N-body Star Cluster Models

Here you will find an overview of a collaboration aimed at producing realistic models of star cluster evolution using the direct N-body method. Simulations are performed on the Swinburne supercomputer, on GPUs, and on GRAPE-6 hardware (the latter through collaboration with the American Museum of Natural History and the Institute of Astronomy). These exhibit the fascinating interplay between stellar, binary and dynamical evolution in a dense stellar environment. By confronting the models with observations (ground-based, HST and Chandra) we aim to further our understanding of the stellar populations and evolution history of star clusters.

bullet Background

bullet Codes

bullet Hardware

bullet Results

bullet Related Links

bullet Data Archive


Most stars are born in a star cluster of some description. The globular clusters of our Galaxy contain a million stars or more and are the oldest stellar aggregates in the Universe. Understanding the complicated evolution of these objects is essential to understanding how our Galaxy formed. Open clusters are smaller and younger. They populate the Galactic Disk and disperse on timescales much less than the age of the Galaxy. Thus stars originating from clusters make a non-trivial contribution to population of stars wandering around in the Galaxy and star cluster evolution leaves an imprint on the observed nature of exotic stars and extra-solar planetary systems. Furthermore, star clusters are dense stellar environments - compared to the solar neighbourhood the density of stars in a cluster can be a factor of 10 million greater. This makes physical collisions and close encounters between stars a reality of life in a cluster. Such activity can radically alter the fates of stars compared to predictions from standard evolution theory and lead to the formation of exotic stars/binaries.

On a basic dynamical level a star cluster is composed of N bodies interacting with each other due to the gravitational force of every other body in the system. The ideal method for following the evolution of such a system is to integrate directly the N individual equations of motion, the N-body approach. In practice this method has only had limited applicability to globular clusters because the required value of N is too large for a simulation to be completed in a reasonable time. A major boost to the usefulness of the N-body method has been the development of special-purpose GRAPE (GRAvity piPE) hardware. The GRAPE has the necessary logic for the force summation hardwired into its chips and is able to calculate forces at Tflops speed. A host computer sends information on all particles to the GRAPE which returns the accelerations (and associated derivatives) for all requested particles. All other operations such as the formation of binaries, stellar and binary evolution, tidal field, etc. must still be performed by the host computer. Prior to the advent of GRAPE the biggest direct N-body simulations performed had N ~2000. Now simulations of clusters with 100,000 stars can easily be completed in a reasonable timeframe using GRAPE-6 hardware. Similar advances have also been made running N-body codes on massively-parallel supercomputers. More recently graphics processing units (GPUs) have been used to replace the GRAPE hardware as the force summation engine. A single Tesla S1070 GPU gives similar performance to a 32-chip GRAPE-6 board and the benefits of a multi-GPU approach are currently being investigated.

Software advances over the same timeframe have also improved the realism of star cluster models. Only a decade ago models did not include stellar evolution and commonly did not even allow a spectrum of stellar masses. Binary evolution was certainly ignored and often simulations would be halted when the first binary formed. Thankfully these shortcuts are now consigned to the past and we now produce models where information on the stellar populations is at the level that allows direct comparison with observations -- and of course the number of stars in the model is now directly comparable to the number of stars in real clusters. Combined with the increased resolution provided by telescopes such as the Hubble Space Telescope and the Chandra X-ray satellite, which can peer into the cores of dense globular clusters, it is an rewarding time to be working in the field of star cluster evolution.


The main code utilized has been NBODY4 which was developed by Sverre Aarseth.
Basic integration of the equations of motion is performed by the Hermite scheme which employs a fourth-order force polynomial and exploits the fast evaluation of the force and its first time derivative by the GRAPE-6. A time-step scheme comprising a series of hierarchical levels allows each star to evolve on its own natural dynamical time-scale while forcing a block of particles to be advanced at each cycle so that efficiency does not suffer. Regularization techniques are used to treat perturbed two-body motion in an accurate and efficient manner with an extension to chain regularization to deal with compact subsystems of up to six bodies. A semi-analytical criterion is utilized to detect and evolve stable hierarchical triple and quadruple systems that otherwise would prove extremely time-consuming by direct integration. Exchange interactions in encounters between single stars and binaries, or binary--binary encounters, where the member of a binary is displaced by an incoming star, are included in this treatment. Direct collisions between stars and the formation of binaries in three- and four-body encounters are also allowed.
Collaborators in the development of NBODY4 include Douglas Heggie, Jun Makino, Rosemary Mardling, Steve McMillan, Seppo Mikkola and Rainer Spurzem.

An important aspect of NBODY4 is that stellar and binary evolution are performed in step with the dynamical evolution so that interaction between these processes is modelled consistently. Stellar evolution is included in the form of the SSE package and binary evolution via the BSE package. These were developed by Jarrod Hurley, Christopher Tout and Onno Pols.

Variants of NBODY4 are also available for use. NBODY6 is the sister code of NBODY4 and replaces the GRAPE-6 interface with a CUDA library to utilise GPUs. It has now become the focus for code development (circa 2009 and beyond). Force calculations and neighbour-finding tasks are performed on the GPU while other tasks are performed on the host CPU (utilising Streaming SIMD Extension in some areas for speed-up).

Also available is NBODY6++ -- a version for massively parallel supercomputers maintained by Rainer Spurzem.


Simulations are performed on GRAPE-6 boards located at the American Museum of Natural History (New York) and the Institute of Astronomy (Cambridge). Each board comprises 32 GRAPE chips and represents 1 Tflop/s of computing power. These became available in early 2001. Simulations with NBODY4 perfomed prior to this were done on a GRAPE-4 board at the Institute of Astronomy.

Simulations are also perfomed with NBODY6 on GPUs at Swinburne University (Melbourne). GPU hardware trialled so far includes the GTX 280 card and the Tesla S1070 series. We can also run NBODY6 without the GPU capability on the supercomputer at Swinburne University (operated by the Centre for Astrophysics & Supercomputing) and other facilities, e.g. at Northwestern in collaboration with Aaron Geller.



[15] N-body models of extended star clusters,
  Hurley J.R. and Mackey A.D.,
  2010, Monthly Notices of the Royal Astronomical Society, 408, 2353
[14] Star cluster evolution in dark matter dominated galaxies,
  Praagman A., Hurley J. and Power C.,
  2010, New Astronomy, 15, 46
[13] Self-consistent simulations of star cluster formation from gas clouds under the influence of galaxy-scale tidal fields,
  Hurley J.R. and Bekki K.,
  2008, Monthly Notices of the Royal Astronomical Society, 389, L61
[12] Deep Advanced Camera for Surveys Imaging in the Globular Cluster NGC 6397: Dynamical Models,
  Hurley J.R., et al.,
  2008, Astronomical Journal, 135, 2129
[11] The Core Binary Fractions of Star Clusters from Realistic Simulations,
  Hurley J.R., Aarseth S.J. and Shara M.M.,
  2007, Astrophysical Journal, 665, 707
[10] Ratios of star cluster core and half-mass radii: a cautionary note on intermediate-mass black holes in star clusters,
  Hurley J.R.,
  2007, Monthly Notices of the Royal Astronomical Society, 379, 93
[9] Dynamical Effects Dominate the Evolution of Cataclysmic Variables in Dense Star Clusters,
  Shara M.M. and Hurley J.R.,
  2006, Astrophysical Journal, 646, 464
[8] A Complete N-body Model of the Old Open Cluster M67,
  Hurley J.R., Pols O.R., Aarseth S.J. and Tout C.A.,
  2005, Monthly Notices of the Royal Astronomical Society, 363, 293
[7] Metallicity Effects on Open Cluster Dynamics,
  Hurley J.R., Tout C.A., Aarseth S.J. and Pols O.R.,
  2004, Monthly Notices of the Royal Astronomical Society, 355, 1207
[6] White Dwarf Sequences in Dense Star Clusters,
  Hurley J.R. and Shara M.M.,
  2003, Astrophysical Journal, 589, 179
[5] Core radius evolution of star clusters,
  Wilkinson M.I., Hurley J.R., Mackey A.D., Gilmore G.F. and Tout C.A.,
  2003, Monthly Notices of the Royal Astronomical Society, 343, 1025
[4] Star Clusters as Type Ia Supernova Factories,
  Shara M.M. and Hurley J.R.,
  2002, Astrophysical Journal, 571, 830
[3] The Promiscuous Nature of Stars in Clusters,
  Hurley J.R. and Shara M.M.,
  2002, Astrophysical Journal, 570, 184
[2] Free-Floating Planets in Stellar Clusters: Not So Surprising,
  Hurley J.R. and Shara M.M.,
  2002, Astrophysical Journal, 565, 1251
[1] Direct N-body Modelling of Stellar Populations: Blue Stragglers in M67,
  Hurley J.R., Tout C.A., Aarseth S.J. and Pols O.R.,
  2001, Monthly Notices of the Royal Astronomical Society, 323, 630



Return to Main page