Rational approximation formula for Chandrasekhar’s H-function for isotropic scattering

Abstract

In this work, 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 combinations of ϖ ₀, the single scattering albedo, and μ, the cosine of the zenith angle θ of the direction of radiation emergent from or incident upon a semi-infinite scattering-absorbing medium, we propose a rational approximation formula with μ ^1/4 and $\sqrt{1-\varpi_{0}}$ as the independent variables. This allows us to reproduce the correct values of H(ϖ ₀,μ) within a relative error of 2.1×10⁻⁵ without recourse to any iterative procedure or root-finding process.