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Pairing Gaps and Neutron Star Cooling
G. Taranto, M. Baldo, G.F. Burgio, H.-J. S., INFN Catania
• Motivation
• Cooling processes
• Pairing gaps
• Cooling scenarios
• Results
PRC 70, 048802 (2004)
PRL 95, 051101 (2005)
PRC 75, 025802 (2007)
PRC 89, 048801 (2014)
MNRAS 456, 1451 (2016)
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Neutron Star cooling:
• Objective: “Explain” the objects in the Temp. vs. Age plot:
320 330 340Age [yrs]
6.14
6.16
6.18
6.20
0 1 2 3 4 5 6 7
log Age [yrs]5
6
7
log
T [K
]
a) sp=1, sn=1, sκ=1
After ∼ 20s SN remnant becomes neutrino transparent
Isothermal after ® 100y, Tcore ≈ (10...100)Tsurface
Neutrino cooling for t ® 105yr, then photon cooling
Data for 19+1 isolated NS from
Beznogov & Yakovlev, MNRAS 447, 1598 (2015)
Klochkov et al., A&A 573, A53 (2015)
Fast cooling of Cas A NS (Disputed !):
Heinke & Ho, ApJ 719, L167 (2010)
Elshamouty et al., ApJ 777, 22 (2013)
Theoretical cooling simulations for fixed NS mass
M/M⊙ = 1.0,1.1, . . . ,2.0
• Major problems:
◦ Stellar atmosphere is unknown, distance not well known
→ Uncertain temperature
◦ Most NS masses are unknown
→ Verification of theoretical models currently impossible
• Models can be falsified when unable to cover all data
• Theoretical input required:
◦ EOS for core, crust, atmosphere
→ composition of stellar matter
◦ Effective masses, Heat capacities and conductivities
◦ Cooling rates for different processes
◦ Pairing gaps for all channels
• We use standard cooling code NSCool of D. Page with
consistent BHF EOS, eff. masses, pairing gaps as input
(checked by independent code of P. Haensel)
• We assume purely nucleonic NS: no hyperons, no QM !
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Cooling Processes:Yakovlev,Kaminker,Gnedin,Haensel, Phys. Rep. 354, 1 (2001)
• Neutrino emissivities without pairing [erg cm−3 s−1] :
◦ Direct Urca n→ p+ + ν ; p+ → n+ ν :
Q(DU) ≈ 4.0× 1027M11T69Θ(
kFp + kFe − kFn)
◦ Modified Urca N+N→ N+N+ +ν ; N+N+ → N+N+ν :
Q(Mn) ≈ 8.1× 1021M31T89αnβn
Q(Mp) ≈ 8.1× 1021M13T89αpβp
(
1− kFe/4kFp)
ΘMp
αp = αn = 1.13, βp = βn = 0.68: in-medium corrections of matrix elements
◦ Bremsstrahlung N+N→ N+N+ ν + ν :
Q(Bnn) ≈ 2.3× 1020M40T89αnnβnn(ρn/ρp)
1/3
Q(Bnp) ≈ 4.5× 1020M22T89αnpβnp
Q(Bpp) ≈ 2.3× 1020M04T89αppβpp
αnn = 0.59, αnp = 1.06, αpp = 0.11, βnn = 0.56, βnp = 0.66, βpp = 0.70
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• Effective mass prefactors:
Mj ≡
(
ρp
ρ0
)1/3(m∗n
mn
)(m∗p
mp
)j
,m∗
m=
k
m
[
de(k)
dk
]
−1
k=kF
BHF results:
0 0.2 0.4 0.60
0.2
0.4
0.6
0.8
1
Mij
CDB + UIX
V18 + UIX
V18 + TBF
M11
(DU)
0 0.2 0.4 0.6
M31
(Mn)
0 0.2 0.4 0.6
M13
(Mp)
0 0.2 0.4 0.6
M40
(Bnn)
0 0.2 0.4 0.6
M22
(Bnp)
0 0.2 0.4 0.6 0.8
M04
(Bpp)
ρ [fm-3
]
M ~
ij
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Effects of Pairing:
Yakovlev,Kaminker,Gnedin,Haensel, Phys. Rep. 354, 1 (2001)
• Damping of DU,MU,BNN reactions:
Q(DU) → Q(DU) × R(n,p) ; =Δ̄(T)
T; R() ≈ e−0
0 =Δ̄(T=0)
T= 1.746
TcT
• A new cooling process: Pair Breaking and Formation:
N→ N+ ν + ν :
Q(PBF) ≈ 3.5× 1021m∗
m
kF
mT79F()
Provides rapid cooling close to below
the critical temperature
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Intermezzo:
• No pairing:
◦ If DU is active (p¦13%), it dominates all other processes
◦ Too fast cooling of most NS
• Yes pairing:
◦ All cooling processes are comparable and must be used
◦ Competition between blocking and PBF
◦ All gaps have to be known
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Gaps in Neutron Star Matter:
X.-R. Zhou, H.-J. S., E.-G. Zhao, Feng Pan, J.P. Draayer; PRC 70, 048802 (2004)
0
1
2
3 Free s.p. spectrumV18
np
1S0
3PF2
0
1
2
3 BHF s.p. spectrumV18
∆ [M
eV]
0
1
2
3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
BHF s.p. spectrumV18 + UIX
ρB [fm-3]
EOS: BHF (V18 + UIX)
• Self-energy effects suppress gaps
• TBF suppress pp 1S0 but strongly
enhance 3PF2 gaps !
• No polarization corrections
included here
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Neutron Star Profile: Particle Densities & Gaps:
EOS: BHF (V18 + UIX + NSC89) , M = 1.2 M⊙
without with
hyperons: hyperons:
0
20
40
60
80
ε,p
[MeV
fm-3
]
ε/10
p
0
0.2
0.4
ρ [fm
-3]
n
p
0
1
2
3
∆ [M
eV]
1S0
3PF2
p nFree s.p. spectrumV18
0
1
2
3
0 2 4 6 8 10 12
∆ [M
eV]
r [km]
BHF s.p. spectrumV18 + UIX
0
50
100
150
200
ε,p
[MeV
fm-3
]
ε/10
p
0
0.5
1
ρ [fm
-3]
n
pΣΛ
0
1
2
3
∆ [M
eV]
1S0
3PF2
p nFree s.p. spectrumV18
0
1
2
3
0 2 4 6 8 10 12
∆ [M
eV]
r [km]
BHF s.p. spectrumV18 + UIX
Polarization effects (including pn interaction) ?
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Proton 1S0 Pairing in Neutron Stars:
M. Baldo, H.-J. S.; PRC 75, 025802 (2007)
• Strong in-medium effects on protons due to large neutron
background
• Consider complete set of medium effects: m*, Z, TBF,
Polarization:
Δ(k′) = −∑
k
Z(k)[V + VTBF + VPo](k′, k)
2√
Ms(k)2 + Δ(k)2Δ(k)
• Weak-coupling approximation:
Δ = cμe1/λ , λ = kFm∗Z2Veff
• Approximation for Landau parameters:
G0 = 0.7 ; F0 = −0.4,−0.6-1
-0.8
-0.6
-0.4
-0.2
-0
0.2
0.4
0.6
0.8
1
1 1.2 1.4 1.6 1.8 2
F0
G
0kF [fm-1]
Ainsworth et al., PLB 222Bäckman et al., PLB 43Jackson et al., NPA 386Schulze et al., PLB 375Schwenk et al., NPA 713
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• Results:
0
1
2k F
[fm
-1] n
p
(a)
0.7
0.8
0.9
1
m∗ /m n
p
(b)
0.4
0.6
0.8
1
Z
n
p
(c)
0
1
2
3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Free s.p., 2BF
BHF s.p., 2BF
BHF s.p., 2BF+3BF
(d)
∆ [M
eV]
ρB [fm-3]
0
0.2
0.4
0.6
0.8
2BF2BF2BF2BF2BF
m
m∗
m∗ ,Z,F0=-0.4
m∗ ,Z,F0=-0.6
-λ
2BF+3BF2BF+3BF2BF+3BF2BF+3BF2BF+3BF
0
1
2
3
0 0.2 0.4
∆ [M
eV]
ρB [fm-3]0 0.2 0.4
Reduction by m∗, Z, TBF; Enhancement by polarization!
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• Pairing gaps used for cooling simulation:
0.5
1
0 0.2 0.4 0.6 0.8 1
ρ [fm-3
]
0
0.5
1
p1S0*
1.0
1.2
1.4
1.6
1.8
p1S0*
2.0
1.6
n3P2*
BHF
0.6
[1015
g/cm3]
1.8
2.0
1.6
1.4
1.2
1.0
APR
n3P2*
p1S0 x1/2
n3P2
1.4
1
1.4
0.2
TC [M
eV]
p1S0 x1/2
n3P2
0.80.4 1.81.41.20.60.2 1.0
[MeV
]
0.6
0.2
∆
1
DU onset:
ρ = 0.82 fm−3
p = 0.140
M/M⊙ = 2.03
ρ = 0.44 fm−3
p = 0.136
M/M⊙ = 1.10
We employ BCS and BCS+m∗ gaps
with global scaling factors s, s∗
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• Nuclear EOS and NS Structure:
Compare APR and BHF(V18+UIX) EOS :
BHF has large p and early DU onset
Mmax > 2M⊙ for both EOSs
DU thresholds: M/M⊙ = 1.10,2.03 (BHF,APR)
0.2 0.4 0.6 0.8 1 1.20
500
1000
1500
2000P
, ε [M
eV fm
-3]
0
0.1
0.2
APRBHF
0.2 0.4 0.6 0.8 1 1.28
10
12
14
R [K
m]
0.2 0.4 0.6 0.8 1 1.2
ρ, ρc [fm-3
]
0
0.5
1
1.5
2
M/M
O ·ε
P
a)
b)
c)
d)
µpx
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Cooling Scenarios:
• Results: BCS gaps, no scaling:
BCS BCS∗
320 330 340Age [yrs]
6.14
6.16
6.18
6.20
0 1 2 3 4 5 6 7
log Age [yrs]5
6
7
log
T [K
]
a) sp=1, sn=1, sκ=1
0 1 2 3 4 5 6 7
log Age [yrs]5
6
7
log
T [K
]
b) sp*=1, sn*=1, sκ=1
◦ Cooling too fast, hot old NS not reproduced
◦ Cas A fast cooling not reproduced
◦ BCS/BCS∗: fast DU cooling blocked for M/M⊙ ® 1.5/1.2
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• Results: Global fit of all data:
No n3P2 cooling, only p1S0 BCS gap
0 1 2 3 4 5 6 7
log Age [yrs]5
6
7
log
T [K
]
a) sp=0.5, sn=0, sκ=1
0 1 2 3 4 5 6 7
log Age [yrs]5
6
7
log
T [K
]
b) sp=1, sn=0, sκ=1
◦ No n3P2 gap, otherwise PBF process cools too much
◦ Magnitude of p1S0 nearly arbitrary
◦ p1S0 gap must extend to large density to inhibit DU for
many sources
◦ Cas A fast cooling not possible
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• Results: Two ways to fit Cas A cooling:
PBF cooling Delayed cooling
320 330 340Age (yrs)
6.14
6.16
6.18
6.20
0 1 2 3 4 5 6 7
log Age [yrs]5
6
7
log
T [K
]
sp=2, sn=0.132, sκ=1
320 330 340Age (yrs)
6.14
6.16
6.18
6.20
0 1 2 3 4 5 6 7
log Age [yrs]5
6
7
log
T [K
]
a) sp*=1, sn*=0, sκ=0.135
Fine-tuned n3P2 PBF cooling
at current age/temperature
of Cas A: Δn3P2 ≈ 0.1MeV
Suppressed thermal conduc-
tivity and delayed heat prop-
agation
Difficult to fit ALL other sources in this case
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Recent Progress on n3PF2 Pairing
• ApJ 817, 6 (2016): J.M. Dong et al.
Role of nucleonic Fermi surface depletion in neutron star
cooling
- No Polarization
• PRC 94, 025802 (2016): D. Ding et al.
Pairing in high-density neutron matter including short- and
long-range correlations
- PNM, No TBF
• PRC 95, 024302 (2017): C. Drischler et al.
Pairing in neutron matter: New uncertainty estimates and
three-body forces
- PNM, No Polarization
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Hans-Josef Schulze INFN Catania
Summary:
• Quantitative knowledge of all pairing gaps is required
• n3P2 PBF cooling clashes with existence of hot old NS
Quantitative theoretical calculation still missing
• DU cooling possible if damped for most NS
→ p1S0 gap must extend to large density
• Rapid Cas A and cooling of all other objects are diffi-
cult to reconcile
• Need masses of cooling NS to verify models !
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Hans-Josef Schulze INFN Catania
Summary:
• Quantitative knowledge of all pairing gaps is required
• n3P2 PBF cooling clashes with existence of hot old NS
Quantitative theoretical calculation still missing
• DU cooling possible if damped for most NS
→ p1S0 gap must extend to large density
• Rapid Cas A and cooling of all other objects are diffi-
cult to reconcile
• Need masses of cooling NS to verify models !
However:
• Purely nucleonic picture is too naive:
Quark matter, hyperons, etc. must be considered . . .
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Hyperon-Nucleon Pairing in Neutron Stars:
Xian-Rong Zhou, H.-J. S., Feng Pan, J.P. Draayer; PRL 95, 051101 (2005)
0
10
20
30
40
50
0 0.5 1 1.5 2 2.5 3 3.5 4
0
1 2
3
4
5
6nΣ− 3SD1
0: NSC891-6: NSC97a-f
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
01-6
nΣ− 1S0
∆ [M
eV]
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
12
3
4
56nΛ 1S0
kF [fm-1]
• NY gaps in symmetric
hyperon-nucleon matter:
YY pairing unknown due to
unknown potentials
Nijmegen potentials predict
very large n− 3SD1 gaps !(no hard core, very attractive)
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• n− 3SD1 pairing in neutron star matter:
0
0.1
0.2
x i = ρ
i/ρB
n/5
pΣ−
Λ
(a)
0
0.5
ρ n+ ρ
Σ [fm
-3]
(b)
0
0.5
α nΣ
(c)
0
10
20
∆ nΣ [M
eV]
nΣ− 3SD1
(d)
0
0.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
∆ NN [M
eV]
ρB [fm-3]
nn 3PF2pp 3PF2
(e)
with V18+UIX+NSC89 BHF EOS
Suppression of nn 3PF2 pairing!Suppression of direct Urca − cooling!
But, at high density many uncertainties:
◦ EOS, composition of matter ?
◦ NY potentials ?
◦ Medium effects on pairing ?
◦ Separation of paired/unpaired phases ?
Presently YN pairing cannot be excluded
