The scaling relation between the mass of supermassive black holes and the kinetic energy of random motions of the host galaxies

Thanks to the angular resolution of modern telescopes and kinematic models, the existence of supermassive black holes (SMBHs) in the inner part of galaxies has been established on quite solid grounds. A possible correlation between the mass of SMBHs and the evolutionary state of their host galaxies is expected. Based on the recent 2D decomposition of mid-infrared Spiter/IRAC images of local galaxies with M_bh measurements, we investigated various scaling laws, studying what the best predictor of the mass of the central SMBHs is. We focused on the M_bh-M_G sigma^2 law, the relation between the mass of SMBHs and the kinetic energy of random motions of the corresponding host galaxies. In order to find the best fit for each of the scaling laws examined, we performed a least-squares regression of M_bh on x for the considered sample of galaxies, x being a whatever known parameter of the galaxy bulge. Our analysis shows that M_bh-M_G sigma^2 law fits the examined experimental data successfully as much as the other known scaling laws and shows a value of chi^2 better than the others, a result which is consistent with previous determinations. This means that a combination of sigma and M_G could be necessary to drive the correlations between M_bh and other bulge properties. This issue has been investigated by a careful analysis of the residuals of the various relations. In order to avoid rushed conclusions on galaxy activity and evolution, the indirect inferring of M_bh from the kinetic energy of random motions should be considered, especially when applied to higher redshift galaxies. This statement is suggested by a reanalysis of the SDSS data used to study the SMBH growth in the nearby Universe. Adopting the M_bh-M_G sigma^2 relation instead of the M_bh-sigma, a radio-quiet/radio-loud dichotomy appears in the SMBH mass distribution of the corresponding SDSS early-type AGN galaxies.