nuclear equation of state and neutron stars · 2021. 7. 15. · nuclear equation of state and...
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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
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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!!
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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
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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.
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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
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Figure: Many body diagrams for nuclear matter calculation (C. Drischler, Phd thesis)
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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)
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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
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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
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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.
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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.
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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)
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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)
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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
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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
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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)
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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.
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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.
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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)
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Domains for a supernova simulation
Figure from Oertel et al., Rev. Mod. Phys. 89, 015007.
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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
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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)
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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
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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
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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 ).
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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
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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
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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)
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Simulation
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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
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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
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Thank you for your attention !
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