Particle Acceleration in Supernova Remnants in the Presence of Streaming Instability and Nonlinear Wave Interactions

The instability in the cosmic-ray (CR) precursor of a SN shock is studied. The level of turbulence in this region determines the maximum energy of accelerated CRs. The consideration is not limited by the case of weak turbulence. It is assumed that Kolmogorov type nonlinear wave interactions together with the ion-neutral collisions restrict the amplitude of random magnetic field. As a result, the maximum energy of accelerated particles strongly depends on the age of a SNR. It can be as high as $10^{17}Z$ eV in young SNR and falls down to about $10^{10}Z$ eV at the end of Sedov stage ($Z$ is the particle charge). This finding may explain why the SNRs with the age more than a few thousand years are not prominent sources of very high energy $\gamma $-rays. The averaged spectrum of ultrarelativistic CR injected in the interstellar medium is close or somewhat steeper than $E^{-2}$.
Comment: 4 pages, 1 figure