The solution of the transfer equation in a homogeneous isotropically scattering plane-parallel medium

Details The solution of the transfer equation in a homogeneous isotropically scattering plane-parallel medium The solution of the transfer equation in a homogeneous isotropically scattering plane-parallel medium

Dec 1994

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994jqsrt..52..857r&link_type=abstract

Journal of Quantitative Spectroscopy & Radiative Transfer (ISSN 0022-4073), vol. 52, no. 6, p. 857-885

Mathematics

Logic

3

Light Scattering, Optical Thickness, Radiative Transfer, Boundary Conditions, Green'S Functions, Integral Equations, Laplace Transformation

Scientific paper

The exact solution of the transfer equation (TE) in a homogeneous plane-parallel medium of any optical extent (i.e., infinite, semi-infinite, or finite) is derived in the case of isotropic local scattering. The sources are not necessarily isotropic. We start from our recent study of the Schwarzschild-Milne (SM) integral equation as derived from the TE with appropriate boundary conditions. Using the Laplace transform on the range of the optical depth variable, the solution of the TE is obtained as the transform, by an introduced logical-operator, of the solution of the SM equation. The latter is then expressed in terms of the sources by means of the Green distribution of the SM equation. We are thus led to calculate the transform of this distribution by the logical-operator, which is a fundamental auxiliary function of plane-parallel media. This transform, namely Gamma, is thoroughly studied in the present article. The transform of the Gamma-function by the logical-operator, i.e., the bi-transform of the Green distribution, yields the solution of the TE in terms of the specified sources. It is thus the Green function of the problem. This function is studied in detail hereafter and written in terms of the Gamma-function by means of simple algebraic formulae. Analytical and numerical studies of the auxiliary functions introduced here and in a former article are required before a numerical evaluation of our solution can be obtained.

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