Thermal evolution of relativistic hyperonic compact stars with calibrated equations of state
Morgane Fortin, Adriana R. Raduta, Sidney Avancini, Constanca Providencia
aa r X i v : . [ nu c l - t h ] F e b Thermal evolution of relativistic hyperonic compact stars with calibrated equations of state
Morgane Fortin ∗ N. Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, 00-716 Warszawa, Poland
Adriana R. Raduta † National Institute for Physics and Nuclear Engineering (IFIN-HH), RO-077125 Bucharest, Romania
Sidney Avancini ‡ Departamento de Fisica, Universidade Federal de Santa Catarina, 88040-900 Florianopolis, Santa Catarina, Brazil
Constanc¸a Providˆencia § CFisUC, Department of Physics, University of Coimbra, Portugal (Dated: February 16, 2021)A set of unified relativistic mean-field equations of state for hyperonic compact stars recently built in [M.Fortin, Ad. R. Raduta, S. Avancini, and C. Providˆencia, Phys. Rev. D , 034017 (2020)] is used to studythe thermal evolution of non-magnetized and non-rotating spherically-symmetric isolated and accreting neutronstars under different hypothesis concerning proton S -wave superfluidity. These equations of state have beenobtained in the following way: the slope of the symmetry energy is in agreement with experimental data; thecoupling constants of Λ and Ξ -hyperons are determined from experimental hypernuclear data; uncertainties inthe nucleon- Σ interaction potential are accounted for; current constraints on the lower bound of the maximumneutron star mass are satisfied. Within the considered set of equations of state, the presence of hyperons isessential for the description of the cooling/heating curves. One of the conclusions we reach is that the criterionof best agreement with observational data leads to different equations of states and proton S -wave superfluiditygaps when applied separately for isolated neutron stars and accreting neutron stars in quiescence. This meansthat at least in one situation the traditional simulation framework that we employ is not complete and/or theequations of state are inappropriate. Another result is that, considering equations of state which do not allow fornucleonic dUrca or allow for it only in very massive NS, the low luminosity of SAX J1808 requires a repulsive Σ -hyperon potential in symmetric nuclear matter in the range U ( N ) Σ ≈ −
30 MeV. This range of values for U ( N ) Σ is also supported by the criterion of best agreement with all available data from INS and XRT. I. INTRODUCTION
The composition of neutron star (NS) interiors is a livelysubject of research in both nuclear physics and astrophysics.Over time the nucleation of various exotic species (hyperons,Delta resonances, pion and kaon condensates and quarks) hasbeen considered, in addition to nucleons. The discovery ofseveral massive pulsars in binary systems with white dwarfs[1–3] brought into focus the issue of hyperonisation of NS in-ner cores. A large number of studies, most of which account-ing for all experimental constraints from nuclear and hyper-nuclear physics, have shown that ≈ M ⊙ NS are not incom-patible with hyperons [4–19]. The way in which the hyperonsmodify other NS observables, e.g. : radius, tidal deformabilityand moment of inertia, has been also addressed in [9, 20].The composition of NS cores also determines the neutrinoemission upon which the surface temperature depends. Neu-trino emission is the dominant heat loss mechanism in the first10 kyr after the star birth in a supernova explosion and duringthe quiescence periods following accretion episodes. Cool- ∗ Electronic address: [email protected],[email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ing tracks of isolated NS (INS) as a function of age [21–27]and heating tracks of X-ray transients (XRT) as a function ofaccretion rate [28, 29] present qualitatively different featuresdepending on whether direct Urca (dUrca) process(es) oper-ate or not. The stars accommodating at least one dUrca pro-cess cool much faster than the stars whose thermal evolutionis regulated by the series of the so-called slow and interme-diate cooling processes (modified Urca, bremsstrahlung and,for paired species, also Cooper pair breaking and formation(PBF)). dUrca reactions exist for all particles which, underdifferent scenarios, are expected to nucleate in NS cores. Inthe case of baryons and quarks, in order for a dUrca process tooperate, the relative abundances of involved baryonic (quark)and leptonic species should verify the triangle inequality ofFermi momenta, p F , i + p F , j ≥ p F , k [30], which typically leadsto a density threshold ( n DU ).The threshold density of nucleonic dUrca depends on therelative neutron and proton abundances in cold catalyzed NSmatter. These are regulated by the density behavior of thesymmetry energy, which is typically expressed in terms of thesymmetry energy of symmetric saturated nuclear matter ( J )and its slope ( L ) and curvature ( K sym ). Correlations betweenthe nuclear symmetry energy and the threshold density of nu-cleonic dUrca have been sought after in [31, 32]. By consid-ering a large number of non-relativistic Skyrme-like and rela-tivistic mean-field (RMF) models Ref. [31] concludes that i)models with high L -values manifest a clear (anti)correlationbetween L and n DU , and ii) no definite behavior can be iden-tified for models with low L -values. Ref. [32] showed thatthe (anti)correlation between L and n DU survives also in hy-peronic stars and nucleation of hyperons does not alter signif-icantly the threshold density of nucleonic dUrca. Λ , Ξ − , Ξ and Σ − , which are the only hyperons that nucleate in NS mat-ter, may - in principle - be involved in dUrca reactions, too.The threshold densities of hyperonic dUrca are determined bythe (poorly constrained) hyperon-nucleon interaction poten-tials [19, 20], in addition to the isoscalar and isovector behav-iors of the nucleonic equation of state (EoS).The neutrino emission from compact star interiors dependssensitively on the magnitude and density dependence of thepairing gaps of fermions, too. The magnitudes of the pairinggaps are controlled by the baryon interactions and by the manyparticle correlations accounted for; their density dependencesare determined by the different particle density profiles, whichare regulated by baryonic effective potentials, with the isovec-tor nucleon potential playing an important role.The consequences of various dUrca processes and the wayin which pairing of different baryonic species in the core im-pact the thermal evolution was discussed at length in [21–24, 27, 33–35] and is qualitatively well understood.Recent progress on the EoS of hyperonic stars and neu-trino emissivities in different channels [36] motivated [37–40]to reconsider the cooling of hyperonic isolated stars. Differ-ent strategies have been used by the different groups of au-thors. [37, 40] focused on the maximum suppression of hy-peronic dUrca reactions, by considering Λ and Ξ S pairinggaps provided by the most attractive bare ΛΛ and ΞΞ inter-action potentials. Based on symmetry arguments, Ref. [40]has also investigated the possibility of high order pairing of Λ hyperons in the core together with its impact on cooling. Inboth cases, the studies employed a bunch of EoS which ex-plore present day uncertainties in the nucleonic sector. Refs.[38, 39] focused on the possibility to obtain, for a given EoS,agreement with observed effective surface temperature versusage data of INS. To this aim, the neutron P − F and pro-ton S pairing and, in the case of [38], also the Λ S pairingin the core were considered as free parameters. [38] used theMKVORH φ [15, 41] RMF model with hadron effective cou-plings and masses dependent on the scalar field; [39] used theFSU2R and FSU2H [42] extensions of the relativistic meanfield model FSU2 [43]. Different modelization of the nucle-onic sector in [15, 41] and [42] makes that the first of thesemodels does not allow for nucleonic dUrca in any stable NS,while the second does. The common conclusion of [37–40]is that thermal states of INS do not rule out hyperon nucle-ation in NS cores, contrary to what was previously claimed by[21, 23, 27].The aim of the present work is to investigate whether ther-mal evolution data of NS may be interpreted such as to getextra information on the EoS, including nucleation of hyper-ons, or, alternatively, assess the most probable composition ofone of the NS for which thermal data are available. To this endevolutionary tracks simulated with a benchmark code and cor-responding to different mass NS built upon realistic EoS [20]are systematically compared with data corresponding to mid- dle age isolated NS and accreting NS in low mass X-ray bina-ries . Uncertainties related to nucleon effective interactions athigh densities, including the isovector channel, are accountedfor by allowing for different EoS models; uncertainties in the Σ -nucleon interaction are accounted for by varying the welldepth of the Σ -hyperon at rest in symmetric nuclear matter;uncertainties related to the proton S pairing in the core, ex-pected to regulate the n → p + l + ˜ ν and Λ → p + l + ˜ ν dUrcaprocesses, are accounted for by considering two extreme pre-scriptions for the dependence of the pairing gap on the Fermimomentum. With the purpose of limiting the parameter spaceand the motivation that hyperon-hyperon interactions are littleknown hyperonic pairing is here disregarded.Let us point out that other approaches have been consideredto describe hyperonic matter. In particular, in [44] the authorshave constrained the density dependence of the meson massesto the 2 M ⊙ condition. A G-matrix approach to the EoS wasconsidered in [45]. In this case it is known that the EoS be-comes too soft with the introduction of hyperons, as shownin [46, 47], and 2 M ⊙ are not described. However, three-bodyforces play an essential role in this approach and are still notconstrained, in particular, the terms involving hyperons, seethe discussion in [48–50].To our knowledge the only attempts done so far in order toconstrain the NS EoS, in particular, the threshold density fordUrca, and/or the nucleonic paring gaps based on systematiccomparison with thermal data have disregarded hyperons andemployed parameterized EoS and/or pairing gaps [33–35, 51].Out of these only [33, 51] performed a best fit analyses ofthermal data of both INS and XRT.The paper is organized as follows. The set of EoS is pre-sented in Sec. II. Properties of NS built on these EoS arediscussed in Sec. III along with the constraints coming fromgravitational wave (GW) detection and mass-radius measure-ments. Observational data are commented in Sec. IV. A shortreview of cooling/heating theory of NS is offered in Sec. V.The results of our simulations are confronted with data inSec. VI. Finally the conclusions are drawn in Sec. VII. II. EQUATIONS OF STATE
A set of phenomenological EoS for hyperonic compactstars has been recently built in Ref. [20] by tuning the Λ -nucleon and Ξ -nucleon interactions on experimental data of Λ and Ξ -hypernuclei, following a procedure described lateron in this section. Uncertainties in the Σ -nucleon interaction,for which poor experimental constraints exist, have been ac-counted for by allowing the well depth of Σ at rest in sym-metric saturated nuclear matter U ( N ) Σ to span a wide range ofvalues.These EoS rely on the covariant density functional the-ory, in which the nucleons interact among each others bythe exchange of scalar-isoscalar ( σ ), vector-isoscalar ( ω ) andvector-isovector ( ρ ) mesons. Interactions between nucle-ons and hyperons are additionally mediated by the hiddenstrangeness vector meson φ .For the nucleonic part, 11 models have been considered:FSU2 [43], FSU2H and FSU2R [39, 42], NL3 [52], NL3 ωρ [53, 54], TM1 [55], TM1 ωρ [9, 53, 56], TM1-2 and TM1-2 ωρ [9], DD2 [57] and DDME2 [58]. The first nine belong tothe category of RMF models with non-linear (NL) terms. Thelast two belong to the category of RMF models with densitydependent (DD) coupling constants. For a general review onRMF models, refer to [59].For each nucleonic EoS, the couplings of the scalar-isoscalar σ meson to the Λ and Ξ -hyperons are determinedfrom the fit of experimental binding energies with the pre-dictions of the single-particle Dirac equations for baryons.The calibration of the EoS to Λ -hypernuclei was performedin [18, 60], where a vast collection of experimental data cor-responding to both hypernuclei in s and p shells was con-sidered. These results are used in the present study. Exper-imental data exist so far only for two Ξ -hypernuclei: Ξ − Be[61] and Ξ − C [62], with the second affected by uncertain-ties related to the state in which one of the daughter nucleiis produced. We use the coupling constants obtained in [20]under the hypothesis according to which Ξ − C is produced inthe first excited state. Based on consistency with data corre-sponding to Ξ − Be, this seems to be the most plausible sce-nario, in agreement with previous conclusion of [63]. Un-certainties related to the interaction between nucleons and the Σ -hyperon are dealt with by allowing U ( N ) Σ to explore the do-main − ≤ U ( N ) Σ ≤
40 MeV. We remind that, according to[64], U ( N ) Σ ≈ ±
20 MeV. The couplings of the vector mesonsto hyperons are expressed in terms of coupling constants ofthe meson under consideration to the nucleon and are derivedby assuming SU(6) flavor symmetry. They are as follows: g ωΛ = / g ω N , g ρΛ = g φΛ = −√ / g ω N , g ωΞ = / g ω N , g ρΞ = g ρ N , g φΞ = − √ / g ω N , g ωΣ = / g ω N , g ρΣ = g ρ N , g φΣ = −√ / g ω N . The hidden strangeness meson φ does notinteract with nucleons, g φ N =
0. For models with DD cou-pling constants, we assume that the coupling constants of thevarious vector fields to the hyperons have the same densitydependence as for the coupling to the nucleons.Out of these EoS models we here select those which satisfythe following conditions: i) the maximum mass of hyperonicstars exceeds the widely accepted lower limit, ≈ M ⊙ , ii) theslope of the symmetry energy of saturated symmetric matteris in agreement with experimental constraints, 40 < ∼ L < ∼ < ∼ L < ∼
86 MeV [66]. Only the followingEoS satisfy the above conditions: DD2 [57], DDME2 [58],NL3 ωρ [53, 54] and FSU2H [39, 42]. With K sym values equalto − . − . − . . − σ confidence level, with therecent constraint K sym = − ±
71 MeV [67] extracted fromLIGO/Virgo and NICER data, or with K sym = − . ± . − < K sym <
16 MeV, with which the firstthree models comply. Others studies obtain wider ranges as[70], where an upper limit at about 68 MeV is obtained. Fi-nally, Bayesian analyses of neutron skin thickness data that in-corporate prior knowledge of the pure neutron matter equationof state from chiral effective field theory calculations predict much smaller values, K sym = − + − MeV [71].
III. PROPERTIES OF HYPERONIC STARS
We now turn to discuss the properties of cold catalyzed NSbuilt upon the core EoS described in Section II. For the crustwe use EoS models which are consistent with the core [20,31, 32]; for the employed approach, see [31]. In Fig. 1 resultscorresponding to extreme values of U ( N ) Σ are illustrated withthick and thin lines for U ( N ) Σ = −
10 and 30 MeV, respectively.The top panel of Fig. 1 illustrates the pressure versus energydensity in the core. Also shown are the constraints obtained by[72], based on NICER and LIGO/Virgo data, corresponding toPSR J0030+0451 [73, 74] and, respectively, GW170817 [75].It comes out that, with the exception of NL3 ωρ which pro-vides a slightly stiffer behavior over the density range 0 . < ∼ n B < ∼ .
28 fm − , all our EOS comply with the constraints in[72]. For the energy density range considered, FSU2H showsthe softest behavior at low energy densities and NL3 ωρ thestiffest behavior at high energy densities. The different behav-iors of the P − ε curves lead to different values for the centralbaryonic number densities for a given mass, as clearly seen inthe second panel of Fig. 1. The lowest central baryonic densi-ties correspond, for all NS masses, to the stiffest EoS NL3 ωρ .As expected, the dispersion among the various EoS increaseswith the NS mass. To give an example, for U ( N ) Σ =30 MeV anda mass of 1 . M ⊙ , n NL3 ωρ c = .
34 fm − , n FSU2Hc = .
48 fm − , n DDME2c = .
47 fm − , n DD2c = .
54 fm − . The reason whythe n DDME2c < n DD2c is attributable to the much higher valueof Q sat , the parameter of the third order term in the Taylorexpansion of the energy per nucleon as a function of the de-viation from the saturation density, whose value is higher forDDME2 than for DD2 [59]. We shall see in Sec. VI that dif-ferent density profiles of the various baryonic species insideNS will result in significantly different thermal evolutions dueto the density dependence of the symmetry energy and EoSstiffness. The different behaviors of P ( ε ) curves are also re-flected in the mass-radius curves for all NS masses, in partic-ular, in the strongly different values of the radii, third panel ofFig. 1. For the canonical 1.4 M ⊙ NS, the dispersion among thepredictions of our models is large ∆ R . = R max1 . − R min1 . = . ωρ , provides the largest radii forall NS masses, and the smallest radii are obtained by DD2,considering NS with M ≥ . M ⊙ . NS with masses exceed-ing the threshold density for nucleation of Σ also manifesta non-negligible dependence of the radii on U ( N ) Σ : the largerthe abundances of Σ , favored by attractive Σ N potentials, thesmaller the NS radii. Also shown are the recent measurementsof the mass and radius of PSR J0030+0451, as obtained by theNICER mission [73, 74]. The large uncertainties neverthelessdo not allow to constraint the EoS. The only astrophysicalconstraints on the NS EoS available so far concern the lowand intermediate density domains and have been provided bythe tidal deformability measurements in the event GW170817[75]. The extent in which our models comply with these datais further considered in the bottom panel, where the tidal de-formability is plotted as a function of the NS gravitationalmass. The vertical error bar corresponds to the Λ = + − constraint in [75]. This constraint is extracted from the effec-tive tidal deformability ˜ Λ , but depends on the analysis under-taken. In particular, it was obtained from a set of EoS thatdid not necessarily describe two solar mass stars, and, there-fore, should be taken with care. FSU2H provides for 1 . M ⊙ a tidal deformability at the upper limit of the observationalconstraint, while the other EoS in this work overshoot theconstraint from GW170817; the largest overestimation cor-responds to the stiffest EoS, NL3 ωρ .Table I summarizes, for each model, several NS proper-ties both for purely nucleonic models (noY) and for hyperonicmodels with different values of U ( N ) Σ : the radius of the canon-ical 1 . M ⊙ NS ( R . ); the maximum mass ( M max ) and respec-tive central baryonic number density ( n c , max ); the onset densi-ties of the hyperonic species that nucleate in stable stars; thethreshold densities of nucleonic and various hyperonic dUrcaprocesses. and corresponding NS masses with these centralbaryonic densities One of the EoS (NL3 ωρ ) provides a max-imum NS mass in excess to the most stringent astrophysi-cal constraint 2 . + . − . M ⊙ [3]. DDME2 agrees, within errorbars, with the commonly adopted lower limit of the maximummass, corresponding to the pulsars PSR J0348 + . ± . M ⊙ [2]. Finally, DD2 and FSU2H predict slightlylower maximum masses, though marginally consistent withthe above constraint from PSR J0348 + n ,• irrespective the nucleonic EoS, the only hyperonicspecies that nucleate in cold catalyzed matter are Λ , Ξ − and Σ − ; the explanation relies on the attractive characterof Λ N and Ξ N interactions and dominance of negativelycharged particles; note that other models [13, 19, 31] ofhyperonic compact stars also allow for Ξ ,• threshold densities for the onset of Σ strongly dependson U ( N ) Σ ; strongly/poorly repulsive potentials favor, asexpected, late/early onset of Σ ,• attractive U ( N ) Σ potentials are responsible for a reduc-tion of M max ; the effect is nevertheless small, as is theabundance of Σ in star matter,• the onset baryonic density of the nucleonic dUrca de-pends on the nucleonic EoS; the two density dependentmodels (DD2 and DDME2) do not allow for nucleonicdUrca to operate in stable stars, while the two modelswith non-linear couplings do allow for this process; thelowest onset density of the nucleonic dUrca occurs forFSU2H, with 0 . ≤ n DU ≤ .
53 fm − , which corre-sponds to stars with masses 1 . ≤ M DU / M ⊙ ≤ .
86, • attractive U ( N ) Σ potentials shift the density thresholdof nucleonic dUrca to lower values; the rationale isthat, by partially replacing the electrons which com-pensate for the positive charge of protons, Σ − alter thebeta-equilibrium condition which determines the rela-tive abundances of neutrons and protons,• depending on the EoS and U ( N ) Σ different hyperonicdUrca processes are active; for the most repulsive (at-tractive) U ( N ) Σ potential, the first hyperonic dUrca whichoperates is Λ → p + l + ˜ ν l ( Σ − → Λ + l + ˜ ν l ). IV. OBSERVATIONAL DATA
Observation of thermal evolution of NS can potentially pro-vide information on NS interiors. Typically two classes ofobjects are considered. The first class corresponds to iso-lated middle-aged (10 − yr) NS, whose effective surfacetemperatures T ∞ s ≈ K or, equivalently, effective photonluminosities 10 < ∼ L ∞γ < ∼ erg s − are negatively corre-lated with the star age. Preeminence of thermal radiation inthe measured total radiation spectrum and thermal relaxationthroughout the volume are the main arguments which allowone to bridge measured temperatures and/or photon luminosi-ties to the multi-source neutrino emission from the core.We here consider the observational data of 19 INS compiledin Table 1 of [33]. They span the age range 0 . ≤ t ≤ . ≤ T ∞ s ≤ T ∞ s corresponds to thermal emission fromthe entire surface. Depending on the particularities of the ob-served spectra and/or the extracted values of the NS radii inthe observational paper(s) related to each source, the NS at-mosphere was assumed to be made of Fe, C or H.The second class of objects corresponds to transiently ac-creting quasi-stationary NS in low-mass X-ray binaries. Con-trary to NS in the first class, NS in this class heat-up becauseof the energy deposited in the bottom layers of the crust by thematerial which is intermittently accreted from the low masscompanion. As such, the thermal evolution is followed asa function of the average accreted mass rate, averaged overperiods of accretion and quiescence, which covers the range10 − ≤ ˙ M / (cid:0) M ⊙ yr − (cid:1) ≤ − . We remind that the heat-ing process is due to a series of nuclear reactions (electroncapture, neutron absorption and emission, pycnonuclear reac-tions) and its rate is estimated at ≈ − L ∞γ = . × − . × erg/s and the averageaccretion rate range ˙ M = . × − − . × − M ⊙ yr − .Despite a large dispersion of data, one can identify a positivecorrelation between the surface photon luminosity and the ac-cretion rate. With the exception of two, particularly faint ob-jects (SAX J1808.4 − TABLE I: Properties of NS built upon relativistic density functional models considered in this work. For each model, we first indicate theproperties of NS without hyperons (noY) and then for different values of U ( N ) Σ , as specified on column 2. R . represents the radius of thecanonical 1 . M ⊙ NS; M max and n c , max refer to the maximum NS mass and the corresponding central baryonic number density. Columns 6-14list the hyperonic species that nucleate in stable stars, their onset densities and the associated NS mass. Columns 15-22 list the density andmass thresholds above which the nucleonic and some hyperonic dUrca processes operate. Particle number densities are expressed in fm − andNS masses are expressed in M ⊙ . np , Λ p , Σ − Λ and Σ − n are abbreviations for the dUrca processes that involve the specified baryons. Y species np Λ p Σ − Λ Σ − n Model U ( N ) Σ R . M max n c , max n Y M Y n Y M Y n Y M Y n DU M DU n DU M DU n DU M DU n DU M DU (MeV) km ( M ⊙ ) (fm − ) (fm − ) ( M ⊙ ) (fm − ) ( M ⊙ ) (fm − ) ( M ⊙ ) (fm − ) ( M ⊙ ) (fm − ) ( M ⊙ ) (fm − ) ( M ⊙ ) (fm − ) ( M ⊙ )DD2 no Y 13.19 2.42 0.849-10 13.07 1.99 1.050 Σ − Λ Ξ − Σ − Λ Ξ − Λ Σ − Ξ − Σ − Λ Ξ − Σ − Λ Ξ − Λ Ξ − Σ − Σ − Λ Λ Σ − Λ Σ − Ξ − ωρ no Y 13.75 2.75 0.688 0.517 2.55-10 13.74 2.27 0.712 Σ − Λ Σ − Λ Ξ − Λ Ξ − Σ − data fill uniformly the domain in L ∞γ − ˙ M .These two faintest objects are of great interest as, accord-ing to the present understanding, they are the only NS whosethermal states require the (unsuppressed) dUrca to operateover important fractions of volume. For 1H 1905+00 only anupper limit on the accretion rate and luminosity exist. SAXJ1808.4 − V. COOLING SIMULATIONS
The thermal evolution of NS is typically studied by con-fronting calculated cooling/heating curves obtained for a cer-tain EoS and given superfluidity (SPF) assumptions with ob-servational data. NS with masses 1 ≤ M / M ⊙ ≤ M max are con-sidered.In the most general case the cooling tracks can be separatedin three families depending on whether dUrca processes op-erate or not. As long as no dUrca is active, NS cool slowly(i). If SPF in the core is disregarded, the cooling curves areindependent of both EoS and NS mass. As soon as one dUrcachannel opens up, the cooling is accelerated. If at least oneof the involved baryonic species is superfluid throughout thewhole core, the cooling is moderated (ii). The efficiency of thecooling increases steeply with the amount of matter which ac-commodates dUrca and the quenching of the SPF gap at highdensities. Cooling curves in this regime manifest increasedsensitivity to EoS, NS mass and SPF properties. Finally, ifat least one dUrca is allowed and it is not regulated by SPFover at least a finite fraction of the whole volume, the coolingis fast (iii). Similarly to (ii), cooling tracks in (iii) manifeststrong dependence on EoS, NS mass and SPF properties.Within the steady state approximation, Yakovlev et al. [28]demonstrated that heating of XRT is equivalent to INS cool- ing. XRT are in the photon or neutrino emission regimes de-pending on whether the energy deposited in the deep layersis transported to the surface from which it is radiated awayby photon emission or, alternatively, spread all over the vol-ume from which it is carried away by neutrino emission. Inthe photon emission regime the surface temperature dependson the accretion rate and is independent of the internal struc-ture of the NS. In the neutrino emission regime the surfacetemperature depends on the internal structure, i.e. neutrinoemission mechanisms and their quenching by SPF. High pho-ton surface luminosities require high accretion rates and slowneutrino emission; low photon surface luminosities requiresmall accretion rates and/or fast neutrino emission. Photonand neutrino emission regimes of XRT correspond to photonand neutrino cooling regimes of INS. According to [28], theonly difference between thermal evolution of INS and XRTconsists in the low dependence of the latter on the heat capac-ity of matter and the thermal conductivity of the isothermalinterior.To perform our cooling simulations we use an upgradedversion of the public domain code NSCool [99] by D. Page.Neutrino emission from the core, which is the dominant cool-ing mechanism for middle-aged INS and XRT, occurs via thefollowing reactions: a) nucleonic and hyperonic dURCA, withemissivities as in [30]; if one or both baryonic species in-volved in dUrca is paired, suppression of emissivity is im-plemented as in [80], b) modified Urca involving nucleons,with emissivities and emissivity suppression as in [81], c)bremsstrahlung from nucleon-nucleon collisions [81], and, incase of paired baryons, d) Cooper PBF in S - and P - wavechannels; emissivities in these two channels are treated as in[82–84] with the vector part of the PBF process strongly sup-pressed and the axial part negligible for the S pairing andreduced for P pairing. Modified Urca involving hyperonsand bremsstrahlung from hyperon-baryon collisions are disre-garded as their emissivities are subdominant to dUrca [36].Baryonic species with attractive interactions are known toexperience pairing. Neutrons in the crust and protons in thecore manifest S pairing, due to their relatively low densi-ties; besides, neutrons in the core manifest P − F pairing.Calculations performed using different interactions and many-body techniques have shown that the magnitude of the pair-ing gaps strongly depends on the nucleon-nucleon interactionsand many-body correlations; for a recent review, see [85].Good constraints on the neutron-neutron interaction comingfrom scattering data determine rather well the extension ofthe neutron pairing gap in the S channel, expressed in termsof the Fermi momentum. The size of this gap and the way inwhich it is quenched as the density increases are affected bythe treatment of many-body effects. For instance, [86] getsa reduction of 1.2 MeV (or a factor of 1.7) when the effectsof both short- and long-range correlations are employed, withrespect to the case in which the calculations are performedwithin the Bardeen-Cooper-Schrieffer (BCS) theory. Proton S and neutron P − F pairing gaps are affected by largererrors. For a review of proton S pairing gaps in NS matter,see [87]. Neutron P − F pairing in pure neutron matterhas been recently addressed in [86], who showed that the sizeof the gap may be modified by a factor of 3 (50) when dif-ferent bare nucleon-nucleon interactions (many-body effects)are employed. When short- and long-range correlations areaccounted for, the largest neutron P − F pairing gap ob-tained in [86] corresponds to the Entem-Machleit N3LO [88]potential; it extends over 1 . < ∼ k F < ∼ . − , with a maxi-mum of ≈ . k F ≈ . − .Neutron S pairing in the crust regulates the thermalizationof the crust and can only have observable effects before thecrust and the core reach thermal equilibrium. As this is notthe case of NS of interest for us, we do not explore its effectsand use only one value, calculated in [89] by accounting forlong-range correlations (polarization effects).Proton S pairing in the core suppresses the neutrino emis-sion from nucleonic and hyperonic Λ → p + l + ˜ ν dUrca re-actions. For internal temperatures lower than the critical tem-perature for pairing, it also leads to neutrino emission fromPBF. Uncertainties related to it are accounted for in this pa-per by considering two extreme scenarios: BCLL from [90]and CCDK from [91]. The BCLL [90] gap is calculated in theframe of the standard BCS theory, using the first-order (bare)particle-particle interaction as the pairing interaction; for thenucleon-nucleon interaction the Argonne v14 potential [92] isused; the gap extends over 0 . < ∼ k F < ∼ − , with a max-imum size of ≈ k F ≈ . − . TheCCDK [91] gap is calculated based on the matrix elementsextracted from Reid-soft-core potential [93]; it extends over0 < ∼ k F < ∼ . − , with a maximum size of ≈ k F ≈ . − .Neutron P − F pairing in the core has a large impacton the thermal evolution, as neutrons represent the dominantspecies in NS cores. For NS with baryonic particle densi-ties smaller than the threshold for nucleonic dUrca, neutron P − F pairing will accelerate the cooling by neutrino emis-sion from PBF. For NS which allow for nucleonic dUrca, itwill contribute to the suppression of the emissivity. This ef- fect is nevertheless expected to be negligible, considering thatnucleonic dUrca is already strongly quenched by proton S pairing, whose gap size is larger than the one of the neutrons.Finally, stars in the photon cooling era will cool much faster,due to the strong reduction of the heat capacity. In order torestrain the parameter space of our study, in this work we as-sume a vanishing pairing gap for neutrons in the P − F channel. All the implications of this assumption are such thatagreement with thermal data of INS is facilitated: larger sur-face temperatures will be obtained for youngest and warmestINS and oldest and coolest INS. We note that this assumptionhas already been employed in cooling studies [38, 94, 95].The measured surface temperatures of NS depend on thecomposition of the atmosphere. Atmospheres made of lightatoms (e.g. hydrogen, helium and carbon) are thought to cor-respond to young and accreting NS while old NS are consid-ered to have atmospheres made of heavy atoms (iron). In theevolutionary stages in which the energy loss is due to neutrinoemission, and is thus dependent on the internal temperature ofthe star, light atom atmospheres lead to higher effective sur-face temperatures than heavy atom atmospheres. When, onthe contrary, the energy loss depends on the surface temper-ature itself higher temperatures are obtained for atmospheresmade of heavy elements. In this work we shall consider, for allsimulations, atmospheres both entirely made of hydrogen andof iron. The relations between the temperature of the outerboundary of the isothermal internal region and the surface areimplemented as in [96]. As such, the effective surface temper-ature will correspond to the two limiting cases correspondingto the absence of light elements (iron atmospheres) and a max-imum amount of them (fully accreted atmospheres),We neglect effects due to rotation, magnetic field and lateheating. VI. RESULTS
As specified in Sec. II, the four selected nucleonic EoS,DDME2, DD2, FSU2H and NL3 ωρ , comply with the twosolar mass constraint on the lower limit of the maximum masswhen the hyperonic degrees of freedom are allowed. In thissection we consider all these models and discuss to what ex-tent the cooling/heating curves predicted by each one are ableto describe the observational data of INS and XRT.For each EoS we solve the Tolman–Oppenheimer–Volkoffequations and we build non-rotating spherically-symmetricconfigurations for masses ranging from just below the massthreshold for the first dUrca process to the maximum mass.Cooling/heating curves of these stars are simulated underthree scenarios for proton S SPF in the core: no SPF; nar-row pairing gap (BCLL) [90]; wide pairing gap (CCDK) [91].As explained in Sec. V, neutron S SPF in the crust is im-plemented according to [89] and neutron P − F SPF in thecore is disregarded. Hyperon SPF is disregarded as well. ForINS we use atmosphere models with the same compositionas the one employed when the temperature was determinedfrom x-ray observations: pure Fe, H and, C for two sources(Cas A and XMMU J1731 − − ≤ ˙ M / (cid:0) M ⊙ yr − (cid:1) ≤ − . A. Nucleonic stars
We briefly discuss the thermal evolution curves of purelynucleonic stars, as predicted by the four EoS models we em-ploy: DDME2, DD2, FSU2H and NL3 ωρ . As specified inTable I, nucleonic dUrca does not operate in stable stars builtupon DDME2 and DD2 models; stars built upon FSU2H andNL3 ωρ models allow for dUrca but only in very massive starswith masses in excess of 2.27 M ⊙ and, respectively, 2.55 M ⊙ .As a consequence, cooling/heating tracks of INS/XRT are ex-pected to be unable to account for the important dispersionmanifested by the thermal data, even if proton superfluidity isaccounted for.This situation is illustrated in Figs. 2 and 3, where the pre-dictions of DDME2 and FSU2H EoS are confronted with thethermal states of INS and, respectively, XRT. Two extreme as-sumptions are made in what regards the proton S -wave SPFgap: vanishing gap (left panels) and wide and large gap (rightpanels); for the latter situation we use the parameterizationCCDK in [91]. Results for the two models not shown, DD2and NL3 ωρ , are similar to those corresponding, respectively,to DDME2 and FSU2H.Independently of how the proton S -wave SPF gap is variedbetween the above mentioned limiting cases, the observationaldata can not be described. Cooling/heating curves producedby EoS models that do not allow for nucleonic dUrca form aunique narrow band. In the case of INS, this band would passthrough most of the data if all stars would have a Fe atmo-sphere, which seems not to be compatible with spectral anal-yses. In the case of XRT, agreement is obtained only for starswith intermediate and high luminosity (high luminosity) andlow (medium to high) accretion rates if H (Fe) atmospheremodels are assumed.Thermal evolution tracks generated by models that allowfor nucleonic dUrca split into two narrow bands separated bya wide gap. The high luminosity bands span the same T s − t and T s − h ˙ M i domains as those spanned by models that donot allow for dUrca; for INS the low luminosity band ex-plores a region where there is no experimental data; for XRT,it allows one to describe the coolest stars, whose average ac-cretion rates cover the whole range of values. Agreementwith these stars nevertheless requires, for the presently con-sidered models, very unlikely masses: ≈ . M ⊙ (2 . M ⊙ ) forFSU2H (NL3 ωρ ). Quantitative differences between resultscorresponding to various scenarios of proton S -wave SPF re-gard the widening of the high luminosity band and a slightshift towards higher luminosities of tracks corresponding tostars which allow for dUrca only in a tiny fraction of the core.The stronger the pairing gap the larger the amplitude of thesemodifications. The first modification is due to the openingof Cooper PBF and the second to the partial suppression ofdUrca reactions by SPF.The task of simultaneously describing the two lots of avail- able data seem to indicate that: i) dUrca is needed and theminimum NS mass that allows for it should be significantlysmaller than the present lower bound on NS maximum mass, ≈ M ⊙ , ii) its efficiency should be regulated by SPF. The caseof purely nucleonic stars has been successfully considered by[33, 35]. As none of the EoS models described in Sec. II al-lows for nucleonic dUrca in stars with masses lower than themaximum measured NS mass, we shall focus on hyperonicdUrca. B. Hyperonic stars
We next discuss the cooling/heating curves obtained for NSwith hyperonic degrees of freedom, whose properties are re-viewed in Sec. III. The same RMF models considered in theprevious section are employed. We will start the discussiontaking DDME2 as reference, and will next complete the dis-cussion looking at FSU2H, NL3 ωρ and DD2.The effect of including the proton S -wave pairing in thecore is illustrated in Fig. 4 for DDME2 with U ( N ) Σ =
30 MeV,for INS (top panels) and XRT (bottom panels). Left, mid-dle and right panels correspond, in this order, to vanishing,BCLL and CCDK gaps. When proton SPF is disregarded,the inclusion of hyperons makes a big difference in the cool-ing/heating curves: it modifies the evolution of stars withmasses > ∼ . M ⊙ , which suffer fast cooling from an age of ≈ yr. The main cooling agent is the Λ → p + l + ˜ ν l and,for M > ∼ . M ⊙ , also Σ − → Λ + l + ˜ ν l (see Table I and com-ments in Sec. III). Agreement with data requires that all NShave masses below 1.4-1.5 M ⊙ . Even so thermal states of INS S -pairing is implemented according toBCLL, no improvement is obtained for the description of INSand XRT. This means that proton Fermi momentum in thecore of NS as small as 1 . M ⊙ already exceeds the maximumvalue for which BCLL provides sizable pairing gaps, whichtranslates into an insufficient suppression of neutrino emis-sion from Λ → p + l + ˜ ν l . If the S -pairing of protons isimplemented according to CCDK, the cooling of stars with M / M ⊙ = . , .
45 is slowed down significantly. Thermaltracks of NS models with 1 . < ∼ M / M ⊙ < ∼ . M / M ⊙ > ∼ . Σ − → Λ + l + ˜ ν l dUrcawhich further enhances their cooling. Not accounting for hy-peronic pairing, this second reaction operates at full power.The best overall agreement with INS data is obtained forCCDK. We notice that if an hydrogen atmosphere is employedfor INS M / M ⊙ < ∼ . M ⊙ , for a large number of stars. The par-ticular case of SAX J1808 requires a more careful discussionand is, thus, left for later.In order to discuss the effect of the magnitude of the Σ N po-tential, Figs. 5 and 6 illustrate thermal evolution tracks of, re-spectively, INS and XRT built upon the DDME2 model whendifferent values of the U ( N ) Σ potential between 10 and 40 MeVare employed. Less repulsive potentials are not shown be-cause they give results even less compatible with observations.For INS the effect of U ( N ) Σ is distinctly seen: the more repul-sive the U ( N ) Σ the better is the temperature-age plane covered.Irrespective the value of U ( N ) Σ and the proton SPF model, INS U ( N ) Σ =
30 MeV and the CCDK model for protonSPF. A higher value, e.g. U ( N ) Σ =
40 MeV, can not be used asit would result in strong constraints on the mass of half of theINS: M > ∼ . M ⊙ .In what regards XRT one notes that the narrow proton SPFgap makes evolutionary tracks more consistent with data inthe sense that the heating curves cover more uniformly theluminosity-accretion rate plane. Large values of U ( N ) Σ improvethe description but, as in the case of INS, they should not ex-ceed 40 MeV. The largest considered value of U ( N ) Σ requiresrelatively small masses for a large number of stars and doesnot allow the observational point U ( N ) Σ = −
10 and 10 MeV Λ → p + e + ˜ ν e and Σ − → Λ + e + ˜ ν e dUrca processes start operating in NS with masses similar tothose obtained when DDME2 is employed; for U ( N ) Σ = Σ − → Λ + e + ˜ ν e dUrca is allowed only in relativelymassive stars; for U ( N ) Σ = −
10 MeV also Σ − → n + e + ˜ ν e is allowed, but only in NS with masses near the largest NSmasses measured so far; the most notable difference with re-spect to DDME2 is that the nucleonic dUrca is now allowed.For the most repulsive U ( N ) Σ potentials it nevertheless oper-ates in NS massive enough to not significantly change thepicture obtained when using DDME2. Thermal evolutiontracks provided by FSU2H are compared with INS and XRTdata in Figs. 7 and 8. Top (bottom) panels correspond to U ( N ) Σ =
20 MeV ( U ( N ) Σ =
30 MeV); left (right) panels corre-spond to BCLL (CCDK) proton SPF gaps. As it was the casewith DDME2, INS data favor U ( N ) Σ =
30 MeV and CCDK.We note that in this case also M < ∼ . M ⊙ ; nar-row proton SPF gaps (BCLL) do now allow thermal states ofstars U ( N ) Σ . Wide proton SPF gaps and re-pulsive U ( N ) Σ potentials would require M / M ⊙ > ∼ .
75 in order to describe the data.We now switch to NL3 ωρ . Data in Table I show that thesame dUrca processes allowed by FSU2H may operate but, ir-respective the value of U ( N ) Σ , they are active in stars more mas-sive than those obtained for FSU2H. Comparison with datacorresponding to INS and XRT is provided in Figs. 9 and,respectively, 10. As it was the case with FSU2H, top (bot-tom) panels illustrate results corresponding to U ( N ) Σ =
20 MeV( U ( N ) Σ =
30 MeV), while left (right) panels consider BCLL(CCDK) proton SPF gaps. As before, agreement with INSdata is favored for CCDK; more precisely, for CCDK the onlyINS whose states are not reproduced by NL3 ωρ are M / M ⊙ > ∼ .
65. Wenote that the mass distribution is similar for U ( N ) Σ =
20 and30 MeV, which seems to be more affected by the superfluidmodel. In what regards XRT, the situation is qualitativelysimilar to those obtained above: better agreement with datais obtained for BCLL and, with the exception of SAX J1808, U ( N ) Σ =
20 and 30 MeV provide similar results.Thermal evolution tracks predicted by DD2 for U ( N ) Σ = U ( N ) Σ , see Table I. Still, as for the previously discussedEoS, the best agreement with INS (XRT) data is obtained forCCDK (BCLL). The only INS impossible to describe whenCCDK is used are U ( N ) Σ and proton SPF gaps (mentioned ineach panel) are accounted for. Inspection of Figs. 12 and 13reinforces the conclusions drawn before, i.e. that the best de-scription of INS is provided by U ( N ) Σ =
30 MeV and CCDK.Inspection of Fig. 14 reveals that if the proton SPF gap islarge, the number of NS which need to have M / M ⊙ > ∼ . ωρ (not shown) are employed. C. The faint SAX J1808
Let us finish this section with a comment on SAX J1808.We have included in the XRT plots results from two differentestimates for the accretion rate and luminosity of this star, alltaken from Ref. [97]. The luminosity corresponding to point i.e. with neutrino emissivity not suppressedby SPF) over a significant fraction of NS volume.In Table II we provide constraints on the minimum mass ofSAX J1808, as obtained by requiring that its thermal state isreproduced by our simulations. Results corresponding to twoextreme hypothesis on the composition of the atmosphere (Hare Fe), different EoS models and values of U ( N ) Σ potentials aswell as proton SPF gaps are reported for each of the two abovecited measurements ( When data , agreement is possible only if a Fe atmosphere is used.The only combination of hyperonic-EoS/proton pairing gapunable to account for ωρ , U ( N ) Σ = ωρ , U ( N ) Σ =
30 MeVand BCLL may also be ruled out because of the unrealisti-cally large mass it predicts. The remaining NS models predictmasses larger than 1 . − . M ⊙ . When data agreement is possible for bothhypothesis on the composition of the atmosphere; in thecase of Fe-atmosphere, the minimum mass of SAX J1808is 1 . − . M ⊙ ; when a H-atmosphere is employed, some ofthe models fail to reproduce the data; they are DDME2 with U ( N ) Σ =
40 MeV, irrespective the proton gap; FSU2H with U ( N ) Σ =
30 MeV and CCDK; NL3- ωρ with U ( N ) Σ =
30 MeVand CCDK; the remaining models predict minimum valuessignificantly larger than those obtained under the assumptionof Fe-atmosphere, i.e. . − . M ⊙ . For both VII. CONCLUSIONS
The main objective of the present study was to verifywhether simultaneous comparison with thermal states of INSand XRT may single out the most probable EoS of hyperoniccompact stars. The candidate EoS in our set have been ob-
EoS core U ( N ) Σ proton set S gap Fe H Fe HM( M ⊙ ) M( M ⊙ ) M( M ⊙ ) M( M ⊙ )DDME2 N - - / / / /DDME2 N - CCDK / / / /FSU2H N - - 2.3 / 2.3 2.4FSU2H N - CCDK 2.3 / 2.3 2.4DDME2 NY 30 - 1.6 / 1.4 1.9DDME2 NY 30 CCDK 1.6 / 1.6 1.9DDME2 NY 30 BCLL 1.6 / 1.6 1.9DDME2 NY 10 CCDK 1.5 / 1.4 1.8DDME2 NY 10 BCLL 1.5 / 1.4 1.8DDME2 NY 20 CCDK 1.5 / 1.5 1.8DDME2 NY 20 BCLL 1.5 / 1.5 1.8DDME2 NY 40 CCDK 1.9 / 1.6 /DDME2 NY 40 BCLL 1.8 / 1.6 /FSU2H NY 20 CCDK 1.6 / 1.5 1.8FSU2H NY 20 BCLL 1.6 / 1.5 1.8FSU2H NY 30 CCDK 1.8 / 1.7 /FSU2H NY 30 BCLL 1.8 / 1.5 2.0NL3 ωρ NY 20 CCDK 1.8 / 1.8 2.1NL3 ωρ NY 20 BCLL 1.8 / 1.8 2.1NL3 ωρ NY 30 CCDK / / 1.9 /NL3 ωρ NY 30 BCLL 2.2 / 1.9 2.0DD2 NY 30 CCDK 1.5 / 1.5 1.7DD2 NY 30 BCLL 1.5 / 1.5 1.7TABLE II: Predictions on the minimum mass of SAX J1808 (in M ⊙ )based on observational data U ( N ) Σ , proton S pairing gapsand models of atmosphere (pure iron and pure hydrogen). ”/” meansthat data reproduction is not possible. tained in [20] and comply with well accepted nuclear and hy-pernuclear data as well as constraints from astrophysical mea-surements, in particular the lower limit ≈ M ⊙ of the maxi-mum NS mass. Uncertainties in the proton S pairing, knownto play a major role in regulating the neutrino emission fromthe core, were accounted for by considering two extreme sce-narios. Uncertainties in the composition of the atmospherehave been accounted for by considering two extreme hypoth-esis (pure H and pure Fe).As previously shown by many studies, agreement with mostdata corresponding to one category of NS, INS or XRT, can beobtained for several combinations of EoS, phenomenologicalpairing gaps of baryons in the core, composition of the atmo-sphere and, in our case, also the well depth of Σ -hyperon atrest in symmetric saturated nuclear matter. Nevertheless noneof the particular combinations we built allows to describe all data. Even more, a tension exists between ”the most suitable”combination EoS plus proton-SPF gap, extracted from the fitof INS or, alternatively, XRT data. This result means that ei-ther at least one of the standard scenarios that we have em-ployed here misses one or more essential ingredients or noneof the EoS is realistic. Use of neutron P − F pairing, heredisregarded, would not improve the overall agreement withdata. The reason is that, given the huge number of neutrons inthe core, even a tiny gap diminishes the NS heat capacity and,0thus, leads to accelerated cooling in the photon cooling era[40], which is not supported by INS data. Pairing of hyperons,here disregarded, would, in principle, improve the agreementwith all data except those of coldest XRT.Finally we have tried to constrain the U ( N ) Σ potential on thethermal state(s) of the faintest XRT, SAX J1808. For the par-ticular set of EoS considered here, that do not allow for nucle-onic dUrca or allow for it only in very massive NS, the condi-tions required by the low luminosity of SAX J1808 can onlybe met under certain hypothesis on the U ( N ) Σ potential, protonpairing gap, composition of the atmosphere and for massesexceeding certain threshold values. In most cases a repulsivepotential, U ( N ) Σ ≈ −
30 MeV, is needed, in agreement withhypernuclear spectroscopic data. This range of values is sup- ported also by the criterion of best agreement with all avail-able data from INS and XRT.
Acknowledgments : This work was supported byFundac¸˜ao para a Ciˆencia e Tecnologia, Portugal, under theprojects UID/FIS/04564/2016 and POCI-01-0145-FEDER-029912 with financial support from POCI, in its FEDER com-ponent, and by the FCT/MCTES budget through nationalfunds (OE), and by the Polish National Science Centre (NCN)under 759 grant No. UMO-2014/13/B/ST9/02621. A. R. R.acknowledges the support provided by the European COSTAction “PHAROS” (CA16214), through a STSM grant aswell as the kind hospitality of the Department of Physics, Uni-versity of Coimbra. CP acknowledges the support of THEIAnetworking activity of the Strong 2020 Project. [1] P. Demorest, T. Pennucci, S. Ransom, M. Roberts, and J. Hes-sels, Nature , 1081 (2010), 1010.5788.[2] J. Antoniadis et al., Science , 6131 (2013), 1304.6875.[3] H. T. Cromartie et al. (2019), 1904.06759.[4] S. Weissenborn, D. Chatterjee, and J. Schaffner-Bielich,Phys. Rev.
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10 MeV) and the dotted ones to purely nucleonicstars. The thin dot-dashed curves in the P ( ε ) plot mark the bor-ders of the domain obtained in [72] from the analyses of NICERand LIGO/Virgo data, corresponding to PSR J0030+0451 [73, 74]and, respectively, GW170817 [75]. The horizontal shaded area in M − n c and M − R plots shows the observed mass of pulsar PSRJ0348+0432, M = . ± . M ⊙ [76]. The experimental constraintson mass-radius relation based on NICER measurements of the mil-lisecond pulsar PSR J0030+0451 are represented in the M − R plot[74] (68% and 95% confidence contours). Finally, the vertical errorbar in the Λ − M plot illustrates the constraints from GW170817, asobtained in [75]. FIG. 2: Thermal states of INS built upon the DDME2 (top) andFSU2H (bottom) EoS and different scenarios for proton S -wave SPF:no SPF (left) and CCDK (right). The colored lines correspond tocalculations employing an iron atmosphere for the masses indicatedon each plot. The gray lines correspond to a hydrogen atmosphereand masses between 1 M ⊙ and the maximum mass, with a step of0.005 M ⊙ . The observational data, taken from [33], are representedwith orange boxes for a hydrogen atmosphere, violet for a carbonone and black for an iron one. We employ 2 − σ error bars, ifthese are available; otherwise a factor of 0.5 and 2 in both the tem-perature and the age (except for upper limits). The sources are 0- CasA NS, 1 - PSR J0205+6449 (in 3C58), 2 - PSR B0531+21(Crab), 3 - PSR J1119 − −
300 (in PupA), 5 -PSR J1357 − −
44, 7 - PSR B0833 −
45 (Vela), 8- XMMU J1731 − + + +
14, 12 - PSR B0633 + − − −
52, 17 - PSR J2043+2740, 18 - RXJ0720.4 − FIG. 3: Thermal states of XRT built upon the DDME2 (top) andFSU2H (bottom) EoS. Different scenarios are considered for proton S -wave SPF: no SPF (left) and CCDK gap (right). The colored linescorrespond to calculations employing a hydrogen atmosphere and amass sequences indicated on each plot. The gray lines correspond toan iron atmosphere and NS masses between 1 M ⊙ and the maximummass, with a step of 0.005 M ⊙ . The observational data, taken from[79], are represented with a factor of 0.5 and 2 in both the luminosityand the accretion rate (except for upper limits). For the source SAXJ1808, we show data from [33] labeled by 0 and [79] designated by1. The other sources are: 2 - 1H 1905+00, 3 - NGC 6440 X −
2, 4 -HETE J1900.1, 5 - EXO 1747, 6 - IGR J18245, 7 - XTE J0929, 8 -Ter 5 X −
1, 9 - Cen X −
4, 10 - XTE J1807, 11 - XTE 2123, 12 - XTEJ1814, 13 - IGR J00291, 14 - SAX J1810.8, 15 - MXB 1659 − − − −
2, 20 - 1RXS J180408, 21 - Swift J1756.9, 22 - XB 1732 − −
1, 25 - 1M 1716 − −
1, 29 - 4U 1730 −
22, 30 -GRS 1747, 31 - SAX J1750.8, 32 - EXO 0748, 33 - XTE J1701, 34- MAXI J0556, 35 - 4U 1608 − FIG. 4: Thermal states of INS (top) and XRT (bottom) for NS builtupon the hyperonic DDME2 EoS with U ( N ) Σ = +
30 MeV. Three su-perfluid scenarios are used for proton S pairing: no gap (left),BCLL gap (middle) and CCDK gap (right). For INS (XRT) the masssequences and color key are as explained in Fig. 2 (3). FIG. 5: Thermal states of INS built upon the hyperonic DDME2 EoS with U ( N ) Σ = , , ,
40 MeV and the BCLL (top) and CCDK (bottom)pairing models. Mass sequences and color key are as explained in Fig. 2. FIG. 6: Thermal states of XRT built upon the hyperonic DDME2 EoS with U ( N ) Σ = , , ,
40 MeV and the BCLL (top) and CCDK(bottom) models for proton S -wave pairing. Mass sequences and color key are as explained in Fig. 3. FIG. 7: Thermal states of INS built upon the hyperonic FSU2H EoSwith U ( N ) Σ =
20 (top) and 30 MeV (bottom) for the BCLL (left) andCCDK (right) models for proton S -wave pairing. Mass sequencesand color key are as explained in Fig. 2.FIG. 8: Thermal states of XRT built upon the hyperonic FSU2H EoSwith U ( N ) Σ =
20 (top) and 30 MeV (bottom) for the BCLL (left) andCCDK (right) models for proton S -wave pairing. Mass sequencesand color key are as explained in Fig. 3. FIG. 9: Thermal states of INS built upon the hyperonic NL3 ωρ EoSwith U ( N ) Σ =
20 (top) and 30 MeV (bottom) for the BCLL (left) andCCDK (right) models for proton S -wave pairing. Mass sequencesand color key are as explained in Fig. 2.FIG. 10: Thermal states of XRT built upon the hyperonic NL3 ωρ EoS with U ( N ) Σ =
20 (top) and 30 MeV (bottom) for the BCLL(left) and CCDK (right) models for proton S -wave pairing. Masssequences and color key are as explained in Fig. 3. FIG. 11: Thermal states of INS (top) and XRT (bottom) built uponthe hyperonic DD2 EoS with U ( N ) Σ = +
30 MeV. The BCLL (left) andCCDK (right) proton pairing models are employed.FIG. 12: Predicted mass range for the observed INS built upon thehyperonic DDME2 EoS with U ( N ) Σ = , ,
40 MeV (left, middleand right panels, respectively). BCLL and CCDK pairing models areconsidered in top and bottom panels, respectively. The color codingis the following: red for a hydrogen atmosphere, violet for a carbonone and gray for an iron one. For a given star, either a vertical bar ora dot on the X-axis is plotted. The dot indicates that the model is notconsistent with the observational data; the vertical bar indicates thepredicted mass range for the star under consideration. FIG. 13: Same as Fig. 12 for the FSU2H EoS with U ( N ) Σ =
20 and30 MeV (left and right panels, respectively) and the BCLL (top) andCCDK (bottom) pairing models. FIG. 14: Analog of Fig. 12 but for XRT. The red (gray) dots or barscorrespond to a H (Fe) atmosphere. The upper (lower) plots cor-respond to the DDME2 (FSU2H) EoS. In each plot the results for U ( N ) Σ ==