The average energy gain, and hence the spectrum of accelerated particles, depends on the distribution of shock crossing angles . 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 (solid line) and density (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 . The case of regular deflection by a large-scale field upstream yielded an only slightly different index . Bednarz & Ostrowski (1998), for various levels of scattering parallel and perpendicular to the average magnetic field direction, found an asymptotic value of 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).
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