Evolution of Star Clusters near the Galactic Center: Fully Self-consistent N-body Simulations

We have performed fully self-consistent $N$-body simulations of star clusters near the Galactic center (GC). Such simulations have not been performed because it is difficult to perform fast and accurate simulations of such systems using conventional methods. We used the Bridge code, which integrates the parent galaxy using the tree algorithm and the star cluster using the fourth-order Hermite scheme with individual timestep. The interaction between the parent galaxy and the star cluster is calculate with the tree algorithm. Therefore, the Bridge code can handle both the orbital and internal evolutions of star clusters correctly at the same time. We investigated the evolution of star clusters using the Bridge code and compared the results with previous studies. We found that 1) the inspiral timescale of the star clusters is shorter than that obtained with "traditional" simulations, in which the orbital evolution of star clusters is calculated analytically using the dynamical friction formula and 2) the core collapse of the star cluster increases the core density and help the cluster survive. The initial conditions of star clusters is not so severe as previously suggested.Comment: 19 pages, 19 figures, accepted for publication in Ap

On the Origin of Density Cusps in Elliptical Galaxies

We investigated the dynamical reaction of the central region of galaxies to a falling massive black hole by N-body simulations. As the initial galaxy model, we used an isothermal King model and placed a massive black hole at around the half-mass radius of the galaxy. We found that the central core of the galaxy is destroyed by the heating due to the black hole and that a very weak density cusp ($\rho \propto r^{-\alpha}$, with $\alpha \sim 0.5$) is formed around the black hole. This result is consistent with recent observations of large elliptical galaxies with Hubble Space Telescope. The velocity of the stars becomes tangentially anisotropic in the inner region, while in the outer region the stars have radially anisotropic velocity dispersion. The radius of the weak cusp region is larger for larger black hole mass. Our result naturally explains the formation of the weak cusp found in the previous simulations of galaxy merging, and implies that the weak cusp observed in large elliptical galaxies may be formed by the heating process by sinking black holes during merging events.Comment: 14 pages with 29 EPS figures; LaTeX; new results added; accepted for publication in Ap

Evolution of Massive Black Hole Binaries

We present the result of large-scale N-body simulations of the stellar-dynamical evolution of a massive black-hole binary at the center of a spherical galaxy. We focus on the dependence of the hardening rate on the relaxation timescale of the parent galaxy. A simple theoretical argument predicts that a binary black hole creates the ``loss cone'' around it. Once the loss cone is formed, the hardening rate is determined by the rate at which field stars diffuse into the loss cone. Therefore the hardening timescale becomes proportional to the relaxation timescale. Recent N-body simulations, however, have failed to confirm this theory and various explanations have been proposed. By performing simulations with sufficiently large N (up to $10^6$) for sufficiently long time, we found that the hardening rate does depend on N. Our result is consistent with the simple theoretical prediction that the hardening timescale is proportional to the relaxation timescale. This dependence implies that most massive black hole binaries are unlikely to merge within the Hubble time through interaction with field stars and gravitational wave radiation alone.Comment: Reviced version accepted for publication in ApJ. Scheduled to appear in the February 10, 2004 issu

Formation of Protoplanets from Massive Planetesimals in Binary Systems

More than half of stars reside in binary or multiple star systems and many planets have been found in binary systems. From theoretical point of view, however, whether or not the planetary formation proceeds in a binary system is a very complex problem, because secular perturbation from the companion star can easily stir up the eccentricity of the planetesimals and cause high-velocity, destructive collisions between planetesimals. Early stage of planetary formation process in binary systems has been studied by restricted three-body approach with gas drag and it is commonly accepted that accretion of planetesimals can proceed due to orbital phasing by gas drag. However, the gas drag becomes less effective as the planetesimals become massive. Therefore it is still uncertain whether the collision velocity remains small and planetary accretion can proceed, once the planetesimals become massive. We performed {\it N}-body simulations of planetary formation in binary systems starting from massive planetesimals whose size is about 100-500 km. We found that the eccentricity vectors of planetesimals quickly converge to the forced eccentricity due to the coupling of the perturbation of the companion and the mutual interaction of planetesimals if the initial disk model is sufficiently wide in radial distribution. This convergence decreases the collision velocity and as a result accretion can proceed much in the same way as in isolated systems. The basic processes of the planetary formation, such as runaway growth and oligarchic growth and final configuration of the protoplanets are essentially the same in binary systems and single star systems, at least in the late stage where the effect of gas drag is small.Comment: 26pages, 11 figures. ApJ accepte

Massive Black Holes in Star Clusters. I. Equal-mass clusters

In this paper we report results of collisional N-body simulations of the dynamical evolution of equal-mass star clusters containing a massive central black hole. Each cluster is composed of between 5,000 to 180,000 stars together with a central black hole which contains between 0.2% to 10% of the total cluster mass. We find that for large enough black hole masses, the central density follows a power-law distribution with slope \rho \sim r^{-1.75} inside the radius of influence of the black hole, in agreement with predictions from earlier Fokker Planck and Monte Carlo models. The tidal disruption rate of stars is within a factor of two of that derived in previous studies. It seems impossible to grow an intermediate-mass black hole (IMBH) from a M \le 100 Msun progenitor in a globular cluster by the tidal disruption of stars, although M = 10^3 Msun IMBHs can double their mass within a Hubble time in dense globular clusters. The same is true for the supermassive black hole at the centre of the Milky Way. Black holes in star clusters will feed mainly on stars tightly bound to them and the re-population of these stars causes the clusters to expand, reversing core-collapse without the need for dynamically active binaries. Close encounters of stars in the central cusp also lead to an increased mass loss rate in the form of high-velocity stars escaping from the cluster. A companion paper will extend these results to the multi-mass case.Comment: 15 pages, 8 figures, ApJ in pres

Pseudoparticle Multipole Method: A Simple Method to Implement High-Accuracy Treecode

In this letter we describe the pseudoparticle multipole method (P2M2), a new method to express multipole expansion by a distribution of pseudoparticles. We can use this distribution of particles to calculate high order terms in both the Barnes-Hut treecode and FMM. The primary advantage of P2M2 is that it works on GRAPE. GRAPE is a special-purpose hardware for the calculation of gravitational force between particles. Although the treecode has been implemented on GRAPE, we could handle terms only up to dipole, since GRAPE can calculate forces from point-mass particles only. Thus the calculation cost grows quickly when high accuracy is required. With P2M2, the multipole expansion is expressed by particles, and thus GRAPE can calculate high order terms. Using P2M2, we implemented an arbitrary-order treecode on GRAPE-4. Timing result shows GRAPE-4 accelerates the calculation by a factor between 10 (for low accuracy) to 150 (for high accuracy). Even on general-purpose programmable computers, our method offers the advantage that the mathematical formulae and therefore the actual program is much simpler than that of the direct implementation of multipole expansion.Comment: 6 pages, 4 figures, latex, submitted to ApJ Letter

Massive Black Holes in Star Clusters. II. Realistic Cluster Models

We have followed the evolution of multi-mass star clusters containing massive central black holes through collisional N-body simulations done on GRAPE6. Each cluster is composed of between 16,384 to 131,072 stars together with a black hole with an initial mass of M_BH=1000 Msun. We follow the evolution of the clusters under the combined influence of two-body relaxation, stellar mass-loss and tidal disruption of stars. The (3D) mass density profile follows a power-law distribution \rho \sim r^{-\alpha} with slope \alpha=1.55. This leads to a constant density profile of bright stars in projection, which makes it highly unlikely that core collapse clusters contain intermediate-mass black holes (IMBHs). Instead globular clusters containing IMBHs can be fitted with standard King profiles. The disruption rate of stars is too small to form an IMBH out of a M_BH \approx 50 Msun progenitor black hole, unless a cluster starts with a central density significantly higher than what is seen in globular clusters. Kinematical studies can reveal 1000 Msun IMBHs in the closest clusters. IMBHs in globular clusters are only weak X-ray sources since the tidal disruption rate of stars is low and the star closest to the IMBH is normally another black hole. For globular clusters, dynamical evolution can push compact stars near the IMBH to distances small enough that they become detectable through gravitational radiation. If 10% of all globular clusters contain IMBHs, extragalactic globular clusters could be one of the major sources for {\it LISA}. (abridged)Comment: 20 pages, 16 figures, ApJ in pres

Accelerating NBODY6 with Graphics Processing Units

We describe the use of Graphics Processing Units (GPUs) for speeding up the code NBODY6 which is widely used for direct $N$-body simulations. Over the years, the $N^2$ nature of the direct force calculation has proved a barrier for extending the particle number. Following an early introduction of force polynomials and individual time-steps, the calculation cost was first reduced by the introduction of a neighbour scheme. After a decade of GRAPE computers which speeded up the force calculation further, we are now in the era of GPUs where relatively small hardware systems are highly cost-effective. A significant gain in efficiency is achieved by employing the GPU to obtain the so-called regular force which typically involves some 99 percent of the particles, while the remaining local forces are evaluated on the host. However, the latter operation is performed up to 20 times more frequently and may still account for a significant cost. This effort is reduced by parallel SSE/AVX procedures where each interaction term is calculated using mainly single precision. We also discuss further strategies connected with coordinate and velocity prediction required by the integration scheme. This leaves hard binaries and multiple close encounters which are treated by several regularization methods. The present nbody6-GPU code is well balanced for simulations in the particle range $10^4-2 \times 10^5$ for a dual GPU system attached to a standard PC.Comment: 8 pages, 3 figures, 2 tables, MNRAS accepte

Evolution of Compact Groups of Galaxies I. Merging Rates

We discuss the merging rates in compact groups of 5 identical elliptical galaxies. All groups have the same mass and binding energy. We consider both cases with individual halos and cases where the halo is common to all galaxies and enveloping the whole group. In the latter situation the merging rate is slower if the halo is more massive. The mass of individual halos has little influence on the merging rates, due to the fact that all galaxies in our simulations have the same mass, and so the more extended ones have a smaller velocity dispersion. Groups with individual halos merge faster than groups with common halos if the configuration is centrally concentrated, like a King distribution of index 10. On the other hand for less concentrated configurations the merging is initially faster for individual halo cases, and slower after part of the group has merged. In cases with common halo, centrally concentrated configurations merge faster for high halo-to-total mass ratios and slower for low halo-to-total mass ratios. Groups whose virial ratio is initially less than one merge faster, while groups that have initially cylindrical rotation merge slower than groups starting in virial equilibrium. In order to test how long a virialised group can survive before merging we followed the evolution of a group with a high halo-to-total mass ratio and a density distribution with very little central concentration. We find that the first merging occurred only after a large number of crossing times, which with areasonable calibration should be larger than a Hubble time. Hence, at least for appropriate initial conditions, the longevity of compact groups is not necessarily a problem, which is an alternative explanation to why we observe so many compact groups despite the fact that their lifetimes seem short.Comment: 15 pages Latex, with 12 figures included, requires mn.sty, accepted for publication in MNRA