Maximum energy of cosmic rays accelerated by supernova shocks

The expression for the cutoff momentum of CR, accelerated by the supernova blast wave is derived. Geometrical factors (finite increase with time shock size, slowing shock speed and CR adiabatic effect in the downstream region) are shown to determine the value of the cutoff momentum. These factors are stronger than the time restriction and have a significant dynamical effect: the supernova blast wave cannot be completely smoothed by the CR backreaction even at very high Mach numbers. The shock transition always includes a pure gas subshock which strongly influences CR acceleration and shock evolution. It is shown that maximum particle energy achievable during CR acceleration by supernova shock can be as large as epsilon(max) approximate to Z x 10(15) eV, if the diffusion coefficient is as small as the Bohm limit. Due to nonlinear effect and adiabatic heating in the downstream region in the free expansion phase the actual value of epsilon(max) is an order of magnitude higher than that from previous estimates based on the plane-wave consideration and is high enough to consider CR diffusive acceleration in SNRs as a main source of galactic CR at least up to the knee energy 3 x 10(15) eV.