15-Digit Accuracy Calculations of Ambartsumian-Chandrasekhar's $H$-Functions for Four-Term Phase Functions with the Double-Exponential Formula

TL;DR

This paper develops a highly accurate iterative method combining Kawabata's approach and the double-exponential formula to compute Ambartsumian-Chandrasekhar's H-functions for anisotropic scattering with four-term phase functions, validated through extensive numerical calculations.

Contribution

It introduces a novel iterative scheme with 15-digit accuracy for H-functions using the DE-formula, enhancing precision in anisotropic scattering calculations.

Findings

Achieved 15-digit accuracy in H-function calculations.

Validated method against existing results for isotropic and Rayleigh scattering.

Demonstrated method's effectiveness with extensive numerical examples.

Abstract

We have established an iterative scheme to calculate with 15-digit accuracy the numerical values of Ambartsumian-Chandrasekhar's H-functions for anisotropic scattering characterized by the four-term phase function: the method incorporates some advantageous features of the iterative procedure of Kawabata (2015) and the double-exponential integration formula~(DE-formula) of Takahashi and Mori (1974), which proved highly effective in Kawabata (2016). Actual calculations of the H-functions have been carried out employing 27 selected cases of the phase function, 56 values of the single scattering albedo $\varpi_0$, and 36 values of an angular variable $\mu(=\cos \theta)$, with $\theta$ being the zenith angle specifying the direction of incidence and/or emergence of radiation. Partial results obtained for conservative isotropic scattering, Rayleigh scattering, and anisotropic scattering due to a full four-term phase function are presented. As a sample application of the isotropic scattering H-function, an attempt is made in Appendix to simulate by iteratively solving the Ambartsumian equation the values of the plane and spherical albedos of a semi-infinite, homogeneous atmosphere calculated by Rogovtsov and Borovik (2016), who employed their analytical representations for these quantities and the single-term and two-term Henyey-Greenstein phase functions of appreciably high degrees of anisotropy, to find that our results are in satisfactory agreement with theirs.