FrederickKLAMBUniversityofIllinoisatUrbanaChampaignDepartmentsofPhysicsandAstronomy1110WGreenStUrbanaIL61801USAflambuiuceduMColemanMILLERUniversityofChicagoDepartmentofAstronomyandAstr ID: 112414
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THEORIGINOFKILOHERTZQPOsANDIMPLICATIONSFORNEUTRONSTARS FrederickK.LAMBUniversityofIllinoisatUrbana-Champaign,DepartmentsofPhys-icsandAstronomy,1110W.GreenSt.,Urbana,IL61801,USA,f-lamb@uiuc.eduM.ColemanMILLERUniversityofChicago,DepartmentofAstronomyandAstrophysics,5640S.EllisAve.,Chicago,IL60637,miller@bayes.uchicago.eduDimitriosPSALTISHarvard-SmithsonianCenterforAstrophysics,60GardenSt.,Cam-bridge,MA02138,USA,dpsaltis@cfa.harvard.edu AbstractOneofthemostdramaticdiscoveriesmadewiththeRossiX-RayTimingExploreristhatmanyaccretingneutronstarsinlow-massbinarysystemspro-ducestrong,remarkablycoherent,high-frequencyX-raybrightnessoscillations.The325{1200Hzquasi-periodicoscillations(QPOs)observedintheaccretion-poweredemissionarethoughttobeproducedbygasorbitingveryclosetotheneutronstar,whereasthe360{600Hzbrightnessoscillationsseenduringther-monuclearX-rayburstsareproducedbyoneortwohotspotsrotatingwiththestarandhavefrequenciesequaltothestellarspinfrequencyorits¯rstovertone.Theoscillationsconstrainthemassesandradiioftheseneutronstars,whicharethoughttobetheprogenitorsofthemillisecondpulsars.Modelingindicatesthatthestarshavespinfrequencies250{350Hzandmagnetic¯elds107{5109G.1.IntroductionThediscoveryofstrongandremarkablycoherenthigh-frequencyX-raybrightnessoscillationsinatleastsixteenneutronstarsinlow-massbinarysystemshasprovidedvaluablenewinformationaboutthesestars,someofwhicharelikelytobecomemillisecondpulsars.OscillationsareobservedbothinthepersistentX-rayemissionandduringthermonuclearX-raybursts(seevanderKlis1997).1 2 Fig.1.PowerdensityspectrumofScoX-1brightnessvaria-tions,showingthetwosimul-taneouskilohertzQPOsthatarecharacteristic.ThesearetwooftheweakestkilohertzQPOsobserved,withrmsam-plitudes1%.Thecontin-uumpowerdensityisconsis-tentwiththatexpectedfromphotoncountingnoise.FromvanderKlisetal.(1997).Thekilohertzquasi-periodicoscillations(QPOs)observedinthepersistentemissionhavefrequenciesintherange325{1200Hz,amplitudesashighas15%,andqualityfactorsº=±ºashighas200.TwokilohertzQPOsarecommonlyobservedsimultaneouslyinagivensource(seeFig.1).AlthoughthefrequenciesofthetwoQPOsvarybyhundredsofHertz,thefrequencyseparation¢betweenthemappearstobenearlyconstantinalmostallcases(seevanderKlisetal.1997andM¶endezetal.1997).The250{600HzbrightnessoscillationsobservedduringtypeIX-rayburstsaredi®erentincharacterfromtheQPOsobservedinthepersistentemis-sion(seeStrohmayer,Zhang,&Swank1997).OnlyasingleoscillationhasbeenobservedduringX-raybursts,andtheoscillationsinthetailsofburstsappeartobehighlycoherent(see,e.g.,Smith,Morgan,&Bradt1997),withfrequen-ciesthatarealwaysthesameforagivensource(comparisonofburstoscillationsfrom4U172834overaboutayearshowsthatthetimescaleforanyvariationintheoscillationfrequencyis�3000yr;Strohmayer1997).Theburstoscilla-tionsin4U172834and4U170242(seeStrohmayer,Swank,&Zhang1998)havefrequenciesthatareconsistentwiththeseparationfrequenciesoftheirkilo-hertzQPOpairs.Theburstoscillationsin4U1636536(Zhangetal.1997)andKS1731260(Smithetal.1997)havefrequenciesthatareconsistentwithtwicetheseparationfrequenciesoftheirkilohertzQPOpairs(Zhangetal.1997;Wijnands&vanderKlis1997).Theevidenceiscompellingthattheburstos-cillationsareproducedbyrotationwiththestarofoneortwonearlyidenticalemittingspotsonthesurface(seeStrohmayeretal.1997).Thefrequenciesoftheburstoscillationsarethereforethestellarspinfrequencyorits¯rstovertone. 3Thefrequencyseparation¢betweenthetwokilohertzQPOsobservedinthepersistentemissionofagivenstariscloselyequaltothespinfrequencyofthestarinferredfromitsburstoscillations(seeMiller,Lamb,&Psaltis1998,hereafterMLP).2.OriginofKilohertzQPOsAlthoughothertypesofmodelshavebeensuggested(Kleinetal.1996;Titarchuk&Muslimov1997),theevidencefavoringbeat-frequencymodelsofthetwokilohertzQPOsisverystrong(seevanderKlis1997),andwethereforefocusonthese.Themagnetosphericbeat-frequencymodelwasdevelopedtoexplainthesingle,15{60Hz\horizontalbranchoscillation"(HBO)observedintheZsources(seeLamb1991).Inthismodel,thefrequencyoftheHBOisthedi®erencebetweentheKeplerianorbitalfrequencyKmatthemainradiuswherethestellarmagnetic¯eldpicksupandchannelsgasfromtheaccretiondiskontothemagneticpolarregionsandthestellarspinfrequencyspin.Strohmayeretal.(1996)appliedthemagnetosphericbeat-frequencyideatothekilohertzQPOpairs,interpretingthefrequencyofthehigher-frequencyQPOinapairasKmandthefrequencyofthelower-frequencyQPOasKmspin.AlthoughitexplainsnaturallywhythefrequencyseparationbetweentheQPOsinapairisnearlyconstantinmostsourcesandequaltotheburstoscillationfrequencyorhalfthisfrequency,therearemanyseriousdi±cultieswiththemagnetosphericbeat-frequencyinterpretationofthekilohertzQPOs(seeMLP).ThemostfullydevelopedandsuccessfulmodelofthekilohertzQPOsistheso-calledsonic-pointbeat-frequencymodel,inwhichthehigherfrequencyinaQPOpairistheorbitalfrequencyofgasattheinneredgeoftheKepleriandisk°owandthelowerfrequencyisthedi®erencebetweenthisfrequencyandthespinfrequencyoftheneutronstar.Thesonic-pointmodelwasdeveloped(MLP)speci¯callytoexplainthekilohertzQPOpairsandisbasedonpreviouswork(Miller&Lamb1996)whichshowedthatthedragforceproducedbyradiationfromacentralstarcanterminateaKepleriandisk°ownearthestar.Inthesonic-pointmodel,someaccretinggasspiralsinwardinnearlycircularKeplerianorbitsuntilitisclosetotheneutronstar,whereradiationforcesorgeneralrelativistice®ectscauseasuddenincreaseintheinwardradialvelocity,whichbecomessu-personicwithinasmallradialdistance(seeFig.2a).Thesharpincreaseintheradialvelocityisusuallycausedbythedragexertedontheaccretinggasbyradi-ationfromthestar,butmayinsteadbecausedbygeneralrelativisticcorrectionstoNewtoniangravityifthegasintheKepleriandisk°owreachestheinnermost 4 Fig.2.Resultsoffullygeneralrelativisticnumericalcomputationsofthegasdynamicsandradiationtransportintheinnerdiskinthesonic-pointmodel,foraneutronstarofradius5M.(a)Inwardradialvelocityv^rofthegasinthediskmeasuredbyalocalstaticobserver,asafunctionoftheBoyer-LindquistradialcoordinateexpressedinunitsofthestellarmassM.(b)Radialopticaldepthfromthestellarsurfacethroughthedisk°owtotheradiusshownonthehorizontalaxis.Eachcurveislabeledwiththeassumedaccretionrate_Mmeasuredinunitsoftheaccretionrate_MEthatwouldproduceanaccretionluminosityatin¯nityequaltotheEddingtoncriticalluminosity.FromMLP.stablecircularorbitwithoutbeingsigni¯cantlya®ectedbyradiation.Asthedisk°owapproachesthesonicpoint,theopticaldepthofthe°owintheradialdirection(measuredfromthestellarsurface)typicallyfallssteeplywithdecreasingradius(seeFig.2b).Thechangefromopticallythicktoopticallythindisk°owoccurswithinafewphotonmeanfreepathsandissomewhatanal-ogoustotheionizationfrontattheboundaryofanHIIregion,exceptthatherethephotonmeanfreepathincreasesbecausetheradiationisremovingangularmomentumandthe°owisacceleratinginward,causingthedensitytofallsharply,whereasinanionizationfrontthemeanfreepathisincreasesbecauseradiationisremovingboundelectronsfromatomsandmolecules,causingtheopacitytofallsharply.Oncetheaccretinggasisexposedtotheradiationfromthestar,itlosesitsangularmomentumtoradiationdraginaradialdistance0:01r.Itthenfallsinwardsupersonicallyalongspiraltrajectoriesandcollideswiththeneutronstararounditsequator,producinganX-rayemittingequatorialring.GasstreaminginwardfromclumpsorbitingnearthesonicradiusalongtrajectorieswiththetightspiralshapeshowninFigure3ageneratesthemoreopenspiraldensitypatternshowninFigure3b.Collisionofthedensergasfrom 5 a b Fig.3.Viewoftheneutronstaranddiskalongtherotationaxisofthedisk,whichisrotatingcounterclockwiseinthisview.(a)Spiraltrajectoryfollowedbyasingleelementofgasasitfallssupersonicallyfromthesonicradiustothestellarsurface.(b)Spiralpatternofhighergasdensityformedbygasstreaminginwardalongspiraltrajectorieswiththeshapeshownin(a),fromaclumporbitingnearthesonicradius.Thespiraltrajectoryanddensitypatternarefromfullygeneralrelativisticcalculations(seeMLP).theclumpswiththestellarsurfacecreatesbeamsofbrighterX-rayemission,likethebeamindicatedbythewhitedashedlinesinFigure3b.Thesebeamsmovearoundthestar'sequator,generatingaquasi-periodicbrightnessoscillationwithfrequencyKs.Thelower-frequencyQPOisgeneratedbyweakX-raybeamsproducedbyfunnelingofpartoftheaccretion°ownearthestarbythestar'sweakmagnetic¯eld.Thesebeamsrotatewiththestarandmodulatetheradiationdragactingonthegasatthesonicradius,modulatingtheinwardmass°uxandtheluminosityatthesonic-pointbeatfrequencyBs(KsspinorKs2spin).Thesonic-pointmodelisconsistentwiththeaccretionrates,stellarmag-netic¯elds,andscatteringopticaldepthsinferredpreviouslyfromEXOSATandGingaobservationsoftheatollandZsourcesandaccountsforthemainfeaturesofthekilohertzQPOs,includingtheirhighandvariablefrequencies,theirhighamplitudesandcoherences,andthecommonoccurrenceofkilohertzQPOsinpairs(seeMLP).ItalsoexplainsnaturallywhythefrequencyseparationbetweenthefrequenciesofakilohertzQPOpairisnearlyconstantandequaltotheburstoscillationfrequencyorhalfthisfrequency.Finally,thesonic-pointmodelcanaccountforthesimilarfrequencyrangesofthekilohertzQPOsinsourceswithverydi®erentaccretionratesandmagnetic¯elds. 63.ImplicationsforNeutronStarsThefrequencyofthehigher-frequencyQPOinakilohertzQPOpairisalmostcertainlytheorbitalfrequencyofgasinKeplerianorbitaroundthestar.Ifso,thehighfrequencyandcoherencetheseQPOsallowonetoderivetightupperboundsonthemassesandradiioftheneutronstarsthatproducesuchQPOsandsigni¯cantconstraintsontheequationofstateofneutronstarmatter.Toseehowsuchboundscanbeconstructed,suppose¯rstthatthestarisnotrotatingandassumethat,forthestarinquestion,QPO2|thehighestobservedvalueofthefrequencyofthehigher-frequency(Keplerian-frequency)QPOinthekilohertzQPOpair|is1220Hz(thisisthehighestQPOfrequencydetectedsofarinanysource;seeMLP).Obviously,theradiusorboftheorbitofthegasproducingtheQPOmustbegreaterthantheradiusofthestar.Thismeansthatthestar'srepresentativepointinthe,MplanemustlietotheleftofthecubiccurveM0(orb)(thedashedcurveshowninFigure4a)whichrelatesthestar'smasstotheradiusoforbitswithfrequency1220Hz.Inordertoproduceawavetrainwithtensofoscillations,orbmustalsobegreaterthantheradiusmsoftheinnermoststablecircularorbit,sotheactualradiusoftheorbitmustlieontheportionoftheM0(orb;ºQPO2)curvethatliesbelowitsintersectionwiththediagonallineM0(ms)(thedottedlineshowninFigure4a)whichrelatesthestar'smasstoms.AsFigure4ashows,thisrequirementboundsthemassandradiusofthestarfromabove.ForQPO2=1220Hz,therepresentativepointofthestarmustlieinthepie-sliceshapedregionenclosedbythesolidlinesinFigure4a.Thus,themassandradiusofanonrotatingstarwiththisQPOfrequencywouldhavetobelessthan1:8Mand16.0km,respectively.Figure4bcomparesthemass-radiusrelationsfornonrotatingstarsgivenby¯veequationsofstatewiththeregionoftheradius-massplaneallowedforthreevaluesofQPO2.Theallowedregionofthe,Mplaneisa®ectedbyrotation(seeMLP).Theparameterthatcharacterizestheimportanceofrotationale®ectsisthedi-mensionlessquantityjcJ=GM2,whereJandMaretheangularmomentumandgravitationalmassofthestar.Forthespinfrequencies300HzinferredinthekilohertzQPOsources,jis0:1{0:3,dependingontheequationofstate.Figure1cillustratesthee®ectsofslowstellarrotationontheallowedregionofthe,Mplane.Theregionallowedforaslowlyrotatingstarisalwayslargerthantheregionallowedforanonrotatingstar,regardlessoftheequationofstate.However,theregionallowedforarapidlyrotatingstarcanbesmallerthanthatforthecorrespondingnonrotatingstar(Miller,Lamb,&Cook1998).IfthefrequencyofakilohertzQPOcanbeestablishedastheorbitalfre-quencyofgasattheinnermoststablecircularorbit,thiswouldbeanimportant 7 Fig.4.(a)Radius-massplane,showinghowtoconstructtheregionallowedforanonrotatingneutronstarwithQPO2=1220Hz(seetext).(b)Comparisonofthemass-radiusrelationsfornonrotatingneutronstarsgivenby¯verepresen-tativeequationsofstatewiththeregionsofthemass-radiusplaneallowedfornonrotatingstarswiththreedi®erentQPOfrequencies.Thelightsolidcurvesshowthemass-radiusrelationsgivenbyequationsofstateA,FPS,UU,L,andM.(c)RegionsallowedforrotatingneutronstarswithvariousvaluesofjandQPO2=1220Hz,when¯rst-ordere®ectsofthestellarspinareincluded.(d)IllustrativeKeplerianQPOfrequencyasafunctionofaccretionluminositygivenbythegeneralrelativisticcalculationsdescribedinx2.Fordetails,seeMLP. 8stepforwardinourunderstandingofstrong-¯eldgravityandthepropertiesofdensematter,becauseitwouldcon¯rmoneofthekeypredictionsofgeneralrel-ativityinthestrong-¯eldregimeand¯xthemassoftheneutronstarinthatsource,foreachassumedequationofstate.Probablythemostconvincingsigna-turewouldbeafairlycoherentkilohertzQPOwithafrequencythatreproduciblyincreasessteeplywithincreasingaccretionratebutthenbecomesconstantandremainsnearlyconstantastheaccretionrateincreasesfurther.Thisbehavioremergesnaturallyfromgeneralrelativisticcalculationsofthegasdynamicsandradiationtransportinthesonic-pointmodel(seeFig.4d).Theconstantfrequencyshouldalwaysbethesameinagivensource.ThisworkwassupportedinpartbyNSFgrantAST96-18524,NASAgrantNAG5-2925,andNASARXTEgrantsattheUniversityofIllinois,andbyNASAgrantNAG5-2868attheUniversityofChicago.ReferencesKlein,R.I.,Jernigan,J.G.,Arons,J.,Morgan,E.H.,&Zhang,W.1996,ApJ,469,L119Lamb,F.K.1991,inNeutronStars:TheoryandObservation,ed.J.Ventura&D.Pines,(Dordrecht:Kluwer),445M¶endez,M.,etal.1997,ApJ,inpress(preprintastro-ph/9712085)Miller,M.C.,&Lamb,F.K.1996,ApJ,470,1033Miller,M.C.,Lamb,F.K.,&Cook,G.1998,inpreparationMiller,M.C.,Lamb,F.K.,&Psaltis,D.1998,ApJ,inpress(MLP)Smith,D.A.,Morgan,E.H.,&Bradt,H.1997,ApJ,479,L137Strohmayer,T.E.,Swank,J.H.,Zhang,W.1998,inTheActiveX-RaySky,ed.L.Scarsi,H.Bradt,P.Giommi,&F.Fiore,NuclearPhys.BProc.Suppl.,inpress(astro-ph/9801219)Strohmayer,T.,Zhang,W.,Swank,J.H.1997,ApJ,487,L77Strohmayer,T.1997,talkpresentedatthe1997HEADMeeting,EstesPark,ColoradoTitarchuk,L.,&Muslimov,A.1997,A&A,323,L5vanderKlis,M.1997,inTheManyFacesofNeutronStars,Proc.NATOASI,Lipari,Italy(Dordrecht:Kluwer),inpress(astro-ph/9710016)vanderKlis,M.,Wijnands,R.,Horne,K.,&Chen,W.1997,ApJ,481,L97Wijnands,R.A.D.,&vanderKlis,M.1997,ApJ,482,L65Zhang,W.,Lapidus,I.,Swank,J.H.,White,N.E.,&Titarchuk,L.1997,IAUCirc.6541