Estimating galaxy cluster magnetic fields by the classical and hadronic minimum energy criterion

We wish to estimate magnetic field strengths of radio emitting galaxy clusters by minimizing the non-thermal energy density contained in cosmic ray electrons (CRe), protons (CRp), and magnetic fields. The classical minimum energy estimate can be constructed independently of the origin of the radio synchrotron emitting CRe yielding thus an absolute minimum of the non-thermal energy density. Provided the observed synchrotron emission is generated by a CRe population originating from hadronic interactions of CRp with the ambient thermal gas of the intra-cluster medium, the parameter space of the classical scenario can be tightened by means of the hadronic minimum energy criterion. For both approaches, we derive the theoretically expected tolerance regions for the inferred minimum energy densities. Application to the radio halo of the Coma cluster and the radio mini-halo of the Perseus cluster yields equipartition between cosmic rays and magnetic fields within the expected tolerance regions. In the hadronic scenario, the inferred central magnetic field strength ranges from 2.4 muG (Coma) to 8.8 muG (Perseus), while the optimal CRp energy density is constrained to 2 per cent +/- 1 per cent of the thermal energy density (Perseus). We discuss the possibility of a hadronic origin of the Coma radio halo while current observations favour such a scenario for the Perseus radio mini-halo. Combining future expected detections of radio synchrotron, hard X-ray inverse Compton, and hadronically induced gamma-ray emission should allow an estimate of volume averaged cluster magnetic fields and provide information about their dynamical state.