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 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 \pm 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 F _ { \nu } / { d } \ln \nu , F _ { \nu } is the radiation flux and \nu 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 non-relativistic case the angular distribution is weighted more towards 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 .