The Slope of the Black Hole Mass versus Velocity Dispersion Correlation

TLDR

Observations of nearby galaxies reveal a tight correlation between black hole mass and host galaxy velocity dispersion, yet published slopes range from 3.75 to 5.3, with low values possibly caused by measurement errors and incorrect Milky Way dispersion assumptions. The authors present a simple model for the Galactic bulge velocity‑dispersion profile. They argue that systematic, dispersion‑dependent measurement errors and the extrapolation of central dispersions to r_e/8 contribute to the observed slope variations. A new determination using 31 galaxies gives a slope of 4.02 ± 0.32, an intercept of 8.13 ± 0.06, and shows that the intrinsic dispersion in log M is no larger than 0.3 dex.

Abstract

Observations of nearby galaxies reveal a strong correlation between the mass of the central dark object M and the velocity dispersion sigma of the host galaxy, of the form log(M/M_sun) = a + b*log(sigma/sigma_0); however, published estimates of the slope b span a wide range (3.75 to 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 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 b=4.02 +/- 0.32, a=8.13 +/- 0.06, for sigma_0=200 km/s. The M-sigma relation has an intrinsic dispersion in log M that is no larger than 0.3 dex. In an Appendix, we present a simple model for the velocity-dispersion profile of the Galactic bulge.