Sverre J. Aarseth Christopher A. Tout Rosemary A. Mardling (Eds.) The Cambridge N-Body Lectures 123 Sverre J. Aarseth Christopher A. Tout University of Cambridge University of Cambridge Institute of Astronomy Institute of Astronomy Madingley Road Madingley Road Cambridge CB3 0HA Cambridge CB3 0HA United Kingdom United Kingdom Preface This book gives a comprehensive introduction to the tools required for direct N-body simulations. The contributors are all active researchers who write in detail on their own special fields in which they are leading international experts. It is their previous and current connections with the Cambridge Insti- tute of Astronomy, as staff or visitors, that gives rise to the title. The material is generally at a level suitable for a graduate student or postdoctoral worker entering the field. The book begins with a detailed description of the codes available for N-body simulations. In a second chapter we find different mathematical for- mulations for special treatments of close encounters involving binaries or multiple systems, which have been implemented. The concept of chaos and stability plays a fundamental role in celestial mechanics and is highlighted here in a presentation of a new formalism for the three-body problem. The emphasis on collisional stellar dynamics enables the scope to be enlarged by including methods relevant for comparison purposes. Modern star clus- ter simulations include additional astrophysical effects by modelling real stars instead of point-masses. Several contributions cover the basic theory and com- prehensive treatments of stellar evolution for single stars as well as binaries. Questions concerning initial conditions are also discussed in depth. Further connections with reality are established by an observational approach to data analysis of actual and simulated star clusters. Finally, important aspects of hardware requirements are described with special reference to parallel and GRAPE-type computers. The extensive chapters provide an essential frame- work for a variety of N-body simulations. During an extensive summer school on astrophysical N-body simulations, held in Cambridge, www.cambody.org, the Royal Astronomical Society en- couraged us to edit a volume on the topic, to be published in The Royal As- tronomical Society Series. Subsequently, we collected the tutorial lecture notes assembled in this volume. We would like to take this opportunity to thank the Royal Astronomical Society for sponsoring the school and the Institute of Astronomy for provision of school facilities. We are grateful to all the authors VI Preface who took time off from their busy schedules to deliver the manuscripts, which were then checked for both style and scientific content by the editors. This collection of topics, related to the gravitational N-body problem, will prove useful to both students and researchers in years to come. Cambridge Sverre J. Aarseth May 2008 Christopher A. Tout Rosemary A. Mardling Contents 1 Direct N -Body Codes Sverre J. Aarseth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Basic Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Data Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 N-Body Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Hermite Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 Ahmad–Cohen Neighbour Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.7 Time-Step Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.8 Two-Body Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.9 KS Decision-Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.10 Hierarchical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.11 Three-Body Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.12 Wheel-Spoke Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.13 Post-Newtonian Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.14 Chain Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.15 Astrophysical Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.16 GRAPE Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.17 Practical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 Regular Algorithms for the Few-Body Problem Seppo Mikkola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Hamiltonian Manipulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 KS-Chain(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.5 Algorithmic Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.6 N-Body Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.7 AR-Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.8 Basic Algorithms for the Extrapolation Method . . . . . . . . . . . . . . . . . 51 VIII Contents 2.9 Accuracy of the AR-Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3 Resonance, Chaos and Stability: The Three-Body Problem in Astrophysics Rosemary A. Mardling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2 Resonance in Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3 The Mathematics of Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4 The Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4 Fokker–Planck Treatment of Collisional Stellar Dynamics Marc Freitag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.3 Fokker–Planck Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.4 Orbit-Averaged Fokker–Planck Equation . . . . . . . . . . . . . . . . . . . . . . . 107 4.5 The Fokker–Planck Method in Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5 Monte-Carlo Models of Collisional Stellar Systems Marc Freitag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2 Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.3 Detailed Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.4 Some Results and Possible Future Developments . . . . . . . . . . . . . . . . 145 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6 Particle-Mesh Technique and SUPERBOX Michael Fellhauer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.2 Particle-Mesh Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6.3 Multi-Grid Structure of Superbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 7 Dynamical Friction Michael Fellhauer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.1 What is Dynamical Friction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.2 How to Quantify Dynamical Friction? . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.3 Dynamical Friction in Numerical Simulations . . . . . . . . . . . . . . . . . . . 175 7.4 Dynamical Friction of an Extended Object . . . . . . . . . . . . . . . . . . . . . 177 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Contents IX 8 Initial Conditions for Star Clusters Pavel Kroupa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 8.2 Initial 6D Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 8.3 The Stellar IMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 8.4 The Initial Binary Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 9 Stellar Evolution Christopher A. Tout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 9.1 Observable Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 9.2 Structural Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 9.3 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 9.4 Radiation Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 9.5 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 9.6 Energy Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 9.7 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 9.8 Evolutionary Tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 9.9 Stellar Evolution of Many Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 10 N -Body Stellar Evolution Jarrod R. Hurley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 10.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 10.2 Method and Early Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 10.3 The SSE Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 10.4 N-Body Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 10.5 Some Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 11 Binary Stars Christopher A. Tout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 11.1 Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 11.2 Tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 11.3 Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 11.4 Period Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 11.5 Actual Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 12 N -Body Binary Evolution Jarrod R. Hurley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 12.2 The BSE Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 12.3 N-Body Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 X Contents 12.4 Binary Evolution Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 13 The Workings of a Stellar Evolution Code Ross Church . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 13.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 13.3 Variables and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 13.4 Method of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 13.5 The Structure of stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 13.6 Problematic Phases of Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 13.7 Robustness of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 14 Realistic N -Body Simulations of Globular Clusters A. Dougal Mackey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 14.2 Realistic N-Body Modelling – Why and How? . . . . . . . . . . . . . . . . . . 347 14.3 Case Study: Massive Star Clusters in the Magellanic Clouds . . . . . . 354 14.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 15 Parallelization, Special Hardware and Post-Newtonian Dynamics in Direct N-Body Simulations Rainer Spurzem, Ingo Berentzen, Peter Berczik, David Merritt, Pau Amaro-Seoane, Stefan Harfst and Alessia Gualandris . . . . . . . . . . . . . 377 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 15.2 Relativistic Dynamics of Black Holes in Galactic Nuclei . . . . . . . . . . 378 15.3 Example of Application to Galactic Nuclei . . . . . . . . . . . . . . . . . . . . . 380 15.4 N-Body Algorithms and Parallelization . . . . . . . . . . . . . . . . . . . . . . . . 381 15.5 Special Hardware, GRAPE and GRACE Cluster . . . . . . . . . . . . . . . . 382 15.6 Performance Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 15.7 Outlook and Ahmad–Cohen Neighbour Scheme . . . . . . . . . . . . . . . . . 386 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 A Educational N -Body Websites Francesco Cancelliere, Vicki Johnson and Sverre Aarseth . . . . . . . . . . . . . 391 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 A.2 www.NBodyLab.org . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 A.3 www.Sverre.com . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 A.4 Educational Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 1 Direct N-Body Codes Sverre J. Aarseth University of Cambridge, Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK 2 S. J. Aarseth 1.2 Basic Features Before delving more deeply into the underlying algorithms, it is desirable to define units and introduce the data structure that forms the back-bone of a general N-body code. From dimensional analysis we first construct ˜ ∗ −5 ∗ 1/2 −1 fiducial velocity and time units by V = 1 × 10 (GM⊙/L ) km s , ˜∗ ∗3 1/2 ∗ 18 T = (L /GM⊙) s, with G the gravitational constant and L = 3×10 cm as a convenient length unit. Given the length scale or virial radius RV in pc and total mass NMS in M⊙, where MS is the average mass specified as in- put, we can now write the corresponding values for a star cluster model as ∗ −2 1/2 −1 ∗ 3 1/2 V = 6.557 × 10 (NMS/RV) km s and T = 14.94(R V/NMS) Myr. Hence scaled (or internal) N-body units of distance, velocity and time are −1 converted to corresponding astrophysical units (pc, km s , Myr) by r˜ = RVr, v˜ = V ∗v, t˜ = T∗t. Finally, individual masses in M⊙ are obtained from m˜ = MSm where MS is now redefined in terms of the scaled mean mass. As the next logical step on the road to an N-body simulation, we consider matters relating to the initial data. Let us assume that a complete set of initial conditions have been generated in the form mi, r˜i, v˜i for N particles, where the masses, coordinates and velocities can be in any units. A standard cluster model is essentially defined by N,MS,RV, together with a suitable initial mass function (IMF). After assigning the individual data, we evaluate the kinetic and potential energy, K and U, taking U < 0. The velocities are scaled 1/2 according to the virial theorem by taking vi = q v˜i, where q = (QV|U|/K) and QV is an input parameter (0.5 for overall equilibrium). Note that, in general, the virial energy should be used; however, the additional terms are not known ahead of the scaling. We now introduce so-called standard units ∑ by adopting the scaling G = 1, mi = 1, E0 = −0.25, where E0 is the new total energy (< 0). Here the energy condition is only applied for bound systems (QV < 1), otherwise the convention E0 = 0.25 is adopted. The final 1/2 1/2 2 scaling is performed by rˆi = r˜i/S , vˆi = viS with √S = E0/(q K + U). ∗ These variables define a standard crossing time Tcr = 2 2T Myr. Many simulations include primordial binary stars for greater realism. Be- cause of their internal binding energies, the above scaling cannot be imple- mented directly. Instead, the components of each binary are first combined into one object, whereupon the reduced population of Ns single stars and Nb binaries are subject to the standard scaling. It then remains for the internal two-body elements, such as semi-major axis, eccentricity and relevant angles to be assigned, together with the mass ratio. The choice of distributions is very wide, but should be motivated by astrophysical considerations. Of special in- terest here are the periods and mass ratios, which may well be correlated for luminous stars (e.g. spectroscopic binaries). More complicated ways of pro- viding initial conditions with primordial binaries can readily be incorporated. Thus, for example, a consistent set of initial conditions that do not require scaling may be uploaded. Such a data set might in fact be acceptable by a well-written code, but this practice is not recommended.