ASTROPHYSICS OF COMPACT STARS: WHITE DWARFS, NEUTRON STARS, AND BLACK HOLES Cesare Chiosi UNIVERSITY OF PADOVA Department of Astronomy ROTAS OPERA TENET AREPO SATOR 2 Contents 1 PREFACE 17 2 INTRODUCTION 19 3 FUNDAMENTAL SCALES 23 3.1 Introduction . 23 3.2 Mass defect . 27 3.3 Hydrostatic Equilibrium . 28 3.4 Normal non degenerate stars . 29 3.5 Energy losses by radiation . 29 3.6 Degenerate stars . 30 4 NOTES ON GENERAL RELATIVITY 33 4.1 What is General Relativity? . 33 4.2 Motion of test particles . 40 4.3 Gravitational redshift . 46 4.4 The weak-field limit . 47 4.5 Geometrical units . 47 4.6 Spherically symmetric gravitational fields . 48 4.7 Spherical stars . 50 5 EQUATION OF STATE BELOW NEUTRON DRIP 53 5.1 Thermodynamic Preliminaries . 53 5.2 Kinetic theory of the equation of state . 56 3 4 CONTENTS 5.3 Fully degenerate (partially relativistic) gas of fermions . 64 5.4 Electrostatic corrections to the Equation of State . 68 5.5 Inverse ¯-decay: corrections to the equation of state . 74 5.5.1 The Harrison-Wheeler equation of state . 77 5.5.2 Baym-Pethick-Sutherland equation of state . 82 6 EQUATION OF STATE ABOVE THE NEUTRON DRIP 85 6.1 From ½drip to ½nuc: the BBP equation of state . 85 6.2 Nucleon-nucleon interaction . 93 6.3 Saturation of nuclear forces . 96 6.4 The Yukawa Potential . 102 6.5 Appearance of new particles . 111 6.6 Unresolved questions . 113 6.7 Modern equations of state . 120 7 POLYTROPES 123 7.1 Lane-Emden equation . 124 7.2 Collapsing polytropes . 127 8 WHITE DWARFS 133 8.1 Introduction . 133 8.2 Equation of state for the inner core . 134 8.3 Hydrostatic structure of the degenerate interiors . 135 8.4 The Chandrasekhar Limit . 140 8.5 External Layers . 142 8.6 Cooling of White Dwarfs . 145 8.7 Revision of the Mass-Radius Relationship . 152 8.7.1 Correction at low densities: electro-static interactions . 153 8.7.2 Correction at high densities: the pycno-nuclear reactions . 154 8.7.3 Correction at high densities: the inverse ¯-decay . 157 8.7.4 Correction at high densities: GR effects . 162 CONTENTS 5 8.8 Effects of Magnetic Fields . 173 8.9 Effects of Rotation . 175 8.9.1 A simple approach . 175 8.9.2 Equilibrium of Rotating Configurations: the MacLaurin Spheroids . 178 8.9.3 Revising the effects of rotation . 181 9 ORIGIN OF NEUTRON STARS 189 9.1 Introduction . 189 9.2 The onset of collapse . 189 9.3 Photo-dissociation . 192 9.4 Neutronization and neutrino emission . 197 9.5 Weak interaction theory . 199 ¡ 9.6 A simple case of electron capture: e + p ! (W) ! ºe + n . 200 9.7 Neutrino opacity and neutrino trapping . 203 9.8 Homologous core collapse and bounce . 207 9.9 Free-Fall reflection: bounce . 211 9.10 Formation of the shock way . 212 9.11 Energy Balance . 215 9.12 Effect of neutrinos . 218 10 NEUTRON STARS 223 10.1 Introductory generalities . 224 10.2 Models for neutron stars . 230 10.3 Realistic neutron star models . 235 10.4 More on the mass-density relationship . 239 10.5 The maximum mass . 240 10.6 The effects of rotation . 246 10.7 Superfluidity . 248 10.8 The Luminosity of a Neutron Star . 260 6 CONTENTS 11 COOLING OF NEUTRON STARS 263 11.1 Introduction . 263 11.2 Neutrino reactions in neutron stars . 264 11.3 Free neutron decay . 266 11.4 Modified URCA rate . 270 11.5 Other reaction rates . 274 11.6 Neutrino transparency . 277 11.7 Cooling curves . 279 12 PULSARS 283 12.1 Discovery . 283 12.2 Which source ? . 284 12.3 Observational properties of Pulsars . 285 12.4 Magnetic Dipole Model for Pulsars . 288 12.5 Braking Index . 294 12.6 Non-vacuum models: the aligned rotator . 295 12.7 Pair creation in the magneto-sphere . 305 12.8 Remark on the pulsar emission mechanisms . 307 12.9 Superfluidity: glitches and star-quakes . 309 13 BLACK HOLES 321 13.1 Simple Theory of Black Holes . 322 13.2 Schwarzschild Black Hole . 329 13.3 Motion of massive test particles . 331 13.4 Motion of massless test particles . 340 13.5 Non Singularity of the Schwarzschild Radius . 343 13.6 Rotating Black Holes: the Kerr solution . 347 13.7 The Area Theorem and Black Hole Evaporation . 359 List of Figures 2.1 Left: stars evolve towards different end states: White Dwarfs, Neutron Stars, and Black Holes. Right: Compact stars are the result of the endpoint in stellar evolution shown as a function of the initial mass. 21 2.2 The masses of Neutron Stars and astrophysical Black Holes. A Black Hole has only two Hairs, the mass and the angular momen- tum represented by the dimensionless spin parameter a. The Neu- tron Stars cluster around 1:4 M¯. Stellar Black Holes are expected to be formed with masses in the range of a few to 100 M¯. Super- massive Black Holes in the centers of galaxies grow by accretion from Black Holes formed at high redshift (very early epochs). 22 4.1 The coordinate basis vector eÁ at A is r2=r1 times longer that the basis vector at B. The lines Á= constant are supposed to be infinitesimally close. 46 4.2 Space-time diagram for gravitational redshift. The vertical lines are the world-lines of a light-ray emitter and a receiver. The dashed lines are the world-lines of two light rays emitted at a coordinate dx0 apart. Note that the coordinates are not inertial (no global inertial frames in the presence of a gravitational field). 48 7 8 LIST OF FIGURES 5.1 Left: Representative equations of state below neutron drip. The labels identify the equation of states discussed in the text: (Ch) the Chandrasekhar ideal electron gas for me = 56=26; (FMT) the Feynman- Metropolis-Teller equation of state for a n ¡ p ¡ e ideal gas; (HW) the Harrison & Wheeler equation of state; (BMP) the Bethe-Pethick-Sutherland equation of state. Right: The adiabatic index Γ = dlnP=dln½ as a function of ½ for the equations of state in the left panel . 82 6.1 Left: Bayam-Bethe-Pathick equation of state. The Harrison-Wheeler (1958) equation of state is shown for comparison. Right: The adiabatic index Γ = dlnP=dln½ as a function of ½ for the Bayam- Bethe-Pathick equation of state. 92 6.2 A plot of the potential energy W , kinetic energy T and total energy E = W + T as a function of radius R for a system of identical nucleons interacting via a purely attractive nuclear potential. 99 6.3 Concentrations nj in a free hyperonic gas a function of total bary- onic density n ............................113 6.4 The equilibrium equation of state of cold degenerate matter. The solid line shows the BPS equation of state in the region ½ · ½drip ' 4:3 £ 1011 g=cm3, matched smoothly to the BBP equation of state 14 3 in the region ½drip · ½ · ½nuc = 2:8 £ 10 g=cm . The dashed line is the Oppenheimer-Volkoff equation of state for comparison fo a free neutron gas. Representative equation of state for the region above ½nuc reside in the box and are shown in Fig.6.5 below . 121 6.5 Representative equations of state for cold degenerate matter above 14 3 ½nuc = 2:8 £ 10 g=cm . 121 7.1 Solutions for different values of the parameter ¸, in the range 0 · ¸ · ¸m. They describe homologously collapsing polytropes with n = 3. 130 LIST OF FIGURES 9 8.1 Internal structure of old C-O White Dwarfs. The core of a cool White Dwarf consists of a C-O crystallized lattice (a kind of gi- gantic diamond), surrounded by a crust consisting of He and H, and a thin H-atmosphere. 134 8.2 Left: Mass - Radius Relationship for White Dwarfs according to the Chandrasekhar theory. Right: Mass-Central-Density Rela- tionship for White Dwarfs according to the Chandrasekhar theory 139 8.3 Cooling sequences of White Dwarfs on the HR Diagram. The diagonal lines are lines of constant radius, which are labelled by the corresponding mass from the Chandrasekhar equation of state for ¹e = 2. The filled dots are observed White Dwarfs of known distances. 142 8.4 Specific heat capacity as a function of temperature (schematic drawing; ions only) . 152 8.5 Electrostatic potential governing the motion of one "incident" nu- cleus relative to an adjacent ”fixed” nucleus in one-dimensional ion crystal lattice. The ions (nuclei) are separated by a distance R0. Zero-point fluctuations (energy E0) in the harmonic potential well near the "incident" ion lattice site can lead to Coulomb barrier penetration and nuclear reactions. 155 8.6 Mass-Radius relations for White Dwarfs whose equation of state and chemical composition change from the ideal Fermi gas to a new one including the effect of inverse ¯-decay. Different equilibrium chemical compositions are considered. 160 8.7 Mass-Central Density relations for White Dwarfs whose equation of state and chemical composition change from the ideal Fermi gas to a new one including the effect of inverse ¯-decay. Different equilibrium chemical compositions are considered. 161 10 LIST OF FIGURES 8.8 Left: The square of the angular velocity along the MacLaurin and Jacobi sequences. The abscissa is the eccentricity e. Right: The angular momentum along the MacLaurin and Jacobi sequences. The abscissa is the eccentricity . 182 8.9 Shape of the potential Φ + Φc along a radial direction located in the equatorial plane (solid line) and in the polar axis (dashed line).