Some analytical results for radiative transfer in thick atmospheres

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Abstract
Singular eigenmode expansions are a convenient analytical tool with which to study problems of monochromatic radiative transfer in thick or semi-infinite atmospheres. Some closed-form solutions are presented for anisotropic scattering with the neglect of polarization effects. A basic ingredient for applications to the semi-infinite medium is Chandrasekhar's H-function, which is best defined through the Wiener-Hopf factorization (Λ(z)]−1 = H(z)H(-z). Here H(z) is required to be regular in the right-half of the complex plane, while Λ(z) is the dispersion function whose zeros are the discrete relaxation lengths. Use of the Busbridge polynomials ql(z), along with H(z), permits the construction of adjoint eigenmodes. A biorthogonality relation follows that may be used to determine the coefficients in eigenmode expansions. Attention is given to the solutions of the Milne and albedo problems in order that the method of matched asymptotic approximations can be used to describe the solution for a thick atmosphere adjacent to a diffusely reflecting ground. Expressions for the emerging distributions are given. A possible extension of the general scheme to problems involving polarization is indicated.