Accurate analytic approximation for the Chapman grazing incidence function

Abstract

A new analytical approximation for the Chapman mapping integral, $${\text {Ch}}$$ Ch , for exponential atmospheres is proposed. This formulation is based on the derived relation of the Chapman function to several classes of the incomplete Bessel functions. Application of the uniform asymptotic expansion to the incomplete Bessel functions allowed us to establish the precise analytical approximation to $${\text {Ch}}$$ Ch , which outperforms established analytical results. In this way the resource consuming numerical integration can be replaced by the derived approximation with higher accuracy. The obtained results are useful for various branches of atmospheric physics such as the calculations of optical depths in exponential atmospheres at large grazing angles, physical and chemical aeronomy, atmospheric optics, ionospheric modeling, and radiative transfer theory.

Analytical Approximation of the Definite Chapman Integral for Arbitrary Zenith Angle

“…Our results (Eqs. ( 11)-( 13)) agree with the approximate formulas tabulated in (Vasylyev, 2021) when evaluated under appropriate limits.…”

“…Several researchers, Green and Barnum (1963); Fitzmaurice (1964); Swider and Gardner (1969); Titheridge (1988); Kocifaj (1996); Huestis (2001), have proposed various analytical approximations of the Chapman function. A comprehensive review and improvement of these approximations were recently offered by Vasylyev (2021). Nonetheless, a straightforward solution applicable to arbitrary path angles and finite integration limits remains elusive.…”