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Probability distributions for photon exit
1. Quanr.
Specrrosc.
Radio,.
Transfer.
PROBABILITY
Vol.
12, pp. 35-58. Pergsmon Press 1972. Printed in Great Britain
DISTRIBUTIONS G.
FOR PHOTON
EXIT
D. FINN
Institute for Astronomy, University of Hawaii, Honolulu, Hawaii 96822, U.S.A. (Received 1 April 1971)
Ab&act-A linear integral equation is formulated for the probability that a photon in a spectral line formed at a specified depth in a scattering atmosphere eventually escapes at a particular frequency in the spectral line. Numerical solutions are obtained for an isothermal atmosphere and their physical meaning discussed under a variety of conditions. 1. INTRODUCTION
WE CONSIDERthe transfer of radiation in a steady-state situation through a plane, semiinfinite, isothermal, and homogeneous atmosphere containing two-level atoms. It is assumed that photons are released throughout the atmosphere with frequency distribution 4,. They transfer through the atmosphere while undergoing absorptions and subsequent emissions. The scattering process is assumed to be isotropic and to produce a complete redistribution of frequency with frequency profile 4,. Eventually, the photons are either thermalized, in which case their energy becomes part of the thermal field, or they escape from the atmosphere. In a previous paper (FINN(‘)), which we will refer to as paper I, we obtained numerical solutions for the probability that a photon formed at a specified depth in an atmosphere, escapes with a particular frequency in the spectral line and in a particular direction. The behavior of this function was discussed in rather general terms. In the present paper we consider probability distribution functions which have been integrated to include all possible directions of escape and attempt to study their behavior in detail. Linear integral expressions for a number of distribution functions are formulated in Section 2. In Section 3 we discuss briefly the numerical methods used for determining these distribution functions. In Section 4 we treat a simple random-walk problem which has a constant step size but which in some respects is similar to the radiative transfer problem. The results are presented and discussed in Sections 5 and 6 for cases where 4, has both Doppler and Voigt forms respectively. 2. FORMULATION We consider the simple case of the transfer of radiation through a plane, semi-infinite, isothermal, homogeneous atmosphere containing model atoms having two energy levels between which electron transitions take place accompanying the absorption and emission of radiation. It is supposed that photon scattering is isotropic, that there is complete redistribution of frequency in the scattering process, and that the frequency profiles 4, of both 35
G. D. FINN
36
the absorption and emission coefficients are identical. For convenience stimulated emission is neglected. The discussion is restricted to problems in which all stochastic variables are invariant with time so that the optical depth scale is fixed. In addition to the scattering process, it is assumed that a photon has a constant probability L of being thermalized after absorption. We suppose that photons in the spectral line are formed throughout the atmosphere with frequency distribution C#J,(normalized to unity) isotropically in direction but not necessarily homogeneously with depth. We measure optical depth on a scale t such that the monochromatic optical depth t,, is given by t, = &t. The cosine of the angle between the direction traveled by a photon and the upward normal to the atmosphere is denoted by cc; a photon is then described by its frequency (v) and direction (r_l).Suppose a (v, p) photon is first emitted in a layer of optical thickness dt, at depth t in the atmosphere. We define p@, t, v, p; v,, pe) to be the probability distribution function for this photon to ultimately emerge from the atmosphere as a (v,, p,) photon. It may easily be shown from first principles--+f. paper I-that this function satisfies the integral equation p(l, t, v, p; v,, p,) = e-‘4~W5(v-
v,)b(p-pC1,) OD
+1
(1) where, for purposes of integration 6 may be considered to be the Dirac b-function. The first term in equation (1) represents a discrete probability pulse for the particular case when (v,, p=)photons are formed at depth t and emerge directly, while the second term represents contributions from the indirect escape of photons. As explained in paper I, the integration over t’ has been written without explicit limits on the understanding that if p is negative the original photon has been emitted downwards into the atmosphere and t’ ranges from t to infinity; if p is positive, on the other hand, the original photon was emitted upwards and t’ ranges from zero to t. In this paper we have adopted the convention of placing the arguments of functions in two groups. The first group inside the parentheses contains variables whose values are specified ; separated from them by a semicolon are the random variables of the probability distribution function. It is convenient to define the distribution function p(& t ; v,) for the probability that a photon which is originally formed at depth t and emitted with arbitrary frequency in the spectral line ultimately emerges from the atmosphere with frequency v,. This function can be calculated in terms of p(il, t, v, p ; v,, ,u,) by multiplying by 4, dv dp/2 dp, and integrating to include all possible frequencies and directions of the original and all possible directions of the emerging photon. Thus we have
p(l,t,v,)
=
j,.jm., 0
j’$p(U,v,ii:v..pi. 0
-1
Probability distributions for photon exit
37
If the same operations are performed on both sides of equation (1) we find the integral equation for p(A,t ; v,) m
$44 t; v,) = %#Q,(t4J+(l-A)
j- ~41,t’;
v,)K(lt-
t’l) dt’
(3)
0
where
03 WY) = 9
s
#&%(~dd dv
(4)
0 is a kernel which arises frequently in the formulation of radiative transfer problems in terms of linear integral equations (cf. FINN and JEFFERIE.@) and E, and E, are the first and second exponential integral functions. Equation (3) may be interpreted in terms of simple physical ideas since the first term on the right represents the probability that a v, photon is formed originally at depth t and escapes directly. Other contributions to p(l, L; v,) coming from the indirect escape of photons are represented by the second term. This term contains the factor K((t- t’l)dt’ which represents the probability that a photon emitted at depth t makes a direct flight to depth t’ and is absorbed there in a layer of thickness dt’-cf. FINNand JEFFERIES.(~) This function is then multiplied by (1 - A), the probability that the photon is scattered, and by ~(1, t’ ; v,) the distribution function for the probability that this photon ultimately emerges at frequency v,. Integration over all depths t’ then includes all possible indirect contributions to p(A t ; VA. In the special case Iz = 0, the atmosphere becomes purely scattering so that there can be no thermalization processes. In this case all photons must eventually escape from the atmosphere, so that m
~(0, t; v,) dv, = 1.
(5)
0
It may be easily shown that this equation is consistent with equation (3). It is useful to consider the distribution functions p(A, t;
P@,t; In+l,v,)
= $#QAt&.)+(l-A)
s
p(l,t’;
(6)
0
where
P(k t ; I 1,v,) = P(J,t ; 1,v3 = 34Q#PvJ.
(7)
The distribution function p(l, t ; n, v,) may be easily obtained by subtraction. Thus, we have p(L,t;n,v,)
= p@,t;
In-l,v&?).
(8)
38
G.
D. FINN
3. NUMERICAL
PROCEDURES
Equation (3) is a Fredholm integral equation of the second kind in the unknown function p(l, t ; v,). With the kernel of equation (4) it is-precisely the integral equation for the source function appropriate to the radiative transfer problem in an atmosphere containing two-level model atoms for which the source term is distributed with depth as ~~YJZZ(t~Ye). Equation (3) may therefore be solved numerically for p(;l, t ; v,) by one of the methods developed for handling problems of this kind (cf. e.g. HUMMERand RYBICKI’~‘). We use the step (2) Also using a step function representafunction method adopted by FINN and JEFFBRIES. tion, the recurrence relation (6) may be solved in conjunction with equation (7) by simple iteration for successive functions in the sequence p@, t ;
4. A SIMPLE RANDOM WALK PROBLEM In this section we discuss a simple random walk problem whose solution is useful in interpreting some aspects of the more complex distribution functions associated with the photon transfer problem. We consider a one-dimensional random walk in which a particle moves along the x axis. We suppose that the particle starts at a positive value of x and takes steps of unit length with equal probability in either direction. We further suppose that the random walk may terminate at any stage resulting in two possible outcomes. We will designate these ‘success’ and ‘destruction,’ and assign to them probabilities p and 1 respectively. The walk continues until the particle is successful, is destroyed, or travels beyond x = 0 where it is absorbed at a perfectly absorbing barrier at x = - 1 when it is deemed to have ‘failed.’ The probability pX that a particle located at depth x eventually succeeds, satisfies the recurrence relation px =
p+(l-A-p)(p .X+1+p X_1) 2
(x 2 0).
The first term on the right represents the probability that the particle succeeds directly, while the second term represents the probability that it neither succeeds nor is destroyed at this stage but moves to the right or left and eventually succeeds. Equation (9) may be solved by standard analytic methods to give the solution P
lpx = A+p i
l-l-p 1 +J@+&/(24--_I
h/@+P)JP-~-P) (
l-l-p
x
11
cx > 0)
(IO)
If the probabilities I and p are small compared with unity, we find the probability of ultimate success pXis given approximately by
We consider, in particular, the following cases :
Probability
distributions
for photon exit
39
(i) If p <
Under these conditions the probability that the particle eventually succeeds is A- ’ times the probability that it succeeds directly.* On the other hand, if the particle starts its walk atx = Owefind
po=J2-$.
(13)
(ii) When 1 = 0, the particle cannot be destroyed. In this case, if the random walk starts at x = 0, equation (11) becomes PO N
Jw.
(14)
The problems of the transfer of a photon through a homogeneous scattering atmosphere and of the simple random walk just discussed may be related as follows : (a) The constant probability of ‘success’ in the simple problem, has as its counterpart in the photon transfer problem, the probability &/2Ez(t&) of a photon being scattered with frequency v, and escaping directly from the atmosphere. For the two problems to have similar solutions this last probability must also be constant. This condition amounts to the requirement t << 4;‘) in which case p becomes identified with &e/2. (b) The term ‘destruction’ in the simple problem is identified with the term ‘thermalization’ in the transfer problem. (c) The term ‘failure’ in the simple problem corresponds to the emergence of a photon from the atmosphere at a frequency in the spectral line other than v,.
5. RESULTS-DOPPLER
PROFILE
4.
In this section we attempt to interpret the behavior of the function p(A, t ; v,) in terms of simple physical ideas and basic probability models. Values of the auxiliary functions p(l, t; sn, v,) and ~(1, t; n, v,) were calculated as an aid in this interpretation. Clearly, on general grounds we should expect the distribution functions p(0, t ; in, v,) and p(l, t ; v,) to be closely related when n equals A-I. In this case any discussion of the behavior of one of these functions applies equally to the other. Specific comparisons between the two functions will be noted in footnotes as they arise. In this application we supposed that the profiles of the absorption and emission coefficients have the Doppler form (15) where the frequency variable v should be interpreted as the magnitude of the frequency departure from the center of the spectral line in units of the Doppler width. * A similar result was obtained in the case of photon transfer by FINN and J~PWRIES.‘~ They found that for a photon located so deep in an atmosphere that it is unlikely to escape at all, the probability of eventual escape was A- 1 times the probability of direct escape.
G. D. FINN
40
1. Purely scattering atmosphere,
rZ= 0
Equation (3) was solved for p(r2,t; v,) in the case of a purely scattering atmosphere, 1 = 0; the results are shown in Fig. 1 for selected values of the frequency of the emergent photon v,. We have marked on the figure the three regions A, B, and C together with
10-7
10-Z
I
I
1
IO2
1
104
I
106
1
108
I
10'0
I
IO"
FIG. 1. The function p(L, t ; v,) in the case of a purely scattering atmosphere for the values of v, indicated+, has Doppler form. The vertical lines mark depths numerically equal to 4;‘. Regions A, B, and C are discussed in the text.
depths numerically equal to Cp,‘. A comparison of the curves of Fig. 1 shows a number of features which we now discuss. (i) In the region A it was found empirically that P(0, t; v,) = 1.&h..
(16)
Since0 N t c f&’ in this region, we may expect the simple random walk problem discussed in Section 4 to have a similar type of solution. It can be seen by comparing equation (16) with equation (14) that-apart from a small constant factor-the two forms agree if p in equation (14) is replaced by &/2. (ii) It was found in the region t >> r$V,‘, that ~(0, t ; v,) varies with depth in proportion to t-l/’ * In addition, if 4,. << 1, it was found that p(0, t ; v,) is distributed with frequency in propbrtion to 4;‘12. We may therefore represent the variation of ~$0, t; v,) in region B of Fig. 1 by the relation 1 P&At ; ve) cc (&-p’ (17) * The same variation with depth under these conditions distribution function.
was noted, but not explained, in paper I for a similar
Probability distributions for photon exit
41
Before attempting to understand physically the depth dependence of p(O, t; v.) in this region, it is necessary to make a digression to investigate the behavior of the associated distribution functions p(0, t ; s n, v,) and p(O, t ; n, v&. Equations (6) and (7) were solved iteratively for values of the successive ‘partial’ distribution function p(0, t ; sn, v,) for n = 1,2,. . . . The results are shown in Fig. 2 for selected I
10-l
10-Z
10-3
10-q
3 c‘ “I 2
10-S
G n
10-G
IO-’
10-e
10-g
lo-‘0 10-z
IO“
.I
IO
IO2
I03
I04
I05
FIG. 2. The function p(A, t ;
values of n in the particular case v, = 0 ; for comparison the corresponding ‘full’ probability function p(0, t ; ve) is shown by a broken line. It can easily be shown from this figure that p(0, t; an, v,) is distributed in proportion to n3/‘te2, provided t 2 n. In addition, comparison of these curves with others for different values of v, reveals that the functions do, t ; 5 n, v,) and p(0, t ; v,) have identical frequency variation. When 4, has Doppler form it is well known that K(t) varies as t -2 for large values of t. Thus, provided 4,. c 1 and 0 <
K(t)n3” _
(18)
Values of p(0, t; n, v,) were then calculated from equation (8) and are shown as functions of n in Fig. 3 for selected values of the depth of formation t, in the particular case v, = 0. It can be seen from this figure (or from the empirical form (18)) that, provided
G. D. FINN
42
0 CCn 5 t, the function ~(0, t; n, v,) varies in proportion to K(t)n1/24v;1’2. For n 2 t, however, the probability of photons escaping after precisely n scatterings is seen to vary in proportion to ne2. IO'
,6 _
I
I
IO
IOC
IOJ
10.
”
FIG. 3. The function p(l, t ; n, v,) as a function of n in the case of a purely scattering atmosphere for v, = 0 and for sejected depths-4, has Doppler form. In the case t = lo4 the probabilities have been multiplied by 100 for convenience.
Equations (6) and (7) were solved for the function p(0, t; an, v,) in cases where c#J,, << 1. We adopted the values v, = 3.5 and v, = 4. Values of ~(0, t ; n, v,) were then obtained from equation (8) and are shown in Fig. 4 for selected values of the depth t. It can be seen that, for depth t x 4,’ and for values of n 5 Q&;‘,the function ~$0,t ; n, v,) is independent of frequency but varies in direct proportion to n and in inverse proportion to t2. It is clear that, under these conditions, p(0, t ; n, v,) varies in proportion to K(t)n and p(0, t ; s n, v,) varies in proportion to K(t)n’. These results may be summarized by the empirical formulae
I1
n2
where 4;’
<< t.
* Continuity of the last two parts of equation must be K(t)fs~2~~‘~2n-2.
n 2 t.
(19) shows that the precise variation
(c) (19)*
of p(0, t ; n, v,) for n 2 t,
Probability distributions for photon exit
43
10-2
I
1
10-2
102
IO ”
FIG.4. The function p(l, f ; n, vJ as a function of IIin the case of a purely scattering atmosphere for the depths indicated-$, has Doppler form. The curves represent the common solutions for the two cases v, = 3.5 and v, = 4.
We are now able to interpret the variation of p(0, t ; v,) with depth in region B of Fig. 1. It can be seen from the general form of equation (19) that a photon formed at depth t in region B i,slikely..~to undergo roughly t scatterings before escaping from the atmosphere with frequency v,. Suppose it remains in the vicinity of t during its first i scatterings. With a probability approximately equal to K(t) dz it then takes a long direct flight upwards to some depth r, also assumed to be in region B, and is absorbed there in a layer of thickness dz. The probability that the photon escapes at frequency v, after fewer than a total of n scatterings is then found by multiplying K(t) dr by ~(0, r;
P@,t;
I&v,)
=
x(t)Igi
p(O,z; in-i-l,v,)dr.
f
(20)
0
For values of T in the range 0 I z 5 n- i- 1 the probability function ~(0,T ; sn- i- 1, v,) has virtually reached its full value p(0, 7; v,). We may therefore substitute equation (17)
G. D. FINN
44
into the right side of equation (20) to give n-i
”
~(0, t ;
sn, v,) a K(t)
(21)
for 0 <
(22)
Equations (21) and (22) agree with the corresponding empirical forms (18) and (19b) respectively indicating that our interpretation of the transfer mechanism is valid. If we let n vary over its range of probable values (0, t) equation (22) may be integrated to give
P(0, t ; v,) K
’ K(t)n”*
I o
dn
(23)
Jh
since K(t) varies in proportion to t-* when t is large. This variation is in agreement with equation (17), the form found empirically to be appropriate to region B of Fig. 1. Evidently a photon which is formed at depth t such that t s 4V,‘, needs about t scatterings to escape with frequency v, . During its passage to the surface it takes a number of long flights. Roughly speaking, each of these flights terminates closer to the surface than an optical distance numerically equal to the number of scatterings still outstanding from the original t. Eventually the photon leaves region B when a long flight terminates closer to the surface than optical distance 4;’ and enters region C. (iii) It was found empirically in region C of Fig. 1 that
pt0,t ; v,) a (4,.t)“*.
(24)
Before attempting to understand physically the frequency dependence of ~(0, t ; v,) in this region it is necessary, once again, to digress. In this case we investigate the behavior of the functions p(0, t; in, ve) and p(0, t; n, v,) for t in the range 0 << t <<4,‘. Equations (6) and (7) were solved for values of the function ~(0, t ; I n, ve) in the case v, = 5. The results are shown in Fig. 5 for selected values of n. From these results, values of the function ~(0, c ; n, v,) were obtained ; they are shown as functions of n in Fig. 6 for selected values oft. It was found for each curve in this figure that the value of p(0, t ; n, vJ is numerically equal to &J2 (in this case 7.84 x lo- ‘*) provided 0 < n 6 t. For n 2 t, on the other hand, ~(0, t; n, v,) varies in proportion to n - ‘I* . A comparison of the values of p(0, t ; n, v,) for different values oft shows that the function is also proportional to ,/t. Values of the function p(0, t ; n, v,) were also calculated for the particular case v, = ,/3 and t = 1, and are shown as a function of n in Fig. 7. It can be seen from this figure that for n >>$V,‘, the distribution function varies approximately in proportion to nm2.
Probability distributions for photon exit
10-s-
I
I
I
I
I
I
I
VI ;
d
x
10-Q
-
IO“’ -
FIG. 5. The function p(A, t ; III, VJ in the case of a purely scattering atmosphere for v, = 5 and for the values of n indicated-#, has Doppler form.
2_
I
I
IO
\
I02
n
FIG. 6. The function p(l, t ; n, v,) as a function of n in the case of a purely scattering atmosphere for Y, = 5 and for the depths indicated-+, has Doppler form.
These results may be summarized by the empirical formulae (= &I2
(0 < n 5 t)
(a)
04 (26)
I
Cc-
provided t <<4,‘.
1
n2
(n k #C:‘)*
(4
l Continuity with the empirical form valid for smaller values of n would require that p(0, t; n, ve)varies in proportion to t14j~~‘~2n-2 for n 2 f#~;‘.
G. D. FINN
46
We are now able to discuss the mechanism of photon transfer in region C of Fig. 1. It can be seen from equation (26a) that a photon formed at depth t has probability &/2 of escaping at frequency v,, provided n 5 t 5 4,‘. This is to be expected since the photon is unlikely to reach the surface layers during PIscatterings, but if scattered in an upward direction with frequency v,, it is virtually certain to escape directly.
IO’
FIG. 7. The function &I, t ; n, YJ as a function of n in the case of a purely scattering atmosphere for v, = ,,/3 and for t = l-4, has Doppler form. The vertical line marks the depth numerically equal to 4;:.
For values of n in the range t 5 n 5 c$~;‘, the value of ~$0,t; n, v,) given by equation (26b) decreases slowly. The reason for this decrease can be traced to the increased probability of the photon reaching the surface layers after fewer than n scatterings and escaping with a frequency closer to the core of the spectral line than v,. Equation (26b) can be simply understood in analogy with the simple random walk problem described in Section 4. The inequality t 6 n 5 4V,‘, has as its counterpart in the simple problem x 6 n 6 p-l. Under these conditions it can be shown-see, for example, FELLERt4)---that a simple random walk,
starting
at position
x where
x is large,
continues
for at least n steps with a probability
,/(2/7+/,/n. The dependence of this probability on n is precisely that given by equation (26b) for the photon transfer problem. The analogy is completed on recognizing that, for a simple random walk, a point is probably displaced a distance x after roughly x2 steps, whereas a typical photon changes its optical depth by roughly t after t scatterings. Hence, the x dependence of the probability for the simple problem is replaced by Jt in the photon problem. If n is further increased the value of ~(0, t ; n, v,) decreases more rapidly. This is caused by the increased a priori probability of escape at frequencies further in the wing of the line than ver as the number of emission absorption processes increases beyond $L’. approximately
equal
to
Probability distributions for photon exit
47
Equation (26) may be integrated to include all likely values of the number of scatterings n, to give the total probability of escape at frequency v,. We find (27) in agreement with the empirical form (24) describing region C of Fig. 1. (iv) It is clear from Fig. 1 that the function p(0, t; v,) has a maximum value at depth t N 4,’ which is essentially independent of v,. The reason for this behavior was discussed in paper I. In Section l(ii) above we discussed the variation of p(0, t; v,) with depth in the region t 2 4”;‘. On the other hand, in Section l(iii), we essentially discussed thefrequency variation of the function in the region t 5 4,‘. Clearly, the appropriate empirical forms given by equations (17) and (24) respectively are correctly matched at t N 4”;‘, provided that the other variable is present as specified in each case. 2. A weakly thermalizing atmosphere, 1 c-c 4,. Equation (3) was solved for p(l, t ; v,) in cases where 1 << &. The results are shown in Fig. 8 for selected values of v, for the case 1 = 1O-8. For comparison we have shown as a broken line a solution for the case 1 = 10e6. We have marked on this figure the regions A, B, C, and D. The first three of these lie above depth LX’, whicn represents the thermalization length of this transfer problem-cf. FINN and JEFFER~ES.(~) Thus, the function ~(1, t ; v,) shows the same behavior as the corresponding regions A, B, and C of Fig. 1 for the case of a purely scattering atmosphere. In the region D of Fig. 8, where &;’ <
(1 - A)nK(t) ‘j’ p(A, 7
; v,)
dr.
(29)
0
* Comparison of equations (18) and (28) shows that in the case n = I-’ the functions p(0, t; in, v,) and p(A,t ; v,) have the same empirical form. This is expected since n >>4,’ and 1-l >>$‘. The relation between the two functions can also be seen by comparing the curves of Fig. 8 with those of Fig. 2.
G. D. FINN
48
At this stage the photon may be considered to belong to region B where the effects of thermalization are insignificant. It then proceeds to the surface by a series of long upward flights as explained in part l(ii) of this section for the case of a purely scattering atmosphere. We therefore substitute the empirical form (17) into the right side of equation (29) to give
(30) in agreement with the empirical form (28).
FIG. 8. The function p(2, t; v,) for the values of v, indicated+, has Doppler form. The solid lines represent the solutions in the case L = 10-s and the broken line represents a solution in the case I = 10e6. Regions A, B, C and D are discussed in the text.
3. A strongly thermalizing atmosphere,
A D 4,.
Values of p(A, t ; ve) were calculated for cases where the probability of thermalization I, is large. The results are shown in Fig. 9 for selected values of v,. The solid lines represent the solutions for the case I = lo-’ and the broken line represents a solution for I = 10-l. We have marked on this figure regions E, F, , F, , G and H as well as depths numerically equal to c#J,‘. We consider the behavior of the function in each of these regions.
Probability distributions for photon exit
49
t FIG.9. The function p(A,r ; v,) for the values of v, indicated-& has Doppler form. The solid lines represent solutions in the case I = 10e2 and the broken line represents a solution in the case ,I = 10-r. Regions E, F,, F2, G and H are discussed in the text.
(i) In the region E, t N 0 and 4,. c 1. The distribution empirically to have the form p(l,t;v,)
4
= *.
function p(L, t; v,) was found
(31)
Apart from a factor of J2, this empirical form agrees with equation (13) for the simple random walk problem under analogous conditions, provided we replace p in that equation by &l2. (ii) It was found empirically from the curves of Fig. 9 that in the region Fi where I-’ << t << qb,‘, th e function p(l, t ; ve) becomes independent of depth and has the form p(l, t; v,) =
$.
(32)
This equation is in agreement with equation (12) for the simple random walk problem under analogous conditions, provided we replace p in that equation by #“e/2.* We may also interpret equation (32) in terms of Bernoulli trials where each scattering of the photon represents a trial. The expression (1 - Iz- #,,)” then represents the probability * Clearly, equation (32) agrees with the form of p(0, t; in, vc)corresponding to equation (26a) for p(0, t ; II,v,). This is expected, since t <<4;’ in each case, and A-’ and n are each smaller than t. The close relation between the two functions &I, t; v.) and p(0, I;
50
G. D. FINN
that the photon is not scattered with frequency v, and is not thermalized during the first n trials. Under the conditions specified here, the probability of a photon being scattered in an upward direction with frequency v, and escaping is 1$,~/2.We may therefore write p(l,t;v,)
‘v
(l-a-&y+ k,
f
(33)
2:
II=0
since &, example, direction 1-i << t
<
PO,t ; v,) =
~W”J).
(34)
It can be seen from Fig. 9 that this variation is in approximate agreement with the behavior of the curves in region FZ where the function is seen to decrease more or less exponentially with increasing depth. (iii) In the region G of Fig. 9, where 1-i <<4,’ << t, it can be seen that the value of ~(1, t ; v,) is independent of v, and varies with 1 and depth in proportion to (At)-‘, so that p(l, t ; v,) cc
F
(35)*
This variation can be readily explained. A photon released under these conditions is likely to escape at frequency v, only if it makes a long direct flight upwards to within an optical distance 4;’ of the surface (the larger of the values of &’ and I- ‘). We suppose the photon remains in the vicinity of depth t for the first n scatterings and then makes a long direct flight upwards to a depth r and is absorbed there in a layer of thickness dr. The probability that the photon eventually escapes with frequency v, is given by ov, p(A, t; v,) N f
(1 -A)nK(t) n=O
I
~(1, z ; v,) dz.
(36)
0
The distribution function p(l, z; v,) appropriate to the right side of equation (36) has been discussed in part 3(ii) above and was represented by equation (32). If equation (32) is substituted into equation (36) we obtain, after a little algebra, the empirical form (35). (iv) In the region H of Fig. 9 it was found that p(l, t; v,) varies with frequency and il in proportion to &J,/IZ. Equation (3) was also solved for the particular case Iz = 10d6, v, = ,/33; the results are shown in Fig. 10. In this figure the region marked H covers a large range of depths and it is clear that the function p(l, t ; vJ varies with depth in proportion to Jt. On the basis of these observations we can write (37) Provided
t << A- ’ cc c&; l, equation
(37) may be obtained
theoretically
by integrating
* This relation is not unexpected since p(O,t : I”, v,) was found to vary as K(t)d for n CC&’ l(ii) of this section, equation (19a).
<
part
Probability distributions for photon exit
51
equation (26) for ~(0, t ; n, v,) over all values of n between zero and 1-i. Since t c A- ’ << &‘, we find (38) By virtue of our earlier remarks the left side of this equation may be replaced by p(l, t ; v,). Equation (38) then reduces to equation (37).
10-j
IO“
;; ; I
9 lo-” f
‘2 II
IO-
10-I
I 2
I
I I02
I
I
I
106
I04
106
IO” D
t
FIG. 10. The function p(l, t; v,) for the case 1 = 10e6 and v, = H is discussed in the text.
6. RESULTS-VOIGT
J334,
PROFILE
has Doppler form. Region
4,
In this section we consider cases where c$, has the Voigt form
(3% with a the broadening parameter. For a equal to the value 0.001, for example, the Voigt and Doppler forms of ~5,are essentially identical out to v N 25 where 4, is roughly numerically equal to a. It would seem that photons released closer to the surface than an optical distance numerically equal to a- ’ and destined to escape at frequencies in the core of the spectral line are unlikely to be scattered with frequencies larger than about 2.5 at any stage. Under these conditions the distribution function p(l, t ; v,) can be expected to show similar behavior for both types of frequency profiles. For the same reasons the functions p(0, t ; in, v,) and p(0, n, v,) show some of the effects peculiar to the type of profile 4, used, only if the number of scatterings n, is larger than a - r. Since it is not practical to perform numerically a large number of iterations of equation (6) we show these effects indirectly. t
;
G. D. FINN
52
1. Purely scattering atmosphere,
A= 0
The function ~(1, t ; v,) was calculated for the case of a purely scattering atmosphere, I = 0, in the case where 4, has Voigt form with broadening parameter a = 0.001. The results are shown in Fig. 11 for selected values of the frequency of the emergent photon v,.
FIG. 11. The function p(l, t; v.) in the case of a purely scattering atmosphere for the values of v, indicated-+, has Voigt form with broadening parameter a = lo- 3. The vertical lines mark depths numerically equal to 4,‘. Regions C, 1 and J are discussed in the text.
We have marked on the figure the regions C, I and J together with depths numerically equal to 4,‘. These values were obtained from tables of the Voigt function prepared by HUMMER.“) We discuss these regions in turn. (i) In the region C, for which t lies in the range 0 << t 5 a- ‘, the function ~(0, t ; vJ varies approximately in proportion to ,/t. This is the same depth variation found for the corresponding region in the Doppler case-cf. Fig. 1. As we remarked earlier such a similarity is to be expected. (ii) In the region I for which 0 <<4; ’ a t, it was found empirically that the distribution function ~(0, t ; v,) has the form 1
P(O, t ; v,) = gzpc.
In order to interpret physically the depth variation of ~(0, t; ve) in this region, we need to know empirical forms for d0, t; n, v,). As remarked earlier we can only obtain these
Probability distributions for photon exit
53
indirectly. It is known by general arguments-& FINN and J~rut#)--that for 4, of Voigt form a typical photon travels roughly a vertical optical distance of n2 (more precisely 10 an2) while undergoing a large number n, of scatterings. It follows that a typical photon formed at depth t escapes from the atmosphere after roughly dt scatterings. By analogy with equation (19) for the Doppler case, it would therefore seem reasonable to assume in the present case, that ~(0, t; n, v,) has the form 0 < n 5 C&i/2
K(t)n
(a) (41)
~(0, r ; n, v,) cc K(t)n1’2 ~,l/4
4yOti2 S n 6 Jt
while decreasing rapidly for larger values of n. In obtaining equation (41) from equation (19) we have arbitrarily replaced &./” by 4:: in part (b) of the equation. The depth dependence of equation (40) can now be interpreted by arguments similar to those used for the Doppler case. We suppose that a photon is formed at depth t and remains in the vicinity of t during the first i scatterings. It then takes a long upward flight to a depth r and is absorbed there in a layer of thickness dr. The probability that the photon escapes from the atmosphere after fewer than n scatterings subsequent to its original emission, can be found by summing over all possible values of i and integrating to include all necessary values of T. We write in this case, n_ 1 W-V p(O,t;
1
p(o,T; In-i,V,)dT.
(42)
b
Equation (42) is similar to equation (20) for the Doppler case except that the upper limit of integration has been changed from n-i to (n- i)2. This has been done in the belief that a photon is likely to escape from the atmosphere during the remaining n - i scatterings allotted, only if the long flight terminates closer to the surface than an optical distance numerically equal to (n-i)‘. Following our arguments for the Doppler case we assume that the function ~(0, T ;
~(0, t ; in, v,) a K(t)
(n - i)*
(43)
The right side of this equation agrees with the functional form of AO, t ; n, v,) specified by equation (41b). Since K(t) varies in proportion to t - 3/2 for large values of t, we may write, in analogy with equation (23),
P(O,t ; v,) a
Pw
54
G. D. FINN
Equation (44) agrees with the observed empirical form (40) indicating that our arguments are valid and that equation (41b) is essentially correct. Evidently, a photon formed at depth t escapes at frequency v, such that t >> 4;’ after undergoing approximately ,/t absorption-emission processes during its passage to the surface. During this transfer the photon makes a number of long flights, each of which terminates roughly closer to the surface than an optical distance numerically equal to the square of the number of scatterings remaining from the original Jt. Eventually the photon leaves region I as it reaches depths closer to the surface than 4,‘. (iii) It was found empirically from Fig. 11 that in the region marked J for which lo3 5 t c q6y;1,the function ~(0, t; v,) varies with depth and frequency according to the form p(0, t ; Ye) cc (P;~P.
(45)
In order to understand the frequency variation of ~(0, t; v,) in this region, we must find empirical forms for d0, t; n, v~). In keeping with the arguments used in part l(ii) of this section, and in analogy with equation (26) for the Doppler case, we shall assume for this region that p(0, t ; n, v,) has the form
1 42
pa
t ; 4 v,)
A. ( Jt I 1/z 2 n
(4
(0 < n 6 Jt)
=Ye
(46) (Jt
5 n 5 &;1/2)
(b)
while decreasing rapidly for larger values of n. Equation (46) was obtained from equation (26) by replacing all quantities interpreted as depths by their square roots.* In analogy with the arguments leading to equation (27) for the Doppler case, we obtain a theoretical expression for ~(0, t ; v,) in the form (47) since t cc 4b,‘. Equation (47) agrees with the empirical equation (45) indicating that equation (46) represents the correct fundamental probability distribution governing the transfer of photons under these conditions. (iv) It can be seen from Fig. 11 that the function ~(0, t ; v,) has a maximum value at r II 4;’ which varies in proportion to ,/c$“= provided c$,, is small. This result can be interpreted in terms of our simple probability model using Bernoulli trials. A photon released at depth &;’ undergoes approximately 4; l/2 scatterings on its way to the surface and, at each stage, has a probability proportional to 4,. of forming a v, photon. The total probability of escape of a v, photon is therefore proportional to 4,.I”. This result may be contrasted with that in the Doppler case. There, the maximum value of p(0, t, v,) was found to be essentially independent of frequency v,. * The empirical form.
forms (46) and (41) have since been confirmed
by calculating
p(0, t ; n, v,) for 4, of Lorentz
Probability distributions for photon exit
55
In general, photons scattered with frequency v, at depth t are likely to escape directly only if t 5 4,‘. We should therefore expect the function p(0, t; v,) to have a maximum value at depth t N 4”;’ regardless of the particular form of &-cf. Figs. 1 and 11. In part l(ii) of this section we discussed the variation ofp(0, t ; v,) with depth for t L 4,‘. On the other hand, in part l(iii) we discussed its variation with frequency for t 5 4,‘. Clearly, the empirical forms given by the right sides of equations (40) and (45) are matched correctly at t = &’ only if the other variable is present as specified in each case. 2. A weakly thermalizing atmosphere, i <<4,. Equation (3) was solved for the function p(l, t ; v,) in cases where 1 <<4,. . The results are shown in Fig. 12 for selected values of v, for the case L = 10V6. For comparison we show as a broken line a solution for the case 1 = 10V5. We have marked on this figure the
t FIG. 12. The function p(l, t; vI) for the values of v, indicated+, has Voigt form with broadening parameter a = 10e3. The solid lines represent solutions in the case 1 = 10e6 and the broken line represents a solution in the case L = lo-‘. The vertical arrows mark depths numerically equal to lOa/A* in the two cases. Regions C, 1, .I and K are discussed in the text.
regions C, I, J and K, and in addition, we have marked by arrows depths numerically equal to iOa/l*, the thermalization length in cases where 1 <
56
G. D. FINN
It was found empirically from the curves of Fig. 11, that provided t >> 10u/A2 and cp,’ <
This variation can be interpreted by arguments similar to those used in part 2 of Section 5 for the corresponding Doppler problem. However, we would expect to alter equation (29) for that case by changing the upper limit of the integration from A- l to Am2(more precisely 10u/A2), the thermalization length for the present case. In analogy with equation (30) we have
(4% in agreement with the appropriate
empirical form of p(A, t ; v,) given by equation (48).
3. A strongly thermalizing atmosphere, I >> 4,. Values of p(A,t ; v,) were calculated for cases where the probability of thermalization 1, is larger than +,,. (a) In a first example we calculated values of p(A, t ; v,) for I >- a. The results are shown in Fig. 13 for selected values of v, for the case I = 10m2. For comparison we have shown as a broken line a solution for the case 1 = 10-r. We have marked on this figure the regions E, F, , F, and L. These cover the same ranges of depths as the corresponding ones in Fig. 9 for the Doppler case. In addition, in regions E, F, and F2, the functions p(A, t ; v,) were found to have precisely the same empirical forms, (31), (32) and (34) respectively, valid in the corresponding Doppler case. This behavior is to be expected since photons emitted in these regions are unlikely to be scattered with a frequency in the wing of the spectral line. However, in the region marked L in Fig. 13 for which II-’ CC4;’ CCt, p(l, t; v,) is found to vary in proportion to (t 3/212)-1-cf. (t2A2)- ’ for the Doppler case. In the present case of a Voigt profile this may be written as p(l, t ; v,) a y.
(50)
This equation is identical in form to equation (35), the empirical form for the Doppler case. This is to be expected since the argument in terms of long flights, leading to equation (35) may be applied equally here. (b) In a final exampie we calculated values of p(lz, t ; v,) in a case where Cp,.<
Probability distributions for photon exit
57
3i‘ 2 1o-s-
n
lo-” -
IQ’3-
IO-'5 10-2
I
I
I
I
IO"
106
,
\,
u
IO10
t
FIG. 13. The function p(A, t; v,) for the values of v, indicated+, has Voigt form. The solid lines represent solutions for the case A = 10m2 and the broken line represents a solution in the case 1 = lo- I. Regions E, F1, F2 and L are discussed in the text.
(ii) In region H, the function p(A, t ; v,) was found to have the empirical form (37), valid for the corresponding Doppler case. Since 0 << t 5 a- l, photons released in this part of the atmosphere, and destined to escape at frequency v, are unlikely to be scattered with frequencies in the wing of the line (except for the final emission). Their behavior is therefore characteristic of transfer problems with 4” of Doppler form. The theoretical arguments used in obtaining equation (37) apply equally here. (iii) We have marked on Fig. 14, the region M covering the range of depths u-l << t c 10u/A2. It can be seen from the figure that the function p(A, t ; v,) increases slowly with increasing depth in this region. Photons released in region M are likely to be scattered with frequencies in the wing of the line on their way to the surface. Their behavior is therefore characteristic of transfer problems with 4, of Voigt form. The rate of increase of p(A, t; v,) with depth in region M is accordingly smaller than it is in region H-cf. regions C and J in Fig. 11. 7. CONCLUSIONS
In this paper we considered the transfer of photons in a spectral line through a simple type of scattering atmosphere with complete redistribution .of frequency in the scattering process. We obtained numerical results for the probability distribution that a photon formed
G.
58
D.
FINN
10-4
10-e
10-a
3.. f x
lo-‘O
IO-'2
IO-l4 10-Z FIG.
102
106 t
10'0
10'4
14. The function p(l, t; v,) for the values of v, indicated for I, = lo-‘-& Regions E, F, , Fz, H, L and M are discussed in the text.
has Voigt form.
at a specified depth ultimately emerges from the atmosphere at a particular frequency in the line. Using these results, the methods of transfer under various conditions were examined. While the distribution functions behave in a variety of ways depending on the form of 4, and on the relative values oft, 4; ‘, and A- 1 we showed that they can be interpreted in terms of basic probability models. These consisted of(i) a random walk problem of constant step size, (ii) a series of Bernoulli trials and (iii) a series of long upward trips in the wing of the spectral line. The author suggests that distribution functions similar to the ones discussed in the present paper may be useful in inferring the structure of stellar atmospheres from observed profiles of spectral lines. Certainly this problem is important in astrophysics and at present is virtually unsolved. I feel that the situation may improve with a proper understanding of the probable behavior of photons transferring through an atmosphere. Acknowledgement-This 25223.
research was supported by the National Science Foundation
Contract Number GP-
REFERENCES 1. G. D. FINN, JQSRT 11,203 (1971). 2. G. D. FINN and J. T. JEFFERIFS, JQSRT8, 1675 (1968). 3. D. G. HUMMJJRand G. RYBICKI, Methods in Computational Physics, Vol. 7, p. 53. Academic Press, New York (1967). 4. W. FELLER, An Introduction 10 Probability Theory and its Applicarion, Vol. I, p. 87. Wiley (1957). 5. D. G. HUMMER,Mem. R. Asrr. Sot. 1, 1 (1965).
Specrrosc.
Radio,.
Transfer.
PROBABILITY
Vol.
12, pp. 35-58. Pergsmon Press 1972. Printed in Great Britain
DISTRIBUTIONS G.
FOR PHOTON
EXIT
D. FINN
Institute for Astronomy, University of Hawaii, Honolulu, Hawaii 96822, U.S.A. (Received 1 April 1971)
Ab&act-A linear integral equation is formulated for the probability that a photon in a spectral line formed at a specified depth in a scattering atmosphere eventually escapes at a particular frequency in the spectral line. Numerical solutions are obtained for an isothermal atmosphere and their physical meaning discussed under a variety of conditions. 1. INTRODUCTION
WE CONSIDERthe transfer of radiation in a steady-state situation through a plane, semiinfinite, isothermal, and homogeneous atmosphere containing two-level atoms. It is assumed that photons are released throughout the atmosphere with frequency distribution 4,. They transfer through the atmosphere while undergoing absorptions and subsequent emissions. The scattering process is assumed to be isotropic and to produce a complete redistribution of frequency with frequency profile 4,. Eventually, the photons are either thermalized, in which case their energy becomes part of the thermal field, or they escape from the atmosphere. In a previous paper (FINN(‘)), which we will refer to as paper I, we obtained numerical solutions for the probability that a photon formed at a specified depth in an atmosphere, escapes with a particular frequency in the spectral line and in a particular direction. The behavior of this function was discussed in rather general terms. In the present paper we consider probability distribution functions which have been integrated to include all possible directions of escape and attempt to study their behavior in detail. Linear integral expressions for a number of distribution functions are formulated in Section 2. In Section 3 we discuss briefly the numerical methods used for determining these distribution functions. In Section 4 we treat a simple random-walk problem which has a constant step size but which in some respects is similar to the radiative transfer problem. The results are presented and discussed in Sections 5 and 6 for cases where 4, has both Doppler and Voigt forms respectively. 2. FORMULATION We consider the simple case of the transfer of radiation through a plane, semi-infinite, isothermal, homogeneous atmosphere containing model atoms having two energy levels between which electron transitions take place accompanying the absorption and emission of radiation. It is supposed that photon scattering is isotropic, that there is complete redistribution of frequency in the scattering process, and that the frequency profiles 4, of both 35
G. D. FINN
36
the absorption and emission coefficients are identical. For convenience stimulated emission is neglected. The discussion is restricted to problems in which all stochastic variables are invariant with time so that the optical depth scale is fixed. In addition to the scattering process, it is assumed that a photon has a constant probability L of being thermalized after absorption. We suppose that photons in the spectral line are formed throughout the atmosphere with frequency distribution C#J,(normalized to unity) isotropically in direction but not necessarily homogeneously with depth. We measure optical depth on a scale t such that the monochromatic optical depth t,, is given by t, = &t. The cosine of the angle between the direction traveled by a photon and the upward normal to the atmosphere is denoted by cc; a photon is then described by its frequency (v) and direction (r_l).Suppose a (v, p) photon is first emitted in a layer of optical thickness dt, at depth t in the atmosphere. We define p@, t, v, p; v,, pe) to be the probability distribution function for this photon to ultimately emerge from the atmosphere as a (v,, p,) photon. It may easily be shown from first principles--+f. paper I-that this function satisfies the integral equation p(l, t, v, p; v,, p,) = e-‘4~W5(v-
v,)b(p-pC1,) OD
+1
(1) where, for purposes of integration 6 may be considered to be the Dirac b-function. The first term in equation (1) represents a discrete probability pulse for the particular case when (v,, p=)photons are formed at depth t and emerge directly, while the second term represents contributions from the indirect escape of photons. As explained in paper I, the integration over t’ has been written without explicit limits on the understanding that if p is negative the original photon has been emitted downwards into the atmosphere and t’ ranges from t to infinity; if p is positive, on the other hand, the original photon was emitted upwards and t’ ranges from zero to t. In this paper we have adopted the convention of placing the arguments of functions in two groups. The first group inside the parentheses contains variables whose values are specified ; separated from them by a semicolon are the random variables of the probability distribution function. It is convenient to define the distribution function p(& t ; v,) for the probability that a photon which is originally formed at depth t and emitted with arbitrary frequency in the spectral line ultimately emerges from the atmosphere with frequency v,. This function can be calculated in terms of p(il, t, v, p ; v,, ,u,) by multiplying by 4, dv dp/2 dp, and integrating to include all possible frequencies and directions of the original and all possible directions of the emerging photon. Thus we have
p(l,t,v,)
=
j,.jm., 0
j’$p(U,v,ii:v..pi. 0
-1
Probability distributions for photon exit
37
If the same operations are performed on both sides of equation (1) we find the integral equation for p(A,t ; v,) m
$44 t; v,) = %#Q,(t4J+(l-A)
j- ~41,t’;
v,)K(lt-
t’l) dt’
(3)
0
where
03 WY) = 9
s
#&%(~dd dv
(4)
0 is a kernel which arises frequently in the formulation of radiative transfer problems in terms of linear integral equations (cf. FINN and JEFFERIE.@) and E, and E, are the first and second exponential integral functions. Equation (3) may be interpreted in terms of simple physical ideas since the first term on the right represents the probability that a v, photon is formed originally at depth t and escapes directly. Other contributions to p(l, L; v,) coming from the indirect escape of photons are represented by the second term. This term contains the factor K((t- t’l)dt’ which represents the probability that a photon emitted at depth t makes a direct flight to depth t’ and is absorbed there in a layer of thickness dt’-cf. FINNand JEFFERIES.(~) This function is then multiplied by (1 - A), the probability that the photon is scattered, and by ~(1, t’ ; v,) the distribution function for the probability that this photon ultimately emerges at frequency v,. Integration over all depths t’ then includes all possible indirect contributions to p(A t ; VA. In the special case Iz = 0, the atmosphere becomes purely scattering so that there can be no thermalization processes. In this case all photons must eventually escape from the atmosphere, so that m
~(0, t; v,) dv, = 1.
(5)
0
It may be easily shown that this equation is consistent with equation (3). It is useful to consider the distribution functions p(A, t;
P@,t; In+l,v,)
= $#QAt&.)+(l-A)
s
p(l,t’;
(6)
0
where
P(k t ; I 1,v,) = P(J,t ; 1,v3 = 34Q#PvJ.
(7)
The distribution function p(l, t ; n, v,) may be easily obtained by subtraction. Thus, we have p(L,t;n,v,)
= p@,t;
In-l,v&?).
(8)
38
G.
D. FINN
3. NUMERICAL
PROCEDURES
Equation (3) is a Fredholm integral equation of the second kind in the unknown function p(l, t ; v,). With the kernel of equation (4) it is-precisely the integral equation for the source function appropriate to the radiative transfer problem in an atmosphere containing two-level model atoms for which the source term is distributed with depth as ~~YJZZ(t~Ye). Equation (3) may therefore be solved numerically for p(;l, t ; v,) by one of the methods developed for handling problems of this kind (cf. e.g. HUMMERand RYBICKI’~‘). We use the step (2) Also using a step function representafunction method adopted by FINN and JEFFBRIES. tion, the recurrence relation (6) may be solved in conjunction with equation (7) by simple iteration for successive functions in the sequence p@, t ;
4. A SIMPLE RANDOM WALK PROBLEM In this section we discuss a simple random walk problem whose solution is useful in interpreting some aspects of the more complex distribution functions associated with the photon transfer problem. We consider a one-dimensional random walk in which a particle moves along the x axis. We suppose that the particle starts at a positive value of x and takes steps of unit length with equal probability in either direction. We further suppose that the random walk may terminate at any stage resulting in two possible outcomes. We will designate these ‘success’ and ‘destruction,’ and assign to them probabilities p and 1 respectively. The walk continues until the particle is successful, is destroyed, or travels beyond x = 0 where it is absorbed at a perfectly absorbing barrier at x = - 1 when it is deemed to have ‘failed.’ The probability pX that a particle located at depth x eventually succeeds, satisfies the recurrence relation px =
p+(l-A-p)(p .X+1+p X_1) 2
(x 2 0).
The first term on the right represents the probability that the particle succeeds directly, while the second term represents the probability that it neither succeeds nor is destroyed at this stage but moves to the right or left and eventually succeeds. Equation (9) may be solved by standard analytic methods to give the solution P
lpx = A+p i
l-l-p 1 +J@+&/(24--_I
h/@+P)JP-~-P) (
l-l-p
x
11
cx > 0)
(IO)
If the probabilities I and p are small compared with unity, we find the probability of ultimate success pXis given approximately by
We consider, in particular, the following cases :
Probability
distributions
for photon exit
39
(i) If p <
Under these conditions the probability that the particle eventually succeeds is A- ’ times the probability that it succeeds directly.* On the other hand, if the particle starts its walk atx = Owefind
po=J2-$.
(13)
(ii) When 1 = 0, the particle cannot be destroyed. In this case, if the random walk starts at x = 0, equation (11) becomes PO N
Jw.
(14)
The problems of the transfer of a photon through a homogeneous scattering atmosphere and of the simple random walk just discussed may be related as follows : (a) The constant probability of ‘success’ in the simple problem, has as its counterpart in the photon transfer problem, the probability &/2Ez(t&) of a photon being scattered with frequency v, and escaping directly from the atmosphere. For the two problems to have similar solutions this last probability must also be constant. This condition amounts to the requirement t << 4;‘) in which case p becomes identified with &e/2. (b) The term ‘destruction’ in the simple problem is identified with the term ‘thermalization’ in the transfer problem. (c) The term ‘failure’ in the simple problem corresponds to the emergence of a photon from the atmosphere at a frequency in the spectral line other than v,.
5. RESULTS-DOPPLER
PROFILE
4.
In this section we attempt to interpret the behavior of the function p(A, t ; v,) in terms of simple physical ideas and basic probability models. Values of the auxiliary functions p(l, t; sn, v,) and ~(1, t; n, v,) were calculated as an aid in this interpretation. Clearly, on general grounds we should expect the distribution functions p(0, t ; in, v,) and p(l, t ; v,) to be closely related when n equals A-I. In this case any discussion of the behavior of one of these functions applies equally to the other. Specific comparisons between the two functions will be noted in footnotes as they arise. In this application we supposed that the profiles of the absorption and emission coefficients have the Doppler form (15) where the frequency variable v should be interpreted as the magnitude of the frequency departure from the center of the spectral line in units of the Doppler width. * A similar result was obtained in the case of photon transfer by FINN and J~PWRIES.‘~ They found that for a photon located so deep in an atmosphere that it is unlikely to escape at all, the probability of eventual escape was A- 1 times the probability of direct escape.
G. D. FINN
40
1. Purely scattering atmosphere,
rZ= 0
Equation (3) was solved for p(r2,t; v,) in the case of a purely scattering atmosphere, 1 = 0; the results are shown in Fig. 1 for selected values of the frequency of the emergent photon v,. We have marked on the figure the three regions A, B, and C together with
10-7
10-Z
I
I
1
IO2
1
104
I
106
1
108
I
10'0
I
IO"
FIG. 1. The function p(L, t ; v,) in the case of a purely scattering atmosphere for the values of v, indicated+, has Doppler form. The vertical lines mark depths numerically equal to 4;‘. Regions A, B, and C are discussed in the text.
depths numerically equal to Cp,‘. A comparison of the curves of Fig. 1 shows a number of features which we now discuss. (i) In the region A it was found empirically that P(0, t; v,) = 1.&h..
(16)
Since0 N t c f&’ in this region, we may expect the simple random walk problem discussed in Section 4 to have a similar type of solution. It can be seen by comparing equation (16) with equation (14) that-apart from a small constant factor-the two forms agree if p in equation (14) is replaced by &/2. (ii) It was found in the region t >> r$V,‘, that ~(0, t ; v,) varies with depth in proportion to t-l/’ * In addition, if 4,. << 1, it was found that p(0, t ; v,) is distributed with frequency in propbrtion to 4;‘12. We may therefore represent the variation of ~$0, t; v,) in region B of Fig. 1 by the relation 1 P&At ; ve) cc (&-p’ (17) * The same variation with depth under these conditions distribution function.
was noted, but not explained, in paper I for a similar
Probability distributions for photon exit
41
Before attempting to understand physically the depth dependence of p(O, t; v.) in this region, it is necessary to make a digression to investigate the behavior of the associated distribution functions p(0, t ; s n, v,) and p(O, t ; n, v&. Equations (6) and (7) were solved iteratively for values of the successive ‘partial’ distribution function p(0, t ; sn, v,) for n = 1,2,. . . . The results are shown in Fig. 2 for selected I
10-l
10-Z
10-3
10-q
3 c‘ “I 2
10-S
G n
10-G
IO-’
10-e
10-g
lo-‘0 10-z
IO“
.I
IO
IO2
I03
I04
I05
FIG. 2. The function p(A, t ;
values of n in the particular case v, = 0 ; for comparison the corresponding ‘full’ probability function p(0, t ; ve) is shown by a broken line. It can easily be shown from this figure that p(0, t; an, v,) is distributed in proportion to n3/‘te2, provided t 2 n. In addition, comparison of these curves with others for different values of v, reveals that the functions do, t ; 5 n, v,) and p(0, t ; v,) have identical frequency variation. When 4, has Doppler form it is well known that K(t) varies as t -2 for large values of t. Thus, provided 4,. c 1 and 0 <
K(t)n3” _
(18)
Values of p(0, t; n, v,) were then calculated from equation (8) and are shown as functions of n in Fig. 3 for selected values of the depth of formation t, in the particular case v, = 0. It can be seen from this figure (or from the empirical form (18)) that, provided
G. D. FINN
42
0 CCn 5 t, the function ~(0, t; n, v,) varies in proportion to K(t)n1/24v;1’2. For n 2 t, however, the probability of photons escaping after precisely n scatterings is seen to vary in proportion to ne2. IO'
,6 _
I
I
IO
IOC
IOJ
10.
”
FIG. 3. The function p(l, t ; n, v,) as a function of n in the case of a purely scattering atmosphere for v, = 0 and for sejected depths-4, has Doppler form. In the case t = lo4 the probabilities have been multiplied by 100 for convenience.
Equations (6) and (7) were solved for the function p(0, t; an, v,) in cases where c#J,, << 1. We adopted the values v, = 3.5 and v, = 4. Values of ~(0, t ; n, v,) were then obtained from equation (8) and are shown in Fig. 4 for selected values of the depth t. It can be seen that, for depth t x 4,’ and for values of n 5 Q&;‘,the function ~$0,t ; n, v,) is independent of frequency but varies in direct proportion to n and in inverse proportion to t2. It is clear that, under these conditions, p(0, t ; n, v,) varies in proportion to K(t)n and p(0, t ; s n, v,) varies in proportion to K(t)n’. These results may be summarized by the empirical formulae
I1
n2
where 4;’
<< t.
* Continuity of the last two parts of equation must be K(t)fs~2~~‘~2n-2.
n 2 t.
(19) shows that the precise variation
(c) (19)*
of p(0, t ; n, v,) for n 2 t,
Probability distributions for photon exit
43
10-2
I
1
10-2
102
IO ”
FIG.4. The function p(l, f ; n, vJ as a function of IIin the case of a purely scattering atmosphere for the depths indicated-$, has Doppler form. The curves represent the common solutions for the two cases v, = 3.5 and v, = 4.
We are now able to interpret the variation of p(0, t ; v,) with depth in region B of Fig. 1. It can be seen from the general form of equation (19) that a photon formed at depth t in region B i,slikely..~to undergo roughly t scatterings before escaping from the atmosphere with frequency v,. Suppose it remains in the vicinity of t during its first i scatterings. With a probability approximately equal to K(t) dz it then takes a long direct flight upwards to some depth r, also assumed to be in region B, and is absorbed there in a layer of thickness dz. The probability that the photon escapes at frequency v, after fewer than a total of n scatterings is then found by multiplying K(t) dr by ~(0, r;
P@,t;
I&v,)
=
x(t)Igi
p(O,z; in-i-l,v,)dr.
f
(20)
0
For values of T in the range 0 I z 5 n- i- 1 the probability function ~(0,T ; sn- i- 1, v,) has virtually reached its full value p(0, 7; v,). We may therefore substitute equation (17)
G. D. FINN
44
into the right side of equation (20) to give n-i
”
~(0, t ;
sn, v,) a K(t)
(21)
for 0 <
(22)
Equations (21) and (22) agree with the corresponding empirical forms (18) and (19b) respectively indicating that our interpretation of the transfer mechanism is valid. If we let n vary over its range of probable values (0, t) equation (22) may be integrated to give
P(0, t ; v,) K
’ K(t)n”*
I o
dn
(23)
Jh
since K(t) varies in proportion to t-* when t is large. This variation is in agreement with equation (17), the form found empirically to be appropriate to region B of Fig. 1. Evidently a photon which is formed at depth t such that t s 4V,‘, needs about t scatterings to escape with frequency v, . During its passage to the surface it takes a number of long flights. Roughly speaking, each of these flights terminates closer to the surface than an optical distance numerically equal to the number of scatterings still outstanding from the original t. Eventually the photon leaves region B when a long flight terminates closer to the surface than optical distance 4;’ and enters region C. (iii) It was found empirically in region C of Fig. 1 that
pt0,t ; v,) a (4,.t)“*.
(24)
Before attempting to understand physically the frequency dependence of ~(0, t ; v,) in this region it is necessary, once again, to digress. In this case we investigate the behavior of the functions p(0, t; in, ve) and p(0, t; n, v,) for t in the range 0 << t <<4,‘. Equations (6) and (7) were solved for values of the function ~(0, t ; I n, ve) in the case v, = 5. The results are shown in Fig. 5 for selected values of n. From these results, values of the function ~(0, c ; n, v,) were obtained ; they are shown as functions of n in Fig. 6 for selected values oft. It was found for each curve in this figure that the value of p(0, t ; n, vJ is numerically equal to &J2 (in this case 7.84 x lo- ‘*) provided 0 < n 6 t. For n 2 t, on the other hand, ~(0, t; n, v,) varies in proportion to n - ‘I* . A comparison of the values of p(0, t ; n, v,) for different values oft shows that the function is also proportional to ,/t. Values of the function p(0, t ; n, v,) were also calculated for the particular case v, = ,/3 and t = 1, and are shown as a function of n in Fig. 7. It can be seen from this figure that for n >>$V,‘, the distribution function varies approximately in proportion to nm2.
Probability distributions for photon exit
10-s-
I
I
I
I
I
I
I
VI ;
d
x
10-Q
-
IO“’ -
FIG. 5. The function p(A, t ; III, VJ in the case of a purely scattering atmosphere for v, = 5 and for the values of n indicated-#, has Doppler form.
2_
I
I
IO
\
I02
n
FIG. 6. The function p(l, t ; n, v,) as a function of n in the case of a purely scattering atmosphere for Y, = 5 and for the depths indicated-+, has Doppler form.
These results may be summarized by the empirical formulae (= &I2
(0 < n 5 t)
(a)
04 (26)
I
Cc-
provided t <<4,‘.
1
n2
(n k #C:‘)*
(4
l Continuity with the empirical form valid for smaller values of n would require that p(0, t; n, ve)varies in proportion to t14j~~‘~2n-2 for n 2 f#~;‘.
G. D. FINN
46
We are now able to discuss the mechanism of photon transfer in region C of Fig. 1. It can be seen from equation (26a) that a photon formed at depth t has probability &/2 of escaping at frequency v,, provided n 5 t 5 4,‘. This is to be expected since the photon is unlikely to reach the surface layers during PIscatterings, but if scattered in an upward direction with frequency v,, it is virtually certain to escape directly.
IO’
FIG. 7. The function &I, t ; n, YJ as a function of n in the case of a purely scattering atmosphere for v, = ,,/3 and for t = l-4, has Doppler form. The vertical line marks the depth numerically equal to 4;:.
For values of n in the range t 5 n 5 c$~;‘, the value of ~$0,t; n, v,) given by equation (26b) decreases slowly. The reason for this decrease can be traced to the increased probability of the photon reaching the surface layers after fewer than n scatterings and escaping with a frequency closer to the core of the spectral line than v,. Equation (26b) can be simply understood in analogy with the simple random walk problem described in Section 4. The inequality t 6 n 5 4V,‘, has as its counterpart in the simple problem x 6 n 6 p-l. Under these conditions it can be shown-see, for example, FELLERt4)---that a simple random walk,
starting
at position
x where
x is large,
continues
for at least n steps with a probability
,/(2/7+/,/n. The dependence of this probability on n is precisely that given by equation (26b) for the photon transfer problem. The analogy is completed on recognizing that, for a simple random walk, a point is probably displaced a distance x after roughly x2 steps, whereas a typical photon changes its optical depth by roughly t after t scatterings. Hence, the x dependence of the probability for the simple problem is replaced by Jt in the photon problem. If n is further increased the value of ~(0, t ; n, v,) decreases more rapidly. This is caused by the increased a priori probability of escape at frequencies further in the wing of the line than ver as the number of emission absorption processes increases beyond $L’. approximately
equal
to
Probability distributions for photon exit
47
Equation (26) may be integrated to include all likely values of the number of scatterings n, to give the total probability of escape at frequency v,. We find (27) in agreement with the empirical form (24) describing region C of Fig. 1. (iv) It is clear from Fig. 1 that the function p(0, t; v,) has a maximum value at depth t N 4,’ which is essentially independent of v,. The reason for this behavior was discussed in paper I. In Section l(ii) above we discussed the variation of p(0, t; v,) with depth in the region t 2 4”;‘. On the other hand, in Section l(iii), we essentially discussed thefrequency variation of the function in the region t 5 4,‘. Clearly, the appropriate empirical forms given by equations (17) and (24) respectively are correctly matched at t N 4”;‘, provided that the other variable is present as specified in each case. 2. A weakly thermalizing atmosphere, 1 c-c 4,. Equation (3) was solved for p(l, t ; v,) in cases where 1 << &. The results are shown in Fig. 8 for selected values of v, for the case 1 = 1O-8. For comparison we have shown as a broken line a solution for the case 1 = 10e6. We have marked on this figure the regions A, B, C, and D. The first three of these lie above depth LX’, whicn represents the thermalization length of this transfer problem-cf. FINN and JEFFER~ES.(~) Thus, the function ~(1, t ; v,) shows the same behavior as the corresponding regions A, B, and C of Fig. 1 for the case of a purely scattering atmosphere. In the region D of Fig. 8, where &;’ <
(1 - A)nK(t) ‘j’ p(A, 7
; v,)
dr.
(29)
0
* Comparison of equations (18) and (28) shows that in the case n = I-’ the functions p(0, t; in, v,) and p(A,t ; v,) have the same empirical form. This is expected since n >>4,’ and 1-l >>$‘. The relation between the two functions can also be seen by comparing the curves of Fig. 8 with those of Fig. 2.
G. D. FINN
48
At this stage the photon may be considered to belong to region B where the effects of thermalization are insignificant. It then proceeds to the surface by a series of long upward flights as explained in part l(ii) of this section for the case of a purely scattering atmosphere. We therefore substitute the empirical form (17) into the right side of equation (29) to give
(30) in agreement with the empirical form (28).
FIG. 8. The function p(2, t; v,) for the values of v, indicated+, has Doppler form. The solid lines represent the solutions in the case L = 10-s and the broken line represents a solution in the case I = 10e6. Regions A, B, C and D are discussed in the text.
3. A strongly thermalizing atmosphere,
A D 4,.
Values of p(A, t ; ve) were calculated for cases where the probability of thermalization I, is large. The results are shown in Fig. 9 for selected values of v,. The solid lines represent the solutions for the case I = lo-’ and the broken line represents a solution for I = 10-l. We have marked on this figure regions E, F, , F, , G and H as well as depths numerically equal to c#J,‘. We consider the behavior of the function in each of these regions.
Probability distributions for photon exit
49
t FIG.9. The function p(A,r ; v,) for the values of v, indicated-& has Doppler form. The solid lines represent solutions in the case I = 10e2 and the broken line represents a solution in the case ,I = 10-r. Regions E, F,, F2, G and H are discussed in the text.
(i) In the region E, t N 0 and 4,. c 1. The distribution empirically to have the form p(l,t;v,)
4
= *.
function p(L, t; v,) was found
(31)
Apart from a factor of J2, this empirical form agrees with equation (13) for the simple random walk problem under analogous conditions, provided we replace p in that equation by &l2. (ii) It was found empirically from the curves of Fig. 9 that in the region Fi where I-’ << t << qb,‘, th e function p(l, t ; ve) becomes independent of depth and has the form p(l, t; v,) =
$.
(32)
This equation is in agreement with equation (12) for the simple random walk problem under analogous conditions, provided we replace p in that equation by #“e/2.* We may also interpret equation (32) in terms of Bernoulli trials where each scattering of the photon represents a trial. The expression (1 - Iz- #,,)” then represents the probability * Clearly, equation (32) agrees with the form of p(0, t; in, vc)corresponding to equation (26a) for p(0, t ; II,v,). This is expected, since t <<4;’ in each case, and A-’ and n are each smaller than t. The close relation between the two functions &I, t; v.) and p(0, I;
50
G. D. FINN
that the photon is not scattered with frequency v, and is not thermalized during the first n trials. Under the conditions specified here, the probability of a photon being scattered in an upward direction with frequency v, and escaping is 1$,~/2.We may therefore write p(l,t;v,)
‘v
(l-a-&y+ k,
f
(33)
2:
II=0
since &, example, direction 1-i << t
<
PO,t ; v,) =
~W”J).
(34)
It can be seen from Fig. 9 that this variation is in approximate agreement with the behavior of the curves in region FZ where the function is seen to decrease more or less exponentially with increasing depth. (iii) In the region G of Fig. 9, where 1-i <<4,’ << t, it can be seen that the value of ~(1, t ; v,) is independent of v, and varies with 1 and depth in proportion to (At)-‘, so that p(l, t ; v,) cc
F
(35)*
This variation can be readily explained. A photon released under these conditions is likely to escape at frequency v, only if it makes a long direct flight upwards to within an optical distance 4;’ of the surface (the larger of the values of &’ and I- ‘). We suppose the photon remains in the vicinity of depth t for the first n scatterings and then makes a long direct flight upwards to a depth r and is absorbed there in a layer of thickness dr. The probability that the photon eventually escapes with frequency v, is given by ov, p(A, t; v,) N f
(1 -A)nK(t) n=O
I
~(1, z ; v,) dz.
(36)
0
The distribution function p(l, z; v,) appropriate to the right side of equation (36) has been discussed in part 3(ii) above and was represented by equation (32). If equation (32) is substituted into equation (36) we obtain, after a little algebra, the empirical form (35). (iv) In the region H of Fig. 9 it was found that p(l, t; v,) varies with frequency and il in proportion to &J,/IZ. Equation (3) was also solved for the particular case Iz = 10d6, v, = ,/33; the results are shown in Fig. 10. In this figure the region marked H covers a large range of depths and it is clear that the function p(l, t ; vJ varies with depth in proportion to Jt. On the basis of these observations we can write (37) Provided
t << A- ’ cc c&; l, equation
(37) may be obtained
theoretically
by integrating
* This relation is not unexpected since p(O,t : I”, v,) was found to vary as K(t)d for n CC&’ l(ii) of this section, equation (19a).
<
part
Probability distributions for photon exit
51
equation (26) for ~(0, t ; n, v,) over all values of n between zero and 1-i. Since t c A- ’ << &‘, we find (38) By virtue of our earlier remarks the left side of this equation may be replaced by p(l, t ; v,). Equation (38) then reduces to equation (37).
10-j
IO“
;; ; I
9 lo-” f
‘2 II
IO-
10-I
I 2
I
I I02
I
I
I
106
I04
106
IO” D
t
FIG. 10. The function p(l, t; v,) for the case 1 = 10e6 and v, = H is discussed in the text.
6. RESULTS-VOIGT
J334,
PROFILE
has Doppler form. Region
4,
In this section we consider cases where c$, has the Voigt form
(3% with a the broadening parameter. For a equal to the value 0.001, for example, the Voigt and Doppler forms of ~5,are essentially identical out to v N 25 where 4, is roughly numerically equal to a. It would seem that photons released closer to the surface than an optical distance numerically equal to a- ’ and destined to escape at frequencies in the core of the spectral line are unlikely to be scattered with frequencies larger than about 2.5 at any stage. Under these conditions the distribution function p(l, t ; v,) can be expected to show similar behavior for both types of frequency profiles. For the same reasons the functions p(0, t ; in, v,) and p(0, n, v,) show some of the effects peculiar to the type of profile 4, used, only if the number of scatterings n, is larger than a - r. Since it is not practical to perform numerically a large number of iterations of equation (6) we show these effects indirectly. t
;
G. D. FINN
52
1. Purely scattering atmosphere,
A= 0
The function ~(1, t ; v,) was calculated for the case of a purely scattering atmosphere, I = 0, in the case where 4, has Voigt form with broadening parameter a = 0.001. The results are shown in Fig. 11 for selected values of the frequency of the emergent photon v,.
FIG. 11. The function p(l, t; v.) in the case of a purely scattering atmosphere for the values of v, indicated-+, has Voigt form with broadening parameter a = lo- 3. The vertical lines mark depths numerically equal to 4,‘. Regions C, 1 and J are discussed in the text.
We have marked on the figure the regions C, I and J together with depths numerically equal to 4,‘. These values were obtained from tables of the Voigt function prepared by HUMMER.“) We discuss these regions in turn. (i) In the region C, for which t lies in the range 0 << t 5 a- ‘, the function ~(0, t ; vJ varies approximately in proportion to ,/t. This is the same depth variation found for the corresponding region in the Doppler case-cf. Fig. 1. As we remarked earlier such a similarity is to be expected. (ii) In the region I for which 0 <<4; ’ a t, it was found empirically that the distribution function ~(0, t ; v,) has the form 1
P(O, t ; v,) = gzpc.
In order to interpret physically the depth variation of ~(0, t; ve) in this region, we need to know empirical forms for d0, t; n, v,). As remarked earlier we can only obtain these
Probability distributions for photon exit
53
indirectly. It is known by general arguments-& FINN and J~rut#)--that for 4, of Voigt form a typical photon travels roughly a vertical optical distance of n2 (more precisely 10 an2) while undergoing a large number n, of scatterings. It follows that a typical photon formed at depth t escapes from the atmosphere after roughly dt scatterings. By analogy with equation (19) for the Doppler case, it would therefore seem reasonable to assume in the present case, that ~(0, t; n, v,) has the form 0 < n 5 C&i/2
K(t)n
(a) (41)
~(0, r ; n, v,) cc K(t)n1’2 ~,l/4
4yOti2 S n 6 Jt
while decreasing rapidly for larger values of n. In obtaining equation (41) from equation (19) we have arbitrarily replaced &./” by 4:: in part (b) of the equation. The depth dependence of equation (40) can now be interpreted by arguments similar to those used for the Doppler case. We suppose that a photon is formed at depth t and remains in the vicinity of t during the first i scatterings. It then takes a long upward flight to a depth r and is absorbed there in a layer of thickness dr. The probability that the photon escapes from the atmosphere after fewer than n scatterings subsequent to its original emission, can be found by summing over all possible values of i and integrating to include all necessary values of T. We write in this case, n_ 1 W-V p(O,t;
1
p(o,T; In-i,V,)dT.
(42)
b
Equation (42) is similar to equation (20) for the Doppler case except that the upper limit of integration has been changed from n-i to (n- i)2. This has been done in the belief that a photon is likely to escape from the atmosphere during the remaining n - i scatterings allotted, only if the long flight terminates closer to the surface than an optical distance numerically equal to (n-i)‘. Following our arguments for the Doppler case we assume that the function ~(0, T ;
~(0, t ; in, v,) a K(t)
(n - i)*
(43)
The right side of this equation agrees with the functional form of AO, t ; n, v,) specified by equation (41b). Since K(t) varies in proportion to t - 3/2 for large values of t, we may write, in analogy with equation (23),
P(O,t ; v,) a
Pw
54
G. D. FINN
Equation (44) agrees with the observed empirical form (40) indicating that our arguments are valid and that equation (41b) is essentially correct. Evidently, a photon formed at depth t escapes at frequency v, such that t >> 4;’ after undergoing approximately ,/t absorption-emission processes during its passage to the surface. During this transfer the photon makes a number of long flights, each of which terminates roughly closer to the surface than an optical distance numerically equal to the square of the number of scatterings remaining from the original Jt. Eventually the photon leaves region I as it reaches depths closer to the surface than 4,‘. (iii) It was found empirically from Fig. 11 that in the region marked J for which lo3 5 t c q6y;1,the function ~(0, t; v,) varies with depth and frequency according to the form p(0, t ; Ye) cc (P;~P.
(45)
In order to understand the frequency variation of ~(0, t; v,) in this region, we must find empirical forms for d0, t; n, v~). In keeping with the arguments used in part l(ii) of this section, and in analogy with equation (26) for the Doppler case, we shall assume for this region that p(0, t ; n, v,) has the form
1 42
pa
t ; 4 v,)
A. ( Jt I 1/z 2 n
(4
(0 < n 6 Jt)
=Ye
(46) (Jt
5 n 5 &;1/2)
(b)
while decreasing rapidly for larger values of n. Equation (46) was obtained from equation (26) by replacing all quantities interpreted as depths by their square roots.* In analogy with the arguments leading to equation (27) for the Doppler case, we obtain a theoretical expression for ~(0, t ; v,) in the form (47) since t cc 4b,‘. Equation (47) agrees with the empirical equation (45) indicating that equation (46) represents the correct fundamental probability distribution governing the transfer of photons under these conditions. (iv) It can be seen from Fig. 11 that the function ~(0, t ; v,) has a maximum value at r II 4;’ which varies in proportion to ,/c$“= provided c$,, is small. This result can be interpreted in terms of our simple probability model using Bernoulli trials. A photon released at depth &;’ undergoes approximately 4; l/2 scatterings on its way to the surface and, at each stage, has a probability proportional to 4,. of forming a v, photon. The total probability of escape of a v, photon is therefore proportional to 4,.I”. This result may be contrasted with that in the Doppler case. There, the maximum value of p(0, t, v,) was found to be essentially independent of frequency v,. * The empirical form.
forms (46) and (41) have since been confirmed
by calculating
p(0, t ; n, v,) for 4, of Lorentz
Probability distributions for photon exit
55
In general, photons scattered with frequency v, at depth t are likely to escape directly only if t 5 4,‘. We should therefore expect the function p(0, t; v,) to have a maximum value at depth t N 4”;’ regardless of the particular form of &-cf. Figs. 1 and 11. In part l(ii) of this section we discussed the variation ofp(0, t ; v,) with depth for t L 4,‘. On the other hand, in part l(iii) we discussed its variation with frequency for t 5 4,‘. Clearly, the empirical forms given by the right sides of equations (40) and (45) are matched correctly at t = &’ only if the other variable is present as specified in each case. 2. A weakly thermalizing atmosphere, i <<4,. Equation (3) was solved for the function p(l, t ; v,) in cases where 1 <<4,. . The results are shown in Fig. 12 for selected values of v, for the case L = 10V6. For comparison we show as a broken line a solution for the case 1 = 10V5. We have marked on this figure the
t FIG. 12. The function p(l, t; vI) for the values of v, indicated+, has Voigt form with broadening parameter a = 10e3. The solid lines represent solutions in the case 1 = 10e6 and the broken line represents a solution in the case L = lo-‘. The vertical arrows mark depths numerically equal to lOa/A* in the two cases. Regions C, 1, .I and K are discussed in the text.
regions C, I, J and K, and in addition, we have marked by arrows depths numerically equal to iOa/l*, the thermalization length in cases where 1 <
56
G. D. FINN
It was found empirically from the curves of Fig. 11, that provided t >> 10u/A2 and cp,’ <
This variation can be interpreted by arguments similar to those used in part 2 of Section 5 for the corresponding Doppler problem. However, we would expect to alter equation (29) for that case by changing the upper limit of the integration from A- l to Am2(more precisely 10u/A2), the thermalization length for the present case. In analogy with equation (30) we have
(4% in agreement with the appropriate
empirical form of p(A, t ; v,) given by equation (48).
3. A strongly thermalizing atmosphere, I >> 4,. Values of p(A,t ; v,) were calculated for cases where the probability of thermalization 1, is larger than +,,. (a) In a first example we calculated values of p(A, t ; v,) for I >- a. The results are shown in Fig. 13 for selected values of v, for the case I = 10m2. For comparison we have shown as a broken line a solution for the case 1 = 10-r. We have marked on this figure the regions E, F, , F, and L. These cover the same ranges of depths as the corresponding ones in Fig. 9 for the Doppler case. In addition, in regions E, F, and F2, the functions p(A, t ; v,) were found to have precisely the same empirical forms, (31), (32) and (34) respectively, valid in the corresponding Doppler case. This behavior is to be expected since photons emitted in these regions are unlikely to be scattered with a frequency in the wing of the spectral line. However, in the region marked L in Fig. 13 for which II-’ CC4;’ CCt, p(l, t; v,) is found to vary in proportion to (t 3/212)-1-cf. (t2A2)- ’ for the Doppler case. In the present case of a Voigt profile this may be written as p(l, t ; v,) a y.
(50)
This equation is identical in form to equation (35), the empirical form for the Doppler case. This is to be expected since the argument in terms of long flights, leading to equation (35) may be applied equally here. (b) In a final exampie we calculated values of p(lz, t ; v,) in a case where Cp,.<
Probability distributions for photon exit
57
3i‘ 2 1o-s-
n
lo-” -
IQ’3-
IO-'5 10-2
I
I
I
I
IO"
106
,
\,
u
IO10
t
FIG. 13. The function p(A, t; v,) for the values of v, indicated+, has Voigt form. The solid lines represent solutions for the case A = 10m2 and the broken line represents a solution in the case 1 = lo- I. Regions E, F1, F2 and L are discussed in the text.
(ii) In region H, the function p(A, t ; v,) was found to have the empirical form (37), valid for the corresponding Doppler case. Since 0 << t 5 a- l, photons released in this part of the atmosphere, and destined to escape at frequency v, are unlikely to be scattered with frequencies in the wing of the line (except for the final emission). Their behavior is therefore characteristic of transfer problems with 4” of Doppler form. The theoretical arguments used in obtaining equation (37) apply equally here. (iii) We have marked on Fig. 14, the region M covering the range of depths u-l << t c 10u/A2. It can be seen from the figure that the function p(A, t ; v,) increases slowly with increasing depth in this region. Photons released in region M are likely to be scattered with frequencies in the wing of the line on their way to the surface. Their behavior is therefore characteristic of transfer problems with 4, of Voigt form. The rate of increase of p(A, t; v,) with depth in region M is accordingly smaller than it is in region H-cf. regions C and J in Fig. 11. 7. CONCLUSIONS
In this paper we considered the transfer of photons in a spectral line through a simple type of scattering atmosphere with complete redistribution .of frequency in the scattering process. We obtained numerical results for the probability distribution that a photon formed
G.
58
D.
FINN
10-4
10-e
10-a
3.. f x
lo-‘O
IO-'2
IO-l4 10-Z FIG.
102
106 t
10'0
10'4
14. The function p(l, t; v,) for the values of v, indicated for I, = lo-‘-& Regions E, F, , Fz, H, L and M are discussed in the text.
has Voigt form.
at a specified depth ultimately emerges from the atmosphere at a particular frequency in the line. Using these results, the methods of transfer under various conditions were examined. While the distribution functions behave in a variety of ways depending on the form of 4, and on the relative values oft, 4; ‘, and A- 1 we showed that they can be interpreted in terms of basic probability models. These consisted of(i) a random walk problem of constant step size, (ii) a series of Bernoulli trials and (iii) a series of long upward trips in the wing of the spectral line. The author suggests that distribution functions similar to the ones discussed in the present paper may be useful in inferring the structure of stellar atmospheres from observed profiles of spectral lines. Certainly this problem is important in astrophysics and at present is virtually unsolved. I feel that the situation may improve with a proper understanding of the probable behavior of photons transferring through an atmosphere. Acknowledgement-This 25223.
research was supported by the National Science Foundation
Contract Number GP-
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