The relativistic fireball model assumes that the event which gives rise to a gamma-ray burst (GRB) deposits an energy into a small baryonic mass M, such that .Typically is assumed to be of order 102 to 103. After an initial acceleration phase, such a fireball reaches a "coasting" stage with Lorentz factor . The fireball drives a blast wave of Lorentz factor into the surrounding medium. The energy expended to shock the surrounding medium eventually decelerates the fireball; in the adiabatic case, the Lorentz factor of the blast wave then decreases with radius as (Rees & Mészáros 1992).
While the GRB emission itself may come from internal shocks (Rees & Mészáros 1994), the afterglow emission is generally thought to be due to electrons accelerated by the blast wave in its deceleration phase (e.g. Mészáros & Rees 1997; Vietri 1997). Fireball models of GRB afterglows (e.g. Sari et al. 1998) usually assume that these accelerated electrons have a power-law spectrum of unspecified index p. Waxman (1995) and Vietri (1995) also suggested that ultra-high-energy cosmic rays (UHECRs), with energies eV, might be accelerated in these relativistic fireballs. In particular, Vietri (1995) argued that in Fermi-type acceleration at an ultra-relativistic shock, a particle could have an energy gain per shock crossing cycle.
Motivated by these considerations, we examine acceleration of the Fermi type at an ultra-relativistic shock in more detail, considering first the spectral index of the accelerated particles, and then the maximum energy attainable by this process.
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