Introduction

We now describe several approximate methods of solution of the radiative transfer equation. Approximate methods play an important role in the subject, because they usually provide more insight than the more accurate methods. Indeed their simple mathematical forms help clarify physical aspects that are not easily discerned from the numerical output of a computer code. Another redeeming feature of approximate methods is that they are often sufficiently accurate that no further effort is necessary. Unlike some of the more sophisticated numerical techniques, these methods also yield approximations for the internal radiation field, including the source function. Of central importance is the two-stream approximation. This class of solutions has been given various names in the past (Schuster–Schwarzschild, Eddington, two-stream, diffusion approximation, two-flow analysis, etc.). In all variations of the method, the intractable integro-differential equation of radiative transfer is replaced with representations of the angular dependence of the radiation field in terms of just two functions of optical depth. These two functions obey two linear, coupled ordinary differential equations. When the medium is homogeneous, the coefficients of these equations are constants, and analytic closed-form solutions are possible. The mathematical forms of these solutions are exponentials in optical depth, depending on the total optical depth of the medium, the single-scattering albedo, one or two moments of the phase function, and the boundary intensities. Some disadvantages are that two-stream solutions maintain acceptable accuracy over a rather restricted range of the parameters; there is no useful a priori method to estimate the accuracy; and one generally needs an accurate solution to obtain a useful estimate of the accuracy.