A high-order spherical harmonics solution to the standard problem in radiative transfer

Abstract

The classical spherical harmonics method is used, in high order, to compute an accurate solution to Chandrasekhar's standard problem in radiative transfer: the illumination of a plane-parallel layer by a solar beam. While the analytical formulation used follows Chandrasekhar's basic approach, several additions to the theory are reported: (1) the solution for the specific case of pure scattering (omega = 1) is developed in detail; (2) a particular solution that is valid even for the case when the direction cosine of the incident beam is equal to one of the eigenvalues is reported; and (3) a final iteration of the equation of transfer is used to obtain an improved analytical expression for the intensity that is, in addition, especially accurate for the case of strong absorption. An explicit presentation of the numerical methods used is given, and the algorithm is shown to be computationally stable even for very high order (N = 499) and for many-term (L = 299) scattering laws.