A(MNLTEXstylefilev1.4)
TheHarmonicandSidebandStructureoftheKilohertz
Quasi-PeriodicOscillationsinScoX-1
MarianoM´endez1,2andMichielvanderKlis1
1
AstronomicalInstitute‘AntonPannekoek’,UniversityofAmsterdamandCenterforHigh-EnergyAstrophysics,Kruislaan403,NL-1098SJAmsterdam,TheNetherlands.2FacultaddeCienciasAstron´omicasyGeof´ısicas,UniversidadNacionaldeLaPlata,PaseodelBosqueS/N,1900LaPlata,Argentina.
arXiv:astro-ph/0006243v1 19 Jun 2000Accepted2000June.Received2000February28;inoriginalform2000February28
ABSTRACT
WeusedatafromtheRossiX-rayTimingExplorertosearchforharmonicsandside-bandsofthetwosimultaneouskilohertzquasi-periodicoscillations(kHzQPOs)inScoX-1.Wedonotdetectanyoftheseharmonicsorsidebands,with95%confidenceupperlimitstotheirpowerbetween∼1%and∼10%ofthepoweroftheupperkHzQPO.Theoscillationsproducedatthesefrequenciesmaybeattenuatedinascatteringcoronaaroundtheneutronstar.Wefindthatupperlimitstotheunattenuatedpowerofsomeofthestrongesttheoreticallypredictedharmonicsandsidebandsareaslowas∼2%oftheunattenuatedpowerofthehigh-frequencyQPOinScoX-1.
Keywords:accretion,accretiondiscs–stars:neutron–stars:ScoX-1–X-rays:stars
1INTRODUCTION
Itisfouryearsnowsincethekilohertzquasi-periodicoscil-lations(kHzQPOs)werediscoveredinthepersistentfluxofScorpiusX-1(vanderKlisetal.1996a)and4U1728–34(Strohmayer,Zhang&Swank1996).Inthemeantime,sim-ilarkHzQPOshavebeenseeninsome20otherlow-massX-raybinaries(LMXBs;seevanderKlis2000forareview).TheseQPOsoftenappearinpairs,withfrequenciesν1andν2(ν2>ν1)between∼400and∼1300Hz,whichinagivensourcecanshiftbyafewhundredHz,apparentlyasafunctionofmassaccretionrate.
MostofthemodelsproposedsofarassumethatoneofthekHzQPOsreflectstheKeplerianorbitalmotionatsomepreferredradiusintheaccretiondisc(e.g.,Miller,Lamb&Psaltis1998;Stella&Vietri1999;Osherovich&Titarchuk1999),butthereareotherexplanationsaswell(Kleinetal.1996a,b;Jernigan,Klein&Arons2000).Inrecentdis-cussions(Lamb&Miller1999;Stella1999;Psaltis1999)itwasemphasizedthatempiricaldiscriminationbetweentwooftheleadingclassesofmodelsispossibleinprinciplebystudyingtheharmonicandsidebandstructureofthekHzQPOs.Hereweconcentrateonlyonthesetwomodelclasses
Inthe‘sonic-point’model(SPM;Milleretal.1998),theQPOatν2(theupperQPO)isproducedattheradiuswheretheradialflowvelocityinthediscturnsfromsubsonictosupersonic(thesonicradius),andtheQPOatν1(thelowerQPO)originatesbyabeatbetweentheupperQPOandthespinfrequencyoftheneutronstar.Inthe‘relativistic-precession’model(RPM;Stella&Vietri1999)theQPOat
c2000RAS
ν2isalsoassumedtobeKeplerian,buttheQPOatν1is
producedbytheapsidalprecessionofaslightlynon-circularinneraccretiondisc.TheQPOfrequenciesintheRPMarecalculatedfortestparticlesinpurelygeodesicrelativisticmotion,i.e.,neglectingthehydrodynamicalandradiativeeffectsoftheaccretionflow.However,Psaltis&Norman(2000)haverecentlyproposedadynamicalmodelinwhichtheQPOsareproducedbyoscillationsintheaccretiondisk.Inthismodel,whichwewillcall‘transition-radius’model(TRM),thereisatransitionradiusintheaccretiondiscthatactsasaband-passfilterwithresonancesneartheorbitalandperiastron-precessionfrequencies.
Besidesthemainpeaksatν1andν2,theSPMandtheTRMpredictother(weaker)harmonicsandsidebandsoftheseQPOs,atspecificfrequencies.Forinstance,theSPMpredictsarelativelystrongharmonicofthelowerQPOat2ν1(seeTable3ofMilleretal.1998foralistofotherside-bandpeakspredictedbytheSPM),whereastheTRMpre-dictsasidebandat2ν2−ν1(cf.eq.[29]inPsaltis&Norman2000).Inprinciple,thedetectionofaQPOat2ν1andanon-detectionofaQPOat2ν2−ν1wouldtendtoruleouttheTRM,whereasthedetectionofaQPOat2ν2−ν1andanon-detectionofaQPOat2ν1wouldtendtoruleouttheSPM(Miller2000).InthisLetter,weusedatafromtheRossiX-rayTimingExplorer(RXTE)tosearchforthepredictedharmonicsandsidebandsinthepowerspectrumofthekHzQPOsourceScoX-1.Becauseofitsveryhighflux,ScoX-1hasextremelysignificantkHzQPOs,andhenceaverysensitivestudyof
2M.M´endezandM.vanderKlis
anyharmonicstructureispossible.WedonotdetectanyofthesesecondaryQPOs.ThefactthatwedetectthekHzQPOsatν1andν2,butnoneoftheseotherpeakssetssevereconstraintsonthemodelscurrentlyproposedtoexplainthekHzQPOsinLMXBs.
2OBSERVATIONS
WeuseddatafromtheProportionalCounterArray(PCA;Jahodaetal.1996)onboardRXTE(Bradt,Rothschild&Swank1993)takenon1996February14,18,19,1996May24–28,1997March15,1997April18–24,1997August22,1998January2–8,1998February27,28,1998May30,31,1998June1,2,1998July2–5,1999January6,8–11and13–16.Eachobservation(i.e.,eachpartofthedatawithauniqueRXTEIDnumber)consistsofdatablocksof∼60sto∼3,700sinterruptedbypassagesofthesatellitethroughtheSouthAtlanticAnomalyandoccultationsofthesourcebytheEarth.Thetotalusabletimewas∼630ks.
Toavoiddetectorsafetytriggers,telemetrysaturation,andtoreducethedead-timeeffectsproducedbythehighcountrateofScoX-1,someobservationswerecarriedoutwiththesourceslightlyoff-axis,withsomeofthefivepro-portionalcounterunitsofthePCAswitchedoff,recordingonlyphotonsdetectedbytheupperanodechainofthePCA,recordingonlyphotonsfromalimitedenergyrange,orus-ingacombinationoftheseconstraints.Inallcases,high-timeresolutiondatawereavailablewithatimeresolutionof0.25msorbetter.Forouranalysisbelowwecombinedalltheavailabledata,irrespectiveofwhethertheywerecollectedusinganyoftheaboveobservationalconstraints.Whensin-gleanddouble-eventdatawererecordedinparallel(seevanderKlisetal.1996b),wecombinedthemoff-linetoenhancethesensitivity.
3ANALYSISANDRESULTS
Wedividedthehightimeresolutiondatainto16ssegments,andproducedapowerspectrumforeachofthesesegmentsuptoaNyquistfrequencyof2048Hz(for∼50%ofthedatawealsoproducedpowerspectrauptoaNyquistfrequencyof4096Hz).For69%oftheobservationsthepowerspectrawerecalculatedusingthefullPCAenergyband.Fortherestofthepowerspectraweuseddatafromselectedenergybands(24%ofthepowerspectrawerecalculatedbetween5−18keV,5%between5−60keV,and2%between2−18keV)becausewefoundthatthekHzQPOsweremoresignificantinthoseenergybands,orbecausethoseweretheonlyenergybandsavailable.Finally,weaveragedtogethergroupsof8contiguous16spowerspectratoproduceaveragepowerspectra.
WesearchedtheseaveragepowerspectraforkHzQPOs,atfrequencies>∼250Hz.WedidthisbyfirstidentifyingthosepowerspectrathatshowedastrongQPO,andvisuallyes-timatingitsfrequency,ν0.WhentwoQPOswerepresentinthepowerspectra,wealwayspickedtheoneathigherfre-quency.ItturnedoutthatinthecaseswhereonlyonekHzQPOwasvisible,itwasalwaystheupperkHzQPO(seebe-low).Wethenfittedthepowerspectra,intherangeν0−100Hztoν0+100Hz,usingafunctionconsistingofaconstant,
Figure1.ThefrequencyseparationbetweenthekHzQPOsinScoX-1asafunctionoftheupperkHzQPOfrequency.
apowerlawandoneLorentzian.WediscardedthepowerspectraforwhichtheQPOswerelessthan3σsignificant.WenotethatinthismannerwemighthavediscardeddatawithweakQPOsthatcouldhavebeendetectedaveragingtogethermoredata.Therewere1,384averagepowerspectrawithsignificantkHzQPOs,equivalentto177,152sofdata.
ForthoseobservationswherewedetectedonlyoneQPOintheaveragepowerspectraweappliedthe‘shift-and-add’technique(M´endezetal.1998a)totryanddetectthesec-ondQPO:BasedonthefrequencyofthedetectedQPO,wealignedandaveragedallthepowerspectraofasingleobser-vation.Inthosecases,thisprocedurerevealedasecondkHzQPO,whichinallcaseswasatalowerfrequencythantheQPOpeakthatweusedtoalignthepowerspectra,show-ingthatalltheinitialfrequencymeasurementsinthe1,384averagepowerspectracorrespondedtotheupperkHzQPO.
Next,wemeasuredthefrequencyseparationbetweenthekHzQPOsasafunctionoftheupperQPOfrequency,ν2.Weproceededasfollows:Wealignedthe1,384powerspectrausingtheupperkHzQPOasareference;wegroupedthedatain49sets,eachofthemcontaining∼40to∼210powerspectra,suchthatν2didnotvarybymorethan5−10Hzwithineachset,andwecombinedthesealignedspectratoproduceanaveragepowerspectrumforeachset.Wefittedthese49powerspectraintherange400−1300Hzusingafunctionconsistingofaconstant,apowerlawandtwoLorentzians.Thefitsweregood,withreducedχ2≤1.1for279degreesoffreedom,andthesignificanceofbothpeakswasalways>3σ.
Fig.1showsthat∆ν=ν2−ν1,thefrequencydifferencebetweenthetwoQPOs,decreasesfrom∼310Hzto∼240Hz,asν2increasesfrom∼840Hzto∼1100Hz.Thisfigurecanbecomparedtofig.3aofvanderKlisetal.(1997),whofirstreportedthedecreasein∆νwithν2inScoX-1.Becauseweincludedmoredata,andbecauseweusedtheshift-and-addtechniquetomeasure∆ν,theerrorsaresmallerinthepresentfigurethaninthepreviousone,andsomestructure,particularlybetween900and1020Hz,becomesapparent(perhapsthisstructureisrelatedtothe‘bump’seeninthe
c2000RAS,MNRAS000,1–5Figure2.Shifted-and-averagedpowerspectraofScoX-1.TheaveragepowerspectrumshownintheleftpanelwascomputedbyfirstaligningtheindividualpowerspectrasuchthataQPOpeakat2ν1wouldalwaysendupat1320Hz,indicatedbythearrow.IntherightpanelaQPOpeakat2ν2−ν1wouldsimilarlyendupat1284Hz(arrow).Inbothpanels,theverticallinesshowthefrequencyrangeusedtofitthedata(seetext).NoticethatthefrequencyalignmentalsoalteredtheapparentshapesandfrequenciesofthestrongQPOpeaks.
plotof∆νvs.ν1of4U1608–52atν1∼700Hz;seefig.3inM´endezetal.1998b).
FromFig.1wecanreadoffν1asafunctionofν2;be-causewealreadyknowν2foreachaveragepowerspectrum,wecancalculatetheexpectedfrequenciesofhypotheticalsignalsatmultiplesofν1andν2,oratfrequenciesthatarecombinationsofthesetwofrequencies,foreachpowerspec-trum.Aswedescribedin§1,someofthesefrequenciesareimportantinthecontextofthemodelsproposedtoexplainthekHzQPOs.
Wecalculatedthefollowingquantities:2ν1,2ν2,ν2−ν1,ν2+ν1,2ν1−ν2,2ν2−ν1,2ν1+ν2,2ν2+ν1and2(ν2−ν1).Wethenaligned,inturn,thepowerspectraoneachoftheabovefrequencies,andaveragedthemtogethertotryanddetectasignalpresentatanyofthesefrequencies.Wefit-tedtheseshiftedandaveragedpowerspectrainasegmentof∼400Hzthatincludedtheexpectedfrequency,usingaconstantplusaLorentzianwiththecentroidfixedineachcaseattheexpectedfrequency,andwithafixedFWHMof200Hz,approximatelyequaltothesumoftheFWHMofeachoftheQPOs(vanderKlisetal.1996b).ThisvalueisabouttwotimeslargerthanthelargestFWHMmea-suredintheQPOsofthisandothersources,andyieldsconservativeupperlimitstotheunobservedharmonicsandsidebands;tighterupperlimitscouldbeobtainedbyfixingtheFWHMtoasmallervalue.NodetectionsresultfromchoosingasmallerFWHMinthecurrentdataset.Inafewcasesweaddedapowerlawtothefittotakeaccountofaslopingcontinuum.Theshapeofthepowerspectrumathighfrequenciesisdominatedbydead-timeeffects(Zhangetal.1995)which,inthecaseofScoX-1atacountratethatexceeds∼25,000cs−1PCU−1,arelargeandnotyetsufficientlywellunderstoodtopredicttheshapeofthehigh-frequencypartofthepowerspectrumaccurately.Wenotethatabetterknowledgeofthiseffectcouldyieldtighterupperlimitsthanthosethatwereportinthispaper.
WedidnotdetectasignificantQPOatanyoftheabovefrequencies.Asanexample,inFig.2weshowtwopower
kHzQPOsinScoX-13
Table1.Upperlimitstotheobservedamplitudesoftheharmon-icsandsidebandsofthekHzQPOsinScoX-1Frequency
Predictedfrequencyrange
Upperlimita
2ν11090–1690Hz0.122ν21690–2170Hz0.13ν2−ν1240–300Hz0.30ν2+ν11390–1930Hz0.082ν1−ν2240–600Hz0.262ν2−ν11140–1330Hz0.122ν1+ν21930–2770Hz0.102ν2+ν12230–3020Hz0.102(ν2−ν1)
480–600Hz
0.21
a
95%confidenceupperlimitstothermsfractionalamplitudeoftheQPOattheindicatedfrequency,inunitsoftheobservedrmsfractionalamplitudeoftheQPOatν2.ThisQPOhasanamplitudebetween0.6%and2.5%(vanderKlisetal.1997).
spectrathatwereshiftedto2ν1andto2ν2−ν1,thefre-quenciesofthemainhigh-frequencypeaks(besidesthekHzQPOs)predictedbythemodelsofMilleretal.(1998)andPsaltis&Norman(2000),respectively.
UpperlimitsareshowninTable1.TocalculatetheseupperlimitswevariedtheamplitudeoftheLorentzianun-tiltheχ2ofthefitincreasedby2.71withrespecttothebest-fittingvalue(95%confidencelevelforasingleparam-eter);theupperlimitswequoteinTable1representthechangeintheamplitudeforthischangeintheχ2.Becauseoftheobservationalconstraintsdescribedin§2,andbecausewecombinedpowerspectracomputedfromdataindifferentenergybands,wecannotgiveupperlimitsinunitsoffrac-tionalrmsamplitude(vanderKlis1995);instead,alltheupperlimitsinTable1areinunitsoftheobservedfrac-tionalrmsamplitudeoftheQPOatν2.Anypeaksatthepredictedfrequenciesarebetween11and156timesweakerthanthepeakatν2,whichtranslatesintormsamplituderatiosbetween0.08and0.30.
4DISCUSSION
WehavemeasuredthefrequenciesofthetwosimultaneouskHzQPOs,ν1andν2,inScoX-1inalargedataset.Wehaveusedthesemeasurementstocalculatetheexpectedfrequen-ciesofhypotheticalsignalsatharmonicsandsidebandsoftheseQPOs,aspredictedbytwoclassesofkHzQPOmodels.WeusedasensitivetechniquetosearchfortheseharmonicsandsidebandsinthepowerspectraofScoX-1,butnonewasdetected.Theirpowerisbetween10and100timeslessthanthepoweroftheupperkHzQPO(95%confidence).
Oscillationsproducedclosetothesurfaceoftheneutronstarcanbeattenuatedinascatteringcorona.Theattenu-ationisgenerallylargerforoscillationsproducedbyapen-cilbeamsweepingthesurroundings(beamingoscillations),thanforoscillationsduetoactualchangesoftheluminositywithtime(luminosityoscillations;Brainerd&Lamb1987;Kylafis&Klimis1987;Kylafis&Phinney1989;Milleretal.1988).Theattenuationfactor(theratiooftheamplitudeA∞oftheoscillationsatinfinitytotheiroriginalamplitudeA0)asafunctionoffrequencyν,forascatteringcoronaofaradiusRandopticaldepthτisgivenby:A∞,lum
4M.M´endezandM.vanderKlis
[H]
Table2.UpperlimitstotheunattenuatedamplitudeoftheharmonicsandsidebandsofthekHzQPOsinScoX-1
Frequency
2ν12ν2ν2−ν1ν2+ν12ν1−ν22ν2−ν12ν1+ν22ν2+ν12(ν2−ν1)
a
τa<16.5<15.7<1<17.37.8<1R(km)a
336202<10262<10352167151<10Upperlimitb
0.150.270.300.110.260.140.370.540.21τa8.79.8<19.1<18.39.810.2<1R(km)a
129109<10121<10138109103<10Upperlimitc
0.300.930.300.350.260.231.432.160.21
OpticaldepthandradiusofapossiblescatteringcloudaroundScoX-1,obtainedbyminimizingeq.[3]ateachfrequency(seetext).
ThesearethevaluesthatwereusedtocalculatethecorrespondingupperlimitsshowninthisTable.
bUpperlimitsfortheunattenuatedamplitudeoftheQPOattheindicatedfrequencyforaluminosityoscillation(seeeq.[1]),inunits
oftheunattenuatedamplitudeoftheQPOatν2.
cUpperlimitsfortheunattenuatedamplitudeoftheQPOattheindicatedfrequencyforabeamingoscillation(seeeq.[2]),inunitsof
theunattenuatedamplitudeoftheQPOatν2.
InbothcasestheQPOatν2isassumedtobeabeamingoscillation(Milleretal.1998)
foraluminosityoscillation,andbyA∞,beam
1+τ
+e−τ,
(2)
forabeamingoscillation,withx=(3πνRτ/c)1/2(Kylafis&Phinney1989).
WecanusetheserelationstocalculatethelargestunattenuatedamplitudesallowedbythedatainTable1.BecauseateachfrequencytheupperlimitontheratioofobservedamplitudesisA∞,y(ν)/A∞,beam(ν2)(Table1),theupperlimitontheratioofunattenuatedamplitudesA0,y(ν)/A0,beam(ν2),canbeobtainedbyminimizingαy(ν)=
A∞,y(ν)/A0,y(ν)
kHzQPOsinScoX-1
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