Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2008 Physics of Compact Stars Jutri Taruna Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] THE FLORIDA STATE UNIVERSITY 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