Mapping the three-body system - decay time and reversibility

In this paper we carry out a quantitative analysis of the three-body systems and map them as a function of decaying time and intial conguration, look at this problem as an example of a simple deterministic system, and ask to what extent the orbits are really predictable. We have investigated the behavior of about 200 000 general Newtonian three body systems using the simplest initial conditions. Within our resolution these cover all the possible states where the objects are initially at rest and have no angular momentum. We have determined the decay time-scales of the triple systems and show that the distribution of this parameter is fractal in appearance. Some areas that appear stable on large scales exhibit very narrow strips of instability and the overall pattern, dominated by resonances, reminds us of a traditional Maasai warrior shield. Also an attempt is made to recover the original starting conguration of the three bodies by backward integration. We find there are instances where the evolution to the future and to the past lead to different orbits, in spite of time symmetric initial conditions. This implies that even in simple deterministic systems there exists an Arrow of Time.Comment: 8 pages, 9 figures. Accepted for publication in MNRAS. Includes low-resolution figures. High-resolution figures are available as PNG

Long-Term Evolution of Massive Black Hole Binaries. III. Binary Evolution in Collisional Nuclei

[Abridged] 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 0.26M 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 simulations that they can not be simply scaled to the large-N regime of real galaxies. The Fokker-Planck model is used to bridge this gap; it includes, for the first time, 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 wave coalescence is shown to occur in <10 Gyr in nuclei with velocity dispersions below about 80 km/s. Formation of a core results 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 coalescence. After the two black holes coalesce, a Bahcall-Wolf cusp appears around the single hole in 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 spheroids.Comment: 21 page

Collisional dynamics around binary black holes in galactic centers

We follow the sinking of two massive black holes in a spherical stellar system where the black holes become bound under the influence of dynamical friction. Once bound, the binary hardens by three-body encounters with surrounding stars. We find that the binary wanders inside the core, providing an enhanced supply of reaction partners for the hardening. The binary evolves into a highly eccentric orbit leading to coalescence well beyond a Hubble time. These are the first results from a hybrid ``self consistent field'' (SCF) and direct Aarseth N-body integrator (NBODY6), which combines the advantages of the direct force calculation with the efficiency of the field method. The code is designed for use on parallel architectures and is therefore applicable to collisional N-body integrations with extraordinarily large particle numbers (> 10^5). This creates the possibility of simulating the dynamics of both globular clusters with realistic collisional relaxation and stellar systems surrounding supermassive black holes in galactic nuclei.Comment: 38 pages, 13 figures, submitted to ApJ, accepted, revised text and added figure

Chaos in the one-dimensional gravitational three-body problem

We have investigated the appearance of chaos in the 1-dimensional Newtonian gravitational three-body system (three masses on a line with $-1/r$ pairwise potential). We have concentrated in particular on how the behavior changes when the relative masses of the three bodies change (with negative total energy). For two mass choices we have calculated 18000 full orbits (with initial states on a $100\times 180$ lattice on the Poincar\'e section) and obtained dwell time distributions. For 105 mass choices we have calculated Poincar\'e maps for $10\times 18$ starting points. Our results show that the Poincar\'e section (and hence the phase space) divides into three well defined regions with orbits of different characteristics: 1) There is a region of fast scattering, with a minimum of pairwise collisions and smooth dependence on initial values. 2) In the chaotic scattering region the interaction times are longer, and both the interaction time and the final state depend sensitively on the starting point on the Poincar\'e section. For both 1) and 2) the initial and final states consists of a binary + single particle. 3) The third region consists of quasiperiodic orbits where the three masses are bound together forever. At the center of the quasiperiodic region there is the periodic Schubart orbit, whose stability turns out to correlate strongly with the global behavior.Comment: 24 pages of text (REVTEX 3.0) + 21 pages of figures. Figures are only available in paper form, ask for a preprint from the author

Interaction of massive black hole binaries with their stellar environment: II. Loss-cone depletion and binary orbital decay

We study the long-term evolution of massive black hole binaries (MBHBs) at the centers of galaxies using detailed scattering experiments to solve the full three-body problem. Ambient stars drawn from a isotropic Maxwellian distribution unbound to the binary are ejected by the gravitational slingshot. We construct a minimal, hybrid model for the depletion of the loss cone and the orbital decay of the binary, and show that secondary slingshots - stars returning on small impact parameter orbits to have a second super-elastic scattering with the MBHB - may considerably help the shrinking of the pair in the case of large binary mass ratios. In the absence of loss-cone refilling by two-body relaxation or other processes, the mass ejected before the stalling of a MBHB is half the binary reduced mass. About 50% of the ejected stars are expelled ejected in a "burst" lasting ~1E4 yrs M_6^1/4, where M_6 is the binary mass in units of 1E6 Msun. The loss cone is completely emptied in a few bulge crossing timescales, 1E7 yrs M_6^1/4. Even in the absence of two-body relaxation or gas dynamical processes, unequal mass and/or eccentric binaries with M_6 >0.1 can shrink to the gravitational wave emission regime in less than a Hubble time, and are therefore "safe" targets for the planned Laser Interferometer Space Antenna (LISA).Comment: Minor revision. 10 pages, 7 figures, ApJ in pres

Rotational Brownian Motion of a Massive Binary

The orientation of a massive binary undergoes a random walk due to gravitational encounters with field stars. The rotational diffusion coefficient for a circular-orbit binary is derived via scattering experiments. The binary is shown to reorient itself by an angle of order (m/M)^1/2 during the time that its semi-major axis shrinks appreciably, where M is the binary mass and m the perturber mass. Implications for the orientations of rotating black holes are discussed.Comment: 16 pages, 3 postscript figures. Accepted for publication in The Astrophysical Journal, vol. 568, 200

Monte Carlo Simulations of Globular Cluster Evolution. III. Primordial Binary Interactions

We study the dynamical evolution of globular clusters using our 2D Monte Carlo code with the inclusion of primordial binary interactions for equal-mass stars. We use approximate analytical cross sections for energy generation from binary-binary and binary-single interactions. After a brief period of slight contraction or expansion of the core over the first few relaxation times, all clusters enter a much longer phase of stable "binary burning" lasting many tens of relaxation times. The structural parameters of our models during this phase match well those of most observed globular clusters. At the end of this phase, clusters that have survived tidal disruption undergo deep core collapse, followed by gravothermal oscillations. Our results clearly show that the presence of even a small fraction of binaries in a cluster is sufficient to support the core against collapse significantly beyond the normal core collapse time predicted without the presence of binaries. For tidally truncated systems, collapse is easily delayed sufficiently that the cluster will undergo complete tidal disruption before core collapse. As a first step toward the eventual goal of computing all interactions exactly using dynamical three- and four-body integration, we have incorporated an exact treatment of binary-single interactions in our code. We show that results using analytical cross sections are in good agreement with those using exact three-body integration, even for small binary fractions where binary-single interactions are energetically most important.Comment: Accepted for publication in ApJ. Minor changes to reflect accepted version. 28 pages, 17 figures; some figures low resolution. Full resolution paper available at http://www.mit.edu/~fregeau/paper3.pd

Four-Body Effects in Globular Cluster Black Hole Coalescence

In the high density cores of globular clusters, multibody interactions are expected to be common, with the result that black holes in binaries are hardened by interactions. It was shown by Sigurdsson & Hernquist (1993) and others that 10 solar mass black holes interacting exclusively by three-body encounters do not merge in the clusters themselves, because recoil kicks the binaries out of the clusters before the binaries are tight enough to merge. Here we consider a new mechanism, involving four-body encounters. Numerical simulations by a number of authors suggest that roughly 20-50% of binary-binary encounters will eject one star but leave behind a stable hierarchical triple. If the orbital plane of the inner binary is strongly tilted with respect to the orbital plane of the outer object, a secular Kozai resonance, first investigated in the context of asteroids in the Solar System, can increase the eccentricity of the inner body significantly. We show that in a substantial fraction of cases the eccentricity is driven to a high enough value that the inner binary will merge by gravitational radiation, without a strong accompanying kick. Thus the merged object remains in the cluster; depending on the binary fraction of black holes and the inclination distribution of newly-formed hierarchical triples, this mechanism may allow massive black holes to accumulate through successive mergers in the cores of globular clusters. It may also increase the likelihood that stellar-mass black holes in globular clusters will be detectable by their gravitational radiation.Comment: Submitted to ApJ Letters (includes emulateapj.sty

Parallelization, Special Hardware and Post-Newtonian Dynamics in Direct N - Body Simulations

The formation and evolution of supermassive black hole (SMBH) binaries during and after galaxy mergers is an important ingredient for our understanding of galaxy formation and evolution in a cosmological context, e.g. for predictions of cosmic star formation histories or of SMBH demographics (to predict events that emit gravitational waves). If galaxies merge in the course of their evolution, there should be either many binary or even multiple black holes, or we have to find out what happens to black hole multiples in galactic nuclei, e.g. whether they come sufficiently close to merge resulting from emission of gravitational waves, or whether they eject each other in gravitational slingshot interactions

Baltic Ecological Recycling Agriculture and Society (BERAS project) - a case of Juva milk system

The aim of the study was to determine the potential, impact and prerequisites of localization and enhanced recycling in a rural food system, illustrated by the case of Juva milk. An interdisciplinary scenario based on the increase of local, organic milk to 50 % of milk comsumption was created and the sustainability was compared, on the basis of the statistics and data collected from the actors, with the present milk system