Theory of synchrotron radiation: II. Backreaction in ensembles of particles

The standard calculations of the synchrotron emission from charged particles in magnetic fields does not apply when the energy losses of the particles are so severe that their energy is appreciably degraded during one Larmor rotation. In these conditions, the intensity and spectrum of the emitted radiation depend on the observation time $T_{obs}$: the standard result is recovered only in the limit $T_{obs}\ll T_{loss}$, where $T_{loss}$ is the time for synchrotron losses. In this case the effects of the radiation backreaction cannot be detected by the observer. We calculate the emitted power of the radiation in the most general case, naturally including both the cases in which the backreaction is relevant and the standard case, where the usual result is recovered. Finally we propose several scenarios of astrophysical interest in which the effects of the backreaction cannot and should not be ignored.