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Global and local Doppler-profile escape factors for plane-parallel geometry
Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 55}63
Global and local Doppler-pro"le escape factors for plane-parallel geometry A.K. Bhatia *, S.O. Kastner q Laboratory for Astronomy and Solar Physics, Goddard Space Flight Center, Code 681, Greenbelt, MD 20771, USA 1-A Ridge Road, Greenbelt, MD 20770, USA Received 1 December 1998
Abstract Global (source- and direction-averaged) and local (direction-averaged) Doppler-pro"le escape factors for plane-parallel sources are calculated and tabulated for a range of line-center optical thicknesses q , and are compared with published or derived small- and large-q approximations. Integration (averaging) over the local escape factor provides a consistency check on both the global and local evaluations. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Global; Local; Doppler-pro"le escape factors
1. Introduction In recent papers [1,2], Doppler-pro"le photon escape factors and monodirectional escape probabilities have been evaluated for the case of cylindrical and hemispherical geometry. The simpler case of plane-parallel `slaba geometry would seem to have been already dealt with. However, on surveying the literature it was observed that while the slab monodirectional escape probability is well-known and has been tabulated (cf. [3]), there exists no tabulation of the global escape factor (i.e. photon escape in all directions, averaged over the source) or of local escape factors (photon escape from a given point in the source). Explicit expressions for these quantities are also well known but have not been numerically evaluated except for some approximate values given by Capriotti [4]. Though few real sources may be well represented by plane-parallel
* Corresponding author. Tel.: 001-301-296-8812; fax: 001-301-286-1753 . q Deceased. E-mail address: [email protected] (A.K. Bhatia). 0022-4073/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 9 9 ) 0 0 1 9 3 - 4
56
A.K. Bhatia, S.O. Kastner / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 55}63
geometry, tabulations for such geometry can be useful, for example in comparing with and checking multilevel calculations of opacity in other geometries. Tables of the global and local Doppler-pro"le escape factors are therefore presented here for a range of optical thicknesses of the plane-parallel source. Checks on the tabulated values are described and a brief discussion of the range of applicability is included.
2. Analysis The line-center optical thickness of the slab, of thickness ¸, will be designated by q ,k ¸,¹, where k is the line-center absorption coe$cient. The notation used below for escape factors and escape probabilities will be that of Kastner and Kastner [3], described also by Kastner and Bhatia [5] in their Appendix A. 2.1. The Doppler-proxle global escape factor SEF (q ) " A discussion of expressions already derived for this quantity, in the literature, was given in Appendix B of Kastner and Bhatia [5] and is summarized here. (a) Capriotti [4] expressed SEF (q ) as " q 1 (1) k 1!exp ! exp(!x) dk dx. e ,SEF (q )" " k (pq \ One must be careful to note that Capriotti's q is one-half the present q . (b) Irons [6] expressed SEF (q ) as the `uniformly excited upper limita: " 1 " k z (2a) h "SEF (q )" ¹ dk dz, " D k where D is the thickness of the slab lying between z"0 and z"D, k is the line-center absorption coe$cient, and ¹(q) is the transmission factor
¹(t)"
\
(x) exp[!t(x)] dx
(2b)
with (x) the Doppler pro"le function [exp(!x)]/(p. (c) The monochromatic escape factor P(q) of Case et al. [7] is 1 P(q)" [!E (q)]. q
(3)
This can be integrated over the Doppler pro"le to give: 1 SEF (q )" " (p
\
P[q(x)] exp(!x) dx,
(4a)
A.K. Bhatia, S.O. Kastner / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 55}63
57
where q(x)"q exp(!x). This results in the integral form 1 SEF (q )" " (pq
\
[!E (q exp(!x)] dx.
(4b)
The three expressions (1), (2) and (4) are of course equivalent. Here we used expression (4b), approximating it by the Gauss}Hermite quadrature:
exp(x) 1 G [!E (q exp(!x))] w SEF K G G " (p q G
(5a)
with N"40 [8]. The exponential integral E (y) was calculated using the recursion relation reproduced by Abramowitz and Stegun [9, expression 5.1.14]. Explicitly, E (t)"+e\R!t[e\R!tE t)],.
(5b)
Two checks on this computation of SEF (q ) are available. Capriotti gave small- and large-q " approximations for e ,SEF (q ), as follows [his expressions (88a), (88b); but note that typo " graphical errors in (88a) have been corrected, as given here, by Kastner and Bhatia [10]]; in these expressions the symbol q,q /2, i.e. Capriotti's q is 1/2 of the presently de"ned q : (!1)IqI> e (q),SEF (q)K1!0.8293q#0.7071q ln(2q)# " k(k#2)!(k#2) I
(6a)
for q42.5, and
1 0.25 K (ln 2q) # #0.14 ln(2q) 2q(p
(6b)
for q'2.5. It is then of interest also to compare values given by these approximate expressions with the exact values obtained from (4b). Another check is available for small q , by integrating (averaging) the CDP monochromatic approximation P "1!(q/2) ln(1/q)!(q/2)(1.5!c)!q/6
(7a)
over the Doppler pro"le, to give: 1 SEF (q )K " (p
exp(x) q exp(!x) 1! ln q 2 \
q exp(!x) q exp(!2x) ! (1.5!c)! exp(!x) dx, 2 6
(7b)
58
A.K. Bhatia, S.O. Kastner / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 55}63
where c"0.577216 the Euler constant. This can be evaluated by the Gauss}Hermite quadrature:
exp(x) q exp(!x) 1 G G ln w 1! SEF (q )K " q 2 (p G G q q ! exp(!x)(1.5!c) ! exp(!2x) . G G 2 6
(7c)
2.2. The local escape factor SEF (y,q ) " 2.2.1. For a distance into the source of y¸, with y dimensionless and of range 04y41, the appropriate expression for the local escape factor in plane-parallel geometry is SEF (y,q )"+K ((pq y)#K [(pq (1!y)],, "
(8)
where the kernel function K ((pq ) is given by K ((pq )"
(x)E (q exp(!x)) dx (9) " \ and from source symmetry we are interested in the half-range 04y40.5. The kernel function K ((pq ) can be calculated using the Pade approximation given by Hummer [11]. This procedure however deserves some discussion because there are two di!erent de"nitions of optical thickness q in the literature, and because the Hummer prescription is not transparent, partly due to typographical ambiguities. The quantity q used here is essentially that de"ned by Ivanov [12], i.e. q ,q'4; whereas Hummer uses the q & de"ned by Avrett and Hummer [13] which is related to q'4 by q &,(pq'4, i.e. q & incorporates the factor (p. Hummer's procedure is then as follows, using the notation q ,q & for convenience: & (a) For the range 04q 438: & q q (9a) K (q )" & ln & #KI (q ), & & (p (2p
where KI (q ) is given by & P (q ) p qG KI (q )" & , G G & & Q (q ) q qG G G & K & with the coe$cients (p , q ) given in Table 1 of Hummer [11]. G G (b) For the range q 538: & q P (m) p mG 2q ln & K (q )" , G G & & Q (m) q mG (p K G G
(9b)
(10)
with m,ln\(q /(p), and with (di!erent) coe$cients (p , q ) given in Table 2 of Hummer [11]. & G G
A.K. Bhatia, S.O. Kastner / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 55}63
59
Checks on the calculated values of the kernel function are available from the earlier tables of Ivanov and Shcherbakov [14] [their Table 1, in which they denote K (q'4) by N (q )] and of Crosbie and Viskanta [15] [their Table 3, in which the appropriate kernel function K (q ),K (q'4) is listed under the pro"le heading `Dopplera]. 2.2.2. An alternative expression for the local escape factor was given by Pham and Hoe [16] as (in the present notation)
yq 1 q y , " SEF exp(!x) exp ! 1# e\V " 2 2 2 2 2(p \ q y #exp ! 1! e\V 2 2 q exp(!2x) # 4(p \
dx
q y y 1# e\V 1# E 2 2 2
y y q 1! e\V # 1! E 2 2 2
dx.
(11)
This expression was not evaluated (or checked) here.
3. The relation between global and local escape factors As in the cylindrical case [2], a valuable consistency check on both the local and global escape factor evaluations is a!orded by the fact that the average of the local escape factor through the slab must equal the global escape factor (as shown analytically e.g. by Kastner and Bhatia [5]):
SEF (y,q ) dy"SEF (q ). (12) " " This is conveniently evaluated using a Gaussian moment quadrature [9, Table 25.8] with k"0, as , LHSK w SEF (y ,q ). G " G J
(13)
4. Results and tabulations Values of the global escape factor SEF (q ) are given and compared in Table 1a as derived from " the CDP small-q approximation (7b), the Capriotti small-q approximation (6a), and as calculated by integration (5) using 40 quadrature nodes. The tabulated values are believed to be accurate to six signi"cant places. Table 1b compares the Capriotti large-q approximation with the corresponding exact values of SEF (q ). "
60
A.K. Bhatia, S.O. Kastner / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 55}63 Table 1 Capriotti e
Exact SEF
(a) Global escape factors for small opacities !3.0 0.997143 !2.8 0.995730 !2.6 0.993641 !2.4 0.990569 !2.2 0.986080 !2.0 0.979562 !1.8 0.970180 !1.6 0.956806 !1.4 0.937967 !1.2 0.911816 !1.0 0.876165 !0.8 0.828647 !0.6 0.767082 !0.4 0.690040 !0.2 0.597340 0.0 0.489133
0.997143 0.995730 0.993641 0.990570 0.986080 0.979562 0.970180 0.956806 0.937968 0.911819 0.876175 0.828688 0.767241 0.690665 0.599769 0.498427
0.997143 0.995730 0.993641 0.990569 0.986080 0.979562 0.970180 0.956806 0.937968 0.911819 0.876175 0.828688 0.767242 0.690667 0.599773 0.498436
Log(q )
Capriotti e
Exact SEF
(b) Comparison of Capriotti approximation with 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
exact SEF for larger opacity 0.498427 0.498436 0.394007 0.394022 0.296053 0.296081 0.212945 0.212996 0.150350 0.148568 0.102806 0.101653 0.695106-1 0.686963-1 0.465991-1 0.460171-1 0.310314-1 0.306190-1 0.205540-1 0.202668-1 0.135545-1 0.133562-1 0.890585-2 0.876993-2 0.583338-2 0.574270-2 0.381073-2 0.375023-2 0.248368-2 0.244252-2
log(q )
CDP small-q approx.
Expression (7b). Expression (6a). Expression (5a).
The Capriotti small-q approximation (6a) is seen to be quite accurate. For values of q larger than about 5, however, the Capriotti large-q approximation (6b) gives less accuracy, deviating from exact values of SEF (q ) by #1% or more. " The check of calculated local escape factors against the corresponding global escape factors, according to relation (13), is shown in Table 2 for four values of opacity q . Excellent agreement is
A.K. Bhatia, S.O. Kastner / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 55}63 Table 2 Comparison of integrated local escape factors with exact SEF Log(q )
Exact SEF
SEF(y, q ) dy
!2.0 !1.0 0.0 1.0
0.979562 0.876175 0.498436 0.101653
0.979562 0.876172 0.498401 0.101307
Expression (13).
Table 3 Local escape factor SEF (y,q ) " y Log(q )
0.1}7
0.1
0.2
0.3
0.4
0.5
!3.0 !2.8 !2.6 !2.4 !2.2 !2.0 !1.8 !1.6 !1.4 !1.2 !1.0 !0.8 !0.6 !0.4 !0.2 0.00 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
0.997320 0.996010 0.994085 0.991273 0.987193 0.981325 0.972970 0.961216 0.944930 0.922784 0.893382 0.855537 0.808768 0.754031 0.694505 0.635903 0.585388 0.548577 0.526238 0.514375 0.508173 0.504758 0.502807 0.501670 0.500999 0.500599 0.500358 0.500210 0.500117 0.500054
0.997205 0.995828 0.993796 0.990815 0.986469 0.980178 0.971155 0.958345 0.940395 0.915634 0.882143 0.837952 0.781453 0.712063 0.631070 0.542240 0.451417 0.364424 0.284265 0.210946 0.145179 0.90766-1 0.51743-1 0.28120-1 0.15502-1 0.88514-2 0.51687-2 0.30570-2 0.18234-2 0.10946-2
0.997143 0.995730 0.993641 0.990569 0.986079 0.979561 0.970177 0.956801 0.937959 0.911802 0.876142 0.828619 0.767091 0.690331 0.599016 0.496749 0.390394 0.288774 0.199778 0.128092 0.75597-1 0.41832-1 0.22892-1 0.12913-1 0.74898-2 0.44125-2 0.26251-2 0.15726-2 0.94724-3 0.57300-3
0.997104 0.995668 0.993543 0.9904014 0.985833 0.979172 0.969562 0.955829 0.936427 0.909395 0.872378 0.822780 0.758144 0.676886 0.579410 0.469450 0.354940 0.247186 0.157448 0.92238-1 0.51014-1 0.27939-1 0.15739-1 0.91181-2 0.53673-2 0.31911-2 0.19107-2 0.11504-2 0.69561-3 0.42213-3
0.997082 0.995633 0.993488 0.990326 0.985694 0.978953 0.969216 0.955283 0.935566 0.908043 0.870265 0.819507 0.753143 0.669400 0.568565 0.454520 0.335926 0.225638 0.136791 0.76350-1 0.41395-1 0.22970-1 0.13172-1 0.77064-2 0.45623-2 0.27229-2 0.16352-2 0.98670-3 0.59773-3 0.36327-3
0.997075 0.995622 0.993470 0.990298 0.985650 0.978882 0.969104 0.955106 0.935287 0.907605 0.869582 0.818449 0.751527 0.666986 0.565080 0.449747 0.329904 0.218927 0.130547 0.71775-1 0.38782-1 0.21656-1 0.12482-1 0.73218-2 0.43415-2 0.25941-2 0.15592-2 0.94146-3 0.57064-3 0.34697-3
61
62
A.K. Bhatia, S.O. Kastner / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 55}63
seen for the lower values of q , with agreement decreasing for the higher values. The latter departure is due to the restricted moment quadrature used for the integration, and is not signi"cant for the present purpose of verifying the general relation and also verifying, through consistency of the independently calculated quantities, the calculated values of both the local escape factor SEF (y,q ) and the global escape factor SEF (q ). " " Table 3 gives values of the local escape factor over the range!3.04log q 42.8, for a location very near the surface of the slab source (y"10\), and at locations up to y"0.5 which is at the midplane of the source. The tabulated values of course apply for corresponding locations in the other half of the source (e.g. the location y"0.8 has the same local escape factor as y"0.2, the location y"0.7 has the same local escape factor as y"0.3, etc.), by symmetry. As expected physically, and as a further veri"cation of the correctness of the numerical calculation, the surface escape factor SEF (y"0, q ) approaches the value 0.5 at high opacity. " 5. Discussion The two principal assumptions underlying the escape probabilities calculated here are the assumption of a spatially constant source function within the source, and the assumption of complete frequency distribution in scattering events. As noted in [1], high source densities/optical thicknesses will favor the second assumption but invalidate the "rst assumption. In particular, the source function will depart signi"cantly from spatial constancy even in an isothermal source, when the line-center opacity q is greater than about 10. The line pro"le then becomes self-reversed and the usual escape probability approach loses validity. However, we have believed that it is useful to compute values of the escape factors for higher values of q (as was done in [1,2]) because they can be used to compare with the results of more detailed calculations in the future, if such results become available.
Acknowledgements This work was supported by NASA-RTOP grant 344-12-53-14.
References [1] [2] [3] [4] [5] [6] [7]
Bhatia AK, Kastner SO. JQSRT 1997;58:347. Bhatia AK, Kastner SO. JQSRT 1998;60:543. Kastner SO, Kastner RE. JQSRT 1990;44:275. Capriotti ER. Astrophys J 1965;142:1101. Kastner SO, Bhatia AK. Astrophys J 1992;401:416. Irons FE. JQSRT 1979;22:1. Case KM, DeHo!man F, Placzek G. Introduction to the theory of neutron di!usion. vol. 1. Washington DC: Los Alamos Scienti"c Laboratory, U.S. Government Printing O$ce. 1953. [8] Stroud AH, Secrest D. Gaussian quadrature formulas. Englewood Cli!s, NJ: Prentice-Hall. 1966.
A.K. Bhatia, S.O. Kastner / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 55}63
63
[9] Abramowitz M, Stegun IA. Handbook of mathematical functions. Washington DC: U.S. Government Printing O$ce. 1965. [10] Kastner SO, Bhatia AK. Astrophys J Suppl 1989;71:665. [11] Hummer DG. JQSRT 1981;26:187. [12] Ivanov VV. Transfer of radiation in spectral lines. NBS Special Publication 385. Washington DC: U.S. Government Printing O$ce. 1973. [13] Avrett EH, Hummer DG. Mon Not Roy Astron Soc 1965;130:295. [14] Ivanov VV, Shcherbakov VT. Astrophysics 1965;1:17. [15] Crosbie AL, Viskanta R. JQSRT 1970;10:487. [16] Pham DW, Hoe N. Phys Lett 1972;38A:521.
Global and local Doppler-pro"le escape factors for plane-parallel geometry A.K. Bhatia *, S.O. Kastner q Laboratory for Astronomy and Solar Physics, Goddard Space Flight Center, Code 681, Greenbelt, MD 20771, USA 1-A Ridge Road, Greenbelt, MD 20770, USA Received 1 December 1998
Abstract Global (source- and direction-averaged) and local (direction-averaged) Doppler-pro"le escape factors for plane-parallel sources are calculated and tabulated for a range of line-center optical thicknesses q , and are compared with published or derived small- and large-q approximations. Integration (averaging) over the local escape factor provides a consistency check on both the global and local evaluations. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Global; Local; Doppler-pro"le escape factors
1. Introduction In recent papers [1,2], Doppler-pro"le photon escape factors and monodirectional escape probabilities have been evaluated for the case of cylindrical and hemispherical geometry. The simpler case of plane-parallel `slaba geometry would seem to have been already dealt with. However, on surveying the literature it was observed that while the slab monodirectional escape probability is well-known and has been tabulated (cf. [3]), there exists no tabulation of the global escape factor (i.e. photon escape in all directions, averaged over the source) or of local escape factors (photon escape from a given point in the source). Explicit expressions for these quantities are also well known but have not been numerically evaluated except for some approximate values given by Capriotti [4]. Though few real sources may be well represented by plane-parallel
* Corresponding author. Tel.: 001-301-296-8812; fax: 001-301-286-1753 . q Deceased. E-mail address: [email protected] (A.K. Bhatia). 0022-4073/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 9 9 ) 0 0 1 9 3 - 4
56
A.K. Bhatia, S.O. Kastner / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 55}63
geometry, tabulations for such geometry can be useful, for example in comparing with and checking multilevel calculations of opacity in other geometries. Tables of the global and local Doppler-pro"le escape factors are therefore presented here for a range of optical thicknesses of the plane-parallel source. Checks on the tabulated values are described and a brief discussion of the range of applicability is included.
2. Analysis The line-center optical thickness of the slab, of thickness ¸, will be designated by q ,k ¸,¹, where k is the line-center absorption coe$cient. The notation used below for escape factors and escape probabilities will be that of Kastner and Kastner [3], described also by Kastner and Bhatia [5] in their Appendix A. 2.1. The Doppler-proxle global escape factor SEF (q ) " A discussion of expressions already derived for this quantity, in the literature, was given in Appendix B of Kastner and Bhatia [5] and is summarized here. (a) Capriotti [4] expressed SEF (q ) as " q 1 (1) k 1!exp ! exp(!x) dk dx. e ,SEF (q )" " k (pq \ One must be careful to note that Capriotti's q is one-half the present q . (b) Irons [6] expressed SEF (q ) as the `uniformly excited upper limita: " 1 " k z (2a) h "SEF (q )" ¹ dk dz, " D k where D is the thickness of the slab lying between z"0 and z"D, k is the line-center absorption coe$cient, and ¹(q) is the transmission factor
¹(t)"
\
(x) exp[!t(x)] dx
(2b)
with (x) the Doppler pro"le function [exp(!x)]/(p. (c) The monochromatic escape factor P(q) of Case et al. [7] is 1 P(q)" [!E (q)]. q
(3)
This can be integrated over the Doppler pro"le to give: 1 SEF (q )" " (p
\
P[q(x)] exp(!x) dx,
(4a)
A.K. Bhatia, S.O. Kastner / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 55}63
57
where q(x)"q exp(!x). This results in the integral form 1 SEF (q )" " (pq
\
[!E (q exp(!x)] dx.
(4b)
The three expressions (1), (2) and (4) are of course equivalent. Here we used expression (4b), approximating it by the Gauss}Hermite quadrature:
exp(x) 1 G [!E (q exp(!x))] w SEF K G G " (p q G
(5a)
with N"40 [8]. The exponential integral E (y) was calculated using the recursion relation reproduced by Abramowitz and Stegun [9, expression 5.1.14]. Explicitly, E (t)"+e\R!t[e\R!tE t)],.
(5b)
Two checks on this computation of SEF (q ) are available. Capriotti gave small- and large-q " approximations for e ,SEF (q ), as follows [his expressions (88a), (88b); but note that typo " graphical errors in (88a) have been corrected, as given here, by Kastner and Bhatia [10]]; in these expressions the symbol q,q /2, i.e. Capriotti's q is 1/2 of the presently de"ned q : (!1)IqI> e (q),SEF (q)K1!0.8293q#0.7071q ln(2q)# " k(k#2)!(k#2) I
(6a)
for q42.5, and
1 0.25 K (ln 2q) # #0.14 ln(2q) 2q(p
(6b)
for q'2.5. It is then of interest also to compare values given by these approximate expressions with the exact values obtained from (4b). Another check is available for small q , by integrating (averaging) the CDP monochromatic approximation P "1!(q/2) ln(1/q)!(q/2)(1.5!c)!q/6
(7a)
over the Doppler pro"le, to give: 1 SEF (q )K " (p
exp(x) q exp(!x) 1! ln q 2 \
q exp(!x) q exp(!2x) ! (1.5!c)! exp(!x) dx, 2 6
(7b)
58
A.K. Bhatia, S.O. Kastner / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 55}63
where c"0.577216 the Euler constant. This can be evaluated by the Gauss}Hermite quadrature:
exp(x) q exp(!x) 1 G G ln w 1! SEF (q )K " q 2 (p G G q q ! exp(!x)(1.5!c) ! exp(!2x) . G G 2 6
(7c)
2.2. The local escape factor SEF (y,q ) " 2.2.1. For a distance into the source of y¸, with y dimensionless and of range 04y41, the appropriate expression for the local escape factor in plane-parallel geometry is SEF (y,q )"+K ((pq y)#K [(pq (1!y)],, "
(8)
where the kernel function K ((pq ) is given by K ((pq )"
(x)E (q exp(!x)) dx (9) " \ and from source symmetry we are interested in the half-range 04y40.5. The kernel function K ((pq ) can be calculated using the Pade approximation given by Hummer [11]. This procedure however deserves some discussion because there are two di!erent de"nitions of optical thickness q in the literature, and because the Hummer prescription is not transparent, partly due to typographical ambiguities. The quantity q used here is essentially that de"ned by Ivanov [12], i.e. q ,q'4; whereas Hummer uses the q & de"ned by Avrett and Hummer [13] which is related to q'4 by q &,(pq'4, i.e. q & incorporates the factor (p. Hummer's procedure is then as follows, using the notation q ,q & for convenience: & (a) For the range 04q 438: & q q (9a) K (q )" & ln & #KI (q ), & & (p (2p
where KI (q ) is given by & P (q ) p qG KI (q )" & , G G & & Q (q ) q qG G G & K & with the coe$cients (p , q ) given in Table 1 of Hummer [11]. G G (b) For the range q 538: & q P (m) p mG 2q ln & K (q )" , G G & & Q (m) q mG (p K G G
(9b)
(10)
with m,ln\(q /(p), and with (di!erent) coe$cients (p , q ) given in Table 2 of Hummer [11]. & G G
A.K. Bhatia, S.O. Kastner / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 55}63
59
Checks on the calculated values of the kernel function are available from the earlier tables of Ivanov and Shcherbakov [14] [their Table 1, in which they denote K (q'4) by N (q )] and of Crosbie and Viskanta [15] [their Table 3, in which the appropriate kernel function K (q ),K (q'4) is listed under the pro"le heading `Dopplera]. 2.2.2. An alternative expression for the local escape factor was given by Pham and Hoe [16] as (in the present notation)
yq 1 q y , " SEF exp(!x) exp ! 1# e\V " 2 2 2 2 2(p \ q y #exp ! 1! e\V 2 2 q exp(!2x) # 4(p \
dx
q y y 1# e\V 1# E 2 2 2
y y q 1! e\V # 1! E 2 2 2
dx.
(11)
This expression was not evaluated (or checked) here.
3. The relation between global and local escape factors As in the cylindrical case [2], a valuable consistency check on both the local and global escape factor evaluations is a!orded by the fact that the average of the local escape factor through the slab must equal the global escape factor (as shown analytically e.g. by Kastner and Bhatia [5]):
SEF (y,q ) dy"SEF (q ). (12) " " This is conveniently evaluated using a Gaussian moment quadrature [9, Table 25.8] with k"0, as , LHSK w SEF (y ,q ). G " G J
(13)
4. Results and tabulations Values of the global escape factor SEF (q ) are given and compared in Table 1a as derived from " the CDP small-q approximation (7b), the Capriotti small-q approximation (6a), and as calculated by integration (5) using 40 quadrature nodes. The tabulated values are believed to be accurate to six signi"cant places. Table 1b compares the Capriotti large-q approximation with the corresponding exact values of SEF (q ). "
60
A.K. Bhatia, S.O. Kastner / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 55}63 Table 1 Capriotti e
Exact SEF
(a) Global escape factors for small opacities !3.0 0.997143 !2.8 0.995730 !2.6 0.993641 !2.4 0.990569 !2.2 0.986080 !2.0 0.979562 !1.8 0.970180 !1.6 0.956806 !1.4 0.937967 !1.2 0.911816 !1.0 0.876165 !0.8 0.828647 !0.6 0.767082 !0.4 0.690040 !0.2 0.597340 0.0 0.489133
0.997143 0.995730 0.993641 0.990570 0.986080 0.979562 0.970180 0.956806 0.937968 0.911819 0.876175 0.828688 0.767241 0.690665 0.599769 0.498427
0.997143 0.995730 0.993641 0.990569 0.986080 0.979562 0.970180 0.956806 0.937968 0.911819 0.876175 0.828688 0.767242 0.690667 0.599773 0.498436
Log(q )
Capriotti e
Exact SEF
(b) Comparison of Capriotti approximation with 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
exact SEF for larger opacity 0.498427 0.498436 0.394007 0.394022 0.296053 0.296081 0.212945 0.212996 0.150350 0.148568 0.102806 0.101653 0.695106-1 0.686963-1 0.465991-1 0.460171-1 0.310314-1 0.306190-1 0.205540-1 0.202668-1 0.135545-1 0.133562-1 0.890585-2 0.876993-2 0.583338-2 0.574270-2 0.381073-2 0.375023-2 0.248368-2 0.244252-2
log(q )
CDP small-q approx.
Expression (7b). Expression (6a). Expression (5a).
The Capriotti small-q approximation (6a) is seen to be quite accurate. For values of q larger than about 5, however, the Capriotti large-q approximation (6b) gives less accuracy, deviating from exact values of SEF (q ) by #1% or more. " The check of calculated local escape factors against the corresponding global escape factors, according to relation (13), is shown in Table 2 for four values of opacity q . Excellent agreement is
A.K. Bhatia, S.O. Kastner / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 55}63 Table 2 Comparison of integrated local escape factors with exact SEF Log(q )
Exact SEF
SEF(y, q ) dy
!2.0 !1.0 0.0 1.0
0.979562 0.876175 0.498436 0.101653
0.979562 0.876172 0.498401 0.101307
Expression (13).
Table 3 Local escape factor SEF (y,q ) " y Log(q )
0.1}7
0.1
0.2
0.3
0.4
0.5
!3.0 !2.8 !2.6 !2.4 !2.2 !2.0 !1.8 !1.6 !1.4 !1.2 !1.0 !0.8 !0.6 !0.4 !0.2 0.00 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
0.997320 0.996010 0.994085 0.991273 0.987193 0.981325 0.972970 0.961216 0.944930 0.922784 0.893382 0.855537 0.808768 0.754031 0.694505 0.635903 0.585388 0.548577 0.526238 0.514375 0.508173 0.504758 0.502807 0.501670 0.500999 0.500599 0.500358 0.500210 0.500117 0.500054
0.997205 0.995828 0.993796 0.990815 0.986469 0.980178 0.971155 0.958345 0.940395 0.915634 0.882143 0.837952 0.781453 0.712063 0.631070 0.542240 0.451417 0.364424 0.284265 0.210946 0.145179 0.90766-1 0.51743-1 0.28120-1 0.15502-1 0.88514-2 0.51687-2 0.30570-2 0.18234-2 0.10946-2
0.997143 0.995730 0.993641 0.990569 0.986079 0.979561 0.970177 0.956801 0.937959 0.911802 0.876142 0.828619 0.767091 0.690331 0.599016 0.496749 0.390394 0.288774 0.199778 0.128092 0.75597-1 0.41832-1 0.22892-1 0.12913-1 0.74898-2 0.44125-2 0.26251-2 0.15726-2 0.94724-3 0.57300-3
0.997104 0.995668 0.993543 0.9904014 0.985833 0.979172 0.969562 0.955829 0.936427 0.909395 0.872378 0.822780 0.758144 0.676886 0.579410 0.469450 0.354940 0.247186 0.157448 0.92238-1 0.51014-1 0.27939-1 0.15739-1 0.91181-2 0.53673-2 0.31911-2 0.19107-2 0.11504-2 0.69561-3 0.42213-3
0.997082 0.995633 0.993488 0.990326 0.985694 0.978953 0.969216 0.955283 0.935566 0.908043 0.870265 0.819507 0.753143 0.669400 0.568565 0.454520 0.335926 0.225638 0.136791 0.76350-1 0.41395-1 0.22970-1 0.13172-1 0.77064-2 0.45623-2 0.27229-2 0.16352-2 0.98670-3 0.59773-3 0.36327-3
0.997075 0.995622 0.993470 0.990298 0.985650 0.978882 0.969104 0.955106 0.935287 0.907605 0.869582 0.818449 0.751527 0.666986 0.565080 0.449747 0.329904 0.218927 0.130547 0.71775-1 0.38782-1 0.21656-1 0.12482-1 0.73218-2 0.43415-2 0.25941-2 0.15592-2 0.94146-3 0.57064-3 0.34697-3
61
62
A.K. Bhatia, S.O. Kastner / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 55}63
seen for the lower values of q , with agreement decreasing for the higher values. The latter departure is due to the restricted moment quadrature used for the integration, and is not signi"cant for the present purpose of verifying the general relation and also verifying, through consistency of the independently calculated quantities, the calculated values of both the local escape factor SEF (y,q ) and the global escape factor SEF (q ). " " Table 3 gives values of the local escape factor over the range!3.04log q 42.8, for a location very near the surface of the slab source (y"10\), and at locations up to y"0.5 which is at the midplane of the source. The tabulated values of course apply for corresponding locations in the other half of the source (e.g. the location y"0.8 has the same local escape factor as y"0.2, the location y"0.7 has the same local escape factor as y"0.3, etc.), by symmetry. As expected physically, and as a further veri"cation of the correctness of the numerical calculation, the surface escape factor SEF (y"0, q ) approaches the value 0.5 at high opacity. " 5. Discussion The two principal assumptions underlying the escape probabilities calculated here are the assumption of a spatially constant source function within the source, and the assumption of complete frequency distribution in scattering events. As noted in [1], high source densities/optical thicknesses will favor the second assumption but invalidate the "rst assumption. In particular, the source function will depart signi"cantly from spatial constancy even in an isothermal source, when the line-center opacity q is greater than about 10. The line pro"le then becomes self-reversed and the usual escape probability approach loses validity. However, we have believed that it is useful to compute values of the escape factors for higher values of q (as was done in [1,2]) because they can be used to compare with the results of more detailed calculations in the future, if such results become available.
Acknowledgements This work was supported by NASA-RTOP grant 344-12-53-14.
References [1] [2] [3] [4] [5] [6] [7]
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A.K. Bhatia, S.O. Kastner / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 55}63
63
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