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 × 10 6 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 σ ∼ < 80 km s −1 , i.e. with binary black hole mass ∼ < 2 × 10 6 M ⊙ . 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. Subject headings:

A recently introduced chaos detection method, the relative Lyapunov indicator (RLI) is investigated in the cases of symplectic mappings and of a continuous Hamiltonian system. It is shown that the RLI is an efficient and easy-to-calculate numerical tool in determining the true nature of individual orbits, and in separating ordered and regular regions of the phase space of dynamical systems. An application of the RLI for stability investigations of some recently discovered exoplanetary systems is also presented.

After being destroyed by a binary supermassive black hole, a stellar density cusp can regrow at the center of a galaxy via energy exchange between stars moving in the gravitational field of the single, coalesced hole. We illustrate this process via high-accuracy N-body simulations. Regeneration requires roughly one relaxation time and the new cusp extends to a distance of roughly one-fifth the black hole's influence radius, with density rho ~ r^{-7/4}; the mass in the cusp is of order 10% the mass of the black hole. Growth of the cusp is preceded by a stage in which the stellar velocity dispersion evolves toward isotropy and away from the tangentially-anisotropic state induced by the binary. We show that density profiles similar to those observed at the center of the Milky Way and M32 can regenerate themselves in several Gyr following infall of a second black hole; the presence of density cusps at the centers of these galaxies can therefore not be used to infer that no merger has occurred. We argue that Bahcall-Wolf cusps are ubiquitous in stellar spheroids fainter than M_V ~ -18.5 that contain supermassive black holes, but the cusps have not been detected outside of the Local Group since their angular sizes are less than 0.1". We show that the presence of a cusp implies a lower limit of \~10^{-4} per year on the rate of stellar tidal disruptions, and discuss the consequences of the cusps for gravitational lensing and the distribution of dark matter on sub-parsec scales.Comment: Accepted for publication in The Astrophysical Journa

A special symmetric configuration of the general four-body problem called the Caledonian Symmetric Four-Body Problem is investigated. Recently, Steves & Roy have discovered a global stability parameter for the system, called the Szebehely constant. The connection between the chaotic behaviour of the phase space and the global stability given by the Szebehely constant is analysed by using the relative Lyapunov indicator (RLI) and smallest alignment indices (SALI) methods of determining regular and chaotic motion. It is found that as the Szebehely constant is increased, making the system hierarchically stable from a global point of view, the corresponding phase space becomes increasingly more regular and less chaotic.