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General Relativistic Collapse of Rotating Stellar Cores in Axisymmetry Technische Universitat¨ Munchen¨ Max-Planck-Institut fur¨ Astrophysik General Relativistic Collapse of Rotating Stellar Cores in Axisymmetry Harald Dimmelmeier Vollst¨andigerAbdruck der von der Fakult¨atf¨urPhysik der Technischen Universit¨atM¨unchen zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigten Dissertation. Vorsitzender: Univ.-Prof. Dr. Franz von Feilitzsch Pr¨uferder Dissertation: 1. Priv.-Doz. Dr. Ewald M¨uller 2. Univ.-Prof. Dr. Manfred Lindner Die Dissertation wurde am 14.09.2001 bei der Technischen Universit¨at M¨unchen eingereicht und durch die Fakult¨atf¨urPhysik am 31.10.2001 angenommen. Max-Planck-Institut fur¨ Astrophysik General Relativistic Collapse of Rotating Stellar Cores in Axisymmetry Harald Dimmelmeier Dissertation Technische Universitat¨ Munchen¨ Physik-Department Abstract Numerical studies of core collapse supernovae are an important field of research in astro- physics. They are concerned with such diverse aims as the investigation of the collapse dynamics, the formation of a high-density neutron star, the propagation of shock fronts and heavy element nuclear synthesis in the shock, the explanation of the light curves, the cooling, spin behavior and oscillation modes of the neutron star, pulsar physics and the interaction of the shock with the interstellar medium and the fate of the supernova remnant. Many of these aspects can be modeled using Newtonian gravity. On the other hand, core collapse of ideal fluids has always been a playground for numerical general relativity. However, numerical simulations of relativistic matter flows which evolve in the presence of strong (and dynamic) gravitational fields are a highly complex problem. To make things worse, the most commonly used formulations of numerical relativity lack long-term stability. Therefore, up to date no numerical simulations of rotational core collapse to a neutron star have been undertaken in general relativity. In this thesis we present a new approach to this problem: In order to simplify the com- plexity of the gravitational field equations of general relativity, Wilson and coworkers have proposed an approximation scheme, where the three-metric γij is chosen to be conformally flat. This reduces the Einstein equations to a set of 5 coupled elliptic equations. We have adopted this approximation for the equations of spacetime, and have combined it with a modern high-resolution shock-capturing scheme to solve the hyperbolic conservation equa- tions for relativistic hydrodynamics, with an equation of state consisting of a thermal and a polytropic contribution. Here we introduce an axisymmetric general relativistic hydrodynamic code which is based upon this approach. We have applied this code to simulations of rotational core collapse. A comprehensive set of tests demonstrates the ability of the code to handle a variety of astrophysical situations, including, among others, the propagation of highly relativistic shocks and the evolution of rapidly differentially rotating neutron stars in equilibrium. We have performed a parameters study of rotational core collapse and have computed the gravitational radiation waveforms for each model. These results extend previous work on rotational supernova core collapse and the resulting gravitational radiation in Newtonian gravity. In most cases, compared to Newtonian simulations the gravitational wave signal is weaker and its spectrum exhibits higher average frequencies, as the inner core is more compact in the deeper gravitational potential in general relativity. The computed wave templates will be useful in the data analysis of the gravitational wave interferometer detectors which are scheduled to become operational in the near future. We present models where relativistic effects qualitatively change the collapse dynamics. We show that in relativistic gravity, core collapse with multiple bounces is only possible for a narrow range of parameters. We further demonstrate that the prospects for detection of gravitational wave signals from supernova core collapse are most likely not enhanced by taking into account relativistic gravity. i ii Contents 1 Introduction 1 1.1 History of Supernova Research . 1 1.2 Supernova Classes . 3 1.3 Physics of Supernova Core Collapse . 5 1.4 Gravitational Radiation . 12 1.5 Simulations of Supernova Core Collapse . 16 1.6 Organization of the Thesis . 19 1.7 Conventions . 20 2 General Relativistic Hydrodynamics 21 2.1 Relativistic Field Equations { 3 + 1 Formalism . 21 2.2 Conservation Equations . .f . .g . 25 2.3 The General Relativistic Riemann Problem . 27 3 Metric Equations 31 3.1 ADM Evolution and Constraint Equations . 33 3.2 Problems with the ADM Equations { Alternative Formulations and Approxima- tions . 34 3.3 Conformally Flat Metric Equations . 37 3.3.1 Conformal Flatness Condition . 37 3.3.2 Conformal Scaling . 40 3.3.3 Derivation of the CFC Metric Equations . 42 3.3.4 Properties of the Elliptic System of Metric Equations . 46 4 Rotational Core Collapse 47 4.1 Physical Model . 47 4.2 Equation of State . 50 4.3 Rotating Relativistic Stars in Equilibrium . 52 4.4 Initial Models and Collapse Parameters . 54 5 Numerical Implementation 57 5.1 Numerical Grid . 57 5.1.1 Grid Setup { Logarithmic Radial Spacing . 57 5.1.2 Symmetry Conditions . 59 5.1.3 Boundary Conditions { Isotropic Schwarzschild Solution . 61 5.2 Hachisu's Self-Consistent Field Method . 63 5.3 Atmosphere Treatment at the Surface of the Star . 64 iii 5.4 High-Resolution Shock-Capturing Methods . 66 5.4.1 Approximate Solvers for the Riemann Problem { The Numerical Flux . 68 5.5 Iteration Scheme { Discretization of the Evolution Equations . 71 5.5.1 Time Evolution of the Conserved Quantities . 71 5.5.2 Recovery of the Primitive Quantities . 78 5.6 Numerical Solution of the Elliptic Metric System . 79 5.6.1 Finite Differencing { Newton{Raphson Iteration . 79 5.6.2 The Linear Problem . 82 5.6.3 Numerical Methods for Solving the Linear Problem . 87 5.6.4 Computational Performance of the Metric Solver { Metric Extrapolation 91 6 Tests 97 6.1 Relativistic Shock Tube Tests . 97 6.2 Rotating Neutron Stars . 101 6.3 Spherical Core Collapse . 106 6.4 Integral Quantities { Conservation of Rest Mass and Angular Momentum . 109 6.5 Excitation of the Thermal Pressure . 115 6.6 Quality of the CFC Approximation during Rotational Core Collapse . 118 6.7 Convergence and Accuracy Tests . 125 6.7.1 Order of Numerical Convergence . 125 6.7.2 Grid and Metric Resolution Tests { Metric Extrapolation Tests . 127 7 Results and Discussion 131 7.1 Collapse and Waveform Types . 131 7.1.1 Type I Collapse Model { Regular Collapse . 132 7.1.2 Type II Collapse Model { Multiple Bounce Collapse . 134 7.1.3 Type III Collapse Model { Rapid Regular Collapse . 134 7.2 Homology of the Infall . 139 7.3 Central Densities and Time of Bounce . 142 7.4 Gravitational Wave Amplitudes . 147 7.5 Multiple Bounces . 151 7.6 Compactness of the Proto-Neutron Star Core . 161 7.7 Rapidly and Highly Differentially Rotating Models . 174 7.8 Gravitational Wave Energy Emission and Spectra . 182 7.9 Evolution of the Rotation Rate . 191 7.9.1 Problems with the Calculation of the Rotation Rate . 199 7.10 Propagation of the Shock Front . 200 8 Summary and Outlook 207 A Characteristic Fields 211 A.1 Characteristic Structure of the Conservation Equations . 211 A.2 Eigenvalues and Eigenvectors . 212 B Source Terms for the Evolution Equations 215 B.1 Energy-Momentum Tensor . 216 B.2 Christoffel Symbols . 217 iv C Gravitational Wave Extraction 221 C.1 Multipole Expansion of the Radiation Field . 221 C.2 Standard Quadrupole Formula . 223 C.3 First Moment of Momentum Density Formula . 224 C.4 Stress Formula . 225 C.5 Ambiguities of the Quadrupole Formula in Relativity . 230 D Density Evolution and Gravitational Wave Signal Catalogue 235 E Acknowledgments 243 References 244 v vi We are all in the gutter, but some of us are looking at the stars. Oscar Wilde, Lady Windemere's Fan (1892). It's the nexus of the crisis And the origin of storms Just the place to hopelessly Encounter time and then came me ... Astronomy { a star Blue Oyster¨ Cult, Astronomy (1974). Ich bin kein Freund großer Worte... Harry Dee. vii viii Chapter 1 Introduction 1.1 History of Supernova Research Rarely occurring spectacular phenomena in the sky have always been fascinating for human beings. Among the most exciting of these are supernovae, although only a few human beings ever have had the opportunity to see one with the unaided eye. In historical times, the ap- pearance of new stars (or guest stars) has often been attributed to some heavenly omen which was supposed to influence mankind's fate for better, or mostly worse [91]. So the director of the Chinese Imperial Astronomical Bureau, Yang Wei-te, who observed the supernova of 1054, the one which should leave behind the much studied Crab nebula, was smart enough to ensure his employer that this sign in the sky could be deciphered as a promise that a person of great wisdom and virtue was to be found within his realm. He certainly did not want to share the ill fate of his two famous predecessors Hsi and Ho who had been beheaded for not being able to fulfil the Emperor's expectations. A few centuries later it was sheer coincidence that the emergence of modern Western science and especially astronomy in the Renaissance period was accompanied by the appearance of two supernovae visible in Europe, and that two of the most famous astronomers ever, Tycho Brahe and Johannes Kepler, each witnessed one of them. They were both, in succession, court astronomers and astrologists of King Rudolph II of Bohemia { a combination of professions which has somewhat gone out of fashion in later times. However, at those times of philosophical and scientific clashes between the geocentric and heliocentric view of the Universe, at least among professionals, the notion of a heavenly phenomena as indicators for famine and other catastrophes (in ancient Japan, astronomers had interpreted the supernova of 1181 as a sign of abnormality, indicating that at any moment we can expect control of the administration to be lost [91]) had already yielded to the idea of scientifically interesting objects of study.
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