Understanding the Solar System with Numerical Simulations and L´evy Flights Thesis by Benjamin F. Collins In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ITUTE O ST F N T I E C A I H N N R 1891 O O L F O I G L Y A C California Institute of Technology Pasadena, California 2009 (Defended May 21, 2009) ii c 2009 Benjamin F. Collins All Rights Reserved iii Acknowledgements This work, literally, would not have been possible without my adviser, Re’em Sari. Collaborating with Re’em was a constantly enlightening process, and I’m very grateful for his attention, patience, and generosity as we worked together for the last six years. Margaret Pan has been a great mentor ever since I joined her and Re’em in studying planetary dynamics. Hilke Schlichting has been an invaluable compatriot through all of our academic rites. While working and traveling with Re’em, Margaret, and Hilke, I have had many terrific adventures that I will always remember fondly. I would like to thank my committee: Marc Kamionkowski, Shri Kulkarni, and Dave Stevenson. In addition to their roles in the preparation and defense of this thesis, I have benefited greatly from interacting with each of them in classrooms, meeting rooms, or impromptu conversations. Finally, I thank my parents, Gale and Malcolm, and my brothers, Steven and Andrew, for their unwavering encouragement of my academic pursuits. iv Abstract This thesis presents several investigations into the formation of planetary systems and the dynamical evolution of the small bodies left over from this process. We develop a numerical integration scheme with several optimizations for studying the late stages of planet formation: adaptive time steps, accurate treatment of close encounters between particles, and the ability to add non-conservative forces. Using this code, we simulate the phase of planet formation known as “oligarchic growth.” We find that when the dynamical friction from planetesimals is strong, the annular feeding zones of the protoplanets are inhabited by not one but several oligarchs on nearly the same semimajor axis. We systematically determine the number of co-orbital protoplanets sharing a feeding zone and the width of these zones as a function of the strength of dynamical friction and the total mass of the disk. The increased number of surviving protoplanets at the end of this phase qualitatively affects their subsequent evolution into full-sized planets. We also investigate the distribution of the eccentricities of the protoplanets in the runaway growth phase of planet formation. Using a Boltzmann equation, we find a simple analytic solution for the distribution function followed by the eccentricity. We show that this function is self-similar: it has a constant shape while the scale is set by the balance between mutual excitation and dynamical friction. The type of evolution described by this distribution function is known as a L´evy flight. We use the Boltzmann equation framework to study the nearly circular orbits of Kuiper Belt binaries and the nearly radial orbits of comets during the formation of the Oort cloud. We calculate the distribution function of the eccentricity of Kuiper belt systems, like the moons of Pluto, given the stochastic perturbations caused by close encounters with other Kuiper belt objects. For Oort cloud comets, we find the distribution function of the angular momentum as it is excited by perturbations from passing stars in the Galaxy. Both systems evolve as L´evy flights. This work unifies the effects of stochastic stellar encounters and the smooth torque from the Galactic potential on Oort cloud comets. v Contents Acknowledgements iii Abstract iv List of Figures viii List of Tables x 1 Introduction and Overview 1 1.1 NumericalCode................................... .... 1 1.2 Co-OrbitalOligarchy. ....... 3 1.3 Shear-Dominated Protoplanetary Dynamics . ............ 3 1.4 L´evyFlightsofCircularBinaryOrbits . ............ 5 1.5 Stellar Perturbations and Galactic Tides Demystified . ................ 6 2 A New Planetary Simulation Code: RKNB3D 8 2.1 TheDifferentialEquations. ........ 9 2.2 TheAlgorithms................................... .... 11 2.3 Enhancements .................................... ... 12 2.4 NumericalTests.................................. ..... 13 2.5 Conclusions ..................................... .... 18 3 Co-Orbital Oligarchy 19 3.1 TheThree-BodyProblem . .. .. .. .. .. .. .. .. .. .. .. ..... 20 3.2 TheDampedN-BodyProblem . .... 24 3.3 OligarchicPlanetFormation. ......... 27 3.4 TheEquilibriumCo-OrbitalNumber . ......... 29 3.5 Isolation....................................... .... 34 3.6 ConclusionsandDiscussion . ........ 35 4 Protoplanet Dynamics in a Shear-Dominated Disk 40 4.1 Shear-Dominated Cooling and Heating Rates . ........... 41 4.1.1 Eccentricity Excitation of Protoplanets . ........... 42 4.1.2 DynamicalFriction. .... 43 vi 4.1.3 PlanetesimalInteractions . ....... 44 4.1.4 Inclinations .................................. ... 44 4.1.5 The Eccentricity Distribution — a Qualitative Discussion ........... 44 4.2 ABoltzmannEquation.............................. ..... 45 4.2.1 TheSolution ................................... 46 4.3 NumericalSimulations . ....... 46 4.3.1 EqualMassProtoplanets . .... 47 4.3.2 MassDistributions . .. .. .. .. .. .. .. .. .. .. .. .. .... 48 4.4 Conclusions ..................................... .... 50 4.5 Appendix: The Analytic Distribution Function . ............. 52 5 Self-Similarity of Shear-Dominated Viscous Stirring 54 5.1 TheTime-DependentBoltzmannEquation . ......... 54 5.2 The Self-Similar Distribution . .......... 56 5.3 The Generalized Time-Dependent Distribution . ............. 57 5.4 NumericalSimulations . ....... 59 5.5 Discussion...................................... .... 62 6 L´evy Flights of Binary Orbits Due to Impulsive Encounters 64 6.1 ASingleEncounter................................ ..... 65 6.1.1 CloseEncounters............................... ... 67 6.1.2 DistantEncounters. .... 67 6.1.3 Collisions .................................... .. 68 6.2 BoltzmannEquation ............................... ..... 68 6.2.1 Eccentricity.................................. ... 68 6.2.2 Inclination ................................... .. 71 6.3 ASpectrumofCollidingPerturbers . ......... 72 6.3.1 γ < 2 ........................................ 72 6.3.2 γ =2 ........................................ 73 6.3.3 2 <γ< 3...................................... 73 6.3.4 CollisionalPerturbations . ....... 74 6.3.5 EccentricityDistributions . ........ 75 6.4 KuiperBeltBinaries .............................. ...... 76 6.4.1 PerturbationsbyaDisk . .... 76 6.4.2 Plutoetal. .................................... 77 6.4.2.1 OrbitalModelofTholenetal. .. 78 6.4.2.2 ADifferentInterpretation. ... 78 vii 6.4.2.3 TheoreticalDistribution. .... 81 6.4.3 OtherInterestingKBOs . .... 83 6.5 OtherBinarySystems .............................. ..... 84 6.6 Conclusions ..................................... .... 85 6.7 Appendix ........................................ .. 87 7 A Unified Theory for the Effects of Stellar Perturbations and Galactic Tides on Oort Cloud Comets 89 7.1 ASingleStellarPassage . ....... 90 7.2 L´evyFlightBehavior. ....... 91 7.3 ConnectiontoGalacticTides . ........ 94 7.4 Conclusions ..................................... .... 101 Bibliography 102 viii List of Figures 2.1 RMS errors in the orbital phase of a test particle with e = 0.25 integrated with RKNB3D and Mercury, using 100 and 300 force evaluations per orbit ........ 14 2.2 RMS error in the orbital phase of a test particle with e =0.5 integrated with RKNB3D and Mercury, using 300 force evaluations per orbit. ............ 16 2.3 RMS error in the eccentricity vector in integrations of two small protoplanets inte- grated with RKNB3D with an average of 1 and 7 force evaluations per orbit; the relative change in total energy of the system in the integration with RKNB3D and withMercury........................................ 17 3.1 Change in semimajor axis after a conjunction of two bodies on initially circular orbits asafunctionoftheinitialseparation. ......... 22 3.2 Histogram of the intragroup and intergroup separations between protoplanets in nu- merical simulations with different initial spacings. ............... 26 3.3 Semimajor axes of protoplanets over time in a simulation of oligarchic growth. 28 3.4 Final N of simulations against the initial N for Σ/σ =0.1. ............ 30 h i h i 3.5 Final average mass ratio, µ , of the protoplanets plotted against the final N for h i h i the ratio of surface densities of Σ/σ =0.1......................... 31 3.6 Equilibrium average co-orbital number N plotted against the surface mass density h ieq ratio of protoplanets to planetesimals, Σ/σ........................ 32 3.7 Equilibrium average spacing between co-orbital groups, x for simulations with h H ieq N = N plotted against the surface mass density ratio Σ/σ............ 33 h ii h ieq 3.8 Average mass of a protoplanet in an equilibrium oligarchy as a function of the surface mass density ratio Σ/σ at a =1AU............................ 34 4.1 Analytic distribution function of protoplanet eccentricities compared with the distri- bution function measured from a numerical simulation and a Rayleigh distribution. 48 4.2 Distribution function of protoplanet eccentricities in a disk with a bimodal mass distribution....................................... ... 50 5.1 Eccentricity distribution of a disk of planetesimals and no dynamical friction at three differenttimes.