Introduction Einstein’s Equations The Tolman-Oppenheimer-Volkov Equation Structure of Compact Objects: Neutron Star and Strange Star Nuclear Model M.I.T Bag Model Experimental Observations and Realistic EoS Results Conclusions Backup Slides References A Brief Introduction to Neutron Stars and Quark Stars Fábio Köpp [email protected] Advisor: Prof. Dr. Cesar Zen Vasconcellos Work presented at 14th International Workshop on Hadron Physics (Hadron Physics 2018), CNUM: C18-03-18 , arxiv:1804.08785 Fábio Köpp Federal University of Rio Grande do Sul Neutron Stars and Quark Stars 1 / 17 Introduction Einstein’s Equations The Tolman-Oppenheimer-Volkov Equation Structure of Compact Objects: Neutron Star and Strange Star Nuclear Model M.I.T Bag Model Experimental Observations and Realistic EoS Results Conclusions Backup Slides References Sumary 1 Introduction 2 Einstein’s Equations 3 The Tolman-Oppenheimer-Volkov Equation 4 Structure of Compact Objects: Neutron Star and Strange Star 5 Nuclear Model 6 M.I.T Bag Model 7 Experimental Observations and Realistic EoS Results 8 Conclusions 9 Backup Slides 10 References Fábio Köpp Federal University of Rio Grande do Sul Neutron Stars and Quark Stars 2 / 17 Introduction Einstein’s Equations The Tolman-Oppenheimer-Volkov Equation Structure of Compact Objects: Neutron Star and Strange Star Nuclear Model M.I.T Bag Model Experimental Observations and Realistic EoS Results Conclusions Backup Slides References Stellar Evolution Figura: Stellar Evolution and their lifetime. Fábio Köpp Federal University of Rio Grande do Sul Neutron Stars and Quark Stars 3 / 17 Introduction Einstein’s Equations The Tolman-Oppenheimer-Volkov Equation Structure of Compact Objects: Neutron Star and Strange Star Nuclear Model M.I.T Bag Model Experimental Observations and Realistic EoS Results Conclusions Backup Slides References Main Features[5, 2] The Neutron Star is formed when a large star which before collapse had a total of 10-29 M . The radius is of order ≈ 10 km and a mass between 1.4-2.6 M ; The NS was proposed by Baade and Zwicky in 1933 at CALTECH and discovered by Joselyn Bell and Anthony Hewish at Cambridge in 1967. The first theoretical model was proposed by J. Robert Oppenheimer and G. M. Volkoff in 1939; The NS are determined by the EoS (equations of state) at densities ranging from 14 3 r0 ≈ 3x10 g/cm (nuclear density) to 10 r0; The EoS (bulk matter: when mass number A ! ¥) for NS is constrained by empirical 1 symmetric nuclear properties near or equal at r0 and extended for non-symmetric matter; In these stars, causality and relativistic propagation are important dynamical considerations; The NS is not made only by neutrons, but also by electron, protons, muons and other particles that can be created at a specific energy. The inclusion of these particles is needed to maintain charge neutrality and the chemical potential in equilibrium; The processes driven by the weak force (b decay, its inverse and quarks decay) must be taken into account; The stability of neutron star is maintained by degeneracy2 pressure of nucleons plus repelling force (short distances) between them. The same occurs for quark star, however, we only have the contribution of degeneracy pressure from the quarks. 1Neutrons = Protons. q 2 2 2 EF T , where EF = kF + m . Fábio Köpp Federal University of Rio Grande do Sul Neutron Stars and Quark Stars 4 / 17 Introduction Einstein’s Equations The Tolman-Oppenheimer-Volkov Equation Structure of Compact Objects: Neutron Star and Strange Star Nuclear Model M.I.T Bag Model Experimental Observations and Realistic EoS Results Conclusions Backup Slides References Einstein’s Equations[5, 2] The Einstein’s equations relate matter-energy to space geometry. The left side tells how the space is curved in presence of matter-energy by the tensor Gµn. This tensor is equal to −8pT µn and is 0 in the absence of matter. 1 µn µn Rµn − g R ≡ G ; (1) 2 We consider a perfect fluid (the pressure is isotropic in each element of the fluid at the rest frame). The shear stress and heat transport are absent in it. T µn = pgµn + (p + e)uµun (2) where p is the pressure, gµn is the metric, e is the energy density and uµ is the quadri-velocity of the fluid. We want to study the case of a static star and spherically symmetric given by the Schwarzschild metric. The invariant line interval is given by: 2 µ n 2n(r) 2 2l(r) 2 2 2 2 2 2 dt = gµndx dx = e dt − e dr − r dq − r sen qdf ; (3) The Schwarzschild metric is diagonal, thus its components are 2n(r) 2l(r) 2 2 2 g00 = e ; g11 = −e ; g22 = −r ; g33 = −r sen q and gµn = 0 (µ 6= n) (4) Solving the Einstein’s equation for this metric and considering a perfect fluid, we obtain the TOV’s equations[1]. Fábio Köpp Federal University of Rio Grande do Sul Neutron Stars and Quark Stars 5 / 17 Introduction Einstein’s Equations The Tolman-Oppenheimer-Volkov Equation Structure of Compact Objects: Neutron Star and Strange Star Nuclear Model M.I.T Bag Model Experimental Observations and Realistic EoS Results Conclusions Backup Slides References The Tolman-Oppenheimer-Volkov Equation [1, 3] These equations are solutions of the Einstein’s equations for static and spherically symmetric star filled with a perfect fluid. −1 dp Ge(r)M (r) p(r) 4pr 3p(r) 2GM (r) = − 1 + 1 + 1 − (5) dr c2r 2 e(r) M (r)c2 c2r dM 4pr 2e(r) = 4pr 2r(r) = (6) dr c2 Where G is the gravitational constant and c is the speed of light. The terms in square brackets are relativistic corrections. These differential coupled equations can be solved using Runge-Kutta method. The evolution of the system occurs in r and the initial conditions for r=0 are: m(r)=0, r(r)= some value and stops every time when p(r6= 0) =0 and jumps to the next r(r) value. The input for solving these equations are the EoS, which may describe nuclear matter, dark matter or quark matter. Fábio Köpp Federal University of Rio Grande do Sul Neutron Stars and Quark Stars 6 / 17 Introduction Einstein’s Equations The Tolman-Oppenheimer-Volkov Equation Structure of Compact Objects: Neutron Star and Strange Star Nuclear Model M.I.T Bag Model Experimental Observations and Realistic EoS Results Conclusions Backup Slides References Structure of Compact Objects Figura: The internal and external layers of neutron star and strange star. In this work, we are considering the entire star made of proton and neutrons for nuclear matter. For quark matter, made of quarks up, down and strange. Charge neutrality and potential chemical equilibrium are not considered here. Fábio Köpp Federal University of Rio Grande do Sul Neutron Stars and Quark Stars 7 / 17 Introduction Einstein’s Equations The Tolman-Oppenheimer-Volkov Equation Structure of Compact Objects: Neutron Star and Strange Star Nuclear Model M.I.T Bag Model Experimental Observations and Realistic EoS Results Conclusions Backup Slides References s − w Nuclear Model [4] Reproduces the nuclear potential features(attractive and repulsive), the binding energy and symmetry energy at saturation point; Failures to reproduce the compressibility factor (K) and the effective mass per nucleon ∗ (m =Mn); The approach used by this model is the Relativistic Mean Field, i.e, the meson field operators are constants in position and time (or, in other words, they are classical field operators). The equations of states (EoS) are: Baryonic density (rB ≡ B=V ) g 3 rB = k 6p2 F Energy Density g2 m2 g Z kF = v 2 + s (M − M∗)2 + d3k(k2 + M∗2)1=2 e 2 rB 2 3 2mV 2gS (2p) 0 Pressure g2 m2 1 g Z kF k2 P = v r2 − s (M − M∗)2 + d3k 2 B 2 3 2 ∗2 1=2 2mV 2gS 3 (2p) 0 (k + M ) Effective Mass 2 ∗ gS M = M − rS mS Fábio Köpp Federal University of Rio Grande do Sul Neutron Stars and Quark Stars 8 / 17 Introduction Einstein’s Equations The Tolman-Oppenheimer-Volkov Equation Structure of Compact Objects: Neutron Star and Strange Star Nuclear Model M.I.T Bag Model Experimental Observations and Realistic EoS Results Conclusions Backup Slides References s − w Nuclear Model [4] - Results 2 2 M2 2 2 M2 The couplings are: Cs ≡ gs ( 2 ) = 357:4 and Cv ≡ gv ( 2 ) = 273:8. The nucleon mass is 938 MeV. ms mv Nuclear Properties of Symmetric Matter (g = 4) ∗ −3 −1 m =Mn K[MeV] B/A [MeV] a4 [MeV] r0 (fm ) KF0 (fm ) Waleckas’s 0.557 535.28 -15.749 22.01 0.193 1.419 Model Experimental 0.564-658 200-300 -15.75 21.96 0.193 1.41 values Sigma-Omega Nuclear Model Sigma-Omega Nuclear Model 600 3 Neutron Matter Neutron Matter Symmetric Matter 2.75 Symmetric Matter 500 Causal Limit 2.5 · · 2.25 400 2 ] 3 1.75 Solar 300 /M 1.5 star M P [MeV/fm 1.25 200 1 0.75 100 0.5 0.25 0 0 100 200 300 400 500 600 700 800 900 1000 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 3 ε[MeV/fm ] ρ / ρ0 Sigma-Omega Nuclear Model 3 PSR J1614-2230 2.75 Neutron Matter Symmetric Matter · · 2.5 Neutron Matter 2.25 2 1.75 Solar Yellow line obtained from Annals Phys 179, 272(1987) /M 1.5 Star M 1.25 1 0.75 0.5 0.25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 R[Km] Figura: EoS, Densities at Maximum Mass star and M-R relation, respectively.