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it has been argued that strange quark matter , consisting of @xmath4- , @xmath5- and @xmath6-quarks , is energetically the most favorable state of quark matter  @xcite . @xcite suggested that there are two ways to form strange quark matter : the quark - hadron phase transition in the early universe and conversion of neutron stars into strange ones at ultrahigh densities . if this strange matter hypothesis is correct , then it has profound impact on physics and astrophysics . pulsars could be strange stars and not neutron stars as previously thought , and there could even be many strange dwarfs and strange planets in the universe  @xcite . several mechanisms have been proposed for the formation of strange quark stars . for example , strange stars are expected to form during the collapse of the core of a massive star after a supernova explosion  @xcite . another possibility is that some rapidly spinning neutron stars in low - mass x - ray binaries ( lmxbs ) can accrete sufficient mass to initiate a phase transition to become strange stars  @xcite . some of the millisecond pulsars may be strange stars , because lmxbs are believed to be the progenitors of millisecond pulsars . strange stars have also been proposed as sources of unusual astrophysical phenomena , such as soft @xmath7-ray repeaters  @xcite , pulsating x - ray bursters  @xcite , cosmological @xmath7-ray bursts  @xcite , sax j1808.4 - 3658  @xcite etc . the discovery of khz quasi - periodic oscillations in lmxbs  @xcite implies that the compact stellar object must have a very soft equation of state ( eos ) , which is consistent with that of strange stars  @xcite . recently , hong et al . @xcitehave argued that the first order phase transition of the color superconductivity occurring at the end point of some massive stars can release sufficient energy to produce hypernovae , which are considered as progenitors of cosmological @xmath0-ray bursts . however , important as this strange - matter hypothesis maybe , it remains notoriously difficult to be proven or refuted on either observational or theoretical grounds . the binding energy per nucleon for strange matter is estimated to be close to that of fe@xmath8 , but it has not been possible to calculate it from qcd to within the few percents level required to address the question . model calculations of the binding energy are clearly inadequate and unreliable  @xcite .    lacking an accurate and reliable method to calculate the strange matter eos , the structure and stability of strange stars have been addressed  @xcite typically in simplified models such as the mit bag model  @xcite . the quarks are treated as relativistic free particles confined in an impenetrable bag . approximately then , the pressure comprises of the fermi gas pressure of the quarks minus a bag pressure , which mimics the strong interaction that holds the quarks together . perturbative corrections to such a simple eos can be made based on one - gluon exchange calculations  @xcite . with this eos , it has been shown that self - bound stars exist with virtually no lower limit on the stellar mass  @xcite ; these stars are basically giant hadrons , held together by strong interaction rather than gravity .    it is arguable whether meaningful refinements of theoretical studies of strange stars can be made at this stage . uncertainties in the strong interaction calculations easily make or kill strange stars , and arguments over details in model - dependent eos s are probably futile . nevertheless , one could ask what kinds of effects commonly accepted ingredients of strong interaction have on the stability and structure of strange stars . whereas such studies are not meant to be quantitatively accurate , one may possibly gain insight into the physics of compact stars .    with this in mind , we study the implications of chiral symmetry on the structure of strange stars . chiral symmetry is widely recognized as an important driver of the strong interaction dynamics , witness the celebrated experimental success of partially conserved axial current ( pcac ) , which signifies the importance of chiral su(2)xsu(2 ) symmetry  @xcite . phenomenologically , the partial breaking of chiral symmetry is associated with the emergence of pion as a pseudo - goldstone boson and hence related to the small pion mass  @xcite ; the coupling of pions to quarks / hadrons is thus a natural result of chiral dynamics and an essential ingredient of strong interaction physics . there are many hadron models that incorporate various features of qcd  @xcite . whereas the mit bag model has the asymptotic freedom and the confinement of quarks built in , it is well known to violate chiral symmetry badly . by coupling the pions to the quarks , thus savaging the axial current at the bag , one can restore chiral symmetry . there are many variants of such chiral bag models  @xcite , and one can imagine these as either modifying the original bag model to include a meson cloud at the bag , or extending the sigma model to a nucleon with quark structure . for our purpose , a simple , representative chiral bag model suffices , and we have thus chosen the cloudy bag model  @xcite . the idea to model a hadron by putting a few relativistic quarks in a cavity has led to much phenomenological success  @xcite . in the simplest version , one considers free dirac fermions of mass @xmath9 confined in a static spherical bag of radius @xmath10 , and the ground state spatial wavefunction is simply  @xcite @xmath11 \ \ , \ ] ] where @xmath12 , @xmath13 , @xmath14 , @xmath15r^3j_0 ^ 2(x)}}$ ] and @xmath16 satisfies @xmath17 } } } $ ] . however , it was soon realized that the original mit bag boundary condition leads to severe violation of chiral symmetry  @xcite . the physical picture is that when a quark with a certain helicity collides with the impenetrable wall , its momentum is reversed but not its spin , leading to a change of the helicity . mathematically , this is expressed by the non - conservation of the quark axial current @xmath18 , where the @xmath0 s are the dirac gamma matrices and @xmath19 is the isospin operator . one resolution of the problem is to introduce a meson field , so that the total ( quark plus meson ) axial current is conserved  @xcite . physically , when a quark hits the bag , it produces mesons that carry away some of the axial current . in this line of thought , the hadron bag is always surrounded by a pion cloud , hence the name cloudy bag model . for example , for a ground state nucleon , the pion field that will restore the chiral symmetry of a bare mit bag is @xmath20 where @xmath21 is the pauli matrix , and the spatial part of the wavefunction @xmath22 is given by @xmath23 here the pion - quark coupling constant @xmath24 is determined by the continunity of axial current at @xmath25 , which yields @xmath26 with @xmath27 is the total number of @xmath4 and @xmath5 quarks in the bag , @xmath28 is the meson mass , @xmath29 is the pion decay constant and @xmath30 . the cloudy bag model lagrangian is a simple extension of that of the mit bag model . here we use the linearized version :    @xmath31 \theta _ v \nonumber \\ & & - { 1 \over 2 } { \bar q } \left ( 1 + i{\bf \tau } \cdot { \bf \phi } \gamma _ 5 /f \right ) q \delta _ s + { 1\over 2 } \left [ \partial _ { \mu } { \bf \phi } ( x ) \cdot \partial ^ { \mu } { \bf \phi } ( x ) - \mu ^2   { \bf \phi}^2 \right]\bar \theta_v \ \ , \end{aligned}\ ] ]    where @xmath32 inside ( outside ) the bag , @xmath33 , @xmath34 is the surface delta function , and @xmath35 is the bag constant . note that the quarks are completely confined in the bag , and the mesons only exist outside the bag . they are free particles andare coupled to the quarks only at the bag surface , through the term proportional to @xmath36 , and are free outside or inside the bag . the chiral bag model has essentially the same langrangian but differs from this one in that it excludes the mesons from the inside of the bag  @xcite . > from the lagrangian , one can obtain the stress - energy tensor : + @xmath37 $ ] , which gives the pressure and density of the system immediately , @xmath38 , @xmath39 . with some straightforward algebra , we get : @xmath40 -b \right\ } \theta _ v \ \ , \label{quarkp}\ ] ] @xmath41 + b \right\ } \theta _ v \label{quarkrho}\ ] ] inside the bag , note that if one ignores the quark mass , the eos becomes @xmath42 which is exactly the one used in previous studies of strange stars  @xcite . we assume that for a macroscopic object like a strange star , the pressure outside the bag due to the meson cloud is dominated by real and free pions , given by @xmath43 at the boundary , continunity of pressure is required . therefore , when the quark pressure equals to the pion pressure just outside the bag , the eos is truncated to the pion one eq . [ peos ] .    in order to deduce the eos and apply it to study the properties of strange star , we need to determine the values of the bag constant ( @xmath35 ) and the strange quark mass ( @xmath44 ) . for simplicity we have taken the masses of up and down quarks to be zero . first , the hadron mass can be expressed as @xmath45 where @xmath10 is the radius of the hadron , @xmath46 is the eigenvalue of quarks with @xmath47 u , d and s. @xmath10 is fixed by minimizing the mass , i.e. , @xmath48 . then the hadron mass can be determined . since @xmath35 and @xmath44 are independent parameters , we first determine @xmath35 by fitting mass of the baryons which consist of only up and down quarks . in this simple model , hadrons with spin 1/2 and 3/2 are degenerate , and we minimizing the averaged root - mean - square error of @xmath49 and @xmath1 . table  [ table : b ] is the result of our fitting . we obtain @xmath50mev . after @xmath35 is fixed , @xmath44 can be determined by fitting baryons which consist of one or more strange quarks . > from table . [ table : ms ] , we obtain @xmath44 = 220 mev .    fig . [ fig : eos ] shows the eos from the cloudy bag model . we have extrapolated it into high - density region . in low - density region , since the pion field decay exponentially outside the bag , the pion ` tail ' in the eos is not important in calculation . hence , it is not shown in the graph . the eos s for different parameters were also shown in the figure as for comparison . note that pion does not come into account in this part of the eos , therefore the one with @xmath51mev and @xmath52mev reduced to the mit bag case , which was used in previous studies of strange star . assuming the pion cloud decay exponentially when leaving the stellar surface , it only exists within an effective range of scale @xmath53 ( @xmath54fm ) . the pion ` shell ' is so thin that has negligible effect on the stellar radius . using typical parameters : @xmath55 cm , @xmath56fm , , mass of the pion cloud surrounding the star can be estimated : @xmath57 which is completely negligible as compared to the stellar mass . the pion cloud surrounding the star is a consequence of chiral symmetry consideration , it has no significance on global stellar structure such as radius and mass .    using the eos of this model and the tolman - oppenheimer - volkoff equations , the properties of strange stars can be obtained . the global stellar structure is very similar to that of mit models  @xcite . > from fig . [ fig : mrhoc ] , maximum mass of cloudy bag strange stars using the best - fit parameters is about@xmath58 . as a comparison , the maximum mass of mit strange stars is 2.0@xmath59 . these two values only differ slightly , @xmath60 . similar to the case of mit strange stars , there is no minimum mass for the cloudy bag strange stars . the mass - radius relation is shown in fig . [ fig : mr ] . from the figure , radius of the cloudy bag strange star is in general larger , this is due to a softer eos obtained by a smaller value of @xmath35 . the stellar density profile of @xmath61 strange stars is shown in fig . [ fig : rhor14 ] , the density is quite uniform inside the star . [ fig : z ] is the gravitational redshift ( @xmath62 ) versus central density ( @xmath63 ) for different parameters , where @xmath62 is given by @xmath64 . from the figure , redshift depends sensitively on strange quark mass @xmath44 . this result can be understood qualitatively from  @xcite : since both the mass and radius of the star depend on the same factor of @xmath65 , their ratio , @xmath66 will be independent of @xmath35 . then the only parameter remaining is @xmath44 . this may provide a possible way to estimate the strange quark mass @xmath44 . if the gravitational redshift of strange stars is observed , @xmath44 may be inferred from the graph . rotation properties of strange star is important in observation , since the spin rate of pulsars can be measured very accurately . due to different visocity of strange matter and neutron matter , the maximum rotation frequency of strange stars could be much higher than neutron stars  @xcite . @xcite derived the approximate mass and radius formulae for static and rotating strange stars using the general relativistic conditions of thermodynamic equilibrium and fitting with values obtained from numerical integration of structure equations . from the formulae , the maximum mass and radius of strange stars can be predicted by fitting only one set of static configuration data .    in our calculation , we will adopt the formulae . the equatorial radius of a rotating strange star can be expressed as a function of static mass ( @xmath67 ) and static radius ( @xmath68 ) : @xmath69^{-1/2},\ ] ] where @xmath70 is the constant of gravitation and @xmath71 is the angular velocity of rotation . gravitational mass of a rotating strange is assumed to be in the form @xmath72 where @xmath73 is the eccentricity , defined by the ratio between polar and equatorial radii , @xmath74 . we used the approximation as given by  @xcite : @xmath75 now the gravitational mass is a function of rotation frequency . the maximum frequency is given by the stability condition @xmath76    fig . [ fig : mw ] shows the gravitational mass ( @xmath77 ) versus the rotation frequency ( @xmath71 ) for different strange stars with maximum masses . according to the graph , the maximum rotation frequency of the cloudy bag strange stars with best - fit parameters is 6300s@xmath78 , which corresponds to the maximum mass of @xmath79 . for mit strange stars , the kepler limit is about 9400s@xmath78 and the maximum mass is @xmath80 . the maximum rotation frequency of a mit strange star is higher than the cloudy bag one by @xmath8130% . this is due to different mass - radius properties . a mit strange star in general more compact , having a smaller radius and larger mass . hence the rotation can be faster . in order to have a better understanding on the evolution of gravitational mass and equatorial radius as the spin rate increases , fig .  [ fig : rotat ] is the plot of @xmath77 against @xmath82 and @xmath71 for the best - fit cloudy bag strange stars with different static masses . the thick solid line on the @xmath77-@xmath82 plane is the mass - radius relation of the static stars , i.e. fig . [ fig : mr ] . then stars with static masses of @xmath83 @xmath84 , @xmath61 , @xmath85 and @xmath58 were investigated as rotation frequency increases . > from the graph , @xmath82 increases monotonically with @xmath71 , while the mass attends a maximum value and then drops with @xmath71 . the change in mass is not so obvious for a low mass strange star , e.g. @xmath83 . for the case of a @xmath61 star , the maximum mass is @xmath86 which corresponds to the kepler frequency of 5000s@xmath78 and the equatorial radius of @xmath87 cm . the change in mass is about 20% and in radius is about 40% . for the star with maximum static mass , the kepler limit is about 6300s@xmath78 , with @xmath88 and @xmath89 cm . the evolution of mass and radius can be investigated by the projection of the curves on the @xmath77-@xmath82 plane . they are shown by the dotted lines . the gravitational mass as well as equatorial radius increase with rotation frequency , until the maximum mass is reached . projection of the curves on the @xmath71-@xmath77 plane is also shown by the dotted lines . they demonstrate clearly the relationship between maximum rotation frequency and static stellar mass . from the graph , the kepler frequency increases with static stellar mass . hence the maximum possible mass of a stable star is the one with largest static mass and rotating at kepler frequency . the graph indicates that if a strange star with period less than 1ms is found , it probably has a mass greater than 1.4@xmath59 . table  [ table : summary ] gives a summary of the results for different parameters . for a rotating strange star , the mass and equatorial radius lie in different region from the static case . using the formulae , for given any two stellar parameters from the gravitational mass , equatorial radius or rotation frequency , the remaining one can be determined from the graph . this may help to determine the size or mass of a strange star since rotating period of a pulsar can be measure very accurately . since the proposal of strange stars , much effort has been devoted to find    observational properties that distinguish strange stars from neutron stars . although many stellar properties are similar in both kinds of stars , their cooling behaviour is very different . energy lost rate of strange matter is much higher than that of neutron matter , therefore the temperature of a young strange star is sufficiently lower than a neutron star of the same age  @xcite .    by chiral symmetry consideration , we suggested that a strange star should be surrounded by a pion field . the thin pion cloud has negligible effect on global stellar structure such as mass and radius . however , it plays an important part in strange star cooling . the decay of pion is a rapid cooling mechanism in our model . as a result , the surface temperature of a cloudy bag strange star is lower than ordinary mit strange star significantly . this may provide a hint to distinguish cloudy bag strange stars and verify the theory . in addition , when the star is cooled by emission of pions , the huge amount of energy released within a short period may be a possible energy source of @xmath0-ray bursts .      in this section , we considered the cooling of a non - rotating bare strange star with typical mass 1.4@xmath59 , which is the typical mass of neutron star . indeed , there is no reason why a strange star and a neutron star should have the same masses . also it is unnecessary that there are identical twin between strange star and neutron star . however , the radius of the strange star is insensitive to the stellar mass . for example , the radius variation of strange star with masses between 1.4 to 2 solar masses is only 3@xmath90 . furthermore , if the internal properties of the strange stars are the same , the cooling curves are almost identical . of course , the density profiles of strange stars with different mass are different . as we will point out in next section , strange stars with different masses will have different electron density due to the color - flavor locked ( cfl ) phase  @xcite . this has a very important effect in cooling calculations . since our focus in this paper is on the role of pion cloud surrounding a strange quark star , which is a natural consequence of chiral symmetry , and in particular , its effect on the cooling properties on strange stars . we are not primarily interested in the detail of other effects , i.e. cfl of color superconductivity , in this paper . therefore , in this paper we treat electron density as a parameter but the mass of the strange star is fixed as @xmath91 in the cooling calculations . for simplicity we assume that all compact star models are non - rotating . as argued by  @xcite , when a strange star is formed , it is very hot and most of the normal matter in the curst is ejected . therefore , it is expected that new born strange stars do not have crusts . besides , we assumed that all pions will decay once they leave the stellar surface beyond the length scale of @xmath53 .    for simplicity , we assumed the star of uniform density and isothermal . > from figure  [ fig : rhor14 ] , the uniform density profile is a good approximation . also , the isothermal approximation is generally used in simplified models . as it is a bare strange star in our case , the core and surface temperature are equal . note that the effects of magnetic field are neglected in this simple model . this will be carried out in further works . superfluidity ( superconductivity ) of quark matter is considered in the model . it is generally believed that quarks can form colored cooper pairs near the fermi surface and become superfluid ( superconducting ) . an early estimation of the magnitude of the gap @xmath1 was @xmath920.1 - 1  mev  @xcite . on the other hand , some recent studies considered instanton - induced interaction between quarks and estimated @xmath93mev  @xcite . furthermore , recent works ( @xcite ) demonstrate the possibilities of diquark condensates characterized by large pairing gaps in quark cores of some neutron stars and in quark stars and discuss different possible phases of quark matter , for examples , the color - phase - locked u - d - s phase ( @xcite ) and two flavor u - d color - superconducting phase ( @xcite ) . in this paper , our main focus is on the role of the pion cloud surrounding a quark star and its effect on the cooling properties of strange stars . we are not primarily interested in the detailed effects of color superconductivity , which has been discussed in some details in ref . ( @xcite ) . furthermore , there are large uncertainties as to the phase as well as the parameter values of color superconductivity in quark stars . therefore we will assume that there are three flavors of u - d - s quarks in a strange star for both small and large color superconductivity gaps . another main difference of these phases is the electron concentration , which plays a very important role in cooling calculations and will be discussed later . from bcs theory , the critical temperature @xmath94 is related to the pairing gap by @xmath95 where @xmath96 is the boltzmann constant . when the temperature is below @xmath94 , it is in the superfluid state and there are some modifications to the specific heat and emissivity .      in the cooling of cloudy bag strange stars , the main cooling processes include quark urca process , pion emission and blackbody radiation . the thermal evolution of the system is determined by the equation : @xmath97 where @xmath98 is the specific heat , @xmath99 , @xmath100 , @xmath101 are neutrino emissivity , pion emissivity and blackbody radiation respectively .    in normal state , we adopted the results of  @xcite , where the quark fermi momentum was approximated by : @xmath102 the specific heat of quark matter is then given by : @xmath103 in superfluid state , the specific heat is modified , we adapted the formula by  @xcite and  @xcite :    @xmath104 \\ & & \mathrm{\;for } \quad 0.2\,t_c\leq t\leq t_c \\ & = & 0 \quad \mathrm{\;for } \quad t < 0.2\,t_c . \end{array } \right\ }     \label{eq : csf}\ ] ]    it has been shown that the goldstone excitations in the quark - gluon plasma contribute to the specific heat of the strange star (  @xcite ,  @xcite ) . @xmath105 where @xmath106 is the number of available massless gluon - photon states , which are present even in the color superconducting phase . when temperature is low , the specific heat of quark matter vanishes . however , electrons are not affected . therefore , the specific heat of electrons becomes important . we also considered this factor @xmath107 where @xmath108 is the electron fraction . recently , rajagopal and wilczek  @xcite have shown that the quark matter in the color - flavor locked phase of qcd , which occurs for large gaps @xmath109 ) , is rigorously electrically neutral , despite the unequal quark masses , and even in the presence of electron chemical potential . in other words , @xmath108 = 0 , and hence @xmath110 = 0 . however , in order to have this electrical neutral situation the following condition must be satisfied @xmath111 where @xmath112 is the baryon chemical potential and @xmath113 is the strange quark mass . realistically , a strange star must have a density profile . if the above condition is satisfied in the core , it may not be satisfied near the surface , where the density is equal to 4b . taking the typical values of baryon density near the strange star surface @xmath114@xmath115 , the baryon chemical potential is @xmath116mev and the right hand side of the equation gives @xmath117 with @xmath44 = 220 mev . therefore we believe that @xmath108 will not be zero throughout a strange star even if its core is in color - flavor locked phase . for simplicity , we treat @xmath108 as a parameter . neutrino is emitted from quark urca process : @xmath118 using the assumption of eq . ( [ eq : pf ] ) , @xcite obtained the neutrino emissivity : @xmath119 where @xmath120 is the strong coupling constant . in superfluid state , i.e. @xmath121 , it is suppressed by a factor of @xmath122 . note that eq . ( [ eq : ev ] ) is an approximation to the exact form @xcite : +    @xmath123     +    where @xmath124 is the fermi weak coupling constant , @xmath125 is the cabibbo angle , @xmath96 is the boltzmann constant , @xmath126 , @xmath127 , @xmath128 , @xmath129 are fermi momenta of @xmath4 , @xmath5 quarks and electrons respectively , @xmath130 is the angle between the @xmath4 and @xmath73 momenta , which satisfies : @xmath131 the condition for reactions  ( [ eq : dbeta1 ] ) and  ( [ eq : dbeta2 ] ) to occur is @xmath132 . we solved the equations numerically and found that in the density range we are interested , the reactions can always occur .    on the other hand , we also considered the similar reactions for @xmath6 quarks , @xmath133 numerical results showed that the value of @xmath134 , which is defined by @xmath135 is smaller than 1 only if the density is as high as @xmath136 . therefore , reactions  ( [ eq : sbeta1 ] ) and  ( [ eq : sbeta2 ] ) do not exist in the star . here , we have also ignored the neutrino emissivity due to the photon - gluon mixing , which can generate the a massive photon - gluon excitation  @xcite . it is because this excitation is only important for the temperature higher than 70 mev and our cooling calculations of quark stars start with temperature about 10mev . the most important feature of cloudy bag strange stars is the emission of pions . due to the collision between quarks and the bag , pions are created at the stellar surface , which represents a conversion of the quark kinetic energy to the mass of the pion cloud . some of the more energetic pions may even escape from the star carrying away substantial energy . the pions are assumed to be decayed once left the stellar surface for a distance larger than @xmath53 . the decay channel is @xmath137 as the decay rate is nearly 100% , it is an efficient cooling mechanism and the final product is a fireball of @xmath138 pairs and photons . we estimated the escape velocity for pions to leave the region of @xmath53 from stellar surface . numerical results showed that this value is very small such that almost every pion can escape and decay . hence , we can assume all pions produced will decay . the pion emissivity is given by : @xmath139 where @xmath140 is the energy density of pion field at stellar surface , which is fixed by the axial current conservation in cloudy bag model . speed of pions emitted , @xmath141 , is estimated by @xmath142 , where @xmath28 is the pion mass . therefore , @xmath143 put in @xmath144 , @xmath145 cm , we get : @xmath146 in superfluid state , since the collision between quarks and the bag is suppressed . by a similar argument , pion emissivity is also reduced by the factor of @xmath122 .    in eq.(34 ) , we show that pions can carry away as much as @xmath147 erg / s . however , the boundary condition requires the density of pion is only @xmath148 , from which we have estimated that the entire strange star will be covered by a pion cloudy with mass @xmath149 . the key reasons why pions with such low masses can carry away that much energy are : ( 1)the thermal conductivity of compact object like a strange star is extremely high , therefore it is a good approximation to assume that the entire strange star is isothermal . in fact the characteristic time scale for the thermal equilibrium is @xmath150 where @xmath10 is the radius of the star and @xmath151 is the speed of sound , which is @xmath152 . this time scale is @xmath153 . therefore the thermal energy can quickly transport from core to the surface . ( 2)in order to carry away @xmath147erg / s , the pion production rate @xmath154 must be larger than @xmath155 . since pions are produced when quarks collide the boundary of the bag , we can estimate that the collision rate between quarks and the boundary is @xmath156 , where @xmath157 is the quark number density and @xmath158 is the quark velocity . we have assumed that the quark mass density is 4b and quarks are moving relativistically . we can see that most pions will be re - absorbed by the boundary in order to maintain the pion density at the boundary . only a small fraction of pions can escape . the last cooling process we considered in cloudy bag strange stars is blackbody radiation . it is given by the well - known formula : @xmath159 where @xmath160 is the stefan - boltzmann constant . however , it has been pointed out that bare strange stars are very poor radiators of thermal photons with energy less than 20mev . this is because the plasma frequency ( @xmath161 ) that is related to particle density of quarks is very high even at the surface of strange stars ( alcock et al . on the other hand , chmaj et al . ( 1991 ) argue that low - energy photons with energy less than the plasma frequency can still leave the strange star surface due to the non - equilibrium quark - quark bremstralung radiation in the surface layer with the thickness of @xmath162fm . nevertheless the efficiency of this non - equilibrium blackbody radiation is only @xmath163 of @xmath101 . as it is a bare strange star in our case , the surface temperature is high and the energy of the blackbody radiation is large . besides , in superfluid state , other cooling processes are suppressed , so this non - equilibrium blackbody radiation will be a very important cooling mechanism at the later stage .    here , we want to remark that eq.34 and eq.35 represent two completely different physical processes . the production of pion is the requirement of chiral symmetry , which is believed to be a good symmetry for strong interaction , and they are produced outside the star due to the collisions between quarks and the bag boundary . on the other hand , the reduced blackbody radiation ( chmaj et al . 1991 ) results from the collisions among quarks . they are electromagnetic waves produced very near the stellar surface but yet inside the star and these em waves leak out the star through the bag boundary .    in order to have a more realistic calculation , the effects of thermal equilibrium radiation and emission of @xmath138 pairs , as suggested by  @xcite , were also discussed . in this theory , hot strange matter is filled with electromagnetic waves in thermodynamic equilibrium with quarks . the energy flux of thermal equilibrium photons radiated from a bare strange star surface is given by  @xcite : @xmath164 where @xmath165mev , @xmath166 is the surface temperature , @xmath167 and @xmath168 is the coefficient of radiation transmission from strange matter to vacuum . @xmath169 @xmath170.\ ] ]    it was pointed out by  @xcite that the strong electric field at the surface of a strange star may be a powerful source of @xmath138 pairs . the production rate is  @xcite @xmath171j(\xi ) \,\mathrm{s^{-1}},\ ] ] where @xmath172 @xmath173 where @xmath174 is the fine - structure constant and @xmath17518mev . the energy flux of @xmath138 pairs from unit surface is given by @xmath176 where @xmath177 is the mean energy of created particles . the cooling mechanisms of mit strange stars are the same as those of cloudy bag strange stars except without the pion emission . as for comparison , the cooling of neutron stars is also discussed . following the work by  @xcite , we considered a simple case which only has neutrino emission and blackbody radiation for cooling . @xmath94 is taken to be @xmath178k for the transition to superfluid state  @xcite . the detail relation between the core temperature @xmath179 and the surface temperature @xmath166 can be obtained by solving the energy transport equations numerically ( e.g. tsuruta 1979 ) and is found to be approximately related by a simple power law @xmath180(see shapiro and teukolsky 1983 , p.330 ) . obviously , this simple relation can not take into account the gravitational effect . @xcite have carried out a more detail analysis and found that : @xmath181 where @xmath182 is the surface gravity in the unit of @xmath183 . specific heat of neutron matter is given by  @xcite : @xmath184 where @xmath185 is the stellar mass in the unit of solar mass , @xmath186 . in superfluid state , it is modified to be : +    @xmath187 \ \mathrm{for } \ t\leq t_c,\ ] ]     +    which is exactly the same formula as eq . ( [ eq : csf ] ) . specific heat of electrons is @xmath188    neutrino luminosity consists of two parts : the first part is    electron - proton coulomb scattering @xmath189 in the crust  @xcite : @xmath190 where @xmath191 is the fractional mass of the crust @xmath192 . the second part is neutrino bremsstrahlung process @xmath193 , which is given by :    @xmath194    another case of neutron star is similar , but with the addition of a rapid cooling mechanism . direct urca process @xmath195 is known to be the most efficient cooling way in neutron stars . at high temperature , the neutrino emissivity can be higher than that from pion condensation and quark urca process by one to two orders of magnitude . @xcite suggested that this process may occur in neutron stars if the proton concentration exceeds a certain value . they estimated the neutrino emissivity to be @xmath196 where @xmath108 is the electron fraction , it is taken to be 0.1 in our calculation and @xmath197 is the nuclear saturation density . we also include the pion condensation ( see shapiro and teukolsky 1983 p.323 ) as anther possible rapid cooling mechanism for neutron stars . the cooling behaviors and the surface luminosities of 1.4@xmath59 strange stars and neutron stars are shown from fig.8 to fig.14 . the cloudy bag strange star is the one obtained using best - fit parameters , i.e. @xmath198mev and @xmath199mev ; the mit one is using the linear eos and the neutron star is the one with typical mass of 1.4@xmath59 and radius of 10 km . strange stars are bare while the neutron star has a crust of mass about 5% of the total stellar mass . the initial temperature was chosen to be @xmath200k , which is a typical temperature when the star is born . due to gravitational redshift , the surface temperature and luminosity observed at infinity ( @xmath201 and @xmath202 ) are related to their value at stellar surface ( @xmath166 and @xmath203 ) by  @xcite : @xmath204 where the surface redshift factor is @xmath205 @xmath77 is the gravitational mass of the star , @xmath10 is the stellar radius and @xmath70 is the gravitational constant . figure  [ fig:8 ] is the luminosities of various emission mechanisms observed at infinity as a function of temperature . the calculations are based on cloudy bag strange star with @xmath206mev , @xmath207 . as a comparison , two cooling mechanisms for neutron stars are also shown . we considered the direct urca process with @xmath207 and @xmath208k . for pion condensation model , @xmath207 and @xmath209k . according to the graph , energy released in direct urca process ( short dashed line ) is highest at high temperature . then the luminosity by pion condensation ( solid line ) follows . quark urca emission follows . when temperature drops below @xmath210k , pion emission ( dash - dot - dotted line ) becomes the most efficient cooling mechanism and dominates over a wide range of temperature . when temperature drops further , the star is in superfluid state . most of the emissions are suppressed and only non - equilibrium blackbody radiation dominates . although thermal equilibrium radiation and @xmath138 pair emission remain unaffected , their values are small and negligible in low temperature . the surface temperature of different kinds of stange stars observed at infinity is plotted in figure  [ fig:9 ] . we first considered a cloudy bag strange star with @xmath206mev and @xmath207 ( the dash - dot - dotted line ) . as pion emission is an efficient cooling mechanism , temperature of the cloudy bag strange star drops rapidly at the beginning . from the figure , temperature drops from @xmath200k to @xmath211k within 1ms . a rough estimation of energy released : @xmath212 where @xmath213 is the total baryon number of the star . it has been suggested that the conversion of neutron stars to strange stars could be the origin of @xmath0-ray bursts  @xcite and 40% of the deconfinment energy ( @xmath214erg ) will be converted into @xmath138 pairs and photons fireball via neutrino - antineutrino annihilation . however , in our model , the evaporation of pions is a more efficient heat dissipation mechanism . the pion decay products form a fireball of @xmath138 and photons directly instead of via neutrino - antineutrino annihilation . therefore the fireball can carry away 100% of the deconfinment energy . since both the time scale and order of magnitude of energy released is similar to typical @xmath0-ray bursts  @xcite , we suggested that the rapid cooling of cloudy bag    strange stars may be a possible energy source of @xmath0-ray bursts . when the core temperature drops below @xmath215k , the cooling slows down . in this stage , the strange matter becomes superfluid . therefore , emissivity of neutrino and pion are reduced by a factor of @xmath122 . finally , the star is cooled by blackbody radiation only .    for a mit strange star with @xmath1=1mev and @xmath207 ( the dash - dotted line ) , it is cooled by neutrino emission , thermal equilibrium radiation , @xmath138 emission and blackbody radiation . thus , the cooling rate is much slower than that of a cloudy bag strange star , which emitts neutrinos as well as pions . similarly , while temperature is lower than @xmath215k , all cooling mechanisms become negligible except blackbody radiation . this is why the long term thermal evolution is the same in both mit and cloudy bag strange stars .    in the case of @xmath1=100mev , temperature of the stars can never exceed the critical temperature @xmath94 . in other words , they are already in superfluid state since born . hence pion and neutrino emissivity are very small . the cooling behaviour of cloudy bag and mit strange stars are similar , since they are mainly cooled by the smae mechanisms . note that for @xmath1=100mev , stars with @xmath216 cool faster than those with @xmath207 . it is because in superfluid state , specific heat mainly comes from the contribution of electrons . therefore , if @xmath108 is smaller , the star can cool faster . it is interesting to see that for mit strange stars , although the one with @xmath1=100mev cools slower , its surface temperature is lower than that of @xmath1=1mev after 1s . this can be explained by the specific heat , since it depends on @xmath1 . under the same temperature , the specific heat is smaller for a larger gap . hence , the temperature drops faster .    for the long - term behavior , since it is dominated by the non - equilibrium blackbody radiation . they only depend on @xmath108 , which is related to the specific heat . figure  [ fig:10 ] shows the surface temperature of strange stars and neutron stars observed at infinity . since neutron stars are covered by crusts , which act as thermal blankets to reduce the energy lost rate . at the beginning , although the core temperature of neutron stars is higher , their surface temperature is lower than strange stars by a lot . this is due to eq . ( [ eq : tstc ] ) , surface temperature is different from core temperature by several orders of magnitude . since neutron stars cool slowly , the surface temperature is eventually higher than that of strange stars at @xmath217s for @xmath1 = 100mev and at @xmath218s for @xmath1 = 1mev respectively .    to investigate the observational properties , the luminosities of electromagnetic radiation from various emission mechanisms for cloudy bag strange stars and mit strange stars were plotted in figure  [ fig:11 ] and  [ fig:12 ] respectively . photon luminosity plotted in the graphs consists of thermal equilibrium photons and non - equilibrium blackbody radiation . note that in both graphs , photon luminosity and @xmath138 pair production start at the same point . since they are independent of superfluidity , they are the same for all kinds strange stars . they become different in later time because of different evolution of the star temperature .    for cloudy bag strange stars with @xmath206mev and @xmath207 ( fig . [ fig:11 ] ) , pion emission ( short dashed line ) dominates over a wide range of time until @xmath219s . hence electromagnetic radiation mainly comes from the decay of pion in this case .    in the case of cloudy bag strange stars with @xmath220mev and @xmath216 , pion emission ( dash - dotted line ) is suppressed by superfluidity and becomes negligible . therefore electromagnetic radiation mainly comes from blackbody radiation , equilibrium radiation and annihilation of @xmath138 pairs .    for mit strange stars with @xmath206mev and @xmath207 ( fig . [ fig:12 ] ) , electromagnetic radiation mainly comes from @xmath138 pairs ( dashed line ) from @xmath221s to @xmath219s , while photon emission ( dash - dot - dotted line ) dominates after @xmath219s . similar for the case of @xmath220mev and @xmath216 , but the photon luminosity ( solid line ) dominates after @xmath222s .    the total electromagnetic radiation from different strange stars and neutron stars are plotted in figure  [ fig:13 ] and  [ fig:14 ] . strange stars in figure  [ fig:13 ] were calculated using @xmath206mev and @xmath207 . from the figure , the cloudy bag strange star ( dotted line ) is most luminous at the beginning . then after 1ms , it transits into superfluid state and the pion luminosity drops drastically . at this time , the luminosity of mit strange star with @xmath206mev , @xmath207 ( solid line ) is highest . at @xmath223s , it also drops due to superfluidity . around @xmath224s , the luminosity of strange stars drop below neutron stars .    for strange stars with @xmath220mev and @xmath216 ( figure  [ fig:14 ] ) , both mit ( solid line ) and cloudy bag ( dotted line ) strange stars behave similarly , as pion emission is highly suppressed . in this case , the luminosity drops below that of neutron stars at @xmath225s . as compared to previous graph , @xmath108 is smaller for strange stars , therefore specific heat is smaller and the stars cools faster . hence the luminosity drops below that of neutron stars at an earlier time . we have used the cloudy bag model to describe the properties of strange stars . the model parameters , i.e. , the bag constant @xmath226 and the strange quark mass @xmath227 , are fixed by comparing with hadron masses . we found that the star is surrounded by a thin pion cloud , which is not significant in the stellar mass and radius . hence , stellar properties are similar in the cloudy bag and mit case , except that the former has slightly larger radius and smaller mass due to a larger @xmath35 . the maximum mass of cloudy bag strange stars is about @xmath228 .    for the rotation properties of cloudy bag strange stars , the approximate formulae were employed . we found that the kepler limit for cloudy bag strange stars with the best - fit parameters is about 6500s@xmath78 , which is lower than the mit one by @xmath229 . the maximum mass of a stable rotating star is about @xmath79 , similar to the value @xmath80 of a mit strange star . the cooling properties of bare strange stars sensitively depend on the gap energy . for small gap energy ( @xmath1 = 1mev ) , the cooling properties of mit strange stars and cloudy strange stars are very much different because the pion emission can cool the strange star rapidly . for the large gap energy ( @xmath1 = 100mev ) , the quarks become superconducting immediately after the star is formed , and the pion emission is totally suppressed . mit strange stars and cloudy strange stars become no difference . furthermore , large gap energy also creates the effect of color - flavor locked , which significantly reduces the electron concentration and its capacity . this makes the cooling curves lower than those with small gap energy . from fig13 and fig14 , we can see that the surface luminosity of a bare strange star is higher than that of a neutron star until @xmath2 and @xmath3 for ( @xmath1 = 100mev ) and ( @xmath1 = 1mev ) respectively . after this period , the surface luminosity of a bare strange star becomes lower than that of a neutron star even rapidly cooling mechanisms , e.g. direct urca process or pion condensation , exist in the core of neutron stars . hence , the cooling behavior may provide a possible way to distinguish a compact object between a neutron star , mit strange star and cloudy bag strange star in observations . it is interesting to note that although the mit and cloudy strange stars are hotter than neutron stars before @xmath217s for @xmath1 = 100mev and @xmath218s for @xmath1 = 1mev respectively(cf . fig10 ) , the thermal radiation of neutron stars are still stronger than those of strange stars until until @xmath2 and @xmath3 for ( @xmath1 = 100mev ) and ( @xmath1 = 1mev ) respectively . it is because the cooling mechanism at the later stage is dominated by the black body radiation , which is strongly suppressed for strange stars as pointed in section ivb . > from fig13 and fig14 , we can see that the cooling curves of a mit strange star and a cloudy strange star are the same when the cooling mechanism is dominated by blackbody radiation . however , the radiation properties of a mit strange star and a cloudy strange star are very much different in the early stage ( t @xmath230 ) for small gap energy case . the cloudy strange star is first cooling very rapidly by emitting pions , which decay to 100mev photons and its temperature drops quickly below that of mit strange star . therefore the mit strange star is cooling mainly by emitting electrons / positrons . the total energy of a cloudy strange star carried away by 100mev photons is @xmath231 , which makes the evaporation of pions a very efficient @xmath0-ray burst mechanism . finally , we want to remark that the cooling curves of strange stars are lower than those cooling curves of neutron stars , which are consistent with the current observed data  @xcite . therefore we actually suggest that the cooling curves may be one possible way to differentiate strange stars from neutron stars . for example , strange stars can behave as pulsars if they have strong magnetic field  @xcite  @xcite . therefore their ages can be estimated by their period and period derivative . if these pulsars are found to have thermal emission lower than that of the standard cooling neutron stars , then we can further to see if there are any e@xmath232 annihilation lines , which are red shifted larger than those of neutron stars ( cf . fig . 5 ) . the features of thermal x - ray emitted from these pulsars associated with bare strange stars are also very different from those of neutron stars  @xcite .
as the chiral symmetry is widely recognized as an important driver of the strong interaction dynamics , current strange stars models based on mit bag models do not obey such symmetry . 
 we investigate properties of bare strange stars using the cloudy bag model , in which a pion cloud coupled to the quark - confining bag is introduced such that chiral symmetry is conserved . 
 the parameters in the model , namely the bag constant and strange quark mass are determined self - consistently by fitting the mass spectrum of baryons . 
 then the equation of state is obtained by evaluating the energy - momentum tensor of the system . 
 we find that the stellar properties of the cloudy bag strange stars are similar to those of mit bag models . 
 however , the decay of pions is a very efficient cooling way . 
 in fact it can carry out most the thermal energy in a few milliseconds and directly convert them into 100mev photons via pion decay . 
 this may be a very efficient @xmath0-ray burst mechanism . 
 numerical results indicate that temperature of a cloudy bag strange star is sufficiently lower than a mit one for the small gap energy of color superconductivity(@xmath1 = 1mev ) . on the other hand 
 , large gap energy ( @xmath1 = 100mev ) can suppress the pion emissivity and hence the cooling curves of cloudy model and mit model are almost identical . 
 the long term cooling behaviors of both mit model and cloudy model are determined by the color - flavor locked phase . 
 the surface luminosity of a bare strange star is higher than that of a neutron star until @xmath2 and @xmath3 for ( @xmath1 = 100mev ) and ( @xmath1 = 1mev ) respectively . 
 after this period , the surface luminosity of a bare strange star becomes lower than that of a neutron star even rapidly cooling mechanisms , e.g. direct urca process or pion condensation , exist in the neutron stars . 
 hence , the cooling behavior may provide a possible way to distinguish a compact object between a neutron star , mit strange star and cloudy bag strange star in observations .