Particle acceleration in ultra-relativistic parallel shock waves

Monte-Carlo computations for highly relativistic parallel shock particle acceleration are presented for upstream flow gamma factors, $\Gamma=(1-V_{1}^{2}/c^{2})^{-0.5}$ with values between 5 and $10^{3}$. The results show that the spectral shape at the shock depends on whether or not the particle scattering is small angle with $\delta \theta < \Gamma^{-1}$ or large angle, which is possible if $\lambda > 2r_{g} \Gamma^{2}$ where $\lambda$ is the scattering mean free path along the field line and $r_{g}$ the gyroradius, these quantities being measured in the plasma flow frame. The large angle scattering case exhibits distinctive structure superimposed on the basic power-law spectrum, largely absent in the pitch angle case. Also, both cases yield an acceleration rate faster than estimated by the conventional, non-relativistic expression, $t_{acc}=[c/(V_{1}-V_{2})] [\lambda_{1}/V_{1}+\lambda_{2}/V_{2}]$ where '1' and '2' refer to upstream and downstream and $\lambda$ is the mean free path. A $\Gamma^{2}$ energy enhancement factor in the first shock crossing cycle and a significant energy multiplication in the subsequent shock cycles are also observed. The results may be important for our understanding of the production of very high energy cosmic rays and the high energy neutrino and gamma-ray output from Gamma Ray Bursts (GRB) and Active Galactic Nuclei (AGN).