Cooling of neutron stars with diffusive envelopes
M. V. Beznogov, M. Fortin, P. Haensel, D. G. Yakovlev, J. L. Zdunik
aa r X i v : . [ a s t r o - ph . H E ] A ug MNRAS , 1–8 (2016) Preprint 27 September 2018 Compiled using MNRAS L A TEX style file v3.0
Cooling of neutron stars with diffusive envelopes
M. V. Beznogov ⋆ , M. Fortin , P. Haensel , D. G. Yakovlev , J. L. Zdunik St Petersburg Academic University, 8/3 Khlopina st., St Petersburg 194021, Russia Nicolaus Copernicus Astronomical Center, Bartycka 18, Warsaw 00-716, Poland Ioffe Institute, 26 Politekhnicheskaya st., St Petersburg 194021, Russia
Accepted . Received ; in original form
ABSTRACT
We study the effects of heat blanketing envelopes of neutron stars on their cooling.To this aim, we perform cooling simulations using newly constructed models of theenvelopes composed of binary ion mixtures (H–He, He–C, C–Fe) varying the mass oflighter ions (H, He or C) in the envelope. The results are compared with those calcu-lated using the standard models of the envelopes which contain the layers of lighter(accreted) elements (H, He and C) on top of the Fe layer, varying the mass of accretedelements. The main effect is that the chemical composition of the envelopes influencestheir thermal conductivity and, hence, thermal insulation of the star. For illustration,we apply these results to estimate the internal temperature of the Vela pulsar andto study the cooling of neutron stars of ages of 10 − yr at the photon coolingstage. The uncertainties of the cooling models associated with our poor knowledge ofchemical composition of the heat insulating envelopes strongly complicate theoreticalreconstruction of the internal structure of cooling neutron stars from observations oftheir thermal surface emission. Key words: dense matter – plasmas – diffusion – stars: neutron – stars: evolution
The long-standing problem of modern studies of neutronstars is to investigate the properties of superdense mat-ter in their cores. One of a few methods to achievethis goal is to study the thermal evolution of neutronstars (particularly, cooling of isolated stars) and com-pare the theoretical models with the available observa-tional data (e.g., Yakovlev & Pethick 2004; Page et al. 2009;Potekhin, Pons & Page 2015 and references therein). Differ-ent models of superdense matter predict different rates ofneutrino cooling of neutron star interiors and, therefore, dif-ferent surface temperatures as they are directly related tothe internal temperatures. This allows one to select mostsuitable models of superdense matter from observations. Itis a challenging task for many reasons, but we mainly focuson one of them. In order to explore the properties of super-dense matter one needs to calculate (constrain) the internaltemperature of the star from observations of its thermal sur-face emission.We will restrict ourselves to not too young neutronstars (of age t & −
100 yr) which are thermally equili-brated and isothermal inside. The initial internal equilibra-tion mainly consists in the equilibration between the crust ⋆ E-mail: [email protected] and the core of a star due to distinctly different microphysicsthere (e.g., Lattimer et al. 1994; Yakovlev et al. 2001 andreferences therein). After the equilibration the main tem-perature gradient in these stars still persists in a thin outerheat blanketing envelope with rather poor thermal conduc-tion. It is usually sufficient to assume that this envelope ex-tends from the atmosphere bottom to the layer of the density ρ b ∼ g cm − . Its thickness does not exceed a few hun-dred meters, and its mass is . − − − M ⊙ . On the otherhand, its mass cannot be smaller than the mass of the neu-tron star atmosphere (typically ∼ − − − M ⊙ ). Let T b be the temperature at the bottom density ρ = ρ b . Althoughstrong magnetic fields in the envelopes of neutron stars canaffect the insulating properties of the envelopes and createan anisotropic distribution of the effective surface temper-ature T s (e.g. Potekhin et al. 2003, 2015), we neglect sucheffects in the present paper. Our results cannot be used tostudy the thermal structure and evolution of neutron starswith very strong fields, particularly, magnetars. Even in thiscase the relation between T s and T b , required for coolingsimulations and data analysis, is uncertain, mainly becauseof uncertain thermal conduction in the blanketing envelopesdue to their unknown chemical composition.Although the chemical composition of the heat blan-keting envelopes is really uncertain, there are some natu-ral limitations (see, e.g., Potekhin et al. 1997 and references c (cid:13) M. V. Beznogov et al. therein). For instance, at high temperatures T and/or den-sities ρ hydrogen transforms into helium due to thermo- orpycno-nuclear reactions and beta captures. Approximately,this happens at T & × K and/or ρ & g cm − .At higher T & K and/or ρ & g cm − , helium, inits turn, transforms into carbon. At still higher T & K and/or ρ & g cm − carbon transforms into heavierelements.The standard model of heat blanketing envelopes is themodel developed by Potekhin, Chabrier & Yakovlev (1997)and elaborated by Potekhin et al. (2003). In this model(hereafter, the PCY97 model), the envelope is onion-like,composed of shells of pure elements with abrupt boundariesbetween the shells. The number of the shells is determinedby a single parameter, ∆ M , the accumulated mass of lightelements (H, He, C). The width of each shell is limited bynuclear transformations of light elements. In the absence oflight elements, the envelope is purely iron. In the presenceof a thick layer of light elements, the envelope consists ofconsecutive shells of hydrogen, helium, carbon and iron. Al-though this model has proved to be useful, it relies on theassumption of abrupt boundaries between the layers of dif-ferent chemical species and employs a specific dependenceof the composition on the accumulated mass (governed bynuclear reactions and beta-captures).Recently we have developed new models of diffusiveheat blanketing envelopes (Beznogov, Potekhin & Yakovlev2016) which consist of binary ion mixtures (either H–He,or He–C, or C–Fe) with a variable mass ∆ M of light el-ements (either H, or He or C, respectively). They can bediffusively equilibrated or not. Since the ions have a ten-dency for separation, such an envelope consists of a top layerof lighter ions, a bottom layer of heavier ions, and a tran-sition layer in between. While considering the equilibratedenvelopes we have used proper relations for the ion diffusivecurrents taking into account the Coulomb coupling of theions and the presence of temperature gradients. Note thatBeznogov et al. (2016) have also introduced a characteristictransition density ρ ∗ from a lighter element to a heavier oneinstead of the mass of light elements ∆ M . There is a one toone correspondence between ∆ M and ρ ∗ .Let us stress that according to Beznogov et al. (2016)the separation between the H and He ions, as well as be-tween the C and Fe ones is rather strong (due to the gravi-tational force) leading to a narrow separation layer. In con-trast, the separation between the He and C ions (which havealmost the same charge-to-mass ratios) is slower (due to theCoulomb force), resulting in a wider transition layer. In anycase, even at not very small deviations from diffusive equilib-rium, the T s − T b relation is rather insensitive to the struc-ture of the transition zone and depends almost solely on∆ M . The T s − T b relations have been calculated assumingdifferent values of ρ b = 10 , 10 and 10 g cm − (which aresuitable for different cooling problems; see Beznogov et al.2016). For convenience of using in the computer codes, nu-merical results have been approximated by analytic expres-sions. To visualize the properties of the new heat blanket-ing envelopes, Beznogov et al. (2016) plotted (their figs. 1–7) representative profiles of particle fractions and temper-atures in the envelopes as the functions of the density aswell as the appropriate T s − T b and T b − ∆ M relations. Notethat those figures corresponded to a neutron star model with mass M = 1 . ⊙ and radius R = 10 km; the same modelwill be used to interpret the observations of the Vela pulsarin Section 3.Here we apply new envelope models to simulate cool-ing of the isolated middle-aged ( t ∼ − yr) neutronstars. We will use the PCY97 model as a reference for com-parison. In Section 2 we present some formation scenariosof accreted envelopes. In Sections 3–5 we outline some cal-culations of internal structure and cooling of neutron starswith new envelope models. Our conclusions are formulatedin Section 6. The composition of heat blanketing envelopes of neutronstars is uncertain. It can greatly vary depending on the for-mation history of the star and its evolutionary scenario.Initially, it was thought that the envelopes (as well asneutron star atmospheres) contain heavy elements like iron,as a result of the formation of the envelope in a hot andvery young star where light elements burnt-out into heav-ier ones. However, a more detailed analysis of the observedspectra of the thermal radiation originating from the atmo-spheres of neutron stars has shown that some spectra arebetter described by blackbody models (of iron atmospheremodels) while others are better approximated by hydrogenatmosphere models (see, e.g., Potekhin 2014 and referencestherein). Moreover, the spectrum of the neutron star in theCas A supernova remnant is well described by a carbon at-mosphere model (Ho & Heinke 2009). The same is true forthe neutron star in the supernova remnant HESS J1731–347(Klochkov et al. 2013).Therefore, the surface composition of thermally emit-ting neutron stars can be different. Naturally, the composi-tion of the underlying envelopes can also be different. Thecomposition of the surface layers can be affected by the fall-back of material on the neutron star surface after the su-pernova explosion, by the accretion of hydrogen and heliumfrom interstellar medium or from a binary companion (ifthe neutron star is or was in a compact binary), by theion diffusion and the nuclear evolution in the neutron starenvelope, and by other effects. A neutron star can directlyaccrete hydrogen and helium (e.g., Blaes et al. 1992). Alter-natively, helium can be produced in nuclear reactions afterthe accretion of hydrogen. Helium can also burn further intocarbon (see, e.g., Rosen 1968; Chang et al. 2010). In somecases the reverse process of spallation of heavier elementsinto lighter ones is possible. There are also indications thatsome transiently accreting neutron stars in low mass X-raybinaries in quiescent states have their outer envelopes com-posed of H and He which are left after an active accretionphase (Brown et al. 2002).All in all, the composition of neutron star envelopes islargely unknown. It seems instructive to consider differentmodels of the envelopes and to analyse observational mani-festations of such models.
MNRAS , 1–8 (2016) ooling of neutron stars g T M K log ∆ M/ M ⊙ H–HeHe–C C–FePCY97 M = 1 . ⊙ R = 10 km T ∞ s = 0 .
68 MK
STANDARD COOLINGMODERATELY ENHANCED COOLING l og f ℓ log ∆ M/ M ⊙ H–HeHe–C C–FePCY97 M = 1 . ⊙ R = 10 km T ∞ s = 0 .
68 MK
Figure 1.
Internal temperature e T (left-hand panel) and neutrino cooling function f ℓ (right-hand panel) of the Vela pulsar for H–He,He–C, and C–Fe models of heat blanketing envelopes versus mass ∆ M of the lighter element in the binary mixture. For comparison, wepresent also e T and f ℓ obtained using the envelope model of Potekhin et al. (1997) (PCY97) versus the mass ∆ M of accreted elements.The level of the standard neutrino cooling f ℓ = 1 on the right-hand panel refers to a non-superfluid star which cools via the modifiedUrca process. The level f ℓ ≈ of moderately enhanced neutrino cooling can be provided by neutrino emission due to moderatelystrong neutron superfluidity in the core. See text for details. l og g T [ K ] log T ∞ s [K] FeCHe∆ M [M ⊙ ]=10 − − − − − M = 1 . ⊙ R = 10 km T ∞ s = . M K C – F e H e – C l og g T [ K ] log T ∞ s [K] FeAcc∆ M [M ⊙ ]=10 − − − − − M = 1 . ⊙ R = 10 kmPCY97 T ∞ s = . M K Figure 2.
Thermal states ( e T versus T ∞ s ) of the Vela pulsar with M = 1 . ⊙ and R = 10 km in the past and the future (on theright and left of the vertical dotted lines, respectively) for different chemical compositions of heat blanketing envelopes. Left-hand panel:He–C and C–Fe envelopes. The thick lines correspond from top to bottom to pure Fe, C and He envelopes, respectively. The thinnerdifferent dashed lines are for binary mixtures with different mass ∆ M of lighter elements. Right-hand panel: the PCY97 envelope. Thethick lines are for pure Fe and pure accreted matter; the thinner dashed lines are for different mass of the accreted matter. For givenenvelope models at a fixed T ∞ s the internal temperature e T increases when ∆ M decreases.MNRAS , 1–8 (2016) M. V. Beznogov et al.
Having a theoretical T s − T b relation and the values of T s in-ferred from observations one can calculate the internal tem-perature of a neutron star.Let us illustrate this by taking the Vela pulsar as anexample. It is a middle-aged pulsar (with the characteristicpulsar age ≈
11 kyr). Its internal thermal relaxation oc-curred long ago, so that its redshifted internal temperature e T is constant over the pulsar interior (excluding the heatblanketing envelope). Using the magnetic hydrogen atmo-sphere model and taking a gravitational mass M = 1 . ⊙ and a circumferential radius of the star R = 10 km [withan apparent radius R ∞ = R/ √ − x g = 13 km, where x g =2 GM/ ( Rc )] Pavlov et al. (2001) inferred the redshifted ef-fective surface temperature T ∞ s = 0 . ± .
03 MK (at 68 percent confidence level). To be specific, we employ T ∞ s = 0 . T s = T ∞ s / √ − x g = 0 .
888 MK.Using a T s − T b relation we can immediately calculate T b andthe redshifted internal temperature e T = T b √ − x g . Notethat the Vela pulsar possesses a magnetic field B ∼ × G, while we, following Beznogov et al. (2016), neglect theeffects of magnetic fields on the heat blanketing envelope(as already mentioned above). We do it for simplicity andillustration. In addition, such magnetic fields do not affectstrongly the T s − T b relations (Potekhin et al. 2003).The left-hand panel of Fig. 1 shows the inferred inter-nal Vela’s temperature e T determined for a number of en-velope models. The short-dashed line corresponds to e T forthe He–C envelope with ρ b = 10 g cm − as a functionof ∆ M = ∆ M He . The long-dashed line is the same for theC–Fe envelope versus mass of carbon, ∆ M = ∆ M C . Thedot-dashed line is for the H–He envelope with ρ b = 10 g cm − (at higher ρ b He starts to burn in pycno-nuclear re-actions) versus mass of hydrogen, ∆ M = ∆ M H . The lineis extended only to ∆ M H ∼ − M ⊙ to which hydro-gen can survive in dense matter (e.g. Section 1). Finally,the solid PCY97 curve shows e T for the H–He–C–Fe enve-lope of Potekhin et al. (1997) versus the mass ∆ M of ‘ac-creted’ elements (H+He+C). The line for the H–He envelopeis plotted up to ∆ M ∼ − M ⊙ , for the He–C envelope– up to ∆ M ∼ − M ⊙ , two other lines are plotted up to∆ M = 10 − M ⊙ , an approximate value of ∆ M to whichcorresponding envelopes can survive; see Section 1.For larger ∆ M we have a more heat transparent en-velope with smaller internal temperature e T for the samesurface temperature T ∞ s . An exception from this rule is pro-vided by the H–He envelope where the situation is invertedas explained by Beznogov et al. (2016). One can see that thevariations of e T with ∆ M , indeed, prevent the accurate deter-mination of e T if the envelope composition is unknown. Thestrongest variations are seen to occur for the PCY97 model,which takes into account a wider range of elements. In thecase of binary mixtures, the variations become smaller, andthey are especially small for the H–He and He–C mixtures.Since the Vela pulsar is at the neutrino cooling stagewith an isothermal interior, its internal temperature e T de-termines (e.g., Yakovlev et al. 2011; Weisskopf et al. 2011)the fundamental parameter of superdense matter in its core,which is the neutrino cooling function ℓ ( e T ) = L ∞ ν ( e T ) /C ( e T ) , (1) where L ∞ ν is the redshifted neutrino luminosity of the star,and C is its heat capacity; both quantities are mainly deter-mined by the star’s core. The convenient unit of ℓ ( e T ) = ℓ ( e T ) SC ∝ e T is provided by the so-called standard neu-trino candle. It corresponds to a non-superfluid star whichcools via the modified Urca processes of neutrino emission.Yakovlev et al. (2011) as well as Ofengeim et al. (2015) ob-tained analytic approximations for ℓ ( e T ) SC calculated for anumber of neutron star models with different masses andnucleonic equations of state (EOSs) in the core. These ap-proximations are universal (almost independent of the EOS)and more or less equivalent (Ofengeim et al. 2015). Theypermit a model-independent analysis of the thermal statesof neutron stars. Such an analysis has been performed pre-viously for the Crab pulsar (Weisskopf et al. 2011); for theneutron star in the Cas A supernova remnant neglectingand including its possible rapid cooling in the present epoch(Yakovlev et al. 2011; Shternin & Yakovlev 2015); and forthe neutron star in the HESS J1731–347 supernova remnant(Ofengeim et al. 2015).Let us perform similar analysis for the Vela pulsar. Tak-ing possible values of e T from the right-hand panel of Fig. 1we can reconstruct ℓ ( e T ). Using the theoretical relations de-rived by Yakovlev et al. (2011) or Ofengeim et al. (2015) weassume that the neutrino cooling function of the Vela pulsarbehaves as ℓ ( e T ) ∝ e T . Then we can determine ℓ ( e T ) for anyvalue of e T and find f ℓ = ℓ ( e T ) /ℓ ( e T ) SC , (2)which is the Vela’s neutrino cooling function expressed interms of standard candles. This analysis is valid for a widerange of physical scenarios including (i) the standard candlecooling, f ℓ = 1; (ii) slower cooling through nucleon-nucleonbremsstrahlung of neutrino pairs if the modified Urca pro-cess is suppressed by strong neutron or proton superfluidity(0 . . f ℓ < < f ℓ . ).It is well known that the Vela pulsar cools some-what faster than the standard candle (e.g., Page et al. 2004;Yakovlev & Pethick 2004) so that f ℓ >
1. The values of f ℓ derived from the values of e T are plotted on the right-handpanel of Fig. 1 as a function of ∆ M . In order to infer f ℓ wehave used theoretical formulae from Ofengeim et al. (2015),but the formulae from Yakovlev et al. (2011) would give sim-ilar results. As on the left-hand panel, the lines of differenttypes refer to the different models of heat insulation in theVela’s envelope. The dependence of f ℓ on the chemical com-position of the envelope is seen to be very strong. By in-creasing the mass of accreted matter in the PCY97 modelto the maximum possible value ∆ M ∼ − M ⊙ we increase f ℓ from about 10 to 2 × . Similarly, increasing the massof carbon in the C–Fe envelope we vary f ℓ from about 10 to 2 . × . Finally, by increasing the mass of He in theHe–C envelope or the mass of H in the H–He envelope wecan vary f ℓ within about one decade around f ℓ ∼ .These results indicate once more that the chemical com-position of the envelope is of great importance for studyingthe internal structure of neutron stars. In addition, the re-sults for the Vela pulsar allow us to draw an important con-clusion. Specifically, let us assume that we wish to describethe cooling of isolated neutron stars using the minimal cool- MNRAS , 1–8 (2016) ooling of neutron stars ing theory (Page et al. 2004; Gusakov et al. 2004). In thistheory, neutron stars have a nucleon core, the powerful di-rect Urca process of neutrino emission (Lattimer et al. 1991)is forbidden, and the cooling enhancement over the standardneutrino candle is provided by the neutrino emission dueto a moderately strong triplet-state Cooper pairing of neu-trons. The minimal cooling theory states that in this casethe enhancement is limited to f ℓ . . Within the mini-mum cooling paradigm, according to the right-hand panelof Fig. 1, the Vela pulsar cannot possess H–He or He–C en-velopes. Its envelope should be mostly composed of iron.Its core should mainly contain moderately strong neutronsuperfluidity to ensure the maximum neutrino emission en-hancement f ℓ ∼ . The Vela pulsar is the isolated neutronstar coldest for its age. Its cooling regime should be similarto that of the Cas A neutron star if the current rapid cool-ing of the Cas A star is real (Heinke & Ho 2010; Page et al.2011; Shternin et al. 2011; Elshamouty et al. 2013; see, how-ever, Posselt et al. 2013 for the alternative view on the rapidcooling of the Cas A star). The Cas A neutron star is justyounger but could become as cold as Vela in about 10 kyr.On the other hand, we can adopt another cooling the-ory which would allow for the existence of stronger neutrinoemission (e.g., Yakovlev & Pethick 2004) in the Vela pulsar,for instance, due to direct Urca process or due to similar pro-cesses enhanced, for instance, by pion condensation. Thenthe composition of the envelope would become again ratheruncertain which would strongly complicate theoretical anal-ysis of the internal structure of the Vela pulsar.In Fig. 2 we show thermal states of the Vela pulsar in thepast and the future for the same assumptions of M = 1 . ⊙ and R = 10 km as in Pavlov et al. (2001) and for differentmodels of the envelopes from Fig. 1. The thermal statesare characterized by the values of e T versus T ∞ s . The ver-tical dotted lines refer to the present epoch ( t = 11 kr, T ∞ s = 0 .
68 MK). The left-hand panel corresponds to theC–Fe and He–C envelopes. The thick lines are for the en-velopes of pure elements : Fe (long-dashed line), C (solidline), and He (dash-dotted line). The thin dashed lines withdifferent dash separations refer to binary mixtures with dif-ferent masses of lighter elements, ∆ M/ M ⊙ = 10 − , 10 − ,10 − , 10 − and 10 − . The lowest ∆ M corresponds to avery thin outer layer of lighter element while the largest∆ M to the envelopes with a very thin bottom layer of heav-ier element. One can observe the evolution of e T ( T ∞ s ) withincreasing ∆ M from the e T ( T ∞ s ) dependence for pure heav-ier to pure lighter element. The older the star (the smaller T ∞ s ) the smaller mass of lighter element affects e T .The right-hand panel of Fig. 2 shows the thermal statesof the Vela pulsar for the PCY97 envelopes. The upper thickline is again for pure Fe while the lower line is for purelyaccreted matter. The thin lines refer to different masses ofaccreted matter. Note that the normalized neutrino coolingfunction f ℓ (in units of standard candles) does not evolve intime as long as ℓ ( e T ) ∝ e T but remains the same as plottedon the right-hand panel of Fig. 1. Detailed calculations of neutron star cooling with new en-velope models are outside the scope of this paper. Let usoutline some selected results.To be specific, we restrict ourselves to the minimal cool-ing paradigm (Page et al. 2004; Gusakov et al. 2004) men-tioned above. Then the main regulators of neutron star cool-ing are (i) the neutrino emission level, f ℓ (which can varyfrom ∼ − to ∼ depending on superfluidity of neu-trons and protons in the core) and (ii) the composition ofthe envelope. Fixing f ℓ and the envelope model but vary-ing ∆ M , we obtain a sequence of the cooling curves T ∞ s ( t ).As a rule these curves are almost ‘universal’, independentof the EOS in the core and of the star’s mass M . This isdemonstrated, for instance, in figs. 24–26 of Yakovlev et al.(2001) for the case of standard neutrino candles ( f ℓ = 1, nosuperfluidity) and iron envelopes.For illustration, let us assume f ℓ = 1 and focus on the ef-fect of the envelopes. Let us choose one neutron star model ofmass M = 1 . ⊙ with the BSk21 EOS (Goriely et al. 2010;Pearson et al. 2012; Potekhin et al. 2013). The stellar radiuswill then be R = 12 .
60 km, and the direct Urca process ofneutrino emission will be forbidden. The four panels of Fig.3 show bands of the cooling curves for such a star having dif-ferent envelopes (from left to right: H–He, He–C, C–Fe, andPCY97, respectively). Computations have been done usingour general relativistic cooling code (Gnedin et al. 2001) andcross checked with the ‘NScool’ cooling code by D. Page .The results of comparison are satisfactory: calculated cool-ing curves differ slightly only at t & t . yrs refer to the initialthermal relaxation within the star (e.g., Yakovlev & Pethick2004).A band on each panel of Fig. 3 is restricted by upperand lower cooling curves (corresponding to almost pure Heand H; He and C; C and Fe; acc and Fe, respectively; ‘acc’refers to a fully accreted PCY97 envelope). Short-dashedlines show some intermediate cooling curves for a few valuesof ∆ M to demonstrate that the bands are actually filled bycooling curves with different ∆ M . Note that different binarymixtures are considered at different ρ b (the same as used inSection 3). Accordingly the He cooling curve for the H–Heenvelope is somewhat different from the He curve for theHe–C envelope.Varying ∆ M for the same envelope model, we change T ∞ s ( t ). This can be treated as the ‘broadening’ of the coolingcurve (because, as a rule, ∆ M is unknown). As seen fromFig. 3, for the H–He and He–C envelopes this broadeningseems weak. However, for the C–Fe and PCY97 envelopes(where the properties of various ion species, particularly,their thermal insulation differ stronger) the broadening islarge and prevents the determination of the internal tem-perature e T from observations. ‘NScool’ cooling code is available at .MNRAS , 1–8 (2016) M. V. Beznogov et al.
HeH l og T ∞ s [ K ] log t [yr] . × − ∆ M H / M ⊙ = H – He
HeC log t [yr] . × − ∆ M He / M ⊙ = He – C
CFe log t [yr] . × − . × − ∆ M C / M ⊙ = C – Fe
AccFe log t [yr] . × − . × − . × − ∆ M acc / M ⊙ = PCY97
Figure 3.
Cooling curves (redshifted effective surface temperature T ∞ s versus age t ) for a 1.4 M ⊙ non-superfluid neutron star with theBSk21 EOS, different chemical compositions of heat blanketing envelopes (from left to right: H–He, He–C, C–Fe, PCY97) and differentaccumulated masses of lighter elements. The curves for the envelopes with maximum ∆ M , containing almost entirely lighter elements,correspond (from left to right) to ∆ M H ∼ − M ⊙ , ∆ M He ∼ − M ⊙ , ∆ M C ∼ − M ⊙ and ∆ M acc ∼ − M ⊙ , respectively. Thecurves for the envelopes with virtually no lighter elements are calculated for ∆ M ∼ − M ⊙ . See text for details. Another important feature of the cooling curves for different∆ M in Fig. 3 is their inversion at certain t when the band ofthe curves becomes thin and then wider again. The inversionis accompanied by the interchange of the cooling curves.For instance, before the inversion on the right-hand panelthe lowest cooling curve is for the iron envelope while afterthe inversion the iron envelope produces the highest coolingcurve. The inversion epoch changes from t ≈ yr for thelighter H–He and He–C envelopes to (2 − × yr for theheavier C–Fe and PCY97 envelopes.These inversions are well known in the literature (e.g.Yakovlev & Pethick 2004). They manifest the transitionfrom the neutrino cooling stage to the photon cooling stage.The transition period is relatively short. The transition hasa dramatic impact on the cooling process. At the neutrinocooling stage, a star cools via neutrinos from the interiorand looks colder for a more insulating envelope composedof heavier elements. At the photon cooling stage, the starcools via photons from the surface. The neutrino emissionbecomes insignificant for the cooling process, and the cool-ing is governed by the heat capacity of the core and the heattransparency of the envelope. More insulating envelopes ofheavier elements produce hotter stars.Fig. 4 shows a selection of theoretical cooling curvesfrom Fig. 3 that are obtained for a 1.4 M ⊙ nonsuperfluid starwith the BSk21 EOS in the core. The four thick curves cor-respond to the envelopes made of pure Fe (the long-dashedline), C (the solid line), He (the dot-dashed line) and of pureaccreted matter in the envelope model of PCY97 (the short-dashed line). The space between the Fe and C curves is filledby the cooling curves for the C–Fe envelope with different∆ M . The space between the C and He curves is filled by thecooling curves for the He–C envelope. The space between theFe and acc curves is covered by the cooling curves for thePCY97 envelope. To simplify Fig. 4 we do not present theresults for the H–He envelope which can be easily visualizedfrom the left-hand panel of Fig. 3.In addition, Fig. 4 presents the observational data onthe isolated middle-aged neutron stars. The data are the
12 3 45 67 89 1011 1213141516 171819 l og T ∞ s [ K ] log t [yr] BSk21 1.4 M ⊙ FeCHePCY97 Acc
Figure 4.
Cooling curves for a 1.4 M ⊙ non-superfluid (standardneutrino candle) neutron star with the BSk21 EOS. The thicklines correspond to envelopes of pure Fe, C, He, as well as of pureaccreted matter in the PCY97 model. The filled area between theFe and C curves is covered by cooling curves for C–Fe envelopeswith different ∆ M ; similarly, the space between the C and Hecurves can be covered with cooling curves for He–C envelopes.The area between the Fe and acc curves is covered by coolingcurves for the PCY97 envelopes. The cooling curves are comparedwith the observations of isolated neutron stars. See the text fordetails. same as those presented in Beznogov & Yakovlev (2015a,b);Ofengeim et al. (2015); references to original publicationscan also be found there. Neutron star labels are as fol-lows, (1) PSR J1119–6127; (2) RX J0822–4300 (in Pup A);(3) PSR J1357–6429; (4) PSR B0833–45 (Vela); (5) PSRB1706–44; (6) PSR J0538+2817; (7) PSR B2334+61; (8)PSR B0656+14; (9) PSR B0633+1748 (Geminga); (10) PSRB1055–52 ; (11) RX J1856.4–3754; (12) PSR J2043+2740; MNRAS , 1–8 (2016) ooling of neutron stars (13) RX J0720.4–3125; (14) PSR J1741–2054; (15) XMMUJ1732–3445; (16) Cas A neutron star; (17) PSR J0357+3205(Morla); (18) PSR B0531+21 (Crab); (19) PSR J0205+6449(in 3C 58).First let us outline briefly the sources which are at theneutrino cooling stage ( t . yr). Recall that for a stan-dard neutrino candle with an iron envelope we would haveone Fe (thick long-dashed ) cooling curve which cannot ex-plain many data. As seen from Fig. 4, even varying thecomposition of the envelope of the standard candle we canexplain much more sources, although not all of them. Theexplanation of other sources at the neutrino cooling stagewould require deviations from the standard neutrino candle(from f ℓ = 1). For instance, the hottest XMMU J1732–3445source can be explained assuming nearly maximum amountof carbon in the envelope and strong proton superfluidity inthe core ( f ℓ ∼ .
01, Klochkov et al. 2015; Ofengeim et al.2015). The coldest sources at the neutrino cooling stage,like the Vela pulsar, can be interpreted, for instance, as neu-tron stars with moderately strong triplet-state neutron pair-ing in the core which increases the neutrino cooling level to f ℓ ∼ (Section 3).Now let us focus on the sources at the photon cool-ing stage (after the inversion of the cooling curves). It isnot a surprise (Fig. 4) that all these sources are compatiblewith the standard neutrino cooling candle (see below). Thevariety of such sources can be explained by different com-position of their envelopes. The hottest neutron stars at thephoton cooling stage (like PSR B1055–52, Pavlov & Zavlin2003; PSR J2043+2740, Zavlin 2009; RX J0720.4–3125,Motch et al. 2003) should have their envelope made pre-dominantly of iron while the coldest stars (like Geminga,Kargaltsev et al. 2005; RX J1856.4–3754, Ho et al. 2007;Potekhin 2014) may have envelopes of lighter elements (forinstance, carbon).Note that although the evolution of neutron stars at thephoton cooling stage does not depend directly on their neu-trino emission, the observed sources with t & yr shouldnot have f ℓ ≫
1. Otherwise they would cool rapidly at theneutrino cooling stage, transit to the photon cooling stageearlier and would become very weak at t & yrs. There-fore, the hottest neutron stars at the photon cooling stageseem to be those which have f ℓ . T ∞ s ( t ) for non-superfluid neutron stars of differentmasses which cool via modified Urca process ( f ℓ = 1) andhave iron heat blanketing envelopes but no strong magneticfields, merge in almost one and the same cooling curve. Thesame is also true for the stars with fully accreted PCY97heat blanketing envelopes although the curve becomes dif-ferent. We have checked that this property survives for thenew envelope models of Beznogov et al. (2016).The above analysis has neglected possible mechanisms of neutron star reheating, for instance, due to ohmic decayof magnetic fields, possible violations of beta-equilibrium,etc.; e.g., Yakovlev & Pethick (2004); Page et al. (2006) andreferences therein. Were these mechanisms operative theywould be able to keep neutron stars warmer. No such mech-anisms seem to be required for ordinary cooling middle-agedneutron stars. We have outlined the effects of our new models for heat blan-keting envelopes (Beznogov et al. 2016) of neutron stars onthe cooling and thermal structure of isolated middle-agedneutron stars. The new envelopes are composed of binaryionic mixtures (either H–He, or He–C, or C–Fe) with anyallowed mass ∆ M of lighter elements. The results are com-pared with the standard PCY97 models of the envelopescontaining shells of H, He, C, and Fe with any possible massof ‘accreted’ (H+He+C) elements (Potekhin et al. 1997). Asdiscussed in Beznogov et al. (2016), the new models allowone to consider wider classes of the envelopes.In Section 2 we have outlined some formation scenariosof the envelopes. In Section 3 we have considered the effectsof the envelopes on inferring the internal temperatures e T ofneutron stars from observations and on constraining theirneutrino cooling function f ℓ (the fundamental parameter ofsuperdense matter in a neutron star core). We have takenthe Vela pulsar as an example. The results confirm previ-ous conclusions (e.g., Yakovlev et al. 2011; Weisskopf et al.2011; Klochkov et al. 2015; Ofengeim et al. 2015) that thecomposition of the heat blanketing envelope is a major in-gredient for the correct interpretation of observations. Theuncertainty in the envelope composition translates into a fac-tor of ∼ uncertainty in f ℓ . Nevertheless, since the Velapulsar is sufficiently cold (Pavlov et al. 2001), we have beenable to conclude that within the minimal cooling scenariothe pulsar should have f ℓ ∼ and the envelope predomi-nantly made of iron.In Sections 4 and 5 we have performed some cooling cal-culations for a 1.4 M ⊙ neutron star with different envelopesand compared the results with observations. A special em-phasis has been made on neutron stars of ages t ∼ . − f ℓ .
1) but possess various envelopes mostlycontaining carbon and iron.Our consideration is definitely not complete. More workis required to overcome the problem of heat blanketing en-velopes in the theory of thermal evolution of neutron stars.In particular, more complicated models of the envelopes canbe constructed taking into account multicomponent ion mix-tures in and out of diffusive equilibrium; the dynamical evo-lution of the diffusive equilibrium can also be modelled. Inaddition, one can elaborate the existing models of diffusivenuclear burning in the envelopes (e.g., Chang & Bildsten2003, 2004; Chang et al. 2010) which is neglected here. Itwould also be very important to include the effects of mag-
MNRAS , 1–8 (2016)
M. V. Beznogov et al. netic fields (Potekhin et al. 2015) on the envelopes of var-ious types. Any additional reliable information on the for-mation history and evolution of the envelopes would also bemost welcome. However, all these problems go far beyondthe scope of the present investigation.
ACKNOWLEDGEMENTS
The work of MB was partly supported by the Dynasty Foun-dation, the work of DG by the Russian Foundation for BasicResearch (grants 14-02-00868-a and 16-29-13009-ofi-m), andthe work of MF, PH, and LZ was supported by the PolishNCN research grant no. 2013/11/B/ST9/04528.
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