Exact Solutions for One-Dimensional Gray Media

Abstract

Methods for finding the exact (semi-analytical) solution to the radiative transfer equation (RTE) for gray one-dimensional media are outlined in this chapter. First, the general solution is developed for a plane-parallel medium, and final solutions are then derived for specific situations, such as nonscattering media, media enclosed by black walls, media with isotropic scattering, media at radiative equilibrium, media with a variable refractive index (graded index), and others. The Fredholm integral equation, which appears in the solution to many radiative transfer problems, is introduced. Next, the formulations are rederived for one-dimensional spherical and cylindrical media with radiative transfer only in the radial direction. The final section of the chapter summarizes numerical methods that may be used to solve the integral equations resulting from some of the aforementioned scenarios.