Analytical fits for the synchrotron emission from a power-law particle distribution with a sharp cutoff

Abstract

In several high-energy astrophysical sites, shocks are assumed to produce a power-law distribution of accelerated charged particles (e.g., electrons, protons) and to generate mild to strong magnetic fields, which favours synchrotron emission. For such environments and conditions, we have performed and present here four practical formulas, with different levels of accuracy, for fitting the synchrotron spectral power radiated by a pure power-law particle distribution, with isotropic pitch angle distribution. The first three ones can be useful compared to the fourth one, because of their simplicity, in the case of particle distribution with no high-energy cutoff. However, the fourth formula, the more accurate one, can be of great interest for astrophysical applications (even though it is more complicated) for the more general case of a power-law distribution with high-energy sharp cutoff. The latter is derived for index p within the range, 1 < p < 6, with maximum relative error of less than 0.5 per cent in the case of infinite energy range and of less than 8.2 per cent in the more general case of particle energy with sharp cutoff. The latter is expressed in terms of parameters as functions of index p. These parameters have been fitted with adopting the Levenberg-Marquardt algorithm in log-log scale. According to these formulas, initially derived for the total spectral power, we have then derived (1) the degree of polarization, (2) cooling spectra for a broken power-law distribution and (3) synchrotron self-absorption spectra for both the broken and non-broken power-law distributions. The proposed expressions are relevant for non-thermal astrophysical sources in the sense that by using them one avoids usually complicated and long CPU time calculations, without performing any integration. © 2014 The Authors. Published by Oxford University Press on behalf of the Royal Astronomical Society.