The spin parameter of uniformly rotating compact stars

0.2 MB·English

by  Ka-Wai Lo

Draftversion January24,2011
PreprinttypesetusingLATEXstyleemulateapjv.03/07/07
THE SPIN PARAMETER OF UNIFORMLY ROTATING COMPACT STARS
Ka-Wai Lo∗ and Lap-Ming Lin†
DepartmentofPhysicsandInstituteofTheoretical Physics,TheChineseUniversityofHongKong,HongKong,China
(Dated: January 24, 2011)
Draft version January 24, 2011
ABSTRACT
1 Westudythedimensionlessspinparameterj (=cJ/(GM2))ofuniformlyrotatingneutronstarsand
1 quarkstarsingeneralrelativity. We shownumericallythatthe maximumvalue ofthe spinparameter
0 ofaneutronstarrotatingattheKeplerianfrequencyisjmax ∼0.7forawideclassofrealisticequations
2 of state. This upper bound is insensitive to the mass of the neutron star if the mass of the star is
n largerthan about 1 M⊙. Onthe other hand, the spinparameterof a quarkstar modeled by the MIT
bagmodelcanbelargerthanunityanddoesnothaveauniversalupperbound. Itsvaluealsodepends
a
J strongly on the bag constant and the mass of the star. Astrophysical implications of our finding will
be discussed.
1
Subject headings: Dense matter—stars: neutron—stars: rotation
2

]
E 1. INTRODUCTION imum value for f (i.e., the Keplerian frequency fK)
has been one of the most studied physical quanti-
H The general stationary vacuum solution (the Kerr
ties for relativistic rotating stars (see, e.g., Cook et al.
spacetime)oftheEinsteinequationsisspecifieduniquely
.
h by the gravitational mass M and the angular momen- (1994); Haensel et al. (1995); Koranda et al. (1997);
p tum J (see, e.g., Wald (1984)). If J ≤ GM2/c, we Benhar et al. (2005); Haensel et al. (2009)). However,
o- have a rotating black hole. However, if J >GM2/c, the incontrasttothesepreviousworks,weshallfocusexten-
sively on the spin parameter j.
r Kerrspacetimewouldhaveanakedsingularitywithouta
t horizon. One could then consider closed timelike curves It has been known that the spin parameter for a
s maximum-mass neutron star lies in the range j ∼
a and causality would be violated (Chandrasekhar 1983).
[ While its validity has not yet been proven, the cosmic- 0.6−0.7 for most realistic equations of state (EOS; e.g.,
Cook et al. (1994) and Salgado et al. (1994)). In this
censorship conjecture (Penrose 1969) asserts that naked
2 work, we first extend the previous works and show that
singularities cannot be formed via the gravitational col-
v the upper bound for the spin parameter j ∼ 0.7 is
lapseofabody. Forthisreason,itisbelievedthatastro- max
3
6 physicalblackholesshouldsatisfytheKerrboundj ≤1, essentiallyindependentofthe massoftheneutronstarif
5 where j =cJ/GM2 is the dimensionless spin parameter. thegravitationalmassofthestarislargerthan∼1M⊙.
Next we study the spin parameter of self-bound quark
3 While the valueofthe spinparameterj playsafunda-
stars,whichwasnotconsideredpreviouslyinCook et al.
. mental role in black-hole physics, it appears that this is
1 (1994) and Salgado et al. (1994). We find that the be-
notthecaseforotherstellarobjects. Inparticular,there
1 haviorofthespinparametersofneutronstarsandquark
is no theoretical constraint on the value of j for stars.
0 stars is very different. In contrastto the case of neutron
It is known that the spin parameter of main-sequence
1
stars, the spin parameter of quark stars does not have
: stars depends sensitively on the stellar mass and can be
v much larger than unity (Kraft 1968, 1970; Dicke 1970; a universal upper bound. It also depends sensitively on
i the parameterofthe quarkmatter EOSand the mass of
X Gray 1982). On the other hand, the spin parameter of
thestar. Furthermore,thespinparameterofquarkstars
compactstarshasnotbeenstudiedindetail(seebelow).
r can be larger than unity. This leads us to propose that
a As we shall discuss in more detail in Section 3, the spin
thespinparametercouldbeausefulindicatortoidentify
parameterofcompactstarsisinterestinginitsownright
rapidly rotating quark stars.
for two reasons. (1) It plays a role in our understand-
Theplanofthis paper isasfollows. Section2presents
ingofthe observedquasi-periodicoscillations(QPOs)in
themainnumericalresultsofthiswork. InSection3,we
disk-accreting compact-star systems. (2) It determines
discusstheastrophysicalimplicationsofourresults. Our
the final fate of the collapse of a rotating compact star.
conclusions are summarized in Section 4.
Ever since the seminal work of Hartle (1967) who
considered the limit of slow rotation, rotating compact
stars have been studied extensively in general relativ- 2. NUMERICAL RESULTS
ity. In the past two decades, various different numer-
2.1. Numerical method and EOS models
ical codes have been developed to construct rapidly
rotating stellar models in general relativity. We re- We make use of the numerical code rotstar from the
fer the reader to Stergioulas (2003) for a review. As C++ LORENE library1 to calculate uniformly rotat-
the rotation frequency f is a directly measurable quan- ing compact star models in general relativity. The code
tity for pulsars, it is thus reasonable that the max- uses a multi-domain spectral method (Bonazzola et al.
1998) to solve the Einstein equations in a station-
∗Currentaddress: DepartmentofPhysics,UniversityofIllinoisat ary and axisymmetric spacetime with a matter source
Urbana-Champaign,Urbana,IL61801, USA.
†Emailaddressforallcorrespondence: [email protected] 1 http://www.lorene.obspm.fr/
2
2.5 2500
A
APR
AU
2 BBB2
FPS
2000 SLY4
UU
)n1.5 A WS
Msu APR Hz) QMB60
M ( 1 ABBUB2 f(K1500 QMB90
FPS
SLY4
UU
0.5 WS
QMB60
QMB90
1000
0
1 2 3 4
ρ (1015g/cm3) 1 1.5 M (M ) 2 2.5
c sun
Fig. 1.— Gravitationalmassasafunctionofthecentralenergy Fig. 2.— Keplerianfrequencyasafunctionofthegravitational
densityfornon-rotatingcompactstarsconstructedwiththechosen massforrotatingcompactstarsconstructed withthechosenEOS
EOSmodelsinthiswork. modelsinthiswork.
1
(Bonazzola et al. 1993; Gourgoulhon et al. 1999). The
0.9
code has been tested extensively and compared with a
0.8
few different numerical codes (Nozawa et al. 1998).
As theoretical calculations for dense matter at 0.7
supranuclear densities are poorly constrained, the EOS 0.6 A
idnertshteoohdigh(s-edee,nsei.tgy.,coWreeboefrcoetmapla.c(t2s0t0a7r)s;isHnaoentsweleleltuanl-. jmax0.5 AAUPR
0.4 BBB2
(2007) for reviews). In this work, we employ eight
FPS
realistic nuclear matter EOS to model rotating neu- 0.3 SLY4
tron stars: model A (Pandharipande 1971), model APR 0.2 UU
WS
(Akmal et al. 1998), model AU (the AV14+UVII model
0.1
in Wiringa et al. (1988) is joined to Negele & Vautherin
0
(1973)), model BBB2 (Baldo et al. 1997), model FPS 1 1.5 2 2.5
(Pandharipande & Ravenhall 1989; Lorenz et al. 1993), M (M )
sun
model SLY4 (Douchin & Haensel 2000), model UU (the Fig. 3.— Maximum spinparameter as a functionof the gravi-
UV14+UVII model in Wiringa et al. (1988) is joined tationalmassforrotatingneutronstars.
to Negele & Vautherin (1973)) and model WS (the
UV14+TNI model in Wiringa et al. (1988) is joined to
Lorenz et al. (1993)). For the quark star models, we use
the simplest MIT bag model with non-interacting mass- 2.2. Neutron stars
less quarks (Chodos et al. 1974). Two different values, Nowweturntothemainfocusofthiswork: thedimen-
60 MeV fm−3 and 90 MeV fm−3, are chosen for the bag sionless spin parameter j. Having seen that f depends
K
constant B. These values correspond approximately to stronglyonthe EOSandthe mass of the star,it may be
the range of B within which the hypothesis of strange quite surprising to learn that the maximum value of the
matter is valid (Haensel et al. 2007). spin parameter j (as set by the Kepler limit) is quite
max
To illustrate the diversity of the EOS models used in universalforrotatingneutronstars. InFigure3, weplot
thiswork,weplotthegravitationalmassM againstcen- j against the gravitational mass M for the selected
max
tral energy density ρc for non-rotating stars constructed nuclearmatter EOSmodels. In the figure,eachline cor-
withthechosenEOSmodelsinFigure1. Thequarkstar respondsto one particular EOSandeachpoint ona line
models QMB60 and QMB90 in the figure correspond to corresponds to a star model with a fixed M rotating at
the cases B = 60 MeV fm−3 and 90 MeV fm−3, respec- its Keplerian frequency. Note that each sequence in the
tively. The maximum mass of compact stars depends figure is terminated at the stellar model with the same
quite sensitively on the EOS models and it ranges from total particle number as the stable maximum-mass non-
about 1.5 M⊙ to 2.2 M⊙. In Figure 2, we plot the Kep- rotating configuration. In the figure, we see that jmax
lerian frequency f against the gravitationalmass M of lies in a narrowrange∼0.65−0.7 for the eight different
K
rotatingcompactstarsbasedonourchosenEOSmodels. nuclear matter EOS models. In particular, the values
Similar to the maximum mass for non-rotating compact do not depend sensitively on the mass of the star. This
stars, Figure 2 shows clearly that f depends strongly extends previous works (Cook et al. 1994; Salgado et al.
K
on the EOS model. It is also sensitive to the mass of 1994) which focus on maximum-mass neutron star mod-
the star. This is the well-known reason why searching els.
for rapidly rotating compact stars can provide us con- Whilethespinparameterofanastrophysicalblackhole
straints on the EOS of dense matter. is constrained by j ≤1, we find that the spin parameter
3
0.6 0.6
0.5 0.5
0.4 0.4
j j
0.3 0.3
0.94 M APR
sun
0.2 1.43 M 0.2 FPS
sun SLY4
1.73 M
0.1 sun 0.1
00 0.2 0.4 0.6 0.8 1 00 0.2 0.4 0.6 0.8 1
f / f
K f / f
K
Fig. 4.— Spinparameter isplotted against thescaledrotation
Fig. 5.— Spinparameter isplotted againstthe scaledrotation
frequency for neutron stars constructed with the FPS EOS. Each
frequency for neutron stars constructed with three different EOS
line represents a sequence of fixed total particle number. Each
sequence is labeled by the gravitational mass of its non-rotating models. ThebaryonicmassisfixedatMB=1.6M⊙.
configuration.
For the QMB60 model, j is decreased by about 24%
max
of a neutron star is bounded by jmax ∼ 0.7. The upper as M increases from 1 M⊙ to 2 M⊙. Comparing to the
bound j is quite universal for different EOS models caseofneutronstars,itisalsoseenthatj hasamore
max max
and gravitational mass larger than ∼ 1 M⊙. For lower significant dependence on the EOS parameter, namely
massneutronstars, M <1M⊙,we findthatj decreases the bag constant in the MIT bag model. Analogous to
with decreasing M. However, we shall only focus on Figure 4 for neutron stars, we plot j against f/f for
K
mass M >1 M⊙ in this work as the observed masses of the QMB60 model in Figure 7. Each line represents a
neutron stars are typically larger than 1 M⊙. We refer sequence of fixed total particle number. As in Figure 4,
the reader to Steiner et al. (2010) for a recent review on each sequence is labeled by the gravitationalmass of its
the observed masses of neutron stars. non-rotating configuration M . We show three different
0
So far we have studied the maximum spin parameter sequences in the figure: M0 =0.82 M⊙ (solid), 1.27 M⊙
jmax ofneutronstarsrotatingattheirKeplerianfrequen- (dashed), and 1.55 M⊙ (dash-dotted). We see that the
cies f . However, realistic neutron stars in general ro- differences between the curves increase significantly as
K
tate slower with frequencies f < f . Is the spin pa- the scaled frequency increases. At f/f = 1, the spin
K K
rameter j still insensitive to the EOS and mass of the parameter is increased by about 27% when M changes
0
star? In Figure 4, we plot the spin parameter j against from 1.55 to 0.82 M⊙.
the scaled rotationfrequency f/f for the FPS EOS. In We see that the spin parameter of quark stars can be
K
the figure, each line represents a sequence of fixed total significantly larger than the upper bound (j ∼ 0.7)
max
particle number (the so-called evolutionary sequence). for neutron stars. It can also be larger than the Kerr
Each sequence is labeled by the gravitational mass of bound j = 1 for black holes. This suggests that the
its non-rotating configuration M . The solid line corre- spin parameter could be a useful indicator to identify
0
spondstoM0 =0.94M⊙,thedashedlinecorrespondsto rapidlyrotatingquarkstars. Discoveringevenonesingle
M0 = 1.43 M⊙, and the dashed-dotted line corresponds compact star with spin parameter j & 0.7 will provide
to M0 = 1.73 M⊙. Note, however, that the gravita- a strong evidence for the existence of quark stars. Fi-
tionalmassincreaseswithrotationfrequency. Thefigure nally, it should be pointed out that the fact that the
shows that the spin parameter changes only by at most spin parameter of quark stars can be larger than 0.7 is
10% (depending on f/f ) when M changes from 0.94 also evident from the results of Stergioulas et al. (1999).
K 0
to 1.73 M⊙. It is also interesting to note that the differ- Using the values of the gravitationalmass M and angu-
encesbetweenthecurvesdecreaseasthescaledfrequency lar momentum J for Keplerian quark stars presented in
f/f tends to 1. This is quite different from the case of Table 1 of Stergioulas et al. (1999), it is easy to check
K
quark stars which will be studied shortly. In Figure 5, that the quark stars considered by the authors all have
we plot j against f/f for three different EOS models. j >0.7. In particular, their results also suggest that
K max
The stellarmodels onthe threesequenceshavethe same j decreaseswithincreasingM aswehaveseeninFig-
max
total particle numbers such that their baryonic masses ure 6.
are fixed at MB = 1.6 M⊙. It is seen that, for a fixed
scaled frequency, the spin parameter of neutron stars is 3. ASTROPHYSICAL IMPLICATIONS
essentially independent of the EOS models. We have studied the spin parameter of uniformly ro-
tating compact stars in general relativity. Our numer-
2.3. Quark stars
ical results show that the behavior of the spin parame-
Let us now consider self-bound quark stars using the ter of quark stars is quite different from that of neutron
MITbagmodel. InFigure6,weplotj againstM for stars. In particular, the spin parameter of neutron stars
max
the two quark matter models QMB60 and QMB90. In is bounded above by j ≈ 0.7, while quark stars can
max
contrast to the case of neutron stars (see Figure 3), we have a value larger than unity. In this section, we shall
seethatj dependssensitivelyonthemassofthestar. discuss (in our view) the astrophysical implications of
max
4
1.5
1.2
1
1 0.82 M
sun
max0.8 QQMMBB6900 j 11..2575 MMsun
j 0.6 sun
0.5
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
0
1 1.5 2 2.5 f / f
M (M ) K
sun
Fig. 7.— Spinparameter isplotted againstthe scaledrotation
Fig. 6.— Maximum spinparameter as afunction ofthe gravi- frequencyforquarkstarsconstructedwiththeQMB60EOS.Each
tational massforrotatingquarkstars. line represents a sequence of fixed total particle number. Each
sequence is labeled by the gravitational mass of its non-rotating
configuration.
our results.
First, how could the spin parameter of a compact star
be measured? Unfortunately, so far there is no gen-
eral technique to infer the spin parameter j of compact flat solution of the Einstein-Maxwell system obtained
stars directly. As far as we are aware, the spin param- by Manko et al. (2000) and study the geodesics in this
eter of a compact star could be potentially measured in spacetime. If there is no charge and magnetic moment,
disk-accreting compact-star systems. In particular, the this analytic solution depends only on the mass, angu-
neutron stars (or quark stars) in low-mass X-ray bina- lar momentum, and quadrupole moment of the space-
ries (LMXBs) provide the most natural cosmic labora- time. Furthermore, this solution only involves rational
tories for studying the spin parameter. In order to un- functions, which helps to simplify the analytical study
derstand how the spin parameter might be inferred in of geodesic motions. Berti & Stergioulas (2004) have
disk-accreting systems, it should be noted that the spin demonstrated that this analytic solution can describe
parameterofthecentralcompactstardirectlyaffectsthe the exterior spacetime of a rapidly rotating neutron star
particle motion around the star. For example, to first (e.g., j >0.5) very well. Nevertheless, further investiga-
order in j, the orbital frequency of a point particle in a tion is needed to check whether this analytic solution is
prograde orbit around a compact star is given by (see, also valid for rapidly rotating quark stars.
e.g., van der Klis (2006)) One of the well-observed features of LMXBs is the
high-frequency (∼ kHz) QPOs. To date, QPOs have
GM 3/2 GM 1/2 been observed in more than 20 LMXBs. These QPOs
2πν = 1−j , (1)
φ " (cid:18) rc2 (cid:19) #(cid:18) r3 (cid:19) osyftsetnemcsomine winhipcahirtshweistphinfrefqreuqeunecniecsieνsuofantdheνlc.omInpaacltl
where r is the orbital radius. For infinitesimally tilted stars ν have been measured, the frequency separa-
star
and eccentric orbits, the disk particles will have radial tion ∆ν = ν − ν is approximately equal to ν or
u l star
(ν )andvertical(ν )epicyclicfrequencieswhichalsode- ν /2. We refer the reader to van der Klis (2006) and
r θ star
pend on j (see van der Klis (2006) for the expressions). Lamb & Boutloukos(2008)forrecentreviews. Whilethe
Furthermore, the combination ν −ν also gives rise to physical mechanism responsible for producing the high-
θ r
the periastron frequency of the orbit. These frequencies frequency QPOsis not knownyet, mostphysicalmodels
in general depend on M and j, and hence their obser- involve orbital motion and disk oscillations. Hence, the
vations (possibly needed to be combined with the mea- frequencies ν , ν , ν and various combinations of them
r θ φ
surement of other stellar parameters such as the mass) areofteninvoked(eitherdirectlyorindirectly)toexplain
wouldinprincipleprovideusefulinformationonthespin high-frequencyQPOs. This is the reasonwhy one might
parameter. Infact, therearestrongevidencesthatthese hope to obtainuseful informationon the spinparameter
frequencies have already been observed in LMXBs. of the central compact star in an LMXB by observing
It should be noted that existing algebraic relations, its QPOs. For example, if the higher frequency of the
such as Equation (1), which relate various orbital fre- QPO pair (ν ) is identified with the orbital frequency
u
quencies to stellar parameters are in general only valid (ν ), one can then obtain a relation between the spin
φ
for small spin rate j ∼ 0.1. The difficulty of obtaining parameter and an upper bound on the mass of the com-
the corresponding algebraic relations for rapidly rotat- pact star (Miller et al. 1998). On the other hand, based
ing stars (j ∼ 1) lies in the fact that there is no exact on the so-called relativistic precession model of QPOs
analytic representationof the vacuum spacetime outside (Stella & Vietri1998,1999),T¨or¨ok et al.(2010)havere-
a rapidly rotating compact star. Having such an ana- cently derived a constraint relating the mass and spin
lytic representation of the spacetime metric will allow parameter of the compact star in Cir X-1. These works
onetoobtainthedesiredalgebraicrelationsforarapidly thus demonstrate the possibility of measuring the value
rotating compact star. A starting point along this di- of j with an independent determination of the mass and
rection would be to take the closed-form asymptotically vice versa.
5
The abovebrief review ofhigh-frequency QPOs serves of the bursts, depends on the mass of the disk, the col-
to point out the astrophysical relevance of the spin pa- lapseofquarkstarsmightthusbeamechanismforshort
rameter and in what situations could it be potentially gamma-raybursts.
measured. Now we are ready to discuss how one could On the other hand, what if the collapse process does
make use of the finding in this work to obtain useful in- not lead to any (or little) mass ejection as in the col-
formationregardingthecentralcompactstarinLMXBs. lapse of neutron stars? In fact, Bauswein et al. (2009)
Suppose a single well established model for the high- haverecentlysimulatedquarkstarmergersbasedonthe
frequency QPOs can be agreed upon in the future, then MIT bag model and the conformally flat approximation
it is possible that rapidly rotating quark stars could be to general relativity. Their results show that the mass
identified from the inferred spin parameters. As shown ejection depends sensitively on the bag constant B. In
inournumericalresults,iftheinferredspinparameterof particular,they find that there are some binary mergers
the central star is larger than ∼0.7, then the star could without mass ejection. Could this conclusion, namely
be a quark star. Of course,the star can either be a neu- theabsenceofmassejectionforsomevaluesofB,alsobe
tron star or quark star if the spin parameter is less than true for the rotational collapse of quark stars? For the
0.7. collapse of rotating neutron stars to black holes, apart
On the other hand, for the present situation where from the small amount of total mass energy M and an-
many models are available, our finding could still be gular momentum J carried away by gravitational radi-
usedtoputconstraintsonthephysicalmodelsforQPOs. ation (Baiotti et al. 2005b), the final black holes have
For example, let us consider the resonantly excited disk- essentially the same M and J, and hence the same spin
oscillationmodelforQPOsproposedbyKato(2008). In parameter j, as the initial star. For a rapidly rotating
the model, Kato suggests that the high-frequency QPOs quark star with initial spin parameter j > 1, if there is
are inertial-acoustic oscillations on a deformed disk that no mass ejection, how could the spin parameter be re-
are resonantly excited by nonlinear couplings between duced efficiently in order to form a regular black hole
the oscillation modes and the disk deformation. Kato that satisfies the Kerr bound j ≤ 1 at the end of the
applies the model to Cir X-1 and finds that it describes collapse? IfaKerrblackholecouldnotbe formedinthe
the observed QPOs quite well if the mass of the cen- process,thenwhatwouldbethefinalfateofthecollapse?
tral star is about 1.5−2.0 M⊙ and the spin parameter These questions deserve further investigation using fully
is j ∼ 0.8. However, our work suggests that uniformly general relativistic modeling. The hope is that studying
rotating neutron stars cannot have j & 0.7. If Kato’s the collapseofquarkstarsmightleadto the discoveryof
model is correct, then our finding implies that the cen- some new phenomena which are not seen in the collapse
tral star in Cir X-1 could be a quark star. On the other of neutron stars. This might then help to shed more
hand, if other measurements in the future (such as the light on the highly nonlinear dynamics of gravitational
mass-radius relation or cooling property of the central collapse in general relativity.
star) suggest that the compact star in the system is in- Theastrophysicalimplicationsdiscussedabovedepend
deed a traditional neutron star, then our finding would cruciallyon the existence of rapidly rotatingquark stars
rule out the resonantly excited disk-oscillation model. with j > 0.7. However, it is known that rapidly ro-
Afterdiscussinghowonemightusethespinparameter tating compact stars may be subject to different kinds
to distinguish between neutron stars and quark stars in of secular or dynamical non-axisymmetric mode insta-
LMXBs, let us now turn to a different issue concern- bilities (Andersson 2003). One might thus worry that
ing the collapse of a rotating star to black hole. In rotating quark stars (if they exist) may be limited
the past decade, general relativistic simulations of ro- to small spin rates due to various instabilities, and
tational collapse of neutron stars modeled by the poly- hence rendering a possible distinction between neutron
tropic EOS have been performed (Shibata et al. 2000a; stars and quark stars unlikely to happen. However,
Shibata 2003a,b; Duez et al. 2004; Baiotti et al. 2005a). Gondek-Rosin´ska et al. (2003) have shown that, taking
These simulationsshow thatno stable massivedisks can into account realistic values of shear viscosity, viscosity-
beformedaroundtheresultingblackholes. Theimplica- drivenbarmodeinstabilitycannotdevelopinquarkstars
tion is that the collapse of a uniformly rotating neutron modeled by the MIT bag model in any astrophysically
star (with j <1) to black hole cannot lead to the black- relevant temperature windows. Even taking the unre-
holeaccretionmodelofgamma-rayburst,whichrequires alistic assumption of infinite shear viscosity, the insta-
arotatingblackholesurroundedbyanaccretiondisk(see bility can develop only if the ratio of the rotational ki-
Piran(2005)fora review). It isexpected thatthe initial netic energy to the absolute value of the gravitational
spin parameter of the collapsing star must be j & 1 in potential energy T/W is larger than 0.1375. The ex-
order to form a massive disk around the final black hole act value depends on the stellar mass. For compar-
(Shibata 2003a,b; Duez et al. 2004). As rapidly rotating ison, a quark star modeled by the MIT bag model,
quark stars can have j > 1, it is thus possible that the with B = 60 MeV fm−3, M = 1.146 M⊙, and j =
collapse of a rapidly rotating quark star could be a pro- 0.8, has the value T/W = 0.126. We have followed
genitor for the black-hole accretionmodel of gamma-ray the samenumericalprocedureofGondek-Rosin´ska et al.
burst. Furthermore,it is noted that the collapse of mas- (2003) to check that this quark star is indeed sta-
sive stars to black holes (the so-called collapsars) could ble against the viscosity-driven instability. We refer
formmassivedisksaroundtheblackholesandhencepro- the reader to Gondek-Rosin´ska & Gourgoulhon (2002)
duce long gamma-ray bursts. Since quark stars have and Gondek-Rosin´ska et al. (2003) for more details
smaller mass (∼ M⊙) comparing to massive stars, the on the numerical procedure. On the other hand,
collapse of quark stars might only produce small disks. Gondek-Rosin´ska et al. (2003) have also discussed that
Asthedurationoftheaccretion,andhencethetimescale
6
the gravitational-radiation driven r-mode instability of the star if the mass of the star is larger than about
seemstobeunimportantforquarkstarsinLMXBsifthe 1 M⊙. On the other hand, the spin parameter of quark
strangequarkmassis m c2 ∼200MeV(standardvalue) starsbehavesquite differently. We findthatthe spinpa-
s
or higher. The r-mode instability can develop and may rameter of quark stars modeled by the MIT bag model
limit quark stars to small spin rates only if the strange can be larger than unity. It also depends sensitively on
quark mass takes the relatively low value m c2 ∼ 100 theEOSparameter(i.e.,thebagconstant)andthemass
s
MeV. of the star.
Even if the viscosity-driven and gravitational- We have discussed (in our view) the astrophysical im-
radiation-driven instabilities (which are both secular ef- plications of our finding in detail in Section 3. We have
fects) cannot develop in a quark star, the star may still discussedhowthespinparameterofcompactstarscould
subject to dynamical bar-mode instability which occurs be potentially measured in LMXBs and its relevance in
at a higher spin rate. This kind of instability has not the physicalmodels for high-frequency QPOs. As a first
been studied for rotating quark stars. However, we can application,our finding implies that the compact star in
still obtain some insightfrom the Newtonian theory of a Cir X-1 could be a quark star if the resonantly excited
rotatingincompressiblestar,whichisagoodapproxima- disk-oscillation model for QPOs is correct (Kato 2008),
tion to quark stars because of their rather uniform den- since the model requires that the spin parameter of the
sity profile. The onset of dynamicalinstability occurs at central star to be j ∼ 0.8 in order to fit the observed
T/W ≈ 0.27 for an incompressible star (Chandrasekhar QPOs. We havealsospeculatedonhowthe collapseofa
1969). Forcomparison,aquarkstarmodeledbytheMIT rotating quark star might be different from the collapse
bag model, with B = 60 MeV fm−3, M = 1.4 M⊙ and ofaneutronstar. AsexplainedinSection3,thecollapse
j = 1.11, has the value T/W = 0.23. Although general ofa rapidlyrotatingquarkstarwithj &1 mightleadto
relativity may change the critical value T/W for the on- the formation of a stable disk around the resulting Kerr
set of the instability to a somewhat smaller value than black hole. This implies that the collapse process might
that is suggested by the Newtonian theory2, it is still formthe centralengine ofa gamma-rayburst. However,
verylikelythatrapidlyrotatingquarkstarswithj >0.7 a fully general relativistic dynamical calculation (which
canexist. SomeofthemmayevenbreaktheKerrbound has not been done in this work)is requiredto shed light
j =1 for black holes. on the issue.
In conclusion,our work suggeststhat discoveringeven
4. CONCLUSIONS one single compact star with spin parameter j & 0.7
will provide a strong evidence for the existence of quark
In this paper, we have studied the dimensionless spin
stars, and hence verifying the hypothesis that strange
parameter j of uniformly rotating neutron stars and
quark matter could be absolutely stable (Witten 1984).
quark stars in general relativity. We find that the max-
imum value of the spin parameter (as set by the Kepler
limit) of neutron stars is bounded above by j ∼ 0.7. ACKNOWLEDGMENTS
max
This upper bound is essentially independent of the EOS This work is supported by the Hong Kong Research
of the neutron star. It is also insensitive to the mass Grants Council (grant no: 401807).
2 Shibataetal. (2000b) have shown that the effects of general
starsmodeledbyapolytropicEOS.
relativityonly change the critical value to T/W ∼0.24−0.25 for
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