Mass and Mean Velocity Dispersion Relations for Supermassive Black Holes in Galactic Bulges

TL;DR

This paper models the relationship between supermassive black hole mass and stellar velocity dispersion in galactic bulges using a self-similar polytropic approach, providing insights into their co-evolution and observed correlations.

Contribution

It introduces a self-similar polytropic model to reproduce the empirical $M_{BH}-\sigma$ relation and explores the physical link between bulge evolution and SMBH formation.

Findings

Reproduces the observed $M_{BH}-\sigma$ power law.

Examines properties of bulges and SMBHs.

Discusses scatter and evolution scenarios of SMBHs.

Abstract

Growing evidence indicate supermassive black holes (SMBHs) in the mass range of $M_{\rm BH}$$\sim 10^6-10^{10}M_{\odot}$ lurking in central bulges of many galaxies. Extensive observations reveal fairly tight power laws of $M_{\rm BH}$ versus the mean stellar velocity dispersion $\sigma$ of the host bulge. The dynamic evolution of a bulge and the formation of a central SMBH should be physically linked by various observational clues. In this contribution, we reproduce the empirical $M_{\rm BH}-\sigma$ power laws based on a self-similar general polytropic quasi-static bulge evolution and a sensible criterion of forming a SMBH surrounding the central density singularity of a general singular polytropic sphere (SPS) \cite{loujiang2008}. Other properties of host bulges and central SMBHs are also examined. Based on our model, we discuss the intrinsic scatter of the $M_{\rm BH}-\sigma$ relation and a scenario for the evolution of SMBHs in different host bulges.