Alfvenic reacceleration of relativistic particles in galaxy clusters: MHD waves, leptons and hadrons

There is a growing evidence that extended radio halos are most likely generated by electrons reaccelerated via some kind of turbulence generated in the cluster volume during major mergers. It is well known that Alfvén waves channel most of their energy flux in the acceleration of relativistic particles. Much work has been done recently to study this phenomenon and its consequences for the explanation of the observed non-thermal phenomena in clusters of galaxies. We investigate here the problem of particle-wave interactions in the most general situation in which relativistic electrons, thermal protons and relativistic protons exist within the cluster volume. The interaction of all these components with the waves, as well as the turbulent cascading and damping processes of Alfvén waves, are treated in a fully time-dependent way. This allows us to calculate the spectra of electrons, protons and waves at any fixed time. The Lighthill mechanism is invoked to couple the fluid turbulence, supposedly injected during cluster mergers, to MHD turbulence. We find that within this approach the fraction of relativistic hadrons in the intracluster medium (ICM) is severely constrained to contain a rather small fraction of the energy density of the ICM. When this constraint is violated, the damping of waves on protons is too strong and not enough energy is left for the acceleration of electrons. When these constraint are not violated, the results of our calculations compare well with the observed spectra and brightness profiles of the non-thermal radiation.