Reduction of the Equations of Radiative Transfer for a Plane-Parallel, Planetary Atmosphere

A presentation of mathematical techniques necessary in unique solution of equations of radiative transfer in a homogeneous atmosphere with Rayleigh scsattering. The singular linear integral equations for the X-and Y-matrices appearing in the azimuth-independent terms of the reflection and transmission matrices for the emerging radiation from a homogeneous, plane-parallel atmosphere with Rayleigh scattering are reduced to four pairs of scalar equations of the forms discussed by T. W. Mullikin. The nonuniqueness in the solutions of one of these pairs of equations is removed by using linear constraints. The solutions are finally expressed as linear combinations of Chandrasekhar's functions and are brought to the same form as that derived by Chandrasekhar, but by completely different methods. The new forms of the solutions are particularly suitable from the computational point of view when large optical thicknesses are involved. Mathematical expressions corresponding to physical quantities such as directional radiation and total fluxes emerging from a Rayleigh scattering atmosphere are also derived.