Analytical Fits to the Synchrotron Functions

TL;DR

This paper develops highly accurate simple fitting formulae for the synchrotron functions $F(x)$ and $G(x)$, with errors less than 0.26% and 0.035%, respectively, facilitating efficient calculations in astrophysics and laboratory physics.

Contribution

It introduces new, simple, and highly accurate fitting formulae for the synchrotron functions based on asymptotic forms and the Levenberg-Marquardt algorithm, improving computational efficiency.

Findings

Relative errors less than 0.26% for F(x) and 0.035% for G(x)

Fitted functions based on asymptotic forms with adjustable parameters

Useful for computing spectral powers and polarization in synchrotron radiation

Abstract

Accurate fitting formulae to the synchrotron function, $F(x)$, and its complementary function, $G(x)$, are performed and presented. The corresponding relative errors are less than $0.26\%$ and $0.035\%$ for $F(x) $ and $G(x)$, respectively. To this aim we have, first, fitted the modified Bessel functions, $K_{5/3}(x)$ and $K_{2/3}(x)$. For all the fitted functions, the general fit expression is the same, and is based on the well known asymptotic forms for low and large $x$-values for each function. It consists of multiplying each asymptotic form by a function that tends to unity or zero for low and large $x$-values. Simple formulae are suggested in this paper, depending on adjustable parameters. The latter have been determined by adopting the Levenberg-Marquardt algorithm. The proposed formulae should be of great utility and simplicity for computing spectral powers and the degree of polarization for the synchrotron radiation, both for laboratory and astrophysical applications.