Particle acceleration in extragalactic sources: the role of synchrotron losses in determining the spectrum

We calculate the observed synchrotron emission from electrons in the vicinity of a plane strong shock front, assuming a steady-state diffusive shock-acceleration mechanism and a finite extent of the emission region, fixed either by telescope beam or a physical boundary. Synchrotron losses are taken into account throughout the acceleration process and the advective transport downsteam. We present analytic results for the electron distribution at arbitrary distances from the shock and the integrated distribution over space. The emitted synchrotron spectrum for a strong, non-relativistic shock is flat at low frequencies (flux density |${S}_{v}\,\propto{v}^{-0.5}$)|⁠, steepens to |${S}_{v}\,\propto{v}^{-1}$| above some break frequency vb, and cuts off at a high-frequency limit vc. For emission regions much larger than the mean free path of the electrons, the break frequency vb is fixed by the losses suffered by the electrons in crossing the emission region, and is thus independent of the precise details of the acceleration process, depending only on the length of the region, the magnetic field and the fluid speed. The cut-off frequency and shape of the observed spectrum near the cut-off are determined by the balance between losses and acceleration gains. Accordingly, information about the mean free path and its momentum-dependence could be obtained from detailed observation.