1. the particle spectra diverge from a power-law in the full energy range;
2. before the spectrum cut-off a harder spectral component can appear;
3. the exact shape of the spectrum depends on both the amplitude of the magnetic field perturbations and the wave power spectrum.
One may note that a power-law part of the particle spectrum steeping with increasing amplitude of the field perturbations, more for the Kolmogorov perturbations, panel (b).
Niemiec and Ostrowski (2003a) note that the spectral indices obtained are consistent with previous numerical calculations of Ostrowski (1991, 1993) and with the analytic results obtained in the limit of small perturbations (Kirk and Heavens, 1989). The non power-law character of the obtained particle spectra results from the limited dynamic range of magnetic field perturbations. In the energy range within which the approximately power-law spectrum forms, particles can be effectively scattered by the magnetic field inhomogeneities. The character of the spectrum changes at the highest particle energies where kres < kmin and particles are only weakly scattered. Then the anisotropic distributed upstream particles can effectively reflect from the region of compressed magnetic field downstream of the shock, leading to the spectrum flattening (Ostrowski, 1991). The cut-off in the spectrum is formed mainly owing to very weakly scattered particles escaping from the shock to the introduced upstream free escape boundary.
In Fig. 4.29.2 are presented the spectra obtained for super-luminal shocks with
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