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

Growing evidence indicate supermassive black holes (SMBHs) in the mass range of $M_{\mathrm{B}\mathrm{H}}$$\sim {10}^6-{10}^{10}{M}_{\odot }$ lurking in central bulges of many galaxies. Extensive observations reveal fairly tight power laws of $M_{\mathrm{B}\mathrm{H}}$ 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 $M_{\mathrm{B}\mathrm{H}}-\sigma$ 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 $M_{\mathrm{B}\mathrm{H}}-\sigma$ relation and a scenario for the evolution of SMBHs in different host bulges.