1 Particle acceleration at relativistic blast waves

The relativistic fireball model assumes that the event which gives rise to a gamma-ray burst (GRB) deposits an energy ${\cal E}$ into a small baryonic mass M, such that $\eta \equiv {\cal E}/ (M c^2) \gg 1$.Typically $\eta$ is assumed to be of order 10² to 10³. After an initial acceleration phase, such a fireball reaches a "coasting" stage with Lorentz factor $\Gamma \sim \eta$. The fireball drives a blast wave of Lorentz factor $\Gamma_{\rm sh}\approx \sqrt{2}\, \Gamma$ 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 $\Gamma_{\rm sh}\propto R_{\rm sh}^{-3/2}$(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 $E_{\rm f}/ E_{\rm i}\sim \Gamma_{\rm sh}^2$ 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.