Relativistic shocks and particle acceleration

In this paper, we investigate the fluid dynamics of relativistic shock waves, and use the results to calculate the spectral index of particles accelerated by the Fermi process in such shocks. We have calculated the distributions of Fermi-accelerated particles at shocks propagating into cold proton–electron plasma and also cold electron–positron plasma. We have considered two different power spectra for the scattering waves, and find, in contrast to the non-relativistic case, that the spectral index of the accelerated particles depends on the wave power spectrum. Our results may be summarized in terms of the spectral index of the synchrotron emission from accelerated electrons ($$S_\nu\propto\nu^{-\alpha}$$): For shocks propagating into cold gas, with speeds up to 0.98c, $$0.5\leqslant\alpha\leqslant0.6$$ for pair plasma, and $$0.34\leqslant\alpha\leqslant0.57$$ for proton–electron plasma. On the assumption of thermal equilibrium both upstream and downstream, we present some useful fits for the compression ratio of shocks propagating at arbitrary speeds into gas of any temperature.