A Log-Quadratic Relation for Predicting Supermassive Black Hole Masses from the Host Bulge Sérsic Index

We reinvestigate the correlation between black hole mass and bulge concentration. With an increased galaxy sample (totaling 27) and updated estimates of galaxy distances, black hole masses, and Sérsic indices n—a measure of concentration—we perform a least-squares regression analysis to obtain a relation suitable for the purpose of predicting black hole masses in other galaxies. In addition to the linear relation, log M_bh = (7.81 ± 0.08) + (2.69 ± 0.28) log(n/3) with _intrinsic = 0.31 dex, we investigated the possibility of a higher order M_bh-n relation, finding the second-order term in the best-fitting quadratic relation to be inconsistent with a value of zero at greater than the 99.99% confidence level. The optimal relation is given by log M_bh = (7.98 ± 0.09) + (3.70 ± 0.46) log(n/3) - (3.10 ± 0.84) [log(n/3)]², with _intrinsic = 0.18 dex and a total absolute scatter of 0.31 dex. When the quadratic relation is extrapolated, it predicts black holes with masses of ~10³ M_☉ in n = 0.5 dwarf elliptical galaxies, compared to ~10⁵ M_☉ from the linear relation, and an upper bound on the largest black hole masses in the local universe equal to 1.2 × 10⁹ M_☉. In addition, we show that the nuclear star clusters at the centers of low-luminosity elliptical galaxies follow an extrapolation of the same quadratic relation, strengthening suggestions for a possible evolutionary link between supermassive black holes and nuclear star clusters. Moreover, we speculate that the merger of two such nucleated galaxies, accompanied by the merger and runaway collision of their central star clusters, could result in the late-time formation of some supermassive black holes. Finally, we predict the existence of, and provide equations for, an M_bh-μ₀ relation, in which μ₀ is the (extrapolated) central surface brightness of a bulge.