Star Cluster Simulations: The State of the Art

This paper concentrates on four key tools for performing star cluster simulations developed during the last decade which are sufficient to handle all the relevant dynamical aspects. First we discuss briefly the Hermite integration scheme which is simple to use and highly efficient for advancing the single particles. The main numerical challenge is in dealing with weakly and strongly perturbed hard binaries. A new treatment of the classical Kustaanheimo-Stiefel two-body regularization has proved to be more accurate for studying binaries than previous algorithms based on divided differences or Hermite integration. This formulation employs a Taylor series expansion combined with the Stumpff functions, still with one force evaluation per step, which gives exact solutions for unperturbed motion and is at least comparable to the polynomial methods for large perturbations. Strong interactions between hard binaries and single stars or other binaries are studied by chain regularization which ensures a non-biased outcome for chaotic motions. A new semi-analytical stability criterion for hierarchical systems has been adopted and the long-term effects on the inner binary are now treated by averaging techniques for cases of interest. These modifications describe consistent changes of the orbital variables due to large Kozai cycles and tidal dissipation. The range of astrophysical processes which can now be considered by N-body simulations include tidal capture, circularization, mass transfer by Roche-lobe overflow as well as physical collisions, where the masses and radii of individual stars are modelled by synthetic stellar evolution.Comment: Accepted by Cel. Mech. Dyn. Astron., 12 pages including figur

Nbody2: A Direct N-Body Integration Code

We give a full description of the code NBODY2 for direct integration of the gravitational N-body problem. The method of solution is based on the neighbour scheme of Ahmad & Cohen (1973) which speeds up the force calculation already for quite modest particle numbers. Derivations of all the relevant mathematical expressions are given, together with a detailed discussion of the algorithms. The code may be used to study a wide variety of self-consistent problems based on a small softening of the interaction potential.Comment: Latex (uses macros elsart.cls, harvard.sty. Accepted for publication in New Astronomy

Post-Newtonian N-body simulations

We report on the first fully consistent conventional cluster simulation which includes terms up to post^{5/2} Newtonian in the potential of the massive body. Numerical problems for treating extremely energetic binaries orbiting a single massive object are circumvented by employing the special ``wheel-spoke'' regularization method of Zare (1974) which has not been used in large-N simulations before. Idealized models containing N = 10^5 particles of mass 1 M_sun with a central black hole of 300 M_sun have been studied on GRAPE-type computers. An initial half-mass radius of r_h = 0.1 pc is sufficiently small to yield examples of relativistic coalescence. This is achieved by significant binary shrinkage within a density cusp environment, followed by the generation of extremely high eccentricities which are induced by Kozai (1962) cycles and/or resonant relaxation. More realistic models with white dwarfs and ten times larger half-mass radii also show evidence of GR effects before disruption. Experimentation with the post-Newtonian terms suggests that reducing the time-scales for activating the different orders progressively may be justified for obtaining qualitatively correct solutions without aiming for precise predictions of the final gravitational radiation wave form. The results obtained suggest that the standard loss-cone arguments underestimate the swallowing rate in globular clusters containing a central black hole.Comment: 8 pages, 6 figures, submitted to MNRA

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

A Complete N-body Model of the Old Open Cluster M67

The old open cluster M67 is an ideal testbed for current cluster evolution models because of its dynamically evolved structure and rich stellar populations that show clear signs of interaction between stellar, binary and cluster evolution. Here we present the first truly direct N-body model for M67, evolved from zero age to 4 Gyr taking full account of cluster dynamics as well as stellar and binary evolution. Our preferred model starts with 12000 single stars and 12000 binaries placed in a Galactic tidal field at 8.0 kpc from the Galactic Centre. Our choices for the initial conditions and for the primordial binary population are explained in detail. At 4 Gyr, the age of M67, the total mass has reduced by 90% as a result of mass loss and stellar escapes. The mass and half-mass radius of luminous stars in the cluster are a good match to observations although the model is more centrally concentrated than observations indicate. The stellar mass and luminosity functions are significantly flattened by preferential escape of low-mass stars. We find that M67 is dynamically old enough that information about the initial mass function is lost, both from the current luminosity function and from the current mass fraction in white dwarfs. The model contains 20 blue stragglers at 4 Gyr which is slightly less than the 28 observed in M67. Nine are in binaries. The blue stragglers were formed by a variety of means and we find formation paths for the whole variety observed in M67. Both the primordial binary population and the dynamical cluster environment play an essential role in shaping the population. A substantial population of short-period primordial binaries (with periods less than a few days) is needed to explain the observed number of blue stragglers in M67.Comment: 32 pages, 17 figures, submitted to MNRA

Direct N-body Modelling of Stellar Populations: Blue Stragglers in M67

We present a state-of-the-art N-body code which includes a detailed treatment of stellar and binary evolution as well as the cluster dynamics. This code is ideal for investigating all aspects relating to the evolution of star clusters and their stellar populations. It is applicable to open and globular clusters of any age. We use the N-body code to model the blue straggler population of the old open cluster M67. Preliminary calculations with our binary population synthesis code show that binary evolution alone cannot explain the observed numbers or properties of the blue stragglers. On the other hand, our N-body model of M67 generates the required number of blue stragglers and provides formation paths for all the various types found in M67. This demonstrates the effectiveness of the cluster environment in modifying the nature of the stars it contains and highlights the importance of combining dynamics with stellar evolution. We also perform a series of N = 10000 simulations in order to quantify the rate of escape of stars from a cluster subject to the Galactic tidal field.Comment: 26 pages, 18 figures, accepted for publication in MNRA