Dissertation submitted to the Combined Faculties for the Natural Sciences and for Mathematics of the Ruperto-Carola University of Heidelberg, Germany for the degree of Doctor of Natural Sciences presented by Diplom-Physiker Patrick Glaschke Born in Erbach, Germany Oral examination: 24. May 2006 Studying the Formation of Protoplanets: A new Hybrid Code for Planetesimal Dynamics Referees: Prof. Dr. Rainer Spurzem Prof. Dr. Werner M. Tscharnuter Zusammenfassung | Abstract Diese Arbeit pr¨asentiert neuartige Simulationen zur Entstehung von Planeten, wobei der Schwer- punkt auf der Entstehung von Protoplaneten aus Planetesimalen liegt. In diesem Stadium wird die Entwicklung des System von Kollisionen und Zwei–K¨orper{Streuungen bestimmt, die eine sorgf¨altige Modellierung der Entwicklung der Geschwindigkeitsdispersion und der Gr¨oßenverteilung ub¨ er den ganzen Gr¨oßenbereich erfordern. Zu diesem Zweck wurde ein neues Hybridprogramm entwickelt, genannt Nbody6Disc, um den vollen Gr¨oßenbereich von kilometergroßen Planetesimalen bis zu Protoplaneten abzudecken. Es vereint die Vorteile eines N–K¨orper{Ansatzes zur Integration der gr¨oßeren K¨orper mit einem zus¨atzlichen statistischen Modell zur Beschreibung der kleineren Planetesimale. Dieses Programm wird auf die Entstehung von Protoplaneten angewendet. Die Simulatio- nen zeigen kein Beenden der protoplanetaren Akkretion durch die Entstehung von Luc¨ ken in der Planetesimalverteilung, sondern nur schwache Fluktuationen in der Oberfl¨achendichte. Darub¨ er hinaus sind diese Strukturen nur schwach mit den Positionen der Protoplaneten korreliert. Die Untersuchung verschiedener Schlagfestigkeiten zeigt, daß die Fragmentation von Planetesimalen haupts¨achlich den gesamten Massenverlust kontrolliert, der aber w¨ahrend des fruhen¨ Wachstums der Protoplaneten relativ gering ist. Es zeigt sich, daß Fragmentation in Verbindung mit dem schnellen Verlust der Fragmente durch Gasreibung eine obere Grenze fur¨ die Masse der Proto- planeten in Abh¨angigkeit vom Abstand zum Zentralstern setzt. This work presents novel simulations of the formation of planets, with a focus on the formation of protoplanets out of planetesimals. The evolution of the system at this stage is driven by encounters as well as direct collisions, and requires a careful modelling of the evolution of the velocity dispersion and the size distribution over the whole size range. To serve this purpose, a new hybrid code called Nbody6Disc has been developed to cover the full size range from kilometre-sized planetesimals to protoplanets. It combines the advantages of a pure N{body approach for the integration of the larger bodies with an additional statistical model to cover the smaller planetesimals. This new code is applied to the formation of protoplanets. The simulations show no termination of the protoplanetary accretion due to gap formation, since the distribution of the planetesimals is only subjected to small fluctuation. Moreover, these features are weakly correlated with the positions of the protoplanets. The exploration of different impact strengths indicates that frag- mentation mainly controls the overall mass loss, which is less pronounced during the early runaway growth. It is shown that fragmentation in combination with the effective removal of collisional fragments by gas drag sets an universal upper limit of the protoplanetary mass as a function of the distance to the host star. v Love is an act of endless forgiveness, a tender look which becomes a habit. Peter Ustinov (1921{2004) vii Contents 1 Introduction 1 2 Planet Formation 5 2.1 Early Stages . 5 2.2 Formation of Planetesimals . 5 2.3 Formation of Protoplanets . 7 2.4 Oligarchic Growth . 7 2.5 Migration . 8 2.6 Summary . 9 3 Theory 11 3.1 Solar Nebula . 11 3.2 Kepler Orbits . 12 3.3 Hill's Problem . 13 3.4 Protoplanet Growth . 15 4 Integrator 19 4.1 NBODY6++ . 19 4.2 Individual Time Steps . 20 4.3 Ahmad{Cohen Neighbour Scheme . 21 4.4 Hermite Scheme . 21 4.5 Hermite Iteration . 22 4.6 Extended Hermite Scheme . 23 4.7 KS{Regularisation . 24 4.8 Additional Forces . 24 4.8.1 Central Potential . 25 4.8.2 Drag Force . 25 4.8.3 KS{Pairs . 26 5 Optimising the Code 27 5.1 Introduction . 27 5.2 Disc Geometry and Neighbour Scheme . 27 5.3 Optimal Neighbour Criterion . 28 5.4 Neighbour Changes . 30 5.5 Neighbour Prediction . 30 5.6 Communication Scheme . 31 5.7 Block Size Distribution . 32 5.8 Optimal Neighbour Number . 34 ix CONTENTS CONTENTS 6 Collisional Model 37 6.1 Introduction . 37 6.2 Concepts . 37 6.3 Theory . 38 7 Collisional Cascades 41 7.1 Introduction . 41 7.2 Self{similar Collisions . 41 7.3 Size{dependent Strength . 43 7.4 Perturbation of Equilibrium . 44 7.5 Migration and Collisions . 47 7.6 Coagulation . 49 7.7 Models for Mred . 50 8 Statistical Model 51 8.1 Fokker{Planck Equation . 51 8.2 Distribution Function . 52 8.3 Dynamical Friction . 53 8.4 High Speed Encounters . 54 8.5 Low Speed Encounters . 57 8.6 Distant Encounters . 58 8.7 Gas Damping . 58 8.8 Unified Expressions . 59 8.9 Inhomogeneous Disc . 59 8.10 Diffusion Coefficient . 61 8.11 Coagulation Equation . 62 8.12 Collisional Damping . 63 8.13 Correlation . 64 8.14 Discretisation . 66 8.15 Integrator . 67 9 Hybrid Code 69 9.1 Introduction . 69 9.2 Mass Transfer . 70 9.3 Disc Excitation . 71 9.4 Pseudo{Force . 71 9.5 Spatial Structure . 72 9.6 Transition Mass . 73 9.7 Boundary Conditions . 74 9.8 Validating the Code . 75 9.8.1 Energy Balance . 75 9.8.2 Coagulation Equation . 77 9.8.3 Complete Code Test . 77 9.8.4 Statistical Code . 81 10 Results 83 10.1 Initial Conditions . 83 10.2 Simulations . 84 10.2.1 Fragmentation Models . 85 10.2.2 Spatial Distribution . 86 10.2.3 Resolution . 93 10.2.4 Surface Density . 94 11 Discussion 101 x CONTENTS CONTENTS A Symbols 105 B Central Force { Derivatives 107 C Plummer Model 109 D Scalable Collisions Flux 111 E Coagulation Equation 113 F Computing Facilities 115 G Rendering Planetesimals 117 H Files 119 I New Subroutines 121 J New Variables 123 Bibliography 124 Acknowledgments / Danksagung 133 xi List of Figures 2.1 Initial planetesimal disc . 6 2.2 Evolved planetesimal disc . 8 3.1 Hill sphere . 15 4.1 Neighbour sphere . ..