A Brief Introduction to Neutron Stars and Quark Stars

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

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 ρ0 ≈ 3x10 g/cm (nuclear density) to 10 ρ0; The EoS (bulk matter: when mass number A → ∞) for NS is constrained by empirical 1 symmetric nuclear properties near or equal at ρ0 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 (β 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 Gµν. This tensor is equal to −8πT µν and is 0 in the absence of matter.

1 µν µν Rµν − 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 µν = pgµν + (p + ε)uµuν (2)

where p is the pressure, gµν is the metric, ε 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 ν 2ν(r) 2 2λ(r) 2 2 2 2 2 2 dτ = gµνdx dx = e dt − e dr − r dθ − r sen θdφ , (3)

The Schwarzschild metric is diagonal, thus its components are

2ν(r) 2λ(r) 2 2 2 g00 = e , g11 = −e , g22 = −r , g33 = −r sen θ and gµν = 0 (µ 6= ν) (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 Gε(r)M (r) p(r) 4πr 3p(r) 2GM (r) = − 1 + 1 + 1 − (5) dr c2r 2 ε(r) M (r)c2 c2r

dM 4πr 2ε(r) = 4πr 2ρ(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, ρ(r)= some value and stops every time when p(r6= 0) =0 and jumps to the next ρ(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

σ − ω 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 (ρB ≡ B/V )

γ 3 ρB = k 6π2 F Energy Density

g2 m2 γ Z kF = v 2 + s (M − M∗)2 + d3k(k2 + M∗2)1/2 ε 2 ρB 2 3 2mV 2gS (2π) 0 Pressure g2 m2 1 γ Z kF k2 P = v ρ2 − s (M − M∗)2 + d3k 2 B 2 3 2 ∗2 1/2 2mV 2gS 3 (2π) 0 (k + M ) Effective Mass 2 ∗ gS M = M − ρS 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

σ − ω 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 (γ = 4) ∗ −3 −1 m /Mn K[MeV] B/A [MeV] α4 [MeV] ρ0 (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. Fábio Köpp Federal University of Rio Grande do Sul Neutron Stars and Quark Stars 9 / 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

M.I.T Bag Model

It was proposed[9] in order to take into account hadronic masses in terms of their constituents; Here we are just considering u, d e s quarks; The quarks are inside a bag which reproduces the asymptotic freedom (inside the bag, quarks behave as being free) and conﬁnement; The lower and upper limits of B constant is obtained by considering two quark or three quarks inside it. Assuming this quark matter as stable as iron atom(the most stable atom). The B constant assuming this assumption[10] 145.0 MeV < B1/4 < 162.76 MeV; The masses of u, d and s quarks are 5 MeV, 7 MeV and 150 MeV[5], respectively; −3 We consider ρ0 = 0.25 fm which is the actual value used in the literature;

The equations of states (EoS) are : Baryonic Density

1 ρ = k 3 2π2 F Energy Density

3 1 1 µf + kf = B + [µ k (µ2 − m2) − m4 ln( )] ε ∑ 2 f f f f f u,d,s 4π 2 2 mf

Fábio Köpp Federal University of Rio Grande do Sul Neutron Stars and Quark Stars 10 / 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

M.I.T Bag Model

Pressure

1 5 3 µf + kf P = −B + [µ k (µ2 − m2) + m4 ln( )] ∑ 2 f f f f f u,d,s, 4π 2 2 mf q 1/4 2 2 where the B =145 MeV. The chemical potential is deﬁned as µf = mf + kf . The difference in signs for B and fermions for the pressure equation, show us this counterbalance between them.

Figura: An ilustrative way to look the quarks and the bag.

Fábio Köpp Federal University of Rio Grande do Sul Neutron Stars and Quark Stars 11 / 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

M.I.T Bag Model - Results

M.I.T Bag Model M.I.T Bag Model 1000 3 B1/4=145 MeV B1/4=145 MeV 900 Causal Limit 2.75 2.5 800 2.25 700 2 Quarks u, d and s

] 600 · 3 1.75 Solar

500 /M 1.5 star M P [MeV/fm 400 1.25 1 300 0.75 200 0.5 100 0.25 0 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 5.5 6 3 ρ ρ ε[MeV/fm ] / 0

M.I.T Bag Model 3 PSR J1614-2230 2.75 T= 0 MeV, B1/4= 145 MeV T=50 MeV , B1/4= 145 MeV 2.5

2.25 Yellow line obtained from PRD 51,1440 (1995) 2 1.75 · Solar

/M 1.5 Star

M 1.25 1 0.75 0.5 0.25 0 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.

Fábio Köpp Federal University of Rio Grande do Sul Neutron Stars and Quark Stars 12 / 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

Experimental Observations and Realistic EoS Results

Figura: Neutron Mass observations and M-R relations from realistic EoS[12].

Fábio Köpp Federal University of Rio Grande do Sul Neutron Stars and Quark Stars 13 / 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

Conclusions

We disregarded some essential physics as chemical equilibrium and the local charge neutrality. Our M-R relation for Sigma-Omega model is almost the same obtained in [7]. The latter considered all physics mentioned before. For the M.I.T model, our M-R results are the same as obtained in [6]. We are working on numerical calculations to include all aspects we mentioned previously in order to be physicaly consistent. . However, the sigma-omega model could describe, in principle, the observation of PS1614-2230. The next step of this work is use some realistic model for the nuclear (outer crust) like Boguta-Bodmer and consider the core of star made of quarks. Use free codes like RNS and Lorene to compare the same models with and without rotation, magnetic ﬁeld and tidal effects. We quote[13, 14] as excellent articles for undergraduate students and for those who search for introductory articles.

"A man provided with paper, pencil, and rubber, and subject to strict discipline, is in effect

Fábio Köppa universal Federal University machine." of Rio Grande do Sul Neutron Stars and Quark Stars Alan Turing 14 / 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

Backup Slides - Sigma-Omega Model

For the nuclear model studied here, we have the following decay: − n → p + e + ¯νe− (7) The chemical equilibrium for this decay is: − µ = µ + µ + µ¯ (8) n p e νe− The condition of electric charge neutrality reads3:

k − = kF (9) Fe p Since the time of weak decays is very short when compared to neutron star formation time, all neutrinos leaked from star (µ¯ = 0). In this way, the chemical equilibrium reads νe− q q q k 2 + m2 = k 2 + m2 + k 2 + m2 (10) Fm n Fe p Fe e

Figura: Relative particle population in term of baryonic density [7].

3 Qb − Qe− = 0 ∑i i ρbi ρe− Fábio Köpp Federal University of Rio Grande do Sul Neutron Stars and Quark Stars 15 / 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

Backup Slides - M.I.T Bag Model

For the M.I.T bag model, we have the following decays[8]:

− d → u + e + ¯νe− (11) − s → u + e + ¯νe− (12) − s → c + e + ¯νe− (13) − − µ → +e + ¯νe− + νµ− (14) (15)

And the reactions:

s + d ←→ d + u (16) c + d ←→ u + d (17) (18)

contribute to the equilibrium of ﬂavors. Hence, from the relation above one gets:

µd = µu + µe− , µc = µu , µd = µs, µe− = µ− (19) As we have done before, using the condition of electric charge neutrality4 and the chemical equilibrium relations above.

4 Qb − Qe− − Qµ− = 0 ∑i i ρbi ρe− ρµ− Fábio Köpp Federal University of Rio Grande do Sul Neutron Stars and Quark Stars 16 / 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

Backup Slides - M.I.T Bag Model

The equations above can be simultaneously solved(numerically) for each baryonic density to get the relative particle population. The relative population for the M.I.T bag model with u,d,s and c quarks plus electrons and muons are shown in the below ﬁgure.

Figura: Relative particle population in terms of density energy[8].

Fábio Köpp Federal University of Rio Grande do Sul Neutron Stars and Quark Stars 17 / 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

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[14] R. R. Silbar and S. Reddy, Am. J. Phys. 72, 892 (2004) Erratum: [Am. J. Phys. 73, 286 (2005)] doi:10.1119/1.1852544 [nucl-th/0309041].

Fábio Köpp Federal University of Rio Grande do Sul Neutron Stars and Quark Stars 17 / 17