Mathematical aspects of the plane-parallel transfer equation

• Published in: Journal of quantitative spectroscopy & radiative transfer 1997-09, Vol.58 (3), p.355-373
• Main Authors: Efimov, G.V, von Waldenfels, W, Wehrse, R
• Format: Article
• Language: English
• ISSN: 0022-4073; 1879-1352
• DOI: 10.1016/S0022-4073(97)00025-3

The modelling of high-quality spectra from laboratory experiments or astronomical observations often requires the consideration, for example, of media with strong density inhomogeneities that can be treated only statistically, of complicated redistribution functions and/or of very many lines. Since the numerical methods available for solution of the corresponding radiative transfer equations are not efficient enough to deal with such situations, basically new methods have to be deviced. In order to provide a sound foundation for such algorithms, the relevant mathematical properties of the radiative transfer equation for static, plane-parallel media with coherent, isotropic scattering are investigated in this paper. The starting point is the solution in terms of the matrix tangent hyperbolic function. It is shown that this operator is the sum of a diagonal operator plus the difference between two positive operators of finite trace, which have the completely unexpected property that the highest eigenvalue practically coincides with the trace so that, in a zeroth approximation, the other eigenvalues can be neglected and the operators can be represented in the form λ max ¦V max 〉〈V max ¦ . This observation allows an explicit approximate solution. By means of a non-local representation of the transfer equation, we deduce that the equation has a variational principle and we prove that the outgoing intensities are positive whenever the ingoing intensities and the de-excitation parameter are positive, as is required by physics.