The $M_{\rm BH}$$-$$R_{\rm b}$ relation and the high-mass end of the $M_{\rm BH}$$-$$\sigma$ relation

Abstract

Using a sample of 151 galaxies with dynamically measured black hole (BH) masses ($M_{\rm BH}$), we investigate the scaling relations between $M_{\rm BH}$ and the stellar velocity dispersion, $\sigma$, and, for a subsample of 30 core-S\'ersic galaxies, between $M_{\rm BH}$ and the size of the partially depleted core, $R_{\rm b}$. Core-S\'ersic galaxies, identified using high-resolution $Hubble ~ Space ~ Telescope$ imaging and spanning both the normal-core $(R_{\rm b}<0.5$ kpc) and large-core ($R_{\rm b}>0.5$ kpc) regimes, define an updated $M_{\rm BH}$$-$$R_{\rm b}$ relation of the form $M_{\rm BH} \propto R_{\rm b}^{1.16 \pm 0.10}$, with an rms scatter of $\Delta_{\rm rms} \simeq 0.28$ dex in $\log M_{\rm BH}$. We find that S\'ersic and normal-core galaxies together follow a common log-linear $M_{\rm BH}$-$\sigma$ relation with a slope of $4.95 \pm 0.29$ and a scatter $\Delta_{\rm rms} \simeq 0.46$ dex. Deviations from this relation arise at the highest BH masses, where large-core galaxies, including six with direct $M_{\rm BH}$ measurements, drive a significant upturn. We find that these galaxies typically host ultramassive black holes whose masses scale more strongly with $R_{\rm b}$ than $\sigma$, and lie $\sim (1-4$) $~\times$ the intrinsic scatter (0.39 dex) above the relation defined by S\'ersic and normal-core galaxies. The $M_{\rm BH}$$-$$R_{\rm b}$ relation shows $\sim 30$-$47\%$ less scatter in $\log M_{\rm BH}$ than the corresponding $M_{\rm BH}$$-$$\sigma$ relation for the same sample. We interpret the high-mass upturn in the $M_{\rm BH}$$-$$\sigma$ diagram as a consequence of successive major, dry mergers, a scenario that naturally explains the observed flattening of the $\sigma-L_V$ relation at $M_V < -23.5$ mag.