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coalescing neutron star binaries are considered among the strongest and most likely sources of gravitational waves to be seen by virgo / ligo interferometers @xcite . due to the emission of gravitational radiation
, binary compact stars decrease their orbital separation and finally merge .
we separate the evolution of a binary system into three phases : _ point - like inspiral _ where orbital separation is much larger than the neutron star radius , _ hydrostationary inspiral _ where orbital separation is just a few times larger than the radius of the neutron star so that hydrodynamics play an important role , and _ merger _ in which the two stars coalesce dynamically .
gravitational waves emitted during last orbits of inspiral phase and the merger phase could yield important information about the equation of state ( eos ) of dense matter @xcite . up to now
, all relativistic calculations ( except those of @xcite ) of the last orbits of inspiral phase have been done for the simplified eos of dense matter - the polytropic eos . in the paper we present results of our studies on late stage of inspiral of binary systems containing equal mass compact stars described by different realistic eos of dense matter .
the calculations are performed in the framework of _ isenberg - wilson - mathews _ approximation to general relativity ( see ref .
@xcite for a review ) .
we consider binary systems consisting of two identical stars .
we choose the gravitational mass of each star to be @xmath4 at infinite separation in order to be consistent with recent population synthesis calculations @xcite and with the current set of well - measured neutron star masses in relativistic binary radio pulsars @xcite .
we assume that the velocity flow in the stellar interiors is irrotational ( the fluid has zero vorticity in the inertial frame ) since the viscosity of neutron star matter ( or strange star matter ) is far too low to ensure synchronization during the late stage of the inspiral @xcite .
in fig . 1 we show gravitational mass versus stellar radius for sequences of static compact stars .
we limit ourselves to neutron stars consisting of nucleons and hyperons and strange quark stars described by the mit bag model . depending on the eos
we obtain the radius of a 1.35 @xmath5 star in the range 10 - 14 km .
we perform calculations for three nuclear eos of dense matter based on modern many - body calculations .
the differences in the m - r relation for stellar models of neutron stars ( solid lines ) shown in fig . 1 reflect the uncertainties in the existing theories of the interactions in nuclear matter .
we consider one soft ( * bpal12 * , @xcite ) and one stiff ( * akmalpr * , @xcite ) eos of matter composed of nucleons , electrons and muons .
we considered also one eos in which hyperons are present at high densities ( * glendnh3 * , @xcite ) .
the neutron star crust is described by means of a realistic eos obtained in the many - body calculations ( see @xcite for details ) .
we note that the equations of state of nuclear matter can be considered static , as the timescales for bulk processes such as viscosity is much longer than the last stages of inspiral considered here .     versus stellar radius @xmath6 for sequences of static neutron stars described by three different nuclear equations of state ( solid lines ) and strange quark stars described by different mit bag models ( dashed lines ) .
, width=491 ]    strange stars are currently considered as a possible alternative to neutron stars as compact objects ( see e.g. @xcite and references therein ) .
typically , they are modeled with an eos based on the mit - bag model in which quark confinement is described by an energy term proportional to the volume ( e.g. @xcite ) in which quark confinement is described by an energy term proportional to the volume @xcite .
there are three physical quantities entering the mit - bag model : the mass of the strange quarks , @xmath7 , the bag constant , @xmath8 , and the strength of the qcd coupling constant @xmath9 . in the framework of this model the quark matter is composed of massless u , d quarks , massive s quarks and electrons .
strange stars are self - bound objects , having high density ( in the range @xmath10 $ ] @xcite ) at the surface . dashed lines in fig . 1 correspond to sequences of static strange quark stars described by three different sets of parameters of the mit - bag model : * sqs56 * - the standard mit bag model : @xmath11 , @xmath12 , @xmath13 ; * sqsb60 * - the simplified mit bag model with @xmath14 , @xmath15 ; @xmath16 ; * sqsb40 * - the  extreme  mit bag model ( relatively low strange quark mass and @xmath8 but high @xmath9 ) : @xmath17 , @xmath18 , @xmath19 .
in the inspiral phase , since the timescale of orbital shrinking due to the emission of gravitational waves is longer than the orbital period , one may consider a binary system to be in a quasi - equilibrium state ( helical killing vector approximation ) .
for each eos , we construct so called an _ evolutionary sequence _ by calculating a sequence of quasi - equilibrium configurations with fixed baryon mass and decreasing orbital separation .
we assume that the timescale in which the neutron star adjusts to the current gravitational potential is much shorter that the timescale of the orbital shrinking .
the present computations of close binary neutron star or strange star systems rely on the assumption of irrotational flow of the fluid and a conformally flat spatial 3-metric ( isenberg - wilson - mathews approximation ) . in order to calculate the last orbits of inspiral phase of binary compact stars we use a numerical code which solves the
five coupled , nonlinear , elliptic equations for the gravitational field , supplemented by an elliptic equation for the velocity potential of irrotational flows ( see@xcite for a discussion on different boundary conditions in the case of strange stars and neutron stars ) . the code has been already used successfully for calculating the final phase of the inspiral of compact stars @xcite .
this code is built upon the c++ library lorene ( http://www.lorene.obspm.fr ) .
the complete description of the resulting general relativistic equations , the whole algorithm , as well numerous tests of the code can be found in @xcite .
additional tests have been presented in sect .
iii of @xcite .
in fig . 2 we show the evolution of equal mass binary neutron stars ( solid lines ) and strange stars ( dashed lines ) having total gravitational mass @xmath20 at infinity .
the binding energy @xmath21 is defined as the difference between @xmath22 ( arnowitt - deser - misner mass - the total mass - energy of a binary system @xcite ) and the total mass of the system at infinite separation .
this energy is equal to total energy emitted by a binary system in gravitational waves .
the frequency of gravitational waves is twice the orbital frequency .
comparison of our numerical results with 3rd order pn point masses calculations @xcite reveals a good agreement for small frequencies ( large separations ) .
the deviation from pn curves at higher frequencies ( smaller separation ) is due to hydrodynamical effects , which are not taken into account in the pn approach .
a turning point of @xmath21 along an irrotational evolutionary sequence indicates the orbital ( dynamical ) instability @xcite .
this instability originates both from relativistic effects and hydrodynamical effects . in the case where no turning point of @xmath21 occurs along the sequence ,
the mass - shedding limit ( roche lobe overflow ) marks the end of the inspiral phase of the binary system , since recent dynamical calculations for @xmath23 polytrope have shown that the time to coalescence was shorter than one orbital period for configurations at the mass - shedding limit ( i.e. see @xcite ) .
thus the physical inspiral of binary compact stars terminates by either the orbital instability ( turning point of @xmath21 ) or the mass - shedding limit . in both cases
, this defines the _ innermost stable circular orbit ( isco)_. the end of inspiral phase strongly depends on eos - for irrotational neutron star binaries a quasi - equilibrium sequence terminates by mass - shedding limit ( circles at the end of each line in fig 2 ) and for strange stars by orbital instability ( shown as diamonds in fig .
the differences in the evolution of binary strange stars and neutron stars stem from the fact that strange stars are principally bound by additional force than gravitation : the strong interaction between quarks .
they are self - bound objects having very high adiabatic index at the stellar surface ( see @xcite ) .
although the crust of a @xmath24 neutron star contains only a few percent of the stellar mass , this region is easily deformed under the action of the tidal forces resulting from the gravitational field produced by the companion star .
the end of inspiral phase of binary stars strongly depends on the stiffness of matter in this region .
the frequency of gravitational waves at the isco is one of potentially observable parameters by the gravitational wave detectors .
in addition so called `` break frequency '' , a characteristic frequency where the power emitted in gravitational waves decreases measurably @xcite , could be an observable quantity . _ the frequencies of gravitational waves at the depature point from the 3pn aproximation and at the isco strongly depend on the eos .
_ for irrotational equal mass ( of 1.35 @xmath25 at infinite separation ) binaries the isco frequency is @xmath26 for strange stars described by the mit bag model and between @xmath27 and @xmath28 for neutron stars described by nuclear eos ( see also @xcite ) .
the 3rd pn approximations for point masses derived by different authors are giving isco at very high frequencies of gravitational waves @xmath29 ( @xcite ) .
it should be mentioned that the frequency at isco found in quasi - equilibrium approximation could differ from that obtained by the numerical integration of the full set of time dependent einstein equations .
the only comparison ( according to the authors knowledge ) between these two approches for binary neutron stars can be found in marronetti et al .
( 2004 ) who showed that for a polytropic eos with @xmath30 and @xmath31 the frequency at the isco at quasi - equilibrium approximation is @xmath32 higher than the dynamical isco .
thus full relativistic hydrodynamical calculations are required in order to obtain detailed predictions for particular forms of the nuclear matter equations of state .
this work was partially supported by the `` ayudas para movilidad de profesores de universidad e invesigadores espanoles y extranjeros '' program of the spanish mec , by the grants 1 p03d 005 30 , pbz - kbn-054/p03/2001 and 5 p03d 01721 , by the `` bourses de recherche 2004 de la ville de paris '' and by the associated european laboratory astro - pf ( astrophysics poland - france ) .
dgr is grateful to the cospar meeting organizers for support .
mb was partially supported by the marie curie intra - european fellowship meif - ct-2005 - 023644 .    c.  alcock , e.  farhi , and a. olinto , astrophys .
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coalescing compact star binaries are expected to be among the strongest sources of gravitational radiation to be seen by laser interferometers . 
 we present calculations of the final phase of inspiral of equal mass irrotational neutron star binaries and strange quark star binaries . 
 six types of equations of state at zero temperature are used - three realistic nuclear equations of state of various softness and three different mit bag models of strange quark matter . 
 we study the precoalescing stage within the isenberg - wilson - mathews approximation of general relativity using a multidomain spectral method . 
 the gravitational - radiation driven evolution of the binary system is approximated by a sequence of quasi - equilibrium configurations at fixed baryon number and decreasing separation . 
 we find that the innermost stable circular orbit ( isco ) is given by an orbital instability for binary strange quark stars and by the mass - shedding limit for neutron star binaries . 
 the gravitational wave frequency at the isco , which marks the end of the inspiral phase , is found to be @xmath0  hz for two @xmath1 irrotational strange stars described by the mit bag model and between @xmath2 and @xmath3 for neutron stars .    ,    ,    ,    ,    
 ,    gravitational waves - relativity- stars : binaries- stars : neutron- equation of state