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 (Kirk, J.G., Schneider, P., Astrophysical Journal 315, 425 (1987)) to apply to shocks of arbitrarily high Lorentz factor. The power-law index of accelerated particles undergoing isotropic small-angle scattering at an ultrarelativistic, 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. 
Axel W. Guthmann, John G. Kirk, Yves A. Gallant and Abraham Achterberg