MATHEMATICAL ASPECTS, OF THE PLANE-PARALLEL TRANSFER EQUATION

The modelling of high-quality spectra from laboratory experiments or a stronomical observations often requires the consideration, for example , of media with strong density inhomogeneities that can be treated onl y statistically, of complicated redistribution functions and/or of ver y many lines. Since the numerical methods available for solution of th e 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 rele vant mathematical properties of the radiative transfer equation for st atic, plane-parallel media with coherent, isotropic scattering are inv estigated in this paper. The starting point is the solution in terms o f the matrix tangent hyperbolic function. It is shown that this operat or is the sum of a diagonal operator plus the difference between two p ositive operators of finite trace, which have the completely unexpecte d 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 lambda(max )\V-max][V-max\. This observation allows an explicit approximate solut ion. By means of a non-local representation of the transfer equation, we deduce that the equation has a variational principle and we prove t hat the outgoing intensities ate positive whenever the ingoing intensi ties and the de-excitation parameter are positive, as is required by p hysics. (C) 1997 Elsevier Science Ltd.