A Fokker-Planck Study of Dense Rotating Stellar Clusters A dissertation presented by John Andrew Girash to The Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Harvard University Cambridge, Massachusetts November 2009 c 2009 by John Andrew Girash All rights reserved. Dissertation Advisor: Professor George B. Field Author: John Andrew Girash A Fokker-Planck Study of Dense Rotating Stellar Clusters Abstract The dynamical evolution of dense stellar systems is simulated using a two-dimensional Fokker-Planck method, with the goal of providing a model for the formation of supermassive stars which could serve as seed objects for the supermassive black holes of quasars. This work follows and expands on earlier one-dimensional studies of spherical clusters of main- sequence stars. The two-dimensional approach allows for the study of rotating systems, as would be expected due to cosmological tidal torquing; other physical effects included are collisional mergers of individual stars and a bulk stellar bar perturbation in the sys- tem’s gravitational potential. The 3 Myr main-sequence lifetime for large stars provides an upper limit on the allowed simulation times. Two general classes of initial systems are studied: Plummer spheres, which represent stellar clusters, and “γ = 0” spheres, which model galactic spheroids. At the initial densities of the modeled systems, mass segregation and runaway stellar collisions alone are insufficient to induce core collapse within the main-sequence lifetime limit, if no bar perturbation is included. However, core collapse is not a requirement for the formation of a massive object: the choice of stellar initial mass function (IMF) is found to play a crucial role. When using an IMF similar to that observed for dense stellar clusters (weighted towards high masses but with a high-mass cutoff of Mmax . 150M⊙) the simulations presented here show, in all cases, that the stellar system forms massive (250M⊙) objects by collisional mergers of lower-mass stars; in almost all such cases the presence of a stellar bar allows for sufficient additional outward transport of angular momentum that a core-collapse state is reached with corresponding further increase in the rate of formation of massive objects. In contrast, simulations using an IMF similar to that observed for field stars in general (which is weighted more towards lower masses) produce no massive objects, and reach core collapse only for initial models which represent the highest-density galactic spheriods. Possible extensions of the work presented here include continuing to track stellar populations after they evolve off the main sequence, and allowing for a (possibly changing) nonspherical component to the overall system potential. iii Contents Abstract iii Acknowledgements viii List of Figures x List of Tables xvi 1 Introduction 1 1.1 Astrophysical Motivation: Quasar Massive Black Holes . ........... 1 1.1.1 StellarSystems.............................. 2 1.1.2 Scenarios for Massive Black Hole Formation in Dense Clusters . 3 1.1.3 Rotation due to Tidal Torquing . 4 1.2 MotivationforTechnique . 6 1.2.1 Comparison of Fokker-Planck and N-body Methods . 6 1.2.2 Orbit Averaging and the Third Integral . ... 7 1.2.3 ChoiceofCanonicalVariables. 7 1.3 Overview of Resulting Model and of Remaining Chapters . ........ 8 2 Dynamical and Gravitational Aspects of the Model 11 2.1 Overview ..................................... 11 2.2 Dynamics ..................................... 11 2.2.1 Numerically Solving for Orbital Endpoints . ...... 13 2.2.2 Calculating Orbital Frequencies and other Dynamic Quantities . 14 2.3 CalculationoftheDensity . 14 2.3.1 Using the Distribution Function in (E, J 2)Space........... 14 2.3.2 Using the (E, J 2)-space Distribution Function in Action Space . 15 iv 2.3.3 Calculating the Density in Action Space . 15 2.4 Rotational Velocity and Orbital Inclination . ......... 16 2.4.1 OrbitalInclination . 17 2.4.2 TheMeanRotationalVelocity . 18 2.4.3 TheCoordinateGrid........................... 19 2.4.4 TheVelocityDispersion . 20 2.5 GravitationalPotential. 20 2.6 InitialConditions............................... 21 2.6.1 Potential-DensityPairs . 21 2.6.2 IntroducingRotation. 23 2.6.3 InitialMassFunction .......................... 24 3 Derivation of the Fokker-Planck Diffusion Coefficients 27 3.1 The Orbit-Averaged Fokker-Planck Equation . ..... 27 3.2 ThePerturbingPotential . 28 3.2.1 Generalformoftheexpansion . 28 3.2.2 EffectofOrbitalInclination . 30 3.3 FieldStarPerturbers. 31 3.4 TheDiffusionCoefficients . 32 3.4.1 MassSegregation............................. 34 3.5 The Stellar Bar: General Considerations . ....... 35 3.6 TheStellarBar: Implementation . 36 3.6.1 Bar Speed Determined by Angular Momentum Conservation .... 38 3.6.2 Bar Speed Determined by Angular Frenquency Conservation . 39 3.6.3 BarPerturbation............................. 39 3.7 FiniteDifferencingScheme. 41 3.7.1 NumericalStability. 41 3.7.2 TimeSplitting .............................. 42 3.7.3 Ensuring a Positive-Definite Distribution . ..... 42 3.7.4 Numerical Boundary Conditions . 43 3.8 StellarMergers .................................. 44 3.8.1 RatesofLossduetoMergers . 44 3.8.2 RatesofGain............................... 45 3.8.3 MassBookkeeping ............................ 47 v 3.8.4 The Delta-function Approximation . 47 3.8.5 TheCrossSection ............................ 48 3.9 BinaryMergers,BinaryHeating . 48 4 Validity Tests and Model Parameter Choices 50 4.1 Overview ..................................... 50 4.2 Potential-calculatingTests. ..... 51 4.3 DynamicalTests ................................. 51 4.3.1 OrbitalFrequencies. 51 4.3.2 OrbitalAngles .............................. 52 4.4 DiffusionCoefficientTests . 56 4.4.1 Low-levelCalculations . 56 4.4.2 Diffusion Coefficients: Reproducing the Potential . ....... 56 4.4.3 TheBarPerturbation .......................... 57 4.5 MergerLosses&Gains.............................. 59 4.6 TestingtheDifferencingScheme. 59 4.7 ModelParameters ................................ 60 4.7.1 Gridsize:actionspace. 60 4.7.2 Gridsize: radialcoordinate . 62 4.7.3 Timestepsize............................... 63 4.7.4 Numberofexpansionterms . 64 4.7.5 Mass Spectrum: Discretizing the Initial Mass Function........ 69 4.8 BinaryHeating .................................. 71 5 Results 73 5.1 GeneralConsiderations. 73 5.1.1 InitialModels............................... 73 5.1.2 PhysicalEffects.............................. 73 5.1.3 Overview ................................. 74 5.2 Plummer-sphereModels . 75 5.2.1 Model E4B, with Kroupa IMF: Core of a Giant Elliptical Galaxy . 75 5.2.2 ModelE4B,Arches-styleIMF. 78 5.2.3 Model E2A, Arches-style IMF: Nuclear Cluster . 83 5.2.4 Model E2B, Arches-style IMF: Nuclear Cluster . 83 vi 5.2.5 Model E2A, Kroupa: Bulgeless Spiral or Dwarf Elliptical Nucleus . 87 5.2.6 EffectofCollisionalMergerRates . 88 5.3 γ =0SphereModels............................... 89 5.3.1 Model G2A, Kroupa IMF: Galactic Spheroid / Spiral Bulge..... 89 5.3.2 Models G3A (Larger Galactic Spheroid) and G3C, Kroupa IMF . 90 5.3.3 ModelG2A,ArchesIMF ........................ 92 6 Discussion 96 6.1 Timescale Arguments and Expectations . ..... 97 6.1.1 MassSegregation............................. 97 6.1.2 CriticalDensity.............................. 97 6.2 Summary of Simulation Results . 98 6.2.1 Plummer-SphereSimulations . 99 6.2.2 “γ =0Sphere”Simulations . 101 6.2.3 Summary ................................. 102 6.3 Future Observations, Future Work . 103 A Non-Spherical Gravitational Potentials 105 A.1 Potential Calculations in Homeoidal Coordinates . .......... 105 A.2 Laplacian Solution in Ellipsoidal Coordinates: Numerical Aspects . 106 B Tables of Symbols 108 Bibliography 117 vii Acknowledgements The full list of people to whom I am thankful for helping over the course of this project is too long to mention completely, so first off I would like to express my great appreciation to everyone in the Department of Physics and at the Center of Astrophysics while I was there. Of individual thanks, first and foremost I’m grateful for my thesis advisor George Field; ever since we started this project I’ve been amazed by your knowledge; as it progressed, was also impressed and aided greatly by your wisdom; and now as we approach the end, am very thankful for your patience. To my thesis committee, Professors Christopher Stubbs, Melissa Franklin and Lars Hern- quist, I give my thanks for your unhesitating willingness to join me in this, and for your wise thoughts and feedback. No less impressive than George Field’s patience is that of my wonderful spouse Reebee, who without complaint put up with countless late-night coding sessions, weekends spent debugging, and holidays taken over by writing. I couldn’t have done this without your support and encouragement. Both for their friendship and their specific help in writing and reading this dissertation I am grateful and indebted to Paul Janzen, Warren Brown, J.D. Paul and Ken Rines. My long-term officemates along the way – Dan Koranyi, Andi Mahdavi and Vit Hradecky – all provided welcome, supportive friendship, lessons on the niceties of caffeine, and healthy distractions both intellectual and not. Thanks for not letting me take myself too seri- ously! When not busy at our respective computers Mike Westover and Dan Green gave me all I could handle on the tennis court, while David Charbonneau, Pauline Barmby, Ian Dell’Antonio, Saurabh Jha, Hannah Jang-Condell,