In galactic nuclei with sufficiently short relaxation times, binary supermassive black holes can evolve beyond their stalling radii via continued interaction with stars. We study this “collisional” evolutionary regime using both fully self-consistent N-body integrations and approximate Fokker-Planck models. The N-body integrations employ particle numbers up to 0.26×106 and a direct-summation potential solver; close interactions involving the binary are treated using a new implementation of the Mikkola-Aarseth chain regularization algorithm. Even at these large values of N, two-body scattering occurs at high enough rates in the N-body simulations that the binary is never fully in the diffusively-repopulated (i.e. large-N) loss cone regime, which precludes a simple scaling of the results to real galaxies. The Fokker-Planck model is used to bridge this gap; it includes, for the first time in this context, binary-induced changes in the stellar density and potential. The Fokker-Planck model is shown to accurately reproduce the results of the N-body integrations, and is then extended to the much larger N regime of real galaxies. Analytic expressions are derived that accurately reproduce the time dependence of the binary semi-major axis as predicted by the Fokker-Planck model. Gravitational radiation begins to dominate the binary’s evolution after a time that is always comparable to, or less than, the relaxation time measured at the binary’s gravitational influence radius; the observed correlation of nuclear relaxation time with velocity dispersion implies that coalescence in ≤ 10 Gyr will occur in nuclei with s∼< 80 km s−1, i.e. with binary black hole mass ∼< 2×106M⊙. The coalescence time depends only weakly on binary mass ratio. Formation of a core, or “mass deficit,” is shown to result from a competition between ejection of stars by the binary and re-supply of depleted orbits via two-body scattering. Mass deficits as large as ∼ 4 times the binary mass are produced before the gravitational radiation regime is reached; however, after the two black holes coalesce, a Bahcall-Wolf cusp appears around the single hole in approximately one relaxation time, resulting in a nuclear density profile consisting of a flat core with an inner, compact cluster, similar to what is observed at the centers of low-luminosity elliptical galaxies. We critically evaluate recent claims that binary-star interactions can induce rapid coalescence of binary supermassive black holes even in the absence of loss cone refilling.

Publication Date



This is the pre-print of an article published by the American Astronomical Society. The final, published version is available here: https://doi.org/10.1086/522691

© 2007 The American Astronomical Society

Also archived in: arXiv:0705.2745 v1 May 18 2007

Note: imported from RIT’s Digital Media Library running on DSpace to RIT Scholar Works in February 2014.

Document Type


Department, Program, or Center

School of Physics and Astronomy (COS)


RIT – Main Campus