Adiabatic Black Hole Growth in Sérsic Models of Elliptical Galaxies

Abstract

We have examined the effect of slow growth of a central black hole on spherical galaxies that obey Sérsic or R 1/m surface-brightness profiles. During such growth the actions of each stellar orbit are conserved, which allows us to compute the final distribution function if we assume that the initial distribution function is isotropic. We find that black-hole growth leads to a central cusp or "excess light", in which the surface brightness varies with radius as R −1.3 (with a weak dependence on Sérsic index m), the line-of-sight velocity dispersion varies as R −1/2 , and the velocity anisotropy is β −0.24 to −0.28 depending on m. The excess stellar mass in the cusp scales approximately linearly with the black-hole mass, and is typically 0.5-0.85 times the black-hole mass. This process may strongly influence the structure of nuclear star clusters if they contain black holes. Fig. 1.-Surface-brightness profile of the dwarf galaxy NGC 4459, an excess-light elliptical. The curve shows the bestfitting Sérsic profile, with index m = 3.17 (Kormendy et al. 2009). Reproduced by permission of the AAS.serve their adiabatic invariants or actions. If the galaxy began with a Sérsic surface-brightness profile, this process will naturally produce excess light relative to the Sérsic profile near the center of the galaxy, and the corresponding excess stellar mass will be simply related to the mass of the BH and the properties of the initial Sérsic profile. The main goal of the paper is to work out these relations and to compare the predictions of this simple model to the observations.Pioneering studies of this process were carried out by Peebles (1972) andYoung (1980). These studies assumed that the galaxies were spherical, as do we, but in contrast they assumed that the stellar distribution function (DF) near the center of the galaxy was Maxwellian before the formation of the BH. They found that as the BH grew the stellar DF developed a cusp in which the density varied as ρ ∼ r −3/2 , corresponding to a surface brightness I(R) ∼ R −1/2 . However, a Sérsic profile does not have a Maxwellian DF, and the properties of the cusp formed by

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