i0. V. G. Gurzadyan and A. A. Kocharyan, Dokl. Akad. Nauk SSSE, 289, 60 (1986)~ ii. V. I. Arnol'd, Catastrophe Theory [in Russian], Izd. MGU, Moscow (].983). 12. R. Gilmore, Catastrophe Theory for Scientists and Engineers, New York (1981)o 13. H. Poincar~, Selected Works [Russian translations], Nauka, Moscow (i97!). 14. B. Lindblad, Mon. Not. R. Astron. Soc., 94, 231 (1934). 15. V. G. Gurzadyan, Astron. Astrophys., i14, 71 (1982). 16. A. N. Tikhonov and V. Ya. Arsenin, Methods of Solving Improperly Posed Problems [in

Russian], Nauka, Moscow (1979). 17. J. M. T. Thompson, Philos. Trans. R. Soc. London, Sero A: 292, 1 (1979). 18. J. M. T. Thompson, Instabilities and Catastrophes in Science and Engineering (1982)o 19. A. A. Samarskii, S. P. Kurdyumov, T. S. Akhromeeva, and Go G. Malinetskii, Vestn.

Akad. Nauk SSSR, No. 9, 64 (1985). 20. A. P. Lightman and S. L. Shapiro, Rev. Mod. Phys., 50, 437 (1978). 21. J. Katz, Mon. Not. R. Astron. Soe., 189, 817 (1979). 22. J. Katz, Mon. Not. R. Astron. Soc., 190, 497 (1980).

REVIEW ARTICLE

FORMATION OF SPECTRAL LINES WHEN THERE IS PARTIAL

FREQUENCY REDISTRIBUTION

D. I. Nagirner

The review consists of the following sections: I) Introduction. Z) Redistribution Functions. 3) Transfer Equations, their Consequences, and Methods of Solution. 4) Radiation Fields. Model Problems. 5) Global Characteristics. Comparison of Different Forms of Scattering. 6) Partial Redistribution in Moving and Nonplanar Media. 7) Applica- tions of the Theory of Partial Frequency Redistribution.

i. Introduction

The theory of radiative transfer in a line with allowance for frequency redistribu- tion has a history of about 40 years. Although the first redistribution functions were obtained already in the thirties and forties, their complexity ruled out the construc- tion of a theory of multiple scattering in a line. A powerful stimulus fo the develop- ment of this theory was the introduction in the forties of the assumption of complete frequency redistribution (CFR) in each scattering event. This assumption, in accordance with which the profiles of the emission and absorption in the line are the same, made it possible to create an analytic theory of multiple scattering, which is presented, for example, in the books of Sobolev [i] and Ivanov [2] (see also the review [3]) and, fol- lowing the development of computers, to create numerical methods and programs for solv~ ing spectral line formation problems (see the books of Mihalas [4] and Athay [5] and the review of Hummer and Rybicki [6]). In the framework of the CFR approximation, solution of (exact, asymptotic, approximate, or numerical) model problems made it possible to investigate many nontrivial problems of multiple scatter in K in lines; important applications were also made.

At the beginning of the seventies, it became clear that the assumption of CFR is not sufficiently accurate for strong resonance lines. In the wings of such lines, the scattering is nearly monochromatic, and this leads to a deviation of the line profiles from those calculated in accordance with the CFR theory; in particular, it is not pos, sible to explain the change of the profiles over the disk of the star. Since then, and especially recently, great successes have been achieved in both the numerical and the analytic investigation of scattering in a line with frequency redistribution different from CFR and known as partial frequency redistribution (PFR) o

No monographs have yet been devoted special!yto partial frequency redistribution.

LeningradState University. Translated from Astrofizika, Vol, 26, No. !, pp. !57- 195, January-February, 1987.

90 0571-7132/87/2601-0090512.50 O 198.7 Plenum Publishing Corporation

Individual questions have been discussed in the books of Athay [5] and especially Mihalas and the reviews [7-9, 3]. Summaries of the development of the theory of radiative transfer in a line in the case of PFR are given in the proceedings [10] of a conference (Trieste, 1984) that to a considerable degree was devoted to this theory. However, many investiga- tions have not yet found a reflection in the books, reviews, and conference proceedings mentioned above. In addition, there are no reviews in Russian, and it was this that stimulated the writing of the present review.

We begin with the notation to be used throughout the complete paper. Much of the notation is standard.

Besides the ordinary frequency v we use the dimensionless x = (v -- v0)/A~D, where v0 is the central frequency and Av D is the Doppler width of the line. By nl and n 2 we denote the concentrations of the atoms in the lower and upper states of the considered line, and by gl and g~ the statistical weights of these states. The Einstein coefficients are denoted by A21 , B21 , and B12, and the coefficients of the probabilities of collisional transitions (with inclusion of the electron concentration) by C21 and C12.

The absorption coefficient in the line in the system of the atom, f(x), is normalized to make the integral of it over all x equal to I. The macroscopic absorption coefficient, calculated per absorbing atom, is represented in the form k(v) = k(~0)~(x)/A = hv0B12~(x)/ 4~AVD, where

+ (x) : j ~ (x -- nv) ['1 (v) d~v ( 1 )

is the profile averaged over the distribution of the absorbing atoms with respect to the velocities v measured in units of the thermal (most probable) velocity of the atoms: v t = (2kTk/M) .u= (T k is the kinetic temperature, and M is the mass of the atom); A = #(0), and n is the direction of propagation of the photon. Usually, the Maxwell dis- tribution FI(v) = ~-3/2 exp(_v 2) is taken.

We use the optical depth averaged over the line, ~; in a homogeneous medium, �9 = ~0/A, where T o is the depth at the line center; T = T0/A is the optical thickness of the medium. The absorption in the continuum is characterized by the coefficient kc, which does not depend on the frequency within the line, and by the ratio 8 = Akc/k(v~)n I = ABe. Other notation will be explained as it appears.

2. Redistribution Functions

i) General Relations. Excellent reviews of redistribution functions with deriva- tion of formulas can be found in Hummer's [7] and the books [4, 5]~ We give here a brief summary, adding results that are not included in the quoted works.

We denote by r(x, x') the redistribution function in the coordinate system asso- ciated with the atom. Here, x and x' are the frequencies of the photon before and after scattering, respectively. The probability that a photon of frequency x" is absorbed and after re-emission has frequency between x and x + dx is r(x, x')dx, so that the integral over all x is equal to f(x'). The redistribution function in the coordinate system asso- ciated with the gas is obtained by averaging over the distribution of the absorbing atoms with respect to the velocities, this distribution generally being taken to be Maxwellian:

R ( x , x ' , ~) = t~r (x - -nv , x ' - - n ' v ) Fl(v)d3v.

Here, n and n' are the unit vectors of the directions of the momenta of the photons with frequencies x and x' (i.e., after and before scattering), and 7 is the scattering angle, cos ~ ~ nn'.

Under the assumption that the scattering in the system of the atom is isotropic, averaging of R(x, x', 7) over the scattering angle X gives the redistribution function RA(X, x'). If we assume dipolar scattering with phase function X(X) = XD(~) = 3(1 + cos 2 y)/4 (instead of X(~) = i for isotropic scattering), then as a result of averaging the product of the phase function and R(x, x', 7) we obtain a function that we denote by RB(X, x'). It is obvious that

(2)

91

t R(x, x') = (x'). (3) dx

The equations r(x, x') = r(x', x) and R(x, x', 7) = R(x', x, 7) express the reciprocity property of optical phenomena. The CFR law is r We now consider special cases of redistribution functions.

2) The Redistribution Function for Radiative Line Broadening. We give the standard classification of redistribution functions [7, ii].

a) Infinitely narrow line and monochromatic scattering <or com~ plete redistribution, which is here equivalent) in the system of the atom, Then f(x)=g(x), r(x, x')=g(x--x')f(x')=/(x)/(x'), ~(x)==-~'2exp(--x 2) (Doppler profile),

RI (x, x', 7) = (1/= sin 4) exp [-- (x ~ -~ x 'e - 2xx' cos 7 )/sin~ 7]. ( 4 )

Formula (4) was obtained by Thomas [12]; the expression for RIA(X, x')~ by Unno [131; and the expression for RIB(X, x'), by Field [14].

In [15], R I is represented by the expansion

R,(~, ~', ~) = ~] cos" ~,,(~)~,,(~'), (5) n ~ O

where %q(x) = exp(--x2)Hn(x)(2n~n!) -z12 and Hn(x) are Hermite polynomials~ The powers of cos 7 can be expanded in the Legendre polynomials Pn(cOs ~), and we then obtain

Ri(x, x', ~ ) = ~ ( n + 1 /9 )P . ( c o s T ) R . ( x , x'). (6 )

One can u se t h e c o m p o s i t i o n t h e o r e m f o r Pn and r e p r e s e n t R I in t h e form o f a s e r i e s in cos m?, where m = 0 , 1, 2 ..... ~ i s t h e a z i m u t h , and a s s o c i a t e d Leg en d re f u n c t i o n s . Such an e x p a n s i o n was o b t a i n e d in [16] by F o u r i e r t r a n s f o r m a t i o n o f R I w i t h r e s p e c t t o t h e two v a r i a b l e s x and x ' ; i t was a l s o shown t h e r e t h a t

R. ~x, x') = 2~ -lj2 ~ e-y" P. (x/y) P. (x'/y) dy =

X m

F (n/2 + m + 1/2) P(n/2 + m + l) ~=o m! r ( . + m +31~) %~ ~" (~)~ .+2,. (=) , (7)

where xM = max([x[, Ix'l). An expansion of RIA(X, x') = R(x, x')12 was obtained in [17]o By means of this expansion it was shown in [18] that the successive convolutions of this function converge rapidly to ~0(x)~0(x') = r162 i.e., to the fiFR law,

b) Damping profile and monochromatic scattering in the system of the atom, i.e., f(x)=fz(x, a)=(a/~)(x~@a2) -t, rlt(x, x')=~(x--x')f(x'), ~(x~= U(a, x) is the Voigt function. Then

RI! (x, x' , 4) = (1/~112 sin 4) exp [-- uel~in = (4/2)] U(a/cos (7/2), s/cos (7/2)), (g )

where u = (x -- x ' ) / 2 , s = (x + x ' ) / 2 . The e x p r e s s i o n (8) was o b t a i n e d by L e v i c h [191 and Henyey [20]. From it we obtain directly the expression found in [21, 22]:

0

Nikogosyan [23], using the connection that he found between RII and RI,

Rtl(x, x', ~ ) = i /L(t ' a ) R ~ ( x § t, x' ~ t , ' i ) d t , (10) )

expanded Rii(x , x', 7) in powers of cos 7 and expressed the coefficients in terms of the functions c~n(x) and

92

~ . ( x , u ) = fL(t ' a ) ~ . ( x + t ) d t = ( _ l ) . ( 2 . . ! ) - - 1 / 2 d~U(a, x). (ii) dx"

The f u n c t i o n s ~n(X, a) have been s t u d i e d by va r i ous au thor s ( s ee , fo r example, [7, 24] ) .

In [25], RIIA(X, x') was expanded in powers of the Voigt parameter a. Procedures for the numerical calculation of RII A were proposed in [26, 27]. For the calculation of RIIA, cubic splines were used separately in [28] in the core and wing of the line. Also given in [28] are matrices of the coefficients for calculating integrals containing RII A. An expression for RIIB(X, x') was obtained in ~7].

c) Dispersion profile and complete redistribution in the sys- t e m o f t h e a t o m: f(x)=/L (x, u), rill(x, x') =/(x)~x'), '~(x) = U(a, x). This case was considered for the first time in [7], where Hummer obtained

.Rlll(X, x t, 7) = o. " i e-U'-dY ~3/2 sin 7 (y - - x ) 2 + a2

--oo t he averaged r e d i s t r i b u t i o n f u n c t i o n

J/~ll, A (X, X ' ) = ' g - - 5 / 2 f e - 9 ' d Y l a r c t g X + Y - - a

o

sin ~ s in ~ t g 7

a G .1

and an expansion of RIIIA(X, x') with respect to an(X, a). In [23], Ri11 was expressed in terms of R I and an expansion was found for RIII:

R~H (x, x', "~) = /L (t, a) dt /L (t', a) d~'R~ (x § t, x' + t', ~') =

oo ~, cos"Ta. (x, a) =. (x'; a).

Methods for calculating RIII were proposed in [27, 29-31].

d) The case when both levels are not infinitesimally thin. Sup- pose the widths of the lower and upper levels are 71 and 72, respectively. Thus, this redistribution function also applies to subordinate lines. Here, the line width is a = 71 + Y2, ~(x) = U(a, x) (in [32] it is incorrectly stated that r is different). The redistribution function in the system of the atom is

(1~)

rv Ix, x ' ) = (~l/a)2/L (x, a~/L (x'. a) [(2~2~1~1) (a + ~) /L (x' --- ~, 2~) +

1] + (~2/2a)/L (~' -- x, 2~1)[i~ (x, ~) +/~ (x', ~/], (15)

A simple d e r i v a t i o n of t h i s f u n c t i o n based on d e t a i l e d ba lance can be found in [33], and a l s o in t h e book [34]. I t s d e r i v a t i o n i s a l so g iven in [35, 11]. As r V, Hummer [7] took a different redistribution function, riv , asymmetric with respect to the the argu- ments x and x'. It was shown in [36] that the correct redistribution function is (15). In the second edition of the book [4] an r V of the form (15) is given and the history of the question is presented. Expressions for RVA are obtained in [31, ii]. In [11], re- lations between r v and R V and other redistribution functions are given.

The redistribution function r V is the most general; all the others are special cases of it [37]. Indeed, for u = 0 it goes over into riI; for Y2 = 0, into riii; and for 71 = 72 = 0, into r I. The same is true of the redistribution function R. For 71 ~ ~2, the first term in (15) can be ignored; we denote the second by r~. To calculate the corresponding redistribution functions RVA and R~A we can use the methods proposed in [29, 31, Ii, 27].

A review and evaluation of methods of calculating the redistribution functions are given in [38].

93

3) Redistribution Under the Influence of Collisions. Although the redistribution functions given above were also used when collisions could not be ignored, this, strictly speaking, is invalid, since these redistribution functions take into account only the radiative damping. Allowance for collisions greatly complicates the redistribUtion func- tions.

The first rigorous treatment of photon redistribution in the case of scattering in a line under the influence of collisions was given in [36] by Omont et al. Using the formalism of quantum electrodynamics and quantum statistics in the impact approximation, they obtained an expression for the redistribution function with allowance forlinelastic and elastic collisions, rc(X, x'), in the form of a combination of dispersion profiles with frequency shifts. When the collisional processes are ignored, their r e goes over into r V. It was shown subsequently [39, 40] that in their derivation of r c these authors took into account incorrectly the inelastic collisions, and their conclusion that there is no frequency redistribution due to quenching collisions is incorrect~ In [40] the redistribution function for zero radiative width, ~l = 0, and no coliisional broadening of the lower level of the line was found to be

r (x, x') ~ [ ( ! - - b) ~ (x-- x ' ) + bf~(x-- A, a) ]A(x ~ - ~, a). (16)

General expressions were also found for the shift ~ and line width a = ~z + ~I + ~E, where 71 and YE are the widths of the upper level due to inelastic and elastic collisions. The value of b depends on which levels can enter into collisional interactions. If the lower level is also broadened, then the redistribution function is much more complicated. It was shown in [41] that collisions have the consequence that the angular dependence of the scattering is no longer necessarily a dipole dependence. S~miiar results were ob ~, tained in [42].

Attempts to derive redistribution functions in the presence of collisions were made long ago (see [43-46]). However, only the modern studies can pretend to rigor.

After the averaging of (16) over the Maxwellian velocity distribution we obtain the redistribution function (i - b)Rii + bRii I, where RII and RIII are taken with shifts. Thus, the redistribution function RIII introduced in [7] found here an application.

In [47], an attempt was made to derive a general expression for the redistribution function in the framework of a unified theory that takes into account simultaneously collisional and quasistatic broadening by means of certain heuristic arguments. The expression obtained is very complicated, and it is only in the special case when the broadening of the lower level of all types can be ignored that one obtains a function of the form (16) with Lorentz profiles replaced by Stark profiles calculated in accordance with the same unified theory [48].

The studies of [40, 47] were continued by the series of papers [49~ 50i, which developed a general theory of redistribution functions in the framework of a unified theory of line broadening by ions and electrons with allowance for the possible degen- eracy of the levels and the overlapping of the lines, and also polarization of the photons and all virtual processes important in the scattering. In [51]~ redistribution in the line L~ under the conditions of the solar chromosphere was considered~ Various approx- imate expressions were given for the redistribution function for the cases when x (or x ~ ) corresponds to the line wing, the redistribution function is averaged over the~angles in the system of the atom, etc. In [52], the redistribution functions from [39] were averaged over the velocity distribution of the absorbing atoms and the angles. A redis- tribution function averaged over the angle and represented as a linear combination of RIIIA(X , x') -- U(a, x)U(a, x') and RII A -- RIIIA was obtained in [53] for the case when degeneracy can occur only with respect to the magnetic quantum number.

In [42], Heinzel and Hubeny noted that the decay and population of levels by in- elastic collisions are taken into account in astrophysical investigations by introduc~ ing a proton survival probability and corresponding terms in the stationarity equations. Therefore, as redistribution function in the presence of collisions they took the func- tion obtained in [36] only for elastic collisions (in the impact approximation) and ex- pressed it in terms of collisionless redistribution functions: r e = (I -- b)r V § brll I where the frequencies of the redistribution functions are taken with shifts and their parameters contain the collisional widths.

94

To conclude this subsection, we mention one further redistribution function r c. Basko [54] showed that for ions with large nuclear charge Z collisions do not give rise to an additional broadening of the profile but influence the frequency redistribution, since the width due to the most important mechanism of collisional broadening - the Stark effect - is proportional to Z -s, while the radiative width is Z 2. The collisions limit the duration of the scattering process, and the redistribution function has the form of the second term in (15), i.e., r~, where Yl is replaced by ~p/2, the width due to collisions with protons, and 7p ~ a.

Frequency redistribution under the influence of collisions is considered in detail in the review [55].

4) The Effect of Recoil. Other Redistribution Functions. None of the redistribu- tion functions we have so far considered reflect the effect of recoil, i.e., the decrease in the frequency of the photon due to transition of its energy into kinetic energy of the moving atom (analogous to the Compton shift). Recoil was taken into account for the first time in [14]. In [56] it was shown qualitatively that recoil must be taken into account for optical thicknesses T > 5.6"10 I~ of the medium. In [57], a nonstationary equation of photon diffusion was derived with allowance for recoil for redistribution corresponding to RII, and it was shown that the recoil has an effect at times t > 4(0) (hv0)~/(2kTkMc2)2acnlk(90). For the line L~, this corresponds to an optical thickness T ~ i014 of a hydrogen plasma cloud.

We mention some other redistribution functions derived by different authors. The redistribution function R x with allowance for jumps of photons from one line of a set of lines with a common upper level to another (cross redistribution) was obtained in [58].

In [26], redistribution functions for the cases I, II, and III were derived under the assumption that the absorption coefficient f(x') and the conditional probability r(x, x~)/f(x ' ) of re-emission are averaged independently over the Maxwellian distribu- tions of the atoms in the ground and excited states, respectively, the redistribution function being then obtained as the products of the results of the averaging. The un- satisfactory nature of these redistribution functions was noted in [6]. We note, in passing, that in [26] there is a list of properties of RI, RII, and RIII and methods of calculating them.

Redistribution functions were introduced in [59] by averaging r(x -- n~, x'--n'v') over the velocities of the atoms at the absorption, 9, and emission ~', of a photon with weight function ~z(9, ~'). Two forms of velocity correlation were considered: complete, when~Z(v, v')=~(~--v')~-3J2exp(--v ~) and the matter reduces to (2), and the absence of cor- relations, ~7(v, 9') = ~-3exp(--v ~- v'2). Then in cases I and Ill we obtain CFR, and in case II the redistribution function Rii(x, x', ~/2), which has properties close to those

of RII A.

In [37], Hubeny considered the general two-photon process of transitions between three levels broadened by damping as a result of radiation and elastic collisions. The relative position of the levels was arbitrary. The corresponding probabilities of transi- tions in the atom, which depend on the frequencies of the two photons, are expressed in terms of a single function, which contains the widths of all three lines. If the initial and final levels are the same and lie below the intermediate level, then the ordinary re- distribution function is obtained. In [60, 61], this approach was generalized to the case when the total number of transitions involved in the process is more than two.

One further variant of redistribution function was obtained in [62], in which it was assumed that the gas consists of turbulent elements, in each of which CFR occurs, their velocities having a Maxwell distribution.

5) Approximate Representations of Redistribution Functions. Besides the exact ex- pressions and expansions for the redistribution functions, frequent use is made of various asymptotic and approximate expressions, which, as a rule, are valid in the wings of the line.

For Ri(x , x', y) as [x[, Ix'[ + m, ]x/x'[ ~ 1 we obtain the representation [63]

~i (x, x', ";) .-~ R~ (x. x') ~ (1 -- cos ~. sign (xxt)), (17)

95

as 1 0 _ 1 / 2 --I q where RIA (x, x') ~ tz., x~) exp (-- x3~ ) as by a f ac to r 3/2. totic value of R IB is greater than RIA

Several different asymptotic expressions are known for RII. it was found in [54] that

where

is the asymptotic behavior of RIA k2~.] The asymp-

For a << i and I is ~ I

RIIA (X, X')'"RIA (X, X')"~ J~,~(x, X/), (18)

I~] oo

x.,_ 2o .... . R~(x,

o 1~,1

.(19)

Here, the second equality is true for Is[ m 1 [21]. For JX~[x !x'l < XD' where x~ exp (-X6) = a~-li2, we canreplace RII A by RIA, while for [xl '[ > x D we can take the func- tion R a (as in the case of the Voigt profile, one can change the Doppler and Lorentz pro- files). At large x, x' and as lul + ~ the function R a decreases exponentially. There- fore, if Ixl m i, the substitution

i a a O ~1 ~ j ' . , (,7 <2o)

which was first used by Harrington [64] although the idea can already be found in [65], is possible.

The approximate form

R.A(x, ~') = b(x) U(a, x)~(~--~') + [ 1 - - a ( ~ , ~')] Li(a, x) U(a, x') (21)

was proposed in [66] by Jefferies and White, who took b(x) = a(x, x') = a(x). In this form, formula (21) was criticized in [67], where it was proposed to take a(x, x') = a(x) = i - exp[--(x/2 - i) =] for Ixl > Ix'l, a(x, x') = a(x') for Ixl < !x'l, = = 2 and

a

b(~) = ! a(x, x ')U(a, x')dx'.

Such an approximation does not destroy the symmetry and normalization properties~ It gives good agreement with the exact values of RII A and is frequently used in calculations It was noted in [68] that it is better to take = = 0.8. The form of [66] is still used (see, for example, [69]). A representation of RIIA intermediate between (18) and (21) was used in [70]: (18) for

R~= -- ga~-Jl--2xe--~" jey~dyj ~ ( x . x ' ) . 0

Finally, in [71], on the basis of calculations made there, it was proposed to use formula (18) but in the expression (19) for R a to replace s in the denominator by s+ = max(S, 3/2 - min(i/2, 13/2 + log al)'sign(3/2 + log a)).

The function Rill(X , x', ~) is very close to the first term of its expansion (14), i.e., to the redistribution function for CFR, and asymptotically, in the line wings, agrees with it [72]. Therefore, one frequently takes Rlll(X, x', ~) U(a, x)U(a, x').

The function RVA(X, x')/U(a, x) has two maxima [33]: at x' = x and x' = 0. In [34], this ratio was replaced by a sum of two 6 functions. This redistribution function was also approximated as follows [63]:

RvA (~, ~') = (l/a) U (a, x) [~U (a, ~') + ~ (~ - - x')l, (22) It was noted in [73] that good results are given by the representation of RVA in the form (21) with a(x) and b(x) acquiring a factor 72/a. At the same time~ the exponential can be dropped from a(x), i.e., this function can be taken to be a step function [74]~

It was noted in [54] that after averaging of r~ over the velocities of the atoms the redistribution function RIIA(X, x') (71 ~ a) is obtained in the core and the near wing, while in the far wing, where lul ~ I, the averaging over the rapidly decreasing Gaussian

96

function does not change the slowly varying function r~ (for the same reason, U(a, x) is asymptotically equal to the Lorentz profile fL(x, a)).

Finally, as redistribution function R c wide use is made (see, for example, [4, 69]) of the expression (i -- b)RllA(X, x') + bU(a, x)U(a, x'), where b = 7E/(TR + 71 + u and YR, YE, and YI are the line widths due to radiation and elastic and inelastic col- lisions, respectively.

3. Transfer Equations t their Consequences t and

Methods of Solution

i) Transfer Equations. We derive the equations that describe the multiple scatter- ing of radiation in a line when there is partial frequency redistribution in the simplest case of a strictly two-level atom. For simplicity, we assume that the redistribution function does not depend on the directions. The stationary transfer equation for the radiation intensity I in a plane layer has the form

hv~ n.'~ ( x ) [ A o t ' Boj], (23) ~0~0I _ k(~)nJ+ 4 ~ A ~ - - - - - - - D - - " - ~ - "

where z is the geometrical depth, ~ is the cosine of the angle between the direction n of the photon and the z axis, ~(x) is the radiation profile, and the second term in the square bracket takes into account stimulated radiation. A characteristic feature of PFR is that ~(x) (this function may also depend on the coordinates) is not identical to the absorption coefficient profile ~(x), i.e., it is not given in advance but must be determined from the equations (see below). We denote q(x) = ~(x)/~(x).

The optical depth T with allowance for stimulated emission, which leads to a decrease in the depth, i.e., to the medium becoming more transparent, is introduced by the equa- tion

d:(=, x) = k (v0) dz[n, -- n~g:q (x)/g2]/A, (24)

and the source function by

5 ( - , =) = %q (x)t[g.~.llg,,,.~-- q (=)] . ( 2 5 )

Then the transfer equation can be rewritten in the form

Fd[/d: = - - qb {x) [ 1 ( ' : , . , x ) - - S(':, x)]. ( 2 6 )

The transfer equation is used in this form by all authors. It must be solved simul- taneously with the equation of statistical equilibrium,

n, ( B12.],, -4:- Cn) = n~. ( A a + ]Pa.]. 4- Co,), ( 2 7 )

where

L+ 0

i n which J ( x ) i s t h e i n t e n s i t y a v e r a g e d o v e r t h e d i r e c t i o n s . To d e t e r m i n e t h e e m i s s i o n p r o f i l e , an a d d i t i o n a l r e l a t i o n i s r e q u i r e d , T h r e e methods o f i n t r o d u c i n g t h i s f u n c - t i o n have been proposed.

A. Thomas's method [75] reduces to the formula

G.-~ (x) + BI= [R (x, x') J (x') & '

+, (x) = G=~ B,=J= (28)

(The f u n c t i o n ~ ( x ) was a l s o d e t e r m i n e d in [76] w i t h r e p l a c e m e n t in t h e s t i m u l a t e d emis - s i o n o f ~ ( x ) by ~ ( x ) . ) Then t h e s o u r c e f u n c t i o n can be e x p r e s s e d in t e r m s o f t h e r a d i a - t i o n intensity as follows:

3(% x) = ~, (x) / R (x, x') J (=5 dx'/+ (x) + (1 -- ~o) B~ (~)/~0, (29)

where B is Planck's function,

), (x) = [1/)0 q- E(x)/%] -~, (30)

97

).o ---- A~.x [A2~ ,-k C o~ (1 - - exp ( - - hvo/k Tk))]-~,.

E (x) = (C~./B~.)[1 -- q (x)] 4- J - q( x) j...

(31)

(32)

It is readily noted that the integral over x of the function ~ defined in this manner is automatically equal to I.

A consequence of (27) and (28) is the equation

,,, c~~ (~) + B~.. R (~. x ') . /(~') a~' - ,,~.~ (x) [A~ + BM, + C~. ( 33 )

Here, we do not indicate the dependences of J(x), ~(x), q(x), %(x), and E(x) on the depth~ In the general case, when the redistribution function is not averaged over the angles, the integral in (33) is replaced by the triple

�9 ~ R (x, x', 7) l('%. rt, x') dx'd2n/4=,,

and ~, q, l, and E also depend on the direction. For CFR R(x, x') = ~(x)#(x~)~ and then ~(x) = ~(x), q(x) = i, E(x) = 9, I = I 0, ~, and S do not depend on the frequency or direction. A scheme of iterative calculation of ~(x) for a two-level atom with continuum was proposed in [77].

B. A different definition of ~ was given by Oxenius [78]. He took into account the possible difference between the velocity distribution ~(~v) of the excited atoms and the Maxwell distribution. The function ~(x) was defined as the emission profile averaged over F 2 in the system of the atom (as above, we omit the dependence on n): ~(x)=

~F~(~) '~(x, v) dSv, where

I'd2 n C,2/(x -- g , ) + B12[rj (x -- ~w, x ' - - n'~) J (x ~) dx'd~n'/4= (~, ~) ( 34 )

4= C12 -~ BI~ ~/(x' -- ,'v) J (x') d2'd~n'/4=

It follows from (34), as noted in [78], that in the case of CFR in the system of the atom~ r(x, x') = f(x)f(x'), and a Maxwellian F~(~) we obtain r = ~ i.e., CFR in the system associated with the gas. For the function F2(~ ) in the stationary case we obtain the equa ~ tion

nxF~ (v) [ Cx2 + Ba2~f (x--nv) j (x) dxd~n/4~l =

I f (35) i s i n t e g r a t e d o v e r t h e v e l o c i t i e s , we a r r i v e a t ( 2 7 ) . T h i s q u e s t i o n was con ~ s i d e r e d in more d e t a i l in [79] w i t h a l l o w a n c e f o r t h e d e p e n d e n c e o f ~ on t h e d i r e c t i o n o f t h e p h o t o n ' s momentum and f o r d e g e n e r a t e l e v e l s . A k i n e t i c e q u a t i o n f o r F2(~) w i t h allowance for the change in the coefficients due to the fact that F 2 differs from the Maxwellian distribution was derived in [80]. In [81], the effect of diffusion of the excited two-level atoms on the radiation field in the line was estimated for the redis- tribution function RIA; in principle, this effect can be appreciable. Kinetic equa- tions for the velocity distributions of multilevel atoms in different states were de- rived in [60].

Calculations of the functions F2(v) have not yet been made. It was shown in [18] that after the first excitation of the atoms for the redistribution function RIA by mono- chromatic radiation with frequency x 0 within the line (graphs for x 0 = 0 and 1 are given) a strongly non-Maxwellian velocity distribution of the atoms is produced. How- ever, it rapidly relaxes with increasing number N of scatterings to the Maxwell distribu- tion, so that already for N = 2 the differences are small.

C. A third definition of ~ has been used by Mihalas and his collaborators [4j 82~ 83]. Using the fact that n2~(x)dx is the number of excited atoms that emit photons with frequencies from x to x + dx, they derived the equation of statistical equilibrium

98

I f ] nl C12+(x)+ Bl~ R(x, x ' )J (x ' )dx ' = n2~(x)[C21+ A: ,+ B~J(x)], (36)

which can s e r v e f o r t h e d e f i n i t i o n of ~ ( x ) . The same e q u a t i o n was o b t a i n e d in [84] . Equa t ion (36) does no t e n s u r e n o r m a l i z a t i o n of ~(x) and i t must be so lved s i m u l t a n e o u s l y with (27).

It was subsequently recognized in [85] that this method of defining ~(x) is incor- rect, although in many cases the differences in the final results obtained by adopting definitions A and C are small. The same conclusions were drawn in [79]. It was noted in [55, 86] that for a consistent formulation of the transfer equation and the equation of statistical equilibrium of the level the stimulated emission profile in these equa- tions must be chosen in different forms.

In the calculations in the majority of studies stimulated emission is treated as negative absorption, i.e., the profile ~(x) is replaced for the term B2xl in the bracket in (23) by r Then the optical depth x defined by Eq. (24) ceases to depend on the frequency, since q(x) in (24), as in the denominator of the source function (25), is re- placed by i. Such an assumption strongly simplifies the calculation of the radiation field in the line. Arguments for its use are given in [55, 77, 87]. A quantitative justification of this assumption is given in [88].

It is obvious that allowance for the stimulated emission makes the problem nonlinear, since the scale of the optical depth and, therefore, the optical thickness and the man- ner in which the source power varies with T, like the quantities X(x), q(x), etc., depend on the previously unknown radiation field.

It was shown in [89] that, as in the case of CFR, the nonlinear problem of calculat- ing the radiation field in a homogeneous planar medium reduces to a linear problem when the stimulated emission is treated as negative absorption.

For completeness, we note that the integral equation for the emissivity of an atom (with a definite velocity), which also depends on the frequency and depth in the plane layer, is given without derivation in [90] for the redistribution function Rll(X, x', 7) without allowance for stimulated processes.

2) EnerKy Balance. We now consider a different situation, in which allowance is made for absorption and emission in the continuum and a dependence of the redistribution function on the angles but stimulated processes are ignored. Then the equation of radiative transfer has the form

~d~d~: - - [~(x ) +p][(~, n, x ) + ~ ( x ) S ( : , n, x). (37)

The source function S is determined by formula (25), in which the term with q(x) in the denominator is ignored, and it is related to the intensity as follows (cos7 = n-~'):

S(% n, x) = S0(z, n, x)-F O/4=*(x)) dx' d2n'R(x, x', ~) I{:, n', x'). (38)

The validity of the assumptions that lead to Eq. (38) was discussed in [6, 53, 91].

Suppose optical depths 0 and T > 0 correspond to the boundaries of the plane layer, that no diffuse radiation is incident on these boundaries, and that the possible ex- ternal sources are included in S o . Then the boundary conditions for the intensity are [(0, n. x)=0 for ~ > 0 and 1(~ ,, x) =0 for ~ < 0.

From Eqs. (37) and (38) it is possible to obtain an integral equation for the source function:

S(=, n, x )= S0( : , . , x ) + > dx' �9 o

T

d~-n' Id*:'~ (x') 0

If S o does not depend on n. and the redistribution function is replaced by the function averaged over the angles, the resulting equation is somewhat simpler:

(39)

99

z'

,}

s(=', x,) R(x, x')E1(I =--='I[~ ~x') + :q):~ (x~ <4o)

We introduce the following quantities: E 0 is the total energy emitted by the primary sources (during unit time in a column of unit cross section) in the plane layer, E R is the total energy emitted in the layer with inclusion of the energy accumulated in multiple scatterings, E L is the energy of photons that perish in the line, i.e., are not re-emitted and are transformed into heat, E c is the energy absorbed in the continuum from the photons in flight outside the scattering, and, finally, E e is the energy that emerges from the layer. These quantities are fourfold integrals (over ~ from 0 s T, over x from --~ to +~, and over the directions n) of the products ~(x)S0, ~(x)S, (i - X)~(x)I, ~I, and ~dl/d~, respectively. It follows from (39) that E R = E 0 + ~EL/(i "- X), and from the transfer equation the energy balance equation E 0 = E L + E c + E e follows. We shall need the quantities we have introduced later.

3) Invariance Principles. These principles, known also as the layer addition method, were introduced into the theory by Ambartsumyan and developed by Chandrasekhar (see [i~ 2]). The invariance principles are frequently used in conjunction with Sobolev's escape- probability method, which gives many relations, including the basic integral equation(39) for the source function, a probability meaning [I]. These methods have great generality and can be applied to many problems of the theory, especially linear problems and, in par- ticular, line formation problems in the case of PFR.

Equations for the probability of photon escape and the coefficients of reflection and transmission by a one-dimensional medium (two-flux approximation) for arbitrary redistribution function R(x, x') were obtained by Sobolev [92]. Systems of integrodif- ferential (with derivatives with respect to the optical depth) equations for theplane layer reflection and transmission functions were also derived for any redistribution func- tion R(x, x', 7) in [93].

The Byurakan astrophysicists have made wide use of the invariance principles and probability methods. They have obtained various forms of equation for the probability of photon escape from a medium, for the reflection and transmission coefficiens and also the Green's functions [94-100].

In the work of the Byurakan theoreticians [94, 95, 97] separation of the variables of the unknown functions and the reduction of these functions to functions of a smaller number of arguments play an important part.

All these results are valid for general redistribution functions (actually, for R I and RIII), but only for homogeneous media, i.e., for constant parameters A~D, a~ X, and B.

4) Numerical Methods. Methods of numerical solution of transfer theory problems, including methods suitable for the calculation of line formation in the case of PFR, are presented in the review [I01] and the books [4, 5]. Therefore, we shall here only mens these methods.

For problems with PFR modifications of the discrete ordinate method have been used [13, 102, 12, 103, 4]. An improved formulation is given in [I04]. In [91], the solutions were expanded with respect to Chebyshev polynomials.

Methods based on equations that are consequences of the invariance principles have been used by the Byurakan school [94, 105-107].

The methods so far discussed are applicable only for homogeneous media. We shall now mention more general methods.

Feautrier's method has been widely used to calculate the radiation fields in the case of PFR. Besides its ordinary formulation, Rybicki's modification has also been used (see, for example, [74]), and also this method in combination with variable Edding- ton factors (for a discussion of all this, see Mihalas's book [4]).

The perturbation method proposed in [108] has proved to be effective. The

100

redistribution function is represented in the form R(x, x ~, Y) = @(x)@(x') + [R(x, x', 7) -- -- @(x)@(x')], and the second term is regarded as a perturbation. The equations are solved iteratively, it being necessary in each stage to solve equations of the same form as for CFR. The core saturation method is adapted for problems with PFR in [109]. Scharmer's effective method, which is a combination of the core saturation and perturbation methods, has recently become popular. It is particularly suitable for the solution of multilevel problems [ii0].

In calculations with allowance for stimulated emission and a solution of multilevel problems, complete linearization of the equations is used [84, iii, 112]. An improved formulation of the method of complete linearization for multilevel problems is given in [68]. A scheme for calculating the radiation fields in the lines of a multilevel system on the basis of the method of equivalent two-level atoms is proposed in [113]. Many calculations have been made by the Monte Carlo method. A modeling of the redistribution functions RI,II,III A is given in [114], and of RI,II,III in [115, 116]. The authors of each paper give their own modeling variant [117, 118]. It is particularly convenient to use the Monte Carlo method to calculate global (integrated) characteristics such as the average number of scatterings and so forth [117]. In [119], a combination of the dis- crete ordinate method in the core of the line with the Monte Carlo method in the wing at x > 2 is used.

Methods of solution of more complicated problems will be discussed in Sec. 6.

4. Radiation Fields. Model Problems

i) Calculations for Homogeneous Media. To investigate the characteristic features of different forms of scattering in the case of PFR the radiation fields and source functions in the line have been calculated in homogeneous (with constant parameters) plane-parallel media. Table i summarizes the information on such calculations, arranged in chronological order. In the table, the following abbreviations are used: MC, the Monte Carlo method; DO, the discrete order method; Chebyshev, expansions with respect to Chebyshev polynomials; i iter., one iteration of the solution for CFR; int. eq., direct discretization of the basic integral equation. In No. I0, the redistribution func- tions R I and RII were multiplied by the Rayleigh phase function 3(1 + cos 2 77)/4. In Nos. i, 8, 9, 14, 17, and 24 the calculation was made for a one-dimensional medium (for- ward--backward scattering). The values of 60 and T o are given in the corresponding col- umns in the brackets. The sources 6(~) denote external illumination of the medium, and in the brackets in this column we give S0@(x), which corresponds to the sources in the continuum.

Almost all the calculations were made after the development of fast computers. Be- fore this occurred, some approximate solutions not given in the table were obtained (for example, [17, 120]). However, they gave only an approximate picture of scattering with PFR.

The most detailed calculations of the source functions and emerging radiation have been made by Hummer [103]; his results are reproduced and discussed in the books [4, 5]. According to [103], the difference between the results for RIA and RIB is maximal in the wing, where it reaches 40%, but it is normally 2-3% of the calculated value. The devia- tion from CFR is greater. However, if in the transfer equation one makes one iteration of the functions calculated for CFR a very good accuracy is obtained. For a > 0, sig- nificant differences appear, namely, for CFR the line profile in the wing approaches the Planck value, but for RII it decreases, i.e., the radiation in the wing is trapped, and monochromatic scattering occurs there.

We shall also say something separately about a number of studies. In [106], Nikoghossian and Yengibarian considered the one-dimensional Schuster problem, i.e., the formation of absorption lines in a one-dimensional medium through which continuum radiation passes. The problem of a steady point source in an infinite medium was con- sidered in [136]. The connection between the intensities of two problems, in one of which the power of the primary sources is the derivative with respect to �9 of the power of the sources in the other, was obtained in [96, 127].

The problem of the emission of an infinite medium (the lower limit of ~' in (39) and (40) equal to --~, T = ~) under the influence of steady uniformly distributed sources

I01

TABLE i. Summary of PFR Calculations for Homogeneous Media~

~" Year a I--). i~(3,,) ]Redistr. fu. T ( T f ) S~(S, 'I)) Method !Literature

7

8

9

10

11

12

13

14

1955

1964

1967

1968

1969

1970

1970

1972

1975

1976

1976

1976

1978

1978

1980

1~8o

1980

1981

1981

1984

1984

1984

1985

1985

0

0

1o-3

10 '~ 4 .3 .10 -4

0 10-3

10 -2

0 8.10 - 4 ' 10 - 2

10 -3

0

0 (), 10 -3

10-3

10 -3

10-3

0, 10 -3

0

0

0 10-3

5.10 - 4

C

2. I0 -3

!0 -3

10 4

2:10 -3

2.10 3

1 . 1 . 1 0 - 3

2.10 . 3

10-i 3

0

0

0

0 10 -6

0

0 10 -4

10 -4

10 -4

0

0 10-4

0

0 .3 10-4

10-4

1 . 5 . 1 0 - 4 ,

5.10 -5

0, 10 -4

0.1, 0 .3

0.02, 0.05

0.1, 0.2, 0 .3

0

5. I0 -6

0 .0 l

0

0 10 -4

0, 10 --3

0 , 1 0 -3

10-4

10-.~ 10 3, 10-6

0.1

0.02, 0.05, 0.1

15

1 6

17

18

19

20

21

22

23

24

0 RI. 1

0 R t , RIA

0 RtltA 0 R . A

0 RII A

0 RIA, RtB

0 g i l a

0 RUA

0 R I 0 R u

0 Rq (x, x ' . ;:/2)

(0.02) RE1

0 R H

0 R ) , R H, R e l .

0 R I I I , R I I I A

0 RIM

0 RIM

0 RIA, Rtt A

0 RIA

0 RIA 0 RIA

0 R ti

(10 -9 , (l--.b)Rll A --

5.10 - u ) - b'l '(x)'I)(x')

(0;01) Rin

Rtf

0 Rip RI! A

0 Rll A

0,10 - 3 RII A, Rl lbl

0, 10- 3 R w ~

0 RvA

0 R VA

0, 10 - a RII

0 R k,l

0 R I A

(lO)

(I0"), n --1(1)3 co

2.10:1),n -3,4,5.61

(2.10 a, 2.10 ~)

10. 10 ~, 10 (, oo

10:, 104, 10 6

10-t

10 a, 10 ~

10 a. 105

DO

(1, 5 10, 15)

c /

10"L ~,

10:, I0 a, eo

oo

(!0"), n = 0(1)4

(2.5), ~,

(10"), n - - 0 ( 1 ) 6

oo

b - 1 0 - 1 , 10 - 2 , 1

,~. o n e - d i m .

( I0" ) . n=- -1 ,0 ,2 ,5

1, 10 "~, 10 4

(10 a)

156

156

104, co

10L oa

oo

0.2 (0.2) 1.2.5,

I, one-dim.

I

1

(6(':0--" To. 2))

(~(=o-- T,).2))

1

1

I

(~ (:)~(:,.- 1))

(~ ( : ) ~ ( ~ - 1 ) )

I

(~(--)), one-dim.

(~,(-.)), one-dls.

1

I 1 _o.o

exp ( - -5 .19 -.4 ~a.s)

(~,i~.)~(~-1))

(6(z)), one-dim.

1, one-dim.

1, one-dim.

(~ (:) ~ (~ -- 1))

1

. , ~- 0[01.0 .1 , 0.5 i

6 (~) ~ (~ -- 1) ~' ( * - * o ) l I

x.----O.1, 1 (1)5

1, ~ (% To/2)

1

1. (~, frO)

1. (~, ( : ) )

I

1

l

(6 ( :)) , one-dim.

1, one-dim.

Int. eq. Chebyshev

I iter.

JMC q-

+DO

DO

Iiter.

DO

[I05]

[122]

Feautrier

Perturb.

Feautriar

MC

[1271

MC

112

113ol MC

MC

Core sat.

1133!

Fsautrier

Rybicki

11151

[22]

172]

i119]

[1o31

i9ol

[Io51

|122{

[1231

| |241

[125l

[126]

i1271

!1281 I129]

113o1 11311

1132]

[lo9]

i134]

[73]

[741

[135!

that radiate isotropically in the continuum was solved in [84]. Analytic solutions of this problem for arbitrary frequency dependence of the source power were obtained for

X = i, 8 > 0 in [14] and for X < i, 8 = 0 in [138].

Calculations have also been made for different cases of inhomogeneity of the medium~

A study was made in [109] of the influence of an optical-depth dependence of A9 D) in [1251: of a and S o , and in [74] of e = 1 -- I and S o (the remaining quantities as in Table 1 under the numbers 20, 12, and 23, respectively). In [139] there was a �9 dependence

of E for the redistribution function RII A with a = 2"10 -a, T o = i0 ~, ~ = 0, S o = i, and S o corresponding to external illumination; a Lambert bottom was also taken into account. In [68], all the parameters were varied in the case of the redistribution function (i --

b)RIi A + bRIIIA-

2) Nonstationary Emission of an Infinite Medium. As we have already said, exact solutions of the transfer equations in the case of PFR can be obtained only in excep- tional cases. The asymptotic behaviors and approximate solutions that we shall discuss

in the following subsections were therefore obtained analytically.

102

TABLE 2 . Asymptotic Behaviors of Xm(t) , I(x, t), N(t), ~(t), and A.

..~(t)

I *~ < ~,.(t) I xl >x,.(t)

N(t)

| :o l<to<T. '-tl. (:/a),'a

(In to) 1/2

(ln ~o)-1:'/2 t o exp (--xZ)/2x

1/2 (In t) 1/2

t/In t

In 4 (1/4a)/4a < t < 571,'e ~

( . t f :4 (2Wat)lf4/T(1/4)

(2=/at)l/%xp ( - =x4/Sat)/F(1/4) 29/2~.1/4t3/4/3~ (1,'4) a 1~

t 3/4 a-- 1/4

t > Yh/a 3

"a:o@ 4: =a/~ (tl~tOIP-/Z

(.h/a)~/2 t

A, ~<< 1 A, ~ << 1

1/~ I . (I/~) 1/~ [In (I/Q] 1:~

2a/=x o < z

a-l/4 :~-3/4 I/~-

a--l/z:' 3/2_ ~ : 3 71 ~'z"-a.'xD

~ /2 a--I/2~--2

z < a - - 112"[31/2

To investigate the effects of PFR in, so to speak, pure form a study was made of the problem of the spreading with time of a line in a conservative infinite medium in the case of a source power that depends only on the time and the frequency and not on the coordi- nates and angles. Then the radiation intensity will also depend only on x and t and is determined by the equation

al/at = - r ( x) I (x, t) + j R ix, x') l (x', t) dx ' (41)

with initial condition I(x, 0) = 6(x). }{ere, t is measured in units of t z = ~(0)[ cnlk(v0), i.e., the mean time taken by a line photon in traveling between two successive scatterings. Another possible mechanism of photon decay, the time they spend in the absorbed state, was not taken into account.

The first exact solution of a problem with PFR was found by Field [14] for the re-

distribution function RIA:

l (x, t) = t f~P (x') exp (-- t(b (x')) dx' + exp ( - - to) r (x) , ( 4 2 ) ,d

where ~(x) = ~-in exp(_x2), and t o = t~ "In. From (42) the large-t asymptotic behaviors given in the first column of Table 2 follow. The frequency Xm(t) separates the regions of the core and wing of the line, and the asymptotic behaviors of I(x, t) are put in the second and third rows.

Basko [54] obtained a solution of (41) for the redistribution function RII A in the asymptotic region of the line wing, where the relation (20) holds, i.e., when (45) goes

over into

OtOl __ a2v~ OxO ( . ~ O_~x )" (43)

This equation is solved by

l (x, l) = (2= /a t ) aI4 exp ( - - ~x'/8al)/F (1/4). ( 4 4 )

The asymptotic behaviors of l(x, t) for this case, and also for the redistribution func- tion rv, found in the same [54], are given in the second and third columns of Table 2.

The solutions of Eq. (41) have a characteristic general property, namely, the pre- sence of a flat part whose extension with respect to the frequency, i.e., Xm(t), increases with the time. The asymptotic behaviors for RIA agree completely with those for the Dop- pler profile and CFR [140], while for r~ they differ only by a factor ~sm/4 from the asymptotic behaviors for CFR with Lorentz profile [54, 141].

The redistribution law R 6 was considered in [54]. As follows from what was said about this redistribution fun6tion in Sec. i, it is close in this case in the core of the line to RIA, in the near wing to RII A z R a, and in the far wing to the unaveraged r~. Therefore, with increasing time the spreading of the line occurs in accordance with the columns of Table 2, passing successively from the first to the third. The first and

103

second regions do not join up. In [54], the case a m 1 was also considered.

In [142] a study was made of the problem of nonstationary emission of an infinite plane homogeneous medium for the redistribution function RIA but for t 2 = 0 (velocity of light infinite) under the assumption that the photons are delayed only by the tin~ that they remain in the absorbed state with average time t z ~ 0. In [138], also for the re- distribution function RIA, the present author found an exact solution to the problem of homogeneous emission of an infinite medium for t 2 = 0, ~ = 0, all X and t I and an arbitrary dependence of the sources on the frequency and time. For flashes (6-function sources) the asymptotic behaviors of the radiation intensity in the line wing and at long times after a flash were obtained.

3) Other Analytic Results. Only a few papers have been devoted to analytic study of PFR in planar media, this being a reflection of the difficulty of such investigations.

For the redistribution function RII A and in Eddington's approximation with respect to the angle, Harrington [64] obtained a diffusion equation for the mean radiation in- tensity in a plane layer using the approximation (20) that he had proposed:

O:" ' 2 ~ x 2 x ~xx = 3r (=-j - - SO). ( 4 5 )

Here, r = U(a, x) on the left was replaced by the asymptotic behavior ~(x) ~ a/~x 2 in the wing, while ~2(x) on the right was replaced by ~(x). By introduction of the new variable

x

= = (~ /3 ) ''2 [d~/U(a, ~) 0

the equation for E ~ i was transformed to

O=J. 4. O~f = 61/2g ( ~ ) ( e j - &) (46)

and was solved by separation of the variables. Solutions were obtained in the form of series in eigenfunctions with respect to ~ for S 0 = 1 and S o = 6(~ - T/2). The series were summed approximately. The frequencies Xma x at which J takes the maximal value were found, and approximate expressions were obtained for J averaged over the frequency and for the mean number of scatterings when T m 1. In the second part of [143] the reflec- tion and transmission functions of a plane layer were obtained in the same approximation.

An approximate solution to the problem of the emission of a plane layer under th~ influence of a source that flashes suddenly at the center of the layer was found in [142], in which the damping times for the redistribution function RIA were also ob ~ tained. The results were compared with experiment (the line X 1048 Ar)o

Van Trigt [16] investigated the eigenvalues Xn of the kernel of Eq. (39) for ~ = 0, T + ~ and redistribution function Ri(x, x', ~). He showed that the leading terms of the asymptotic behaviors of Xn -- 1 in this case and for CFR with Doppler profile are iden- tical.

Frisch [63] made a detailed investigation of the asymptotic behavior of the source functions in a plane infinite medium (Eq. (39) with T = ~ and lower limit in the integral over ~' equal to -~) for almost conservative scattering (~ = i -- % ~ 1), ~ = 0, and S o = S0(z). She used the formalism of the Fourier transform with respect to the optical depth, S(u, x), and a selection of ~-dependent scale factors for the optical depth, the absorption profile r and other quantities in such a way as to obtain finite limits as ~ + 0 in the equations. For CFR one would have R(x, x') = ~(x)r and S(u) = S0(u) [I -- XV(u)] -z where

v iu) = (]/~) t d~+~ (-~) ~-~tg ~/+ (~), (47)

and if u * +0 then I -- V(u) ~ u(ln I/u) -I12 for the Doppler profile and I -- V(u) ~ (~au/ 2) z12 for the Voigt profile. For the derivation of the asymptotic behaviors for PFR the approximate representations of the redistribution function given in Sec. 2 were employed:.

104

For the functions R I and Ri11 one can use the perturbation method, i.e., represent the source functions in the form S(%, x) = S~ + Sz(T, x), where S o does not depend on x and S l is a correction. The functions S o are simply equal to the corresponding func- tions for CFR with Doppler and Voigt profiles, while Sz/S ~ are proportional to I/in(i/g) and E, respectively. The dependence of S z on x for the redistribution function RIA was discussed. Somewhat more complicated is the situation with regard to RVA, which asymp- totically (in accordance with Eq. (22)) is a combination of monochromatic scattering and CFR. In this approximation, the difference

s(~, .0 - [~.~u (a, =)/2.] ~ a~' s (~', ~) E~ ( I ~ - ~' I U(~, ~))

is independent of the frequency. The Fourier transform of S(x, x) with respect to �9 can be found in the explicit form

- ~ (48) F(x, a)= 1--07e/au) U(a, x)arctg(u/U(a, x)).

I n t h e l i m i t u + O, t h e f u n c t i o n i n t h e c u r l y b r a c k e t i s e q u i v a l e n t t o ( ~ i / a ) ( ~ a u / 2 ) u2 + E, w h i c h d i f f e r s f rom CFR by t h e p r e s e n c e o f t h e f i r s t f a c t o r i n t h e f i r s t t e r m -- t h e CFR fraction. Monochromatic scattering does not affect the asymptotic behavior, since the terms corresponding to it decrease more rapidly with the distance.

The biggest difference from CFR is obtained for the redistribution function RII. In [63], an asymptotic equation for S essentially identical to Harrington's equation (45) was derived. We consider separately the cases when the photon is created in the core and in the wing of the line. In the equation the variables are renormalized (scale factors are introduced), but solutions of it are not given. An analogous equation for S for the redistribution function RII A was obtained in a further paper of Frisch [144] for the case when the photons perish only in flight as a result of absorption in the continuum (e = 0,

§ 0), A detailed exposition of these results and their generalization can be found in [145 , 1 4 6 ] .

5.__Global Characteristics. Comparison of Different

Forms of Scattering

I) Mean Number of Scatterings and Mean Photon Path. In this section, we discuss global characteristics of the radiation field in the case of PFR such as the mean number of scatterings N = ER/E 0 (or, in accordance with the definition of [103], <N> = %EL/(1 -- %)E 0 = N -- I), the mean photon path s = Ec/~E0, and the thermalization length. In this subsection, we consider the first two quantities, whose determination has been considered in a number of studies. Note that N and s still have a meaning for % = i, ~ = 0.

Since all the results for the redistribution functions RI, RIII, and R V are rather similar to the analogous results for CFR, in almost all studies the function RII has been considered.

In [147] approximate expressions were obtained for N, which was calculated for dif- ferent T for the redistribution functions RIA and RII A (a = 4o3"10-4), and it was sug- gested that the scattering in the line wing for the function RII has a diffusion nature. This idea was developed in [148], which introduced the effective frequency of scatter- ing in accordance with which the diffusion occurs.

In [119] calculations of N and s were made for the redistribution functions R I and RII not averaged over the angle for T ~ l0 S , S o = i, e = 0, ~ = 0, a = 0, 4.7"10 -4 , and 4.7"10 -3 It was found that for T ~ 102 the value of N is the same as for CFR, while at large T it is proportional to T and not T 2. For a plane layer of optical thickness T 2"107 with a source in the line at the center of the layer and the redistribution func- tion RII A (a = 10 -2 , 4.3"10-4), % = i, ~ = 0 the value of N was also calculated in [117] and compared with the results for CFR. The quantity <N> was calculated for RIIA, g = 10 -4 , T ~ 108 , S o = i, a = 10 -3 , and 10 -2 in [103]. It was found to be greater than in the case of CFR.

105

In [149] Feautrier's method for the function RII A for a = 4.7"10 -3 and 4o7o!0 -~, e = ~ = 0 was used to calculate N and the frequency Xma x at which the radiation flux in the line emerging from a plane layer with sources in the line in the center of the layer takes its maximum value. It was shown that N ~ CT with N independent of a. In [1501, the same calculations were used to determine 2s It was found that 2Z/T is propor- tional to T u3. By direct application of Eddington's method, and also from the ~esuits of [64], it was found that 2s = c(9aT/2~) 11a.

The most detailed calculations of N and s were made in [151] by Feautrier's method. The sources were taken in the center of the layer or were uniformly distributed over the layer and the redistribution function was RII A with a = 4.6"i0 -n, n = I, 2 .... , 5, T from I to 109 , and e = 0. It was shown that for B = 0 and aT > I0 ~ the value of s can be re- presented with very good acuracy in the form Z(T) = C(aT)(aT)U3T, where C(z) is a slowly varying function of order unity that depends on the form of the sources. The ratios N/I tend to constants that agree very well with the ones obtained in [64] for aT e 5.103~

In [118, 152], the Monte Carlo method was used to calculate Z(T, ~) and N(T, ~) for a layer with a source in the middle for the redistribution function RII A with a = 4.7"10 -4.

Using qualitative arguments, Frisch [63] derived the functional dependence of N on T for the redistribution function RIIA, i = i, 8 = 0 for photons created in the line core, N ~ T, the result agreeing with the result of [64] and [151], and in the wing N ~ (aT) u3.

The mean number of photon scatterings in an infinite conservative medium during time t was obtained in [54] from the definition

t

d 0 - - ~

where l(x, t) is the solution of Eq. (41).

The asymptotic behaviors of N(t) at large t in the case of the redistribution func- tions RIA, Re, and r~ are given in Table 2. Basko [54] also found the dependence of the mean photon displacement ~(t) in an infinite conservative medium on the position of its creation in the line center during time t. We have given all the formulas without the numerical factors, which are estimated in [54].

With increasing t, the quantities N(t) and ~(s go over from one asymptotic be- havior to another, the time intervals of their validity being the same as for the cor- responding solutions l(x, t). In [153], the Monte Carlo method was used to calculate N(t) and ~(t) for 71 = 0, a = 0, 0,01, 0,i, I and t from I to 106 for two initial con- ditions: l(x, 0) = 6(x) and l(x, 0) = ~(x). It was noted that the asymptoti c behaviors are valid when at > 103, and the numerical factors were found more accurately. The transition to the asymptotic behavior for the second initial condition when a = I occurs much more slowly.

2) Thermalization Length A

This is the mean distance between the position at which the photon arises and the position at which it leaves the scattering process (thermalization) due to true absorp- tion in the line or in the continuum. An extensive literature (see [2, 61) has been de- voted to its determination, asymptotic behavior, and numerical estimates in the case of CFR. For PFR, the thermalization length was introduced much later.

In [150] the condition that for thermalization length A due to absorption in the continuum the relation s = i must hold led for the redistribution function RII A to the relation A ~ a-U4~ -a1~.

Estimates for the thermalization length for the redistribution function R~A were found in [54] on the basis of the concept of the thermalization time tth. If i = I and the thermalization is determined by absorption in the continuum, then tth is determined by the equation tth'~ = I; if, in contrast, absorption of photons in the line plays the main part, then from the condition N(tth) = I/E. The thermalization length is defined as follows: A = ~(tth), where T(t) is the mean photon displacement during time t. Esti- mates of N(t) and ~(t) have been given above. The dependences of A on ~ for ~ = lland on e = i -- % for ~ = 0 are given in Table 2. The boundaries for ~ are the opposite of

(49)

106

of the boundaries for t, and the boundaries for e are given in the last row of the table; x D is the frequency that separates the Doppler core and the Lorentz wing of the Voigt pro- file.

Exactly the same functional dependences of A on e were obtained in [63], where I/A was defined as the scale factor of the optical depth that leads to a finite limit in the asymptotic equations as ~ + 0. For the redistribution function RIII as for RV, A ~ I/e 2. The concept of the thermalization frequency x c was also introduced; ~(xc)A = i for all redistribution functions except RII, and U(a, xc)A = x c for RII. The estimates of x c for RII agree with the estimates in [64] of the frequencies Xma x ~ (aT) u3 at which the mean intensities J of the radiation emerging from a plane layer with a central source in the line have a maximum and with the estimates [149] of the photon diffusion frequency x.~ if A in x c is replaced by T. In [63] it is also shown that if for the redistribution func- tion RII a photon is produced in the line wing then A ~ a'ie -3/2 and x c ~ e-u2. And in the case % = i, ~ ~ 1 an expression was obtained for A in [144] that agrees with the asymptotic behaviors of [54], while the thermalization frequency was found to be x c ~ a l / 4 1 ~ - ] /~ �9

3) Comparison of Scattering in The Case of PFR and CFR. Let us draw some conclu- sions. As follows from the numerical calculations, multiple scattering in the case of angle-dependent redistribution functions takes place in almost the same way as for redis- tribution functions averaged over the angle. Further, both the analytic and the numerical results show that the estimates of N, A, and other global characteristics for PFR are the same for the redistribution functions R I and RIII as the corresponding estimates for CFR and for the function R V are very close to them (if 71 is not <<a). The source functions and intensities are also nearly the same. Of course, this conclusion is only qualitative. One can always find a situation, i.e., a definite arrangement of the sources, a definite inhomogeneity of the medium, etc., for which CFR will give quantitatively inaccurate re- sults. However, it appears that for these redistribution functions it is always suf- ficient to take a few iterations of the solutions obtained under the assumption of CFR [68].

For some quantitites there can be qualitative differences even between RIA and CFR. For example, Arutyunyan and Nikogosyan [154] found the mean number of scatterings N.~(x) of photons of frequency x incident on a semi-infinite medium from without and then dif- fusely reflected .from it for ~ = 0 and % < i. If x + =, then for the redistribution function RIA they found N,(x) § I, whereas for CFR N,(x) + I/e. For the mean number of scatterings of all photons, including those that are absorbed in the medium, there is no such difference.

The greatest difference from CFR is obtained for the redistribution function RII, for which the scattering in the line wing has a diffuse nature. Therefore, the mean num- ber of scatterings and the mean photon path are greater than for CFR (N is proportional to T and not T I/2, and s ~ T 41~ and not T), while the thermalization length is less (A ~ 6 -3/4 and i/e and not i/6 and I/e2). However, when collisions are taken into account the redistribution function is a linear combination of RII and RIII for a resonance line and of R V and RIII for a subordinate line, so that the scattering is closer to CFR the greater the part played by collisions -- a conclusion drawn earlier from general considerations. A criterion of applicability of CFR is established in [155] for the atmospheres of stars, namely, the collisional width 7c must be greater than the radiative width 7R at depth T o - 8000 in the line center. The approximation of CFR is always valid for weak lines and the core and far wing of strong lines; the criterion applies only to the near wing. In [155] the criterion is applied to resonance lines in the spectrum of stars of class A, and the influence of the element abundances and rotation of the star is studied.

In the following section, we discuss studies that take into account motion of the medium and deviation of its geometry from the planar situation.

~. Partial Redistribution in Moving and Nonplanar Media

i) Moving Planar Media. If the matter moves with a certain macroscopic velocity V (we shall measure it in units of the thermal velocity), then in the observer system the absorption coefficient for photons in the line will be ~(x--nV), where n is the unit vector that specifies the direction of flight of the photon in this system. Accordingly,

107

the redistribution function of an atom with the given velocity v will be r(x--~v--~V, x~-- n'v--n'V), and averaged over v it will be R(x--nV, xr--n 'V, ~). In [!56]~ this expression was averaged over the directions and the redistribution function RA(X , x', V) was obtained. For cases I, II, III it is given, respecitvely, by Eqs. (7) (for n = 0), (9), and (12)~ itbeing necessary in the integrands for I and II to add the factor exp(--V=)sinh(2yV)/2yu and for II the factor exp(--V2)sinh[2(y + [ul)V]/2(y + [ul)V. These factors become equal to unity for V = 0. The obtained functions were investigated in [156].

In [157], Feautrier's method, which was adapted to problems with motion in [1581, was used with the redistribution function RIA(X, x', V) to solve the equation of radia- tive transfer in a line in a semi-infinite (solar) atmosphere. Further, for a planar medium (semi-infinite and with T = 10) the source functions and profiles of the emerg- ing radiation were calculated in [159]. Both studies found large differences between the results for CFR and PFR if V varies with T (a constant V leads merely to a shift of the frequency scale).

However, it was shown, in [160], that these large differences are in fact fic- titious. They are due to averaging of the redistribution function over the directions in the observer coordinate system. If one takes the redistribution function in the ob- server system in the form RA(x--Vn, x' -- Vnr), i.e., if one averages the redistribution function in the system associated with the gas, then a large difference should not be obtained.

In this connection, the transfer equation is frequently written down for the system associated with the gas, i.e., the comoving system. Making calculations in this way, Mihalas et al. [161] confirmed the conclusion of [160]. In [161], they describe the method of calculation and show that for the cases considered in [157] the results differ from CFR no more than they do for V = 0. The same conclusions were reached in [162~.

In [163] Vardavas recognized the validity of the criticism in [160] of his papers [157, 159] and confirmed it by calculations. Scattering with the redistributionfuncs RII in a moving medium was investigated in [164] for T = I0 and 200, E = 10 "4, a = i0 -s, S O = I, V(~) = V0(l -- 2~/T), V 0 = 1 and 3. The differences in the fluxes from CFR are small at not too large T and increase with increasing T and velocity gradient.

An analogous calculation for the redistribution functions RI! and Ril I with complete angular and frequency dependence was made in [125]. A method of solving the problem is given. The power of the primary radiation sources imitated the rise of the temperature in the chromosphere. The velocity of the motion was taken equal to the thermal velocity, V(~) = vt(~), in accordance with the model of the chromosphere, like the damping con- stant a(T) (together with a = 10-3). The remaining parameters were the same as for the calculations with V = 0 (see No. 12 in Table i). It was shown that the redistribution function RIII gives hardly any differences compared with CFR, i.e., absorption profiles with one minimum are obtained. For the function RII , the intensity in the core of the absorption line is also fairly close to the corresponding CFR intensity. Because of the monochromaticity of the scattering in the wings the radiation intensity is there much weaker than for CFR. Therefore, in both fixed and moving media use of the redistribu- tion function RII leads to profiles with two sharp peaks on either side of the central minimum.

A model of a homogeneous layer expanding with constant velocity gradient without absorption and with a central isotropic source was considered in [118], in which s 0) and N(T, 0) were calculated for a number of velocity gradients and optical thicknesses from 30 to 3"107 for the redistribution function RII A.

A method was proposed in [16'5] for solving multilevel problems of transfer in lines with allowance for dependence of the redistribution function on the angles and macroscopic velocity and dependence of the parameters on two spatial coordinates (two-dimensional plane problem).

2) Movin K Nonplanar Media. Line formation in the case of PFR in spherically sym- metric media has been investigated in a number of studies.

An integral equation for the source function and an integrodifferential equation for the scattering function, which occurs in the equation for the intensity of the emergent radiation (also an integrodifferential equation) were obtained in [166];

108

The resolvent of the equation for the source function, escape probabilities, and a number of auxiliary functions were introduced in [167], in which a probability treatment of the problem was given. A numerical method for solving line formation problems in a spherical shell was proposed in [168].

Usually, the extension of a long-lived atmosphere and stellar envelopes are coupled with matter outflow or, conversely, accretion. In such cases not only the extension of the medium but also the macroscopic motion of the matter in it is important. Therefore, as a rule, both these effects are taken into account at once.

3) Spherically Symmetric Motions. Methods of numerical solution of transfer prob- lems in a radiation line in moving spherical media have been developed by a number of investigators.

Very general methods have been proposed by Mihalas and his collaborators. A method of solution of the transfer equation in the comoving coordinate system for angle-averaged redistribution functions is described and used in [112, 169]. It employs variable Edding- ton factors and moment equations. A method of solution of the transfer equation in the comoving system for the total redistribution functions based on the general Feautrier scheme is given in [170].

One further method of solution of such problems was proposed by Peraiah [133]. The case of the redistribution function R I was investigated in detail [171, 172]. The ob- tained profiles are complicated; some of them are similar to P Cyg type profiles. In general, the difference from CFR is large, though the profiles averaged over the disk (the observed profiles) do not differ so strongly. The influence of averaging over the angles and replacement of a dipole phase function by a spherical phase function was investigated in [133]. In [173] it was shown for the example of scattering with the re- distribution function RIA in an isothermal spherical shell that an expansion velocity Vma x ~ 60v t has little influence on the degree of excitation of a two-level atom.

The mean numbers N of scatterings and the fractions P = Ee/E 0 of photons emerging from spherical shells with R2/R l = 3 and i0 for the redistribution functions RIA and RIB, $ = 0, e = 10-s--10 -8 (varies with ~) expanding with a constant velocity gradient were calculated in [174]. With increasing expansion velocity, N decreases but P in- creases.

Several papers (see, for example, the review [175]) have pointed out an error in the calculations of Peraiah's group for the redistribution function RII in [176, 177], where this function was assumed to be invariant with respect to a change in sign of the fre- quency, but this is not the case.

A method for solving problems with cylindrical symmetry was proposed in [178].

The only work of an analytic nature in this field is Chugai's paper [179]. In the approximation (20), he considered the problem of line formation for the redistribution function RII A in a homogeneous infinite isotropically expanding medium when i m a m 70 = dV/d~ = const and the sources at every position are at the line center. He found the profile of the resulting line and estimated the frequency x d = (a/y0) I/3 that separates the region of diffusion of photons with respect to the frequency as a result of scatter- ings, Ixl < x~, and the region of drift of the photons in frequency due to expansion of the medium: |x I > x d. On the average, a photon undergoes N ~ i/70 scatterings, almost all of them in the line core. When displaced from the position of its creation by ~d = 1/70 (analog of the thermalization length), the photon enters the region of frequencies Ixl > x d. In the region x < --x d, the photon is scattered x~ ~ N times. Scatterings still occur at a displacement A = a/7~ m ~d, where A is the thermalization length for CFR (see [180]).

Some of the studies mentioned here are discussed in the review [175].

7. Applications of the Theory of Partial Frequency Redistribution

i) The Line L~ in Nebular Spectra. In this section, we briefly consider some of the most important objects in which PFR plays an important role (see also the reviews [55, 181]).

Among the main objects in which the theory of line formation is applied we have

109

gas and above all planetary nebulas (see the review [182]).

It was in connection with the problem of the pressure of the L~ photons exerted on the matter of a planetary nebula, determining thereby its dynamics, that the investi ~ gation of PFR was begun [13, 22, 102]. The results of [13, 102] were compared in [148] with the results obtained there by the approximate method of [147], and the density of the L= radiation and its pressure in nebulas were estimated. The part played by dust and two-photon transitions was estimated in [119].

The profiles of the emerging L~ radiation, the total intensities, and the profiles of the mean intensity were calculated as functions of the distance from the central star of the planetary nebula in [183] with allowance for expansion of the nebula and in [184] allowance for dust as well. We should mention that these calculations may be subject to the error mentioned above.

2) Bowen Mechanism. Another problem that requires PFR theory is the so-called Bowen mechanism of emission of the allowed lines 11 3454, 3133, 3322 O III in the spectra of high-excitation planetary nebulas due to the He II line Le, The wavelengths of the lines O III I 303.799 and He II I 303.780 are very nearly equal. After excitation of the upper level of the O III line there is fluorescence in the Bowen lines.

A quantitative treatment of this mechanism with allowance for PFR is given in [65], where an approximate solution for the redistribution function RII isobtained. The formation of the He II line L= is considered separately from the O III line.

A more rigorous treatment of the question is given in [1851, where the exact re- distribution function RII is taken for the He II line L~ and CFR for O i!i, absorption by H and He I is taken into account, and the photon diffusion in these lines is calculated simultaneously. The source of the He II emission is assumed to be the recombination mechanism due to the radiation of central stars with temperatures 63000~ and 10s~ The same calculation is made for the nuclei of Seyfert galaxies with central source spectrum -~v -~, ~ = 1.24 and 0. It is shown that the efficiency of the mechanism is fairly high (40% for ~ = 1.24, 20% for ~ = 0). In [186], a simplified theory of this effect is given.

3) Line Profiles in the Spectra of the Sun and Stars. At the present time, the calculation of the profiles of resonance lines serves as a means of constructing and refining models of the upper atmosphere of the Sun and the chromospheres of stars.

Under the assumption of CFR, the choice of a model of the solar chromosphere makes it possible to calculate line profiles in agreement with the observations for the center of the disk, but it is not possible to match the variation of the profiles of strong lines with absorption and emission components over the disk. One of the first studies of this subject was the paper [187]. Reviews of this work can be found in [4~ 8~ 9, 181]. In a series of papers, Mihalas and collaborators showed that only allowance for PFR can improve the situation. In all such calculations, the redistribution function was taken in the form (i - b)RiiA(X, x ~) + bU(a, x)U(a, x')~ and for transitions from the sublevels of one level R X was taken instead of RII.

The method of calculation of [IIi] was used for illustrative calculations of the profiles of the line Le, the Mg II h and k lines [188], and the Ca II H and K lines [189] in the spectrum of the Sun. The L~ profile with depth-dependent Av D and redistribution function in accordance with a model of the chromosphere was calculated in [82] for a two-level hydrogen atom. The profile agrees well with the observations~while the pro- file calculated for CFR is 5--6 times stronger in the wing.

The L~, H and K Ca II, h and k Mg II solar profiles were calculated in [190] (improved expansion constants). The need to take into account CFR in the calculation of resonance lines in the spectrum of the Sun was demonstrated in [191] for the example of the line ~ 2852 Mg I. In [192, 193] it was shown that to obtain agreement with the observations of the variation of the profile from the center to the limb of the solar disk and improve the model of the atmosphere it is also necessary to take into account CFR in the formation of weaker lines such as i 4554 and ~ 5854 Ba !I.

Models of quiescent and active regions of the solar chromosphere based on com- parison of calculations for PFR with rocket observations of L~ were constructed in [194] using the redistribution function of [39]. The redistribution function of [51~, which

Ii0

is valid over the complete line, was used in [69]. With the standard chromosphere model, agreement with the observations was obtained in the core and in the shape of the wings, but the wings were found to be four times weaker than observed.

Partial frequency redistribution was also taken into account in calculations of line profiles in the spectra of solar formations. It was noted in [70] that scattering with the redistribution function RII is nearly monochromatic in the line wing for cal- culation of the L~ profile in the spectrum of a chromospheric filament. It was shown in [195] that for scattering of solar radiation by optically thin formations above the disk lines containing information about the redistribution function are formed. The L~ profile of a protuberance for T0(L =) = 2"105 was calculated in accordance with CFR and PFR in [196]. It was shown that the total intensity of the L~ line and the ratio of the L= and H~ intensities are two times greater for CFR, the observations confirming PFR.

Three series of papers by Linsky and collaborators (the most recent of them [197], [198], and [199]) have been devoted to the construction of chromosphere models of cer- tain stars of late types, dwarfs, giants, and supergiants on the basis of calculations of profiles for various lines, the Ca II and Mg II lines being calculated with allowance for PFR by the methods of [iii, 112]. It was shown in [200] that for correct calcula- tion of model atmospheres of type A and their ultraviolet spectrum it is necessary to take into account PFR in the lines L= -- L6.

A line profile was calculated in the spectrum of a red giant with allowance for the effect of stellar wind in [201]. The influence of microturbulence on profiles was studied in connection with the Wilson--Bappu effect in [129].

4) Other Applications. The deviations from local thermodynamic equilibrium were investigated with allowance for PFR for scattering in lines in [202, 203]. The approach adopted there was criticized in [204].

A theory of line formation in the case of PFR was developed in [17] in connection with the investigation of turbulent motions in the lower chromosphere.

In [205], the Monte Carlo method was used to calculate scattering of the radiation of the O I resonance triplet (taken as one line) in the Earth's atmosphere. In [139], the auroral emission of Jupiter's atmosphere was investigated. In [54], it was shown that the results of the study can be applied to the investigation of Fe XXVI x-ray lines in the spectra of gas clouds in the neighborhood of an x-ray star. Finally, in [179] the obtained estimates were applied to the shells of supernovas in the stage a year after the explosion.

Sum/narizing, we may say that in recent years the theory of scattering with partial frequency redistribution has developed rapidly and been enriched by applications. The redistribution functions in the case of collisions have been obtained, asymptotic esti- mates of many quantities have been made, the nature of this form of scattering has been investigated, numerical methods of solution of complicated problems have been created and implemented in computer programs and applied to calculations for astrophysical objects. The time has now come to apply the redistribution functions corresponding to collisions to specific lines of multilevel atoms under definite conditions.

LITERATURE CITED

i. V. V. Sobolev, Transfer of Radiative Energy in the Atmospheres of Stars and Planets [in Russian], Gostekhizdat, Moscow (1956).

2. V. V. Ivanov, Radiative Transfer and the Spectra of Celestial Bodies [in Russian], Nauka, Moscow (1969).

3. D. I. Nagirner, Itogi Nauki Tekh., VINITI, Astronomiya, 22, 220 (1983). 4. D. Mihalas, Stellar Atmospheres, San Francisco (1970). 5. G. Athay, Radiation Transport in Spectral Lines, D. Reidel, Dordrecht, Holland (1972). 6. D. G. Hummer and G. B. Rybicki, Ann. Rev. Astron. Astrophys., 9, 237 (1971). 7. D. G. Hummer, Mon. Not. R. Astron. Soc., 125, 21 (1962). 8. G. B. Rybicki, in: The Energy Balance and Hydrodynamics of the Solar Chromosphere

and Corona, Clermont Ferrand, France (1977), p. 191. 9. R. W. Milkey, in: Interpretation of Atmospheric Structure in the Presence of Inhomo-

geneities, Univ. Sydney, Australia (1976), p. 55.

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