An eigenfunction method for particle acceleration at ultra-relativistic shocks

We adapt and modify the eigenfunction method of computing the power-law spectrum of particles accelerated at a relativistic shock front via the first-order Fermi process [6] to apply to shocks of arbitrarily high Lorentz factor. The power-law index of accelerated particles undergoing isotropic small-angle scattering at an ultra-relativistic, unmagnetized shock is found to be s=4.23±0.2 (where s=d ln f/d ln p, with f the Lorentz-invariant phase-space density and p the momentum), in agreement with the results of Monte-Carlo simulations. We present results for shocks in plasmas with different equations of state and for Lorentz factors ranging from 5 to infinity.