Notes on probabilistic radiative transfer

NOTE NOTES ON PROBABILISTIC RADIATIVE TRANSFER GERARDD. FINN Institutefor Astronomy, University of Hawaii, 2680Woadlawn Drive, Honolulu, HI 96822,U.S.A. (Received 3 Dcccm6er 1975) AbdraeGSome further applications are made of Friacha’ theorem on the surface values of functions satisfying linear integral equations of transfer. The meaning of the surface values is discussed in terms of random walks. I. APPLICATION

OF FBISCHS’ THEOREM

THE INTEGRAL equation for the transfer of radiation in a spectral line through a plane-parallel, semi-infinite atmosphere has the form

S(T)= M(T) + (1 - A) l- S(f)K(lt - 71)dt,

(1)

where B(T) is the Planck function, 7 is the optical depth into the atmosphere at the center of the line, A the constant thermalization probability, and K(lt 1)is a kernel whose precise form depends on the nature of the scattering process but is assumed normalized to unity over the entire real axis. The kernel can be identified with the probability that a photon emitted at depth t travels directly to depth T where it is absorbed. FRWH and FRISCH”have proved that, for equations of the form (l), 1/2(s*@)

For the case of constant

- s*(o)) = A L- s(T)?

B(T),

dr + l/2(1 - A)S*(=‘).

(2)

they immediately deduced the well-known relation S(0) = j/AB

04

since it is known that S(a) = B.

(W

It is interesting to apply the theorem to some other integral equations occurring in radiative transfer theory. (a) FINN(*) formulated the equation p(A,T;

ve)=

1/2~“.~2(T~,)+(l-A)~-p(A,

t; v.)K(lt -Tj)dt,

which is appropriate to the case of complete redistribution in frequency during the scattering process, for the distribution over frequency of the probability that a photon emitted at depth T eventually excapes from the atmosphere with frequency u, in a spectral line. Here, I& is the profile of the absorption and emission coefficients. Since p(A, CQ; v.) is zero, application of the theorem yields (5) 615

616

D. FINN

GERARD

The expression on the right side of this equation is the compound probability distribution function that a downward directed photon released from the surface has frequency ve, and excapes after one or more scatterings with frequency v,. Apart from the factor (1 - A), this equals the square of the probability distribution for the eventual escape, with frequency v,, of photons released at the surface. (b) FINN”’formulated the integral equation

I

w

o p.(~W(lf--l)dt

pe(~)=A(~)+(l-A)

(6)

for the probability P=(T) that a photon of arbitrary frequency and direction released at depth T eventually escapes, where A(T) = IrnK(t)dt.

(7)

T

Application of Frischs’ theorem to eqn (6) gives

Here, the quantity on the right can clearly be identified as the probability pD that a photon initially directed down into the atmosphere from the surface eventually escapes. Apart from the factor (I - A), it equaIs the square of the total probability of eventual escape of photons emitted from the surface. The certainty of the escape of upward directed photons from the surface gives rise to the probability relation P,(O) = (1/2)PD+ l/Z so that eqn (8) becomes (1-A)p,2(0)=2P=(0)-1

Pa)

p.(O) = l/(1 + VA).

(9b)

with solution

In view of the relation B - S(t)

Pe(f) =

(1 -

A)B

[cf. FINNand JEFFERIES”‘], we can immediately derive eqn (3a) once again. 2. MEAN NUMBEROF SCATTERINGS In an earlier paper, FINN(~) introduced ii.(t), the mean number of scatterings (counting the first emission as a scattering) undergone by photons emitted at depth t and destined to escape from the atmosphere. It is convenient to define here A(t), the mean number of scatterings undergone by UNphotons emitted at depth t. By consideration of eqns (lo)-(14) of that paper, it is easy to show that A(t) satisfies the integral equation g(T)=l+(l-A)

_ fi(t)K(lt

- T[)dt.

(10)

This expression may also be derived by heuristic arguments following a photon through its first absorption and subsequent re-emission as [e.g., FINN”‘].For homogeneous, isothermal atmos-

Notes on probabilisticradiativetransfer

617

pheres we can see, by comparing this oquation with eqn (l), that S(T) = A&i(r).

(11)

This equation has a very simple physical meaning: the rate of omissions (including both created and scattered photons) from an element of thickness dr is given by S(T) d7 and is equal to the rate of creation of photons in the element, AB dr, times the mean number of scatterings these photons undergo before either escaping or being thermal&l. Since eqns (1) and (10) may be trivially modified to cover cases of finite atmospheres, eqn (11) is also valid in these cases. An approximate empirical relation of the same form as eqn (11) was found by ATHAY and SKUMANI&’ and FINN”’using different definitions for the “mean number of scatterings”. If we combine eqns (3) and (1l), we find R(m) = l/A,

(12a)

A(0) = l/VA.

(12b)

3. THE d/A RESULT

Instead of discussing eqn (3a) for S(O), we study the equivalent results for p.(O) and ii(O) contained in eqns (9b) and (12b) and the simple relationship between probabilities embodied in eqns (5) and (8). As a first step in the understanding of these results, we consider a simple, one-dimensional random-walk model of the transfer problem in which the tlight of a photon between omission and subsequent absorption is modelled by the photon taking unit step in either direction with equal probability.? If we include the probability A for destruction of the photon at each step, we may write for the probability px that a photon released at depth x (relative to the surface at x = 1) eventually reaches x = 0 (when it is considered to have escaped) as p x

=(1-A) -+Px--L-Px+l) po= 1.

(13)

These equations have the solution p =

WW

x

In particular, (1 -

Alp,* = (1 - Alpz.

(14b)

The right side of this equation equals the probability that a model photon released downward from the surface at x = 1, is re-emitted (at x = 2) and eventually escapes. Apart from a factor (1 - A), this probability is equal to the square of the probability of escape of all model photons released at the surface. This relationship is completely analogous to that in eqns (5) and (8) for actual photon transfer. The probability p, of eventual excape of a model photon released at the surface is found from eqn (14a) to be 1 P”1+~A~2 for small values of A. Apart from a factor of d2, oqn (15) agrees with eqn (9b).S tA constantstep modelof the threedimensionsbecomesan increasinglymoreaccuraterepresentativeof photontransfer as the conditionsof the scatteringprocessmorenearlyresembk coherentscattering.In the surface.layersof the atmosphere, the modelshouldbe approximatelycorrecteven for completeredistributionof frequenciessince most photonsremainin the core of the spectralline. SltwassuggestedbyJ.T.Jefferiesthattheappearanceofthe~2maybeduetothereplacingoftransferinthreedimen~~sby a one-dimensionalrandomwalk model.

GERARD

618

D.

FINN

By an analogous argument, we have for ii,, the expected number of scatterings undergone by a photon about to be released at depth x before escape or destruction, ii,

=(l-h)+q+fix_,-ex+,)

(16)

subject to the boundary condition go = 0. Since eqn (16) has the solution

fix+

1_ [

l-d/nV/(2--A) l-h

(

X >1

we have for fi,, the mean number of scatterings undergone by particles released at the surface, nl -

2

J h

for small values of A, in essential agreement with eqn (12b).

REFERENCES 1. U. FRISCH and H. FRISCH,Mon. Not. R. Astr. Sot. 173, 167 (1975). 2. G. D. FINN, Jt.@tT 12, 35 (1972). 3. G. D. FINN, JQSRT 13,683 (1973). 4. G. D. FINN and J. T. JEFFERIE& JQSRT 8, 1675(1968). 5. G. D. FD(N,JQSRT 22, 149 (1972). 6. GRANT R. ATHAY and A. SKUMANICH,Astrophys. J. 170,605 (1971).

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