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f,:i)raryNBS SPECIAL PUBLICATION

332

Spectrum Formation

in Stars With

Steady-State

Extended

Atmospheres

ITED STATES DEPARTMENT OF COMMERCE Maurice H. Stans, Secretary

NATIONAL BUREAU OF STANDARDS Lewis M. Branscomb, Director

Spectrum Formation in

Stars With Steady - State

Extended Atmospheres

Proceedings of the International Astronomical

Union Colloquium No. 2, Commission 36

Munich, Germany, April 16-19, 1969

Edited by

H. G. Groth and P. Wellmann

Universitats-Sternwarte Munchen

Institut fur Astronomie und Astrophysik

Miinchen, Germany

\) , National Bureau of Standards Special Publication 332

* • » Nat. Bur. Stand. (U.S.), Spec. Publ. 332, 342 pages (August 1970) CODEN: XNBSA

Issued August 1970

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD Catalog No. C13.10:332), Price $1.75
NATIONAL BUREAU OF STANDARD

FEB 1 2 19/1

GENERAL ABSTRACT ' r> J ~<

Commission 36 of the International Astronomical Union

sponsored a symposium on Spectrum Formation in Extended Stellar Atmospheres held 16 - 19 April, 1969.

The host was the Observatory of the University of

Munich. A major problem is the definition of what

is meant by an extended stellar atmosphere. There

are intuitive notions in the literature, but the

question of specific definitions in various kinds of

objects was discussed. Questions of what specifically cause the anomalous extent were largely bypassed. Attention focused mainly on the type spectrum to be expected in various situations. The

spectral features to be expected in both static and

dynamic atmospheres, including and excluding departures from LTE, were discussed. The symposium was divided into four sections: A. Type of Problems

Which Exist; B. Theoretical Methods for Handling

Non-LTE Problems; C. Chromospheres and Coronae of

Stars; D. Summary. This volume contains the mms of

the formal papers that were presented plus an edited

and abridged version of the discussions following

each paper.

Key words: extended atmospheres; non-LTE; chromospheres; coronae; non-classical atmospheres.
Foreword

The planning of this colloquium on Extended

Stellar Atmospheres was carried out by Dr. A. B. Underhill, President of Commission 36, and the

Organizing Committee of Commission 36, comprised

of Drs. K. H. Bohm, R. Cayrel, J. T. Jefferies,

V. V, Ivanov, R. N. Thomas, and S. Ueno in close

collaboration with Drs. P. Wellmann and H. Groth

whose kind invitation gave rise to the idea. This

colloquium is one of a series that have been sponsored by Commission 36 on problems in stellar

atmospheres. The aim of these colloquia has been

to hold a working session among a number of people

vitally interested in the problem, and to provide

an inexpensive and readily accessible volume of

the proceedings, which can be used by all interested in the subject, especially students and

young researchers. A number of United States federal laboratories having strong programs in astronomy have been hosts to these colloquia and editorspublishers of the proceedings. These include the

National Bureau of Standards through the Joint Institute for Laboratory Astrophysics, the Smithsonian Astrophysical Observatory, and now the Goddard

Space Flight Center. These proceedings, published

by NBS for JILA, follow the approach explored with

the publication of the proceedings of the Symposium

on Wolf-Rayet Stars, hosted by JILA in 1968. Dr.

R. N. Thomas, aided by Alice Levine , of the JILA

staff have provided the link between the formal

editors, Drs. Wellmann and Groth, and the NBS publications group, under Betty L. Oberholtzer. The

typescript was prepared by Breta Triolo.

Lewis M. Branscomb, Director

iii
CONTENTS

Preface vii

Participants viii

Part A. TYPE OF PROBLEMS THAT EXIST 1 Chairmen: P. Swings P. Wellmann

A. B. Underhill Definition of the Types of Problems that Exist in Steady-state

Extended Atmospheres (introductory

paper) 3

R. N. Thomas Definition of the Physical Problems Connected with Extended

Atmospheres 3 8

K. H. Bohm and

J. Cassinelli:

Extended Atmospheres of Planetary

Nuclei 54

H. Nussbaumer: The N IV X5820 Multiplet in WN

Stars 61

R. Grant Athay

and R. C.

Canfield

:

A Self-Consistent Model Atmosphere

Program with Applications to Solar

01 Resonance Lines 65

Part B. THEORETICAL METHODS FOR HANDLING NON-LTE

PROBLEMS 8 5 Chairmen: R . N. Thomas^ A. G. Hearn

G. Rybicki Theoretical Methods of Treating

Line Formation Problems in Steadystate Extended Atmospheres (introductory paper) 8 7

C. Magnan: Application of Monte Carlo Methods

in Transfer Problems 119
W. Kalkofen: Line Formation in Moving Atmospheres 120

D. H. Menzel Laser Action in Non-LTE Atmospheres 134

H. P. Jones and

A. Skumanich:

Line Formation in Multi-Dimensional Media 138

H. Gerola and

N. Panagia:

The Continuous Spectrum of Hydrogen in a Low-Density envelope 171

R. Grant Athay Bandwidth Requirements in Spectral

Line Transfer Calculations 1 7 9

I. P. Grant and

G. E. Hunt:

Comparison of Discrete Space and

Differential Equation Methods in

Non-LTE Transfer Problems 18 9

T. R. Carson: Coherent Line Formation with Depth

Dependent Parameters 226

Part C. CHROMOSPHERES AND CORONAE OF STARS 23 9 Chairmen: j. c. Pecker^ D. G. Hummer

F. Praderie What Do We Know Through Spectral

Information on Stellar Chromospheres and Coronas? (introductory

paper) 241

R. N. Thomas What Should We Do To Know More

About Chromospheres and Coronae of

Stars? (introductory paper) 259

H. G. Groth: Observations of c Aurigae Stars

and Their Interpretation 283

P. Wellmann The Ionization in Nova Atmospheres 290

v
A. H. Vaughan, Jr. Circums tellar Call K lines in G,

and A. Skumanich: and M Giants and Supergiants 29 5

R. W. Hillendahl: A Physical Mechanism for the Generation of Extended Stellar Atmospheres 300

Part D. SUMMARY 3 21

J. C. Pecker: Concluding Remarks 323

vi
PREFACE

This colloquium was one of similar colloquia

of Commission 36 held in the past few years. The

president of Commission 36, Professor A. B. Underhill,

planned this colloquium, while the staff members of

the Universitats-Sternwarte Munchen were responsible

for the local organization. There was an urgent need

to discuss problems connected with the structure of

extended atmospheres and line formation in these

atmospheres

.

Each session of the colloquium had one or two

introductory papers followed by discussion. The

number of contributed papers was limited.

There was no case where the chairman was forced

to cut off the discussion because of lack of time,

but there was an important criticism by J. C. Pecker

and R. N. Thomas: "This colloquium has absolutely no

point, if it is so formal that no one ventures to say

anything except the older people who think they know

what they are doing anyway. It is very important

that the younger people take part in the discussion."

When compiling these proceedings we have tried

to report the discussion remarks as far as possible,

when they were relevant to the problem. The paper

by G. Rybicki arrived only as the manuscript of this

conference was going to press. We were forced to

omit the discussion of this paper from the proceedings because the discussion could not be understood

without the paper.

We are very grateful to Anne B. Underhill for

the decision to hold this colloquium at Munich. We

thank all participants of the conference for coming

to our institute.

H. G. Groth

P. Wellmann

vii
PARTICIPANTS

R. C. ALTROCK, Sacramento Peak Observatory, Sunspot,

New Mexico 88349

R. G. ATHAY, High Altitude Observatory, Boulder,

Colorado, 80302

K. H. BOHM, Department of Astronomy, University of

Washington, Seattle, Washington 98105

R. C. CANFIELD, Sterrewacht Sonnenborgh, Servaas

Bolwerk 13, Utrecht, Netherlands

F. CAPUTO, Laboratorio di Astrofisica, Casella

Postale 67, 00044 Frascati (Roma), Italia

T. R. CARSON, University Observatory, Buchanan

Gardens, St. Andrews, Scotalnd, U.K.

J. M. le CONTEL, Observatoire de Paris, 61 Avenue

de 1'Observ. Paris 14 e , France

Y. CUNY, Observatoire de Paris, Section d 'Astrophysique, 92 Meudon, France

A. M. DELPLACE, Observatoire de Meudon, Meudon 92,

France

S. DUMONT, Observatoire de Meudon, 9 2 Meudon, France

G. H. ELST, Sterrewacht, Ringlaan 3, Brussel 18,

Belgium

H. GEROLA, Universidad de Buenos Aires, Facultad de

Ciencias Exactas y Naturales , Depto de Fisica,

Buenos Aires, Argentina

N. GOKDOGAN, University Obs . Astronomi Kursiisii,

Beyazit, Istanbul, Turkey

G. GONCZI, Laboratoire d ' Astrophysique , Faculte des

Sciences, Av. Valrose, 06 Nice, France

I. P. GRANT, Atlas Computer Laboratory, Chilton,

Didcot Berkshire, England

H. G. GROTH, Universitats-Sternwarte Munchen, 8 Miinchen 80, Scheinerstr . 1 , Germany

A. G. HEARN, Culham Laboratory, Abingdon, Berkshire,

England

R. HERMAN, 7 bis rue Trudon, 9 2 Antony, France

T. HIRAYAMA, Tokyo Astr. Obs. Univ. of Tokyo, Mitaka,

Tokyo, Japan

R. W. HILLENDAHL, Lockheed Palo Alto Research

Laboratories 3251 Hanover Street, Palo Alto,

California 94304

M. HOTINLI, University Obs. Astronomi Kiirsusu,

Beyazit, Istanbul, Turkey

L. HOUZIAUX, Department d 'Astrophysique , Faculte

des Sciences, Plaine de Nimy, Mons , Belgium

D. G. HUMMER, JILA, University of Colorado, Boulder,

Colorado 80302

K. HUNGER, Institut fur theoretische Physik,

Technische Univ. Berlin, 1 Berlin 10, ErnstReuter-Platz , Germany

viii
G. E. HUNT, Atlas Computer Laboratory, Chilton,

Didcot, Berkshire, England

W. KALKOFEN, Harvard and Smithsonian Observatories,

Cambridge, Massachusetts 02138

H. LAMERS , Sterrewacht Sonnenborgh, Servaas Bolwerk

13, Utrecht, Netherlands

A. MARTINI, Laboratorio di Astrofisica, Casella

Postale 67, 00044 Frascati , Italy

CHR. MAGNAN, Institut d

1 Astrophysique , 98 bis BD

Arago, Paris (14 e ), France

D. H. MENZEL, Harvard College Observatory, Cambridge,

Massachusetts 02138

D. MUGGLESTONE, Department of Theoretical Physics,

University of Queensland, Australia

L. NEVEN, Sterrewacht, Ringlaan 3, Brussel 18,

Belgium

H. NUSSBAUMER, Physics Department, University

College, Gower Street, London W.C.I. , England

D. E. OSTERBROCK, Department of Astronomy, University

of Wisconsin, Madison, Wisconsin

N. PANAGIA, Laboratorio di Astrofisica, Casella

Postale 67, 00044 Frascati, Italy

J. C. PECKER, Observatoire de Meudon, Place Jules

Janssen, 92 Meudon, France

A. PETON, Observatoire de Meudon, Meudon 92, France

E. PEYTREMANN, Observatoire de Geneve, 129 0 Sauverny,

Switzerland

F. PRADERIE, Observatoire de Meudon, 9 2 Meudon,

France

L. PUPPI, Laboratorio di Astrofisica, Casella Postale

67, 00044 Frascati, Italy

F. QUERCI, Observatoire de l'Universite Toulouse, 1.

Avenue ^Cami lie , Flammarion

M. RUDKJOBING, Ole Romer Observatory, 800 Aarhus C,

Denmark

G. RYBICKI , Smithsonian As trophysical Observatory,

60 Garden Street, Cambridge, Massachusetts 02138

A. SKUMANICH, High Altitude Observatory, Boulder,

Colorado 80302

D. W. STIBBS, University Observatory, Buchanan

Gardens, St. Andrews, Fife, Scotland

P. SWINGS, Institut d 'Astrophysique , Cointe Sclessin,

Belgique

R. N. THOMAS, JILA, University of Colorado, Boulder,

Colorado 80302

A. B. UNDERHILL, Sterrewacht, Servaas Bolwerk 13,

Utrecht, Netherlands

R. VIOTTI, Laboratorio di Astrofisica, Casella

Postale 67, 00044 Frascati, Italy

P. WELLMANN, Universitats-Sternwarte Miinchen 8 Miinchen 80, Scheinerstr . 1 , Germany

ix
LOCAL PARTICIPANTS

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USW = University Observatory of Munich

MPI = Max Planck Institut for Astrophysics Munich

x
PART A

TYPE OF PROBLEMS THAT EXIST

Chairmen: P. Swings, P. Wellmann

DEFINITION OF THE TYPES OF PROBLEMS

THAT EXIST IN STEADY-STATE

EXTENDED ATMOSPHERES

by

Anne B. Underhill*

Astronomical Institute

Utrecht 3 N ether lands

ABSTRACT

In section 1 practical details concerning the

equivalence of observational and theoretical descriptions of stellar spectra are reviewed, particularly

the difficulty of identifying the observed reference

level (continuum) with the theoretical continuum in

the case when many lines are present. In this con- nection thought must be given to how integrals over frequency should be normalised and evaluated because

the effective continuous absorption coefficient does

not remain constant over the range from 0 to 00

. The

choice of spectroscopic details by which to determine

Teff' 1°9 9 an<3 abundances requires careful consideration .In section 2 the factors by which an extended

atmosphere are recognized are summarized and the

question is posed do all stars have extended atmospheres. Another question requiring an answer is

whether the concepts microturbulence and macro turbulence are physically real concepts or whether they

are merely fitting parameters to make a simple LTE

theory account for the observed spectra of supergiants in which rather wide lines occur and many

multiplets show rather steep gradients. In section

3 the types of line sensitive to non-LTE conditions

are described. These are resonance lines, lines

arising from metastable levels, subordinate lines for

which the upper level is sufficiently separated from

the continuum and other levels that this upper level

is chiefly populated by radiative processes from the

ground or other low lying levels and lines which go

Now at NASA Goddard Space Flight Center,

3
into emission in low density atmospheres as a result

of optical-pumping (fluorescent) processes. Such

lines should not be used for abundance determinations

by means of LTE theory though this is frequently

done

.

Theoretical considerations are discussed in

Section 4 where first the problem of the two-level

atom is sketched and then the problem is generalised

to a many-level atom. The parameter A which gives

the probability that a photon is lost from the line

by de-excitation processes other than spontaneous

emission is defined and it is pointed out that non- LTE physics has the effect of adding a scattering

term to the expression for the source function. One

example is given of the effect of changing the line source function from the Planck function to a form

suitable for isotropic coherent scattering. The line

becomes deeper and wider for the same number of

atoms. Interpretational problems in stellar spectra

are discussed in section 5. It is noted that many

lines in main-sequence early type spectra show the

effects of departures from LTE. These effects are shown to a conspicuous degree by the spectra of shell

stars. The example of He I 5876 in 10 Lacertae, 09V,

is discussed and the implication for interpreting

the He I lines in all B type main-sequence stars are touched upon. Helium-weak and helium-strong spectra

probably indicate variations in density of the outer

atmosphere rather than true abundance differences.

The spectra of supergiants are also considered and

it is pointed out that the la supergiants of type B

may be hydrogen-poor.

Finally in section 6 the problem of choosing

simplified physical representations of line forming

when non-LTE physics must be used is discussed. Some

relevant points concerning the observed spectral

lines used for spectral classification are illustrated by means of partial energy-level diagrams.

Key words: interpretation of stellar spectra,

extended atmospheres, line formation.

I. INTRODUCTION

This colloquium is concerned with finding a physical and mathematical description of how the

stellar spectrum is formed in an atmosphere that is

"extended" but in a steady state. We first need to
decide what are the important physical characteristics implied by the word "extended." Secondly a definition of "steady state" should be agreed upon,

and thirdly some description should be given of what

is meant by spectrum. The spectrum of a star com- prises strong absorption lines, weak absorption

lines, and continuous spectrum as well as emission

lines; each of these characteristic parts of a stellar spectrum has properties determined by the

interaction of each atom, ion, or molecule with the

radiation field to produce the stellar spectrum.

The ideas we use and the words by which these

ideas are transmitted are in many cases rather vague

and contradictory. Something of value will have been

accomplished at this colloquium if we are able to

make our ideas and the words used to describe them

more precise. Even more will have been accomplished

if we are able to see how existing methods of analysis can be used, and to define rather sharply the

physical as well as the mathematical aspects of our problems. In what follows I am going to speak as a stellar spectroscopist anxious to obtain information

about the physical conditions in stellar atmospheres

from a quantitative study of the parts of the stellar

spectrum available to me. The standard way of proceeding is to compare observed spectroscopic observations with theoretical predictions resulting from an explicitly defined physical and mathematical model.

If agreement is found then one says the parameters

of the model are characteristic of that part of the

stellar atmosphere involved in forming the part of

the spectrum under consideration. If no agreement

is found then the model must be changed.

One must consider carefully the meaning of the

word "agreement" used in this context. Clearly the

variation with wavelength of any predicted intensity

distribution which is the same as the observed spectral distribution within the uncertainties of the

measurements must be considered to offer a possible

solution to our problem. A choice between several

(at first glance possible) models can be made by

using a well thought-out selection of spectral details for the comparison.

With stars one observes the total radiation

field from the part of the star facing the observer;

thus the variation of an intensity distribution with

wavelength corresponds to the variation of the theoretical quantity F v with wavelength. Furthermore

because of practical difficulties the observed intensity distributions are not given in absolute

energy units but are expressed as relative inten5
sities. In the case of line profiles, the observed

intensities are expressed as fractions of the intensity that would have been available if no line was present. The reference level of intensity is

known as the continuum; it is found by drawing a smooth curve through the intensity level of parts of

the spectrum where no absorption lines appear to be.

When one interprets the observed variation with wavelength of the intensity in the continuous spectrum,

which means comparing intensities over a range of

several thousand angstroms, the intensity at some particular wavelength is chosen as reference point.

It is important to keep these practical definitions in mind when comparing theoretical and observed

spectra because often what seems to be an obvious

theoretical level of reference is not what is in

practice used. For instance the simple concept of a continuous spectrum and a few superimposed absorption

lines is useful only when the lines are too few in

number to obscure the trend of the continuous spectrum with wavelength. This is so over much of the

normally observed spectral range for 0 and B stars.

In these cases it appears to be straightforward to

compare observed and predicted spectra of early type

stars. However, if one considers the far ultraviolet

spectral range for OB stars, 912& < X < 1900&, the

spectra are full of strong lines and the definition

of a theoretical continuum that corresponds to the

observed datum line selected as the continuous spectrum may be difficult. This problem is familiar to

those who attempt to interpret the spectra of stars

of types F and later in the spectral region X < 5000A.

Because the observed quantity is a relative

variation of intensity with wavelength, one theoretical parameter of interest is (&v+ K v)/ Ko where £ v represents the total absorption coefficient due to

all possible lines and k v represents the absorption

coefficient due to all possible sources of continuous

absorption at frequency v. Here k q is the total

absorption coefficient due to continuous sources of

opacity at the reference frequency v Q . In all theoretical studies of line formation done to date it

has been assumed that one may replace k v by k0 , whatever the value v - v Q . With non-coherent processes

the appropriate expressions for the emergent intensity contain integrals over v from 0 to 00 with

(& v + k v ) as a varying parameter. When these expressions are evaluated (typical expressions are the

kernel functions of Avrett and of Hummer) use is made

of the fact that the variation with v of fc v is given

6
by a function which is normalised over the range 0 to

00 . However in this range k v is also normalised and

the shape function varies from zero to a maximum

value to zero again. With non-coherent processes,

which indeed are what occur in stellar atmospheres,

the transport of radiation at different frequencies

rather far from the line center depends quite sen- sitively on what sort of exchanges may take place in

the wings of the lines. It may be a minor point,

but it seems to me that we might consider whether

in some of the lines it is reasonable to replace

l v + k v by £ v + K o> that is, to assume that we can write the total absorption as & 0 (<|>v + 3) where 3 is

a constant which is k 0/& 0 and <£ v is a normalised

shape function. The rate of change of k v with v might be closely the same as that of £ v over an important part of the range and furthermore both quantities may have comparable values over part of the

range. Whether such considerations are important

depends upon the ratio of i v to k v at the line center. If this ratio is large, l v and k v will not

attain comparable values until v - v Q is large. Then

the difference between k v and k q may be significant

and an asymmetry might occur between the two sides of

the line profile as it is defined observationally

.

These considerations should be held in mind when deciding upon the appropriate quadrature formula for

the frequency integrals, a point which will certainly

come to discussion later in the conference.

The parameters that are sought to describe the

stellar atmosphere are factors such as the temperature, pressure, and density as functions of depth in

the atmosphere and the composition of the atmosphere.

The parameters that describe the star are effective

temperature and surface gravity. In principle these

can be found by studying spectral features that are sensitive to the temperature distribution in the

stellar atmosphere and to the density distribution

so long as the temperature distribution is determined

by the total radiative flux passing through the atmosphere and the pressure structure is determined by

the need to maintain hydrostatic equilibrium against

the acceleration of gravity. It is by no means obvious which spectral features are most suited for

these purposes at each spectral type though the well

known spectroscopic type and luminosity criteria

serve as an empirical starting selection.

A star with an extended atmosphere usually

refers to a supergiant or a shell star. However this

concept requires more detailed consideration which

will be given below. So far as the words "steady

7
state" are concerned, the intent is to restrict the

discussion to stars having atmospheres in which spectroscopic changes do not occur, or occur so slowly

that the process of change does not need to be con- sidered. We have arbitrarily excluded from discussion rapidly changing, evolving situations. Furthermore in the case of pulsating variables we are not

seeking at this time to find the cause of the observed regular changes in light and spectrum so much

as to find out what sets of physical variables correspond to the various recurring sets of spectroscopic

phenomena. This problem has a known solution if one may assume that the physical variables and the observable spectroscopic details are bound together as though thermodynamic equilibrium existed at each

place in the atmosphere. However, the values of

temperature, pressure and density found in this way

are not consistent with the hypothesis of local

thermodynamic equilibrium. Furthermore the motion of

the atmosphere may have a significant effect on the

line shapes and strengths.

II. FACTORS BY WHICH AN EXTENDED ATMOSPHERE

IS RECOGNISED

The concept "an extended atmosphere" grew up

empirically as a result of comparing details in the

spectral type, thus effective temperature, and interpreting the differences in spectrum in terms of

LTE theories applied to a single layer of gas open

to interstellar space and irradiated from below by

the flux emerging from the rather dense layer known

as the photosphere. The gas pressure in supergiant

reversing layers was found to be approximately 100

times less than that in main sequence reversing

layers. Simple geometric considerations indicate

that the reversing layer of a supergiant has a con- siderably greater size than that of a main sequence

star of the same effective temperature. Hence the

origin €>f the term "extended."

Many of the extra absorption lines, which appear

in the spectra of some B type main sequence stars

and lead to the classification of these stars as shell stars, are very like lines in the spectra of

A or late B type supergiants. From this observed

fact has come the idea that the atmospheres of shell

stars have even greater extension and lower density

than the atmospheres of supergiants.

The conclusion that the pressure is lower in a

8
supergiant atmosphere than in a main sequence atmosphere comes from the facts that in supergiant spectra the Stark-broadened wings are not strikingly

present for H or He I lines and the forbidden He I lines do not appear. Also the lines in the second

spectra of the metals are stronger in supergiant

spectra relative to the lines from the first spectra

of the metals than is the case for main sequence

stars of about the same temperature. Further differences between absorption lines in supergiant spectra

and those in main sequence stars are that in supergiants the lines have rather wide, steep-sided profiles and that when the spectra are compared multiplet by multiplet it is seen that the supergiant

lines, in spite of their greater equivalent width

appear to lie on the Doppler part of the curve of

growth rather than on the transition part where many

of the lines in main sequence spectra lie. This

effect is known as the gradient effect; it was interpreted by Struve and Elvey 1 in terms of microturbulence. When Struve 2 noticed that the profiles

of the strong lines in supergiant spectra were wider

than could be accounted for by the velocity fields

deduced from curves of growth and the hypothesis of

microturbulence , the concept "macro-turbulence" was introduced to account for the extra width. These

attempts to reconcile a simple LTE theory of line

formation with what is observed in the case of stars

having extended atmospheres have lead to the introduction of physical concepts that are difficult to

justify in detail. It is time to ask whether the

observed phenomena—steep gradients and rather wide

lines in supergiant spectra—should not be interpreted in terms of an improved and more realistic

theory of line formation.

In the case of shell stars, absorption lines

of the second spectra of the metals very like those

in supergiant spectra may be found but the hydrogen

lines have much deeper cores, in some cases going

nearly to zero intensity, and emission wings are seen for the first few members of the Balmer series.

The strong lines of Fe II also often have emission

wings. In some stars the relative intensities of

the He I lines from the levels 2

3 S, 2

l S, 2

3 P and 2

2 P are not the same as for main sequence stars, the

lines from 2

3 S and 2

X S being stronger than normally

expected according to the observed strengths of lines

from 2

3 P and 2*P. Struve and Wurm 3 named such

changes in relative line strengths dilution effects

and they showed that the relative populations would

change in a low density gas in which the radiation

9
density was reduced from its thermal equilibrium

value in such a way that the atoms tend to accumulate

in the metastable levels. Consequently whenever the

lines arising from metastable levels are seen to be

unusually strong in a stellar spectrum, one speaks

of an extended atmosphere. The theory of Struve

and Wurm is a simple version of the type of theory

one must consider when it is not appropriate to assume LTE

.

In summary, the qualitative characteristics of

a stellar spectrum that lead to the inference of the

presence of an extended atmosphere are (1) broad,

steep-sided absorption lines, (2) little or no Stark

broadening of the H and He I lines and the absence

of [He I] lines, (3) a steep gradient in multiplets

of strong lines, and (4) exceptionally great relative

strength of absorption lines arising from metastable

levels. In an extended atmosphere the density is

sufficiently low that one cannot assume that collisional processes will establish LTE level populations .A phenomenological description of these ideas

coupled with pertinent references to the details seen

in stellar spectra has been given by Struve. 4 He

asked nearly the same questions as those facing us now.

1. Why do some stars possess tenuous outer

atmospheres or shells while other stars, apparently

of identical physical characteristics, do not have

such shells?

2. What is the origin of a shell and how is it

supported in apparent violation of the laws of

mechanics?

3. How can we account for the remarkable

tendency of nearly all shells to vary either periodically or, more often, in an irregular manner?

4. Why do some shells expand while others are stationary?

The first question should be modified to enquire

whether all stars have extended atmospheres and to

ask what factors make these extended atmospheres

visible in the normally accessible spectral region.

It is essential to define accurately criteria that

are sensitive to the presence of a tenuous outer

atmosphere. The low density of particles in an ex- tended atmosphere surrounding a star and the . departure of the radiation field from the black-body

distribution appropriate to the electron temperature

make it necessary to abandon the hypothesis of LTE.

We would like to have a calibration of the strength

and shape of suitable lines in terms of the density,

10
electron temperature, and size of the extended atmosphere. Another relevant factor will be the velocity

field in the line of sight. An extended atmosphere

can be detected only if spectral lines of adequate

sensitivity to the conditions in an extended atmosphere happen to fall in the spectral region under

.study. Observation has already shown that extended

atmospheres of widely different properties exist

around many stars. The corona of the sun is an example of an extended atmosphere that can be detected only by very special observations, whereas the

extended atmospheres of Wolf-Rayet stars can be detected by simple low-dispersion observations. All

stars may possess extended atmospheres in the general

sense; we should like to find out what spectroscopic

phenomena at each spectral type give the clearest

evidence of the presence of the extended atmosphere

and of its physical properties. The complementary

question (what spectroscopic details in a stellar

spectrum can be interpreted reliably in terms of

simple LTE theories of spectrum formation?) is also

not without interest. Such features can be used with

the existing methods of analysis to obtain an estimate of the physical conditions and the abundances

of the elements in the parts of the stellar atmosphere where the density is high enough that LTE is

a reasonable hypothesis.

III. THE TYPES OF SPECTRAL LINE SENSITIVE TO

NON-LTE CONDITIONS

Spectral classification and the recognition of

stars with extended atmospheres is based on the apparent relative strengths of selected absorption

lines. These relative strengths are chiefly determined by the central intensities of the lines; the

width of the line and the exact shape of the profile

are less important factors. The central intensity

of a strong line is determined by the value of the source function at the edge of the atmosphere. In

the case of a two-level atom one can write

S^ oc n 2 n 1 (1)

where n 2 is the population of the upper level and n 2 is the population of the lower level. In the case of

a normal atmosphere the balance of radiative and

11
collisional processes results in certain values of

n 2 /n 1 through the atmosphere, thus in a certain cen- tral intensity for the line. If in an extended

atmosphere, owing to the lower density and the

changed radiation field, the ratio n 2 /n x is reduced,

a deeper line will result and the line will be said

to have become stronger. If the ratio n 2 /n 1 is increased strongly, the line may appear in emission.

Lines that go into emission are those for which the

upper level can be populated by absorption of line

radiation generated in an intrinsically strong line

such as Lyman a of H or He II 303. In the literature

of the 1930 's and 1940 's this process was called

fluorescence. Its occurrence is a sure indication

that non-LTE effects are important in at least part

of the stellar atmosphere. (In this discussion it

is implicitly assumed that the continuous absorption

is small with respect to that in the line center and

that it remains essentially unchanged by changes in

the density of the atmosphere.)

However, it is not sufficient to focus attention

on the changes in population of only one level; the

change in source function is brought about by the

relative change of n 2 to n 1# The populations of levels 1 and 2 can vary at different rates under different circumstances. The type of variation depends

explicitly upon the spectroscopic description of levels 1 and 2 and the sizes of the collision and

radiative cross sections between these levels and all

other levels in the atom or ion including the con- tinuum and doubly excited states lying above the primary ionization limit. All such doubly excited

states do not autoionize with high probability.

The types of line which will be sensitive to

non-LTE conditions are (1) resonance lines, (2)

lines arising from metastable levels, (3) subordinate

lines for which the upper level is sufficiently

separated from the continuum and other levels that

this upper level is chiefly populated by radiative

processes from the ground or other low lying levels,

and (4) lines that go into emission in low density

atmospheres as a result of the optical pumping

(fluorescent) processes mentioned above. Resonance

lines are sensitive to non-LTE conditions because the

spontaneous transition probabilities from neighbouring levels into the ground level are large for permitted dipole transitions whereas often the energy

differences from the ground level to neighbouring

levels are sufficiently great that the probability

of collision-induced transitions is small when

electrons having energies corresponding to tempera12
tures of 10,000 to 20,000 degrees are considered.

Electron temperatures of this size are expected in

the atmospheres of A and B stars. Somewhat similar

arguments hold for metastable levels, although it may

happen, as with He I, that the lowest metastable

level is within easy "collision distance" of the continuum. In that case, when the density is not too

low, the population of the metastable level may

approach its thermal equilibrium value for the local

electron temperature because the level is strongly

coupled through collisions to the continuum.

Lines sensitive to non-LTE conditions include

many of the lines that are used for classifying stars

of types A, B, 0 and Wolf-Rayet. Some examples are

the Balmer lines of hydrogen; the He I lines from

2

3 S, 2

J S, 2

3 P and 2

l V; the He II lines from n = 3 and n = 4; C III 4647, 50, 51 and 5696; C IV 5801,

12; N III 4634, 40, 41; N IV 3478, 82, 84 and 4057;

N V 4603, 19; O 17771, 74, 75; Ca II H and K; Mg II

4481; Si II 4128, 30 and 4200; Si III 4552, 68, 74

and Si IV 4088 and 4116. In the cooler stars and

in shell stars, strong lines arising from the metastable levels of the ground configuration of Cr II,

Mn II, Fe II, and Ni II influence strongly our classification of stars. In each case both the lower and

the upper level of the line are sufficiently isolated

that radiative processes as well as collisional processes are important in establishing the population

ratio n 2 /nj . In a few cases (for example Ha, He II

4686, C III 5696, and N III 4634, 40, 41) particular

radiative processes are known to generate emission

lines by causing an overpopulation of the upper level

of the line. In other cases the ratio n 2 /n 1 de- creases in extended atmospheres relative to its value

in normal atmospheres with the result that deeper

(stronger) than normal absorption lines are seen. Most of these changes are due to the decreased density. An increased density in the atmosphere (more

collisions) forces the population ratios towards the

LTE values. The intensities of none of the above

lines should be interpreted in terms of abundance

without first making some investigation of the effects of non-LTE and how the relevant level populations depend on the density, temperature, and the

available geometric path length.

That the level populations of H and He are sen- sitive to non-LTE conditions in those parts of the

atmosphere where the strong lines are formed, that

is in regions where the electron density is less

than 10, has been shown by Strom and Kalkofen, 5 ' 6 Kalkofen and Strom, 7 Kalkofen, 8 by Mihalas 9

' 10

13
and by Mihalas and Stone. 11 Most of these studies

are concerned with the continuous spectrum, of earlytype stars. The effects on the line spectrum have

not been investigated in detail. Hearn (in press)

has studied some effects of non-LTE conditions on

the strengths of the He I lines. All of these investigations lead to the inference that the strengths

of the classification lines are very sensitive to

the density in the atmosphere. The sensitivity to

temperature changes is much less. These facts are used empirically for recognising supergiants and

shell stars. The calculations referred to were made with models believed to represent main sequence

stars. It is clear that non-LTE effects are important for understanding the meaning of the spectral

classification of main sequence A, B, and 0 stars as well as that of supergiants.

IV. THEORETICAL CONSIDERATIONS

In the case of an atom with two bound levels

only and no continuum, the equation of transfer within the line can be written for plane parallel layers

as ydl (t ,y)

with

S(t) = [1 - A ( x ) ] f <j> v (T)J

v (x)dv + X(t) B

v

( t ).(3)

0 Here t is the optical depth at the center of the

line, <J> v(t) is a normalised shape function for the

line, B v

(x) is the Planck function and y,I v and Jv have their usual meaning. The quantity X(t) is the

relative probability that a line photon is lost to

the line as a result of collisional de-excitation,

X(t) = n

eC 21 / (A 21 + ne

C 21 ), (4)

where C 21 is the cross section for collisional deexcitation, ne is the electron density at level

and A 21 is the Einstein probability for spontaneous

14
depend upon the

X is a function of

emission. Since both ne and C 21 electron temperature at level t,

T . In the case of many-level atoms, re-emission at

frequencies within the line may occur because of

spontaneous emission and from the energy field

associated with the electrons. This possibility can

be taken into account formally by writing the source

function in line frequencies as

S(t) = [1 - X(x)]

v

(t) J

v (x)dv + X Bv

(t) , (5

where

X(T)=(n

eC 2 i+ I A +D)/(A 2 i+ n C 2 i+ I k k

2k (6

and X is suitably defined. In equation (6) D represents all possible transitions from level 2 that are generated by the radiation field and all collisional

transitions except from level 2 to level 1'. No

general rules can be given for defining X*; it represents photons that are emitted at line frequencies as

a result of the original line photons going into the

energy of the electron gas on one of the alternative

routes out of level 2. The index k in equation (6)

refers to all levels below level 2 into which the

atom can decay be spontaneous emission except level

1. For the lines used to classify stellar spectra

k is a small integer, usually less than four.

Other terms can be added to equations (2) and

(5) to take account of sources of continuous absorption and blending lines. Once the source function

is known, the emergent flux from the plane parallel

layers can be found as the $ - transform for the

case x = 0.

The well known difficulties of this transfer

problem arise because equation (2) is an integrodifferential equation owing to the occurrence of the

quantity

J

V

(T) + 1 - 1 i v (x,y dy (7

in equation (5) and because X and X depend on the

radiation field as does x which depends on the popu15
lation of the lower level on the line, ni (z) . We

have

where aQ is the atomic absorption coefficient at the

center of the line. The level populations are re- lated to the radiation field and to each other by the

demand that a steady state exist. The quantity A may

be defined to be independent of the radiation field

by neglecting transitions out of level 2 generated by

the radiation field. In the case of some extended

atmospheres, photo-excitation processes and photoionizations may be neglected. Then the problem be- comes simpler.

One advantage of writing the source function in

the form of equation (5) is that one sees that there

is a "scattering term" —the first term--and a term

from all other processes. When collisional deexcitation is much more important for the upper level

than spontaneous emission, X 1 and the source function becomes identical with the Planck function.

Then the equation of transfer has the simple form

which is used in all LTE studies. One reason why LTE

calculations usually are unsuitable for resonance

lines, for lines arising from metastable levels, and

for subordinate lines from low-lying levels is that

for these lines A may be significantly different from

unity over much of the atmosphere owing to the low

density in the line-forming regions and the diluted

radiation field.

Finally it should be noted that if the scattering term in the source function dominates, i.e., A and A* are both small, a deeper and stronger line

will result for the same number of atoms than is

found when A and A* are large. A simple example is

shown in Figure 1 where the calculated profile of

Si III 4552 is displayed for the case that the re- emission in line frequencies is according to

Kirchhoff's law (thin line) and when it is given by

coherent, isotropic scattering (thick line) . The

model atmosphere used (Guillaume 1

3

) is a lineblanketed model of approximate type B1.5V, having

Teff = 23 255°K and log g = 4.0. The line absorption

coefficient corresponds to thermal Doppler broadening

and a damping coefficient 10 times the classical

damping constant; the fractional abundance by weight

of silicon is 1.206 x 10" 3. This example is merely

an illustration of the very significant changes in

t(z) a dz

o

(8)

16
100

80

60

20

1

1 1 1 1 1 1 11 1 1 1 1 1 1 1

1 1 1 1

1

-

-

\\ fl

1 \ /

1

11 —

\ / Sill 4552

\ / in Model B13 \^/ scattering LTE III i i 1 i i i i I i i i i I 1.0 0.5 0.0 0.5 1.0 A Figure 1. Predicted profiles of Si III 4552 in a line-blanketed model of type B1.5V for the cases (a)

that the line is formed by coherent, isotropic

scattering (thick line) and (b) that the line is

formed in LTE (thin line)

.

line shape and depth which can occur as the source

function varies owing to changes in A and A*. In

the example the level populations were calculated

according to the Saha and Boltzmann laws. An optical

depth of 1.0 in the center of the line has been

reached when x, the characteristic optical depth in

which the model is defined, is 0.01. The electron

density at the level t = 0.01 is 5.0 x 10 13

.

V. INTERPRETATION PROBLEMS IN STELLAR SPECTRA

Main- sequence Stars

Model atmospheres in radiative and hydrostatic

equilibrium can be constructed that give a continuous

spectrum very like what is observed for B and early

A type main sequence stars. The most satisfactory

models include absorption in the strongest lines as a source of opacity as well as absorption in the con- tinua of H~ , H, He and He+ and electron scattering.

What may be termed the classical method for computing

model atmospheres adopts the hypothesis of LTE in

order to simplify predicting the spectrum. More

17
elegant methods of finding the continuous spectrum

that allows for non-LTE have been developed by

Kalkofen, Mihalas and Strom and by Feautrier 1

[ but

so far as B stars are concerned, the difference of

the predicted continuous spectrum in the normally

observed spectral range from that of a classical main

sequence model is too little to be detected using the

presently available observational material (cf.

Mihalas 1

) . The classical models have been used, wit

an LTE theory of line formation, to predict the

equivalent widths and profiles of absorption lines

in the spectrum and to find abundances of the elements. The method is to choose those abundances that

give a best overall fit between the predicted equivalent widths and the observed values. The technique

is familiar and no detailed comments on this procedure are necessary here.

This type of analysis is a somewhat more sophisticated way to find abundances than simple curve-ofgrowth analysis in that account is taken of the

variation of pressure and temperature with depth in

the atmosphere. However, because the hypothesis of

LTE is used, the chief free parameter that remains

by which to obtain a fit between observed and predicted equivalent widths is the abundance. No

attempt is made to compare computed and observed profiles. A further fitting parameter that is used with

lines on the flat part of the curve of growth is

microturbulence

Sets of abundances in a number of B and early A

type stars have been found by model-atmosphere and

curve-of-growth studies. The observed fact that

about ten percent of the main sequence stars between

types A5 and B3 have absorption line spectra that

differ from each other and from the spectra of most

stars having the same distribution of energy in the

continuous spectrum over the range 3500 to 6700A

leads to differing sets of abundances. Ingenious

attempts have been made to explain these anomalous

abundances in terms of nucleogenesis in the center

of the star and/or spallation on the surface of the

star. However, the whole chain of reasoning is not

very satisfactory and it is time to consider seriously whether we do not have anomalous atmospheres

rather than anomalous abundances; an idea put forward by the author in 1964 (Underhill 1

) . By

anomalous atmospheres it is meant that the process of

line formation, in the case of the lines that lead

to deviating abundances, is not an LTE process as has been assumed. That classical model atmospheres

and the LTE theory of line formation are not truly

18
satisfactory for explaining many details in the

observed spectra of normal sharp-lined main sequence

stars can be demonstrated by comparing observed and

predicted line profiles (Underhi 11 1

7

' ) . It is

easy to show that the radiation in the profiles of

all strong and moderately strong lines in 0 and B

type spectra comes from such high levels in the

atmosphere that the density is too low to ensure that

LTE be established as a result of collisions for the

lines that are studied.

It is chastening to note that in the peculiar

main sequence stars the so-called anomalous abundances have usually been estimated from lines of the

types listed in Section 3 as being particularly

sensitive to departures from LTE. The H, He I, 0 I,

and Si III spectra as well as the second spectra of

the metals and of the rare earths contain metastable

levels. The occurrence of these levels affects the

populations of all the lower levels of the relevant

atom or ion while closely coupled to the metastable

levels. The abundances of Li, Be, Ca, Sr, and Ba are usually estimated from resonance lines. The ionization balance of P II ^ P III is particularly sen- sitive to collisions with He I atoms in the metastable 2

3 S state (Underhill 1

9

) , while Ba II can be

ionized efficiently by La quanta. Furthermore, LTE

theories of line formation and the use of the best

available gf values for Si II lines do not give a satisfactory understanding of the strengths of the

Si II lines in what are usually called normal stars

(Underhill 20 ). The reason for the observed discrepancies in the Si II spectrum is not understood at

present. Clearly many problems remain in the interpretation of the lines from apparently sharp-lined

main sequence stars. Any abundance anomalies of less

than a factor 100 should be scrutinized carefully to see if they do not reflect departures from LTE rather

than true abundance anomalies.

Mention has been made of normal stars. These

are stars with spectra like the spectra of most of

the spectral classification standard stars. It is

going too far to assume without proof that normal

also means that the stronger lines, that is the

classification lines, can be interpreted correctly

by means of the hypothesis of LTE.

The ad hoc fitting parameter microturbulence

may well be an indication that in the case of the

stronger lines A < 1 and consequently a scattering

term must be added to the source function. Such a term gives stronger lines for a given number of * atoms than does a purely LTE source function. Con19
20
cern regarding the interpretation of microturbulence

has been expressed by Menzel 21 and by Jefferies and

Thomas

.

2 2 An extreme example of what sort of discrepancies occur between observed profiles and profiles

predicted assuming LTE is shown in Figure 2 where

the predicted profile for He I 5876 obtained with a line-blanketed model which should represent 10

Lacertae, 09V, well (Underhill 1

7

) is compared with the

observed profile. This observed profile is from only

one IIIM spectrogram, 30 A/mm at 5876A, obtained at

the Dominion Astrophysical Observatory, thus it is

somewhat uncertain, but the uncertainties are not

greater than ± 5 percent, particularly in the center

of the line. Observational error is not the source

of the discrepancy. The predicted profile has been

found assuming that the line profile is given by the

Voigt function with thermal Doppler broadening and a damping constant 100 times the classical damping constant. Such a value for the damping constant is

close to what a more exact representation of the

Stark effect of the 2

3 P°- 33 D line of He I will give.

The predicted equivalent width is 0.344& which is

comparable with the value predicted by Mihalas 23 using the correct Stark broadening theory and a model

of similar pressure-temperature structure. The

equivalent width of the observed profile in Figure 2 is 1.14A; graving 21* has found 1.23A and Mihalas 25

gives 0.90A. No reasonable changes in the model nor adjustments of the line absorption coefficient will

compensate for this discrepancy. It is necessary to use a theory of line formation that takes into

account the fact that X ^ 1 for He I 5876 in the

outer layers of an O type star. The electron density is less than 1.7 x 10 14 in the layers of model

PPB40 which are important for forming the parts of

the profile between ± 1. oA.

Another indication that changes of the density

in the atmosphere (thus changes of the significance

of non-LTE physics for the strength of the He I absorption lines in B type main sequence stars)

rather than gross changes in the helium abundance

may be the explanation of "helium-weak" and "heliumstrong" B stars of the same colour is afforded by

the spectroscopic changes of HD 125823. Bidelman 26

noted that when the He I lines changed strength the

rest of the spectrum remained essentially unchanged.

This was confirmed by Thackeray. 27 The changes are more precisely described by Jaschek, Jaschek,

Morgan, and Slettebak 28 who find that the variations

in the spectrum of HD 125823 can be represented as an

21
oscillation between MK types B2V and B7IV. On high

dispersion spectrograms the Si II and Si III lines are sensibly unchanged in intensity throughout the

variations

Another observational confirmation of the different sensitivity of lines seen in late B type

spectra to non-LTE physics is the variation observed

by Struve and Swings 2 ^ and by Struve 30 of the

strengths of the Ti II, Mn II, Fe II and Ni II lines

in the spectrum of Pleione. Most of these lines are used with unjustified confidence in LTE theories of

line formation to deduce anomalous abundances in

main sequence stars. The spectra of shell stars show

to an enhanced degree the line intensity changes that

have been mentioned above as indicating that non-LTE

physics should be used. In addition the geometrical

arrangement of the extended atmosphere affects the

profiles that are observed.

Supergiants

The chief differences between the spectra of

supergiants and those of main sequence stars having

about the same intensity distribution in the continuous spectrum are that in supergiants (1) the Stark

broadening of the H and He lines is greatly reduced,

(2) all other lines are stronger and wider, and (3)

multiplets generally show a greater gradient. Since

the electron pressure is lower in the atmospheres of

supergiants than in the atmospheres of main sequence

stars, there is even less reason to believe that the

hypothesis of LTE will be adequate for a theory of

line formation.

Similar discrepancies between LTE theory and

observation occur as have been noted for the main

sequence stars. Simple LTE analysis of the equivalent widths of lines in the spectra of B type supergiants (for example Unsold, 31 Voigt 32

) has always

lead to a somewhat larger abundance of helium relative to hydrogen than is found for the main sequence

B stars. This result is at least partly due to the

fact that the observable lines of the He I spectrum

are not in LTE in supergiant atmospheres and departures from LTE tend to make the He I lines deeper.

The hydrogen lines in B type supergiants do not

look like the lines predicted using LTE theory and

classical model atmospheres. Some example of the

discrepancies that occur are shown in Figure 3 which displays profiles taken from the observations

of van Helden, 3 ^ Lamers

,

3 h and Smit 35 of Hy in 3

22
t—i—i—

|—i—i—i—i—

|—i—i—i—i—|—i—i—

r

BO la

J I I I I I I I I I I I I I I I L -5 0 *5 A

Figure 3. Observed HA profiles in B type supergiants

compared with HA profiles predicted by Mihalas . Profile (1) is from a model with Te ff = 10,080°, log g

= 1.0; profile (2) from a model with Te ff = 10,080 , log

g = 2.0; profile (3) from a model with Teff = 14,000 , log g = 3.5; profile (4) from a model with Te ff = 24,000°, log g = 3.5. The wavelength unit indicated

is one angstrom.

23
Orionis, B8Ia,o 2 Canis Majoris, B3la, and e Orionis,

BOIa, compared with profiles computed by Mihalas

2

Mihalas used the best available LTE theory and a normal hydrogen-helium composition. The theoretical

profiles are from unblanketed models with the following characteristics:

Profile T ff ( K) log g

1 10 080 1.0

2 10 080 2.0

3 14 000 3.5

4 24 000 3.5

These models have the lowest values of log g of each

group computed at each effective temperature. It is

generally considered that log g in B type supergiant

atmospheres is of the order of 2.0 although at effective temperatures near 24,Q00°K the condition of

hydrostatic equilibrium cannot be met for log g smaller than about 3.0. If the effective temperatures of the selected supergiants are higher than the

values listed above, say by 2000°, the predicted

profiles for the same value of log g will be a few

percent (< 5) less deep.

The observed profiles of Hy do not have wings

like the predicted Stark-broadened wings. From con- sideration of the fit in the wings alone one would

postulate that the electron densities in the atmosphere are like those given by a model with log g of

the order of 1.5 or less. However, the Balmer series

of hydroaen breaks off at n = 25 in $ Orionis, at

n = 24 o

2 Canis Majoris and at n = 21 in e Orionis.

These values indicate electron densities of the order

of 5.8 x 10 12 to 2.3 x 10 1

. The Mihalas model with

Teff = 10,0 80 °K and log g = 2.0 has electron densities of this order in its outer layers while the

model with log g = 1.0 has electron densities between

10 10 and 10 12

. Consequently one concludes that log

g in the atmospheres of the B type supergiants of

luminosity class la is probably not smaller than 2.0.

The apparent fit of computed profile (1) with the

observed profile for $ Orionis must be fortuitous.

It is clear that the observed Hy profiles are

24
less deep than the predicted LTE profiles over their

whole range. (An extrapolation to lower log g must

be made in the case of the two high temperature

models.). The computed equivalent widths are too

large. Another indication that the profiles do not

have the Stark wings which are expected is that the

break off of the Balmer series is sharp in each of

these supergiants. The wings of the last members of

the Balmer series do not merge noticeably. Struve

and Chun 36 have noticed that the He I lines in the

spectrum of 55 Cygni , B3Ia, give evidence of Stark

broadening indicating that the electron density is

not particularly low in the atmosphere of this

supergiant.

The discrepancies in the centers of the hydrogen

lines are as interesting as the lack of Stark wings.

In each case the line is less deep by 10 to 20 percent of the continuum than what is predicted. In the

center of a very strong line the central intensity

is proportional to the ratio of the source function

at the edge of the atmosphere to the source function

at the depth where the continuous spectrum is formed.

The predicted profiles displayed in Figure 3 have

been computed using LTE theory, thus the computed

residual intensity at the center of Hy represents

approximately the ratio of the Planck function at

the edge of the atmosphere to the Planck function

at the depth where the continuum is formed. The predicted central depths are a representation of the

adopted temperature law in the atmosphere. The work

of Kalkofen, of Strom and Mihalas using main sequence

models indicates that the hydrogen atom will not behave in the atmospheres of supergiants as if it were

in LTE, for the density is too low to maintain SahaBoltzmann populations in all levels. Consequently

the source function in the center of Hy at the outer

edge of the atmopshere may be reduced significantly

from its LTE value. In any case, the LTE value is

the maximum value which can be attained. Let us postulate that the source function in the continuous

spectrum, which in an atmosphere of normal composition is dominantly due to absorption in the Paschen

continuum of hydrogen, is not significantly changed

from its LTE value when more correct non-LTE calculations are made. It follows that acknowledging

that LTE is not valid in supergiant atmospheres leads

us to expect even deeper Hy profiles than have been

predicted, provided that we do not change the temperature structure of the atmosphere greatly. Thus

allowing for departures from LTE will not resolve

the discrepancy found for Hy

.

Motion of the atmosphere such as rotation of

25
the star or large scale random motions in the atmosphere will not resolve the problem either, because

although one may obtain a shallower profile in this

way, the equivalent width is not reduced. The

observed discrepancy is not only one of shape of

profile, but also one of equivalent width.

Another possible solution is to introduce a rising temperatute in the outermost parts of the

model. Such an ad hoc procedure would require re- jection of the condition that the transfer of energy

through the atmosphere is regulated only by the con- dition that radiative equilibrium exist. The fact

that emission is observed at Ha in the la supergiants is, perhaps, an encouragement for proceeding

in this manner, but care would be required not to destroy the rather satisfactory interpretation of the

colours of B type supergiants provided by models

with a temperature that decreases outwards. Furthermore it should be noted that lines of Na I and Fe II are visible in the spectrum of 3 Orionis while the

spectra of o

2 Canis Majoris and e Orionis do not

contain lines suggesting significantly higher levels

of excitation than are seen in B3 and BO main se- quence stars respectively. Another objection

against adding to the model a high temperature outer

layer in which the core of Hy is formed is that at a short distance from the line center, say 5A, one would be outside the Hy line absorption coefficient

of this layer and thus would be able to see through

to the cooler "normal" atmosphere. According to the

LTE predictions one should then see the extended

Stark-broadened wings of the Hy profile formed in the

normal reversing layer. Such wings are not seen. The ad hoc solution of adding a hot layer to the

model seems unsatisfactory, but it must be admitted

that no details have been worked out.

A more promising solution to the problem of the

weak hydrogen lines in B type supergiants of luminosity class la is to postulate that these atmospheres

are hydrogen-poor (Underhill 3

) . If the abundance

of hydrogen is so small that hydrogen is no longer

the chief source of opacity in the region 3650 - 7000A, then the strengths of the Balmer lines should

reflect the hydrogen abundance, cf. the calculations

of Bohm-Vitense

3

Strong Stark-broadened wings will

not be seen because the low abundance of hydrogen

prevents the wings from attaining observable depths

at distances from the line center great enough that

Stark effect dominates the shape of the line absorption coefficient. The rather steep-sided deep cores may still be due to the effect of departures from LTE.

26
If the atmospheres of the B type supergiants are hydrogen-poor, but perhaps not quite so poor as the

helium stars like HD 124448, the chief source of

opacity in the region 3650-700 will be absorption

from the n = 3 levels of He I. These continuous

absorption coefficients vary as v -3

, thus no significant difference in UBV colour is expected between model atmospheres of normal, hydrogen-rich

composition and those of hydrogen-poor composition.

The temperature-pressure structure may well be different since the opacity in the region 50 4 - 912A"

will differ greatly between hydrogen-rich and

hydrogen-poor atmospheres.

The model-atmosphere calculations and lineprofile predictions for hydrogen-poor stars with

effective temperatures of 12,900°, 9,900° and 7350°K

and log g = 2.0 and 4.5 presented by Bohm-Vi tense 3

8

point in the direction of the conclusions sketched

above. The value of the suggestion that the la

supergiants of type B are hydrogen-poor cannot be

fully assessed until more models such as those of

Bohm-Vi tense are available at a greater range of

temperature and gravity and unti'l an improved theory

for predicting line profiles which takes into account

the major effects of departures from LTE is developed. Some models for hydrogen-poor stars in the

needed range of Te ff and log g have been computed

by Klinglesmith 3

9

but they are not generally available.

The results of Bohm-Vitense indicate that

hydrogen profiles of the observed shape can be obtained by reducing the hydrogen abundance in the

atmospheres of the la supergiants of type B by a factor of at least 1000. If the atmosphere is

hydrogen-poor, presumably the whole star is hydrogenpoor. To reach such a state would require mixing

the outer parts of the star with the core in which

hydrogen had been depleted as the result of nuclear

reactions. The hydrogen lines in the spectra of the

lb supergiants of type B are stronger than those of

the la supergiants. It may be that the lb supergiants are hydrogen-rich stars at the beginning of

their departure from the main sequence while the la

supergiants are much further evolved.

VI. THE CHOICE OF SIMPLIFIED PHYSICAL REPRESENTATIONS

OF LINE FORMING IN NON-LTE

The full problem of line formation in a stellar

27
atmosphere when the hypothesis of LTE cannot be made

is too complex to be solved in the most general

manner, for it involves the simultaneous solution of

coupled transfer equations in all frequencies of

interest and the equations of statistical equilibrium. Furthermore the atmosphere must be taken to

consist of at least three species of atom: hydrogen,

helium and the atom under consideration in several

stages of ionization. This last requirement occurs because in some spectra the observed line intensities are affected by energy coincidences between the

observed energy levels and radiations such as Lyman

a of hydrogen or of He+ or because collisions with

abundant excited atoms such as metastable helium

atoms can cause particularly strong ionization or excitation of certain species. Furthermore hydrogen

and helium serve as the chief sources of continuous

opacity

A first simplification may be obtained by postulating that all the levels down to some particular

energy below the primary ionization limit have LTE

populations with respect to the population of the

ion. This step is justified because the higher

levels are relatively close together in energy and

if the electron density is not very low, Boltzmann

populations will be set up as a result of collisions.

The chief problem is to decide how far down LTE populations extend. The populations of all levels below

the selected level must be calculated using the

equations of statistical equilibrium. The number

of terms in these equations can be reduced by a judicious selection of the significant processes

causing transitions between the various energy states

of the atom or ion.

If the spectral lines under study occur between

levels that have excitation energies so high that at

the local electron densities LTE populations prevail,

then adopting the LTE theory of line formation is

probably a fairly good approximation. However, most

of the significant lines in early type spectra do not

fall into this class and the energy-level diagram of

the atom or ion must be studied together with estimates of the radiative and collision transition

probabilities to see what processes are important

in establishing the level populations and the effective value of A. Some examples of the distribution of levels in

essentially one-electron spectra are shown in Figure

4. The ordinate is excitation potential/ionization

potential; the ionization potential of each atom or ion is given. Metastable levels are indicated by

28
an M. More levels occur above the levels which have

been drawn. The transitions that give the characteristic absorption lines observed in stellar spectra

are indicated. Two types of transitions are observed: (1) resonance lines (Li, Ca+

, Ba+)

, and (2)

subordinate lines from the second, third, or fourth

lowest level (H, C+ 3

, N+ \ Si +3, Mg+

) . Clearly resonance lines are particularly sen- sitive to non-LTE physics since they are formed between levels that are well isolated from all other

levels. In the case of Ca+ and Ba+ metastable levels occur between the levels of the resonance lines, and

the whole set of low lying levels makes an isolated

group.

In the case of the subordinate lines, in each

case the lower level is sufficiently isolated that

non-LTE physics is important in determining its

population. Whether the population of the upper

level is seriously affected by departures from LTE

or not is something that requires detailed investigation in each case. The transitions indicated in Figure 4 are Ha, Li I 6708, C IV 5801, 12, N V 4603, 20,

Si IV 4088, 4116, Mg II 4481, Ca II 3933, 68 and

Ba II 4554, 4934.

In the case of essentially two-electron spectra

somewhat similar patterns emerge. Some partial

energy-level diagrams are shown in Figure 5. More

levels occur above the levels that are shown. In the

case of the light elements the intersystem transitions are rather weak and the sets of energy levels

of different multiplicity are pretty well independent

of each other. In the heavier elements, for instance

Ga+

, the intersystem radiative transitions are fairly

strong. In spectra from ions and atoms like those

shown in Figure 5 the presence of metastable levels

will play a considerable role in establishing the

> Figure 4 (pp. 30-31). Scaled diagrams of the lowest

energy levels of some effectively one-electron spectra. The ordinate is excitation potential/ionization

potential. The ionization potential is given above

each column. The transition that gives the characteristic lines observed for each atom or ion in stellar

spectra is indicated. Metastable levels are marked

by an M. >

Figure 5 (pp. 32-33). Scaled diagrams of the lowest

energy levels of some effectively two-electron spectra. Ordinate and notation as in Figure 4.

29
IT) ^"<fr -<*

CM <\l C\J

10 0°CL

-<fr ro CO <M

1

Q-Mo: o

30
Q co M CVJ °CL

• 1^

Q- O CO

Ml

2 £ v

QtO^-L. CO

CVJ C\J CNJ CVJ CVJ

UJl 1— ^

31
n n «

! 8

ro <0 ro

IH

Q_ ft.

CO

ro ro ro H

o o Of) <*> (*) ro fO OJ CM 1

*rVro "cm'oj

a-irr o

32
o1*1in o q

to** <o

CL CL Q CO m ro ro ro

j^to

"* in

1

O)

CO *

ro

ro

^co J2) in'V tj- ro ^ ro

I

I

CVJ

Q to CL ro ro ro

QJICL L

33
level populations. No resonance lines are observed

from spectra of this sort. The chief subordinate

transitions of He I are shown as well as C III 5696

(3 1 ?

0 - 3J D) and 4647, 50, 51 (3 3 S - 3

3 P°); N IV 4057

(3 1 ?

0 - 3

X D) and 3478, 82, 84 (3 3 S - 3

3 P°); 0 I 8446

(3 3 S° - 3

3 P) and 7771, 4, 5 (2 5 S° - 3

5 P); Si III

4552, 68, 74 (4 3 S - 4

3 P°) and Ga II 4251 etc.

(4 3 D - 4

3 P°). The chief absorption lines of Ga II

which are observed (Bidelman and Corliss 40

) come from

the 4

3 D levels.

It is quite clear that the chief lines that are observed in the spectra of He and the ions shown in

Figure 5 come from levels sufficiently isolated that

one cannot assume that the level populations are in

the Saha-Boltzmann ratios and that A = 1 for lines

like those shown. Adoption of LTE methods of analysis to estimate the populations of the upper and

lower levels of triplet lines (quintet in the case

of 0 I) may be justified to some extent because it

appears quite possible that the triplet levels are rather tightly bound to the continuum by collisions.

The difficulty arises in relating these populations

to the total abundance of the element and the populations of the singlet levels (singlet and triplet

levels for O I) . Because of the rather large energy

gaps in the lower part of the energy level diagram,

the populations of the lower singlet levels are most likely not in LTE at electron densities of the

order of 10 13 - 10 14 which are the electron densities found in the atmospheres of main sequence stars.

The 3^° - 3

J D transitions of C III and N IV

appear selectively in emission in Of stars and have

flat-topped profiles in Wolf-Rayet stars. The 3

l D

level is believed to be selectively populated by a radiative process in the case of C III and by a collisional process in the case of N IV.

No procedure can be given here for how to solve

for the desired level populations and line intensities. However the data displayed in Figures 4 and 5 should make it clear that the full non-LTE problem

must be solved and that it is not necessary to con- sider anything like an infinite number of levels or

of transitions between these levels.

One question that remains is how significant

are transport problems for determining the radiation

field that enters into the equations of statistical

equilibrium which determine the level populations.

One way of advancing with the problem might be to

simplify considerably the transfer problem in most

radiative transitions and just retain a full transport representation for the line in question and for

34
one or two related lines. If there is a differential

field of motion in the atmosphere such that over a geometric length corresponding to unit optical depth

in the line the Doppler shift places the relevant

line radiation outside the extent of the line absorption coefficient, the transfer problem can be simplified. Then the atmosphere, or at least well defined

: parts of it, can be treated as if optically thin.

Such a manipulation is done, for instance, in order

to estimate the line shapes to be expected from a spherically expanding atmosphere. However if the

radiation field is completely separated from the

populations of the energy levels, the strengths of

the observed spectral lines can tell nothing about

the physical processes going on or the abundances of

the elements.

VII. SUMMARY

The interpretational problems posed by the

spectra of main sequence stars, shell stars, and

supergiants of types A5 and earlier have been re- viewed. It is indicated that the hypothesis of LTE

is untenable for the interpretation of most lines

that are characteristic for the classification of

these stars. A sketch has been given of how an improved theory might be developed. Particular problems that are encountered are listed in the abstract.

A significant result of these yet qualitative considerations is the realisation that the atmospheres of

the la supergiants of type B may be hydrogen-poor.

In no other way can one obtain weak hydrogen lines

such as are observed and still maintain electron

densities of sufficient magnitude to cause the Balmer

series to break off between n = 21 and n = 25.

ACKNOWLEDGMENTS

I am indebted to the Kitt Peak National

Observatory for the privilege of obtaining the high

dispersion spectra of B type supergiants which

spurred this investigation and to R. C. P. van

Helden, H. J. Lamers , and A. B. M. Smit of the

Utrecht Observatory for deriving line profiles from

these spectra. The theoretical considerations given
here have benefited greatly from many discussions

with A. G. Hearn who spent three months at the

Utrecht Observatory on a fellowship from the

Netherlands Organization for the Advancement of Pure

Research

REFERENCES

1. 0. Struve and C, T. Elvey, A4 th.opky-6 . J. 79 , 409 (1934).

2. 0. Struve, Ai>th,opkyi> . J. 104 , 138 (1946).

3. O. Struve and K , Wurm, Ai>th,ophyi> . J. 88 , 84 (1938).

4. O. Struve, At>th.opkyi> . J. 9_5, 134 (1942).

5. S. E. Strom and W. Kalkofen, At>th.ophy6 . J. 144 , 76 (1966)

6. S. E. Strom and W. Kalkofen, A6th.ophy6 . J. 149 , 191 (1967)

7. W. Kalkofen and S. E. Strom, J. Quant.

Spo.cth.obc. and Rad. lh.anb{ch.. 6_, 653 (1966).

8. W. Kalkofen, Ahth.opkyh . J. 151, 317 (1968).

9. D. Mihalas, Abth.oph.y6 . J. 149 , 169 (1967).

10. D. Mihalas, Abth.ophyb. J. 150 -, 909 (1967).

11. D. Mihalas and M. E. Stone, AbtKophyb . J. 151 , 293 (1968).

12. A. G. Hearn, Won. Not. Roy. Abtnon. Soc.

(in press)

13. C. Guillaume, Bull. Abtnon. Inbt. Ncththlandb

18, 175 (1966)

14. P. Feautrier, Ann. Ap . 31, 257 (1968).

15. D. Mihalas, Abtnophyh . 77 153 , 317 (1968).

16. A. B. Underhill, ?h.oc. I. XT"U. Symp. No. 26

(ed. Hubenet) Academic Press, London (1966)

p. 118.

17. A. B. Underhill, Bull. Action. Inbt. N zthdhlandb

500 (1968).

18. A. B. Underhill, Vtbtab In AbtKonomy (in press).

19. A. B. Underhill, Bull. Ai>th.on. Inbt. Ndtkdh.lo.ndb

19, 537 (1968)

20. A. B. Underhill, Abth.opkyb . J. 151, 765 (1968).

21. D. H. Menzel, Pop. Abth.on. 66 (1939).

22. J. T. Jefferies and R. N. Thomas, AAth.oph.yA . J. 127 , 667 (1958)

23. D. Mihalas, Abth.ophyb . J. Supp. _9 , 321 (1965).

24. G. Traving, 1. AbtKophyb . 41, 215 (1957).

25. D. Mihalas, Abth.ophyb . J. 140 , 885 (1964).

26. W. P. Bidelman, AbtKon. J. 70, 667 (1965).

36
27. A. D. Thackeray, Mow. Not. AA&ion. Soc. S.

A^ica 25, 7 (1966)

.

28. C. Jaschek, M. Jaschek, W. W. Morgan and A.

Slettebak, At>tn.ophyt> . J. 153, L 87 (1968).

29. 0. Struve and P. Swings, Abtnopkub , J. 97/

426 (1943).

30. 0. Struve, AbtKopky*. J. 99, 205 (1944).

Ill:

A. Unsold, Z. Attiopky*. 23, 100 (1944).

H. H. Voigt, Z. k6th.opkyt>. 31, 48 (1952).

33. R. C. P. van Helden, unpublished material kindly

communicated. 1 34. H. J. Lamers , in preparation.

35. A. B. M. Smit, Action, and A&t>iophyi> . (in

press)

,

36. 0. Struve and H. Chun, KhtKophyh . J. 107,

109 (1948).

37. A. B. Underhill, Abtnon. and Ap . 1, (1969).

38. E. Bohm-Vitense, At>tnopky6 . J. 150, 483 (1967).

39. D. A. Klinglesmith, AbtKon. J. 72., 808 (1967).

40. W. P. Bidelman and C. H. Corliss, A^tKopky^. J. 135, 968 (1962)

.

37
DEFINITION OF THE PHYSICAL PROBLEMS CONNECTED

WITH EXTENDED ATMOSPHERES

by

R. N. Thomas

Joint Institute for Laboratory Astrophysics

Boulder3 Colorado

ABSTRACT

The necessity of carefully defining the phenomenological basis for classification of atmospheres

as being "extended" is emphasized, and four alternative bases for such classification are suggested

(1) the necessity to include curvature terms; (2)

the presence of an ejected shell surrounding a cen- tral star; (3) an observational discrepancy between

predicted and observed density gradient; (4) an anomaly between predicted and observed phenomena in

stars with "dynamic" atmospheres such as cepheids

A number of physical problems connected with the

presence of an extended stellar atmosphere are then

categorized according to these alternative bases.

Key words: extended stellar atmosphere, classical

atmosphere model.

The general subject of this conference is

"spectrum formation in stars with extended steadystate atmospheres." The theme of this first day is

"definitions of the problems that exist." Because

we use the term "extended atmosphere," we imply that

these stars differ from the usual kind of star, which

have atmospheres that are "nonextended . " But we must

recognize that most of our experience and physical

intuition rests on our experience with these ordi-

* Of the National Bureau of Standards and the

University of Colorado.

38
nary, nonextended atmospheres. There is the very

real possibility that we may assess erroneously some problem on an extended atmosphere by applying an 'intuition that was developed in the study of nonj extended atmospheres. So at the beginning of this

first day I think we must be very explicit, even

| pedantic, in defining the terms we use to describe

'both the general subject and the problems that we

; think exist.

On this basis, I am not quite satisfied with the

definitions Anne Underhill used, and the implications

| of some of her terms, because it seems to me that her

approach is indeed wholly intuitive. For example,

her most explicit definition is: "A star with an extended stellar atmosphere usually means a supergiant or a shell star." She then defines each of

, these stellar types in terms of certain spectral

features, which we think we can interpret. But most

of this interpretive process rests on our experience

as to how these spectral features would be produced

in ordinary, nonextended atmospheres. Again, Anne

states: "We assume Tef f is the same for main se- quence and supergiant stars. Then a spectral difference between these two classes reflects a pressure

difference in the atmospheres, which implies a difference in radius, thus an atmospheric extension."

and "We have certain types of lines in main sequence

stars, and certain types in supergiants; some stars

combine features of each—these we call shell stars."

There are strong implications in these definitions, but we are so accustomed to making them, when

discussing ordinary stars, that we gloss over them.

The balance of Anne's talk, logically enough, is devoted to a discussion of these spectral features and

her reasons for thinking that our conventional,

classical atmosphere [CA] models cannot describe

them.

I myself think that there are many features

that cannot be interpreted by the classical atmosphere models in stars whose atmospheres are neither

"supergiant" nor "shell." Some of these features

may arise because some parts of the atmosphere are more "extended" than the classical atmosphere model

predicts. Other features may be anomalous on the

basis of the classical atmosphere model, yet their

appearance may have nothing to do with atmospheric

extension.

So I would prefer to have a definition of an extended stellar atmosphere that is more directly

tied to a pictorial, geometrical notion of "atmospheric extent." There are several possibilities

39
for setting up such a definition, and I think we should consider them all. During this symposium

we can use these categories as a reference frame.

Perhaps we will find one particular category is

preferable; perhaps we will find none are very

satisfactory. I will outline my suggested alternatives; you may object and change them as you

like.

A. DEFINITION OF AN EXTENDED ATMOSPHERE

We have several possibilities for this definition: one conceptual, from an a priori standpoint, within the framework of our ordinary models;

one based on some circumstance that produced a configuration of an "unusual" type but to which

we try to apply our "usual" thinking; one based on observations that directly contradict what we would

expect from our "usual" models; and one based on

some configuration we expect to occur but which we expect to lie outside the scheme of our "usual"

models. I will summarize these, then ask which we might adopt.

For reference, first recall that the classical

atmosphere (CA) is defined by: RE (radiative equilibrium) or CE (convective equilibrium) ; HE (hydrostatic equilibrium) ; and LTE (local thermodynamic

equilibrium) . Then we have four alternative classification schemes, each based on some distinction

between the category and some property of the CA.

1. Distinction between an atmosphere where

curvature terms need be included and an atmosphere

where they need not be included. Other than this

difference, we could hold to the CA model, if such

a CA model could give an extended atmosphere.

Note that this was essentially the type of atmosphere considered by Kosirev, Chandrasekhar , and

others during the 1930's when they were trying to

find the change in the spectral distribution of emer- gent radiation due to the effects of extent and

curvature. Note that they did not apply the CA

assumptions literally, but adopted arbitrary density

distributions to get the atmospheric extent; but

they did retain, at least implicitly, the "physics"

of the CA model.

It may be that to get such an atmosphere under

CA, we need to go to small values of ge ff = ^dynamical " 9radiative) / and problems of stability

arise. But this is the direction of approach.

40
2. A distinction based on the presence of an

I "ejected" atmosphere or shell surrounding a central

star and atmosphere where the configuration is considered to be now steady, or at least quasi-steady,

but where there is no pretense that the configuration

j occurred as a CA evolution, just that a CA model,

I only slightly fudged, may now describe it.

A variety of objects could possibly be placed

in this category: shell stars of all sorts; various

nebulae including planetary nebulae, T Tauri shells,

etc.

3. The wholly observational distinction between

the predicted extent of a star with given g and Te ff

I and its "observed " extent. Then, assuming the paraI meters necessary to compute the CA model--g and Te ff

j are reasonably accurate, the essential question is

how does one specify the "observed extent." (We bypass for the moment the question of the implication

on the model of the "microturbulent" parameter re- quired for stellar spectral fit to the CA models)

.

The question of specifying the "observed extent"

obviously underlies this whole possibility for defining extended stellar atmospheres. There are a number of possibilities for defining the observed

extent:

(a) Direct approach . "Tangential" observations, i.e., eclipse

studies, that directly measure extent. Solar

studies are obvious. Studies of stars like

31 Cyg, C Aur, and V444 Cyg give direct evidence

of atmospheric extent.

(b) Indirect studies of "extent" parameters , Studies of spectral features that appear to

imply large geometrical extent: (1) "dilution"

effects; (2) observation of an abnormally large

number of Balmer lines; (3) some forbidden line

effects (These last two are a combination of

low-density and large geometrical extent phenomena.); and (4) interpretation of emission

lines as coming from an extended stellar atmosphere . (c) Indirect studies based on the predicted effect of

other parameters

.

Micro- and macro-turbulence values that are comparable to the compression-disturbance

velocity in the atmosphere would be expected to

distend the atmosphere.

Excitation effects that suggest anomalously

high excitation, hence higher electron temperature, suggest greater atmospheric extent. (Note

41
that some of these effects were what led to

studies trying to find a UV excess in an ex- tended atmosphere.)

Some years ago, the sun would have been included

as the classical example of this type. After all,

people thought that there was absolutely nothing in

the non-eclipse spectrum to imply anything "anomalous." But now, with hindsight, we recognize that

there are many things in the visual disk spectrum

that suggest the following classification: e.g., the

H and K line profiles, the A10830 He line, the presence of the solar granulation.

4. Distinction between the "normal" CA models

that we expect most stars to satisfy and whatever

models are required to describe pulsating stars,

ejecting stars, and other dynamically unstable stars

which do, however, exhibit at least a quasi-steadystate configuration.

While we know what we mean by stars that exhibit

an overall divergence from the strict CA model, we

do not know in detail which these are. For example,

we would now include the solar chromosphere-corona

in the class of such atmospheres that the CA model

will not satisfy, but several years ago we would not

have done so. Further, the sun as a whole is thought

by most people, to satisfy the CA, and it is often

argued that the chromosphere-corona are an essen- tially unimportant part of the solar atmosphere.

(With which viewpoint some of us would take issue.)

It is often argued that at each phase of the

cepheid pulsation the atmospheric configuration can

be mimicked by a CA model with the proper choice of

Te ff and ge ff

It is sometimes argued that supergiant atmospheres satisfy the CA model. Other people disagree.

The current satellite observations of their far UV

spectrum show large mass outflow, and almost cer- tainly a chromosphere-coronal phenomenon.

Despite the problem of identifying the stars, or the parts of the stellar atmosphere that might

satisfy the distinction of being in the extended

stellar atmosphere class, all seem to agree that the

effect of a momentum input or a mechanical energy

dissipation or both is to make the atmosphere more extended than it would be in the CA model. But in

any event, a basic point is that the CA assumptions

are violated. Thus in discussing an extended stellar

atmosphere, we must carefully decide which of the

above criteria we adopt in defining an extended

stellar atmosphere (ESA)

42
B. SUMMARY OF THE PHYSICAL PROBLEMS CONNECTED

WITH THE IDEA OF AN ESA

Clearly, the problems we will list and emphasize

depends upon which definition of an ESA we adopt.

Let us consider some of these problems according to

the various alternatives given above:

1: "Curvature" criterion

(a) The main question is, Under what conditions

can a CA model give an ESA?

(b) The second question is, What structural

difference comes from such an ESA model?

1. What difference does it make in a Te

(i)

model?

2. What difference does it make in the

predicted spectrum: I (X, continuum);

line profiles. We should consider this

problem according to LTE , to be wholly

consistent.

3. What difference does such a model induce, in the expectation of non-LTE

effects? Since such effects come wholly from inhomogeneity induced by

the boundary, and the "kind" of

boundary behavior changes with these

curvature terms, it is an important

question

.

2. "Ejected" atmosphere or shell

(a) A major problem is to deduce the geometric

distribution of the ejected material. One

approach is to try to follow in detail the

distribution of matter to be expected from

a nova outburst, checking the inferred distribution, based on some sort of calculations, against the spectrum that should be

produced by such a distribution. Another

approach is to assume various degrees of

"connectedness" of the distribution of

material, and determine the "stratification"

spectrum.

(b) It is equally important to ask whether

the CA model or a simple modification can be

applied. Clearly, here the non-LTE aspects

associated with "dilute" radiation are important, but it is also important to ask

whether such things as RE, HE, etc., can be

taken over. So it comes down to asking what

kinds of steady-state equilibrium can be

stable for such configurations, taking into

43
account various kinds of interaction with

the "parent" star—radiation, continuous

small-scale mass ejection, etc.

(c) There are also many problems connected with

predicting the spectrum from such an ESA.

Again, these are connected with Session II

of this symposium.

3. "Observational discrepancy" with prediction of

CA model

(a) We made the point above that the primary

problem is one of inferring parameters that

either support or negate the CA predictions.

Again the problem can be broken down:

1. Eclipse studies. Just as there was

no real understanding of the implications of solar eclipse studies on the

specific details of the outer solar

atmosphere until a complete non-LTE

diagnostics was developed, so there

exists the same situation for eclipse

studies of 31 Cyg, etc. Mainly, we

have curve-of-growth studies. We need

to see what is needed, in terms of

data available, and develop the diagnostic methods.

2. Non-eclipse studies. Many of these

studies are simply spectral studies

and spectroscopic diagnostics, and

thus come under Session II. Some of

these studies are of the physical

implication type and resemble solar

studies, where, e.g. "observed" supersonic turbulence required high Te . Thus we ask for studies of the physical

consistency between the simultaneous

"inferred" presence of various parameters. It is sometimes difficult to

separate the above point and this

point, just as in the solar problem,

and they must be studied together.

E.G., does an inferred supersonic turbulence imply a faulty diagnostics or

a faulty inference on Te

? (b) Emission lines: general approach. While

this is logically part of the spectroscopic

studies it should be broadened in implication to be mentioned separately. Basically

the question we ask about emission lines is

whether their presence can be interpreted to

44
imply: (1) simply an extended atmosphere,

and "classical" grad Te , (2) the presence

of a "specialized" mechanism, such as the

Schuster mechanism, and again essentially

a "classical" atmosphere, (3) the presence

of a chromosphere-corona, thus mechanical

energy supply, and definite departure from

RE.

Thus the presence of emission lines is a class

by itself when discussing spectroscopic diagnostics.

It is also a bridge to alternative (4)

.

4. "A priori" rejection of CA models

The foremost problem is a conceptual one: What

are the parameters that fix the state of the atmosphere of such a star? There may, of course, be

several cases. But overall, the big difference between a classical aerodynamical situation and a classical stellar atmospheric configuration is the

coupling of velocity field to internal degrees of

freedom. In the aerodynamic case, coupling is

through Te (electron temperature) , whose value is

fixed by the kinetic temperature, whose value is

fixed by the dynamical flow problem directly. In the

stellar case, the coupling is through the radiation

field, whose value is indirectly affected by the

velocity field. The question is, In such cases as this model type 4 distinction between ESA and

"normal" atmospheres, does this situation change

with respect to the static case? The "classical"

arguments on cepheids , etc., would say it does not

change: you give geff and Te ff at each phase, with

ge ff being the thing fixed by the dynamical problem

but the radiation field still fixes the local value

of Te , and the internal degrees of freedom, and so the spectral properties. Thus the atmospheric extent

would arise from dynamical properties, not from increased Te due to mechanical energy dissipation,

in this "classical," RE, cepheid picture. The CA

would not apply in its entirety, of course, only in

this modified version.

But it is also possible that mechanical energy

dissipation must also be included. In this case, we must determine the effect on Te , and the coupling to

internal degrees of freedom. In this case, the basic

question posed is: How do you really describe the

dependence of the local value of Te on the quantity

and quality of the radiation field and the local

mechanical energy dissipation?

45
DISCUSSION

Pecker: It should be noted that, even in

steady state, we should write time-dependent equilibrium equations. At a given location in the atmosphere conditions are fluctuating; they are steady

only in a statistical way. Depending on the relaxation time of the medium, as compared with the

characteristic time of fluctuations, it may be

necessary to solve the time-dependent equations

before integration over the time. The result could

be different from the solution of time-independent

equilibrium equations.

A. B. Underhill made a guess of "what lines are sensitive to non-LTE conditions." She proposes

three categories. This seems to me very dangerous.

As soon as we suspect that non-LTE physics is necessary, we have to investigate, in each particular

case, whether LTE might be used as a sufficiently

good approximation.

The Balmer lines should be added to the list.

The studies. by Feautrier, for example, show that

b lr b 2 are far from equilibrium.

In the case of the hydrogen lines in supergiants even a non-LTE theory does not solve all

problems. A, B. Underhill mentions that we could

consider a smaller hydrogen abundance to reconcile

observations and theory (i.e. 3 Ori and others).

But what about departures from hydrostatic and ra- diative equilibrium? These could give rise to an increase in temperature and variations of ge ff in

the outer layers of the atmosphere and should be

seriously considered.

Und.zn.kill: If the temperature increases with

height in an atmosphere, you may find emission

components in some lines or indications for higher

excitation temperature in the lines. But the spectra

of B-type supergiants seem to indicate that there

is a very low excitation temperature.

M&nz&Z: Are there some similarities to the

solar chromosphere?

Undzfih-lZl: It is very difficult to observe

chromospheric effects in the light integrated over

the disk, even in the case of the sun.

PecfceA: If the lines show low excitation temperature, you can only say that there is no increase

of the source-function. It is wrong to conclude

there is no increase of excitation temperature.

UndzfiksLZZ: There is no known mechanism for

heating a chromosphere, for hydrogen and helium are already completely ionized in the outermost layers.

46
Katko fain: The model calculations by Mihalas

and Feautrier do show an increase of temperature in

the UV.

ttnd<LKk<LLL'. An increase of 2000° for an average

of 20,000° K is too small to show conspicuous

effects

.

Menze.^-* I disagree with the statement that the

assumption of LTE gives only absorption lines.

Strictly, if the temperature of the gas equals that

of the photosphere, the spectrum will contain neither

absorption nor emission lines. If the photospheric

temperature is greater, we obtain absorption lines;

if the temperature of the gas is greater, the lines

appear in emission.

I am not implying that departures from LTE do

not exist. Cyclic transitions, perhaps involving

metastable levels, must cause such departures. However, they may not be as great as certain non-LTE

theories seem to require. Such neglected phenomena

as non-static atmospheres, flares, "star spots,"

magnetic fields, atmospheres with non-uniform temperatures and dynamical flow can produce what appear

to be significant spectral anomalies, as interpreted

by conventional theories.

Most proponents of non-LTE theories of stellar

atmospheres attribute the supposed spectral anomalies

to the interaction of the radiation and collision

fields, a source function differing from that of

Planck radiation for a given temperature, or non- Maxwellian distribution of velocities of colliding

electrons, plus the interactions and resonances

real or fancied—between different atomic constituents .We don't have to go very far to see that this

model simply does not work. The sun is an excellent

example, as I first demonstrated about 40 years ago,

not from any model but from the observations alone.

I have always insisted that observations are fundamental. Models are important, but they must be

subservient to the observations rather than to traditional or classical concepts of how stellar atmospheres should be constructed.

Just look at the sun! The normal Fraunhofer

spectrum is consistent—at least roughly-- with that

from a gas at a temperature of about 5000°K. The

chromosphere, a layer of gas contiguous to the re- versing layer, contains strong lines of ionized

metals, of helium, and of ionized helium. In the

old days-- and by "old days" I mean the very ancient

pre-DHM era, which no one here other than I can recall —some astrophysicists claimed that the chro47
mospheric spectrum was entirely in accord with the

existing theory. They pointed qualitatively to the

Saha formula and naively suggested that low pressure

was responsible for the presence of ionized helium

in the chromosphere, a fact that I later showed could

be explained only on the basis of high temperature.

Miss Underhill calls attention to many lines she

doesn't "like," because they "behave anomalously."

Anomalously, that is, according to her models. But I think we need to postulate the existence of truly

distended atmospheres —stellar coronas, if you will.

In such regions we shall expect to find numerous

"anomalous" conditions, such as shock waves, magnetohydrodynamic flow, and atmospheric irregularities.

No wonder the spectra seem to be anomalous. But the

fault lies in our models, not in the stars.

Und&sih<LlZ: What I don't like is calculating

those lines with LTE-theories . I do like those lines

because they do give us a clue to what is going on

in particular processes. I cannot agree with Dr.

Menzel that it may be possible to heat the extended

atmospheres in layer sections, giving each section

a temperature and then going on with LTE calculations. The critical lines do come from levels so widely separated that you cannot justify SahaBoltzmann relations. The reason is that in early

type stars the particle densities are too low (of

the order of 10 -10 14

) even in main sequence stars.

Saha-Boltzmann relations are reasonably valid if the

particle density is of the order of 10 . The difference between the atmospheres of early

type stars and the sun is, that in early type stars

the density of the photosphere is comparable with the

density of the solar chromosphere. In extended atmospheres the densities must be even lower.

WzlZmann: It seems that the lack of adequate

mathematical methods to treat the NLTE case is a severe handicap for the explanation of many observations related to stars ranging in type from the

supergiants with minor deviations from LTE to extreme

NLTE-objects like planetary nebulae, as well as cer- tain Be stars, WR stars, and novae. The complete

problem is to solve the (infinite) system of linear

equations Nj_ = const together with the transfer

equations for the radiation densities ui^. It might

be possible to introduce new special functions

adapted to this task. This is a challenge to the

theoreticians

Undznkill: It should be possible to reduce the

infinite system to only a few equations by neglecting

many transitions for physical reasons or by tying the

48
higher levels to the continuum. In addition it might

not be too bad to assume black-body radiation density

for the secondary lines and restrict the transfer

problem to one or two lines. How to proceed is a question to be answered by the theoreticians.

Whitman: I should like to point out that the

formation of emission lines might be strongly influenced by the velocity distribution in the atmosphere . Hltlo.nda.hl: In calculations recently completed

for supergiant atmospheres the line shapes (core

depth, width, wings and equivalent width) were found

to be quite sensitive to the velocity distribution

in the atmosphere. Because of this sensitivity it

is difficult to see how one can draw conclusions

about the importance of non-LTE unless the velocity

dependence has been taken into account.

Thomab: Let's be sure that questions on symbiosis are not questions of the method of description

rather than physical facts. Since Menzel's work in

the 1930 's it is clear that the sun is a symbiotic

object in the above sense: T = 3400 for NaD-lines,

25000° for He II. But recent work using non-LTE

and non-radiative equilibrium has shown that this

"apparent 11 symbiosis is not symbiosis at all; it is

just a deficiency in our methods.

Pecfee^i: If there are objects between the supergiants and the stars with large shells, i.e. planetary nebuale, it is not a priori necessary that we

find the same mechanism. The only reason for intermediate objects could come from stellar evolution.

{rizllman: The gap is filled by novae in different states and by Be stars of different types.

Rijbicki: The question of the appearance of

emission lines is a question of the behaviour of the source function. Independent of LTE or non-LTE

conditions it is the projected area that gives rise

to emission lines. So it is important to consider

the geometric structure.

Tkoma* : The point is not whether an extended

atmosphere can produce an emission line only by

geometry—but whether an observed emission line implies such a geometrical origin uniquely. It is

possible to have an intrinsic emission line in a non-extended atmosphere.

Rybicki: If you are going to obtain the mechanism of emission from the observations you must be sure what kind of mechanism you mean. In the case

of supergiants you cannot get an emission line, if

you do not have an increase in the source function.

Gfiotk: A. B. Underhill postulated that the

49
atmospheres of some B-type supergiants of luminosity

class la are hydrogen-poor. Our studies of supergiants (a Cyg A2 la, <j) Cas FO la) show that the

atmospheres of these stars do have normal hydrogen

abundances. The Balmer series of hydrogen breaks

off at n = 32-34, as one would expect for supergiants of low electron density. Using an LTE model

with variable microturbulence, I found excellent

agreement between calculated and observed hydrogen

lines (Hy-H 10 ) for a Cyg.

Estimates show, that the luminosity of a Cyg

is very high, Mbol being of the order of -8m . <J> Cas

is one of the most luminous stars of our galaxy

( Mbol = -8m .9). We don't yet have a model calculation of its atmosphere, but the appearance of hydrogen lines is quite normal. The high luminosities of

a Cyg and $ Cas imply that these stars are evolved

stars, probably further evolved, than the B type la

supergiants. The mass of a Cyg and <J> Cas can be

estimated to be 30-40 solar masses.

HsL&t&ndcLkZ: In the calculations mentioned above

it is unnecessary to assume a turbulent velocity.

The dispersion in atmospheric velocities, which occur

as a result of hydrodynamic motion, fulfills the role

of the turbulent velocity.

Houziaux: To fit the Balmer line profiles you

introduced a change in He/H abundance. Did you find

an influence on the shape of the continuum?

Undtsiklll: I used the schematic models by BohmVitense, who has only calculated the continuum. Line

calculations have not been done.

Giotk: At Munich we are working on the analysis

of spectra of a number of supergiants (la, lb) with

spectral types A and F. In two cases (a Cyg, n Leo) we already have model calculations.

The observed equivalent widths of metallic lines

cannot be explained, if we assume an LTE model with

constant microturbulence. The abundances of weak and

strong lines, even within one multiplet, differ by

a factor of 10-100. If we introduce an increasing

microturbulence with decreasing optical depth we get

excellent agreement for the abundances of weak and

strong lines.

Using a curve-of-growth analysis, we find for

all supergiants a damping parameter log 2a of the

order of -1 to -1.5. We know that there is only one effective damping process for low density atmospheres, namely radiation damping which should give

log 2a approximately -2.5 to -3.0. Again there is a discrepancy by a factor of 10 to 100 between observed

and predicted damping constants, which can be ex50
plained by the variation of microturbulence

.

UndzikZZZ: Do you have spectra of high dispersion, so that you can compare the predicted turbulent velocities with the line profile?

Gfiotk: Normally it is impossible for supergiants to prove the predicted microturbulence by

the profiles. In all stars studied the shape of the

profile is due to rotation or macroturbulence with a Doppler width larger than that of microturbulence.

Menze£: The departures from LTE must not be

very large. If you assume changes in atmospheric

structure, it is possible to fit the observations

using only very small deviations from LTE.

Mug gZ&A tone,: Does the dependence of microturbulence on optical depth change the line core or the wings?

Gnoth: The central depth of the lines becomes

smaller and the wings get broader, as in the case of

higher damping, if you assume that the equivalent

width is the same as in the case of constant turbulence velocity.

HZZZzndahZ: Using the microturbulence in the

calculation of line profiles does mean that you introduce an additional parameter. I propose to take

another free parameter. Using a linear velocity

distribution you get the gradient of velocities as a new parameter. I believe that you will find the

same result as before.

Gfiotk: It is possible that a velocity gradient

in the atmosphere could give similar line profiles

to those which you get with microturbulence depending

on optical depth. But a velocity gradient must give

rise to differential displacements of lines, which

are formed in different optical depths.

Pecker: Describing a velocity field by a given

number of parameters is an ambigious procedure, as long as one does not define the shape of the function. For instance the functions

(a) £ = £j o < t < t

£ (step function)

S = £ 2 T

£ 1 T <

00

(b) and £ = a+bx new line have a completely different shape and are not equivalent.

If one tries to represent a solar line at the

center of the disk, one can get a fit either using

(a) or (b) . This has been shown clearly by Roddier

and Gonczi for lines of Sri.

51
UndznkiZZ: To get a good fit of the observations, it is necessary to use as many lines as possible. It is important to use lines of different

excitation and ionization potential to find unique

solution

H4.ZZznda.kZ: My proposal to use a linear velocity distribution comes from the result of calculations for supergiants type la where I got this

result

Thoma.6 : Do you know the mechanism producing

the velocity field?

UZZZzndakZ: The velocity field is produced by

a rarefaction wave, which is excited by a shock

front. (Detailed description of the processes in

Hillendahl's paper.)

Vzzkzi'> D. H. Menzel has rightly attracted

attention to the fact that improvement of fitting

the observations by theory can be done either by

working on non-LTE analysis with a classical model,

or by using LTE and an improved model (i.e. an improved Te and ne distribution)

I do not challenge this fact. Actually a given

set of observations can be accounted for by a variety

of situations. This means there are indeterminacies

in the problem. One must realize that the several

discrepancies we have are not necessarily of the same nature. If we consider the theory of line formation, we do know the physics well, the main

difficulty being either numerical or bad knowledge

of physical parameters. I cannot say anything about

the models. We don't know the physics of heating

processes, or the effects of gravity waves etc.

Therefore in a first approximation one should always

use NLTE analysis for which the physics is unambiguous. The next step in learning more about the

physical mechanism involved in the departures from

radiative and hydrostatic equilibrium must be the

comparison between the observations and the result

of theoretical non-LTE treatment.

Bo km: in the case of plane-parallel atmospheres

the progress of the non-LTE theory has been very

large. Recently we have made NLTE calculations for

spherical atmospheres. We found these calculations

to be very complicated, so that they can be done only

in extreme cases, i.e. planetary nebulae.

Hmmmzn.: It is possible to do non-LTE calculations in spherical geometry. If the velocity gradients are very large, then this becomes fairly simple.

This has been worked out by J. Castor at JILA and

has been applied to He II in Wolf-Rayet stars. Work

on small or zero gradient has been performed by

52
Mathis at Wisconsin and Skumanich at Boulder.

Pecker: I agree that non-LTE problems can be

handled mathematically for plane-parallel and spherical atmospheres, but the main problem is to understand the physics of extended atmospheres.

ThomaA : The physical principles for non-LTE

atmospheres are well known. It is important to use them in the right way.

Bo km: I am not sure that the physics is so simple if you take a large number of coupled terms.

53
EXTENDED ATMOSPHERES OF PLANETARY NUCLEI

by

K. H. Bohm and J. Cassinelli

University of Washington

Seattle

ABSTRACT

Exploratory calculations on nongray, hydrostatic-equilibrium model envelopes for central stars

of planetary nebulae of high temperature and possibly

near the instability limit are reported. It is con- jectured that these may be related to Wolf-Rayet type

nuclei; it appears possible to obtain an OV absorption, and OVI emission, spectrum even in an LTE

calculation.

Key words: nongray, central stars of planetary

nebulae, Wolf-Rayet nuclei, instability limit.

There are a number of theoretical and empirical

arguments that indicate that central stars of planetary nebulae in the range 5.5 x 10 4 °K < Te f f

< 9.0 x io h °K have extended atmospheres (in the sense that curvature effects become very important)

From a theoretical point of view we must realize

that:

(1) Because of the high effective temperature of

these objects, the scale height

H = RT/yg (1)

tends to be rather large; and

(2) The objects in the temperature range quoted

are fairly close to the instability limit due to ra54
diation pressure effects (Harman and Seaton 1964) as defined by

grad' = c

{ k +<j 0 } F d & — a 0 (2) v e& v v ^ c e£

with g = surface gravity of the star, grad

= radiative acceleration, k v = monochromatic absorption co- efficient, a p = electron scattering coefficient.

This fact leads to a rather low effective surface

gravity in these stars which in turn leads to an additional considerable increase of the scale heights.

As pointed out earlier (Bohm 1969) , one gets hydrostatic atmospheres with a thickness larger than one stellar radius if one approaches the instability limit (2) to a point where

grad l % 0.9 X |g| (3)

In such an atmosphere curvature effects become very

important. This point will be discussed below.

From an empirical point of view it is important

to note (O'Dell 1968) that central stars showing

Wolf-Rayet spectra occur practically only on the part

of the Harman-Seaton sequence that lies close to the

instability limit.*

Since Wolf-Rayet spectra can be formed only in a rather extended atmosphere (or envelope) we may also

conclude that the observational evidence indicates

that stars in this region of the Te ff-g-plane

have extended envelopes . One might even hope that a thorough study of atmospheres in the indicated range

of Tef f" and g-values will. eventually lead to a theIn view of the considerable uncertainties in the

empirical determination of Teff and g, it is difficult to be absolutely certain about a statement like

this (Cf . diagram no. 7 in Bohm 196 8) . However, the

evidence given by O'Dell (1968) certainly shows that

the above statement at least does not contradict the

present observational evidence.

55
oretical understanding of the Wolf-Rayet phenomenon.**

(A study of planetary nuclei has one great advantage as compared to the investigation of WR field

stars: We know Te ff and g before we start the

study.

)

The problem of constructing model envelopes for

central stars is being approached in three steps:

(1) in order to get some basic orientation,

Bohm (1969) calculated some nongrey hydrostatic models near the instability limit using a plane-parallel

approximation. Of course these results can be con- sidered at best only as a qualitative approximation.

However, even these crude calculations offer some interesting insight. They show the very strong dependence of the geometrical thickness of the atmosphere and also of the energy distribution of the

surface flux F v

(0) on g (Bohm 1969). In this connection it is also instructive to see how strongly the

density stratification is changed by a rather small

change in the surface gravity. This is illustrated

by Figure 1.

(2) J. Cassinelli is trying to construct hydrostatic nongrey model envelopes for these objects taking into account curvature effects. He showed that

a generalization of Lucy's (1964)' temperature connection procedure is possible though the setting up of

a relation between the zero-order moment Jv and the

second-order moment Kv of the intensity I v (y) offers

some difficulties in the spherically symmetric case. As in the plane-parallel case the determination of

the monochromatic J v for a given Bv

(x v

,r) has to be

carried out for every frequency between two successive temperature correction steps. This can be ac- complished e.g. by the application of Carlson's S n method (Cf. Carlson and Lathrop 1968). It is obvious that the determination of nongrey model en- velopes is a laborious task (even if one makes the

drastic simplification of assuming LTE) . It is

therefore important to ask whether the effort is

worthwhile and what one may hope to accomplish. We

do hope that a detailed study of these envelopes will

eventually lead to an understanding of central star

i

i J As J. Schmid-Burgk (1968) has pointed out stars

close to the instability limit (2) (but not yet in

the actual instability region) easily generate a "strong stellar wind." This would make the mass

loss of these Wolf-Rayet stars understandable.

56
T >

Figure 1. The density stratification p (t) for the

models with Te ff = 6.3 x 10 h °K, g = 1.26 x 10 k cm

sec" 2 and Teff = 6.3 x lo

1* °K, g = 0.97 x 10 1* cmsec" 2

. spectra showing emission lines. Preliminary calculations indicate that it might, for instance, be possible to understand why certain central stars, especially of the type NGC 246 (see Greenstein and

Minkowski 1964) , show OVI lines in emission while the

OV lines appear in absorption. Figure 2 shows that

with a simple LTE assumption for the model with T e ff

= 6.3 xio 1*, g = 0.97 x ±o k cm sec

-2 we can get a situation where in the outer very extended part of the

atmosphere (t < 3 x 10" 3

) the oxygen exists mostly

in the form of OVI whereas in deeper layers down to

t = 0.45 the OV dominates. It is obvious that such

an extended atmosphere could give rise to an OVI

emission spectrum (because the OVI exists in an "extended envelope" that is optically thin in the con- tinuous radiation) and an OV absorption spectrum.

57
IxlC4

IxlO"3 IxlO

2

I I 10 f >

"igure 2. The ratio x of the number of ions in a riven ionization stage to the total number of oxygen

ons as a function of t for the envelope model with

'eff = 6.3 x io 4 °K, g = 0.97 x 10 4 cm sec" 2

.

r hether this model could explain the observed spec- rum in detail has not yet been checked. We felt

uch a detailed check would be worthwhile only after

really self-consistent spherically symmetric raditive equilibrium model has been constructed.

(3) For the reasons indicated above it is obious that stellar envelopes with a continuous outlow of matter and not in hydrostatic equilibrium

hould be investigated. Such a study is being caried out by J. Schmid-Burgk . However, in this case

he combined hydrodynamics—radiative transfer probem becomes so complex that one has to restrict oneelf to a grey approximation of the radiative transer problem. Curvature effects have been included

n this treatment.

We feel that the two lines of approach described

nder (2) and (3) should be followed simultaneously

n order to learn something about the nongrey raditive transfer and the hydrodynamic aspects of the

roblem. One may hope, of course, that eventually

e shall be able to treat the whole problem combinng the hydrodynamics and a realistic treatment of

he nongrey radiative transfer problem.

58
REFERENCES

Bohm, K. H. 1968, I.A.U. Symposium No, 34 "?'lan&tarty

N&bulaz," eds. D. R. Osterbrock and C. R. O'Dell,

Dordrecht: D. Reidel Publ. Co., p. 297.

Bohm, K. H. 1969, A&trion. and khtriopkyh . 1, 180.

Carlson, B. G. , and Lathrop, K. D. 1968, ~£omputtng

Wztko&h In Rzactori Vkyktdb , eds. H. Greenspan,

C. N . Kelber, D. Okrent, New York: Gordon and

Breach , p. 167.

Harman, R. J., and Seaton, M. J. 1964, Ap. J. 149 , 824.

O'Dell, C. R. 1968, I.A.U. Symposium No. 34

"Planztariy Nzbulaz," eds. D. E. Osterbrock and

C. R. O'Dell, Dordrecht: D. Reidel Publ. Co.,

p. 361.

Schmid-Burgk , J. 196 8, unpublished.

DISCUSSION

Bo km: I would like to ask whether there are better methods to solve the nongrey spherical problem.

Hammzri: There are papers by Mathis and

Skumanich who treat these problems.

Rybtckt: Perhaps the method of Feautrier can

be generalized to solve the spherical problem.

Thomas : Are the shells you have calculated

close to the instability limit?

Bo km: The observations show a very large

scatter, so that it is not possible to find a clear

decision. O'Dell, using new observations, does believe that for all planetary nebulae with central

stars that are of the WR type, the shells are close

to the limit of instability; but I think the scattering in absolute magnitude (AM = 1^5) is too large for one to make this conclusion. Some of the central

stars at the lower end of the sequence are Of stars.

Humme.ri: The nebulae that are connected with Of

stars are very large in general. The calculated

Zanstra temperatures do not correspond with the observed Of spectra.

Und&rikill: Some years ago I estimated roughly

the temperatures of WR stars. I concluded from these

temperatures that the difference between WC and WR

stars is not due to differences in abundance but it

is differences in excitation.

Whitman: I have tried to solve the transfer

problem in spherical atmospheres by using the follow59
ing procedure: The radiation is split up in two

parts, (1) the radiation that is coming directly from

the star, and (2) the diffuse radiation of the shell.

If you integrate each part independently, you get a simple solution of the problem.

Bo km: That is the same procedure we followed,

but we feel uncertain about it. In the nongrey case

the radiation in different frequencies is coupled,

and we should solve the general transport equation.

60
THE N IV A5820 MULTIPLET IN WN STARS

by

H. Nussbaumer*

Department of Physios

University College

London

ABSTRACT

Evaluating transition probabilities of N IV

2p3d 3 P° to 2s4s 3 S and 2p3p 3 P and an emission

feature at A7410 it is shown that a disputed

emission at A5810 in WN stars may not be attributed

to N IV.

Key words: HD 192163 WN6 , N IV transition

probabilities

.

The interpretation of stellar spectra assumes

the knowledge of atomic data, energy differences for

identification, oscillator strengths and cross sections for explaining the spectral intensities.

A group under the direction of M. J. Seaton at

University College, London, has completed programs

for the calculation of atomic structures, oscillator

strengths and collision cross sections for complex

atoms. The atomic structure program is essentially

based on the work of Condon and Shortley (1935) but

the inclusion of configuration interaction is built

in as an essential feature. I shall apply some of our first results to solve a puzzle in the spectra

of Wolf-Rayet WN stars. If any of you have some particularly pressing needs for oscillator strengths

or collision cross sections we shall do our best to

provide them.

* On an E.S.R.O. Fellowship.

61
log I

5600 5800' 6000 6200 6400 6600 6800 A

Log I tracing of the spectrum of. HQ 192 163 WN 6 Figure 1

Figure 1 shows part of the spectrum of HD

192163 WN6, taken by Underhill (1959). The emission feature at A5810 is usually attributed to C IV.

But the case in favor of the attribution to N IV

2p3p 3 P-2p3d 3 P

0 has again been argued by Hiltner and

Schild (1966) ; this transition gives a line centered

at A5820. Laboratory investigations of N IV by

Hallin (1966) have failed to produce that line.

Figure 2 shows the allowed transitions from 2p3d 3 P°

with the corresponding transition probabilities.

They are from Table 3 of Nussbaumer (1969) where gf

values for all the transitions between the 3 4 lowest levels of N IV have been calculated. The level

2p3d 3 P° will be depopulated mainly to 2p 2 3 P. But

in an extended atmosphere that radiation might be

reabsorbed and the relative intensities of

2p 2 3 P-2p3d 3 P° A298 and 2p3p 3 P-2p3d 3 P 0 A5820 need

therefore not be the same in laboratory experiments

and in astronomical observations. For this reason

the absence of N IV A5820 in the laboratory cannot

be invoked to ban that line from the Wolf-Rayet

spectrum.

Figure 2 shows that at A7413 there is a line

with a transition probability three times higher

than for N IV A5820. Swings and Jose (1950) find a faint emission feature at A7410 of intensity 1 against an intensity of 20 for Ha A6563. They attribute A7410 to N III 2s2p3s 4 P°-2s 2 4s 2 S. But even if we attribute that emission to the two-elec62
TRANSITIONS FROM N IV 2p 3d 3 P°

Figure 2.

tron transition 2s4s 3 S-2p3d 3 P

0 it is obvious from

the data of Figure 2 that the emission at X5 810 is

far too strong to be attributed to N IV 2p3p 3 P2p3d 3 P°. I think these arguments based on transitions from a common upper level will finally

dispose of the temptation to attribute the emission feature at A5810 to N IV.

The transition 2s4s 3 S-2p3d 3 P° is an example

of a two-electron jump; such transitions are naturally accommodated in our approach based on the

idea of configuration interaction.

63
REFERENCES

Condon, E . U. , and Shortley, G. H. 1935, Tkc Tkzoiy

o I Atomic Spcc.th.CL (Cambridge: Cambridge

University Press)

Hallin, R. 1966, Alk. Ft/4., 32, 201-211.

Hiltner, W. A., and Schild, R. E. 1966, Ap. J., 143 , 770-773.

Nussbaumer, H. 1969, Mon. Not. R. astr. Soc .

,

145 , 141-150.

Swings, P., and Jose, P. D. 1950, Ap, J., Ill , 513-529.

Underhill, A. B. 1959, Pub. Vom. Ap. 0b6 . Victoria,

11, 209-234.

DISCUSSION

TJic^tz: We have made the same calculation and

it would be interesting to compare the results. For

Si II we were able to compare our values with those

of other authors and we found very large differences

for weak and strong lines. The result does depend

on the number of transitions, which are included in

the calculations.

Hvl66 baumcJi : In our calculation all known transitions are included.

Wcllmann: May I remind you, that Dr.

Nussbaumer asked to know which transition probabilities of astrophysical interest should be calculated. I would like to remind you that the exact

calculations of collision cross sections for forbidden lines by Seaton and his group were very

important for the determination of electron densities and temperatures in diffuse nebulae. They

should be continued.

UndcJihill: Observations show that the N IV

line at X6200A, which should be much stronger than

the A5820A line, is only a very weak feature. This

also leads to the conclusion that the observed emission at X5810 is not due to N IV.

Swing* : Could you calculate gf-values for Fe

II lines?

NLL6 6bcLu.me.Ji: I am not sure because Fe II is a very complicated ion.

64
A SELF-CONSISTENT MODEL ATMOSPHERE PROGRAM WITH

APPLICATIONS TO SOLAR 01 RESONANCE LINES

by

R. Grant Athay and Richard C. Canfield*

High Altitude Observatory

National Center for Atmospheric Research

Boulderj Colorado

ABSTRACT

Profiles and total intensities are computed for

solar 01 resonance lines at A1302 and X1305 using a model atmosphere program that includes non-LTE

effects in both hydrogen and oxygen and that includes

microturbulence both as a line broadening mechanism

and as a contribution to the gas pressure. Good

agreement is obtained between computed and observed

intensities. The computed profiles appear to have

too much 'self -reversal.

Key words: resonance lines, non-LTE line formation,

solar UV lines.

I. INTRODUCTION

The development of adequate methods for com- puting profiles and intensities of strong stellar

lines where non-LTE (local thermodynamic equilibrium)

effects are of overriding importance has provided a powerful new means for studying the outer envelopes

of extended stellar atmospheres. In principle, at

least, such studies are capable of yielding thermodynamic models and the statistical characteristics

of the fluid motions at the depths where the lines

are formed. In practice, we are barely beginning

*Currently at Utrecht Observatory.

65
to make use of such methods, partly because we are still developing the diagnostic techniques and

partly because we still lack adequate data.

The solar chromosphere provides an excellent

proving ground for the development of the diagnostic

methods of the non-LTE approach to line formation.

Although the thermodynamic and hydrodynamic state

of the chromosphere are known only approximately,

a few properties of the chromosphere and its adjacent

layers are known with relatively high precision.

Eclipse data, for example, provide spatially averaged

values of the quantity <nenp

T~V 2 >, where ne = electron density, np

= proton density, and T = electron temperature, that are accurate to about

a factor of two in the height range 500 km to 2500 km

above the solar limb. It is not known exactly what

the "average" refers to, but present evidence indicates that at depths near 500 km the chromosphere

is not grossly inhomogeneous . Thus, one may assume

that the eclipse value of <nenp T~ 3

' 2 >, a quantity

we shall subsequently denote by EPT, is representative of the mean chromosphere near 500 km. Above

about 10 00 km the chromosphere is evidently dominated

by inhomogeneities (such as spicules) and the value

of EPT is not representative of the mean chromosphere. Below 500 km EPT is not well known.

The known values of EPT near 500 km provide very

useful checks on model atmosphere calculations.

Additional valuable boundary conditions are provided

by photospheric models and coronal models, which are reasonably well known. It seems well established,

for example, that in the low corona the electron

density is of the order 3 x 10 8 cm" 3 and the electron

temperature is of the order of 2 x 10 6 °K. Assuming

that np

= ne , we find that the gas pressure in the

low corona is approximately 0.2 dyne cm-2

. The gas

pressure at a continuum (A5000) optical depth, t 0 , of 10~ 5

, which corresponds to a geometrical height

of about 500 km above the limb, is of the order of

2 x io 2 dyne cm

-2

. Thus, the gas pressure decreases

by a factor of only about 10 3 between a height of

500 km in the chromosphere and the base of the

corona. Within this same interval T increases from

about 6000°K to 2 x 10 6 °K.

In addition to the stronger Fraunhofer lines

that are formed in the chromosphere, profiles are available for the Lyman-a and Lyman- 3 lines of

hydrogen and for the C II line at A1336 (Berger and

Bruner 1

) and the 01 lines at X1302 and A1305 (Bruner

and Rense 2 ). The 01 lines are of interest in that

they are formed in the middle chromosphere where the

66
quantity EPT is reasonably well known.

Oxygen and hydrogen have similar ionization

potentials and similar excitation potentials for

their respective resonance lines. This leads to

the result that the two atomic species ionize at

the same depths- in the atmosphere. The ionization

of hydrogen largely determines both the mean molecular weight in the atmosphere and the electron

density and therefore critically influences the model

atmosphere. In addition, part of the ionization of

01 is produced by the Lyman continuum radiation of

hydrogen. It is necessary, therefore, to solve the

ionization equilibrium and Lyman continuum radiative

transfer problems for hydrogen in order to compute

profiles of the 01 lines.

II. CONSTRUCTION OF SELF-CONSISTENT

MODEL CHROMOSPHERES

We propose to compute a model chromosphere

using a minimum number of arbitrary parameters,

i.e., we maximize the constraints on the model.

The constraints exercised are the following:

1. The quantity nenp T

37

2

= EPT given by

the model must agree within a factor of two with

eclipse values derived by Henze 3 in the height range

500-1000 km, and it must not exceed the eclipse

values at heights above 1000 km.

2. The gas pressure in the chromosphere must

exceed 0.2 dyne cm" 2

. 3. At height zero, i.e., at the limb as defined by a tangential continuum opacity of unity

at X5000, we adopt T = 4600° and a hydrogen density,

nH of 1.6 x 10 16

. 4. We assume that the hydrostatic equation

dp = - p g dh (1)

is valid and that

p = nkT + ± c

2 ^ ni mi , (2)

where C is the microturbulent velocity and ni and mi

are the number density and mass of particles of type

i. The other symbols have their usual meaning. We

67
adopt a helium: hydrogen number density ratio of 1:10

•5. We require that the flux in the Lyman con- tinuum produced by the model chromosphere be within

fifty percent of the value reported by Noyes and

Kalkofen 4. With these constraints there is very little

freedom in the temperature model, but there is

essentially complete freedom in C(x).

Equation (2) may be rewritten in the form

where

Also, we may write

p = nkT F (3)

C

2 ? n.m. F = 1 + —1 . ( 4 ) 3nkT

p = 1.4 nR

m

H

, (5)

1.1 n

R

+ n

e

, (6)

and

I =

n

H nTT + n

H e where nn is the number of neutral hydrogen atoms plus

protons and itih is the hydrogen mass. Equations (1)

and (3) then yield

1.4 m g I din (1.1 n +n ) = - din T dh . (8) e k F T (1+0. II)

The ionization parameter I has limits 0 . 5 <_ I £ 1 and can be determined for a given model atmosphere only by solving the radiative transfer

equations for the Lyman continuum. The transfer

solutions depend upon ne and nH as well as Te

. Thus, we must proceed iteratively to find a self -consistent

solution. We start by assuming T(h), F(h) and 1(h).

We then solve equation (8) to obtain preliminary

68
values of n^Ch) and ne (h). We use these initial

values, denoted by nfj(l) and n e

(l) to obtain initial

values of all collision and recombination rates and

initial values of t in the Lyman continuum. Since

t depends upon n , the ground state population, and

n 1 depends upon t through the transfer equation, the

transfer equation must be solved iteratively with n 2 to obtain consistent values of ni(l) and t. The

converged values of ni(l) are then combined with

nH(D to give ne (l') and 1(2). From 1(2) we obtain

nH

(2) and ne (2) from equation (8) and repeat the

iteration on t and m . This double iterative

procedure is continued until final convergence is

achieved for t, tir, m and n e

. For an arbitrary choice of T(h) and F (h) , we

do not satisfy the constraints (1) , (2) and (5)

.

Hence, for a given F(h), we must adjust T(h) until

the constraints, are satisfied. Each change in

either T(h) or F(h) requires a new iterative so- lution for Yin, ni and ne

. We have constructed by this technique twelve

self-consistent model atmospheres. Most of the

ionization of hydrogen occurs in the regions where

Lyman continuum transfer is important and where the

Lyman lines are in radiative detailed balance.

Hence, it is not essential to consider the transfer

problem in the Lyman lines. Furthermore, most of

the ionizations in hydrogen occur from the n 2 level,

which is populated mainly from the ni level, and a fairly accurate solution is obtained using a model

atom with two bound energy levels plus a continuum.

III. LYMAN CONTINUUM TRANSFER SOLUTIONS

In treating the Lyman continuum problem we make

the following simplifying assumptions: (1) There is no opacity other than Lyman continuum opacity from

X912 to approximately A450. (2) Stimulated emissions

may be ignored. (3) Lyman-a is in radiative detailed balance, and we ignore bound levels above

n=2. (4) The scattering is non-coherent.

It is convenient to work with the b n parameters

rather than energy level populations. The Lyman

continuum source function for the problem we are considering is

0 , 3 -hv/kt

v C 2 b,

69
The only unknown in this equation is b :

, We could,

therefore, formulate the problem in such a way as to

solve for bi" 1 or the product of bi" 1 and any known

function. For convenience, we define a dimensionless

frequency parameter y such that

vVi (10)

where vi is the frequency at the Lyman limit. We may

then write

S y

= y

3 exp - [hv a (y-l)/kT] S lr , (11

where

Si

2hvj hvi/kT

(12)

This divides Sy into two factors, one frequency dependent and one frequency independent. We shall

formulate the transfer equation to solve for the

frequency independent term, Si.

It follows from the equations of statistical

equilibrium and from the frequency dependence of the

photoionization cross section for hydrogen that

-hvi/kT°°

where

Ei

dy + e* B

S = (13

1 + e

•hv/kT

dv , (14)

J = mean intensity,

Bi = Planck function at y = 1

70
R R

E * = £ + W A R ' U5)

C C 1 2 2

4. R R

t 21 C2 /tan £ "

e + w-5 R ' (16)

C CI 2 2 r - c

1

= collisional recombination rate

~~ A radiative recombination rate

c 1 -hv . /kT

R. . = W. P. . = S. e

1 P. . , ID i ID i 1 D

£k = statistical weight of level i,

hv . = energy of level i measured from the ground

state

,

P. . = transition rate i to j ID J

P ii = sum rates out °f level i.

We write the transfer equation in the flux

derivative form

dH

-r-^ = J - S (17)

and eliminate the term in J from equations (17) and

(13) . From equation (17) , we have

-hvi/kT °° hvi/kT 00 y -t e

. 4 & 1

n dH

JL y

i+ dx

y y

dy + Si . (18)

71
It follows from equation (13) that

e* B hvi/kT oo

, dH — -g— dy 4 di

y y

(19)

In symbolic form, equation (17) gives

dH

y _ (A - I)S. (20)

where A is the mean intensity operator and I is

the unit operator. Using equation (11), we may

rewrite equation (20) as

dH

dx * - y

3 1 ( a ) exp - jjivi (y-1) /kT (21)

y

3 $ (y,T) Si (22)

Thus, we have finally

e* B

i -hvi/kT

e E

y

1 $ (y,T)Si dy (23)

This equation is solved by the method of Athay

and Skumanich 5 by writing

Si = (1 - a) 1 e* B

(24)

where a is a matrix operator defined by

hvi/kT

£ E

y

1 0 (y,T) dy (25)

72
The y and t spaces are discretized into 9 points in

y between y = 1 and y = 2 and 42 points in t. We

have tried increasing the number of y points and increasing the range in y to y = 3. Neither change

produces significant changes in Si.

The iterative process of solving equations (8)

and (25) to obtain self -consistent solutions for a

1 given T(h) and F(h) generally requires 13 iterations,

j The complete program requires approximately 3 minutes on a CDC 6600. The solutions appear to be

very stable even when large gradients in T(h) and

F (h) are introduced.

The assumption that Lyman-a is in detailed

balance is not necessary of course. The problem

1 could be solved by including transfer solutions in

\ Lyman-a, but trial calculations showed that this

! greatly increases the number of iterations required

and did not result in any important changes in the

model atmosphere for the region of interest for

the 01 lines.

IV. CHARACTERISTICS OF MODELS

Constraint (1) from Section II effectively fixes

T(h) between 500 and 1000 km subject somewhat to the

assumed form of F (h) . However, it follows from

equation (4) that the ratio of C to the mean thermal

velocity of hydrogen atoms is given by

/ n / |r = 1-04 (F-l) l/2 (1.1 + -

e

-) l/2

. (26)

Thus, if ne /nH

<< 1, £/V(H) = .5 for F ^ 1.2 and

C/V(H) = .75 for F % 1.5. In the low chromosphere

C is a few km/sec at most, and F is therefore close

to unity. In the mid-chromosphere even if we want

% 10 km/sec, which is larger than required for the

01 lines, we need only F 1.5. In practice, therefore, we cannot change F(h) very much without

introducing unreasonable values of £. Between heights of zero and 500 km we have

adopted a rather arbitrary model for T(h) and we have set F = 1. The T(h) model has a slow rise in

T to 4700°K at 200 km followed by a more rapid rise

to 5200°K at 400 km and & 5600°K at 500 km. The

73
value of T at 500 km is dictated by the eclipse

value of n e npt~ 3 /

2 at that height. It would make

little difference to the model at 500 km and higher

if, for example, we decreased T to 4000°K at 200 km

followed by a rapid rise to % 5600°K at 500 km. This

would effectively change the density scale height

by about 20 percent for one scale height and would

result in a small density decrease at 500 km. The

constraints we have set on the model do not, therefore, lead to any practical restrictions on the

temperature minimum region.

Between 500 km and 1000 km constraint (1)

imposes a gradual rise in T to % 6200° at 1000 km

(cf. Athay 6 ). If T is constant in this height

interval the computed values of EPT decreases

much too rapidly with height. If T increases only

somewhat more rapidly than indicated EPT increases

with height.

The reason for the strong sensitivity to T in

the height interval 500-1000 km results from the

fact that hydrogen is still only partially ionized.

In fact at 1000 km hydrogen is only about 2 percent

ionized. The ionization does not reach the 50 percent level until T reaches about 8000°K. This again means that constraint (1) limits the kinds of models

that may be considered. A rapid increase of T with

height cannot be permitted in a model of the mean

chromosphere at depths where T < 8000° without

introducing an increase in EPT with height, which

is unacceptable.

At 1000 km the gas pressure in the chromosphere

is approximately 7 dyne cm

-2

. Therefore, constraint

(2) becomes very important in determining the nature

of the model at heights above 1000 km. The gas

pressure cannot fall by more than a factor of 40

before the coronal temperature rise occurs, and a sharp temperature rise cannot occur until T > 8000°.

This requires that T continue to rise rather gradually until it reaches about 8000°.

We have somewhat arbitrarily adopted a T(h)

distribution above 1000 km that gives d log 10 T/dh

% 10" 9 cm-1

up to T 7200°. This is the same gradient that exists between 500 and 1000 km, and

is about the maximum gradient allowed by EPT. The

value T = 7200° is reached at approximately 1600 km

and the gas pressure has decreased to about 0.5 dyne

cm

-2

. If we continued increasing T at this same

rate, the gas pressure would reach 0.2 dyne cm-2

before T reached 8000°. It is necessary, therefore,

to steepen the gradient in T between 7200° and 8000°.

Hydrogen is sufficiently ionized at 7200° to permit

74
a steepening of the temperature gradient without

making EPT too large. The larger T becomes the more

the gradient may be steepened.

Restrictions on d log 10 T/dh are also imposed

by constraint (5) - the observed Lyman continuum

intensity. The type of model under consideration

tends to give too much flux in the Lyman continuum.

I Steepening the temperature gradient between T = 7 500°

and T = 10000° reduces the Lyman continuum flux to

within the limits set by constraint (5)

.

A model that satisfies all of the constraints

in Section II is given in Table 1. It is possible

to construct a family of such models with different

assumed values for F(h). Such models differ from

one another primarily in the height at which the

sharp temperature rise occurs. The differences

amount to only a few hundred kilometers, however,

: even for relatively large changes in C. A total of 12 models were tested by comparing

! computed and observed profiles for the 01 lines.

Each of the 12 models gave Lyman continuum intensities within a few percent of the observed values

and each gave total intensities in the 01 lines

that agreed within a factor of two with the intensities observed by Hinteregger 7

. The model in Table

1 gave the best agreement with the 01 profiles. Details of the oxygen calculations are discussed in

the following sections.

V. OXYGEN TRANSFER SOLUTIONS

Source functions for the resonance lines of 01 are computed using the technique described by Athay

and Canfield 8 for the Mg b and Na D lines. The 01

transitions occur between 2p

1+ 3 P and 3s 3 S°. There

appears to be no interlocking transitions of importance other than transitions to and from the

continuum. This conclusion was reached after trial

calculations that included 3s 5 S°, 3p 5 P° and 3p 3 P

levels showed that the source functions for the resonance triplet were essentially the same as when

these levels were omitted.

Both ionization and recombination occur mainly

from and to the ground state, 2p h 3 P. Also, the

ground state serves as the primary source of electrons for the 3s 3 S° level. This accounts for the

lack of important interlocking levels.

Photoionization rates from the 2p 4 3 P level

are computed using the Lyman continuum intensities

computed for the model atmosphere and photoionization

75
in in n h o id H

OOOO^^lOhOOOKTiOni/)

in ooooHHojnronnnfon

"^OOCNOCTirOOrOr^OCNCNCNlCN 0(Ninomoo<*(Nint^cooooooo ^ fO OJ (N H I I I I I I I ^ o n

rH CN "vT <T> O O I S oo

in e

o CO J- to CM ro LO O rH O o o O

rH rH H

rH CN ro "^F LO CN O O X X X

CN LO O rH l> LO

CN CT> H rH rH LO

rH lo CN LO O o CT\ CT> ^ CN CO o LO o PI \o LO CN O 00 O o> o LO 0 LO in m LO LO LO LO "sT ro ro ro

rH

(D O CN LO lo ro <nF 00 CO VO CN o

CN ro ro CN iH o CO LO LO U0 CN rH rH rH rH rH rH rH o O o o o o

rH H rH H rH rH rH rH H rH H rH rH

c ro o CN ro CO CO LO cn CN O

LO r- CN ro CN rH o CO LO ID LO LO &>0 O o rH rH r—

rH H rH o O o o o O H rH rH rH rH H rH H rH rH rH rH rH 1—

rH ocNCNoon^oo^ror^r-cTi^o

cN^^ovoooror-<NCT\cor--LO^r

LOLO^^rOCNCNrHrHOOOOO

oooooooooooooo OOOOOLOOOOOLOOOO

sLD>CMrs>(T\CNLnO^'^(N^, CTi(N j-sT'sfLOLOLOLOLor^-r^-oocriO'^o

i—I rH CN o O o o o O O O LO LO ro 00 r- o O LO o o o o ro r» CO o

CN LO o CN LO a> o

rH rH rH rH rH rH rH rH CN

76
cross sections measured by Cairnes and Samson 9

. For

|the 3s 3 S° photoionization rates we use a cross

jsection equal to .19Tra0

2 (v Q/v) 3 obtained from the

Burgess and Seaton 10 quantum defect method together

with quantum defects computed by Peach 11

. We use

a solar radiation temperature of 5350° to give the mean intensity for photoionization.

Transition probabilities for the resonance

triplet are obtained from E of = .19 as given by

Parkes, Keyser, and Kaufman. Electron collisional

excitation cross sections, Q±jr are not well known

for 01. We adopt a mean value of Q = . lua^ for each

I of the resonance transitions (cf. Stauffer and

McDowell 13

). However, we vary this Q to ascertain

its effect upon the 01 lines. For transitions between J substates of 2p 4 3 P we use Q = lOiraQ.

i Collisional ionization rates are unimportant.

However, we use Q = . 15-rraQ for ionizations from

2p h 3 P and Q = 7.5naQ for ionizations from 3s 3 S°.

The source functions for the triplet are evaluated iteratively. Stimulated emissions are ignored

and the scattering is assumed to be non-coherent.

We adopt an oxygen abundance of 6 x 10" 4

. The three members of the 01 triplet appear to

be of about equal intensity, .01 ergs cm2

sec

-1 at

the orbit of earth (Hinteregger 7

) . Assuming that

the intensity is relatively uniform across the solar

disk, we find a flux at the center of the solar disk

of approximately 35 ergs cm

-2 sec

-1 ster -1

. Computed intensities of the 01 lines are sensitive to the model atmosphere, to the oxygen abundance, to the gf values, and to line broadening

parameters. However, they are not particularly

sensitive to the collisional excitation cross

section. For example, an increase in Q by a factor

of four for each of the triplet lines increases the

line intensity by a factor of approximately 1.5 only

(see Table 2)

.

The chromosphere is effectively thick in the 01

triplet and the line intensities are proportional

to the product of the degradation length, Tdeg> and

the mean source term for photon generation <e*B> in

the layer 0 < t £ T deg- An increase in the collisional excitation cross section increases e* and

decreases T^e g. The two effects nearly cancel.

The line broadening parameters affect x^gg, and

hence the line intensities. We have used a Voigt

profile with both radiation and collision damping.

Since the cross section for collision damping is not

accurately known, we pick a value for the collision

damping half width that gives best agreement with

77
TABLE 2 LINE FLUXES (ergs sec" 1 cm

2 ster ) AND RATIOS OF

PEAK INTENSITY TO CENTRAL INTENSITY

FOR X1302 AND X1305

line X1302 X1305

Q VW Flux Peak : Center Flux Peak: Center

1 17 2.6 16 2.3

1 3 24 2.7 22 2.2

10 39 3.1 37 2.7

1 25 2.2 23 2.1

4 10 60 2.5 56 2.2

30 41 2.4 40 2.6

the observed profile. ' The adopted form for the half

width in angstroms is 6.4 x 10" 21 njj (VW) where the

parameter VW may take on different values. The

value of VW affects the intensity of the line as well

as its profile. Intensities of the X1302 and A1305

members of the multiplet are given in Table 2 as a function of VW and the electron excitation cross section Q. The intensities would be influenced

also by changes in the gf values.

The shapes of the wings of the lines depend

upon the damping parameters and can be fitted rea- sonably well by adjusting VW. The shapes of the

Doppler cores of the lines depend upon both the

magnitude and gradient of C- In general, the com- puted profiles show strong self reversal whereas the

observed profiles show weak self -reversal . The

amount of self-reversal in the computed profiles

decreases as dc/dh is increased. However, it does

not appear to be possible to obtain complete agreement between observed and computed profiles without

excessively large values of d^/dh. The self-reversal

is not markedly sensitive to either the collision

damping or the collisional excitation cross section

as evidenced by the ratios of peak intensity to

central intensity given in Table 2.

78
Log

4.5 44 - 25000 —

15 43 -20000

4 2 -15000

4 1 T40

-12000

- 10000

- 10

(km/sec)

3 9

3 8

3 7

- 8000

- 7000

- 6000

- 5000^^

/

s

IS-—- \V00 V9(2:|00

- 5

36 lr' 1 1 1 D 500 1000 1500 2000 2500

Height (km) Figure 1 . A temperature model deduced from the con- straints in Section IV and a C (h) model that gives

best agreement with the position of peak intensity

and the ratio of peak intensity to central intensity

in the 01 lines. The optical depth at the center of

A1305 is indicated by t 0 , and the optical depth at

the head of the Lyman continuum is indicated by t 912 . Figure 1 exhibits C (h) and T(h) from Table 1.

Figure 2 exhibits the X1305 profile computed from

the model in Table 1 for three values of VW together

with an observed profile. The observed profile

shown is reduced from observations by Bruner and

Rense 2

. Improved observations have been made with

increased spectral resolution and are in process of

reduction at the time of this writing. It does not

appear that the improved profiles will differ greatly

from the one shown. The source function for the

X1305 line and the non-LTE departure coefficients

for the ground state, bi, and the 3s 3 S° level, b 3 , are shown in Figure 3. A more complete tabulation

of the 01 calculations is given in Table 3. The

adopted collision cross section between different

J states in the ground level is sufficiently large

to give relative populations that conform very nearly

to a Boltzmann distribution.

Although the computed and observed profiles do

not show good agreement near line center there are several possible explanations. The sharp emission

79
! U

i w ro m ro o o CO m rH

CM ro CM m CM o o o

! w rH rH r—

in

| 1 00

i

i w

i

I

1

i

1 1—

II

! II i—

f ^

i J- to 1 cm CO O O ;

1 l-M CM 00 CM LO InO j |

j !,Q m X X

j

j CM CM ro CO i»o o oo r-~ I

i j. , | QQ 2^

i

-

i TM I !

1

i

! 1

j

! cT

j £ co. lD in LO O CM CO r- i

' 00 *sO rH Cn VO CM in ro 1

i 0 cn 1—1 rH rH rH H i rH CM CM CM CM !

1 1 l 1 1

i

LO.

i

j

l i—

! in

£_) n

I i—

iH ro ro ^ r» o r> o ro

! <y\ ko r- as ro. CO in ro

r ) ,_q rH rH O CO 00 CO

l i—j j- rH rH rH

r i o. i

! CM

»•-••) H

1 X cm o o

O o rH • 'o | x ro O CM o\ o in (Ti H rH 1 i—

ro 00 CO CO VD rO rH rH 00 ro x ! rH m 1 to rH ro I

1-1

!

1 i-i in o o o

o rH . 'oj

ro x ro rH CM CTl rH CO CM CO CM a\ CO rH I rH ro 00 00 G> 00 in ro rH rH rH X ! m : CO rH ro

j

in j- CM <N CM

s in o o o O o o

o rH rH rH rH t—rH ro | ro r- ro ^ ' iH X X X X X X r*« rH rH rH o

CM H CM o in O o o o j 0 rH . H ro CO LO rH vo rH

-P o o o O o o o O o in in ro co } o o LD o o o o r- ro r- CO <3\ o ! •H ^ CM in o CM m r-> cr» en cr> o

(D — H rH rH rH rH rH rH rH CM

80
I VW=I

Figure 2. Comparison of observed X1305 profile

(preliminary reduction, see text for explanation) and

computed profiles for VW =1, 3 and 10.

peak in the computed profiles at the edge of the

Doppler core has a width of the same order as the

instrumental resolution and is perhaps partially

"lost" in the observed profile. Also, macroturbulence and atmospheric inhomogenieties may substantially alter the "average" profile observed with low

spatial resolution.

The tendency for non-LTE source functions to

give central intensities that are somewhat lower

than observed appears to be a common difficulty.

In computations of Mg II and Ca II profiles (Athay

and Skumanich 1

"* ) and Na D profiles (Athay and

Canfield 8

) it has been necessary to resort to

rather large values of d£/dh in order to obtain

central intensities as large as those observed.

Broadly speaking, the values of C in Table 1 and

Figure 1 are consistent with values obtained from

the Ca II and Na D lines. The 01 lines are formed

81
Figure 3* Source function and non-LTE departure co- efficients for A1305. Source functions for the other

two members of the triplet are closely equal to the

A1305 source function (see Table 3)

somewhat higher than the Ca II lines, but the Doppler

cores of these lines are strongly overlapped in their

depths of formation. It should be possible to com- bine analyses of the Na D, Ca II, Mg II, and 01 lines

to obtain a single £ (h) model provided the concept

of a spherically symmetric chromosphere has a valid

meaning in this context at the depths where the lines

are formed. Additional help will come from analysis

of C II profiles recently observed by Berger and

Bruner 1

. The authors are indebted to W. Frye for most of

the programming and computing and especially to W. A.

Rense and E. Bruner for allowing them the use of the

01 profiles prior to publication.

82
REFERENCES

1. R. A. Berger and E. C. Bruner, Jr. 1969, Ap. J. 155 , L115.

2. E. C. Bruner, Jr., and W. A. Rense 1969, Ap. J.,

in press.

3. W. B. Henze 1968, Thesis, Univ. of Colorado,

Boulder (also Solar Physics, in press).

4. R. W. Noyes and W. Kalkofen 1969, Paper

presented at Solar Physics Meeting of AAS

,

Pasadena.

5. R. G. Athay and A. Skumanich 1967, Ann. d'Ap.

30, 66-9. 6. R. G. Athay 1969 , Solan. Phytic*, in press.

7. H. E. Hinteregger 1965, The Solan. Spectrum

,

Ed. C. de Jager (Reidel: Dordrecht), 179.

8. R. G. Athay and R. C. Canfield 1969, Ap. J. 156 , 000.

9. R. B. Cairnes and J. A. R. Samson 1965, Phy*

.

Rev. 139, A1403.

10. A. Burgess and M. Seaton, 1960 M.N. 120 , 121.

11. G. Peach 1962, M.N, 124, 371.

12. D. A. Parkes , L. F. Keyser and F. Kaufman 1967,

Ap. J. 149, 217.

13. A. D. Stauffer and M. R. C. McDowell 1966,

Pk.oc. Phy*. Soc. Lon. 89_, 289.

14. R. G. Athay and A. Skumanich 1968 , Solan. Phytic*

3, 181.

DISCUSSION

Thoma* : You find a very steep temperature rise

for t 912 = 1. How is the fit of other observation

such as the Balmer and Paschen jump? Can you explain

the observed helium lines? Also, I worry at your

neglect of Ha in solving the LyC transfer problem.

Including it, as I did in the chromosphere book we

wrote, will raise b lf b 2 , Te , dTe/dh if that solution

is a guide.

Athay: If I change the microturbulence and the

pressure gradient I shall get the increase of temperature at smaller heights and the observations fit

very well. The helium lines can be explained if you

take an asymmetric atmosphere.

Haggle* tone: Are your results very sensitive to

the dependence of microturbulence with height?

Athay: The increase of turbulence has different

effects. It makes the line broader, increases the

83
central intensity, and it decreases the intensity of

the emission. The observed relation of emission to

absorption is 2:1.

Undzikill: The resolution of the observed

spectra is low, of the order of 20 or 5A.

Atkay: There are new observations with a resolution of 0.015A (Bruner and Rense Ap. J. 157 , 417,

1969) . But even with this resolution you cannot

observe the reversal in the line center with sufficient accuracy. We do find similar problems with

reversal in the line centers for Lya, Ca II K and

Na I D lines.

Kalkoian: What about Ly3?

Atkay: Ly3 is very weak. The Ly-continuum is

strong, approximately 10 times the intensity of Lya;

Ly3 has only 1% of the intensity of Lya.

KalkoA&n: Did you test your theory with observed

lines of high excitation and with the observed center

to limb variations?

Atkay: We find agreement within a few percent

with the observed limb darkening and limb brightening . Wzllmann: If one observes 01 lines in stellar

spectra and can do a temperature determination with

these lines, can you say in which layer the lines

are formed?

Atkay- No.

84
PART B

THEORETICAL METHODS FOR HANDLING NON-LTE PROBLEMS

Chairmen : R. N. Thomas, A. G. Hearn

THEORETICAL METHODS OF TREATING LINE FORMATION

PROBLEMS IN STEADY-STATE EXTENDED ATMOSPHERES

by

George B. Rybicki

Smithsonian Astrophy sioal Observatory

Cambridge 3 Massachusetts

ABSTRACT

Theoretical methods applicable to the study of

line formation in steady-state extended atmospheres

are reviewed. The formal solution of the transfer

equation is considered, as well as numerical and

analytical methods of determining the source function. Topics discussed include: the local frequency

transformation, geometrical effects, and the case of

large velocity gradients. A new plane-parallel

approximation for spherically symmetric moving

atmospheres is given that takes account of transverse

velocity gradients.

Key words: spectral line formation, spherical

geometry, radiative transfer, stellar atmospheres,

moving atmospheres

.

INTRODUCTION

Methods of handling line formation problems

have advanced markedly in recent years for atmospheres that are static and for which the planeparallel approximation is reasonably valid. However,

two major difficulties may arise when extended

atmospheres are considered: first, the geometrical

extension may be so large that the plane-parallel

approximation is no longer valid; and second,

macroscopic velocity fields may need to be taken

into account. While only the first of these is

strictly implied by the term "extended atmosphere,"

the second is included because so many of the

87
astronomical examples of extended atmospheres have

such macroscopic velocity fields.

In this paper the theoretical work relating

to these two difficulties will be reviewed. In

doing so primary consideration has been given to

work in the spirit of the modern approach (see, e.g.

Jefferies, 1968) to line formation, in which the

equations of statistical equilibrium are solved in

conjunction with the transfer equations. Many of

the results of the early work that depend on the

assumption of coherent scattering, for example,

must be viewed with suspicion for the problem of

line formation. However, many of the methods and

techniques developed in these papers are relevant,

and it has been the intention here to extract such

useful information whenever possible.

After introducing the basic equations, the

formal solution of the transfer equation will be

discussed. Then analytical and numerical methods

of determining the source function will be considered, including Sobolev's theory of moving atmospheres . One new result presented here is a formulation

of the plane-parallel approximation for moving

atmospheres having spherical symmetry that takes

account of transverse velocity gradients.

BASIC EQUATIONS

The line formation problem is defined by the

simultaneous solution of the equations of statistical

equilibrium and the transfer equations for the atom

or ion under consideration. The assumption of

complete redistribution is usually made to account

for the noncoherent nature of the scattering. While

this assumption has proved of great utility in

static, plane-parallel atmospheres, there are reasons

to believe that it may not be as good when large

velocity gradients are present. This is because

the mechanism of trapping radiation, which usually

produces isotropic, frequency independent intensities

in the line core, will not operate so effectively,

since radiation can escape more readily and because

of the anisotropy introduced by non-uniform expansion. The only work relating to this point is

Magnan ' s (1968). He found only fairly small errors

due to the complete redistribution assumption in an atmosphere with velocities of the same order as

88
the Doppler velocity. However, the question deserves

further investigation under a variety of conditions

before any final conclusions can be drawn. In this

paper complete redistribution will be assumed,

and the modifications necessary to treat Doppler re- distribution will merely be indicated.

The effect of extended geometry on the equations

of statistical equilibrium is simply to make all

quantities depend on a general spatial point r, instead of the simple height or optical depth variable

of the plane-parallel case. The transfer equations

must be similarly modified by considering specific

intensities which are functions of position r, frequency v, and direction defined by a unit vector

In geometries having certain symmetries, such as spherical symmetry, the equations may depend on fewer variables, of course, and in the limit of a thin spherically symmetric shell, the plane-parallel

equations are recovered. With few exceptions (e.g.,

Bappu and Menzel, 1954) all of the work done on the

problem of extended geometries has assumed spherical

symmetry. In cases where Sobolev's theory of moving

atmospheres is applicable it is not strictly bound

by any particular assumption on the geometry, but

in the actual applications of this theory spherical

or plane-parallel geometry have been used.

The primary effect of velocity gradients is

the Doppler shift of the radiation field as seen in

a local frame of reference moving with the material

at any point. This manifests itself in those terms

in the equations accounting for the interaction between matter and radiation, namely, in the emission

and absorption coefficients. For complete redistribution, where the source function is frequency independent, the modification to the usual equations

is simply to replace the absorption profile <J)(r,v) v by 4>(r,£,v) = <j>(r,v- — £*v(r)), where v is the

*~ c — **-' O

line center frequency, v(r) is the velocity of the

material at point jr, ancT "c is the velocity of light.

This correctly describes the Doppler effect to lowest

order inj//c. For the radiation field this effect

is important because it can radically change optical

depth relations along a ray, as the absorption at

any point depends sensitively on any displacement

of the profile. Similarly the matter can be brought

to quite different states of excitation by motion

which causes it to absorb far more or far less in

any transition, as it absorbs in different parts of

the highly frequency-dependent radiation field.

89
A secondary effect of velocity fields is to

add streaming terms to the statistical equilibrium

equations, which describe the changes to the populations from convective transport of material.

Although this might be an important effect in some physical situations, it would not seem that it will

be so for any stars discussed here. This may be seen by considering an extreme physical example

with velocities on the order of 10 3 km s -1 and a characteristic length of 10 6 km, which are typical

for some Wolf-Rayet stars. This leads to an effective rate coefficient or order 10~ 3 s" 1 which is

very small compared with other typical rate co- efficients entering the statistical equilibrium

equations. Therefore this effect will be neglected

here

The equations of statistical equilibrium are statements that the populations ni (r,) at a point

corresponding to the various relevant levels of

excitation energy Ej_ are independent of time

Typically i ranges from 1 to N for N bound levels,

with the addition of values for adjacent stages of

ionization

The rate coefficients r^j may be separated into

rate- coefficients due to radiation and to collision

with other particles (probably mainly electrons, but

perhaps high energy protons or alpha particles may

contribute)

For the radiative transitions between bound levels,

(1)

(2)

E . > E . i 3

. . = B . . J. , (3)

90
where the A's and B's are the Milne form of the

Einstein coefficients and

J . . = -t— 1 j 4tt dv dl <j> (r , v jP- l-v(r) ) I (r,£) . (4)

Here I^(r ,JL) is the specific intensity of radiation

at frequency v at point r in direction %. The nor- malized profile function^is defined by

(r,v) = k^ (r)/k. (r) , (5

(£) where k v is the line opacity for the transition

i->j , in the rest frame of the material, and where

the integrated line opacity k 0 is

U) dv (6)

Thus

(r,v) dv=l (7)

The line center frequency v^_. is computed from

hv.. E . -E . i 3 (8)

h being Planck's constant.

The appearance of the velocity field in the

profile function in Eq. (4) is responsible for the

velocity effects associated with the equations of

statistical equilibrium. It should be noted that

the profile function cannot now be taken outside of

the angular integration, as in the static case.

91
The equation of transfer for radiation in the

line transition i^-j is

= k (r) (j) (r,v

v . . + S(r)

(9)

where

S(r) =

-l

Cio)

is the line source function and gj_ is the statistical

weight of level i. The line opacity k p is given by

The boundary conditions are usually given by

specifying the incident intensities on all boundaries

of the region, or, in some cases, certain linear

relations (reflection conditions) relating incident

and emergent intensities.

In writing Eq. (9) continuum absorption has been

taken into account by means of the continuum source function Bc and the continuum opacity k c . Also,

complete redistribution has been assumed in this

equation. In order to account for Doppler redistribution properly, the absorption and emission of a number ni(r,w)dw of particles in level i in velocity

range dw would Tiave to be considered, rather than

simply 'the total number in each level. This would

also necessitate writing a more complete equation

of statistical equilibrium for ni(r,w). Another way

to take Doppler redistribution into 'account is by use of redistribution functions (Hummer, 1962, 1968)

although this method assumes that the velocity

distribution of the lower level is known to be

Maxwellian, and that stimulated emissions are

(n.B. . - n.B. .\

\ 3 ]i i ID/

(11)

92
negligible; these assumptions will certainly apply

to resonance lines of high excitation potential, but

perhaps not necessarily for other lines. The relationship between non-Maxwellian velocity distributions of the excited levels and Doppler redistribution has been emphasized by Oxenius (1965)

.

It is somewhat more convenient to work m terms

of a normalized frequency variable defined by

v-v . . 12

A. . ID

(12)

where

v. -v..

and where vth i- s a typical Doppler velocity

characterizing the atmosphere, given by

Here ^]^± nis a typical kinetic temperature, which may

include contributions from "microturbulence . " For

Doppler profiles the profile function then becomes

<J> (r ,x)

x 2 /6 2 (r) )15)

where

6(r) = tn ~^ th

and

vth£ ; f2kTv

. (r) 1 i kin ^

L m J 17)

is the Doppler velocity characterizing the profile at

the point r. A dimensionless variable u(r) is also

93
defined in units of v

fch by

v(r)

u(r) = —

th

With these variables Eqs . (4) and (9) become

l-g- I (r,A) = k (r)

(18)

.00

f dx d£ (f)(r,x - £«u(r))I (r,A) (19)

- OO

+ k

c

( 5i [" X x

+ B

c] (20)

while in Eqs. (5), (6), and (7) the variable v is

replaced by x. Finally in Eq. (11) an additional

factor of A^! appears on the right-hand side. The

range of the variable x may be taken as - 00 to + 00 with inconsequential error. The advantage of these dimensionless variables

is that material velocities of the order of the

typical thermal velocity correspond to |u| ^ 1, and

it is in these cases that velocity effects begin to

be important.

THE FORMAL SOLUTION

Since the pioneering work of Beals (1931) many

investigators have assumed various structures for

extended atmospheres, including the specification

of the source function, and then have computed the

emergent flux using the formal solution of the

transfer equation. Such calculations avoid the

difficult problems of actually determining the source function as a consistent result of the

specification of the fundamental atmospheric

parameters. Despite their possible inconsistencies,

however, the importance of these calculations for

the development of the subject should not be underestimated. It is probably fair to say that the

present-day conception of extended stellar atmo94
spheres is largely based upon such calculations,

or upon intuition based upon knowledge of the

formal solution. It is not yet possible to solve

for the source function including all those effects

that are known or suspected to be important, so

the calculations based on the formal solution are still likely to be important for some time. In any

case such calculations will be necessary as a final

step in a more detailed calculation and in some

cases the primary effects of velocity fields are to

be found in this step. For example, the redshift

found by Hummer and Rybicki (1968) for an expanding

atmosphere is, as pointed out by them, solely due

to the change in optical depth relations in various

parts of the line, and not to any substantial change

in the source function from the static case. The formal solution of the transfer equation is

well-known (see, e.g., Chandrasekhar (1934). Write

Eq. (20) as

31 (S = K (K) [-1 +SU (21) 9£ x x x'

where £ measures path length along a ray from some convenient point, and where

kx U) = k

£

(£) fU,x) + k

c U)

k Q a) 4>U,x) SU) + k U) B U)

J U) = — ~ " (22)

k U) <J>U,x) + K U)

All quantities are expressed as functions of £ along

the ray. Then the formal solution is

-l\ Kx

(C')d5'

I X U> = I x

(0) e

f C -/!. < (5")d5"

+ K (5') (V) e * dC (23)

' o

95
The intensity at a given point is then expressed as a quadrature involving the total opacity k , the total source function <J / and a value of intensity at

another point on the ray I x (0), which is known if

£ = 0 is a boundary point and I x (0) is known incident

boundary condition. In some cases the zero point can

be taken so that the exponential factor multiplying.

I x (0) is essentially zero, and the value of lx(0) is

not needed.

For a given set of values for the function kx and U x the emergent intensity along any ray can be

computed. A convenient measure of the emergent flux

which would be received by a distant observer over

the area of a projected disk of the atmosphere. This

quantity might be called the specific luminosity of

the star in that direction and at the given frequency; in cases where the star radiates isotropically it is simply the total luminosity per unit frequency range divided by 4tt. The actual flux at the

point of observation is proportional to the specific

luminosity, the factor being the inverse square of

the observation distance. The observed shape of

the spectrum at any distance from the star is

correctly given by the specific luminosity as a function of frequency.

The introduction of any symmetries , such as spherical symmetry, will simplify the problem

considerably. In plane-parallel geometry the formal

solution is given by Eq. (23) except all quantities

depend on the depth variable z alone, so that £ becomes z and d£ becomes dz/y, and similarly for

the various primed variables. In spherically

symmetric geometry all quantities in Eq. (23) depend

on the radius r alone, but the relationship between

£ and r is much more complex. It is sometimes con- venient in this case to use a new variable p which

represents the distance of closest approach of the

ray to the center of spherical symmetry, and to

express all quantities in terms of r and p rather

than r and u. This latter procedure was first introduced by Chandrasekhar (1934) . It is also

possible to use £ and p as the basic variables and

this has essentially been done by Castor and Van

Blerkom (1970) . These two choices of variables

are useful because the transfer equation can be

written in terms of a single partial derivative,

rather than the two of the r,y representation.

This is already clear from Eq. (21)

Another kind of simplification arises for

stars with large velocity gradients. For each

96
frequency the radiation in a spectral line originates

in relatively localized regions that lie very close

to the so-called surfaces of constant radial velocity

defined by x = u^ where U£ is the component of

velocity along the line of sight. For any given

velocity field these surfaces can be determined;

then assuming that the continuum opacity is small,

the only region where any emission in the line can take place at frequency x is very near the appropriate surface. The line source function S can be

assumed to be constant over this region and the

intensity along the ray is equal to

-T -T

Ix

(£) = I x

(0) e X

+ S(l-e X

) , (24)

where x x is the optical thickness of the region along

the ray. The computation of t x will be postponed

until the following section when local frequency

variables are introduced. When more than one such

surface is cut by a single ray corresponding formulas

can be similarly derived. For example, when two

surfaces are involved,

(T +T r ) X X

— T ' -T -T ' + S e x

(l-e X

) + S'(l-e x

) . (25)

Here the ray starts at £ = 0, first cuts the surface

corresponding to the unprimed quantities , and then

cuts the surface with the primed quantities. For

further details of such calculations see Rublev

(1961, 1964), Lyong (1967), and Castor (1970).

ANALYTICAL METHODS FOR DETERMINING

THE SOURCE FUNCTION

While most methods for actually determining the source function are numerical ones, there are some topics of an analytical nature that are relevant to

the line formation problem beyond the simple consid97
erations of the formal solution. Among these are

the local frequency transformation, special geometries, the infinite medium with constant velocity

gradient, and Sobolev's theory of moving atmospheres.

These will be discussed in this section and the

numerical methods in the next.

a. Local Frequency Transformation

Milne (1930) introduced a transformation to

a new frequency variable x

1 related to x by

x* = x - u(r) -A . (26)

Originally this transformation was formulated in the

plane-parallel case; Eq. (26) is the generalization

to arbitrary geometry. Clearly x' is the frequency

seen in a frame of reference moving with the local

material at the pointy when radiation travelling

in direction I has frequency x in the stationary

frame

If intensities and other variables are written

as functions of x

1 instead of x then the transfer

equation becomes

31 , 31 ,

where Q is the following quadratic form in the

components of the vector £ : 3u

Q (r , £) = 7 ^-5 I l a = I G 0 i i D (28)

Here a and 3 are tensor indices ranging from 1 to

3, which label the components of the corresponding

vectors. Since the antisymmetric part of Ga 3 does

not contribute to Q, it may be defined as

98
This may be recognized as the rate of strain tensor

of fluid dynamics. There are several special forms

f or ja, Ga £, and Q corresponding to cases of various

symmetries

:

1. Plane-parallel symmetry

Let a=3 be the index for the direction normal

to the planes of symmetry. Then

u =u =0 ; u =u(z) 12 3 G =u'(z) (30)

3 3 Q = y

2 u' (z)

2. Spherical symmetry

Let the indices take the values r, 6, and <$>

,

corresponding to the radial, polar, and azimuthal

directions at the given point. Then

u

e =V° ; vu(r)

G

ee=G ^

=u(r)/r ; Grr=u ' (r)

Q = y

2 u' (r) + (l-y 2 )u(r)/r .

Note that here and in the preceding the values of

Ga $ not given are zero, and that y is the usual

direction cosine of the ray.

There are two special cases that frequently

occur in the spherical case. When u(r) is constant

there results Q = (l-y 2 )u/r, and when u(r) is proportional to r (uniform expansion) there results

Q = u' . One advantage of such a transformation is that

absorption and emission in the local frame are usually close to being isotropic, so that the vari99
ation of intensity with angle can be expected to be

smoother at fixed x' than at fixed x. For example,

consider a rapidly moving region, [u|>>1. As the

direction of a ray is varied slightly the absorption

in the region can vary tremendously when the right

condition of line of sight velocity is met for the

particular frequency x. This leads to rapidly

varying functions of angle, which are hard to handle

numerically

The local frequency transformation makes obvious

the fact that only velocity gradients affect the line

formation problem, since only gradients appear in Q.

This is also quite obvious on physical grounds, but

some of the implications are quite deep. If the

opacities and profile function are independent of

positioner, and if CHr,^£) is likewise independent of

r, then Eq. (27) is a' integro-dif ferential equation

with spatial translational invariance. It is

therefore susceptible to a host of techniques of

solution more or less familiar in the static case,

but now referring to a constant velocity gradient.

This fact forms the basis of Sobolev's analytical

solution, to be discussed later in this section.

The problem left in the last section, that of

computing the value of x x in Eq. (25) , will now

be completed. It is given by

where r = r(£') along the ray and where kc is assumed

negligible. Changing the variable of integration to

the local frequency variable gives

where the value r = rQ , the position of the surface

of constant velocity^ has been inserted in those

functions that are slowly varying over the range

in which the integrand is not negligible. These

functions have further been taken from under the

integral. The origin of the factor in the denominator is the transformation of differentials,

T X

(J) (r ,x

1 ) dx 1 ,w O

r

f (32)

100
and

noting that

h = w- (34)

The absolute value is to account for the correct

ordering of the limits of integration, making the

lower limit smaller than the upper limit. From

the normalization of <$> it follows that

T = Q(r,i)

o

(35

since the range of integration for large velocity

gradients is essentially over the entire line.

There are further uses of the local frequency

transformation, one of which will appear presently.

It will merely be pointed out here that, in such a local frequency description, redistribution functions

can be used in their static form, rather than having

to take the macroscopic motions directly into account

in the redistribution functions themselves.

b. Special Geometries

The two special geometries of interest are the

plane-parallel (pp) and spherically symmetric (ss) ones. The directional derivatives for these two cases are

3Ix

3Ix 1 • 3F

1 = "

( P?> (36)

and

31, 91

9rx

+ 1-y

r

91.

3lT (ss) (37)

101
in terms of the frequency x. In terms of the local

(1) frequency variable x

1 = x - yu(r) they are

91

9r

x' 9x' x - y

2 = y u' (r) (IPP) (38)

X + !-^ 2 = y ay

+ r r

[y

2 u'(r) + (1- y

2 )Hl£)]

91

(lss) (39)

An interesting fact emerges from these expressions.

The plane-parallel equations are usually obtained

by dropping the term involving the partial derivative

with respect to y, as in Eqs . (36) and (37), the

argument being that this term is of order of the

ratio of a mean free path to the radius of curvature

of a layer. By the same argument the corresponding

term in Eq . (39) may be dropped, but this still

leaves an extra term involving u/r when compared to

Eq. (38) . This term cannot be dropped on any

reasonable grounds since in extended atmospheres it

is on the same order as the term in u

1 (r) , which is

clearly not negligible since it accounts for all the

velocity gradient effects. Thus a paradox has

appeared: Making the plane-parallel approximation

first and performing the local frequency transformation second does not give the same results as these

operations performed in the reverse order.

We feel that the reason for this paradox is

that the derivative 9I x/9y at constant x is not

small and cannot be dropped, while the derivative

9I x «/9y at constant x

1 is small and can be dropped.

This point was discussed in the preceding subsection.

As a consequence Eq . (36) is probably not the best

formulation of the plane-parallel approximation

for moving spherically symmetric atmospheres. A

more correct formulation (c) in local frequency

variables would be

102
- [y 2 u'(r) + (1-u 2

) H^>J^2Ll (clpp) (40)

rather than Eq . (38)

.

In order to understand why this new planeparallel approximation differs from the usual one,

it is useful to distinguish between two types of

spherical divergence effects: First, there is the

usual divergence of the rays , which may be neglected

for cases where the thickness of the layer in

question is much smaller than the mean free path.

Second, there is the divergence of the velocities , which may be quite important because of the extreme

sensitivity of the absorption coefficient to slight

shifts in frequency. The terms corresponding to

divergence of the velocities have been retained in

this new formulation while the terms corresponding

to the divergence of the rays have been dropped.

It is interesting to note that precisely the same arguments have been used by McCrea and Mitra (1936)

for including Doppler shifts in the moving atmosphere equations while dropping aberration effects,

even though these are of the same order in v/c

.

This idea may be illustrated by some examples

giving some further physical insight. Suppose

first of all that u(r) is proportional to r, the

velocity field for a uniformly expanding atmosphere.

The local velocity field as viewed from a local frame

of reference moving with the material at any point

is isotropic; in fact it appears to be a uniform

expansion away from that point. This simple fact

is correctly represented by Eq. (40) where the co- efficient of 9I x «/3x' is independent of u, while in

Eq. (38) it is proportional to \i 2

, implying a strong

directional effect. Another case of interest is

that of constant outflow, u(r) = const; the term in

Eq. (38) now vanishes, which implies the absence of

any velocity effects. This is clearly not so, however, since there is a transverse velocity gradient

in this case, which, for example, makes it easier

for photons to escape the atmosphere along a tangent.

These examples show the superiority of Eq. (40) over

Eq. (38).

103
It remains to write the corrected formula in

terms of the variable x,

31,

dr = V

31,

Jr

2xU(r) x

dx (cpp: 41"

There would not seem to be any particular advantage

in using Eq. (41) over Eq. (40) since a derivative

with respect to frequency still appears. However,

this derivative can be eliminated by performing a different sort of frequency transformation to a frequency x" defined by

x" = x + (*-.-) J ^

dr (42)

where any indefinite integral can be chosen. Then

81

9r

31 „ x x , s = y ^_ (ppp 8r 43)

which is exactly in plane-parallel form, but in terms

of the new frequency variable. This will be called

the pseudo-plane-parallel formulation (ppp) , and may

be of particular use for methods, such as those using

integral equations, where frequency derivatives can- not be easily handled.

It is interesting to write the (ppp) equation

in full for the case of a uniform expansion. Taking

dr = u

for this case yields

31 „ ST = k

* (r

' 4'x

" " ^)[- v + s]

+ k

c

(r) I x

„ + B

c

J (44

104
! This would be precisely the usual plane-parallel

I equation, except that in the argument of the profile

I function u is divided, rather than multiplied by p.

o. Infinite Medium With Constant Velocity Gradient

Sobolev (1957) showed that in an infinite planeparallel medium with a constant velocity gradient

the solution for the source function of a two-level

atom with no continuum could be reduced to the

solution of the integral equation (in a notation

which differs from Sobolev' s)

S(t) = (1-e) K( t-t" |)S(T , )dT' + eB(x) (45)

Here e is a constant, the ratio of the collisional

de-excitation rate to the total de-excitation rate

and the kernel function K is defined by

K(t)

1

2 dx du

(x) 0 (x + yut) exp (x + yuz) dz>

.

(46)

The variable x is an equivalent integrated line

optical depth, defined for a medium at rest. The

velocity gradient is a constant and is defined by

Y =

du

9t (47)

The factor A that appears in Sobolev 's paper is to be

taken as unity here because of the use of integrated

line absorption to define the optical depth scale

rather than the line center absorption that Sobolev

uses. The function B(t) is the Planck function at

the local electron temperature and at the line frequency .

105
The trans lational mvariance of the problem

is clear, since the kernel function of this integral

equation is a function of the difference |t - T

1

| alone. Such integral equations have been extensively

studied, but usually in the static case. Much is

known about the solutions of such equations and how

they depend on e and on B(t) . A very useful physical picture is obtained by

regarding the quantity S(t) in this equation as the

probability of emission of a single photon, rather

than as the average number of such emissions. Then

eB(x) gives the probability density of creation of

a photon at the point x, K(t) gives the single-step

distribution function for the free propagation

distance of the photon before absorption, and e is

the probability that an absorption will be followed

by the destruction of the photon. A very useful

concept associated with this probabilistic viewpoint

is that of thermalization length , an average distance

traveled by the photon between its creation and

its destruction (see Rybicki and Hummer, 1969)

By use of the concept of thermalization length much

can be said about the solution S(t) without actually

solving the entire problem, and it is instructive

to do so in the present simple case of a moving

atmosphere

There is a very important physical distinction

between the usual static case of thermalization

and the present one, however, which is due to a change in the normalization of the kernel function.

Ordinarily the normalization would be

K(|T|)dx = 1 (static case) (48)

since any photon eventually is absorbed somewhere.

But in the present case this is to be replaced by

. OO K(

t I )dT =1-3 (49)

— oo with 0<B<1. This implies that a photon need not be

absorbed, but can escape the medium entirely with

escape probability 3. This is an entirely new phenomenon which might be called intrinsic escape ,

106
since it does not depend on the presence of boundaries, but rather on the rapidly decreasing opacity

seen by a photon as it becomes further and further

removed into the line wings by virtue of the Doppler

shift of the material. This opacity decreases

rapidly enough so that the optical thickness in

the direction of a velocity gradient is bounded no matter how long the geometrical path length is.

From Eq. (46) it follows that

yI y

2 dy (50

For details of this reduction the reader is referred

to Sobolev (1957)

.

In spite of this normalization the equations can again be brought to the form

S(t) = (1 e) K(|t-t* I )S(T')dT f + e B(t) 51)

where the kernel K is now normalized,

,00 k(|t| dx = 1 (52

This is accomplished by the definitions

1 K(t) =

1 - K(T)

e = 1 -(1-6) (1-e) = £ + B(T) = § B(T

£3

(53)

Since Eq. (51) is now in the usual form, the

thermalization length may be discussed. First of

all, it should be noted that when $>>e the value of

£ is insensitive to £, and in fact £ ^ 3. This means that the mechanism of loss of photons by

107
intrinsic escape dominates the loss by collisional

de-excitation, and therefore the thermalization

process will be determined by 3 rather than e when

3>>e. Secondly, it should be noted that the kernel

K(t) is very much more sharply cut off for large

values of x than in the static case, at least for

Doppler profiles. This is because it is precisely

those photons which would have traveled a long

distance in the static case that now escape

and do not appear in the distribution K(x) at all.

The actual cutoff of the kernel can be roughly

estimated to be at distances of the order of

t ^ 1/Yz since this is roughly the scale over which

the profile shifts through its own width, and photons

that travel to larger distances will escape com- pletely.

Rybicki and Hummer (1969) have shown that when

the distribution of single flights is sharply cut

off, the thermalization length A is of the coherent

type, and is of the order of the width of the kernel

times (e)- 1/2

, that is, A ^ y-Me)- 172

. When

properties vary slowly on the scale of the thermalization length an approximate solution to Eq. (51)

may be obtained by removing S(t) from under the

integral. Then

S(t) = B(t) , (54)

or, when $>>£,

S( T ) = . (55)

The source function is in this case simply determined

by the creation rate e B(t) and the escape probability 3.

It is important to note that for Eq. (55) to

be valid it is not necessary to have very large

gradients, y>>l. For example, a fairly small y can still lead to a 3 satisfying 3>>e, thus the sole

requirement is one concerning thermalization length

and the scale of variation of properties L. For

small y it follows that 3 ^ y/3 (see Sobolev 1957)

,

and 3>>e implies e ^ 3, so that the thermalization

length is A ^ y

_3/2 . The condition of validity is

simply A>>L, and if L is sufficiently large this can be met for arbitrarily small y.

The reason that so much space has been given

here to the discussion of Eqs . (45) and (51) from

108
\ the point of view of thermalization theory is that,

i within the framework of the simplifying assumptions

| made, it has been possible to derive solution (55) , which is identical with the one found in Sobolev's

1 theory of moving atmospheres. Furthermore (and

; most importantly) it has been possible to fix

conditions of validity of this solution, which

relate the magnitude of the velocity gradient to

scales of variation of the physical parameters.

In particular, it has been pointed out that this

solution may be valid for "small" gradients,

whereas Sobolev's theory of moving atmospheres has

often been regarded only as a "large" gradient

theory. These considerations will be useful in

the following discussion.

Castor (1970) also discusses the validity

of Sobolev's theory by deriving diffusion-like

corrections and estimating their magnitude. This

is roughly analogous to the above procedure, since

the use of the coherent type of thermalization

length is essentially a diffusion theory result.

However, the criterion Castor obtains is that the

typical mean thermal velocities must be small in

comparison to typical macroscopic velocities.

By estimating the macroscopic velocity as the

gradient times a scale length we may put this

into the form L>>y_1

, which is a less restrictive

condition than the one obtained here. This point

needs further investigation.

d. Sobolev's Theory of Moving Atmospheres

In general, velocity gradients present great

difficulties in the problem of line formation.

However, the remarkable theory of moving atmospheres

developed by Sobolev (1947, 1957) demonstrates that

velocity gradients are actually a simplifying

feature when they are of sufficient magnitude. In

favorable cases an entire coupled multi-level

transfer problem, a formidable problem even in the

static case, can be reduced to a set of algebraic

equations for the populations at each point in the

atmosphere. A brief discussion of Sobolev's theory

will now be given to show how the concept of escape

probability can be extended to multi-level problems.

The continuum opacity kc will be assumed negligible

in the neighborhood of each line.

109
The basic idea of the theory is as follows:

consider a region of the atmosphere in which the

properties are more or less homogeneous, and which

is sufficiently large in the sense to be discussed

below. In the static case the radiative transition

rate between any two levels, say i to j , is exactly

the same as the rate in the reverse transition, j to

i. This is because any transition leading to the

emission of a photon will in turn produce the reverse transition when that photon is absorbed somewhere

else in the same region. The two transition rates

averaged over the region must then be equal, and

if the conditions are homogeneous the equality

applies also at a single point. The condition on

the size of the homogeneous region can be seen to be

that every photon emitted in the region must also be

absorbed there, to a good approximation.

When there is a velocity gradient a photon can escape the atmosphere entirely, and in this case

the downward radiative transition rate will exceed

the upward rate. The difference in these two rates

is simply niAi$ji, where 3ji is the probability of

escape of the photon due to the Doppler shift of

the profile.

The escape probability 3 ji can be easily found

by assuming a constant velocity gradient over the

region of interest. Suppose a photon is emitted

at frequency x and direction & at a certain point.

It is convenient to use a frame of reference such

that this point is at rest at the origin. The

velocity field is then

(56

The total optical thickness of the medium from the

origin in the direction I at frequency x is

U) = k. . f cMx-u(W d£ (57)

Setting x' = x - u = x - £Q U) , there results

T (£) = X —

ID QU) (x' )dx' (58)

110
The integrated line opacity for the transition i to

j is denoted by ki j

.

The probability of escape in a frequency range

dx about x and in an angular range d£ about I is

dj^(f)(x) dx exp [-t

x U)J . (59)

The net probability of escape is thus

e ji = 4¥ 1 dx * (x) exP [" T§f |

X

*< x '> dx ']-

— 00 —00

Changing the variable of integration from x to

j

X

(Mx' dx

yields the final result

' j i 4tt dl (60

The escape probability is therefore independent

of the profile function under the assumption of com- plete redistribution. The restriction to rectangular

profiles in the original work of Sobolev (1947) is

unnecessary, as pointed out by Sobolev (1957)

.

The expression (60) is the generalization of the

escape probability formula for arbitrary velocity

gradients. Substituting in the appropriate Q for

plane-parallel geometry, namely Eq . (30), the

expression (50) is again obtained, since u' (z)/k = 3u/3t = y. Similarly Eq. (31) for spherical geometry

111
leads to the formula

ji( r) y

2 u» (r)+(l-y 2 )u(r)/r|

ID

An equivalent formula was derived by Castor

For the special case of uniform expansion, u' = const. , and the corresponding formula is

The determination of the escape probability

for each transition depends on local parameters

in the atmosphere, namely the integrated line

opacity and the velocity gradient. The equations

of statistical equilibrium can be written in terms

of net radiative rates, which are simply related to

the escape probabilities, as shown above. It follows

that an entire multi-level problem can be reduced to

an algebraic set of equations, which can be solved

locally at each point in the atmosphere, without

regard for conditions at other points. This un- coupling of the various parts of the atmosphere is

due to the Doppler shift between material at separated points, which causes them to absorb and emit

in quite different parts of the spectrum.

As an example of such a formulation of a multir

level problem, consider the case of an atom in the

dilute radiation field p^ c of a star. The statistical equilibrium equations are

exp

(|y 2 u'

3-D (r)+(l-y 2 )u(r)/r

dy . (61)

(1970)

= u/r

(62)

n.( J A., 3i + B. p. ) Hk^i ic ic/

Z

, n

kAki 6 ik + nen+C i

( V ' (63

k=i+

112
I where all collisional processes except recombination

have been neglected. The local physical parameters

fix all the constants in this equation, which may

then be solved for the ni . For details see Sobolev

(1947) . A more recent and more sophisticated example

I of the formulation and solution of multi-level

problems using Sobolev' s method is given by Castor

| and Van Blerkom (1970) , who solve a 30-level He atom

i in spherical geometry. Collision processes are included and the effects of continuous absorption are taken into account approximately.

An important area for future investigation is

the determination of criteria for the validity of

Sobolev' s theory. The crucial question seems to be,

as in the simplified two-level case, how small a scale of variation of the physical parameters is

allowable while still maintaining the local homogeneity that is necessary for the use of the escape

probability concept. Possible approaches are to

extend the two-level thermalization arguments or Castor's diffusion correction terms to the multilevel case.

NUMERICAL METHODS

Most numerical calculations have treated the

effects of velocity gradients for two-level atoms

in plane-parallel geometries. These calculations

have been exploratory in nature, to discover effects

rather than to obtain accurate solutions to a physical problem. This seems to be appropriate at

the- present stage of development of the subject.

Kulander (1964, 1968) assumed an atmosphere

which consisted of several layers in which all

physical properties were constant. This allowed

a semi-analytic approach to be taken, since the

solution for the discrete ordinate intensities in

each layer was a linear combination of elementaryexponential functions. Boundary conditions at

the interface between two layers were simply that

all intensity components must be continuous. In

this way Kulander was able to solve for the source function and the emergent intensities for many

different cases. This work showed that the source function was generally discontinuous across the

boundary between layers and sometimes large in- creases in source function near the surface could

be obtained in this way. These increases simply

113
mean that the radiation field near the boundaryis quite weak in the line core for material at

rest, but as the material moves it can absorb the more intense radiation in the line wings. Examples

were shown where emission features could actually

result from these increases in source function.

In general, line profiles are now much more complex

and, of course, asymmetrical.

Kulander (1967) also developed a numerical

method of solution using differential equations,

which is directly applicable to media with continuous

variation of properties. However, it would seem

that this method must be unstable for large optical

thicknesses for reasons given by Hummer and Rybicki

(1967) in their discussion of the fundamental matrix

method. The method of slabs, on the other hand, can

be used for arbitrarily thick atmospheres, including

semi-infinite ones, if not too many slabs are taken.

When large numbers of slabs are taken in order to

model a continuous distribution of properties then

the choice of a stable method of solving the relevant

equations becomes important. The number of frequency

and angular components that can be treated poses

another limitation of the method. Kulander used a single-point angle quadrature, which is roughly

equivalent to the Eddington approximation. Numerical

problems would probably limit the total number of

frequency components to perhaps 100. Up to 13

components were used by Kulander (1967)

,

It should perhaps be mentioned here that the

methods used by Abyankhar (1964a,b; 1965) are closely

related to the above slab method, but they have been

formulated in terms of coherent scattering so that

the details are not directly relevant here.

Hummer and Rybicki (196 8) have used a differential equation method based on an extension of the

Riccati method (Rybicki and Hummer, 1967) , which is

applicable to continuous variation of properties.

Using this method they show how a uniformly expanding

atmosphere can produce a red-shifted emission line.

The reason for this shift is simply an optical depth

effect, the actual change in the source function due

to the motion being irrelevant. The emission and

absorption of the material closest to the observer

is shifted to the violet and therefore optical depth

unity on the violet side of the line occurs much

closer to the surface, where the excitation is

smaller. This reduces the intensity of the violet

emission, and the line appears red-shifted.

The small step size which must be used in the

numerical integration to avoid multiscale instabil114
ities limits the use of the Riccati method to atmospheres having optical depths perhaps of a few

hundred (Rybicki and Hummer, 196 7) . There is a way

of treating semi-infinite atmospheres if variation

of properties is confined to a layer near the surface, again of no more than a few hundred optical

depths in thickness. The Riccati method is also

limited by the number of discrete ordinates in angle

and frequency that can be reasonably handled, no more than perhaps 60 components in one hemisphere.

Another method which no doubt can be employed

to advantage in this problem is that of Feautrier

(1964) which should avoid the difficulties of the

Riccati method as far as the multiscale instability

is concerned.

One modification of the method is necessary

if the usual form of the equations in the method is

not to be changed, namely, the quantities J and F must now be defined by

J(x,y,x) = i [I (t,u,x) + I (t,-u,-x) ]

F(x,y,x) = ^ [I(x,y,x) - I(t,-u,-x)] . (64)

Changing the sign of x, as well as that of y , takes

advantage of the symmetry of the profile function

(f)(T,y,x) under the joint interchanges y -> -y and

x -> -x.The methods employing integral equations that

have been used in the static case can be modified

to include velocity gradients, as Kalkofen (1970)

has shown. One advantage of this method is that

a much larger number of angle and frequency points

can be taken than in the above methods, since

the computation time increases linearly rather

than quadratically or cubically with the number of

components chosen. This advantage may be very

important for cases of large velocity gradients.

Calculations of line formation have been

performed by Mathis (196 8) and Magnan (196 8) in

spherical geometries. Mathis used an iteration

scheme to solve for the source function and emergent

intensities for a uniformly expanding, spherically

symmetric atmosphere . and he found a red-shifted

emission line of the same type as the plane-parallel

calculation of Hummer and Rybicki (196 8) . Since

his method appears to be equivalent to A - iteration

115
in the plane-parallel case, its rate of convergence

will be slow when the mean number of scatterings is

large

Magnan (1968) used a Monte Carlo method for a spherically symmetric atmosphere with a constant

velocity of outflow. This macroscopic velocity

was the same Order of magnitude as the thermal

velocity. He treated several cases, which included

effects of dipole scattering and of various boundary

conditions. A very interesting and important com- parison is that between the calculations made with

and without the assumption of complete redistribution. The difference between the emergent profiles

for these two cases is that the complete redistribution profile seems more smoothed-out than the exact

calculation, but the differences are quite small.

It would be unwise, however, to generalize about the

adequacy of the assumption of complete redistribution

from this calculation, since the physical case

is not a severe one, the macroscopic velocity not

being very large in comparison to thermal velocities.

An outstanding advantage of the Monte Carlo

method is the ease of formulation in complex situations, which derives from the functional matching

of a probabilistic numerical method to a basically

probabilistic physical process. However, as is well

known, there is a heavy penalty in that the accuracy

of the calculation grows as the square root of the

computation time so that the method is only useful

when the time to calculate one "event" is very small.

This requires that the mean number of scatterings

be small. In a calculation such as Magnan ' s the

advantages are clear, since the case is not severe

and the aims are limited. The range of problems for

which the method is useful has yet to be determined.

It might be pointed out that the effects found

by Magnan are entirely due to the transverse

velocity gradient, which no plane-parallel calculation, as presently formulated, accounts for properly.

Therefore, there is no plane-parallel calculation

that can be compared with this one in order to

determine the effect of spherical geometry. By use

of the improved plane-parallel approximation given

in this paper such a comparison would be possible.

ACKNOWLEDGMENTS

The author wishes to express his thanks to

116
D. G. Hummer and W. Kalkofen for several helpful

discussions, and to J. Castor for sending manuscripts

prior to publication.

REFERENCES

Abyankar, K. C. 1964a, Ap. J., 140, 1353.

Abyankar, K. D. 1964b, Ap. J., 140 , 1368.

Abyankar, K. D. 1965, Ap. J., 141, 1056.

Bappu, and Menzel, D. J. 1954, Ap. J . , 119 , 508.

Beals, C. S. 1931, Monthly Ho tic. 0,6 Roy. kbtnon. Soc,

91, 966.

Castor, J. I. 1970, Monthly Notice Roy. kbtKon. Soc.

(to be published)

.

Castor, J. I., and Van Blerkom, D. 1970 (to be

published)

.

Chandrasekhar , S. 1934, Monthly Notices Roy. kbtnon.

Soc, 94, 522.

Feautrier, P. 1964, Compt. R<ind. Acad. Set. VaKlh

258 , 3189.

Hummer, D. G. 1962, Monthly Notice* Roy. Khtnon.

Soc, 125 , 21.

Hummer, D. G. 1968, Monthly Hotlceh Roy. kbtKon.

Soc, 141, 479.

Hummer, D. G. , and Rybicki , G. B. 1967, in Method*

o{ Computational Vhyt>lct> , Eds. B. Alder, S. Fernbach, M. Rotenberg, Academic Press, New York,

p. 53.

Hummer, D. G. , and Rybicki, G. B. 196 8, Ap. J.,

153 , L107.

Jefferies, J. T. 1968, Spectral Line Formation,

Blaisdel, Waltham, Mass.

Kalkofen, W. 1970, proceedings this colloquium, see page 120.

Kulander, J. L. 1964, Thesis, University of Colorado

(unpublished)

.

Kulander, J. L. 1967, Ap. J., 147 , 1063.

Kulander, J. L. 1968, J. Quant^t. Spectno&c Radlat.

lnanh\<ih., 8_, 273.

Lyong, L. V. 1967, Soviet khtnon. , 11, 224.

Magnan, C. 1968, KhtKophyh . letter, 2, 213.

Mathis, J. 1968, in Resonance Line* In kbtnophyblcb , Manuscripts presented at a conference held in

Boulder, Colorado 9-11 September, 196 8, National

Center for Atmospheric Research, Boulder,

Colorado

.

117
McCrea, W. and Mitra, K. 1936, Z. khtnophyb .

,

11, 359.

Milne, E. A. 1930, I. KbtKophyt* . 1, 98.

Oxenius, J. 1965, J. Quantlt. Sptctnobc . Radlat.

TJLan&te.tL, _5, 771.

Rublev, S. V. 1961, Sov. kt>tKon., 4, 780.

Rublev, S. V. 1964, Sov. IkbtKon. , 7, 492.

Rybicki, G. B. , and Hummer, D. G. 1967, Ap. J.,

150 , 607.

Rybicki, G. B. and Hummer, D. G. 1969, Monthly

Notice,* Roy. khtKon. Soc. , 144 , 313.

Sobolev, V. V. 1947, Moving Envelope.* o& Stan*

,

Leningrad State University (in Russian) ; Transl.

By S. Gaposchkin, Harvard University Press,

Cambridge, Mass., 1960.

Sobolev, V. V. 1957, Sov. A&tKon., 1, 678.

Sobolev, V. V. 1958, in Thtotiztical AAtlopkyAicA

,

Ed. V. A. Ambartsurayan , Pergamon Press, New York,

Chapter 28.

DISCUSSION

The manuscript of Rybicki arrived very late

and consequently it was not possible to prepare

a condensed version of the lively discussion

following this paper. One aspect of the problem

of methods was discussed by C. Magnan who has

submitted the following summary of his remarks.

118
APPLICATION OF MONTE CARLO METHODS

IN TRANSFER PROBLEMS

by

C . Magnan

Institut d ' As trophy sique

Paris

I have not prepared any formal communication.

Nevertheless I would like to mention the types of

calculation that are now in progress in Paris.

I am handling transfer problems in moving atmospheres by using Monte Carlo techniques. A twolevel atom and an isothermal atmosphere are always

assumed. Two types' of geometry are considered: a spherical atmosphere with a constant velocity of

expansion, or a flat disk with constant velocities

of expansion and rotation. Non-coherent scattering

(either entirely incoherent or partially coherent)

is assumed with respect to the "local frequency"

(as seen by the absorbing atoms)

.

Concerning the "creation" of photons, two

problems have been handled: the photons are created

via absorption of a continuous spectrum produced by

an underlying photosphere (the Schuster problem) or they are created within the envelope via electronic

collision or recombination from ionized states.

The most typical results concern the Schuster

problem. P Cygni type profiles are very well re- produced with the adopted models; symmetrical

emission features on both sides of a central absorption line are found in the case of a rotating disk.

At the present time we are limited to cases where the mean number of scatterings is of the order

of twenty. The order of magnitude of this number

is given by t/V (where x is the optical depth of

the envelope and V the macroscopic velocity)

.

119
LINE FORMATION IN MOVING ATMOSPHERES

by

W. Kalkofen

Smithsonian As trophy si cal Observatory

and

Harvard College Observatory

Cambridge , Massachusetts

ABSTRACT

We discuss an integral equation method that

permits the calculation of the line source functions

and of the emergent profiles in finite and semiinfinite atmospheres with macroscopic motion normal

to the surface. Solutions are presented for a semiinfinite atmosphere with a temperature rise in the

outward direction and with a flow that decays with

increasing depth. The computed profiles have the

form of P Cygni lines.

Key words: line formation, moving atmospheres,

line-profile computation.

I . INTRODUCTION

Several authors have obtained numerical so- lutions of the equations of statistical equilibrium

and radiative transfer. Magnan (1968) has employed

the Monte Carlo method to determine the source function in a model of a planetary nebula; Hummer and

Rybicki (196 8) have solved a similar problem by means of differential equations with the Riccati

transformation; and Kulander (1968) has used

differential equations with the Eddington approximation in the solution of the equation of transfer for

atmospheres with simple velocity fields. We describe

here an integral equation method and apply it in the

calculation of the source function in a model of an extended atmosphere.

120
The various methods have certain limitations.

In the Monte Carlo method the history of many photons

is computed. Since the accuracy of the solution in- creases only slowly with the number of photons con- sidered, the desired accuracy must be balanced

against the thickness of the medium. In practice,

this method is useful only for media with moderate

optical thickness. However, the geometry of the

medium and the physical processes taken into account

may be quite complicated. Thus, incomplete redistribution of the photons over the line can be treated

relatively easily. In the Riccati method the integration steps must be comparable with the shortest

photon mean free path. Therefore, this method is

also restricted to media with relatively small

optical thickness. Kulander's technique permits the

calculation of the source function in atmospheres

with large optical thickness. But the use of the

Eddington approximation in flow problems in which

the Doppler shift in the normal direction may be

large whereas the horizontal shift is zero can lead

to large errors. The method is therefore suitable

only for exploratory calculations.

The integral equation method is formulated for

the case of a plane-parallel medium that may be

finite or semi-finite and that has. arbitrary flow

along the normal to the surface. Anistropic microturbulence can be taken into account if it can be

expressed in terms of normal and horizontal com- ponents. Complete frequency and angle redistribution of the photons is assumed so that the line

component of the source function is frequencyindependent and isotropic.

The conditions that we have imposed in the

formulation of the integral equation method will

frequently be realized approximately in stellar

atmospheres. Thus, for lines we may usually assume

that the atmosphere is plane-parallel over regions

whose horizontal extent is large compared with

the thermalization length. The gas flow will

usually be directed along the normal to the surface.

Other flow patterns could, in principle, be treated

by the same method. But differential motion within

horizontal layers could not be dealt with easily.

In general, however, the Doppler shift due to

horizontal differential velocities that might occur,

for example, in a uniform expansion should be small

over a thermalization length. One may then compute

the source function for a vanishing horizontal

velocity gradient but must take its effect into

account in the calculation of the emergent flux.

121
The requirement of complete redistribution will

usually be nearly satisfied, as Magnan has shown for

a planetary nebula. In this context we note that

the case of a semi-infinite stellar atmosphere is

less severe than that of a planetary nebula, where

the intensity outside the line becomes quite small,

whereas in a semi-infinite medium the background

radiation sets a limit on the intensity. We may

therefore assume that the line source function is

virtually frequency-independent and isotropic.

Because of these limitations of the methods,

one would solve transfer problems for finite media

with a complicated geometry using the Monte Carlo

method, with a differential equation technique such

as that of Hummer and Rybicki or that of Feautrier

(1964) , which is described in detail by Cuny (1967) , or with the integral equation method of Jones and

Skumanich (1968) . For finite or semi-infinite media

with a plane-parallel geometry, one would use either

our integral equation or Feautrier' s differential

equation method. If the transfer is characterized

by incomplete redistribution, one would choose the

differential equation; if a large number of frequency

and angle points has to be taken into account because

of high flow velocities, one would choose the integral equation method. If both complications are present, one could attempt to reformulate the equations in terms of the local frequency variable that

has been used by Magnan in the Monte Carlo method.

For typical problems arising in extended atmospheres,

both Feautrier' s technique and our integral equation

method could be used successfully, but the latter

might give somewhat better results.

2. THE BASIC EQUATIONS

We seek the solution of the simultaneous

equations of statistical equilibrium and radiative

transfer. Following Thomas (1957) , we express the

statistical equilibrium equation in the form

S = e(S - B) , (1)

where S is the frequency- and angle-independent line

component of the source function, J is the mean integrated intensity, e is the probability of true

122
absorption, and B is the Planck function that depends, in general, on the electron temperature and

on the radiation field exclusive of the line radiation. We write the equation of radiative transfer

as J(t) - S(t) = H(t,t')S(t») + c(t) , (2)

where C is the contribution of the background radiation to the line intensity J, and H is an integral

operator. In the special case of vanishing background radiation, zero flow velocity, and isotropic

Doppler broadening, H is related to the well-known

A-operator

.

In the integral equation method we combine

equations (1) and (2) into a single equation for the

line component S of the source function. The resulting equation, in matrix form, is solved for S by

means of a matrix inversion,

(e - H)

1 (e B + C) . (3)

This procedure, with perhaps minor alterations, is

standard for the integral equation method. For

depth- and frequency-dependent but angle-independent

absorption profiles, this method was developed by

Athay and Skumanich (196 7) . The new feature that

we want to discuss is the calculation of the matrix

operator H when the absorption profile of the line

depends on depth, frequency, and angle so that the

monochromatic A-operator cannot be used.

We assume that the gas in the atmosphere is

moving with the velocity q along the outward normal

of the plane-parallel atmosphere; q is an arbitrary

function of the depth. The quantity in the transfer

equation that is modified by the flow is the absorption profile. If the line is broadened by the

Doppler effect and by damping, the profile is given

by the Voigt function,

(t) a Av '^ /it A ^

71 e

J dy

(4)

D

-°° a + (v - y

where a is the damping parameter, AvD

is the Doppler

123
width due to thermal and random microturbulent

motion, and v is given by

Av -(v /c\q(x)y

r L ( z>\ Av

D

(x,y) w

The frequency displacement Av is measured in the rest

frame of the atom and Vq is the frequency at the line

center. We note that the Doppler width depends on direction when the microturbulence is anisotropic.

An important property of the absorption profile

<j> is its symmetry with respect to the frequency displacement Av and the direction cosine y

,

Because of this symmetry in $ it is convenient to

define an average intensity I:

*Av,p< T) = I [*Av, + ^Av,-^'] ' V > °' <7)

in which the intensity in the forward direction

along the ray (Av,y) is combined with the intensity

in the opposite direction along the ray (-Av,-y).

_The integral form of the transfer equation

for I is given by the equation

T

f v

'" J

where J is the total source function,

f kL

(x)<|>. ( T ) S ( T ) k

C

. (t)B(t) J (T) =

c ' (9)

^(T)*. ( T ) + kC

(T)

LA V ,\X

t r* with k and k the opacities of the line and of the

background continuum, respectively, and B the source

124
function of the background continuum, and where

t « is the specific monochromatic optical distance

A v , u along the ray (Av,y); T. is the corresponding

zw , y distance between the two boundaries if the medium

has finite optical thickness.

The integral kernel K is given by the equation

K(T,t) = \ e

-|-r-t|

_ (1Q)

The kernel has the following asymptotic behavior:

K(x,t) = 6(T,t)|l + — + Cr \e~ T )+ (^[e" (T " T)

] , (11)

I dT 2i

where 6 is the Dirac 6-function. At large distances

from the boundaries of the medium, therefore, the

kernel K becomes essentially the second-derivative

operator. The mean integrated intensity J is defined by the expression

J(t) = d(Av) dy i Av,y (T) J Av,y (T (12

In order to solve the equations numerically,

we choose a discrete set of depth points and expand

the total source function in terms of piecewise

quadratic segments. Instead of the integral operator

involving the kernel K, we obtain for equation (8)

the product of a matrix operator and the vector of

the total source function v/ . By combining equations

(8), (9), and (12) and regrouping terms, we finally

obtain the transfer equation in the form of equation

(2) . 3. SOLUTIONS

Our aim is to study the influence of macroscopic

flow on the line source function and on the emergent

monochromatic flux, and hence to investigate the

formation of "P Cygni lines." To keep the interpretation relatively simple, we have worked with a single model of a semi-infinite plane-parallel atmos125
COCO

0.15

0.12 -

0.09 -

0.06

0.03 -

0.00

-2.00 6.00

PLANCK FUNCTION AND SOURCE FUNCTION FOR

2 THE VELOCITY v(t) = Figure 1.

I + r/O.I

phere with a temperature rising in the outer layers

from 8000 °K - 10,000°K. In this medium we compute

a line source function with the frequency of the

Ha line for e = 10" 2

, a = 0, and constant thermal

Doppler width. The function B in equation (1) of

statistical equilibrium is equal to the source function B of the background continuum. The function

B(t), divided by 2h v

3 /c 2

, is plotted in Figures 1 and 3. The ratio of the opacity of the line at the

center to that of the background continuum is con- stant with kL/kc = 10 1 *. The velocities q(x) are given in units of the constant thermal Doppler width

by the equation

where

V(t) =

V =

V(0)

1 + t/T (13)

The optical depth t is measured along the local line

center

In a preliminary investigation we have deter126
TABLE 1.

ACCURACY OF THE ANGLE QUADRATURE

Angle Error (in percent)

approximation S (0) S(0.1) I (0.3)

7 0.04 0.2 0.06

6 0.1 0.5 0 . 16

5 0.2 0.9 0.3

4 0.3 1.4 0.6

3 0.5 2.3 1.1

2 1.3 4 2.5

1 1.4 8 9

mined the dependence of the computed source function,

the emergent normal intensity, and the emergent mono- chromatic flux on the angle quadrature. For this

purpose we have solved the equations with V(0) = 2 and T = 0.1. This flow velocity produces a shift of

the absorption profile by two Doppler widths at the

surface, and the velocity gradient is large only in

the outermost layers. The source function has a minimum at a depth of t = 0.1, and the minimum of

the intensity is displaced in the violet direction

by 0.3 thermal Doppler width. Table 1 gives the

accuracy in percent of the solutions for the first

to the seventh angle approximations measured against

the solution in the eighth approximation. All

solutions were obtained with 63 frequency points

in the range (-4,4) of the frequency in Doppler

units and 40 points in the depth range (0,10 6 ). An

accuracy of at least one percent in the quantities

listed requires at least five discrete angle points.

All subsequent solutions were therefore computed

with the fifth angle points. All subsequent solutions were therefore computed with the fifth angle

approximation. In the calculation of the monochromatic emergent flux, five angles did not give

sufficiently accurate results. This quantity was therefore computed in the 2 4th approximation.

127
In order to investigate the effects of flow in

a moving atmosphere, we have solved the equations

for four cases of macroscopic flow and for the

static atmosphere. For the flow problems we have

chosen

V(0) = 10 and T = 10 , 10 2

, 10 3, 10 lf .

Thus the velocity is largest at the surface and

decays essentially as t . The gradient is large

only near t = T. The computer plot of these

velocities is given in Figure 2.

We note that for e = 10" 2

, the " thermalization

length" of a Doppler-broadened line is t 0 = 100.

The line that is formed in the atmosphere with the

flow parameter T = 10 h

, therefore, sees only the

constant flow velocity corresponding to 10 Doppler

widths in the line formation region. Hence the source function should be identical with that of a static atmosphere. This is indeed the case, as can

be seen from Figure 3, where the solutions for the

stationary medium and for the atmosphere with

-2.00 0.00 2.00 4.00 6.00

log t

MACROSCOPIC VELOCITIES

v(0) 2 3 4

v(t)= -j-^p , v(0)=IO, T=I0, 10 , 10 , 10 Figure 2.

128
do

0.15 B

0.12 —

0.09

0.06

0.03 ^/ 2 3 4

-"^n in in in 0.00 1 1 1 1 1 1 1 -2.00 0.00 2.00 4.00 6.00

log t PLANCK FUNCTION AND LINE SOURCE FUNCTIONS

FOR € = I0

2 AND v(t) = j^j:,T.= 10, I0 2

, 10

3

, lO4

THE

STATIC CASE IS LABELLED BY 0. Figure 3. constant expansion velocity in the significant surface layers coincide. For T = 10 3

, there is already

a noticeable gradient in the macroscopic flow in

the region of the atmosphere in which the line

radiation reaches saturation. Because of the frequency shift of the line opacity, photons emitted

near the maximum of S can escape more easily than

they can in the static case. The source function

maximum is therefore reduced. The surface layers

below t = 100, however, still have a nearly constant

flow velocity. The surface value of S is therefore

still virtually identical with that of the static

solution. The same effects are seen in the source function for T = 100, for which the velocity change

is large in the relaxation region and small near

the surface. Thus the value of the source function

at its maximum is further reduced but its surface

value is nearly unchanged. For T = 10 , the velocity

gradient is small in the relaxation region and

large near the surface. Near the source function

maximum, the solution for S, therefore, moves back

toward the static solution. But in the surface

layers where V is varying rapidly, the line absorption coefficient is now large in a spectral range

where the background radiation can exert a stronger

129
influence on the line radiation. The surface value

of S is consequently raised above that of the static

case

If the region in which the velocity gradient

is large is pushed sufficiently close to the surface

by reducing T further, or if e is reduced significantly below 10" 2

, the source function can be forced

by the background radiation to rise near the surface.

This effect can, indeed, be seen in Figure 1, where

V(0) = 2 and T = 0.1.

The most striking result of these calculations

is the similarity in the solutions for a wide range

of flows, which indicates that the source functions

are not very sensitive to the Doppler shift. The

emergent radiation field, however, is strongly

influenced by the flow.

The emergent normal intensities are shown in

Figure 4. For the static atmosphere the intensity

is symmetric. For the flow parameter T = 10 k

, the

intensity is blue-shifted by 10 Doppler widths.

The profile is nearly symmetric, only the slight

enhancement of the red peak indicating the flow.

For smaller values of the flow parameter T, the

peaks on the low-frequency side of the line, center

are more and more shifted to the red, the amplitudes

having heights that correspond to the values of the

maximum in the source functions. For T = 10, the

intensity minimum is shifted to Av = 8.75.

130
The emergent monochromatic fluxes are plotted

in Figure 5. Since the fluxes result from the

weighted superposition of intensities for normal

emergence, for which the frequency shift amounts to

10 Doppler widths, down to those for grazing emergence, for which the line is centered on Av = 0, the

flux profiles are nonsymmetrical even for the uniform

expansion given by the case of T = 10 k

. Only the

static atmosphere produces a symmetrical flux profile. The flux curves for the flows with T = 10 2 and

T = 10 have peaks on the low-frequency side of the

line profile. The condition for the appearance of

such emission features is that the maximum of the source function fall into a region of the atmosphere

for which the flow velocity is sufficiently small.

The minima of the fluxes are blue-shifted by

Av = 8.25 thermal Doppler widths for the flow parameters T = 10* and 10% by Av = 7.75 for T = 10 2

, and by Av = 4.5 for T = 10. The two maxima occur

at Av = 0.25 for T = 10 2 and at Av = 1.75 for T = 10.

The separation of the extrema is therefore Av = 7.5

Doppler widths for T = 10 2 and Av = 6.25 for T = 10.

Thus, the blue shift of the central absorption

feature is smaller than might have been expected on

the basis of a normal velocity displacement of 10 4 Doppler widths , even when the outer layers expand

uniformly as, in effect, they do for T = 10 \- and

the separation of emission and absorption features

I.25T 10

1.00 x

ZD 0.75

0.50

0.25 -4.00 0.00 4.00 8.00 12.00 FREQUENCY

THE EMERGENT MONOCHROMATIC FLUX FOR

10 ? ^ 4

v (t) = , T = 10, 10 , 10 , 10 , AND FOR v = 0.

1+ t/T

Figure 5,
is not simply related to the flow velocity in the

line formation region. The thermal Doppler width

can be easily determined only in the line of the

static atmosphere.

4. CONCLUSIONS

The profiles that we have computed have the

general appearance of "P Cygni lines," from which

they differ mainly in the relatively low emission

intensities. The emission can be increased by means of a higher temperature rise in the outer

layers of the star. Our model atmosphere was chosen more in accordance with the model calculations

of Feautrier (1968) and of Auer and Mihalas (1969)

,

who found temperature increases of 1000° - 2000 °K

for their models of early-type atmospheres in

radiative and statistical equilibrium. Some mechanical mode of energy transport could give a much

larger temperature rise as, for example, it does in

the Sun. Such a temperature increase has been postulated for the supergiant a Cygni by Groth (1960)

,

who furthermore placed the emission into an inwardfalling outer shell at a high temperature. Such

an elaborate model atmosphere is clearly not

necessary, as our calculations show. The only re- quirements for the appearance of P Cygni lines are

a temperature inversion in the atmosphere or envelope

and flow in the outward direction.

It is a pleasure to thank Deane Peterson for

discussions of the numerical method, Charles Whitney

for useful comments on the interpretation of the

results, and Lane Emerson for his help in computing

and plotting with the aid of an electronic computer.

REFERENCES

Athay, R. G. , and Skumanich, A. 1967, Ann. d'ap.,

30, 669.

Auer, L. H. , and Mihalas, D. 1969, Ap . J. [LtttdUb]

,

156 , L151.

Cuny, Y. 1967, Ann. d'ap., 30, 143.

Feautrier, P. 1964, C. R . , 258 , 3189; 1968, Ann.

d'ap., 31, 257.

Groth, H. G. 1960, Z. Az>tKopkyt> . , 51, 231.

132
Hummer, D. G. , and Rybicki, G. B. 196 8 , Psioc. Con&.

on Resonance. Lin&t> In A^tKopky^., National Center

for Atmospheric Research.

Jones, H. P., and Skumanich, A. 1968, Pioc. Con&. on RmonoLnco. llnzh In AAtiopky* . , National Center for

Atmospheric Research.

Kuhlander, J. L. 1968, J.£.S.R.T., 8_, 273.

Magnan, C. 1968, Ap. J. [LzttzA.*] , 2, 213.

Thomas, R. N. 1957, Ap. J., 125 , 260.

DISCUSSION

UvideJihi.JLLi How many levels did you include in

your calculations?

KaZko & 2.n: We included only two levels to show

the principle of the problem.

UndnKkllt: The observations do not show the

long tails which you have calculated.

KaZko jjen: But there is also an increase in the

line core, which is in agreement with the observation. If you apply the theory to the lines in the

spectrum of e Ori the agreement is fairly good.

Und&sikZll: .The observations show only very

weak P Cyg profiles in Ha.

133
LASER ACTION IN NON-LTE ATMOSPHERES

by

Donald H. Menzel

Harvard College Observatory

and

Smithsonian Institution As trophy sioal Observatory

The radiative transfer equation is written in

microscopic form, and from some simplifications on the ratio of occupation numbers for upper and lower

level, a laser action is suggested.

Key words: radiative transfer, laser action.

I like to write the equation of radiative

transfer for line absorption in a plane-parallel

atmosphere in the following exact form:

where I is the specific intensity and a is the

atomic absorption coefficient at frequency, v. The

quantities m and r\2 are the atomic populations of

the lower and upper levels, whose respective statistical weights are oil and w 2 . In thermodynamic equilibrium, a single parameter, the absolute temperature T, governs the

Boltzmann and Planck formulas, so that

ABSTRACT

(1)

n 2 u) 2 , m -hv/kT

n ' =

a, e and I = 2hv 3 c

2 ehv/kT _ ±

' 1 (2)

134
When these equations are substituted into (1)

the right-hand side vanishes, as it must for thermodynamic equilibrium, since the intensity is isotropic

and independent of position.

In equation (1) , the second term in the parenthesis represents the stimulated emission or, as it

should more properly be called, the negative absorption. Too many astrophysicists either combine it

with the random emission, the second term on the

right-hand side, or neglect it altogether. The true

source function takes the form of the second term on

the right-hand side of (1) , which in no sense resembles a Planck function except when hv >> kT.

The quantities ni and n 2 are to be calculated

from the equations of statistical equilibrium, which

involve collisional excitation and de-excitation as well as radiational processes.

Many years ago, in problems related to gaseous

nebulae, I introduced a dimensionless parameter, b,

to indicate the degree of departure of a gas from

thermodynamic equilibrium at temperature T of the

electron gas. This parameter equalled unity for

thermodynamic equilibrium. Thus I could write

n 2 nT b i co •hv/kT (3)

with which expression (1) becomes

dl

dh cos0 = - n i

a

+ 2hv 3 D 2 hv/kT

2 Dl

(4)

As long as hv/kT >> 1, we can usually neglect

the term representing the stimulated emissions. And,

as long as b 2 /bi does not depart too far from unity,

the second term on the right-hand side" is approximately equal to the source function.

Many studies have shown that the b's exhibit

the following behavior for nebulae and, presumably,

also for stars with highly distended atmospheres.

First of all, b's for ground or metastable levels

135
tend to be high. Second, the first excited levels

directly above the ground level tend to have b's

much less than unity. Third, the b's for still

higher levels slowly tend to unity at the series

limit

For all lines except those from the ground or metastable levels, then, b 2 /bi will exceed unity.

And when the temperature is high enough, the stimulated emission may exceed the ordinary absorption

so that

dl o« -hv/kT/2hv t1 /c , ^ cos0 ^ ma £7 e ( — + I) . (5) fc

1

* )

Since this term is essentially positive, the emission

line increases in intensity with depth.

This process is truly a laser action, analogous

to those responsible for the high-level radio emission of atomic hydrogen. I believe this process can explain many of the anomalies referred to by Miss

Underhill as occurring in stars of exceptionally

high temperatures.

The process is self -limiting , however, since an increase in the intensity of the incident radiation

causes the medium to approach local thermodynamic

equilibrium. However, as long as the energy is

"diluted," some sort of laser action will occur.

DISCUSSION

Huann: It is important to know the electron

density. In the case of the higher levels you get

an equilibrium distribution because of electron

collisions. If the electron density is large

enough even the lower levels will have a Boltzmann

population. For the lower levels the spontaneous

emissions are much more important than the induced

emissions. In the case of Ha the induced emission

is very small. If the electron density is of the

order 10 10 or 10 12 the lower levels follow the

Boltzmann distribution and there is no induced

emission at all.

Me.nze£: There is a cooling effect by forbidden

136
transitions, which will also cause strong deviations

from LTE.

Unde.JLk>LZZ: In the main part of the atmospheres

of WR stars, the electron densities are of the order

of 10 11 to 10 12 and the electron temperatures are

of the order of 30000° to 50000° K. These values

are not suitable for efficient laser action, which

you have described.

RybZcksL: To my knowledge, no non-LTE solutions

in multi-level atoms have shown inverted populations

for stellar atmospheres extended or not, even when

stimulated emissions have been properly included.

Of course one should always be aware of this possibility, since, if it occurs, it can be a dominant

effect in the formation of the spectrum. It should

be noted that any laser action will build up intensities that will tend to destroy the action by re- turning the population ratios to their normal order.

This is particularly true in a strict plane-parallel

atmosphere where any negative opacity in a region between two planes would build up infinite intensities,

since infinite path lengths exist within the negative

opacity region.

137
LINE FORMATION IN MULTI -DIMENSIONAL MEDIA

by

H. P, Jones and A. Skumanich

High Altitude Observatory and University of Colorado

Boulder, Colorado

ABSTPACT

The flux divergence technique of Athay and

Skumanich (1967) is generalized for application to

media whose properties vary in more than one spatial

dimension. In this method, the flux divergence is

viewed as an integro-diff erential functional of the source function. The source function is then ex- panded in terms of basis functions along characteristic paths, and, with the help of various interpolations, the flux divergence is converted to an approximate linear algebraic operator on a discrete

spatial grid. A large but finite set of linear,

inhomogeneous , simultaneous algebraic equations

with known matrix coefficients is thus generated

and is solved by direct matrix inversion for the source function at each point of the spatial grid.

Some aspects of the accuracy, stability, and

computational convenience of the technique are discussed. Sample solutions for depth dependent,

axially symmetric variations of temperature are

shown.

Key words: radiative transfer in inhomogeneous

media, line formation in multi-dimensional media,

numerical methods in transfer, matrix methods for

integro-dif f erential operators.

I. INTRODUCTION

A casual inspection of a spectroheliogram at

almost any wavelength reveals that the radiation

field of the solar atmosphere has considerable horizontal structure. The ordinary plane-parallel

138
idealization is thus at best an average description

of the medium, and a more realistic treatment can be

provided by model atmospheres whose properties are allowed to vary in two or three dimensions.

The formation of spectrum lines in such media

has, until recently, received little attention.

Rybicki (1965) has developed Fourier transform

techniques suitable for media in which the absorption coefficient is constant with position. Wilson

(1968) has published results obtained with a trial

and error scheme making use of a three-dimensional

equivalent of the Eddington approximation. Avery

and House (1968) have adapted the Monte Carlo

technique for use in line formation problems and

have paid particular attention to spicular geometries .In the present paper, we will present a new numerical technique for dealing with line formation

in multi-dimensional media. The method is a generalization of a one-dimensional technique developed

independently by Kuhn (19 66) and by Athay and

Skumanich (1967) . In contrast with the Monte Carlo

scheme, it is "deterministic" or non-statistical.

The method may be used with variable absorption

coefficient and is, in principle, non-iterative for

linear problems.

II. BASIC EQUATIONS

We write the time independent transfer equation

as —-Kr- n-V J (x,n) = S (x) - I (x,n) . (1)

K (x) v v v

We have used the following notation: x is the position vector of a point in the region of interest; n

is a unit direction vector; K v is the absorption co- efficient at frequency v; S v is the source function;

I v is the specific intensity.

In the one-dimensional case it is customary to use a geometric position variable which increases

outward (i.e., away from the region of interest)

and an optical depth scale increasing in the opposite

sense; this results in the multiplication of the

right hand side of (1) by -1. We have not adopted

139
this convention here. As a result certain sign

differences will appear when the following analysis

is compared with the equations of Athay and

Skumanich (1967).

We define two angular moments of the specific

intensity by

I v J (x) = -r- (x,n) dft (2) v 4tt and

H

v

(x) _1_

4tt n I^(x,n) dft (3)

where the integrations are over the full domain of

the solid angle Q. If one integrates Eq. (1) over

solid angle, one obtains

V~-H (x) S (x) - J (x V v

4)

We wish to consider the equation of transfer

at frequencies centered about a spectrum line with

maximum absorption coefficient at v = v Q . We split

the absorption coefficient into the sum of the part

arising from bound-bound processes in the line

(K (v) ) and the part arising from all other processes occurring at^the same frequency (K (v) )

Thus, at any point x, we write

K

v = K

L

(v) + K

c

(v) . (5)

To a good approximation K (v) is independent of

frequency at frequencies where KL(v) is appreciable.

Defining Kc = Kc

(vQ ) and r^ = Kc /Kl(v) we have

K = K (v) (1 + r ) . (6)

V ±j v

KT (V)

Letting rQ = r , K = K ( v ) , and $ = -i we have o L o v K

q

140
K

v

(x) = K

q

(x) (* v

CxJ + r Q

(x) ) (7)

Separating the emission coefficient in similar

fashion and noting that the source function is the

ratio of emission to absorption one obtains

(J) (x)S T (x) + r S (x) V

$ v(x) + r Q(x)

where S Lv and S cv are the line and continuum source

functions. For this paper, we will take S c v

(x") = B v

(T e (£) ) = B(x); i.e., for frequencies ' around

the line, the continuum source function is the

Planck function at frequency v Q and local electron

temperature Te (x") . If one assumes "complete re- distribution" in frequency of scattered radiation

in the rest frame of the atom, the line source

function is frequency independent. For a two-level

atom the statistical equilibrium equation may be

written in the form

J + sB

1 + e

(9)

where

Jo v v dv (10)

and

UL

(1 - exp hv

Q/kT

e

y ) .

Here, $ is the profile coefficient normalized to

unity (v V | o Vv)' UL is the collisional rate

per atom from the upper to lower levels in the atom, and AUL is the Einstein A coefficient.

141
To obtain a single equation for the source function, one can use the transfer equation to eliminate

J from Eq. (9) . Operating on (1) with

dvO , v' we have that

S dv - J.

V v (12

Substituting (8) and (9) into (12) one has that

fir

'- 3

v dv = s J

V V dv

LJ AY +r

v o (l+e)ST + eB + B

r $ dv

o v +r

v o

(13

We define

6 =

r 0 dv

o v

<bY + r

v. o

and note that since

i + r

v o

1 -

i + r

v o

$ dv

v v + r

v o

= 1 - 6 . (14

Putting (14) into (13) and combining terms, one finally has that —- V'H dv = B

K V (15

142
We will use (15) as the basic equation in

developing our multi-dimensional technique. It is

incomplete, however, until we relate V»SV to S L

and B i.e., until we express H v as a functional

of S L and B. To find the needed relation we must

turn to the formal solution of the equation of

transfer. ^ At any^point x we form characteristics or paths , p(x,n), which are straight lines passing

through x in direction n. If s represents a geometric length along the path, measured from some arbitrary origin and increasing in direction n, the

vector xP(s) of any point on the path is given by

x

P (s) = x + n (s-s ) (16) o where s Q is the distance from x to the point at which

s = o. Figure 1 illustrates the path geometry.

S = 0

Figure 1. Geometry of a path p(x,n) passing through

x in direction n. Geometric length s is measured

from an arbitrary origin and increases in the same sense as n. s Q is the length from the origin to x, and xP(s) is the position vector of an arbitrary

point on the path.

143
We define an optical path length, tv , by

rS t (s) = K

v (5p (s')) ds« (17)

and adopt the following conventions: a quantity

evaluated along the path will be denoted by a script

letter. If it is regarded as a function of geometric

path length, a superscript s will be included, while

if no superscript appears, the quantity is understood

to be a function of optical path length at frequency

v

Thus

S

v (5P (s))=^(s) =*fv (t v (s)), (18)

I v(xP (s)) =^(s) =A/tv (s)), (19)

K (xP (s)) = k

S

(s) = k (t (s)). (20)

It must be remembered that our path notation implies

that a script quantity is an implicit function in x and n and explicit in path length.

With these definitions and conventions, the

equation of transfer (1) may be rewritten as

k

v(s) ds dt

v

Equation (21) has the well-known solution for ^ v in

terms of^v J v

(t v

) = e"Sv

(o) + e"

tv

f ^v (t;)e

tv

dt; . (22)

We will assume that xp (s=0) is at a boundary and

either ^v(O) = 0 or is specified by appropriate

boundary conditions. Since we shall be considering

half-spaces we shall assume that -S) v (0) is suffi144
ciently bounded so that e

t^J(0)->0 for tv

->°°. In Eq. (15) we are interested in the quantity

OO <|) ^ m dv .

We note that

AM

K v = f4tt

dJJ dt dft (23)

Note that dj v /dtv depends implicitly on direction

and location. Our final functional relation is that

r

1 *'K" v

4tt o Jfl K

v dt v\- Jo v v 7

^dv.

(24)

An alternate expression can be obtained by

transforming the integration in (24) to a volume

integration. For example, one can show (cf. among

others, Rybicki 1965) that

H

v

(x) _1_

4tt K (x')S (x')e

T (X ,X 1 ) + . V (x-x 1 x-x • I 3 d

3 x' (25)

where V is the volume of interest. Here, t v (x,x') is

the optical path length along the characteristic

between the "field point" x and the "source point"

x"

f . In this case, Eq. (24) becomes

00 (J) ^ v-a dv K v -00 \)

.00 $

4 TT J K

o v

v (x,x'

)

S

v (x')Kv (x')e l2=?H d

3 x'. x-x 1 (26)

145
In either case, we see that the weighted frequency

integral of the flux divergence is an integrodifferential operator on the total monochromatic

source function. We call this operator av

if (26)

is to be used and ap if (24) is used; more precisely,

we write

foo

<|>

-oo \) c

Note particularly that a is a linear operator, i.e.,

a(a + b S*) = aa(Sj) + ba(S^) (27)

where a and b are arbitrary constants, a is avor ap

and and are arbitrary functions of appropriate

arguments. The linearity of a holds only so long as k v is independent of S v , a condition that is approximately satisfied in some, but by no means all, situations of astrophysical interest. If the operator

is non-linear, then iterative techniques must be

employed

In general form, then (15) may be written

* Using (8) and (27) we have that

Letting o£ (S L

) = c^^j-jand ^ (B) =<V^?-^1_j,

we have

(1 + c^) S L

= (1 - c^)B = B< (30)

Note that (29) holds even if a is non-linear

since it remains linear in the factor S v

.

146
where 1 is the unit operator. We regard, e, 6, and

B' as kno

Formally,

known functions and wish to solve for S .

o S T (x) = (1 + a ) 'B' (x) . (31)

We now seek an algebraic representation of a. The basic step is to assume that we may represent

S v in terms of its values on a finite grid of space

points {x ; n = 1,2, . . . N } , i.e., we assume

N

(32)

where the functions {P v

(x) ; n = 1,2, N } are presumed known but are left unspecified for the

moment. We let

a\i) = JIt-It— p ) = « (rlr— p <*P <t ))\ n V\V r

o

v ' n/ P\*v+ r

o

v ' n v /

and

°—

p (JP(t

r v ,n v

Then (29) becomes

N

S_ (x) + I a (x)ST (x^ ,) =B'(x) (33) L

n' = l with

If the P v n

(x) are known in terms of the spectral

functions of the operator a, then the truncation

error in (32) is, in principle, known.

147
Setting B ' (xn ) = B' n , SL

(xn ) = Sn/ and

evaluating (33) at each of the grid points one obtains the N-fold set of equations

n

?=1

an' (5n)S

n' = Bn

n N. (35)

o o Letting a.nn' = a. (x ) , we have

7 6 ..+ a , S , = B' L . nn 1 nn 1 n

1 n

n

1 (36)

where 6 nn » is the Kronecker delta. Clearly, (36)

is the algebraic analog of (30). Defining Qnn » =

6 nn' + ann» one finally has

We have thus expressed aS at any point of our grid as a linear combination of S's at every other

point on the grid; i.e., we have formed a set of

N inhomogeneous , simultaneous, linear, algebraic

equations with known coefficients (provided we can evaluate agi (xn )

) which may be cast in matrix-vector

form and solved by some suitable standard matrix

inversion technique. The method is non-iterative

if a direct inversion scheme such as Gauss elimination is used.

The method is limited in practice by the size

of the matrices that can be calculated, stored, and

inverted. Even a modest grid may produce quite a large matrix; for example, for a 30 x 30 twodimensional grid, the matrix Q has order of approximately 1,000. If the matrix is full, roughly 10 6 numbers must be calculated and stored and a 1,000

x 1,000 matrix must be inverted. A high speed,

large-scale computer is thus a necessity.

In view of the scale of the computations, a desirable characteristic of the method is that it be

spatially stable, that is, the solutions should be

relatively insensitive to the "fineness" of the

discrete grid. One reason for choosing (15) as the

basic operator equation is that previous experience

in one-dimension (cf. Athay and Skumanich 1967)

indicates just this kind of stability.

= B* n

(37)

148
III. EVALUATION OF THE FLUX DIVERGENCE OPERATOR

The method outlined in the preceding section

illustrates the form of the procedure we have used

but masks considerable detail. In particular, we have not said how to evaluate a(Pv> n (x)); nor have

we specified what functions Pv,n(x) are suitable

for our purposes. We proceed to examine both of

these points in more detail.

To evaluate oc(P v ^ n ) for any known function

p v,n(x ) we maY use either (24) or (26). Equation

(26) has the more concise analytic form and may

appear to be more suitable at first sight. However,

useful closed-form expressions for ot n (3) are difficult to obtain even for simple functions P v n and standard two-dimensional geometries. They

are nearly impossible to obtain if one considers

a wide class of problems with variable absorption

coefficient (possibly given in tabular form, for

example) . We have tried approximating volume integrals of the form in (25) by certain types of

quadrature formulae and have met with some success.

However, we have not yet been able to satisfactorily

carry out the full set of operations in (26)

.

Further, the methods so far devised have proven

extremely inefficient with regard to computing time.

Some of the difficulty may be seen in the behavior

of the kernel

x (x,x' ) ,+ v (x-x

'

x-x 1 3

which has a singularity (although integrable) at

x = x

1 and whose variation differs considerably in

different parts of the range of x - x

1 . We have found equation (24) to be more suitable

for numerical purposes. The operations involved here

are fairly simple and are in a form where variable

absorption coefficient can be handled more conveniently.

The disadvantage in the use of (24) is that the

dependence of the operators on the basic spatial

coordinates x is now implicit. One must thus be

careful to preserve the order of the operations and

must be prepared to relate functions evaluated along

a path to those evaluated in the basic coordinate

system.

149
Though much of what follows is applicable to

general geometries, we now confine our attention to

a two-dimensional, axi symmetric medium. The atmosphere is bounded by a plane at z = 0 and is semiinfinite in the z-coordinate . We take z to increase

into the atmosphere. The axis of symmetry is perpendicular to the boundary, distance from the axis

is denoted by r, and 0 <. r < 00 . Directions n at

points (r,z.) in the medium are specified by two

angles, 0 and <f> , such that n*r = sin 9 cos <J> and

n • z = cos 9, where r and z are local unit vectors

in the direction of increasing coordinate.

We assume, as we have indicated before, that we can write along a path

N

38)

where the f ^ are the quadratic basic functions of

Avrett and Loeser (1963)

,

f^t) = 1

f

£ (t)

fN

(t '

(1 - _L) 2 O < t < t

£

0t.

t > t

£=2,3, .. . ,N-1

(39)

We choose the constants c„ such that

(40)

Thus

m=l

(41)

or

jdi (t ) ^

v v —

(42)

150
For a given set of tm , the F

i

m are easy to

compute (cf . Athay and Skumanich 1967) . We shall

use 'the same set of tm for each frequency. The

geometric point along the path corresponding to a given tm is therefore a function of frequency and we relate ^v

(tm) to ^^(s^) for a fixed set of sk by an interpolatory transformation due to Kalkofen (1967)

,

i.e. ,

where the transformation depends on the frequency v and the path p(x,n). We note here that any spatial

variations in k v are introduced via Vmk

.

Similarly, we transform^ ^ (s k ) to S v (xn) by an interpolatory routine such that

^v (s k

) =

I Tkn' (x,n)S^,(xnl ) . (44) n

1 Thus we have that

x,,m,k,n' v

I p..n.(^) )s M

(5ril )= I p xi n ,(5

p (tj)s (xn

,).

(45)

, v , n ' v v n ' £ , v , n v

•

The elements of V and T will be discussed in a later

section. We now proceed to evaluate ap

(P v r n

1 (t v ) )

.

According to (24) the first step is to find

t

t

V V

V t

J (t ") = e

" V (t')e V 'dt\ . (46)

V V V

0

To accomodate the semi-infinite geometry more con- veniently, we will now restrict our paths to have

directions n with 0 <_ 9 <. tt/2 and will evaluate

J v

(t v ) in both the +6 and -ft directions. We therefore define

151
J) +

(t+

) = s (t ) v

v v' vv v' (x,

0 < 9 < j (47)

where the subscripts give the arguments defining the

path. The origin of geometric and optical path

length is taken to be at the intersection of the

path and the boundary plane. The notation ty implies

optical path length measured in the above way and

increasing into the medium, while tv on the right

hand side of (47) implies optical path length increasing in the same sense as implied in the subscripts. Thus

a& (t )\ ai)-(t+

) \

V V ] _ V V

J ^ dt

v /(x,7T-e,<HTT) , + / (x,e,<f>)

' V

We will henceforth drop the + from t v , always understand that our paths are now restricted, and remember

that we must use

dt

in evaluating our operator

Now Eq. (46) yields

J 0 (48)

and

(49)

Letting Q) t (t ) =X (f ,) , we have that

36 V V 36

152
(50)

where

6>v>= i

-

0<t <t 0

2e

- (t -t 0

H = 2,3, . . .N-l (51)

tS'v = e

v + \

- 1 N

and

6l<V = 1

2e r^y. w^v ^

6n ( V = 1 + t

£ = 2,3, . . .N-l (52)

We note that G is the "path" analog of the Gfunction introduced by Athay and Skumanich (196 7)

153
We can now take the derivative of (50) by

differentiating (51) and (52) . Note that the f £ in

(38) are well suited for this procedure since both

the f& and their first derivatives are continuous

over the entire range of the argument. Thus, we have that, letting

at

Jo ,m,k,n'

(53)

-T

= e (54)

\

2,3, . . .N-l

-t. 6*<V = 1 - 6

and that

[e' (VV 1,1/, Ml- 0 £

fc v 1 4

0; t

t

< t

v

< »

«. = 2,3, .. . ,N-1 (55)

154
As indicated previously (cf . pg. 151) we have

transformed <Jt defined on an "internal" space of

path grid points t = {t^, I = 1,...,N} to an "external" space s = {s-, = 1,...N} via $ = V*f

S

. If we were to proceed in strict analogy to the one- dimensional case, we would evaluate 0 ~ (t^) at each

of the internal points {t * n = 1,...,N}. Then,

± - 1 r

. + - i letting (6 F ) = ) G 0 (t )F 0

, we would transform

n , m ^ 36 n 36m

- ± — 1 /S the vector (6 F ) VjS , which is defined on the internal grid to a vector on the external grid by some other (inverse) interpolatory scheme represented by

a matrix W. In other words,

(6-F-') ext

= W(6-F-') intV

(cf . Kalkofen 1968) . Because of the simple nature

+ of the dependence of £,^v

^ on t

v

' we nave not found

this necessary. Instead, we calculate t^(x,n) for

every v, x, and n and proceed analytically. This is

done conveniently by precalculating the following

quantities:

m it 2 ^

\

t

l t„ 2 /

u+

= o im n FT* ~

(t £~ fcn+i)

U

+

= 2 T lm e

N-i F 7

1 "<VW u = 2 y Im e nm n . 2 i=n+i t^

N-i FT

1 V

+ = 2 T nm A=n+i t n 2

155
+ = 2 I -M (l +

i

) £=n+i t

l \ Hi

W = W

+

- 2V+ nm nm nm

- - + + + + nu =U = v =v =w~ = w~ = o . N-i,m N,m N-i,m N,m N-i ,m N,m

Using the above quantities, one can show that

(G+F~

1

) = I <jt(tv )F-m= e"

tv (V F m " *Nm)

v,m £=i

-e

- ( w

u+ +tv+ nm v nm

-W

+

+ F

* nm Nm

and

v,m £=i

W + F

* n,m N,m

where t < t < t , , . Thus (> F K ^ n v — n+i v t can be

v ,m evaluated for any t by calculating three exponentials, finding n (i.e., the interval in which t

lies) , and performing a small number of multiplications and additions.

The remaining operations are integrations over

solid angle and frequency. We first consider

156
Noting that a characteristic passes through the same

r and z points for both + <J> and -<J> , we have

V-H

J n

d(j) sin9 d (57)

(x,0,<|>)

We approximate both integrals by quadrature formulae.

Although many schemes are possible, we have used

Gauss-Legendre quadrature for the 0 - integration

and Gauss-Chebyshev quadrature for the <J> -integration

(cf. Handbook o £ Ma.tkzmcLtie.CLJt Function* , AMS 5_5,

1964) . Thus

n ML 63 \

K

v

2 * p£i X=i " dt v/ r -i, (x,cos , (y if ) ,

dt 7 .

) (58)

(x, COS (~Y V ) '^X'J '

In (58) 2M is the order of the Legendre

quadrature, L is the order of the Chebyshev quadrature, Yy are the appropriate roots of Legendre

polynomials, and cos^x are roots of Chebyshev

polynomials. Noting that cos _1 (-Y u

) = 77 ~ cos~ l (y ), we have

-> -.ML

\f«H = ^- 7 J W W,

2 * y=i A=i y X

dJ> dt0

(x,cos (Y

y

) ,<$> x

)

dt

(X,C0S (Y ) , TT — 4> x

(59)

Recalling that -=-r dt

v- is even in <J> and noting that

( 2 A- 1

)

tt 2l we see tnat TT ~ (^x = + (^L+i-A* The W

A are

157
equal (W^= —) and since we sum over all X, (57) becomes i V^H =7^ J WW, K 2tt l , y X y,A dt dt Jy,

where

(£)v y , A

7 foo(t JfT

1 V . (v,x,n JT, , (x fi JS (x . ^ £ v &m mk ' ' yX kn ' yX v n'

£ ,m, k ,

(60)

The frequency integral is reduced to the following

quadrature formula

4\f (y)dy = I c f (y )+ c^f^ (61

where foo is the asymptotic value of f (y) as y > °°. The weights c^ are computed following a quadrature

procedure developed by one of us (Skumanich 1966)

in which f (y) is approximated by piecewise linear

segments with the subsequent integrals performed

analytically. We have simply chosen yx+1 to be

sufficiently large so that

x+i

f (y)dy = I cf

C =1

(62)

Here

y = ( v_v0

)/ Av

D

(63)

Thus

D

yy (64)

dv

158
where

Then

n = <J> dy . (6 5)

.00 <f) ,00

(J) X+1C

«v^*v§ ^^H-i^^H- (66)

Recalling the definitions of , (x) and , (x) we

have

a°(x)= — I I c(x")w,W

7rn(e(x)+ 5(x)) C,y,X l,m,k ^ A y

X F £m V

Eik (v ' x ' A

yX )Tkn' (x ' A

yX } 7^^~7 ^ ~

' * (xn

.) + r

Q

(x )

Note that a^, (x) is given by the same formula with

C (xn'>

+

replaced by rQ

(xr

, ) / ( <J> (xn, ) + rQ

(xn

,). We then

evaluate a

0

, and a

1

, at all points (x ) and proceed

n n n to form and solve the linear algebraic equations as outlined in §11.

159
IV. INTERPOLATORY TRANSFORMATIONS

For the transformation of functions of a single

variable we have used a subroutine, MAPPAR, written

by Kalkofen (1967) . The basic step is to suppose

that a function in a given domain is well represented

by an interpolation function. A "backward" parabola

is fitted such that for a function f(x') on the

interval x. < x

1 < x.,,

3 - D+ 1

f(x , ) J

X, -X

i )(X, -Xi^ ) f + ^-x._ i)( x'-x.

+ l)

(x

j . 1 -xj

)(x

j _ 1 -xj+1 ) j-i

(Xj . x ._ i)(Xj .Xj+i) 3

(x'-x ) (x'-x.)

+ -7 r-7 r- f . (68)

For x' in the same interval a "forward" parabola is

given by

(x '~ x

i+i) (x '" x i+2 ) (x'-x.) (x'-x. ) f(x '^

(x -x

3 x -x

1 V Tx -x ) (x -x T

f {x i + i j x

j + i Mx

j x

j + 2) 3 < x

j +1 *jMxj + 1 xj + 2

J 3 + 1

(x' Xj) (x' (69;

(x j+2 -x.) (x. +2 -x

In MAPPAR, except for the end intervals, a weighted

sum of the two is taken; the backward weight is P,

the forward Q = 1-P. Then, for x. < x! < x.

f(x!) - I V. . ,f . , ID D (70)

160
where

V. (x?-x.) (x|-xj+1

< xjLi

:- xj>«xj-l

(71)

V

1 D

(x! -x. )(x.,-x., » (x! -x., )(x.-x.

(x^-x.. , ) (x.-x. xl )

' v (x^ -x_. + x

) (x^-x^ , (72)

V

ij +1

(x. ) (x! -x .

)

(^ j+1

- x

j-i) (x + Q

(73)

(x!-x.)(x: -x.

+ 2

) (x. + 1 -x.) (x. + 1 -x. + 2)

'

and

v. .

, l] + 2 x .

, (x .

, -x . ) (x ., -x .

, (74)

Note particularly that the V's can be expressed in

matrix form and depend only on the relation of x^ to

x_. and not on the f 's. Also, the x| and x_. must

refer to the same variable so that in our problem,

where we wish to find

either s (t^) or t^(s), where

(s) , we must first find

s(t ) v dt

o K

v (x*(f

v

))

(75)

t

v

(s)

rS K (xP (s'))ds' . J n (76)

A similar but somewhat simpler formula is used

for finding T, the transformation between*/ S

(s) and

S(r,z). We assume that it is approximately valid to

write

161
^

S

(s) - S(rP (s),z P ts)) = ^-^^^ s

(rp (s)-r.

+ i

) ( Z p (s ) -z .

)

+

ir i-r

i + i^ z j+ i- z j)

S i,j +

(rF (s)-r, ) (z p (s)-z.^

2±L

(r i + r

ri^ z j- z j+1 ) " b i+i,j

(rP (s)-r. ) (zP (s)-z.)

+ -7- J_ ^

(77)

whenever r. < rp (s)< r. +i < ^ and z <zp (s) <z . + f <z Letting n = i + I(j-l) we have 3

S(xP (V> =

^

TknS n (78

where the non-zero elements of T are given from (77).

In the cylindrical geometry which we have chosen

rP (s) =

[r 2 +(z tan 0-s sinG) 2 -2r(z tan 9-s sin0) cos*] 17 2 (79)

and

zP (s) = s cosG (80)

where r and z are coordinates through which the path Plr,z,8,<|>) passes. To change geometries, one must carefully choose the s = 0 point with respect to the boundary configuration and appropriately alter the

xP (s) functions.

162
Boundary conditions on the present geometry are that S L

(r , z) ^B^ (z) as z -> °°, where B^fz) = Lim B(r,z)

and is independent of r. Also S L

(r , z ) -*S L

(°° , z ) as

r -> 00

, i.e., for large r the source function reverts

to the one-dimensional case. These conditions are used in the mappings by assuming that S is linear

Li (as is B) in t for z > z and S T (r T ,z) for r > r T

. C — J J_i J. — -L Thus we must be sure to make r^ amd Zj sufficiently

large

.

V. RESULTS AND CONCLUSIONS

Some sample solutions obtained with the above

procedure are shown in Figures 2-7. In each case,

all parameters except temperature are constant with

Model I B= 1+4.5 Tc +IO exp(-K)

3 Tc )[mo exp(-i0yOc )] € = I0

2 a=io

3

r0 =io

4

3-1

Figure 2. Planck function surface for Model I. Log B(p0

,T 0 ) is plotted vs. log p Q and log t q . t q

is the total optical depth at line center and p Q is

the total line center radial optical distance from

the axis.

163
Model I Source Function

to

CPo

Figure 3. Source function surface, S (p 0 /T 0 ) for

Model I.

_ 2 position. We have arbitrarily chosen e = 10

a = 10~ 3, and rQ = 10" 4 while the Planck functions

has form

CT -(dp J

m

B(p

c

,t c

) = 1 + Ae (1 + Be ) + 3t

where x c is the continuous vertical optical depth

and p c is the continuous radial optical depth

measured from the axis. The absorption coefficient

at line center is assumed to be 1.

We have in mind a temperature structure designed

to mimic a hot "chromospheric" column (pq<l/d). imbedded in a cooler ambient "chromosphere"" (p c >l/d)

and overlying a homogeneous , plane-parallel "photosphere" t c >>l/c with linear vertical gradient in B. The model is merely for numerical testing, however,

and we will not attempt to draw from our results

any direct conclusions about lines actually observed

in the solar atmosphere. The models are nevertheless

not unlike possible choices one might make in

attempting to treat certain solar lines.

164
Model 1

B=| + 4.5 T +10 expHO3 T )[\ + \0 expHOyO ) 50 n c c 6 = I0

2 a = IC? rQ

= l6

4

Figure 4. Planck function for Model II

Model H

Source Function

CO

cnO

2

Log

To Figure 5. Source function for Model II

165
The figures show B and Sl surfaces with the

vertical axis being log B (or log Sl) and the

horizontal axis being the logarithm of the respective

total optical coordinate at line center. In all cases A = B = 10, and c = 10 3 (corresponding to a "chromospheric" optical thickness of about 10 at line

center) . The parameter d controls the radial thickness of the hot inhomogeneity while m and d control

the "edge scale of the radial variation of B. In the model I solution, the radial thickness

is 10 3 at line center and m = 1 so that the edge

scale is also of order 10 3

. All horizontal scales

are thus larger than a thermalization length (^10 2 ).

It is not surprising that the two-dimensional solution for any value of p is very nearly that which

would be obtained from a strictly plane-parallel

atmosphere with the vertical behavior of the Planck

function the same as at the given radius in the

two-dimensional case. Model II differs from Model I in that m = 50, or the edge scale is ^20. Detailed examination of

the solution shows that the source function reaches

its asymptotic radial behavior at a distance of

roughly a thermalization length on either side of

the region of rapid radial variation of the Planck

function. (This is not evident in Figure 5 due to

the large scale of the p-axis compared to a thermalization length) . Even though the edge scale for B

is less than a thermalization length, the radial

dependence of Sl is suprisingly similar to that of

B. It is possible that the solution is incorrect.

However, both the radial thickness and the edge scale

are larger than the vertical thickness of the

"chromosphere." Since both scales are coupled in

the term H , a possible explanation of the result

is that the vertical scale still dominates. In terms

of photon diffusion, a photon originating at a given

radius can escape from the surface of the atmosphere

or diffuse to the homogeneous photosphere more readily than it can diffuse to a different radius

with noticeably different temperature.

In Model III, m = 1 and the radial thickness is

1. The radial scale should dominate, then, at least

at intermediate depths. The solution, however, is

unstable in the sense that oscillations appear which

have no apparent physical cause. One would also

expect the source function near the axis to be less

for a radially thin region than for a radially thick

one. However, the sharp peak in Sl in Model III at

To ^10 near the axis is about twice the value of Sl

for Model I in the corresponding region.

166
Model III

Figure 6. Planck function for Model III.

The source function behavior can be made

smoother with some adjustment of grids, but the

peak remains. Apparently, then, no physically

realistic solution has been obtained for the third

model

.

Some simple tests of the method have been made.

First, we must obtain the correct solution in the

plane-parallel case. We have compared our technique

with independent plane-parallel calculations for

-CT

B = 1 + Ae C + ex c with c = 10 3

, 3 = 4.5, the same e, a, and r Q as before, and A = 10, 100, and

1000. The two-dimensional code with a 10 * 10 p * x grid agrees to better than 10 percent in all cases with a typical error of less than 4 percent.

For a thick inhomogeneity , the solutions near

the axis and at very large radii must also reduce to

the corresponding plane-parallel limit. Again, our solutions satisfy this requirement to about the

accuracy indicated above.

With the exception that points must be nested

167
Model IE Source Function

2i

..Figure 7. Source function for Model III.

so that a reasonable number lie in regions of

Planck function variation, the nesting of the spatial

grid has not proven critical. Different types and

orders of angular quadratures have been tried and

again, do not have much effect on the solution to

within the sort of accuracy indicated above.

The selections of the t& (the path grid) does seem to be important in determining the solution.

In every case that has been so far attempted,

oscillatory solutions appear at intermediate depths

(t 0 > 2) whenever the radial behavior of the Planck

function is sharp enough to dominate; these are just

the solutions which are sensitive to choice of path

grid. Numerical experiments to determine more

exactly how different choices of path origin and path

grid affect the numerical stability of the technique

are in progress.

The technique is quite convenient for media with

an absorption coefficient which is constant or varies

168
only with depth. Central processing time for such

a problem with a 10 x 10 grid is about 45 seconds

on the CDC 6600 and increases linearly with the total

number of space points (i.e., to about 90 seconds

for a 14 x 14 grid) . Fully variable opacity will of course cause an increase in computing time.

In spite of the stability problems encountered

to date, we feel the method holds great promise. In

principle, it is able to deal with quite arbitrary

spatial behavior and boundary configurations and is

particularly well-suited for media with very large

optical extent.

The authors wish to thank Professor R. D.

Richtmyer for valuable discussion in the early

stages of this work.

REFERENCES

Athay, R. G. , and Skumanich, A. 1967, Ann. d'Ap.,

30, 669.

Avery, L. W. , and House, L. L. 1968, Ap. J.,

in press.

Avrett, E. H. , and Loeser, R. 1963, J. Quant. Spdct.

and Rad. TfianAi., 3, 201.

Davis, P. J., and Polosnky, I. 1964, "Numerical

Interpolation, Differentiation, and Integration"

in Handbook o & Mathematical Function* , ed. M.

Abramowitz and I. Stegun (Washington, D. C:

U. S. Government Printing Office) p. 875.

Kalkofen, W. 1967, private communication.

Kalkofen, W. 1968, Bould&i Conscience on Resonance

Line* In Ah tKo phytic* , in press.

Kuhn, W. R. 1966, Thesis, University of Colorado.

Rybicki, G. B. 1965, Thesis, Harvard University.

Skumanich, A. 1966, Ai>tn. J. 11, 871.

Wilson, P. R. 1968, Ap. J. 15JL, 1029 .

DISCUSSION

Undcxhlll: Did you use depth-dependent functions in your calculations?

Skumanich: We considered only horizontal

169
changes in the examples shown. But we have also

calculated other models, as indicated in the text.

Uvi&ZKkLLL' Can you include the granulation?

SkuLmOLYiiah : We have thought it best to calculate

simple nonstatistical structures initially. The

granulation problem, if modeled by periodic structures, can and will be included ultimately.

170
THE CONTINUOUS SPECTRUM OF HYDROGEN

IN A LOW-DENSITY ENVELOPE

by

1 H. Gerola

Universidad de Buenos Aires, Departamento de Fisica,

Faoultad de Cienoia s Naturales y Exaotas 3 Buenos Aires, Argentina

and

N. Panagia

Laboratorio di Astrofisica, Frasoati (Roma), Italy

ABSTRACT

A progress report on theoretical work on the

formation of a continuum in low-density stellar en- velopes. Preliminary results are given on a comparison between the recombination case and the case for

a semi-empirical approach including the effect of

collisions from the ground state.

Key words: recombination case, collisional excitation, low-density envelope.

This note is a progress report on the theoretical work concerning the problem of formation of the

continuum in low density stellar envelopes excited

essentially by collisions. By low density envelope

is meant an envelope in which emission lines are formed

.

When collisions are important for ionization

and excitation essentially two facts must be considered, which may give the following significant differences with respect to the normal radiative recom- bination theory:

(1) the metastable level of hydrogen, 2s, may be

171
populated directly from the ground level, which is b

far the most populated level in typical nebular con- ditions. This is the difference with respect to the

solely radiative excitation situation, when the 2s

level may be reached only from the upper levels;

(2) for the same values of electron temperature

and density, the product of the electron density and

neutral hydrogen concentration is much higher in the

collisional case than in the radiative case. These facts lead one naturally to consider the

effects of two emission processes:

(i) the two-photon continuous emission in the

decay of the 2s level;

(ii) the continuous emission in the reaction for

the formation of the negative hydrogen ion.

The effect of two-photon emission in the case

of pure collisional excitation was studied by the

authors in a previous paper (Gerola and Panagia,

196 8) , with the usual approximation of cases A and B

due to Baker and Menzel (1938) . If the degree of ionization is considered, it

is found that, for the same values of electron density and electron temperature, the ionization in the

collisional case is smaller than in the radiative

case. This fact opens the possibility that another

process may lead to an important contribution to the

continuous spectrum only in the collisional excitation case: this process is the continuous emission

in the reaction for the formation of the negative

hydrogen ion. The rate of this reaction is proportional to the product of electron density and neutral

hydrogen density and this product is larger in envelopes when collisions are responsible for ionization.

It is to be expected that the efficiency of this

process will be greater at temperatures lower than

10 ho K.Greenstein and Page (1951) found that H emissivity is unimportant for planetary nebulae; it

should be remembered that typical planetary nebulae

seem to be definitely radiatively excited.

It has been verified that in low density con- ditions an abundance of H~ ions high enough to give

a finite opacity for the continuum of the Balmer and

higher series and to modify the neutrality condition

is never reached. What is found instead is that the

H~ concentration is of the order of 10 _7 Ne and can therefore be neglected.

So the continuum is simply the sum of recombination plus two-photon plus H~ free-bound emission.

To see the problem, let us write the statisti172
cal equilibrium equation for the levels with n >^ 3 in the form:

oo ,00 ^ n1

Y F „ + F, dV 3- + F = I F

where £ F

n" n

""" S t^ie ra<3iative cascade rate

n" = n+i

from upper levels to level n;

F, dv is the recombination rate from

v kn

n the continuum;

3*a is the collisional rate of excitation

in

from ground level for the pure collisional

case ;

F, is the radiative rate of excitation in

from ground level,

n-i

and V F , is the radiative rate of decay

n'= 1 from level n. Now we shall discuss the factor, which was introduced in 1953 by Chamberlain in his first study of

the collisional Balmer decrement, where it was pointed out that all intermediate cases between = 0 and

% = 1 are physically possible. This factor assumes the value 0 in the radiative case and 1 in the

collisional case and represents the amount of colli173
sional ionization from ground level relative to the

total amount of ionization.

In reality a combination of the two excitation

mechanisms is possible and the effective value of 'li depends not only on the present geometry and the

ultraviolet radiation field, but also on the past

history of the system nebula plus the exciting star.

So, for example, the envelope of a star that has re- cently flared actually has an energy input due to

the present radiation of the star and to the energy

i r

J L 1/a 3 2 1 Figure 1 . The dashed curve represents the recombina

tion spectrum; for the other continua (solid lines)

the number attached to each curve refers to the

values of which are listed in Table 1. All the

continuous spectra were computed for Ne = 10 4 cm" 3 and R = 10 1

cm.

174
TABLE 1.

VALUES OF H

Number of

the curve

1

2

3

45

6

7

8

9 10

5000

(Fig.l)

1.00 x io°

6.67 x 1CT 1 1.22 x lo _1

1.22 x 10" 2 1.22 x icr 3 1.22 x ICT 4 1.22 x icr 5 1.22 x 10" 6 1.22 x 10~ 7 1.22 x 10" 1

T

e

(°K)

10000

(Fig. 2)

1.00 x io°

6.21 x icr 1 2.01 x 10" 1 2.01 x 10 -2

2.01 x 10" 3 2.01 x 10~ 6 2.01 x 10 _1

0

20000

(Fig. 3)

1.00 x io°

5.70 x 10" 1 1.26 x 10" 1 1.26 x 10" 2 1.26 x 10" 6

injected into the envelope by the recent flare of

the star. Only a detailed model of the evolution

of this system would permit the determination of 'H* from other physical parameters. We have chosen an empirical approach, being interested here on the

actual appearance of the continuum, that is in the

spectroscopic problem; so % was let free to run as an independent parameter, and the computed spectra were compared with observations.

A detailed description of the calculations will

be found in a paper that will be published soon. Figures 1, 2 and 3 represent some computed con- tinua for different values of electron temperature.

Dashed curves correspond to the recombination spectrum while the other curves represent the total con- tinuum for the values of 9^ given in Table 1. It is to be noted that even with a small value

of the behaviour of the continuum is very different

from the pure recombination spectrum. This is so

175
+2

+g>+1

0

-1

-2

1/a 3 2

*1

Figure 2. As in Figure 1.

because the ground level is so highly populated that

a small collisional contribution is sufficient to

affect profoundly the 2s level population, with all

the consequences for the continuous spectrum.

Clearly for very small values of '$0 , that is, for

those values, corresponding to the radiative case, our results are very similar to those obtained by

Spitzer and Greenstein (1951) and by Seaton (1960)

Regarding the comparison with observations,

Pagel (1969) and Viotti (1969) found that the con- tinuous emission spectrum of the peculiar object n Car and the intensity of the Ha-line can be ex- plained in terms of a collisional case spectrum.

176
Figure 3. As in Figure 1 Another object with similar features is MHa

328-116. Up to 1964 its appearance was that of a normal M star with some emission features superimposed on the M spectrum; in 196 5 this star abruptly

rose in luminosity, changing also the spectral

characteristics

.

0' Dell's measurements of the continuum (1967)

and those independently obtained by Caputo et al.

(1969) can be fitted very well with a continuous

spectrum corresponding to a mixed case of excitation (30 - . 6) . What is to be noted is that these spectra

cannot be explained by means of any other emission

mechanism

.

!

177
Studies of the Balmer decrement, taking into

account the reabsorption from 2s and 2p levels, in

the cases of a mixed excitation, are being considered.

REFERENCES

Baker, J. G. , and Menzel, D. H. 1938, Ap. J. 8J3, 52

Caputo, F. , Panagia, N .

, and Gerola, H. 1969, to be

published

Chamberlain, J. W. 1953, Ap. J. 117 , 387.

Gerola, H. , and Panagia, N. 1968"^ ~Kp . Space, Sex.

2, 285.

Greenstein, J. L. , and Page, T. 1951, Ap. J. 114 , 106.

O'Dell, C. R. 1967, Ap. J. 149, 373.

Pagel, B. E. J. 1969, Natuiz 22JL, 325.

Seaton, M. J.' 1960, Rep. Piog. Pky6 . 2_3, 313.

Sptizer, L., Jr., and Greenstein, J. L . 1951, Ap. J 114, 407.

DISCUSSION

Su)Zng& : Collision terms were already introduced in the statistical equilibrium equation by

Menzel

Gzsiola: The notation I used is due to

Chamberlain

Ed<Ltoh.<Lal KzmoL>i\i'. In the paper by Gerola and

Panagia the definition of $ ^ n is not explained

in detail although the reason for introducing this

term can be seen. Confusion exists about what is

meant by the factor chC , which was not introduced

in precisely this manner by Chamberlain (1953 Ap. J. 117 , 387) and about what the expression is, for

"the collisional rate of excitation from the ground

level for the pure collisional case." In the

examples presented the parameter 3c is used only as an arbitrary multiplying factor. To understand

fully the meaning of the results one must know the

precise definition of ff*n> It is presumed that this

definition will be given in the unpublished work by

Gerola and Panagia to which reference is made.

178
BANDWIDTH REQUIREMENTS IN SPECTRAL LINE

TRANSFER CALCULATIONS

by

R. Grant Athay

High Altitude Observatory

National Center for Atmospheric Research

Boulder, Colorado

ABSTRACT

Accurate evaluation of a line source function,

S, requires that the frequency bandwidth be sufficiently large to include properly transfer effects

in the line wings. The bandwidth required to achieve

a given level of accuracy in the evaluation of S can

be specified, in units of the Doppler width, in terms

of three parameters: the ratio of continuum to line

opacity, rQ / the probability for collisional deexcitation, e, and the Voigt wing parameter a. Bandwidths required to give S to an accuracy of 2 percent are given for values of e and rQ from 10" 2 to 10" 8 and for values of a from 10" 2 to 10" 5

.

Key words: line source function, bandwidth requirements .

I . INTRODUCTION

In computing a line source function, S, by

numerical techniques, it is generally necessary to

discretize the frequency variable and to limit the

frequency range to some finite band centered on the

line. The accuracy with which S can be evaluated

depends upon both the nesting of the frequency points

and the total bandwidth considered.

The bandwidth required to achieve a given level

of accuracy in S depends upon the extent of the line

wings and, in addition, upon departures from local

thermodynamic equilibrium, LTE . In LTE one frequency

179
can be treated independently of any other frequency

and the required bandwidth reduces to zero. In nonLTE S depends upon radiation transfer effects over a band of frequencies that, for some lines, may be very

broad. Generally, if a line has negligible wings

the bandwidth can safely be restricted to the Doppler

core and the bandwidth is sufficiently narrow that it

presents no particular difficulties. When, however,

the line wings are strongly developed the problem of

properly treating the entire bandwidth of the line

with a set of discrete frequency points may lead to

serious difficulties. It is essential to know,

therefore, how far into the line wings the bandwidth

must extend in order to give an accurate evaluation

of S.

The strength and extent of line wings are determined by two parameters : the Voigt parameter a and the ratio of continuum to line opacity, rQ =

t 0 . If we define the shape factor for the line

absorption coefficient to be (j) v , where y is a dimensionless frequency measured in units of the Doppler

width, i.e., y = Av/AvD

, we may approximate cj> v

for a Voigt profile by

= e y < 1

/TTy : y > 1 (1)

An element of opacity at frequency y is then given by

dx = <J> dT + dT

y y o c

For constant rQ , Ty(line) = t c for a value of y

such that cj)y = rQ and Ty(line) = 10~n t c for cf>y = 10~n rQ . One may reasonably argue that the effects

of the wings are adequately included if Ty(line)

10

y of

Equation (1) gives for this case a limiting

Yl = 7.5 (a/r 1/ 2 (2)

180
In extreme cases such as the H and K lines of Ca II or Lyman-a in the solar spectrum (a/r Q ) 1 /2 % 10 4

. Hence, we find y 1 10 5

. To properly treat such large values of y is both

difficult and time consuming. Furthermore, it is

totally unnecessary in cases where the wings are formed in LTE.

The purpose of the calculations presented here

is to evaluate the bandwidth requirements for a range

of the relevant parameters. We treat the simple case

of a two-level atom, which requires only one parameter in addition to a and rQ

. The added parameter

is the ratio of collisional to radiative de-excitation rates, which we define as e = C 21 /A 21 where

the subscripts designate the upper and lower levels

respectively. For a given set of e , a and rQ , we evaluate the minimum value of y for which S is

accurate to 2 percent at each depth. The parameters

e, a and rQ together with the temperature and Doppler

width are held constant with depth.

II. CALCULATIONS AND RESULTS

The line source function is computed using the

flux divergence method of Athay and Skumanich (1967) . In their formulation for a two-level atom with non- coherent scattering, the frequency independent source function is given by

OM r <p ah

S = B + — [ T-5— 3"* ^ <

< 2 ' e+S d) +r dx 11 J \r o n

where B is the Planck function, H is the monochromatic net flux,

N = (2 dy) 1 * tt 1/2 (3)

and

2tt'

> +r

y o dy (4

181
We note from the form of equation (3) that two

general conclusions can be stated about the required

bandwidth as follows:

(1) If 6 > e, Yj must be sufficiently large to

give an accurate value of

y, + r dH

dr dy

This quantity would be independent of y l if were constant with y, which is true

In

dHy/dTQ were constant witn y

in LTE but not in the general case. non-LTE the quantity dH /dx0 is relatively

large near line center and decreases to a relatively small, constant value at some point in the line wings. . At the surface t = 0, S/B <* 6 l/2

. Thus, if we considered that only 6 depended upon y x we would require 4 percent

accuracy in 6 to attain 2 percent accuracy

in S.

Also, we note that in the case,

6 >> e, the required bandwidth is independent of e.

(2) If e >> 6, the bandwidth will depend upon

e. For large e, the non-LTE effects will

be limited to the Doppler core and y = 3 will suffice. For small e, the required

bandwidth will extend into the wings, but

not so far as in case 1 for a value of 6 equal to the e of case 2.

It is clear from case 1 that an extreme limit

on y 1 for 2 percent accuracy in S is given by the

condition that 6 be accurate to 4 percent. Thus, if we write

5 = 2tt" i/2

1 +

/rrr

dy

O 2 ¥

= 2tt " 3A (a r o

) lA tan" 1 Yj (/FrQ

/a) l/

(5)

182
(a rQ

) 1/2

Yi = 00

the 4 percent criterion for 6 gives

Yl = 11.5 (a/r

Q

) 1/2(6)

which is about a factor 1.5 greater than the value

given by equation (2) . The limit on y 1 set by equation (2) corresponds to approximately 6 percent

accuracy in 6, or 3 percent accuracy in S if 6 >> e. To evaluate y 2 more explicitly for a given com- bination of e, r Q and a, we solve equation (3) for

a set of values of Yl . We increase y lf step-wise,

until an increase in y x by a factor of ten changes

S by less than 2 percent at each optical depth

point. We assume that S has converged to a correct

solution when this condition is met.

The values of y : tested are 3, 4 10 k , 3 x 10 k , and 10k+i for k > 1. The spacings of the y points

for y < 10 are as follows:

0 <_ y £ 3 3 < y < 4 4 < y _< 10

For y _> 10^, k >^ 1, four points in y are used per

decade. These are spaced at 1.5 x 10 k , 3 x 10k,

6.5 x iok and 10k+1 . As an example, if y = 10 3 the number of points used is:

0.25 0.5 3

0 < y < 3 3 < y < 4 4 < y < 10

13 2 2

10 < y < 100 100 < y < 1000

4 4

183
After yi is carried far enough to show convergence a minimum value of y 2 is determined that gives

S accurate to 2 percent. These minimum values are obtained by interpolation between the tested values

of yj, They should be accurate to at least a factor

of two.

It is possible, of course, that our convergence

criterion is not strict enough. When both e and rQ are small and when a is large the convergence is very

slow. For example, with e = rQ = 10" 8 and a = 10" 2 the 2 percent criterion gives y 1 2 x 10 4 whereas a

10 percent criterion gives yi ^ 500. We note, however, that y 1 = 2 x io ** is about a factor of two

greater than the value given by equation (6) . Hence,

we conclude that our convergence test is valid in

this case, which is the case of slowest convergence

treated here. In all other cases, y 1 is less than

or equal to the value given by equation (2) and the

convergence is reasonably fast.

The greatest likelihood of error in determining

y i by the method described is in underestimating y x

. Thus, it is more likely that the value of y x given

for a particular set of values of e, ro and a is too

low, rather than too high.

The convergence of S to its correct solution as

y increases occurs differently in different parts

of the atmosphere and for different values of the

various parameters. The most notable effects are related to the wing parameter a. For t q

<< a

-1

, the

convergence is from below, i.e., if Yi is too small

S is too small. For x 0

>> a" 1

, however, the con- vergence is from above. Also, if (e + rQ ) < a < 10

(e + rQ

) the convergence of S is much more rapid for

t 0

>> a" 1

. These two effects are illustrated in

Figure 1.

Values of y 1 for the 2 percent criterion are shown in Figure 2 for different combinations of

values of r Q and e between 10" 2 and 10" 8 and for

values of a between 10" 2 and 10" 5

. Computations were

made each two decades in e and rQ (10~ 2

, 10" k

, 10" 6 and 10" 8

) and each decade in a (10~ 2

, 10~ 3

, 10" 1* and

10~ 5 ). The curves shown, therefore, are drawn

through four points in each case. We emphasize again

that the absolute errors in y 1 may be as large as a factor of two. On the other hand, the curves drawn

through the points show a large measure of continuity

from one curve to another. Thus, the errors do not

appear to be so large as to obscure the nature of the curves

,

There are threshold effects around Yi — 3 tor

a < 10" 3 and y x = 15 for a = 10" 2 that strongly

184
LogT0 Figure 1. Illustration of approach to proper so- lution as y 2 is increased for two cases a = 10" 3 and

a = 10- 2 influence the shapes of the curves for small values

°f Y i • However, certain systematic effects are evident in the curves whenever y x is large or when

either e or rQ is small.

In the plot of log y 1 versus log 6 in Figure 2,

it is evident that for 6 >> e y 2 « 6" 1

. Within the

range of 6 and a where this is evident, viz.,

^ £ 10" 3 and a _> 10" 3 line wings are well developed.

Equation (5) shows that, for this case, 6 « tq

1 ^

.

Thus, we find a region where y 1 « rQ

1/2 as predicted

by equations (2) and (6) . The plots of log y x versus

log rQ exhibit the rQ

" 1/2 dependence for r Q = 10" 6 and 10" 8

. Also, when 6 << e the results in Figure 2 show y 1 to be independent of 6. The same effect is

again evident in the plot of log y 1 versus log r Q for e = lo- 2

.

185
-

« ° / y —

(-6-2) o

-> f

(-6-3) / —

—

(-4-2) ~

(-4-3) //A

r I

| t-2-3) | (-2-2) |

CM O

(m m t

0) +J0 D z

0)M

1 •H

4—1C

•H c

n5 V

U) V

1 (J u H0

'4-1 to

1 C

,Q

> 601 6 601 O (D U U

M-2-31 1

&0~1

O -H a

-< V

>l v o

M

0) o

c

<D w

-P G U

CJ 0)

Q) M-l Q <DC

. O

-P

Sh ^

s 3

fd

186
Similarly, in the plot of log y 1 versus log a we find that when a < 10" 3 and yi is well away from

its threshold values y x « a

1 /2 as expected. There

is a tendency in all four curves of log y 1 versus

log a to have more nearly y x « a for a > 10~ 3

. To

preserve the a

1 /2 dependence it would be necessary

to reduce all of the y 1 values for a = 10~ 2 by about

a factor of two. While this is permissable for individual points it seems unlikely that all of the

values of y l for a = 10" 2 would be overestimated by

the same amount.

On the plots of log y x versus log e in Figure

2 the plus signs indicate the points e = 6 and the

open circles indicate the points e = rQ . It is clear

that the dependence of y 1 on e weakens at about the

point e = 6 and that Vj becomes essentially independent of e for e < 10 6. For e > 6 and y 1 well

away from its threshold values (i.e., on the curves

rQ = 10~ 8

) it appears that y 1 « e" 1

. To an accuracy of about a factor of two and for

y 1 > 10, the results in Figure 2 may be approximated

by:

a l/2

Yi % ^~E~ ' e » 6

and

Yl % 10(a/ro ) 1/2 ^ 13 | , -6 >> e .

The first of these limits on y x corresponds to

T Y ^

£l

T o' or to TY

= 1 at T o

= e ~ 2

' tne thermalization length for tne pure dispersion case.

It should be remembered that the above values

of y x are valid for the 2 percent criterion on the

accuracy of S. When line wings are well developed

and when 6 >> e the required values of y x for a given accuracy in 6 can be estimated from equation

(5). For 20 percent accuracy in 5, equation (5)

gives values of y 1 a factor of five below those re- quired for 4 percent accuracy and for 2 percent

accuracy in 6 a factor of two increase in y 1 is re- quired. However, accuracy in 6 is not the only

criterion for accuracy in S.

187
Estimates of y 1 for 10 percent accuracy in S were made along with the estimates for 2 percent

accuracy. A few checks were made on y 1 for 1 percent accuracy in S. In almost all cases where

6 >> e the changes in y x indicated in the preceding

paragraph give an accuracy in S equal to twice the

accuracy in 6, i.e., 20 percent accuracy in 6 corre- sponds to 10 percent accuracy in S . In a few cases much larger changes in y 1 are indicated. For ex- ample, with a = 10" 2 and e = rQ

= 10~ 8 the estimated values of y x are 500, 2 x 10 4 and > 10 5 for

accuracies in S of 10, 2 and 1 percent, respectively.

In the case of e >> 6, 10 percent accuracy in

S requires values of y x about a factor of five below those given in Figure 2 and 1 percent accuracy

in S requires values of y about a factor of two

larger than those given in Figure 2.

A few additional calculations were made in which

both B and AvD varied with depth. The depth variation of B mimicked the solar atmosphere and AvD was equated to the thermal Doppler width. This did not

produce any substantial change in y! . However, rapid

depth variations in any one of the parameters influencing S, viz., B, AvD , e, r^ and a may very well

lead to significant changes in y 2

. It is perhaps

wise in using the results in Figure 2 and equation

(7) to use the minimum values of e and rQ and maximum

values of a for a particular problem.

Although calculations have not been made for

multilevel atoms, I see no particular reason why the

inclusion of added levels should increase the values

of y 1

. However, if the added levels increase the

effective value of e the required values of y : may be

substantially reduced. It would seem entirely safe,

therefore, to use the results found here for multilevel problems.

REFERENCE

Athay, R. G. , and Skumanich, A. 1967, Ann. d'Ap.,

30, 669.

DISCUSSION

llndzftki.lt'' Have you investigated what happens

if you encounter a neighbouring line in your necessary bandwidth y 1 (100 Doppler-widths ) for the frequency integration?

Athay: No, I don't know what happens to yi in

that case.

188
COMPARISON OF DISCRETE SPACE AND DIFFERENTIAL

EQUATION METHODS IN NON-LTE LINE TRANSFER PROBLEMS

by

I. P. Grant and G. E. Hunt

Science Research Council, Atlas Computer Laboratory

Chilton, Didcot, Berkshire, England

ABSTRACT

Numerical methods are essential to the treatment

of line formation in inhomogeneous non-LTE atmospheres. The new methods due to Hummer and Rybicki

and to Feautrier now make it possible to make such

calculations, although these are often quite timeconsuming.

We shall describe an alternative approach using

discrete space techniques depending on concpets of

invariance. The solution algorithm is closely re- lated to the method of Hummer and Rybicki, whose

equations are obtained as a limiting case. The

stability and errors of our algorithm are susceptible

to mathematical analysis, and make it possible to

identify the critical parameters in the calculation

with precision. The results for a two-level problem

will be compared with those from an implementation

of the Rybicki -Hummer equations and a comparison will

be made of the performance of the two procedures in

respect of speed of computation and storage requirements .

Key words: line formation, inhomogeneous non-LTE

atmospheres, numerical methods.

1. INTRODUCTION

In this paper, we shall be concerned with com- putational problems arising in the calculation of the

line spectrum of a non-LTE model stellar atmosphere.

This field has recently been treated in book form by

189
Jefferies, 1 and Hummer and Rybicki 2 have reviewed

the state of the art of computation as it was at

October 1966. At the time that these articles were

written, the best methods available for atmospneres

in which the parameters vary with depth were the

Riccati method 2

' 3 and that of Feautrier

2 ' 4 ' 5 Since

then, we have developed a theory of radiative transfer, based on principles of invariance, which is

particularly suited to computation. 6 ' 7 The equations

of this theory have a simple structure which .is easy

to interpret in physical terms, and their solution

is a relatively rapid and simple matter. Their form

is sufficiently universal to make generalization

easy, and it is possible to make a sufficiently

precise analysis of error propagation and stability

to give us confidence in the numerical results

obtained.

We shall present an application of this general

theory to the problem of determining the line profile

due to non-coherent scattering in an inhomogeneous

plane-parallel atmosphere. We have confined our- selves for the present to problems already treated

by the Riccati method, 3 and we have therefore assumed

that we are dealing with a model atom with two discrete levels only. One of our objects has been to see if our methods are in any way competitive with

others already in the field; astrophysicists will

want solutions to more elaborate model problems in

the future, and reliable methods promising any

economy while maintaining accuracy are highly desirable. We shall therefore make a detailed comparison

of our technique with the Riccati method, and, in so doing, deal with some problems that we have encountered in carrying out their implementation.

2. GENERAL PRINCIPLES

Our computational procedures depend upon a conservation principle which may be expressed in the

form 7u

+

(y) = t(y,x)u+

(x) + r(x,y)u~(y) + Z +

(y,x) u (x) = r(y,x)u (x) + t(x,y)u (y) + Z (x,y)

where x,y are depth co-ordinates in the planeparallel atmosphere such that a <. x < y <. b . The

quantities u

+ (x) , u~ (x) , u+ (y) , u~(y) are the

190
specific intensities of radiation at the corresponding levels x, y, the superscripts referring to the

sign of the cosine, y, of the angle between the

direction in which the radiation travels and the

normal to the layers in the sense of increasing

[depth, x. In the present case, u

+ (x) , for example,

will take values u+(x)(y,0 = I(u,£;x), where

I(y,£;x) is the specific intensity for some particular value of p, 0 < u <. 1, and £ is a scaled frequency variable, £ = (v-v Q )/A s , at depth x. The

terms £+(y,x), Z~(x,y) represent the emission of

radiative energy within the layer [x,y] in the pos-

| itive (negative) direction at y (x) . Finally, the

quantities t(y,x), t(x,y), r(y,x), r(x,y) represent

operators that describe the transmission and reflection of radiation by the layer. In line transfer

problems they will involve, in general, redistribution of radiant energy both in direction and in frequency. Thus, for example, we may write

[r(x f y)u (y) ] (y,U = 1_

2y dy' dC R(x,y|y,C;-y ' ,£') [u (y)(-y',C')] (2

for the radiation diffusely reflected from negative

directions into the positive direction y at the

surface y of the layer [x,y] . This representation

is intractable for computation as it stands, but if,

as is common, in practice we use a discrete-ordinate

approximation both with regard to y and to £ , the

intensities u

+ (x) , u~ (x) , and sources Z +

(y ,x) , I~ (x ,y

)

can be written as vectors in a finite-dimensional

linear vector space, and the quantities t(x,y),

t(y,x), r(x,y), r(y,x) become matrix operators mapping this space onto itself. We can then apply familiar computational techniques to deal with equations (1) . Suppose, for the moment, that we know the operators and ( source vectors for each layer of an arbitrary finite partition of [a,b] into N layers by

planes at x = xn , a = x 2 <x 2 <. . . . <x^+ 1 =b. Equations

(1) then provide a set of 2N equations for the 2N

intensity vectors u^, l<.n<N and uj, 2<n<N+l in terms

of the boundary values u+ = u+(a), u^+i = u~(b) and

internal sources. We may write these equations in

the form

191
(I L) U = E + I

«K-> ^> S (3)

where the vectors U, Z and I are defined by

+ u 2 £ +

(2,l)

- — t(2,l)ut

uT E"(lf 2) r(2,l)uj

+U 3 L (3/2) U U 2

•

E~(2,3)

•

' ~S

0

E + (N+1,N) r (N,N+l)u~+i

U5

Z~ (N,N+1) t(N,N+l)U-+1

8

(4)

and where the matrix L has the block form

L =

AS"

o o o r(l,2)

o o o tU,2) o

t(3,2) o o o o r(2,3)

r(3,2) o o o o t(2,3)

t(4,3) o o o o r(3,4)

r(4,3) o o o o t(3,4)

o

r (N+1,N) o t(N+l,N) o

(5)

192
We recognize that this is a particular fprm of blocktridiagonal matrix; the method of dealing with equations involving such matrices is well-known, and a good account is given in the text book by Isaacson

and Keller. 8 When this process is written out, we get an algorithm which may be set out as follows:

Define r(l,n+l), , V~ , n = 1,2, , N by the equations n+1/2 n+1/2

r(l,l) = 0, r(l,n+l) = r(n,n+l) + + t(n+l,n)r (l,n) [i-r (n+l,n)r (l,n) ]t

-1

(n,n+l) (6)

V

+

7 = ut, V

+

+

=

i/2 1 n+1/2

t (n+l,n)V^_

i/

2

+ I + (n+l,n) + Rn+ J>~

(n ,n+l) (7)

V

n+1 / 2 -

r < n+1 ' n > V

n- 1/2 + T

n+1/2 Z "( n ' n+1 >

where

Tn+ 1 / 2 = [I-r(n+l,n)r(l f n)] =

I+r (n+l,n) [I-r (1 ,n) r (n+1 ,n)

] " *r (1 ,n)

t(n+l,n) = t(n+l,n) [I-r (l,n)r (n+l,n)

]

-1

(8)

r(n+l f n) = r (n+1 ,n) [I-r (1 ,n) r (n+1 ,n)

]

1

and

Rn+i/ 2 = t(n+1 ' n > r d' n ) These equations can be solved for increasing values

193
of n, and the final solutions u~ can be

obtained from n n

u* = r(l,n+l)u , + V

+

, n+i n+i n+1/2

un

= t(n,n+ l)un+i + V

n+i/2 ,

(9)

starting with the specified boundary value of u^+1 , for n = N, N-l, ... , 1 in succession. The matrix

t (n,n+l) = T . t(n,n+l)

n+ 1/ 2 may be computed at the same time as those of

equations (8) . More information about these algorithms and their physical interpretation is con- tained in our earlier papers.

6

' Our strategy is therefore to put the equations

of the problem into this canonical form, and thence

to solve them by the standard technique outlined

above. One way of doing this is to remark that in

the limit as y tends to x, equations (1) must

converge to the conventional differential equation

of radiative transfer. In this way, we may identify

the leading terms in an expansion of the matrix

operators in terms of the layer thickness (y-x)

they can be expressed in terms of the usual known

physical quantities point-by-point, such as absorption coefficients, scattering coefficients and so

on. The source terms may be identified in the same

way. The approximations so defined have rather poor

truncation errors, and it is better to start with a difference scheme which is known to have higher

accuracy (6) . That is the approach that we shall

adopt in this paper. Of course, the operators and sources obtained in this way must have expansion

in powers of the layer thickness agreeing with

those obtained from the simpler procedure through

first order terms.

3. DISCRETE EQUATIONS FOR LINE TRANSFER PROBLEMS

The purpose of this section is to write the

integro-diff erential equations of transfer for the

194
I formation of spectrum lines in the required form,

j using a notation rather similar to that of Hummer

and Rybicki

.

2

t 3 For a two-level atom, assuming

complete redistribution, they write

y It

= te+cMU] (s

5y

-i

5

) (10)

where I^y is the specific intensity at a frequency

specified by

S = (v-v

q

)/A

s

, (11)

in a direction specified by u where A s is, as usual,

some standard frequency interval. Here <j>(£) is the

line profile, normalized so that

<MS)d£ = 1. (12)

J -00 The mean optical depth is defined in terms of

the geometrical depth z by

hv

dx = kT (z)dz t—?- (NiB 12 -N 2 B 21 )dz (13)

where Bi 2 (z) , B 2 i(z) are the Einstein coefficients,

and Ni(z),N 2 (z) the population densities of the lower

and upper states respectively. The quantity 3 is

the ratio kc/kL of opacity due to continuous absorption per unit interval As to that in the line, and in

this event

Vz) - frfW s l

(z) + wmr s c

(z) (14)

where S c (z) is the continuum source function, which,

for present purposes, we shall write in the form

S c

(z) = p(z) B(vQ

,T

e

(z)) (15)

where B is the Planck function for frequency v Q at

195
electron temperature Te , and both p and B are assumed

to be independent specified functions of z. The line source function for complete redistribution, Sl(z),

is given by

S. (z) = A 2i N 2 (z)/[B 12 Ni (z) -B 2 iN 2 (z)] / (16

J_i

and, when combined with the statistical equilibrium

equation

Ni [Bi 2 UJ^ + C 12 ] =

N 2 [A 2 1 + C 2 i + B 2 1

it becomes

S L

(z) = (JL-'e) (17)

where J r is the mean intensity, B is the Planck

function and

2 1 C 2 i+A 2 i [l-exp(-hv

o/kT

e

)

]

(is;

is the probability per scatter that a photon will be

destroyed by collisional de-excitation. We are, as usual, interested in cases with e < < 1, and we

shall assume that this parameter is specified in

advance

Our computational problem is therefore to solve

equation (10) with a source function defined by

equations (14), (15), and (17) regarding B(z), p(z),

e(z), <J>(£) E <j)(z,0, and B(z) as specified functions

In addition we need boundary conditions specifying

the fluxes incident on the external surfaces. This

problem will be solved by a suitable discretization

in frequency, direction, and space. For the frequency discretization, we choose division points

196
and weights {a-j_} such that

<|>U)f (C)d£ Jfc E a.f (E. ) i=-I

Z a.=l (19)

i=-I 1

and for the directional discretization we choose

abscissae {y-j} and weights {bj} such that m m

f (y)dy % I b.f (y.) , E b.=l

o j=i 11 j=i 11 (20)

We define the mxm matrices

b = [b. 6. .] , m = fy. 6. .

and the m-vectors

h

i ,n

Klil,q?Tn

)

u

.

l ,n

I( -

ym^i ;T n

: (21)

197
Then, if we integrate equation (10) from t to t , we may write the result as

n n+i

+ - x a

2 n+!/ 2 n+i/2^1 ,n+i/ 2 x

T a. , , (h h

T )b(u++u~) . , i'=-I

x . (3+<j>. ) A u

+

. (22) n+i/ 2 i n+i/ 2 ifH+i/2

+

2 T n+i/ 2 a n+i/ ,n+i/ 2

T a. , , (h h

T )b(u+

+ u ) . , . i

i ' ,n+i/ 2 ^ i',n+i/ 2

T . (3+0. ) . u. . , n+1/2 i n+1/2 i/n+1/2

Here we have used superscript T to denote transposition, x , = t , , - x > o, and other subn+ 1/2 n+ 1 n scripts (n+1/2) to denote averages for the layer.

Also a , = 1-e , n+1/2 n+1/2

Corresponding to each pair of subscripts (i,j), we now define an index k so that

(i,j) e k = j + (i - l)m, 1 < k <L K = ml. (23)

198
For each value of k, we define coefficients

W. = W, , , by k k,n+i/ 2 2

), W, = a . b . k k 1 j

(24

Let

u~ (L

*-n

i f n

l,n

, u+(R) ^n

i ,n

(25

be K-vectors corresponding to frequencies to left

(L) and right (R) of the line center. Let (J)

. and — n+i/ 2 g , be K-vectors, where the elements of g . are defined by

9n (p3+£<K ) .. B , ^k,n+i/ 2 k n+;L/2 n+ 1/2

(26

and let A, W and M be K x K matrices defined by

A , = [A, . , ] = ^n+i/ 2 kk 1

t ( ^k) n+1/2 6 kk' ] '~W

n+1/2

= [Wk,n+1/2 6kk' ] ' (27

and

M =

M

m

~~M

m Ow

o M— m

199
Then equations (22) reduce to

M[u"\ (r)-u

+

(R)] + t A / u"!" (R)

~- ~n+i ~n n+i/ 2^n+i/2^n+i/

J- a ((J) (j) T

) , W . [u+

(L) +

2 n+i/ 2 ~ ~ n+i/ 2-"n+i/ 2 —

n+i/ 2

+ u

+ (R)+u (L)+u (R)]„. ^ n+ , 1/2

+ T n+1/2Sn+1 /2(R)

and (28) : M[u (R)-u (R) J + t A , u , , (R) v^n+j ^n n+i/ 2"*-n+ 1/ 2^n+ 1/

To. 0 : ((J) ^ T

) . W _,_ [U+

(L) + 2n+i /2 n+1/2 ~ ~ n+i/ 2~n+i/ 2

L-~

+ u

+ (R)+u (L)+u (R) ] , ~ ~~ ~~ n+1/2

+ T _l g ^ (R) n+i/ 2 ^n+i/ 2 and two similar equations with L and R interchanged

everywhere. When the line profile is symmetric,

we have u± (L) = u± (R) , g(L) = g(R), and equations

(24) simplify, "in either caseT the structure of the

equations is similar to those of equations (2.10) of our earlier paper, 6 and hence the same procedures

may be used to re-write them in the form of equations

(1) . The details are presented in the Appendix. In

the symmetric case, to which we shall confine our- selves in this paper, we find

^

+n+ i = £<n+1 ' n ^n + £ (n ' n+1 ii?n+1 +i%+l,n)

(29) u~ = r(n+l,n)u+

+ t(n,n+l)u~ + Z~(n,n+1)

200
where

r(n,n+l) = r(n+l,n)

T

r = y (C C ) M W ^n+1/2 n+1/2 ~* n+1/2^ ""n+1/2 (30)

and

t(n,n+l) = t(n+l,n)= t , + r (31) /^n+ 1/2 n+ 1/2

in which the first term is a diagonal matrix,

t = [I-ir , M

1 A , ] [I+ir , M *A , ] 1 ^ n+1/2 2 n+1/2""' ^n+1/2 ^ 2 n+1/2^ ^n+1/2

In these definitions I is the K x K identity matrix,

C j = [I + ir . U

1 A , ] 1

. M

1

(f) ,(32) ^n+ 1/2 ^ 2 n+ 1 / 2^ ~,n+ 1 / 2 —' ^n+ 1/2

and

where

Yn+I/2 =

a n+1 / 2 T n+1 / 2 /[1 - 26 n+1 / 2

] (33)

6 = a t , x n+1/2 n+1/2 n+1/2

2 x Y W

k/n+i/ 2^^+1/2 (34)

£ 2u +t A k=i zyk n+1/2 kk, n+1/2

The following properties of the matrices are easy

but important consequences of the definitions:

(i) Jrn+ /

is a matrix approximation to

exp [-t . M

1 v A , ] , and so represents n+1/2 '-n+1/2 v the unscattered attenuation in the layer.

201
(ii) JFn+

i s unsymmetric, but the matrix

S defined by ~-n+i/ 2

S , = W ^ Mr ^ -n+i/2 -n+i/ 2-^n+i/ 2 (35)

(iii) r , has non-negative elements (written

-~n+ 1/ 2 r ^. o) . Indeed, from (34) , taking *~n+i/ 2 account of (24) and (27) , we have

6 = a t n+!/ 2 n+1/2 n+1/2

I M a .b .d). v v i iTi, n+1/2

!=!-,= ! H

-j n+i/ 2

v n+1/2 y i,n+i/2 ;

I M

<an+i/2-. , T 7 a.ilb .= ~- a , , 2 n+i/

so that

o <y < gn+i/ 2 T n+i/ 2 n+1/2 j—

n+1/2

(iv) A sufficient condition for t , * > o is n+1/2

2yk

2y

T n+i/ 2 < m

kn = e+T(oTn+i/

say. This is more stringent than one needs

to ensure that t(n,n+l) and t(n+l,n) are non-negative. A better sufficient con- dition is

t < t mm

if j

2(l-e) a. <fr. . b.

i i / n+j/ 2 3 (B+cj). ) . i n+1/2 crit

(36)

202
Let

M

| u | I = 2tt V a. y b .u .

| u. (u . =-I 1 j=i J J 1 3 (37)

2* f W

k

|uk

Then | \u\ | is a vector norm satisfying the

usual norm conditions, and a matrix norm,

| | , consistent and subordinate to this norm is 7

max I W

k

|A, kl | W, , . (38)

k ' k i ,—

Now define the cell matrix £ (n,n+l)

associated with the layer under consideration by

r (n,n+l)

t:(n+l,n) r,(n,n+l)

r(n+l,n) t(n,n+l)

By analogy with (38) , and assuming (36)

holds

,

|

| T (n,n+l) |

| = max { |

1 1 (n+l,n) +

+r (n+l,n) t (n,n+l) +r (n,n+l)

1 1 , + 2r _,_ ~* n+ 1 / 2 r* n+ 1 / 2 = a. n+i / 2

+ 0(t 2 n+i / 2

{(e+ecf),) , }

min k n+1 ^ 2 +

k

y k

)

(40)

< 1.

203
We have shown 7 that the conditions set out above

are necessary and sufficient to ensure the existence

of a unique bounded non-negative solution to our equations

To complete the specification, we need the source vectors, which are given by

Z + (n+l,n) = E (n,n+l)

E , = ttT [I + t ^ +2r^l M" 1 ^n+1/2 g 2 n+1/2 -vi/ '^-n+i/2 ^n+1/2 ^n+1/2

4. IMPLEMENTATION OF THE PROCEDURES FOR LINE TRANSFER

The structure of equations (30) and (31) , which

represents the reflection operators for a thin layer

as a Kronecker product, allows a great simplification

of the algorithm. We suppose that we require only

to determine the source function and the emergent

intensity vector, so that we shall set down only

those equations which are directly relevant to the

computation in this section. The remaining equations

not needed for this part of the computation are set

out in the Appendix.

We write

S , = M W r(l,n) , (42) --n+1/2 ~~n+i/ 2 ~- and define the scalars

T

A = C S C n+1/2 —n+i/ 2~-n+i/ 2*"n+i/ 2 and

Yn+1/2

n+1/2 i-y„ , , A n+1/2 n+1/2

(43)

Then equations (6) become —

S , n+ 3 / 2 ^n+i/ 2^n+ 3 / 2 ^n+i/ z^n+i/ 2~n+i/ 2 +An+i/ 2~n+i/ 2 *n+i/ 2

^

' (44)

204
to be evaluated recursively for n = 1,2, ••• , N,

starting with J> 3/2= 0, where the K-vectors X

and X j_ , are~~defined by

~~ n 1/2 —n+1/2 J

—

X . = (M W , + t S )C n+1/2 — n+1/2 ~n+i/ 2-n+i/ 2 ^n+i/ 2

i?n+1/ r -n+1/ 2

(^n+1/ 2 £n+1/ 2 + J? ^n+1/ 2

}

Notice that X ^ will equal X

T

^ if S L —n+1/2 -n+1/2 n+1/2

is symmetric. This is only the case if J#n+

is

independent . of depth, that is to say if' 1/2

(a^/cj^) is independent of n. This depends on the

choice of quadrature formula; J£n+is independent

of n for most choices. ~ 1/2

We also define

V 7 = M W V* (45) ^n+1/2 *~ "*n+i/ 2 ~n+ 1/ 2 and the K-vector

V + S ^ Z (46) ^ n+1/2 ~-n-i/ 2*-n+i/ 2~n-i/ 2 ~n+i/ 2 ~n+i/ 2 Then the first of equations (7) becomes ~ n+1/2 -~n+ 1 / 2^n+ 1 / 2 + Bn+i/ 2 ^n+i/ -^n+i/ 2 ^n+i/ 2

'(47

to be evaluated recursively for n = 1,2, ••• ,N,

starting with

J1/2= M 2? l/2 Si ffi l/2= W

3/2 being assumed), where

T

n+i/ 2 = A

n+i/25n+i/2-n+i/ 2

* (48

205
Then, given iJN+1 we can evaluate for n=N, N-l, . ..,]

the equation '

u = t , u , +

*~ n '^n-r i / 2"*n+

+ [B ^ + A , (X , , u~ ) ] C , + Z ^ (49 n+1/2 n+1/2 Vn+i/2-n+i 7 ~n+i/ 2 ~~n+i/ 2

v

to give the upward directed intensities at each

interface. The line source function, defined by

(17) , requires the calculation of

^|(J)

T

, W . u

+

+ <j> T

, , W , , u~l ^ l^n-i/2«^n- i/2*M, n ~-n-i- i/ 2 ^n+ i/z^n J (50)

T T - 1 ~ Z u + d> , M V a- n ^n n-i/2 -~ where

= / M

1 (W * / / S . , > ) + / W , — n J^n=i/ /

2 ^ --n+ i/2'^n- i/2'v-n+i/2 '-n+i/ 2 ^n+ 1/2

(51)

may be computed along with X , , , X , , and Y , , . ^n+1/2 ^-n+1/2 —n+1/2

Notice that if ui and the line source function are all that we require, it is unnecessary to compute

u* directly.

We may also wish to compute the emission in some other direction u than that prescribed by the rule

for angular quadrature. Now that we have the line- source function this is straightforward. Writing

the required frequency distribution as an I-vector,

ui • (y) , we see that

u, (y) =

T N+i

e-TA<T)/y

[ (1 _ e(T) ) (

f

) ( T )J(T)+gl ,(T)]dT (52

0

206
where A(t) is an I x I diagonal matrix, and (j>y(x),

gy(x) are I-vectors defined for the single value of

% as in equations (27) and (26) respectively. The

discrete representation of (52) may be written

Uj (y) %

n= i — 2 n+i/ 2 n+i/2-n+i/ 2

+ .2n+!/ 2

] T n+!/ 2 (53)

The choice of J , , and of t(l,n+^-) is slightly

n+ 1/2 2 arbitrary. We have used

t(l,n4-i) = |[t(l,n) + t(l,n+l)]

where

n- i t(l,n) = n t , / (u) , k=i ^n+ ^2 (54)

with

~"n+ 1/ 2

i — ix / A./y

2 n+i/ 2 r

p

2 n+1/2 1 and

n+ 1/2 2 n+ 1 n

(55)

It would be possible to use more accurate formulae,

bu*t we have not found it necessary to do so.

5. THE RICCATI METHOD

If we pass to the limit as ~t n+i/ 2 o, our equations reduce to a set equivalent to those of the

207
Riccati method of Rybicki and Hummer. 2

' 3 Equation

(44) becomes

- S(t)M 'a(t) + o(t)X(t)X(t) (56)

with initial condition S (o) = 0,

X(t) = [W(t) + S(t)M *] <()(t) X(t) = (|) T

(t) [M *S(t) + W(t)]

also

dV(x) _ 1

. . dW(x)

W (t) V(t) - [A(t) + dx ^ dx

(57)

a(T)X(T)(}) ,r (T)]M"'

1 V(T) + (W(T) + S(t)M

1 )g(t)

corresponding to (47), with V(o) = MW(o) u

+ (o), and

du (t) - 1 dx M [A(t) - q(t)4)(t)X(t) ] u (t) +

o(t)M

1 ^)(t)<J) T (t)M(

J V(t) - M \j(t) (58)

while (50) and (51) give

J(t) = X(t) u (t) + (J) T (t)M

r v(T). (59)

Apart from superficial differences arising from

a difference of notation, the set (56) - (59) differ

from those of Rybicki and Hummer by terms involving

- 1 <jW JJ -g^. The matrix Jtf is defined by (24) . In most

applications, the weights for the frequency integra208
tion, ai , have been chosen proportional to the

corresponding value of <J>. Thus the element

a.b,

W

k

=

1 1

in such cases is independent of t, and the terms

- 1 dW involving jj, disappear. This has a number of

useful consequences both for the discrete and the

Riccati methods; in particular the operator S becomes symmetric, a fact which can be used to reduce

both the work and the storage required. Other

aspects of the choice of quadrature formulae will

be discussed in the following section.

We can also solve equations similar to (52) to

determine the emissions in directions other than

those used in the quadrature formula. This may be

conveniently incorporated in the procedure by way

of differential equations which can be integrated

along with equation (58)

.

6. COMPUTATIONAL ASPECTS

We have generated solutions of the two-level

model using both techniques on the ICT Atlas computer

at the S.R.C.'s Chilton Laboratory. The discrete

method involves extensive use of a matrix multiplication routine which computes scalar products of

two vectors double length and rounds the result

correctly to single length. In the Riccati method,

we use Merson's version of the fourth-order RungeKutta method, which provides an estimate of error

that can be used to control the step- length. It is

also necessary in the Riccati method to interpolate

in tables of Xn and y n = <J>T M

-1 generated on the

forward sweep"' to provide ^Ene necessary intermediate

values for Runge-Kutta integration in the reverse

direction. We have done this by constructing the

coefficients of a Newton interpolation polynomial

for a given set of nodes for each interval in turn.

The coefficients may be preserved, and used to

evaluate the interpolation polynomial for each of

the required intermediate points. The coefficients

are changed only when the nodes change, and this

enables us to deal very simply with the end intervals .

209
6.1 Storage Allocation

Both programs require basic data to generate

the following information in the general inhomogeneous case:

i?* T n'

e n+ 1 / 2

' Y n+ 1 / 2 'Jn+ 1 / 2 'in+ 1 / 2 'in+ 1 / 2 Total: 3(K+1)N + 2K+1 cells.

The discrete method requires storage of

A

n+i/ 2

' Bn+i/ 2 '-n+1/2 '-n+i/ 2

r -n+i/ 2

' £ n+i/ 2 '^n+i/ 2 '~n on the forward sweep, and of J on the reverse sweep.

Total: (6K+3)N+1 cells.

The Riccati method requires storage of

n ^n

on the forward sweep, and of J on the reverse sweep.

Total: (K+2)N cells.

These requirements must be added to the storage

needed for program and system routines and for working space. For a typical problem with I = 10 , m = 2 (giving K = 20) and N = 50, the additional storage

required by the discrete method is about 8 blocks of

512 words, less than 20% of the total. It would be

possible, with small modifications of the discrete

method, to reduce this excess considerably.

[ 6. 2 Timing

We have estimated the amount of computing time

required by counting multiplications and divisions

needed in both methods. This does not allow for

storage references and other operations, but it is

reasonable to assume that they are roughly proportional to the number of multiplications and divisions

in a problem of this type, and we have found no evidence to contradict the assumption. On this

basis, the discrete calculation uses {5K 2 + 28K + 4}

210
operations per layer, and the Riccati method {—= + 125K 1 —2— + 2P(p+ll) (K + 1)} operations per layer, where

p is the order of interpolation required on the reverse sweep. For practical values of K of about

20, this means that the discrete method is between

5 and 6 times faster than the Riccati method per

layer. We have found that it is often necessary

in the Riccati method to subdivide the layers in

order to maintain accuracy, particularly near the

boundaries. This is, of course, done automatically

by the step controller. There is no such problem

in the discrete method for which we only need to

satisfy the inequality (36) , so that the practical

increase in speed is often larger, although bookkeeping overheads diminish the effect. For example

in a typical problem with constant Doppler width,

having t=50, 3=0, e = 10" 6

, K = 30, the discrete

method completed the computation in 30 seconds,

whereas the Riccati method took over 6 minutes.

6.3 Angular Quadrature

There are few problems involved here; we have

found that the line profiles and source functions

are insensitive to our choice. We have usually

employed the abscissae and weights for Gauss quadrature of order m = 2 over [0,1], using an additional

integration to compute the normal emergent line profile. We have also used Radau quadrature of order 3 with an abscissa specified at u = 1. This simplifies

the program, but, in addition to increasing the

storage requirement, it is rather expensive in com- puting time, firstly because it increases K by 50%,

and secondly because the smallest cosine, \i lr is nearer to zero than with the Gauss rule. This causes

a decrease in the critical step size in the discrete

method

.

6.4 Frequency Quadrature

The choice of frequency quadrature presents a more difficult problem. For pure Doppler broadening,

we can employ the finite bandwidth approximation used

by Avrett and Hummer. 9 In our implementation, we

have truncated the frequency interval to become

-a <. £ <. a, and used Gauss quadrature over this finite range. The accuracy depends on the number of

211
points, and on the value of « , and we have selected

a value of I such that the normalization condition

(19) is satisfied to prescribed accuracy. For

example, choosing a = 4, we require I = 10, so that

the error in normalization may be less than one part

in 10 6. We renormalize the integral to ensure that

(19) is satisfied to within the round-off accuracy

of the computer, about 11 decimal digits. It is

important to do this so as not to introduce spurious

sources and sinks in the calculation. 7 Hummer and Rybicki have pointed out 2

' 3 that this

finite bandwidth approximation is unsatisfactory for

Voigt profiles, for the bandwidth a has to be increased substantially, particularly when $>o. They

have discussed methods of overcoming this difficulty

which involve splitting the range of integration

and renormalizing each segment separately. While it

certainly eliminates difficulties arising from lack

of normalization, there is an arbitrariness about

the procedure which is a little unsatisfactory, and

it is not clear what effect changes in the method of

splitting have on the final results. The difficulty

will be more acute in variable property media, and it seems clear that this problem needs further investigation.

7. NUMERICAL RESULTS

Our main effort has been directed towards the

comparison of the discrete and Riccati methods in a few idealized situations. We have computed the line source function Sl(t) at a selected set of optical

depths in slabs of various thicknesses together with

the normal emergent intensity for Doppler broadened

lines

In the case of a homogeneous slab, the results

from the two programs are closely similar, as Figures

1 and 2 indicate. However, there are small differences which are best demonstrated by an examination

of Table 1.

(a) Symmetry of S^Ct) About the Mid-Plane . The

discrete solution is symmetric to the num- ber of decimal places printed out. The

Riccati solution is asymmetric by up to

5% in the worst case of linear interpolation, and up to 2% when cubic or quintic

polynomial interpolation is used on the

backward sweep.

212
o si2

•rH g Q u0m • C 1 3 O

rH

fd

II • d

•H CO > C T3 H 0 fl 0 H rO CO +J OJ 0 - H G rH

+J

M-l || 0g

d) Q. U 0) 5H - u 3 rH fd 0W II COG

<U CQ 0 G H H x! +J J -M H 0

CO O H

0) o +J

?H rH id s u ^ II uH

213
o 2 3 4

NORMAL EMERGENT INTENSITY FROM A UNIFORM MEDIUM

OF OPTICAL THICKNESS T-IOO, WITH €*IO"6 AND B-l, AS A

FUNCTION OF DOPPLER WIDTH. Figure 2. Normal emergent intensities corresponding

to the line source functions shown in Figure 1.

Discrete and Riccati solutions are not resolved.

214
(b) Choice of Angular Quadrature . The results

are insensitive to this choice. Comparing

Gauss and Radau solutions, we see that the

greatest difference is in the fourth

significant figure, both for the line source function and for the normal emergent

intensity

.

(o) Variation of Step Length. The layer thick- nesses are presented in the discrete

method. We have done some experiments to

examine the truncation errors involved,

which are summarized in Table 2.

It is clear that we can be confident of reduci ing the discretization error from this source to a reasonable level if we take T<x crit. Further work

to be published elsewhere shows that this error,

for the operator approximations we have used, is

proportional to the square of the layer thickness.

This is verified by the results of Table 2.

In the Riccati method we have asked for an absolute accuracy of 10 _lt in the S-integration , and

a relative accuracy of 10" h in the V and u~ integrations. The program chooses its own step length to

meet these criteria. This is always reduced most

near the boundaries at the start of a sweep, sometimes as small as 0.036 (where 6 is the Doppler

width) though for the most part it is close to the

value of 0.76 quoted as the stability limit by

Rybicki and Hummer. 3 The very small step size

needed near the boundaries is almost certainly

associated with the fact that the source function has

a logarithmically infinite slope there, 9 so that the

Taylor expansion about t = 0, on which the RungeKutta process is based does not exist in the first

interval.

(d) Effect of Order of Interpolation in the

Riccati Method. Table 2 shows that the

biggest improvement comes from replacing

linear by cubic interpolation. The use of

a fifth-order polynomial produces a less

marked effect, and it is clear that there

is little point in going further than thirdorder with step sizes of the order used in

this problem.

(e) The Line Fro file. The most marked discrepancy between the two calculations is at the

line center, which is formed in the outermost layers of the slab. There is a deficiency at £ = 0 of about 6% in the Riccati

215
u

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216
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217
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CD W

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218
solution with linear interpolation falling

to 2% if cubic interpolation is used. This

difference decreases towards the wings of

the line as may be expected.

We have also computed some results with variable

I line-profile. The results, shown in Figures 3 and 4

\ are closely similar to those of the corresponding

Figures 6 and 7 of Rybicki and Hummer's paper. 3

8. CONCLUSIONS

The discrete method has a number of advantages

compared with the Riccati method which may be summa- rised as follows:

1. Much greater speed - an order of magnitude.

2. Better control of discretization and roundoff errors.

3. Ease and simplicity of programming.

Its greatest disadvantage is that it requires

I more storage, at least in the current version. It

might be possible to circumvent this by reprogramming so that it is unnecessary to retain data for

all depths other than certain specified ones. This

may be done by computing and the scalar product

T - i ~

<J> / M V . required for (49) for selected points

n = n 2 , n 2

, on the forward sweep, together with

vectors £ ^ n £' n

£+ a

^ and matrices t (n^ ,n^

+ 1

) required

to solve the equation

u— n = t !l Vn A+il*n V

These vectors and matrices are obtained by simple recurrence forumlae derivable from equation (49)

.

A similar dodge has been used to limit the

storage requirement in our version of the Riccati

method, but the larger the intervals chosen, the more strain is thrown on the interpolation polynomial in the backward sweep. There is no such

constraint in the discrete method. The accumulation

of round-off error in calculating scalar products

in the Riccati method also needs some investigation,

but it could largely be eliminated at the cost of a little inconvenience.

219
I9

2

f

<9

I

F8

220
NORMAL EMERGENT INTENSITIES FROM A UNIFORM MEDIUM OF OPTICAL THICKNESS T- IOO. AS A FUNCTION OF DOPPLER WIDTH Figure 4. Normal emergent intensities at x = 0 cor- responding to the line source functions shown in

Figure .3

.

We have not attempted any solutions for semiinfinite media so far, but it would not be difficult

to do so using the Asymptotic theory as described by

Rybicki and Hummer. 3 Alternatively, since the

asymptotic region is of necessity assumed to be

homogeneous, the doubling method could be applied. 6

/ Finally, we may remark that Feautrier's method

leads to a set of matrix equations which, like our

own equations (3) , have a block tridiagonal form

\ and can be solved using the same algorithm. 8 We

have tried to see if it is possible to interpret the

Feautrier algorithm in terms of our theory. We have

not been able to do so because the way in which the

positive and negative intensities are combined by

i Feautrier makes it impossible to disentangle the

operators. We have therefore been unable to make

any fair comparison of his method and our own. 221
ACKNOWLEDGMENT

We are indebted to Mrs. Elizabeth Gill for her

assistance with the programs.

REFERENCES

1. J. T. Jefferies, Spo.ctn.al Line. Formation

,

Blaisdell (1968).

2. D. G. Hummer and G. Rybicki, MctkodA o fa Computational Vkyt>lc6 7, 53-127 (1967) . 3. G. B. Rybicki and D. G. Hummer, AAtn.opkyA . J. 150 , 607-635 (1967)

4. P. Feautrier, C. R . Acad. Sal. Van.lt> , 258 , 3189 (1964).

5. D. Mihalas, Attiopky* . J. 150 , 909-925 (1967).

6. I. P. Grant and G. E. Hunt, Hon. Not. R. A&tn.. Soc. 141, 27-42 (1968)

7. I. P. Grant and G. E. Hunt, ?n.oc. Roy. Soc.

A313 , 183-198; ibid. A313 , 199-216, (1969).

8. E. Isaacson and H. B. Keller, Knalyhlh o£

Hmm<in.l(ial ttttkod*, pp. 58-62. Wiley (1966).

9. E. H. Avrett and D. G. Hummer, Mon. Hot, R. Attn.. Soc. 130, 295-331 (1965).

222
APPENDIX

A. The results of Sections 3 and 4 depend heavily

on the following Lemma Let X. and Y be any two

K-vectors and let

T

z = y:x.

Then if |z| < |a|

,

[al - X.YT ]" A = a *[I + (a-z) *X.r]

Proof . By the binomial expansion, the result follows

directly

.

B. Thin Layer Operators and Sources

The approximations given in our paper (6) for

thin-layer operators make use of matrices

Z = A - a<H

T

W, Y = aq>4> T W (Bl

and

[M + it Z]" 1

. (B2)

If the "diamond" weights are taken to minimize the

local truncation error, equations (A. 7) and (A. 8) of

reference 6 may be written for the present problem

in the form

r = p(I. - p

2

> _1

6. , t 2(1 - P

2 )"6 - I (B3

where

ttT A Y , 8 = 1- yT A Z CB4)

223
From (B2) and (Bl) = [JA + |t A. - ^tJj^W]

1

= (M + |tA)

1 [I - jH^J 1

where

D = ~tW CM + ixA)

1

is a diagonal matrix. Applying the lemma, we get

A = (M + IT A)"

J [I +

1

cJ)(J) T

Dl (B5) ~S ~" Z ^ 1-0 Z-Z" r*S where

6 = (j) TD4)

is the quantity written out in equation (34) . Thus

and an additional application of the lemma gives

(I - p

2

) 1 = I +

6

. (M + ixA) ^^(ioTW) (B6)

Equations (30) to (33) now follow in a straightforward manner.

C. Discrete Equations which are not used in the

present problem

We first list the local operators that we do not

use explicitly:

T = I + A . C ^ C

T

^ S , ^n+1/2 ^ , n+i/ 2^n+i/ 2"n+i/ 2^n+i/

224
t(n+l,n)- tn+i/2+ A

n+i/2M-V;

i/2

Xn+l/2 C^+i/2l?Wn+l/2

|<n+l,n) = A

n+i/2£n+i/2 C^+i/2

M W

n+i/2

(CI)

Jn+l/2= «" 1 Sn+ 1/2Pn+ 1 / 2 + An+W2-n+ 1 /2^+i/z\3n+ l / 2 t(n,n+l) = tn+ + A

n+ l/2.£n+i/ 22n+ 1 / 2

The equation that is not used explicitly on the

forward sweep is

V . = I ^ + B ^ C . , (C2) - n+ 1 / 2 1/2 n+ i / 2<n+ 1 / 2 and that which is not used explicitly on the backward sweep is = M

-1

W~^ S , u~ + M^w" * \T . (C3) ~«n+i ~> «^n+3/ 2T-'n+ 3/ 2^n+i ^ ^n+ 1 / 2^^+ i / 2

225
COHERENT LINE FORMATION WITH DEPTH-DEPENDENT

PARAMETERS

by

T. R. Carson

University Observatory

St. Andrews^ U.K.

ABSTRACT

A discrete ordinate method is developed for

solving the equation of coherent line formation,

for arbitrary given variations with depth in an atmosphere of the temperature (or Planck function,

assuming local thermodynamic equilibrium) and the

line absorption and scattering coefficients. The

direct solution thus obtained can then be used as the starting point of an iterative procedure.

Results obtained for an exactly soluble case indicate the utility of the method.

Key words: numerical methods, coherent line

formation.

INTRODUCTION

The equation of coherent line formation can be

solved exactly only in a number of special cases where certain restrictions are put upon the problem

(see references) . Such restrictions may include the

imposition of simplifying boundary conditions and/or

assuming particular variations with optical depth of

parameters such as the temperature and the coefficient of absorption and scattering in the line. When

these restrictions are invalid, or otherwise un- acceptable, a rigorous solution may still be obtained

by numerical iteration from an initially guessed

226
trial solution, but this method converges very slowly

! when the scattering coefficient is large compared

'with the absorption coefficient. In the method of

I the present paper none of the above mentioned simj plifications is assumed. The equation of transfer

! is written in the form of an integral equation for

the source function for which a solution is obtained

directly by reduction of the integral equation to a

. set of simultaneous linear equations. Iteration

, from the direct solution then leads to more accurate

results, as indicated by applying the method to a known soluble case.

The equation of radiative transfer, including

line absorption, in plane parallel geometry may be

written

where v is the frequency, x is the geometrical depth

measured in the direction of the outward normal to

the atmosphere, y is the cosine of the angle between

the direction of transfer and the outward normal,

and p is the mass density. I (x,u) is the specific

intensity of radiation, J v

(x) the mean intensity,

B V (T) the Planck function for temperature T, k v and

are the continuous and line mass absorption co- etficient and (1-e) is the fraction of radiation

absorbed which is scattered. Introducing an optical

depth t v defined in terms of some absorption coefficient kv by the relation

THE EQUATION OF TRANSFER

dl (x,y)

pdx

+ (l-e)£

vJ

v

(x) + (k v + * v )B

v (T(x))

the equation of transfer takes the form

227
y = a (t )I (t ,y) +

v v v v dx

V

: (t )J (t ) - y (t )B (t ) V V V v 'v v v v

k + I V V

V K.

(1 - e)l

k + I y = ^

1

Defining the source function

S (x ) = 3 (t )J (t ) + y (t )B (t ) V V V V V V V V V V

the equation of transfer becomes

dl (t ,y)

y —3—- = a (t )I (t ,y) - S (x ) dx^ v v v v' v v

for which the solution may be formally written down

exp

v d T

" dx*

a .(t'\.) —^- S ( t ' ) v

V V

T' v v y

v rJ T exp

dx

a Cx") — v v' y ax 1 S Cx')

V V

228
Using the relations

J (t ) v v + 1 I v

(T v ,u)dy

S (t') Ei

V V

(T M )dT M

\> dT' ,

where E (x) is the exponential integral of index n defined by

E (x) =

n -xt -n

J. e dt i -xs n2

s ds

e

the source function S_(t_) then satisfies the integral equation v v

13 (t, :) r r 00 s (x ) =

V V

V V VT ; )E1 a (T")dT n lldT'

t

, V V V

1 J V

J

+ Y (T )B (l

' V V V V

SOLUTION FOR THE SOURCE FUNCTION

Suppressing the indicated frequency dependence

of all quantities, the integral equation for the source function may be written

f oo S(T) = |

[J

S(T') E 2 |f(T,T') [dT'J + YB(t)

where

f (t,t ') = | [

T a

' T T")dT"

229
Choosing a set of points of subdivision with

appropriate weights a^ , the integral over t ' may be

replaced by a summation, thus transforming the integral equation into a set of simultaneous linear

equations

S.1 y.B. '1 1

where the indices i,j denote that the quantities are evaluated for Tj_,Tj. The calculation of the co- efficient matrix elements is straightforward as long

as the range of integration does not include an interval or subdivision one end of which corresponds to

j = i. For j = i, fij vanishes and hence E 1 {f$_^} is

infinite. We must therefore find an alternative

expression for the integral over those ranges of integration that include the singularity viz.,

i f(Ti TV) S (T' )dT

and

i+i

f (t S (x' )dT

Since as f -> o

00 n n

E x (f) -> -Y- I f

- fcnf = EJ (f) -Jlnf

1 n . n

where y = 0.577215665.... we may isolate the singularity in the inf term and write

f ( t

1 ) r S ( t

1 ) dx 1 I f( TMi S(T')dT' +

Jlnf (T')S(T')dT'
The integrand of the first term on the right-hand

side is now everywhere finite and so the integral

may be represented by a quadrature formula in the

usual way, as a weighted sum of the values of the

integrand at the points of subdivision. The second

integral may also be so represented provided we make some assumptions about the variation of the

i integrand. Thus expanding (J>(t" ) about x" = t.

ot ( t " ) = a. + a ! ( x

" - x . ) +

1 1 1 where a! = (da (t" ) /dt " ) we have up to the terms

indicated i

(t T . 1

T ' < T

f (T

ou (t 1 - T a!

l

(T T . 1 T" > T . — 1

and

£na

± + £n(T

±

' t

1 ) + £n

a!2a.

l

(T i" x')

£nf(x') =

£na

i + £n(T

'

V + in

a!

1 + 2aT T i

} T

'

Putting S(t') = S(T

i

) = S. and taking it outside

the integral sign we have

T . 1 Inf (t')S(t')cIt i[Unai t + l- i 231
+ Fi (t) + F 2 (t, - b)J , t = T ± - x

i _ i

[ £nf (t 1 ) S (t 1 ) dx ' =S

i [(£na.)t +

+ Pi (t) + F 2 (t, + b)J , t = T i+1

-

T i+1

1

where b = a! /2a. and

Fj (t) £nxdx = [xUnx - l)"]

1 = t Unt - 1) •o

F 2 (t,b) =

| &n(l+bx)dx = ( 1+bx) j An ( 1+bx) -l)

|

1 =

= i[ci+bt) &n(l+bt)-l -1

Now replacing the derivative a? by the divided

differences 1

a! = (a.- a.^/CT.- T± ) , t._

x < t' < x.

a

i = (ai + a" a

i

) /< Ti+ i- V' t ±

< t' < t. + i we obtain an expression for the integral through

the singularity at x • in terms of S., thus completing the reduction of the integral equation to a set

of simultaneous linear equations.

THE EMERGENT INTENSITY

Given the source function, the specific intensity at any depth and in any direction may then be

232
obtained. To facilitate the quadrature we write

f (t') = f

i + f

'

(t' - x i

)

S (t' ) = S

± + S' (t ' - T i

where f is as defined, f. = f (t i

) , S.^ = S(t^) and

the primes denote derivatives. The contribution to

I v due to the source function in any interval (ti,t 2 ) is then

A12I

t 2 exp

(in

[-

I f (tMJ /y] S(T')dx'

e"bt

) -^; 2 te-bt

]

,

where t = t 2 - T lf a = fj/p, b = f J 2 /y , f{ z =

(f 2 -f 1 )/t 2 = (S 2 -S I )/t. The intensity emergent

from the atmosphere in the direction corresponding

to y is simply I v (o,y) from which the equivalent

width of a line in wavelength units is given by

W

x (y) = Vo,y)j dX

where the residual intensity

1 y (o,y)

R

A (0, p) = _|

I

A (o,y)

and I^(o,y) is the background or continuous emergent

intensity in the absence of line absorption. When

the integrated width, and not the variation with the

angle of emergence, is required we have instead

R

x (o)}d\

233
with

R, (o)

A 1

1 yl, (o,y)dy

yl, (o,y)dy

J o

NUMERICAL CALCULATIONS

To test the feasibility of our method we apply

it to a situation where the solution is already ac- curately known. For k v = k v (so that x v is now the

optical depth in the continuum ) and with n = & v/kv

constant with depth, e - o and a temperature distribution such that B v

(t v

) = BQ

(1 + 3/2 t v ) the exact

solution has been given by Chandrasekhar (1947) in

the form

R(o,y) H(y) £| + y6 + | 6 1/2 (1-6 ) ouj 6

1

R(o) = 3^| 6a 2 + I

ai + \ 6 1/2 (1-6 ) a \ f 1 ]

/2 /(y +

f)

where 6 = l/(l+n) and

a =

n

y H(y)dy

H(y) being the solution of the integral equation

H(y) 1 + 6) yH.(y) H(y ' )dy

y+y

In applying the method of solution developed in

the present paper, the number and distribution of the

points of subdivision in optical depth have to be

specified. It seemed desirable to have a distribution with zero optical depth as one end point and

for which the points of subdivision were closer together at small optical depth than at large optical

234
depth. The points of subdivision were therefore

chosen according to the formula

t r = A(n-l) 2, n = 1, N

with A and N as variable parameters. Because of the

unequal intervals the trapezoidal rule was used in

performing the relevant quadratures.

Calculations were carried out for -N = 25 and

with A taking the values 0.01, 0.02 and 0.04. The

results for (o) for various values of l/(l+n)

obtained from both the direct and iterated solutions,

are shown in Table 1 together with the exact results.

It may be seen that the direct solution is very good

for small n but can be considerably in error for

large n . However iteration of the solution leads to

a very marked improvement, but for any value of n the best direct solution does not necessarily lead

to the best iterated solution. Thus while A = 0.02

gives the best iterated solution for small n , that

for large r\ is obtained with A = 0.01, indicating

that small optical depths are more important for

strong lines than for weak lines. The results have

also been used to calculate the equivalent width

Vl\ of a line, for various values of the Voigt profile

parameter a which is the ratio of the Lorentz width

to the Gaussian width and for various values of n Q/

the value of n at the line centre. Table 2 gives

the values of log (VI\/\) . It will be observed that

both the direct and iterated solutions give good

agreement with the exact results. In particular the

iterated solution for A = 0.02 gives results which

are indistinguishable from the exact results for all

values of the parameters considered. This convergence of the calculated values of the equivalent

width must be due to the fact that the important

contributions come from regions of the line where

n is small, for which values both the direct and

iterated solutions are most accurate.

ACKNOWLEDGMENT

The author wishes to express his thanks to

Professor D. W. N. Stibbs for initiating and sus- taining his interest in this problem. Most of the

work described in this paper was carried out during

the author's tenure of a 1967-1968 Visiting Fellow235
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237
ship at the Joint Institute for Laboratory Astrophysics of the National Bureau of Standards and

the University of Colorado

REFERENCES

Chandrasekhar , S. 1947, Ap. J., 106 , 145. Also

Radiative. T'nanh {tn. (Oxford: Clarendon Press,

1950), §84.

Eddington, A. S. 1929, M.N.R.A.S., 89, 620.

Hulme, H. R. 1939, M.W.R.A.5., 99_, 730.

Krook, M. 1938, M.W.R.A.5., 98, 477.

Siewert, C. E. , and McCormack, N.J. 1967, Ap. J.,

149 , 649.

Spitzer, L. , Jr. 1938, Ap. J., 87_, 1.

Swings, P., and Dor, L. 1938, Ap. J., 88_, 516.

DISCUSSION

Kalko faun to Giant'- Can you apply your method to

semi-infinite media?

Giant: We did not try that, but it may be

possible to handle these cases.

Pecker: As we conclude today's session I would

like to remind you once more that many phenomena that

can be observed very clearly on the solar disk are practically invisible in stellar spectra or can be

found only in the far UV. This means that many of

the important phenomena that occur in extended atmospheres give only very small observable effects.

238
PART C

CHROMOSPHERES AND CORONAE OF STARS

Chairmen : J. C. Pecker, D. G. Hummer
i
WHAT DO WE KNOW THROUGH SPECTRAL INFORMATION ON

STELLAR CHROMOSPHERES AND CORONAS?

by

Franchise Praderie

Observatoire de Paris

Section d

' Astrophy sique de Meudon

ABSTRACT

Four problems in interpreting spectra to infer

chromospheres-coronas are summarized. (1) The ,a_ priori difficulties in interpreting spectra lie in

uncertainty on the range of possible models, coming

from uncertainty as to which conservation equations

may be applied, and from lack of an exhaustive list

of spectral indicators that may be used for uniqueness tests. (2) As spectral indicators we consider:

emission lines, self-reversed emission cores, the

presence of He I lines in stars not of early type,

coronal-type high ionization, excess continuum

emission in the rocket UV and the far infrared.

(3) To determine what we can infer from observations,

we summarize information i inferred by comparison of

models to data, on velocity fields, and on spectral

variability which might suggest chromospheric activity. (4) We summarize the evidence for chromospheres

in A stars, as being those where convection-induced

acoustic heating is marginal.

Key words: chromosphere, corona, spectral indicators, conservation equations.

In the framework R. N. Thomas proposed at this

colloquium* for approaching the problem of extended

atmospheres, stellar chromospheres and coronas come both under case 3 and case 4.

Case 3: Spectral features that disagree with

* See pp. 38-45.

241
classical atmosphere (CA) predictions appear in

stellar spectra. For a few stars, eclipse observations are possible; for many others even non-eclipse

observations imply the existence of a chromosphere

or corona. At this point, the last sentence says no more than: "CA assumptions are insufficient."

However it leads to a first definition of a stellar

chromosphere

Taking account of the very common occurrence

of H and K reversals in stars later than FO , one can give a symptomatic definition of one type of

stellar chromosphere: an outer layer giving rise

to emission in the H and K lines of ionized calcium.

This definition applies to otherwise "normal" and

rather cool stars even if they are variable in light,

and suggests an analogy with the sun. But exceptions

to this statement appear immediately. Stars with a majority of emission lines, like Wolf-Rayet stars,

or with a certain number of emission lines, recurrent

novae, P Cyg or Be, or peculiar systems like 3 Lyr, are not included in such an empirically based, and

necessarily limited, definition. However some of

these stars show certain chromospheric or coronal

indications like the He I A 10830 line (3 Lyr, P

Cyg). On the other hand, the only coronal lines of

highly ionized atoms observable in stars appear in

repeating novae like T CrB, in variable stars like

CI Cyg, AG Peg, RX Pup, or in stars like Z And. Thus,

the symptomatic definition, based only on the K line

(or on other lines, whatever they are), is not

satisfactory

Case 4 is the logical consequence of case 3;

it stresses the search for the physical causes that

act to violate the CA assumptions so as to produce

peculiar spectral features attributable to an ex- tended atmosphere. A causal definition of a chromosphere or corona is then more satisfying: chromosphere and corona are hot (relative to the photosphere) , tenuous extended parts of the atmosphere,

transparent in all but a few wavelengths, produced

by the dissipation of mechanical energy waves;

Cayrel's mechanism (1963) appeals to a purely radiative source, and may be efficient in the lowest

chromospheric layers as a subsidiary cause for the

increase of electron temperature. One must admit

that the heating mechanisms can act even if no striking observational consequences appear at first

sight. They produce a rise of the electron temperature Te , after some minimum obtained in the low

photospheric layers. As a result, the excitation and

ionization conditions in the chromosphere (or corona)

242
and the spectroscopic state of the gas cannot be

described by a simple LTE theory, or even a non-LTE

theory that imposes radiative equilibrium.

The plan of this talk will be:

I. Introduction: What are the difficulties in

interpreting spectral information about stellar

chromospheres and coronas?

II. Spectral indicators of chromospheres and coronas

.

III. Interpretation of chromospheric (and coronal)

spectral features.

IV. Do the A stars have chromospheres?

Neither the heating mechanisms nor the generation process of a stellar wind will be reviewed, and

I will completely exclude the solar case, only re- ferring to it as a powerful guide toward the study

of the outer layers of stars.

I . INTRODUCTION

a. Model Problem

The basic problem in interpreting spectral information in chromospheres and coronas is not essentially different from what it is in any atmosphere:

one wants to deduce the complete physical state of

the gas from the observed distribution of radiation

intensity with the frequency v, and from the shape

of spectral lines. That is what Jefferies (1968)

calls the analytical problem in interpreting the

observations

.

The main difficulty in that problem has often

been underlined by Thomas and Jefferies and by

others: to solve this analytical problem, one has

to go first through a preliminary and unavoidable

synthetic approach; namely, the theoretical computation of the radiation emitted by a gas specified

by the distribution of density, kinetic temperature,

and chemical composition. So one needs a good

theory of line and continuum formation.

Moreover, and this distinguishes the chromospheric and coronal situation from the classical

photospheric case, the equation of energy conservation includes a mechanical contribution in addition

to the radiative one. The energy is transported

and dissipated under several qualitatively well243
known forms, and eventually the whole heated material

of the corona expands hydrodynamically , as first

suggested by Parker (1963) . Delache has shown (1967)

that the equation of momentum conservation must include a wave-radiation pressure term, which at least

in the transition region between the solar chromosphere and corona prevents the hydrostatic equation

from being a good approximation. Therefore the a_ priori models that are built as a preliminary step

to compute, even by a good theory, the emergent

spectrum from a chromosphere or corona are not com- plete .It makes no sense to infer the physical con- ditions in the chromospheric or coronal gas from the

spectral information unless one checks that the con- servation laws are not violated at each height.

Omitting this check would imply that the transport

and dissipation phenomena are completely known and

that one knows where the mechanical energy is generated (Lighthill 1967) , which is far from true.

Thus the conservation laws can at best be used to

put boundary conditions on the run of Te and Ne . Finally, the very few models that have been

computed in the stellar case are stationary and assume that the material is homogeneous . The last

assumption is known to be far from reality in the

solar case.

b. Uniqueness of the Models

Due to these theoretical limitations and to the

scarcity of easily accessible spectral features

(i.e., in the near UV or visible part of the spectrum) in stars other than the sun, one cannot be

sure, at the present time of being able to derive

unique interpretations from the chromospheric and

coronal observations. Each observed characteristic

in the spectrum, especially in the line spectrum,

depends on the model via several parameters, which

are all depth dependent. As an example, let us quote the systematic study of the H and K lines by

Athay and Skumanich (196 8b) . For stars not too

different from the sun, they show the dependence

of the intensity I 2 and of the width w 2 of the K 2 emission with (1) the Doppler width and its gradient through the atmosphere, (2) parameters describing the run of Bv (Te ) , the Planck function, with x c , the continuous optical depth, and (3) characteristics

of the line such as e = C 21 /A 21 , the optical depth

x 0 at the line center, the damping constant a, each

244
depending on Ne , Te . We adopt

-(Ct ) X -(300 t ) 1/2

B (T ) = 1 + 3t + A e - a e

v e c

„

A W . + r -< d V

Z D = -r-^—7 r = 1 + C e AAd

(t c=i)

dT , r = = r x 1/3 + 10" 10 o dx

Q

0,1 o With several restrictions on a, rQ 2

, and C, Athay

and Skumanich give the following expressions for I 2 and W 2

:

T n Ae 1

2 (r Ca 2

) lA (1 +. c)

l/2 o / 1

w 3 ^i- -) 1/k

(1 + C)

o, 1

Of course, all these parameters cannot be left

free a priori if one has at one's disposal only the

observed profile of the K line, i.e., essentially 3 measurable quantities: I 2 , w 2 , and their product,

which varies like the energy loss in the chromosphere.

Therefore it is urgent both to search for more observable chromospheric and coronal spectral indicators and to improve the model theories.

II. THE KNOWN CHROMOSPHERIC AND CORONAL INDICATORS

Normal stars as well as variable stars show

strong observational evidence of the presence of

chromospheres and coronas. One often speaks of

chromospheric "activity," even for normal stars,

because of the variation in the structure of the H

and K lines from night to night.

245
I will list the spectral features that indicate

the presence of a chromosphere or a corona and in

which stars they appear; I will omit the peculiar

case of eclipsing systems, which are reviewed in

Groth 1 s paper

a. Line Spectrum

Prominent chromospheric stellar lines lie

in the visible, ultraviolet, and near infrared

spectral regions. A self-reversed emission core in

a strong resonance line is often a signature of a chromosphere because it reflects a map of the source function with depth, showing a maximum for small

values of t 5000 *. But the cores of all strong lines

are formed in the low part of the chromosphere and even in absorption they give insight to the properties of the corresponding layers

(1) H and K reversals are observed in numerous

G, K, M giants and supergiants and in many mainsequence stars (Wilson and Bappu 1957) . The K 2 intensity, I 2

, takes very different values for stars

of the same luminosity and spectral type. For mainsequence stars, the brighter the emission, the closer

the position of the star to the lower boundary of the

zero-age main sequence (Wilson and Skumanich 1964)

The width of the emission, w, is correlated with the

visual luminosity (Wilson-Bappu effect, w % Lv

1/6

) independently of I 2 and of the spectral type. This

relation does not apply to cepheids (Kraft 1957) nor

to T Tauri stars (Kuhi 196 5)

For stars of earlier spectral type than the sun,

the spectral type in which the H-K emission ceases

seems to be a matter of dispersion of the spectrograms, as long as the photospheric brightness is not

too high. With a dispersion of 10 A/mm, no emission

is found for b-y < 0.30 (type F5) , but weak emission

appears on high dispersion spectra of o a CMi , F5 IV

(Kraft and Edmonds 1959, 3. 2 and 4.8 A/mm) a Car

F0 lb (Warner 1966, 6.8 A/mm and y Vir B, F0 V

(Warner 1968, 4.7 A/mm).

(2) Traces of variable chromospheric activity

have been searched for in 139 stars by Wilson (1968)

,

who observed the flux at the center of the H and K

lines with a two-channel photometer over one year.

The results are not clearly in favour of a variability. Deutsch (1967) found large changes of K 2

* See R. N. Thomas, pp. 38-45.

246
emission in two K giants: a Tau (K5 III) and y

I Aql (K3 II)

.

H-K emission is the only chromospheric emission

feature observable in the visible spectrum of a great

majority of the stars studied by Wilson. In T Tauri

stars, however, the near infrared triplet of Ca II

| seems to be characteristic (Herbig, quoted by

McConnell, 1967) . Due to the presence of many other

emission lines in the spectrum of T Tauri stars, to

their irregular variability, and to the peculiarities

of their UV spectrum, one cannot oafely compare their

I chromospheric problem to that of normal stars, even

if one can be sure that there is one. Ha is often observed in absorption in stars with

H and K reversals. Suspecting that its central part

is formed in the same region as the K 2 emission,

Kraft, Preston, and Wolff (1964) tried to correlate

the width of the core with the luminosity of stars.

The relation between this width and the absolute

ultraviolet magnitude is not as good as the relation

discovered by Wilson and Bappu.

He I X 10830 was first observed in emission in

P Cyg and in carbon Wolf-Rayet stars (Miller 1954^,

with IZ emulsion and a very low dispersion (1300 A/mm

at lu) ; then Kuhi (1966) made photoelectric scans around the helium line in Wolf-Rayet stars, and

observed it in emission in all Wolf-Rayet stars. The

development of image tubes and the Lallemand camera now allows one to observe the helium line in absorption and in late type stars. Vaughan and Zirin (19 6 (1968) looked for this line at 8.4 A/mm in 86 stars,

the majority being of G and K type. About 30 of them

show the He line in absorption; in 5 others the line

appears in emission. Among the 12 B, A, and F stars

of the sample, only 3 Ori (B8 la) and a CMi (F5 IV)

show the line, in absorption. For 2 stars an activity is detected by variations of the helium line with

time. In several cases, the line shows a structure.

Although it does not constitute a simple case

of a star with a chromosphere, 3 Lyr presents a broad

emission feature at A 10830 (observation by Knappenberger and Fredrick, 196 8, with a mica window image

tube, 58 A/mm.

In the ultraviolet spectrum of stars, the reso- nance doublet of Mg II at 2800 & has been reported

(Heinze et al. 1967) in absorption in the spectrum

of aCMa. One would think it would be observed, with

an emission, in all stars where H and K emission

exists, since in the sun the emission in Mg II is

stronger than that in Ca II (Dumont 1967, Kandel

1967, Athay and Skumanich 1968a).

247
Ly a observed in six Orion hot stars by Morton

et al. (1968) is probably of interstellar origin

only. The same difficulty in separating the stellar

Lyman lines from the interstellar ones is quoted by

Smith (1969) in a paper on rocket observations of a Vir Bl V)

Anway, it seems preferable to search for Ly a in late type stars, where the maximum of the photospheric flux is very far from the spectral range of

the Lyman lines and where ultraviolet emission, if

it exists at all, can only be of chromospheric

origin. Predictions have been made by Oster and

Patterson (196 8) , who conclude that chromospheric

Ly a could be detectable with a 10 cm reflector for

very near cold stars (distance less than 3 parsec)

Coronal line observations in stellar spectra

are rare. Coronal lines were discovered by Adams and

Joy (1933) in RS Oph, a repeating nova, but were not

identified until 1945. Other repeating novae show

lines of highly ionized atoms: T Pyx, T CrB. For

this last star, Bachonko and Malville (1968) measured

the equivalent widths of the following lines: A 5303

[Fe XIV] ; A 6374 [Fe Xj ; A 5536 [A X] , on spectra

obtained during the 1946 outburst of the star, and

analyzed the green coronal line, from which they

deduced a relation between the radius of the corona

and the density necessary to produce the observable

green line: Ne = 3-5 • 10 cm" 3

, Te is assumed to be

10 6 °K.

Other coronal lines have been observed in CI Cyg

and AG Peg (lines of [Fe X]), in RX Pup (lines of

[Ca VII]), in Z And (lines of [Fe VII]). Sahade

(1960) has discussed these peculiar emission-line

stars, and further work is due, among others, to

Bloch (196 4) , Boyarchuk (1966) , and Boyarchuk et al.

(1964, 1967)

Coronal ions could be observed in ultraviolet

spectra, but severe limitations exist. (1) Below

912 A, all stellar flux is absorbed by interstellar

hydrogen even for the nearest stars. (2) The total

light flux emitted by the solar corona is several

powers of ten lower than the visible flux. A stellar corona would have to be much more powerful than the

solar one to be detected with the availaole space

equipment, except for very hot stars.

b. Continuous Spectrum

For the same reason as in the sun, it is possible to see the low chromospheric layers of a star by

observation of the continuous spectrum in the ultra248
violet and infrared regions, where = 1 corresponds

to very small values of T5000 (Noyes et al. 1966).

But the lack of angular resolution in the observations is a very strong limitation, and the derivation

of semi-empirical models by inversion of limb-darkening curves is excluded thus far.

Nevertheless, absolute intensities in the ultraviolet spectrum below 2000 A, especially in A and F

stars where the opacity due to metals (Mg, Si, Al)

and to carbon dominates that of hydrogen (H I and H ) and is very strong, will help to check the validity

of theoretical model atmospheres in the very superficial layers. But it seems impossible to expect

that these observations will be of as much help in

determining stellar chromospheric models as those

coming from the study of the cores of strong lines.

Towards the longer wavelengths, observation of

an intense infrared emission near lOy has been re- ported first in 1965 for 3 late-type stars: a Ori

,

a Tau, and y Cep (Low 1965). Gillett et al. (1968)

have confirmed this feature for 4 cool stars (a Ori,

y Cep, 0 Cet, x Cyg) / hut nothing appears for a hot

star like a CMa; they suggest two interpretations,

one implying a chromospheric temperature, the other

an emission by circumstellar matter around the star.

Other authors now favour the last item, on the basis

of the general appearance of such circumstellar

envelopes around several types of cool stars (Stein

et al. 1969) , and of the resemblance of the emission

to that of solid particles (see Wolf and Ney, 1969)

.

The evidence of a stellar wind in a Ori (Deutsch

1959, Weymann 1962) led Weymann and Chapman (1965) to

compute the theoretically predicted free-free emission of a hydrodynamic hot flow emitted by a Ori.

They concluded that between 1 mm and 3 cm the absolute flux is just at the limit of detectability for

radio astronomers. Kellermann and Pauliny-Toth

(1966) searched for that emission at X = 1.9 cm, but

could not detect it.

All in all, there is very little indirect evidence for chromospheres and coronas in the stellar

case. The only exception consists of the K-line

emission, which has been detected for more than a hundred stars

.

III. WHAT IS INFERRED FROM THE OBSERVATIONS?

The observed quantities are essentially (1)

relative continuous intensities, and (2) line para249
meters: central intensity I Q , emission width, halfwidth AA 1/2 , equivalent width W, and in rare cases,

structure of the whole profile.

Due to the significant results obtained in re- cent years on the solution of the transfer problem

(Cuny, Dumont) which allow one to consider the

"synthetic problem" in line formation to be fairly

well solved, I will focus my attention on three main

fields where physical information has been gained

from stellar chromospheric observations.

a. Construction of Models

(i.e. Jefferies' "analytical problem")

Given my restricted definition of stellar chromospheres , I will review only the work done by Kandel

on late-type emission dwarfs. Groth will report on chromospheres in K giants belonging to eclipsing

binaries. In the first case, deduction of the model

from the observations is mainly based on the Ca II K

line. Other chromospheric indicators have not yet

been intensively used for the purpose of models; we will consider them later.

(1) Kandel (1967) constructed chromospheric

models to interpret the strong K line emission observed in K and M dwarfs that showed no Ha emission.

He specifies the chromosphere by a temperature profile Te

(N) , electron temperature versus the number

of hydrogen nuclei, N, in a column of 1 cm2 above a reference level corresponding to the surface of a photospheric model; the parameters are the temperature Tc and the mass Nc of a large plateau, and the

gradient C = -d (log Tc ) /d (log N) of the temperature

law just under the plateau. Other basic assumptions

in Kandel 's computation are hydrostatic equilibrium

and LTE ionization balance. They are questionable,

but are adopted for simplicity.

A series of such a priori chromospheric models

was put at the top of convenient photospheric models

that were built to reproduce the visible absorption

spectrum, Kandel computed the Ca II emission equivalent width, Efc, by using the non-LTE theory of

Dumont (1967) and traced iso-E curves in a (Nc , Tc ) diagram. Computing then the radiative energy loss

by these theoretical chromospheres, Kandel eliminated

those models that are thermally unstable and those

that should produce emission in Ha. A very limited

region remains acceptable, with Tc

< 10,000°K.

Despite its evident limitations, Kandel'

approach is a physically well-grounded one. One can

250
suspect his photospheric models, as he does himself,

i because of the importance of convection; the mechan-

! ical flux he pumps in the convection zone to heat the

chromosphere is as uncertain as the application of

Lighthill's theory to stars. But, given the present

set of developed theories, he tries to derive the

; most complete model. If one remains unsatisfied by

I all he approximations involved in such an exercise,

one 1 Jst return to the problem of improving the

energy generation and transport theories.

( 2) Rough indications about chromospheric con-

| ditions have been derived from other lines. The He

I A 10830 line has been shown by Athay (1963) to be

very sensitive to the temperature between 10 h and

i 2*\§ h °K. When it is observed, the He I absorption

line indicates a hot chromosphere or more explicitly,

a chromosphere hotter than that of stars with only

| K reversal (T > 20, 000°K) . Let us recall that

X 10830 appears in emission in three supergiants, including e Ori (BO la) (Vaughan and Zirin 1968) ; its

presence in absorption in 3 Ori might be used as a test of Underhill's hypothesis on helium overabundance in B supergiants if the non-LTE problem is

solved uniquely.

The Ha core has been suggested by Cuny (1968)

to be a test for low chromospheric solar models. She

computed the non-LTE Ha profile with two solar models, HAOl and the interspicular model of Athay and

Thomas. With the latter the residual intensity is

larger than observed; with the former, smaller.

Collisions have a greater contribution to the source function in the interspicular model, which is denser,

so she proposed that the central part of the Ha line

could be used, by comparison with observations at

high spectral resolution, at least to check plausible

chromospheric models, if not to establish them.

In a rather similar way, Kandel has used the

absence of Ha emission in 61 Cyg B to limit the

possible range of his chromospheric parameters for

that star.

The Ly a and Ly $ lines, if they can be ever observed in cool stars, will be an excellent tool

for studying stellar chromospheres.

(3) Purely theoretical non-grey, radiativeequilibrium atmosphere-models were computed by

Feautrier (1968). Taking account of"H~ departures

from LTE, Feautrier 1 s models present a rise of temperature above the purely photospheric layers; these

models constitute a good set of a priori models to

be used in studying chromospheric conditions from

spectral features in stars hotter than the sun.

251
b. Velocity Fields

Although this subject is not completely separate

from the model problem, because of coupling between

dynamical phenomena, transfer, and physical structure

of the atmosphere, one can only be very short and

restrict one's attention to qualitative or first

approximation theory results. The whole theoretical

problem lies indeed in a premature state. Velocity

fields are however deduced from two types of observations .( 1 ) Line displacements. Stellar chromospheric

lines often show an absorption component displaced

towards the violet, the emission component being

shifted longward. The case of the K line in cepheids

is typical, but the phenomenon is more general. For

instance, He I A10830 in the most luminous stars

observed by Vaughan and Zirin presents the same feature. P Cygni type profiles are observed also in

extended atmospheres which may not be chromospheres

in the restricted sense given above. In the expanding envelopes discovered by Morton (1967) for 3 supergiants, the absorption components of the strong

resonance lines of Si IV, C IV, N V, and Si III indicate velocities in excess of the estimated escape

velocity. A P Cygni profile is characteristic of

the He I A10830 line in 3 Lyr (Knappenberger and

Fredrick 1968)

(2) Line asymmetries. As in the sun, where both

the Ca II and Mg II resonance lines show asymmetry

in the reversed part of the core, large asymmetries

are observed in stellar K 2 components (Wilson and

Bappu 1957, Deutsch 1969), which vary with time, so that it seems difficult to think of any interpretation of the K-line profile that would ignore motions

and inhomogeneities in the atmosphere of those

stars.* The most prominent feature of chromospheric

line profiles is broadness, so that their halfwidth

cannot be interpreted as being due to a purely thermal Doppler effect. Large turbulent motions are present in chromospheric layers. Several authors

have argued that these motions are macroturbulent

ones, arising from large mass motions. Moreover,

there seems to be a correlation between K-line emission and photospheric turbulence: (1) In cepheids,

Kraft (1967) reported that the turbulence increases

after minimum light, at the same time as the K emis-

* I am indebted to Dr. A. J. Deutsch for having

stressed this point to me.

252
sion appears. (2) Bonsack and Culver (1965), on the

other hand, have produced evidence that the K-line

emission width w is correlated with the halfwidth

AA1/2 of photospheric V I lines in a sample of G,

K, and M stars. The correlation being less clear

between w and the curve-of-growth velocity, they

concluded that macroturbulent motions exist in the

photosphere of the stars studied, and stressed a possible common origin of the K reversal width and

broadening of photospheric lines.

o. Activity

One gets a very qualitative insight to chromospheric activity from the following facts:

(1) Emission in the K line has been found to be

variable in cool giants, first by Griffin (1963) and

later by Deutsch (1967) and Liller (1968).

(2) The K-line reversal seems to be associated

with magnetic field in the hottest star in which it

has been found (Warner 1968)

.

(3) The A10830 He I line shows intensity variations which recall the increase of intensity of that

line in solar plages and prominences.

Wilson (1963) suggested that the existence of a chromosphere and the strength of the K-line emission

are correlated with the presence of magnetic activity

and that they decline with time as do the magnetic

fields

.

A very special case is that of the transitory

character of the K-line emission in cepheid atmospheres; the appearance and disappearance of the

emission is connected with the phase of the pulsation, and may be interpreted by periodic modifications of the excitation in the atmosphere, which are probably not of the same nature as those which

appear, periodically or not, in the above quoted

examples. Indeed, the change in excitation is generally associated with the passage of a shock wave

through the atmosphere

.

To end this paragraph, I would only mention the

problem of correlation of chromospheric properties

with general properties of the stars. The fundamental explanation of the Wilson-Bappu effect seems

to be still unknown; on the other hand, stars like

the cepheids or T Tauri stars do not obey the WilsonBappu relation, supporting the idea that the K emission may have a different origin in stars where the

chromosphere is heated by different mechanisms. If one tentatively accepts the idea that the type of

253
variability that gives rise to light variation in

cepheids or in T Tauri stars, and that probably

differs between both types of stars, generates mechanical energy in the outer layers of these stars, one is prepared to admit that the Wilson-Bappu effect

is relevant only to stars in a certain stage of their

evolution, which stage is characterized by a certain

kind of heating. As pointed out by Athay and

Skumanich (1968), "W 2 reflects on the nature of the

energy conversion mechanism. " Of course the interpretation of chromospheric observations in stars can

be considered as being directed towards discovering

the mechanical energies at work in a star.

IV. DO A STARS HAVE A CHROMOSPHERE?

In A stars, no chromosphere has been identified

with confidence; on theoretical grounds, it is usu- ally thought that since ionization convection zones become thin for stars hotter than spectral Fo- F5, no strong acoustic energy may be generated. Thus

the conditions for a strong chromosphere are not

present. But Wilson (1966) has claimed that he sees no good reason why chromospheres should cease towards

the uper edge of main-sequence stars. Leaving for

the moment the question of B stars, and by close

continuity with the observations already available,

I will give some arguments in favour of chromospheres

in A stars.

(1) There are favorable observational indices

for chromospheres in these stars, although at this

time they do not constitute indicators:

(a) K reversal in y Vir B (FO V) , which is a magnetic star. Many A stars are magnetic

stars

(b) Variability in the K line. Henry (1967)

observed 4 Ap stars already known to be

variable in the K line, confirmed their

variability, and discovered 4 others.

Baglin et al. (1968) discovered a large

variation of that type in y Boo (A7 III),

which is a 6 Scuti variable. The variation

seems to have nothing to do with the light curve period. This is under further study.

(c) The A stars show a large microturbulent

parameter (6.9 km/s) and this feature is

not limited to Am stars, as was established

by Baschek and Reimers (1969)

254
(2) What else should one look at to detect chromospheres in A stars?

(a) The core of the K and of the Ha line. If

the K line, collision dominated, presents

even a small reversal, this will imply a chromospheric temperature gradient. The

Lallemand camera is the best detector to use

for such a study of the cores of very strong

lines because of its lack of threshold at

low fluxes and the linearity of its response . (b) 0 1, C I, Si II, Al II, Mg II resonance

lines in the ultraviolet spectrum.

(c) There is almost no chance of detecting Ly a, even if its emission in an A star would be

the same fraction as in the sun of the

acoustic flux heating the chromosphere.

An estimate made for a typical A star with

Te ff = 8000°, log g = 3.9 indicates a flux at the

ground of 5. 10" 4 photons cm_2 s~

1

, to compare with the

value of 6 for the sun.

(3) How could one heat a chromosphere in A type

stars?

A model study of an Am star (Praderie 1967)

shows that if the observed microturbulent parameter

is not due to other physical effects (such as NLTE

effects of the source function for lines on the

plateau of the curve of growth) , the atmosphere must

be very convective to produce the value of derived

from the curve of growth. But an adequate convective

model fails to reproduce the wings of strong lines,

which, for A stars, are formed in the convectively

unstable layers of the photosphere.

With the help of Lighthill's theory of generation of acoustic noise from convective turbulence,

and a convection efficiency compatible with line-wing

observations, the acoustic flux produced in the con- vection zone may be estimated. For a star of Te ff = 8000°K, log g = 3.9, it turns out to be one-tenth of

the solar value (assumed' to be 10 8 ergs cm" 2 s _1

)

.

Thus it seems doubtful that one could heat, in that

star, a chromosphere as hot as the solar chromosphere . But the problem of the high £t parameter remains. The acoustic flux produced in the convection

can be increased by enhancing the helium abundance

and then the depth of the He I and He II parts of

the ionization convection zone (Mariai 1969) . However, in Am stars, no He I line is observable. In some silicon Ap stars, work by Searle and Sargent

(1964) and by Hyland (1967) gave observational

255
evidence for helium depletion in the atmosphere. So no conclusions can be drawn about helium abundance;

but significant progress will arise from the search

for He I X10830 in Ap stars with strong magnetic

fields, both to decide if a temperature of 20 / 000°K

can be obtained in a possible chromosphere for these

stars and to see if the helium abundance distribution

in these stars is uniform with depth. In some of

those stars, Zirin (1968) has already found evidence

of the presence of He 3

. To conclude, I stress that several factors that

might have a strong relation to the presence of chromospheres in stars have not been reviewed here. The

role of a companion in enhancing the chromospheric

spectral features has often been mentioned. The

problem of generation of stellar winds was out of

the scope of the colloquium, but its importance

should not be minimized.

ACKNOWLEDGMENTS

I am indebted to R. N. Thomas and J. C. Pecker

for useful discussions, and to A. J. Deutsch and

L. V. Kuhi who made me aware of several references

that I would have missed otherwise.

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Wilson, 0. C, and Bappu, M. K. V. 1957 , Ap. J. 125 , 661.

Wilson, O. C. 1963, Ap. J. 138^, 832.

Wilson, O. C, and Skumanich, A. 1964 , Ap. J. 140 , 1401.

Wilson, O. C. 1966, Ap. J. 144 , 695.

Wilson, O. C. 1968, Ap. J. 153 , 221.

Woolf, N. J., and Ney, E. P. 1969, Ap. J. 155 , L181

Zirin, H. 1968, Ap. J. 152, L177.

258
WHAT SHOULD WE DO TO KNOW MORE ABOUT CHROMOSPHERES

AND CORONAE OF STARS?

by

Richard N. Thomas

Joint Institute for Laboratory Astrophysics*

Boulder 3 Colorado

ABSTRACT

Chromospheres-coronas satisfy the last two of

the proposed classification schemes: inadequacy of

the classical atmosphere (CA) model to represent

observations and a priori rejection of the CA model.

So we survey the question of what is required for more knowledge from the standpoint of asking what

conceptual modifications will increase knowledge

and what new observations are required. We stress

that continued progress requires a continual interchange of ideas between the solar situation and

the range of stellar situations.

Key words: chromospheres, coronas, classical atmosphere model.

The general subject of today's discussion is:

"What do we know about chromospheres and coronae

of stars?" Thus, as Francoise Praderie emphasized

in the preceding summary paper, we restrict attention

today to a particular kind of extended stellar atmosphere, one which can fall. into either classification

3 or 4 of the alternatives I proposed the first day

of this conference. I would like to emphasize to

you that we are, at least implicitly, making a much

stronger restriction of the kind of stellar configuration to be discussed than we would have, with

the same literal title, in a discussion of extended

stellar atmospheres 25, or even 15, years ago. The

* Of the National Bureau of Standards and the

University of Colorado.

259
fact that it is a stronger restriction means, of

course, that we think we know more today about

chromospheres and coronae than we did 25, or even

15, years ago. I think we all generally understand

that by "chromosphere" and "corona" in today's thinking, we mean an extended stellar atmosphere that

arises as a consequence of a departure from strictly

radiative or convective equilibrium because of a local dissipation of mechanical energy. In the

chromosphere and corona, we have a reversal of the

photospheric temperature gradient: dTe/dh > 0.

Overall, the evolution of thought during the last 25

years has centered on how deep in the atmosphere

this region dTe/dh > 0 can be pushed, and how many

phenomena arise because of this outward rise in Te . In this symposium where we ask what the problems of

extended stellar atmospheres are, the important thing

is not just the fact that we know more about, this

particular kind of extended stellar atmosphere, but

also the way in which that knowledge has evolved.

The path has not been simply one of acquiring new

data, but also one of more critical analysis of old

data and old concepts. So as we explore my topic

today—What should we do to know more about chromospheres and coronae? —we might follow the same

path: examining concepts as well as data.

Even 40 years ago, most astronomers would have

said that stellar chromospheres and coronae satisfied

classification 3 of extended stellar atmospheres--

inadequacy of the CA model to represent observations —even though neither observations nor the idea

of the CA model were as clear as they are today.

Twenty-five or even 15 years ago, most astronomers

would have admitted that a stellar chromosphere

probably satisfied the category 4 criterion: a priori rejection of the CA model. But the conceptual

basis for rejecting the CA model 40 and 25 years ago

differed considerably from that which has evolved

over the last 25 years. At that time, the admission

of inadequacy rested upon the ad hoc construction of

a phenomenon outside either our laboratory experience

or conceptual understanding—the "turbulent" models

of 40, 25, 15 and even more recent years ago—and the

inability to see how it could arise from the CA model

because we didn't understand what it was. So the

chromosphere probably departed from the CA conditions, but probably because HE was invalid, while RE

and LTE probably remained valid. Today, however, we profess to a very explicit physical understanding of

why the atmosphere is extended, relative to the

physical picture underlying the CA model. We have

260
simply dropped the notions that either RE or CE

i provide all the energy transport in the atmosphere,

! and that LTE under the RE distribution of Te de-

; scribes the state of the gas. We have admitted the

possibility that various kinds of instabilities

arising even in the CA model may amplify, producing

a local dissipation of mechanical energy, and that

! excitation conditions may differ from the CA strictures. By loosening the strictures placed on the

diagnostic framework, we permit a tightening on the

physical picture of the model.

The dropping of LTE—which has loomed large

,

sometimes almost irrelevantly so—in the discussions

this week has a different kind of significance than

that of RE. The RE, HE questions relate to the aero- dynamical phenomena and give us the extended or non- extended configuration of the atmosphere; they tell

' us how the configuration occurs. Dropping LTE simply

lets us state, unbiasedly and correctly, what the

configuration and state of the gas is.

So we restrict attention to a kind of extended

stellar atmosphere, whose general physical cause we think we understand. But when we ask what we know

about chromospheres and coronae, it is the details

of the phenomenon and the variation in details from

spectral type to spectral type that we must investigate. In this connection, I have stressed the

points in the preceding paragraphs for two reasons.

First, we have reached the stage of being willing to accept the above implications on what we mean

by stellar chromospheres and coronae through a com- bination of two kinds of investigation: on the one

hand, an analysis of physical self-consistency of

atmospheric model and of spectroscopic diagnostics;

on the other hand, an application of this analysis

to solar observations and models. To this, we have

added also observationally-less-detailed , exploratory

investigations of other stellar atmospheres. But

by and large, the situation is as I said: a compound

of general theory and solar interpretive-observational work. Then, we have generalized and extrapolated to various stellar cases —testing, substantiating, modifying as we have been able.

Second, Francoise Praderie has given us a very

complete summary of what we know about chromospheres

and coronae of stars. In addition, in the process

of asking what the spectral indicators of the presence of chromospheres and coronae are, she has gone

a long way toward answering the question of what we must do to learn more: How do we extend the observation of these spectral indicators. It would be

261
pointless for me, here, to follow her discussion by

simply commenting on the various indicators she has

summarized

.

So, in considering this path of evolution of our knowledge, to answer the question of what we must do

to learn more about chromospheres and coronae , I lean

heavily on the evolution of thinking about the solar

case. And, in the context of the specific features

discussed by Francoise, we continue this extrapolation-generalization from the solar case.

I would stress that we must adopt a conceptual

generalization from solar considerations, not nec- essarily a literal one. In such a literal approach,

we would assume that all chromospheres must originate

from the detailed causes underlying the solar chromosphere, and must have the detailed structure of

the solar chromosphere. Such a literal approach, for

example, is found in the 1955 IAU Jo-int Va,A culAAZo n on Tu/ibul£nc& In Stttlan. ktmoi, pk&si.e,A , held at Dublin. 3 On the other hand, my own thinking has always cen- tered around the notion that the common feature of

chromospheres and coronae is simply a mechanical

energy supply, which may arise in many different ways

in many different spectral classes. I have repeatedly cited the sun and the Wolf-Rayet stars as probably lying near the two extremes of the kind of

steady-state atmospheres produced by aerodynamical

effects. In the sun, we consider only departures

from RE; in the WR stars, departures from both RE

and He. I consider to be most misleading the kinds

of arguments cited that expect chromospheres-coronae

only in stars of spectral class FO and later, based

on the notion of acoustic waves from an atmospheric

convection zone as the source of chromospherescoronae. Such arguments confuse the existence of sources of mechanical instability with the details

of one such source. I have stressed this point in

the first paper of the series: Superthermic Phenomena in Stellar Atmospheres (Thomas 19 48) and in the

introduction to our survey of the physics underlying

the solar chromosphere (Thomas and Athay 1961)

.

Parker makes the same point in the concluding chapter

of his monograph on the solar wind (Parker 1963)

.

The two symposia on cosmical gas dynamics devoted

to aerodynamical phenomena in stellar atmospheres

developed this outlook (Thomas 1961, 1967). The

symposium on Wolf-Rayet stars held in 196 8 brought

together much material that strengthened the case

for considering the line-forming region of the WR

atmosphere simply as an extended chromosphere (Gebbie

and Thomas 196 8) . So what I would stress to you

262
here, when we ask what we should do to know more about stellar chromospheres and coronae , is that it

is not only a matter of getting more and better

observations of known indicators. It is equally

. a matter of becoming very clear about the kind of

evolution in our thinking that led us to focus on the true nature of the chromosphere-coronal phenomj enon, hence the kind of parameters that might tell

1 us more about them.

Tables I and II summarize the evolution of our

I thinking on the solar chromosphere-corona. Table I

J concentrates on the "classical" problem, referring

to the concepts of 40 and 25 years ago that I stressed above. Points 1 and 2 of Table II [corresponding

to points 1 and 2 of Table I] refer to the resolution

of these problems.

The dilemma and resolution summarized in the

tables in points 1 and 2 require little discussion.

In essence, the "classical" dilemma of the extended

I solar atmosphere comes from looking at an apparent

| gross anomaly—the atmospheric extent—which is

significant enough to vitiate the CA model, through

a set of glasses tinted with the CA predictions:

| the excitation state of the outer atmosphere must

not exceed that of the photosphere, and the outer

atmosphere must be so thin as to be transparent,

certainly in its effect on the disk spectrum,

likely in considering the eclipse line spectrum.

Given these observations, and given these strictures

on interpretation, the only resolution was to "invent" a new kind of phenomenon, "astronomical turbulence," whose physical properties were constructed

explicitly to (1) satisfy the observations as interpreted within the diagnostic strictures, and (2) beg

the question of physical consistency by postulating

unknown physical interactions. (1) refers to the

requirement that emission gradient be interpreted as density gradient. (2) refers to the requirement that

"astronomical turbulence," with characteristic random

velocity exceeding the thermal velocity of the medium, somehow have any mechanical energy dissipation

suppressed. The "secondary" classical dilemma, that

associated with the symbiotic behavior of excitation

state, was relegated to secondary status mainly because we retained the CA-tinted glasses, even in

studying this non-CA phenomenon. The final resolution of these two aspects of the dilemma came when

the diagnostic approach was freed from all a priori

imposition of the strictures of the CA model, under

the principle that no one of the CA assumptions

should be more sacred than any other, when it became

263
TABLE I, SCHEMATIC EVOLUTION

How Astronomers Viewed the Solar

Original Observational Original

Problem Interpretation

1. Anomalous extent "Turbulent" momentum

of atmosphere: support.

a * Apparent scale-height No energy dissipation

of order 1000 km. by such turbulence

(i) Differential scaleheights from

different lines,

and continua

b. Isothermal scaleheight of order

100 km.

2. Symbiotic excitation

phenomena

a. Presence of He I,

He II

UV excess in sun

b. Tex as low as 3000°

for some metals

c. Excitation increases

outward

Line-blanketing

depression of

T [boundary]

(i) Spectral lines of

"coronium"

necessary to relax any one of them. Lifting the

restriction on excitation state and opacity of the

atmosphere permitted the density gradient to depart

from the emission gradient, and a distribution of

Te to be sought which made the observations coherent.

It is interesting to note that the path of resolution

was not really so logical as stated, but came as a shocked necessity to permit dTe/dh > O from the

264
OF THINKING ON SOLAR CHROMOSPHERE

Chromospheric Problem 25 Years Ago

Original Physical

Problem Perturbing Questions

Physical model of How to tell T , T, such "turbulence" from V.turb

How to avoid mechanical

energy dissipation,

hence rise in T, , T

Origin of such excess Self-consistent

calculation

Why should excitation

increase outward in

such a model?

Why such different

values from different

lines?

identification of the coronal lines. But it took

some time to admit this situation to the region

where the dilemma was quantitatively the most

evident—the chromosphere —and to recognize that

these regions could have an effect on the disk

spectrum.

Points 3-5 of Table II refer to observational

details that arose later, somewhat after the classi265
TABLE II, SCHEMATIC EVOLUTION OF

Change in "Observational" View

Original Observational

Problem Refined Observation

Anomalous extent

of atmosphere

:

"True" scale-heights

differ from observed

a. Apparent scale-height

of order 1000 km.

(i) Differential scaleheights from

different lines,

and continua

b. Isothermal scaleheight of order

100 km.

2. Symbiotic excitation

phenomena

Differs from line to

line

Atmosphere not isothermal, but HE not

bad approximation up

to 1000 km. Any

turbulence definitely

subsonic

Symbiotic appearance

exists

a. Presence of He I,

He II

b - Tex as low as 3000

. for some metals

Same

Same

c. Excitation increases

outward

(i) Spectral lines of

"coronium"

3. Emission lines in

rocket UV, on disk

Same; identification ol "coronium" lines

Same form as Ca+ cores

4. Correlation between

H and K lines and

magnetic field

Non-spherical ly

symmetric emission

Profile changes over

disk

Same

266
THINKING ON SOLAR CHROMOSPHERE

of Solar Chromosphere Today

Physical Interpretation Questions Remaining

Compound of outward in- crease in excitation,

coming from dTe/dh > 0 , and

of self-absorption, non-LTE

Above

Refined observation

still needed

Mechanical energy supply

dTe/dh > 0 Non-LTE ionization equilibrium

Source, and interpre

tation, of any turbulence

Compound of dTe/dh > 0 and non-LTE coming from

boundary

dTe/dh > 0; collisions + radiative excitation

Non-LTE coming from

boundary

Line-blanketing effect

small on non-LTE theory

Consequence of dT

e/dh > 0

Better observations

More detailed non-LTE

calculations for

complex ions

Influence of inhomogeneous atmosphere

Detailed calculation

More refined observation

Detailed model

Consequence of dTe/dh > 0 Detailed calculation

Questions of linebroadening

Non-uniform mechanical

dissipation under influ- ence of magnetic field

Detailed mechanism

and model

Inhomogeneous mechanical

heating; inhomogeneous

mechanical structure

Detailed observation

Detailed model

267
cal dilemma represented by points 1 and 2 was well

on the way toward resolution. But had the observational material underlying these points 3-5 been

available 25, possibly even 15 years ago, we must

be honest and realistic enough to admit that the same CA-tinted glasses would probably have inhibited

their correct interpretation. This suggests that a useful way to compare the path of evolution of

thinking on the solar chromosphere-corona to that

on the stellar, would be to compare the full range

of "anomalous" phenomena for each, using always as reference the CA model. After all, when we discuss

"extended stellar atmospheres," we do so relative to

the "non-extended" CA model atmosphere.

If we adopt this approach, we recognize that we

can group the apparent anomalies under three headings: "apparent" macroscopic structural anomalies;

symbiotic excitation anomalies, usually microscopic,

at least in implication; and basic conceptual anomalies. Consider these, specifically.

THE OUTER SOLAR ATMOSPHERE

A. "Apparent " Structural Anomalies

1. Overall extent of the atmosphere:

Emission scale-heights large relative to the

scale-height of an isothermal atmosphere,

using a temperature derived from either a CA theory or a CA-interpreted observation.

Differing emission scale-heights from different lines and differing elements.

2. Gross structural inhomogeneities

:

Mainly apparent in those observations capable

of resolving small fractions of the solar

disk.

3. Differential brightness correlated with differential values of local magnetic fields:

Again, mainly apparent in those observations

capable of resolving small fractions of the

solar disk.

4. Differential macroscopic velocity fields:

Two kinds of measures exist; those depending

on resolving small fractions of the solar

disk, and those not so depending,

a. Non-dependent on disk resolution:

Inferences of differential macroscopic

fields from conclusions on line-broadening

268
mechanisms; either for a line-profile, or

a total energy lying in the line.

Inferences based on asymmetries of line

profile, which are interpreted in terms of

differential motions,

b. Dependent on disk-resolution:

Indirect inferences in the Doppler cores of

spectral lines based on limb-darkening

data

.

Direct measures based on line-shifts on different parts of the disk.

I have grouped items 3 and 4 in this heading of

"structural" anomalies because of the evident temptation to associate structural behavior with kinematic behavior of the atmosphere, on the one hand;

and the problem of separating thermal motion from

other velocity fields, on the other hand. Uncer-

; tainty on thermal motions introduces uncertainty

in what the extent of the atmosphere should be, even in HE.

B. Symbiotic Excitation Anomalies

1. Low-excitation indicators:

a. The low residual intensities, or low values

of Texr in most strong lines of the disk

spectrum; values much below the CA-derived

boundary values for Te . b. The persistence to great heights of a significant population of the second quantum

level of hydrogen.

2. High-excitation indicators:

a. The presence of HE I 10830 in the disk

spectrum, and of lines of He I and He II

in the eclipse spectrum as low as 1300 km

above the limb

.

b. The intensity in the continuum, on the

disk, in the rocket UV and in the radio

regions

.

c. The presence of highly-ionized elements

in the eclipse spectrum, and in the rocket

UV spectrum of disk and limb.

3. Mixed: + The self-reversed emission cores of Ca and

Mg+ in the disk spectrum coupled with the

low values of T^x ; the absence of such

emission cores in hydrogen Balmer lines and

in other strong lines; self-reversed emission lines for hydrogen Lyman [and possibly

269
other strong] lines in rocket UV disk spectrum.

C. Basic Conceptual Anomalies

1. The presence of a wide variety of emission

lines in the solar rocket UV disk spectrum.

2. Evidence for a continuous outward mass flow

from the sun. REMARKS

I have deliberately omitted features of the

"non-quiet" sun, and associated high-energy phenomena, to remain in keeping with the "steady-state"

restriction of this meeting. Such an attitude may

well be as subject to the same criticism that has

been directed to the attitude that holds too strongly

to an attempt to retain the CA model, which I have

criticized here. But, I adopt it.

At first sight, some items in the categories

A-C might equally well, or even better, be placed

in another. For example: emission lines is placed

in C rather than in B; the magnetic-correlation item

(A3) is in A rather than in B. But consider the

logic of the categories.

In category A, I tried to group those features

bearing directly—either diagnostically or causally

with the atmospheric extent. Thus all material re- lating to velocity fields comes there, regardless of

whether the velocity field is implied to extend the

atmosphere by a momentum action, as in a flow field

or "turbulence," or whether it might be, or might

produce, a thermal field. I include item A3 here

because, although there have been suggestions in

the literature that the magnetic field somehow

directly modifies the source function, it seems much more likely that it affects the velocity fields

present—either macroscopic or thermal—and the

energy dissipation from them.

In category B, I tried to group those features

giving direct empirical evidence on the excitation

when properly intepreted, of course. The point of

using the term "symbiotic" is that it describes very

aptly what we have in such an atmosphere. There is

a competition between the processes controlled by

radiative effects —the photospheric radiation field

corresponding to RE and the radiative transfer effects of the boundary—and those processes introduced by the mechanical dissipation of energy.

270
In category C, I have placed those features

; which, when present in stellar spectra, immediately

I suggest the presence of chromospheres and coronae,

; simply by their presence. The inference is not com-

! pletely certain; some further criteria are required.

I But their presence stands as a stimulating and em- barrassing beacon into our inquiry as to the adequacy

| of the CA model for any particular star. Embarrassing, because on a literal CA model, they cannot

exist; stimulating, because of the attempts to see

i how little of the CA structure can be changed to

; accomodate them.

In a broad sense, we have now two questions to answer. (1) What can we say about the broad array

I of data, suggesting anomalies, that exists for stellar chromospheres-coronae? (2) In what ways are we trying to make these data more coherent for the

solar case, in order to make the solar picture more quantitatively detailed; and how is this applicable

to the stellar situation, again in the broad sense?

Consider these in turn; my remarks on the first need

only be a summary, drawing on the material already

presented by Francoise Praderie.

THE EXTENDED STELLAR ATMOSPHERE

A. "Apparent" Structural Anomalies

1. Overall extent of the atmosphere:

Apparent density gradient inferred from transmission characteristics of the eclipsing

atmosphere in binaries.

"Cool" atmospheres such as £ Aur, 31 Cyg.

"Hot" atmospheres such as V444 Cyg.

Differential size of continuum-emitting, and

line-emitting , atmosphere, from inter ferometric studies.

2. Macroscopic velocity fields:

a. Curve-of-growth studies:

•Large "turbulent" velocities, some superthermic, from data on supergiants.

Superthermic relative to a temperature

derived from CA-theory or CA-interpretation

.

"Turbulent" velocities from curve-ofgrowth eclipse studies.

b. Line-profile studies:

Large "turbulence" values necessary to

match line-profiles.

271
c. Evidence on systematic flow fields from

line-displacements, either from element to

element, or between emission and absorption components. Visual and rocket UV.

B. Symbiotic Excitation Anomalies

1. Low-excitation indicators:

a. Low residual intensities, or low values of

Tex , in strong spectral lines, relative

to Te inferred from continuum.

b. Apparent indications of anomalously high

number of second-quantum level population

of hydrogen.

2. High-excitation indicators:

a. Presence of X10830 of He I in cool stars.

b. UV and IR excesses in certain spectral

classes, especially in supergiants

.

c. The Russell-Adams effect.

d. A higher ionization and excitation level

in the line spectrum than in the continuum.

3. Mixed:

a. All the features of the "standard" symbiotic stars.

b. The self-reversed emission cores of Ca H

and K; in addition, there are other lines

that seem to show such self-reversal.

c. The appearance of He I emission lines in

the carbon, cool star R Cor Bor.

C. Basic Conceptual Anomalies

1. The presence of emission lines generally,

both in the visual and in the rocket UV.

2. Evidence, in some stars, of a continuous outward mass flow.

When we compare the two tabulations for the

solar and the stellar cases, we are struck by the

similarities of "anomalous phenomena." When we discuss interpretations, we are struck by the

repetition of the "inertia" encountered in the

early days of solar work now in the stellar case. Consider the kinds of attempts being made to clarify the above anomalies, with respect to these two

points of similarity. Again, I would stress that

all these considerations are based on similarity of

general problem—the evidence that a mechanical

dissipation of energy produces an outward rise in

272
Te —not on any required identity of details of

mechanical energy supply.

RESOLUTION AND INTERPRETATION OF ANOMALOUS FEATURES

1. Distribution of Te We have remarked, at the outset, the generally

accepted interpretation of a chromosphere-corona

today is an outer extended atmosphere arising because of an outward increase in Te , coming from a mechanical dissipation of energy. So, a primary

problem is to establish, for any given star, whether

Te does indeed increase outward, and the details of

its distribution.

a. Empirical

For the sun, we still do not have a complete

specification of the value of Te [min] and the details of its behavior in this region. A great deal

of effort is being devoted to the problem. Some

aspects can be duplicated in the stellar case. From the Continuum . In those spectral regions

where iv does not reach 1 near Te [min] , limb-darkening studies are required for precise work. Eclipse

studies are required, generally, to go far into the

atmosphere, except in those spectral regions of

great opacity, such as the rocket UV and the radio

region. For intermediate regions, where t v ^ 1 does

correspond to Te [min] , such as is —apparently—the case in the submm region and that near X ^ 1200A,

unresolved disk studies can be used in a scheme of

successive approximation. All these procedures can

be followed in the solar case; presently the most

fruitful lines of further effort appear to lie in

higher geometrical resolution studies. In most

stellar cases, we^are confined to integrated-disk

results. So, one is forced to use F[X] rather than

Ix[y]f and an iterative scheme. But indications of

stars having both UV and IR excesses already suggest,

as in the solar case, the presence of an outward

rise in Te as the simplest explanation. Extending

observations to the rocket UV, and farther into the mm and radio regions, increases the supply of data.

For two classes of stars, we gain additional information, of the limb-darkening type. One class is the

eclipsing binaries, for which change in spectral features during ingress and egress can be studied. A

second class consists of those stars for which the

new refinements in interferometry can be applied

(e.g., Hanbury Brown 1968).

273
From the lines. The interpretation of such

lines as Ca+ H and K, the Mg+ lines, the Balmer

lines of hydrogen, the Lyman lines of hydrogen,

and the NaD lines are more and more discussed in

the literature. Difficulties still appear to exist

equally in obtaining very high resolution profiles

for the stellar case, and being absolutely sure of

the details of the theory for both solar and stellar

cases. Here, the solar case—where good spectral

resolution exists —provides absolutely invaluable

"calibration" of the theory against a solar 'model

coming from the continuum. The self-reversed

emission core of, e.g., Ca+ now seems universally

accepted as an indicator of the presence of a chromosphere—for, say, the period of the last 12

years. It is however not quite so clear how we interpret the details of the K 2 and H 2 regions

outside the emission cores in terms of Te [min]

.

In addition to those complications arising be- cause the observed profiles represent the integral

over sizeable portions of the disk, more problems

arise in the theory: the accuracy of the assumption

of complete redistribution for scattering, the

problem of line broadening (which couples this

problem of the Te distribution to that of velocity

fields) , and the question of interlocking with other

levels. All these are theoretical problems, but

their solution clearly underlies the empirical analysis. And equally clearly, they are the same problem

for sun and stars, so that solar and stellar investigations can hardly be separated.

b. Theoretical

We have already remarked on those theoretical

aspects underlying the diagnostic spectroscopy here;

consider those aspects dealing with theoretical model

atmospheres

.

One aspect is the problem placed into focus by

Cayrel (1963) for the sun, but of long-standing for

the planetary nebulae: the balance between quantity

and quality of radiation in fixing Te , and the question of getting some nontrivial outward rise in Te due to it alone. Clearly, such a rise could introduce confusion in deciding on the presence of a chromosphere. So, the problem must be completely

clarified; it is not so, at present (cf. Jordan

1969a, 1969b)

.

A second aspect is the Te-distribution arising

from the mechanical heating. I do not propose here

to summarize all the aspects of the problem, aero- dynamical and astronomical. One recent summary for

the solar case lies in Jordan's thesis. The papers

274
by Lighthill and Moore in the 1965 Cosmical Gas

Dynamics Symposium focus attention on the problem

of whether the particular acoustic heating in the sun arises in the convective, or in the overshoot,

region of the solar atmosphere—a most important

concept in extending solar thinking to the stellar

case. Again, I hope it is clear to you that the solar

case can be utilized as a guide to stellar problems.

More often than not, we are trying to extend to

the astronomical environment, or develop completely

afresh, physical ideas for which the theory is com- plex and for which we have only little intuition.

The solar case provides a useful testing ground

because of the wealth of geometrical and spectral

resolution.

2. Velocity Fields

We have four main interrelated questions we need to answer on velocity fields. (a) How to

separate thermal from macroscopic velocity fields?

(b) How to separate the effects of increased kinetic

temperature, and of momentum input causing departure

from HE, on the atmospheric extent? (c) How to

separate the required velocity field from radiation

transfer effects in line-broadening mechanisms? (d)

What is the origin and aerodynamic behavior of ve- locity fields?

Questions (a) and (c) overlap in their concern

with specifying just what* random, macroscopic velocity field must be introduced in addition to the

thermal field. If we already knew the thermal

field—as is the a priori assumption in the CA

approach, or as we might possibly learn if we could

get a complete atmospheric model from the preceding

point (l)--it would be easier to infer the macroscopic field.' Also, in principle, the separation

of thermal and random macroscopic fields should

be straightforward because of the differential mass

dependence. Unfortunately there are two severely

complicating factors: radiation transfer effects,

and the effects of atmospheric inhomogeneities

.

Examples of each of these, for the sun, lie in

Redman's [1942] attempts to use eclipse profiles

of hydrogen, helium, and metals to infer Te ; and

early attempts to identify the yellow coronal line

by comparison of its width with that of known coronal

iron lines (cf., the summary by Billings 1966). We

have already commented on the coupling between un275
certainties in line-formation theory and derived

velocities, even under the non-LTE approach. The

LTE approach gives too shallow lines , hence interpretation of observed line profiles or equivalent

widths by an LTE diagnostics gives spurious microturbulent velocities. So we need to pay careful

attention to comparison between theory and observation in a known atmosphere before we can be certain

of results on an unknown atmosphere. We note that

certain stellar eclipsing binaries provide line

profiles and equivalent widths that change during the course of the eclipse, and so can presumably be used

in a way similar to solar studies. Groth will be

reviewing one aspect of this, at this symposium.

Kuhi has summarized the situation for the WR stars

(1968) . So we can advance our knowledge of stellar

chromospheres in this respect, by obtaining better

observations, and devoting attention to the interpretive problems I have just summarized.

Items (a) and (b) overlap in their concern with

what fixes the density gradient of the atmosphere.

The problem couples closely to that of specifying

the excitation state of the atmosphere, as a function

of height. Again, the solar studies provide a useful guide.

Item (d) is almost wholly a theoretical problem

at the present time, using as a boundary condition

the mechanical energy and momentum supply required

to satisfy the results of aspect (1) and (a) -(c)

above. Although considerable work has been done on solar-type chromosphere-coronal aerodynamics, much more remains to be done.. And the field is essen- tially virgin on most other chromosphere-coronal

types. For example, although much has been done on the aerodynamics of the nonatmospheric regions of

cepheids , the atmosphere has thus far been treated

in only cursory detail. Possibly Hillendahl's

remarks later in the session will cause this comment

to be revised.

3. Inhomogeneities 3 With and Without Magnetic Fields

I do not intend to comment in any way on this

most important point, except to say that the only

direct information we have, comes from solar studies

based on resolution of the disk. Possibly stellar

eclipse studies can somehow be interpreted to give

indirect information; certainly, we have evidence

that the atmospheres are not spherically symmetric.

276
So what we need are ingenious ideas for empirical

inference; or theoretical generalization from experience with the solar case.

4. Excitation Anomalies

There is a temptation to spend a great deal

of time on this subject; indeed, I would like to see several days devoted to it. Clearly, to map out

excitation stratification effects we need a detailed

study of line profiles, studies of lines of different

regions of origin in the atmosphere, and stellar

eclipse studies. Again, I refer you to the evolution

of our thinking on the solar chromosphere as the

best example of how an apparently complex situation

can be untangled (even though I would be the last to

claim that we have done so, in detail, as yet) . And

I refer you to the WR atmosphere, as the best example

of a situation where we cannot yet agree completely

on the direction of change of excitation with height

in the atmosphere, in more than an overall way. So

I think it best to simply say: Consider the excitation problem, in all its aspects, observational

and theoretical, when we ask what we can do to know

more about stellar chromospheres and coronae. And

I would emphasize very heavily any stellar spectrum

that shows symbiotic effects as a strong candidate

for having a chromosphere-corona.

5. Basic Conceptual Anomalies

In a discussion of a number of points above,

we emphasized how to pin down the details of the

chromosphere-corona. At the present time, in the

great majority of the stellar situations, our first

problem is simply to identify which star falls into

the extended-atmosphere class, and then of thes^,

which have chromospheres - coronae . Thus, we would

like to develop some criteria, based on outstanding

features that would help this identification. In

the last sentence of (4) above, I suggested that

symbiotic spectral features might well be one such

criterion. This is a conjecture. Certainly, stars

with chromospheres-coronae exhibit some symbiotic

features; it is not clear that the converse is true,

that all symbiotic stars have chromospheres-coronae.

The problem remains to be investigated.

In just this category fall those features that

I have labelled basic conceptual anomalies. We know

277
that stars with chromospheres-coronae exhibit emis^

sion lines, and probably exhibit a steady-state massloss. The basic properties of each of these two

phenomena are directly tied to a high-temperature

outer atmosphere. The question is, can we apply the

converse, and use these two phenomena as indicators

of chromospheres-coronae.

Consider the emission lines. We must distinguish between intrinsic emission lines, and those

arising wholly from a geometrical effect—a situation

where the opacity in the lines so greatly exceeds

that in the continuum over a much larger disk that

an emission line results. Then we would first need

a method to distinguish an intrinsic emission line

from a geometrically induced one. Next, we would

need to determine whether intrinsic emission lines come only from a reversal of a Te -gradient, a chromosphere-corona. We know that in certain situations,

fluorescence processes can produce emission; Anne

Underhill has discussed examples of these. So we would need a means for discarding such fluorescentproduced lines. Then we are left with the Schuster

mechanism which, in the literature, seems to be

the favored mechanism, especially among those trying

to retain the CA model. Katharine Gebbie and I think we have shown this mechanism can be rejected, in all

but exceptionally unlikely cases,' in stellar atmospheres (196 8) . So, we are left to devise methods

for distinguishing between intrinsic and "geometrical" emission lines. One approach to this has been

described by us some years ago (C. Pecker-Wimel and

Thomas 1963) ; the group under Rense at the University

of Colorado has been testing its utility in the solar

rocket UV spectrum. Mrs. Gebbie and I are continuing

with this problem of trying to decide the use of

emission lines as an a priori chromosphere indicator.

You have heard from Rybicki of the work by him and

Hummer on investigations of the properties of intrinsic emission lines, which can be adapted to

chromospheric situations. A number of people in

JILA have been working on these problems including,

beside Hummer, Mrs. Gebbie, and myself, Castor,

Paczynski, Lindsey Smith, van Blerkom. By the time

this symposium is over, you will have heard from, or about, a number of other workers elsewhere.

Again, reference should be made to Francoise

Praderie's summary.

Consider the steady-state mass loss. The

physics of the problem are summarized in Parker's

book (1963) . Again, we see the utility of the solarstellar comparison. The recent observations of

278
supergiants (Morton 1967; Stecher and West 1968)

exhibiting evidence of expansion in the rocket UV but

not in the visual—thus implying stratification of

velocity field—coupled with the observed suggestion

of UV and IR excesses in the continuum present the

kind of observational situation we face. Clearly,

the point is to try to put all these observations

together into a coherent picture, in the same kind

of way Delache has tried for the sun. Again, we

come together with Francoise Praderie's summary.

I hope, therefore, I have n clear to you why

I believe the best answer to the quescion: What can we do to know more about stellar chromospheres and coronae lies in continuing this parallel investigation of stellar and solar situation—depending

upon the solar studies for high resolution of detail

in one specific situation, and upon stellar studies

for extrapolating and expanding the varieties of

possible causes and configurations of chromospherescoronae

.

REFERENCES

Billings, D. E. 1966 , A Guide, to tkt Soldi Corona,

Academic Press.

Cayrel, R. 1963, Ck % Ac. Scl. Vah.li> , 251_, 3309.

Gebbie, K. B. and Thomas, R. N. 196 8, Ap. J. 15 4 , 285.

Gebbie, K. B. and Thomas, R. N. 1968, ?H.o ce.tdlng-6

o^ tke. Symposium on WoZ&-Ray&t Stah.& , National

Bureau of Standards Special Publication 307.

Hanbury Brown, R. 196 8, in Gebbie, K. B. and Thomas,

R. N. , Vno (Kidding b o{ tke. Symposium on ^ol^-Rayzt

StaiA , National Bureau of Standards Special

Publication 307, p. 79.

IAU, 1957, Joint VI* cuaaIo n on Tuibulznce. In ST zllan.

AtmoAph.2.lZA , Transactions of the IAU, 1955, IX , 727, Cambridge University Press.

Jordan, S. D. 1969a, in press, Ap. J., 1969b, in

press, Ap. J., 1968, Thesis, University of

Colorado.

Kuhi, L. V. 196 8, in Gebbie, K. B. and Thomas, R. N .

,

Vnoczzdlng^ o\ tke, SympoAtum on {JJoli-Rayzt Static

,

National Bureau of Standards Special Publication

307, p. 103.

Morton, D. C. 1967, Ap. J. 147, 1017.

Parker, E. N. 1963, I ntth.ploLndta.ny Dynamical

P/ioc£44 e-6 , Interscience Publishers.

279
Pecker-Wimel, C. and Thomas, R. N. 1963, Ap. J. 137 , 967.

Redman, R. 0. 1942, MWRAS 102, 134.

Stecher, T. P. and West, D. K. 1968, in Gebbie, R. B. and Thomas, R. N. , Vn.o czzdi,ngA o\ tkz Sympo&i.um

on WoZ^-Rayzt Stant> , National Bureau of Standards

Special Publication 307, p. 65.

Thomas, R. N. 1948, Ap. J. 108 , 130.

Thomas, R. N. 1$61, Vn,oc.zzd*.nq& o< tkz fouitk

Sympo6i.um on CoAmZcaZ Gtu Dynamic* , Supp. to

Nuovo Cimento XXII . Thomas, R. N. and Athay, R. G. 1961, Vhy&lc* oft tkz

So tan. Ckn.om&opkzn.z t Interscience Publishers.

Thomas, R. N. 1967, Vn.Oddzdi.nQh o£ tkz Vl^tk

Symposium on Co&mtzaZ Ga& Vynami.c.6 , Academic

Press

.

DISCUSSION

ti<Llle.ndakl: For y Boo (period 6 n ) one would

expect the K line emission to change with a "period"

of about ln . This will be explained in my talk

later.

Vn.ado.fii. z: We observed this star with an ex- posure time of 3m at intervals of half an hour.

With this high time resolution we found that the

Ca II changes sometimes, but not periodically.

Llndzn.ki.ZZ: Can we find from the spectral lines

available for analysis in stars of type A and earlier clear indications for an increase of temperature

by a significant factor, that is 1.5 times or larger? I doubt it. Consequently it seems to me we have no need for the early type stars to think

in terms of chromospheres with a large difference

between temperature in the photosphere and the

chromosphere

.

You have mentioned the often quoted remark that

the velocity of outward motion is larger for ions

of higher ionization potential in P Cygni stars.

This remark is without sound observational basis.

In a study of P Cygni to appear in BAN Vol. 20 M.

de Groot shows that the clearest monotonic relation

is obtained by plotting the expansion velocity versus

(IP + EP) . The meaning of this empirical relationship cannot be deduced satisfactorily by any simple

theoretical considerations.

280
Thomcu> : I would like to call your attention

to the proceedings of a symposium of Wolf-Rayet

stars held at Boulder in 1968 (NBS Special Publication 307, 1968). One of its aims was to consider

that the atmospheres of these stars are simply

chromospheres

.

P^iacfe^e : The observation of P Cygni profiles

in the UV spectrum of early type stars does show a dependence of expansion velocity on ionization

potential.

Und£Siki.Zl: The UV observations are not

sufficient to permit such a general conclusion.

Hea/tn: In the solar spectrum or in the spectrum

of late type stars, the observation of He I A10 830

in absorption is a clear indication of a chromosphere. But for early type stars special 'conditions

are indicated only if this line appears in emission.

Atkay: It is important in discussing stellar

chromospheres to remember that a chromosphere

represents some type of equilibrium between a mechanical energy input and an energy output, most

probably radiative. The temperature structure of

a chromosphere will depend on both input and output. The temperature of the solar chromosphere is

determined largely by the fact that hydrogen is an efficient radiator at temperatures a few thousand

degrees above the photospheric temperature. In

early type stars hydrogen is highly ionized and no

longer an efficient radiator. Thus a given mechanical energy input will produce a chromosphere that

differs greatly in its general properties from the

solar chromosphere. It is possible that even small

amounts of mechanical energy could produce important

chromospheric phenomena in such stars.

0£>t2,Kbtiocki Observations in the near UV have

contributed much to our knowledge of the solar

chromosphere. I would like to call attention to

the OAO UV measurements made by the University of

Wisconsin group. They now have results extending

down to about 1500A for over 100 stars, including

several late type stars , and though the resolution

is low it seems likely, that these observations

will be very useful in trying to understand stellar

chromospheres

.

SkumanZck : Observations show a change of the

Ca II emission line of 61 Cyg in an interval of

5 years.

Uzydfi: In the case of the sun a combination

of magnetic fields and shock waves is a sufficient

mechanism to produce the heating of the chromosphere

and corona. If there are similar magnetic fields

281
in stars this mechanism could be applied.

Hlllzvidcikl: There is a thesis by D. M. Pyper

on magnetic A-type stars. She gets indications for

shock waves

.

Hzaivi: Are there serious arguments against

the hypothesis that al l stars or most stars have

chromospheres?

Thomas : We do know that the sun has a chromosphere, which is produced by the mechanical energy

of the convection zone. We further know that all

late type stars do have convection zones. Heating

of chromospheres by mechanical energy is the only

known process. We don't know a specific heating

mechanism for early type stars, but that doesn't

mean one doesn't exist. Personally, I have pushed

studies of aerodynamic phenomena in al l stellar

atmospheres simply to try to identify such mechanical

heating mechanisms; I have always argued that we must be prepared for chromospheres in all stars

but we must find the specific source of their energy

supply.

282
OBSERVATIONS OF £ AURIGAE STARS

AND THEIR INTERPRETATION

by

H. G. Groth

Universitats-Sternwarte Munohen

Institut fur As tronomie und As trophy sik

ABSTRACT

A summary of information on the extended atmosphere of the K component coming from three eclipsing

binaries consisting of a K supergiant and a main

sequence B star. Various anomalies exist: (1)

Discrepancy between electron density and metallic

density inferred from observations, noting that the

electrons should come from the metals. This suggests

a cloud structure in the atmosphere. (2) Anomalous

excitation temperatures and populations of the second

quantum level of hydrogen suggests a solar-chromosphere type behavior for these K atmospheres. (3)

Significant changes in the profiles of the Ca+ lines

from one eclipse to another. The necessity to distinguish between effects coming from radiation of

the B star on the K atmosphere and chromosphericlike effects is emphasized.

Key words: chromosphere, super-excitation, RussellAdams effect, chromospheric clouds.

It is very difficult to get direct information

about the outermost layers of stars other than the

sun. The information comes mainly from spectral

pecularities and the absorption or emission in the

centers of strong lines. But there is a group of

stars that permits us to make detailed analysis of

the extended outer atmospheres of a few late type

supergiants. These are the eclipsing and spectroscopic binaries with long period (972 d - 3800 d )

.

I shall discuss three of them: C Aur, 31 Cyg, and

283
32 Cyg. Each system consists of a K type supergiant

(K3 - K5 lb) and a main sequence B star (B3-B8 V)

.

The diameter of the B type component is small com- pared with the dimensions of the extended atmospheres

of the K supergiants. Both components show equal

intensity near A 4000, so that outside eclipse

the spectrum is the sum of a K and a B type spectrum.

Some time before or after the total eclipse of the

B star by the K star, the light of the B star is

absorbed in the chromosphere (the outer layers of

the atmosphere) of the K supergiant. These additional absorption lines have been observed and

analysed by several authors at different heights

above the "limb" of the star, giving valuable

information of the physical state of these layers

(see, i.e., O. C. Wilson 1960).

But in spite of a long series of observational

results and great effort in the analysis of the

observations in the last thirty years, there is no satisfactory interpretation.

At a first glance the problem seems to be quite

simple. There is a light source and some absorbing

material, a nearly ideal situation for spectroscopic

analysis. But unfortunately a B star is a very hot

light source and the separation of the two components

is not very large. For £ Aur the mean separation

of the K star and the B star is only three times the

radius of the K component. As a consequence we must

expect an influence of the B radiation on the K atmosphere, and so we get into trouble.

What information can we get from the observa*

tions? If we measure the chromospheric part of the

absorption, we can derive the number of atoms and its

gradient for different elements in a column across

the chromosphere by using the curve-of-growth method

for pure absorption. By assuming spherical symmetry

it is easy to calculate the number density from the

solution of an Abel-type integral equation. If the

assumption of spherical symmetry is correct, we have

the number of atoms per cm 3 and its dependence

on height, that is on the distance from the limb of

the K star, for some atoms: Hydrogen (Balmer quanta),

Ca II, Ca I, (Fe II), Fe I, Ti II, Ti I, and others.

From the curve-of-growth analysis we have in addition

excitation temperatures, which are of the order of

4000 to 4500°K. Some authors have deduced an increasing temperature with increasing height. The

microturbulence comes out to be 10-20 km/sec.

If we employ normal abundance ratios of the

metals to hydrogen, we get a total hydrogen density

of 10 10 atoms/cm 3 at a height of 4 x 10 6 km above

284
the limb for C Aur. The decrease of density can be

described by an exponential law.

with a of the order of a few times 10" 7 km_1

. All authors agree (for all three stars) that

the radiation of the B star is not able to ionize

hydrogen in the K chromosphere. Some detailed calculations by Mr. Pollitsch of our Institute have

shown for the system £ Aur that the Stromgren limit

would be in the atmosphere of the B component. We

should expect to have only neutral hydrogen in the

K type atmosphere and as a consequence all electrons

would come from the metals. In the case of the first

ionisation of the metals (5-8 eV) we can neglect

the optical dilution and take only pure geometrical

dilution of the B radiation (the dilution factor is

of the order of 10~ 6 to 10~ 7 ). We can use a modified

Saha equation and calculate the electron density

assuming an appropriate temperature for the B star.

We find that the local temperature calculated from

the diluted B radiation is approximately the same as the excitation temperature derived by the curve-ofgrowth method.

Using the Saha equation for the metallic ionisation, we are surprised to find* a very large

electron density of 10 9 -10 10 per cm 3

, the number

of metallic atoms being of the order of 10 6 per cm 3

. With the hydrogen completely neutral we must come

to the conclusion that the material in the chromosphere of the K star must be concentrated in clouds

that have a density 10 3 -10 i+ times larger than we calculated for a smooth density distribution. The

number of clouds in the line of sight must be large

enough to produce rather small density fluctuations.

There are observations that show some fluctuations

especially in the K line of Ca II. It seems that

we can explain the ionisation of metals by the

radiation of the B star, if we propose a cloudy

structure for the K atmosphere. In this connection,

there is only a problem with the second ionisation

of calcium which has an ionisation potential of

11.8 eV. A larger number of Ca III atoms could be

formed than has been considered, but this is of

minor importance here.

The main question and, in my opinion, the key

for the whole problem is the excitation of the Balmer

lines

.

285
From the observed number of metallic atoms,

assuming normal abundances, we get the total number

of hydrogen atoms. We observe the Balmer lines in

absorption and get the number of atoms in the

second state. Using the normal Boltzmann equation,

we find in any case a temperature of 6000-7000°K for

the excitation of hydrogen, which is considerably

higher than the excitation temperature of the metals.

The Lya radiation is responsible for the population

of the second state of hydrogen. To get enough

Lya quanta for the observed Balmer lines we have

to consider several possibilities:

1. Direct Lya radiation from the B star.

a. If the Lya line is an absorption feature,

the number of Lya quanta is too small unless

the star has a temperature of the order of

100,000°K.

b. If Lya shows up in emission, there could

be enough quanta coming from the B star

itself, but we do not know how strong the

Lya radiation of a main sequence B star can

be.

2. Lya quanta coming from the H II zone around

the B star. It is very difficult to calculate the Lya radiation field in this geometrically very complicated, case . If the H II zone ends up in the B atmosphere we have case

lb.

3. Increase of temperature in the K chromosphere, that means a transition from chromosphere to corona as in the case of the sun. The most favourable possibilities are cases 2 and 3. If we have an H II zone around the B star,

there would be a transfer of Ly continuum radiation

to Lya radiation in a manner similar to the cascade

processes which occur in diffuse nebulae. Any Ly

continuum quantum can end up as a Lya quantum, so these quanta can be available in a considerable

number for the excitation of the second state in

the chromosphere.

But when considering this process we must

remember

:

1. the ionisation is going on during the whole

orbital period.

2. long wavelength quanta are also produced

which should be observed as emission com- ponents in the Balmer lines, especially Ha.

In this connection it is important to mention

that the orbits of all stars are very eccentric. For

C Aur the eccentricity is 0.4. If there was a con- siderable amount of emission from the H II zone

286
around the B star, this emission should show a periodic change, because the density of the material

must change with the orbital period. It would be

very interesting to observe Ha and other Balmer

lines during the whole orbital period and to look for

intensity fluctuations with this period. The

amplitude of these intensity changes may be very

small, so that I am not quite sure whether the

changes would be observable or not.

As mentioned earlier, the high excitation of

the second state of hydrogen could be produced in

the K chromosphere itself. Wellmann and I tried

to calculate this effect which we called "superexcitation," in the case of C Aur (P. Wellmann

1939; H. G. Groth 1957) . The result was that there

must be an increase of temperature with increasing

excitation or ionisation potential. For hydrogen

excitation and ionisation we found a temperature

of approximately 6000°K. The chromosphere of the

K supergiant would be in a similar state as the

solar chromosphere.

There are three observational effects in favor

of this last hypothesis. The first one is the long

known Adams-Russell phenomenon which is the fact

that the hydrogen lines in all late type supergiants

are too strong for the spectral types. This effect

can be explained by the superexcitation . For the

system of C Aur I could show that one-half of all

hydrogen atoms that contribute to the formation of

the Balmer lines in the K spectrum will be excited

in the chromospheric layers.

The second observation is by Odgers and Wright

(1964) . They showed that far outside eclipse there

are considerable changes in the emission and absorption profile of the Ca II K line.

These changes are very similar to the changes

of the K line on the solar disk, but the scale must

be much larger. I would be quite surprised if these

changes could be explained- as an effect of B star

radiation.

The third effect that shows changes in the K

chromosphere is the different extent of the chromosphere. The height at which the first trace of

chromospheric absorption of the K line has been

observed changes by nearly a factor '3. For t, Aur

this normally happens 10-12^ before the beginning

or after the end of total eclipse. Sometimes the

first trace of a Ca II line was observed 40 d from

second or third contact, sometimes the beginning was between these extremes. The chromosphere is not

necessarily symmetrical: the extent can be different

287
at ingress and egress. A relation of these changes

with the period of the system has not been found.

Summarising I can say there are some effects

that show that the outermost atmosphere of the K

supergiants can have a structure similar to that

of the solar chromosphere and perhaps the solar

corona. The influence of the ultraviolet radiation

of the B star cannot be completely neglected.

In my opinion two independent studies are necessary. The first is for the theoreticans

,

namely to study the ionisation effects of the

diluted B radiation in detail and in addition to

study the non-LTE effects in the atmospheres of K

supergiants

.

The second task is the spectroscopic observation

of Balmer lines and the Ca II K line during the whole

period and further observations of chromospheric

eclipses with high dispersion.

REFERENCES

Groth, H. G. 1957, Z. tf. k&tKoph. 43, 185.

Odgers, G. J., and Wright, K. 0. 1964, VkO Victoria

ContK. 89 . Wellmann, P. 1939, Ve.tLoHe.ntl. d. Unlve.n.*ltati>-

Ste.h.nwa>Lte. Babe.l6be.xg KIT, 4.

Wilson, 0. C. 1960, StaKh and Ste.llatL Sy&te.mA, 6, 436

(Compilation of all papers known to 1960)

.

DISCUSSION

Pe.cke.fL: Is the apastron close to the eclipse

or not? This could be important for effects that

depend on the distances.

GtLOtk: For all three stars the total eclipse

is very near to the periastron.

Hllle.ndakl: Does the magnitude of the K star

change?

GtLotk: Observations during total eclipse show

irregular changes up to Om .l in the blue.

Hllle.ndah.1: Did you use a velocity distribution

in the chromosphere or only turbulent velocities?

GtLOth: I have used only turbulent velocities.

We.llmann: I have tried to represent the

observed curve of growth introducing a velocity

gradient. But the calculations show that only a very small gradient might be possible. A larger

288
gradient would change the curve of growth completely.

ThomaA : The large turbulence shows that the

dissipation of mechanical energy is sufficient to

I heat the chromosphere. If this works, you are not

forced to postulate clouds of high densities.

GfLOtk: If you assume that the source for the

high excitation is in the K atmosphere, it is not

| necessary to have clouds at all. That was already

j shown by Wellmann (1939) and in my paper (Groth

I 1957) .Undo,Jihi.£Jt: There exist fluctuations of the

] intensity of the Ca II K line. I tried to find

fluctuations of the intensities of the Fe I lines,

but without success.

Gfioth' The fluctuations of the Ca II

I intensities are found mainly in the highest layers.

These fluctuations may be due to inhomogeneities in

the chromosphere of the K type star. The Fe I

\ lines can be observed only near the limb of the K

star, where the density fluctuations are very small.

Mag nan: Another idea that can be put into the

discussion is the possibility of a mechanical heating

of the chromosphere of the K star. This could

explain the ionisation equilibrium without the

assumption of a cloudy atmosphere (Magnan, 196 5,

Ann. d' KbtKopky* . 28, 512.

Gn.otk: In the paper by P. Wellmann

(Veroffentl. d. Universitats-Sternwarte Babelsberg

XII, 4, 1939) it was already shown that the dissipation of the mechanical energy is sufficient to

explain the "superexcitation" of the chromosphere

of the K type star.

Htlldndakl: There is no source for the mechanical energy and no mechanism to transport it

to the chromosphere.

W&Zlmann: The mechanical energy is supported

and transported by convection. The large turbulence

velocity shows that it can reach the higher layers.

289
IONIZATION IN NOVA ATMOSPHERES

by

P. Wellmann

Universitats-Sternwarte Munchen

Institut fur Astronomie und As trophy sik

ABSTRACT

Calculations on the expansion of novae shells

are presented, from which the degree of ionization

is obtained using a simplified form of a non-LTE

ionization equation. From this, estimates of the

time of vanishing of the Balmer absorption spectrum

can be made.

Key words: novae shell, ionization, expansion.

Atmospheres of novae are strictly speaking not

in a steady state. However, relaxation effects are not observed before the later nebular stage. Therefore the development of a nova shell may be understood as a sequence of different extended atmospheres

of the type discussed in this colloquium.

The nova phenomenon varies considerably in amplitude, time scale, and light curve details, but it

is possible to reduce the data to an average nova

by using the time t 3 (the interval between maximum

brightness and mmax + 3^0) as a time unit. A nova

that followed the average rather closely, and which

is appropriate for an investigation because the

absolute intensities of emission lines were measured,

is DK Lac 1950. It will be used here as an example.

During the first phases from 0.1 t3 to about 6 t3 the magnitude m of this star was strongly correlated to Ts (radiation temperature) as well as to

the spectral type (intensities of absorption and

emission lines) . Hence it was possible to eliminate

the effects of light and temperature fluctuations and

of secondary eruptions. We disregard here the dif290
ferent components of the diffuse enhanced spectra

i and the Orion spectrum because they belong to the

! secondary processes or originate in masses of gas,

which are negligible compared with the main shell.

The principal spectrum showed an increase of

radial velocity from about 800 to 1300 km/sec during

the absorption line stage and a nearly constant

j value in the later development. An estimate shows

that the observed RV changes cannot be interpreted

i as accelerations. Instead we have to assume that

J velocities between certain limits were present all

! the time, and that the measured line shift is due to

a motion of the effective layer of absorption. Each

substantial atmospheric element had a constant outward velocity between vmi n and vmax . The basic model is a shell with radius and geometrical depth increasing linearly with time. Hence

the density of each substantial element decreases

with r~

3

, and the velocity field is

v(r,t) = r/t-t

Q

, t being the time of the initial magnitude rise.

During the stages with forbidden lines, Ne , the

electron density, can be found from the intensities

of these lines. According to a well-known theory,

we have, for instance, for [0 III] , intensity (N x + N 2—) = f (N ,T ) intensity (A 4363) e' e' with a given function f. Unfortunately f has a form

that forbids the solution for N and T of two such

e e equations for different ions, with a reasonable

accuracy. However, because of the "thermostatic"

action of the forbidden transitions, an electron

temperature of about 10000 °K during the whole period

can be accepted. The corresponding Ne , proportional

to N (total density) is well defined and changes as

(t-t

Q

)~ 3

. Using the hydrodynamic force equation,

|x + v |z = k%0 , and the equation of continuity,

9N . 1 9 (N v r

2

) _ n 9t r

2 9r" ° ' it is very easy to verify that the observed relations
agree and that the density field is given by

N

N(r,t) =

(t-t ) 2 r The principal shell crossed a radius R during an interval of time,

At = J-i l_)

V v . V / \ mm max /

and that gives At in the range of 10 1 to 10 2 min, if

R is of the order of the solar radius. The corre- sponding density is of the order of 10 20 atoms/cm 3

. The nova phenomenon started with a short time "explosion" of a layer not very far below the original

photosphere.

We return to the problem of the formation of the

spectra of different excitation. The computation of

the population of atomic or ionic levels from the

steady state equations

d N( X± ) d t

is rather complicated. So we use the first approximation for the equation of ionization under non-LTE

conditions^= -OS! K W T (T , >/>eX»ATs

. N

(n-i) B(n_l)

The dilution factor is W ^ (R^/2r) 2 e

T where x is

the optical depth between the photosphere and a point at distance r. This is to be combined with

the definition of dx

dx = (l-x)eN(r) a dr,

where

(n) /V,, T (n

)

x = N /EN

Here a is the absorption coefficient averaged over

v from the series limit to infinity. It is conve- nient to write a = 0.5 a (series limit), which can be

292
verified a posteriori . e is the abundance of the

element investigated. This differential equation

has been solved numerically for t (r) and x(r), using

arbitrary N(r). Of course, the Stromgren theory of

ionization zones is the special case N = const,

e = 1.

Under certain conditions, which are fulfilled

in many cases, x changes rapidly from 1 to 0 at a

: distance, r s

. These "Stromgren radii," if of finite

I size, are very sensitive to Ts

: a small change in

| temperature makes rs jump through the whole main

' shell and then causes a very quick spectral change

! with m(Ts ). The motion of rs for hydrogen through

| the shell explains the width and the radial velocj ities of the Balmer lines. At first, the whole

j shell belongs to the H I region, and the lines are very wide (width corresponding to vmax - vmi n )

,

with RV ^ 0.5 (vmax + vmi n ) . Since r s increases

faster than rraax , the effective center of the absorptions moves to vmax* the width becoming normal.

When rs = rmax/ the whole of the shell belongs to the

H II region and the Balmer apsorptions vanish from

the spectrum. Using Ts from different observations

(spectrophotometry of continuum and Zanstra methods)

and adjusting the parameter NQ to give the correct

absolute intensities of the hydrogen emissions, it was possible to predict the times of disappearance

of the He I, 0 I, N II, [N II] and [0 III] emissions

sometimes within a few days. The main source of

uncertainty is the still inadequate information on

Ts- It can be shown that the condition for a change

of spectral type, r s equal to the average shell

radius, is fulfilled for specific Ts , independently

of time, if the degree of ionization follows the temperature without delay. Only in the later nebular

stage, when the electron density is below the predicted limiting value of Ne = 10 7 - 6

, a delay of one

day or more between temperature changes and the

appearance and disappearance of the nebular lines

Nj,2 has been observed.

REFERENCES

Hiller, I. 1965, Thesis, Hamburg.

Wellmann, P. 1951, Z4 . . Ap., 29, 112.
DISCUSSION

OAtzJiblock: Are the time-dependent effects of

the finite rate of emission of ionizing radiation by

the nova important to your problems?

WzlZmann: The retardation of the ionization

changes did not occur before 4.t3, that is in the

later nebular stage, and was observed mainly in

[0 III]

.

NiU 4 bailment: I want to give two warnings: (1)

Atomic data: Seaton's group has recalculated some of

the [0 III] collision cross sections. Changes in some of these are important. For all interpretational work where cross sections or oscillator

strengths are used it is worthwhile to make sure that you have the latest calculations. (2) Line

widths: Combined effects of radial velocities and

temperature gradients may influence the line widths

considerably. Suppose a radial expansion but unique

temperature, then you have the normal Doppler broadening. Now suppose a radial drop in temperature sets

in and increases the abundance of your ion. The displaced line component will then be more intense than

the undisplaced component. Thus the radial drop in

temperature produces a line width that makes you believe in a temperature increase.

W&llmann'. Every time Seaton's group has published new values of the cross sections, we have re- peated our calculations for the forbidden lines.

Fortunately the fundamental result has not changed

very much. But there is another difficulty. The

relations between Ne and T

e for different forbidden

lines should give a unique solution for electron density Ne and electron temperature Te

. But we don't

find such a solution. It may be that this effect is

due to inaccurate cross sections.

294
CIRCUMSTELLAR CA II K LINES IN G, K AND M GIANTS

AND SUPERGIANTS

by

A. H. Vaughan, Jr., and A. Skumanich

Mt. Wilson and Palomar Observatories

California Institute of Technology and

Carnegie Institute of Washington

Pasadena s California

and

High Altitude Observatory

National Center for Atmospheric Research

Boulder 3 Colorado

ABSTRACT

Tentative evidence, -based on photoelectric

observations of the Ca+ emission core, is presented

for the existence of circumstellar envelopes in

several G and K type "giants." A significant

emission asymmetry in a Tau may imply a chromospheric

rather than a circumstellar source.

Key words: Ca emission core, circumstellar

absorption lines, late-type giants.

We present here a preliminary report of photoelectric scans of the cores of the Call K line in

several. giant and supergiant stars. As Deutsch

(1960) has pointed out, M "giants" - this term includes supergiants - have K emission cores on which

are superimposed circumstellar (CS) lines. These

are distinguishable from the K 3 chromospheric core

The National Center for Atmospheric Research is

sponsored by the National Science Foundation.

295
by "their very low central intensities . . . and sometimes by their radial velocities as well." We believe that we have evidence for such CS lines in G

and K "giants.

In Figures 1, 2, and 3 we present uncorrected

scans averaged over several nights 1 observations of

five "giants." We have plotted photon number (for

a given count in the neighboring continuum) versus wavelength in arbitrary units (220 units = 1 A)

.

The instrumental resolution, instrumental half-width

at half-maximum, was 0.14 A (30 units).

The K emission profile is seen to be asymetric

in both giants and supergiants. However, only in

the supergiants does one find intensities for the

"central" absorption feature which fall below the Ki

value. If one reflects the emission core about its

bisector (i.e., if one "symmetrizes" the emission

core) one obtains the dashed line indicated in the

figures . ** The ratio of the observed intensity to that defined by the dashed line yields a measure of the

strength of what we interpret as the CS line. These

"absorption" depths are indicated in the top of the

diagrams. Table 1 summarizes the equivalent width

and wavelength shift for these CS lines.

The tabulated equivalent widths agree in magnitude and show the same increase with luminosity

TABLE 1

Star Sp W(A)

-V

km/sec

-V(W+B)

km/sec

a Tau K5 III 0.15 20 4

3 And MO III 0.20 21 16

3 Peg M2 II-III 0.31 4 6

£ Gem G 8 I b 0.65 17 10

£ Peg K 2 I b 0.47 26 28

In the case of £ Gem and 3 Peg the circumstellar

line was also symmetrized to obtain the complete

dashed curve.

296
0

s f

1 I 0 ; CIRCUMSTELLAR '. J CIRCUMSTELLAR \ LINE 0.2 •-, ,• " LINE 0.2- •, } DEPTH '•} DEPTH *-, 1 0 4^ V - 04-

\J 06-

Figure 1 . Averaged photoelectric scan of the K

emission core (uncorrected for instrumental broadening) . Photon count (for a given count in the neighboring continuum) versus scanner setting is represented. Wavelength increases to the right.

Figure 2. Data similar to Figure 1.

297
320

280

240

200

160

120

80

40

CIRCUMSTELLAR

LINE DEPTH

£ Peg

(Avgof2)

0

0.2-

0.4-

0.6-

0.8-

I A

J I I L

i

1

.< i i

i i

J L J L 42 44 46 48 5000 52 54

Figure 3. Data similar to Figure 1 as in the later-type stars, Deutsch (1960) . The

wavelength shifts (relative to the emission bisector)

indicate a nearly constant expansion of - 20 km/sec

except for 3 Peg. For comparison, we have given the

shifts measured from spectral plates by Wilson and

Bappu (19 57) . The agreement, except for a Tau, is

fairly good.

An alternative explanation for the a Tau profile would be to allow an appreciable velocity

gradient through the emission forming layers, cf.

Weyman (1963) . Such a gradient introduces an asymetry with one side of the K 2 peak reduced (or eliminated) relative to the other, cf. Vaughan (1968)

and Hummer and Rybicki (196 8) . Such an explanation

would satisfy the statistical results (based on wavelength shifts) of Wilson (1960) which indicate no mass loss in class III stars earlier than MO. A

complicating feature here is the evidence in the individual scans of a Tau of multiple weak dips in the

298
depressed violet side. This raises the question of

the significance of a mean profile. Thus, our re- sults on a Tau should be viewed as tentative and may

be changed by the more detailed investigation in

progress

.

REFERENCES

Deutsch, A. J. 1960, Sto.lla.ti Ktmot> pkcno.* : Stan.* and

StzttaK Syt>tnm{>, VI, ed. J. L. , Greenstein

(Chicago: Univ. of Chicago Press), p. 543.

Hummer, D. G. , and Rybicki , G. B. 1968, Ap. J.,

153 , L107.

Vaughan, A. H. , Jr. 1968, Ap. J., 154 , 87.

Weyman, R. 1963, Ann. Rev. k&tl. and Ap. , 1, 97.

Wilson, O. C, and Bappu, M. K. V. 1957, Ap. J.,

125 , 661.

Wilson, 0. C. 1960, Ap. J., 132 , 136.

DISCUSSION

Pecker: Is it possible to distinguish between

the effects of a chromosphere and those of circumstellar envelopes?

Skuma.nX.ck : I would prefer an explanation by

circumstellar envelopes. This is much simpler and

will give a better fit of the line profile. This has

already been shown by Deutsch.

Wnd2.nkX.tt' Have you observed the Na I D-lines

by the same technique? If your interpretation of the

absorption dips as circumstellar lines is correct,

similar dips should be found in Na I, without the

additional complication of emission components.

SkumanX.dk' A slight doubling of the Na I Dlines is observed in the spectrum of a Ori; it can

be interpreted as a circumstellar effect. In the

spectra we have observed the effect is probably too

small because the mass of the shell is not large

enough. We have not tried to observe' the effect.

We.ttman: Several years ago I could show that

the line doubling in a Ori is due to chromospheric

effects

.

SkuLma.nX.dk i The doubling of the Na I D-lines

should be interpreted by circumstellar clouds.

Chromospheric effects tend to wipe out the structure

as Hummer has shown.

299
A PHYSICAL MECHANISM FOR THE GENERATION

OF EXTENDED STELLAR ATMOSPHERES*

by

R. W. Hillendahl t Department of Astronomy

University of California at Berkeley

and

Lockheed Palo Alto Research Laboratory

3251 Hanover Street

Palo Alto 3 California

ABSTRACT

A physical mechanism that can result in the

generation of extended expanding atmospheres is

discussed. The process involves the unloading of

stellar material following the arrival of a shock

wave at the edge of the star. The basic principles

are developed from a discussion of a simplified

case that has been studied in the laboratory; they

are then applied to the atmosphere of a star. A

radiation-hydrodynamics computation of a model

cepheid is then used to obtain quantitative atmospheric profiles. The computed continuum and

spectral lines during the unloading process are then examined. A discussion of the possibility

that the unloading process occurs in stars other

than cepheids suggests the existence of a shock

visibility factor associated with ionization or dissociation in the region behind the shock front

and leads to a possible alternate interpretation

of the variable star instability strips in the

H-R diagram.

Key words : extended atmospheres , shock waves

,

hydrodynamics, cepheids.

* This work was made possible by a grant of computer

time made available by the Berkeley Astronomy Department and by a Lockheed Independent Research

Grant.

^Now at Lockheed Palo Alto Research Laboratory,

3251 Hanover Street, Palo Alto, California 93404.

300
INTRODUCTION

Many stars in the upper part of the HertzsprungRussell Diagram appear to have extended atmospheres

i in which an outwardly directed flow of matter is ob-

; served. While a number of probable physical causes

! of this phenomenon have been studied (cf. Weymann 1

)

,

j it is clear that our present understanding is far

', from satisfactory. In the course of a recent study 2 of atmospheric phenomena in classical cepheids , a

, mechanism was encountered that can cause atmospheric

characteristics similar in many respects to those

ascribed to some types of these "steady-state ex-

; tended atmospheres .

It is the purpose of this presentation to discuss the basic concepts associated with this extension mechanism as studied both theoretically and in

the laboratory, and then to apply these concepts to

stellar atmospheres. Specific results are then given

for a model classical cepheid. Finally, it is

1 suggested that the mechanism may not be confined to

cepheid atmospheres, but may be of more general

applicability. Some of the consequences of this

suggestion are then explored.

THE BASIC PHENOMENA

The phenomena to be discussed are associated

with the processes that occur when a shock wave progressing through a gaseous medium arrives at an interface between the material and a vacuum. Previous theoretical studies of these phenomena (cf

.

Bird 3

, Sakurai 1*) as they apply to stars have either

employed simplifying assumptions or have failed to

establish the relationship between the theoretical

results and the associated observable characteristics. Recently however, Zel'dovich and Raizer 5 have

reviewed both theoretical and laboratory studies of

a much simpler configuration, which provide us with

a clearer understanding of the dominant processes

that occur. Their studies concern the emergence of a planeparallel shock front from a solid, for the case in

which the strength of the shock wave is sufficient

to cause a conversion of the solid into a gaseous

state as the shock front advances through the material. As the shock front arrives in the surface

layer, the momentum associated with the shock wave

causes the surface layer to "blow-off" into the

301
vacuum. The rapid expansion of this layer of material lowers the pressure in the surface layer and

gives rise to an outwardly directed pressure gradient between the surface layer and the next interior

layer of material. This newly created pressure

gradient then causes the outflow of additional

material. This "unloading" process continues into

the deeper layers in the form of a rarefaction wave

progressing backward into the material, while the

unloaded matter itself is accelerated outward in the

direction of the initial motion of the shock wave.

The unloading (rarefaction) wave travels backward

into the material with the speed of sound that corresponds to the thermodynamic state behind the shock

front. Its velocity is directed opposite to and is

smaller in magnitude than that of the shock front.

If the original extent of material is large, the

unloading process can continue for an extended

period of time. The duration of this process can therefore be much greater than the time interval

associated with the passage of the shock front

through the photosphere. Under suitable circumstances, it can resemble a steady state process.

The unloaded material flows outward with a relatively large velocity which is the result of

the outward acceleration experienced by the matter

in crossing the shock wave and the additional outward acceleration caused by the pressure gradient

in the rarefaction wave. The terminal velocity of

a given particle after it unloads to (essentially) zero pressure can be calculated by noting that its

terminal kinetic energy is equal to the sum of its

kinetic and thermal energy prior to its engulfment

by the rarefaction wave. For the simplified case

of a nearly weak shock emerging from a material

having no pressure gradient prior to arrival of

the shock, Zel'dovich and Razier 3 obtain the

"velocity doubling law," i.e., the terminal velocity

of an unloaded particle is twice the velocity imparted to it, by acceleration in the shock front.

This result is indicative of the importance of the

acceleration in the rarefaction wave, but it apparently represents only a special case. The equations

for the general case do not limit the velocity in- crease to a factor of two, and suggest in fact that

much larger increases in velocity may occur if

ionization or dissociation is occurring within the

rarefaction wave. The passage of the shock front sets the entire

mass of material into motion in the direction in

which the shock travels. Thus, while the rarefaction

302
wave moves "inward" in terms of the mass of the

material, it and all parts of the material move "outward" with respect to the laboratory frame of reference.

APPLICATION TO STELLAR ATMOSPHERES

While the situation is much more complicated at

the edge of a star due to the presence of the preexisting motion and/or pressure gradients, the basic

physical laws that apply to the laboratory situation

should still be operative. Thus, when a shock front

arrives at the surface of the star, the unloading of

stellar material is to be anticipated. It is further

to be expected that the unloading process lasts

longer and has therefore a higher probability of

observation than the passage of the shock through

the photospheric layer. This follows because the

unloading process involves the motion of a sonic

disturbance into an essentially infinite supply of

material, while the shock emergence involves the

motion of a supersonic disturbance through a relatively narrow photospheric layer.

Neither the outward flow of the unloaded material

nor the inward progress of the rarefaction wave can continue indefinitely in a star. The outward moving

particles are accelerated to a finite terminal velocity and are subject to the star's gravitational

field. Should the terminal velocity exceed the

escape velocity, continued outward motion and mass

loss could occur. Particles achieving a lesser than

escape velocity will reach a maximum altitude and

then be accelerated downward. Differential motion

should then occur, which under favorable circumstances, might be observable in terms of the shifting

and/or asymmetrical contours of spectral lines.

Were the material returning under gravitational

acceleration to collide with unloading material still

in the process of moving outward, a temperature

inversion could occur which would result in emission

lines. • An interesting phenomenon can also occur in the

rarefaction wave as it moves inward in terms of the mass of the star. As has been noted above, the

terminal velocity achieved by a given particle in

the unloading process is the sum of the velocity

achieved due to passage through the shock front and

the velocity achieved due to acceleration in the

pressure gradient of the rarefaction wave. This

303
latter acceleration depends primarily upon the

thermal energy of the particle prior to the arrival

of the rarefaction wave. As this wave progresses

deeper into the mass of the star, the temperature

and hence the thermal energy increases. Particles

in the deeper layers will thus undergo a greater

outward acceleration than the particles in layers

initially at lar.ger radii. If these particles from

successively deeper layers overtake the particles

already unloaded, conditions may be favorable for

the formation of a secondary shock wave. When the

head of the rarefaction wave passes through a region

where ionization or dissociation is occurring, a very rapid increase in particle velocity takes place.

Conditions for the formation of a secondary shock are therefore most favorable in such regions of the star.

Such a secondary or "blow-off" shock wave would

then progress outward through the star in a manner

similar to the original shock wave; the blow-off

shock should itself be followed by an inward rare- faction wave similar to that which follows the

original shock. Thus a shock wave emerging from a stellar atmosphere should be followed by a series of

additional shock waves, reasonably well separated

in time, until the process becomes the victim of a damping mechanism or is interrupted by the gravitational return of previously unloaded material.

These secondary shock waves provide a possible

explanation 2 for multiple and irregular periods that

are well known in variable stars (cf. Arp 6

, Abt 7 ).

The frequency with which the secondary shocks might

be anticipated would appear to depend upon the instantaneous atmospheric structure of the star, which

in turn must ultimately depend upon the structure of

the deep interior. However, it is not clear that

there exists any direct connection between the interior structure and the periodicity of the secondary

shocks, such as, for example, the relationship that

defines the fundamental periodicity of a cepheid.

Thus, at the present state of our understanding,

one cannot assume that the secondary shocks are strictly periodic or that their period is related

to the fundamental period of the star.

It should also be noted that the secondary shocks

need not be associated with a pulsating star, but may

occur in a localized region of any star so long as the diameter of the region from which a shock wave

emerges is large compared to the scale height of

the atmosphere. For example, the multiple outward

waves seen in certain solar disturbances 8 may be

related to the secondary shock process, however, it

304
is not clear in this instance whether or not the

magnetic field merely contributes to or actually

causes the observed effect.

If spectral absorption lines originate in layers

of a star where unloading of material is occurring,

the spectroscopic gravity deduced from the analysis

of these lines may be significantly lower than the

dynamic gravity derived from the mass and radius of

the star. This effect results from the lower pressure gradient that occurs everywhere in a rarefaction

wave except in the immediate vicinity of the leading

edge of the wave, where the pressure gradient is

very steep. The unloading process may therefore

offer a possible explanation for the low values of

spectroscopic gravity associated with stars of the

higher luminosity classes. This mechanism can far

outweigh the effects of radiation pressure.

THE CLASSICAL CEPHEID AS AN EXAMPLE

The phenomena discussed above are all governed

by a well-defined set of physical equations and can therefore be studied in detail. The relevant

equations are the familiar equations of stellar

structure, but without the restrictive assumption of

hydrostatic equilibrium normally used in stellar

evolution computations. As is the case for static

stellar structure, analytic solutions are made

possible by simplifying assumptions, particularly

with regard to the equation of state and the opacity.

One can deduce general properties from such solutions, but it is difficult to obtain quantitative

models for comparison with observation. One therefore resorts to numerical methods , thereby allowing

the inclusion of virtually any desired amount of

detail in expressing the thermodynamic and optical

properties of the gas.

For the present application, a radiationhydrodynamics code 9

' written especially for cepheid

computations was employed 2. The code is a onedimensional, spherical, lagrangian, LTE, transport

code employing an augmented 9 gray absorption coefficient. Convention and stellar rotation "are not

taken into account. The methods used are a generalization of and are patterned directly after the

methods of Henyey, et al. 11 The final results are independent of the choice

of the initial configuration so long as the computations are carried far enough in time so as to be305
l_J I 1 I L I L I I I 1 I l_l 0.6 0.8 0 0.2 0...4 0.6 0.8

PHASE

Figure 1. Radius vs time history in the atmosphere

layers of a model cepheid.

come reasonably repetitive. The cepheid is therefore

an ideal choice for exploratory computations both

because the model is relatively free from arbitrary

assumptions and because the wealth of available

observational data permits a detailed verification

of the results. The initial configuration used in

present computations consisted of an augmented 2 version of a 7.6-day cepheid model supplied by

Christy 12

. In the present context we are interested in the

dynamic and optical properties primarily during those

phases of the pulsation when unloading occurs. However, for purposes of orientation, Fig. 1 shows the

radius versus time history over the entire cycle for

every fifth mass point in the computed results. The

dotted line represents the photospheric radius. SI

indicates the location of a shock originating in the

hydrogen ionization zone which travels inward in

terms of radius, but outward in terms of the mass

306
70

30

,'S2 /S3

T7 1 r-pr /S2 /sa

SHOCK

RAREFACTION \ 7=1 \ _1 i I I 1 1

70

30

0.0 0.2 0.4 0.6 0 •PHASE

0.0 0.2

Figure 2. Atmospheric phenomena in a model cepheid.

of the star. This shock causes the bump in the light

output during the rise to maximum light, but it is

not important to the unloading process. S2 is the

primary shock originating in the He II ionization

zone which is responsible for the pulsation of the

star. It reaches the edge of the star just after

maximum light and initiates the unloading process

which lasts until. a blow-off shock occurs at about

phase 0.2.

These events are shown more clearly in Fig. 2 in which the solid line shows the motion of the

photosphere throughout the cycle. The short time

required for the principal shock S2 to traverse the

photosphere, compared to the rarefaction-unloading

phase R2, is clearly evident. The process repeats

itself as the first blow-off shock S3 traverses the

photosphere near phase, 0.2 and the resultant rarefaction R3 follows. At about phase 0.35, a bump in

the rarefaction R3 can be seen in Fig. 2. This can

be identified in Fig. 1 with the reversal of direction

307
of the photospheric layers as they reach maximum

altitude, and corresponds to the observed secondary

maximum in the light output. A second blow-off

shock, S4, occurs at phase 0.4, which is soon joined

by a reflection S5 of the primary shock off the

center of the star. Following this, the atmosphere

is characterized by a gravitational collapse. The

properties of the model during the rarefaction phase

R2 are of primary interest. Fig. 3 shows representative density and temperature profiles during this

process. The shaded area serves to locate the photosphere, which occurs at mass zone 54 at this phase.

The strange looking density spike is not a shock wave,

but is a result of the rarefaction process. As the

shock approached the edge of the atmosphere, the

rarefaction process had already been initiated in the

region behind the shock front. This causes the outward acceleration of material which then overtakes

material at larger radii causing a density build-up.

The net result is an extended transparent region of

relatively high density. The corresponding temperature profile, which has been superposed below the

density profile in Fig. 3, shows an extremely steep

gradient in the photospheric region, and an extended

temperature plateau in the outer atmosphere. The

geometrical thickness of this region is approximately

308
one solar radius at this phase, but becomes almost

a factor of 10 larger during the pulsation cycle.

The type of atmospheric profile shown in Fig. 3 results in the emission of a continuum and spectral

lines which are different from those which are observed for a hydrostatic atmosphere. Oke has

published detailed spectral scanner measurements

of the continuum at the appropriate phase (0.1)

for the 7.2 day cepheid Eta Aquilae. Six color

measurements have also been published by Stebbins,

Kron, and Smith 1

4

. Attempts to match these observations by hydrostatic model atmosphere continua 1 3,15

have shown that such models cannot be made to fit

accurately at both the red and violet extremes of

the spectrum.

Using the method of dynamic atmospheres, the

author 2 was able to obtain improved correspondence

with the observed continuum. This method utilizes

a model consisting of a black body source screened

by a semi-transparent layer having a given density,

temperature, and thickness. The dynamic model fitted

to the observations at phase 0.1 consisted of an 8400°K black body screened by a layer of density 10" 9 gm cm" 3

, temperature 550 0°K, and thickness 1.25 x 10 11 cm. The detailed composition of the layer is

given in Table 1. The profiles in Fig. 3, from the

photospheric layer (zone 54) outward show a close

resemblance to this model.

As compared with a hydrostatic atmosphere, the

unloading atmosphere has a lower flux in the ultra- 0 violet but has an excess spectral flux near 10,000 A.

The extended layer has maximum opacity in the UV and

most of the observed radiation is emitted in this

layer, while at 10,000 K the layer is more transparent and most of the radiation originates in the

higher temperature photospheric region. The actual

spectral continuum computed from the atmospheric profiles given in Fig. 3 agrees qualitatively with the

observations, and can be made to agree quantitatively

if the temperature in the plateau region is raised

about 250°K. The continuum is especially sensitive

to this temperature as it determines the population

; of the Balmer ground state upon which the spectral

absorption coefficient is critically dependent.

The absorption lines formed in the atmosphere

during the unloading process also show properties

that are of interest. Both the unusual temperature

and density profiles and the differential motion

within the plateau region influence the character of

the lines. For a velocity distribution that can be

represented analytically, Underhill 16 has demon309
TABLE l t DETAILED COMPOSITION AND THERMODYNAMIC PROPERTIES

OF DYNAMIC MODEL LAYER FOR ETA AQUILAE

AT PHASE 0.1

Tnfal NnTnh(=>T" Df=>n c;"i t"V 4 . 30 x 10 ll+ /cm 3 Electron Density 3 . 80 X 10 11 /cm 3 HI 3 . 77 X 10 li+ /cm 3 HII 3 . 39 X 10 11 /cm 3

TT —

ri -L • / O X TO 5 /r>m 3 -Lu / cm

Ht 2.44 X 10 5 /cm 3 H 2 9.28 X 10 8 /cm 3 Hel 5.28 X 10 13 /cm 3 Hell 1.63 X 10 1 /cm 3 Helll —

Metals Composition GMA

Metals to Hydrogen Ratio 2.0 x

10*" 3 Helium to Hydrogen Ratio 0.167

Temperature 5500 • °K

Total Pressure 3.289 x 10 2 dynes/cm : Radiation Pressure 2.3 dynes/cm 2 Gas Pressure 3.266 x 10 2 dynes/cm

Electron Pressure 0.289 dynes/cm 3 Molecular Weight 1.40 4 gm/mol

Density 10~ 9 gm/cm 3

strated that a uniformly expanding atmosphere causes

the spectral lines to become shallower and broader.

For the present application, a general program

was written to accept an arbitrary velocity distribution in the atmosphere. Specific results for se- lected spectral lines are discussed below. The

primary purpose of these calculations is to show the manner in which the dynamic properties of the atmosphere influence the shapes of the spectral lines.

No attempt has been made to determine the extent to

which these lines agree quantatively with observation.

310
Figure 4 shows schematically the instantaneouslyfixed geometry of a model star consisting of spherical shells of material (70 in the model used here)

whose radii, temperatures, densities, and velocities

are known from the radiation - hydrodynamics computation. Assuming LTE and a GMA chemical composition,

the thermodynamic properties of each shell, including

the concentrations of all species, follow from the

equation-of-state . The continuum 2 and line absorption 17 coefficients can then be computed directly

from the atomic and molecular properties. The Voigt

function was evaluated using a routine due to

Rybicki 18 while Stark broadening was evaluated using

the method of Edmonds, Schulter, and Wells 19

. To obtain the emergent intensity 1^ x f°r a given

radiation frequency, an integration is carried out

along a ray path through the star defined by the

impact parameter x. Values of x are selected so as

to cause the path of integration to bisect each

spherical shell as indicated in Fig. -4. The observed

flux is then given by

2 n

.A

'V ,x xdx

311
while the emergent intensity is given by

B

1^ = f B ( y z) e'U Z d(y z)

V , X J V A

where y = yi +y

c

y = continuum absorption coefficent

yi = line absorption coefficient

z = geometrical distance measured along the

ray path

B = Planck function.

v

Since the model results from a radiationhydrodynamics computation, one is not free to choose

the zoning structure so as to maintain a small optical thickness for all of the individual shells of

material. The source function is therefore interpolated in optical space 9 to achieve greater accuracy. The radial motion of each shell is reduced to

a velocity along the ray path in the evaluation of

the line absorption coefficient.

The X 4508 A line of Fe II was used as a test

line to study the effects of the atmospheric motion

upon the line shape. Two computed profiles for this

line are shown in Fig. 5. The narrow line with

steep sides and flat bottom was produced by arbitrarily setting all of the shell velocities to zero. The broadened line results from the velocity distribution given by the radiation-hydrodynamics computation. It is clear from a comparison of these re- sults that the motion of the atmospheric layers

broadens the line and increases its equivalent

width. The central flux is increased slightly.

Further numerical experimentation, in which the

entire atmosphere was assigned a uniform expansion

velocity revealed that both the uniform expansion

and the differential expansion make significant

312
contributions to the line shape. The differential

expansion causes an appreciable increase in the

equivalent width of the line. The feature on the

violet side of the profile is caused by a region of

abnormally high outward velocity occurring near the

head of the rarefaction wave. The asymmetry of the

line appears to be consistent with the analytical

results of Van Hoof and Deurinck 20

. The broadened contour of the computed A 4508

line is similar to the shape for the cepheid SV Vul

as observed by Kraft 21

, et al. However, in the

present computation, neither stellar rotation nor

a "turbulent velocity" were assumed in order to

produce the line broadening.

The profiles of the first four Balmer lines were also computed. The H^ profile is shown in

Fig. 6. The computed profiles for the four lines

are all similar, but show a lower central flux

and broader wings with increasing principal quantum

number. The depression of the continuum over a

313
wide region of the spectrum is to be anticipated

on the basis of strong wings of these lines. The

effect should be particularly evident near the

series limit where the wings of the higher members

of the Balmer series should overlap and effectively

extend the Balmer continuum well to the red of the

series limit.

The computed profile for the Ca II K line is

also shown in Fig. 6. The line is quite broad and

has a weak red shifted emission core. This emission

core results from a slight temperature inversion in

the outermost shells of the cepheid model as shown in

Fig. 3. The dashed line in Fig. 6 shows the change m the K line contour when the outer shell of the cepheid model is arbitrarily removed during the line profile computation. It is interesting to note that

the emission core is red shifted, while the absorp- tion profile is blue shifted, even though all parts

ot the atmosphere are moving outward and might there- tore be expected to produce a blue shift. While the reason for this effect has not been positively identified, it apparently is confirmed by the obser- vations of Jacobsen 22

. an,n^He ^ comPuted Profiles have not as yet been quantitatively compared with densitometer profiles, it is clear that th«* nni^n — _ v \T~ # .ar ty

314
GENERALIZATION TO OTHER STARS

A number of lines of evidence indicate the

possibility that the unloading mechanism might be

operating in stars other than cepheids . Kraft 23

,

j in his extensive discussion of the spectra of supergiants and cepheids, concludes that, "the classical

cepheids have spectra that, in detail, are virtually

identical with those of the non-variable supergiants

at all phases of the variation." Abt 7

, in his study

of the variability of supergiants, finds that "probably all stars in the H-R diagram above Mv = + 1 and

to the right of the main sequence are variable in

light and radial velocity." In addition, he notes

that many of the supergiants show a multiple or

; irregular periodicity in velocity that suggests that

secondary shocks might be present.

Taken at face value, these lines of evidence

raise the interesting possibility that all luminous

stars above the main sequence could actually be variable stars. However, since the stars we normally

refer to as "variables" seem to occur in well defined

"instability strips" in the H-R diagram, such an interpretation would require that a selection process

be operative. Such a selection process could hardly

be observational, and would have to have a physical

basis

.

Historically, the various classes of variable

stars have been identified by their relatively large

light variations rather than their velocity variations. Abt 7 found that the "non-variable" supergiants have light amplitudes of 0.0 5 - 0.10 mag.,

whereas the "variable" stars have light amplitudes

on the order of 10 times this amount. The velocities

of the "non-variables" have not been studied in

sufficient detail to establish any correlation with

these light variations. Thus, while it is clear that

the existence of instability strips must ultimately

be associated with the interior structure of the

star, the manner in which this structure manifests

itself in terms of observables is not well understood.

A possible cause for the selection phenomenon

can be deduced from an analysis of the model cepheid

computation discussed earlier. The author 2 found

that the large increase in light amplitude in the

cepheid resulted from an abrupt spatial density

decrease on the back side of the shock front. After

shock transparency, this rapid density decrease

causes the photosphere to move rapidly inward into

315
the star, thereby producing a rapid temperature in- crease in the photospheric region.

i't was further found that in the cepheid the

steep density decrease behind the shock results

from thermal ionization of hydrogen in these layers.

If one follows the progress of a given shock from

the deep interior of the star to the surface, one

finds that the density profile just behind the shock

front changes rapidly. Whenever ionization (or dissociation) is occurring behind the shock, a deep

density minimum occurs in this region. The density

at this minimum may be as low as a factor of 10 below

the shock front value. However, when no ionization

occurs, the minimum density is rarely more than 30%

below the shock front value.

This suggests that the production of a large

light amplitude requires that ionization or dissociation occur just behind a shock front during the

relatively short time interval when the shock trav- erses the photosphere. It also implies that the

passage of a shock front through the photosphere

of a star does not produce large observable effects

unless ionization or dissociation is in progress

just behind the shock front. This "shock visibility

factor" might well be the cause of the apparent instability strips in the H-R diagram.

If one compares the emergence of the primary

shock and secondary shocks in the cepheid computation, one finds support for the above interpretation.

The shock velocities are nearly equal, but the

secondary shocks are followed by only a shallow

density minimum. The increase in the computed light

output is about 1 mag. for the primary shock emergence, but only about 0.1 mag. for the secondary

shock emergence.

There is some observational evidence from

variable stars to support this computed result.

Bernheimer 2

4

observed a light increase in Eta Aquilae

of 0.15 mag. in 40 minutes. Preston 25 has observed

similar rapid low amplitude light variations in RR

Lyrae which occur at phases where the emergence of

shocks are expected from the radiation-hydrodynamics

computations

.

A further test of the hypothesis can be made by

plotting the traces in the H-R diagram where various

ionization and dissociation processes are 50% complete. This can be accomplished by using the data 26

to convert Mv values to log Pe values , and the

equation of state. The 1 temperature values associated

with these traces are the temperatures in the region

behind the shock front. The effective temperature

316
of the reversing layer just prior to the shock emergence should be lower than this temperature, and

systematically related to it. If the major classes

of variable stars are superposed on this H-R diagram,

it is found that their reversing layer temperatures

are about 2/3 of the temperatures where He I ionization, H ionization, and H 2 dissociation are 50%

complete. However, this result should be considered

as suggestive/ rather than conclusive, since the

variables occupy rather broad strips in the H-R

diagram and it is difficult to establish the Mv

- P e relationship in some parts of the diagram due to the

lack of stars. In any event, the unloading process

appears to be interesting and should receive further

study.

In conclusion, I would like to thank Drs . L. Henyey, R. K - M « Landshoff, R. E. Meyerott, and M.

Walt IV for their continued support and encouragement .

REFERENCES

1. R. Weymann, At>tn.ophyt> . J., 136 , 476 (1962).

2. R. W. Hillendahl, Dissertation, Univ. of

California at Berkeley (1968)

.

3. Ya. B. Zeldovich and Yu. P. Raizer, Phytic* o{

Shock Wave.6 and Hlgh-Te.mpe.fiatu.sic HydKodynamlc Phe.nom2.na, Academic Press, New York

and London (1966), pp. 101, 716, 762.

4. G. A. Bird, Atttiophy* . J. , 139 , 675 (1964).

5. A. Sakurai, J. fluid Mech. , 1, 436 (1956).

6. H. C. Arp, Astronomical J., 60, 1 (1955).

7. H. A. Abt, A*tn.ophyt>. J., 126 , 138 (1957).

8. G. E. Moreton, Sky and Telescope , 21, 145

(1961)

.

9. R. W. Hillendahl, "Approximation Techniques for

Radiation-Hydrodynamics Computations ,

Defense Atomic Support Agency Report 152 2 (1964)

.

10. R. W. Hillendahl, Proceedings oft the. Workshop

on the. Interdisciplinary Aspect* ofi Radiative.

Transfer, ed. R. Goulard Joint Institute

for Laboratory Astrophysics, Boulder,

Colorado (1965).

11. L. G. Henyey, R. LeLevier, R. D. Levee, K. H. Bohm, and L. Willets, Astrophys . J., 129, 628

(1959)

.

12. R. H. Christy, California Institute of

Technology, private communication (196 8)

.

317
13. J. B. Oke, kbtKophy* . J., 133, 90 (1961).

14. J. Stebbins, G. F. Kron, and J. L. Smith,

UtlopkyA. J., 115, 292 (1952).

15. C. Whitney, Dissertation, Harvard University

(1955)

.

16. Anne B. Underhill, kbtnopky* . J., 106 , 128

(1948)

.

17. L. H. Aller, A-6 tnophy&lab , Ronald Press, New

York (1953)

.

18. G. B. Rybicki, "Routine for Evaluation of the

Voigt Function," forwarded by D. G. Hummer.

19. F. N. Edmonds, Jr., F. M. Schluter, and D. C.

Wells, III, Mo,moi>u 9 Roy, St. Soc. 7JL, 271

(1967) . 20. A. Van Hoof and R. Deurinck, Ki> tHophy* . J.,

115 , 166 (1952) . 21. R. P. Kraft, D. C. Camp, J. D. Fernie, C.

Fujita, and W. T. Hughes, Ah trophyh . J.,

129 , 50 (1959)

.

22. T. S. Jacobsen, Pub. Dominion A6t. Ob*. X,

No. 6, 145 (1956) . 23. R. P. Kraft, in Stdlldh. Atmo* pkztio,* , ed. J.

Greenstein, Univ. of Chicago Press (1960)

.

24. w. E. Bernheimer, Lund Medd. , Ser. II, 61, 11

(1930) . 25. G. W. Preston, AttKophy* . J., 134, 633 (1961).

26. C. W. Allen, Ahtno physical Quan title,A , Univ. of

London, Athlone Press (1963), pp. 201, 208.

DISCUSSION

Wcllmann: If you extrapolate a nova outburst

backward, you will find that a large fraction of

the mass has been thrown out during a very short

time. This is like the unloading of a shock front.

There are also secondary outbursts. Can these be

explained by your secondary shock waves? Would your

model give the right densities and time scale?

Hille.nda.hZ: My model contains many possibilities. It can probably also explain a nova outburst.

The production of secondary shock waves by a first

shock is a very common physical phenomenon. It is

not limited to cepheids, but it is observed in the

solar atmosphere as well as in the laboratory; it is

also observed with supersonic airplanes and during

the re-entry of Gemini space-vehicles.

Lament* : can your mechanism explain the standing

waves in P Cygni type stars, which exist according to

a thesis by de Groot?

318
Hlllzndahl: Perhaps.

Unde.t1k4.ZZ: It is pleasing to see that your

predicted profile for Fe II A4508 resembles more

closely the observed profiles than is the case for

outward and inward moving layers. In particular

note the difference in asymmetry between the profile

j from the model with the computed velocity distribuI tion and the profile formed for layers moving with

i a constant velocity.

StsLbbA : I should like to ask, what is the basic

j physical reason why the proposed shock mechanism

leads to a violation of Wesselink 's method?

HlZZzndakZ: If you consider the difference

between the motion of the masses and the motion of

I the photospheres you will find a radius that differs

! by 20 percent from the radius found by Wesselink'

s

| method. I think my method is a modified Wesselinkmethod.

?tiado.tiA.e.: Are your results contradictory to

j Kippenhahn's calculation?

HlZZe.nda.kZ: I didn't say anything about the

exciting processes.

?fiad2.fil<i' But how are the shocks produced?

HlZZe.nda.kZ: They are produced by the mechanism

given by Christie.

319

PART D

SUMMARY
:
CONCLUDING REMARKS

by

J. C. Pecker

Observatoive de Meudon 3 France

We have seen from the talks of A. B. Underhill

and R. N. Thomas how the viewpoints of the many

kinds of astronomers who came to this meeting differ,

not only with respect to what they mean by "extended

atmospheres" but also concerning methods for studying extended atmospheres.

For a long time spectroscopists and theoreticians of stellar atmospheres have not moved exactly

in the same universe. This conference is an attempt

to fill the gap, we all have seen that it is not

easy, but progress has been made. I shall try to

summarize the progress starting from the general

conceptual approach and trying to come back to the

particular cases that puzzle observers.

I. THE CONCEPTS

The problem, as I said, is that for a long time

it was found to be very difficult to build models

of the good classical type, that is in RE (radiative

equilibrium) , HE (hydrostatic equilibrium) , LTE

(local thermodynamic equilibrium) , which might

represent the most honest stars that theoreticians

might be tempted to like. Most of the theoreticians

were sufficiently aware of this fact not to give

more than very qualitative suggestions on how to

explain monsters. On the other hand, however,

observers, i.e.., spectroscopists, have had a tendency

to concentrate on stranger and stranger objects, to

have more fun, probably, and they seemed to have

been satisfied with zero-order-approximation interpretations —often being so crude as to assume LTE!

In a way they were like the entomologists or zoologists of the eighteenth century (thus proving that,

contrary to a widely distributed opinion, physical

323
sciences might follow in their development the

biological sciences) . Clearly it is no longer pos^ ? sible to be satisfied by this state of affairs. The

age of computers has finally come and theoreticians

are now able to challenge the analytical or numerical difficulties that one meets when trying to

understand nonclassical situations.

In this theoretical approach there are two

phases: (a) The first problem to solve, as usual, is:

that of diagnostics . This is essentially what we have tried to do during these three short days. We

have tried to interpret the observed features in

terms of models, in the broader sense of this word.

I shall come back to this point.

(b) The second problem is the physical interpretation or understanding of models that fit the observations better. We barely touched this problem and we hardly mentioned the physical choice among the

variety of models that can represent a given set of

observed features. Clearly this approach is linked

with stellar evolution, for stellar evolution might

affect more rapidly the nondense external layers

than it does the general structure of a star. It is

only after we shall be able to enter this second

phase of the theoretical approach that we may be

able to put some unity, or at least some logic, into

the description and understanding of our zoological | garden.

I shall come back to this point at the end of

these final remarks. In a general sense, we should

keep in mind the few questions asked a long time ago

by Otto Struve, and of which Anne Underhill has very

wisely reminded us. We should remember they are still basically unsolved, and are still the essential

questions. Slightly reformulating the first ones, one has to reply to the questions: Why have some stars extended atmospheres—whatever meaning each

of us gives to the words—and some other stars,

otherwise apparently identical, have none? What,

in the stars with extended atmospheres, is the cause

of the departures from RE, HE, and LTE that one needs to introduce in order to interpret the obser- r vations?

It seems that to the last question at least we \ have a partial reply, which we all seem to agree

upon, and about which Anne Underhill and several

other speakers have been insisting very much and

very rightly. It seems indeed that we do know why

there are departures from LTE in a given configuration—or to say it better, why we should not

expect LTE to apply. For instance, as stressed by I

324
Anne Underhill, we know that in low density atmospheres we should not expect enough collisions to

maintain the extreme situation of LTE . That this

is indeed true for all atmospheres has been stressed

by R. N. Thomas. This is clear, and I will not come back to this point.

But why are there departures from RE or HE?

What heating mechanisms can possibly alter the

electron temperature distribution of a classical

atmosphere? What is sustaining or pushing the atmosphere in such a way as to produce low effective

gravities in the outer layers? We do not know! Our

ideas are too vague to allow us to compute these

effects in a stellar atmosphere defined by a few

observable parameters such as its location in a color-magnitude diagram, or even for a stellar atmosphere much better, defined, such as that of the sun. One can hope that our improvements in the nonLTE atrophysics will enable us to describe in a better way the observations, one cause of the indeterminacy being thus taken away. Figure 1 describes the logic of the methods in use. In addition, the kind of wrong and artificial

determinations that might have been given to the

problem by the use of LTE astrophysics and that might

have led in some cases to wrong conclusions about the

geometry or the velocities, should be now easily

avoided. We know that LTE is an assumption that one

cannot' make a priori, that one cannot use this

assumption unless it has been unambiguously proved

in a given particular case, that it may, indeed, be

used. We know that this is very rarely the case,

especially for stars with extended atmospheres.

How to include non-LTE in the transfer equations

and how to do it without neglecting curvature terms

or velocity terms, or even inhomogeneities has been

discussed here at length and in a very clear way by

George Rybicki, and after him by Kalkofen, Magnan

,

Grant, and Skumanich. However, the theory has still

a long way to go. Even in cases with simple velocity fields, we have seen that it is difficult, sometimes, to take into account the proper depth variation of the physical parameters. We have also seen

with Nussbaumer that severe doubts still exist con- cerning the physical data needed for the astrophysical theory. Precise physical data is obviously

a serious need of non-LTE astrophysics.

But we have seen also that the theory is sometimes satisfying. The range of possible models not

only has decreased but some absurd ones cannot be

considered anymore. We have seen that the continuum

325
COi

o

—1 Q CO ZlUJ Ld <> <r

LlqUu co h-m Ld WO Ll

CO

COLd PQI

CO

12 Ld u U

ET %

U) z

a

<\RA BUN Cl <

z

o

£

z

to

COu u

O

3

326

1
spectrum of a star like Eta Carinae is well represented by the nonclassical continuum proposed by

Gerola. We have seen, in Kalkofen's paper for

example, good (but possibly still not unique) representations for P Cygni lines; Rybicki gave us some examples of profiles of which the asymmetry computed

was comparable to those of actual supergiants.

Magnan has succeeded in mimicking the Be stars

reasonably well; the analysis of WR lines with some non-LTE profiles in plausible geometrical dynamical

situations has been noted several times . We have

also seen that chromospheric spectral features might

be described by some reasonable models, such as those of Kandel, reported this morning by Francoise

Praderie. We have seen also the notable progress

described by Groth on the chromosphere of the K

type component of Zeta Aurigae. Even in the peculiar case of novae, we are coming to a better understanding of the spectrum by making a careful discussion of the ionization, as shown by Wellman.

Certainly even when the crude approximations

sometimes made are taken away, a unique description

is not yet reached. This has been shown this morning

by Groth, but was clear also from several examples

given by F. Praderie, and quite clear from

Skumanich's interpretation of the K-line reversal in

a few red giants.

Coming to this last controversy, we must understand the indeterminacy. A given line is indeed an image, through some convolution, of the trend of the source function with the optical depth. To go

from that law to the distribution of physical

quantities, we need a model. The critical relation

in it might be t (h) , the relation between the opacity

and the altitude in the atmosphere. Eclipses can

lead us to guesses about t (h) but in other cases we

are left without much indication of this behaviour.

The relation t (h) is strongly coupled with the set

of temperatures or densities as a function of the

altitude which are coherent, through non-LTE theory,

with the source function itself.

This difficulty, I believe, is the chief

difficulty we shall meet in the stellar cases, the

very origin of the indeterminacy mentioned above.

Possibly only careful solution of the simultaneous

problem of several different lines can help us. The

profile of a line is an indication about S (x) , but

the profile is not dependent on one function only

but upon relations such as Te

(h) and ne (h). How

then to solve for this uncertainty of the diagnostics, an uncertainty which is quite general?

327
In given cases we get a set of possible behaviours of Te

(x), x(h) and v(h) that depart more or less strongly from the radiative equilibrium temperatures, from the hydrostatic equilibrium relation between t and h , and from the non-moving atmosphere.

The choice is to be determined by the physical plausibility of the models. However we know very little

about how to make this choice —as has been said many

times during these three days

!

II. APPLICATIONS

The methodology being thus clear, but still

insufficient, let us see if we can describe in some slightly coherent way the various phenomena we have

been talking about. We have considered several

types of extended atmospheres following more or

less Thomas' classification scheme. It is somewhat

difficult or even meaningless to locate some of the

types of object we are considering on a colormagnitude diagram, but let us attempt it (see Figure

2) . (1) We have talked about stars in which the

extended character is rather severe (WR stars, supergiants, Be stars). These stars are generally highly

luminous, in rapid evolution, and some of them are relatively young. It is remarkable, as in the case

of the nuclei of planetary nebulae, to see that they

are located in the HR diagram near the instability

limit set up by the radiation-pressure negative gravity. In other words, the physical phenomenon that

appears important is essentially a mechanical support

of the outer layers. Clearly the difference between

objects with and without an extended atmosphere

(everything else being alike) might be a difference

of age --a difference that can be indeed linked also

with really circumstellar phenomena such as dust

clouds. The kind of theory brilliantly developed

today by Hillendahl could be compared to this case. Through shock waves reaching the surface, a low

effective gravity atmosphere is created—even relatively far from the above quoted instability limit.

(2) Later we turned our attention to more moderate examples, for which we suspect that phenomena similar to those shown in an embryonic way in

the sun are also present, possibly more conspicuously. This subject was extensively discussed by

F. Praderie.

Different mechanisms that act in different ways

can be invoked. In the sun and similar stars, the

328
heating (due to what?) is probably more important

than the mechanical support. But in active younger

stars, such as T Tauri > very active phenomena integrated over the stellar disk might appear to give

strong chromospheric-like phenomena. The heating

there is closely linked with the magnetic properties

329
of the stars —as it may also be in the case of some A stars.

Thomas has shown how the study, not of the sun itself, but of the evolution of thinking in the case i of the sun could be mimicked, mutatis mutandi , in

stellar cases, account being taken of the fact that

the phenomena might be physically different or of a different order of magnitude.

I would like at this point to repeat what Thomas

said about the fact that the measurement of continua

in UV or IR would be very useful to get the location

in the atmosphere of the minimum temperature. To

his comments I would like to add that the mass of

matter that is above (or outside) the location of

the minimum temperature is a parameter which may

vary within very large limits; and it might be an essential parameter. Having this remark in mind, one can understand why the UV solar spectrum- looks

like the visible spectrum of some of the monsterstars; just because the mass of the extended atmosphere —or call it chromosphere if you like —is

bigger, much bigger, than that of the sun quite

irrespective of what the physical process is which

keeps supporting the extended atmosphere. This

really explains why the discussion of some features

of the UV spectrum of the sun (0 I lines) by Athay

fitted into the program better on Monday together

with the discussion of the chromospheric phenomena.

If we limit ourselves to the observation of the

visible integrated light of a star, the chromospheric;

phenomena in objects that might be low in the HR diagram can be observed, as has been told by F. Praderie,

only in the center of some very strong lines. This

is naturally a severe difficulty.

When dealing with this kind of very fine and

difficult spectroscopy one cannot be satisfied with

the kind of rough theory that was described yesterday. It became clear today, at several moments in

the discussion, that we should improve the theory

greatly to take into account inhomogeneities and to

introduce wavelength-dependent source functions,

i.e., better frequency redistribution in lines than

is generally assumed. Clearly a lot of progress has

yet to be made along these lines.

i III. CONCLUSION

Clearly observers should observe and theoreticians theorize, and they should incubate together!

330
We now have a better idea which lines are sensitive

to what and what theories are likely to be applicable

in a particular case. Let us all remember the phi-

; losophy of two famous thinkers: "Things are not what

they seem" (Thomas) and "I was pleased—and then I started reading" (A. B. Underhill) . In no better way

can the need for controversy and for colloquia be

expressed.

IV. ACKNOWLEDGMENTS

Yesterday, when Dr. Wellmann was replying to

the expression of our • gratitude that had been given

; by Anne Underhill, he said that it might be too

early, the meeting being not yet over. I may say

i today that, the meeting being over, we know perfectly

well that Anne Underhill was completely right; it is

a great pleasure for me to again thank very heartily

Dr. Wellmann and Dr. Groth for all that they have

done in order to allow us to take the utmost benefit

out these three days, not only through scientific

discussions, but through our contacts with our German

friends. I would like, Dr. Wellmann, to ask you,

also, to convey our thanks to the secretarial staff,

which has been so helpful for all of us, and also to

your colleagues and students who have been helping

so much in many ways, and especially by recording

the whole of this conference, and by taking accurate

notes during it.

It is customary to limit the thanks to the local

organisers, but I think we should also express our gratitude to Anne Underhill, who has been so active

in organizing this meeting, and who, fortunately,

let it be sufficiently unplanned. It was indeed

an excellent thing that this meeting was unplanned,

because it gave us the opportunity of extensive

discussions about the various points raised by the

speakers. I thank also all those who have contributed to the discussion, especially (for different

reasons) the younger ones, and our old friend

Dick Thomas.

I have said earlier that we are able, now, to

express our gratitude to Dr. Wellmann and Dr. Groth,

the meeting being finished. Actually, we should

possibly wait a little bit, because they have still

an enormous amount of work to do with the proceedings

of the conference! This is an enormous task, but

judging by what I have seen these days, I think it

331
is quite safe that we can express our gratitude in

advance to Dr. Wellmann, Dr. Groth, and their colleagues, for what they will do for us in the near

future.

The meeting is adjourned.

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