Cosmic rays X. The cosmic ray knee and beyond: diffusive acceleration at oblique shocks

¹ Institut für Physik, University of Dortmund, Otto-Hahn-Str. 4, 44227 Dortmund, Germany e-mail: meli@physik.uni-dortmund.de
² Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany
³ Department of Physics and Astronomy, University of Bonn, Germany e-mail: plbiermann@mpifr-bonn.mpg.de

Received: 3 February 2006
Accepted: 21 April 2006

Abstract

Our purpose is to evaluate the rate of the maximum energy and the acceleration rate that cosmic rays acquire in the non-relativistic diffusive shock acceleration as it could apply during their lifetime in various astrophysical sites, where highly oblique shocks exist. We examine numerically (using Monte-Carlo simulations) the effect of the diffusion coefficients on the energy gain and the acceleration rate, by testing the role between the obliquity of the magnetic field to the shock normal, and the significance of both perpendicular cross-field diffusion and parallel diffusion coefficients to the acceleration rate. We find (and justify previous analytical work – Jokipii 1987, ApJ, 313, 842) that in highly oblique shocks the smaller the perpendicular diffusion gets compared to the parallel diffusion coefficient values, the greater the energy gain of the cosmic rays to be obtained. An explanation of the cosmic ray spectrum in high energies, between 10¹⁵ eV and about 10¹⁸ eV is claimed, as we estimate the upper limit of energy that cosmic rays could gain in plausible astrophysical regimes; interpreted by the scenario of cosmic rays which are injected by three different kind of sources, (a) supernovae which explode into the interstellar medium, (b) Red Supergiants, and (c) Wolf-Rayet stars, where the two latter explode into their pre-supernovae winds.