Mon.Not.R.Astron.Soc.000,000–000(0000) Printed5February2008 (MNLATEXstylefilev2.2) The correlations between the twin kHz QPO frequencies of LMXBs 1 1 1 2 2 C.M. Zhang , H.X. Yin , Y.H. Zhao , F. Zhang , L.M. Song 1.NationalAstronomicalObservatories,ChineseAcademyofSciences,Beijing100012,China,[email protected] 2.AstronomicalInstitute,InstituteofHighEnergyPhysics,ChineseAcademyofSciences,Beijing,China 6 0 5February2008 0 2 n ABSTRACT a J We analyzed the recently published kHz QPO data in the neutron star low-mass X-ray 6 binaries(LMXBs),inordertoinvestigatethedifferentcorrelationsofthetwinpeakkilohertz 1 quasi-periodic oscillations (kHz QPOs) in bright Z sources and in the less luminous Atoll sources. We find that a power-law relation ν ∼ νb between the upper and the lower kHz 1 1 2 QPOs with differentindices: b ≃1.5for the Atoll source 4U 1728-34and b ≃1.9 for the Z v sourceScoX-1.TheimplicationsofourresultsforthetheoreticalmodelsforkHzQPOsare 8 1 discussed. 3 Keywords:accretion:accretiondisks–stars:neutron–binaries:close–X-rays:stars 1 0 6 0 / 1 INTRODUCTION peakseparationsarefoundtobeeithercloseto(butnotconsistent h p with)thespinfrequencyνsortoitshalfνs/2(see,e.g.,Wijnandset - Withtheadvent oftheRossiX-rayTimingExplorer(RXTE),our al.2003;vanderKlis2004;Lamb&Miller2001).Theaboveob- o knowledgeofthepropertiesoftheaperiodicvariabilityofneutron servationsofferstrongevidenceagainstthesimplebeat-frequency r t star(NS)lowmassX-raybinaries(LMXBs)tookasubstantialstep model,inwhichthelower-frequencyisthebeatbetweentheupper- as forward,especiallyinitiatedbythediscoveryofthekilohertzquasi- frequencyandtheNSspinfrequencyνs(seee.g.Strohmayeretal. : periodic oscillations (kHz QPOs) in about twenty more LMXBs 1996; Zhang et al. 1997; Miller et al. 1998), i.e. ν1 = ν2 −νs. v (see van der Klis 2000, 2004, for a review). The kHz QPOs in Nonetheless,withthediscoveryofpairsof30–450HzQPOsfrom i X the power spectra of these systems cover the range of frequency afewblack-holecandidateswithfrequenciesratios3:2(see,e.g., fromsomehundredHztomorethanonekHz,andtheyoftenoccur vanderKlis2004),Abramowiczetal.(2003ab)reportedthatthera- r a inpairsinthepersistentemission:theupperkHzQPOfrequency tiosoftwinkHzQPOsinScoX-1tendtocumulatearoundavalue (ν2, hereafter the upper-frequency) and the lower kHz QPO fre- of3:2,andtheyinterpreteditasanevidenceofanear∼3:2res- quency (ν1, hereafter the lower-frequency). The kHz QPOswere onance.ThiswasfurtherarguedbyAbramowiczetal.(2003ab)to soon found to behave in a rather regular way and the study of beapromisinglinkwiththeblack-holehigh-frequencyQPOs(see theirphenomenologyledtothediscoveryoftightcorrelationsbe- e.g.vanderKlis2004). Moreover,theproductionmechanismsof tween.theirfrequenciesandotherobservedcharacteristicfrequen- kHzQPOsarestillopenissues:theyhavebeenidentifiedwithvar- cies(see,e.g.,Psaltisetal.1998,1999ab;Stellaetal.1999;Belloni iouscharacteristicfrequenciesintheinneraccretionflow(seee.g. etal.2002).Withoutanydoubt,RXTEhasprovidedaprobeintothe Stella&Vietri1999;Titarchuketal.1998;Titarchuk&Osherovich accretionflowinthenon-Newtonian stronggravityregimewhere 2000;Psaltis&Norman2000;Lamb&Miller2001;Zhang2004). Einstein’sGeneral Relativity might be tested (van der Klis2000, 2004). The correlation between the upper-frequency and the lower- frequencyacrossdifferentsourcesorforaparticularsource,such asScoX-1,canberoughlyfittedbyapowerlawfunction(see,e.g., In this paper, in order to check the predictions of the kHz Psaltisetal.1998,1999a),butalsobyalinearmodel(seeBelloni QPOmodels,weanalyzetherecentlypublishedkHzQPOdataby et al. 2005). The kHz QPO peak separation ∆ν ≡ ν2 −ν1 be- RXTE,whichhavebeenusedbyseveralauthors(see,e.g.,Belloni tweentheupper-frequencyandlower-frequencyinagivensource etal.2005;Me´ndez&vanderKlis1999,2000;Psaltisetal.1998, isgenerallyinconsistentwithaNSspinfrequency.Insomesources, 1999ab;vanderKlis2000,2004,andoriginalreferencestherein). ∆ν islowerorhigher thantheNSspinfrequency (whendirectly Most of the data are provided by T. Belloni, M. Me´ndez and D. measured,seee.g.Me´ndez&vanderKlis1999;Jonker,Me´ndez, Psaltis, and the others are extracted from the references listed in &vanderKlis2002b) orthanthenearlycoherentoscillationfre- Table 1. Therefore, the data we analyzed here constitute a larger quency(νburst)observedduringtypeIX-rayburstthatisidentified samplethanthatpresentedbyBellonietal.(2005).Insection2,we tobethestellarspinfrequency(Muno2004;Strohmayer&Bild- criticallydiscuss thetwinkHz QPOcorrelation. Theconclusions sten2003;Wijnandsetal2003;vanderKlis2004).Theaveraged anddiscussionsaregiveninthelastsection. 2 C.M. Zhang et al. 2 THECORRELATIONSBETWEENTHETWINKHZ QPOS Figure1showsvariouscorrelationsofthetwinkHzQPOs,suchas ν1vs.ν2,∆νvs.ν2andν2/ν1vs.ν2,obtainedusingthesimulta- 1000 (a) neouslydetectedtwinkHzQPOdataofLMXBslistedinTable1, Sco X-1 Z exclude Sco X-1 Wefittedpower-lawrelation, Hz) 800 A4Uto l1l 7e2x8cl-u3d4e 4U 1728-34 ν1 =a(cid:16)100ν02Hz(cid:17)b Hz, (1) Hz QPO ( 600 ZA tfo=itl3tli n0fig0tti nHgz forthekHzQPOsamplesofAtollandZsourcesseparately.Asim- wer k 400 =3:2 ilarrelationwasdiscussedbyPsaltisetal.(1998), withasmaller Lo setofQPOdatapointsforScoX-1.Thenormalizationcoefficient 200 a and thepower-law index b for various cases arelistedinTable 2.Wefindthat theν1 vs.ν2 correlationsfor theAtollsource4U 0 1728-34 and theZsource ScoX-1aresomewhat distinctintheir (b) power-lawindices,∼1.5and∼1.9,respectively.Furthermore,for 400 ssd8laeli4iwfeg0fmehiHrtnselzydnn)ito,lcatewrratsgoene(gfoira.ebeχvst.oa2o2ri-nf.vt2hadν0lii2usfaefbinesnrrd.oeFZkn1iet.g7nsnuo9poru)eor.rmwc2Heaaeoslrsiw-z(hliaaeo.wvtewi.eosrcnν,oχ2tcrh2ro<eeetlefaχfis8t2itc4osi0tnetehnsbHattestzciacnaanounfidrsrdgeepusooνrpwf2eotenh2>rdea- Peak Separation (Hz)230000 to the relation (1), where in many cases the minimum χ2-values 100 are larger than ∼ 3.0 for the Z and Atoll samples, relatively too highvaluesconsiderthemgoodfits.However,usingonlythedata 0 pointsforScoX-1and4U1728-34,theminimumχ2-valuesforre- 3.5 (c) lation(1)become≃0.6and≃0.9respectively,lessthanthecor- 3.0 respondingvalue∼2forScoX-1previouslyobtainedbyPsaltiset 2.5 1aklHa(zn1d9Q94P8UO).s1T7wh2ie8trh-e3fd4oi,rfefwe,refeonortbtnthaoeirnminpadoliiwvziaedtrui-olaanlwZcocoeorfrfirAecltiaoetnliltossnopsuobrwceetewsr-eSlaecwnotiXhne-- Upper/Lower12..50 dicesbetweenthetwosources.Thechoiceofapower-lawrelation todescribethekHzQPOdataisarbitrary.Otherfunctionalforms 1.0 canalsofitthedataequallywell,asthelinearcorrelationadopted 0.5 byBellonietal.(2005)withasmallersampleofQPOdata.Ifrela- 0.0 tionEq.(1)isvalid,itimpliesanon-monotonicchangeofthepeak 0 200 400 600 800 1000 1200 1400 separation, with a maximum at an upper kHz QPO frequency of Upper kHz QPO (Hz) ∼700Hz,asshowninFigure1b. Figure 1. Plots of (a) ν1 vs. ν2, (b) ∆ν vs. ν2 and (c) ν2/ν1 vs. ν2. The Z [Atoll] fitting line represents the fitted correlation be- 2.1 Testingtheconstanttwinpeakseparation tween the pair kHz QPO frequencies for the Z [Atoll] sources as ν1 = (724.99±2.52 Hz)(ν2/1000Hz)1.86±0.03 [ν1 = (683.48± Infact,itisnotjustforScoX-1(vanderKlisetal1997;Me´ndez 3.01Hz)(ν2/1000Hz)1.61±0.04]. &vanderKlis2000)thatthepeakseparationisknowntobenot constant but also for other Z sources, e.g., GX 17+2 (Homan et al. 2002), GX 340+0 (Jonker et al. 2000) and GX 5-1 (Jonker χ2-values are very high, and we therefore conclude that there is et al. 2002a), and for several Atoll sources, e.g., 4U 1728–34 notaconstantpeakseparationeitherforZsourcesorAtollsources. (Migliari, van der Klis, & Fender 2003; Me´ndez & van der Klis Nevertheless, this also confirms the previously known result that 1999), with ∆ν always significantly lower than the burst oscilla- theScoX-1dataareinconsistentwithaconstantpeakseparation tion frequency νburst, 4U 1636–53 (Jonker, Me´ndez, & van der (seealsovanderKlisetal.1997;Psaltisetal.1998;Me´ndez&van Klis2002b;Me´ndez,vanderKlis,&vanParadijs1998b)with∆ν derKlis2000). varyingbetweenbeinglowerandhigherthanνburst/2,4U1608-52 (Me´ndezetal.1998c)and4U1735-44(Fordetal.1998),etc. In order to test the hypothesis of a constant peak separation 2.2 Testingapreferred3:2ratiointhetwinkHzQPOs betweenthetwinkHzQPOs,weinspectoursampleofsimultane- ously detected twin kHz QPOs over a wide range of frequencies From Figure 1b, one can see that a constant ratio relation ν2 = separatelyforAtollsources(110pairs)andZsources(158pairs). (3/2)ν1, shown as a dash-dotted line, is not consistent with the Infigure1b,weshowthatthepeakseparationinindividual Atoll observeddata.Moreover,Figure1cshowsthatthefrequencyratio orZsources, asforinstanceScoX-1and4U1728-34, decreases systematically decreases from 3.2 to 1.2 with increasing the kHz systematicallywithincreasingupperfrequencyifν2islargerthan QPO frequency. As a further investigation, we also performed a ∼700 Hz, a fact that has been reported in the literatures before χ2 testagainst aconstant ratio,showninfigure2c.Theobtained (see, e.g. van der Klis 2000, 2004 for recent reviews). Figure 2b χ2-valuesaretoohightoconfirma3:2peakratiofortheallkHz shows the results of a χ2 test against a constant peak separation QPOdata.Inaddition,theincompatible3:2ratiopeakdistribution oftwinkHzQPOsinAtollandZsources.Theresultingminimum has been also studied by Belloni et al. (2005) for many sources: Thecorrelationsbetween thetwin kHz QPOfrequencies 3 100 1000 (a) Atoll Z 800 10 Hz) =3=030/2 Hz /d.o.f. 1 AAZS4GGUtlcrrloooo l1uul X7pp-2ee18dd- 3A Z4toll Lower kHz QPO ( 460000 AZRA tWPfoiMtlOtli n fMoigttf i nomgf =A2=.00.7 (a) Z <840 Hz Z >840 Hz 200 0.1 0 1 2 3 4 Power law index b 0 500 1000 (b) atoll z 400 Hz) /d.o.f. 100 Separation ( 300 Peak 200 (b) 10 100 200 300 400 Peak Sepa ration (Hz) 100 100 3.5 (c) 3.0 2.5 /d.o.f. 10 Upper/Lower2.0 atoll 1.5 z (c) 1 1.0 0 1 2 3 Upper/Lower 0.5 0 200 400 600 800 1000 1200 1400 Figure2. Chi-squaretestsfor(a)thehypothesisofapower-lawcorrelation, Upper kHz QPO (Hz) (b)aconstantpeakseparationand(c)aconstantratiobetweenthetwinkHz QPOsofLMXBslistedinTable1. Figure3. ThepanelsarethesameasforFigure1.Forreasonofclarity,we dividedallkHzQPOdataintobinsof50Hzintervalinν2,andthenaver- agethecorrespondingkHzQPOdataineachbin.TheZ(Atoll)fittingline theyshowedthatthedistributionofQPOfrequenciesinScoX-1, represents thefittedcorrelation betweenthetwinkHzQPOsforgrouped 4U1608–52,4U1636–53,4U1728–34,and4U1820–30ismulti- Z(Atoll)samplesasν1 = (727.88±6.63 Hz)(ν2/1000Hz)1.91±0.05 peaked,withthepeaksoccurringatthedifferentν2/ν1 ratios,not (ν1 = [681.42±5.39 Hz][ν2/1000Hz]1.65±0.05). AWOM(RPM)is thetheoreticalcurveofthemodelbyZhang(2004)withtheaveragedstel- allratiosappearinginallsources. larmassdensityparameterA=0.7(byStella&Vietri[1999]withthemass parameterm=2.0M⊙). 2.3 TestingotherkHzQPOmodels In order to account for a variable peak separation of twin kHz QPOs,someviablemodelshavebeenproposed.Thevariablepeak bin.Weplotthesegroup-averagedvaluesinFigure3,withthesame separationwaspredictedintherelativisticprecessionmodel(Stella panelsasFigure1.Wepointoutthatneithertherelativisticpreces- & Vietri 1999) and the Alfve´n wave oscillation model (Zhang sionmodelnortheAlfve´noscillationmodelcanexplainthedistinc- 2004),whereforbothmodelstheupper-frequencywasascribedto tionsofkHzQPOsforboththeAtollandZsources,eventhough theKeplerianorbitalfrequency,whilethelower-frequencywasas- both models are in good agreement with the observed kHz QPO cribedtotheperiastronprecessionfrequencyandtheAlfve´nwave data,asshowninFigure3,oncethemodelparametersaretuned. oscillationfrequency,respectively.InFigure1,theQPOdatapoints Themotivationforperforminga50Hzbinningtodiscussthe arescattered,whichmakesitdifficulttoestimatethekHzQPOre- twinkHzQPOcorrelationswasthatwewanttoshowtheaveraged lationsby readingthe figuredirectly. Inorder tocompare clearly effects.Thedispersionsoftheaverageddatapointsarebiggerthan themodelswiththetrendsoftwinkHzQPOs,wedividedalldata theiraveragederrorbars(Figure3),sotheminimumχ2-valuecor- points into 50 Hz bins, and then averaged the quantities inevery responding toeach group isusually muchlarger than1.0. There- 4 C.M. Zhang et al. fore,weremarkthattheirerrorbarsunderestimatethetrueuncer- thanks to the critical comments from the anonymous referee that tainties,andthecalculationsshowthattheminimumχ2-valuesfor greatlyimprovedthequalityofthepaper. thefittedlinesofthegroupedsamples,asshowninthecaptionof Figure3,arealsotoohighforthefitstobeacceptable. REFERENCES Abramowicz,M.A.,Bulik,T.,Bursa,M.,&Kluz´niak,W.2003a, 3 CONCLUSIONSANDDISCUSSIONS A&A,404,L21 Abramowicz,M.A.,Karas,V.,Kluz´niak,W.,Lee,W.H.,Rebusco, Wehaveanalyzedanupdatedsampleoffrequenciesofthesimulta- P.,2003b,PASJ,55,467 neouslydetectedtwinkHzQPOsinLMXBs.Ourmainconclusions Belloni,T.,Psaltis,D.,&vanderKlis,M.,2002,ApJ,572,392. are the following. (1) The power-law correlations were analyzed by means of χ2 tests: the sources 4U 1728-34 and Sco X-1 are Belloni, T., Mendez, M. & Homan, J. 2005, A&A, 437, 209 foundtoyieldgoodpower-lawfits,withminimumχ2-valueslower (astro-ph/0501186) DiSalvo,T.,Me´ndez,M.,&vanderKlis,M.2003,A&A,406, than1.Thepower-lawindicesareb≃1.5fortheAtollsource4U 177 1728-34andb ≃ 1.9fortheZsourceScoX-1.Asimilarpower- Ford, E.C., van der Klis, M., van Paradijs, J., Me´ndez, M., Wi- lawindex waspreviously obtained withaslightlyhigh minimum χ2-value ∼ 2 by Psaltis et al. (1998). As it is known, Atoll and jands,R.,etal.1998,ApJ,508,L155 Hasinger,G.,&vanderKlis,M.1989,A&A,225,79 Zsourcesshow distinctpropertiesintheirspectraandluminosity Homan,J.,vanderKlis,M.,Jonker,P.G.,Wijnands,R.,Kuulkers, (Hasinger&vanderKlis1989;vanderKlis2000),andwedonot E.,etal.2002,ApJ,568,878 yetknowwhatpropertiescausethedifferencesintheirpower-law Jonker, P.G., van der Klis, M., Wijnands, R., Homan, J., van indices.Nevertheless,ifthepower-lawrelationswithdifferentin- Paradijs,J.,etal.2000,ApJ,537,374 dices for Atoll and Z sources areconfirmed by future detections, Jonker, P. G., van der Klis, M., Homan, J., Me´ndez, M., et al. thenanykHzQPOmodelsdiscardingthedistinctionsoftheAtoll 2002a,MNRAS,333,665 andZsourceswillconfrontseverearguments.(2)Clearly,obeying Jonker, P.G.,Me´ndez, M.,&vanderKlis,M.2002b, MNRAS, suchapower-lawrelationwouldcontradicttheconstantpeaksepa- 336,L1 rationandconstant(3:2)peakratiobetweenkHzQPOs,andinfact Lamb,F.K.,&Miller,M.C.2001,ApJ,554,1210 theplottedcurvesinFigure1andFigure3areincompatiblewith Markwardt, C.B., Strohmayer, T.E., & Swank, J.H. 1999, ApJ, these constant relations. These conclusions have been previously 512,L125 inferred with smaller samples of kHz QPO data (see, e.g. Psaltis Me´ndez, M., van der Klis, M., Wijnands, R., Ford, E.C., van et al. 1998; Psaltis et al. 1999ab; Belloni et al. 2005), but con- Paradijs,J.,&Vaughan,B.A.1998a,ApJ,505,L23 trarytothe suggestions by thesimple beat model and any model Me´ndez,M.,vanderKlis,M.,&vanParadijs,J.1998b,ApJ,506, thatpredict∆ν=constantandν2 =(3/2)ν1,respectively.Inaddi- L117 tion,basedontheupdatedkHzQPOdataofLMXBswefindthat Me´ndez,M.,vanderKlis,M.,vanParadijs,J.,Lewin,W.H.G.,& thereisnoextremelysharpconcentrationata3:2peakratioasindi- catedbytheχ2testinFigure2c;theratiosarebroadlydistributed Vaughan,B.A.1998c,ApJ,494,L65 Me´ndez,M.,&vanderKlis,M.1999,ApJ,517,L51 from∼1.2to∼3.2overafrequencyrangeofsomehundredHz, Me´ndez,M.,&vanderKlis,M.2000,MNRAS,318,938 asshowninFigure1c.Therefore,thenon-linearresonancemodel Migliari,S.,vanderKlis,M.,&Fender,R.2003,MNRAS,345, (seee.g.Abramowiczetal.2003b;Rebusco2004) canbeconsis- L35 tentwiththisdistributionofpeakratios.Nevertheless,itisshown Miller,M.C.,Lamb,F.K.,&Psaltis,D.1998,ApJ,508,791 inFigure1cthatthevalueofν2/ν1 decreasessystematicallywith Muno, M.P. 2004, “Millisecond Oscillations During Thermonu- increasingQPOfrequency.(3)Inaroughapproximation,thekHz clearX-rayBursts”,reviewarticlefor“X-RayTiming2003:Rossi QPOfrequencycorrelationseemstobeconsistentwiththepredic- andBeyond”,eds.P.Kaaret,F.K.Lamb,&J.H.Swank(Melville, tionsbythemodelbasedonthebasicgeneralrelativisticfrequen- NY:AmericanInstituteofPhysics) ciesaroundacompact object (Stella&Vietri1999) andthoseby O’Neill,P.M.,Kuulkers,E.,Sood,R.K.,&vanderKlis,M.2002, themodelbasedontheAlfve´nwaveoscillation(Zhang2004),with MNRAS,336,217 properlyselectedparameters.Bothmodelspredictedapeaksepa- Psaltis,D.,Mendez,M.,Wijnands,R.,Homan,J.,Jonker,P.,van ration decreasing with increasing QPO frequency, but increasing whentheupperfrequencyislow,forinstancelessthan∼700Hz. derKlis,M.,Lamb,F.K.,Kuulkers,E.,vanParadijs,J.,&Lewin, W.H.G.1998,ApJ,501,L95 Therefore,morekHzQPOdetectionsareneededtoconfirmthepre- Psaltis,D.,Wijnands,R.,Homan,J.,Jonker,P.,vanderKlis,M., dictionsofthemodels.Inconclusion,iffuturedatastillsupportthe Miller,M.C.,Lamb,F.K.,Kuulkers,E.,vanParadijs,J.,&Lewin, conclusions obtained inthepaper, theywillpose new constraints W.H.G.1999a,ApJ,520,763 onmodelsforexplainingkHzQPOs. Psaltis,D.,Belloni,T.&vanderKlis,M.1999b,ApJ,520,262 Psaltis,D.,&Norman,C.2000,(astro-ph/0001391) Rebusco,P.,2004,PASJ,56,553 Stella,L.,&Vietri,M.,1999,Phys.Rev.Lett.,82,17 4 ACKNOWLEDGEMENTS Stella,L.,Vietri,M.,&Morsink,S.1999,ApJ,524,L63 WearegratefulforT.Belloni,M.Me´ndezandD.Psaltisforpro- Strohmayer,T.,&Bildsten,L.2003,toappearinCompactStellar viding the QPO data, and many helpful discussions with T.P. Li, X-Ray Sources, eds. W.H.G. Lewin and M. van der Klis, Cam- M.Abramowicz,T.Belloni,Kluz´niak,W.,P.RebuscoandJ.Petri bridgeUniversityPress(astro-ph/0301544) arehighlyappreciated.Thisresearchhasbeensupportedbythein- Strohmayer, T., Zhang, W., Smale, A., Day, C., Swank, J., novativeprojectofCASofChina.Theauthorsexpressthesincere Titarchuk,L.,&Lee,U.1996,ApJ,469,L9 Thecorrelationsbetween thetwin kHz QPOfrequencies 5 Titarchuk,L.,Lapidus,I.,&Muslimov,A.1998,ApJ,499,315 Table2.ThetwinkHzQPOcorrelationν1=a(ν2/1000Hz)b Titarchuk,L.&Osherovich,V.2000,ApJ,537,L39 vanderKlis,M.,Wijnands,A.D.,Horne,K.,&Chen,W.1997, Sources∗ a(Hz) b ApJ,481,L97 AllZsamples 724.99±2.52 1.86±0.03 vanderKlis,M.2000,ARA&A,38,717(astro-ph/0001167) Z(ν2 <840Hz) 812.09±2.59 2.20±0.10 vanderKlis,M.2004,toappearinCompactstellarX-raysources, Z(ν2 >840Hz) 722.72±1.45 1.79±0.03 eds. W.H.G. Lewin & M. van der Klis, Cambridge University ScoX-1 721.95±0.69 1.85±0.01 Press(astro-ph/0410551) AllAtollsamples 683.48±3.01 1.61±0.04 van Straaten, S., Ford, E. C., van der Klis, M., Me´ndez, M., & 4U1728-34 667.86±5.59 1.51±0.07 Kaaret,P.2000,ApJ,540,1049 vanStraaten,S.,vanderKlis,M.,DiSalvo,T.,&Belloni,T.2002, AllZandAtollsamples 699.13±2.23 1.68±0.02 ApJ,568,912 ∗:DataaretakenfromthereferenceslistedinTable1. vanStraaten,S.,vanderKlis,M.,&Me´ndez,M.2003,ApJ,596, 1155 Wijnands, R., van der Klis, M., Homan, J., Chakrabarty, D., Markwardt, C.B., & Morgan, E.H. 2003, Nature, 424, 44 (astro-ph/0307123) Zhang,C.M.2004,A&A,423,401(astro-ph/0402028) Zhang, W., Strohmayer, T. E., & Swank, J. H. 1997, ApJ, 482, L167 6 C.M. Zhang et al. Table1. ListofLMXBswiththesimultaneouslydetectedtwinkHzQPOs,togetherwiththerelevantquantities Source∗ ν1(1) ν2(2) ∆ν(3) h∆νi(4) νν21(5) hνν21i(6) νs(7) h∆νsνi(8) Refs. name (Hz) (Hz) (Hz) (Hz) (Hz) Millisecondpulsar SAXJ1808.4-3658 499 694(725) 195 195 1.39 1.39 401 0.49 W Zsource ScoX-1 544-852 844-1086(1130) 223-312 287 1.26-1.57 1.44 - - M,B,K GX17+2 475-830 759-1078(1080) 233-308 282 1.28-1.60 1.46 - - B,K,P,1 GX340+0 197-565 535-840(840) 275-413 334 1.49-2.72 2.04 - - B,K,P,2 GX349+2 712-715 978-985(1020) 266-270 268 1.37-1.38 1.38 - - B,K,3 GX5-1 156-634 478-880(890) 232-363 318 1.38-3.06 2.05 - - B,K,P,4 CygX-2 532 857(1005) 324 324 1.61 1.61 - - B,K,P Atollsource 4U0614+09 153-823 449-1162(1330) 238-382 322 1.38-2.93 1.57 - - B,K,P,5,6 4U1608-52 476-876 802-1099(1099) 224-327 287 1.26-1.69 1.44 619 0.46 M,B,K,7 4U1636-53 644-921 971-1192(1230) 217-329 286 1.24-1.51 1.35 581 0.49 B,K,P,8,9 4U1702-43 722 1055(1085) 333 333 1.46 1.46 330 1.01 K,P,10 4U1705-44 776 1074(1074) 298 298 1.38 1.38 - - B,K,P 4U1728-34 308-894 582-1183(1183) 271-359 327 1.31-1.89 1.50 363 0.90 B,K,P,6,11 KS1731-260 903 1169(1205) 266 266 1.29 1.29 524 0.51 B,K,P 4U1735-44 640-728 982-1026(1160) 296-341 313 1.41-1.53 1.45 - - B,K,P 4U1820-30 790 1064(1100) 273 273 1.35 1.35 - - B,K,P 4U1915-05 224-707 514-1055(1265) 290-353 338 1.49-2.3 1.71 270 1.25 B,K,P XTEJ2123-058 849-871 1110-1140(1140) 261-270 266 1.31-1.31 1.30 - - B,K,P *: SourceswiththesimultaneouslydetectedtwinkHzQPOs. However,wedonotincludeXTEJ1807–294becausethe discoveryofitspairkHzQPOswasreportedbutunpublished(seee.g. vanderKlis2004). (1): therangeofν ;(2): the 1 rangeofν ,themaximummeasuredν isshowninparenthesis;(3): therangeof∆ν;(4): theaverageof∆ν;(5): the 2 2 rangeoftheratioofν toν ;(6): theaverageoftheratioofν toν ;(7): thespinfrequencyortheburstfrequency 2 1 2 1 identifiedasthespinfrequency(seee.g. Strohmayer&Bildsten2003;Muno2004;vanderKlis2004,andreferences therein); (8): theratiobetweentheaveraged∆ν andthespinfrequency. W:Wijnandsetal. 2003; B:Bellonietal. 2005,Bellonietal. 2002; P:Psaltisetal. 1999ab; K:vanderKlis2000,vanderKlis2004; M:M´endezetal. 1998, M´endez&vanderKlis1999,2000. 1. Homanetal. 2002;2. Jonkeretal2000;3. O’Neilletal. 2002;4. Jonkeretal. 2002a;5. vanStraatenetal. 2002;6. vanStraatenetal. 2000;7. vanStraatenetal. 2003;8. DiSalvoetal. 2003;9. Jonkeretal. 2002b;10. Markwardtetal. 1999;11. Migliarietal. 2003.