Particle Acceleration at Ultra-Relativistic Shocks and the Spectra of   Relativistic Fireballs
  We examine Fermi-type acceleration at relativistic shocks, and distinguish between the initial boost of the first shock crossing cycle, where the energy gain per particle can be very large, and the Fermi process proper with repeated shock crossings, in which the typical energy gain is of order unity. We calculate by means of numerical simulations the spectrum and angular distribution of particles accelerated by this Fermi process, in particular in the case where particle dynamics can be approximated as small-angle scattering. We show that synchrotron emission from electrons or positrons accelerated by this process can account remarkably well for the observed power-law spectra of GRB afterglows and Crab-like supernova remnants. In the context of a decelerating relativistic fireball, we calculate the maximum particle energy attainable by acceleration at the external blast wave, and discuss the minimum energy for this acceleration process and its consequences for the observed spectrum. 
Yves A. Gallant, Abraham Achterberg, John G. Kirk and Axel W. Guthmann