Smithsonian

Contributions to Astrophysics

VOLUME 5, NUMBER 11

ON THE Λ AND Φ OPERATORS

OF RADIATIVE TRANSFER THEORY

by S. S. KUMAR

SMITHSONIAN INSTITUTION

Washington, D.C.

1962

Publications of the Astrophysical Observatory

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II

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Contents

Publications of the Astrophysical Observatory

On the Λ and Φ Operators of Radiative Transfer Theory

Introduction

The formulas used in the computations

Acknowledgment

References

Abstract

On the Λ and Φ Operators of Radiative Transfer Theory

By S. S. KUMAR[1]

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 Λ and Φ operators an outline is given here.

We approximate the monochromatic source function Sv at any depth τv by an expression of the type

where am, bn, cp, dp are unknown parameters and En(τv) is the n-th exponential integral function defined by

The monochromatic source function satisfies the following integral equation:

where σs is the coefficient of electron scattering, κv the coefficient of continuous absorption, and Bv the Planckian function. The operator Λ is defined by

If we put the assumed solution (1) for the source function in equation (3), we get

[1] Smithsonian Astrophysical Observatory.

The parameters am, bn, cp, 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:

where the operator Φ is defined by:

In order to use this method we must know the Λ and Φ operators of the functions τm, En, e–dpτ. 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).

The Λ and Φ operators are also very useful in obtaining the temperature distribution for nongray model atmospheres (Swihart, 1956; Böhm, 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 Φ operators, we present here a generalization of Swihart’s method. The initial approximate temperature distribution is represented by

To obtain the temperature distribution for the nongray model we use the Φ operator and get the following expression for the correction ΔT:

where Δa_0, Δa_1, … are obtained by solving the following set of algebraic equations:

where ΔF is the known difference between the desired flux and the flux corresponding to the gray model. Since the Φ operator is probably insufficient to give the correct temperature distribution, we must also use the Λ operator.

The formulas used in the computations

The following formulas, given by Kourganoff (1952), were used to prepare tables 1 and 2 for the Φ and Λ operators of e–dτ:

The functions E_1(τ) and E_1(τ—τd) are given by

and E_2(τ) is given by

For the special case of d=1, the following formulas were used for the Φ and Λ operators:[2]

For computing the function E_1(x), where x stands for either τ or (τ—dτ), subroutine NUEXPI,[3] distributed by SHARE, was used.

The procedure for computing E_1(x) follows. Actually, the subroutine computes the function Ei(x), which is defined by

For a different range of values of x, the following formulas were used to obtain Ei(x):

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

where a’s and b’s have been tabulated by Hastings.

[2] The operators were computed on an IBM-7090 at the Smithsonian Astrophysical Observatory.

[3] 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 Λ and Φ operators.

For this range,

The quantities cn’s have been computed by Cooley.

For this range the following expansion is used in the subroutine:

where

For each value of n, x_0 is a known quantity.

The subroutine uses the following formula:

The accuracy of the Λ and Φ operators is dependent upon that of the function E_1(x). The subroutine gives E_1(x) correct up to seven digits. The operators in table 2 are computed and correct up to six digits.

TABLE 1.—Values of Φ_0(e–dτ) and Λ_0(e–dτ)

Since E_1(τ—dτ) was computed for a large range of the argument, varying from –87.00 to +9.4, we have also tabulated E_1(τ–dτ) along with the Λ and Φ operators. For very small values of d and τ table 2 gives the rounded-off values for (τ–dτ). For some values of d, we have plotted the Λ and Φ operators of the function (e–dτ). Figures 1, 2, 3, 4, 5, 6 show the operators for d=0.1, 0.4, 1.0, 5.0, 10.0, 30.0.

For τ = 0 and for any value of d the following simple expressions were used

Table 1 gives the values of Φ_0 (e–dτ) and Λ_0(e–dτ) for several values of d.

For the function τm the following formulas were used:

where

where

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 E_2, E_3, E_4, E_5, E_6, E_7, and E_8, and used the tables published by Placzek (1947).

FIGURE 1.—The Λ and Φ operators of e–dτ, with d=0.1.

FIGURE 2.—The Λ and Φ operators of e–dτ, with d=0.4.

FIGURE 3.—The Λ and Φ operators of e–dτ, with d=1.0.

FIGURE 4.—The Λ and Φ operators of e–dτ, with d=5.0.

FIGURE 5.—The Λ and Φ operators of e–dτ, with d=10.0.

FIGURE 6.—The Λ and Φ operators of e–dτ, with d=30.0.

FIGURE 7.—The Λ and Φ operators of τ^2.

FIGURE 8.—The Λ and Φ operators of τ^4.

The Λ and Φ operators of τm are correct to six decimal places. Figure 7 shows the operators of the function τ^2. Figure 8 gives the operators of the function τ^4.

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

BÖHM, K. H.

1954. Die Temperaturschichtung der Sonnen-atmosphäre im nichtgrauen Strahlungs-gleichgewicht. Zeitschr. Astrophys., vol. 34, pp. 182–208.

HASTINGS, C.

1955. Approximations for digital computers. Princeton University Press, Princeton, 201 pp.

KOURGANOFF, V.

1952. Basic methods in transfer problems. Clarendon Press, Oxford, 281 pp.

KUMAR, 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 Λ and Φ operators for the problem of transfer of radiation in nongray stellar atmospheres is discussed, and extensive tables of these operators of the functions e–dτ and τm are given.

TABLE 2.—The Λ and Φ operators of the exponential function

TABLE 3.—The Λ operators of (τM)

TABLE 4.— The Φ operators of (τM)