Differential Approximations to the Radiation Transfer Equation by Chapman–Enskog Expansion

Direct numerical solution of the radiation transfer equation is often easier than implementation of its differential approximations with their cumbersome boundary conditions. Nevertheless, these approximations are still used, for example, in theoretical analysis. The existing approach to obtain a differential approximation based on expansion in series of the spherical harmonics is revised and expansion in series of the eigenfunctions of the scattering integral is proposed. A system of eigenfunctions is obtained for an arbitrary phase function, and explicit differential approximations are built up to the third Chapman–Enskog order. The results are tested by its application to the problem of a layer. The third-order Chapman–Enskog approximation is found to match the boundary conditions better than the first-order one and gives considerably more accurate value for the heat flow. The accuracy of the both first- and third-order heat flows generally increases with the optical thickness. In addition, the third-order heat flow tends to the rigorous limit value when the optical thickness tends to zero.