Particle acceleration at ultrarelativistic shocks: An eigenfunction method

We extend the eigenfunction method of computing the power-law spectrum of particles accelerated at a relativistic shock front to apply to shocks of arbitrarily high Lorentz factor. In agreement with the findings of Monte Carlo simulations, we find that the index of the power-law distribution of accelerated particles, which undergo isotropic diffusion in angle at an ultrarelativistic, unmagnetized shock, is s = 4.23 +/- 0.01 (where s = -d ln f/d ln p with f the Lorentz invariant phase-space density and p the momentum). This corresponds to a synchrotron index for uncooled electrons of alpha = 0.62 (taking cooling into account alpha = 1.12), where alpha = -d ln Fv(alpha)/d ln alpha, F-nu is the radiation flux, and nu is the frequency. We also present an approximate analytic expression for the angular distribution of accelerated particles, which displays the effect of particle trapping by the shock: compared with the nonrelativistic case the angular distribution is weighted more toward the plane of the shock and away from its normal. We investigate the sensitivity of our results to the transport properties of the particles and the presence of a magnetic field. Shocks in which the parameter sigma (the ratio of Poynting to kinetic energy flux) upstream is not small are less compressive and lead to larger values of s.