Rational Approximation Formula for Chandrasekhar's H-function for Isotropic Scattering

TL;DR

This paper introduces a rational approximation formula for Chandrasekhar's H-function in isotropic scattering, achieving high accuracy with a simple, non-iterative method based on numerical integration and rational approximation.

Contribution

The authors develop a closed-form integral representation and a rational approximation formula for the H-function, enabling accurate computation without iterative procedures.

Findings

Achieved 11-figure accuracy using Gauss-Legendre quadrature.

Proposed a rational approximation with less than 2.1% relative error.

Provided a simple, non-iterative method for computing the H-function.

Abstract

We first establish a simple procedure to obtain with 11-figure accuracy the values of Chandrasekhar's H-function for isotropic scattering using a closed-form integral representation and the Gauss-Legendre quadrature. Based on the numerical values of the function produced by this method for various values of the single scattering albedo and the cosine of the azimuth angle of the direction of radiation emergent from or incident upon a semi-infinite scattering-absorbing medium, we propose a rational approximation formula, which allows us to reproduce the correct values of the H-function within a relative error of 2.1/100000 without recourse to any iterative procedure or root-finding process.