Atmospheric Thermal Emission Effect on Chandrasekhar’s Finite Atmosphere Problem

Abstract

The solutions of the diffuse reflection finite atmosphere problem are very useful in the astrophysical context. Chandrasekhar was the first to solve this problem analytically, by considering atmospheric scattering. These results have wide applications in the modeling of planetary atmospheres. However, they cannot be used to model an atmosphere with emission. We solved this problem by including the thermal emission effect along with scattering. Here, our aim is to provide a complete picture of the generalized finite atmosphere problem in the presence of scattering and thermal emission, and to give a physical account of the same. For that, we take an analytical approach using the invariance principle method to solve the diffuse reflection finite atmosphere problem in the presence of atmospheric thermal emission. We established general integral equations of the modified scattering function S(τ; μ, ϕ; μ 0, ϕ 0), transmission function T(τ; μ, ϕ; μ 0, ϕ 0), and their derivatives with respect to τ for a thermally emitting atmosphere. We customize these equations for the case of isotropic scattering and introduce two new functions V(μ), and W(μ), analogous to Chandrasekhar’s X(μ), and Y(μ) functions, respectively. We also derive a transformation relation between the modified S and T functions and give a physical account of the V(μ) and W(μ) functions. Our final results are consistent with those of Chandrasekhar’s at the low emission limit (i.e., only scattering). From the consistency of our results, we conclude that the consideration of the thermal emission effect in the diffuse reflection finite atmosphere problem gives more general and accurate results than considering only scattering.