Finding the Ambartsumyan functions and some of their analogs

Abstract

New methods are proposed for finding the Ambartsumyan functions φ(η) for a half space and φ(η, τ) and ψ(η, τ) for finite layers, as well as their analogs with complete frequency redistribution, X (z, τ) and Y (z, τ). Substantial simplifications are obtained for monochromatic conservative scattering. Besides the Ambartsumyan functions, expressions for several of their angular moments are obtained directly in terms of the basis functions u ±. A system of differential equations is obtained for the basis functions. A system of equations without the characteristic pseudosingularities is obtained for φ(η, τ) and ψ(η, τ) instead of the classical system of nonlinear equations. Some aspects of the numerical realization of the proposed method are also discussed.