Particle acceleration by ultrarelativistic shocks: theory and simulations

We consider the acceleration of charged particles near ultrarelativistic shocks, with Lorentz factor Gamma (s) >> 1. We present simulations of the acceleration process and compare these with results from semi-analytical calculations. We show that the spectrum that results from acceleration near ultrarelativistic shocks is a power law, N(E) proportional to E-s, with a nearly universal value s approximate to 2.2-2.3 for the slope of this power law. We confirm that the ultrarelativistic equivalent of the Fermi acceleration at a shock differs from its non-relativistic counterpart by the occurrence of large anisotropies in the distribution of the accelerated particles near the shock. In the rest frame of the upstream fluid, particles can only outrun the shock when their direction of motion lies within a small loss cone of opening angle Delta theta similar to Gamma (-1)(s) around the shock normal. We also show that all physically plausible deflection or scattering mechanisms can change the upstream flight direction of relativistic particles originating from downstream by only a small amount: Delta theta similar to Gamma (-1)(s). This limits the energy change per shock crossing cycle to DeltaE similar to E, except for the first cycle where particles originate upstream. In that case the upstream energy is boosted by a factor similar to Gamma (2)(s) for those particles that are scattered back across the shock into the upstream region.