Keplerian frequency of uniformly rotating neutron stars and quark stars

0.25 MB·English

by  P. Haensel

Astronomy & Astrophysics manuscript no. fmax (cid:13)c ESO 2009
January 23, 2009
Keplerian frequency of uniformly rotating neutron stars and quark
stars
P. Haensel1, J.L. Zdunik1, M. Bejger1, and J.M. Lattimer2
9 1 N.Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka18, PL-00-716 Warszawa, Poland
0 2 DepartmentofPhysicsandAstronomy,StateUniversityofNewYorkatStonyBrook,StonyBrook,NY11794-3800,
0 USA
2 [email protected], jlz@[email protected], [email protected], [email protected]

n Received xxxAccepted xxx
a
J ABSTRACT

9
Aims. We calculate Keplerian (mass shedding) configurations of rigidly rotating neutron stars and quark stars with

] crusts. We check the validity of empirical formula for Keplerian frequency, fK, proposed by Lattimer & Prakash,
R fK(M) = C (M/M⊙)1/2(R/10 km)−3/2 , where M is the (gravitational) mass of Keplerian configuration, R is the
S (circumferential) radius of the non-rotatingconfiguration of thesame gravitational mass, and C =1.04 kHz.
. Methods. Numerical calculations are performed using precise 2-D codes based on the multi-domain spectral methods.
h
Weuse a representative set of equations of state (EOSs) of neutron stars and quark stars.
p Results. We show that the empirical formula for f (M) holds within a few percent for neutron stars with realistic
K
o- EOSs, provided 0.5M⊙ <M <0.9Mmstaaxt, where Mmstaaxt is the maximum allowable mass of non-rotating neutron stars
r for an EOS, and C = CNS = 1.08 kHz. Similar precision is obtained for quark stars with 0.5 M⊙ < M < 0.9 Mmstaaxt.
st For maximal crust masses we obtain CQS = 1.15 kHz, and the value of CQS is not very sensitive to the crust mass.
a All our C’s are significantly larger than the analytic value from the relativistic Roche model, CRoche = 1.00 kHz.
[ For 0.5 M⊙ < M < 0.9 Mmstaaxt, the equatorial radius of Keplerian configuration of mass M, RK(M), is, to a very
good approximation, proportional to the radius of the non-rotating star of the same mass, RK(M) = a R(M), with
1 aNS ≈aQS≈1.44. The valueof aQS is very weakly dependenton themass of the crust of thequark star. Both a’s are
v smaller than the analytic value aRoche =1.5 from therelativistic Rochemodel.
8
6 Key words.densematter – equation of state – stars: neutron – stars: rotation
2
1
. 1. Introduction the EOS. In the present paper we calculate precise 2-D
1
0 models of rapidly rotating neutron stars and quark stars
9 Because of their strong gravity, neutron stars can be very with different EOSs. We use the relativistic Roche model
0 rapid rotators. In view of the high stability of pulsar fre- (Shapiro et al.,1983)tomotivatetheempiricalformulafor
v: quency (even the giant glitches produce relatively small fK(M)proposedby Lattimer & Prakash(2004).We calcu-
i fractional change of rotation frequency, <∼ 10−5), one can latetheoptimalvalueofprefactorC andweestablishlimits
X treat pulsar rotation as rigid. The frequency f of stable to the validity of the empirical formula.
r rotation of a star of gravitational mass M and baryon Asofthiswriting,themaximumrotationfrequencyofa
a mass lower than the maximum allowable for non-rotating pulsar is 716 Hz (PSR J1748−2446ad,Hessels et al. 2006).
stars is limited by the (Keplerian) frequency f of a test Kaaret et al. (2007) reported a discovery of oscillation fre-
K
particle co-rotating on an orbit at the stellar equator. quency1122HzinanX-rayburstfromtheX-raytransient,
The relation between f and stellar gravitational mass XTE J1739-285, and concluded that ”this oscillation fre-
K
M, f = f (M), depends on the (unknown) equation quency suggests that XTE J1739-285 contains the fastest
K K
of state (EOS) at supranuclear densities. Both quanti- rotating neutron star yet found”, but this observation has
ties, M and f, are measurable, and the condition im- not been confirmed or reproduced.
plied by a measured frequency fobs of a pulsar of mass The problem of the constraint fmEOaxS(M)>fobs was al-
M, f < f (M), could be used to constrain theoretical readyconsideredby Shapiro et al.(1983)after the epochal
obs K
models of dense matter (for a recent review of theory of discovery of the first millisecond pulsar, PSR 1937+214
dense matter see Haensel et al. 2007). Numerical calcula- with f = 641Hz (Backer et al., 1982). Shapiro et al.
obs
tion of fK(M) requires precise, time consuming 2-D cal- (1983)usedaformulaforfmax(M)basedontherelativistic
culations of stationary rotating configurations in general Roche model. After the announcement of ill-fated discov-
relativity.Therefore,thesearchforasufficientlypreciseap- ery of a 2 kHz pulsar, this formula was used to show that
proximatebutuniversalformulaforfK(M)isofgreatinter- nearly all EOSs of dense matter existing at that time were
est. Lattimer & Prakash (2004) proposed an approximate ruled out by this observation (Shapiro et al., 1989).
empirical formula fK(M) ≈ C(M/M⊙)1/2(R/10 km)−3/2, In Sect. 2 we summarize results obtained with the rel-
where R = R(M) is the circumferential radius of static ativistic Roche model. Sect. 3 contains description of real-
star of mass M, and C = 1.04 kHz does not depend on istic EOSs of nuclear matter, whereas Sect. 4 provides the
2 P.Haensel et al.: Keplerian frequency of neutron stars
assumptionsandmethods usedtocalculaterotatingstellar Table 1. Equations of state of neutron star core. N - nu-
models.InSect.5wecheckthevalidityofempiricalformula cleons and leptons. NH - nucleons, hyperons, and leptons.
against results of precise 2-D calculations for ten realistic Exotic states of hadronic matter are indicated explicitly.
EOSforneutronstars.Hypotheticalself-boundquarkstars Maximumallowablemassfornon-rotatingstars,Mstat,and
max
withnormalcrustareconsideredinSect.6.Sect.7presents the circumferential radius of non-rotating stars of 1.4 M⊙,
the static and rotating configurations in the mass-radius R , are given in last two columns, respectively.
1.4
plane. In Sect. 8 we derive approximate relations between
thecircumferentialradiusofastaticconfigurationandthat
ofaKeplerianconfigurationofthesamegravitationalmass, EOS model ref. Mmstaaxt R1.4
[M⊙] [km]
for neutron starsand quarkstars.Discussionof our results
N,energy
is presented in Sect. 9. FPS a 1.800 10.85
density functional
N,relativistic
GN3 b 2.134 14.22
2. Relativistic Roche model mean field
N,energy
DH c 2.048 11.69
There exists an instructive model of neutron stars for density functional
which an analytic formula for f (M) can be obtained N,variational
K WFF1 d 2.136 10.47
(Shapiro et al.1983,1989).ItisarelativisticRochemodel, theory
in which the mass of the star is assumed to be strongly APR N,variational d′ 2.212 11.42
centrallycondensed.Consideracontinuoussequenceofsta- theory
tionary configurations of constant gravitational mass M, NH,energy
BGN1H1 e 1.630 12.90
and rotation frequencies ranging from zero to f . Let the density functional
K
circumferential radius of non-rotating configuration be R. BBB N, Bruecknertheory f 1.920 11.13
NH,relativistic
Under the assumption of an extreme central mass conden- GNH3 g 1.964 14.20
sation, Shapiro et al. (1983, 1989) found an equation sat- mean field
isfied by the coordinates of the stellar surface (Eq. (2) of N + mixed
GMGS-Km h 1.422 9.95
Shapiro et al. 1989). Inthe special caseof the stellarequa- N-kaon condensed
tor, this equation implies that for normal equilibrium con- N + pure
GMGS-Kp i 1.420 13.20
figurations rotating uniformly at f, the equatorial circum- kaon condensed
ferential radius R satisfies
eq
ReferencesfortheEOS:a-Pandharipande & Ravenhall(1989);
2GM 2GM b - Glendenning(1985); c- Douchin & Haensel (2001);d - A14
+(4πf)2R2 = , (1) ArgonneNNpotentialandUrbanaVIIthreebodyNNNpoten-
Req eq R tial,fromWiringa et al.(1988);d′ -A18ArgonneNNpotential
with relativistic corrections and Urbana modified UIX∗ NNN
where the extreme central concentration of matter implies
potential model, from Akmal et al. (1998); e - Balberg & Gal
thegravitationalmasscanbetreatedasconstantandequal (1997); f - Paris two-body NN potential and UrbanaUIX three
tothe staticvalueM.The left-hand-sideofEq.(1)reaches bodyNNNpotential,from Baldo et al.(1997);g-Glendenning
a minimum at R = (GM/4π2f2)1/3. For the solution to (1985); h, i - kaon condensate models with Ulin = −130 MeV,
eq K
exist for a given f, the value of the left-hand-side at this Pons et al. (2000)
minimum shouldnot exceed2GM/R,which implies a con-
dition on f (Shapiro et al., 1983, 1989),
particle orbiting at r = R in the Schwarzschild space-
K
1 2 3/2 GM 1/2 time around a point mass M at r = 0. An approximate
f ≤fK = 2π 3 R3 . (2) equality fK(M) ≈ foSrcbhw.(M,RK) was shown to be valid
(cid:18) (cid:19) (cid:18) (cid:19) within a few percent for normal neutron stars and quark
Therefore, the Keplerian frequency is stars (Bejger et al., 2007). This relation holds strictly for
the relativistic Roche model,
1/2 −3/2
M R
fRoche(M)=1.00 kHz . (3) 1/2
K (cid:18)M⊙(cid:19) (cid:18)10 km(cid:19) fKRoche(M)=foSrcbhw.(M,RK)= 21π GRM3 . (5)
(cid:18) K (cid:19)
As stated in Shapiro et al. (1989), “the Relativistic Roche
modelprovidesasurprisinglyaccurateestimateofthemax-
imumrotationratealongconstant-restmasssequences”for 3. Realistic EOSs of hadronic matter
many realistic EOSs.
In view of a high degree of our ignorance concerning the
It is easy to show that an additional relation be-
EOS of dense hadronic matter at supranuclear densities
tween Keplerian and static configuration can be obtained.
(ρ > 3 × 1014 g cm−3), it is common to consider a set
Namely,usingEq.(1),oneobtainsaformulaexpressingR
eq of EOSs based on different dense matter theories (for a
for Keplerian configuration, R , in terms of R for static
K review, see Haensel et al. 2007). We used ten theoretical
configuration of the same mass M,
EOSs. These EOSs are listed in Tab. 1, where the basic
3 informations(label of anEOS,theory ofdense matter,ref-
RK(M)= R(M) . (4) erence to the original paper) are also collected.
2
Six EOSs are based on realistic models involving only
The formula for f (M), Eq. (2), and that for R (M), nucleons (FPS, BBB, DH, WFF1, APR, GN3). Four re-
K K
implythatf (M)isequaltotheorbitalfrequencyofatest maining EOSs are softened at high density either by the
K
P.Haensel et al.: Keplerian frequency of neutron stars 3
appearanceofhyperons(GNH3,BGN1H1),oraphasetran-
sitionto a kaon-condensedstate (GMGS-Km, GMGS-Kp).
For GMGS-Km and GMGS-Kp models, the hadronic
Lagrangian is the same. However, to get GMGS-Kp one
assumes that the phase transition takes place between two
pure phases and is accompanied by a density jump, cal-
culated using the Maxwell construction. The GMGS-Km
EOS is obtained assuming that the transition occurs via a
mixed state of two phases (Gibbs construction). A mixed
stateisenergeticallypreferredwhenthesurfacetensionbe-
tween the two phases is below a certain critical value. As
the value of the surface tension is very uncertain, we con-
sidered both cases.
In all EOSs models (except FPS), the core EOS was
joined with the DH EOS of the crust (Douchin & Haensel,
2001). For the FPS model, the FPS EOS of the core was
supplemented with the FPS crust EOS of Lorenz et al.
(1993).
Our set of EOSs includes very different types of mod-
els. This is reflected by a large scatter of the maximum
allowable masses of non-rotating stars, 1.42M⊙ <∼Mmax <∼
2.21M⊙,andarangeofcircumferentialradiiofnon-rotating
stars with M = 1.4M⊙, 9.95km <∼ R1.4 <∼ 14.22km (Table
1). Fig.2. (Color online) Same as Fig.1 but for BBB,WFF1,
and DH EOSs.
Fig.1.(Coloronline)PrecisevaluesofKeplerianfrequency
Fig.3. (Color online) Same as Fig. 1 but for BGN1H1,
f (solid line) and those calculated using Eq. (6) (dashed
K GNH3, GMGS-Km and GMGS-Kp EOSs. Notice that due
line), assuming C = C = 1.08 kHz, versus stellar mass
NS to a very strong softening by the kaon condensate and si-
M. We consider masses 0.5 M⊙ < M < 0.9 Mmstaaxt and multaneousconstraintM <0.9Mstat,the GMGS-Kmand
APR, GN3, and FPS EOSs. max
GMGS-Kpcurvesdonotcontainkaon-condensedsegments.
Therefore, the curves for both these EOSs coincide.
4. Calculating stationary rotating configurations 1999 for the complete set of partial differential equa-
tions to be integrated). The numerical computations
The stationary configurations of rigidly rotating neutron have been performed using the rotstar code from the
stars have been computed in full general relativity by LORENE library (http://www.lorene.obspm.fr). The
solving the Einstein equations for stationary axisymmetric code implements a multi-domain spectral method
spacetime (see Bonazzola et al. 1993; Gourgoulhon et al. introduced in Bonazzola et al. (1998). A descrip-
4 P.Haensel et al.: Keplerian frequency of neutron stars
Fig.4. (Color online) Same as Fig. 1 but for three EOSs Fig.5.(Coloronline) Gravitationalmass,M,versusequa-
of quark stars. Quarks stars possess maximal crust, with torial radius, R , for static and rigidly rotating neutron
eq
bottom density ρ = ρ . To get dash lines, we used Eq. stars, based on the DH EOS. Solid line S: static mod-
b ND
(6) with C =C =1.15. els (i.e., R(M)). Solid line K: Keplerian (mass-shedding)
QS
configurations (i.e., R (M)). The area, bounded by the
K
S, K curves and a dash line S K , consists of
max max
tion of the code can be found in Gourgoulhon et al. points corresponding to stationary rotating configurations.
(1999). The accuracy of the calculations has been Configurationsbelongingtoashadedtriangularareaabove
checked by evaluation of the GRV2 and GRV3 virial the dotline S −K0 havebaryonmassM largerthan
max max b
error indicators (Gourgoulhon & Bonazzola, 1994; themaximumallowablebaryonmassfornon-rotatingstars,
Bonazzola & Gourgoulhon, 1994), which showed val- Mstat . Three lines corresponding to neutron stars rotat-
ues lower than ∼10−5. b,max
ing stably at f =641 Hz,716 Hz, and 1122 Hz, are labeled
with rotation frequencies. The nearly horizontal dot line
S −K correspondstoconfigurationswithfixedbaryon
5. Maximum frequencies for realistic EOSs of 1.4 1.4
number equal to that of the non-rotating star of gravita-
neutron stars
tional mass M =1.4 M⊙. Generally, solid lines connecting
InFigs.1−3wecomparepreciselycalculatedKeplerianfre- filled circle with filled square correspond to M = const.,
quencies with those given by the empirical formula whiledotlinesconnectingfilledcirclewithopensquarecor-
respondtostarswithM =const..Forfurtherexplanations
b
M 1/2 R −3/2 see the text.
f (M)≈C , (6)
K
M⊙ 10 km
(cid:18) (cid:19) (cid:18) (cid:19)
whereM isthe gravitationalmassofrotatingstarandRis exceedingtheneutrondripdensityρND ≈4×1011 g cm−3.
the radius of the non-rotating star of mass M, R=R(M). Maximum mass of normal crusts is reached for ρb =ρND.
The optimal value of the C prefactor is C = 1.08 kHz. First we consider quark stars with a maximum crust.
NS
Theprecisionoftheempiricalformulaforf staysremark- As we see in Fig.4, precision of empirical formula within
K
ably high for 0.5M⊙ <M <0.9Mmstaaxt. Relative deviations the mass range 0.5 M⊙ < M < 0.9 Mmstaaxt is as high as for
are typically within 2%, with largest deviations of at most neutron stars (typical relative deviation within 2%, largest
6% for the highest masses. deviation of about 4% at highest masses). However, the
value of C is larger than for neutron stars, C =1.15.
QS
Let us consider now quark star models with less
6. Quark stars
massive crusts. These models were constructed assuming
ρ < ρ . The effect on the optimum value C ≈
Thecaseofstrangestars,built ofself-boundquarkmatter, b ND QS
isdifferentfromthatofordinaryneutronstars.Matterdis- fK(M)(M/M⊙)−1/2(R/10 km)3/2 turned out to be very
tribution within quark stars has very low density contrast small. At a fixed M, decrease of ρ leads to an increase
b
between the quark core edge and its center. We considered of f (more compact star). Simultaneously, however, the
K
three EOSs of self-bound quark matter, based on the MIT static value of R(M) decreases, and therefore both effects
Bag Model (Farhi & Jaffe, 1984; Zdunik, 2000). Model pa- cancel out to a large extent. Consequently, C depends
QS
rametersaregiveninTable2.Quarkstarsarelikelytohave rather weakly on the crust mass, and in principle one may
a thin normal matter crust, with bottom density, ρ , not use C =1.15 for any crust.
b QS
P.Haensel et al.: Keplerian frequency of neutron stars 5
Table 2. Parameters of the bag models for quark stars. B non-rotating star (filled circle - S ) and Keplerian con-
1.4
-MITbagconstant,m -strangequarkmass.Forallmod- figuration (filled square - K ), in Fig.5. They are joined
s 1.4
els the QCD coupling constant equals α =0.2. Maximum by a solid horizontal line. At a fixed baryon mass, M , M
s b
allowable mass for non-rotating stars, Mstat, and the cir- increases with increasing rotation frequency. Dot line con-
max
cumferential radius of non-rotating stars of 1.4 M⊙, R1.4, necting filled circle (S1.4) and an open square near K1.4
are given in last two columns, respectively. contains configurations with fixed M , equal to that of
b
a non-rotating star with 1.4 M⊙. Deviation of solid line
from the dot one visualizes the rotationalincrease of M at
EOS [MeVBfm−3] [mMsecV2] M[Mmst⊙aax]t [Rk1m.4] eaqfiuxaeldtoM1.b7.%F,oarnad1fo.4r Mthe⊙mstaaxri,mtuhme fsrtaacttiiconmaalsisn(cdreoatsleinies
SQM1 56 200 1.90 11.27
S →K0 ) it reaches 3.8%.
SQM2 45 185 2.02 11.86 max max
SQM3 67 205 1.65 9.94
8. Relation between R(M) and R (M)
K
Let us consider a family (sequence) of stationary configu-
7. Static and rotating configurations in the
rationsrotatingstablyatafrequencyf.Theyformacurve
mass-radius plane inthe M−R plane(seeexamplesinFig.5).Thecurveis
eq
bound at R = R (f) by the axisymmetric instability,
The formulae for f (M) are based on a one-to-one corre- eq min
K implying star collapse into a Kerr black hole. The largest
spondence between a static configuration S, belonging to
circumferentialradiusis reachedfor the Keplerianconfigu-
static boundary S of the region of rotating configurations,
ration,R (f)=R (M).Bejger et al.(2007)haveshown
and the rotation frequency of a Keplerian configuration K max K
thatR (f)is(within 2%)equalto the radiusofanorbit
on the K boundary. This correspondence is visualized in max
of a point particle moving in the Schwarzschild space-time
Fig. 5, based on the numerical results obtained for the DH
around a point (or a spherical) mass M. This implies
EOS.ThefrequencyofrotationofaKeplerianconfiguration
K is obtained via the mass and radius of a static configu- 1/3
GM
ration S with same M. Both configurations are connected R (f)≈ . (8)
by a horizontal line in the Req−M plane. max (cid:18)4π2f2(cid:19)
The empirical formula for the absolute upper bound
For convenience we introduce a frequency f ,
on f of stably rotating configurations for a given EOS, 0
fEOS (Haensel & Zdunik, 1989; Friedman et al., 1989;
Smhaaxpiro et al., 1989; Lattimer et al., 1990; Haensel et al., f = 1 GM⊙ =1.8335 kHz . (9)
1995), is of a different character. It results from an (ap- 0 2πs(10 km)3
proximate but precise) one-to-one correspondencebetween
the parameters of two extremal configurations,static S Validity ofthe empiricalformula,Eq.(6), suggeststhen an
max
and Keplerian K (filled circles), and reads approximate proportionality
max
fmEOaxS ≈C (cid:18)MMms⊙taaxt(cid:19)1/2 R10Mstakmsttmaaxt!−3/2 , (7) RK(M)≈aR(M) , a=(cid:18)fC0(cid:19)−2/3. (10)
FortheextremerelativisticRochemodel,R (M)isstrictly
where C is to a very good approximation independent of K
proportional to R(M), and a =1.5, Eq. (4). For neu-
the EOS. We have C ≈ C = 1.22 kHz (Haensel et al., Roche
NS QS tron stars and quark stars with crusts, with masses within
1995). This value is noticeably higher than C or C ,
which determine fK(M) for 0.5M⊙ <M <0.9MNSmstaaxt. QS a0.5feMw⊙p<ercMent<, a0s.9sMhomswtaaxnt,othneFpirgo.p6o.rtHioonwaelviteyr,htohldesbwesitth-fiint
The functional form of Eq. (7) is, in fact, exact in gen-
proportionality factors are smaller than 1.5 of the Roche
eral relativity for uniform rotation of stars with the so-
model, a ≈a ≈1.44.
calledminimumperiodEOSofKoranda et al.(1997).This NS QS
The dependence of a on the crust mass, M , is very
EOS contains the single parameter, ǫ , which is the tran- QS cr
c weak.ThiscanbeexplainedviatheeffectsofM onR(M)
sition energy density between the low-density EOS with cr
and f (M). These effects oppose themselves: at fixed M,
P = 0 and the high-density EOS with P = ǫ −ǫ . The K
c R(M) increases, and f (M) decreases, with an increasing
value of C for the maximum mass case is 1.35 kHz. K
M .Thecancellationofbotheffectsresultsinanonlyslight
In Fig. 5 we also displayedthe correspondencebetween cr
decrease of a with increase of M (see Fig. 6).
the stellar configurations of the same baryon number. The QS cr
Lasota et al. (1996) derived an approximate relation
line S →K0 separates”supramassive”configurations
max max between equatorial radius of a maximally rotating con-
from the ”normal”ones, which canbe reachedby spinning
figuration, Rrot and the radius of non-rotating neutron
upanon-rotatingstar.The maximumrotationalfrequency fmax
for the ”normal” sequences (reached at point K0max) has rstoatratiwnigthcomnfiagxuimrautmiona,llsotwabalbelebomthassw,itRhMstrametasxp.ecMtatxoimmaallsys
been discussed by Cook et al. (1994a,b) for polytropic and
shedding and axisymmetric perturbations, is actually very
realisticEOSs(theirTables3and7respectively).Itshould
close to that with largest mass, Mrot (in Fig. 5 they
be noted that this value cannot be estimated using our max
are indistinguishable). The approximate proportionality
formula for f (M), Eq. (6), because our formula is valid
within a restrKicted mass range 0.5M⊙ <M <0.9Mmstaaxt. found by Lasota et al. (1996) for neutron stars is Rfromtax ≈
Formula (6) connects configurations of the same grav- 1.32 Rstat . This relation connects two extremal config-
Mmax
itational mass M. For example, it connects M = 1.4 M⊙ urations. They have different masses, related by Mmroatx ≈
6 P.Haensel et al.: Keplerian frequency of neutron stars
than for the relativistic Roche model, C = 1.00 kHz
Roche
(Shapiro et al., 1983).
Using an approximate but quite precise Schwarzschild-
like formula, relating M, R , and f (Bejger et al., 2007),
K K
we show that to a very good approximation the mass-
shedding radius at a given M is proportional to the static
radius R(M), provided0.5M⊙ <M <0.9Mmstaaxt. For neu-
tron stars and quark stars we obtain the best-fit propor-
tionality factor a ≈ a ≈ 1.44. These proportionality
NS QS
factors are smaller than the exact factor 1.5 obtained for
the relativistic Roche model.
Concluding, we derived a set of empirical formulae,
expressing Keplerian frequency and equatorial radius of
Keplerian configuration in terms of the mass and radius
of normal configuration of the same mass. These formu-
lae can be used for masses 0.5M⊙ < M < 0.9Mmax. The
formulae are approximate but quite precise, and therefore
might be useful for constraining the EOS of dense matter
by the observations of pulsars.
Acknowledgements. This work was partially supported by the
Polish MNiSW grant no. N20300632/0450 and by the US DOE
grant DE-AC02-87ER40317. MB was partially supported by Marie
Curie Fellowship no. ERG-2007-224793 within the 7th European
CommunityFrameworkProgramme.
Fig.6. (Color online) Equatorial circumferential radius of
theKeplerianconfiguration,R (M),vs.circumferentialra-
K
dius ofthe static configurationR of the same gravitational References
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(solid lines) and quark stars with crust (dash lines). Three
1804
straight cyan lines correspond: upper line to aRoche = 1.5, BackerD.C.,KulkarniS.R.,HeilesC.,etal.1982,Nature,300,61
middle line to a = a = 1.44, and bottom line a = 1.39. BalbergS.,GalA.,1997,Nucl.Phys.A.,625,435
NS
BalbergS.,LichtenstadtI.,CookG.B.,1999,ApJ,121,515
Neutron stars: Color of a curve for a given EOS coincides
BejgerM.,HaenselP.,ZdunikJ.L.,2007,A&A,464,L49
with that of the EOSlabel(APR,...,WFF1). Quark stars: BaldoM.,BombaciI.,BurgioG.F.1997,A&A,328,274
nearly straight green, blue, and red solid lines, located in BonazzolaS.,GourgoulhonE.,1994,Class.QuantumGrav.,11,1775
Bonazzola S., Gourgoulhon E., Salgado M., Marck J.-A., 1993, A & A,
the lower bundle, correspond to the SQM1, SQM2, and 278,421
SQM3EOSsofTable2withamaximumsolidcrust.Green BonazzolaS.,GourgoulhonE.,MarckJ.-A.,1998,Phys.Rev.D,58,104020
CookG.B.,ShapiroS.L.,TeukolskyS.A.1994a,ApJ,422,227
dash lines in this bundle describe results obtained for the
CookG.B.,ShapiroS.L.,TeukolskyS.A.1999,ApJ,424,823
SQM1EOSofquarkcoreandlow-masscrusts:0.3Mcr,max DouchinF.,HaenselP.2001,A&A,380,151
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mumdeviationoccurringforhighestM notexceeding6%).
Quark stars can reach larger f (M) than neutron stars.
K
With a maximum crust on quark stars, we get C =
QS
1.15kHz.ThevalueofC doesnotdependsignificantlyon
QS
the crust mass, and can be used also for bare quark stars.
We notice that both C and C are significantly larger
NS QS