Origin of the Correlations Between Supermassive Black Holes and Their
  Host Galaxies
  Observations have shown that supermassive black holes in nearby elliptical
galaxies correlate tightly with the stellar velocity dispersion (the $\MBH -
\sigma$ relation) and the stellar mass (the $\MBH - \Mhost$ relation) of their
host spheroids. However, the origin of these correlations remains ambiguous. In
a previous paper by Zhu et al., we proposed a model which links the M-$\sigma$
relation to the the dynamical state of the system and the $\MBH - \Mhost$
relation to the self-regulation of galaxy growth. To test this model, we
compile a sample of observed galaxies with different properties and examine the
dependence of the above correlations on these parameters. We find that galaxies
that satisfy the the $\MBH - \sigma$ correlation appear to have reached virial
equilibrium, as indicated by the ratio between kinetic energy and gravitational
potential, 2K/U $\sim$ 1. Furthermore, the ratio of black hole accretion rate
to star formation rate remains nearly constant, BHAR /SFR $\sim$ $10^{-3}$, in
active galaxies over a wide range of mass in the redshift range z=0 - 3. These
results confirm our theoretical model that the observed correlations have
different origins: the $\MBH - \sigma$ relation may result from galaxy
relaxation, while the $\MBH$ - $\Mhost$ relation may be due to self-regulated
black hole accretion and star formation in galaxies.
