M∙ − σ relation in spherical systems

Abstract

To investigate the M∙−σ relation, we consider realistic elliptical galaxy profiles that are taken to follow a single power-law density profile given by Ï (r)=Ï 0(r/r0)−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 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∙∠σp 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∙=fMb, and applying this to several galaxies, we find the individual best fit values of p as a function of f; also by minimizing χ2, we find the best fit global p and f. For Nuker profiles, we find that p=3.81±0.004 and f=(1.23±0.09)×10−3 which are consistent with the observed ranges.