The Nuclear Symmetry Energy and Properties of Neutron Stars

Lie-Wen Chen ( 陈列文 )(Department of Physicsa and INPAC, Shanghai Jiao Tong U

niversity. lwchen@sjtu.edu.cn)

十三届全国中高能核物理大会暨第七届全国中高能核物理专题研讨会 ,2009 年 11 月 5-7 日，中国科学技术大学，合肥，中国

Collaborators ：Wei-Zhou Jiang (SEU)Che Ming Ko and Jun Xu (TAMU)Bao-An Li (TAMU-Commerce) Hong-Ru Ma (SJTU)Gao-Chan Yong (IMP,CAS)De-Hua Wen (SCUT)Zhi-Gang Xiao and Ming Zhang (Tsinghua U)Bao-Jun Cai, Rong Chen, Peng-Cheng Chu, Zhen Zhang (SJTU)

Outline

The nuclear symmetry energy

Constraining the density dependence of the nuclear symmetry energy in heavy-ion collisions

The nuclear symmetry energy and neutron stars

Summary and outlook

Main References: L.W. Chen, C.M. Ko, B.A. Li, and G.C. Yong, Front. Phys. China 2(3), 327 (2007)

[arXiv:0704.2340]B.A. Li, L.W. Chen, and C.M. Ko, Phys. Rep. 464, 113-281 (2008) [arXiv:0804.3580]J. Xu, L.W. Chen, B.A. Li, and H.R. Ma, Phys. Rev. C 79, 035802 (2009) [arXiv:0807.4477]J. Xu, L.W. Chen, B.A. Li, and H.R. Ma, Astrophys. J. 697, 1549-1568 (2009) [arXiv:0901.2309]Z.G. Xiao, B.A. Li, L.W. Chen,G.C. Yong, and M. Zhang, Phys. Rev. Lett. 102, 062502 (2009)D.H. Wen, B.A. Li, and L.W. Chen, Phys. Rev. Lett., in press [arXiv:0908.1922]

Liquid-drop model

(Isospin) Symmetry energy term

W. D. Myers, W.J. Swiatecki, P. Danielewicz, P. Van Isacker, A. E. L. Dieperink,……

Symmetry energy including surface diffusion effects (ys=Sv/Ss)

The Nuclear Symmetry Energy

EOS of Isospin Asymmetric Nuclear Matters

2 4ym ( )( , ) ( ), ( ),0) /( n pE OE E

(Parabolic law)

The Nuclear Symmetry Energy2

sym 2

1 ( , )( )

2

EE

The Nuclear Matter Symmetry Energy

0

sym

sy

2

0 0

0 0

s

sy

0

msym 0

0

0

y

m

m

, ( )3 18

30 MeV (LD mass formula: )

( )3 (Many-Body Theory:

( )

: ; Exp: ???

( )

50 200 e )M

( )

V

E Myers & Swiatecki, NPA81; Pomorski & Du

EL

KL

dek, PRC67

L

K

E E

0

2sym2

0 sym2

asy sym

sym

(Sharma et al

isobaric incompressiblity

( )9 (Many-Body Theory: : ; Exp: ???)

The isospin part of the of asymmetric nuclear matter

700 466 MeV

320 180 6 GMR( : M eV

K

K K K

K

L

E

Shlomo&Youngblood,PRC47,529(93);

., PRC38, 2562 (88));

566 1350 34 159 M eV

550 1

(

(T. Li et al, PRL99,162503(2007))00 MeV )

Symmetry energy term

(poorly known)

Symmetric Nuclear Matter

(relatively well-determined)

Equation of State of symmetric nuclear matter is relatively well determined

(1) EOS of symmetric matter around the saturation density ρ0

Giant Monopole Resonance 0

22

0 0 2Incompressibility: K =9 ( )

d E

d

K0=231±5 MeVPRL82, 691 (1999)Recent results:K0=240±20 MeVG. Colo et al. U. Garg et al.

__

GMR 0Frequency f K

(2) EOS of symmetric matter for 1ρ0< ρ < 3ρ0 from K+ production in HIC’sJ. Aichelin and C.M. Ko,

PRL55, (1985) 2661C. Fuchs,

Prog. Part. Nucl. Phys. 56, (2006) 1

Transport calculations indicate that “results for the K+ excitation function in Au + Au over C + C reactions as measured by the KaoS Collaboration strongly support the scenariowith a soft EOS.”

C. Fuchs et al, PRL86, (2001) 1974

Equation of State of symmetric nuclear matter is relatively well determined

See also: C. Hartnack, H. Oeschler, and J. Aichelin,

PRL96, 012302 (2006)

(3) Present constraints on the EOS of symmetric nuclear matter for 2ρ0< ρ < 5ρ0 using flow data from BEVALAC, SIS/GSI and AGS

Use constrained mean fields to predict the EOS for symmetric matter

• Width of pressure domain reflects uncertainties in comparison and of

assumed momentum dependence.

P. Danielewicz, R. Lacey and W.G. Lynch, Science 298, 1592 (2002)

2Pressure P( )s

E

The highest pressure recorded under laboratory controlled conditions in nucleus-nucleus collisions

y

px

High density nuclear matter2 to 5ρ0

Equation of State of symmetric nuclear matter is relatively well determined

Isospin Physics with Heavy-Ion Collisions

Density Dependence of the Nuclear Symmetry Energy

HIC’s induced by neutron-rich nuclei (CSR/Lanzhou,FRIB,GSI,RIKEN……)

Most uncertain property of an asymmetric

nuclear matter

Isospin Nuclear PhysicsWhat is the isospin dependence of the in-medium nuclear effective interactions???

Neutron Stars …

Structures of Radioactive Nuclei, SHE …

Isospin Effects in HIC’s …

Many-Body Theory

Many-Body Theory

Transport Theory General Relativity

Nuclear Force

EOS for Asymmetric

Nuclear Matter

On Earth!!! In Heaven!!!

Most recent review (169 pages): Bao-An Li, Lie-Wen Chen, and Che Ming Ko, Physics Reports 464, 113-281 (2008)

The Symmetry Energy

The multifaceted influence of the nuclear symmetry energy A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys. Rep. 411, 325 (2005).

Isospin physicsIsospin physics

Science 312, 190 (2006)

LHC

Liquid

Hadron gas

Nucleon gas

DeconfinementChiral symmetry restoration

QCD Phase Diagram

isospin

Heavy-Ion Accelerator Facilities

1. HIRFL, CSR/HIRFL (China)2. GANIL (France)3. GSI (Germany)4. NSCL/MSU,FRIB/MSU5. RIKEN (Japan)

1 50 100 150 200 250 300

10

102

103

104

A

E (

MeV

/u)

HIRFL

GANIL

NSCL/MSU

CSR/HIRFL

RIKEN

GSI

Dubna, LBL, ORNL, TAMU,INFN, KVI,…

High Energy HI Accelerator Facilities:

AGS,RHIC/BNLSPS,LHC/CERN

Medium Energy HI Accelerator Facilities

Nuclear Matter EOS: Many-Body Approaches

Microscopic Many-Body Approaches Non-relativistic Brueckner-Bethe-Goldstone (BBG) Theory Relativistic Dirac-Brueckner-Hartree-Fock (DBHF) approach Self-consistent Green’s Function (SCGF) Theory Variational Many-Body (VMB) approach …… Effective Field Theory Density Functional Theory (DFT) Chiral Perturbation Theory (ChPT) …… Phenomenological Approaches Relativistic mean-field (RMF) theory Relativistic Hartree-Fock (RHF) Non-relativistic Hartree-Fock (Skyrme-Hartree-Fock) Thomas-Fermi (TF) approximations Phenomenological potential models ……

Chen/Ko/Li, PRC72, 064309(2005) Chen/Ko/Li, PRC76, 054316(2007)

The Nuclear matter symmetry energy

Z.H. Li et al., PRC74, 047304(2006) Dieperink et al., PRC68, 064307(2003)

BHF

Constraining the density dependence of the nuclear symmetry energy in heavy-ion collisions

Promising Probes of the Esym(ρ) in Nuclear Reactions (an incomplete list !)

At sub-saturation densities (亚饱和密度行为) Sizes of n-skins of unstable nuclei from total reaction cross sections Proton-nucleus elastic scattering in inverse kinematics Parity violating electron scattering studies of the n-skin in 208Pb n/p ratio of FAST, pre-equilibrium nucleons Isospin fractionation and isoscaling in nuclear multifragmentation Isospin diffusion/transport Neutron-proton differential flow Neutron-proton correlation functions at low relative momenta t/3He ratio Hard photon production

Towards high densities reachable at CSR/China, FAIR/GSI, RIKEN, GANIL and, FRIB (高密度行为) π -/π + ratio, K+/K0 ratio? Neutron-proton differential transverse flow n/p ratio at mid-rapidity Nucleon elliptical flow at high transverse momenta n/p ratio of squeeze-out emission

Solve the Boltzmann equation using test particle method Isospin-dependent initialization Isospin- (momentum-) dependent mean field potential

Isospin-dependent N-N cross sections a. Experimental free space N-N cross section σexp

b. In-medium N-N cross section from the Dirac-Brueckner approach based on Bonn A potential σin-medium

c. Mean-field consistent cross section due to m* Isospin-dependent Pauli Blocking

0 sym

1(1 )

2 z CV V V V

Phase-space distributions ( , , ) satify the Boltzmann equation

( , , ) ( , )p r r p c NN

f r p t

f r p tf f I f

t

Isospin-dependent BUU (IBUU) model

Transport model for HIC’s

EOS

Isospin- and momentum-dependent potential (MDI)

30

0

0

0

0.16 fm

( ) / 16 MeV

MDI Interaction

( ) 31.6 MeV

211 MeV

*/ 0.6

g( o )

8

G ny

sym

E A

E

K

m m

Chen/Ko/Li, PRL94,032701

(2005)Li/Chen, PRC72, 064611

(2005)

Das/Das Gupta/Gale/Li,

PRC67,034611 (2003)

Transport model: IBUU04

1.05

00

0.69 31.6( /31.6( / ) )( )

Symmetry energy constrained at -saturation densities

between the and lines, agrees extremely well with the

sub

-1 0 PR

AsymE

x x

(IBUU04)

(ImQMD)

X=-1

Tsang et al., PRL 102, 122701 (2009)

Symmetry Energy at Sub-saturation Densities

Chen/Ko/Li, PRL 94, 032701 (2005)

(Subsaturation ： 0.2-0.3<ρ/ρ0<1.2)

IBUU04, Xiao/Li/Chen/Yong/Zhang, PRL102, 062502(2009) A Supersoft Esym at supra-saturation densities !!!

M. Zhang et al., PRC80,034616(2009)

Symmetry energy at High Densities

Lattimer/Prakash, Science 304, 536 (2004)

Neutron star has solid crust over liquid core.Rotational glitches: small changes in period from sudden unpinning of superfluid vortices.

Evidence for solid crust. 1.4% of Vela moment of in

ertia glitches. Needs to know the transitio

n density to calculate the fractional moment of inertia of the crust

Link et al., PRL83,3362 (99)

The Nuclear Symmetry Energy and Neutron Stars

core-crust transition

2. Thermodynamic approach

Or , similarly one can use 3. the RPA

If one uses the parabolic approximation (PA)

Then the stability condition is:

>0

Onset of instability in the uniform n+p+e matter

1. Dynamical approachk0 (neglecting Coul.)

Stability condition:

Core-Crust Transition Density: Parabolic Law fails!

(1) It is NOT enough to know the symmetry energy, one almost has to know the exact EOS of n-rich matter

Why?Because it is the determinant of the curvature matrixthat determines the stability condition Example:

Not so surprise:

Zhang/Chen, CPL 18 (2000) 142Steiner, Phys.Rev. C74 (2006) 045808

Higher-order term effects on direct URCA

Xu/Chen/Li/Ma, PRC79, 035802 (2009)

(2) Locating the inner edge of neutron star crust

Kazuhiro Oyamatsu, Kei Iida

Phys. Rev. C75 (2007) 015801

pasta

Xu/Chen/Li/Ma, PRC79, 035802 (2009)

Xu/Chen/Li/Ma, ApJ 697, 1547 (2009), arXiv:0901.2309

Parabolic Approximation has been assumed !!!

Significantly less than their fiducial values:ρ t=0.07-0.08 fm-3 and Pt=0.65 MeV/fm3

(3) Constraints on M-R relation of NS

EOS of Neutron Star Matter

(3) Constraints on M-R relation of NS

(Isospin Diff)

(Empirical estimate Link et al., PRL83,3362(99))

LattimerPrakash

(4) Properties of neutron star crusts

Xu/Chen/Li/Ma, ApJ 697, 1549 (2009), arXiv:0901.2309

Larger L leads to thicker neutron-skin , but thinner neutron star crust !!!

core crust total

Neutron skin v.s. Esym:Chen/Ko/Li, PRC72,064309 (2005)

The softest symmetry energythat the TOV is still stable is x=0.93 giving M_max=0.11 solar mass and R=>28 km

For pure nucleonic matter???New Physics???Supersoft symmetry energy at HD ?

K0=211 MeV is used, higher incompressibility

for symmetric matter will lead to higher masses systematically

?

(5) HD Esym and properties of neutron stars

Supersoft symmetry energy at high densities

Supersoft symmetry energy at high densities

Among 119 Skyrme forces, there are 55 forces are consistent with the flow data !!

Among the 55 forces, there are: 18 forces giving supersoft symmetry energy19 forces giving soft symmetry energy18 forces giving normal symmetry energy

Supersoft HD Symmetry Energy: Neutron Stars

and Fifth Force ???Supersoft symmetry energy at HD

non-Newtonian gravity in neutron stars???Wen/Li/Chen, PRL, in press, arXiv:0908.1922

A neutral spin-1 boson: 1. Very light; 2. Weakly coupled to baryons(Fayet et al: U-soson?)

The Yukawa term is simply part of the matter system in general relativity. Consequently, the Einstein equation remains the same and only the EOS is modifed. -----Y. Fujii: in Large Scale Structures of the Universe, page 471-477, Eds. J. Audouze et al. (1988), International Astronomical Union.

Supersoft HD Symmetry Energy: Neutron Stars and Fifth Force ???

2 2 2For MDI with x=1, we have in order to support a neutron star with mass

larger than 1.4 solar mass without violating the causality and the observed rotation

/ 50 15

al constraint.

0 GeVg

Wen/Li/Chen, PRL, in press, arXiv:0908.1922

The isospin diffusion data, Isoscaling and Isotope dependence of GMR seem to give a stringent constraint for the sub-saturation density behavior of the symmetry energy (L=86±25 MeV and Kasy=-500±50 MeV)

Probing the high density behavior of the symmetry energy remains a big challenge and the pion ratio data from FOPI favor a supersoft Esym at high densities.

Significant constraints on inner edge and crust properties of neutron stars have been already obtained from present knowledge on symmetry energy at subsaturation density region.

Supersoft Esym at high densities may give constraints on violation of the inverse-square-law of gravity

IV. Summary and Outlook

Thanks ！

(5) Inner Crust EOS Dependence

Xu/Chen/Li/Ma, ApJ 697, 1549 (2009), arXiv:0901.2309

The mass is insensitive to the inner crust EOSThe radius is sensitive to the inner crust EOS for a softer symmetry energyThe inner crust EOS has tiny effects on the ΔI/I when ΔI/I is small