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

To investigate the $M_\bullet -σ$ relation, we consider realistic elliptical galaxy profiles that are taken to follow a single power law density profile given by $ρ(r) = ρ_{0}(r/ r_{0})^{-γ}$ 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 $M_{\bullet} \propto σ^{p}$ for different profiles. An analytical relation $p = (2γ+ 6)/(2 + γ)$ is found which is in reasonable agreement with observations (for $γ$ = 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 $χ^{2}$, 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.