sign in sign up
<<
Home , Black dwarf, Urca process, White dwarf

ASTROPHYSICS OF COMPACT : DWARFS, NEUTRON STARS, AND BLACK HOLES

Cesare Chiosi

UNIVERSITY OF PADOVA

Department of

ROTAS OPERA TENET AREPO SATOR 2 Contents

1 PREFACE 17

2 INTRODUCTION 19

3 FUNDAMENTAL SCALES 23 3.1 Introduction ...... 23 3.2 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 33 4.1 What is General Relativity? ...... 33 4.2 Motion of test particles ...... 40 4.3 ...... 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 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 β-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 ρdrip to ρnuc: the BBP equation of state ...... 85 6.2 -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

7 123 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 ...... 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 : 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 β-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 emission ...... 197 9.5 Weak interaction theory ...... 199

− 9.6 A simple case of capture: e + p → (W) → νe + 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 ...... 218

10 NEUTRON STARS 223 10.1 Introductory generalities ...... 224 10.2 Models for neutron stars ...... 230 10.3 Realistic neutron models ...... 235 10.4 More on the mass- relationship ...... 239 10.5 The maximum mass ...... 240 10.6 The effects of rotation ...... 246 10.7 Superfluidity ...... 248 10.8 The of a ...... 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

12 283 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 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 ...... 329 13.3 Motion of massive test particles ...... 331 13.4 Motion of massless test particles ...... 340 13.5 Non Singularity of the ...... 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 shown as a function of the initial mass...... 21

2.2 The 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 grow by from Black Holes formed at high redshift (very early epochs). . . . 22

4.1 The coordinate basis vector eφ at A is r2/r1 times longer that the basis vector at B. The lines φ= 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 -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/dlnρ as a function of ρ 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/dlnρ as a function of ρ for the Bayam- Bethe-Pathick equation of state...... 92 6.2 A plot of the W , T and total energy E = W + T as a function of radius R for a system of identical 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 . The

solid line shows the BPS equation of state in the region ρ ≤ ρdrip ' 4.3 × 1011 g/cm3, matched smoothly to the BBP equation of state 14 3 in the region ρdrip ≤ ρ ≤ ρnuc = 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 ρnuc reside in the box and are shown in Fig.6.5 below . . . 121 6.5 Representative equations of state for cold above 14 3 ρnuc = 2.8 × 10 g/cm ...... 121

7.1 Solutions for different values of the parameter λ, in the range 0 ≤

λ ≤ λm. 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 consists of a C-O crystallized lattice (a kind of gi- gantic ), 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 µe = 2. The filled dots are observed White Dwarfs of known distances...... 142

8.4 Specific 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 to a new one including the effect of inverse β-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 β-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 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 axis (dashed line).

Horizontal lines K1 and K2 correspond to different values of the constant K in eqn.(8.216) ...... 183

9.1 Left: Path of central conditions. Right: Stratification of nuclear burning shells in a typical massive star in which the core has been just formed...... 193

9.2 The chemical composition in the interior of a highly evolved model

of 25 M¯ star of Population I. The mass concentrations of a few

important elements are plotted against the mass Mr. Below the abscissa the location of shell sources and typical values of temper- ature (in K) and density (in g/cm3) are indicated...... 193

9.3 Schematic picture of the velocity profile vr in a collapsing stellar

core originally of about 1.4 M¯. Note the two regimes: on the left the velocity increases with r; it correspond to the homologous part; on the right, the velocity decreases outward, it roughly corresponds to the free-fall regime...... 210

9.4 Behaviour of the during the formation of the shock wave...... 213 LIST OF FIGURES 11

9.5 Schematic picture of a collapsing at bounce. The ar- rows corresponds to the velocity field. At the sphere labelled core shock, the shock is formed inside which the matter is almost at rest. Above the shock there is a still collapsing shell in which neu- trinos are trapped. But on the top there is a shell from which neutrinos can escape. One can define a neutrino phot-sphere in analogy to the in a . Part of the neutrinos can be trapped in the collapsing matter thus giving the outward moving shock front enough energy to reach the Si-burning layer...... 220

10.1 Mass - central density relationships for fully degenerate White Dwarfs and Neutron Stars. The transition from the maximum

mass for White Dwarfs (∼ 1.2 M¯) to the Neutron star domain is calculated with the Harrison-Wheeler equation of state...... 231

10.2 Cross sectional slices of the 1.4 M¯ neutron stars by Reid (left and TNI (right) illustrating the various regions described in the text.

The moments of inertia of the entire crust, Ic and of the superfluid

region, Is are also given...... 236 10.3 Left: Gravitational mass vs. central density for various equations of state. The ascending portions of the curves represent stable neutron stars. Right: Gravitational mass versus radius for the same equations of state as in the left panel ...... 236

10.4 Observational determinations of neutron star masses. The most probable value for each determination is given by the filled circles. The shaded region represents the smallest range of neutron star masses consistent with all the data...... 239

10.5 The stability domain in the uniform density approximation. The

curve separating the stable and unstable regions is Γc of eqn.(10.62). The dashed line shows the adiabatic index Γ of a free neutron gas. 244 12 LIST OF FIGURES

10.6 Mimicking the rigid body rotation of a superfluid by a large number of vortex lines...... 254

− 11.1 Feynman diagrams for electron-neutrino scattering νe + e → νe + e−; (a) charged current reactions; (b) neutral current reactions . 266

11.2 Schematic neutron star cooling curves of interior temperature ver- sus time for various processes, were they operate alone ...... 280

12.1 Chart record of individual pulse for one of the first discovered pulsar PSR 0329+54. They are recorded at the frequency of 410 MHz. The pulses occur at regular intervals of about 0.714 s. . . . 286

12.2 The pulse period of PSR 0833-45 (Vela) as function of time (from 1968 to 1980). The four large jumps are enumerated and the time span in years between jumps is noted...... 287

12.3 Left: Galactic distribution of pulsars. In the adopted coordinates 0◦ latitude corresponds to the galactic plane, while 0◦ longitude, 0◦ latitude corresponds to the direction of the Galactic Center. Pulsar share the distribution of young luminous OB stars and Remnants. Right: Distribution of pulsars in Galactic latitude. Latitude 0◦ corresponds to the Galactic plane...... 288

12.4 Oblique rotating magnetic dipole ...... 289

12.5 Rotating magnetic dipole whose magnetic axis is aligned to the rotational axis...... 297

12.6 Light cylinder ...... 301 LIST OF FIGURES 13

12.7 Spatial distribution of charges in the magneto-sphere The pulsar has parallel magnetic and rotation axes. Particles that are at- tached to closed magnetic field lines co-rotate with the star and form the co-rotating magnetosphere. The magnetic field lines that pass through the light cylinder (where the velocity of co-rotation equals the velocity of light) are open and are deflected back to form a thoroidal field component. Charged particles stream out along these lines. The critical field line is at the same electric potential as the exterior . This line divides regions of positive and negative current flow from the star and the plus and minus indicate the charge of particular regions of space. The di-

agonal dashed line is the locus of Bz = 0, where the space charge changes sign...... 302

12.8 Twisting of magnetic field-lines near the light cylinder...... 303

12.9 Schematic view of a rotator whose magnetic dipole is not aligned with the rotation axis. It shows the magnetosphere of a rotating neutron star. The rotating magnetic field lines define a light cylin-

der with radius RL = c/Ω ' 50000(P/sec) km. At the light cylin- der, the dipolar magnetosphere gets distorted. Field lines from the polar caps cross the light cylinder (open field lines), generated near the polar caps can escape along these magnetic field lines. The polar caps act as a kind of discharge tubes, where strong parallel electric fields, generated by the rapid rotation, ex- tract charges from the surface of the neutron star and accelerate them to high energies ...... 304

12.10Curvature radiation and pair cascading ...... 306

12.11Time dependence of the angular velocity before and after a . 312

13.1 Illustration of light-cones at different distances r from the central

singularity, inside and outside the Schwarzschild radius rs . . . . 326 14 LIST OF FIGURES

13.2 The radial infall into a Black Hole for a test particle starting at

distance 5rs with zero velocity. Il motion is shown in terms of the particle proper time τ, and in terms of coordinate time t of an observer at infinity ...... 326 13.3 Fall from rest toward a Schwarzschild Black Hole as described (a) by a comoving observer (proper time τ and (b) a distant observer (time t). In the one description, the pont r = 0 is attained and quickly. In the other description, r = 0 is never reached and even r = 2M is attained only asymptotically...... 335 13.4 Sketch of the effective potential profile for a particle with non-zero rest mass orbiting a Schwarzschild Black Hole of mass M. The three horizontal lines labelled by different values of E˜2 correspond to (1) unbound, (2) capture, and (3) bound orbit, respectively. . . 336 13.5 The effective potential profile for non-zero rest.mass particles of various angular momenta ˜l orbiting a Schwarzschild Black Hole of mass M. The dots at the local minima locate radii of static √ circular orbits. Such orbits exist only for ˜l > 2 3M...... 337 13.6 Impact parameter b for particle with trajectory r = r(φ) about mass M...... 339 13.7 Sketch of the effective potential profile for a particle with zero rest mass (photons) orbiting a Schwarzschild Black Hole of mass M. If √ the particle falls from r = ∞ with impact parameter b > 3 3M √ it is scattered back out to r = ∞. If, however, b < 3 3M the particle is captured by the Black Hole...... 342 13.8 (a) The angle ψ between the propagation direction of a photon and the radial direction at a given point P . (b) Gravitational capture of radiation by a Schwarzschild Black Hole. Rays emitted from each point into the interior of the shaded conical cavity are captured. The indicated capture cavities are those measured in the ortho-normal frame of the local static observer...... 343 LIST OF FIGURES 15

13.9 Kruskal diagram of the ...... 345 13.10Kruskal diagram of the Schwarzschild metric showing the relation between Schwarzschild (t, r) and Kruskal (u, v) coordinates . . . 346 13.11Kruskal diagram for . Only the unshaded region is physically meaningful, The remainder of the diagram must be replaced by the space-time geometry inside the star (Black Hole)...... 347 13.12Event horizons, singular ring, ergo-sphere of the . . . 350 13.13Ergo-sphere of a Kerr Black Hole: the region between the static limit (the flattened outer surface with r = M + (M 2 − a2cos2θ)1/2) and the , i.e. the inner sphere with radius r = M + (M 2 − a2)1/2...... 351 13.14Spatial diagram of the Kerr solution on the equatorial plane. Note the progressive change in the inclination of the light cones. . . . 354 16 LIST OF FIGURES Chapter 1

Preface

These lecture notes are intended to provide students of the Master and the PhD Programs in Astronomy at the University of Padova a complete support to the Course on ”Astrophysics of Compact Objects: White Dwarfs, Neutron Stars and Black Holes”. The course requires the knowledge of elementary stellar evolution theory, in particular as far as the structure and evolution of the progenitor stars of White Dwarfs, Neutron Stars and Black Holes are concerned. Writing these notes, I have extensively used several classical text books on the same subject, very often literally taking the original text, however presenting the material either in different order or in a more concise fashion. The books I refer to, which are highly recommended to students for consultation and deeper reading, are 1. ”Black Holes, White Dwarfs and neutron Stars. the physics o compact objects”, Stuart L. Shapiro & Saul A. Teukolsky, 1983, John Wiley & Sons, Inc. 2. ”Compact objects in astrophysics: White Dwarfs, Neutron Stars and Black Holes”, Max Camenzind, 2007, Springer, Berlin 3. ”Theoretical Astrophysics”, Thanu Padmanabhan, 2001, Cambridge Uni- versity Press 4. ” and Evolution”, Rudolf Kippenhahn & Alfred Weigert, 1989, Springer-Verlag, Berlin, Heidelberg, New York 5. ”Quarks, , and the Big Bang2, Jonathan Allday, 1999, Institute of

17 18 CHAPTER 1. PREFACE

Physics Publishing, Bristol & Philadelphia 6. ”Weisse Zwerge, Neutronensterne und Schwarze L¨ocker: Physik kompak- ter astrophysikalischer Objecte”, 2005, Ewald M¨uller,Max-Planck Institut f¨ur Astrophysik, Garching bei M¨unchen, Germany

Finally these lecture notes are for internal use only.

Padova, January 2008 Chapter 2

Introduction

As a class of astronomical objects, compact objects include White Dwarfs, neu- tron stars, and Black Holes. They are the endpoints of stellar evolution and they are today important constituents of galaxies. Our is populated by bil- lions of White Dwarfs, a few hundred million Neutron Stars and probably a few hundred thousand Black Holes. Of all these objects only Black Holes can appre- ciably grow in mass. In the form of supermassive Black Holes, these objects live in practically every center of a galaxy. Our Galaxy harbor a Black Hole of about 6 3.8 × 10 M¯, whereas M87 in the Virgo Cluster encloses a Black Hole of about 9 3 × 10 M¯. These supermassive Black Holes are the most extreme objects found in the . While Neutron Stars and Black Holes have been only recently discovered by means of their companion radio and X-ray emission, White Dwarfs have already been detected about a century ago by their optical emission. Compact stars present very extreme physical conditions and therefore are unique laboratories of Fundamental Physics and General Relativity. White Dwarfs are the end product of the evolution of low and intermediate mass stars that past core burning cannot proceed to the next nuclear step, i.e. -burning, but losing the whole envelope liberate the central core made of Carbon and . There is also a parallel channel in which low mass stars cannot reach the stage of central Helium burning, but losing the whole envelope (by mass removal in a binary system) liberate the cental Helium

19 20 CHAPTER 2. INTRODUCTION core. In both cases the density are so high that are strongly degenerate. The study of compact stars begins with the discovery of White Dwarfs and the successful description of their properties by the Fermi-Dirac statistics, assuming that they are held up against gravitational collapse by the degeneracy pressure of the electrons, an idea first proposed by Fowler in 1926, and thoroughly ex- plored by Chandrasekhar who discovered that a maximum mass for white dwarfs ought to exist due to relativistic effect. He realized that, if the White Dwarf was sufficiently massive, the relativistic electrons will become relativistic thereby rendering the star susceptible to further gravitational collapse. Chandrasekhar deduced that a White Dwarf would undergo gravitational collapse if the mass exceeds MCh ' 1.4 M¯. However, White Dwarf have still unprocessed nuclear material inside, in fact most of them are made of Helium, or Carbon and Oxy- gen. This will prevent them from ending up as a black Hole, because when the gravitational instability sets in at MCh, this will cause explosive nuclear burning and complete disruption of the star.

In 1932 Chadwick discovered the neutron. Immediately the ideas formulated by Fowler for electrons were generalized to neutrons. The existence of a new class of compact objects, with a large core of degenerate neutrons, was predicted, the Neutron Stars. The first models of Neutron Stars were developed by Oppenheimer & Volkoff and Tolman in 1939. Their calculations have shown the existence of a maximum mass, like in the case of White Dwarfs, above which the star is not stable and collapses to a Black Hole. They predicted a maximum mass of about

0.75 M¯ and only 30 years later the first neutron star was discovered, in fact a strange object pulsating in the radio range (radio pulsars) which was quickly identified as a fast rotating neutron star. Detailed studies on the maximum mass of Neutron Stars have shown the extreme sensitivity of this value to the underlying equation of state and roughly doubled the old value. Nowadays the maximum mass is expected to fall in the range 1 to 3 M¯, the exact value being much dependent of physics of the equation of state. The progenitors of Neutron Stars are massive stars with initial mass in the range 10 to 40-50 M¯ which proceed 21

Figure 2.1: Left: stars evolve towards different end states: White Dwarfs, Neu- tron Stars, and Black Holes. Right: Compact stars are the result of the endpoint in stellar evolution shown as a function of the initial mass. all the way up the formation of an Iron core, followed by photo-dissociation and neutronization of matter, pressure failure of the electrons, core collapse and if the infalling material made of neutron can develop enough pressure to contrast the , and the formation of a central object made of free neutrons. This is at the base of the the explosion of type II supernovae. However, if the neutrons cannot rebind themselves in a stable configuration, the collapse cannot be halted, and a Black Hole with typical stellar mass is formed. This is expected to occur for massive stars in the mass interval 50 to 100 M¯. The interior of a Black Hole is very enigmatic. Its surface is formed by a kind of semipermeable membrane forbidding any classical emission from its surface. The very source of the gravitational field of Black Holes is a kind of curvature singularity, which is hidden behind the membrane. It is expected that quantum effects will smooth these singular mass currents in the center of a . As already said Black Holes span an enormous range of masses, going 10 from about 3 to 10 M¯ and those whose mass exceed the initial typical mass of a massive stars cannot be of direct stellar origin. They likely originate from mass accumulation onto an original seed. Already in 1964 Black Holes have been 22 CHAPTER 2. INTRODUCTION

Figure 2.2: The masses of Neutron Stars and astrophysical Black Holes. A Black Hole has only two Hairs, the mass and the angular momentum represented by the dimensionless spin parameter a. The Neutron 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¯. Supermassive Black Holes in the centers of galaxies grow by accretion from Black Holes formed at high redshift (very early epochs). proposed as the ultimate source for , and currently are at the base of many high energy phenomena taking place in galaxies. The different end states of stars are schematically shown in Fig 2.1. The complex variety of phenomena that are attributed to the presence of compact stars, either neutron stars or Black Holes, is schematically shown in Fig. 2.2. Chapter 3

Fundamental scales

3.1 Introduction

In there are four fundamental forces: gravitational, electromagnetic, strong nuclear, and weak-electro-nuclear. The first two are long distance in- teractions, the other two are very short distance interactions. Gravitational vs Electro-magnetic. The electro-magnetic and gravita- tional forces between two of mass mp and charge e at the distance r are expressed by

e2 Gm2 F = F = p (3.1) em r2 G r2 from which 2 ³ 2 ´.³ 2 ´ Fem e e Gmp α 36 = 2 = = ' 10 (3.2) FG Gmp hc¯ hc¯ αG where ³ e2 ´ 1 ≡ α ' (3.3) hc¯ 137 is the Sommerfield fine structure constant, and ³Gm2 ´ p ≡ α ' 6 × 10−39 (3.4) hc¯ G is the gravitational fine constant, h =h ¯ × 2π is the , and c is the (in vacuum). It follows from this, that the gravitational force can be neglected in the description of matter properties at small scales ( and nuclei).

23 24 CHAPTER 3. FUNDAMENTAL SCALES

Atomic length scale: the Bohr radius a0. Consider an electron of mass me moving with velocity v along an orbit of radius a0 around a ( ). Electromagnetic and parallel each other

2 2 e mev 2 = (3.5) a0 a0 Introducing the angular momentum L we write

e2 L2 2 = 3 (3.6) a0 mea0 which imposing quantization of L (L = nh¯) becomes

e2 h¯2 2 = 3 (3.7) a0 mea0 It follows from this

h¯2 ³hc¯ h¯ 1 h¯ a0 = 2 = 2 ) → a0 = (3.8) mee e mec α mec whereh/m ¯ ec is the Compton wave length of electrons. Atomic energy scale. The binding energy of an electron to a proton is

2 e 2 2 Eb ' = α mec (3.9) a0

2 where mec is the rest mass energy of the electron. The atomic distance and energy scales are at work in the electromagnetic inter- action and the Heisemberg Indetermination Principle. Where is the gravitational interaction important? The gravitational force is important in neutral and massive objects. What are the associated dis- tance and energy scales? Consider a body of mass M and radius R containing N atoms with mass number A Z

M = NAmu (3.10) where mu ≡ (1/12)m12C ' mp; mu is the atomic mass units and m12C is the mass of the Carbon atom with mass number A = 12. 3.1. INTRODUCTION 25

The radius of the star can be expressed as

a R = N 1/3 0 (3.11) Z

The is

2 ³ 2 2 ´ ³ 2 ´ GM 2 GA mp 5/3 2 Gmp EG = − = −N ' −N A Z (3.12) R R a0 or

5/3 2 2 EG ' −N αG α mec A Z (3.13)

The internal energy due to all atoms is

2 2 3 Ei ' NEb(Z) ' Nα mec Z (3.14)

2 where Eb(Z) is the binding energy of an atom with charge Z, Eb(Z) = (Ze )Z/a0. The total energy is

5/3 2 2 3 2 Etot = EG + Ei = (−N αG α A Z + Nα Z )mec (3.15)

Assuming A/Z = 2 (typical nuclei in stellar composition), the total energy is

Etot = 0 when

5/3 NmaxαG = Nmaxα (3.16)

Therefore, Etot < 0 (bound state) if the following relation is satisfied

³ ´ α 3/2 54 Etot < 0 if N < Nmax = ' 10 (3.17) αG

Objects corresponding to the case under examination are characterized by M and R given by

30 Mmax ' Nmaxmp ' 10 g (3.18)

1/3 10 Rmax ' Nmaxa0 ' 10 cm (3.19) 26 CHAPTER 3. FUNDAMENTAL SCALES

They correspond to the mass and size of a like Jupiter1. Objects with mass

M ≤ Mmax are stabilized against the gravitational force by nucleon-nucleon forces 2 which are much stronger than the electromagnetic interaction by αs = 10 α

³ ´ ³ ´ s αs 3/2 α 3/2 3 54 57 Nmax = ' 10 × 10 ' 10 (3.20) α αG

s 33 Mmax ' Nmaxmp ' 10 g (3.21)

s 1/3 −5 6 Rmax ' (Nmax) a0 × 10 ' 10 cm (3.22) i.e. a Neutron Star. The factor 10−5 comes from the different intensities of the constants, the masses and radii of nucleons. Schwarzschild Radius. A particle of mass m at any distance from the surface of a body of mass M and radius R, in order to leave it must posses a kinetic energy equal to the gravitational potential energy

mv2 GMm = (3.23) 2 R The critical velocity, called escape velocity, is

2Gm v2 = (3.24) R

Assuming v = c, the maximum possible velocity, for a certain mass M we get a critical radius called the Schwarzschild Radius

2GM 2G hcmi R = where = 1.5 × 10−28 (3.25) s c2 c2 g

If a body of mass M is compressed to its Schwarzschild Radius, the escape velocity is the speed of light.

Objects with R < Rs are called Black Holes. The mean density of a Black Hole of mass M is

1 27 8 Masses and radii: M⊕ = 5.9 × 10 g, R⊕ = 6.4 × 10 cm; MJ = 332M⊕, RJ = 11.9R⊕ 3.2. MASS DEFECT 27

4π ρ = M/( R3) (3.26) 3 s from the definition of Schwarzschild radius we obtain 3 1 c6 ρ > ( ) (3.27) 32π M 2 G3 or ³ ´ h i 16 M −2 g ρ > 1.84 × 10 3 (3.28) M¯ cm 16 −3 In a Black Hole with M = 1 M¯. the mean density is ρ = 2 × 10 g cm , 9 about 100 times larger than nuclear density. In Black Hole of 10 M¯, ρ = 0.02 g cm−3. The existence of a Black Hole does not necessarily imply high densities.

3.2 Mass defect

When a star is formed by a gas cloud (spherical and with mass M, gravitational energy is liberated by contraction. The variation in gravitational energy EG by accreting mass onto a spherical distribution of material m(r) is Gm(r) dE = − dm with dm = 4πρr2dr (3.29) G r The total gravitational energy Z R m(r) EG = −G dm (3.30) 0 r or Z R h Z r i 1 2 2 EG = −G 4π ρ(ξ)ξ dξ 4πρr dr (3.31) 0 r 0 If the density is constant, we get 3 GM 2 4π E ' − 0 M = ρR3 (3.32) G 5 R 0 3 The factor 3/5 comes from the spherical and the incompressibility of the mass distribution. In general, this factor is ' 1. GM 2 E ' − 0 (3.33) G R 28 CHAPTER 3. FUNDAMENTAL SCALES

The liberated gravitational energy is emitted as photons and/or neutrinos. The star mass M0 is therefore smaller than the initial mass M |E | GM 2 ³ 1 R ´ M = M − G ' M − 0 = 1 − s (3.34) 0 c2 0 Rc2 2 R The mass defect is given by ³ 1 R ´ ∆M = M − M = M − M 1 − s (3.35) 0 0 0 2 R The relative mass defect is ∆M R ' s (3.36) M0 R In the and all other stars down to White Dwarfs, the gravitational mass defect is very small, less than the mass defect due to nuclear reactions (∆Mnuc ' 1%).

3.3 Hydrostatic Equilibrium

Consider a sphere of gas of mass M and radius R in hydrostatic equilibrium: at each point the pressure gradient is balanced by the gravitational force

dP Gm(r) = − ρ(r) (3.37) dr r2 The order of of the pressure gradient is dP P ' − (3.38) dr R The order of magnitude of the gravitational force Gm(r) GM ρ(r) ' ρ (3.39) r2 R2 It follows that P R ' s (3.40) ρc2 R The ratio between the Schwarzschild radius and the stellar radius is nearly equal to the ratio between the mean pressure and the rest mass energy of the star. 3.4. NORMAL NON DEGENERATE STARS 29 3.4 Normal non degenerate stars

Stellar matter is composed by a perfect gas with equation of sate k P = ρT or PVmol = RT (3.41) µmH where µ is the mean molecular weight of the elemental species present, Vmol is 23 the molar volume Vmol = NaMA/ρ, Na = 1/mH = 6.023 × 10 , MA = AmH is the mass of atoms with mass number A, mH is the atomic mass unit, and R is the universal gas constant R = 8.32 × 107erg/K/Mol. Equation (3.41) can be written as

P R T T = = k (3.42) ρ Na MA MA In hydrostatic equilibrium M T ∝ (3.43) R and Rs P kT = 2 = 2 (3.44) R ρc MAc which correlates the relativist effect (l.h.s.) to the temperature of the star (r.h.s). For hydrogen burning

Rs kT 1Kev −6 ' 2 ' ' 10 (3.45) R MAc 1Gev In normal stars the size of the relativistic effects is determined by nu- clear physics. Their size does not depend on the G.

3.5 Energy losses by radiation

Stars lose energy into the surrounding medium by emitting radiation- The lumi- nosity of a star, energy emitted by the surface per unit time, can be approximated to that of a at a certain temperature T . Therefore we have

|E˙ | = L = 4πσR2T 4 (3.46) 30 CHAPTER 3. FUNDAMENTAL SCALES where σ = 5.67 × 10−5 erg/cm2/s/K4 is the Stefan-Boltzmann constant. The constant σ comes from the Black Body theory σ = (2π5k4)/(15c2h3). If a star has no internal sources of nuclear energy, the star will cool down with the -Helmholtz time scale

E GM 2 τ = G ' ' R−1T −2 (3.47) KH L R(4πσR2T 4)

7 For the Sun τKH ' 10 yr. Stars made of an ideal gas can exist in hydrostatic equilibrium for a time longer than τKH only if in their interior another source of energy is active (for instance a nuclear source). This conclusion eventually follows from the fact that

¯ ¯ ∆M ¯ Rs ∆M ¯ ¯ ' ' 10−6 << ¯ (3.48) M0 grav R M0 nuc 3.6 Degenerate stars

While the pressure in a non degenerate star is driven by the kinetic energy of the gas particles, in a degenerate star the pressure is a consequence of the Pauli Principle.

P = f(T ) for an ideal gas ρc2 P = f(ρ) for a degenerate gas of fermions (3.49) ρc2

For an ideal gas when T → 0, P → 0, no equilibrium is possible. For a degenerate gas of electrons (Fermions and the first particle to feel de- generacy) the pressure can be estimated as follows. Consider an electron in a cube of edge d and volume d3. The Heisemberg Principle says that the moment pF of the electron obeys

pF × d ' h¯ (3.50)

The kinetic energy of the electron is 3.6. DEGENERATE STARS 31

2 2 pF h¯ ²F = ' 2 (3.51) 2me med If ² << kT the kinetic energy of the electron is not determined by the temperature but the density 2 P ²F h¯ 2 ' 2 ' 2 2 (3.52) ρc MAc med MAc Note that at given density, the lightest the particle the larger is its and hence its contribution to pressure. Therefore we can approximate

P (e−,A) ' P (e−) (3.53)

The mass density is given by the most massive particles m + M M ρ ' e A ' A (3.54) d3 d3 Using these approximations in eqn. (3.52) we obtain

P hρ¯ 2/3 ³ m ´³ ρ ´2/3 ' = e (3.55) ρc2 5/3 2 M ρ meMA c A c where MA 7 g ρc ≡ 3 = 3 × 10 [ 3 ] (3.56) (¯h/mec) cm is the density at which the particle mean distance is equal to the Compton wave- −11 length of electrons λ/2π =h/m ¯ ec = 3 × 10 cm.

For ρ > ρc, it follows from the Heisenberg Principle h¯ h¯ pF ≥ ≥ ≥ mec (3.57) d λe i.e. the electrons are relativistic. The Fermi energy is ²F ' pF c (it follows from 2 ²F ' pF /2me) and ³ ´ P me ρ 1/3 2 ' if ρ > ρc (3.58) ρc MA ρc The equation of state for a gas of fermions is

5/3 P ∝ ρ if ρ < ρc (3.59)

4/3 P ∝ ρ if ρ > ρc (3.60) 32 CHAPTER 3. FUNDAMENTAL SCALES

As a consequence, thanks to quantum effects a gas will exerts pres- sure even if T = 0. In the White Dwarfs the pressure is given by degenerate relativistic electrons. In neutron stars by the degenerate non relativistic gas of neutrons. White Dwarfs and Neutron Stars do not have an internal energy source, yet they can be in hydrostatic equilibrium. Chapter 4

General Relativity

Modifications to the gravitational field intensity due to General Relativity become important when considering the stability of White Dwarfs, and the equilibrium and stability of Neutron Stars and Black Holes. In this chapter we present a short summary of General Relativity and its implications, leaving to specialized textbooks the rigourous and complete presentation of the subject.

4.1 What is General Relativity?

The General Relativity Theory (GRT) is a relativistic theory of grav- itation. What problems arise in trying to make the Newton gravitation theory relativistic? The Newtonian gravitation is described by a field theory for a scalar field Φ satisfying the Poisson Equation

2 ∇ Φ = 4πGρ0 (4.1) where ρ0 is the mass density of matter. The gravitational acceleration of any objects in this field is given by −∇Φ

In Relativity all forms of energy are equivalent to mass, so that a relativistic theory of gravity would contain all forms of energy as sources of the gravitational field, not just ρ0. In particular, the energy density of the gravitational field itself is proportional to (∇Φ)2 in the Newtonian

33 34 CHAPTER 4. NOTES ON GENERAL RELATIVITY limit. Therefore, taking the l.h.s. of eqn.(4.1), we expect a relativistic theory to involve nonlinear differential equations for the gravitational field. Symbolically we may write

F (g) ∼ GT (4.2) where g → gravitational field → Φ in the weak limit; F → differential operator → ∇2 in the weak limit;

T → some quantity describing all non gravitational forms of energy → ρ0 in the weak limit. Einstein’s great insight: to make GR a geometric theory of gravitation. Let us start by examining the geometry of the Special Relativity Theory (SRT). In SRT space-time is the basic concept. Space-time consists of events, which require four numbers for their complete specification. Three numbers to fix the location in the chosen frame of reference, and one number to give the time. Geometrically, the space-time is represented by a four-dimensional manyfold (”surface”) each point corresponding to an event. An observer in space-time makes measurements, i.e. assigns coordinates to the events. There is a preferred family of observes in SRT: the inertial observers for whom a free particle moves with uniform velocity. All inertial observers are related by Lorentz transformations. The coordinate system of an inertial observer is called ”an inertial coordinate system” or ”Lorentz Frame”. The interval (”distance”) between two events in space-time is given by

ds2 = −c2dt2 + dx2 + dy2 + dz2 (4.3) where dt, dx, dy, and dz are the differences between the coordinates of events in the ”Lorentz Frame”, and c is identified with the speed of light. The interval ds does not depend on which inertial frame is adopted. It is Lorentz invariant. Writing x0 = ct, x1 = x, x2 = y, and x3 = z we can write eqn.(4.3) as 4.1. WHAT IS GENERAL RELATIVITY? 35

2 α β ds = ηαβdx dx (4.4) where

ηαβ = diag(−1, 1, 1, 1) (4.5) is a 4x4 matrix and there is an implied sum from 0 to 3 for the repeated indices in eqn.(4.4). The quantity ηαβ is called the Metric Tensor of the space-time in SRT. It completely specifies the space-time geometrically. This space-time is said pseudo-Euclidean metric space (that is Euclidean but for the minus sign in eqn.(4.5). This geometry is also called Minkowski Space. Using non-Lorentzian Coordinates. One can use non Lorentzian (non- inertial) coordinates to describe the space-time. For instance one could adopt polar coordinates for the spatial part, or one could adopt the coordinates of an accelerated observer. If the relationship (transformation) between the inertial coordinates xα and the non-inertial coordinates yα is

xα = xα(yγ) (4.6) eqn.(4.4) becomes

2 γ α β ds = gαβ(y )dy dy (4.7) where by the chain rule

∂xλ ∂xσ g (yγ) = η (4.8) αβ ∂yα ∂yβ λσ Even though the metric is now very complicated with all kinds of off-diagonal and position dependent coefficients, space-time is always flat: there exists a transfor- mation of coordinates such that the metric takes the pseudo-Euclidean form (4.5) everywhere. In GR, space-time is still a 4-dimensional manifold of events but now the interval between events is given by 36 CHAPTER 4. NOTES ON GENERAL RELATIVITY

2 γ α β ds = gαβ(x )dx dx (4.9) where no choice of coordinates xα can be found to reduce the metric to the form of eqn.(4.5) everywhere. The space-time is curved. The functions gαβ are used to represent the gravitational field variables: the gravitational field determines the geometry! The interval ds is still an invariant so that the transformation for the components α α gαβ of the metric tensor passing from the coordinates x to x is

∂xλ ∂xσ g = g (4.10) αβ ∂xα ∂xβ λσ As in SRT, when evaluated along the world-line of a particle (fixed spatial coor- dinates), ds measures the proper time interval dτ along the line ds2 = −c2dτ 2. Internal (dot)-product. A metric allows one to define dot-products be- tween vectors. Equation (4.9) can be written as

ds2 = dx · dx (4.11)

In general, the dot product between two vectors A and B is

α β A · B = gαβA B (4.12)

Such vectors, with four components, are called four-vectors and are indicated with the ”bar” to distinguish them from the three-vectors. We can also write

β Aα = gαβA (4.13) or α αβ A = g Aβ (4.14)

αβ where ||g || is the inverse matrix of ||gαβ||. α α It is worth recalling the geometrical meaning of A and Aα. A is often referred to as the contravariant component of a vector when the basis vectors are 4.1. WHAT IS GENERAL RELATIVITY? 37

tangent to the coordinate line, whereas Aα is called the covariant compo- nent of a vector when the basis vectors are orthogonal to the coordinate surfaces. The following relationships hold

α α α,β A · B = AαB = A Bα = AαBβg (4.15)

In SRT, the coordinates of an event always correspond to the result of some physical measurements. Even the case of non-inertial reference systems, the co- ordinates can be interpreted by finding their relationship to an inertial coordinate system, where t, x, y, and z have their standard meaning (measurements made with clocks and rods). In GRT, there is no preferred coordinate system, in principle any system that smoothly labels events in the space-time can be adopted. We must look at the question of measurements more closely. Local Inertial Frames. The physical interpretation of GRT depends on the concept of local inertial frames. Although in general gαβ in eqn.(4.9) cannot be diagonalized by a coordinate transformation to ηαβ everywhere in space-time, we can choose any event in space-time to be the origin of the coordinates and then at that point we can diagonalize gαβ. Furthermore, one can find a coordinate transformation that set the first derivatives of gαβ at the origin to zero. In other words the Taylor transformation of the metric about the origin would look like

2 2 α β ds = [ηαβ + O(|x| ]dx dx (4.16) where O → 0. Any small region of the coordinate frame in which the metric takes the form (4.16) is called a local inertial frame. To highlight the reason for such designation consider the local inertial frame of an observer in space-time. An observer is represented by the world-line of all events he/she experiences. Choose some particular event on the world-line as the origin of the system (4.16). Erect a unit vector e˜t tangent to the t-coordinate line, and similarly construct e˜x, e˜y, and e˜z. Since gα,˜ β˜ = ηα,β this is an orthonormal tetrad; therefore e˜t · e˜t = −1, 38 CHAPTER 4. NOTES ON GENERAL RELATIVITY

e˜x · e˜x = 1, e˜t · e˜x = 0 and so on. The tilde indicates an orthonormal tetrad. The important result is that the geometry is the same of the SRT case. The departures from SRT will be noticed on the scale of the second derivatives of gαβ, the stronger the gravitational field the shorter is the scale. Principle of Equivalence. An observer carrying a local orthonormal tetrad is called ”Local Inertial” or ”Local Lorentzian Observer”. GRT goes beyond this statement by simply asserting that measurements in a local inertial frame are carried out as in SRT. It further asserts that all non gravitational laws of physics are the same in a local inertial frame as in SRT. This law is called the . It has its origin in the equivalence of inertial and gravitational mass. The classical example is the elevator thought- experiment. Elevator Experiment. Consider an observer performing experiments in a closed box that is being accelerated upward with uniform acceleration. The results would be indistinguishable from those of an observer inside a stationary closed box in a uniform gravitational field. Conversely, in a box freely falling in a uniform gravitational field, there would be no observable effect of the gravitational field (e.g. astronauts in a satellite in orbit about the ). This example also clarifies the physical meaning of Local Inertial Frame: it is the reference frame of an observer freely falling in a gravitational field. The Equivalence Principle is a generalization of a principle which says that the Laws of Mechanics do not allow one to detect a gravitational field locally to a principle that no laws of physics allow one to detect a gravitational field locally. The effects of gravitation disappear in a freely falling frame. The Equivalence Principle tells us how to formulate non-gravitational laws in presence of a gravitational field. Start with any law of Physics: for instance the conservation of energy-momentum

αβ ∇αT = 0 (4.17) 4.1. WHAT IS GENERAL RELATIVITY? 39

∂ ∇ = (4.18) α ∂xα where T αβ is the stress-energy tensor and eqn.(4.17) says that its 4-divergence vanishes. By the Equivalence Principle eqn.(4.17) must be valid in a local inertial frame where the metric is (4.16). Now we want to write eqn.(4.17) in a general coordinate system with metric (4.9). To do this we need to define a more general derivative operator (covariant derivative) than eqn.(4.1). Even in curvilinear coordinates in flat space, one knows that there are extra terms in the r.h.s. of eqn.() when differentiating vectors. For example if in spherical coordinates the vector has components

r θ φ A = A er + A eθ + A eφ (4.19) the divergence is not simply

r θ φ ∂rA + ∂θA + ∂φA (4.20)

because we must also differentiate er, eθ, eφ. The basis vectors are not con- stant in space because the components gαβ are not constant in this coordinate system. Similarly in a curved space-time, there must be extra terms in the covariant derivative ∇α that come from the variation of gαβ. These terms cannot be removed everywhere by a coordinate transformation, the derivatives of Gαβ represent the effect of the gravitational field. The mathematical implementation of the Equivalence Principle is called the Principle of General Covariance: one write covariant equations in SRT and imposes that they are covariant under general coordinate transformations. Acceleration at a point. There is no analog in GRT of the Newtonian concept of ”gravitational acceleration at a point”: such local acceleration is re- movable by going to a freely falling coordinate system. However, the ”difference” between the gravitational accelerations of two nearby test bodies is nor remov- able in general. Thus the true gravitational field in GRT has as its Newtonian 40 CHAPTER 4. NOTES ON GENERAL RELATIVITY analog te ”tidal” gravitational field ∂2Φ/∂xi∂xj, where Φ is the Newtonian po- tential. This is because the relative Newtonian gravitational acceleration of two test bodies is

∂ ³ ∂Φ ´ ai = ai(x + ∆x) − ai(x) = ∆xj (4.21) rel ∂xj ∂xi The term Gαβ. So far, we have discussed how gravitation affects all the other phenomena of Physics and how the geometry is related to the physical measure- ments in a local inertial frame. To complete the picture we need to determine how the mass-energy distributions determines the geometry, gα,β by means of equations like eqn.(4.2). The goal is achieved by the fundamental Einstein’s relation

8πG Gαβ = T αβ (4.22) c4

αβ where G is a second-order non linear differential operator acting on gαβ. The source term for the Einstein equation is the stress-energy tensor T αβ. This com- plicated equation reduces to the Poisson eqn.(4.1) in the newtonian limit and αβ guarantees the conservation of the energy-momentum since ∇αG = 0.

4.2 Motion of test particles

A test particle is an idealization of a material object. It is supposed to be small (does not perturb the space-time around) uncharged (does not respond to electro- magnetic forces), spherical (feels no torques). It simply moves freely in the grav- itational field. In SRT (no gravitational field), test particles move with uniform velocity. Their equation of motion can be derived from the variational principle that ex- tremizes the distance (interval along the world-line:

Z δ ds = 0 (4.23) 4.2. MOTION OF TEST PARTICLES 41

To verify this, write the integrand in the form

ds = (−ηx˙ αx˙ β)1/2dλ (4.24) where dxα x˙ α ≡ (4.25) dλ Here λ is any parameter along the world-line. The expression (4.24) for ds is in- variant under a change of the parameter λ → λ(λ0). The lagrangian for eqn.(4.23) is L = (−ηx˙ αx˙ β)1/2 (4.26)

The Euler-Lagrange equations of motions obtained from eqn.(4.23) are

d ³ ∂L ´ ∂L = (4.27) dλ ∂x˙ α ∂xα where the r.h.s. is zero since L is independent of xα. Since

∂L = −L−1η x˙ β (4.28) ∂x˙ α αβ we get 1 dL η x˙ β − η x˙ β = 0 (4.29) αβ L dλ αβ By re-scaling λ → λ(λ0) we can make L be constant along the world-line (the parameter λ is also called the affine parameter). One can always parameterize the world-line of a particle by the length s along the curve, i.e. the proper time τ, actually s = cτ. In this case, λ = s, L = 1 along the curve and so eqn.(4.29) becomes

β ηαβx¨ = 0 (4.30)

αβ Multiplying by the matrix inverse of ηαβ, denoted η , we obtain

d2xγ x¨γ = = 0 (4.31) dτ 2 which is the equation for uniform velocity in a straight line. 42 CHAPTER 4. NOTES ON GENERAL RELATIVITY

Geometrically, curves of extremal length are called geodesics. The geodesics of the Minkowski space in SRT are straight lines. Time-like geodesics. The above case, where ds2 < 0, describes the time-like geodesics, the world-lines of free material particles. Null geodesics. Photons and other mass-less particles, travel at the speed of light, ds2 = 0. free photons or other mass-less particles are said to move along null geodesics. In the this case the parameter λ cannot be the proper time. It is convenient to choose it such as

dxα = pα (4.32) dλ α β the four-momentum of photons. Since ηαβp p = 0 at all times for a photon, this is a valid choice for λ: ds2 = 0 (in the case of a particle with mass m the corresponding λ would be λ = τ/m). Equation (4.31) now is dpα = 0 → pα = constant (4.33) dλ Space-like geodesics. We can also have space-like geodesics for which ds2 > 0. They correspond, for example to straight lines in Euclidean 3-space at some instant x0 = constant (two simultaneous events in the space-time). The point with this complicated discussion of a simple problem in SRT (uni- form velocity in absence of gravitational fields) is that it immediately goes over the case of GRT. By the Equivalence Principle, eqn.(4.23 must be a variational principle for the motion of test particles in GRT: free particles move along geodesics of the space-time. Now eqn.(4.26) becomes

γ α β 1/2 L = [−gαβ(x )x ˙ x˙ ] (4.34) so eqn.(4.27) gives 1 g x¨β + g x˙ γx˙ β − g x˙ γx˙ β = 0 (4.35) αβ αβ,γ 2 γβ,α The second term comes from d ∂g dxγ g = αβ (4.36) dλ αβ ∂xγ dλ 4.2. MOTION OF TEST PARTICLES 43

∂g g = αβ (4.37) αβ,γ ∂xγ We have assumed an affine parameterization such that L = constant. Now write

1 g x˙ βx˙ γ = (g + g )x ˙ βx˙ γ (4.38) αβ,γ 2 αβ,γ αγ,β so that eqn.(4.35) becomes

β α γ gαβx¨ + Γαβγx˙ x˙ = 0 (4.39) where 1 Γ = (g + g − g ) (4.40) αβγ 2 αβ,γ αγ,β γβ,α Multiplying by the matrix inverse of the metric tensor, denoted gλα and the relabelling λ ←→ α we get

α β γ x¨ + Γβγx˙ x˙ = 0 (4.41) where α αλ Γβγ = g Γλβγ (4.42)

The Γ-symbols are named ”Christoffel symbols”. Equation (4.41) is the final form of the geodesic equation in GRT. The Equivalence Principle is satisfied: in a local inertial frame, one may choose coordinates so that gαβ,γ → 0, i.e. the Γ-symbols vanish. Thus the motion of a test particle in a local inertial frame is uniform velocity in a straight line. Requiring this statement to be true in any inertial frame at any point in space-time leads to eqn.(4.41), where the Γ-s represent the effect of the gravitational field. Note the difference between the Principle of General Covariance adopted in GRT and the Principle of Lorentz Covariance adopted in SRT: – In Lorentz Covariance, transformations from one inertial frame to another leave the form of the physical law unchanged. The velocity of transformation must drop out the final equation. This requirement restricts possible laws of physics quite severely. 44 CHAPTER 4. NOTES ON GENERAL RELATIVITY

– The principle of covariance in GRT does not restrict the possible laws of physics. One can formulate a law in a local inertial frame, transform it to a general coordinate system, and say that the extra terms describe the gravitational field. Only experiments can then decide if the law is true. One may demonstrate that if λ is the affine parameter, the lagrangian expressed by eqn.(4.34 is equivalent to

γ α β 1/2 α β L = [−gαβ(x )x ˙ x˙ ] ≡ gαβx˙ x˙ (4.43) for geodesics. Canonically Conjugate Momentum. As usual, define the canonically conjugate momentum of the coordinate xα by

∂L p = (4.44) α ∂x˙ α From eqn.(4.43) and eqn.(4.32) we see that

β β α αβ pα = gαβx˙ = gαβp or p = g pβ (4.45)

αβ where g in the matrix inverse of gαβ. Note that if L is independent of the 1 coordinate x say, the p1 is a constant of motion. For example, the metric of two-dimensional euclidean space is

ds2 = dr2 + r2dφ2 (4.46) leading to the Lagrangian

1 L = (r ˙2 + r2φ˙2) (4.47) 2 The equation of motions are r¨ − rφ˙2 (4.48)

r2φ˙ = constant (4.49)

If λ = t (time) we recognize the classical Newtonian equations in empty space. 4.2. MOTION OF TEST PARTICLES 45

Thanks to eqn.(4.44) the constant in eqn.(4.49) is pφ (the angular momentum per φφ φ ˙ unit mass. Since g = 1/gφφ, eqn.(4.45) gives p = φ. The physically measured value of the φ-components of the momentum is the projection of the vector p along a unit vector in the φ-direction. Now the coordinate er and eφ basis satisfy

2 er · er = grr = 1, er · eφ = grφ = 0, eφ · eφ = gφφ = r (4.50)

We can understand the last equation of (4.50) recalling that eφ is tangent to the φ-coordinate line. This means that it connects two radial line φ = constant (see

Fig. 4.1). It is clear that eφ at radius r1 and eφ at radius r2 have lengths in the 2 ratio r1/r2, therefore eφ · eφ ∝ r with the proportionality constant depending on 2 the scaling of the φ coordinate. One can always assume eφ · eφ = r In general one has the relation

eα · eβ = gαβ (4.51)

α which results from eqn.(9) and eqn.(11), noting that dx = dx eα. Special cases are

ei · ej = δij 3D − cartesian

eα˜ · eβ˜ = ηα˜β˜ 4D − orthonormal (4.52) We can now choose the orthonormal set of basis vectors to be

e˜r = er (4.53)

1 e = e (4.54) φ˜ r φ so that 1 1 e · e = e · e = g = 1 (4.55) φ˜ φ r2 φ φ r2 φφ then 1 pφ˜ = p = p · e = p = rφ˙ (4.56) φ˜ φ˜ r φ a familiar result for the momentum per unit mass (i.e. the velocity) along eφ˜. 46 CHAPTER 4. NOTES ON GENERAL RELATIVITY

Figure 4.1: The coordinate basis vector eφ at A is r2/r1 times longer that the basis vector at B. The lines φ= constant are supposed to be infinitesimally close.

4.3 Gravitational redshift

The simplest way to explain the physical reasons of the gravitational redshift is given by this thought example. Consider an emitter and a receiver of electro- magnetic waves (photons) at two fixed positions in the space-time in a static gravitational field. The frequency ν of the radiation at the emitter is just the inverse of the proper time between two wave crests as measured in the frame of the emitter; that is

1 c νem = = (4.57) α β 1/2 dτem (−gαβdx dx )em Since the emitter stays at fixed position dx1 = dx2 = dx3 = 0. An expression similar to eqn.(4.57) holds for the receiver, so we have

1/2 0 νrec [(−g00) dx ]em = 1/2 0 (4.58) νem [(−g00) dx ]rec The coordinate time dx0 between two wave crests is the same at the emitter and receiver because the gravitational field is static: nothing depends on x0. Whatever the world-line of one photon is from emitter to receiver, the next photon follows the same path simply displayed by dx0 at all points (see Fig.4.2) 4.4. THE WEAK-FIELD LIMIT 47

1/2 νrec (−g00)em = 1/2 (4.59) νem (−g00)rec

The gravitational redshift is due to the local variation of g00 between emitter and receiver.

4.4 The weak-field limit

How can we estimate the weak-field limit of GRT? This can be obtained by con- sidering an alternative derivation of the redshift based on the energy conservation. In virtue of the basic Einstein’s relation E = mc2 a photon of frequency ν has an effective mass hν m = (4.60) c2 Its total energy in a Newtonian gravitational field wi potential Φ(x) is hν+mΦ(x). Equating the energy at the emitter to that at the receiver we obtain

2 νrec (1 + Φ/c )em = 2 (4.61) νem (1 + Φ/c )rec which is usually written as

∆ν ∆Φ = − 2 (4.62) νem c since Φ/c2 << 1 in a Newtonian limit. Comparing eqn.(4.59) and eqn.(4.61) we find

2Φ g ' −(1 + ) (4.63) 00 c2 in the Newtonian limit.

4.5 Geometrical units

It is very convenient in GRT to choose units so that c = 1 and G = 1. In other words –Time is measured in cm (1 s = 3 × 1010 cm) 48 CHAPTER 4. NOTES ON GENERAL RELATIVITY

Figure 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).

–Mass is measured in cm (1 g = 0.7425 × 10−28 cm). The numerical factor is G/c2.

A convenient conversion factor is 1 M¯ = 1.4766 km. Be careful when using geometrized units!

4.6 Spherically symmetric gravitational fields

A spherically symmetric metric can depend on time, radial coordinates and an- gles, but the latter only in the combination

dΩ2 = dθ2 + sin2θdφ2 (4.64)

The most general symmetric metric can be written as

ds2 = −A(r, t)dt2 + B(r, t)dr2 + 2C(r, t)dt dr + D(t, r)dΩ2 (4.65) where A(r, t), B(r, t), C(r, t), and D(r, t) are generic functions that we do not need to specify here. Look now for simplifications! 4.6. SPHERICALLY SYMMETRIC GRAVITATIONAL FIELDS 49

Choose a new radial coordinate

r0 = D1/2(r, t) (4.66)

Substituting in eqn.(4.64) and dropping the prime, we get

ds2 = −E(r, t)dt2 + F (r, t)dr2 + 2G(r, t)dt dr + r2dΩ2 (4.67)

Also in this case the generic functions E, F, and G do not need to be specified. The are simply related to A, B, and C in some way. The form of eqn.(4.67) suggests the following change of coordinates

dt0 = E(r, t)dt − G(r, t)dr (4.68)

In general, the r.h.s. of eqn.(4.68) will not be a perfect differential. Since there are only two variables, we know that there always exist a function H(r, t) such that

dt0 = H(r, t)[E(r, t)dt − G(r, t)dr] (4.69) is a perfect differential. Making the substitution in eqn.(4.67) leaves two metric functions, the coefficients of (dr0)2 and dr2. Dropping the prime, we write

ds2 = −e2Φdt2 + e2λdr2 + r2dΩ2 (4.70) where Φ and λ are functions of t and r. The exponential for the coefficients is for later use (this representation of the coefficients is legitimate owing to the very generic nature of all functions in use here). Important results of the Newton theory of gravitation are: 1) At any point outside a spherical mass distribution, the gravitational field depends only on the interior to that point. 2) Even if the mass interior is moving spherically symmetric, the field outside is constant in time 3) We have simply Φ = −GM/r or Φ = −M/r in geometrized units. 50 CHAPTER 4. NOTES ON GENERAL RELATIVITY

This result is also true in GRT (the Birkoff Theorem): the only vacuum, spheri- cally symmetric gravitational field is static. It is called the Schwarzschild metric

³ 2M ´ ³ 2M ´−1 ds2 = − 1 − dt2 + 1 − dr2 + r2dΩ2 (4.71) r r The word ”vacuum” denotes a region of the space-time where the gravitational effect of any matter present is negligible. The constant M appearing in eqn.(4.71 is the mass of the source. We can see this by considering the weak-field limit, r >> M, which shows that the potential is −M/r; that is, M is the mass. The Schwarzschild metric applies everywhere outside a spherical star, right up to its surface.

4.7 Spherical stars

The metric given by eqn.(4.70) also describes the gravitational field inside a spherical star. For a star in hydrostatic equilibrium we can take Φ and λ to be independent of t. We assume that the stellar material can be described as perfect fluid with some equation of state

ρ = ρ(n, s) (4.72) where n is the number density of particles and s is the entropy. Since c = 1 we are not distinguishing between total energy density ² and total mass density ρ = ²/c2. The pressure follows from the first law of thermodynamics and can be written as

P = P (n, s) (4.73)

Although a perfect fluid is adiabatic (the entropy s of a fluid element remains constant), it is not necessarily isoentropic (s does not need to have the same value everywhere). In the case of White Dwarfs and Neutron Stars, the temperature is 0 essentially zero everywhere (more precisely kT << EF ) so that s = 0 everywhere. 4.7. SPHERICAL STARS 51

Therefore the equation of state is simply of the form

P = P (ρ) (4.74)

The equation of stellar structure in GRT can be found in standard textbooks. We write them in a form that highlight the similarity to Newtonian equations. First we define a new metric function m(r) by

³ 2m´−1 e2λ = 1 − (4.75) r The Einstein equations give dm = 4πr2ρ (4.76) dr

dP ρm³ P ´³ 4πP r3 ´³ 2m´−1 = − 1 + 1 + 1 − (4.77) dr r2 ρ m r

dΦ 1 dP ³ P ´−1 = − 1 + (4.78) dr ρ dr ρ For P << ρ and m << r we get the Newton equations. Equation (4.77) is called the ”Oppenheimer-Volkoff” equation of hydrostatic equi- librium. The quantity m(r) has the interpretation of ”mass inside radius r. Equation (4.76) gives

Z R M = 4πr2ρdr (4.79) 0 for the total mass of the star. Note that this includes all contributions to the mass, including the gravitational potential energy. This fact is somewhat obscured by the simple appearance of eqn.(4.79), but recall that the proper volume element is not 4πr2dr; it is

³ 2m´−1/2 dV = (g )1/2dr × 4πr2 = 1 − 4πr2dr (4.80) rr r Thus eqn.(4.80) does not simply add up ρdV, the local contribution to the total mass-energy, but also includes a global contribution from the negative gravita- tional potential energy of the star. 52 CHAPTER 4. NOTES ON GENERAL RELATIVITY

Inserting now the natural constants c and G the eqn.(4.76) through (4.78) are

dM = 4πr2ρ (4.81) dr

dP GM ³ P ´³ 4πP r3 ´³ 2GM ´−1 = − ρ 1 + 1 + 1 − (4.82) dr r2 ρc2 Mc2 r

dΦ 1 dP ³ P ´−1 = − 1 + (4.83) dr ρ dr ρc2 In the Post-Newtonian approximation we have

dP GM ³ P 4πP r3 2GM ´ = − ρ 1 + + + (4.84) dr r2 ρc2 Mc2 r

dΦ 1 dP ³ P ´ = − 1 − (4.85) dr ρ dr ρc2 Chapter 5

Equation of State below Neutron Drip

5.1 Thermodynamic Preliminaries

Conventionally, all thermodynamical quantities refer to a number of particles N confined in a volume V . Thermodynamics is relativistically invariant. This can easily understood if all quantities describe measurements made in a local inertial frame comoving with the fluid. We can imagine a local Lorentzian frame moving with the same velocity of the fluid. Let n be the number density of as measured in this reference frame, and let ² be the total energy density (including the rest-mass energy). The ²/n is the energy density per . It is convenient to define quantities on a per- baryon basis as the the baryon number is a conserved quantity. This is not always true because there are many baryon and non-conserving reactions that may occur at super-high energies (> 1015) Gev. In this case it would be required to introduce explicitly the volume of a given fluid element. Whenever necessary this will be taken into account. The First Law of Thermodynamics takes the general form

² 1 dQ = d( ) + P d( ) (5.1) n n

1 where dQ is the heat gained per baryon, and n is the volume per baryon. If a

53 54 CHAPTER 5. EQUATION OF STATE BELOW NEUTRON DRIP process occurs in a fluid element that is at equilibrium at all times, then

dQ = T ds (5.2) where s is the entropy per baryon. Combining eqn.(5.1) and eqn.(5.2) we obtain ² 1 d( ) = −P d( ) + T ds (5.3) n n In writing eqn.(5.3) we tacitly assume that ² = ²(n, s). In general in a fluid containing different types of particles, the energy density is a function also of their abundances (or concentrations). The concentration of species i is defined as n Y = i (5.4) i n where ni is the number density of particles i. Therefore

² = ²(n, s, Yi) (5.5)

The most general form of eqn.(5.3) is ² 1 X d( ) = −P d( ) + T ds + µidYi (5.6) n n i where ∂(²/n) ∂(²/n) P ≡ − = n2 (5.7) ∂(1/n) ∂n

∂(²/n) T ≡ (5.8) ∂s

∂(²/n) µi ≡ (5.9) ∂Yi

The quantity µi is called the chemical potential of the species i. It represents the change in energy density for a unit change in number density of species i, while volume, entropy and all other number densities are kept constant.

Since ² contain the rest-mass energy, so does µi. Note that eqns. (5.7), (5.8), and

(5.9) can be considered as the definition of P , T and µi. Furthermore, the system of the three relations can be seen as a general representation of the equation of 5.1. THERMODYNAMIC PRELIMINARIES 55 state of a multi-component fluid, which requires only the knowledge of the energy density per baryon ²/n. In equilibrium, reactions between particles can occur each of which balanced by its inverse, so that the concentration of the species i remain constant. There- fore the Yi are not independent of the other thermodynamic variables. We deter- mine the equilibrium relations for four typical physical situations: 1) System close to equilibrium. Consider first the special case where the system is infinitesimally close to equilibrium. Reactions occur to bring the system to equilibrium but the system is kept thermally isolated (dQ = 0) and at constant volume V (dn = 0) so that no work is done on the system P d(1/n) = 0. In this case d(²/n) = 0 the energy remains constant. Entropy is generated by the reactions but because entropy is maximum at equilibrium (Second Law), ds = 0. The equilibrium condition is X µidYi = 0 (5.10) i Suppose we consider the reaction

− e + p ↔ n + νe (5.11)

in equilibrium. Then the following relations hold dYe = dYp = −dYn = −dYνe , which imply

µe + µp = µn + µνe (5.12)

Similar relations hold for all other reactions producing equilibrium. If the µi are known as a function of the relevant values of n and s or n and ², we can derive the equilibrium concentrations. 2) System not thermally isolated and on which work can be done. Now consider the general case where the system is not thermally isolated and work can be done on the system. If the system were to achieve equilibrium by quasi static reactions, then T ds = dQ. In general the Second Law implies

dQ ≤ T ds (5.13) 56 CHAPTER 5. EQUATION OF STATE BELOW NEUTRON DRIP and therefore ² 1 d( ) + P d( ) ≤ T ds (5.14) n n If equilibrium is reached with n and s constant, eqn.(5.14) yields

d² ≤ 0 (5.15)

The equilibrium state corresponds to no further changes in ² (i.e. d² = 0), and is a minimum of ² at fixed n and s. Using eqn.(5.6) we recover eqn.(5.10). 3) System with constant T and n. Similarly if T and n are kept constant, eqn.(5.14) yields df ≤ 0 (5.16) where ² f ≡ − T s (5.17) n is the Helmholtz free energy per baryon. 4) System with constant T and P. Finally, if T and P are kept constant (the most frequent case), then dg ≤ 0 (5.18) where ² + P g ≡ − T s (5.19) n is the Gibbs free energy per baryon. Equilibrium corresponds to a minimum in this energy. The expression of the equilibrium condition is useful when there are phase-transitions accompanied by discontinuities in n and continuity in T and P . Using eqn.(5.6) and eqn.(5.19) we get 1 X dg = dP − sdT + µidYi (5.20) n i Again the condition that g is minimum at constant T and P implies eqn.(5.10).

5.2 Kinetic theory of the equation of state

In kinetic theory, the number density in the phase space of each species of particle dN /d3xd3p, provides a complete description of the system. One can define the 5.2. KINETIC THEORY OF THE EQUATION OF STATE 57 dimensionless distribution function in phase space, f(x, p, t) as dN g = f (5.21) d3xd3p h3 where h3 is the volume of a cell in phase-space, and g is the statistical weight, that is the number of states of a particle with a given momentum p. For massive particles g = 2S + 1 (S= spin), for photons g = 2, for neutrinos g = 1. The function f gives the average occupation number of a cell in phase-space. The product d3xd3p is Lorentz invariant (scalar under Lorentz transformations) and hence f is also Lorentz invariant. The number density of each species of particle is given by Z dN n = d3p (5.22) d3xd3p The energy density is given by Z dN ² = E d3p (5.23) d3xd3p where E = (p2c2 + m2c4)1/2 (5.24) where m is the rest-mass of the particle. The pressure of a system with an isotropic distribution of momenta is given by 1 Z dN P = pv(p) d3p (5.25) 3 d3xd3p where v(p) is the velocity-momentum relationship (depends on relativistic condi- tions), and the factor 1/3 comes from isotropy. The velocity-momentum relation- ship is v = pc2/E. Equation (5.25) simply states that pressure is a momentum flux. For an ideal gas in equilibrium, f has the form

1 f(E) = (5.26) exp[(E − µ)/kT ] ± 1 where the upper sign refers to fermions (Fermi-Dirac statistics) and the lower sign to bosons (Bose-Einstein statistics), k is the Boltzmann constant, and µ 58 CHAPTER 5. EQUATION OF STATE BELOW NEUTRON DRIP is the chemical potential. Often instead of the chemical potential µ one uses the quantity η = µ/kT (also known as the degeneracy parameter). It is worth recalling that the quantities µ or η and 1/kT stem from the conditions of total particle number conservation and total energy conservation via the so- called Lagrangian multipliers (see any textbook of Statistical Mechanics). It means that when the total number of particles is not conserved (like for photons that are continuously emitted and absorbed), the terms η or µ are zero. Summary. With the above definitions we may write the Number Density, the Energy Density and the Pressure in a very general way N g Z p2dp n = = 4π V h3 exp[(E − µ)/kT ] ± 1 E g Z p2dp ² = = 4π E V h3 exp[(E − µ)/kT ] ± 1 1 g Z p2dp P = 4π pv(p) 3 h3 exp[(E − µ)/kT ] ± 1 Depending on the type of particles and relativistic conditions (i.e. the relation v(p)), many different cases can be derived from the three relations above. The chemical potential µ and/or the degeneracy parameter η drive the problem. Both can be negative or positive depending on the physical conditions. In the following we will limit ourselves to the case of Fermions (electrons in most circumstances) and present some typical situations that occur in stellar interiors. The case of Bosons will be limited to photons. 1) Non degenerate perfect gas. For low particle densities and/or high temperature, the corresponding µ and η are large and negative (µ → −∞, η → −∞) therefore the exponential term in eqn.(5.26) is much larger than 1, and the distribution function f(E) reduces to µ − E f(E) ' exp( ) (5.27) kT with f(E) << 1, i.e. the Maxwell-Boltzmann distribution. Using the relation defining the number density of particles we get µ h3 n n = ln[ ] ≡ ln[ ] (5.28) kT (2πmkT )3/2 g n∗ 5.2. KINETIC THEORY OF THE EQUATION OF STATE 59 i.e. µ/kT < 0 if the density fall below a critical value n∗ g n∗ ≡ (2πmkT )3/2 (5.29) h3

Inserting m = Amu and n = ρ/Amu, we obtain µ h3 1 1 ρ = ln[ 3/2 3/2 ] (5.30) kT (2πAmuk) g Amu T or µ 3.2 × 10−3ρ = ln[ ] (5.31) kT gA5/2T 3/2 For the air (A = 30) at temperature T = 273 K one obtains µ/kT = −30. Energy density and pressure depend on the relativistic conditions. 1a) Non Relativistic. We take the expression for the particle energy E = [p2c2 + m2c4]1/2 and since in this case p/mc << 1 we expand it in series of (p/mc)2 so that 1 p E ' mc2[1 + ( )2 + ...] 2 mc The particle (kinetic) energy is simply given by p2 E = E − mc2 ' NR 2m (the rest-mass energy of the particle has been subtracted). The number density of particles is

Z ∞ 2 Z ∞ 2 g p dp g η p dp n = 4π p2 ' 4πe p2 (5.32) h3 0 h3 0 exp[ 2mkT − η] + 1 exp[ 2mkT ] posing x = p2/2mkT and mkT dx = pdp, we obtain

g Z ∞ x1/2dx n = 2π(2mkT )3/2eη (5.33) h3 0 exp[x] 1 3 √ The integral is the Γ( 2 + 1) function, Γ( 2 ) = π/2. Therefore g n = (2πmkT )3/2eη (5.34) h3 which is exactly eqn.(5.28) above. Applying the same procedure to the energy density and pressure we obtain 3 ² = nkT (5.35) 2 60 CHAPTER 5. EQUATION OF STATE BELOW NEUTRON DRIP

2 P = nkT ≡ ² (5.36) 3 1b) Fully Relativistic. For non degenerate fully relativistic particles p/mc >> 1 and E ' pc. In this case we derive the following relations

³kT ´3 n = 8πg eη (5.37) hc

³kT ´3 ² = 24g kT eη ≡ 3nKT (5.38) hc

1 P = nkT ≡ ² (5.39) 3 It worth noting that for non degenerate fermions, the pressure in extreme rela- tivistic conditions is the same as in the non relativistic ones, whereas the number density and energy density do not. 2) Partially degenerate perfect gas of fermions. This is the most com- plicated situations because we cannot simplify the distribution function f(E). We start from the general expressions for number density, energy density, and pressure but we retain the parameter η under the integrals. Also in this case we distinguish between non relativistic and fully relativistic conditions.

g Z p2dp n = 4π h3 exp[(E − µ)/kT ] + 1

g Z p2dp ² = 4π E h3 exp[(E − µ)/kT ] + 1 1 g Z p2dp P = 4π pv(p) 3 h3 exp[(E − µ)/kT ] + 1 2a) Non Relativistic. The particle energy (kinetic) can be approximated to E ' p2/2m. Immediately we get: The number density

g Z ∞ p2dp n = 4π p2 (5.40) h3 0 exp[ 2mkT − η] + 1 5.2. KINETIC THEORY OF THE EQUATION OF STATE 61 the energy density

g Z ∞ p2 p2dp ² = 4π p2 (5.41) h3 0 2m exp[ 2mkT − η] + 1 and finally the pressure

g Z ∞ p2 p2dp P = 4π p2 (5.42) h3 0 m exp[ 2mkT − η] + 1

Introducing the auxiliary variable x = p2/2mkT (and mkT dx = pdp), all the integrals acquire the form

Z ∞ xndx Fn(η) = (5.43) 0 exp[x − η] + 1 They are called the Fermi Integrals. The suffix n stands for the power of the variable x in the numerator of the above integrals. In the expression for the number density n = 1/2 in that for the energy and pressure n = 3/2. The calculation of the Fermi integrals is a cumbersome affair but for a few extreme cases, like complete degeneracy. With the aid of the Fermi integrals we obtain

g n = 2π(2mkT )3/2F (η) (5.44) h3 1/2 This is an implicit relation between the number density, temperature and degen- eracy parameter η. It can be used to derive η at given n and T . This equation shows that degeneracy increases at increasing density and decreasing tempera- ture. The solution of this equation is not straightforward and numerical methods must be used.

g ² = 2π(2mkT )3/2kT F (η) (5.45) h3 3/2 or F (η) ² = nkT 3/2 (5.46) F1/2(η) and finally 62 CHAPTER 5. EQUATION OF STATE BELOW NEUTRON DRIP

2 F (η) 2 P = nkT 3/2 ≡ ² (5.47) 3 F1/2(η) 3 2b) Fully Relativistic. In fully relativistic conditions we have p/mc >> 1 and E ' pc. the number density is given by

g Z ∞ p2dp n = 3 4π pc−η (5.48) h 0 exp[ kT ] + 1 Posing x = pc/kT one gets Z ³kT ´3 ∞ x2dx n = 4πg (5.49) hc 0 exp[x − η] + 1 or

³kT ´3 n = 4πg F (η) (5.50) hc 2 Similarly g Z ∞ p2dp ² = 3 4π pc pc+η (5.51) h 0 exp[ kT ] + 1 and

² = nkT F3(η) (5.52)

For the pressure we have

1 g Z ∞ p2dp P = 3 4π pc pc−η (5.53) 3 h 0 exp[ kT ] + 1 and

1 1 P = ² ≡ nkT F (η) (5.54) 3 3 3 3) Fully degenerate perfect gas. This is the simplest case to analyze. For completely degenerate fermions (T → 0 and/or µ/kT → ∞, η → ∞) µ is called the Fermi energy EF and the distribution function becomes

f(E) = 1 E ≤ EF = µ 5.2. KINETIC THEORY OF THE EQUATION OF STATE 63

f(E) = 0 E > EF = µ (5.55)

The major difference is related to the Fermi integrals. It can be shown that

Z η n x 1 n+1 lim Fn(η) = dx = η (5.56) η→∞ 0 exp(x − η) + 1 n + 1 3a) Non Relativistic g 2 n = 2π(2mkT )3/2 η3/2 (5.57) h3 3

g 2 ² = 2π(2mkT )3/2kT η5/2 (5.58) h3 5 and replacing η with the aid of eqn (5.57)

3h2 ³ 3 ´3/2 ² = (n)5/3 (5.59) 10m 4πg from which

5/3 5/3 P ∼ n ∼ (YF ρ) (5.60) where YF is the concentration of fermions per baryon. 3b) Fully Relativistic ³kT ´3 1 n = 4πg η3 (5.61) hc 3

³kT ´3 1 ² = 4πg kT η4 (5.62) hc 4 or 3³ 3 ´1/3 ² = hc(n)4/3 (5.63) 4 4πg and 4/3 4/3 P ∼ n ∼ (YF ρ) (5.64)

The most remarkable result here is that in a fully degenerate gas, the pressure does not depend on temperature. This is of paramount importance in relation to stellar structure and evolution. 4) Partially degenerate and partially relativistic gas of fermions. This is the most complicate case to deal with. As it would be of marginal interest 64 CHAPTER 5. EQUATION OF STATE BELOW NEUTRON DRIP in the present context we leave it aside. The reader is referred to specialized books of Statistical Mechanics and Stellar Structure Theory. Later we will consider the case of fully degenerate partially relativistic gas of fermions which is the typical situation of White Dwarfs and Neutron Stars. 5) Radiation (the fully relativistic bosons). Finally, we derive the energy density and pressure of a gas of photons (radiation). In this case µ = 0, η = 0, g = 2, and the distribution function is

1 f(E) = (5.65) exp[E/kT ] − 1 Repeating the usual procedure we derive

Z ∞ x3dx ² = 8π(kT )3kT (5.66) 0 ex − 1 The integral is π4/15. Finally

8π5k4 ² = T 4 ≡ aT 4 (5.67) 15(hc)3 a is the radiation constant (see the theory of Black Body). The pressure is

1 1 P = ² = aT 4 (5.68) 3 3 5.3 Fully degenerate (partially relativistic) gas of fermions

Even if there is nothing special in this case compared to the previous ones, we treat it separately because in many circumstances we will use the relations for the number density, energy density and pressure of this case. Since it is fully degenerate F (E) = 1 for 0 ≤ E ≤ EF and F (E) = 0 for E > EF , where EF is the Fermi energy. So a great deal of simplification is possible. Thereinafter we will limit ourselves to consider only the case of electrons. The corresponding Fermi momentum is defined by

2 2 2 4 1/2 EF = (pF c + mec ) (5.69) 5.3. FULLY DEGENERATE (PARTIALLY RELATIVISTIC) GAS OF FERMIONS65

Since the gas is considered partially relativistic no simplifications can be made for the energy. The number density is Z pF 2 2 8π 3 ne = 4πp dp = pF (5.70) h3 0 3h3

Introducing the dimensionless Fermi momentum x = pF /mec then

1 3 ne = 2 3 x (5.71) 3π λe where λe =h/m ¯ ec is the Compton wave length of electrons. The pressure is given by Z PF 2 2 1 p c 2 Pe = 2 2 2 4 1/2 4πp dp (5.72) 3 0 (p c + mec )

4 5 Z x 4 8πmec x dx Pe = (5.73) 3h3 0 (1 + x2)1/2 or 2 mec 25 −2 Pe = 3 φ(x) = 1.4218 × 10 dyne cm (5.74) λe where 1 φ(x) = {x(1 + x2)1/2(2x3/3 − 1) + ln[x + (1 + x)1/2]} (5.75) 8π2 Similarly, for the energy density we obtain Z pF 2 2 2 2 4 1/2 2 ²e = (p c + mec ) 4πp dp (5.76) h3 0

2 mec 25 ²e = 3 χ(x) ' 1.4 × 10 erg/baryon (5.77) λe where 1 χ(x) = {x(1 + x2)1/2(1 + 2x2) − ln[x + (1 + x2)1/2]} (5.78) 8π2 Even if electrons contribute very much to pressure, the mass density is dominated by the rest-mass of ions. This density is

X ρ0 = niMi (5.79) i 66 CHAPTER 5. EQUATION OF STATE BELOW NEUTRON DRIP

where ni and Mi are the number density and mass of the species i. If we define the mean rest-mass of baryons P 1 X niMi mB = niMi = P (5.80) n i niAi then

nemB ρ0 = nmB = (5.81) Ye where Ye is the concentration of electrons (mean number density of electrons per 12 −24 baryon). For example, for pure fully ionised C, mB ≡ mu ≡ 1.66057 × 10 6 3 3 and Ye = Z/A ' 0.5, ρ0 ' 1.95 × 10 x g/cm . Sometimes it is useful to introduce the quantity

mB µe = (5.82) muYe called the mean molecular weight per electron, so that

6 3 −3 ρ0 = µemune = 0.97395 × 10 µex g cm (5.83) or ρ x = 1.088 × 10−2( 0 )1/3 (5.84) µe −3 where ρ0 is in g cm . Often the distinction between mB and mu can be dropped. Similarly, a quantity µ (the mean molecular weight) is introduced so that

X ρ0 = (ne + ni)µmu (5.85) i and 1 X mu = (Ye + Yi) (5.86) µ mB

Except for very accurate evaluations mu/mB ≡ 1. The mean molecular weight is particularly useful in a non degenerate gas, when the pressure is

X ρ0 P = (ne + ni)kT ≡ kT (5.87) i µmu 5.3. FULLY DEGENERATE (PARTIALLY RELATIVISTIC) GAS OF FERMIONS67

The relations (5.74) and (5.83) yield the equation of state of an ideal fully degen- erate gas P = P (ρ0) parametrically in terms of x, that determines the relativistic 2 2 conditions. Note that ρ = ρ0 + ²e/c and that usually the term ²e/c is negligible. Useful approximations. There are useful approximations for φ(x) and χ(x) that we list below limited to the first leading term (see specialized textbooks for more details): For x << 1 (non relativistic)

1 1 φ(x) → x5 χ(x) → x3 (5.88) 15π2 3π2

For x >> 1 (fully relativistic)

1 1 φ(x) → x4 χ(x) → x4 (5.89) 12π2 4π2

In both cases the equation of state can be written in the polytropic form that we have already found.

Γ P = Kρ0 (5.90) where K and Γ are constants, in the two limiting cases 6 3 For x << 1 and/or ρ0 << 10 g/cm (non relativistic)

5 32/3π4/3 h¯2 1.0036 × 1013 Γ = ,K = 5/3 5/3 = 5/3 cgs (5.91) 3 5 memu µe µe

6 3 For x >> 1 and/or ρ0 >> 10 g/cm (fully relativistic)

4 31/3π2/3 hc¯ 1.2435 × 1015 Γ = ,K = 4/3 4/3 = 4/3 cgs (5.92) 3 4 mu µe µe The case of neutrons. The above results can be scaled trivially with particle mass mi and statistical weight gi to determine the equation of state of an arbitrary species of ideal fermions. For neutrons the basic relations are: 6 3 For x << 1 and/or ρ0 << 10 g/cm (non relativistic)

pF xn = (5.93) mnc 68 CHAPTER 5. EQUATION OF STATE BELOW NEUTRON DRIP

2 mnc 38 3 ²n = 3 χ(xn) = 1.625 × 10 erg/cm (5.94) λn

mn 1 3 15 3 3 ρ0 = mnnn = 3 3 x = 6.1067 × 10 xn g/cm (5.95) λn 3π

Γ P = Kρ0 (5.96)

15 3 For x << 1 and/or ρ0 << 6 × 10 g/cm (non relativistic)

2/3 4/3 2 5 3 π h¯ 9 Γ = ,K = 8/3 = 5.3802 × 10 cgs (5.97) 3 5 mn

15 3 For x >> 1 and/or ρ0 >> 10 g/cm (fully relativistic)

1/3 2/3 4 3 π hc¯ 15 Γ = ,K = 4/3 = 1.2293 × 10 cgs (5.98) 3 4 mn

2 In this case the mass density ρ = ²n/c is entirely due to neutrons and greatly exceed ρ0 whenever the neutrons are relativistic.

5.4 Electrostatic corrections to the Equation of State

The ideal Fermi gas equation of state presented in the previous section was used by Chandrasekhar (1931) in his pioneering study of the equilibrium structure of White Dwarf that will be presented in chapter (8). There are two important cor- rections to be applied to this equations of state: (a) the electrostatic interactions among the ions and the electrons; and (b) the effect of inverse β-decay. The electrostatic correction is the subject of this section. The electrostatic correction arises because the positive charges are not uniformly distributed in the gas but are concentrated in nuclei (ions) of charge Z. This decreases the energy and pressure in turn of the ambient electrons: the repelling electrons are, on the average, further apart than the mean distance between nuclei and electrons, so repulsion is weaker than attraction. 5.4. ELECTROSTATIC CORRECTIONS TO THE EQUATION OF STATE69

In a non degenerate gas, the Coulomb effects become more important as the density increases. The ratio between the Coulomb and thermal energy is given by E Ze2/ < r > Ze2n1/3 C = ' e (5.99) kT kT kT −1/3 which increases with ne. Here < r >' ne is the characteristic electron-ion separation. For a degenerate gas the comparison is made with the Fermi energy 0 2 in non relativistic approximation EF ' pF /2me,

2 EC Ze / < r > 0 = 2 (5.100) EF pF /2me

3 Using the ne ∝ pF relation for fully degenerate non relativistic gas of electrons we get EC 1 2/3 Z 1 ne −1/3 0 = 2( 2 ) 1/3 = ( 3 22 −3 ) (5.101) EF 3π a0 ne Z × 6 × 10 cm 2 2 0 where a0 =h ¯ /mee is the Bohr radius. Thus EC << EF in most astrophysical degenerate gases. We may derive an approximate expression for the correction to the degenerate equation of state of an ideal Fermi gas using the fact that to a first approximation ne is uniform. As the temperature tends to zero, the ions tend to be located in a lattice that maximizes the inter-ion separation (minimizes the energy of ion-ion electrostatic interactions). Around each nucleus of the lattice consider a spherical 3 cell with volume 4πr0/3 = 1/nN where nN is the number density of nuclei. The gas is imagined to be divided in neutral spherical cell of radius r0 each of which contains the Z electrons closest to the nucleus (Wigner-Seitzer model). The total electrostatic energy of any one cell is the sum of the potential energy due to electron-electron (e−e) interaction and the electron-nucleus (e−n) interaction. To assemble a uniform sphere of Z electrons requires the energy Z r0 qdq E(e−e) = (5.102) 0 r where r3 q = −Ze 3 (5.103) r0 70 CHAPTER 5. EQUATION OF STATE BELOW NEUTRON DRIP is the charge inside the sphere of radius r. It is supposed that the charge distri- bution in the sphere is uniform. The integration gives 3 Z2e2 E(e−e) = (5.104) 5 r0 To assemble the electron sphere around the nucleus require the energy Z r0 dq E(e−n) = Ze (5.105) 0 r 3 Z2e2 E(e−n) = − (5.106) 2 r0 The total energy is

9 Z2e2 EC = E(e−e) + E(e−n) = − (5.107) 10 r0 The electrostatic energy per electron is E 9 4π C = − ( )1/3Z2/3e2n1/3 (5.108) Z 10 3 e where we have replaced r0 with the aid of 3Z ne = 3 (5.109) 4πr0 The numerical factor of eqn.(5.108) is 1.45079 very close to the value of 1.44423 for a body-centered cubic lattice (BCC). The corresponding pressure is

2 d(EC /Z) 3 4π 1/3 2/3 2 4/3 PC = ne = − ( ) Z e ne (5.110) dne 10 3 The contribution to the total pressure is negative! First, let us examine the consequences for the extremely relativistic case of fully degenerate electrons. There the ideal Fermi gas has pressure hc¯ P → (3π2)1/3n4/3 (5.111) 0 4 e

The ratio of the total pressure P0 + PC to P0 is P P + P 25/3 3 = 0 C = 1 − ( )1/3αZ2/3 (5.112) P0 P0 5 π 5.4. ELECTROSTATIC CORRECTIONS TO THE EQUATION OF STATE71 where α = e2/hc¯ = 1/137 is the fine structure constant. Although the above Coulomb correction is relatively small, it is important in high density White Dwarfs and low density Neutron Stars.

In the non relativistic limit, where P0 is

5/3 2 2 2/3 ne P0 =h ¯ (3π ) (5.113) 5me we obtain 2/3 P0 + PC Z = 1 − 1/3 1/3 (5.114) P0 2 πa0n which predicts P = 0 when Z2 ne = 3 3 (5.115) 2π a0 corresponding to a density

2 3 ρ0 ' 0.4Z g/cm (5.116)

3 for A = 2Z. This equation gives ρ0 ' 250 g/cm for iron, instead of the lab- oratory value of 7.86 g/cm3. The reason for the discrepancy is that at low densities the gas of electrons is not uniformly distributed. The electron shell effects mask the simpler statistical effects. The simplest statistical treatment of atomic structure is the Thomas-Fermi model. Thomas-Fermi model. One assumes that within each Wigner-Seitzer cell, the electrons move in a slowly varying spherically symmetric potential V (r). Because the potential is nearly constant, we can use free-particle Fermi-Dirac statistics. This means that we are taking the interaction energy between electrons to be much less than the kinetic or the potential energies of an individual electron.

Therefore at any r all states up to E = EF are occupied. The energy EF does not depend on r, otherwise electrons would migrate to a region of smaller EF . Thus 2 pF (r) EF = −eV (r) + = constant (5.117) 2me 72 CHAPTER 5. EQUATION OF STATE BELOW NEUTRON DRIP

where pF is the maximum momentum of electrons at r. The corresponding num- ber density of electrons is 8π 8π n = p3 = {2m [E + eV (r)]}3/2 (5.118) e 3h3 F 3h3 e F The potential V (r) is determined by Poisson’s equation

2 ∇ V = 4πene + nuclear contribution (5.119)

The nuclear contribution is a delta function about the origin, so we can ignore it at r > 0 and impose the condition

lim rV (r) = Ze (5.120) r→0

The boundary condition at the cell surface r0 is that the electric field vanish ¯ dV ¯ ¯ = 0 (5.121) dr r0 Equations (5.118) and (5.119) give the relation 1 d2 32π2e (rV ) = [2m (E + eV )]3/2 (5.122) r dr2 3h3 e F to be solved with the boundary conditions (5.120) and (5.121). It is convenient to change to dimensionless units by putting

r = µx (5.123) and Ze2 E + eV (r) = Φ(x) (5.124) F r where ³ 9π2 ´1/3 µ = a (5.125) 128Z 0 Equation (5.122) becomes d2 Φ3/2 Φ = (5.126) dx2 x1/2 with boundary conditions

0 Φ(x0) Φ(0) = 1 and Φ (x0) = (5.127) x0 5.4. ELECTROSTATIC CORRECTIONS TO THE EQUATION OF STATE73

This equation has to be solved numerically. The results are as follows: (a) Starting at the origin with Φ(0) = 1 there is a special value of Φ0(0) = −1.588 for which the solution is asymptotic to the x-axis at large x and the surface 0 condition (5.127) is satisfied by having Φ (x0) → 0, Φ(x0) → 0 for x0 → ∞. This corresponds to P → 0 and ρ → 0 for x0 → ∞. The result is not physically consistent because it would imply atoms with infinite radius. The model can be corrected. (b) For Φ0(0) > −1.588 the solution does not vanish anywhere and diverges for x → ∞. The surface condition (127) is satisfied at some finite value x0. It corresponds to atoms under pressure. (c) We are not interested here to the case with Φ0(0) < −1.588 for which Φ(x) = 0 at finite x0 corresponding to free positive ions. We derive now the pressure at the border of the cell using the free-particle ex- pression Z pF 2 1 2 p 2 P = 3 4πp dp (5.128) 3 h 0 me

8π 5 P = 3 pF (r0) (5.129) 15h me Note that since dV/dr = 0 at the cell boundary, no force is exerted on a cell by neighbouring cells. Finally, we recast the pressure as a function of Φ(x0) and x0. With the aid of eqn.(5.118), (5.123) - (5.125), eqn.(5.129) becomes

2 2 1 Z e Φ(x0) 5/2 P = 4 [ ] (5.130) 10π µ x0

The mass density is simply given by the total rest mass inside the cell

3Amb ρ0 = 3 3 (5.131) 4πµ x0 The equations (5.130) and (5.131) represent the equation of state parametric in x0.

For low densities x0 → ∞ the asymptotic solution of eqn.(5.126) is

144 Φ(x) ' for x → ∞ (5.132) x3 74 CHAPTER 5. EQUATION OF STATE BELOW NEUTRON DRIP

Thus

−10 10/3 P ∝ x0 ∝ ρ0 (5.133)

The equation of state at low densities gets rather stiff with Γ ' 3.3. At high densities the Thomas-Fermi equation of state recovers the eqn.(5.114). The Thomas-Fermi equation of state can be used up to densities of 104 g/cm3, for higher densities one can use the ideal gas description with the corrections for the Coulomb lattice.

5.5 Inverse β-decay: corrections to the equation of state

At high densities the most important correction to the equation of state is due to inverse β-decay e− + p → n + ν (5.134)

The protons and neutrons are generally bound to nuclei. In this section we will explore the effects of inverse β-decay in a gas made of free electrons, protons and neutrons. We assume that the neutrinos generated in eqn.(5.134) escape from the system (very low interaction cross section). Later we will consider the more realistic case of bound nuclei. The reaction (5.134) can proceed whenever the electrons acquire enough energy to balance the mass difference between proton and neutron,

2 (mn − mp)c = 1.29 Mev

The inverse β-decay is effective in transforming protons into neutrons if the direct β-decay n → p + e +ν ˜ (5.135) does not occur. 5.5. INVERSE β-DECAY: CORRECTIONS TO THE EQUATION OF STATE75

Reaction (5.135) is blocked if the density is high enough that all electron energy levels in the Fermi sea are occupied up to the one that the emitted electron would fill. Thus there is a threshold density for the onset of reaction (5.134). We study the properties of this mixture of particles assuming they are in equilibrium, thus the relation between the chemical potential can be used

µe + µp = µn (5.136)

µν = 0 as the neutrinos leave the system (their number density is set zero). Define

e n p pF pF pF xe = xn = xp = (5.137) mec mnc mpc As

e 2 2 4 1/2 µe = [(pF c) + mec ] and similar for p and n, eqn.(5.136) becomes

2 1/2 2 1/2 2 1/2 me(1 + xe) + mp(1 + xp) = mn(1 + xn) (5.138)

Charge neutrality ne = np implies that

1 3 1 3 2 3 xe = 2 3 xp (5.139) 3π λe 3π λp that is

mexe = mpxp (5.140)

The equation of state can now be easily obtained as a function, for instance, of the parameter xe. The procedure is as follows: choose a value of xe; eqn.(5.140) gives xp and eqn.(5.138) gives xn. Then the pressure is

2 2 2 mec mpc mnc P = 3 φ(xe) + 3 φ(xp) + 3 φ(xn) (5.141) λe λp λn the energy density is

2 2 2 mec mpc mnc ² = 3 χ(xe) + 3 χ(xp) + 3 χ(xn) (5.142) λe λp λn 76 CHAPTER 5. EQUATION OF STATE BELOW NEUTRON DRIP and the number density is

1 3 1 3 n = 2 3 xp + 2 3 xn (5.143) 3π λp 3π λn

The minimum density at which neutrons appear can be derived by setting xn = 0 in eqn.(5.138). At these densities the protons are not relativistic so that xp << 1. Therefore from eqn.(5.138) we have

2 1/2 me(1 + xe) = Q = mn − mp (5.144)

Solving for xe we obtain from eqn.(5.139) and eqn.(5.143) that

1 Q 2 3/2 30 −3 n = 2 3 [( ) − 1] = 7.37 × 10 cm (5.145) 3π λe me and hence 7 3 ρ0 = nmp ' 1.2 × 10 g/cm (5.146)

At higher densities, the equilibrium composition contains an increasing percent- age of neutrons. The composition can be determined by substituting eqn.(5.140) into eqn.(5.138)

2 2 2 1/2 2 1/2 2 1/2 (me + mpxp) + mp(1 + xp) = mn(1 + xn) (5.147)

Squaring eqn.(5.147) twice and simplifying we obtain

2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4mnmpxp(1+xn) = (Q −me)[(mn +mp) −me]+2mnxn(mn −mp −me)+mnxn (5.148) Thus

n ³ m x ´3 p = p p nn mnxn 2 2 2 2 2 2 2(mn−mp−me) (Q−me)[(mp+mn) −me] h1 + 2 2 + 4 4 i3/2 1 mnxn mnxn = 1 (5.149) 8 1 + 2 xn

Now Q and me are both << mn so

h 2 2 2 2 4 i np 1 1 + 4Q/mnxn + 4(Q − me)/mnxn 3/2 ' 2 (5.150) nn 8 1 + 1/xn 5.5. INVERSE β-DECAY: CORRECTIONS TO THE EQUATION OF STATE77

Thus, the ratio np/nn initially decreases as xn increases (as density increases). It reaches a minimum value of

n h Q (Q2 − m2)1/2 i3/2 p ' + e = 0.0026 (5.151) nn mn mn at 3/2 ³ 2 2 ´ 2 Q − me 3/4 nn = 2 3 2 (5.152) 3π λn mn 11 3 ρ0 = mnnn = 7.8 × 10 g/cm (5.153) and then rises monotonically to the limit value of 1/8 for xn → ∞ or ρ0 → ∞. Note that the equilibrium determined here is stable because it corresponds to a minimum of ². The results of this section cannot be applied to real stars, because thermodynamic equilibrium cannot be achieved in open systems, the n- p-e composition must be derived, and the matter is initially organised in nuclei with bound protons and neutrons.

5.5.1 The Harrison-Wheeler equation of state

We now turn to a quantitative treatment of the inverse β-decay in realistic stellar conditions for the density range 107 < ρ < 1011 g/cm3. In a star of about 57 1 M¯ there are about 10 baryons (protons and neutrons) locked up in nuclei 33 57 (A ' M/mu ' 10 /mu ' 10 ) immersed in a bath of ' A/2 electrons. We want to determine the lowest energy state for a system with the above number of baryons in β-equilibrium with a relativistic electron gas. We must determine (a) which nuclei are present, i.e. the values of A and Z that minimize the energy, and (b) the corresponding pressure. We assume that past nuclear burnings, the material will cool down to complete thermodynamics equilibrium. The matter composition and equation of state will then be determined by the lowest energy state. Suppose that the assembly of baryons has A ≤ 90, the state of lowest energy 56 consists in a single nucleus, with the 26F e the nucleus with the tightest binding. 78 CHAPTER 5. EQUATION OF STATE BELOW NEUTRON DRIP

For A ≥ 90 the state of lowest energy corresponds to more than one nucleus; the tightest binding energy is obtained for A values integral multiples of 56. Thus at increasing A, it becomes more and more appropriate to describe the matter 56 of minimum energy as made of pure 26F e. The story changes when A increases above A ' 1057 and the self-gravitation becomes important. For baryons in hydrostatic equilibrium, the density exceeds ρ ' 107 g/cm3. Consequently, the electrons become degenerate and relativistic. They can combine with bound nuclear protons to form neutrons, altering the equilibrium nuclear composition 56 step-by-step from 26F e to more neutron-rich material. The overall physical picture can be visualized as follows. If nuclear forces alone determine the structure of equilibrium matter, nucleons would accumulate in nuclei of unlimited size. However, the Coulomb repulsive forces (among protons) become so strong that such large nuclei would undergo nuclear fission. For low densities, these two opposing effects balance at A = 56. However, when the relativistic electrons enter the picture, the balance shift. In nuclei that contain a larger proportions of neutrons to protons (because of the inverse β-decay), the Coulomb forces play a smaller role. Hence there is a natural tendency for larger nuclei to form. When the density increases to ∼ 4 × 1011 g/cm3, the ratio n/p reaches a critical level. Any further increase of density lead to ”neutron drip”, i.e. a two phase system in which nuclei, free electrons, and free neutrons co-exist and together determine the lowest energy state. Above this density threshold more and more free neutrons are present. Finally when the density is above 4 × 1011 g/cm3 more pressure is provided by free neutrons. The neutron gas so controls the situation that one can describe the medium as a vast nucleus with lower-than-normal nuclear density. Given these general considerations, we start by writing the energy density of a mixture of nuclei, free electrons, and free neutrons in the form

0 ² = nN M(A, Z) + ²e(ne) + ²n(n) (5.154) where M(A, Z) is the energy of a nucleus (A, Z) including the rest mass of the 5.5. INVERSE β-DECAY: CORRECTIONS TO THE EQUATION OF STATE79 nucleons. It is conventional in nuclear physics to include also the rest-mass of the electrons associated to the M(A, Z) nucleus. Therefore we must subtract 2 the term nemec from the energy density of the electrons in eqn.(5.154). This 0 is why we have denoted it as ²e(ne). The quantity nN is the number density of nuclei, while nn is the number density of free neutrons. The baryon density n and electron density ne are given by

n = nN A + nn and ne = nN Z (5.155) that is

1 = AYN + Yn and Ye = YN Z (5.156) where the Y ’s are the concentrations. Thanks to these relations the energy density which is usually a function ²(n, YN ,Ye,Yp) becomes ²(n, A, Z, Yn). The quantity M(A, Z) is poorly known experimentally for very neutron-rich nuclei produced above 1010 g/cm3 and must be inferred theoretically. This usually done by means of semi-empirical mass expressions, among which a popular one is

2 2 M(Z,A) = [(A − Z)muc + Z(mp + me)c − AEb] 1 Z Z2 = m c2[b A + b A2/3 − b Z + b A( − )2 + b ] (5.157) u 1 2 3 4 2 A 5 A1/3 where Eb is the mean binding energy per baryon and where the five coefficients are

b1 = 0.99175, b2 = 0.00084, b3 = 0.00076

b4 = 0.01911, b5 = 0.10175 (5.158)

This expression is based on the drop model of a nucleus. The terms have the following interpretation

(i) The dominant contribution to Eb is proportional the volume (hence A because 1/3 the nuclear radii scale as R ∝ A ). This is the meaning of the large value of b1.

(ii) The term b2 is the surface energy.

(iii) The term b3 is just (mn − mp − me)/mu 80 CHAPTER 5. EQUATION OF STATE BELOW NEUTRON DRIP

(iv) The term b4 is the symmetry term: it favors nuclei with equal number of protons and neutrons.

(v) The term b5 is the Coulomb energy; the inter-proton mean distance is taken A1/3. (vi) Pairing and shell effects have been neglected. Recalling that n Z n = (1 − Y ) n = n(1 − Y ) n = nY (5.159) N A n e n A n n eqn.(5.154) becomes M(Z,A) ² = n(1 − Y ) + ²0 (n ) + ² (n ) (5.160) n A e e n n and furthermore,

0 0 d²e ²e + Pe 2 = = EFe − mec (5.161) dne ne d²n ²n + pn = = EFn (5.162) dnn nn Although A and Z vary in discrete steps, Harrison & Wheeler considered them as continuous variables. Equilibrium conditions: Start with ∂² ∂M = 0 → = −(E − m c2) (5.163) ∂Z ∂Z Fe e This is the continuum limit of the β-stability condition: M(Z − 1,A) in equilibrium with M(Z,A), the free electron being at the top of the Fermi sea. Similarly ∂² ∂ ³M ´ = 0 → A2 = Z(E − m c2) (5.164) ∂A ∂A A F e e This is the continuum limit of (A−1) atoms of type (Z,A) in equilibrium with A atoms of type (Z,A − 1). An additional Z free electrons must be created with energies at the top of the Fermi sea as the nuclear charge increases from (A − 1)Z to AZ. 5.5. INVERSE β-DECAY: CORRECTIONS TO THE EQUATION OF STATE81

Equation (5.163) and eqn.(5.164) can be combined to give ∂M ∂M Z + A − M = 0 (5.165) ∂Z ∂A The condition ∂² ∂M = 0 → = EFn (5.166) ∂Yn ∂A which is the continuum version of the equilibrium between M(Z,A) and M(Z,A − 1) and a free neutron. Equation (5.163) gives ³ ´ h i 2Z Z 2 1/2 me b3 + b4 1 − − 2b5 1/3 = (1 + xe) − 1 (5.167) A A mu with xe = pe/mec. Equation (5.165) gives b Z = ( 2 )1/2A1/2 = 3.54A1/2 (5.168) 2b5 while eqn.(5.166) yields

−1/3 2 2 2A 1 Z Z 2 1/2 mn b1 + b2 + b4( − 2 ) − b5 4/3 = (1 + xn) (5.169) 3 4 A 3A mu where xn = pn/mnc. Note that while Z increases with A the ratio Z/A decreases with A−1/2 under these high density conditions. How to construct the equation of state from these relations? With start with a value of A > 56, the eqn.(5.168) gives Z. Test whether neutron drip has been reached – that is whether eqn.(5.169) yields xn > 0. If so, compute nn,

Pn and ²n from equations (5.70), (5.74), and (5.77) with mn replacing me (ideal

Fermi gas). Otherwise set these quantities to zero. Equation (5.167) gives xe and 0 therefore ²e, Pe and ne. Finally we derive

² n M(A, Z)/Z + ²0 + ² ρ = = e e n (5.170) c2 c2

P = Pe + Pn (5.171) A n = n + n (5.172) e Z n 82 CHAPTER 5. EQUATION OF STATE BELOW NEUTRON DRIP

Figure 5.1: Left: Representative equations of state below neutron drip. The la- bels identify the equation of states discussed in the text: (Ch) the Chandrasekhar ideal electron gas for me = 56/26; (FMT) the Feynman- Metropolis-Teller equa- tion 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/dlnρ as a function of ρ for the equations of state in the left panel

56 The resulting equation of state is shown in Fig.5.1 for a pure 26F e composition. Deviations from the ideal Fermi gas occur at ρ > 107 g/cm3. The onset of neutron drip occurs at ρ ∼ 3.2×1011 g/cm3 for (A, Z) ∼ (122, 39.1) – i.e. near the element

Yttrium – and EFe ∼ 23.6 Mev. Above this density, free neutrons begin to supply a larger contribution to total pressure: at ρ ∼ 4 × 1012 g/cm3 where (A, Z) ∼ (187, 48.4) the neutrons provide about 66% of the pressure and nuclei are rapidly becoming unimportant.

5.5.2 Baym-Pethick-Sutherland equation of state

The Harrison-Wheeler equation of state has been derived assuming a simple an- alytical representation of M(Z,A). However, the rest nuclei have discrete values of A and Z, and shell effects play an important role in determining the bind- ing energy of nuclei. Furthermore, the fact that at low temperatures and high densities the nuclei tend to organize in lattice structure has been neglected. We have already pointed out that the lattice energy was important in determining 5.5. INVERSE β-DECAY: CORRECTIONS TO THE EQUATION OF STATE83 the equilibrium composition even though it was a small correction to electron pressure. The reason is that the equilibrium nuclide is determined largely by a competition between the nuclear surface energy and the Coulomb energy. This latter reduces the positive nuclear Coulomb energy by ∼ 15% and this affects the composition. Salpeter (1961) first took these effects into account. Baym-Pethick-Sutherland improved on Salpeter’s treatment adopting a better semi-empirical expression for M(Z,A) and introducing the lattice energy. The new expression for the energy density is M(Z,A) ² = n(1 − Y ) + ²0 (n ) + ² (n ) + ² (5.173) n A e e n n L For the lattice energy we adopt the same expression used for the electrostatic corrections 2/3 2 ²L = −1.444Z e ne4/3 (5.174)

The condition (∂²/∂Yn) = 0 now gives M(Z,A) + Z(E − m c2) + 4Z² /3n E = Fe c L e (5.175) Fn A Neutron drip begins when the l.h.s of the above equation equals the rest mass of 2 neutrons mnc .

We shall assume here we are always below the neutron drip and set ²n = 0 in the following discussion. The equation of state is derived by this procedure: choose a value for the number density n and try a pair of A and Z. The associated M(Z,A) are known either from experiments or sophisticated models. Since nN = n/A and ne = nZ/A, one can compute ². We try all possible values combinations of A and Z (values of M(Z,A)) and look for the pair (A, Z), the nuclide, that minimizes ² and yields the equilibrium nucleus. The pressure is given by ¯ 2 ∂(²/n)¯ P = n ¯ = Pe + PL (5.176) ∂n A,Z where Pe is the Fermi gas of electrons and 84 CHAPTER 5. EQUATION OF STATE BELOW NEUTRON DRIP

Table 5.1:

Equation of State Density Regime Composition Theory

Chandrasekhar 0 ≤ ρ ≤ ∞ e− Non interacting e− Ideal electron gas Nuclei specified by µe Ideal n − p − e− gas 0 ≤ ρ ≤ 1.2 × 107 e−, p Equilibrium Matter 1.2 × 107 ≤ ρ ≤ ∞ n, p, e− Equilibrium Matter 4 − 56 Feynmann-Metropolis-Teller 7.9 ≤ ρ ≤ 10 e and 26F e Thomas-Fermi-Dirac Atomic model 4 − 56 Harrison-Wheeler 7.9 ≤ ρ ≤ 10 e and 26F e Thomas-Fermi-Dirac Atomic model 4 7 − 56 10 ≤ ρ ≤ 10 e and 26F e Non interacting electrons 107 ≤ ρ ≤ 3 × 1011 e− and Equilibrium Nuclide Semiempirical Mass and equilibrium matter 3 × 1011 ≤ ρ ≤ 4 × 1012 e−, n and Equilibrium Nuclide Semiempirical Mass and equilibrium matter 4 × 1011 ≤ ρ ≤ ∞ Same as ideal n − p − e− Equilibrium Matter 4 − 56 Baym-Pethick-Sutherland 7.9 ≤ ρ ≤ 10 e and 26F e Thomas-Fermi-Dirac Atomic model 4 9 − 56 10 ≤ ρ ≤ 8 × 10 e and 26F e Ideal Electrons with Coulomb Lattice 8 × 109 ≤ ρ ≤ 4.3 × 1011 e− and Equilibrium Nuclide Laboratory Nuclear Energies Coulomb Lattice Energy Equilibrium Matter

1 P = ² (5.177) L 3 L Whenever there is a phase transition from one stable nuclide to another, there is a discontinuity in n and ρ = ²/c2. This occurs because the pressure inside a star must be a continuous function of the radius. We can estimate the size of the discontinuity in the following way: Pe depends only on ne (which is essentially continuous) and PL << Pe. But ne = n(Z/A), therefore we get

∆ρ ∆n ∆(Z/A) = = − (5.178) ρ n Z/A For instance passing from 56F e (Z/A=0.4643) to 62Ni (Z/A=0.4516), the density increases by about 3%. Because of these discontinuities, some special method must be envisioned to determine the pressure at which the transition actually occurs (see specialized textbooks on the subject). A summary of the equations of state we have presented is given in Table 5.1 and their pressure-density relations are shown in Fig.5.1. Chapter 6

Equation of State above the Neutron Drip

In this chapter we continue our analysis of the equation of state of cold, dense 11 3 matter in the density range from above the neutron drip, ρdrip = 4.3×10 g/cm , 14 3 to nuclear (ρnuc = 2.8 × 10 g/cm ), and above nuclear. At ρnuc the nuclei begin to dissolve and merge together. In this density interval the properties of dense matter and the equation of state are rather well understood. At density above

ρnuc, the physical properties of matter and the equation of state are still highly uncertain. We must emphasize that despite the effort the equation of state at very high densities, in particular above ρnuc is still uncertain with the consequence that the structure of high density neutron stars or part of them is still a vivid topic of investigation.

6.1 From ρdrip to ρnuc: the BBP equation of state

In the density interval ρdrip to ρnuc, matter is composed of nuclei, electrons, and free neutrons. As the density tends to ρnuc the nuclei disappear because their binding energy decreases with the increasing density. We can understand this in part since the attractive force between two unlike nucleons in 3 the S1 state which is crucial in binding the deuteron (see later) does not act on neutrons because of the Pauli Principle. So as the density increases and the nuclei

85 86 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP become more and more neutron rich, their stability decreases until a critical value of neutron number is reached, at which point the nuclei dissolve, essentially by merging together. In this section we shall describe the equation of state of Bayam-Bethe- Pethick (1971, thereinafter BBP) as a sort of reference case to illustrate the general methodology in this kind of problems. The BBP equation of state greatly improves and extends the results found by Harrison & Wheeler for lower density intervals. There are a number of reasons for the difference with respect to the lower density case. First, since the nuclei are really neutron-rich, the matter inside nuclei is very similar to the free neutron gas outside. Second the surface energy of nuclei is calculated taking into account that there is matter around. In fact, the free neutrons outside decrease the surface energy of nuclei. Third BBP takes into account that cold dense matter can arrange itself in crystalline lattices due to Coulomb interactions. Finally, the treatment is based on the compressible liquid drop model of nuclei. The starting point is the total energy density which is expressed as

² = ²(A, Z, nN , nn,VN )

= nN (WN + WL) + ²n(nn)(1 − VN nN ) + ²e(ne) (6.1)

Where

- nN is the number density of nuclei;

- nn the number density of neutrons outside of nuclei (neutron gas);

- VN is the volume occupied by a nucleus. This volume decreases because of the pressure of the free neutrons;

- WN is the energy of a nucleus inclusive of the rest-mass. it depends on A,

Z, nn and VN ; 6.1. FROM ρDRIP TO ρNUC : THE BBP EQUATION OF STATE 87

- WL is the lattice energy, while ²n and ²e are the energy density of the free neutrons and electrons;

- VN nN is the fraction of volume occupied by nuclei and (1 − VN nN ) is the fraction occupied by free neutrons.

In terms of these quantities we have

ne = ZnN charge neutrality (6.2) total baryon density

n = AnN + (1 − VN nN )nn (6.3)

Note that nn is defined in terms of the number of free neutrons Nn in the volume

Vn outside the nuclei Nn Nn nn = = (6.4) Vn V (1 − VN nN ) where the volume V contains Nn neutrons and nN V nuclei. Equilibrium is determined by minimizing ² at fixed n. Since ² depends on five variables, there will be four independent conditions. (1) The first condition is derived by considering a unit volume with fixed number ZnN of protons, a fixed number of neutrons nN (A − Z) in nuclei, a fixed fraction nN VN occupied by nuclei, and a fixed number nn(1 − VN nN ) of neutrons outside the nuclei. What is the optimal value of A of the nuclei? This is determined by minimizing ² with respect to A at fixed nN Z, nN A, nN VN and nn.

This implies that ²n is fixed, and since ne is fixed, so is ²e. Define the variable

Z x = (6.5) A

Since nN = const/A in this variation, eqn.(6.1) gives

∂ ³W + W ´ N L = 0 (6.6) ∂A A x,nN A,nN VN ,nn

This means that the energy per nucleon inside nuclei must be mini- mum. 88 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP

(2) The second condition is that the nuclei must be stable to β- decay - that is changes in Z. Thus ² must be minimum with respect to variation in Z at fixed A, nN ,VN , nn. Note that

∂ d²e ∂ne ²e(ne) = = µenN (6.7) ∂Z dne ∂Z by the well known relation

d²(ne) 2 = EF,e − mec dne

(EF,e is the Fermi energy of electrons) and eqn.(6.2). In eqn.(6.7), µe is the chemical potential of electrons. Therefore we obtain ∂ µ = − (W + W ) (6.8) e ∂Z N L A,nn,VN ,nn Equation (6.8) can be rewritten in terms of the nuclear chemical potentials. The (N) chemical potential of the neutrons in the nuclei, µn is the minimum energy required to add a neutron to the nucleus, that is ∂ µ(N) = (W + W ) (6.9) n ∂A N L Z,nN ,VN ,nn Similarly, the proton chemical potential involves a fixed neutron number (A − Z) ∂ µ(N) = (W + W ) p ∂Z N L A−Z,nN ,VN ,nn ∂ ∂ = (W + W ) + (W + W ) (6.10) ∂Z N L a,nN ,VN .nn ∂A N L Z,nN ,VN ,nn since ∂A/∂Z = 1. The β-stability condition eqn.(6.8) can be written as

(N) (N) µe = µn − µp (6.11)

(3) The third condition is that the gas of free neutrons be in equilib- rium with neutrons in the nuclei; that is there is no energy cost to transfer a neutron from the gas to the nucleus. This requires minimizing ² with respect to A at fixed Z, nN , VN and n. Differentiating eqn.(6.3) with respect to A under these conditions one obtains ∂n n n = − N (6.12) ∂A 1 − VN nN 6.1. FROM ρDRIP TO ρNUC : THE BBP EQUATION OF STATE 89

In differentiating (WN + WL) in eqn.(6.1) we can use

¯ ¯ ¯ ∂ ¯ ∂ ¯ ∂nn ∂ ¯ ¯ = ¯ + ¯ (6.13) ∂A Z,nN ,VN ,n ∂A Z,nN ,VN ,nn ∂A ∂nn Z,nN ,VN ,A

Since WL is independent of nn, recalling eqn.(6.9) we find h ¯ i (N) nN ∂WN ¯ d²n nN µn − nN ¯ + (1 − VN nN ) = 0 (6.14) 1 − VN nN ∂nn Z,A,nN ,VN dnn or (N) (G) µn = µn (6.15) where ¯ (G) nN ∂WN ¯ d²n µn = ¯ + (6.16) 1 − VN nN ∂nn Z,A,nN ,VN dnn is the free neutron chemical potential. The term d²/dnn is the usual term cor- responding to a change in the bulk energy of the neutron gas. The other term corresponds to a change in the nuclear surface energy when a neutron is added to the gas. In fact, the energy of the nuclei per unit of the volume occupied by the outside neutron gas is (see eqn.(6.4))

(nN V )WN nN = WN (6.17) Vn 1 − VN nN which is exactly the term differentiated in eqn.(6.16). (4) The fourth equilibrium condition is that there is pressure bal- ance between the neutron gas and the nuclei

P (G) = P (N) (6.18)

This follows from minimizing ² with respect to VN at fixed Z, A, nN , and

Nn/nn(1 − VnnN ). Since nn = constant/(1 − VN nN ), we obtain

∂n n n n = n N (6.19) ∂VN 1 − VN nN

The partial derivative ∂/∂VN can be split into

¯ ¯ ¯ ∂ ¯ ∂ ¯ ∂nn ∂ ¯ ¯ = ¯ + ¯ (6.20) ∂VN Z,A,nN ,nn(1−VN nN ) ∂VN Z,A,nN ,nn ∂VN ∂nn Z,A,nN ,VN 90 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP

Differentiating eqn.(6.1) and writing (1 − VN nN ) = nn(1 − VN nN )/nn we derive ¯ ∂ ∂nn ∂WN ¯ 0 = nN (WN + WL)Z,A,nN ,nn + nN ¯ ∂VN ∂VN ∂nn Z,A,nN ,Vn ³ ´ ∂nn ∂ ²n +nn(1 − VN nN ) (6.21) ∂VN ∂nn nn The pressure due to nuclei is

(N) ∂ P = − (WN + WL)Z,A,nN ,nn (6.22) ∂VN and so using eqn.(6.16) and eqn.(6.19), eqn.(6.21) gives eqn.(6.18), where

(G) (G) P = nnµn − ²n (6.23)

The total pressure is ∂ ³ ² ´ P = n2 = P (G) + P + P (6.24) ∂n n e L where ³ ´ ¯ 2 ∂ ²e 2 ∂WL ¯ Pe = ne PL = nN ¯ (6.25) ∂ne ne ∂nN Z,A,VN ,nn

To calculate the equation of state, one must specify WN , WL, ²n, and ²e. The latter two are known, the problem is with WN and WL.

The nuclear energy term WN . Bayam-Bethe-Pathick use a compressible liquid drop model of nuclear matter and adopted for WN the expression

2 2 WN = A[(1 − x)mnc + xmpc + W (k, x)] + WC + WS (6.26) where WC is the Coulomb energy, WS the surface energy, and W (k, x) the energy per nucleon of bulk nuclear matter of nucleon number density 2k3 n = (6.27) 3π2 which defines k. The meaning of the parameter x will become clear later on. The bulk energy W (k, x) includes the effects of nucleon-nucleon interactions but excludes surface effects and Coulomb interactions. Inside the nucleus the number density is n = A/VN . 6.1. FROM ρDRIP TO ρNUC : THE BBP EQUATION OF STATE 91

For a consistent description, the same function W (k, x) is used for the neutron gas, with x = 0, that is

2 ²n = nn[W (kn, x) + mnc ] (6.28) where 2k3 n = n (6.29) n 3π2 The quantity W (k, x) must be determined. Empirically, it is evaluated by smoothly interpolating the results of many-body calculations done in various limits of k and x. Theoretically it is derived from assuming a nuclear potential as it will be dis- cussed below.

The surface energy WS is constructed to vanish explicitly when the density of the neutrons gas equals the density of the nucleus.

The energies WC and WL are jointly considered by means of a suitable expression. 2 2 The leading term in WC is 3Z e /5rN , the Coulomb energy of a uniformly charged 3 sphere of radius rN , where VN ≡ 4πrN /3. Bayam-Bethe-Pathick combine this leading term with the lattice energy WL to get

2 2 ³ ´ 3 Z e rN 2 rN WC+L = 1 − (1 + ) (6.30) 5 rN rc 2rc where

4π n r3 ≡ 1 (6.31) 3 N c The electrons are an ideal degenerate Fermi gas and so

hc¯ ² = (3π2n )4/3 (6.32) e 4π2 e

Given explicit functions for WS and W (k, x), the equation of state can be derived using the equilibrium conditions (6.6), (6.11), (6.15) and (6.18) and determining the pressure from eqn.(6.24). Note that the condition for neutron drip to occur is (G) 2 µn ≥ mnc 92 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP

Figure 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/dlnρ as a function of ρ for the Bayam-Bethe-Pathick equation of state.

The resulting equation of state is shown in Fig.6.1. Remarks. The key features of the Bayam-Bethe-Pethick equation of state are the as follows (i) The free neutrons progressively contribute to pressure: at ρ ' 1.5 × 12 3 13 3 10 g/cm Pn/P ' 0.2; at ρ ' 1.5 × 10 g/cm , Pn/P ' 0.8. (ii) As neutron drip is approached at ρ ' 3 × 1011 g/cm3,Γ ' 4/3 (extremely relativistic degenerate gas). Slightly above the neutron drip, Γ abruptly drops to small values. This occurs because the low density free neutrons contribute very much to density ρ and little to pressure. Γ does not increase above 4/3 until ρ ≥ 7 × 1012 g/cm3. This result has the important consequence that no stable stars can have central densities in the range explored here. (iii) Bayam-Bethe-Pathick find that nuclei survive up to densities ρ ' 2.4 × 1014 g/cm3. Above this limit, the nuclei get in contact each other, the lattice disappears, and one has a nuclear liquid. (iv) At densities ρ ≥ 2.4 × 1014 g/cm3 the electron chemical potential satisfy the condition µe ≥ 104 Mev. This energy corresponds to mass mµ of the ””. Therefore ”” can appear in the electron bath. Muons must be included in the equation of state and will contribute to pressure. The Bayam-Bethe-Pathick 6.2. NUCLEON-NUCLEON INTERACTION 93 equation of state cannot go beyond ρ ' 5 × 1014 g/cm3 as the standard nuclear theory can no longer be applied.

6.2 Nucleon-nucleon interaction

This is the most difficult and uncertain part of the derivation of the equation of state. We begin our analysis with a general discussion of the nuclear force and the associated potential and the kind of constraints (properties) it must satisfy in order to reproduce the observational data on nuclei and nuclear interaction. (1) As for electromagnetism, in the non relativistic limit we may assume that the nuclear force is conservative and independent of the velocity of the nucleus. The force may therefore be derived from a static potential. (2) Nuclear forces, however, do not obey the superposition principle: the total interaction in a many-body system does not reduce to the sum of the interaction among all pairs. However below and near ρnuc three-bodies and higher order interactions can be neglected and considering only pairs is sufficient. (3) The potential energy of the interaction between two nucleons depends not only on their relative distance r but also on their spins. (4) The static potential V can depend on three vectors: n, a unit vector along the radial direction between the two nucleons, and s1 and : the spin vectors of the nucleons. (5) We assume that the nuclear force is invariant with respect to rota- tion, inversions, and time reversal. Therefore the potential is a scalar under rotations. Furthermore, it cannot contain the gradient vector because this would be equivalent to a velocity dependent force.

(6) With the three vectors n, s1, s2, we can construct only two scalars: s1 · s2 and (n·s1)(n·s2). Thus the most general spin dependent potential is of the form

Vord = V1(r) + V2(r)(σ1 · σ2) + V3(r)[3(σ1 · n)(σ2 · n) − σ1 · σ2] (6.33) 94 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP

where σi = 2si are called the Pauli spin operators. The third term is arranged in such a way that it goes to zero on averaging over all directions of n: the forces given by this term are called ”tensor forces” and the dependence on n means that they are non-central. (7) The suffix ord means that these forces are the so-called ordinary contri- bution, in other word they are the same for two protons, two neutrons, and a pair neutron-proton. In the technical language the ordinary forces do not change the charge state of the nucleon. The charge symmetry is called iso- topic invariance. Formally, one may consider protons and neutrons as two different charge states of the nucleon (do not mistake this kind of charge with the standard electrical charge). This symmetry under interchange of pro- tons and neutrons can be treated as a formalism completely analogous to the rotation group. Mathematically, the nucleon is represented by a two component spinor in an abstract group space. This property similar to the ordinary spin is called isotopic spin or isospin indicated by t. The proton has isospin +1/2, the neutron -1/2. Define now the operator τ = 2t as the Pauli spin matrix acting on spinors in isospin space. The total isospin of a system of nucleons is

T = t1 + t2 + ... with z-component T3 = (t1)3 + (t2)3 + .... Since the eigenvalue of the operator t3 is 1/2 for protons and -1/2 for neutrons, we have T3 = Z − A/2 for a system of Z protons and A − Z neutrons.

(8) For a two fermion system, the total wave function ψ(r1, s1; r2, s2)ω(t1, t2), where ω(t1, t2) is the isospin part, must be antisymmetrical with simultaneous interchange of the r’s, s’s and t’s. The total isospin T = t1 + t2 determines the symmetry of ω, and S = s1 + s2 determines the symmetry of the spin part. For two nucleons T = 0 or 1. The triplet state T = 1 is symmetrical, with

T3 = 1, 0, −1. It can describe pp, pn or nn systems. The singlet state T = 0, ω is antisymmetrical, T3 = 0 and one has the system np only. (9) Since the value of T determines the symmetry of ω and hence because of the antisymmetry of the total wave function, the symmetry of ψ, conservation of T is equivalent to the existence of a definite symmetry for ψ. This appears to be 6.2. NUCLEON-NUCLEON INTERACTION 95

an exact symmetry of strong interactions. The conservation of T3 is equivalent to conservation of charge for a fixed number of baryons, and so is an exact symmetry even in presence of Coulomb forces. (10) Interchange operators: Heisenberg Operator. We can construct τ an interchange operator called the Heisenberg Operator P that interchanges r1, τ 2 τ σ1 and r2, σ2 for two particles. Since (P ) = 1, the eigenvalues of P are ±1, depending on whether the operator acts on a symmetrical or antisymmetrical wave function ψ(r1, σ1; r2, σ2). Now ψsym corresponds to antisymmetric ω, that is T = 0. Similarly ψant corresponds to a symmetric ω with T = 1. Thus we can write P τ in a form that acts on the isospin variables of the wave function if it has the properties τ τ P ω0 = +ω0 P ω1 = −ω1 (6.34) where the subscripts denote the value of T . Since T2 has eigenvalues T (T + 1) we see that 1 P τ = 1 − T2 = 1 − (t + t )2 = − − 2t · t (6.35) 1 2 2 1 2 2 2 where we have used the fact that t1 and t2 have the same definite values t(t+1) = 3/4. Thus finally 1 P τ = − (1 + τ · τ ) (6.36) 2 1 2 (11) Interchange operators: Bartlett Operator. It exchanges the spins of two particles, leaving their coordinates (positions) unchanged. The Bartlett operator P B has eigenvalues ±1 so that

B B P ψS=0 = −ψS=0 P ψS=1 = +ψS=1 (6.37)

Comparing with eqn.(6.34) we see 1 P B = S2 − 1 = (1 + (σ · σ )) (6.38) 2 1 2 (12) Interchange operators: Majorana Operator. It exchanges the par- ticle positions only, leaving the spins unchanged. 1 P M = P BP τ = − (1 + σ · σ )(1 + τ · τ ) (6.39) 4 1 2 1 2 96 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP

Interchange interactions imply a contribution to the potential. In fact, we will see that interchange interactions are necessary to achieve ”saturation” of nuclear forces. Interchange interactions are not mere formalism but a substan- tial ingredient of the complex nuclear interaction. The most general velocity-independent exchange potential can be written as

V (r) = Vord(r) + Vexch(r) (6.40) where

τ Vexch(r) = {V4(r) + V5(r)(σ1 · σ2) + V6(r)[3(σ1 · n)(σ2 · n) − (σ1 · σ2)]}P

It can be shown that S2 but not S commutes with V (r). Therefore only the modulus of S not its direction is conserved in the interaction. It follows from this that while the total angular momentum J = L + S is conserved, L needs not to be separately conserved due to the tensorial nature of the force. We can classify the allowed states of a two-nucleon system according to the values of T and S. For example, if T =1 and S = 1, the coordinate part of the wave function must be antisymmetric under exchange of particles (odd parity), and so L must 3 be odd. If L = 1 we have three possibilities J = 0, 1, 2. The the states are P0, 3 3 3 3 3 P1, or P2. If L = 3 we have J = 2, 3, 4 giving states F2, F3, F4. Because 3 3 only J and not L separately, is in general conserved, the states P2 and F2 will 3 3 3 be mixed, that is ( P2 + F2). The states P0 and 3P1 have no states with which to mix. In this case L can be conserved separately. The lowest few states of the two-nucleon system are listed in Table 6.2.

6.3 Saturation of nuclear forces

Experimental data show that, apart from Coulomb and surface effects, the en- ergy and volume of nuclei increase proportionally to A. This property is called saturation of nuclear forces. It imposes severe constraints of the potential V (r). For example, looking at eqn.(6.40), one might suggest that a 6.3. SATURATION OF NUCLEAR FORCES 97

Table 6.1: States of a two-nucleon system (N-N)

TS Parity Possible States Possible (N-N) 3 3 3 1 1 odd P0, P1, ( P2 + 3F2), 3F3, .. nn, pp, 1 1 1 0 even S0, D2, 1G4, ... or np 3 3 3 0 1 even ( S1 + 3D1), D2, ( D3 + 3G3), ... 1 1 0 0 odd P1, F3, ... np

simple attractive potential of the form

V (r) = V1(r) + V3[3(σ1 · n)(σ2 · n) − σ1 · σ2] (6.41) is appropriate to fit all data for N-N systems in the S state. However it cannot be the basic law for nuclear forces. A qualitative explanation is given below. ♠ Type of potential for saturation. The total energy of a nucleus is

E = W + T (6.42) where T is the kinetic energy of nucleons and W the potential energy. The potential energy would be a sum over all pairs of nucleons of V (|ri − rj|), that is proportional to A(A − 1)/2 times a negative function. The kinetic energy T , because of the Pauli Principle, is the given by the Fermi statistics

A T ∼ AE0 ∼ An2/3 ∼ A( )2/3 = A5/3R−2 (6.43) F R3

0 where R is the radius of the nucleus and EF is the Fermi energy per nucleon. The ground-state value of R can be found from minimizing E. For large values of A, W ∝ A2 >> T and so it is the minimum of W that determines R. The minimum will be at the scale distance of nuclear forces, independent of A (this argument in given in more detail below). Moreover, the binding energy 2 will be Eb = −Emin, that is proportional to A . This is in contradiction with experimental data: R ∼ A1/3 (constant density of nuclei independent of A), and

Eb ∼ A. 98 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP

Nuclear forces must have a property that leads to saturation! That is Eb ∼ A. This occurs in nuclei as well as in nucleon assemblies of large density. Intuitively, the force must be attractive for a small number of nucleons and become repulsive for a large number. Chemical forces are a typical example of saturation: two atoms of H can combine to form H2. In the H2 molecule two electrons can occupy the same orbital with opposite spins, but a third electron cannot be accepted because of the Pauli Principle. In the case of nuclear forces, saturation is the result of several concomitant factors: the Pauli Principle, the exchange forces that originate from the P B and P τ operators, and the existence of a ”repulsive core”, i.e, a repulsive potential V (r) at small distances. ♠ Simple demonstration of the existence of a repulsive core. We will show that a merely attractive potential leads to nuclear collapse for large A’s, whereas a potential with a repulsive core leads to saturation. The average kinetic energy of a system of A nucleons, treated as an ideal degen- erate non relativistic gas, is

2 3 3 ³3π2 ´2/3 h¯ T = AE0 = An2/3 (6.44) 5 F 10 2 m where the nucleons are considered as identical particles having (2t+1)(2s+1) = 4 spin states to disposal for each momentum state. We approximate the interaction between two nucleons to be an attractive square well of range b and depth −V0 (where V0 is positive). If p is the probability for any two nucleons to be less than b apart, the total potential energy is

A(A − 1) W = − pV (6.45) 2 0

Suppose that the nucleons are confined in a spherical nucleus of radius R with uniform and uncorrelated spatial distribution. The probability p ≡ p(b, R) is defined as 1 ZZ p(b, R) = H(b − |r − r |)dr dr (6.46) Ω2 1 2 1 2 6.3. SATURATION OF NUCLEAR FORCES 99

Figure 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. where

H(x) = 1 for x > 0 (6.47) = 0 for x < 0 (6.48) is the step function and 4πR3 A Ω = n = (6.49) 3 Ω Clearly if b > 2R, the step function is always unity and p(b, R) = 1. In general

³ b ´3h 9 b 1 ³ b ´3i b p(b, R) = 1 − + for R > R 16 R 32 R 2 b = 1 for R < (6.50) 2 The total energy E = W + T is plotted in Fig.6.2 as a function of R. Clearly there is no minimum for R < b/2. For R > b/2 the total energy can be written as 100 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP

αA5/3 A(A − 1) b h 9 b 1 ³ b ´3i E = − V b2 1 − + (6.51) R2 R2 0 2R 16 R 32 R where 2 3 ³9π ´2/3 h¯ α = ' 30 Mev fm2 (6.52) 10 8 m

2 The order of magnitude of the term V0b can be estimated for the deuteron

2 2 V0b ' 100 Mev fm (6.53)

Since the coefficient of the square bracket is much bigger than T , even for moderate values of A, the minimum will occur for R only slightly greater than b/2. In fact setting dE/dR = 0, with A − 1 ' A, we get

3V b2A1/3 b ³ 3 b 1 b3 ´ 0 1 − + = 1 (6.54) 4α R 4 R 16 R3 which can be solved for R. In addition to the minimum at R ' b/2, where the bracketed quantity is small, there is also a maximum at R = R1 >> b, where b/R is small. The conclusion is straightforward: with a potential given by eqn.(6.45), nuclei 3 compressed to densities above n = 3A/4πR1 collapse to a stable state of radius R ∼ b/2, where b is the range of the nuclear interactions. This state has binding 2 energy Eb = −E ∝ A(A − 1) ∼ A in full disagreement with experimental data. ♠ Repulsive core. Now suppose that there is a repulsive core at short ranges. This means that around each nucleon there is a forbidden region where the wave function must vanish. As for any given density the available volume is decreased, the moment and kinetic energy are increased. By increasing T with respect to W the minimum of E is shifted to more realistic values. Suppose that

2ro is the mean distance between nucleons in the nucleus, then

1/3 R = r0A (6.55) 6.3. SATURATION OF NUCLEAR FORCES 101

If rc is the radius of the repulsive core, we can modify the expression for the kinetic energy αA5/3 αA αA T = 2 = 2 → 2 (6.56) R r0 (r0 − rc)

The value for W is not significantly changed because r0 > rc. Minimizing the modified version (different kinetic energy) of eqn.(6.51) with respect to r0, we obtain

2 ³ 3 ´³ ´ 3V0b b 3 b 1 b rc 3 1 − 1/3 + 3 1 − = 1 (6.57) 4α r0 4 r0A 16 r0A r0

Numerically solving for r0, one finds b = 1.8 fm and rc = 0.4 fm, a value consis- tent with data,

0.9 ≤ r0 (fm) ≤ 1.5 for 4 ≤ A ≤ 216 (6.58)

Experiments yield r0 of about 1.2 fm and more important show that r0 varies little with A. The repulsive core produces saturation! ♠ To summarize, whatever expression is adopted for the nuclear in- teraction to calculate the equation of state, an important experimental constraint is that the chosen potential reproduce the observed proper- ties of nuclear matter at saturation. Specifically, there are four important parameters to be reproduced: (i) the density at which saturation occurs is

−3 n0 = 0.16 nucleons fm (6.59)

(ii, iii) The energy and compressibility of symmetric nuclear matter. These are defined in terms of the function W (k, x) that we have already intro- duced for the bulk energy of nuclear matter per nucleon. Symmetric nuclear matter has Z = A/2 and x = 1/2 (this the meaning of the parameter x). In the vicinity of the saturation density, we can write

³ 1´ 1 ³ k ´2 W k, = −WV + K 1 − (6.60) 2 2 k0 102 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP where experimentally

2 ¯ ∂ W (k, 1/2)¯ 2 ¯ WV = 16 Mev K ≡ k 2 ' 240 Mev (6.61) ∂k k=k0

−1 Here k0 = 1.33 fm follows from eqn.(6.27) and (6.59). K is called the com- pression modulus. It is determined experimentally (∼ 300 Mev).

(iv) Finally the volume symmetry coefficient SV , which measures the ”cur- vature” of W (k, x) due to changes in x

2 ¯ 1 ∂ W (k0, x)¯ SV = ¯ ' 30 Mev (6.62) 8 ∂x2 x=1/2

Accordingly we can write for k near k0 and x near 1/2

1 ³ k ´2 ³ 1´2 W (k, x) = −WV + K 1 − + 4SV x − (6.63) 2 k0 2 In recent times, the above expressions have been refined both theoretically and experimentally. These refinements are beyond our simple approach.

6.4 The Yukawa Potential

We are left with important issue of the correct dependence of the dominant N-N potential on the separation of the nucleons. To illustrate the issue in a simple fashion, we start from the general expression for the potential given by relations

(6.33) and (6.40), where we retain only the first term V1(r) and all the others are set zero for the purposes of this discussion. In 1935 Yukawa made the suggestion that as the electromagnetic force arises from the exchange of virtual photons, the nuclear force might arise from the exchange of virtual particles called mesons. To explain the finite range of nuclear forces, the mesons would have to have a nonzero mass, in contrast with the long-range electromagnetic force mediated by the massless photons. The classical version of massive scalar and vector field theories tells us that the scalar field (one field component) describes quanta of spin 0, whereas the vector theory (3 independent field components) describes quanta 6.4. THE YUKAWA POTENTIAL 103 of spin 1. Furthermore, in the limit of slowly moving particles of ”charge” g interacting via scalar or vector fields, the interaction energy between two particles is given by e−µr V = ±g2 (6.64) 1,2 r where µ is the inverse of the Compton wavelength of the field quanta. This energy corresponds to the term V1(r). The plus sign (repulsive force) applies to the vector field, while the minus sign (attractive force) applies to the scalar field. As the typical distance range of the attractive nuclear forces is about 1.4 fm, this requires the field quanta to have a mass ∼ 140 Mev. This is exactly the mass of the meson-π or pion. The pions have spin 0. Therefore exchange of pions is a major component of the attractive nuclear force. Fitting expression (6.64) to low energy data, we find g2/hc¯ ∼ 10. This is the origin of the name ”strong nuclear force”. For electromagnetism we have e2/hc¯ ∼ 1/137. Simple derivation of the Yukawa potential. Consider the electrostatic potential about a charged point particle. This is given by

∇2Φ = 0 (6.65) which has the solution e Φ = (6.66) 4π²0r where e is the charge and ²0 is the dielectric constant of vacuum. This describes the potential for a force mediated by mass-less particles, the photons. For a particle with mass, the relativistic equation E2 = p2c2 + m2c4 can be converted into a wave equation by the substitutions ∂ ∂ ∂ ∂ E → ih¯ ; p → −ih¯ , p → −ih¯ , p → −ih¯ (6.67) ∂t x ∂x y ∂y z ∂z hence ∂2Φ −h¯2 = (m2c4 − h¯2c2∇2)Φ (6.68) ∂t2 which in the static case yields ³ m2c2 ´ ∇2 − Φ = 0 (6.69) h¯2 104 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP which gives ∇2Φ = 0 for the mass-less case as required. For a point source with spherical symmetry, the differential operator can be written as 1 d2 d2 m2c2 ∇2Φ → (rΦ) so (rΦ) = rΦ (6.70) r dr2 dr2 h¯2 with solution e−r/R Φ = g2 (6.71) r where g is a constant (the coupling strength) and R =h/mc ¯ is the range of the force. This is known as the Yukawa form of the potential, and was originally introduced to describe the interaction between protons and neutrons due to pion exchange. In our notation Φ = V1,2(r) and R = 1/µ. This potential corresponds to the exchange of a virtual boson whose rest-mass is m. The sign ± and association of it to the vector and scalar components of the field – ”the plus sign (repulsive force) applies to the vector field, while the minus sign (attractive force) applies to the scalar field” – cannot be derived from these simple arguments but require the full theory of quantum fields. We will show now that the Yukawa potential e−µr Φ = ±g2 (6.72) r can be used to derive the equation of state of nuclear matter. This of course is a simplified picture of the whole problem. Zel’dovich (1962) model. We start with this simple classical analysis. The total energy of a system of particles is derived from summing up over the interactions of all pairs. For simplicity, the particles are supposed to be uniformly distributed, thereby neglecting the effect of interaction on the mean inter-particle separation. Finally the number of particles is sufficiently large that we can replace summations by integrals, and the characteristic size of the assembly R is R >> 1/µ. The interaction energy in a volume V is ZZ X −µrij 1 1 2 2 e EV = Vij = ± n g dVidVj (6.73) 2 i6=j 2 rij 6.4. THE YUKAWA POTENTIAL 105

To evaluate the integrals consider a particle rj as the origin and the integrate over the spherical shells of radius r + rij Since R >> 1/µ, no sizable differences arise from extending the integration to infinity.

Z ∞ e−µr 4π 4πr2dr = (6.74) 0 r µ2 integrating now over rj we obtain

1 4π E = ± n2g2 V (6.75) V 2 µ2

Thus the total energy density is

2πn2g2 ² = ² ± (6.76) kin µ2

The term of kinetic energy can be taken equal to that of a degenerate non rela- tivistic or relativistic gas

3 h¯ ² = nmc2 + (3π2)2/3 n5/3 (NRD) kin 10 m (9π)2/3 = cn4/3 (ERD) (6.77) h¯

This model provide a crude estimate of the bulk energy of nuclear matter W = ²/n − mc2. The equation of state follows from

d ³ ² ´ P = n2 (6.78) dn n giving 2πn2g2 P = P ± (6.79) kin µ2 where Γ Pkin = KΓρ (6.80) and where Γ = 5/3 for NRD and Γ = 4/3 for ERD. The constant KΓ depends on Γ (kind of ). 106 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP

For low densities (ρ ' ρnuc), where the nuclear force is expected to be attractive, the pressure is softened by the inclusion of the N-N inter- actions, while at very high densities, the equation of state is hardened by the dominance of the repulsive core (vector fields). As ρ = ²/c2 → ∞, the equation of state satisfies

P → ρc2 (6.81)

The sound speed approaches

³dP ´1/2 c = → c (6.82) s dρ in contrast to an ideal relativistic gas for which

1 2 1 P → ρc cs → √ c (6.83) 3 3 Hartree Analysis. The above classical estimate can be recovered in the zero-order quantum mechanical analysis of Hartree in the non relativistic limit. In this approach, the many-body nucleon system is described by a product of wave functions

Ψ = u1(r1)u2(r2)...... uN (rN ) (6.84) where each nucleon i is described by its own (normalized) wave function ui(ri). The assumed form of Ψ does not include the effects of spin, nor of particle correla- tions since each ui is function of ri regardless of the position of the other nucleons. The effects of spin and correlations will be included in the Hartree-Fock analysis to be discussed below. In the simple Hartree analysis, the energy of the system

< H > = < Ψ|H|Ψ > X Z 2 ∗ h¯ 2 = dVui (r)(− ∇ )ui(r) i 2m X ZZ 2 2 + dV1dV2V12|ui(r1)| |uj(r2)| (6.85) i6=j 6.4. THE YUKAWA POTENTIAL 107

where H is the Hamiltonian and V12 is the potential energy. Note that in the Hartree analysis, the condition Z 2 |ui(r)| dV = 1 (6.86)

is used, but different ui are not necessarily orthogonal. This generates a set of equations

h¯2 − ∇2u + V u = ² u (6.87) 2m i i i i i where Vi(r1) is the so-called effective potential on particle i, which is defined by

X Z 2 Vi(r1) = dV2V12|uj(r2)| (6.88) i6=j Instead of solving the system of equations (6.88), we proceed in the following way. We assume that the lowest energy level can be described by free-particle, plane wave states of momentum p = hk

1 u = eik·ri (6.89) i V1/2

Consistent with the Fermi statistics we assume that the lowest energy levels are occupied up to p = pF or k = kF . The sum over i becomes integrals over k up to kF . The product wave function (6.94) represent a uniform density degenerate system. We can therefore think of the ui’s as a rather restricted set of trial functions in eqn.(6.85) only appropriate for weak interaction potentials. This the lowest order quantum mechanical analog of the Zel’dovich model. Equation (6.85) becomes with the aid of eqn.(6.89)

X 2 X ZZ p 1 2 exp(µr12) < H >= ± 2 dV1dV2g (6.90) k 2m V k,k0 r12

From which we derive (the double integral can be calculated exactly)

X p2 X 2πg2 < H >= ± 2 (6.91) k 2m k,k0 Vµ 108 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP

Now X Z Z 1 2 3 3 3 → 3 d p = 2 d k(2π) (6.92) V k h After some manipulations we derive

< H > 2πn2g2 ² = + nmc2 = ² ± (6.93) V kin µ2 exactly the same as the classical result in the non relativistic limit. Complete Hartree-Fock analysis. The above procedure suggests the way in which spin and exchange effects can be included. This is made in the com- plete Hartree-Foch model by requiring that the wave function of system of N Fermions is antisymmetric under interchange of any pair of particles by means of the Slater Determinant and using for the wave functions of individual particles the representation

ui(j) = ui(rj)χi(σj) (6.94) where χ(σ) is either χ1 (spin up) or χ2 (spin down)

³1´ ³0´ χ1 = χ2 = (6.95) 0 1

¯ ¯ ¯ ¯ ¯u1(1)...... u1(N) ¯ ¯ ¯ ¯ ¯ 1 Ψ = ¯...... ¯ (N!)1/2 ¯ ¯ ¯ ¯ ¯uN (1)...... uN (N)¯ (6.96)

Carrying out the calculations with this more complicated wave function, the contribution to the energy density due to nucleon-nucleon interaction corrected for nucleon exchange effects is

g2πn2V ² = ∓ (6.97) int µ2 This is opposite in sign and precisely 1/2 of the eqn.(6.75) determination. The reason is that only particle with opposite spins can get close enough 6.4. THE YUKAWA POTENTIAL 109 to interact by the Pauli Principle, so the total interaction energy is 1/2 of what it would be without spin effects. The physical effect of the exchange contribution is to lessen the role of nucleon interactions and thus to lower the energy for a repulsive force and to increase the energy for an attractive force. Hence the exchange term contributes an effective attraction (²exch < 0) for repulsive forces and an effective repulsion (²exch > 0) for attractive forces. Recalling that ρ ≡ ²/c2 we have

3 h¯2 πn2g2 ρ = nm + (3π2)2/3 n5/3 ± (6.98) 10 mc2 µ2c2 πn2g2 P = Kn5/3 ± (6.99) µ2

They constitute the equation of state obtained from a Yukawa-type interaction potential in the Hartree-Fock model. Correlations. The above equation of state cannot properly account for cor- relations between nucleons. In fact, in many-nucleon systems, the nucleons are not totally independent but tend to establish correlations. This is taken into account suitably constructing a wave function for the whole fermion system in which two-body correlations are included. Bethe-Johnson. In this equation of state the interaction potential is Yukawa- like. We have already said that the exchange of vector mesons induces repulsive nucleon-nucleon interactions, while the exchange of scalar mesons induces attrac- tive forces. There are three vector mesons of lowest mass: ρ (769 Mev), ω (738 Mev) and φ (1019 Mev). Among them the strongest coupling constant between 2 nucleons is for ω with gω ' 10±2 according to experimental data. Bethe & John- son have considered only this interaction whose range is abouth/m ¯ ωc ' 0.25 fm. Since ω is an iso-scalar, the repulsive core should be independent of total iso-spin T in the nucleon-nucleon system. The Bethe-Johnson potential has the form

X e−jx V (r) = Cj + VT (r) (6.100) j x 110 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP where m c x = µr µ = π = 0.7 fm−1 (6.101) h¯

The term VT (r) is the tensorial component. The coefficients Cj for j 6= 1 are chosen to fit the experimental data, C1 and VT are from theory: the one-pion exchange potential model. In eqn.(6.100), the exchange of (pseudo) scalar π- mesons is used to get the long-range attractive part of the potential (range j−1 × −1 −1 µπ ' 5.5µω , where the main attraction is due to 2-π exchange, i.e. j = 2, which results in Yukawa terms with negative Cj. The dominant repulsive term is given by e−µωr V ≡ g2 (6.102) ω ω r 2 2 where gω/hc¯ = 29.6 Mev. The value for gω is disturbingly high, about three times the value from high energy data. This is because of the particular fit to low energy data made by Bethe & Johnson. Their equation of state can be summarized as follows (n is in fm−1);

² = W (k, 0) + m c2 n n W (k, 0) = 236 na Mev/particle (6.103) d(²/n) P = n2 = 364 na+1 Mev fm−3 dn = 5.83 × 1035na+1 dyne cm−2 (6.104) dP na c2 = = c2 (6.105) s dρ 1.01 + 0.648na where

a = 1.54 0.1 ≤ n ≤ 3 fm−3 or 1.7 × 1014 ≤ ρ ≤ 1.1 × 1016 g/cm3 (6.106)

The most important point of this equation of state is the stiffness corresponding to an adiabatic index Γ = 2.54. This will reflect onto the neutron star mass. Bethe & Johnson have also investigated the case of a hyperonic liquid made of n, p, Λ, Σ, and ∆ particles. They found indeed that the light with masses < 1250 Mev appear at the typical neutron star densities. In such a case 6.5. APPEARANCE OF NEW PARTICLES 111 the resulting equation of state is not too different from that of a pure neutron equation of state. In general, there is a slight softening of the equation of state due to the availability of new cells in the phase space and the corresponding lowering of the Fermi sea. The inclusion of these effects in the equation of state is quite complicate and still uncertain. In any case the appearance of new par- ticles does change the equation of state, involving different contributions to the total pressure, new interactions between hyperons in addition to those between nucleons. The issue is worth being expanded in some detail.

6.5 Appearance of new particles

To show how the appearance of new particles is incorporated in the equation of state, let us first consider the appearance of muons in the ideal gas of n, p and e. Normally, muons decay to electrons via

− − µ → e + νµ +ν ˜e (6.107)

When the Fermi energy of the electrons is high enough, it is energetically more convenient for electrons to turn into muons, so that muons and electrons are in equilibrium µ− ↔ e− (6.108) we assume here that neutrinos leave the system (extremely low cross section). The condition for chemical equilibrium is expressed by the chemical potentials

µµ = µe (6.109) making sure that appropriate quantities are conserved (in this case, charge). Equilibrium between n, p and e requires

µn = µp + µe (6.110) while charge neutrality imposes

np = ne + nµ (6.111) 112 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP

Given expressions for the chemical potential and number densities in terms of the densities, then eqn (6.109) through (6.111) and the equation for the density, are a sufficient set of equations to solve for all the gas properties as a function of the density, in other words to derive the equation of state. The above equations can be written as

2 2 1/2 2 2 1/2 mµc (1 + xµ) = mec (1 + xe) (6.112)

2 2 1/2 2 2 1/2 2 2 1/2 mnc (1 + xn) = mpc (1 + xp) + mec (1 + xe) (6.113)

3 3 3 (mpxp) = (mexe) + (mµxµ) (6.114)

mn mp me mµ ρ = 3 χ(xn) + 3 χ(xp) + 3 χ(xe) + 3 χ(xµ) (6.115) λn λp λe λµ where the x0s are the dimensionless Fermi momenta. The threshold conditions for muons to appear is nµ = 0 and xµ = 0. Since the electrons are extremely relativistic we can take xe >> 1, the eqn (6.112) through (6.114) become

mµ = mexe (6.116)

2 1/2 2 1/2 mn(1 + xn) = mp(1 + xp) + mexe (6.117)

mpxp = mexe (6.118) from which

nh 2 2 1/2 i o (mp + mµ) + mµ 2 1/2 xn = − 1 = 0.4986 (6.119) mn

14 3 and so xp = 0.1126, xe = 206.8 and ρ = 8.21 × 10 g/cm . Remarks. An example of equation of state for a hyperonic liquid containing n, p, Λ and Σ particles is shown in Fig.6.3 after Canuto (1975). The neutrons remain the dominant species in neutron star interiors. Even though the Bethe- Johnson equation of state suffered of many limitations that are not mentioned here, it long remained one of the best. 6.6. UNRESOLVED QUESTIONS 113

Figure 6.3: Concentrations nj in a free hyperonic gas a function of total baryonic density n

6.6 Unresolved questions

There are a number of problems requiring many difficult refinements at densities around 2ρnuc: ♠ ∆ resonances. Among others is the appearance of the ∆ resonance, an excited state of the nucleon with mass 1236 Mev and quantum numbers t = 3/2 and J = 3/2. Pion exchange between nucleons can produce virtual intermediate states which are NN, N∆, and ∆∆. As a consequence, some of the attractive pion exchange processes could be inhibited in a dense nuclear medium due to modification of the intermediate state energy and the Pauli Principle. Therefore, the equation of state could be stiffer than that given by the free-space potentials. ♠ Pion condensation. Ignoring the effects of strong interactions between pions and nucleons, the criterion for the formation of negatively charged pions in dense nuclear matter via the reaction

n → p + π− (6.120)

− is that µn − µp = µe exceeds the π rest mass, µπ = 139.6 Mev. At ρ ∼ ρnuc µe is about 100 Mev, so one may expect π− to form at somewhat higher densities. 114 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP

This would have two important consequences: the equation of state is softened (and the rate of neutrino cooling is enhanced). Pions are hadrons and interact strongly. Therefore their properties are influenced by the surrounding medium. Pions have spin 0, so they form a Bose-Einstein gas that can give rise to a Bose condensate for sufficiently low temperatures. As they do not obey the Pauli Principle, they can condensate to the lowest energy, momentum state. An ideal condensate consists of large number of particles in the zero kinetic energy level. To find the critical temperature Tc at which this might occur, let us recall that the maximum chemical potential for bosons of mass m is µ = mc2. A larger value would imply negative occupation numbers for some momentum states, that is not possible. For a given density n, the temperature Tc is determined by setting µ = mc2 in

Z g 1 3 n = 2 d p (6.121) h3 e(E−mc )/kTc − 1 A low temperatures we can make the non-relativistic approximation p2 E − mc2 = (6.122) 2m Defining the dimensionless variable p2 z = (6.123) 2mkTc we find Z ∞ 1/2 g 3/2 z dz n = 3 (mkTc) (6.124) h¯ 21/2π2 0 ez − 1 The integral has the value π1/2ζ(3/2)/2 (ζ is the Riemann function) and so

3.31³n´2/3 T = h¯2 (6.125) c mk g

For T < Tc, particles with positive kinetic energy are distributed according to 3/2 eqn.(6.121) with Tc replaced by T . According to eqn.(6.124), n ∼ T , so

³ T ´3/2 n(z > 0) = n (6.126) Tc 6.6. UNRESOLVED QUESTIONS 115

The remaining particles are all in the lowest state with z = 0

h ³ T ´3/2i n(z = 0) = n 1 − (6.127) Tc The particles in the z = 0 state have no momentum and hence do not contribute to pressure. As T → 0 virtually all the bosons are in the z = 0 state. Thus it is soon clear why the presence of the pion condensate softens the equation of state. The effect can be quite important. To evaluate the effect of pion condensation let us consider the simple case of an ideal n − p − e gas at T = 0 and allow the pions π− to be produced above threshold. The equilibrium implies

µn − µp = µe = µπ (6.128) from which

2 1/2 2 1/2 2 1/2 mn(1 + xn) − mp(1 + xp) = me(1 + xe) (6.129) 2 1/2 me(1 + xe) = mπ (6.130)

In eqn.(6.130) we have used the fact that at T = 0 pions have zero kinetic energy. Charge neutrality requires

ne + nπ = np (6.131) so 1 3 1 3 2 3 xe + nπ = 2 3 xp (6.132) 3π λe 3π λp We can derive the baryon density the mass density and the pressure

1 3 1 3 n = 2 3 xp + 2 3 xn (6.133) 3π λp 3π λn me mp mn ρ = 3 χ(xe) + 3 χ(xp) + 3 χ(xn) + mπnπ (6.134) λe λp λn 2 2 2 mec mpc mnc P = 3 φ(xe) + 3 φ(xp) + 3 φ(xn) (6.135) λe λp λn

Given ρ from the above equations we may derive xn, xp, xe, and nπ. From − eqn.(6.130) for xe >> 1 (ultra-relativistic) we see that the threshold for π 116 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP production occurs at Mπ xe = = 273.2 (6.136) me

At threshold nπ = 0 so eqn.(6.132) gives

mπ xp = = 0.1488 (6.137) mp the eqn.(6.129) gives

xn = 0.5843 (6.138) and finally from eqn.(6.134)

15 3 ρ ≡ ρπ = 1.36 × 10 g/cm (6.139)

For ρ < ρπ the equation of state is the same as in chapter 2, while for ρ >

ρπ, xe remains constant so ne and Pe as ρ increases. At increasing density, an increasing fraction of the negative charge consists of pions that contribute to rest-mass but not to pressure. The equation of state is particularly soft, much softer than in absence of pions. Pion condensation may also enhance the possibility that neutron star matter at high densities solidifies. In fact, the short-range repulsive core of the NN potential might be strong enough to lock up neutrons into a regular lattice. As a consequence, neutron stars could have solid cores as well as solid external layers (crust). It is clear that an infinite hard-core potential by construction gives rise to a solid structure as soon as the inter-particle spacing approaches the core radius. An infinitely strong repulsion would indeed cage each neutron in its own lattice site. The question is whether a more realistic Yukawa like potential with soft repulsive core can result in solidification. In fact, if the potential is too soft, allowing particles to tunnel trough it, solidification may never occur in practice.

♠ Ultra-high densities. At densities as high as 10 × ρnuc, it is no longer possible to describe nuclear matter in terms of a non relativistic Schr¨odingerequation or in terms of a potential. The ”meson clouds” surrounding nucleons begin to overlap and the picture of localized individual 6.6. UNRESOLVED QUESTIONS 117 particles interacting via two-body forces fails. A typical approach in this density regime is to construct a relativistic Lagrangian that allows nucleons to interact attractively via scalar meson exchange and repulsively via the exchange of more massive vector mesons ω. In the non relativistic limit, both the classical and quantum field theories yield Yukawa-like potentials. It is found that at the highest densities the vector meson exchange dominates and the Zel’dovich model is still valid 2 P → ρc , cs → c (6.140)

Hagedorn model. In alternative, one can suppose that the whole spectrum of resonant states arise at high densities. A baryon resonance mass spectrum is assumed to appear according to

N(m)dm ∼ maem/m0 dm (6.141) where N(m)dm is the number of resonant states in the mass interval m and m + dm. Existing data on baryonic resonances can be fitted to eqn (6.141) with m0 = 160 Mev and −3.5 ≤ a ≤ 2.5. 2 In equilibrium, the threshold condition for a mass m to appear is mc = µn so the neutron chemical potential sets the limit on the maximum resonance mass at every density. For asymptotically high densities, eqn.(6.141) gives Z µn a µn/m0 n = N(m)dm = m0µne (6.142) 0

Since the resonances are appearing exponentially fast, the highest mass states are non-relativistic.Thus the density is asymptotically Z µn a+1 µn/m0 ρ ∼ mN(m)dm ∼ m0µn e (6.143) 0

∼ nµn (6.144)

The pressure is d ³ρc2 ´ dµ P = n2 ∼ n2c2 n ∼ n2c2dn/dµ (6.145) dn n dn n 118 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP

Equation (6.142) gives dn n a µn/m0 ∼ µne ∼ (6.146) dµn m0 while eqn. (6.143) gives µ ln ρ ∼ n (6.147) m0 Thus the pressure is ρc2 ρc2 P ∼ ≡ (6.148) ln ρ ln(ρ/ρ0) 15 3 with ρ0 = 2.5×10 g/cm . The Hagedorn equation of state is particularly soft as it allows the continuous creation of new particles with increasing density, rather then pushing up the Fermi sea of a single species. In this model no repulsive interactions have been included. The sound velocity of this kind of matter is

2 h i 2 dP c 1 cs = = 1 − (6.149) dρ ln(ρ/ρ0) ln(ρ/ρ0)

For ρ/ρ0 → ∞ cs → 0 in striking contrast to the mean-field treatment that gives cs → c, This equation of state requires a deep revision. However, there is no hard experimental or theoretical evidence to decide between these two extremes! ♠ Quark matter. The building block of all particles feeling the strong interaction (called hadrons) are quarks1. If this is true, any description of nuclear matter at high density must involve quarks. Nuclei get in touch at baryon density 3 −1 ∼ (4πrn/3) ∼ few ρnuc; here rn ∼ 1 fm, the characteristic nucleus radius. Above this density, one may argue that matter should undergo a phase transition at which quarks begin to drip out of the nucleons. The result would be quark matter, a degenerate Fermi liquid. Since free quarks have never been detected in experiments, they are believed to be permanently confined to the interior of hadrons by a force that increases as one tries to separate the quarks. However current theories on the quark-quark in- teractions (”Quantum Chromo-”) suggests that the interaction becomes arbitrarily weak as the quarks are squeezed together (”asymptotic freedom”). It

1See appendix A for a quick summary of the families of elementary particles. 6.6. UNRESOLVED QUESTIONS 119 is conceivable that at high densities quarks may drip out the nucleons to form a quark environment which to a first approximation can be treated as an ideal relativistic Fermi gas made of quarks. What would be the corresponding equation of state? The weak interaction may change quarks into other types of quarks (new flavors). In neutron stars the density is just above the threshold only for the existence of the lightest quarks u, d and s. The following decays are possible

− d → u + e +ν ˜e

− d → u + µ +ν ˜µ

− s → u + e +ν ˜e

− s → u + µ +ν ˜µ (6.150)

Assuming β-equilibrium and neglecting neutrinos as usual we have

µd = µu + µe

µd = µu + µµ (6.151)

µs = µu + µe

µs = µu + µµ (6.152)

For an extremely relativistic Fermi gas

3 ni ∝ giµi (6.153) where

− − gi = 2 for i = e , µ , gi = 6 for i = u, d, s (6.154)

The 6 for quarks is the product of two spin states and three ”color” states. We obtain

µd = µs nd = ns (6.155)

Equilibrium between µ− and e− gives

µµ = µe nµ = ne (6.156) 120 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP

Quarks carry a different fractionary charge. This is (u: 2/3, d: - 1/3, s: - 1/3). Charge neutrality implies 2 1 1 n − n − n − n − n = 0 (6.157) 3 u 3 d 3 s µ e which reduces to

nu − ns − 3ne = 0 or nu − ns − 3nµ = 0 (6.158)

Using eqn.(6.152) and eqn.(6.153)

³n ´1/3 ³n ´1/3 ³n ´1/3 s = u + e 6 6 2 ³n ´1/3 ³n ´1/3 ³n ´1/3 s = u + µ (6.159) 6 6 2 Defining

n n x = e y = u (6.160) ns ns we find from eqn.(6.158) and eqn.(6.159)

y − 1 = 3x 1 = y1/3 + (3x)1/3 (6.161)

The only real solution of system (6.161) is y = 1, x = 0. Thus

nu = ns = nd ne = nµ = 0 (6.162)

The key feature of this model is that at high density the extreme relativistic free particle results apply, that is

1 P → ρc2, ρ → ∞ (6.163) 3 This is a relatively soft equation of state. In Figs.6.4 and 6.5 and Table 6.2 we show a summary of the equation of state discussed in this chapter.

6.7 Modern equations of state 6.7. MODERN EQUATIONS OF STATE 121

Figure 6.4: The equilibrium equation of state of cold degenerate matter. The solid 11 3 line shows the BPS equation of state in the region ρ ≤ ρdrip ' 4.3 × 10 g/cm , matched smoothly to the BBP equation of state in the region ρdrip ≤ ρ ≤ ρnuc = 2.8 × 1014 g/cm3. 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 ρnuc reside in the box and are shown in Fig.6.5 below

Figure 6.5: Representative equations of state for cold degenerate matter above 14 3 ρnuc = 2.8 × 10 g/cm 122 CHAPTER 6. EQUATION OF STATE ABOVE THE NEUTRON DRIP

Table 6.2:

Equation of State Density Regime Composition Theory

Chandrasekhar 0 ≤ ρ ≤ ∞ e− Non interacting e− Ideal electron gas Nuclei specified by µe Ideal n − p − e− gas 0 ≤ ρ ≤ 1.2 × 107 e−, p Equilibrium Matter 1.2 × 107 ≤ ρ ≤ ∞ n, p, e− Equilibrium Matter 4 − 56 Feynmann-Metropolis-Teller 7.9 ≤ ρ ≤ 10 e and 26F e Thomas-Fermi-Dirac Atomic model 4 − 56 Harrison-Wheeler 7.9 ≤ ρ ≤ 10 e and 26F e Thomas-Fermi-Dirac Atomic model 4 7 − 56 10 ≤ ρ ≤ 10 e and 26F e Non interacting electrons 107 ≤ ρ ≤ 3 × 1011 e− and Equilibrium Nuclide Semiempirical Mass and equilibrium matter 3 × 1011 ≤ ρ ≤ 4 × 1012 e−, n and Equilibrium Nuclide Semiempirical Mass and equilibrium matter 4 × 1011 ≤ ρ ≤ ∞ Same as ideal n − p − e− Equilibrium Matter 4 − 56 Baym-Pethick-Sutherland 7.9 ≤ ρ ≤ 10 e and 26F e Thomas-Fermi-Dirac Atomic model 4 9 − 56 10 ≤ ρ ≤ 8 × 10 e and 26F e Ideal Electrons with Coulomb Lattice 8 × 109 ≤ ρ ≤ 4.3 × 1011 e− and Equilibrium Nuclide Laboratory Nuclear Energies Coulomb Lattice Energy Equilibrium Matter Chapter 7

Polytropes

A polytrope is the most simple model of a star and goes back to Lane (1970) and Emden (1907). In a polytropic star, the pressure is related to density by the general relation

n+1 P (r) = Kρ(r) n (7.1) where K is the so-called polytropic constant and n the polytropic index. In principle, relation (7.1) is valid across the whole star as K and n do not depend on the radius. The polytropic relations does not necessarily coincides with the equa- tion of state. There are two possibilities: (1) The equation of state is polytropic. For instante, the equation of state of fully degenerate electrons in the non relativistic (P ∝ ρ5/3, with n = 1.5) and fully relativistic regime (P ∝ ρ4/3, with n = 3) is polytropic. (2) The dependence of pressure and density are coupled. There are several examples of it. For instance

(i) The isothermal stratification of perfect gas: P = (k/µmu)ρT with

T = T0 = constant. In the this case P ∝ ρ, n = ∞. The constant K depends on

T0 and µ. It may change from star to star, but it is constant inside a star. (ii) A fully convective star (or region of a star) in which convection is nearly adiabatic: (dlnT/dlnP ) = ∇A. In the case of a mono-atomic perfect gas 2/5 ∇A = 2/5, i.e. T ∝ P . If µ is constant

123 124 CHAPTER 7. POLYTROPES

n+1 1 − ∇A P ∝ ρ n with n = (7.2) ∇A with n = 3/2. (iii) Equation of state of a perfect gas plus radiation pressure k 1 P = ρT + aT 4 (7.3) µmH 3

Introducing the ratio β = Pg/P and taking into account the well known result that β ' const in a star of mass M, one can easily recast the equation of state as follows h i k 4 3 1 − β 1/3 4/3 P = ( ) ( ) 4 ρ (7.4) µmH a β a polytrope with n = 3. In general, the quantity β is not constant, it varies from the center to the surface and with the star mass. The first dependence can be neglected. Thus in a star of mass M, β is a constant and eqn.(7.4) is a real polytrope. Comment: If the equation of state is of polytropic type, the constant K is assigned and depends only on physical constants. In all other cases, K is a sort of free parameter, constant inside a star, but variable from star to star.

7.1 Lane-Emden equation

We start from the equation of hydrostatic equilibrium and mass conservation dP GM dM = − r ρ r = 4πr2ρdr (7.5) dr r2 dr Combined together they give the Poisson equation 1 d ³r2 dP ´ = −4πGρ (7.6) r2 dr ρ dr For a polytrope dP 1 1 dρ = K(1 + )ρ n (7.7) dr n dr from which ³ ´ n + 1 1 d 2 1−n dρ K r ρ n = −4πGρ (7.8) n r2 dr dr 7.1. LANE-EMDEN EQUATION 125

Introducing the dimensionless variables

n ρ = ρcφ and r = αξ (7.9) with 1−n 2 K(n + 1) n 2 α = ρc [cm ] (7.10) 4πG we obtain the non linear differential equation of Lane-Emden for the function φ(ξ) 1 d ³ dφ´ ξ2 = −φn (7.11) ξ2 dξ dξ with obvious conditions

¯ dφ¯ φ(ξ = 0) = 1 and ¯ = 0 (7.12) dξ ξ=0

Analytical solutions exist only for n = 0, 1, 5. The surface of the polytrope is reached when φ(ξ1) = 0 for the first time. Total Radius 1−n n hK(n + 1)ρc i1/2 R = αξ = ξ (7.13) 1 4πG 1 Total Mass

Z Z R ξ1 2 3 2 n M = 4πr ρdr = 4πρcα ξ φ (ξ)dξ (7.14) 0 0 or Z ξ1 ³ ´ 3 d 2 dφ M = −4πρcα ξ dξ (7.15) 0 dξ dξ and finally ¯ 3 3 dφ¯ M = −4πρcα ξ1 ¯ (7.16) dξ ξ1 or h i3/2 (3−n) K(n + 1) 2n 2 0 M = −4π ρc [ξ φ (ξ )] (7.17) 4πG 1 1

Attention. For n=3 the mass does not depend on ρc

³ K ´3/2 M = 4π [−ξ2φ0(ξ )] (7.18) 3 πG 1 1 n=3 126 CHAPTER 7. POLYTROPES

Mass-Radius relationship. This is easily obtained. The Mass-Radius rela- tionship is univocally determined by K and n

1 hK(n + 1)inh n+1 in−1 R3−n = − ξ n−1 φ0(ξ ) M 1−n (7.19) 4π G 1 1

For n = 3 the mass M does not depend on R. For n = 1 the radius R does not depend on M. Given K and M there is only one solution (one radius). Potential Energy. This is given by

3 GM 2 E = − (7.20) G 5 − n R

Mass-Density Radius relationships. From the Mass-Radius and Mass-Central- Density relationships of polytropes one obtains

1−n 2n R ∝ M 3−n and ρc ∝ M 3−n (7.21)

For a fully degenerate, non relativistic gas of electrons we have n = 3/2 and

K = k3/2, i.e.

−1/3 2 R ∝ M together with ρc ∝ M (7.22)

Since dρ dη > 0 → > 0 (7.23) dM dM Degeneracy of electrons increases with the mass of the polytrope until the elec- trons become relativistic. The polytrope n = 3 has singular Mass-Radius rela- tionship, there is a limit mass M3 independently of R and ρc. For n = 3 − ² with ² << 1 we have

1− 2 6 −2 R ∝ M ² together with ρc ∝ M ² (7.24)

Therefore

lim R = 0 and lim ρc = ∞ (7.25) ²→0 ²→0 7.2. COLLAPSING POLYTROPES 127 7.2 Collapsing polytropes

Up to now we have only considered polytropic gaseous spheres in hydrostatic equilibrium. One can also find solutions for polytropes with n = 3 for which the inertial term in the momentum equation are important. The equation of motion in general form and spherical symmetry is

∂v ∂v 1 ∂P ∂Φ r + v r + + = 0 (7.26) ∂t r ∂r ρ ∂r ∂r where vr = ∂r/∂t. Let us consider a relativistic degenerate gas n = 3 or γ = 4/3. In analogy to the static case, we define a dimensionless length-scale ξ by

r = a(t)ξ vr =a ˙(t)ξ (7.27) such that ξ is time independent, the whole time dependence being contained in the function a(t). The form (7.27) describes a homologous change. Introducing now the function ψ (called the velocity potential) defined by

∂ψ v = (7.28) r ∂r we can write

∂ψ ∂ψ 1 av = aaξ˙ = a = and ψ = aaξ˙ 2 (7.29) r ∂r ∂ξ 2 where we have fixed the constant of integration in the velocity potential by ψ = 0 at ξ = 0. Note that the time derivative ψ in the comoving frame of reference is

dψ ∂ψ ∂ψ ∂ψ = + v = + (˙aξ)2 (7.30) dt ∂t r ∂r ∂t With the new variables, the Poisson equation

1 d ³ dΦ´ r2 = 4πGρ (7.31) r2 dr dr becomes 128 CHAPTER 7. POLYTROPES

1 ∂ ∂ψ (ξ2 ) = 4πGρa2 (7.32) ξ2 ∂ξ ∂ξ while the continuity equation in its general form

∂ρ 1 ∂(ρr2v) = − (7.33) ∂t r2 ∂r becomes

1 dρ 1 ∂ ξ2∂ψ 1 dρ a˙ + ( ) = + 3 = 0 (7.34) ρ dt ξ2a2 ∂ξ ∂ξ ρ dt a This means that ρ ∝ a−3 in the comoving frame. 3 As in the static case, we define the function φ(ξ) by ρ = ρcφ (ξ). The function φ(ξ) will turn out to be related to the Lane-Emden function of index n = 3. Note that ρc is a function of time. In order to stay as close as possible to the formalism of the static case, we fix a = r/ξ by the relation

1 πG = ρ2/3 (7.35) a2 K c where K is the polytropic constant. In such a way

K 1 ρ = ρ φ3(ξ) = ( )3/2 φ3(ξ) (7.36) c πG a3 We now come to the equation of motion and define

Z dP h = = 4Kρ1/3 (7.37) ρ Inserting ψ and h in the equation of motion (7.26) gives

∂2ψ 1 ∂ ∂ψ ∂Φ ∂h + ( )2 + + = 0 (7.38) ∂r∂t 2 ∂r ∂r ∂r ∂r which can be integrated with respect to r. If we set the integration constant to ∂ψ zero, replace ∂r byaξ ˙ , and consider eqn.(7.30), we find that

dψ 1 = a˙ 2ξ2 − Φ − h (7.39) dt 2 7.2. COLLAPSING POLYTROPES 129 and therefore with eqn.(7.29)

1 aaξ¨ 2 = −Φ − h (7.40) 2 From eqn.(7.36) and eqn.(7.37) follows

K3/2 1 h = 4Kρ1/3 = 4 φ(ξ) (7.41) (πG)1/2 a We try a similar dependence of Φ on time t and write

K3/2 1 Φ = 4 g(ξ) (7.42) (πG)1/2 a which defines the dimensionless function g(ξ). If we insert these two latter rela- tionships in eqn.(7.40) we find

1 K3/2 1 a2a¨ = −4 (g + φ) (7.43) 2 (πG)1/2 ξ2 Since the l.h.s. of this equations depends only on time and the r.h.s. only on space, both sides must be equal to a constant λ, i.e.

3 (πG)1/2 a2a¨ = −λ (7.44) 4 K3/2

g + φ 6 = λ (7.45) ξ2 The first of these equation can be integrated twice. After multiplication with a/a˙ 2, the first integration gives

8 K3 1 a˙ 2 = λ( )1/2 (7.46) 3 πG a where the constant of integration is set to zero (assuming a zero velocity when the sphere is expanded to infinity). Multiplied by a, it gives

2 d h8λ K3 i1/2 a1/2a˙ = (a3/2) = ± ( )1/2 (7.47) 3 dt 3 πG 130 CHAPTER 7. POLYTROPES

Figure 7.1: Solutions for different values of the parameter λ, in the range 0 ≤ λ ≤ λm. They describe homologously collapsing polytropes with n = 3. where the sign ± indicates explosion or contraction, respectively. This equation can immediately be integrated, yielding for a collapse (˙a < 0) that starts at a0 for t = 0

3h8λ K3 i1/2 a3/2(t) = a3/2 − ( )1/2 t (7.48) 0 2 3 πG This expression gives the time dependence of the scaling factor a(t) and therefore of the density. We now investigate the spatial dependence of our solution. In particular the function φ(ξ) must be determined. For this purpose we write the Poisson equation in the dimensionless variable ξ

1 ∂ ∂Φ (ξ2 ) = 4πGρa2 (7.49) ξ2 ∂ξ ∂ξ If we replace Φ, g(ξ) and ρ in the expressions we have found, we obtain

1 d dφ (ξ2 ) + φ3 = λ (7.50) ξ2 dξ dξ For λ = 0 we recover the Lane-Emden equation. The solutions for λ 6= 0 deviate from hydrostatic equilibrium, and λ measures this deviation. The numerica in- 7.2. COLLAPSING POLYTROPES 131 tegration of this equation shows that physically significant solutions are possible only for small values λ, precisely in the interval λ < λm = 0.0065. For larger val- ues, the function φ(ξ) and density ρ do not vanish at finite values of ξ, but after reaching a minimum they diverge to infinity. The situation is shown in Fig.7.1. The solution we have found can be interpreted in the following way. Consider a polytrope with index n = 3 in equilibrium. The equilibrium does not depend on the radius. The total energy of the system is zero. In fact, the gravitational potential and internal energies of a polytrope are

3 GM 2 Ω = − (7.51) 5 − n R and 1 U = − Ω (7.52) 3(γad − 1) from which

3 h 1 iGM 2 W = Ω + U = − 1 (7.53) 5 − n 3(γad − 1) R

In our case, γad = 4/3 so that W = 0. The polytrope is insensitive to changes of radius. But if for any reason the pressure is reduced (for instance because the polytropic constant K decreases) then the star starts contracting according to eqns.(7.44) and (7.45). The first gives the temporal behaviour, the second one shows the deviation from the Lane-Emden caused by the inertial terms. This class of polytropes will be used to describe the collapse of the central core of a massive star that gives rise to a supernova explosion. 132 CHAPTER 7. POLYTROPES Chapter 8

White Dwarfs

8.1 Introduction

White Dwarfs are the first example in theory of stellar structure of macroscopic consequences of quantum and relativistic effects taking place at microscopic level. White Dwarfs are characterized by too low and too small radii for their mass. The White Dwarf B has: L = 1/400L¯, M = 1M¯, R =

1/40R¯ and Teff = 10000 K. At least one white dwarf has a radius nearly equal to that of the Moon. This implies that the density of white dwarfs is very high going from 105 to 108 g cm−3. Therefore the gas of electrons must be strongly degenerate. As a consequence of this, White Dwarfs have structure and evolution completely different from that of normal stars. To formulate the theory of White Dwarfs we assume that the material is fully ionized, and made of free electrons and nuclei. Total pressure and total energy density are given by the sum of the contributions by nuclei and electrons. Radi- ation can be neglected. The electron gas is assumed to fully degenerate everywhere in the star but for a thin layer close to the surface. Since at the typical density of 106 g cm−3 the kinetic energy of electrons is about 0.15 Mev, whereas that of nuclei at the same density and temperatures of about 106 − 107 K (typical values in white dwarfs interiors) is only about 1 - 10 Kev, we can neglect the pressure due

133 134 CHAPTER 8. WHITE DWARFS

Figure 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 gigantic diamond), surrounded by a crust consisting of He and H, and a thin H-atmosphere. to nuclei (ions). At such high densities and low temperatures, the core of a typical old and cool Carbon-Oxygen White Dwarf consists of a crystal lattice (a kind of gigantic dia- mond), surrounded by a crust consisting of He and H, and a thin H-atmosphere. An artist view of this type of White Dwarf is shown in Fig. 8.1

8.2 Equation of state for the inner core

The equation of state of fully degenerate and partially relativistic electrons is given by the parametric relations

P = Pe = Af(x) (8.1)

ρ = Bx3 (8.2) where

πm4c5 A = = 6.002×1022 dynes cm−2 (8.3) 3h3 8.3. HYDROSTATIC STRUCTURE OF THE DEGENERATE INTERIORS135

3 3 8πm c 5 −3 B = 3 µe = 9.736×10 µe g cm (8.4) 3h N0

f(x) = x(x2 + 1)1/2(2x2 − 3) + 3ln[(1 + x2)1/2 + x] (8.5)

3 pF 3h ne 1/3 N0ρ x = , pF = ( ) , ne = (8.6) mc 8π µe where x is the dimensionless Fermi momentum which measures the relativistic conditions: x → 0 non relativistic regime, x → ∞ fully relativistic regime. Kinetic energy density per unit volume is

u = ue = Ag(x) (8.7) where

√ g(x) = 8x3( 1 + x2 − 1) − f(x) (8.8) and

P P f(x) = e = (8.9) u ue g(x)

8.3 Hydrostatic structure of the degenerate in- teriors

In the following we adopt the Newtonian formulation for the hydrostatic equilibrium and mass conservation. Combining the equation of hydrostatic equilibrium and mass conservation and substituting the pressure and density with the above relations we obtain

A 1 d r2 df(x) [ ] = −4πGBx3 (8.10) B r2 dr x3 dr given that 136 CHAPTER 8. WHITE DWARFS

1 df(x) d = 8 (x2 + 1)1/2 (8.11) x3 dr dr we derive 1 d h d i πGB2 r2 (x2 + 1)1/2 = − x3 (8.12) r2 dr dr 2A Define now the dimensionless quantity

z2 = x2 + 1 (8.13) whose physical meaning is

E z = ( F ) + 1 (8.14) mc2 where EF is the Fermi energy. Define also the auxiliary variables

r = α ζ and z = zcΦ (8.15) with

2A 1 α = ( )1/2( ) (8.16) πG Bzc which has the dimensions of a length. Introducing these definitions in eqn.(8.12) we derive

³ ´ 1 d 2 dΦ 2 1 3/2 2 (ζ ) = − Φ − 2 (8.17) ζ dζ dζ zc whose boundary conditions at the center are

dΦ ζ = 0 Φ = 1 = 0 (8.18) dζ Note that this equation becomes that of a polytrope of index n = 3 when z → ∞ (x → ∞) i.e. in the fully relativistic case, and to that of a polytrope of index n = 3/2 when z → 1 (x → 0), i.e. in the case of non relativistic gas. In all other cases, this equation is parametric in zc. 8.3. HYDROSTATIC STRUCTURE OF THE DEGENERATE INTERIORS137

The surface is obviously reached when the density drops to zero i.e. when the following relations hold

1 ζ = ζ1 x1 = 0 z1 = 1 Φ(ζ1) = (8.19) zc It follows from this that the density distribution given by the function Φ(ζ) is univocally determined by zc which in turn depends on ρc. This means that the structure of a white dwarf is determined by its central density via the relationship between x and this latter. In general, the il radius and the mass of a White Dwarf are univocally determined by the choice for zc (ρc) and µe. The total radius R is given by

2A 1/2 1 R = αζ1 = ( ) ( )ζ1 (8.20) πG Bzc

This suggests us to introduce a typical scale length l1 so defined

³ 2A ´1/2³ 1 ´ l = αz = (8.21) 1 c πG B or 1/2 ³ 3 ´ 3 6 1 h 1/2 N0 7.776 × 10 l1 = = km (8.22) 8π m cG µe µe

The radius of a White Dwarf made of carbon and oxygen (µe ' 2) is about 3300 km. Furthermore, given µe and zc, the total radius is fixed. The total mass M is easily derived from the definition of mass inside a sphere of radius ζ

Z ζ M(ζ) = 4πα3 ρ ζ0 2dζ0 (8.23) 0 from which we obtain

³ 2A ´3/2 1 ³ dΦ´ M(ζ) = 4π − ζ2 (8.24) πG B2 dζ The total mass is obtained by evaluating the term (−ζ2Φ0) at the surface 138 CHAPTER 8. WHITE DWARFS

Table 8.1: Numerical results for White Dwarf models (Cox & Juli 1968)

1 2 dΦ ρc 2 2 xc ζ1 −(ζ ) µ M µeR zc dζ µe e 0.0 ∞ 6.8968 2.0182 ∞ 5.84 0.000 0.01 9.95 5.3571 1.9321 9.48(e8) 5.60 4.170 0.02 7.00 4.9857 1.8652 3.31(e8) 5.41 5.500 0.05 4.36 4.4601 1.7096 7.98(e7) 4.95 7.760 0.1 3.00 4.0690 1.5186 2.89(e7) 4.40 10.000 0.2 2.00 3.7271 1.2430 7.70(e6) 3.60 13.000 0.3 1.53 3.5803 1.0337 3.43(36) 2.99 16.000 0.5 1.00 3.5330 0.7070 9.63(e5) 2.04 19.500 0.8 0.50 4.0446 0.3091 1.21(e5) 0.89 28.200 1.0 0.00 ∞ 0. 0. 0. ∞

2A 1 M = 4π( )3/2( )(−ζ2Φ0) (8.25) πG B2 1

2 0 where B depends on µe and (−ζ Φ )1 depends on zc. Given µe and ρc the total mass M is fixed. The ratio between the central and the mean density is given by

3 ρc 1 3/2 ζ1 = (1 − 2 ) 2 0 (8.26) < ρ > zc 3(−ζ Φ )1 which also depends on zc (ρc). In Table 8.1 we present the main properties of the solutions for White Dwarfs with increasing x and/or zc. The solutions show that fully degenerate configu- rations with assigned values of ρc and µe, have radius, mass and ratio of central to mean density univocally determined. This means that there exists a univocal relationship R = R(µe,M), i.e. the radius of a fully degenerate struc- ture depends only on the mass and chemical composition (mass-radius relationship). This is in contrast to the case of normal stars whose radius is essentially determined by the energy balance condition and the mechanism of energy transport. In addition to this we may note that in analogy with ordinary polytropes, the 8.3. HYDROSTATIC STRUCTURE OF THE DEGENERATE INTERIORS139

Figure to be replaced

Figure 8.2: Left: Mass - Radius Relationship for White Dwarfs according to the Chandrasekhar theory. Right: Mass-Central-Density Relationship for White Dwarfs according to the Chandrasekhar theory mass-radius relationship of White Dwarfs has dR/dM < 0. However the mass exponent changes with the radius. The theoretical mass-radius and mass-central-density relationships for a fully degenerate configuration according the Chandrasekhar Theory reasonably agree with observational data for White Dwarfs (see Fig. 8.2). Perhaps the most important property of fully degenerate stars is that the have a limit mass as the central density ρc → ∞ (fully rela- tivistic condition). As ρc → ∞, xc → ∞, zc → ∞, and 1/zc → 0, Φ(ζ) tends to that of a polytrope of index 3 and the corresponding total mass is given by

³ 2A ´3/2 1 M = 4π ( )(−ξ2θ0 ) (8.27) πG B2 3 1 or 5.836 M = 2 M¯ (8.28) µe where the mass in in solar units. This limit mass is named the Chandrasekhar mass. With the typical chemical composition of White Dwarfs (4He, 12C and 16O), the 140 CHAPTER 8. WHITE DWARFS

mean molecular weight of electrons is µe = 2, and the Chandrasekhar mass is

MCh = 1.46M¯ (8.29)

The corresponding Radius tends to zero (at least in mathematical sense). In reality such a situations is never reached as at densities of about 109 − 1010g cm−3 two effects are known to intervene: (a) the neutronization of mat- ter by inverse β-decay that would modify the equation of state so that the simple assumption that pressure is sustained by the sole degenerate electrons can no longer be applied. These effect would change the chemical composition and the equation of state in turn; (b) general relativity instability that would affect the condition for hydrostatic equilibrium. Both would concur to change the Mass-Radius relationship as ρc → ∞. We will come back to this later on.

8.4 The Chandrasekhar Limit

The existence of a mass limit for fully degenerate stars is such an important results that we should try to understand it in as simple way as possible. The demonstration is due to Landau (1932) and applies both to White Dwarfs and Neutron Stars in chapter (10). Suppose there are N fermions in a star of radius R, so that the number density of fermions n ∝ N/R3. The volume per fermion is ∼ 1/n (Pauli Principle), and by the Heisenberg Principle the momentum of a fermion is ∼ hn¯ 1/3. Thus the Fermi energy of a gas particle in relativistic regime is hcN¯ 1/3 E ∼ hn¯ 1/3c ∼ (8.30) F R The gravitational energy per fermion is GMm E = − B (8.31) G R where M = NmB (Note that even if most of the pressure comes from electrons, most of the mass is in baryons.) The equilibrium can be reached at the a minimum of the total energy E 8.4. THE CHANDRASEKHAR LIMIT 141

hcN¯ 1/3 GNm2 E = E + E = − B (8.32) F G R R Both terms scale with 1/R. When the sign of E is positive (i.e. N is small) E can be decreased by increasing R. This decreases EF and the electrons tend to 2 2 become non relativistic with EF ∼ pF ∼ 1/R . Thus eventually EG dominates over EF with increasing R, and E becomes negative, increasing to zero as R → ∞. There will therefore be a stable equilibrium at a finite value of R. On the other hand, when the sign of E is negative (i.e. N large) E can be decreased without bound by decreasing R – no equilibrium exists and gravitational collapse sets in. The maximum baryon number for equilibrium is therefore determined by setting E = 0:

³ ´ hc¯ 3/2 57 Nmax ∼ 2 ∼ 2 × 10 (8.33) GmB Mmax ∼ NmaxmB ∼ 1.5M¯ (8.34)

With the exception of composition dependent numerical factors, the maximum mass of a degenerate star depends only on fundamental constants.

The equilibrium radius associated with masses approaching Mmax is deter- mined by the onset of relativistic degeneracy

2 EF ≥ mc (8.35) where m refers to either electrons or neutrons. Using eqn.(8.30) and (8.34) this conditions gives

h¯ ³ hc¯ ´1/2 R ≤ 2 (8.36) mc GmB 8 ∼ 5 × 10 cm for m = me (8.37)

5 ∼ 3 × 10 cm for m = mn (8.38) 142 CHAPTER 8. WHITE DWARFS

Figure 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 µe = 2. The filled dots are observed White Dwarfs of known distances.

There are two regimes for collapse: one for densities above White Dwarf values and another for densities above nuclear densities. In both cases M ∼ Mmax.

8.5 External Layers

In modelling the structure of White Dwarfs we have assumed the pressure to be given by an ideal gas of electrons and to hold everywhere in the star from the center to the surface. Furthermore, we have not taken into account the source of the luminosity. First of all, we recall that White Dwarfs of given mass and chemical composition have a fixed radius and therefore they are located in the Hertzsprung-Russell Diagram (HRD) along lines of constant radius as shown in Fig. 8.3. According to the Mass-Radius relationship, the higher the 8.5. EXTERNAL LAYERS 143 mass and central density, the smaller is the Radius. In Fig. 8.3 the theoreti- cal sequences are compared with observational data for White Dwarfs and their immediate progenitors (the nuclei of Planetary Nebulae). Second, as the interior of a White Dwarf is completely degenerate, the electrons have large mean free path because of the filled Fermi sea, implying strong and hence uniform temperature. The isothermal interior is covered by non degenerate surface layers which are in radiative equilibrium. In these surface layer, the pressure is no longer given by the sole electrons but follows the normal equation of a perfect gas in which both ions and electrons contribute. The energy flows by diffusion of photons and the luminosity is regulated by

c d L = −4πr2 (aT 4) (8.39) r κρ dr

Combining the equation of hydrostatic equilibrium (Newtonian approximation) and that of radiative transfer, and assuming for the opacity a Kramers-like law −3.5 24 2 (κ = κ0ρT , where κ0 = 4.34 × 10 Z(1 + X) cm /g and Z and X are the and hydrogen mass-abundance) we obtain

dP 16πacG M T 6.5 = (8.40) dT 3 L κ0ρ where M e L are the total mass and luminosity. As the gas in the surface layers is surely non degenerate with negligible radiation pressure, we can substitute the density ρ by means of the equation of state [P = k/(µmH )ρT ] to obtain a differential relation between P and T , which upon integration gives

2 16πacG k M T 8.5 P 2 = + C (8.41) 8.5 3 mH L κ0µ We may reasonably assume as surface boundary conditions T = 0 and P = 0 at r = R, thus C = 0. Substituting P with ρ (via the equation of state) we get a relation between T and ρ that can be used from the surface down to the layer at 144 CHAPTER 8. WHITE DWARFS which degeneracy begins.

³ 2 4ac 4πGM µm ´1/2 ρ = H T 3.25 (8.42) 8.5 3 κ0L k

At the transition layer pressure must be continuous: equality of the non degener- ate gas pressure (consider only electrons so that µ = µe) to that of the degenerate electrons. This leads to

ρ kT ³ρ ´5/2 tr tr = 1.0 × 1013 tr . (8.43) µemH µe or

−8 3/2 3 ρtr = 2.4 × 10 µeTtr g/cm (8.44)

Eliminating the density between this relation and eqn.(8.42) and introducing the 24 value of κ0 = 4.34 · 10 · Z(1 + X), we derive

³ ´ 3.5 −27 µ MTtr L = 1.3×10 2 (8.45) µe Z(1 + X) where L e M are in solar units. An approximate expression for X = 0, Y = 0 and Z = 1 so that µ = µe = 2 is

−28 3.5 7/2 L = 6.6 × 10 MTtr ≡ LoMTtr (8.46)

−28 −2 where L and M are in solar units and L0 = 6.6 × 10 . Typically L ' 10 − −5 7 6 3 3 −10 L¯ corresponding to Ttr ' 10 − −10 K and ρtr ' 10 g/cm << ρc. From this relation, derived the luminosity and mass of the White Dwarf from observations, one estimates Ttr, which thanks to the high electronic con- duction is also the temperature of the degenerate interior. The thickness of the external layer can be derived from the T (r) relation obtained from the integration of radiative equilibrium condition,

1 µ mH GM R Ttr = ( − 1) (8.47) 4.25 k R rtr 8.6. COOLING OF WHITE DWARFS 145

Table 8.2: Characteristic quantities of White Dwarfs (M = 1M¯, R = 0.0083 7 −3 R¯, ρc = 2.8 × 10 gcm .

L/L¯ Ttr logRtr (R − rtr)/R η tcool (106 K) (Gyr) 10(e-2) 17 3.0 0.011 600 0.3 10(e-3) 9 2.6 0.006 1000 1.6 10(e-4) 4 2.1 0.003 2500 8.0

from which we get the relative thickness (R − rtr)/R reported in Table 8.5 for a few illustrative cases. Very thin indeed! Therefore, neglecting the surface layer in modelling the White Dwarf is amply justified as the transition density is small compared to the central value and the thickness of the external region is very small with respect to the total radius.

8.6 Cooling of White Dwarfs

Elementary theory of cooling. In a White Dwarf the only significant source of energy is the thermal energy of ions. Simple considerations allow us to conclude that (i) no nuclear energy sources can be present because they would lead to a catastrophic explosion; (ii) very little gravitational energy can be released because the star has already reached a degenerate state; (iii) the thermal energy of the electrons cannot be released because all the lowest energy states are occupied in a degenerate gas; (iv) the energy losses by neutrino emission is only important in the very early stages. The only available source of energy is the thermal energy of the ions, which for a mono-atomic non-degenerate gas is given by

3 k TM U = (8.48) 2 mH µi where µi is the mean molecular weight of ions. 146 CHAPTER 8. WHITE DWARFS

For temperatures of about 107 K the total energy reservoir is about 1048 ergs, which is comparable to the total energy output of some supernovae in the visual part of the spectrum. The cooling rate is given by −dU/dt which must balance the irradiated luminosity

dU d 3 k L = − = − ( TM) (8.49) dt dt 2 µimH which, introduced in the relation giving the luminosity as a function of the mass M and temperature T (the transition value) gives

d 3 k 7/2 − ( ) = L0T (8.50) dt 2 µimH T which integrated becomes

3 k −5/2 −5/2 (T − T0 ) = L0(t − t0) (8.51) 5 µimH where T0 and t0 are the initial temperature (much higher) and the initial time −5/2 from which cooling begins. Neglecting the term T0 , and replacing L0 we get

3 k TM ³ L ´−5/7 tcool ' ∝ (8.52) 5 µimH L M This relation has been obtained setting the integration constant equal to zero, which means that the cooling time has been calculated from the instant in which the temperature was much higher (virtually infinite). In practice this is equivalent to calculate the cooling time from the stage in which the temperature was two times higher than the present one as the time scale associated to earlier stage was very short (mostly because of neutrino cooling). Therefore the cooling time can be considered as the time elapsed from the last nuclear episode to the present.

Typical values of tcool are given Table 8.5. They must be taken as merely indica- tive estimates because many important phenomena have been neglected in this over-simplified formulation of the problem. 8.6. COOLING OF WHITE DWARFS 147

General remarks based on the . For a system in hydro- static equilibrium the Virial Theorem requires that the following relation between internal energy Ei and gravitational energy EG is satisfied

˙ ˙ ζEi + EG = 0 (8.53) where Ei is the internal energy, EG is the gravitational energy, and ζ is a pa- rameter of the order of one that depends on the equation of state. The internal energy is the sum of the ions and electron contributions, Ei = Eion + Eel. The parameter ζ in an average of the parameter ζ0 defined by

P ζ0u = 3 (8.54) ρ where u is the internal energy per unit mass. For instance for highly degener- ate electrons ζ0 = 2 for non relativistic conditions, ζ0 = 1 for fully relativistic conditions; for ions ζ0 = 2 for an ideal gas. If there is , the con- tribution uC of the Coulomb energy and up of the lattice oscillations have to be considered (see below). In the case of the Coulomb part we will obtain that 1/3 0 uC = neEC /ρ ∼ ρ ; in this case PC /ρ = uC /3, i.e. ζ = 1. The situation is more complicate with up, but this has a small effect. Summing up all effects, the average ζ for the whole White Dwarf will be in the range 1 < ζ < 2. As in ”normal stars” we have a simple relation between Ei and

EG, the absolute value of both being of the same order.

The total energy will be W = Ei + Eg. The energy conservation requires ζ − 1 L = −W˙ = − E˙ = (ζ − 1)E˙ (8.55) ζ g i ˙ Therefore L > 0 requires contraction (EG < 0) and an increase of the internal ˙ energy (Ei > 0). So far, it is the same as in normal, non degenerate stars. The crucial question is that Ei is distributed among ions and electrons.

In the case of non degenerate ions and electrons, there is equipartions with Eion ∼ ˙ ˙ Eel ∼ T , such that Ei ∼ T . Ei > 0 means T > 0. Thus the loss of energy 148 CHAPTER 8. WHITE DWARFS

(L > 0) leads to a heating T˙ > 0. This corresponds to saying that the star has a negative gravo-thermal specific (see any textbook on stellar structure, e.g. Kippenhahn & Weigert). In the case of White Dwarf the behaviour is different. To clarify the point let us suppose that electrons are non-relativistic degenerate and the ions are described by a perfect gas. Then ζ = 2 and L = −EG/2, i.e. the star must 1/3 contract releasing twice the energy lost bay radiation. Since −EG ∼ 1/R ∝ ρ ˙ we have EG/EG ∼ (1/3)ρ/ρ ˙ . The compression, however, increases te Fermi 2 2/3 energy EF of electrons. Their internal energy is Ee ' EF ∼ pF ∼ ρ such that ˙ Ee/Ee = (2/3)ρ/ρ ˙ . So we have the simple relation

˙ Ee ˙ Ee ˙ Ee ' 2 EG = − EG (8.56) EG Ei

The internal energy Ei is introduced via the Virial Theorem (EG = 2Ei), where

Ei = Eion + Ee. If the White Dwarf is already cool, Eion << Ee and Ei ' Ee. This means that ˙ ˙ Ee ' −EG ' 2L (8.57) and nearly as much energy as released by contraction is used up by ˙ ˙ raising the Fermi energy of the the electrons. With Ee ' −EG, the energy balance

˙ ˙ ˙ L = −Eion − Ee − EG (8.58) becomes ˙ ˙ L ' −Eion ∼ −T (8.59)

Therefore, the ions lose as much energy by cooling as the White Dwarf loses by radiation. The contraction is then a consequence of the decreasing ion pressure even if Pion is only a small fraction of P . In spite of the decreasing ˙ ˙ ion energy, the whole internal energy rises, since Eion + Ee ' L. This evolution tends finally to a cold , the contraction stops and all the internal energy is in the form of Fermi energy 8.6. COOLING OF WHITE DWARFS 149

Refinements of cooling theory. The above consideration suggest some refinements of the cooling theory. We start from the energy equation

dLr = ²G (8.60) dMr with the gravitational energy generation per unit mass (²N = 0 and ²ν = 0)

T ³∂P ´ ² = −c T˙ + ρ˙ (8.61) G v ρ2 ∂T where cv is the specific heat at constant unit volume and the second term at the r.h.s. can be neglected. Upon integration we obtain

Z M ˙ ˙ −L = cvT dMr ' cvT0M (8.62) 0 given that White Dwarf interiors are nearly isothermal T = T0 with T0, however, changing with time. If the ions are an ideal gas

ion 3 k 3 k cv = ' (8.63) 2 µimH 2 AmH where A is the mass number of the dominant chemical species present in the White Dwarf.

The specific heat cv of electrons is more complicate to derive (Chandrasekhar 1939). A useful expression is √ 2 2 2 el π k Z 1 + x cv = 2 2 T [x = pF /mc] (8.64) mec AmH x π2k Z kT = for x << 1 (8.65) 2 AmH EF The ratio (for x << 1) el 2 cv π kT ion = Z (8.66) cv 3 EF ion is very small if Z is small and kT << EF . Therefore cv ∼ cv . Therefore eqn.(8.62) describes L as given by the change of the internal energy of ions. Combining now relation eqn.(8.28) and eqn.(8.62), we would obtain an expression for the cooling time which is essentially similar to the one we have already derived 150 CHAPTER 8. WHITE DWARFS

7 4.7 × 10 M/M¯ 5/7 tcool ' ( ) years (8.67) A L/L¯

It is evident that the correct determination of cv strongly affect the cooling time. ˙ Large values of cv give slower cooling rates (T ∼ 1/cv), i.e. a longer cooling time ion tcool ∼ cv). The simplest assumption would be cv = cv as we did above. However for small M (i.e. moderate ρc and large Z and T ) one cannot neglect the electrons. 7 e ion At T = 10 K, M = 0.5M¯ and a C-O mixture, cv = 0.25cv . For small T the ions dominate. However when T gets very low the ions arrange themselves in a crystal lattice and the specific heat is influenced by this phenomenon. The theory of crystallization and associated specific heats is very complicate. In the following we will limit to a few general considerations. At low temperature and high densities, the Coulomb interactions of the ions become comparable to their mean kinetic energy. The ions no longer move freely but tend to form a rigid lattice, which minimize their total energy (minimize the mutual repulsion). This occurs when the thermal energy 3kT/2 becomes comparable to the Coulomb energy per ion of charge Ze. If we define a volume

Vion per ion by the relation nion × Vion = 1 (nion is the number density of ions), 3 and a mean separation rion between the ions, we have Vion = 4πrion/3. The ratio

Γc of Coulomb energy to kinetic energy of the ions is given by

2 2 1/3 (Ze) −3 Z nion Γc = = 2.7 × 10 in cgs (8.68) rionkT T is a measure of the importance of this effect. For Γc ' 100 a heated crystal will melt or a cooling plasma will crystallize, which determines the melting tempera- ture

2 2 ³ ´ Z e 4πρ 1/3 3 2 −1/3 1/3 Tm ' = 2.3 × 10 Z µi ρ in cgs (8.69) Γck 3µimH Ions in the lattice are not at rest but oscillate around their equilibrium position with some ion plasma frequency

2Ze 1/2 Ωp = (πρ) (8.70) AmH 8.6. COOLING OF WHITE DWARFS 151

The problem is similar to that encountered with pycno-nuclear reactions where the interionic potential can be approximated to

2 2 1 2 4Z e V (x) = Kx with K = 3 2 R0 where 2R0 is the interionic separation and x is the displacement from the mid point of the interionic distance and x < R0. The zero point of the oscillation 1 energy is E0 = 2 h¯Ωp. We can now define the Debye temperature Θ as kΘ =h ¯Ωp or

he Z Z Θ = √ ρ1/2 ' 7.8 × 103 ρ1/2 K (8.71) kmH π A A

Therefore, kΘ is the characteristic energy of the lattice oscillations which cannot be excited for T/Θ < 1. In White Dwarfs one has Θ < Tm.

As a consequence of all this the specific heat cv per ion changes with the T .

(i) Starting with large T (Γc << 1) the ions form an ideal gas. Each degree of freedom contributes kT/2 to the energy (i.e. k/2 to cv) and cv = 3k/2. (ii) With decreasing T one find an increasing correlation for the ion positions owing to growing importance of Coulomb forces in the range 1 < Γc < 10. This gives additional degrees of freedom, since energy can go in lattice oscillations, and cv increases above 3k/2 with the maximum of cv = 3k reached when the plasma crystallizes at T = Tm. (iii) With further decreasing T gradually fewer oscillations are excited, and 3 the ion specific heat cv drops below 3k/2 around T = Θ. Finally cv ∼ T .

The variation of cv/k as a function of temperature is shown in Fig. 8.4. For more details see Shapiro & Teukolsky (1983) where formal derivation of the trend described above is given. The large variations of cv (increase by a factor of 2, then decrease to zero) strongly influence the cooling time. In addition there is a release of latent heat of about kT when the plasma crystallizes. The critical 7 temperature at which cv rapidly decreases is about 10 K. 152 CHAPTER 8. WHITE DWARFS

Figure 8.4: Specific heat capacity as a function of temperature (schematic draw- ing; ions only)

8.7 Revision of the Mass-Radius Relationship

The mass-radius relationship we have just obtained needs suitable corrections of the theory both at the low and high mass end.

In the case of stars with M → 0 we expect the radius to decrease (R → 0) and the density to become nearly constant (ρ ' const) like in the case of . In contrast, here we have R → ∞ and ρ → 0. The solution is given by the correction to the equation of state due to the coulomb interaction between ions and electrons that we have already examined in chapter (5) and are shortly summarized below.

In the case of stars with mass tending to the limit value (M → MCh) and density tending to infinity (ρ → ∞), the radius tends to zero (R → 0). The solution comes from considering the effects on the equation of state and star structure due to inverse β-decay, possible presence of pycno-nuclear reactions, and General relativity. While the effect of inverse β-decay and companion neu- tronization have already been presented in chapter (6), those of the other two have been postponed up to here. In the following we shortly summarize the effects of inverse β-decay and discuss at some extent those of pycno-nuclear reactions and General Relativity instabilities. 8.7. REVISION OF THE MASS-RADIUS RELATIONSHIP 153

8.7.1 Correction at low densities: electro-static interac- tions

Consider a plasma made of nuclei of type (Z,A) at low temperature and number density n0 and degenerate electrons with number density ne. At low temperature the nuclei settle to a crystalline lattice where they oscillate at the lattice nodes. The electrons are uniformly distributed inside the lattice. Let us now consider the lattice as made of elemental spheres each of these containing one ion with positive charge Ze located at the centre and Ze electrons uniformly distributed in each elemental. The radius of a sphere is

0 1/3 R = Z rea0 (8.72) where a0 is the Bohr radius and re is the mean separation between the electrons in units of a0.

To calculate the coulomb energy ZEC of the sphere, we take concentric shells of radius y and charge −3Zey2dy/R03 and remove them to infinity, thereby over- coming the potential difference Ze(1 − y3)/R03. An integration over the whole sphere gives the energy per electron as

2 2/3 2 9 Ze 9 Z e Z 1/3 −EC = 0 = ' 2 1/3 ρ6 Kev (8.73) 10 R 10 rea0 A 6 −3 where ρ6 = ρ/10 g cm . Even for T → 0 the ions with mass m0 = Amu cannot sit at rest on their nodes in the lattice, but oscillate with the ion plasma 2 2 2 frequency ωE ∝ Z e n0/m0 where n0 and m0 are the number density and mass of ions, respectively. The zero point of the the ion energy is ZEzp = (3/2)¯hωE per ion. With ρ = n0Amu (mu the atomic mass unit) we have per electron

³ ´ 3 4π 1/2 he 1/2 0.6 1/2 Ezp = ρ ' ρ6 Kev (8.74) 2 3 2πAmu A 12 6 −3 For C (Z = 6, A = 12) and ρ = 10 g cm , the energies are −EC ' 5.2 keV 2/3 −1/6 and Ezp ' 0.05 keV. The ratio −EC /Ezp ' ZA ρ varies only very little with ρ and increases with Z and A (heavier elements). 154 CHAPTER 8. WHITE DWARFS

Therefore cold configurations (cold White Dwarfs) are crystallized. The ions form a regular lattice that minimizes the energy; they perform low-energy oscillations around their average positions, where they are kept by mutual repulsive forces. The energy per electron is now given by the sum

E = E0 + EC + Ezp ' E0 + EC < E0 (8.75)

where E0 is the mean energy of an electron in an ideal fermi gas. The effect of

EC on the pressure is derived from the thermodynamic definition of pressure

∂E ∂E ∂E P = − ' − 0 − C < P (8.76) ∂(1/n) ∂(1/n) ∂(1/n) 0 where the derivatives are taken at constant entropy and P0 is the pressure of the ideal Fermi gas. The lowering of E and P due to EC < 0 stems from the concentration of all positive charges into the nucleus, while the negative charges are more uniformly distributed. The average electron-electron distance is somewhat larger than the average electro-nucleus distance and therefore repulsion is smaller than attraction. This effect dominates at low values of M and relatively low densities. The pressure decrease engenders a decrease of the star radius, which goes back to finite values.

8.7.2 Correction at high densities: the pycno-nuclear re- actions

Pycno-nuclear reactions may occur at very high densities, like those encountered in White Dwarfs of high mass, even if the temperature is extremely low and in addition to altering the equation of state by changing the chemical composition they are a potential source of thermal instability with catastrophic consequences for the White Dwarf. These reactions depend almost exclusively on density and they arise because the zero-point energy of the nuclei oscillations in a lattice is different from zero thus allowing them to tunnel through the potential barrier to the neighbouring site and induce nuclear reactions. We have already recalled 8.7. REVISION OF THE MASS-RADIUS RELATIONSHIP 155

Figure 8.5: Electrostatic potential governing the motion of one ”incident” nucleus 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. that at low temperatures and high densities, the nuclei arrange themselves in a crystal lattice that minimizes the Coulomb electrostatic repulsion. The nuclei are not at rest at each node of the lattice but oscillates with some frequency. These reactions suddenly start for densities above a limit value ρpyc that depends on 6 −3 1 9 the type of nuclear fuel. One finds that ρpyc ' 10 g cm for H, ρpyc ' 10 g −3 4 10 −3 12 cm for He, and ρpyc ' 10 g cm for C. Let us approximate the potential felt by an ion in the crystal lattice by that for an ion (test particle) located between two fixed identical ions separated by the distance 2R0 along any one direction. The situation is illustrated in Fig.8.5. This potential is given by

Z2e2 Z2e2 2Z2e2 2Z2e2 1 V (x) = + − = 2 (|x| < R0 − Rn) R0 − x R0 + x R0 R0 [(R0/x) − 1] (8.77) where Rn is the nuclear radius (range of nuclear forces). Close to the nucleus we will smoothly join this potential to the attractive nuclear potential of width Rn.

For x << R0 the above potential can be approximated as an harmonic oscil- lator with 2 2 1 2 2 2 4Z e V (x) ' MΩ0x , Ω0 = 3 (8.78) 2 MR0 156 CHAPTER 8. WHITE DWARFS where M is the ion rest mass. The ion has the ground state energy 1 E = h¯Ω (8.79) 0 2 0 and is confined to a volume of radius

³ 2¯h ´1/2 r0 = (8.80) MΩ0 The ground state wave function can be approximated as

2 1 |ψ| ' 3/2 3 (8.81) π r0 near the origin. The oscillation energy E0 allows the ion to have a finite prob- ability of tunnelling the potential barrier to the adjacent sides. The tunnelling probability is Z b 2 1/2 T = exp{− dx[2MV (x) − E0] } (8.82) h¯ a where a and b are the limits between which tunnelling occurs. taking a = r0 and b = R0 − Rn and using V (x) of eqn.(8.77) we get

h ³ 2 2 ´ Z 1−R /R ³ 2 ´ i 4MZ e R0 1/2 0 n u 1/2 T = exp − 2 2 2 − α du (8.83) h r0/R0 1 − u where u = x/R0 and

E R ³ h¯ ´1/2³R3 ´1/4 α = 0 0 r = 0 (8.84) 2Z2e2 0 2Ze M

In the limit of Rn → 0 and (r0/R0) << 1, this integral can be simplified to give ³ 2 ´ R0 R0 T = exp − 2 2 (8.85) r0 r0 The reaction rate per ion pair is given by T S(E) P = v|ψ|2 (8.86) E q 2 3 3/2 −1 where v is the velocity of the ion (v = 2E/m) and |ψ| ' (r0π ) , S(E) is the standard reaction cross section factor for nuclear reactions. Putting all together we get

³ 2 ´1/2 (Z2e2M)3/4 h (MR )1/2 i P = 4 S(E) exp − 4Ze 0 (8.87) 3 2 5/4 π (¯h R0) h¯ 8.7. REVISION OF THE MASS-RADIUS RELATIONSHIP 157

If nA is the number density of ions evaluated when one ion is assigned to each sphere of radius R0/2, the the number of reactions per unit volume per second is

³ ρ ´ R = n P = A2Z4S(E)γλ5/4exp(−²λ−1/2) (8.88) A A where A is the atomic weight, γ is a numerical factor γ ' 1.1 × 1044, ² = 2.85 is also a numerical factor, and

2 ³ ´ 2 ³ ´ h¯ nA 1/3 h¯ 3 1/3 1 λ = 2 2 = 2 2 (8.89) 2MZ e 2 2MZ e π R0

To determine the critical density at which these reactions become important, we can define a time scale by writing

n R × T = n or T = A (8.90) A R and must specify the function S(E). This process converts 1H to 4He, 4He to 12C, and 12C to 24Mg on a time scale of about 105 yr at densities above 5 × 104, 8 × 108, and 6 × 109 g/cm3, respectively. Therefore, pycno-nuclear reactions can be the major sources of instability at White dwarf densities. These rates are still a matter of debate because of the many uncertainties in modelling the crystalline structure of the star and the effect of impurities.

8.7.3 Correction at high densities: the inverse β-decay

This phenomenon becomes important at high densities. Consider a nucleus (Z − 1,A), unstable to β-decay, that transforms to the stable nucleus (Z,A) + e− + ν with decay energy Ed. If the nucleus (Z,A) is surrounded by a gas of degenerate electrons with kinetic energy at the

2 2 1/2 EF = mec [(1 + x ) − 1] (8.91) such that EF > Ed, then the nucleus (Z,A) becomes unstable to the capture of electrons, i.e. the inverse β-decay 158 CHAPTER 8. WHITE DWARFS

(Z,A) + e− → (Z − 1,A) + ν (8.92) may occur In general, we have to deal with particularly stable even-even nuclei

(Z,A) and then Ed(Z − 1,A) < Ed(Z,A).

If EF > Ed(Z,A) then also EF > Ed(Z − 1,A) and the inverse β-decay proceed further to the nucleus (Z − 2,A). The new nuclei are now stabilized by the Fermi sea, i.e. they cannot emit an electron with Ed(< EF ) as there are no free levels for it in the phase space. The nucleus enrich in neutrons.

The Fermi energy EF increases with ρ; therefore for each type of nucleus (Z,A) there will be a threshold density ρn above which the process otherwise known as neutronization takes place. In the case of 1H and 4He the threshold densities are 7 −3 7 −3 ρn = 1.2 × 10 g cm and ρn = 1.2 × 10 g cm , therefore of no interest as pycno-nuclear reactions have already occurred before neutronization can begins. 12 12 12 10 3 Even for the decay of 6 C →5 B →4 B one has ρn = 3.9 × 10 g/cm > ρpyc 56 56 56 though this is reversed for heavy nuclei. The decay of 26Fe →25 Mn →24 Cr, for 9 −3 example has a threshold density ρn = 1.14 × 10 g cm < ρpyc. Chemical equilibrium. To proceed further, we must clarify some aspects of the nature of the chemical composition which is gradually built up in stars. In normal stars the chemical composition is considered as a sort of free parameter or as a frozen quantity. This sounds reasonable as long as the time scale over which the chemical composition changes by nuclear reactions is very long, so that the momentarily chemical composition can be considered fixed in the star. At each instant, the star has a certain chemical composition dictated by the dominant group of nuclear reactions. Thus the chemical composition can be very far from the so-called equilibrium value that it would have if a large number of (possible all) could occur simultaneously, thus establishing in the star the so-called nuclear statistical equilibrium chemical composition. This is the typical situation in very late stages of massive stars when the so-called statistical equilibrium at the iron-group elements is reached or during explosive 8.7. REVISION OF THE MASS-RADIUS RELATIONSHIP 159 nuclear burnings. White Dwarfs are manly made by helium or carbon-oxygen and fully degenerate electrons. Their chemical composition is very far from that at equilibrium. How we apply the concept of equilibrium composition in this case? We start noting that at very high densities, the presence of pycno-nuclear reaction and inverse β-decay that occur on very short time scales destroys the notion of frozen com- position. The natural question arises: what the equilibrium composition would be if we let these processes to occur at constant total baryon number? Leaving aside pycno-nuclear reactions, the equilibrium composition can be found starting from a certain type of nuclei in β-interaction with degenerate electrons and by varying Z and A look for the minimum energy configuration. This would yield the chemical composition. For isolated nuclei, the competition between attractive nuclear forces and coulomb repulsive forces gives taht the maximum binding energy among nucleons is reached at 56F e. Therefore 56F e is the equi- librium compositions for densities up to ρ < 8 × 106g cm−3. At increasing ρ the balance is shifted to heavier nuclei with a larger number of neutrons. Replacing a proton with a neutron in nuclei by inverse β-decay weakens the coulomb repulsion inside nuclei. The direct β cannot occur owing to the high degeneracy of the external electrons. Furthermore, the energy of the crystalline structure, while does not appreciably change the pressure, it alters the coulomb energy at the surface of nuclei. At increasing density the equilibrium composition is shifted toward nuclei with large A and N. At densities ρ > 4 × 1011g cm−3, the situation of minimum energy is that in which neutrons leave the nuclei (this is called neutron drip). The composition consists of two phases: the lattice of nuclei (with sufficient electrons for charge neutrality) plus a population of free neutrons. Their number as well as the pressure Pn they exert increase with density ρ. At ρ ' 4 × 12 −3 10 g cm the neutron pressure exceeds the electron pressure Pn > Pe. At ρ ' 2 × 1014g cm−3 the nuclei completely dissolve leaving behind a gas of degenerate neutrons with a little percentage of protons and electrons. The neutrons obey 160 CHAPTER 8. WHITE DWARFS

Figure 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 β-decay. Different equilibrium chemical compositions are considered. their own equation of state. Once the equation of in this extreme conditions (equilibrium composition in presence of neutronization) is known, one can integrate the me- chanical equations (mass conservation and hydrostatic equilibrium) to get new Mass-Radius and Mass-Central-Density relationship, which conceptually extend those derived for normal equation of state for White Dwarfs. The new Mass- Radius relationship is shown in Fig. 8.6, while the new Mass-Central-Density relation is shown in Fig.8.7. The new Mass-Radius relation predicts the existence of a new limit mass at decreasing radius. The new limit mass is smaller than the Chandrasekhar limit and it is reached for radii larger than the zero. In reality this limit mass will never be reached by real stars because of General Instability effects (see below). The new Mass-Central-Density relation predicts the existence of a new limit mass at increasing central density. The new limit mass is smaller than the Chan- drasekhar limit and it is reached for finite densities (instead of ρc → ∞). In reality this new limit mass can be hardly reached by real stars because of Gen- 8.7. REVISION OF THE MASS-RADIUS RELATIONSHIP 161

Figure 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 in- cluding the effect of inverse β-decay. Different equilibrium chemical compositions are considered. eral Instability effects (see below).

Do White Dwarfs become Neutron Stars? To avoid misunderstand- ing, we stress the following fact: White Dwarfs originate as remnants of stars of a certain mass range in which the nuclear reactions have not proceeded to the maximum possible limit. For example most of the White Dwarfs are made of helium or carbon and oxygen in which further nuclear reactions have not taken place. In this sense, they do not constitute the lowest energy state of an object with a certain number of baryons. They are, however, quite stable even in cosmological time scales, as there is no possibility of further nuclear reac- tions taking place in the low-density isolated White Dwarfs. If nuclear reaction were possible, owing to the strong degeneracy and positive gravo-thermal specific heat this would induce a with catastrophic consequences (ex- plosion). Neutron stars, on the other hand, arise as the stellar remnant, in which the nuclear reactions could have produced 56F e in the core, which represent the lowest energy state at moderate densities. Com- puting the structure of a neutron star at much higher densities we will always use 162 CHAPTER 8. WHITE DWARFS an equation of state in which the lowest energy state was explicitly taken into ac- count. Because of this, neutron stars can be considered as true minimum-energy configurations. It is therefore clear that no stable structures can exist in the intermediate regime that roughly correspond to the transition from White Dwarf to neutron star structure. White Dwarfs and Neu- tron stars are two distinct regimes of matter behaviour that cannot communicate each other.

8.7.4 Correction at high densities: GR effects

The discussion of White Dwarfs so far has been based on Newtonian gravity. At high densities, however, it is necessary to take into account general relativity effects which could also lead to instability. The physical reason for this instability is the following: in the Newtonian theory an equilibrium configuration is possible if the gravitational field of the mass den- sity balances the forces generated by the pressure gradient. In principle, we can arrange the equation of state such that the pressure is arbitrarily high for any given density. In General Relativity, however, pressure also contributes to effective mass and hence increasing the pressure will also increase the effective mass and the grav- itational force in turn. Therefore we cannot ensure that a system is stable by increasing the pressure arbitrarily. To analyse this problem we can proceed as follows: We have seen that White Dwarfs are governed by a polytropic equation of state (but for the very thin external layer) in which the central density ρc is the parameter, and that all properties can be expressed as functions of this parameter. This means that also the total energy is E(ρc). At the equilibrium dE = 0 dρc 8.7. REVISION OF THE MASS-RADIUS RELATIONSHIP 163 and the stability of the system is related to the sign of d2E 2 dρc 2 2 If d E/dρc > 0, the system is in a minimum-energy configuration and is stable. 2 2 2 2 If d E/dρc < 0, the system is in a local maximum and unstable. If d E/dρc = 0, the system is in a neutral condition.

The method we intend to follow is straightforward: derive the function E(ρc) for a polytrope of index n, including the first order effect of General relativity, and use the above condition to determine the stability. The total energy of a polytrope with P = KρΓ (K is the polytropic constant), is the sum of the internal energy Ei ∝ PV and the gravitational energy EG ∝ 2 −(GM /R). Denoting the various constants of proportionality as k0, k1, .... we get

GM 2 GM 2 E = E +E = k PV −k = k P Mρ−1−k = k MρΓ−1−k GM 5/3ρ1/3 i G 0 1 R 0 1 R 2 c 3 c (8.93) where the volume V has been replaced by M/ρ.

The equilibrium condition dE/dρc = 0 leads to the relation

M ∝ ρ(3/2)(Γ−4/3) (8.94)

For stability, an increase in mass must lead to an increase of central density; that is we need d ln M 4 = (Γ − ) > 0 (8.95) d ln ρc 3 The second derivative at the turning point is proportional to 2 ³ ´ d E 5/3 −5/3 4 2 ∝ GM ρc Γ − (8.96) dρc 3 The system is stable (second derivative greater than zero) if (Γ > 4/3). Both are long known results! To proceed from Newtonian Theory to General Relativity we need the corresponding expression for E(ρc) in general relativity. The lowest 164 CHAPTER 8. WHITE DWARFS order relativistic correction can be derived as follows: in Newtonian gravity, the 2 gravitational energy is EG = −α1GM /R where α1 is a numerical factor that can be determined. In general relativity this is equivalent to an effective mass

2 meff = |EG|/c (8.97)

The coupling of this mass with M will provide the lowest-order correction

GMm G2M 3 E = −α eff = −α α (8.98) G,corr 2 R 1 2 c2R2

3 where α2 is a known numerical factor. Writing ρc = α3M/R and eliminating R we obtain ³ ´³ ´ α1α2 G 2 7/3 2/3 EG,coor = − M ρc (8.99) α3 c where α3 is another numerical factor that can be derived if the mass distribution in the star is known. The exact value is not relevant here. Inserting this correction in the expression for the total energy we obtain

G E = k MρΓ−1 − k GM 5/3ρ1/3 − k ( )2M 7/3ρ2/3 (8.100) 2 c 3 c 4 c c

The definition of k4 is obvious. Repeating the analysis by computing the second 1/3 1/3 derivative and writing ρc = α3(M R) we obtain

4 2 k4α3 GM Γ > + ( )( 2 ) (8.101) 3 3 k3 c R

This shows that General Relativity changes the critical Γ to a larger value. In the case of polytrope of index n = 3 the correction to Γ has the value 2.25(GM/c2R). There is another correction to be applied because we have adopted (Γ−1) for the internal energy the polytropic formulation (the term k2Mρc ), which would strictly hold for fully degenerate and fully relativistic electrons. In the case of a White Dwarf this would correspond to an infinite central density when the structure is that of the polytrope with index n = 3. But in general electrons are not fully relativistic and the White Dwarf 8.7. REVISION OF THE MASS-RADIUS RELATIONSHIP 165 is not a polytrope. This would immediately reflect on the real value of Γ and the stability conditions in turn. As the correcting procedure is quite elaborate, we break it in steps. First we derive the actual Γ starting from the general expression for pressure (fully degenerate case) Z pF 8π 2 Pe = pv(p)p dp (8.102) 3h3 0

We expand v(p) in Taylor series of mec/p

³ m2c2 ´−1/2 m2c2 v = c 1 + e ' c(1 − e ) (8.103) p2 2p2 to derive ³ 2 2 ´ 2πc 4 mec Pe = 3 pF 1 − 2 (8.104) 3h pF 3 1/3 The density scales as ρ ∝ ne ∝ pF thus pF = Kρ . Take the logarithm of (8.104)

2 2 2 2 mec 4 mec 2/3 ln Pe ' 4 ln pF − 2 + const ≡ ln ρ − 2 ρ + const (8.105) pF 3 K Therefore from the definition of Γ

2 2 dlnPe 4 2 mec −2/3 4 2 mec 2 Γ = = + 2 ρ = + ( ) (8.106) dlnρ 3 3 K 3 3 pF Comparing the two conditions for Γ we have derived, stability requires

2 GM mec 2 2 < ( ) (8.107) Rc pF

3 which replacing pF in terms of ρ ∼ M/R becomes

GM 5/3 ³m c´2 < e (µ m )2/3 (8.108) R3c2 h e H or ³ ´ 8 M 5/9 −2/9 R > 10 × µe cm (8.109) M¯

For a White Dwarf close to the Chandrasekhar limit (µe = 2, M = 1.4 M¯), and 6 2 3 ρ > ρrel = 10 µe g/cm (with ρrel the density at which relativistic effects begin) 9 1/3 the radius scales R ' 10 (ρrel/ρ) cm. The radius will fall below the value set 166 CHAPTER 8. WHITE DWARFS

by the inequality (8.109) if ρ > 600 ρrel; the white dwarf will become unstable because of relativistic effects when ρ > 1.2 × 109 g/cm3. More rigourous formulation. We shall now provide a more detailed and rigourous evaluation of these results. The idea is to derive an equation of the form

E = ENewt + ∆EGRT (8.110) where ENewt is the Newtonian energy of the star and ∆EGRT is the correction due to General Relativity. Minimizing the energy will give the equilibrium con- figuration, and the second derivative will give stability information. The total energy of a general relativistic, spherical mass distribution with total mass M and baryon number N, excluding the rest energy, is

2 2 E = Mc − mBc N (8.111)

where mB is the mass of individual baryons (mB = AmU ' AmH ). Taking for the metric describing the space-time associated to spherically sym- metric distribution of mass the form

ds2 = −e2φdt2 + e2λdr2 + r2dΩ2 (8.112) where the function m(r) is defined by the relation

³ 2m´−1 e2λ = 1 − (8.113) r and geometrized units (c = G = 1) are used. The total mass M is

Z R Z R Z R 2 2 2 M = ρ4πr dr = ρ0(1 + u)4πr dr = mB n(1 + u)4πr dr (8.114) 0 0 0 and the total number of baryons is

Z q Z Z R R R h 2m(r)i−1/2 N = n 3gd3x ≡ ndV = n 1 − 4πr2dr (8.115) 0 0 0 r 8.7. REVISION OF THE MASS-RADIUS RELATIONSHIP 167

where ρ is the total mass density ρ0 is the rest mass density, u is the internal energy density (both contribute to mass) so that

ρ = ρ0(1 + u) (8.116) n is the baryon number density, and finally

q ³ 2m´−1/2 3gd3x ≡ dV = (g )1/2dr × 4πr2 = 1 − 4πr2dr (8.117) rr r is the proper volume element in the 3-space. The total energy can be written Z R h ³ 2m´1/2 i E = ρ 1 − − ρ0 dV (8.118) 0 r

Substituting ρ = ρ0(1 + u) and considering u and m/r as small quantities the second order Taylor expansion gives Z R h m m 1³m´2i E = ρ0 u − − u − dV (8.119) 0 r r 2 r

The product ρ0dV is an invariant and is not expanded. The corresponding expression for the total energy in Newtonian Theory is

Z R Z M 0 m 0 ENewt = ρ0udV − dm (8.120) 0 0 r0 where 0 dm = ρ0dV (8.121) ³3V ´1/3 r0 = (8.122) 4π Note that the functions m’(V) and r’(V) differ from their relativistic counterparts because of eqn.(8.116) and eqn.(8.117). We are trying to compute the energy of a star, first according to General Relativity, then according to Newton Theory, and then calling the difference ∆EGRT . How can we be sure because of the ambiguity in the meaning of coordinates in General Relativity, of what exactly is the same in the two cases? Equivalently, if given two identical stars and asked to compute 168 CHAPTER 8. WHITE DWARFS

one’s energy as E, and the other’s as ENewt, how do we know that the two stars are identical? The answer is that the two stars have the same number of baryons in a given proper volume (coordinate independent statement!) in other words ρ(V) is the same function in General Relativity and Newtonian Theory. Subtracting eqn.(8.120) from eqn.(8.119) we obtain

Z R h m 1³m´2 m0 mi ∆EGRT = ρ0 − u − + − dV (8.123) 0 r 2 r r0 r Now we proceed to transform eqn.(8.123) in such a way that using newtonian relations for ρ0, r etc we can calculate ∆EGRT . It can be shown that to the first order

4πr3 ³ 3 Z r ´ V = 1 + mr dr (8.124) 3 r3 0 Equations (8.124) and (8.122) give

1 Z r r0 − r = mr dr (8.125) r2 0 Working again to the first order we find

Z V h ³ ´ i 0 2m 1/2 m (V) − m(V) = dV ρ0 − ρ 1 − (8.126) 0 r Z V ³ m´ = − ρ0dV u − (8.127) 0 r Now in eqn.(8.123) we may write m0 m m0 − m m(r0 − r) − = − (8.128) r0 r r0 rr0 and substitute the results into eqn.(8.125) and eqn.(8.127). Since we have re- tained all second-order terms consistently, if we now calculate the integrals with the Newtonian relations for ρ0, r, etc.. we are making an error only at the third order. The gravitational correction can be expressed as

∆EGRT = I1 + I2 + I3 + I4 + I5 (8.129) 8.7. REVISION OF THE MASS-RADIUS RELATIONSHIP 169 where Z M m I1 = − u dm (8.130) 0 r Z 1 M ³m´2 I2 = − dm (8.131) 2 0 r Z M dm Z m I3 = − udm (8.132) 0 r 0 Z M dm Z m m I4 = dm (8.133) 0 r 0 r Z M mdm Z r I5 = − mr dr (8.134) 0 r4 0 This result is true for any given equation of state; given a Newtonian theory, the relativistic corrections are given by these terms correct to the second order in u and m/r. Polytropic Equation of state. In this case we can simplify the above integral using the relations (remember that we are still using geometrized units)

P 1 dP m u = n = − 2 (8.135) ρ0 ρ0 dr r where n is the polytropic index. With the aid of these relations and standard expressions from polytropic theory, we can relate the integrals to each other,

1 2 3 2n I = I ,I = 2I − I − I ,I = I − (I + I ) (8.136) 5 n 1 4 2 n 1 n 3 3 1 n + 1 2 4 and we can express ∆EGRT as 5 + 2n − n2 n − 1 ∆E = 2I + 3I = −kM 7/3ρ2/3 (8.137) GRT n(5 − n) 1 5 − n 2 c where k is a constant expressible in terms of the Lane-Emden function θ(ξ) that satisfies the equation 1 d ³ dθ ´ ξ2 = −θn (8.138) ξ2 dξ dξ with central boundary conditions ξ = 0, θ(0) = 1, and θ0(0) = 0. Note that the correction term of eqn.(8.137) has the same form as determined above from simple arguments. 170 CHAPTER 8. WHITE DWARFS

The constant k derived from the polytropic theory is

Z 2/3 h ³ 2 ´ ξ1 (4π) 5 + 2n − n 3 0 n+1 k = 2 0 7/3 − 2 ξ θ θ dξ (5 − n)[ξ1 |θ (ξ1)|] n + 1 0 Z 3 ξ1 i + (n − 1) ξ4θ02θndξ (8.139) 2 0

The practical use of the constant k of eqn.(8.139) requires the restoration of the natural units, i.e. k must be multiplied by (G/c)2. We shall now apply this result to the case of a relativistic White Dwarf corre- sponding to a polytrope of index n = 3. The total energy of a polytrope with P = Kρ(n+1)/n, where K is the polytropic constant, is given by

E = Eint + Egrav + ∆Eint + ∆EGRT (8.140)

The internal thermal energy is Z Z Z ξ1 P 1/n n 2 n+1 1/n Eint = udm = n dm = Kρc M 2 0 ξ θ dξ = k1Kρc M (8.141) ρ |ξ1 θ | 0 The gravitational energy is Z Z 5/3 ξ1 m 1/3 GM 3 0 n 1/3 5/3 Egrav = −G dm = (4πρc) 2 0 5/3 ξ θ θ dξ = −k2Gρc M (8.142) r |ξ1 θ | 0 with

2 0 2 0 1/3 n(n + 1) |ξ1 θ | 3 |4πξ1 θ | k1 = = 1.75579 k2 = = 0.639001 (8.143) 5 − n ξ1 5 − n ξ1

31/3π2/3 hc¯ 1.2435 × 1015 K = 4/3 4/3 = 4/3 cgs (8.144) 4 mp me µe for n = 3. The coefficient of the GRT correction has the value 0.918294 in geometrized units, so that

³G´2 ∆E = −k M 7/3ρ2/3, k = 0.918294 (8.145) GRT 4 c c 4 8.7. REVISION OF THE MASS-RADIUS RELATIONSHIP 171

The correction term ∆Eint represents the contribution of the electrons to the internal energy arising from the departure from the extreme relativistic limit of n = 3, which is of comparable order with general relativistic corrections and therefore must be retained. The internal energy density of a relativistic gas can be expanded in a Taylor series of x = pF /mec. Recalling the kinetic theory of gas pressure, the electron energy density is

³ 2 ´ ³ 2 ´ E 1 mec 1 mec 4 3 2 = 2 3 χ(x) = 2 3 (6x − 8x + x ....) (8.146) V 24π λe 24π λe and the number density of electrons is

1 1 3 n = 2 3 x (8.147) 3π λe This gives for the energy per particle

E 3 ³ 4 1 ´ = m c2 x − + + ... (8.148) nV 4 e 3 x correspondingly, the energy per unit mass is

3 m c2 ³ 4 1 ´ u = e x − + + .... (8.149) 4 µemp 3 x

The first term is the one that gives Eint, the second term is the negative of rest 2 mass energy -mec N and can be dropped with the understanding that E now measures the total energy of the system including the rest mass. So the first non trivial correction term is the third one which contributes an amount

2 Z ³ 2 3 ´ 3 mec 1 3π ρλe 1/3 ∆Eint = dm, x = (8.150) 4 µemp x µemp This gives 2 3 mec −1/3 ∆Eint = k3 2/3 Mρc (8.151) h¯(µemp) with Z ξ1 3 1 1 2 2 k3 = 1/3 2 0 ξ θ dξ = 0.519723 (8.152) 4 (3π) |ξ1 θ | 0 Putting together all the contributions to energy for a polytrope with n = 3 we get 172 CHAPTER 8. WHITE DWARFS

5/3 1/3 −1/3 7/3 2/3 E = (AM − BM )ρc + CMρc − DM ρc (8.153) with

2 3 2 mec G A = k1K,B = k2G, C = k3 2/3 ,D = k4 2 (8.154) h¯(µemp) c

The equilibrium configuration is determined by the condition (dE/dρc = 0) lead- ing to

1 1 2 (AM − BM 5/3) ρ−2/3 − CMρ−4/3 − DM 7/3ρ−1/3 = 0 (8.155) 3 c 3 c 3 c To the first order ignoring the terms proportional to C and D we get

³ A ´3/2 ³ 2 ´2 5.86 M = = 1.457 M¯ = 2 M¯ (8.156) B µe µe which shows that equilibrium condition for relativistically electrons is possible only at the Chandrasekhar mass. The terms C and D provide only small correc- tions to this limit. 2 2 The stability is determined by (d E/dρc ≥ 0) and the stability limit is for 2 2 (d E/dρc = 0), which is 2 4 2 − (AM − BM 5/3)ρ−5/3 + CMρ−7/3 + DM 7/3ρ−4/3 = 0 (8.157) 9 c 9 c 9 c

Substituting for (AM −BM 5/3 from eqn.(8.155), all the terms will be of the same order and M can be replaced with (A/B)3/2. Then we get the critical density

2 2 2 2 ³ ´ CB 16k3k2 mpµe 10 µe 2 ρc = 2 = 2 2/3 2 3 = 2.9 × 10 (8.158) DA (3π ) k4k1 λeme 2 Note that the existence of this density is a purely relativistic effect and will not occur if ∆EGRT (proportional to D) is not present. White Dwarf with densities higher than this limit become unstable by General Relativity. Surprisingly, the numerical value of the critical density does not depend on G. 56 10 3 For F e with µe = 2.154 we have ρc = 3.07×10 g/cm which is higher than the neutronization threshold of 1.4×109 g/cm3 making this instability irrelevant. For 8.8. EFFECTS OF MAGNETIC FIELDS 173

10 3 helium and carbon, ρc ' 2.65×10 g/cm which is lower than the neutronization thresholds of 1.37 × 1011 g/cm3 and 3.9 × 1010 g/cm3, respectively. Thus in helium-carbon White Dwarfs, the general relativistic effects provide the limit on the stability. The ratio between General Relativity and Newtonian contributions is

∆EGRT k4 G 2/3 1/3 −3 R ≡ = 2 M ρc ' 6.57 × 10 (8.159) ENewt k2 c which secures that the above perturbative method is valid. Finally, the realistic effects described in this section show that the limiting configuration for White Dwarfs is set by different physical processes rather than the Chandrasekhar mass, which is of only theo- retical significance.

8.8 Effects of Magnetic Fields

Let us consider the case of non rotating White Dwarf with a magnetic field of intensity B and apply the virial condition for equilibrium,

EG + Ei + Em = 0 (8.160)

where EG is the gravitational energy, Ei = 3M < P/ρ > is the internal energy and Em is the magnetic energy

DB2 E4πR3 E = (8.161) m 8π 3 The brackets denote an average. In the case of high conductivity, the magnetic flux

2 Φm ∼< B > R (8.162)

2 is conserved therefore the magnetic energy can be written as Em ∼ Φm/R. Considering now the two cases of non relativistic and fully relativistic electrons we have 174 CHAPTER 8. WHITE DWARFS

GM 2 M 5/3 Φ2 0 = −α + β + γ m (NRD) (8.163) 3/2 R 3/2 R2 3/2 R GM 2 M 4/3 Φ2 0 = −α + β + γ m (ERD) (8.164) 3 R 3 R 3 R where αn, βn and γn are positive constant that can be derived from the polytropic theory. For both types of degeneracy, the effect of a magnetic field is to expand the star; in short adding magnetic flux is equivalent to reduce the gravitational constant G

2 h i 0 γΦm Em G = G − 2 = G 1 − (8.165) αM |EG|

For the NRD case we can solve eqn.(8.163) for the radius of the equilibrium configuration

β3/2 R0 R = 0 1/3 = (8.166) α3/2G M 1 − Em/|EG| where R0 is the radius when B = 0. For small ratios δ = Em/|EG| the radius increases by a small amount. For the ERD case the situation is different: the limit mass is not severely affected by the magnetic field whereas the radius is. The new limit mass is

³ 2 ´ ³ 2 ´ 2/3 β3 γ3Φm β3 γ3Φm M = 1 + 4/3 ' 1 + 2 (8.167) α3G β3M α3G α3GM if δ << 1. Therefore

Mmax = MCh(1 + δ) (8.168)

MCh is the standard limit. The radius can increase significantly even for small values of δ. The reason i simply that for δ << 1 the star is close to n = 3 polytrope and the energy satisfies 8.9. EFFECTS OF ROTATION 175

3 − n E = − |E | << |E | (8.169) 3 G G A small change in E can result in a large change in R. ∆E ∆R = − with ∆E = ∆E (8.170) E R m ∆R 3 ∆E 3 = m = ∆δ (8.171) R 3 − n |EG| 3 − n Integrating, assuming n constant as δ increases, we find 3 R = R exp( δ) (8.172) 0 3 − n The radius increases significantly even for small δ. Although the result is of theoretical significance, we have no evidences for strong magnetic fields such that Em ' EG in White Dwarfs.

8.9 Effects of Rotation

8.9.1 A simple approach

Let us consider the effect of rotation on the structure of White Dwarfs. Let M and R be the mass and radius, Ω the angular velocity, I ∝ MR2 the momentum of inertia (the proportionality constant depends on the mass distribution), J = IΩ the conserved angular momentum, and T the rotational kinetic energy

1 J 2 T = IΩ2 ∝ Ω2MR2 ∝ (8.173) 2 MR2 Let us also consider only the two extreme cases: (i) fully degenerate non rela- tivistic gas of electrons (P ∝ ρ5/3), structure described by a polytrope with index n = 3/2 (NRD); (ii) degenerate and relativistic gas of electrons (P ∝ ρ4/3) de- scribed by a polytrope with index n = 3 (ERD). With aid of the Virial Theorem the equilibrium is given by Egrav + Eint + Erot = 0

GM 2 M 5/3 J 2 0 = −α + β + k (NRD) (8.174) 3/2 R 3/2 R2 3/2 MR2 GM 2 M 4/3 J 2 0 = −α + β + k (ERD) (8.175) 3 R 3 R 3 MR2 176 CHAPTER 8. WHITE DWARFS

where αn, βn, and kn are suitable constants that we do not need to determine. The NRD case is not of much significance because it provides a small correction to the standard mass-radius relationship. The situation is different for the ERD case. In absence of rotation (J = 0) eqn.(8.175) can be satisfied only for a specific value of M because the two terms have the same dependence on R. With rotation, however, the rotational term has a steeper dependence on R (∝ R−2) than the other two; it is then possible to obtain an equilibrium model for any mass by decreasing the radius sufficiently. This is in sharp contrast to the non rotating case that leads to the Chandrasekhar limit. However, very small radii would imply high densities which would fall above the threshold set by general relativity effect that we have discussed in previous sections. Nevertheless, this analysis shows that the mass bound is increased by ro- tation. Writing the Virial equation in terms of the total kinetic energy T and 2 gravitational potential energy with |EG| = | − GM /R| we have

GM 2 ³ 2T ´ M 4/3 0 = −α3 1 − + β3 (8.176) R |EG| R

Solving for M we find

h β i3/2 M = 3 (8.177) α3G(1 − 2T/|EG|) which can be expressed in terms of the Chandrasekhar mass in absence of rotation

MCh as MCh MCh,R = 3/2 (8.178) (1 − 2T/|EG|) For small angular velocities it can be approximated to

³ 3T ´ MCh,R ' MCh 1 + (8.179) |EG|

An important question arises. Does rotation change MCh or not? It depends on several factors. Studies of uniformly rotating White Dwarfs obeying the Chandrasekhar equation of state demonstrated that the 8.9. EFFECTS OF ROTATION 177 mass limit is increased by only 3.5%. The reason is that uniform ro- tation in centrally concentrated bodies requires the ratio T/|EG| to be less than 0.007. The demonstration is as follows. In a polytropic with n = 3 the central density is about 54 times the mean density and so the object is highly centrally concen- trated. Consider first a spherical star rotating with break-up velocity at the equator GM v2 = Ω2R2 = (8.180) R For a polytrope with n = 3

3 GM 2 3 GM 2 |E | = = (8.181) G 5 − n R 2 R and 1 T = IΩ2 (8.182) 2 where Z Z M 2 ξ1 2 2 2 2 2 MR n 4 I = M < r >= r dm = 4 0 θ ξ dξ (8.183) 3 3 0 3 ξ1 |θ (ξ1)| 0 The integral has the value 10.851, and so

< r2 >= 0.11303R2 (8.184)

Therefore T 1 2 M < r2 > Ω2 = 2 3 = 0.025 (8.185) 3 2 |EG| 2 GM /R By contrast, for an incompressible fluid (n → 0)

2 3 GM 2 < r2 >= R2, |E | = (8.186) 3 G 5 R and so at break-up T 1 = (8.187) |EG| 3 This illustrates the effect of the mass concentration on the rotational mass limit. 178 CHAPTER 8. WHITE DWARFS

The above discussion has ignored the fact in reality rotating stars are not spherical but spheroidal. Taking this into account we arrive to the result that in uniformly rotating stars ³ T ´ ' 0.00744 (8.188) |EG| max There are many other possible situations leading to significantly different values for T/|EG|. In general a range of values from 0.14 to 0.20 is obtained. This means that the limit mass could be increased by as much as a factor of 2. This factor can be even larger in some special cases. However, observational data exist indicating that White Dwarfs do not rotate very rapidly (typical rotational velocity vrot ' 10 − 40 km/s) which implies that a large fraction of the angular momentum in the precursor stars has been lost in earlier stages. In conclusion, the effect of rotation on MCh is still highly uncertain. The demonstration of the limit T/|EG|max ≤ 0.00744 is postponed to the section below.

8.9.2 Equilibrium of Rotating Configurations: the MacLau- rin Spheroids

The above discussion on the effects of rotation is not particularly satisfactory as it neglects many important properties of rotating configurations, the deviation from spherical symmetry in particular. Most of our intuition about rotating, self-gravitating configurations comes from studying uniform density ellipsoids, which can be analyzed relatively sim- ply. The simplest such homogeneous ellipsoids are the MacLaurin spheroids, which rotate with uniform angular velocity Ω. The gravitational potential at any point (x, y, z) inside a uniform spheroid is of type 2 2 2 Φ = −πGρ[A − A1x − A2y − A3z ] (8.189)

0 where the A s depend on the shape of the ellipsoid and A1 + A2 + A3 = 2. This follows from the Poisson equation

∇2Φ = 4πGρ (8.190) 8.9. EFFECTS OF ROTATION 179 with ρ constant. For the derivation of the A0s see Chandrasekhar. The formal solution of eqn.(8.190) is Z d3x0 Φ = −Gρ (8.191) |x − x0| In spherical coordinates

X∞ l 1 r< 0 0 = l+1 Pl(cos θ)Pl(cosθ ) + [φ − dependent terms] (8.192) |x − x | l=0 r> where the Pl’s are the Legendre polynomial and r< (r>) are the lesser (greater) of r and r0. By azimuthal symmetry the φ-terms do not contribute to eqn.(8.191). The surface of the spheroid has the polar equation R − R(θ), where

sin2 θ cos2 θ 1 + = (8.193) a2 c2 R2 where a and c are the semi-major and semi-minor axes. Thus

X∞ Z π 0 0 0 Φ = −2πGρ Pl(cos θ) sin θ dθ Pl(cos θ ) l=0 0 ³ Z r (r0)l+2dr0 Z R (r)ldr0 ´ × + (8.194) 0 rl+1 r (r0)l−1

Upon integration eqn.(8.194) becomes

n 1 Z 1 dx Φ = −2πGρ − r2 + 3 0 1/a2 + (1/c2 − 1/a2)x2 r2 Z 1 h 1 1 1 io − (3 cos2 θ − 1) dx(3x2 − 1) log ( − )x2 (8.195) 4 0 a2 c2 a2

(1 − e2)1/2 1 − e2 A = A = sin−1 e − 1 2 e3 e2 2 2(1 − e2)1/2 A = − sin−1 e 3 e2 e3 2a2(1 − e2)1/2 A = sin−1 e (8.196) e where x = cosθ0 and the eccentricity e is defined by

c2 e2 ≡ 1 − (8.197) a2 180 CHAPTER 8. WHITE DWARFS

A uniformly rotating spheroid in hydrostatic equilibrium satisfies

dv 1 = − ∇P − ∇Φ (8.198) dt ρ with v = Ω × r (8.199)

Now, choosing Ω along the z-axis we can write

dv = −Ω2(xe + ye ) = Ω × (Ω × r) (8.200) dt x y which is the centripetal acceleration of the fluid. Thus the z component of eqn.(8.198) gives 1 ∂P ∂Φ 0 = − − (8.201) ρ ∂z ∂z while the x component gives

1 ∂P ∂Φ −Ω2x = − − (8.202) ρ ∂x ∂x

Since Φ is quadratic function of the coordinates, P must be a quadratic function too. Since P vanishes on the surface of the spheroid, we have

³ x2 + y2 z2 ´ P = P 1 − − (8.203) c a2 c2 where Pc is the central pressure. Using eqn.(8.189) we get from eqn.(8.201)

2 2 Pc = πGρ c A3 (8.204) and then from eqn.(8.202) that

³ A c2 ´ Ω2 = 2πGρ A − 3 1 a2 h(1 − e2)1/2 3(1 − e2)i = 2πGρ (3 − 2e2) sin−1 e − (8.205) e3 e2

The moment of inertia of the spheroid about the rotation axis is

2 I = Ma2 (8.206) 3 8.9. EFFECTS OF ROTATION 181 where 2 M = πa3(1 − e2)1/2ρ (8.207) 5 The angular momentum is J = IΩ (8.208)

The kinetic energy is 1 T = IΩ2 (8.209) 2 The gravitational potential energy is Z −1 3³4π ´2 sin e W = ρ Φd3x = − Gρ2a5 (1 − e2) (8.210) 5 3 e Finally, the ratio T/|W | is a useful parameter to describe the MacLaurin spheroids T 3 ³ e(1 − e2)1/2 ´ = 1 − − 1 (8.211) |W | 2e2 sin−1 e The analysis for stability reveals that (i) there are two instabilities of interest, both setting in via toroidal modes with azimuthal dependence exp(±2iφ). The pair of modes goes dynamically unstable for e <∼ 0.953 corresponding to T/|W | > 0.2738. (ii) Secular instability sets in at e ' 0.812, T/|W | ∼ 0.137. Actually this is point a point of bifurcation, from which two sequences of equilibrium configurations appear: MacLaurin and Jacobi sequences. The angular velocity and angular momentum of the two sequences are shown in Fig.8.8.

8.9.3 Revising the effects of rotation

The analysis carried out in previous sections has ignored the fact that the rotating configuration is indeed a spheroidal with radius R1 > R at the equator. We can analyze the shape approximately by adopting the Roche Model. This model assumes that the distribution of the bulk mass is unaffected by rotation, which is a good approximation because of the central condensation. In the outer layers, the gravitational potential remains Φ = −GM/r. For constant angular velocity about the axis z we may introduce the centrifugal potential 1 1 Φ = Ω2(x2 + y2) = Ω2r2 sin2 θ (8.212) C 2 2 182 CHAPTER 8. WHITE DWARFS

Figure 8.8: Left: The square of the angular velocity along the MacLaurin and Jacobi sequences. The abscissa is the eccentricity e. Right: The angular momen- tum along the MacLaurin and Jacobi sequences. The abscissa is the eccentricity

Then dv = ∇Φ (8.213) dt c and so the hydrostatic equilibrium condition becomes

1 0 = ∇P + ∇(Φ + Φ ) (8.214) ρ c or

h + Φ + Φc = K (8.215) where K is a constant and Z dP Γ P h = = (8.216) ρ Γ − 1 ρ is the enthalpy per unit mass. We assume that K in eqn.(8.215) has the same value as in the non rotating case. Since h(R) = 0 we have GM K = − (8.217) R

The effective potential Φeff = Φ + Φc is shown in Fig.8.9. Along an equatorial 2 1/3 radius Φeff has a maximum at rc = (GM/Ω ) , where Φ = −3GM/(2rc). Note that eqn.(8.216) has a meaningful solution only when h goes to zero at some r = R1 the surface of the star. Since h(r) is the distance from Φeff (r) to the line Φ = K in Fig.8.9, we see that for a meaningful solution K must be less than 8.9. EFFECTS OF ROTATION 183

Figure 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). Horizontal lines K1 and K2 correspond to different values of the constant K in eqn.(8.216)

Φmax. At equality R1 has the maximum value, i.e, rc = 3R/2. Therefore the maximum expansion of a uniformly rotating star along the equator is a factor of 3/2. The corresponding maximum angular velocity Ω is

³GM ´1/2 ³ ´3/2³GM ´1/2 Ω = 3 = 2/3 3 (8.218) rc R a factor 0.544 smaller than the maximum value given by v2 = GM/R of the spherical case. Hence

T ³2´3 = × spherical case = 0.00744 (8.219) |W | max 3 Models of rotating White Dwarfs have been calculated by Ostriker and Boden- heimer (1968) in the mass range 0.5 − 4.1 M¯, to whom we refer for details. Mass limit for rapidly rotating White Dwarfs. Given the above premises, we now examine the mass limit for rapidly rotating White Dwarfs assuming that in any case they are both secularly (T/|W | ≤ 0.14) and dynamically stable (T/|W | ≤ 0.26). To this aim we adopt the standard energy method. We make several simplifying assumptions: (i) the density is constant in these spheroids; (ii) that a spherical surface of constant density in the non-rotating case transforms into a spheroidal surface enclosing the same volume; (iii) the equation of state is the same of a non rotating star; (iv) the density as a a function of the mass 184 CHAPTER 8. WHITE DWARFS inside a given layer in the spheroidal star is the same function as in a spherical star. The internal energy is Z 1/3 Eint = u dm = k1Kρc M (8.220) where k1 and K are known from the polytropic theory. Now the gravitational potential energy of a sphere of constant density (n = 0 polytrope) is

3 GM 2 3³4π ´1/3 W = − = − GM 5/3ρ1/3 (8.221) 5 R 5 3 The corresponding result for a spheroid of constant density is

−1 −1 3 GM 2 sin e 3³4π ´1/3 sin e W = − = − GM 5/3ρ1/3 (1 − e2)1/6 (8.222) 5 a e 5 3 e For a spherical polytrope n = 3 the potential energy is

5/3 1/3 W = −k2GM ρc (8.223)

Under our assumptions, we show how W for the polytrope n = 3 is modified by rotation in the same way as it is modified for the homogeneous star, namely sin−1 e W = −k GM 5/3ρ1/3 (1 − e2)1/6 (8.224) 2 c e The result follows from Newton’s theorem that the potential inside an ellipsoidal shell of constant density is constant. Imagine constructing a spheroidal star by starting with the outermost spheroidal layer of constant density. Inside this layer, place the next layer of higher constant density, and continues in this way. Each layer is placed in a cavity where the potential is constant. The total potential energy will be smaller than for a spherical star by the same factor as the potential energy inside the spheroidal cavity is smaller than for the spherical cavity of the same volume and with the same external mass - that is, by the factor sin−1 e (1 − e2)1/6 (8.225) e 8.9. EFFECTS OF ROTATION 185

This proves the result (8.224). It will be convenient to introduce the oblateness parameter ³ c2 ´1/3 λ ≡ = (1 − e2)1/3 (8.226) a2 Then eqn. (8.225) becomes

g(λ) ≡ λ1/2(1 − λ3)−1/2 cos−1(λ3/2) (8.227) and

5/3 1/3 W = −k2GM ρc g(λ) (8.228)

Finally we need to evaluate the rotational energy

J 2 T = (8.229) 2I

Now I ∝ Ma2 for a spheroid. Hence, compared with a sphere of the same volume

I a2 1 = 2 2/3 = (8.230) Imax (a c) λ

From eqn (8.183) and (8.229) we find

2 −5/3 2/3 T = k5λJ M ρc (8.231) where 2/3 2 0 5/3 3(4π) ξ1 |θ (ξ1)| k5 = R = 1.2042 (8.232) ξ1 n 4 4 0 θ ξ dξ Thus

1/3 5/3 1/3 2 −5/3 2/3 E = k1KMρc − k2GM ρc g(λ) + k5λJ M ρc (8.233)

The corrections ∆Eint and ∆EGT R can be ignored since they are small and only relevant testing radial stability.

Equilibrium is determined by setting ∂E/∂ρc = 0 and ∂E/∂λ = 0, keeping M and J fixed, the condition ∂E/∂λ = 0 yields

2 1/3 0 k5J ρc T g(λ) g (λ) = 10/3 + (8.234) k2GM |W | λ 186 CHAPTER 8. WHITE DWARFS

Using eqn.(8.227) we obtain

T 1h 3λ3 3λ3/2 i = 1 + − (8.235) |W | 2 1 − λ3 (1 − λ3)1/2 cos−1 λ3/2 which is equivalent to relation (8.211) between e and T/|W | for a MacLaurin spheroid.

The condition ∂E/∂ρc = 0 yields 1 1 2 k KMρ−2/3 − k Gg(λ)M 5/3ρ−2/3 + k λJ 2M −5/3ρ−1/3 = 0 (8.236) 3 1 c 3 2 c 3 5 c or 1 ³ 2T ´ k KMρ−2/3 − k Gg(λ)M 5/3ρ−2/3 1 − = 0 (8.237) 1 c 3 2 c |W | Thus

³ k K ´3/2 M = 1 k2Gg(λ)(1 − 2T/|W |) M = ch (8.238) [g(λ)(1 − 2T/|W |)]3/2

2 where Mch = 1.457(2/µe) M¯. The maximum equilibrium mass for secularly stable rotating config- urations follows from setting T/|W | = 0.14. We find

λ = 0.693 g(λ) = 0.974 ³ 2 ´2 M = 1.70 Mch = 2.5 M¯ (8.239) µe The dynamically stable mass limit is found by setting T/|W | = 0.26. We find

λ = 0.475 g(λ) = 0.902 ³ 2 ´2 M = 3.51 Mch = 5.1 M¯ (8.240) µe The relevant dissipative mechanism for T/|W | ≥ 0.14 is gravitational radia- tion which acts on timescales of 103 − 107 yr. 8.9. EFFECTS OF ROTATION 187

Observational studies seem to indicate that White Dwarfs do not rotate ap- preciably (vrot ≤ 40 km/s and probably below 10 − 20 km/s ). Apparently a large fraction of the angular momentum of the progenitor star is transported outward before the formation of White Dwarfs. 188 CHAPTER 8. WHITE DWARFS Chapter 9

Origin of Neutron Stars

9.1 Introduction

There is by now a wealth of circumstantial evidence linking the birth of Neutron Stars to core collapse and supernova explosion of massive stars. Evolutionary calculations of non-rotating stars with initial masses in the range 10 ≤ M/M¯ ≤ 60 − 70 all exhibit the development of an unstable core, containing about 1.5 M¯. These stars complete the whole nuclear sequence from H to Si burning and their cores become electron degenerate only at the stage of the iron-core construction. Subsequent photo-dissociation of iron, will decrease the pressure support and start the collapse, which is further strengthened by the photo-dissociation of alpha particles and the neutronization process taking place in the infalling material. The companion supernova explosion give rise to the class of Supernovae named Type II.

9.2 The onset of collapse

The path of central conditions of a massive star core in logρc - logTc plane is such that no important electron degeneracy is encountered up to the very latest stages (Fig.9.1). Therefore, the core heats up passing from a nuclear burning to the subsequent one. For stars with M < 40M¯ or MHe < 40 M¯ (mass of the original He-core), the region of (e+, e−) pair instability is also avoided, whereby γ < 4/3.

189 190 CHAPTER 9. ORIGIN OF NEUTRON STARS

After passing though the whole nuclear sequence, the core reaches the stage of Sil- icon burning. The nuclear burning in shell located at different radii has produced the typical onion skin structure shown in Fig.9.1. Eventually, the central region has reached such high temperatures that the abundances of chemical elements are in nuclear statistical equilibrium conditions and a nucleus of 56F e (the species with the highest binding energy) is formed. The temperature is T ∼ 9 × 109 K. Virtually all calculations have shown that all massive stars develop a central core of 1.5 M¯ consisting of iron group nuclei supported by electron degeneracy pressure. The typical chemical structure of a 25 M¯ Population I star is shown in Fig.9.2 just after Si-burning and before core collapse. The convergence of the core to a common mass can be explained with the same kind of argument which leads to the minimum mass for the stars, and the common core mass for ignition of He

(∼ 0.5 M¯) and C (∼ 1.4 M¯) in degenerate conditions. The explanation is due to Arnett (1979). Because of the large temperature gradient (entropy gradient), heat is transported through the core largely by convection. When convection is present, motions appear that tend to equalize the temperature and mix the gas. The core is relatively homogeneous in composition. Start from the equation of state for a gas of nuclei and electrons (degenerate)

P YekT γ γ−1 = + KγYe ρ (9.1) ρ mB here Ye = ne/n is the number of electrons per baryon, and Kγ is a constant in the non-relativistic γ = 5/3 and extreme relativistic γ = 4/3 degeneracy limits. For a spherical configuration of mass M and radius R, the hydrostatic equilibrium condition requires Pc GM 2/3 1/3 ' ' fGM ρc (9.2) ρc R where the subscript c denotes the central values and f is a suitable factor de- pending on the density profile. Combining (9.2) and (9.1) one obtains

YekTc 2/3 1/3 γ γ−1 = fGM ρc − KγYe ρc (9.3) mB 9.2. THE ONSET OF COLLAPSE 191

Consider now the maximum temperature a configuration of given mass M can achieve. The result will indicate the mass required to ignite a given nuclear fuel. 1/3 For large M the first term on the r.h.s. of eqn. (9.3) dominates so Tc ∝ ρc . Consequently, continued contraction lead to even higher temperature so that the fuel ignites eventually. However, for small M and γ > 4/3, Tc falls to zero when

ρc increases to ³fGM 2/3 ´1/(γ−4/3) ρcrit = γ (9.4) KγYe

1/3 2/3 For ρc < ρcrit we have Tc ∝ ρc M , while larger ρc are unphysical (Tc < 0).

Thus the configuration will pass trough a maximum Tc as it contracts, and then cools as the degenerate electrons provide the dominant support. The two types of behaviour are divided roughly by the Chandrasekhar limiting mass, for which Tc = 0 and γ = 4/3 in eqn. (9.3)

³K ´3/2 M = 4/3 Y 2 ' 5.83Y 2 M (9.5) ch fG e e ¯

9 2 Now Si-burning, the final nuclear stage, requires Tc ∼ 3×10 K or kTc ≥ 0.6mec . It is not surprising that to reach such high maximum temperatures and ignite

Si, masses MSi,ign ' Mch are required.

If after the exhaustion of the lighter nuclei in the core, Mcore < MSi,ign, the ignition of Si in the core must wait until shell-burning of lighter nuclei increases

Mcore. Since Ye ' 0.42 at the onset of Si-burning, the requirement that nuclear burning go to completion in the core is

Mcore ≥ MSi,ign ' Mch(Ye = 0.42) ' 1.2 M¯ (9.6)

If instead, Mcore > MSi,ign at some earlier epoch in the evolution of a massive star, hydrostatic contraction will proceed to rather high temperatures. Between subsequent burning stages, neutrino cooling which varies with a high power of temperature, will become important near the center. Neutrino cooling reduces the entropy, thereby generating a positive entropy gradient. The net effect will be to inhibit the growth of the convective core at each nuclear stage, thereby shrinking 192 CHAPTER 9. ORIGIN OF NEUTRON STARS the region over which chemical mixing occurs. The thermonuclear ashes will encompass a smaller mass than contained in the original unburned core.

The end result of either scenarios – that is Mcore/Mch smaller or larger than unity prior Si-burning – is to drive the Mcore toward Mch (result confirmed by detailed numerical calculations). The ashes of Si burning consist of neutron-rich iron group elements.

Thus a typical 15 M¯ star will have a core of about 1.5 M¯ at the onset of collapse with initial temperature Tc,i, density ρc,i, electron concentration Ye,i, entropy per baryon si approximately given by

9 Tc,i = 8 × 10 K = 0.69 Mev/k

9 3 ρc,i = 3.7 × 10 g/cm

Ye,i = 0.42

si/k = 0.91 (9.7)

These physical conditions are very peculiar in many aspects. The pressure is dominated by fully degenerate electrons, which at temperatures of 8 × 109 ◦K 2 are relativistic (kT ' 1.7 mec ). The adiabatic exponent γad is close 4/3. Two different effects combine to drive the core to a collapse: (i) initially the collapse is triggered by the partial photo-dissociation of iron nuclei. This partial dissocia- tion uses up the binding energy of nuclei, thus lowering the pressure. (ii) As the collapse proceeds, the density rises, increasing the chemical potential of the elec- trons. This causes the neutronization of the core (electrons are captured by heavy nuclei while the inverse β-decay cannot occur because of the strong degeneracy). This lower the electron contribution to the total pressure, and the collapse pro- ceeds even further. Both photo-dissociation and neutronization lower the adiabatic index of stellar material below 4/3, thus leading to collapse.

9.3 Photo-dissociation

The partial photo-dissociation of nuclei into α particles at high T is well un- derstood. At high T and ρ, reactions involving the strong and electromagnetic 9.3. PHOTO-DISSOCIATION 193

Figure 9.1: Left: Path of central conditions. Right: Stratification of nuclear burning shells in a typical massive star in which the iron core has been just formed.

Figure 9.2: The chemical composition in the interior of a highly evolved model of 25 M¯ star of Population I. The mass concentrations of a few important elements are plotted against the mass Mr. Below the abscissa the location of shell sources and typical values of temperature (in K) and density (in g/cm3) are indicated. 194 CHAPTER 9. ORIGIN OF NEUTRON STARS interactions are sufficiently rapid that they proceed essentially in equilibrium with their inverses. The dissolution of 56F e is typical

56 γ + 26F e ⇔ 13α + 4n (9.8)

The energy required for this process is

2 Q = c (13mα + 4mn − mF e) = 124.4 Mev (9.9)

In equilibrium, the chemical potentials obey the relation

µF e = 13µα + 4µn (9.10)

For temperatures and densities of interest, the nuclei and nucleons are non de- generate, so that the Maxwell-Boltzmann statistics apply, so that for each species we have 2 µ − m c2 hn ³ 2πh¯ ´3/2i i i = ln i (9.11) kT gi mikT Recall that for a system with internal degree of freedom, such as the nucleus that has exited states, the statistical weight gi is the nuclear partition function X −Er/kT gi = (2Ir + 1)e (9.12) i where Ir is the spin of the r-th excited state and Er is the energy above the ground state. For temperatures ≤ 1 Mev, one may set gα = 1 (ground state,

I = 0), gn = 2 (free fermion, I = 1/2), and gF e = 1.4 (ground state with I = 0 plus lowest excited states). Substituting eqn.(9.11) in eqn.(9.10) yields the Saha equation for the equilibrium ratio of α particles and neutrons to F e nuclei

13 4 13 4 ³ ´ ³ 13 4 ´ nα nn gα gn kT 24 mα mn 3/2 −Q/kT = 2 e (9.13) nF e gF e 2πh¯ mF e With an accuracy better than one percent, we can replace the mass of a species with atomic weight A by Amu, so that 13 4 43 ³ ´ nα nn 2 mukT 24 −Q/kT = 3/2 2 e (9.14) nF e (56) (1.4) 2πh¯ 9.3. PHOTO-DISSOCIATION 195

If we assume the 56F e is the most abundant nucleus, the reaction (9.8) implies

4 n = n (9.15) n 13 α

Equations (9.14) and (9.15) allow one to calculate the degree of dissociation at any value of T and ρ.

The dissociation of 50% of the iron content at densities near ρc,i requires temper- atures near T9 ≈ 11 which is not much higher than Tc,i and a density 39.17 log ρ = 11.62 + 1, 5 log T9 − (9.16) T9

Hence it is not surprising that collapse sets in when the 1.5 M¯ core enter this regime. At somewhat higher temperatures (for the same density) the α particles photo- dissociate into free neutrons and protons

γ +4 He ⇔ 2p + 2n (9.17)

The energy required for this process is Q0 = 28.3 Mev. The companion Saha equation is 2 2 ³ ´ 0 npnn mukT 9/2 −Q = 2 2 exp( ) (9.18) nα 2πh¯ kT Since the required energy per new particle created via reaction (9.17) (∆N = 4− 1 = 3; Q0/∆N = 9.5 Mev) is larger than the energy per new particle created via reaction (9.8) (∆N = 13+4−1 = 16; Q/∆N = 7.7 Mev) higher temperatures are needed to photo-dissociate 4He than 56F e. Thus there exists a temperature domain in which 56F e has dissociated into 4He, but 4He has not yet dissociated into free nucleons. In any realistic calculation, one must treat the entire set of nuclei, stable and unstable, which exist at high temperature and density in nuclear statistical equilibrium (NSE). Typical reactions are

(Z,A) + p ⇔ (Z + 1,A + 1) + γ (Z,A) + n ⇔ (Z,A + 1) + γ (9.19) 196 CHAPTER 9. ORIGIN OF NEUTRON STARS as well as reactions such as (α, γ),(α, n), (α, p), (p, n) and so on. At the onset of collapse, the nuclei are often approximated as an ideal non-relativistic Maxwell- Boltzmann gas so that eqn.(9.11) applies for each species. For each reaction in the network, the equilibrium conditions for the chemical potentials µi and concentrations Yi applies

X µidYi = 0 i

However, because all possible reactions of the form (9.19) are in equilibrium, we have only two independent chemical potentials, corresponding to conservation of baryon number and charge. Choosing these to be µp and µn, we have

µ(Z,A) = Zµp + (A − Z)µn (9.20) as the condition for nuclear statistical equilibrium. Equation (9.11) then gives

g A3/2 Q n = n = (Z,A) nZ nA−Z exp[ (Z,A) ] (9.21) i (Z,A) 2AθA−1 p n kT where 2 Q(Z,A) = c [Zmp(A − Z)mn − M(Z,A)] (9.22) is the nuclear binding energy, g(Z,A) is the nuclear partition function (eqn.(9.12)), and ³m kT ´3/2 θ = u (9.23) 2πh¯2

To specify np and nn we invoke baryon and charge conservation X ρ niAi = = n (9.24) i mu X niZi = nYe (9.25) i where Ye is the mean number of electrons per baryon. It is thus evident from eqns.(9.21) – (9.25) that in NSE, the composition, equation of state, specific entropy, and other quantities can be determined once ρ, T , and

Ye are specified. 9.4. NEUTRONIZATION AND NEUTRINO EMISSION 197 9.4 Neutronization and neutrino emission

As the core density increases, the high electron Fermi energy drives electron capture onto nuclei and free protons (”neutronization”). This reduces Ye and decreases the contribution of degenerate electrons to the total pressure support- ing the core against gravitational collapse. Eventually, the Chandrasekhar mass

Mch(Ye), the maximum mass that can be supported by degenerate electrons, falls below the core mass. From this moment on, the core collapse proceeds in earnest: as the density increases further, more and more electron captures occur, the pressure and the difference (Γ − 4/3 < 0) decrease further, and the collapse accelerates. In the neutronization process, neutrinos are emitted and, at least initially, many of them escape. Neutrinos are generated during collapse both by neutronization and by ther- mal emission. Thermal mechanisms proceed via the annihilation of real and virtual e+e− pairs, forming νν˜ pairs. If neutrino pairs escape the lepton number in the core is unchanged. Thermal emission processes. The most significant thermal emission pro- cesses are 1. Pair annihilation

e+ + e− → (W, Z) → ν +ν ˜ (9.26)

The symbols W, Z indicate the charged and neutral currents present in the weak interactions according to the Weinberg, Salam & Glasgow theory (see chapter 11 for more details). 2. Plasma decay

(plasma excitations) → (W, Z) → ν +ν ˜ (9.27)

A plasmon is a quantized electro-magnetic wave propagating in a dense dielectric plasma. It behaves like a relativistic bose particle with rest mass mplasmon = hω¯ p/c where ωp is the plasma frequency. Unlike a free photon, a plasmon is 198 CHAPTER 9. ORIGIN OF NEUTRON STARS energetically unstable to decaying into a neutrino-antineutrino pair by virtue of its rest-mass energy excess. 3. Photo-annihilation

e− + γ → (W, Z) → e− + ν +ν ˜ (9.28)

4. Bremsstrahlung

e− + (Z,A) → (W, Z) → (Z,A) + e− + ν +ν ˜ (9.29)

All of the above reactions can proceed both by charged currents via the exchange of W± vector bosons and by the neutral currents via the exchange of Z vector mesons. In the case of e+e− annihilation, for example, an electron neutrino pair ± νeν˜e can be produced either by Z or by W exchange. However, muon and tau neutrino pairs νµν˜µ and ντ ν˜τ are only produced by Z exchange. Typically, thermal neutrino processes dominate in the core of a massive star until the collapse is fully underway. They are also important during the late stages of the collapse when they help carry thermal energy away from the shock heated outer regions of the core. The thermally generated neutrinos have energies E(ν) near kT . Neutronization reactions. The most important neutronization reactions are 1. Electron capture by nuclei

− e + (Z,A) → (W) → νe + (Z − 1,A) (9.30)

2. Electron capture by free protons

− e + p → (W) → νe + n (9.31)

Both reactions proceed via the charged current of the weak interac- tion. Once collapse gets underway neutronization becomes the most important mechanism for generating neutrinos. Neutronization reduces Ye decreasing the electron pressure and influencing nuclear statistical equilibrium. The mean en- ergy of neutrinos generated via electron capture is comparable to the 9.5. WEAK INTERACTION THEORY 199 electron Fermi energy

1/3 hE(ν)i ∼ hEe− i ∼ µe = 51.6(Yeρ12) Mev (9.32)

12 where ρ12 = ρ/10 . This equation follows from the energy relation for degenerate relativistic particles with x >> 1.

9.5 Weak interaction theory

Our current understanding of weak interactions such as β-decay is provided by the Weinberg-Salam-Glashow (WSG) Theory, in which the weak force between fermions is mediated by the exchange of massive bosons, much as the electromagnetic interactions are mediated by the exchange of photons (mass-less bosons). In the standard WSG model, there are ”two charged intermediate bosons” W + and W − and one ”neutral intermediate boson” Z0. They are predicted to have masses given by

2 37.3 Gev mW c = = 78.1 ± 1.7 Gev sin θW 2 2 mW c mZ c = = 88.9 ± 1.4 Gev (9.33) cos θW where the Weinberg angle θW is experimentally determined to be

2 sin θW = 0.228 ± 0.010 (9.34)

2 4 2 2 At the low-interaction energies E /c << mW , mZ , which characterize all processes occurring in neutron star interiors, the WSG hamiltonian encom- passes the old V −A hamiltonian proposed by Feynman and Gell-Mann theory (1958), where V stands for the vector part of the interaction and A the axial vector part. In both theories the effective hamiltonian density has the form G H = √F J †J µ (9.35) 2 µ 200 CHAPTER 9. ORIGIN OF NEUTRON STARS

where Jµ is the 4-current density of interacting fermions. The Universal Fermi coupling constant GF appearing in the low-energy Hamiltonian is related to the WSG parameters (¯h = c = 1)

1 α √ GF = 2 2 4 2 mW sin θW = 1.16632 ± 0.00004 × 10−15 Gev−2 = 1.43582 ± 0.00005 × 10−49 erg cm3 1.0 × 10−5 ' 2 (9.36) mp where α = e2/hc¯ = 1/137 is fine structure constant. An important difference between the old and the new theory at low energy is that in the old one only charged current reactions were allowed. In the language of the new theory, these are the reactions that are mediated by the charged vector bosons W + and W −. In the WSG model, additional neutral current reactions, mediated by Z0 can exist. 2 2 Cross sections in the old theory typically go as GF E where E is the center- of-mass energy in a two-body interaction. Clearly as E → ∞ the cross section diverge. In the WSG theory the cross sections converge faster than (ln E)n with n ≤ 2 so the divergence is ruled out. Finally, WSG theory unifies in a single lagrangian both the weak and electromagnetic interactions. More precisely, the field equation obtained from the lagrangian relate electromagnetic to weak fields just as Maxwell equations relate E to B. Origin of the Grand Unification Theory.

9.6 A simple case of electron capture: e− + p → (W) → νe + n

We now compute the electron capture rate onto free protons, that is eqn.(9.31). This case is easier to calculate and it gives the correct order of mag- nitude of the total capture rate. The method is the same as used to calculate − 9.6. A SIMPLE CASE OF ELECTRON CAPTURE: E +P → (W) → νE+N201 the cooling rate by various neutrino processes (see chapter 11). To illustrate the method we need some introductory ideas on the Weak Interactions Theory and definition of capture rate. The capture rate for given initial proton and electron states is

³ X ´ 2π 1 2 dΓ = |Hfi| (1 − fν)ρνdEνδ(Eν + Q − Ee) (9.37) h¯ 2 spin where the matrix element is given by

X ¯ ¯ 1 2 2 2 2 ¯ CA ¯ |Hfi| = GF CV (1 + 3a ) |CV | = 0.9737 a = ¯ ¯ = 1.253 2 spin CV and using the energy conservation condition

Eν = Ee − Q (9.38) the density of final neutrino states is (Q − E )2 ρ = e ν 2π2h¯3c3

The quantity 1 − fν is the fraction of unoccupied phase space for neutrinos. For both electrons and neutrinos we assume Fermi equilibrium distributions 1 fj = (9.39) exp[(Ej − µj)/kT ] + 1 Equation (9.39) will not be valid for the neutrinos until they become trapped in the core and their density builds up. However, even if neutrinos escape from the core easily prior to this stage and hence are not thermalized, we expect fν << 1, so that the details of their distribution are not significant. Assembling the various factors we obtain 2π E2 dΓ = G2 C2 (1 + 3a2)(1 − f ) ν δ(E + Q − E )dE (9.40) h¯ F V ν 2π2h¯3c3 ν e ν

To find the total rate we integrate over all initial electron states and over dEν. For the density of the electron states we take

2 4πp dp 4πpEe ρe = 3 = 2 3 h dEe c h 202 CHAPTER 9. ORIGIN OF NEUTRON STARS and get 2π G2 C2 (1 + 3a2) Z Γ = F V E2dE SE (E2 − m2c4)1/2dE δ(E + Q − E ) (9.41) h¯ (2π2h¯3c3)2 ν ν e e e e ν e where

S = fe(1 − fν) (9.42)

Using the δ-function to integrate over dEν we find 2π G2 C2 (1 + 3a2) Γ = F V I (9.43) h¯ (2π2h¯3c3)2 where Z ∞ 2 2 4 1/2 2 I = dEeEe(Ee − mec ) (Ee − Q) S (9.44) Q

At the beginning of collapse, the electrons are extremely relativistic (Ee >> 2 mec ; µe >> kT ) and fe = 0 for Ee > µe, and neutrinos escape freely (fν << 1). the integral is Z µe 2 2 1 5 I = dEeEe (Ee − Q) = µe (9.45) Q 5 In late stages of collapse, neutrinos are trapped in the core. In this regime 2 Ee << mec , µe >> kT , µnu >> kT . The integral is Z µe 2 2 2 1 5 5 Q 4 4 Q 3 3 I = dEeEe (Ee−Q) = [µe−(Q+µν) ]− [µe−(Q+µν) ]+ [µe−(Q+µν) ] Q+µν 5 2 3 (9.46) The rate per unit volume of electron captures onto free protons is found by multiplying Γ by np: dn dn e = − p = −n Γ (9.47) dt dt p From eqn.(9.43) and eqn.(9.45) we derive

dY 2π G2 C2 (1 + 3a2) µ5 e = −Y F V e (9.48) dt p h¯ (2π2h¯3c3) 5 The mean energy of the neutrino emitted following electron capture is estimated to be

5 10 hEνi = = hEei (9.49) 6µe 9 9.7. NEUTRINO OPACITY AND NEUTRINO TRAPPING 203 9.7 Neutrino opacity and neutrino trapping

Each neutrino emission process has an inverse process corresponding to absorption. Both absorption and scattering impede the free escape of neutrinos from a collapsing core. The most important processes are 1. Free nucleon scattering

ν + n → (Z) → ν + n ν + p → (Z) → ν + p (9.50)

2. Coherent scattering by heavy nuclei

ν + (Z,A) → (Z) → ν + (Z,A) (9.51)

3. Nucleon absorption

ν + n → (W) → p + e− (9.52)

4. Electron-neutrino scattering

e− + ν → (W, Z) → e− + ν (9.53)

Similar processes for antineutrinos. The first two opacity sources exist by virtue of the neutral currents and were not considered before the WSG theory of weak interactions. Their total cross sections as measured in the matter rest frame are given by ³ ´ 1 Eν 2 2 σn = σ0 2 Eν << mnc (9.54) 4 mec for free neutrons scattering, and ³ ´ h i coh 1 Eν 2 2 Z 2 Z 2 −1/3 σA ' σ0 2 A 1 − + (4 sin θW − 1) Eν << 300A Mev 16 mec A A (9.55) for coherent scattering, where

4 h¯ −4 GF 2 −44 2 σ0 = ( ) ( 2 ) = 1 − 76 × 10 cm (9.56) π mec mec and θW = 0.228 ± 0.010 is the experimental Weinberg angle. 204 CHAPTER 9. ORIGIN OF NEUTRON STARS

2 (i) For low energy neutrinos (Eν << mnc ) scattering from nucleons and nuclei is elastic: the initial and final neutrino energies are nearly equal. (ii) Neutrinos scatter also coherently from nuclei and the cross sec- tion goes as A2. The scattering is coherent in that the nucleus reacts nonlinearly as a single particle and not as A separate nucleons. The persistence of heavy nu- clei at high densities makes coherent scattering a dominant opacity source during ”hot” core collapse. (iii) The significance of neutral current reactions is that they in- crease the neutrino opacity without changing the emission rate appre- ciably. Thus their inhibit neutrino transport and favour neutrino trapping. (iv) In contrast to scattering from nucleons and nuclei, electron- neutrino scattering is non conservative and changes the neutrino en- ergy in the star rest frame. The inelastic nature of the ν −e scattering results 2 from the low rest-mass energy of the electrons, so that Eν > mec . Collision be- tween neutrinos and degenerate electrons can lead to appreciable neutrino energy loss. High energy neutrinos (Eν >> µe) can interact with degenerate electrons residing deep in the Fermi sea. For such interactions the total energy is roughly equal to Eν and this energy is shared between the elec- tron and the neutrino following scattering. For the low energy neutrinos

(Eν << µe) the effects of electron degeneracy are very important. One may expect that neutrinos would gain some energy from such an interaction leading to energy equilibration. However an electron with Ee < µe cannot lose energy. As a result, in order for the interaction to proceed at all, a low energy neutrino with Eν << µe can lose even more than half its energy in a scattering event. This process is very important in thermalizing the neutrinos and helps drive them to local equilibrium once they are trapped in the core. The cross section is given by Eν 2 Eν σ = χσ0( 2 ) (9.57) mec EF (e) where σ0 is the same as before and χ = 0.1 for the V-A theory and χ = 0.06 for the WSG theory. 9.7. NEUTRINO OPACITY AND NEUTRINO TRAPPING 205

(v) Nucleon absorption (the inverse of neutronization) also con- tributes to the thermalization process. The cross section for the absorption of a neutrino by free non relativistic neutrons is 2 2 ³ ´ ³ ´h³ ´ ³ 2 ´ i (1 + 3a )CV Eν 2 Q Q 2 mec 2 1/2 σa = σ0 2 1 + 1 + − I(Eν + Q) 4 m − ec Eν Eν Eν (9.58) where 1 I(Eν + Q) = 1 − (9.59) exp[(Eν + Q − µe] + 1

For extremely relativistic electrons the factor I goes to exp(−µe/kT ) for Eν <<

µe and to unity for E ≥ µe, reflecting the important role of electron degeneracy in blocking low energy absorptions. Mean free path. As the density increases during core collapse and the opacity rises, the neutrinos experience greater difficulty escaping from the star before being dragged along with the matter. At densities above

11 3 ρtrap = 3 × 10 g/cm (9.60) the neutrinos are trapped, comove with matter and build up a semi-degenerate

Fermi sea. By definition at ρ ∼ ρtrap the timescale for neutrinos to diffuse becomes comparable to the collapse timescale

−1/2 −3 −1/2 tcoll = (Gρ) ∼ 4 × 10 ρ12 s (9.61) where ρ ∼ M/(4πR3/3) is the mean density of the collapsing core and M ∼

M¯. The diffusion time scale can be estimated by assuming that the coherent scattering is the dominant opacity source, so that λcohN t ∼ A scat (9.62) diff c

coh where λA is the mean free path of a typical neutrino in the sea of heavy nuclei

(Z,A) and Nscat >> 1 is the number of scatterings experienced by the neutrino prior to escape. Coherent scattering induces a random-walk trajectory for the neutrino without appreciably changing its energy before reach- ing the surface. Therefore Nscat can be determined from the relation 206 CHAPTER 9. ORIGIN OF NEUTRON STARS

coh 1/2 λA Nscat ∼ R (9.63)

The mean free path can be derived from eqn. (9.55) ³ ´ coh −1 coh ρ coh (λA ) = nAσA = σA (9.64) Amb where we assume that all nucleons are heavy nuclei and take 56F e to be repre- sentative of these nuclei. Using eqn. (9.32) and (9.49), we derive the energy of the neutrino upon emission following electron capture onto a proton

1/3 Eν ' µe = 33ρ12 Mev (9.65)

Substituting (9.64) into (9.63) yields

coh −1 −5 5/3 −1 (λA ) ∼ 3.9 × 10 ρ12 cm (9.66)

The mean free path is now of the order of 105 cm (i.e. of the order of 12 km) for densities close to 10 . Solving eqn.(9.62) for Nscat and substituting the result together with eqn.(9.65) into eqn.(9.61) we obtain

tdiff ∼ 0.08ρ12 s (9.67)

Equations (9.60) and (9.66) show that at high densities tdiff >> tcoll. The two timescales are nearly equal

11 3 tdiff ∼ tcoll ⇒ ρ ∼ ρtrap ∼ 1.4 × 10 g/cm (9.68)

Neutrino trapping has enormous implications for the core collapse.

For ρ > ρtrap most of the neutrinos from electron captures remain inside the star. The neutrino distribution approaches the equilibrium Fermi state so that the entire physical state of the system can be specified by three quantities: T , ρ and Ye. The neutrino luminosities are greatly reduced by trapping. Con- sider the luminosity once the centre of the core achieves nuclear densities ρnuc = 2.8 × 1014 g/cm3. Above this densities, thermal pressure and nuclear forces cause 9.8. HOMOLOGOUS CORE COLLAPSE AND BOUNCE 207 the equation of state to stiffen, preventing further collapse. Now most of the gravitational binding energy of the core is ultimately released in the form of neutrinos. In the absence of neutrino trapping, the total gravitational binding energy would be emitted as neutrinos in a collapse timescale, 3 the time for the core to contract from 2Rnuc to Rnuc, where ρnuc = M/(4πRnuc/3) giving Rnuc ∼ 12 km for M ∼ M¯. The corresponding maximum value for the luminosity would be

2 GM /Rnuc 57 Lν,max ∼ ∼ 10 erg/s (9.69) tcoll

In reality, neutrino trapping forces the liberated gravitational potential energy to be emitted on the timescale of diffusion tdiff >> tcoll at ρ ∼ ρnuc. The actual neutrino luminosity is

2 GM /Rnuc 52 Lν ∼ ∼ 10 erg/s (9.70) tdiff which is within an order of magnitude of more detailed model calculations which all exhibit the strong inequality Lν << Lν,max. This inequality simply shows the inability of the neutrinos to stream freely out of the core during the advanced stages of collapse. The bulk of the liberated gravitational energy must be converted into other forms of internal energy (thermal energy, energy of excited nuclear states, bounce kinetic energy) rather being immediately ra- diated in the form of escaping neutrinos. On dynamical timescales, neutrino trapping thus causes the late stages of core collapse to pro- ceed adiabatically to high approximation.

9.8 Homologous core collapse and bounce

10 Suppose we have a core at the onset of collapse with central values ρc ' 10 −3 10 ◦ g cm and Tc ' 10 K. The electrons are relativistic and degenerate; the equation of state is polytropic and can be written as 208 CHAPTER 9. ORIGIN OF NEUTRON STARS

P = K0ρ4/3 (9.71)

0 4/3 where K = K4/3/µe and n = 3, whose properties and collapse have already been amply described in chapter 7. Goldreich & Weber (1980) have shown that if the adiabatic index of a New- tonian gas sphere satisfies Γ = 4/3, then the collapse of the inner fraction of the configuration will be homologous that is the position and velocity of a given mass point in this homologous inner core will vary as

r˙ a˙ r(t) = a(t)r = (9.72) 0 r a where r0 is the initial position. The density, meanwhile, will evolve self-similarly according to −3 ρ(r(t), t) = a ρ0(r0) (9.73)

Therefore we can apply the formalism and the results found for the collapsing polytrope already described in chapter 7. The parameter λ appearing in the modified Lane-Emden equation is a measure of the deviation from hydrostatic equilibrium (λ = 0) allows solutions with

finite radius only for values 0 < λ < λmax = 0.006544.

Let ψ(ξ) be the solution for the space part. Indicate with ξ3 the value of di ξ at the surface of the collapsing core, where ψ(ξ3) = 0. For λ = 0 one has ξ3 = 6.897, whereas for λmax one has ξ3 = 9.889. The limit value of λ corresponds to the case of collapse with free-fall acceleration. From relation (7.40) applied to the surface we have

0 3/2 4 (K ) ξ3 ξ3a¨ = − λ √ (9.74) 3 πG a2 while from relation (7.46) we obtain

³ ´ ρ 3 1 d 2 dψ = ψ = λ − 2 ξ (9.75) ρc ξ dξ dξ

Posing r = aξ e Rc = aξ3 and using 9.8. HOMOLOGOUS CORE COLLAPSE AND BOUNCE 209

Z Rc 3 2 ρ = 3 ρr dr (9.76) Rc 0 we obtain

ρ h3³dψ ´i = λ − (9.77) ρc ξ dξ ξ=ξ3

Applying this to the limit case λ = λmax in which dψ/dξ is zero at the surface, we obtain

ρ = λmax (9.78) ρc The core starts from the equilibrium condition with λ = 0: Here the actual acceleration at the surface is zero because gravity and pressure gradient cancel each other. But if the pressure is slightly decreased, the core will start collapsing (λ > 0). The numerical integration for different values of λ in the range 0 ≤

λ ≤ λmax gives values for ξ3 and ρ/ρc in the ranges 6.897 ≤ ξ3 ≤ 9.889 and

0.01846 ≤ ρ/ρc ≤ 0.0654. The mass of collapsing polytropes is derived from the expression

3 3 3 ³ 0 ´ 4πa ξ ρc ρ 4πξ K 3/2 ρ Mc = = (9.79) 3 ρc 3 πG ρc This relation for λ = 0 yields the Chandrasekhar mass. In general, the masses obtained for 0 ≤ λ ≤ λmax fall in the interval MCh ≤ Mc ≤ 1.0499MCh. At increasing λ, i.e. deviation from hydrostatic equilibrium, the limit core mass slightly increases with respect to the Chandrasekhar value, if the electron molec- ular weight is kept constant. Only core masses in this small interval can collapse homologously. How- 2 ever, on one hand MCh ∝ µe, on the other electron captures during the collapse lower µe, so that the mass limit for homologous collapse decreases during the collapse itself. If at the beginning µe = µe0 and MCh = MCh0, after some time only the mass 210 CHAPTER 9. ORIGIN OF NEUTRON STARS

Figure 9.3: Schematic picture of the velocity profile vr in a collapsing stellar core originally of about 1.4 M¯. Note the two regimes: on the left the velocity increases with r; it correspond to the homologous part; on the right, the velocity decreases outward, it roughly corresponds to the free-fall regime.

µe0 2 Mc = 1.0499( ) MCh0 (9.80) µe

can collapse homologously.

The radial velocity of matter during the collapse changes with r and/or M(r) as shown in Fig.9.3. The maximum separates the homologously collaps- ing inner core (left) from the free-falling outer part (right). During the collapse the boundary between the two regions does not remain constant but moves to smaller values of r and/or M(r).

The collapse is extremely short lived: it takes a time which is of the order of −1/2 10 −3 the free-fall time τff = (Gρ) : for the typical value of ρ = 10 g cm the duration is about 40 milliseconds. It decreases to 0.4 milliseconds when the density increases to about 1015 gcm−3. 9.9. FREE-FALL REFLECTION: BOUNCE 211 9.9 Free-Fall reflection: bounce

Because of the collapse, the density finally approaches that of a neu- tron star (nuclear densities of the order of 1014 g cm−3). The equation of state changes and becomes stiff, i.e. the matter becomes incompress- ible. This terminates the collapse. If the whole process were completely elastic, then the kinetic energy of the col- lapsing matter would be sufficient to bring it back after reflection to the state just before the collapse began. The energy in question is essentially the gravitational energy liberated by shrink- ing the nucleus from the dimension of a White Dwarf to that of a Neutron Star

³ ´ 2 2 1 1 GMc 53 Eg ' GMc − ' ' 3 × 10 erg (9.81) Rn Rwd Rn where Mc is the mass of the collapsing core and Rn e Rwd are the typical radii of a Neutron Star and a White Dwarf. We compare this with the energy required to expel the envelope, i.e. the layers that had no time to follow the central core collapse

Z M 2 GM(r)dM(r) GM 52 Eexp = << ' 3 × 10 erg (9.82) Mwd r Rwd for a typical star with total mass of about M = 10M¯. More realistic estimates 50 bring Eexp down to about 10 erg. Therefore only a small fraction of the energy involved in the collapse of the core is sufficient to blow away the envelope. What happens after the reflection of the outer infalling layers (bounce) depends on which fraction of the liberated gravitational energy is trans- ferred to the external layers (i.e. the mantle or, roughly speaking the layers comprised the original F e core and the Si- burning shell) in form of kinetic energy of outward explosive motion. Remember that the energy estimated in (9.81) would suffice only to bring back the whole collapse to its original position and no energy would be left for expelling 212 CHAPTER 9. ORIGIN OF NEUTRON STARS

the envelope. But if a remnant (Neutron Star) of mass Mn remains in the condensed state, the energy of its collapse is available. The question is how this can be used for accelerating the rest of the material outwards. A possible mechanism would be a shock wave moving outwards. The remnant is somewhat compressed by inertia beyond the equilibrium state and afterwards, acting like a spring, it expands pushing back the infalling material above. This creates a pressure wave that gets steeper moving across layers of lower density. The kinetic energy stored in such a wave may be sufficient to lift the envelope into space.

9.10 Formation of the shock way

In this section we examine in some detail the formation mechanism and main properties of a shock wave. As already mentioned, when the center of the star reaches and goes above nuclear densities, the material opposes to further compression and develop a strong pres- sure gradient. The inward free-fall motion is halted an a pressure front propagates outwards. If the free fall motion were suddenly halted, the variation in velocity at a certain layer would be given by

∆u = −uinf (9.83)

where uinf is the free-fall velocity. According to the basic laws of acoustics, this change in velocity is accompanied by a change in pressure

∆P = ρvs∆u (9.84) where vs is the local sound speed. The change in pressure in turn is ac- companied by a change in density

∆ρ u = (9.85) ρ vs 9.10. FORMATION OF THE SHOCK WAY 213

Figure 9.4: Behaviour of the radial velocity during the formation of the shock wave

This relation is valid as long as the r.h.s. is small with respect to unity (this situation is referred to as approximately sonic).

It can be demonstrated that the changes in pressure ∆P and density ∆ρ are the same when the change of velocity occurs gradually. Therefore, the result is of general validity.

The collapse solution near the center of the star gives a velocity u < vs and therefore the variation ∆P is modest. the pressure variation propagates outwards and in the meantime the ratio u/vs increases (u increases while vs decreases), until the layer whereby u = vs is reached sonic point. Thereinafter the sonic approximation cannot be applied and shock wave is formed.

The wave front present a discontinuity in the physical variables that is described by the Hugoniot equations. Let us indicate with the suffix 1 the quantities in front (more external) of the shock way and with suffix 2 those behind the shock front (more internal). Let U be the velocity of the shock front with respect to the center of the star. Let u1 e ²1 be the velocity and internal energy per unit mass of the material above the shock front, and u2 e ²2 those of the material below the shock front. 214 CHAPTER 9. ORIGIN OF NEUTRON STARS

The Hugoniot equations express the conservation of the mass flow, (ρu), momen- 2 1 2 tum flow (P + ρu ), kinetic and internal energy flow (ρu[ 2 u + ²]). Adapted to our case the Hugoniot equations are

ρ1(U − u1) = ρ2(U − u2) (9.86)

2 ρ2 P2 − P1 (U − u1) = (9.87) ρ2 − ρ1 ρ1 ³ ´ 2 1 1 (u2 − u1) = (P2 − P1) − (9.88) ρ1 ρ2 1 ³ 1 1 ´ ²2 − ²1 = (P2 + P1) − (9.89) 2 ρ1 ρ2

If the shock way is strong, P2 >> P1, then from eqn.(9.88) and (9.89) one obtains the relation

2 2(²2 − ²1) = (u2 − u1) (9.90)

If we adopt ²1 = 0 and the equation of state of a perfect gas, the energy ²2 is given by

αP2 ²2 = (9.91) ρ2 so that

ρ 2 = 2α + 1 (9.92) ρ1 where α = 3/2 for a perfect gas. Since eqn.(9.90) is valid for any equation of state, it can be used to estimate the velocity of the material behind the shock front. The effect of the compressional front passage on the velocity as a function of the radial distance is shown in Fig.9.4. The dynamical and thermodynamical effects caused by the passage of a shock way throughout the stellar material can be summarized as follows: (1) The motion is initially supersonic, but becomes subsonic behind the shock front. 9.11. ENERGY BALANCE 215

(2) The density behind the shock is larger that the density of the unperturbed medium. (3) The pressure behind the shock is higher than the pressure of the unperturbed medium. (4) The temperature behind the shock is higher than that of the unperturbed medium.

(5) In a strong shock wave (P2 >> P1), the ratios P2/P1 and T2/T1 tend to diverge, whereas the ratio ρ2/ρ1 remains finite and small, see eqn.(9.92).

9.11 Energy Balance

With the scheme just described a problem soon arises. The newly formed Neutron

Star has a mass of the order of the final Chandrasekhar mass MCh,F . The rest of the collapsing material is still made of 56F e. This means that a large fraction of the energy of the compression front will be spent to photo-dissociate 56F e. Therefore only a small fraction of the original energy will remain (none in the most extreme case) to energize and lift the layers above the original iron core. It is soon clear that the success of this mechanism depends on the mass of the newly born Neutron Star (which determine the time, the stage of re-bounce) and the original mass of the 56F e core. If the mass of the Neutron Star is large and therefore close to the mass of the original 56F e core, there will be little 56F e mass to photo-dissociate after re-bounce. The opposite, if the mass of the Neutron Star is small. This simply means that the layer at which re-bounce occurs and success and failure of our mechanism ultimately depend on the equation of state inside the Neutron Star, and how quickly it becomes stiff enough to halt the collapse. How much energy is dissipated by photo-dissociation? The process

γ +56 F e ⇒ 134He + 4n (9.93) absorbs 216 CHAPTER 9. ORIGIN OF NEUTRON STARS

2 Q = (13mα + 4mn − m56)c ' 124.4 Mev (9.94) whereas the process

γ +4 He ⇒ 2p + 2n (9.95) absorbs

2 Q = (2mp + 2mn − mα)c ' 23 Mev (9.96)

56 Expressing these energies in a more practical form, each 0.1M¯ of F e requires 1.5 × 1051 erg of energy to fully photo-dissociate. Neutronization of stellar matter by electron captures and compan- ion neutrino emission is another energy sink to be considered in the balance. Just to summarize the key steps of the reasoning: in normal condition a neutron is unstable (with life time of about 10.25 min)

− n → p + e + νe (9.97) generating electrons with energies up to 1.3 Mev and antineutrinos. However, the neutron decay cannot occur if electrons of these energies are already present in the medium. In fact at high electron densities many electrons with energies well above 1.3 Mev are present which inhibit neutron decay. The newly created neutrons survive. There also the electron capture process (otherwise known as neutronization)

− e + p → n + νe (9.98)

This constitutes an efficient energy loss mechanism by neutrino emission. The maximum energy loss by electron captures is about 1.6 × 1052 erg. In brief, in 57 a core of about 1 M¯ there are about 10 degenerate electrons each of which carries an energy of about 10 Mev (the Fermi energy at the densities of interest). 9.11. ENERGY BALANCE 217

There is immediately a problem of energy balance. In fact, the gravitational energy liberated by the collapse of the core is about 3 × 1053 erg. Of this energy, 51 52 1.5×10 ×∆MF e/0.1×M¯ erg or equivalently ' 1.5×10 erg is spent to photo- dissociate the whole 56F e. About 1.6 × 1052 is dissipated in electron captures via the emission of neutrinos. Other energy losses to be considered are the kinetic energy of the expelled material, the energy in the light curve of the companion supernova explosions. All these are at least one or two orders of magnitude smaller. Summing up all the contribution there remains an amount of energy comparable to the gravitational energy to be dissipated. This means that there must be an intermediate stage between the collapse and the formation of a Neutron Star, during which are lost about 3 × 1053 erg of energy.

This stage is the formation of a hot Neutron Star losing energy in form of neutrinos. Only a small fraction comes from electron captures. The others have a different origin. The temperature is so high that copious (ν, ν˜) are generated by many processes: annihilation of (e+, e−) pairs, plasmon neutrinos, photo-neutrinos, brems-strahlung neutrinos. Three types of neutrinos can in principle be produced

(ν, ν˜e)(νµ, ν˜µ)(ντ , ν˜τ ) (9.99)

In many cases there is not enough energy to activate the third channel (τ neutri- nos.

Therefore, the real signal of the gravitational collapse of a nucleus is the emission of a large quantity of energy in form of neutrinos. The fraction spent in the spectacular phenomenon of a supernova explosion as seen in the visible and other pass-bands is only marginal with respect to the neutrino luminosity. 218 CHAPTER 9. ORIGIN OF NEUTRON STARS 9.12 Effect of neutrinos

The energy balance we have just outlined shows that in most cases the shock wave has little chances of expelling the envelope as nearly all its energy is dissipated in the photo-dissociation process. This is also confirmed by most recent detailed and accurate numerical models. The way out is perhaps given by neutrinos. In so far, we have considered neutrinos as a net energy loss for a star as due to their small cross section they can freely leave a star without interacting with the stellar medium. This is no longer true in the case of the final collapse of the core of a massive star. The typical energy of the neutrinos released during the collapse is of the order of the Fermi energy of relativistic electrons

³ ´ ³ ´ ³ ´ Eν EF pF 3 1/3 h ρ 1/3 −2 ρ 1/3 2 ' 2 = = ' 10 (9.100) mec mec mec 8πH mec µe µe

If heavy nuclei are present, the neutrinos interact predominantly through coherent scattering rather than scattering by free nucleons

ν + (Z,A) → ν + (Z,A) (9.101)

The cross section of this process is of the order of

³ ´ Eν 2 2 −45 2 σν ' 2 A × 10 cm (9.102) mec or

³ ´ 2 ρ 2/3 −49 2 σν ' A × 10 cm (9.103) µe

The mean free path lν through a medium with number density of nuclei (Z,A) equal to n = ρ/AH is

³ ´ 1 1 ρ −5/3 25 lν = = × 1.7 × 10 cm (9.104) nσν µeA µe 9.12. EFFECT OF NEUTRINOS 219

9 It is an easy matter to check that with µe = 2, A = 100, and ρ/µe = 3.6 × 10 −3 7 g cm , the mean free path is lν ' 10 cm, i.e. comparable to radius of the core itself. In other words, the interactions of neutrinos with the core matter cannot be ignored.

At increasing density, lν gets smaller, so that matter becomes opaque to neutrinos. These can escape only after undergoing many scattering events. At very high densities, it may happen that the diffusion velocity of the neutrinos becomes smaller than the collapse velocity. Detailed calculations show that at densities 11 −3 ρ ≥ 3 × 10 g cm , the neutrinos cannot escape by diffusion during the τff time scale. The neutrinos are trapped in a sphere whose radius is about 40 km, ρ ' 1011 g cm−3, and T ' 5 Mev. For comparison, the shock front is much more far out, at about r ' 200 km. Figure 9.5 schematically shows the structure of the nucleus in this phase. In the most internal part (named core), the matter is practically at rest and at its surface the in-falling layers bounce. Above the core, there is a still collapsing layer, inside which the neutrinos are trapped (the free-fall velocity is nearly equal to the escape velocity of neutrinos). Finally, there is a further out region from which neutrinos escape freely. Is it possible to use an even small fraction of the trapped neutrino energy to energize the shock wave ? Even a fraction as small as 1% would be sufficient. The problem is very complicate because one has to derive an equation for neutrino transport (much similar to that for photons). It is also necessary to derive the neutrino distribution as a function of their energy instead of using their mean energy. This is obvious because the cross section depends on the energy of the neutrinos: those with low energy can escape more easily than those of high energy. Furthermore, the neutrinos trapped in the core influence the neutronization pro- cess inside this. At increasing density the neutrinos become degenerate with very high Fermi energy. Electron captures become less probable because the neutrinos must be emitted with energies above the Fermi sea. 220 CHAPTER 9. ORIGIN OF NEUTRON STARS

Figure 9.5: Schematic picture of a collapsing stellar core at bounce. The arrows corresponds to the velocity field. At the sphere labelled core shock, the shock is formed inside which the matter is almost at rest. Above the shock there is a still collapsing shell in which neutrinos are trapped. But on the top there is a shell from which neutrinos can escape. One can define a neutrino phot-sphere in analogy to the photosphere in a stellar atmosphere. Part of the neutrinos can be trapped in the collapsing matter thus giving the outward moving shock front enough energy to reach the Si-burning layer. 9.12. EFFECT OF NEUTRINOS 221

12 −3 At the density of about 3 × 10 g cm , the neutronization stops and γad = 4/3. This corresponds to a fully degenerate relativistic conditions. The collapse goes on up to densities of about ρ > 1014 g cm−3 whereby the equation of state becomes stiff and γad > 5/3. Hydrodynamical models of collapse show that most likely part of the neutrino energy is transferred to the shock wave which can proceed further into the upper layers. The passage of the shock wave increases the temperature into the so-called mantle of the star, the region comprised between the original 56F e core at the beginning of collapse and the original He-core. The temperature increase render more vigorous all the nuclear reactions among all elements thereby present. On a ver short time scale the elements are processed to nuclear statistical equilibrium. The nuclear energy release increases the internal energy above the gravitational binding energy of the mantle. This is the material expelled in the supernova explosion together with what remains of the original envelope after mass loss by 56 stellar winds in earlier stages. In particular, the 28Ni produced in this latest stages is the energy source powering the light curve via radioactive decay

56 56 56 28Ni → 27Co → 26F e (9.105) the first step with decay lifetime of τ = 6.5 days, and the second one with decay lifetime of τ = 77 days. In concluding this section, we like to recall that the above picture refers to the evolutionary scheme according to which at the end of the hydrostatic sequence of 56 nuclear burnings, a F e core with mass close to the Chandrasekhar mass MCh is formed. Current stellar model calculations show that solutions in which 56F e is much bigger than the Chandrasekhar limit MCh are also possible. In such a case the collapse of the 56F e nucleus leads to the formation of a Black Hole, i.e. an object with mass grater than MCh and therefore forced to complete collapse with no possibility of a new state of gravitational equilibrium. The initial dimension of the 56F e core, among other factors, depends on the neutrino cooling history in previous stages, the carbon burning in particular, 222 CHAPTER 9. ORIGIN OF NEUTRON STARS which in turn depends on the efficiency of the reaction 12C(α, γ)16O during the Helium burning phase. A low efficiency of this reaction leaves lots of carbon, the carbon burning phase last very, so that neutrinos can cool the carbon burning 56 core very efficiently. As a consequence of this, the final F e core is close to MCh, and after the collapse a Neutron Star is formed. The opposite occurs for a high efficiency of the 12C(α, γ)16O reaction: the collapse proceeds to forming a Black Hole. The initial star mass separating the two regimes, i.e. formation of either a

Neutron Star or a Black Hole, is about 40 M¯. Chapter 10

Neutron Stars

The existence of Neutron Stars was first proposed by Baade and Zwicky in 1934. They pointed out these stars would be at very high density and small radius and would be much more gravitationally bound than ordinary stars. Remarkably, they also suggested that Neutron Stars would be formed in supernova explosions. The first model of Neutron Stars was calculated by Oppenheimer and Volkoff (1939) who assumed matter to be composed of an ideal Fermi gas of free neutrons at high density. Work on neutron stars at that time focused mainly on the idea that neutron star cores in massive stars might be a source of energy. When this idea failed as the detail of become understood, neutron stars were nearly ignored by the astronomical community for the next thirty years but for a few exceptions. In this chapter we shall present equilibrium models for non-rotating cold neutron stars. They are constructed by assuming a certain equa- tion of state and solving the Oppenheimer-Volkoff equation of hydro- static equilibrium. Indeed the Newtonian description is no longer valid at the physical condition of Neutron Stars. One can construct a sequence of mod- els parameterized by the central density ρc: models with dM/dρc > 0 are stable and those with dM/dρc < 0 are unstable. The key uncertainty in Neutron Stars models is the equation of state of 14 3 nuclear matter, particularly above the nuclear density (ρnuc ' 2.8×10 g/cm ). We are indeed facing an unusual situation: for White Dwarfs, observations of

223 224 CHAPTER 10. NEUTRON STARS masses and radii are used as confirmations of the astrophysical models; for Neu- tron Stars, because of the uncertainties in the equation of state, observations of masses and radii are instead used to test theories of nuclear physics!

10.1 Introductory generalities

The dimensional arguments of Landau presented in chapter 8 dedicated to White Dwarfs has already pointed out that assuming an ideal degenerate gas of neutrons gives values for the maximum mass and corresponding radius of about

Mmax ∼ 1.5 M¯ and R ' 3 km

Similar values are indicated by observational data on pulsars which according to current wisdom are the manifestation of Neutron Stars.

MNS ∼ 1M¯ and RNS ∼ 10 − 15 km

These values of mass and radius imply extraordinarily high densities of about 1014 − 1015 g/cm3. The Neutron Stars are the result of core collapse in massive stars inducing the Type II supernova explosions. They are born with very high temperatures, say in the range 109 to 1010 K, but very rapidly they cool down to a few 106 K, within about 105 years. Given their density, the equation of state is that of a degenerate gas with high Fermi energy (EF ∼ 1 Mev and above) so that the effect of temperature is virtually zero. A fully degenerate gas at T ∼ 0. (1) A naive model. Suppose now for the sake of argument that neutron stars can be treated with the same formalism we have adopted for White Dwarfs. The only change would be to replace me with mn and µe with µn = 1 in the equation γ of state, Pn = Kγρ0 , where γ = 5/3 or 4/3 according to whether neutrons are non relativistic or fully relativistic, Kγ is the corresponding polytropic constant, and

ρ0 is the density based on the rest-mass. The transition from non relativistic to 15 3 fully relativistic neutrons would occur at ρ0 ∼ 6.5 × 10 g/cm . The results are 10.1. INTRODUCTORY GENERALITIES 225 the same as those obtained for White Dwarfs, i.e. the same M(R) relationship but for the two above replacements, and the same limit mass for fully degenerate, fully relativistic neutrons, i.e.

Mmax,NS = 5.73 M¯ and R → 0

However, these results do not correspond to a real Neutron Star for a number of reasons: (i) The model of non-interacting particles does no longer apply to high density neutrons. The equation of state is different from that derived for non interacting particles. To a first approximation γ = 5/3 still holds good for non relativistic low density neutrons, but γ → 1 for the relativistic ones. (ii) In the relativistic regime the actual density to consider is ρ = 2 ρ0 + ²/c with ² the energy density of neutrons. This correction was not necessary in the case of electrons in White Dwarfs. The distinction between ρ and ρ0 is very important as it reflects onto the final structure of a Neutron Star. (iii) The Newtonian hydrostatic equilibrium equation is no longer applicable, but it must be replaced by that in general relativity. So this kind of solution is not viable in the case of Neutron Stars. Despite these warnings some aspects of this model can still hold for Neutron Stars in the low density regime, and hence low mass range. (2) Neutronization, Equation of State, GR hydrostatic equilibrium. Although these subjects have already been amply discussed in previous sections it might be worth recalling here the salient points for the sake of completeness. Neutronization is due to inverse β-decay between nuclei and electrons at the top of the Fermi sea when direct β-decay ca no longer occur as impeded by electron degeneracy. The processes are of type (Z,A)+e− → (Z−1,A)+ν, where the (Z − 1,A) nucleus is potentially unstable according to (Z − 1,A) → (Z,A) + e− +ν. However, the second reaction cannot occur because all the states available to the electrons are occupied. This leads to a slow decrease of the number of free electrons, and increase of the number of neutrons in nuclei. At increasing 226 CHAPTER 10. NEUTRON STARS density the neutronization processes become more and more frequent until the binding energy of nuclei become positive and the nuclei melts into free neutrons and protons. The result is that matter is transformed at the expenses of the gravitational energy of the star. In brief, the neutron rich nuclei already releasing 11 −3 free neutrons (neutron drip) at densities of about ρdr ' 4 × 10 g cm . Matter now is composed of nuclei (possibly structured in a crystal lattice), electrons in the right number to ensure the charge neutrality and free neutrons. The number density of free neutrons increase so that they start contributing to the total 11 −3 pressure. At density ρdr ' 4 × 10 g cm one has P ' Pe >> Pn, at ρ ' 12 −3 13 −3 4×10 g cm Pn ' P/2, at ρ ' 1.5×10 g cm Pn > 0.8P , and finally at even larger density P = Pn. It is worth recalling that the values of ρ just mentioned depend on the model for the nucleon-nucleon interaction that is assumed. At increasing efficiency of the neutron drip, the nuclei dissolve. This approximately occurs at ρ ' 2.4×1014 g cm−3. A gas (liquid) neutrons, plus e− and p is left over. The concentration of electrons, protons and neutrons is set by the equilibrium condition for the reaction n ↔ p + e− (neutrinos and/or antineutrinos do not interact with matter and escape the star). The equilibrium condition is expressed by relations between the Fermi energies and the number densities

n p e EF = EF + EF ne = np (10.1)

This implies that np = 0.01 × nn for a large range of ρ. As this latter increases up to nuclear density values ' 1015 g cm−3, The number densities of neutrons, protons and electrons are in the ratios nn : np : ne = 8 : 1 : 1.

Equation of state. Up to ρ ' ρdr, The equation of state is that of de- generate relativistic electrons (eventually corrected for various processes), that is 4/3 P ' Pe ∝ ρ . After the onset of neutron drip, the equation of state changes at increasing ρ as nn increases at expenses of ne, so that the increase dP in pres- sure is small. The gas becomes more compressible (soft equation of state). In other words Γad = (dlnP/dlnρ)ad < 4/3. Only when the neutrons will contribute 12 −3 significantly to pressure Γad > 4/3 again. This occurs at ρ ≥ 7 × 10 g cm . 10.1. INTRODUCTORY GENERALITIES 227

When Pn becomes the dominant source of pressure, to a first approximation the neutrons can be considered as an ideal gas of non interacting, fully degenerate fermions. The equation of state is

Γ Pn = KΓρ0 (10.2) with

³ ´ 2 5 1 3 2/3 h 15 −3 Γ = K5/3 = 8/3 ρ0 << 6 × 10 g cm (10.3) 3 20 π mn in the non relativistic case and

³ ´ 4 1 3 1/3 hc 15 −3 Γ = K4/3 = 4/3 ρ0 >> 6 × 10 g cm (10.4) 3 8 π mn in the relativistic one. In eqn.(10.2) the density corresponds to the rest mass and thus ρ0 = nnmn. However for relativistic conditions, instead of ρ0 one has to use 2 the relationship between the the total mass and energy density ρ = ρ0 + u/c .

Such a distinction is not necessary for electrons as in this case ρ0 (that stems from non degenerate nuclei) is always dominant with respect to the energy density u/c2 2 that derives from degenerate electrons. Now both ρ0 and u/c are given by the same particles (degenerate neutrons). For degenerate non relativistic neutrons 2 2 ρ0 >> u/c so that ρ ' ρ0. For degenerate, relativistic neutrons ρ0 << u/c and ρ ' u/c2. Recalling now that for relativistic particles P = u/3 and P = ρc2/3, it follows that

k Pn ∝ ρ (10.5) with k = 5/3 in the non relativistic case and k = 1 in the relativistic one. The distinction between ρ and ρ0 cannot be ignored as it influences very much the structure of a Neutron Star. At the densities we are considering, the interactions between particles cannot be ignored. They in fact become important long before 15 −3 reaching the density limit of 6 × 10 g cm at which pF = mnc. To derive correctly the effect of interactions, we must determine the interaction potential. 228 CHAPTER 10. NEUTRON STARS

Starting from the nucleon interaction inside the nuclei, we may say that the interactions between free neutrons will be strongly attractive as long as their mutual distance is relatively large (relatively low density), and strongly repulsive when they come very close each other (high densities). Attraction will give rise to a soft equation of state, whereas repulsion will produce a rigid equation of state. Another cause of great uncertainty in the equation of state is that at very high densities new particles come into the scene. For example the appearance of hyperons that will much contribute to ρ but little P . See chapters 5 and 6 for a detailed discussion of the equation of state. Hydrostatic equilibrium in GRT. The strong gravitational fields in neu- tron stars are described by the Einstein Equations that we have already presented in chapter 4 and are repeated here for the sake of completeness. The Einstein field equations are

1 κ 8πG R − g R = T κ = (10.6) ik 2 ik c2 ik c2 where Rik is the Ricci tensor, gik the metric tensor, R the Riemann curvature, and Tik the energy-momentum tensor, which for and ideal gas has non-zero com- 2 ponents T00 = ρc , T11=T22=T33=P , where ρ is the density inclusive of the energy density, and P the pressure. Density and pressure are related by the equation of state P (ρ). We are now interested to a static (independent of time) and spherically sym- metric solution. The line-element ds (distance between two events) in spherical coordinates is given by

ds2 = eνc2dt2 − eλdr2 − r2(dθ2 + sin2θdφ2) (10.7) with ν = ν(r) and λ = λ(r). With these definitions for Tik and ds, the field equations (10.6) reduce to a system of three ordinary differential equations

κP ³ν0 1 ´ 1 = e−λ + − (10.8) c2 r r2 r2 10.1. INTRODUCTORY GENERALITIES 229

κP 1 ³ 1 ν0 − λ0 ν0λ0 ´ = e−λ ν00 + ν02 + − (10.9) c2 2 2 r 2

³λ0 1 ´ 1 κρ = e−λ − + (10.10) r r2 r2 where the prime indicates the derivative with respect r. Multiplying eqn.(10.10) by 4πr2 and integrating one obtains

κm = 4πr(1 − e−λ) (10.11) where m is the gravitational mass inside the sphere of radius r

Z r m = 4π r2 ρ dr (10.12) 0 for r = R, m is the gravitational mass M of the star. This is the mass that a distant observer would measure from the the gravitational effects. This is not the mass given by the product of the total number of baryons by the baryonic (atomic) mass unit, as it contains not only the baryon rest mass but also the total energy (divided by c2). This in turn contains the internal as well as the grav- itational energy. This latter is negative so that the gravitational mass is reduced (exactly the same as for the binding energy of a nucleus that reduced the total mass of nucleons giving rise to the mass defect). Finally, eqn.(10.12) should not be mistaken with the newtonian counterpart. In- 2 deed ρ = ρ0 + u/c where ρ0 is the rest-mass density and u is the whole energy density. Differentiating eqn.(10.8) with respect to r one derives P 0 = P (λ, λ0, ν, ν0, r) and with the aid of eqn.(10.9) and eqn.(10.10) one may arrive to the equation of hydrostatic equilibrium, mass conservation, and gravitational potential in Gen- eral Relativity

dP GM ³ P ´³ 4πP r3 ´³ 2GM ´−1 = − ρ 1 + 1 + 1 − (10.13) dr r2 ρc2 Mc2 r 230 CHAPTER 10. NEUTRON STARS

dM = 4πr2ρ (10.14) dr

dΦ 1 dP ³ P ´−1 = − 1 + (10.15) dr ρ dr ρc2 10.2 Models for neutron stars

Given the equation of state, we may solve the above equations and get the struc- ture of the Neutron Star at varying central density ρc. The method is much similar to that followed for White Dwarfs. Using the pure, ideal neutron star equation of state and the GRT hydrostatic equilibrium equations, Oppenheimer - Volkoff calculated models for neutron stars whose mass-central density relationship is shown in Fig.10.1. Note that

15 −3 Mmax = 0.7 M¯,R = 9.6 km ρc = 5 × 10 g cm (10.16) are the parameters of the maximum mass configuration. The stability analysis 15 3 shows that configurations with ρc > 10 g/cm are unstable to gravitational collapse. The Chandrasekhar mass limit for a neutron star (i.e. Newtonian n = 3 polytropes of infinite density) is 5.73 M¯. Relativity lowers this limit for two main reasons:

(i) the maximum mass occurs at a finite value of the density ρc where the neutrons are becoming relativistic: but are not extremely relativistic;

(ii) 5.73 M¯ is the rest mass of the neutrons. The total mass is lower than this because of the gravitational binding energy of the star. Low density neutron stars with the ideal neutron gas equation of state can be approximated to n = 3/2 Newtonian polytropes, so that using the polytropic theory we can express radius and mass as a function of the central density ρc 10.2. MODELS FOR NEUTRON STARS 231

Figure 10.1: Mass - central density relationships for fully degenerate White Dwarfs and Neutron Stars. The transition from the maximum mass for White Dwarfs (∼ 1.2 M¯) to the Neutron star domain is calculated with the Harrison- Wheeler equation of state.

ρ R = 14.64( c )−1/6 km (10.17) 1015 g cm−3 ρ M = 1.102( c )1/2 M (10.18) 1015 g cm−3 ¯ and the Radius-Mass relationship as 15.12 km M = ( )3 M (10.19) R ¯ Thus there is no minimum mass in the Oppenheimer-Volkoff solution:

M → 0, R → ∞ as ρc → 0. In reality this is not likely to occur because at decreasing ρc neutrons become unstable to β-decay and transform into protons. We can give an approximate evaluation of the Oppenheimer-Volkoff results using the energy variational method. Since ρc at Mmax is at the transition from non-relativistic to relativistic neutrons, it is not clear what kind of polytrope approximates a neutron star near Mmax. The stability analysis shows that using n = 3 in the variational principle is inconsistent; we therefore adopt n = 3/2. The results accurately describe low-density neutron stars and give an estimate of

Mmax. 232 CHAPTER 10. NEUTRON STARS

Using the polytropic theory we can express the internal Eint and gravitational Egrav energies as

2/3 Eint = k1Kρc M k1 = 0.795873 (10.20) and 1/3 5/3 Egrav = k2Gρc M k2 = 0.760777 (10.21) where K is the polytropic constant for n = 3/2.

Now we apply a correction ∆Eint to the internal energy to take into account that neutrons may enter the relativistic regime. To this aim, first we compute the energy density ² − m c2n u = n n n (10.22) ρ0 where 3 mnx ρ0 = mnnn = 2 3 (10.23) 3π λn where x = p/mnc is the usual relativistic parameter. Expressing u with the aid of the function χ(x) for neutrons, we find ³ 3 3 ´ u = c2 x2 − x4 (10.24) 10 56 the first term is Eint the second one gives the correction ∆Eint 3 Z ∆E = − c2 x4dm (10.25) int 56 Adopting the usual substitution for a polytrope in the integral we get 4 h¯ 4/3 ∆Eint = −k3 Mρc (10.26) 16/3 2 mn c where k3 can be calculated by the polytropic theory Z ξ1 3 2 4/3 1 3.5 2 k3 = (3π ) 2 0 θ ξ dξ = 1.1651 (10.27) 56 |xi1θ (ξ1)| 0 We must also correct the gravitational energy by the term of self- gravity (the gravitational energy is a source of mass in GRT and there- fore contributes to the total gravitational energy). The correction is G2 ∆E = −k M 7/3ρ2/3 where k = 0.6807 (10.28) grav,GRT 4 c2 c 4 10.2. MODELS FOR NEUTRON STARS 233

The total energy of the star is then

2/3 5/3 1/3 4/3 7/3 2/3 E = AMρc − BM ρc − CMρc − DM ρc (10.29) where

4 2 k3h¯ G A = k1KB = k2GC = D = k4 (10.30) 16/3 2 2 mn c c

Equilibrium is achieved when ∂E/∂ρc = 0. this relation simplifies to

−1/3 2/3 −2/3 1/3 4/3 −1/3 2Aρc − BM ρc − 4Cρc − 2DM ρc = 0 (10.31)

Keeping only the first two terms, one recovers the results for the n = 3/2 poly- trope. Keeping all the terms gives a better approximation to the M(ρc) relation of Oppenheimer and Volkoff. 2 2 The onset of instability occurs when ∂E /∂ρc = 0. This relation be- comes

−1/3 2/3 −2/3 1/3 4/3 −1/3 −2Aρc + 2BM ρc − 4Cρc + 2DM ρc = 0 (10.32)

Add eqns. (10.31) and (10.32) to get BM 2/3 ρ = (10.33) c 8C Substitute this back into eqn.(10.31) and write

y = M 4/9 (10.34)

The resulting equation is cubic in y

2A − 3B2/3C1/3y − 2Dy3 = 0 (10.35)

The positive root is y = 6.605 × 1014 g/cm3, so that

15 3 M = 1.11 M¯, ρc = 7.4 × 10 g/cm (10.36)

Note that the mass from eqn.(10.30) is really the rest mass of neutrons. If we substitute eqn.(10.30) we find E = −0.08 M (10.37) c2 ¯ 234 CHAPTER 10. NEUTRON STARS

So the maximum total mass of the Neutron Star is 1.03 M¯, about 40% higher than the value calculated by Oppenheimer and Volkoff. If we assume that a Neutron Star is always supported by the ideal degenerate 5/3 neutron in non relativistic conditions at all densities (P = K5/3ρ0 at all ρ0) and adopt the post newtonian approximation for hydrostatic equilibrium, we find that the M(ρc) relation agrees with eqn.(10.18) in the low-density range but the maximum mass would tend to M ∼ 3.38 M¯ as ρc → ∞. Furthermore, dM/dρc would be > 0 all the way up to the maximum value (stable configurations). The fully relativistic equation of hydrostatic equilibrium yields value of 0.84 M¯ at 15 3 ρc = 5 × 10 g/cm with dM/dρc > 0, and for all ρc > ρc,max dM/dρc < 0 (unstable). A more realistic equation of state considers an equilibrium mixture of nonin- teracting neutrons, protons and electrons. Recall that neutrons are present only at densities exceeding 1.2×107 g/cm3. The neutron/proton ratio reaches a maxi- 11 3 mum at about 7.8×10 g/cm and then decreases to 8 as ρ → ∞ (nn : np = 8 : 1). As always for an ideal gas P → ρ/3 as ρ → ∞. Stellar models calculated with this recipe do not differ much from the Oppenheimer - Volkoff results. For example

15 3 Mmax = 0.72 M¯ R = 8.8 km ρc = 5.8 × 10 g/cm (10.38)

11 3 Star with ρc ≤ 7.8 × 10 g/cm really belong to the high density branch of White Dwarfs and are therefore unstable. There must be a minimum in the M(ρc) relationship. This is the minimum mass for neutron stars. Using eqn.(10.18) we get

Mmin = 0.03 M¯ R = 48 km (10.39)

More realistically, using the Harrison-Wheeler equation of state, we would obtain

13 3 Mmin = 0.18 M¯ R = 300 km ρc = 2.6 × 10 g/cm (10.40) 10.3. REALISTIC NEUTRON STAR MODELS 235

Table 10.1: Equations of state (source and type) and associated Maximum Mass.

Equation of State Type Mmax/M¯ π Soft 1.5 R Soft 1.6 BJ Stiff 1.9 TNI Stiff 2.0 TI Stiff 2.0 MF Stiff 2.7

10.3 Realistic neutron star models

In chapter 6 we have presented a variety of realistic equations of state for cold, dense matter to each of which actual neutron star are associated. All the models are calculated using the Oppenheimer-Volkoff equations, so that GRT is fully taken into account. For the sake of illustration we present here some general properties of the models and look for general trends. We start summarizing in Table 10.1 the source of the equation of state and the corresponding maximum mass. In Fig. 10.2 we show cross sectional slices of two 1.4 M¯ stars computed from the soft Reid and stiff TI equation of states. The layering of the configurations is a consequence of the onset of different regimes in the equation of state as one proceeds to higher densities. Note the large inner region in superfluidity regime (see below). Finally in Fig.10.3 we show the gravitational mass vs. central density for neutron stars with the various equations of state (left panel), and the gravitational mass ve. radius for the same models (right panel). The ascending branches of the gravitational mass vs. central density curves correspond to stable configurations. The different layers composing a Neutron Stars can be identified as follows: (1). The surface (ρ ≤ 106 g/cm3), a region in which the temperature and the magnetic fields expected for neutron stars may significantly affect the equation of state. The external atmosphere is very hot and compressed. In fact, the temperature is about 106 K and the thickness is very small because of the high 236 CHAPTER 10. NEUTRON STARS

Figure 10.2: Cross sectional slices of the 1.4 M¯ neutron stars by Reid (left and TNI (right) illustrating the various regions described in the text. The moments of inertia of the entire crust, Ic and of the superfluid region, Is are also given.

Figure 10.3: Left: Gravitational mass vs. central density for various equations of state. The ascending portions of the curves represent stable neutron stars. Right: Gravitational mass versus radius for the same equations of state as in the left panel 10.3. REALISTIC NEUTRON STAR MODELS 237

14 −2 , g0 ' 1.3 × 10 cm s . For comparison in the Sun, the surface 4 −2 8 −2 gravity is g0 ' 2.7 × 10 , cm s , and in a White Dwarf g0 ' 10 cm s . The pressure scale hight at the surface of a Neutron Star is about 1 cm. (2). The outer crust (106 ≤ ρ ≤ 4.3 × 1011 g/cm3), a solid region in which a Coulomb lattice of heavy nuclei coexists in β-equilibrium with a relativistic degenerate electron gas (see the case of White Dwarfs). (3). The inner crust (4.3 × 1011 ≤ ρ ≤ (2 − 2.4) × 1014 g/cm3) which consists of a lattice of neutron-rich nuclei together with a superfluid neutron gas and an electron gas.

14 3 (4). The neutron liquid (2 − 2.4) × 10 ≤ ρ ≤ ρcore g/cm ) which contains superfluid neutrons with a smaller concentration of superfluid protons and normal electrons.

(5). A core region (ρ > ρcore), which may or may not exist in some stars and depends on whether or not pion condensation occurs, or whether there is a transition to a neutron solid or quark matter or some other phase physically distinct from a neutron liquid at densities above some critical value. Looking at various models, several important features emerge: (a) Stars calculated with a stiff equation of state have greater maximum masses than stars derived from a soft equation of state. (b) Stars derived from a stiff equation of state have a lower central density, a larger radius, and a much thicker crust than do stars with same mass calculated with a sift equation of state. (c) Pion condensation, if it occurs, tends to contract neutron stars of a given mass as well as decrease Mmax. Some general considerations. If the equation of state is stiff, the central 15 3 density of a relatively massive 1.4 M¯ neutron star is ≤ 10 g/cm ; in fact, even 15 3 the most massive, stable neutron stars have ρc ≤ few × 10 g/cm . Thus the possibility of a transition to quark matter or some other exotic form of matter seems unlikely. However, we cannot exclude the existence of a third stable branch of ”quark matter stars” on the M(ρc) plane beyond White Dwarfs and Neutron 238 CHAPTER 10. NEUTRON STARS

Stars.

Neutron stars with masses near the Chandrasekhar limit of 1.4 M¯ may be favoured in nature. Such stars with moderately stiff equations of state are not likely to form pion condensates which requires ρ ≥ 2ρnuc, In any case give the uncertainties these conclusions are not definitive. The minimum mass of a stable Neutron Star is determined by setting the mean value of the adiabatic index Γ equal to 4/3 for stability against collapse. As discussed in chapter 6, Γ(ρ) falls below 4/3 at neutron drip (ρ ' 4.3×1011 g/cm3) and does not rise above 4/3 again until the density ρ exceeds ' 7 × 1012 g/cm3. The resulting minimum Neutron Star mass is

14 3 Mmin = 0.0925 M¯ ρc = 1.55 × 10 g/cm R = 164 km (10.41)

The reason that ρc for the minimum mass configuration is much larger than 7 × 1012 g/cm3 is that (i) it is the mean value of Γ that is relevant for stability; (ii) that Γ must be greater than 4/3 in General Relativity. Since the equation of state is reasonably well understood for all ρ ≤ ρc, we regard the above parameters for the minimum mass configuration as well established. The same statement cannot be made for the maximum mass equilibrium configuration, due to uncertainties in the equation of state above ρ ' 2.4 × 14 3 10 g/cm . At present most equations of state give Mmax < 3 M¯. Observational determinations of Neutron Star masses. We have seen that global Neutron Star parameters, such as masses, radii, moments of inertia and so on are very sensitive to the adopted microscopic model of the nucleo- nucleon interaction. Thus astronomical determinations of these parameters can shed light on hadronic physics. Recent determinations of the Neutron star masses have been obtained from (i) X-ray binaries. In such a case masses can be determined via the Kepler Third Law (see any textbook of celestial mechanics or Basic Astronomy). (ii) Binary pulsars. The case of PSR 1913*16 is te the prototype. The available mass estimates for the pulsar and the companion are 10.4. MORE ON THE MASS-DENSITY RELATIONSHIP 239

Figure 10.4: Observational determinations of neutron star masses. The most probable value for each determination is given by the filled circles. The shaded region represents the smallest range of neutron star masses consistent with all the data.

Mpulsar = 1.41 ± 0.06 M¯ (10.42)

Mcompanion = 1.41 ± 0.06 M¯ (10.43)

Recent determinations of neutron star masses are shown in Fig.10.4

10.4 More on the mass-density relationship

Three are the salient aspects of the M(ρc) relationship:

(i) The existence of a minimum mass Mmin after the White Dwarf 14 −3 regime. This is corresponds to ρc ' 2×10 g cm . Mmin ∼ 0.05M¯.

(ii) The existence of maximum value of the mass, Mmax, at very high densities. The maximum mass is determined by the transition of neutrons from non-relativistic to relativistic conditions. The existence of Mmax does not depends on the particular equation of state in use, but its value does.

Currently Mmax ∼ 1 − 3 M¯ (see Fig.10.1 or 10.4). Finally, Neutron Stars with mass M > Mmax must inevitably collapse to a Black Hole. The existence of Mmax caused by the transition non-relativistic to relativistic conditions can be justified 240 CHAPTER 10. NEUTRON STARS with dimensional arguments. As in the case of normal hydrostatic equilibrium P ∝ M 2/R4; eliminating ρ by means of ρ ∝ M/R3 we obtain

P ∝ M 2/3ρ4/3 (10.44)

Introducing now a polytropic-like equation of state ρ ∝ P 1/Γ, replacing ρ and solving for M we derive we derive

M ∝ ρ3(Γ−4/3)/2 (10.45)

In the non-relativistic case Γ = 5/3 from which M ∝ ρ1/2 and hence dM/dρ > 0. In the fully relativistic case Γ = 1 from which M ∝ ρ−1/2 and hence dM/dρ < 0.

Somewhere there must be dM/dρ = 0, that is the Mmax. The transition to the relativistic situations occurs when the energy density u/c2 exceeds the rest- 2 2 mass density ρ0c . In contrast, using always ρ0 and therefore setting u/c ∼ 0 in the total density ρ, the maximum possible mass is the Chandrasekhar limit

MCh = 5.73M¯ (for neutrons) which corresponds to Γ = 4/3). Therefore, the following inequality must be verified Mmax < MCh. (iii) Intimately related to the existence of the low and upper bound- aries in the M(ρc) relationship of Fig. 10.1 is the problem of stability.

The M = M(ρc) relationship can be considered as a series of equilibrium models described by the parameters ρc. Starting from the low density end (typical of 5 −3 planets (ρc ' 10 g cm ) toward increasing densities the curve shown four sig- nificant values for the mass, i.e. Mmin and Mmax for White Dwarfs and Neutron

Stars. Stable branches are those with dM/dρc > 0, unstable the others. Note that M(ρc) of White Dwarfs may overlap that of Neutron Stars. Note that no stable configuration exist in the density interval ∼ 109 − −1012 g cm−3.

10.5 The maximum mass

It is clear from our discussion that neutron stars are the most com- pact configurations of matter that can possibly withstand gravitational 10.5. THE MAXIMUM MASS 241 force. Hence the maximum mass of neutron stars provides an upper bound on the stable, lowest-energy configurations of matter in its most compact form. For this reason, it is very important to estimate, as accurately as possible, the value of this mass. Given the equation of state of matter, the maximum mass can be derived from the integration of the Oppenheimer-Volkoff equation. This will lead to an M(ρ) curve from which the maximum mass can be read off. Unfortunately the equation of state is not yet firmly established in the highest density range. The problem of the maximum mass must be addressed in an indirect manner. Rhoades and Ruffini (1974) made the following reasoning and assumptions:

(1) General relativity is the correct theory of gravity; this means that the Oppenheimer- Volkoff equations are the ones to use.

(2) The equation of state satisfies the microscopic stability condi- tion dP ≥ 0 (10.46) dρ If this conditions were violated, small elements of matter would spontaneously collapse.

(iii) The equation of state must satisfy the causality condition

dP ≤ c2 (10.47) dρ

that is the speed of sound must be less than the speed of light.

(iv) The equation of state below some matching density ρ0 is known.

They performed a variational analysis to determine which equation of state above ρ0 subject to conditions (10.46) and (10.47) minimizes the mass. They 242 CHAPTER 10. NEUTRON STARS found that the equation of state for which equality holds in relation (10.47) yields plausible result. The equation of state is

2 P = P0 + (ρ − ρ0)c ρ ≥ ρ0 (10.48)

14 3 They choose ρ0 = 4.6 × 10 g/cm and adopted the Harrison-Wheeler equation of state below ρ0. The choice of ρ0 is not crucial; the resulting Mmax can be −1/2 scaled roughly as ρ0 . A numerical integration of the Oppenheimer-Volkoff equation with the above recipe for the equation of state yields

³ ρ ´−1/2 M = 3.2 0 M (10.49) max 4.6 × 1014 g/cm3 ¯

We can recover this result with the aid of heuristic arguments ap- plied to a uniform density sphere in General Relativity. Assume that n and ρ are constant in the the interior of the star. The equation of state is defined by ρ(n) and the pressure is given by the First Law of Thermodynamics

dρ ρ + P/c2 = (10.50) dn n

The mass of the star is

Z R 4 M = 4π ρr2dr = πR3ρ (10.51) 0 3 and the total baryonic number is (with c = G = 1)

Z R nr2dr A = 4π 0 [1 − 2m(r)/r]1/2 ³ 3 ´3/2 = 2πn (χ − sin χ cos χ) (10.52) 8πρ see eqn.(4.80) in chapter 4. Here we have defined χ by

³8πρ´1/2 sin χ = R (10.53) 3 so that 2M sin2 χ = (10.54) R 10.5. THE MAXIMUM MASS 243

Now according to the minimization technique an equilibrium configuration must extremize the energy at fixed number of baryons

¯ ¯ ¯∂M ¯ ¯ ¯ = 0 (10.55) ∂χ A

(here χ is the independent parameter). Using eqn.(10.50) and eqn.(10.53) in eqn.(10.55) we find P = ζ(χ) (10.56) ρ where 6 cos χ ζ(χ) = − 1 (10.57) 9 cos χ − 2 sin2 χ/(χ − sin χ cos χ) For a given equation of state ρ(n), eqn.(10.50), (10.56), (10.53), (10.51), and (10.52) determine P , χ, R, M, and A as a function of n. It can be demonstrated that (i) The newtonian limit of the above equation is PV = |W |/3, where |W | is the gravitational potential energy of a uniform density sphere of volume V ; (ii) In the newtonian limit M ∝ P 3/2/ρ2, and hence M increases with n provided that P increases faster than ρ4/3. In the high density limit where P/ρ →∼ constant, one obtains instead M ∝ ρ−1/2, the scaling we have already mentioned at the beginning of this discussion. This shows that M must achieve a maximum at some intermediate values of n. As usual, the existence of a maximum is related to

¯ 2 ¯ ¯∂ M ¯ ¯ ¯ > 0 (10.58) ∂χ2 A

This condition can be written as a condition on the adiabatic index

Γ > Γc(χ) (10.59) where ∂ ln P ³ P ´dP Γ ≡ = 1 + (10.60) ∂ ln n ρ dρ The critical adiabatic index is

n (3ζ + 1)h(ζ + 1) io Γ (χ) = (ζ + 1) 1 + tan2 χ − 1 (10.61) c 2 6ζ 244 CHAPTER 10. NEUTRON STARS

Figure 10.5: The stability domain in the uniform density approximation. The curve separating the stable and unstable regions is Γc of eqn.(10.62). The dashed line shows the adiabatic index Γ of a free neutron gas.

In the Newtonian limit 4 89 M Γ = + (10.62) c 3 105 R

The dependence of Γc on ζ = P/ρ is shown in Fig.10.5. The adiabatic index Γ for any equation of state can be plotted and the intersection with Γc determines the point of instability. Also shown in this figure is causality constraint dP/dρ ≤ c2. This requires ζ ≤ 0.364, χ ≤ 64◦, 19 and so

M ≤ 0.405 (causality + stability constraints) (10.63) R We can now proceed to determine the maximum mass of a Neutron Star.

Since we assume we know the equation of state for ρ ≤ ρ0 with P0 = P (ρ0), the causality constraint dP/dρ ≤ c2 allows us to write

2 P ≤ P0 + v (ρ − ρ0) ρ ≥ ρ0 (10.64)

where v ≤ 1 (in units of c) is the maximum sound speed above ρ0. For a fixed ρ, M increases with P and hence v. The maximum occurs when P/ρ reaches a critical value, denoted ζc, at which Γ = Γc(χc), that is

ζc + 1 2 v = Γc (10.65) ζc 10.5. THE MAXIMUM MASS 245

Table 10.2: Maximum mass of stable Neutron Stars (in M¯)

14 3 15 3 ρ0 = 5 × 10 g/cm ρ0 = 1 × 10 g/cm 33 2 34 2 v/c P0 = 7 × 10 dyne/cm P0 = 5 × 10 dyne/cm Baym-Bethe-Pathick Reid 1.0 3.6 2.6 0.75 3.0 2.2 0.50 2.0 1.6 0.25 0.68 unstable

from eqn.(10.60). Substituting the limit value for P/ρ from eqn.(10.65) into eqn.(10.64) sets an upper limit on the mean density of the star

2 ρ0 − P0/v ρc ≥ (10.66) 1 − (ζc + 1)/Γc

This fixes an upper mass limit via eqn.(10.51) and (10.53)

³ ´ ³ ´ 1 3 1/2 1 − (ζc + 1)/Γc 1/2 2 M ≤ 2 sin χc (10.67) 2 8π ρ0 − P0/v The resulting mass limit has been determined by Nauenberg and Chapline using 14 3 the Baym-Bethe-Pathick equation of state to fix ρ0 = 5 × 10 g/cm . The maximum mass for v = 1 is

Mmax ∼ 3.6 M¯ (10.68)

Other evaluation have been made using the Reid equation of state. The values of Mmax for different values of v/c are given in Table 10.2. Finally, we note that even abandoning the causality constraint, still leads to a severe mass limit, assuming General Relativity to be valid. Letting v → ∞ we 2 seen from eqn.(10.65) that Γc → ∞. From eqn(10.61) we see that Γc ∼ ζ and ζ → ∞ as χ → 80◦.03. Thus eqn.(10.67) implies

³ ´ ³ ´ 1 3 1/2 1 1/2 3 Mmax ∼ sin χ 2 8π ρ0 ³4.6 × 1014 g/cm3 ´1/2 = 6.05 M¯ (10.69) ρ0 246 CHAPTER 10. NEUTRON STARS

In fact, the exact limit in General Relativity is attained for an incompressible star. This limit is M 4 ( ) = (10.70) R max 9 and eqn.(10.51) gives

8 ³ 3 ´1/2 ³4.6 × 1014 g/cm3 ´1/2 Mmax = = 5.03 M¯ no causality 27 4πρ0 ρ0 (10.71)

In summary, stiff equations of state currently predict Mmax ∼ 1.5 −

2.7 M¯. Depending on the equation of state at very high densities, Gen- eral Relativity predict absolute upper limits in the range 3−5 M¯. The higher values are only possible if one abandons the causality constraint dP/dρ ≤ c2.

10.6 The effects of rotation

In the above discussion we have always assumed that the configurations were non- rotating and spherical. This is not true in general, in fact we know that pulsars rotate. Moreover, we know that in the case of White Dwarfs rotation may substantially increase the maximum mass. We shall show in this section that, in contrast to White Dwarfs, rotation cannot increase the maximum Neutron Star mass appreciably (≤ 20%). Also, even this increase should occur only for rotational velocities much higher than observed. Rapidly rotating configurations in General Relativity are techni- cally difficult to deal with. In addition no simple stability criteria are known. Existing calculations assume (i) slow rotation; or (ii) uniform rotation and homogeneous configurations; or (iii) post newtonian gravity and an ideal Fermi gas equation of state. In the following we shall adopt the qualitative analysis of Lightman and Shapiro to highlight the problem. With the aid of the polytropic theory, the 10.6. THE EFFECTS OF ROTATION 247 total energy of a rotating Neutron Star can be written as

2/3 5/3 1/3 4/3 7/3 2/3 2 −5/3 2/3 E = AMρc −Bg(λ)M ρc −CMρc −DM ρc +k5λJ M ρc (10.72)

Here, E is the energy of a rotating Neutron Star, assumed to be close to a polytrope with n = 3/2, in the post-Newtonian limit. The quantities A, B, C, D have been defined in eqn.(10.29), while the oblateness parameter λ and the function g(λ) are defined in eqn (8.226) and (8.227). The quantity k5 is the defined in eqn (8.232) and has the value 1.926 for a polytrope with n = 3/2. Finally J is the angular momentum (conserved). The equilibrium condition ∂E/∂λ = 0 at fixed M and J gives the usual re- lation between T/|W | and eccentricity – se eqn.(8.234) and (8.235). The second equilibrium condition is ∂E/∂ρc = 0 (fixed M and J). Solving these two equa- tions simultaneously gives the equilibrium relation between M and ρc for various 2 2 values of J. The maximum mass is determined from the condition ∂ E/∂ρc = 0 (condition for the onset of radial instability). It can be demonstrated that the above three conditions can be written as

k J 2ρ1/3 T g(λ) 5 c = (10.73) BM 10/3 |W | λ Bg(λ)M 2/3 ρ = (10.74) c 8C ³ 4T ´ 2A + (Bg(λ))2/3C1/3 − 3 + M 4/9 − 2DM 4/3 = 0 (10.75) |W | The above three equations can be solved as follows: choose a value of λ, and hence find g(λ) and the ratio T/|W |. Then solve eqn.(10.75), a cubic in M 4/3, for M. Equation (10.74) gives ρc and eqn.(10.73) gives J. Illustrative results are given in Table 10.3. Since no general criteria are known in General Relativity for the onset of non axisymmetric instabilities, we assume that the approximate Newtonian criteria are still valid, i.e. T/|W | ≤ 0.14 for secular stability and T/|W | ≤ 0.26 for dynamical stability. Friedman and Schutz (1975) have shown that the typical radiation timescale for neutron stars with T/|W | ∼ 0, 14 is ≤ 1 yr, extremely 248 CHAPTER 10. NEUTRON STARS

Table 10.3: Maximum masses for rotating Neutron Stars

J Mass ρc 49 2 15 3 (10 g cm /s) T/|W | (M¯) (10 g/cm ) 0. 0. 1.03 7.41 0.21 0.02 1.06 7.57 0.32 0.04 1.09 7.73 0.42 0.06 1.12 7.88

short. From the entries of Table 10.3 the maximum increase in Mmax due to rotation (T/|W | = 0.14) is

1.24 − 1.03 ' 20% increase (10.76) 1.103

Finally, we can understand the difference between Neutron Stars and White Dwarfs from eqn.(10.75) which can be rewritten as

³2A − 2DM 4/3 ´9/4h 1 i9/4 M = (10.77) 3B2/3c1/3 g2/3(1 − 4T/3|W |)

The term in square brackets represents the increase due to rotation and is equally effective in White Dwarfs and Neutron Stars. The term proportional to D repre- sents the decrease in M due to relativistic gravity. The ratio of this term to the Newtonian term, 2A is ∼ GM/Rc2 and so it is negligible in White Dwarfs but not for Neutron stars.

10.7 Superfluidity

There is an interesting complication that needs to be taken into account in deter- mining the structure of Neutron Stars. A collection of fermions (like protons or neutrons) can, under certain circumstances, become a superconductor or a super- fluid. In a super-conducting state, charged fermions will exhibit zero resistance; a superfluid will similarly exhibit zero for its flow. The most dramatic feature of a superfluid is its ability to flow without any viscos- ity. We shall first provide a simple criterium that must be satisfied by a system 10.7. SUPERFLUIDITY 249 if dissipative processes are not to be effective, then we shall discuss a microscopic model from which this criterium is derived. Let us consider a fluid moving along a tube with velocity v. The friction against the walls as well as internal friction will dissipate the kinetic energy of the fluid over a period of time, will the flow gradually becoming slower. It is more convenient to discuss this flow in the frame in which the fluid is at rest, and the tube is moving with velocity -v. The effect of viscosity now will be to make the fluid begin to move, eventually achieving the same speed of the tube, when this happens the flow has ceased in the original laboratory frame. Microscopically the movement of the liquid arises because of the gradual excita- tion of the internal degree of freedom of the fluid. Quantum mechanically, these elementary excitations can be thought as some kind of pseudo-particles, each of which has momentum p and energy ²(p). When a single excitation appears in the

fluid, the energy E0 and momentum P0 of the liquid (in the coordinate frame in which it was initially at rest) will be equal to those of the elementary excitation: that is E0 = ²(p) and P0 = p. Going back to the original frame in which the tube was at rest we obtain for the energy E and momentum P of the fluid 1 E = E + P · v + Mv2 P = P + Mv (10.78) 0 0 2 0 where M is the mass of the fluid. Substituting E0 = ² and P0 = p we get 1 E = ² + p · v + Mv2 (10.79) 2 In this frame the liquid had originally the kinetic energy (1/2)Mv2, hence the energy change due to the appearance of an excitation is ² + p · v. Because the fluid actually slows down we must have ² + p · v < 0. The minimum value of the left-hand side is for p and v antiparallel. Therefore we need to satisfy the condition ²(p) v > (10.80) p 2 for some value of p. Simple excitation of the kind ²(p) = (p /2meff ) will lead to the condition v > (²(p)/p) = (p/2meff ) for sufficiently low values of p. 250 CHAPTER 10. NEUTRON STARS

In contrast suppose that the energy -momentum relation for excitations has the form h³ p2 ´2 i1/2 ²(p) = + ∆2 (10.81) 2m in this case the right hand side of eqn.(10.80) given by ²(p) ³ p2 ∆2 = + )1/2 (10.82) p 4m2 p2 √ has a minimum at pmin = 2m∆. Hence unless the velocity is larger than (2∆/m)1/2 condition (10.80) cannot be satisfied and the elementary excitation cannot be produced. In other words, no dissipation can take place in such a liquid, for flow velocities below a critical value. It is clear that whether a system will exhibit superfluidity (flow without dissi- pation) crucially depends on the ²(p) relation for elementary excitations. The relation (10.81) is said to have a band-gap or energy-gap, that is we need to supply an energy ∆ even at zero momentum to produce the excitation. Whether the elementary excitation of a system possess a band-gap depend on dynami- cal interactions. Systems exhibiting superfluidity and superconductivity possess interactions that lead to a band-gap. Inside a neutron star, the attractive nuclear interaction among neutrons makes them superfluid at all densities above the neutron drip. The protons are super- conducting throughout the core. In fact the many body fermion system with interactions that favors the formation of pairs of particles in two-body states, a phase transition to superfluid state can take place. If the particles are charged the state will be also super-conductive. This is indeed the explanation for super- conductivity in : here electrons in the with momentum k and spin s pair with electrons having momentum -k and spin −s. Since the basic nuclear interaction is attractive at large distances, the same long range pairing mechanism may arise in dense hadronic matter, for example neu- tron star matter. Although two neutrons cannot be bound in a vacuum, they can be bound when they are in the field of other nucleons. In highly degenerate Fermi systems, pairing occurs mainly between states near the Fermi surface. In 10.7. SUPERFLUIDITY 251

neutron stars, where the ration nn/np is high, one needs to consider only n-n and n-p pairing; states of opposite momentum are at different Fermi surfaces for n and p, so that n-p pairing does not occur. Pairs of neutrons are bosons, so their behaviour is presumably similar to that of 4He atoms in liquid helium at T = 2.19 K. In neutron stars similar behaviour may occur when the ther- mal motion energy kT is less that the latent heat ∆ associated with the phase transition to a paired state with band-gap ∆. The band-gap depends on the strength of the pairing interaction and on the density. At nuclear densities we know from laboratory nuclei that both neutrons and protons have undergone a pairing transition in cold nuclear matter and that ∆ ∼ 1 − 2 Mev. Hence at the relatively low temperatures (≤ Kev) expected for all but the youngest neutron stars superfluidity is effective. One expects superfluidity to occur in the crust and interiors of neutron stars. Furthermore, one expects the remaining protons in the interior to be paired and hence superconductive. Finally, electrons are expected not to pair. Modern theories predict that at least three distinct hadronic super-fluids exist inside a neutron star (i) In the inner crust (4.3 × 1011 < ρ < 2 × 1014 g/cm3), the free neutrons 1 may pair in the S0 state to form a superfluid amid the neutron-rich nuclei still present. (ii) In the quantum liquid regime (ρ ≥ 2 × 1015 g/cm3), where the nuclei have dissolved into a degenerate fluid of protons and neutrons, the neutron fluid is 3 likely to be paired in the P2 state. (iii) The protons in the quantum liquid are expected to be paired and super- 1 conductive in the S0 state. The existence of an energy band-gap ∆ between the ground state and the first excited state has a number of important consequences. To begin with, it implies that the ground state acts as a coherent condensate with the condensation energy 2 per particle being of the order of Ec ' (∆ /²F ). Therefore, the presence of superfluidity will not appreciably affect the equation of state because the Ec is 252 CHAPTER 10. NEUTRON STARS much lower than the Fermi energy. At any finite temperature we can describe the fermion systems as made by superfluid components of particles that are in the ground state with density n0 (these particles contribute to the mass but not to pressure because they are in the zero momentum level) and a normal component of excitations with density nex ∝ exp(−∆/kT ). The latter component is negligible in neutron star interiors. Thermal effects. The heat capacity of a superfluid or superconductor is much lower than the the thermal capacity of a normal degenerate gas at low temperature. In fact the heat capacity varies as exp(−∆/kT ), because to excite baryons one has to break the pairs. The reduced heat capacity will shorten the cooling time scale for neutron stars and pulsars. However the other components of the fluid (electrons) still have their full heat capacity, so the effect of the superfluid is somewhat reduced. Magnetic effects. A laboratory super-conductor exhibits the Meissner ef- fect: a magnetic field is expelled from any region that becomes superconductive. Free electrons are however present in the case of a neutron star, and so we expect the magnetic field to thread the entire configuration. As a result the charged components of the crust and core are tied together by the magnetic field and co- rotate. The superfluid neutrons are, however, only weakly coupled to the crust and the charged components. Now the charged particle components are con- stantly decelerated by the radiation reaction torque transmitted by the magnetic field. The superfluid region must then always rotate faster than the rest of the stars. This will have effects on several properties of pulsars. Hydrodynamic effects: neutral particles. Because of the coherence forced on the system by the energy gap, the particles are now described by a single wave function 1/2 ψ ∝ n0 exp(iφ) (10.83) where |ψ|2 is related to the density of particles and the phase φ is related to the macroscopic velocity of the particle by

v ∝ ∇φ (10.84) 10.7. SUPERFLUIDITY 253

The existence of a single wave function for all particles with a well defined φ has interesting implication when any superfluid rotates. Rigid body rotation with angular velocity Ω requires that

v = Ω × r → ∇ × v = 2Ω 6= 0 (10.85)

On the other hand, if the macroscopic velocity field is proportional to the gradient of the wave function we would expect

v ∝ ∇φ → ∇ × v = 0 (10.86)

This suggest that a superfluid cannot maintain rigid-body rotation as envisaged above. When a superfluid rotates, it breaks down into a series of microscopic vortices, each carrying a certain amount of angular momentum. The total angular velocity is provided by a large number of such vortices rather than by rigid rotation of all particles. In this case the velocity field v becomes singular on the vortex, allowing ∇ × v to vanish everywhere except at the origin. To see this, consider a single vortex line treated as infinitely long along the z axis. Suppose that the microscopic velocity field along this vortex is given by h¯ h¯ ˜e v = ∇φ = φ (10.87) 2mn 2mn r Because there is a singularity at r = 0, the phase of the wave function can in general change by 2πn (n= 1,2, 3...) as we go around the singularity. Therefore the line integral of v around a circle gives I ³ πh¯ ´ πh¯ h v · dl = n = nK K = = ' 2 × 10−3 cm2/s (10.88) mn mn 2mn In differential form, this is equivalent to the relation

(2) ∇ × v = nKδD (r) (10.89)

(2) where δD (r) is the two-dimensional Dirac delta-function. A number of such vortices can mimic the macroscopic rigid-body rotation along the lines shown in Fig.10.6. In this case we have I Z Z X (2) v · dl = (∇ × v) · dS = K δD (r − ri)dS i Z r = K n(r0)2πr0dr0 ≡ KN(r) (10.90) o 254 CHAPTER 10. NEUTRON STARS

Figure 10.6: Mimicking the rigid body rotation of a superfluid by a large number of vortex lines. where n(r) is the number of vortices per unit area and N(r) is the total number of vortices enclosed by a circle of radius r. When v(r) is along the direction eφ the equation simplifies to

1 ∂ 2πrv(r) = 2πr2Ω(r) = KN(r) (r2Ω) = Kn(r) (10.91) r ∂r

In the case of rigid body rotation (∂Ω/∂r) = 0 and the number density of vortices is given by n = (2Ω/K). Thus a uniform vortex density proportional to the rotational angular velocity is set up in a superfluid when it is rotating rigidly. The average spacing between vortices is given by

³ Ω ´−1/2 d ' n−1/2 ' 3 × 10−3 (10.92) vortex 100 rad/s

Hydrodynamic effects: charged particles. The situation is more compli- cated as regards protons as they are charged particles. In this case the momentum is e p = m v + ( )A (10.93) p c Where A is the vector potential. Therefore, for protons, the phase gradient of the wave function (here indicated by χ) is related to velocity by

h¯ e v = ∇χ − A (10.94) 2mp mpc 10.7. SUPERFLUIDITY 255

If we attempt to keep the phase χ non singular, the taking the curl of this equation, we obtain e e 2Ω = ∇ × v = − ∇ × A = − B (10.95) mpc mpc This result is unrealistic for a neutron star as it requires an angular velocity 15 Ω = (eB/2mpc) ' 3 × 10 B12 rad/s which is enormous. It is therefore necessary to consider the option of singular phases. It is now possible to minimize the kinetic energy by taking v = 0 and setting h¯ e ∇χ = A except at singularities (10.96) 2mp mpc Because the curl of the r.h.s. is nonzero, we must again have a singular behaviour for ∇χ. Evaluating the line integral of both sides over a circle and using the fact that when ∇χ is integrated around a singularity it can change by (2πn) with n = 1, 2, 3...., we get I hc¯ I hc¯ hc Φ = A · dl = ∇χ · dl = × 2πn = n (10.97) 2e 2e 2e

This shows that the magnetic flux must be quantized in units of Φ0 ≡ (hc/2e) ' 2 × 10−7 G/cm2. The quantity B/2 is related to the flux-tube lines in exactly the same way as Ω is related to the distribution of vortex lines. If nf (r) is the density of flux-tubes then the analogous of eqn(10.91) is 1 ∂ ³1 ´ Br2 = Φ n (r) (10.98) r ∂r 2 o f The spacing between the flux lines is

−1/2 −10 −1/2 lf ' nf ' 5 × 10 B12 cm (10.99) for a uniform magnetic field. Translating to band-gap The singular phases obtained in the previous paragraphes are mathematical simplifications of reality. A vortex line is not 2 physically singular at the microscopic level. When the kinetic energy (1/2)mnv arising from the velocity field v = K2πr of the single vortex line is comparable 256 CHAPTER 10. NEUTRON STARS

2 with the condensation energy per particle Ec = (∆ /²F ) the particle will get out the condensed phase.

Equating the kinetic energy to Ec we define a coherence scale length ξ through the equation ∆2 1 K2 Ec ' = mn 2 (10.100) ²F 2 4πξ or ³ 2 ´ ³ 2 ´ mnK 1/2 πh¯ ²F 1/2 ξ ' 2 ²F ' 2 (10.101) 8π∆ 8mn ∆ Numerically ξ ' (10 − 30) fm for a neutron star. Microscopic origin of the Band-Gap. For the sake of completeness we shall now provide a microscopic quantum-mechanical model illustrating how the basic relation (10.81) arises. Consider a set of particles such as neutrons in a neutron star. The behaviour of such a system will be governed by the complex many-body interaction be- tween large numbers of particles, and the system will try to occupy a state of minimum energy. Under suitable circumstance there could arise weak, attractive interactions between pairs of particles. Such forces are of fairly long range in real space and then will correspond to nearly zero momentum states in fourier space (remember the indetermination principle (∆x ∆p ≤ h¯). In the case of neutron stars these effects are due to nuclear forces. When such small attractive forces exist between the fermions in the system, the ground state can exhibit certain peculiarities that we shall now discuss. We begin by writing the Hamiltonian for such a system by adding to the Hamiltonian Ho for non interacting particles, the

Hamiltonian Hint due to the interaction

H = H0 + Hint (10.102)

If we have nk particles each of energy ²k andh ¯k momentum, the Hamiltonian H0 is

X X † H0 = ²Knk = ²kakak (10.103) k K 10.7. SUPERFLUIDITY 257

† where ak and ak are the creation and the annihilation operators, respectively

(classical Hamiltonian), whereas Hint is

X † † Hint = −G aka−ka−k0 ak0 (10.104) k,k0>0 where G is the coupling constant of the interaction. The combination a−k0 ak0 0 0 † † annihilates a pair with momenta (k , −k ) and the combination aka−k creates the pair with momenta (k, −k). This can be thought of as a scattering of a pair from the state k0 to the state k. Although this coupling is very simple and perhaps unrealistic, it captures the essential feature of the problem. The original ground state |0i of the system is destroyed by all the annihilation operators ak, and the expectation value of H0 in this state is zero. To find the new ground state in presence of Hint we minimize the expectation value for the total Hamiltonian H = H0 = +Hint subject to some conditions. A trial is given by the expression ³ ´ Y † † |ψ >= uk + vkaka−k |0i (10.105) k>0 where uk and vk are two well-behaved functions related to each other by the normalization condition 2 2 uk + vk = 1 (10.106) Because of this condition, the quantum state is parameterized by one set of numbers, say vk. We now need to minimize hψ|H|ψi subject to the constraint that the total number of particles is conserved: that is we must keep

D ¯ ¯ E ¯ X † ¯ ψ¯ akak¯ψ = N (10.107) k while minimizing the energy. Imposing the condition by means of a Lagrange multiplier (λ) and simplifying the variational equation we get D ¯ ¯ E δ ¯ X † ¯ ψ¯H − λ akak¯ψ = 0 (10.108) δvk k leads to the result 1 v2 = {1 − (² − λ)[(² − λ)2 + ∆2]−1/2} (10.109) k 2 k k 258 CHAPTER 10. NEUTRON STARS where ∆ is given by X ∆ = G ukvk (10.110) k Since the lagrange multiplier corresponds to the conservation of the total number of particles it can be interpreted as the chemical potential or equivalently, the 2 † Fermi energy ²F . Similarly because vk = hψ|akak|ψi, we can identify it as the 2 occupation probability of a given state. Determining uk from the normalization condition of eqn.(10.106) and substituting in eqn.(10.110) we obtain the relation for the energy gap ∆ (we also replace λ with ²F )

X 1 2 2 −1/2 ∆ = G∆ [(²k − ²F ) + ∆ ] (10.111) 2 k The trivial solution is ∆ = 0 which corresponds to the normal, zero-temperature Fermi distribution. The other solution is

X 2 2 2 −1/2 = [(²k − ²F ) + ∆ ] (10.112) G k

2 Assuming ²k = (k /2m) and constant density of states, we convert the summation to an integral and obtain

G Z b 4πk2dk 1 3 2 2 2 1/2 = 1 (10.113) 2 a (2πh¯) [(k /2m − ²F ) + ∆ ] To determine the nature of the integral we note that we can write

³ k2 ´2 ³ 1 ´2 − ² + ∆2 = (k − p )2(k + p )2 + ∆2 (10.114) 2m F 2m F F

If our integral has the dominant contribution around k ' pF this expression can 2 2 2 be approximated as [(k − pF ) vF + ∆ ] where vF = pF /m. Choosing the limits of the integral such that ∆ << vF |pF − k| << ²F leads to the approximate evaluation of the integral (10.113)

Z 2 2 Z G4π k dk G p ²F dq ' F 3 2 2 2 1/2 2 3 2 2 1/2 2(2πh¯) [vF (pF − k) + ∆ ] 4π h¯ vF ∆ (∆ + q ) G ³2p2 ´ ² ' F ln( F ) (10.115) 2 3 4π h¯ vF ∆ 10.7. SUPERFLUIDITY 259 where the definition of q is obvious. Condition (10.113) can now be used to derive ∆ ³ 2π2h¯3 ´ ³ 1 ´ mp ∆ = ² exp − ≡ ² exp − η = F (10.116) F F 2 3 GmpF Gη 2π h¯ The effect of paired correlations is to lower the ground state energy by the amount

X X ³ X ´ 2 2 2 ∆E = ²k − (²k − ²F )vk + G vkuk ' η∆ (10.117) ²k<²F k k The physical meaning of the lower-energy ground state can be understood along the following lines. Consider a transformation from the original creation and † † annihilation operators (ak, ak) to a new set (αk, αk), where

† † α = vkak + ukak (10.118) etc. These quantities can be thought of as creation and annihilation operators of some well-defined quasi-particles whose energies are

2 2 1/2 Ek = [(²k − ²F ) + ∆ ] (10.119)

Note that the first excited state for the quasi-particle has a band-gap ∆ with respect to the ground state. In contrast, ordinary quantum particles of a free- field theory can have a continuous set of energies arbitrarily close to the rest mass energy mc2 with no band gap. The existence of a band-gap implies that any dissipative process with a characteristic energy less than ∆ cannot excite the quasi-particles and change the system from the ground state to an excited state.

Finally, the band-gap disappears at some critical temperature Tc given by kTC ' 0.57∆ (the proportionality constant is empirically estimated). For kTc << ∆, the band-gap is given by

h ³ ´ ³ ´i 2πkT 1/2 ∆0 ∆ = ∆0 1 − exp − ∆o ≡ ∆(T = 0) (10.120) ∆0 kT whereas near Tc is given by

s ³ T ´ ∆ = 3.06kT 1 − (10.121) Tc 260 CHAPTER 10. NEUTRON STARS

In general at any finite temperature, the system is made of a superfluid component with density ρs and a normal component with density ρn, this latter being

ρ ³2π∆ ´1/2 ³ ∆ ´ n = 0 exp o (for kT << ∆) ρ kT kT ³2T ´ = − 1 (for T ' Tc) (10.122) Tc

Although there is no attractive force between two free neutrons, such a force arises in the context of many-body interactions. It is similar to the lattice providing an attractive residual force between the electrons in a solid. The typical energy is of the order of 3 Mev. In the inner crust, the temperatures (kT ' 1 − 10 Kev) are significantly lower than the pairing energy and therefore we expect the neutron to be superfluid. By the same reasoning the protons in the inner regions are superconductive. The electrons will be a normal Fermi gas as they do not feel the nuclear force.

10.8 The Luminosity of a Neutron Star

The luminosity of Neutron Stars is so low that they would be hardly detectable. However many of them are sufficiently close by and even more important they possess several important properties: (1) strong magnetic fields; (2) rapid ro- tation, (3) emission of radiation in the radio wave length interval in a narrow cone around the magnetic axis. In general the rotation axis does not coincide with the magnetic axis so that for suitable orientations of the star, an observer may receive the radiation emitted in the cone at regular intervals. The pulsars are indeed interpreted as rapidly rotating neutron stars (the rotational periods are of order of sec or even shorter). For instance, the pulsars observed in the Crab (typical ) has a period of about 1/30 sec. The shortest observed period is a few millisec. The origin of the collimated radiation in the cone that is observable even in the optical and x-ray pass-bands is not completely clarified. The energy emitted in the cone is eventually taken from the 10.8. THE LUMINOSITY OF A NEUTRON STAR 261 the rotational energy reservoir. The pulsar slows down and the period increases with time. 262 CHAPTER 10. NEUTRON STARS Chapter 11

Cooling of Neutron Stars

11.1 Introduction

The determination of surface temperatures of Neutron Stars can in principle yield important information about the state of interior hadronic matter and Neutron Star structure. It is generally believed that Neutron Stars are formed at very high interior temperatures (T ≥ 1011 K in the core of a supernova explosion). The dominant cooling mechanism immediately after formation is neutrino emission, with an initial timescale of seconds. After about one day the internal temper- ature has dropped to 109 − 1010 K. Photon emission overtakes neutrino emission when the temperature is about 108 K, with a corresponding surface temperature roughly two orders of magnitude smaller. Neutrino cooling dominates during the first ∼ 103 years. Thermal evolution is sensitive to the adopted nuclear equation of state, the neutron star mass, the possible existence of superfluidity, pion condensation, quark matter and other physical ingredients. Typically the surface temperatures fall to about 106 K for objects approxi- mately 300 years old and remain in the vicinity of about (0.5 − 2) × 106 K for at least 104 years. The subject of this chapter is to present in some detail the dominant cooling mechanisms when the Neutron Stars has already reached an internal temperature

263 264 CHAPTER 11. COOLING OF NEUTRON STARS of 109 K and less.

11.2 Neutrino reactions in neutron stars

We shall interested in the thermal history of neutron stars after it has already cooled to an interior temperature below 109 K. The relatively short epoch during which the temperature drops from 1011 K to 109 K has already been described in Chapter 9. For internal temperature below a few times 109 K, any neutrino emitted during the cooling process escapes freely from the star, with- out interacting with the Neutron Star matter. At very high temperatures (T ≥ 109 K) found in the cores of evolved, massive stars, the dominant mode of energy loss via neutrinos is from the so-called URCA reactions

− − n → p + e +ν ˜e e + p → n + νe (11.1)

These reactions also dominate during the collapse. In both cases the nucleons in the hot interior are non-degenerate. However, when the nucleons become degenerate as in a Neutron Star that has cooled below 109 K, these reactions are highly suppressed. We demonstrate here this important result. Matter in the degenerate interior satisfies the the β-equilibrium condition

µn = µp + µe (11.2) where to a good approximation the chemical potential are just the Fermi energies. Thus

EF (n) = EF (p) + EF (e) (11.3) where at nuclear densities 2 2 pF (n) EF (n) = mnc + 2mn 2 2 pF (p) EF (p) = mpc + 2mp EF (e) ' pF (e)c (11.4) 11.2. NEUTRINO REACTIONS IN NEUTRON STARS 265

Charge neutrality requires

pF (p) = pF (e) (11.5)

So eqn.(11.3) becomes

2 ³ ´ pF (n) pF (p) ' pF (e)c 1 + − Q (11.6) 2mn 2mpc

2 where Q = (mn − mp)c = 1.293 Mev is small compared to the other terms in eqn.(11.6). From this equation we see that the neutron Fermi energy (minus the rest-mass) is very nearly equal to the electron Fermi energy:

2 0 pF (n) EF (n) ≡ ' pF (e)c = EF (e) (11.7) 2mn and so

pF (e) = pF (p) ¿ pF (n) (11.8)

0 0 EF (p) ¿ EF (n) (11.9)

Now consider the possibility of a reaction such as neutron decay, eqn.(11.1). The only neutrons capable of decaying are those within 0 ∼ kT of the Fermi surface EF (n). Hence, by energy conservation, the final proton and electron must also be within ∼ kT of their Fermi surfaces; the energy of the escaping neutrino must also be ∼ kT . Now according to inequality (11.8), the proton and electron must have small momenta compared to neutron. But this is impossible: the decay cannot conserve momentum if it conserves energy. In order for the process to work, a bystander particle must be present to absorb momentum. The process can be modified in the following way (modified URCA)

− n + n → n + p + e +ν ˜e (11.10)

− n + p + e → n + n + νe (11.11)

This is likely important for Neutron Star cooling. 266 CHAPTER 11. COOLING OF NEUTRON STARS

− − Figure 11.1: Feynman diagrams for electron-neutrino scattering νe+e → νe+e ; (a) charged current reactions; (b) neutral current reactions

Accompanying these reactions are the muon-neutrino emitting reactions

− n + n → n + p + µ +ν ˜µ (11.12)

− n + p + µ → n + n + νµ (11.13)

2 14 3 which occur whenever µe > mµc (ρ ≥ 8 × 10 g/cm ). Corresponding re- actions involving τ-neutrinos do not occur at typical Neutron Star 2 interior densities, since mτ c = 1784 Mev À µe. Reactions (11.1) proceed by a charged current interaction (the neutrons change to charged protons), while the scattering processes

ν + n → ν + n ν + p → ν + p (11.14) proceed by a neutral current. Some processes, such as neutrino-lepton scattering

− − − − νe + e → νe + e νµ + µ → νµ + µ (11.15) occur via both charged and neutral currents (see Fig.11.1 ).

11.3 Free neutron decay

The modified URCA reactions – eqn.(11.10) and (11.11) – involve both strong and weak interactions. For instance, pions are exchanged between the colliding neutrons in eqn.(11.10) before one of the neutrons decay into a proton. 11.3. FREE NEUTRON DECAY 267

Before calculating the rate of these processes, we consider a simpler case: pure neutron decay in vacuum, where strong interaction effects are very small. Much of this discussion also applies to β-decay in nuclei. The energy released in a typical β-decay is small (Q ∼ Mev) compared to the rest-mass of the nucleons. It can be demonstrated that the decay rate can be expressed as ³ X ´ 2π 1 2 dΓ = |Hfi| ρeρν˜dEedEν˜δ(Eν˜ + Ee − Q) (11.16) h¯ 2 spin

− where ρe and ρν˜ are the densities of the final states of the e andν ˜, respectively, per unit energy interval, Ee is the electron energy. The quantity Hfi is the weak interaction matrix element in the non relativistic limit Z Z ∗ ∗ ∗ ∗ Hfi = GF Ψf ΨidV = GF ψpψe ψν˜ψndV (11.17)

∗ ∗ ∗ where ψp, ψe , ψν˜, ψn are the proton, electron, anti-neutrino, and neutron wave function, respectively. We imagine that the decaying neutron is in a box of unit volume and normalize all wave functions accordingly. The integral in eqn.(11.17) is the probability to find all the particles at the same point in space. Therefore, the interaction can be described by a contact potential Z ∗ 0 0 0 Ψf (r)VW (r, r )Ψi(r )dVdV (11.18) where 0 0 VW (r, r ) = GF δ(r − r ) (11.19)

If we neglect the Coulomb distortion on the electron due to the proton, we can write ψe and ψν˜ as plane-wave states. In addition, we note that the wavelengths of these states are much longer than the nuclear radius: λ 1 h¯ hc¯ = = = ∼ 10−11 cm (11.20) 2π k p E for E ∼ Mev. We can thus expand the spatial part of the lepton wave functions as ∗ ∗ ψe ψν˜ = exp[−i(ke + kν˜) · r] = 1 − i(ke + kν˜) · r + ..... (11.21) 268 CHAPTER 11. COOLING OF NEUTRON STARS and retain only the first term of the expansion. Provided that the remaining nucleon integral does not vanish, it is said to describe an allowed transition. Higher order terms give rise to forbidden transitions. Clearly, no orbital angular momentum is carried out by an allowed transition. The spin part of the combined lepton wave function can be either o singlet state (total spin zero) or a triplet state (total spin unity). In singlet transitions, the total nucleon wave function does not change its spin or its total angular momentum J, so ∆J = 0. Triplet transitions require a spin flip, and total angular momentum can be conserved if ∆J = ±1 or 0. The singlet transition selection rule ∆J = 0, is called the Fermi selection rule. The triplet transitions obey the Gamow-Teller selection rule. The spin flip in triplet transitions is effected by the axial part of the weak interaction, while the singlet transitions occur via the pure vector part. We can now write

X 1 2 2 2 2 2 2 |Hfi| = GF [CV |MV | + 3CA|MA| ] (11.22) 2 spin

Here MV and MA are the vector and axial nuclear matrix elements, which are determined by the overlap integral of the initial and final nuclear states. CV and CA are coupling constants that would be unity if strong interaction had no effect on the interaction, and the factor 3 arises from the statistical weight of the triplet transition. Experimentally, it is found that

|CV | = 0.9737 ± 0.0025 ¯ ¯ ¯ CA ¯ ¯ ¯ ≡ a = 1.253 ± 0.007 (11.23) CV For neutron decay, it is a good approximation to take

MV ' MA ' 1 (11.24) since the neutron and proton wave functions are very similar. Thus we get

X 1 2 2 2 2 |Hfi| = GF CV (1 + 3a ) (11.25) 2 spin 11.3. FREE NEUTRON DECAY 269

On integration over dEν˜ using the δ-function, eqn.(11.16) becomes 2π dΓ = G2 C2 (1 + 3a2)ρ ρ dE (11.26) h¯ F V e ν˜ e For an electron with a definite spin orientation we have

2 2 3 dp 4πpEe ρe = 4πρ p h¯ = 2 3 (11.27) dEe c h where (E2 − m2c4)1/2 p = e e (11.28) c Since the neutrino is massless, we have by energy conservation

Eν˜ = Q − Ee = ρν˜c (11.29) and so (Q − E )2 ρ = e (11.30) ν˜ 2π2h¯3c3 Thus G2 C2 (1 + 3a2) dΓ = F V (E2 − m2c4)1/2E (Q − E )2dE (11.31) 2π3h¯7c6 e e e e e Defining the dimensionless energies

Ee Q ² ≡ 2 ²0 ≡ 2 = 2.5312 (11.32) mec mec we get m5c4G2 C2 (1 + 3a2) dΓ = e F V df (11.33) 2π3h¯7 where the energy spectrum of the decay electron is given by

2 1/2 2 df = (² − 1) ²(²0 − ²) d² (11.34)

Integrating over all allowed range of ² from 1 to ²0 gives 1 1 f = (²2 − 1)1/2(2²4 − 9²2 − 8) + ² ln[² + (²2 − 1)1/2] = 1.6369 (11.35) 60 0 0 0 4 0 0 0 Finally, the neutron decay rate is m5c4G2 C2 (1 + 3a2)f 1 Γ = e F V = s (11.36) 2π3h¯7 972 270 CHAPTER 11. COOLING OF NEUTRON STARS

The experimental value of the rate is 1/(925 ± 11) s: the discrepancy arises because we ignored the Coulomb effects in the calculation of f (which increases to about 1.7), and other more subtle quantum electrodynamic effects (radiative corrections).

11.4 Modified URCA rate

Having evaluated the weak interaction matrix element for free neutron decay, we can now calculate the rate of the modified URCA reactions

− n + n → n + p + e +ν ˜e

− n + p + e → n + n + νe

− n + n → n + p + µ +ν ˜µ

− n + p + µ → n + n + νµ

We shall follow the original approach by Bahcall and Wolf (1965). Let indicate with 1, 2, 1’, p, e and ν the two initial neutrons, the final neutron, the proton, the electron and the antineutrino, respectively. The reaction rate for given initial states 1 and 2 is

2π dΓ = δ(E − E )|H |2ρ ρ ρ dE dE dE (11.37) h¯ f i fi p e ν˜ p e ν˜

2 where |Hfi| must be summed over final spins and averaged over initial spins. We shall explicitly retain the same normalization volume V, so that in the empty space for each species j we would have

d3p ρ dE ≡ d3n = V j (11.38) j j j h3

However the reactions take place in a dense gas and most of the low energy 3 cells in phase space are occupied. Each factor d nj must be multiplied by the factor (1 − fj) where 1 fj = (11.39) exp[(Ej − µj)/kT ] + 1 11.4. MODIFIED URCA RATE 271

is the fraction of phase space occupied at energy Ej. There is another thing to note in eqn.(11.37): the number of phase-space factors is one less the number of final particles. We can get a more symmetric expression by adding a factor

3 3 3 3 3 3 h 3 3 (2π) δ (p − p )d p 0 = δ (p − p )d n 0 = δ (k − k )d n 0 (11.40) f i 1 f i 1 V f i 1 V by integrating over pl0 gives unity. The total antineutrino luminosity in a volume V is obtained by integrating

(11.37) over all initial states, multiplied by Eν˜

4 Z (2π) 3 3 3 3 3 3 3 2 L = d n d n d n 0 d n d n d n δ(E − E ) × δ (k − k )S|H | E ν˜ h¯V 1 2 1 p e ν˜ f i f i fi ν˜ (11.41) in which

S = f1f2(1 − f10 )(1 − fp)(1 − fe) (11.42) where the factors f1 and f2 account for the distribution of initial states. All important matrix elements for the interaction have the form

Hfi = hn, p, e, ν˜|VW |n, ni (11.43) where VW is the weak interaction ”contact” Hamiltonian given previously for the free-neutron decay [eqn.(11.19)]. Now it is again appropriate to represent the leptons as free particles states. However it is not possible for the nucleons because the total nucleon hamiltonian is

Hnuc = Hfree + Hs + VW ≡ H0 + VW (11.44)

where Hfree is the free particle contribution (kinetic energy of each particle) and

Hs is the strong interaction hamiltonian. Therefore, eqn.(11.43) gives the lowest order transition rate only if the nucleon wave functions are already eigenfunctions of H0. The solution of the Schr¨odingerequation 272 CHAPTER 11. COOLING OF NEUTRON STARS

including Hs is a cumbersome affair. We shall here give a qualitative attempt to estimate the effects of Hs on the two-body nucleon wave function. Write Z 0 ∗ ∗ ∗ 0 0 Hfi = dVdV ψnp(r)ψe (r)ψν˜(r)VW (r, r )ψnn(r ) G Z = F dVψ∗ (r)ψ (r) (11.45) V nn nn here we have used eqn.(11.19) and set

1 1 ψ (r) = eike·r → (11.46) e V1/2 V1/2 and similarly for ψν˜. We assume that the interaction in the initial n − n state is dominated by s-wave scattering (i.e. L = 0). Then the n − n state must have total spin S = 0. The V part of the weak interaction couples this state to an S = 0 n − p state, while the A part couples it to an S = 1 n − p state. Thus [see eqn.(11.21)]

X 2 2 4GF 2 2 2 2 |Hfi| = 2 (CV |MV | + 3CA|MA| ) (11.47) spin V where Z 0 ∗ 0 MV = dV(ψnp) ψnn Z 1 ∗ 0 MA = dV(ψnp) ψnn (11.48)

The factor 4 occurs in eqn.(11.47) because either neutron in the n − n pair can become a proton, giving a factor of 2 in amplitude and 4 in probability.

Since the range of strong force is of the order of λπ =h/m ¯ πc, we expect that the relative wave functions in eqn.(11.48) will overlap in a small volume of 3 3 the order of (λπ/2π) . Thus M ∼ (λπ) /V. Defining non-dimensional matrix elements ˜ VMV ˜ VMA MV = 3 MA = 3 (11.49) (λπ/2π) (λπ/2π) we obtain from eqn.(11.41)

4 2 −1 −9 2 ˜ 2 2 ˜ 2 Lν˜ = 64π VGF h¯ (λπ/2π) (CV |Mv| + 3CA|MA| )P (11.50) 11.4. MODIFIED URCA RATE 273 where the dimensionless factor P is

Z Y6 15 3 3 P = V−6(λπ/2π) d njSEν˜δ (kf ki)δ(Ef − Ei) (11.51) j=1

Numerically, we have for the neutrino emissivity ²ν˜ L ² ≡ ν˜ = (5.1 × 1048erg/cm3/s)P (|M˜ |2 + 4.7|M˜ |2) (11.52) ν˜ V V A The calculation of the factor P gives

³ ´ −30 ρ 2 8 P = 2.1 × 10 T9 (11.53) ρnuc 14 3 9 where ρnuc = 2.8×10 g/cm , and T9 = T/10 K. The 8 powers of T originate as follows: for each degenerate species, only a fraction ∼ kT/EF can effectively contribute to the cooling rate. There are two such initial species and three such final species. The antineutrino phase space is 2 proportional to Eν˜ and the energy loss rate gives another factor Eν˜.

Since Eν˜ ∼ kT , we have altogether 8 powers of T . ˜ 2 ˜ 2 We are left with the matrix elements |MV | and |MA| to determine. We define a fundamental length scale from the Fermi momentum of the dominant neutrons by h¯ ³ ρ ´−1/3 l = ∼ 0.4(λπ/2π) (11.54) pF (n) ρnuc ˜ ˜ Bahcall and Wolf argue that both MV and MA are expected to vary proportional 3 to (l/(λπ/2π)) , that is to be of the order of unity at ρ ∼ ρnuc and to decrease as 1/ρ. They find ³ρ ´4/3 |M˜ |2 = |M˜ |2 + 1.0 nuc (11.55) A V ρ Thus eqn.(11.52) becomes

³ ´ 19 3 ρ 2/3 8 ²ν˜ = (6.1 × 10 erg/cm /s) T9 (11.56) ρnuc The inverse reaction gives the same energy loss rate. 2 The muon-neutrino reactions must be considered when µe > µµc , i.e.

ρ > 2.9ρnuc. The νµ reactions differ from the νe reactions only in the factor P , 274 CHAPTER 11. COOLING OF NEUTRON STARS

2 2 where a factor pµdpµ appears instead of pedpe. Thus the ratio of νµ to νe energy loss rate is 2 pµdpµ F = 2 (11.57) pedpe The ratio F is given by

F = 0 ρ ≤ 2.9ρnuc (11.58) h ³ 2 ´ i mµc 2 1/2 F = 1 − ρ ≥ 2.9ρnuc (11.59) EF (e) Finally, multiplying eqn.(11.56) by 2(1 + F ) gives the total energy loss rate by modified URCA reactions

³ ´ URCA 20 3 ρ 2/3 8 ²ν = (1.2 × 10 erg/cm /s) T9 (1 + F ) (11.60) ρnuc For a neutron star of mass M and uniform density ρ, this gives a luminosity

³ ´ URCA 38 M ρnuc 1/3 8 Lν = (8.5 × 10 ) (1 + F )T9 erg/s (11.61) M¯ ρ The rate has been subsequently revised by Friman and Maxwell who adopted a better description of the NN interaction and gave

³ ´ URCA 39 M ρnuc 1/3 8 Lν = (5.3 × 10 ) T9 erg/s (11.62) M¯ ρ 11.5 Other reaction rates

Having sketched the rate of energy losses by the modified URCA pro- cess, we can now discuss briefly the rates of other possible cooling reactions. (a) Nucleon Pair Bremsstrahlung. The most significant cooling mecha- nism that becomes possible when neutral currents are considered is nucleon pair bremsstrahlung

n + n → n + n + ν +ν ˜ n + p → n + p + ν +ν ˜ (11.63) 11.5. OTHER REACTION RATES 275

The rate depends on T 8 and is about 30 times smaller than the modified URCA rate. (b) Neutrino Pair Bremsstrahlung. If the neutrons are locked in a superfluid state, the rates discussed so far are cut down by the factor ∼ exp(−∆/kT ), where ∆ is the superfluid energy gap. In this case, only cooling from neutrino pair bremsstrahlung from nuclei in the crust can be important. The rate for the process

e− + (Z,A) → e− + (Z.A) + ν +ν ˜ (11.64) generates a luminosity which is estimated to be

brems 39 Mcr 6 Lν = (5 × 10 erg/s)( )T9 (11.65) M¯ 6 where Mcr is the mass of the crust. Since the luminosity scales as T9 , it decreases less rapidly than reaction (11.61) as the star cools. (c) Pionic Reactions. As Bahcall and Wolf originally pointed out, pion condensation can dramatically increase the cooling rate in neutron star interiors. If pion condensates exist, the ”quasi-particle” β-decay can occur via

0 − N → N + e +ν ˜e (11.66) and its inverse. Here the particle N and N 0 are linear combinations of neutron and proton states in the pion sea. The pion condensate allows both energy and momentum conservation in the reaction, which is the analogue of the ordinary . Let us consider a simplified version of eqn.(11.66), i.e. cooling by decay of free pions

− − π + n → n + e +ν ˜e

− − π + n → n + µ +ν ˜µ (11.67) and the inverse process

− − n + e → n + π + νe 276 CHAPTER 11. COOLING OF NEUTRON STARS

− − n + µ → n + π + νµ (11.68)

As in the URCA process, the total rate for all four reaction is essentially four times the rate of the first of (11.67) alone. The muons are already present when and if the pions appear. We can make an estimate of the reaction rate as follows. Since there are two fewer fermions participating in the reaction than in the modified URCA reactions, the factor P varies as T 6 rather than T 8. Therefore, we expect the total rate to be ³ 0 ´ π URCA EF (n) 2 nπ Lν ∼ Lν × (11.69) kt nn where nπ/nn is the ratio of the number densities of pions and neutrons. Since 0 2/3 EF ∼ 60(ρ/ρnuc) Mev eqn. (11.60) and (11.69) give

π 44 M ρ nπ 6 Lν ∼ (8 × 10 ) T9 erg/s (11.70) M¯ ρnuc nn

π These estimate has been subsequently slightly revised. Since Lν is larger than URCA Lν by a large factor, pion cooling will dominate at all temperature of interest, provided pion condensation occurs. (d) Quark Beta Decay. If the core of neutron stars consists of quarks, then other neutrino emissions are possible. Unlike ordinary neu- tron star matter, the simple β-decay (normal URCA process) can occur. Recalling the discussion on the equation of state, let us suppose that three type of quarks (u, d, s) are present in β-equilibrium. There it was shown that ignoring the quark masses and quark mutual interactions, the equilibrium composition was given by

nu = nd = ns = n ne = nµ = 0 (11.71) where n = (nu + nd + ns)/3 is the baryon number density. Each quark species has Fermi momentum given by

³ n ´1/3 Mev pF (q) = 235 (11.72) nnuc c

−3 where nnuc = ρnuc/mb ' 0.17 fm . 11.6. NEUTRINO TRANSPARENCY 277

The existence of quark masses likely alters the above conditions. The quark s is rather massive (ms ∼ 100 − 300 Mev, and if present it is not likely to be relativistic. The u and d quarks are much lighter mu ∼ md ∼ 5 − 10 Mev so they are highly relativistic and their masses can be still ignored. The simplest process is

− d → u + e +ν ˜e (11.73)

− u + e → d + νe (11.74)

Iwamoto (1980) studied in detail these reactions and proposed the following ex- pression for the neutrino emissivity

quark 24 3 n 1/3 6 ²ν = (8.8 × 10 erg/cm /s) αs Ye T9 (11.75) nnuc where Ye = ne/n is the number of electrons per baryon (∼ 0.01). The parameter

αs ∼ 0.1. The corresponding luminosity is

³ ´ quark 44 M 6 Lν ' (1.3 × 10 erg/s) T9 (11.76) M¯ Finally, there will be also contributions from reactions like

− s → u + e +ν ˜e (11.77)

− u + e → s + νe (11.78) which are not described here. In any case the presence of quark matter in Neutron Star cores would cool the star at a rate faster than an ordinary Neutron Star and comparable to that of a star with a pion-condensed core.

11.6 Neutrino transparency

Once the neutrino luminosities are known, we can compute the cool- ing timescales. Fundamental to the discussion below, is the assumption 278 CHAPTER 11. COOLING OF NEUTRON STARS that neutrinos, once created, escape from Neutron Stars without un- dergoing further interactions or energy loss. We now show that for T ≤ 109 K, this assumption is valid. Before the discovery of neutral currents, the argument was as follows. Reactions of the type

− νe + n → p + e

+ ν˜e + p → n + e

+ ν˜e + p + n → n + n + e (11.79) and so on are all forbidden in Neutron Star interiors by conservation of energy and momentum. The most important interaction for neutrino energy loss would then be inelastic scattering off electrons (for νe andν ˜e) and off muons (for νµ and − ν˜µ). In a degenerate gas, the cross section for νe − e scattering, assuming only charged current interactions, is

³ ´ Ee 2 Ee σe ' χσ0 2 (11.80) mec EF (e) where ³ ´ ³ ´ 4 h¯ −4 GF 2 −44 2 σ0 ≡ 2 = 1.76 × 10 cm (11.81) π mec mec and where

χ ' 0.10 (V − A theory : only charge current interactions) χ ' 0.06 (WSG theory : charged and neutral current interactions)

The cross sections for the other three scattering processes are comparable. The mean free-path of an electron neutrino is then

−1 λe = (σene) (11.82) which can be translated into

³ ´ ³ ´ 7 ρnuc 4/3 100 Kev 3 λe ∼ (9 × 10 km) (11.83) ρ Eν 11.7. COOLING CURVES 279

Since λe >> 10 km the star is transparent to neutrinos. With the possibility of neutral currents, the effective absorption mean free path is significantly reduced because of the n − ν elastic scattering reactions, such as

n + νe → n + νe n + νµ → n + νµ (11.84) which can occur via neutral currents. Although elastic scattering does not degrade the neutrino energy, it does prevent neutrinos from escap- ing directly after emission. The neutrino scatters many times off neutrons in the interiors, until it finally encounters an electron (or muon) and scatters inelastically, or else it reaches the surface and leaves the star with its energy unchanged. The cross section for elastic scattering off neutrons is ³ ´ 1 Eν 2 σn = σ0 2 (11.85) 4 mec and the associated mean free path is ³ ´ 1 ρnuc 100 Kev 2 λn = ' 300 km (11.86) σnnn ρ Eν The effective mean free path for inelastic scattering off electrons, taking into account the increased probability of encountering an electron in the interior due to the zig-zag path of the neutrino, is ³ ´ 1/2 5 ρnuc 100 Kev 5/2 λeff ∼ (λnλe) ∼ 2 × 10 km (11.87) ρ Eν

So although λeff is significantly less than λe, it is larger than the radius of a Neutron Star. The transparency of a Neutron Star to low energy neutrinos is still valid.

11.7 Cooling curves

The temperature of a neutron star can now be calculated as a function of time. The thermal energy resides almost exclusively in degenerate 280 CHAPTER 11. COOLING OF NEUTRON STARS

Figure 11.2: Schematic neutron star cooling curves of interior temperature versus time for various processes, were they operate alone fermions (neutrons or quarks). Neglecting interactions, the heat capacity of N such particles of mass m and relativity parameter x = pF /mc is ¯ 2 2 1/2 dU ¯ π (x + 1) kT Cv = Ncv ≡ ¯ = Nk( ) (11.88) dT N,V x2 mc2 where cv is the specific heat per particle (for more details see Chandrasekhar). Integrating eqn.(11.88) we can derive the internal energy for a normal neutron star of mass M, density ρ and temperature T . One can adopt x << 1 ³ ´ 47 M ρ −2/3 2 Un = 6 × 10 ( ) T9 erg (11.89) M¯ ρnuc

For the quarks (using nu = nd = ns = n) we derive ³ ´ 47 M ρ −1/3 2 Uq = 9 × 10 ( ) T9 erg (11.90) M¯ ρnuc The temperature T is the interior temperature, which is nearly constant. Neu- tron star interiors are nearly isothermal thanks to the high thermal conductivity of the degenerate electron gas. A significant temperature gradient is present only in the crust. The cooling equation is dU dT = C = −(L + L ) (11.91) dt v dt ν γ where Lν is the total neutrino luminosity and Lγ is the photon luminosity. As- suming black-body radiation from the surface at an effective temperature Te we 11.7. COOLING CURVES 281 have R L = 4πR2σT 4 = 7 × 1036( )2T 4 (11.92) γ e 10 km e,7 The effective temperature is in units of 107 K. Inserting the appropriate luminosi- ties into eqn.(11.91) gives the time for the star to cool from an initial temperature T (i) to a final value T (f). It may be worth of interest to summarize here the cooling laws for the various processes we have presented

n h i o ρ −1/3 −6 T9(f) 6 ∆t(URCA) = [1 yr] ( ) T9 (f) 1 − (11.93) ρnuc T9(i)

n h i o −2 ρ 1/3 −4 T9(f) 4 ∆t(pions) = [20 s] θ ( ) T9 (f) 1 − (11.94) ρnuc T9(i) where θ ∼ 0.3.

n h i o n −1/3 −4 T9(f) 4 ∆t(quarks) = [1 hr] ( ) T9 (f) 1 − (11.95) nnuc T9(i)

n h i o ρ −2/3 −4 T9(f) 4 ∆t(brems) = [2 yr] ( ) T9 (f) 1 − (11.96) ρnuc T9(i) Detailed calculations give the following relation between the surface and interior temperature T e ∼ α−2 0.1 ≤ α ≤ 1 (11.97) T or equivalently ³ ´ ³ ´ Te −2 1/8 M 1/4 R −1/2 ' 1 × 10 T9 (11.98) T M¯ 10 km Finally, for photons

n h io 3 2 M 1/3 −2 Te,7(f) ∆t(photons) = [2 × 10 yr] α ( ) Te,7 (f) 1 − (11.99) M¯ Te,7(i) where α is that of eqn.(11.97). The above results are shown in Fig. 11.2, where the importance of the different cooling mechanisms is plotted as a function of time for a neutron star with M = 282 CHAPTER 11. COOLING OF NEUTRON STARS

2 M¯, ρ = ρnuc and θ = 0.1. Each curve gives T (t) for each process separately assuming that all the other are absent. The cooling time scales are calculated assuming a normal fluid. Su- perfluidity would modify the results in two different ways: (i) the specific heat capacity increases discontinuously as the temperature falls below the transition temperature, and then decreases exponentially at lower temperatures. Thus im- mediately above the transition temperature the cooling timescale increases, while at lower temperatures it decreases. (ii) Neutrino production is suppressed in a superfluid, thus increasing the cooling time scales. The above predictions of cooling timescale, internal and surface temperature fairly agree with current available observational informa- tion of Neutron star effective temperature, ages, and radii. Chapter 12

Pulsars

12.1 Discovery

In 1967 Jocelyn Bell of the Cambrigde group of astronomers headed by Antony Hewish detected a radio source emitting radiation at 81.5 Mhz with period of 1.377 s. Since then many more objects were detected which were named ”pulsars”. Key properties of the pulsars are (i) Pulsars have periods going from 1.6 ms to 4.3 s. (ii) Pulsar periods increase very slowly, and never decrease (except for occa- sional ”glitches”). (iii) Pulsars are extraordinarily precise clocks. Some pulsar periods have been measured to 13 significant digits. Very soon it was realized that the best candidates to explaining the pulsar phe- nomenon were the neutron stars, whose existence was long before predicted by Baade & Zwicky (1934) and Oppenheimer & Volkoff (1939). It was also sug- gested that they should occur in supernova explosions (Baade & Zwicky 1934, Colgate & White 1966) and that initially they would be rapidly rotating, with strong magnetic fields, and that the energy source of the might be a rotating neutron star. Preliminary models of a simple magnetic dipole capa- ble of converting neutron star rotational energy into electromagnetic radiation were already known at the time of the pulsar discovery. So the discovery by the Cambridge group did not occur in a theoretical desert. Immediately Gold (1969)

283 284 CHAPTER 12. PULSARS suggested that the discovered object could be a rotating neutron star with surface magnetic fields of around 1012 . He pointed out that such an object could explain the remarkably stable period and also predicted a small increase of the period as the neutron star slowly loses rotational energy. Shortly thereafter, the slowdown of the Crab pulsar was detected.

12.2 Which source ?

The three key properties listed above, pose strong constraints to the physical nature and dimensions of the emitter. In time scale of 1.6 ms the light travels about 500 km. This is an upper limit to the size of the emitting region. Indeed no consistent models have been proposed in which the emitting region is part of much bigger source. In particular, it is hard to have such a good clock mechanism unless the emitting region is closely coupled to the whole source. So we must take 500 km as the maximum dimension of the source. Is it a White Dwarf, a Neutron Star or a Black Hole? White Dwarfs. There are three possibilities for the underlying clock mech- anism: rotation, pulsation, or binary system. The shortest period for a rotating White Dwarf occurs when it is rotating at the break up velocity GM Ω2R = (12.1) r2 where Ω is the angular velocity. The above equation can be written in term of the density Ω2 ∼ Gρ (12.2) the universal relation between the dynamical time scale and the density for a self-gravitating system. Taking the maximum mean density of 108 g/cm3, we get for the period 2π P = ≥ 1 s (12.3) Ω Similar reasoning can be made for pulsating White Dwarfs that for the shortest period of the fundamental mode would predict about ∼ 2 s. Higher harmonics are 12.3. OBSERVATIONAL PROPERTIES OF PULSARS 285 unlikely because there would need special conditions to excite a harmonic without exciting the fundamental mode. Any small non linearity would immediately destroy the sharpness of the periods. Furthermore, loss of energy by vibration usually lead to a decrease in the period. Binary White Dwarfs would satisfy the condition GM Ω2r ∼ (12.4) r2 where r is the orbital radius, Since r ≥ R we would recover the same results as above. Thus White Dwarfs are definitely ruled out! Neutron stars. A pulsating neutron stars has a density ∼ 106 higher than a White Dwarf. The fundamental period is ∼ 10−3 s, typically too short. By adjusting the parameters one can arrange the period of a binary neutron star to fall in the range 10−3 − 4 s. However in such a case one expect a strong emission of gravitational waves, with a lifetime

³ P ´8/3 τ ' 10−3 yr (12.5) 1 s while the observational lifetime of pulsars P/P˙ ' 107 yr. Furthermore, the period is expected to decrease. Black Hole. There is no chance here to attach a periodic emitter to the surface. Rotating Black Holes are axisymmetric. Any mechanism depending on accretion would not be periodic to such a precision. The conclusion is that a rotating neutron star is a good candidate.

12.3 Observational properties of Pulsars

In the following, we present a brief summary of the salient properties of this class of celestial objects. (a) Pulse shape and spectra All known pulsars exhibit broad-band radio emission in form of periodic pulses (see Fig.12.1). Pulse intensities vary over a wide range, and sometimes pulses are missing. The basic pulse record, however, is periodic. The duty cycle (fraction 286 CHAPTER 12. PULSARS

Figure 12.1: Chart record of individual pulse for one of the first discovered pulsar PSR 0329+54. They are recorded at the frequency of 410 MHz. The pulses occur at regular intervals of about 0.714 s. of period with measurable intensity) is about 1-5 %. On time scales shorter than 1 ms, the pulse shape is quite complex. A typical pulse consists of two or more sub-pulses with complicated sub-structures on timescales as short as 10 µs. However, the average of several hundred pulses is found to be remarkably stable. By measuring the arrival time of the same feature in successive averaged pulses, one can verify that the rotating neutron star is a an excellent clock. Some pulsar periods are known to a part in 1013.

α The specific radio intensity is a steep power law, Iν ∝ ν , where a typical value of α is α ∼ −1.5 for ν < 1 GHz and even steeper at higher frequencies. A typical intensity (averaged over a period and integrated over solid angle) at 400 MHz is ∼ 0.1 Jy (1 Jy = 10−23 erg/s/cm2/Hz). Many pulsars show a high degree of linear polarization, up to 100 % in some cases. The amount and position angle of the linear polarization frequently vary with time across the pulse. is not as frequent and as strong as linear polarization. (b) Periods Observed periods lie between 1.558 ms (PSR 1937+214) and 4.308 s (PSR 1845- 19). The second shortest period is that of Crab, PSR 0531+21 (0.0331 s). The third shortest period is that of PSR 1913+16 (0.059 s), and the forth shortest 12.3. OBSERVATIONAL PROPERTIES OF PULSARS 287

Figure 12.2: The pulse period of PSR 0833-45 (Vela) as function of time (from 1968 to 1980). The four large jumps are enumerated and the time span in years between jumps is noted. period is for the Vela pulsar, PSR0833-45 (0.089 s). The median period is 0.67 s. In all cases where accurate measurements have been made, pulsar pe- riods are found to increase in a steady way but for the presence of ”Glitches” (see below). The situation is shown in Fig.12.2 for the pulsar PSR 0833-45 (Vela). Typically P˙ ∼ 1015 s s−1. The implication from pulsar models is that pulsars are probably younger than a characteristic time T ≡ P/P˙ ∼ 107 yr. Shorter period pulsars tend to have larger slowdown rates and shorter character- istic times. For the Crab T ∼ 2486 yr. However, the pulsar PSR 1913+16 has T ∼ 2.17 × 108 yr. The most dramatic period irregularities in pulsars are the sudden spinups, or glitches, observed in the Crab (|∆P/P | = 10−8) and Vela (|∆P/P | = 10−6) pulsar periods. In each of the four giant Vela glitches observed so far (see Fig.12.2), the decrease in period was accompanied by an increase in period derivatives (∆P/˙ P˙ ∼ 0.01), which decayed away in about 50 days. (c) Distribution and Association with Supernova Remnants We know that pulsars originate in our own Galaxy because (i) they are concen- trated in the Galactic Plane (Fig.12.3) and (ii) they show dispersion characteristic of signal propagation over Galactic Distances. 288 CHAPTER 12. PULSARS

Figure 12.3: Left: Galactic distribution of pulsars. In the adopted coordinates 0◦ latitude corresponds to the galactic plane, while 0◦ longitude, 0◦ latitude corresponds to the direction of the Galactic Center. Pulsar share the distribution of young luminous OB stars and Supernova Remnants. Right: Distribution of pulsars in Galactic latitude. Latitude 0◦ corresponds to the Galactic plane.

The angular distribution of pulsars in the Galactic Plane is similar to that of supernova remnants and their presumed progenitors, OB stars, though the mean scale height above the plane is somewhat larger ∼ 300 pc. This is consistent with the hypothesis that pulsars arise in supernova explosions, possibly acquiring a relatively high velocity. More impressive is the association of several pulsars with known supernova remnants. Classical examples are PSR 0531+21 and PSR 0833-45 associated with the Crab and Vela supernova remnants, respectively.

12.4 Magnetic Dipole Model for Pulsars

The simplest model of a Pulsar that can account for many of the observed proper- ties is the oblique rotator magnetic dipole model which explicitly demon- strates how pulsar emission is derived from the kinetic energy of a rotating neutron star. Rotating magnetic dipole. Consider a neutron star of mass M and radius R, rotating with angular velocity Ω, and possessing a magnetic dipole moment m oriented at angle α to the rotation axis. The star is in vacuum (no matter around). 12.4. MAGNETIC DIPOLE MODEL FOR PULSARS 289

Figure 12.4: Oblique rotating magnetic dipole

The rotation is assumed to be sufficiently slow that non-spherical distortions due to rotation can be neglected. The geometry is shown in Fig.12.4. Independently of the internal field geometry, a pure magnetic dipole field at the magnetic pole of the star, Bp or simply B, is related to m by BR3 |m| = (12.6) 2 where R is the stellar radius. Such a configuration, when observed from infinity, has a time-varying dipole moment, and so radiates energy at the rate 2 E˙ = − |m¨ |2 (12.7) 3c2 Decompose now the magnetic moment along the three directions given by the 0 unitary vectors ek, e⊥ and e ⊥ as indicated in Fig.12.4, where ek is parallel to the 0 rotation axis, and e⊥ and e ⊥ are fixed mutually orthogonal and perpendicular to ek, we find 1 m = BR3(e cosα + e sin α cos Ωt + e0 sin α sin Ωt) (12.8) 2 k ⊥ ⊥ and 290 CHAPTER 12. PULSARS

B2R6Ω4 sin2 α E˙ = − (12.9) 6c3 Note that radiation is emitted at frequency Ω. Equation (12.9) leads to several important consequences. First the energy carried out by radiation originates from the rotational energy of the star 1 E = IΩ2 (12.10) 2 where I is the moment of inertia. Thus

E˙ = IΩΩ˙ (12.11)

If E˙ < 0, Ω˙ < 0 that is the star slows down. If we define a characteristic time scale at the present epoch by

³Ω´ 6Ic3 T = − = (12.12) 2 6 2 2 Ω˙ B R sin αΩ0 we can integrate eqn.(12.11) and (12.12) to obtain

³ 2 ´ 2Ωi t −1/2 Ω = Ωi 1 + 2 (12.13) Ω0 T where Ωi is the initial angular velocity, and Ω0 is the present-day angular velocity.

Setting Ω = Ω0 in eqn.(12.13), gives the present age of the pulsar

³ 2 ´ T Ω0 T t = 1 − 2 ' (12.14) 2 Ωi 2 for Ω0 << Ωi. For the Crab, in 1972 it was estimated T ∼ 2486 yr and t = 1243 yr, which is in good agreement with the historical age 1972−1054 (recorded time of supernova explosion) = 918 yr. It is worth noticing that this estimate (first made by Gunn & Ostriker 1969) does not depend on detailed properties of the underlying neutron star. It depends only on the general behavior of Ω(t) due to magnetic dipole radiation. Energetics. The magnetic dipole model provides also a quantitative account of the energetic of pulsars (the Crab as a prototype). Following 12.4. MAGNETIC DIPOLE MODEL FOR PULSARS 291

Gunn & Ostriker (1969), let us assume that the Crab is a spherical neutron star 45 2 with mass of 1.4 M¯, radius of 12 km, moment of inertia of I = 1.4 × 10 g cm . Then eqn.(12.10) and (12.11) give

E ' 2.5 × 1045 erg E˙ ∼ 6.4 × 1038 erg/s (12.15)

Note that eqn.(12.10) and eqn.(12.11) do not depend on the detailed mechanism of energy loss, but only on the assumption that the pulsar is a neutron star with rotation as the energy source. It is remarkable that E˙ is comparable with the energy requirements of the Crab nebula, which are approximately 5×1038 erg/s. Note also that E˙ of eqn.(12.15) is much greater than the observed ratio in the radio pulse, which is ∼ 1031 erg/s. Using E˙ for the Crab from eqn.(12.15) and adopting the magnetic dipole model, eqn.(12.9) gives

B = 5.2 × 1012 Gauss (sin α = 1) (12.16)

Such value arises naturally from the collapse of a main sequence star with a typical ”frozen in” surface magnetic field of 100 Gauss. The decrease in radius by a factor of ∼ 105 leads to an increase in B by a factor of ∼ 1010. This value is also indicated by observational data on X-ray binary pulsars, for instance

B = 4 − 6 × 1012 Gauss in Her x − 1 = 2 × 1012 Gauss 4U 0115 − 63 (12.17)

These very intense magnetic fields are presumably produced at the time of pulsar formation. An important question to be asked is: are they stable? can they decay away? The decay time of a magnetic field is roughly σL2 t ∼ (12.18) d c2 where L is characteristic length (say the radius L = R), σ is the conductivity for which we may have an estimate from dimensional arguments m c3 σ ∼ e ∼ 1023 s−1 (12.19) e2 292 CHAPTER 12. PULSARS

6 thus for a typical neutron star td ∼ 10 yr >> age of the Crab. The decay time scale is fairly long, and the decay of the pulsar magnetic field is small during the pulsar lifetime. Rotation versus Gravitational Waves. Our previous estimate of the Crab pulsar age based on rotation was reasonable but not particularly satisfactory. We introduce here the alternative mechanism of gravi- tational wave emission and argue that a combination of the two may lead to fully satisfactory results. Suppone that a Neutron Star slightly deviates by rotation from perfect spherical symmetry and takes the shape of an ellipsoid with moment of inertia I and ellipticity ² difference in equatorial radii a − b ² = = (12.20) mean equatorial radius (a + b)/2 The power emitted by gravitational radiation (the lowest order is quadrupole) by a rotating ellipsoidal Neutron Star is given by

32 G E˙ = − I2²2Ω6 (12.21) 5 c5 ³ I ´2³ P ´−6³ ² ´2 = 1.4 × 1038 erg/s 1.45 × 1045g cm2 0.0331 s 3 × 10−4

According to eqn.(12.21) even small ellipticities can generate the emis- sion required to account for the Crab deceleration, eqn.(12.15). Interior anisotropic magnetic fields of ∼ 1015 Gauss, corresponding to interior main se- quence fields of ∼ 105 Gauss can generate such small ellipticities. The deceleration law for ² constant is

˙ ˙ 6 EGW = IΩΩ ∝ Ω (12.22)

Defining ³Ω´ TGW = − (12.23) Ω˙ 0 we can integrate eqn.(12.22) to obtain

³ 4 ´ 4Ωi t −1/4 Ω = Ωi 1 + 4 (12.24) Ω0 TGW 12.4. MAGNETIC DIPOLE MODEL FOR PULSARS 293

where Ωi is the initial angular velocity at t = 0 and Ω0 is the present value.

Setting Ω = Ω0 we obtain ³ 4 ´ TGW Ω0 t = 1 − 4 4 Ωi 2486 < = 621 yr (12.25) 4 The gravitational radiation cannot alone be responsible for the Crab slowdown. However a combination of gravitational and dipole radia- tion can be found that yields both the correct age and the observed slowdown. In the combined model, the time derivative of the total energy is

˙ ˙ ˙ ˙ 4 6 E = EROT + EGW ≡ IΩΩ = −βΩ − γΩ (12.26) where β and γ are defined by eqn.(12.9) and eqn.(12.21). The

¯ 2 4 1 Ω˙ ¯ βΩ + γΩ = ¯ = 0 0 (12.27) T Ω 0 I Integrating eqn.(12.26) we obtain the age t

³ λ + µ´ 2t (1 + λ) 1 − µ + λ log = (12.28) λ + 1 T where 2 2 γΩ0 Ω0 λ = µ = 2 (12.29) β Ωi The total energy radiated electromagnetically is Z βΩ4 ∆Eem = dΩ (12.30) Ω˙ 4 −1 Assuming Ωi = 10 s one obtains

50 ∆Eem = 5.9 × 10 erg (12.31) while 1 ∆E = (Ω2 − Ω2) − ∆E GW 2 i 0 em 1 ' IΩ2 = 7 × 1052 erg (12.32) 2 i 294 CHAPTER 12. PULSARS

The power emitted at the present time is

˙ 38 ˙ 38 EGW = 1.4 × 10 erg/s Eem = 5.1 × 10 erg/s (12.33)

4 −1 while at the beginning (always for Ωi = 10 s )

˙ 48 ˙ 45 EGW = 2.9 × 10 erg/s Eem = 3.9 × 10 erg/s (12.34)

The gravitational radiation dominates the energy loss in early stages of pulsar evolution (roughly during the first 130 years).

12.5 Braking Index

For any power law deceleration model, such as the magnetic dipole, the slow-down of the angular velocity can be expressed as

Ω˙ = −(constant)Ωn (12.35) where the parameter n is called the braking index. For the magnetic dipole n = 3. In general the index n can be defined as

ΩΩ¨ n ≡ − (12.36) Ω˙ 2 In principle, the braking index can be estimated from the experimental data (pulsar frequency and derivatives). In case of the Crab the measured braking index is n = 2.515 ± 0.005 (12.37)

A number of factors can be responsible for the deviation from the expected value n = 3. They are not discussed here. If the gravitational radiation alone were the braking mechanism, then n = 5. The combined mechanism (magnetic dipole + gravitational waves) would predict 3 + 5λ n = ' 3.4 now (12.38) 1 + λ 12.6. NON-VACUUM MODELS: THE ALIGNED ROTATOR 295

Solution of the braking equation. The equation

ΩΩ¨ = −nΩ˙ 2 (12.39) has solution ³ t ´−1/(n−1) ³ t ´−γ Ω(t) = Ω 1 + ≡ Ω 1 + (12.40) 0 τ 0 τ where γ = 1/(n − 1) and Ω0 and τ are constants. From eqn.(12.9) it follows that B2(t) ∝ ΩΩ˙ −3 varies as

³ t ´− 1 ( n−2 ) ³ t ´γ−(1/2) B(t) = B 1 + 2 n−1 ≡ B 1 + (12.41) 0 τ 0 τ

When n > 3 (so that γ < (1/2)), such a model could represent a pulsar whose magnetic field is decaying in time because of some process. If, for example, n = 4, then

³ ´ ³ ´ ³ ´ 3 t −1/3 t −1/6 2 2I c 1 Ω = Ω0 1 + B = B0 1 + B0 ≡ 6 2 2 (12.42) τ τ R Ω0τ sin α

12 Assuming that the pulsar is born with period P = 1 ms, magnetic field B0 ' 10 Gauss, moment of inertia I ' 1045 g/cm3, and radius R = 10 km, then we can estimate τ to be τ ' (4/ sin2 α) yr. If we take α ' π/4 as a typical value for the inclination angle (sin2α = 0.5), the decay time scale is τ ∼ 8 yr. This result does not contradicts the one we have estimated above for the decay-time-scale of the magnetic field, which was derived from a different reasoning and in the context of a pure magnetic dipole model (n = 3).

12.6 Non-vacuum models: the aligned rotator

The model for the pulsar we have just presented is based on several assumptions and approximations: (1) We have assumed that the region surrounding the pulsar is a vac- uum and that radiation can escape freely. (2) We have ignored the effect of General Relativity near the neutron star. 296 CHAPTER 12. PULSARS

(3) We have not derived the electromagnetic fields inside and outside the neutron star nor we have secured their matching at the surface. We can obtain a more realistic model for the pulsar and its magnetosphere by treating the Neutron Star as a conducting, rotating sphere of mass M, radius R angular velocity Ω and with a magnetic field B inside. If we take the outside region to be a vacuum, then we are required to solve the Maxwell equation both inside a spherical rotating conductor and on the outside and to match the solutions at the surface. Because the surface gravity of a Neutron Star is g = (GM/R2) ∼ 1.9 × 1014 cm s−2, the scale height of the atmosphere (approximated to pure hy- drogen) at a temperature T is

kT ³ T ´ H ' ' 0.4 6 cm (12.43) mH g 10 This is much smaller than any other relevant length scale in the prob- lem and hence the surface between the conductor and the vacuum can be thought as a sharp discontinuity. The simplest internal magnetic-field configuration corresponds to (1) a uni- formly magnetized sphere with B = B0ez or (2) a point dipole with its magnetic moment along the z-axis. Because the rotation is along z there will be no radi- ation field coming from time-varying multi-pole moments. We will develop case (1) in detail and quote the results for case (2). The geometry of the configuration we are going to study is illustrated in Fig.12.5 The velocity at any point r in the conductor, corresponding to uniform rota- tion with angular velocity Ω is given by

v = Ω × r = (Ωr sin θ)eφ (12.44) where we are using the standard polar coordinates (r, θ, φ). When the neutron star rotates, a charge q in the interior feels the Lorentz force (q/c)v × B and moves until an electric field E is generated, which can compensate for the mag- 12.6. NON-VACUUM MODELS: THE ALIGNED ROTATOR 297

Figure 12.5: Rotating magnetic dipole whose magnetic axis is aligned to the rotational axis. netic force. Thus vanishing of the net force implies

v × B Ω × r ΩB r sin θ E = − = − × B = − 0 (sin θe + cos θe ) (12.45) c c c r θ

This electric field satisfies the condition ∇ × E = 0 and hence can be expressed as E = −∇φin(r, θ). Integrating l ≡ r sin θ from the origin to the surface we can determine φin(r, θ) as

³ΩB ´ ³ΩB ´ φ = 0 r2 sin2 θ + costant = − 0 [P (cos θ) − 1] + φ (12.46) in 2c 2c 2 0 where P2(cos θ) is the Legendre polynomial and φ0 is a constant. Because E · B = 0 inside the star, the magnetic field lines are equi-potentials labelled by the voltage, which in turn is determined by the location at which the particular field line emerges on the star’s surface. Outside the star, the electric

field is given by E = −∇φout(r, θ), where φout(r, θ) satisfies the Laplace equation 2 ∇ φout = 0 Taking the general solution of the form

∞ X al φout = l+1 Pl(cos θ) (12.47) l=1 r 298 CHAPTER 12. PULSARS and matching the potential at surface of the star r = R, we see that the outside solution must be ΩB R5 φ = 0 (3 cos2 θ − 1) (12.48) out 6cr3 2 with the constant φ0 in eqn.(12.46) set to φ0 = −(ΩB0R /3). The corresponding electric field on the outside is given by ΩB R5 ΩB R5 E(r, θ) = − 0 (3 cos2 θ − 1)e − 0 sin θ cos θe (12.49) 2cr4 r cr4 θ

3 The magnetic field outside is of course that of a dipole with moment m = B0R B R3 cos θ B R3 sin θ B(r, θ) = 0 e + 0 e (12.50) r3 r 2r3 θ These solutions ensure that the radial component of the magnetic field

Br and the tangential component of the electric field Eθ are continuous at the surface of the star. The equation for the field lines corresponding to this magnetic dipole will be of interest later on. These lines r(θ) satisfy the equation (Br/Bθ) = (dr/rdθ) = (2 cos θ/ sin θ), which can be integrated to give

r = K sin2 θ (12.51) where K is a constant identifying the field line. Before proceeding further, it is important to determine the charge and the current distribution that give rise to these fields by using the equations ³∇ · E´ ³ c ´ ρ = and J = ∇ × B (12.52) 4π 4π Inside the star, these relations give ³∇ · E´ ³ΩB ´ ρ = = 0 and J = 0 interior (12.53) in 4π 2πc in On the outside, both charge and current vanish by construction ³∇ · E´ ρ = = 0 and J = 0 exterior (12.54) out 4π out To determine the charge and the current on the surface, we have to calculate the discontinuities in the electric and magnetic fields. The charge density on 12.6. NON-VACUUM MODELS: THE ALIGNED ROTATOR 299

the surface is given by ρs = [Er]/4π, where [Er] denotes the discontinuity in the radial component of the electric field across the surface. Using the gradients in the potential φ we have

1 h ∂φout ∂φin i ΩB R ρ = − r + r = 0 [2 − 5P (cos θ)] (12.55) s 4π ∂r ∂r 12π 2

Similarly for the surface current we have

c cB J = [Bout − Bin] = 0 sin θ e (12.56) s 4π 8π θ

These results show that an aligned rotator requires (1) a charge density on the inside as well as on the surface; (2) a current flowing on the surface. Light cylinder. Given these conditions, we should also check whether the conditions for the validity of the Magneto-Hydro-Dynamics (MHD) approximation are satisfied. It can be directly verified that the mean charge density and the mean current density are

ΩB cB ρ ' 0 and J ' 0 (12.57) c R

To satisfy the condition ρc << J we need ΩR << c. This condition also ensures that v << c and E << (ΩR/c)B0 << B0, thereby allowing MHD approximation to be used. For typical values of a neutron star, (ΩR/c) ' 10−2 therefore MHD is secured. The same analysis, however, shows that very soon difficulties will arise at distances of the order of r ≥ RL = c/Ω. The radius RL defines a cylinder aligned with the magnetic dipole, called the ”light cylinder” (see below). We shall show now that for realistic values of neutron star param- eters, the region outside the star is unlikely to a vacuum as assumed above. This is due to the extremely high values of the surface electric field and voltages near the surface, In fact the surface potential and electric field are

ΩB R2 ³ Ω/2π ´³ B ´³ R ´2 φ ' 0 ' 3 × 1016 0 V (12.58) s 2c 30 s−1 1012G 10 km 300 CHAPTER 12. PULSARS

ΩR ³ B ´³Ω/2π ´³ R ´ E ' B ' 2 × 108 0 esu (12.59) c 0 1012G 1s−1 10 km The electric force that is due to this field on a charged particle is very much stronger than the gravitational force (by a factor ∼ 108 for a proton), and hence charged particles are stripped from the neutron star surface to create a magneto- sphere around the star. Whether this actually occurs depends on detail of the solid-state physics at the surface. Assuming for the moment that this happens, we can ask how the structure of the magnetosphere and the nature of the problem change because of this process. The charged particles stripped from the surface will spiral along the magnetic field lines and will drift along it. The spiralling motion cause the charge to radiate away the kinetic energy corresponding to transverse motion so that the main residual motion of the charged particle will be along the magnetic-field line. Furthermore, moving along the magnetic-field line will be accelerated by the electric field component Ek = (E · B)/|B| parallel to the direction of the magnetic field. Near the surface Ek varies as E · B ΩR E = = − B cos3 θ (12.60) k |B| c 0 The global motion of the charged particle will depend on whether the magnetic field lines to which they are attached closes back before reaching the light cylinder with radius RL = c/Ω or not (see Fig. 12.6). For a dipole, the field lines are described by the (sin2 θ/r) = const. Field lines starting out with within an angular region θ > θp will loop back before reaching the light cylinder. The critical line, starting at (r = R, θ = θp) should have 2 r = RL at θ = π/2. The constancy of (sin θ/r) = const implies that sin θp = 1/2 (R/RL) . The corresponding radius of the ”polar cup” region is given by ³ ´ ³ ´ ³ ´ R 1/2 4 R 3/2 P −1/2 RP ' R sin θP = R = 1.4 × 10 6 cm (12.61) RL 10 cm 1 s Charged particles are stripped off the polar cap region, and redis- tribute around the star, forming a co-rotating magneto-sphere. Likely 12.6. NON-VACUUM MODELS: THE ALIGNED ROTATOR 301

Figure 12.6: Light cylinder the particles will redistribute in such a way that no net electromagnetic force acts on them: that is ³v ´ E + × B = 0 (12.62) c where v = Ω × r. From the resulting electric field E we can derive the charge density from the equation ∇ · E = 4πρ. This gives

4πcρ = ∇ · [B × (Ω × r)] = −2Ω · B (12.63)

This equation has an important consequence: that is the spatial segregation of the charge:

(i) regions with Ω · B = ΩBz > 0 will have negative charges;

(ii) regions with Ω · B = ΩBz < 0 will have positive charges. The number density of charges is approximately

³ P ´−1 n = 7 × 10−2B cm−3 (12.64) e z 1 s

The regions are shown in Fig.12.7. The charged particles stripped from the polar cap (θ < θp) move along the field-lines and hit the light cylinder where relativist effects become important and the ordinary MHD is no longer applicable

(the plasma cannot be kept co-rotating with the star). Near r = RL the structure 302 CHAPTER 12. PULSARS

Figure 12.7: Spatial distribution of charges in the magneto-sphere The pulsar has parallel magnetic and rotation axes. Particles that are attached to closed mag- netic field lines co-rotate with the star and form the co-rotating magnetosphere. The magnetic field lines that pass through the light cylinder (where the veloc- ity of co-rotation equals the velocity of light) are open and are deflected back to form a thoroidal field component. Charged particles stream out along these lines. The critical field line is at the same electric potential as the exterior interstellar medium. This line divides regions of positive and negative current flow from the star and the plus and minus indicate the charge of particular regions of space. The diagonal dashed line is the locus of Bz = 0, where the space charge changes sign. of the field lines changes, the field line are swept back near the light cylinder and stream to infinity outside the light cylinder. They are called open lines. Some details are shown in Fig.12.8. Particles will flow away from the pulsar along the open field lines which originate from a region near the polar cap. The potential difference ∆φ between the centre ad the edge of the polar cap can be estimated from our solution

3 ³ ´³ ´ ΩB0R R 12 B P −2 ∆φ = = 6 × 10 12 V (12.65) 2c RL 10 G 1 s

The potential difference along the magnetic field lines over a distance R, given by ∆φ ' (E·B/B)R, will also be of the same order as that given above. In other words the particles that flow along open lines, will be accelerated by a voltage of this order somewhere along their path, thus acquiring energies of the order 12 2 6 × 10 eV (B12/P ). The electrons become highly relativistic with a gamma 7 2 factor of γ ' 10 (B12/P ) (where P is in seconds). 12.6. NON-VACUUM MODELS: THE ALIGNED ROTATOR 303

Figure 12.8: Twisting of magnetic field-lines near the light cylinder.

Do energy losses really occur? Because of the axisymmetric configuration, we may believe that there cannot be any energy-loss mechanism. This is not true because of the existence of the light-cylinder and open field lines. There is a flow of electromagnetic energy through the light cylinder (and a consequent torque on the pulsar) that can be estimated as follows. The magnetic field intensity at the light cylinder is approximately

3 B0R 3 2RL Because of the torsion of the field lines near the light cylinder we would expect

B R3 B ' B ' √0 r φ 3 2 2RL The moment of the rφ component of the magnetic stress tensor

B B T = r φ r,φ 4π will lead to the torque

dΩ B B K I = r φ (4πR2 )R ' − (B R3)2Ω3 (12.66) dt 4π L L 8c3 0 where K is a numerical factor of the order of unity. Apart from a numerical factor, this will give the same spin-down rate for the pulsar as for the rotating dipole discussed above. 304 CHAPTER 12. PULSARS

Figure 12.9: Schematic view of a rotator whose magnetic dipole is not aligned with the rotation axis. It shows the magnetosphere of a rotating neutron star. The rotating magnetic field lines define a light cylinder with radius RL = c/Ω ' 50000(P/sec) km. At the light cylinder, the dipolar magnetosphere gets distorted. Field lines from the polar caps cross the light cylinder (open field lines), plasma generated near the polar caps can escape along these magnetic field lines. The polar caps act as a kind of discharge tubes, where strong parallel electric fields, generated by the rapid rotation, extract charges from the surface of the neutron star and accelerate them to high energies 12.7. PAIR CREATION IN THE MAGNETO-SPHERE 305

The final model: oblique rotator. Finally, we may drop the hypothesis of a rotator aligned with the magnetic dipole and move to the more general one in which the rotation axis and magnetic dipole axis are not aligned. This is the present-day credited model. The results we have obtained do not change significantly apart from a more complicated formalism. The typical situation is shown in Fig. 12.9.

12.7 Pair creation in the magneto-sphere

Even if copious extraction of charged particles does not occurs, a magneto-sphere made of electron-positron pairs can still arise around the pulsar for the following mechanism. Any electron in the magnetosphere, spiralling along a magnetic

field line, is accelerated by the parallel component of the electric field, Ek =

(E · B/|B|). Near the surface, Ek is

ΩR E = − B cos3θ k c 0 whose value is so strong that electrons can be accelerated to very high energies, 7 2 γ ' 10 (B12/P ). These electrons spiral around the lines with drift motion along the direction of the lines. The kinetic energy of the spiralling motion, trans- verse to the local direction of the field lines, is quickly lost through synchrotron radiation. The lifetime for such synchrotron radiation is

³ ν ´−1/2 c τ ' 1.5 × 10−11B−3/2 c s with ν = γ3 (12.67) 12 1 GHz c 2π% where % is the curvature radius of the orbit.

Parallel to the magnetic field, however, the Ek component accelerate the par- ticles. Since the magnetic field lines are curved, the forward motion is also along a curved track with curvature radius % = (rc/Ω)1/2. Such a relativistic particle, with charge e, mass m, and energy E = γmc2 moving in an orbit with curvature radius %, radiates a synchrotron energy with spectrum

e2 c ³ ν ´1/3 I(ν) ' γ for (ν ≤ νc) (12.68) 2πc % νc 306 CHAPTER 12. PULSARS

Figure 12.10: Curvature radiation and pair cascading with

c γ3 ³cΩ´1/2 ³ γ ´3³ % ´−1 ν = γ3 = ' 1023 Hz c 2π% 2π r 107 108 cm ³ γ ´3³ % ´−1 ' 108 (eV/h) (12.69) 107 108 cm The primary charges around the polar region will radiate γ-rays tangentially, and these γ-rays, if they are energetic enough, will give rise to electron-positron pairs. These secondary charges will be accelerated in opposite directions and will, in turn, give rise to more electron-positron pairs, thereby establishing a cascade (see Fig.12.10). A photon in vacuum cannot convert itself into an electron-positron pair with- out violating the conservation of energy and momentum. The photon, however, can convert itself into a virtual pair of particles of mass m and charge q for a short period of time ∆t ' (¯h/mc2), after which the charged particles recombine. The situation is different in presence of a strong electric field, as the electric field can accelerate the virtual particles sufficiently to allow the virtual pair to become a real one. This can occur if the work done by the field in moving the charged particles over a distanceh/mc ¯ is comparable with the rest mass of the particles 2 2 mc . This requires an electric field with intensity Ec such that qEc(¯h/mc) = mc 2 3 or Ec = (m c /qh¯). Because magnetic fields cannot work on charged particles, such an effect normally does not occur in a pure magnetic field. In fact this effect requires that the Lorentz invariant condition (E2 − B2) > 0 be satisfied in any 12.8. REMARK ON THE PULSAR EMISSION MECHANISMS 307 reference frame. Although a pure magnetic field or a pure photon cannot lead to the production of a pair, the existence of both together does. This arises because the total electric field now is due to that of the photon (treated as electromag- netic radiation), whereas the total magnetic field is the sum of the pulsar and 2 2 photon magnetic fields. It is possible that (Etot − Btot) > 0 thereby leading to pair production. The energyhω ¯ of the initial photon goes into producing a pair of particles with γ factor γ ∼ (¯hω/2mc2). Photon mean free path. The mean free path l = κ−1 for photons with energyhω ¯ > 2mc2 moving at an angle θ to the magnetic field lines can be estimated from theory. We find ³ ´³ ´³ ´ 4.4 h¯ Bq l = 2 exp(3/3χ) (12.70) e /hc¯ mec B sin θ where

2 3 hω¯ B sin θ mec 13 χ = 2 Bq = ' 4.4 × 10 G (12.71) mec Bq eh¯ (the expressions are valid only for χ << 1). For B ' 1012 G, we have l−1 = κ = 2 × 106 cm−1. The exponential shows that a small change in χ leads to large variations of l. The magnetosphere behaves as a opaque solid with attenuation varying from zero to unity over an extremely short distance. We can estimate the location r of this surface by simply ignoring the spatial variations of B and θ and setting κr = 1. This equation defines a thin spherical layer surrounding the pulsar. The photons with energy γ inside this layer are degraded to (e−, e+) pairs. It can be demonstrated that if this mechanism is at work, a correlation between the intensity of the magnetic field (B) and period of the pulsar (P ) can be predicted and compared with observational data. The agreement is satisfactory.

12.8 Remark on the pulsar emission mechanisms

From all we have been saying so far, the most credited model for pulsars is the ”polar cap” scheme in which radiation and particles are emitted in a cone aligned with the axis of the magnetic dipole (one cone per pole) and located within the 308 CHAPTER 12. PULSARS co-rotating magnetosphere. The magnetic dipole may have different inclinations with respect to the rotation axis. There exists another model named ”light cylinder emission” in which the axis of the emission cone is tangent to the ”light cylinder” and perpendicular to the rotation axis. In this model the emission occurs at RL. As this model has not received worldwide acceptance by the scientific community, it is left aside here. In any case the data to our disposal show that the emission mechanism must satisfy some fundamental requirements. They are (i) The radiation must be emitted in a relatively narrow beam, fixed in orien- tation with respect to the neutron star. The beam must be ≤ 10◦ in longitude as seen by a distant observer and this width must remain constant for long periods of time. Moreover the beam shape and longitude must remain stable for many rotation periods. (ii) The radiation mechanism must produce rather broad-band radiation at both radio and optical frequencies. The radio pulses have bandwidths ≥ MHz. (iii) The radiation process must generate the observed luminosities and bright- ness temperatures in the radio, optical, and X-ray bands. (iv) At radio wavelengths, the emission should show strong linear polarization, approximately independent of frequency, and stable for long time intervals. The ”brightness temperature” of an emitting region is defined by

Iν ≡ Bν(Tb) (12.72)

−1 −2 −1 −1 where Iν is the specific intensity (erg s cm Hz sr ) and Bν is the Planck function. For hν << kTb we have the Rayleigh-Jeans approximation 2πν2 I ' kT (12.73) ν c2 b Now observed pulsar radio luminosities, assuming conical beams and rea- sonable distances, range from 1025 − 1028 erg s−1. If we assume that the emitting area is ∼ (ct)2 ' 1015 cm3 (with t ∼ 10−3 s) we derive

4 7 2 Iν ' 10 − 10 erg/s/cm /Hz/sr 12.9. SUPERFLUIDITY: GLITCHES AND STAR-QUAKES 309

23 26 Tb ' 10 − 10 K

17 22 kTb ' 10 − 10 eV (12.74)

These results pose serious questions and constraints. For incoherent emission, thermodynamics implies kTb ≤ Ep where Ep is the particle energy. This is simply a restatement of the fact that a black-body is the most efficient radiator. But the particle energies required by eqn.(12.74) are absurdly high. Even if such high energy particles were available, they would radiate most of their energy at very high frequencies and not in the radio band. We conclude that in the radio band a coherent mechanism is required, in which the total intensity is ∼ N 2 times the intensity radiated by a single particle, where N is the number of particles emitting coherently. Such coherence is not required for the observed X-ray or optical pulsed emission.

12.9 Superfluidity: glitches and star-quakes

It is remarkable that timing data on pulsars following their sudden spin-ups may provide evidence for superfluidity in neutron stars. These spin-ups are called glitches. In the following most of data refer to Crab and Vela pulsars, but the discussion is quite general. When Vela showed the first sudden change in Ω,˙ ∆Ω˙ /Ω˙ ' 10−3, together with a small increase in Ω, ∆Ω/Ω ' 2 × 10−6, Bayam and collaborators (1969) soon proposed a simple ”two components” model for neutron stars to explain this phenomenon. The model we are going to present is largely phenomenological, but close to reality. A neutron star is supposed to consists of two components. The first one is the crust and charged particles of moment of inertia Ic, weakly coupled to the second one made of superfluid neutrons, of moment of inertia

In. The charged component is assumed to rotate at the observed angular velocity Ω(t) as the charged particles are assumed to be strongly coupled to the magnetic field. The rotation of the neutron superfluid is assumed to be quasi-uniform with average angular velocity Ωn(t). The coupling between the two components is 310 CHAPTER 12. PULSARS

described by a a single parameter τc, i.e. the relaxation time for frictional dissipation. In the model, it is imagined that the speed-up is triggered by a ”star- quake” occurring in the crust. The details of the star-quake are unimportant here. What is relevant are the assumptions (i) The crust spin-up is rapidly communicated to the charged particles in the interior by the strong magnetic field (time scale of about 100 s). (ii) The response of the neutron superfluid to the speed-up is much slower (about one year) due to a much weaker frictional coupling between the normal and superfluid components. The spin-up of the crust is communicated to the charged particles in the interior by the magnetic field which threads them. Each distortion of the magnetic field generates ”magnetic sound waves” or Alfv´enwaves which travel with a speed

³P ´1/2 B2 v ∼ B P ∼ (12.75) A ρ B 8π with time scale of about 50 s. The glitch-free interaction between the two components after a star-quake is governed by two equations

˙ Ic(Ω − Ωn) IcΩ = −α − (12.76) τc ˙ (Ω − Ωn) InΩn = Ic (12.77) τc where α is the external braking torque on the crust due to magnetic dipole radiation-reaction forces. Taking α and τc to be constant over the time scales of interest, the above equations can be integrated to give

α I Ω = − t + n Ω e−t/τ + Ω (12.78) I I 1 2 −t/τ ατ Ωn = Ω − Ω1e + (12.79) Ic where τ I I = I + I τ = c n (12.80) c n I 12.9. SUPERFLUIDITY: GLITCHES AND STAR-QUAKES 311

and Ω1 and Ω2 are two integration constants that depend on the initial conditions. The steady-state solution (t/τ → ∞) is

ατ α In In τc Ωn − Ω = = τc = Ω (12.81) Ic I Ic Ic T where 1 Ω˙ α = − = (12.82) T Ω IΩ is the characteristic age we have already define in eqn.(12.12). Observational data show that τ goes from months for Vela to weeks for Crab. Thus, crudely assuming In ∼ Ic would predict

(Ω − Ω) n ∼ 10−5 Ω for Vela and Crab. Equation (12.78) can be used to fit the post-glitch timing data of pulsars (Crab and Vela). Suppose that a glitch occurs instantaneously in the observed angular velocity at time t = 0: Ω(t) → Ω(t) + ∆Ω0. Replace Ω1 and Ω2 in eqn.(12.78) and define the parameter Q in such a way that eqn.(12.78) takes the form

−t/τ Ω(t) = Ω0(t) + ∆Ω0[Qe + 1 − Q] (12.83)

The parameter Q describes the degree to which the angular velocity relaxes back to its extrapolated values: if Q = 1, Ω(t) → Ω0(t) at t → ∞. Here Ω0(t) = Ω0−αt is the pulsar angular velocities in absence of glitches. The behaviour of eqn.(12.83) is shown in Fig.12.11. Can the data provide physical insight on neutron stars assuming the model to be correct? Consider the parameter Q and τ and define

∆Ω(t) = Ω(t) − Ω0(t) (12.84) we find from eqn.(12.83) ∆Ω(˙ t = 0) Q = − τ (12.85) ∆Ω0 312 CHAPTER 12. PULSARS

Figure 12.11: Time dependence of the angular velocity before and after a glitch. and ∆Ω(˙ t = 0) τ = (12.86) ∆Ω(¨ t = 0) Thus Q and τ can be derived from post-glitch data. Now the parameter Q can be related to the momenta of inertia of the various components. To this aim, assume that the star-quake responsible for the glitch gives rise to changes ∆Ic, ∆Ω, ∆In, ∆Ωn, and so on, but leaves τc and α constant. Then on the star-quake time scale, the angular momentum of each component is separately conserved, implying

∆I ∆Ω c = − (12.87) Ic Ω ∆I ∆Ω n = − n (12.88) In Ωn

Differentiating eqn.(12.76) gives

˙ α∆Ic ∆Ω − ∆Ωn ∆Ω = 2 − (12.89) Ic τc But eqn.(12.87) and eqn.(12.82) give

α∆Ic α∆Ω I ∆Ω 2 = − = − (12.90) Ic IcΩ Ic T Typically I T c >> 1 (12.91) I τc 12.9. SUPERFLUIDITY: GLITCHES AND STAR-QUAKES 313 so the first term in eqn.(12.89) can be ignored in comparison with the second. Therefore, eqn.(12.89) becomes, using eqn.(12.82),

∆Ω˙ ∆Ω T ³ Ω∆Ω ´ = 1 − n (12.92) Ω˙ Ω τc ∆Ω

Now eqn.(12.92) and (12.82) give for eqn.(12.85) (with ∆Ω = ∆Ω0) τ ³ ∆Ω ´ Q = 1 − n (12.93) τc ∆Ω Recalling eqn. (12.80), (12.87) and (12.88) we get I ³ ∆I /I Ω ´ Q = n 1 − n n n (12.94) I ∆Ic/Ic Ω

Now Ωn/Ω << 1 so typically we expect I Q = n (12.95) I

So measuring Q one can infer the ratio In/I. Even if the two component model is correct, the ratio In/I depend on the equation of state and the mass of the neutron star. Some examples of ratio In/I are given in Table 12.1 for different equation of sate and mass of the neutron star. Typically, stiff equations of state predict smaller masses and lower In/I. Origin of the pulsar glitches. Numerous models have been presented to explain the origin of glitches. As a prototype of the interplay between micro- physical properties and macrophysical observable properties we present here the star-quake model. Here a sudden cracking of the neutron star crust decreases the moment of inertia and hence increases Ω. This model provides a reasonable explanation for the Crab spin-ups, but does not, without modifications, apply to Vela. The star-quake mechanism is based on the key idea that the nuclei in the crust of the neutron star (ρ ≤ 2 − 2.4 × 1014g/cm3) form a solid, Coulomb lattice. The crust is oblate in shape because of rotation. As the star slows down, centrifugal forces on the crust diminish and a stress arises to drive the crust to a less oblate equilibrium shape. However, the rigidity of the crust resists this stress and the 314 CHAPTER 12. PULSARS

Table 12.1: Properties of neutron stars sensitive to the hadron equation of state and mass. Hardness of the equation of state and value of the neutron star mass increase from top to bottom.

EOS Mass (in M¯) In/I tq (yr) Reid 1.33 0.96 2100 Bethe-Johnson 1.33 0.77 130 Mean Field 1.33 0.66 25 Tensor Interaction 1.33 0.44 10 Tensor Interaction 0.10 0. 0.003 Tensor Interaction 0.29 0. 0.16 Tensor Interaction 0.73 0.06 0.92 Tensor Interaction 1.08 0.29 3.9 Tensor Interaction 1.33 0.44 10 Tensor Interaction 1.85 0.70 71 Tensor Interaction 1.93 0.66 600

shape remain more oblate than the equilibrium value. Finally, when the crust stress has reached a critical value, the crust cracks. Some stress is relieved. and the oblateness excess is reduced. As a result the moment of inertia of the crust is suddenly, and so by angular momentum conservation Ω is suddenly increased. This gives the observed spin-up. We define a time dependent oblateness parameter ² according to

I = I0(1 + ²) (12.96)

where I is the moment of inertia and I0 is the non rotating, spherical value. The parameter ² is related to the deformation parameter λ and the eccentricity e of a spheroid in the following way. Suppose we have a spheroid with major semi-axis a and minor semi-axis c. The deformation λ and eccentricity e are defined as

³ c2 ´1/3 c2 λ = and e2 = 1 − (12.97) a2 a2 respectively. They are related each other by

1 + ² = λ−1 = (1 − ²2)−1/3 (12.98) 12.9. SUPERFLUIDITY: GLITCHES AND STAR-QUAKES 315

For small deviations from sphericity, we have

e2 ² ' (12.99) 3

Consider now the total energy of a rotating neutron star in Newtonian description

E = Eint + W + T + Estrain (12.100) where Eint is the internal energy, W the gravitational energy, T the rotational energy, and Estrain the strain energy of the crust. For the internal, gravitational, and rotational energies we may use polytropic approximations. Internal energy: this is simply given by

1/3 Eint = k1Kρc M (12.101)

where k1 is a numerical factor of the order of unity and K is the polytropic constant. Both are known from the polytropic theory and the equation of state. Gravitational energy: this is the gravitational energy of a non rotating poly- trope corrected by a function g(λ) due to rotation

1/3 5/3 W = −k2Gρ M g(e) (12.102)

where k2 is a constant of the order of unity derived from the polytropic theory. The function g(e) is sin−1 g(λ) = (1 − e2)1/6 (12.103) e This function is taken from the theory of rotating polytropes and accounts for the deformation of equipotential surface (with respect to spherical ones) induced by rotation. Introducing the parameter λ, we get

g(e) ≡ g(λ) = λ1/2(1 − λ3)−1/2 cos−1(λ3/2) (12.104)

For λ ' 1 (1 − λ3)2 ²2 g(λ) ' 1 − ' 1 − (12.105) 5 5 316 CHAPTER 12. PULSARS

Rotational energy: this is given by

J 2 T = = k J 2M −5/3ρ2/3λ (12.106) 2I 5 where J is the angular momentum, k5 is of the order of unity (' 1.2), and λ is given by the ratio of the moment of inertia of a spheroid with major semi-axis a and minor semi-axis c to that of a real sphere of the same volume

I a2 1 = 2 2/3 = (12.107) Isphere (a c) λ

With the aid of the above relations, we have

2 2 (1 − ²) Eint + W + T = Eint + W0 + A² + J (12.108) 2I0 where W0 is the gravitational potential energy for anon rotating neutron star,

A = |W0|/5, and the last term is the contribution due to rotation. Finally Eint is independent of ² (rotation).

Strain energy. We are left with Estrain. The calculation of this term is more complicated. The strain energy arises from the compression of a solid (Coulomb lattice). We know that the crust is composed by neutron-rich nuclei locked in a lattice structure. The strain energy can be evaluated as follows: suppose that the star has some initial deformation ²0. At this reference oblateness, the star is strain-free. As Ω decreases ² falls below ²0, straining the crust. Consider, one dimension lattice like that imagined for pycno-nuclear reactions. The Coulomb potential between two nuclei can be approximated to an elastic potential. Com- pression of the lattice as the star readjusts its shape results in a strain or elastic energy 1 E ∼ N( Kx2) (12.109) strain 2 where 3 N ∼ nAR (12.110)

3 is the number of lattice sites (nuclei) in the crust, nA ∼ 1/R0 is the mean ion 1/3 density there, and R0 ∼ R/N is the ionic separation, K is the lattice spring 12.9. SUPERFLUIDITY: GLITCHES AND STAR-QUAKES 317

2 2 3 constant K = 4Z e /R0, and the displacement x can be evaluated as δR Rδ² R x R ∼ ∼ ∼ |(² − ² )| (12.111) δ 0 N 1/3 N 1/3 N 1/3 0 Therefore, eqn.(12.109) becomes

2 2 Z e 3 2 Estrain ' nAR (² − ²0) (12.112) R0 A more accurate treatment gives for an unscreened BCC Coulomb lattice

2 Estrain = B(² − ²0) (12.113) ³4π ´ Z2e2 B = 0.42 R3 n (12.114) 3 A a 2 a = ( )1/3 (12.115) nA The mean stress of the crust is defined as

¯ ¯ ¯ 1 ∂Estrain ¯ 2B σ ≡ ¯ ¯ = µ(²0 − ²) (12.116) Vc ∂² Vc where Vc is the volume of the crust and µ = 2B/Vc. Finally, using eqn.(12.108) and eqn.(12.113), for the total energy we have

2 2 (1 − ²) 2 E = E0 + A² + J + B(² − ²0) (12.117) 2I0 where E0 is the energy in absence of rotation. Minimizing eqn.(??) with respect to ², at fixed M, ρ and J we obtain the equilibrium condition I Ω2 B² ² = 0 + 0 (12.118) 4(A + B) A + B

AS Ω2 >> Gρ² and A >> B we have I Ω2 ² = 0 (12.119) 4A This means that the rigidity of the crust results in a small deviation of ² from the value it would have if the star were a perfect liquid.

When the mean stress exceeds a critical value σc, the crust will crack and the net stress, strain and oblateness will suddenly be reduced. The crust quake leads 318 CHAPTER 12. PULSARS

to a sharp decrease ∆²0 in the reference oblateness and a corresponding decrease ∆² in the actual oblateness. In other words if we differentiate eqn.(12.118) we obtain B B ∆² = ∆² ' ∆² << ∆² (12.120) A + B 0 A 0 0 This change is directly observable, since by eqn.(12.113)

∆I ∆Ω ³∆Ω´ ∆² = = − = −(1 − Q) (12.121) I Ω Ω 0

Where we have used the definition of Q; and ∆Ω in the second equality is the difference between the stable angular velocity before the glitch and following complete relaxation after the glitch. In the case of Vela, eqn.(12.119) implies an equilibrium oblateness ² ' 10−4, while eqn.(12.121) gives for the spin-up ∆² ' 10−6. The correspondent values for the Crab are ² ' 10−3 and ∆² ' 10−8 − 10−9. An observable effect arises from the shrinkage of the crust by a fraction of millimeter! Following a quake, the pulsar continues to slow down in the usual way until the stress builds up to the critical value. The time scale tq between quakes is then given by |∆σ| t = (12.122) q σ˙ From eqn.(12.116) and (12.120) the stress relieved in a quake is

∆² ∆σ = µ(∆² − ∆²) ' µA (12.123) 0 B

The subsequent build-up of the stress occurs at steady rate

2 2B 1 ˙ 2B 1 Ω σ˙ = −µ²˙ = − I0ΩΩ = ( ) I0 (12.124) Vc 2A Vc 2A T We have used eqn.(12.119) for² ˙. Thus we obtain

ω2 t = T q |∆²| (12.125) q Ω2 where 2 2 2A ωq = (12.126) BI0 12.9. SUPERFLUIDITY: GLITCHES AND STAR-QUAKES 319

2 The parameter ωq greatly varies with the mass and so does tq. They also depend on the equation of state. For example for the Crab with M = 1.3 M¯ (EOS-TI), tq ∼ 10 years in agreement with data. The same time interval, but with EOS-BJ, would require M ∼ 0.5 M¯. There are problems with Vela that perhaps suggest the existence of core − quakes that might occur if a neutron star has a solid core. This might be possible for suitable equations of state. 320 CHAPTER 12. PULSARS Chapter 13

Black Holes

Neutron Stars have an upper mass limit. If the nucleus of the collapsing nucleus of a massive has a mass exceeding this limit or the incipient neutron star cannot develop enough pressure to contrast gravity, the collapse cannot be halted and the nucleus gives rise to a Black Hole. A Black Hole is the end point of the evolutionary history of a star. It can be defined as a region of the Space-Time enclosed by an horizon of events; a region of the space-time with so an intense gravitational field that neither matter nor radiation can escape from the surface. A Black Hole can be detected only via the gravitational interaction exerted on nearby objects (the most direct method). The fundamental parameter of a Black Hole is its mass M, which determines also the dimensions of the Black Hole. There is a simple way for describing a Black Hole. Consider a particle of mass m subject to the gravitational field of a star of mass M, which escapes from the surface with escape velocity such that at infinity the velocity is zero. The total energy of the particle is

1 GmM mv2 − = 0 (13.1) 2 R or the minimum escape velocity is

2GM v2 = (13.2) R

321 322 CHAPTER 13. BLACK HOLES

As no particle can travel with velocities larger than the speed of light, we obtain the Schwarzschild relation for the radius RS of the Black Hole:

2GM R = (13.3) S c2

A Black Hole with the mass of the Sun has RS ' 3 km. Physically RS gives at each instant the surface of events of the Black Hole. In addition to the mass, a Black Hole may have other properties such as: rotation and electric charge. The first one derives from the fact that if the collapsing nucleus is rotating, so will be the Black Hole. The second derives from the possibility that the matter forming the Black Hole may not be electrically neutral. In such a case the black Hole will be charged and will exert electrical forces on the matter outside the event horizon. However, further acquisition of matter in general containing charged particles, will rapidly neutralize the charge. Any other property of the original material is lost when falling into a Black Hole.

13.1 Simple Theory of Black Holes

Given this brief introduction, we present here a simplified version of the theory of Black Holes to get some familiarity with the this types of objects. To this aim we will limit ourselves to the case of non-rotating Black hole. The obviously necessary framework is that of General Relativity. Consider the gravitational field surrounding a very condensed object of mass M, radius R and spherical symmetry. The vacuum solution (no matter around) of the Einstein field equations was found as early as 1916 by Karl Schwarzschild. It gives the line element ds, i.e. the distance between two neighbouring events in the 4-dimensional space as

2 i j ds = gijdx dx (13.4)

³ r ´ ³ r ´−1 ds2 = 1 − s c2dt2 − 1 − s dr2 − r2dθ2 − r2sin2θdφ2 (13.5) r r 13.1. SIMPLE THEORY OF BLACK HOLES 323

³ r ´ ds2 = 1 − s c2dt2 − dσ2 (13.6) r where one has to sum from 0 to 3 on the indices i and j, and where the usual spherical coordinates r, θ and φ are taken as the spatial coordinates x1, x2, x3 0 and x = ct. The critical parameter rs is the Schwarzschild radius we have already introduced above. −1 The second component of the metric tensor (1 − rs/r) becomes singular at r = rs. However, this singularity is not physical as it would disappear when other suitable coordinates are adopted (see below). The proper time τ, as measured by an observer carrying a standard clock is related to the line element ds along the his world line by the relation

1 dτ = ds (13.7) c For a stationary observer (dr = dθ = dφ=0) at infinity (r → ∞) the proper time

τ∞ coincides with t according to the metric given by eqn.(13.6). Consider now two stationary observers, one at the position r, θ, φ, the other at infinity. Their proper times τ and τ∞ are related to each other by

dτ ³ r ´1/2 = 1 − s (13.8) dτ∞ r Suppose that the first observer operates a light source emitting a signal at regular intervals dτ (for instance an atom emitting with the frequency ν0 = 1/dτ. The other observer will receive the signal and measure the time interval according to his clock dτ∞, i.e. will measure the frequency ν = 1/dτ∞. The resulting red-shift due to the gravitational field is given by the relation

ν − ν ν dτ ³ r ´−1/2 z = 0 = 0 − 1 = ∞ − 1 = 1 − s − 1 (13.9) ν ν dτ r which gives z → ∞ for r → rs. The metric components in eqn.(13.6) show that the 4-dimensional space-time is curved, and this also holds for the 3-dimensional space. 324 CHAPTER 13. BLACK HOLES

At the surface of a mass configuration of mass M and radius R, the Gaussian Curvature K of the position space is given by

GM 1 r 1 K = − = − s (13.10) c2R3 2 R R2 This is usually very small compared with the curvature R−2 of the 2-dimensional surface. For example −K ' 2 × 10−6R−2 at the surface of the Sun, but it grows to −K ' 0.15R−2 at the surface of a neutron star. The two curvatures become comparable at the surface of a Black Hole with R = Rs Consider a test particle small enough for the gravitational field not to be disturbed by its presence that freely moves in the field from point A to B. Its world line in the 4-dimensional space is a geodesic, i.e. the length sAB is an extremum. In other words any infinitesimal variation does not change the length

Z B δsAB = δ ds = 0 (13.11) A If the test particle moves locally with velocity v over a spatial distance dσ, the proper time interval will be the smaller, the larger v. It becomes

dτ = ds = 0 per for v = c (13.12) i.e. for photons and other particles of zero rest-mass: they move along null geodesics. For material particles, the requirement v < c of special relativity (which is locally valid) means dτ 2 and ds2 > 0. Such separations are called time-like. The world line of material particles must be time-like. Separations with ds2 < 0 or dτ 2 < 0 would require v > c. They are called space-like. For example the distance between two simultaneous events (dt = 0) is space-like. The null geodesics (ds2 = 0) giving the propagation of photons, describe Hyper- Cones in space-time which are called Light Cones. In order to see their prop- erties near r = rs, we introduce a time coordinate t given by 13.1. SIMPLE THEORY OF BLACK HOLES 325

¯ ¯ rs ¯ r ¯ t = t + ln ¯ − 1¯ (13.13) c rs which transforms (13.6) to

³ r ´ r ³ r ´ ds2 = 1 − s c2dt2 − 2 s cdrdt − 1 + s dr2 − r2dθ2 − r2sin2θdφ2 (13.14) r r r which is no longer singular at r = rs. Consider only the radial boundaries of the light cones, i.e. the path of radially emitted photons (dφ = dθ = 0). Then eqn.(13.14) yields for ds2 = 0, and after division c2dr2, the quadratic equation

³ r ´³ dt ´2 2r dt 1 ³ r ´ 1 − s − s − 1 + s = 0 (13.15) r dr cr dr c2 r This equation has the solutions

³ dt ´ 1 ³ dt ´ 1 1 + r /r = − and = s (13.16) dr 1 c dr 2 c 1 − rs/r These derivatives are the inclinations of the two radial boundaries of the light- cones in the r − t plane of Fig. 13.1. The first derivative corresponds to an inward motion with constant velocity c. The second derivative changes sign at r = rs, being positive per r > rs, where the photons can be emitted outwards (dr > 0). With decreasing r the second derivative becomes larger such that the light cone narrows and its axis turns to the left towards the r axis. At r = rs the light cone is such that no photon can be emitted to the outside (dr > 0).

This is the reason for calling a configuration with R = rs ”a Black Hole” and for speaking of a Schwarzschild radius rs of a Black Hole of mass M.

For r < rs both solutions are negative and the whole light cone is turned inwards. Therefore for r < rs, radiation and all material particles (that 326 CHAPTER 13. BLACK HOLES

Figure 13.1: Illustration of light-cones at different distances r from the central singularity, inside and outside the Schwarzschild radius rs

Figure 13.2: The radial infall into a Black Hole for a test particle starting at distance 5rs with zero velocity. Il motion is shown in terms of the particle proper time τ, and in terms of coordinate time t of an observer at infinity 13.1. SIMPLE THEORY OF BLACK HOLES 327 can move only inside the light cone) are drawn inexorably toward the centre.

This means that there is no static solution (dr = dθ = dφ = 0) inside rs, since it would require a motion vertically upwards in Fig. 13.1, i.e. outside the light cone. In order to study the motion of a material particle, we consider all variables to depend on the parameter τ, the proper time, varying monotonically along the world line dτ = ds/c. Dots denote derivatives with respect to τ. For examplex ˙ α = dxα/dτ is the α-component of a 4-velocity. Introducing dxα =x ˙ αdτ into (13.6) gives the useful identity

³ r ´ ³ r ´−1 c2 = g x˙ ix˙ j = c2 1 − s t˙2 − 1 − s r˙2 − r2(θ˙2 + sin2θφ˙2) (13.17) ij r r

The condition that the world line is a geodesic means that the variation δs = δτ = 0, which yields the Eulero-Lagrange equations

d ³ ∂L ´ ∂L − = 0 (13.18) dτ ∂x˙ α ∂xα with the Lagrangian L given by

h i i j 1/2 2cL = gijx˙ x˙ (13.19) or h ³ r ´ ³ r ´−1 ³ ´i1/2 2cL = c2 1 − s t˙2 − 1 − s r˙2 − r2 θ˙2 + sin2θφ˙2 (13.20) r r From eqn.(13.18; 13.19; 13.20) follows the value L = 1/2. For x0 = ct eqn.(13.19) becomes

d h³ r ´ i ³ r ´ 1 − s t˙ = 0 1 − s t˙ = constant = A (13.21) dτ r r We confine ourselves to the discussion of a radial infall (θ˙ = φ˙ = 0) starting from r0 with null velocity at time τ = 0. Instead of also deriving the equation of motion for x1 = r from the Euler-Lagrange equation, we simply introduce the second eqn.(13.21) into eqn.(13.17) and solve forr ˙. We obtain 328 CHAPTER 13. BLACK HOLES

h r i1/2 r˙ = c A2 − 1 + s (13.22) r 2 For our purposes we set A − 1 = −rs/r0. This means that the particle starts with zero velocity at r0. Upon integration we derive

s 1 r r τ = 0 0 (sin η + η) (13.23) 2 c rs where η = arccos(2r/r0 − 1), as can be verified by differentiation. The function

τ = τ(r) is shown in Fig. 13.2 for r0 = 5rs. Nothing special occurs when the particle reaches r = rs. The total proper time for reaching r = 0 is given by

³ ´ π rs r0 3/2 τ0 = (13.24) 2 c rs

For r0 = 10rs we have τ0 = 49.67rs/c, which for M = M¯ corresponds to about 9.8 × 10−6 s. The motion in terms of the coordinate time t as seen by an observer at infinity is quite different. The relation between t and τ is given by dτ/dt =

(1−rs/r)/A, which goes to zero when r → rs. From this relation and eqn.(13.22) one obtains a differential relation for t(r) which is integrated to give

t ξ + tan η/2 r h r i = ln | | + ( 0 − 1)1/2 η + 0 (η + sin η) (13.25) rs/c ξ − tan η/2 rs 2rs 1/2 where η = arccos(2r/r0 − 1) and ξ = (r0/rs − 1) . The t = t(r) is shown in Fig.

13.2 for r0 = 5rs.

The fact that the observer sees the τ clock of the particle to slow down for r → rs implies that a r → rs the time t = t(r) → ∞.

Events inside rs are completely shielded for the distant observer at infinity by the coordinate singularity at the Schwarzschild radius acting as an event horizon. These considerations may suffice to illustrate some important proprieties of Black Holes. 13.2. SCHWARZSCHILD BLACK HOLE 329

First of all we must note that the Schwarzschild metric is a vacuum solution, which is not valid inside the mass configuration, but holds only from the surface outwards. As observed from the in-falling surface of the Black Hole (proper time τ), the collapse proceeds fairly rapid and smoothly through the

Schwarzschild radius rs. Once inside rs a static configuration is no longer possible and the collapse inevitably proceeds towards the cen- tral singularity. This is shown by the fact that material particles have world line inside the local light-cone, and this is open only towards r = 0 (even radia- tion falls to r = 0). Note that it would not help to to invoke an extreme pressure exerted by unknown physical effects, since pressure would contribute to the grav- itational energy. The singularity at r = 0 is an essential one with infinite gravity. Quantum effects should have to be included and one can speculate whether they might remove the singularity. con gravit`ainfinita. The collapse of a star will present itself quite differently to an observer located at infinity. In his coordinate time t he will see the collapse of the stellar surface to slow down more and more, the closer it comes to rs. In fact he will find that this critical point cannot be reached in a finite time t; for the collapsing surface seems to become stationary there. In addition the approach to the surface rs strongly affects the light re- ceived by the distant observer. He receives photons in ever increasing intervals and ever decreasing energy as the redshift z → ∞: the collapsing star will finally ”disappear” for the distant observer. Only a strong gravitational field is left. From now on the presence of a Black Hole can be revealed only by radiation emitted in its vicinity by in-falling particles or the motion of a visible companion if the Black Hole belongs to a binary system.

13.2 Schwarzschild Black Hole

In this section we discuss the theory of a non rotating (J = 0) and neutral (Q = 0) Black Hole in the Schwarzschild solution. We adopt the geometrized fundamental 330 CHAPTER 13. BLACK HOLES units (G = c = 1) and the signature (-,+,+,+). The Schwarzschild metric is

2M 2M ds2 = −(1 − )dt2 + (1 − )−1dr2 + r2dθ2 + r2 sin2 θdφ2 (13.26) r r

A static observer in this gravitational field is one who is at fixed r, θ and φ. The proper time for this observer is

³ 2M ´ dτ 2 = −ds2 = 1 − dt2 (13.27) r or ³ 2M ´1/2 dτ = 1 − dt (13.28) r This shows the gravitational time dilatation for a clock in the grav- itational field compared to a clock at infinity. As already pointed out eqn.(13.26) diverges at r = 2M which is the event horizon (surface of the Black Hole). It is also known as the static limit, because static observers cannot exist inside r = 2M. They are inexorably drawn the central singularity. A static observer can make measurements with his ortho-normal tetrad which is related to the metric tetrad by the relations

2M e = (1 − )−1/2e (13.29) t˜ r t 2M e = (1 − )1/2e (13.30) r˜ r r 1 e = ( )e (13.31) θ˜ r θ 1 e = ( )e (13.32) φ˜ r sin θ φ

This is an ortho-normal system since

2M 2M e · e = (1 − )−1e e = (1 − )−1g = −1 (13.33) t˜ t˜ r t t r tt 13.3. MOTION OF MASSIVE TEST PARTICLES 331 13.3 Motion of massive test particles

Let us consider the motion of a freely moving test particle with mass m. Such particle moves along geodesics of the space-time. The geodesic equations are derivable from the Euler-Lagrange equations with lagrangian given by 2M 2M 2L = −(1 − )t˙2 + (1 − )−1r˙2 + r2θ˙ + r2 sin2 θφ˙ (13.34) r r where t˙ = dt/dλ is the t-component of the quadri-momentum, the same for the remaining three. The parameter λ is chosen in such a way that λ = τ/m. The Euler-Lagrange equations are d ∂L ∂L ( ) − = 0 with xα = (t, r, θ, φ) (13.35) dλ ∂x˙ α ∂xα For θ, φ and t they are d (r2θ˙) = r2 sin θ cos θφ˙2 (13.36) dλ

d (r2 sin2 θφ˙) = 0 (13.37) dλ

d h³ 2M ´ i 1 − t˙ = 0 (13.38) dλ r Instead of writing the equation forr ˙ we note that

α β 2 gαβp p = −m (13.39) therefore L = −m2/2. Now eqn.(13.35) shows that if we orient the coordinate system in such a way that initially the particle is moving in the equatorial plane (θ = π/2, θ˙ = 0), then the particle will remain in the equatorial plane. This result follows from the uniqueness theorem of such differential equation, since θ = π/2 for all λ. Furthermore, this result is obvious from the spherical symmetry. With θ = π/2, eqn.(13.37) and eqn.(13.38) become

2 ˙ pφ = r φ = constant = l (13.40) 332 CHAPTER 13. BLACK HOLES

³ 2M ´ −p = 1 − t˙ = constant = E (13.41) t r i.e. the constants of motion corresponding to the coordinates φ and t in the Lagrangian (13.34). Measuring the energy. To understand their physical meaning, consider a measurement of the particle energy made by an observer in the equatorial plane. The locally measured energy is the t-component of the 4-momentum as measured by the observer local ortho-normal tetrad.

³ 2M ´−1/2 E = pt˜ = −p = −p · e = p · 1 − e (13.42) loc t˜ t˜ r t 2M ´−1/2 = −(1 − p (13.43) r t

³ 2M ´1/2 E = 1 − E (13.44) r loc

For r → ∞ E → Eloc. So the conserved quantity E is called ”energy at infinity” It is related to the local energy by a redshift factor. Measuring the angular momentum. Similar reasoning can be followed to understand the physical meaning of l. Consider the tangential velocity component vφ measured by the local observer with his local ortho-normal tetrad

φ˜ p p ˜ p · e ˜ p · e /r p /r vφ˜ = = φ = φ == φ = φ (13.45) t˜ t˜ p p Eloc Eloc Eloc and so φ˜ l = Elocrv (13.46)

Comparing with the newtonian expression mvφ˜r we see that l is the conserved angular momentum. For particle with non-zero rest mass it is convenient to express E and l as quan- tities per unit mass

E l E˜ = and ˜l = (13.47) m m Recalling that λ = τ/m from eqn.(13.39), (13.40) and (13.41) we derive 13.3. MOTION OF MASSIVE TEST PARTICLES 333

³ dr ´2 ³ 2M ´³ ˜l2 ´ = E˜2 − 1 − 1 + (13.48) dτ r r2

dφ ˜l = (13.49) dτ r2

dt E˜ = 2M (13.50) dτ 1 − r Equations (13.48), (13.49), and (13.50) give the r(τ), φ(τ) and t(τ) relations. The motion of the test particle. It is particularly interesting to consider orbits just outside the event horizon. The locally measured value of vr˜, radial velocity component, is given by

pr˜ p p · e p (1 − 2M/r)1/2 pr vr˜ = = r˜ = r˜ = r = (13.51) t˜ p pt˜ Eloc Eloc E Recalling that pr = mdr/dτ and eqn.(13.48) we derive

dr h 1 ³ 2M ´³ ˜l2 ´i1/2 vr˜ = = 1 − 1 − 1 + (13.52) Edτ˜ E˜2 r r2 For r → 2M, vr˜ → 1. A local static observer at r sees the particle to approach the event horizon along a radial geodesic at the speed of light independent of ˜l. The same observer sees the particle to approach the event horizon with

³ 2M ´1/2 ˜l vφ˜ = 1 − (13.53) r 2E˜

For r → 2M, vφ˜ → 0. The simplest geodesics are those for radial infall, φ =constant. This occurs for ˜l = 0 and eqn.(13.48) becomes

dr ³ 2M ´1/2 = − E˜2 − 1 + (13.54) dλ r By considering the limit r → ∞ of eqn.(13.54) we have three possibilities (i) E˜ < 1, the particle fall from rest at r = R (ii) E˜ = 0, the particle falls from rest at infinity; 334 CHAPTER 13. BLACK HOLES

(iii) E˜ > 1, the particle falls with finite inward velocity from infinity where v = v∞ If we integrate eqn.(13.54) for the case E˜ < 1 so that 1 − E˜2 = 2M/R, and assume τ = 0 ar r = R, we get

³ R3 h ³ r r2 ´1/2 ³2r ´i τ = )1/2 2 − + cos−1 − 1 (13.55) 8M R R2 R Introducing the ”cycloid parameter” η

R r = (1 + cos η) (13.56) 2 we obtain ³ R3 ´1/2 τ = (η + sin η) (13.57) 8M Integrating now eqn.(13.49) for the coordinate time t in terms of η (t = 0 at r = R)

¯ 1/2 ¯ ³ ´ h i t ¯(R/2M − 1) + tan(η/2)¯ R 1/2 R = ln ¯ ¯+ −1 η+ (η+sin η) (13.58) 2M (R/2M − 1)1/2 − tan(η/2) 2M 4M

These solutions show two important results of radial infall: 1) The proper time τ to fall from rest at r = R > 2M to r = R = 2M is finite. The proper time to fall from r = R = 2M to r = 0 is also finite. In general the proper time to fall from a generic position r to r = 0 is finite. 2) The coordinate time t (proper time of an observer at infinity) to fall from the generic position to r = R = 2M is infinite. At r = 2M tan(η/2) = (R/2M −1)1/2. The results are shown in Fig.13.3. This is exactly the same result we have dis- cussed when presenting the simple theory of Black Holes. The effective potential. Consider now non-radial motions. Instead of integrating eqn.(13.48) through (13.50), we try to get a general picture of the orbits by considering and analyzing the ”Effective potential”

³ 2M ´³ ˜l2 ´ V (r) = 1 − 1 + (13.59) r r2 13.3. MOTION OF MASSIVE TEST PARTICLES 335

Figure 13.3: Fall from rest toward a Schwarzschild Black Hole as described (a) by a comoving observer (proper time τ and (b) a distant observer (time t). In the one description, the pont r = 0 is attained and quickly. In the other description, r = 0 is never reached and even r = 2M is attained only asymptotically.

Eqn. (13.48) becomes ³ dr ´2 = E˜2 − V (r) (13.60) dτ For a fixed value of ˜l, the effective potential V (r) is shown in Fig. 13.4. In this diagram there are three horizontal lines corresponding to different values of E˜2. According to eqn.(13.60) the distance from the horizontal line to V (r) gives (dr/dτ)2. Orbit 1. The horizontal line labelled 1 corresponds to a particle coming in from infinite with energy E˜2. When the particle reaches the value of r corresponding to point A, dr/dτ passes through zero and changes sign; the particle return to infinity. Such an orbit is unbound and A is called the turning point. Orbit 2. This is ”capture orbit”: the particles plunges into the Black Hole.

Orbit 3. This is bound orbit, with two turning points A1 and A2. The point B corresponds to a stable circular orbit. If the particle is slightly perturbed away from B, the orbit remains close to B. The point C is an unstable circular orbit: a particle placed in such an orbit will, upon experiencing the slightest inward motion, falls back toward the Black Hole and be captured. If it is perturbed 336 CHAPTER 13. BLACK HOLES

Figure 13.4: Sketch of the effective potential profile for a particle with non-zero rest mass orbiting a Schwarzschild Black Hole of mass M. The three horizontal lines labelled by different values of E˜2 correspond to (1) unbound, (2) capture, and (3) bound orbit, respectively. outward, it flies to infinity. Orbits of type 1 and 3 exist also in Newtonian Mechanics, whereas orbits of type 2 are possible only in General Relativity. There are several interesting properties that can by inferred from studying the effective potential: (1) One can easily verify that ∂V/∂r = 0 when

Mr2 − ˜l2r + 3M˜l2 = 0 (13.61) √ therefore there are no maxima or minima of V (r) if ˜l < 2 3M. (2) V (r) = 1 for ˜l = 4M The variation of v(r)1/2 as a function of r/M are shown in Fig. 13.5. (3) Circular orbits occur when

∂V (r) ∂r = 0 and = 0 (13.62) ∂r ∂τ

From eqn.(13.60) and (13.61) we may derive

Mr2 ˜l2 = (13.63) r − 3M 13.3. MOTION OF MASSIVE TEST PARTICLES 337

Figure 13.5: The effective potential profile for non-zero rest.mass particles of various angular momenta ˜l orbiting a Schwarzschild Black Hole of mass M. The dots at the local√ minima locate radii of static circular orbits. Such orbits exist only for ˜l > 2 3M.

(r − 2M)2 E˜2 = (13.64) r(r − 3M) Therefore, circular orbits can exist down to r = 3M which corresponds to the case of a photon (E˜ = E/m → ∞). The circular orbits are stable if V (r) is concave, that is if ∂2V (r)/∂r2 > 0, and unstable if V (r) is convex, that is ∂2V (r)/∂r2 < 0. (4) Circular Schwarzschild orbits are stable if r > 6M, unstable if r < 6M. (5) The binding energy per unit mass of a particle in the last stable circular orbit at r = 6M is

m − E E ³8´1/2 E˜ = = 1 − = 1 − = 5.72% (13.65) binding m m 9 where we have used eqn.(13.64) to calculate E˜ = E/m for r = 6M. This is the fraction of rest-mass energy released when a particle originally at rest at infinity spirals slowly towards a Black Hole to the innermost stable circular orbit, and then plunges into the Black Hole. This mechanism for converting rest-mass into other forms of energy is much more efficient than nuclear burning where only 0.9% of the rest mass is used passing from H to Fe. 338 CHAPTER 13. BLACK HOLES

Mass accretion onto a Black Hole can be a powerful source of energy. It is at the base of many models seeking to explain astronomical observations of huge energy outputs form compact regions. (6) The velocity of a particle in the innermost stable circular orbit as measured by a local static observer is vφ˜ = 1/2 (c=1). The result comes from eqn.(13.53) and (13.64) with r = 6M. (7) Suppose that a particle is moving in the last stable circular orbit and emitting monochromatic radiation with frequency νem in its rest frame. The frequency received at infinity varies periodically in the interval √ 2 √ ν < ν < 2ν 3 em ∞ em (8) Since dφ˜ = rdφ, the proper circumference is 2πr. The period as measured at the star and at infinity are

1/2 −4 Tstar = 24πM and T∞ = Tstar/(2/3) = 4.5 × 10 s(M/M¯)

(9) The angular velocity as measured at infinity Ω = dφ/dt is the same in the Schwarzschild geometry as in the Newtonian theory

³M ´1/2 Ω = r3 Impact Parameter. An important parameter to be derived is the ”capture cross section” for particles falling in from infinity. This is simply given by the relation 2 σcapture = πbmax (13.66) where bmax is the maximum impact parameter of a particle that is captured.

The parameter bmax has the dimension of a length and σ of an area. Consider a particle of mass m along its trajectory r = r(φ) around a Black Hole. At each given time the particle is at certain point of the trajectory. The tangent to the orbit at that form an angle φ with the polar radius center on the Black Hole and given the position of the particle. The geometrical situation is shown in Fig.13.6. Drawing a straight line parallel to the tangent and passing through 13.3. MOTION OF MASSIVE TEST PARTICLES 339

Figure 13.6: Impact parameter b for particle with trajectory r = r(φ) about mass M. the coordinate centre (Black Hole). The distance between the two straight lines is the impact parameter b. The mathematical definition of the impact parameter is b = lim r sin φ (13.67) r→∞ We want to express b as a function of ˜l and E˜. For r → ∞, eqn.(19) and (21) yield

1 ³ dr ´2 E˜2 − 1 ' (13.68) r4 dφ ˜l2 Substituting r = b/φ we identify

1 E˜ − 1 = (13.69) b2 ˜l2 ˜ 2 −1/2 or in terms of the velocity at infinity, E = (1 − v∞)

˜ 2 −1/2 l ' bv∞(1 − v∞)

' bv∞ for v∞ << 1 (13.70)

˜ Consider a non relativistic particle moving toward the Black Hole (E ' 1, v∞ << 1). We know from property (2) above that a particle is captured if ˜l < 4M. Thus 4M bmax = (13.71) v∞ 340 CHAPTER 13. BLACK HOLES which gives the capture cross section 4π(2M)2 σcapture = 2 (13.72) v∞ This value should be compared with the geometrical capture cross section of a particle by a sphere of radius R in Newtonian Theory ³ ´ 2 2M σNewt = πR 1 + 2 (13.73) v∞R A Black Hole captures non relativistic particles like a Newtonian sphere of radius R = 8M.

13.4 Motion of massless test particles

In the case of photons (m = 0) we have

dt E = (13.74) dλ 1 − 2M/r

dφ l = (13.75) dλ r2

³ dr ´2 l2 ³ 2M ´ = E2 − 1 − (13.76) dλ r2 r By the Equivalence Principle, the particle’s world-line should be independent of its energy. We can see this by introducing a ne parameter

λnew = lλ (13.77)

Writing l b ≡ (13.78) E and dropping the subscript ”new” we find dt 1 = (13.79) dλ b(1 − 2M/r)

dφ 1 = (13.80) dλ r2 13.4. MOTION OF MASSLESS TEST PARTICLES 341

³ dr ´2 1 1 ³ 2M ´ = − 1 − (13.81) dλ b2 r2 r The world-line depends only on the parameter b i.e. the particle’s impact param- eter, and not l or E separately. Taking the limit for m → 0 of eqn.(13.69) we see that b is the same as that defined for material particles. Also for the photons we can study the orbits by examining the effective potential

1 ³ 2M ´ V (r) = 1 − (13.82) phot r2 r so that eqn.(13.81) becomes

³ dr ´2 1 = − V (r) (13.83) dλ b2 phot The geometry of the photon potential is shown in Fig.13.7. Clearly the distance 2 2 from an horizontal line of hight 1/b to Vphot gives (dr/dλ) . 2 Vphot(r) has a maximum of 1/(27M ) at r = 3M. The critical impact parameter separating capture from scattering orbits is given by 1/b2 = 1/(27M 2) or √ bc = 3 3M (13.84)

The capture cross section for photons from infinity is

2 2 σphot = πbc = 27πM (13.85)

Propagation angle. To calculate the emission properties of gas near a Black Hole we must know the propagation directions, as measured by a static observer, of photons emitted at radius r and propagating to infinity. Looking at Fig.13.7, a photon ar radius r > 3M escapes only if the following conditions are satisfied

√ (i) vr˜ > 0 or (ii) vr˜ < 0 and b > 3 3M (13.86)

Take a velocity vector v and unitary modulus |v| = 1 identifying the propagation direction. If ψ is the angle between the vector and the radial direction

vφ˜ sin ψ vr˜ = cos ψ (13.87) 342 CHAPTER 13. BLACK HOLES

Figure 13.7: Sketch of the effective potential profile for a particle with zero rest mass (photons) orbiting a Schwarzschild Black√ Hole of mass M. If the particle falls from r = ∞ with impact√ parameter b > 3 3M it is scattered back out to r = ∞. If, however, b < 3 3M the particle is captured by the Black Hole. but according to eqn.(13.39) and (13.53) with b = l/E

b ³ 2M ´1/2 vφ˜ = 1 − (13.88) r r

Thus the inward-moving photon escape the Black Hole if √ 3 3M ³ 2M ´1/2 sin ψ > 1 − (13.89) r r

At r = 6M, escape require ψ < 135◦; at r = 3M ψ < 90◦ so that all inward mov- ing photons are captured (50% of the radiation emitted by a stationary isotropic emitter is captured); in the interval 2M < r < 3M, emitted photons can escape if √ 3 3M ³ 2M ´1/2 sin ψ < 1 − (13.90) r r Finally as the emitter approaches r = 2M, only the outward emitted photons strictly moving along the radial direction can escape. The various cases are illus- trated in Fig.13.8, the directions along which photons are captured are visualized by the black areas. 13.5. NON SINGULARITY OF THE SCHWARZSCHILD RADIUS 343

Figure 13.8: (a) The angle ψ between the propagation direction of a photon and the radial direction at a given point P . (b) Gravitational capture of radiation by a Schwarzschild Black Hole. Rays emitted from each point into the interior of the shaded conical cavity are captured. The indicated capture cavities are those measured in the ortho-normal frame of the local static observer.

13.5 Non Singularity of the Schwarzschild Ra- dius

The Schwarzschild metric of eqn.(13.26) becomes singular at r = 2M, the co- efficient of (dt)2 goes to zero and the coefficient of dr2 gets infinite. However we cannot immediately conclude that this behaviour represents a true physical singulary. The coefficient of dφ2 becomes zero at θ = 0 but this is a coordinate singularity that can be easily removed changing the coordinate system. There is already some clue that the Schwarzschild radius r = 2M is also a coordinate singularity. As a matter of facts, an infalling particle does not notice anything strange at r = 2M. There is nothing special about the r(τ) law at this point. However the coordinate time t becomes infinite at r = 2M. This strongly suggests the presence of a coordinate singularity rather than a physical singularity.

There are many different coordinate transformations that can be used to show 344 CHAPTER 13. BLACK HOLES that r = 2M is not a physical singularity. Here we shall use the Kruskal coordi- nate system. It is defined by the transformation

³ r ´1/2 t u = − 1 er/4M cosh (13.91) 2M 4M

³ r ´1/2 t v = − 1 er/4M sinh (13.92) 2M 4M The inverse transformation is ³ r ´ − 1 er/2M = u2 − v2 (13.93) 2M t v tanh = (13.94) 4M u The metric takes the form 32M 3 ds2 = e−r/2M (−dv2 + du2) + r2dθ2 + r2 sin2 θdφ2 (13.95) r where r is defined implicitly in terms of u and v by eqn.(13.93). This metric has no singularity at r = 2m. However it is still singular at r = 0. This is a physical singularity that cannot be eliminated by changing coordinates. The gravitational field has infinite strength there. From eqn.(13.93) we note that r = 0 is for v2 − u2 = 1 or v = ±(1 + u2)1/2, i.e. there are two singularities! Note also that r ≥ 2M corresponds to the region u2 ≥ v2, i.e. u ≥ |v| or u ≤ −|v|. There are two regions with r ≥ 2M! This means that the Schwarzschild coordinate system covers only part of the space-time. The Kruskal coordinates give an analytical extension of the solution of the field equations that extends to the whole space time. The situation is shown in Fig.13.9. In the Kruskal coordinates, the radial light rays travel on 45◦ straight lines. The Kruskal diagram is a space-time diagram with the time coordinate v plotted vertically and the spatial coordinate u plotted horizontally. It can interpreted as a diagram of space-time in Special Relativity as the light cones are at 45◦ at each point and the particle world-lines must lie inside the light cones. 13.5. NON SINGULARITY OF THE SCHWARZSCHILD RADIUS 345

Figure 13.9: Kruskal diagram of the Schwarzschild metric.

There are four regions in this diagram. What is their physical interpretation? – Region I is ”our universe”, the original region with r > 2M. – Region II is the ”interior of the Black Hole”, the region with r < 2M. – Regions III and IV are the ”Other Universe”. Region III is asymptotically flat, whereas region IV corresponds to r > 2M. Checking the signs of u and v in the four quadrants, one finds the relationship between Kruskal and Schwarzschild coordinates in the various regions

³ r ´1/2 t u = ± − 1 er/4M cosh (13.96) 2M 4M ³ r ´1/2 t v = ± − 1 er/4M sinh (r ≥ 2M) (13.97) 2M 4M and

³ r ´1/2 t u = ± 1 − er/4M sinh (13.98) 2M 4M ³ r ´1/2 t v = ± 1 − er/4M cosh (r ≤ 2M) (13.99) 2M 4M

Where the upper sign refers to ”our universe”, the lower sign the ”other universe”. 346 CHAPTER 13. BLACK HOLES

Figure 13.10: Kruskal diagram of the Schwarzschild metric showing the relation between Schwarzschild (t, r) and Kruskal (u, v) coordinates

Equation(13.93) holds everywhere, while the r.h.s of eqn.(13.94) becomes u/v for r ≤ 2M. Equation (13.94) shows that the line of constant t are straight lines. These relationships are shown in Fig. 13.10. The singularity at the top of the diagram at r = 0 occurs inside the Black Hole. Clearly any time-like world-line at r ≤ 2M (region II) must strike the singularity. The singularity at the bottom of the diagram represents a ”” from which anything can come spewing out! It is important to remark that the full analytically continued Schwarzschild metric is merely a mathematical solution of the Einstein equations. In reality there are only two regions, part of which are physically permitted. For a Black Hole formed by gravitational collapse, part of space-time must contain the collapsing matter. According to the Birkoff theorem, the geometry of the space-time outside the collapsing star is still described by the Schwarzschild metric. Therefore, the world-line of a point on the surface of the star will be the boundary of the physically meaningful part of the Kruskal diagram. The situation is shown in Fig.13.11. The ”White Hole” and the ”Other Universe” are not present in real Black Holes. 13.6. ROTATING BLACK HOLES: THE KERR SOLUTION 347

Figure 13.11: Kruskal diagram for gravitational collapse. Only the unshaded region is physically meaningful, The remainder of the diagram must be replaced by the space-time geometry inside the star (Black Hole).

The Kruskal diagram makes clear the three important properties of Black Holes: (1) Once an object crosses the surface r = 2M, it must inexorably go to the singularity r = 0. (2) Once inside the region r = 2M, it cannot send signals back out to infinity. (3) There is no local test for this horizon. An observer does not notice anything significantly different in crossing the horizon.

13.6 Rotating Black Holes: the Kerr solution

Stars rotate during their life so that when central nuclei of massive stars collapse to a Black Hole they likely do it in presence of rotation. How is the theory of Black Holes modified by this occurrence? The solution of the Einstein equation discovered by Kerr in 1963 was not initially recognized to be the solution for rotating Black Holes. The properties became more transparent when presented in the Boyer-Lindquist (1967) coordi- 348 CHAPTER 13. BLACK HOLES nates (a generalization of the Schwarzschild coordinates). The most general stationary metric for Black Holes characterized by the pa- rameters M (mass), J (angular momentum), and Q (charge) is called the Kerr- Newman metric. Special cases are: the Kerr metric (Q = 0), the Reissner- Nordstrom metric (J = 0), the Schwarzschild metric (J = Q = 0). The Kerr metric in the Boyer-Lindquist formalism is ³ 2M ´ 4aMr sin2 θ Σ ds2 = − 1 − dt2 − dtdφ + dr2 (13.100) Σ Σ ∆ ³ 2Mra2 sin2 θ ´ +Σdθ2 + r2 + a2 + sin2 θdφ2 Σ Here the Black Hole is rotating in the φ direction and J a = ∆ = r2 − 2Mr + a2 Σ = r2 + a2 cos2 θ (13.101) M The metric is stationary (metric coefficients independent of t) and axisymmetric about the polar axis (metric coefficients independent of φ). The angular momen- tum per unit mass a is measured in cm when expressed in units with G = c = 1. Setting a = 0 we recover the Schwarzschild metric. The parameter M is the mass of the Black Hole. One can note that the Kerr metric is symmetric to the transformations

t → −t ∧ φ → −φ (13.102) t → −t ∧ a → −a (13.103)

The first one secures that the Kerr field is generated by a rotating source. The second one means that a is truly the angular momentum. Singularity. The Kerr metric is singular only at Σ = 0, i.e.

r2 + a2 cos2 θ = 0 (13.104) which is satisfied by r = 0 and cos θ = 0. This means that there is a ring of radius a on the equatorial plane in which the metric is singular. Horizon. The horizon occurs when ∆ = 0. This occurs first at the larger root of the quadratic equation ∆ = 0 13.6. ROTATING BLACK HOLES: THE KERR SOLUTION 349

2 2 1/2 r+ = M ± (M − a ) (13.105)

Later we will see that another horizon is possible. The above result is a particular case of the more general condition that has to be used to determine the surface at which redshift is infinite (the horizon) r2 − 2Mr + a2 cos2 θ g = = 0 (13.106) 00 Σ Solving this equation we obtain

2 2 2 1/2 r± = M ± (M − a cos θ) (13.107)

The two values of r define two surfaces S+ and S−, which have the following properties

(i) In the Schwarzschild limit for a → 0 the surface S+ is for r = 2M (the

Schwarzschild event horizon) and S− at r = 0 (ii) The two surfaces are axially symmetric.

(iii) The surface S+ at the equator has radius r = 2M, at the poles r = M + (M 2 − a2)1/2. This is possible only if a2 < M 2. Cosmic Censorship. Note that a must be less than M for a Black Hole to exist. If a exceeds M one would have a gravitational field with a ”naked” singularity, i.e. not clothed by an event horizon. A major unsolved problem in General Relativity is Penrose’s Cosmic Censorship Conjecture, that gravitational collapse from well behaved initial conditions never gives rise a . Certainly no mechanism is known to take a Kerr Black Hole with a < M and spin it up so that a > M (see below). A Black Hole with a = M is called a maximally rotating Black Hole.

(iv) The surface S− is fully internal to S+

(v) The existence of S+ and S− implies the existence of two event horizons. From the definition of horizon (hyper-surface with r = const and of light-type with g11 = 0) we have ∆ r2 − 2Mr + a2 g11 = − = − = 0 (13.108) Σ r2 + a2 cos2 θ 350 CHAPTER 13. BLACK HOLES

Figure 13.12: Event horizons, singular ring, ergo-sphere of the Kerr metric which to be satisfied requires ∆ = 0 whose solution gives

2 2 1/2 r± = M ± (M − a ) (13.109)

Therefore, there are two event horizons!

(vi) In the Schwarzschild limit (a → 0) r+ = 2M and r− = 0.

(vii) The event horizon r+ lies internal to S+. (viii) The solution is regular in the three regions

(1) r+ < r < ∞ where ∆ > 0

(2) r− < r < r+ where ∆ < 0

(3) 0 < r < r− where ∆ > 0 (13.110)

This complicate situation is shown in Fig. 13.12. Ergo-Sphere. In the case of the Schwarzschild metric we have made use of static observers to study its properties. We can generalize to rotating Black Holes by introducing stationary observers, who are at fixed r and θ but rotate with a constant angular velocity dφ uφ Ω = = (13.111) dt ut The condition u · u = −1 (i.e. the observers follow a time-like world-line) is

t 2 2 −1 = (u ) [gtt + 2Ωgtφ + Ω gφφ] (13.112) 13.6. ROTATING BLACK HOLES: THE KERR SOLUTION 351

Figure 13.13: Ergo-sphere of a Kerr Black Hole: the region between the static limit (the flattened outer surface with r = M + (M 2 − a2cos2θ)1/2) and the event horizon, i.e. the inner sphere with radius r = M + (M 2 − a2)1/2.

The quantity in square bracket must be negative. Since gφφ is positive , this is true only if Ω lies between the roots of the quadratic equation given by setting [...] = 0. Thus

Ωmin < Ω < Ωmax (13.113) where

2 1/2 2 1/2 −gtφ + (gtφ − gttgφφ) −gtφ − (gtφ − gttgφφ) Ωmax = Ωmin = (13.114) gφφ gφφ

2 2 2 Note that Ωmin = 0 when gtt = 0, i.e. when r −2Mr +a cos θ = 0. This occurs at

2 2 2 1/2 r0 = M + (M − a cos θ) (13.115)

Observeres between r+ and r0 must have Ω > 0: no static observer (Ω = 0) exist within r+ < r < r0. The surface r = r0 is called the static limit.

Note that the horizon r+ and the static limit are distinct for a 6= 0 . The situation is shown in Fig.13.13 2 From eqn.(13.114) we see that stationary observers fail to exist when gtφ−gttgφφ <

0, that is¡ when ∆ → 0. This can occur only for r < r+. This is the generalization 352 CHAPTER 13. BLACK HOLES of the static observers in the Schwarzschild metric who cannot exist inside the horizon.

The Kerr metric can be extended inside r+ in the same way as we did with the Kruskal continuation of the Schwarzschild metric. However, this interior solution is of scarce interest for two reasons. First, part of it must be replaced by the interior of the collapsing object that formed the Black Hole. Second, there is no Birkoff’s theorem for rotating collapse. The Kerr metric is not the exterior metric during the collapse; it is only the asymptotic form of the metric when all the dynamics has ceased. Therefore we will limit ourselves to consider only regions with r ≥ r+, adopting the viewpoint that anything entering inside that region becomes casually disconnected from the rest of the universe. Geodesics in the Kerr metric. A complete description of the geodesics of the Kerr Black Hole is quite complicated because of the lack of spherical symmetry. We will present only a few salient aspects of the problem. (1) No null radial geodesics. As a consequence of the lack of spherical sym- metry, null radial geodesics do not exist (case of photons and mass-less particles). The rotating source drags with it the space around and geodesics. This effect is known as frame dragging. While in the Newtonian Theory it is always possible to find a rotating coordinate system in which the source is at rest, this is no longer possible in General Relativity as there is frame of reference in which the Kerr metric coincides with the Schwarzschild metric. (2) Null non radial geodesics. On the hyper-surface θ = const (axial sym- metry) the equations of null geodesics are

l t˙ = (r2 + a2) (13.116) ∆ r˙ = ±l (13.117) l φ˙ = a (13.118) ∆ where the meaning of the parameter l will become clear later on. Using the three regions defined by relations (13.110), in region (1) where ∆ > 0, dr/dt > 0, that isr ˙ = +l describes a null geodesics towards the exterior, Similarly,r ˙ = −l 13.6. ROTATING BLACK HOLES: THE KERR SOLUTION 353 describes null geodesics towards the interior. The bundle of inward and outward null geodesics has in the Kerr metric the same meaning and role as the radial null geodesics of the Schwarzschild metric. They contain the information on the change of the light cones along r.

At event horizon, r = r+, t and φ are singular. Like in the case of the Schwarzschild metric, these singularities can be removed passing to another system of coordi- nates (e.g. Eddington-Finkelstein). Stationarity limit. Consider photons moving around the Black Hole with assigned r and θ (dr = dθ = 0) in region (1) with line-elements ds2 = 0. With the Boyer-Lindquist metric we obtain ∆ sin2 θ (dt − a sin2 θdφ)2 − [(r2 + a2)dφ − adt]2 = 0 (13.119) Σ Σ Solved in dφ/dt one gets dφ a sin θ ± ∆1/2 = (13.120) dt (r2 + a2 sin θ ± a∆1/2 sin2 θ These curves are not geodesics. However they touch the world-lines of photons moving around the Black Hole with fixed r and θ. The positive sign means dφ/dt > 0 the photons travel along the rotation direction (co-rotating or direct motion). Is the negative sign permitted and when? Let us examine when (13.120) can be negative. In region (1) we have

2 2 2 1/2 2 r > r+ ⇔ (r + a ) sin θ − a∆ sin θ > 0 (13.121)

As the denominator of eqn.(13.120) is positive dφ/dt ≤ 0 if

1/2 a sin θ − ∆ for r ≥ r+ (13.122)

At the surface S+ the derivative dφ/dt = 0, thus any particle on this surface trying to travel around the Black Hole in counter-rotating orbits must travel at the speed of light and remain stationary with respect to an observer at infinity. In the ergo-sphere, the light cones twist toward the φ direction so that photons and particles are forced to travel around the Black Hole in co-rotating mode. For 354 CHAPTER 13. BLACK HOLES

Figure 13.14: Spatial diagram of the Kerr solution on the equatorial plane. Note the progressive change in the inclination of the light cones.

this reason the surface S+ is called the stationarity limit. This surface is time-like and can be crossed by particles moving both inward and outward. Effective Potential: Orbits on the equatorial plane. To get some in- sight into the effect of rotation on the geodesics, we shall consider the motion of particles in the equatorial plane. Setting θ = π/2 in the metric of eqn (13.101) we derive the Lagrangian

³ 2M ´ 4aM r2 ³ 2Ma2 ´ 2L = − 1 − t˙2 − t˙φ˙ + r˙2 + r2 + a2 + φ˙2 (13.123) r r ∆ r where t˙ = dt/dλ and the same for the other coordinates. We can soon obtain two first integrals ∂L p = = constant = −E (13.124) t ∂t˙ ∂L pφ = = constant = l (13.125) ∂φ˙ Using eqn.(13.123) we obtain (r3 + a2r + 2Ma2)E − 2aMl t˙ = (13.126) r∆ (r − 2M)l + 2aME φ˙ = (13.127) r∆ We can get a third integral of motion as usual by setting

α β 2 gαβp p = −m (13.128) 13.6. ROTATING BLACK HOLES: THE KERR SOLUTION 355 and thus m2 L = − (13.129) 2 From eqn.(13.126) and (13.127) we derive

³ dr ´2 r3 = E2(r3 + a2r + 2Ma2) − 4MEal − (r2 − 2M)l2 − m2r∆ (13.130) dλ = F (E, l, r) (13.131) with obvious definition of F (E, L, r). The function F (E, l, r) can be considered as an effective potential for radial motion in the equatorial plane. For example, circular orbits are possible when

dr ∂F (E, l, r) = 0 → F (E, l, r) = 0 = 0 (13.132) dλ ∂λ

Solving eqn.(13.132) for E and l we obtain √ r2 − 2Mr ± a Mr E˜ = √ (13.133) r(r2 − 3Mr ± 2a Mr)1/2 √ √ Mr(r2 ∓ 2a Mr + a2)1/2 ˜l = ± √ (13.134) r(r2 − 3Mr ± 2a Mr)1/2 where the upper sign refers to co-rotating orbits (the orbital angular momentum of the particle is parallel to the spin angular momentum of the Black Hole). The lower sign is for counter-rotating orbits. ♠ Circular orbits exist for r = ∞ all the way down to the limit circular orbit, when the denominator of eqn.(13.133) vanishes.

√ r2 − 3Mr ± 2a Mr)1/2 = 0 (13.135) which is cubic equation in r1/2. Solving this equation we find for the photon orbit

n h2 a io r = 2M 1 + cos cos−1(∓ ) (13.136) ph 3 M

For a = 0, rph = 3M; for a = M there are two cases: rph = M (co-rotating orbit) and rph = 4M (counter-rotating orbit). 356 CHAPTER 13. BLACK HOLES

For r > rph not all circular orbits are bound. An unbound circular orbit has E/m > 1. Given a small perturbation, a particle in an unbound orbit will escape too infinity in an asymptotically hyperbolic orbit.

♠ Bound circular orbits exist for r > rmb where rmb is the radius of the last marginally bound orbit with E/m = 1. One finds

1/2 1/2 rmb = 2M ∓ a + 2M (M ∓ a) (13.137)

Note that rmb is the minimum periastron of all parabolic orbits (E/m = 1). In astrophysical problems, particles infall from infinity is nearly parabolic as v∞ << c. For a = 0, rmb = 4M. For a = M, rmb = M (co-rotating orbits) or rmb = 5.83M (counter-rotating orbits). ♠ Orbit stability. Even bound circular orbits are not all stable. Stability requires

∂2F (E, l, r) ≤ 0 (13.138) ∂r2 From the definition of F (E, l, r) we derive 2 (1 − E˜2) ≥ M (13.139) 3 Substituting E˜ with eqn.(13.133) one obtains a quartic equation in r1/2 for the limit case of equality. The radius of the last marginally stable circular orbit rms is

1/2 rms = M{3 + Z2 ∓ [(3 − Z1)(3 + Z1 + Z2)] (13.140) where

³ a2 ´1/3h³ a ´1/3 ³ a ´1/3i Z = 1 + 1 − 1 + + 1 − (13.141) 1 M 2 M M ³ a2 ´1/2 Z = 3 + z2 (13.142) 2 M 2 1

For a = 0, rms = 6M. For a = M, rms = M (co-rotating orbit) and rms = 9M (counter-rotating orbit). 13.6. ROTATING BLACK HOLES: THE KERR SOLUTION 357

Table 13.1: Summary Table

Case r r r E˜ 1 − E˜ ph mb ms q a = 0 3M 4M 6M 8/9 0.057 q a = M (l > 0) 1M 1M 1M 1/3 0.423 q a = M (l < 0) 4M 5.8M 9M 25/27 0.038

♠ Binding energy as a source of energy. A quantity of great interest for the efficiency of a Black Hole as an energy source is the binding energy of the marginally stable circular orbit. If we eliminate the radius from eqn.(13.133) with the aid of eqn.(13.139) we find √ a 4 2(1 − E˜)1/2 − 2E˜ = ∓ √ (13.143) M 3 3(1 − E˜2) Examining this relation we have the following results: q q (i) The quantity E˜ decreases from 8/9 for a = 0 to 1/3 for a = M in the q q case of co-rotating orbits. It decreases from 8/9 for a = 0 and to 25/27 for a = M in the case of counter-rotating orbits. (ii) The maximum binding energy 1 − E˜ for a maximally rotating Black Hole √ (a = M) is 1 − 1/ 3, i.e. the 42.3 % of the rest-mass energy. This is the amount of energy released by matter spiraling toward a Black Hole through a succession of equatorial quasi circular orbits. In contrast little energy is released during the the last plunge from rms into the Black Hole. The results obtained for the various cases are summarized in Table 13.1. ♠ Trajectories with negative energy. An interesting property of rotating Black Holes is the existence of test particle trajectories with negative energy. If we solve the eqn.(13.131) for E we derive

2aMl + (l2r2∆ + m2r∆ + r3r˙2)1/2 E = (13.144) r3 + a2r + 2Ma2 The sign of the square root is determined by letting r → ∞, To have E < 0 we require a retrograde orbit (l < 0) with 358 CHAPTER 13. BLACK HOLES

l2r2∆ + m2r∆ + r3r˙2 < 4a2M 2l2 (13.145)

The boundary of the region with negative energy orbits can estimated by mini- mizing the l.h.s. of eqn.(13.145), that is taking m → 0 (ultra-relativistic particles) andr ˙ →< 0. The boundary is at r = 2M = r0(θ = π/2) One can in fact show that the static limit r0 is the boundary of the region containing negative energy orbits for all values of θ. A particle can only be injected into such orbit inside the static limit; it then plunges into the Black Hole. ♠ The Penrose experiment. A Black Hole can be an enormous reservoir of energy as shown by Penrose (1968) thought experiment. Suppose we have a particle coming from infinity with initial energy Ein. The trajectory is chosen in such a way that the particle penetrates inside the static limit. The particle is also ”programmed” to split in two once inside. One piece goes into a negative energy trajectory and the other inside the Black Hole with energy Edown < 0. The other goes back to infinity with energy Eout > 0. The energy conservation requires

Ein = Eout + Edown → Eout > Ein (13.146)

Even though some rest-mass energy has been lost in the Black Hole, there is a net gain of energy at infinity. This energy gain is at expenses of the rotational energy of the Black Hole which slows down when the retrograde negative energy particle is captured, The region r+ < r < r0 where energy extraction is called the Ergo-Sphere. Unfortunately the Penrose experiment is of little practical interest because the breakdown of the two particles inside the ergo-sphere must happen with relative velocity of at least (1/2)c; it is hard to imagine an astrophysical process that produce such large relative velocities. ♠ Maximum extractable energy. The maximum amount of energy that can be extracted from the Black Hole by the is given by the following expression

∆Mmax = M0 − Mirr(M0, a0) (13.147) 13.7. THE AREA THEOREM AND BLACK HOLE EVAPORATION 359

where M0 is the initial mass of the Black Hole and a0 its angular momentum.

Mirr is called the irreducible mass given by

s q h ³ ³ ´2´i 1 2 2 1 a0 Mirr = r+a0 = M0 1 + 1 − (13.148) 2 2 M0 2 2 For a0 = M0 we derive ³ 1 ´ ∆Mmax = 1 − √ M0 ' 0.29M0 (13.149) 2 13.7 The Area Theorem and Black Hole Evap- oration

Hawking (1973) proved a remarkable theorem about Black Holes: in any inter- action the surface area of a Black Hole can never decrease. If several Black Holes are present, it is the sum of the surface areas that can never decrease. We calculate the surface area of a Kerr Black hole from the metric of eqn.(13.101) 2 2 1/2 setting t = constant and r = r+ = constant, and using r+ = M + (M − a ) . We obtain the line element ds2

2 2 2 2 2 2 (2Mr+) 2 2 ds = (r+ + a cos θ)dθ + 2 2 2 sin θdφ (13.150) r+ + a cos θ The area of the Horizon is ZZ √ A = gdθdφ ZZ = 2πr+ sin θdθdφ = 8πM[M + (M 2 − a2)1/2] ≡ 8πA˜ (13.151) where g is the determinant of the metric coefficient in eqn.(13.150), and the definition of A˜ is obvious. Note that for a = 0, A = 4π(2M)2, as we would expect. We can use the area theorem to show that one cannot make a naked singularity by adding particles to a maximally rotating Black Hole in an effort to spin it up. From eqn.(13.151) we find that δA > 0 implies 360 CHAPTER 13. BLACK HOLES

[2M(M 2 − a2)1/2 + 2M 2 − a2]δM > Maδa (13.152)

As a → M, this becomes

MδM > aδa → δ(M 2) > δ(a2) (13.153)

Thus M 2 always remain greater than a2, and the horizon is not destroyed. The capture cross for particles that increase a/M goes to zero as a → M. Thermodynamics of Black Holes The law of increase of area looks very much like the Second Law of Thermodynamics for the increase of entropy. Basing on this, Bekenstein (1973) tried to develop a thermodynamics of Black Hole interactions. However, in classical General relativity, there is no equilibrium state involving Black Holes. If a Black Body is placed into a radiation bath, it continuously absorbs radiation without ever coming into equilibrium. If a hot body falls into a Black Hole, the entropy of the Universe decreases. As this would contradict the second Law of Thermodynamics, Black Holes must possess entropy. As radiation or matter fall into Black Holes, their increase the mass, the angular momentum and the entropy. Is the Black Hole entropy related to the horizon area? The situation given by the lack of equilibrium states for Black Holes with respect to the surrounding medium was changed by the Hawking discovery that when quantum effects are taken into account, Black Holes radiate with a thermal spectrum. The Hawking radiation: creation of particle pairs in proximity of the hori- zon by means of the quantum field theory in a strong gravitational field. The idea is as follows: Vacuum has a complex structure that allows the creation, interaction and destruction of virtual particles (of extremely short lifetime, which is inversely proportional to the energy budget (∆E × ∆t ≤ h¯, the Heisemberg Principle). Vacuum in normal conditions is stable, that is no real particles (long lived) can be created. However, in presence of external force fields (for example, B, E, and G), virtual particles can receive energy from the surrounding field and 13.7. THE AREA THEOREM AND BLACK HOLE EVAPORATION 361 materialize as real particles. Hawking (1973) states that vacuum is unstable in the strong gravitational field of a Black Hole. It can be argued and proved the Black Holes radiate with thermal spectrum and we may state from now that mainly massless particles (photons, neutrinos, gravitons) are emitted. The expectation number of particles of a given species in a mode with frequency ω is

Γ < N >= (13.154) exp[¯hω/kTH ] ∓ 1 where Γ is the absorption coefficient for the mode incident on the Hole. It is a slowly varying function of ω depending on the kind of particle emitted and is close to unity for wavelengths much less than M (mass length). Thereinafter we adopt Γ = 1.

The temperature TH is the Hawking temperature for a Black Hole emitting as a black body. It is given by

hκ¯ T = (13.155) H 2πck where k is the Boltzmann constant and κ is the intensity of the gravitational field at the surface (horizon) of the Black Hole (remind the gravitational force in a star of mass m and radius r, g = −mG/r2). The corresponding quantity for a rotating Black Hole in our units and notation is

4π q κ = M 2 − (J/M)2 (13.156) A or r+ − M κ = 2 (13.157) r+ + a

The surface gravity of a Black Hole (even a rotating one ) and hence TH are constant at the horizon and are determined by the mass and angular momentum.

For a maximally rotating Black Hole (a = M) and κ = 0, thus TH = 0. For the sake of simplicity thereinafter we shall consider only Schwarzschild Black Holes (J = 0). The results can be easily extended to the case with rotation. 362 CHAPTER 13. BLACK HOLES

Therefore 1 hc4 i κ = (13.158) 4M G and the corresponding TH is hc¯ 3 T = (13.159) H 8πGMk It is inversely proportional to the mass M and its value is fixed by a combination of physical constants, Inserting the Planck mass s hc¯ m = ' 2.177 × 10−5 g (13.160) P l G we obtain m c2 ³m ´ ³M ´ kT = P l P l ≡ 10−7 ¯ K (13.161) H 8π M M

From a dimensional point of view, TH is derived from equating the thermal wave- length hc/kT¯ to the Schwarzschild radius. To create a particle of mass m it must be

2 kTH ∼ mc that is the Schwarzschild radius must be nearly equal to the Compton wavelength of the particle

λc ∼ h/mc¯

We can now compute the entropy of the Black Hole, since the area is (c’s and G’s are restored)

³2GM ´ A = 4π (13.162) c2 we have c6 1 d(Mc2) = dA ≡ T dS (13.163) G2 32πM where the entropy S is identified with kc3 1 S = A (13.164) Gh¯ 4 13.7. THE AREA THEOREM AND BLACK HOLE EVAPORATION 363

The ratio of the macroscopic quantity A to the microscopic quantityh ¯ ensures that the entropy S is very large. This is in accord with the idea that many different internal states of the Black Hole corresponds to the same external grav- itational field, and that information is lost from the outside world once the Black Hole is formed (Black Holes have no hair). Because of the emission of thermal quanta, M decreases by energy conservation, and thus so does A and S. This violates the Hawking’s area Theorem. To clarify the issue we need to consider the Key assumptions on which the area theorem is based. One of the postulates is that matter obeys the ”strong” energy condition, which requires that a local observer always measures positive energy densities and that there are no space-like energy fluxes. The Black Hole evaporation can be understood as pair creation in the gravitational field of the Black Hole, one member of the pair going down the Black Hole and the other going to infinity. The particle pair materialize with a space-like separation, thus there is an effective space-like flux of energy. The area theorem of classical General Relativity is then replaced by a generalized Second Law of Thermodynamics: ”in any interaction, the sum of the entropies of all Black Holes plus the entropy of matter outside Black Holes never decreases”. Simple modelling of the Hawking process. To better understand the Hawking process, consider the case of pair production (e+, e−) in a strong elec- tric field E. In , the vacuum is continuously undergoing fluctuations, where a pair of virtual particles is created and the annihilated. The electric field tends to separate the charges. If the field is strong enough, the par- ticles tunnel the quantum barrier and materialize as real particles. The critical field strength is achieved when the work done by the field in separating them by a Compton wavelength equals the energy necessary to create the particles

2 eEλc ∼ 2mc (13.165) where mc2 is the rest-mass energy, that is the minimum energy to create the particles. 364 CHAPTER 13. BLACK HOLES

In a Black Hole, the tidal gravitational force across a distance λc is GmM λ (13.166) r3 c

2 The work done is the product of this with λc. Setting r ∼ GM7c , since the maximum field intensity is nera the horizon and equating the work to 2mc2 gives

GM λ ∼ (13.167) c c2

The particles are created when their Compton wavelength equals the Schwarzschild radius. These considerations must be adjusted to the case of mass-less particles, which have no barrier to penetrate. Their production rate is controlled by the phase space available. It we equate the mean separation of black body photons,hc/kT ¯ , to the only scale associated with the Black Hole, GM/c2, we get a dimensional es- timate of the black body temperature we have already presented, see eqn.(13.160). Evaporation. The rate at which Black Holes evaporate their mass by means of the Hawking process can be expressed with the black body emission

dM dE ≡ ∼ σT 4 A (13.168) dt dt H where σ = π2k4/60¯h3c2. We mar write it as

dM ³m ´2³m ´ − ∼ B P l P l N (13.169) dt M τP l

3 −6 where B = 1/(30π8 ) ' 2.6 × 10 , mP l is the Planck mass, τP l is the Planck q 5 −44 time, τP l = hG/c¯ ' 5.6 × 10 s, and N is the number of states and particles emitted. The lifetime of a Black Hole is

E M ³ M ´ 1 ³ M ´3 tBH ≡ ≡ ∼ −2 ' τP l (13.170) E˙ M˙ M 36 mP l or ³ M ´3 t ' 1o10 N −1 yr (13.171) BH 1015 g 13.7. THE AREA THEOREM AND BLACK HOLE EVAPORATION 365

65 15 For Black Holes, TBH ∼ 10 yr. Those with mass of about 10 g, tBH ∼ 10 Gyr. Presumably these ”mini” Black Holes could only have been formed from density fluctuations during the . Their Schwarzschild radius is about 1 Fermi. If mini Black Holes were formed over a large mass range, those with mass M << 1015 g would long since have terminated their life.

What is expected to be the final stage of a Black Hole? As the mass decrease, TH and N increase so that a sort of explosion is like to occur with an energy release of

dE ³1015 g ´2 ∼ 1020 erg s−1 (13.172) dt M producing quanta with energy ³1015 g ´ hω¯ ∼ 100 Mev (13.173) M Can this emission be detected? Detailed predictions of the energy spectrum by a 1015 g Black Hole has been made. The observed energy density at about 100 Mev is 1018, i.e. ∼ 10−38 g cm−3. Calculations have shown that about 10% of the energy emitted by an exploding Black Hole is in form of photons (as opposed to neutrinos, gravitons or particle with mass). Thus the density of 1015 g Black Holes must be less than 1037 g/cm3, which is about 10−8 times the critical density to close the Universe.

Question: Does the Black Hole evaporation leave a remnant mass Mrem ' mP l? The question is still unanswered because one needs a theory of which not yet available. First Law of Thermodynamics for Black Holes. Inverting the relation- ship giving the surface area A(M,J) of a Black Hole, one gets

A 2 2 hπ( ) + 4J i1/2 M = M(A, J) = 4π (13.174) A It follows from this relation that the internal energies of two stationary Black Holes that differ in the area and angular momentum by δA and δJ, respectively, will also differ in the mass (energy) of the quantity 366 CHAPTER 13. BLACK HOLES

κ dM = dA + Ω dJ (13.175) 8π BH where ΩBH = 4πJ/(MA) is the angular velocity and ΩBH dJ is the rotational energy of the Black Hole. This relation is identical to the First Law of Thermodynamics.