D. BHATTACHARYYA
Articles written in Journal of Astrophysics and Astronomy
Volume 39 Issue 1 February 2018 Article ID 0004
$M_{\bullet}$ − $\sigma$ 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, $M_{\bullet} = f M_{\rm b}$, 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 $f = (1.23 \pm 0.09) × 10^{−3}$ which are consistent with the observed ranges.
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