Instability of the Gravitational N-Body Problem in the Large-N Limit

We use a systolic N-body algorithm to evaluate the linear stability of the gravitational N-body problem for N up to 1.3 x 10^5, two orders of magnitude greater than in previous experiments. For the first time, a clear ~ln N-dependence of the perturbation growth rate is seen. The e-folding time for N = 10^5 is roughly 1/20 of a crossing time.Comment: Accepted for publication in The Astrophysical Journa

Evolution of Binary Supermassive Black Holes via Chain Regularization

A chain regularization method is combined with special purpose computer hardware to study the evolution of massive black hole binaries at the centers of galaxies. Preliminary results with up to N=260,000 particles are presented. The decay rate of the binary is shown to decrease with increasing N, as expected on the basis of theoretical arguments. The eccentricity of the binary remains small.Comment: 8 pages. To appear in "Nonlinear Dynamics in Astronomy and Physics, A Workshop Dedicated to the Memory of Professor Henry E. Kandrup", ed. J. R. Buchler, S. T. Gottesman and M. E. Maho

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

Core Formation by a Population of Massive Remnants

Core radii of globular clusters in the Large and Small Magellanic Clouds show an increasing trend with age. We propose that this trend is a dynamical effect resulting from the accumulation of massive stars and stellar-mass black holes at the cluster centers. The black holes are remnants of stars with initial masses exceeding 20-25 solar masses; as their orbits decay by dynamical friction, they heat the stellar background and create a core. Using analytical estimates and N-body experiments, we show that the sizes of the cores so produced and their growth rates are consistent with what is observed. We propose that this mechanism is responsible for the formation of cores in all globular clusters and possibly in other systems as well.Comment: 5 page

The correlation structure of dark matter halo properties

We investigate the correlation between nine different dark matter halo properties using a rank correlation analysis and a Principal Component Analysis for a sample of haloes spanning five orders of magnitude in mass. We consider mass and dimensionless measures of concentration, age, relaxedness, sphericity, triaxiality, substructure, spin, and environment, where the latter is defined in a way that makes it insensitive to mass. We find that concentration is the most fundamental property. Except for environment, all parameters are strongly correlated with concentration. Concentration, age, substructure, mass, sphericity and relaxedness can be considered a single family of parameters, albeit with substantial scatter. In contrast, spin, environment, and triaxiality are more independent, although spin does correlate strongly with substructure and both spin and triaxiality correlate substantially with concentration. Although mass sets the scale of a halo, all other properties are more sensitive to concentration.Comment: 5 pages, 2 figures. Accepted by MNRAS Letters. Minor change

Shadowing unstable orbits of the Sitnikov elliptic 3-body problem

Errors in numerical simulations of gravitating systems can be magnified exponentially over short periods of time. Numerical shadowing provides a way of demonstrating that the dynamics represented by numerical simulations are representative of true dynamics. Using the Sitnikov Problem as an example, it is demonstrated that unstable orbits of the 3-body problem can be shadowed for long periods of time. In addition, it is shown that the stretching of phase space near escape and capture regions is a cause for the failure of the shadowing refinement procedure.Comment: 9 pages, 13 figures, accepted in MNRA

Systolic and Hyper-Systolic Algorithms for the Gravitational N-Body Problem, with an Application to Brownian Motion

A systolic algorithm rhythmically computes and passes data through a network of processors. We investigate the performance of systolic algorithms for implementing the gravitational N-body problem on distributed-memory computers. Systolic algorithms minimize memory requirements by distributing the particles between processors. We show that the performance of systolic routines can be greatly enhanced by the use of non-blocking communication, which allows particle coordinates to be communicated at the same time that force calculations are being carried out. Hyper-systolic algorithms reduce the communication complexity at the expense of increased memory demands. As an example of an application requiring large N, we use the systolic algorithm to carry out direct-summation simulations using 10^6 particles of the Brownian motion of the supermassive black hole at the center of the Milky Way galaxy. We predict a 3D random velocity of 0.4 km/s for the black hole.Comment: 33 pages, 10 postscript figure

Henry Kandrup's Ideas About Relaxation of Stellar Systems

Henry Kandrup wrote prolifically on the problem of relaxation of stellar systems. His picture of relaxation was significantly more refined than the standard description in terms of phase mixing and violent relaxation. In this article, I summarize Henry's work in this and related areas.Comment: 11 pages. To appear in "Nonlinear Dynamics in Astronomy and Physics, A Workshop Dedicated to the Memory of Professor Henry E. Kandrup", ed. J. R. Buchler, S. T. Gottesman and M. E. Maho

Self consistent model for the evolution of eccentric massive black hole binaries in stellar environments: implications for gravitational wave observations

We construct evolutionary tracks for massive black hole binaries (MBHBs) embedded in a surrounding distribution of stars. The dynamics of the binary is evolved by taking into account the erosion of the central stellar cusp bound to the massive black holes, the scattering of unbound stars feeding the binary loss cone, and the emission of gravitational waves (GWs). Stellar dynamics is treated in a hybrid fashion by coupling the results of numerical 3-body scattering experiments of bound and unbound stars to an analytical framework for the evolution of the stellar density distribution and for the efficiency of the binary loss cone refilling. Our main focus is on the behaviour of the binary eccentricity, in the attempt of addressing its importance in the merger process and its possible impact for GW detection with the planned Laser Interferometer Space Antenna ({\it LISA}), and ongoing and forthcoming pulsar timing array (PTA) campaigns. We produce a family of evolutionary tracks extensively sampling the relevant parameters of the system which are the binary mass, mass ratio and initial eccentricity, the slope of the stellar density distribution, its normalization and the efficiency of loss cone refilling. We find that, in general, stellar dynamics causes a dramatic increase of the MBHB eccentricity, especially for initially already mildly eccentric and/or unequal mass binaries. When applied to standard MBHB population models, our results predict eccentricities in the ranges $10^{-3}-0.2$ and $0.03-0.3$ for sources detectable by {\it LISA} and PTA respectively. Such figures may have a significant impact on the signal modelling, on source detection, and on the development of parameter estimation algorithms.Comment: 15 pages, 9 figures, accepted for publication in the Astrophysical Journa