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					Probing properties of neutron stars with heavy-ion reactions

    Bao-An Li                   & collaborators:
                                Plamen G. Krastev, Will Newton, De-Hua Wen and Aaron Worley,
                                Texas A&M University-Commerce
                                Lie-Wen Chen and Hongru Ma, Shanghai Jiao-Tung University
                                Che-Ming Ko and Jun Xu, Texas A&M University, College Station
                                Andrew Steiner, Michigan State University
                                Zhigang Xiao and Ming Zhang, Tsinghua University, China
                                Gao-Chan Yong and Xunchao Zhang, Institute of Modern Physics, China
                                Champak B. Das, Subal Das Gupta and Charles Gale, McGill University

Outline:

•   Symmetry energy at sub-saturation densities constrained by heavy-ion
    collisions at intermediate energies
    Imprints of symmetry energy on gravitational waves
    (1) Gravitational waves from elliptically deformed pulsars
    (2) The axial w-mode of gravitational waves from non-rotating neutron stars

•   Symmetry energy at supra-saturation densities constrained by the FOPI/GSI data
    on the π-/π+ ratio in relativistic heavy-ion collisions
    Disturbing/Puzzling(Interesting?) implications for neutron stars
     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 latest results: talks by Bill Lynch, Hermann Wolter and Pawel Danielewicz

                                    Recent progress and new challenges in
                                    isospin physics with heavy-ion reactions:
                                    Bao-An Li, Lie-Wen Chen and Che Ming Ko
                                    Physics Reports, 464, 113-281 (2008)
                                    arXiv:0804.3580
      The Esym (ρ) from model predictions using popular interactions
                       1 2 E
Esym (  )          E (  ) pure neutron matter  E (  )symmetric nuclear matter
                       2  2


   Examples:

                                                            23 RMF
                                                            models
ρ




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

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




                                      soft
                                                              Default: Gogny force

                          Density ρ/ρ0
MDI single nucleon potential within the HF approach using a modified Gogny force:
                                  '                                             B   1
U (  ,  , p, , x )  Au ( x )        Al ( x )      B( ) (1  x )  8 x
                                                                            2
                                                                                                '
                                  0              0      0                           1 0
                                                                                            


  2C ,                  f ( r, p ')          2C , '            f ' ( r, p ')
    0                                           0 
            d3p'                                       d3p'
                    1  ( p  p ') 2 /  2                    1  ( p  p ') 2 /  2


             1                     2 Bx                    2 Bx
 , '       , Al ( x )  121       , Au ( x )  96       , K  211MeV
             2                      1                     1 0
The momentum dependence of the nucleon potential is a result of the non-locality
of nuclear effective interactions and the Pauli exclusion principle

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).
  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:
                                         P.E. Hodgson, The Nucleon Optical Model, World
U n / p  U isoscalar  U Lane          Scientific, 1994

U Lane  (U n  U p ) / 2  V1   R  Ekin ,   G.W. Hoffmann and W.R. Coker, PRL, 29, 227 (1972).
                                                 G.R. Satchler, Isospin Dependence of Optical Model
V1  28  6MeV, R  0.1  0.2                Potentials, in Isospin in Nuclear Physics,
for E kin  100 MeV                              D.H. Wilkinson (ed.), (North-Holland, Amsterdam,1969)
     Constraints from both isospin diffusion and n-skin in 208Pb
Isospin diffusion data:                                  MDI potential energy density
M.B. Tsang et al., PRL. 92, 062701 (2004);
T.X. Liu et al., PRC 76, 034603 (2007)


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


                              124Sn+112Sn




            implication



                          PREX?                                             ρρ

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


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)
  Symmetry energy constrained at sub-saturation densities
  31.6(  /  0 ) 0.69  Esym (  )  31.6(  /  0 )1.05
  between the x=0 and x=-1 lines, agrees extremely well with the APR

L.W. Chen, C.M. Ko and B.A. Li, Phys. Rev. Lett 94, 32701 (2005)
                                                                                (ImQMD)
                                                                     (IBUU04)




                                                                        For more details
                                                                        Talk by Bill Lynch




                                                                   Courtesy of M.B. Tsang
Partially constrained EOS for astrophysical studies


                           Danielewicz, Lacey and Lynch,
                           Science 298, 1592 (2002))
             Constraining the radii of NON-ROTATING neutron stars

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




                        ●                          .




APR: K0=269 MeV.
The same incompressibility for symmetric nuclear
matter of K0=211 MeV for x=0, -1, and -2
Astronomers discover a neutron-star spining at 716

                   RNS code by Stergioulas & Friedman




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




                                                              Science 311, 1901 (2006).
    Gravitational waves from elliptically deformed pulsars
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

                      Breaking stain of crust

Mass quadrupole moment




                                         EOS

  B. Abbott et al., PRL 94, 181103 (2005)
  B.J. Owen, PRL 95, 211101 (2005)
     Constraining the strength of gravitational waves
          Plamen Krastev, Bao-An Li and Aaron Worley, Phys. Lett. B668, 1 (2008).




                                                     Compare with the upper limits of 76
                                                     pulsars from LIGO+GEO observations
                                                        Phys. Rev. D 76, 042001 (2007)




It is probably the most uncertain factor




  B.J. Owen, PRL 95, 211101 (05)
Spin-down estimate for fast-spinning NS
Aaron Worley, Plamen Krastev and Bao-An Li (2009)




   The moment of inertia is calculated from RNS
   instead of using the




                                                    ellipticity
Testing the standard fudicial value of the moment of inertia




   Aaron Worley, Plamen Krastev and Bao-An Li,
   The Astrophysical Journal 685, 390 (2008).
(completely due to general relativity)
                                          The EOS of neutron-rich matter enters here:
The first w-mode
The frequency is inversely proportional
to the compactness of the star




MNRAS, 299 (1998) 1059-1068


MNRAS, 310, 797 (1999)
Imprints of symmetry energy on the axial w-mode
De-Hua Wen, Bao-An Li and Plamen G. Krastev (2009)



          8.8
          8.6                           wI         MDIx0
          8.4
                                                   MDIx-1
                                                   APR
          8.2
(kHz)




          8.0
          7.8
          7.6
          7.4
          7.2



           5
                            wII
           4
 (kHz)




           3

           2

           1

           0
                1.0   1.2         1.4        1.6   1.8      2.0
                                   M(Msun)
    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)
De-Hua Wen, Bao-An Li and Plamen G. Krastev (2009)
               0.45                                                          0.30
                            MDIx0         wI                                            MDIx0        wII
                            MDIx-1                                           0.25       MDIx-1
               0.40                                                                     APR
                            APR




                                                                    Re(M)
                            FIT                                              0.20
      Re(M)




               0.35
                                                                             0.15
               0.30                                                          0.10

               0.25                                                          0.05

                                                                             0.00

                                                                             0.60
               0.24

                                                                             0.55
               0.22
      Im(M)




                                                                    Im(M)
                                                                             0.50
               0.20

                                                                             0.45
               0.18

                                                                             0.40
                      0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24
                                                                                    0.12 0.14 0.16 0.18 0.20 0.22 0.24
                                       M/R
                                                                                                  M/R
      The Esym (ρ) from model predictions using popular interactions
                       1 2 E
Esym (  )          E (  ) pure neutron matter  E (  )symmetric nuclear matter
                       2  2


   Examples:
                                                              EOS of pure neutron matter
                                                              Alex Brown,
                                                               RMF
                                                            23PRL85, 5296 (2000).
                                                            models
ρ




                                                                                    ???
                                ???
                                                                           APR



                                                                               -
                                                Density
Can the symmetry energy becomes negative at high densities?
Yes, due to the isospin-dependence of the nuclear tensor force
The 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

                                                        Why?
                                                        Can the modern effective field theory verify this?


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

 Negative symmetry energy  Isospin separation instability
because of the Esym 2 term,
for symmetric matter,
it is energetically more favoriable to write  =0=1-1,
i.e., pure neutron matter + pure proton matter



  x  0.048[ Esym (  ) / Esym ( 0 )]3 (  / 0 )(1  2 x )3
  Pion ratio probe of symmetry energy               GC
                                                                       
                                                                               0
                                                                                       
                                                    Coefficients2
        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)
               5 N 2  NZ
                                         (           N
                                                           )2
        
                 5Z     2
                            NZ                        Z

    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 ]
   

                                      n   m 1 1 3 m m
 n   p  (V  V )  VCoul  kT {ln       bm (  T ) (    )}
               n     p                                           m
              asy   asy
                                      p m m       2         n   p



  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
Is π-/π+ ratio really a good probe of the symmetry energy at supra-normal densities?




                                                                            XL=XH=1




                                                                           XL=XH=-2




    X L  X for   0
    X H  X for   0
                                                                  1    2
                                                            0  N *0
Sub-saturation density: 5%                 
                                                                                     
Supra-saturation densities: 25%         (  )like                3    3      
                                                                              t 
                                                                                   
                                                                                     
                                                                 1  2 *
                                                              N
                                                                  3    3
Isospin asymmetry reached in heavy-ion reactions

                                                    Symmetry energy
                    48    48




                                                     density
                                     E(  ,  )  E(  , 0)  Esym (  ) 2


                                                                 E/A=800 MeV,
                   124   124                  197     197        b=0, t=10 fm/c
            t=10 fm/c

t=10 fm/c




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


                              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 (K0=211 MeV
                              is used in the results shown here)


             (distance from the center of the reaction system)
π-/π+ ratio as a probe of symmetry energy at supra-normal densities
  W. Reisdorf et al. for the FOPI/GSI collaboration , NPA781 (2007) 459

                                  IQMD: Isospin-Dependent Quantum 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




                                                                          100                3 0 
                                         corresponding to Esym (  )            (22 / 3  1) EF ( ) 2 / 3
                                                                           8 0               5    0
                                          Need a symmetry energy softer than the above
                                          to make the pion production region more neutron-rich!
                                          E(,  )  E(,0)  Esym (  ) 2
                                          low (high) density region is more neutron-rich
                                          with stiff (soft) symmetry energy
Near-threshold π-/π+ ratio as a probe of symmetry energy at supra-normal densities
  W. Reisdorf et al. for the FOPI collaboration , NPA781 (2007) 459

                                         IQMD: Isospin-Dependent Quantum 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



                                                                         100                3 0 
                                         corresponding to Esym (  )           (22 / 3  1) EF ( ) 2 / 3
                                                                          8 0               5    0
                                          Need a symmetry energy softer than the above
                                          to make the pion production region more neutron-rich!
                                          E(,  )  E(,0)  Esym (  ) 2
                                          low (high) density region is more neutron-rich
                                          with stiff (soft) symmetry energy
N/Z dependence of pion production and effects of the symmetry energy
Zhi-Gang Xiao, Bao-An Li, L.W. Chen, G.C. Yong and. M. Zhang
PRL (2009) in press.




                        400 MeV/A
Excitation function




                      Central density
IF the conclusion is right,                    For pure nucleonic matter
Disturbing implications?

K0=211 MeV is used, higher incompressibility
for symmetric matter will lead to higher
masses systematically




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




                                ?
 n  e     
     Summary

   • The symmetry energy at sub-saturation densities is constrained to


      31.6(  / 0 )0.69  Esym (  )  31.6(  / 0 )1.05
      L=86  25 MeV
      It agrees extremely well with the APR prediction



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

				
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