A VLT-ULTRACAM study of the fast optical quasi-periodic oscillations in the polar V834 Centauri

4.2 MB·

by  M. Mouchet

Astronomy&Astrophysicsmanuscriptno. 30166_hk_Mouchet_V834Cen (cid:13)cESO2017
January11,2017
A VLT-ULTRACAM study of the fast optical quasi-periodic
oscillations in the polar V834 Centauri
M.Mouchet1,J.-M.Bonnet-Bidaud2,L.VanBoxSom3,4,2,E.Falize2,3,D.A.H.Buckley5,J.B.Breytenbach5,R.P.
Ashley6,T.R.Marsh6,andV.S.Dhillon7,8
1 LUTH, Observatoire de Paris, PSL Research University, CNRS, Université Paris Diderot, Sorbonne Paris Cité, 5 Place Jules
Janssen,F-92195Meudon,France(e-mail:[email protected])
2 CEASaclay,DSM/Irfu/Serviced’Astrophysique,F-91191Gif-sur-Yvette,France
3 CEA-DAM-DIF,F-91297Arpajon,France
4 LERMA,ObservatoiredeParis,PSLResearchUniv.,CNRS,SorbonneUniversités,UPMCUniv.Paris06,F-75005,Paris,France
7 5 SouthAfricanAstronomicalObservatory,POBox9,Observatory7935,CapeTown,SouthAfrica
1 6 DepartmentofPhysics,UniversityofWarwick,CoventryCV47AL,UK
0 7 DepartmentofPhysicsandAstronomy,UniversityofSheffield,SheffieldS37RH,UK
2 8 InstitutodeAstrofisicadeCanarias,E-38205LaLaguna,Tenerife,Spain

n
Received....;accepted...
a
J

0 ABSTRACT
1
Quasi-periodicoscillations(QPOs)ofafewsecondshavebeendetectedinsomePolars, thesynchronisedsubclassofcataclysmic

] systemscontainingastronglymagnetisedwhitedwarfwhichaccretesmatterfromareddwarfcompanion. TheQPOsarethoughtto
E berelatedtoinstabilitiesofashockformedintheaccretioncolumn,closetothewhitedwarfphotosphereabovetheimpactregion.
H WepresentopticalobservationsofthepolarV834CentauriperformedwiththefastULTRACAMcameramountedontheESO-VLT
simultaneouslyinthreefilters(u’,Heiiλ4686,r’)tostudytheseoscillationsandcharacterisetheirpropertiesalongtheorbitwhen
.
h the column is seen at different viewing angles. Fast Fourier transforms and wavelet analysis have been performed and the mean
p frequency,rmsamplitude,andcoherenceoftheQPOsarederived;adetailedinspectionofindividualpulseshasalsobeenperformed.
- TheobservationsconfirmtheprobableubiquityoftheQPOsforthissourceatallepochswhenthesourceisinahighstate, with
o observedmeanamplitudeof2.1%(r’),1.5%(Heii),and0.6%(u’). TheQPOsarepresentinther’filteratallphasesoftheorbital
tr cycle,withahigherrelativeamplitudearoundthemaximumofthelightcurve. TheyarealsodetectedintheHeiiandu’filtersbut
s atalowerlevel. Trainsofoscillationsareclearlyobservedinther’lightcurveandcanbemimickedbyasuperpositionofdamped
a
sinusoidswithvariousparameters. TheQPOenergydistributioniscomparabletothatofthecyclotronflux,consistentforther’and
[
Heiifiltersbutrequiringasignificantdilutionintheu’filter. New1Dhydrodynamicalsimulationsofshockinstabilities,adaptedto

1 thephysicalparametersofV834Cen,canaccountfortheopticalQPOamplitudeandX-rayupperlimitassumingacrosssectionof
v theaccretioncolumnintherange∼(4−5)×1014cm2.However,thepredictedfrequencyislargerthantheobservedonebyanorder
7 ofmagnitude. ThisshortcomingindicatesthattheQPOgenerationismorecomplexthanthatproducedinahomogeneouscolumn
7 andcallsforamorerealistic3Dtreatmentoftheaccretionflowinfuturemodelling.
6
Keywords. –magneticwhitedwarfs–X-rays:binaries—accretion—instabilities–shocks
2
0
.
11. Introduction radiation in the optical/infrared. These two cooling processes
0
leadtoastratificationindensityandtemperature. Attheshock,
7
Cataclysmic variables are close interacting binaries consisting the temperature reaches a few tens of keV, depending mainly
1
:ofawhitedwarf(WD)accretingmatterfromareddwarfcom- on the mass of the white dwarf (see reviews by Cropper 1990
vpanion via Roche lobe overflow; an accretion disk is generally andWarner1995). Earlytheoreticalstudies(Langeretal.1981,
Xiformed around the white dwarf. In magnetic cataclysmic vari- Chevalier & Imamura 1982) demonstrated that the accretion is
ables (MCVs), the white dwarf is strongly magnetised and the thermallyunstableifitisdominatedbybremsstrahlungcooling.
araccretionflowiscapturedfromtheorbitalplanealongthemag- Shockheightoscillationsareexpectedwithperiodsoftheorder
neticfieldlinesandformsanaccretioncolumntowardstheWD ofthecoolingtimeneartheshock,typicallyafewtenthsofasec-
magnetic poles. Among MCVs, Polars are a subclass charac- ondtoseconds.Wu(2000)presentedacomprehensivereviewof
terisedbya(quasi)-synchronisationofthewhitedwarfwiththe stationaryandnon-stationarymodelsavailableatthatdate. Nu-
orbitalrotation,whileinIntermediatePolars,thecompactobject mericalsimulationsshowthattheseoscillationsareexpectedin
isrotatingfasterthanthebinarysystem. InPolars,noaccretion X-raysandtheoptical,andthattheyaredampedwhencyclotron
disk is expected to form. Material falling down the column is coolingisnon-negligible(Saxtonetal.1998, Saxton1999, Wu
acceleratedandashockisformednearthesurfaceofthewhite 2000,Busschaertetal.2015).
dwarfclosetothemagneticpoles. Belowtheshock,theplasma Based on this cooling instability model, quasi-periodic os-
cools by emitting bremsstrahlung in the X-rays and cyclotron cillations(QPOs)wereexpectedtoberathercommoninPolars.
However,afteraninitiallysuccessfuldiscoverytimewhenrapid
Sendoffprintrequeststo:M.Mouchet QPOs(∼1-3s)weredetectedintheopticallightoffivepolars,
Articlenumber,page1of14
no further detections were reported despite the growing num- whether QPOs might also partly come from line emitting re-
berofnewlyidentifiedpolars,nowamountingtomorethanone gions as suggested by AN UMa observations (Bonnet-Bidaud
hundred. TheoriginoftheseQPOsisthereforestillasubjectof etal.1996).
controversy.AfterthefirstdiscoveryofQPOsinthetwosystems In this paper we test the standard accretion model, which
V834CenandANUMa(Middleditch1982),threemoreobjects predicts an oscillating shock with modulated cyclotron emis-
were found to exhibit oscillations: EF Eri (Larsson 1987), VV sion. An additional aim is to test whether the observed broad
Pup(Larsson1989), andBLHyi(Middleditchetal.1997), the frequency distribution of the QPO is due to a superposition of
lastonebeingdiscoveredtwodecadesago. simultaneous oscillations with different frequencies or cumula-
Theoscillationsareallfoundduringhighstatesandseema tive contribution over time of individual oscillations at a given
ratherpersistentfeatureforV834CenandANUMa. Theperi- frequency. Wealsocomparethepropertiesoftheopticaloscilla-
odsareintherange(0.3-1)Hzwithrmsamplitudesofafewper tionswiththepredictionsof1Dsimulationsinthenon-stationary
cent. From high time resolution spectroscopy, QPOs were also regime.
reportedintheHeiiλ4686andH linefluxesofANUMaduring In Section 2, the observational data are described, and the
γ
ahighstate(Bonnet-Bidaudetal.1996). different observational results are presented in Sect. 3 (light
Oscillations have also been searched for in the X-ray band curves)andinSect. 4(periodograms). InSect. 5wediscussthe
withdifferentsatellites(Beardmore&Osborne1997,Wolffetal. energydistributionoftheoscillationsandcomparetheobserva-
1999,Steiman-Cameron&Imamura1999,Christian2000,Pan- tions with the predictions of 1D numerical simulations from a
del et al. 2002, Ramsay & Cropper 2007). Recently, a more radiationhydrodynamiccode.
complete survey of a meaningful set of polars observed with
theXMM-NewtonsatellitealsofailedtodetectsignificantX-ray
2. Observations
QPOs(Bonnet-Bidaudetal.2015). Sofar,theinterpretationof
thepolarQPOsintermsofshockcoolinginstabilitiesstilllacks V834Cenwasobservedon2005May16,usingthehigh-speed
adefinitiveobservationalconfirmation. CCDcameraULTRACAM1 (Dhillonetal.2007). Thecamera
Inthispaperwepresenthigh-qualityopticaldataofthepolar wasmountedasavisitinginstrumentontheUT3Unit(Melipal)
V834CenobtainedattheVLTobservatoryusingthetriple-beam oftheEuropeanSouthernObservatory(ESO)8.2mVeryLarge
high-speedCCDcameraULTRACAM,inanefforttobetterde- Telescope (VLT). ULTRACAM is a triple-beam CCD camera
scribethecharacteristicsoftheopticalQPOs. designed to provide three-colour imaging photometry at high
The system V834 Cen was discovered as a bright X-ray temporal resolution down to 0.002s. The observations were
source by Mason et al. (1983), and identified as a polar on the made simultaneously in the two filters u’ (effective wavelength
basisofastrongpolarisedopticalandinfraredflux(Baileyetal. 3557Å)andr’(effectivewavelength6261Å)oftheSloanDigi-
1983). Photometricdatarevealedanorbitalperiodof101.5min.
talSkySurvey(SDSS)andinanarrowbandfilter(centralwave-
Thedistanceisestimatedatd∼116pc(Barrettetal.1999)and length4662Å,FWHM108Å),coveringtheemissionlinesHeii
thecyclotronpolarmagneticfieldatB∼23MG(Ferrarioetal.
4686Å andtheBowencomplexCiii-Niiiat4640-50Å anddes-
1992). Theopticalfluxandpolarisationmodulationscanbein-
terpreted mainly as an effect of the beaming of the cyclotron ignatedasthe‘Hefilter’inthefollowing.
The total duration of the observation is 5.58h, covering
radiation. Atminimumoftheopticallightcurve, thecyclotron
slightly more than three orbital cycles, with nearly continuous
radiatingpoleisseennearlyfaceonwithaminimumfluxcorre-
coverage,exceptforthreesmallgapsof2sandoneof40s.Data
sponding to a minimum cyclotron opacity while at maximum
were obtained in a two-window drift mode with a 4x4 binning
the radiating column above the pole is seen edge on and the
configuration. The exposure time was of 0.0473s with a dead
cyclotron opacity and flux are maximum. In the case of V834
timeof0.0039s,givingaresolutionof0.05122s. Theexposure
Cen, thegeometryofthesystemissuchthattheaccretingpole
time was selected to give a good sampling of the expected 2s
isnoteclipsedandisalwaysinthehemisphereofthewhitedwarf
quasi-periodicpulses. Theobservationtimeswerecorrectedfor
facing us with an orbital inclination and magnetic colatitude of
thelighttraveltimetothesolarsystembarycentre.
i ∼ (45−50)◦ and β ∼ (15−25)◦, respectively (Bailey et al.
The weather was good with typical seeing varying between
1983,Cropper1989,Ferrario&Wickramasinghe1990,Mauche
0.5 and 1 arcsec and good atmospheric transparency. The air
2002,Potteretal.2004).
mass was always lower than 1.33. The data were reduced us-
After their discovery, QPOs with frequencies (0.2-1.2) Hz
ingthestandardULTRACAMpipelinesoftware2,correctingfor
andamplitude(1-3)%werealsoreportedinV834CenbyMid-
biasandflatfield. Tocheckforvariationoftheskytransmission
dleditchetal.(1991),Larsson(1992),andImamuraetal.(2000).
andthestabilityofthewholeinstrument,weperformedaperture
The relative QPO amplitude was found to be modulated with
photometry on the target and a comparison star in the field ac-
the orbital period, with zero modulation coinciding with min-
quiredsimultaneously. ThemeanlevelofV834Cenwasabout
imum brightness. Different colour dependencies of the QPOs
0.14, 0.28, and 1.60 times the level of the comparison star, re-
weregivenfromnon-simultaneousobservations,withmaximum
spectively in the r’, He, and u’ filters. During the first hour of
amplitude either in the R or B band (Middleditch et al. 1991,
theobservation, abiasproblemoccurredforther’channeland
Larsson 1992). Upper limits for QPOs in X-rays were also re-
the data were rescaled using the comparison star. No absolute
ported(Bonnet-Bidaudetal.1985,Beardmore&Osborne1997,
photometry was derived since we were mostly concerned with
Imamuraetal.2000)withthemoststringentlimitprovidedbya
thestudyofrapidvariability. Thehighcountrateandthetypical
longXMM-Newtonobservation(Bonnet-Bidaudetal.2015).
shape of the light curve indicate that the source was in a high
In order to test whether the QPO characteristics are com-
accretion state. After correction from the extinction, the mean
patible with a cyclotron origin, we obtained high time resolu-
countratesgiveareasonableestimateoftheaveragemagnitude
tion data simultaneously at three different wavelengths, includ-
of15.0and15.8forther’andu’filters,respectively.
ing two broadband filters in u’ (3557Å) and r’ (6261Å), and
a narrowband filter covering the Heii λ4686 line. The choice 1 http://www.vikdhillon.staff.shef.ac.uk/ultracam/
of the Heii emission line was motivated by the attempt to test 2 http://deneb.astro.warwick.ac.uk/phsaap/software/ultracam/html/
Articlenumber,page2of14
M.Mouchetetal.: QPOvariabilitiesinV834Cen
mum is slightly shifted at 0.906. Similar deep minima are also
2
visibleintheU-bandlightcurvesshowninMasonetal.(1983)
r' and Middleditch et al. (1991). The two dip positions given in
1,5 Middleditchetal.(1991)correspondtoaspectroscopicphaseof
0.900and0.902(±0.014)givenbytheSchwopeetal.ephemeris
usedinthepresentpaper,thuswithaconsistentphasewithinthe
1 errorbarsofthepresentobservations.
Forther’andHefilters,themid-timeofthebroadminimumwas
determinedbyfittingaparabolatodatauptohalfamplitude,be-
0,5
tween phase 0.75 and 1.05. The mean weighted phases of the
four minima are consistently found at 0.899 ±0.006 and 0.909
±0.019respectivelyforther’andHelightcurves.
0
We note that observations obtained at different dates revealed
e He different white light curves, changing from a double-humped
at
r 1,5 to a saw-tooth shape (Osborne et al. 1987, Cropper 1989). In
nt ourobservationinther’andHefilters,aftera512ssmoothing,
u
o the second cycle appears double-humped, due to significant in-
c
d 1 creasedflaringactivity, whilethetwoothercycleshaveamore
e
z pronouncedsaw-toothshape.
ali
m 0,5
r
o
n 4. Fastquasi-periodicvariability
0
In addition to the orbital modulation and flaring variations, a
u' close look at the light curves at high temporal resolution re-
1,5 veals clear variations on a timescale of a few seconds. Figure
2showsatypicaldatasegmentofther’lightcurveatthehighest
0.05sresolutionduringtheorbitalbrightphase. Quasi-periodic
1 oscillationsaredirectlyseendisplayingasinusoidalshapewith
strongvariableamplitudesatperiodaround∼2.9s. Suchoscil-
lationsarealsopresentinthebrightestpartsoftheHelightcurve
0,5
andarebarelyvisibleintheu’lightcurve.
TostudythecharacteristicsoftheseQPOs,fastFouriertrans-
0 form (FFT) analysis of the light curves were performed for the
0,5 1 1,5 2 2,5 3 3,5 4 4,5 three filters. The data set is almost continuous in time, with
Phase only four short gaps, the longest being of 40s. A continu-
ous light curve at a constant step of 0.05122s was built, filling
Fig.1: Normalisedlightcurvesinther’(top),He(middle),and the short gaps with values extrapolated from the signal imme-
u’ (bottom) filters versus orbital phase at a resolution of 5.1s diately surrounding the gaps with typical random noise. Stan-
(rebinof100points).Thecontinuousobservationcoversslightly dard FFTs were computed for consecutive segments of 1024
morethanthreecycles. datapoints(52.45seconds)usingthepowspeccommandofthe
XronospackageoftheHEASOFTsoftware3,resultinginatotal
of383FFTsforeachfilter. Theobtainedpowerdensityspectra
3. Orbitallightcurves
(alsocalledFFTsinthefollowing)extendupto9.76Hz(Nyquist
frequency)withabinsizeof0.01906Hz.
Photometriclightcurveswerefoldedaccordingtotheephemeris
given by Schwope et al. (1993), where phase 0 corresponds to
theblue-to-redcrossingtimeofthesecondaryradialvelocities.
4.1. Meanpowerspectra
Figure 1 shows the light curves for the three filters, plotted at
a5.122sresolutionasafunctionoftheorbitalphasewherethe To define the overall properties of the fast variability, mean
fluxhasbeennormalisedtotheaveragevalue. quadratic-summed FFTs over the full observation were first
The three cycles are similar, showing a nearly sinusoidal computed for the three filters and are shown in Fig.3 in loga-
variation with a pronounced minimum. A strong and variable rithmic scale. A broad (∼ 0.25−2 Hz) bump, signature of the
flickering/flaring is also visible during the bright phase. The r’ quasi-periodicoscillationsisclearlydetectedforthethreefilters
andHelightcurveshaveasimilaroverallshape;thelattershows with decreasing amplitudes from r’ to u’. In addition, a signif-
aslightlybroaderminimum. Theamplitudeoftheorbitalmod- icant power is seen at low frequencies (red noise due to flick-
ulation decreases from r’ to He with a peak-to-peak amplitude ering/flaring)aswellastypicalwhitenoiseathighfrequencies.
varying from 1.9 mag (r’) to 1.1 mag (He). The u’ light curve Whenthecountratestatisticsarebetter, asinthecaseofther’
hasadifferentshapewithasmaller∼0.5magamplitude,aless filter,asignificantQPOexcessoverwhitenoiseisvisibleupto
pronounced broad minimum, and the presence of a narrow dip ∼5Hz.
aroundφ ∼ 0.9superimposedonthisminimum. Thisfeatureis Tocomputethemeanamplitudesoftheoscillations,wefirst
alsopresent,butmuchlessvisible,inthetwootherfilters. Four subtractedbestfitpowerlawsdefinedatlowfrequenciesin the
pronounceddipsaredetectedinu’withthelastthreedipsposi- range(0.03-0.2)Hz(rednoise)andhighfrequenciesintherange
tioned at phases 0.892, 0.891, and 0.891 (±0.001) with widths
between20and60s,whilethefirstdipisbroaderanditsmini- 3 https://heasarc.gsfc.nasa.gov/docs/xanadu/xronos/xronos.html
Articlenumber,page3of14
V834Cen QPOs (r' filter) Mean period = 2.91 s Mean amplitude = 4.9 %
1.1
e
d 1.05
u
plit
m
A 1
e
v
ati 0.95
el
R
0.9
3840 3850 3860 3870 3880 3890 3900
Time (s)
Fig. 2: Typical QPOs seen in the r’ light curve in a 60s data segment. Low-frequency variations have been removed and data
normalisedbydividingbya5smovingaverage. Individualoscillationpulsesareseenwithvariableamplitudes. Abestsinefitto
thedatayieldsameanperiodof2.91sandmeanamplitudeof4.9%.
Table1: MeanpropertiesoftheQPOs in the two ranges ν -ν and ν -ν (median). The compu-
low m m high
tations were performed on the power spectrum after red noise
andwhitenoisesubtractionandintheselectedfrequencyrange
r’ He u’ (0.25-5)Hz.
Amplitude(rms%) 2.09(0.01) 1.48(0.07) 0.55(0.17) The mid-frequency ν is listed in Table 1 with values 0.52,
m
Mid-freq. νm(Hz)* 0.52(0.01) 0.61(0.03) 0.67(0.20) 0.61,and0.67Hzrespectivelyforther’,He,andu’filters. The
FWHM(Hz) 0.25(0.03) 0.26(0.03) 0.26(0.07) asymmetryoftheQPObumpwithapositiveskewnessprovides
lower values for ν , with respectively 0.40, 0.46, and 0.50 Hz
p
(*)seetextfordefinition forther’,He,andu’filters;instead,theweightedfrequencyν
w
yields higher values at 0.68 and 0.76 for the r’ and He filters
andgivesunreliableresultsfortheu’filter,duetolowsignal-to-
(5-9)Hz(whitenoise). ThesefitsareshowninFig.3. Theam- noiseratio. Thetypicalfrequencyvaluesareallintherange0.4-
plitudesoftheQPOswerethenevaluatedbycomputingthein- 0.8Hz.Regardlessofthemethod,thereisatendencyforslightly
tegratedpowerintherangefrom0.25to5Hz. Confidenceinter- smallervaluesinther’filterthanintheHefilter,butastheerror
valswerecomputedbypropagatingtheuncertaintyinthebestfit barsarecomputedherefrompurestatistics, thedifferencemay
power law parameters. Results for the three filters are listed in notbesignificantbecauseofsystematicuncertainties.
Table1.QPOsaredominantinther’filterreachinganamplitude The QPO widths – defined as full width at half maxi-
of2.09%,andshowadecreasingvaluefortheHefilter(1.48%) mum (FWHM) – were measured for the power spectra slightly
andu’filter(0.55%). smoothedbyarebinningof0.04Hz. Atypicalwidthof0.26Hz
Asacheck,wehavealsoevaluatedtheQPOamplitudeswith isfoundforthethreefilters(Table1).Thecorrespondingquality
a more straightforward measurement by detrending at low fre- factor Qisthereforeonlyoftheorderof2whenaveragedover
quencyusingathird-orderpolynomialandsubtractingaconstant thefull∼5.5hrobservation.
background evaluated between 5 and 9Hz at high frequency.
Thepowerspectrumwasthenintegratedbetween0.25and5Hz.
4.2. Periodogramtemporalvariations
Confidence intervals were obtained by varying the subtracted
background by plus or minus three times the standard devia- Inspection of the 52.4s segment light curves and their associ-
tion on the mean background value. The corresponding ampli- atedFFTsrevealsstrongtimevariabilityforthethreefilters. To
tudesforther’, He, andu’filtersarerespectively2.19±0.01%, visualise this temporal variability, we built 2D time-frequency
1.72±0.06%,and0.83±0.11%. Theresultsarecomparablewith images of these FFTs after normalising the power value to the
the values given in Table 1, but as expected, this background whitenoiselevelevaluatedintherange5-9Hz.
subtractingmethodslightlyoverestimatestheQPOamplitudes. TheimagethatshowstheQPOpowerdensity(relativeamplitude
ThebroadexcessoftheQPOs,inlogscale,inthepowerden- squared)forther’filterisdisplayedinFig.4intherange[0.019-
sity spectrum is clearly asymmetric so that the mean QPO fre- 5]Hz. Theimage,builtfrom383powerspectrawithnooverlap,
quencycanbedefinedindifferentways.Threetypesofmeasure- has been slightly smoothed by a two-point Gaussian function.
mentscanbecompared:thefrequencyν ofthehighestpeak,the TheQPOsareclearlyseenaround0.5Hzduringbrightphases.
p
meanfrequencyν weightedbypowervalues(statisticalmean), Therelativeamplitudeisstronglymodulatedwiththesourceflux
w
and the mid-frequency ν which gives equal integrated powers withadeepminimumcentredaroundphase0.9, corresponding
m
Articlenumber,page4of14
M.Mouchetetal.: QPOvariabilitiesinV834Cen
10-3
r'
z)
H
2/s
m
er (r 10-4
w
o
P
10-5
He
z) 10-3
H
2/s
m
er (r
w
o
P
10-4
u'
10-3
Hz) Fig.4: Two-dimensionaltime-frequencyimageofthefractional
2s/ rms amplitude for the r’ light curve. The X-axis is the orbital
m
er (r p5hHazse. aCnodlothuer-cYo-daxedissiqsuthareedfreaqmupenlictuydienitsheshroewstnricatcecdorradningge 0to-
w
o the scale at the right (in units of 1.4 × 10−5 rms2.Hz−1). The
P
individual(52.4s)spectrahavebeennormalisedtotheiraverage
levelintherange5-9Hzandtheimagehasbeensmoothedwitha
10-4 two-pointGaussianfilter. Thecorrespondingnormalisedr’light
0.1 1 10 curveisshownatthetop.
Frequency (Hz)
Fig.3: QPOfrequencyprofiles. Meanpowerdensityspectraof
thethreelightcurves(top:r’,middle:He,bottom:u’)forthefull
∼5.6hrobservationareshowninlogarithmicscaleintherange
frequencyandallowsustoscanshortdurationsignalsaswellas
0.03-9.76Hz. TheYscaleisthesquaredfractionalamplitudein
longerones.
units of rms2.Hz−1. Two best fit power laws are shown repre-
The resulting wavelet 2D map for the r’ filter is shown in Fig.
sentingtheadditionallow-frequencyrednoise(dottedline)and
5wherethesignalsquaredamplitudeisshownasafunctionof
high-frequency white noise (dashed line); their sum (full line)
frequency.Significantamplitudesaredetectedaroundperiodsof
hasbeensubtractedtocomputetheQPOpower(seetext).
∼2scorrespondingtotheQPOs,andalsointhewiderange50-
1000s. Thethinblacklinerepresentstheconeofinfluencethat
correspondstothemaximumperiodforwhichsignificantinfor-
to the light curve broad minimum. At bright phases, a signif- mationcanbederivedatagiventime. Periodsgreaterthanthis
icant power is seen up to ∼4-5 Hz. The maximum power is limitaresubjecttoedgeeffects(seeTorrence&Compo1998).
concentrated around ∼ 0.5Hz, showing trailing structures cor-
responding to significant changes in amplitude and frequency
For comparison with the mean power spectrum obtained
overminutetimescales. The2DimagefortheHefiltershowsa
fromtheFFTanalysis(seeSect.4.1),wealsobuilta1Dwavelet
comparableshapebutwithaloweramplitude. Acoherence2D
transformbyaveragingthewaveletsquaredamplitudesoverthe
spectrum produced with the r’ and He light curves indicates a
wholeobservation. TheresultsareshowninFig. 6forthethree
highdegreeofcoherencebetweenthesetwobands. filtersr’,He,andu’. Thefull(5×10−5-10Hz)frequencyrange
As significant red noise is present due to the source (∼100 is covered in logarithmic scale in Fig. 6 (main frame), reveal-
s) flaring activity, we also investigate the source variability us- ingthefastQPOsasabumpwitha∼2s(0.5Hz)timescaleand
ingIDLroutinesforwaveletanalysisprovidedanddescribedby a strong power at the orbital period at 101.5 min (1.64×10−4
Torrence & Compo (1998). This allows us to efficiently exam- Hz).Asmallerbutsignificantexcessisalsoclearlyvisibleinthe
inevariabilityatvarioustimescales.WeusetheMorletGaussian intermediate range of minutes. The insets in Fig. 6 also show
waveletbasisfunction: asinusoidatfrequencyν, modulatedin theamplitudeinthesubset(0-2Hz)frequencyrangeinthemore
amplitudebyaGaussianenvelopewithanextentτ suchthatthe conventionallinearrepresentation,revealingthebroadQPOex-
d
productτ ·νisconstant. Themodulationisthusadjustedtothe cesswithadecreasingamplitudeinthethreefilters.
d
Articlenumber,page5of14
3 108
r'
14
2 108
wer) 12 Power1 108
o
P
log ( 10 00 0.F5reque1ncy (H1z.)5 2
r'
8
12
1.5 106
11
He
10 1 106
wer
er) 9 Po5 105
w
o
P
g ( 8 00 0.5 1 1.5 2
o Frequency (Hz)
Fig.5: Wavelettransform(scalogram)ofther’filterlightcurve, l 7 He
mappingthetime-frequencyevolutionofthesignal. Significant
6
QPOamplitudeisvisiblearound∼2s(delimitatedbyhorizon-
talwhitelines)butvariabilityisalsopresentintherange∼100s. 5
Highflaringactivityisalsovisiblearound1000sinthesecond
1 106
orbitalcycle. Thethinblacklineatthetopofthefiguredelim- 10 u'
its the cone of influence below which the information is fully
usable. 9 wer5 105
er) Po
w
o 8
P
4.3. QPOorbitalvariability g ( 00 0.5 1 1.5 2
o 7 Frequency (Hz)
To investigate the variations of the QPO characteristics along l
u'
the orbital cycle, FFTs computed over 52.4s time intervals
6
were phased and averaged in 20 orbital phase bins with each
phase bin including typically 19 FFTs. Quantitative QPO
5
measurements were derived for the 20 phase bins in the three
-4 -3 -2 -1 0 1
filters. As the signal statistics are lower than for the orbital
averaged spectra, the fractional rms amplitude was computed log (Frequency) (Hz)
here by subtracting a constant background at high frequency
(5-9Hz), as described in Sect. 4.1. The power spectrum was Fig. 6: Main frame: Mean wavelet spectra of the light curves
integrated between 0.25 and 5Hz. As shown in Sect. 4.1, this for the three filters (top: r’, middle: He, bottom: u’) shown in
gives amplitudes nearly consistent with the more complete red logarithmic scale in the range (5×10−5 - 10)Hz. The peak at
andwhitenoisecorrections. TheQPOcharacteristicfrequency 1.6 × 10−4 Hz corresponds to the orbital frequency and some
wasalsodeterminedbyevaluatingthemid-frequency,asdefined powerisalsovisibleintherange(10−3-10−2)Hz. Insets: Subset
inSect. 4.1. in the range (0-2)Hz is shown in linear scale to emphasise the
fastQPOs.
The phase averaged QPO amplitudes are shown in Fig. 7.
Given the different filters, QPOs could not always be detected.
Inparticular,nosignificantQPOsweredetectedfortheu’filter phase bins, built by averaging 4096-point FFTs corresponding
in most orbital bins except one at phase 0.85, and so this filter totimedurationof210s,forabetterfrequencyresolution. The
is not shown. In the He filter, amplitudes in the phase interval overallQPOprofilesarebroad,butevenafteraveragingseveral
(0.85-1.00)arenotstatisticallysignificant. individual narrow peaks are still detected in most phase bins.
TheQPOfractionalrmsamplitudesmeasuredforther’and Forr’, QPOsarestrongestaroundφ ∼0.1and0.7, andarenot
Hefilter(seeFig.7)arenotconstantalongtheorbit,butroughly distinguishableatφ∼0.9atthisscale.
behaveasthelightcurveswithabroadminimumaroundphase
0.9. Thehighestvaluesaremeasuredforbothfiltersintherising
(phase ∼0.1-0.2) and the decreasing (phase ∼0.7) parts of the
4.4. QPOshort-termvariability
lightcurvewithnoobviouscorrelationwiththeflaringactivity.
The width of the broad minimum is also slightly larger in He InspectionofFig.1andFig. 4showsthatthesourcebehaviour
thaninr’filter,similarinshapetothecorrespondinglightcurve is not totally reproducible from cycle to cycle; for instance,
(seeFig. 1). thesecondcycleshowsincreasedflaringactivity,andenhanced
To illustrate the orbital variability of the power density QPOsarealsoseenatdifferentphasesfromcycletocycle. We
spectra in the r’ filter, in Fig. 8 we show power spectra in ten therefore analysed separately the 383 FFTs of 52.4s duration
Articlenumber,page6of14
M.Mouchetetal.: QPOvariabilitiesinV834Cen
4
10 10
r'
3
3 s)
al rms (%) 2 2 3 c/e (unit 10 r' flu
action 1 mplitud 1 1 x (units
Fr 1 e rms a 10 c/s5
0 He olut )
s
b
A
0
0 0,5 1 1,5 2
Orbital phase
0.1 0.1
Fig. 7: Fractional rms amplitude variation as a function of the
orbitalphaseforther’(reddots)andHe(greendots)filters. The %) 4
differentright(r’)andleft(He)scaleshavethesameamplitude, e (
ud 3
butareshiftedby0.5toavoidoverlap. plit
m 2
A
O
P 1
0.04 Q
0
0.5 1 1.5 2 2.5 3 3.5 4 4.5
Orbital phase
φ = 0.8 Fig.9: Top: AbsoluteQPOrmsamplitudeversusorbitalphase
computed for the52.4 s data segments inthe r’ filter, shown in
0.03 logarithmic scale (in red). The r’ light curve (in black) is also
superposed, with the scale given at right. Bottom: Fractional
QPOrmsamplitude(inpercent)versusorbitalphase.
φ = 0.6
obtainedusingthepreviouslydescribedbackgroundsubtracting
method.
r
e 0.02
w Figure9(top)showstheabsoluteamplitudesoftheQPOsin
o
P the r’ filter along the observation. For comparison, the corre-
φ = 0.4 spondingr’lightcurveisalsoshownatthesame52.4sresolu-
tion. Thedataareplottedinlogarithmicscaletoshowtheratio
moreclearly. Atfirstsight,theabsoluteamplitudeoftheQPOs
seemstocloselyfollowtheoveralllightcurvevariations,reach-
ingmaximavaluesaround ∼ 4×103 count/sduringthesource
0.01 bright orbital phases, corresponding to a meanratio ∼ 0.025 of
φ = 0.2
the source flux. Some of the flaring activity visible in the sec-
ond cycle (φ ∼ 2.5) also has a counterpart with corresponding
higherQPOamplitudes. However,thefractionalrmsamplitude
is not strictly constant (Fig.9, bottom). At low level, the frac-
tional amplitude steadily increases with flux, before reaching a
plateauaround2.4%, wheresystematiccorrelationisnolonger
φ = 0.0
0 observed.
The QPO amplitudes for the r’ filter clearly show that sig-
0 0.5 1 1.5 2
nificant oscillations are present during the whole observation,
Frequency (Hz) including the faint phase. At minimum flux, around φ ∼ 0.9,
the amplitude reaches small but still marginally significant val-
Fig. 8: Phase-resolved power spectra for the r’ light curve in ues (see Fig. 7 and Fig. 9). At this phase, the oscillations are
the range (0-2)Hz, built from individual 210s FFTs. Phase in- verymarginallyvisibleintheHelightcurveandessentiallynot
creasesfrombottomtotopbystepsof0.1inphase. Powerisin seenintheu’lightcurve,duetolowercountrate(weaksignal-
unitsofrms2.Hz−1. Eachphasespectrumisshiftedverticallyby to-noiseratio). Wenotethatthemeansourcecountratesforthe
aconstantvalueof0.004. Heandu’filtersaresimilarandabout10timessmallerthanthat
ofr’.
Articlenumber,page7of14
4Count rate (unit 10) 112680 r' Normalized rate (c/s)01..90155 r' Power spectrum 123000000000 r'
14 0
1.6
4Count rate (unit 10) 11..124 He Normalized rate (c/s) 01..90155 He Power spectrum 1000 He
4e (unit 10) 11..46 u' d rate (c/s)1.015 u' pectrum 16 u'
Count rat 1.2 Normalize0.95 Power s 08
1
1.270 1.272 1.274 1.276 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5
Orbital phase Phase (P=2.556 s) Frequency (Hz)
Fig.10: Left: ExampleofindividualQPOsina52.4ssegmentofthelightcurvesaroundφ = 1.273,forr’(top,red),He(middle,
green), u’(bottom, blue). Middle: Pulseprofilefoldedwiththebestsinusoidalfitperiodof2.556s; pulsesarereportedtwicefor
clarityandthephaseoriginisarbitrary. Right: Correspondingpowerspectraintherange0-1.5Hz,usingLeahynormalisation(see
text). The90%significancelevelforamono-frequencysignalisatapowervalueof17foreachfilter.
4 c/s)Count rate (unit 10 122802 122802 122802 122802
2.500 2.505 2.510 2.515 2.520 2.525 2.530 2.535
mplitude 1.02 P=1.692 s 1.02 P=1.852 s 1.02 P=2.306 s 1.02 P=2.254 s
malized A 0.981 0.981 0.981 0.981
Nor
0.0 Power0.5 1.0 1.5 2.0 0.0 Power0.5 1.0 1.5 2.0 0.0 Power0.5 1.0 1.5 2.0 0.0 Power0.5 1.0 1.5 2.0
4000
3000
wer 500 500 2000 500
Po
1000
0 0 0 0
0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5
Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz)
Fig.11: Top: r’lightcurveoffourcontiguous52.4sdatasegmentswithnooverlap(coveringtheorbitalphaseinterval2.5to2.535
with no interruption). Middle: Mean pulse shape. The QPO light curves are folded at the best sinusoidal period indicated and
repeated twice for clarity. Bottom: Corresponding power spectra using Leahy normalisation. The 90% significance level is at a
powervalueof17. TheY-scaleforpowerspectraisthesameforallsegmentsexceptthethird.
4.5. QPOtrainsofoscillations power in r’ and in He. Clear individual oscillations are seen in
ther’andHelightcurves,butaremuchlessvisibleinu’. When
The high quality of the ULTRACAM data obtained at VLT al- superimposed, the r’ and He light curves are similar in shape,
lows us to isolate data segments where flux oscillations are di- andcross-correlationshowsnosignificanttimelag. Theassoci-
rectlyvisibleinthelightcurve. Thisallowsadetailedstudyof ated power spectra are also shown in Fig. 10. To evaluate the
theQPOfrequenciesaswellasthepulseshapeoftheoscillations significance, the power is normalised here according to Leahy
thatcould,inprinciple,becomparedtonumericalsimulations. etal.(1983),withP =2(a)2/N ,wherea istheFourierampli-
i i γ i
Forillustration, weshowinFig.10, a52.4ssegmentofthe tudeatfrequencyν andN thetotalcounts. Withthisnormal-
i γ
light curves for the three filters obtained at an orbital phase isation, the power is independent of the source counts and the
φ = 1.273. This part is chosen as having the most significant
Articlenumber,page8of14
M.Mouchetetal.: QPOvariabilitiesinV834Cen
meanvalueofaPoissoniannoiseisequalto2. Allspectrashow 5.1. Energydistributionoftheoscillations
asimilarcomplexshapewithmultiplesignificantpeaks,indicat-
According to the shock instability model, optical QPOs origi-
ing a superposition of independent oscillations. The r’ and He
nate from the cyclotron emission of the accretion column and
filters shows the same main peaks at frequencies 0.40 and 0.48
should scale with this emission. However, contrasted with the
Hz,whilepeaksintheu’filterareat0.48and0.53Hz. Wenote,
hardX-raysrangewherethebremsstrahlungemissionfromthe
however,thattheu’peaksareonlymarginallysignificant. With
columnisthedominantprocess,inpolarsystemstheopticalcy-
the adopted Leahy normalisation, the 90% confidence level for
clotronfluxmaybestronglydilutedbyothercontributions. For
apuresinusoidal(mono-frequency)signalisatapowerlevelof
example,theWDheatedpolarcap,theaccretionstream,orthe
17.
secondarystaralsoradiateintheoptical. ThetrueQPOpulsed
To derive a mean pulse shape, the data were folded at the
fractionshouldthereforereferonlytothecyclotronfluxandnot
mean best fit r’ QPO period (2.556s) determined from a stan-
dardχ2 test. Thelightcurvesfoldedwiththismeanperiodand to the total flux. As our observations are not flux calibrated, to
determine the absolute energy spectrum of the QPOs we first
normalised by their mean values are shown in Fig. 10. They
makethereasonableassumptionthatthesourcespectrumduring
show an almost sinusoidal shape and the best sinusoidal fit for
ourobservationissimilartothatobservedinanequivalenthigh
each filter is displayed as a thin line. In this interval, the com-
statebyMiddleditchetal.(1991). Forthisconversion,wemake
putedamplitudesforr’,He,andu’arerespectively4.30±0.26%,
the approximation that our two u’ and r’ filters are not signifi-
4.25±0.45%, and1.13±0.64%(errorbarsat90%confidence
cantlydifferentfromtheUandRcinMiddleditchetal.(1991),
level).
and indeed the estimated mean magnitudes for r’ (15.0) and u’
Closerinspectionofthe52.4sr’lightcurverevealsthatthe
(15.8) (see Sect. 2) are in agreement with their U and Rc light
individual∼2.5soscillationsarenotregular,eitherinshapeor
inamplitudeandduration.Forinstance,thetwofirstoscillations curves. TheBfilter,centredon4400Å,isalsoclosetothecen-
inFig. 10arestrong,butthethirddoesnotdroptoitsminimum, tralwavelengthofourHefilter(4662Å).
andsubsequentpulsesarenotwelldefined. Inaddition,thetwo The non-cyclotron contribution to the optical flux was then
firstpeaksareseparatedbyashortertimethantheaveragebest evaluatedfromthespectrumreportedforthesourceinlowstate
periodof2.556s(0.391Hz). Ashiftisalsoseenatorbitalphase byPuchnarewiczetal.(1990),includingmainlythecontribution
1.2744andapulseisessentiallymissingatphase1.2757. ofthetwostars. Thedifferencebetweenthehigh-andlow-state
Significant variations are also often present from one data spectra should give a reasonable estimate of the accretion col-
segmenttothenextandtherearesomecaseswherenarrowpeaks umncyclotronemissionalone. Bysubtractingthenon-cyclotron
appear or disappear on timescales of the order of the segment flux,thefractionalQPOamplitudeinthedifferentfiltersisthen
duration. As an example, Fig. 11 shows the light curves and revisedas2.58%,2.00%,and0.73%respectivelyforther’,He,
powerspectraforther’filterinfourconsecutive52.4sdatain- andu’filters. Inthecontextofshockradiativeinstabilities, the
tervalswithnooverlap. Complextrainsofoscillationsareseen QPOamplitudesareexpectedtobeaconstantfractionofthecy-
in the first two intervals followed by a highly coherent single- clotronfluxateachwavelength,aresultthatdifferssignificantly
frequency oscillation with large amplitude in the third segment from the observed amplitudes. However, in the narrow He fil-
andthendegradedagaininamorecomplexsuperpositioninthe ter,thefluxisalsoaffectedbythecontributionofthebroadHeii
lastinterval.Allmajorpeaksinthepowerspectraarewellabove λ4686emissionline. Avalueof37Å fortheequivalentwidth
the99%detectionlevelatapowervalueof22. Thistypeofbe- of the line has been reported during high states (Rosen et al.
haviouroccursconsistentlythroughouttheobservation. 1987). Asthefilterwidthisabout110Å,thelinecontributionto
Figure 12 shows the 2D time-frequency image correspond- thetotalfluxinthefilteristhen∼25%. Ifthecontinuumalone
ingtothis∼210sinterval. Tobettershowthetemporalevolu- isoscillating,therelativeamplitudeofQPOsmustthereforebe
tion,sliding52.4sFFTshavebeencomputedwitha1.28sstep, corrected by a factor (1/0.75), leading to a corrected value of
resultingina97%overlap. Theevolutionofthepowerspectrum ∼2.68%. Takingintoaccounttheuncertaintyontheexactvalue
isclearlyseenwithcomplexmulti-peaksofequivalentstrengths of the line equivalent width at the time of the observation, this
spreadoverawiderangeoffrequenciesandprogressivelyevolv- value appears sufficiently close to the r’ value so that only the
ing into a narrower frequency range as time progresses. Some continuumisprobablyresponsiblefortheoscillations. Contrary
significantpowerisalsoseenclosetotwicethemainfrequency towhatissuspectedforANUMa(Bonnet-Bidaudetal.1996),
inthelastpartoftheinterval. Atthetoptheoscillationsinthe theHeii4686Å emissionlinedoesnotappearheretocontribute
sourcefluxarealsoshown,normalisedbydividingbya5smov- appreciably.
ingaverageandapplyinga0.5ssmoothing.Theydisplayimpor- QPO amplitudes are thought to be a direct measure of the
tantvariationsinshapeandamplitude. purecyclotronenergydistribution.Figure13showstheQPOen-
ergydistributionandalsotheexpectedaccretionfluxdefinedas
thedifference(high-low)state. Alsoshownisthereconstructed
5. Discussion
cyclotronemissionassumingaconstantfractionof2.58%asob-
Quasi-periodicoscillationsweredetectedthroughoutour∼5.6h served in the r’ filter. The reconstructed distribution is roughly
observationcovering3.3orbitalcycles. Previously, oscillations compatiblewiththeexpectedcyclotronemissionfortheHefil-
werealsonearlyconstantlyobservedduringthesourcehighstate ter,butalargediscrepancyexistsintheu’filter. Thisdifference
(Larsson 1985, Middleditch et al. 1991, Larsson 1992). The can be explained if, in addition to the column, other sources
rangeofQPOfrequency(0.3-0.8Hz)isalwayssimilar,whilethe of light also contribute at this short wavelength, as noted ear-
QPOamplitudes,expressedintermsoffractionofthetotalflux, lier (Sambruna et al. 1991). Among the possible contamina-
varysomewhatfromdatetodateandalsowithwavelength. Our tions is the heated surface of the white dwarf surrounding the
observationsarethefirsttoprovidesimultaneousQPOmeasure- base of the column. Assuming a 18 eV blackbody spot as de-
mentsatthreedifferentopticalwavelengths,withhighsignal-to- rived from EUVE high-state observations (Mauche 2002), the
noise ratio, and therefore provide interesting constraints on the additionalfluxwillrequireaspotwithafractionalemittingarea
energydistributionoftheoscillations. f ∼ 0.02 with respect to the white dwarf surface. This is sig-
Articlenumber,page9of14
Fig. 12: Bottom: Two-dimensional time-frequency image of the rms fractional amplitude in a 210s time interval for the r’ filter
(sameintervalasinFig.11).TheX-axisistheorbitalphaseandtheY-axisisthefrequencyintherestrictedrange0-1Hz,increasing
fromtoptobottom.Colour-codedrmsamplitudeisshownaccordingtothescaleatright(inunitsof10−6rms2.Hz−1).Sliding52.4s
FFTshavebeencomputedwitha1.28sstep. Top: Correspondingfractionalr’lightcurveshownafternormalisationbydividing
theoriginaldatabya5smovingaverageandaftera0.5ssmoothingtobettershowtheshapeofoscillations.
nificantlylargerthanthetypicalvalue f ∼ 0.001deducedfrom bynomeansunique,itshowsthattheobservedtrainsofpulses
the EUVE observations and may indicate more dilution in this can result from the superposition of individual damped pulsa-
bandbyothercontributionssuchastheBalmerdiscontinuityin tions.
emissionortheheatedhemisphereofthecompanion.
Withsuchbehaviour,thenotionofpulsecoherencyishardto
5.2. Oscillationsfromsimulateddata define. Ifthecoherencyisdefinedastheratioν/δν,whereδνis
thepeakwidthinthepowerspectra,thenitisclearthatonlong
The shape of the oscillations and the characteristics of the
timescalesthecoherencyisalwaysverylow,withavalueclose
powerspectra,mostoftenshowingmultiplepeaksevenonshort to∼2derivedfromthemeanQPOcharacteristics(seeTable1).
timescales, are suggestive of a superposition of modes. We at-
Even on shorter time intervals, narrow peaks are only seen to
tempted to reproduce the trains of pulses by performing sim- persistfornotmuchmorethanoneofourtypical∼52.4sinter-
ulations of light curves. We chose to add damped sinusoids vals,leadingtoaν/δνmaximumcoherencyof∼25.
withfrequencies,amplitudes,phases,anddampingtimechosen
randomlyfromGaussiandistributionsaroundvaluesandwidths
typical for QPOs. The start times of the sinusoids were also Inthisrespect,wenotethatthedescriptiongiveninLarsson
randomly chosen. We compare these simulations with selected (1992)issomewhatmisleading. Larsson(1992)chosetodefine
portions of the smoothed r’ light curve obtained after dividing coherencyasaratio P/|P˙|where P˙ issupposedtoresultfroma
theoriginaldatawitha5smovingaverageandapplyinga0.5s regularfrequencyshiftwithtime. Withthisdefinition, Larsson
smooth to the result. Figure 14 shows a typical simulation in a (1992)determinedacoherencyofP/<|P˙|>≈620s. Wedonot
157sinterval(3x1024points). Detailsofthe15inputsinusoids observe any regular systematic frequency drift in our data, nor
are shown (top), as well as the resulting light curve (middle) could we define a significant P˙ value. In fact, Larsson (1992)
comparedtoadatainterval(bottom)wherethemostsignificant also mentioned that variations in P˙ were erratic resulting in an
oscillationsareseen,withamplitudereaching∼10%(atorbital averagevalueofzero. Thecoherencydefinedasν/δνseemsto
phaseφ=1.08−1.11).Similarfeaturesarevisibleinbothsimu- be a much more reliable measure, and it indicates that pulses
lationanddatawithastrongmodulationoftheQPOamplitude. are erratically variable in amplitude and frequency on a typical
The resulting mean power spectra for data and simulations are maximum timescale of ∼ 20s. Inspection of Fig. 14 indeed
also structured with several similar peaks in the same range of revealsthattrainsofpulseshavetypicaldurationof(10-20)sor
frequencies. Though this reconstruction is rather arbitrary and (5-10)cycles.
Articlenumber,page10of14