$$M_{\bullet } - \sigma $$ 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 the 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, \(M_{\bullet } =f M_\mathrm{b}\), 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.