MILNE PROBLEM FOR A NON-CAPTURING MEDIUM - ACCURATE ANALYTIC APPROXIMATIONS FOR PARTICLE DENSITY AND EMERGENT ANGULAR-DISTRIBUTION

An approximate solution of Milne's integral equation for a non-absorbi ng medium is deduced by using variational calculus. A three-parameter expression for the particle density, much superior to the existing alt ernatives, is presented; besides reproducing the extrapolated endpoint to almost 1 part in 10(8), it gives, correct to 0.064%, the particle density, psi0(z), and the angular distribution, phi (mu), of the emerg ing particles. The maximum error in psi0(z) occurs at z = 0; that in p hi (mu), at mu = 0. The discrepancies diminish rapidly as z and mu inc rease, and become quite small (<0.02%) already when the arguments equa l 0.01.