Modeling the interaction between ICM and relativistic plasma in cooling flows and application to the Perseus cluster

We argue that the presence of diffuse synchrotron emission forming the so-called radio mini-halos in some cooling flow clusters can be explained by re-acceleration of relativistic electrons, the necessary energetics being supplied by the cooling flows themselves. In particular, the re-acceleration due to MHD turbulence has the correct radial dependence on the parameters to naturally balance the radiative losses. As an application we show that the main properties of the radio mini-halo in the Perseus cluster (brightness profile, total radio spectrum and radial spectral steepening) can be accounted for by the synchrotron radiation from relic relativistic electrons in the cluster, which are efficiently re-accelerated by MHD turbulence via Fermi-like processes due to the compression of the cluster magnetic field in the cooling flow region. Since the presence of an observable radio mini-halo in a cooling flow region critically depends on the combination of several physical parameters, we suggest that the rarity of radio mini-halos found in cooling flow clusters is due to the fact that the physical conditions of the ICM are intermediate between those which lead to the formation of extended radio halos and those holding in cooling flows without radio halos. The basic results of our model remain unchanged even if the cooling flow is stopped somewhere in the innermost region of the cluster.