Cooling of Small and Massive Hyperonic Stars

Cooling of Small and Massive Hyperonic Stars

Abstract

We perform cooling simulations for isolated neutron stars using recently developed equations of state for their core. The equations of state are obtained from new parametrizations of the FSU2 relativistic mean-field functional that reproduce the properties of nuclear matter and finite nuclei, while fulfilling the restrictions on high-density matter deduced from heavy-ion collisions, measurements of massive 2 neutron stars, and neutron star radii below 13 km. We find that two of the models studied, FSU2R (with nucleons) and in particular FSU2H (with nucleons and hyperons), show very good agreement with cooling observations, even without including extensive nucleon pairing. This suggests that the cooling observations are compatible with an equation of state that produces a soft nuclear symmetry energy and, hence, generates small neutron star radii. However, both models favor large stellar masses, above , to explain the colder isolated neutron stars that have been observed, even if nucleon pairing is present.

neutron stars, stellar cooling, mass-radius constraints, equation of state, hyperons
\affiliation

Department of Physics, Kent State University, Kent OH 44242 USA

1 Introduction

The study of the thermal evolution of neutron stars has been a prominent tool for probing the equation of state (EoS) and composition of these objects (TSURUTA1965; Maxwell1979; Page2006; Weber2007; Negreiros2010). The reason behind is that the thermal properties that govern the cooling down of neutron stars, particularly neutrino emission processes, depend strongly on the particle composition and, thus, on the EoS of dense matter (Page2006; Page2009). Furthermore, recent observations have produced a wealth of neutron star data that can be used to constrain the properties of the underlying microscopic models used to describe these objects.

Several works have addressed the thermal evolution of neutron stars, comparing it to observed data, under the light of different phenomena, such as deconfinement to quark matter (Horvath1991; Blaschke2000a; Shovkovy2002; Grigorian2005; Alford2005a), stellar rotation (Negreiros2012; Negreiros2013; Negreiros2017), superfluidity (Levenfish1994; Schaab1997; Alford2005; Page2009; Fortin:2017rxq), magnetic fields (Aguilera2008; Pons2009; Niebergal2010a; Negreiros2017b), among others (Weber2005; Alford2005; Gusakov2005; Negreiros2010). A main point of contention in the thermal evolution studies is whether or not fast cooling processes take place, since if the star cools down too fast, it will yield to disagreement with most observed data (Page2004). The most prominent fast cooling process is the direct URCA (DU) process, i.e., the beta decay of a neutron and the electron capture by a proton. These processes are present if the proton fraction is high enough as to allow for momentum conservation. This leads to a direct connection between the thermal behavior and the symmetry energy of nuclear matter, as the latter is directly related to the proton fraction (Horowitz:2002mb; Steiner:2004fi; Than:2009ct; Beloin:2016zop).

In this work, we investigate the thermal evolution of neutrons stars whose composition is described by the microscopic models developed in Tolos:2016hhl; Tolos:2017lgv, which produce EoS’s for the nucleonic and hyperonic inner core of neutron stars that reconcile the mass observations (Demorest:2010bx; Antoniadis:2013pzd) with the recent analyses of stellar radii below 13 km (Guillot:2013wu; Guillot:2014lla; Guver:2013xa; Heinke:2014xaa; Lattimer:2014sga; Lattimer:2013hma; Nattila:2015jra; Ozel:2015fia; Ozel:2016oaf; Lattimer:2015nhk). Moreover, the properties of nuclear matter and of finite nuclei (Tsang:2012se; Chen:2014sca) are reproduced together with the constraints on the EoS extracted from nuclear collective flow (Danielewicz:2002pu) and kaon production (Fuchs:2000kp; Lynch:2009vc) in heavy-ion collisions (HICs). The study is performed within the relativistic mean-field (RMF) theory for describing both the nucleon and hyperon interactions and the EoS of the neutron star core.

Two models have been formulated, denoted by FSU2R (with nucleons) and FSU2H (with nucleons and hyperons), based on the nucleonic FSU2 model of Chen:2014sca. For the FSU2H model, the couplings of the mesons to the hyperons are fixed to reproduce the available hypernuclear structure data. The impact of the experimental uncertainties of the hypernuclear data on the stellar properties was analyzed in Tolos:2017lgv. The main differences between the two models were found in the onset of appearance of each hyperon. However, the values of the neutron star maximum masses showed only a moderate dispersion of about 0.1. Note that a broader dispersion of values may be expected from the lack of knowledge of the hyperon-nuclear interaction at the high densities present in the center of 2 stars. Hopefully, advances in the theoretical description of hyperon-nucleon interactions in dense matter from chiral effective forces (Haidenbauer:2016vfq) and possible constraints from future HIC experiments (Ohnishi:2016elb) will help narrowing down these uncertainties.

In the present study, we focus on how the hyperons as well as the symmetry energy of the microscopic model are correlated with the cooling history of neutron stars. We also consider the influence from nucleonic pairing on the cooling results. However, we do not consider hyperonic pairing, stellar rotation nor magnetic fields (Negreiros2012; Negreiros2013b; Negreiros2017), which are left for future study. We note that after hyperons were first proposed as one of the possible components of neutron stars (Glendenning:1982nc), their possible influence on the cooling of neutron stars was investigated in Prakash1992, and included in most thermal evolution studies thereafter. Most recently, 2018MNRAS.475.4347R revisited the topic by analyzing the cooling of massive stars described by different relativistic density functional models including nucleonic and hyperonic pairing. We will show that the microscopic models proposed here, given their underlying properties (especially the density slope of the symmetry energy), provide a very satisfactory agreement with observed cooling data. Particularly interesting is the fact that even without resorting to strong proton pairing, the results agree with most cooling observations for a large range of neutron stars.

2 The Models

2.1 Underlying Lagrangian

Our models are based on two new parametrizations of the FSU2 RMF model (Chen:2014sca). The Lagrangian density of the theory reads (Serot:1984ey; Serot:1997xg; Glendenning:2000; Chen:2014sca)

(1)

with the baryon (), meson (=, , and ), and lepton (=, ) Lagrangians given by

(2)

where and are the baryon and lepton Dirac fields, respectively. The field strength tensors are , , and . The strong interaction couplings of a meson to a certain baryon are denoted by and the baryon, meson and lepton masses by . The vector stands for the isospin operator.

The Lagrangian density (2) incorporates scalar and vector meson self-interactions, as well as a mixed quartic vector meson interaction. The nonlinear meson interactions are important to describe nuclear matter and finite nuclei. The scalar self-interactions with coupling constants and (Boguta:1977xi) are responsible for softening the EoS of symmetric nuclear matter around saturation density and allow one to obtain a realistic value for the compression modulus of nuclear matter (Boguta:1977xi; Boguta:1981px). The quartic isoscalar-vector self-interaction (with coupling ) softens the EoS at high densities (Mueller:1996pm), while the mixed quartic isovector-vector interaction (with coupling ) is introduced (Horowitz:2000xj; Horowitz:2001ya) to modify the density dependence of the nuclear symmetry energy. From the Lagrangian density (2), one derives in the standard way the equations of motion for the baryon and meson fields, which are solved in the mean-field approximation (Serot:1984ey).

To compute the structure of neutron stars, one needs the EoS of matter over a wide range of densities. The core of a neutron star harbours chemically-equilibrated (-stable), globally neutral, charged matter. We compute the EoS for the core of the star using the Lagrangian of Eqs. (1)–(2). As in Tolos:2016hhl; Tolos:2017lgv, for the crust of the star we employ the crustal EoS of Sharma:2015bna, which has been obtained from microscopic calculations. Once the EoS is known, the solution of the Tolman-Oppenheimer-Volkoff (TOV) equations (Oppenheimer:1939ne) provides the mass and radius of the neutron star.

2.2 Parametrizations

Models FSU2 FSU2R FSU2H
(MeV) 497.479 497.479 497.479

(MeV)
782.500 782.500 782.500

(MeV)
763.000 763.000 763.000

108.0943 107.5751 102.7200

183.7893 182.3949 169.5315

80.4656 206.4260 197.2692

3.0029 3.0911 4.0014

0.000533 0.001680 0.013298

0.0256 0.024 0.008

0.000823 0.045 0.045
0.1505 0.1505 0.1505

16.28 16.28 16.28

238.0 238.0 238.0


(MeV)
37.6 30.7 30.5

(MeV)
112.8 46.9 44.5


Table 1: Parameters of the models FSU2 (Chen:2014sca), and FSU2R and FSU2H (Tolos:2016hhl; Tolos:2017lgv) [here we use the slightly updated version of the FSU2R and FSU2H parameters given in (Tolos:2017lgv)]. The mass of the nucleon is MeV. Also reported are the values in nuclear matter at the saturation density for the energy per particle (), incompressibility (), symmetry energy (), and slope parameter of the symmetry energy ().

We first consider the nucleonic RMF parametrization FSU2 of Chen:2014sca. The parameters of the model were fitted by requiring an accurate description of the binding energies, charge radii and monopole response of atomic nuclei across the periodic table, and, at the same time, a limiting mass of in neutron stars. The resulting FSU2 parameter set (Chen:2014sca) (the parameters and saturation properties of the models can be found in Table 1) provides a stiff enough EoS in the region of large baryon densities and, as a consequence, reproduces heavy neutron stars. Figure 1 shows the pressure of -stable neutron star matter as a function of the baryon density for the different models considered in the present work. It can be seen that the pressure from FSU2—and also the pressure from all the other models that we use—passes through the region constrained by the study of flow data in experiments of energetic HICs (Danielewicz:2002pu). The predicted mass-radius (M-R) relation of neutron stars is represented in Fig. 2. FSU2 reaches a maximum mass of and, thus, accomodates the observed masses of in pulsars PSR J1614–2230 (Demorest:2010bx) and PSR J0348+0432 (Antoniadis:2013pzd). The radius of the star at maximum mass is of 12.1 km in FSU2 (see Table 2). For a neutron star with a mass of , FSU2 predicts a stellar radius of 14.1 km.

Figure 1: Pressure of -stable neutron star matter as a function of baryon number density for the models used in this work. The colored area depicts the region compatible with collective flow observables in energetic heavy ion collisions (Danielewicz:2002pu) (we note that although this constraint was deduced for pure neutron matter, it is useful here because at the implied densities the pressures of neutron matter and -stable matter are very similar).

Recent progress in astrophysical estimates of neutron star radii—see for example Ozel:2016oaf for a review—suggests that radii may be smaller than previously thought. Particularly, the extractions of radii from quiescent low-mass X-ray binaries (QLMXBs) and X-ray bursters are pointing toward values no larger than approximately 13 km (Guillot:2013wu; Guillot:2014lla; Guver:2013xa; Heinke:2014xaa; Lattimer:2014sga; Lattimer:2013hma; Nattila:2015jra; Ozel:2015fia; Ozel:2016oaf; Lattimer:2015nhk). The first observation of a binary neutron star merger by the LIGO and Virgo collaborations also appears to indicate that neutron stars cannot have excessively large radii (TheLIGOScientific:2017qsa). It is worth mentioning that the advent of accurate data on neutron star radii should allow one to probe the EoS of neutron-rich matter in a complementary way to the heavy masses. This is due to the fact that, while the maximum stellar mass depends on the high-density sector of the EoS, the strongest impact on the radius of the star comes from the EoS in the low-to-medium density region of 1–2 times the saturation density (Lattimer:2006xb; Ozel:2016oaf). Given that the nuclear symmetry energy governs the departure of the energy of neutron matter from symmetric matter, it means that data on neutron star radii pin down the density dependence of the symmetry energy around and the slope parameter , defined as , which is intimately related with the isospin properties of atomic nuclei, although its value is still uncertain (Li:2014oda). Larger values (stiffer symmetry energy) favor larger radii in neutron stars, whereas smaller values (softer symmetry energy) favor smaller radii. Therefore, an astrophysical confirmation of small neutron star radii would imply that the nuclear symmetry energy cannot be overly stiff around saturation density.

Figure 2: Mass-radius relations for neutron stars predicted by the models used in this work. The accurate mass measurements of in pulsar PSR J1614–2230 (Demorest:2010bx) and in pulsar PSR J0348+0432 (Antoniadis:2013pzd) are also shown.
Models (km) (km)
FSU2(nuc) 2.07 12.1 5.9 14.1
FSU2R(nuc) 2.05 11.6 6.3 12.8
FSU2H(nuc) 2.38 12.3 5.4 13.2
FSU2R(hyp) 1.76 11.6 6.5 12.8 2.4
FSU2H(hyp) 2.02 12.1 5.8 13.2 2.2
Table 2: Neutron star properties from the models used in this work. Results are shown for nucleonic (nuc) and hyperonic (hyp) stars. The quantity is the central baryon number density at star with maximum mass normalized to the saturation density , whereas is the onset density of appearance of hyperons normalized to .

In order to account for the possible existence of massive stars with small radii in our theory, yet without compromising the agreement with constraints from the properties of atomic nuclei and from HICs, in Tolos:2016hhl; Tolos:2017lgv we developed the FSU2R parametrization, which produces a soft symmetry energy and a soft pressure of neutron matter for densities . This can be seen in Fig. 1 by comparing the EoS’s of FSU2R (solid black line) and FSU2 (dashed blue line). For a given neutron star, in the FSU2R EoS up to densities there is less pressure to balance gravity, thereby leading to increased compactness of the star and smaller stellar radius. In the high-density sector of the EoS, the FSU2R and FSU2 EoS’s are close to each other (cf. Fig. 1), and, thus, FSU2R also reproduces heavy neutron stars, as can be seen in Fig. 2. Smaller radii within the range of 11.5–13 km are obtained in FSU2R for neutron stars between maximum mass and (see Fig. 2 and Table 2), owing to a softer symmetry energy, while reproducing the properties of nuclear matter and nuclei. Our conclusions are in keeping with the results of recent studies with RMF models with a soft symmetry energy (Chen:2015zpa; Chen:2014mza). Indeed, FSU2R predicts MeV and MeV (Tolos:2017lgv), which are in good accord with the limits of recent determinations (Lattimer:2012xj; Li:2013ola; Roca-Maza:2015eza; Birkhan:2016qkr; Oertel:2016bki).

Since the FSU2 and FSU2R parametrizations assume nucleonic (non-strange) stellar cores, we shall often use the notation FSU2(nuc) and FSU2R(nuc) to refer to these models. We also analyze in the present work the consequences of the appearance of hyperons inside neutron star cores. As described in Tolos:2016hhl; Tolos:2017lgv, the couplings of the hyperons to the vector mesons are related to the nucleon couplings by assuming SU(3)-flavor symmetry, the vector dominance model and ideal mixing for the physical and mesons, as, e.g., employed in recent works (Schaffner:1995th; Banik:2014qja; Miyatsu:2013hea; Weissenborn:2011ut; Colucci:2013pya). The values of the hyperon couplings to the scalar meson field are determined from the available experimental information on hypernuclei, in particular by fitting to the optical potential of hyperons extracted from the data. Finally, the coupling of the meson to the baryon is reduced by 20% from its SU(3) value in order to reproduce bond energy data (Ahn:2013poa).

The EoS of neutron star matter and the M-R relation from the FSU2R model with inclusion of hyperons—dubbed as FSU2R(hyp) model—are plotted, respectively, in Figs. 1 and 2. As expected, due to the softening of the high-density EoS with hyperonic degrees of freedom (compare the FSU2R(hyp) and FSU2R(nuc) EoS’s in Fig. 1), we obtain a reduction of the maximal neutron star mass below 2 in the FSU2R(hyp) calculation. However, we may readjust the parameters of the nuclear model by stiffening the EoS of isospin-symmetric matter for densities above twice the saturation density, i.e., the region where hyperons set in, simultaneously preserving the properties of the previous EoS for the densities near saturation, which are important for finite nuclei and for stellar radii. The couplings of the hyperons to the different mesons can be determined as before. The parameters of the new interaction (Tolos:2016hhl; Tolos:2017lgv), denoted as FSU2H, are displayed in Table 1, along with the predicted symmetry energy at saturation density and its slope , which are safely within current empirical and theoretical bounds (cf. Fig. 4 of Tolos:2016hhl and Fig. 1 of Tolos:2017lgv). The neutron star calculations with the FSU2H model with allowance for hyperons in the stellar core—FSU2H(hyp) model—successfully fulfill the 2 mass limit with moderate radii for the star (see Fig. 2 and Table 2), while the base nuclear model FSU2H still reproduces the properties of nuclear matter and nuclei. In isospin-symmetric nuclear matter, FSU2H leads to a certain overpressure in the EoS at high densities (Tolos:2016hhl), but the EoS in neutron matter satisfies the constraints from HICs (Danielewicz:2002pu), as can be seen in Fig. 1.

We also draw in Fig. 2 the M-R relation predicted by FSU2H if one neglects hyperons—FSU2H(nuc) model. As expected, since the hyperonic FSU2H(hyp) EoS is softer than the FSU2H(nuc) EoS after hyperons appear (see Fig. 1), the neutron star calculations with FSU2H(nuc) lead to a higher maximum mass than FSU2H(hyp). In the next section, comparisons between cooling calculations performed with FSU2H(hyp) and FSU2H(nuc) will be used to discuss the influence of the occurrence of hyperons on neutron star cooling.

3 Cooling from low to high-mass neutron stars

Once the microscopic models for the EoS and the resulting properties of neutron stars have been discussed, we proceed to calculate the thermal evolution of such stars. We recall that the cooling of a neutron star is driven by neutrino emission from its interior, as well as photon emission from the surface. The equations that govern their cooling are those of thermal balance and of thermal energy transport (2006NuPhA.777..497P; 1999Weber..book; 1996NuPhA.605..531S), given by ()

(3)
(4)

The cooling of neutron stars depends on both micro and macroscopic ingredients, as can be seen in eqs. (3) and (4), where all symbols have their usual meaning and the thermal variables are the neutrino emissivity , the thermal conductivity , the specific heat , the luminosity , and the temperature .

In addition to Eqs. (3) and (4), one also needs a boundary condition connecting the surface temperature to that in the mantle (Gudmundsson1982; Gudmundsson1983; Page2006), as well as the condition of zero luminosity at the center in order to satisfy the vanishing heat flow at this point. In this study, we make use of all neutrino emissivities allowed for the EoS’s and the corresponding compositions, that is, all processes involving nucleons and, when pertinent, hyperons, as well as the appropriate specific heat and thermal conductivity. A thorough review of such processes can be found in reference Yakovlev2000.

3.1 Cooling of neutron stars without nucleon pairing

We first consider the thermal evolution of neutron stars without taking into account any sort of pairing (neither in the core nor in the crust). This is done with the ultimate goal of determining how the different models for the EoS describe the thermal behaviour of neutron stars. Such results are shown in Figs. 37, where the temperature is drawn as a function of age for each model. In these figures, we depict several cooling curves for low, medium and high neutron star masses for each model, in order to investigate the cooling behaviour for different central density regimes. The figures also display the observed surface temperature versus the age of a set of prominent neutron stars, including that of the remnant in Cassiopeia A (Beloin:2016zop; SafiHarb2008; Zavlin1999; Pavlov2002; Mereghetti1996; Zavlin2007; Pavlov2001; Gotthelf2002; McGowan2004; Klochkov2015; McGowan2003; McGowan2006; Possenti1996; Halpern1997; Pons2002; Burwitz2003; Kaplan2003; Zavlin; Ho2015).

Figure 3: Surface temperature as a function of the stellar age for different neutron star masses in the FSU2 model. Also shown are different observed thermal data.
Figure 4: Same as in Fig. 3 but for the FSU2R (nuc) model.
Figure 5: Same as in Fig. 3 but for the FSU2H (nuc) model.
Figure 6: Same as in Fig. 3 but for the FSU2R (hyp) model.
Figure 7: Same as in Fig. 3 but for the FSU2H (hyp) model.
Models DU threshold hyp DU threshold
(fm) (fm) (fm) cooling (fm) cooling (fm) cooling
FSU2 (nuc) 0.21 0.35 fast 0.47 fast 0.64 fast
FSU2R (nuc) 0.61 0.39 slow 0.51 slow 0.72 fast
FSU2H (nuc) 0.52 0.34 slow 0.39 slow 0.45 slow
FSU2R (hyp) 0.57 0.37 0.40 slow 0.87 fast
FSU2H (hyp) 0.52 0.34 0.34 slow 0.44 slow 0.71 fast
Table 3: Thermal behaviour of the different models studied. The DU threshold indicates the density at which the URCA process of nucleons becomes effective. Similarly, the hyp DU threshold indicates the density at which the hyperonic DU processes, given by the , begin to act. Also shown is the central density for three selected neutron star masses, as well as whether such stars exhibit slow or fast cooling.

The cooling behaviour from each microscopic model is summarized in Table 3 for low-mass to high-mass neutron stars. When DU reactions of neutrino production are allowed according to the model, they lead to an enhanced cooling of the star. We summarize here the DU processes that may take place inside the neutron star:

We stress that the presence of the particles is not enough for such processes to take place, and one must also account for momentum conservation, as previously discussed. We also recall that all inverse reactions also take place as to maintain (on average) chemical equilibrium. In Table 3 we indicate the density threshold for the nucleonic DU process and, when applicable, the threshold for the hyperonic DU processes, determined by the particle as the first hyperon to appear. We also show the central density for three selected neutron star masses, as well as whether such stars exhibit slow or fast cooling scenarios. The information supplied in Table 3 will help to understand the thermal evolution of neutron stars, displayed in Figs. 3 to 7 for the different models and discussed in the following:

FSU2 (Fig. 3): All neutron stars whose microscopic composition is described by this model allow for pervasive DU process, even for low-mass stars with low central densities of around 2, leading to a cooling which is too fast in comparison with the observed data.

FSU2R(nuc) (Fig. 4): For this model most stars (up to ) exhibit slow cooling. This model could, in principle, explain most of the observed data. However, it would mean that all colder stars with K have masses higher than , which seems unlikely.

FSU2H(nuc) (Fig. 5): For this model the DU process is absent in all stars studied, from low-mass stars with lower central densities to high-mass stars with higher central densities. Therefore, all stars exhibit slow cooling, which allows for agreement with some of the observed data but fails to explain most of the observed cold stars.

FSU2R(hyp) (Fig. 6): The results of this model exhibit a relatively similar pattern to that of the FSU2R(nuc) model, with the exception that the highest mass possible in this model is , and that only stars with masses similar to that one exhibit fast cooling.

FSU2H(hyp) (Fig. 7): This particular model shows the most promising results, with stars with lower masses exhibiting slow cooling (without DU) and higher masses displaying fast cooling (with DU), while intermediate masses show a behavior in between these extremes. This model explains most of the observed data, including a reasonable agreement with Cas A, without the need of resorting to extensive pairing which is subject to uncontrolled uncertainties.

The different cooling pattern for each model can be understood from the microscopic differences between the models. We start by analyzing the case of low-mass stars of with central densities around twice nuclear saturation density. The different cooling pattern of a star observed for FSU2 (fast cooling) and FSU2R(nuc) (slow cooling) is mainly due to the different density dependence of the symmetry energy around saturarion in these models and, hence, to the different value of the symmetry energy slope parameter (), which can be read in Table 1 ( MeV in FSU2 and MeV in FSU2R). The larger the value of is, the more protons are produced and, thus, the DU process appears at lower densities, making the cooling of the star more efficient. Indeed, even if a star in the FSU2 model has a smaller central density than in the FSU2R(nuc) model, the cooling is faster in FSU2 as the DU threshold for FSU2 is at a much smaller density, as seen in Table 3. This conclusion is corroborated by the cooling behaviour of FSU2R(nuc) and FSU2H(nuc), which show a similar qualitatively slow cooling for neutron stars with , as both models have an alike value (cf. Table 1). Therefore, one finds that in low-mass stars, where the central density does not go much above 2, large stellar radii (stiff nuclear symmetry energy near ) are associated with fast cooling, whereas small stellar radii (soft nuclear symmetry energy near ) imply slow cooling, as seen in Dexheimer:2015qha.

As for high-mass stars (), we find that the different behaviours exhibited by the cooling curves of FSU2 (fast), FSU2R(nuc) (fast) and FSU2H(nuc) (slow) are correlated with the different values of the central densities in these stars, as seen in Table 3. This is understood by considering the fact that the FSU2H(nuc) model produces a stiffer EoS in the region of high densities than the FSU2 and FSU2R(nuc) models (see Fig. 1). Therefore, for the same heavy stellar mass, neutron stars obtained within the FSU2H(nuc) model have much lower central densities than in FSU2 and FSU2R(nuc), making the DU process less efficient and, thus, leading to a slower cooling. For these high stellar masses, the densities reached are much higher than saturation density and, therefore, the slope of the symmetry energy at saturation is not determinant for the cooling behaviour.

Figure 8: Particle fractions as functions of the baryonic density for the models discussed in the text.

With regards to the models that include hyperons, as their EoS’s are softer than their nucleonic counterparts, they produce stars with higher central density values, which may then overcome the DU threshold, as one can clearly see in Table 3. As a consequence, stars with that cooled more slowly without hyperons in the FSU2R(nuc) model, change to a faster cooling pattern in the presence of hyperons, as seen for FSU2R(hyp). We also observe in Table 3 that hyperonic models activate the DU process at similar or even lower densities than the models without hyperons. This is mainly due to the fact that the presence of hyperons (mostly particles) reduces the neutron fraction at a given baryon number density and, consequently, the DU constraint , with , and being the Fermi momenta of the neutron, proton and electron, respectively, can be fulfilled at a lower density. This is seen in Fig. 8, where the particle fractions are shown as functions of the neutron star matter density, for the various interaction models explored in this work. It is clear that the appearance of the hyperons between 0.3 and 0.4 fm for the FSU2R(hyp) and FSU2H(hyp) models (middle and lower panels) induces a substantial decrease in the neutron fraction compared to the purely nucleonic models collected in the upper panel. In summary, the presence of hyperons in the cores of medium to heavy mass stars speeds up their cooling pattern. When comparing the cooling curves of the FSU2R(hyp) and FSU2H(hyp) models, we notice that higher stellar masses are needed in the case of FSU2H(hyp) to reach a fast cooling behavior. This is due to the fact that the EoS of the FSU2R(hyp) model is softer and the central densities achieved are larger compared to the ones for FSU2H(hyp), hence the DU threshold is overcome more easily, even if it appears at slightly higher densities. Finally, it may be noticed that the cooling pattern seen in Fig. 7 for FSU2H(hyp) is similar to that in Fig. 4 for FSU2R(nuc), so that it could argued that it is unnecessary to resort to a hyperonic model such as FSU2H(hyp) to explain most of the observed cooling data. However, purely nucleonic models are forcedly omitting the presence of hyperons, which are physically allowed at intermediate densities when the appropriate chemical equilibrium conditions are fulfilled.

3.2 Neutron superfluidity and proton superconductivity effects on the cooling of neutron stars

We now turn our attention to investigate the neutron superfluidity and proton superconductivity on the cooling of neutron stars. Pairing effects have been considered a key factor for the thermal evolution of neutron stars (Beloin:2016zop; Yakovlev2000; Gnedin2008; Weber2009a; Page2004; Weber2007), as well as of extreme importance to find agreement between the theoretical models and the observed data, particularly for Cas A (Page2011a; Ho2009; Yang2011; Heinke2010; Ho2015; Shternin2011a; Blaschke2012; Ho2009; Negreiros2013b).

As indicated before, one of the major cooling channels in neutron stars, especially during the first  years, is neutrino production reactions. Most of these reactions involve baryons, and chief among those is the DU process, which if present is the leading cooling mechanism in neutron stars. The introduction of a superfluidity (conductivity) gap in the energy spectrum of such baryons reduces the reaction rates. One notes that the reduction factor depends on the temperature and its relation to the corresponding superfluidity (conductivity) critical temperature (or the gap), and leads to a sharp drop of neutrino emissivity after the matter temperature drops below the pairing critical temperature. The calculation of the reduction factor for each neutrino emission process is a complicated procedure, which can be obtained by the study of the phase-space of the emission processes (see Yakovlev2000 for a comprehensive calculation of such factors).

There is, however, a great deal of uncertainties regarding proton and neutron pairing in neutron stars (Yakovlev2000; 2006NuPhA.777..497P; Page2004), specially at high densities, not to mention the possibility of pairing among hyperons, which is even more uncertain. For that reason we have chosen to analyze different pairing scenarios for protons. These scenarios reflect three assumptions: a) shallow proton pairing (limited to 2–3 ), b) medium proton pairing (extending up to 4 ), and c) deep proton pairing (extending beyond 5 , deep in the inner core). In this manner, we can estimate how extensive the proton pairing must be so as to allow for a good comparison with the observed data. In Fig. 9 we show the critical temperature for proton singlet () pairing as a function of density for the three different scenarios.

Figure 9: Critical temperature for proton singlet () pairing as a function of normalized number baryon density for the three pairing scenarios studied.

As for neutron pairing, somewhat less uncertain than that of protons (particularly for the neutron singlet pairing in the crust), we chose the standard approach: we allow for extensive neutron singlet () pairing at subnuclear densities (crust) and for a limited neutron triplet () pairing in the core, with a maximum critical temperature K, similarly to the pairing used to explain the observed temperature of Cas A in Page2011a. The critical temperature of both neutron singlet and triplet pairings as a function of density can be seen in Fig. 10.

Figure 10: Critical temperature for neutron pairing (singlet and triplet) as a function of normalized number baryon density.

Apart from the suppression of neutrino emission reactions that involve paired particles, a consequence of pairing is the appearance of a new transient neutrino emission process, commonly known as pair breaking-formation (PBF) process. This process can be represented as , with denoting paired baryons. It can be also understood as the annihilation of two quasi-baryons with similar anti-parallel momenta into a neutrino pair (1976ApJ...206..218F). This process, also comprehensively described in Yakovlev2000, is transient, reaching a maximum near the superfluidity (conductivity) onset and decreasing afterwards.

For the analysis of pairing effects on the cooling of neutron stars, we concentrate our study on the two most relevant cases, the FSU2R(nuc) and the FSU2H(hyp) models (Tolos:2016hhl; Tolos:2017lgv), which are the ones that best reproduce the observed data on cooling in the previous section. We also show the predictions from the FSU2(nuc) model (Chen:2014sca) for comparison. The main features of the results are as follows:

FSU2 (Fig. 11): The original FSU2 model, as discussed before, has a small DU threshold, leading to fast cooling for all stars studied. The inclusion of shallow and medium proton pairing (in addition to neutron pairing, common to all simulations) is ineffective in slowing down the thermal evolution. Although part of the DU (among other processes) is suppressed, there is still a relatively large region at high densities in which the DU takes place, thus leading to a cooling behaviour very similar to that of Fig. 3. The deep proton pairing, on the other hand, is effective in slowing the cooling, and leads to a slow cooling behaviour, as shown in Fig. 11. While the first “knee” in the cooling curves, which usually happens between 50–150 years, is associated with the core-crust thermal coupling, the second “knee” around years is linked to the onset of neutron pairing and the subsequent production of neutrinos coming from the PBF process.

Figure 11: Surface temperature as a function of the stellar age for different neutron star masses in the FSU2 model subjected to deep proton pairing as well as neutron pairing.

FSU2R(nuc) (Fig. 12): As previously seen, this model exhibits slow cooling only for stars below as the DU appears at high densities of fm. Medium pairing allows for a satisfactory agreement with data, with the caveat that all observations with K would need to be stars of relatively high mass (above ) within this model, as seen in Fig. 12. Deep proton pairing leads to the complete absence of DU processes, thus leading to a slow cooling scenario for neutron stars of all masses.

Figure 12: Same as Fig. 11 but for the FSU2R(nuc) model subjected to medium proton pairing as well as neutron pairing.

FSU2H(hyp) (Fig. 13): This model, which was in good agreement with the observed thermal data without the inclusion of pairing, also benefits from pairing. The inclusion of shallow/medium pairing leads to an improved agreement with the observed data. Our study shows that shallow to medium pairing leads to an optimal agreement with the data, in particular with Cas A data, as shown in Fig. 13. We note that within this model the best fit to Cas A is obtained for a star. This is due to the change of slope of the cooling curve by the implementation of the onset of neutron triplet pairing and the PBF process within the FSU2H(hyp) model, as seen in the inset of Fig. 13. For a star the onset happens a little later ( years), so that the agreement with the slope of the cooling curve of the data is not as good.

Figure 13: Same as Fig. 11 but for the FSU2H(hyp) model subjected to medium proton pairing as well as neutron pairing.

Summarizing, while including neutron pairing together with shallow or medium proton pairing is inefficient in slowing the thermal evolution of the stars predicted by the FSU2 model, deep proton pairing does improve the agreement with data but still does not explain the cooling of intermediate and low temperature stars. The FSU2R(nuc) and FSU2H(hyp) models already perform well without consideration of pairing. Nevertheless, including medium proton pairing, in addition to the neutron pairing, improves the agreement of the cooling curves of the FSU2R(nuc) and FSU2H(hyp) models with data, especially concerning the slope for the cooling of Cas A in the case of the FSU2H(hyp) model, as can be seen in the inset of Fig. 13. This is our preferred model as the colder stars are described with masses a little below those for the FSU2R(nuc) model, and also because it allows for the presence of hyperons when the chemical equilibrium conditions are fulfilled, being at the same time able of supporting neutron stars with 2.

We note that in our work hyperon pairing has not been taken into account. This was addressed recently by 2018MNRAS.475.4347R where the cooling of hyperonic stars was studied, although following a somewhat different perspective than ours. The authors of 2018MNRAS.475.4347R, employing different relativistic density functional models, focused their investigation on the role of hyperonic DU processes on cooling, as well as on the effect of pairing on such processes. They have found that the is the dominant hyperonic DU process. Furthermore, they have found that observed data for colder objects can be explained by stars with whereas the hotter ones are well explained by stars with . Our results are in agreement with their conclusions regarding the dominant hyperonic DU process. However, in our preferred FSU2H(hyp) model, we have found that hotter stars can be described by objects with masses of up to , as opposed to . One must note that for the aforementioned results 2018MNRAS.475.4347R have considered pervasive pairing such that the nucleonic DU process is completely suppressed. The possibility of nucleonic DU was also considered, but did not produce agreement with observed data. In this regard our studies differ considerably from those of 2018MNRAS.475.4347R, as we have not completely excluded the nucleonic DU process and we have not considered hyperonic pairing. For a future investigation, we wish to expand our study to also include hyperonic pairing.

4 Conclusions

We have performed cooling simulations for isolated neutron stars with recently developed equations of state for the core. These are obtained from new parametrizations of the FSU2 relativistic mean-field functional (Chen:2014sca), which reproduce the properties of nuclear matter and finite nuclei, while fulfilling the restrictions on high-density matter deduced from heavy-ion collisions, measurements of massive 2 neutron stars, and neutron star radii below 13 km.

The analysis of the cooling behavior has shown that the underlying microscopic model is successful in explaining most of the observed cooling data on isolated neutron stars. Particularly satisfactory is the parametrization FSU2H(hyp), which allows for slow cooling for stars with , moderate or fast cooling otherwise. This indicates that most of the observed data, as well as Cas A, could be explained, without all observed colder stars being constrained to relatively high mass. A similar behavior is exhibited by the parametrization FSU2R(nuc), however with the caveat that all colder observations would have higher masses above and that hyperons are not present.

We have also investigated the role of pairing for the cooling behavior of these two models together with the original FSU2 model. For that purpose, we have considered singlet and triplet neutron pairing, as traditionally assumed for thermal evolution calculations, as well as proton pairing in the stellar core. Due to the current uncertainties regarding proton pairing, we have taken into account different “depths” of proton singlet pairing, i.e., we have allowed the protons in the core to pair up to different densities in the core, going from low densities (shallow pairing) up to higher densities (deep pairing). As compared to FSU2 model, we have found that the FSU2R(nuc) and, especially, FSU2H(hyp) only need shallow to moderate proton pairing for a very satisfactory agreement with observed cooling data, particularly with Cas A.

Our calculations indicate that if stellar radii are large (stiff symmetry energy), neutron stars cool down fast for all masses, unless deep proton pairing is active. If stellar radii are small (soft symmetry energy), only heavy neutron stars cool down fast, and just shallow to mild proton pairing is needed for improving the comparison with the cooling data. The better agreement with the data in the calculations using the FSU2R(nuc) and FSU2H(hyp) models suggests that the cooling observations are more compatible with a soft nuclear symmetry energy and, hence, with small neutron star radii. It is nevertheless to be mentioned that there is a tendency in the present calculations to favor rather large stellar masses for explaining the observed colder stars with surface temperatures K.

As for future perspectives, apart from the inclusion of hyperonic pairing, we intend to extend our study of the thermal evolution of neutron stars within this microscopic model to the analysis of the influence of rotation and magnetic fields. Both these effects are known to break the spherical symmetry of the star and could influence the microscopic, macroscopic and thermal properties of the star (Negreiros2012; Negreiros2013b; Negreiros2017).

Acknowledgements

R.N. acknowledges financial support from CAPES and CNPq, as well as that this work is a part of the project INCT-FNA Proc. No. 464898/2014-5. L.T. acknowledges support from the Ramón y Cajal research programme, FPA2013-43425-P and FPA2016-81114-P Grants from Ministerio de Economía y Competitividad (MINECO), Heisenberg Programme of the Deutsche Forschungsgemeinschaft under the Project Nr. 383452331 and PHAROS COST Action CA16214. M.C. and A.R. acknowledge support from Grants No. FIS2014-54672-P and No. FIS2017-87534-P from MINECO, and the project MDM-2014-0369 of ICCUB (Unidad de Excelencia María de Maeztu) from MINECO.

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