Dependence of the Spectrum of Shock-Accelerated Ions on the Dynamics at the Shock Crossing

Diffusive shock acceleration (DSA) of ions occurs due to pitch-angle diffusion in the upstream and downstream regions of the shock and multiple crossing of the shock by these ions. The classical DSA theory implies continuity of the distribution at the shock transition and predicts a universal spectrum of accelerated particles, depending only on the ratio of the upstream and downstream fluid speeds. However, the ion dynamics at the shock front occurs within a collision-free region and is gyrophase dependent. The ions fluxes have to be continuous at the shock front. The matching conditions for the gyrophase-averaged distribution functions at the shock transition are formulated in terms of the transition and reflection probabilities. These probabilities depend on the shock angle and the magnetic compression as does the power spectrum of accelerated ions. Their spectral index is expressed in terms of the reflectivity. The spectrum is typically harder than the spectrum predicted by the classical DSA theory.