A new Monte Carlo code for star cluster simulations - I. Relaxation

We have developed a new simulation code aimed at studying the stellar dynamics of a galactic central star cluster surrounding a massive black hole. In order to include all the relevant physical ingredients (2-body relaxation, stellar mass spectrum, collisions, tidal disruption, ...) , we chose to revive an umerical scheme pioneered by H enon in the 70's (H enon 1971b,a; H enon 1973. It is basically a Monte Carlo resolution of the Fokker-Planck equation. It can cope with any stellar mass spectrum or velocity distribution. Being a particle-based method, it also allows one to take stellar collisions into account in a very realistic way. This rst paper covers the basic version of our code which treats the relaxation-driven evolution of a stellar cluster without a central BH. A technical description of the code is presented, as well as the results of test computations. Thanks to the use of a binary tree to store potential and rank information and of variable time steps, cluster models with up to 2 10 6 particles can be simulated on a standard personal computer and the CPU time required scales as Np ln(Np) with the particle number Np. Furthermore, the number of simulated stars needs not be equal to Np but can be arbitrarily larger. A companion paper will treat further physical elements, mostly relevant to galactic nuclei.

[1]  Reinhard E. Schielicke,et al.  Reviews in modern astronomy , 2003 .

[2]  U. Bern,et al.  A new Monte Carlo code for star cluster simulations - II. Central black hole and stellar collisions , 2002, astro-ph/0204292.

[3]  M. Freitag.  Monte Carlo cluster simulations to determine the rate of compact star inspiralling to a central galactic black hole , 2001 .

[4]  Bern University,et al.  Monte Carlo cluster simulations to determine the rate of compact star inspiraling to a central galactic black hole , 2001, astro-ph/0107193.

[5]  H. Baumgardt.  Scaling of N-body calculations , 2000, astro-ph/0012330.

[6]  M. Giersz.  Monte Carlo simulations of star clusters – II. Tidally limited, multimass systems with stellar evolution , 2000, astro-ph/0009341.

[7]  Cody P. Nave,et al.  Monte Carlo Simulations of Globular Cluster Evolution. II. Mass Spectra, Stellar Evolution, and Lifetimes in the Galaxy , 1999, astro-ph/9912155.

[8]  Andreas Just,et al.  Dynamics of Star Clusters and the Milky Way , 2001 .

[9]  Riccardo Giacconi,et al.  Black holes in binaries and galactic nuclei, diagnostics, demography, and formation : proceedings of the ESO workshop held at Garching, Germany, 6-8 September 1999, in honour of Riccardo Giacconi , 2001 .

[10]  E.P.J. van den Heuvel,et al.  Black holes in binaries and galactic nuclei , 2001 .

[11]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[12]  T. Zeeuw.  Evidence for Massive Black Holes in Nearby Galactic Nuclei , 2000, astro-ph/0009249.

[13]  M. Milosavljevic,et al.  Dynamics of Black-Hole Nuclei , 2000, astro-ph/0008497.

[14]  D. Heggie.  Mass Loss from Globular Clusters , 2000, astro-ph/0007336.

[15]  J. Makino.  Direct Simulation of Dense Stellar Systems with GRAPE-6 , 2000, astro-ph/0007084.

[16]  E. Athanassoula,et al.  Central cusp caused by a supermassive black hole in axisymmetric models of elliptical galaxies , 2000, astro-ph/0004214.

[17]  Andrew Gould,et al.  A Cluster of Black Holes at the Galactic Center , 2000, astro-ph/0003269.

[18]  M. Lewerenz,et al.  Molecular Dynamics On Parallel Computers , 2000 .

[19]  C. Pichon,et al.  Stellar dynamics in the Galactic centre: Proper motions and anisotropy , 2000, astro-ph/0001428.

[20]  F. Rasio,et al.  Thermal and Dynamical Equilibrium in Two-Component Star Clusters , 1999, astro-ph/9912457.

[21]  K. Gebhardt,et al.  Canada-France-Hawaii Telescope Adaptive Optics Observations of the Central Kinematics in M15 , 1999, astro-ph/9912172.

[22]  H. Rix,et al.  Binary Black Hole Mergers from Planet-like Migrations , 1999, The Astrophysical journal.

[23]  M. Giersz,et al.  A stochastic Monte Carlo approach to model real star cluster evolution - II. Self-consistent models and primordial binaries , 1999, astro-ph/9911504.

[24]  D. Heggie,et al.  The time-scale of escape from star clusters , 1999, astro-ph/9910468.

[25]  F. Rasio,et al.  Monte Carlo Simulations of Globular Cluster Evolution. I. Method and Test Calculations , 1999, astro-ph/9909115.

[26]  A. Kugel,et al.  Towards the million body problem on the computer - no news since the three-body problem? , 1999, astro-ph/9906155.

[27]  R. Spurzem.  Direct Simulation of Dense Stellar Systems with GRAPE-6 , 2000 .

[28]  T. Alexander,et al.  The Distribution of Stars near the Supermassive Black Hole in the Galactic Center , 1999, astro-ph/9907241.

[29]  UK IoACambridge,et al.  Symbiotic starburst–black hole active galactic nuclei – I. Isothermal hydrodynamics of the mass‐loaded interstellar medium , 1999, astro-ph/9907198.

[30]  G. Drukier,et al.  Anisotropic Fokker-Planck Models for the Evolution of Globular Star Clusters: The Core-Halo Connection , 1999 .

[31]  R. Spurzem.  Direct N-body simulations , 1999, astro-ph/9906154.

[32]  L. Ho.  Supermassive Black Holes in Galactic Nuclei , 1998, astro-ph/9803307.

[33]  George K. Miley,et al.  GALAXY INTERACTIONS AT LOW AND HIGH REDSHIFT , 1999 .

[34]  L. Ho.  in Observational Evidence for Black Holes in the Universe , 1999 .

[35]  M. Davies,et al.  The destructive effects of binary encounters on red giants in the Galactic Centre , 1998 .

[36]  E. Becklin,et al.  High Proper-Motion Stars in the Vicinity of Sagittarius A*: Evidence for a Supermassive Black Hole at the Center of Our Galaxy , 1998, astro-ph/9807210.

[37]  T. Lauer,et al.  M32 ± 1 , 1998, astro-ph/9806277.

[38]  D. Heggie,et al.  Basic N-body modelling of the evolution of globular clusters — I. Time scaling , 1998, astro-ph/9805344.

[39]  S. Zwart,et al.  The Disruption of Globular Star Clusters in the Galaxy: A Comparative Analysisbetween Fokker-Planck and N-Body Models , 1998, astro-ph/9805310.

[40]  D. Vokrouhlický,et al.  STELLAR DYNAMICS IN A GALACTIC CENTRE SURROUNDED BY A MASSIVE ACCRETION DISC : I. NEWTONIAN DESCRIPTION , 1998, astro-ph/9805011.

[41]  Andrea M. Ghez,et al.  High Proper Motion Stars in the Vicinity of SGR A*: Evidence for a Supermassive Black Hole at the Center of Our Galaxy , 1998 .

[42]  M. Giersz.  Monte Carlo simulations of star clusters - I. First Results , 1998, astro-ph/9804127.

[43]  D. Vokrouhlický,et al.  Stellar capture by an accretion disc , 1997, astro-ph/9710169.

[44]  S. Tremaine,et al.  The Demography of Massive Dark Objects in Galaxy Centers , 1997, astro-ph/9708072.

[45]  Y. Sofue.  The central regions of the Galaxy and galaxies , 1998 .

[46]  H. Ford,et al.  HST Detections of Massive Black Holes in the Centers of Galaxies , 1997, astro-ph/9711299.

[47]  Eyal Maoz.  10.9. Dynamical constraints on alternatives to massive black holes in galactic nuclei , 1997 .

[48]  Reinhard Genzel,et al.  On the nature of the dark mass in the centre of the Milky Way , 1997 .

[49]  Koji Takahashi.  Fokker-Planck Models of Star Clusters with Anisotropic Velocity Distributions III. Multi-mass Clusters , 1997, astro-ph/9703190.

[50]  Reinhard Genzel,et al.  Stellar proper motions in the central 0.1 PC of the galaxy , 1997 .

[51]  J. Makino.  Merging of Galaxies with Central Black Holes. II. Evolution of the Black Hole Binary and the Structure of the Core , 1996, astro-ph/9608161.

[52]  Alfred Krabbe,et al.  The Dark Mass Concentration in the Central Parsec of the Milky Way , 1996 .

[53]  K. Danzmann.  LISA: laser interferometer space antenna for gravitational wave measurements , 1996 .

[54]  G. Meylan,et al.  Internal dynamics of globular clusters , 1996, astro-ph/9912495.

[55]  A. Eckart,et al.  Observations of stellar proper motions near the Galactic Centre , 1996, Nature.

[56]  J. Makino.  Postcollapse Evolution of Globular Clusters , 1996, astro-ph/9608160.

[57]  G. Quinlan.  The time-scale for core collapse in spherical star clusters , 1996, astro-ph/9606182.

[58]  W. Zurek,et al.  Red Giant--Disk Encounters: Food for Quasars? , 1996, astro-ph/9605137.

[59]  S. Aarseth,et al.  Direct collisional simulation of 10000 particles past core collapse , 1996, astro-ph/9605003.

[60]  G. Quinlan.  The dynamical evolution of massive black hole binaries i , 1996, astro-ph/9706298.

[61]  Koji Takahashi.  Fokker-Planck Models of Star Clusters with Anisotropic Velocity Distributions II. Post-Collapse Evolution , 1995, astro-ph/9507040.

[62]  D. Heggie,et al.  Statistics of N-body simulations — III. Unequal masses , 1995, astro-ph/9506143.

[63]  Jeremiah P. Ostriker,et al.  Dynamical Evolution of Globular Clusters , 1996 .

[64]  Kevin Patrick Rauch.  Dynamical evolution of star clusters around a rotating black hole with an accretion disc , 1995 .

[65]  R. Spurzem,et al.  Comparison between Fokker-Planck and gaseous models of star clusters in the multi-mass case revisited , 1995 .

[66]  Hyung-Mok Lee.  Evolution of galactic nuclei with 10-M⊙ black holes , 1994, astro-ph/9409073.

[67]  L. Hernquist,et al.  Models of Galaxies with Central Black Holes: Simulation Methods , 1994, astro-ph/9409014.

[68]  L. Hernquist,et al.  Models of Galaxies with Central Black Holes: Adiabatic Growth in Spherical Galaxies , 1994, astro-ph/9407005.

[69]  John Kormendy,et al.  Inward Bound—The Search for Supermassive Black Holes in Galactic Nuclei , 1995 .

[70]  Reinhard Genzel,et al.  The nucleus of our Galaxy , 1994 .

[71]  D. Heggie,et al.  Statistics of N-Body Simulations - Part Two - Equal Masses after Core Collapse , 1994, astro-ph/9403024.

[72]  M. Giersz,et al.  A comparison of direct N-body integration with anisotropic gaseous models of star clusters , 1993, astro-ph/9305033.

[73]  D. Heggie,et al.  Statistics of N-body simulations – I. Equal masses before core collapse , 1993, astro-ph/9305008.

[74]  R. Genzel,et al.  The nuclei of normal galaxies : lessons from the Galactic Center , 1994 .

[75]  M. Lee.  N-body evolution of dense clusters of compact stars , 1993 .

[76]  Stephen L. W. McMillan,et al.  An O(N log N) integration scheme for collisional stellar systems , 1993 .

[77]  S. Shapiro,et al.  Collisions and close encounters between massive main-sequence stars , 1993 .

[78]  Piet Hut,et al.  Binary-single-star scattering. V: Steady state binary distribution in a homogeneous static background of single stars , 1993 .

[79]  Georges Meylan,et al.  Structure and dynamics of globular clusters , 1993 .

[80]  E. S. Phinney,et al.  BINARIES IN GLOBULAR CLUSTERS , 1992, astro-ph/9710262.

[81]  W. Benz,et al.  Three-dimensional hydrodynamical simulations of colliding stars. III: Collisions and tidal captures of unequal-mass main-sequence stars , 1992 .

[82]  C. Pethick,et al.  Tidal capture of stars by a massive black hole , 1992 .

[83]  P. Hut,et al.  A post-core-collapse model for the nucleus of M33 , 1991, Nature.

[84]  Rainer Spurzem,et al.  Anisotropic gaseous models for the evolution of star clusters , 1991 .

[85]  C. J. Clarke,et al.  Star–disc interactions near a massive black hole , 1991 .

[86]  K. M. Crenell.  Molecular dynamics on parallel computers , 1991 .

[87]  Richard H. Durisen,et al.  Dynamical and luminosity evolution of active galactic nuclei - Models with a mass spectrum , 1991 .

[88]  H. Richer,et al.  Multicomponent Models for the Dynamic Evolution of Globular Clusters , 1991 .

[89]  M. Rees.  "Dead Quasars" in Nearby Galaxies? , 1990, Science.

[90]  W. Benz.  Smooth Particle Hydrodynamics: A Review , 1990 .

[91]  J. R. Buchler,et al.  The numerical modelling of nonlinear stellar pulsations: problems and prospects. Proceedings. , 1990 .

[92]  David H. Porter,et al.  A tree code with logarithmic reduction of force terms, hierarchical regularization of all variables, and explicit accuracy controls , 1989 .

[93]  M. Lee,et al.  Adiabatic growth of a black hole in a rotating stellar system , 1989 .

[94]  Hyung-Mok Lee,et al.  Stellar encounters in the Galactic center - formation of massive stars and close binaries , 1988 .

[95]  C. Norman,et al.  The Evolution of Starburst Galaxies to Active Galactic Nuclei , 1988 .

[96]  G. Voit,et al.  X-ray induced stellar mass loss near active galactic nuclei , 1988 .

[97]  Martin J. Rees,et al.  Tidal disruption of stars by black holes of 106–108 solar masses in nearby galaxies , 1988, Nature.

[98]  D. Heggie,et al.  Two homological models for the evolution of star clusters , 1988 .

[99]  Hyung-Mok Lee.  Dynamical effects of successive mergers on the evolution of spherical stellar systems , 1987 .

[100]  R. Durisen,et al.  The Evolution of Active Galactic Nuclei. II. Models with Stellar Evolution , 1987 .

[101]  R. Durisen,et al.  The Evolution of Active Galactic Nuclei. I. Models without Stellar Evolution , 1987 .

[102]  Piet Hut,et al.  Use of Supercomputers in Stellar Dynamics , 1986 .

[103]  H. Cohn,et al.  Realistic models for evolving globular clusters: core collapse with a mass spectrum , 1986 .

[104]  W. Saslaw,et al.  Gravitational Physics of Stellar and Galactic Systems: Finite spherical systems – clusters of galaxies, galactic nuclei, globular clusters , 1985 .

[105]  S. Inagaki,et al.  Equipartition in multicomponent gravitational systems , 1985 .

[106]  D. Sugimoto,et al.  Gravitational N-body problem and the validity of concepts of the gas model , 1985 .

[107]  S. Shapiro.  Monte Carlo Simulations of the 2+1 Dimensional Fokker-Planck Equation: Spherical Star Clusters Containing Massive, Central Black Holes , 1985 .

[108]  W. C. Saslaw.  Gravitational Physics of Stellar and Galactic Systems: Evaporation and escape , 1985 .

[109]  W. Saslaw.  Gravitational Physics of Stellar and Galactic Systems: Introduction , 1985 .

[110]  D. Sugimoto,et al.  Post-collapse evolution and gravothermal oscillation of globular clusters , 1984 .

[111]  H. Cohn,et al.  Late core collapse in star clusters and the gravothermal instability , 1980 .

[112]  S. Tremaine,et al.  A reinvestigation of the standard model for the dynamics of a massive black hole in a globular cluster , 1980 .

[113]  P. Young.  Numerical models of star clusters with a central black hole. I - Adiabatic models. , 1980 .

[114]  R. Mathieu,et al.  Random gravitational encounters and the evolution of spherical systems. VIII. Clusters with an initial distribution of binaries , 1980 .

[115]  M. Rees,et al.  Massive black hole binaries in active galactic nuclei , 1980, Nature.

[116]  P. P. Eggleton,et al.  On the consequences of the gravothermal catastrophe , 1980 .

[117]  M. Bailey.  On the fate of stellar mass loss in galactic nuclei , 1980 .

[118]  H. Cohn,et al.  Numerical integration of the Fokker-Planck equation and the evolution of star clusters , 1979 .

[119]  S. L. Shapiro,et al.  Star clusters containing massive, central black holes. III - Evolution calculations , 1979 .

[120]  R. Kulsrud,et al.  Stellar distribution around a black hole: Numerical integration of the Fokker-Planck equation , 1978 .

[121]  S. L. Shapiro,et al.  Star clusters containing massive, central black holes: Monte Carlo simulations in two-dimensional phase space , 1978 .

[122]  S. Shapiro,et al.  The distribution and consumption rate of stars around a massive, collapsed object , 1977 .

[123]  Richard A. Wolf,et al.  Star distribution around a massive black hole in a globular cluster , 1976 .

[124]  S. Shapiro,et al.  The distribution of stars around a massive black hole , 1976, Nature.

[125]  Niklaus Wirth,et al.  Algorithms + Data Structures = Programs , 1976 .

[126]  L. Spitzer,et al.  Random gravitational encounters and the evolution of spherical systems. VII. Systems with several mass groups. , 1975 .

[127]  L. Spitzer,et al.  Random gravitational encounters and the evolution of spherical systems. VI. Plummer's model. , 1975 .

[128]  M. Hénon,et al.  Two Recent Developments Concerning the Monte Carlo Method , 1975 .

[129]  Donald E. Knuth,et al.  The Art of Computer Programming: Volume 3: Sorting and Searching , 1998 .

[130]  M. Hénon,et al.  Dynamical structure and evolution of stellar systems , 1973 .

[131]  W. Saslaw.  THE DYNAMICS OF DENSE STELLAR SYSTEMS , 1973 .

[132]  Donald E. Knuth,et al.  Sorting and Searching , 1973 .

[133]  P. J. E. Peebles,et al.  Star Distribution Near a Collapsed Object , 1972 .

[134]  L. Spitzer,et al.  Random Gravitational Encounters and the Evolution of Spherical Systems. IV Isolated Systems of Identical Stars , 1972 .

[135]  M. Hénon.  Monte Carlo models of star clusters , 1971 .

[136]  Stuart L. Shapiro,et al.  Random Gravitational Encounters and the Evolution of Spherical Systems. III. Halo , 1971 .

[137]  R. Larson.  The Evolution of Star Clusters , 1970 .

[138]  R. Larson.  A Method for Computing the Evolution of Star Clusters , 1970 .

[139]  L. Spitzer,et al.  Equipartition and the Formation of Compact Nuclei in Spherical Stellar Systems , 1969 .

[140]  Richard M. West,et al.  Highlights of astronomy , 1968 .

[141]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .