Numerical Simulations of Neutron Star - Black Hole Mergers by Frank L¨offler Max–Planck–Institut f¨ur Gravitationsphysik Albert–Einstein–Institut Universit¨at Potsdam thesis supervisor: Professor Bernard F. Schutz October 2005 To my parents. Contents Acknowledgments 9 1 Overview 11 1.1 Theblackhole/neutronstarproblem . 11 1.2 Previousresults .......................... 13 1.3 Outline............................... 14 1.4 Conventionsandunits . 15 2 Physical foundations 17 2.1 GeneralRelativity . .. .. 17 2.2 Analyticsolutions. 19 2.2.1 Minkowski......................... 20 2.2.2 Schwarzschild . .. .. 20 2.2.3 Kerr ............................ 21 2.3 Neutronstars ........................... 22 2.3.1 Equationsofstateforneutronstars . 23 2.3.2 Non-perfectfluids. 24 2.3.3 PerfectFluid ....................... 24 2.3.4 Polytropicequationofstate . 25 2.4 The3+1Formalism........................ 25 2.5 InitialData ............................ 28 2.5.1 TOVstars......................... 29 2.5.2 Michelsolution . .. .. 30 2.5.3 The York-Lichnerowicz conformal decomposition . 32 2.6 Evolution ............................. 35 2.6.1 The evolution system: BSSN . 35 2.6.2 Boundaryconditions . 39 2.6.3 TheRiemannProblem . 40 2.6.4 The Valencia formulation for relativistic hydrodynamics 41 2.7 Gaugeconditions ......................... 43 2.8 Summary ............................. 44 5 6 CONTENTS 3 Discretisation 47 3.1 Finite difference and finite volume methods . 48 3.1.1 TheCFLcondition . 49 3.1.2 Cactus / CactusEinstein ............... 50 3.1.3 Mesh refinement - Carpet ................ 50 3.2 SpectralMethods ......................... 51 3.3 Themethodoflines........................ 52 3.4 Hydrodynamical discretisation . 52 3.4.1 Reconstructionmethods . 54 3.4.2 Riemannsolvers. 59 3.4.3 Treatmentoftheatmosphere . 60 3.4.4 Shocktubetests ..................... 61 3.4.5 Whisky ........................... 61 3.5 Excision applied to spacetime variables . 63 3.6 Convergencetests. .. .. .. 65 3.7 Numericaltestswithstars . 66 3.8 Summary ............................. 69 4 Excision methods for HRSC schemes 71 4.1 Motivation............................. 71 4.2 Modelandequations . .. .. 72 4.3 Numericalmethods .. .. .. 72 4.3.1 MPPM........................... 73 4.4 Excisionboundaries. 74 4.4.1 Slope-limitedTVD . 77 4.4.2 ENO............................ 77 4.4.3 PPM............................ 78 4.4.4 MPPM........................... 78 4.5 Numericaltests .......................... 78 4.5.1 Shockwavetests . .. .. 79 4.5.2 Michelsolution . .. .. 80 4.5.3 NeutronStarcollapse. 82 4.6 Summary ............................. 86 5 3D simulations of rotating NS collapse 87 5.1 Motivation............................. 87 5.2 Basic equations and their implementation . 91 5.3 Initialstellarmodels . 92 5.4 Dynamicsofthematter . 94 5.4.1 Slowly rotating stellar models . 98 5.4.2 Rapidlyrotatingstellarmodels . 100 CONTENTS 7 5.4.3 Disc formation and differential rotation . 108 5.5 Dynamicsofthehorizons. .112 5.5.1 Measuring the Event-Horizon Mass . 112 5.5.2 Measuring the angular momentum of the black hole . 116 5.5.3 Black-Hole Mass from the Christodoulou Formula . 120 5.5.4 Reconstructing the global spacetime . 123 5.6 Summary .............................125 6 Black hole - neutron star systems 129 6.1 Introduction............................129 6.2 Obtainingagoodinitialguess . .130 6.3 Solving the York procedure with matter . 133 6.4 Using BAM astheellipticsolver. .135 6.5 Using TwoPunctures asellipticsolver . .135 6.5.1 TheMethod........................135 6.5.2 Results...........................139 6.6 Evolutionsofmixedbinarydata . 154 6.7 FutureWork............................172 6.8 Summary .............................175 Summary 177 Zusammenfassung 179 Bibliography 181 8 CONTENTS Acknowledgments First of all I want to thank Ed Seidel for giving me the opportunity to start my studies at the Albert-Einstein-Institute and being available for discussions despite his very busy schedule. I would like to thank Bernard Schutz and the Max-Planck-Society for general support. Most of the numerical and hydrodynamical things and tricks I have learned from Ian Hawke, who is a great tutor in many ways. I also thank him for the massive and excellent proof-reading of this thesis. Denis Pollney re- ceives my thanks for, among other things, his support as head of our research group, for help with any problems and little issues and also for proof-reading. I would like to thank all the people I have worked with and whose pro- grams I was allowed to use. Enumerating them would generate a far too long list. Nevertheless some people stand out for various smaller and bigger reasons. They are, in alphabetical order, Marcus Ansorg for very useful dis- cussions and permission to use and modify parts of his work, Luca Baiotti and Christian Ott for their amazingly extensive proof-reading of this the- sis, Thomas Radke for his assistance in Cactus, compiler and OpenDX issues, Luciano Rezzolla for his help with any upcoming physics-question, for his invitations to SISSA, his support with a fellowship of EGO and for carefully proof-reading this thesis, Erik Schnetter for all his advice concerning Carpet in special and Cactus in general, Jonathan Thornburg for all the time he has spent in answering my numerous big and little questions and also for proof- reading my thesis and Steve White for hacking our terrifying cvs-scripts, help and insight into CSS and nice discussions in between. The results of this thesis have been obtained by using computer time allocations at the Albert Einstein Institute (AEI, Germany), the National Center for Supercomputing Applications (NCSA, USA), the University of Parma (Italy), the Leibniz-Rechenzentrum M¨unchen (LRZ, Germany), the Pittsburgh Supercomputing Center (PSC, USA) and the Rechenzentrum Garching (RZG, Germany). This work was supported in part by DFG grant “SFB Transregio 7: Gravitationswellenastronomie”, by NSF grants PHY-02- 18750, PHY-02-44788 and PHY-03054842, by the MIUR and EU Programme 9 10 CHAPTER 0. ACKNOWLEDGMENTS “Improving the Human Research Potential and the Socio-Economic Knowl- edge Base” (Research Training Network Contract HPRN-CT-2000-00137), by NASA grant NNG04GL37G and by DGAPA-UNAM grants IN112401 and IN122002. Last, but surely not least, I want to thank my wife Katja for so many things, that it could fill pages. I would not have written this thesis without her. Chapter 1 Overview 1.1 The black hole / neutron star problem The behaviour of matter is not well known in very high density regimes or in very strong gravitational fields. Some reasons for this are first the technical problems to create such or similar environments on Earth or in nearby space, and second that regions where such conditions prevail are so far away that they are hard to observe with current astrophysical instruments. Prime examples of objects with very strong gravitational fields are neutron stars and black holes. Neutron stars are the collapsed remains of formerly massive, “usual” (nu- clear burning) stars with cores reaching nuclear densities (very approximately 1014g/cm3; for a more detailed discussion of the birth of neutron stars, see e.g. [1]). As a first approximation a neutron star may be thought of a very big (ca. 10km in radius) atomic nucleus. This, however, is only the simplest possible approximation. It neglects its constituents, because a neutron star does not only consist of neutrons, but also electrons, protons and possibly more exotic matter. This mix, which can be modeled as fluid, might also be a superfluid due to the high pressure. The above picture also neglects the neutron star crust (see e.g. [2–4]). This crust is a very thin lattice of neutrons at the surface of the star. However, it is, e.g., an important component for modelling pulsar glitches (see e.g. [5–7]). It also neglects any magnetic fields, which are potentially very strong and may have a crucial influence on the structure and dynamics of neutron stars. Obviously, this is not a good way to think about a neutron star in general. However, as a first approximation and in certain situations it is possible to model a neutron star in such a way (see section 2.3). There are many definitions as to what a black hole is. It may be thought 11 12 CHAPTER 1. OVERVIEW of as the end state of a stellar collapse in which no internal forces, such as a pressure force, is able to halt the collapse and prevent the formation of a singularity in spacetime [8, 9]. However, at least in the classical theory, no observer that could “see” such singularity would be able to communicate this information to another observer outside the black hole. This is called the “cosmic censorship hypothesis” [10]. The boundary of a black hole is the event horizon. Nothing can escape from within this horizon, not even light. This is also the definition of this horizon: a closed surface from which not even light can escape outwards (a more precise definition follows in section 2.2.2). In classical General Relativ- ity, it is impossible for an observer to find out what is inside and come back out. In this context, anything happening inside of a black hole is irrelevant to the exterior and therefore physically not interesting. However the interior may have mathematically interesting features and may also be relevant in other theories of gravity. Another way to look at black holes is as being collapsing stars (in the case of black holes formed by collapsing stars; there are also discussions about so called primordial black holes, which formed