Energy spectra of cosmic rays accelerated at ultrarelativistic shock waves

Energy spectra of particles accelerated by the rst-order Fermi mechanism are investigated at ultrarelativistic shock waves, outside the range of Lorentz factors considered previously. For particle transport near the shock a numerical method involving small amplitude pitch-angle scattering is applied for ows with Lorentz factors from 3 to 243. For large shocks a convergence of derived energy spectral indices up to the value 1 2:2 is observed for all considered turbulence amplitudes and magnetic eld con gurations. Recently the same index was derived for -ray bursts by Waxman [Astrophys. J. Lett. 485, L5 (1997)].