Energy Spectra of Cosmic Rays Accelerated at Ultrarelativistic Shock Waves

Energy spectra of particles accelerated by the first-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 flows with Lorentz factors γ from 3 to 243. For large γ shocks a convergence of derived energy spectral indices up to the value σ_∞≈2.2 is observed for all considered turbulence amplitudes and magnetic field configurations. Recently the same index was derived for γ-ray bursts by Waxman [Astrophys. J. Lett. 485, L5 (1997)].