The singular eigenfunction method: the Milne problem for isotropic and extremely anisotropic scattering

Abstract

In the C N method of solving the third form of the transport equation, the medium as a result of Placzek lemma is extended to infinity. Infinite medium Green function which is obtained by the Fourier transform technique is used and the method is applied to one velocity problems in plane and cylindrical geometries. As the result of physical applications of the C N method in different geometries, it is seen that the only difficulty lies in writting the expression of the Green function in a form easy to handle. In the new method of solving the third form of the transport equation (that we have generated recently), three methods, namely, C N, F N and the method of elementary solutions are considered, compared and the Green function in terms of the singular eigenfunctions is used. This method yields simple analytical expressions that can be solved numerically more efficiently than the C N method because the expression of the Green function is in the form easy to handle. Here this new method is applied to calculate the extrapolation length for the Milne problem which is a classical problem in astrophysics concerned with the diffusion of radiation through a stellar atmosphere for both isotropic and anisotropic scatterings. It is shown that the numerical results which are tabulated for selected cases are accurate even in the approximations of the lowest order and are in good agreement with the numerical results obtained by the other methods.