On the S- and T-functions of S. Chandrasekhar in arbitrary stratification

Abstract

Introducing a new approach, based on the invariance principle, to the study of radiative transfer, Ambarzumian [I] solved exactly the problem of the diffuse reflection and transmission of axially symmetric radiation by a finite medium. Generalizing extensively the technique, Chandrasekhar [2] first formulated a complete set of the principles of invariance in conjunction with the transfer equation and resolved a number of outstanding problems with the aid of it. Thereafter the problem of diffuse reflection and transmission in a finite homogeneous atmosphere with isotropic scattering was reconsidered by several authors (cf. Miss Busbridge [3], Sobolev [4], Ueno [5]). Recently, based on the principle of invariant imbedding, Bellman and Kalaba [6] derived exactly the emergent intensity of axially symmetric radiation diffusely reflected by a finite inhomogeneous medium at the lower boundary. Making use of the probability method, Sobolev [7] also obtained the exact solution of the transfer equation for diffuse reflection in a semiinfinite inhomogeneous medium. Later, extending the method of invariance due to Chandrasekhar to the case of finite, nonseparable plane-parallel media, Preisendorfer [8] established the functional relations for the reflectance and transmission operators on flat layer of arbitrary stratification; see also Bellman et al. [9, lo]. In the preceding paper [ll], with the aid of the probabilistic method, we obtained the exact solution of the equation of transfer for diffuse reflection and transmission of parallel beam by a plane-parallel inhomogeneous atmosphere of the finite optical thickness or with isotropic scattering. Because of