A Log‐Quadratic Relation for Predicting Supermassive Black Hole Masses from the Host Bulge Sersic Index

We reinvestigate the correlation between black hole mass and bulgeconcentration. With an increased galaxy sample, updated estimates of galaxydistances, black hole masses, and Sersic indices `n' - a measure ofconcentration - we perform a least-squares regression analysis to obtain arelation suitable for the purpose of predicting black hole masses in othergalaxies. In addition to the linear relation, log(M_bh) = 7.81(+/-0.08) +2.69(+/-0.28)[log(n/3)] with epsilon_(intrin)=0.31 dex, we investigated thepossibility of a higher order M_bh-n relation, finding the second order term inthe best-fitting quadratic relation to be inconsistent with a value of zero atgreater than the 99.99% confidence level. The optimal relation is given bylog(M_bh) = 7.98(+/-0.09) + 3.70(+/-0.46)[log(n/3)] -3.10(+/-0.84)[log(n/3)]^2, with epsilon_(intrin)=0.18 dex and a total absolutescatter of 0.31 dex. Extrapolating the quadratic relation, it predicts blackholes with masses of ~10^3 M_sun in n=0.5 dwarf elliptical galaxies, comparedto ~10^5 M_sun from the linear relation, and an upper bound on the largestblack hole masses in the local universe, equal to 1.2^{+2.6}_{-0.4}x10^9M_sun}. In addition, we show that the nuclear star clusters at the centers oflow-luminosity elliptical galaxies follow an extrapolation of the samequadratic relation. Moreover, we speculate that the merger of two suchnucleated galaxies, accompanied by the merger and runaway collision of theircentral star clusters, may result in the late-time formation of somesupermassive black holes. Finally, we predict the existence of, and provideequations for, a relation between M_bh and the central surface brightness ofthe host bulge