Rainer Spurzem

EFFICIENT MERGER OF BINARY SUPERMASSIVE BLACK HOLES IN NON- AXISYMMETRIC GALAXIES

Binary supermassive black holes form naturally in galaxy mergers, but their long-term evolution is uncertain. In spherical galaxies, N-body simulations show that binary evolution stalls at separations much too large for significant emission of gravitational waves (the “final pars ec problem”). Here, we follow the long-term evolution of a massive binary in more realistic, triaxial and rota ting galaxy models. We find that the binary does not stall. The binary hardening rates that we observe are suffici ent to allow complete coalescence of binary SBHs in 10 Gyr or less, even in the absence of collisional loss-con e refilling or gas-dynamical torques, thus providing a potential solution to the final parsec problem. Subject headings:

SUPERBOX – an efficient code for collisionless galactic dynamics

We present SUPERBOX, a particle-mesh code with high resolution sub-grids and an NGP (nearest grid point) force-calculation scheme based on the second derivatives of the potential. SUPERBOX implements a fast low-storage FFT-algorithm, giving the possibility to work with millions of particles on desk-top computers. Test calculations show energy and angular momentum conservation to one part in 10(5) per crossing-time. The effects of grid and numerical relaxation remain negligible, even when these calculations cover a Hubble-time of evolution. As the sub-grids follow the trajectories of individual galaxies, the code allows a highly resolved treatment of interactions in clusters of galaxies, such as high-velocity encounters between elliptical galaxies and the tidal disruption of dwarf galaxies. Excellent agreement is obtained in a comparison with a direct-summation N-body code running on special-purpose GRAPE 3 hardware. The orbital decay of satellite galaxies due to dynamical friction obtained with SUPERBOX agrees with Chandrasekhars treatment when the Coulomb logarithm ln Lambda approximate to 1.5

Direct N-body simulations

Abstract Special high-accuracy direct force summation N-body algorithms and their relevance for the simulation of the dynamical evolution of star clusters and other gravitating N-body systems in astrophysics are presented, explained and compared with other methods. Other methods means here approximate physical models based on the Fokker–Planck equation as well as other, approximate algorithms to compute the gravitational potential in N-body systems. Questions regarding the parallel implementation of direct “brute force” N-body codes are discussed. The astrophysical application of the models to the theory of relaxing rotating and non-rotating collisional star clusters is presented, briefly mentioning the questions of the validity of the Fokker–Planck approximation, the existence of gravothermal oscillations and of rotation and primordial binaries.

FAST COALESCENCE OF MASSIVE BLACK HOLE BINARIES FROM MERGERS OF GALACTIC NUCLEI: IMPLICATIONS FOR LOW-FREQUENCY GRAVITATIONAL-WAVE ASTROPHYSICS

We investigate a purely stellar dynamical solution to the Final Parsec Problem. Galactic nuclei resulting from major mergers are not spherical, but show some degree of triaxiality. With N-body simulations, we show that equal-mass massive black hole binaries (MBHBs) hosted by them will continuously interact with stars on centrophilic orbits and will thus inspiral—in much less than a Hubble time—down to separations at which gravitational-wave (GW) emission is strong enough to drive them to coalescence. Such coalescences will be important sources of GWs for future space-borne detectors such as the Laser Interferometer Space Antenna (LISA). Based on our results for equal-mass mergers, and given that the hardening rate of unequal-mass binaries is similar, we expect that LISA will see between ∼10 and ∼ few × 10 2 such events every year, depending on the particular massive black hole (MBH) seed model as obtained in recent studies of merger trees of galaxy and MBH co-evolution. Orbital eccentricities in the LISA band will be clearly distinguishable from zero with e 0.001–0.01.

Long-Term Evolution of Massive Black Hole Binaries. II. Binary Evolution in Low-Density Galaxies

We use direct-summation N-body integrations to follow the evolution of binary black holes at the centers of galaxy models with large, constant-density cores. Particle numbers as large as 0.4 × 106 are considered. The results are compared with the predictions of loss-cone theory under the assumption that the supply of stars to the binary is limited by the rate at which they can be scattered into the binarys influence sphere by gravitational encounters. The agreement between theory and simulation is quite good; in particular, we are able to quantitatively explain the observed dependence of binary hardening rate on N. We do not verify a recent claim that the hardening rate of the binary stabilizes when N exceeds a particular value or that Brownian wandering of the binary has a significant effect on its evolution. When scaled to real galaxies, our results suggest that massive black hole binaries in gas-poor nuclei would be unlikely to reach gravitational wave coalescence in a Hubble time.

A comprehensive Nbody study of mass segregation in star clusters: Energy equipartition and escape

We address the dynamical evolution of an isolated self-gravitating system with two stellar mass groups. We vary the individual ratio of the heavy to light bodies, μ from 1.25 to 50 and alter also the fraction of the total heavy mass M h from 5 to 40 per cent of the whole cluster mass. Clean-cut properties of the cluster dynamics are examined, like core collapse, the evolution of the central potential, as well as escapers. We present in this work collisional N-body simulations, using the high-order integrator NBODY6++ with up to Ν * = 2 x 10 4 particles improving the statistical significancy of the lower-N * simulations by ensemble averages. Equipartition slows down the gravothermal contraction of the core slightly. Beyond a critical value of μ ≈ 2, no equipartition can be achieved between the different masses; the heavy component decouples and collapses. For the first time, the critical boundary between Spitzer-stable and Spitzer-unstable systems is demonstrated in direct N-body models. We also present the measurements of the Coulomb logarithm and discuss the relative importance of the evaporation and ejection of escapers.

BINARY BLACK HOLE MERGER IN GALACTIC NUCLEI: POST-NEWTONIAN SIMULATIONS

This paper studies the formation and evolution of binary supermassive black holes (SMBHs) in rotating galactic nuclei, focusing on the role of stellar dynamics. We present the first N-body simulations that follow the evolution of the SMBHs from kiloparsec separations all the way to their final relativistic coalescence, and that can robustly be scaled to real galaxies. The N-body code includes post-Newtonian corrections to the binary equations of motion up to order 2.5; we show that the evolution of the massive binary is only correctly reproduced if the conservative and terms are included. The orbital eccentricities of the massive binaries in our simulations are often found to remain large until shortly before coalescence. This directly affects not only their orbital evolution rates, but has important consequences as well for the gravitational waveforms emitted during the relativistic inspiral. We estimate gravitational wave amplitudes when the frequencies fall inside the band of the (planned) Laser Interferometer Space Antennae (LISA). We find significant contributions?well above the LISA sensitivity curve?from the higher-order harmonics.

Compact binaries in star clusters – I. Black hole binaries inside globular clusters

We study the compact binary population in star clusters, focusing on binaries containing black holes, using a self-consistent Monte Carlo treatment of dynamics and full stellar evolution. We find that the black holes experience strong mass segregation and become centrally concentrated. In the core the black holes interact strongly with each other and black hole–black hole binaries are formed very efficiently. The strong interactions, however, also destroy or eject the black hole–black hole binaries. We find no black hole–black hole mergers within our simulations but produce many hard escapers that will merge in the Galactic field within a Hubble time. We also find several highly eccentric black hole–black hole binaries that are potential Laser Interferometer Space Antenna (LISA) sources, suggesting that star clusters are interesting targets for space-based detectors. We conclude that star clusters must be taken into account when predicting compact binary population statistics.

The dragon simulations: globular cluster evolution with a million stars

Introducing the dragon simulation project, we present direct N-body simulations of four massive globular clusters (GCs) with 106 stars and 5 per cent primordial binaries at a high level of accuracy and realism. The GC evolution is computed with nbody6++gpu and follows the dynamical and stellar evolution of individual stars and binaries, kicks of neutron stars and black holes (BHs), and the effect of a tidal field. We investigate the evolution of the luminous (stellar) and dark (faint stars and stellar remnants) GC components and create mock observations of the simulations (i.e. photometry, colour–magnitude diagrams, surface brightness and velocity dispersion profiles). By connecting internal processes to observable features, we highlight the formation of a long-lived ‘dark’ nuclear subsystem made of BHs, which results in a two-component structure. The inner core is dominated by the BH subsystem and experiences a core-collapse phase within the first Gyr. It can be detected in the stellar (luminous) line-of-sight velocity dispersion profiles. The outer extended core – commonly observed in the (luminous) surface brightness profiles – shows no collapse features and is continuously expanding. We demonstrate how a King model fit to observed clusters might help identify the presence of post core-collapse BH subsystems. For global observables like core and half-mass radii, the direct simulations agree well with Monte Carlo models. Variations in the initial mass function can result in significantly different GC properties (e.g. density distributions) driven by varying amounts of early mass-loss and the number of forming BHs. (Less)

Dynamical evolution of rotating stellar systems – II. Post-collapse, equal-mass system

ABSTRA C T We present the first post-core-collapse models of initially rotating star clusters, using the numerical solution of an orbit-averaged 2D Fokker ‐ Planck equation. Based on the code developed by Einsel & Spurzem, we have improved the speed and the stability and included the steady three-body binary heating source. We have confirmed that rotating clusters, whether they are in a tidal field or not, evolve significantly faster than non-rotating ones. Consequences for the observed shapes, density distribution and kinematic properties of young and old star clusters are discussed. The results are compared with gaseous and 1D Fokker‐ Planck models in the non-rotating case.