Turbulence and particle acceleration in collisionless supernovae remnant shocks

II. Cosmic-ray transport

¹ Centre d'Études Spatiales et du Rayonnement, CNRS, Université Paul Sabatier Toulouse 3, 9 avenue du Colonel Roche, 31028 Toulouse Cedex, France e-mail: Alexandre.Marcowith@cesr.fr
² Institut d'Astrophysique de Paris, UMR 7095 CNRS, Université Pierre & Marie Curie, 98bis boulevard Arago, 75014 Paris, France e-mail: lemoine@iap.fr
³ Laboratoire d'Astrophysique de Grenoble, CNRS, Université Joseph Fourier II, BP 53, 38041 Grenoble, and Institut Universitaire de France, France e-mail: Guy.Pelletier@obs.ujf-grenoble.fr

Received: 20 December 2005
Accepted: 16 March 2006

Abstract

Supernovae remnant shock waves could be at the origin of cosmic rays up to energies in excess of the knee (eV) if the magnetic field is efficiently amplified by the streaming of accelerated particles in the shock precursor. This paper follows up on a previous paper (Pelletier et al. 2006, A&A, in press) which derived the properties of the MHD turbulence so generated, in particular its anisotropic character, its amplitude and its spectrum. In the present paper, we calculate the diffusion coefficients, also accounting for compression through the shock, and show that the predicted three-dimensional turbulence spectrum (with and the wavenumber components along and perpendicular to the shock normal) generally leads to Bohm diffusion in the parallel direction. However, if the anisotropy is constrained by a relation of the form , which arises when the turbulent energy cascade occurs at a constant rate independent of scale, then the diffusion coefficient loses its Bohm scaling and scales as in isotropic Kolmogorov turbulence. We show that these diffusion coefficients allow to account for X-ray observations of supernova remnants. This paper also calculates the modification of the Fermi cycle due to the energy lost by cosmic rays in generating upstream turbulence and the concomittant steepening of the energy spectrum. Finally we confirm that cosmic rays can produced an amplified turbulence in young SNr during their free expansion phase such that the maximal energy is close to the knee and the spectral index is close to 2.3 in the warm phase of the interstellar medium.