Correlation of the magnetic field and the intra-cluster gas density in galaxy clusters

We present a correlation between X-ray surface brightness and Faraday rotation measure in galaxy clusters, both, from radio and X-ray observations as well as from modeling of the intra-cluster medium. The observed correlation rules out a magnetic field of constant strength throughout the cluster. Cosmological, magneto-hydrodynamic simulations of galaxy clusters are used to show that for a magnetic field of cosmic origin this correlation is expected and excellently reproduces the observations showing that the rms scatter of the Faraday rotation increases linearly with the X-ray surface brightness. From the correlation between the observable quantities, rotation measure and X-ray surface brightness, we infer a relation between the physical quantities: magnetic field and gas density. For the best available observations, those of A119, we find $B \propto n_{\rm e}^{0.9}$