Time-dependent treatment of cosmic-ray spectral steepening due to turbulence driving

Abstract

Cosmic-ray acceleration at non-relativistic shocks relies on scattering by turbulence that the cosmic rays drive upstream of the shock. We explore the rate of energy transfer from cosmic rays to non-resonant Bell modes and the spectral softening it implies. Accounting for the finite time available for turbulence driving yields a much smaller spectral impact than found earlier with steady-state considerations. Generally, for diffusion scaling with the Bohm rate by a factor $\eta$, the change in spectral index is at most $\eta$ divided by the Alfv\'enic Mach number of the thermal sub-shock. For small $M_A$ it is well below this limit. Only for very fast shocks and very efficient cosmic-ray acceleration the change in spectral index may reach 0.1. For standard SNR parameters it is negligible.