Solution of radiative transfer in anisotropic plane-parallel atmosphere

Abstract

We solve the radiative transfer problem analytically in the anisotropic, plane-parallel atmosphere. Chandrasekhar formalized the radiative transfer process as a simultaneous, two-variable, non-linear, integral equation and obtained the analytical solution of the second approximation for the isotropic plane-parallel atmosphere. We obtain the second approximation for the anisotropic atmosphere, by integrating the first approximation multiplied by weighting functions which are products of the scattering phase functions. We truncate the second approximation and obtain the radiance at the top of the atmosphere as a quadratic equation with a logarithmic term in the optical thickness. We evaluate the second approximation of the radiance at the top of the atmosphere for Rayleigh scattering and the maritime aerosol atmosphere and compare them with both the exact solution and the single scattering approximation.