Fast spinning strange stars: possible ways to constrain interacting quark matter parameters

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by  Sudip Bhattacharyya

MNRAS000,1–14(2015) Preprint25January2016 CompiledusingMNRASLATEXstylefilev3.0
Fast spinning strange stars: possible ways to constrain
interacting quark matter parameters
Sudip Bhattacharyya1⋆, Ignazio Bombaci2,3,4, Domenico Logoteta3, Arun V. Thampan5,6
1Department of Astronomy and Astrophysics, TataInstitute of Fundamental Research, Mumbai 400005, India
2Dipartimento di Fisica, Universita`di Pisa, Largo B. Pontecorvo, 3 I-56127 Pisa, Italy
3INFN, Sezione di Pisa, Largo B. Pontecorvo, 3 I-56127 Pisa, Italy
6
4European Gravitational Observatory,Via E. Amaldi, I-56021 S. Stefanoa Macerata, Cascina Italy
1
5Department of Physics, St. Joseph’s College, 36 Lalbagh Road, Bangalore 560027, India
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6Inter-UniversityCentre for Astronomy and Astrophysics (IUCAA), India
2

n
a
AcceptedXXX.ReceivedYYY;inoriginalformZZZ
J

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2
ABSTRACT

Forasetofequationofstate(EoS)modelsinvolvinginteractingstrangequarkmatter,
]
E characterizedby an effective bag constant (Beff) and a perturbative QCD corrections
H term(a4),weconstructfullygeneralrelativisticequilibriumsequencesofrapidlyspin-
ning strange stars for the first time. Computation of such sequences is important to
.
h study millisecondpulsarsand other fastspinning compactstars.Our EoSmodels can
p support a gravitational mass (MG) and a spin frequency (ν) at least up to ≈ 3.0M⊙
o- and ≈ 1250 Hz respectively, and hence are fully consistent with measured MG and
r ν values. This paper reports the effects of Beff and a4 on measurable compact star
t properties,whichcouldbeusefultofindpossiblewaystoconstrainthesefundamental
s
a quark matter parameters, within the ambit of our EoS models. We confirm that a
[ lower Beff allows a higher mass. Besides, for known MG and ν, measurable parame-

ters, such as stellar radius, radius-to-massratio and moment of inertia, increase with
1
v the decreaseofBeff.Ourcalculationsalsoshow thata4 significantlyaffectsthe stellar
0 rest mass and the total stellar binding energy. As a result, a4 can have signatures in
2 evolutionsofbothaccretingandnon-accretingcompactstars,andtheobserveddistri-
1 bution of stellar mass and spin and other sourceparameters.Finally, we compute the
6 parameter values of two important pulsars, PSR J1614-2230and PSR J1748-2446ad,
0 whichmayhaveimplicationstoprobetheirevolutionaryhistories,andforconstraining
1. EoS models.
0 Key words: equation of state – methods: numerical – pulsars: individual: PSR
6
J1614-2230,PSR J1748-2446ad– relativity – stars: neutron – stars: rotation
1
:
v
i
X
r 1 INTRODUCTION mentalproblemofphysics,andamajorefforthasbeenmade
a during the last few decades to solve this problem by mea-
Despite more than 80 years since the proposal of
suring the stellar bulk properties. A number of these mea-
neutron stars as a new class of astrophysical objects
surement methods for low-mass X-ray binaries (LMXBs),
(Baade & Zwicky 1934)andabout50yearssincetheirfirst
based on thermonuclear X-ray bursts, regular X-ray pulsa-
discovery as radio pulsars (Hewish et al. 1968), their true
tions,high-frequencyquasi-periodicoscillations, broadrela-
natureandinternalcompositionstillremainoneofthemost
tivistic spectral emission lines, quiescent emissions and or-
fascinating enigma in modern astrophysics.
bitalmotions,havebeenreviewedinBhattacharyya (2010).
The bulk properties and the internal constitution of
Spectral and timing properties of non-accreting millisecond
compactstars(neutronstars)primarilydependontheequa-
radiopulsarscanalsobeusefultomeasurestellarmassand
tion of state (EoS) of strong interacting matter, i.e. on the
radius(e.g.,Bogdanov & Grindlay (2009);Bogdanov et al.
thermodynamicalrelation between thematter pressure,en-
(2008)).However, untilrecently theoretically proposed EoS
ergydensityandtemperature.DeterminingthecorrectEoS
models could not be effectively constrained because of sys-
model describing the interior of compact stars is a funda-
tematic uncertainties(e.g., Bhattacharyya (2010)).
The recent precise measurement of the mass (1.97
±
⋆ E-mail:[email protected] 0.04M⊙)ofthemillisecondpulsarPSRJ1614-2230hasruled
(cid:13)c 2015TheAuthors
2 Bhattacharyya et al.
out all EoS models which cannot support such high values Along these lines, for example, a model of the equa-
of masses (Demorest et al. 2010). However, in response to tion of state (EoS) of strange quark matter (SQM)
this discovery, new realistic EoS models, that support high (Farhi& Jaffe 1984) inspired by the MIT bag model of
mass, have been proposed. So essentially all types of EoS hadrons (Chodos et al. 1974) has been extensively used
models,suchasnucleonic,hyperonic,strangequarkmatter, by many authors to calculate the structure of strange
hybrid,stillsurvive,anditisstillaveryimportantproblem stars(Witten 1984;Alcok et al. 1986;Haensel et al. 1986;
to constrain compact star EoS models. Li et al. 1999a,b;Xu et al. 1999),orthestructureoftheso
Inthispaper,weareinterestedinfastspinningcompact calledhybridstars,i.e.compactstarswithanSQMcore.In
stars. So far the spin frequencies of a number of such stars thismodelSQMistreatedasanidealrelativistic Fermigas
havebeenmeasured(e.g.,Watts (2012);Patruno& Watts ofup(u),down(d)andstrange(s)quarks(togetherwithan
(2012); Smedley et al. (2014)). Some of these sources are appropriatenumberofelectronstoguaranteeelectriccharge
binary millisecond radio pulsars, and the masses of a frac- neutrality and equilibrium with respect to the weak inter-
tion of them have been relatively precisely measured (e.g., actions),thatreside inaregion characterized byaconstant
Smedley et al. (2014)). In order to constrain EoS models, energy density B. The parameter B takes into account, in
three independentbulk parameters of a given compact star a crude phenomenological manner, the nonperturbative as-
aretobemeasured,andthethirdobservableparameter(af- pectsofQCDandisrelatedtothebagconstantwhichinthe
terthemassandspin)couldbethestellarradius(Loet al. MITbagmodel(Chodos et al. 1974)givestheconfinement
2013; Bogdanov & Grindlay 2009) or the moment of iner- of quarkswithin hadrons.
tia (Morrison et al. 2004). Here we note that a very high The deconfinement phase transition has been also
observedspinfrequencycouldconstrainEoSmodels,butso described using an EoS of quark gluon plasma derived
far the highest spin frequency observed is 716 Hz from a within the Field Correlator Method (FCM) (Dosh 1987;
radio pulsar PSR J1748-2446ad (Hessels et al. 2006). This Dosh & Simonov 1988; Simonov 1988; DiGiacomo et al.
spin frequency is allowed by almost all proposed EoS mod- 2002) extended to finite baryon chemical potential
els, and hence this spin measurement alone is not a useful (Simonov & Trusov 2007a,b; Simonov 2005, 2008;
property toconstrain these models. Nefediev et al. 2009). FCM is a nonperturbative approach
From a fundamentalpoint of view, theEoS of strongly to QCD which includes from first principles, the dynamics
interacting matter should be derived by numerically solv- of confinement. The model is parametrized in terms of the
ingquantumchromodynamics(QCD)equationsonaspace- gluon condensate G2 and the large distance static quark-
time lattice (lattice QCD). Since the central density of antiquark (QQ¯) potential V1. These two quantities control
compact stars can significantly exceed the saturation den- the EoS of the denconfined phase at fixed quark masses
sity ( 2.8 1014 g/cm3) of nuclear matter and their and temperature. The main constructive characteristic of
∼ ×
temperature could be considered equal to zero after a FCM is the possibility to describe the whole QCD phase
few minutes of their formation (Burrows & Lattimer 1986; diagram as it can span from high temperature and low
Bombaci et al. 1995; Prakash et al. 1997), these compact baryon chemical potential, to low T and high µb limit.
stars can be viewed as natural laboratories to explore Recently,Bombaci & Logoteta (2013)haveestablished
the phase diagram of QCD in the low temperature T that the values of gluon condensate G2 extracted from the
and high baryon chemical potential µb region. In this re- measured mass M = 1.97 ±0.04M⊙ of PSR J1614-2230
gion of the QCD phase diagram a transition to a phase (Demorest et al. 2010) are fully consistent with the values
with deconfined quarks and gluons is expected to oc- ofthesamequantityderivedwithinFCM,fromlatticeQCD
cur and to influence a number of interesting astrophys- calculations of the deconfinement transition temperature
ical phenomena (Perez-Garcia et al. 2010; Sotaniet al. at zero baryon chemical potential (Borsanyi et al. 2010;
2011; Berezhiani et al. 2003; Bombaci et al. 2007, 2011; Bazavov et al. 2012).FCMthusprovidesapowerfultoolto
Weissenborn et al. 2011; Nishimuraet al. 2012). link numerical calculations of QCD on a space-time lattice
Recent high precision lattice QCD calculations at zero with measured compact star masses (Logoteta & Bombaci
baryon chemical potential (i.e. zero baryon density) have 2013).
clearlyshownthatathightemperatureandforphysicalval- In this paper we make use of a more traditional ap-
ues of the quark masses, quarks and gluons become the proach to compute the EoS of SQM. In fact, the simple
most relevant degrees of freedom. The transition to this version of the MIT bag model EoS can be extended to in-
quark gluon plasma phase is a crossover (Bernard et al. cludeperturbativecorrectionsduetoquarkinteractions,up
2005;Cheng et al.2006;Aokiet al. 2006)ratherthanareal to the second order (O(α2s)) in the strong structure con-
phase transition. Lattice QCD calculations in this regime stantαs (Freedman & McLerran 1977,1978;Baluni 1978;
have also distinctly demonstrated the importance of taking Fraga et al. 2001;Kurkelaet al. 2010)1.Withinthismodi-
intoaccounttheinteractionsofquarksandgluonssincethe fiedbagmodelonecanthusevaluatethenon-idealbehaviour
calculated EoS significantly deviates from that of an ideal of theEoS of cold SQMat high density.
gas. The modified bag model EoS has already been used
Unfortunately, current lattice QCD calculations at fi- to calculate the structure of non-spinning strange stars
nitebaryonchemicalpotentialareplaguedwiththesocalled (Fraga et al. 2001; Alford et al. 2005; Weissenborn et al.
“signproblem”,whichmakesthemunrealizablebyallknown
lattice methods. Thus, to explore the QCD phase diagram
at low temperatureT and high µ , it is necessary toinvoke
b
some approximations in QCD or to apply some QCD effec- 1 InFarhi&Jaffe (1984)theEoSforstrangequarkmatterwas
tivemodel. calculateduptotheorderO(αs).
MNRAS000,1–14(2015)
Fast spinning strange stars 3
2011; Fraga et al. 2014) and hybrid stars (Alford et al. (Berezhiani et al. 2002,2003;Bombaci et al. 2004).Inthis
2005; Weissenborn et al. 2011). scenario quark stars could be formed via a conversion pro-
As already mentioned, there are two types of compact cessofmetastablenormalneutronstars(Bombaci & Datta
stars containing SQM. The first type is represented by the 2000; Berezhiani et al. 2002, 2003).
socalledhybridstars,i.e.compactstarscontainingaquark- Several experimental searches for strangelets (small
hadron mixed core and eventually a pure SQM inner core. lumps of absolutely stable SQM) have been undertaken us-
Thesecondtype,strangestars,isrealized whenSQMsatis- ing different techniques. For example the Alpha Magnetic
fiestheBodmer–Wittenhypothesis(Bodmer 1971;Witten Spectrometer(AMS-02)(Tomasetti 2015),on board of the
1984). According to this hypothesis SQM is absolutely sta- InternationalSpaceStationsinceMay2011,couldbeableto
ble, in other words, its energy per baryon (E/A) is less detectstrangelets incosmicrayswithexcellentchargereso-
uds
thanthatofthemostboundatomicnuclei(56Fe,58Fe,62Ni) lution up to an atomic number Z 26. Strangelets search
∼
which is 930.4 MeV. The absolute stability of SQM does inlunarsoil,broughtbackbytheNASAApollo11mission,
∼
not preclude the existence of “ordinary”matter (Bodmer have been performed using the tandem accelerator at the
1971; Witten 1984). In fact, under this hypothesis, atomic Wright Nuclear Structure Laboratory at Yale (Han et al.
nuclei can be considered as metastable states (with respect 2009)
tothedecaytoSQMdroplets)havingamean-lifetimewhich Itisimportanttoemphasizethatallthepresentobser-
ismanyordersofmagnitudelargerthantheageoftheUni- vational data and our present experimental and theoretical
verse. knowledgeofthepropertiesofdensematter,donotallowus
In the last few decades, many researchers (see e.g. to accept or to exclude the validity of the Bodmer–Witten
Xu et al. (2001); Weber (2005) and references therein) hypothesis.
havetriedtoidentifypossibleclearobservationalsignatures In this paper, we assume the validity of the Bodmer–
to distinguish whether a compact star is a strange star, a Witten hypothesis and we compute equilibrium sequences
hybridstar or a“normal”neutron star (nucleonicstar). ofrapidlyspinningstrangestarsingeneralrelativity.In 2,
§
The mass-radius (M-R) relation is one of the most wedescribethemodifiedbagmodelEoS(Fraga et al. 2001;
promisingcompactstarfeaturestosolvethispuzzle.Infact, Alford et al. 2005;Weissenborn et al. 2011)forSQMused
“low mass” (i.e. M . 1M⊙) strange stars have M R3, in our calculations and the two parameters, effective bag
∼
whereas normal neutron stars, in the mass range between constant (Beff) and perturbativeQCD corrections term pa-
0.5M⊙and∼0.7Mmax(whereMmaxisthestellarsequence rameter (a4), which characterize this model. In § 3, we dis-
maximummass),havearadiuswhichisalmostindependent cuss the method to compute parameters of fast spinning
on the mass (Lattimer& Prakash 2001; Bombaci 2007) compactstarsinstableconfigurations.Herewealsodescribe
ThisqualitativedifferenceintheM-Rrelation isduetothe variouslimitsequences.In 4,wepresentthenumbersfrom
§
factthatstrangestarsareself-boundobjectswhereasnormal our numerical calculations using tables and figures. In 5,
§
neutronstarsareboundbygravity.ConstraintsfortheM-R wediscusstheimplicationsofourresults,especiallyforcon-
relation,extractedfromobservationaldataofcompactstars strainingBeff anda4 usingobservations.In 6,wesumma-
§
intheX-raysourcesSAXJ1808.4-3658(Li et al. 1999a)and rize thekey pointsof this paper.
4U1728-34(Liet al. 1999b)seemtoindicatethattheseob-
jects could be accreting strange stars. Note that the obser-
vationsofthermonuclearX-raybursts,whicharebelievedto 2 THE MODIFIED BAG MODEL EQUATION
originate from unstable thermonuclear burning of accumu- OF STATE FOR STRANGE QUARK
lated accreted matter on the compact star surface, cannot MATTER
preclude these LMXBs from having strange stars. This is
because,accordingtoaproposedmodel(Stejner& Madsen The EoS for strange quark matter including the effects of
2006),theburstscouldhappenonathincrustof“ordinary” gluon mediated QCD interactions between quarks up to
matter (i.e. a solid layer consisting of a Coulomb lattice of O(α2s) can be written in a straightforward and easy-to-use
atomic nuclei in β-equilibrium with a relativistic electron form similar to the simple and popular version of the MIT
gas, similar to the outer crust of a neutron star) separated bagmodelEoS.Thegrandcanonicalpotentialperunitvol-
fromthemainbodyofthestrangestarbyastrongCoulomb ume takes the form (we use units where ~ = 1, and c = 1)
barrier (Alcok et al. 1986; Stejner& Madsen 2005).
stituOtiothnerosfigcnoifimcpaancttobsstearrvsatmioanyalicnofmoremaotuitonisnonththeencoexnt- Ω=i=uX,d,s,eΩ0i + 4π32(1−a4)(cid:16)µ3b(cid:17)4+Beff. (1)
few years from the expected detection of gravitational
waves from compact stars with ground-based interferome- whereΩ0 is thegrand canonical potential for u, d,s quarks
i
ters, such as Advanced VIRGO and Advanced LIGO. In andelectronsdescribedasidealrelativisticFermigases. We
fact, it has been shown by different research groups, e.g. take m = m = 0, m = 100 MeV and m = 0. The
u d s e
(Andersson et al. 2002;Benharet al. 2007;Fuet al 2008; secondtermontherighthandsideofEq.(1)accountsforthe
Andersson et al. 2011; Rupaket al. 2013), that gravita- perturbativeQCDcorrections to (α2)(Fraga et al. 2001;
O s
tional waves driven by r-mode instabilities or from f-, p- Alford et al. 2005; Weissenborn et al. 2011) and its value
andg-modeoscillationscouldbeabletodiscriminateamong representsthedegreeofdeviationsfromanidealrelativistic
different typesof compact stars. Fermi gas EoS, with a4 =1 corresponding to the ideal gas.
Another interesting possibility is that both “nor- Thebaryonchemicalpotentialµ canbewrittenintermsof
b
mal” neutron stars (nucleonic stars) and quark stars theu, d and s quark chemical potentials as µ =µ +µ +
b u d
(i.e. strange stars or hybrid stars) could exist in nature µ . The term B is an effective bag constant which takes
s eff
MNRAS000,1–14(2015)
4 Bhattacharyya et al.
intoaccountsinaphenomenologicalwayofnonperturbative plottedin themiddlepanelof Fig. 1,one sees thattheEoS
aspects of QCD. in the form P(µ ) = Ω(µ ) has a sizeable dependence on
b b
−
Using standard thermodynamical relations, the energy theparameter a4.
density can be written as: Theinfluenceoftheparametera4 ontheEoS(Eqs.(1),
(2))ofSQMismoreclearin thecase ofmassless quarks.In
3 µ 4
ε=i=uX,d,s,eΩ0i +4π2(1−a4)(cid:16) 3b(cid:17) +i=uX,d,s,eµini+Beff, fpaacrta,minettrhicisalcafosremonienctaenrmsshoowf tthheatbtahreyoEnoSnufmorbSerQdMe,nsinitya,
can be written as:
(2)
where ni is the number density for each particle species ε=Kn4b/3+Beff
which can becalculated as P = 1Kn4/3 B , (7)
3 b − eff
∂Ω
n = (3)
i −(cid:18)∂µi(cid:19)V where
and thetotal baryon numberdensity is 9π2/3
K = (8)
1 4a14/3
n = (n +n +n ). (4)
b 3 u d s and eliminating thebaryon numberdensityn onegets
b
Equilibrium with respect to the weak interactions implies
1
the following relations between the quarks and electron P = 3(ε−4Beff), (9)
chemical potentials:
whichdoesnotdependona4.Thusthestellargravitational
µs =µd =µu+µe, (5) mass MG versus the central energy density εc, or the stel-
theelectric charge neutrality condition requires: lar radius R versus εc, in the case of massless quarks, will
not depend on the perturbative QCD corrections term a4.
2n 1n 1n n =0. (6) These stellar properties will have a tiny dependence on a4
3 u− 3 d− 3 s− e inourcaseduetothefinitevalueofthestrangequarkmass,
Sinceinthepresentpaperwestudythecaseofspinning ms=100MeV.Nevertheless,stellarpropertieslikethetotal
strange stars, we consider values of the EoS parameters a4 rest mass of the star M0(εc), the relation between MG and
andBeff sothatSQMsatisfiestheBodmer–Wittenhypothe- M0 and consequently the value of the total stellar binding
sis.Next,toguaranteetheobservedstabilityofatomicnuclei energy B = M0 MG (Bombaci & Datta 2000) will sig-
−
with respect toa possibledecay toadroplet of non-strange nificantly be affected by the parameter a4. This ultimately
(i.e. u, d) quark matter, we require that the energy per should affect theenergetics of explosive phenomenalike su-
baryon(E/A) ofnon-strangequarkmattershould satisfy pernovae or the conversion of normal neutron stars (nucle-
ud
thecondition(E/A) >930.4MeV+∆,where∆ 4MeV onic stars) to strange or hybrid stars (Bombaci & Datta
ud
∼
accounts for finite size effects of the energy per baryon of a 2000; Berezhiani et al. 2003).
dropletofnon-starngequarkmatterwithrespecttothebulk To illustrate how the parameter a4 affects the stellar
(A ) case (Farhi & Jaffe 1984). rest mass M0 and binding energy B, let us consider for the
→∞
The values for the EoS parameters considered in the momentthecaseofnon-spinningconfigurations.Thestellar
calculations reported in this work, are listed in Table 1. In rest mass is given by
particular, to explore the effects of the perturbative QCD
R 2Gm(r) −1/2
cthorereficxteiodnvsaolunetBhe1/p4r=op1e3rt8ieMs eoVf ,spwinencionngsisdtrearntgweosdtaiffrse,refnort M0 =muZ0 4πr2nb(r)(cid:20)1− c2r (cid:21) dr (10)
eff
values of the parameter a4 (= 0.61 and 0.80); a4 = 0.61 wherem =931.494 Mev/c2 istheatomic mass unit,n (r)
u b
gives a larger deviation from theideal gas than a4 =0.80. is the baryon number density at the radial coordinate r,
The EoS, for the three parametrizations used in the
andm(r)isthestellargravitationalmassenclosedwithinr.
presentwork,areplottedinFig.1.NoticethatatfixedB ,
eff Now,inaddition,considerthecaseofmasslessquarks.Using
thevalueofa4 hasaverysmall effectontheEoS expressed Eq.(7) and (8), we get thefor the baryonnumberdensity
as P = P(ε), i.e. pressure P as a function of the energy
densityε(leftpanelinFig.1),whichentersinthestructure n (r)= 2 3/2 1 a1/4 ε(r) B 3/4. (11)
equations for non-spinningand spinning stars. To highlight b (cid:18)3(cid:19) √π 4 h − effi
the influence of the perturbative QCD corrections term a4
on the EoS, in comparison with the EoS 1 and EoS 2, we SubstitutingthisexpressioninEq.(10)andconsideringthat
also plot in Fig. 1 that for the case B1/4 = 138 MeV and thefunctionsm(r)andε(r)donotdependontheparameter
eff
a4 =1.0(idealrelativisticFermigasplusbagpressure).We a4, one obtains the following scaling relation for the stellar
do not, however, use this EoS in the calculations for spin- rest mass
nfoirngthsitsracnhgoeicsetaorfstrhepeopraterdamheetreer,ssiBnce1ffe/4waenhdavae4,vethriefieadtotmhaict M0(εc;a4)=(cid:18)aa4′ (cid:19)1/4M0(εc;a′4) (12)
nuclei are unstable with respect to decay to a droplet of 4
non-strangequark matter. where a4 and a′4 are two different values of the perturba-
The results plotted in the left panel of Fig. 1, how- tiveQCDcorrectionsparameterandε isthecentralenergy
c
ever, do not imply that the perturbative QCD corrections densityof thestar. Therefore, for a star with a given gravi-
are unimportant. Clearly, looking at Eq.(1) and the results tationalmassM ,thestellarbindingenergyscaleswiththe
G
MNRAS000,1–14(2015)
Fast spinning strange stars 5
600
(138, 1.0)
EoS 1 (138, 0.8)
EoS 2 (138, 0.61)
500
EoS 3 (125, 0.5)
400
]
3
m
f
V/
300
e
M
[
P
200
100
0
0 500 1000 1500 800 1200 1600 0 0.3 0.6 0.9 1.2 1.5
ε [MeV/fm3] µb [MeV] n [fm-3]
b
Figure 1.PressureP versus energy density ε (leftpanel), P versus baryon chemical potential µb (middlepanel) and P versus baryon
numberdensitynb (rightpanel)forthethreestrangequarkmatter EoSparametersets usedinthepresentwork(see§2andTable1).
TohighlighttheinfluenceoftheperturbativeQCDcorrectionsterma4 ontheEoS,incomparisonwiththeEoS1andEoS2cases,we
alsoplottheEoSwithBe1ff/4=138MeV anda4=1.0(idealrelativisticFermigasplusbagpressure).
parameter a4 as withthetwocentralenergydensitiessatisfyingthecondition
B(a4)=(cid:18)aa′44(cid:19)1/4B(a′4). (13) Eεcq,1u/aεtci,o2n=s (B1e4ff),1a/nBdeff(1,25.) give the scaling law for the m(a1s6s)-
radius relation. In particular they hold (Witten 1984;
In Fig. 2 we report our results for the rest mass M0 of Haensel et al. 1986) for themaximum mass configuration.
thestarasafunctionofitscentraldensity(leftpanel)andas
Finally, the perturbative QCD corrections to the EoS
afunctionofthecorrespondinggravitationalmassM (right
G make it possible to fulfil the Bodmer–Witten hypothesis in
pstaanresl)w.iRthestuhletsEionSF1ig(.a42=are0.r8e0g)aardnidngEonSon2-s(pai4n=nin0g.6s1t)rawnitghe awirtehgiaonmoafxtihmeuBme1ff/g4r–aav4itpatlaionneawlmhearsesosnigenciafincagnettlystlraarnggeerstthaarns
B1/4 = 138 MeV in both cases. As we can see, decreasing
eff 2 M⊙ (see Fig. 1 in Weissenborn et al. (2011)), and thus
the value of the parameter a4, i.e. increasing the deviation in agreement with current measurements of compact star
from the ideal relativistic Fermi gas (a4 = 1), results in a masses like that of PSR J1614-2230 (M = 1.97 0.04M⊙;
reductionofthestellarrestmassatagivencentraldensityor ±
Demorest et al. (2010)).
at agiven M .Noticethattheresultsplottedin Fig. 2, for
G
a strange quark mass m = 100 MeV, are well reproduced
s
by the scaling relation given in Eq. (13). As we will see in
thefollowing, thisscalingrelationwillbealsoveryusefulin 3 COMPUTATION OF RAPIDLY SPINNING
thecase of fast spinningstrange stars. STELLAR STRUCTURES
Simple scaling relations for the properties of non-
For computing the stationary and equilibrium se-
spinning strange stars have been obtained in terms of the
quences of strange stars, we make use of the for-
effective bag constant B (Witten 1984; Haenselet al.
eff malism mentioned in Cook et al. (1994); Dattaet al.
1986; Bombaci 1999; Haensel et al. 2007) in the case of
(1998);Bombaci et al. (2000);Bhattacharyyaet al. (2000,
the EoS given by Eq. (9). As discussed in AppendixA, the
2001a,b,c);Bhattacharyya (2002,2011).Thegeneralspace-
gravitationalmassandtheradiusofthestar,fortwodiffer-
time for such a star is (using c = G = 1; Bardeen (1970);
entvaluesB andB oftheeffectivebagconstant,are
eff,1 eff,2 Cook et al. (1994)):
related by
ds2 = eγ+ρdt2+e2α(dr2+r2dθ2)+eγ−ρr2sin2θ
B 1/2 −
MG(εc,1;Beff,1)=(cid:18)Beeffff,,12(cid:19) MG(εc,2;Beff,2), (14) (dφ−ωdt)2, (17)
whereγ,ρ,αaremetricpotentials,ωistheangularspeedof
the stellar fluid relative to the local inertial frame, and t, r
B 1/2 andθ aretemporal,quasi-isotropicradialandpolarangular
R(εc,1;Beff,1)=(cid:18)Beeffff,,12(cid:19) R(εc,2;Beff,2), (15) coordinates respectively. Assuming the matter comprising
MNRAS000,1–14(2015)
6 Bhattacharyya et al.
Figure 2. Stellar rest mass M0 versus central density (left panel), and M0 versus stellar gravitational mass MG (right panel) for
non-spinningstrangestars.
thestartobeaperfectfluidwithenergymomentumtensor: ture are obtained, it becomes feasible to compute gen-
eral quantities exterior to the compact star like the Kep-
Tµν =(ε+P)uµuν+Pgµν (18)
lerian angular speed and specifically, the radius rISCO of
the innermost stable circular orbit (ISCO). These quan-
and a unit time-like description for the four velocity vec-
tities depend on the effective potential that has a maxi-
torwecandecomposetheEinsteinfieldequationsprojected
onto the frame of reference of a Zero Angular Momentum mum at rISCO. Here is how rISCO is calculated. The ra-
dial equation of motion around such a compact star is
Observer (ZAMO) to yield three elliptic equations for γ,
r˙2 e2α+γ+ρ(dr/dτ)2 =E˜2 V˜2, where, dτ is the proper
ρ and ω and a linear ordinary differential equation for α tim≡e,E˜ isthespecificenergy,−whichisaconstantofmotion,
(Cook et al. 1994). The elliptic equations are then con-
and V˜ is the effective potential. This effective potential is
verted to integral equations using the Green’s function ap-
l2/r2
proach.The hydrostaticequationsare derivedfrom therel- given by V˜2 = eγ+ρ[1+ ]+2ωE˜l ω2l2, where l is
eγ−ρ −
ativistic equations assuming a linear spin law (in our case
the specific angular momentum and a constant of motion.
the spin law integral vanishes for the rigid spin condition). rISCO is determined using the condition V˜,rr = 0, where
The solution for the hydrostatic equilibrium equation re-
a comma followed by one r represents a first-order partial
ducestosolvingofonealgebraicequationateachgridposi-
derivative with respect to r and so on (Thampan & Datta
tion and matching these with the equatorial values for self-
1998). We consider rorb (=rISCO) to be the smallest possi-
consistency.Foradesiredcentraldensityρcandpolarradius bleradiusof theaccretion disc.But Ris theabsolutelower
toequatorialradiusratio(rp/re),amulti-iterationrunofthe limit of the disc radius. Therefore, if the star extends be-
program to achieve self-consistency yields a 2-dimensional
yondtheISCO,wesetr =R(Bhattacharyyaet al. 2000;
orb
distribution of the metric potentials, density and pressure
Bhattacharyya 2011).
repesenting an equilibrium solution. This equilibrium so-
lution is then used to compute compact star parameters,
Sinceobservationsofsomepulsarshaveprovidedamea-
such as gravitational mass (MG), rest mass (M0), equato- sure for MG and ν, we compute constant MG and constant
rialcircumferentialradius(R),totalangularmomentum(J),
ν equilibriumsequencesofacoupleofpulsars.ForeachEoS
spin frequency (ν), moment of inertia (I), total spinning
model,wealsocomputeanumberofconstantM0sequences.
kinetic energy (T), total gravitational energy (W), surface
These sequences would represent the evolution of isolated
polar redshift (Zp), and forward (Zf) and backward (Zb) compact stars conserving their rest mass. Such sequences
redshifts for tangential emission of photons at the equator ∂J
arestabletoquasi-radialmodeperturbationsif <0;
(Cook et al. 1994; Dattaet al. 1998). ∂ρc|M0
Once the equilibrium parameters describing the struc- we calculate the limit where this inequality does not hold.
MNRAS000,1–14(2015)
Fast spinning strange stars 7
In addition to this instability limit, there are three other s−1 and 1250 HzrespectivelyforEoS 3(seeTables3and
∼
limits: (1) thestaticornonspinninglimit, whereν 0and 4).ButnotethatthesethreenumbersforagivenEoSmodel
→
J 0; (2) the mass-shed limit, at which the compact star arefor threedifferentconfigurationson themass-shed limit
→
spins too fast to keep matter bound to the surface; and (3) sequence.Tables3and4,whichdisplaythestableparameter
thelow-masslimit,belowwhichacompactstarcannotform. valuesfor maximum valuesof MG and J respectively, show
These four limits together define the stable stellar parame- thatthereisagapofalmost 2kmbetweenthestarandthe
terspacefor an EoS model(Cook et al. 1994).Here,apart ISCO,and T/W is roughly 0.21 for each EoS model.
from the instability limit, we calculate thestatic and mass- Table 5 displays three configurations on the rest mass
shed limit sequences, but we do not attempt to determine M0 =2.00M⊙ sequencefor EoS 1and EoS 2. These config-
the low-mass limit, where the numerical solutions are less urationsarealsomarkedwithfilledcirclesinFigs.3–8.This
accurate (Cook et al. 1994). table shows that, for the same ρc, most of the parameters
have significantly different values for the two EoS models.
This difference is around or more than one order of magni-
tudefor J, ν and T/W for thehighest ρc value considered.
4 RESULTS
After this general characterisation of the SQM EoS
The results of our computations are summarized in Figs. 3 models for rapidly spinning stars, we compute the stable
through11andTables2through7.Figs.3–5areforEoS1, structures of two well known pulsars. We show the param-
Figs.6–8areforEoS2,andFigs.9–11areforEoS3.Figs.3, eter values for the massive pulsar PSR J1614-2230 (MG =
6and 9show MG versusρc curves.The staticlimits, which 1.97M⊙,ν =317.5Hz;§1)forthethreeEoSmodelsinTa-
cansupportmaximumMG valuesof2.09M⊙,2.07M⊙ and ble6.Thispulsarismarkedwithasquaresymbolineachof
2.48 M⊙ for EoS models 1, 2 and 3 respectively, are shown Figs.3–11.ThemassofthefastestknownpulsarPSRJ1748-
by solid curves. As ρc increases, more inward gravitational 2446ad (ν = 716 Hz; § 1) is not yet known. We, therefore,
pull,i.e.,moreMGisrequiredtobalancethepressure.Apart computetheν =716Hzequilibriumsequencesforthethree
from thestaticlimit,themass-shedandinstabilitylimit se- EoSmodels,andshowthestablestellarparametervaluesfor
quencesarealsoshowninthesefigures(see 3).Thestaris anumberofconfigurationsinTable7.Themaximumvalues
not stable for the parameter space to therig§ht of thecurve ofMG supportedbyν =716 Hzare 2.18 M⊙,2.16 M⊙ and
definingtheinstabilitylimit.Anumberofconstantrestmass 2.64 M⊙ for EoS models 1, 2 and 3 respectively. Figs. 3–11
sequences,and theMG =1.97 M⊙ sequenceare also shown showtheν =716 Hzsequencewith long-dashedcurvesand
in Figs. 3, 6 and 9. On a constant rest mass sequence, the themaximumMG configurationwith adiamondsymbolon
stellar J increases from right to left, and so does MG to each of thesecurves.
balancetheextracentrifugalforce. Therest masssequence,
whichjoins themaximumMG onthestaticlimit, separates
the supramassive sequence region (above) from the normal
5 DISCUSSION
sequenceregion(below).Inthesupramassiveregion,acom-
pactstarissomassivethatitcanbestableonlyifithassuf- In this paper, we confirm that strange stars with interact-
ficient J value. When J decreases (say, via electromagnetic ing quark matter can support > 2M⊙ gravitational mass,
and/or gravitational radiation) to the value corresponding evenwhentheyarenotspinning.Sincethehighestprecisely
to the instability limit, the compact star collapses further measured masses ofcompact starsare 2M⊙ (see 1;also
≈ §
to become a black hole. Another surprising aspect of the Antoniadiset al. (2013)), this provides a possibility of the
supramassive region is, as J decreases, I can decrease at a existence of strange stars. In order to explore this possi-
higher rate, and hence ν can increase. So the star can spin bility, here we consider three EoS models based on MIT
up,as it loses angular momentum. bag model with perturbative corrections due to quark in-
Figs. 4, 7 and 10 show MG versus R curves for various teractions ( 2). These EoS models are characterized by an
§
sequences.ThesefiguresshowthatRusuallyincreaseswith effective bag constant (B ) and a perturbative QCD cor-
eff
MG. This property distinguishes strange stars from “nor- rections term (a4). We, for the first time, compute rapidly
mal” neutron stars (see also 1). Figs. 5, 8 and 11 show spinning strange star equilibrium sequences for these mod-
ν versus dimensionless J (cJ/§GM2) curves. These figures els, which are essential tostudy millisecond pulsars, as well
0
clearlyshow,whileν decreaseswiththedecreaseofJ inthe as non-pulsar fast spinning compact stars in LMXB sys-
normal sequence region, it can have opposite behaviour in tems( 1).Wefindthatthemaximummassessupportedby
§
thesupramassiveregion.Finally,wenotethatallthecurves ourEoS models are in therange 3.0 3.6M⊙ for themass-
−
of the Figs. 3–11 are qualitatively consistent with those for shed limit, and in the range 2.2 2.6M⊙ for the spin
≈ −
other SQM EoS models (see, for example, Bombaci et al. frequency (ν =716 Hz) of thefastest known pulsar. Hence,
(2000)). a precise measurement of a high mass will be required to
Table2displaysthevaluesofρc,M0,R,R/rg androrb reject theseEoS modelsbased on mass measurement alone.
for the maximum MG configurations in static limit. This The maximum spin frequency of 1250 1500 Hz can be
≈ −
tableshowsthat thestellar radiusis3.74rg (3.75rg,3.75rg) supportedbyEoS 1 3 ( 4). Therefore, spin-down mecha-
− §
andtheISCOis 7( 7, 8) km abovethecompact star nisms, such as those due to disk-magnetosphere interaction
∼ ∼ ∼
forEoS1(2,3).Here,rgistheSchwarzschildradius.Forthe (Burderiet al. 1999; Ghosh 1995), electromagnetic radia-
mass-shed limit, the maximum values of MG, J and ν are tion(Ghosh 1995)and/orgravitationalradiation(Bildsten
3.03 M⊙,7.17 1049 gcm2 s−1 and 1500 Hzrespectively 1998),mayberequiredtoexplainthetheabsenceofanob-
for EoS 1, 3.00×M⊙, 7.01 1049 g cm∼2 s−1 and 1501 Hz served spin frequency above 716 Hz. The spin-down due to
respectively for EoS 2, an×d are 3.60 M⊙, 9.98 1∼049 g cm2 disk-magnetosphereinteractioncanhappenforbothaccret-
×
MNRAS000,1–14(2015)
8 Bhattacharyya et al.
Figure 3. Gravitational mass versus central density plot of Figure 4. Gravitational mass versus equatorial radius plot of
strangestarsforEoS1(see§2andFig.1).Thesolidcurveisfor strange stars for EoS 1 (see § 2 and Fig. 1). The meanings of
anon-spinningstar,theshort-dashedcurveshowsthemass-shed curvesandsymbolsaresameasinFig.3.
limit, the long-dashed curve is for a spin frequency of 716 Hz,
the dotted curves are for evolutionary (i.e., constant rest mass)
sequences,thedash-dotcurvegivestheinstabilitylimittoquasi- nismscanalsoworkforstrangestars(Ahmedovet al. 2012;
radialmodeperturbations,andthedash-triple-dotcurveisforthe Andersson et al. 2002).
gravitationalmass=1.97M⊙.Theconstantrestmassvalues,for Wenotethatthemaximumangularmomentumappears
dotted curves from bottom to top, are 1.71 M⊙, 2.00 M⊙, 2.63 at a lower central density than that for themaximum mass
M⊙, 2.72 M⊙, 2.92 M⊙, 3.15 M⊙, 3.33 M⊙, 3.66 M⊙ and 3.82 (Tables 3 and 4). This is in agreement with calculations
M⊙. The triangle and cross symbols are for the maximum mass for other EoS models (Bombaci et al. 2000). A compari-
(Table 3) and the maximum total angular momentum (Table 4)
son between EoS 1–2 and EoS 3 for the maximum mass
respectively for the mass-shed sequence, the three filled circles
or maximum angular momentum configurations gives some
markedwith‘1’,‘2’and‘3’ontherestmasssequenceM0=2.00
M⊙correspondtothe‘No.’columnofTable5,thesquaresymbol idea about the extent of the dependence of stellar param-
isfortheobservedmass(=1.97M⊙)andspinfrequency(=317.5 eter values on the EoS model parameter Beff (Tables 2–4).
Hz)ofPSRJ1614-2230(Table6),andthediamondsymbolisfor Here we list some of the points. (1) The central density is
themaximummasswhichcanbesupportedbythefastestknown significantlylowerforthelowereffectivebagconstant(B )
eff
pulsar PSR J1748-2446ad spinning at 716 Hz (Table 7). See § 4 value (i.e., EoS 3). This general behaviour is in agreement
foradescription. with the scaling law Eq. (16) mentioned in 2 (see also
§
A). (2) A lower effective bag constant value can support
§
a larger mass, and the corresponding radius is also higher.
ing “normal” neutron stars and strange stars, if the spin- TheseareexpectedfromthescalinglawEqs.(14)and(15),
down in the propeller regime is more than the spin-up in especiallyformaximummassconfigurationsofnon-spinning
theaccretionregime(Burderi et al. 1999).Theelectromag- compactstarswithEoSmodelgiveninEq.(9)( 2;seealso
§
netic radiation and the gravitational radiation exert nega- A).Wefindthat,forourEoS 2andEoS3,themassratio
tive torques ν3 and ν5 respectively on the compact a§nd the radius ratio expected from Eqs. (14) and (15) hold
∝ ∝
star(Ghosh 1995;Bildsten 1998).Thesetwolattermecha- within a few percent not only for non-spinning maximum
MNRAS000,1–14(2015)
Fast spinning strange stars 9
Figure 5.Spin frequency versus dimensionless angular momen- Figure 6. Gravitational mass versus central density plot of
tumplotofstrangestarsforEoS1(see§2andFig.1).J isthe strange stars for EoS 2 (see § 2 and Fig. 1). The meanings of
totalangularmomentumandM0 istherestmass.Themeanings curvesandsymbolsaresameasinFig.3.Theconstantrestmass
ofcurvesandsymbolsaresameasinFig.3. values, fordotted curves frombottom to top, are2.00M⊙, 2.50
M⊙,2.71M⊙,2.92M⊙,3.13M⊙,3.34M⊙,and3.55M⊙.
mass configurations (Table 2), but also for mass-shed limit
maximum mass configurations (Table 3). As a result, the and R is also very important for the X-ray emission, be-
stellar compactness, which is the mass-to-radius ratio, and causetheboundarylayeremissiontotheaccretiondiscemis-
hence the surface redshift values, are somewhat similar for sionratiodependsonthislength(e.g.,Bhattacharyyaet al.
all Beff values. (3) T/W weakly depends on Beff, and the (2000)). We find that rISCO is greater than R for the max-
value is 0.21 for maximum mass and maximum angular imum mass and maximum angular momentum configura-
∼
momentum configurations. At this value, the compact star tions, and the gap-length for EoS 1–2 is somewhat lower
couldbesusceptibletotriaxialinstabilities(Bombaci et al. than that for EoS 3. This is expected from the scaling law
2000). (4) The oblateness of the compact star, as inferred equations14and15( 2;seealso A),becauserISCO MG.
§ § ∝
fromRp/R( 0.53 0.55),isaweakfunctionofBeff.(5)J Let us now examine how Beff can be constrained from
∼ −
andIstronglydecreases,whileνsignificantlyincreases,with observations, especially when MG and ν have been mea-
the increase of B . The strong B -dependence of I is ex- sured. Comparing EoS 2 (B1/4 = 138 MeV) and EoS 3
eff eff eff
pectedfromthescalinglawequations14and15( 2;seealso (B1/4 = 125 MeV), we find significant differences for the
§wiAth),Bb1e/c2au(TseabIle∼3)M,aGsRe2x.pWecteedve(rsiefye thAat).νSirnocue§gJhly sIcνa,leJs me1eaff7s%uraabnlde par3a1m%etreersspeRc/tirvge,lyR(TanabdleI,6)w.hNicohteatrhea≈t, s1i7n%ce,
isexpecefftedtoscalewithBe−ff1,whichw§everify(Table∝3).(6) t≈hese param≈eters depend on a4 only very weakly (see row
Thetwodifferentsituations,viz.,rISCO >RandrISCO <R 1 and row 2 of Table 6), the relatively small a4 difference
(rISCO isISCOradius)haveimportantconsequencesforob- between EoS 2 and EoS 3 does not prohibit us to study a
served X-ray features, and hence on the measurements of B dependence.
eff
compact star parameters (e.g., Bhattacharyya (2011)). In In order to examine if theabove quoted differences are
the former situation, the length of the gap between rISCO in agreement with the scaling laws mentioned in A, even
§
MNRAS000,1–14(2015)
10 Bhattacharyya et al.
Figure 7. Gravitational mass versus equatorial radius plot of Figure 8. Spin frequency versus dimensionless angular momen-
strange stars for EoS 2 (see § 2 and Fig. 1). The meanings of tumplotofstrangestarsforEoS2(see§2andFig.1).J isthe
curvesandsymbolsaresameasinFig.3. totalangularmomentumandM0 istherestmass.Themeanings
ofcurvesandsymbolsaresameasinFig.3.
forafastspinningstarforwhichtheTOVequations(A4and odstomeasureR/rg arealsoavailable(e.g.,Bhattacharyya
A5)arenotexactlyvalid,firstwementiontheratiosofR/rg, (2010), Bhattacharyyaet al. (2005)). Possible measure-
R and I for B1/4 =138 MeV and B1/4 =125 MeV. These ments of R for LMXBs and the related difficulties have
eff eff
are [R/rg]EoS2/[R/rg]EoS3 =0.84, REoS2/REoS3 =0.84, and been discussed in a number of papers (e.g., Gu¨veret al
IEoS2/IEoS3 = 0.73 (Table 6). Now, since radius is a weak (2012a,b); Steineret al. (2010, 2013); Suleimanov et al.
function of central density for most of the observationally (2011a,b); Guillot & Rutledge (2014)). Modelling of burst
relevant portion of the parameter space (say, MG > 1M⊙; oscillations observed with a future large area X-ray tim-
see, for example, Figs. 3 and 4), we can assume R B−1/2 ing instrument can tightly constrain R of a compact star
∝ eff
from Eq. A7, and hence we expect REoS2/REoS3 0.82, in an LMXB (Lo et al. 2013). The pulsed X-ray emis-
≈
which is close to theabovementioned value. Notethat MG sion from radio millisecond pulsars can also be useful to
is constant for all EoS models in Table 6. Therefore, while constrain R (Bogdanov & Grindlay 2009; Bogdanov et al.
comparingparametervaluesforEoS2andEoS3,weexpect 2008; O¨zel et al. 2015). In fact, the upcoming NICER
R/rg ∝ R ∝ Be−ff1/2 and I ∝ R2 ∝ Be−ff1. Hence from § A space mission is expected to measure the R of the nearest
we expect [R/rg]EoS2/[R/rg]EoS3 = 0.82 and IEoS2/IEoS3 = and best-studied millisecond pulsar PSR J0437-4715 with
0.67, which are close to theabove mentioned values. 5% accuracy. Besides, a measurement of I of the binary
So let us now see if some of the above quoted percent- pulsar J0737-3039A with 10% accuracy has been talked
age differences of R/rg, R and I (Table 6) can be obser- about (Morrison et al. 2004). Apart from R/rg, R and I,
vationally measured. A fortuitous discovery of an atomic [r R] of LMXBs, which can be inferred from spec-
orb
−
spectral line can constrain R/rg with better than 5% accu- tral and timing studies of observed X-ray emission (e.g.,
racy,evenwhenthestarisrapidlyspinningmakingtheline Bhattacharyyaet al. (2000)), also has a significant differ-
broadandskewed(Bhattacharyyaet al. 2006).Othermeth- ence ( 98%) between EoS 2 and EoS 3. Therefore, since
≈
MNRAS000,1–14(2015)