Nuclear Equation of State and Neutron Stars

Yeunhwan Lim

Department of Science Education Ewha Womans University

JUL 14, 2021

APCTP Focus Program in Nuclear Physics 2021

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 1 / 32 Neutron Stars

Formed after core collapsing supernovae. Suggested by Walter Baade and Fritz Zwicky (1934) - Only a year after the discovery of the neutron by James Chadwick Jocelyn Bell Burnell and Antony Hewish observed pulsar in 1965. Neutron star is cold after 30s 60s of its birth - inner core, outer core, inner∼ crust, outer crust, envelope - R : 10km ∼ +0.1 +0.07 - M : 1.2 2.x M (2.14 0.09(2.08 0.07) M PSR J0704+6620; 2.01 0∼.04 M PSR J0348+0432− − ; 1.97 0.04 M PSR J614-2230 ) - 2 10±11 earth g General relativity ± - B× ﬁeld : 108 10→12G. ∼ - Central density : 3 10ρ0 Nuclear physics!! ∼ →

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 2 / 32 Inner structure of neutron stars Envelope

8 3 ρ ρs(= 10 g/cm ) Outer crust Z, e ∼ ρ 1010g/cm3 BCC lattice ∼ ρ ρ (= 4 1011g/cm3) Inner crust ∼ d × n, Z, e 1 n S0 superﬂuidity

ρ 0.5ρ (= 2 1014g/cm3) Outer core ∼ 0 × n, p, e, µ Uniform nuclear matter 8 3 1 ∼ n P2, p S0 superﬂuidity 15km ρ 2ρ0 Inner core ∼ Hyperons? Bose (K−, π−) condensation 1 Quarks? Hyperon S0 superﬂuidity Color superconductivity

Neutron Stars: - Dense nuclear matter physics

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 3 / 32 TOV equations for macroscopic structure

dp G(M(r) + 4πr 3p/c2)( + p) = , dr − r(r 2GM(r)/c2)c2 − (1) dM = 4π r 2, dr c2 Nuclear physics provide the information for and p.

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 4 / 32 Nuclear matter properties

Nuclear equation of state at T = 0 MeV

20 PNM

(MeV) 10

E/A S 32 MeV 0 v ' SNM 10 −

20 − 0.00 0.08 0.16 0.24 0.32 3 ρ(fm− )

Figure: Energy per baryon for symmetric nuclear matter and pure neutron matter

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 5 / 32 Figure: Many body diagrams for nuclear matter calculation (C. Drischler, Phd thesis)

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 6 / 32 Most neutron matter results can be ﬁtted using the quadratic expansion.

60 Chiral EFT Fit 40

(MeV) 20 PNM E/A 0 SNM

20 − 0.00 0.05 0.10 0.15 0.20 0.25 0.30 3 n (fm− )

1 1 2 2 (n, x) = τn + τp + (1 2x) fn(n) + 1 (1 2x) fs (n) , (2) E 2m 2m − − − 3 3 X (2+i/3) X (2+i/3) fs (n) = ai n , fn(n) = bi n (3) i=0 i=0

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 7 / 32 Neutron Star EOS constraints

Exp. Theory

EDF (n, x) E

Obs.

Experiments - SNM properties, Neutron skin thickness, binding energies Theory - Neutron matter calculations (QMC, MBPT, ..) Observation Gravitation wave : tidal deformabilities, Moment of inertia, Nicer (mass-radius), maximum mass(Mmax > 2.0M )

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 8 / 32 EOS constraints

Nuclear EOS constraints - Microscopic calculation(Pure neutron matter) - Nuclear structure: Neutron skin, binding energies of nuclei - Maximum mass of neutron stars(Mmax > 2.0M )

- Gravitational wave: tidal deformabilities(Λ1.4) - (Moment of inertia) - NICER(Neutron Star Interior Composition Explorer): mass and radius

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 9 / 32 EOSs and MR

Statistical uncertainties from EOSs (Theory + Experiment)

1e-03 1e-02 1e-01 1

1.0 500 1

60 0.9 Prior 2.0 400 40 0.8

20 0.7 1e-01 1.5 )

300 0 0.6 M (

(MeV) 20 − 0.5 200 0.00 0.05 0.10 0.15 0.20 0.25 0.30 M 1.0 E/A 1e-02 0.4

100 0.5 0.3

0.2 Posterior 0 1e-03 0.0 0.1 0.0 0.2 0.4 0.6 0.8 1.0 9 10 11 12 13 14 15 3 n(fm− ) R (km)

Y. Lim & J.W. Holt, PRL 2018.

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 10 / 32 Tidal deformability from EOSs

+169 +380 Λ1.4 = 350 114(EOSs) vs Λ1.4 = 190 120 (LIGO). − −

1000

800

600 Λ

400

200

0 1.0 1.2 1.4 1.6 1.8 2.0 2.2 M(M )

Y. Lim & J.W. Holt, PRL 2018.

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 11 / 32 Probability distribution of central density I

1.2 1.2

1.0 1.0 M = 1.1M M = 1.2M

3 3 3 3 0.8 0.362 fm− nc σ 0.514 fm− 0.8 0.388 fm− nc σ 0.55 fm− ) )

c ≤ ± ≤ c ≤ ± ≤

n 3 3 n 3 3

( 0.303 fm− nc 2σ 0.698 fm− ( 0.327 fm− nc 2σ 0.761 fm− ± ±

P ≤ ≤ P ≤ ≤ 0.6 3 0.6 3 n˜c = 0.416 fm− n˜c = 0.443 fm−

PDF, 0.4 PDF, 0.4

0.2 0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 3 3 nc (fm− ) nc (fm− )

1.2 1.2

1.0 1.0 M = 1.3M M = 1.4M

3 3 3 3 0.8 0.415 fm− nc σ 0.589 fm− 0.8 0.444 fm− nc σ 0.632 fm− ) )

c ≤ ± ≤ c ≤ ± ≤

n 3 3 n 3 3

( 0.353 fm− nc 2σ 0.832 fm− ( 0.381 fm− nc 2σ 0.91 fm− ± ±

P ≤ ≤ P ≤ ≤ 0.6 3 0.6 3 n˜c = 0.47 fm− n˜c = 0.497 fm−

PDF, 0.4 PDF, 0.4

0.2 0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 3 3 nc (fm− ) nc (fm− )

Figure: Lim & Holt, Eur. Phys. J. A 55, 209 (2019)

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 12 / 32 Probability distribution of central density II

1.2 1.2

1.0 1.0 M = 1.5M M = 1.6M

3 3 3 3 0.8 0.475 fm− nc σ 0.682 fm− 0.8 0.503 fm− nc σ 0.749 fm− ) )

c ≤ ± ≤ c ≤ ± ≤

n 3 3 n 3 3

( 0.41 fm− nc 2σ 1.004 fm− ( 0.424 fm− nc 2σ 1.144 fm− ± ±

P ≤ ≤ P ≤ ≤ 0.6 3 0.6 3 n˜c = 0.53 fm− n˜c = 0.569 fm−

PDF, 0.4 PDF, 0.4

0.2 0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 3 3 nc (fm− ) nc (fm− )

1.2 1.2

1.0 1.0 M = 1.7M M = 1.8M

3 3 3 3 0.8 0.541 fm− nc σ 0.819 fm− 0.8 0.585 fm− nc σ 0.894 fm− ) )

c ≤ ± ≤ c ≤ ± ≤

n 3 3 n 3 3

( 0.459 fm− nc 2σ 1.266 fm− ( 0.498 fm− nc 2σ 1.387 fm− ± ±

P ≤ ≤ P ≤ ≤ 0.6 3 0.6 3 n˜c = 0.608 fm− n˜c = 0.653 fm−

PDF, 0.4 PDF, 0.4

0.2 0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 3 3 nc (fm− ) nc (fm− )

Figure: Lim & Holt, Eur. Phys. J. A 55, 209 (2019)

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 13 / 32 3 2 1 3 2 1 0 3 2 1 3 2 1 0 10− 10− 10− 10− 10− 10− 10 10− 10− 10− 10− 10− 10− 10

1.6 600 PNM32 PNM25 PNM32 PNM25 )

2 1.4

400 g cm 4 . 45 1 1.2 Λ (10 338 .

1 1.0

200 I

0.8 0 5 10 15 20 25 30 35 10 15 20 25 30 35 5 10 15 20 25 30 35 10 15 20 25 30 35 3 3 3 3 p2n0 (MeV fm− ) p2n0 (MeV fm− ) p2n0 (MeV fm− ) p2n0 (MeV fm− )

2 1 2 1 0 10− 10− 10− 10− 10 100 0.11 800 3.5 Λ = 37.109 + 181.540 (I45) 0.6 − R = 0.999986 0.10 xy 600 ) 3 − ) 0.5

3 0.09

− 1

Λ 10− 400 (fm

t 0.08 (MeV fm n 0.4 t p 200 0.07 M = 1.338M 0.3

0.06 0 10 2 30 40 50 60 70 80 30 40 50 60 70 80 1.0 1.2 1.4 1.6 − 3 3 45 2 L (MeV fm− ) L (MeV fm− ) I (10 g cm )

Figure: Lim & Holt, EPJA 2019

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 14 / 32 2 1 0 1 0 10− 10− 10 10− 10

3.0 Smooth high-density prior Modiﬁed high-density prior Miller 2.5 Riley )

2.0 M ( 1.5

M 1.0 0.5 0.0 Posterior : Mmax > 2.59M Posterior : Mmax < 2.59M 3.0 2.5 )

2.0 M ( 1.5

M 1.0 0.5

0.0 9 10 11 12 13 14 15 9 10 11 12 13 14 15 R (km) R (km)

Figure: Mass radius conﬁdence intervals, NICER, PNM, SNM, GW170817, GW190425, arXiv:2007.06526

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 15 / 32 Mass radius of neutron stars using various constraints (Y.Lim and A. Schwenk in preparation)

3.0 Prior Posterior 2.5

2.0 )

( 1.5 M 1.0 d = 1.0 d = 3.0 d = 5.0 0.5 d = 7.0 kF expansion d = ∞ 0.0 9 10 11 12 13 14 9 10 11 12 13 14 15 R (km) R (km)

Figure: Mass radius conﬁdence intervals, NICER(J003+0451), PNM, SNM, GW170817, Mmax > 2.01, NICER2(J0704+6620)

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 16 / 32 Nuclear Equation of State for Hot Dense Matter

What is it and why is it important? - Nuclear EOS is thermodynamic relation for given ρ, Ye , T with wide range of variables. (1 MeV 1010 K) 4 14 ' 3 (ρ = 10 10 g/cm , Ye = 0.01 0.65, T = 0.1 200MeV) ∼ ∼ ∼ - core collapsing supernova explosion, proto-neutron stars, and compact binary mergers involve neutron stars.

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 17 / 32 How can we construct EOS table ?

How can we construct EOS table ? We need nuclear force model and numerical method. Nuclear force model Numerical technique Skyrme Force model Liquid Drop(let) approach (LDM) (non-relativistic potential model) Relativistic Mean Field model (RMF) Thomas Fermi Approximatoin (TF) Finite-Range Force model Hartree-Fock Approximation (HF) Nuclear Statistical Equilibrium (NSE)

LS EOS Skyrme force + LDM (without neutron skin) ⇒ STOS RMF + Semi TF (parameterized density proﬁle) ⇒ SHT RMF + HARTREE ⇒ HSB RMF + NSE ⇒ Nuclear force model should be picked up to represent both ﬁnite nuclei and neutron star observation + Neutron matter calculation.

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 18 / 32 Representative EOSs

LS EOS (Lattimer Swesty 1991) Use Skyrme type potential with Liquid droplet approach - Consider phase transition, several K STOS EOS (H. Shen, Toki, Oyamastu, Sumiyoshi 1998), new version (2011) Use RMF with TF approximation and parameterized density profile (PDP) - Old : awkward grid spacing +, ,0 - New : finer grid spacing, adds Hyperon(Λ, Σ − ) SHT EOS (G. Shen, Horowitz, Teige 2010) Use RMF with Hartree approximation HSB (M. Hempel and J. Schaffner-Bielich). 2010, 2012 - Use Relativistic mean field model (TM1, TMA, FSUgold) - Nuclear statistical equilibrium (alpha, deutron, triton, helion)

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 19 / 32 Domains for a supernova simulation

Figure from Oertel et al., Rev. Mod. Phys. 89, 015007.

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 20 / 32 Items in EOS tables

1 - Total pressure P 2 - Total free energy per baryon f 3 - Total entropy per baryon s 4 - Total internal energy per baryon e 5 - Neutron chemical potential 6 - Proton chemical potential 7 - Neutron mass fraction (external to nuclei) 8 - Proton mass fraction 9 - Alpha particle mass fraction 10 - Baryon pressure 11 - Baryon free energy per baryon 12 - Baryon entropy per baryon 13 - Baryon internal energy per baryon 14 - Nuclei ﬁlling factor u 15 - Baryon density inside heavy nucleus 16 - dP/dn 17 - dP/dT 18 - dP/dYe 19 - ds/dT 20 - ds/dYe 21 - Mass number of heavy nucleus 22 - Proton fraction of heavy nucleus 23 - Number of neutrons in neutron skin of heavy nucleus 24 - Baryon density of nucleons external to heavy nucleus and alpha particles 25 - Proton fraction of nucleons external to heavy nucleus and alpha particles 26 - Out of whackness: µn µp µe +1.293 MeV − −

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 21 / 32 Calculation of EOS

Schematic picture of inhomogeneous nuclear matter(neutron star crust)

n p H α

Liquid Drop Model(Fast and accurate) Most diﬃcult part: inhomogeneous matter, low temperature Adopt state-of-the-art neutron matter results -ex) MBPT(Drischler et al., PRL 2019), QMC(Tews et al., PRC 2016)

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 22 / 32 New energy density functional for the nuclear EOS. (Y. Lim, S. Huth, and A. Schwenk in preparation)

30 Hebeler+ NNLOsim N3LO 20 (MeV)

E/N 10

300 Unitary gas 2σ ± 20 (MeV)

E/N 10

0 0 0.04 0.08 0.12 0.16 0.20 3 n (fm− )

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 23 / 32 Free energy

Total free energy density consists of

F = FN + Fo + Fα + Fd + Ft + Fh + Fe + Fγ (4)

where FN , Fo , Fα, Fe , and Fγ are the free energy density of heavy nuclei, nucleons out the nuclei, alpha particles, electrons, and photons.

FN = Fbulk,i + Fcoul + Fsurf + Ftrans

Fo = Fbulk,o α, d, t, h particles : Non-interacting Boltzman gas e, γ : treat separately

For Fbulk,i , Fbulk,o , and Fsurf , we use the same force model. Fsurf from the semi inﬁnite nuclear matter calculation The is the modiﬁcation of LPRL (1985), LS (1991, No skin) - Consistent calculation of surface tension - Deuteron, triton, helion - The most recent parameter set

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 24 / 32 Free energy minimization

For ﬁxed independent variables (ρ, Yp, T ), we have the 11 dependent variables (ρi , xi , rN , zi , u, ρo , xo , ρα, ρd , ρt , ρh). where i heavy nuclei, o nucleons outside, x proton fraction, u ﬁlling factor, and νn neutron skin density.

From baryon and charge conservation, we can eliminate xo and ρo . Free energy minimization, ∂F = ∂F = ∂F = ∂F = ∂F = ∂F = ∂F = ∂F = ∂F = 0. ∂ρi ∂xi ∂rN ∂zi ∂u ∂ρα ∂ρd ∂ρt ∂ρh Finally, we have 6 equations to solve and 6 unknowns. z = (ρi , ln(ρno ), ln(ρpo ), xi , ln(u), zi ).

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 25 / 32 Results : Relative pressure diﬀerence bewteen the current code & LS SkM∗

5 5

Yp = 0.01 Yp = 0.10 3 3 1 1 10 Phase boundaries 10 Phase boundaries

1 1 (MeV) (MeV) 1 1 T − T −

100 100 3 3 − −

6 5 4 3 2 1 0 5 6 5 4 3 2 1 0 5 10− 10− 10− 10− 10− 10− 10 − 10− 10− 10− 10− 10− 10− 10 − 3 3 n (fm− ) n (fm− )

5 5

Yp = 0.40 Yp = 0.50 3 3 1 1 10 Phase boundaries 10 Phase boundaries

1 1 (MeV) (MeV) 1 1 T − T −

100 100 3 3 − −

6 5 4 3 2 1 0 5 6 5 4 3 2 1 0 5 10− 10− 10− 10− 10− 10− 10 − 10− 10− 10− 10− 10− 10− 10 − 3 3 n (fm− ) n (fm− )

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 26 / 32 Phase Boundary

20 20 SRO LS220 SRO LS220 YL LS220 YL LS220

15 Yp =0.005 15 Yp =0.105

Phase boundaries Phase boundaries

10 10 (MeV) (MeV) T T

5 5 Homogeneous Homogeneous

Inhomogeneous Inhomogeneous 0 6 5 4 3 2 1 0 6 5 4 3 2 1 10° 10° 10° 10° 10° 10° 10° 10° 10° 10° 10° 10° 3 3 n (fm° ) n (fm° )

20 20 SRO LS220 SRO LS220 YL LS220 YL LS220

15 Yp =0.405 15 Yp =0.655

Phase boundaries Phase boundaries

10 10 (MeV) (MeV) T T

5 5 Homogeneous Homogeneous

Inhomogeneous Inhomogeneous 0 6 5 4 3 2 1 0 6 5 4 3 2 1 10° 10° 10° 10° 10° 10° 10° 10° 10° 10° 10° 10° 3 3 n (fm° ) n (fm° )

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 27 / 32 npaH 16 LS 220, Yp = 0.4 npaH+Zeﬀ Yp = 0.40 15 14 STOS , Yp = 0.4 npaH+Zeﬀ+dth SHT, Yp = 0.4 12 Model C 10 10 Phase boundaries (MeV)

(MeV) 8 T T 6 5 4 Homogeneous Inhomogeneous 2

0 8 7 6 5 4 3 2 1 10− 10− 10− 10− 10− 10− 10− 10− 0 6 5 4 3 2 1 10− 10− 10− 10− 10− 10− 3 3 n (fm− ) ρ (fm− )

Figure: Phase boundaries using Model C from EOS table (Left) and representative EOSs (Right)

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 28 / 32 Simulation

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 29 / 32 Particle fraction

Yp = 0.01 Yp = 0.1 Yp = 0.4 Yp = 0.65 100 Xn 1 Xp 10−

i 2 Xd 10− X Xt 3 = 1 MeV 10−

Xh T 4 10− Xα 5 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 100 1 10−

i 2 10− X

3 = 5 MeV 10− 4 T 10− 5 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 100 1 10−

i 2 10− X

3 = 10 MeV 10− 4 T 10− 5 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 100 1 10−

i 2 10− X

3 = 50 MeV 10− 4 T 10− 5 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 100 1 10−

i 2 10− X 3 10− = 100 MeV

4 T 10− 5 10− 6 5 4 3 2 1 0 5 4 3 2 1 0 5 4 3 2 1 0 5 4 3 2 1 0 10−10−10−10−10−10− 10 10−10−10−10−10− 10 10−10−10−10−10− 10 10−10−10−10−10− 10 3 3 3 3 n (fm− ) n (fm− ) n (fm− ) n (fm− )

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 30 / 32 Summary

Neutron star and Nuclear equation of state Nuclear Model : Energy density functional (Exp. + Theory + Obs.) - should be consistent with ﬁnite nuclei, - neutron matter calculation - maximum mass of neutron stars, gravitational wave, MR(NICER), (moment of inertia in the future) Numerical Method: Liquid Drop Model - fast and accurate - surface tension, critical temperature, eﬀective charge (Screening from outside particles) - deuteron, triton, helion

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 31 / 32 Thank you for your attention !

Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 32 / 32