On the Nature of X-ray Variability in Ark 564

13 Pages·2002·0.92 MB·English

Astronomy & Astrophysics manuscript no.
(will be inserted by hand later)
On the Nature of X-ray Variability in Ark 564
1,2 3 3 4,5 6 3
M. Gliozzi , W. Brinkmann , C. R¨ath , I.E. Papadakis , H. Negoro , and H. Scheingraber
1 George Mason University, Department of Physics and Astronomy, MS 3F3, 4400 University Dr., Fairfax, VA
22030-4444
2 Max-Planck-Institut fu¨r extraterrestrische Physik,Postfach 1312, D-85741 Garching, Germany
3 Centre for Interdisciplinary Plasma Science, Max-Planck-Institut fu¨r extraterrestrische Physik, Postfach 1312,
D-85741 Garching, Germany
4 Foundation for Research and Technology{Hellas, 711 10 Heraklion, Crete, Greece
5 Physics Department,University of Crete, 710 03 Heraklion, Crete, Greece
6 Cosmic Radiation Laboratory, RIKEN,2-1 Hirosawa, Wako{shi, Saitama 351-0198, Japan
Received: ; accepted:
Abstract. We use data from a recent long ASCA observation of the Narrow Line Seyfert 1 galaxy Ark 564
to investigate in detail its timing properties. We show that a thorough analysis of the time series, employing
techniquesnot generally applied to AGN light curves, can provide useful information to characterize theengines
ofthesepowerfulsources.Wesearchedforsignsofnon{stationarityinthedata,butdidnot(cid:12)ndstrongevidences
for it. We (cid:12)ndthat theprocess causing thevariability is very likely nonlinear, suggesting that variability models
based on many active regions, as the shot noise model, may not be applicable to Ark 564. The complex light
curve can be viewed, for a limited range of time scales (as indicated by the breaks in the structure and power
density spectrum), as a fractal object with non{trivial fractal dimension and statistical self{similarity. Finally,
using a nonlinear statistic based on the scaling index as a tool to discriminate time series, we demonstrate that
thehighandlowcountratestates,whichareindistinguishableonthebasisoftheirautocorrelation,structureand
probability density functions, are intrinsically di(cid:11)erent, with thehigh state characterized by higher complexity.
Key words.Galaxies: active { Galaxies: fundamental parameters { Galaxies: nuclei { X-rays:galaxies
1. Introduction
this technique uses only the (cid:12)rst two moments (namely
themeanandthevariance)oftheprobabilitydistribution
Active Galactic Nuclei (AGN) are variable in every ob-
functionassociatedwiththephysicalprocessesunderlying
servablewaveband.TheX-rayfluxexhibitsvariabilityon
the signal. However, only a Gaussian distribution can be
time scales shorter than any other energy band, indicat-
completely describedby the (cid:12)rsttwo moments, andthere
ing that the emission occurs in the innermost regions of
are several indications that probability density functions
the central engine. Therefore, a study of the X-ray vari-
associated with X-ray light curves both in AGN and in
ability provides an additional powerful tool to probe the
Galacticblackholesystems,arenotGaussian(e.g.Leighly
extreme physical processes operating in the inner parts
1999, Greenhough et al. 2002). Finally, the light curves,
of the accretion flow close to the accreting black hole.
i.e. the starting point for any kind of temporal analysis,
Although X-ray variabilityhas been observedin AGN for
have been often considered as a by-product of the spec-
morethantwodecades,itsoriginandnatureisstillpoorly
tral analysis, which still catalyzes most of the attention
understood. Until recently, there were no su(cid:14)ciently long
and e(cid:11)orts, in particular now that high resolution X-ray
observations of AGN with good signal to noise to allow spectroscopy of AGN is possible thank to Chandra and
a thorough analysis. Secondly, the temporal analysis is XMM-Newton. Spectral analysis proved to be very useful
frequently limited to calculations in the Fourier domain
inprovidingconstraintsonthe physicalparametersofthe
of the power density spectrum (or, equivalently, in the
accretion flow around black holes but it must be pointed
time domain of the structure function) which, although
out that, due to low signal-to-noise data, spectral models
useful for detecting possible periodicities or typical time
areoftenappliedtotime-averagedspectra,despitethefact
scales in a signal, does not exploit all the potential in-
thatsourcesshowrapidvariability.Thustheadditionalin-
formation contained in a time series. The reason is that
formation provided by timing observations can be crucial
to break the degeneracy among spectral models.
Send offprint requests to: [email protected]
2 M. Gliozzi et al.: Nature of X-ray variability
Fig.1. ASCA SIS0 light curve of Ark564. The dot{dashed
thick line divides the light curves in two halves used to inves-
tigate stationarity. The dashed lines de(cid:12)ne two intervals used
to characterize the timing behavior during the low and high
count rate states.
One ofthe mostcriticalopenquestions,relatedto the
X-rayvariabilityinAGN,concernsthe natureofthe vari-
ability: is it linear or nonlinear? In a mathematical sense
linearitymeansthatthevalueofthetimeseriesatagiven
time can be written as a linear combination of the values
at previous times plus some random variable. A positive
detection of nonlinearity would have an immediate and
very important consequence for the modeling of the re- Fig.2.ASCASIS0lightcurvesofArk564duringthehigh(top
gionproducingX-rays:allthe variabilitymodelsbasedon panel)andthelow(bottompanel)countratestate.Timebins
many independent active regions,as the shotnoise model are 16s. Both light curves contain 10000 data points each.
(Terrel 1972) or magnetic flares (e.g. Galeev et al. 1979)
wouldberuledoutinfavorofinherentlynonlinearmodels was at a high and low count rate state, respectively. We
as the self-organized criticality disk model (Mineshige et use a new technique basedon the constrainedrandomiza-
al. 1994) or the emission of X{ray radiation from a puta- tion of a time series and the nonlinear prediction error as
tive jet. an indicator of nonlinearity, in order to test the hypothe-
In an attempt to answer this question and, more gen- sis that the light curve is a realizationof a linear process.
erally,toinvestigatethenatureoftheX-rayvariability,we In Sect. 5 we (cid:12)rst investigate the variability behavior of
focus on a prominent object of a particular class of AGN, thesource,utilizingbothstandard(excessvariance,prob-
the Narrow-Line Seyfert 1 galaxies (NLS1), which often abilitydensityfunction)andnon{standard(fractals)tech-
display rapid, large amplitude X-ray variability as well niques. Then, in order to investigate further whether the
as extreme long-term changes (Forster & Halpern 1996, behavior of the source is identical in the two states (high
Boller et al. 1997, Brandt et al. 1999), and therefore rep- and low) we used techniques like the phase space recon-
resentthe idealobjectsforanX-raytemporalanalysis.In struction and the scaling index method. Finally, in Sect.
particular we analyze a recent long ASCA observation of 6 we draw the main conclusions.
Ark 564, the brightest NLS1 in the 2-10 keV band, us-
ing non-standard (at least for X-ray astronomy) analysis 2. Data
techniques.
Theoutlineofthispaperisthefollowing.InSect.2we Ark 564 was observed by ASCA from 2000 June 01
describe the data used for the timing analysis. In Sect. 3 11:51:27to2000July0523:57:54.WeextractedtheX-ray
we investigatethe important issue of the stationarity of a data from the public archive at ISAS and applied stan-
timeseries,bydividingthelightcurveintotwoequalparts dard criteria for the data analysis in order to create light
andbycomputingandcomparingthemean,the variance, curves. Source counts were accumulated from a (cid:24) 80(cid:2)80
theauto-correlationandstructurefunctionsandthepower rectangle in detector coordinates to maximize the X-ray
spectraofthetwohalves.Sect.4dealswiththesearchfor signal for the SIS0. The background was estimated from
nonlinearity.To this end, we choosetwo parts of the light the source free region on the same chip and its level was
curve with reasonablelength ((cid:24) 4 days),when the source less than 3% of the source count rate. During the analy-
M. Gliozzi et al.: Nature of X-ray variability 3
sis it became apparent that all instruments su(cid:11)ered from Table 1. Statistical comparison between (cid:12)rst and second half
unexpected light leaks.This is probably due to an expan- of thelight curve
sionoftheatmospherecausedbyrecentsolaractivityand
to the low satellite orbit during the observation. We in- Property First half Second half
vestigated this light leak e(cid:11)ect using energy spectra, low
energy light curves, and DFE curves for each of the indi- Mean 1.893±0.004 1.727±0.003
vidualdatasets(eachFRF(cid:12)le).Wefoundthatinthedata Variance 0.600±0.004 0.440±0.003
screening the BR EARTH criterion for the SIS data had to
bechangedfrom20o to35o forthe0.55{10keVdata.The Skewness 0.663±0.013 0.625±0.013
background-subtractedlightcurveisplottedinFig.1.The Kurtosis 0.732±0.026 0.696±0.025
reasonfortheselectionoftwosub-intervalswithhighand
lowmeancountrateisexplainedinSection3.Fig.2shows
a blow-up of the light curves during the high (top panel)
andlowcountratestate(bottompanel).Inthe following,
we restrict our analysis to the energy range 0.7{10 keV,
completely free of the light leak problem, and use data
1
only from the SIS0 detector, which is the most sensitive
(and therefore has the highest count rate).
3. Stationarity Check
F)
S
A time series, as any other scienti(cid:12)c measurement,has to g(
o
provideenoughinformationto allowthe determination of l
thequantityofinterestunambiguouslyanditisusefulonly
if reproducible, at least in principle. In other words, all
statisticalpropertiesderivedfromthe time seriesanalysis
should be invariant under a shift of the time origin. This
0
is the de(cid:12)nition of a completely stationary process.
However for most of the physical processes, in partic-
ular in astrophysics, this severe requirement is not appli-
cable and two weaker conditions (which nevertheless de- 1000 10000 100000 1000000
scriberoughlythesamephysicalbehavior)areapplied.As log(t) (sec)
(cid:12)rstrequirement,the time seriesshouldcovera stretchof
Fig.3. Structure function of Ark 564. The black error bars
time which is much longer than the characteristic time
(upper curve) correspond to the (cid:12)rst half of the light curve,
scale of the system. A second important requirement is
the gray ones (lower curve) to the second half. The dotted
that the properties of the system generating the signal
lines show the best (cid:12)tting \broken power law" model to the
must not change during the observation period. This can
data.
be checked in principle by measuring statistical proper-
ties, as the mean and the variance for the (cid:12)rst and the
second half of the data available and verifying that their curve can be entirely ascribed to the weak nonstationary
variations are within their statistical fluctuations. inherent in the red noise.
However, in case of AGN X-ray light curves, these re- The long light curve of Ark 564 allows an accurate
quirements are usually not ful(cid:12)lled for the following two investigation of the stationarity of the source. For this
reasons. First, X-ray observations of AGN have typically reason,wedividedthelightcurveintotwoequalparts,and
a duration ranging between a few hours to a few days (in computednotonlytheusualparameters,asthemeanand
the best cases). This time interval is usually su(cid:14)cient to the variance,but also the skewnessandkurtosis (listed in
accumulateenergyspectrawithgoodsignal-to-noiseratio, Table1)andestimatedthestructureandauto{correlation
but notto revealthe characteristictime scaleofanAGN. functions of the two halves.
Second,theAGNpowerdensityspectrumischaracterized TheerrorsquotedinTable1werecomputedassuming
by a \rednoise"power law componentP(f)/f−α (with that the data distribution is normal and the data are a
(cid:11)(cid:24)1:5; e.g. Lawrence & Papadakis1993), indicating the collection of independent measurements. However, in the
presence of temporal correlations. Therefore, even if the timeseriesconsidered(andingeneralinalltheAGNlight
parametersof the power density spectrum do not change, curves)themeasurementsarenotindependent,butcorre-
other parameters as the mean can change. However, as lated.Inthiscase,theactualerrors,whichmightbemuch
demonstratedbyPress(1978)and,morerecentlyforNLS1 larger than the errors computed as explained above, can-
galaxies,by Leighly (1999), the di(cid:11)erences in excess vari- not be easily quanti(cid:12)ed, since they depend on unknown
ance between the (cid:12)rst and the second half of the light quantities. For example, the actual error of the observed
4 M. Gliozzi et al.: Nature of X-ray variability
two halves of the light curve seems also to be reflected
in the di(cid:11)erence of the auto{correlation functions, esti-
mated by using the Discrete Correlation Function (DCF)
of Edelson & Krolik (1988). The rate at which the auto{
correlation function decays to zero may be interpreted as
a measure of the \memory" of the process and thus Fig 4
indicatesthatthe\memory"ofthesecondpartofthelight
curve is shorter than that of the (cid:12)rst part.
Likethemeanandvariance,thestructurefunctionand
autocorrelation function of the (cid:12)rst and second half of
the lightcurveshowdi(cid:11)erences.However,before reaching
the conclusion that the light curve of Ark 564 is non{
stationary, we should caution that it is very di(cid:14)cult to
determine appropriate errors for the structure and au-
tocorrelation functions and thus to assess the statistical
signi(cid:12)cance of the apparent di(cid:11)erence. (the reported er-
rors basically give information about how many points
contribute to each time scale.) We therefore tried to in-
vestigate further the stationarity using the power density
spectrum (e.g. Papadakis& Lawrence 1995) which, being
the counterpart of the structure function in the Fourier
Fig.4. Autocorrelation function of Ark 564. The (cid:12)lled circles domain,shouldprovidethe sameinformation.Infact,de-
correspondtothe(cid:12)rsthalfofthelightcurve,theopensquares tailed analysis of the power spectrum of the whole ASCA
to thesecond half. light curve of Ark 564 (Papadakis et al. 2002), together
with the results of Pounds et al. (2001), shows that the
power spectrum actually starts to flatten below (cid:24) 10−5
mean and variance depends on the true variance, on the Hz, in agreement with the structure functions plotted in
true autocorrelation function of the underlying process Fig 3. However, things are a bit more complicated when
and on the value of the power density spectrum at zero we divide the total light curve in two parts and compute
frequency (see Priestley 1989). Therefore, since the dif- theirpowerspectra.Thepowerspectrumisamuch\nois-
ference of the mean and variance between the two halves ier"function,whencomparedtothestructurefunction.As
couldbe the resultofthe factthatred-noiseprocessesare a result, signi(cid:12)cant binning of the \raw" power spectrum
weakly non-stationary, we consider this di(cid:11)erence as an is needed in order to investigate its shape. Without go-
indication, not as a proof, for stationarity. ing into details ofhow the powerspectrum wascomputed
To further investigate this issue we performed an (see Papadakis et al. 2002), we simply mention that for a
analysis based on the structure function (e.g. Simonetti timeseriesof15dayslength(correspondingtoaminimum
et al. 1985, Hughes et al. 1992), which has the ability to frequency of 7:7(cid:2)10−7 Hz), after the necessary binning
discern the range of time scales that contribute to the of the spectrum, the lowest accessible frequency is higher
variations in the data set. It is well known that structure than 10−5 Hz. Therefore, all information about the pos-
functions are a(cid:11)ected by the strong correlation between sible presence of a break above 105s is inaccessible and
the points in the light curve. As with other statistics in we cannot investigate if the location is di(cid:11)erent between
thetimedomain,likeautoandcrosscorrelationfunctions, the two parts of the light curve. Nevertheless, the overall
the points in the structure function are not independent, power spectra of the two parts are consistent within the
and the degree of their dependence is not easy to be de- errors, a result that argues in favor of the the light curve
terminedinadvance.However,thestructurefunctionsare being stationary.
not a(cid:11)ected by the presence of missing parts in the light Concluding, we can say that we (cid:12)nd no strong evi-
curves, therefore, their use can provide us with useful in- dence for non-stationarity. The mean, variance, structure
formationinthecaseofunevenlysampleddata,likeinour function and autocorrelation functions of the two parts
case.Forthatreason,wecomputedthestructurefunctions of the light curve do show di(cid:11)erences, but these could
forthe(cid:12)rstandsecondhalfofthelightcurve.Theresults be the result of the red-noise character of the observed
are plotted in Fig 3 (black and grey curve, respectively). variations. In fact, the power spectra of these parts show
Themostimportantdi(cid:11)erenceinFig3isthetimeatwhich nostatisticallysigni(cid:12)cantdi(cid:11)erences.However,sincenon-
the structure functions reacha plateau state. By (cid:12)tting a stationaritycana(cid:11)ecttheidenti(cid:12)cationofgenuinenonlin-
broken power law model to both structure functions, the earity present in the data, and because of the presence of
best(cid:12)tbreaktimescaleis5:6−+21..78(cid:2)104 sforthe(cid:12)rstpart indicationsfornon-stationarity(althoughnotstatistically
of the light curve and 1:8+1.2 (cid:2)104 s during the second signi(cid:12)cant) we decided not to utilize the whole data set
−0.8
interval(the errorscorrespondto a68%con(cid:12)dencelevel). foraninvestigationofthequestionoflinearityofthelight
This intrinsic di(cid:11)erence in the temporal properties of the curve. Instead, we chose two parts that are \locally" sta-
M. Gliozzi et al.: Nature of X-ray variability 5
A relatively simple but sensitive method to test the
presenceofnonlinearityinasignalisthesocalledmethod
ofsurrogatedata(e.g.Theiler etal.1992),whichisbased
onthefollowingideas.First,onehastoconceiveanullhy-
pothesis (for example, that the surrogates have the same
linear properties as the real data) that we want to test
against. Second, a level of signi(cid:12)cance for the test must
be speci(cid:12)ed (a test valid at 95% signi(cid:12)cance level means
that with a chance of (cid:11) = 0:05 the null hypothesis is er-
roneously rejected). Then one creates a number of sur-
rogate data according to the chosen null hypothesis (
Nsurr =1=(cid:11)−1arecreatedforaone{sidedtest,wherethe
nullhypothesisisrejectedifthedatadeviatefromthesur-
rogates in a speci(cid:12)ed direction, whereas Nsurr = 2=(cid:11)−1
arecreatedincaseofatwo-sidedtest,wherethe datacan
deviatefromthesurrogatesoneitherside).Finallyonehas
tocomparetherealdatawiththesurrogatetimeseriesus-
ing nonlinearstatistics.Ifthe realdata di(cid:11)er signi(cid:12)cantly
from the surrogates, nonlinearity can be inferred.
Standard surrogate methods (Theiler et al. 1992),
make use of the Fourier transformation to conserve the
auto{correlations of the original data: the original time
series is rescaled to a Gaussian distribution (cid:12)rst; then,
the Fourier phases are randomized and the Fourier trans-
formation is inverted. Finally, a rescaling to the original
Fig.5. Normalized real time series (top panel; mean = 0,
σ =1) for the high state of Ark 564 and corresponding surro- distribution is performed. However, such a method can-
gate time series (bottom), which preserves the linear proper- not be applied to unevenly sampled data (as the ASCA
tiesoftheoriginaldataasmean,varianceandauto{correlation light curve of Ark 564), because it utilizes Fourier trans-
function. formations and their inverses. Therefore, a more general
approach based on the constrained randomization of a
timeseries(Schreiber1998)wasused.Inthisnewmethod,
tionary (meaning that after dividing them in two halves, the desired properties (e.g. auto{correlations) of the sur-
nosigni(cid:12)cantdi(cid:11)erencewasfoundbetweenthemean,vari- rogatesareimposedbyconstraints:thesurrogatesstartas
ance, autocorrelation and structure functions of the (cid:12)rst a randomized distribution of the original data, and then
and second part), have reasonable length ((cid:24)4 days), and are annealed until they match the ACF of the original
have di(cid:11)erent mean count rates. The selected \high" and data,atwhichpointthey areacceptedassurrogates.The
\low"intervals,plottedinFig2,contain10000datapoints constraints are implemented as a cost function E (e.g.
each (using a bin size of 16 s) with mean count rates of Schmitz & Schreiber 1999), which is constructed to have
2:14(cid:6)0:01 cts s−1 and 1:32(cid:6)0:01 cts s−1, respectively. a globalminimum whenthe constraintis ful(cid:12)lled (in sim-
In this way, apart from checking whether the light curve ple words, if the constraint chosen is that surrogate data
is nonlinear or not, we can also test if the source behaves havethesameauto{correlationastherealdata,E willbe
di(cid:11)erently in di(cid:11)erent flux states. a function of the di(cid:11)erence between the auto{correlations
of real and surrogate data). Combinatorial minimization
by complete enumeration is not feasible in case of long
4. Search for nonlinearity timeseries,sincethecomputationale(cid:11)ortgrowsexponen-
tially with the length of the time series. Instead, we use
The search for nonlinearity in a time series is not only the so called simulated annealing (e.g. Metropolis et al.
importantbutunavoidableinorderto progressinthe un- 1953,Kirkpatricketal.1983),apowerfulmethodformin-
derstanding of the origin of X-ray variability in AGN. In imization in presence of many false (local) minima, that
fact, variouskinds of models (Gaussian,non-Gaussian,or isexpectedto(cid:12)ndanapproximatesolutioninpolynomial
nonlinear) have been proposed to explain the AGN vari- time.
ability.Allofthemareabletoexplainthered{noisepower Themainideabehindthemethodofsimulatedanneal-
spectrum and other temporal characteristics such as the ing is to interpret the cost function E as the energy in a
auto- and cross-correlation functions. However, it is im- thermodynamic system. Minimizing E then is equivalent
portantto stressthat neither the power density spectrum to (cid:12)nding the ground state of the system, and typically,
nor the auto-/cross-correlationor structure functions are foraglassysolid,thisstateisreachedby(cid:12)rstheatingand
abletodistinguishwhetheraprocessislinearornonlinear subsequently cooling it (a procedure called \annealing",
(e.g. Vio et al. 1992). hence the name of the method). In order to anneal the
6 M. Gliozzi et al.: Nature of X-ray variability
weuse suchsurrogatestotestfornonlinearityinthe orig-
inal data. In principle, any nonlinear statistic might be
used, however, many statistics useful for evenly sampled
timeseriescannotbeeasilygeneralizedtounevenlyspaced
data. First we tried a simple but rob(cid:80)ust statistic which
measures the time reversibility : γ / [yn−yn−1=(tn−
tn−1)]3, where yn denotes the count rate at time tn (e.g.
Schmitz&Schreiber1999).Foratimeseriesgeneratedby
a linear process, as in the case of surrogates, we expect
γ ’0.Sincethemeasureoftimereversibilityofatimese-
riesgeneratedbyanonlinearprocesscanbeeithersmaller
or bigger than zero, we have performed a two{sided test.
We found no statistically signi(cid:12)cant di(cid:11)erences between
Fig.6. Nonlinear prediction errors calculated from the surro-
real data and surrogates for both the low and high count
gate data (shorter lines) and from the real data (longer thick
ratestates,i.e.,the analysisdoesnotsupportthehypoth-
dashedline)duringthehighstate.Thehorizontalerrorbarin-
dicatesthestandarddeviationaroundthemeanofthestatistic esis of a non-linear time series.
computed from thesurrogates. However,itmustbepointedoutthatasymmetryunder
time reversal is a su(cid:14)cient but not a necessary condition
for nonlinearity. In addition, it has been demonstrated
system to the \ground state", that is, to the minimum
(Schreiber & Schmitz 1997) that, although the time re-
cost function, we (cid:12)rst \melt" the system at a high tem-
versibility is usually a good indicator of nonlinearity, in
perature T, and then decrease T slowly. In practice, we
cases of very noisy data it can fail completely. According
started with a random permutation of the original time
to Schreiber & Schmitz (1997), who systematically evalu-
seriesandde(cid:12)ned the starting costfunction in the follow-
atedthe abilities ofdi(cid:11)erentobservablesto detectnonlin-
ing way:
earity,oneofthebestindicatorsisthenonlinearprediction
(cid:40)(cid:88) (cid:41)1/2 error (e.g. Kantz & Schreiber 1997), which relies on the
E = [ACFreal(i)−ACFsurr(i)]2 ; (1) time delay embedding ofa scalartime series,described in
section 5.3.
i
The results of a one{sided test (the nonlinear predic-
where ACFreal is the binned auto-correlation function of tion error for a nonlinear time series is expected to have
the real data and ACFsurr of surrogates. The surrogate lower values compared to the case of a linear time series)
is successively modi(cid:12)ed by exchanging randomly chosen for the high count rate state are shown in Fig 6, where
pairs of elements and thus alsothe value of the costfunc- the value of the nonlinear prediction error for the origi-
tion changes. The modi(cid:12)cation will be accepted with a nal data is plotted with a longer thick dashed line, the
probability p = 1 if the energy decreases ((cid:1)E < 0) and meanandthe standarddeviationofthe statistic obtained
with p = exp(−(cid:1)E=T), where T is a free parameter, if from the surrogates are represented by a cross and an er-
(cid:1)E (cid:21) 0. Such updating scheme, proposed by Metropolis ror bar, respectively. It is evident that the original data
and collaborators (1953), allows the system to be close aresingledoutbytheestimateofthenonlinearprediction
to the \thermodynamic equilibrium" at each stage. To errorand,therefore,thenullhypothesisofalinearstation-
summarize: the simulated annealing is performed by ex- ary process has to be rejected at 97.5% signi(cid:12)cance level.
changing pairs of points (in the time domain) in each it- Applying the same procedure to the interval of the light
eration step. Therefore, the surrogates have exactly the curvecorrespondingto the low state ofArk 564,we reach
same distribution as the real data, i.e. they are \ampli- a similar conclusion: nonlinearity is detected, although at
tude adjusted".By keepingtheir autocorrelationfunction a slightly lower (95%) signi(cid:12)cance level.
close to that of the original data all linear properties of It must be noticed that also Edelson and collabora-
the initial time series are preserved. tors (2002) searched for nonlinearity in this source and
We repeated the whole procedure several times and claimed a weakly signi(cid:12)cant(less than 1.5 (cid:27)) detection of
created 39 independent surrogate data sets, necessary to nonlinearity. However, as the authors themselves remark,
perform a two{sided test for nonlinearity valid at 95% the methodthey used(the standardsurrogatemethod) is
signi(cid:12)cance level or a one-sided test at 97.5% signi(cid:12)cance not the best suited for the ASCA data of Ark 564.
level(seebelow).Fig5showsthecomparisonbetweenthe
realnormalizedtime series(normalizedtoamean=0and
5. Time Series Analysis of Ark 564
(cid:27) =1)duringthehighstateofArk564andacorrespond-
ing surrogate time series (bottom), which preserves the 5.1. Standard Analysis
linear properties of the original data like mean, variance
and auto{correlationfunction. Detailed temporal and spectralvariability analysesof the
After having produced randomized versions of un- 35 day ASCA observation of Ark 564, based on power
evenly sampled time series with given linear correlations, density spectra, cross{correlation functions are reported
M. Gliozzi et al.: Nature of X-ray variability 7
Fig.7. Autocorrelation function of Ark 564 during the high
((cid:12)lled circles) and low (open squares) count rate states. Note
that the range of the horizontal axis is a factor (cid:12)ve smaller
than that in Fig. 4.
elsewhere (e.g. Turner et al. 2001, Edelson et al. 2002,
Papadakis et al. 2002) and are beyond the scope of this
work. Here, we investigate whether there are di(cid:11)erences
in the variability behavior of the source during the low
andhighstate.Weusestandardtechniqueslikethe auto{
correlation function and excess variance analysis, linear
techniques that are not frequently applied to AGN light
curves like the probability density function and, (cid:12)nally, Fig.8. Probability density function, normalized with respect
fractal and nonlinear analysis techniques. to the average count rate < cts > and standard deviation σ
fortimeseries inthehigh (top)andlow(bottompanel)count
An analysis of the discrete auto{correlation functions
rate states.
(see Fig. 7), for the two chosen intervals seems to indi-
cate that the high state ((cid:12)lled circles) is characterizedby
a slightly shorter characteristic time scale than the low the average count rate <cts> and standard deviation (cid:27)
countratestate(opensquares),accordingtothenarrower duringthehighandlowstate(seeFig8).Thetwodistribu-
coreoftheformerACF.However,themostrelevantquan- tions look quite similar and di(cid:11)er from a Gaussian distri-
tity, the time lag at which the ACF decays to zero, is the bution (directconsequence oftime correlationsin the sig-
same ((cid:28) (cid:24)2:5(cid:2)104 s) for both states. nal): both are characterized by a rather normal left-hand
To compare the degree of variability during the high tail (related to the small amplitude events in the time se-
and the low state (Fig.(cid:112)2) we used the fractional variabil- ries) and a right-hand tail following a power law trend. It
ity amplitude Fvar = (S2−h(cid:27)e2rri)=hXi2, where S2 is is worth noticing that a qualitatively similar probability
the total variance, (cid:27)2 the mean error squared and hXi densityfunctionisdisplayedbytwoGalacticaccretingob-
err
the mean count rate (e.g. Edelson et al. 2002). We found jects,themicro-quasarGRS1915+105andthe blackhole
that during the low state the variabilityof the sourcewas candidate Cygnus X-1 and not by a non{accreting object
higher, Fvar =(0:373(cid:6)0:003),than during the high state as the Crab nebula (Greenhough et al. 2002).
(0:300(cid:6)0:002).Itisworthnoticingthatfromarecenttem-
poral analysis based on Fvar recently carried out on the 5.2. Fractal Analysis
ASCA light curve of Ark 564, Turner et al. (2001) found
no clear correlationbetween the fractionalvariability and Fractals are now widely used in the modeling and in-
the X-ray flux. terpretation of many di(cid:11)erent natural phenomena (see,
We have compared the probability density functions e.g.,Mandelbrot1982,Pietronero&Tosatti1986),includ-
for the time series of Ark 564 normalized with respect to ing astrophysical phenomena (see, e.g., Heck & Perdang
8 M. Gliozzi et al.: Nature of X-ray variability
Fig.9. Intervalof the light curveof Ark 564 shown with high
resolution (top panel)and low resolution (bottom).Both axes
are scaled bya factor of 5.
Fig.10. Flare of Ark 564 shown with increasing time resolu-
tion. The data points are connected with a line to show the
averaged flare shapes.
1991). In this section, we use methods tailored for fractal
analysis only as tools for extracting information from the
signal, without considering the dynamical implications of that in this case the scaling factor is limited not by in-
thefractality.Ourpurposeistogodeeperintothecompar- trinsicphysicalreasonsbutbyinstrumentalreasons(gaps
isonbetweenthehighandlowcountratestates,startedin due to earth occultation and low sensitivity). Moreover,
theprevioussectionwithstandardtechniques,byutilizing approaching the resolution limit, the omnipresent noise
conceptscharacteristicoffractalanalysis.Inparticular,af- component would hide the small-scale structures and the
tershowinginaratherqualitativewaythatthelightcurve fluctuations wouldbe washedoutby measurementerrors.
ofArk564canbeconsideredasafractalobject(character- It must be noted that fractal properties, as the frac-
ized by self-similarity and fractal dimension), we perform tal length and dimension of the X-ray light curve of an
aquantitativecomparisonoftheirfractalpropertiesbased AGN (the Seyfert 2 galaxy NGC 5506) were already in-
on the Hurst exponent (e.g. Kantz & Schreiber 1997) vestigated by McHardy and Czerny (1987), who derived
AGN lightcurves,sampledwith su(cid:14)ciently highreso- some constraints on the emission mechanisms producing
lution,canbeextremelycomplexandthescalarsignalcan X-rays.Here,ourmainaimisnottorepeatasimilarkind
beconsideredasafractal,inthesensethatitsgraph,asa ofanalysis,buttoinvestigatethesource’sbehavioratlow
function of time, has a nontrivial dimension. This means and high flux state, by comparing their fractal properties
that the length of the graph seen at a (cid:12)nite resolutionin- (i.e. their Hurst exponent, see below).
creases \forever" when the resolution on the time axis is Inadditiontoanontrivialdimension,fractalsarechar-
increased, since more and more fluctuations are resolved. acterized by self-similarity. A structure is said (strictly)
A priory there should not be a minimum time scale of self-similarifitcanbebrokenintoarbitrarilysmallpieces,
variability in an AGN, apart from causality reasons. For each of which is a small replica of the entire structure.
example, a process which is the result of adding expo- However, there are several variants of the mathematical
nential shots will have a power density spectrum with a de(cid:12)nition of self{similarity. Dealing with erratic signals
slope −2 extending towards in(cid:12)nite positive frequencies. typical of AGN X-ray light curves, we are mainly inter-
Itsdimensionis,however,notanintegerbutliesanywhere ested in the \statistical self-similarity" and self{a(cid:14)nity,
between one and two,depending on the resolution.To vi- wherethesmall\replica"maybesomewhatdistorted(for
sualize this concept let us consider a portion of the light example skewed) with respect to the whole.
curveofArk564scaledintimeandspacebythesamefac- In the case of the light curve of Ark 564 this prop-
tor(see Fig 9).Inthe limit ofhighresolution(top panel), erty can be evidenced by considering the same portion of
thegraphtendsto(cid:12)lltheplaneandapproachesdimension the light curve with di(cid:11)erent resolution factors (i.e. with
two,whereasinthe opposite limit, itdegeneratestowards di(cid:11)erent time binning). In Fig 10 we focus on one of the
a horizontal line and thus to a dimension of one. Note manyflareswhichoccurredduring the ASCAobservation
M. Gliozzi et al.: Nature of X-ray variability 9
0 (for functions constant over time) and 1 (for functions
increasing or decreasing linearly with time). Intermediate
values of H are generated by fractal functions, random
Gaussian noise (H ’ 0:2) and Gaussian random walk
(H ’ 0:5; typical for Brownian motion and normal di(cid:11)u-
sion process), whose increments possess a (cid:12)nite variance
forevery(cid:1)tandareuncorrelatedatsuccessivetimesteps.
It is worth noticing that very di(cid:11)erent processes (e.g.
hydrodynamicalturbulence, standardand anomalous dif-
fusion)areabletoproducesignalscharacterizedbyscaling
laws and that they can be discriminated on the basis of
their Hurst exponent. For example, this method has been
used to quantify solar magnetic complexity (Adams et al.
1997), and the persistence of solar activity (Lepreti et al.
2000). Fig 11 shows the growth of range of h(cid:1)ctsi for
Ark 564 during the high ((cid:12)lled circles) and the low (open
diamonds) state. The slopes, obtained from a linear least
square (cid:12)t, are very similar for the high (0:29(cid:6)0:03) and
the low state (0:28(cid:6)0:03), indicating that the processes
at work during the high and low state are of the same
nature.
Fig.11. Logarithmic plot of the growth of range (cid:104)(cid:1)cts(cid:105). The
(cid:12)lled circles refer to the high state while the diamonds to the A similar analysis was performed by Greenhough and
low state. collaborators (2002) on very long RXTE light curves of
three galactic objects: the Crab nebula, Cygnus X-1 and
the micro-quasar GRS 1915+105. Interestingly enough,
of Ark 564. It is shown that a rather smooth flare at low
only the latter, which is thought to be a scaled downver-
resolution (we started with a time binning equal to the
sion of an AGN, had a Hurst exponent (H ’0:3) consis-
orbital period) looks more and more complex when seen
tent with the values found for Ark 564.
with increasing time resolution:eachsingle flare observed
at higher resolution is composed of several sub{flares. It
canbe seenthat from1408s bins to 704s bins the points 5.3. Nonlinear Analysis
start to cluster, indicating that the amount of variabil-
ity on time scales shorter than 1000s is smaller. This is Linear methods intrinsically assume that the dynamics
also reflected in the power density spectrum (Papadakis of the system is governed by the linear paradigm: small
etal.2002),whichshowsabreaktoasteeperslopearound causes lead to small e(cid:11)ects. Since linear equations can
(cid:23) =10−3 Hz. The frequent gaps caused by earth occulta- onlyleadtoexponentiallygrowing(ordecaying)solutions
tion and the statistical fluctuations associated with mea- or to periodic oscillations, an irregular behavior of the
surementdonotallowtousesmallertimebinsinthescal- system has to be attributed to some random external in-
ingdownprocess.Itis worthnotingthatalsoTanihataet put.Howevernonlinear,chaoticsystemscanproducevery
al. (2001) noticed a similar behavior in a recent study irregular data with purely deterministic equations of mo-
on time variability of blazars: none of the observed large tion. In other words, the apparently irregular and unpre-
flares show a smooth rise or decay but exhibit substruc- dictable behavior typical of X-ray light curves of AGN
tures, with smaller flares having shorter time scales. is not necessarily due to stochastic dynamics (i.e the ac-
Statistical self-similarity, although only for a limited tion of a large number of excited degree of freedom). It
time interval (as indicated by the breaks in the structure might be also generated by chaotic dynamics of a limited
function and power density spectrum), seems to occur in- numberofcollectivemodes:dissipation,couplingbetween
dependently ofthe stateofthe source.Inorderto investi- di(cid:11)erent degrees of freedom and the action of an exter-
gate quantitatively the scaling behavior of the light curve nal (cid:12)eld may lead to collective behavior (e.g. Eckmann&
on short time scales (which are not accessible to visual Ruelle 1985). The presence of nonlinearity in the X{ray
inspection) during the high and low count rate states, we light curveof Ark 564 leavesopenthe possibility that the
have calculated, for di(cid:11)erent time intervals (cid:1)t (ranging light curve is produced by a chaotic system. However, it
from 32 to 1600 s), the di(cid:11)erence between the maximum must be noted that nonlinearity is a necessary but not
and the minimum count rates (cid:1)cts. Running this proce- su(cid:14)cient condition for a light curve to be produced by
dure overthe portions of the light curve correspondingto a chaotic system (Vio et al. 1992). In any case our main
the high and the low states for each (cid:1)t, an arrayof (cid:1)cts aimis nottodiscriminatechaosfromstochasticity,butto
is created from which the mean h(cid:1)ctsi is found. For self{ compare the source’s behavior in the low and high state
similar data h(cid:1)ctsi / (cid:1)tH, where H is called Hurst ex- using nonlinear statistics. The scaling index method, the
ponent(e.g.Kantz&Schreiber1997)andrangesbetween nonlinear statistics that we describe below, could be used
10 M. Gliozzi et al.: Nature of X-ray variability
Fig.13.SpectrumofthescalingindexdistributionofArk564
during thehigh (solid line) and low (dashed line) states.
X[ti+(cid:1)t],...,X[ti+(d−1)(cid:1)t], where d is the dimension
of the embedding space (the so called\embedding dimen-
sion"). The resulting d-dimensional arti(cid:12)cial phase space
represents all topological properties of the system in the
real phase space, as long as the embedding dimension is
twicethedimensionoftherealphasespace(Takens1981).
For illustrationpurposes we show in Fig 12 the three{
dimensional pseudo phase space portraits of the light
curve of Ark 564 during the high (top panel) and the low
Fig.12.3D{phasespaceportraitsoftheinvestigatedportions (bottom) state. Roughly speaking, the location of each
of the light curve of Ark 564, with 10000 data points each. point in the 3{D pseudo phase space (which is the sim-
The time delay used to generate the arti(cid:12)cial phase space is
plestcase,fora visualcomparisonbetweentwodatasets)
100timesthetemporalresolutionofthetimeseries.Theunits
is de(cid:12)ned by three data points in the time series, which
representthenumberofphotonscontainedinthesmallesttime
are separated by 100 and 200 steps, respectively (Xi+100
bin (i.e. in 16s).
means an element displaced by 100 positions in the time
serieswithrespectXi).Thetimedelayisalmostarbitrary.
even in the case of linear time series, since its ability to However, using too small time delays causes a clustering
discern underlying structures in noisy data is not directly ofthestatevectorsaroundthediagonal(duetothestrong
based on the nonlinear nature of the underlying process. time correlations existing between successive elements of
When methods of nonlinear dynamics are applied to the vectors) and thus hamper the discrimination between
time series analysis, the concept of phase space recon- the two data set under examination. On the other hand,
struction represents the basis for most of the analysis the choice of a large delay value would reduce the num-
tools. In fact, for studying chaotic deterministic systems ber of points in the phase space, lowering the statistical
it is important to establish a vector space (the phase signi(cid:12)cance of the analysis. Note that discontinuities in
space) such that specifying a point in this space speci- the time series due to earth occultation do not represent
(cid:12)es a state of the system, and vice versa. However, in a problem for this kind of analysis, as far as the number
order to apply the concept of phase space to a time se- of data points is su(cid:14)ciently large for the reconstruction.
ries, which is a scalar sequence of measurements, one has In order to quantify the di(cid:11)erence between the phase
to convert it into a set of state vectors. This is called space portraits in the high and low states, a suitable
phase space reconstruction and it is technically solved quantity is the correlation integral C(r), which basically
via the method of time delays and embedding. Starting countsthenumberofpairsofpointswithdistancessmaller
from the originaltime series X[ti] and introducing a time than r. This algorithm was introduced by Grassberger &
delay (cid:1)t, one can construct d − 1 additional data sets Procaccia (1983) and was (cid:12)rst used in astronomy to an-