The correlations between the twin kHz QPO frequencies of LMXBs

0.33 MB·English

by  C. M. Zhang

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.
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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.