New probability distributions in astrophysics: IV. The relativistic Maxwell-Boltzmann distribution

TL;DR

This paper analyzes two relativistic generalizations of the Maxwell-Boltzmann distribution, deriving key statistical properties and exploring astrophysical applications like synchrotron emission with relativistic electrons.

Contribution

It introduces detailed derivations of normalization constants, moments, and asymptotic behaviors for the relativistic MB and MJ distributions, with practical astrophysical applications.

Findings

Derived normalization constants and moments for relativistic distributions.

Provided asymptotic and approximate expressions relating temperature and average value.

Applied distributions to model synchrotron emission in astrophysics.

Abstract

Two relativistic distributions which generalizes the Maxwell Boltzman (MB) distribution are analyzed: the relativistic MB and the Maxwell-J{\"u}ttner (MJ) distribution. For the two distributions we derived in terms of special functions the constant of normalization, the average value, the second moment about the origin, the variance, the mode, the asymptotic behavior, approximate expressions for the average value as function of the temperature and the connected inverted expressions for the temperature as function of the average value. Two astrophysical applications to the synchrotron emission in presence of the magnetic field and the relativistic electrons are presented.