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the theoretical possibility that strange quark matter  made up of roughly equal numbers of up , down and strange quarks  may be more stable than atomic nuclei ( specifically iron , which is the most stable atomic nucleus ) constitutes one of the most startling predictions of modern physics ( see @xcite ) , which , if true , would have implications of greatest importance for laboratory physics , cosmology , the early universe , its evolution to the present day , and massive astrophysical objects ( cf .
unfortunately it seems unlikely that lattice qcd calculations will be accurate enough in the foreseeable future to give a definitive prediction on the absolute stability of strange matter , so that one is presently left with experiments and astrophysical studies ( cf .
@xcite ) to either confirm or reject the absolute stability of strange matter .
this letter , dealing with the second item , compares the cooling behavior of neutron stars with the one of their hypothetical strange counterpars  strange stars ( cf .
the theoretical predictions are compared with the body of observed data taken by rosat and asca .
there have been investigations on this topic prior to this one ( e.g. , see @xcite ) .
these , however , did not incorporate the so - called standard cooling scenario that turns out to be possible not only in neutron star matter but in strange quark matter too , altering some of the conclusions made in the earlier investigations significantly .
we use the mit bag model including @xmath1-corrections ( cf .
@xcite ) to model the properties of absolutely stable strange matter .
its equation of state and quark - lepton composition , which is governed by the conditions of chemical equilibrium and electric charge neutrality , is derived for that range of model parameters , that is , bag constant @xmath2 , the strange quark mass @xmath3 , and strong coupling constant @xmath4 , for which strange matter is absolutely stable ( energy per baryon @xmath5 less than the one in iron , @xmath6  mev ) . in the limiting case of vanishing quark mass ,
the electrons are not necesaary to maintain charge neutrality . in the realistic case of finite strange quark masse @xmath7 , the electrons can nevertheless vanish above some density which depends on @xmath8 .
it was pointed out by duncan et al .
( 1983 ) ( see also @xcite ) that the neutrino emissivity of strange matter depends strongly on its electron fraction , @xmath9 .
for that reason we introduce two different , complementary parameter sets denoted sm-1 and sm-2 ( see tab .
[ tab : parameter ] ) , which correspond to strange matter that contains a relatively high electron fraction ( sm-1 ) , and @xmath10 ( sm-2 ) for the density range being of interest here .
the quark _ direct _ urca processes @xmath11 and @xmath12 as well as their inverse ones are only possible if the fermi momenta of quarks and electrons ( @xmath13 , @xmath14u , d , s;e@xmath15 ) fulfill the so - called triangle inequality ( e.g. , @xmath16 for process ( [ eq : dtou ] ) ) .
this relation is the analogue to the triangle inequality established for nucleons and electrons in the nuclear matter case ( direct urca process , @xcite ) .
if the electron fermi momentum is too small ( i.e. , @xmath9 is too little ) , then the triangle inequality for the above processes ( [ eq : dtou ] ) and ( [ eq : stou ] ) can not be fulfilled , and a bystander quark is needed to ensure energy and momentum conservation in the scattering process .
the latter process is known as the quark _ modified _ urca process , whose emissivity is considerably smaller than the emissivity of the direct urca process .
if the electron fraction vanishes entirely , as is the case for sm-2 , both the quark direct and the quark modified urca processes become unimportant .
the neutrino emission is then dominated by bremsstrahlung processes only , @xmath17 where @xmath18 , @xmath19 denote any pair of quark flavors . for the emissivities associated with the quark direct urca , quark modified urca , and quark bremsstrahlung processes
, we refer to refs .
price ( 1980 ) , iwamoto ( 1982 ) and duncan et al .
( 1983 ) .
it has been suggested ( see @xcite ) that the quarks eventually may form cooper pairs .
this would suppress , as in the nuclear matter case , the neutrino emissivities by an exponential factor of @xmath20 , where @xmath21 is the gap energy , @xmath22 boltzmann s constant , and @xmath23 the temperature .
unfortunately , there exists up to now neither a precise experimentally nor theoretically determined value of the gap energy .
so to provide a feeling for the influence of a possibly superfluid behavior of the quarks in strange matter too , we choose @xmath24  mev as estimated in the work of bailin & love ( 1979 ) .
( such a @xmath21 value is not too different from the nuclear - matter case , where the proton @xmath25 gap , for instance , amounts @xmath26  mev ( cf .
@xcite ) , depending on the nucleon - nucleon interaction and the microscopic model . )
the outcome of our superfluid strange matter calculations will be labeled sm-1@xmath27 and sm-2@xmath27 .
among the x - ray observations of the 14 sources which were identified as pulsars , the rosat and asca observations of psrs 0833 - 45 ( vela ) , 0656 + 14 ,
0630 + 18 ( geminga ) and 1055 - 52 ( see tab . [ tab : observations ] ) achieved a sufficiently high photon flux such that the effective surface temperatures of these pulsars could be extracted by two- or three - component spectral fits ( cf .
the obtained effective surface temperatures , shown in figs .
[ fig : nsf ] and [ fig : sf ] , depend crucially on whether a hydrogen atmosphere is used or not .
since the photon flux measured solely in the x - ray energy band does not allow one to determine what kind of atmosphere one should use , we consider both the blackbody model and the hydrogen - atmosphere model , drawn in in figs .
[ fig : nsf ] and [ fig : sf ] as solid and dashed error bars , respectively .
the kind of atmosphere possessed by a specific pulsar could be determined by considering multiwavelength observations ( see @xcite ) .
all error bars represent the @xmath28 error range due to the small photon fluxes .
the pulsars ages are determined by their spin - down times assuming a canonical value of 3 for the braking index . in reality
the braking index may be quite different from 3 .
its variation between 2 and 4 , for instance , would change the age of geminga as indicated by the horizontal error bar shown at the bottom of figs .
[ fig : nsf ] and [ fig : sf ] .
the thermal evolution of strange stars and neutron stars was simulated using the evolutionary numerical code described in schaab et al .
( 1996b ) ( see also @xcite ) .
the neutron star models are based on a broad collection of eoss which comprises relativistic , fieldtheoretical equations of state as well as non - relativistic , schroedinger - based ones ( see @xcite for details ) . as a specific feature of the relativistic models , they account for all baryon states that become populated in dense neutron star matter up to the highest densities reached in the cores of the heaviest neutron stars constructed from this collection of equations of state .
neutron stars are known to loose energy either via standard cooling or enhanced cooling . both may be delayed by superfluidity .
consequently all four options have been taken into account here .
these are labeled in figs .
[ fig : nsf ] and [ fig : sf ] as ns-1 ( enhanced cooling ) and ns-2 ( standard cooling ) for normal neutron star matter , and ns-1@xmath27 and ns-2@xmath27 ( delayed cooling ) for superfluid neutron star matter .
the parameters of ns-1@xmath27 and ns-2@xmath27 are listed in tab .  4 of ref .
schaab et al .
( 1996b ) . in analogy to this ,
the corresponding strange - star cooling curves are sm-1 ( enhanced cooling ) and sm-2 ( standard cooling ) for normal strange quark matter , and sm-1@xmath27 and sm-2@xmath27 ( delayed cooling ) for superfluid quark matter .
all calculations are performed for a star mass of @xmath29 , about which the observed pulsar masses tend to scatter .
the band - like structure of the cooling curves is supposed to reflect the uncertainties inherent in the equation of state of neutron - star and strange - star matter .
these have their origin , in the case of neutron stars ( bands filled with dots ) , in the different many - body techniques used to solve the nuclear many - body problem , and the star s baryon - lepton composition . in the latter case ,
strange - star matter , the solid bands refer to different bag values , @xmath2 , which vary from 137 to 148 mev for sm-1 , and from 133 to 146 mev for sm-2 .
all values correspond to absolutely stable strange matter .
one might suspect that the large gap between the cooling tracks of the sm-1 and sm-2 models in fig .
[ fig : nsf ] can be bridged steadily by varying the strong coupling constant @xmath8 in the range 0.10.15 .
however it turns out that the gap can be filled only for @xmath8-values within an extremely small range .
this is caused by the sensitive functional relationship between @xmath8 and the neutrino luminosity @xmath30 , which is rather steep around that @xmath8-value for which the electrons vanish from the quark core of the star .
all other values of @xmath8 give cooling tracks which are close to the upper or lower bands , respectively .
this behavior might be compared with the case of neutron stars , where the neutrino luminosity depends sensitively on the star s mass .
one sees from figs .
[ fig : nsf ] and [ fig : sf ] that , except for the first @xmath0 years of the lifetime of a newly born pulsar , both neutron stars and strange stars may show more or less the _ same _ cooling behavior , provided both types of stars are made up of either normal matter or superfluid matter .
( we will come back to this issue below . ) this is made possible by the fact that both standard cooling ( ns-2 ) as well as enhanced cooling ( ns-1 ) in neutron stars has its counterpart in strange stars too ( sm-2 and sm-1 , respectively ) .
the point of time at which the surface temperature drop of a strange star occurs depends on the thickness of the nuclear crust that may envelope the strange matter core ( cf .
@xcite ) . in the present calculation ,
strange stars possess the densest possible nuclear crust , which is about 0.2 km thick .
thinner crusts would lead to temperature drops at even earlier times , and thus an earlier onset of the photon cooling era .
figures [ fig : nsf ] and [ fig : sf ] indicate that the cooling data of observed pulsars do not allow to decide about the true nature of the underlying collapsed star , that is , as to whether it is a strange star or a conventional neutron star .
this could abruptly change with the observation of a very young pulsar shortly after its formation in a supernova explosion . in this case
a prompt drop of the pulsar s temperature , say within the first 30 years after its formation , could offer a good signature of a strange star ( see @xcite ) .
this feature , provided it withstands a rigorous future analysis of the microscopic properties of quark matter , could become particularly interesting if continued observation of sn 1987a would reveal the temperature of the possibly existing pulsar at its center .    finally , we add some comments about the possibility that only the neutron star is made up of superfluid matter but not the strange star . in this case one
has to compare the models sm-1 and sm-2 ( see fig .
[ fig : nsf ] ) with models ns-1@xmath31 and ns-2@xmath31 ( see fig . [
fig : sf ] ) yielding to an overall different cooling history of neutron stars and enhanced - cooling strange stars ( sm-1 ) . therefore , the standard argument pointed out quite frequently in the literature that strange stars cool much more rapidly than neutron stars applies only to this special case .
.bag constant , @xmath2 , strange quark mass , @xmath3 , and the qcd coupling constant , @xmath4 , for the two sets of parameters denoted sm-1 and sm-2 .
the energy per baryon , @xmath5 , for two and three flavor quark matter is given too . [
tab : parameter ] [ cols="<,^,^ " , ]                                                                    schaab , c. , voskresenski , d. , sedrakian , a.  d. , weber , f. & weigel , m.  k. 1996 , impact of medium effects on the cooling of non  superfluid and superfluid neutron stars , to be published in a&a , prepint astro - ph/9605188
the general statement that hypothetical strange ( quark matter ) stars cool more rapidly than neutron stars is investigated in greater detail . 
 it is found that the direct urca process could be forbidden not only in neutron stars but also in strange stars . in this case 
 , strange stars are slowly cooling , and their surface temperatures are more or less indistinguishable from those of slowly cooling neutron stars . furthermore the case of enhanced cooling is reinvestigated . 
 it shows that strange stars cool significantly more rapidly than neutron stars within the first @xmath0 years after birth . 
 this feature could become particularly interesting if continued observation of sn 1987a would reveal the temperature of the possibly existing pulsar at its center .