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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