Nuclear Equation of State and Neutron Stars
Yeunhwan Lim
Department of Science EducationEwha 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 thediscovery 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- M : 1.2 ∼ 2.x M� (2.14+0.1
−0.09(2.08+0.07−0.07)M� PSR J0704+6620;
2.01± 0.04M� PSR J0348+0432 ; 1.97± 0.04M� PSR J614-2230 )- 2× 1011 earth g → General relativity- B field : 108 ∼ 1012G.- 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
Inner coreHyperons?
Quarks?
Outer coren, p, e, µ
Inner crustn, Z, e
Outer crust Z, e
Envelopeρ ∼ ρs(= 108g/cm3)
8 ∼15km
ρ ∼ 2ρ0
ρ ∼ 0.5ρ0(= 2× 1014g/cm3)
ρ ∼ ρd(= 4× 1011g/cm3)
ρ ∼ 1010g/cm3
Bose (K−, π−) condensation
Hyperon 1S0 superfluidity
Color superconductivity
Uniform nuclear matter
n 3P2, p 1S0 superfluidity
n 1S0 superfluidity
BCC lattice
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
dr= −G (M(r) + 4πr3p/c2)(ε+ p)
r(r − 2GM(r)/c2)c2,
dM
dr= 4π
ε
c2r2,
(1)
Nuclear physics provide the information for ε 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
0.00 0.08 0.16 0.24 0.32
ρ(fm−3)
−20
−10
0
10
20
30
40
E/A
(MeV
)PNM
SNM
Sv ' 32 MeV
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.
0.00 0.05 0.10 0.15 0.20 0.25 0.30
n (fm−3)
−20
0
20
40
60
E/A
(MeV
)
PNM
SNM
Chiral EFT
Fit
E(n, x) =1
2mτn +
1
2mτp + (1− 2x)2fn(n) +
[1− (1− 2x)2
]fs(n) , (2)
fs(n) =3∑
i=0
ain(2+i/3) , fn(n) =
3∑i=0
bin(2+i/3) (3)
Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 7 / 32
Neutron Star EOS constraints
Exp. Theory
Obs.
EDFE(n, x)
Experiments- SNM properties, Neutron skin thickness, binding energies
Theory- Neutron matter calculations (QMC, MBPT, ..)
ObservationGravitation 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)
0.0 0.2 0.4 0.6 0.8 1.0
n(fm−3)
0
100
200
300
400
500
E/A
(MeV
)
Posterior
1
1e-01
1e-02
1e-03
11e-011e-021e-03
0.00 0.05 0.10 0.15 0.20 0.25 0.30
−20
0
20
40
60
Prior
9 10 11 12 13 14 15
R (km)
0.0
0.5
1.0
1.5
2.0
M(M�
)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
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
Λ1.4 = 350+169−114(EOSs) vs Λ1.4 = 190+380
−120 (LIGO).
1.0 1.2 1.4 1.6 1.8 2.0 2.2
M(M�)
0
200
400
600
800
1000
Λ
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
0.2 0.4 0.6 0.8 1.0 1.2 1.4
nc (fm−3)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
PD
F,P
(nc)
M = 1.1M�
0.362 fm−3 ≤ nc±σ ≤ 0.514 fm−3
0.303 fm−3 ≤ nc±2σ ≤ 0.698 fm−3
nc = 0.416 fm−3
0.2 0.4 0.6 0.8 1.0 1.2 1.4
nc (fm−3)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
PD
F,P
(nc)
M = 1.2M�
0.388 fm−3 ≤ nc±σ ≤ 0.55 fm−3
0.327 fm−3 ≤ nc±2σ ≤ 0.761 fm−3
nc = 0.443 fm−3
0.2 0.4 0.6 0.8 1.0 1.2 1.4
nc (fm−3)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
PD
F,P
(nc)
M = 1.3M�
0.415 fm−3 ≤ nc±σ ≤ 0.589 fm−3
0.353 fm−3 ≤ nc±2σ ≤ 0.832 fm−3
nc = 0.47 fm−3
0.2 0.4 0.6 0.8 1.0 1.2 1.4
nc (fm−3)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
PD
F,P
(nc)
M = 1.4M�
0.444 fm−3 ≤ nc±σ ≤ 0.632 fm−3
0.381 fm−3 ≤ nc±2σ ≤ 0.91 fm−3
nc = 0.497 fm−3
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
0.2 0.4 0.6 0.8 1.0 1.2 1.4
nc (fm−3)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
PD
F,P
(nc)
M = 1.5M�
0.475 fm−3 ≤ nc±σ ≤ 0.682 fm−3
0.41 fm−3 ≤ nc±2σ ≤ 1.004 fm−3
nc = 0.53 fm−3
0.2 0.4 0.6 0.8 1.0 1.2 1.4
nc (fm−3)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
PD
F,P
(nc)
M = 1.6M�
0.503 fm−3 ≤ nc±σ ≤ 0.749 fm−3
0.424 fm−3 ≤ nc±2σ ≤ 1.144 fm−3
nc = 0.569 fm−3
0.2 0.4 0.6 0.8 1.0 1.2 1.4
nc (fm−3)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
PD
F,P
(nc)
M = 1.7M�
0.541 fm−3 ≤ nc±σ ≤ 0.819 fm−3
0.459 fm−3 ≤ nc±2σ ≤ 1.266 fm−3
nc = 0.608 fm−3
0.2 0.4 0.6 0.8 1.0 1.2 1.4
nc (fm−3)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
PD
F,P
(nc)
M = 1.8M�
0.585 fm−3 ≤ nc±σ ≤ 0.894 fm−3
0.498 fm−3 ≤ nc±2σ ≤ 1.387 fm−3
nc = 0.653 fm−3
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
5 10 15 20 25 30 35
p2n0(MeV fm−3)
0
200
400
600
Λ1.
4
PNM32
10−110−210−3
10 15 20 25 30 35
p2n0(MeV fm−3)
PNM25
10010−110−210−3
5 10 15 20 25 30 35
p2n0(MeV fm−3)
0.8
1.0
1.2
1.4
1.6
I 1.3
38(1
045g
cm2)
PNM32
10−110−210−3
10 15 20 25 30 35
p2n0(MeV fm−3)
PNM25
10010−110−210−3
30 40 50 60 70 80
L (MeV fm−3)
0.06
0.07
0.08
0.09
0.10
0.11
nt
(fm−
3)
10−110−2
30 40 50 60 70 80
L (MeV fm−3)
0.3
0.4
0.5
0.6
p t(M
eVfm−
3)
10010−110−2
1.0 1.2 1.4 1.6
I (1045 g cm2)
0
200
400
600
800
Λ
Λ = −37.109 + 181.540 (I45)3.5
Rxy = 0.999986
M = 1.338M�
100
10−1
10−2
Figure: Lim & Holt, EPJA 2019
Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 14 / 32
0.0
0.5
1.0
1.5
2.0
2.5
3.0
M(M�
)
Smooth high-density prior
Miller
Riley
Modified high-density prior
9 10 11 12 13 14 15
R (km)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
M(M�
)
Posterior : Mmax > 2.59M�
9 10 11 12 13 14 15
R (km)
Posterior : Mmax < 2.59M�
10010−110−2 10010−1
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)
9 10 11 12 13 14
R (km)
0.0
0.5
1.0
1.5
2.0
2.5
3.0M
(M�
)Prior
kF expansion
9 10 11 12 13 14 15
R (km)
Posterior
d = 1.0
d = 3.0
d = 5.0
d = 7.0
d =∞
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 ofvariables. (1MeV ' 1010 K)(ρ = 104 ∼ 1014g/cm3, Ye = 0.01 ∼ 0.65, T = 0.1 ∼ 200MeV)
- core collapsing supernova explosion, proto-neutron stars, and compact binarymergers 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 techniqueSkyrme 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 andneutron 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- New : finer grid spacing, adds Hyperon(Λ,Σ+,−,0)
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 f3 - Total entropy per baryon s 4 - Total internal energy per baryon e5 - Neutron chemical potential 6 - Proton chemical potential7 - Neutron mass fraction (external to nuclei) 8 - Proton mass fraction9 - Alpha particle mass fraction 10 - Baryon pressure11 - Baryon free energy per baryon 12 - Baryon entropy per baryon13 - Baryon internal energy per baryon 14 - Nuclei filling factor u15 - Baryon density inside heavy nucleus 16 - dP/dn17 - dP/dT 18 - dP/dYe19 - ds/dT 20 - ds/dYe21 - Mass number of heavy nucleus 22 - Proton fraction of heavy nucleus23 - Number of neutrons in neutron skin
of heavy nucleus24 - Baryon density of nucleons external
to heavy nucleus and alpha particles25 - Proton fraction of nucleons external
to heavy nucleus and alpha particles26 - 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)
H
e
α
n
p
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)
0
10
20
30
E/N
(MeV
)
Hebeler+NNLOsimN3LO
0 0.04 0.08 0.12 0.16 0.20
n (fm−3)
0
10
20
30
E/N
(MeV
)
Unitary gas±2σ
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, nucleonsout 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, andνn neutron skin density.
From baryon and charge conservation, we can eliminate xo and ρo .
Free energy minimization,∂F∂ρi
= ∂F∂xi
= ∂F∂rN
= ∂F∂zi
= ∂F∂u = ∂F
∂ρα= ∂F
∂ρd= ∂F
∂ρt= ∂F
∂ρh= 0.
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 currentcode & LS SkM∗
10−6 10−5 10−4 10−3 10−2 10−1 100
n (fm−3)
100
101
T(M
eV)
Phase boundaries
Yp = 0.01
−5
−3
−1
1
3
5
10−6 10−5 10−4 10−3 10−2 10−1 100
n (fm−3)
100
101
T(M
eV)
Phase boundaries
Yp = 0.10
−5
−3
−1
1
3
5
10−6 10−5 10−4 10−3 10−2 10−1 100
n (fm−3)
100
101
T(M
eV)
Phase boundaries
Yp = 0.40
−5
−3
−1
1
3
5
10−6 10−5 10−4 10−3 10−2 10−1 100
n (fm−3)
100
101
T(M
eV)
Phase boundaries
Yp = 0.50
−5
−3
−1
1
3
5
Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 26 / 32
Phase Boundary
10°6 10°5 10°4 10°3 10°2 10°1
n (fm°3)
0
5
10
15
20
T(M
eV)
Inhomogeneous
Homogeneous
Phase boundaries
Yp = 0.005
SRO LS220
YL LS220
10°6 10°5 10°4 10°3 10°2 10°1
n (fm°3)
0
5
10
15
20
T(M
eV)
Inhomogeneous
Homogeneous
Phase boundaries
Yp = 0.105
SRO LS220
YL LS220
10°6 10°5 10°4 10°3 10°2 10°1
n (fm°3)
0
5
10
15
20
T(M
eV)
Inhomogeneous
Homogeneous
Phase boundaries
Yp = 0.405
SRO LS220
YL LS220
10°6 10°5 10°4 10°3 10°2 10°1
n (fm°3)
0
5
10
15
20
T(M
eV)
Inhomogeneous
Homogeneous
Phase boundaries
Yp = 0.655
SRO LS220
YL LS220
Yeunhwan Lim (EWHA) Nuclear Equation of State and Neutron Stars JUL 14, 2021 27 / 32
10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1
n (fm−3)
0
5
10
15
T(M
eV)
Inhomogeneous
Homogeneous
Model C
Phase boundaries
Yp = 0.40npaHnpaH+Zeff
npaH+Zeff+dth
10−6 10−5 10−4 10−3 10−2 10−1
ρ (fm−3)
0
2
4
6
8
10
12
14
16
T(M
eV)
LS 220, Yp = 0.4
STOS , Yp = 0.4
SHT, Yp = 0.4
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
10−4 10−3 10−2 10−110−5
10−4
10−3
10−2
10−1
100
Xi
Yp = 0.01
Xn
Xp
Xd
Xt
Xh
Xα
10−4 10−3 10−2 10−1
Yp = 0.1
10−4 10−3 10−2 10−1
Yp = 0.4
10−4 10−3 10−2 10−1
T=
1M
eV
Yp = 0.65
10−4 10−3 10−2 10−110−5
10−4
10−3
10−2
10−1
100
Xi
10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1
T=
5M
eV
10−4 10−3 10−2 10−110−5
10−4
10−3
10−2
10−1
100
Xi
10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1
T=
10M
eV
10−4 10−3 10−2 10−110−5
10−4
10−3
10−2
10−1
100
Xi
10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1
T=
50M
eV
10−610−510−410−310−210−1100
n (fm−3)
10−5
10−4
10−3
10−2
10−1
100
Xi
10−510−410−310−210−1100
n (fm−3)
10−510−410−310−210−1100
n (fm−3)
10−510−410−310−210−1100
n (fm−3)
T=
100
MeV
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), (momentof inertia in the future)
Numerical Method: Liquid Drop Model- fast and accurate- surface tension, critical temperature, effective charge (Screening fromoutside 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