2018MNRAS.478.4922Z

Galaxy cluster merger shocks are the likely source of relativistic electrons, but many observations do not fit into the standard acceleration models. In particular, there is a long-standing discrepancy between the radio derived Mach numbers M_ radio_ and the Mach numbers derived from X-ray measurements, M_ X_. Here, we show how superdiffusive electron transport and superdiffusive shock acceleration (SSA) can help to solve this problem. We present a heuristic derivation of the superlinear time growth of the mean square displacement of particles, <Δx²> ∝ t^β, and of the particle energy spectral index in the framework of SSA. The resulting expression for the radio spectral index α is then used to determine the superdiffusive exponent β from the observed values of α and of the compression ratio for a number of radio relics. Therefore, the fact that M_ radio_> M_ X_ can be explained by SSA without the need to make assumptions on the energy spectrum of the seed electrons to be re-accelerated. We also consider the acceleration times obtained in the diffusive case, based both on the Bohm diffusion coefficient and on the quasi-linear diffusion coefficient. While in the latter case the acceleration time is consistent with the estimated electron energy loss time, in the former case it is much shorter.