Probing properties of neutron stars with heavy-ion reactions

transcript

Outline:

• Symmetry energy at sub-saturation densities and its impacts on astrophysics

Example: Core-crust transition density in neutron stars

• Symmetry energy at supra-saturation densities and its impacts on astrophysics

Example: Saving neutron stars with negative symmetry energy at supra-saturation densities with the light-weakly interacting U boson and its implications for cosmology

& collaborators:Joshua Edmonson, M. Gearheart, Will Newton, Justin Walker, De-Hua Wen, Chang Xu and Gao-Chan Yong, Texas A&M University-CommerceLie-Wen Chen and Hongru Ma, Shanghai Jiao-Tung UniversityPlamen G. Krastev, San Diego State UniversityChe-Ming Ko and Jun Xu, Texas A&M University, College StationWei-Zhou Jiang, Southeast University, Nanjing, ChinaZhigang Xiao and Ming Zhang, Tsinghua University, ChinaXunchao Zhang and Wei Zuo, Institute of Modern Physics, ChinaChampak B. Das, Subal Das Gupta and Charles Gale, McGill UniversityAndrew Steiner, Michigan State University

Bao-An Li

• Questions need your help !

The multifaceted influence of the isospin dependence of strong interaction

and symmetry energy in nuclear physics and astrophysics

J.M. Lattimer and M. Prakash, Science Vol. 304 (2004) 536-542. A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys. Rep. 411, 325 (2005).

The Esym (ρ) from model predictions using popular interactions

pure neutron matter symmetric nuclear matter2

1( ) ( ) ( )

EE E E

Examples:

Density

23 RMFmodels

Range of symmetry energy from isospin diffusion

More examples of microscopic model predictions

M.B. Tsang et al., PRL 92, 062701 (2004)M.B. Tsang et al., PRL 92, 062701 (2004)L.W. Chen, C.M. Ko and B.A. Li, PRL 94, 32701 (2005)L.W. Chen, C.M. Ko and B.A. Li, PRL 94, 32701 (2005)M. A. Famiano et al., PRL 97, 052701 (2006).M. A. Famiano et al., PRL 97, 052701 (2006).

(Colo)

(M.B. Tsang)(MDI)

Pressure of pure neutron-matter at ρ0

Masses of nuclei(Danielewicz)

Isospin diffusionChen, Li & Ko

(ImQMD Analyses

Tsang et al.)

Pigmy Dipole Resonance (PDR)Land/GSI, PRC76, 051603 (2007)

Latest constraints on the symmetry energy at sub-saturation densities

Gianluca Colo, arXiv:0902.3739M. B. Tsang, Yingxun Zhang, P. Danielewicz, M. Famiano, Zhuxia Li, W. G. Lynch, and A. W. Steiner, PRL 102, 122701 (2009).M. Centelles, X. Roca-Maza, X. Vias, and M. Warda, PRL 102, 122502 (2009).G. Lehaut, F. Gulminelli, and O. Lopez, PRL 102, 142503 (2009).

46 MeV < L < 111 MeV

Astrophysical impacts of the partially constrained symmetry

energy• Nuclear constraints on the moment of inertia of neutron starsarXiv:0801.1653

Aaron Worley, Plamen Krastev and Bao-An Li, The Astrophysical Journal 685, 390 (2008).

• Constraining properties of rapidly rotating neutron stars using data from heavy-ion collisions arXiv:0709.3621

Plamen Krastev, Bao-An Li and Aaron Worley, The Astrophysical Journal, 676, 1170 (2008)

• Constraining time variation of the gravitational constant G with terrestrial nuclear laboratory data arXiv:nucl-th/0702080

Plamen Krastev and Bao-An Li, Phys. Rev. C76, 055804 (2007).

• Constraining the radii of neutron stars with terrestrial nuclear laboratory data

Bao-An Li and Andrew Steiner, Phys. Lett. B642, 436 (2006). arXiv:nucl-th/0511064

• Nuclear limit on gravitational waves from elliptically deformed pulsars

Plamen Krastev, Bao-An Li and Aaron Worley, Phys. Lett. B668, 1 (2008). arXiv:0805.1973

• Nuclear constraints on properties of neutron star crust Jun Xu, Lie-Wen Chen, Bao-An Li and HongRu Ma, Phys. Rev. C79, 035802 (2009),

arXiv:0807.4477;

and The Astrophysical Journal 697, 1549 (2009), arXiv:0901.2309

Neutron Star Crust• Rotational glitches: small changes in period from sudden unpinning of superfluid vortices.– Evidence for solid crust.– 1.4% of Vela moment of

inertia glitches.– Needs to know the

density and pressure at the transition to calculate the fractional moment of inertia of the curst

Can one extract transition density from heavy-ion collisions?Chuck Horowitz at WCI3, TAMU, 2005

Yes, the symmetry energy constrained by intermediate energy heavy-ion experiments is in the same density range of the inner crust

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

Dynamical approach Thermodynamic approachK0

Similarly one can use the RPA

Stability condition:

If one uses the parabolic approximation (PA)

Then the stability condition is:

Pasta phases

3D-Hartree-Fock method for the pasta phase

in the inner crust of neutron stars• 3D Hartree-Fock calculations with Skyrme energy-density functional• Assume one can identify (local) unit cubic cells of matter at a given

density and temperature, calculate one unit cell containing A nucleons (A up to 3000)

• Periodic boundary conditions enforced by using FTs to take derivatives Periodic boundary conditions enforced by using FTs to take derivatives and obtain Coulomb potentialand obtain Coulomb potential

φφ(x,y,z) = (x,y,z) = φφ(x+L,y+L,z+L) (x+L,y+L,z+L) • Impose parity conservation in the three dimensions: Impose parity conservation in the three dimensions:

tri-axial shapes allowed, but not asymmetric ones. tri-axial shapes allowed, but not asymmetric ones.

Solution only in one octant of cell.Solution only in one octant of cell.• Quadrupole constraint placed on neutron density > self consistently

explore deformation space (energies of nuclear pasta shapes)• Method self-consistently incorporates the nuclear clusters at the bottom

of the inner crust together with their surface and curvature energies, and the unbound neutrons

William G. Newton, Ph.D thesis, University of Oxford, 2008William G. Newton and Jirina R. Stone, Physical Review C79, 055801

(2009)

• By performing calculations at increasing density one can observe the density at which matter becomes uniform (the energy density converges to that of uniform matter)– Above calculations for the SkM* Skyrme

parameterization and 500 nucleons in the unit cell.

Transition Density with 3D-Hartree-Fock

Transition Density with 3D-Hartree-Fock:

Comparison with Dynamical Method• 3DHF method used to calculate

transition density for 4 Skyrmes so far.• Consistently about 0.002 fm-3 higher

than estimating when uniform matter becomes unstable to small-amplitude long wavelength density perturbations (dynamical method)

• Dynamical method exact if transition was second order, gives lower limit if transition is first order

Jun Xu, Lie-Wen Chen, Bao-An Li and HongRu Ma, Phys. Rev. C79, 035802 (2009), and The Astrophysical Journal 697, 1549 (2009),

Core Crust Total radius

The Esym (ρ) from model predictions using popular interactions

1( ) ( ) ( )

EE E E

Examples:

Density

23 RMFmodels

EOS of pure neutron matterAlex Brown, PRL85, 5296 (2000).

Pion ratio probe of symmetry energy

at supra-normal densities

nn 0 1 5 a) Δ(1232) resonance model pp 5 1 0 in first chance NN scatterings: np(pn) 1 4 1 (negelect rescattering and reabsorption)

R. Stock, Phys. Rep. 135 (1986) 259. b) Thermal model: (G.F. Bertsch, Nature 283 (1980) 281; A. Bonasera and G.F. Bertsch, PLB195 (1987) 521)

exp[2( ) / ]n p kT

H.R. Jaqaman, A.Z. Mekjian and L. Zamick, PRC (1983) 2782.

c) Transport models (more realistic approach): Bao-An Li, Phys. Rev. Lett. 88 (2002) 192701, and several papers by others

31 1( ) {ln ( ) ( )}

m mn p mnn p asy asy Coul m T n p

mV V V kT b

GCCoefficients2

W. Reisdorf et al. for the FOPI/GSI collaboration , NPA781 (2007) 459

IQMD: Isospin-Dependent Quantum Molecular DynamicsC. Hartnack, Rajeev K. Puri, J. Aichelin, J. Konopka, S.A. Bass, H. Stoecker, W. Greiner Eur. Phys. J. A1 (1998) 151-169

π-/π+ ratio as a probe of symmetry energy at supra-normal densities

lowlow (high)(high) density region is more neutron-rich density region is more neutron-rich withwith stiff stiff (soft)(soft) symmetry energysymmetry energy

2( , ) ( ,0) ( )symE E E

Need a symmetry energy softer than the above to make the pion production region more neutron-rich!

2/3100 3(2corresponding t 1) ( )

5o ( )

8 FsymE E

E/A=800 MeV, b=0, t=10 fm/c

124 124 197 197

Isospin asymmetry reached in heavy-ion reactions2( , ) ( ,0) ( )symE E E

Symmetry energy

density

FRIB/M

CSR/IMP

Radioactive

Facilitie

N/Z dependence of pion production and effects of the symmetry energyZhi-Gang Xiao, Bao-An Li, L.W. Chen, G.C. Yong and. M. ZhangPRL 102, 062502 (2009).

FAIR/G

400 MeV/A

Excitation function

Central density

The most important contributions of nuclear force

At saturation densityUsing Paris potential

1( ) ( ) ( )

EE E E

I. Bombaci and U. LombardoPRC 44, 1892 (1991)

Using the Reid93 interaction

PRC68, 064307 (2003)

What will happen if the short-range repulsive tensor force is included at high densities?

Can the symmetry energy becomes negative at high densities?Yes, due to the isospin-dependence of the nuclear tensor forceThe short-range repulsion in n-p pair is stronger than that in pp and nn pairs At high densities, the energy of pure neutron matter can be lower than symmetric matter leading to negative symmetry energy

Example: proton fraction with 10 interactions leading to negative symmetry energy

3 30 0(0.048[ / ( )] ( / )() 1 2 )symsymE Ex x

symmetry energy

because of the term,

for symmetric matter,

it is energetically more favoriable to write =0=1

Negative Isospin separati

i.e., pure neutron m

on insta

atter +

re proton

matter

Is the negative symmetry energy

“unpleasant” or unphysical?Unpleasant !E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R.Schaeffer, NPA627, 710 (1997); NPA635, 231 (1998).Repeated by several others in other papers

Unphysical !

Quoted by several people

Why ?The only reason seems to be that “ neutron stars will then collapse while they do exist in nature”

How neutron stars are stablized?

TOV equation

P(r+dr)

P(r) Gravity

Nuclear pressure

Do we really know gravity at the Fermi distance?

So far, down to the 10 fm level, there is NO violation of the ISL

Extra dimension at short length or a new Boson?

String theorists have published TONS of papers on the extra dimension

In terms of the gravitational potential

Repulsive Yukawa potential due to the exchange of a new boson proposed in the super-symmetric extension of the Standard Model of the Grand Unification Theory

The neutral spin-1 gauge boson U is a candidate, it can mediate the interaction among dark matter particles, e.g., Pierre Fayet, PLB675, 267 (2009), C. Boehm, D. Hooper, J. Silk, M. Casse and J. Paul, PRL, 92, 101301 (2004).

Influences of the U-boson on Neutron stars

1xdxdxn

starsneutron for relevant 10km,Rfor 2

420) of mass of nucleus HYPER (a fm 10Rfor 2

10025.0 2

It has NO effect on finite nuclei

M.I. Krivoruchenko, et al., ep-ph/0902.1825v1,De-Hua Wen et al. (2009)

EOS of MDIx1+WILB

22 / g

M-R relation of neutron star with MDIx1+WILB

The moment of inertia provides a sensitive probe to determine g2/2

Questions and possible answers?

• What is causing the uncertain symmetry energy at high densities? (short-range tensor force??)

• How can one trace back to the underlying force from observables in nuclear reactions?

• Effective interactions, such as Skyrme and Gogny can lead to various high-density behavior of the symmetry energy, but they do not have the explicit Tensor force, ??

• None of the transport models using explicit tensor force, but yet, …

• ….

Symmetry energy and single nucleon potential used in the IBUU04 transport model

'0 0 0 0

, 3 , ' 3 '2 2 2 2

1 2 2,

( , , , , ) ( ) ( ) ( ) (1 ) 81

2 2( , ') ( , ')' '1 ( '

' , ( ) 121 , ( ) 96 ,

) / 1 ( ') /

2112 1 1

BU p A A B

C Cf r p f r pd p d p

B BA A

x x x x x

xK MeVx

SSSSSSSSSSSSSS

C.B. Das, S. Das Gupta, C. Gale and B.A. Li, PRC 67, 034611 (2003).

B.A. Li, C.B. Das, S. Das Gupta and C. Gale, PRC 69, 034614; NPA 735, 563 (2004).

softsoft

Single nucleon potential within the HF approach using a modified Single nucleon potential within the HF approach using a modified Gogny force:Gogny force:

Density ρ/ρ0

The momentum dependence of the nucleon potential is a result of the non-localityof nuclear effective interactions and the Pauli exclusion principle

The x parameter is introduced to mimic various predictions on the symmetry energy by different microscopic nuclear many-body theories using different effective interactions

Default: Gogny force

Two-body force: Gogny force

One-body potential

HF calculations

, , , ', ', ' | ( ) [ ]

(1 ) | , , , ', ', '

, , , ', ', ' | (1 ) ( )(1 ) | , , , ', ', '

rp p W BP HP MP P Exp

P P P p p

p p t P r r P P P p p

SSSSSSSSSSSSSSSSSSSSSSSSSSSS

SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS

Astronomers discover a neutron-star spining at 716

Science 311, 1901 (2006).

Plamen Krastev, Bao-An Li and Aaron Worley, APJ, 676, 1170 (2008)

RNS code by Stergioulas & Friedman

Gravitational waves from elliptically deformed

pulsars

Mass quadrupole moment

Breaking stain of crust

B. Abbott et al., PRL 94, 181103 (2005)B.J. Owen, PRL 95, 211101 (2005)

Solving linearized Einstein’s field equation of General Relativity, the leading contribution to the GW is the mass quadrupole moment

Frequency of the pulsar

Distance to the observer

Constraining the strength of gravitational wavesPlamen Krastev, Bao-An Li and Aaron Worley, Phys. Lett. B668, 1 (2008).

Compare with the latest upper limits from LIGO+GEO observations

It is probably the most uncertain factor

B.J. Owen, PRL 95, 211101 (05)

Phys. Rev. D 76, 042001 (2007)

0.12 0.14 0.16 0.18 0.20 0.22 0.24

MDIx0 MDIx-1 APR

0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24

MDIx0 MDIx-1 APR FIT

Scaling of the frequency and decay rate of the w-mode

MNRAS, 299 (1998) 1059-1068

MNRAS, 310, 797 (1999)

L. K. Tsui and P. T. Leung, MNRAS, 357, 1029(2005) ; APJ 631, 495(05); PRL 95, 151101 (2005)

Can the symmetry energy becomes negative at high densities?Yes, due to the isospin-dependence of the nuclear tensor forceThe short-range repulsion in n-p pair is stronger than that in pp and nn pairs At high densities, the energy of pure neutron matter can be lower than symmetric matter leading to negative symmetry energy

Example: proton fraction with 10 interactions leading to negative symmetry energy

3 30 0(0.048[ / ( )] ( / )() 1 2 )symsymE Ex x

symmetry energy

because of the term,

for symmetric matter,

it is energetically more favoriable to write =0=1

Negative Isospin separati

i.e., pure neutron m

on insta

atter +

re proton

matter

W. Reisdorf et al. for the FOPI collaboration , NPA781 (2007) 459

IQMD: Isospin-Dependent Molecular Dynamics C. Hartnack, Rajeev K. Puri, J. Aichelin, J. Konopka, S.A. Bass, H. Stoecker, W. Greiner Eur.Phys.J. A1 (1998) 151-169

Near-threshold π-/π+ ratio as a probe of symmetry energy at supra-normal densities

lowlow (high)(high) density region is more neutron-rich density region is more neutron-rich withwith stiff stiff (soft)(soft) symmetry energysymmetry energy

2( , ) ( ,0) ( )symE E E

Need a symmetry energy softer than the above to make the pion production region more neutron-rich!

2/3100 3(2corresponding t 1) ( )

5o ( )

8 FsymE E

Momentum and density dependence of the symmetry (isovector) potential

Lane potential extracted from n/p-nucleus scatterings and (p,n) charge exchange reactions provides only a constraint at ρ0:

Lane 1

( ) / 2 ,

28 6 MeV, 0.1 0.2

for E 100 MeV

n p R kin

U U U V E

P.E. Hodgson, The Nucleon Optical Model, World P.E. Hodgson, The Nucleon Optical Model, World Scientific, 1994 Scientific, 1994

G.W. Hoffmann and W.R. Coker, PRL, 29, 227 (1972).G.W. Hoffmann and W.R. Coker, PRL, 29, 227 (1972).

G.R. Satchler, Isospin Dependence of Optical Model G.R. Satchler, Isospin Dependence of Optical Model Potentials, Potentials, in Isospin in Nuclear Physics, in Isospin in Nuclear Physics, D.H. Wilkinson (ed.), (North-Holland, Amsterdam,1969)D.H. Wilkinson (ed.), (North-Holland, Amsterdam,1969)

/n p isoscalar LaneU U U

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 matterAstrophysical implications

K0=211 MeV is used, higher incompressibility

for symmetric matter will lead to higher masses systematically

0.6931.6( / ) ( ) 31.6( / )

( ) 500 50 MeV,

L=86 25 MeV

Summary

• Based on model analyses of intermediate energy heavy-ion collision data, the symmetry energy at sub-saturation densities is constrained to

• The FOPI/GSI pion data indicates a symmetry energy at supra-saturation densities softer than the APR prediction

Is Is ππ--//ππ++ ratio really a good probe of the symmetry energy at supra-normal densities? ratio really a good probe of the symmetry energy at supra-normal densities?(Tetsuya Murakami and possibly many others) (Tetsuya Murakami and possibly many others)

( )like

1 23 31 23 3

Sub-saturation density: 5% Supra-saturation densities: 25%

0X X for

X X for

XL=XH=1

XL=-2, XH=1

XL=1, XH=-2

XL=XH=-2π

t=10 fm/c

Correlation between the N/Z and the π-/ π+

(distance from the center of the reaction system)

t=10 fm/c

Another advantage: the π-/ π+ is INsensitive to the incompressibility of symmetric matter and reduces systematic errors, but the high density behavior of the symmetry energy (K

0=211 MeV

is used in the results shown here)

Asymmetric nuclear matter

In hyperonic matter

What we found about the core-crust transition density

It is NOT accurate 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:

Thermodynamical method

Jun Xu, Lie-Wen Chen, Bao-An Li and HongRu Ma, arXiv:0807.4477

Constraint on the core-crust transition density

Kazuhiro Oyamatsu, Kei Iida Phys. Rev. C75 (2007) 015801

Need to reduce the error barswith more precise data and calculations!

Transition pressure

Jun Xu, Lie-Wen Chen, Bao-An Li and HongRu Ma, arXiv:0807.4477

• Gravitational Waves = “Ripples in space-time”What are Gravitational Waves?

Amplitude parameterized by (tiny) dimensionless strain h: h(t) = DL/L

Lx Lx[1 + h(t)]

Traveling GW

1 cost F t; h cos ( ) F t; h cos sin ( )

2h t t

F+ and Fx : plus and cross polarization, bounded between -1 and 1

h0 – amplitude of the gravitational wave signal, – polarization angle of signal

– inclination angle of source with respect to line of sight, (t)- phase of pulsar

The expected signal has the form (P. Jaranowski, Phys. Rev. D58, 063001 (1998) ):

proper separation between two masses

GravityJ.B. Hartle

• Test General Relativity:– Quadrupolar radiation? Travels at speed of light?– Unique probe of strong-field gravity

• Gain different view of Universe:– Sources cannot be obscured by dust / stellar

envelopes– Detectable sources are some of the most

interesting, least understood in the Universe

– Opens up entirely new non-electromagnetic spectrum

Why do we need to study Gravitational Waves?

Michael LandryLIGO Hanford Observatoryand California Institute of Technology

58 Gravitational Waves

LISAGravitational

Wave Interferometer

Projects

LIGO, GEO, TAMA; VIRGO taking data; LISA is a ESA-NASA project

Michelson-Morley IFO

Compact binary inspiral: “chirps”

Possible sources of Gravitational Waves:

Supernovae / GRBs: “bursts”

Elliptically deformed pulsars: “periodic”

Examples

Orbital decay of the Hulse-Taylor binary neutron star system (Nobel prize in 1993)is the best evidence so far.

Non-radial oscillations of neutron stars

• Solid black lines: LIGO and GEO science requirement, for T=1 year

• Circles: upper limits on gravitational waves from known EM pulsars, obtained from measured spindown

• Only known, isolated targets shown here

The LIGO Scientific Collaboration, Phys. Rev. D 76, 042001 (2007)

Estimate of gravitational waves from spinning-down of pulsarsAssumption: spinning-down is completely due to the GW radiation

“Standard fiducial value”

Testing the standard fudicial value of the moment of inertia

Aaron Worley, Plamen Krastev and Bao-An Li, The Astrophysical Journal 685, 390 (2008).

The ellipticity of pulsars

Plamen Krastev, Bao-An Li and Aaron Worley, Phys. Lett. B668, 1 (2008).

Formation of dense, asymmetric nuclear matter

SoftSoft

Soft Esym

Stiff Esym

density

Symmetry energy

n/p ratio at supra-normal densities

Central density

π-/ π+ probe of dense matter

2( , ) ( ,0) ( )symE E E

RIKEN-MSU TPC?

Zhigang Xiao, Bao-An Li, Lie-Wen Chen, Gao-Chan Yong, Ming Zhang

arXiv:0808.0186

Excitation function

Central density

Momentum dependence of the isoscalar potentialCompared with variational many-body theory

Constraints from both isospin diffusion and n-skin in 208Pb

Neutron-skin from nuclear scattering: V.E. Starodubsky and N.M. Hintz, PRC 49, 2118 (1994);

B.C. Clark, L.J. Kerr and S. Hama, PRC 67, 054605 (2003)

Isospin diffusion data:M.B. Tsang et al., PRL. 92, 062701 (2004); T.X. Liu et al., PRC 76, 034603 (2007)

Hartree-Fock calculationsA. Steiner and B.A. Li, PRC72, 041601 (05)

J.R. Stone

implication

Transport model calculationsB.A. Li and L.W. Chen, PRC72, 064611 (05)

Symmetry energy from isoscaling analyses

0.690( ) 31.6( / )

453 MeV, 65 MeV

D.V. Shetty, S.J. Yennello and G.A. Souliotis Phys. Rev. C75 (2007) 034602; Phys. Rev. C76 (2007) 024606

Range of symmetry energy from isospin diffusion

Gravitational Radiation from Rotating Neutron Stars (Pulsars)

Wobbling neutron star

R-modes

“Mountain” on neutron star

Accreting neutron star

Constraints from both isospin diffusion and n-skin in 208Pb

Neutron-skin from nuclear scattering: V.E. Starodubsky and N.M. Hintz, PRC 49, 2118 (1994);

B.C. Clark, L.J. Kerr and S. Hama, PRC 67, 054605 (2003)

Isospin diffusion data:M.B. Tsang et al., PRL. 92, 062701 (2004); T.X. Liu et al., PRC 76, 034603 (2007)

Hartree-Fock calculationsA. Steiner and B.A. Li, PRC72, 041601 (05)

implication

Transport model calculationsB.A. Li and L.W. Chen, PRC72, 064611 (05)

Partially constrained EOS for astrophysical studies

Danielewicz, Lacey and Lynch, Science 298, 1592 (2002))

0.6931.6( / 31.6(( ) ) / )

between the and linx=0 sx e-1=

Symmetry energy constrained at sub-saturation densities

L.W. Chen, C.M. Ko and B.A. Li, Phys. Rev. Lett 94, 32701 (2005)L.W. Chen, C.M. Ko and B.A. Li, Phys. Rev. Lett 94, 32701 (2005)

(IBUU04)

For more detailsTalk by Bill Lynch

(ImQMD)

Constraining the radii of NON-ROTATING neutron stars

APR: K0=269 MeV.

The same incompressibility for symmetric nuclear matter of K0=211 MeV for x=0, -1, and -2

Bao-An Li and Andrew W. Steiner, Phys. Lett. B642, 436 (2006)

Nuclear lim

(completely due to general relativity)

MNRAS, 299 (1998) 1059-1068

The first w-mode The frequency is inversely proportional to the compactness of the star

The EOS of neutron-rich matter enters here:

MNRAS, 310, 797 (1999)

1.0 1.2 1.4 1.6 1.8 2.00

MDIx0 MDIx-1 APR

M(Msun

Imprints of symmetry energy on the axial w-modeDe-Hua Wen, Bao-An Li and Plamen G. Krastev (2009)

Probing properties of neutron stars with heavy-ion reactions

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