Symmetry Energy, Tensor Force & Maximum Rotation of Neutron
Stars
ECT* EMMI Workshop “Neutron-rich matter & Neutron Stars” September 30th – October 5th 2013, Trento (Italy)
Isaac Vidaña CFC, University of Coimbra
Take away messages
ü Tensor force plays a crucial role in Esym(ρ)
ü Neutron star rotation can constraint Esym(ρ)
details in: Phys. Rev. C 84, 062801 ( R) (2011)
Phys. Rev. C 85, 045808 (2012)
Why Esym(ρ) is so uncertain ?
Still limited knowledge of the nuclear force particularly of its spin and isospin dependence
Lattimer & Lim (2013) Brown(2000)
using: ü BHF with Av18 + UIX
ü Hellmann-Feynman Theorem
Symmetry Energy & Tensor Force
“Analysis of the contribution of the different terms of the NN force to Esym and L=3ρ0dEsym/dρ”
BHF approximation of ANM in a Nutshell
" Bethe-Goldstone Equation Partial sumation of pp ladder diagrams
ü Pauli blocking ü Nucleon dressing
" Energy per particle
Infinite sumation of two-hole line diagrams
€
EA
(ρ,β) =1A
2k 2
2mτ
+12
Re Uτ ( k )[ ]
%
& '
(
) *
k≤kFτ
∑τ
∑
€
G ω( ) =V +V Qω − E − E ' + iη
G ω( )
€
Eτ (k) =2k 2
2mτ
+ Re Uτ (k)[ ]
€
Uτ (k) = k k ' G ω = Eτ (k) + Eτ ' (k ')( )
k k '
Ak '≤kFτ '
∑τ '∑
€
Un ~U0 +Usymβ
€
Usym =Un −Up
2β
BHF nucleon mean field in ANM
Symmetry potential
Isospin splitting of mean field in ANM
€
Up ~U0 −Usymβ
G-matrix gives access to in-medium NN cross sections
€
σττ ' =mτ*mτ '
*
16π 242J +14π
Gττ '→ττ 'LL 'SJ 2
, ττ '= nn, pp,npLL 'SJ∑
ü microscopically based
ü density dependence (Pauli blocking) ü isospin dependence (ρn different from ρp)
Brueckner-Hatree-Fock:
ü gives total energy
ü does not gives separately neither <T> nor <V> because it does not provide the correlated many-body wave function
But
However
Hellmann-Feynman theorem can be used to calculate <V>
Hellmann-Feynman theorem
€
dEλ
dλ=ψλ
d ˆ H λdλ
ψλ
ψλ ψλ
Proven independently by many-authors: Güttinger (1932), Pauli (1933), Hellmann (1937), Feynman (1939)
§ Writting the nuclear matter Hamiltonian as:
€
ˆ H = ˆ T + ˆ V § Defining a λ-dependent Hamiltonian:
€
ˆ H λ = ˆ T + λ ˆ V
è
€
ˆ V =ψ ˆ V ψψ ψ
=dEλ
dλ$
% &
'
( ) λ=1
H. Hellmann R. P. Ferynman
ENM ESM Esym L
<T> 53.321 54.294 -0.973 14.896
<V> -34.251 -69.524 35.273 51.604
Total 19.070 -15.230 34.300 66.500
§ Kinetic contribution to
ü L: smaller than FFG one (~29.2 MeV) Figure from Xu & Li, arXiv:1104.2075v1 (2011)
ü Esym: very small and negative in contrast to FFG result (~14.4 MeV) è strong isospin dependence of short range NN
correlations
§ Potential contribution to
ü L: ~78% of the total L
ü Esym: almost equal to total Esym
Kinetic and potential energy contributions
S(ρ) = Cs,k
2ρρ0
!
"#
$
%&
2/3
+Cs,p
2ρρ0
!
"#
$
%&
γ
Kinetic (FG) Potential
S i n c e k i n e t i c contribution to Esym is small
Potential contribution in the parametrization currently used in transport models must be larger
è
Can be 0 (typically ~ 25 MeV) Must be different from current values
(S,T) ENM ESM Esym L
(0,0) 0 5.600 -5.600 -21.457
(0,1) -29.889 -23.064 -6.825 -17.950
(1,0) 0 -49.836 49.836 90.561
(1,1) -4.362 -2.224 -2.138 0.450
<V> -34.251 -69.524 35.273 51.604
Spin-Isospin channel & partial wave decomposition
ENM ESM Esym L
3S1 0 -45.810 45.810 71.855
3D1 0 -0.981 0.981 -3.739
Σrest up to J=8 -34.251 -22.733 -11.518 -16.512
ü Largest contribution from S=1, T=0 channel
ü Similar T=1 channel contributions to ENM and ESM which almost cancel out in Esym
ü Main contribution from 3S1-3D1 p.w. (not present in NM)
Few words on the NN and NNN forces used …
§ Argonne V18 (Av18) NN potential
€
Vij = Vp (rij )Oijp
p=1,18∑ Oij
p=1,14 = 1,σ i ⋅σ j( ),Sij,
L ⋅S,L2,L2
σ i ⋅σ j( ),
L ⋅S( )
2"#$
%&'⊗ 1, τ i ⋅
τ j( )"# %&
€
Oijp=15,18 = Tij ,
σ i ⋅ σ j( )Tij ,SijTij , τ zi + τ zj( )[ ]
§ Urbana IX (UIX) NNN potential
€
VijkUIX =Vijk
2π +VijkR
€
Vijk2π : Attractive Fujita-Miyazawa force
π
π
€
Vijk2π = A Xij ,X jk{ } τ i ⋅
τ j , τ j ⋅ τ k{ } +
14Xij ,X jk[ ] τ i ⋅
τ j , τ j ⋅ τ k[ ]
%
& '
(
) *
cyclic∑
€
Xij =Y mπ rij( ) σ i ⋅ σ j + T mπ rij( )Sij
€
Y (x) =e−x
x1− ex
2( )
€
T(x) = 1+3x
+3x 2
"
# $
%
& ' e−x
x1− ex
2( )2
€
VijkR = B T 2 rij( )T 2 rjk( )
cyclic∑
€
VijkR : Repulsine & Phenomenological
€
UNNeff r ij( ) = VUIX r i,
r j , r k( )n r i,
r j , r k( )d3 r k∫
Reduced to an effective density-dependent 2BF
Oijp=1,3 =1,
σ i ⋅σ j( ) τ i ⋅
τ j( ),Sij
τ i ⋅τ j( )
Contributions from different terms of the NN force
ü Largest contribution from tensor components
ü Contributions from other terms negligible (e.g. charge symmetry b re a k i n g t e r m s ) o r almost cancel out
§ Esym: 36.056 (Total: 34.4)
§ L: 69.968 (Total: 66.5)
Ωc/Ω
Kep
ler r-mode unstable
due to GW emission
r-mod
e da
mpe
d by
sh
ear v
isco
sity
r-mod
e da
mpe
d by
bu
lk v
isco
sity
Symmetry Energy & Maximum Rotation of NS
“Study the role of L on the NS r-mode instability”
ü Microscopic Models • BHF with Av18 + UIX
• APR
• AFDMC
ü Phenomenological Models • Skyrme
• Relativistic Mean Field
Using:
Bulk & shear viscosities
€
ηn = 2 ×1018 ρ1015gcm−3
&
' (
)
* +
9 / 4T
109K&
' (
)
* + −2
€
ηe = 4.26 ×10−26 xpnb( )14 / 9
T−5 / 3
ü n scattering 1
ü e- scattering 2
ü Modified URCA
€
N + n→ N + p + l + ν l , N + p + l→ N + n + ν l
ü Direct URCA
€
n→ p + l + ν l , p + l→ n + ν l
€
ξ = ξMURCA + ξDURCA =λNlω 2
∂P∂Xl
∂ς l∂nbNl
∑ +λlω 2
∂P∂Xl
∂ς l∂nbl
∑
€
η =ηn +ηe
Haensel et al., AA 357, 1157 (2000); AA 372, 130 (2001) 1 Cutler & Lindblom., ApJ 314, 234 (1987). 2 Shternin & Yakovlev, PRD 78, 063006 (2008)
L dependence of ξ and η
€
ξ = Aξ LBξ ,η = AηL
Bη
L dependence described by simple power laws
ü ξ increases with L for all densities
ü η increases with L for nb> n0 & decreases with L for nb < n0
consequence of lepton fraction dependence
Dissipative time scales of r-modes
€
1τ i
= −1
2EdEdt
$
% &
'
( ) i
ü τGW larger for models with larger L
€
1τξ
=4π690
Ω2
πGρ
'
( )
*
+ ,
2
R2l−2 ρr2l+2dr0
R∫[ ]
−1
ξrR'
( )
*
+ ,
0
R∫
2
1+ 0.86 rR'
( )
*
+ ,
2/
0 1
2
3 4 r2dr
€
1τη
= l −1( ) 2l +1( ) ρr2l+2dr0
R∫[ ]
−1
ηr2ldr0
R∫
€
1τGW
=32πGΩ2l+2
c 2l+3
l −1( )2l
2l +1( )!![ ]2l + 2l +1
&
' (
)
* +
2l+1
ρr2l+2dr0
R∫
ü τξ & τη smaller for models with larger L
ü τGW, τξ & τη decrease when increasing M
Larger L è stiffer EoS è less compact star è τGW larger
τξ (τη) decrease with ξ (η) but ξ(η)increase with L
Given an EoS: the more massive the star the denser it is è τGW, τξ ~(ρ/ξ)R2 & τη ~(ρ/η)R2 decrease
€
1τ Ω,T( )
= −1
τGW Ω( )+
1τξ Ω,T( )
+1
τη T( )
~ e−iωt−t/τ
time dependence of an r-mode
€
1τ Ωc,T( )
= 0
r-mode instability region è
€
Ω <Ωc
unstable
€
Ω >Ωc
stable
ü instability region smaller for models with aaalarger L (increase of ξ & η with L)
ü instability region larger for more massive aaastar (time scales decrease when M aaaincreases)
Critical angular velocity
Constraining L from LMXBs
ü No constraint on L if
observational constraints from pulsar in LMXB 4U 1608-52
ü estimated core temperatureT ~ 4.55 x 108 K ü measured spin frequency 620 Hz
radius: 10, 11.5, 12 or 13 km mass: 1.4 or 2 M¤
€
ΩKepler ≈ 7800 MMsun
$
% &
'
( )
10kmR
$
% &
'
( )
3
s−1
• R4U 1608-52 < 11 km ( Ωc > Ω ) • R4U 1608-52 > 12(13) km & M=1.4(2)M¤ (Ωc < Ω)
ü L > 50 MeV if (assuming 4U 1608-52 stable)
• R4U 1608-52 is 11.5-12(11.5-13) km & M=1.4(2)M¤
Take away messages (once again)
ü Tensor force plays a crucial role in Esym(ρ)
ü Neutron star rotation can constraint Esym(ρ)
details in: Phys. Rev. C 84, 062801 ( R) (2011)
Phys. Rev. C 85, 045808 (2012)
Collaborators
C. Providência A. Polls
Sponsors
and of course you for your attention …
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