The Slope of the Black Hole Mass versus Velocity Dispersion Correlation

Abstract

Observations of nearby galaxies reveal a strong correlation between the mass of the central dark object M BH and the velocity dispersion σ of the host galaxy, of the form log( M BH / M ☉ ) = α + β log(σ/σ 0 ); however, published estimates of the slope β span a wide range (3.75-5.3). Merritt & Ferrarese have argued that low slopes (≲4) arise because of neglect of random measurement errors in the dispersions and an incorrect choice for the dispersion of the Milky Way Galaxy. We show that these explanations and several others account for at most a small part of the slope range. Instead, the range of slopes arises mostly because of systematic differences in the velocity dispersions used by different groups for the same galaxies. The origin of these differences remains unclear, but we suggest that one significant component of the difference results from Ferrarese & Merritt's extrapolation of central velocity dispersions to r e /8 ( r e is the effective radius) using an empirical formula. Another component may arise from dispersion-dependent systematic errors in the measurements. A new determination of the slope using 31 galaxies yields β = 4.02 ± 0.32, α = 8.13 ± 0.06 for σ 0 = 200 km s -1 . The M BH -σ relation has an intrinsic dispersion in log M BH that is no larger than 0.25-0.3 dex and may be smaller if observational errors have been underestimated. In an appendix, we present a simple kinematic model for the velocity-dispersion profile of the Galactic bulge.