Last orbits of binary strange quark stars

PHYSICAL REVIEW D 71, 064012 (2005)

Last orbits of binary strange quark stars

Francois Limousin,1,* Dorota Gondek-Rosinska,1,2,3,4,† and Eric Gourgoulhon1,‡

1Laboratoire de l’Univers et de ses Theories, UMR 8102 du C.N.R.S., Observatoire de Paris, Universite Paris7, F-92195 Meudon Cedex, France

2Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716 Warszawa, Poland3Departament de Fisica Aplicada, Universitat d’Alacant, Apartat de correus 99, 03080 Alacant, Spain

4Institute of Astronomy, University of Zielona Gora, Lubuska 2, 65-265, Zielona Gora, Poland(Received 26 November 2004; published 14 March 2005)

*Electronic†Electronic‡Electronic

1550-7998=20

We present the first relativistic calculations of the final phase of inspiral of a binary system consisting oftwo stars built predominantly of strange quark matter (strange quark stars). We study the precoalescingstage within the Isenberg-Wilson-Mathews approximation of general relativity using a multidomainspectral method. A hydrodynamical treatment is performed under the assumption that the flow is eitherrigidly rotating or irrotational, taking into account the finite density at the stellar surface—a distinctivefeature with respect to the neutron star case. The gravitational-radiation driven evolution of the binarysystem is approximated by a sequence of quasiequilibrium configurations at fixed baryon number anddecreasing separation. We find that the innermost stable circular orbit (ISCO) is given by an orbitalinstability both for synchronized and irrotational systems. This contrasts with neutron stars for which theISCO is given by the mass-shedding limit in the irrotational case. The gravitational wave frequency at theISCO, which marks the end of the inspiral phase, is found to be �1400 Hz for two irrotational 1:35 M�

strange stars and for the MIT bag model of strange matter with massless quarks and a bag constant B �60 MeV fm�3. Detailed comparisons with binary neutrons star models, as well as with third order post-Newtonian point-mass binaries are given.

DOI: 10.1103/PhysRevD.71.064012 PACS numbers: 04.40.Dg, 04.25.Dm, 04.30.Db, 97.10.Kc

I. INTRODUCTION

One of the most important predictions of general rela-tivity is gravitational radiation. Coalescing neutron starbinaries are considered among the strongest and mostlikely sources of gravitational waves to be seen byVIRGO/LIGO interferometers [1,2]. Because of the emis-sion of gravitational radiation, binary neutron stars de-crease their orbital separation and finally merge.Gravitational waves emitted during the last few orbits ofinspiral could yield important informations about the equa-tion of state (EOS) of dense matter [3–6]. With accuratetemplates of gravitational waves from coalescing binarycompact stars, it may be possible to extract informationabout physics of neutron stars from signals observed by theinterferometers and to solve one of the central but alsomost complex problem of physics—the problem of theabsolute ground state of matter at high densities. It is stillan open question whether the core of a neutron star consistsmainly of superfluid neutrons or exotic matter like strangequark matter, pions, or kaons condensates (see, e.g.,Ref. [7] for a recent review). The possibility of the exis-tence of quark matter dates back to the early seventies.Bodmer [8] remarked that matter consisting of deconfinedup, down, and strange quarks could be the absolute groundstate of matter at zero pressure and temperature. If this istrue then objects made of such matter, the so-called strange

address: [email protected]: [email protected]: [email protected]

05=71(6)=064012(11)$23.00 064012

stars, could exist [9–11]. Strange quark stars are currentlyconsidered as a possible alternative to neutron stars ascompact objects (see e.g. [12–14] and references therein).

The evolution of a binary system of compact objects isentirely driven by gravitational radiation and can beroughly divided into three phases : pointlike inspiral, hy-drodynamical inspiral, and merger. The first phase corre-sponds to large orbital separation (much larger than theneutron star radius) and can be treated analytically usingthe post-Newtonian (PN) approximation to general relativ-ity (see Ref. [15] for a review). In the second phase theorbital separation becomes only a few times larger than theradius of the star, so the effects of tidal deformation, finitesize, and hydrodynamics play an important role. In thisphase, since the shrinking time of the orbital radius due tothe emission of gravitational waves is still larger than theorbital period, it is possible to approximate the state asquasiequilibrium [16,17]. The final phase of the evolutionis the merger of the two objects, which occur dynamically[18–21]. Note that quasiequilibrium computations fromthe second phase provide valuable initial data for themerger [5,18,20,22].

Almost all studies of the final phase of the inspiral ofclose binary neutron star systems employ a simplified EOSof dense matter, namely, a polytropic EOS [3,4,17,22–28].There are only two exceptions: (i) Oechslin et al. have useda pure nuclear matter EOS, based on a relativistic meanfield model and a ‘‘hybrid’’ EOS with a phase transition toquark matter at high density [5]; (ii) Bejger et al. havecomputed quasiequilibrium sequences based on three nu-clear matter EOS [6]. In this article we present results on

-1 2005 The American Physical Society

LIMOUSIN, GONDEK-ROSINSKA, AND GOURGOULHON PHYSICAL REVIEW D 71, 064012 (2005)

the hydrodynamical phase of inspiraling binary strangequark stars described by MIT bag model. The calculationsare performed in the framework of Isenberg-Wilson-Mathews approximation to general relativity (seeRef. [29] for a review). We consider binary systems con-sisting of two identical stars. We choose the gravitationalmass of each star to be 1:35 M� in infinite separation inorder to be consistent with recent population synthesiscalculations [30] and with the current set of well-measuredneutron star masses in relativistic binary radio pulsars[31,32]. We compare the evolution of a strange star binarysystem with a neutron star binary in order to find anycharacteristic features in the gravitational waveform thatwill help to distinguish between strange stars and neutronstars. We consider two limiting cases of velocity flow instellar interior: the irrotational and the synchronized casein order to exhibit the differences between these two ex-treme states. The irrotational case is more realistic sincethe viscosity of neutron star matter (or strange star matter)is far too low to ensure synchronization during the latestage of the inspiral [33,34]. Because of the finite density atthe surface of bare strange stars, we had to introduce atreatment of the boundary condition for the velocity po-tential (in the irrotational case) different from that ofneutron stars, where the density vanishes at the stellarsurface.

The paper is organized in the following way: Sec. II is abrief summary of the assumptions upon which this work isbased; Sec. III is devoted to the description of the EOSused to describe strange stars and neutron stars. In Sec. IVwe briefly describe the basic equations for quasiequili-brium and derive the boundary condition required forsolving the fluid equation of irrotational flow with finitesurface density, which is relevant for strange stars. InSec. V we present the numerical results for co-rotatingand irrotational strange stars binaries and compare theirquasistationary evolution with that of neutron stars, as wellas with that of post-Newtonian point-masses. Section VIcontains the final discussion. Throughout the paper, we usegeometrized units, for which G � c � 1, where G and cdenote the gravitational constant and speed of light,respectively.

II. ASSUMPTIONS

The first assumption regards the matter stress-energytensor T, which we assume to have the perfect fluid form:

T � �e� p�u � u� pg; (1)

where e, p, u and g are, respectively, the fluid properenergy density, the fluid pressure, the fluid 4-velocity,and the spacetime metric. This is a very good approxima-tion for neutron star matter or strange star matter.

The last orbits of inspiraling binary compact stars can bestudied in the quasiequilibrium approximation. Underthis assumption the evolution of a system is approximated

064012

by a sequence of exactly circular orbits. This assumptionresults from the fact that the time evolution of an orbit isstill much larger than the orbital period and that the gravi-tational radiation circularizes an orbit of a binary system.This implies a continuous spacetime symmetry, calledhelical symmetry [35,36] represented by the Killing vector:

‘ �@@t

� �@@’

; (2)

where � is the orbital angular velocity and @=@t and @=@’are the natural frame vectors associated with the timecoordinate t and the azimuthal coordinate ’ of an asymp-totic inertial observer.

One can then introduce the shift vector B of co-orbitingcoordinates by means of the orthogonal decomposition of ‘with respect to the �t foliation of the standard 3 � 1formalism:

‘ � Nn�B; (3)

where n is the unit future directed vector normal to �t,N iscalled the lapse function, and n B � 0.

We also assume that the spatial part of the metric (i.e.,the metric induced by g on each hypersurface �t) is con-formally flat, which corresponds to the Isenberg-Wilson-Mathews (IWM) approximation to general relativity [37–39] (see Ref. [36] for a discussion). Thanks to this approxi-mation we have to solve only five of the ten Einsteinequations. In the IWM approximation, the spacetime met-ric takes the form:

ds2 � ��N2 � BiBi�dt2 � 2Bidtdxi � A2fijdxidxj; (4)

where A is some conformal factor, fij the flat spatialmetric, and Latin indices run in f1; 2; 3g (spatial indices).The comparison between the IWM results presented hereand the nonconformally flat ones will be performed in afuture article [40].

The fourth assumption concerns the fluid motion insideeach star. We consider two limiting cases: synchronized(also called co-rotating) motion and irrotational flow(assuming that the fluid has zero vorticity in the inertialframe). The latter state is more realistic.

We consider only equal-mass binaries consisting ofidentical stars with gravitational masses M1 � M2 �1:35 M� measured in infinite separation. The main reasonfor choosing these particular masses is that five out of sixobserved binary radio pulsars have mass ratio close to unityand gravitational masses of each star �1:3 � 1:4 M�

[31,32]. In addition, population synthesis calculations[30] have shown that a significant fraction of the observedbinary neutron stars in gravitational waves will containstars with equal masses �1:4 M� and systems consistingof a low and a high mass neutron star.

-2

8 9 10 11 12R [km]

0.8

1

1.2

1.4

1.6

1.8

2

M [

Mso

l]

FIG. 1 (color online). Gravitational mass M versus areal radiusR for sequences of static strange quark stars described by thesimplest MIT bag model (solid line) and neutron stars describedby polytropic EOS with � � 2:5 and � � 0:0093m0n

�1:5nuc

(dashed line). The two sequences are crossing at the point M �1:35 M� and R � 10:677 km (marked by a circle).

LAST ORBITS OF BINARY STRANGE QUARK STARS PHYSICAL REVIEW D 71, 064012 (2005)

III. THE EQUATION OF STATE ANDSTELLAR MODELS

It has been shown [3,4] that the evolution of equal-massbinary neutron stars depend mainly on the compactnessparameter M=R, where M and R are the gravitational massmeasured by an observer at infinity for a single isolatedneutron star and the stellar radius, respectively. It is there-fore interesting to check if the properties of inspiralingstrange stars can be predicted by studying binaries consist-ing of polytropic neutron stars having the same mass andthe same compactness parameter. Therefore we performcalculations for two different equations of state of densematter: a strange quark matter EOS and a polytropic EOS.

Typically, strange stars are modeled [10,11] with anequation of state based on the MIT bag model of quarkmatter, in which quark confinement is described by anenergy term proportional to the volume [41]. The equationof state is given by the simple formula

p � a�� 0�; (5)

n�p� � n0

�1 �

1 � aa

p�0

�1=�1�a�

; (6)

where n is the baryon density and a; �0; n0 are someconstants depending on the three parameters of the model(the bag constant B, the mass of the strange quarks ms, andthe strength of the QCD coupling constant �). In generalthis equation corresponds to a self-bound matter with massdensity �0 and baryon density n0 at zero pressure and witha fixed sound velocity (

��a

p) at all pressures. It was shown

that different strange quark models can be approximatedvery well by Eqs. (5) and (6) [42,43].

In the numerical calculations reported in the presentpaper we describe strange quark matter using the simplestMIT bag model (with massless and noninteracting quarks),for which the formula (5) is exact. We choose the value ofthe bag constant to be B � 60 MeV fm�3. For this modelwe have a � 1=3, �0 � 4:2785 1014 g cm�3, and n0 �0:28 665 fm�3. In general for the MIT bag model thedensity of strange matter at zero pressure is in the range�3 1014 � 6:5 1014 g cm�3 and a between 0.289 and1=3 (for 0 � � � 0:6 and 0 � ms � 250 MeV [49]). Thehigher value of a and of �0 the higher compactness pa-rameter for a star with fixed gravitational mass.

Up to now, the majority of calculations of the hydro-dynamical inspiral phase [3,4,17,22–28] and all calcula-tions of the merger phase [18–21] have been performed forbinary systems containing neutron stars described by apolytropic EOS:

p � �n�; (7)

where � and � coefficients are some constant numbers: �represents the overall compressibility of matter while �measures the stiffness of the EOS. The total energy densityis related to the baryon density by

064012

e�n� ��

�� 1n� ��0n; (8)

where �0 is the chemical potential at zero pressure.In order to compare results for strange stars with those

for neutron stars, we determine the values of � and �whichyield to the same radius for the gravitational mass M �1:35 M� as that obtained for a static strange star. It wasshown [6] that the properties of inspiraling neutron starsdescribed by realistic EOS can be, in a good approxima-tion, predicted by studying binaries with assumed poly-tropic EOSs with � � 2 or 2.5. For a 1:35 M� strange starwe have a high value of compactness parameter M=R �0:1867 so we have chosen � � 2:5, for which we found� � 0:0093m0n

�1:5nuc , with m0: � 1:66 10�27 kg and

nnuc � 0:1 fm�3.In Fig. 1 we present the mass-radius relation for a

sequence of static stars described by the simplest MITbag model (solid line) and the polytropic EOS (dashedline) parametrized by central density. For small massstrange stars M� R3 since density is almost constant in-side a star ��0. In the top panel of Fig. 2 we show the massdensity distribution inside the strange star (solid line) andthe neutron star described by polytropic EOS (dashed line)having gravitational mass 1:35 M� and areal radius10:667 km (the configurations corresponding to the cross-ing point on Fig. 1). The huge density jump at the surfaceof the strange star corresponds to �0 � 4B. The value ofdensity at the surface describes strongly or weakly boundstrange matter, which in every case must be absolutelystable with respect to 56Fe.

An important quantity relevant for evolution of binarycompact stars is the adiabatic index

-3

0 2 4 6 8 10 12r [km]

0

2

4

6

8

10

12

14 ρ

[1014

g cm

-3]

0 2 4 6 8 10 12r [km]

0

5

10

15

20

25

30

γ

FIG. 2 (color online). Mass density (top panel) and the adia-batic index � (bottom panel) versus the areal radial coordinate rfor a static strange quark star (solid line) and a polytropicneutron star (dashed line), having both a gravitational mass of1:35 M� and an areal radius R � 10:677 km (resulting in thecompactness parameter M=R � 0:1867). The vertical dotted linecorresponds to the stellar surface.


� � d lnp=d lnn: (9)

We assume that matter is catalyzed so the adiabatic indexcan be calculated directly from EOS (see Refs. [44,45] fordiscussion on different kind of adiabatic indices and cor-responding timescales). Note that for the polytropic EOSgiven by Eq. (7) the index � coincides with the adiabaticindex of a relativistic isentropic fluid. Dependence of theadiabatic index � on stellar radii for both EOS is shown inthe bottom pannel of Fig. 2. The adiabatic index of strangematter is qualitatively different from the adiabatic index forpolytropic EOS or for realistic EOS. The values of � in theouter layers of strange stars are very large and for �! �0

we have � � a� �=�� 0� ! 1. The EOS of neutronstars for densities lower than �1014 g cm�3 (the crust) iswell established [7]. In the outer crust of an ordinaryneutron star the pressure is dominated by the ultrarelativ-istic electron gas, so we have � � 4=3. The values of the

064012

local adiabatic index in the inner crust of a neutron stardepends strongly on density and varies from � ’ 0:5 nearthe neutron drip point to � ’ 1:6 in the bottom layers nearthe crust-core interface.

In our calculations we use equation of state in the form:

n � n�H�; e � e�H�; p � p�H�; (10)

where H is pseudoenthalpy (the log-enthalpy) defined by:

H�n�: � ln�e� pnE0

�; (11)

where the energy per unit baryon number is E0 � m0 for apolytropic EOS, m0 is the rest mass of relativistic particlesand E0 � �0=n0 � 837:26 MeV for strange quark modeldescribed above. For our model of strange quark matter wehave:

� � �0�3e4H � 1�=4; p � �0�e4H � 1�=4;

n � n0e3H:

(12)

IV. EQUATIONS TO BE SOLVED

We refer the reader to Ref. [46] for the derivation of theequations describing quasiequilibrium binary stars withinthe IWM approximation to general relativity. After recall-ing these equations, we mainly concentrate on the equationfor the velocity potential of irrotational flows. Actually thisequation has a different structure for strange stars than forneutron stars. This results from the nonvanishing of thedensity at the stellar surfaces of strange stars (cf. the toppanel in Fig. 2).

A. The gravitational field equations

The gravitational field equations have been obtainedwithin the 3 � 1 decomposition of the Einstein’s equations[47,48], taking into account the helical symmetry of space-time. The trace of the spatial part of the Einstein equationscombined with the Hamiltonian constraint results in twoequations:

�! � 4"A2�E� S� � A2KijKij �ri!r

i%; (13)

�% � 4"A2S�3

4A2KijKij �

1

2�ri!r

i!�ri%ri%�;

(14)

where ri stands for the covariant derivative associatedwith the flat 3-metric fij and �: � riri for the associatedLaplacian operator. The quantities ! and % are defined by!: � lnN and %: � ln�AN�, and Kij denotes the extrinsiccurvature tensor of the t � const hypersurfaces. E and Sare, respectively, the matter energy density and the trace ofthe stress tensor, both as measured by the observer whose4-velocity is n� (Eulerian observer):

-4


E: � T�!n�n!; (15)

S: � A2fijTij: (16)

In addition, we have also to solve the momentum con-straint, which writes

�Ni �1

3ri�rjN

j� � �16"NA2�E� p�Ui

� 2NA2Kijrj�3%� 4!�; (17)

where Ni: � Bi � ��@=@’�i denotes the shift vector ofnonrotating coordinates, and Ui is the fluid 3-velocity.

B. The fluid equations

Apart from the gravitational field equations, we have tosolve the fluid equations. The equations governing thequasiequilibrium state are the relativistic Euler equationand the equation of baryon number conservation. Bothcases of irrotational and synchronized motions admit a firstintegral of the relativistic Euler equation:

H � !� ln!0 � ln! � const; (18)

where !0 is the Lorentz factor between the co-orbitingobserver and the Eulerian observer and ! is the Lorentzfactor between the fluid and the co-orbiting observers (! �1 for synchronized binaries).

For a synchronized motion, the equation of baryonnumber conservation is trivially satisfied, whereas for anirrotational flow, it is written as

)H�" �riHri" � A2h!nUi0riH

� )H�ri"ri�H � %�

� A2hUi0ri!n�; (19)

where " is the velocity potential, h: � exp�H�, ) thethermodynamical coefficient:

) : �d lnHd lnn

; (20)

and !n denotes the Lorentz factor between the fluid and theEulerian observer and Ui

0 is the orbital 3-velocity withrespect to the Eulerian observers:

Ui0 � �

Bi

N: (21)

The fluid 3-velocity Ui with respect to the Eulerianobserver is equal to Ui

0 for synchronized binary systems,whereas

Ui �1

A2!nhri" (22)

for irrotational ones.

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C. Boundary condition for the velocity potential

The method of solving the elliptic Eq. (19) for thevelocity potential is different for neutron stars and strangestars. In the case of neutron stars, the coefficient )H infront of the Laplacian vanishes at the surface of the star soEq. (19) is not merely a Poisson type equation for ". Ittherefore deserves a special treatment (see Appendix B in[26] for a discussion). In the case of strange stars, thecoefficient )H � 1=3 in whole star so we have to dealwith a usual Poisson equation and consequently we have toimpose a boundary condition for the velocity potential atthe stellar surface.

We can define the surface of the star by njsurf � n0 �constant. The surface of the fluid ball is obviously Lie-dragged along the fluid 4-velocity vector u, so that this lastcondition gives

�Lun�jsurf � 0; (23)

where Lu is the Lie derivative along the vector field u. Letus decompose u in a part along the helical Killing vector ‘and a part S parrallel to the hypersurface �t:

u � +�‘� S�: (24)

The condition (23) is then equivalent to

+�L‘n�LSn�jsurf � 0: (25)

Now, if the fluid flows obeys to the helical symmetryL‘n � 0; inserting this relation into Eq. (25) leads to�LSn�jsurf � 0 or equivalently (since S is a spatial vector):

�Sirin�jsurf � 0: (26)

Now, let us express S in terms of the spatial vectors U andB. First, Eq. (24) implies n u � +n ‘. Secondly, thefluid motion u can be described by the orthogonal decom-position u � !n�n� U� which yields n u � �!n.Finally, from Eq. (3), we have n ‘ � �N so that thefactor + can be expressed as + � !n=N and Eq. (24)becomes

u �!nN

�‘� S�: (27)

Now, combining Eq. (27) and Eq. (3), we have

u � !n

�n�

1

N�S� B�

�: (28)

Comparing with the orthogonal decomposition u �!n�n� U�, we deduce that S � NU�B. Inserting thisrelation into Eq. (26) leads to the boundary condition

�NUirin� Birin�jsurf � 0: (29)

Now, using Eq. (22), we obtain a Neumann-like boundarycondition for ":

-5


�rinri"�

��surf� �

�!nhA2

NBirin

��surf: (30)

Considering the elliptic Eq. (19) for " we see that theboundary condition we have obtained is consistent with thecase n � 0 (or equivalently )H � 0) at the stellar surfacesince, from Eq. (20), riH � )H

n rin.

400 600 800 1000 1200fGW

[Hz]

-0.03

-0.025

-0.02

Ebi

nd [

Mso

l]

SQS MITPolytrope γ =2.53PN (Blanchet 2002)

FIG. 3 (color online). Binding energy as a function ofgravitational-wave frequency along evolutionary sequences ofcorotating binaries. The solid line denotes strange quark stars,the dashed one neutron stars with polytropic EOS, and the dottedone point-mass binaries in the 3PN approximation [54]. Thediamonds locate the minimum of the curves, corresponding tothe innermost stable circular orbit; configurations to the right ofthe diamond are secularly unstable.

V. NUMERICAL RESULTS

A. The method

The resolution of the above nonlinear elliptic equationsis performed thanks to a numerical code based on multi-domain spectral methods and constructed upon theLORENE C �� library [50]. The detailed description ofthe whole algorithm, as well as numerous tests of the codecan be found in [46]. Additional tests have been presentedin Sec. III of [4]. The code has already been used success-fully for calculating the final phase of inspiral of binaryneutron stars described by polytropic EOS [4,17,27,51,52]and realistic EOS [6]. It is worth to stress that the adapta-tion of the domains (numerical grids) to the stellar surface(surface-fitted coordinates) used in this code is particularlyuseful here, due to the strong discontinuity of the densityfield at the surface of strange stars (cf. the top panel inFig. 2). Adapting the grids to the stellar surface allows toavoid the severe Gibbs phenomenon that such a disconti-nuity would necessary generate when performing polyno-mial expansions of the fields [53].

The hydrodynamical part of the code has been amendedfor the present purpose, namely, to solve Eq. (19) for thevelocity potential " subject to the boundary condition(30). Let us recall that in the original version of the code,the treatment of Eq. (19) was different due to the vanishingof the density field at the stellar surface (see Appendix B ofRef. [46]).

We have used one numerical domain for each star andthree (respectively four) domains for the space aroundthem for a small (respectively large) separation. In eachdomain, the number of collocation points of the spectralmethod is chosen to be Nr N, N’ � 25 17 16,where Nr, N,, and N’ denote the number of collocationpoints ( � number of polynomials used in the spectralmethod) in the radial, polar, and azimuthal directions,respectively. The accuracy of the computed relativisticmodels has been estimated using a relativistic generaliza-tion of the virial theorem [36] (see also Sec. III A ofRef. [4]). The virial relative error is a few times 10�5 forthe closest configurations.

B. Evolutionary sequences

For each EOS we construct an evolutionary sequence,i.e., a sequence of quasiequilibrium configurations withfixed baryon mass and decreasing separation. Such a se-

064012

quence is expected to approximate pretty well the trueevolution of binary neutron stars, which is entirely drivenby the reaction to gravitational radiation and hence occur atfixed baryon number and fluid circulation.

For a given rotational state we calculate evolutionarysequences of binary system composed of two identicalneutron stars or two identical strange stars. The evolutionof inspiraling corotating (irrotational) binaries is shown inFig. 3 (Fig. 4). Figure 3 and upper panel of Fig. 4 show thebinding energy Ebind versus frequency of gravitationalwaves fGW and lower panel of Fig. 4 shows the totalangular momentum of the systems as a function of fGW.The binding energy is defined as the difference between theactual ADM mass (see e.g. Ref. [52]) of the system,MADM,and the ADM mass at infinite separation (2:7 M� in ourcase). The frequency of gravitational waves is twice theorbital frequency, since it corresponds to the frequency ofthe dominant part l � 2, m � �2. Solid and dashed linesdenote quasiequilibrium sequences of strange quark starsbinaries and neutron stars binaries, respectively. Dottedlines in Figs. 3 and 4 correspond to the 3rd PN approxi-mation for point masses derived by [54]. Finally, in sub-section V D, we compare our results with third order post-Newtonian results for point-mass particles obtained in theeffective one body approach by Damour et al. 2000 [55],Damour et al. 2002 [56], and in the standard nonresummedpost-Newtonian framework by Blanchet 2002 [54].

A turning-point of Ebind along an evolutionary sequenceindicates an orbital instability [36]. This instability origi-nates both from relativistic effects (the well-known r �6M last stable orbit of Schwarzschild metric) and hydro-

-6

1000 1200 1400fGW

[Hz]

-0.04

-0.038

-0.036

-0.034

-0.032E

bind

[M

sol]

SQS MITPolytrope γ = 2.53PN (Blanchet 2002)

1000 1200 1400fGW

[Hz]

0.86

0.88

0.9

J / M

0 2 [

G/c

]

SQS MITPolytrope γ = 2.53PN (Blanchet 2002)

FIG. 4 (color online). Binding energy (top panel) and angularmomentum (bottom panel) as a function of gravitational-wavefrequency along evolutionary sequences of irrotational binaries.The solid line denotes strange quark stars, the dashed linepolytropic neutron stars, and the dotted line point-mass binariesin the 3PN approximation [54]. The diamonds correspond todynamical orbital instability (the ISCO).


dynamical effects (for instance, such an instability existsfor sufficiently stiff EOS in the Newtonian regime, see e.g.[51] and references therein). It is secular for synchronizedsystems and dynamical for irrotational ones.

In the case where no turning point of Ebind occurs alongthe sequence, the mass-shedding limit (Roche lobe over-flow) marks the end of the inspiral phase of the binarysystem, since recent dynamical calculations for � � 2polytrope have shown that the time to coalescence wasshorter than one orbital period for configurations at themass-shedding limit [19,28]. Thus the physical inspiral ofbinary compact stars terminates by either the orbital insta-bility (turning-point ofEbind) or the mass-shedding limit. Inboth cases, this defines the innermost stable circular orbit(ISCO). The orbital frequency at the ISCO is a potentiallyobservable parameter by the gravitational-wave detectorsand thus a very interesting quantity.

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C. Corotating binaries

Quasiequilibrium sequences of equal-mass corotatingbinary neutron stars and strange stars are presented inFig. 3. For both sequences we find a minimum of thebinding energy. In the present rotation state, this locates asecular instability [36]. The important difference betweenneutron stars and strange stars is the frequency at whichthis instability appears. Indeed, there is a difference ofmore than 100 Hz: 1020 Hz for strange stars and1140 Hz for neutron stars. The binding energy is the totalenergy of gravitational waves emitted by the system: acorotating binary strange star system emits less energy ingravitational waves and loses less angular momentumbefore the ISCO than a binary neutron star one with thesame mass and compaction parameter in infiniteseparation.

Comparison of our numerical results with 3rd order PNcalculations reveals a good agreement for small frequen-cies (large separations) (see Fig. 3.) The deviation from PNcurves at higher frequencies (smaller separation) is due tohydrodynamical effects, which are not taken into accountin the PN approach.

D. Irrotational binaries

In Fig. 4 we present the evolution of the binding energyand angular momentum for irrotational sequences of bi-nary neutron stars and strange stars. We also verify thatthese sequences are in a good agreement with PN calcu-lations for large separations.

We note important differences in the evolution of binarysystems consisting of strange stars or neutron stars. Thestrange star sequence shows a minimum of the bindingenergy at fGW ’ 1390 Hz, which locates a dynamical in-stability [36] and thus defines the ISCO. The minimum ofEbind coincides with the minimum of total angular momen-tum J. This is in accordance with the ‘‘first law of binaryrelativistic star thermodynamics’’ within the IWM ap-proximation as derived by Friedman, Uryu, and Shibata[36] and which states that, along an evolutionary sequence,

/MADM � �/J: (31)

The surface of strange stars at the ISCO is smooth (seeFig. 5). On the contrary the neutron star sequence does notpresent any turning point of Ebind, so that the ISCO in thiscase corresponds to the mass-shedding limit (final point onthe dashed curves in Fig. 4). The gravitational-wave fre-quency at the ISCO is much lower for neutron star binariesthan for strange star binaries.

As already mentioned the adiabatic index in the outerlayers of a compact star in a binary system plays a crucialrole in its evolution, especially in setting the mass-shedding limit. Although the crust of a 1:35 M� neutronstar contains only a few percent of the stellar mass, thisregion is easily deformed under the action of the tidal

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23 24 25 26 27 28 29 30d [km]

7.6

8

8.4

8.8

Rx, R

z [km

]

FIG. 6 (color online). Coordinate ‘‘radius’’ (half the coordi-nate size of a star in fixed direction) versus coordinate separationfor irrotational quasiequilibrium sequences of binary strangestars (solid line) and neutron stars (dashed line). Upper linescorrespond to equatorial radius Rx (radius along the x axis goingthrough the centers of stars in a binary system) and lower linesare polar radius Rz (radius along the rotation axis).

FIG. 5. Internal velocity fields of irrotational strange quarkstars binaries at the ISCO. upper panel: velocity U in the orbitalplane with respect to the ‘‘inertial’’ frame (Eulerian observer);lower panel: velocity field with respect to the corotating frame.The thick solid lines denote the surface of each star.


forces resulting from the gravitational field produced bythe companion star. The end of inspiral phase of binarystars strongly depends on the stiffness of matter in thisregion. It has been shown that the turning-point orbitalinstability for irrotational polytropic neutron stars binariescan be found only if � � 2:5 and if the compaction pa-rameter is smaller than certain value ([4,25]). In fact, asshown in Fig. 31 of paper [4], they did not find ISCO for

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irrotational binary neutron stars with � � 2:5 or � � 3 forcompaction parameter as high as M=R � 0:187.

In Fig. 6 we present the evolution of two different stellarradii: the equatorial radius Rx, defined as half the diameterin the direction of the companion and the polar radius,defined as half the diameter parallel to the rotation axis. Forspherical stars Rx � Rz. We see that at the end of theinspiral phase, neutron stars are, for the same separation,more oblate (more deformed) than strange stars. Binaryneutron stars reach the mass-shedding limit (the point atwhich they start to exchange matter—a cusp form at thestellar surface in the direction of the companion) at coor-dinate separation d� 25 km. We do not see any cusps forstrange stars even for distances slightly smaller than thedistance corresponding to the ISCO �23:5 km.

It is worth to remind here the results on rapidly rotatingstrange stars and neutron stars. The Keplerian limit isobtained for higher oblateness (more deformed stars),measured, for example, by the ratio of polar and equatorialradius, in the case of strange stars than in the case ofneutron stars [42,57–61].

The differences in the evolution of binary (or rotating)strange stars and neutron stars stem from the fact thatstrange stars are principally bound by another force thangravitation: the strong interaction between quarks.

As already mentioned the frequency of gravitationalwaves is one of potentially observable parameters by thegravitational-wave detectors. We can see from Fig. 7 thatthe 3rd PN approximations for point masses derived bydifferent authors are giving ISCO at very high frequenciesof gravitational waves >2 kHz. Since in the hydrodynam-ical phase of inspiral the effect of a finite size of the star

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0 500 1000 1500 2000 2500 3000fGW

[Hz]

-0.05

-0.04

-0.03

-0.02

-0.01

0E

bind

[M

sol]

SQS MIT irrot.SQS MIT corot.Polytrope γ = 2.5 irrot.Polytrope γ = 2.5 corot.3PN irrot. (Blanchet 2002)3PN corot. (Blanchet 2002)3PN irrot. (Damour et al. 2002)3PN corot. (Damout et al. 2002)

FIG. 7 (color online). Binding energy along evolutionary sequences of corotational (thick dashed lines) and irrotational (thick solidlines) equal mass of 1:35 M� strange stars and polytropic neutron stars binaries compared with analytical results at the 3rd post-Newtonian order for point-masses by Damour et al. 2000 [55], Damour et al. 2002 [56], and Blanchet 2002 [54].


(e.g. tidal forces) is very important we see deviation of ournumerical results from point-masses calculations. The fre-quency of gravitational waves at the ISCO strongly de-pends on equation of state. For irrotational equal-mass (of1:35 M� at infinite separation) strange stars binaries de-scribed by the simplest MIT bag model this frequency is�1400 Hz and for neutron stars binaries described byrealistic EOS it is between 800 Hz and 1230 Hz [5,6].

VI. SUMMARY AND DISCUSSION

We have computed evolutionary sequences of irrota-tional and corotating binary strange stars by keeping thebaryon mass constant to a value that corresponds to indi-vidual gravitational masses of 1:35 M� at infinite separa-tion. The last orbits of inspiraling binary strange stars havebeen studied in the quasiequilibrium approximation and inthe framework of Isenberg-Wilson-Mathews approxima-tion of general relativity. In order to calculate hydrody-

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namical phase of inspiraling irrotational strange starsbinaries, i.e., assuming that the fluid has zero vorticity inthe inertial frame, we found the boundary condition for thevelocity potential. This boundary condition is valid forboth the case of nonvanishing (e.g., self-bound matter)and vanishing density at the stellar surface (neutron starmatter). In our calculations strange stars are built bystrange quark matter described by the simplest MIT bagmodel (assuming massless and noninteracting quarks).

We have located the end of each quasiequilibrium se-quence (ISCO), which corresponds to some orbital insta-bilities (the dynamical instability for irrotational case orthe secular one for synchronized case) and determined thefrequency of gravitational waves at this point. This char-acteristic frequency yields important information about theequation of state of compact stars and is one of the poten-tially observable parameters by the gravitational-wave de-tectors. In addition, the obtained configurations providevaluable initial conditions for the merger phase. We found

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the frequency of gravitational waves at the ISCO to be�1400 Hz for irrotational strange star binaries and�1000 Hz for synchronized case. The irrotational case ismore realistic since the viscosity of strange star matter isfar too low to ensure synchronization during the late stageof the inspiral. For irrotational equal mass (of 1:35 M�)neutron star binaries described by realistic EOS [5,6] thefrequency of gravitational waves at the ISCO is between800 Hz and 1230 Hz, much lower than for a binary strangequark star built of self-bound strange quark matter. In thepresent paper we considered only strange quark stars de-scribed by the simplest MIT bag model with massless andnoninteracting quarks. In order to be able to interpret futuregravitational-wave observations correctly it is necessary toperform calculations for different strange star EOS pa-rameters (taking also into account the existence of a thincrust) and for large sample of neutron stars described byrealistic equations of state. For some MIT bag modelparameters one is able to obtain less compact stars thanconsidered in the present paper and have lower values ofthe frequency of gravitational waves at the end of inspiralphase. It also should be taken into account that stars in abinary system can have different masses [30]. The case ofbinary stars (with equal masses and different masses)described by different strange quark matter models willbe presented in a separate paper [62].

We have shown the differences in the inspiral phasebetween strange quark stars and neutron stars describedby polytropic equation of state having the same gravita-tional mass and radius in the infinite separation. It wasalready shown by Bejger et al. 2005 [6] that the frequencyof gravitational waves at the end point of inspiraling neu-tron stars described by several realistic EOS without exoticphases (such as meson condensates or quark matter) can bepredicted, in a good approximation, by studying binaries

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with assumed polytropic EOSs with � � 2 or 2.5. Forrealistic EOS and polytropes with � � 2:5 [4,25] a qua-siequilibrium irrotational sequence terminates by mass-shedding limit (where a cusp on the stellar surfacedevelops).

We found that it was not the case for inspiraling strangestar binaries which are self-bound objects having verylarge adiabatic index in the outer layer. For both synchro-nized and irrotational configurations, we could always finda turning point of binding energy along an evolutionarysequence of strange quark stars, which defines an orbitalinstability and thus marks the ISCO in this case. In theirrotational case for the same separation strange stars areless deformed than polytropic neutron stars and for thesame ratio of coordinate radius Rx=Rz their surfaces aremore smooth. A cusp is not appearing on the surface of astrange star in a binary system even for separation corre-sponding to orbital instability. The frequency of gravita-tional waves at the end of inspiral phase is higher by300 Hz for the strange star binary system than for thepolytropic neutron star binaries.

The differences in the evolution of binary (or rotating)strange stars and neutron stars stem from the fact thatstrange stars are principally bound by an additional force,strong interaction between quarks. Strange stars wouldexist stably even if gravity were switched off.

ACKNOWLEDGMENTS

Partially supported by the KBN GrantNos. 5P03D.017.21 and PBZ-KBN-054/P03/2001; by‘‘Ayudas para movilidad de Profesores de Universidad eInvesigadores espanoles y extranjeros’’ from the SpanishMEC; by the ‘‘Bourses de recherche 2004 de la Ville deParis’’; and by the Associated European Laboratory Astro-PF (Astrophysics Poland-France).

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Last orbits of binary strange quark stars