3 Spectrum of Fermi-accelerated particles

The average energy gain, and hence the spectrum of accelerated particles, depends on the distribution of shock crossing angles $\theta$. This distribution, which will be highly anisotropic in the case of an ultra-relativistic shock, depends in turn on the assumed deflection mechanism, and may be obtained by numerical simulation (Bednarz & Ostrowski 1998; Gallant et al. 1998).

  

Figure 1: Asymptotic angular distribution of the particles at shock crossing in the case of direction-angle scattering upstream, expressed both in terms of flux $F(\mu')$ (solid line) and density $n(\mu')$ (dashed line), each normalised to unity

Figure 1 shows the distribution obtained by Gallant et al. (1998) for the case of scattering in random magnetic fields both upstream and downstream, which yielded a spectral index $p \sim 2.25$. The case of regular deflection by a large-scale field upstream yielded an only slightly different index $p \sim 2.3$. Bednarz & Ostrowski (1998), for various levels of scattering parallel and perpendicular to the average magnetic field direction, found an asymptotic value of $p \sim 2.2$ for sufficiently relativistic shocks.

It is noteworthy that the values of p obtained in these simulations are compatible with those inferred from observations of the afterglows of GRB 970228, GRB 970402 (Waxman 1997) and GRB 970508 (Galama et al. 1998).