×

The Chandrasekhar function revisited. (English) Zbl 1360.82006

Summary: The need for an accurate (better than 10 significant digits) and fast algorithm for calculating the Chandrasekhar function, \(H(\mu, \omega)\), has stimulated the present analysis of different solutions of the relevant integral equation. It has been found that a very accurate analytical solution can be derived that is conveniently used in the range of small arguments, \(\mu\) and \(\omega\). In a limited range of arguments, the \(H\) function can be expressed in terms of a rapidly converging series of Bernoulli constants. For example, the \(H\) function for \(\mu = 1\) and \(\omega = 1\) was readily calculated with an accuracy of 31 digits. A new algorithm, derived from an integral representation, is proposed for efficient calculations. Together with an algorithm published by D. W. N. Stibbs and R. E. Weir [Mon. Not. R. Astron. Soc. 119, 512–525 (1959; Zbl 0092.22403)], this algorithm was used in calculations of extensive tables of the \(H\) function with an accuracy of 21 significant digits. Based on the above analysis, a mixed algorithm optimized with respect to the execution time was designed.

MSC:

82-04 Software, source code, etc. for problems pertaining to statistical mechanics
82-08 Computational methods (statistical mechanics) (MSC2010)
65R20 Numerical methods for integral equations
65D20 Computation of special functions and constants, construction of tables
82C70 Transport processes in time-dependent statistical mechanics

Citations:

Zbl 0092.22403
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Chandrasekhar, S., Radiative transfer, (1960), Dover Publications, Inc. New York · Zbl 0037.43201
[2] Hovenier, J. W.; van der Mee, C. V.M.; de Heer, D., Differential properties of \(H\)-functions with applications to optically thick planetary atmospheres and spherical clouds, Astronom. Astrophys., 207, 194-203, (1988)
[3] Das, R. N., Solution of a Riemann-Hilbert problem for a new expression of chandrasekhar’s \(H\)-function in radiative transfer, Astrophys. Space Sci., 317, 119-126, (2008) · Zbl 1161.85309
[4] Case, K. M.; Zweifel, P. F., Linear transport theory, (1967), Addison-Wesley Reading, MA · Zbl 0132.44902
[5] Tilinin, I. S.; Werner, W. S.M., Escape probability of Auger electrons from noncrystalline solids: exact solution in the transport approximation, Phys. Rev. B, 46, 13739-13746, (1992)
[6] Tilinin, I. S.; Jablonski, A.; Tougaard, S., Path-length distribution of photoelectrons emitted from homogeneous noncrystalline solids: consequences for inelastic background analysis, Phys. Rev. B, 52, 5935-5945, (1995)
[7] Jablonski, A.; Tilinin, I. S.; Powell, C. J., Mean escape depth of signal photoelectrons from amorphous and polycrystalline solids, Phys. Rev. B, 54, 10927-10937, (1996)
[8] Davidović, D. M.; Vukanić, J.; Arsenović, D., Two new analytic approximations of the chandrasekhars’s H function for isotropic scattering, Icarus, 194, 389-397, (2008)
[9] Basko, M. M., K-fluorescence lines in spectra of X-ray binaries, Astrophys. J., 223, 268-281, (1979)
[10] Lichtman, A. P.; Rybicki, G. B., Inverse Compton reflection: the steady-state theory, Astrophys. J., 236, 928-944, (1980)
[11] Hapke, B., Bidirectional reflectance spectroscopy: 1. theory, J. Geophys. Res. Solid Earth, 86, 3039-3054, (1981)
[12] Jablonski, A.; Tilinin, I. S., Towards a universal description of elastic scattering effects in X-ray photoelectron spectroscopy, J. Electron Spectrosc. Relat. Phenom., 74, 207-229, (1995)
[13] Jablonski, A.; Tougaard, S., Escape probability of electrons from solids. influence of elastic electron scattering, Surf. Sci., 432, 211-227, (1999)
[14] Tilinin, I. S.; Werner, W. S.M., Angular and energy distribution of Auger and photoelectrons escaping from non-crystalline solid surfaces, Surf. Sci., 290, 119-133, (1993)
[15] Werner, W. S.M., Influence of multiple elastic and inelastic scattering on photoelectron line shape, Phys. Rev. B, 2964-2975, (1995)
[16] Karanjai, S.; Karanjai, M., Polynomial approximations of chandrasekhar’s \(H\)-function for isotropic conservative scattering, Astrophys. Space Sci., 178, 331-333, (1991)
[17] Kawabata, K.; Limaye, S. S., Rational approximation for chandrasekhar’s \(H\)-function for isotropic scattering, Astrophys. Space Sci., 332, 365-371, (2011) · Zbl 1237.85025
[18] Chandrasekhar, S.; Breen, F. H., On the radiative equilibrium of a stellar atmosphere. XIX, Astrophys. J., 106, 143-144, (1947)
[19] Stibbs, D. W.N.; Weir, R. E., On the \(H\)-functions for isotropic scattering, Mon. Not. R. Astron. Soc., 119, 512-525, (1959) · Zbl 0092.22403
[20] Placzek, G., The angular distribution of neutrons emerging from a plane surface, Phys. Rev., 72, 556-558, (1947) · Zbl 0029.33501
[21] Bosma, P. B.; de Rooij, W. A., Efficient methods to calculate chandrasekhar’s \(H\)-functions, Astronom. Astrophys., 126, 283-292, (1983)
[22] Hovenier, J. W.; van der Mee, C. V.M.; de Heer, D., Differential properties of \(H\)-functions with applications to optically thick planetary atmospheres and spherical clouds, Astronom. Astrophys., 207, 194-203, (1988)
[23] Hiroi, T., Recalculation of the isotropic H functions, Icarus, 109, 313-317, (1994)
[24] Jablonski, A., An effective algorithm for calculating the Chandrasekhar function, Comput. Phys. Commun., 183, 1773-1782, (2012) · Zbl 1305.82007
[25] Jablonski, A., Improved algorithm for calculating the Chandrasekhar function, Comput. Phys. Commun., 184, 440-442, (2013) · Zbl 1306.82002
[26] Program CHANDRAS_v2, Computer Physics Communications Program Library, Programs in Physics and Physical Chemistry. web address: http://cpc.cs.qub.ac.uk/summaries/AEMC_v2_0.html.
[27] Rutily, B.; Bergeat, J., The Neumann solution of the multiple scattering problem in plane-parallel medium-iia. semi-infinite spaces and \(H\)-function, J. Quant. Spectrosc. Radiat. Transfer, 38, 47-60, (1987)
[28] Press, W. H.; Teukolski, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical recipes. the art of scientific computing, (2007), Cambridge University Press Cambridge · Zbl 1132.65001
[29] Ivanov, V. V., Transfer of radiation in spectral lines, U.S. department of commerce, (National Bureau of Standards Special Publication, vol. 385, (1973), U.S. Government Printing Office Washington DC), 128
[30] Dudarev, S. L.; Whelan, M. J., Temperature dependence of elastic backscattering of electrons from polycrystalline solids, Surf. Sci., 311, L687, (1994)
[31] Adams, E. P., Smithsonian mathematical formulae and tables of elliptic functions, 120, (1922), Smithsonian Institution Washington
[32] Gradshteyn, I. S.; Ryzhik, I. M., Table of integrals, series, and products, 17, (2007), Amsterdam Elsevier · Zbl 1208.65001
[33] Jablonski, A.; Powell, C. J., Improved analytical formulae for correcting elastic- scattering effects in X-ray photoelectron spectroscopy, Surf. Sci., 604, 327-336, (2010)
[34] Jablonski, A.; Powell, C. J., Elastic photoelectron scattering effects in quantitative X-ray photoelectron spectroscopy, Surf. Sci., 606, 644-651, (2012)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.