The intra-cluster magnetic field in the double relic galaxy cluster Abell 2345

Magnetic fields are ubiquitous in galaxy clusters, yet their radial profile, power spectrum, and connection to host cluster properties are poorly known. Merging galaxy clusters hosting diffuse polarized emission in the form of radio relics offer a unique possibility to study the magnetic fields in these complex systems. In this paper, we investigate the intra-cluster magnetic field in Abell 2345. This cluster hosts two radio relics that we detected in polarization with 1-2 GHz JVLA observations. X-ray XMM-Newton images show a very disturbed morphology. We derived the Rotation Measure (RM) of five polarized sources within $\sim$ 1 Mpc from the cluster center applying the RM synthesis. Both, the average RM and the RM dispersion radial profiles probe the presence of intra-cluster magnetic fields. Using the thermal electron density profile derived from X-ray analysis and simulating a 3D magnetic field with fluctuations following a power spectrum derived from magneto-hydrodynamical cosmological simulations, we build mock RM images of the cluster. We constrained the magnetic field profile in the eastern radio relic sector by comparing simulated and observed RM images. We find that, within the framework of our model, the data require a magnetic field scaling with thermal electron density as $B(r)\propto n_e(r)$. The best model has a central magnetic field (within a 200 kpc radius) of $2.8\pm0.1$ $\mu$G. The average magnetic field at the position of the eastern relic is $\sim$0.3 $\mu$G, a factor 2.7 lower than the equipartition estimate.