Particle acceleration and relativistic shocks

Observations of both gamma-ray burst sources and certain classes of active galaxy indicate the presence of relativistic shock waves and require the production of high energy particles to explain their emission. In this paper we first review the basic theory of shock waves in relativistic hydrodynamics and magnetohydrodynamics, emphasizing the astrophysically interesting cases. This is followed by an overview of the theory of particle acceleration at such shocks. Whereas, for diffusive acceleration at non-relativistic shocks, it is the compression ratio which fixes the energetic particle spectrum uniquely, acceleration at relativistic shocks is more complicated. In the absence of scattering, particles are simply `compressed' as they pass through the shock front. This mechanism - called shock-drift acceleration - enhances the energy density in accelerated particles, but does so without changing the spectral index of upstream particles. Scattering due to MHD waves leads to multiple encounters between the particles and the shock front, producing an energetic particle population which depends on the properties of the shock front and the level and nature of particle scattering. We describe the method of matching the angular distributions of the upstream and downstream distributions at the shock front which leads to predictions of the spectral index. Numerical simulation of particle transport provides an alternative means of calculating spectral indices, and has recently been extended to cover ultra-relativistic shocks. We review these calculations and summarize the applications to the astrophysics of relativistic jets and fireball models of gamma-ray-bursts.