ABSORPTION AND EMISSION PROFILES IN A NONLINEAR PROBLEM OF RESONANCE SCATTERING

A.

Kh. Khachatryan and A. A. Akopyan

The absorption and emission profiles are compared for a nonlinear problem of radiative transfer. Particular attention is devoted to the differences between the profiles of the absorption and emission coefficients due to deviation of the velocity distribution of the atoms from the Maxwellian distribution. The results of some numerical calculations in the case of Doppler line broadening are given. i. Introduction The nonlinear radiative transfer problem of monochromatic scattering in a three-dimensional medium was considered for the first time in [I]. Subsequently, during recent years, there have been several studies [2-5] devoted to nonlinear radiative transfer problems for general frequency redistribution laws. When the nonlinear effects are taken into account, the local optical properties of the medium come to depend on the state of the radiation field. When the effects of incoherence of the elementary scattering effect are taken into account, the problem becomes even more complicated, since in t h i s c a s e the absorption and emission profiles are not identical. The nonlinear problem of incoherent scattering in a spectral line was considered in [4] under the assumption that the absorption and stimulated emission profiles are identical. This assumption was called a "first approximation." Use of the method of [i] made it possible in [4] to linearize the Corresponding equation. By a variety of analytic constructions the problem was reduced to comparatively simple numerical calculations. Some numerical results obtained by this method are given in the paper [5] of the present authors. The question of the accuracy of the "first approximation" and the further refinement of the solution to the problem arises naturally. The aim of the present paper is to find a measure of the deviation of the absorption and emission profiles and establish the extent to which the "first approximation" of [4] is valid. A similar question was considered in [3];

we shall discuss it below.

2. One-Dimensional Semi-Infinite Isothermal Medium We consider a one-dimensional semi-infinite isothermal medium consisting of two-level atoms and free electrons. Following [2], we denote the velocity distribution function of the atoms in the ground and excited states by f1(~) and f2(~), respectively, and the numbers of atoms in unit volume in the corresponding states by n I and n 2. The velocity distribution of all the atoms is assumed to be Maxwellian, i.e., we assume that the presence of a strong radiation field changes the distribution of all the atoms little from the Maxwellian distribution:

(i)

(2) where the functions fk(~) are normalized as follows:

f A (~) day = 1,

(k = O, 1, 2).

(3)

Institute of Applied Physics Problems, Armenian Academy of Sciences; Computational Center of the Ministry of Communications, Armenian SSR. Translated from Astrofizika, Vol. 25, No. i, pp. 189-195, July-August, 1986. Original article submitted July 26; 1985; accepted for publication March 20, 1986. 0571-7132/86/2501-0465512.50 9 !987 Plenum Publishing Corporation

465

The equation of stationarity has the form

hal (~) B,2 i q (x, ~) [I+ (=, x) + I- (z, x)] ax

=

nlf~G){ a~x+ All + BI* I E(x, v)[l+(z, x)+ l-(z, x)]dx.

(4)

o,J

%

--00

Here I+-(z, x) a r e t h e i n t e n s i t i e s of t h e r a d i a t i o n p r o p a g a t i n g t o t h e l e f t and t h e r i g h t , r e s p e c t i v e l y , x = ( v - - v 0 ) / A v D i s t h e d i m e n s i o n l e s s f r e q u e n c y , a:z i s t h e c o e f f i c i e n t of electron collisions of the second kind, Bl2, B2z , All are the Einstein coefficients, and q(x, ~) and E(x, ~) are the microscopic absorption and emission profiles. In accordance with the results of [2], the microscopic emission profile is given by

BI~ ~[ R.. (x, x') [; ~ (z, x') + ] - (z, x')]dx'

..) -~ |

E(x, ; ) =

B,.,

v

,

[

---

,~ (.,~,

(5)

g) [,f§ (z, x) + I-(~, ~)la~

I$0

where R ~ (x, x') is the frequency redistribution function for an atom moving with veloc-> v ity v. Integrating Eq. (4) over all velocities, we obtain

,,

B~ [ ~(~)[t+(z, ~)+I-(~. x ) l d x = tJ --OO

n2/.2t nt- AI, q- B,x ~ ,~(x)[/+

(z, x)+ I-(z, x)]tdx.

(6)

--OO

Here,

?(x) and $(x) are t h e a b s o r p t i o n and e m i s s i o n p r o f i l e s : ,3

d

In the general case of incoherent scattering, the source function depends on the frequency:

S(z, x)=

.2Aa~(x)

(8)

, ~B , ~ ( ~ ) - . ~B~,~ ( ~ ) When one is treating the nonlinear problem of the formation of a spectral line, the mathematical difficulties which arise when allowance is made for the incoherence of the elementary scattering event usually lead to a simplifying assumption being made about the source function, namely, that it does not depend on the frequency. This assumption is equivalent to assuming complete frequency redistribution (for which ?(x)=~(x)). This last is satisfied under the following assumptions: a) complete frequency redistribution in the frame of reference of the atom; b) the distribution of the atoms in the ground and excited states is Maxwellian. Condition a) is satisfied in the majority of cases when one is considering problems in a spectral line (see [6]). Then

R.(x, x') = q(x, ~q(x', ~); E(x, ~) = q(x,v). v

Condition b) is equivalent to fulfillment of the equations

J 466

(9)

However, at high radiation densities the velocity distribution of the absorbing and emitting atoms will be non-Maxwellian. Let ~fk be the deviation of fk(~) from the localequilibrium function f0(~), i.e.,

(ii) Then, taking into account (9), we obtain from (7)

(x) = ~ (x) + ~ (x); ~ (x) = ~ (x) + ~+ (x). Solving the system of equations (1) and (4) simultaneously, we obtain

A (C,) =

10 (~) _ .0 - -=

(13)

2

9

(12)

I g.

B,I

2 A,,

/o

~" ( 1 + g~ B*"r,

(i4)

(~).

Here, gk is the statistical weight of the k-th level,

~(=, ~)=~

q(x, ;)[7+ (z, ~) + 7- (~, x)l ~. J

We denote by i l l , fi~, I+(z, x) the level populations and radiation i n t e n s i t y obtained on the basis of " f i r s t approximation." The finding of the i n t e r i o r regime, and also the degree of e x c i t a t i o n of the atoms in the " f i r s t approximation" is given in f a i r d e t a i l in [4, 5], and therefore we s h a l l not dwell on i t here. We merely note that when the numerical calculations were made in the " f i r s t approximation" i t was assumed that the medium is illuminated by external radiation with no frequency dependence. Taking into account (I0), we can obtain from (2) and (6) the ratios n0/fiI and n0/fi2:

1+ ( 1+ ~% ) "~S. . . . . . .g l . . .. . .r/O

1T' ( 1-v-' ~',..),.,-

~0

-n I

1

+x~

'

(15)

-

n,

gl

OO

S = >. j ~(x)[7+(z, x)-FT-(z, x)]dx,

(16)

where ~v

X

(e,, )

Ael

3. Results of Numerical Calculations We give below the results of numerical calculations for a line with zero natural width. In this case,

m:-

2kT

.

.

--a~

A (~) = \2~k T/ (17)

rz(x, x')=

i

exp ( -- l~) dt.

max(I-d,Ix'l)

Then, substituting (17), (13), and (14) in (7) and integrating over the velocities (for gs = g2), we find

467

TABLE i. Relative Deviation of the Profiles of the Absorption and Emission Coefficients.

k

~

)

0.99

I i

0.995

"~"

0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3

~ (=)

0.25

0.5

O.043 0.044 0.045 3.053 0.075 0.101 0.094

D.046 0.046 0.049 0.058 0.084 0.112 0.I08

0.040 O.040 O. 042 0.050 ).074 3.102 ~.0:8

0.047 3.047 3.04~ O. 053 3.069 0.076 9.I01 0.115 0.100 0.114

0.040 0.040 0.041 0.046

1

0.044 ).044 ).045 3.050

,"o .,

2

0.25

0.5

I

2

3.028 I0.028 3.029 D.035 0.053 0.076 0.071

--0.137 --0.138 --0.144 --0.171 --0.239 --0.313 --0.302

--0.098 --0.099 -0.103 --0.120 --0.176 --0.231 --0.227

--0.062 --0.063 --0.065 --0.082 --0.115 --0.155 -0.152

--0.036 --0.036 --0.038 --0.044 --0.081 --0.100 --0.091

0.028 0.02' O.g2S 3.031 0.056 9.047 0.105 0.076 0.103 0.075

--0. 138 --0.139

--0.098 -0.098 --0.140 -0.099 --0.152 --0.107 --0.216 --0.157 --0.314 --0.223 --0.315 --0.237

--0.062 -0.062

I

I

~_+ ;i~[___j~+_ (z; =~+ 7- (=, = l [~ + 2>.~[7+(~, =)+ z-(~, =)1 }

= ao_ o-"I i;

--0.035 -0.035 --0.063 --O.03d -0.041 --0.071 --0.101 --0,05 c. --0.157 i -0.09~ -o.i58 --0.09:

~t +

+

2~,-.,

I

[7 + (=, x) + 7- (., =)If

Table 1 gives the function ratios g ~ ( x ) and g~{x!, calculated in accordance with (18) and (19) on the boundary z = 0 of the medium for different values of k and x. In [3], the results are given of analogous calculations for finding the ratio of the absorption and emission profiles for z = 2 (for which n 2 ~ nz). The problem was solved in [3] by means of an iterative process, the solution to the linear problem being taken as the first approximation. We note that, in contrast to [3], we have used as first approximation the solution of the nonlinear problem (under the assumption that the absorption and stimulated emission profiles are identical). For x ~.~--2, the relative deviation is on the average 5-I0%, in fairly good agreement with the results of [3]. As is noted by the authors of [3], their method cannot be used when x > 2 because the iterative process does not converge. It turns out that for x > 2 the deviation becomes even less. We also made some numerical calculations for z < 2 . From the numbers given it can be seen that decrease in the value of x has the result that the deviation of the emission profile becomes greater. Thus, for x ~ I--2 the relative deviation is comparatively small. This indicates that in nonlinear problems the assumption of identity of the profiles of the absorption and stimulated emission coefficients does not lead to serious deviations from the exact solution of the problem. We thank Professor N. B. Engibaryan for discussions. LITERATURE CITED I. N. B. Engibaryan, Astrofizika, i, 297 (1965). 2. L. Oxenius, J. Quant. Spectrosc. Radiat. Transfer, ~, 771 (1965). 468

(18)

(19)

3. 4. 5. 6.

R. N. A. D.

Steinitz and R. A. Shine, Mon. Not. R. Astron. Soc., 162, 197 (1973). B. Engibaryan and A. Kh. Khachatryan, Astrofizika, 23, 145 (1985). Kh. Khachatryan and A. A. Akopyan, Astrofizika, 23, 569 (1985). Mihalas, Stellar Atmospheres, San Francisco (1970).

OPTICAL PROPERTIES OF CYLINDRICAL CORE--MANTLE DUST GRAINS A. E. ll'in The efficiency factors are calculated for cylindrical two-layer dust grains with silicate core and ice mantle. For dielectric grains it is found that the core does not have a significant influence on the optical properties of the grain if the ratio of the core radius to the mantle radius is less than 0.I. Expressions are obtained for the efficiency factors in the Rayleigh approximation, and their region of applicability is determined. i. Introduction Much of the information about the properties of interstellar dust grains is obtained by analyzing observations of interstellar absorption and interstellar linear and circular polarization. For the modeling of these phenomena, it is necessary to know the optical properties of dust grains. In the interpretation of the interstellar extinction curve, a model of spherical grains is usually considered [i]. However, this model is inapplicable for investigation of interstellar polarization, which arises when radiation passes through an ensemble of nonspherical oriented particles. From the computational point of view, it is easiest to consider interstellar grains in the form of circular cylinders. In the monographs [i, 2] it was noted that if the ratio of the length of the cylinder to its radius is greater than four its optical characteristics can be calculated using the formulas obtained for the solution to the problem of the diffraction of a plane electromagnetic wave by an infinite circular cylinder [3]. A model of such particles was used for simultaneous interpretation of observations of interstellar absorption and interstellar linear and circular polarization [2]. According to modern ideas [4], it is believed, as the most probable scenario, that dust grains grow in interstellar clouds by the accretion of light elements on initial condensation nuclei ejected from the atmospheres of giants and supergiants of late spectral classes. The currently most attractive model is the model proposed by Hong and Greenberg [5] of cylindrical two-layer grains consisting of a core and a concentric mantle. It is actively used to interpret observations of interstellar absorption and polarization [5-7]. In the present paper, we investigate the optical properties of such grains, give the results of calculations of the efficiency factors, obtain the formulas of the Rayleigh approximation, and determine the region in which they are valid. 2. Basic Relations We consider a two-layer cylinder in which a c is the radius of the core, a is the radius of the mantle, and m I = n I -- kli and m 2 = n 2 -- k2i are the complex refractive indices of the matter of the core and mantle, respectively. Let the incident radiation of wavelength % make angle ~/2 -- ~ with the axis of the particle. Instead of a c and a we use, as usual, the dimensionless parameters x 0 = 2~ac/% and x = 2za/%. We introduce the complex efficiency factor Q = Qext + iQq. The dimensionless quantities Qext and Qq are the efficiency factors of extinction and phase retardation, and for cylindrical particles of length L they are determined in terms of the corresponding cross sections Cex t and Cq as follows [2]: Leningrad State University. Translated from Astrofizika, Vol. 25, No. i, pp. 197205, July-August, 1986. Original article submitted December 27, 1985; accepted for publication April 15, 1986.

0571-7132/86/2501-0469512.50

9 1987 Plenum Publishing Corporation

469