ON THE RATE OF CONVERGENCE OF THE SPHERICAL-HARMONICS METHOD IN THE MILNE PROBLEM

Abstract

The extrapolated end point, z0, and the ratio of current to flux at the free surface, j(0)/ρ(0), for the Milne problem are evaluated by the spherical-harmonics method in the PN approximation, N being odd. It is shown that, for N large, the approximations to these quantities are related to the exact values by[Formula: see text]in which c is the number of secondaries per collision, f1(c) and f3(c) are independent of N, and ζ1 and ζ2 are the absolutely lesser and greater roots of[Formula: see text]κ being the exact inverse diffusion length. The relation (A) is subject to the restrictions[Formula: see text]