Milne's problem for a non-capturing medium: Accurate analytic approximations for particle density and emergent angular distribution

Abstract

An approximate solution of Milne's integral equation for a non-absorbing medium is deduced by using variational calculus. A three-parameter expression for the particle density, much superior to the existing alternatives, is presented; besides reproducing the extrapolated endpoint to almost 1 part in 10 8, it gives, correct to 0.064%, the particle density, ψ 0( z), and the angular distribution, φ(μ), of the emerging particles. The maximum error in ψ 0( z) occurs at z = 0; that in φ(μ), at μ = 0. The discrepancies diminish rapidly as z and μ increase, and become quite small (<0.02%) already when the arguments equal 0.01.