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 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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