On the radiative equilibrium of a stellar atmosphere. XXI.

Abstract

In this paper the theory of diffuse reflection and transmission by a plane-parallel atmosphere of finite optical thickness is considered under conditions of (I) isotropic scattering with an albedo ϖ₀≤1, (II) scattering in accordance with Rayleigh's phase function, (III) scattering in accordance with the phase function λ(1+x cosθ), and (IV) Rayleigh scattering with proper allowance for the polarization of the radiation field. In all cases considered, it has been possible to eliminate the constants of integration (which are twice as many as in the case of semi-infinite atmospheres) and express the solutions for the reflected and the transmitted radiations in closed forms in a general nth approximation. It is further shown how a pair of functions, X(μ) and Y(μ), depending only on the roots of a characteristic equation and the optical thickness of the atmosphere, play the same basic role in this theory as H(μ) does in the theory of semiinfinite atmospheres. The passage to the limit of infinite approximation and the determination of the exact laws of diffuse reflection and transmission are thus made possible.