Smithsonian

Contributions to Astrophysics

VOLUME 5, NUMBER 11

ON THE A AND $ OPERATORS

OF RADIATIVE TRANSFER THEORY

by S. S. KUMAR

SMITHSONIAN INSTITUTION

Washington, D.C.

1962
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FEED L. WHIPPLE, Director,

Astrophysical Observatory,

Smithsonian Institution.

Cambridge, Mass.

II

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington 25, D.C. - Price 30 cents
On the A and O Operators

of Radiative Transfer Theory

By S. S. KUMAR1

Introduction

For some time I have been working on an

analytical method for computing fluxes in

model stellar atmospheres (Kumar, 1961).

The method will be discussed in greater detail

in another paper, but since it makes use of

the A and $ operators an outline is given

here.

We approximate the monochromatic source

function S, at any depth T, by an expression of

the type

M

0

r, (i)

where am, bn, cp, dp are unknown parameters

and En(Tr) is the n-th exponential integral

function defined by

(2)

The monochromatic source function satisfies the

following integral equation:

where a, is the coefficient of electron scattering,

K, the coefficient of continuous absorption, and

B, the Planckian function. The operator A is

defined by

}=lj J(t) Erdt-xDdt^AU (x)]. (4)

If we put the assumed solution (1) for the source

function in equation (3), we get

1

 Smithsonian Astrophysical Observatory.

(5)

The parameters am, bn, cv, dp are obtained by

solving a set of transcendental equations obtained from equation (5).

The monochromatic flux at any depth and for

any frequency is given in terms of these parameters by the following expression:

2 * (En)

(6)

where the operator * is defined by:

-2$j(t)E2{x-t)dtm*{f(x)\. (7)

In order to use this method we must

know the A and $ operators of the functions

r

m

, En, e~

d

p

T

. The operators of the function

En have been computed by Kourganoff (1952),

but since no tables are available for the operators of the other two functions, the object of

this paper is to present them (tables 1-4).

1.5.3
154 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

The A and $ operators are also very useful

in obtaining the temperature distribution for

nongray model atmospheres (Swihart, 1956;

Bohm, 1954), not at present known exactly.

We start, therefore, with a temperature distribution that corresponds approximately to the

nongray model, and we employ a procedure for

correction. To show the importance of the <£

operators, we present here a generalization of

Swihart's method. The initial approximate

temperature distribution is represented by

(8) 2E2+b3Ei+. . .

To obtain the temperature distribution for the

nongray model we use the <£ operator and get

the following expression for the correction AT:

=|7 1 [Aao+(Aa1)r + . . .(Ab2)E2

+ (Ab3)E3+.. . Acoedr+ (9)

where Aao, Aalt . . . are obtained by solving the

following set of algebraic equations:

=^ T\

-...] , (io)

where Ai*' is the known difference between the

desired flux and the flux corresponding to the

gray model. Since the $ operator is probably

insufficient to give the correct temperature distribution, we must also use the A operator.

The formulas used in the computations

The following formulas, given by KourganofF

(1952), were used to prepare tables 1 and 2 for

the $ and A operators of e~

dT

:

d+1

\d-l\

— (11)

(12)

The functions E^T) and E^r—rd) are given by

JS;1(T) =—0.5772156—log |i

and E2(T) is given by

i • i\

(13)

E2(j) = e

 T—TEI(T). (14)

For the special case of d=l, the following

formulas were used for the $ and A operators:2

-log, 2r}

A(e-')=|

(15)

[log, 2r+0.577215]+i Ex(r). (16)

For computing the function E^x), where x

stands for either T or (r—dr), subroutine

NUEXPI,3 distributed by SHARE, was used.

The procedure for computing Ex{x) follows.

Actually, the subroutine computes the function

Ei{x), which is defined by

E^^—Eii—x).

For a different range of values of x, the following

formulas were used to obtain Ei{x):

i)

For this range the subroutine uses the following formula given by Hastings (1955):

where o's

Hastings.

and 6's have been tabulated by

1

 The operators were computed on an IBM-7090 at the Smithsonian

Astrophysical Observatory.

> Written by James W. Cooley at the AEC Computing and Applied

Mathematics Center of the New York Institute of Mathematical Sciences

and slightly changed by Owen Gingerich at the Smithsonian Astrophysical Observatory to permit its use with the main program for computing

the A and * operators.
A AND * OPERATORS OF RADIATIVE TRANSFER THEORY 155

ii)

For this range,

16

n=0

(18)

The quantities cn's have been computed by

Cooley.

iii) 5<z<24.5

For this range the following expansion is used

in the subroutine:

where

(19)

(19a )

For each value of n, x0 is a known quantity,

iv) 24.5 O .

The subroutine uses the following formula:

The accuracy of the A and $ operators is

dependent upon that of the function Ei{x).

The subroutine gives Et(x) correct up to seven

TABL E

d

0.01

0.05

0. 10

0. 20

0. 40

0. 60

0.80

1.00

2.00

3.00

4.00

5.00

6. 00

7.00

8. 00

9.00

10. 00

15.00

20.00

25.00

30. 00

1.— Values o/*o(c~dT)

1. 01060

0. 968400

0. 937820

0. 883950

0. 794100

0. 722203

0. 663162

0. 613706

0. 450695

0. 358602

0. 298820

0. 256658

0. 225228

0. 200838

0. 181337

0. 165367

0. 152042

0. 108688

0. 0847774

0. 0695741

0. 0590355

and A0(e~dr)

Ao(e~dT)

0. 497474

0. 487895

0. 476554

0. 455802

0. 420590

0. 391671

0. 367368

0. 346574

0. 274652

0. 231047

0.201180

0. 179176

0. 162159

0. 148531

0. 137326

0. 127921

0. 119895

0. 0924198

0.0761130

0. 0651620

0. 0572329

digits. The operators in table 2 are computed

and correct up to six digits.

Since EX{T—(IT) was computed for a large

range of the argument, varying from —87.00 to

+ 9.4, we have also tabulated E^r—dr) along

with the A and $ operators. For very small

values of d and T table 2 gives the rounded-off

values for (r—dr). For some values of d, we

have plotted the A and $ operators of the function ie'"*). Figures 1, 2, 3, 4, 5, 6 show the

operators for d=0A, 0.4, 1.0, 5.0, 10.0, 30.0.

For T= 0 and for any value of d the following

simple expressions were used

(21)

(21a)

Table 1 gives the values of *0(«

 dt) an d Ac

(e~dt) for several values of d.

For the function r

m

 the following formulas

were used:

(

m _* \

2 h &m+2-*)+(-i)mEm+3(T),

where

0 if /3= rn.+2—k is even,

{ is odd.

0 if /3

2/0 if 0

(22)

(23)

where

TO if a

*~ \2/aif a

= m-\-l — k is even,

is odd.

(24)

(25)

Tables 3 and 4 were computed for six values

of m=0, 1, 2, 3, 4, 5. For these computations

we needed the exponential integrals E2, E3, E4,

Eh, E6, E7, and .Eg, and used the tables published

by Placzek (1947).
156 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

FIGURE 1.—The A and <£ operators of i~dr, with d=0.1.

FIGURE 2.—The A and # operators of e~

dr

, with d=0A

FIGURE 3.—The A and * operators of e~

dr

, with </=1.0.
A AND * OPERATORS OF RADIATIVE TRANSFER THEORY 157

FIGURE 4.—The A and * operators of (~dT, with d=5.0.

FIGURE 6.—The A and $ operators

T

, with d=30.0.

FIGURE 5.—The A and * operators of /"**, with d= 10J0.

FIGURE 7.—The A and * operators of r*.
0.0 2.0 4.0 6.0 8.0 10.0

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

FIGURE 8.—The A and $ operators of T4.

The A and <£ operators of r

m

 are correct to six

decimal places. Figure 7 shows the operators

of the function r2. Figure 8 gives the operators

of the function r\

Acknowledgment

I am extremely grateful to Dr. Charles A.

Whitney and Dr. L. H. Aller for their helpful

discussions on the problem of nongray stellar

atmospheres.

References

BOHM, K. H.

1954. Die Temperaturschichtung der Sonnenatmosphare im nichtgrauen Strahlungsgleichgewicht. Zeitschr. Astrophys., vol.

34, pp. 182-208.

HASTINGS, C.

1955. Approximations for digital computers.

Princeton University Press, Princeton,

201 pp.

KOUKGANOFF, V.

1952. Basic methods in transfer problems.

Clarendon Press, Oxford, 281 pp.

KTTMAB, S. S.

In press. A method for computing monochromatic

fluxes in model stellar atmospheres. Astron. Journ.

PLACZEK, G.

1947. The function En(x). Nat. Res. Council

Canada, Atomic Energy Project, Chalk

River, Ontario.

SWIHART, T. L.

1956. Model atmospheres. Astrophys. Journ.,

vol. 123, pp. 139-142.

Abstract

The importance of A and 3> operators for the problem of transfer of radiation in nongray stellar atmospheres is

discussed, and extensive tables of these operators of the functions e~

dT and r* are given.
11

T

0*001

0.005

0.01O

0,050

0.100

0.200

0.300

0*400

0,500

0.600

0.700

0.800

0*900

l.OOO

1.100

1.200

1.300

1.400

1.500

2.000

2.500

3.OOO

3.500

4.000.

4.500

5.000

5.500

6.000

6.500

7.000

7.500

8.000

8.500

9.000

9.500

a

0.010

O.OlO

O.OlO

0.010

0.010

O.OlO

O.OlO

O.OlO

O.OlO

0.010

O.OlO

0.010

0.010

0.010

O.OlO

0.010

0.010

0.010

0.010

O.OlO

O.OlO

0.010

0.010

0.010

0*010

0.010

0.010

O.OlO

0.010

O.OlO

O.OlO

O.OlO

O.OlO

0.010

O.OlO

A AND #

TABLE 2.

T-dT

0.001

0.005

O.OlO

0.049

0.099

0.198

0.297

0.396

0.495

0.594

0.693

0.792

0.891

0.990

1.089

1. 188

1.287

1.386

1.485

1.980

2.475

2.970

3.465

3.960

4.455

4.950

5.445

5.940

6.435

6.930

7.425

7.920

8.415

8.910

9.405

OPERATORS OF RADIATIVE TRANSFER THEORY

—The A and * operators of the exponential function

. , -d T y

A(e )

0.501175E 00

0.511803E 00

0*522633E 00

0*583327E 00

0.635665E 00

0.709162E OO

0.760968E 00

0.300062E 00

0.830608E OO

0.854994E 00

0.874747E 00

0.890912E 00

0.904238E 00

0.915283E 00

0.924470E 00

0.932130E 00

0.938524E 00

0.943860E 00

0.948309E 00

0.961313E 00

0.965362E 00

0.965112E 00

0.962712E 00

0.959209E 00

0.955I33E 00

0.950759E 00

0.946234E 00

0.941636E 00

0.937008E 00

0.932373E 00

0.927745E 00

0.923130E 00

0.918533E 00

0.913956E 00

0.909400E 00

159

0.634158E 01

0.473610E 01

0.404788E 01

0.247746E 01

0.183202E 01

0.123089E 01

O.913133E 00

0.70-9131E 00

0.565885E 00

0.459912E-00

0.37B777E-00

0.315131E-00

0.264289E-00

0.223100E-00

0.189355E-00

0.161454E-00

0.138208E-00

0.U8715E-00

0.102279E-00

0.502744E-01

0.257503E-01

0.135564E-01

0.727902E-02

0.396717E-02

0.218760E-02

0.121774E-02

O.683152E-O3

0.385759E-03

0.219043E-03

0.124975E-03

0.716037E-04

0.411759E-04

0.237556E-04

0.137453E-04

0.797402E-05

0.99121 IE 00

0.983398E 00

0.973633E 00

0.902344E 00

0.825195E 00

0.695557E 00

0.591553E 00

0.505127E 00

0.433350E-00

0.373047E-00

0.321533E-00

0.277649E-00

0.240234P-00

0.207886E-00

0.180145E-00

0.156036E-00

0.135162E-00

0.117065E-00

0.101303E-00

0.476608E-01

0.198021E-01

0.500107E-02

-0.297737E-02

-0.732327E-02

-0.969696E-02

-0.109971E-01

-0.116990E-01

-0.120682E-01

-0.122508E-01

-0.123282E-01

-0.123459E-01

-0.123298E-01

-0.122945E-01

-0.122483E-01

-0.121960E-01

605361—62-
160

TABLE

T

0*001

0.005

0*010

0*050

0*100

0*200

0*300

0*400

0*500

0*600

0*70O

0*800

0*900

1*000

1*100

1*200

1*300

1*400

1*500

2*000

2*500

3.000

3.500

4*000

4*500

5*000

5*500

6*000

6*500

7.000

7.500

8.000

8*500

9*000

9*500

d

0*050

0*050

0*050

0*050

0.050

0.050

0.050

0.050

0.050

0.050

0.050

0.050

0.050

O.050

0.050

0.050

O.050

0.050

0.050

0.050

0.050

0.050

0.050

0.050

0.050

0.050

0.050

0.050

0.050

O.050

0,050

0.050

0.050

0.050

0*050

T-dT

0*001

0.005

0.009

0.047

0.095

0. 190

0.285

0.380

0.475

0.570

0.665

0.760

0,855

0.950

1.045

1.140

1.235

1.330

1*425

1.900

2.375

2*850

3*325

3*800

4.275

4.750

5.225

5.700

6.175

6.650

7.125

7.6OO

8.075

8.550

9.025

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS v

2.— The A and * operators of the exponential function (continued)

„ / . v , / -dTv , , -dTv

E,(T-dT; A(e ) i(e )

0.638278E 01 0.491S43E-00 0.965759E 00

0.477714E 01 0.502087E 00 0.957794E 00

0.408873E Ol 0.512816E 00 0.947906E 00

0*251675E 01 0*572643E 00 0.875443E 00

0.186945E 01 0.623790E 00 0.796051E 00

0.126486E 01 0.694653E 00 0.663750E 00

0.943965E 00 0.743593E 00 0.556892E 00

0.737112E 00 0.779655E 00 0.468811E-00

0*591280E 00 0.807047E 00 O.395298E-00

0.482960E-00 0.828187E 00 0.333370E-00

0.399695E-00 0.84462IE OO 0.280857E-00

0.334115E-00 0.857413E OO 0.236092E-00

0.281519E-00 0.867324E 00 '0* 197763E-00

0.238738E-00 0.87492IE 00 0.164850E-00

0.203547E-00 0*880639E 00 0*l36471E-00

0.174335E-00 0.884813E 00 0*U1959E-00

0.149898E-00 0.88771 IE OO 0.907354E-01

0.129325E-00 0.889545E 00 0.723267E-01

0.11I909E-00 0.890489E 00 0.563364E-01

0.562044E-01 0.886040E 00 0.246143E-02

0.294022E-01 0.872909E 00 -0.248973E-0I

0.158054E-01 0.855872E 00 -0.388108E-01

0.866412E-02 0.837128E 00 -0.457149E-01

0.482025E-02 0.817743E 00 -0.488949E-01

0.271303E-02 0.798253E 00 -0.500728E-01

0,154136E-02 0.778930E 00 -0.501674E-01

0.882491E-03 0.759912E 00 -0.496766E-01

0.508547E-03 0.741270E 00 -0.488726E-01

0.294680E-03 0.723036E OO -0.479058E-01

0.171569E-03 0.705222E 00 -0.468599E-QI

0.100307E-03 O.687832E 00 -0.457812E-0I

0.588588E-04 0.670662E 00 -0.446958E-01

0.346496E-04 0.654305E 00 -0.436182E-01

0.204571E-04 0*638154E 00 -0.425562E-01

0.121093E-04 0.622401E 00 -O*41S141E-Q1
11

7

0*001

0*005

0*010

0*050

0*100

0*200

0*300

0.400

0.500

0*600

0.700

0*800

0*900

1*000

1*100

1*200

1*300

l«40O

1*500

2*000

2*500

3*000

3*500

4*000

4*500

5*000

5*500

6*000

6*500

7,000

7*500

8*000

8*500

9*000

9*500

A

TABLE

a

0*100

0.100

0*100

0*100

0*100

0*100

0*100

O.lCO

0.100

0*100

O.lCO

0*100

0.2 00

0*100

0*100

0.100

0*100

0. ICO

0*100

0*100

0.100

0*}00

0*100

0.100

0*100

0*100

0*100

0*100

0*100

0*100

0*100

0*100

0*100

0*100

0*100

AND *

2 —The

T-dT

0*00]

0*004

0*009

0.045

0.090

0*180

0*270

0.360

0*450

0.540

0*630

0.720

0.810

0.900

0*990

1*080

1*170

1*260

1.350

1*300

2*250

2*700

3*150

3*600

4*050

4*500

4.950

5.400

5*850

6*300

6*750

7*200

7*650

8*100

6*550

OPERATORS OF RADIATIVE TRANSFER THEORY

A and <J> operators of the exponential function (continued)

161

0.643680E 01

0.483096E 01

0.414229E 01

0.256838E 01

0.191874E 01

0.1309S0E Ol

0.984933E 00

0.774462E 00

0.625331E 00

0.514004E 00

0.427997E-00

O.359918E-00

0.305043E-00

0.260184E-00

0*223lOOE-00

0.192160E-00

0.166150E-00

0.144142E-00

0.125417E-00

0.647131E-01

0.347621E-01

0.191819E-01

O.107913E-01

0*616041E-02

0*35574 IE-02

0.207340E-02

O.121774E-02

0.719804E-03

0*427812E-03

0.255471E-03

0,153187E-03

0*921881E-04

0.556577E-04

0*33699SE-04

0.204571E-04

A(e )

0*480168E-00

0*490618E-00

0.501225E OO

0.56001 IE 00

0.609725E 00

0.677436E 00

0.722963E 00

0.755436E 00

0.779119E 00

0*796467E 00

0.809054E 00

0.817964E 00

0.823976E 00

0*82767IE 00

0.829491E 00

0.829782E 00

0*828817E 00

0.8268I6E 00

0.823956E 00

0.801066E 00

0.770623E 00

0*737493E 00

0*703879E 00

0.670817E 00

0.638792E 00

0.608018E 00

0.578576E 00

0*550477E 00

0*523697E 00

0*498194E-00

0*473918E-00

0.450817E-00

0.428837E-OO

0*407927E-00

0.388034E-00

0.935883E

0.927650E

0.917603E

0.843365E

0.762093E

0*626484E

0.516979E

0.426765E

0.351521E

0.288218E

0.234637E

0.189049E

0.150113E

0.116764E

0.8B1205E

0.634723E

0.422299E

0.239000E

0.807023E

-0*440844E

-0.688586E

-0.798308E

-0.836924E

-0.838608E

-0.821236E

-0*794313E

-0.762951E

-0.729909E

-0.696678E

-0.6640S0E

-0.632437E

-0.602036E

-0.572930E

-0.545135E

-0.518634E

00

00

00

00

00

00

00

00

00

00

00

00

00

00

01

01

01

01

02

01

01

01

01

OJ

01

01

01

01

01

01

01

01

01

01

0I
162

TABLE

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

2.—The A and * operators of the exponential function (continued)

T

0*001

0*005

0*010

0.050

0*100

0*200

0*300

0*400

0*500

0*600

0*700

0.800

0*900

1.000

1*100

1*200

1*300

1*400

1*500

2*000

2*500

3*000

3*500

4*000

4*500

S.OCO

5.50O

6*000

6.500

7*000

7.500

8.000

8*500

9.000

9.500

d

0.200

0.200

0*200

O.2O0

0*200

0*200

0*200

0.200

0*200

0.200

0.200

0*200

0*200

0.200

0.200

0.200

0.200

0*200

0.200

0.200

0*200

0*200

0*200

0*200

0.200

0*200

0.200

0.200

C.200

0.200

0.200

0*200

0*200

0.200

0.200

T-dT

0.001

0.004

O.OO8

0.040

0.080

0.160

0.240

0.320

0.400

0.480

0.560

0.&40

0.720

C.800

0.880

0.960

1 .040

1.120

1.200

1.600

2.000

2.400

2.800

3.200

3.600

4.000

4.400

4.800

5.200

3.600

6.000

6*400

6.800

7.200

7.600

S1(T-dT)

0.655448E 01

0.494824E 01

0.425908E Ol

0.268126E 01

0.202694E 01

0.140919E 01

0.107624E 01

0.858335E 00

0.702380E 00

0.584784E 00

0.493020E-00

0.419652E-00

0.359918E-00

0.310597E-00

0.269413E-00

0.234708E-00

0.205238E-00

0.180052E-00

0.158408E-00

0.863083E-01

0.489005E-01

0.284403E-OI

0.168553E-01

0.101330E-01

0*616041E-02

0.377935E-02

0.233601E-02

0.145299E-02

0.908622E-03

0.570840E-03

0.360082E-03

0.227948E-03

0.144758E-03

0.921881E-04

0.588588E-04

A(e"UI)

0.459378E-O0

0.469649E-O0

0.480030E-00

0.53686IE 00

0.583889E 00

0.645714E 00

0.684922E 00

0.710821E 00

0.727784E 00

0.738344E 00

0.744131E 00

0.746269E CO

0*745570E 00

0*742639E 00

0.737938E 00

0.731827E 00

0.724590E 00

0.716456E 00

0.707609E 00

0.657094E 00

0.602956E 00

0.54991OE 00

0.499870E-00

0.453534E-00

0.4 11046E-00

0.372301E-00

0*337077E-00

0*305115E-00

0.276145E-00

0*249903E-00

0.226142E-00

0.204634E-C0

0.185167E-00

0.167550E-00

0.151608E-00

He" ')

0.881756E 00

0.873201E 00

0.862730E 00

0.785427E 00

0.700750E 00

0.55951 IE 00

0.445692E-00

0.352230E-00

0.274622E-00

0.209691E-00

0.155079E-00

0.108976E-00

0.699561E-01

0.368747E-01

0.880396E-02

-0.150200E-01

-0.352288E-01

-0*523499E-0l

-0.668260E-01

-0*110826E-00

-0.126477E-00

-0.128386E-00

-0*123716E-00

-0*116079E-00

-0*107322E-00

-0.983870E-01

-0*897422E-01

-0.816078E-01

-0.740721E-01

-0.671546E-01

-0.608392E-01

-0.550927E-01

-0.498748E-01

-0.451430E-01

-0.408554E-01
1

T

0*001

0*005

0*010

0*050

0*100

0*200

0,300

0*400

0*500

0.600

0*700

0*800

O.900

1*000

1*100

1*200

1*300

1*400

1*500

2*000

2*500

3*000

3*500

4*000

4*500

5*000

5*500

6*000

6*500

7*000

7,500

0.000

8*500

9*000

9*500

A

TABLE

d

0.300

0.300

0*300

0.300

0.300

0*300

0.300

0.300

0*300

0.300

0.300

0.300

0.300

0.300

0*300

0*300

0.300

0*300

0.300

0.300

0.300

0*300

0*300

0*300

0*300

0.300

0*300

0.300

0.300

0.300

0,300

0*300

0*300

0*300

0*300

AND *

2.—The

T-dT

0,001

0.003

0*007

0,035

0,070

0, 140

0,210

0,280

0,350

0,420

0,490

0,560

0,630

0,700

0,770

0,840

0,910

0,980

1,050

1 ,400

1.750

2.100

2.450

2.300

3*150

3*500

3.850

4.200

4.550

4.9O0

5.250

5.600

5.950

6,300

6,650

OPERATORS OF RADIATIVE TRANSFER THEORY

A and * operators of the exponential function (continued)

163

E1(x-dx)

0.66879IE 01

0.508127E 01

0.439162E 01

0.280989E 01

0.215084E 01

0.152415E 01

0«U8290E 01

0.9573C8E 00

0.794215E 00

0.669997E 00

0.572C89E 00

0.493020E-00

0.427997E-00

0.373769E-00

0.328032E-00

0.289103E-00

0.255714E-00

0,226891E-00

0,201873E-00

0,116219E-00

0.694887E-01

0,426143E-Ol

0.266156E-C1

0.168553E-01

0.107913E-01

0,6970i4E-02

0.453499E-02

0.296876E-02

0.195366E-02

0,l«29148E-02

0,857127E-03

0.570840E-03

0.381353E-03

0.255471E-03

0.171569E-03

-dr.

0.440807E-00

0,450915E-00

0,461087E-00

0.516112E 00

0.560661E 00

0.617089E 00

0.650575E 00

0.670604E 00

0.681663E CO

0.686359E 00

0.686377E 00

0.682878E 00

0.676702E 00

0.668474E 00

0.65867IE 00

0.647662E OO

0.635738E 00

0.623132E 00

0.610028E 00

0.541423E 00

0.474174E-00

0.412342E-00

0.357136E-O0

0.308589E-00

0.266260E-00

0.229532E-O0

0.197761E-O0

C.170326E-00

0*146665E-00

0.126271E-OO

0*108703E-00

0.935731E-01

0.805458E-01

0.693302E-01

0*596753E-0l

0.834108E

C825262E

0.814429E

0.734429E

0.646829E

0.501005E

0.383985E0.288434E0.209636E0.1442S1E0.897859E'

0.443194E0.633478E-

-0.253869ET-

-0.518381E-

-0.738354E-

-0.920550E-

-0.107062E-

-0.119332E-

-0.151716E-

-0.156079E-

-0.147912E-

-0.134654E-

-0.119924E-

-0.105449E-

-C920046E-

-0.798880E-

-C.691552E-

-0*597467E-

-0.515523E-

-0.444447E-

-0.382960E-

-0.329858E-

-0.284052E-

-0»244566E00

CO

00

00

CO

00

00

00

00

00

01

01

02

01

01

01

01

00

C0

00

00

00

00

00

00

01

01

01

01

C1

01

01

01

01

01
164

TABLE

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

2.—The A and * operators of the exponential function (continued)

T

0*001

0*005

0*010

0*050

0*100

0*200

0*300

0*400

0*500

0*600

0*700

0*800

0*900

1*000

1*100

1*200

1*300

1*400

1*SOO

2*000

2*500

3*000

3*500

4*000

4*500

5*000

5,500

6*000

6.500

7*000

7*500

a.ooo

8*500

9.000

9*500

d

0*400

0*400

0*400

0*400

0*400

0*400

0*400

0*400

0*400

0*400

0*400

0*400

0*400

0*400

0*400

0*400

0*400

0*400

0*400

0*400

0*400

0.400

0*400

0*400

0,400

0*400

0*400

0*400

0*400

0*400

0*400

0.400

0*400

0*400

0*400

T-dT

0*00]

0*003

0,006

0.030

0,060

0,120

0, 180

0,240

0,300

0.360

0.420

0.480

0.540

0.600

0.660

0.720

0.780

0.840

0.900

1 ,200

1,500

1,800

2,100

2,400

2,700

3,000

3,300

3,600

3,900

4,200

4,500

4,800

5,100

5,400

5.7OO

E1(T-dT)

0.684197E Ol

0.523492E Ol

0.454477E Ol

0.295912E 01

0*22953IE 01

0.165954E 01

0.130980E 01

0.107624E 01

0*9O5677E 00

0.774462E 00

0.669997E 00

0.584784E 00

0.514004E 00

0.454379E-00

0.403586E-00

0.359918E-00

0.322088E-00

0.289103E-00

0.260184E-00

0.158408E-00

0.10002CE-00

0.647131E-01

0.426143E-01

0.284403E-01

0*191819E-0l

0.130484E-01

0*893904E-02

0.616041E-02

0.426715E-02

0.296876E-02

0.207340E-02

0*145299E-02

0.102130E-02

0.719B04E-03

0.508546E-03

A(e )

0.424087E-00

C434043E-00

0,44402lE-00

0.497368E-00

0,539615E 00

0,591069E 00

C619347E 00

0,634115E 00

0,63997lE 00

0.639593E 00

0.6347J2E 00

0.626526E 00

0.615894E 00

0.603456E 00

C589697E 00

0.574989E 00

C559624E 00

0.543832E 00

0.527793E 00

0.448048E-00

0*374779E-00

0.310948E-00

0.256753E-00

0.211380E-00

0*173700E-00

0. 142565E-00

0,1 16917E-00

0.958329E-01

0.785229E-01

C643240E-01

0,526838E-0l

0,431452E-0l

0.3S330BE-01

0,289301E-01

0.236881E-01

0.791787E 00

0.782677E 00

0.771518E 00

0.6B9134E 00

0.599017E CO

0.449474E-00

0.330160E-00

0.233443E-00

0.154374E-00

0.894336E-01

0.359809E-01

-0.802445E-02

-0.441983E-01

-0.738395E-01

-0.980079E-01

-0.U7575E-00

-0.133263E-00

-0.145675E-00

-0.155319E-00

-0.174863E-00

-0,167985E-00

-0.150747E-00

-0,130573E-00

-0,110824E-00

-0.929052E-01

-C772756E-01

-0,639484E-01

-0,527411E-0I

-C433992E-01

-0.356568E-01

-0.292646E-01

-0.240007E-01

-0,196737E-0t

-0*16121XE-OJ

-0.132067E-01
1

T

0*001

0.005

0*010

0*050

0*100

0*200

0*300

0*400

0*500

0*600

0*700

0*800

0*900

1*000

1*100

1*200

1*300

1*400

1*500

2*000

2*500

3*000

3*500

4*000

4*500

5*000

5*500

6*000

6*500

7*000

7*500

8*000

8*500

9*000

9*500

A

TABLE

a

0.500

0.500

0.500

0.500

0.500

0.500

0.500

0*500

0*500

0.500

0.500

0.500

0.500

0.500

0.500

0.500

0.500

0.500

0.500

0.500

0.500

0*500

0.500

0*500

0.500

0.500

0.500

0*500

0.500

0*500

0.500

0*900

0*500

0*500

0.500

AND *

2.—The

T-dT

0*000

0.002

0.005

0.025

0.050

0. 100

0.150

0.200

0.250

0.300

0.350

0.400

0.450

0.500

0.550

0.600

0,650

0.700

0,750

1.000

1.250

1.500

1.750

2.000

2.250

2*500

2*750

3*000

3.250

3*500

3.750

4*000

4*250

4*500

4.750

OPERATORS OF RADIATIVE TRANSFER THEORY

A and * operators of the exponential function (continued)

165

E1(T-dT)

0.702419E 01

0.541675E 01

0.472610E 01

0.313651E 01

0.246790E 01

0.182292E 01

0.146446E 01

0.122265E 01

0.104423E 01

0.905677E 00

0.794215E 00

0.702380E 00

0.625331E 00

0.559774E 00

0.503364E 00

0.454379E-00

0.411517E-00

0.373769E-00

0.340341E-00

0.219384E-00

0. 146413E-00

0.100020E-00

0.694S87E-01

0.489005E-01

0,347621E-01

0.249149E-01

0.179789E-01

0«130484E-01

0.951651E-02

0.697014E-02

0.512410E-02

0.377935E-02

0.279565E-02

0.207340E-02

0.154136E-02

., -dT v A(e )

0.408927E-OO

0.418742E-00

0.428539E-00

C.480318E-00

C.520419E 00

0.567266E 00

0.590787E 00

0.600826E 00

0.602086E 00

0.59731OE 00

0.588274E 00

0.576199E 00

0.561961E 00

0.546206E 00

0.529419E 00

0.511971E 00

0*494146E-00

0.476166E-00

0.458202E-00

0.37235IE-00

0.297724E-00

C.235865E-00

0.185805E-00

0*145842E-00

0.114202E-00

0.892827E-01

0.697235E-01

0.544071E-01

0.424323E-01

0.330802E-01

0.257822E-01

0.200902E-01

0.156526E-01

0.121939E-01

0.949871E-02

0.753909E

0.744559E

0.733109E

0.648596E

0.556306E

0.403767E

0.282906E

0.185765E

0.107146E

0.433322E

-0.847313E

-0.504379E

-0.842809E

-0.111386E

-0.132880E

-0.149690E

-0.162585E

-0.172205E

-0.179084E

-0.185906E

-0.1*68948E

-0.144443E

-0.119456E

-0.968498E

-0.775401E

-0.615678E

-0.486118E

-0.382335E

-0.299890E

-0.234767E

-0.183531E

-0.143333E

-0.111857E

-0.872471E

-0.680249E

00

00

00

00

00

00

00

00

00

01

02

01

01

00

00

00

00

00

00

00

00

00

00

01

01

01

01

01

01

01

01

01

01

02

02
166

TABLE

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

2.—The A and * operators of the exponential function (continued)

T

0.001

0.005

0.010

0.050

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0*800

0*900

1*000

1*100

1*200

1*300

1*400

1*500

2*000

2*500

3*000

3*500

4*000

4*500

5*000

5*500

6*000

6*500

7,000

7*500

8.000

8*500

9*000

9*500

a

0*600

0*600

0.600

0.600

0.600

0.600

0.600

0.600

0.600

0.600

0.600

0.600

0.600

0.600

0*600

0*600

0.600

0*600

0*600

0*600

0*600

0.600

0*600

0*600

0*600

0.600

0*600

0*600

0*600

0*600

0*600

0*600

0*600

0*600

0*600

T-dT

0*000

0*302

0.004

0.020

0.040

0.080

0.120

0.160

0.200

0.240

0.280

0.320

0.360

0.400

0.440

0.480

0.520

0.560

0.600

0.800

l.OOO

1*200

1 .400

1.600

1.800

2.000

2.200

2.400

2.600

2*800

3*000

3.200

3.400

3.600

3.800

E1(T-dT)

0.724723E 01

0.563939E Ol

0.494824E Ol

0.33547IE Ol

0.268126E 01

0.202694E 01

0.165954E 01

0.140919E 01

0.122265E 01

0.107624E 01

0.957308E 00

0.858335E 00

0.774462E 00

0.702380E 00

0.639733E 00

0.584784E 00

0.536220E 00

0.493020E-00

0.454379E-00

0.310597E-00

0.219384E-00

0.158408E-00

0.U6219E-00

0.863083E-01

0.647131E-01

0.489005E-01

0.371911E-01

0.284403E-01

0.218502E-01

0.168552E-01

0.130484E-01

0.101330E-01

0.789097E-02

0.616041E-02

0.482025E-02

A(e )

0.395099E-O0

0.404782E-00

0.414409E-00

0.464717E-00

0.502806E 00

0.545373E 00

0.564534E 00

0.57031OE 00

0.567503E 00

0.558917E 00

0.54636IE 00

0.531074E 00

0.513940E 00

0.495603E-00

0.47654AE-00

0.457121E-00

0.437609E-00

0.418213E-O0

0.399090E-00

0.310745E-00

0.237740E-00

0. 180014E-GO

0.135416E-00

0.101426E-00

0.757425E-01

0.564443E-01

0.42000lE-Oi

0.312181E-01

0.231853E-01

0.172092E-01

0.127677E-01

0.946927E-02

0.702116E-02

0.520495E-02

0.385796E-02

0.719778E

0.710209E

0.698494E

0.612075E

0.517904E

0.362979E

0.24 1188E

0. 144224E

0.666202E

0.444975E

-0.452491E

-0.847772E

-0.115959E

-0.140263E

-0.158885E

-0.172805E

-0.182835E

-0.189648E

-0.193804E

-0.188789E

-0.163390E

-0.133572E

-0.105736E

-0.820490E

-0.628419E

-0.477034E

-0.359852E

-0.270233E

-0.202266E

-C.151024E

-0.U2557E

-0.837729E

-0.622841E

-0.462704E

-0.343527E

00

00

00

00

00

00

00

00

01

02

01

01

00

00

00

00

00

00

00

00

00

00

00

01

01

01

01

01

01

01

01

02

02

02

02
T

0*001

0*005

0*010

0*030

0*100

0*200

0*300

0*400

0*500

0*600

0*700

0*800

0*900

1*000

1*100

1*200

1*300

1*400

1*500

2*000

2*500

3*000

3*500

4*000

4*500

5*000

5*500

6*000

6*500

7*000

7*500

8*000

8*500

9*000

9*500

605361—1

TABLE

d

0,700

0.700

0,700

0,700

0.700

0,700

o,-»oo

0.700

0.700

0.700

0.700

0,700

0.700

0,700

0.700

0.700

0.700

0,700

0.700

0.700

0.700

0.700

0,700

0,700

0,700

0.700

0.700

0,700

0,700

0,700

0,700

0.700

0,700

0,700

0,700

52 3

A AND *

2.—The

T-dT

0,000

0.001

0.003

0.015

0.030

0*060

0.090

0. 120

0. 150

0.180

0,210

0,240

0,270

0.300

0.330

0.360

0.390

0,420

0.450

0.600

0.750

0,900

1*050

1*200

1 .350

1.500

1 .650

1.800

1.950

2.100

2,250

2.400

2*350

2,700

2*850

OPERATORS OF RADIATIVE TRANSFER THEORY

A and * operators of the exponential function (continued)

167

0,753481E 01

0.592657E 01

0.523492E 01

0.363743E 01

0.295912E 01

0,22953IE 01

0.191874E 01

0.165954E 01

0,146446E 01

0.130980E 01

0.U8290E 01

0.107624E 01

0,984933E 00

0.905677E 00

0,B36l0lE 00

0.774462E 00

0.719437E 00

0.669997E 00

0,62533IE 00

0.454379E-00

0,340341E-00

O,26O184E-OO

0.201873E-00

0,158408E-00

0.12S417E-00

0 ,100020E-00

0.802476E-01

0.647131E-01

0.524144E-01

0.426143E-01

0,347621E-01

0.284403E-01

0.233293E-01

0.191819E-01

0.158054E-01

. / -dT» A(e )

0*382419E-00

0.391978E-00

0,40l444E-00

0,450367E-00

0.486565E-00

0.525140E 00

0.540295E 00

0.54222IE 00

0,535813E 00

0.523926E 00

0.508396E 00

0.490472E-00

0.471036E-00

0*450725E-00

0,430006E-00

0.409222E-00

0*388628E-00

0,36B412E-00

0.348711E-00

0.260428E-00

0.190858E-00

0.138286E-00

C994534E-01

0,711627E-0l

0.507360E-01

0.360774E-01

O.256O37E-O1

0.181436E-0I

0.128425E-01

0*908214E-0H

0.641841E-02

0.453344E-02

0*320066E-02

0,225891E-O2

0.159382E-02

-dr

*(e

0*688838E

0*679070E

0*667113E

0,578980E

0*483178E

0.326387E

0.204182E

0.107881E

0.317348E

-0*284057E

-C756745E

-0.112508E

-0»140838E

-0.162216E

-0.177904E

-0.188933E

-0.196152E

-0,200262E

-0.201847E

-0*186275E

-0.154187E

-0.120860E

-0.917758E

-0*682955E

-0.501325E

-0.364472E

-0*263130E

-0.188979E

-0.135190E

-0*964180E

-0.686036E

-0.487226E

-0*345522E

-0*244743E

-0.173196E

00

00

00

00

•00

•00

•00

•00

•01

•01

•01

-00

•00

•00

•00

-oc

-00

•oc

-00

-00

-00

-00

•01

-01

•01

-01

•o;

•01

•01

•02

•02

•02

•02

-02

-02
168 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 2.^-The A and * operators of the exponential function (continued)

T

0,001

0*005

0*010

0*050

0*100

0*200

0*300

0*400

0*500

0*600

0*700

0*800

0*900

1*000

1*100

1 *2C0

1*300

1 *400

1*500

2*000

2*500

3*000

3*500

4*000

4*500

5*000

5*500

6*000

6*500

7*000

7*500

6*000

6*500

9*000

9.500

d

0*800

0*800

0*800

0,800

0*800

0*800

0*800

0.800

0*800

0,800

0*800

0*600

0*600

0*800

0*600

0,600

0.800

OkBOO

o»ooo

o»aoo

0*800

O»800

0,800

0*800

0,800

0*800

0*800

0.800

0*800

0.800

0*800

0*600

0.600

0*600

0*800

T-dT

0,000

0,001

0.002

0*010

0.020

0*040

0.060

OiOaO

0. 100

0. 120

0* 140

0* 160

0. 18O

0.20O

0*220

0.24O

0.260

O.28O

0.30O

0.400

O.50O

0.600

0,700

0*800

0.900

l.OOO

i. too

1.20O

1 .300

1.400

1.500

1 .600

1.700

1 .800

I .900

E1(T-dT)

0.794018E 01

0.633154E 01

0.563939E Ol

0.403793E 01

0.335471E 01

0.268126E 01

0.22953IE Ol

0.202694E 01

0.182292E 01

0.165954C Ol

0.152415E 01

0.140919E 01

0.130980E 01

0.122265E 01

0.114538E Ol

0.107624E 01

0.101389E 01

0.957308E 00

0.905677E 00

0.702380E 00

0.559774E 00

0.454379E-00

0.373769E-00

0.310597E-00

0.260184E-00

0.219384E-00

0.185991P-00

0.158408E-00

0,l3545lE-00

0.116219E-00

O.1OOO2OE-OO

0.363033E-01

0.746546E-O1

0.647131E-01

0.562044E-01

A(e ~")

0.370737E-00

0.380178E-00

0.389494E-00

0.437105E-00

0.471520E-00

0.506362E 00

0.517826E 00

0.516270E 00

0.506672E 00

0.491930E-00

0.473898E-00

0.453827E-00

0.432589E-00

0.41O805E-O0

0.388924E-00

0.367267E-00

0.346067E-00

0.325487E-00

0.305641E-00

0.219190E-00

0* 154075E-00

0.106972E-00

0.736591E-01

0.504265E-01

0.343754E-01

0.233586E-01

0.156335E-01

0* 107119E-01

0*72356'4E-02

0.488172E-02

0.329018E-O2

0.221567E-02

0*149105E-02

0.100284E-02

0.674163E-03

?(e

 l )

0.660645E 00

0.650695E 00

0.638514E 00

0.548834E 00

0.451618E-00

0.293406E-00

0.171219E-00

0.759772E-01

0.163180E-02

-0.561951E-01

-0.100815E-00

-0.134801E-00

-0.160192E-00

-0.178620E-00

-0.191407E-00

-0.199631E-00

-0.204173E-00

-0.205760E-00

-0.204988E-00

- 0 . 180304E-00

-0.143192E-00

-0.107877E-00

-0.787500E-01

-0.563173E-01

-0.397049E-01

-0.277058E-01

-0.191848E-01

-0.132064E-01

-0.904922E-02

-0.617807E-02

-0.420550E-02

-0.285591E-02

-0.193562E-02

-0*130975E-02

-0.885051E-03
A AND * OPERATORS OF RADIATIVE TRANSFER THEORY

TABLE 2.— The A and * operators of the exponential function (continued)

169

T

0*001

0.005

0*010

0.050

0*100

0*200

0*300

0*400

0*500

0*600

0*700

0,800

0,900

1,000

1,100

1,200

1.300

1 .400

1.500

2.000

2.500

3.000

3.500

4.000

4.500

5.000

5.500

6,000

6*500

7.000

7,500

8,000

8,500

9,000

9.500

a

0.900

0.900

0.900

0.900

0.9C0

0.9CO

0.900

0.900

0,900

0.900

0.900

0.900

0,900

0.900

0.900

0.900

0.900

0.900

0.9C0

0.9C0

0.900

0.9C0

0.9CO

0.900

0.900

0.900

0.900

0,900

0,900

0.9C0

0,f»C0

0,900

0,900

0,900

0.900

T-dT

0.000

0.000

0.00]

0.005

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0.080

0.090

0.100

0,110

0. 120

0.130

0. 140

0.150

0.200

0.250

0.300

0.350

0.400

0.450

0.500

0.550

0.600

0.650

0.700

0,750

0,800

0,850

0.900

0,950

E1(T-dT)

0,863322E 01

0.702419E 01

0.633154E 01

0.472610E 01

0.403793E 01

0.335471E 01

0.295912E 01

0.268 126E 01

0.246790E 01

0.229531E 01

0.215084E 01

0.202694E 01

0.191874E 01

0.182292E 01

0.173711E 01

0.165954E 01

0.158890E 01

0.152415E 01

0.146446E 01

0.122265E 01

0.104428E 01

0.905677E 00

0.794215E 00

0.702380E 00

0.625331E 00

0.559774E 00

0.503364E 00

0.454379E-00

0.411517E-00

0.373769E-00

0.340341E-00

0.310597E-00

0.284019E-00

0.260184E-00

0.238738E-00

0.359928E-00

0.369258E-00

0.378431E-00

0.424800E-00

0.457527E-00

0.488870E-00

0.496928E-00

0.492217E-00

0.479794E-OO

0.462589E-00

0.442463E-00

0.420662E-00

0.398040E-00

0.375199E-00

0.352560E-00

0.330419E-00

0.308981E-00

0.288384E-00

0,268716E-00

0.185283E-00

0,125106E-00

0,833691E-01

0,550621E-01

0.361337E-01

0.235988E-01

0*153553E-0l

0.996225E-02

0*644813E-02

0.416555E-02

0.268668E-02

0,173051E-02

0,111336E-02

0.715610E-03

0.459569E-03

0.294922E-03

0.634833E 00

0.624714E 00

0.612328E 00

0.521247E 00

0.422807E-00

0.263557E-00

0.141751E-00

0.478886E-01

-0.243903E-01

-0.797006E-01

-0*121534E-00

-0.1S2601E-00

-0.175041E-00

-0.190564E-00

-0.200S49E-00

-0*206112E-00

-0.208163E-00

-0.207444E-00

-0.204562E-00

-0*172230E-00

-0*131578E-00

-0.954871E-01

-0*671597E-01

-0.462628E-01

-0.3I4032E-01

-0*2l0a68E-0l

-0.140430E-01

-0.929191E-02

-0*611655E-02

-0.400945E-02

-0*261914E-02

-0.I70599E-02

-0.110849£-02

-0.718762E-03

-0.465224E-03
170

TABLE

T

0*001

0.005

0*010

0.050

0*100

0*200

0.3C0

0*400

0«5C0

0.6C0

0.T00

0*8C0

0.9C0

1*000

1*100

1.200

1.300

1 .400

1*500

2*000

2*500

3*000

3*500

4*000

4*500

5*000

5*500

6*000

6.500

7.000

7.500

6*000

8*500

9.000

9.500

a

1 .000

1 .000

1.000

1 .OCO

1 .000

l.OCO

1*000

1.000

1.000

1.000

1.000

1 .000

l.OCO

1.000

1.000

1 .000

1.000

1.000

1 .000

1 .000

1 .000

1 .000

1 .000

I .000

1 .000

1.000

1 .000

1 .000

1.000

1*000

1.000

1 .000

1 .000

1.000

1 .000

T

0 .

0 .

0 .

0 .

0 .

0 .

0 .

o.

0 .

0 .

0 .

0 .

0 .

0 .

0 .

0 .

0 .

0 .

0 .

0 .

0 .

0 .

0 .

0 .

c.

0 .

0 .

0 .

0 .

0 .

0 .

0 .

0 .

0 .

0 .

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

2.—The A ind 4> operators of the exponential function (continued)

T-dr

VOL. S

E^T-d-r)

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0«577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

-0.577216E 00

*(e~dT)

0.349891E-00

O*359115E-0O

0*368152E-00

0.413338E-00

0*444465E-00

0.472520E-00

0.477430E-00

0.469861E-00

0.454936E-00

0*435611E-00

0.413746E-00

0*390571E-00

0.366919E-00

0.343362E-00

0.320292E-00

0*297974E-00

O.276584E-OO

0.256230E-00

0.236974E-00

0*157316E-00

0.102203E-00

0*654963E-0l

0.415810E-01

0.262189E-01

0.164473E-01

0.102761E-01

0*639976E-02

0.397516E-02

0.246374E-02

0.152417E-02

0*941430E-03

0.580700E-03

0*3S7768E-03

0.220191E-03

0.135394E-03

0*61 U01 E 00

0.600825E 00

0.588250E 00

0.495896E-00

0.396398E-00

0.236443E-00

0.11S323E-00

0.231008E-01

-0.469102E-01

-0.995660E-01

-O*138539E-00

-0.166673E-00

-O.186207E-00

-0.198922E-00

-0.206246E-00

-0.209328E-00

-0*2091OOE-00

-0.206312E-00

-0.201S76E-00

-0*162993E-00

-O*120068E-00

-0.841210E-01

-0.571381E-01

-0.380094E-01

-0.249107E-01

-0.161456E-01

-O.103753E-01

-0.662216E-02

-0.420350E-02

-0.265618E-02

-0*167209E-02

-0*104922E-02

-0.656575E-03

-0.409891E-03

-0.255360E-03
A AND # OPERATORS OF RADIATIVE TRANSFER THEORY 171

TABLE 2.— The A and * operators of the exponential function (continued)

T

0.001

0.005

0.010

0.050

0.100

0.200

0.300

0.400

0.500

0.600

0.70O

0.800

0.900

1.000

1.100

1.200

1.300

1.400

1.500

2.000

2.500

3.000

3.500

4.000

4.500

5.000

5.500

6.000

6.500

7.000

7.500

8.000

d

2.000

2.000

2.000

2.000

2.OOO

2.000

2.000

2.000

2.000

2.000

2.000

2.000

2.000

2.000

2.000

2.000

2.000

2.000

2.000

2.000

2.000

2.000

2.000

2.000

2.000

2.000

2.000

2.000

2,000

2.000

2.000

2.000

T-dT

-O.OOl

-0.005

-0.010

-0.050

-0.100

-0.200

-0.300

-0.400

-0.500

-0.600

-0.700

-O.8OO

-O.9OO

-1.000

-t.100

-1.200

-1.300

-1.400

-1.500

-2.000

-2.500

-3.000

-3.500

-4.000

-4.500

-5.000

-5.500

-6.000

-6.500

-7.000

-7.500

-8.000

K^T-dT)

0.632954E 01

0.471610E 01

0.401793E 01

0.236788E Ol

0.16228 IE 01

0.82176 IE 00

0.302669E-00

-0.104765E-00

-0.454220E-00

-0.769e81E 00

-0.10649 IE Ol

-0.134740E Ol

-0.162281E Ol

-0.189512E 01

-0.216738E 01

-0.244209E 01

-0.272140E 01

-0.300721E 01

-0.330129E 01

-0.495423E 01

-0.707377E 01

-0.993383E Ol

-0.139254E 02

-0.196309E 02

-0.279337E 02

-0.401853E 02

-0.584655E 02

-0.859898E 02

-0.127747E 03

-0.191505E 03

-0.289388E 03

-0.440380E 03

0.277766E-00

0.286152E-00

0.294105E-00

0.329853E-00

0.348436E-00

0.352057E-00

0.335625E-00

0.310773E-00

0.282757E-00

0.254290E-00

0.226822E-00

0.201109E-00

0.177508E-00

0. 156135E-OO

0.136968E-O0

0.119903E-00

0.104794E-00

0.914736E-01

0.797694E-01

0.399406E-01

0. 199950E-01

0.100988E-01

0.516756E-02

0.268333E-02

0. 141407E-02

0.755645E-03

O.4O8937E-O3

0.223793E-03

0.123666E-03

0.689092E-04

0.386729E-04

0.218369E-04

0.447803E-00

0.436414E-00

0.422517E-00

0.322134E-00

0.218044E-00

0.623246E-01

-0.427419E-01

-0.112256E-0O

-0.156399E-00

-0.182375E-00

-0.195396E-00

-0.199277E-00

-0.196822E-00

-0.190096E-00

-0.180611E-00

-0.169475E-00

-0.157487E-00

-0.145217E-00

-0.133065E-00

-0.807841E-01

-0.463118E-01

-0.258820E-01

-0.143132E-01

-0.789396E-02

-0.436001E-02

-0.241696E-02

-0.134615E-02

-0.753555E-03

-0.423956E-03

-0.239665E-03

-0.136087E-03

-0.775863E-04
172 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

T

0,001

0*005

0*010

0*050

0*100

0*200

0*300

0*400

0*500

0*600

0*700

0*800

0*900

1*000

1*100

1*200

1.300

1*400

1*500

2*000

2*500

3*000

3*500

4*000

4*500

5.000

5.500

6*000

6*500

7*000

7.500

8.000

TABLE

d

3*000

3*000

3*000

3.O0O

3*000

3.000

3.000

3.000

3.000

3.000

3.000

3.000

3.000

3.000

3.000

3*000

3.000

3.000

3.000

3.000

3.000

3.000

3.000

3.0CO

3.0C0

3.030

3.000

3.000

3.000

3.000

3.000

3.000

; 2.—The A

T-dT

-0.002

-0.010

-0.020

-0.100

-0.200

-0.400

-0.600

-0.800

-1.000

-1.200

-1.400

-1.600

-1.800

-2.000

-2.200

-2.400

-2.600

-2.800

-3.000

-4.000

-5.000

-6.000

-7.000

-a.ooo

-9.000

-10.000

-11.000

-12.000

-13.000

-14.000

-15.000

-16.000

and * operators of the exponential function (continued)

E1(T-dT)

0.563539E Ol

0.401793E 01

0.331471E 01

0.162291E 01

0.821761E 00

-0.104765E-00

-0.76988IE 00

-0.134740E 01

-0.189512E 01

-0.244209E 01

-0.300721E 01

-0.360S32E Ol

-0.424987E 01

-0.495423E Ol

-0.573261E 01

-0.660067E Ol

-0.75761 IE Ol

-0.867930E Ol

-0.993383E Ol

-0.196309E 02

-0.401S53E 02

-0.859898E 02

-0.191505E 03

-0.440380E 03

-0.103788E 04

-0.249223E 04

-0.607141E 04

-0.149595E 05

-0.371977E 05

-0.931925E 05

-0.234956E 06

-0.59556IE 06

A(e )

0*234017E-00

0*241B02E-00

0*248975E-00

0.277955E-00

0.287941E-00

0.276759E-00

0.250083E-00

0.219497E-00

0.189549E-00

0.162105E-00

0*137817E-00

0.116757E-00

0.987303E-01

0.834251E-01

0.704989E-01

0.596171E-01

0*504728E-0i

0*427941E-0l

0.363458E-01

0.165465E-01

0*792069E-02

0.395765E-02

0.204376E-02

0.108157E-02

0.582873E-03

0.318481E-03

0*175897E-03

0.979877E-04

0.549733E-04

0.310243E-04

0.175972E-04

0* 100248E-04

§(e )

0*355538E-00

0.343492E-00

0.328847E-00

0*225115E-00

0.122140E-00

-0.200636E-01

-0.104095E-00

-0.150649E-00

-0.172988E-00

-0.179864E-00

-0*177tl2E-00

-0.168620E-00

-0.156969E-00

-0*143848E-00

-0.130342E-00

-0*117127E-00

-0.104605E-00

-0.929913E-01

-0.823829E-01

-0.437798E-01

-0.230219E-01

-0.122069E-01

-0.655623E-02

-0.356605E-02

-0*196113E-02

-0.108855E-02

-0.608891E-03

-0.342802E-03

-0.1940S7E-03

-0*1I0371E-03

-0*630311E-04

-0.361248E-04
A AND * OPERATORS OF RADIATIVE TRANSFER THEORY 173

TABLE 2.—The A and * operators of the exponential function (continued)

T

O.OOl

0.005

0.010

0.050

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1 .000

1.100

1 .200

1.300

1 .400

1.500

2.000

2.500

3.000

3.500

4.000

4.500

5.000

5.500

6.000

6.500

7.000

7.500

8.000

d

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

4.000

T-dT

-0.003

-0.015

-0.030

-0.150

-0.300

-0.600

-0.900

-1.200

-1.500

-1.800

-2.100

-2.400

-2.700

-3.000

-3.300

-3.600

-3.900

-4.200

-4.500

-6.000

-7.500

-9.000

-10.500

-12.000

-13.500

-15.000

-16.300

-18.000

-19.500

-21.000

-22.500

-24.000

E1(T-dT)

0.522892E 01

0.360743E 01

0.289912E 01

0.116409E 01

0.302669E-00

-0.769881E 00

-0.162281E 01

-0.244209E 01

-0.330129E 01

-0.424987E 01

-0.533324E 01

-0.660067E 01

-0.811035E 01

-0.993383E 01

-0.121610E 02

-0.149063E 02

-0.183157E 02

-0.225774E 02

-0.279337E 02

-0.859898E 02

-0.289388E 03

-0.103788E 04

-0.388374E 04

-0.149595E 05

-0.588270E 05

-0.234956E 06

-0.949S81E 06

-0.387790E 07

-0.159596E 08

-0.661272E 08

-0.275573E 09

-0.115412E 10

A(e )

O.2O4O34E-OO

0.211351E-00

0.217911E-OO

0.24 1632E-00

0.245307E-00

0.224764E-00

0.193539E-00

0.162320E-00

0.134461E-00

0. 1 10782E-00

0.911433E-01

0.750596E-01

0.619683E-01

0.513356E-01

0.426960E-01

0.356609E-01

0.299136E-01

0.251996E-01

0.213158E-01

0.973978E-02

0.475954E-02

0.242856E-02

0.127500E-02

0.682860E-03

0.371168E-03

0.204072E-03

0.U3237E-03

0.633098E-04

0.356211E-04

0.201506E-04

0. 1 14523E-04

0.653519E-05

0.295639E-00

0.283157E-00

0.268043E-00

0.163182E-00

0.637406E-01

-0.625350E-01

-0.126903E-00

-0.155108E-00

-0.162447E-00

-0.158156E-00

-0.147807E-00

-0.134723E-00

-0.120847E-00

-0.107268E-00

-0.945592E-01

-0.829832E-01

-0.726198E-01

-0.634466E-01

-0.553875E-01

-0.281714E-01

-0.146130E-01

-0.774338E-02

-0.417512E-02

-0.228186E-02

-0.126050E-02

-0.702305E-03

-0.3Q4075E-03

-0.222439E-03

-0.126194E-03

-0.719055E-04

-0.4 11286E-04

-0.236040E-04
174 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

T

0*001

0.005

O.OlC

0.050

0.100

0.200

0.30C

0.400

0.500

0.600

0.700

0.80C

0.900

l.OOC

1.100

1.200

1 .300

1,400

1 .500

2.000

2.500

3.000

3.500

4.000

4.500

5.000

5.500

6.000

6.500

7.000

7,500

8.COO

TABLE

a

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.C00

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.000

5.C00

2 —The A

T-dT

-0.004

-0.020

-0.040

-0.200

-0.400

-0.800

-1.200

-1.600

-2.000

-2.40O

-2.80O

-3.200

-3.600

-4.000

-4.400

-4.800

-5.200

-5,600

-6.000

-8.300

-10.000

-12.000

-14.000

-16.000

-18.000

-20.000

-22.000

-24.000

-26.000

-28.000

-30.000

-32.000

and * operators of the exponential function (continued)

0.494024E 01

0.331471E 01

0.260126E 01

0.821761E 00

-0.104765E-00

-0.134740E 01

-0.244209E 01

-0.360532E 01

-0.495423E 01

-0.660067E 01

-0.86793CE 01

-0.H3673E 02

-0.149063E 02

-0.196309E 02

-0.260090E 02

-0.346979E 02

-0.466249E 02

-0.631018E 02

-0.859898E 02

-0.440380E 03

-0.249223E 04

-0.149595E 05

-0.931925E 05

-0.59556 IE 06

-0.387790E 07

-0.256157E 08

-0.171145E 09

-0.U5412E 10

-0.784294E 10

-0.53645IE 11

-0.368973E 12

-0.255004E 13

,/ -dT»

A(e )

0.181936E-00

0.188868E-00

0.194923E-00

0.214369E-00

0.213239E-00

0.186749E-00

0.154105E-00

0.124518E-00

0.999724E-01

0»803194E-0l

0.648105E-O1

0.526222E-01

0.430282E-01

0,354388E-0l

0.293941E-01

0.245421E-01

0.206158E-01

0. 174130E-01

0. 147803E-01

0.689121E-02

0.342041E-02

0.176247E-02

0.931021E-03

0.500690E-03

O.27295OE-O3

0.150404E-O3

0.836030E-04

0.468080E-O4

0.263673E-04

0.149306E-O4

0.849273E-05

0.484991E-O5

0.253392E-00

0.240600E-00

0.225177E-00

0.1204 12E-00

0.256150E-01

-0.847761E-01

-0.132426E-00

-0.147093E-00

-0.144968E-00

-0.134899E-00

-0.121669E-00

-0.107786E-00

-0.944970E-01

-0.823589E-0I

-0.715583E-01

-0.620923E-01

-O.538681E-01

-0.467569E-01

-0.406221E-01

-0.204904E-01

-0.106524E-01

-0.566650E-02

-0.306555E-02

-0.167984E-02

-0.929838E-03

-0.518911E-03

-0.291554E-03

-0.164749E-03

-0.93S519E-04

-0.533484E-04

-0.305353E-04

-O•175350E-04
A AND * OPERATORS OF RADIATIVE TRANSFER THEORY 175

T

0,001

O.0O5

0.010

0.050

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1 .000

1 »1OO

1 .200

1.300

1 .400

1 .500

2.000

2.500

3.000

3.500

4.000

4.500

5.000

5.500

6.000

6.500

7.000

7.500

8.000

TABLE

d

6.000

6.000

6.000

6.000

6.000

6.000

6.000

6.000

6.000

6.000

6.000

6.000

6.000

6.000

6.000

6.000

6.000

6.000

6.COO

6.000

6.C00

6.000

6.000

6.COO

6.OOO

6.000

6.COO

6.000

6.000

6.COO

6.000

6.000

2.—The A

T-dT

-0.005

-0.025

-0.050

-0.250

-0.500

-I•0O0

-I.50 0

-2.000

-2.500

-3.000

-3.500

- 4 . "300

-4.500

-5.000

-5.500

-6.000

-6.500

-7.000

-7.500

-10.000

-12.500

-15.000

-17.500

-20.000

-22.500

-25.000

-27.500

-30.000

-32.500

-35.000

-37.500

-40.000

and # operators of the exponential function (continued)

E1(T-dT)

0.471610E 01

0.308651E 01

0.236788E 01

0.542543E 00

-0.454220E-00

-0.189512E 01

-0.330129E 01

-0.495423E 01

-0.707377E 01

-0.993383E 0 1

-0.139254E 02

-0.196309E 02

-0.279337E 02

-0.401853E 02

-0.584655E 02

-0.859S98E 02

-0.127747E 03

-0.191505E 03

-0.289388E 03

-0.249223E 04

-0.235651E 05

-0.234956E 06

-0.242401E 07

-0.256157E 08

-0.27S573E 09

-0.300595E 10

-0.331533E 11

-0.368973E 12

-0.413743E 13

-0.466906E 14

-0.52978 IE 15

-0.603972E 16

, / -dT -.

A(e )

0.164843E-00

0.171445E-00

0.177068E-00

0.192936E-00

0.188072E-00

0.157899E-00

0.125583E-00

0.985285E-01

0.773923E-01

0.6t2502E-0l

0.489694E-01

0.395769E-01

0.323224E-0J

0.266523E-01

0.22I653E-01

0»185715E-0J

0.156609E-01

0.132798E-01

0.U3145E-01

0.535128E-02

0.267697E-02

0.138556E-02

0.734013E-03

0.395531E-03

0.215946E-03

0. 1 19132E-03

0.662820E-04

0.371389E-04

0.209341E-04

0.118605E-04

0.674966E-05

0.38561OE-05

0.221894E-00

0.208873E-00

0.193241E-00

0.893097E-01

-0.355303E-03

-0.958703E-0I

-0.129894E-00

-0.134996E-00

-0.127285E-00

-0.114679E-00

-0.100965E-00

-0.878487E-01

-0.760052E-01

-0.656142E-01

-0.566303E-01

-0.4891SOE-Ol

-0.423160E-0J

-0.366666E-01

-0.318277E-01

-0.160748E-01

-O.838368E-O2

-0.447101E-02

-0.242331E-02

-0.132976E-02

-0.736862E-03

-0.411578E-03

-0.23M14E-03

-0.130845E-03

-0.743378E-04

-0.424103E-04

-0.242840E-04

-0.139499E-0*
176 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

T

O.OO1

0*005

O.Ol G

o.cso

0*100

0*200

0*300

0*400

0*500

0*600

0*700

0*800

0.900

1*000

1.100

1.200

1.300

1*400

1*500

2.000

2*500

3*000

3*500

4.000

4.500

5.000

5.500

6.C0C

6.500

7.C00

7*500

B.OOC

TABLE 2.—The A

d

7.000

7.000

7.000

7.000

7.COO

7.COO

7.000

7.000

7*000

7.000

7.COO

7.000

7.000

7.000

7.COO

7.COO

7.000

7.000

7.COO

7.000

7.000

7.000

7.000

7.000

7.000

7.000

7.000

7.000

7.000

7.000

7.C00

7.000

T-dT

-0.006

-0.030

-0.060

-0,300

-0.500

-1.200

-1.800

-2.400

-3.000

-3.600

-4.200

-4.800

-5.400

-6.000

-6.600

-7,200

-7.800

-8.400

-9.000

-12.000

-15.000

-18.000

-21.000

-24.000

-27.000

-30.000

-33.000

-36.000

-39.000

-42.000

-45.000

-48.000

and 4> operators of the exponential function (continued)

0.453277E Ol

0.289912E 01

0.217528E 01

0.302669E-00

-0.769881E 00

-0.244209E 01

-0.424987E Ol

-0.660067E 01

-0.993383E 01

-0.149063E 02

-0.225774E 02

-0.346979E 02

-0.541935E 02

-0.859898E 02

-0.138426E 03

-0.225688E 03

-0.372006E 03

-0.618919E 03

-0.103788E 04

-0.149595E 05

-0.234956E 06

-0.387790E 07

-0.661272E 08

-0.U5412E 10

-0.2C4965E 11

-0.368973E 12

-0.671464E 13

-0.123285E 15

-0.228045E 16

-0.424480E 17

-0.794392E 18

-0.149363E 20

, / -dT y

A(e )

0.151147E-00

0.157463E-00

0.162710E-00

0.175524E-00

0.167721E-00

0.135415E-00

0.104381E-00

0.800901E-01

0.620312E-01

0.487301E-01

0.388597E-01

0*314263E-0l

0.257306E-01

0.212899E-01

0.177717E-01

0.149445E-01

0.126445E-01

O.107540E-01

0.918621E-02

0.438143E-02

0.220105E-02

0.1 14206E-02

0.606020E-03

0.326954E-03

0.178669E-03

0.986384E-04

0.549122E-04

0.307833E-04

0.173588E-04

0.983834E-05

0.560056E-05

0.320046E-05

./ -dT N §(e )

0.197453E-00

0.184258E-00

0.168485E-00

0.658553E-01

-O.18519OE-O1

-0.100525E-00

-0.123704E-00

-0.122265E-00

-0.1U518E-00

-0.981866E-01

-0.850781E-01

-0.73231OE-Ol

-0.629122E-01

-0.540719E-01

-0.465483E-01

-0.401552E-01

-0.347175E-01

-0.300820E-01

-0.261195E-01

-0.132273E-01

-0.691421E-02

-0.369315E-02

-0.200398E-02

-0.11006IE-02

-0.610295E-03

-0.341070E-03

-0.191858E-03

-0.108521E-03

-0.616750E-04

-0*351961E-04

-0.201582E-04

-0.115824E-04
A AND * OPERATORS OF RADIATIVE TRANSFER THEORY 177

7

0*001

0.005

0.010

0.050

0.100

0.200

0.300

0.400

0*500

0.600

0.700

o.eoo

0.900

1*000

1*100

1*200

1.300

1 .400

1.500

2*000

2.500

3.0C0

3.500

4.000

4.500

5.000

5.500

6.000

6.500

7,000

7.500

8.CCO

TABLE

a

e.ooo

e.ooo

8.000

8.000

8*000

8.000

8.000

8*000

8*000

8.000

8.000

8.000

8.000

8.000

8*000

8*000

8.000

8.00C

8.000

8.000

8.000

8.000

8.000

8.000

8.000

8.000

8.000

8.000

8.000

8.000

8.000

8.000

2.—The A

T-dT

-0.007

-0.035

-0.070

-0.350

-0.700

-1.400

-2.100

-2.800

-3.500

-4.200

-4.900

-5.600

-6.300

-7.000

-7.700

-8.400

-9.100

-9.800

-10.500

-14.000

-17.500

-21.000

-24.500

-28.000

-31.500

-35.000

-38.500

-42.000

-45.500

-49.000

-52.500

-56«000

and 4> operators of the exponential function (continued)

E^T-dx)

0.437762E 01

0.273988E 01

0.201080E 01

0.894340E-01

-0.10649IE 01

-0.300721E 01

-0.533324E 01

-0.867930E 01

-0.139254E 02

-0.225774E 02

-0.373324E 02

-0.631018E 02

-0.1C8916E 03

-0.191505E 03

-0.342040E 03

-0.618919E 03

-0,113204E 04

-0.208905E 04

-0.388374E 04

-0.931925E 05

-0.242401E 07

-0.661272E 08

-0.166216E 10

-0.536451E 11

-0.157210E 13

-0.466906E 14

-0.140163E 16

-0.424480E 17

-0.129500E 19

-0.397544E 20

-0.122694E 22

-0.380433E 23

A(e )

C139882E-00

0*145944E-O0

0* 150857E-00

0.161026E-00

0.150896E-00

0* 117533E-00

0.882685E-01

0.666507E-01

0.512143E-01

0.401409E-01

0.320468E-01

0.259913E-O1

0.213554E-01

0.177319E-01

0.148491E-01

0.125215E-01

0.1C6193E-01

0.904930E-02

0.774273E-02

0*371175E-02

0*186945E-02

0*971548E-03

0.516107E-03

0.278670E-03

0.152378E-03

O.B41658E-04

0.468743E-04

C262861E-04

0.148271E-04

0.840549E-05

0.478591E-05

0.273544E-05

$(e~dT)

0.177908E-00

0.164577E-00

0.148712E-00

O.476886E-O*1

-0.314199E-01

-0.101369E-00

-0.116054E-00

-0.110286E-00

-0.981103E-01

-0.850016E-01

-0.729112E-01

-0.623778E-01

-0.534055E-01

-0.458221E-01

-0.394195E-01

-0.340029E-01

-0.294058E-01

-0.254903E-01

-0.221435E-01

-0.112394E-01

-0.588415E-02

-0*314626E-02

-0.170B53E-02

-0.938893E-03

-0.520863E-03

-0.291200E-03

-0.163857E-03

-0.927074E-04

-0.526998E-04

-0.300802E-04

-0.172311E-04

-0.990213E-05
178 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 2.—The A and * operators of the exponential function (continued)

T

0*001

0*005

0*010

0*050

0*100

0*200

0.3O0

0*400

0*500

0*600

0.7O0

0.800

0.900

1 .000

1.100

1.200

1.300

1 .400

1.500

2*000

2*500

3.000

3.500

4*000

4*500

5*000

5.500

6*000

6*500

7.000

7.500

8.000

a

9*000

9*000

9*000

9*000

9.OOO

9*000

9*000

9.000

9.000

9.000

9.000

9.000

9.0OO

9.000

9.000

9.000

9.000

9.000

9*00C

9.000

9.000

9.OOO

9.000

9.000

9.000

9.000

9.000

9.000

9.000

9.000

9.000

9.000

T-dT

-0*008

-0*040

-0*080

-0*400

-0*800

-1.600

-2.400

-3*200

-4.000

-4.800

-5.600

-6.400

-7.200

-8*000

-8.800

-9.600

-10.400

-1 1.200

-12.000

-16.000

-20.000

-24.000

-28.0O0

-32.000

-36.000

-40*000

-44.000

-48*000

-52.000

-56.000

-60.000

-64.000

0.424308E 01

0.2601266 01

0.186688E 01

-0.104765E-00

-0.13474OE 01

-0.360532E 01

-0.660067E 01

-0.H3673E 02

-0.196309E 02

-0.346979E 02

-0.631018E 02

-0.H7935E 03

-0.225688E 03

-0.440380E 03

-0.872895E 03

-0.175214E 04

-0.355306E 04

-0.726534E 04

-0.149595E 05

-0.59556IE 06

-0,256157E 08

-0.115412E 10

-0.53645IE I 1

-0.255004E 13

-0.123285E 15

-0,603<572E 16

-0.299045E 18

-0.149363E 20

-0.751479E 21

-0.380433E 23

-0.1936I8E 25

-0.989965E 26

A(e )

0*13O423E-OO

0.136257E-00

0.140870E-00

0*148721E-00

0.136748E-00

0.103083E-00

0.757930E-01

0.566152E-01

0.433518E-01

0*340057E-Ol

0.272251E-01

0.221562E-01

0.182643E-01

0.152088E-01

O.127666E-01

0,107864E-01

0.916229E-02

0.781772E-02

0.6696O4E-02

0.322061E-02

0*162493E-02

0.845421E-03

0.449457E-03

0.242824E-03

0.132838E-03

0*733991E-04

0.408901E-04

0.229359E-04

0.129400E-04

0.733706E-05

0.417322E-05

0.238843E-05

-dT,

$(e

O.1619O3E

0*148466E

0.132545E

0.333287E

-0.406447E

-0.999486E

-0.108065E

-0.99S447E

-0.869176E

-0.7448O5E

-0.634944E

-0.541491E

-0.462946E

-0.397036E

-0.341587E

-0*294747E

-0.255008E

-0.221152E

-0.192200E

-0.977232E

-0.512168E

-0.274061E

-0.148907E

-0.818640E

-0.454304E

-0.254059E

-0.142991E

-0*809176E

-0.460056E

-0.262631E

-0.150465E

-0.864767E

00

00

00

01

01

01

00

01

01

O1

01

01

01

01

01

01

0 1

01

01

02

02

02

02

03

03

03

03

04

04

04

04

05
A AND # OPERATORS OF RADIATIVE TRANSFER THEORY 179

TABLE 2.—The A and * operator* of the exponential function (continued)

T

0.001

0.005

0.010

0.050

0.10O

0.200

0*300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

1*100

1*200

1*300

1 .400

1*500

2*000

2*500

3*000

3*500

4*000

4*500

5*000

5*500

6*000

6*500

7,000

7,500

8,000

d

10,000

10.000

10.000

10.000

10.000

1C.0OO

10.000

10.000

10.000

10.000

10.000

10.000

10*000

10.000

10*000

10*000

10.000

10.000

10,000

10*000

10*000

10*000

10*000

10*000

10*000

10,000

10,000

10,000

10,000

10.000

10,000

10,000

T-dT

-0,009

-0,045

-0,090

-0,450

-0,900

-1,800

-2,700

-3,600

-4,500

-5,400

-6,300

-7.200

-8.100

-9.000

-9.900

-10.800

-11,700

-12*600

-13*500

18*000

-22*300

-27,000

-31*500

-36,000

-40,500

-45,000

-49*500

-54,000

-58*500

-63,000

-67*500

-72,000

0.412429E Ol

0.247837E 01

0.173866E 01

-0.284855E-00

-0,16228IE 01

-0,424987E 01

-0,811035E 01

-0.149063E 02

-0.279337E 02

-0.541935E 02

-O,lO8916E 03

-C225688E 03

-0.479322E 03

-0.103788E 04

-0.228158E 04

-0.507596E 04

-0,114018E 05

-0,258137E 05

-0,588270c 05

-0,387790E 07

-0.275573E 09

-0.204965E 11

-0.157210E 13

-0.123285E 15

-0.983159E 16

-0.794392E 18

-0.648677E 20

-0.534303E 22

-0.443298E 24

-0.37C065E 26

-0.31C569E 28

-0.261838E 30

A(e"QT)

0,122348E-OO

CI27974E-O0

0*132315E-00

0.138119E-00

0,124687E-00

0.912483E-01

0.659729E-01

0.489537E-01

0.374671E-01

0,294605E-0l

0»236635E-0l

0.193187E-01

0*159681E-Ol

0*133256E-Ol

0*M2050E-0i

0.947987E-02

0*806117E-02

0.688422E-02

0.590075E-O2

0.284467E-02

0*1437l0E-02

O*748318E-03

0.398068E-03

0.215156E-03

0.117742E-03

0*650757E-04

0*362613E-04

0,203434E-04

0,114792E-04

0.650967E-05

0.370749E-05

0*211957E-05

0*148547E-00

0.135026E-00

0.119075E-00

0.217977E-01

-0.472321E-01

-0.972053E-01

-0.100297E-00

-0.901288E-01

-0.776204E-01

-0.660295E-01

-0.560901E-01

-0.477636E-01

-0.408187E-01

-0.350112E-01

-0.301315E-01

-0,260103E-01

-0*225127^-01

-0.I95313E-01

-0.169803F-01

-0.864472E-02

-0*453438E-02

-0.242771E-02

-0.131961E-02

-0.725708E-03

-0*402835E-03

-0*225324E-03

-0.12684OE-O3

-O,7I78B9E:-O 4

-0.408207E-04

-0.233058E-04

-0.133535E-04

-0.767536E-05
180 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

T

0*001

0*005

0*010

0*050

0*100

0*200

0*300

0*400

0*500

0*600

0.700

0*800

0*900

1*000

1*100

1*200

1*300

1*400

1*500

2*000

2*500

3.000

TABLE 2.—The A

d

15*000

15*000

15*000

15*000

15*000

15.000

15*000

15*000

15*000

15*000

15*000

15*000

•15.000

15*000

15*000

15*000

15*000

15*000

15*000

15*000

15*000

15.000

T-dT

-0.014

-0*070

-0*140

-0*700

-1*400

-2*800

-4*200

-5.600

-7*000

-8*400

-9*800

-11*200

-12*600

-14*300

-15.400

-16.800

-18.200

-19.600

-21.000

-28.000

-35*000

-42*000

and * operators of the exponential function (continued)

-dTv

0,367743E 01

0.201080E 01

0.124384E 01

-0.10649IE 01

-0.300721E 01

-0.867930E 01

-0.225774E 02

-0.631018E 02

-0.191505E 03

-0.618919E 03

-0.208905E 04

-0.726534E 04

-0.258137E 05

-0.931925E 05

-0.340645E 06

-O*12576OE 07

-0.468093E 07

-0.175426E 08

-0.661272E 08

-0.536451E 11

-0.4669O6E 14

-0.424480E 17

A(

0.946800E-01

0.994824E-01

0.102743E-00

0*10]133E-00

0.841239E-01

0.553805E-01

0.385991E-01

0.286375E-01

0*221922E-0i

0.176926E-01

0.143766E-01

0*118412E-0l

0.9B5245E-02

0.826306E-02

0.697473E-02

0*591872E-02

0.504527E-02

0.431737E-02

0*370692E-02

0.179735E-02

0*911047E-03

0.475449E-03

,/ -dT»

0*1050936-00

0.9J3520E-01

0.755014E-01

-0.113824E-01

-0.592710E-01

-0*780517E-01

-0.698794E-01

-0*588914E-01

-0.493229E-01

-0.41S096E-01

-0*331527E-01

-0.299363E-01

-0.256142E-01

-0.220028E-01

-0.189641E-01

-0.163922E-01

-0.142048E-01

-0.123366E-01

-0*107353E-01

-0.548386E-02

-0*288264E-02

-0.154571E-02
A AND * OPERATORS OF RADIATIVE TRANSFER THEORY 181

T

0*001

0*005

0«01O

0*050

0*100

0*200

0*300

0*400

0*500

0*600

0,700

0*800

0*900

1*000

1*100

1*200

1*300

1*400

1*500

2*000

2*500

3*000

TABLE

d

20*000

20*000

20*000

20.000

20*000

20*000

20*000

20*000

20*000

20.000

20.000

20*000

20,000

20*000

20*000

20*000

20.000

20*000

20.000

20.000

20*000

20.000

2.—The A

T-dT

-0*019

-0.095

-0*190

-0*950

-1.900

-3.800

-5.700

-7.600

-9.500

-11.400

-13.300

-15.200

-17,100

-19,000

-20,900

-22*800

-24*700

-26*600

-28*500

-38.300

-47.500

-57.000

and * operators of the exponential function (continued)

0.33670IE 01

0.167936E 01

0.884095E 00

-0.175915E 01

-0.459371E 01

-0.170948E 02

-0.681350E 02

-0.314572E 03

-0.160503E 04

-0.869781E 04

-0.489620E 05

-0.282878E 06

-0.166563E 07

-0.995091E 07

-0.601369E 08

-0.366840E 09

-0.225517E 10

-0.139546E 11

-0.868329E 11

-0.861639E 15

-0.915684E 19

-0.10T564E 24

A(e )

0.782326E-01

0.824277E-01

0.849009E-01

0,787968E-0l

0,61454OE-Oi

0.384396E-01

0.268704E-O1

0.201985E-01

0.158162E-01

0.126955E-01

O.1O3621E-O1

0.856076E-02

0.713879E-O2

0.599736E-02

0.506915E-02

0.430643E-02

0.367432E-02

0.314670E-02

0*270363E-02

0.131403E-02

0.667026E-03

0.348443E-03

*(e )

0.811263E-01

0.673437E-01

0,517989E-0i

-0.249669E-01

-0»574782E-01

-0.614449E-01

-O,517898E-Ol

-0,429094E-0t

-0.358185E-01

-0*30l563E-0t

-0*255670E-01

-0.217973E-01

-0.186S82E-01

-0.160490E-01

-0.138419E-01

-0.119717E-O1

-0.103794E-0I

-0.901833E-02

-0.785080E-02

-0.401623E-02

-0.2I1318E-02

-0.U3388E-02
182 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

T

0.001

0.005

0.010

0.050

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

1.100

1.200

1.300

1 .400

1.500

2.0C0

2.500

3.000

TABLE 2 —The A

d

25.000

25.000

25.000

25.000

25.000

25.000

25.000

25.000

25.0C0

25.000

25.000

25.000

25.000

25.000

25.000

25.000

25.000

25.000

25.O00

25.000

25.000

25.000

T-dT

-0.024

-0.120

-0.240

-1.2OC

-2.400

-4.30C

-7.200

-9.600

-12.000

-14.400

-16.8OC

-19.200

-21.600

-24.000

-26.4OC

-28.800

-31.200

-33.60C

-36.000

-48.0O0

-60.000

-72.000

and $ operators of the exponential function (continued)

E1(T-dx)

0.312834E 01

0.I41935E 01

0.594697E 00

-0.244209E 01

-0.660067E 01

-0.346979E 02

-0.225688E 03

-0.175214E 04

-0.149595E 05

-0.134809E 06

-0.125760E 07

-0.120195E 08

-0.U6962E 09

-0.U5412E 10

-0.115154E 11

-0.U5943E 12

-0.U7623E 13

-0.120092E 14

-0.123285E 15

-0.149363E 20

-0.193618E 25

-0.261838E 3C

A(e )

0.671701E-01

0.708832E-01

0.727423E-01

0.638100E-01

0.474262E-01

0*291396E-01

0.206109E-01

0.156386E-01

0.123105E-01

0.991236E-02

0.810694E-02

0.670741E-02

0.559945E-02

0.470824E-02

0.398237E-02

0.338516E-02

0.288972E-02

0.247583E-02

0»212801E-02

0. 103563E-02

0.526132E-03

0.274995E-03

0.658889E-01

0.521476E-01

0.369957E-01

-0.305956E-01

-0.522582E-01

-0«495203E-01

-0.407385E-01

-0.336444E-01

-0.281006E-01

-0.236806E-01

-0.200929E-01

-0.171413E-01

-0»146882E-01

-0.126330E-01

-0.108997E-01

-0.942995E-02

-0.817800E-02

-0.710733E-02

-0.618854E-02

-0.316344E-02

-0«166800E-02

-0.895353E-03
A AND * OPERATORS OF RADIATIVE TRANSFER THEORY 183

TABLE 2.—The A and * operators of the exponential function (continued)

-dTN

T

O.OO1

0.005

0.010

0.050

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1 .000

1.100

1.200

1 .300

1 .400

1 .500

2.000

2.500

3.000

Q

30.000

30.000

30.000

30.000

30.000

30.000

30.000

30.000

30.000

30.000

30.000

30.000

30.000

30.000

30.000

30.000

30.000

30.000

30.000

30.000

30.000

30.000

T-dT

-0.029

- 0. 145

-0.290

-1.450

-2.900

-5.800

-8.700

-11.60C

-14.500

-17.400

-20,300

-23.200

-26.IOC

-29.COO

-31.900

-34.800

-37.700

-40.60C

-43.500

-58.000

-72.500

-87.00C

E1(T-dT)

0.293403E 01

0.120338E 01

0.348202E-00

-0.31531 IE 01

-0.928602E 01

-0.736008E 02

-0.800749E 03

-0.104 170E 05

-0.147866E 06

-0.220683E 07

-0.340365E 08

-0.537353E 09

-0.863313E 10

-0.140599E 12

-0.231486E 13

-0.384536E 14

-0.643544E 15

-0.108381E 17

-0.183516E 18

-0.271234E 24

-0.428678E 30

-0.706613E 36

A(e

0.591490E-01

0.624624E-01

0.638230E-01

0.531055E-01

0.381428E-01

0.234209E-01

0.167418E-01

0. 127731E-01

0.100834E-01

0.313316E-02

0.665962E-02

0.551471E-02

0.460684E-02

0.387568E-02

0.327959E-02

0.278880E-02

0.238138E-02

0.204085E-02

0.175455E-02

0.854593E-03

0.434387E-03

0.217473E-03

0.553283E

0.416739E

0.269547E

-0.325190E

-0.466171E

-0.410723E

-0.334901E

-0 .276601E

-0.231207E

-0.194967E

-0.165511E

-0.141254E

-0.121079E

-0.104165E

-0.898935E

-0.777878E

-0.674722E

-0.586477E

-0.510733E

-0.261623E

-0.137777E

-0.738458E

01

01

01

01

01

01

01

01

01

01

01

01

01

01

02

02

02

02

02

02

02

03
184 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 3.—The A operators of

m=O m=2 m=3

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.12

0.14

0.16

0.18

0.20

0.24

0.26

0.32

0.36

0.40

0.44

0.48

0.52

0.56

0.60

0.64

0.72

0.80

0.90

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70

1.80

1.90

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

9.50

.500000E+00

•525165E+00

.543448E+00

.559164E+00

•573230E+00

.586083E+00

.597977E+00

.609082E+00

.619519E+00

.629378E+00

.638728E+00

•656112E+00

.672011E+00

.686663E+00

.700246E+00

•712900E+00

•735834E+00

.756131E+00

.774259E+00

.790565E+00

.805316E+C0

.818723E+00

.8309566+00

•842157E+00

.852441E+00

.861908E+00

.870642E+00

•886194E+00

•899574E+00

•913798E+00

.929752E+00

.9358&9E+C0

.944448E+00

.951777E+00

•958055E+00

.963450E+00

.968098E+00

.972115E+00

•975592E+00

.97861CE+00

•981233L+00

.994679E+00

.998401E+00

.999502E+00

.999841E+00

,999948E*00

.999983E+00

.999994E+00

.999997E+00

.250000E+0O

.255138E+OO

.260484E+00

.265999E+00

.271662E+00

.2774*9E+OO

.283380E+CO

.289416E+00

.295560E+00

.3O18O5E+OO

.308146E+00

.321097E+00

.334380E+00

•347969E+00

•361840C+00

.375963E+00

.404957E+00

.434804E+00

•465419E+00

•496721E+00

.528643E+00

.561128E+OO

•594125E+00

.627591E+00

.661486E+00

.695775E+OO

.730429E+00

.8OO718E+OO

.872162E+00

.962851E-I-00

.105485E+01

•114794E+01

.124197E+01

•133679E+01

.143229E+01

.152837E+01

•162495E+01

•172197E+01

.181936E+01

.191707E+01

•2O15O7E+O1

.300446E+01

•400138E+01

.500044E+01

.600014E+01

.700005E+01

.800002t.+01

.900000E+01

.950000E+01

.333333E+00

.338384E+00

•343540E+00

.348805E+00

.354181E+00

•359672E+00

.365280E+00

.371008E+00

.376858E+00

.382831E+00

.388930E+00

.401514E+00

.414623E+00

.428269E+00

.442464E+00

.457219E+00

.488450E+00

.522036E+00

.558040E>00

.596521E+00

.637531E+00

.681119E+00

.727326E+00

.776191E+00

.827752E+00

.682040E+00

.939085E+00

.10.6156E+01

.119538E+01

.137885E+01

.158060E+01

.180066E+01

.203984E+01

.229770E+01

.257460E+01

.287066E+01

.318598E+01

.352067E+01

.387480E+01

.424843E+01

.464164E+01

.965900E+01

.166642E+02

.256659E+02

.366664E+02

.496666E+02

.646666E+02

.816666E+02

.909167E+02

.750000E+00

•760075E+00

.770304E+00

.780689E+00

.791233E+00

.801941E4-00

.812815E+00

.823859E+00

•835077E+00

.846472E4-00

.858048E+00

.881758E+00

.906240E+00

.931524E+00

.957643E+00

.984631E+00

.104135E+01

.110195E+01

.116673E+01

.123598E+01

.131000E+01

.138909E+01

.147357E+01

.156376E+01

.165997E+01

.176253E+01

.187177E+01

.211162E+01

.238222E+01

.276790E+01

.321136E+01

.371812E+01

.429375E+01

.494391E+01

.567428E+01

.649059E+01

.739860E+01

.840412E+01

.951295E+01

.107309E+02

.120640E+02

.330220E+02

.720065E+02

•135002E+03

.228001E+03

.357000E+03

.528000E+03

.747000E+03

.876375E+03

.240000E+01

.243020E+01

.246081E+01

•249183E+01

.252326E+01

.255513E+01

.258742E+01

.262016E+01

.265333E+01

.268696E+01

.272105E+01

.279064E+01

.286216E4-01

.293566E+01

.301122E+01

.308891E+01

.325094E+01

.342234E+01

.360378E+01

.379594E+01

.399955E+01

.421541E+01

.444435E-«-01

•468726E+01

.494507E+01

.521878E+01

.550944E+01

.614600E+01

.686415E+01

.789234E+01

.9086186+01

.104699E+02

.120698E+02

.139148E+02

.160357E+02

.184657E+02

.212404E+02

.243976E+02

.279774E+02

.320225E+O2

.365775E+02

.121729E+03

.324777E+03

.729792E+03

.144480E+04

.260180E+04

.435680E+04

.688980E+04

.851086E+04

.100000E+02

•1O12O8E+O2

•102430E+02

.103668E+02

.104922E+C2

.106192E+02

•107477E+02

.108779E+02

.110098E+02

.111433E+02

.112785E+02

.115540E+02

.118366E+02

.121265E+02

.124238E+02

.127288E+02

.133627E+02

.140298E+02

.147323E+02

.154721E+02

.162314E+02

.170727E+02

.179385E+02

.188514E+02

.198144E+02

.208305E+02

•219030E+02

.242315E+02

.268307E+02

.305134E+02

.347506E+02

.396312E+02

.452565E+02

.517418E+02

.592173E+02

.678290E+C2

.777404E+02

•891332E+02

.102206E+03

.117188E+03

.134316E+03

.495320E+03

.154677E+04

.407837E+04

.936001E+04

.192617E+05

.363733E+05

.641250E+05

.833219E+05
A AND # OPERATORS OF RADIATIVE TRANSFER THEORY 185

TABLE 4.— The * operators of (rM)

m=O m=3

o.op

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.12

0.14

0.16

0.18

0.20

0.24

0.28

0.32

0.36

0.40

0.44

0.48

0,52

0.56

0.60

0.64

0.72

0.80

0.90

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70

1.80

1.90

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

9.50

.lOOOOGE+01

.980553E+00

•961936E+00

.943995E+00

.926646E+00

.90983w8E+00

.893522E+00

•877665E+00

•862239E+00

.847219E+00

.832583E+00

.8G4387E+00

.777521E+00

•751876E+00

.727359E+00

•703891E+00

.659828E+00

.619217E+00

.581675E+00

•546883E+00

.514573E+00

.484513E+00

.456502E+00

.4303646+00

.405943E+00

.383101E+00

•361714E+00

.322872E+00

•286648E+00

.251406E+00

•219384E+00

.191762E+00

.167869E+00

.147153E+00

.129151E+00

•113479E+00

.998114E-01

.878734E-01

.77431AE-01

.682860E-01

.602668E-01

•178612E-01

.552272E-02

•175560E-02

.569200E-03

.187320E-03

.623600E-04

•209600E-04

.121800E-04

.666667E+00

•676568E+00

.66628QE+0Q

.695810E+00

•705162E+00

.714344E+00

.723361E+00

.732216E+00

•740916E+00

.749462E+00

•757861E+00

.774228E+00

•790046E+00

•80533at+00

.820128E+00

•834439E+00

.861701E+00

.887271E+00

•911279E+00

.933842E+00

.955063E+00

.975038E+00

.993851E+00

.101158E+01

.102830E+01

.104408E+01

.105897E+01

.108632E+01

• H1075E+01

.113771E+01

.116121E+01

.118173E+01

.119968E+01

.121541E+01

.122920E+01

•124132E+01

.125197E+01

.126134E+01

.126959E+01

.127687E+01

.128329E+01

.131800E+01

.132849E+01

•133177E+01

.133282E+01

.133316E+0X

.133328E+01

.133331E+01

•133332E*01

.100000E+01

.101343E+01

.102706E+C1

.104088E+01

.105489E+01

•106909E+01

.108546E+01

.109802E+01

.111275E+01

.112766E+01

.114273E+01

•117337E+01

.120466E+01

.123657E+01

.126906E+01

.130217E+01

.137003E+01

.144000E+01

.151195E+01

.158577E+01

.166133E+01

.173854E+01

.181731E+01

.189753E+01

.197913E+01

.206204E+CX

.214616E+01

.231783E+01

.249363E+01

.271854E+01

.294348E+C1

.318282E+01

.342100E+01

.366255E+01

.390704E+01

.415411E+01

.440347E+01

.465482E+01

.490793E+01

.516259E+01

.541862E+01

.802679E+01

.106753E+02

.133362E+02

.160001E+02

. 186670E+02

.213334E+02

.240000E+02

.253334E+02

.240000E+01

.243020E-t-01

•246081E+01

•249183E+01

•252326E+01

.255512E+01

•258741E+01

.262013E+01

.265329E+01

.268690E+01

.272095E+01

.279043E+01

.286177E+01

.293500E+01

.301017E+01

.308731E+01

.324762E+01

.341620E+01

.359330E+01

.377914E+01

.397395E+01

.417793E+01

,43912ec+0i

.461414E+01

•484673E+01

.508918E+01

.534166E+01

•587726E+01

•645455E+01

.723624E+01

.808616E+01

.900577E+01

.999626E+01

•110587E+02

.121941E+02

•134032E+02

.146868E+02

.160455E+02

.174798E+02

,189904E-»-02

.205775E+02

.407287E+02

.687767E+02

•104792E+03

.148797E+03

.200799E+03

.260800E+03

.328800E+03

.365800E+03

.80C000E+01

.809660E+01

.819442E+01

•829347E+01

.839377E+01

.849534E+01

.859819E+01

.&70234E+01

.eao7ai£+oi

.691461E+01

.902276E+01

.924321E+01

.946929E+01

•970114E+01

.993893E+01

.101828E+02

.106895E+02

.112225E*02

.117831E+02

.123728E+02

•129929E+02

.136450E+02

•143304E+02

.150507E+02

.158074E+02

.166022E+02

.174365E+02

.192304E+02

.212024E+02

.239383E+02

.270005E+02

.304165E-t-02

.342146E+02

.384231t+02

.430712E+02

•481882E+02

.538037E+02

.599477E+02

.666502E^02

.739417E+02

.818527E+02

.201856E+03

.418218E+03

.762695E+03

•126721E+04

.196374E+04

.288427E+04

.406060E+04

.475507E+04

.342857E+02

.346681E+02

.35C954E+02

.355076E+02

.359248E+02

,363*70E*02

.367743E+02

.372068E+02

.376446E1-02

.380876E+02

.385360E+02

.394493E+02

.403649t+02

.413434E+02

.423253E+02

.433314E+02

.454182E+02

.476089E+02

.499090E+02

.523241E+02

.548602E+02

.575234E+02

.603204E+02

.632579E+02

.663431E+02

.695834E+02

•729866E+02

.U03142E+02

.883946E>02

.996668E+02

.112387E+03

.126726E+03

.142868E+03

.161009E+03

.181364E-^03

.204159E+03

.229635E+03

.25805CE+03

•289676E+03

•324799E+03

.363721E+03

•103941E+04

.254285E4-04

.543511E+04

.104365E+05

.1S42 72E+05

.304472E+05

.476966E+05

.587010E+05