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SPH Simulation of the Formation and Evolution of Protoplanetary Disks SPH Simulation of the Formation and Evolution of Protoplanetary Disks Annabel Cartwright A thesis submitted to Cardiff University, for the degree of Doctor of Philosophy September 2006 UMI Number: U584879 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI U584879 Published by ProQuest LLC 2013. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 Acknowledgements I am so grateful to Professor Anthony Whitworth for giving me the chance, rather late in life, to discover whether I can do scientific research, and for his patience and good humour when I repeatedly went off in orthogonal directions to those he had in mind. His extremely businesslike approach to supervision kept me going. I also acknowledge the financial support of PPARC. I would also like to thank the cohabitants of my office, particularly Simon Goodwin, Dimitris Stamatellous, Rhianne Attwood and David Hubber, who have helped me through many a sticky patch. Thanks to Spiros Kitsionas for asking the question which made me think about whether clustering analysis could be performed on grey scale data. Finally, this would have been impossible without the enthusiastic encouragement of my husband Joe Cartwright, and his assistance with everything from childcare to in- depth discussions of fluid dynamics. Abstract The formation and evolution of protoplanetary disks is simulated by computer modelling, using the Smoothed Particle Hydrodynamics (SPH) method. The suitability of SPH for modelling disks is investigated, and problems are identified with the SPH implementation of Artificial Viscosity in disks with Keplerian velocity profiles. Analytical and experimental results reveal that the resultant viscous force for a simu­ lated Keplerian disk is in the opposite direction to that produced by linear shear. Applying Artificial Viscosity only to approaching particles results in a radial force four times larger than the force in the direction of the orbit. The viscous force can change direction if temperature, and therefore sound speed, decreases. Techniques for activating Artificial Viscosity only when convergence is detected are found to fail in differentially rotating disks. Both the Balsara Switch and Time Dependent Artificial Viscosity use the SPH estimate of V • v \s p h , which has a low freqency time varying component which is independent of h, and so cannot be removed by increasing the number of SPH particles. An alternative method, based on pattern recognition, is shown to reduce the viscous spread of a differentially rotating ring by an order of magnitude. We also identify problems associated with the gravitational field of disks. The use of an annulus to represent a portion of a much larger, continuous disk, may yield unrepresen­ tative results. The edge effects can cause preferential accretion zones, where the Toomre Q parameter is not the same as it would be for the same region of an extended disk. SPH simulations of Protoplanetary disks produce condensations which do not persist long enough to collapse. The high tidal shearing forces in a Keplerian accretion disk disrupt the condensations before they accumulate enough mass to collapse. Including a more realistic treatment of the thermal physics, and reducing the effective shear viscosity, makes the situation worse. DECLARATION This work has not previously been accepted in substance for any degree and is not being concurrently submitted in candidature for any degree. Signed ................ Date RfeA... STATEMENT 1 This thesis is the result of my own investigations, except where otherwise stated. Other sources are acknowledged by explicit references. A bibliography is appended. Signed r , Date . 3s .... .. STATEMENT 2 I hereby give consent for my thesis, if accepted, to be available for photocopying and for inter-library loan, and for the title and summary to be made available to outside organisations. Signed D ate.......)3CY>.. .Qik A.W.... ..- Contents 1 Introduction 1 1.1 Introduction and Thesis P la n ......................................................................... 1 1.2 Mathematical Conventions and Definitions and Physical Constants ........... 4 2 The evolution of circumstellar disks. 5 2.1 Introduction ......................................................................................................... 5 2.2 Observational evidence for protoplanetary disks and extra-solar planets. 5 2.3 Analytical predictions for protoplanetary disks .............................................. 7 2.4 Simulations of protoplanetary d is k s ................................................................ 8 2.5 Accuracy of the SPH simulation m ethod ....................................................... 12 2.6 Conclusion ............................................................................................................ 13 3 Smoothed Particle Hydrodynamics 15 3.1 Introduction ......................................................................................................... 15 3.2 SPH Operation ................................................................................................... 16 3.2.1 Initialisation .............................................................................................. 16 3.2.2 Advancing the particles : the equation of motion ............................. 17 3.2.3 O utput ....................................................................................................... 17 3.3 Treebuilding ......................................................................................................... 18 3.4 Calculating h,p and pressure forces ................................................................. 21 3.4.1 The kernel function ................................................................................. 23 3.4.2 Energy equation ........................................................................................ 23 3.5 Gravity calculations .............................................................................................. 24 3.5.1 Gravity Softening .................................................. 25 3.5.2 Tree Gravity .............................................................................................. 26 3.6 Multiple Particle Time Stepping ....................................................................... 28 3.7 Time trials............................................................................................................. 28 3.8 Sink P articles ..................................................................................................... 29 i CONTENTS 3.9 Parameters ............................................................................................................. 29 3.10 Conclusion ................................................................................................................ 29 4 Disk Configuration 31 4.1 Disk initialisation ................................................................................................... 31 4.2 Self gravitating d is k s ............................................................................................. 32 4.2.1 The effect of density gradient on the rotation c u r v e ........................... 33 4.2.2 Outer edge effects........................................................................................ 36 4.2.3 Inner edge effects ........................................................................................ 37 4.2.4 Combining disks and central point m asses ........................................... 39 4.2.5 The effect of density gradients on the Toomre Qparam eter .................... 40 4.3 Disk Settling .......................................................................................................... 42 4.4 Setting the height of particles ............................................................................. 47 4.5 Setting the disk r o ta tin g ...................................................................................... 48 4.6 Conclusion ................................................................................................................ 48 5 Artificial Viscosity 51 5.1 Artificial Viscosity in S P H ................................................................................... 51 5.2 Velocity variation in linear and Keplerian shear flow ...................................... 52 5.3 Artificial Viscosity: Keplerian shear flo w ......................................................... 56 5.4 Artificial Viscosity: Linear shear flow ................................................................ 59 5.5 Numerical simulations of Artificial Viscosity in Linear and Keplerian Shear flow .......................................................................................................................... 61 5.6 Including all neighbours in the calculation of Artificial Viscosity ....................63 5.7 The inclusion of the term in /r2 64 5.8 Conclusion ...............................................................................................................
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