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:
ρ
23 RMF
models
Density
-
Symmetry energy and single nucleon potential used in the IBUU04 transport model
ρ
The x parameter is introduced to mimic
various predictions on the symmetry energy
by different microscopic nuclear many-body
theories using different effective interactions
soft
Default: Gogny force
Density ρ/ρ0
MDI single nucleon potential within the HF approach using a modified Gogny force:
 '

 
B   1
2
U (  ,  , p, , x )  Au ( x )
 Al ( x )
 B( ) (1  x )  8 x
 '
0
0
0
  1  0
2C ,
2C , '
f ( r , p ')
f ' ( r , p ')
3
3

d
p
'

d
p
'
0 
1  ( p  p ') 2 /  2
0 
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).
V1  28  6MeV, R  0.1  0.2
for E kin  100 MeV
G.R. Satchler, Isospin Dependence of Optical Model
Potentials, in Isospin in Nuclear Physics,
D.H. Wilkinson (ed.), (North-Holland, Amsterdam,1969)
Constraints from both isospin diffusion and n-skin in 208Pb
Isospin diffusion data:
M.B. Tsang et al., PRL. 92, 062701 (2004);
T.X. Liu et al., PRC 76, 034603 (2007)
MDI potential energy density
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 first w-mode
The frequency is inversely proportional
to the compactness of the star
MNRAS, 299 (1998) 1059-1068
MNRAS, 310, 797 (1999)
The EOS of neutron-rich matter enters here:
Imprints of symmetry energy on the axial w-mode
De-Hua Wen, Bao-An Li and Plamen G. Krastev (2009)
8.8
wI
8.6
MDIx0
MDIx-1
APR
8.4
(kHz)
8.2
8.0
7.8
7.6
7.4
7.2
5
wII
(kHz)
4
3
2
1
0
1.0
1.2
1.4
1.6
M(Msun)
1.8
2.0
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
0.35
0.30
wI
0.25
Re(M)
Re(M)
0.40
MDIx0
MDIx-1
APR
FIT
0.30
MDIx0
MDIx-1
APR
wII
0.20
0.15
0.10
0.05
0.25
0.00
0.60
0.55
0.22
Im(M)
Im(M)
0.24
0.20
0.50
0.45
0.18
0.40
0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24
M/R
0.12 0.14 0.16 0.18 0.20 0.22 0.24
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:
ρ
EOS of pure neutron matter
Alex Brown,
23PRL85,
RMF 5296 (2000).
models
???
???
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 E sym 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
at supra-normal densities
GC
Coefficients2
a) Δ(1232) resonance model
in first chance NN scatterings:
(negelect rescattering and reabsorption)

5 N 2  NZ

 (
2

5Z
 NZ

nn
pp
np(pn)
N
Z


0
5
1
1
1
4

0


5
0
1
)2
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 ]


m
n
m 1 1 3 m m
n   p  (V  V )  VCoul  kT {ln  
bm (  T ) (    )}
n
p
p m m
2
n
asy
p
asy
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
Sub-saturation density: 5%
Supra-saturation densities: 25%

(  )like 


1
2
       0  N *0


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
124
124
197
197
E/A=800 MeV,
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
corresponding to Esym (  ) 
100 
3

 (22 / 3  1) EF0 ( ) 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
corresponding to Esym (  ) 
100 
3

 (22 / 3  1) EF0 ( ) 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,
Disturbing implications?
For pure nucleonic matter
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