nuclear equation of state and neutron stars · 2021. 7. 15. · nuclear equation of state and...

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!!


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


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.


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


Figure: Many body diagrams for nuclear matter calculation (C. Drischler, Phd thesis)


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)


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


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.


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.


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)


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)


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


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


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)


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.


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.


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)


Domains for a supernova simulation

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


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


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)


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


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


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 ).


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


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


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)


Simulation


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


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


Thank you for your attention !


nuclear equation of state and neutron stars · 2021. 7. 15. · nuclear equation of state and...

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