Harmony in Electrons: Cyclotron and Synchrotron Emission by Thermal Electrons in a Magnetic Field

TL;DR

This paper provides a comprehensive calculation of cyclotron and synchrotron emission spectra for thermal electrons in magnetic fields, covering a wide temperature range and offering accurate fitting formulas for practical use.

Contribution

It introduces a complete, numerically exact solution for electron cyclotron-synchrotron emission without approximations, including all relevant harmonics and angles, and extends the results with new fitting formulas.

Findings

Emission spectra for various electron temperatures are calculated.

Fitting formulas accurately represent the detailed spectra.

Asymptotic synchrotron formula applies at very high temperatures.

Abstract

We present a complete solution to the cyclotron-synchrotron radiation due to an isotropic distribution of electrons moving in a magnetic field. We make no approximations in the calculations other than artificially broadening the harmonics by a small amount in order to facilitate the numerics. In contrast to previous calculations, we sum over all relevant harmonics and integrate over all particle and observer angles relative to the magnetic field. We present emission spectra for electron temperatures $T=5\times10^8$ K, $10^9$ K, $2\times10^9$ K to $3.2\times10^{10}$ K, and provide simple fitting formulae which give a fairly accurate representation of the detailed results. For $T\geq3.2\times10^{10}$ K, the spectrum is represented well by the asymptotic synchrotron formula, which is obtained by assuming that the radiating electrons have Lorentz factors large compared to unity. We give an improved fitting formula also for this asymptotic case.