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Cooling of Hybrid Neutron Stars and Hypothetical Self-Bound Objects with Superconducting Quark Cores

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Cooling of Hybrid Neutron Stars and Hypothetical Self-Bound Objects with Superconducting Quark Cores A&A 368, 561–568 (2001) Astronomy DOI: 10.1051/0004-6361:20010005 & c ESO 2001 Astrophysics Cooling of hybrid neutron stars and hypothetical self-bound objects with superconducting quark cores D. Blaschke1,2,H.Grigorian1,3, and D. N. Voskresensky4 1 Fachbereich Physik, Universit¨at Rostock, Universit¨atsplatz 1, 18051 Rostock, Germany 2 European Centre for Theoretical Studies ECT∗, Villa Tambosi, Strada delle Tabarelle 286, 38050 Villazzano (Trento), Italy 3 Department of Physics, Yerevan State University, Alex Manoogian Str. 1, 375025 Yerevan, Armenia e-mail: [email protected] 4 Moscow Institute for Physics and Engineering, Kashirskoe shosse 31, 115409 Moscow, Russia Gesellschaft f¨ur Schwerionenforschung GSI, Planckstrasse 1, 64291 Darmstadt, Germany e-mail: [email protected] Received 19 September 2000 / Accepted 12 December 2000 Abstract. We study the consequences of superconducting quark cores (with color-flavor-locked phase as repre- sentative example) for the evolution of temperature profiles and cooling curves in quark-hadron hybrid stars and in hypothetical self-bound objects having no hadron shell (quark core neutron stars). The quark gaps are varied from 0 to ∆q = 50 MeV. For hybrid stars we find time scales of 1 ÷ 5, 5 ÷ 10 and 50 ÷ 100 years for the formation of a quasistationary temperature distribution in the cases ∆q =0,0.1MeVand∼>1 MeV, respectively. These time scales are governed by the heat transport within quark cores for large diquark gaps (∆ ∼> 1 MeV) and within the hadron shell for small diquark gaps (∆ ∼< 0.1 MeV). For quark core neutron stars we find a time scale '300 years for the formation of a quasistationary temperature distribution in the case ∆ ∼> 10 MeV and a very short one for ∆ ∼< 1 MeV. If hot young compact objects will be observed they can be interpreted as manifestation of large gap color superconductivity. Depending on the size of the pairing gaps, the compact star takes different paths in the log(Ts)vs.log(t) diagram where Ts is the surface temperature. Compared to the corresponding hadronic model which well fits existing data the model for the hybrid neutron star (with a large diquark gap) shows too fast cooling. The same conclusion can be drawn for the corresponding self-bound objects. Key words. dense matter – stars: interiors – stars: evolution – stars: neutron 1. Introduction allow for unpaired quarks of one color whereas in the color- flavor locking (CFL) phase all the quarks are paired. The interiors of compact stars are considered as systems Estimates of the cooling evolution have been per- where high-density phases of strongly interacting matter formed (Blaschke et al. 2000) for a self-bound isother- do occur in nature, see Glendenning (1996) and Weber mal quark core neutron star (QCNS) which has a crust (1999) for recent textbooks. The consequences of differ- but no hadron shell, and for a quark star (QS) which has ent phase transition scenarios for the cooling behaviour of neither crust nor hadron shell. It has been shown there in compact stars have been reviewed recently in comparison the case of the 2SC (3SC) phase of QCNS that the con- with existing X-ray data, see Page (1992), Schaab et al. sequences of the occurrence of gaps for the cooling curves (1997). are similar to the case of usual hadronic neutron stars (en- A completely new situation might arise if the scenarios hanced cooling). However, for the CFL case it has been suggested for (color) superconductivity (Alford et al. 1998; shown that the cooling is extremely fast since the drop in ∼ Rapp et al. 1998) with large diquark pairing gaps (∆q the specific heat of superconducting quark matter dom- ÷ 50 100 MeV) in quark matter are applicable to neutron inates over the reduction of the neutrino emissivity. As star interiors. Various phases are possible. The two-flavor has been pointed out there, the abnormal rate of the tem- (2SC) or the three-flavor (3SC) superconducting phases perature drop is the consequence of the approximation of homogeneous temperature profiles the applicability of Send offprint requests to:D.Blaschke, which should be limited by the heat transport effects. Page e-mail: [email protected] et al. (2000) estimated the cooling of hybrid neutron stars Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20010005 562 D. Blaschke et al.: Cooling of hybrid neutron stars (HNS) where heat transport effects within the supercon- belowpTcrit,q we use the interpolation formula ∆(T )= ducting quark core have been disregarded. Neutrino mean ∆(0) 1 − T/Tcrit,q, with ∆(0) being the gap at zero tem- free paths in color superconducting quark matter have perature. been discussed in Carter & Reddy (2000) where a short The contribution of the reaction ee → eeνν¯ is very period of cooling delay at the onset of color superconduc- small (Kaminker & Haensel 1999) tivity for a QS has been conjectured in accordance with ee =2.81012 Y 1/3u1/3T 8 erg cm−3 s−1, (3) the estimates of Blaschke et al. (2000) in the CFL case for ν e 9 small gaps. but it can become important when quark processes are In the present paper we want to consider a more de- blocked out for large values of ∆q/T in superconducting tailed scenario for the thermal evolution of HNS and quark matter. QCNS which includes the heat transport in both the quark and the hadronic matter. We will demonstrate how long it takes for the HNS and for the QCNS to establish a quasis- 2.1.2. Specific heat tationary temperature profile and then we consider the in- For the quark specific heat we use an expression (Iwamoto fluence of both the diquark pairing gaps and the hadronic 1982) gaps on the evolution of the surface temperature. 21 2/3 −3 −1 cq ' 10 u ζS T9 erg cm K , (4) ' 5/2 − 2. Processes in HNS and QCNS where ζS 3.1(Tcrit,q/T ) exp( ∆q/T ). Besides, one should add the gluon-photon contribution (Blaschke et al. 2.1. Processes in quark matter 2000) 13 3 −3 −1 2.1.1. Emissivity cg−γ =3.010 Ng−γ T9 erg cm K , (5) A detailed discussion of the neutrino emissivity of quark where Ng−γ is the number of available massless gluon- matter without the possibility of color superconductivity photon states (which are present even in the color super- has been given first by Iwamoto (1982). In his work the conducting phase), as well as the electron one, quark direct Urca (QDU) reactions d → ueν¯ and ue → dν 19 2/3 2/3 −3 −1 ce =5.710 Ye u T9 erg cm K . (6) have been suggested as the most efficient processes. Their emissivities have been obtained as1 2.1.3. Heat conductivity QDU ' 26 1/3 6 −3 −1 ν 9.410 αsuYe ζQDU T9 erg cm s , (1) The heat conductivity of the matter is the sum of partial where at a compression u = n/n0 ' 2 the strong cou- contributions (Flowers & Itoh 1981) ≈ pling constant is αs 1 and decreases logarithmically X 1 X 1 at still higher densities. The nuclear saturation density κ = κi, = , (7) −3 κi κij is n0 =0.17 fm , Ye = ne/n is the electron fraction, i j 9 and T9 is the temperature in units of 10 K. If for some- where i, j denote the components (particle species). For what larger density the electron fraction was too small quark matter κ is the sum of the partial conductivities ' −8 (Ye <Yec 10 ), then all the QDU processes would of the electron, quark and gluon components (Haensel & be completely switched off (Duncan et al. 1983) and the Jerzak 1989; Baiko & Haensel 1999) neutrino emission would be governed by two-quark re- actions like the quark modified Urca (QMU) and the κ = κe + κq + κg, (8) → quark bremsstrahlung (QB) processes dq uqeν¯ and where κ ' κ is determined by electron-electron scatter- → e ee q1q2 q1q2νν¯, respectively. The emissivities of the QMU ing processes since in superconducting quark matter the and QB processes have been estimated as (Iwamoto 1982) partial contribution 1/κeq (aswellas1/κgq) is addition- QMU ∼ QB ' 19 8 −3 −1 ally suppressed by a ζQDU factor, as for the scattering on ν ν 9.010 ζQMU T9 erg cm s . (2) impurities in metallic superconductors. For κee we have Due to the pairing, the emissivities of QDU processes 23 −1 −1 −1 −1 κee =5.510 uYe T9 erg s cm K , (9) are suppressed by a factor ζQDU ∼ exp(−∆q/T )andthe emissivities of QMU and QB processes are suppressed and by a factor ζQMU ∼ exp(−2∆q/T )forT<Tcrit,q ' κq ' κqq 0.4∆q whereas for T>Tcrit,q these factors are equal r to unity. The modification of Tcrit,q(∆q)relativetothe 23 4π −1 −1 −1 −1 ' 1.110 uζQDUT9 erg s cm K , (10) standard BCS formula is due to the formation of correla- αs tions as, e.g., instanton- anti-instanton molecules (Rapp where we have accounted for the suppression factor. The et al. 2000). For the temperature dependence of the gap contribution of massless gluons we estimate as 1 Please notice that the numerical factors in all the esti- κ ' κ mates below depend essentially on rather unknown factors and g gg ' 17 2 −1 −1 −1 thereby can be still varied. 6.010 T9 erg s cm K . (11) D. Blaschke et al.: Cooling of hybrid neutron stars 563 2.2. Processes in hadronic matter gap which we took as function of density from Fig.
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