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

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.