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Non-coherent scattering in subordinate line—V
J Qwnf S,w!ro\< Hud~ur Pnntcd I” the U.S.A.
7rurnlvr
Vol. 32. lua. 2. PP. 159-168.
1984 0
NON-COHERENT
SCATTERING
SOLUTIONS
IN SUBORDINATE
OF THE TRANSFER
WJ22-l073/84 1984 Pergamoo
$3.00 + MJ Press Ltd.
LINES-V
PROBLEM
I. HUBEN+ and P. HEINZEL Astronomical
Institute of the Czechoslovak Academy of Sciences, 251 65 Ondiejov, Czechoslovakia (Received 5 October 1983)
Abstract-We present several examples of the numerical solution of the radiative transfer in subordinate lines. Using a simplified physical model that yields the line source function analogous to the usual two-level-atom form modified by the presence of the redistribution function R, in the scattering integral, we have solved the transfer problem for isothermal, plane-parallel atmospheres, both finite and semi-infinite. For finite atmospheres, we have found substantial differences between the solutions with R, and those with complete redistribution. On the other hand, for semi-infinite atmospheres the complete redistribution appears to be a good approximation, at least for a,= a, (damping parameters for the lower and upper levels, respectively). It is shown that the effect of R, becomes more pronounced with increasing ratio ~,/a,. Finally, it is demonstrated that an approximate form for R, analogous to that of Kneer for R,, serves as a very good approximation for computing the line profiles, particularly in the line wings.
1. INTRODUCTION In the previous papers of this series’-3 (hereafter referred to as I, II, and IV, respectively), we have demonstrated that the resonance scattering in isolated subordinate lines can be described in terms of a superposition of Hummer’s4 redistribution function RllI and newly introduced function Rv. In IV, an efficient numerical method for an evaluation of Rv has been presented. As a next step, we employ this redistribution function in simple radiative transfer calculations in order to demonstrate its astrophysical significance. For such exploratory purposes, a simplified treatment of the atomic model as well as the physical properties of a line-forming medium will be adopted. Attention will be primarily devoted to a comparison of the transfer solutions using the redistribution function R, to those following from traditional approaches, assuming either complete redistribution (CRD) or a coherent scattering in the atom’s rest frame described by the function R,,. Finally, because the function Rv is somewhat costly to calculate,3 we test its simple approximate form, which is a generalization of the Kneer’s’ approximation to R,,. 2. PHYSICAL
MODEL
The simplest atomic model needed to study subordinate line formation is a three-level atom. However, even in this simple case, the line source function is, in principle, extremely complicated.6 Therefore, for the purposes of model calculations, and in order to obtain a direct comparison with previous similar studies of resonance line formation,7,8 we make the following simplifying approximations: (i) Velocity distributions of atoms in the ground state and both excited states are Maxwellian. (ii) We neglect all photon correlations between different lines (e.g., a photon emitted in the line 3cr 2 is supposed to be uncorrelated to a photon previously absorbed in the line 1 t) 3, etc.). In other words, we assume the generalized redistribution functions’ for all transition sequences except 2+3+2 to be simply given by the products of the corresponding natural excitation absorption profiles.6 (iii) Stimulated emissions are neglected.7 (iv) We assume correlated re-emission6 in the line 3 (--t2, i.e. we neglect velocity-changing collisions occuring during the scattering process 2+3+2. (v) We adopt the isotropic approximation,* i.e., the exact angletit may seem emissions can be (see below) which emission is much
that this approximation is too crude. However, assuming, as is usually done, that stimulated treated as negative absorptions would yield a formal modification of the parameters 1 and c are here taken as ad hoc parameters. On the other hand, a more exact treatment of stimulated more complicated6 and beyond the scope of the present study. 159
160
I. HUBERT and P. HEINZEL
dependent redistribution functions are replaced by their angle-averaged counterparts IV and references cited therein). Under these approximations, the line source function can be written1° as
(see
cc
n3A32pv
+ n2B23a32
[S
R(v’, v)J,,, dv’ - &,,
n2&3qv,
(2.1)
0
where p3 = A,, + A32 + c,,
a32 = A32/P3y
+ c32,
(2.2)
and
s cc
J=
JAP, dv.
(2.3)
0
Here, cpvis the (normalized) natural excitation absorption profile for the line 2 c* 3, R(v’, v) is the appropriate angle-averaged redistribution function, A and B are the usual Einstein coefficients, C denotes inelastic collisional rates and n, and n3 are corresponding level populations. The redistribution function is given by2
R(v’, v) = Y&(v’, v) + (1 - YPMv’, v),
(2.4)
where Y = P3AP3
(2.5)
+ Q37
with QE being the rate of elastic collisions. In Eq. (2.1), the populations can be eliminated by means of the statistical equilibrium equations (see, e.g., Mihalas”). For simplicity, we shall assume that both resonance lines are optically sufficiently thick in the region of formation of the subordinate line and, consequently, they can be assumed to be in detailed radiative balance. Further, we replace RIII(v’, v) by q,,cP,. Neither of these approximations is substantial and may easily be relaxed. On the other hand, they are quite adequate for our exploratory work [see also the discussion above Eq. (2.1 l)], Performing straightforward algebraic manipulations, we obtain
s
*
m
S(x) = (1 -6
m
R,(x’, x)J,, dx’ + cB + 6, Jxqc, dx’ + L cpxs -co -cc
-2)
(2.6)
where (2.7)
x = (v - vo)/Avo,
c= 1
=
Ya3,
6=
= A32(C2,
A321(A32 -
C32/cA32 +
+ A3,
~,G,l~,W23(A32
+
(2.8)
c32). c32
+
C3, +
+ c32>l~
Q&
(2.9) (2.10)
B is the Planck-Wien function at the line-center frequency vo, and Av, is the Doppler width. It is worthwhile to inspect Eq. (2.6). The scattering term splits into two parts: the first corresponds to complete redistribution, while the second describes partial redistribution effects. In the limit of low densities, where collisions can be neglected, the branching ratio + A,,), which may not be close to unity, in contrast 1 approaches the value i +A,,/(A,,
Non-coherent
scattering in subordinate lines-V
161
to the case for the two-level atom. The physical reason for this is obvious: the transition 3 - 1 provides an extra channel for destroying and the creating photons in the subordinate line 3 - 2, so that the relative importance of the resonance scattering (and hence a correlation between absorbed and remitted photons) is diminished. Nevertheless, since we are interested in a comparison between radiative transfer solutions with various redistribution functions rather than in solving any specific transfer problem, we neglect the latter effect and write formally I = 1 - 6. Although this is a rather unphysical assumption that amounts to neglecting the terms A,,, C3,, and QE while retaining (usually comparable to smaller) term C 32 in Eq. (2.9), it is suitable for our purposes. For the same reason, we neglect 6 in Eq. (2.6). The source function then becomes S(x)
l--E = -
a2
R,(x’, x)J,, dx’ + tB.
(2.11)
cpx s --m For other types of the redistribution functions (i.e., R,,,,,,,,), this frequency-dependent source function has been discussed in Refs. 8 and 11 (Sec. 13-4). The line source function of Eq. (2.11) will be used in subsequent calculations. However, in a more realistic case, the source function and emergent radiant intensity will lie between values given by Eq. (2.11) and those for complete redistribution, depending on the value of the branching ratio 1. Thus, our results should be viewed as an upper limit for departures from complete redistribution induced by resonance scattering in subordinate lines. 3. RESULTS
AND
DISCUSSION
We now present numerical results for isothermal, plane-parallel, semi-infinite and finite atmospheres. The line source function is given by Eq. (2.11) with the absorption profile ‘P, = H(a, + a,,, x)/G,
(3.1)
where H(a, x) is the Voigt function, a, and a, are the damping parameters for the lower and upper levels, respectively. The angle-averaged redistribution function Rv (and RI,) was computed according to the procedure described in IV, using a 16-point Gaussian quadrature over scattering angles; the angle-dependent functions with an isotropic phase function were calculated according to I. The transfer equation was solved by using the Feautrier-Auer Hermitian technique,12 with a logarithmically spaced mean optical depth mesh of 5 points per decade. The mean optical depth is defined by the relation dr = -k, dz = - (huo/4u)nzBz3 dz.
(3.2)
The angular dependence of the radiation field was treated by using the variable Eddington factors technique (VEFT);14 the corresponding angular integrations were performed with a 3-point Gaussian quadrature. Typically, 3-4 iterations of VEFT ensure sufficiently converged solutions. In evaluating the scattering integral, the natural cubic spline representation for the radiation field” was used, with a 25-point frequency grid between x = 0 and x,,, = 24. We have slightly modified the original cubic splines computer code of Adams, Hummer, and Rybicki13 in order to account for the central peak of Rv. Furthermore, the resulting matrix R, was calculated using pretabulated values of Rv(x’, x) (or R,,) in a sufficiently fine frequency mesh. In subsequent calculations, the Planck function was set equal to unity and e was taken to be 10m4. For clarity, no overlapping continuum was assumed. Our computer program was tested against the results given in Refs. 7 and 8 for CRD and R,, and satisfactory agreement was found. We have also tested the accuracy of our results by increasing the number of frequency and/or depth points and found that the chosen quadrature parameters yield results accurate to 3sf. Nevertheless, mode1 calculations for larger values of damping parameters (e.g., a Z 10m2,which are not presented here) would require larger values of x,, (cf Ref. 7).
162
T !!:~:a
0.23 r
log
and P.
HEINZEL
5
!
I
-2.0
I
-1
I
0
I
I
1 log l-
I I
2
3
Fig. 1. Source functions S,r for an isothermal slab atmosphere of optical thickness T = 104. Full lines: partial redistribution (PRD) with R,, a, = a, = 10m3.The curves are labeled with emission frequencies in units of Doppler width. Dashed lines: frequency-independent source function for complete redistribution (CRD), a = a,+ a, = 2 x IO-‘. The atmosphere is self-radiating with c = 10d4 and without incident radiation. The abscissa is mean optical depth r.
Emergent line profiles were found to be insensitive to small inaccuracies in the evaluation of R, discussed in I and IV. Therefore, in transfer calculations, one can use a less accurate integration in evaluating the E-term [see Eq. (Cl) of I, where M 2: 25 and h z 0.2 seem to be sufficient], as well as a lower-order quadrature in the angle-averaging procedure @-point Gaussian quadrature). This choice of parameters reduces time and storage requirements considerably. Nevertheless, our results are based on a relatively precise Rv (using M = 100, h = 0.05, and 16-point Gaussian quadrature). (a) Slab atmospheres The depth-dependence of the line source function for various frequencies within a slab of optical thickness T = 104, with no incident radiation at the bottom, is shown in Fig. 1. The corresponding frequency-independent source function for the case of complete redistribution is also presented. The damping parameters were set equal to al = a,, = 10m3. The corresponding normally emergent intensity is displayed in Fig. 2, together with those for the case of R,,and CRD (with a = a, + a, = 2 x 10w3).As the line is symmetric around the center, only halves of profiles are drawn. As was demonstrated in the literature,*.’ ’ there exist large differences between calculated profiles assuming CRD and R,,, particularly in the line wings. This well-known effect is caused by the high degree of coherence of the scattering process described by the function R,,. As expected, the intensity for Rv lies between those for R,,and CRD. This result can be readily explained on the basis of the general properties of resonance scattering in subordinate lines. Indeed, in contrast to the case of R,,where a photon absorbed at frequency x’ in the line wing (x’ 2 3) will most probably be re-emitted in the vicinity of x = I’ (the difference between x and x’ in the laboratory frame is caused by the Doppler redistribution), there is an additional probability for a photon to be emitted in the vicinity of the line center for R,. The ratio of the probability of (almost) coherent re-emission to the probability of re-emission near the center is proportional to au/a,, as was demonstrated in IV. We note that similar behavior is encountered also for resonance lines and is attributed to elastic collisions. I5 To avoid misunderstandings, we stress that the above discussed property of Rv is valid regardless of the presence of collisions and is caused by broadening of the lower state of a given transition. The collisional redistribution is properly accounted for by Eq. (2.4).
Non-coherent
scattering in subordinate
I r log
0.0
/ 0
I 2
,
I
4
x
I
6
163
lines-V
I 8
I
1 10
Fig. 2. Normally emergent intensities are shown for the source functions displayed in Fig. 1. Full line: PRD with R,,, a,= ~1, = 10e3; dashed line: CRD with (I = 2 x 10m3. The case with R,, (dash-dotted line), LI= 2 x IO-j, is added for comparison. Abscissa: emission frequency in units of Doppler width.
The physical explanation for the behavior of the transfer solutions with Rv is similar to that for RI*. In the line core, photons can only partly be scattered towards the wings and, therefore, the thermalization in the core is more rapid than for CRD but slower than for RI,. In the wings, the source function lies below the corresponding CRD values in the region of the monochromatic optical depth near unity because photons are less efficiently fed into the wings from the core than for CRD. Consequently, the corresponding emergent intensity is lower. The effect of enhancing the ratio ~,/a, is illustrated in Figs. 3 and 4. The emergent intensity now appears to depart relatively more from CRD.
0.0
r
log s
-1.0
-2.0
Fig. 3. As in Fig. 1 but for (I,= 10e4, a,= 1O-3 (a = 1.1 x 10m3).
I. HUBE& and P. HEINZEL
164
Fig. 4. Normally emergent intensities are shown for the source functions displayed in Fig. 3. n, = lo-*, a, = lo-’ (a = 1.1 x 10m3).Otherwise, the same models are used as in Fig. 2.
(b) Semi-infinite atmospheres In Fig. 5, the line source function is shown as a function of optical depth for various frequencies within a semi-infinite atmosphere, along with the corresponding frequencyindependent source function for CRD. The damping parameters were set equal to al = a, = 10e3. The overall behavior of the line source function is analogous to that of R,, discussed by Hummer,’ the departures from CRD being much smaller because of diminished influence of wing coherence. The corresponding normally emergent intensities for Rv, RII, and CRD are shown in Fig. 6. Differences between Rv and CRD are very small
0.0
-1.0
T:m
1 -1
0
I 1
/
I
I 3
2
i
I
1
5
6
log 7Fig. 5. The source functions S, are shown for an isothermal, semi-infinite atmosphere with t = 10m4.Full lines: PRD with R,,a,= a,= IO-'; the curves are labeled with emission frequencies. Dashed line: frequency-independent source function for CRD, D = 2 x lo-‘. Abscissa: the mean optical depth T.
165
Non-coherent scattering in subordinate lines-V
t
3
0
10
x
Fig. 6. Normally emergent intensities are shown for the source functions displayed in Fig. 5. Full line: PRD With Rv. a, = a, = lo-); dashed line: CRLI with II = 2 x lo-‘. The case with R,, (dash-dotted line) is added for comparison. Abscissa: the emission frequency in Doppler-width units.
and are therefore
of little practical
importance
in interpreting
subordinate
line profiles.
An interesting effect is the departure from CRD in the line core, where the intensities for both Rv and R,, are about 20% lower than for CRD. This difference may be of some interpretational significance. However, in actual stellar atmospheric applications, the shape of the line core is affected by many other factors that are not considered here (temperature and density stratification, effects of multi-level coupling, etc.) and also by other physical uncertainties (e.g. microturbulent velocity, rotation, etc.) which may well have comparable or greater effects on the observed line profiles. As for finite atmospheres, departures of the transfer solutions with Rv from those for CRD are much more pronounced for larger values of the ratio a,/~, (see Figs. 7 and 8).
I
-1
1
0
I
/ 1
I
I
2
log r
I 3
I
I 4
I
I 5
Fig. 7. As in Fig. 5 but for a,= lo-‘, u, = 10m3(a = 1.1 x lo-‘).
1
6
I. HUBEN% and
166
P. HEINZEL
Fig. 8. Normally emergent intensities are shown for the source functions displayed in Fig. LI,= 10e4, a, = 10m3, a = I. 1 x 10-l. Otherwise, the same models are used as in Fig. 6.
7:
(c) Approximate form for Rv
We suggested in IV an approximate form for R, which represents a simple generalization of that for RI, introduced by Jefferies and WhiteI and modified by Kneer.’ The function Rv is approximated by RV(x’, x) z a(x’, x)cp,b(x’
where 6(x’ -x)
is the Dirac delta-function
-x)
+ [l - a(x’, x)]q,q,,
and a(x’, x) is defined by
4x’, x) = [au/h + aJaW, XL where aK(x’, x) is the function introduced
(3.3)
(3.4)
by Kneer,’ viz.,
aK(x’, x) = aK(x,x’) = a [max (Ix’I, /xl)],
(3.5)
Z aK(x, x) = a(x)=
a K(~, x ‘)cp,, dx ‘,
s -*
1 -exp{-[(x
-A)/x,]~},
for x > A
(3.6) (3.7)
and a(x)=O,
for x6A.
This approximation to Rv satisfies the requirements of normalization and symmetry. We have tested several values of x,, and A and found that Kneer’s choice, namely, x0 = 2, A = 2, yields the most accurate results for subordinate lines.? Comparisons of normally emergent intensities, calculated with exact and approximate forms for R, for both finite and semi-infinite atmospheres (with a, = a, = 10e3 or a, = lo-‘, a, = lo-‘) are presented in Figs. 9 and 10, respectively. It is readily seen that the agreement tin fact, the approximation (3.3) does not allow for the Doppler diffusion in the line wings, due to the presence of the 6-function in its second term. Consequently, this approximation is effectively more suitable for R, than for R,,, because the former redistribution contains always a non-zero contribution of complete redistribution. Mathematically, a(x)-+ I for large x in the case of R,,, while n(x)+o,,/(a, + u,) for R,.
Non-coherent
scattering in subordinate
lines-V
167
-0.5
-1.0
-1.5
-2.0
Fig. 9. A comparison of normally emergent intensities is shown for the exact R, (dashed line) and an approximate form of Rv (full tine); a, = a, = 10m3.The model atmospheres are the same as in
previousfigures.
between the two approaches is very good, particularly for a, N ~1,.The approximation (3.3) thus offers a very simple and easy way of calculating radiative transfer in subordinate lines without explicit evaluation of the complicated Rv. 4. CONCLUSIONS
The results presented here were aimed at a demonstration of effects of partial redistribution on the subordinate line formation. For the sake of simplicity, we have chosen an idealized physical model that allows us to use the usual two-level-atom form for the line source function, modified by the presence of the redistribution function R, in the scattering integral. Line transfer calculations were performed for plane-parallel, isothermal atmospheres. This approach shows the importance of the redistribution function Rv and allows direct comparison with similar studies of resonance-line formation.
-0.5
Fig. 10. As in Fig. 9 but for u,= 10m4, II, = 10-3.
168
I. HUBERTand P. HE~NZEL
The principal conclusions may be summarized as follows: (1) Generally, the emergent line profiles calculated with the function Rv lie between those for complete redistribution and R,, (coherent scattering in the atom frame). Furthermore. the larger the ratio a,, a,, the greater are departures from complete redistribution” (2) For semi-infinite atmospheres, complete redistribution appears to be a good approximation for subordinate lines, at least for a, z a,. (3) For finite slab atmospheres, the effect of Rv is, in general, more pronounced than in semi-infinite atmospheres, particularly for the line wings. (4) Our approximate form for R, is a generalization of those of Jefferies and White and Kneer for RI, and appears to be a satisfactory approximation for transfer calculations, particularly for a, z a,. Because of our choice of L ‘v 1, the results represent an upper limit for actual departures from complete redistribution for subordinate lines using the simplified physical model specified by assumptions (i)-(v) (see Section 2). On the other hand, the line source function of Eq. (2.6) still underestimates departures form CRD because the only resonance scattering considered here is 2+3-+2. In principle, we should allow also for other possible sequences ending with the transitions 3-2 and 2+3 (see Ref. 6). Appropriate model calculations must be performed before a more complete understanding of subordinate line formation will be achieved. are indebted to Dr. H. Frisch (Observatoire de Nice) for providing us with her copy of a natural cubic splines computer code of Adams, Hummer, and Rybicki. The numerical calculations were performed with the EC 1040 computer of the Ondiejov Observatory Computing Centre.
Acknowledgements-We
REFERENCES 1. P. Heinzel, JQSRT25, 483 (1981). 2. P. Heinzel and I. Hubeny, JQSRT 27, 1 (1982). 3. P. Heinzel and I. Hubeny, JQSRT 30, 77 (1983). 4. D. G. Hummer, Mon. Not. R. Astr. Sot. 125, 21 (1962). 5. F. Kneer, Astrophys. J. 200, 367 (1975). 6. I. Hub&, J. Oxenius, and E. Simonneau, JQSRT 29, 477 and 495 (1983). 7. E. H. Avrett and D. G. Hummer, Mon. Nor. R. Astr. Sot. 130, 295 (1965). 8. D. G. Hummer, Mon. Not. R. Astr. Sot. 145, 95 (1969). 9. I. Hubeny, JQSRT 27, 593 (1982). 10. I. Hubenjr, Bull. Astron. Inst. Czechosl. 32, 271 (1981). 11. D. Mihalas, S&lur Atmospheres, 2nd Edn. Freeman, San Francisco (1978). 12. L. H. Auer, JQSRT 16, 931 (1976). 13. T. F. Adams, D. G. Hummer, and G. B. Rybicki, JQSRT 11, 1365 (1971). 14. L. H. Auer and D. Mihalas, Mon. Not. R. Astr. Sot. 149, 60 (1970). 15. A. Omont, E. W. Smith, and J. Cooper, Asrrophys. J. 175, 185 (1972) 16. J. T. Jefferies and 0. White, Astrophys. J. 132, 767 (1960).
7rurnlvr
Vol. 32. lua. 2. PP. 159-168.
1984 0
NON-COHERENT
SCATTERING
SOLUTIONS
IN SUBORDINATE
OF THE TRANSFER
WJ22-l073/84 1984 Pergamoo
$3.00 + MJ Press Ltd.
LINES-V
PROBLEM
I. HUBEN+ and P. HEINZEL Astronomical
Institute of the Czechoslovak Academy of Sciences, 251 65 Ondiejov, Czechoslovakia (Received 5 October 1983)
Abstract-We present several examples of the numerical solution of the radiative transfer in subordinate lines. Using a simplified physical model that yields the line source function analogous to the usual two-level-atom form modified by the presence of the redistribution function R, in the scattering integral, we have solved the transfer problem for isothermal, plane-parallel atmospheres, both finite and semi-infinite. For finite atmospheres, we have found substantial differences between the solutions with R, and those with complete redistribution. On the other hand, for semi-infinite atmospheres the complete redistribution appears to be a good approximation, at least for a,= a, (damping parameters for the lower and upper levels, respectively). It is shown that the effect of R, becomes more pronounced with increasing ratio ~,/a,. Finally, it is demonstrated that an approximate form for R, analogous to that of Kneer for R,, serves as a very good approximation for computing the line profiles, particularly in the line wings.
1. INTRODUCTION In the previous papers of this series’-3 (hereafter referred to as I, II, and IV, respectively), we have demonstrated that the resonance scattering in isolated subordinate lines can be described in terms of a superposition of Hummer’s4 redistribution function RllI and newly introduced function Rv. In IV, an efficient numerical method for an evaluation of Rv has been presented. As a next step, we employ this redistribution function in simple radiative transfer calculations in order to demonstrate its astrophysical significance. For such exploratory purposes, a simplified treatment of the atomic model as well as the physical properties of a line-forming medium will be adopted. Attention will be primarily devoted to a comparison of the transfer solutions using the redistribution function R, to those following from traditional approaches, assuming either complete redistribution (CRD) or a coherent scattering in the atom’s rest frame described by the function R,,. Finally, because the function Rv is somewhat costly to calculate,3 we test its simple approximate form, which is a generalization of the Kneer’s’ approximation to R,,. 2. PHYSICAL
MODEL
The simplest atomic model needed to study subordinate line formation is a three-level atom. However, even in this simple case, the line source function is, in principle, extremely complicated.6 Therefore, for the purposes of model calculations, and in order to obtain a direct comparison with previous similar studies of resonance line formation,7,8 we make the following simplifying approximations: (i) Velocity distributions of atoms in the ground state and both excited states are Maxwellian. (ii) We neglect all photon correlations between different lines (e.g., a photon emitted in the line 3cr 2 is supposed to be uncorrelated to a photon previously absorbed in the line 1 t) 3, etc.). In other words, we assume the generalized redistribution functions’ for all transition sequences except 2+3+2 to be simply given by the products of the corresponding natural excitation absorption profiles.6 (iii) Stimulated emissions are neglected.7 (iv) We assume correlated re-emission6 in the line 3 (--t2, i.e. we neglect velocity-changing collisions occuring during the scattering process 2+3+2. (v) We adopt the isotropic approximation,* i.e., the exact angletit may seem emissions can be (see below) which emission is much
that this approximation is too crude. However, assuming, as is usually done, that stimulated treated as negative absorptions would yield a formal modification of the parameters 1 and c are here taken as ad hoc parameters. On the other hand, a more exact treatment of stimulated more complicated6 and beyond the scope of the present study. 159
160
I. HUBERT and P. HEINZEL
dependent redistribution functions are replaced by their angle-averaged counterparts IV and references cited therein). Under these approximations, the line source function can be written1° as
(see
cc
n3A32pv
+ n2B23a32
[S
R(v’, v)J,,, dv’ - &,,
n2&3qv,
(2.1)
0
where p3 = A,, + A32 + c,,
a32 = A32/P3y
+ c32,
(2.2)
and
s cc
J=
JAP, dv.
(2.3)
0
Here, cpvis the (normalized) natural excitation absorption profile for the line 2 c* 3, R(v’, v) is the appropriate angle-averaged redistribution function, A and B are the usual Einstein coefficients, C denotes inelastic collisional rates and n, and n3 are corresponding level populations. The redistribution function is given by2
R(v’, v) = Y&(v’, v) + (1 - YPMv’, v),
(2.4)
where Y = P3AP3
(2.5)
+ Q37
with QE being the rate of elastic collisions. In Eq. (2.1), the populations can be eliminated by means of the statistical equilibrium equations (see, e.g., Mihalas”). For simplicity, we shall assume that both resonance lines are optically sufficiently thick in the region of formation of the subordinate line and, consequently, they can be assumed to be in detailed radiative balance. Further, we replace RIII(v’, v) by q,,cP,. Neither of these approximations is substantial and may easily be relaxed. On the other hand, they are quite adequate for our exploratory work [see also the discussion above Eq. (2.1 l)], Performing straightforward algebraic manipulations, we obtain
s
*
m
S(x) = (1 -6
m
R,(x’, x)J,, dx’ + cB + 6, Jxqc, dx’ + L cpxs -co -cc
-2)
(2.6)
where (2.7)
x = (v - vo)/Avo,
c= 1
=
Ya3,
6=
= A32(C2,
A321(A32 -
C32/cA32 +
+ A3,
~,G,l~,W23(A32
+
(2.8)
c32). c32
+
C3, +
+ c32>l~
Q&
(2.9) (2.10)
B is the Planck-Wien function at the line-center frequency vo, and Av, is the Doppler width. It is worthwhile to inspect Eq. (2.6). The scattering term splits into two parts: the first corresponds to complete redistribution, while the second describes partial redistribution effects. In the limit of low densities, where collisions can be neglected, the branching ratio + A,,), which may not be close to unity, in contrast 1 approaches the value i +A,,/(A,,
Non-coherent
scattering in subordinate lines-V
161
to the case for the two-level atom. The physical reason for this is obvious: the transition 3 - 1 provides an extra channel for destroying and the creating photons in the subordinate line 3 - 2, so that the relative importance of the resonance scattering (and hence a correlation between absorbed and remitted photons) is diminished. Nevertheless, since we are interested in a comparison between radiative transfer solutions with various redistribution functions rather than in solving any specific transfer problem, we neglect the latter effect and write formally I = 1 - 6. Although this is a rather unphysical assumption that amounts to neglecting the terms A,,, C3,, and QE while retaining (usually comparable to smaller) term C 32 in Eq. (2.9), it is suitable for our purposes. For the same reason, we neglect 6 in Eq. (2.6). The source function then becomes S(x)
l--E = -
a2
R,(x’, x)J,, dx’ + tB.
(2.11)
cpx s --m For other types of the redistribution functions (i.e., R,,,,,,,,), this frequency-dependent source function has been discussed in Refs. 8 and 11 (Sec. 13-4). The line source function of Eq. (2.11) will be used in subsequent calculations. However, in a more realistic case, the source function and emergent radiant intensity will lie between values given by Eq. (2.11) and those for complete redistribution, depending on the value of the branching ratio 1. Thus, our results should be viewed as an upper limit for departures from complete redistribution induced by resonance scattering in subordinate lines. 3. RESULTS
AND
DISCUSSION
We now present numerical results for isothermal, plane-parallel, semi-infinite and finite atmospheres. The line source function is given by Eq. (2.11) with the absorption profile ‘P, = H(a, + a,,, x)/G,
(3.1)
where H(a, x) is the Voigt function, a, and a, are the damping parameters for the lower and upper levels, respectively. The angle-averaged redistribution function Rv (and RI,) was computed according to the procedure described in IV, using a 16-point Gaussian quadrature over scattering angles; the angle-dependent functions with an isotropic phase function were calculated according to I. The transfer equation was solved by using the Feautrier-Auer Hermitian technique,12 with a logarithmically spaced mean optical depth mesh of 5 points per decade. The mean optical depth is defined by the relation dr = -k, dz = - (huo/4u)nzBz3 dz.
(3.2)
The angular dependence of the radiation field was treated by using the variable Eddington factors technique (VEFT);14 the corresponding angular integrations were performed with a 3-point Gaussian quadrature. Typically, 3-4 iterations of VEFT ensure sufficiently converged solutions. In evaluating the scattering integral, the natural cubic spline representation for the radiation field” was used, with a 25-point frequency grid between x = 0 and x,,, = 24. We have slightly modified the original cubic splines computer code of Adams, Hummer, and Rybicki13 in order to account for the central peak of Rv. Furthermore, the resulting matrix R, was calculated using pretabulated values of Rv(x’, x) (or R,,) in a sufficiently fine frequency mesh. In subsequent calculations, the Planck function was set equal to unity and e was taken to be 10m4. For clarity, no overlapping continuum was assumed. Our computer program was tested against the results given in Refs. 7 and 8 for CRD and R,, and satisfactory agreement was found. We have also tested the accuracy of our results by increasing the number of frequency and/or depth points and found that the chosen quadrature parameters yield results accurate to 3sf. Nevertheless, mode1 calculations for larger values of damping parameters (e.g., a Z 10m2,which are not presented here) would require larger values of x,, (cf Ref. 7).
162
T !!:~:a
0.23 r
log
and P.
HEINZEL
5
!
I
-2.0
I
-1
I
0
I
I
1 log l-
I I
2
3
Fig. 1. Source functions S,r for an isothermal slab atmosphere of optical thickness T = 104. Full lines: partial redistribution (PRD) with R,, a, = a, = 10m3.The curves are labeled with emission frequencies in units of Doppler width. Dashed lines: frequency-independent source function for complete redistribution (CRD), a = a,+ a, = 2 x IO-‘. The atmosphere is self-radiating with c = 10d4 and without incident radiation. The abscissa is mean optical depth r.
Emergent line profiles were found to be insensitive to small inaccuracies in the evaluation of R, discussed in I and IV. Therefore, in transfer calculations, one can use a less accurate integration in evaluating the E-term [see Eq. (Cl) of I, where M 2: 25 and h z 0.2 seem to be sufficient], as well as a lower-order quadrature in the angle-averaging procedure @-point Gaussian quadrature). This choice of parameters reduces time and storage requirements considerably. Nevertheless, our results are based on a relatively precise Rv (using M = 100, h = 0.05, and 16-point Gaussian quadrature). (a) Slab atmospheres The depth-dependence of the line source function for various frequencies within a slab of optical thickness T = 104, with no incident radiation at the bottom, is shown in Fig. 1. The corresponding frequency-independent source function for the case of complete redistribution is also presented. The damping parameters were set equal to al = a,, = 10m3. The corresponding normally emergent intensity is displayed in Fig. 2, together with those for the case of R,,and CRD (with a = a, + a, = 2 x 10w3).As the line is symmetric around the center, only halves of profiles are drawn. As was demonstrated in the literature,*.’ ’ there exist large differences between calculated profiles assuming CRD and R,,, particularly in the line wings. This well-known effect is caused by the high degree of coherence of the scattering process described by the function R,,. As expected, the intensity for Rv lies between those for R,,and CRD. This result can be readily explained on the basis of the general properties of resonance scattering in subordinate lines. Indeed, in contrast to the case of R,,where a photon absorbed at frequency x’ in the line wing (x’ 2 3) will most probably be re-emitted in the vicinity of x = I’ (the difference between x and x’ in the laboratory frame is caused by the Doppler redistribution), there is an additional probability for a photon to be emitted in the vicinity of the line center for R,. The ratio of the probability of (almost) coherent re-emission to the probability of re-emission near the center is proportional to au/a,, as was demonstrated in IV. We note that similar behavior is encountered also for resonance lines and is attributed to elastic collisions. I5 To avoid misunderstandings, we stress that the above discussed property of Rv is valid regardless of the presence of collisions and is caused by broadening of the lower state of a given transition. The collisional redistribution is properly accounted for by Eq. (2.4).
Non-coherent
scattering in subordinate
I r log
0.0
/ 0
I 2
,
I
4
x
I
6
163
lines-V
I 8
I
1 10
Fig. 2. Normally emergent intensities are shown for the source functions displayed in Fig. 1. Full line: PRD with R,,, a,= ~1, = 10e3; dashed line: CRD with (I = 2 x 10m3. The case with R,, (dash-dotted line), LI= 2 x IO-j, is added for comparison. Abscissa: emission frequency in units of Doppler width.
The physical explanation for the behavior of the transfer solutions with Rv is similar to that for RI*. In the line core, photons can only partly be scattered towards the wings and, therefore, the thermalization in the core is more rapid than for CRD but slower than for RI,. In the wings, the source function lies below the corresponding CRD values in the region of the monochromatic optical depth near unity because photons are less efficiently fed into the wings from the core than for CRD. Consequently, the corresponding emergent intensity is lower. The effect of enhancing the ratio ~,/a, is illustrated in Figs. 3 and 4. The emergent intensity now appears to depart relatively more from CRD.
0.0
r
log s
-1.0
-2.0
Fig. 3. As in Fig. 1 but for (I,= 10e4, a,= 1O-3 (a = 1.1 x 10m3).
I. HUBE& and P. HEINZEL
164
Fig. 4. Normally emergent intensities are shown for the source functions displayed in Fig. 3. n, = lo-*, a, = lo-’ (a = 1.1 x 10m3).Otherwise, the same models are used as in Fig. 2.
(b) Semi-infinite atmospheres In Fig. 5, the line source function is shown as a function of optical depth for various frequencies within a semi-infinite atmosphere, along with the corresponding frequencyindependent source function for CRD. The damping parameters were set equal to al = a, = 10e3. The overall behavior of the line source function is analogous to that of R,, discussed by Hummer,’ the departures from CRD being much smaller because of diminished influence of wing coherence. The corresponding normally emergent intensities for Rv, RII, and CRD are shown in Fig. 6. Differences between Rv and CRD are very small
0.0
-1.0
T:m
1 -1
0
I 1
/
I
I 3
2
i
I
1
5
6
log 7Fig. 5. The source functions S, are shown for an isothermal, semi-infinite atmosphere with t = 10m4.Full lines: PRD with R,,a,= a,= IO-'; the curves are labeled with emission frequencies. Dashed line: frequency-independent source function for CRD, D = 2 x lo-‘. Abscissa: the mean optical depth T.
165
Non-coherent scattering in subordinate lines-V
t
3
0
10
x
Fig. 6. Normally emergent intensities are shown for the source functions displayed in Fig. 5. Full line: PRD With Rv. a, = a, = lo-); dashed line: CRLI with II = 2 x lo-‘. The case with R,, (dash-dotted line) is added for comparison. Abscissa: the emission frequency in Doppler-width units.
and are therefore
of little practical
importance
in interpreting
subordinate
line profiles.
An interesting effect is the departure from CRD in the line core, where the intensities for both Rv and R,, are about 20% lower than for CRD. This difference may be of some interpretational significance. However, in actual stellar atmospheric applications, the shape of the line core is affected by many other factors that are not considered here (temperature and density stratification, effects of multi-level coupling, etc.) and also by other physical uncertainties (e.g. microturbulent velocity, rotation, etc.) which may well have comparable or greater effects on the observed line profiles. As for finite atmospheres, departures of the transfer solutions with Rv from those for CRD are much more pronounced for larger values of the ratio a,/~, (see Figs. 7 and 8).
I
-1
1
0
I
/ 1
I
I
2
log r
I 3
I
I 4
I
I 5
Fig. 7. As in Fig. 5 but for a,= lo-‘, u, = 10m3(a = 1.1 x lo-‘).
1
6
I. HUBEN% and
166
P. HEINZEL
Fig. 8. Normally emergent intensities are shown for the source functions displayed in Fig. LI,= 10e4, a, = 10m3, a = I. 1 x 10-l. Otherwise, the same models are used as in Fig. 6.
7:
(c) Approximate form for Rv
We suggested in IV an approximate form for R, which represents a simple generalization of that for RI, introduced by Jefferies and WhiteI and modified by Kneer.’ The function Rv is approximated by RV(x’, x) z a(x’, x)cp,b(x’
where 6(x’ -x)
is the Dirac delta-function
-x)
+ [l - a(x’, x)]q,q,,
and a(x’, x) is defined by
4x’, x) = [au/h + aJaW, XL where aK(x’, x) is the function introduced
(3.3)
(3.4)
by Kneer,’ viz.,
aK(x’, x) = aK(x,x’) = a [max (Ix’I, /xl)],
(3.5)
Z aK(x, x) = a(x)=
a K(~, x ‘)cp,, dx ‘,
s -*
1 -exp{-[(x
-A)/x,]~},
for x > A
(3.6) (3.7)
and a(x)=O,
for x6A.
This approximation to Rv satisfies the requirements of normalization and symmetry. We have tested several values of x,, and A and found that Kneer’s choice, namely, x0 = 2, A = 2, yields the most accurate results for subordinate lines.? Comparisons of normally emergent intensities, calculated with exact and approximate forms for R, for both finite and semi-infinite atmospheres (with a, = a, = 10e3 or a, = lo-‘, a, = lo-‘) are presented in Figs. 9 and 10, respectively. It is readily seen that the agreement tin fact, the approximation (3.3) does not allow for the Doppler diffusion in the line wings, due to the presence of the 6-function in its second term. Consequently, this approximation is effectively more suitable for R, than for R,,, because the former redistribution contains always a non-zero contribution of complete redistribution. Mathematically, a(x)-+ I for large x in the case of R,,, while n(x)+o,,/(a, + u,) for R,.
Non-coherent
scattering in subordinate
lines-V
167
-0.5
-1.0
-1.5
-2.0
Fig. 9. A comparison of normally emergent intensities is shown for the exact R, (dashed line) and an approximate form of Rv (full tine); a, = a, = 10m3.The model atmospheres are the same as in
previousfigures.
between the two approaches is very good, particularly for a, N ~1,.The approximation (3.3) thus offers a very simple and easy way of calculating radiative transfer in subordinate lines without explicit evaluation of the complicated Rv. 4. CONCLUSIONS
The results presented here were aimed at a demonstration of effects of partial redistribution on the subordinate line formation. For the sake of simplicity, we have chosen an idealized physical model that allows us to use the usual two-level-atom form for the line source function, modified by the presence of the redistribution function R, in the scattering integral. Line transfer calculations were performed for plane-parallel, isothermal atmospheres. This approach shows the importance of the redistribution function Rv and allows direct comparison with similar studies of resonance-line formation.
-0.5
Fig. 10. As in Fig. 9 but for u,= 10m4, II, = 10-3.
168
I. HUBERTand P. HE~NZEL
The principal conclusions may be summarized as follows: (1) Generally, the emergent line profiles calculated with the function Rv lie between those for complete redistribution and R,, (coherent scattering in the atom frame). Furthermore. the larger the ratio a,, a,, the greater are departures from complete redistribution” (2) For semi-infinite atmospheres, complete redistribution appears to be a good approximation for subordinate lines, at least for a, z a,. (3) For finite slab atmospheres, the effect of Rv is, in general, more pronounced than in semi-infinite atmospheres, particularly for the line wings. (4) Our approximate form for R, is a generalization of those of Jefferies and White and Kneer for RI, and appears to be a satisfactory approximation for transfer calculations, particularly for a, z a,. Because of our choice of L ‘v 1, the results represent an upper limit for actual departures from complete redistribution for subordinate lines using the simplified physical model specified by assumptions (i)-(v) (see Section 2). On the other hand, the line source function of Eq. (2.6) still underestimates departures form CRD because the only resonance scattering considered here is 2+3-+2. In principle, we should allow also for other possible sequences ending with the transitions 3-2 and 2+3 (see Ref. 6). Appropriate model calculations must be performed before a more complete understanding of subordinate line formation will be achieved. are indebted to Dr. H. Frisch (Observatoire de Nice) for providing us with her copy of a natural cubic splines computer code of Adams, Hummer, and Rybicki. The numerical calculations were performed with the EC 1040 computer of the Ondiejov Observatory Computing Centre.
Acknowledgements-We
REFERENCES 1. P. Heinzel, JQSRT25, 483 (1981). 2. P. Heinzel and I. Hubeny, JQSRT 27, 1 (1982). 3. P. Heinzel and I. Hubeny, JQSRT 30, 77 (1983). 4. D. G. Hummer, Mon. Not. R. Astr. Sot. 125, 21 (1962). 5. F. Kneer, Astrophys. J. 200, 367 (1975). 6. I. Hub&, J. Oxenius, and E. Simonneau, JQSRT 29, 477 and 495 (1983). 7. E. H. Avrett and D. G. Hummer, Mon. Nor. R. Astr. Sot. 130, 295 (1965). 8. D. G. Hummer, Mon. Not. R. Astr. Sot. 145, 95 (1969). 9. I. Hubeny, JQSRT 27, 593 (1982). 10. I. Hubenjr, Bull. Astron. Inst. Czechosl. 32, 271 (1981). 11. D. Mihalas, S&lur Atmospheres, 2nd Edn. Freeman, San Francisco (1978). 12. L. H. Auer, JQSRT 16, 931 (1976). 13. T. F. Adams, D. G. Hummer, and G. B. Rybicki, JQSRT 11, 1365 (1971). 14. L. H. Auer and D. Mihalas, Mon. Not. R. Astr. Sot. 149, 60 (1970). 15. A. Omont, E. W. Smith, and J. Cooper, Asrrophys. J. 175, 185 (1972) 16. J. T. Jefferies and 0. White, Astrophys. J. 132, 767 (1960).