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 pr oduction of high energy particles to explain their emission. In this paper we first review the basic theory of shock waves in relativistic hydrodynami cs and magnetohydrodynamics, emphasizing the astrophysically interesting ca ses. This is followed by an overview of the theory of particle acceleration at such shocks. Whereas, for diffusive acceleration at non-relativistic sh ocks, it is the compression ratio which fixes the energetic particle spectr um uniquely, acceleration at relativistic shocks is more complicated. In th e absence of scattering, particles are simply 'compressed' as they pass thr ough the shock front. This mechanism-called shock-drift acceleration-enhanc es the energy density in accelerated particles, but does so without changin g the spectral index of upstream particles. Scattering due to MHD waves lea ds to multiple encounters between the particles and the shock front, produc ing an energetic particle population which depends on the properties of the shock front and the level and nature of particle scattering. We describe t he method of matching the angular distributions of the upstream and downstr eam distributions at the shock front which leads to predictions of the spec tral index. Numerical simulation of particle transport provides an alternat ive means of calculating spectral indices, and has recently been extended t o cover ultra-relativistic shocks. We review these calculations and summari ze the applications to the astrophysics of relativistic jets and fireball m odels of gamma-ray-bursts.