Abstract. Radio and far infrared luminosities of star-forming galaxies follow a tight linear relation. Making use of ASCA and BeppoSAX observations of a well-defined sample of nearby star-forming galaxies, we argue that tight linear relations hold between the X-ray, radio and far infrared luminosities. The effect of intrinsic absorption is investigated taking NGC 3256 as a test case. It is suggested that the hard X-ray emission is directly related to the Star Formation Rate. Star formation processes may also account for most of the 2-10 keV emission from LLAGNs of lower X-ray luminosities (for the same FIR and radio luminosity). Deep Chandra observations of a sample of radio-selected star-forming galaxies in the Hubble Deep Field North show that the same relation holds also at high (0.2 < ∼ z < ∼ 1.3) redshift. The X-ray/radio relations also allow a derivation of X-ray number counts up to very faint fluxes from the radio Log N-Log S , which is consistent with current limits and models. Thus the contribution of star-forming galaxies to the X-ray background can be estimated.

The radio spectral index map of the Coma halo shows a progressive steepening of the spectral index with increasing radius. Such a steepening cannot be simply justified by models involving continuous injection of fresh particles in the Coma halo or by models involving diffusion of fresh electrons from the central regions. We propose a two‐phase model in which the relativistic electrons injected in the Coma cluster by some processes (starbursts, AGNs, shocks, turbulence) during a first phase in the past are systematically reaccelerated during a second phase for a relatively long time (∼1 Gyr) up to the present time. We show that for reacceleration time‐scales of ∼0.1 Gyr this hypothesis can well account for the radio properties of Coma C. For the same range of parameters which explain Coma C we have calculated the expected fluxes from the inverse Compton scattering of the Cosmic Microwave Background (CMB) photons, finding that the hard X‐ray tail discovered by BeppoSAX may be accounted for by the stronger reacceleration allowed by the model. The possibility of extending the main model assumptions and findings to the case of the other radio haloes is also discussed, the basic predictions being consistent with the observations.