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 (P-R) versus size of the radio emitting region (R-H), and P-R versus total cluster mass (M-H) contained in the emitting region and cluster velocity dispersion (sigma(H)) 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 P-R and R-H and a very tight `geometrical' scaling between M-H and R-H. From these correlations P-R is also expected to scale with M-H and s(H) 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 M-H. A byproduct correlation between R-H and sigma(H) is also found, and can be further tested by optical studies. In addition, we find that observationally R-H 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.