31 research outputs found
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 -body simulations. Over the
years, the 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 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