New Astronomy 15 (2010) 515–519

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New Astronomy

journal homepage: www.elsevier .com/locate /newast

Signal of quark deconfinement in thermal evolution neutron starswith deconfinement heating

Miao Kang a,*, Xiao-Dong Wang a, Xiao-Ping Zheng b

a The College of Physics and Electronics, Henan University, Kaifeng 475004, PR Chinab The Institute of Astrophysics, Huazhong Normal University, Wuhan 430079, PR China

a r t i c l e i n f o

Article history:Received 3 August 2009Received in revised form 16 December 2009Accepted 16 December 2009Available online 23 December 2009

Communicated by E.P.J. van den Heuvel

Keywords:Stars: neutronStars: rotationEquation of state

1384-1076/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.newast.2009.12.009

* Corresponding author. Fax: +86 0378 3881602.E-mail address: kangmiao07@gmail.com (M. Kang

a b s t r a c t

As neutron stars spin-down and contract, the deconfinement phase transition can continue to occur,resulting in energy release (so-called deconfinement heating) in case of the first-order phase transition.The thermal evolution of neutron stars is investigated to combine phase transition and the related energyrelease self-consistently. We find that the appearance of deconfinement heating during spin-down resultin not only the cooling delay but also the increase of surface temperature of stars. For stars characterizedby intermediate and weak magnetic field strength, a period of increasing surface temperature could exist.Especially, a sharp jump in surface temperature can be produced as soon as quark matter appears in thecore of stars with a weak magnetic field. We think that this may serve as evidence for the existence ofdeconfinement quark matter. The results show that deconfinement heating facilitates the emergenceof such characteristic signature during the thermal evolution process of neutron stars.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Fundamental properties of supranuclear matter in the cores ofneutron stars, such as the chemical composition and the equationof state, are still poorly known. Simulations of the thermal evolu-tion of neutron stars confronted with soft X-ray, extreme UV, andoptical observations of thermal photon flux emitted from their sur-face provide the most valuable information about the dense matterin the interior of these stars.

As the interior density gradually increases of neutron stars,deconfinement phase transition continuously takes place inducingnot only structural changes but also energy release in case of afirst-order phase transition. The generation of energy increasesthe internal energy of the star which is called deconfinement heat-ing (Haensel and Zdunik, 1991; Yu and Zheng, 2006; Kang andZheng, 2007). The thermal evolution of neutron stars is connectedwith their spin-down and the resulting changes in structure andchemical composition (from nucleon matter to deconfined quarkmatter) have been investigated in our work. We have investigatedthe thermal evolution of neutron stars with such a deconfinementphase transition (Kang and Zheng, 2007). The results show thatdeconfinement heating delays dramatically the cooling of neutron

ll rights reserved.

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stars, which have a higher surface temperature compared with tra-ditional cooling for the same age stars.

Many efforts are devoted to explore the observational signal of adeconfinement phase transition which have been suggested in theform of characteristic changes of observables, such as thepulse timing (Glendenning et al., 1997; Chubarian et al., 2000;Poghosyan et al., 2001), brightness (Dar and DeRújula, 2000) andsurface temperature (Schaab et al., 1996; Blaschke et al., 2001;Yuan and Zhang, 1999; Stejner et al., 2008) of the pulsars duringtheir evolution. The surface temperature changes when the quarkmatter appears in the cores of neutron stars. We explore thedeconfinement signature by studying the changes of surface tem-perature in the thermal evolution process of neutron stars.

In this paper, we reinvestigate the thermal evolution of neu-tron stars and look for a characteristic change in the surface tem-perature with deconfinement heating. The released energy canalso be estimated as a function of the change rate of the decon-finement baryon number using the parameterized approach. Theneutron stars containing quark matter are called hybrid stars.We take Glendenning’s hybrid stars model (Glendenning, 1997)based on the perturbation theory developed by Hartle (1967) tostudy the rotational evolution structure of stars. The ArgonneV18þ dtþ UIX� model (APR) (Akmal et al., 1998) of hadronicmatter and the MIT bag-model of quark matter are used to con-struct the model of stars, but the medium effect of quark matterhas been considered in quasi-particle description (Schertler et al.,1997).

Fig. 1. For beta-stable matter, the energy per baryon for the ArgonneV18þ dtþ UIX� model, and quark effective mass MIT bag-model are shown byfull and dashed curves (from top to bottom bag constant B = 136,108 and85 MeV fm�3, respectively); the dotted lines correspond to neutral mixtures ofcharged hadron and quark matter.

Fig. 2. Gravitational mass M in solar masses as a function of the central density forrotating neutron star configurations with a deconfinement phase transition. Thelower curve correspond to static configurations. the upper one to those withmaximum rotation frequency mk . The lines between both extremal cases connectconfigurations with the same total baryon number. The dotted lines indicate thatthe quark matter is produced in the core of neutron stars. The bag constant of thequark matter is B ¼ 108 MeV fm�3.

516 M. Kang et al. / New Astronomy 15 (2010) 515–519

2. Deconfinement phase transition and neutron stars structure

Early works on deconfinement phase transition, the Maxwellconstruction, show a sharp transition taking place between thetwo charge neutral hadron and the quark phases (Baym and Chin,1976). In the 1990s, Glendenning (1992, 1997) pointed out thatthis assumption was too restrictive. More generally, the transitioncan occur through the formation of a mixed phase of hadron mat-ter and quark matter, with the total charge neutrality beingachieved by a positively charged amount of hadronic matter anda negatively charged amount of quark matter. Following Glenden-ning’s model, we use a standard two-phase description of theequation of state (EOS) through which the hadron and quarkphases are modelled separately. The resulting EOS of the mixedphase is obtained by imposing Gibbs’s conditions for phase equilib-rium with the constraint that the baryon number and the electriccharge of the system are conserved to the neutron star matter.

The Gibbs condition for chemical and mechanical equilibrium atzero temperature between the two phases reads

pHPðln;leÞ ¼ pQPðln;leÞ ð1Þ

where pHP is the pressure of confined hadron phase and pQP is thepressure of deconfined quark phase.

The conservation laws can be imposed by introducing the quarkfraction v defined as v ¼ VQ=V . Only two independent chemicalpotentials remain according to the corresponding two conservedcharges of the b-equilibrium system. The total baryon number den-sity qB is

qB ¼NB

V¼ vqQP þ ð1� vÞqHP ð2Þ

the total electrical charge is

0 ¼ QV¼ vqQP þ ð1� vÞqHP ð3Þ

and the total energy density is

� ¼ EV¼ v�QP þ ð1� vÞ�HP ð4Þ

Using the Eqs. (1)–(4), we can obtain the EOS of mixed phase mat-ter. In describing the hadronic part of the neutron star, we adopt theAPR model (Akmal et al., 1998). For the EOS, it is based on the mod-els for the nucleon interaction with the inclusion of a parameterizedthree-body force and relativistic boost corrections. We use the EOSof an effective mass bag-model for the quark matter part of the neu-tron star (Schertler et al., 1997).

In Fig. 1, we show the model EOS with deconfinement transi-tion, which is the typical scheme of a first-order transition. Thephase transition construction in a two-component system leadsto a continuous increase in energy per baryon in the mixed phasewith increasing density. It is well known that hadron matter is themost stable phase at lower densities, and that quark matter is themost favorite phase at higher densities. Meanwhile, the mixedphase has the lowest energy at intermediate densities. We choosethe parameters for quark matter EOS with s quark massms ¼ 150 MeV, coupling constant g ¼ 3 and different bag constantB ¼ 85 MeV fm�3

; B ¼ 108 MeV fm�3; B ¼ 136 MeV fm�3,

respectively.With the EOS presented above, we are ready to study the struc-

tural evolution of the rotating neutron stars. In this paper, we applyHartle’s approach (Hartle, 1967) as in Kang and Zheng (2007) toinvestigate the structure of the stars. By treating a rotating staras a perturbation on a non-rotating star and by expanding the met-ric of an axially symmetric rotating star in even powers of theangular velocity X, we can obtain the structure of the rotatingstars.

The resulting gravitational masses of neutron stars forB ¼ 108 MeV fm�3 are shown in Fig. 2, as a function of central bar-yon density for static stars as well as for stars rotating with themaximum rotation frequency mk. The solid almost horizontal con-nect configurations with the same total baryon number. In orderto explore the increase in central density due to spin-down, wecreated sequences of neutron star models. A Model in particular se-quence has the same constant baryon number, increasing centraldensity and decreasing angular velocity.

3. Deconfinement heating

There is deconfinement heating production due to spin-down inneutron stars due to the nuclear matter continuously convertinginto quark matter. The released energy had been estimated as afunction the rate of change the deconfinement baryon number

Fig. 3. Central density as a function of rotational frequency for rotating neutronstars of different gravitational mass at zero spin. All sequences are with constanttotal baryon number. Dotted horizontal lines indicate that deconfined quark matteris produced and dashed horizontal lines indicate that the nucleon direct Urcaprocess is triggered.

M. Kang et al. / New Astronomy 15 (2010) 515–519 517

using the parameterized approach (Kang and Zheng, 2007). Re-cently we studied the mechanism of energy release in detail (Kanget al., 2007). Through studying a random process of infinitesimalcompression for the mixed phase region, we can calculate the en-ergy release per baryon using the following formula

d~e� de ¼ d~edqB� de

dqB

� �dqB ð5Þ

where d~edqB

denotes the enthalpy change per baryon .The deconfinement heating is coupled with the rotation evolu-

tion of neutron stars. Combining the energy change with the evo-lutionary structure of neutron stars, we get the total heatluminosity (Kang et al., 2007, 2008)

Hdec ¼Z

dedv

_vðtÞqBdV ð6Þ

where v is the rotation frequency of the star. The spin-down of starsis due to the magnetic dipole radiation. The rotation frequency is gi-ven by

_v ¼ �16p2

3Ic3 l2v3 sin2 h ð7Þ

where I is the stellar moment of inertia, l ¼ 12 BR3 is the magnetic

dipole moment, and h is the inclination angle between the magneticand rotational axes. We now combine the energy release behaviorwith spin-down. Our recent work shows that exact calculationsagree well with the earlier order of magnitude estimates and sup-ports the previous parameterized approach. Because the change ofrotating stars is sufficiently slow, the formula of heat luminositycan be replaced by the simple parameterized form (Kang and Zheng,2007),

Hdec ¼ �qdNQ

dv_vðtÞ ð8Þ

where �q, about 0.1 MeV, is the mean value of energy release in themixed phase and NQ represents the baryon number of quarks in theinterior of the star.

4. Signal of quark deconfinement and thermal evolution ofneutron stars

The cooling of neutron stars could take place via two channels –neutrino emission from the entire star and thermal emission ofphotons through the transport of heat from the internal layers tothe surface. Neutrino emission is generated in numerous reactionsin the interior of neutron stars, e.g. as reviewed by Page et al.(2005). For the calculation of cooling of the hadronic part of theneutron star, we use the main processes including the nucleon di-rect Urca (NDU) and the nucleon modified Urca (NMU) and the nu-cleon bremsstrahlung (NB). For the quark matter, we consider themain process: the quark direct Urca (QDU) processes on unpairedquarks, the quark modified Urca (QMU) and the quark bremsstrah-lung (QB). For pure neutron stars, the direct Urca reaction (themost efficient) is allowed only at very high densities because it isimpossible to satisfy the conservation of momentum unless theproton fraction exceed the value where both the charge neutralityand the triangle inequality can be observed (Lattimer et al., 1991).However, for neutron stars that contains quark matter, this is notso because charge neutrality does not have to be conserved locallyfor a mixed phase. Hence the NDU process is active in the mixedphase.

The calculation of the evolution of the thermal energy of neu-tron stars (heating and cooling) is achieved by coupling their rotat-ing structure and deconfinement phase transition. Fig. 3 displaysthe central density of different masses neutron stars for

B ¼ 108 MeV fm�3, as a function of their rotational frequency. InFig. 3, the dotted horizontal line indicates the deconfined quarkmatter produced and dashed horizontal line indicate the NDU pro-cesses triggered. In the interior of these stars, the emerging ofdeconfined quark matter accompanies the gradual energy releaseleads to rising of surface temperature, and appearance of theNDU process can result in a rapid decrease of the temperature dur-ing the spin-down. For M ¼ 1:5;1:55;1:6M� neutron stars, thedeconfinement phase transition occurs during the thermal evolu-tion process which may lead to appearance of a characteristic sig-nature. We will now discuss in detail the quark deconfinementsignal.

We combine the equation of thermal balance with the rotatingstructure equations of the stars (Kang and Zheng, 2007; Hartle,1967) and rewrite the energy equation in the approximation ofan isothermal interior (Glen and Sutherland, 1980)

CV ðTi; vÞdTi

dt¼ �L1m ðTi; vÞ � L1c ðTs;vÞ ð9Þ

CV ðTi; vÞ ¼Z RðvÞ

0cðr; TÞ 1� 2MðrÞ

r

� ��1=2

4pr2dr ð10Þ

L1m ðTi;vÞ ¼Z RðvÞ

0eðr; TÞ 1� 2MðrÞ

r

� ��1=2

e2U4pr2dr ð11Þ

where Ts is the effective surface temperature, TiðtÞ ¼ Tðr; tÞ is theredshifted internal temperature; Tðr; tÞ is the local internal temper-ature of matter, and UðrÞ is the metric function(describing gravita-tional redshift) (Yakovlev and Haensel, 2003). Furthermore,L1m ðTi;vÞ and CV ðTi; vÞ are the total redshifted neutrino luminosityand the total stellar heat capacity, respectively, which are functionsof rotation frequency and temperature; cðr; TÞ is the heat capacityper unit volume. L1c ¼ 4pR2ðvÞrT4

s ð1� Rg=RÞ is the surface photonluminosity as detected by a distant observer (Rg is the stellar grav-itational radius). The effective surface temperature which is de-tected by a distant observer is T1s ¼ Ts

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� Rg=R

p. Ts is obtained

from the internal temperature by assuming an envelope model(Gudmundsson et al., 1983; Potekhin et al., 1997). Using to Eqs.(7)–(11), we can simulate the thermal evolution of neutron starswith deconfinement heating.

In Figs. 4 and 5, we present thermal evolution behavior of a1:6M� neutron star for different magnetic fields ð109—1012 GÞ.

Fig. 4. Thermal evolution curves of a 1:6M� neutron star with deconfinementheating for various magnetic field strengths (solid curves) and the curves withoutdeconfinement heating (dotted curves).

Fig. 5. Surface temperature change of a 1:6M� neutron star with rotationalfrequency for various magnetic field strengths with deconfinement heating (solidcurves) and the curves without deconfinement heating (dotted curves).

Fig. 6. Thermal evolution curves of neutron stars with deconfinement heating for differRectangles in the left-hand panel indicate observational data on cooling neutron stars w

518 M. Kang et al. / New Astronomy 15 (2010) 515–519

Due to the coupling of thermal evolution and spin-down, all curves(with deconfinement heating and without deconfinement heating)show clear magnetic field dependence. During the star’s spin-down, deconfined quark matter appears in the core of the star ata spin frequency of v ¼ 1123 Hz, the surface temperature dropsrapidly when the NDU process (enhanced cooling) occurs at a spinfrequency of v ¼ 492 Hz. It is evident that the temperature of thecurves with deconfinement heating (solid curves) are higher thanfor the standard cooling scenario without deconfinement (dottedcurves). We can observe a competition between cooling and heat-ing processes from the heating curves, where deconfinement heat-ing can produce a characteristic rise of surface temperature andeven dominate the history of thermal evolution. Eventually, theyreach a thermal equilibrium, where the heat generated is radiatedaway at the same rate from the star surface. We find the weakermagnetic fields have the larger change of temperature. The lowmagnetic field ð109 GÞ produces a sharp jump in surface tempera-ture as soon as the deconfinement quark matter appearing duringspin-down. Intermediate magnetic field ð1010; 1011 GÞ lead toslight changes in the temperature, but high magnetic field formonly the temperature plateau at a time.

In Fig. 6, we present the cooling behavior of different massesstars for magnetic field B ¼ 1012;1011 G (left panel) and magneticfield B ¼ 109;108 G (right panel) with deconfinement heating.The observational data, taken from Tables 1 and 2 in Page et al.(2004), have been shown in left panel. Comparing with previousinvestigation (Kang and Zheng, 2007), we find the thermal evolu-tion curves of our present work are more compatible with theobservational data (left panel). In our previous work it seemedthat, the NDU processes should be triggered easily in the modelof relativistic mean field (the critical mass for fast cooling occur-ring is being low) (Glendenning, 1997). In our present study, usingthe APR EOS, NDU processes can not be triggered easily in starswhich lead to the higher temperatures of the evolution curves thanin the previous cases. For example, the NDU reactions only appearabove 1.56M� in present model. In the cases of weak magneticfield, stars have high temperatures >ð105 KÞ at older ages>ð109 yrsÞ. We thus think that high temperature of some millisec-ond pulsars with low magnetic fields (Kargaltsev et al., 2004),especially for PSR J0437-4715, can be explained using the decon-finement heating model of neutron stars. We can observe that1.5M� neutron stars follow a similar thermal evolution track as1:6M�, but there is not a period increasing in temperature for1:7M�. Through comparing Fig. 6 with Fig. 3, we find the quark

ent stars masses and B ¼ 1012 ; 1011 G (left panel) and B ¼ 109;108 G (right panel.ith strong magnetic fields.)

M. Kang et al. / New Astronomy 15 (2010) 515–519 519

matter to appear at the birth of star for 1:7M�. For 1:5M� and1:6M� stars, quark deconfinement occurs when the central densitygradually increases during spin-down, which results in the tem-peratures of the stars to increase rapidly. This is a characteristicsignal as quark matter arises during the rotational spin-down ofstars for weak magnetic case.

5. Conclusions and discussions

The chief aim of the present work is to explore the signal ofquark matter appearing through theoretical simulation of the ther-mal evolution curves of neutron stars with deconfinement heating.We have constructed models of rotating neutron stars that followthe mixed phase investigation of Glendenning based on Hartle’sperturbative approach. The total thermal luminosities have beenobtained using the parameterized approach.

Recently, Stejner et al. (2008) have investigated the signature ofdeconfinement with spin-down compression in cooling neutronstars. A period of increasing surface temperature can be producedwith the introduction of a pure quark core for strongly superfluidstars of strong and intermediate magnetic field strength and the la-tent heat of deconfinement reinforces the signature only but is it-self relatively less significant. Contrary to their studies, our resultsshow that deconfinement heating can drastically affect the thermalevolution of neutron stars. The rise of surface temperature of cool-ing stars, as a signature of quark deconfinement, is derived fromthe deconfinement heating. It is noteworthy that a significant riseof the temperature accompanies the appearance of quark matter atolder ages for low magnetic field stars. This may be a evidence forexistence of quark matter, if a period of rapid heating is observedfor a very old pulsar. Deconfinement heating provides a new wayto study the signal of deconfinement.

We found that the deconfined signal appears for neutron starsof mass 1:4M� K M K 1:64M�. The influence of different EOSof the hadron phase and the model parameters of the quark phase(bag constant B, coupling constant g) on the phase transition den-sities, rotational structure of neutron stars and correspondinginternal structure etc have been studied by many investigators

(Schertler et al., 2000; Pan et al., 2006). The deconfinement heatingrate and mass range of a deconfinement sinal emerging can bechanged with varying of these parameters. In future, we willsystematically investigate the effect of these parameters whichwill be the subject of our future investigations.

Acknowledgment

This work is supported by NFSC under Grant Nos. 10747126 and10773004.

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