15-Digit Accuracy Calculations of Chandrasekhar's $H$-function for Isotropic Scattering by Means of the Double Exponential Formula

TL;DR

This paper demonstrates that the Chandrasekhar $H$-function for isotropic scattering can be computed with at least 15-digit accuracy using the double exponential formula, improving precision over previous methods.

Contribution

It introduces a numerical scheme utilizing the double exponential formula for highly accurate computation of the $H$-function, addressing digit loss and computational errors.

Findings

Achieved 15-digit accuracy in $H$-function calculations.

Compared the DE-formula method with Gauss-Legendre quadrature.

Provided detailed results for various scattering parameters.

Abstract

This work shows that it is possible to calculate numerical values of the Chandrasekhar $H$-function for isotropic scattering at least with 15-digit accuracy by making use of the double exponential formula (DE-formula) of Takahashi and Mori (Publ. RIMS, Kyoto Univ. Vol. 9, 721, 1974) instead of the Gauss-Legendre quadrature employed in the numerical scheme of Kawabata and Limaye (Astrophys. Space Sci. Vol. 332, 365, 2011) and simultaneously taking a precautionary measure to minimize the effects due to loss of significant digits particularly in the cases of near-conservative scattering and/or errors involved in returned values of library functions supplied by compilers in use. The results of our calculations are presented for 18 selected values of single scattering albedo $\varpi_0$ and 22 values of an angular variable $\mu$, the cosine of zenith angle $\theta$ specifying the direction of radiation incident on or emergent from semi-infinite media.