Past exam of the astrophysics course of the University of Oxford / 2022 / 4 / e / Solution

One explanation is an upper electron energy imposed by the accelerator: finite age, finite confinement size, escape, or a balance of acceleration and losses limits $\gamma$ to ${\gamma }_{\max }$. Since the synchrotron critical frequency scales as ${\nu }_c\propto B{\gamma }^2\sin \alpha$, the absence of higher-energy electrons produces a corresponding high-frequency cutoff, broadened by the single-particle spectrum.

The other is synchrotron cooling, with inverse Compton scattering often contributing: $\dot{\gamma }=-b{\gamma }^2$ gives $t_{\mathrm{cool}}=1/(b\gamma )$. High-energy electrons disappear fastest, so a population of age $t$ develops a cooling scale ${\gamma }_b\sim 1/(bt)$ and ${\nu }_b\propto B/(b^2{t}^2)$. A burst population can develop a strong cutoff. With continuous power-law injection, the simplest steady cooling model instead steepens the electron energy index by one and the synchrotron radiation spectral index by $1/2$, giving a break rather than necessarily an exponential cutoff. The detailed spectral shape therefore distinguishes finite acceleration from aging and injection history.