Nonlinear processes in cosmic-ray precursor of strong supernova shock: Maximum energy and average energy spectrum of accelerated particles

The instability in the cosmic-ray precursor of a supernova shock is studied. The level of turbulence in this region determines the maximum energy of accelerated particles. The consideration is not limited by the case of weak turbulence. It is assumed that the 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. The average spectrum of cosmic rays injected in the interstellar medium in the course of adiabatic SNR evolution takes the approximate form E−2 at energies larger than 10–30 GeV/nucleon with the maximum energy that is close to the position of the knee in cosmic-ray spectrum at 4 × 1015 eV. At an earlier stage of SNR evolution – the ejecta-dominated stage, the particles are accelerated to higher energies and have a rather steep power-law distribution. These results suggest that the knee may mark the transition from the ejecta-dominated to the adiabatic evolution of SNR shocks which accelerate cosmic rays.