQ)e —t / z di/z (7.7)

As this integral has already been evaluated in (5.1.44), we find

HCz)H(z ) 0-

z + z

0 0 (7.8)

Thus the solution to our problem is contained in (7.6) and (7.8), which

express Ij^(0,z,zo) in terms of H(z) and its derivatives for any integral

n > 0. By using the equation satisfied by H(z) it is easy to express the

derivatives of the H-function in terms of H(z) and integrals involving it.

PARTICULAR CASE. ALTERNATIVE FORM OF THE RECURRENCE RELATION. As an example,

we shall find 1^(0, z,Zq). From (7.6) and (7.8) we have

I (0,z,z ) = ZnH(z)-- 1 1 = z.z — + z. — . (7.9)

1 0 0 dzoV z + Zq / (z + Zq)2 0 z + z^ dzQ

From (5.4.18) it follows that

dH(z ) dzQ 0

oo A f zH(

0 ^ Q

z) G(z)dz . (7.10)

Substituting (7.10) into (7.9), we finally obtain

1^(0, z,Zq) = H(z)H(Zq) • 2

z + z. + A z H(z )f^(z ) z+zq 2 ^ ^

^ (7.11)

where

fi(z)

00 H(z')

z+z' )

G(z')dz' . (7. 12)

A table of the function 9.(z) for the Doppler profile and X = 1 has been given

by V. V. Ivanov and D. I. Nagirner (1965).

The recurrence relation for Ij^(0,z,zo) may be written in a form more convenient for calculations. According to (5.1.45),

lj^(0 ,z ,Zq)

00

0

—t/z

S(T,z)dT/z (7.13)

Multiplying (5.1.35) by t e and integrating over x from 0 to <=° , we find,
6.8 MEAN NUMBER OF SCATTERINGS 285

after simple rearrangement,

^n^O'^'^O^ = 777" ^n-l^O'^'^O^ ' TTT "^^^^^^0^ ' ^ = ... , (7.14)

Z+Zq ^ Z + Zq

where

°n^^0^ = 7 / I^(0,z,ZQ)G(z)dz . (7.15)

0

Substituting (7.14) into (7.15), we get the following relation for the determination of D (Zn):

n ^ 0

00

0 0

Since Iq(0,z,Zq) is known, (7.15) and (7.16) make it possible to find I^, I2,

etc., sequentially. Specifically, substituting (7.8) into (7.14) and (7.16), we recover (7. 11)

6.8 MEAN NUMBER OF SCATTERINGS

BASIC EXPRESSION. It is clearly of interest to obtain the mean number of

scatterings N experienced by a photon. The main reason for this interest is

that the product of N, the excited-state atomic lifetime and the number of

photons created per unit of time by the primary sources , gi_ves the total num- ber of excited atoms in the medium. Moreover, by knowing N, for media with

negligible continuous absorption (3 = 0), one can find the energy carried out

of the medium in the line under consideration. Let Eg be the energy expended

in primary excitation, so that Eg = hvgNj^, where Nj^ is the total number of

primary excitations to the upper level k. Then the energy lost by radiation

escaping in the line i ^ k is clearly

E = [i_(i_x)N]Eq .

Hence the quantity 1— (1—A)N is the mean escape probability.

Let us now proceed to obtain N. Consider a cylinder of geom.etrical

thickness dl , having a cross section of 1 cm^ . If, for simplicity, stimulated

emission is disregarded, the optical thickness of this cylinder is

dt = kj^j^ (vo)ni^dl , where k^-. (vg) is the atomic absorption coefficient at line

center, and n^ is the population of the lower level. _^Within the cylinder the sources create Aj^^n^dl photons per unit time, where n^, is the population of

the upper level attributable solely to primary sources, ignoring photo-excitation caused by the medium's own radiation field_j_in the line considered (see

Sec. 2.4). Using the relation between n^ and S given by (2.4.3), we find

\i\^^^

= \i ^i^ 2hv^ gi

^
286 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

The number of photons created by the sources per unit time in a column of 1 cm^ cross section extending through the whole semi - infinite layer is

/OO OO 00

n. dl = A, .— I Sn.dl = A, .— / S rildT

0 2h^0 ^1 0 2hvo^ gi kij,(Vo) J

However, the total number of photons emitted within the same column is greater

than Nj^ because of multiple scattering. If the population of the upper level

is nj^ , then the total number of photons emitted in a column of height dl is

Aj^-^nj^dl. Using the known relation between the line source function S and the

level populations, we find

c^

A, ,n, dl = A, . — ^ n. Sdl . 'ki^'k"" ^ki 2hv^ gi ^

Therefore the total number of photons emitted in the whole column per sec is

equal to

00

which can also be written as

c2 Sk

2hv3 gi k.j^CVg)

00 — f \ = A^.

^—

S(x IT)dT . (8.3)

A —

It is obvious that the ratio Nj^/Nj^ gives the mean number N of scatterings

experienced by a photon. Hence

00 / S(T)dT

N = ^—— . (8.4)

/ S*(T)dT

0"

Thus, determining the mean number of scatterings involves the calculation

of the inte_gral of the source function. [Ed. note. In many cases of interest

for which N is finite, the expressions (8.1) and (8.3) have only formal significance since the integrals diverge. In these cases (8.4) must be interpreted

in terms of an appropriate limiting process.]

N EXPRESSED IN TERMS OF THE RESOLVENT FUNCTION. We shall show that for an arbitrary S (t) the mean number of scatterings is simply expressed in terms

of the res'olvent function $(t). Let the sources be concentrated within an infinitely thin layer lying at depth x = ti, so that S*(t) = 6 (t—t^) . The source function will then be equal to the Green's function G(t,T2), and to

determine the mean number of s catterings- it is necessary to evaluate the

inte gral
6.8 MEAN NUMBER OF SCATTERINGS 287

oo oo y G(T,T-^)dT =1+1' rd.T^dl . (8.5)

Integrating the relation (see Sec. 5.1)

9r(T,T^) 3r(T,T^)

8t 3t;l

^ *(t)<I'(tJ (8.6)

over T from 0 to <», we obtain

r 9r(T,Tj 3r(T,T ) r r J — J

r(.,x^)dx = .(x^) J

(x)dx . (8.7)

0 "'-1-0 0

Using the fact that

and

we obtain

1 + / $(x)dx = H(«>) = —^

(8.8)

r(»,T-^) = 0 , (8.9)

^ y r(x,x^)dx = (1-A) "'"$(Xt) , (8.10) ,-1/2,

and hence

1 0

Xt

1 + y* r(x,x^)dx = (1-A)~^/^M + y $(x)dxj . (8.11)

Denoting the mean number of scatterings for S (x) = 6 (x—x^) as N(x-|^), we have

N(x^) = (l-A)-^/^nTp . (8.12)

Because of the linearity of the basic integral equation, the function S(x) is

obtained from G(x,x-]^) by integration over the distribution of sources:

oo S(x) = y S*(x-^)G(x,x^)dx^ , (8.13)

522-519 O - 74 - 20
288 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

and therefore the mean number of scatterings N is found to be

oo / S*(T)N(T)dT

S (T)dT

(8.14)

or CO / S (T)^(T)dT

0

(8.14')

IVe note that, according to (8.12), the mean number of scatterings of a photon that began its journey through the medium at a depth is equal to,

within a constant factor, the source function for a medium with a uniform

distribution of primary sources (see Sec- 6.1 and 6.3). Therefore the analysis given in Sec. 6.3 can be applied directly to the present problem. In

particular, it is found that as long as is much less than the thickness of

the boundary layer t]-, , the mean number of scatterings increases rather rapidly with T]^. When the sources are concentrated at the boundary (t-i^ = 0), the mean numbe_r of scatterings attains its minimum value of (1—A)—1/2. The rate

at which N(t]^) increases with becomes smaller as t-, approaches the order

of , and for T]^ >> t-^ , N(t]^) is practically constant with a value close to

that for an infinite medium, (1 — X)~l.

The increase of N(t]^) with xi is readily understood. There are two rea- sons for the loss of photons in the process of multiple scattering. One is

the destruction of photons during the scattering process by inelastic processes, the transformation into photons of other lines, etc. Since X is considered to be independent of x, this mechanism is equally effective at all

depths. The second factor limiting the number of scatterings is the escape

of photons. It is obvious that the effectiveness of this mechanism for remov- ing photons from_the scattering process decreases with the distance from the

boundary. Thus N increases with the depth of the sources.

For pure scattering N is infinite. In reality, the mean number of scatterings is always finite, since there are always processes destroying photons

as they are scattered (collisions of the second kind, photo- ioni z ation from

the excited state, etc.), so that the strictly conservative case never occurs

in nature.

PARTICULAR CASES. For those cases in which the intensit_y of the emergent

radiation can be expressed in terms of the H- function, N may also be similarly

expressed. As an example we take the exponential distribution of sources:

S (x) = e .

' (8.15)

* Substituting this expression for S (x) into (8.14') and making some simple

transformations, we obtain
6.9 INTEGRAL PROPERTIES OF THE ESCAPING RADIATION 289

N= Cl-A)"^^'^^l + J

e"^/^$CT)dTj . (8.16)

0

Invoking the relation between the Laplace transform of $(t) and the H- function

(see Sec. 5.1), we finally find

N = (l-X)~^/^H(z) . (8.17)

This expression has already been obtained in Sec. 6.4 by another means. Now

we have found it_as a consequence of the general result (8.14). The expression (8.17) for N is very convenient, since the values of H(z) for the most

important profiles are known (see Sec. 5.6 and 5.7). For

S*(t) = Te~^/^ (8.18)

the mean number of scatterings is found to be equal to

N = (l_X) ^'"H(z)|^l + ^zH(z)fi(z)J , (8.19)

where n(z) is given by (7.12). We shall not pause to prove this result.

All of the relations obtained in this section are valid for an arbitrary

profile, including the rectangular one (monochromatic scattering), since the

specific form of the function G(z) is nowhere used.

In Sec. 8.9 we shall turn once again to the problem of calculating the mean number of scatterings, alloiving for the effects of continuous absorption

and finite optical thickness.

6.9 INTEGRAL PROPERTIES OF THE ESCAPING RADIATION

PHYSICAL SIGNIFICANCE OF THE HOPE FUNCTION. As we have seen in Chapter III,

in problems of monochromatic scattering in a semi - infini te conservative atmosphere, an important role is played by the Hopf function q(T), which is probably the most widely known of the special functions of the theory of multiple

light scattering. The properties of q(T) were studied in detail in Sec. 3.8.

We shall show now that the Hopf function has an imm.ediate physical interpretation: q(T) is the mathematical expectation of the cosine of the angle of

emergence of the photons that began their random walks in the medium at the

depth T. Incidentally, from this it immediately follows that the values of

q(T) lie in the interval (0,1). The well-knom monotonic increase of q(T)

also has a_simple physical explanation. We recall that q(T) increases from

q(0) = 1//3 = 0.5 77... to q

(oo) = 0.710

In order to demonstrate this interpretation of the Hopf function, we

note that q(T) is the bounded solution of the equation

00 q(T) = I J

E^(|t-t' |)q(T')dT' + I

E3(T) . (9.1)

0
290 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

This equation is obtained by substituting (3.8.9) into (3.8.8). Let p(T,y)

be the bounded solution of the equation

p(T,y) = i / E (|t-t' |)P(T' ,y)dT' + ^ e

'/l'

. (9.2)

2 y 4iT

Since

J. £3(1) = J

e~^/^ydy , (9.3)

0

it follows from (9.1) and (9.2) that q(T) is the superposition of the functions p(T,y) for different values of y,

/

q(T) = 2tj Jp(T,y)ydy . (9.4)

0

The probabilistic considerations of Sec. 5.2 show that p(T,y)dco is the

probability for a photon absorbed at depth t to escape from the medium (in

general, after multiple scattering) within an element of solid angle dw around

the direction making an angle arccos y with the outward normal. Since we are considering conservative scattering, the total probability that the photon

will escape is unity, i.e.

1

2tt y p(T,y)dy = 1 . (9.5)

0

Hence from (9.4) it follows that q(T) is indeed the mean value of the cosine

of the angle of emergence:

1 f p(T,y)ydy

q(T) = y(T) = (9.6)

/ p(T,y)dy

0 A COUNTERPART OF THE HOPF FUNCTION FOR LINE TRANSFER PROBLEMS. The physical

significance of the function q(T) expressed by (9.6) can be regarded as a physical definition of q(T). This definition has the merit of not being re- stricted to the case of conservative monochromatic scattering. The function

q(T) defined according to (9.6) preserves, for the general case of 0 < A < 1 and an arbitrary line profile, the specific features of the Hopf function:

it is a monotonically increasing function that varies within narrow limits.

Let us consider multiple scattering in a line with an arbitrary profile

a(x). Let p(T,y,x) be the probability of photon escape from depth t with a given frequency and direction (see Sec. 5.2). We introduce
6.9 INTEGRAL PROPERTIES OF THE ESCAPING RADIATION 291

P(t,P) = J

p(T,p,x)dx . (9.7)

00

According to (5.2.4), p(T,y) can also be represented as

CO p(T,y) = A / a(x)S(T,y/a(x)) dx , (9.8)

4tt J00

where S(t,z) is the bounded solution of the auxiliary equation

oo S(t,z) = A J

K^(|t-t' |)S(T' ,z)dT' + e~^/^

0

(9.9)

The quantity p(T,y)dco = 27Tp(T,y)dy is obviously the probability that a photon

absorbed at a depth t will escape from the medium within an element of solid

angle dw around the direction making an angle arccos y with the outward normal. Substituting (9.8) into (9.6), we get by definition

1 kj\xd\xf (x)S(T,y/a(x))dx

0 -00

1 ~ f I

A a(x)S(T,y/a(x))dx

0 In particular, for conservative scattering

1

q(T) =|Ay*ydyy* a(x)S(T,y/a(x))dx , A = 1 . (9.11)

0 -~

Let us consider this latter case in a little more detail.

From (9.11) and (9.9) we find that q(T) is the solution of the equation

00 q(T) =i f

K^(|t-t' |)q(T')dT' + i K3^(T) , (9.12)

where

oo

'

Kj^d) = aJa(x)E3(a(x)T)dx . (9.13)
292 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

In Sec. 2,6 it was shown that the function K^-|^(t) can be represented as

CO K3^(t) = y e~^/^zG2(z)dz , (9.14)

0

where

00 G^Cz) = 2A J

a"'^^(t)dt , m = 0,1,2, .. . , / (9.15)

x(z)

with x(z) = 0 for z < 1 and a(jc(z)) = 1/z for z > 1. Therefore (9.11) can be

rewritten as

00 q(T) = i J

S(T,z)zG2(z)dz . (9.16)

0

A few words about the values of q(T) for t = 0 and x = oo seem to be in

order. From (9.16) we have

q(0) = J

' ^^-^^^

where

00 j H(z)G^(z)z^dz , i,k = 0,1, . "ik ^ J

H(z)Gi^(z) z^dz , i,k = 0,1, ... . (9.18)

0

The values of these weighted moments of the Doppler H- function found by numerical integration are given in Table 31. A§ ^regards q(~), it can be shown that

for Y < 1

TABLE 31

THE MOMENTS a., OF THE CONSERVATIVE DOPPLER H-FUNCTION

ik

k i=0 i=l i=2 i=3

1 2.0000 - - -

2 1.1798 1.0935

3 0.9318 0.6442 0.7516

4 0.7962 0.5085 0.4390 0.5673
6.9 INTEGRAL PROPERTIES OF THE ESCAPING RADIATION 293

q(~) =

-"-^^^

. (9.18)

2(1+Y)

For the proof of this result, see V. V. Ivanov (1970a). For example, in the case of the Doppler pro fi le, qp

(

t) increases, according to (9.17) and (9.18),

from qD(0) = 0.547 to qD(°=) = 0.66 7. FREQUENCY SPREAD OF THE ESCAPING RADIATION. The function q(T) gives us some general idea of the angular distribution of the radiation escaping from depth

T. Similarly, the frequency spread of radiation escaping from depth t can be

characterized by the function

1 f a(x)dx fJ p(T,y,x)dy

a(T) = ^

j-^ . (9.19)

J dx f p(T,y,x)dy

0

As T increases, ct(T) decreases, since the width of the line increases with the

depth at which photons are born. The larger the average value of |x| , of

course, the smaller is the corresponding value of a(x).

In the conservative case the integral in the denominator of (9.19) is

evidently (Zir)"!, and (9.19) can be rewritten in the form

a(T) = 27T / ct(x)dx J

p(T,y,x)dy , X = 1 , (9.20)

whence

a(T) = \ j

S(T,z)G2(z)dz , X = 1 . (9.21)

0

Combined with (9.9), (9.21) shows that a(T) for X = 1 is the bounded solution

of the equation

oo a(T) = I j

K^(|t-t' I)a(T')dT' +|k22(t) , (9.22)

0

where

00 00 ^ K22(t) =A f

a2(x)E2(a(x)T)dx = /

e~^/^G2(z)dz . , (9.23)

-i 0

There is a simple relation between a(T) , q(T) and the solution of the

generalized Milne problem S(t) which follows from (9.22) and (9.12>r, namely
294 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

q(T) = q(0)S(T) -/ a(T' )dT' (9.24)

In the particular case of the rectangular profile (monochromatic scattering) Kj^]^(t) = Ej, (t) , k,n = 1,2,..., and (9.22) gives a(T) = 1. This re- sult follows also from simple physical £onsiderations . Moreover, for mono- chromatic scattering we have q(0) = 1//3, and (9.24) assumes the form

q(T) -4= S(t) - T /3

(9.25)

This is a well-known relation (see Sec. 3.8).

Equation (9.24) is of interest in two respects. First, it shows that the

simple relation between q(T) and the solution of the Milne problem (and hence,

the resolvent function $(t)) is the special feature of monochromatic scattering, for in the general case this relation also involves a. For this reason

the function q(T) for non- rectangular profiles, unlike that for monochromatic

scattering, cannot play an important role in the general theory. Second,

equation (9.24) enables one to find immediately the^asymptotic form of a(T)

for large t. Since q(T) for t ^ «> is bounded, and S(t) diverges (see Sec.

6.1), we conclude that

ad) ~ q(0)<I>(T) = - a^2^'^'^^ > ^ (9.26)

To more_easil^y visualize this result, we introduce the frequency x(t) such

that ot(x) = a(T). Then, e.g., for the Doppler profile (9.26) gives

Xpd) i?.n 3/4 1/2, „ ,-1/4

T\ T (ilnx)

'12

1/2

(9.27)

The growth of x with t characterizes broadening of the line as the depth at

which photons are created increases. In the opposite extreme case when the sources are concentrated at the boundary, we have from (9.21)

a(0)

1

2 02 (9.28)

Specifically, for the Doppler profile ctp(O) = 0.590.

6.10 THE EFFECT OF MULTIPLE SCATTERING ON THE ESCAPING RADIATION

INTRODUCTORY REMARKS. According to the assumption of complete frequency re- distribution, the probability that an excited atom will emit a photon of a given frequency is proportional to the line absorption coefficient. On the

other hand, numerous examples cons i dered. ab ove have shown that the radiation

emerging from the medium has a spectral distribution quite different from

that of the absorption coefficient. These changes in the spectral composition

of the radiation are caused by the strongly selective nature of the interaction between the line radiation and the gas. Let us trace how the spectral

composition of the radiation changes.
6.10 THE EFFECT OF MULTIPLE SCATTERING ON THE ESCAPING RADIATION 295

The mean free path of a photon with a frequency close to the line-center

frequency is much smaller than that of a wing photon. Having absorbed a photon, an atom can re-emit it anywhere in the line, including the far wing. If

this latter possibility is realized, the re-emitted photon most probably will

not excite an atom for a rather long time, since its mean free path is large,

while if the atom re-emits the photon in the line core, it will soon excite

an atom. Therefore, spectral composition of radiation will change with time.

Broadly, the change amounts to a gradual pumping of photons from the line core into the wings, so that the line becomes wider. Similar changes occur if we consider the propagation of radiation in space. As the distance from the source increases, the line broadens, a region of nearly constant intensity

appears in its core, and then the central depression develops. These considerations suffice to explain the wide variety of the profiles encountered in

the preceding sections.

PHOTON ESCAPE PROBABILITY. Let us now consider the problem quantitatively.

More precisely, let us discuss the properties of the function expressing the

probability that the excitation of an atom at depth t will finally lead to

the escape of a photon in a given frequency and direction. This probability,

denoted in Sec. 5.2 as p(T,y,x), was shown to be the solution of the equation

oo p(T,y,x) =A

/

K^(|T-T'|)p(T',y,x)dT' + Aa(x)e-"WT/y ^ (Iq-.I)

0 Hence,

p(T,y,x) = Aa(x)S(T,y/a(x)) ,' (10.2)

4lT

where S(t,z) is the solution of the auxiliary equation

00 S(T,z) = I y K^(|t-t' |)S(t' ,z)dT' + e~^/^ . (10.3)

0

First we assume that excitation occurred at the boundary t = 0. The medium

acts as a reflecting screen, since a photon emitted by the excited atom in

the direction of the medium can be reflected from it. Consequently the

probability that the photon will ultimately be moving outward, i.e., from the

medium, increases. Since in multiple scattering the frequency may change, the

reflecting properties of the medium are very distinctive. It would be even

more appropriate to consider the medium not as a screen, but as a kind of am- plifier, with a gain which depends on both frequency and direction. According

to (10.2), the probability of photon escape from the boundary is

p(0,y,x) = Aa(x)H(y/a(x)) , (10.4)

4Tr

since S(0,z) = HCz) (see Sec. 5.1). If there were no multiple scattering, the

probability would be (see Sec. 5.2)

47r Hence the gain is H(y/a(x)), which elucidates the physical meaning of the

H- f unct ion
296 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

In Fig. 36 we show the probability of photon escape from the boundary

along the normal as a function of frequency (Doppler profile) . As A increases , multiple scatterings become more important, and the pumping of photons

from the core into the wings causes the line width to increase.

Now let an excited atom appear at an arbitrary depth t. The mean numbei

of scatterings increases with t, and the influence of pumping becomes more important. According to (10.2) and (5.1.36), the probability of photon es- cape can be represented as

p(T,y,x) = ~ Aa(x)H(z)^e~^/^ + e"*^ ' ^ ^ ^

<i> (t ' ) di ' ^

, (10.5)

where z = y/a(x). Comparison with (10.4) makes it clear that the factor in

brackets allows for these extra scatterings. As t increases, the line be- comes wider (Fig. 37, after G. D. Finn, 1971). For large t there is a typical s e 1 f - re ve rs al

ESCAPE OF PHOTONS FROM GREAT DEPTHS. Substantial simplifications occur if a photon is born at a great depth. It can be shown that for z << t and large

T

S(t,z) ~ zH(z)$(t) , z << 1 , t >> 1 , (10.6)

and hence, according to (10.2),

p(T,y,x) ~ M yH(y/a(x)) $(t) , t » 1 , t >> 1 . (10.7)

Fig. 36. Probability of photon escape along the normal from the boundary

(Doppler profile).
6.10 THE EFFECT OF MULTIPLE SCATTERING ON THE ESCAPING RADIATION 297

Fig. 37. Probability of photon escape along the normal from various depths

(Doppler profi le , conservative scattering).

In the opposite limiting case z >> t we have, from (10.5),

p(T,p,x) ~ ^ a(x)H(y/a(x))T(T) , t « 1 . (10.8)

Let us discuss the results (10.7) and (10.8). Since H(z) increases with

z, (10.7) shows that the probability o£ photon escape from a great depth

considered as a function of x increases with |x| for a(x)T/y >> 1, i.e. in

the line core. In the wing of the line, i.e. for a(x)T/p << 1, the photon

escape probability decreases with |x|. Hence a maximum is attained for

a(x)T/y ~ 1.

So far we have assumed only that the photon is released at a depth

T >> 1, with no restrictions on t as compared to the thickness of the boundary

layer t^^ . In the limiting cases of t << t-^ and t >> tj^ further simplifications arise.

If the photon is born at a depth x >> t^^ , the asymptotic form (5.5.26)

can be substituted for $(t) in (10.7), so that — K (t)

p(T,y,x) ~ ^ yH(y/a(x))^-J—- , ^» 1 , x » . . (10.9)

Using the results of Sec. 5.4, one can show that the factor H(ij/a(x)) in this

expression approaches its limiting value H(o°) = (1 — X)~l/2 at such values

of |x| that the inequality a(x)T/y still holds. Hence in (10.8) we can safely replace H(y/a(x)) by (1 - X)~^'^. Considering further, (5.5.35), we

find that in the wings

p(T,y.x)~^^, (10.10)

4tt 1—a y D
298 SEMI- INFINITE MEDIUM: STANDARD PROBLEMS

Figure 38 shows the general features of p(T,y,x) for t >> t^^ . As t increases,

the plateau becomes wider, while the depth and the shape of the central selfreversal in this limiting case do not depend on t.

Let us now turn to the opposite limiting case of >> t >> 1. The left

inequality implies that we are dealing with a nearly conservative medium (see

Sec. 6.2). Obviously the limiting forms of (10.7) and (10.8) caii be used here

as well. However, for t^^ >> t >> 1 a more general asymptotic expression for

P(t,1j,x) can be given that applies for all x. According to (10.5), to get the

asymptotic form of p(T,y,x) for large x one has only to consider the asymptotic behavior of the function appearing in brackets in this expression. We

de fine

X F(x,z) = e~^/^ + J

e"'^^"^' ^/^$(x' )dx' . (10.11)

0

Setting z = x/t and x' = xy , we get

F(x,l) = e-'L.H^) f

^

dy

Using (5.5.20), we find

0

F(x,^) ~ x$(x)e ^ j e^^y^ "^dy , X|^ >> x >> 1 ,

Fig. 38. Qualitative behavior of the probability of photon escape from

depth X >> X, .
6.10 THE EFFECT OF MULTIPLE SCATTERING ON THE ESCAPING RADIATION 299

pCT,y,x) ~ ^ Aa(x)H(y/a(x)) T$(T)f^^^ . >> t >> 1 , (10.12)

4tt

where

± £Ct) = e"''^ J

e^^y^~^dy . (10.13)

Since f(0) = l/y, t£(t) -> 1 as t «> and t$(t) ~ y'I'(t) for Tu^ >> t >> 1 (see

Sec. 5.5), the result (10.12) reduces to (10.7) and (10.8), m the limiting

cases o£ a(x)T/y >> 1 and a(x)T/iJ << 1, respectively.

A simple approximate form for p(T,y,x) in the conservative region, i.e.

for T << Tb , is suggested by (10.12) and (10.4), namely

p^(T,y,x) = A Aa(x)H(y/a(x))S(T)Yf(^^T^ , (10.14)

where S(t) is the solution of the Milne problem for the profile a(x), and the

subscript a stands for "approximate." This expression correctly describes

the behavior of p(T,y,x) for large t, since yS(t) ~ t$(t) (see Sec. 5.5), and

i| gives the exact result (10.4) for x = 0. Hence it may be expected that

'I (10.14) is a reasonable approximation for all t' << t, .

CHAPTER VII

CONTINUUM CONTRIBUTIONS TO LINE FORMATION

1 We have previously assumed, when solving the transfer equation, that linefrequency photons disappear from the multiple scattering process either by

j escaping from the medium or by being absorbed in a scattering event (because

I! of collisions of the second kind, re-emission in another line, etc.) . The

possibility that a photon can be destroyed in flight has not yet been considered. The most important process of this kind for the destruction of photons

is the photo-ionization of atoms by radiation in the line under consideration.

Another mechanism is the absorption of line radiation by macroscopic particles,

such as interstellar dust grains.

Another important process that we have also previously disregarded in

solving the transfer equation is the emission of continuum radiation by the

medium. This process leads to the formation of a continuum against which the

lines are usually seen. Since this radiation can be absorbed by atoms, caus- ing transitions between discrete levels, it constitutes an additional source

of atomic excitation.

The transfer equation allowing for emission and absorption processes in

!| the continuum was derived in Sec. 1.6. Particular forms of this equation,

I specialized to media with plane and spherical geometries, were considered in

irSec. 2,3. Finally, the integral form of the transfer equation that directly

j expresses the condition of statistical equilibrium was introduced in Sec. 2.4.

ijNow we shall solve it for several of the simpler cases. Throughout this

chapter we will assume that the following are independent of position: the

jprobability X that a photon survives the scattering; the ratio B of the con-

' tinuum absorption coefficient to that at line center; and the profile a(x) of

the line absorption coefficient. Stimulated emission is assumed to be negligible . Many of the results in this chapter originally appeared in the dissertaijjtion of D. I. Nagirner (1966, unpublished); see also D. I. Nagirner (1968).

'jilmportant results have also been given by D. G. Hummer (1968).

/
302 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

7.1 KERNELS OF THE INTEGRAL EQUATIONS AND ASSOCIATED FUNCTIONS

It is convenient to study the properties of the kernel of the integral

equation for the line source function and several related functions before

turning to the solution of the transfer equation for line radiation allowing

for absorption and emission in the continuum.

BASIC FORMULAE. The following two functions were introduced in Sec. 2.4:

OO K^(t,6) = Ay* a^(x)E^((a(x) + 3)T)dx , (1.1)

OO

00 K^^(t,B) = a(x)E^((a(x) + 3)T)dx . (1.2) — oo

The kernel of the basic integral equation for media with plane and spherical

geometries is expressed in terms of the first of these functions, while the

second appears in the term of the equation that describes the excitation of

atoms by continuum radiation. We also define

00 ^ K.(t,3) = A / ^-iiSl-E ((a(x) + 3)T)dx , (1.3)

^ J a(x) +3

OO

00 K.n(T,B) = A ( -^M^E2((a(x) + 3)T)dx . (1.4)

J a(x)+3 ^

00

As 3 0 the functions Ki(t,3) and K2(t,3) become the functions K]^(t) and

K2(t) respectively, which have been studied in detail in Sec. 2.6 and 2.7;

K]^i(t,3) reduces to K-[^-]^(t); and K2o(t,3) tends to infinity.

The line of reasoning used in Sec. 5.2 to explain the physical significance of K2(t) and K2(t) is also valid for media with continuous absorption.

We find that ( A/2 ) Ki

(

t , 3 ) dx is the probability that an excited atom will

emit a photon and that this photon will then be absorbed for the first time

in a layer of thickness dx, located at a distance x from the emitting atom,

causing the excitation of an atom in this layer. Thus it follows that the

quantity

00 00

A = \j K-^(|Tl,3)dx = A y

K^(x,3)dx , (1.5)

-00 0 . which is equal, according to (1.1), to

oo

X = XA f -Vr^dx , (1-6)

J a(x)+3
7.1 KERNELS OF THE INTEGRAL EQUATIONS AND ASSOCIATED FUNCTIONS 30 3

represents the probability that an atom originally excited in an infinite

medium will emit a photon that subsequently somewhere excites another atom.

This quantity should play an important role in all problems of the transfer

of line radiation in which continuous absorption is considered, since it

(along with the geometry of the medium) determines the role of multiple scatterings. The dependence of A on g for various forms of the absorption coefficient is discussed in detail in Sec. 7.3.

The kernel function K]^(t,6) can be represented as a superposition of

exponentials, namely,

z c I

K^(t,3) = / e ^/^G(Odz/z , (1.7)

whe re

z = i , ^ = —^ . (1.8)

c 3 ' ^ l-3z

Differentiating (1.1) with respect to t and considering the relation

E^(t) = -e"^/t ,

we find

9K^(t,B)

^ _g

oo

9t -e tA/ 2..—a(x)Tj —la (x)e ^ ^ dx , ^ J—00

or

9t di

But from (2.6.16) it follows that

dK, (t)

3K (t,3) dK (t)

^ - e"^^ — . (1.9)

- -f e-^/^G(c)di;/c^ . (1.10)

dx

0 Substituting this expression into (1.9) and integrating, we obtain

oo

0

The substitution c = z/(l-6z) in (1.11) leads to the form (1.7).

Ki;|[(t,8) may also be represented as a superposition of exponentials. It

has been shown in Sec. 2.6 that the function

522-519 O - 74 - 21
304 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

can be represented as

00 K^^(t) = a(x)E-|^(a(x) T)dx (1.12)

OO

oo K-^^(T) = J

e~^/^GQ(c)dc/^ , (1.13)

0 where Gq(z) is defined by (2.6.12) and (2.6.13). Starting from this, and

repeating literally the reasoning that led us to (1.11) and (1.7), we find

00 G ( ) K^^(t,S) = e-^^y* e"^/^ ^_^dc/C , (1.14)

or

z

K;^^(t,3) = j e~^^^G^U)dz/z , (1.15)

where and ^ are given, as before, by (1.8)

It is obvious that

K2(t,3) = yK^(t' ,3)dT' . (1.16)

T The same relation also exists between the functions K, -, and K^„. Therefore,

we obtain

z.c K2(T,3) = e-^^ f e-^/^-^I^dc = f e-^/^G(c)dz ,' (1-17)

K^.(t,3) = e"^^ / e-^/^-^^ilLd? = f e-^/^G„(Odz . (1.18)

' (1 + 30^ ' 0

^''^ 0 Setting T = 0 in (1.17) and taking account of (1.16) and (1.5), we find

z

-

X = AK,(0,3) = xl —^i^^ d? = xl G(^)dz . (1.19)

/ (1-3C)2 {

THE CASE OF THE WEAK CONTINUUM ABSORPTION. The absorption coefficient in the

continuum is usually much smaller than the absorption coefficient at line

center, i.e. 3 << 1. Values of 3 on the order of 10"8 _ 10~4 seem to be more

the rule than the exception. The smallness of 3 leads to considerable simplification. Specifically, it allows the functions introduced above to be ex- pressed in terms of their limiting values, corresponding to 3 = 0. For
7.1 KERNELS OF THE INTEGRAL EQUATIONS AND ASSOCIATED FUNCTIONS 305

6 << 1 the continuous absorption may be considered a minor perturbation. It is true, as we shall soon verify, that the effects caused by this minor perturbation are often very large.

Let us consider the kernel function K,(t,3) for 3 << 1. The substitution

t/c; = y transforms (1.11) into

oo J y + 3T

. K^(t,3) = e-P^ / e"^ ''^^'/jj dy . (1.20)

0 Using (2.6.42), we find that i f t ^ and 6 0 s o that 3t = const, then

00 V r r.-^'^ 2k 1 I -y x' (x/y) , K (t,3) ~ e I e ^ — ' J dy . (1-21)

-L 2y + 1 T y y + 3T

Considering (2.6.35), we have

00 K rT,3) - e-^-"

-^^^

f

^ih-llLdy . (1.22)

1 2y+1 t J y+3T ^

^

^ 0

But, as has been shown in Sec. 2.6, for x ^ =°

2y+1

Therefore (1.22) can be rewritten in the form

K rx) ~ 2A Il^r^ 2^:111 . (1.23)

i 2 Y+ 1 T

K^(t,3) ~ K^(T)e ^^k^(6T) , 3 << 1 , (1.24)

where

Y + 1) J

-y 2y+l

kAs) =

^ I ^-1 dy . (1.25)

1 r(2Y + l) J y + s

ji When 3 << 1, the expression (1.24) is valid for all t. This can be verified

by showing that (1.24) is satisfied for all t consistent with the inequality

Bt << 1, and not only for 1 << x << 3"!. For 3t << 1, (1-24) and (1.25) give

K^(x,3) ~ K^(t) , X << 3"^ , (1.26)

j;j which may also be obtained directly from (1.11). This then proves our asser-

' tion. In the opposite limit 3x >> 1 we find from (1.24) and (1.25)

K^(x,3) ~ (2y + 1)-^7 e , x >> 3 . (1.27)

J- 3 T
306 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

Thus, i£ even weak absorption is present, for sufficiently large t the kernel

function decreases exponentially; more precisely, it tends to zero for t

-> as (x' [T)/T2)e-BT.

The result (1.24) shows that for small 3 the kernel Ki(t ,B) , a function

of two variables, may be expressed in terms of functions of one variable.

Here, however, one stipulation must be made. For the Voigt profile the representation (1.24) with kj(BT) given by (1.25) for y = 1/4 is applicable only

when 6t is of order unity for t in the region where the Voigt asymptotic forms can be used for Kx(t). This region is defined by the inequality t >> a-1,

where a is the parameter of the Voigt function (see Sec. 2.7). It therefore

appears that for a Voigt profile the expression (1.24) is applicable only when

3 << a (in practice, when 3 ^ O.Ola),

Proceeding in a similar manner, we obtain from (1.14)

where

whereas (1.17) gives

whe re

We note that

K^^(t,3) ~ K^^(T)e ^^k^^(3T) ,

• (1.28)

00

11 r(2Y) J y+s

0

K^d.B) ~ K2(T)e ^'k2(3T) , (1.30)

oo "277/

-y 2y+1,

e ^ y ' dy k,(s) = I ^ ^ . (1.31)

r(2Y) J (y,3)2

k^^(s) = ^[l-k-^(s)] . (1.32)

For the Doppler profile

k^(s) = l-s+s^e^E-^(s) ; k^^(s) = 1-s e^ E-|^ (s ) ;

k2(s) = l-2se^E^(s)+se^E2 (s) .

A graphic representation of the behavior of the kernel Kj^(t,3) for various values of 3 appears in Figs. 39-41, which are taken from the work of

E. H. Avrett and R. Looser (1966). In this paper tables are given of

AK2(At, 3/A) and Kii(At, 3/A) for th6 Doppler (a = 0) and Voigt (a = 10-3 and

10-2) profiles, along with the coefficients of approximate representations

of these functions as a sum of exponentials.
7.1 KERNELS OF THE INTEGRAL EQUATIONS AND ASSOCIATED FUNCTIONS 30 7

Fig. 39. The kernel functions K^(t,B) for the Doppler profile.

THE FUNCTIONS U AND V. Along with the functions K;^ and K2 , an important role

is also played by the two functions

00 u,6) = y* K^(t,3)cos TudT , (1.33)

where K]^(s,B) is the one-sided Laplace transform of the kernel K^^Cx.g) with

respect to the variable t:

00 /

K^(s,3) = I

e ^""K^Cx.BjdT . (1.35)

^ 0 These functions are related by

V(u,3) = U-,3 . (1.36)
308 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

Using (1.7) we find that

z

-Q l + u^z^ ^ (l + 6C)^ + u2(;2

z

U(.,e) = z^/ GU')-^ - z^/

^ ^

, (1.38) ^ z^-z'^ z^ + • ) ^-C'

while substituting (1-1) into (1.33) and (1.34) gives

00 V(u.6) = ^ / a^(x)arc tg

" dx , (1.39) —oo

oo

CO

For real values o£ z satisfying \z\ < the argument of the logarithm must
7.1 KERNELS OF THE INTEGRAL EQUATIONS AND ASSOCIATED FUNCTIONS 309

be replaced by its modulus, and the integrals in (1.38) are to be understood

as Cauchy principal values.

To study the asymptotic behavior o£ solutions o£ the transfer equation,

one needs the asymptotic forms of V(u,3) and U(z,6) for u 0 and z ^ °°

,

respectively. For a Voigt profile they may, for example, be obtained as follows. From (1.37) we have

oo • V(u,6) = I - f

^3^'-(e^^^^)^'^ (z')dz' . (1.41) ^ (l+6z' ) 2+u2z'

When both 3 and u are small, the main contribution to the integral on the

right comes from the values of the integrand at large z'. Therefore G(z') may

be replaced by the leading term of the asymptotic expansion (2.7.27), so that

oo V,,(u.6) ~ L - l(aU(a^\l/2 r

2Bz ' ^ (3^^u^) z

'

,

, 3/2 _ ^^^^^^

V 3\ ^ / y (-1 + 32' )2+u2z'2

Evaluating the integral, we finally get
310 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

Vy(u,6) ^(^aU(a,0))l/2 (1.43)

This expression is valid for u << 1, 3 << 1, and an arbitrary value o£ the

ratio B/u. As 6/u 0 it gives the leading term of the expansion of 1 — Vy(u)

for u << I given in Sec. 2.7 (see (2.7.34))- In an analogous manner we find

from (1.38) that for z >> 1 and 3 << 1, subject to the condition 3z ^ 1,

Uy(z,B) 1 I iir n^N^^^ C1 + 3Z) -^/^ -(TTaU(a,0))

^ /I

(1.44)

For the Lorentz profile the asymptotics of the functions V(u,3) and U(z,3) can be obtained from (1.43) and (1.44) by taking the limit a In this procedure the limit aU(a,0) l/ir as a ^ « must be taken into account.

We shall not give the asymptotic forms corresponding to the Doppler profile, since they are very cumbersome and are not very useful in studying the

asymptotic behavior of the solutions of the transfer equation.

7.2 INFINITE MEDIUM

THE GREEN'S FUNCTION. As the first standard problem of multiple scattering

with continuous absorption, let us consider an infinite, homogeneous medium

with an isotropic point source whose frequency distribution is proportional

to a(x) , i.e. let us find the infinite medium Green's function. The calculation of the Green's function S (t) involves the solution of the equation

(see Sec. 2.4) P

S (t) = P 4tt /

exp{-3 Ii-t' I }M2 (

I T-T '

I

)

t-t'

S (T')dT' + —

P - 47T -3x^2^^^

.2 (2.1)

where x = |^| and the integration extends over the entire x_'-space. For 3=0,

(2.1) reduces to the equation studied in detail in Chapter IV. The main

object will be to discuss the new effects that appear when continuous absorption is considered, as well as to determine the limits of applicability of the

theory developed in preceding chapters, in which it was assumed that there is no continuous absorption.

As in the case 3 = 0 , we introduce the function $ , related to S by CO ' P

S (x)

P

1 , . 2Trx dx °° (2.2)

and satisfying the equation

00 f^(T) = \ j

K^(|x-x' i ,3)<f„(x')dx' + |k^(|x|,3) . (2.3)

For the physical significance of $^(x), see Sec. 3.6.

Taking the two-sided Laplace transform of (2.3), and using the convolution

theorem, we obtain
7.2 INFINITE MEDIUM 311

where

$ (s) + $ C-s) = 00^'' , , , - 1 , (2.4) oo^-^ Tfl/sl

T(l/s) = 1 - |k^(s,B) - |k^(-s,6) , (2.5)

and the bar denotes, as in the preceding chapters, the one-sided Laplace tranS'

form with respect to the spatial variable.

Let us examine the function on the right side of (2.4). By virtue of

(1.11)

00 00

) K.(s,3) E f e-^^K.(T,6)dT = f f^'-.l^l r • (2.6)

^ J ^ J (l+3z) (l+Bz+sz)

0 0 Substitution of t = (1 + 3z)/z gives

00 K(s,3) = / G(l/(t-3)) _ ^2.7) J t(t+s)

t5 Thus K;l(s,3) is a Cauchy-type integral. As in Sec. 4.1, we assume that a(x)

is a monotonic continuous function of |x|, 0 < |x| < <». Then G(z) does not

vanish for all z, 0 < z < o°, and from (2.7) we conclude that the function

K]^(s,3) is regular in the plane of the complex variable s cut along the real

axis from -°o to -3. It can now be seen from (2.5) that T(l/s) is regular on the s plane with branch cuts along the real axis from -«> to -3 and from 3 to

oo. We shall show that within the domain of regularity T(l/s) does not vanish

The roots of T(l/s) can lie only on the real axis. Actually, from (2.5) and

(2.6) we have

00 T(l/s) = 1 - A / ^

. (2.8)

(l + 3z)^-s2z^

For s = X + iy with x 7^ 0 and y 7^ 0 , we conclude that the imaginary part of

the integral cannot vanish because G(z) is positive. Moreover for s = iy it

follows from (2.8) that T(l/s) > 1 — X > 0, so that our assertion is proven.

In order to show that T(l/s) 7^ 0 as well for real values of s lying in the

domain of regularity (-3 < s < 3), we rewrite (2.8) in the form

T(l/s) = 1 - X I

G(z)_dz

_ ^2.9)

/ l+23z+z2 (32-s2)

Since G(z) > 0 , in the region of s-values we are considering, for 3 > 0 , we

have

00 00 00 / , < / fifif < / G(z)dz = 1 . / l+23z+z2(32_s2) J l+23z J
312 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

Consequently, T(l/s) > 1 — X > 0, which therefore proves the non-existence o£

a root (D. I. Nagirner, 1966, unpublished; Yu. Yu. Abramov, A. M. Dykhne

,

A. P. Napartovich, 1967).

Since ^'^(t) tends to zero as x -> °°, the function *^(s) is regular for

Re s > 0 . However, because of the properties of the function T(l/s) just

demonstrated, the right side of (2.4) is regular in the strip -3 < Re s < 3.

Hence $^(s) is regular in a wider region as well, that is, in the half-plane

Re s > -3. Therefore $,^(-5) is regular for Re s < 3 • From (2.4) we have

^oo(T)

+ i oo

— 1°°

1 T(l/s) - 1 - -•] e ds (2.10)

In order to obtain $ (x) for x > 0 , we note that the function in square

brackets is regular m the left half-plane cut along the real axis from -°° to

-3. Therefore the contour of integration may be shifted to the left and

deformed, replacing integration along the imaginary axis by integration

along the contour shown in Fig. 42. The integral along the large circular

arc of radius R centered at the origin tends to zero as R ^- «>. The integral

over the small circle of radius r centered at the point s = -3 also tends to zero as r ^ 0 (these parts of the contour are shown in the figure by broken

lines). Therefore the integral along the imaginary axis is equal to the

integral al^ong the sides of the cut (-<»,-3]. Taking into account the regularity of <I>^(-s) for Re s < 0 , we find

Fig. 42. Path of integration in the evaluation of the Green's function allowing for continuous absorption.
7.2 INFINITE MEDIUM 313

or

^ r ^ r 1 TX, , 1 / 1 TX,

f (t) = r I

;—e dx + / —-e dx , 2tti J T_(l/x) 2TTi J T+(l/x)

_00 _g

CO

2ttiJ [_T_(-1/x) T^(-1/x)J

where

±Vx/ \x±i(

(2. 12)

For X > 6 (2.5) gives us

1 - AK^(-x±iO,B) - |k^(x,3)

Keeping (2.7) in mind, and using the well-known expressions for the limiting

values of Cauchy-type integrals, we get

or

where Ll(l/x,3) is given by (1.38). The integral in (1.38) is to be understood

as the principal value. Substituting (2.13) into (2.11), setting z = 1/x,

and taking into account the evenness of $^(t), we get

z

^^(t) e-^l''l/"^R(z)G(?)dz/z , (2.14)

0 where

1-1

R(z) |[l - AU(z,8)]^ + [a^zG(0]^[ . (2.15)

U(z,6) is given by (1.38), and and r, are defined according to (1.8). From

(2.14) and (2.2) we find, finally,

z e"^/^R(z)G(c)dz/z2 . (2.16)
514 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

This result was first obtained by D. I. Nagirner. For B 0, (2.14) and (2.16)

reduce to (4.1.18) and (4.1.20), respectively.

The expressions obtained here can scarcely be called simple. Using them

to calculate, for example, ^^^(t) is not much simpler than the direct numerical

solution of the basic integral equation for $^(t). However, they are important in the sense that they make it rather simple to study the dependence of

the solution on the parameters X and 3 and to identify those cases in which

simplifications arise. In the following two sections we shall discuss this

prob lem.

INTEGRAL RELATIONS. From (2.4), one can easily obtain the even-order moments

of $ (t) , They can be expressed in terms_of the moments of the kernel function

K]^(t,6). In the domain of regularity of $^(s), i.e. for Re s > -6, we have

CS) E

00 /

e (T)dT = T$ (t) di +

CO

T $ ( t) dx

Therefore, in the strip -3 < Re s < 3

,(s) ,(-s)

00 /

(T)dT + S

00 /

T $ (t) dx + (2.17)

On the other hand,

K^(s 3) =

00 /

e ^^K, (T,3)dT

CO 00 2 ^ y* K^(x,3)dx -sj xK^(x,3)dx + ^^f T^K-^(T,6)dT - .

0 0

so that according to (2.5)

Thus

oo 00 A) = 1 - A y* K^(x,3)dx - ^^\f T^K^(x,3)dx +

0 0

OO Ay x^K^(x,3)dx

^ + s^ — + ... , (2.18)

TCl/s) i_x (i-x)2

where X is given by (1.5) or (1.6). Substituting (2.17) and (2.18) into (2.4)

and equating the coefficients of identical powers of s on the left and right

s i des , we ob t ain
7.3 THERMALIZATION LENGTH 315

00

2 /<D^(T)dT = J_ - 1 , (2.19)

CO f

X

2 / T2K^(T,3)dT

(T)dT = — . (2.20)

(1-A)2

Similar expressions may be obtained for higher order moments as well. We note

that by using (2.2) the relation (2.19) can be rewritten in the form

oo 1+4^/ S (T)T^dT = — . (2.21)

0 ^

If a line has infinitely extended wings, in the limiting case 6=0 all

moments of $^(t) except the zeroth diverge, because of the slow decrease of

K]^(t) as T ^ °°. Of course, this does not apply in the case of a rectangular

profile, which refers to the usual monochromatic scattering. The relations

(2.19) and (2.20) become, respectively, (3.3.9) and (3.3.10).

7.3 THERMALIZATION LENGTH

DEFINITION OF THE THERMALIZATION LENGTH. As we explained in Sec. 4.5, for

scattering in media without continuous absorption (3 = 0) there is a certain

characteristic length — the so-called the rmali zation length — that represents

the optical distance from the place where a photon is born to the place where

it dies. The definition of the thermalization length used in Sec. 4.6 can be

extended in an obvious way to a medium with continuous absorption. We define

^^(t) = 1 + 47T / Sp(T')T'^dT' . (3.1)

The function increases monotonically from unity at t = 0 to

^JT) = ,

- (3.2)

1-A

where X is given by (1.6). The last equation is simply (2.21) with different

notation. We define, as in Sec. 4.6,

Sco^-^) = 'i'ao(^) , A = 1 , 3. = 0 . (3.3)

We then take for the thermalization length the root of the equation

S^(T^) = (l-A)-^ . (3.4)
316 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

The thermali zation length is, roughly speaking, the optical distance from the source at which H'^(t) approaches its asymptotic value for t = As we shall

see shortly, it would have been desirable to refer to as the migration

length; but we prefer to follow the more common terminology.

MEAN OPTICAL PATH PER SCATTERING. Since X depends on both X and 6, the thermalization length is a function of both of these parameters. However, limiting cases do exist in which the dependence on one of these quantities practically disappears. Let us consider this question in a little more detail.

From (1.6) we have

X = X - X36 (B) (3.5)

whe re

00

5 (B) 7 A /

^^^^^ dx . (3.6) a(x) +3

Therefore the problem is essentially a study of the function 6(B)- The physical significance of this function follows from the relation

oo /0

K^(T,B)dT = 1 - B6(B) , (3.7)

which is a consequence of (3.5) and (1.5). Since K-|^(t,B) is the probability

of direct radiative transfer of excitation (see Sec. 7.1), we conclude that

B6(6) is the probability that a photon is destroyed in its first flight.

But B may be regarded as the probability of photon destruction per unit line

center optical path. Therefore 6(3) is the mean optical path of a photon per

scattering

Substituting a(x) = 1/t in (3.6), we find

oo xj X' (t)dt

6(B) = 2A

Bt

' 1 where the function x(t) is such that a(x(t)) = 1/t, with x(t) > 0. Integrating by parts we obtain

oo 6(3) = 2AB / ^'^^^^^

, (3.8)

/ (l+3t)^

Let us study the behavior of 6(3) for small values of 3, which, from

(3.8) is evidently determined by the behavior of x(t) as t We rewrite

(3.8) in the form

oo 6(5) = 2A f (3.9)

/ (1 + u)^
7.3 THERjMALIZATION LENGTH 317

and use the fact that as 3 0 (see Sec. 2.6)

where y is the characteristic exponent. Substituting (3.10) into (3.9), we obtain

0 ^1"^^

from which we finally arrive at

oo 6CB) ~ ZaT ILl^^fl] , (3.11)

/ (l+u)2 \3/

6(B) ~ ^^^^^"^^^ X I ^) , 8 - 0 . (3.12)

s mZiTY

Let us consider in a little more detail the most important specific cases Going to the limit y 1/2, we find that for the Doppler profile

^D^^^ ^(^n (3.13)

v^V By/

As the asymptotic expansion of 62(3) is expressed in inverse powers of £n(l/3)

rather than of 3, it is therefore very helpful to have not only the leading

term, but also all subsequent terms of the asymptotic series. They may be

found in the following manner.

For the Doppler profile the expression (3.9) assumes the form

6,(3) = ^f^niy/' f (1 . ^Y'^ . (3.14)

^ /^V B/ ^ ^ my (i+u)2

Hence as 6 ^ 0

6i,(3)

\ B/ j = i t^lJ.. J (l+u)2 (^^1)

(3.1?)

But

00 /

[In u) ^ du ^

n (1+U)2

j = 2n - 1 ,

2(2^^-^-l)u2^|B2^| , j = 2n ,

where B^^ are the Bernoulli numbers, n = 1,2, Therefore (3.15) gives
318 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

l\ 1/2 2n

) S (in i\ 2n

wi th 1 and

C3.16)

2n

= _2(4n-3):: 2n-l

(4n):: ^

^ 2n

"2n , n = 1,2,.. (3.17)

The first few values of 6^ are:

2n

5^ = -0.41123; 6^ = -1.77569; 6^ = -29.1046 .

For a Voigt profile the asymptotic form (3.12) can also be refined. It

is found that

U(a,0)

(3.18)

For the Lorentz profile the integral (3.6) can be expressed in terms of the

elementary functions, that is,

5^(3)

/3(1 + B)

(3.19)

Values of the function 6(3) for the Doppler (a = 0), Voigt (a = 0.001; 0.01

and 0.1) and Lorentz (a = °°) profiles are shown in Table 32. D. G. Hummer

TABLE 32

MEAN OPTICAL PATH PER SCATTERING 6(B)

3 a = 0 a = 0 . 001 a = 0 01 a = 0 1 a = oo

10 9 346 -2 9 . 346 -2 9 349 -2 9 374 -2 9 .535 - 2

8 1 150 1 . 150 1 150 1 154 1 .179 -

6 1 494 1

494 1 494 1 501 1 .543 -

4 2 132 2 . 132 2 134 2 148 2 .236 -

2 3 738 3. 738 3 743 3 789 4 .082 -

1

1

6 049 6 . 051 6 066 6 206 7 .071 -

8 10" 6 925 6 . 927 6 949 7 145 8 .333 -

6 10' 1 8 124 8. 127 8 160 8 455 1 .021 0

4 10" 1 9 898 9 . 905 9 961 1 047 0 1 . 336 0

2 10" 1

1 297 0 1 . 299 0 1 312 0 1 428 0 2 .041 0 (Continued)
7.3 THERMALIZATION LENGTH

TABLE 32 (Continued)

319

QP 3. = nu a. — n u U i a. —

— nu . U i a. — n 11 a — oo

1 10-1 1 .588 0 1 59 1 0 1 .620 0 1 . 86 1 0 3 . 015 0

8 10-2 1 .676 0 1 680 0 1 .716 0 2 .017 0 3 . 402 0

6 10-2 1 785 0 1 79 1 0 1 . 838 0 2 . 234 0 3 . 965 0

4 10-2 1 .931 0 1 939 0 2 .009 0 2 .581 0 4 . 903 0

2 10-2 2 159 0 2 174 0 2 . 309 0 3 .337 0 7 . 00 1 0

1 10-2 2 . 364 0 2 394 0 2 . 646 0 4 . 409 0 9 . 950 0

8 10-^ 2 426 0 2 462 0 2 . 770 0 4 . 847 0 1 . 114 + 1

6 10-^ 2 504 0 2 55 1 0 2 947 0 5 . 496 0 1 . 287 + 1

4 10-^ 2 609 0 2 677 0 3 .239 0 6 . 597 0 1 . 578 + 1

2 10-^ 2 778 0 2 907 0 3 9 10 9 .128 0 2 . 234 + 1

1 10-^ 2 937 0 3 177 0 4 .923 0 1 .276 + 1 3. 161 + 1

8 10-4 2 986 0 3 280 0 5 . 344 0 1 .423 + 1 3. 534 + 1

6 10-4 3 048 0 3 429 0 5 .989 0 1 .639 +

4. 081 + 1

4 10-4 3 133 0 3 681 0 7 . 097 0 2 . 003 + 1 4. 999 + 1

2 10-4 3 273 0 4 278 0 9 .689 0 2 . 825 + 1 7 . 070 + 1

1 10-4 3 407 0 5 210 0 1 346 + 1 3 .991 + 1 1 . 000 + 2

8 10-^ 3 449 0 5 612 0 1 . 500 + 1 4 . 461 + 1 1 . 118 + 2 6- 10-^ 3 502 0 6 . 222 0 1

726 + 1 5 150 + 1 1 . 291 + 2 4- 10-^ 3 576 0 7. 290 0 2 106 + 1 6 306 + 1 1 . 581 + 2

2

• 10-^ 3 698 0 9 . 834 0 2 969 + 1 8 915 + 1 2 . 236 + 2

1

• 10-^ 3. 816 0 1 . 358 + 1 4 192 + 1 1 261 + 2 3 . 162 + 2 8- 10-6 3 854 0 1

512 + 1 4 686 + 1 1 409 + 2 3 . 536 + 2

6

• 10-6 3 901 0 1 738 + 1 5 409 + 1 1 627 + 2 4 . 082 + 2 4- 10-6 3 967 0 2 . 119 + 1 6 623 + 1 1 993 + 2 5 . 000 + 2

2

• 10-6 4 077 0 2 . 985 + 1 9 364 + 1 2 819 + 2 7 . 071 + 2

1

• 10-6 4 185 0 4 . 214 + 1 1 324 + 2 3 986 + 2 1 . 000 + 3 8- 10-7 4 219 0 4 , 710 + 1 1 . 480 + 2 4 457 + 2 1 . 118 + 3 6- 10-7 4 262 0 5 . 437 + 1 1

709 + 2 5 146 + 2 1 . 29 1 + 3

4

• 10-7 4 323 0 6 . 65 7 + 1 2 093 + 2 6 . 303 + 2 1 . 581 + 3

2

• 10-7 4 424 0 9 . 411 +

2 . 960 + 2 8 913 + 2 2 . 236 + 3

1

< 10-^ 4 523 0 1

331 + 2 4 . 186 + 2 1 . 261 + 3 3 . 162 + 3 8- io-« 4 554 0 1

488 + 2 4 . 681 + 2 1409 + 3 3 . 536 + 3 6- io-« 4 594 0 1

718 + 2 5 405 + 2 1

627 + 3 4 . 082 + 3 4- io-« 4 650 0 2 104 + 2 6 619 + 2 1 . 993 + 3 5 . 000 + 3

2 lo-s 4 745 0 2 975 + 2 9 361 + 2 2 . 819 + 3 7 . 07 1 + 3

1 io-« 4 837 0 4 . 208 + 2 1 324 + 3 3. 986 + 3 1

000 + 4

522-519 O - 74 - 22
320 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

(1968) has recently published a five- figure table of (1/A)6(3/A) for

log3 = 10.0 (0.5) 0.0 and -loga = o° ; 4.0 (0.5) 0.5; -oo. it should be noted

that the function (1/A)6(3/A) is the Menzel "curve of growth" for the

Schus ter-Schwarzs child model (D. H. Menzel, 1936).

THERMALIZATION LENGTH. LIMITING CASES. Now that we have studied the behavior

of 6(3), we can consider the dependence of on A and 3. When the inequality

1-X >> A36(3) (3.20)

is satisfied, the quantity 1-A, as may be seen from (3.5)

independent of 3 and is close to 1-A. According to (3.4)

length in this limiting case is almost independent of 3 and

value for 3=0 (see Sec. 4.6). The physical significance

as follows. As has been shown in Sec. 7.1, A is the probab

creation of an excited atom will result in the emission of

subsequently excite another atom. Consequently 1-A is the

per scattering that the photon will be destroyed, and

is practically

the thermalization

approaches its

of this result is

ility that the

a photon that will

total probability

1-A

is the mean number of scatterings of_a photon in an infinite medium. When

the inequality (3.20) is satisfied, N is practically independent of 3 and is

almost equal to its value for 3=0. Consequently when (3.20) is satisfied,

most of the photons are destroyed in the scattering event, while their death

in flight (because of continuous absorption) is insignificant. We note in

passing that the mean length of a photon path in an infinite medium is obvious ly

T = 6 (3)N = 6(6)

1-A+A36 (3)

Using the asymptotics of 6(3) just obtained, we find that for 3 << 1 the

condition (3.20) has the following forms for the most important special cases

of the absorption coefficients:

Doppler: 1-A >> A —3 ( iln 1/3 ) "'"^^ (3. 20a)

Voigt: 1-A >> A(^U(a,0))^/^(a3)

,

1/2 Lorentz: 1-A >> A3^ .

It follows from (3.18) that (3.20b) is valid only for values of

small in comparison with a. For 1-A << 1 the inequality (3.20) may be reversed:

(3.20b)

(3.20c)

that are

1-A << 36(3) (3.21)
7.3 THERMALIZATION LENGTH 321

[the factor X on the right can be replaced by unity, since 1-A << 1), In this

limiting case the basic mechanism for removing photons from the process of

multiple scattering is absorption in the continuum (occurring in flight) , and

the death of photons in the scattering event can be disregarded. The mean number of scatterings N and the thermali zation length can be considered to

be independent of X, as though A = 1.

It must be emphasized that for a Voigt profile the destruction of photons

during scattering is insignificant compared to their death due to continuous

absorption, when 1 - A is much less than /aS" (and not 31). This important

conclusion, which' follows from a reversal of the inequality (3.20b), is valid

for 3 << a.

It is now simple to find the dependence of the thermali zati on length on

3 for those 3 << 1, which satisfy the condition (3.21). Since 1-A << 1, one

can obtain x-j- for Doppler, Voigt and Lorentz profiles from the expressions

(4. 6 . 13a) - (4. 6 . 13c) , by substituting 1-A for 1-A. This is clear from the

definitions of x^ for 3=0 and 3 > 0, given by (4.6.10) and (3.4), respectively. Since, when the inequality (3.21) is satisfied, the quantity 1-A

can be replaced by 36(3), we can use the asymptotic forms of 6(3) given above

for 3 << 1 to obtain

, 1-A << 3 << 1 , (3,22a)

(1-A)^ << a3 << a^ << 1 , (3.22b)

Doppler: x^ ~ l6(^^^|)

Voigt: x^

-1

9

Lorentz: x^ ~ |-3 , (1-A)^ << 3 << 1 . (3.22c)

Thus, for Voigt and Lorentz profiles, the thermali zation length is a quantity on the order of 3"-'-, while for the Doppler profile, x^ is approximately In 1/3 times smaller than 3"-^. At first sight this result might seem

strange. The effect of continuous absorption is to limit the path length of

the photon through the medium to 3"-'-. It might therefore seem that if

continuous absorption is the main reason for the death of photons, the thermalization length should differ from 3~-'- only by a factor on the oider of

unity. In fact this is not so, since the thermali zation length does not

express the path length covered by a photon in the medium, but, rather, the mean displacement from its birthplace to its place of death. The photons do

not follow straight trajectories, but instead describe zigzag paths, so that

the mean displacement of the photons can be significantly smaller than the

length of the path they have followed in the medium. Because of these jagged

trajectories the thermalization length for the Doppler profile is on the order

of iln 1/3 times smaller than B"-*-. For monochromatic scattering this effect

is even more telling. Since monochromatic scattering is a diffusion process,

we should expect that the displacement of a photon will be on the order of

the square root of its path length. Therefore the thermalization length, to

within a factor on the order of unity, should equal 1//3. It is easy to

verify that this is indeed so. For the rectangular profile we have from (3.6)

1 1+3

Consequently

,

1_X = 1-A + . (3.2 3)

1+3
322 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

Since for monochromatic scattering S^(t) (3/2)t^ when t >> 1 (see Sec. 3.9), we find from (3.4) and (3.23)

which agrees completely with the estimate just given.

Comparing these results with those found in Sec. 4.6, one can arrive at

the following general conclusion. If the destruction of photons in scattering

is negligible compared to that caused by continuous absorption, and if the

line absorption coefficient decreases rapidly enough in the wings so that the

accumulation effect is significant, then the the rmali zation length is substantially less than 3~1. The greater the accumulation effect, the greater the

difference of from g~l.

It has been shown in Sec. 4.6 that no accumulation occurs if the characteristic exponent y is less than 1/2. In the scattering process the redistribution of photons between the core of the line and its peripheral regions

is found to be more important in this case than the change in the direction

of their motion. Consequently if the inequality (3.21) is satisfied and

Y < 1/2, the thermali zation length is equal to 6~1 to within a factor on "the

order of unity.

If neither of the inequalities (3.20) or (3.21) is satisfied, the thermalization length depends on both A and 3. However, as follows from (3.4)

and (3.5), for two media (1 and 2) in which values of X and 6 are such that

and the profiles of the absorption coefficient are the same, the thermalization lengths are equal.

GENERAL ANALYSIS. The Green's function for an infinite medium will be ana- lyzed only for the most interesting case, that of the Voigt profile. Moreover,

from the beginning we shall assume that 1-X << 1. This condition implies

that the two inequalities 1-X << 1 and B << 1 are simultaneously fulfilled.

From physical considerations, it might be expected (and this will soon be

verified by direct calculation) that in this case a region around the source will exist in which the death of photons has almost no effect. The Green's

function in this regi on^di f fers little from that in a conservative medium.

The function Sr)(T) for X = 1 was discussed in detail^in Chapter IV. Therefore an investigation of the Green's function for 1-X << 1 involves mainly

a study of its behavior for large t. Setting y = t((1/z)-3), we get from (2.16)

1-X << 3 << 1 (3.22d)

1 - X^ + X^3-,^6(3-^) = 1 - X^ + X232'5(62) ,

7.4 ANALYSIS OF THE INFINITE MEDIUM GREEN S FUNCTION

(4.1)

Because of the exponential factor, the contribution to this integral from the

region of large y is insignificant. -Therefore for a Voigt profile, when t is
7.4 ANALYSIS OF THE INFINITE MEDIUM GREEN'S FUNCTION 323

sufficiently large we find from (4.1), considering (2.7.27), (2.15) and (1.44)

and the fact that 1-X << 1,

where

1 R(y)

SV.) 2/aUXa^y/^e^ r ^-J^^^^^^^Z

^ ^^^^^

Hill ^0

^ ^

l /TTaU(a.O) y^^ (y + 23T)^/^ ]^ ^ 3\ T ; y +3T J

,

['l /.aU(a,Q) V/^ y^l'

(4.3)

The asymptotic expression (4.2) is rather complicated. However, there are two

important limiting cases in which it becomes substantially simpler.

6-SOLUTION. If

1-X << (^U(a,0)a3)^^^ , (4.4)

the dependence of the Green's function Sp on A practically vanishes. Consequently, as follows from (4.3), one can set A = 1, and

S^(T) ~ -i- ? Ul^l^^'^ ' ^^-^^

P 4^t2 2tt (U(a,0)aT)l/'^ P

where

sV(3T) = -ie-^^ r\-y ^l!^!(zW_ dy . (4.6)

P /tt / (y + 23T) 3 + y

0

The representation (4.5) is valid when ax >> 1 and a >> 3. However, if (4.5)

is rewritten in the form

S^(T) ~ S^(t,1)s^(3t) , (4.7)

i where Sp(T,l) is the Green's function for a conservative medium, i.e. a medium

' with 3=0 and X = 1 (see Chapter IV), then we obtain a representation which,

i in contrast to (4.5), is valid for all x, and not only for x >> . This

' result is applicable when the inequalities a >> 3 and (4.4) are satisfied.

We have here a tremendous simplification — a function of three variables (x

,

I X, 6) is, in the limiting case being considered, expressed in terms of two

jj functions, each of which depends on only one argument.

From (4.6) we find

Sp(3x) ~ 1 , 3x << 1 , (4.8)
324 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

V 3 e"^^ Sp(BT) - I

^ , 3t >> 1 , (4.9)

so that for t << 1/3 (4.7) gives

Sp(T) ~ S^(t,1) , T << 1/3 . (4.10)

In other words, at distances from the source which are small in comparison

with the thermalization length, the function S^(t) for A 1 is close to the source function in a conservation medium.

We shall call the source function corresponding to X = 1 and 3 0 the

3-solution . When X 7^ 1 and the condition (4.4) is fulfilled, the source function is close to the 3-solution for all t. X-SOLUTION. We shall call the source function corresponding to 3 = 0 the

X- s olution . In preceding chapters we have investigated this same X-solution.

From (4.3) it follows that for

1 - X >> (^U(a,0)aB) (4.11)

the dependence of the right side on 3 nearly vanishes, as the values of R(y) are practically equal to its values for 3=0. Therefore in this limiting

case one can set 3 = 0 in (4.3) and obtain from (4.2)

S^(t) ~ S^(T,X)e-^'' , (4.12)

where Sp(T,X) is the Green's function for a medium with 3 = 0. In other words

,

when the condition (4.11) is satisfied, the Green's function can be obtained

by multiplying the X-solution by e'^"^. Since for 1-X << 1 Sp(T,X) can be

expressed in terms of functions of one variable (see Sec. 4.4), in the limiting case (4.11) the Green's function for X / 1 and 3 7^ 0 may also be expressed

in terms of functions of a single variable.

A MORE GENERAL CASE. If neither of the inequalities (4.4) and (4.11) is

satisfied, the great simplifications found above do not occur. As follows

from (4.2) and (4.3), when 1-X << 1, for an arbitrary relation between 1-A

and a3, the Green's function can be represented as \

S^(t) ~ S^(T,l)s^(t,3T) , (4.13)

where

V

s P

(t,3T) = ^e-^^ ( .-^ y'^'hy^^^^'-^y

, (4.i4)

^R ^ [t(y + 3T) + (y + 23T) 3/2]2+y3

t = . (4.15)

(TiaU(a,0)) ""^
7.5 SEMI- INFINITE MEDIUM 325

Hence, in this case, we conclude that for t << 3~ the Green's function can

be assumed to be equal to S^(t,X), i.e. we can set 3=0 and use the results

obtained in Chapter IV.

7.5 SEMI-INFINITE MEDIUM

TRANSFER EQUATION. Without any loss of generality the transfer equation for

line frequencies can be written in the form

^d^T^^ = Ca(x)+3)I(T,y,x) -

dx

1 (5.1) - |Aa(x) J

a(x')dx' J

I (t , y

' , x' ) dy ' - a(x)S*(T)

1

The quantity I(T,y,x) in this equation is the intensity of the diffuse line

radiation, i.e. the radiation emitted in discrete radiative transitions. To

get the total radiation intensity one has to add to I(T,y,x) the intensity of

the unscattered continuum radiation of the medium and the intensity of the

direct external radiation attenuated by the medium. The boundary condition

for equation (5.1) is

1(0, y,x) = 0 , y < 0 . (5.2)

The primary line s ource fun ct ion S]^(t) can be rep res en ted as (see Sec. 2.4)

CO S*(t) = 3^ J

K^^(|t-t' I ,3)S^(T')dT' + S*(T) . (5.3)

0

The integral term in (5.3) accounts for atomic excitation by the medium's own continuum radiation, and S (x) describes all other mechanisms of primary

excitation (electron impact, recombination, radiative excitation by the external radiation, etc.).

If we introduce the line source function

S(x) = \^ J

a(x')dx' JI(x,y' ,x')dy' + sj(x) , (5.4)

_oo _1

(5.1) can be rewritten in the form

a(x)+3 dx\a(x) / a(x)

from which it is evident that the function ( (a (x) +3 ) /a (x) ) I (x , y , x) depends

only on two variables, namely, x and
326 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

z = (5.6)

We define

I(t,z) = ^/ I(T,y,x) . (5.7]

a(x)

Then the transfer equation (5.5) assumes the classical form

^dl(_r^ = I(T.z) - S(t) . (5.8)

ax

We must emphasize that the function I(t,z) for 6 7^ 0 is not the radiation

intensity, but just an auxiliary function related to the intensity by equation

(5.7).

Substituting (5.7) into (5.4) and changing the order of the x' and y'

integrations, we find that the line source function S(t) can be expressed in

te rms of I(t,z) as

z

S(t) = I J

I(T,z')G(?')dz' + S*(t) , (5.9) — z c where

z = 1/3 . (5.10)

1-3 I z'

I ' "c

The boundary condition (5.2) assumes the form

l(0,z) = 0 , z < 0 . (5. 11)

Equations (5.8) to (5.11) are the basic equations of the problem of radiative

transfer in line frequencies in a s emi - infini te medium with continuous absorption.

Substituting into (5.9) the formal solution of the transfer equation

(5.8) subject to the boundary condition (5.11), we arrive at the integral

equation for the line source function

S(T) - I

0 wh ere

00 J

K^(|t-t' I ,6)S(T')dT' + sJ(T) , (5.12)

z c :i(t,3) = f

e"''/"' G(c')dz'/z' (5.13)
7.5 SEMI- INFINITE MEDIUM 327

As we have seen in Sec. 7.1, the kernel function K-|^(t,3) can also be written

as

00

K-lCt,6) = a^Cx)E-^((a(x)+B)T)dx . (5.14)

In the next subsection we discuss briefly the properties of the basic

special functions in terms of which the solution of (5.12) is expressed. For

a more detailed discussion, see D. I. Nagirner (1968).

THE RESOLVENT FUNCTION AND THE H- FUNCTION. Equation (5.1) may be solved by

the method utilized in Chapter V for the case 3=0. We will omit all the

calculations and give only a summary of the results. The resolvent function

$(t) , which is the solution of the equation

oo $(t) = K^(|t-t' I ,8)$(T')dT' + |k^(t,3) (5.15)

0

and is related to the resolvent r(T,T') of equation (5.1) by the expression

r(T,T') = $(|t-t'|) + / $(T-t) ^(t' -t) dt , T-^ E min(T,T') , (5.16)

0 is equal to

z /c

e-^/"R(z)G(i;)-^ , (5.17)

zH ( z)

0

where R(z) is given by (2.15), and H(z) satisfies the equation

z

H(z) = 1 + ^zH(z) /

tililiG(c' )dz' . (5.18)

^ J z+z

0

In view of the importance of the H- function, we shall consider it in a little more detail. Let us define

z c

a^^ = / H(z)G(Oz''dz , n = 0,1, (5.19)

The quantity is the n-th weighted moment of the function H(z), with the

weight function G(c). Further, let
328 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

/

z

,c

, GCOz'^dz , n = 0,1, C5.20)

From (5.18) it is easy to find that

2n .

j = 0

> >

For the proof of this relation, see I. W. Busbridge (1960). For n = 0, (5.21)

gives

Xa^ = A + — ,

from which, considering the boundedness of as X ->- 0, we obtain

ag = |(l - /T^) . (5.22)

For 6=0 this expression reduces to (5.4.16), since in this case A = X. It

should be mentioned that in line transfer problems with g = 0, usually all

of the moments except ag diverge. In many problems, in addition to ctj^

,

we encounter the moments

z

.c

^ik y* H(z)Gj^(c)z''dz , i,k = 0,1,... , (5.23)

0

where the function G]^(z) is defined by the expressions (2.6.12) and (2.6.13).

It is obvious that ani ^ a^. The moments must be found numerically (with

the exception of ^ s a^) .

It follows from (5.5) and (5.9) that as z increases from 0 to °° the function H(z) increases monotonically from H(0) = 1 to

H(-) = (1-X)-^/^ . (5.24)

The explicit expression for H(z) when z > 0 is
7.5 SEMI- INFINITE MEDIUM 329

H ( z ) = e xp

V n

lTv\\ - AVCu,6)] du )

l + z^u^ j

[5 .25)

where V(u,3) is given by (1.37) or (1.39). When 6=0, (5.25) reduces to

(5.4.11) .

Let us consider one important limiting case of (5.25). When the line ab

sorption coefficient has a Voigt profile and the inequalities 1-A << 1 and

1-X << (aB)-'-'^ are satisfied, then X can be replaced by unity in (5.25)

(3-solution) . The H-function is then much simplified. For z << g"l it is

practically independent of 3 and is asymptotically equal to ihe conserva-tive

H-function, corresponding to the case 1-A =3=0, which was tabulated in

Sec. 5.7. When z becomes of the order of 3"-'-, the continuous absorption is

"switched on." The H-function for values of z in this region can be found

from the asymptotic expression

Hy(z) - H^(z,l)h^(3z) , (5 . 26)

where

£n h^(s)

OO

£n- /t^+s^+2s dt

(/t2+s2+s) 1/2 i+t2

(5.27)

and Hv(z,l) is the conservative H-function. This expression is obtained from

(5.25) by the use of (1.43) and is valid when

(1-A)^ << a3 <<

Values of the function h-|^(3z) are given in Table 33.

The functions Hv(z) and related quantities were recently tabulated by

D. I. Nagirner for a large number of values of A and 3 and several values of

the Voigt parameter a. Tables 34 and 35, which give values of the H-function,

the moment oqq, and the function

W(z)

J z + z

-Go(c')dz' (5.28)

I encountered in the next section, are taken from the much more extensive unpublished tables which are being prepared for publication by D. I. Nagirner.
330 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

TABLE 33

THE FUNCTION h (gz)

6 z h 1 (Bz) 13z h 1 (gz)

1

tIz ^1 (ez) ; z hj(6z)

0 . 000 1 . 0000 0 .075 0 .9025 0 ,. 19 0 . 8297 0 . 475 0 ., 7322

0 . 005 0 .9870 0 .080 0 . 8984 0 ,. 20 0 . 8248 0 . 500 0 ,,7261

0. 010 0 .9774 0 .085 0 . 8944 0 ,.22 0 . 8156 0 . 525 0 ,,7204

0 . 015 0 .9690 0 . 090 0 . 8905 0 ,. 24 0 . 8069 0 . 550 0 .,7148

0 . 020 0 .9615 0 . 095 0 .8867 0 ,. 26 0 . 7988 0 . 575 0 ,, 7095

0 . 025 0 .9545 0 . 10 0 .8830 0 ., 28 0 . 7911 0 . 60 0 ., 7044

0 . 030 0 .9481 0 . 11 0 .8760 0 ,. 30 0 . 7838 0 . 65 0 ,, 6947

0 . 035 0 .9420 0 . 12 0 . 8692 0 .,32 0 . 7768 0 . 70 0 .,6857

0 . 040 0 .9363 0 . 13 0 .8628 0 ., 34 0 . 7702 0 . 75 0 ,,6772

0 . 045 0 .9308 0 . 14 0 . 8567 0 ,, 36 0 . 7639 0 . 80 0 ,,6693

0 . 050 0 .9 256 0 . 15 0 . 8509 0 ,. 38 0 . 7578 0 . 85 0 ,, 6618

0. 055 0 .9206 0 . 16 0 . 8453 0 ., 400 0 . 7520 0 . 90 0 ,,6547

0 . 060 0 .9 158 0 . 1 7 0 . 8399 0 ., 425 0 . 745 1 0 . 95 0 .,6480

0 . 065 0 .9112 0 .18 0 . 8347 0 ,. 450 0 . 7385 1

00 0 .,6417

0 . 070 0 .9068

TABLE 34

THE FUNCTIONS H(z) AND W(z) FOR THE VOIGT PROFILE

,

X =1 AND 3 = 10" 4

a = 0 . 001 a = 0 . 0 1 a = 0 . 1

7 H(z) W(z) H(z) WCz) H(z) W(z)

z

z

"

l + 3z

0 . 0 1 . 000 0 .000 1 .000 0 . 000 1 .000 0 , 000

0 . 1 1 . 156 0 . 4 76 1 . 154 0 .486 1 . 141 0 .524

0 . 2 1 .273 0 . 791 1 . 268 0 .811 1 . 243 0 . 889

0 . 3 1 .377 1.044 1 .370 1 .076 1 . 331 1 .19 5

0 . 4 1 .473 1 . 259 1 . 463 1 . 302 1 .411 1 . 464

0 . 5 1 . 563 1 . 448 1 .550 1 . 503 1 .485 1 . 708

0 . 6 1 .649 1.616 1 . 633 1 .683 1 . 553 1 .932

0 . 7 1.731 1 . 770 1 .712 1 . 849 1 .618 2 . 141

0 . 8 1.810 1.911 1 . 788 2 .002 1 .679 2 . 338

0 .9 1 . 886 2.041 1 . 860 2 . 144 1 . 737 2 .525

1 . 0 1.960 2.163 1 .931 2 .278 1 . 792 2 . 703

(Continued)
7.5 SEMI- INFINITE MEDIUM

TABLE 34 (Continued)

331

a=0.00 1 a=0.01 a=0.01

HC z) WCz) H(z) W ( z) H ( z) W ( z)

1 X z = a(x)+3

0 . 00 1

960 2 163 1 9 31 2 278 1.792 2 703

0

25 2006 2 237 1 9 74 2 359 1.823 2 . 802

0 . 50 2

158 2 470 2 116 2 617 1.920 3 117

0 75 2 . 459 2 89 4 2 39 4 3 09 3 2.101 3 70 7

1 00 3

000 3 565 2 882 3 871 2.391 4 69 2

1

25 39 52 4 5 84 3 710 5 115 2.823 6 2 88

11 . o u

rD . ^; 0 c D Z 0/ID 1 Z 8 D n 7 n / J. 0 4

QOQ C y1 o D 4

I 75 8 . 5 3 7 s 5 3 3 7 180 16 8 4.154 1 2 9 2

2 . 00 1 3 . 39 1 2 4 4 10 1 2 1 7 1

4.972 1 8 9 7

2 . 2 5 20 . 60 10 3 1 3 5 6 2 8 89 5.772 26 96

2 50 29 . 04 30 0 4 1

82 4 7 80 6.466 35 89

2 75 35 . 89 46 0 1 19 24 70 34 7.022 44 5 2

300 39 . 4 8 6 1

4 8 20 69 8 8 6 4 7.462 5 2 34

3 25 40 . 79 69 . 88 2 1 49 100 6 7.824 59 46

3 50 4 1 . 7 3 0 4 2 1 9 9 10 8 g 8.139 6 6 15

3 75 4 1

3 7 7 4 3 7 2 2 36 115 4 8.423 72 5 8

4 46 7 5 1 7 22 fS 7 1 ? 1 8.683 7 8 84

4 25 4 1

54 75 77 22 9 4 125 2 8.926 84 . 9 7

4 50 4 159 76 2 7 23 1 7 130 8 9.153 90 9 7

4 75 41

64 76 69 23 38 135 1 9.366 96 86

5 0 4 1 \J o 7 7 1 3 8 9 9.567 10 2 6

5 5 41

75 77 63 23 87 145 5 9.938 113 8

6 0 41

79 78 06 24 11 15 1 3 10.27 124 6

6 5 41

83 78 40 24 32 156 0 10,57 134 9

7 0 41 . 86 78 67 24 48 160 1 10 . 85 144 7

7 5 41

88 78 89 24 62 163 6 11.10 154 1

8 .0 41 . 90 79 07 24 74 166 6 11.33 163 1

8 . 5 41 . 92 79 22 24 84 169 1 11.54 171 7

9 .0 41 . 93 79 34 24 93 171 4 11.74 179 8

9 . 5 41. 94 79 45 25 00 173 3 11.92 187 6 10 .0 41 . 95 79 54 25 07 175 0 12.08 194 9

a 00 102 3 246 4 448. 9
332 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

TABLE 35

THE FUNCTIONS H(2) AND FOR THE VOIGT PROFILE, A = 1 AND 6=10 -6

a = U . (J U 1 a = 0.01 a = 0 . 1 z H(z) W(z) H(z) W(zl H Cz) W(zl

z

z

l + 6z'

n n 1 . 000 nU

n n n . U U U 1 n n n . U U U 0 . 000 1 .000 0 .000

u

1

. 1 1 . 156 U

/l Q C

. 4 O D 1 .15 4 0 . 500 1 • 141 0 .5 46

nu 9 1 .273 U . o U i 9 ^ Q . 2 o y 0 .839 1 .243 0 .9 34

u

•7

• o 1 .378 11 . U / U 1 t 9 n

1 . 116 1 . 332 1 .26 3

u

A 1 .474 11 . / y 4 1

/I A /I

. 4 O 4 1 . 357 1 .411 1 .555

nu

c

• 0 1 . 565 1

/I Q 0 . 4 y z 1 .551 1 .571 1 .4 85 1 .822

u , o 1 .651 1 . D oy 1 ,634 1 . 765 1 .55 4 2 .069

n • 7 1 . 733 11 . O O 1 7 1 1 .944 i A 1 o

. 0 1 o

927 n 1

. oO 1 0 . 8 1 .813 1 .981 1 . 789 2 . 110 1 .680 2 .521

0 . 9 1 . 889 2 . 119 1 . 862 2 . 266 1 . 738 2 . 730

1 .00 1 .963 2 . 250 1 .933 2 .413 1 . 793 2 .931

X

a

1 (x) + B

u n n . u u 1 .963 9 9 c n . / o U 1 0 7 7

. y o o 2 .413 1 . yy 3

72 n 7 1 .931

u . ^ D 2 . 0 10 9 7 9 O i . y / 6 2 .503 1 . 0 2 4 3 .043

u . 0 U 2 . 163 9

i. . 5 0 i 92 . 1 i y 2 . 789 i n o o

3 .40 1

u

7 c 2 . 466 7J . U 4 4

927 Q Q

. oy 0 3 . 327 9

2

1 n T

. i U o 4 . U 0 1

1i n n 3 .013 •2 o7 Q A 92 . oy u 4 .230 7

2

tC\ A .39 4

r5

7 7 O

• ZOO

1 9 C 3 .982 4

Q O 9

. y 0 2 7 9

. / 2d 5 . 735 72

o o o

. O Z 8

7 1 <^ 7 .16 2

i

c n . 0 U 5 . 706 6

o n r\ .89 9 5 .10 9 8 . 355 3 .426 1

. 3 7

1 . / 0 8 . 783 1. 2 1 7

O O 7 .283 13 .28 4 .175 1 5 . 6 6

2. . (JO 14 .21 1

. 50 1

. 39 23 .28 5 .013 2 3 . 9 8

I

o c

. Z o 23 . 26 29 o c\ , 89 1

. 2 2 44 .22 5 . 845 35 . 56

2 . 50 36 .51 6 1 . 44 1 8 . 2 3 84 . 76 6 .583 49 . 32

I . 75 52 . 14 13 8 . 0 2 1 . 7 7 145 . 7 7 .188 6 3 . 4 1

•z

. U 0 66 . 81 o n ^ 2 9 6 . 2 24 . 3 7 208 . 9 7

^ "7 O

. 6 7 o 7 6 . 6 3

•7o

9 C 77 . 85 CAT 5 0 o7

. 2 2 6 . 0 8 259 . 4

o0 n n o

. u y Z oy C 7. 6 2

7

. o 0 84 . 15 ^ r o 6 5 8 . 5 2 7 . 3

300 . 0

o8 , 4oU 10 2 . 1 87 .62 7 5 5 . 3 2 8 . 34 336 . 5 8 . 79 8

11/1 114 . 6

4 .00 90 . 11 829 . 3 29 .27 372 . 0 9 . 116 127 . 2

4 .25 92 .27 896 . 7 30 .15 407 . 3 9 .419 140 . 1

4 . 50 94 .27 961 . 7 30 .99 442 . 8 9 . 709 153,. 1

4 . 75 96 . 14 1025 31 . 80 478 . 5 9 .988 166 ,. 3 (Continued)
7.6 LINE FORMATION IN AN ISOTHERMAL ATMOSPHERE

TABLE 35 (Continued)

333

H(z)

0.001

W(z) H(z)

0.01

W(z) H(z)

0 . 1 W(z)

a (x) +B

5 0 9 7 9 2 1087 32 59 514 4 10 26 179 7

5 5 101 2 120 8 34 07 586 6 10 77 207 1

6 0 104 2 1322 35 47 659 3 11 26 235 3

6 5 106 9 1432 36 80 732 5 11 72 264 2

7 0 109 4 1537 38 06 805 9 12 16 293 6

7 5 111 6 1636 39 27 879 5 12 59 323 8

8 0 113 7 1730 40 42 953 2 13 00 354 5

8 5 115 6 1820 41 52 1027 13 39 385 8

9 0 117 3 1905 42 57 1100 13 78 417 5

9 5 118 9 1986 43 59 1173 14 15 449 7 10 0 120 4 2062 44 57 1246 14 51 482 4

00 4607 8213 142.5- 10'

7.6 LINE FORMATION IN AN ISOTHERMAL ATMOSPHERE

FORMULATION OF THE PROBLEM. We shall apply the results of the preceding sec- tion to the problem of absorption line formation in stellar spectra. Our

discussion will be brief. A detailed review of work in this area has been

given by A. Unsold (1955) and K.-H. Bohm (1960).

Strong lines are formed in the outermost layers of the atmosphere, where

the continuum optical depth is small. To a good approximation one can regard

such lines as being formed in an isothermal atmosphere.

Originally it was assumed that the frequency of a photon remains constant

during scattering (so-called coherent scattering). Using this assumption,

A. S. Eddington (1929) discussed the line formation problem with absorption*

in the continuum and scattering taken into account, and obtained an approximate solution. An exact solution was later found by S. Chandrasekhar (1950).

However, as was explained in Sec. 1.5, the assumption that the frequency

remains constant during scattering cannot be accepted. J. Houtgast (1942)

formulated the same problem with the assumption of complete frequency redistribution. The exact solution of this problem was obtained by V. V. Sobolev

(1949; 1954; see also V. V. Sobolev, 1956 , Chapter VIII), but only the line

profiles were found. The problem of determining both the source function and

the line profile has been discussed by J. T. Jefferies and R. N. Thomas (1958,

1959), M. P. Savedoff (1952) and others, who used approximate methods to solve

the problem. Much more complete and accurate results have been obtained by

D. G. Hummer (1968) who used the best available numerical methods to solve the

transfer equation. A number of Hummer's results will be given in this chapter.

Below we consider the simplest model of strong line formation in stellar

spectra. The results obtained here from the solution of this model problem
334 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

are of more theoretical than practical interest, and attempts to use them

directly to interpret observed profiles of strong lines would hardly be justified. However, some general conclusions about the way in which the solution

depends upon the parameters may be useful.

Let us now turn to the mathematical formulation of the problem. We con- sider an isothermal, semi- infinite atmosphere with no external illumination.

The atmosphere is composed of a mixture of two- level atoms (line frequency vg)

and atoms of other types that can be ionized by radiation of frequency vq.

The kinetic temperature of the gas T is such that hvQ >> kT, so that stimulated emission in the line may be ignored. The survival probability A of a photon during scattering and the ratio 3 of the continuum absorption coefficient caused by photo- ioni zation of atoms of "the contaminants," to the line

center absorption coefficient are assumed to be independent of depth. It is

further assumed that the atoms of the "contaminants" are in LTE. Under these

assumptions the evaluation of the radiation field in the gas and the determination of the degree of excitation as a function of depth involves essen- tially the solution of the transfer equation

^dlir^^ = (a(x)+6)l(T,y,x) -

•Aa(x)

J

a(x')dx'y I ( x , y

' , x ' ) dy ' - [( 1- X) a (x) +B] (T)

with the boundary condition

1(0, y,x) = 0 , y < 0 . (6.2)

Here I(T,y,x) is the total intensity of radiation (i.e. the sum of the intensity of the diffuse line radiation and the intensity of unscattered continuum

radiation), Bvq(T) is Planck's function, and X is the survival probability of

a photon during scattering, which in the case at hand is equal to

^2 1 X = 21 e 2

Throughout the rest of this section it will be assumed that

B^, (T) = 1 ; (6.3)

in other words, the intensity and the line source function will be expressed

in units of the Planck intensity.

Introducing the dimensionless line source function

1 S(t) = y* a(x')dx' y I (T,y' ,x' )dy' + 1-X , _oo _1

which is related to level populations by
7.6 LINE FORMATION IN AN ISOTHERMAL ATMOSPHERE 335

hvQ/kT

we obtain from (6.1) — (6.3) the following integral equation for S (see Sec,

2.4 and 7.5) :

oo S(t) = I y K^(|t-t' I ,6)S(T')dT' + S*(t) , (6.4)

where

S*(t) = 1 - X + 3-| y* K^-^(|t-t' I ,B)dT' . (6.5)

0

In addition to the source function, the intensity of emergent radiation is

also of great interest. It is given by (see Sec. 2.3)

oo 1(0, y,x) = y* (a(x)S(T)+3)e"(^'^''^''^)^/^dT/y . (6.6)

0

LIMITING CASES. We shall begin our analysis of the above equations by con- sidering the source term in (6.4), i.e. the function S]^(t). Having rewritten

(6.5) in the form

/go 00 \ sJ(t) = 1 - a + 8|f2y K^^(t,3)dt - y* K^^(t,3)dtj ,

we can use the relations appearing in Sec. 7.1 to obtain

S*(t) = 1 - X - |3 K2o(t,3) (6.7)

The function K2o(t,3) decreases monotoni cally from K2o(0>3) = 6(3) to zero

at T = 00. Correspondingly, S*(t) increases monotonically from

S*(0) = 1 - A + |-A36(3) (6.8)

to

sJ(t) = 1 - a + A36(3) , (6.9)

i.e. it changes by no more than a factor of two (Fig. 43; after D. G. Hummer,

196 8).

522-519 0 - 74 - 23
336 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

Fig. 43. Depth dependence of the primary source function in an isothermal

atmosphere with 1 - A = 10"6. The ratio of S|[t) /Sj^C^^) is plotted

on the ordinate.

As has been shown in Sec. 7.3, for 1 - X >> XB6(3) the destruction o£

line photons by continuous absorption is insignificant compared to their destruction during scattering. From (6.7) and (6.8) it follows that in this case

S^(t) 1-A (6.10)

Thus in the primary atomic excitation, electron impacts dominate photoexcitation by continuum radiation. Since neither the birth nor the death of photons

is controlled by the continuum, the source function S(t) should be close to

the corresponding function for 6=0, studied in Sec. 63, i.e.

S(t) (1_X)1/2^(t) (6.11)

where

I'(t) = 1

T $(t) dt

and the resolvent function $ refers to 3 = 0. Although in this limiting case continuous absorption has practically no effect on the level populations, the
7.6 LINE FORMATION IN AN ISOTHERMAL ATMOSPHERE 337

difference of B from zero should be taken into account in calculating the

intensity of the emergent radiation. Substituting (6.11) into (6.6), we obtain, after simple rearrangement, the following expression for l(0,)j,x) in

terms of the H-function for 3=0:

1(0, y,x) 1 -

a(x)

1 1/2

a(x).3 L- ^'-'^

For the parts of the line where a(x) >> 3, we find

1/2

Va(x)+3 yj

(6.12)

1(0, y,x) ~ (1-X) (6.13)

As 0° the intensity of the emergent radiation, as might have been ex- pected, tends to unity, i.e. to (T) . When 3=0, the results (6.12) and

(6.13) reduce to (6.3.19) and become exact rather than asymptotic relations.

This limiting case can be called the A-solution for a semi - in fini te , isothermal medium, since the source function here depends only on X and not on

Let us now consider the opposite limit, 1 - A << 36(3). The death of

photons during scattering through collisions of the second kind can, in this

instance, be ignored in comparison with their death in flight (see Sec. 7.3).

Also, the initial "pumping" results from the underlying continuum self- radiation of the medium rather than from collisional processes, as may be seen from

(6.7) and (6.8), since

1 - A << ^36(3) 36(3) - j3K2q(t,3)

Since the role of collisional processes is small in both the birth and the

death of photons, we can set A = 1. The solution in this case will be called

the B-solution. The line source function is close to the 3-solution if

1 - A << 36 (3) . For the 3-solution the primary source function

exactly half of its value at infinity:

Sl(T)

S*(0)

s*(^ 0

at the boundary is

(6.14)

This result follows from (6.8) and (6.9), and also directly from physical

considerations: at infinite depths the continuum radiation causing the initial excitation is isotropic, and its intensity is equal to (T) , while at

the boundary the continuum radiation, with the same intensity, strikes atoms

only from the side toward the medium (i.e. within solid angle 2tt, and not 4tt). Without solving the equation for the source function, we find from (6.14)

that asymptotically as 3 0

S(0) ~ b/36(3) A = 1 << 1 (6.15)

where b is a constant depending on the profile, but in all cases lying between

unity and one-half. Actually, we have
338 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

OO

;^(t) + J

S(t) = S,(t) + s^i

J

S, (T')r(T' ,T)dT' , C6.16)

0

where r(T',T) is the resolvent of equation (6.4). Hence

oo

;^(0) + J

S(0) = S-, (0) + y St (T')$(T')dT'

0

A Considering the monotonicity of , we obtain

oo \ / 00 S^(0)[l + y

<J)(T')dT'j <S(0) < S*(«)fl + y $(T')dT'j , (6,17)

with equality attained only for 3=0. But

oo

1 + y $(T')dT' = HM = (1-X)"^/^ = (1-X+X36 (3))"^^^ (6.18)

0

Therefore, setting X = 1 in (6.17) and using (6.8) and (6.9), we find

'|/36 (3) < S(0) < /36 (3) ,

from wh-ich the above assertion follows on letting 6^0.

The expression (6.15) gives an estimate of the departure from LTE in the

surface layers for small 3. This estimate applies both for A = 1 and for

X 1, as long as 1 - X << 36(3).

BOUNDARY LAYER. When 1-X and 36(3) are of the same order, both S* and S

depend on x , X and 3- However, for an arbitrary relation between 1-X and

36(3) rather important information on the depth dependence of the source function may be obtained from a simple analysis of the basic equation for S,

without actually solving it.

From physical considerations it is obvious that in the present problem

the source function should be a monotonically increasing function of t. This

conclusion follows also from (6.16). Since the atmosphere is assumed to be

isothermal, at some distance from the boundary the conditions should approach

those of thermodynamic equilibrium. Therefore

S(~) = 1 . (6.19)

This follows also from (6.4) and (6.9). The most important feature of the
7.6 LINE FORMATION IN AN ISOTHERMAL ATMOSPHERE 339

solution is the large variation in S(t). From (6,17) we can obtain the estimates for S(0)

|(1-X)^/^ < S(0) < (l-A)^/^ (6.20)

with equality attained for 3=0.

One interesting consequence follows from (6.20). If as X -> 1 the ratio

(l-X)/g6(8) tends to a limit, then the value of

(1-X) ^/^S(0) (6.21)

will tend to some constant^ The value of this constant depends upon the value

of the limit l-X/66(3) as X ^- 1, and also, of course, upon the profile. However, in all cases the limiting value of (6.21) lies between 1/2 and 1. Therefore for sufficiently small 1-X

S(0) ~ b(C)(l-X) 1/2 (6.22)

where

K - 66(3)

and b(5) is a function depending on the form of the profile

in narrow 'limits:

(6.23)

It varies with-

^

< b(f) 1

with b (<») = 1. The expression (6.15) is a special case of (6.22) for ? = 0.

If desired, values of the function b(5) may be found empirically by analyzing

the results of a numerical solution of the equation for the line source function . From (6.19) and (6.20) it follows that as t increases from 0 to <» , the source function increases by a factor not less than (1/2) (1-X)" 1/2 and not

more than (l-X)"l/2. Thus when 1-X is small (the case of greatest interest),

the line source function changes by a large factor. Here the situation is

similar to that for the X-solution considered in detail in Sec. 6.1-6.3.

Qualitatively the picture may be described in the following way. In a homogeneous, isothermal gas there exists a boundary layer within which the source function changes by a factor of. order (1-X)~ 1/2. The smaller the

probability 1-X per scattering that the photon dies, the thicker is this_

layer. In the most important case of nearly conservative scattering (1-X<<1)

,

the structure of the boundary layer is very simple. In fact, if X tends to

unity (this means that X -> 1 and 3 0 at the same time), the equation (6.4)

reduces to the homogeneous equation

S(t)

OO (|t-t' |)S(T')dT' (6.24)
340 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

whose solution we shall consider so normalized that S(0) = 1. In other words,

Jim Ifll = SCt) , (6.25)

i.e. the line source function, normalized to unity at the boundary o£ the

medium, tends to the solution of the conservative Milne problem as A 1 (see

Sec. 6.1). From the general result obtained in Chapter VI from the study of

the A-solution, one can take as the thickness of the boundary layer, by

definition, the value t = such that

S(T^) = (1-A)-^/^ . (6.26)

Then for x much less than

whereas for t >> t,b

S(t) ~ S(0)S(t) , t << t, ,. (6.27)

S(t) ~ 1 , t >> t, . (6.28)

The expression (6.27) shows that when 1-A is small the structure of

the boundary layer is very simple: right down to a depth of the order of

T\) the degree of excitation increases for all A and 3, just as in the generalized conservative Milne problem. Saturation sets in as t approaches t]-, . For values of x in this region the behavior of the source function is not

universal — it depends on A and g. The results of a numerical solution of the

transfer equation provide a good illustration of these conclusions (Table 36

and Figs. 44 and 45; after D. G. Hummer, 196 8). Thus, the data of Table 36

show that for the Doppler profile, throughout the region in which S (x) <0 . 2S (<») the maximum error of the result (6.27) is about 1 percent.

TABLE 36

THE FUNCTION S(t)/S(0) IN AN ISOTHERMAL ATMOSPHERE (DOPPLER PROFILE, 1-A=10"^)

X /tT 0 10' 7 10' 6 10" 5 10" 4 10" 3

1 1 68 0 1 68 0 168 0 1 68 0 1 .69 0 1 70 0 10 4 83 0 4 84 0 4 . 84 0 4 85 0 4 86 0 4 78 0 10^ 1 78 + 1 1 78 + 1 1 . 79 + 1 1 78 + 1 1 68 + 1 1 28 + 1

6 33 + 1 6 33 + 1 6 . 26 + 1 5 74 + 1 3 89 + 1 1 65 + 1 10-^ 2 10 + 2 2 07 + 1 1 . 87 + 2 1 18 + 2 4 63 + 1 1 66 + 1

5 75 + 2 5 26 + 2 3 . 41 + 2 1 34 + 2 4 65 + 1 10^ 9 32 + 2 7 48 + 2 3. 73 + 2 10^ 9 97 + 2 7 75 + 2 3. 74 + 2 10« 1 00 + 3 7 76 + 2
7.6 LINE FORMATION IN AN ISOTHERMAL ATMOSPHERE 341

0

-

'^^////

CO -2

log

-3

0

i-X = io'^

-4 1 1 1 1 1 1 1 1 - -2 -1 0 1 2 3 4 5 6

log VF T

Fig. 44. Depth dependence of the source function in an isothermal atmosphere

for the Doppler profile and 1 - A = 10"^.

From Figs. 44 and 45, and also from (6.22), it follows that for 1-X<<1

even very weak continuous absorption strongly decreases the departures from

LTE . This is perhaps one of the most important results in this chapter and

has the following physical interpretation. The electrons have a Maxwellian

velocity distribution. Therefore the radiation field in a line, and along

with it the degree of atomic excitation will become closer to equilibrium as the radiation interacts more strongly with the electron gas. These interactions are of two types: (1) collisional excitation followed by line frequency

emission and de- exci tation of atoms that have been photo-excited; (2)the

emission during recombination of line frequency photons and continuous absorption, i.e. photo- ioni zation of atoms of the "contaminant" by line radiation.

A sharp drop in the departure from LTE as the continuum is "switched on" indicates that processes of the second type are very effective mechanisms for the

interaction of the electron gas with the radiation field.

LINE PROFILES. The profile of the line depends on the depth dependence of the source function and is very sensitive to the value of 6 . As 6 increases, the

line becomes narrower, and the central intensity grows rapidly (Fig. 46; after

D. G. Hummer, 1968)

An order of magnitude estimate of the central intensity can be obtained

from the following considerations. The central parts of the line are formed

in the layers nearest the surface, where the line source function is of the

order of S(0). Moreover, the intensity of the emergent radiation at each

frequency is close to the source function at the depth where the corresponding

part of the line is formed. In other words, the central intensity measured

in units of the Planck intensity should be of the order of S(0), or (l-A)^/2.

For 1-X >> B6(6) it is ' independent" of 3 (X-solution) ; for 1-X << 65(B) it
342 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

-2-1012345 67

log t/A

Fig. 45. Depth dependence of the source function in an isothermal atmosphere

for a Voigt profile and 1 - A = 10~^.

Fig. 46. Line profiles formed in an isothermal atmosphere. Doppler profile,

1 - X = 10-6.
7.6 LINE FORMATION IN AN ISOTHERMAL ATMOSPHERE 343

depends on ly on 3 C3-solution) . For the 8-solution the central intensity is

of order /36 (3) • In the case o£ a Voigt profile with 3 << a, the main te'rm of the asymptotic form (3.18) may be taken for SyiQ) , giving a central intensity of roughly (a3)-^/^. Therefore, for, let us say, a = 10-2, g = 10-^^ and

1 - X << 10-4 the residual central intensity is of the order of 0.01, i.e.

rather large. From Figs. 44 and 46 it appears that the estimates just given

agree well with available numerical data. We note that most of the curves in

Figs. 44 and 46 correspond to the 3-solution.

The effect of the parameters A and 3 on the intensity in line cores was

first discussed by V. V. Sobolev (1954; see also V. V. Sobolev, 1956), who

obtained the above estimates. Sobolev's procedure was quite different from

ours

In the present problem the intensity of the emergent radiation can be

expressed in terms of the H-function. Let us rewrite (6.6) in the form

l(0,y,x) = + ^sf^^I^") , (6.29)

a(x)+3 y \ y /

where the bar over S, as before, indicates the Laplace transform with respect

to the spatial variable. According to (6.7) and (1.18) the source term in

the equation for the source function can be written as

z

S*(t) = 1 - X - 3jJe-^/"GQ(Odz (6.30)

0

Therefore, by virtue of the superposition principle

z c S(t) = (1-X)S(t,oo) -3t/ S(T,z)GQ(c)dz , (6.31)

where S(t,z) is the solution of the auxiliary equation

00 S(t,z) = ly K^( |t-t' I ,3)S(t' ,z)dT' + e"''/^ . (6.32)

0

We recall that S(t,z) can be expressed in terms of the H-function and the

resolvent function $ as (see Sec. 5.1)

S(t,z) = H(z)(e-''/^ + e"^^"^'^/^$(T')dT'] . (6.33)

We now apply the Laplace transform to equation (6.31). Using (6.33) and (5.24)

and employing the relation
344 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

OO H(z) = 1 +• J

e '^/^$(T)dT

0

(6.34)

we find from (6.29), after minor rearrangements,

1(0, y,x) = a(x)+6

1 3 + a(x)H(z) (1-A) 1/2

2 00 AKCz)]j (6 . 35)

Here ago a-^^d W(z) are defined by (5.23) and (5.28), respectively, and

z = a(x)+3

The quantities H(z), W(z) and ago appearing in this expression are tabulated

for a great many values of A and 3 and for several values of the Voigt parameter a. Therefore the result (6.35) can easily be used for the practical

calculation of the line profiles.

The result (6.35) was obtained by V. V. Sobolev (1954). For the special

case X = 1 an expression differing from (6.35) only in its notation was found

earlier by the same author (V. V. Sobolev, 1949) by the use of invariance

principles. Subsequently the result was extended to the case of arbitrary A and 3 (V. V. Sobolev, 1954) by a generalization of the probabilistic approach

outlined in Sec. 5.2. The contents of these papers are given in detail in a monograph by V. V. Sobolev (1956).

CONCLUDING REMARKS. Although throughout this section it has been assumed that

the atmosphere is isothermal, in normal stellar atmospheres the temperature

increases with depth. The following procedure is often used to allow approximately for this circumstance. In the transfer equation (6.1),ByQ(T) is

regarded as a given specific function of depth, while the effect of temperature (and density) variations on the frequency dependence of the line absorption coefficient and on the parameters A and 3 is ignored. When the depth

dependence of Bvg(T) is particularly simple (linear, for example), the intensity of the emergent radiation can be expressed in terms of the H-function.

Such expressions are given by V. V. Sobolev (1954, 1956), I. W. Busbridge

(1953), S. Ueno (1955-1956), and others.

We conclude with one final observation. We have so far assumed that photons are destroyed during scattering only by collisions of the second kind.

However, under conditions typical of a stellar atmosphere, photo- i oni z ation

from the excited state is often found to be more important than collisions of

the second kind (B . Stromgren, 1935). Similarly, recombination can be more effective than collisional excitation in populating the upper level. In the case in which recombination and ionization from the upper level dominate

collisional processes, the transfer equation continues to have the form of

(6.1), with the probability A that a photon survives the scattering now equal

to

where t^z^Ic^Ic '^^^ number of photo- ioni zations from the upper level per

unit volume per unit time. Lines for which collisional population and deA = 2c^2c
7.7 STRUCTURE OF THE BOUNDARY LAYER 345

population o£ the upper level dominate the recomb inati on- i oni zation mechanism

are sometimes referred to as collisi onally controlled. If, on the contrary,

recombinations and ionizations from the upper level play a more important role

than collisional processes, we then speak of photo- i oni zation- controlled

lines. Other variants are also possible (say, collisional population and depopulation by photo- ioni zation , etc.). These questions are discussed in detail by R. N. Thomas (1965a, 1965b); also see R. N. Thomas and R. G. Athay

(1961) .

7.7 STRUCTURE OF THE BOUNDARY LAYER

We have seen above that in a rarified, homogeneous gas occupying a halfspace, the degree of excitation falls monotonically toward the boundary. Thus

near the boundary between the gas and a vacuum, there exists a boundary layer

from which radiation escapes.

It is appropriate to borrow the term "boundary layer" from hydrodynamics,

since there in essence, the effect of a boundary on the distribution of particles among the trans lat ional degrees of freedom is considered, while in the

present case, we are concerned with the distribution of atoms over states of

excitation. We note in passing that in the dynamics of rarified gases it is

necessary in studying boundary effects to base the analysis not on the hydrodynamic equations, but on Boltzmann's equation. Boltzmann's equation for the

Bhatnagar- Gross - Krook model is very similar in its mathematical properties to

the transfer equation discussed in this book. In particular, the so-called

slip coefficient of rarified gas dynamics is an exact counterpart of the Hopf

constant q(°°). Some of the results above may be directly applied to problems

arising in the dynamics of rarified gases. For problems in this area, con- sult, for example, M. N. Kogan (1967). Further discussion of these questions

is far beyond the scope of the present book.

In the preceding section it was found that the structure of the boundary

layer in a homogeneous, isothermal gas is universal in the sense that for

nearly conservative scattering, throughout the greater part of this layer, the

line source function is proportional to the solution S(t) of the corresponding

conservative Milne problem. Actually, the proportionality of the source function and S(t) in the surface layers holds under much more general assumptions.

Strictly speaking, it is just this universal behavior of the source function

near the boundary which makes it possible to speak of the boundary layer as a characteristic phenomenon.

Let us first of all consider the kinds of conclusions that can be drawn

from the proportionality of the source function to S(t) in the surface layers

of a homogeneous . is othermal gas. In the case of a X-solution the primary

source function S]^(t) is independent of depth (it is equal to ( 1-X) Bvq (T) ) . For the 3-solution representing the opposite extreme case, the primary source

function doubles within the boundary layer. However, in both instances the source function in the boundary layer is proportional to the solution of the

generalized Milne problem. Hence we conclude that within some limits the

details of the behavior of S]^(t) near the boundary do not affect the depth

dependence of the source function throughout the boundary layer. Of course,

here (and throughout this section) we are talking about media with nearly

conservative scattering (1-X << 1).

Now let us consider a medium with constant A and 3 and some arbitrarily^

specified primary source function sf(T). One might ask what properties of S]_ are required to insure that the source function in the surface layers will be
346 CONTINUUM CONTRIBUTIONS TO LINE FORMATION

proportional to S(t). The answer to this question may be obtained from simple

physical considerations. The solution S(t) o£ the generalized Milne problem

corresponds to the case in which the excitation in a conservative medium

originates from a source of infinite strength, lying infinitely deep in the

medium, with no sources at finite distances from the boundary. If the con- servative m.edium (A = 1) is replaced by a medium with 1-X << 1, then in the

surface layers the destruction of photons is unimportant compared to their

escape through the boundary. If, moreover, the strength of the sources located in the surface layers is small enough in comparison with the strength

of the sources at great depths, the conditions are close to those characteristic of the Milne problem. In this case, therefore, we expect the source function in the surface layers to be approximately proportional to S(t) , with a proportionality coefficient determined by the value of A and the depth dependence of the source strength.

Proceeding from these considerations, it can be verified that in the

boundary layer the source function , will be proportional to S^t), if, first,

1-X << 1; second, the function sJ(t) does not vary too greatly within the

boundary layer; and, third

T, J

S*(T)dT << / S*(T)dT .

0 0 Calculations fully support these qualitative considerations, as may be seen from the following two examples. The first example is taken from the . paper by D. G. Hummer (1968) which presents the results of the numerical

solution of equation (6.4) for S* = const. Values of S(t)/S(0) computed for

the case of a Doppler profile with 1-X = 10"^ and various values of 3 are shown in Table 37. A comparison of the data of Tables 36 and 37 shows at a glance the similarity in structure of the boundary layers represented there.

TABLE 37

THE FUNCTION S(t)/S(0) FOR A UNIFORM SOURCE DISTRIBUTION

(DOPPLER PROFILE, 1-A=10"^)

T /F 10

-7

10

-6

10

-5

1 1.68 0 1

.68 0 1

.68 0 10 4 ,, 83 0 4 ,. 83 0 4 ,. 82 0 102 1

, 78 + 1 1 ,,78 + 1 1 ,, 75 + 1 10^ 6 ,.31 + 1 6 ., 19 + 1 5 ..55 + 1 . 10^ 2 ,.06 + 2 1 ,, 82 + 2 1 ,. 10 + 2 10^ 5 .. 18 + 2 3,,23 + 2 1

,22 + 2 10^ 7,, 27 + 2 3.,48 + 2 10^ 7,.48 + 2
7.7 STRUCTURE OF THE BOUNDARY LAYER 347

As the second example we can use the problem o£ diffuse reflection from a conservative medium (1-A =3=0), discussed in Sec. 6.4. In this case S^Ct)

is proportional to exp(-T/zo), where Zq = yg/^'C^o) ^^0 cosine of the

angle of incidence of the radiation, xg its frequency). When zq is large,

the external radiation penetrates deep into the medium, i.e. photons of the

diffuse radiation field are "born" mainly in deep layers. We there fore^expect

that in the boundary layer the source function will be proportional to S(t).

That this is precisely the case, we can prove from an analysis of the line

profiles. The central parts of lines are formed in the surface layers, and

their form reflects the depth dependence of the source function near the

boundary. For the Milne problem the intensity of the emergent radiation is

expressed in terms of the H- function for conservative scattering (see Sec.

6.1):

1(0, y,x) =

As was shown in Sec. 6.4, an expression differing from this only by a constant

factor is also obtained for the central parts of the line in the problem of

diffuse reflection with zg >> 1 (see (6.4.15)). This implies J-hat. in

fact the source function in the surface layers is proportional to S(t).

An important conclusion can be drawn from this discussion: the universality of the structure of the boundary layer implies that the form of the

central parts of absorption lines and of sufficiently deep central depressions

of emission lines is also universal. This circumstance, which deserves further and more detailed quantitative study, may, it seems, be used to obtain

information about the physical conditions in the outermost layers of stellar

atmospheres from an analysis of profiles of strong lines.

In conclusion, we point out that in our discussion of the structure of

the boundary layer, the assumption that X and 3 are independent of depth is

not essential. The conclusions drawn above should remain valid in the more general case, in which X and 3 vary with depth in such a way that the inequalities 1-A << 1 and 3 << 1 are satisfied to sufficiently great depths. How- ever the requirement that the Doppler width (or another characteristic unit

of frequency) is constant does appear to be indispensable.

CHAPTER VI I I

PLANE LAYER OF FINITE THICKNESS

Infinite and semi - infini te media are apparently the only geometries for

which the solution of the transfer equation can be obtained in closed form.

As it is not possible to solve the transfer equation for a plane layer of

finite optical thickness, we must be satisfied with much more modest results.

The most we can hope for is to develop a rigorous asymptotic theory for the case of an optically thick layer. For monochromatic scattering, this is not

too difficult. However, -the difficulty increases sharply for problems of

radiative transfer in spectral lines. A rigorous asymptotic theory has not

yet been completely developed, although many results of practical interest

have been obtained.

In Sec. 8.1 and 8.2 various equations and expressions are introduced

which are valid for arbitrary values of the parameters characterizing the

optical properties of the medium. There are now three such parameters -- the

survival probability X of a photon during scattering, the ratio B of the con- tinuum absorption coefficient to the absorption coefficient at line center

and, finally, the line center optical thickness tq of the medium. Simplifications that occur for tq >> 1 are investigated in Sec. 8.3 - 8.5, i.e. an asymptotic theory is constructed. Sec. 8.6 - 8.10 contain a detailed discussion of several model problems. The majority of them have previously been

studied in connection with a semi - infini te medium. Finally, at the end of the

chapter (Sec. 8.11) an approximate solution of the basic integral equation for

the line source function is given.

THE RESOLVENT AND THE RESOLVENT FUNCTION. Let us begin with the integral equation for the line source function S(t), which for a plane layer of optical

thickness tq has the form

8.1 BASIC EQUATIONS

(1.1)

0
350 PLANE LAYER OF FINITE THICKNESS

A where S]^(t) is a given function representing the strength of the primary sources. We shall first study the general properties of (1.1), without introducing a specific form of Sf(T). We recall that the kernel function K]^(t,B) is

oo K^(t,6) = ^ j

a^(x)E^^(a(x)+B)T^dx . (1.2)

00

Its properties are studied in Sec. 7.1.

We denote by r(T,T';To) the resolvent of (1.1), i.e. the solution of the

equati on

r(T,T';TQ)=| j

K^(|T-t| ,3)r(t,T' ;TQ)dt + I

K^(|t-t' 1 ,3) . (1.3)

0

The function S(t) is expressed in terms of the resolvent by

S(t) = sJ(t) + Jr(T,T' ;TQ)S*(T')dT' . (1.4)

0

As in the case of a s emi - infini te medium, the resolvent is a symmetrical function of T and T '

r(T,T' ;Tq) = rfT- ,t;Tq) , (1.5)

which follows from the symmetry of the kernel of equation (1.3). Moreover,

r(T,T';TQ) = r(TQ-T,Tg-T' ;Tq) . (1.6)

This relation, easily proven with the help of (1.3), expresses the invariance

of the resolvent relative to the choice of direction in which the optical

depth is measured, i.e. its independence of the boundary from which x is

measured.

The resolvent r(T,T' ;Tg) can be expressed in terms of a function of a smaller number of variables $(t;tq), which will be referred to as the resolvent function. This function is a special value of r(T,T';Tg):

$(T;Tn) = r(T,0;Tj = r(0,T;Tj , (1.7)

and therefore satisfies the equation
8.1 BASIC EQUATIONS 351

t;Tq) = y J

K^(|T-t| ,6)$(t;TQ)dt + A K^(t,6) . (1.8)

In fact, proceeding as in Sec. 5.1, we find

^ /

K^(|T-t| ,6)f(t)dt = J

K^(lT-t| ,6)f' (t)dt + ^0 0

+ f(0)K^(T,6) - f(TQ)K-^(TQ-T,B) .

Using (1.6) and (1.7), we now obtain from (1.3)

f

K,(,.-.|.e,(iI.-y,)dt _3r ^ ^

\

0

+ $(t';tq) I

K^(t,6) - $(tq-t';tq) |

K^(tq-t,6) .

(1.9)

(1.10)

Comparison with (1.8) gives

|I. . _^ = Kt;To)$(t';To) - $ (xq-t ' ; t^) $ (tq-t ; x^) , (1.11)

from which it can be found that

r(T,T' ;Tq) = 0(|t-t' I ;Tq) +

+ / [$(T-t;TQ)<&(T'-t;TQ) - ^>(TQ-T+t;TQ)$(TQ-T'+t;TQ)]dt , (1.12)

0

1 1 where t-^ is the smaller of x and x'. For tq = «> the expressions (1.11) and

f (1.12) reduce, respectively, to (5.1.29) and (5.1.31).

I We can obtain yet another relation satisfied by the resolvent function

$(x;Xq). Differentiating (1.3) with respect to x^ , we find

9r(x,x';Xo) }^ 8r(t,x';xo) ,

0 0

" (1.13)

From a comparison of (1.13) and (1.8) it follows that

522-519 0 - 74 - 24
352 PLANE LAYER OF FINITE THICKNESS

9r(T,T' ;Tq)

^0

Setting t' = 0 here, we get

3$(t ;tq)

8Tq

$(Tq-t;TqJ$(Tq-t' ;Tq) . (1.14)

$(Tq;tq)$(Tq-t;tq) . (1.15)

This relation will be very useful later. The only property of the kernel

K-j^( I T—t' I ,B) used in deriving (1.12) and (1.15) is its dependence on | t—t '

| . Consequently, the results (1.12) and (1.15) are valid for a wide class of

integral equations with symmetrical displacement kernels. These equations were originally found by V. V. Sobolev (1958a, 1958b).

AUXILIARY EQUATION. X AND Y FUNCTIONS. As before, we begin with the representation of the function K-,(t,3) as a superposition of exponentials (see

Sec 7.1):

z c K,(T,B) . / e-^/^G(c)% , . i , c E . (1.16)

0

Let us introduce the auxiliary function S(t,z;Tq) satisfying the equation

^0

S(t,z;Tq) = A J K^(|t-t' I ,3)S(T' ,z;TQ)df + e'^^^. (1.17)

0

According to (1.16) the free term in (1.8) is a superposition of the free

terms of equation (1.17). By virtue of the linearity of these equations, we conclude that

z c $(t;Tq) = A J

s(t,z;Tq)G(0^ . (1.18)

0

From (1.17), with the help of (1.9), we can find that S(t,z;Tq) also satisfies

the equation

(1.19)

8S(t,z;t„) , ^—^ = _£s(t,z;tq) + X(z,Tq)$(t ;Tq) -

- Y(z ;Tq) $(Tq-t ;Tq) ,

where

X(z;Tq) = S(0,z;Tq) , (1.20)

Y(z;Tq) = S(Tq,z;Tq) . (1.21)
8.1 BASIC EQUATIONS 353

Thus

S(t,z;tq) = X(z;TQ)^e + J

e" ^^^"^ ' ^ $ (x ' ; ) dx

T

0

0 (1.22)

Substituting (1.22) into (1.18), we arrive at the following equation for

$(t ;tq) :

L $(t;tq) = N(t;tq) + f

[$(x 'Tq)N(t-t ' ;Tq) -

0

- $(Tq-t' ;Tq)M(t-t' ;Tq)] dx ' ,

(1.23)

where

z c

^^^'-"O^ = J f

e-^/"'x(x';xQ)G(c')^ , (1.24)

2 0

z c ^^"'^0^ = 7 /

e-"/"'Y(z';xQ)G(c')^ , (1.25)

0

with C = z'/(l-6z').

RESOLVENT FUNCTION EXPRESSED IN TERMS OF THE X- AND Y- FUNCTIONS. As has been

shown in Sec. 5.3, in the case of a semi - infini te medium, the resolvent function $(x) can be expressed in terms of '^'^^(x) and the H- function. In a similar way, $(x;xo) can be expressed in terms of 'l><„(x) and X- and Y-functions,

00 /

^(x;xq) = $^(x) - I $^(x+x')N(x' ;Xg)dT' -

- f

$^(Xq-x+t')M(t' ;xQ)dT'

(1.26)

0

Actually, the equation for r (x,x') ,

00 r^(x,x') = I y* K^(|x-x"| ,3)r^(x",x')dx" + I

K^(|x-x' I ,B) ,
354 PLANE LAYER OF FINITE THICKNESS

can be ivritten in the following form for 0 < t,t' <

"0

(t,t') = J K^(|t-T"| ,3)r^(T",T')dT" +

0

oo

+ |k^(|t-t' I ,6) +y J K^(T+T",B)r^(-T",T')dT" + (1.27)

0

00

+ 7 f

K^(T"-T,3)r^(T",T')dT" .

^0

Comparing (1.27) with (1.3) and (1.17) and recalling (1.16), we get

r^(T,T') = r(T,T' ;Tq) +

z

^ J

S(t,z;tq)G(c)^ J

+ e-^"/"r^(-T",T')dT" + (1.28)

2 0

z C - ^1 / S(to-t.z;To)G(0^ J

e-^"/^r^(TQ.T",T')dT"

0

Let us now set t = 0 and then replace t' by t. Taking into account the notation introduced in (1.20) - (1.21) and (1.24) - (1.25), we obtain

oc /

(T) = $(t;Tq) + J

r^(-T' ,t)N(t' ;TQ)dT'

0 (1.29)

/

+ f r^(T„+T' ,t)M(t' ;T.)dT' . 0 y-J'^y- ' ^0-

0

Equation (1.26) follow.s at once from this, if we recall that r^(Ti, xi) = (t2—Ti) (see Sec. 3.6). It is interesting that (1.26) may also be obtained

by simple probabilistic considerations (V. V. Ivanov, 1964a).

When X = 1, the function $0,(1) may not exist; this occurs, for example,

in the case of the Doppler profile (see Sec. 4.7). However, $(t;to) can be

expressed in terms of the resolvent r(T,T') for a semi-infinite (and not

infinite) medium and the X- and Y- functions;
8.1 BASIC EQUATIONS 355

00 -/ r(Tn+T' ,t)M(t' ;Tn)dT' , (1.30)

00 T;Tq) = r(TQ,TQ-T) - J

r(TQ+T' ,Tq-t)N(t' ;TQ)dT (1.31)

These expressions are derived similarly to (1.26), but are free o£ diverseness for X = 1. The relations given here show that to obtain an expression in closed form

for the Green's function for a layer of finite optical thickness, it is sufficient to obtain explicit expressions for the functions X(z;to) and Y(z;to).

Although it has not been possible to do this, it is clearly possible to develop many important properties of these functions.

ALTERNATIVE REPRESENTATION OF THE SOLUTION. Along, with the usual expression

for the solution in terms of the resolvent (Eq. (1.4)), an alternative representation, which involves only the resolvent function and not the resolvent

itself, seems to be useful. The solution of (1.1) depends on Tq as a parameter; we shall here write it as an extra argument. Differentiating (1.1)

with respect to and using (1.7), we obtain (c£. (1.15))

9S(t;Tq)

" ^'^^O'^O^^'^^O'^'^O^ ' (1.32)

from which

S(t ;t^) = S(t;t)

u /

S(t' ;T')$(T'-T;T')dT' (1.33)

Now, letting t„ = t in (1.4), we find

T S(t,t) = S*(t) + J

S*(t' )$(t-t' ;T)dT' . (1.34)

0

Equations (1.33) and (1.34) are the desired representation of the source function in terms of the resolvent function.

THE TRANSFER EQUATION, EMERGENT INTENSITIES, AND FLUX RELATIONS. It is more

common to formulate problems of multiple light scattering in terms of the

integro-differential transfer equation for the radiation intensity rather than

in terms of the integral equation for the source function. Having this in

mind, we conclude this section with a brief summary of the basic equations

for the radiation intensity and related quantities.

The radiative transfer equation for the intensity of diffuse line radiation in a plane geometry is
356 PLANE LAYER OF FINITE THICKNESS

^ dI(T y,x) ^ (a(x)+B)lCT,y,x)

dT

1 I

Aa(x) Ja(x')dx' J

I ( t , y ' , x ' ) dy ' -a(x)S*(T) , (1.35)

-1

and the boundary conditions are

1(0, y,x) = 0 , y < 0

I (Tq ,y ,x) = 0 , y > 0

The substitution

whe re

a(x) +6

reduces (1.35) and (1.36) to (for details, see Sec. 7.5)

z

z dI(T,2) _ dT — z c

I (0 ,z) = 0 , z < 0

I(Tq,z) = 0 , z > 0

where

(1.36)

I(T,z) = ^Ml§.i(T,y,x) , (1.37)

a(x)

(1.38)

I(T,z)-| J

I(T,z')G(c')dz' - S*(T) (1.39)

(1.40)

z = 1 , ^' = JllA— . (1.41)

c 6 a(x)+6

The line source function S(t), which is the solution of the basic integral equation (1.1), is expressed in terms of I(t,z) as follows:

S(T) = A y l(T,z')G(c')dz' + sJ(t) . (1.42)

I — z c
8. 1 BASIC EQUATIONS 357

The intensities of the emergent diffuse radiation are

^0

1(0, y,x) = f SCT)e~^/^a(x)^ , U

> 0 (1.43)

0

^0r -(Tq-t)/|z| , I(T.,y,x) = / S(T)e ct(x)^ , y < 0 , (1.44)

0

'

'

where z is given by (1.38).

Further, let ttF(t) be the total flux of line radiation along the normal

(it is positive when the energy flows in the direction of decreasing t) :

^F(t) = 2tt / dv' f I (t ,y' ,x' )y' dy' . (1>45)

6^ -A

Using (1.37), one can show that

z

uF(t) = Av^ J

I(t,z')[G(C')+6Gq(c')] z'dz' , (1.46) — z c where the function Gq is defined by (2.6.12) - (2.6.13). Substituting into

(1.46) the formal solution of the transfer equation (1.39), we express the

total flux in terms of the source function:

uF(t) = Av^ y [K2(|t-t' I ,3)+BK2q(|t-t' I ,3)]sgn(T'-T)S(T')dT' , (1.47)

0

where the functions K- and K-„ are defined by (7.1.3) and (7.1.4) respectively. ^

The total density of the line radiation at depth t is

P(t) = 27ri J

dv' / I(T,y' ,x')dy' , (1.4-8)

0 -1

from which

z

Ac J

p(t) = Av^ / I(T,z')G„(c')dz' , (1.49)

0 — z

I
358 PLANE LAYER OF FINITE THICKNESS

or

p(x) = Av— f K,, (It-t- I ,3)SCT')dT'

Ac /

(1.50)

For 3 = 0 and = °° the expressions given in this subsection reduce to those

found in Se c . 5.1.

8.2 X- AND Y-FUNCTIONS

PRELIMINARIES. The functions X and Y play the same role for a layer of finite optical thickness as the H-function does for a semi-infinite medium.

As TQ °° the function X(z;Tg) becomes H(z) and Y(z;tq) tends to zero:

X(z,oo) = H(z) ; Y(z,c») = 0 . (2.1)

This result follows from (1.20) and (1.21).

The X- and Y- functions were first introduced by V. A. Ambartsumian (1943;

see also V. A. Ambartsumian, 1960) in a study of isotropic monochromatic

scattering. The corresponding functions for problems with anisotropic mono- chromatic scattering were studied by S. Chandrasekhar (1950), I. W. Busbridge

(1960) and others. The X- and Y- functions for line transfer problems were introduced and studied by V. V. Ivanov (1963, 1964b), D. I. Nagirner (1967),

M. A. Heaslet and R. F. Warming (1968b), F. Fuller and B. Hyett (1968). Most

of the relations given in this section are a simple transformation to the case of line- frequency scattering of results that had been obtained earlier

in the study of monochromatic scattering (see S. Chandrasekhar, 1950; I. W.

Busbridge, 1960; V. V. Sobolev, 1956, 1957b). It would be wrong, however, to

think that the specific properties of line- frequency radiative transfer problems, which arise from the possibility that a photon changes frequency during

scattering, have little effect on the properties of X- and Y-functions. In

studying the asymptotic behavior of X(z;tq) and Y(z;Tq) for large Tq these

properties are found to be dominant. However, a large number of relations do

exist that are valid under very general assumptions about the nature of the

interaction between radiation and matter. This section is primarily devoted

to a study of such general relations.

EQUATIONS SATISFIED BY THE X- AND Y-FUNCTIONS. According to (1.4) and (1.17)

/ ^0 ,

, — T/Z — t'/z

S(t,z;Tq) = e + J

r(T.T';Tg)e di ' . (2.2)

0

We now successively set t = 0 and x = tq. By virtue of (1.6) and (1.7)

r(To,T';To) = $(Tg—t';to). Keeping this in mind, and also considering (1.20)

and (1.21), we arrive at the important relations

X(z;Tq) = 1 /

•t/z $(t ; Tq) dx (2.3)
8.2 X- AND Y- FUNCTIONS 359

Y(z;Tq) = e ^0/2 + J

Q (^0 t) /

z ^ . _ ^2.4)

0

From these relations we can develop many vital properties of the X- and Yfunctions and obtain equations from which they may be determined.

The quantity z in (1.17), and consequently in the relations (2.3) and

(2.4) as well, can have any (generally, complex) values. Substituting -z for

z in (2.3), multiplying both sides by exp(— Tq/z)., and comparing the result

with (2.4), we find that the X- and Y- functions are related by

Y(z) = e '^0/^X(-z) . (2.5)

We shall now derive equations for X(z) and Y(z). First we obtain the

following nonlinear equations, the generalized Ambartsumian- Chandrasekhar

equations

X(z) = 1 . Az r

X(z)X(z;)-Y(z)Y(z')

^

2 J z ' +z

Y(z)

(2.6)

where

z = I z' = z'/(l-3z') . c p

Multiplying (1.22) first by exp_(-T/z'), then by exp(-(TQ-T) /z ' ) , integrating

over T from 0 to tq and taking into account (2.3) and (2.4), we obtain, after

minor transformations,

/ S(T,z;x„)e-^/''dx . X[z]X(z')-Y(z)Ytz') _

^ (2.7)

?

J

S(T,z;T,)e-^^0-^^/^'dT = X(z)Y(zJ)--X(z')Y(z) ^

0

!j Multiplying these equations by ( A/2) G ( c

' ) /z ' , integrating over z' from 0 to

f z^, and using (1.18) and (2.3) - (2.4), we arrive at (2.6).

I Other equations for the X- and Y- functions can be obtained from (2.3)

i and (2.4). Differentiating these expressions with respect to Tq and keeping

j (1.15) in mind, we find
PLANE LAYER OF FINITE THICKNESb

+ / e ^/'$(TQ-T;TQ)dT)$(Tg;To) , 0

3Y(z;t )

(•/ -(Tq-t)/z

-T ; Tq) dxj $(Tq ^0^

or, finally,

3X(z;Tq)

9t 0 Y(z ;tq) $(Tq ;Tq) ,

8Y(z;t.) , = Y(z;Tq) + X(z;Tq)$(tq;Tq) , 3t 0 where, according to (1.18) and (1,21),

z

^^^O'^O^ = I / nz';To)G(c')^

(2.8)

(2.9)

From (2.3) and (2.4) the boundary conditions for the equations (2.8) are seen to be

X(z ;0) = Y(z ;0) = 1 . (2 . 10)

From (2.8) it is evident that the X- and Y-functions can be found if the

function $(to;to) is known. This fact is essential for the investigation in

Sec. 8.5 of the asymptotic behavior of X(z;tq) and Y(z;tq) for large tq.

Equations (2.8) - (2.9) with the boundary conditions (2.10) are equivalent to the following coupled nonlinear integral equations for X and Y:

X(z;Tq) = 1 + Ay Y(z;T)dT JY ( z

' ; t) G ( C ' ) ^ ,

(2.11)

Y(z;Tq) = e " + A

(Tq-t)/z

X(z;T)dT y Y(z' ;t)G(C')^

Using the method described by V. V. Sobolev (1957b) we can obtain yet another

system of nonlinear integral equations for the X- and Y-functions:
8.2 X- AND Y- FUNCTIONS 361

X(z ;Tq) = 1 + a y X(z;T)dT f

e V' ^7 XCz';t

z

Y(z;Tq) = e + J

Z Y(z;T)dT / e

°

(2.12)

)/z'

X(z' ;t)G(c')^

z

The X- and Y- functions also satisfy coupled linear integral equations

which are obtained from the equation

[^l-XU(z,6)j S(t,z;Tq) = e~^/^ +

(2.;

^7/ ; 0_ G(^')dz' - e ^0/" A / LG(^')dz'

2 J z'-z 2y z +z

by setting x = 0 and t = x

0

z r -1 . r'' X(z' ;x ) l-AU(z,B) X(z;Xq) = 1 ^ ^ z / ___^G(c')dz' -

0

2 7 z' +

Y(z' ;x.) —G(c')dz' , z

z c [i-XU(z,6)]y(z;tq) = e ^Q/^ + Az J

c ( C ) d z

' 2 0

z c -Tn/z > /* ' ; x^)

-e 0' Az / ^G(c')dz' .

(2.14)

2 0

Here U(z,6) is defined by (7.1.38) or (7.1.40). The equation (2.13) is an extension of the equations obtained in problems of monochromatic scattering

by E. G. Yanovitskii (1964) and T. W. Mullikin (1964).

INTEGRAL RELATIONS. We shall now obtain some integral relations satisfied by

X(z) and Y(z) . We define
362 PLANE LAYER OF FINITE THICKNESS

z c \(^q) = f X(z;TQ)z^GCOdz , n = 0,1,... .

0 (2.15)

z c 3j^(tq) = J

Y(z;Tg)z^G(Odz , n = 0,1, ... .

0 The quantities and Q-^ are the n-th order weighted moments of the functions

X(z) and Y(z). We also define

z c

= f

z'^GCOdz . (2.16)

According to (7.1.7)

00

^n = ^

/Ki(T,6)T-dx , (2.17)

0 i.e. nign is the n-th moment of the kernel function of the basic integral

equation. It must be stressed that the integrals gj^ may diverge when n ex- ceeds a certain value. In this case the corresponding and 3^ also diverge

The most important example of such divergence is that of line scattering with

out continuous absorption (3=0). If a line has infinite wings, then only

the zeroth moments exist.

From (2.8) we find, taking (2.15) and (2.9) into consideration,

da^dg)

dT,

dgo^^o^

dTo

'0^ ^0

= [-f*-o^-o^] + an(Tn)h(TQ;TQ) ,

(2.18)

from which

daQ = 3Qd3o . (2.19)

Setting n = 0 and tq = 0 in (2.15) and keeping (2.10) and (7.1.19) in mind, we obtain

aQ(0) = 3n(0) 0 G(c)dz = - . X (2.20)

Therefore (2.19) gives

(2.21)
8.2 X- AND Y- FUNCTIONS ^6

In the conservative case, i.e. for 6 = 1 — A = 0, this expression becomes

+ 3q = 2 , X = 1 . (2.22)

Equation (2.21) is a special case of the general relation

^2n = g2n ^ f . 1/-^) ' ^"Zn-j "j-^2n-j ^

j

^ ' ^ = 0,1,2,... , (2.23)

which can be obtained from the first of the equations (2.6). We shall not

pause to give its proof, which is essentially the same as for monochromatic

scattering (see I. W. Busbridge, 1960). This relation is valid, of course,

only when all of the moments appearing in it exist. For t^^ = <» it reduces

to (7.5.21). We note, in addition, that for 0 < X < 1

z C 2" y [1 - (l-X)^/2H(z)]zG(Odz = I J

6^(T)dT . (2.24)

0 0

This result holds even when gi diverges. If g^ is finite (specifically, for

monochromatic scattering)

,

00 / [1 - ^Q{T)dT = g^ . (2.25)

0

The proofs of these relations are omitted.

For X = 1 the following relations are also valid:

y [X(z;TQ)-Y(z;TQ)]zG(z)dz = Tq6q(Tq) , (2.26)

0 f

{2-[X(z;TQ)+Y(z;TQ)]3o(TQ)}zG(z)dz = f

3Q(T)dT , (2.27)

0 0

00 y {(TQ + 2z)BQ(TQ)[X(z;TQ)+Y(z;TQ)]-4z}zG(z)dz = 0 . (2.28)

We stress that the integral in (2.26) cannot, generally speaking, be represented as the difference of and Bi, since these moments separately may

diverge. This is true of the integrals appearing in (2.27) and (2.28) as well.
364 PLANE LAYER OF FINITE THICKNESS

Turning to the proof o£ (2.26), we introduce the notation

00 ZCtq) = f

[X(z;TQ)-Y(z;TQ)]zG(z)dz . (2.29)

0

Differentiating this expression and keeping (2.8) in mind, we find

dZ(T.)

^f- - 3o(To) - nTo;TQ)Z(To) . (2.30)

But it follows from (2.18) and (2.22) that for X = 1

^Tq;! ) =

' A

>

' (2.31)

0 0 6o(To) diQ

so that (2.30) can be written in the form

3o(TQ)dZ(To)-Z(TQ)d3o(TQ)

r = dx^ . (2.32)

^0^-0^

From (2.29) and (2.10) we have Z(0) = 0. Therefore (2.32) gives Z(tq) = igBoC'i^O^' Q.E.D. The relations (2.27) and (2.28) are proven in a similar way.

In situations in which the first and second moments of the X- and Y- functions exist for X = 1 (for example, in monochromatic scattering), the integrals in (2.26) - (2.28) can be expressed as sums of integrals. The expression (2.28) is then found to be a special case of (2.23) in which n = 1 and

X = 1.

The generalization of (2.26) and (2.27) to the case of arbitrary X, 0 <

X < 1, has the form

z c 0 AX(°o;tq) J

[X(z;Tq)-Y(z;Tq)] zG(Odz = - (1-X) j x2(»;T)dT , (2.26')

X

0 0

"^0 dx

z Xn

c 0

(2.27')

J

[2X(=o;xQ)-X(z;xQ)-Y(z;xQ)]zG(Odz = X(°°;Xq) J ^ 0

Q X («>;x)

- (1-X)XqX(oo;Tq) .

The proofs of these results follow the same line as the proof of (2.26).

SERIES EXPANSIONS OF X AND Y. For z 7^ 0 series expansions in inverse powers

of z for X(z) and Y(z) follow from (2.3) and (2.4):

(-1)' Z ' , (2.33)

i = 0 ^'
8.2 X- AND Y- FUNCTIONS 365

i = 0

(2.34)

Here

(2.35)

and for i > 1 ^0

(2.36)

(2 . 3.7)

so that

j = 0 where are binomial coefficients. In particular,

(2.38)

X^do) ^ Yi^q) = TqX(co;Tq) (2.39)

Substituting (2.33) and (2.34) into (2.8) and equating the coefficients

of zO and in both sides of the resulting expansions we find

dX(oo;TQ)

dif. X(«>;To)Kto;to) (2.40)

and

dX-^(TQ)

dx, yi(To)*(To;TQ)

= X(oo;Tq) + X;^(Tq)$(To;Tq) ,

(2.41)

from which we obtain

^0

. 0 0

(2.42)

The expression (2.42) will be useful later.
366 PLANE LAYER OF FINITE THICKNESS

X(°°) AND ITS ESTIMATES. As z increases from 0 to «> the functions X(z) and

Y(z) increase monotonically from 1 and 0 respectively to

0 X(co;Tq) = Y(~;Tq) = 1 + y $(T;TQ)dT (2.43)

This follows from (2.3) and (2.4).

The value of X(<») is encountered in the solution of many model problems

For its physical significance, see Sec. 8.9. Going to the limit z •> » in

any of the equations (2.6), we find

x(<»;tq) ao(TQ)-BQ(TQ)r

or, if (2.21) is used.

X(~;Tq) 2^0

-1

Specifically, in the conservative case

{2.44)

(2.45)

X(-;tJ = , X = 1 0 PoC-^o)

Let us show, following V. V. Sobolev (1967b) , that

^0 0f K2(T,6)T(TQ-T)dT| < X^(oo;Tq) < (l-X + A K2 ( T

g

, 3

)

)

3 /

-1

(2.46)

(2.47)

The upper bound is obtained in the following manner. Multiplying (2.4) by

G(^) and integrating over z from 0 to z^, we find

6o(tq) = K2(tq,3) + f

K2(TQ-T,3)<I'(T;TQ)dT (2.48)

Since K2(t,3) decreases monotonically as x increases, we have

SqCTq) > K2(Tq,3)[1 + f

$(T;Tg)dT^ ,

or
8.2 X- AND Y- FUNCTIONS 36 7

Qq(^o) > K2(To,B)X(o°;Tq) . (2.49)

Combining this inequality with the relation

(1-X)x2(=o;tq) + AX(oo;Tq)3^(Tq) = 1 , (2.50)

which is a consequence of (2.34) and (2.21), we get an upper bound on X(o°;tq),

i.e. the right inequality of (2.47).

The derivation of the lower bound is a little more complicated. For

z = equation (1.17) takes the form

S(t.»;Tq) = A y k^(1t-t' I ,B)S(T' ,-;TQ)dT' + 1 . (2.51)

0

Replacing t by Tq—t here, we find

S(t,'»;Tq) = S(Tq-t ,«>;Tq) . (2.52)

Multiplying (2.51) by t(tq—t), integrating over t from 0 to tq , and using

(2.52), we obtain

^0

y S(t ,~;Tq) T(TQ-T)dT =

0

^0 T ^3

= X f

S(T,«';TQ)dT f

K^(t' ,3)(T-T')(TQ-T + T')dT' + .

0 0

Thus

^0

(1-X) y S(T,«>;TQ)T(TQ-T)dT +

0

^0 T

+ 2X y S(T,«';TQ)dT y K2(t' ,6)T'dT' = (2.53)

0 0 ^0 T ,3

0 0

= X y S(t,«>;Tq) (2T-TQ)dT y K2(T',3)dT' +-

0

Further, (1.22) gives

522-519 O - 74 - 25
PLANE LAYER OF FINITE THICKNESS

S(t,oo;Tq) = X(=o;TQ)fl + f

$(T';TQ)dT' - f $ ( T q-t ' ; Tq ) dx ' ]

, (2.54)

from which

s(t ,~;tq) < s(tq/2 ,<=°;tq) =

Tn/2

(l + J$(t' ;TQ)dT' - J

\ 0 0 )

(Tq-t' ;TQ)dT'J < (2.55)

0

< X(oo;Tq)(i + f $(T' ;TQ)dT' j = x2(<»;Tq)

The integral on the right side of (2.53) is positive. Discarding it and

majorizing the function S(T,°°;Tf^) on the left by x2(oo;t ), we arrive at the

left inequality of (2.47). ^

From (2.47) we have, for X = 1,

71 / K2(T)T(To-T)dT

LOO

<X(°°;tq) < (2.56)

From the asymptotic properties of the function ^2(1) proven in Sec. 2.6, it can be shown that for y < 1

r ^0^2 ^^0^

/ K.(T)T(T.-T)dT — , ^ - , (2.57) J 2^ ^ ^ 0 ^ (2-2y)(3-2y) ' 0

'

where y is the characteristic exponent (see Sec. 2.6). Considering the mono- tonicity of X(oo;to), we may conclude from (2.56) and (2.57) that in the con- servative case

^^"''"o^ ~ T—-V77 . ^ = 1 > ^0 " ~ ' [k2(t)J1/2

where c^ is a certain (at present unknown) constant. In Sec. 8.5 it will be

shown that

Specifically, c^^ -> 1 as y 0, ci = 0.914 for y = 1/^ (Voigt and Lorentz profiles) and ci = 0.707 for y = 1/2 (Doppler profile).
8.2 X- AND Y-FUNCTIONS 369

The asymptotic expression (2.58) can be rewritten in the form

^^°°'"0) ~ ^^i^^^/'o^y^^'^O^ , X = 1 , - , (2.60)

if use is made of (2.6.68). This representation is more general than (2.58)

and is valid for arbitrary values of the characteristic exponent y, including

Y

= 1. Specifically, for the most important particular profiles (2.60) as- sumes the form

Milne: Xj^(°°;Tq) ~ y Tq , (2.60a)

Doppler: Xj^(~;Tq) ~ u'^ ^ ^ t

^'^ ^ ^ (Inx q)'^^ ^

, (2.60b)

Lorentz; (»; t„) ~ ^5^101 t„ (2.60d)

Higher accuracy asymptotic expressions for X(<»;to) are available for

the Milne rectangular profile (see Sec. 8.4) and in the Doppler case. It may

be shown (V. V. Ivanov, 1970b) that in the latter case

^D^-'^O^ = .^/^o^/^£nT,)l/4j^l.^ . OUZnr^y') (2.61)

where

X-, = i(l + 2Y*-4£n2) = -0.07727 . (2.62)

Error estimates in (2.60c) and (2.60d) remain unknown.

The accuracy of the asymptotic forms (2 . 60b ) , (2 . 6 1) and (2.60d)

may be evaluated from the data presented in Table 38, where the ratio

X(°°;TQ)/X^g (°°;to) is tabulated as a function of tq for the Doppler and Lorentz

profiles. Here X(<»;to) is the numerically exact value of X(oo) which was calculated from the numerical data of A. L. Crosbie and R. Viskanta (1970a) , and

X^5(o°;to) is its asymptotic value as given by equations (2.60b) (column labelled Doppler I), (2.61) (Doppler II) and (2.60d) (Lorentz).

If only an estimate of X(°°) is needed, one may use the approximation

X^(-;To) = [l-X.XK2(To)]"'^' E TJtq) , ^2-^3)

which is valid for arbitrary X, 0<A<1, 6=0 and y < 1. As a rule, the

accuracy of this approximation increases as y decreases.

There have been many tabulations of the X- and Y-functions for monochromatic scattering. The most extensive tables are those of Y. Sobouti (1963)
370 PLANE LAYER OF FINITE THICKNESS

TABLE 38

r 00 • T 1 "FOR THF nOPPLER AND LORENTZ PROFILES

U U U U 1 C L

T O T) "f" 7

^0 I I I

10 0 . 960 0.992 0 .983

20 0 .940 0 . 964 0.990

50 0 . 939 0.958 0 .996

100 0 .946 0.962 0 . 999

and J. L. Carlstedt and T. W. Mullikin (1966). Tables o£ X and Y for the

Doppler profile were published by F. B. Fuller and B. J. Hyett (1968). These

tables must be used with caution, since not all of the entries are correct.

The authors seriously overestimate the accuracy of the tables, especially for

large values of tq . A. L. Crosbie and R. Viskanta (1970b) give graphs of the

conservative X and Y functions for several forms of the line absorption coefficient.

One may hope that extensive tables of X and Y for the most important

absorption profiles, with a wide range of the parameters Tq , A and 6, will be- come available in the near future.

8.3 ROLE OF DISSIPATIVE PROCESSES

DISSIPATION MEASURE OF THE MEDIUM. Because of the great number of parameters

that can appear in the solution of the transfer equation for a plane layer,

all of the limiting cases in which some simplification occurs are very welcome. In this section we shall study the nature of the simplifications that occur in the most interesting case, that of nearly conservative scattering

(1—X<<1) in an optically thick layer (tq>>1).

There are three mechanisms for eliminating photons from the random walk

process in the medium -- destruction in the scattering event, absorption in

flight, and escape. The first two processes convert the energy of the radiation field into other forms, i.e. the radiant energy is dissipated. The

destruction of photons in scattering and their absorption in flight are therefore conveniently lumped together under the general name of dissipative p roc- esses . The escape of radiation from the medium is obviously not a dissipative

process, since the energy remains in the form of radiation.

The importance of dissipative processes per scattering is characterized

by the value of 1 — X. However, it does not follow that if 1 — X is small,

the energy dissipated in the medium as a whole will also be small. The total

energy dissipated is determined not only by the value of 1 — A, but also by

the optical thickness of the medium tq , and increases with tq. It seems appropriate to introduce a quantitative measure of the importance of dissipative

processes in a medium. We call this quantity the dissipation measure of the
8.3 ROLE OF DISSIPATIVE PROCESSES 371

medium and define it as

oj = —^ ~

/\ \

, (3.1)

where tq is the optical thickness o£ the layer, and V(u) is the cosine transform of the kernel function for 6=0 (it was studied in detail in Sec. 2.6

and 2.7). Media with 03 << 1 will be called weakly dissipative. The limiting

case of a weakly dissipative medium is the conservative medium for which tjj = 0.

If (jo >> 1, we shall say that the medium is strongly dissipative.

In order to understand why the quantity oj characterizes the dissipative

properties of the medium as a whole, it is easiest to refer to the special

cases which are of practical interest.

For monochromatic scattering

V^(u) = il£_t£ii = 1 _ h! ^ .... , ^ u 3

'

and (3.1) gives for g = 0

OJ ~ 3(l-X)Tn ~ (kT.)2 , (3.1a)

where k is the diffusion length (see Sec. 3.2). The relation (3.1a) agrees

completely with what would be expected from the results of Chapter III: the

role of energy dissipation is determined by the value of the characteristic

parameter kxg . For Doppler, Voigt, and Lorentz profiles, we use the asymptotic forms of V(u) found in Sec. 2.7 and obtain from (3.1)

Doppler: ~ —^ Tq ( £nTQ ) ^ ( 1-X) , (3.1b)

771/ ^

1 Voigt: 03 ~ {^) (1-X) , T» ± , (3.1c)

(27rU(a,0))l/2 \ a/ U a

Lorentz: co ~ — (1-X) . (3. Id)

^2

^

Let us consider a little more closely what these expressions give when

one dissipative process dominates the other. First, we will assume that in

the destruction of photons the role of continuous absorption is insignificant

compared to that of scattering, i.e. that 1 — X >> 66(3) (see Sec. 7.3). In

this case 1 — X can be replaced by 1 — X. From a comparison of (2.6.40) and

(2.6.48) we find that (for y < 1)

2r (2y) s iniTY

K2(To)

Substituting this expression into (3.1) we find that for 1 — X >> 66(6)
372 PLANE LAYER OF FINITE THICKNESS

03 ~ ir(2Y)sin^Y-^ . (3.2)

Thus, i£ Y is not too close to unity, the dissipation measure of the medium,

to within a factor of order unity, is in this case equal to ( 1—X) /K2 (tq )

From the preceding chapters it is clear that this ratio is the characteristic

quantity that determines the importance of the destruction of photons in

scattering relative to their escape from the medium, i.e. the role of the

dissipative processes.

In the opposite limiting case 1 — X << 36(3), continuous absorption dominates scattering in destroying photons. At what value of tq should the effect

of dissipative processes begin to be felt in this case? The answer is obvious: when the disp lacement of the photon through a distance of order tq

corresponds to a continuum optical path length of order unity measured along

the trajectory of the photon. For y < iT^ (in particular, for Voigt and

Lorentz profiles) the zigzag nature of the trajectories can be ignored (see

Sec. 7.3). Therefore for displacement through a distance tq , the photon

covers an optical path in the continuum of order Btq. Consequently, for

Voigt and Lorentz profiles the dissipative processes become significant when

Tq becomes of order S"-'-. With the Doppler profile the situation is different,

since the zigzag nature of the trajectories must be taken into account. From

the discussion in Sec. 7.3 it follows that continuous absorption becomes

significant when is of the order of ( 3Jin ( 1/ 3) ) •

Let us compare the results of these physical considerations^with the

implications of the expressions (3.1b) - (3. Id). Replacing 1— X=l— X+

X36(6) in (3.1b) and (3. Id) by 36(3) and taking the leading terms of the as- ymptotic expansions of 6(3) from Sec. 7.3, we find for the case under consideration

„ 1/2 / xl/2

Doppler: to ~ ^T^ilni^) ^ ( I ' (3.3a)

3 1 /2 Voigt and Lorentz: w b [^'^nJ • (3.3b)

/2

"

Thus it follows that the value of co increases with tq, and becomes of

order of magnitude unity when tq = (^lnl/B)~^ for the Doppler profile and

Tq = 3~1 for the Voigt and Lorentz profiles. This behavior of o) is just that

expected of a quantity that characterizes the importance of dissipative processes in a medium.

In summary, it may be said that the value of the parameter co , defined by

(3.1), should in all instances be a good measure of the role of dissipative

processes. For oj << 1 dissipation is small so that almost all of the energy

generated in the line escapes. For oj >> 1, on the contrary, only a small

fraction of the energy expended in the excitation of the upper level is

carried out of the medium by line radiation.

ANALYSIS OF THE LIMITING CASES. Now we can turn directly to a discussion of

the simplifications that arise in the soj-ution of the transfer equation in

various limiting cases. The greatest simplification occurs when the dissipation measure of the medium is small (co << 1). In this case as a first

approximation we may assume that no energy at all is dissipated, i.e. the

medium can be regarded as conservative (to = 0). Correspondingly, one can set

X = 1 and 3 = 0 in the integral term of the equation for the line source function S(t)
8.3 ROLE OF DISSIPATIVE PROCESSES 373

"^0

S(t) = ^ JK-^(|t-t' I ,3)S(T')dT' + S*(t) , (3.4)

0

and instead o£ solving (3.4), one can solve

S(t) =1 f

K^(|t-t' |)S(T')dT' + S*(t) . (3.5)

0

A It must be stressed that the source term Si(t) in (3.5) is the same as in

(3.4). Usually sJ(t) depends on the parameters \ and B, and this dependence

must be retained. Thus, in the limiting case o£ a weakly dissipative medium,

the source function S(t) depends on \ and 3 entirely through the source term

S-[^(t), which, o£ course, is a great simplification.

Now let us turn to media with significant dissipation. Here we also can distinguish cases in which certain simplifications arise. If 1 — X >> 36(3),

the destruction of photons by continuous absorption can be disregarded in

comparison with that occurring in the scattering process. Consequently, (3.4) can be replaced by

S(t) = A JK^(|t-t' |)S(T')dT' + S*(t) , (3.6)

0

i.e. we can set 3 = 0 in the integral term. For 1 — X << 36(3), on the con- trary, photons die primarily because of continuous absorption; and instead

of using (3.4), we may find the source function S(t) from the equation

^0

S(t) = i J

K^(1t-t' I ,6)S(T')dT' + S*(t) . (3.7)

0

Until now X and 3 have been considered independent of position. The line

of reasoning followed in this section shows that these rather stringent conditions can, in certain circumstances, be relaxed substantially. Thus, if the

dissipative processes are negligible in a medium with variable X and 3, the source function can be found, as before, from equation (3.5). Variability of

X and 3 affects only the form of the function sf (x) . However, it seems un- likely that a simple, yet sufficiently general, criterion can be found, for

ignoring the energy dissipation in a medium with variable X and 3.

Classification of media according to the importance of the dissipative

processes has, in essence, been known for a long time in transfer theory, but

no common terminology has been developed. E. H. Avrett and D. G. Hummer

(1965) and D. G. Hummer (1968) use the expression "effectively thin" for

weakly dissipative media, and "effectively thick" for strongly dissipative

media. It has been suggested by other -authors that media with co << 1 be

! called "optically thin," despite the fact that tq >> 11 A quantitative mea-

\ sure of the importance of dissipative processes has apparently not been introi duced until now. It is, however, necessary to emphasize the close connection
374 PLANE LAYER OF FINITE THICKNESS

that exists between the concepts of the dissipation measure o£ the medium,

thermalization length, and the thickness of the boundary layer.

8.4 ASYMPTOTIC SOLUTIONS FOR MONOCHROMATIC SCATTERING

RESOLVENT FUNCTION. Let us proceed now to discuss the asymptotic behavior of

solutions of the transfer equation for an optically thick layer. We shall

first consider monochromatic scattering with 3=0. This is the simplest

case for the following reason. In the case of monochromatic scattering the

Laplace transform of the half-space resolvent function $(t) has a branch line

(—<», —1) and a pole s = —k(0 < k < 1), where k is the root of the characteristic equation (see Chapter III)

Jl an ll^ = 1 . (4.1)

2k 1-k

The fact that the right-most singularity of $(s) is the pole (and not the end

of the branch line, as occurs for line- frequency scattering) greatly simplifies matte rs .

An investigation of the asymptotic behavior of solutions of the transfer

equation for large tq involves essentially the determination of the asymptotic

form of the resolvent function $(t;to) for tq >> 1. Since exact solutions of

the transfer equation are known for the half-space, it suffices to express

$(t;to) for large tq in terms of quantities referring to a semi - infinite

medium.

For monochromatic scattering the function (|>(t;to) satisfies the equation

^^^'''O^ = J / E^(|t-t' |)$(t' ;TQ)dT' + jE^(T) . (4.2)

0

It can be shown that

^as'^^''^0^

= *(t) + aD(T) - bD(TQ-T) , (4.3)

where

shk(T„ + 2T ) and D(t) is the function related to the solution S(t) of the non- cons ervative

Milne problem

00

) = S(t

I y E^(1t-t' |)S(T')dT' , S(0) = 1 , (4.5)

0

by
8.4 ASYMPTOTIC SOLUTIONS FOR MONOCHROMATIC SCATTERING 375

S(t) = H(l/k)[e^''-D(T)] . (4.6)

An explicit expression for D(t) was given in Sec. 3.8.

To prove (4.3) it suffices to substitute this expression into the basic

equation (4.2) for $(t;to) and make use of (3.7.12), (3.9.16) and the fact

that $(t) satisfies the equation

00 $(t) = - y* E^(|t-t' |)$(T')dT' + -E-i^(t) (4.7)

0

and D(t) is the solution of

1 D(t) = I /

E^(|t-t' |)D(T')dT' ^ ^ f

e~^/^-^ (4.8)

0

2 ^ 1+ky

Equation (4.8) is readily obtained by substituting (4.6) into (4.5).

Letting k tend to zero, it can easily be shown that in the conservative

case (4.3) assumes the form _ T+q(T)^(Tf.-T)+q(<»)

^as^^'^O^ = - ^3 7^-^ , A = 1 , (4.9)

as u TQ+2q(~)

where q(T) is the Hopf function. Obviously equation (4.9) is valid not only

for A = 1, but also for values of X < 1 for which kxg << 1.

Let us consider more closely the limiting form which (4.3) assumes far

from the boundaries, i.e. when t >> 1, xg — t >> 1. Expression (3.8.40)

shows that for large t -k(T+2T ) D(t) = e

^ + 0(e ^) , T -> ~ . (4. 10)

Substituting (4.10) into (4.3) and using (3.9.16), we find after minor manipulation that far from the boundaries 4)(t;Tq) approximately equals

sh k (ip,—T + T ) $,(t;t.) = 2kB

^

. (4.11)

0. yj sh k (TQ + 2Tg)

In particular, if the layer is weakly dissipative (kr^ << 1), _ Tf.-T+q(°o)

<D,(T;Tn) = /3 "

, , ,

. (4.12)

d^ 0^ TQ + 2q(oo)

The function '^'j(t;Tq) satisfies the diffusion equation
376 PLANE LAYER OF FINITE THICKNESS

dT

with the boundary conditions

- k"$^(T;TQ) = 0 (4.13)

$^C-t^;To) = 2kB

(4.14)

i.e. $^(T;Tg) is the resolvent function in the diffusion approximation.

Some comments here will be helpful in understanding the nature of the

asymptotic solutions of the line transfer problems that will be given in the

next section.

It is well known that the diffusion approximation involves essentially

setting the mean free path of a photon equal to zero. Physically this means

that the approximation is valid when distances from the boundaries are large

compared to the mean free path. It is natural to ask whether there is an approximation of the same physical nature available for line transfer problems.

The answer is yes, although the generalization is non-trivial, for the diffusion equation breaks down and its counterpart in line transfer problems is an integral rather than a differential equation, etc. Nevertheless, asymptotic

solutions that generalize (4.11) to the case of an arbitrary line profile do

exis t

Let us rewrite the conservative diffusion solution in a somewhat less

accurate form. Setting t = xtq in (4.9) (or in (4.12)) and then letting

TQ tend to infinity, we obtain

lim $(xTq;Tq) = /3(l-x) , X = 1 . (4.15)

Thus, for finite but large Tq far from the boundaries we have

$(xTq;Tq) ~ /3 (1-x) , 0 < x < 1 , X = 1 , Tq >> 1 . (4^16)

This expression elucidates the large-scale behavior of the conservative resolvent function when distances from the boundaries are large compared to the mean free path of a photon. In the next section it will be shown that (4.16)

is a special case (corresponding to y = 1 and <}) = 1/3) of the general largescale asymptotic form

$(xT.;Tn)

° "^(1-x)^ , 0 < X < 1 , A = 1 , >> 1 , (4.17)

" " r(Y)/*(l/TQ) "

which may, therefore, be regarded as a generalization of the diffusion approximati on.

X- AND Y- FUNCTIONS. The X- and Y- functions for large tq can be asymptotically

expressed in terms of the H-function, namely, for 0 < y < 1
8.4 ASYMPTOTIC SOLUTIONS FOR MONOCHROMATIC SCATTERING 377

as (4.18)

In the conservative case (A = 1) (4.18) assumes the form

X^3(y,To) = H(y)

y

TQ + 2q(oo)

J

yH(y)

TQ + 2q (°°)

(4.19)

The asymptotic forms (4.19) apply also for \ ^ 1, provided that kr^ << 1.

The asymptotic expressions (4.18) - (4.19) can be verified, for example,

by direct substitution into the nonlinear Ambartsumian- Chandrasekhar equations, i.e. equations (2.6) with G(c) = 1,0 < <; < 1 , G(?) = 0 ,

i; > 1 . It

should be noted that the solution of the Ambartsumian- Chandrasekhar equations

is not unique. The solution that has the necessary physical significance

must satisfy the constraints

i X f

X(y')

2 J 1-ky'

-kT

dy ' + e

0 X

2 I

Y(y')

J l+ky'

dy' 1 ,

(4.20)

f Wi . e

0 A f

ny^i dy' = 1 ^ l + ky- 2 ^ 1-ky'

which are obtained from (2.14) by setting z = 1/k and requiring X(l/k) and

Y(l/k) to be finite. Using (3.7.12) and (3.7.21) - (3.7.21"), one can easily

show that the asymptotic forms (4.18) satisfy these constraints.

We have so far assumed that 0 < y < 1. For some problems, values of

X(z) and Y(z) for z greater than unity are also of interest. It can be shown

that for z > 1 and Tq >> 1

X,3(z;Tq) = ^[l - kz cth k(To-2x^)] -

li^e-'''^schk(x,.2x^) ,

(4.21)

^as(^'^o)

_ H(z) kz csch k(Tn+2T„) + 1-kz 0 ^

1+kz

[l+kZ cth k(TQ + 2Tg)]

(4.22)
378 PLANE LAYER OF FINITE THICKNESS

Specifically, in the conservative case,

as U \^ TQ + 2q(oo)/ To + 2qCoo)

Y^,(z;tJ = -iilll) + HC-z)^ + ^ ^^e . (4.24)

For z << tq terms containing the factor exp(—Tq/z) can be discarded, and the

expressions (4.21) - (4.24) reduce to (4.18) - (4.19).

If, in either (4.23) or (4.24), z tends to infinity and (3.7.26) is used,

it can be verified that

^as^-^'^o) = ^as^^'^o) = Y[^0^2q(o=)] , x - 1, t

^ ^ ^ . (4.25)

Disregarding 2q(oo) in comparison with tq , we arrive at an expression that is

also obtained as a special case of (2.60). It follows from (4.25) and (2.40)

that

»as(^o;-0^ - , X . 1 , T„ » 1 . (4.26)

IS

The asymptotic form of X(°°;Tq) for Tq >> 1 and arbitrary X, 0 < X < 1,

X^3(~;To) = th . tJ . (4.27)

As kiQ ^ 0, (4.27) is transformed into (4.25), as we would expect. The ex- pression (4.27) can be obtained, for example, by taking the limit z 0° in

(4.21). We note that the function H(—z) for z > 1 appearing in (4.21) - (4.24) is expressed in terms of H(z):

H(z)H(-z)(l - — Jin —\ = 1 , Izl > 1 . (4.28)

V 2 z-1/

This relation is a consequence of (3.7.20) and (3.7.11).

LIMITING FORMS OF THE ASYMPTOTIC EXPRESSIONS. In the next section we shall

consider the asymptotic behavior of the X- and Y-functions for an arbitrary

line profile. If monochromatic scattering is regarded as a special case of

the general theory, we get asymptotic forms that are somewhat less accurate

than those just obtained. Let us derive these less accurate asymptotic re- sults from the above asymptotic expressions. From (4.23) - (4.25) it can be

shown that in the conservative case the following limits exist

Xdg/t ;tq)

(4.29)

^^"^ 7=77 7

= ^10^^) '

' Tq-«» /ttX(~; Tq)

Y(TQ/t;TQ)

Tq-x» /7TX(<»;Tg)

lim — = ^20"^^^ '
8.4 ASYMPTOTIC SOLUTIONS FOR MONOCHROMATIC SCATTERING 379

and equal

fio(t) =

. ,1 . (t-l+e"^)

^1/2^2

^20^^) = ^2 [l-(t.I)e-^] .

(4.30)

The approach o£ Y (tq /t ; Tq ) /X(=° ; tq ) to the limiting form /iT£2o('t) is illustrated in Fig. 47. In constructing the curves for tq = 1 and tq = 2 . 5 we have used the values of Y given by Y. Sobouti (1963) . In the next section it

will be shown that limits similar to (4.29) exist for an arbitrary line profile, with the functions i.^Q{t) , i = 1,2, depending on the profile (more precisely, on the characteristic exponent).

We note also that the asymptotic form (4.15) of the resolvent function

$(xtq;to) for tq °° can be easily obtained from (4.29), (4.30), and (4.25).

Indeed, (2.3) can be rewritten as X(TQ/t;TQ)

^

"^0 /* -tx - ^

, / e ^^$(xT^;Tjdx , (4.31)

X(oo;Tq) X(oo;t^) X(«>;Tq) »;t. J

^ 0' 0-

0 from which, in the case of monochromatic scattering, as t

1 -1 (t-l+e""^) = -1 J

e-^^<I'^3(xTQ;TQ)dx , (4.32)

0 where ^'as ^^''^0 ' '^0 ^ limiting form of $(xtq;to) ""^0 ^ °°* '^^^ relation

(4.32) may be regarded as an integral equation for <l>a^s (^"^0 J'^O^ • solution

is given by (4.15), as may be verified by direct substitution into (4.32).

An important general conclusion may be drawn from the last observation.

If, for a given line profile, we know the asymptotic behavior of X(<»;Tp,) and
380 PLANE LAYER OF FINITE THICKNESS

the limits (4.30), the relation (4.31) will give us, in the limit as tq ^

an integral equation for the limiting form of the resolvent function. The

solution of this equation will, in turn, give us the asymptotic form of

$(xto;tq) which is a counterpart of (4.16). This approach will be used in

the next section to obtain the general asymptotic form (4.17) that holds for

an arbitrary line profile.

So far we have considered conservative scattering. Let us now turn to

the non- conservative case. For a given, fixed X < 1 the functions fio(t)

and f2o(t) defined by (4.29) are equal to 1 and , respectively. This is

easily shown with the help of (4.21) and (4.27). We may, however, proceed

in a somewhat different manner, by letting X tend to 1 as tq °° in such a way that Rtq (and hence the dissipation measure oo) remains constant. The

limiting functions f^g and f20 will then depend on t and oj . Their explicit

expressions are easily obtained from (4.21) and (4.27) and are not reproduced

here

Proceeding from the equation (4.31) in a manner similar to that used for

conservative scattering, we may prove that the function tq $ (xtq ; tg ) /X (=» ; tq ) of the three variables tq , x and A, tends to a function of only two variables,

X and CO, in the limit as X -> 1, tq °°, co = const. It turns out that in this

limit $(xto;to) tends to _ sh[kT (1-x)]

i.e. to (4.11) with 2kB replaced by /3 and Tq set equal to zero. Generalization of this expression to the case of an arbitrary line profile seems to be

impossib le

The asymptotic forms of $(t;tq) for tq >> 1, and of X(ij;to) and Y(y;Tg)

for large ig and y < 1 were obtained by V. V. Sobolev (1964). The derivation

of the asymptotic expressions for X and Y for z > 1, quoted here without

proof, may be found in V. V. Ivanov (1964b). The asymptotic forms of the

functions X(y;Tg) and Y(y;Tr,) for 0 < y < 1 are discussed in many papers, e.g.

V. V. Sobolev (1957b), H. C: van de Hulst (1964), H. C. van de Hulst and F. Therhoeve (1966), V. V. Ivanov and V. V. Leonov (1965). The most complete

investigation of the asymptotic behavior of the X- and Y-functions that arise

in problems of monochromatic (generally speaking, anisotropic) scattering, is

that of T. W. Mullikin (see J. C, Carlstedt and T. W. Mullikin, 1966). A

great deal of useful information on the problems discussed in this section

has been given by M. A. Heaslet and R. F. Warming (1968a). It seems 0.

Halpern and R. K. Luneburg (1949) were the first to investigate the asymptotic

behavior of the exact solution of the transfer equation in an optically thick

layer. Although their work was done in the late thirties, it was not published for ten years. Specifically, in this work, asymptotic expressions

were obtained for the intensity of diffusely- reflected and diffusely- trans

mitted radiation for an optically thick layer illuminated by col limate d li ght

The forms (4.18) and (4.19) follow directly from expressions given by Halpern

and Luneburg.

8.5 LINE TRANSFER PROBLEMS: ASYMPTOTIC SOLUTIONS

X- AND Y-FUNCTIONS (CONSERVATIVE CASE). Let us turn to scattering in line

frequencies and begin by determining the asymptotic behavior of the X- and

Y-functions for the conservative case, X = 1, i.e. 1 — X = 3 = 0. As has

been shown in Sec. 8.3, this condition can be regarded as satisfied if the

dissipation measure oj is much less than unity.
8.5 LINE TRANSFER PROBLEMS: ASYMPTOTIC SOLUTIONS 381

The basis o£ the investigation is the asymptotic expression for the

function <5(tq;tq) as Tq °°. From (2.40) we have

d In X(oo;t ) Ht,;t^) - . (5.1)

Inserting (2.60) here, we find that for X = 1

* (tq ;tq) ~ JL , Tq ^ =0 . (5.2)

^0

It must be emphasized that the value of the numerical coefficient in (2.60)

is unimportant. The asymptotic behavior (5.2) is determined entirely by the

functional form of X(o°;to) as tq ^ °°.

For any fixed finite z, as Tq ^ °= the function X(z;tq) tends to H(z).

However, this limiting process is non-uniform, since X(<=°;tq) is finite, while

H(z) oo as z 00. Therefore, generally speaking, one cannot expect that from

the limiting values

X(z;oo) = H(z); Y(z;oo) = 0 (5.3)

one can obtain asymptotic formulae for X(z;tq) and Y(z;tq) which will be valid for all z. It is, however, possible to use (5.3) to obtain asymptotics

for values of z that are not too great in comparison with tq. Conversely,

the condition that X(oo;xq) is finite will correspond to the case in which z is not very small in comparison with tq . In the intermediate region, i.e.

when z is approximately equal to tq , both expressions should give the same result with comparable accuracy. In this way the asymptotic value of X(°o;xo)

could be obtained (although we shall use another method) . We note that the

functional form of X(o°;to) for tq ^ °° is not in question, as it is given by

(2.60). We are now concerned only with the evaluation of the numerical coefficient in (2.60).

The preceding discussion may be reformulated as follows. There are two

types of asymptotic expansions of the X- and Y-functions for large tq. The

first type corresponds to a fixed z , 0 < z < «>

^ and tq ^ °° • Expansions of

the second type are obtained when tq and z are allowed to approach infinity

simultaneously with their ratio t = tq/z, 0 < t < <»

, remaining constant. They

are similar to the well-known Debye expansions of the Bessel functions

J'^(vx) for V ^ 0°, X = const. Expansions of the second type are very important, and we shall derive them first.

We wish to find asymptotic expressions for X(z;to) and Y(z;to) of the

second type (tq is large and z is not too small in comparison with tq) in _the

form

X(z;tq) ~ X(«.;to) Xl^-^^' 4^ z"' . (5.4)

i = 0 ^'

0°

. ^

i Y(z;Tq) - X(~;Tq) ^(-1)^ 4^ z^^ , (5-5)

. i=0

^'
382 PLANE LAYER OF FINITE THICKNESS

where a. and b. for i > 1 are unknown constants,

1 1

' and

ag = = 1

Substituting (5.4) and (5.5) into the equations (see Sec. 8.2)

3X(z;Tq)

8t 0 Y(z ;tq) <|)(Tq ;Tq) ,

8Y(z;t.) , = --i Y(z;Tq) + X(z;Tq)$(tq;Tq)

3t

0

(5.6)

(5.7)

and using (5.2) and (2.40), we find that a- and b. are determined by the recurrence relations

(Y + i)a^ = yb^ ,

rY+i)b. = ib . , + ya. ^ ' ^1 1—1 ' 1

i = 1,2,

from which, considering (5.6), we find

y(y + 1) . . . (Y + i-1)

a

1

b.

1

(2Y+l)(2Y+2)...(2Y+i)

(y + 1) (y + 2) . . . (Y + i)

(2Y + l)(2Y + 2)...(2Y-^i)

i = 1,2,.

We note by comparing (5,4) and (5.5) with (2.33) and (2.34) that

x.(To) ~ a.T;x(oo;TQ) ,

>^i^^0) ~ b.TjX(oc;TQ)

Introducing (5.8) and (5.9) in (5.4) and (5.5), we obtain

X(z;Tq) ~ X(-;TQ)^F^^Y,2Y + l;-^j ,

Y(z;Tq) ~ X(c»;tq)-^F^^y + 1,2y-^1;-^^ ,

(5.8)

(5.9)

(5.10)

(5.11)

(5.12)

where ^F^(a,c;t) is the confluent hypergeometric function
8.5 LINE TRANSFER PROBLEMS: ASYMPTOTIC SOLUTIONS 383

,F,(a,c;t) = 1 . i 1- + ^dlHl ii ^ 11 c i: c(c+l) 21

^ a(a+l) (a+2) ^ c(c+l)(c+2) 3: (5.13)

The function ]^F]^Ca,c;t) satisfies the functional relations (see, e.g., I. S.

Gradshtein and I. M. Ryzhik, 1962):

-^F^(a,c;t) = e^ (c-a, c ;-t) , (5.14)

^F^(a+l,c+l;t) = ^F^(a+l,c;t) - ^F^(a,c;t) (5.15)

a^F^(a+l,c+l;t) = (a-c) ^F^ (a, c+1 ; t) + c^F^(a,c;t) ,

and is also related to the modified Bessel function I by

.-t/2 ^F^0+v,l + 2v;t^ , ^\2/ 22vr(v+l)

Using (5.14) - (5.17), we obtain from (5.11) and (5.12)

(5.16)

(5.17)

where

X(z;Tq) ~ 2

Y(z;tq)

tl/2-Ye-t/2

>-i/2(i)^^-)^"^..i/2(f); , i = 1,2

(5.18)

(5.19)

(5.20)

The quantity X(o°;to) that appears in these expressions is still unknown,

Although it was found in Sec. 8.2 that X(o°;to) ~ C]^[ K2 (t ) ] ~l/2 as tq °°

,

we have not until now been able to determine the value of c^. We proceed

from the following relation, which is obtained from (2.26) and (2.46):

00 /

[X(z;TQ)-Y(z;Tn) ]zG(z)dz 0

0- X(«';tq)

As Tg » the right side increases indefinitely (for y < 1) . Thus it follows

that the main contribution to the integral on the left comes from the region

of large z. Therefore we can replace X(z;to) and Y(z;tq) by their asymptotic

representations (5.11) and (5.12) to obtain

522-519 O - 74 - 26
384 PLANE LAYER OF FINITE THICKNESS

0 0 (5.21)

zG(z)dz ~ 1 .

Let us now introduce t = Tp./z. From the results obtain in Sec. 2.6, it

follows that

Using this result and (5.15), we reduce (5.21) to the form

00 X^~;Tq)K2(Tq)— J

t^^-\F^(Y + l,2Y + 2;-t)dt ~ 1 , (5.22)

from which

X(-;Tq) ~ , (5.23)

/K2(To)

where

^1 = (f(^ /t^^-\F,(yn,2y.2;-t)dtp .

The integral appearing in this expression for c^ can be expressed in terms of

the r-function (see I. S. Gradshtein and I. M. Ryzhik, 1968; p. 872). It is

readily shown that

= r(Y) /_iiiiIDL_) ^

. (5.24)

1 \2Trr(2Y)/

This result has already been given without proof in Sec. 8.2 (however, the

value of ci has played no role in the derivations given so far).

The expressions (5.18) and (5.19), in which X(«';to) is given by (5.23)

and (5.24) or (2.60), are the sought-for asymptotics of the X- and Y-functions , for large tq and values of z that are not very small in comparison with tq. , Setting z = Tg/t in (5.18) and (5.19) and using (2.60), we finally find for

j the leading terms of the expansions of X(TQ/t;Tg) and Y(TQ/t;TQ) for «> and t = const, 0 < t < °°

,

T X(To/t;T,) ~ f f,o(t) , (5.25)

^ /4)(1/tq)
8.5 LINE TRANSFER PROBLEMS: ASYMPTOTIC SOLUTIONS 385

YCTo/t;T,) ~ ^ ^^

f,o(t) , (5.26)

where the functions f^Q(t) are given by (5.20).

We shall now obtain asymptotics of the first type, which are valid when

z is not too much larger than tq. Substituting (5.2) into (5.7), setting

t = tq/z, and denoting differentiation with respect to t by the prime symbol,

we have

Y ' = _Y + 1 X .1

(5.27)

The solution of these coupled equations that satisfies the condition X(z;co) = H(z) and the relation (2.5) gives the desired asymptotic forms of X- and

Y- functions. It can be shown (we omit the proof) that

X(z;Tq) ~ H(z)f^(t) + H(-z)g^(t) , Y(z;Tq) ~ H(z)f2(t) + H(-z)g2(t) ,

(5.28)

(5.29)

where

£i(t) Zlltl ^-t/2j /tV I /t\

2 ® j Y-l/2\2j- Y + l/2(^2j

COSTTY

Vl/2(l)"Sn/2(l)]j.

(5 . 30)

(5.31)

with the upper signs corresponding to i = 1 and the lower ones to i = 2.

Here Iy,(x) and Ky(x) are the Bessel functions of imaginary argument, with

Ky(x) defined as

I (x)-I (x) K (X) = I

s inVTT

(5.32)

For scattering in line frequencies in media with no continuous absorption

(6 = 0) the function H(z) has a branch line on the negative real semi-axis.

The quantity H(— z) appearing in (5.28) and (5.29) is understood for z > 0 as the mean of the values of the H- function on the sides of the cut, and can be

found as follows. When z does not lie on the real axis, (5.4.19) can be

written (for X = 1) in the form

H(z)H(-z) [l-U(z) ] = 1 , (5. 33)
386 PLANE LAYER OF FINITE THICKNESS

where

U(z} = z'

00 f

G(z'

J 2

' dz (5.34)

Thus, using the formulae for the limiting values of Cauchy-type integrals,

we find that for z > 0

H(-z+iO)

l-U(z)+i|zG(z) H(z)

(5 . 35)

with U(z) being understood as a principal value. The value of H(—z) for

z > 0 is

H(-z) = j[H(-z+iO) + H(-z-iO)] , (5. 36)

which, according to (5.35), becomes

H(-z) = [l-U(z)] R(z)

H(z)

(5.37)

whe re

R(z) = < [l-U(z)] ^ + -zG(z)

2 11

-1

(5 . 38)

The expressions (5.28) — (5.31), in which H(—z) is given by (5.37), are the desired asymptotics of the X- and Y-functions for tq >> 1 and z that is

not too large in comparison with tq • We note that in those cases in which

H(z) behaves asymptotically as a power of z as z «>, the expressions (5.28) —

(5.31) are valid for all z > 0. In particular (5.18) and (5.19) then follow

from them.

In order to make practical use of the asymptotic expressions just obtained, tables of Iv(5c) and Ky(x) are necessary. The values of these functions for V = 0 and 1, appearing in the asymptotics of the X- and Y-functions

for the Doppler profile (y = 1/2), are available in all treatises on the

theory of Bessel functions, and also in many reference books. For a Voigt

profile (y = 1/4), I^ and for v = —1/4 and 3/4 appear. Tables of I_i/4(x);

and l3/4(x) are given by G. Watson (1945), and the functions K_]^/4(x) and

K3/4(x) have been tabulated by H. R. F. Carsten and N. W. McKerrow (1944).

We shall develop 'further the limiting forms assumed by the above asymptotic expressions for z << tq , i.e. for the central parts of the line. Using

the fact that for t >> 1
8.5 LINE TRANSFER PROBLEMS: ASYMPTOTIC SOLUTIONS 387

it is easy to show from (5.28) and (5.29) that for Tq >> 1 and z << Tq

X(z;Tq) ~ H(z)^l-Y^-^+ ...^ , (5.39)

Y(z;tq) ~ zH(z)J^ + . . . . (5.40)

Let us consider in a little more detail two important specific cases. First we will discuss monochromatic scattering. Since for this case y = 1,

the functions l^(t/2) and K^(t/2) of half-integral order appear in the expres

sions for fi(t) and gi(t). They can be expressed as elementary functions,

specifically

,

/(«) (I)

If we denote the X- and Y-functions for monochromatic scattering as Xj^(z;tq)

and Ym(z;to), and the H- function as Hj^(z), we find from (5.28) and (5.29)

that for TQ >> 1

X

M

(z;tq) ~ Hj^(z)^l-:^^- Hj^(-z):Ae ^ , (5.41)

Y^(z;To) ~ H^(z)^ . H^(-z)fl.:i-^e . (5.42)

We note that the function H^(z) is regular in the z-plane cut from 0 to -1.

Therefore for z > 1 the function U^i-z) is related to H(z) by (4.28), and not

by (5.37). In the present case, (5.37) was to be used only for 0 < z < 1. However, as the terms containing H-^(-z) in (5.41) - (5.42), for 0 < z < 1, are negligible, there is no need to evaluate this function.

We see that (5.41) - (5.42) and (4. 23) - (4.24) are identical if 2q(oo)

is negligible compared to tq. Therefore, as in the Milne problem, the general asymptotic expressions valid for an arbitrary line profile, in the specific case of monochromatic scattering, give the same accuracy as the uncorrected diffusion approximation, i.e. the diffusion approximation with the

extrapolation length set equal to zero.
388 PLANE LAYER OF FINITE THICKNESS

The second specific case we will discuss is that of the Doppler profile.

In this case the asymptotic forms of the X- and Y- functions can be somewhat

improved. It can be shown that for ^ «> and t = const, 0 < t < «>

,

^d("7'^o)

i/4V^ li^^^

j = 0

. To \ 1/4 1/2. „ .1/4X"^ 21

(5.43)

(5.44)

where the functions f^Q(t), i = 1,2, are given by (5.20) for y = 1/2, and

t £il(t) = |e ^/^[lQ(t/2) + (-l)^^^I^(t/2)][2x3, -|/ Ii(x/2)KQ(x/2)dx]

0 (5.45)

|e ^/^[KQ(t/2) + (-l)^K^(t/2)] [l^(t/2)-l] . i = 1,2 ,

where Xi is given by (2.62). Explicit expressions for fj^- for j > 1 are un- known, and it is unlikely that these functions may be expressed in terms of

known functions. Fortunately there is very little need for such expressions

The asymptotic forms (5.39) — (5.40) in the case of the Doppler profile

are refined as follows (z = const, t

0 0

zHp(z) m

X^(z;Xo) ~ H^(z) -

^^0

^. = 0 (ilnTQ)J

(5.46)

Yp(z;To)

zH„(z)

2^0 p^C^n^o^^

(5.47)

where m^ = n^ = 1 and

n . = oj . , J 3

. = 7 oj.co. • — (1—l)m. , , 3 1 3-1 ^ j-1 '

j = 1,2, .

i = 0

(5.48)

Here the constants oj^ are the coefficients of the asymptotic expansion

(to;to)

1 v;

2^0 (^nTo)J

(5.49)

I
8.5 LINE TRANSFER PROBLEMS: ASYMPTOTIC SOLUTIONS 389

In particular,

oJq = 1 , oj^ = I

, 0)^ = -^(l+ 2Y*-45-n2) = 0.0386 . (5.50)

The values of to. for j > 2 are unknown. Substituting (5.50) into (5.48), we obtain

m = 1 ; = -0.4409

1 (5.51)

n, = - ; n^ = 0.1545 . 1 2 2

•The proofs of (5.43) — (5.51) are omitted because of space limitations.

RESOLVENT FUNCTION. LARGE-SCALE DESCRIPTION (CONSERVATIVE CASE). Following

the derivation sketched in Sec. 8.4 we can now easily obtain the asymptotic

expression for the resolvent function. We proceed from (2.3), which we re- write in the form

^^'^q/^'^q) — 1 / -tx ^i^^V^ol, ^77 ^— = r + / e T„ dx . (5.52)

X(~;to) X(»;t^) J 0 X(c»;Tq)

Letting tq tend to infinity, we discover, with the aid of (5.25) and (2.60),

that in the limit as tq =° the left side is independent of tq. Since the

first term on the right tends to zero as t^. °° , we conclude that a limit

exists

TQ$(XTg;Tg)

F(x) = lim — ~ . (5.53)

I Consequently, in the limit «> (5.52) assumes the form

i Direct substitution into (5.54) verifies that the solution of this integral

? equation for F(x) is

F(x) = i^li^ x^-^l-x)^ . (5.55)

r2(Y)

Thus it follows from (5.53) that for tq >> 1 the resolvent function far from

the boundaries asymptotically equals

T ^..(^^^n'^n) =

°

. . . . , 0 < X < 1 . (5.56)

" " r(Y)v^TT7T^

This expression is a generalization of the unimproved diffusion solution

(4.16) to the case of an arbitrary line profile. As in the case of the diffusion solution, it applies only when the distances from the boundaries are
390 PLANE LAYER OF FINITE THICKNESS

much larger than the mean free path o£ a photon. However, as we shall see shortly, it may be used to construct asymptotic expressions that are valid

for arbitrary t, 0 <t < Tq .

Let us describe the general physical picture of radiative transfer in

a medium of large optical size. If the medium is conservative, we have two

characteristic lengths: (1) the photon mean-free path (which is unity in our notation) and (2) the size of the medium (tq, in the present case). If the

second length is much larger than the first one, i.e. ig >> 1, it should be

possible to describe the radiation field deep in the medium in a way analogous to the hydrodynamic approximation of gas kinetic theory. However, when

the distances from the boundaries are of the order of a photon mean-free path

such a description must break down, just as the hydrodynamic equations are inapplicable near the walls (Knudsen layer).

The problem of gas flow for small Knudsen numbers (which corresponds to

that with large tq in radiative transfer) is always split into two problems:

the hydrodynamic solution for the inner parts of a medium with given geometry

and the gas-kinetic solution near the walls. If the curvature of the wall is

small compared to the mean free path of a molecule, the wall may be considered plane. Hence the kinetic description is needed only for the simplest

geometry, namely, the half-space. At distances from the wall of the order of

several mean free paths, the hydrodynamic description becomes valid. The

hydrodynamic solution for a medium of a given geometry is fitted near the wall

to the gas kinetic solution for a half-space, i.e. kinetic theory is used

only to provide exact boundary conditions for the hydrodynamic equations (see,

e.g., M. N. Kogan, 1967)

The situation in radiative transfer is exactly the same. The half-space

solutions near the boundary have been thoroughly studied in the preceding

chapters. Now we shall consider in detail solutions similar to hydrodynamic

solutions, which we will call large-scale solutions . The expression (5.56)

is an example of a large-scale solution.

We have to consider two important problems: first, fitting the kinetic

half-space solutions to the large-scale solutions and, second, deriving largescale equations for radiative transfer in spectral lines. These equations

should be simpler than the "exact" transfer (kinetic) equation. We note that

the large-scale expression (5.56) was found from the "exact" transfer equations as the large-scale equations are yet to be derived.

To illustrate how the first problem is solved let us fit the large-scale

solution (5.56) to the exact half-space solutions. Near the boundary t = tq

the T-degendence of the resolvent function $(t;to) must be nearly the same as that of S(tq—t). For a thick layer, in which the primary source (in the case under consideration, the exciting radiation incident upon the boundary t = 0)

is located far from the boundary t = tq , this assertion is proven by the

discussion in Sec. 7.7. Therefore, for tq >> 1 and values of t satisfying

TQ - T << TQ ,

$(t;tq) - $(Tq ;Tq)S(Tq-t) , Tq >> 1 , Tq-t << Tq . (5.57)

The value of $(to;to) may be obtained by fitting this result to the largescale solution (5.56). Let us consider those t for which tq >> tq — t >> 1. Using (5.5.41) we can rewrite (5.57) as
8.5 LINE TRANSFER PROBLEMS: ASYMPTOTIC SOLUTIONS 391

*(t;tq) ~ Kto;tq)—-—== , t»1 , Tq»To-t»1 , (5.58)

while (5.56) gives

(t —t)^

r (y) /(1)(i/t^) ^0

from which

$(Tq;Tq) ~ y/Tq , Tq>>1 . (5.60)

Thus, as the boundary t = tq is approached, the large-scale solution is to be

replaced by (5.57), with $(tq;tq) given by (5.60).

Similarly, near the boundary x = 0 o£ a layer of large optical thickness

TQ , the influence of the boundary t = tq is weak, and in the first approximation may be neglected. Therefore,

«'(t;Tq) ~ $(t) , Tq>>1 , t<<Tq . (5.61)

As T increases, $(T;Tg) calculated according to (5.61) approaches <^'as ( ^ 5 t^O ) as given by (5.56), and we enter the region where (5.61) is to be replaced

by the large-scale expression (5.56).

An important by-product of this discussion is an asymptotic expression

for the resolvent function $(t;tq) in terms of half-space solutions

S(T -T)

which is valid for tq >> 1 and for all x, 0 < x < xq. For x satisfying the

inequalities x >> 1, xq — t >> 1, this expression reduces to the large-scale

asymptotic form (5.56); near the boundary x = 0 it gives (5.61); and for

values of x close to xq we recover (5.57). This remarkable representation

was found by Yu. Yu. Abramov, A. M. Dykhne , and A. P. Napartovich (1969a).

Let us now turn to the derivation of the basic equation of the largescale approximation. We proceed directly from the basic integral equation

for the source function

^0

S(x) = I y K^(|x-x' |)S(x')dx' + S*(x) . (5.63)

0

Using the identity

^0

i f

K^(|x-x' |)dx' = 1 - |k2(x) -

f

K2(Xo-x) , (5.64)

L
392 PLANE LAYER OF FINITE THICKNESS

we can rewrite it as

S(t) (K2(t)+K2(Tq-t)) = Tq (5.65)

= J

K^(|t-t' I) [S(t')-S(t) ]dT' + 2S*(t)

Let us consider the inner region of the layer where neither t nor tq — x is

small. It is clear that the main contribution to the integral in (5.65) comes from the values of t' that are far from t (for x' = t the integrand

vanishes). Thus we can replace K]^(|x—x'|) in the integrand by its asymptotic

form. Using the asymptotic relations (tq <» , x = const > 0)

K2(xXq) ~ x~^'^K2(Xq) ,

-2y-1 ^^2 '^0-^ (xT^) ~ 2yx —

which follow from (2.6.67) and (2.6.68), and substituting x = xTq and

t' = x'Tq in (5.65), we obtain

S(xTo)K2(to)(x ^^^-Hd-x) ^Y) =

/• S(x' T„)-S(XX^) ^

0 1^ ^ I

From this equation we may conclude that in the limit as Tq -> «> the product

S(xtq)K2(Xq) tends to a limiting function

s(x) = lim S(xXq)K2(Xq) , (5.66)

T ->-00 0 which is the solution of the equation

1 s(x)(x~2^+(l-x)"2^) = 2y f

s(x)-s(x') ^ ^ (S.67)

0 I ^-^

where

s*(x) = lim 2S*(xXq) .

' (5.68)

Equation (5.67) is the desired large-scale equation. It is valid for

0 < Y < 1/2 and may be regarded as a generalization of the diffusion equation.

We note that the integral operator in (5.67) belongs to a class of pseudo-
8.5 LINE TRANSFER PROBLEMS: ASYMPTOTIC SOLUTIONS

differential operators which have properties akin to those of differential

operators

The advantages of the large-scale equation (5.67) over the usual equation (5.63) are, first, that it does not contain a large parameter (Tq) and,

second, that the kernel of (5.67) is much simpler.

A thorough study of the large-scale behavior of the solutions of the

basic equation (5.63) has been made recently by Yu. Yu. Abramov, A. M. Dykhne

,

and A. P. Napartovich (1969b) and by A. P. Napartovich (1969). In particular,

they obtained the following large-scale approximation to the resolvent of

(5.63) :

^as'^^'^0 "^0 "''^0'' " r(xTQ (1-x' ) ,x' Tq (1-x)) , Tq^°>, x,x'=const, 0<x,x'<l , (5.69)

where r(T,T') is the half-space resolvent, for which they found the following,

highly accurate approximations (for t' > t) :

T^[T,Tn = S^-^-^^^^f^)$(T'-T)F(Y;2Y;l+Y;T/T') , y < j , (5.70)

r (t,t') = ^S(t)^(t') (T'/T)^^^£n '^^'^

, Y

= 1/2 , (5.71)

r^(T,T') = S(t)$(t')F(1;1-y;1+y;t/t') , y > 1/2 . (5.72)

Here F(a;b;c;z) is the hypergeometric function. Unfortunately we lack the

space to derive these expressions. Important additional information on the

asymptotic behavior of $(t;tq) and related quantities is given by D. I. Nagirner (1969) .

NON- CONSERVATIVE CASE. Let us now turn our attention to media that dissipate

significant amounts of energy. Consideration of the general case would hardly be justified because the resulting expressions would be extremely cumber-

^ some. Instead we shall discuss a particular situation in which results can

I be obtained in an intelligible form. All of these results pertain to the case in which the absorption of photons in flight is negligible compared to

their death in scattering; that is, we assume that 3 = 0 and X 7^ 1 . As has

1 been shown in Sec. 7.3, it is thereby assumed that the inequality A66(B) <<

1 — X is satisfied.

Let the medium be strongly dissipative (oj >> 1). Under the above conditions this implies that

XK2 (Tq) << 1 - X . (5.73)

We shall now prove that in this case, for Tq >> 1,
394 PLANE LAYER OF FINITE THICKNESS

For T < Tq equation (5,1.32) can be rewritten in the form

S(T,z) = A

J

K^(|t-t' |)S(t' ,z)dT' +

0

00 ""^^ + I f

K^(t'-t)S(t' ,z)dT' ,

or

+ e

z T

0

(5.75)

00 oo

0 0

By comparing this equation with (1.17) for 3=0 and using the superposition

principle we can conclude that

S(t,z;tq) = S(t,z) -

(5.76) 00 00 ^ ' -

\ J

S(TQ + t,z)dt J

e-^/"'s(TQ-T,z' ;Tg)G(z')^ .

0 0

^

Multiplying this equation by (A/2)G(z)/z and integrating over z from 0 to <»

,

we obtain

$(t;tq) = $(t) -

00 00 -

\ f *(To+t)dt y

(5.77)

$(T.+t)dt / e ^/^'s(T.-T,z' ;tJG(z').'^^'

0 0

In particular, we have

^(Tq-.Tq) = $(Tq) -

"

2 y*^^0''^)<^^ /e-^/"'x(z' ;Tq)G(z')^ ,

(5.78)

0 0

which can be used to prove (5.74)
8.5 LINE TRANSFER PROBLEMS: ASYMPTOTIC SOLUTIONS 395

The inner integral on the right side o£ (5.78) has a logarithmic singularity at t = 0 and decreases rapidly as t increases. At the same time, for

tq >> 1 the function ^Cxg + t) decreases relatively slowly as t increases.

Therefore when to is large the integral appearing in (5.78) equals, to within

the accuracy of the leading term of the expansion.

00 00

I y*(TQ + t)dt ye-^/"'x(z' ;Tq)G(z')^

0 0 (5.79)

and (5.78) gives

$(Tq;Tq) ~ $(Tq)|^1 - |aQ(TQ)j . (5.80)

From (2.21) for 6 = 0 we find

-S^^o^ = 1 - (i-^^r^o^-^oV • ^'-'^^

But from (2.47) and (2.45) it follows that if the condition (5.73) is fulfilled, then

1 - X » ^3o(Tq) . (5.82)

Therefore (5.80) and (5.81) give

Htq;tq) ~ (1-X)^/^$(tq) . (5.83)

It was shown in Sec. 5.5 that when the inequality (5.73) is satisfied,

(T ) Ktq) ~ ^ \' . (5.84)

U ^_x)3/2

The result (5.74) follows from the last two relations.

The expression (5.74) can be used to obtain a number of useful results.

Thus, it is easy to prove that

X ,-1/2

X(«>;Tq) ~ (l-X)--^^^ll - ^ : J

. (5.85)

This follows from (2.40) and (5.74) if allowance is made for the fact that
396 PLANE LAYER OF FINITE THICKNESS

—1/2

X(<»;Tg) tends to H(<=o) = (1—A) as tq °°. Further, combining (5.85) with

(2.45) and keeping (5.82) in mind, we find

3o(Tq) ~ (1-X)-^/^K2(tq) . (5.86)

Combining this result with (5.81), we obtain

an(Tn) ~ -fl " ^~ - —

^^^^^'^^^

J

• (5.87)

° 0 A 8 (i_A)3/2 /

With the help of (5.74) we can also obtain the asymptotic forms of the

functions X(z;to) and Y(z;tq) for tq >> 1 when z satisfies the inequality

Tg >> z. Setting t = tq in (5.76), we obtain

Y(z;Tq) = S(Tq,z) -

00 00 S(Tn + t,z)dt / e t/2'x(z' ;Tq)G(z')^

0 0

In Sec. 6.10 it was shown that for t >> 1 and z << t

There fore

Y(z;Tq) ~ zH(z) U(tq) -

00

(5 . 88)

S(t,z) ~ zH(z)$(t) , t >> 1 , z << t . (5.89)

00 oo ~

2 / '^'^^O^^^^^ f

e"^/''x(z' ;Tq)G(z')^1 ,

0 0

^

or, considering (5.78),

Y(z;Tq) ~ zH(z)$(Tg;TQ) , Tq » 1 , z « Tq . (5.90)

The first of equations (5.7), together with (5.3), gives

X(z;Tn) = H(z) - J

Y(z;t)$(t ;T)dT .

- (5.91)

^'o Substituting (5.90), we get
8.6 THE SCHUSTER PROBLEM 39 7

oc /

X(z;t,) ~ H(z) - zH(z) I (T;T)dT , >> 1 , z << . (5.92)

Finally, introducing $(t;t) from (5.74) into (5.92) and (5.90), and using the

asymptotic properties of K]^(t) derived in Sec. 2.6, we find that when the

inequalities (5.73) and z << are satisfied.

X(z;t.) ~ H(z) - zH(z).

jk^Ctq)

n2

4y + 1 1-X

(5.93)

Y(z;t„) ~ zH(z)

1-X (5.94)

It is worth mentioning that the condition (5.73) was not assumed in deriving (5.90) and (5.92). Therefore these results are valid for media with an arbitrary dissipation measure. Specifically, for co << 1 we have $(to;to) ~ y/tq; (5.90) and (5.92) reduce to the expressions (5.39) and (5.40), obtained

previously by other means.

This section is mainly based on the work of the author (V. V. Ivanov,

1963, 1964b, 1970b) and of D. I. Nagirner (1967). The derivations of the

refined asymptotic forms for the Doppler profile given here without proof may

be found in V. V. Ivanov (1970b). The large-scale description was first

clearly formulated in a remarkable paper by Yu. Yu. Abramov, A. M. Dykhne

,

and A. P. Napartovich (1969), although it was implicitly used earlier (e.g.,

V. V. Ivanov, 1966). For an alternative derivation of the large-scale asymptotic forms and of various other results presented in this section, see C. van Trigt (1969), (1970), and especially (1971).

8.6 THE SCHUSTER PROBLEM

BASIC EQUATIONS. For the first standard problem of multiple scattering in a

! plane layer we shall consider the well-known Schuster problem, defined as

• follows. We consider a plane layer of a gas whose optical thickness at line

j center equals tq. The layer is completely transparent in the continuum (3 =

I 0) and contains no internal sources of radiation. Incident on the boundary

ij T = TQ is isotropic continuum radiation, whose intensity is independent of

I frequency and equal to Iq. It is usually assumed that pure scattering (X =

jj 1) occurs within the medium; however, in the interests of generality we shall

I consider X to be arbitrary (X < 1) . In the classical statement of the prob1 lem it is assumed that the frequencies of the photons do not change during

I scattering (so-called coherent scattering). As was shown in Sec. 1.5, this

I assumption is unrealistic. We shall assume complete frequency redistribu1 tion, i.e. that a photon "forgets" its frequency each time it is scattered.

We wish to find the intensity of the radiation l(0,ij,x) emerging through the

boundary t = 0. We shall also obtain the intensity I(TO,y,x) of the diffuse-

' ly reflected radiation, which is of interest for a number of problems, and

' shall discuss the so-called "curve of growth" for the Schuster problem.

Under the assumptions just stated, the Schuster problem involves the

I solution of the transfer equation
398 PLANE LAYER OF FINITE THICKNESS

dl (t .y ,X) / ^ T A- ^ y

^ = a(x)I(T,y,x) -

-^AaCx) / a(x')dx' I I (t , y

' , x ' ) dy

(6.1)

-1

with the boundary conditions

1(0, y,x) = 0 I(TQ,y,x) =0, y < 0 , )

In. y > 0 . ) (6.2)

I£ we introduce the line source function

00 1 S(t) = |a y

a(x')dx'y* I(T,y' ,x')dy' , (6.3)

-1

we have for the intensity of the emergent radiation

T,0 — (a (x) /y) r ( ( ^ / ^ 1(0, y,x) = I^e / ^^^^^^^^^a(x)dT/y , y > 0 , (6.4)

0

}^ _(a(x)/|y|)(T -T)

I(TQ,y,x) = J

S(T)e a(x)dT/ly| , y < 0 . (6.5)

0

The integral equation for the line source function in the present case has

the form

S(t) K^(1t-t' |)S(T')dT' + lofl^Z^^Q-^) ' f^-^^

0

which is easily derived from (6.3) — (6.5). Comparing this equation with

the auxiliary equation (1.17) for the function S(t,z) and recalling that

00 :^o-^ = /

(t.-t)/z'

K7(Tn-T) = / e G(z')dz' , 0

we can write, by virtue of the superposition principle,
8.6 THE SCHUSTER PROBLEM 399

00 S(T) = y S(TQ-T,z')GCz')dz' . (6.7)

0 Of course, the functions S(t) and S(t,z) also depend on Tq ; however, to keep

the notation as concise as possible, we shall not state this explicitly.

EMERGENT INTENSITIES. Substituting (6.7) into (6.4) and (6.5) and taking

account of (2.7), we obtain

1(0, y,x) = Iq|^|6oX(z) + ^1 - laQ^Y(z)j , y > 0 , (6.8)

I(TQ,y,x) = Io|l - ^l-^aQjX(z) - j3qY(z) , y < 0 , (6.9)

where

z = a(x)

Here ag and 3o are the zero-order moments of the X- and Y- functions, defined

by (2.15). The expressions (6.8) and (6.9) give the solution of the Schuster

problem.

For pure scattering the above expressions can be simplified slightly.

Using (2.22) and (2.46), we find that for X = 1

1(0, y,x) = L!_[X(z)+Y(z) ] , y > 0 , (6.10)

2X(oo)

^0

I(T.,y,x) = u_[2X(~)-X(z)-Y(z) ] , y < 0 . (6.11)

" 2X(oo)

The result (6.10) shows that in the radiation emerging at t = 0 , an absorption line is seen on a continuum of intensity Iq. Let us estimate the

j central intensity of the line assuming the optical thickness tq of the layer

I to be sufficiently large. For radiation emerging along the normal (y = 1),

ii we have z = 1 at line center (x = 0). Further, for tq >> 1, the quantity

X(l) + Y(l) is asymptotically equal to H(l), where H(z) is the conservative

ji H-function. Using the asymptotic form of X(<») for tq >> 1, given by (2.58),

1 we find from (6.10) that the intensity of the radiation emerging at line cen- ter along the normal to the boundary t = 0 is asymptotically equal to

1 1(0,1,0) ~ I^HIII /K^(T^ , Tq >> 1 , (6.12)

^^1

ij where ci is given by (2.59). Specifically, for a Voigt profile, with aiQ >> 1

' we find, using (2.7.29),

1(0,1,0) ~ I, H^(l)

2iT-^U(a,0)

r(i/4) 9 (6.13)

522-519 O - 74 - 27
400 PLANE LAYER OF FINITE THICKNESS

LINE SOURCE FUNCTION. In this subsection we restrict ourselves to the conser- vative case. Using the easily proven identity

-^0

J f

K^(|t-t' |)dT' = l-iK2(T) -iK2(TQ-T) , (6.14)

0

we can show from (6.6) that for X = 1 the source function satisfies the relation

S(t) + S(Tq-t) = Iq . (6.15)

As follows from (6.7), the source function increases monotonically with t between the values

8(0) = \^ ; S(Tq) = - IBq^ . (6.16)

Specifically, for large t

0

S(0) ~ I±/K^; S(To) - Io(l -^^/IJlT^y (6.17)

The values of S(t)/Iq for the Doppler and Lorentz profiles and several

values of tq are given in Table 39 (after A. L. Crosbie and R. Viskanta,

1970a). They were obtained by an iterative solution of the integral equation

(6.6) with X = 1. We note that in the conservative case S(xTQ)/Ig tends- as Tq -> 00 to a limiting function

f(x) = lim — (6.18)

^0"°° ^0

which describes the large-scale behavior of the source function. It can be

shown that

£(x) = f

yY-l(l-y)^-ldy . (6.19)

or

f(x) = ^^i^ x^f(i-y;y;y + i;x) , (6. 20)

Yr2(Y)

where y is the characteristic exponent and F is the hypergeometric function.

In particular, for the rectangular profile (y = 1) we have

£,^(x) = X , (6.20a)
8.6 THE SCFIUSTER PROBLEM 401

TABLE 39

THE SOURCE FUNCTION S(t)/I^ FOR THE CONSERVATIVE SCHUSTER PROBLEM

T/T^

0

= 1 TL o

= 2 T

o

T — O

1 n T = 0

Z U

D L D L D L D L D L

0 00 . 7092 . 6569 . 7751 . 7074 . 8541 . 7695 .8997 . 8080 .9 320 .8396

0 05 .6796 .6334 .7351 . 6738 . 8005 . 7182 .8361 .7388 . 8587 .7486

0 10 . 6560 .6152 . 7040 . 6490 .7599 .6834 .7891 .6962 .8061 .6994

0 15 .6 344 .59 88 /' ~t C /I .01 bo . 62 70 . 7231 .6541 .7471 .6623 .7602 .6627

0 20 .6139 .5834 . 6487 . 6068 .6885 . 6281 . 7080 .6333 . 7182 .6326

0 25 . 5941 .5687 . 6228 .5876 .6554 . 6042 .6710 . 6075 .6788 . 6064

0 30 .5748 . 5545 . 59 75 . 5693 .6233 .5818 .6354 .5839 .6412 .5828

0 35 . 5558 .5406 . 5728 .5516 . 5919 .5605 . 6007 . 5617 .6049 . 5609

0. 40 .5371 .5269 .5484 . 5342 . 5610 . 5400 .5668 .5406 .5695 . 5400

0 45 .5185 .5134 .5241 .5170 .5304 .5199 .5333 .5202 . 5346 .5199

0. 50 . 5000 . 5000 . 5000 . 5000 .5000 . 5000 . 5000 . 5000 . 5000 . 5000

D = Doppler; L = Lorentz

while for the Doppler profile

2 fj^(x) = — arcsin/x . (6.20b)

THE MILNE PROBLEM AS A LIMITING CASE OF THE SCHUSTER PROBLEM. The generalized

Milne problem studied in detail in Sec. 6.1 can be regarded as a limiting

case of the Schuster problem for tq °°. Actually, let us increase both tq

and the intensity Ig of the exciting radiation in such a way that the relation

In = 2/6o - 2X(a') holds. For A = 1 the free term in equation f6.61 is then

! found to be equal to X(oo) K2 (tq-t) . Since for pure scattering X(<») -Cj [K2 (tq)]"^ as Tg > 00, for any finite t the free term tends to zero when xg increases

indefinitely. Consequently, for A = 1 and Tg = °° equation (6.6) reduces to

the homogeneous equation corresponding to the generalized Milne problem.

According to the first of the expressions (6.16), the solution is normalized

I to unity at t = 0 . Further, setting Ig = 2X(<») in (6.10), we find that the

intensity of the emerging radiation is, in the limit as xg , equal to

1(0, y,x) = H(z), which agrees with the result obtained by another method in

Sec. 6.1.

It is interesting to see the results of analogous arguments for non- conservative scattering. As was shown in Sec. 8.5, in this case 6p~(l-X)-l/2K2 (xg) for tq -> ~. Therefore, setting Ig = (2/X)6q^, we find that

when Xp, ->- «> equation (6.d) takes the form
402 PLANE LAYER OF FINITE THICKNESS

S(T) = I /k^(It-t' |)S(T')dT' + (l-X)^/2 , (6.21)

0 and (6.8) gives

1(0 ,y,x) = H(z) . (6.22)

We have arrived at the problem of the radiation field in a medium with a uniform source distribution, which was studied in detail in Sec. 6.1 — 6.3.

However, in the present model the primary atomic excitation occurs not becaus

of electron impact, as was assumed in Sec. 6.3, but because of the radiation

in the far wings of the line coming from an infinitely powerful source that

radiates in the continuum at infinite depths. The result (6.22) shows that

the function H(z) gives the frequency and angular dis tribjation of radiation

transmitted by an infinitely thick layer.

CURVE OF GROWTH. Let us return to the Schuster problem. As we have found,

an absorption line is present in the radiation emerging at t = 0 . We shall now find its equivalent width as a function of tq • For simplicity we assume

that pure scattering occurs (X = 1). The equivalent width of the line (in

the flux) is defined as

oo W = y* (l-r^)dv , (6.2 3)

0 where r^ is the residual flux, i.e. the ratio of the flux at frequency v in

the line to the flux in the adjoining continuum. Since in the present model

the emergent radiation in the continuum is isotropic and its intensity is

equal to Iq , the corresponding flux is ttIq. Therefore

1 r = — /l(0,y,x)y dp . (6.24)

0

0

We recall that although we write x as the frequency variable, the radiation

intensity is calculated per unit interval of the usual frequency v. Substituting (6.10) into (6.24), inserting the resulting expression into (6.23),

and transforming the variable of integration from v to x, we find

p

1 W = j f

[2X(«>)-X(z)-Y(z) ]y dy , (6.25)

0 X(~)

where Av is the characteristic frequency interval used to define the dimensionless frequency x (usually, Avq) . Setting y = a(x)z and changing the orde

of integration, we obtain

00 /

W =

/ t2X(oo) - X(z) - Y(z)]zG(z)dz , (6.26)

AX(a>)
8.6 THE SCHUSTER PROBLEM 403

where A is the usual normalization constant, A_£ a{x)dx = 1. From (2.27) and

(2.46) we obtain an alternative expression for W:

W = Av

^0 -/A J ' dT

Ay x2(«.;t)

(6.27)

The values of AW/Av for the Doppler and Lorentz profiles found from (6.26)

by numerical integration are given in Table 40 (after A. L. Crosbie and R. Viskanta, 1970a).

The most interesting case is that of an optically thick layer (Tq >> l).

It follows from (6.27) and (2.58) that for the Doppler profile

W AVjp2/iT J' K2(T)dT ~ AVp2/£n Tq , Tq >> 1 , 0 whereas for a Voigt profile

"^0

(6.28)

W - AVp^-^ f K^(T)dT ~ Avp _ , JL^=(aTn)"'" , ax^ >> 1 , (6.29) 1/2

cJU(a,0) J

and for the Lorentz profile

3c|/U(a,0)

TABLE 40

VALUES OF AW/Av FOR THE DOPPLER AND LORENTZ PROFILES

Dopp ler Lorent z Doppler Lorentz

0, 0 0 0000 0 0000 9.0 1 430 2 547

0 1 0 0880 0 . 0911 10 1 473 2 698

0 3 0 2259 0 . 2441 15 1 663 3 358

0 5 0 335 1 0 3740 20 1 741 3 910

0 7 0 4259 0 4884 25 1 823 4 395

1 0 0 5385 0 6396 30 1 888 4 832

1 5 0 6849 0 8542 35 1 941 5 232

2 0 0 7982 1 037 40 1 986 5 605

2 5 0 8900 1 199 45 2 025 5 954

3 0 0 9667 1 345 50 2 059 6 284

4 0 1 089 1 602 60 2 118 6 897

5 0 1 185 1 827 70 2 166 7 460

6 0 1 262 2 030 80 2 207 7 983

7 0 1 327 2 215 90 2 242 8 474

8 0 1 .382 2 386 100 2 274 8 937
404 PLANE LAYER OF FINITE THICKNESS

, 4v^ 1/2 , (6. 30)

Here is given by (2.59) with y = 1/4 (precisely, ci = 0.914), and Av^ is

the collisional line width.

The numerical data presented in Table 41 make it possible to estimate

the accuracy of the asymptotic expressions (6.28) and (6.30). In Table 41 we give the ratio W^s/W for the Doppler and Lorentz profiles as a function

of TQ. Here W is the numerically exact value of the equivalent width, and

is the asymptotic value as given by (6.28) and (6.30) for the Doppler

and Lorentz profiles, respectively.

We note that the asymptotic expression (6.28) in the case of the Doppler

profile can be improved somewhat, for it may be shown that (V. V. Ivanov,

19 70b) ,

W = Avj^2/£n Tq 1 + Q

/In

0 . 15454

fcn T„

(6.31)

where

(6. 32)

Unfortunately, the numerical value of Q is unknown; it may only be asserted

that Q < 0.

The expressions obtained so far in this subsection refer to the conser- vative case (X = 1) . How sensitive is the equivalent width to the value of

X? In order to answer this question, we shall consider the opposite limiting

case — a purely absorbing medium (X = 0). Obviously, in this case the intensity of the emergent radiation is

1(0, y,x) ^0^

-Tq/z

(this also follows from (6.8) for X = 0) . Substituting this expression into

(6.24), we find from (6.23), exactly as in the derivation of (6.26), that

TABLE 41

THE RATIO W /W FOR THE DOPPLER AND LORENTZ PROFILES

as

0 Dopp 1 e r Lorent z

10

20

50

100

1. 162

1.121

1.084

1.065

1.056

1.031

1.014

1.008
8.7 DIFFUSE REFLECTION AND TRANSMISSION 405

W = Av zG(z)dz

0

or

W = Av^ (6. 33)

0

A comparison of this expression with (6.28) shows that for the Doppler

profile, when tq >> 1, the leading term of the asymptotic expansion of W(tq)

is the same for X = 1 and X = 0. In a similar way, we conclude from a com- parison of (6.33) with (6.29) that for the case of a Voigt profile, when

axQ >> 1, the equivalent width for X = 0 is asymptotically larger than for

X = 1 by the factor 2c^ = 1.67. The fact that W is smaller for X = 1 than

for X = 0 is physically quite understandable: for pure scattering line photons

that have been scattered partially escape from the boundary x = 0 causing W

to be smaller than for X = 0. For a Doppler profile the importance of this

effect is so slight that it does not affect the leading term of the asymptotic expansion of W(Tg). For a Voigt profile and X = 1 the effect of the

diffusely- transmitted radiation reduces the equivalent width by a factor of

1.67. It is obvious that for X < 1 this effect should be still less significant. We conclude that in the Schuster problem the equivalent width of the

line is not very sensitive to the value of X.

The dependence of the equivalent width of the line on the number of

ij absorbing atoms is known as the curve of growth. In constructing a family of

- theoretical curves of growth the equivalent width W should be plotted as a function of the quantity Tq/A, rather than of tq since the latter depends not

, only on the number of absorbing atoms and the transition probability, but al-

, so on the form of the line (say, upon the value of the Voigt parameter a),

i' The quantity Tq/A is independent of a (for more details, see final part of

I Sec. 2.3).

! One final observation is necessary. Until now we have always treated

Doppler and Voigt asymptotics separately. However, when a is sufficiently

I small, for a certain range of t the Doppler asymptotics are valid for the

Voigt profile as well. If we speak of asymptotics with respect to t, then

ij this range is defined by the inequalities 1 << x << 1/a. For asymptotics

.' .with respect to z, "the Doppler region" is given by the inequalities

' 1 << z << 1/a. For ax >> 1 and az >> 1 the usual Voigt asymptotics hold.

Ij In using the Doppler asymptotics with respect to x for the Voigt case, one should, strictly speaking, allow for the effect of the line profile on the

I x-scale (the factor Ay = U(a,0); see preceding paragraph). However, since

the Doppler asymptotic region exists only for a << 1 (in practice for a ^

\ 10-2), ^this effect can be ignored, since U(a,0} ~ U(0,0) = tt-^/^ when a << 1 .

, DIFFUSE REFLECTION AND TRANSMISSION. Let us now consider the so-called

'J problem of diffuse reflection and transmission which is defined in the following way. Parallel beams of radiation are incident on the boundary at x = 0 of a medium of optical thickness xq . The total flux through a unit area peril pendicular to the direction of inci-dence (integrated over all frequencies v)

r is Iq. The incident radiation is considered to be monochromatic with dimen8.7 DIFFUSE REFLECTION AND TRANSMISSION.

A MEDIUM WITH UNIFORMLY DISTRIBUTED SOURCES
406 PLANE LAYER OF FINITE THICKNESS

sionless frequency xq , and the angle of incidence is arc cos ijq . The medium

contains no internal sources of radiation, so that its luminescence is caused

only by external illumination. The problem is to determine the intensity of

the radiation emerging from the medium.

The intensity I(t,ij,x) of the diffuse radiation in this case is determined by the transfer equation

^dlXj^iiiiO = (a(x)+3)I(T,M,x) - ^a(x) / aCx')dx' f I (t , y

' , x ' ) dy ' -

_c» _/i (7,1)

, X A , , , , -(a(xQ)^6)T/yo

with the boundary conditions

1(0, y,x) = 0 , y < 0 , I I(TQ,y,x) = 0 , y > 0 . )

From (7.1) and (7.2) it follows that the line source function

1

(7.2)

S(t) = ^A / a(x')dx' / I(T,y' ,x')dy' + I.-^ —a(x.)e %7.3)

satisfies the integral equation

lo

0

whe re

r , Aa(x^) -t/z^

S(T) = \ JK^(|t-t' I ,3)S(T')dT' . Iq^ -^e , (7.4)

^0

0 a(xQ)+f

Comparison of (7.4) and (1.17) gives

(7.5)

^^'^ ~- ^Oi7-Mr^S(x.z,;To) . (7.6)

Because the basic equations are linear, we can, without loss of generality,

assume from now on that

= 4TTAV _ ^7^7)

0 XAa(xQ)
8.7 DIFFUSE REFLECTION AND TRANSMISSION 407

The intensities o£ diffusely reflected and diffusely transmitted radiation can be expressed in terms of the X- and Y- functions. Taking into account

(7.6) and (7.7), we have

1(0, y,x) = J

S(T,ZQ;TQ)e ^/^dx^^ , y > 0 , (7.8)

0

""-^ -(T.-T)/Z

I(TQ,y,x) = J

S(T,ZQ,TQ)e ' ° "^""iTf

, y < 0 , (7.9)

whe re

z = —Li^— . (7.10)

a(x) +6

Using (2.7), we finally obtain

X(z )X(z)-Y(z )Y(z)

1(0, y,x) =

^- ^17^ , y > 0 , (7.11)

Zq+z u y

X(z)Y(z )-X(z )Y(z)

I(Tn,y,x) = zz„^^ , y < 0 , (7.12)

where zq and z are defined by (7.5) and (7.10). These expressions give the

solution of the problem of diffuse reflection and transmission. They are a generalization of the results first obtained by V. A. Ambartsumian (1943)

for the problem of diffuse reflection and transmission with isotropic mono- chromatic scattering. It should be noted that for isotropic monochromatic

scattering with B 0 , the X- and Y-functions can be expressed simply in

terms of the corresponding functions for 3=0. Since for monochromatic

scattering G(z) = 1 when z < 1 and G(z) = 0 when z > 1, we obtain from (2.6)

Xj^(z;Tq,X,3) =» Xj^^z(1 + 3) ;To(1 + 3) . > o)

Yj^(z;to.A,3) = Y^j(z(1.3);Tq(1-3) > ^ ^ ^)

(7.13)

Thus allowing for continuous absorption, in this particular instance, turns

-out to be very simple, as it involves just the introduction of certain scaling factors into the arguments

In the limit as Tq °° (7.11) assumes the form

H(z)H(z^) , . 1(0, y, X) = 0_zz^. (7.14)

2+zq ^ y

This expression gives the solution of the problem of diffuse reflection from

a semi- infini te medium in which both continuous absorption and the destruc-
PLANE LAYER OF FINITE THICKNESS

tion o£ photons in scattering occur. For 3=0 (7.14) reduces to the expres

sion obtained in Sec. 6.4, exce] we are taking Iq = 47rAv/XAa (xq ) sion obtained in Sec. 6.4, except that there we assumed Ig = 1, whereas here

An analysis of the line profiles described by expressions (7.11) and

(7,12) can be performed in the same way as in Sec. 6.4 for lines formed by

diffuse reflection from a semi - infinite atmosphere without continuous absorption. Unfortunately the X- and Y-functions are still far from being completely tabulated (see Sec. 8.2). However, they can be easily found for weakly

dissipative atmospheres by using the asymptotic expressions obtained in Sec.

8.5.

The value of expressions (7.11) and (7.12) is not just that they give

the solution to the problem of diffuse reflection and transmission; they

describe as well the line profiles formed in a medium with an exponential

distribution of primary sources. In problems of practical interest, the

primary source function S^(t) can often be approximated by an expression of

the form

A ^ ^ A — / 2 • S^(t) = 2^^i^ , (7.15)

i = l where Si and are constants. If S-i^(t) has this form, the intensities of

the emergent radiation are equal to

aX(z)X(z. )-Y[z)Y(z. ) r ^ c

1

^ J y -^J ^a(x) ^ n S. z. , y > 0 , 1 z^ +z 1 y

IU^, X(z)X(z.)-Y(z)Y(z.)

(7.16)

.;,X(z)Y(z. )-X(z. )Y(z)

^^^^

Zi +z

i = l ^

n I(T ,y.x) = y S i i ^i^T^ , y < 0 ,

where z, as before, is given by (7.10). Let us consider one important particular case in greater detail.

UNIFORMLY DISTRIBUTED SOURCES. We assume that the internal sources are uniformly distributed, i.e. in (1.1) S*(t) = S* = const. Setting n = 1, Z]^ = and S* = S* in (7.15) and (7,16), we find that

1(0, y,x) = I(T.,-y,x) = S*X(c») [X(z)-Y(z) ]z^^ , y > 0 , (7.17)

The line profile strongly depends upon the values of the parameters X,

B, and tq, and also, of course, upon the absorption coefficient profile. Let

us discuss the case of 6 = 0. In Figs. 48-51 line profiles are shown for the

case of a Doppler absorption coefficient with various values of tq and X, as obtained numerically by A. G. Hearn (1962), Figure 52 shows profiles calculated by E, H, Avrett and D. G. Hummer (1965) (also by a numerical solution

of the equation for S(t) and subsequent calculation of the emergent intensity) for several values of the Voigt parameter a, 1 — A = 10—4, a^d a series

of values of tq. It is evident from the figures that the line widens as Tq

increases. When X = 0 and tq >> 1, for an absorption coefficient with an arbitrary profile the line will show no reversal. As 1 — X decreases, a characteristic self- reversal appears at line center because the escape of

radiation causes the excitation to decrease toward the boundaries.
8.7 DIFFUSE REFLECTION AND TRANSMISSION 409

A

-3 I I I I \ ^

0 1.0 2 0 3.0 4.0

X Fig. 48. Line profiles formed in a plane layer with uniformly distributed

sources. Doppler absorption coefficient, A = 0, 6=0.

-4l__J I I I 3^

0 LO 2.0 3.0 4.0

X Fig. 49. Line profiles formed in a plane layer with uniformly distributed

sources. Doppler absorption coefficient, A = 0.9, 6 = 0,

Comparison of Figs. 50 and 51 shows that for small 1 — X a range of tq

exists within which the line profile for a given absorption coefficient is

entirely determined by the value of tq and is independent of X (curves cor- responding to TQ ^ 10). This indicates that a medium is weakly dissipative.

In the limiting case of a conservative medium, the line profiles in the

Doppler case have the form shown in Fig. 53 (after A. G. Hearn, 1964a). When

Tq is large, line profiles may be calculated from (7.17), using the asymptotic

forms of the X- and Y- functions obtained in Sec. 8.5. The results are com- pletely in agreement with those obtained by numerical methods (such a comparison has been made by D. I. Nagirner, 1967; and by M. A. Heaslet and R. F. Warming, 1968b).

The characteristic feature of the profiles appearing in Fig. 53 is the

trough at line center for large tq. The location of the intensity maxima can easily be estimated (V. V. Ivanov, 1964b, 1966). Let us consider the case of

the Doppler profile as an example. From (7.17), (5.18), and (5.19) we find
410 PLANE LAYER OF FINITE THICKNESS

A ^ A

0 1.0 2 0 3.0 4.0 0 1.0 2.0 3.0 4.0

Fig. 50. Line profiles formed in a Fig. 51. Line profiles formed in a plane layer with uniformly distributed plane layer with uniformly distributed

sources. Doppler absorption coeffi- sources. Doppler absorptibn coefficient, A = 0.99, e = 0. cient, X = 0.999, 6=0.

that for those values of x for which the quantity

t = ^a(x) (7.18)

is not too large, i.e. rather far from the line center,

1(0, y,x) ~ 2S*x2(-)e~^/^I^(t/2) (7.19)

2 (in the present case y = 1/2 and a(x) = e

''^ ) . Differentiating this expression with respect to x and equating the derivative to zeto to determine

t = —a(x ) , max y max
8.7 DIFFUSE REFLECTION AND TRANSMISSION 411

2 3 4 5

Fig. 52. Line profiles formed in a plane layer with uniformly distributed

sources, 1 — X = 10"^ and g = 0, for three values of the Voigt parameter

a: a = 0; 0.001 and 0.01. Numbers on curves are values of = t^/A.

we arrive at the equation

or, using the well-known properties of Bessel functions,

= 0

The value of t given by the root of this equation is equal to 3.09, so

that "^^^

2 l2.e '''"^^ ~ 3.09 . y

Thus

max

1.13)

\l/2

(7.20)
412 PLANE LAYER OF FINITE THICKNESS

This asymptotic expression increases in accuracy with tg and becomes exact as

TQ In practice it may be safely used for tq > 10. From this expression

it appears, incidentally, that when line broadening is caused by the Doppler

effect, the line width provides a poor basis for determining the optical

thickness of the medium in which the line is formed. The depth of the central

reversal is significantly more sensitive to the value of tq. However, in the case of a Voigt profile, with tq >> 1/a, the situation is reversed. In this

case the separation of the maxima increases in proportion to tq^/^, whereas

the ratio of the intensity at the maxima to that at line center increases

only as ^^1/4 (cf . Sec. 6.5).

The most important physical problem which involves the solution of the

transfer equation with a uniform distribution of primary sources is the calculation of the radiation field and the degree of excitation in a homogeneous,

isothermal layer of gas of two-level atoms. If the upper level is populated

by electron impact and by photo - exci tation arising from the self - radiation

of the gas, and is depopulated by downward radiative transitions and collisions of the second kind, and no absorption or emission in the continuum

occurs (3 = 0) , then

S* = C1-A)B (T) , ^0(7.21)
8.8 LINE FORMATION IN AN ISOTHERMAL ATMOSPHERE 413

X = A

21 e 21

(7.22)

In this case the quantity (1 — X)I(0,l,x)/S , the logarithm of which appears

as the ordinate in Figs. 48-52 , is obviously equal to I (0 *1 , x) /ByQ.(T) . It. is

evident from the figures that for weakly dissipative media the intensity in

all frequencies is much smaller than Planckian. Consequently, at all depths

the degree of excitation is small compared to the value given by the Boltzmann

formula. As the dissipation measure oj increases, the maximum value of the

intensity approaches the Planck intensity and the degree of excitation in the

center of the layer becomes close to the Boltzmann value. For w >> 1 there

is a region at the center of the layer where conditions are close to LTE

(Figs. 54-56, after E. H. Avrett and D. G. Hummer, 1965).

A more detailed discussion of this problem, particularly its physical

aspects, and additional numerical results are given by A. G. Hearn (1962,

1964b) and E. H. Avrett and D. G. Hummer (1965); see also the next two sec- tions .Recently F. B. Fuller and R. F. Warming (1970) considered the radiation

field in a medium with a source distribution of the form Si (t) = t^, n = 0,1,... We note in passing that the main results of this paper are readily

obtained by expanding (7.4), (7.8) — (7.9) and (7.11) — (7.12) in powers of

BASIC EQUATIONS. Let us return to the problem studied in Sec. 7.6, retaining

all the assumptions used in that section except that we now take the 'line

center optical thickness of the layer to be equal to tq , instead of being

infini te

If Planck's function is set equal to unity, the integral equation for"

the line source function S(t) for the problem at hand has the form

1/z

8 8 LINE FORMATION IN AN ISOTHERMAL ATMOSPHERE

(8.1)

0

A where the primary source function S, is equal to

(8.2)

0 The intensity of the emergent radiation is given by

dr

y

y > 0 (8.3)

0
414 PLANE LAYER OF FINITE THICKNESS

log VtF T

Fig. 54. The source_ function in a plane layer of optical thickness

TO = (l/v'TTjlO^ with uniformly distributed sources. Doppler

absorption coefficient. Dotted line — source function for

T„ = and 1 - X = 10-4.

A line appears superimposed on a continuum whose intensity can be obtained

from (8.3) by setting x = °o

-6T„/y

l(0,p,=°) = l(TQ,-y,oo) = 1 - e , y > 0 . (8.4)

We note that Qtq/\i is just the continuum optical thickness o£ the layer mea- sured along the beam.

We wish to find the profile of the line and to determine the source

function, i.e., the depth dependence of the degree of excitation. A great

deal of important information can be obtained directly from an analysis of

the basic equation for the source function, without actually solving it.
8.8 LINE FORMATION IN AN ISOTHERMAL ATMOSPHERE 415

'00

I

o

10^ 10=

DOPPLER PROFILE

VOIGT PROFILE a^O.OI

l-X=IO-^

J \ \ L_

2 3 4 5 log r/A

7 8

Fig. 55. The source function in a plane layer with uniformly distributed

sources for the Doppler and Voigt (a = 0.01) profiles with

6=0, 1 — X = 10-4 for several values of Tj/A, as indicated

on the curves. Source functions are plotted only for 0 < x < Tq/2.

THE SOURCE TERM. We shall start with a consideration of the source term Sj.

The expression (8.2) can be rewritten in the form

sJ(t) = 1-X-b|k2o(t,S) -6|k2o(to-t,6) , (8.5)

or

s*(T) = i-x+e| J

K^^(t,B)dt + 6- f

K^^(t,3)dt (8.6)

It is obvious that S-j^ (and therefore S) is symmetrical relative to t = Tq/2 :

. S*(t) = S*(Tq-t) . (8.7)

Since Kii(t,6) is positive and decreases monotonically with t, the function

S*, as follows from (8.6), reaches its maximum at the center of the layer

(at T = To/2) . It is readily shown that for < «

522-519 O - 74 - 28
416 PLANE LAYER OF FINITE THICKNESS

CO

-I

I -2

o

-3

CD

— . 10

i-X = io-^

1 1

1 1 1 1 1 1 III*

2 -10 12345678

log TTT

Fig. 56. The source function for the same values of the parameters as in

Fig. 55, for the Lorentz profile.

S^(Tq/2) < 2S^(0) . (8.8)

Actually, by virtue of the monotonicity of K^-j^(t,B) we have

T /

K^^(t,3)dt < I K^^(t,6)dt + /K^^(t,3)dt <

/0

To/2

.< 2 / K-^^(t,B)dt < 2 I K^-^Ct,B)dt , 0

(8.9)

which in combination with (8.6) leads to (8.8)

From (8.6) and (8.9) it follows that if

0

1 - A >> XB / K-^^(t,B)dt , (8.10)
8.8 LINE FORMATION IN AN ISOTHERMAL ATMOSPHERE 417

excitation of atoms by the continuum radiation is negligible compared to collisional excitation, and if

1 - A «

I y

K^^(t,6)dt , (8.11)

0

the initial population of the upper level is excited by continuum radiation.

When the optical thickness of a medium is large (xg >> 1) , rather simple sufficient conditions can be obtained for the relation (8.10) to be valid.

Since for Tq < °° J

K^^(t,3)dt < jK^^(t,6)dt = 6(g) , 0 0

the inequality (8.10) will be satisfied if

1 - X >> XB6(6) . (8.12)

We can obtain another sufficient condition if we note that for 6 > 0

Therefore, if

K^^(T,e) < K^^(t,0) e K^^(t) .

1 - A >> XB J

K^-j^(t)dt , (8.13)

0

the inequality (8.10) will also be satisfied. According to (2.6.38), for

sufficiently large values of t,

K^l(T) ~ A^I|lix'(T) , (8.14)

and, consequently, if a line has infinite wings,

J

K^^(t)dt ~ A^i|llx(TQ) .

0

Therefore, when is large enough, the condition (8.13) assumes the form

1 _ X » X6AlIilix(xQ) . (8.15)
418 PLANE LAYER OF FINITE THICKNESS

Specifically, for the most important profiles:

Doppler: 1 - X >> X6

/In Tq (8.15a)

/it Voigt: 1 _ X >> X64 (U(a,0)aTg)-'-/^ , ai^ >> 1 , (8. 15b)

Lorentz: 1 - X >> ^6 4^ "^q

"'^^^

* (8.15c) 7

In the limiting case of a strongly dissipative medium (co >> 1), the condition

(8.12) is identical to (8.10). However, if the dissipation measure is not

large (oj < 1), the inequality (8.12) imposes upon 1 — X more stringent limitations than (8.10). In this case it is preferable to use (8.15) instead of

(8.12). In the limiting case of a weakly dissipative medium (o) << 1) of large

optical thickness, (8.12) is identical to (8.10).

LIMITING CASES. If the inequality (8.10) is satisfied, the situation is

greatly simplified, since the primary source function may be taken as equal

to 1 — X. Thus we recover the problem of calculating the radiation field in

a medium with a uniform distribution of sources, which was studied in the

preceding section. It must be noted that the intensity of the emergent radiation should be found from the expression (8.3), i.e. to the intensity of the

diffuse radiation, found in Sec. 8.7, one must add the intensity of the con- tinuum radiation that escapes from the medium without scattering. The final

expression is

1(0, M,x) = I(TQ,-y,x) = i|(l-X)X(cc)[X(z)-Y(z)]a(x) +

+ Bll-e

'^^^

l{ , M > 0 ,

(8.16)

where

a(x)+3

Until now we have focused our attention on the simplifications that

arise from the domination of one of the mechanisms for the primary population

of the upper level (collisional excitation or excitation from continuum) over

the other. Now we shall study simplifications that arise for other reasons.

If the dissipation measure of the medium is small (to << 1) , then to a good degree of approximation the medium can be regarded as conservative (see

Sec. 8.3). In this case the line source function S(t) can be found from the

following equation instead of (8.1):

S(

}0

t) = J y

K^(|t-t' nS(T')dT' + S*(t) , 03 << 1 , (8.17)

where the primary source function S2(t) is given by (8.2). Further, since

the medium is weakly dissipative, then Tq << 1/B and 1 - X << 1. Therefore
8.8 LINE FOI^IATION IN AN ISOTHERMAL ATMOSPHERE 419

in (8.2) the function AK]_x(|t-t' | ,6) can be replaced by Kii(|t-t' | ,0) e Kii(|t-t'|) (see Sec. 7.1), so that (8.2) assumes the form

S*(t) = 1-X+l J

K^^(t)dt + 1/ K^^(t)dt , to << 1 . (8.18)

0 0

Substituting this expression into (8.17), we find that the source function

S(t) can be rewritten in the form

S(t) = (I-A)S^(t) + BS2(t) , 0) << 1 , (8.19)

where S-|^(t) and S2(t) are the solutions of the equations

^0

= I J* K^(|t-T' |)S^(T')dT' + 1 , 0

(8.20)

S2(0 = y /k^(|t-t' |)S2(T')dT' + I J

K^^(t)dt + I f

K^^(t)dt . (8.21)

It is very important that Si(t) and S2(t) depend only on t (and xg), and not

on X and 6. Thus when the medium is weakly dissipative, the dependence of

the source function on the parameters X and 3 is found to be extremely simple.

If, in addition to the condition oj << 1, either the inequality (8.15) or

its opposite is satisfied, further simplifications arise. In the first of

these cases one may set

S(t) = (I-X)S^(t) , (8.22)

and in the second

S(t) = aS^d) . (8.23)

It should also be noted that when tq is large the ratio Si(t)/S2(t) will depend very weakly on x since the free term of equation (8.21) varies in this

case within narrow limits.

Now let us discard the assumption that the dissipation measure w of the

medium is small. As before, we shall consider the dissipation per scattering

1 — X to be small, and co to be arbitrary. In this case, too, it is found that

for XQ >> 1 we can obtain important information about the depth- dependence of

the source function without solving the equation for S(x). As was shoAvn

above, the primary source function S*(x) has its maximum at x = xo/2, so that

for X << xq
420 PLANE LAYER OF FINITE THICKNESS

T

J.0 f

sJ(T)dT << J

sJ(T)dT

0

The considerations of Sec. 7,7 therefore make it possible to assert that when

T is much smaller than the thickness of the boundary layer T\y (and if tq < n^,

then when t is significantly smaller than tq/2), the source function S(t)

should be asymptotically proportional to the solution of the generalized Milne

probl em S

(

t) , i.e.

S(t) = S(Tq-t) - S(0)S(t)

^ (8.24)

for

T << min

The fact that the source function increases in proportion to S(t) from

the boundary down to rather large depths allows us to draw a useful qualitative conclusion about the form of the central parts of the line. That is to

say, it can be asserted that in the central region of the line the intensity

should depend on frequency in approximately the same way as in the generalized

Milne problem, i.e. the intensity should increase with the distance from the

line center.

NUMERICAL DATA. Let us proceed to a discussion of the available numerical

data. All of this material is from the work of D. G. Hummer (1968) and was obtained from a numerical solution of the transfer equation by the method of

discrete ordinates. The depth dependence of the line source function S(t)

for media with t o = fT— 10 ^ and to = ''t— 1/2 10 4 is shown in Fig. 57. We recall

that we have set Bvf,(T) = 1 for simplicity. These results are for the

Doppler profile and 1 — X = 10—6. por all values of g shown on the curves

for the case of To = 'n—1/2 10 2 , the dissipation measure w of the medium is small

and decreases as B decreases. The curves for To = 'n—l/2lo4 show the transition

from weakly dissipative (3 ~ /tt 10—5) to strongly dissipative (3 ~ /ir 10—4)

media.

—1/2 2 For Tq = -n '10 the continuum optical thickness Btq of the medium is

small for all values of 3 appearing in Fig. 57. Nevertheless the continuum

exerts a tremendous influence on the line source function because when 1 — A is small the primary excitation by continuum radiation dominates the electronimpact excitation (curves with 3 ~ /tt 10—6). has been explained above,

in this case the line source function is given by (8.23). It should be proportional to 3, i.e. the shape of the graph of S(t) should not depend on 3.

These conclusions are well illustrated by the curves corresponding to 3 =

/rf 10—5 ^ ^ 10—4 (for 3 = /rT 10—3 dissipative processes begin to be

felt, and for 3 = /rf 10—6 the role of electron impact in primary excitation

becomes significant). The similarity in the shape of the curves for 3=0

and 3 = /tt 10—5 is in accordance with the approximately constant value of the

ratio S^(t)/S2(t) discussed above.

—1/2 4 A consideration of the three lower curves for tq = tt 10 once again

illustrates that the dependence of S(t)/S(0) upon X and 3 is very weak for

media with co << 1. When the dissipation measure is large the source function

S(t) for T < tq/2 coincides almost exactly with the corresponding source func-
8.8 LINE FORMATION IN AN ISOTHERMAL ATMOSPHERE 421

0

o -3

-4

AA B

-10^3^/ / ///'

/

— 1 u

J 1 1 1 1 -2-1 0 I -2-1 0 I 2 3 log VtF T

Fig. 57. The source function in a homogeneous isothermal layer with a Doppler absorption coefficient, t /ri" = 10^ and 10^ and

1 - X = 10-6. 0

tion for a semi-infinite medium (in Fig. 44, p. 341, and Fig. 57 the curves

for B = /tt 10—3 ape indistinguishable).

In Table 42 values of S(t)/S(0) are given for tt-^^^tq = 10^ and 10"^, with

several values of 6 and 1 — X = 10—6 (Doppler profile). A comparison of

Table 36 (p. 340) and Table 42 shows that near the boundary the source function is proportional to the solution of the generalized Milne problem, in

agreement with the conclusion reached earlier.

The intensity of the radiation in the direction y = 1 , corresponding to

each source function shown in Fig. 57, is given in Fig. 58. The line profiles are complicated, and in many cases the distinction between an absorption and an emission line is meaningless. The equivalent widths of such

lines have very little physical significance.

The depth dependence of the source function for a Voigt profile, with

several values of tq and 6, is shown in Figs. 59-61. When the Voigt parameter

a is small and tq << 1/a, we expect that the source function will differ

little from the corresponding source function for a = 0, i.e. for the Doppler

profile (see end of Sec. 8.6). The closeness of the curves for a = 0 and

a = 10~3 in Fig. 59 shows that even for aiQ of order unity, the deviation of

the absorption coefficient from the Doppler form has little effect. However,

when aT„ is large the form of the profile has a significant effect on the
422 PLANE LAYER OF FINITE THICKNESS

TABLE 42

THE FUNCTION S(t)/S(0) IN A HOMOGENEOUS ISOTHERMAL LAYER

(DOPPLER PROFILE, 1-A=10"^)

T /ir t/tt

0

0 10

- 7 10

-6

10

-5

10

- 4 10"

1 1

67 1 . 68 1 . 70 1 . 71 1 . 71 1.71

10 4. 54 4. 63 4 . 78 4. 83 4 . 83 4 . 76

50 8. 12 8 . 33 8. 71 8. 83 8 . 80 8.37

10^ 1 1. 68 1 . 68 1 . 69 1 . 69 1

69 1 . 70

10 4, 81 4 . 83 4 . 87 4. 88 4. 87 4 . 79

100 16 . 9 17. 1 17 . 4 17. 4 16 . 7 12 . 8 500 30 . 3 30 . 8 31

5 314 28. 1 16 . 0

A B

0 I 2340 I 234

X

Fig. 58. The intensity of normally emergent radiation from a homogeneous

isothermal layer (Tq>^ = 10^ and 104, 1 — A = lO"^, Doppler

profi le)
8.9 MEAN NUMBER OF SCATTERINGS 423

o

^

A/3 =10"

.41-2

/A= 10-

= 10"

a =0_

a = 1 0_ a^lO

0 I 2 log r/A

Fig. 59. The source function in a homogeneous isothermal layer with

tq/A = LO-^, 1 — X = 10-6, fQj. various values of the Voigt

parameter a.

depth dependence of the source function. A detailed comparison and discussion of the features of the curves shown in Fig. 45 (p. 342) and Figs. 59-61

has been given by D. G. Hummer (1968). The most outstanding feature of these

curves is the fact that as a increases, the values of S(t) near the boundary

increase for strongly dissipative media (see Fig. 45, p. 342) and decrease

for weakly dissipative media (Fig. 59). The intersection of the curves in

Figs. 60 and 61 indicates the transition between these limiting cases.

8.9 MEAN NUMBER OF SCATTERINGS

BASIC RELATIONS. The mean number N of scatterings experienced by photons of

the diffuse radiation field, for a given primary source function S^(t) with

the normalization

/S-^(T)dT = 1 (9.1)
424 PLANE LAYER OF FINITE THICKNESS

-2-10 1 2 3 4

log r/A

Fig. 60. The source function in a homogeneous isothermal layer with

tq/A = 104, 1 _ A = 10"6 for various values of the Voigt

parameter a.

is obviously equal to (for details, see Sec. 6.8)

^0 ^ ^ y S(T)dT . (9.2)

0 This expression (and all the following analysis) is valid for arbitrary 3^0.

We shall consider the primary emission of a line photon as its first scattering.

At depth T = T, a plane isotropic source is emitting in line frequencies.

Then

I

S*(T) = 6(T-T^) ,

and the source function S(t) is equal to the Green's function:
8.9 MEAN NUMBER OF SCATTERINGS 425

-2 -10 1 2 3 4 5 6

log t/A

Fig, 61. The source function in a homogeneous isothermal layer with

tq/A = 10^, 1 — X = 10"^, for various values of the Voigt

parameter a.

S(t) = r(T,T^;TQ) + 6(t-t-^) . (9.3)

Denoting the mean number of scatterings in this case as 17(1 -t^), we have

from (9.2) and (9.3)

N(t^;tq) = 1 + y

r(T,T^;TQ)dT (9.4)

Setting z = oo in (2.2), we find

N(t^;Tq) = S(t^,oo;Tq) .

' (9.5)
426 PLANE LAYER OF FINITE THICKNESS

Therefore according to (1.17) N(t;to) satisfies the equation

^0

N(t;Tq) = I

J'k^CIt-t- I ,B)NCt' ;TQ)dT' + 1 , (9.6)

i.e., is equal to the source function in a medium in which Si = 1. Its explicit expression in terms of X(«>) and the resolvent function $(t;to) follows

from (9.5) and (1.22) :

N(t ;Tq) = X(oo;Tg)h + f

$(t;TQ)dt - J

<I'(TQ-t;TQ)dtj . (9.7)

^ When the source distribution is described by the primary source function

S^(t), then

^0

S(t) = S*(t) + J

sJ(T^)r(T,T^;TQ)dT^ , (9.8)

0

* and if we assume that S, (t) is normalized according to (9.1), then (9.2) and

(9.4) give

f

N sJ(T)N(T;TQ)dT . (9.9) =

0

Thus, if equation (9.6) is solved, i.e. if the source function is known for a medium with a uniform source distribution, the mean number of scatterings

for any source distribution can be found by simple integration. In this con- nection it is worth recalling that numerical solutions of (9.6) have been

published by several authors (for references, see Sec. 6.7).

From (9.9) it follows that whatever the dis^tribution of sources, the mean number of scatterings cannot be less than N(0,Tq), nor can it exceed

N(Tq/2; Tq), i.e.

N(0;Tq) < N < N^Tq/2; Tq^ , (9.10)

or, if (9.5) and (2.55) are used,

X(oo;Tq) < N < x2(»;Tq) . (9.11)

The more strongly the sources are concentrated toward the center of the layer,

the closer N is to N(tq/2;tq). However, even for a uniform distribution of sources the mean number of scatterings differs from N(tq/2;tq) by less than a factor of two (V. V. Sobolev, 1967b). Moreover, it is only when the sources
8.9 MEAN NUMBER OF SCATTERINGS 427

are much weaker deep within the medium than near the boundary that the mean number of scatterings approaches its minimum value o£ N(0;Tg) = X(o°;tq).

Incidentally, the relation N(0;Tq) = X(oo;Tq) elucidates the physical significance o f X (o°) .

We note that for 1 — X << 1 and Tq >> 1

N(T;Tg) = N(Tq-t;Tq) - X(oo;tq)S(t) , t << min(T^,TQ/2) . (9.12)

The large-scale behavior of N(t;to) in the conservative case (physically,

for weakly dissipative optically thick media) is

T ^^xY(l-x)^

N(xTr,;T^) , w << 1 , >> 1 . (9.13)

0 0 r(2Y + l)Kl/To) '

' 0

This expression is valid for distances from the boundaries that are large

compared to the mean free path of a line center photon. It is readily obtained from (9.7) with the help of (5.56).

Comparison of (9.12) and (9.13) suggests an asymptotic expression for

the mean number of scatterings that is valid for an arbitrary x, 0 < x - Tq , provided that Xq >> 1 and co << 1:

2 N(x;x.) ~ lLIXls(x)S(x -X) . (9.14)

" 2r(2Y) ^

This simple expression, which was found by A. P. Napartovich (1969), properly

describes both the boundary regime_(9 . 12 ) and the large-scale behavior (9.13)

of the mean number of scatterings N(x;xq).

UNIFORMLY DISTRIBUTED SOURCES. For a medium with uniformly distributed

sources (S* = const.), the mean number of scatterings, which we shall henceforth denote as Ng , can be expressed in terms of X(oo) . From (9.9) and (9.5) we have

^0

Nq = Js(x,oo;xQ)dx . (9.15)

° 0 Setting z = 0° in (2 . 7) , we find

^0

J*e"^/^S(x,<»;Xg)dx = X(oo ; x^ ) [X ( z ; x^ )-Y ( z ; Xq) ] z . (9.16)

0 letting z go to infinity and using (2.33) and (2.34), we get

J

S(x,-;xQ)dx = X(-;Xq) [y^(xQ)-x^(xQ)] , (9.17)
428 PLANE LAYER OF FINITE THICKNESS

so that

X(«';t ) Nq = —[y^(T^)-x^iTQ)] . (9.18)

"^0

^0

Finally, using (2.42) we have

Nq = ^ y X^=°;T)dT . (9.19)

° 0

We note in passing that by combining (9.19) with (2.26') we obtain an alternative representation of Ng in terms of the X- and Y-functions. We shall not

reproduce it here since it is not used below.

Let us discuss the most important special cases in a little more detail,

beginning with monochromatic scattering in an optically thick layer.

In Sec. 8.4 it was shown that '

-1/2 ^'^^O'^^^e^ Xj^^(co;tq) = (1-X) ^/"th ^y—^ + ••• (9-20)

Hence, having rewritten (9.19) in the form

N,^ = ^(1_X) ^ J

th^ \

dx ^ J

(x^(oo;,) __l_th2-_^jdx

for >> 1 we get

N ^

where

(l_A)-l|^l - J- ^thll^l^^^Iil-th kTg - |(1-X)Q^ + ...j , (9.21)

n 2 12 k(T+2T )\

0

-1.

The neglected terms in square brackets in (9.21) are o(to ) as tq ~. In

particular, for kig >> 1 and 1 — A << 1

Nq^ = (1-A)~^fl - ^ + ...1 , (9.23)

and for kXp -> 0

2 Nq^ = ^+|q(-)x + 3q2(-)+l22[a3q(co)_aJ ^ ... , (9.24)
8.9 MEAN NUMBER OF SCATTERINGS 429

where the a- are the moments of the conservative Ambartsumian H- function:

and q(<») = 0.710446 ... is Hopf's constant.

The curves of Ng^ versus tq are shown i_n Fig. 62 for several values of

X. For weakly dissipative media (o) << 1) , Nq grows with tq in approximately

the same way as for the conservative medium. For larger values of tq, when

to becomes of t_he order of unity, saturation sets in, and for strongly dissipative media Nq is close to its maximum possible value (1 — X)—1. The general features of the behavior of Ng discussed in the preceding

paragraph are clearly valid for an arbitrary_line profile. Although the

general asymptotic (tq >> 1) expression for Ng , valid for an arbitrary dissipation measure and an arbitrary line profile, is unknown, its limiting forms

for strongly and weakly dissipative media are easily obtained.

Let us first consider strongly dissipative media (w >> 1). We shall

assume that continuous absorption is negligible (B = 0). From (9.19) and

(5.85) we get

Fig, 62. Mean number of scatterings in a layer with uniformly distributed

sources (monochromatic scattering).
430 PLANE LAYER OF FINITE THICKNESS

r

'0

-1 1 _^ J_ /"k + Od/Tp) . (9.25)

I£ a line has infinitely extended wings, the integral appearing in this formula diverges as Tq <». Since, according to (2.6.40),

^0

j'K2(t)dT ~ 2AnilJx(To) , 0

(9.25) can be rewritten in this case as

,-1 Nq = (1-X) ^ZaUIiI + 0(l/j ) . (9.26)

-A 2y+1 ^0 J

0

1 -

1-A 2y

Of much greater interest is the case of a weakly dissipative optically

thick medium. From (9.19) and (2.60) (or directly from (9.13)) it is readily

shown that

where

r = . (9.28)

0 24y-' ^ ^

' + 2^2^2Y + 3 j

Specifically, for the most important line profiles:

Milne : N^^ ~ ^ ^ ^ ' (9.27a)

Doppler: Nq° ~ —Iq/ItTT^ , (9.27b)

Lorentz: n/ ~ ^TT^^o'^' • 2 TT For the Milne (rectangular) and Doppler profiles, the asymptotic forms (9.27a)

and (9.27b) can be refined. The refined form of (9.27a) is given by (9.24).

For the Doppler profile it may be shown (V. V. Ivanov, 1970b) that

N„°^ (./4,l/^„(.„x„,l/2[l 0((.nx„)-^)] -

(9.27e)

TT-*-/^ rn ^l/2ri 0.40454 ^ ,-2^ = - T^iln Tq)

1^1 - - 0((£n Tq) )
8.9 MEAN NUMBER OF SCATTERINGS 431

Estimates for the higher-order terms in [9.27c) and (9.27d) are unknown.

Table 43 gives values of Nq found by E. H. Avrett and D. G. Hummer (1965)

from a numerical solution of the equation for the source function for 3=0

and Si = const, using (9.2). Values of Nq for the limiting cases tu << 1 and

0) >> 1 agree well with those obtained from the expressions derived above.

HOMOGENEOUS SPHERE WITH UNIFORMLY DISTRIBUTED SOURCES. Let us now find the

m.ean number of scatterings of a photon in a homogeneous sphere with uniformly

distributed sources (V. V. Ivanov, 1969). If the optical diameter of the

sphere is tq, the line source function is the solution of the equation

S(T) =

4tt

exp(-6|T-T' |)M2(|t-t' I) S(T')dT' + 1 (9.29)

T—

in which integration extends over all t_' with J_x' mean number of scatterings will be denoted by Nq

^0

To/2

24

T 3

0 0/

S

(

t) T dx

< Tq/2. The corresponding

C lear ly

,

(9, 30)

We set, for 0 < t < Tq/2,

TABLE 43

MEAN NUMBER N^ OF SCATTERINGS FOR A UNIFORM DISTRIBUTION OF SOURCES AND B = 0 log(l-X)

a

- 4 -6 - 8

0 2 8 20 + 1 8 28 + 1 9 09 + 1

4 5 52 + 3 1 34 + 4 1 36 + 4

6 9 88 + 3 6 10 + 5 1 71 + 6

8 1 00 + 4 9 90 + 5 6 45 + 7

0. 001 2 7 95 + 1 8 05 + 1

4 2 71 + 3 3 77 + 3

6 7 22 + 3 2 60 + 4

8 9 60 + 3 2 15 + 5

0.01 2 6 51 + 1 6 56 + 1

4 8 27 + 2 9 03 + 2

6 4 51 + 3 8 30 + 3

8 8 87 + 3 7 71 + 4

00 2 8 14 —- 4 8 31 + 1

6 7 72 + 2

8 4 52 + 3

522-519 O - 74 - 29
432 PLANE LAYER OF FINITE THICKNESS

S^(t) = (T-TQ/2)S(T-Tg/2) (9.31)

and extend Sj^(t) to Tq/2 < t < Tq in such a way that

S^Ctq-t) = -S^(t) . (9.32)

It is readily shown (see Sec. 2.4) that S^(t) is the solution o£ the equation

S;^(T) = I

/ K^(|t-t' I ,S)S^(T')dT' + t-Tq/2 . (9.33)

Using (9.30) — (9.32), one can show that

% = J

S^(T)TdT , (9.34)

^0 0

or, if use is made of (9.33),

_*

0 '0 0 where s^(t), i = 0,1, are the solutions of the equations

s.(t) = ^ J

K^(|t-t' I ,B)s. (T')dT' + , i = 0,1 . 0

(9.36)

The integrals appearing in (9.35) can be expressed in terms of the standard functions of radiative transport theory. Indeed, we have

2 9 Sq(t) = S(t,";Tq) ; s^(t) = limz — S(t,z;Tq) , (9.37)

where S(T,z;Tg) is the usual auxiliary function defined by (1.17). Proceeding from either of the equations (2.7) and using (9.37) and (2.33) — (2.34),

after some manipulation we obtain —

^0

=

^ :r3[^1^^0^y2^'^0^-^l^'^0^^2f^0^ - i^f~'^0^(y3^^0^-^3^^0)) "

-|TQX(cc;TQ)(y2(Tg)-X2(To))j .

(9.38)

This expression for N„ is valid for arbitrary t^. A, and 3 and an arbitrary

line profile.
8.9 MEAN NUMBER OF SCATTERINGS 433

In the conservative case we find from (9.38), using (2.60) and (5.8) —

(5.10), that

A

^0 ~ iTirn^o'" > - = 0 « -0 ~ ' ^9.39) K1/Tq)

where

C* = 3^2 ^'^ ^(2y + 3) "(^-V^) . (9.40)

Comparison of (9.39) and (9.27) shows that the asymptotic value of the mean number of scatterings for a homogeneous sphere of optical diameter tq >> 1 and for a plane layer of thickness tq has the same functional^ dependence on

Tg. As one would expect, the difference in geometry causes Ng* to differ

from No by a constant factor of order of unity, namely.

In the particular case of the Doppler profile the asymptotic form (9.39)

can be refined. It can be shown (V. V. Ivanov, 1970b) that in the conservative case for Tq >> 1

* 3 1/2

^0 = 16 " ^0

r * >l/2 -, 3+8iln2-4Y ^ r^fm ^-2\

(InTf.) 11 i— + 0 ((£n Tf.) ) L 8£n Tq ^

.

=

3 1/2/ XQ(£nTQ)^^/^L 0. 77954 ^ nfror. -r ^'h

^1 - . 0((to Tq) )

(9.42)

The derivation of this expansion is omitted.

CONCLUDING REMARKS. In a similar way it is possible to estimate the mean number of scatterings in other cases as well. Usually the exact value of

N is of less interest than an estimate of its order of magnitude. Sometimes

one can obtain this estimate without determining the source function, as was

done for a medium with uniformly distributed sources. As an example, we shall

estimate N in the problem considered in Sec. 8.8. As has already been noted,

the primary source function S]^*(t) in this case has its maximum at t = to/2

and its minimum at t = 0. Consequently, from (9.9) we obtain the following

rigorous estimates for N:

S*(0)

0)dT

or

* 5^(0)

<
434 PLANE LAY'ER OF FINITE THICKNESS

Since SjCt) varies within narrow limits, this inequality imposes rather narrow limits on N/Nq . We note that the estimates o£ N for this problem obtained by D. G. Hummer (1968) are incorrect, since his original expression for N

in media with 3 / 0 is erroneous. Estimates of the mean number of scatterings have been discussed by V. A.

Ambartsumian (1948, 1960), V. V. Sobolev (1966, 1967b, c), D. G. Hummer (1964),

D. E. Osterbrock (1962), E. R. Capriotti (1965) and others to whom we direct

the reader for further information.

ENERGY BALANCE IN A MEDIUM. Energy released within a medium in line frequencies is either absorbed or escapes. For media with negligible continuous

absorption (3 = 0) the relative importance of the two processes can be estimated when the mean number of scatterings is known. When 3=0, the probability 1 — X of photon destruction times the mean number of scatterings N

gives the fraction of the energy expended in primary atomic excit^ation that

is reabsorbed in multiple scatterings. The quantity 1 — (1 — A)N is the mean

escape probability, i.e. the fraction of the released energy that escapes

from the medium.

For media with 3 7^ 0 the situation is different, since photons are destroyed both in scattering processes and in fl_ight. Hence, to obtain the mean es£ape probability one has to know both N and the mean photon path

length T. Since the probability of photon destruction per unit optical path

is 3, the total probability of photon dest_ruction in flight is 3T, and the mean escape probability is 1 — (1—A)N — 3T.

BASIC RELATIONS. To find T we can proceed as follows. In the stationary

case the ratio of the total radiant energy present in the medium to the energy released by primary sources per unit time is clearly the mean time that

the photon spe_nds in flight. Multiplying this ratio by the velocity of light

c, we obtain T. Since T depends only on the distribution of primary sources, and not on their absolute strength, without loss of generality we can assume that

Let us denote by p(t) the radiation density integrated over all line frequencies at depth t. From the argument of the preceding paragraph we have

8.10 MEAN LENGTH OF A PHOTON PATH

1 (10.1)

0

T

(10.2)

0

or, using (1.50),

(10.3)

0 0
8.10 MEAN LENGTH OF A PHOTON PATH 435

Therefore, i£ the source function is known, one can obtain T by simple integration. One can also express T in terms of the intensities of the emergent radiation

T = 1/2 J[2N - (l(0,z')H-I(TQ,-z')z')]GQ(c')dz' , (10.4)

0

where N is the mean number of photon scatterings and I(t,z) is related to the

intensity of radiation I(t,p,x) by (1.37). We omit the derivation of this

express i on

It can be shown that to find T in a layer with given optical properties

and an arbitrary source distribution it suffices to solve the basi^c integral

equation (1.1) with a particular source term S2(t). Indeed, let T(t-|^;tq) be

the mean path length of a photon in a layer with the source at depth t = t-^. The source term in this case is S]^(t) = 6 (t— T]^) , where 6 is the Dirac deltafunction, and the source function is

S(t) = 6(t-t^) + r(T,T-^) , (10.5)

where r(T,T]^) is the resolvent of the basic integral equation. The mean

length of a photon path in a layer with an arbitrary source distribution nor- malized according to (10.1) is obviously

_ ? . _

T = J

S^*(T)T(T;TQ)dT . (10.6)

0

Hence to find T for an arbitrary source distribution it_ is sufficient to obtain T(t;Tq). Let us derive the integral equation for T(t;Tq).

We proceed from the identity

y* K^^(|t-t' I .6)dT = y*(2-e-^'/"'-e ^

""^^ )GQ(i;')dz' (10.7)

0 0

which is easily verified with the help of (7.1.13). Let us consider, further,

the usual auxiliary function S(T,z;Tg), which is the solution of the equation

S(t,z;To) = ^ y

K^(|t-t' I ,3)S(T' ,z;TQ)dT' + e . (10.8)

0

It is related to the resolvent V by

lo

S(t,z;Tq) = e--^/^ + j

e"^ ' / (t , x ' ) dx ' (10.9)
PLANE LAYER OF FINITE THICKNESS

Substituting (10.7) and (10.5) into (10.3) and using (10.9), we get

T(t;tq) = I y* [2S(t,~;Tq)-S(t,z' ;Tq)-S(tq-t,z' ;Tq)]Gq(^ ')dz' • (10.10)

0

Comparing (10.10) and (10.8), using the superpos_ition principle and recalling

(7.1.13) we arrive at the desired equation for T(t;tq):

T(t;Tq) Kj(It-t' I ,6)T(T' ;TQ)dT' +T*(t;Tq) , (10.11)

where

^.0 lo-"^

T*(t;Tq) = K^^(t' ,3)dT' + K^^(T',6)dT' . (10.12)

0 0

A is Setting A = 0 in (10.11) we find T(t;to) = T (t-.tq); thus T is the mean optical path length of the first flight of a photon born at a depth t. If

(10.11) is solved, we can use (10.6) to obtain T for a given source distribution Sj(t). There is no need to find in advance the source function corre- sponding to S*(t).

ESTIMATES AND ASYMPTOTIC_FORMS . Equation (10.11) can be used to impose close

bounds on T(T;Tg). Let N(t;tq) be the mean number_of scatterings of a photon

born at a depth t. In Sec. 8.9 it was shown that N(t;to) satisfies the equation

N(t;Tq) = I

/ K^(|t-t' I ,B)N(T' ;TQ)dT' + 1 . (10.13)

0 _*

Let us compare (10.12) and (10.13). According to (10.12), T (t;Tq) varies

within narrow limits:

^.0 lo

i J

K^^(t' ,3)dT' < T*(t;Tq) < f

3)dT' T (t;tJ < / K^^(T',6)dT' . (10.14)

Hence

,

T T(t;Tq) = t(T;TQ)N(T;TQ) J

K^^(T',6)dT' , (10.15)

where

1/2 < t(T;TQ) < 1 . (10.16)
•1

8.10 MEAN LENGTH OF A PHOTON PATH 437

Therefore_j_ knowledge o£ N(t;Tq) enables one to obtain rather accurate estimates of T(t ; Tq) • For a layer with an arbitrary source distribution we obtain from (10.6),

(9.9), and (10.15)

T = Nt / K^^(t' ,6)dT'

0

(10 .17)

where

1/2 < t < 1 (10. 18)

We note in passing that for monochromatic scattering with 3=0

XT(t;tJ = N(t;tJ - 1 , (10 . 19)

which follows from (10.11) and (9.6). Hence, for an arbitrary source distri'

bution

AT = N - 1 . (10 . 20)

The physical significance of this result is that for monochromatic scattering, the mean optical path length per scattering is unity.

Let us consider in a little more detail the case of a layer with uniformly distributed sources. The corresponding mean photon path length will_be

denoted by Tg . Combining (10.4) with (7.17) and (9.19) we can express Tg in

terms of X- and Y- functions, namely

Tg = (1/

0 r

J

x2(=o;T)dT-X(oo;TQ)[x(z' ;Tq)-Y(z' ;Tg)]z'

0

Gg(c')dz' .(10.21)

This expression can be used to find the asymptotic form of Tg for large tq

in the conservative case. Vie quote the result, omitting the proof. It can

be shown that if a line has infinitely extended wings,

0 (1-Y) (2Y-M) ^^^0^

4y ^'(^0^

where y is the characteristic exponent.

In particular, for the Doppler profile

7?T D

, 0) = 0 , Tp, (10.22)

0 (10. 22a)
438 PLANE LAYER OF FINITE THICKNESS

while for the Voigt and Lorentz profiles

t/ ~ T^ ~ C9/4)Tq (10.22b)

For 0 < Y < 1/2 equation (10.22) can be rewritten in the form

. ^ (1-Y)(2y-^1) ^

0 4y(1-2y) 0 (10 .23)

The functional dependence of Tq on tq is in this case universal. The fact

that Tg grows proportionally to tq implies that for y < 1/2, the zigzag

nature of the trajectories can be ignored. This conclusion was already

reached in Sec. 7.3 from other considerations; cf. also Sec. 4.5. According

to (10.22a), for the Doppler profile (y = 1/2), Tq increases slightly faster

than TQ which means that the zigzags become important. For conservative

monochromatic scattering (10.20) and (9.27a) give

M

^0 ToV4 , (10.22c)

so that essentially in this case the displacement of a photon is proportional

to the square root of its path length. This is the well-known result of the

usual random-walk theory.

Finally, we mention that the function

S(t) = (1-X)N(t;Tq) + X3T(t;Tq)

satisfies the basic integral equation with the source term

S^*(t) = 1-^ + 47 K^^(|t-t' I ,3)dT' .

The value of 1 — S(t) is evidently the mean escape probability from depth t. At the same time, as we have seen in Sec. 8.8, S(t) is the source function

for a homogeneous isothermal layer. Hence the values of the source function

for this particular problem given in Sec. 8.8 can be used to discuss the

energy balance in a layer with an arbitrary source distribution.

This section is based on the author's paper (V. V. Ivanov, 1970c) where

further details can be found.

' 8.11 APPROXIMATE SOLUTION OF THE BASIC EQUATION

BASIC FORMULA. In this section an approximate expression is given for the

resolvent function $(t;to) of the basic integral equation for the line source function in a layer with negligible continuous absorption (3 = 0)

(V. V. Ivanov, 1972). This approximation is a natural outgrowth of the above

study of the asymptotic solutions of the integral equation for the line source

function. The approximation is sufficiently accurate for use in practically

all the physical applications of the theory.
8.11 APPROXIMATE SOLUTION OF THE BASIC EQUATION 439

The resolvent function <I)(t;t^) is the solution of the equation

.0 T,

0 where

Ct;tq) = A y k^(|t-t' hnx' ;TQ)dT' + |k^Ct) , (11.1)

00 K^(t) = ^ J

Cx)E^(a(x) T)dx . (11.2)

00

We postulate the following approximate form for 'I'(t;Tq):

(A/2)K^(t) [I-A+AK2 (tq) ]^^^

*.(t;t„) = — , (11.3)

[1-A+AK2(t) ]3/2[i_a+xK2(Tq-t) ]l/2

where

00 K2(t) = j

K^(T)dT . (11.4)

T

As a rule, the accuracy of this approximation increases as the value of the

characteristic exponent decreases.

This approximation is based on the following considerations, Yu. Yu.

Abramov, A. M. Dykhne , and A. P. Napartovich (l-969a, 1969b) have found (see

also Sec. 8.5) for the limiting cases of weakly dissipative (1 — X << K2(tq))

and strongly dissipative (1 — A >> K2(to)) media of large optical thickness

the asymptotic relation

T(T -t)

$ At\x^) = $(t) , (11.5)

where $(t) e $(t,°°) is the half-space resolvent function, and

T

H'(t) = 1 + J

$(t)dt . . (11.6)

It may be expected that the right side of (11.5) will give rather a good

approximation to $(t;to) for a layer of arbitrary thickness tq with an arbitrary dissipation measure. To obtain (11.3) it suffices to substitute into

the right side of (11.5) the approximate expression for <I>(t) derived in Sec.

5.5

(A/2)K (t)

$.(T) = 3/2 • [I-X+AK2 (t) ]

t
440 PLANE LAYER OF FINITE THICKNESS

Let us consider some limiting forms of (11.3). Setting t = Tq , we get

(X/2)K^(tq)

'a^^O'-^O^ = l-X.XK^(Xo) •

Hence, for 1 — A >> K^dg) (strongly dissipative medium)

(X/2)K (t.)

*3(to;to) ^-i , (11.9)

which coincides with the exact asymptotic form (5.74) of $(TQ;Tg) for an optically thick, strongly dissipative medium. In the opposite limiting case of

1 - X << (tq) , (11.8) gives

When TQ is large, we have Ki(to) ~ (2y/to)K2 (tq) (see Sec. 2.6), and (11.10)

gives for $(tq;tq) the asymptotically exact value of y/tq (cf. (5.2)).

It follows directly from the integral equation (11.1) for the resolvent

function, that as tq ^ °° the function $(t;to) diverges as (X/2)K^(t). The

approximation (11.3) has the same property.

We also note that substituting (11.8) into the exact equation

dX(co;TQ)

= X(~;TQ)$(Tg;TQ) , (11.11)

found in Sec. 8.2 (cf. (2.40)), leads to the approximation

Xg(~;TQ) = [1-X+XK2(Tq) ]~1/^ . (11.12)

This approximation was obtained in Sec. 8.2 from different considerations.

UNIFORMLY DISTRIBUTED SOURCES. Let us use the approximate form (11.3) to find

an approximate solution for the line source function in a plane layer with

uniformly distributed sources. Let us set S* = 1. Then the corresponding

s_ource function S(t) is numerically equal to the mean number of scatterings

N(t;Tq) of a photon born at depth t (see Sec. 8.9) and satisfies the equation

N(t;tq) = I JK^(|t-t' |)N(T' ;TQ)dT' + 1 . (11.13)

0

As we have seen in Sec. 8.9, N(t;to) can be expressed in terms of X(°°,to) and

the resolvent function $(t;t^) as

N(t;Tq) = X(oo;tq)M + f

$(t;TQ)dt - ^ $ (T^-t ;Tq) dtj . (11.14)

j V n n

'
8.11 APPROXIMATE SOLUTION OF THE BASIC EQUATION 441

To obtain an approximate expression for NCt;!^) let us consider the

integral ^

'a(t;To)dt

0 with ^At;t^) given by (11.3). Integrating by parts, we obtain

T '3(t;TQ)dt = (11.15)

/ i_x + AK2(Tq) t

I — J + / $^ (T^-t ; T^) dt \[1-X + XK2(t) ] [l-X + AK.dn-T) ] / / ^ °

°

Substituting (11.15) into (11.14) and replacing X(oo;t„) by its approximate

form (11.12), we finally get

r T-l/^r -,-1/2

N^(t;Tq) = |^1-A + AK2(t)J |^1-A + AK2 (Tq-t)J . (11.16)

This approximate expression has qualities that seldom appear together: simplicity, high accuracy and a wide range of validity. We note that the

function K2(t) appearing in this expression may be regarded as known (see

Sec. 2.6 and 2.7).

The accuracy of the approximate form (11.16) can be slightly improved

by replacing the K2-functions by l/S^ (cf. Sec. 6.3), where S(t) is the

solution of the corresponding generalized Milne problem. The disadvantage

of the resulting approximation

S(t)S(t -T)

^a = ^2 T72 ^2 T7T (11.17)

[l+(l-A)SnT) ]-^/Ml+(l-X)S^(Tg-T)

is that it can be used in practice only when the solution of the corresponding

Milne problem has been tabulated.

Let us discuss the accuracy of the approximation (11.16). It can be

shown that for A = 1 and -> °°

TT^(to/2;to) NJO;Tq) \

2ijT(2y)/

From these and other analytical considerations one may expect that in cases

of practical interest the error of the approximation (11.16) does not exceed

a factor of 2. This estimate is in perfect agreement with the direct numerical tests of the accuracy of (11.16). Relevant data made available by D. G. Hummer are lis_ted in Tables 44 and 45 (the third digits are unce rtain)_j^ In

these tables N(t;to) is the numerically exact solution of (11.13) and N^(t;tq)

is the approximation (11.16). We can conclude that, in general, the accuracy

of the approximate form (11.16) is similar to that of the basic assumption of

complete frequency redistribution.
442 PLANE LAYER OF FINITE THICKNESS

TABLE 44

ACCURACY OF THE APPROXIMATION (11.16) (DOPPLER PROFILE, 1-X=10"^)

/ttt /tt'TTT N (t,t ) /ttt /ttt o o N(t,t^) o 0 a 0 N(t,t ) o

0 1. 47 1. 55 1 05 10^ 0 5 08 + 2 6 .39 +2 1 26

1 _l 1. 55 1. 62 1 05 1 -1 5 57 +2 6 .90 + 2 1. 24

5 _l 1. 67 1. 69 1 01 5 -1 7 08 + 2 8 .30 +2 1. 17

0 2. 59 3. 47 1 34 1 8 54 + 2 9 .87 +2 1. 16

1 _l 2. 80 3. 69 1 32 5 1 69 + 3 2 .21 +3 1. 31

5 _l 3. 37 4. 20 1 25 1 + 1 2 46 + 3 3 .50 + 3 1. 42

1 3. 80 4. 63 1 22 5 + 1 6 12 + 3 9 .09 +3 1. 49

0 3. 65 5. 48 1 50 1 +2 9 05 + 3 1 .39 +4 1. 48

1

_ I 3. 97 5. 88 1 48 5 + 2 2 22 +4 3 .21 +4 1 45

5 _l 4. 90 6. 89 1 41 1 + 3 3 20 +4 4 .65 +4 1, 45

1 5. 70 7. 92 1 39 5 + 3 7 45 +4 1 .07 +5 1. 44 . 5 8. 17 1. 20 + 1 1 47 1 +4 1 03 +5 1 .49 +5 1 45

0 1. 33 + 1 2. 10 + 1 1 58 5 +4 1 75 +5 2 .50 +5 1. 43

1 _l 1. 46 + 1 2. 26 + 1 1 55 10^ 0 9 21 +2 9 .39 + 2 1. 02

5 _l 1. 85 + 1 2. 72 + 1 1 47 1 -1 1 01 +3 1 .01 +3 1. 00

1 2. 22 + 1 3. 23 + 1 1 45 5 -1 1 28 + 3 1 .22 +3 0. 95

5 4. 30 + 1 7. 06 + 1 1 64 1 1 55 +3 1 .45 +3 0. 93

1 + 1 6. 05 + 1 1. 08 +2 1 79 5 3 07 + 3 3 .26 + 3 1. 06

5 +

1. 09 + 2 2. 03 +2 1 86 1 + 1 4 45 + 3 5 . 15 +3 1. 15

0 4. 84 + 1 7. 31 + 1 1 51 5 + 1 1 11 +4 1 .34 +4 1. 21

1 _1 5, 30 + 1 7. 90 + 1 1 49 1 +2 1 64 +4 1.97 +4 1. 20

5 -1 6. 73 + 1 9. 51 + 1 1 41 5 +2 4 02 +4 4 .74 +4 1. 18

1 8. 12 + 1 1. 13 +2 1 39 1 +3 5 81 +4 6 .87 +4 1. 18

5 1. 61 +2 2. 53 +2 1 57 5 +3 1 37 +5 1 60 +5 1. 17

1 +

2. 32 + 2 3. 99 + 2 1 72 1 + 4 1. 93 +5 2 26 +5 1. 17

5 +

5. 67 + 2 1. 01 + 3 1 78 5 + 4 4. 06 + 5 4 69 + 5 1. 16

1 +2 8. 15 +2 1. 45 + 3 1. 76 1 +5 5. 24 + 5 5 96 +5 1. 13

5 + 2 1. 47 + 3 2. 54 + 3 1, 73 5 +5 7. 37 +5 7 86 +5 1. 06

0 1. 67 +2 O

Z . 41 + Z

1i . A A 00 0 1 00 + 3 1 .00 + 3 1. 00

1 " 1 1. 83 +2 Z . dU + 2

1 A 1 4 Z

1 -1 1. 10 + 3 1 .08 + 3 0. 98

5 "1 2

32 +2 O . 1 A 14 + 2 1 . 7 tr 5 -1 1 39 + 3 1 .30 + 3 0. 94

1 2

80 + 2 o

7 X 1 J + 2

1I , oo 1 1 68 + 3 1.55 + 3 0. 92

5 5

55 + 2 8. 35 + 2 1 50 5 3 33 + 3 3 .47 + 3 1 04

1 +8

05 +/ 1. 32 + 0 1. 64 1 + 1 4 83 + 3 5 48 + 3 1. 13

5 +

2. 01 + 3 3. 43 + 3 1. 71 5 + 1 1 21 + 4 1 .42 + 4 1. 17

1 + 2 2. 96 + 3 5. 03 + 3 1 70 1 + 2 1 78 + 4 2 . 10 + 4 1. 17

5 + 2 7. 12 + 3 1. 19 + 4 1. 67 5 + 2 4 37 +4 5 .04 + 4 1. 15

1 + 3 1. 00 +4 1. 67 + 4 1. 67 1 + 3 6 31 + 4 7 .31 + 4 1. 16

5 + 3 1. 78 + 4 2. 89 + 4 1. 63 5 + 3 1 49 +5 1 .70 +5 1. 14

1 + 4 2 09 +5 2 .41 +5 1 15

5 + 4 4 43 +5 5 .00 +5 1 13

1 + 5 5 73 +5 6 .39 +5 1 11

5 + 5 8 62 +5 8 .86 +5 1.03

1 +6 9 31 + 5 9 .39 +5 1 01

5 +6 9 92 +5 9 .87 + 5 0. 99

1 + 7 9 94 + 5 9 .94 + 5 1. 00

10

10

10'

10
8.11 APPROXIMATE SOLUTION OF THE BASIC EQUATION 443

TABLE 45

ACCURACY OF THE APPROXIMATION (11.16) (VOIGT PROFILE WITH a=10 1-X=10'^)

N(

o N

a 0 a ' 0

T

0 T N( T, T 0 N

a

(t, t N

a

T, T A

T, T N(

' o A A 0 N( T, T

V

n1 . 40 i . C /I D 4 i . U 0 i u Urb . 4 Z + 2

rb

7 C

. / b + z 1 , U D

1

t1 . 5 3 1

6 1 1 . 0 5 i - i b , + 2

/;D

9 1

. 2 1 + Z i A C. U b

5 1 . 6 4 1

69 1 . 0 3 5 - i

•7 A O 4 o + 2 / A 7 . 4 / + 2 i A A 0 3 . 0 4

rD . 4d i . D U i y

u y + Z

QO . 8 0 + Z UQ Q . y o

1 3 . n 7 rD . 0 c

1 . 4 y

rO 1

oU . 7 + o 1 O Q J. 7 11 .XX

5 4 . 8 4 6 . 86 11 . 4 Z i + i 2 . b y

. 7 7b

1 A

. i 4

, 7

i 9 1. Z X 1 5 . 69 7 . 8 7 1 . 38 5 +1 6 . o o z o

. 7 + 3 / . y 0, 7

i 9 7 . Z / 5 8 . 1

1

19 + 1 1 . 46 1 +2 y

A A U 4

, 7 + 3 i

1 A

. I 4 + 4 i 9 A . Z 0

0 1 . 3 1 +

1 . 9 8 +1 . 5 1 5 +2 1 . y U + 4 2

7 n . bU + 4

1I 9 1

. Z X

1 1

42 +2 . 1

+

1 . 5 1

1 I 7

1 +3 2 . 4 4

t A + 4 2

0 7 + 411 . X 0

5 ~ ^ 1 . 79 +

2 . 5 7 +1 . A A 44 r 7 5 +3 73 . D 4 + 4 4 . U U

J. A + 4 i . X u

1 2 . 1 7 +

3 . 0 4 +

1 . 40 1 +4 4 . 1 3

I A + 4

A4 . 40 J. A + 4 i A Q

. U 0

5 4 . 20 +

6 . 6 7 +

1 . 59 5 +4 5 . 6 2

• A + 4 6 . U 0 + 4 1 A 7

. U / 1 + 1 5 . 90 +

1 . 0 3 + 2 1 . 75 1 +5 6 . 5 5 + 4

-7/ . (J 4 + 4 i A 7

. U / 5 +

1 . 05 + 2 1 . 9 1 + 2 1 . 82 5 +5 9 . 5 8 + 4 1 . 0 4 + 5 1 A A

. U y

0 4 . 00 +

4 . 9 8 +

1 . 2 5 i + D 1 . i o + b

11 9 7 . Z J + b 1 n Q . u y

1 - 1 4 . 35 +

5 . 38 +

1 . 2 4 5 +6 1 . 6 4 + b 1 7 0 + b 1 A Q . u y

5 - 1 5 . 5 2 +

6 . 47 +

1 . 1 7 1 +7 1 . o n 89 + b 2 . U b + b

nX A C

1 6 . 70 +

7 . 69 +

1 . 15 5 +7 2 . 7 £ 36 + b 2

C 7

. b / + b

11 A Q . u y

5 1

32 + 2 1 . 7 2 + 2 1 . 30 oo 0 1 . U 0

. 7

i . U U

, 7

1 A A . u u

1 +1 . 9 1 + 2 2 . 7 1 + 2 1 . 42 1 - 1 1 . u y

. 7 + 3 i . U o

. 7 + b U . y y 5 +

4 . 56 + 2 6 . 80 + 2 1 . 49 5 - 1 1 . 7 0 J O

. 7 + b i . bU . 7 + b A

U

Q A

. y 4

1 + 2 6 . 46 + 2 9 . 59 + 2 1 . 4 8 i

11 . 0 o. 7 1i . b 4

. 7 + O Au Q 9 . y z 5 + 2 1

09 + 3 1

60 + 3 1 . 4 7 b

7 ^ 1 J 1

. 7 7o . 4 b

. 7 + J

11 . U 4

0 7 . 26 + 1 7 . 75 +

1 . 0 7 1 +1 4 . 1 0

/ o

. 7 + b

rb

A A . 4 0

. 7 i

1 A . X 4

1 - 1 7

89 +

8 . 37 +

1 . 06 C J. 1 b + i

1 1 0 + 4 1 X Q . oy + 4 1 . z

9 nu

5 - 1 1

00 + 2 1 . 0 1 + 2 1 . 0 1 1 +2 11 . O / + 4 1 . y o + 4

1

. X

1 Qy

1 1

2 2 + 2 1 . 20 + 2 0 . 9 8 D +2 7o . b U + 4

A4

n n . u u + 4

11

1 A

5 2 . 4 1 + 2 2 . 68 + 2 1

1

1 _L 7

1 +3 A4 . b U

J. /I + 4

/I 4

Q fl

. y 0 + 4 1 . X X

1 + 1 3 . 48 + 2 4 . 2 3 + 2 1

2 2 0 + o

£:D . 7 9 + 4 0 • Qy J

4. A 11 5 +

8 . 42 + 2 1 . 0 7 + 3 1

2 7

1 J- /I i +4 •7/ . D ZJ.+ A4

7 7 c: 4.+ 4 11 n 9

1 + 2 1 . 2 1 + 3 1 , 5 3 + 3 1

26 5 +4 11 . n /I U 4 + b

11 . U 4 + 0 X n n

5 + 2 2 . 52 + 3 3 . 0 8 + 3 1

2 2

1 J. c i + b11 . 9 1 Z 1 + o 1 9 9 • Z Z4. 11 . U X

1 + 3 3 . 2 2 + 3

•7 o , o r\ o U

, 7 11 . 1 O 0 ^ o

1± • 7 7 + 5 . 8 1 + 5 1 . 0 2 5 + 3 4 . 3 2

, 7 + 3

A4 . Q 7 , 7 11 . 1 i. 1 + 6 2 . 09 + 5 2 . 14 + 5 1 . 0 2 0 1 . 9 5 + 2 I . 1 A i 4 + 2 1 . 1 u O TO n 4 + 5 3 . 1

+ 5 1 . 0 2

1 - 1 2 . 1 Z + 2 Z . + ^

11 . n Q 1 + 7 3 . 5 5 + 5 3 . 6 4 + 5 1 . 0 3 5 - 1 2 . 69 + 2 z . / / + Z

11 . n X u o +7 4 _ 9 6 + 5 5 . 0 7 + 5 1 . 02

1 3 . 2 7 + 2

•7o . 7 n + L

1

i. , n 1 U 1 1 +8 5 . 64 + 5 5 . 75 + 5 1 . 02

5 6 . 46 + 2 7 . t "7 + 2

11 . i 4 o^ +8~ o 7 ^ 2 8 + 5 7 . 3 3 + 5 1 . 0 1

1 +

y

T 7 3 3 + 2 1 . 1 7 . 7 + o 1 . z o 1 +9 7

9 3 + 5 7 . 9 4 + 5 1 . 00

5 + 1 2 . 26 + 3 2 . 96 + 3 1

31 5 +9 9 . 07 + 5 9 .07 + 5 1 .00

1 + 2 3. 25 + 3 4 . 23 + 3 1

30 1 + 10 9 . 41 + 5 9 .41 + 5 1 .00

5 + 2 6 . 82 + 3 8. 54 + 3 1

25

1 + 7, 8. 77 + 3 1

06 + 4 1

21

5 + 3 1

31 + 4 1 . 48 + 4 1

13

1 + 4 1

48 + 4 1 . 65 + 4 1

1

5 + 4 1 99 + 4 2 . 20 + 4 1

11

1 + 5 2 . 29 + 4 2 . 55 + 4 1

11

5 + 5 2 . 92 + 4 3. 27 + 4 1

12

10

10'

10"

10

10

REFERENCES

Abramov, Yu. Yu

1967a

1967b

1969a

1969b

Pykhne, A. M. , Napartovich, A. P.

Transfer o£ high-intensity resonance radiation.

i Tear. Fiz., 52_, 536.

Transfer of resonance radiation in a half-space

3, 459.

Radiative transfer of excitation in a finite volume. Zh

Eksperim. i Tear. Fiz., _56_, 654.

Stationary Problems of Radiative Trans fer of Excitation

Zh, Eksperim

Astrofizika.

Abramov, Yu. Yu . , Napartovich, A. P. 1968 Excitation wave caused by a light flare Astrofizika, 4., 195.

Allen, C. W.

1963 Astrophysical Quantities, 2nd ed. Athlone Press, London.

Ambartsumian , V. A.

1942 Light scattering by planetary atmospheres. Astron. Zh.,l_9, 30

1943 On the problem of diffuse reflection of light by a turbid

medium. Dokl. Acad. Nauk SSR, 38_, 257.

1948 On the number of scatterings of photons diffusing in a turbid

medium. Dokl. Akad. Nauk- Arm. SSR, 8., 101.

1960 Scientific Works, Vol. I. Akad. Nauk. Arm. SSR Press, Erevan.

1964 On radiation- induced decrease of optical density of a medium.

Dokl. Akad. Nauk Arm. SSR, 39, 159.

1966 On some non-linear problems of radiative transfer theory, in

Theory of Stellar Spectra, V. V. Sobolev, V. G. Gorbatskii,

V. V. Ivanov, eds. Nauka Publ. Co., Moscow. P. 91.

Armstrong, B. H,

1967

Avrett, E. H,

1965

Spectrum line profiles: the Voigt function,

Radiative Transfer, 7, 61.

J. Quant. Speotry

Avrett, E. H,

1965

Solutions of the two-level line transfer problem with complete

redistribution, in Proceedings of the Second Harvard-Smithsonian Conference on Stellar Atmospheres, Smithsonian Astrophysical Observatory Special Report No. 174, Cambridge. P. 101.

Hummer, D. G.

Non-coherent scattering. II. Line formation with a frequency

independent source function. Monthly Notices Roy. Astron.

Soc, 130 , 295.
446 REFERENCES

Avrett, E. H., Loeser, R. 1966 Kernel representations in the solution o£ 1 ine - trans £er problems. Smithsonian As trophys ical Observatory Special Report

No. 201, Cambridge

Bates, D. R. (editor)

1962 Atomio and Molecular Processes. Academic Press, New York.

Biberman, L. M.

1947 On the theory of the diffusion of resonance radiation.

Zh . Ekspevim. i Teor. 'Fiz., 17, 416.

1949 Departures from thermodynamic equilibrium in a plasma due

to the escape of radiation. Zh . Ekspevim. i Teor. Fiz.,

19_, 584.

Bohm, K.-H.

1960 "Basic theory of line formation," in Stellar Atmospheres

,

J. L. Greenstein, ed. University of Chicago Press, Chicago.

Chap. 3.

Braginskii, S. I., Budker, G. I. 1958 Physical phenomena in gas discharge flaming up during incomplete ionization, in Plasma Physics and the Problem of Controlled Thermonuclear Reactions , U.S.S.R. Academy of Science

Press, Moscow. Vol. I. Busbridge , I . W

1953 Coherent and non-coherent scattering in the theory of line

formation. Monthly Notices Roy. Astron. Soc, 113 , 52.

1960 Mathematics of Radiative Trans fer . Cambridge Univers ity

Press, Cambridge.

Capriotti , E

1965 Mean escape probabilities of line photons. Astrophys. J.,

142 , 1101.

Carlstedt, J. L., Mullikin, T. W.

1966 Chandrasekhar ' s X- and Y-functions. Astrophys . J. Suppl.,

12, N113.

Carsten, H. R. F., McKerrow, N. W

The tabulation of

order. Phil. Mag., (7) 35, 812

1944 The tabulation of some Bessel functions K^(x) of fractional

Case, K. M.

1957a On Wiener-Hopf equations. Ann. Phys., 2, 384.

1957b Transfer prolDlems and reciprocity princTple. Rev. Mod. Phys.,

29_, 651.

I Case, K. M. , Hoffmann, F. de , Placzek, G. 1953 Introduction to the Theory of Neutron Diffusion . U. S. Govern

ment Printing Office, Washington, D. C.
REFERENCES 447

Case, K. M. , Zweifel , P. F. 1967 Linear Transport Theory. Addison-Wesley , Reading, Mass

Chandrasekhar , S.

1950 Radiative Transfer. Clarendon Press, Oxford.

Chamberlain, J. W. , McElroy, M. B.

1966 Diffuse reflection by an inhomogeneous planetary atmosphere.

Astro-phys. J., 1_44_, 1148.

Crosbie, A. L., Viskanta, R. 1970a Effect of band or line shape on the radiative transfer in a nongray planar medium. J. Quant. Speotry Radiative Transfer,

10_, 487.

1970b Nongray radiative transfer in a planar medium exposed to a collimated flux. J. Quant. Speotry Radiative Transfer',

10_, 4 65 . Cuperman, S., Engelmann, F., Oxenius , J.

1963 Nonthermal impurity radiation from a spherical plasma.

1964

Davison, B. 1951

1958

Phys. Fluids, 6_, 108

Nonthermal impurity radiation from a spherical plasma.

Phys. Fluids, 7, 428.

Influence of a black sphere and of a black cylinder upon the

neutron density in an infinite non - capturing medium.

ProQ. Phys. Soo., A64 , No. 382, 881,

Neutron Transport Theory. Clarendon Press, Oxford.

Dobrowolny, M., Engelmann, F. 1965 Absorption and emission profiles of line radiation in non- equilibrium. Nuovo Cimento, 39_, 965.

Eddington, A,

1929

S.

Finn, G, D. 1967

1971

The formation of absorption lines. Monthly Notices Roy.

Astron. Soo., 89, 620.

Frequency redistribution on scattering. Astrophys. J.,

147 , 1085.

Probabilistic radiative transfer. J. Quant. Speotry

Radiative Transfer, 11, 203.

Finn

D., Jefferies, J. T. 1968 Studies in spectral line formation. I. Formulation and simple

applications. J. Quant. Speotry Radiative Transfer, 8_, 1675.

Feller, W.

1966

Fock, V. A.

1944

An Introduction To Probability Theory and Its Applications

Vol. II. J. Wiley, New York.

On some integral equations of mathematical physics

Mat. Sb., 14 (56) , N 1-2, 3.

522-519 O - 74 - 30
448 REFERENCES

Fuller, P. B., Hyett, B. J. 1968 Calculations o£ Chandrasekhar ' s X- and Y-£unctions and their

analogs for noncoherent isotropic scattering. NASA Technical

Note D-4712.

Fuller

Gakhov

F . B . , Warming , R. F

1970

F. D. 1963

Germogenova, T 1960

1966

Gradshtein , I . 1962

Grinin, V. P. 1969

An invariant imbedding approach to spectral-line formation by

noncoherent scattering. Astrophys. Space Soi., 7, 93.

Boundary Value Problems . Fizmatgiz, Moscow.

, A.

Spherical Milne problem: extrapolated length and density near

the boundary, in Some Mathematical Problems of Neutron Physics

Moscow University Press, Moscow. P. 80.

Diffusion of radiation in a spherical shell surrounding a point source. Astrofizika, 2_, 251.

S. , Ryzhik, I . M.

Tables of Integrals, Sums, Series and Products . Fizmatgiz,

Moscow.

On the relative intensities of the hydrogen lines in spectra

of nebulae. Astrofizika, 5, 213.

Halpern, 0., Luneburg, R. K 1949 Multiple scattering of neutrons

of finite thickness. Phys. Rev.

II. Diffusion in a plate

76, 1811.

Halpern, 0

1938

Hasted, J. B. 1964

Hearn, A. G.

1962

1964a

1964b

Heaslet, M. A,

1966

1968a

1968b

Luneburg, R. K., Clark, 0.

On multiple scattering of neutrons. I. Theory of the albedo

of a plane boundary. Phys. Rev., 53, 173.

Physics of Atomic Collisions . Butterworth, London

Radiative transfer of Doppler broadened resonance lines.

U.K. Atomic Energy Authority, Report CLM-P-16.

Radiative transfer of Doppler broadened resonance lines. II.

Proc. Phys. Soa., 84_, 11.

Radiative transfer of Doppler broadened resonance lines.

Proc. Phys. Soc, 81_, 648.

, Warming, R. F. Theoretical predictions of radiative transfer in a homogeneous

cylindrical medium. J. Quant. Spectry Radiative Transfer,

6, 751.

Radiative source-functions predictions for finite and semiinfinite non-conservative atmospheres. Astrophys. Space Sci.,

1, 460.

Theoretical predictions of spectral line formation by nonco- herent scattering. J. Quant. Spectry Radiative Trans fer,

8, 1101.
REFERENCES 449

Heitler, W.

1954

Henyey, L. G, 1940

Holstein, T. 1947

1951

Quantum Theory of Radiation, Clarendon Press, Oxford,

The Doppler effect in resonance lines

26, 50.

Proo. Nat. Acad, Sai

Hopf, E. 1934

Imprisonment of resonance radiation in gases

72_, 1212.

Imprisonment of resonance radiation in gases

ifey.^ 83, 1159.

Mathematiaal Problems of Radiative Equilibrium.

University Press, Cambridge.

Phys . Rev .

,

II. Phys.

Cambridge

Houtgast, J. 1942

Huang, Su-shu

1952

The Variations in the Profiles of Strong Fraunhofer Lines

Along a Radius of the Solar Disk. Thesis, University of

Utrecht

The variational method for problems of neutron diffusion and

of radiative transfer. Phys. Rev., 88, 50.

Hulst, H. C. van de

1964

Hulst

Hummer

1968

H. C. 1966

, D. G.

1962

1963

1964

1965a

1965b

1968

1969

Diffuse reflection and transmission by a very thick planeparallel atmosphere with isotropic scattering. laarus,

3, 336.

A^symptotic fitting, a method for solving anisotropic transfer

problems in thick layers. J. Comput. Phys., 3_, 291. van de , Therhoeve, F. The escape of radiation from a semi - infinite nonconservative

atmosphere. Bull. Astron. Inst. Neth., 18, 377.

I. The redistribution functions

Monthly Notices Roy. Astron. Soa.

Non-coherent scattering

with Doppler broadening,

125 , 21.

Numerical solution of the transfer equation with noncoherent

scattering. J. Quant. Speatry Radiative Transfer, 3_, 101.

The mean number of scatterings by a resonance - 1 ine photon.

Astrophys . J., 140 , 276.

The emission coefficient, in Proceedings of the Second.

Harvard-Smithsonian Conference on Stellar Atmospheres

,

Smithsonian Astrophys ical Observatory Special Report No. 174,

Cambridge. P. 13.

The Voigt function. An eight - s ignificant - figure table and

generating procedure. Mem. Roy. Astron. Soc., 70_, 1.

Non-coherent scattering. III. The effect of continuous

absorption on the formation of spectral lines. Monthly

Notices Roy. Astron. Soc., 158 , 73.

Non-coherent scattering. VI. Solutions with a frequencydependent source function. Monthly Notices Roy. Astron. Soc,

145, 95.

ll,
450 REFERENCES

Hummer, D. G 1966

1967

Hummer, D. G,

1966

Ibragimov , I 1965

Rybicki, G. B. Non-LTE line formation with spatial variations in the Doppler

width. J. Quant. Speatry Radiative Transfer, 6_, 661.

Computational methods for non-LTE line transfer problems, in

Methods in Computational Physios, Vol. 7, p. 51. Academic

Pres s , New York

Stewart, J. C. Thermalization lengths and the homogeneous transfer equation

in line formation. Astrophys. J., 146, 290.

A. , Linnik, Yu . V.

Independent and Stationary Related Quantities

Mos cow

Fizmatgiz

,

Ivanov, V. V.

1960

1962a

1962b

1963

1964a

1964b

1965

1966

1967

1968

1969

1970a

1970b

1970c

1972

Ivanov, V. V.

1965

Diffusion of radiation with frequency redistribution in a one -dimens ional medium. Vestn. Leningr . Univ., No. 19, 117.

Diffusion of radiation with frequency redistribution in a semi - infinite medium. Uoh. Zap. Leningr. Univ., No. 307 , 52 . Diffusion of resonance radiation in stellar atmospheres and

nebulae. I. Astron, Zh . , 39_, 1020.

Diffusion of resonance radiation in stellar atmospheres and

nebulae. II. Astron. Zh . , 40_, 257 . Light scattering in a plane layer. Astron. Zh . , 41_, 44.

On the problem of light scattering in an atmosphere of finite

optical thickness. Astron. Zh., 4]^, 1097.

On the transfer of line radiation. Uah. Zap. Leningr. Univ.,

No. 328, 44.

Populations of excited levels in an optically thick gas layer,

in Theory of Stellar Spectra, V. V. Sobolev, V. G. Gorbatskii,

V. V. Ivanov, eds . , Nauka Publ. Co., Moscow, P. 127.

Time variation of the profile of a Doppler broadened resonance

line. Bull. Astron. Inst. Neth., 19_, 192.

On the Milne problem in the theory of line formation.

Astrofizika, 4^, 5. Mean number of photon scatterings in a homogeneous optically

thick sphere, in Stars, Eehulae, Galaxies . Akad. Nauk

Arm. SSR Press, Erevan. P. 27.

Transfer of resonance radiation in purely scattering media. I. Semi - infinite medium. J. Quant. Speatry Radiative Trans fer,

10_, 665.

Transfer of resonance radiation in purely scattering media.

II. Optically thick layer. J. Quant. Speatry Radiative

Transfer, 10_, 681.

Mean photon path length in a scattering medium. As tro fizika,

6, 643.

An approximation to the solution of the radiative transfer

equation in line frequencies. Astron. Zh . , 49_, 115.

Leonov, V. V.

Light scattering in an optically thick atmosphere with a non- spherical phase function. Izv. Akad. Nauk SSSR., ser. Phys

of the Atmosphere and Ocean, 1, 803.
REFERENCES 451

Ivanov, V. V.

1965

1966

Nagirner , D.I.

H-£unctions o£ the theory of transfer of resonance radiation.

Astro fizika, 143.

Transfer of resonance radiation in an infinite medium. II.

Astro fizika, 2, 147.

Ivanov, V. V., Sabashvili, S 1972

A,

Ivanov , V. V.

1965a

1965b

Jef feries , J

1968

Jefferies, J.

1958

1959

Jefferies , J

1960

Transfer of resonance radiation and photon random walks.

Astrophys. Spaoe Soi., 17_, 13.

Shcherbakov, V. T.

Tables of functions encountered in the theory of transfer of

resonance radiation. I. Astrofizika, I^, 22.

Tables of functions encountered in the theory of transfer of

resonance radiation. II. Astrofizika, 1, 31.

Spectral Line Formation. Blais-dell, Waltham, Mass.

T. , Thomas , R.N.

The source function in a non-equilibrium atmosphere. II.

Depth dependence of source function for resonance and strong

subordinate lines. Astrophys. J., 121 , 667.

The source function in a non-equilibrium atmosphere. III.

Influence of a chromosphere. Astrophys. J., 129 , 401.

T. , White , 0. R. The source function in a non-equilibrium atmosphere. VI.

The frequency dependence of the source function for resonance

lines. Astrophys. J., 132 , 767.

King, J. I. F. 1965

Kogan, M. N.

1967

Kourganoff, V,

1952

Krein, M. G.

1958

, Sillars, R. V., Harrison, R. H.

Hopf q-function evaluated to 8-digit accuracy,

142, 1655.

As trophy s . J .

,

V V Kuscer

I

1953

Rarefied Gas Dynamics. Nauka Publ. Co., Moscow.

A.

Basic Methods in Transfer Problems. Clarendon Press, Oxford,

Integral equations on the half-line with displacement

kernels. Usp. Mathem. Nauk., L3, N5 , 3.

Penetration of radiation through a thick slab of isotropically

scattering material. Can. J. Phys., 3j^, 1187.

Levich, V. G,

1940 On the theory of resonance fluorescence

Teor. Fiz., 10, 1293.

Zh . Eksperim. i
452 REFERENCES

Lotz, W.

Mark, C,

Marshak

Menzel

Mihalas

1967 Electron- impact ionization cross -sections and ionization

rate coefficients for atoms and ions. Astrophys. J. Suippl.,

14_, 207.

1947 Neutron density near a plane surface. Phys. Rev., 72, 558.

, R. 1947

D.1936

1937

, D. 1970

Milne, E. A.

1930

Minin

Morse

I . N.

1958

The variational method for asymptotic neutron densities

Phys. Rev., 71, 688.

The theoretical interpretation of equivalent breadths of

absorption lines. Astrophys. J., 84_, 462.

Absorption and emission of radiation. Astrophys. J., 85_, 330

Stellar Atmospheres . Freeman, San Francisco

Thermodynamics of the stars, in Handbuah der Astrophys.,

Bd. Ill, 1, Chapter II. J. Springer, Berlin.

On the theory of the diffusion of radiation in a semi-infinite

medium. Dokl. Akad. Nauk SSSR., 120, 63.

P. , Feshbach, H.

1953 Methods of Theoretical Physios . Part I Co. , New York.

McGraw-Hill Book

Mullikin, T. W,

1964

Munch, G.

1960

Radiative transfer in finite homogeneous atmospheres with

anisotropic scattering. Astrophys. J., 139 , 379.

Theory of model stellar atmospheres, in Stellar Atmospheres,

J. L. Greenstein, ed. Univ. of Chicago Press, Chicago. Chap. I

, N. I. Singular Integral Equations, 2nd ed. Fizmatgiz, Moscow,

Muskhelishvili

1962

Nagirner , D . I

1964a On the solution of the integral equations of the theory of

light scattering. Astron. Zh . , 42_, 669.

On multiple light scattering in a spectral line. Vestn.

Leningr. Univ., No. 1, 142

1964b

1965

1967

1968

1969

On light scattering in a spherical nebula of large optical

thickness. Uoh. Zap. Leningr. Univ., No. 328, 66.

On the asymptotic form of the X- and Y-functions of the

theory of multiple light scattering in line frequencies.

Astrofizika, 3_, 293.

Multiple light scattering in a semi - infinite medium. Uoh.

Zap. Leningr. Univ., No. 337, 3.

Transfer of resonance radiation in an optically thick layer,

Astrofizika, 5, 507.
REFERENCES 453

Nagirner, D. I., Ivanov, V. V.

1966 Transfer of resonance radiation in an infinite medium. I. Astro fizika, 2_, 5. Nagirner, D. I., Schneeweis, A. B. 1973 Functions characterizing resonance scattering in an infinite

medium. Uah. Zap. Leningr. Univ., No. 363, 16.

Napartovich, A. P.

1969 Transfer of resonance radiation in a weakly ionized plasma.

Ph. D. thesis.

Osterbrock, D. E.

1962 The escape of resonance - line radiation from an optically

thick nebula. Astrophys. J., 135 , 195.

Oxenius , J.

1965 Emission and absorption profiles in a scattering atmosphere.

J. Quant. Speotry Radiative Transfer, 5_, 711.

1966 Radiative transfer and irreversibility. J. Quant. Speotry

Radiative Transfer, 6_, 65.

Phelps, A. V.

1959 Diffusion, de -excitation and three-body collision coefficients

for excited neon atoms. Phys. Rev., 114 , 1011.

Phelps, A. v., McCobrey, A. 0.

1960 Experimental verification of the "incoherent" scattering

theory for the transfer of resonance radiation. Phys. Rev.,

118 , 1561.

Placzek, G. , Seidel, W.

1947 Milne's problem in transport theory. Phys. Rev., 72_, 550.

Regemorter, H. van

1962 Rate of collisional excitation in stellar atmospheres.

Astrophys. J., 1 36 , 906.

Rybicki, G. B., Hummer, D. G.

1967 Spectral line formation in variable -property media. I. The Riccati method. Astrophys . J., 150, 607.

1969 Non-coherent scattering. V. Thermalization distances and

their distribution functions. Monthly Notices Roy. Astron.

Soo., 1_4_4, 313.

Sampson, D. 1969 Some aspects of radiative transport in the outer layer of a hot star. J. Quant. Speotry Radiative Transfer, 9_, 825 .

Savedoff, M. P. 1952 Formation of absorption lines by noncoherent scattering.

Astrophys. J., 115 , 509.
454 REFERENCES

Smith, M. G. 1964

Sobel ' man , I . 1963

Sobolev, V. V. 1949

1951

1954

1955

1956

1957a

1957b

1958a

1958b

1959a

1959b

1962

1964

1965

1966-1967

1967a

1967b

Sobolev, V. V.

1962

Sobouti , Y

1963

Stibbs, D. W. 1963

The isotropic scattering o£ a concentrated ray pencil from a point source. Proo. Cambridge Phil. Soo., 60, 105.

I Introduction to the Theory of Atomic Spectra. Fizmatgiz,

Moscow.

Non-coherent light scattering in stellar atmospheres.

Astron. Zh . , 26_, 129.

A new method in the theory of light scattering. Astron. Zh.,

i8, 355.

Line formation by non-coherent scattering. Astron. Zh., 31,

231.

Diffusion of radiation with frequency redistribution. Yestn.

Leningr. Univ., No. 5, 85.

Transfer of Radiant Energy in Stellar and Planetary Atmospheres. Gostech izdat , Moscow.

Diffusion of radiation in a semi - infinite medium. Dokl. Akad.

Nauk SSSR., 116 , 45.

Diffusion of radiation in a medium of finite optical thickness. Astron. Zh . , 34, 336.

On the theory of the^Jif fusion of radiation. Izv. Akad. Nauk

Arm. SSR., ser Fiz.-Mat., 1_1, 39,

Diffusion of radiation in a plane layer. Dokl. Akad. Nauk.

SSSR., 120 , 69.

On some problems of the theory of the diffusion of radiation.

Dokl. Akad. Nauk SSSR., 1_29_, 1265.

On the theory of the diffusion of radiation in stellar atmospheres. Astron. Zh . , 3_6_, 576.

Hydrogen lines in the spectra of prominences. Astron. Zh .

,

39, 632.

Diffusion of radiation in a plane layer of large optical

thickness. Dokl. Akad. Nauk SSSR., 155 , 316.

Diffusion of radiation in a nebula of large optical thickness,

in Kinemati cs and Dynamics of Stellar Sy stems and Physios of

Interstellar Medium. Nauka Publ. Co., Alma-Ata. P. 285.

Number of scatterings of diffusing photons. I-III.

Astrofizika, 2_, 135 , 1966 ; 2_, 239 , 1966 ; 3 , 5 , 1967.

A Course in Theoretical Astrophysics . Nauka Publ. Co.,

Mos cow

Number of scatterings of diffusing photons. IV. Astrofizika,

3, 137.

, Ivanov, V. V.

On the intensities of hydrogen emission lines in stellar

spectra. Vch. Zap. Leningr. Univ., No. 307, 3.

Chandrasekhar ' s X-, Y- and related functions. Astrophys. J. Suppl., 7, No. 72.

N.

Scattering functions for planetary atmospheres. Mem. Soc.

Roy. Sci. Liege, 7, 169.
REFERENCES 455

Stibbs, D. W,

1959

Stromgren, B,

1935

Terebizh, V. 1967

Titchmarsch , I 1937

Thomas, R. N.

1957

1965a

1965b

Thomas , R . N

1961

N. , Wier, R, E.

On the H-£unctions for isotropic scattering. Monthly Notices

Roy. Astron. Sac, 119, 512.

The influence of electron capture on the contours of Fraunhofer lines. Z. Astrophys., 10, 237.

Yu.

On some non-linear problems of the theory of transfer of line

radiation. Astro fizika , 3, 281.

Introduction to the Theory of Fourier Integrals

Press, London.

Oxford Univ,

The source function in a non- equil ibrium atmosphere, I. The resonance lines. Astrophys. J., 12 5 , 260.

Some Aspects of Non-Equilibrium Thermodynamics in the Presence

of a Radiation Field. Univ. of Colorado Press, Boulder, Colo

Computation of the line source function. A review of the

physical problem, in Proceedings of the Second Harvard-Smithsonian Conference on Stellar Atmospheres , Smithsonian Astrophysical Observatory Special Report No. 174, Cambridge. P. 71.

Athay, R. G.

Physics of the Solar Chromosphere

London

Interscience , New York,

Tomatsu, T. , Ogawa, T 1966

Trigt, C. van

Theoretical studies of the airglow resonant emission transfer.

Report on Ionosphere and Space Research in Japan, 20, 418.

1969 Analytically solvable

Phys. Rev., 181, 97.

prob lems in radiative transferI

1970 Analytically solvable

Phys. Rev., Al , 1298.

prob lems in radiative trans fer

II . 1971 Analytically solvable

Phys. Rev., A4 , 1303.

prob lems in radiative transferIll

Ueno , S

1955-1956

Unno, W.

1952a

1952b

Usold, A.

The formation of absorption lines by coherent and noncoherent

scattering. I-IV. Contr. Inst. Astrophys. Kyoto, No. 58,

1955; No. 62, 1956; No. 63, 1956; No. 64, 1956.

On the radiation pressure in a planetary nebula. Puhl. Astron

Soc. Japan, 3_, 178.

Note on the Zanstra redistribution in planetary nebulae. Puhl

Astron. Soc. Japan, 4, 100.

1955 Physik der Sternatmospharen . 2 Auf 1 . Springer, Berlin,
456 REFERENCES

Veklenko, B. A.

1957 On the Green's function o£ the equation o£ the diffusion of

radiation. Zh. Eksperim. i Teov. Fiz.^ 33, 817.

1959 On the Green's function 'of the equation oT the diffusion of

resonance radiation. Zh. Eksperim. i Tear. Fiz.^ 36_, 204 . Walsh, P.

1959 Effect of simultaneous Doppler and collision broadening and

hyperfine structure on the imprisonment of resonance radiation.

Phys. Rev., 116 , 511.

Warming, R. F. 1970a The direct calculation of the H-£unction for completely non- coherent scattering. Astrophys . J., 159 , 593.

1970b Asymptotic source-function expansions for resonance lines in

a semi- infinite medium. J. Quant. Speatry Radiative Transfer,

10, 1219.

Watson, G. N.

1945 A Treatise on the Theory of Bessel Functions , 2nd ed.

Cambridge University Press, Cambridge.

Weinstein, M.

1962 Effect of reflecting boundaries on transport of resonance

radiation. J. Appl. Rhys., 33_, 587.

Wiese, W. L., Smith, M. W. , Glennon, B. M.

1966 Atomia Transition Probahilities , Vol. I. U. S. Government

Printing Office, Washington, D. C.

Wiese, W. L., Smith, M. W. , Miles, B. M.

1969 Atomic Transition Prohahilities , Vol. II. U. S. Government

Printing Office, Washington, D. C.

Wing, G. M.

1962 An Introduotion to Transport Theory. J. Wiley, New York.

Woolley, R. v.d. R. , Stibbs, D. W. N.

1953 The Outer Layers of a Star. Clarendon Press, Oxford.

Yanovitskii, E. G.

1964 On the diffusion of radiation in a plane isotropically

scattering layer. Astron. Zh . , 41_, 898.

Yengibarian, N. B.

1966 On a non-linear problem of radiative transfer. Astrofizika,

1» 297.
ii

NOTATION

A necessary (though not sufficient) condition for a symbol to appear in

the following list is that it be used in more than one section of the book.

Occasionally a symbol .appearing below may be used in another sense elsewhere

in the book (for example, the letter a, which we use extensively as a parameter of the Voigt function, is also used as the constant in the StefanBoltzmann law in §1.1, etc.). In all such cases the meaning is either men- tioned in the appropriate place or is clear from the context.

As a rule, in the following list the meaning of each symbol is explained

and the page number is given where the symbol first appears. Occasionally

the page numbers where the quantity in question is discussed more fully are aiso appended. When for any reason it is not feasible to explain a symbol

in this list, reference is made to the appropriate part of the book. Commonly

used symbols (such as h for Planck's constant and c for the velocity of light,

etc.) are not included below.

The subscripts D, V, L and M are widely used to indicate quantities referring to Doppler, Voigt and Lorentz profiles and to monochromatic scattering,

respectively. The subscript p indicates that a quantity refers to a point

source in an infinite medium while the subscript <» indicates functions referring to an infinite medium.

a constant, normalizing to unity the integral of the profile

of the absorption coefficient over the entire range of dimensionless frequencies, x, 28

Einstein coefficient- for spontaneous emission, 13

the number of recombinations onto the level i per second per

ion and unit electron density, 16

ratio of collisional and radiative damping width to Doppler

width, 26

80

80

Planck's function, 3

ci

a

B (T)
NOTATION

Einstein coefficients for absorption and stimulated emission

(calculated for mean intensity of radiation rather than radiation dens ity) ,13

number of photoioni zations from the level i per second per

atom in that level, 16

number of stimulated photo - recomb inat ions into the level i per

second per ion and a unit electron density, 16

Menzel parameter, 8 number of i-k transitions induced by electron impact per second

per atom in the level i and unit electron density, 10

number of collisional ionizations from the level i per second

per atom in that level and unit electron density, 12

number of three-body recombinations into the level i per second

per ion and unit electron density, 12

coefficient in the asymptotic form of the resolvent function

$(t) for conservative scattering in a spectral line (for a semi - inf ini te medium), 226

coefficient in the asymptotic form of the Green's function

for an infinite homogeneous medium, (conservative case), 166

coefficient in the asymptotic form of X('»,t ) as t for

conservative scattering, 368

first exponential- integral function, 11

n-th exponential- integral function (n = 1,2,...)> 77

78

. 78

statistical weight of the level i, 6

the H-function for scattering: (a) in media with no continuum absorption, 201, 212, 222; (b) in media with continuum

absorption, 327

function appearing in the asymptotic form of H(z) for z >> 1

,

221

intensity of radiation, 2, 50

mean intensity of radiation, 2 mean intensity of radiation in the i->k line weighted with the

profile of the absorption coefficient, 13, 15

the kernel of the basic integral equation for media without

continuum absorption, 68, 76-82

77-82
NOTATION 459

76-78

the kernel o£ the basic integral equation for media absorbing

in the continuum, 67, 302-306

67, 302-306

root o£ the characteristic equation (monochromatic scattering),

101-105

atomic absorption coefficient in the i^k line, 14

atomic absorption coefficient in the i-th continuum, 16

.66

mean number of scatterings of a photon, 107, 285-289, 423-434

population of the level i, 6 electron density, 6 density of ions in the ground state, 6

the probability of photon escape from depth x, 205, 295-299

Hopf's function, 139-140, 289-294

102

115

line source function, 40, 51, 54

continuum source function, 43

primary line source function, 51, 54

primary line source function for media with continuum radiation, 66 , 325

auxiliary functions, 201, 205, 352

source function for the Milne problem: (a) monochromatic

scattering, 139; (b) line scattering, 253

146, 178

temperature, electron temperature, 3, 11

excitation temperature, 9 radiation temperature, 4

78- 83

307-310
460 NOTATION

U(a,x) normalized Voigt function, 25

V(u) 79-83

V(u, 3) 307- 310

X(z) , X(z,T ) generalized Ambartsumian- Chandrasekhar X- function, 352,

° 358-370

X dimensionless frequency, measured from the line center, 26, 27

x(z) 78

X. (t ) 364- 365

1 ^ o-^ Y(z) , Y(z,T^) generalized Ambartsumian- Chandrasekhar Y-function, 352, 358-370

y- (t ) ^ 1

^ 365

z = 194

a(x)

z = . \ . 326 a(x)+3

= ^ 303

c 6 a(x) profile of the line absorption coefficient normalized so that

a(0) = 1, 28

* a.,a_-| weighted moments of the H- and X- functions, (i = 0,1,... )>

^ 128-130, 361-363

3 ratio of the continuum to the line center opacities, 58

3^ weighted moment of the Y-function, 362

r(s) Euler's gamma- function , 81

r(T,T') resolvent of the source function equation for a semi- infinite

medium: (a) monochromatic scattering, 126; (b) line

scattering, 198-199

Y characteristic exponent, 83-86

Y* = 0.577216 Euler's constant

Av characteristic frequency interval in units of which the dimensionless frequency x is measured, 25-26

6(3) 316

(v), e (v) line emission coefficient, 40

1 K

p

£ (v) continuum emission coefficient, 42-43

C = —^

303

l-3z
NOTATION .461

X

X

y = cos

V

V .IC

0

T

^0

$Ct)

in problems with plane geometry, the angle between a certain

direction and the external normal to the boundary x = 0,

57-58

survival probability pe-r scattering o£ a photon, 50, 53

survival probability per scattering of a photon in media with

continuous absorption, 302, 316

57-58

central frequency of the line, 10, 24

threshold frequency for ionization from the level i, 8

(a) monochromatic scattering, 110; (b) line scattering, 167

(a) monochromatic scattering,

(a) monochromatic scattering,

225-226

230

117; (b) line scattering, 186

134; (b) line scattering,

volume line absorption coefficient, 40

volume continuum absorption coefficient, 42

optical depth at line center, 58-61

optical thickness, 58

boundary layer thickness, 258, 340

diffusion length (monochromatic scattering) , thermalization length, 178-180, 315-322

resolvent function for semi - infinite medium:

matic scattering, 126; (b) line scattering.

resolvent function for layer of thickness t

137, 229-231

dissipation measure of a medium, 370-372

0'

103

(a) monochro199, 223, 228

350-355
FORM NBS-n4A (1-71)

1 U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS SP-3852. Gov't Accession No.3. Recipient's Accession No.

4. TITLE AND SUBTITLE

TRANSFER OF RADIATION IN SPECTRAL LINES by V. V. Ivanov

English-language Edition of Radiative Transfer and the

Spectra of Celestial Bodies

5. Publication Date November 1973

6. Performing Organization Code

7. AUTHOR(S)

D. G. Hummer (translation editor)

8. Performing Organization

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Pro;ect/Task/Work Unit No.

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

Same as No. 9

.

13. Type of Report & Period Covered

r Lna L

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

This book is a revised and somewhat extended version of V. V, Ivanov'

s

Radiative Transfer and the Spectra of Celestial Bodies , published in

Moscow in 1969. The principal subject is the transfer of radiant

energy through a gas composed of atoms with two discrete levels.

Although the emphasis of the book is on analytical methods, extensive

numerical and graphical results are presented.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Astrophysics; energy loss; photons; radiative transfer; spectral line profiles;

stellar atmospheres.,

18. AVAILABILITY STATEMENT 19. SECURITY CLASS

(THIS REPORT)

21. NO. OF PAGES

[X] UNLIMITED.

UNCL ASSIFIED

480

1 1 FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

20. SECURITY CLASS

(THIS PAGE)

22. Price

UNCLASSIFIED

$7. 45

USCOMM-DC 66244.P71

U.S. GOVERNMENT PRINTING OFFICE : 1974 O - 522-519