Neutron star cooling within the equation of state with induced surface tension
NNeutron star cooling within the equation of state with induced surface tension
S. Tsiopelas, V. Sagun ∗ CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal
We study the thermal evolution of neutron stars described within the equation of state withinduced surface tension (IST) that reproduces properties of normal nuclear matter, fulfills the pro-ton flow constraint, provides a high-quality description of hadron multiplicities created during thenuclear-nuclear collision experiments, and is equally compatible with the constraints from astrophys-ical observations and the GW170817 event. The model features strong direct Urca processes for thestars above 1 . M (cid:12) . The IST equation of state shows a very good agreement with the availablecooling data, even without introducing nuclear pairing. We also analysed an effect of the singletproton/neutron and triplet neutron pairing on the cooling of neutron stars of different mass. Wedemonstrate a full agreement of the predicted cooling curves with the experimental data. Moreover,the IST EoS provides a description of Cas A with both paired and unpaired matter.Keywords: neutron stars, equation of state, cooling I. INTRODUCTION
Born out of supernova explosions, neutron stars (NSs)are considered to start their life having very high in-ternal temperatures and cool down through a combina-tion of thermal radiation from their surface and neutrinoemission from their interior. From the first day of theirlives, when the temperature of their interior has alreadydropped from ∼ K to ∼ − K making it trans-parent to neutrinos, up until the first million years oftheir existence, thermal energy is carried away mainly inthe form of neutrino radiation. During this time, mea-surements of surface temperature and luminosity of thestars can provide significant information about the prop-erties of matter in their depth, since the thermal evolu-tion depends on factors such as the internal compositionand the thermodynamic properties of matter are definedby the Equation of State (EoS), the chemical abundancesof the envelope and the type of pairing between the con-stituent particles. It is, therefore, necessary for any the-oretical calculations aiming to describe the cooling pro-cess of NSs to consider the various combinations betweenthose factors. The set of cooling curves extracted fromthese simulations can then be tested against the observ-able features of NSs in the X-ray part of the spectrum.The particle composition of the core of NSs plays aprominent role in their cooling process, since it is the de-cisive factor of whether the direct Urca (DU) process ofneutron β -decay and its inverse process are allowed to oc-cur in the interior of the star. These are the most efficientneutrino-emitting reactions that can happen in hadronicmatter and, when permitted, they lead to a rapid coolingof the NS, substantially different from the case they areforbidden [17, 19]. For these reactions to be possible, theFermi momenta of the participating particles must satisfythe kinematic restriction of triangle inequality. Takinginto account charge neutrality and the relation betweenthe Fermi momenta and the number density of each par- ∗ Emails: [email protected], [email protected] ticle, this constraint leads to the requirement that theproton fraction should be higher than ∼
11% of the totalbaryon density [17].Since the composition ultimately depends on the sym-metry energy of nuclear matter, the choice of the EoSstrongly affects the calculations regarding the thermalevolution of NSs. In this paper, we will use the EoS for-mulated within the framework of induced surface tension(IST) with its original purpose being the description ofexperimental data referring to nuclear and hadron mat-ter. The recent results show it can be equally applied tomodel the properties of symmetric nuclear matter, anal-ysis of hadron yields created at heavy-ion collision fromthe AGS to ALICE energies, as well as to describe com-pact astrophysical objects [22–24, 27]. The IST EoS is,furthermore, free of any restraints on the number of in-cluded particle species, which makes it highly appropriatefor studying the properties of strongly interacting mat-ter at high densities. A comprehensive analysis of themodel can be found in Refs. [13, 22]. For a realistic de-scription of the outer layers of the NS the IST EoS issupplemented by the Haensel-Zdunik (HZ) EoS for theouter crust and the Negele-Vautherin (NV) EoS for theinner crust [11, 15].An additional important factor worth considering isthe superfluidity (superconductivity) of neutrons (pro-tons) in the NS interior, since they are able to alter thethermal evolution of the star [32]. Despite suppressingthe rates of neutrino emission from the rest of the pro-cesses, pairing between protons or neutrons introduces anew ν e − emitting mechanism due to the permanent for-mation and breaking of Cooper pairs, known as PBF[19]. This neutrino emitting process is activated in eachbaryon species as soon as the temperature reaches therespective critical temperatures T c,n , T c,p . Nevertheless,the details such as the onset T, the form of the gap pa-rameter and the peak of the PBF emissivity are stronglysensitive to the gap model adopted to describe the matterin this supercritical state. As a result, applying differentmodels on the investigation of NS cooling provides alsothe possibility of gaining a further insight on the behav-ior of paired matter. a r X i v : . [ a s t r o - ph . H E ] J un Based on the IST EoS, in this work we study the ther-mal evolution of NSs, aiming to describe the availablecooling data. Investigation is focused on the implicationsof employing various gap models to describe neutron su-perfluidity and proton superconductivity. In addition, weexamine the effect of envelope composition on the coolingprocess of compact stars. Modeling the thermal evolutionof NSs was performed using the
NSCool code [17, 18].The paper is organized as follows: In Sec. II we presenta brief description of the EoS model. In Sec. III wediscuss the neutrino emission mechanisms taking place ina NS as well as the pairing models used in the simulations.Sec. IV is dedicated to the results of the calculationswhile Sec. V contains the summary and the conclusions.
II. EQUATION OF STATE
The IST EoS is formulated to include neutrons, pro-tons and electrons. It accounts for strong short rangerepulsion and relatively weak long-range attraction be-tween nucleons, while electrons are treated as an idealFermi gas. The former part of the nucleon-nucleon inter-action is modelled with the hard core radius, similarly tothe famous Van der Waals EoS. The hard core repulsionof nucleons leads to an appearance of excluded volume.However, contrary to the Van der Waals approximation,the IST EoS instead of the constant excluded volume hasa density-dependent one. The key element of the modelis the IST coefficient. It accounts not only for the correctvalues of four virial coefficients of hard spheres, but alsoextends the causality range of the model to the densityrange typical for the NSs [22, 27]. Moreover, the long-range attraction and asymmetry between neutrons andprotons are accounted via the mean-field potentials, pa-rameters of which were fitted to the properties of matterat saturation density. More detailed information aboutthe model and its application to the NSs can be found inRefs. [22, 23, 27].We adopt the softer parameterisation of the IST EoSthat corresponds to the set B of Ref. [21] (for simplicitywe keep referring to this parameterisation of the modelas set B). The set B was chosen due to the better agree-ment with the astrophysical constraints and results com-ing from the GW170817 [1]. It gives the values of thesymmetry energy E sym = 30 . L = 93 . K = 201 . M max = 2 . M (cid:12) is consistent with the recent measure-ments of the most massive NSs, i.e. PSR J0348+0432 [4]and PSR J0740+6620 [9]. III. COOLING PROCESSES
The thermal evolution of NSs can be divided in twostages. During the first one, known as neutrino cooling
GW170817
PSR J0348+0432PSR J0740+6620 M / M ⊙ FIG. 1: Mass-radius relation for non-rotating NSs calculatedfor set B of the IST EoS [21]. Horizontal bands correspond tothe two most massive NSs, e.g. PSR J0348+0432 [4] (magentaband) and PSR J0740+6620 [9] (blue band). The shaded greyarea represents the M-R constraint taken from Refs. [29, 30],while the constraint depicted as a cyan area was taken fromRef. [16]. The red line represents the allowed range of NSradius, according to GW170817 event. era, ν e emission generated from a plethora of emissionmechanisms throughout the whole interior of the stardominates the cooling process [17, 19]. On a timescale ofa million years after the NS formation, when T core hasdropped below 10 K , neutrino emission from the core isless efficient due to the strong temperature dependence,and photon emission from the surface overtakes as theleading heat loss mechanism. Beyond this point, theneutrino cooling era is over and the photon coolingera begins. This shift is marked by an abrupt declineof the total luminosity and surface temperature of thestar, with more massive stars exhibiting steeper dropsof those quantities than less massive ones, since photonluminosity follows the typical blackbody radiation law.During the neutrino cooling era, the leading neutrino-generating process varies over the regions of the starand over time. Overall, the main factors regulatingthe neutrino emissivity of each process are density,temperature and the existing degree of Cooper pairingbetween particles. For example, in the envelope of thestar pair annihilation is the most productive mechanism,while in the outer crust neutrino emission is dominatedby plasmon decay until its temperature reaches a fewtimes 10 K , replaced by electron-ion bremsstrahlungbeyond that point [19, 32]. The latter remains the mostefficient energy loss mechanism throughout the innercrust as well, as long as the neutrons of this region arepaired [14]. Even in their supercritical state though,inner crust neutrons contribute to the total neutrinoemission through the PBF process over a narrow rangeof densities and temperatures [19].In the core of the star, apart from the bremsstrahlungprocesses between the free particles, the main neutrino-emitting processes that can occur are the DU process ofthe neutron β -decay and its inverse: n → p + e + ¯ ν e , p + e → n + ν e . (1)As it was mentioned above, the realization of these re-actions depends on the proton fraction in the star inte-rior. For the IST EoS adopted in this work, these fastprocesses can only proceed in the core of stars with cen-tral densities n c higher than n DU = 0 . f m − . ForNSs with lower n c , where the Fermi momenta of the in-volved particles fail to satisfy the kinematic restriction of p F,n ≤ p F,p + p F,e , neutron β -decay and its inverse canstill proceed with the help of a spectator proton or neu-tron that provides the extra p F,i needed for the conser-vation of momentum. In this case, though, the efficiencyof these so-called modified Urca (MU) processes in re-moving heat from the star is lower than that of the DUones, resulting in an overall slower cooling of the star.Nevertheless, neutrino emission from MU processes sur-passes the rest of the emissivities in the core, unless nand p are in a paired state, which in turn results in afurther suppression of the neutrino emission rates. How-ever, pairing partially compensates for the delay in thecooling of the NS core as it introduces PBF, an additionalefficient channel for carrying the heat away in the formof neutrino-antineutrino pairs [19].Cooper pairing of core protons and crust neutrons setsin a few years after the NS birth, while core neutronspairing is viable at a later time. As regards the typesof pairing, all of them follow the standard BCS pattern,with free neutrons of the inner crust and protons of thecore undergoing singlet-state pairing ( S ), while neu-trons of the core are expected to pair in the triplet-state( P ) [5, 6, 17]. Although a complete, precise descriptionof the effect for densities relevant to NS matter is stillpending, it is widely applied for investigating its imprinton the thermal evolution of compact stars. The transi-tion of neutrons(protons) to superfluid(superconducting)state inflicts a suppression on their specific heat as well.Most studies agree that once it switches on for a certainspecies, c υ,i is reduced by a factor R ∼ e − ∆ /T , where∆ is the gap parameter, linked to the respective criti-cal temperature T c via the standard BCS pairing rela-tion T c ≈ .
57∆ [5, 6, 17]. A further drop in tempera-ture, causes a further reduction of heat capacity to thepoint of being equal to that of leptons [19]. Concern-ing the neutrino emissivity of processes involving pairedbaryons, suppression is induced as well, since particlesin such a supercritical state have to overcome the en-ergy gap, in order to interact with another particle. Inthe range of temperatures T (cid:28) T c , this behavior is il-lustrated numerically using a yet another set of controlfunctions R χ ( T /T c ), which differs between each particle -3 -2 -1 n b /n T c / [ K ] SFBAOT72 n b /n T c / [ K ] AOCCDK
FIG. 2: Density dependence of the critical temperature forthe considered singlet and triplet neutron gaps (upper panel)and proton singlet pairing gaps (lower panel). species χ [32].The onset temperatures T c,n ( T c,p ) of superfluidity (su-perconductivity), the associated ∆ and the profile of PBFneutrino emissivity vary over the several models devel-oped to describe the paired matter. Therefore, when in-cluding nucleon pairing for simulating the thermal evo-lution of NSs, the effect is subject to the gap models em-ployed. Our choice for the simulations were the followingmodels: SFB [28] for the S superfluidity of neutrons,T72 [31] and AO [2] for their P superfluidity, and AO[3], CCDK [8] for the proton S channel in different com-binations. The critical temperatures T c were calculatedaccording to the phenomenological formula suggested byKaminker et al. and the parameters used by Ho et al.[12, 14].The relation between the critical temperature and thebaryon density for all the adopted gap models is shown onFig.2. According to the SFB model, S pairing of crustneutrons appears first at densities around 0 . n , oncethe crust temperature has dropped below ∼ · K .As the crust keeps cooling, the region of superfluid crustneutrons expands both outwards and towards the crust-core interface. Regarding superconductivity of protonsin the core, CCDK model suggests that the proton pair-ing occurs initially at a temperature not greater than7 · K in the layer corresponding to ∼ n . On thecontrary, AO model proposes that the onset temperatureis a few times lower, at 2 · K , while the associatedbaryon density only slightly lower. Finally, as claimedby the T72 model, the threshold for the onset of P su-perfluidity of core neutrons is below 10 K and pairingoccurs in a narrow, symmetric region centered around thenormal nuclear density n . The respective AO model, onthe other hand, although it does not differ considerablyin terms of T c,n , implies that neutron pairing can occurthroughout the whole region of the core.The composition of the envelope is another decidingfactor for the surface photon luminosity of the star, whichis ultimately the quantity of observational importance.Namely, heavier elements tend to delay heat transportfrom the outer crust to the surface, since in this casethe electron thermal conductivity is reduced [33]. Thestandard approach of thermal evolution codes is to set anenvelope model as an outer boundary condition that linksthe temperature at the bottom of the envelope ( T b ) to thetemperature of the stellar surface ( T s ). The fundamentalassumptions behind this method are that the envelopehas a thermal relaxation timescale which is much shorterthan that of the crust, and that the neutrino emissivityin the envelope is negligible [17]. In this work, we usedtwo distinct envelope models: one composed of heavyelements (Fe) and a hydrogen-rich one that contains thefraction of light elements η = ∆ M/M = 10 − [20]. IV. RESULTS t (yr)5.25.45.65.86.06.26.46.66.87.0 l o g T ∞ s ( K )
12 3 4 5 678 910 1112 1314 15161718000000000 M fl M fl M fl M fl M fl FIG. 3: Cooling curves for stars of different mass
M/M (cid:12) =1 . , . , . , . , . T ∞ S de-notes the surface temperature at infinity. The solid curvescorrespond to the light-element envelope ( η = 10 − ), whilethe dashed curves were obtained for the Fe envelope. Thedata points are taken from [7]. At the beginning we focused on the thermal evolutionof NSs without any sort of pairing between the nucle-ons. As you can see on Fig.3 all the cooling curves de-picted in color exhibit a slow cooling due to the dom-ination of the MU processes, since the DU ones are not kinematically allowed. Finally, at central densitiesof over n DU = 0 . f m − of beta-stable and chargeneutral matter, which correspond to NSs with masses M ≥ . M (cid:12) , the DU processes are switched on in thecore of the star so that it undergoes enhanced cooling(see the black curves on Fig.2). In order to model theuncertainties of the heat-blanketing effect of the enve-lope, we compare the thermal evolution of NSs with anon-accreted envelope containing heavy elements (dashedcurves on Fig.3) with the envelope containing light ele-ments (solid curves on Fig.3).Remarkably, the obtained cooling curves for unpairedmatter describe the experimental data very good. In ad-dition, we would like to stress the description of the NS inthe Cassiopeia A (Cas A) supernova remnant, that cor-responds to the star noted as 0 on Fig.2. Namely, we findthat Cas A can be equally described by both a rapidlycooling 2 M (cid:12) star with a light-elements envelope and aslow cooling low-mass star with a Fe envelope. t (yr)5.25.45.65.86.06.26.46.66.87.0 l o g T ∞ s ( K )
12 3 4 5 678 910 1112 1314 15161718000000000 M fl M fl M fl M fl M fl M fl M fl t (yr)5.25.45.65.86.06.26.46.66.87.0 l o g T ∞ s ( K )
12 3 4 5 678 910 1112 1314 15161718000000000 M fl M fl M fl M fl M fl M fl M fl FIG. 4: The same as Figure 3, but considering the effect ofneutron superfluidity in the S channel via the SFB model[28] and proton superconductivity in the S channel withthe AO model [2] (upper panel) and CCDK model [8] (lowerpanel). This result thus leaves an equal possibility for bothcooling scenarios. As a matter of fact, Cas A is a highlydebatable object; as was discussed in many papers in-cluding Refs.[34-37], some data of Cas A indicate a rapid t (yr)5.25.45.65.86.06.26.46.66.87.0 l o g T ∞ s ( K )
12 3 4 5 678 910 1112 1314 15161718000000000 M fl M fl M fl M fl M fl t (yr)5.25.45.65.86.06.26.46.66.87.0 l o g T ∞ s ( K )
12 3 4 5 678 910 1112 1314 15161718000000000 M fl M fl M fl M fl M fl FIG. 5: The same as lower panel of Figure 4, but supple-mented with the triplet neutron pairing in the core of thestar described by the T72 model [31] (upper panel) and theAO model [3] (lower panel). cooling with a few percent drop of the surface temper-ature across 10 years of observations, while other datasuggest a slower cooling.Furthermore, we studied the effect of neutron superflu-idity and proton superconductivity on the thermal evo-lution of NSs in two stages. First, we considered n S superfluidity with the SFB model [28] together with ap S superconductivity described by AO [3] and CCDK[8] models (see Fig.4). For both considered combinationsof gaps, i.e. SFB+AO (see the upper panel on Fig.4) andSFB+CCDK (see the lower panel on Fig.4), the IST EoSis in very good agreement with the experimental data.As in the non-superfluid case, the model predicts the sur-face temperature for the Cas A both with the fast coolingcurve for M DU = 2 . M (cid:12) , as well as with the curves forlow-mass stars with a heavy-elements envelope. For theSFB+CCDK gaps the Cas A is only described with acurve for M = 1 . M (cid:12) also with an Fe envelope.Finally, we studied the effect of the neutron tripletpairing on the cooling of NSs by considering a shallow gap with a maximum of the critical temperature at thesaturation density n (T72 model [31]) and an extendedgap with a maximum of T c at 2 n (AO model [3]). Asit is shown on Fig.5 the combination of models SFB(for n S )+T72(for n P )+CCDK (for p S ) (upper panel)and SFB(for n S )+AO(for n P )+CCDK (for p S )(lower panel) give qualitatively similar result. Addingthe n P pairing in the core of NSs leads to more rapidcooling and makes it incompatible with most of the ob-servational data. Therefore, we conclude that within ourmodel, neutron pairing in the triplet channel is inconsis-tent with the experimental data. V. CONCLUSIONS
We studied the cooling of NSs with the novel IST EoSthat was previously applied to the analysis of nuclearmatter properties, heavy-ion collision experimental data,and was recently generalized for the description of matterinside NSs. The considered model parameterisation is ingood agreement with the GW170817 and the recent mea-surements of the most massive NSs. For the stars with M (cid:62) . M (cid:12) the model allows for the occurrence of DUprocesses that leads to much faster cooling. An effect ofneutron pairing in S and P channels, as well as pro-ton pairing in S channel is thoroughly analysed withdifferent superfluidity/superconductivity models. In ad-dition, effects of different types of envelopes on the ther-mal evolution of NSs were also compared. We show thatfor unpaired matter the obtained cooling curves are fullyconsistent with the currently existing observational data.Remarkably, the IST EoS with unpaired matter predictsthe surface temperature of Cas A both with the fast cool-ing curve for M DU = 2 . M (cid:12) with the accreted envelopeof light-elements and the low-mass star with the heavy-elements envelope.The presence of the proton(neutron) superconductiv-ity(superfluidity) in the singlet channel slightly slowsdown the cooling and makes the agreement with the dataeven better. The two considered scenarios of the tripletneutron pairing in the core of the star (a shallow and anextended neutron superfluidity) lead to too rapid coolingof all NSs, such that the old stars ( t > yr) cannot bereproduced. Thus, we can conclude that the calculationsfavours the vanishing neutron triplet pairing gap. VI. ACKNOWLEDGEMENTS
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