Radiative transfer in one-dimensional inhomogeneous atmospheres

We use Ambartsumian's method of addition of layers to show that various problems, including the standard ones, of radiation transfer in a plane-parallel inhomogeneous atmosphere may be reduced to the solution of the Cauchy problems for linear differential equations. This allows avoiding the known difficulties arising in solving the boundary-value problems to which the classical approach leads. For the purpose of exposition, the paper deals with the simplest one-dimensional problem of multiple scattering for an atmosphere of finite optical thickness. The idea of the approach is that we start with determining the reflection and transmission coefficients of an atmosphere by solving the initial-value problem for a set of linear differential equations of the first order. After that the internal radiation field is found immediately without solving any new equation. The approach is applied to several classical problems of astrophysical interest. In particular, we evaluate the mean number of scatterings undergone by different types of photons. The transfer of radiation in an atmosphere with arbitrarily distributed internal sources is considered. Analytical solutions for these problems are obtained. Simple recursion formulas are derived to find the radiation intensity emitted by a multicomponent atmosphere. The problem of multiple scattering of radiation with partial redistribution over frequencies is discussed to demonstrate the generalization of the approach to the matrix case. The results of numerical calculations are given.