Particle transport in tangled magnetic fields and Fermi acceleration at relativistic shocks

This paper presents a new method of Monte-Carlo simulations of test particle Fermi acceleration at relativistic shocks. The particle trajectories in tangled magnetic fields are integrated out exactly from entry to exit through the shock, and the conditional probability of return as a function of ingress and egress pitch angles is constructed by Monte-Carlo iteration. These upstream and downstream probability laws are then used in conjunction with the energy gain formula at shock crossing to reproduce Fermi acceleration. For pure Kolmogorov magnetic turbulence upstream and downstream, the spectral index is found to evolve smoothly from s=2.09 +/- 0.02 for mildly relativistic shocks with Lorentz factor Gamma=2 to s=2.26 +/- 0.04 in the ultra-relativistic limit Gamma >> 1. The energy gain is ~Gamma^2 at first shock crossing, and ~2 in all subsequent cycles as anticipated by Gallant & Achterberg (1999). The acceleration timescale is found to be as short as a fraction of Larmor time when Gamma >> 1.