Synchrotron radiation

Remarks

Example: for an electron of charge $-e$ and mass $m_e$, the radiated power averaged over isotropic pitch angles is $P = \frac{4}{3} \sigma_T c \beta^2 \gamma^2 U_B$. Here $\beta = v/c$, $\sigma_T$ is the Thomson cross-section and $U_B = B^2/(2\mu_0)$ is the energy density of the field in SI units. The spectrum of one electron peaks near the critical frequency $\nu_c = \frac{3}{2} \gamma^2 \nu_B \sin\alpha$, where $\nu_B = eB/(2\pi m_e)$ is the non-relativistic gyrofrequency. If the electrons have an energy distribution $N(E) \propto E^{-p}$, the emitted spectrum in the optically thin regime is $F_\nu \propto \nu^{-(p-1)/2}$. Non-example: radiation of slow electrons with $\gamma \approx 1$ in a magnetic field is called cyclotron radiation; it is emitted at the gyrofrequency and its harmonics, not over a broad band.