Mass deficits, stalling radii, and the merger histories of elliptical galaxies

A binary supermassive black hole leaves an imprint on a galactic nucleus in the form of a mass deficit, a decrease in the mass of the nucleus due to ejection of stars by the binary. The magnitude of the mass deficit is in principle related to the galaxy's merger history, but the relation has never been quantified. Here high-accuracy N-body simulations are used to calibrate this relation. Mass deficits are shown to be M-def approximate to 0.5M(12), with M-12 the total mass of the binary; the coefficient in this relation is found to depend only weakly on M-2/M-1 or on the galaxy's preexisting nuclear density profile. Hence, after N mergers, M-def approximate to 0.5 N M-center dot, with M-center dot the final ( current) black hole mass. When compared with observed mass deficits, this result implies 1 less than or similar to N less than or similar to 3, in accord with hierarchical structure formation models. Implications for binary stalling radii, the origin of hypervelocity stars, and the distribution of dark matter at the centers of galaxies are discussed.