Particle reacceleration by compressible turbulence in galaxy clusters: effects of reduced mean free path

Abstract: Direct evidence for in situ particle acceleration mechanisms in the inter-galactic-medium (IGM) is provided by the diffuse Mpc--scale synchrotron emissions observed from galaxy clusters. It has been proposed that MHD turbulence, generated during cluster-cluster mergers, may be a source of particle reacceleration in the IGM. Calculations of turbulent acceleration must account self-consistently for the complex non--linear coupling between turbulent waves and particles. This has been calculated in some detail under the assumption that turbulence interacts in a collisionless way with the IGM. In this paper we explore a different picture of acceleration by compressible turbulence in galaxy clusters, where the interaction between turbulence and the IGM is mediated by plasma instabilities and maintained collisional at scales much smaller than the Coulomb mean free path. In this regime most of the energy of fast modes is channeled into the reacceleration of relativistic particles and the acceleration process approaches a universal behaviour being self-regulated by the back-reaction of the accelerated particles on turbulence itself. Assuming that relativistic protons contribute to several percent (or less) of the cluster energy, consistent with the FERMI observations of nearby clusters, we find that compressible turbulence at the level of a few percent of the thermal energy can reaccelerate relativistic electrons at GeV energies, that are necessary to explain the observed diffuse radio emission in the form of giant radio halos.