A Limit on Axion from the Cooling Neutron Star in Cassiopeia Aevents.kias.re.kr/ckfinder/userfiles/201901/files/yanagi... · 2019-01-15 · MU by extensive proton superconductivity

A Limit on Axion from the Cooling Neutron Star in Cassiopeia A

Keisuke Yanagi (University of Tokyo)

Based on K. Hamaguchi, N. Nagata, K.Y., J. Zheng [1806.07151]

The 8th KIAS workshop on particle physics and cosmologyOct. 30 2018

Short summary

Within the Standard Model

Cooling theory can fit the observed surface temperature of Cas A neutron star (NS)

SM + Axion

If the axion couplings to nucleons are too large, cooling is enhanced and cannot fit Cas A NS

fa > O(108) GeV

Te ’ Te0ðT=108 KÞ!K; (7)

where Te0 # 106 K and !# 0:5. The evolution of Te ishence similar to that of T, and the internal cooling curvesof Fig. 2 map onto analogous models of Figs. 1, 3, and 4.The scale Te0 and the exponent! in Eq. (7) both depend onthe chemical composition of the envelope. The presence oflight elements, e.g., H, He, C, and/or O, increases Te0 andreduces! compared to the case of heavy elements, e.g., Fe,depending on the total mass !Mlight of light elements [22].

Using Eq. (7), the slope s ¼ dlog10Te=dlog10t of thetransit cooling curve from Eq. (4) is

s ¼ !dlog10T

dlog10t¼ % !

6

ft=tC1 þ fðt % tCÞ=tC

; (8)

whereas the slopes of the asymptotic trajectories, Eqs. (2)and (6), are both s ¼ % !=6# % 1=12. As long as t is onlyslightly larger than tC, the transit slope is larger than thoseof the asymptotic trajectories by a factor#f. The observedslope over a 10 yr interval is sobs ’ % 1:4. Note, however,that the model ‘‘0.5’’ of Fig. 1 does not exhibit such a largeslope. We are thus led to investigate the origin of therapidity of Cas A’s cooling.

Several factors influence the rapidity of the transit phase.First, LPBF depends on the shape of the Tcnð"Þ curve. Aweak " dependence, i.e., a wide Tcnð"Þ curve, results in athicker PBF neutrino emitting shell and a larger LPBF than astrong " dependence. Second, the T dependence of Te, i.e.,the parameter ! in Eq. (7), also affects the slope in Eq. (8).Third, protons in the core will likely exhibit superconduc-tivity in the 1S0 channel. Most calculations of the protoncritical temperature, Tcpð"Þ, are larger than Tcnð"Þ at low

densities. Proton superconductivity suppresses the MUprocess in a large volume of the core at a very early age,reducing LMU [23]. In our analytical model, this reductiontranslates to a lower L9 and, hence, to a larger f. Theanalytical model as well as our calculations reveal thatproton superconductivity significantly accelerates coolingduring transit and results in a large slope. This feature,essential to account for Cas A’s cooling rate, is illustratedin the right panel of Fig. 2.By varying the relevant physical ingredients, such as the

density range of proton 1S0 superconductivity, the shape ofthe Tcnð"Þ curve, the chemical composition of the enve-lope, and the star’s mass, many models can reproduce theaverage observed Te of Cas A. These models yield slopesranging from # % 0:1 (no rapid cooling and no constrainton TC) up to % 2. A typical good fit to the rapid coolingof Cas A is shown in Fig. 3, where the large slope resultsfrom the strong suppression of LMU by extensive protonsuperconductivity. Figure 4 demonstrates that the resultTC ’ 0:5 ' 109 K does not depend on the star’s mass,but that the slope during the transit is very sensitive tothe extent of proton superconductivity. Models successfulin reproducing the observed slope require superconductingprotons in the entire core. Although spectral fits [5] seemto indicate that Cas A has a larger than canonical mass(1:4M(), a recent analysis [6] indicates compatibility, towithin 3#, with a smaller mass, 1:25M(. The need forextensive proton superconductivity to reproduce the largeobserved slope favors moderate masses unless supercon-ductivity extends to much higher densities than currentmodels predict (see, e.g., Fig 9 in [14] for a large sampleof current models).The inferred TC ’ 0:5 ' 109 K, either from Figs. 1, 3,

and 4 or from Eq. (3), appears quite robust and stems fromthe small exponent in the relation TC / ðC9L

% 19 t% 1

C Þ1=6.Assuming L9 is not very strongly affected by protonCT = 10 K

T = 0CCT = 5.5x10 K8

9

FIG. 3. A typical good fit to Cas A’s rapid cooling for a 1:4M(star, built from the EOS of APR [25] with an envelope mass!Mlight ¼ 5 ' 10% 13M(. The two dotted curves, with indicated

values of TC, are to guide the eye. The three models have aproton 1S0 gap from [26] (the model ‘‘CCDK’’ in [14]) whichresults in the entire core being superconducting. The insertshows a comparison of our results with the five data points of[7] along with their 1# errors.

0.510.520.57

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M/M T [10 K]C9

FIG. 4. Cooling curves with different masses and values ofTC as indicated. For the 1:9M( star, !Mlight ¼ 5 ' 10% 11M(.For the other two masses shown, !Mlight ¼ 5 ' 10% 13M(. Theassumed proton 1S0 gap is the same as in Fig. 3. The slopes, atthe current age of Cas A, are % 1:4, % 0:9, and % 0:5 for the 1.3,1.6, and 1:9M( models, respectively: the decrease, with increas-ing mass, directly reflects the decrease of the core’s fractionalvolumes in which protons are superconducting.

PRL 106, 081101 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending

25 FEBRUARY 2011

081101-3

[Page et al., Phys. Rev. Lett. 106, 081101 (2011)]

[This work]comparable to the SN1987A limit

Outline

• Observation of neutron star in Cassiopeia A

• Cooling theory of a neutron star

• Axion emission from neutron star

Observation of neutron star in Cassiopeia A

Neutron star in the Cassiopeia A

• Supernova remnant in the Cassiopeia constellation

• Distance:

• Birth date from remnant expansion:1681 ± 19[Fesen et al., Astrophys. J. 645, 283 (2006)]

Cassiopeia A

[Wikipedia]

t ≃ 340 yrAge:

Neutron star in Cas A (Cas A NS)

d = 3.40.3−0.1 kpc

• Point source in the center of Cas A

• Thermal X-ray spectrum detected by the Chandra

• Carbon atmosphere fit: M = (1.4 ± 0.3) M⊙

[Reed et al., Astrophys. J. 440, 706 (1995). ]

[Tananbaum (1999)]

Cooling of Cas A NS

• First direct observation of NS cooling

• Temperature decrease:

TESTS OF THE NUCLEAR EQUATION OF STATE AND . . . PHYSICAL REVIEW C 91, 015806 (2015)

the spectra using the specific NS mass and radius implied by theEOS under consideration, and then we perform a least squaresfit to the observed temperature decline. Thus our derived massand radius consistently fit both the spectra and temperatureevolution of the Cas A NS.

Despite the many possible combinations, we find only afew combinations that match the observed spectra and coolingrate. One solution yields M = 1.812 MSun (BSk20 EOS,TToa triplet gap, iron envelope). Other solutions yield M =1.582 MSun (BSk21 EOS, TTav triplet gap, iron envelope),M = 1.441 MSun (BSk21 EOS, TToa triplet gap, iron enve-lope), M = 1.441 MSun (BSk21 EOS, TToa triplet gap, carbonenvelope with 10−15 MSun), and M = 1.582 MSun (BSk21EOS, TToa triplet gap, carbon envelope with 10−8 MSun).Only three of these solutions give a good χ2

ν value forthe least squares fit of all the temperatures: χ2

ν = 0.55 forM = 1.441 MSun and BSk21 EOS with iron envelope, χ2

ν =0.47 for M = 1.441 MSun and BSk21 EOS with 10−15MSuncarbon envelope, and χ2

ν = 0.94 for M = 1.812 MSun andBSk20 EOS with iron envelope, all using the TToa tripletgap; the fit also requires the supernova that produced theNS to have occurred in the years 1674, 1669, and 1653,respectively, which matches well with the determination fromthe expansion of the supernova remnant of 1681 ± 19 [16].The other two fits require the supernova to have occurred inthe years 1617 and 1586, respectively. We show the best-fitsolution (M = 1.441 MSun) in Fig. 14. Given the currentsystematic uncertainties, including absolute flux calibration ofthe observations (see [4,17]), we estimate a mass uncertaintyof approximately ∼0.03 MSun for a given EOS and gap model.

FIG. 14. (Color online) Redshifted surface temperature T ∞s as a

function of year, with redshift 1 + zg = 1.229. Crosses and 1σ errorbars are the observed Chandra ACIS-S Graded temperatures of theCas A NS. Cooling curve is for a M = 1.441 MSun and R = 12.59 kmNS built using the BSk21 EOS with an iron envelope and SFB neutronsinglet, CCDK proton singlet, and TToa neutron triplet gap models.Inset: Expanded view of temperature evolution as a function of time.

V. DISCUSSION

For the first time, we successfully obtain consistent fitsbetween the nine epochs of Chandra ACIS-S Graded spectraand the derived temperature evolution. Our best-fit yields aNS mass M = 1.44 MSun and radius R = 12.6 km using theBSk21 EOS, TToa neutron triplet superfluid and CCDK protonsinglet superconductor gap models, and an iron envelope orthin carbon layer (with $M ≈10−15 MSun) on top of an ironenvelope. Because there still exist large observational andtheoretical uncertainties, we cannot absolutely rule out theother EOSs or some of the other superfluid and superconduct-ing gap models considered here. What we show is that it ispossible to accurately measure the mass of a NS using themethod described. Future work will examine what constraintsare implied for the case where the Cas A NS is not coolingsignificantly or is cooling at a lower rate, as suggested by theanalyses of [17] and [4], respectively.

While the parametrization of the gap energy [see Eq. (2)] isan approximation, we demonstrate the features that gap modelsshould possess if they are to fit the Cas A NS observations.In particular, the proton singlet gap should be large enough topermit a large fraction of the core to become superconductingearly in the age of the NS in order to suppress early neutrinocooling. The neutron triplet gap also needs to extend to a largefraction of the core but with a maximum critical temperaturethat is just at the right level so that rapid cooling does notinitiate too early or too late in order to explain the Cas Aobservations [11,12]. For the neutron singlet gap, its effecton the temperature evolution occurs early on (age !102 yr),during the thermal relaxation phase when the NS interior isstrongly non-isothermal [14]. The Cas A data do not provideuseful constraints for this gap. However we note that someneutron singlet gap models (e.g., AWP2, SCLBL, and SFB)extend beyond the inner crust into the core. While suchbehavior has no distinctive effect on the cooling behavior ofan isolated NS, it may affect observable phenomena such aspulsar glitches [70–73].

There are other possible explanations for the coolingbehavior of the Cas A NS besides the onset of core superfluidityand superconductivity, e.g., heating by r-mode oscillations[74] or magnetic field decay [75], very slow thermal relaxation[69], rotationally induced neutrino cooling [76], and transitionto quark phases [77,78]. It would be interesting to see whatconstraints on some of these models could be obtainedby performing consistent fitting of the Cas A spectra andtemperature evolution similar to the one performed here.

Finally, we note that it is desirable to use a single nucleartheory calculation to obtain consistent EOS and superfluid andsuperconductor gap energies. However, this is not possibleat the present time. Our work is, in part, to motivate such acalculation. A second purpose is to motivate the production ofanalytic approximations to the detailed calculations performedby the nuclear physics community, not just of the EOS [e.g.,pressure as a function of density P (n)], but also nucleoneffective masses and superfluid and superconducting gapenergies [i.e., meff(n) and $(kFx)]. Analytic approximationsare vital for modeling of astrophysical sources, and we note thevaluable contributions of [79,80] for SLy and [26,27] for BSk.

015806-9

Surface temperature detected for 14 yearsT∞

s = Ts 1 − 2GM/R

[Ho et al. (2015)]

( ΔT∞s

T∞s )

CasA

= 3 − 4 % /10 year

Can we explain this decrease?

Yes! (blue line: theory)

Cooling theory of neutron star

Cooling theory of NS without Cooper pairing

CdTdt

= − Lν − Lγ

Thermal balance equation

Internal temperature

Heat capacity (n, p, e, μ)

Neutrino luminosity

Photon luminosity:Lγ = 4πR2σBT2s

C ∝ T


CdTdt

= − Lν − Lγ




Neutrino luminosity

Photon luminosity:

negligible at Cas A age ~ 340 yr

Lγ = 4πR2σBT2s

C ∝ T


CdTdt

= − Lν − Lγ




Neutrino luminosity

Photon luminosity:


Lγ = 4πR2σBT2s

C ∝ T

Neutrino emission • Direct Urca process

• Modified Urca processn → p + ℓ + νℓ p + ℓ → n + νℓ

n + N → p + N + ℓ + νℓ p + N + ℓ → n + N + νℓ

Neutrino emissione!

_!_

e!_

n

np

n

p

e e !

n

Modified Urca BremsstrahlungDirect Urca

n n n n

nn

These processes occur near the Fermi surface.

If the direct Urca process can occur, the neutrino emission is significantly increased.

pF ≫ T, mn−mp

π


CdTdt

= − Lν − Lγ




Neutrino luminosity

Photon luminosity:


Lγ = 4πR2σBT2s

C ∝ T


• Modified Urca process

LDUν ∝ T6

LMUν ∝ T8

n → p + ℓ + νℓ p + ℓ → n + νℓ

n + N → p + N + ℓ + νℓ p + N + ℓ → n + N + νℓ

Neutrino emissione!

_!_

e!_

n

np

n

p

e e !

n


n n n n

nn



pF ≫ T, mn−mp

π


CdTdt

= − Lν − Lγ




Neutrino luminosity

Photon luminosity:


Lγ = 4πR2σBT2s

C ∝ T



LDUν ∝ T6

LMUν ∝ T8

forbidden for M ≲ 2 M⊙n → p + ℓ + νℓ p + ℓ → n + νℓ

n + N → p + N + ℓ + νℓ p + N + ℓ → n + N + νℓ

Neutrino emissione!

_!_

e!_

n

np

n

p

e e !

n


n n n n

nn



pF ≫ T, mn−mp

π

Cooling theory of NS with Cooper pairing

CdTdt

= − Lν




• Pair-breaking and formation (PBF)

LDUν ∝ T6

LMUν ∝ T8

n → p + ℓ + νℓ p + ℓ → n + νℓ

n + N → p + N + ℓ + νℓ p + N + ℓ → n + N + νℓ

× (exponential supression)

N + N → [NN] + ν + ν

PB

F

10 P

32

ν_

ν

n n

1.0

T/T

Contr

ol

functi

on R

S

0.5

c

0.0 0.2 0.4 0.6 0.8 1.00.0

∆

Figure 17: Left panel: Feynman diagram for ⌫⌫ emission from the formationof a nn Cooper pair (pair breaking and formation, PBF, process). Right panel:control functions RPBF for the PBF process.

Temperature dependence of the PBF neutrino emissivity

The temperature dependence of the PBF process (left panel of Fig. 17) can beascertained from Eq. (45) according to the following T -power counting:

✏PBF

/ T3· T

3· T · 1 ·

1

T·R(�/T ) · T = T

7R(�/T ) , (47)

where the two T3 and the first T factors arise from the phase space integrations

of the neutrino pair and the first participating nucleon, respectively. The fac-tor 1 results from the phase space integration of the second nucleon. As thereare only two degenerate fermions in this process (in contrast to the Urca andbremsstrahlung processes that involve 3, 4, or 5 degenerate fermions), the mo-menta of the neutrino pair and the first nucleon are chosen the momentum ofthe second nucleon is fixed by the three-momentum conserving delta function.Thus, this second nucleon does not introduce any T dependence. The T

�1 de-pendence arises from the energy conserving delta function. The last T factor isfrom the neutrino pair’s energy, whereas the T and � dependence of the matrixelement of the reaction are included in the function R(�/T ), which is shown inthe right panel of Fig. 17.

An alternative way of looking at the PBF process is simply as an interbandtransition of a nucleon [117]. Considering the particle spectrum in a paired state(the right panel of Fig. 5), the lower branch (with ✏ < ✏F ��) corresponds topaired particles whereas the upper branch to excited ones, i.e., the “brokenpair” leaves a hole in the lower branch. A transition of a particle from theupper branch to a hole in the lower branch corresponds to the formation of aCooper pair.

37

dominant for T < Tc

T < Tc, neutrons and protons form Cooper pairsϵN(k) = μF,N + sign(k − kF,N) Δ2

N + (k − kF,N)2

k

ε

Normal Fermi Liquid Superfluid Fermions

ε

Fk

ε

2∆

ε(k)

kF

(k)

ε

k

F εF

Figure 5: Comparison of quasi-particle spectra, ✏(k), for a normal and a super-fluid Fermi liquid. The reorganization of particles at ✏ ⇠ ✏F into Cooper pairsresults in the development a gap 2� in the spectrum so that no particle canhave an energy between ✏F �� and ✏F +�.

(T)∆(T)∆

T

∆

Tc Tc

(T)∆∆

T

∆

T

First Order Phase Transition Second Order Phase Transition Smooth State Transition

Figure 6: Temperature evolution of the state of a system parametrized by an“order” parameter, �(T ).

is a continuous change of �(T ) with no critical temperature (see the right panelof Fig. 6). (Examples: liquid $ gas above the critical point; atomic gas $

plasma.)

A simple example

A simple model can illustrate the di�culty in calculating pairing gaps. Con-sider a dilute Fermi gas with a weak, attractive, interaction potential U . Theinteraction is then entirely described by the corresponding scattering length5 , a,which is negative for an attractive potential. In this case, the model has a singledimensionless parameter, |a|kF , and the dilute gas corresponds to |a|kF ⌧ 1.Assuming the pairing interaction is just the bare interaction U (which is calledthe BCS approximation), the gap equation at T = 0 can be solved analytically,

5The scattering length a is related to U by a = (m/4⇡h2)U0 with Uk =Rd3r exp(ik ·

r)U(r).

14

Energy gap O(0.1 - 1) MeV ~ O(109-10) KIn NS core • Proton singlet pairing (1S0) • Neutron triplet pairing (3P2)

[Page et al. (2013)]

Cooling theory vs. Cas A observation

Neutron triplet pairing explains the rapid cooling of Cas A NS!

Te ’ Te0ðT=108 KÞ!K; (7)



s ¼ !dlog10T

dlog10t¼ % !

6


; (8)






% 19 t% 1


T = 0CCT = 5.5x10 K8

9



0.510.520.57

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M/M T [10 K]C9



25 FEBRUARY 2011

081101-3


Onset of neutron 3P2 pairing

T (n)c ∼ 5 × 108 K fits Cas A NS

Rapid cooling by neutron PBF

First observation of neutron superfluidity in NS

Proton 1S0 pairing • Large Tc is necessary to avoid overcool by modified Urca before t ~ 330 yr

T (p)c ≳ 109 K

Modified Urca suppressed by proton pairing

Axion emission from neutron star

Axion couplings to nucleons

• Axion: a Nambu-Goldstone boson associated with Peccei-Quinn symmetry

• Axion couplings to nucleons

• Model dependence

Next, we consider SM + Axion

ℒ =12 (∂μa)

2+

1fa

αS

8πGμνGμν + ∑

q

Cq

2faqγμγ5q∂μa + ⋯

ℒ = ∑N=n,p

CN

2faNγμγ5N∂μa + ⋯

(Cq = 0)KSVZ model:

DFSZ model:

(Cu,c,t =13

cos β , Cd,s,b =13

sin β2)

Cp = − 0.47(3) , Cn = − 0.02(3)

Cp = − 0.182(25) − 0.435 sin β2

Cn = − 0.160(25) + 0.414 sin β2

can be 0

[Pecci and Quinn (1977); Weinberg (1978); Wilczek (1978)]

[Kim (1979); Shifman, Vainshtein, Zakharov ((1980)]

[Zhitnitsky (1980); Dine, Fischler, Srednicki (1981)]

Axion emission from neutron star

• Axion Bremsstrahlung

• Axion PBF

Neutrino emissione!

_!_

e!_

n

np

n

p

e e !

n


n n n n

nn



pF ≫ T, mn−mp

π

a

PB

F

10 P

32

ν_

ν

n n

1.0

T/T

Contr

ol

functi

on R

S

0.5

c

0.0 0.2 0.4 0.6 0.8 1.00.0

∆

Figure 17: Left panel: Feynman diagram for ⌫⌫ emission from the formationof a nn Cooper pair (pair breaking and formation, PBF, process). Right panel:control functions RPBF for the PBF process.

Temperature dependence of the PBF neutrino emissivity

The temperature dependence of the PBF process (left panel of Fig. 17) can beascertained from Eq. (45) according to the following T -power counting:

✏PBF

/ T3· T

3· T · 1 ·

1

T·R(�/T ) · T = T

7R(�/T ) , (47)

where the two T3 and the first T factors arise from the phase space integrations

of the neutrino pair and the first participating nucleon, respectively. The fac-tor 1 results from the phase space integration of the second nucleon. As thereare only two degenerate fermions in this process (in contrast to the Urca andbremsstrahlung processes that involve 3, 4, or 5 degenerate fermions), the mo-menta of the neutrino pair and the first nucleon are chosen the momentum ofthe second nucleon is fixed by the three-momentum conserving delta function.Thus, this second nucleon does not introduce any T dependence. The T

�1 de-pendence arises from the energy conserving delta function. The last T factor isfrom the neutrino pair’s energy, whereas the T and � dependence of the matrixelement of the reaction are included in the function R(�/T ), which is shown inthe right panel of Fig. 17.

An alternative way of looking at the PBF process is simply as an interbandtransition of a nucleon [117]. Considering the particle spectrum in a paired state(the right panel of Fig. 5), the lower branch (with ✏ < ✏F ��) corresponds topaired particles whereas the upper branch to excited ones, i.e., the “brokenpair” leaves a hole in the lower branch. A transition of a particle from theupper branch to a hole in the lower branch corresponds to the formation of aCooper pair.

37

a

CdTdt

= − Lν − La

Even if Cn ~ 0, axion emission is sizable due to the proton contribution!

N + N → [NN] + a

Extra cooling by axion

Limit on axion decay constant

Limit on fa of KSVZ (DFSZ) model

fa ≳ 5 (7) × 108 GeV

comparable to the limit from SN1987Afa ≳ 4 × 108 GeV

For fixed , we vary the neutron gap profile to fit the Cas A NS temperature If the fit fails for any gap parameter, we exclude that axion model

fa

:a certain neutron gap fit the datafa = 1 × 109 GeV

fa = 4 × 108 GeVthe fit fails for any gap profile NS is overcooled before neutron Cooper pairing

stronger axion emission

Uncertainty from envelope

Envelope: composed of light element (H, He, …) and heavy element (Fe)

CoreT

Ts

for a given Tb if they are present in sufficient amounts. Thelarger the amount of light elements present, the higher thetemperature at which their effect will be felt due to the tem-perature dependence of the location of the sensitivity strip. Butat very high temperatures, the light elements have practicallyno effect because they cannot penetrate deep enough. Theresulting T1

e -Tb relationships for various amounts of lightelements are shown in Figure 14.

The presence of a magnetic field can also affect the structureof the envelope (Greenstein & Hartke 1983). The effect is toenhance heat transport along the field and inhibit transportalong directions perpendicular to the field. This results in anonuniform surface temperature distribution, with a very coldregion in which the field is almost tangential to the surface as,e.g., around the magnetic equator for a dipolar field, and acorresponding modification of the T1

e -Tb relationship (Page1995). However, the overall effect is not very large but issomewhat sensitive to the presence of strongly nondipolarsurface fields (Page & Sarmiento 1996). For a field of the orderof 1011–1012 G, one obtains a slight reduction of T1

e comparedto the field-free case, whereas for a higher field T1

e beginsto be enhanced. The enhancement of T1

e is, however, muchsmaller than what is obtained by the presence of light elements(Potekhin et al. 2003). Moreover, there are possible insta-bilities due to the nonuniformity of the temperature (Urpin2004) that have not yet been taken into account in magnetizedenvelope calculations and may somewhat affect these results,but we do not expect significant changes. Hence, the importantcase for our purpose would be the maximal reduction of T1

eobtained for a pure heavy element envelope at B ¼ 1011 G,which is illustrated in Figure 14.

One must finally mention that our calculations are based onthe assumption of spherical symmetry in the interior and thatthe only asymmetries considered, due to the presence of amagnetic field, are within the envelope and hence includedinto this outer boundary condition. However, this assumptionis questionable in some magnetic field configurations wherethe field is confined to the stellar crust. As shown by Geppert

et al. (2004), the crust is highly nonisothermal in such casesand this can affect the thermal evolution because the resultingphoton luminosity is lowered compared to the isothermal crustcase.

5. A GENERAL STUDY OF NEUTRON STAR COOLINGWITHIN THE ‘‘MINIMAL SCENARIO’’

In this section, we will consider the individual effects ofthe chief physical ingredients that enter into the modeling ofthe cooling of an isolated neutron star. Our purpose here istwofold:

1. to determine the sensitivity of results to uncertainties ininput physics in order to obtain a broad range of predictionsthat, we hope, encompasses all possible variations within theminimal cooling scenario;2. to provide us with the means to identify the types of

models that will result in the coldest possible neutron starswithin this paradigm.

Theoretical refutations of the critical physical ingredientsneeded for these coldest models could allow us to raise thetemperature predictions and possibly provide more, or stron-ger, evidence for ‘‘enhanced cooling.’’ The task of identifyingthe minimally cooling coldest star will be taken up in x 6. Anobject colder than such a star could be considered as evidencefor the presence of physics beyond the minimal paradigm.All results in this section use stars built using the APR EOS,

except for x 5.8, where the effects of the EOS are studied for astar of 1.4 M", and for x 5.7, where effects of the stellar massare studied.

5.1. Neutrino vversus Photon Coolingg Erasand the Effect of the Envvelope

The basic features of the thermal evolution of a neutron starcan be easily understood by considering the global thermal en-ergy balance of the star

dEth

dt# CV

dT

dt¼ $L!$ L" ; ð35Þ

where Eth is the total thermal energy content of the star andCV its total specific heat. This equation is accurate when thestar is isothermal, which is the case for ages larger than a fewdecades. Since the dominant neutrino processes all have a T 8

temperature dependence, the neutrino luminosity can be ex-pressed as

L! ¼ NT8: ð36Þ

Furthermore, most of the specific heat comes from the de-generate fermions in the core for which

CV ¼ CT ð37Þ

in the absence of pairing interactions. The photon luminositycan be written as

L" # 4#R2$SBT4e ¼ ST 2þ 4% ; ð38Þ

where Te, the effective temperature, is converted into the in-ternal temperature T through an envelope model with a power-law dependence: Te / T 0:5þ % with %T1 (see eq. [34] and

Fig. 14.—Relationship between the effective temperature T1e and the in-

terior temperature Tb at the bottom of the envelope assuming various amountsof light elements parametrized by & # g2

s14!ML=M (!ML is the mass in lightelements in the envelope, gs14 the surface gravity in units of 1014 cm s$1, andM is the star’s mass), in the absence of a magnetic field (Potekhin et al. 1997).Also shown are the T1

e -Tb relationships for an envelope of heavy elementswith and without the presence of a dipolar field of strength of 1011 G followingPotekhin & Yakovlev (2001).

PAGE ET AL.636 Vol. 155Ts − T relationCharacterized by

η = g214ΔM/M

mass of light elements

We do not know η

fa > O(108) GeV

We evaluate the uncertainty from envelope

Cas A surface T

Cas A core T

Tcore from theory at Cas A age

Summary

Within the Standard Model

Cooling theory can fit the observed surface temperature of Cas A neutron star (NS)

SM + Axion

If the axion couplings to nucleons are too large, cooling is enhanced and cannot fit Cas A NS

fa > O(108) GeV

Te ’ Te0ðT=108 KÞ!K; (7)



s ¼ !dlog10T

dlog10t¼ % !

6


; (8)






% 19 t% 1


T = 0CCT = 5.5x10 K8

9



0.510.520.57

1.91.61.3

M/M T [10 K]C9



25 FEBRUARY 2011

081101-3

[Page et al., Phys. Rev. Lett. 106, 081101 (2011)]

[This work]comparable to the SN1987A limit

Neutron triplet pairing is important

O(1) uncertainty from envelope profile

Backup

6.5

W

O

5.5

2 4 log t (yrs)

2 4 6 log t (yrs)

L 1

-

1 1 1 1 1 1 1 1 1 1

(d)J ^ 1 4 :

[^^^^^^ 1 ^ ^ '* -1 ~^&^^P^^i8 1 ^ " ^ kft" ^ ? > ^ ^ - :

1 7^^ i . l s^ KWI 2 4 6

log t (yrs) 2 4

log t (yrs)

FIGURE 2. Observational limits on surface temperatures of isolated neutron stars compared to theoretical cooling curves. (a) Cooling curve for a nonsuperfluid 1.3MQ star. Any other panel shows 88 curves fromM = 1.1 to 1.97MQ with step 0.0 IM©: (b) No superfluidity, sharp direct Urea threshold; (c) proton superfluidity Ip and (d) 2p which broaden the threshold.

O-M O

(a)

OJ

o a " o

^ • ^ h m OJ fe. f^

T = c o n s t

/

' 1 tn 1 " 1 0 (u

/ ^ y ^ -n t u ^

fast

^- j

bfl O

slow

(b) T = const

s^ 1 ^ 1

sB cd "

si / j r a J

^ap m R m75

n • ^ CD

Pi Pe p MjM, MM,

FIGURE 3. Very schematic view of (a) neutrino emissivity profile in a neutron star core and (b) neutrino luminosity versus star's mass at a fixed core temperature T. Units are arbitrary.

for all stars with 1. IM© <M < MQ, and for a wide class of EOSs [24]. This universal curve goes through the scat-ter of observational points but cannot explain the sources which are hottest and coolest for their age. These sources seem to have neutrino luminosities lower and higher than the modified Urea level.

Now consider a set of nonsuperfluid neutron star mod-els of masses from 1.1 to 1.97MQ, with the mass step of O.OIM© (Fig. 2b). The first 26 models (M < l.SSM© < MD) belong to the class of low-mass stars and show the universal slow cooling behavior The next 1.36M© star has a small central kernel with the enhanced neutrino emission and shows much faster coohng (is much colder than all observed sources). AU other more massive stars cool even faster via the direct Urea process and belong to the family of cold massive rapidly coohng stars. Evi-dently, these cooling curves carmot explain the data.

BASIC CONCEPT

A majority of reahstic coohng scenarios, constructed up to now, satisfy the following basic phenomenological concept:

(1) Low-mass stars undergo slow neutrino cooling. (2) Massive stars cool much faster via neutrino emis-

sion from their irmer cores. (3) There is a family of medium-mass stars whose

cooling is intermediate between the slow and fast ones. Accordingly, the density profile of the neutrino emis-

sivity in a neutron star core can look like that plotted in Fig. 3a; pi andp2 mark the density range, where the slow cooling transforms into the fast one. This Qv{p) profile translates into a characteristic dependence of the neutrino luminosity Zy on the stellar mass shown in Fig. 3b; the transition between the slow coohng to the fast one takes place in the mass range from Mi to M2.

Thus, at the present stage of investigation, the cooling theory has potential to test four main parameters of the neutrino emissivity function Qv{p) or the luminosity functionZy (A/)- They specify the lower and upper levels of neutrino emission (either 2siow and gfast or Zgiow and Zfast) and the position of the transition zone between the slow and fast emission (either p\ and p2 or Mi and M2).

EXAMPLES OF PHYSICAL MODELS

Two examples [52] of successful realization of the above scheme are given in Figs. 2c and d. The figures show the cooling curves for neutron stars of M =(1.1-1.97)M© with strong proton superfluidity (Ip or 2p) in the core (but with normal neutrons). The density profiles 7 (p) of the proton critical temperature are shown in Fig. la; they are phenomenological and can be regarded as illustra-

383

Cas A NS: 14 years data

Other NS: • About 30 observations of surface temperature

• Single measurement of for each NS

Why Cas A NS is special?

{(t0, T∞s,0), (t1, T∞

s,1), …}

(t, T∞s )

In Cas A NS, temperature decline rate is available More powerful to constrain cooling theory

[Yakovlev et al. (2008)]

available, are shown in Tables 1–3. The data in these tablesshow that, in general, there is a large discrepancy between thespin-down age and the kinematic age in cases where both aregiven. In most cases, the spin-down age is longer, but in thecase of the Vela pulsar, it is shorter. Typical discrepancies areof order 3 or larger. For this reason, we have used the kinematicage in Figures 1 and 2 where available, and otherwise haveassigned an uncertainty of a factor of 3 in each direction to thespin-down ages.

2.3. Distances and Luminosities

The distances are estimated from pulsar dispersion mea-sures, estimated distances to the related supernova remnants, orobservations of interstellar absorption to other stars in prox-imity. In three cases, parallax estimates are available. Thedetails are discussed in Appendix A. Uncertainties in the dis-tances are in many cases rather large. Since the inferred lu-minosities of the stars are proportional to the square of theassumed distances, it is usually the case that the inferred stellarluminosity has greater relative error bars than the inferred stel-lar temperature. However, in cases in which the compositionof the stellar atmosphere is uncertain, but the distance to thesource is accurately known, the inferred stellar luminositymight be more accurately estimated. A consistency check ofthe measurements of T1, L1, and the distance d is that therelation

L1 ¼ 4!R21"SBT

41 ð1Þ

should give a radius at infinity R1 comparable to the radius ofa neutron star. This is the case for the measurements listed inTable 1, whereas for the measurements based on BB spectral fitslisted in Table 2 only 1055$ 52 has a possibly acceptable R1,but with very large errors due to the uncertainty in d. However,BB models are overly simplistic. Nonmagnetic heavy-elementdominated atmospheres tend to have values of R1 factors of2 to 3 larger than a BB (Romani 1987), so that essentially allthe sources listed in Table 2 satisfy this consistency check. Forthe objects listed in Table 3, this consistency test is only mar-ginally possible for J0205+6449 (3C 58).Theoretical cooling calculations also give an effective tem-

perature T1e and a luminosity L1 that are related to each other

by the equation (see also eq. [B9])

L1 % e2!(R)L(R) ¼ 4!R12"SBT14e ; ð2Þ

where we have used the superscript ‘‘1’’ to denote the the-oretical values and subscript ‘‘1’’ for the observed values atinfinity in order to emphasize the difference. It is only for astar for which the measured T1 and L1 satisfy equation (2)and for which an accurate measurement of d exists (implying asmall error bar on L1) that comparison of cooling curves withdata in terms of T or L are equivalent. For the stars listed inTable 2 that do not pass the above consistency test, the mea-sured T1 is thus not an effective temperature and cannot bedirectly compared with the calculated T1

e . In these cases, theluminosity L1 is more representative of thermal emissionand should be used for comparison with L1. For this reason,we have chosen to tabulate luminosities as well as tem-peratures in Tables 1 and 2, and we have plotted both tem-perature and luminosity in Figure 1. In Table 3, we havereported only upper limits to L1, which is the quantity thatobservation can usefully constrain and plotted them separatelyin Figure 2.One feature notable in Figures 1 and 2 is the sizes of the error

boxes, particularly in the age dimension. These uncertaintiesrepresent an inherent difficulty in using these observations tofirmly constrain the details of neutron star cooling. For this rea-son, instead of attempting to detail properties of the equationof state, superconductivity, and/or neutrino emissivities fromthe observations, our approach will be to model a reasonably

Fig. 1.—Inferred temperature T1 (top) and luminosity L1 (bottom) vs. agefor neutron stars with thermal emission. Data from Table 1 are marked as‘‘H atmosphere fits’’ and data from Table 2 as ‘‘Blackbody fits.’’

Fig. 2.—Inferred upper limits on thermal luminosity L1, vs. age, for thecompact objects listed in Table 3.

PAGE ET AL.626 Vol. 155

available, are shown in Tables 1–3. The data in these tablesshow that, in general, there is a large discrepancy between thespin-down age and the kinematic age in cases where both aregiven. In most cases, the spin-down age is longer, but in thecase of the Vela pulsar, it is shorter. Typical discrepancies areof order 3 or larger. For this reason, we have used the kinematicage in Figures 1 and 2 where available, and otherwise haveassigned an uncertainty of a factor of 3 in each direction to thespin-down ages.

2.3. Distances and Luminosities

The distances are estimated from pulsar dispersion mea-sures, estimated distances to the related supernova remnants, orobservations of interstellar absorption to other stars in prox-imity. In three cases, parallax estimates are available. Thedetails are discussed in Appendix A. Uncertainties in the dis-tances are in many cases rather large. Since the inferred lu-minosities of the stars are proportional to the square of theassumed distances, it is usually the case that the inferred stellarluminosity has greater relative error bars than the inferred stel-lar temperature. However, in cases in which the compositionof the stellar atmosphere is uncertain, but the distance to thesource is accurately known, the inferred stellar luminositymight be more accurately estimated. A consistency check ofthe measurements of T1, L1, and the distance d is that therelation

L1 ¼ 4!R21"SBT

41 ð1Þ

should give a radius at infinity R1 comparable to the radius ofa neutron star. This is the case for the measurements listed inTable 1, whereas for the measurements based on BB spectral fitslisted in Table 2 only 1055$ 52 has a possibly acceptable R1,but with very large errors due to the uncertainty in d. However,BB models are overly simplistic. Nonmagnetic heavy-elementdominated atmospheres tend to have values of R1 factors of2 to 3 larger than a BB (Romani 1987), so that essentially allthe sources listed in Table 2 satisfy this consistency check. Forthe objects listed in Table 3, this consistency test is only mar-ginally possible for J0205+6449 (3C 58).Theoretical cooling calculations also give an effective tem-

perature T1e and a luminosity L1 that are related to each other

by the equation (see also eq. [B9])

L1 % e2!(R)L(R) ¼ 4!R12"SBT14e ; ð2Þ

where we have used the superscript ‘‘1’’ to denote the the-oretical values and subscript ‘‘1’’ for the observed values atinfinity in order to emphasize the difference. It is only for astar for which the measured T1 and L1 satisfy equation (2)and for which an accurate measurement of d exists (implying asmall error bar on L1) that comparison of cooling curves withdata in terms of T or L are equivalent. For the stars listed inTable 2 that do not pass the above consistency test, the mea-sured T1 is thus not an effective temperature and cannot bedirectly compared with the calculated T1

e . In these cases, theluminosity L1 is more representative of thermal emissionand should be used for comparison with L1. For this reason,we have chosen to tabulate luminosities as well as tem-peratures in Tables 1 and 2, and we have plotted both tem-perature and luminosity in Figure 1. In Table 3, we havereported only upper limits to L1, which is the quantity thatobservation can usefully constrain and plotted them separatelyin Figure 2.One feature notable in Figures 1 and 2 is the sizes of the error

boxes, particularly in the age dimension. These uncertaintiesrepresent an inherent difficulty in using these observations tofirmly constrain the details of neutron star cooling. For this rea-son, instead of attempting to detail properties of the equationof state, superconductivity, and/or neutrino emissivities fromthe observations, our approach will be to model a reasonably

Fig. 1.—Inferred temperature T1 (top) and luminosity L1 (bottom) vs. agefor neutron stars with thermal emission. Data from Table 1 are marked as‘‘H atmosphere fits’’ and data from Table 2 as ‘‘Blackbody fits.’’

Fig. 2.—Inferred upper limits on thermal luminosity L1, vs. age, for thecompact objects listed in Table 3.

PAGE ET AL.626 Vol. 155


TESTS OF THE NUCLEAR EQUATION OF STATE AND . . . PHYSICAL REVIEW C 91, 015806 (2015)

the spectra using the specific NS mass and radius implied by theEOS under consideration, and then we perform a least squaresfit to the observed temperature decline. Thus our derived massand radius consistently fit both the spectra and temperatureevolution of the Cas A NS.

Despite the many possible combinations, we find only afew combinations that match the observed spectra and coolingrate. One solution yields M = 1.812 MSun (BSk20 EOS,TToa triplet gap, iron envelope). Other solutions yield M =1.582 MSun (BSk21 EOS, TTav triplet gap, iron envelope),M = 1.441 MSun (BSk21 EOS, TToa triplet gap, iron enve-lope), M = 1.441 MSun (BSk21 EOS, TToa triplet gap, carbonenvelope with 10−15 MSun), and M = 1.582 MSun (BSk21EOS, TToa triplet gap, carbon envelope with 10−8 MSun).Only three of these solutions give a good χ2

ν value forthe least squares fit of all the temperatures: χ2

ν = 0.55 forM = 1.441 MSun and BSk21 EOS with iron envelope, χ2

ν =0.47 for M = 1.441 MSun and BSk21 EOS with 10−15MSuncarbon envelope, and χ2

ν = 0.94 for M = 1.812 MSun andBSk20 EOS with iron envelope, all using the TToa tripletgap; the fit also requires the supernova that produced theNS to have occurred in the years 1674, 1669, and 1653,respectively, which matches well with the determination fromthe expansion of the supernova remnant of 1681 ± 19 [16].The other two fits require the supernova to have occurred inthe years 1617 and 1586, respectively. We show the best-fitsolution (M = 1.441 MSun) in Fig. 14. Given the currentsystematic uncertainties, including absolute flux calibration ofthe observations (see [4,17]), we estimate a mass uncertaintyof approximately ∼0.03 MSun for a given EOS and gap model.

FIG. 14. (Color online) Redshifted surface temperature T ∞s as a

function of year, with redshift 1 + zg = 1.229. Crosses and 1σ errorbars are the observed Chandra ACIS-S Graded temperatures of theCas A NS. Cooling curve is for a M = 1.441 MSun and R = 12.59 kmNS built using the BSk21 EOS with an iron envelope and SFB neutronsinglet, CCDK proton singlet, and TToa neutron triplet gap models.Inset: Expanded view of temperature evolution as a function of time.

V. DISCUSSION

For the first time, we successfully obtain consistent fitsbetween the nine epochs of Chandra ACIS-S Graded spectraand the derived temperature evolution. Our best-fit yields aNS mass M = 1.44 MSun and radius R = 12.6 km using theBSk21 EOS, TToa neutron triplet superfluid and CCDK protonsinglet superconductor gap models, and an iron envelope orthin carbon layer (with $M ≈10−15 MSun) on top of an ironenvelope. Because there still exist large observational andtheoretical uncertainties, we cannot absolutely rule out theother EOSs or some of the other superfluid and superconduct-ing gap models considered here. What we show is that it ispossible to accurately measure the mass of a NS using themethod described. Future work will examine what constraintsare implied for the case where the Cas A NS is not coolingsignificantly or is cooling at a lower rate, as suggested by theanalyses of [17] and [4], respectively.

While the parametrization of the gap energy [see Eq. (2)] isan approximation, we demonstrate the features that gap modelsshould possess if they are to fit the Cas A NS observations.In particular, the proton singlet gap should be large enough topermit a large fraction of the core to become superconductingearly in the age of the NS in order to suppress early neutrinocooling. The neutron triplet gap also needs to extend to a largefraction of the core but with a maximum critical temperaturethat is just at the right level so that rapid cooling does notinitiate too early or too late in order to explain the Cas Aobservations [11,12]. For the neutron singlet gap, its effecton the temperature evolution occurs early on (age !102 yr),during the thermal relaxation phase when the NS interior isstrongly non-isothermal [14]. The Cas A data do not provideuseful constraints for this gap. However we note that someneutron singlet gap models (e.g., AWP2, SCLBL, and SFB)extend beyond the inner crust into the core. While suchbehavior has no distinctive effect on the cooling behavior ofan isolated NS, it may affect observable phenomena such aspulsar glitches [70–73].

There are other possible explanations for the coolingbehavior of the Cas A NS besides the onset of core superfluidityand superconductivity, e.g., heating by r-mode oscillations[74] or magnetic field decay [75], very slow thermal relaxation[69], rotationally induced neutrino cooling [76], and transitionto quark phases [77,78]. It would be interesting to see whatconstraints on some of these models could be obtainedby performing consistent fitting of the Cas A spectra andtemperature evolution similar to the one performed here.

Finally, we note that it is desirable to use a single nucleartheory calculation to obtain consistent EOS and superfluid andsuperconductor gap energies. However, this is not possibleat the present time. Our work is, in part, to motivate such acalculation. A second purpose is to motivate the production ofanalytic approximations to the detailed calculations performedby the nuclear physics community, not just of the EOS [e.g.,pressure as a function of density P (n)], but also nucleoneffective masses and superfluid and superconducting gapenergies [i.e., meff(n) and $(kFx)]. Analytic approximationsare vital for modeling of astrophysical sources, and we note thevaluable contributions of [79,80] for SLy and [26,27] for BSk.

015806-9

108 109 1010

Tcore [K]1032

1034

1036

1038

1040

1042

1044

L[e

rg/s

]

Modified UrcaPBFPhoton

Summary of neutrino emission

• Direct Urca process is powerful, but forbidden in Cas A NS

• Weaker Modified Urca dominates before Cooper pairing

• After pairing, the PBF process dominates the cooling

Proton pairing

Neutron pairing

Proton singlet gap modelT

[

10

K

]9

c

BS

F−1

1.0 1.4 1.8 2.0

Cru

st−

Core

1.50 0.5 1.00

2

4

6

8

10

AO

T

CCY BCLL

EEHO

CCDK

2BF+3BF

k [fm ]

BS 2BF[M

eV]

1.5

0.5

1.0

∆

0

Figure 11: Some theoretical predictions of Tc and �, vs proton kF , for theproton 1S0 gap in �-equilibrium uniform neutron-proton matter. The value ofkF corresponding to the transition from the crust to the core is indicated: valueson the right of this line correspond to the neutron star core but values on theleft are not realized since protons in the crust are confined within nuclei whichare finite size systems while this figure presents results for infinite matter. Onthe top margin are marked the values of the proton kF at the center of a 1.0,1.4, 1.8, and 2.0 M� star built with the APR EOS [16]. See text for description.

BCS for the proton 1S0 gap. Among the latter, we show results from [46]:these authors used either only two body forces in the interaction kernel, curve“BS2BF”, or two body forces supplemented by the inclusion of three body forces,curve “BS2BF+3BF” which shows that three body forces are repulsive in the 1S0channel. These “BS” results also include e↵ects of medium polarization. Recallthat for the 1S0 pairing of neutrons in pure neutron matter, polarization has ascreening e↵ect and quenches the gap. However, in neutron star matter, wherethe medium consists mostly of neutrons, the strong np-correlations result inmedium polarization inducing anti-screening [47] for the 1S0 pairing of protons.

The anisotropic3P-F2 neutron (and proton) gap

The 1S0 neutron gap vanishes at densities close to the crust-core transitionand the dominant pairing for neutrons in the core occurs in the mixed 3P-F2

channel. Uncertainties in the actual size and the range of density in whichthis gap persists are, however, considerable. As previously mentioned, a majorsource of uncertainty is the fact that even the best models of the N-N interactionin vacuum fail to reproduce the measured phase shift in the 3P2 channel [48].Also significant are the e↵ects of the medium on the kernel and 3BF, even at

21

Theoretical calculations are relatively certain

We use CCDK model because

• Large proton gap is more favorable for Cas A

• Axion emission from proton is suppressed

derived bound on is conservative

fa

Neutron triplet gap modelthe level of the BCS approximation. It was found in [49] that 3BF at the Fermisurface are strongly attractive in the 3P-F2 channel in spite of being repulsive inthe bulk. Moreover, due to medium polarization a long-wavelength tensor forceappears that is not present in the interaction in vacuum and results in a strongsuppression of the gap [50].

1.4 2.01.8 1.0 1.4 2.01.8

SF

3bf

HGRR

NijI

CDB

AV182bf

AO T

Cru

st

Co

re NijII

a

b

c

a2

Cru

st

Co

re

1.0

Figure 12: Left panel: some theoretical predictions of Tc for the neutron 3P-F2

gap in uniform pure neutron and �-equilibrium matter. See text for description.Right panel: some phenomenological models of Tc for the neutron 3P-F2 gapused in neutron star cooling simulations. Models “a”, “b”, and “c” are from[51] and [52], model “a2” from [53]. On the top margin are marked the valuesof kFn at the center of a 1.0, 1.4, 1.8, and 2.0 M� star built with the APR EOS[16].

Figure 12 shows examples of theoretical predictions of Tc for the neutron3P-F2 gap. The three dotted lines show some of the first published models:“HGRR” from [54], “T” from [55] and “AO” from [29]. The four continuouslines show results of models from [48] calculated using the Nijmegen II (“NijII”),Nijmegen I (“NijI”), CD-Bonn (“CDB”), and Argonne V18 (“AV18”) potentials(displayed values are taken from the middle panel of Figure 4 of [48]). Theresults of these four models start to diverge at kFn above 1.8 fm�1 and illustratethe failure of all four N-N interactions models to fit the 3P2 laboratory phase-shifts above Elab ' 300 MeV. All of these calculations were performed for pureneutron matter using the BCS approximation.

In the case of the 1S0 gap, medium polarization is known to result in screen-ing and to reduce the size of the gap. In the case of a 3P2 gap, polarizationwith central forces is expected to result in anti-screening and to increase thesize of the gap. However, Schwenk & Friman [50] showed that spin-dependentnon-central forces do the opposite and strongly screen the coupling in the 3P2

channel, resulting in a Tc lower than 107 K: this “SF” value is indicated in the

22

Theoretical calculations are highly uncertain

We model the gap by the Gaussian

3 parameters • Height • Width • Center

T (n)c

How to constrain the axion model?

If SM + axion cannot fit the Cas NS, the model is excluded

constraint on fa ℒ = ∑N=n,p

CN

2faNγμγ5N∂μa + ⋯

• Neutron 3P2 gap model

• Proton 1S0 gap model

}SFB model

APR

M = 1.4 M⊙

not so sensitive to these choices

vary to fit Cas A NS (3 parameter Gaussian model)

does not matter as long as large enough(we use CCDK model)

[Chen et al. (1993)]

Cooling model parameters

• Neutron 1S0 gap model

• Equation of state

• NS mass

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