We present a model of cosmic ray heating of clusters' cores that reproduces
the observed temperature distribution in clusters by using an energy balance
condition in which the emitted X-ray energy is supplied by the hadronic cosmic
rays, which act as warming rays (WRs). The temperature profile of the IC gas is
correlated with the WR pressure distribution and, consequently, with the
non-thermal emission (radio, hard X-ray and gamma-ray) induced by the
interaction of the WRs with the IC gas and magnetic field. The temperature
distribution of the IC gas in both cool-core and non cool-core clusters is
successfully predicted from the measured IC gas density distribution. Under
this contraint, the WR model is also able to reproduce the thermal and
non-thermal pressure distribution in clusters, as well as their radial entropy
distribution. The WR model provides other observable features: a correlation of
the pressure ratio (WRs to thermal IC gas) with the inner cluster temperature
T_{inner}, a correlation of the gamma-ray luminosity with T_{inner}, a
substantial number of cool-core clusters observable with the GLAST-LAT
experiment, a surface brightness of radio halos in cool-core clusters that
recovers the observed one, a hard X-ray emission from cool-core clusters that
is systematically lower than the observed limits and yet observable with the
next generation HXR experiments like Simbol-X. The specific theoretical
properties and the multi-frequency distribution of the e.m. signals predicted
in the WR model render it quite different from the other models proposed for
the heating of clusters' cool-cores. Such differences make it possible to prove
or disprove our model as an explanation of the cooling-flow problems on the
basis of multi-frequency observations of galaxy clusters.Comment: 19 pages, 17 figures, A&A in pres
