The dynamical evolution of massive black hole binaries I. Hardening in a fixed stellar background

Abstract

We use a hybrid N-body program to study the evolution of massive black hole binaries in the centers of galaxies, mainly to understand the factors affecting the binary eccentricity, the response of the galaxy to the binary merger, and the effect of loss-cone depletion on the merger time. The scattering experiments from paper I showed that the merger time is not sensitive to the eccentricity growth unless a binary forms with at least a moderate eccentricity. We find here that the eccentricity can become large under some conditions if a binary forms in a galaxy with a flat core or with a radial bias in its velocity distribution, especially if the dynamical friction is enhanced by resonances as suggested by Rauch and Tremaine (1996). But the necessary conditions all seem unlikely, and our prediction from paper I remains unchanged: in most cases the eccentricity will start and remain small. As a binary hardens it ejects stars from the center of a galaxy, which may explain why large elliptical galaxies have weaker density cusps than smaller galaxies. If so, the central velocity distributions in those galaxies should have strong tangential anisotropies. The wandering of a binary from the center of a galaxy mitigates the problems associated with loss-cone depletion and helps the binary merge.

Binary black holes and tori in AGN

“…Therefore a primary BH with mass M 1 ejects more mass if it merges with N black holes of mass M 1 /N than merging with another BH of mass M 1 . This is the same result which has been obtained by Quinlan (1996) in scattering experiments.…”

“…Quinlan (1996) obtains for the timescales needed for merging once the binary has become hard (ã ≈r cusp ), about 4 × 10 7 yr. This is very similar to our result, which is supposed to be a lower limit since in the simulation the shrinking of the distance between the black holes has not been taken into account.…”

“…The kinetic energy is lost to these stars, resulting in a heating of the core, but the stars do not gain enough energy as to leave the system. Eventually, after some 10 7 yr, the distance between both the BHs shrinks to the cusp-radius of M 1 , r cusp = r core M 1 /M core , where the velocity dispersion of the stars of the core equals the keplerian velocity in the potential of M 1 (Begelman et al 1980;Mikkola & Valtonen 1992;Vecchio et al 1994;Quinlan 1996). Inside r cusp the kinematics of the stars and M 2 is dominated by the potential of M 1 , and the stars are now interacting with both BHs rather than with M 2 only.…”

Evolution of massive black hole binaries in rotating stellar nuclei: Implications for gravitational wave detection

“…Eccentricity evolution. In the regime where binary hardening is driven by interaction with stars, as opposed to GW emission, eccentricity evolution has been shown to be modest, at least in spherical nonrotating nuclei [25]. On this basis, most discussion of the stochastic GW spectrum have assumed zero eccentricities.…”

Astrophysical N-Body Simulations: Algorithms and Challenges

“…Frank & Rees (1976) examined the interplay between mass and energy transport by two-body relaxation and loss-cone accretion of stars on orbits with low angular momentum by the black hole; their results were confirmed by Monte-Carlo numerical models of Marchant & Shapiro (1980), later followed by multi-mass direct numerical solutions of the 1D Fokker-Planck equation for isotropic stellar cusps of Murphy, Cohn & Durisen (1991). Only recently the first self-consistent N -body models of massive black holes including a sufficient number of stars in their surrounding cusps were done by use of hybrid N -body algorithms (Quinlan 1996, Quinlan & Hernquist 1997 or a high-speed special purpose computer for a direct summation algorithm (Makino & Ebisuzaki 1996, Makino 1997. However, the latter work was occupied mainly with the question of dynamical friction of black hole binaries in a galactic nucleus after a merger event.…”