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.
