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 $\rho(r) = \rho_{0}(r/ r_{0})^{-\gamma}$ 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 $p = (2\gamma + 6)/(2 + \gamma)$ 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.