Energy spectrum of particles accelerated in relativistic collisionless shocks

We analytically study diffusive particle acceleration in relativistic, collisionless shocks. We find a simple relation between the spectral index s and the anisotropy of the momentum distribution along the shock front. Based on this relation, we obtain s = (3β_u - 2β_uβ _d² + β_d³)/(β_u - β_d) for isotropic diffusion, where β_u (β_d) is the upstream (downstream) fluid velocity normalized to the speed of light. This result is in agreement with previous numerical determinations of s for all (β_u β_d), and yields s = 38/9 in the ultrarelativistic limit. The spectrum-anisotropy connection is useful for testing numerical studies and constraining anisotropic diffusion results. It suggests that the spectrum is highly sensitive to the form of the diffusion function for particles traveling along the shock front.