Abstract
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 at supernova-remnant shocks yields a smaller spectral impact than found earlier with steady-state considerations. Generally, for diffusion scaling with the Bohm rate by a factor η, the change in spectral index is at most η divided by the Alfvénic Mach number of the thermal sub-shock. For M A ≲ 50 it is well below this limit. Only for very fast shocks and very efficient cosmic-ray acceleration can the change in spectral index reach 0.1. For standard SNR parameters, it is negligible. Independent confirmation is derived by considering the synchrotron energy losses of electrons: if intense nonthermal multi-keV emission is produced, the energy loss, and hence the spectral steepening, is very small for hadronic cosmic rays that produce TeV-band gamma-ray emission.
Highlights
We explore the rate of energy transfer from cosmic rays to nonresonant Bell modes and the spectral softening it implies
For diffusion scaling with the Bohm rate by a factor η, the change in spectral index is at most η divided by the Alfvenic Mach number of the thermal subshock
Bell et al (2019) studied cosmic-ray acceleration at nonrelativistic shocks using a tensor expansion of the Vlasov equation, and they found a steepening of the cosmic-ray spectrum at supernova remnants (SNR) arising from energy transfer from the cosmic rays to turbulence in the precursor of the shocks
Summary
Bell et al (2019) studied cosmic-ray acceleration at nonrelativistic shocks using a tensor expansion of the Vlasov equation, and they found a steepening of the cosmic-ray spectrum at supernova remnants (SNR) arising from energy transfer from the cosmic rays to turbulence in the precursor of the shocks. Assuming the turbulence in question is nonresonant, the so-called Bell mode (Bell 2004), and taking an estimate for its magnetic-field energy density at the saturation level, UδB. Where vsh denotes the shock speed and Ucr the energy density in cosmic rays immediately upstream of the shock, they find a spectral steepening by
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