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 bulge concentration. With an increased galaxy sample ( totaling 27) and updated estimates of galaxy distances, black hole masses, and Sersic 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 is an element of(intrinsic) =0: 31-(+0: 09)(0.07) 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 is an element of(intrinsic)=0: 18-(+0. 07)(0.06) dex and a total absolute scatter of 0.31 dex. When the quadratic relation is extrapolated, it predicts black holes with masses of similar to 10(3)M(circle dot) in n=0. 5 dwarf elliptical galaxies, compared to similar to 10(5)M(circle dot) from the linear relation, and an upper bound on the largest black hole masses in the local universe equal to 1.2(-0.4)(+2 6) x10(9) M-circle dot. 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-mu(0) relation, in which mu(0) is the ( extrapolated) central surface brightness of a bulge.