Ima j Appl Math 1973 Peraiah 75 90

/ . Inst. Maths Applies (1973) 12, 75-90

Numerical Solution of the Radiative Transfer Equationin Spherical Shells

A. PERAIAH t AND I. P. GRANT

Mathematical Institute, 24-29 St. Giles, Oxford

[Received 19 July 1972 and in revised form 25 September 1972]

A method of solving numerically problems of radiative transfer in shells with sphericalsymmetry is proposed. The difference equations are derived by discrete ordinate methodsand solved using algorithms due to Grant & Hunt (1968,1969). We have shown that themethod is stable, and will yield non-negative solutions, provided the optical thicknessof an elementary shell and the ratio of its thickness to the radius satisfy certain restric-tions. Two simple problems are presented to illustrate the theory.

1. Introduction

THE solution of the radiative transfer equation in plane parallel stratification has beenextensively investigated in fields such as astrophysics, neutron transport and meteoro-logy. However, this is not an adequate representation for studying radiative transferin the extended atmospheres of supergiant stars, and a proper study of these shouldincorporate the effect of curvature (cf. Underhill, 1970; Vardya, 1971).

Chandrasekhar (1934) and Kosirev (1934) were the first to examine the radiativetransfer equation with spherical symmetry. Recently Castor (1970),Cassinelli(1971)andHummer & Rybicki (1971) have examined this problem in some detail. The lastreference describes the "variable Eddington factor" method, in many ways the mostgenerally satisfactory approach for spherical systems published to date. However,it does seem to be rather time-consuming, and there is certainly room for alternativeslike the one described in this paper.

Here we shall discuss a method to obtain diffuse radiation in the extended atmo-spheres of supergiant stars. It is based on the discrete space theory of Grant & Hunt(1968, 1969a, 19696) which can be applied in principle to any medium which can bepartitioned by a suitable one parameter family of surfaces. So far only the planeparallel case has been treated in detail, but there is no reason why similar methodsshould not be applied in other simple geometries.

We have tried several methods of discretizing the transfer equation in sphericalsymmetry (Peraiah, 1971). The major problems come from the fact that a ray changesits angle with respect to the radius vector as it traverses the medium. Our first attemptto approximate the (so-called) curvature term used an orthogonal polynomial approxi-mation. This failed to give non-negative transmission and reflection operators, andwas therefore abandoned. We also abandoned an approximation based on the CSN

method proposed by Lathrop & Carlson (1971) for a similar reason. However, the"cell" approximation described in this paper does give usable results.

t Present address: Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay-5, India.

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76 A. PERAIAH AND I. P. GRANT

Since the purpose of this paper is to understand the diflBculties which arise in treatingproblems with curved geometry, we have confined our attention to the simplest typeof scattering problem with no time or frequency dependence.

2. Reflection and Transmission Operators in a Basic Cell

The equation of radiative transfer in divergence form is

J ' , p, (i')I(r, fi') «foj (2.1)

where co(r) is the albedo for single scattering, I{r, fi) is the specific intensity, r the radius,H = cos 0, o{r) is the absorption coefficient, b represents the sources inside the cellandp is the phase function. The functions a{r), a>(r), b(r) and p(r, fi, pi') are prescribed(generally piecewise continuous) functions of their arguments subject to the conditions

b{r) > 0, o(r) > 0, 0 s£ a)(r) ^ 1,and

= 1, p{r,fi,fi') 5= 0. - ]- l

If we write A(r) = 4nr2,

U(r, fi) = A(r)I{r, fi)B(r) = A(r)b(r)

then we can rewrite the equation of transfer in the form

j ^ + [ ( 111 f f1 1 (23)

= <x(r)|[l -a)(r)]B(r) + \(o(r) j ^ p(r, n, fi')U{r, fi') dp'j

for outward-going rays, and

for inward-going rays, where we have restricted fi to lie in the interval [0, 1].In the "cell" method of deriving difference equations one formally integrates (2.3)

and (2.4) over an interval [rn, rn+1] x [fij-^ /f/+$] defined on a two dimensional grid(Carlson, 1963; Lathrop & Carlson, 1967). The choice of the set {/•„} will be discussedin Section 2. The choice of {fij+i} is dictated by convenience. We shall employ theroots, fij, and weights, CJt of the Gauss-Legendre quadrature formula of order J over[0, 1], and define the cell boundaries by writing fi± = 0 and taking

= t Ck, j = l,2,...,J. (2.5)fc=i

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RADIATIVE TRANSFER EQUATION 77

It is easy to see that /*,_j. < \ii < fij+^l this follows immediately from the interpolatorycharacter of the Gauss formula.

We do the fi integration first. This replaces (2.3) by

= ff(r)C,j[l-a,(r)]B(r) (2.6)

where Uf(r) = U(r, nj), Uj(r) = U(r, ~n}), p++(r)jr = p(r, ̂ ^ . ) , p-+(r)jr =p(r, —/Xj, fly), and so on. We get a similar equation from (2.4). The reason for thechoice (2.5) should now be obvious; it permits us to evaluate the scattering integralterm with the maximum accuracy assuming that the solutions Uf, Uj are sufficientlysmooth. Provided we consider the diffuse field we can be sure that this is the case.However, we have not yet defined l/f+i, and we do this, with some loss of accuracy,by writing

TJ± - (Pj+i-Vj+iWf+toj+i-PjWf+i ,• - i ? T 1uj+i — 5 J — I, Z, . . ., J — I

and, for convenience, define U$ = U$ by interpolation

l ; . (2.8)Writing

U*(r) = lUHr), • • -, U*(r)T, (2-9)and making use of (2.7), (2.8) we can write (2.6) in matrix form

M^+[AUor r

= <x(r){[l -a)(r)]B+(r)+|a)(r)[p+ +(r)CU+(r) + p+ -(r)CU"(r)]and similarly

r+(r)CU+(r)+p--(r)ClT(r)]}. (2.10)Here C and M are diagonal matrices with elements [Cjdjy], Uijdjy] respectively,B+ and B~ are vectors with a similar structure to (2.9) and A+ and A~ are square JxJmatrices defined by the equations

, ; = 1,2,...,J-1

k =j,j = 1 ,2 , ...,J

fc.;_!,;. 2 , 3 , . . . , / (2.11)

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andCjAjk = - # y . A f l . (2.12)

These we call the curvature matrices.To perform discretization with respect to the radial co-ordinate we integrate

equations (2.10) from rn to rn+1 giving

(2.13)

Here Un+ = U+(rn), whilst variables subscripted with n+$, for example Uf+i, rn+i,

con+i must be associated with some suitable average over the cell. We define Arn+i =rn+i-rn, 1,,-i = on+iArn+i, and p = Arn+i/rn+i, where rn+i is a suitable mean radius,for example \{rn+1+rn). A convenient definition of U++i, U~_£ is

V:+i = KUn+

+1 + Un+), U"+i = i(U-+1 + Un-); (2.14)

this is the conventional "diamond" difference scheme which was used in the planeparallel case (Grant & Hunt, 1968).

It is now straightforward, but very tedious, to rearrange (2.13) with the aid of (2.14)in the canonical form

rt(n + l« ) r («n + l)TU+ ~ | | T V | (2 l5)

(Grant & Hunt, 1969). The r and t matrices and the vectors S* can be expressed interms of the matrices and vectors appearing in (2.13). Because of the lengths of theseexpressions they are given in the Appendix.

The matrices r and t appearing in the canonical form (2.15) have a physical inter-pretation as operators for diffuse reflection and transmission, respectively, of radiationincident on the spherical shell between radii rn, rn+1. Similarly the vectors Sn + i can beregarded as representing the radiation which emerges from the surfaces of the bareisolated shell due to internal emission sources. For further details see Grant & Hunt(1969a).

At this point we should, perhaps, say a little about the two formulations we rejected.In each case this was done because of properties of the A operators which appeared.The first method tried approximated the /i-variation of U by a polynomial expression

l/(r, ix) ~ f ak(r)pk(ji) (2.16)o

where the pk(ji) are the orthonormal polynomials associated with Gauss-Legendrequadrature on [0, 1], so that if p* is the vector

) . • • •> Pk(Mm)~\

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then= 5kl, k, I ^ m. (2.17)

By differentiating (2.16) with respect to /i and using the properties of the pk(jj), onearrives at an equation of the same form as (2.10) but with different matrices A+, A~.These matrices are full matrices rather than band matrices, with elements of bothsigns, and their structure is such that it is impossible to ensure that the t and r matricesare non-negative for any choice of radial mesh—an essential requirement discussedin the next section.

The CSN method proposed by Lathrop & Carlson (1971) was also tried in an attemptto find a less crude approximation than the one we adopted finally. The essential ideais to approximate U(r, fi) in each cell [rn, rtt+1] x [/ly, nj+1\ by a bilinear spline

U(r, n) ~ A + BZ+Cfi + DZij (2.18)where

2r-rn-rn+1t] =

We insert (2.18) in (2.1) and average over each cell to give an equation in A, B, C, Dwhich may conveniently be rewritten in terms of cell boundary values like U(rn, Hj).The result has a similar form to (2.13), but with more complicated matrices on theright hand side. The same difficulties as with the polynomial approximation (2.16)and the added algebraic complexity ensured its rejection.

3. Stability, Non-negativity of Operators and Source VectorsThe analysis of the canonical form (2.15) has been described in detail in the second

of the papers by Grant & Hunt (1969). The theory makes use of a modified vector norm

| | U | | D = | | D U | | 1 = ^ d y | « 7 | (3.1)

where D = 27iMc is a positive diagonal matrix, and where U denotes U+ or U~.Thus ||U||j) has the physical meaning of the total flux, ingoing or outgoing, crossinga given spherical surface. Since there is no ambiguity induced by omitting the suffix D,we shall leave it out from now on. The matrix norm consistent and subordinate with(3.1) is given by

||A|| = max £ |(DAD-%|. (3.2)k j=i

We need to extend these definitions a little before applying them to equations (2.15).We do this in the obvious way by writing

Define the cell operator matrix S(n, n+1) by the equation)lJ (3-4)

where we require S(«, n+1) to have non-negative elements, S(«, n+1) ^ 0. Then||S(n, n +1) || = max (max (Sk, S'k)) (3.5)

k

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where

S'k= £We say that S(«, n+l) is

strictly dissipative if ||S(«, n+l) | | < 1dissipative if ||S(«, n+l) | | = 1 but some of Sk, S'k < 1 (3.6)conservative if ||S(n, n+l) | | = 1 and a// 5 t , St< = 1.

The theory shows how to construct cell matrix operators for composite adjacentlayers (m, n), («, p), say, using the star-product.

S(w, p) = S(m, n) * Sin, p) (m < n < p) (3.7)where the r and t operators appearing in the cell operator of the composite layer are

t(p, m) = t(p, «)[I-r(ff7, n)r(p, n)]-H(n, m)t(m, p) = t(m, «)[I-T(p, n)r(m, n)]-H(n, p)r(p, m) = r(«, m)+t(m, ri)t(p, n)[I-r(m, ri)r(p, n)]-H(n,p) (3.8)t(rn,p) = i(n,p)+t(p, n)r(m, n)\I-T(j>, n)r(m, n)]-H(n,p).

The cell matrix operators play the role of Green's functions in the discrete theory,and it is clear that whatever algorithm is used to solve the difference equations (2.15)the results can be expressed by some equation of the form

l ( m - p ) - ( 3 9 )

The properties of cell matrices with respect to the star product are therefore crucialto our understanding of the behaviour of our numerical method. The key theorem isas follows (Grant & Hunt, 19696):

THEOREM 1. Let the cell matrices S(m, ri), S(n,p), m < n < p be non-negative andeach satisfy one of the conditions listed in (3.6). Then their star-product, S(m, p), definedby equations (3.7), (3.8) exists; it is also non-negative, and

(a) strictly dissipative if both S(m, n) and S(«, p) are strictly dissipative;(b) dissipative if either S(w, n) or S(«, p) is dissipative and the other at most

conservative;(c) conservative if both S(m, n) andS(n,p) are conservative.

To understand these results, we note first that the specific intensity, by its physicaldefinition is a non-negative quantity. The requirement that S(m, n) be a non-negativeoperator is therefore needed to ensure that the response of the layer to arbitrary non-negative inputs is a non-negative output. The norm ||S(w, n)||, gives essentially themaximum ratio of the total output flux to that incident on the layer, and so, in theabsence of internal radiation sources should not exceed unity. The theorem showsessentially that composite layers whose components themselves have operators whichare non-negative and satisfy one of conditions (3.6) inherit the same desirable pro-perties. We therefore wish to assure ourselves that the finite difference approximationsof equation (2.15) and the appendix have these properties. We shall find that this

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imposes restrictions both on the optical thickness, x, of a layer and on its "aspect-ratio", p = Ar/r.

When internal sources are present, we have to include a source vector, £(m, p), inequation (3.9), which may be defined in terms of the vectors £(m, n), £(n, p),m<n<p,by

S(m, p) = \(m, n; p)L{m, n)+\'(m; n, />)£(«, p) (3.10)where

andt(w, «)[I-r(p, n)r(m, n)]~lr(p, n)

( ' >JP) " L0 Km, n)\l-r(p, n)x{m, n)\-* JIt is clear that the behaviour of source vectors and the star product is closely boundup with that of cell matrices, and it is not difficult to prove (Grant & Hunt, 19696)

THEOREM 2. Let E(j,j+1) 3* 0 be bounded, j = m, m+1, ...,p-1. Then £(m,/>) isalso bounded and non-negative.

We therefore turn to examine the discrete ordinate approximations for cell matricesand source vectors in spherical geometry given in the Appendix. The main complicationis provided by the curvature operators A+, A" (2.11), (2.12) which contain largeelements of both signs (see Table 1). The result is that the aspect ratio of a layer, p,

TABLE 1

Curvature matrices, [A/J,[AjjJ. Matrices of order / = 4, abscissae ftj = 006943, 0-33001,0-66999, 0-93057 for Gauss-Legendre quadrature on [0,1]

(a)A/ t

1234

0-46494-1-78139

00

2-23590-004258-1-15005

0

Ajk= -2-87476

0115005

-0-75945-0-73228

00

0-58343-109379

must be small to compensate for this. From equation (A.4) et seq., it is clear that weneed A+ ^ 0, A~ $s 0, S+- 5= 0, S-+ > 0, and we can achieve this if

Tcrit =HI -

for diagonal elements, and for off-diagonal elements

— < min I milT j b=7i

where pJk is either pfk+ or pjk~. This gives us

6

(3.11)

(3.12)

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THEOREM 3. When <x> > 0, it is possible to construct discrete ordinate approximationsfor cell matrix operators and source vectors for single layers of maximum thickness andaspect ratio given by (3.11) and (3.12).

This gives us limited comfort. We cannot hope to get non-negative operators innon-scattering media, and even when it is possible to construct them, the values ofp permitted are very small. Nevertheless there exists a range of problems which canbe attempted successfully so that the method is not without its uses.

To complete this Section, we note that the algorithms (3.7), (3.8) and (3.10) do notalways provide the most economical method of solving radiative transfer problemswith the aid of equations (2.15). The different possibilities are set out in Grant & Hunt(1969a). However, one place where the algorithms of this section are important is theconstruction of cell matrices and source vectors which do not satisfy the limitations ofthe inequalities (3.11) and (3.12). In particular, if we have a shell which is subdividedinto thinner shells each having the same normal optical thickness and nearly the samecurvature, the very fast doubling algorithm (Grant & Hunt, 1969a) may be invokedwith little loss of accuracy.

4. Flux ConservationA convenient test of the efficacy of our methods is to study the case of a spherical

shell with to = 1. On this case, the physical system must neither create nor destroyenergy, and we wish to show that the same thing is true of the finite difference approxi-mation to within acceptable precision. We already know that we can find values ofT and p satisfying inequalities (3.11) and (3.12) in such cases, and it remains to demon-strate that the matrices S(n, n+l) are conservative in the sense of (3.6), when T issufficiently small.

Examination of the formulae given in the Appendix shows that for small xn+i,

t(n+1, n) = I-rB+

+VB+i+0(Tn+i)

r(n +1 , B) = rB-+Vn+i+0(rn+i)where, in the notation of (A.I),

r-++* = M - ' Q ; ^ . l • }

Similar formulae hold for t(w, n+ l),r(n, n+1) with + and — superscripts interchanged.The proof that equations (4.1) lead to a conservative cell matrix depends on the identity

£ / ; 7 = 0 (4.3)

which is an immediate consequence of (2.11) and (2.12). Moreover, the normalizationof the phase function p{r, \i, n') equation (2.1), leads to the restriction

| i = 1- (4-4)

Thus a typical sum, Sk, in equation (3.5), can be written

t ,n)+r(n + l,n)]D-% (4.5)

= l ,2 J.

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In practice, one finds that Sk = 1 to within the rounding error of the computer,provided care has been taken to ensure that (4.3) and (4.4) are satisfied to the sameaccuracy. Our algorithm is therefore conservative in a practical sense, although it israther hard to to give a rigorous proof.

5. Numerical Tests of the MethodIn many problems of interest the geometrical situation can be represented by Fig. 1.

Here the region A < r < B is the domain of interest, either the extended atmosphereof a giant star, or the atmosphere of a planet. In this situation, we usually wish toknow the radiation field at internal levels as well as the amount emerging from the

WU-

FIG. 1. Description of the diffuse radiation field in spherically symmetric atmospheres.

atmosphere, and for this purpose, the most efficient algorithm is based on the observa-tion that equations (2.15) can be written as a linear system with a block tridiagonalmatrix of rather simple form. The algorithm, described by Grant & Hunt (1968),assumes that the domain of interest is divided into N shells as shown in Fig. 1. Onefirst computes, sequentially for n = 1, 2 , . . . , N, the matrices r(l, ri) and vectorsV*+t, V*+i from

(«, n+1) (5.1)

with the initial conditions r(l, 1) = 0, V^ = U+(B). This requires calculation of theauxiliary quantities

t(n+1, ri) = t(n+1, n)[I-r(l, «)r(n+1, w)]-1

t(n+1, n) = r(n+1, n)[I-r(l, «)r(«+1, n)]-1

)r(l,n) (5.2)

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which do not need to be preserved from one step to the next. Next one computessequentially for n = N, N-1, N-2,..., 2, 1

u,-=i(» , B + i )u . - t ;^} <">with the initial condition UN+I = U~(A), say, and where

t(»,w + l) = Tn+it(«,n+l). (5.4)If, instead of prescribing the flux incident at r = A, one has a reflecting surface withan operator rG,

U N + I = T G U + + 1 (5.5)

then equations (5.3) need modifying for n = N. We find easily thatU^+1 = [ I - r ( l , N + l f r e ] - 1 ^ (5.6)

from which UJV+I can be obtained using (5.5) and thence all remaining fluxes can becalculated using (5.3).

In both sets of calculations reported here we have used a discretization with constantAr = (B—A)/N, and have ignored source terms, setting Z,?+i = £,T+i = 0 for allvalues of n. We have not taken too much care to optimize the choice of aspect ratiop = Ar/r, although this has a minor effect on the truncation error. We have simplychosen pn+i = Arn+i/rn (note that the discretization shown in Fig. 1 implies rn > rn+1)and have in all cases used N = 100. The atmosphere has been chosen to be homo-geneous for simplicity and the radial optical thickness is denoted by x.

TABLE 2

Global energy conservation for a conservative isotropically scattering shell illuminated iso-tropically on the inner boundary, r = A. The columns give the total flux emerging from the shellat radii A and B, and verify the equation F+(A)+F~(B) — F~(A) = %. Calculations are based

on 8 point Gauss-Legendre quadrature

x = a(B-A):

B/A

101-31-51-720

F~(B)l2n

0195030-246170-273250-296110-32439

2

F+(A)/2n

0-304970-253830-226750-203890-17561

F-(B)/2n

010383013354015177016881019227

5

F+(A)/2n

0-396170-366460-348230-331190-30773

10

F-(B)/2n

005387007550008650009716011254

F+(A)l2n

0-441630-424500-413500-402840-38746

The first series of calculations prescribe Ut+ = 0 and assume an isotropic flux,

F~(A) = n in the outward direction at the inner boundary. Table 2 analyses the globalenergy balance. Because we have a conservative isotropic scattering domain, and noflux is incident on the outer boundary, we expect to find

F~(A) = F~(B)+F+(A) (5.7)where F~(B) is the total energy emerging at r = B, and F+(A) is the total energybackscattered into the inner region. Equation (5.7) is satisfied to within rounding

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errors (1 part in 109) provided care is taken to ensure that (4.4) is satisfied to thesame precision. This is readily achieved in the case of isotropic scattering by presentingthe Gauss-Legendre abscissae fij and weights Cs to a sufficient number of significantdigits, though more care has to be taken with other phase functions (Grant & Hunt,19696).

Apart from verifying that the method gives global flux conservation, Table 2 showsa feature we might expect on physical grounds, namely an increase in the relative fluxemerging from the outer surface as B/A increases.

FIG. 2. Angular distribution of the run of specific intensities from bottom (n = 100) to the top(n = 1) of the atmosphere in plane parallel case, T = 10, B/A = 10, N = 100, <o = 1, pjk = 1 forall / and k.

Figures 2 and 3 show the angular distributions at different shell boundaries for thecase of a plane parallel slab, B/A -> 1, and an atmosphere with B/A = 1 -5. A commondifficulty in calculations of this sort arises from the rapid change in the angulardistribution near the outer boundary, which can give rise to unphysical oscillationsin both radial and angular distributions in the first few shells (Grant, 1968). There isno sign of such behaviour in either graph, though it appears immediately inequality(3.11) is violated. Figure 4 shows the emergent angular distribution for different valuesof B/A, exhibiting the well-known tendency for the distribution to approach that ofa parallel beam at large distances from the centre of the system, although the ratioB/A is too small in the test calculations to show this markedly. Figures 5 and 6 displaythe emergent flux as functions of T and B/A respectively.

The second set of problems we have used to test our model are those in which we

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001 b

0005-I

FIG. 3. Angular distribution of the specific intensity from the inside (n = 100) to the outside (n = 1)of a spherical atmosphere, T = 10, B\A = 1-5, N = 100, a> = 10, isotropic scattering.

5r o-3^

o i h

0 0

FIG. 4. Angular distribution of emergent intensity, T = 5, <o = 10, isotropic scattering. (1) B\A20; (2)B\A = 1-5; (3)B\A = 1-3; (4)B/^ = 10.

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0-4

FIG. 5. Emergent fluxes'as function of optical depth, w = 1-0, isotropic scattering. (1) BIA = 2-0;= l-5;(3) BIA = 10.

0 3 5

0-30 -

0-20

EUJ

010-

IO 2 0

FIG. 6. Emergent fluxes as function of BjA. <o = 10, isotropic scattering.

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have illuminated the same system isotropically at the outer boundary and assumedspecular reflection at the inner boundary (rG = I in (5.5)). In such a case the positiveand negatively directed fluxes at every interface should be equal, and in the caseof an isotropic slab should have the same value, n, at all levels. Table 3 shows whathappens in a spherical shell with B/A = 1-5. There is again equality of inward andoutward net fluxes at each shell boundary, although the values rise towards the innerradius r = A. This seems plausible on physical grounds, but it is not easy to find asimple analytical model to check the behaviour.

TABLE 3

Net fluxes, F+ = F~, at shell interfaces, x = 2, BjA = 1-5

n

100502010987654321

F+/27T = F~/2n

1-11460-71390-56980-53140-52770-52410-52060-51710-51360-51010-50670-50330-5000

6. Conclusions

The method described in this paper enables the accurate calculation of radiativetransfer in spherically symmetric shells subject to the limitations imposed by theinequalities (3.11) and (3.12). It is therefore limited in practice to situations in whichthe albedo for single scattering is not too small, and in which the phase function isnot too highly peaked about the forward direction. This covers many of the practicalcases, but it is clear that a radically new departure will be needed if one wants toperform accurate calculations near a centre of spherical symmetry or in atmospherewith a low single scattering albedo.

It seems that the conditions under which the method may be used are applicablein the envelopes of some supergiant stars, and we have made use of it to study a simpleline formation problem (Peraiah & Grant, 1971,1972).

One of us (A. P.) wishes to thank Dr M. S. Vardya for reading the manuscript.

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AppendixFormulae for the Matrices and Vectors Occurring in (2.15)

Define first

and then

and

Write

and

Then

and

Qn+4 = M. + i

QnTi = i<Bn

+ - _QnV* =

S + + =M-K+*(I -Q.V*)

- = CM+K+itf-Q-V*)]"1 J '

A + =

A

r+- = A+S+-, r~+ = A-S-+

t+ = | I -

t-[A-S2t-r-+A+M

Tn+i(l-coB+i)t+|A+B++r+-A-Bi

Tn+i(l-co.+i)t-|A-B-+r-+A+B+|.

REFERENCES

(A.I)

(A.2)

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