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 …