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
2.2. Processes in hadronic matter gap which we took as function of density from Fig. 2 of (Schaab et al. 1997). 2.2.1. Emissivity For T
Next, we take into account the modified Urca (MU) NPBF 28 ∗ 7 1/3 processes nn → npeν¯, np → ppeν¯, and the reverse pro- ν =6.610 (mn/mn)(∆n(T )/MeV) u −3 −1 cesses. The final emissivity is given by Friman & Maxwell ×ξI(∆n(T )/T )ergcm s , (15) (1979); Yakovlev et al. (1999) PPBF 28 ∗ 7 2/3 ν =0.810 (mp/mp)(∆p(T )/MeV) u nMU 21 ∗4 1/3 8 −3 −1 × −3 −1 ν =8.610 mnMU(Yeu) ζnMUT9 erg cm s , (12) I(∆p(T )/T )ergcm s , (16) pMU 21 ∗4 1/3 8 −3 −1 ν =8.510 mpMU(Yeu) ζpMUT9 erg cm s . (13) where q p ∗ ∗2 2 ' − Here mi = mrel,i + pF,i is the non-relativistic quasi- I(∆i(T )/T ) 0.89 T/∆i(T ) exp[ 2∆i(T )/T ] , (17) particle effective mass related to the in-medium one- ξ ' 0.5for1S pairing and ξ ' 1for3P pairing. A particle energies from a given relativistic mean field 0 2 significant contribution of the proton channel is due to the model for i = n, p. We have introduced the abbre- ∗4 ∗ 3 ∗ ∗4 NN correlation effect taken into account in Voskresensky viations mnMU =(mn/mn) (mp/mp)andmpMU = ∗ 3 ∗ & Senatorov (1987). (mp/mp) (mn/mn). The suppression factors are ζnMU = ' {− } ' 2 ζnζp exp [∆n(T )+∆p(T )]/T , ζpMU ζp ,andshould be replaced by unity for T>Tcrit,i when for given species 2.2.2. Specific heat i the corresponding gap vanishes. For neutron and pro- For the nucleons (i = n, p), the specific heat is ton S-wave pairing, ∆ (0) = 1.76 T and for P -wave i crit,i (Maxwell 1979) pairing of neutrons ∆n(0) = 1.19 Tcrit,n. To be conser- vative we have used in (12) the free one-pion exchange 20 ∗ 1/3 −3 −1 ci =1.610 (m /mi) u ζMMU T9 erg cm K . (18) estimate of the NN interaction amplitude. Restricting i ourselves to a qualitative analysis we use here simplified The specific heat for the electrons is determined by (6) exponential suppression factors ζi. In a more detailed anal- and that for the photons for T>Tcp is given by (5) with ysis these ζi-factors have prefactors with rather strong the corresponding number of polarizations Nγ =2. temperature dependences (Yakovlev et al. 1999). At tem- peratures T ∼ Tc their inclusion only slightly affects the 2.2.3. Heat conductivity resulting cooling curves. For T Tc the MU process gives in any case a negligible contribution to the total emissiv- The total conductivity is ity and thereby corresponding modifications can again be omitted. Also for the sake of simplicity the general pos- ' 1 1 1 3 κ κn + κe, = + , (19) sibility of a P2(|mJ| = 2) pairing which may result in a κn κnn κnp power-law behaviour of the specific heat and the emissiv- ity of the MU process Voskresensky & Senatorov (1987); where the partial contributions are (Flowers & Itoh 1981; Yakovlev et al. (1999) is disregarded since mechanisms of Baiko & Haensel 1999) this type of pairing are up to now not elaborated. Even 4 3 more essential modifications of the MU rates may pre- 22 mn zn ζn −1 −1 −1 κnn =8.310 ∗ erg s cm K , (20) sumably come from in-medium effects which could result mn Skn T9 2 3 in extra prefactors of 10 ÷10 already at T ∼ Tc. In order S =0.38 z−7/2 +3.7 z2/5 , (21) to estimate the role of the in-medium effects in the NN in- kn n n teraction for the HNS cooling we have also performed cal- m 2 z2ζ T κ =8.91016 n n p 9 erg s−1 cm−1 K−1, (22) culations for the so-called medium modified Urca (MMU) np ∗ 3 mn zpSkp process (Voskresensky & Senatorov 1986; Migdal et al. −2 2 8 −1 1990) by multiplying the rates (12) by the appropriate Skp =1.83 zn +1.43 zn(0.4+zn) . (23) prefactor Here we have introduced the apropriate suppression fac- 0 MMU MU ' 3 6 e8 ' 10/3 tors ζi which act in the presence of gaps for superfluid ν /ν 10 Γ (g )/ω (k pF) u , (14) hadronic matter and we have used the abbreviation zi = 0 1/3 1/3 where the value Γ(g ) ' 1/[1 + 1.4 u ] is due to the (ni/(4n0)) . The heat conductivity of electrons is given dressing of πNN vertices and ωe ≤ mπ is the effective pion by Eq. (9). 564 D. Blaschke et al.: Cooling of hybrid neutron stars
2.3. Processes in the mixed phase 3. Evolution of inhomogeneous emissivity profiles
In some density interval nH −1 respectively. ' 18 R 4 −3 −1 The flux of energy l(r) per unit time through a spher- γ 210 Ts7 erg cm s , (24) 10 km ical slice at the distance r from the center is proportional to the gradient of the temperature on both sides of the 7 where Ts7 is the surface temperature in units of 10 K, spherical slice, using it as the boundary condition in our transport code. r Φ Thus simplifying we assume that the outer crust and the ∂(T e ) − 2M l(r)=−4πr2κ(r) e Φ 1 − , (25) photosphere are thin enough to approximate rm ' R, ∂r r q where R is the star radius. Then, the luminosity at rm −Φ − 2M corresponds to the surface luminosity of the photons from the factor e 1 r corresponds to the relativistic the star l(rm)=Lγ(Ts). correction of the time scale and the unit of thickness. D. Blaschke et al.: Cooling of hybrid neutron stars 565 1 3 ∆ : S & P The equations for energy balance and thermal energy M = 1.4 MO h 0 2 transport are (Weber 1999) 1 ∆ q = 0 ∂ 1 ∂ 2Φ − 2Φ Φ 0.1 le = νe + cV (T e ) , (26) t = 0 ∂A n ∂t -5 10 yr ∂ 1 leΦ T [MeV] 0.01 T e2Φ = − , (27) 0.1 yr ∂A κ 16π2r4n 1 yr 1 5 yr where n = n(r) is the baryon number density, A = A(r)is 50 yr the total baryon number in the sphere with radius r and 100 yr r 0.1 500 yr ∂r 1 − 2M · T [MeV] = 2 1 (28) ∆ = 0.1 MeV ∂A 4πr n r 0.01 q The total neutrino emissivity ν and the total specific heat 1 cV are given as the sum of the corresponding partial con- tributions defined in the previous section for a composition 0.1 ni(r)ofconstituentsi of the matter under the conditions T [MeV] ∆ of the actual temperature profile T (r, t). The accumulated q = 50 MeV mass M = M(r) and the gravitational potential Φ = Φ(r) 0.01 can be determined by 0 0.5 1 r r /rm ∂M ε 2M = 1 − , (29) ∂A n r Fig. 1. Early evolution of temperature profiles for a HNS of M =1.4 M with large (lower panel), small (middle) and ∂Φ 4πr3p + m 1 = q , (30) vanishing (upper panel) diquark pairing gaps ∂A 4πr2n − 2M 1 r where ε = ε(r) is the energy density profile and the pres- For large quark gaps, the hadron phase cools down on sure profile p = p(r) is defined by the condition of hydro- thetimescalet ' 0.1 ÷ 5 yr due to the heat transport to dynamical equilibrium the star surface whereas the quark core keeps the heat dur- ∂p ∂Φ ing this time. Therefore, if an independent measurement = −(p + ε) · (31) ∂A ∂A of the core temperature was possible, e.g. by neutrinos, then a core temperature stall during first several years of The boundary conditions for the solution of (26) and (27) cooling evolution would be a case for quark core super- read l(r =0)=l(A =0)=0andT (A(r )=A, t)= m conductivity with large pairing gaps. Then the quark core T (rm = R, t)=Tm(t), respectively. begins to cool down slowly from its mixed phase bound- In our examples we choose the initial temperature to ary, the cold from hadrons via mixed phase spreads to the be 1 MeV. This is a typical value for the temperature center, demonstrating that the direct neutrino radiations Topacity at which the star becomes transparent for neu- during all the time are unefficient within the quark core. trinos. Simplifying we disregard the neutrino influence on The homogeneous temperature profile is recovered at typ- < ÷ transport. These effects dominate for t ∼ 1 100 min, ical times of t ' 50 ÷ 100 yr. when the star cools down to T ≤ T and become opacity For ∆ =0.1 MeV the QDU processes and the heat unimportant for later times. q conductivity within the quark core are quite efficient. Therefore at very small times (t 10−5 yr) the tem- 4. Temperature evolution of HNS perature profile within the quark core is a homogeneous one whereas during the next 5 ÷ 10 yr the hadron shell 4.1. Temperature profiles T(r,t) is cooled down from the both sides, the mixed phase With the above inputs we have solved the evolution equa- boundary at n(rH)=nH and the star surface. Thus, for tion for the temperature profile. In order to demonstrate ∆q 1 MeV the interior evolution of the temperature the influence of the size of the diquark and nucleon pair- has an influence on the surface temperature from the very ing gaps on the evolution of the temperature profile we early times. Actually the shortest time scale which deter- have performed solutions with different values of the quark mines the slowing of the heat transport within the crust and nucleon gaps. The representative examples are shown is in this case t ∼ 1 yr. Thus at much smaller times one in Fig. 1. Nucleon gaps are taken the same as in Fig. 6 has Ts(t) ' Ts(0) with a drop of T (r, t) in the vicinity of of Schaab et al. (1997). The quark gaps larger than r/rm = 1. Due to the relatively small heat content of the 1 MeV show the same typical behaviour as for the gap crust this peculiar effect is disregarded here. ∆q = 50 MeV. The gaps much smaller than 1 MeV ex- Qualitatively the same behaviour is illustrated for nor- hibit the typical small gap behaviour as for ∆q =0.1MeV mal quark matter with the only difference that quantita- in our example. tively the quark processes are more efficient at ∆q =0and 566 D. Blaschke et al.: Cooling of hybrid neutron stars 7 1 3 equaton of state for hadron matter and smaller values of ∆ [ S & P ] h 0 2 the gaps in the hadronic phase. The unique asymptotic behaviour at log(t[yr]) ≥ 5for 0.2 ∆ h ∆ 6.5 q = 50 MeV all the curves corresponding to finite values of the quark and nucleon gaps is due to a competition between normal electron contribution to the specific heat and the pho- MMU ton emissivity from the surface since small exponentials 6 switch off all the processes related to paired particles. This [K]) s tail is very sensitive to the interpolation law Ts = f(Tm) used to simplify the consideration of the crust. The curves log(T 5.5 ∆ = 0 coincide at large times due to the uniquely chosen rela- q 2/3 tion Ts ∝ Tm . M = 1.4 MO ∆ The curves for ∆q =0.1 MeV demonstrate an interme- q = 0.1 MeV 5 diate cooling behaviour between those for ∆q =50MeV and ∆ = 0. Heat transport becomes not efficient after t q w first 5÷10 yr. The subsequent 104 yr evolution is governed by QDU processes and quark specific heat being only mod- 4.5 erately suppressed by the gaps and by the rates of NPBF -2 -1 0 1 2 3 4 5 6 7 8 processes in the hadronic matter (up to log(t[yr]) ≤ 2.5). log(t[yr]) At log(t[yr]) ≥ 4 begins the photon cooling era. The curves for normal quark matter (∆ = 0) are gov- Fig. 2. Evolution of the surface temperature Ts of HNS with q 2/3 < M =1.4 M for Ts ∝ Tm . Data points are from Schaab erned by the heat transport at times t ∼ 5yrandthenby et al. (1999) (full symbols) and from Yakovlev et al. (2000) QDU processes and the quark specific heat. The NPBF (empty symbols). tw is the typical time which is necessary for processes are important up to log(t[yr]) ≤ 2, the photon the cooling wave to pass through the crust era is delayed up to log(t[yr]) ≥ 7. For times smaller than tw (see Fig. 2) the heat transport is delayed within the crust area (Lattimer et al. 1991). Since we, for simplicity, the cold thereby traverses the hadron phase more rapidly, disregarded this delay in our heat transport treatment, for ' ÷ at typical times t 1 5yr. such small times the curves should be interpreted as the Tm(t) dependence scaled to guide the eye by the same law 2/3 4.2. Evolution of the surface temperature ∝ Tm ,asTs. A detailed comparison of the cooling evolution (log Ts vs. log t) of HNS for different values of quark and hadron gaps 5. Temperature evolution of QCNS > is given in Fig. 2. We have found that the curves for ∆q ∼ 5.1. Temperature profiles T(r,t) 1 MeV are very close to each other demonstrating typical large gap behaviour. As representative example we again As can be seen from the lower panel of Fig. 3 for our take ∆q = 50 MeV. The behaviour of the cooling curve for large gap example (∆q = 50 MeV), where Ye =0isde-6 t ≤ 50÷100 yr is in straight correspondence with the heat termined from equation of state, the QCNS cools down transport processes discussed above. The subsequent time rather smoothly from its surface during the first 300 yr. evolution is governed by the processes in the hadronic shell All the small gap examples exhibit about the same be- and by a delayed transport within the quark core with haviour as the case ∆q =0.1 MeV which we present in the a dramatically suppressed neutrino emissivity from the upper panel of Fig. 3. Opposite to the large gap case, the color superconducting region. In order to demonstrate this transport processes are very efficient and a homogeneous feature we have performed a calculation with the nucleon temperature profile is recovered at t 10−5 yr. In reality gaps (∆i(n),i= n, p) being artificially suppressed by a in the latter case T (rm,t) ' T (rm, 0) at times 1yrwith factor 0.2. Then up to log(t[yr]) ∼< 4 the behaviour of a drop of T (r) in a narrow interval near r/rm =1and the cooling curve is analogous to the one we would have we disregard this peculiar behaviour as we have explained obtained for pure hadronic matter. The curves labelled this above discussing HNS. “MMU” are calculated with the rates of modified Urca processes of Eq. (12) multiplied by the factor (14), i.e. with 5.2. Evolution of the surface temperature inclusion of appropriate medium modifications in the NN interaction. As can be seen from Fig. 2, these effects have In Fig. 4 we demonstrate the evolution of the surface tem- an influence on the cooling evolution only for log(t[yr]) ∼< 2 perature log Ts vs. log t for representative cases. The curve since our specific model equation of state does not allow for ∆q = 50 MeV (Ye =6 0) shows a 300 yr delayed cooling for large nucleon densities in the hadron phase for the which then is controlled by the electron specific heat and M =1.4 M neutron star under discussion. The effect the photon emissivity from the surface, as it was stated would be more pronounced for larger star masses, a softer in Blaschke et al. (2000). However, in difference with the D. Blaschke et al.: Cooling of hybrid neutron stars 567 1 emissivity and the quark specific heat, both being only moderately suppressed whereas late time asymptotics, as ∆ = 0.1 MeV q in the large gap example, relates to the photon era. A simi- 0.1 lar trend is present for normal quark matter staying below t = 0 . -5 the corresponding curves for ∆q =01 MeV due to the ab- T[MeV] 10 yr sence of the ζ suppression factors in the former case. At 0.01 0.1 yr 8 1 yr t>10 yr both cases have a different photon era asymp- 5 yr totic behaviour whereas all finite gap curves have the same 50 yr 0.001 large time asymptotics when Ye =0.6 1 100 yr 500 yr At Ye = 0 the QDU processes are absent and the large time asymptotics are governed by the gluon-photon spe- cific heat which is a nonlinear function of the temperature and the photon emissivity. Due to this nonlinearity the 0.1 T[MeV] large time asymptotics are different in all the cases under consideration. All the results for Ye = 0 (including case of ∆ = 50 MeV q large gaps) coincide with those obtained in Blaschke et al. (2000) although the sharp fall of the curve for the large 0.01 0 0.5 1 gap case is in reality controlled by the heat conductivity r /rm of the crust that will lead to a cooling delay within the first year of evolution (Pizzochero 1991) as in Fig. 2 the Fig. 3. Early evolution of temperature profiles for a QCNS of curves for t 7.5 6. Conclusion ∆ = 50 MeV = 0 Ye For the CFL phase with large quark gap, which was ex- 7 pected to exhibit the most prominent manifestations of colour superconductivity in HNS and QCNS, we have ∆ = 0.1 MeV found an essential delay of the cooling during the first 6.5 Y = 0 e 50÷300 yr due to a dramatic suppression of the heat con- ductivity in the quark matter region. This delay makes the [K]) s 6 cooling of QCNS not as rapid as one could expect when ignoring the heat transport. In HNS compared to QCNS log(T (large gaps) an additional delay of the subsequent cool- 5.5 ing evolution comes from the processes in pure hadronic matter. In spite of that we found still too fast cooling ∆ = 0 for those objects. Therefore, with the CFL phase of large Ye = 0 5 M = 1.4 M quark gap it seems rather difficult to explain the major- O t w ity of the presently known data both in the cases of the HNS and QCNS, whereas in the case of pure hadronic 4.5 -10 -8 -6 -4 -2 0 2 4 6 8 stars the available data are much better fitted within the log(t[yr]) same model for the hadronic matter we used here. For 2SC (3SC) phases we expect analogous behaviour to that Fig. 4. Evolution of the surface temperature Ts for the QCNS demonstrated by ∆ = 0 since QDU processes on unpaired ∝ 2/3 q with M =1.4 M for Ts Tm quarks are then allowed. We however do not exclude that new observations may lead to lower surface temperatures corresponding curve of Blaschke et al. (2000) we have a for some supernova remnants and will be better consistent much larger typical cooling time scale (t ∼ 300 yr) since with the model which also needs further improvements. in that work the simplifying approximation of a homoge- On the other hand, if future observations will show very neous temperature profile has been used. For the low gap large temperatures for young compact stars they could be limit ∆q 10 MeV in this particular QCNS case there interpreted as a manifestation of large gap color supercon- is no heat transport delay and our results coincide with ductivity in the interiors of these objects. those obtained in Blaschke et al. (2000). Unimportant dif- Acknowledgements. Research of H.G. and D.B. was supported ferences are only due to the inhomogeneous density profile in part by the Volkswagen Stiftung under grant no. I/71 226 and a different value of Ye(n) which follows in our case and by DFG under grant no. 436 ARM 17/1/00. H.G. ac- from an actual calculation for the given equation of state. knowledges the hospitality of the Department of Physics at The curve corresponding to ∆q =0.1 MeV (Ye =6 0) from the University of Rostock where this research has been per- very early times is due to a competition between QDU formed. D.N.V. is grateful for hospitality and support of GSI 568 D. Blaschke et al.: Cooling of hybrid neutron stars Darmstadt. D.B. acknowledges support of the Department of Kaminker, A. D., & Haensel, P. 1999, Acta Phys. Pol. B, 30, Energy for his participation in the INT program on “QCD 1125 at Nonzero Baryon Density” (INT-00-1) at the University of Lattimer, J. M., Pethick, C., Prakash, M., & Haensel, P. 1991, Washington as well as a fellowship at the ECT∗ Trento where Phys. Rev. Lett., 66, 2701 this work was completed. We thank Th. Kl¨ahn for his contri- Maxwell, O. V. 1979, ApJ, 231, 201 butions during early stages of this work; M. Colpi, M. Prakash, Migdal, A. B., Saperstein, E. 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