The $M_{\bullet }-\sigma$ relation in spherical systems

To investigate the $M_{\bullet }-\sigma$ relation, we consider realistic elliptical galaxy profiles that are taken to follow a single power law density profile given by $\rho (r)={\rho }_0(r\mathrm{/}{r}_0{)}^{-\gamma }$ or the Nuker intensity profile. We calculate the density using Abel’s formula in the latter case by employing the derived stellar potential in both cases, we derive the distribution function $f(E)$ of the stars in presence of the supermassive black hole (SMBH) at the center and hence compute the line of sight (LOS) velocity dispersion as a function of radius. For the typical range of values for masses of SMBH, we obtain $M_{\bullet }\propto {\sigma }^p$ for different profiles. An analytical relation $p=(2\gamma +6)\mathrm{/}(2+\gamma )$ is found which is in reasonable agreement with observations (for $\gamma$ = 0.75 - 1.4, $p$ = 3.6 - 5.3). Assuming that a proportionality relation holds between the black hole mass and bulge mass, $M_{\bullet }=f{M}_b$, and applying this to several galaxies we find the individual best fit values of $p$ as a function of $f$; also by minimizing ${\chi }^2$, we find the best fit global $p$ and $f$. For Nuker profiles we find that $p$ = $3.81\pm 0.004$ and $f$ = $(1.23 \pm 0.09)\times 10^{-3}$ which are consistent with the observed ranges.