Constraining relativistic protons and magnetic fields in galaxy clusters through radio and γ-ray observations: the case of A2256

Giant radio halos are the most relevant examples of diffuse synchrotron emission from galaxy clusters. A number of these sources have very steep spectra, of spectral index alpha >= 1.5-1.6 (F(v) alpha v(-alpha)), and are ideal targets for testing current models of the origin of the relativistic particles. A2256 hosts the nearest radio halo with a very steep spectrum, of alpha = 1.61, and a very large population of relativistic protons in the cluster would be necessary if the halo were produced by synchrotron emission from secondary particles. In this case, the 0.1-1 GeV gamma-ray luminosity is expected to be 10-20 times higher than that of clusters hosting radio halos of similar radio power at GHz frequencies but with spectra more typical of the presently observed halo population, alpha similar to 1.2. Based on these assumptions, future FERMI/GLAST observations are expected to detect A2256, provided that the magnetic field in the central cluster region is <= 10-15 mu G. We show that this will provide a prompt test of hadronic models for the origin of radio halos, and complementary constraints on both the cluster magnetic field and the physics of particle acceleration mechanisms.