Integrals of the transfer equation. I. Quadratic integrals for monochromatic, isotropic scattering.
Abstract
Quadratic integrals of the transfer equation are introduced for the case of monochromatic isotropic scattering in a planeparallel atmosphere. These integrals are described as natural generalizations to all depths in the atmosphere of a certain class of results exemplified by the HopfBronstein relation and the squareroot law of Frisch and Frisch (1975). Two quadratic integrals (Q and R) are constructed on the basis of a fundamental equation, the Qintegral is used to derive and generalize the cited relation and law for the type of scattering considered, and the mean intensity is determined at the boundary of two halfspaces having different albedoes and source distributions. The Rintegral, regarded as a generalization of the flux integral to nonconservative atmospheres, is applied to the case of an isotropic point source of radiation situated between two slabs. Some special inhomogeneous source distributions are also examined.
 Publication:

The Astrophysical Journal
 Pub Date:
 April 1977
 DOI:
 10.1086/155141
 Bibcode:
 1977ApJ...213..165R
 Keywords:

 Atmospheric Scattering;
 Entire Functions;
 Isotropic Media;
 Monochromatic Radiation;
 Radiative Transfer;
 Albedo;
 Atmospheric Radiation;
 Atmospheric Stratification;
 Boundary Value Problems;
 Half Spaces;
 Inhomogeneity;
 Quadratic Equations;
 Astrophysics