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Electronic Theses, Treatises and Dissertations The Graduate School

2008 Physics of Compact Stars Jutri Taruna

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COLLEGE OF ARTS AND SCIENCES

PHYSICS OF COMPACT STARS

By

JUTRI TARUNA

A Dissertation submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Degree Awarded: Spring Semester, 2008 The members of the Committee approve the Dissertation of Jutri Taruna defended on March 7, 2008.

Jorge Piekarewicz Professor Directing Dissertation

Ettore Aldrovandi Outside Committee Member

Simon Capstick Committee Member

Paul Eugenio Committee Member

Laura Reina Committee Member

Approved:

Mark Riley, Chair Department of Department of Physics

Joseph Travis, Dean, College of Arts and Sciences

The Office of Graduate Studies has verified and approved the above named committee members.

ii This thesis is dedicated to my parents: Ch. Taruna and Lisna Taruna. May their souls rest in peace in heaven.

iii ACKNOWLEDGEMENTS

I would like to acknowledge everyone who has supported me throughout my studies. My special thanks goes to my advisor Jorge Piekarewicz for being a patient advisor, whose guidance and kindness meant so much for me. I would also like to thank Simon Capstick for the support and advice he has given me all these years at FSU. To Paul Eugenio, Laura Reina, and Ettore Aldrovandi for the time they spent as my graduate committee. To the faculty members of the nuclear theory group at FSU for being very supportive and helpful whenever I need advice and suggestions. To my best friends Alvin Kiswandhi and Suharyo Sumowidagdo for being my personal diary in the ups and downs of my graduate life. Special thanks to Haryo who despite being far away in Fermilab, has managed to keep his presence close and be my living physics dictionary. I couldn’t have done it without you. To my friends at nuclear theory group Tony Sumaryada, Naureen Ahsan and Olga Abramkina for the good time we share that has brightened my days at FSU. Thanks to all my friends Harianto Tjong, Paula Sahanggamu, and others who have enlightened my life in Tallahassee and has made it less lonely. I will miss you all. Last but not least, I owe much gratitude to my sisters whose love and support made it possible for me to achieve my dreams.

iv TABLE OF CONTENTS

List of Tables ...... vii

List of Figures ...... viii

Abstract ...... x

1. INTRODUCTION ...... 1 1.1 Stellar evolution ...... 1 1.2 Neutron stars ...... 3 1.3 Supernova neutrinos ...... 7 1.4 This study ...... 9

2. PEDAGOGICAL INTRODUCTION TO PHYSICS OF COMPACT STARS . 11 2.1 Hydrostatic Equilibrium ...... 11 2.2 Degenerate Free Fermi Gas ...... 13 2.3 White Dwarf Stars ...... 16 2.4 Neutron Stars ...... 23

3. VIRTUES AND FLAWS OF THE PAULI POTENTIAL ...... 28 3.1 Free Fermi Gas ...... 29 3.2 Pauli Potential: A New Functional Form ...... 32 3.3 Simulation Results ...... 35 3.4 Comparison to other approaches ...... 38 3.5 Finite-Size Effects ...... 42

4. EQUATION OF STATE FOR NUCLEAR PASTA ...... 45 4.1 Modeling the Nuclear Pasta ...... 46 4.2 Nuclear Matter Equation of State ...... 48 4.3 Semi Empirical Mass Formula ...... 50 4.4 Simulation Results ...... 52

5. CONCLUSIONS ...... 63 5.1 Introduction to Physics of Compact Stars ...... 63 5.2 Virtues and Flaws of the Pauli Potential ...... 64 5.3 Equation of State for Nuclear Pasta ...... 65

v A. Monte Carlo Simulation ...... 66 A.1 The estimator ...... 66 A.2 The Metropolis Algorithm ...... 67 A.3 Equilibration ...... 68

B. Woods-Saxon Potential ...... 69

BIOGRAPHICAL SKETCH ...... 77

vi LIST OF TABLES

1.1 Parameters for the sun and typical white dwarf, neutron star and black hole parameters...... 2

3.1 Strength (in MeV) and range parameters (dimensionless) for the various components of the Pauli potential...... 36

4.1 Models parameters for the spin-independent term...... 53

4.2 Binding energy per nucleon, charge radii and predictions for neutron skin .. 56

4.3 Semi empirical mass formula best fit results for both parameter sets...... 59

B.1 Table of nuclei and their Fermi momentum...... 71

vii LIST OF FIGURES

1.1 A schematic cross section of a neutron star...... 4

1.2 The neutron EoS for 18 Skyrme parameter sets...... 5

1.3 Theoretical predictions of the mass (in units of solar mass) versus radius of neutron stars for several EoSs...... 6

1.4 The Nuclear Landscape...... 7

2.1 A star in hydrostatic equilibrium...... 11

2.2 The interplay between various physical effects on the “toy model” problem of a M =1M⊙ star...... 20 2.3 Mass-versus-radius relation for white dwarf stars with a degenerate electron gas EoS...... 23

2.4 Mass-radius relations for neutron stars with a degenerate neutron gas EoS . 27

3.1 Average kinetic energy of a system of N = 1000 identical fermions at a temperature of τ =T/TF =0.05...... 37 3.2 Momentum distribution of a system of N = 1000 identical fermions at a temperature of τ = T/TF =0.05 for a variety of densities (expressed in units −3 of ρ0 =0.037 fm )...... 38 3.3 Two-body correlation function for a system of N =1000 identical fermions at a temperature of τ =T/TF =0.05 for a variety of densities (expressed in units −3 of ρ0 =0.037 fm )...... 39 3.4 Comparison between the Pauli potential introduced in this work Eq. (3.18) and earlier approaches based on Eq. (3.16)...... 40

3.5 “Kinematical” velocity distribution of a system of N =1000 identical fermions at a temperature of τ = T/TF =0.05 for a variety of densities (expressed in −3 units of ρ0 =0.037 fm )...... 42

viii 3.6 Finite-size effects on the canonical momentum distribution (left-hand panel) and the two-body correlation function (right-hand panel) ...... 43

4.1 Binding energy predicted by the Semi Empirical Mass formula for various nuclei. 51

4.2 Two-body correlation function g(r) as a function rpF ...... 53

4.3 Energy per particle of symmetric nuclear matter as a function of densities ρ/ρ0. 54

4.4 Energy per neutron of pure neutron matter as a function of densities ρ/ρ0. . 55 4.5 Binding energy per particle versus number of particles A for various nuclei. . 56

4.6 Neutron skin versus total number of particles for various nuclei ...... 57

4.7 Charge radii versus total number of particles ...... 58

4.8 Monte Carlo snapshots of a configuration of N= 800 neutrons and Z= 200 protons at density 0.025 fm−3 (left) and 0.01 fm−3 (right)...... 60

4.9 Neutron-neutron two-body correlation function ...... 61

4.10 Proton-proton two-body correlation function ...... 61

4.11 Proton-neutron two-body correlation function ...... 62

A.1 Monte Carlo thermalization step vs energy per nucleon ...... 68

B.1 Typical shapes for Woods-Saxon potential...... 70

ix ABSTRACT

This thesis starts with a pedagogical introduction to the study of white dwarfs and neutron stars. We will present a step-by-step study of compact stars in hydrostatic equilibrium leading to the equations of stellar structure. Through the use of a simple finite- difference algorithm, solutions to the equations for stellar structure both for white dwarfs and neutron stars are presented. While doing so, we will also introduce the physics of the equation of state and insights on dealing with units and rescaling the equations.

The next project consists of the development of a “semi-classical” model to describe the equation of state of neutron-rich matter in the “Coulomb frustrated” phase known as nuclear pasta. In recent simulations we have resorted to a classical model that, while simple, captures the essential physics of the nuclear pasta, which consists of the interplay between long range Coulomb repulsion and short range nuclear attraction. However, for the nuclear pasta the de Broglie wavelength is comparable to the average inter-particle separation. Therefore, fermionic correlations are expected to become important. In an effort to address this challenge, a fictitious “Pauli potential” is introduced to mimic the fermionic correlations. In this thesis we will examine two issues. First, we will address some of the inherent difficulties in a widely used version of the Pauli potential. Second, we will refine the potential in a manner consistent with the most basic properties of a degenerate free Fermi gas, such as its momentum distribution and its two-body correlation function. With the newly refined potential, we study various physical observables, such as the two-body correlation function via Metropolis Monte-Carlo simulations.

x CHAPTER 1

INTRODUCTION

Compact objects –white dwarf stars, neutron stars, and black holes– are born when normal stars “die”, that is when the stars can no longer generate thermonuclear fusion to counter act gravity [1, 2]. At this point the stars will need to generate another internal mechanism to prevent themselves from collapsing. This will be the focus of my dissertation. We will first examine the simplest mechanism, in which there are no interactions between constituents of the stars other than through the Pauli exclusion principle. Later in the dissertation we will discuss a more realistic model incorporating all the essential interactions.

1.1 Stellar evolution

Thermonuclear fusion drives stars through many stages of combustion; the hot center of the stars allows hydrogen to fuse into helium. Once the core has burned all available hydrogen it will contract until another source of support becomes available. As the core contracts and heats, transforming gravitational energy into kinetic (or thermal) energy, the burning of the helium ashes begins. For stars to burn heavier elements, higher temperatures are necessary to overcome the increasing Coulomb repulsion and allow fusion through quantum-mechanical tunneling. Thermonuclear burning continues until the formation of an iron core. Once iron –the most stable of nuclei– is reached, fusion becomes an endothermic process. As a result, no further energy can be produced by nuclear fusion. When thermonuclear fusion can no longer support the stars against gravitational collapse, either because they are not massive enough or because they have developed an iron core, the stars die and compact objects are ultimately formed.

1 Table 1.1: Parameters for the sun and typical white dwarf, neutron star and black hole parameters.

Massa Radiusb Mean Density Surface Potential Object (M) (R) (g cm−3) (GM/Rc2) −6 Sun M⊙ R⊙ 1 10 −2 6 −4 White dwarf M⊙ 10 R⊙ 10 10 ≤ ∼ −5 ≤ 15 ∼ −1 Neutron star 1–3 M⊙ 10 R⊙ 10 10 a ∼ 33 ∼ ≤ ∼ M⊙ = 1.989 x 10 g b 10 R⊙ = 6.9599 x 10 cm

The primary factor in determining what type of compact objects the stars will form, is their initial mass. White dwarfs are believed to originate from light stars with masses M ≤ 4 solar mass M⊙. A good example of this type of star is our Sun. The Sun is not massive enough to produce fusion of heavier elements. Once all of the hydrogen and helium in the core has been burned, it will die as a white dwarf star [3]. Towards the final stages of burning, the star will expand and expel most of the outer matter to create a planetary nebula. At the beginning, the core contracts and heats up through conversion of gravitational energy into thermal kinetic energy. However, at some point the Fermi pressure of the degenerate electrons begins to dominate, the contraction is slowed down, and the core becomes a compact object known as a white dwarf, cooling steadily towards the ultimate cold, dark, static black dwarf star. Neutron stars, on the other hand, result from one of the most cataclysmic events in the universe, the death of a star with an initial mass much bigger than the mass of our Sun. During the collapse of the core, a supernova shock develops, ejecting most of the mass of the star into the interstellar space and leaving behind an extremely dense core –the neutron star. As the star collapses, it becomes energetically favorable for electrons to be captured by protons, making neutrons and neutrinos. The neutrinos carry away 99% of the gravitational binding energy of the compact object, leaving neutrons behind to support the star against further collapse [4]. The pressure provided by the degenerate neutrons, like degenerate electron pressure for white dwarf stars, has a limit on the mass it can bear. Beyond this limiting mass, there is no source of pressure that can prevent gravitational contraction. If such is the case, then the star will continue to collapse into a black hole.

2 Relative to normal stars of comparable mass, compact stars have much smaller radii and therefore, much stronger surface gravitational field as shown in Table 1.1 [5]. Hence, to understand the physics of compact objects other than white dwarfs we need to go beyond Newtonian mechanics and take into account the effect of general relativity [6, 7].

1.2 Neutron stars

Shortly after the discovery of the neutron in 1932 [8], Baade and Zwicky proposed the idea of a neutron star; a gravitationally bound, dense compact object, as the remnant of a supernova explosion [9]. In 1939 Oppenheimer and Volkoff [10], assuming that the star is a gravitationally bound object supported by neutron degeneracy pressure, made their first theoretical calculation of the properties of the star. They obtained a maximum neutron-star 15 3 mass of around 0.7 M⊙, radii of about 10 km and a central density ρc of about 5x10 gr/cm 3 fm−3 which is about 20 times that of normal nuclei ρ 0.15 fm−3. Thus, the predicted ≈ 0 ≃ star was one of the densest form of matter in the universe. However, the star remained but a theoretical conjecture for another twenty eight years. In 1967, a major breakthrough came when Jocelyn Bell and her advisor Antony Hewish [11] discovered the first pulsar–a rapidly rotating neutron star. Since then, observational discoveries and theoretical predictions have made these objects powerful tools in understanding matter on Heaven and Earth. Fig. 1.1 shows a schematic cross section of a typical neutron star [12]. In the outer part of the star, due to Coulomb repulsion, nuclei form a body centered cubic (bcc) lattice immersed in roughly uniform sea of electrons to make it charge neutral. This region is referred to as the outer crust. As the density increases, weak interactions come into play making the nuclei neutron rich via electron capture. At density around 4 x 1011 gr/cm3 nuclei become so neutron rich that the last occupied neutron levels are no longer bound; neutrons begin to drip out of nuclei. This region is known as the inner crust of the star. Below this region, at the core of the star with density around 1014 gr/cm3, nuclei disappear and the system dissolves into uniform neutron-rich matter. In the region between the inner crust and outer core, at densities around 1013 – 1014 gr/cm3 , competition between nuclear attraction and Coulomb repulsion leads to a very complex ground state that involve various shapes. This region is referred to as the “nuclear pasta” phase. In this sub-nuclear density region,

3 Figure 1.1: A schematic cross section of a neutron star. The possible hadronic components of the matter are listed, and the horizontal scale gives estimates of the radial dimension (not drawn to scale)

matter can be described very well in terms of interacting protons, neutrons and electrons. As we go deeper towards the center of the star and thus increasing density and pressure, the strong interactions play an increasingly important role. Hence, we might have to include the contributions from heavier baryons or even their quark and gluon constituents. The extreme density range in a neutron star poses challenging problems, since all four known fundamental interactions (strong and weak nuclear forces, electromagnetic, and gravity) play significant roles. Hence a neutron star serves as a melting pot for nuclear, particle, and astrophysics studies [13, 14, 15, 16]. One of the underlying problems in studying neutron stars is our limited understanding of the equation of state (EoS) for nuclear matter. So far, the available experimental data can only probe stable nucleonic matter around nuclear matter saturation density. As a result, the EoS of nuclear matter at extreme densities (both at sub-saturation and high densities) is not very well constrained, as shown in Fig. 1.2 [17]. This also implies that the symmetry energy, which is the difference in energy between pure neutron matter and symmetric nuclear matter, is also largely uncertain. A consequence of

4 Figure 1.2: The neutron EoS for 18 Skyrme parameter sets. The filled circles are the Friedman-Pandharipande (FP) variational calculation and the crosses are SkX. The neutron density is in units of neutron/fm3.

this uncertainty is that different models predict different masses and radii of neutron stars, as shown in Fig. 1.3 [18]. It has been known that stars calculated with a stiffer symmetry energy EoS will yield greater maximum masses and larger radii than stars derived from a soft equation of state [19]. Thus constraining the EoS of neutron rich matter is essential and remains a fundamental problem in nuclear physics and astrophysics [20, 21, 22, 23, 24]. A potential constraint on the EoS will come from the Lead Radius Experiment (“PREX”) at the Jefferson Laboratory. This experiment will use the parity violating weak neutral 208 interaction to measure the RMS neutron radius Rn of Pb to 1% accuracy. Thus it will provide the first accurate measurement of Rn, and hence of the neutron skin, which is defined as the difference between the root-mean-square radii of the neutron relative to that of the proton [25]. A large neutron skin implies a greater neutron matter pressure (∂E/∂ρ). The same phenomenon also applies to neutron stars as both –the radii of neutron rich nuclei and the radii of neutron stars– are governed by the same equation of state. This strong correlation is understandable since it is the same neutron pressure that support a neutron star against

5 Figure 1.3: Theoretical predictions on mass (in units of solar mass) versus radius of neutron stars for several EoSs. The left hand side is for stars containing nucleons and, in some cases, hyperons. The right hand side is for stars containing more exotic components, such as mixed phases with kaon condensates of strange quarks, or pure strange quark matter stars.

gravitational collapse. Hence, models that predict thicker neutron skins will most likely predict neutron stars with larger radii [26]. Similar events are happening at other facilities as well. The Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory and the Facility for Rare Isotope Beams (FRIB) will also probe large neutron-proton asymmetries, thus providing constraints on the equation of state of asymmetric matter. On the other hand, the operation of more powerful telescopes has also started to turn neutron stars from just theoretical curiosities into powerful observational tools [27, 28]. Since neutron stars contain highly asymmetric matter (N Z) these observations will also bring ≫ a better understanding of nuclear matter beyond the valley of stability. A chart of nuclei is displayed in Fig. 1.4. The black squares represent stable nuclei and nuclei with half- lives comparable to or longer than the age of the Earth. Stable light nuclei lie close to the N=Z line, but heavier nuclei require an excess number of neutrons to be stable. The stable nuclei are surrounded by the yellow area representing the present limits of unstable nuclei for

6 Figure 1.4: The Nuclear Landscape. The black squares represent stable nuclei and the yellow squares indicate unstable nuclei that have been studied in the laboratory. The unstable nuclei yet to be explored are indicated in green (Terra incognita). The red lines show the magic numbers.

which there is experimental information. The green area comprises the unknown, proton and neutron rich regions. A better understanding of neutron stars will be beneficial in unveiling the neutron rich terra incognita.

1.3 Supernova neutrinos

Neutrinos play an important role in supernova explosions due to their weak interaction with matter. As explained in section 1.1, during the collapse of the star, electrons become energetic enough to react with protons through inverse beta decay (electron capture) making neutrons and neutrinos. The neutrinos (νe) escape and carry away most of the energy, while the newly formed neutrons then undergo beta decay emitting away electron anti neutrinos

7 (¯νe), i.e.

p + e− n + ν (electron capture) , (1.1a) → e n p + e− +¯ν (beta decay) . (1.1b) → e

The above processes of absorption and reemission of free electrons are responsible for a major energy loss in the star due to the emission of neutrinos and anti neutrinos [29]. The pairs of reactions in Eq. (1.1a) and Eq. (1.1b) were proposed by George Gamow and Mario Schoenberg in 1941 as the mechanism for stellar collapse and supernova explosions later known as the direct Urca-process. As Gamow recounted “We called it the Urca process, partially to commemorate the casino in which we first met, and partially because the Urca process results in a rapid disappearance of thermal energy from the interior of a star, similar to the rapid disappearance of money from the pocket of the gamblers on the Casino da Urca” [30]. At this stage the star is essentially transparent to the neutrinos, and thus neutrinos can escape effortlessly through the body of the star. However, this process does not last very long, because the stellar core very rapidly becomes opaque to neutrinos. The transparency of matter to neutrinos depends on the ratio of neutrino mean free path λ and the radius R of the system. If λ R then matter is transparent, otherwise matter is opaque. As a ≫ result of the opacity, neutrinos are no longer freely streaming out of the core. Instead, they will start to interact with matter. Because neutrinos are the main transporters of energy in stellar collapse, our understanding of neutrinos and their interaction with matter is essential in determining the mechanism of supernova explosion and hence the type of compact objects formed as the remnant of the collapse. Up until 1973, neutrinos had only been observed to participate in charged-current weak interactions via the exchange of W+ or W− boson. However, in the summer of 1973, the predicted neutral-current interaction was experimentally confirmed. This new type of weak interaction via the exchange of a neutral boson (Z0) opened a new chapter in the study of neutrinos. Neutrinos could merely scatter from nucleons or electrons without changing the charge of the particles. In 1975, Tubbs and Schramm found that neutral-current neutrino scattering might boost the neutrino scattering cross section off nucleons through coherent neutrino scattering [31].

8 Such is the case when the density of the core reaches 1012 gr/cm3, where the neutrino ≃ nucleus elastic scattering is thought to temporarily trap the neutrinos. This trapping is important for the electron-per-baryon fraction Ye of the supernova core. It hinders further conversion of electrons to neutrinos and thus provides the lepton degeneracy pressure that helps support the star from collapsing.

1.4 This study

In this study I focus on two main ideas. The first is a pedagogical introduction to the physics of compact stars focusing on white dwarfs and neutron stars in hydrostatic equilibrium. As will be shown later, the only missing ingredient in this formalism is the equation of state. I will start with the simplest case using the free Fermi gas EoS. This formalism is based on the assumption that there is no interaction between particles, and thus the pressure is solely generated by the Pauli exclusion principle. Using the free Fermi gas EoS, I will study the relations between mass and radius of white dwarfs and neutron stars. While the free Fermi gas EoS works very well for white dwarfs, the same conclusion can not be drawn for neutron stars. Therefore, in the second part of this dissertation I will focus on constructing a more realistic model of the equation of state by incorporating all the essential interactions. This work is an extension of previous studies [32, 33], where I will add a spin-dependent term to the nuclear pasta Hamiltonian. The spin-dependent contribution is introduced via a semi-classical “Pauli potential” that mimics fermionic correlations of a quantum free Fermi gas. The reason for this modification is to generalize our model such that it can include both the vector and axial-vector contribution in studying neutrino–pasta scattering. In the process I need to re-adjust and optimize variable parameters based on the constraints from free Fermi gas and nuclear matter properties. The manuscript is organized as follows. In Ch. 2 I start with a pedagogical introduction to the physics of white dwarfs and neutron stars. The discussion is based on the assumption of a static and spherically symmetric star in hydrostatic equilibrium. I will present results of the mass-radius relationships for white dwarfs and neutron stars using the free Fermi gas equation of state. In Ch. 3 I discuss the formulation of semi-classical “Pauli potential”. I will present the existing Pauli potential models and the problems encountered in these

9 formalisms. Next, I introduce a modified model to overcome these problems. These two chapters are essentially excerpts from papers I wrote with Piekarewicz and collaborators [34, 35]. In Ch. 4 I introduce a “semi-classical” model for nuclear pasta. Nuclear pasta is a frustated system, which is characterized by the inability to minimize all interactions (short- range nuclear interactions and long-range Coulomb repulsions) simultaneously. I will present the full Hamiltonian for the nuclear pasta and will discuss the optimization of parameters sets based on the free Fermi gas two-body correlation function, and constraints from symmetric nuclear matter and finite nuclei properties. Using this refined Hamiltonian, I will also discuss the nuclear pasta two body correlation function via Monte Carlo simulations. Finally a summary and conclusion are presented in Ch. 5. A brief discussion of the Monte Carlo algorithm used in this work is included in Appendix A. Lastly, detailed calculations of the Fermi momentum for finite nuclei are given in Appendix B.

10 CHAPTER 2

PEDAGOGICAL INTRODUCTION TO PHYSICS OF COMPACT STARS

2.1 Hydrostatic Equilibrium

In this section we will focus our attention on a static and spherically symmetric star, the so called Schwarzschild star. We start by considering a small volume element located between radii r and r + ∆r, of cross-sectional area ∆A, and volume ∆r ∆A (see Fig. 2.1). The gravitational force (Fg) acting on the small volume element is

∆ P(r+ r)

∆Α

P(r) F ∆r

r

Figure 2.1: A star in hydrostatic equilibrium. The radial force (both internal pressure and gravity) acting on a small mass element a distance r from the center of the star.

11 GM(r)∆m F = , (2.1) g − r2 where G represents the gravitational constant, ∆m is the mass of the small volume element, and M(r) is the mass enclosed by a spherical shell of radius r which can be expressed in differential form as dM(r)=4πr2ρ(r)dr , (2.2) with ρ(r) denotes the mass density of the star. If the internal pressure (P ) acting on the outer surface of the volume element is not equal to the pressure acting on the inner surface, then the total radial force acting on the volume element is equal to GM(r)∆m d2r F = P (r + ∆r)∆A + P (r)∆A = ∆m , (2.3) r − r2 − dt2 with the minus sign referring to the vector in rˆ direction. Expanding the above equation − to lowest order in ∆r we get GM(r)ρ(r) dP d2r = ρ(r) . (2.4) − r2 − dr dt2 Assuming the star is in hydrostatic equilibrium (¨r =r ˙ 0) we arrive at the fundamental ≡ equations describing the structure of Newtonian stars. That is, dP GM(r)ρ(r) = , P (r =0) P ; (2.5a) dr − r2 ≡ c dM = +4πr2ρ(r) , M(r =0) 0 . (2.5b) dr ≡ It is simple to see that in hydrostatic equilibrium, the pressure of the star is a decreasing (or at least not increasing) function of r; otherwise the star collapses. Further, the enclosed mass is obviously an increasing (or non-decreasing) function of r. Note that the radius of the star R is defined as the value of r at which the pressure vanishes [P (R) = 0] and the mass of the star as M =M(R). However for more massive stars, where general relativity plays an important role, we have to modify Eq. (2.5a) to include corrections from general relativity. For a Schwarzschild star, we are going to use the Tolmann-Oppenheimer-Volkoff (TOV) equations, that is

dP (r) GM(r) (r) P (r) 4πr3P (r) 2GM(r) −1 = E 1+ 1+ 1 , (2.6) dr − c2r2 (r) c2M(r) − c2r  E    where (r) is the energy density. E 12 The first term on the right hand side corresponds to the non-relativistic case, that is when P/ 1 and 2GM/c2r 1. The second and third terms arise from the pressure E ≪ ≪ being a form of energy density, and the last term is the correction due to the curvature of space in the strong gravitational field of the star. Eq. (2.6) together with dM 4πr2 (r) = E . (2.7) dr c2 determine the structure of a relativistic star. The above set of equations, together with their associated boundary conditions, must be completed by an equation of state (EoS), namely a relation P = P (ρ) between the density and pressure. For pedagogical purposes we will start with the EoS of a zero-temperature Fermi gas [36]. Even though stars have high temperature, the zero-temperature Fermi gas assumption is reasonable, based on the following reason. For a compact object such as a white dwarf star, the number density is very high and so is the Fermi energy. Typical electron Fermi energies are of the order of 1 MeV which correspond to a Fermi temperature of T 1010 K. F ≃ As the temperature of the system is increased (say from T =0 to T =106 K) electrons try to jump to a state higher in energy by an amount of the order of kBT , but fail, as most of these transitions are Pauli blocked. Only those high-energy electrons that are within k T 100 eV from the Fermi surface can make the transition, but those represent a tiny B ≃ fraction T/T 10−4 of the electrons in the star. Hence, for the purpose of computing the F ≃ pressure of the system, it is extremely accurate—to 1 part in 104—to describe the electrons as a Fermi gas at zero temperature. The same assumption will be made for neutron stars. Indeed, all of the systems dealt with in this paper will be treated as degenerate Fermi gas at zero temperature. Although core temperatures in neutron stars may increase by as much as two orders of magnitude relative to that in white dwarf stars, the density typically increases by 8-9 orders of magnitude. Thus, it is safe to use a zero temperature Fermi gas of neutrons to model the pressure of the system. However, the density in a neutron star is so large that interactions between nucleons may no longer be ignored. This will be discussed in Chap. 4 2.2 Degenerate Free Fermi Gas

The main assumption behind the Fermi gas hypothesis is that no correlations (or interactions) are relevant to the system other than those generated by the Pauli exclusion principle. To

13 start, the Fermi wavenumber kF is defined as the momentum of the fastest moving fermion and is solely determined by the number density (n N/V ) of the system. That is, ≡ V k3 N =2 Θ(k k )=2 d3k Θ(k k )= V F , (2.8) F −| | (2π~)3 F −| | 3π2~3 k X Z or equivalently 2 3 1/3 kF = 3π ~ n . (2.9)

In Eq. (2.8), Θ(x) represents the Heaviside (or step)
function. Having defined the Fermi wavenumber kF, the energy density of the system is obtained by simply measuring the energy of a system in which all single-particle momentum states are progressively filled in accordance with the Pauli exclusion principle. For a degenerate (spin-1/2) Fermi gas at zero temperature, exactly two fermions occupy each single-particle state below the Fermi momentum pF = ~kF; all remaining states above the Fermi momentum are empty. In this manner one obtains the following expression for the energy density: d3k E/V =2 Θ(k k )ǫ(k) , (2.10) E ≡ (2π~)3 F −| | Z where ǫ(k) is the single-particle energy of a fermion with momentum k and E is the energy. In what follows, the most general free-particle dispersion (energy vs. momentum) relation is assumed, namely, one consistent with the postulates of special relativity. That is, ~k c ǫ(k)= (~k c)2 +(mc2)2 = mc2 1+ x2 , with x F . (2.11) F F F ≡ mc2   p q Performing the integral in Eq. (2.10) we obtain

= (x ) , (2.12) E E0E F where is a dimensionful constant that may be written using elementary dimensional E0 analysis (mc2)4 , (2.13) E0 ≡ (~c)3 and (x ) is a dimensionless function of the single variable x given by E F F 1 xF 1 (x ) x2√1+ x2 dx = x 1+2x2 1+ x2 ln x + 1+ x2 . E F ≡ π2 8π2 F F F − F F Z0 h
q  q i(2.14)

14 The pressure of the system may now be directly obtained from the energy density by using the following thermodynamic relation, which is valid only at zero temperature: ∂E ∂(V ) P = = E = P P. (2.15) − ∂V − ∂V 0  N,T ≡0  N,T ≡0 In analogy to the energy density, dimensionful and dimensionless quantities for the pressure have been defined: (mc2)4 P = = , (2.16a) 0 E0 (~c)3 x ′ P (x ) F (x ) (x ) . (2.16b) F ≡ 3 E F − E F h i Here ′ is the first derivative of the energy density. With an expression for the pressure in E hand, we are finally in a position to compute its derivative with respect to xF (a quantity that is labeled as η). As we shall see in the next section, η — a function closely related to the zero-temperature incompressibility — is the only property of the EoS that Newtonian stars are sensitive to. We obtain

4 dP xF ′′ 2 ′ P0 xF η = P0 (xF) (xF) = 2 , (2.17) ≡ dxF 3 E − 3E 3π 2   1+ xF q where ′′ is the second derivative of the energy density. The above expression has a E surprisingly simple form that depends on the energy density only through its derivatives. Alternatively, we could have bypassed the above derivation in favor of the following general relation valid for a zero-temperature Fermi gas: dP dǫ = n F , (2.18) dxF dxF where the Fermi energy ǫF is defined as the energy of the fastest moving electron. In view of Eq. (2.18), it seems unnecessary to go through the trouble of computing the energy density and the corresponding pressure if all that is required is the dependence of the Fermi energy on xF. However, this is a necessary step to be taken when we want to include general relativity. While Newtonian stars indeed depend exclusively on η, the structure of Schwarzschild stars (such as neutron stars) are highly sensitive to corrections from general relativity. These corrections depend on both the energy density and the pressure.

15 2.3 White Dwarf Stars 2.3.1 Toy Model of White Dwarf Stars

Before returning to attempt a numerical solution to the equations of hydrostatic equilibrium, we consider as a warm-up exercise a simple, yet highly illuminating, toy model of a white dwarf star. Assume a white dwarf star with a uniform, spherically symmetric mass density of the form ρ =3M/4πR3 , if r R ; ρ(r)= 0 ≤ (2.19) (0 , if r > R , where M and R are the mass and radius of the star, respectively. For such a spherically symmetric star, the gravitational energy released during the process of “building” the star is given by R EG = 4πG M(r)ρ(r)r dr , (2.20a) − 0 r Z M(r)= 4πr′2ρ(r′) dr′ . (2.20b) Z0 Further, if the white dwarf star has a uniform mass density as assumed in Eq. (2.19), it becomes straightforward to perform the above two integrals. Hence, the gravitational energy released by “building” such a star is given by 3 GM 2 E (M, R)= . (2.21) G −5 R From the above relation, we conclude here that without a source of support against gravity, a star with a fixed mass M will minimize its energy by collapsing into an object of zero radius, namely, into a black hole. We know, however, that white dwarf stars are supported by the quantum-mechanical pressure from its degenerate electrons, which (at temperatures of about 106 K) are fully ionized in the star (recall that 1 eV 104 K). In what ≃ follows, we assume that electrons provide all the pressure support of the star but none of its mass, while nuclei (e.g., 4He, 12C, ... ) provide all the mass but none of the pressure. The electronic contribution to the mass of the star is inconsequential, as the ratio of electron to nucleon mass is approximately equal to 1/2000. The energy of a degenerate electron gas was computed in the previous section. Using Eqs. (2.9) and (2.12) one obtains,

2 2 (xF) EF(M, R)=3π Nmec E 3 , (2.22) xF

16 where me is the rest mass of the electron. Naturally, the above expression depends on the mass and the radius of the star, although this dependence is implicit in xF. While the toy- model problem at hand is instructive of the simple, yet subtle, physics that is displayed in compact stars, it also serves as a useful framework to illustrate how to scale the equations.

2.3.2 Scaling the Equations

One of the great challenges in astrophysics, and the physics of compact stars is certainly no exception, is the enormous range of scales that one must simultaneously address. For example, in the case of a white dwarf star it is the pressure generated by the degenerate electrons (constituents with a mass of m =9.110 10−31 kg) that must support stars with e × 30 masses comparable to that of the Sun (M⊙ =1.989 10 kg). This represents a disparity in × masses of 60 orders of magnitude! Without properly scaling the equations, there is no hope of dealing with this problem numerically. We start by defining f E /Nm c2 from Eq. (2.22), a quantity that is both dimension- F ≡ F e less and intensive (i.e., independent of the size of the system). That is,

2 (xF) fF(xF)=3π E 3 , (2.23) xF where the dimensionless Fermi momentum xF was introduced in Eq. (2.11) and a closed-form expression for (x ) has been displayed in Eq. (2.14). Note that the scaled Fermi momentum E F xF quantifies the importance of relativistic effects. At low density the Fermi momentum is small (x 1), therefore the corrections from special relativity are negligible and electrons F ≪ behave as non relativistic Fermi gas. In the opposite high-density limit when the Fermi momentum is large (x 1) the system becomes ultra-relativistic and the “small” (relative F ≫ to the Fermi momentum) electron mass may be neglected. We shall see that in the case of white dwarf stars, the most interesting physics occurs in the x 1 regime. F ∼ The dynamics of the star consists of a tug-of-war between gravity, that favors the collapse of the star, and electron-degeneracy pressure that opposes the collapse. To efficiently compare these two contributions, the contribution from gravity to the energy must be scaled accordingly. Thus, in analogy to Eq. (2.23), we form the corresponding dimensionless and intensive quantity for the gravitational energy (f E /Nm c2) in Eq. (2.21). One obtains, G ≡ G e 3 GM M 3 GM m f (M, R)= = N . (2.24) G −5 Rc2 Nm −5 Rc2 Y m    e     e e  17 Note that in the above equations we have assumed that the mass of the star, M = AmN , may be written exclusively in terms of its baryon number A and the nucleon mass mN (the small difference between proton and neutron masses is neglected). This is an accurate approximation, as both nuclear and gravitational binding energies per nucleon are small relative to the nucleon mass. Further, Y Z/A represents the electron-per-baryon fraction e ≡ 4 12 56 of the star (e.g., Ye =1/2 for He and C, and Ye =26/56 in the case of Fe).

The final step in the scaling procedure is to introduce dimensionful mass M0 and radius

R0 quantities that, when chosen wisely, will embody the natural mass and length scales in the problem. To this effect we define

M M/M and R R/R . (2.25) ≡ 0 ≡ 0 In terms of these natural mass and length scales, the gravitational contribution to the energy of the system in Eq. (2.24) takes the following form:

3 GM0 mN M fG(M, r)= 2 . (2.26) − 5 R0c Yeme R      While the dependence of the above equation on M and R is already explicit, the Fermi gas contribution to the energy depends implicitly on them through xF. To expose explicitly the dependence of fF on M and R we perform the following simple manipulation aided by relations derived earlier in Sec. 2.2. One obtains ~k c 3 9π M ~c/m c2 3 M x3 = F = Y 0 e . (2.27) F m c2 4 e m R 3  e  "   N   0  # R While we have already referred earlier to M0 and R0 as the natural mass and length scales in the problem, their values have yet to be determined. Thus, they are still at our disposal. Their values will be fixed by adopting the following choice: let the “complicated” expressions enclosed between square brackets in Eqs. (2.26 and 2.27) be set equal to one. 3 GM m 9π M ~c/m c2 3 0 n = Y 0 e =1 , (2.28) 5 R c2 Y m 4 e m R   0   e e  "   n   0  # This choice implies the following values for white dwarf stars with an electron-to-baryon ratio equal to Ye =1/2: 5 √ −3/2 2 2 M0 = 15παG mnYe = 10.599 M⊙ Ye =2.650 M⊙ , (2.29a) 6 Y−→e=1/2

√15π −1/2 ~c R0 = αG 2 Ye = 17 250 km Ye = 8 623 km . (2.29b) 2 m c Y−→e=1/2  e  18 Here the dimensionless strength of the gravitational coupling between two nucleons has been introduced as Gm2 α = N =5.906 10−39 . (2.30) G ~c × The aim of this toy-model exercise is to find the minimum value of the total (gravitational plus Fermi gas) energy of the star as a function of its radius for a fixed value of its mass. Before doing so, however, a few comments are in order. First, from merely scaling the equations and with no recourse to any dynamical calculation we have established that white dwarf stars have masses comparable to that of our Sun but typical radii of only 10 000 km

(recall that the radius of the Sun is R⊙ 700 000 km). Further, we observe that while R ≈ 0 scales with the inverse electron mass, the mass scale M0 is independent of it. This suggests that neutron stars, where the neutrons provide all the pressure and all the mass, will also have masses comparable to that of the Sun but typical radii of only about 10 km (i.e., 2000 times smaller than those found in white dwarf stars). Now that the necessary “scaling” machinery has been developed, we return to our original toy-model problem. Taking advantage of the scaling relations, the energy per electron in units of the electron rest energy is given by the following remarkably simple expression:

2/3 2 (xF) f(M, xF)= fG(M, xF)+ fF(xF)= M xF +3π E 3 . (2.31) − xF The mass-radius relation of the star may now be obtained by demanding hydrostatic equilibrium: ∂f(M, x ) F =0 . (2.32) ∂x  F M While a closed-form expression has already been derived for the energy density (x ) in E F Eq. (2.14), it is instructive to display explicit non-relativistic and ultra-relativistic limits, both of which are very simple. These are given by

3 2 1+ 10 xF , if xF 1 ; fF(xF)= ≪ (2.33) 3 x , if x 1 . ( 4 F F ≫ To conclude this section, Fig. 2.2 has been included to illustrate how simple, within the present approximation, it is to compute the radius of an arbitrary mass star (see Fig. 2.2).

For the present example, a 1 M⊙ star has been used. First, scales for input quantities

19 Figure 2.2: The interplay between various physical effects on the “toy model” problem of a M =1M⊙ star. The negative of the derivative of the gravitational energy is displayed as the lower horizontal line. The derivative of the Fermi energy in the ultra-relativistic limit is displayed as the upper horizontal line. The (blue) line displays the derivative of the Fermi gas energy in the non-relativistic limit. Finally, the exact Fermi gas expression, which interpolates between the non-relativistic and the relativistic result, is displayed with the (red) curve.

such as the dimensionful mass (M0) and length (R0), are defined. Next, energies and their derivatives are computed and a plot displaying the latter is generated. The derivative of the gravitational energy (actually the negative of it) is constant and is displayed as the lower horizontal line. Similarly, the derivative of the Fermi energy in the ultra-relativistic limit (upper horizontal line) is also a constant equal to 3/4 independent of the mass of the star. The (blue) line with a constant slope displays the derivative of the Fermi gas energy in the non-relativistic limit. Finally, the exact Fermi gas expression, which interpolates between the non-relativistic and the relativistic result, is displayed with the (red) curve. The equilibrium density of the star is obtained from the intersection of the red and blue lines with the gravitational line. In the non-relativistic case the solution may be computed 2/3 analytically to be xF0 =5M /3. However, this non-relativistic prediction overestimates the Fermi pressure and consequently also the radius of the star. The non-relativistic predictions

20 for the radius of a 1 solar-mass white dwarf star is RNR =7162 km. In contrast, the result with the correct relativistic dispersion relation in Eq. (2.11) is considerably smaller RRel =4968 km. Yet an even more dramatic discrepancy emerges among the two models. While the non- relativistic result guarantees the existence of an equilibrium radius for any value of the star’s 1/3 mass (RNR =3/5M ), the correct dispersion relation predicts the existence of an upper limit beyond which the pressure from the degenerate electrons can no longer support the star against gravitational collapse. This upper mass limit, known as the Chandrasekhar mass, is predicted in the simple toy model to be equal to:

3/2 MCh = (3/4) M0 =1.72M⊙ . (2.34)

As it will be shown later in Fig. 2.3, accurate numerical results yield (for Ye = 1/2) a

Chandrasekhar mass of MCh =1.44 M⊙. Thus, not only does the toy model predict the existence of a maximum mass star, but it does so with an 80% accuracy. Note that it is important not to confuse this mass with the original mass of the star forming a white dwarf which is around 4 M , since the collapsing star will loose some of its initial mass to form ≤ ⊙ a white dwarf star.

2.3.3 The Real Model of White Dwarf Stars

We now return to the exact (numerical) treatment of white dwarf stars. While the toy-model problem developed in Sec. 2.3.1 provides a particularly simple framework to understand the interplay between gravity, quantum mechanics, and special relativity, a quantitative description of the systems demands the numerical solution of the hydrostatic equations [Eqs. (2.5a and 2.5b)]. For the present treatment, however, one continues to assume that the equation of state is that of a simple degenerate Fermi gas as discussed in Sec. 2.2. In this case the equation of state is known analytically and it is convenient to incorporate it directly into the the differential equation. In this way Eq. (2.5a) becomes

dx GM(r)ρ(r) F = , (2.35) dr − r2η where the equation of state enters only through a quantity directly related to the zero- temperature incompressibility. This quantity, η = dP/dxF, was defined and evaluated in

Eq. (2.17). Moreover, the density of the system ρ(r) can be expressed in terms of xF. It is

21 given by m c2 3 m ρ = e N x3 . (2.36) ~c 3π2Y F   e At this point all necessary relations have been derived and the equations of hydrostatic equilibrium in Eqs. (2.5a and 2.5b) using the equation of state in Sec. 2.2 may now be written in the following form:

1+ x2 dx GM m M F F = 0 n , x (r =0) x ; (2.37a) dr − R c2 Y m r2 q x F ≡ Fc  0   e e  F −1 dM 3π M ~c/m c2 3 =+ Y 0 e r2x3 , M(r =0) 0 . (2.37b) dr 4 e m R F ≡ "   n   0  # Here the dimensionless distance r defined in Eq. (2.25) and the central (scaled) Fermi momentum xFc have been introduced. The structure of the above set of differential equations indicates that our goal of turning Eqs. (2.5a and 2.5b) into a well-posed problem, by directly incorporating the equation of state into the differential equations, has been accomplished.

But we have done better. By defining the natural mass and length scales of the system (M0 and R0) according to Eq. (2.28), the two long expressions in brackets in the Eqs. (2.37a and 2.37b) reduce to the simple numerical values of 5/3 and 1/3, respectively. Finally, then, the equations of hydrostatic equilibrium describing the structure of white dwarf stars are given by the following expressions: dx F = f(r; x , M) , x (r =0) x ; (2.38a) dr F F ≡ Fc dM = g(r; x , M) , M(r =0) 0 , (2.38b) dr F ≡ where the two functions on the right-hand side of the equations (f and g) are given by

1+ x2 5 M F f(r; x , M) and g(r; x , M)=+3 r2x3 . (2.39) F 2 q F F ≡−3 r xF This coupled set of first-order differential equations may now be solved using standard numerical techniques, such as the Runge Kutta algorithm [37]. As in the toy-model problem, solutions is presented in Fig. 2.3 for the full relativistic dispersion relation as well as the non- relativistic approximation, where, in the latter, case the square-root term appearing in the function f(r; xF, M) is set to one.

22 Figure 2.3: Mass-versus-radius relation for white dwarf stars with a degenerate electron gas EoS. While the non-relativistic calculation guarantees the existence of an equilibrium radius for any value of the stars’ mass, the correct dispersion relation (for Ye =1/2) predicts the existence of an upper mass limit known as the Chandrasekhar mass of MCh =1.44 M⊙.

2.4 Neutron Stars

The escape velocity from the surface of a neutron star is about 50% of the speed of light. This fact alone already suggests the importance of general-relativistic corrections for this group of stars, that also includes the hitherto undiscovered hybrid and quark stars. Still, the calculations for these stars can be patterned after the white dwarf calculations in Sec. 2.3.3 by incorporating a few important differences. First, in a white dwarf star the pressure is produced by the quantum pressure of the degenerate electrons, while in a neutron star it is due to the degenerate neutrons. Neutrons become degenerate at a much larger density so neutron stars have masses comparable to those of white dwarf stars, but significantly smaller radii. Second, in a white dwarf star gravity is weak enough that it can be described equally well by Newtonian or general-relativistic dynamics. In Sec. 2.3.3 Newtonian mechanics was chosen for simplicity. However, the significantly smaller radii of neutron stars demand a general-relativistic treatment. As we will soon show, Eq. (2.5b) remains basically unaltered:

23 the mass density gets replaced by the full energy density that includes rest mass, kinetic, and binding energy effects. In contrast, Eq. (2.5a) picks up three correction factors with all of them working in favor of gravity. Finally, the equation of state for spherically symmetric, general relativistic stars in hydrostatic equilibrium is more difficult than just replacing degenerate electron pressure with degenerate neutron pressure. At the extreme densities encountered in neutron stars, the equation of state is strongly influenced by the strong interactions among the constituents. However, for pedagogical purposes of this exercise, it is sufficient to treat the pressure as though it came only from the degenerate neutrons. We will discuss the full extent of the potential, taking into account the strong interactions in Ch. 4. Before delving into the formalism of Schwarzschild stars, let us derive the appropriate mass and radius parameters for the case of neutron stars. Since in the present approximation neutrons provide both the mass and the pressure, these dimensionful parameters may be easily obtained through the following substitutions into Eqs. (2.29a and 2.29b): Y 1 and e → m m . This yields e → n 5 M = √15πα−3/2m = 10.6 M , (2.40a) 0 6 G n ⊙ √15π ~c R = α−1/2 =9.39 km . (2.40b) 0 2 G m c2  n  As advertised earlier, neutron stars have masses comparable to those of white dwarf stars but considerably smaller radii. In the general-relativistic regime the stellar structure equations— the Tolman-Oppenheimer-Volkoff (TOV) equations—are given by

dP (r) GM(r) (r) P (r) 4πr3P (r) 2GM(r) −1 = E 1+ 1+ 1 , (2.41a) dr − c2r2 (r) c2M(r) − c2r  E    dM(r) 4πr2 (r) =+ E . (2.41b) dr c2

Relative to the corresponding equations for Newtonian stellar structure [see Eqs. (2.5a and 2.5b)], the TOV equations display three corrections, that have been enclosed in square brackets. A more subtle correction involves the replacement of the rest-mass density of the star ρ(r), by its corresponding energy density (r). From general relativity we know that E gravity “couples” to the energy density. This implies, in the particular case of a simple Fermi

24 gas of neutrons, that the overall mass of the star receives contributions not only from the rest mass of the constituent neutrons, but in addition, from their kinetic energy. To make further contact with the Newtonian equations, as well as to introduce some simplification in the notation, the TOV equations in Eqs. (2.41a and 2.41b) are rewritten as dP (r) GM(r)ρ(r) = Γ(r) , (2.42a) dr − r2 dM(r) = +4πr2ρ(r) γ(r) , (2.42b) dr where the corrections from general relativity are embodied in the functions γ(r) and Γ(r), with the latter one being a short-hand notation for the four general-relativistic corrections

Γ(r) γ(r)Γ (r)Γ (r)Γ (r) . (2.43) ≡ 1 2 3 h i The individual corrections have been defined as follows:

γ(r) (r)/c2ρ(r) (x )/(x3 /3π2) , (2.44a) ≡E −→ E F F P (r) P (xF) Γ1(r) 1+ 1+ , (2.44b) ≡ (r) −→ (x )  E  " E F # 3 4πr P (r) 2 3 P (xF) Γ2(r) 1+ 1+9π r , (2.44c) ≡ c2M(r) −→ M(r)   " # −1 2GM(r) −1 10 M(r) Γ (r) 1 1 . (2.44d) 3 ≡ − c2r −→ − 3 r     Formally, Eq. (2.42a) may be regarded as a Newtonian equation for hydrostatic equilibrium with a “slowly varying” gravitational constant of the form G (r) G Γ(r). What results eff ≡ is particularly interesting in that all four relativistic corrections are greater than one, i.e., G (r) 1 for all r. Thus, general relativity enhances the unrelenting pull from gravity. Note eff ≥ that the arrows in the above equations are meant to represent the properly scaled form of the relativistic corrections that must be incorporated into the Newtonian “source terms” displayed in Eq. (2.39). Hence, the suitably scaled TOV equations, amenable to a numerical treatment, are given in complete analogy to the Newtonian case as follows: dx F = (¯r; x , M) , x (¯r =0) x ; (2.45a) dr¯ F F F ≡ Fc dM = (¯r; x , M) , M(¯r =0) 0 , (2.45b) dr¯ G F ≡ 25 where now the source terms are defined through the two functions ( and ), defined as F G follows:

(¯r; x , M)= f(¯r; x , M)Γ(¯r; x , M) and (¯r; x , M)= g(¯r; x , M)γ(x ) . (2.46) F F F F G F F F Recall that the two functions f and g have been previously defined in Eq. (2.39), and that

Γ(¯r; xF, M) is given as the product of the four (scaled) corrections displayed in Eq. (2.44). In Fig. 2.4 results are displayed for a neutron star supported exclusively by the pressure from its degenerate neutrons. To set a baseline, the limiting (Chandrasekhar) mass for a neutron star without general-relativistic corrections is simply equal to 4 times the

Chandrasekhar mass of a Ye =1/2 white dwarf star, or 5.76 M⊙ [recall that the mass of the 2 star scales as Ye ; see Eq. (2.29a)]. To understand the role of the various general-relativistic corrections we incorporate them one at the time, in order of increasing importance. The curve labeled Γ1 [as in Eq. (2.44b)] incorporates only one such correction. Although there are quantitative changes relative to the structure of white dwarf stars, for example the

Chandrasekhar mass has been reduced from 5.76 M⊙ to approximately 3.7 M⊙, they both display qualitatively similar mass-radius relations. In particular, the limiting mass is attained at zero radius. The shape of the curve with the “next” general-relativistic correction, however, is dramatically different. The curve labeled Γ1 +Γ2 attains it maximum mass of M 1.9 M at a finite radius of R 6.6 km. Note that beyond this limiting value the max ≃ ⊙ max ≃ mass-radius relation becomes a double (or even multiple) valued function. While the TOV equations guarantee hydrostatic equilibrium, they do not guarantee that the equilibrium will be stable. Beyond the limiting value the star is indeed in hydrostatic equilibrium, but the equilibrium is unstable; the smallest of perturbations will convert the neutron star into a black hole. As expected, adding more relativistic corrections pushes the limiting mass to smaller and smaller values (recall that all corrections increase the effect from gravity), yet the qualitative shape ceases to change and one ultimately recovers the Oppenheimer-Volkoff result of 1939. That is, a neutron star supported exclusively by the pressure from its neutrons attains its limiting mass of M 0.7 M at a radius of R 9.1 km. max ≃ ⊙ max ≃

26 Figure 2.4: Mass-radius relations for neutron stars with a degenerate neutron gas EoS. The different curves include general-relativistic corrections one at a time.

27 CHAPTER 3

VIRTUES AND FLAWS OF THE PAULI POTENTIAL

Insights into the complex and fascinating dynamics of Coulomb frustrated systems across a variety of disciplines are just starting to emerge (see, for example, [38] and references therein). In the particular case of neutron stars, one is interested in describing the equation of state of neutron-rich matter across an enormous density range using a single underlying theoretical model. In recent simulations the author of Ref. [32, 33, 39] have resorted to a “semi-classical” model that, while exceedingly simple, captures the essential physics of Coulomb frustration and nuclear saturation. The model includes competing interactions consisting of a short-range nuclear attraction (adjusted to reproduce nuclear saturation) plus a long-range Coulomb repulsion. The charge-neutral system consists of electrons, protons, and neutrons, with the electrons (which at these densities are no longer bound) modeled as a degenerate free Fermi gas. So far, the only quantum effect that has been incorporated into this “semi-classical” model is the use of an effective temperature to simulate quantum zero-point motion. The main justification behind the classical character of the simulations is the heavy nature of the nuclear clusters. Indeed, at the low densities of the neutron-star crust, the de Broglie wavelength of the heavy clusters is significantly smaller than their average separation. However, this behavior ceases to be true in the transition region from the inner crust to the outer core. At the higher densities of the outer core, the heavy clusters are expected to “melt” into a collection of isolated nucleons with a de Broglie wavelength that becomes comparable to their average separation. Thus, fermionic correlations are expected to become important in the crust-to-core transition region. Unfortunately, in contrast to classical simulations that routinely include thousands — and even millions — of particles, full quantum-mechanical

28 simulations of many-fermion systems suffer from innumerable challenges (see [40] and references therein). One of the problems with doing the full quantum-mechanical simulations is the need to antisymmetrized the wave functions. For a system of N particles, one needs to deal with N components to the wave function even for a single slater determinant, which can be computationally demanding. In an effort to “circumvent” — although not solve — some of these formidable challenges, classical simulations of heavy-ion collisions and of the neutron-star crust have resorted to a fictitious “Pauli potential”. Within the realm of nuclear collisions, the first such simulations were those of Wilets and collaborators [41]. Other simulations with a more refined Pauli potential have followed [42, 43, 44, 45, 46], but the spirit has remained the same: introduce a momentum dependent, two-body Pauli potential that penalizes the system whenever two identical nucleons get too close to each other in phase space. A goal of the present contribution is to show that the demands imposed by such a Pauli potential are too weak to reproduce some of the most basic properties of a zero-temperature (or cold) Fermi gas. Thus, we aim at refining such a potential in a manner that reproduces three fundamental properties of a cold Fermi gas. These are: (i) the kinetic energy (as others have done before us), (ii) the momentum distribution, and (iii) the two-nucleon correlation function. 3.1 Free Fermi Gas

The zero temperature Fermi gas is the simplest many-fermion system. Such a system displays no correlations beyond those imposed by the Pauli exclusion principle and is described by the following free Hamiltonian: N p2 H = i . (3.1) 2m i=1 X Here m is the mass of the fermions and N denotes the (large) number of particles in the system. As no interaction of any sort exists among the particles, the eigenstates of the system are given by a product of (single-particle) momentum eigenstates, suitably antisymmetrized to fulfill the constraints imposed by the Pauli principle. For simplicity, we assume that the fermions reside in a very large box of volume V = L3 and that the momentum eigenstates satisfy periodic boundary conditions. We will be interested in the thermodynamic limit of N and V , but with their ratio fixed at a specific value of the number density → ∞ → ∞ 29 ρ N/V . ≡ The (“box”) normalized momentum eigenstates are simple plane waves. That is,

1 ip·r ϕp(r)= e . (3.2) √V Given that the eigenvalue problem is solved in a finite box using periodic boundary conditions, the resulting single-particle momenta are quantized as follows: 2π 2π p(n)= n (n , n , n ) , with n =0, 1, 2 , andi = x, y, z . . . (3.3) L ≡ L x y z i ± ± with the corresponding single-particle energies given by ǫ(p)= p2/2m. Up to this point the spin/statistics of the particles has not come into play. We are now interested in describing the ground state of a system of N non-interacting, identical fermions and the resulting many-body correlations. Such a zero-temperature state is obtained by placing all particles in the lowest available momentum state, consistent with the Pauli exclusion principle. Using fermionic creation and annihilation operators satisfying the following anti-commutation relations [47],

† † ′ ′ † Ap,Ap′ = δp,p and Ap,Ap = Ap,Ap′ =0 , (3.4) n o n o n o the ground state of the system may be written as follows:

pF Φ = A† Φ , (3.5) | FGi p| vaci p=0 Y where Φ represents the (non-interacting) vacuum state and the Fermi momentum p | vaci F denotes the momentum of the last occupied single-particle state. Note that henceforth, no intrinsic quantum number (such as spin and/or isospin) will be considered. In essence, one assumes that all intrinsic degrees of freedom have been “frozen”, thereby concentrating on a single fermionic species (such as neutrons with spin up). In what follows, we compute expectation values of various quantities in the Fermi gas ground state ( Φ ). | vaci We start by computing the Fermi momentum pF in terms of the number density of the system ρ=N/V . That is,

3 3 d p pF N = nFD(n) V nFD(p) = V , (3.6) V →∞ (2π)3 T =0 6π2 n −→ X Z

30 or equivalently, 2 1/3 pF = 6π ρ . (3.7)

Note that in, Eq. (3.6), nFD(p) denotes the Fermi-Dirac
occupancy of the single-particle state denoted by p (or n) and the thermodynamic limit has been assumed. As all ground- state observables will be computed over a spherically-symmetric Fermi sphere, we define the Fermi-Dirac momentum distribution f(q) as follows: ∞ 2 f(q)=3q nFD(q) , with f(q)dq =1 , (3.8) Z0 where the dimensionless quantity q p/p is the momentum of the particle in units of the ≡ F Fermi momentum. All classical simulations performed and reported in the next sections must be carried out by necessity at finite temperature. Thus, we now incorporate finite temperature corrections to the various observables of interest. For temperatures T that are small relative to the Fermi temperature T (with T ǫ ), finite-temperature corrections may be implemented F F ≡ F by means of a Sommerfeld expansion [48]. For example, to lowest order in τ T/T the ≡ F momentum distribution becomes 3q2 f(q, τ)= 3q2Θ(1 q) . (3.9) π2 −→τ→0 − exp q2 1+ τ 2 τ +1 − 12   .  Here Θ(x) is the “Heaviside step function” appropriate for a zero-temperature Fermi gas. Similarly, the energy-per-particle of a “cold” Fermi gas may be readily computed. One obtains [48] ∞ 3 5π2 E/N = ǫ q2f(q)dq = ǫ 1+ τ 2 + (τ 4) , (3.10) F 5 F 12 O Z0   2 where the Fermi energy is defined by ǫF =pF/2m. The last Fermi-gas observable that we focus on is the “two-body correlation function” g(r). This observable measures the probability of finding two particles at a fixed distance r from each other. Moreover, the two-body correlation function is a fundamental quantity whose Fourier transform yields the static structure factor, an observable that may be directly extracted from experiment. As such, the two-body correlation function is the natural meeting place of theory, experiment, and computer simulations [49]. The two-body correlation function may be derived from the density-density correlation function [47]. That is, 1 1 g(r)= ρ (x, y)= ψˆ†(x)ψˆ†(y)ψˆ(y)ψˆ(x) , (3.11) ρ2 2 ρ2 D E 31 where ψˆ(x) is a fermionic field operator and the Dirac brackets denote a thermal expectation value. As the two-body correlation function for a non-interacting Fermi gas may be readily evaluated at zero temperature [47], we only provide its extension to finite temperatures. To lowest order in τ T/T (and for a single fermionic species) one obtains ≡ F j (z) 2 π2 g(r)=1 3 1 1 z2τ 2 , (3.12) − z − 12     where z p r and j (z) is the spherical Bessel function of order n=1, namely, ≡ F 1 sin(z) cos(z) j (z)= . (3.13) 1 z2 − z Equations (3.9), (3.10), and (3.12) display the three fundamental observables of a cold Fermi gas that we aim to reproduce in this work via a momentum-dependent, two-body Pauli potential. Note that in most (if not all) earlier studies of this kind, only the kinetic energy of the Fermi gas [Eq. (3.10)] was used to constrain the parameters of the Pauli potential [41, 42, 43, 44, 45, 46]. We are unaware of any earlier effort at including more sensitive Fermi-gas observables to constrain the parameters of the model. Clearly, it should be possible to reproduce the kinetic energy even with an incorrect momentum distribution. Thus, while we build on earlier approaches, we also highlight some of their shortcomings. 3.2 Pauli Potential: A New Functional Form

In the previous section the wave function of a zero-temperature Fermi gas was introduced as follows: pF Φ = A† Φ . (3.14) | FGi p| vaci p=0 Y Essential to the dynamical behavior of the system are the anti-commutation relations [Eq. (3.4)] that enforce the Pauli exclusion principle (i.e., (A† )2 0). As it is often done, p ≡ one may project the above “second-quantized” form of the many-fermion wave-function into configuration space to obtain the well-known Slater determinant. That is,

ϕp1 (r1) ... ϕp1 (rN ) ..... 1 ΦFG(p1,..., pN ; r1,..., rN )= ..... , (3.15) √N! .....

ϕpN (r1) ... ϕpN (rN )

32 where the single-particle wave-functions ϕp(r) are the (“box”) normalized plane waves defined in Eq. (3.2). The Slater determinant embodies important correlations that were discussed in the previous section and that we aim to incorporate into our classical simulations. These are:

(a) As a consequence of the Pauli exclusion principle [(A† )2 0], the probability that two p ≡ fermions share the same identical momentum is equal to zero. Mathematically, this result follows from the fact that the wave-function vanishes whenever two rows of the Slater determinant are equal to each other.

(b) Similarly, the wave-function also vanishes whenever two columns of the Slater determi- nant are identical. This fact precludes two fermions from occupying the same exact location in space.

(c) At zero temperature, only momentum states having a magnitude p less than the Fermi | | momentum pF are occupied; the rest are empty. The first two properties are embedded in the momentum distribution of Eq. (3.9), namely, a quadratic momentum distribution sharply peaked at the Fermi momentum pF (recall that such a momentum distribution emerges after folding the Heaviside step function with the phase space factor). The third property induces spatial correlations that are captured by the two-body correlation function g(r) of Eq. (3.12). Indeed, the two-body correlation function is related to the integral of the Slater determinant over all but two of the coordinates of the particles (e.g., r1 and r2). Clearly, the Slater determinant vanishes whenever r1 = r2, and so does the two-body correlation function at r = r r 0. It is the aim of this contribution to | 1 − 2|≡ build a Pauli potential that incorporates these three fundamental properties of a free Fermi gas. However, before doing so, we will briefly review the Pauli potential introduced by Wilets and collaborators — and used by others with minor modifications — to simulate the collisions of heavy ions and the properties of neutron rich matter at sub-saturation densities. Such a Pauli potential is given by a sum of momentum-dependent, two-body terms of the following form: N V (p ,..., p ; r ,..., r )= V exp( s2 /2) , (3.16) Pauli 1 N 1 N 0 − ij i

33 where V0 >0 and the dimensionless phase-space “distance” between points (pi, ri) and (pj, rj) is given by 2 2 2 pi pj ri rj sij | −2 | + | −2 | . (3.17) ≡ p0 r0

Here p0 and r0 are momentum and length scales related to the excluded phase-space volume that is used to mimic fermionic correlations. That is, whenever the phase-space distance 2 between two particles is such that sij .1, then a penalty is levied on the system in an effort to mimic the Pauli exclusion principle. Although the parameters of this Pauli potential

(V0, p0, and r0) can — and have — been adjusted to reproduce the kinetic energy of a free Fermi gas, it fails (as we show later) in reproducing more sensitive Fermi-gas observables, particularly, the momentum distribution f(p) and the two-body correlation function g(r). Upon closer examination, the above flaws should not come as a surprise. In our previous discussion of the Slater determinant it has been established that the probability of finding two identical fermions in the same location in space “or” with the same momenta must be identically equal to zero. Yet the Pauli potential of Eq. (3.16) fails to incorporate this important dynamical behavior. Indeed, the above Pauli potential imposes a penalty on the system only when both the location “and” momenta of the two particles are close to each 2 other (i.e., sij .1). In particular, no penalty is imposed whenever two fermions occupy the same location in space, provided that their momenta are significantly different from each other, i.e., p p 2 p2. Thus, the Pauli potential of Eq. (3.16) will generate an incorrect | i − j| ≫ 0 two-body correlation function, namely, one with g(r) = 0 as r tends to zero. By the same 6 token, an incorrect momentum distribution will be generated, although not necessarily its second moment. To remedy these deficiencies, a Pauli potential is now constructed so that the three properties [(a), (b), and (c)] defined above are explicitly satisfied. To this end, we introduce the following form for the Pauli potential:

N V (p ,..., p ; r ,..., r ) = V exp( r /r )+ V exp( p /p ) Pauli 1 N 1 N A − ij 0 B − ij 0 i

+ VC Θη(qi) , (3.18) i=1 X

34 where r = r r , p = p p , q = p /p , and Θ is a suitably smeared Heaviside-step ij | i − j| ij | i − j| i | i| F η function of the following form: 1 Θη(q) Θ(q) . (3.19) ≡ 1 + exp[ η(q2 1)] η−→→∞ − − The parameters of the model VA,VB,VC and r0, p0, η will be adjusted to reproduce both the momentum distribution and two-body correlation function of a low-temperature Fermi gas. Note that the phase-space dependence of the Pauli potential has been separated into a “sum” of two-body pieces, with the first acting exclusively in configuration space and the second one only in momentum space. The first term in the potential imposes a penalty as the particles get too close (of the order of r0) to each other. Similarly, the second term in the potential penalizes particles whenever their relative momenta becomes of the order of p0. Finally, the third “one-body” term enforces the low-temperature behavior of the Fermi gas, namely, that the probability of finding any particle with a momentum significantly larger than the Fermi momentum is vanishingly small. Most of the parameters will depend explicitly on the density of the system (see Sec. 3.3). This reflects the complex many-body nature of the Pauli correlations and our inability to simulate them by means of a “simple” (albeit momentum dependent) two-body potential. 3.3 Simulation Results

We start this section by listing in Table 3.1 the parameters of the Pauli potential. For the strength parameters, the following simple density dependence is assumed:

αi V (ρ)= V 0 ρ/ρ , i = A, B, C , (3.20) i i 0 { } −3   where ρ0 = ρsat/4=0.037 fm is the density of a single fermionic species (for example, −3 neutrons with spin up) at nuclear-matter saturation density (ρsat = 0.148 fm ). For the range parameters (r0 and p0) the following scaling relation is adopted:

r0 = βA/pF , (3.21a)

p0 = βBpF . (3.21b)

Monte Carlo simulations for a system of N = 1000 identical fermions at the finite (but small) temperature of τ = T/TF = 0.05 have been performed. Initially, the particles are

35 Table 3.1: Strength (in MeV) and range parameters (dimensionless) for the various compo- nents of the Pauli potential. See Eqs. (3.18), (3.20), and (3.21). These values have been used to simulate a system of N =1000 identical fermions at a temperature of τ =T/TF =0.05.

0 0 0 VA VB VC αA αB αC βA βB η 13.517 1.260 3.560 0.629 0.665 0.831 0.845 0.193 30

distributed randomly throughout the box with momenta that are uniformly distributed up to a maximum momentum of the order of the Fermi momentum. After an initial thermalization phase of typically 2000 sweeps (or 2 million Monte Carlo moves for both coordinates and momenta) data is accumulated for an additional 2000 sweeps, with the data divided into 10 groups to avoid correlations among the data. It is from these 10 groups that averages and errors are generated.

3.3.1 Kinetic Energy

The kinetic energy of a finite-temperature Fermi gas as a function of density in Eq. (3.10) is displayed in Fig. 3.1. The (black) solid line represents the analytic behavior of a free Fermi gas in Eq. (3.10) correct to second order in τ. The (red) line with circles is the result of the Monte Carlo simulations with the Pauli potential defined in Eq. (3.18). The agreement (to better than 5%) is as good as the one obtained with earlier parametrization of the Pauli potential. To our knowledge, reproducing the kinetic energy of a free Fermi gas is the sole constraint that has been imposed on most of the Pauli potentials available to date. However, while a host of Pauli potentials can reproduce such a behavior, it is unclear if these potentials can also reproduce the full momentum distribution. Thus, we now show how the Pauli potential defined in Eq. (3.18) is successful at reproducing two highly sensitive Fermi-gas observables, namely, the momentum distribution and the two-body correlation function.

3.3.2 Momentum Distribution

The momentum distribution obtained from the Monte Carlo simulations is displayed in Fig. 3.2 for a variety of densities. Note that the momentum distribution has been normalized

36 35 N=1000 30 τ=T/T =0.05 F 25

20 /N kin

E 15

10 Monte Carlo 5 Exact

0 0.0 0.5 1.0 1.5 2.0 ρ/ρ0

Figure 3.1: Average kinetic energy of a system of N = 1000 identical fermions at a temperature of τ = T/TF = 0.05. The line with circles is the result of the Monte Carlo simulations with the Pauli potential of Eq. (3.18). The solid line is the exact behavior of a non-relativistic Fermi gas, as given by Eq. (3.10).

−3 to one and that the densities have been expressed in units of ρ0 =0.037 fm . As indicated in Eq. (3.9), the momentum distribution of a cold Fermi gas depends solely on the two dimensionless ratios q = p/pF and τ = T/TF. Thus, all curves must collapse into the exact one — displayed by the (black) solid line — independent of density. It is gratifying to see that this is indeed the case. In contrast, we show in the next section that the standard Pauli potential of Eq. (3.16) fails to reproduce this behavior.

3.3.3 Two-Body Correlation Function

The two-body correlation function of a free Fermi gas is displayed in Fig. 3.3. The two-body correlation function g(r) is related to the probability of finding two particles separated by a distance r. For a free Fermi gas, the probability of finding two identical fermions at zero separation is identically equal to zero [see Eq. (3.12)]. As this short-range (anti-)correlation

37 3.0 ρ=0.2 N=1000 τ=T/T =0.05 2.5 ρ=0.4 F ρ=0.6 ρ=0.8 2.0 ρ=1.0 )

τ ρ=1.2 1.5 ρ=1.4 f(q, Exact 1.0

0.5

0.0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 q=p/p F

Figure 3.2: Momentum distribution of a system of N = 1000 identical fermions at a temperature of τ = T/TF = 0.05 for a variety of densities (expressed in units of ρ0 = 0.037 fm−3). The momentum distribution has been normalized to one [see Eq. (3.8)]. The (black) solid line with no symbols gives the exact behavior of a non-relativistic Fermi gas.

is the sole consequence of the Pauli exclusion principle, the “Pauli hole” disappears for −1 distances of the order of the inter particle separation (pF ). As in the case of the momentum distribution, the correlation function depends only on two dimensionless variables (z = pFr and τ). It is again gratifying that the simulation curves scale to the exact correlation function, depicted here with a (black) solid line. 3.4 Comparison to other approaches

In this section we compare the Pauli potential introduced in Eq. (3.18) to earlier approaches that are based on Eq. (3.16). Such approaches have been very successful in reproducing the kinetic energy of a free Fermi gas for a wide range of densities. Indeed, the kinetic energy displayed in Fig. 3.2 of Ref. [45] is as good — if not better — than the one obtained here. However, a faithful reproduction of the kinetic energy does not guarantee that the

38 1.00 N=1000 τ=T/T =0.05 0.80 F ρ=0.2 ρ=0.4 ) 0.60 τ ρ=0.6 ρ=0.8 g(z, 0.40 ρ=1.0 ρ=1.2 ρ=1.4 0.20 Exact

0.00 0 3 6 9 12 15 z=p r F

Figure 3.3: Two-body correlation function for a system of N = 1000 identical fermions at a temperature of τ = T/TF = 0.05 for a variety of densities (expressed in units of −3 ρ0 = 0.037 fm ). The (black) solid line with no symbols gives the exact behavior of a non-relativistic Fermi gas.

system displays the same phase-space correlations as that of a free Fermi gas. To illustrate this point we compare in Fig. 3.4 the results from the two approaches for the momentum −3 distribution and two-body correlation function at a fixed density of ρ0 =0.037 fm . The left-hand panel in the figure displays the momentum distribution and indicates that while earlier approaches (depicted with a blue line with circles) are accurate at reproducing the second moment of the distribution (i.e., the kinetic energy) the momentum distribution itself shows a behavior that differs significantly from that of a cold Fermi gas. The right- hand panel shows deficiencies that are as — or even more — severe. Two points are worth highlighting. First, the two-body correlation function g(r) generated with the standard form of the Pauli potential does not vanish at r = 0. Second, for distances of the order of the inter-particle separation and beyond, the two-body correlation function develops artificial oscillations. Failure in reproducing the correct behavior of g(r) at r = 0 is relatively simple

39 6.0 1.2 N=1000 5.0 ρ=ρ0 1.0

4.0 0.8 ) ) τ τ 3.0 0.6 f(q, g(z,

2.0 0.4 Maruyama This work 1.0 Exact 0.2

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 5 10 15 20 q=p/p z=p r F F

Figure 3.4: Comparison between the Pauli potential introduced in this work Eq. (3.18) and earlier approaches based on Eq. (3.16). The left-hand panel shows the momentum distribution while the right-hand panel the two-body correlation function. The simulations have been carried out for a system of N = 1000 identical fermions at a density of −3 ρ0 =0.037 fm . The (blue) line with circles is obtained from a Monte Carlo simulation using Eq. (3.16) with the parameters of Ref. [45]. The (red) line with squares displays the results using the Pauli potential introduced in this work; the black line gives the exact behavior of a non-relativistic Fermi gas.

to understand. Potentials based on Eq. (3.16) impose a penalty on the system only if both the positions and momenta of the two fermions are close to each other. Yet the correlations embodied in a free Fermi gas are significantly more stringent than that. Indeed, a Slater determinant vanishes if either the positions or the momenta of the two fermions are equal to each other. In contrast, the development of artificial structure in g(r) is a more subtle effect that is intimately related to the momentum dependence of the potential and that we now address. The artificial oscillations present in g(r) are reminiscent of the structure of liquids and/or crystals. The emergence of crystalline structure in a low-temperature/low-density system would be expected if the energy of localization becomes small relative to the mutual repulsion

40 between the particles. Such, however, is not the behavior of a cold Fermi gas. While spatial anti-correlations would favor the formation of a periodic structure, the large velocities of the particles — resulting from the Pauli exclusion principle — would make their localization extremely costly. What is not evident is the reason for the momentum distribution generated with a standard Pauli potential to lead to crystallization, whereas not that of a real Fermi gas (see left-hand panel in Fig. 3.4). To our knowledge, the answer to this question was first provided by Neumann and Fai [50]. One must realize that any Hamiltonian that contains momentum-dependent interactions — such as most (if not all) Pauli potentials — generates a “canonical” momentum distribution that may (and does!) differ significantly from the corresponding “kinematical” momentum distribution π mr˙ . Indeed, in a Hamiltonian formalism where the Hamiltonian h i≡h i depends on the positions and canonical (not kinematical) momenta of all the particles, namely, H = H(p1,..., pN ; r1,..., rN ) the kinematical velocities must be obtained from

Hamilton’s equations of motion, i.e., r˙ i = ∂H/∂pi. For a Hamiltonian that contains momentum-dependent interactions — as in the case of the Pauli potential — then the kinematical momentum π mr˙ differs from the corresponding canonical momentum i ≡ i pi. This suggests that while a given choice of Pauli potential may produce the correct canonical momentum distribution, it may generate kinematical velocities that are too small to prevent crystallization. Such a distinction between canonical and kinematical momenta is an unwelcome, yet unavoidable, consequence of the approach. A cold Fermi gas — a system subjected to no interactions — is the quintessential quantum system were such an artificial distinction is not required. It is worth noting that although some earlier choices yield an extremely soft kinematical momentum distribution, the Pauli potential introduced in this work is not immune to such a disease. Indeed, we believe that an unrealistically soft kinematical momentum distribution is likely to be a general result of the Hamiltonian approach. Thus, we close this section by displaying in Fig. 3.5 the kinematical velocity distribution obtained with the new choice of Pauli potential introduced in Eq. (3.18). For comparison, the exact distribution of a cold Fermi gas (solid black line) is also included. The dependence of the Pauli potential on the canonical momentum distribution is responsible for generating such a soft velocity distribution, with its peak around 1/10 of the Fermi velocity.

41 9.0

8.0 ρ=0.2 N=1000 ρ=0.4 τ=T/T =0.05 F 7.0 ρ=0.6 6.0 ρ=0.8 ρ=1.0 )

τ 5.0 ρ=1.2 ρ=1.4

h(u, 4.0 Exact 3.0

2.0

1.0

0.0 0.00 0.25 0.50 0.75 1.00 1.25 u=v/v F

Figure 3.5: “Kinematical” velocity distribution of a system of N = 1000 identical fermions at a temperature of τ = T/TF = 0.05 for a variety of densities (expressed in units of −3 ρ0 =0.037 fm ). The velocity distribution has been normalized to one. The (black) solid line with no symbols gives the exact behavior of a non-relativistic Fermi gas.

And while we were able to avoid crystallization with the present set of parameters (see Figs. 3.3 and 3.4), the risk of crystallization looms large (see next section). 3.5 Finite-Size Effects

We have observed that the Pauli potential introduced in Eq. (3.18), with its parameters suitably adjusted, has been successful in reproducing a variety of Fermi-gas observables, such as its kinetic energy, its (canonical) momentum distribution, and its two-body correlation function. However, the first indication of a potential problem — and one that may be generic to all approaches employing momentum-dependent interactions — is the emergence of an unrealistically soft velocity distribution and with it, the possibility of artificial spatial correlations (i.e., crystallization). Fortunately, with the choice of parameters adopted in this work, the problem of crystallization was avoided (see Fig. 3.3). Yet, there is no guarantee

42 that crystallization will not become a problem as one examines the sensitivity of our results to finite-size effects.

3.0 1.0 ρ=ρ0 τ=0.05 0.8 2.0 ) ) τ 0.6 τ f(q, g(z, N=250 0.4 1.0 N=500 N=1000 N=2000 0.2 Exact

0.0 0.0 0.0 0.3 0.6 0.9 1.2 1.5 0 5 10 15 q=p/p z=p r F F

Figure 3.6: Finite-size effects on the canonical momentum distribution (left-hand panel) and the two-body correlation function (right-hand panel) for a system of identical fermions at −3 a temperature of τ = T/TF =0.05 and a density of ρ = ρ0 =0.037 fm . Simulations were carried out for systems containing N =250, 500, 1000 and 2000 particles. The (black) solid line with no symbols gives the exact behavior of a non-relativistic Fermi gas.

In order to estimate the sensitivity of our results to finite-size effects, Monte Carlo simulations were performed for a system containing N = 250, N = 500, N = 1000, and N =2000, identical fermions (note that the results reported so far have been limited to 1000 particles). The conclusions from this study are mixed. First (and fortunately) no evidence of crystallization or of significant finite-size effects were found. These findings are displayed in Fig. 3.6 for both the canonical momentum distribution (left-hand panel) and the two-body correlation function (right-hand) panel. Unfortunately, however, in order to preserve the high quality of the results previously obtained with 1000 particles, a parameter of the Pauli 0 potential [VB in Eq. (3.18)] had to be fine tuned. Specifically, the following scaling with particle number was used: 1000 V 0(N)= V 0(N =1000) , (3.22) B B N  

43 0 with VB(N =1000)=1.26 MeV being the value listed in Table 3.1. This unpalatable fact may be a reflection of the highly challenging task at hand: how to simulate fermionic many-body correlations by means of a “simple” two-body Pauli potential.

44 CHAPTER 4

EQUATION OF STATE FOR NUCLEAR PASTA

As has been discussed earlier, the density range in neutron stars spans from essentially zero at the edge of the star to about 0.8 baryons fm−3 at the center of the star. Due to this extreme density range, modeling a realistic equation of state that works for the whole density region is a difficult task. Therefore, in this chapter we will limit our study to constructing a realistic equation of state for the pasta region in neutron stars. The physics of nuclear pasta is essentially a competition between short range nuclear attraction that tries to correlate nucleons into nuclei, and the long range Coulomb repulsion that prevents protons from getting too close to each other. The first question that comes to mind in dealing with nuclear pasta is whether it should be treated as a classical or a quantum system. From statistical mechanics, we know that a system can be treated classically if the mean inter particle distance (V/N)1/3 in the system is much larger than the mean thermal wavelength λT . That is,

nh3 nλ3 1 , (4.1) T ≡ (2πmkT )3/2 ≪ where n is the baryon density, m is the mass of the particles in the system, k is the Boltzmann 1/2 constant, and h is the Planck constant. The mean thermal wavelength λT = h/(2πmkT ) . For nuclear pasta the density is very low (around 10−3 nucleon/fm3) and the temperature in the star is very high (of the order of 108 K 10−2 MeV). To illustrate this condition, let ≈ 3 56 us take a look at an iron nucleus in nuclear pasta. The value for nλT for a Fe nucleus in this region is around 0.18. Therefore, in studying nuclear pasta we will use semi-classical Monte Carlo simulations by incorporating the fermionic correlation via a semi-classical Pauli potential.

45 4.1 Modeling the Nuclear Pasta

In this section we introduce a semi-classical model that while simple, captures the essential physics of the nuclear pasta. In this model a neutral system consisting of electrons, protons and neutrons is considered. The nucleons interact via a short range nuclear and screened Coulomb potential. The electrons are assumed to form a degenerate free Fermi gas of density ρe=ρp to ensure charge neutrality. The very slight polarization of electrons leads to a screening length λ for the Coulomb interactions between protons. The full Hamiltonian is of the form

H = K + VTot , (4.2a)

VTot = Vnuc(i, j)+ Vc(i, j) , (4.2b) ipotential energy. In this model the potential energy consists of nuclear potential and Coulomb potential. The two-body nuclear potential (Vnuc) consists of spin-independent (VSI ) and spin-dependent(VSD) terms of the following form:

Vnuc(i, j)= VSI(i, j)+ VSD(i, j) , (4.3a) 2 2 2 −r /Λ1 −r /Λ2 −r /Λ3 VSI(i, j) = ae ij + be ij + cτz(i)τz(j)e ij , (4.3b)

(−rij /r0) VSD(i, j)= VA e δσiσj δτiτj , (4.3c) αA 0 VA(ρ)= VA ρ/ρ0 , and r0 = βA/pF . (4.3d)   Here the distance between particles is denoted by r = r r , and p is the Fermi ij | i − j| F momentum of the nucleon as defined in Appendix B. The isospin of the ith particle is τz =

+1 for a proton and τz = -1 for a neutron, while σ denotes the z-component of the spin of the particles.

The two-body spin-independent potential (VSI ) in Eq. (4.3b) represents the characteristic intermediate-range attraction and short-range repulsion of the nucleon-nucleon (NN) force. The inclusion of the isospin dependence in the potential insures that while pure neutron matter is unbound, the symmetric nuclear matter is bound appropriately. The spin- independent potential contains free-parameters (a, b, c, Λ1, Λ2, Λ3) that will be adjusted to

46 reproduce as accurately as possible the following properties: a) the saturation density and binding energy per nucleon of symmetric nuclear matter and b) the binding energy and charge radii for several finite nuclei.

The two-body spin-dependent potential (VSD) in Eq. (4.3c) represents the fermionic correlations between fermions. It is obtained from the Pauli-potential in Eq. (3.18) without taking into account the momentum-dependent term of the potential. This is a necessary step since the potential in Eq. (3.18) due to its momentum-dependence, presents problems in the calibration process. That is, we could not find a best fit that satisfies the above mentioned nuclear matter and finite nuclei properties. Also note that in Eq. (4.3c) we have generalized the Pauli potential in Eq. (3.18) to include different fermionic species.

Finally, a screened Coulomb interaction of the following form is included :

2 e −rij /λ Vc(i, j)= e τp(i)τp(j) , (4.4) rij where τp = (1+ τz)/2 is a proton projection operator and λ is the screening length that results from the slight polarization of the electron gas. The screening length λ defined in a Thomas-Fermi approximation is not significantly smaller than the length L of our simulation box, unless for cases in which a very large number of particles is used. Therefore to control finite-size effect we have arbitrarily adopted a screening length of λ 10 fm . ≡ We perform simulations for a canonical ensemble with a fixed number of particles (A) at a temperature T and a fixed baryon density (ρ). The simulation volume is then simply V = A/ρ. To minimize finite size effects, periodic boundary conditions are adopted. The distance between 2 particles (rij) is then calculated from the x, y, and z coordinates of the ith and jth particles as follows:

r = [x x ]2 + [y y ]2 + [z z ]2 , (4.5) ij i − j i − j i − j q l [l ]= l LR i , i (x,y,z) . (4.6) i i − L ≡   where [li] represents the minimum distance between two particles in the i-th axis. L is 1/3 R li the length of the simulation box which is represented by V and L is a notation to represent rounding off (l /L) to the next integer if (l /L) 0.5 and to the lower integer i i ≥ otherwise.

47 The total energy (E) of the system is made of kinetic (K) and potential (VTot) energy contributions. The average energy can be calculated as a thermal expectation value as follows: 1 E = d3Ar d3ApHe−H/T , (4.7) h i Z(A, T, V ) Z with the canonical partition function given by

Z(A, T, V )= d3Ar d3Ap e−H/T . (4.8) Z However, since the potential energy defined in Eq. (4.2b) is independent of momentum, the partition function for the system factorizes into a product of a partition function in momentum space (ZP ) and a partition function in coordinate space (ZR), such that

Z(A, T, V )= ZRZP , (4.9a)

3 3 3 −VTot/T ZR = d r1 d r2...d rA e , (4.9b) Z 3 3 3 −K/T ZP = d p1 d p2...d pA e . (4.9c) Z Hence, Eq. (4.7) becomes 1 1 E = d3r d3r ...d3r V e−VTot/T + d3p d3p ...d3p K e(−K/T ) . (4.10) h i Z 1 2 A Tot Z 1 2 A R Z P Z The second term in Eq. (4.10), which is the expectation value of kinetic energy, reduces to its classical value 3 K = AT, (4.11) h i 2 while the first term–the expectation value of the potential energy– will be evaluated using Metropolis Monte Carlo simulations. For a detailed discussion on this algorithm please refer to Appendix A. 4.2 Nuclear Matter Equation of State

Infinite nuclear matter is modeled as an infinite system of nucleons (protons and neutrons) interacting via nuclear interactions but with the electromagnetic interaction turned off. De- spite its idealization, understanding the properties of infinite nuclear matter is a prerequisite for any consistent theory of nuclei. In this section we will discuss the equation of state (EoS) of cold nuclear matter, that is, matter at zero temperature and in its lowest energy state

48 [51]. The EoS of cold nuclear matter describes the relationship between the binding energy per nucleon versus density and proton fraction, which can be written in terms of isospin asymmetry parameter δ (N Z)/A as [24] ≡ −

2 4 (E/A)(ρ, δ)=(E/A)(ρ, δ =0)+(Esym/A)(ρ)δ + O(δ ) . (4.12)

The first term in Eq. (4.12) denotes the energy for symmetric nuclear matter (N = Z), whereas the second term is the symmetry energy. To lowest order in δ, the symmetry energy is the difference in energy between pure neutron matter and symmetric nuclear matter. Hence, the symmetry energy may be viewed as the penalty imposed on the system by departing from the symmetric (N=Z) limit.

Symmetric nuclear matter saturates. That is, there exists an equilibrium density ρ0 at which the pressure vanishes. Expanding the energy of symmetric nuclear matter around saturation density ρ0 yields

2 1 ρ ρ0 (E/A)(ρ, δ =0)=(E/A)(ρ0,δ = 0)+ K − + ... (4.13) 18 ρ0   Here the first term represents the binding energy per nucleon for symmetric nuclear matter at saturation density ρ0, and the second term is the compression modulus K. The compression modulus defines the curvature of symmetric nuclear matter at ρ0. That is,

2 2 ∂ (E/A)(ρ, δ = 0) K =9ρ0 2 . (4.14) ∂ρ ρ=ρ0

The larger the value of K, the stiffer the EoS, namely, the faster the energy increases with density. A large value of K corresponds to a stiffer equation of state while a small value of K represents a softer equation of state.

Experimental data on finite nuclei have placed constraints on the properties of symmetric −3 nuclear matter. The symmetric nuclear matter saturation density ρ0 is around 0.15 fm and has a binding-energy per nucleon E/A(ρ ,δ = 0) -16 MeV. The constant density of nuclear 0 ≃ matter and the proportionality of the binding energy to the mass number A represent an important fact about nuclear forces. Nuclear forces show saturation [52]; that is, the forces are attractive for a small number of nucleons but become repulsive for a larger number. The saturation point corresponds to the equilibrium point (at zero temperature) of nuclear matter and is characterized by the vanishing pressure of the system,

49 ∂(E/A) P = ρ2 =0 . (4.15) ∂ρ ρ=ρ0   and positive curvature ∂P/∂ρ 0. This is all that is known about the EoS with a high ≫ degree of experimental confidence. The next question is whether we will be able to probe the vicinity of the saturation point to get a better information about the EoS. The first quantity of interest in this respect is the compression modulus K which determines the shape of the equation of state at zero temperature close to the saturation point. It should be noted that while the linear term in Eq. (4.13) vanishes for symmetric nuclear matter (i.e., the pressure), no such special saturation point exist in the case of the symmetry energy. Rather, it is the symmetry energy at lower density of ρ 0.1 fm−3 that seems to be accurately constrained 0 ≃ (to within 1 MeV) by available ground-state observables [53] 4.3 Semi Empirical Mass Formula

In the previous section, we discussed the binding energy of infinite nuclear matter which is an idealization of actual finite nuclei. We found that the binding energy per nucleon of symmetric nuclear matter at saturation density is around -16 MeV. However, treating the binding energy of actual nuclei requires further corrections due to the Coulomb repulsion between protons and surface effects which have been neglected in the first step. This method was first devised by von Weizs¨acker in 1935 and suggest a binding energy per nucleon of the following form: [54, 55]

B a Z2 (A 2Z)2 = a surf a a − . (4.16) A vol − A1/3 − coul A4/3 − sym A2 The leading term in the mass formula is the volume term, which indicates the constant binding energy per nucleon of symmetric nuclear matter at saturation density. The second term is the correction due to surface effects. The nucleons on the surface of the nucleus are surrounded by less nucleons than those in the interior, and therefore give a smaller contribution to the binding energy. For a spherical nucleus the surface area is related to the volume as S ∝ V 2/3. The volume of the nucleus, under this assumption, will be proportional to A and hence S ∝ A2/3 The third term describes the Coulomb energy, which indicates the reduction in the binding energy due to the Coulomb repulsion between protons. The last term describes the symmetry energy, showing the decrease in binding energy due to unequal

50 16

14 surface

12 Coulomb 10 symmetry 8 /A (MeV) bind

E 6

4 volume Total 2

0 0 50 100 150 200 250 A

Figure 4.1: Binding energy predicted by the Semi Empirical Mass formula for various nuclei. The solid line refers to contribution to the Binding energy from the volume term. Relative contributions to the binding energy after the inclusion of of the various other terms (asurf , acoul, asym) have been identified, leading to the predicted binding energy (Total)

number of protons and neutrons. A best fit of the various coefficients in Eq. (4.16) to many nuclei in the mass table [56] gives the following values:

avol = 15.72MeV (4.17)

asurf = 17.54MeV (4.18)

acoul = 0.71MeV (4.19)

asym = 23.38MeV (4.20)

Fig. 4.1 displayed the relative importance of the various terms (avol, asurf , acoul, asym) to the binding energy. The first parameter (avol) represents the binding energy per nucleon for symmetric nuclear matter. After including corrections from various terms, the binding energy per nucleon (except for the few lightest nuclei) is close to 8 MeV.

51 4.3.1 Spin-Dependent Potential

The spin-dependent potential as stated in Eqs. (4.3c and 4.3d) has three free parameters

(VA, αA, and βA). Using Eq. (3.20) expressed in terms of pF , the spin-dependent potential becomes

1.995 VA(pF )=13.517 pF /pF 0 MeV , (4.21a)   r0 =0.845/pF . (4.21b)

4.4 Simulation Results

We performed Metropolis Monte Carlo simulations for A = 250 identical particles at a fixed temperature T = 1 MeV such that T/T 1 using the parameters in Eq. (4.21). The F ≪ simulation results for the two- body correlation function are shown in Fig. 4.2. Also shown in the figure is the analytic calculation of the free Fermi gas two-body correlation function of Eq. (3.12). We find that our simulations give a reasonable results compare to the analytic prediction.

4.4.1 Spin-Independent Potential

We now return to discuss the choice of spin-independent parameters (a, b, c,Λ1,Λ2,Λ3) that must be used to reproduce various bulk properties of symmetric nuclear matter, such as the saturation density, the binding energy per nucleon at saturation, and the compression modulus. As mentioned earlier, symmetric matter saturates at a density of around 0.15 fm−3 and an energy per nucleon -16 MeV. With this information at hand, we are ready ≃ to calibrate the model parameters. Results from the fit yields the two models listed in Table (4.1). Again, we ran Metropolis Monte Carlo simulations to reproduce the above properties of cold nuclear matter. However, in semi-classical simulations it is impossible to perform calculations at exactly zero temperature. Therefore, for simulation purposes the temperature will be fixed arbitrarily at T = 1 MeV. This T = 1 MeV temperature should be regarded as an additional model parameter. Table (4.1) displays the model parameters obtained from the fits. The first three terms are in MeV while the last three terms are in fm2.

52 ρ/ρ =1 ρ/ρ =0.17 ρ/ρ =0.07 1 0 1 0 1 0

0.8 0.8 0.8

g(r) 0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0 0 1 2 3 4 5 6 7 8 0 1 2 3 4 0 1 2 3 4

rpF

Figure 4.2: Two-body correlation function g(r) as a function rpF for a system of A = 250 identical particles at a temperature of T = 1 MeV for various densities. Only the spin- dependent part of the potential was used.

Table 4.1: Models parameters for the spin-independent term. Strength (in MeV) and range parameters (in fm2) for various components of the spin-independent potential.

Set a b c Λ1 Λ2 Λ3 I 108 -30 27 1.37 2.625 2.380 II 102 -30 30 1.50 2.770 2.590

Fig. 4.3 displays the energy per particle versus density for symmetric nuclear matter using both parameter sets for A=500 particles (N=Z=250). With these model parameters nuclear saturation is achieved at a density of 0.15 fm−3 and has a binding energy/particle ≈ -18 MeV. For comparison, the energy per particle for the soft FSUGold model and the stiff ≈ NL3 model [57, 58, 59] is also shown in Fig. 4.3. These two models based on the mean-field approximation have been very successful in reproducing ground state properties (such as binding energies, charge radii, separation energies, etc) for a variety of nuclei. In addition to symmetric nuclear matter, we also performed simulations for pure neutron matter (N/A

53 50

FSU 40 NL3 SM(set-1) SM(set-2) 30

20

10 E/A (MeV)

0

-10

-20 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 ρ/ρ0

Figure 4.3: Energy per particle of symmetric nuclear matter as a function of densities ρ/ρ0 for various parameter sets. The simulation was performed using A = 500 particles (Z/A = 0.5) at a temperature of T = 1 MeV.

= 1) as shown in Fig. 4.4. Here we display the energy per neutron as a function of density for A = N = 500 particles. Pure neutron matter is predicted to be unbound in all models. Again for comparison, the energy per neutron for the soft FSUGold model and the stiff NL3 model is also shown. At high densities our results for symmetric nuclear matter show a close resemblance to the soft FSUGold model, but for pure neutron matter the simulations tend to resemble the stiff NL3 EoS. However, the behavior of nuclear matter at high densities is largely unconstrained due to the limited amount of experimental data. Thus we will make no attempt to discuss the properties of nuclear matter in the high-density region. Instead we will limit our discussion to the region around saturation density and below. We now proceed to analyzing the density dependence of the EoS. For symmetric nuclear mater, the dynamics of the EoS around saturation density is controlled by the compression modulus (K). Using Eq. (4.14) we evaluated the compression modulus for both parameter sets. Our calculations yield K 249.3 MeV for the first parameter set and K 296.1 MeV ≈ ≈ for the second parameter set. Hence the first parameter set yields a slightly softer EoS for symmetric nuclear matter. For comparison the NL3 and FSUGold models give K = 271

54 MeV and K = 230 MeV respectively.

100 FSU NL3 NM(set-1) 80 NM(set-2)

60

E/A (MeV) 40

20

0 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 ρ/ρ0

Figure 4.4: Energy per neutron of pure neutron matter (N/A=1) as a function of density ρ/ρ0 for various parameter sets. The simulation was conducted for A = 500 particles at a temperature of T = 1 MeV.

In the case of finite nuclei, the full Coulomb potential is included without the need for a screening length. That is, e2 Vc(i, j)= τp(i)τp(j) . (4.22) rij To evaluate the Fermi momentum for finite nuclei as required by Eq. (4.3d), we use a Woods- Saxon density distribution , as explained in Appendix B. Again, Metropolis Monte Carlo simulations (at T = 1 MeV) were used to compute the average potential energy for finite nuclei. In Fig. 4.5 we plot binding energies for the ground state of several finite nuclei. Also shown in the figure are the corresponding experimental values. Results from both models, with the exception of a few finite nuclei, seem to reproduce the global trend of the binding energy curve. For completeness, binding energies per nucleon for various parameter sets have also been collected in Table (4.2). Also shown in the table are the predicted values for the neutron skin and charge radii for several nuclei along with their experimental values.

55 Binding Energy per Nucleon for Finite Nuclei 9.5

90 9 62 Zr 118 48 Ni Sn 40 Ca 150 Ca Sm 8.5 24 Mg 20 Ne 232 8 16 196 Th O Pt

12 208 7.5 C Pb 238 U /A(MeV) 7 bind E 6.5 Exp 6 Set-1 Set-2 5.5

5 0 25 50 75 100 125 150 175 200 225 250 A

Figure 4.5: Binding energy per particle versus number of particles A for various nuclei.

Table 4.2: Binding energy per nucleon and charge radii for several nuclei from simulations using various parameter sets and the experimental data. In addition, predictions are displayed for the neutron skin of these nuclei.

Nucleus Observable Experiment NL3 FSUGold Set-1 Set-2 40Ca B/A(MeV) 8.55 8.54 8.54 8.10 8.43 Rch (fm) 3.45 3.46 3.42 3.48 3.58 Rn-Rp(fm) – -0.05 -0.05 -0.07 -0.07 48Ca B/A(MeV) 8.67 8.64 8.58 8.21 8.57 Rch (fm) 3.45 3.46 3.45 3.68 3.77 Rn-Rp(fm) – 0.23 0.20 -0.09 -0.07 90Zr B/A(MeV) 8.71 8.69 8.68 8.61 8.87 Rch (fm) 4.26 4.26 4.25 4.34 4.53 Rn-Rp(fm) – 0.11 0.09 0.18 0.20 208Pb B/A(MeV) 7.87 7.88 7.89 7.91 8.05 Rch (fm) 5.50 5.51 5.52 5.67 5.98 Rn-Rp(fm) – 0.28 0.21 0.48 0.53

56 Set-1 Set-2 0.4

0.3

0.2 S(fm)

0.1

0

0 25 50 75 100 125 150 175 200 225 A

Figure 4.6: Neutron skin versus total number of particles A for various nuclei.

The neutron skin thickness (S) of a nucleus is defined as the difference between the root- mean-square radii of the neutron (Rn) relative to that of the proton (Rp)[60]. That is

S = R R , (4.23a) n − p 1 N R2 = (r R )2 (4.23b) n N i − CM i=1 X 1 Z R2 = (r R )2 , (4.23c) p Z i − CM i=1 X where RCM is the center of mass coordinate. The preliminary results for neutron-skin calculations are given in Fig. 4.6. The neutron skin is strongly correlated to the slope of symmetry energy [61]. Increasing the slope of the symmetry energy and thus the internal pressure of the system will result in a bigger neutron skin. In our models, the second parameter set has a higher symmetry energy slope compared to the first parameter set. Thus it explains why the neutron skin predicted with the second parameter set is larger than with the first set. Next, we present a comparison between charge radii obtained from the simulations to the experimental values in Fig. 4.7. The charge radii

57 6 238 U 232 208 Th 196 Pb Pt 5 150 Sm

118 Sn

(fm) 90

ch Zr

R 4 62 Ni 40 48 Ca Ca Exp 3 Set-1 Set-2

2 0 25 50 75 100 125 150 175 200 225 250 A

Figure 4.7: Charge radii versus total number of particles from simulations and the corre- sponding experimental data.

is defined as [62] 2 2 2 Rch =(Rp + RP ) , (4.24)

2 2 where RP is the mean square charge radius of an individual proton = 0.64 fm [63]. The solid lines represent a best fit to the data. We find that our simulation results are in good agreement with the experimental data. We are also interested in using our simulations to obtain a best fit to the semi- empirical mass formula. To implement the optimization, the amoeba subroutine from Numerical Recipes [37] was used. The subroutine will find the best empirical parameters 2 (avol, asurf , aCoul, asym) by minimizing the Chi-squared function (χ ) such that

1 X M 2 χ2 = i − i . (4.25) A Wi X   Here Xi are the experimental values for binding energy, Mi are the numerically determined values, Wi is the weight given to the experimental data, and A is the number of nuclei used in the fit. Note that in this calculation the weight has been set equal to one. The fitting results are shown in Table 4.3. The semi-empirical mass formula coefficients obtained

58 Table 4.3: Semi empirical mass formula best fit results for both parameter sets.

Set avol asurf acoul asym (MeV) I 16.80 22.54 0.72 25.67 II 16.55 20.97 0.67 28.59

from both model parameters are in a good agreement with the ones obtained directly from experimental data as given in Eqs. (4.17–4.20).

4.4.2 Two-body Correlation Function

In this section we present simulation results for nuclear pasta using the full Hamiltonian described in Eqs. (4.2a and 4.2b) for both parameter sets given in Table (4.1). We perform

Metropolis Monte Carlo simulations at a fixed electron fraction Ye=0.2 and a temperature of T=1 MeV. In core-collapse supernova the electron fraction starts at Ye around 0.5, but drops with time, as the result of electron capture, to a smaller value of Ye of the order of 0.1.

Hence Ye = 0.2 is a good representation of a typical neutron-rich condition in the star. Since the program is computationally intensive, in this preliminary study we run the simulations for A = 1000 particles. At fixed Ye = 0.2, we run the simulations for 800 neutrons and 200 protons with equal number of spin-up and spin-down nucleons at a density of ρ = 0.025 fm−3, which is typical for the pasta region. At this density the simulation volume has a length of approximately L = 34.19 fm. As we have done before, we fix the electron screening length arbitrarily to λ 10fm . ≃ A sample configuration of A = 1000 particles at density ρ = 0.025 fm−3 is displayed in Fig. 4.8. We found that all the protons and some of the neutrons are clustered into nuclei. In addition there is a low density neutron gas between the clusters. At this density the nuclei seem to cluster into cylindrical-like structures immersed in a dilute neutron gas. The most interesting part of the simulations is that the decision of whether nucleons cluster in nuclei or remain in the gas is being answered dynamically. As a comparison, if we lower the density from ρ=0.025 fm−3 to ρ=0.01 fm−3, as is shown in the right hand side of Fig. 4.8, we see that the system transforms into more conventional spherical shape neutron-rich nuclei immersed in a dilute neutron gas.

59 Figure 4.8: Monte Carlo snapshots of a configuration of N= 800 neutrons and Z= 200 protons at density 0.025 fm−3 (left) and 0.01 fm−3 (right). The following color code has been used: neutron spin up (tan); neutron spin down (white); proton spin up (blue),proton spin down (red)

The two-body neutron-neutron correlation function g(r) is shown in Fig. 4.9. The neutron-neutron correlation function is measured by computing the relative distance between neutron-neutron pairs. At short distances the correlation function is very small due to the hard core in the NN potential. The two-body correlation function shows large broad peaks between r = 2 fm and r = 6 fm. These peaks correspond to the other neutrons with the same spin bound in the same and other clusters. Superimposed on this broad peak we find three sharp peaks corresponding to the nearest, second-nearest, and third-nearest neighbors. At larger distances between 8 and 15 fm, the correlation function shows a modest dip below one, suggesting that the attractive NN interaction has shifted some neutrons from larger to smaller distances to form the clusters. The two body correlation function for proton-proton and neutron-proton are shown in Fig. 4.10 and Fig. 4.11. Again the correlation function is measured by computing the relative distance between proton-proton pairs and proton-neutron pairs respectively.

60 3

Set-1 Set-2 2.5

2

1.5 g(r)

1

0.5

0 0 2 4 6 8 10 12 14 16 r(fm)

Figure 4.9: Neutron-neutron two-body correlation function at a temperature of T = 1 MeV, −3 an electron fraction Ye = 0.2, and density ρ = 0.025 fm .

10

9 Set-1 Set-2 8

7

6

5 g(r)

4

3

2

1

0 0 2 4 6 8 10 12 14 16 r(fm)

Figure 4.10: Proton-proton two-body correlation function at a temperature of T = 1 MeV, −3 an electron fraction Ye = 0.2, and density ρ = 0.025 fm .

61 9

8 Set-1 Set-2

7

6

5 g(r) 4

3

2

1

0 0 2 4 6 8 10 12 14 16 r(fm)

Figure 4.11: Proton-neutron two-body correlation function at a temperature of T = 1 MeV, −3 an electron fraction Ye = 0.2, and density ρ = 0.025 fm .

62 CHAPTER 5

CONCLUSIONS

5.1 Introduction to Physics of Compact Stars

Chapter 2 has been written in the spirit of giving students a pedagogical introduction to the study of compact stars particularly white dwarfs and neutron stars. Students learned several important lessons from this project. One of them relates to the usefulness of scaling the equations. Without scaling, the problem would have been unsolvable. This is due to the tremendous range of scales encountered in this problem; there are more than 60 orders of magnitude between the minute electron mass and the immense solar mass. Another important lesson learned is that, contrary to what seems to happen in the classroom, most problems in physics have no analytic solution. Thus, numerical analysis is a necessary step towards a solution. In this project we relied on the free Fermi gas equation of state in calculating the properties of white dwarfs and neutron stars. While the free Fermi gas EoS seems to be sufficient for white dwarf stars leading to the Chandrasekhar upper mass limit, the same conclusion can not be obtained for neutron stars. The structure of neutron stars poses several additional challenges. First, Newtonian gravity must be replaced by general relativity. This implies that the structure equations must be replaced by Tolman-Oppenheimer-Volkoff equations. Second at higher densities encountered in the interior of neutron stars, the equation of state receives important corrections from the interactions among the neutrons. Thus, Pauli correlations are no longer sufficient to describe the equation of state. This is evident in the upper mass limit for neutron stars which yield too small a mass at around 0.7 solar mass. In reality we have seen stars bigger than this limiting mass. In fact neutron stars have been observed to have mass around 1–3 solar mass. Therefore, corrections from nuclear interactions are crucial in the study of neutron stars. This is a topic of intense research

63 activity in our effort to understand the physics of neutron stars and the structure of exotic compact objects known as hybrid and quark stars. 5.2 Virtues and Flaws of the Pauli Potential

In Chapter 3 we have conducted a systematic study of the standard version of the Pauli potential. While simple and widely used, such a version fails to reproduce some of the most basic properties of a free Fermi gas. We found that the constraints imposed by such a Pauli potential, namely, the suppression of phase-space configurations for having two fermions with both positions and momenta similar to each other, are too weak to faithfully reproduce some basic properties of a free Fermi gas. By examining the well-known behavior of the Slater determinant we suggest that phase-space configurations should be suppressed when either the positions or the momenta of the fermions are close to each other. By incorporating these features into a new form of the Pauli potential — and by carefully tuning the parameters of the model — the momentum distribution and the two-body correlation function of a free Fermi gas were accurately reproduced. This new version — inspired by the properties of a Slater determinant — generated accurate (canonical) momentum distribution and two-body correlation functions, while avoiding crystallization. However, in the course of this study a pathology that is generic to all momentum-dependent Pauli potential in conjunction with the Hamiltonian approach was uncovered. The momentum distribution generated via Monte Carlo (or other) methods may differ significantly from the resulting “kinematical” momentum distribution. This suggests that while the kinetic energy of the free Fermi gas (computed from the canonical momenta) may be accurately reproduced, the distribution of velocities may be grossly distorted. Indeed, we found a distribution of velocities that significantly under-estimates — by a factor of 10 — that of a free Fermi gas. Such “sluggishness” among the particles could have disastrous consequences by inducing artificial ordering in the system (e.g., “crystallization”). The possible appearance of artificial long-range order in the system must be examined on a case by case basis. For example, with the standard version of the Pauli potential [45] the system displays an anomalous two-body correlation function suggestive of crystallization. On the other hand, the Pauli potential introduced in this work faithfully reproduces the two-body correlation function of a free Fermi gas. However, the possibility for generating artificial

64 correlations in the system (e.g., crystallization) remains large. To avoid this problem, we will resort to using a momentum-independent Pauli Potential. Consequently, the potential will not generate the correct momentum distribution and kinetic energy of a free Fermi gas. However, we are still able to impose the correct two-body correlation function with this new potential. 5.3 Equation of State for Nuclear Pasta

In the previous work by Piekarewicz and collaborators [32, 33], the spin-independent (vector part) of the weak neutral-current neutrino pasta scattering has been studied via a spin- independent semi-classical model. However, due to its spin-independent properties this model is not sufficient in calculating the axial-vector (spin) response. Thus, a new model that incorporate the spin-dependent is needed. The search for a new-modified model is the main focus of this chapter. In Chapter 4 we have employed a semi-classical model to simulate the dynamics of the pasta phase of neutron-rich matter. Despite its simplicity, the model describes the crucial physics of nuclear pasta that is the interplay between short-range nuclear attraction and long range Coulomb repulsion. Apart from these features, the model has also incorporated the fermionic correlations via a classical Pauli potential as discussed in Chapter 3. Using the Metropolis Monte Carlo algorithm we obtained two best-fit parameter sets that satisfactorily reproduced various bulk properties of symmetric nuclear matter: such as the saturation density, the binding energy per nucleon at saturation, and the compression modulus, and properties of several finite nuclei. The new parameter sets also captured the clustering of symmetric nuclear matter below saturation density, a phenomena typical of nuclear pasta. With the newly refined potential, we are interested in studying how the nuclear pasta affects the neutrino transport which is a crucial factor in the core-collapse of a supernova. We evaluated the two-body correlation function for nuclear pasta via Metropolis Monte Carlo simulations. Although it is beyond the scope of the current dissertation, we are currently working on the spin-dependent weak neutral current neutrino responses leading to the study of neutrino transport properties in nuclear pasta.

65 APPENDIX A

Monte Carlo Simulation

A.1 The estimator

In our work we need to calculate the expectation value of energy, that is

−βEµ Eµe E = µ . (A.1) e−βEµ h i P µ However, the ideal route to calculate the expectationP value by averaging over all states µ, is impossible for a large system. Thus we resort to using Monte Carlo techniques. These techniques work by choosing a subset of states µ –e.g. (µ1 ...µM )– at random based on some probability distribution p which will be discussed later. The expectation energy E for µ h M i this subset of states becomes

M −1 −βEµi i=1 Eµipµi e EM = M − , (A.2) h i 1 −βEµj P j=1 pµj e where E represents the estimator of EP, which in the limit M , E = E . h M i h i → ∞ h M i h i The next question is how to weight the states M such that EM will be an accurate estimate of E . This is related to choosing the p distributions that represents the correct h i µ distribution of the system. For a classical simulation the best choice for pµ is the Boltzmann probability distribution, that is

−1 −βEµ pµ = Z e . (A.3)

Substituting Eq. (A.3) into Eq. (A.2) yields

1 M E = E . (A.4) h M i M µi i=1 X The tricky part in Monte Carlo simulation is generating an appropriate random set of states according to the Boltzmann probability distribution. All Monte Carlo simulations rely

66 on Markov processes to generate the random set of states. A Markov process is a mechanism which, given a system in one state µ, generate a new state ν in a random fashion. We will use a Markov process repeatedly to generate a Markov chain of states in such a way that eventually the simulation will produce a succession of states with the desired Boltzmann probability distribution. We call the process of reaching the Boltzmann distribution A.2 The Metropolis Algorithm

One type of Markov chain Monte Carlo techniques is the Metropolis algorithm. To generate

Eµi in Eq. (A.4) using Metropolis algorithm we rely on the following procedures. Suppose the simulation starts at an arbitrary point Xn. To generate Xn+1, we make a trial move to a new point Xt. There is no strict rule on how to choose the new point. It can be chosen, for example, uniformly at random within a multi dimensional cube of a small side δ about

Xn. The trial move will then be accepted or rejected according to the ratio

p(X ) R = t = e−β(E(Xt)−E(Xn)) (A.5) p(Xn)

If R is larger than 1, then the move is always accepted and we let Xn+1 = Xt. If R is less than 1, then the following rules apply. If R is bigger than η, an arbitrary random number uniformly distributed in the interval [0,1], then the move is accepted. Otherwise the move is rejected, and we have Xn+1=Xn. This procedure generates Xn+1, and we can generate the next step by following the same process. Notice that by doing this selection, we always accept transition to a new state which has lower or equal energy to the present state. If the new state has a higher energy than the present state, then we may accept it with the probability given above. The next important question is how to efficiently choose the step size (δ). To answer this question, suppose Xn is at the maximum of p, the most likely place to be. If δ is large, then p(Xt) will be very much smaller than p(Xn) and most trial moves will be rejected. However, if δ is small, most trial moves will be accepted, but the random walker will not move very far. This will lead to a poor sampling of the distribution. Therefore, a good rule of thumb is that the size of the trial step should be chosen so that about half of the trial steps are accepted [64].

67 7

/A

6 /A

/A

5

4 E/A (MeV)

3

2

1 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 # of thermalization step

Figure A.1: Monte Carlo thermalization step vs energy per nucleon for symmetric nuclear matter with A=500 particles at T=1 MeV

A.3 Equilibration

Once the selection process has taken place, the next question is when we should start calculating our estimator EM . To answer this question we have to perform two computational steps. The first step is the equilibration step. In the equilibration step, we run our simulation for a suitably long period of time until the system comes to equilibrium. Then we measure the quantity we are interested in –in our case the Energy– over another suitable long period of time and average it, to evaluate the estimator in Eq. (A.4). Fig. A.1 shows one of our equilibration time for symmetric matter simulations with A=500 particles at density ρ 0.4 fm−3 and temperature T =1 MeV. By looking at the plot we can guess that the system has reached equilibrium at around Monte Carlo stept 4000. ∼ Up until this point the energy keeps changing, but after this point it just fluctuates around a steady average value. However, it many cases it is possible that the system gets stuck in some meta-stable state, and gives a roughly constant values for all quantities we are observing for a while. We may mistake this local energy minimum–in which the system stays temporarily– for the global energy minimum. Thus cautionary steps must be taken to avoid this trap[65].

68 APPENDIX B

Woods-Saxon Potential

A good representation of finite nuclei density is provided by the Woods-Saxon model, that is ρ ρ(r)= 0 , (B.1) 1 + exp[(r R)/β] − where ρ is the saturation density 0.16 baryon/fm−3, β is related to the width of the edge 0 ≈ region and is fixed at 0.55 fm. R is the mean nuclear radius and its value (R) is given by normalizing the total number of baryons (A), such that

A = 4πρr2dr . (B.2) Z Fig. B.1 displayed typical shapes for Wood-Saxons potential. It is evident from the figure that even though larger nuclei have a larger mean diameter, the edge (β) region has a similar width in all nuclei.

We are interested in evaluating the Fermi momentum (pF ) for finite nuclei as required by Eqs. (4.21a and 4.21b). To do so, information on average density is needed. We use the Woods-Saxon density distribution to calculate the average density, i.e. 1 ρ = 4πρ2r2dr . (B.3) avg A Z Once the value of average density is known, we can obtain the Fermi momentum by using the following relation 3π2ρ 1/3 p = avg . (B.4) F 2   Table (B.1) shows the finite nuclei and their corresponding Fermi momentum.

69 0.2

40 Ca 118 ρ Sn 0 208 0.15 Pb ) -3 0.1 (fm ρ

0.05

0 0 2 4 6 8 10 12 14 16 18 20 r(fm)

Figure B.1: Typical shapes for Woods-Saxon potential.

70 Table B.1: Table of nuclei and their Fermi momentum.

Nuclei Z N A pF (MeV)

12C 6 6 12 197.330 16O 8 8 16 207.013 20Ne 10 10 20 211.162 24Mg 12 12 24 214.333 40Ca 20 20 40 222.286 48Ca 24 24 48 224.813 62Ni 28 34 62 228.099 90Zr 40 50 90 232.391 118Sn 50 68 118 235.176 150Sm 62 88 150 237.429 194Pt 78 116 194 239.640 196Pt 78 118 196 239.724 208Pb 82 126 208 240.205 232Th 90 142 232 241.062 238U 92 146 238 241.257

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76 BIOGRAPHICAL SKETCH

Jutri Taruna

Education

2001-2008 Ph.D. in Theoretical Nuclear Physics, Department of Physics, Florida State • University.

1997-2000 M.Sc. in Theoretical Nuclear Physics, Department of Physics, University of • Indonesia.

1989-1995 B.Sc. in Physics Education, State University of Jakarta. •

Experience

2003-2008 Research Assistant, Department of Physics, Florida State University. • 2001-2007 Teaching Assistant, Department of Physics, Florida State University. •

Publications

1. Taruna, J. and Piekarewicz, J. and Perez-Garcia, M. A., Virtues and Flaws of the Pauli Potential, J. Phys. A: Math Theor 41 035308 (2008).

2. Jackson, Chris B. and Taruna, J. and Pauliot, S. L. and Ellison, B.W. and Lee, D. D., and Piekarewicz, J., Compact Objects for Everyone: I. White dwarf stars, Eur. J. Phys. 26 695 (2005).

77