  We wish to estimate magnetic field strengths of radio emitting galaxy
clusters by minimising 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% +/- 1% 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.
