The non-linear self-regulation of cosmic ray acceleration at shocks

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Abstract
We describe time-dependent numerical simulations of the acceleration of cosmic rays at shock fronts. When the low-energy particle injection rate is above approximately 3 × 10–4 of the background particles flowing into the shock, the cosmic ray pressure is comparable with or exceeds the post-shock thermal pressure, and the shock becomes cosmic ray (CR) dominated. When the CR pressure is large, the scale-length of the density transition is increased, thereby reducing the acceleration efficiency and limiting the cosmic ray energy to that available from the bulk plasma motions. The cosmic ray energy spectrum is softened by non-linear feedback. Even at high Mach number, the softening is sufficient to account for the spectral indices of synchrotron-emitting electrons in most radio sources. The cosmic rays pass energy to magnetic turbulence at a rate determined by the Alfvén velocity. If the turbulence is damped, the background plasma is heated even in a cosmic ray dominated shock. The dependence of CR spectra on Mach number, injection rate, shock velocity, Alfvén velocity and time evolution are examined.

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