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

TL;DR

This paper derives a theoretical relation between supermassive black hole mass and stellar velocity dispersion in elliptical galaxies using realistic density profiles, aligning well with observed data.

Contribution

It provides an analytical expression for the $M_{\bullet} - \sigma$ relation based on power-law and Nuker galaxy profiles, connecting theory with observations.

Findings

Derived the relation $p = (2\gamma + 6)/(2 + \gamma)$ for different profiles.

Found best-fit $p$ = 3.81 and $f$ = 1.23e-3 for Nuker profiles.

Results are consistent with observed $M_{\bullet} - \sigma$ relations.

Abstract

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.