Compact stars: Neutron stars or quark stars or hybrid stars?

-- PHYSICS REPORTS

ELSEVIER PhysicsReports242 (1994)313—332

Compactstars:Neutronstarsor quark starsor hybrid stars?

Erlend Ostgaard

Fysiskinstitutt, AVH. Universiteteti Trondheim,7055 Dragvo!!, Norway

Abstract

Usingresultsfrom energycalculationsof neutronmatter,we constructvariousneutronstarequationsof state.Fromtheseequationsof state,togetherwith the Tolman—Oppenheimer—Volkoffequations,we calculatequantitiessuchaspressure,massdensity,massenergydensity,total mass,radius,andmomentof inertiafor neutronstars.Comparisonismade with calculationsbasedon other nuclear potentialsand nuclear energy calculations,and our results are inreasonableagreementwith resultsfrom observationaldata.

We alsocalculatemass,radius,momentof inertiaandsurfacegravitationalredshift of “quark stars”describedby threemodels:anon-interactingFermigasmodel,an asymptoticMIT bagmodel,andaperturbativeQCDmodel.We alsogiveresultsof phasetransitioncalculations,andcommenton possibleobservabledifferencesbetweenneutronstarsandquarkstars.

Finally, propertiesof “hybrid stars” consistingof a coreof strangequark mattersurroundedby ordinaryneutronmatter are investigated.We discussstar models basedon phenomenologicalequationsof state including a phasetransitionbetweenthe hadronic phaseandthe quark—gluonplasma.For certainparameters,theseequationsof statesupporttheexistenceof hybrid stars.The identificationof suchobjectscould provide information on the propertiesofstrangequark matter.

1. Introduction

Accordingto theoriesof stellarevolution,the final stagein astar’slife is representedby compactobjects,i.e., compactstars.Themain factordeterminingwhethera starendsup asawhite dwarf,neutronstar,or black hole is then thoughtto be the star’s mass.

White dwarfs are believed to originate from light starswith massesM < SM®, andthere isamaximumallowedmassaround1 .4M® for white dwarfs.Whitedwarfprogenitorstarsprobablyundergoa relatively gentlemassejection,forming planetarynebulaeat the endof their evolutionbefore becomingwhite dwarfs.

Neutron stars andblack holes are then believed to originate from more massivestarswithmassesM > 5M®. However, the separationbetweenthose stars that form neutron stars andthosethat form black holesis very uncertainbecausethefinal stagesof evolutionof massivestarsare poorly understood.Neutron stars also have a maximum mass in the range of 1.4—3M®,but numericalcalculationsof massloss from starsandsupernovaexplosionsare in a primitive

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314 E. Ostgaard/PhysicsReports242 (1994) 313—332

stage.Thus the fate of a starwith amass M > SM® is not clear, but neutronstarsseemmostlikely.

In theneutronstarinterior, theconfigurationis simply aconsequenceof the onsetof differentregimesin theequationof stateaswe go to higherdensities.Thelayers maythenbe identified as[1]; The surfacefor p ~ 106gem3 is a region in which the temperaturesand magneticfieldsexpectedfor most neutronstarscansignificantly affect theequationof state.The outercrust for106gcm3<p < 4.3 x 10” gem3 is asolid region in which a Coulomb lattice of heavy nucleicoexists in f3-equilibrium with a relativistic degenerateelectron gas. The inner crust for4•3 x 10” gem3 < p < 2.4 x 10’4gcm3 consistsof alatticeof neutron-richnuclei togetherwithasuperfluidneutrongasandan electrongas.The neutronliquid for 2.4 x iO’~g cm3 <p < Pcore

containsmainlysuperfluidneutronswith asmallerconcentrationof superfluidprotonsandnormalelectrons.A core region p > Pcore may or may not exist in some other stars,and dependsonwhetherthereis atransitionto pioncondensationor aneutronsolid or quarkmatteror someotherphaseat densitiesabovesomecritical valuePcore~

If theequationof stateis stiff, the centraldensityof arelatively massiveneutronstarof 1 .4M® isp~< 1015 g cm~ and eventhe most massive,stableneutron stars have p~< 3 x 1015 g cmA transition to quark matter or some other exotic form of matter may then seemunlikely.However,theexistenceof athird stablebranchof “quark stars”on thetotalmassM versuscentraldensityp~,diagrambeyondwhite dwarfs andneutronstarsalso remainsapossibility.

2. General theory

Assuming an isotropic mass distribution and the validity of Einstein’s general theory ofrelativity, theoverall structureof aneutronstaris describedby theTolman—Oppenheimer—Volkoff(TOY) equations[2, 3]

dP — — [G/r2][p(r) + P(r)/c2][M(r) + 4itr3(P(r)/c2)] (2 1)dr — 1 — 2GM(r)/rc2

and

= 4icr2p(r), (2.2)dr

whereP(r) is the pressure,p(r) is the massenergydensity,M(r) is the gravitationalmassinside r,G is the gravitationalconstant,and c is the speedof light. Equation(2.2) may be written as anintegral equation,

M(r) = 4m I p(s)s2ds, (2.3)

~Jo

and(2.1) and (2.3) are,togetherwith an equationof state

P = P(p), (2.4)

E. østgaard/PhvsicsReports242 (1994) 313—332 315

the fundamentalstructureequations.The equationof state(2.4) is obtainedfrom the Helmholtz’free energyF through the relation

P = — ~F/~V, (2.5)

where V is the volumeof the system.In neutronstars,the temperatureT is much lower thantheFermi temperatureTF. Hence,we considertheneutronstarsto below-temperaturesystems,andwe get

P = n2[~~] (2.6)

whereE/N is the energyper particle,andn is the particledensityN/V.We alsocalculatethe momentof inertia I for slowly rotatingsphericallysymmetricstarsfrom

~(~i’~ =~i (2.7)\~0�2J~=~Q’

whereJ is theangularmomentumandQ is theangularvelocity measuredin an inertial systematinfinity. The metric outsidea slowly rotatingstaris the Schwarzsehildmetric with an additionalcrossterm [4, 5] — 2wr2sin20 d~dt, wherew(r) is theangularvelocity of the local non-rotatingsystemas measuredby an observerin a far-awayinertial system.

To calculatethetotal mass,radius,andmomentof inertia we usetheequationof state(2.6), and(2.1), (2.3), (2.7), and the input boundaryconditions

= P(nC) , M(0) = 0 , (2.8)

wherethe subscriptc refersto the centerof the star.We thenintegratefrom r = 5 to thesurfaceof

thestardefinedby r = R whereP(R)= 0. (2.9)

In principle, c5 = 0, but in practicea very small, but finite valueis chosen.Thus, for aparticularequationof stateP = P(n), the input parameterin our static or slowly

rotating neutron-starmodel is the central particle density.The describedprocedurealso givespressure,massdensity,and massenergydensityprofiles in the star.

3. Neutron stars

In the neutron star matter calculations, we develop equations of state P = P(n) that arecontinuouslysmoothin differentdensityregions,i.e., they arewithout discontinuities.Otherwisewe would get an infinite pressuregradientandinfinite forcesactingon masselements.

The resultingtotal equationof stateP(n) is thenusedas input in theTOV equations,andfigures1 and2 showresultsfor themassM(nC)andtheradiusR(n~),wheren~is thecentralparticledensity.For the pure AO-5 equationsof state [6, 7], we get a variation of the moment of inertia asa function of the stellarmassas shownin Fig. 3. For thestarwith maximummasswe obtain

I(Mmax) = 1.08 x iO”~gem2 . (3.1)

Correspondingresultsarealso shownin Table 1.

316 E. østgaard/PhysicsReports242 (1994) 313—332

___ Ii’ ~: \\ __-

~, ___ ____

Cf)o.9 — __ __ —

0.? __ __ __ __ — 6 __ __ ____2 4 6 8 2 4 6 8

Central particle density n© (1m3) Central density n0 (fm

3)

Fig. 1. Total massM (in units of the solar massM

0) as function of centralparticledensity n,.

Fig. 2. Total radiusR as function of central particledensity n~.

1.3

0.8 1.2 1.6

M (Me)

Fig. 3. Themomentof inertial asfunctionof total starmassfor theAø-5 model [6,7]. Thecurvecorrespondsto centralparticledensitiesn, between0.4fm

3 and2.4fm3.

More recentresults,including new equationsof state[9, 10], are [11]

1.65M® < Mmax < 2.43M®, 8.8km < R < 12.7km, 1.7 fm3 > n~>0.72fm3.(3.2)

Fromall the calculationswefind that starscalculatedwith astiff equationof statehavegreatermaximummassesthanstarsderivedfrom asoft equationof state,andstarscalculatedfrom astiffequationof statehavea lower centraldensity,a largerradius,andamuch thicker crust thandostarsof the samemasscomputedfrom asoft equationof state.

E. Ostgaard/PhvsicsReports242 (1994) 313—332 317

Table 1Propertiesof themaximummassMJB neutronstarmodels[8] andtheAø-5 model[6, 7]. Mis themaximumstarmass,R is thecorrespondingstarradius,Pm, is thecentralmassdensity,P.is thecentralpressure,andI is the total momentofinertia.

Model Mm~x R p,,,, PC 1(M®) (km) (l0’5gcm3) (lo36dyncm2) (10~’gcm2)

I-H 1.87 9.93 2.14 1.08 1.59Ill-H 1.74 9.05 2.54 1.38 1.26V-H 1.63 9.18 2.61 1.15 1.11V-N 1.78 9.34 2.44 1.26 1.35AO-5 1.65 8.31 2.78 1.42 1.08

Radial stability for compactstarscan be examinedby studyingthe massversuscentraldensitycurveandthe massversusradiuscurve.In general,this kind of work is time-consumingandoftenrathermodel-dependent.This is partly the reasonwhy analysesof radial stability aresomewhatindecisive.But the existenceof afourth classof stablecompactstars,the quarkstars,maypossiblybe ruled out from such analysis.

The generalresult is that equilibrium configurationswith dM/dp~> 0 arestable,while thosewith dM/dpC < 0 areunstable.And,correspondingly,weget dR/dp~< 0 for astableconfigurationanddR/dp~> 0 for an unstableconfiguration.

The observedmassesfor the starsof the Hulse—Taylorbinary pulsarPSR 1913 + 16 are [12]

Mpuisar = 1.442±0.003M®,(3.3)

MCompanion = 1.386±0.003M®

A generallyacceptedlowestlimit on the massfor this pulsaris

M ~ 1.35M® , (3.4)

which in fact should eliminatethe softestequationsof state.Presentobservationsof otherneutronstarsindicatefurther that

1.2M® � M � 1.6M®, (3.5)

which shouldbe valid if all theneutronstarshaveapproximatelythe samemass.If this is not thecase,we should get

1.OM® � M � 2.2M® . (3.6)

Reliablemeasurementsof neutron-starradii do not existat present.But estimatesbasedon otherobservationslead to

R ~ 8.5 km, (3.7)

which, however,maybeafactor 2 too smallbecauseof neglecting,for instance,effectsfrom strongmagneticfields. Our resultsarein good agreementwith theseexperimentalvalues.


The pure Aø-5 model [6, 7] gives the value (3.1) for the moment of inertia of the star ofmaximummass,whichalsocould beanticipatedfrom otherequationsof state.The resultsarecloseto resultsfrom similar calculations,andnot contradictoryto anyobservations.But, althoughourcalculationsreproducereasonablevaluesfor mass,radius, etc., the real matter composition inneutronstarsmaystill bean openquestion,andwhetherneutronstarsactuallyareneutronstarsor“quark stars” or somethingelse hasbeendiscussedalreadyfor sometime.

4. Phasetransition to quark matter

The questionnow is: Is it possiblethat matterat ultrahigh densities

p > 1015g/em3 (4.1)

mayundergoaphasetransition from a“baryon” to a “quark” state?

The phasetransitionis calculatedfrom the Gibbs’ criterionli~(1~~)= liq(Pt), (4.2)

andgiven the equationof stateof the cold matter under consideration,the Gibbs’ energyli perbaryonfollows fromderivatingthetotal energydensitywith respectto baryonnumberdensity.Thefirst orderphasetransitionis thencalculatedby demandingthattheoccurringphasehasthelowestGibbs’ energyper baryon.

In Table 2 we show someresults from calculationswith a perturbativeQCD model and anasymptoticbagmodelasmodelsof quark stars[13]. The five neutronstarequationsof statearethe four Malone—Johnson--Betheequationsof state [8] andthe Arntsen—østgaardequationofstate [6]. These were calculatedin a lowest-orderconstrained-variation(LOCV) schemeandaFermi-hypernetted-chain(FHNC)scheme,respectively.Only for onemodeldo wefind apossiblequarkcoreneutronstar.This happensfor the, presumably,least realisticneutronstarmodelMJBI-H, calculatedin a particularQCD model wherethe cut-off is 300 MeV. Seealso Fig. 4.

FromTables 1 and2 it is clearthat themost “realistic” equationsof statedo not give aquarkmattercorein the star.However, increasingthe densityfurther gives rise to anew branchon themassversuscentraldensitycurvewith apositive derivativewhichcould, in principle, meanquarkstarsas anew classof compactobjects.

5. Quark stars

Freequarkshavenot beenseen,but thereare strongreasonsto believethat the hadronsarecomposedof quarks, and the idea of quark stars has alreadyexisted for some time [14, 15].Calculationsof thepossiblephasetransitionfrom baryonmatterto quarkmatterin modelsof coldcompactstarshavebeenperformedby severalgroups[16—21],but the resultsarenot conclusiveconcerningthe existenceof quark matterinside neutron stars.It hasalso beensuggestedthatstrangematter,i.e., quark matter with strangenessper baryon of order unity, may be the truegroundstateof matter[22]. Thepropertiesof strangematteratzeropressurehavebeenexamined,it hasbeenfound thatstrangemattercan indeedbe stablefor awide rangeof parametersin strong

E. østgaard/PhysicsReports242 (1994)313—332 319

Table 2Thecalculatedbaryon—quarkphasetransition.PT is thetransitionGibbsenergyperbaryon,PT is thetransitionpressure,n1 is the baryonnumberdensity on the“neutron side” of the transition,n2 is the baryonnumberdensityon the“quarkside” of the transition,andp~is the massdensity of the neutronside of the transition.

Model PT PT pmi

(GeV) (lo36dyncm2) (fm3) (fm3) (lO15gcm3)

(a) QCD, AF = 250MeV

I-H 1.64 0.72 1.09 1.44 1.82Ill-H 1.84 1.30 1.49 2.11 2.49V-H 1.96 1.75 1.88 2.59 3.15V-N 1.83 1.26 1.46 2.07 2.44AO-5 1.98 1.84 1.82 2.68 3.05

(b) QCD, ‘iF = 300MeV

I-H 1.88 1.21 1.34 2.19 2.24Ill-H 2.03 1.79 1.68 2.78 2.81V-H 2.14 2.35 2.13 3.32 3.57V-N 2.02 1.76 1.66 2.75 2.78AO-5 2.14 2.33 1.97 3.30 3.30

(c) QCD, ‘iF = 400MeV

I-H 2.28 2.19 1.70 3.85 2.85Ill-H 2.38 2.85 2.02 4.39 3.38V-H 2.49 3.70 2.59 5.08 4.34V-N 2.37 2.78 2.00 4.34 3.35AO-5 2.45 3.39 2.21 4.83 3.70

(d) MIT, a, = 0.549, B = 59.2MeVfm3

I-H 2.82 3.82 2.13 3.47 3.57Ill-H 3.05 5.31 2.58 4.42 4.32V-H 3.40 8.23 3.65 6.11 6.11V-N 3.05 5.25 2.58 4.39 4.31Aø-5 3.16 6.12 2.60 4.91 4.35

interactioncalculations[23], anddetailsof the extensionto finite pressureandso-calledstrangestarsaregiven [24, 25]. Theproblemof theexistenceof strangestarsis, however,still unresolved[26].

A very importantquestionis how it is possibleto distinguishbetweenquarkstarsandneutronstars.It hasbeensuggestedto usemeasurementsof the surfacegravitationalred shift z, sincedifferentequationsof stategive differentresultsfor z(M), M beingthestarmass[27,28]. It is alsopossiblethat theneutrinoflux from quark matteris substantiallylargerthanfrom conventionalneutronstarmatter [29, 30], resulting in a fasterstarcooling, andthis might be detectablebyobservations.The regionof allowedhigh-densityequationsof statemay be narrowedfurther by

320 E. østgaard/PhvsicsReports242 (1994) 313—332

23 1M __L_————~I~

1.2I

0.8 —— — ____________10 20 30

P (iou dyn/cm2)

Fig. 4. The calculatedphasetransitionfor QCD, ‘iF = 300MeV, matchedwith the MJB models I—H andV—H [13,43].

observationsof pulsarperiods [31, 32]. Given a sub-millisecondpulsar, we may arguethat theability of such a fast rotating star to avoid rotational break-upinducessevererestrictionsandaconventionalneutronstarwill not be able to resistthe largecentrifugal forces.The problemofrapid rotation of compact stars has receivedmuch attention [33—36],and although no sub-millisecondpulsarhasbeenseenamongthe about500 pulsarsobservedso far, furtherobservationscould revealsuch an object.

We have performed calculations of the star mass, radius, moment of inertia, and surfacegravitationalred shift for threemodelsof quarkstars[37] to establishresultsfor the discussionofthepossibleexistenceof quarkstars.Also, the Keplerianfrequencycanbecalculatedfrom themassandthe radiusof thenon-rotationstarsin the models.

Our structurecalculationsareperformedfor: (i) anon-interactingmodel [15] which is apara-Fermi statisticsversion of the ideal Fermi gas non-interactingsingle speciesequationof state,(ii) an asymptoticMIT bagstar [38—40],and (iii) aperturbativeQCD model[18,41]. Thephasetransitioncalculationshavebeenperformedfor models(ii) and(iii) for thequark phaseof thestar,andthe four MJB models[8] anda derivedequationof state[6] for the baryonphaseof the star.

The non-interactingmodel can be written as

P = 7.92x 1036 m~f(xq)dyncm2, (5.1)

p C/c2 = (5.34x 10’5 nmq + 8.81 x 10’sm~g(xq))gem’

whereP is the pressure,p is the massenergydensity,a is the totalenergydensity,mq is the quarkmassin units of GeVc2 n is the baryonnumberdensityin units of fm -~,Xq is PF.q /mqc wherePF,q is the quark Fermi momentum,and

f(x) = x(2x2 — 3)(x2 + 1)1/2 + 3 sinh’ x, (5.2)

g(x) = 8x3[(x2 + 1)1/2 —1] —f(x).

E. østgaard/PhvsicsReports242 (1994)313—332 321

The baryonnumberdensityn is relatedto xq by

n = 13.2 m~x~, (5.3)

i.e.,

n \1/3 1x =l—~ — (5.4)

q \\13.2) mq

The asymptoticMIT bagstaris describedby

a = An413 + B , P (1/3)An413— B , (5.5)

wherea is the energydensity,andB is theconfining “bag pressure”of the orderof 50MeVfm ~.

The constantA is for a u, d, s modelgiven by

A = 9It213(1 + ~cc,/m)hc, (5.6)

whereA is aconstantsince in this model thefirst-order effectivequark—gluoncouplingconstantcc, is takento be aconstant.

The perturbativeQCD model canbe formulatedsimply as

a = An413 , (5.7)

whereA is relatedto cc, as in (5.7), but wherenow

it 1cc =— (5.8)C 181n(kF/AF)’

andAF is the infraredQCD cut-off. Introducingtheparameter

x = kF/AF , (5.9)

this modelcan be written as

a=~hc(1 +27~)A~x4,

(5.10)

3 [ 4 / 1\12P=—~hcl1+ (1————JIA~x

4.4it [ 27lnX\ ‘~X/]

Herethe baryonnumberdensityn is relatedto AF and x by

n = A~~3/it2. (5.11)

Results for mass,radius, moment of inertia and surface gravitational red shift for objectsdescribedby these models now follow from numericalintegrationsof the Tolman—Oppen-heimer—Volkoffequations,andthe Schwarzschildmetric hasbeen “perturbed”by an additionalcrossterm dç~dt in the calculationsof I.

We havecalculatedtheMIT bagstarfor differentsetsof the parameters(cc,,B), sincethis set isnot uniquelydefined.With B given in units of [MeV fm 3] oneset is

(cc,,B) = (0.549,59.2). (5.12)

322 E. østgaard/PhysicsReports242 (1994) 313—332

For this set themaximummassMmax,andthe correspondingradiusR andmomentof inertia I isfound to be

Mmax 1.98M® , R(Mmax) 10.79km, I(Mmax) 2.20x i0~~gem2 . (5.13)

The calculationsin the perturbativeQCD model indicatethat both the mass andthe radiusincreaseasAF decreases.The calculationsindicatefor AF = 250MeV that

Mmax ~ 1.32M® , R(Mmax)~ 7.20km, I(Mmax) ~ 0.67 x iO’~gem2 . (5.14)

The results of our calculationsare summarizedin Figs. 5—9. The non-interactingFermi gasmodel resultsin rathersmall radii for largequark massesmq, and the valuesfor total massM,radiusR, andmomentof inertia I arelike thosefor aneutronstarfor aquarkmassof 400MeV/c2.

Themomentof inertia I for the threemodelsis plottedas functionof thestellarmassM in Fig. 7.The form of the curves can be explaineddirectly from the correspondingrelationsbetweenM

andR.The surfacegravitational redshiftz. is calculatedfrom

—1, (5.15)— 2GM/(Rc2)

andz~is plottedin Fig. 8 asafunctionof the stellarmass.A comparisonof the bagstarresultwiththe two neutronstarmodelsI—H and V—H [8] is shownin Fig. 9. For astarmass M ~ 1.4M®measurementsof z, can hardly distinguishbetweenthe differentstars.

_/fl~ _

_ 00 ~OIL1L~2nc (fm~) R (km)

Fig. 5. TotalmassM asfunctionofcentralbaryonnumberdensityn, for thethreequarkstarmodels.(1) theperturbativeQCD model with ‘iF = 300MeV, (2) the asymptotic bag model with a, = 0.549 and B = 59.2MeV fm3, (3) thenon-interactingFermi gasmodelwith input quark massmq = 400MeV/c2.

Fig. 6. Total massM as function of quark starradiusR. (1) the perturbativeQCD model with ‘iF = 300MeV, (2) theasymptoticbagmodelwith a, = 0.549 andB = 59.2MeV/fm3, (3) thenon-interactingFermigasmodelwith inputquarkmass = 400MeV/c2.

E. Østgaard/PhvsicsReports242 (1994) 313—332 323

::~N :::________

/ . —~

01 09 II 13 15 17 19 O0~ 09 11 1.3 15 1.7 19

M [M9] M [MGI

Fig. 7. Total moment of inertia I as function of total quark starmass M. (1) the perturbativeQCD model with‘iF = 300MeV, (2) the asymptoticbagmodel with a, = 0.549 andB = 59.2MeV/fm

3, (3) thenon-interactingFermi gasmodelwith input quark massmq = 400MeV/c2.

Fig. 8. The surfacegravitational redshift z, as function of total mass M. (1) the perturbativeQCD model with‘iF = 300 MeV, (2) the asymptoticbagmodelwith a, = 0.549and B = 59.2MeV/fm3, (3) the non-interactingFermigasmodel with input quark massmq = 400MeV/c2.

(2)00 _____ - ~._____ ~) ‘1

M (M1)

Fig. 9. The surfacegravitationalredshiftz. asfunctionof total massM for theasymptoticquarkbagstarmodel(1) andthe neutronstarmodelsI—H (2) andV—H (3) [13,43].

A simple estimatefrom the centrifugal timescalerelation

T=2it/.~J~, (5.16)

shows that the density of a sub-millisecondpulsar is higher than maximal densities found inplausibleneutronstarmodels.Thus,we shouldat leastconsiderthe possibility that suchpulsarsmay be rapidly spinningquark stars[34].


Thereis no evidencefor theexistenceof asub-millisecondpulsarat present.But the problemoflimiting pulsarperiodsandtheproblemof thecouplingof stronggravityandrapid rotationshouldstill be considered.The existenceof quarkstarsor possiblyhybrid stars(neutronstarswith quarkcores)thenseemsto be an openquestion.

6. Hybrid stars

We now considermodelsfor compactstarshavinga first orderphasetransitionfrom ordinaryhadronicmatter to aquark—gluonplasma.Appropriateequationsof statemay thenindicatetheexistenceof hybrid stars.We calculate their gross propertieslike total mass,radius, surfacegravitationalredshiftandKeplerianfrequency,andwewill discusswhetheronecaninfer from theexistenceof this type of starsconstraintson the underlyingequationof state.

Theequationof statefor neutronmatter,particularlyat highdensities,is still a matterof debate.The uncertaintiesarereflectedin theconsiderablevarietyof significantlydifferentparametrizationswhich areappliedfor the descriptionof the structureof neutronstars[42]. Wehaveexaminedtwostandardapproachesdesignedfor neutron matter, namely that of Bethe and Johnson[8,43](referring to theirmodelI—H) andan equationof stateobtainedfrom Walecka’smean-fieldtheory[44—46].

We define the hadronicequationof stateas the compressionalenergyper particlee,omp(n) intermsof the conservedbaryonchargedensityn which is carriedby severalspeciesof baryons.Wewill refer to two commonlyusedparametrizationsfor thenuclearmatterequationof state,namelythat of Sierk andNix [47],

esN(n) = ~ — 1)2 , (6.1)

andthe quadraticform [48]

K (n — n0)2

eQ(n)= 1~ no~ (6.2)wheren

0 is thenormalnuclearmatterdensityandK is thecompressionconstantcharacterizingthepropertiesof nuclear matter at densities n > n0. Given a parametrizatione,omp(n), a largerK correspondsto a more repulsivenucleon—nucleoninteraction.

The total energydensityfor coldmatter(T = 0) thentakesthe form

a(n) = n[e,omp(n) + W0 + Wsym + m~], (6.3)

where W0 = — 16 MeV is the binding energy per nucleon at normal nuclear matter densityn0 = 0.145fm

3m~= 939MeV is the restmassof aneutron,and 13’ym = 32MeV is the symmetryenergyof neutronmatteratn

0, estimatedfrom theliquid dropmodel.Within thisphenomenologi-calapproach,the symmetryandbindingenergyWsymand W0,respectively,determinetheproper-ties of matterat saturation,while ecomp(n) incorporatesall density-dependenteffects.

At densitiesbelow n0, thepressurein thestaris no longerdeterminedby the nucleon—nucleoninteractionsonly. Below acritical low-densityvalue ~LD we apply apolytropic form for ecomp(n),matchingthe pressureandenergydensityof the nuclearpart of theequationof stateat densitiesaroundnLD ~ 1.2 n0.

E. østgaard/PhvsicsReports242 (1994) 313—332 325

Whenthe totalenergydensityis known,the pressureat zerotemperaturecanbeobtainedfromthe thermodynamicrelation

p(n)=n~4~~_e(n). (6.4)

Similarly, the chemicalpotential is relatedto the energydensityby

öa(n) p(n) + a(n)li(fl)~” n (6.5)

The quark—gluonplasmais takeninto accountby a “bag model” equationof stateof the form

P=4(a—4B), (6.6)

wheretheenergydensityof the quark—gluonplasmais that of amixture of gluonsandmasslessu, dands quarks(Nf = 3) at zero temperature.In first order perturbationtheory the energydensitytakesthe form

= ~ (i — ~ ccs) ~2 + B, (6.7)

wherethechemicalpotentialliB of thebaryonsis relatedtothat of thequarksby RB =3liQ~Thebag

constantB is the difference betweenthe perturbativevacuumand the “true” vacuumenergydensity.Dueto theperturbativecorrectionsin the quark—gluonequationof state,ourcalculationsalsocontain astrongcoupling constantcc

0 which is assumedto be a constant[49, 50] cc0 = 0.4,lowering the pressurein the plasma.It has beenshown, however, that the running couplingconstantin termsof the thermodynamicvariablesli~T and the QCD scaleparameterA;

2 n F 4(1 \2 1~L~’~r21—1g ________ I ~liB) + 1~).UL~I I

ccs(~u,T) = = ~ — 4 N, [In A~ j (6.8)

variesonly little, i.e., typically within the bounds0.3 < cc. < 0.55 for the relevantrangeof densitiesconsideredhere. And the conclusionswill not dependsignificantly on the choice of the strongcouplingconstant.

Dependingon the centraldensityof the star,suchamulti-phaseequationof statewill allow fortheformationof neutronstarsaswell asfor hybrid stars.In principle,alsopurestrangestarscan beobtainedasa limiting case,as also discussedelsewhere[51—57].Givenoneof the aboveformsofa(n) for nuclear matter, we are left with three parameters,namely the compressionconstantK associatedwith the hadronicphase,andthe vacuumenergydensityB andthe strongcouplingconstantcc,, which characterizethe quark—gluonphase.

At centraldensitieshigher thanthe critical densityof the quark—gluonplasmanq,Cr, thecoreofthestarwill consistof strangequarkmatter.Oncethe critical pressureis reached,thedensitydropsdiscontinuouslyfrom nq,,. to ~n,cr, indicatinga first order phasetransition.The densityprofilesshown indicatehybrid starswith astrangemattercoreof about6—7 km radiusandan outerlayerof neutronmatter of about3—4 km thickness.More than60% of the total massis in the quarkmatterphase.A stiffer nuclearequationof statelowers thecritical densitiesboth for the hadronicphaseflfl Cr’ and for the quark—gluonphasenq,,r.


Fig. 10 shows the mass of a staras a function of its central baryondensity for two differentparametrizationsof the nuclearpart of the equationof state.The left figure is obtainedwith the“Sierk—Nix” equationof state,while the right onerefersto a“Quadratic” form of thecompressionenergy. We recognizetwo branchesof solutions; the neutron stars at lower central densitiesn, � nn,,r, andthe hybrid starregionat densities~c � ~q,Cr~ Both branchesare separatedby theregionof instability at ~~n,Cr < n < nq,,r. While the massesof the neutronstarsincreaserapidly asa function of the centraldensity,the massesof the hybrid starsvary relatively slowly with n~.

Taking into account that the stability of hybrid stars under density fluctuations requiresdM/dcC > 0, wefind stablehybrid starsatenergydensitiesarounda, ~ 0.5—1.5GeV/fm3correspond-ing to centralbaryondensitiesn, ~ 4—9n

0,whereasmoredensestarswill collapseinto blackholes.Without a phasetransition,we find stableneutron starsup to densitiesof iOn0, maximum

massesof M ~ 1.8—2.6M®, wherethestarmassesare given in units of the solarmass M®, andcorrespondingradii of R ~ 10—13 km,dependingon the equationof state.Within awide rangeofcompressionenergies,stablehybrid starsat centraldensitiesof 4—9n0,massesof M ~ 1.4—1.7M®

andradii of R ~ 10—13km, arepredicted.Hybrid starsareconsiderablylighter thanneutronstarsfor the samecentraldensitybecausethe

equationof statefor the quark—gluonphaseis much softerthanthat for neutronmatter,which isillustrated by Fig. 11.

Fora neutronstara stiffer equationof stateresultsin largermassesandradii. This dependenceis,however,reversedfor hybrid stars,whereamorerepulsiveinteractiongivesslightly lighterstars.Thisis basicallycausedby two competingfeaturesof thephasetransition:A stiffer equationof stategivesalarger quark corebut alsolower critical baryondensities,thusless massin the hadronicphase.

In thiscontextwewould like to emphasizethat theinsensitivityof thegrosspropertiesof hybridstarson the hadronieequationof stateis alsorecoveredwhenequationsof statedesignedespeciallyfor neutronstarmodelsareapplied.Thisis illustrated by Fig. 12 for thecaseof BetheandJohnson

nc/no nc/no

Fig. 10. Themassesof neutronstarsandhybrid starsin units of the solarmassM0 asfunction of centraldensity n~fordifferentequationsof state.Theupperfigure refersto a“Sierk—Nix” parametrizationof the nuclearequationof stateatK = 380and550MeV, the lower figure to aquadraticequationof stateat K = 170,240and360MeV, respectively,andthe samequark—gluonequationof state(a. = 0.4, B”

4 = 165 MeV). For both parametrizationsthelargestcompressionconstantis given by the lowestcritical hadronicdensity.

E. Osrgaard/PhysicsReports242 (1994) 313—332 327

8 ~214~6

R [1cm]

Fig. 11. Themass—radiusrelationsM(R) for pureneutronstars(full-lines) andhybrid stars(dottedlines). All curvesreferto a“Sierk—Nix” parametrizationplus quark—gluonplasma.The stiffestequationof stategivesthemostmassiveneutronstars(correspondingto largestK). We indicatethe critical pointswherethe stability of thesolutionschanges.(a, = 0.4andB”

4 = 165MeV for the quark—gluonequationof state.K = 380 and550MeV.)

8~i~14 16

R [1cm] R [1cm]

Fig. 12. Comparisonof the relationsM(R) as obtainedfrom equationsof statetypically used for neutronmatterandnuclearmatter.In theleft figure, thedashedline correspondsto acalculationusingtheequationof stateofMalone,BetheandJohnson[13,43], whereasin the right figure an equationof statefrom Walecka’smean-fieldtheoryfor neutronmatter[44—46] hasbeenapplied. Full lines refer to resultsfrom the “Sierk—Nix” parametrizationat K = 380 and550MeV. In all caseswe haveapplied the standardquark—gluonequationof state(a, = 0.4, B”4 = 165MeV). ThesmallestK givesthe largestMmdx.

[13,43], andanequationof statefor neutronmatter obtainedfrom relativisticmean-fieldtheory[44—46].The mean-fieldequationof statehasa strongsimilarity with thehard (K = 550MeV)“Sierk—Nix” parametrizationfor nuclear matter at densitiesabove n

0, whereas the “Bethe—Johnson” parametrizationof the compressionenergy can be well approximatedby a soft(K = 380MeV) “Sierk—Nix” equationof statefor n > nLD ~ 1.2n0.

More “modern” equationsof state [9] give approximatelythe sameresultsas the relativistic“Waleekamean-field”equationof state,the “Bethe—Johnson”equationof state,or the“Quadratic”equationof statefor K = 240 or 360MeV [11].

328 E. østgaard/PhvsicsReports242 (1994) 313—332

Stablehybrid starswill only existat sufficiently low vacuumenergydensitiesdueto the strongincreaseof the critical density of the quark phase flq, Cr with larger B [58]. Higher densitiesaccompaniedby higherpressuresare neededto overcomethe confining vacuumpressureof thequark—gluonplasma.At B’74 = 300MeV, we find that nq~,.~ 23n

0,far abovethe maximumforstableneutronstarsaroundn, < iOn0. With B’

14 reducedto 180MeV, gravity still overtakestheFermi pressureon the hybrid starbranchwhich is illustratedin Fig. 13. The critical valuefor thebag constantat which stable hybrid stars might exist is found to be aroundB’~4= 170MeV.Reducing B further widens the range of stable hybrid stars towards smaller masses.AtB’14 = 160MeV,wefind thatliner ~ 1.6n

0.A furtherreductionof B would makethe quark—gluonplasmathegroundstateof ordinarynuclei, i.e., ~ < n0. For practicalreasonswe will not discussstrangematterat lower densitiesthan normalnuclearmatter,sincethe self-boundstrangestarshavebeendiscussedelsewhere[5 1—57]. Theremainingintervalallowsonly for a smallvariationofthe vacuumenergydensity.Similar conclusionsconcerningthe dependenceon the bagconstanthavealso beendrawn [49].

Varying the strongcouplingconstanteffectively widenstheallowedrangefor thevacuumenergydensity.However,only asmallvariation is admissible:Increasingcc, decreasesthemaximumvalueof B which gives stablehybrid stars. With the original MIT-bag value of B”

4 = 145 MeV asareasonablelower limit, we obtainamaximumvalueof; = 0.75. Similarly, settingcc, = 0 givesanoverall maximumvalue for stablehybrid starsat B’14 = 185 MeV . For reference,cc, = 0.3 givesa maximumbagpressureof B”4 = 175MeV comparedto B”4 = 170MeV for cc, = 0.4. Condi-tions for obtainingstablehybrid stars,defined by the parameterscc, andB in the quark—gluonequationof state,areshownin Fig. 14.

This showsthatdespitethe uncertaintyin cc,, thereis only asmallintervalof valuesfor B whichallowsfor stablehybrid stars,andtheinclusionof anon-zerostrangemasswould result in an evenstrongerrestrictionon thebagconstant[59]. The relativestabilitybetweenstrangequarkmatter

100 1000 10000

~ ~MeV/fm3]

Fig. 13. Thedependenceof thestarmasson thecentralenergydensityc, at differentbagconstantsof B”4 = 160,165 and180MeV for constanta, = 0.4,anda“Quadratic”equationof stateat K = 240MeV. Theuppermostcurvecorrespondstosolutionsin the absenceof a phasetransition, i.e., pureneutronstars.The lowestcurvecorrespondsto the smallestB.

E. øsrgaard/PhysicsReports242 (1994) 313—332 329

and hadronic matter hasalso beendiscussedfor the deconfinementphasetransition in ultra-relativistic heavy-ioncollisions in the contextof the decayof strangelets,leadingto very similarconclusions[60,61].

A harderequationof stategives lower critical transitiondensitiesboth for the hadronieandthequark—gluonphase,which in turn implies a largercoreof strangematterin the hybrid star.Thisaspectis revealedmore clearly in Fig. 15, showing the relative massfraction of strangematterresidingin ahybridstar.Although thetotal massesof the hybrid starsshowntherevary relativelylittle, themassesof thestrangemattercoreschangestronglywith the underlyinghadronicequationof state.Thesemodificationsin theinternalstructureof ahybridstarshouldprimarily influencetheneutrinocooling rates,but could alsomodify otherquantitieslike the heatconduction,magneticfields, etc. An enhancementof theneutrinoemissivity in strangestarsdueto the largernumberof~3-decaychannelsin the quark—gluonphasehasactuallybeenpredicted[29, 30,62,63].

Are thereotherobservableswhich could discriminatea hybrid starfrom aneutronstar?Besidesthe investigation of the rotation frequenciesof pulsars,the surfacegravitational redshift z, ofphotonshasbeenconjecturedasapossiblecandidatefor the identification of strangestars[27].This observableis determinedby the massandradiusof a starfrom the relation(5.16).

Figure 16 illustratesthatz, increasesconsiderablywith the total massof the star.For neutronstarsatlargemasses,z~alsovariessignificantly with the underlyingequationof state.Nevertheless,for agiven starmasscompatiblewith a neutronstar as well as with a hybrid star, the surfacegravitationalredshift of ahybrid staris only slightly largerthanthat of aneutronstar.Hence,the

IC’

.0

145 155 165 175 185 196 205 0 5 10 15 20

B1’~(MeV]Fig. 14. Stability regionfor hybrid stars,definedby theparametersa, = 0.4 andB in theQGPquark—gluonequationofstate,andthe“Quadratic”equationof statefor K = 240MeV. Theshadedareadefinesstablehybrid stars,thelower leftregioncorrespondsto quark stars,andtheupperright regioncorrespondsto black holes.

Fig. 15. Theratiobetweenthemassresidingin thequark—gluonphaseM9 andthetotal massof thestarM1 asfunctionof

its centraldensityn~.Theresultsreferto a“Quadratic” equationof state.At the samecentraldensitythe ratio increaseswith thecompressionconstant.Thecircles indicatethe limiting massesassociatedwith the equationof state.(a, = 0.4,B”

4 = 165MeV. K = 170,240and360MeV.)


measurementof z,will not be conclusiveto provetheexistenceof hybrid stars.Similarconclusionswere drawn for self-bound strangematter stars [24,25,37]. The question, to what extent theobservedangularvelocities of pulsarsimposeconstraintson the various starmodelshas beenwidely discussedso far [31,32,35,50,64—69].Most investigationsreferto the casesof pureneutronand strangestars, respectively.Unfortunately, almost all equationsof state applied so far arecompatiblewith the rotation frequenciesof the fastestobservedrotatingstars.

The (relativistic)Keplerian rotationperiod readsas [32, 64,65,68]

PK = 0.026~/~‘~‘ [ms] . (6.9)

Theminimum rotation period for neutronstarsis ~K mm ~ 0.6ms,varying relatively little underamodificationof the underlyinghadronicequationof state.For hybrid starson the otherhand,

mm amountsto approximately0.7ms,andis thereforealmostcomparableto thatfrom neutronstars.Also in this case,the minimum rotation periods of hybrid starsvary remarkablylittle fordifferentcompressionandbagconstants,aswell as for different parametrizationsof the nuclearequationof state.

The limited interval of starmassesfor which hybrid starscan exist is also reflectedin narrowboundsfor themaximumandminimumangularvelocities.Figure 17 illustratesthat theKeplerianrotation periods TK of stablehybrid stars are limited to the range0.7ms � TK � 1.9ms, whichcould be importantfor the possibility to distinguishapulsarasapossiblehybrid starcandidate.

__ ~00 7d/~ 20

M/M® M/M0

Fig. 16. Surfacegravitational redshiftz, versusthestarmassfor the samecasesas in Fig. 15. Full lines correspondtopureneutronstarswhereasthedashedlines indicatehybrid stars.Foragivenmass,thelargestredshiftis obtainedfor thestiffestequationof statefor thehybrid star,andviceversafor theneutronstar. (a, = 0.4, B”

4 = 165MeV, K = 170,240and360MeV.)

Fig. 17. TheKeplerianangularvelocity ~K versusthestarmassfor thoseequationsof statewhichgive themaximumandminimum frequencies.Thefull line refers to a “Quadratic” equationof stateat K = 170MeV andB”4 = 160MeV, forwhichthefastestrotatinghybrid starsarefound,whilethedashedcurvehasbeenobtainedfroma “Sierk—Nix” equationof stateat K = 550MeV and B’14 = 160MeV, giving the slowestangularvelocities.We get 0.7ms < TK < 1.9ms.

E. Østgaard/PhysicsReports242 (1994)313—332 331

7. Summary and discussion

In summary,wefind stableneutronstarsup to centraldensitiesof n, ~ 10n0,maximummassesof M ~ 1.8—2.6M®, andminimum radii of R ~ 10—13km, dependingon the equationof state.Hybrid starsmayexistat centraldensitiesbetweenn, ~i 4—9n0,correspondingto energydensitiesa, ~ 0.5—1.5GeV/fm

3,massesof M ~ 1.4—1.7M® andradii of R ~ 10—13km.The massrangeof hybrid starscoincideslargely with the massesof themost massiveneutron

starsobservedso far. Furthermore,the rotationalfrequenciesandthe surfacegravitationalredshiftof photonscannotbe usedas uniquesignals to distinguishahybrid starfrom aneutronstar.

Hybrid starsareexpectedto beconsiderablylighterthanneutronstarsatthe samecentraldensity.Thisexpressestheeffectivesofteningof thematterdueto thephasetransitionto thequark—gluonphase.

A supernovaexplosionmusthaveveryspecificcharacteristicsto be compatiblewith thenarrowmassinterval requiredby stablehybrid stars.However,the mostmassivepulsarsobservedso farcould possiblybe either aneutronstaror ahybrid star.

The existenceof stablehybrid starsis, however,very sensitiveto the equationof stateof thequark—gluonphase.Only asmallrangein B allows for stablestarsandreasonablecritical baryondensitiesfor thephasetransition.Onthe otherhand,if hybridstarscould beidentified in thefuture,this could provide detailedinformation on strangequarkmatter.

Theinterpretationof pulsarsas quarkstarsmayrun into difficulties. We have,for instance,theproblemof glitehing. Possiblyconnectedto thisphenomenonis gamma-raybursting,whichwouldremainunexplainedin aquarkstarpicture.It is possible,however,thata neutronstarwith aquarkcore,i.e.,ahybrid star,could berelevant,andthenthe glitching phenomenacould possiblybeusedto distinguishahybrid starfrom a neutronstar.

The cooling history of a compact star is strongly dependenton the underlying emissionprocesses.It hasbeensuggestedthat theneutrinoemissivity from pure quarkmatter is substan-tially larger than that from ordinary neutronstarmatter.Although therehavebeenargumentsopposingthis view, it could possiblybe usedto distinguishahybrid starfrom a neutronstar.

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Compact stars: Neutron stars or quark stars or hybrid stars?