Coupling between QPOs and broadband noise components in GRS 1915+105

2.2 MB·English

by  Thomas J. Maccarone

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed12January2011 (MNLATEXstylefilev2.2)
Coupling between QPOs and broadband noise components in GRS
1915+105
1
Thomas J. Maccarone, Philip Uttley
1
0 UniversityofSouthampton,SchoolofPhysicsandAstronomy,HighfieldCampus,Southampton,Hampshire,SO171BJ
2
Michiel van der Klis, Rudy A. D. Wijnands

n
AstronomicalInstitute“AntonPannekoek”,UniversityofAmsterdam,Kruislaan403,1098SJ,Amsterdam,TheNetherlands
a
J Paolo S. Coppi

0 YaleUniversity,DepartmentofAstronomy,POBox208101,NewHavenCT06520-8101
1

]
E
H
ABSTRACT
.
h
p We explorethe use of the bispectrumfor understandingquasiperiodicoscillations. The
- bispectrumisastatisticwhichprobestherelationsbetweentherelativephasesoftheFourier
o
spectrum at different frequencies. The use of the bispectrum allows us to break the degen-
r
t eraciesbetweendifferentmodelsfortimeserieswhichproduceidenticalpowerspectra.We
s
lookatdatafromseveralobservationsofGRS1915+105whenthesourceshowsstrongquasi-
a
[ periodicoscillationsandstrongbroadbandnoisecomponentsinitspowerspectrum.Weshow

that,despitestrongsimilaritiesinthepowerspectrum,thebispectracandifferstrongly.Inall
1
cases,therearefrequencyrangeswherethebicoherence,ameasureofnonlinearity,isstrong
v
forfrequenciesinvolvingthefrequencyofthequasi-periodicoscillations,indicatingthatthe
9
quasi-periodicoscillationsarecoupledtothenoisecomponents,ratherthanbeinggenerated
7
9 independently.We comparethe bicoherencesfrom the data to simple models,finding some
1 qualitativesimilarities.
.
1 Key words: accretion, accretion discs – methods: statistical – X-rays:binaries –
0 stars:winds,outflows–stars:binaries:close
1
1
:
v
i
X 1 INTRODUCTION than those of the shots. More sophisticated models of variability,
forexample,self-organizedcriticality(e.g.Takeuchi,Mineshige&
r
a Recentstudiesofthevariabilitypropertiesofaccretingblackholes Negoro 1995; Takeuchi & Mineshige 1997), predict correlations
and neutron stars have shown that these systems’ Fourier power betweenthearrivaltimesand/orintensitiesoftheshots.Otherre-
spectra in certain spectral states are well-described by a sum of lated“reservoir”models(e.g.Merloni&Fabian2001;Maccarone
many Lorentzian components (see e.g. Olive et al. 1998; Nowak & Coppi 2002; Malzac, Merloni & Fabian 2004), where the ac-
2000;Belloni,Psaltis&vanderKlis2002;vanStraatenetal.2002; cretiondiskand/ortherelativisticjettapsanenergysupplyeffec-
Pottschmidtetal.2003).Essentiallythesamecomponentsseemto tively enough to reduce the available energy for future emission
appearinnearlyallsources,andatnearlyallluminosities,andtheir also predict variability correlated over many frequencies. Reso-
frequenciestendtobewellcorrelatedwithoneanotherandwiththe nancemodelsforproducingquasi-periodicoscillationsshouldalso
sourcespectralstates(Wijnands&vanderKlis1999;Psaltis,Bel- clearlyproducenon-linearvariability(e.g.Psaltis&Norman2001;
loni&vanderKlis1999).Thisphenomenologysuggeststhatthere Abramowicz&Kluzniak2001;Schnittman&Bertschinger2003;
may be a single origin for most of the variability features in the Maccarone & Schnittman 2005), while non-resonant models for
power spectra of accreting black holes and neutron stars. Similar producingthesameQPOs(e.g.Chen&Taam1992,1995;Rezzolla
correlationsinfrequenciesbetweenthedifferentcomponentsseem etal.2003;Giannios&Spruit2004)could,butneednotshowcou-
toapplyeventoaccretingwhitedwarfs(e.g.Mauche2002;Warner, plingbetweenthedifferentfrequencies.Propagationmodels,where
Woudt&Pretorius2003). disturbancesmovethroughanaccretiondisc,alsoshouldproduce
Coupling between different components in the power spec- non-linear coupling of variability components, as the variability
trum has been suggested in many theoretical contexts. In a shot propertiesaremodifiedateachannulus(Lyubarskii1997).
noise model (e.g. Terrell 1972), variability components on the Akeytoverifyingandunderstandingthispossibleunifiedori-
timescale of the shot will be correlated with one another, result- ginforvariabilityistogobeyondthesimplepowerspectrumand
ingin,e.g.fastriseexponentialdecayorexponentialrise,fastde- beginstudyingtheirnon-linearvariability.Thefirstattemptsatthis
cayprofiles,butthereshouldbenocouplingontimescaleslonger failedtodetectsignaturesofnon-linearity,withthemethodsinclud-
2 Maccaroneet al.
ingthetimeskewness(Priedhorskyetal.1978)andsearchesfora tivedifferences inthepropertiesof thebicoherence asafunction
lowdimensionalchaoticattractor(Lochner,Swank&Szymkowiak ofphotonenergy,sowehaveusedthelowestenergysetofchan-
1989).Ontheotherhand,morerecentworkwithbetterlightcurves nelsinasingledatamodeforeachobservationforconvenience–
didestablishthatthelightcurvesof CygnusX-1arenottimere- wewillnotattempttomakeinterpretationsofthedataatalevelof
versible(Timmeretal.2000;Maccarone&Coppi2002),thatthere detailapproachingthedifferencescausedbychangingenergyband
existsacorrelationbetweenrmsamplitudeandfluxofasourcethat used.ThedataarethenFouriertransformedwith4096elementsper
is inconsistent with pure shot noise models (Uttley & McHardy transform, and time resolution of 1/64 seconds, 1/128 and 1/512
2001), that there is coupling between variability components on seconds, respectively. Thesearechosensothatthequasi-periodic
allobservabletimescales(Maccarone&Coppi2002),andthatin oscillationisnearthemiddleoftherangeofFourierbinsineach
some cases, there is a low dimensional chaotic attractor after all observation(i.e.alowertimeresolutionisusedwhentheQPOhasa
(Misraetal.2004),andtheobservedlightcurvesmaybedescribed lowerfrequency).Theintegrationtimesusedare8640,10240,and
byaLorenzsystem(Misraetal.2006).Withthisinmind,wenow 8160seconds, respectively, withonlythefirstpartofobservation
approachlookingatthepropertiesofthecouplingbetweenquasi- 30184-01-01-000 used. Thisdecision wasoriginally madedue to
periodic components and noise components inGRS 1915+105, a memorylimitationswhenthecalculationswerefirstmade,andthe
bright Galactic X-ray binary with an accreting black hole which decisionnottore-dothecalculationwasmadebecausethereissig-
has been the subject of several long observations with the Rossi nificantfrequencydriftoverthefullobservation,andtheeffectsof
X-rayTimingExplorer(RXTE). frequencydriftareamelioratedbyusingshortertotalintegrations.
We treat this work as a pilot study – the first attempt to ap- TheFouriertransformsarethencombinedtoproducethebispectra
plythebispectrumtoastronomicaldatawithstrongquasi-periodic andbicoherencesofthesedatasets.
oscillations. As such, we aim to illustrate the power of the tech- We have examined the amount of frequency drift inthe dif-
niquebyshowingdataandsimplemodelsthatcangiveverysimilar ferent observations. For observation 10408-01-25-00 and 20402-
powerspectra,andverydifferentbicoherences,butconsideritbe- 01-15-00,thepeakfrequencyinthepowerspectrumvariesbyless
yondthescopeoftheworktotrytofitmodelstothedataprecisely. than 10% over the full integrations – less than the widths of the
We present computations of the bicoherence for several observa- QPOs. For 30184-01-01-000, there is substantial variation of the
tionsofGRS1915+105, discussthemeaningof thebicoherence, peakfrequencyifweconsiderthefullobservation–withpeakfre-
andpresentafewtoymodelsforthebicoherencewhichwecom- quenciesrangingfromabout2.7Hzto4Hz.However,inthepart
parewiththedata. ofthedatasetweconsider,thevariationisonlyfromabout3.3Hz
to4Hz.Someminoreffectsfromthisvariationcanbeseeninthe
computedbicoherences,andwediscussthembelow.
2 DATAUSED
For thispilot study, wepresent the resultsfrom three representa-
3 STATISTICALMETHODS
tiveobservationsofGRS1915+105;ourgoalsinthispaperarenot
tomakeacompletecharacterisationofthebicoherenceproperties Forthiswork,wewillfocusoncomputingthebispectrumandthe
of all sources in all states, but rather to demonstrate of the util- closelyrelatedbicoherence.Thebispectrumcomputedfromatime
ityof thetechnique for identifying phenomenological differences seriesbrokenintoKsegmentsofequallengthisdefinedas:
between apparently rather similar lightcurves with rather similar
K−1
powerspectra,andtoshowafewtheoreticalmodelsthatproduce 1 ∗
bicoherencessimilartosomeoftheobservations.Allthesedataare B(k,l)= K XXi(k)Xi(l)Xi(k+l), (1)
fromvariabilityclassχ intheclassificationscheme of Belloni et i=0
al.(2000). Inthisclass, GRS1915+105 stayssteadilyinstateC, whereXi(k)isthefrequencykcomponentofthediscreteFourier
its“low/hard”state–the“hardveryhighstate”inthenomencla- transformoftheithtimeseries(e.g.Mendel 1991;Fackrell1996
tureofFender&Belloni(2004),orthehardintermediatestatein and references within). The bispectrum is well defined only for
themorerecentclassificationofHoman&Belloni(2005).Inthis k+llessthanorequaltotheNyquistfrequencyofthedataused
state,GRS1915+105 showsastrongpowerlawcomponent inits to compute it, and is defined nontrivially only for k < l, since
X-rayspectrum,andhighrmsamplitudequasi-periodicoscillations B(k,l) = B(l,k)–althoughinthispaper,wewillplotthebico-
atabout0.5-10Hz,oftenwithpowerfulharmonics(seeFender& herencefork > landforl > k tohelpguidetheeyetofeatures
Belloni2004andreferenceswithin).Thebroadbandnoisecompo- intheplots.Theexpectationvalueofthebispectrumisunaffected
nentinthepowerspectrumcontaintsagreaterfractionoftherms byadditiveGaussiannoise,althoughitsvariancewillincreasefora
amplitudeofvariabilitythandotheQPOsandharmonics. noisysignal.Poissonnoisecanaffectthebicoherenceforfrequen-
The RXTE observation identification numbers of the three cies where the Poisson level is high compared with the level of
observations presented are10408-01-25-00 (taken 19 July1996), intrinsicvariability–theamplitudeofthePoissonnoisecorrelates
20402-01-15-00 (taken9Feb1997),and30184-01-01-000 (taken withthecountrate,sothattherewillbephasecouplingbetweenthe
4April1998). Standardscreeninghasbeenappliedtothedatato instrinsicvariabilityofthesource’ssignal,andthenoiselevelinthe
ensurenousageofdatatakenduringEarthocculationsorperiods Poissoncomponent(seee.g.AppendixCofUttleyetal.2005).In
ofhighoffset.Thedatausedarethesinglebitmodedatawiththe thispaper,wefocusonlowfrequencyvariabilityofahighlyvari-
lowest setof energychannels: channels 0-35areusedfor 10408- ablesource,soPoissonnoiseisunimportant.
01-25-00 and30184-01-01-000, whilechannels0-13areusedfor The definition of the bispectrum gives its absolute phase a
20402-01-15-00. These correspond approximately to the energy well-definedmeaning,inconstrasttothephaseoftheFourierspec-
range from 2-13 keV and 2-5 keV, respectively. We have done trumwhich has adependence on anarbitary reference time. One
checksoftheotherdatamodeswithdifferentenergychannelranges canthenattempttodeterminethedegreeofconstancyofthatphase,
andhavefoundnoqualitativedifferencesandonlyminorquantita- makinguseofarelatedquantity,thebicoherence–themagnitude
Couplingbetween QPOsandbroadbandnoisecomponentsin GRS1915+105 3
ofthebispectrum,normalisedtoliebetween0and1.Definedanal- from actual data are plotted in figure 1, and the corresponding
ogously to the cross-coherence function (e.g. Nowak & Vaughan powerspectraareplottedinfigure2.
1996),itisthevectorsumofaseriesofbispectrummeasurements
divided by thesum of themagnitudes of theindividual measure-
ments.Ifthebiphase(thephaseofthebispectrum)remainsconstant
overtime,thenthebicoherencewillhaveavalueofunity,whileif
4.1 The“cross”pattern
the phase is random, then the bicoherence will approach zero in
thelimitofaninfinitenumber of measurements. Specifically,the Whenthesourceisinthe“radioquiet”stateCatlowcountrates,
squaredbicoherence,b2isdefinedas: theQPOfrequencyislow(lessthan2Hz).Here,thebicoherence
∗ 2 showsa“cross”behavior(seetheresultsfromObsID20402-01-15-
2 (cid:12)PXi(k)Xi(l)Xi(k+l)(cid:12) 00plottedinfigure1bforthebicoherenceandfigure2bforthecor-
b (k,l)= (cid:12) 2 (cid:12) 2. (2)
P|Xi(k)Xi(l)| P|Xi(k+l)| respondingpowerspectrum),withlargebicoherenceforfrequency
pairswhereonefrequencyistheQPOfrequencyandtheothercan
ThisnormalizationofthebicoherencewasproposedbyKim
take on any value. Here, in contrast to the web pattern, there is
&Powers(1979).Ifitshowsvariationasafunctionofthefrequen-
ciesused,thetimeseriesanalysedisnonlinear.Ithasbeenshown largebicoherenceevenfornoisefrequencieslessthantwicefQPO.
Additional power isseen at the harmonics and for theharmonics
thatothernormalisationsaremorelikelytodetectnonlinearityun-
interactingwiththenoise.
dercertainspecificcircumstances(Hinich&Wolinsky2004),but
It is interesting to consider whether the cross pattern might
weusetheKim&Powers(1979)normalisationheretoretaincon-
be indicating the effects of very broad wings to the QPO, with
sistencywithpastwork.Inpreviousworkwherewewerestudying
no actual coupling between the QPO and thenoise, but just cou-
variabilitycomponentswhichwereverybroadinthepowerspec-
plingbetweentheQPOanditsharmonics.Thisisunlikelytobethe
trum (Maccarone & Coppi 2002), we binned the data to make a
case.Firstly,thebicoherencefeaturesareasymmetricinfrequency
onedimensionalfunctionofk+l,andsubstantiallyre-binnedthe
space,whiletheQPOsarewell-fitbyLorentzians,indicatingthat
bispectrumvaluesoverfrequenciesandthencomparedtheresults
theyarenotasymmetricinfrequencyspace.Inprinciple,sincethe
withmodelpredictions.Sincethatwork,wehavebecomeawareof
bicoherenceisnormalisedbydividingbyquantitiesrelatedtothe
acorrectionwhichis,inprinciple,importantforstudyingaperiodic
power spectrum, changes inthestrengthof thenoise component,
variabilitywiththebicoherence.Themaximumvalueofthebico-
whichwould dilute thecoupled variability, could be expected. In
herenceissuppressedbysmearingofmanyfrequenciesintoasin-
thiscase,though,thereisbothmorepowerinthenoisecomponent
glebininthediscreteFouriertransform(S.Vaughan,privatecom-
ofthepowerspectrumandstrongerbicoherenceatfrequenciesbe-
munication).Thissuppressioncannotbecalculatedinastraightfor-
lowtheQPOfrequencythanaboveit.Secondly,theharmonicstend
wardway(seee.g.Greb&Rusbridge1988).Wenotealso,though,
tofollow either roughly circular patternsfor their bicoherence, if
thatsincecomparisonsinMaccarone&Coppi(2002) weremade
thequasi-periodicityiscausedbyphasedisconnectionsinanoth-
onlywithmodelcalculationsmadewiththesametimebinningas
therealdata,theseeffects,whatevertheymaybe,arethesamefor erwiseperiodicsignal,orf1 =f2patterns,ifthequasi-periodicity
iscausedbythepresenceofmanyrealfrequenciesduetochanges
therealdataandthesimulateddata,andhencetheconclusionsof
infrequencyovertimeorthesuperpositionofmultiplefrequencies
thatpaperarenotaffectedsubstantially.Therefore,tomakequan-
asmighthappen ifthevariabilityisduetoorbitalmotionsovera
titativecomparisonsofrealdatawithsimulatedbicoherences,itis
range of radii (see Maccarone & Schnittman 2005), and what is
importanttousethesamefrequencybinninginbothcases.
observedisdifferentfromboth.
Anon-zerobicoherencecanbeusedtoruleout,forexample,
modelswherethevariabilityondifferenttimescalescomesfromin-
dependentGaussianmodelcomponentswhicharethenaddedlin-
early. Beyond that, it is often difficult to interpret the results of
bicoherenceanalysis,butcomparisonofobservedresultswithre- 4.2 The“hypotenuse”pattern
sultsfromsimulatedlightcurvescanbeaverypowerful tool.As
Athighercountratesofthe“radioquiet”stateC,thebicoherence
shown already, for example, in Maccarone &Schnittman (2005),
pattern changes dramatically (see the results from ObsID 30184-
lightcurveswithverysimilarpowerspectracanhaveeasilydistin-
01-01-000plottedinfigure1cforthebicoherenceandfigure2cfor
guishablebicoherenceplots.
the corresponding power spectrum) . A high bicoherence is seen
Forthiswork,wewillconsidervariabilityonlyatfrequencies
primarilywherethetwonoisefrequenciesadduptotheQPOfre-
which are low enough that the intrinsic variability of the source
quency–thatis,theregionsoflargebicoherencemakeadiagonal
is strong, so that neither Poisson noise nor dead time affects the
line. Werefer to thisas the“hypotenuse” pattern because there-
Fourierspectrumsubstantially.Wedonotethattheseproblemsmay
gionofstrongbicoherenceformsthehypotenuseofatrianglewith
bemoreimportantathigherfrequenciesandthattheywillneedto
its other two sides being the axes of the plot. The “hypotenuse”
beconsideredinfutureanalyses.
patterntypicallyalsoshowsaroughlycircularspotofhighbicoher-
encewithf1=f2=fQPO,sincethereispowerbothatthefundamen-
talandatthefirstharmonicandthesetwovariabilitycomponents
4 BICOHERENCERESULTS
are coupled. A few other observations at similar count rates and
We have computed the bicoherence for many observations of withsimilarQPOfrequenciesinthenon-plateauχstatehavebeen
GRS1915+105.Forthesakeofbrevity,wepresentonlythreerep- examined,andshowthesamequalitativefeaturesintheirbicoher-
resentativeresults,aimingtopresentthreeobservations withvery ences.Wealsonotethatthediagonalelongationinthebicoherence
similarpowerspectra,butqualitativelydifferentbicoherences.We plotfromlowerlefttoupperrightforf1 = f2 = fQPO islikely
findthreephenomenologicalpatternsofvariabilityhere,whichwe to be related to the frequency drift during the observation that is
describedas“web”,“cross”,and“hypotenuse”. Thebicoherences mentionedabove.
4 Maccaroneet al.
4.3 The“web”pattern theintegralofthepowerdumpedintoitoveralongtimescale,and
hencenotstronglycorrelatedwiththeinstantaneousdrivingpower.
Ahybridclassofbicoherenceplotscanbeseenforsomeoftheob-
In our observations, the Q values of the QPOsare typically
servations,wherefeaturesresemblingthoseseeninthe“cross”and
∼3−10,andtheQPOsareoverresolvedinfrequencyspacebyfac-
ofthe“hypotenuse”patternscanbeseentogether.Whenthesource
torsofabout 10-20.Thisyieldsexpectedmaximaforthesquared
isintheradioplateaustate, asinObsID 10408-01-25-00, thebi-
bicoherence of ∼ 0.05 (with the exact value depending both on
coherenceshowsaratherstrongsignalfortwocombinations(see
whichobservationisconsidered,andwhethermultiplenoisecom-
figure1aandfigure2aforthecorrespondingpowerspectrum)–a
ponents interact nonlinearly toproduce the QPO,or the QPOin-
diagonallinefromupperlefttolowerright,withf1+f2 =fQPO, teractsnonlinearly withone noise component to produce another
forall0 < f1,f2 < fQPO.Additionally, onevertical/horizontal noisecomponent).Largervaluesarepossiblefortheharmonicsof
streak is seen with one of the frequencies equal to the QPO fre-
theQPOs,wherethewidthofthedrivingspectrumissmallerthan
quency,andtheotherafrequencygreaterthanthetwiceQPOfre-
thewidthofthenoisecomponentinfrequencyspace.
quency.Nosignificantvertical/horizontalsignalisseenforthecase
Ifweassumethattheperturbationswhicharecoupledtoone
thatf1 < 2fQPO.Thevalueofthebicoherencetapersoffforfre- anothergiveanX-raycountratewhichislinearlyproportionalto
quencieswherethepowerinthenoisecomponentbecomessmall.
theamplitudeoftheperturbation,wecanthenusethemagnitudeof
We refer to this pattern of behavior as the “web” pattern. A few
thebicoherencetogainsomeinsightintowhethertheinteractions
other observations at similar count ratesin theplateau statehave
canbethroughanarrowresonanceattheQPOfrequency.Ourfind-
beenexamined,andshowthesamequalitativefeaturesintheirbi-
ingsthat themaximum valuesof thesquared bicoherence for the
coherences. caseswhere theQPO and thenoise areinteracting are∼ 10−1.5
would then imply that the coupling between the noise and QPO
isrelativelyclosetoitsmaximalvalue.Thisplacessomeimmedi-
4.4 Afewbriefremarksonotherobservations ateconstraintson theclasses of models whichcan be considered
–modelsmusthavealargefractionofthepowercomingfroman
In some observations not plotted in this paper, with the highest
emission region which behaves as a single “system”, and, if the
count rates seen from GRS1915+105 in itsχ class, the bicoher-
interactionsbetweennoisecomponents produceaQPOthrougha
enceseemstotendtowardszeroentirely.However,atthehighest
resonantinteraction,itmustbeahighlydampedresonance.
count rates, the QPO frequency can also change substantially on
the timescale of a single RXTE observation. The bicoherence is
notwellsuitedtodealingwithdatasetswhicharenon-stationary
5.1 Simulatedlightcurveanalysis
inthismanner.Thisproblemmayexisteveninsomeofthelower
count ratedata sets, but at a much lower level. Wenote that ina We now consider several different mathematical forms for light
fewoftheobservations,thebicoherenceisextendedalongtheline curves which correspond, at least approximately, tophysical sce-
f1 = f2 around the location where the interactions are between nariosfor producing QPOsandnoise components which arecor-
theQPO and itsfirst harmonic – thiseffect can be seen showing relatedinsomemanner. Weshowthat thesedifferent modelscan
upweaklyinFigure1a.ItwasshowninMaccarone&Schnittman producerelativelysimilarpowerspectrawhileproducingbicoher-
(2005)thatthisischaracteristicofaQPOwhichisbroadeneddue encediagramswhicharequalitativelyquitedifferentfromonean-
totheexistenceof power at manyfrequencies, rather thandueto other. Two of the patterns of behaviour found – the “cross” and
phasedisconnections. the “hypotenuse” – are reproduced reasonably well by relatively
straightforwardmodels.Thethird,the“web”isonlyapproximately
reproducedherewitharelativelysimplemodel.
5 DISCUSSION
5.1.1 Bicoherenceintermsofreservoirmodels
The overall values of the bicoherence inthese data can also give
ussomecluesastothetypesofprocessesthatmightbeproducing Wherethebicoherence’svalueislarge,thevariabilitymustbecou-
the phase correlations. The bicoherence relates the instantaneous pled on the three timescales corresponding to the three frequen-
responseofthedrivenoscillationtochangesinthedrivingmodes. cies included for the computation of the bicoherence. A natural
As stated above, Greb & Rusbridge (1988) have shown that the waytocouplevariabilityondifferenttimescalesiswithareservoir
maximum value of the squared bicoherence isδω/Σ, whereδω is model,wheretherearemultiplevariabilitycomponentsthatdrain
thefrequencyresolutionintheobservation(orthewidthofaQPO, thesamereservoir.Inthisway,thedifferentcomponents“compete”
iftheQPOisresolved),andΣisthefrequencywidthofthedriving forpower–whenthenoisecomponentbecomesstronger,therewill
spectrum.Therefore,ifaQPOisdrivenbyinteractionsofallnoise belessenergyinthereservoiravailablefortheQPOandviceversa,
frequencieslessthantheQPOfrequency,thenthelargestpossible leadingtoaphasecoupling.Ananalogousideahasbeenputforth
value of the squared bicoherence is Q−1, where Q is the quality toexplainthecouplingbetweenopticalvariabilityandX-rayvari-
factoroftheQPO. abilityinXTEJ1118+480(Malzac,Merloni&Fabian2004).
Greb&Rusbridge(1988)alsoconsidertheeffectsofhavinga Todeterminewhattheoutputlightcurvewilllooklikeforthe
relativelynarrowresonance.Inthiscase,thesquaredbicoherenceis simulations, wedefine, ineach case,twopower spectratouseas
reducedbyafactoroftheratioofthecoherencetimeinthedriven outputs. One, the noise component, is a broad Lorentzian with a
modetothatinthedrivingmode.Thus,forresonantmodes,which peakfrequencyofzero.Theother,thequasi-periodicoscillation,is
remaincoherent forlongperiodsoftimerelativetothetimescale amuchnarrowerLorentzianwithapeakatahigherfrequency(we
onwhichthedrivingsignalremainscoherent,thebicoherencewill useQ=30here,whereQisthequalityfactoroftheLorentzian).
besubstantiallyreduced.Thiscanbegraspedintuitivelyinaquali- Thetimeseriesforthesecomponents aremadeusingthemethod
tativesense–astrongresonancewillhaveanamplituderelatedto ofTimmer&Ko¨nig(1995).Thismethodgeneratessimulatedtimes
Couplingbetween QPOsandbroadbandnoisecomponentsin GRS1915+105 5
seriesfromapowerspectrumundertheassumptionthattheprocess componentsassumedtobeLorentzianswithparameter valuesset
isGaussian–thuswecanbeassuredthatanynon-linearitywede- tovaluessimilartothoseseeninthedataforvariousobservations.
tectisbecauseofthephysicsweputinaftergeneratingtheinitial For a very small number of time bins, the value of one of these
randomtimeseries. timeserieswillbenegative,inwhichcasewesetittozero.Alter-
Then,areservoirisdefined.Energyisinjectedintothereser- natively,theenergyextractedinatimeintervalmaybelargerthan
voireitherataconstantrate,oratapurelyrandomrate(i.e.aran- thesizeofthereservoir,inwhichcasewesetthepowersuchthat
dom number uniformly distributed between zeroand one timesa thewholereservoirisdrainedinthattimestep.
normalisation).Energyisdrawnfromthereservoirbyeachofthe Whentheamplitudeofvariabilityislarge,thereservoir’slevel
twocomponents.Thefluxofacomponentistakentobethevalue fluctuatesstrongly.Thisleadstotheproductionofharmonicsinthe
ofitstimeseriesmultipliedbythesizeofthereservoiratthatin- QPO(sincetheoscillationsbecomenon-sinusoidalwhenthereser-
stant times some normalization. The flux is then subtracted from voir fluctuates in response to the oscillations themselves). It also
thereservoir.Specifically,wedothefollowing: leadstophasecouplingofalltypesbetweentheQPOandthenoise.
′ Thisproducesa“web-like”patterninthebicoherence;however,the
R(t)=R(t−∆t)+Yr(t), (3) web-likepattern hereisnot identical tothe oneseen inthe radio
whereR′(t)isthesizeofthereservoirattimet,aftertheenergy plateau state.In thereal data, the“cross-like” structures begin to
injection hastaken place, but before thedraining of thereservoir manifestthemselvesonlyfornoisefrequencieslargerthanthefre-
hastakenplace,∆tisthetimestep,R(t−∆t)isthesizeofthe quencyofthefirstharmonic,whileinthesesimulations,thecross
energyreservoirattheendoftheprevioustimestep,andYr(t)is structuresappearatallnoisefrequencies.Furthermore,insomeof
theinjectionrateintothereservoirpertimestep; thesimulations,therearediagonalsfromupperlefttolowerright
′ related to the cross structure which are strong features at all fre-
YQPO(t)=yQPO(t)×R(t)×kQPO, (4) quencies adding up to at least one of the harmonics. These are
whereYQPO istheoutput fluxfromthe QPOcomponent, yQPO not present in the real data. The strong diagonals manifest them-
is the time series of the QPO produced by the Timmer & Ko¨nig selves when the total variability in the reservoir is dominated by
method,andkQPOisanormalizationconstant; thenoisecomponent,ratherthanbytheQPO.Themodelbicoher-
′ enceisshowninFigure3btorepresentacasewherethejetextracts
YN(t)=yN(t)×R(t)×kN, (5) power (i.e.sothattheextractednoise-component power doesnot
whereYN istheoutput fluxfromthenoisecomponent, yN isthe leaddirectlytoobservedemission),andinFigure4bforthecase
timeseriesofthenoiseproducedbytheTimmer&Ko¨nigmethod, where the noise component is added back in to the model light
andkN isanormalizationconstant;and,finally, curve.Figure5bshowsthesimulatedpowerspectrumcorrespond-
ingtothebicoherenceshowninFigure3b.
′
R(t)=R(t)−YQPO(t−∆t)−YN(t−∆t). (6) In the numerical calculations where the noise component is
Two general cases areconsidered, and we search a range of notaddedtothefinallightcurve,theQPOhasameancountrate
parameterspaceforeachcase.Thefirstiswheretheoutput“light of8000,witharootmeansquaredvariabilitylevelof1200,while
curve”isthesumofthefluxesofthetwocomponents.Thismight thenoisecomponenthasameanvalueof20000,withanrootmean
correspondtoacasewhereaccretionpowerisdissipatedeitherin squaredvariabilitylevelof8000.Were-fillthereservoirbyadding
someoscillatingregionwhichproducestheQPO,oranon-resonant arandomnumberdrawnfromauniformdistributionbetween0and
part of the accretion disc, which produces the noise component. 8000. The draining of the reservoir takes place by summing the
ThesecondiswheretheoutputlightcurveisonlytheQPO’sflux. twotimeseries, multiplying bythesizeof thereservoir, dividing
ThismightcorrespondtothecasewherethebroadLorentzianrep- 200000,andsubtractingthatfromthereservoirvalue.
resents a jet which extracts power from the system, but does not Wherethenoisecomponentisaddedtothefinallightcurve,a
emitintheX-rays(seee.g.Malzacetal.2004). slightlydifferentprocedureisadopted.TheQPOandnoisecompo-
Ifthereservoirisdrainedbyacomponentwhichdoesnotcon- nentsareaddedtogetherintheinitialpowerspectrummadebefore
tribute to the X-ray light curve, then the size of the reservoir in theinversionintoatimeseriesusingtheTimmer&Ko¨nigmethod.
“view”oftheX-raylightcurve,willeffectivelybesomequantity TheQPOisgivenanormalizationinthepowerspectrum200times
minus the integral of the power drained by the other component. higher than the noise component’s. The actual value in the final
Asaresult,thereservoiritselfwillhaveasignatureofitsmodula- power spectrum comes fromconverting thispower spectrum into
tion,sowhentheoutputlightcurveisobtainedbymultiplyingthe atimeserieswithmeanvalueof8000andrmsamplitudeof6000.
reservoirsizebythetimeseriesfortheQPO,the“QPO”compo- Allotherproceduresareasabove.
nentwillnowbemodulatedonthetimescaleofthe“noise,”andthe Atlowvariabilityamplitude,onlythecross-likestructuresare
noisepowerspectrumwillaffectfinalpowerspectrum.Asweshow seenstrongly,althoughaweak“hypotenuse”featureisstillpresent
below,thequalitativepropertiesofthebicoherencearelargelythe whenthenoisecomponentisaddedtotheoutput“lightcurve.”The
sameregardlessofwhetherthetwodrainsontheenergyreservoir bicoherenceisshownwithoutthenoisecomponent addedbackin
bothcontributetotheoutputflux,oronlyoneofthesedoes.Again, in Figure 3a, and with it added back in in Figure 4a. The power
the logic we follow here is quite similar to that in Malzac et al. spectrumcorresponding tothelowvariabilityamplitudemodelis
(2004). Giventhat,aswewillshow below, thesemodels seemto shown inFigure5a. Thesimulated bicoherence plots inthiscase
match the data for the plateau states more closely than the more look quite a bit like the bicoherence plot seen from observation
radioquietχstates,itwouldnotbesurprisingifenergyextraction 20402-01-15-00(comparetheobservationaldatainFigure1bwith
byajetwereanimportantfactorindeterminingthepropertiesof thesimulationsinFigure3a).
thelightcurve. Theprocedureforconvertingthemodelintoasimulatedlight
After the basicmodel equations aredefined, atimeseries is curveisthesameasabove, except forsomechanges invaluesof
producedforthemodel.TheQPOandnoisecomponentsarepro- parameters. For the case where the noise component is not seen
ducedastimeseriesusingtheTimmer&Ko¨nigmethod,withboth in the output simulated light curve, the noise component in this
6 Maccaroneet al.
casehasanrmsamplitudeof1000insteadof8000,withallother theonewhosepowerspectrumandbicoherenceareshowninFig-
parametervaluesthesame.Wherethenoisecomponentisaddedto ures 45b and 3b, respectively, matches many of the properties of
thefinalsimulatedlightcurve,thermsamplitudeofthesimulated thedata, but not all;however, here, the problem isprimarilythat
lightcurveis2500insteadof6000. themodelshowstoostrongabicoherenceinstreakswherethereis
onlynoise,ratherthanpowerintheQPOoritsharmonics.
5.1.2 Bicoherenceintermsofdampedforcedharmonic
oscillators 5.1.3 Caveats
Wehavealsoconsideredthecaseofadamped,forcedharmonicos- Therearefurtherstepswhichwillbeneededtomakemodelswhich
cillator.Thisisconceptuallysimilartothemechanismsuggestedby matchallthestatisticalpropertiesoftheobservedlightcurves.For
Psaltis&Norman(2000)forproducingbothquasi-periodicoscil- example,inmostcases,theobservedfluxdistributionswillfollowa
lationsandnoisecomponentsfromasingleperturbationspectrum log-normaldistribution(Uttleyetal.2005)-wedonotattempthere
–the quasi-periodic oscillation appears at the resonant frequency toenforcethisrequirement.WedonotethatitwasfoundbyUtt-
ofthesystem,whileawhitenoiseinputspectrumisturnedintoa leyetal.(2005)thatexponentiatingthefluxvaluesinalightcurve
rednoisecomponentbythefactthatdampedoscillatorsactaslow- canpushafluxdistributiontowardsalog-normaldistribution.We
passfilters.However,theresponseofthesystemwasconsideredto havedonethisinafewcases,andhavefoundthatthebicoherence
belinear inthat work, meaning that non-linear featureslike non- plotschangeverylittlewhenthefluxvaluesareexponentiated.Ut-
zerobicoherencecannotbeexplainedwithoutsomemodifications. tley et al. (2005) have already shown that power spectra are not
Whenmodifyingthesystemofequations,toallowforanon-linear typicallyaffected strongly bythis transformation. Therefore, flux
restoringforce,wefindthattheresultingpatternofbicoherencethat distributionsandbicoherenceplotsarelargelyindependenttestsof
resultsfromthisscenarioisrelativelysimilartothe“hypotenuse” thevariability’scoupling,althoughthefluxdistributionsproduced
pattern, although we do have trouble reproducing a strong bico- inthesesimulateddatasetsdonotdeviatestronglyfromlog-normal
herence with a realistic power spectrum. An alternative physical distributions in any event. Furthermore, in cases where there are
mechanismforproducing suchaquasi-periodicoscillationwould strongQPOswithverylowfrequencies,therearecleardeviations
be to have variations in the accretion rate excite inertial-acoustic ofthefluxdistributionfrombeinglog-normal(Misraetal.2006).
modesintheinneraccretiondisc(Chen&Taam1992;1995). Wehaveadditionallynotensuredthatthepowerspectraofrealob-
Quantitatively,thisscenarioismodelledinthesamewayasa servations are matched exactly by these simulations. In essence,
spring which does not follow Hooke’s Law (i.e. F = −kx),but sincetheinputmodelsdonothaveanyradiativetransferinthem,
ratherhasanasymmetric,higherorderrestoringforce,F whichis we are assuming that the radiative efficiency of any perturbation
givenbyF =−k1x+k2x2,wherexisthedisplacement,andk1 will be constant when we call the output time series “simulated
and k2 are constants parameterizing the strength of the restoring lightcurves.”Deviationsfromthisassumptionarelikelytoaffect
force. If k2 = 0, this does reduce to Hooke’s Law, and the lack thepowerspectra,and,especially, thefluxdistributions,farmore
of a non-linear term leads to a lack of observable features in the seriouslythantheyaffectthemorphologyofthebicoherenceplots.
bicoherenceplot. Suchdeviationsarequiteeasytoimagineinsituationswherethere
Thefullequationsofmotionforthis“oscillator”are: existsaresonance.
2 Anadditionalpotentialproblemexistswithconnectingthetoy
F(t)=−k1x(t)+k2x (t)−γv(t)+YWN(t), (7)
modelspresentedheretoreality.Muchofthephenomenologyap-
whereγ isthedampingcoefficientandYWN isthemagnitudeof pearstobesimilarbetweenthedifferentsub-typesofχclasses.It
thedrivingforce,whichisassumedtobeawhitenoiseprocess; wouldthereforebesurprisingiftherewereaqualitativechangein
themechanismforproducingtheQPOs.Itwouldthusbeusefulto
v(t)=v(t−∆t)+∆t×F(t)/m, (8)
identifyameanstounitethedifferentmechanismsdescribedhere,
wheremisthemassoftheoscillator,whichwesettounityforthese ortodevelopanentirelynewphysicalmodeltoexplainalltheob-
calculations,and∆tis0.001.Thecalculatedsimulatedlightcurve servedphenomenologyinamoreunifiedway,withjustthechange
isthetimeseriesx(t),andhas1048576points.Thebicoherenceis of one or two parameters causing the broad range of phenomena
computedfrom256chunksof4096points. seenintermsofbicoherencepatterns.
Forthefirstcalculation,wesettheparametervaluestok1 =
50000.0,k2 = 0.2,andγ = 7.0. Weshowtheplotofsimulated
datain3c.Wenotethereisonesubstantialdifferencebetweenthe
6 CONCLUSIONS
modeldataandtherealdatafor30184-01-01-000,whichisthatthe
modeldatahasstrongerbicoherencealongthe“hypotenuse”when We have shown clearly that the variability in the noise compo-
thetwofrequenciesaremostdifferentfromoneanother,whilethe nents and quasi-periodic oscillations of GRS 1915+105 are cor-
real data shows the opposite trend. This trend makes the model related with one another. The bicoherence provides a good dis-
valuesmoreseriouslydeviant fromthereal dataforthestrongest criminatoramongvariabilitymodelswhichproducesimilarpower
dampingofthemotionoftheoscillator. spectrathroughquitedifferentphysicalprocesses.Wehavefound
Whenthedampingfactorisverylow,thentheresultingbico- that it is plausible that in the “plateau” state of GRS 1915+105,
herenceplotscanlookquitesimilartothe“web”pattern.Infigure thevariabilityiscausedbyareservoirofenergybeingdrainedbya
6, we show the power spectrum and bicoherence plot for a low noisecomponent(whichcouldbetheradiojet)andaquasi-periodic
dampingcase.Thecalculationsaredoneinthesamemannerasthe component,whileinthebrighterpartoftheχstate,thevariability
onesdescribedinthepreviousparagraph,exceptwithk2=0.25and isconsistent withawhitenoiseinput spectrumdrivingadamped
γ =0.6.Forvaluesofγintermediatebetween0.6and7.0,the“hy- harmonic oscillator with a non-linear restoring force. While the
potenuse”canbecomemuchbroaderwithoutcreatingsuchstrong modelspresentedherearealmostcertainlynotuniquesolutionsfor
additionalharmonicsinthepowerspectrum.Thissimulation,like whatisoccuringphysicallyinthesesystems,itisquiteclearthat
Couplingbetween QPOsandbroadbandnoisecomponentsin GRS1915+105 7
4
3
2
1
0
0 1 2 3 4
Frequency 1 (Hz)
6
4
2
0
0 2 4 6
Frequency 1 (Hz)
6
4
2
0
0 2 4 6
Frequency 1 (Hz)
Figure 1. The bicoherence plots from the observations. (a) the data for Figure2.Thepowerspectrafortherealdata:(a)observation10408-01-25-
observation 10408-01-25-00, the prototypical “web” source. The colour 00(b)observation20402-01-15-00(c)observation30184-01-01-000.
schemeisgivenfordifferent values ofthelog10b2 asfollows:red:-1.0;
green:-1.25;darkblue:-1.5;lightblue:-1.75(b)thedataforobservation
20402-01-15-00,theprototyical“cross”source.Thecolourschemeisgiven
thebicoherencewillprovideexcellent constraintsonmorephysi-
fordifferentvaluesofthelog10b2asfollows:red:-1.0;green:-1.25;dark callymotivatedmodelsforthevariability,andthatwecandefinitely
blue:-1.5;lightblue:-1.75(c)thedataforobservation30184-01-01-000,
identifycaseswherethepropertiesofthepower spectraarequite
theprototypical“hypotenuse”source.Thecolourschemeisgivenfordif-
ferentvaluesofthelog10b2asfollows:red:-1.0;green:-1.25;darkblue: similar,butthepropertiesofthelightcurvesarequitedifferent.
-1.5; light blue: -1.75. The equality ofthe values of the bicoherence for
reflectionsaboutx=yistrivial.
7 ACKNOWLEDGMENTS
WearegratefultoRonElsnerforsendingusanunpublisheddraft
writtenbyhimself,R.BussardandM.Weisskopfmanyyearsago
discussingsomeof thefundamental propertiesofthebispectrum,
8 Maccaroneet al.
andtoPatriciaAre´valo,MikeNowakandSimonVaughanforuse- Timmer,J.,Schwarz,U.,Voss,H.U.,Wardinski,I.,Belloni,T.,Hasinger,
fuldiscussionsrelatedtovariabilitystatistics,andtoMarcKlein- G.,vanderKlis,M.&Kurths,J.,2000,PhRvE,61,1342
Wolt, Mariano Me´ndez and Luciano Rezzolla for more general, Uttley,P.&McHardy,I.M.,2001,MNRAS,323L,26
butequallyinterestingdiscussions,andtoRobFenderforhelping UttleyP.,McHardyI.M.,VaughanS.,2005,359,345
vanStraaten,S.,vanderKlis,M.,DiSalvo,T.,Belloni,T.,2002,ApJ,568,
toclarifysomeofthenomenclature surrounding GRS1915+105,
912
andTomaso Bellonifor useful comments onadraftof thepaper.
WarnerB.,WoudtP.A.,PretoriusM.L.,2003,MNRAS,344,1193
Wealsothanktheanonymousrefereeforsomeusefulsuggestions
Wijnands,R.&vanderKlis,M.,1999,ApJ,514,939
whichhavehopefullyimprovedtheclarityofthepaper.TJMalso
wishestothankDaveSmithoftheQuantumandFunctionalMatter
group in the Southampton School of Physics and Astronomy for
usefuldiscussions.
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Couplingbetween QPOsandbroadbandnoisecomponentsin GRS1915+105 9
4 4
3 3
2 2
1 1
0 0
0 1 2 3 4 0 1 2 3 4
Frequency 1 (Hz) Frequency 1 (Hz)
4 4
3 3
2 2
1 1
0 0
0 1 2 3 4 0 1 2 3 4
Frequency 1 (Hz) Frequency 1 (Hz)
2 Figure 4. The model bicoherence plots for reservoir models where the
noisecomponentcontributestotheobservedflux.Forallplots,thecolour
schemesareasfortheprevious plotofsimulateddata.Theplotsare:(a)
lowvariabilityamplitude(b)highvariabilityamplitude.Theequalityofthe
1.5
valuesofthebicoherenceforreflectionsaboutx=yistrivial.
1
0.5
0
0 0.5 1 1.5 2
Frequency 1 (Hz)
Figure3.Themodelbicoherence plots.Forallplots,thecolourschemes
areasfortherealdata, exceptthatforthemodelplots,thereisanaddi-
tionalcontour,inpurple,atlog10b2 = −2.0.Theplotsare:(a)thecase
of“jet” extraction withahighvariability amplitude (b)thecase of“jet”
extractionwithalowvariabilityamplitude(c)dampedforcedoscillations.
Theequalityofthevaluesofthebicoherenceforreflectionsaboutx=yis
trivial.
10 Maccaroneet al.
10
8
6
4
2
0
0 2 4 6 8 10
Frequency 1 (Hz)
Figure 6. (a) the power spectrum for the low-damping forced oscillator
(b)thebicoherence forthelowdampingforcedoscillator, withthesame
contourvaluesasusedintheothersimulations’plots.Theequalityofthe
valuesofthebicoherenceforreflectionsaboutx=yistrivial.
Figure5.Thepowerspectraforthesimulateddata:(a)web-like(b)cross-
like(c)dampedforcedoscillations–notethatthesearethepowerspectra
forthesametimeseries,inthesameorder,aspresentedinFigure3.