New scaling relations in cluster radio haloes and the re-acceleration model

In this paper we derive new expected scaling relations for clusters with giant radio haloes in the framework of the re-acceleration scenario in a simplified, but physically motivated, form, namely: radio power (PR) versus size of the radio emitting region (RH), and PR versus total cluster mass (MH) contained in the emitting region and cluster velocity dispersion (σH) in this region. We search for these correlations by analysing the most recent radio and X-ray data available in the literature for a well-known sample of clusters with giant radio haloes. In particular we find a good correlation between PR and RH and a very tight ‘geometrical’ scaling between MH and RH. From these correlations PR is also expected to scale with MH and σH and this is confirmed by our analysis. We show that all the observed trends can be well reconciled with expectations in the case of a slight variation of the mean magnetic field strength in the radio halo volume with MH. A byproduct correlation between RH and σH is also found, and can be further tested by optical studies. In addition, we find that observationally RH scales non-linearly with the virial radius of the host cluster, and this immediately means that the fraction of the cluster volume which is radio emitting increases with cluster mass and thus that the non-thermal component in clusters is not self-similar.