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Statistical analyses of long-term variability of AGN at high radio frequencies

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T. Hovatta, M. Tornikoski, M. Lainela, H. J. Lehto, E. Valtaoja, I. Torniainen, M. F.Aller, and H. D. Aller. 2007. Statistical analyses of long-term variability of AGN athigh radio frequencies. Astronomy & Astrophysics, volume 469, number 3, pages899-912.© 2007 European Southern Observatory (ESO)Reprinted with permission.http://dx.doi.org/10.1051/0004-6361:20077529
A&A 469, 899–912 (2007)DOI: 10.1051
/
0004-6361:20077529
c
ESO 2007
Astronomy
&
Astrophysics
Statistical analyses of long-term variabilityof AGN at high radio frequencies
T. Hovatta
1
, M. Tornikoski
1
, M. Lainela
2
, H. J. Lehto
2
,
3
, E. Valtaoja
2
,
3
, I. Torniainen
1
, M. F. Aller
4
, and H. D. Aller
4
1
Metsähovi Radio Observatory, Helsinki University of Technology, Metsähovintie 114, 02540 Kylmälä, Finlande-mail:
tho@kurp.hut.fi
2
Tuorla Observatory, University of Turku, Väisäläntie 20, 21500 Piikkiö, Finland
3
Department of Physics, University of Turku, 20100 Turku, Finland
4
Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USAReceived 23 March 2007
/
Accepted 23 April 2007
ABSTRACT
Aims.
We present a study of variability time scales in a large sample of Active Galactic Nuclei at several frequencies between 4.8 and230 GHz. We investigate the di
ﬀ
erences of various AGN types and frequencies and correlate the measured time scales with physicalparameters such as the luminosity and the Lorentz factor. Our sample consists of both high and low polarization quasars, BL Lacertaeobjects and radio galaxies. The basis of this work is the 22 GHz, 37 GHz and 87 GHz monitoring data from the Metsähovi RadioObservatory spanning over 25 years. In addition, we used higher 90 GHz and 230 GHz frequency data obtained with the SEST-telescope between 1987 and 2003. Further lower frequency data at 4.8 GHz, 8 GHz and 14.5 GHz from the University of Michiganmonitoring programme have been used.
Methods.
We have applied three di
ﬀ
erent statistical methods to study the time scales: the structure function, the discrete correlationfunction and the Lomb–Scargle periodogram. We discuss also the di
ﬀ
erences and relative merits of these three methods.
Results.
Our study reveals that smaller ﬂux density variations occur in these sources on short time scales of 1–2 years, but largeroutbursts happen quite rarely, on the average only once in every 6 years. We do not ﬁnd any signiﬁcant di
ﬀ
erences in the time scalesbetween the source classes. The time scales are also only weakly related to the luminosity suggesting that the shock formation iscaused by jet instabilities rather than the central black hole.
Key words.
galaxies: active – methods: statistical
1. Introduction
Long term multifrequency monitoring data of a large sample of Active Galactic Nuclei (AGNs) provides an e
ﬃ
cient means forstudying the physical processes behind the variability behaviourof individual objects. It is also a useful tool for studying di
ﬀ
er-ences between various AGN classes. Hughes et al. (1992) usedthe structure function (SF) to study the time scales of variabilityin a large sample of sources at frequencies 4.8, 8 and 14.5 GHz,using data from the University of Michigan monitoring pro-gramme. The SF was also used by Lainela & Valtaoja (1993),hereafter Paper I, to study the time scales of the Metsähovi mon-itoring sample. Since then the amount of data has more thantripled. In this paper we analyse the updated extensive databaseand compare the results to Paper I.The discrete correlation function (DCF) and theLomb–Scargle periodogram (LS-periodogram) were alsoused to search for the variability time scales at several frequencybands. Both of these have been used previously to study peri-odicities and time scales in individual sources (e.g. Villata et al.2004; Ciaramella et al. 2004; Raiteri et al. 2003, 2001; Royet al. 2000). Aller et al. (2003) studied the Pearson-Readhead
Table 5 is only available in electronic form at
http://www.aanda.org
extragalactic source sample, monitored at the University of Michigan, by using the LS-periodogram without ﬁnding anysigniﬁcant periodicities.We have used these methods to look for the typical ﬂare oc-currence rates and other variability properties of this large sam-ple of sources. By using more than one method we hoped toensure that the time scales obtained are real. Furthermore, theexperience we gain in using di
ﬀ
erent methods will let us choosein the future the proper methods for speciﬁc analysis needs.The DCF was used to study the radio-optical correlations in theMetsähovi monitoring sample by Tornikoski et al. (1994) andHanski et al. (2002).In thepresentpaperwe have,however,usedthe autocorrelationat each radio frequencyband instead of crosscorrelation between di
ﬀ
erent frequencies.The long term variability time scales can tell us about howoften certain objects, or certain classes of objects, are in a ﬂaringstate and how long do these ﬂares typically last. We also learnabout ﬂare evolution from one frequency domain to the other.This helps us in developingradio shock models. The knowledgeof typical ﬂare time scales is also importantfor the workdoneonextragalactic foreground sources for the ESA Planck Surveyorsatellite
1
mission, to be launched in 2008. The Planck satellitewill be used to study the cosmic microwave background (CMB)
1
http://www.rssd.esa.int/Planck
Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20077529
900 T. Hovatta et al.: Statistical analyses of long-term variability of AGN at high radio frequencies
emission, and all the foregroundsources, including AGNs, mustbe removed from the results. Therefore it is important to under-stand the characteristic time scales of AGN variability at highradio frequencies. This paper is part of our broader study of var-ious AGN types that a
ﬀ
ect the foregroundof Planck.This paper is organisedas follows: in Sect. 2 we describe thesource sample and the data. The methods used for the analysesare described in Sect. 3 and the results are presented in Sect. 4.Finally, we will discuss the scientiﬁc outcome of the results inSect. 5. In Sect. 6 we will draw the conclusions.
2. The sample and observations
The sample consists of 80 sources which have been selectedfrom the Metsähovi monitoring sample. We included objects forwhich we have data from a time window of over 10 years in atleast two frequency bands. However, most of the sources havebeen monitored up to 25 years, enabling a search for longertime scales. Sources which are included in the monitoring listare bright sources with a ﬂux density of least 1 Jy in the ac-tive state. The present sample is shown in Table 5. The columnsshow the observing frequency, the length of the time series andthe number of points for each source.The sample consists of di
ﬀ
erent types of AGNs. 24 of thesources are BL Lacertae objects (BLOs), 23 Highly PolarizedQuasars (HPQs), 28 Low Polarization Quasars (LPQs) and5 Radio Galaxies (GALs). Quasars are considered to be highlypolarized if their optical polarization has exceeded 3 percent atsome point in the past. It is possible that some of the low polar-ized objects are in reality HPQs but they have not been observedinanactivestate.For5objectswehadnoinformationabouttheiroptical polarization, and they are considered LPQs in this study.When we study the statistical di
ﬀ
erences of the various groups,only BLOs, HPQs and LPQs will be considered, because of thelow number of GALs in our sample. The variability of BLOswill also be studied in more detail in a forthcoming paper byNieppola et al. (2007, in preparation for A&A).The core of this work is the monitoring data from the 14 mMetsähovi Radio Telescope. We have been monitoring a sampleof AGNs for over 25 years at frequencies 22, 37 and 87 GHz(Salonen et al. 1987; Teräsranta et al. 1992, 1998, 2004, 2005).Our study also includes unpublished data at 37 GHz fromDecember 2001 to April 2005. The data for BL Lacertae objectsfrom this period are published in Nieppola et al. (2007). Theobservation method and data reduction process are describedin Teräsranta et al. (1998). The Swedish-ESO SubmillimetreTelescope (SEST) at La Silla, Chile, was used in our monitor-ing campaign to sample the high frequency, 90 and 230 GHzvariability of southern and equatorial sources (Tornikoski et al.1996;Tornikoskiet al. 2007,inpreparationforA&A).The mon-itoring campaign at SEST lasted from 1987 to 2003. High fre-quency data at 90 and 230 GHz were also collected from theliterature (Steppe et al. 1988, 1992, 1993; Reuter et al. 1997).The lower frequency data at frequencies 4.8, 8 and 14.5 GHzwere providedby the Universityof MichiganRadio Observatory(UMRAO) monitoring programme. Details of calibration anddata reduction are described in Aller et al. (1985). We had suf-ﬁciently well-sampled 230 GHz data for our analysis for only7 sources and therefore that frequency band is not used whenaverage time scales and di
ﬀ
erences between the source classesare studied.The median interval between the observations in individualsourcesvariedfrom31daysto47daysat22and90GHz,respec-tively. At 37 GHz the median sampling rate was 41 days, butthe value depends on the source. The minimum average value,6.8 days, was at 37 GHz for the source 3C 84, which is used asa secondary calibrator in the Metsähovi observations. The mini-mum average for a source not used as a calibrator was 8.9 daysfor the source 3C 273 at 37 GHz. The maximum average value,186.4 days, was for the source 2234
+
282 also at 37 GHz. Wealso compared the sampling rates in Paper I for the 40 sourcesin common in our samples with sampling rates from data af-ter 1993. At 22 GHz the di
ﬀ
erence is larger with a median of 39 days in PaperI and 19days after 1993.At 37GHz the mediansamplingrate is 31daysforbothdatasets. Forthe 40sourcesnotincluded in Paper I, the median sampling rate during the wholeperiod was 40 days at 22 GHz and 61 days at 37 GHz.
3. Methods
Three methods were used to study the characteristic time scalesof di
ﬀ
erent types of AGNs: SF, DCF and LS-periodogram. Wechose to use three di
ﬀ
erent methods because we also wanted tostudy these methods in more detail, as well as the di
ﬀ
erencesbetween them. An additional reason for using the SF analysiswas to compare the results with those of the analysis done inPaper I. This way we can study how 13 years of additional dataa
ﬀ
ect the time scales.
3.1. The structure function
The general description of the structure function is given bySimonetti et al. (1985). We will use only the ﬁrst-order SF de-ﬁned in Eq. (1),
D
1
(
τ
)
=
[
S
(
t
)
−
S
(
t
+
τ
)]
2
(1)where
S
(
t
) is the ﬂux density at time
t
and
τ
is the time lag. Ouranalysis follows the descriptions in Paper I and Hughes et al.(1992). Here we will only shortly describe the method.An ideal structure function is presented in Fig. 1. It consistsof two plateaus and a slope between them. The
x
-axis showslogarithm of the timelag,
τ
, and the
y
-axis shows the logarithmof the structure function,
D
(
τ
). We can identify a time scale atthe point
T
max
where the structure function reaches its second,higherplateau.Thistimescale is themaximumtimescaleofcor-related behaviour. For lags longer than
T
max
, we have a plateauwith amplitude equal to twice the variance of the signal. Thelower plateau at short timelags is equal to twice the averagevari-ance of the measurement noise for a single data point.Inaddition,wecanﬁndoutthenatureoftheprocessfromtheslope b between the two plateaus. If the lightcurve can be mod-elled as a whiteora rednoiseprocess,thena slopeofunityinthestructure function implies shot noise, and a slope close to zeroimplies ﬂicker or white noise. Usually the process is a mixtureof these processes and the slope is something between 0 and 1.See Hufnagel & Bregman (1992) for details. If one large out-burst dominates the time series, the slope may be steeper than 1.A strong linear trend or a strong periodic oscillation in the datais expected to produce a slope of 2.In our analysis the timelag runs from 1 week to the length of the light curve.Many of the sources have been observedapprox-imately once a week, and therefore we have chosen the lowerlimit to be one week. Higherfrequencies(90 and 230GHz) haveusually been monitored for shorter times. We ﬁrst calculated thedi
ﬀ
erences squared for all the two point pairs, and then to createthe structure function we averaged all the samples into 0.1 dexwide bins.
T. Hovatta et al.: Statistical analyses of long-term variability of AGN at high radio frequencies 901
T
max
2
noise2
2σ2σ
signal
log
l o g ( D )
(τ)
( τ )
Fig.1.
Ideal structure function.
We estimated the error caused by observationaluncertaintiesto the SF by an independent bootstrap method. For each sourceand frequency we created a model light curve by running overthe light curve a boxcar with a length of 10 days and averaging.We tested also other averaging lengths but using much shortervalues we would have not provided enough non-zero residu-als for scrambling. Much longer averaging lengths would havestarted to average over signiﬁcant variability in some sources.We then subtracted this 10-day model from each light curve andcreated a bank of residuals. A new simulated light curve wasmade by adding to each point in the model a randomly selectedresidual. The time sampling of the srcinal light curve was thuspreserved. This was repeated so that each residual was selectedonce. Using this new light curve we recalculated a new simu-lated SF. The procedure was repeated 1000 times. This enablesus to put conﬁdence limits to the SF, e.g. the 99% conﬁdencelimit at a given time scale was at the value where only 5 pointswere above or below the value. This method clearly does notrequire the conﬁdence limits to be symmetric.Results of the analysis are presented in Sect. 4 and compari-son with Paper I in Sect. 5.
3.2. Discrete correlation function
Discrete correlation function was ﬁrst introduced by Edelson& Krolik (1988). Hufnagel & Bregman (1992) generalized themethod to include a better error estimate. The advantage of DCFcompared to other correlation methods is that it is suitable forunevenlysampleddata,whichis usuallythecase inastronomicalobservations. Here we will describe only brieﬂy the method andformulaeused,andrefertoTornikoskietal.(1994)andHufnagel& Bregman (1992) for details.First we need to calculate the unbinned correlations for thetime series. Note thatthe formulationbelowallows forcross cor-relations. This is done using Eq. (2), where
a
i
and
b
j
are individ-ual points in the time series
a
and
b
, ¯
a
and ¯
b
are the means of thetime series, and
σ
2
a
and
σ
2
b
are the variances. After calculatingthe UDCF the correlation function is binned. The method doesnot deﬁne a priori the bin size so we have tested several val-ues. If the bin size is too large, information is lost. On the otherhand, if the bin size is too small, we can get spurious correla-tions, and the time scales may be di
ﬃ
cult to interpret. We havechosen a bin size of 50 days for all autocorrelations. For severalsources we also tested smaller bin size of 25 days but this didnot make noticeable changes to the results.
UDCF
ij
=
(
a
i
−
¯
a
)(
b
j
−
¯
b
)
σ
2
a
σ
2
b
·
(2)By binning the UDCF we obtain the DCF using Eq. (3). Here
τ
is the time of the centre of the time bin and
n
is the number of points in each bin. We can also calculate the error in each binby using Eq. (4). This represents the standard deviation of theUDCF estimates within the bin.
DCF
(
τ
)
=
1
n
Σ
UDCF
ij
(
τ
) (3)
σ
dcf
(
τ
)
=
1
n
−
1
Σ
UDCF
ij
−
DCF
(
τ
)
2
0
.
5
.
(4)A disadvantage of this method is that it does not give any ex-act probability value for the calculated results. The only way wecan study the reliability of the method is to use simulations. Theerror caused by observational uncertainties have been estimatedwith the same bootstrapmethodas forSF, describedin the previ-ous section. Here we have used 10000 simulations which meansthat the 99% conﬁdence limit was at the value where 50 pointswere above or below the value. One should note that these con-ﬁdence limits represent the ambiguity caused by observationaluncertainties and do not address possible questions posed by thesampling of the data. The errors obtained with this method aresimilar to those calculated using Eq. (4).We also used simulated periodic data to test the capabilityof DCF to ﬁnd real time scales and found out that it could de-tect all real time scales well. For our simulations we created ﬂuxdensity curves with strict periodicities by multiplying ﬂares of real sources to extend over a period of 25 years. The DCF coulddetect the period for the simulated data with good precision.Results of the DCF analysis are presented in Sect. 4.
3.3. Lomb-Scargle periodogram
Fourier-based methods can be used for studying periodicities inlight curves. We tested if these methods are also suitable forstudying the characteristic variability time scales of AGNs. Wehave chosen the commonly used method of Lomb-Scargle peri-odogram for this study (Lomb 1976; Scargle 1982). It is basedon the discrete Fourier-transform which has been modiﬁed forunevenlysampleddata. Themethodsearchesforsinusoidalperi-odicities in the frequency domain. This turns out to be problem-atic because our light curves are not well represented by a sumof sinusoidal functions. Usually the most signiﬁcant spike of theperiodogramturned out to be at the time scale of the total lengthof the time series, and other spikes were its harmonics.We have taken the formulae as they appear in Press et al.(1992). First we need to calculate the mean and the standarddeviation of the time series. We calculated the periodogramwitha samplingintervalof1
/
4
T
in the frequencyspace.Here
T
is thetotal length of the time series. The upper limit to which the peri-odogram was calculated was
N
/
2
T
, where
N
is the total numberof observations. In evenly spaced data this would correspond to
902 T. Hovatta et al.: Statistical analyses of long-term variability of AGN at high radio frequencies
the Nyquist frequency. Now we can calculate the Lomb-Scargleperiodogram by using Eq. (5),
P
N
(
ω
)
=
12
σ
2
Σ
j
(
a
j
−
¯
a
)cos
ω
(
t
j
−
τ
)
Σ
j
cos
2
ω
(
t
j
−
τ
)
+
12
σ
2
Σ
j
(
a
j
−
¯
a
)sin
ω
(
t
j
−
τ
)
Σ
j
sin
2
ω
(
t
j
−
τ
)(5)where
t
j
is the date of an individual observation and
ω
is thefrequency at which we are calculating the periodogram.
a
j
isan individualdata point of time series
a
, and ¯
a
is the mean of thetime series.
τ
can be calculated from Eq. (6).tan(2
ωτ
)
=Σ
j
sin2
ω
t
j
Σ
j
cos2
ω
t
j
·
(6)A false alarm probability level of
z
≈
ln(
N
/
p
) for a 99.9%for the most signiﬁcant spike can be calculated (Scargle 1982).Unfortunately, this does not tell anything about the signiﬁcanceof other spikes in the periodogram and therefore only the mostsigniﬁcant one is used in this analysis. In radio data the annualgaps in the data are much shorter than in the optical and donot contribute to aliasing in a signiﬁcant way creating spuriousspikes.
4. Results
4.1. Results of the structure function analysis
We used the structure function to study the characteristic timescales of 80 sources at frequencies 4.8, 8, 14.5, 22, 37, 90and 230 GHz.In total we calculated 411 structure functions from whichwe could determine 447 time scales. In 39 cases we could notdetermine a variability time scale because the function was tooﬂat or because the errors in the structure functionwere too large.In 60 cases we could only get a lower limit for the time scale.We could also determine more than one time scale in 69 cases.The relative number of sources for which we could not de-termine a time scale depended on the frequency, for exampleat 14.5 GHz we had only two such cases (sources 1147
+
245and 0446
+
112), but at 90 GHz one third of the sources failedto provide us with a good estimate for the time scale. This wasmainly due to the undersampling at 90 GHz and especially at230 GHz, which caused the structure functions to have large er-rors and therefore to be di
ﬃ
cult to interpret. Also there wereonly ﬁve sources with data at 230 GHz.Also the number of the lower limit estimates varied betweenthe frequency bands and the source classes. In Table 1 the num-ber of such sources is written in parenthesis for each frequencybandandclass. Therelativepercentageoflowerlimittime scalesin LPQs, BLOs and HPQs were 24%, 13% and 11%.We have plotted the distributions of the time scales at dif-ferent frequencies and source classes in Fig. 2. The mediantime scales of each class in the histograms are marked by ver-tical lines. The average and median time scales are presented inTable 1.Because a substantial number of time scale estimates arelower limits, a much better representative for a characteristictime scale is the median time scale of the group. Practically allmedian time scales have value of less than 5 years. The mediantime scales also shorten on average with increasing frequency.To provide a measure of the slope of the structure functionbetween the plateaus we calculated the local slope in each struc-ture function using trains of 2, 11 and 20 points. By comparingthe plots of slope vs. time scale we estimated the actual slope foreach light curve. The 2 point slope provided a local measure of the slope, while for the overall slope between the plateaus, the11 point or the 20 point trains gave more reliable results. Whichof the two was better depends on the distance of the plateausin log
τ
. Thiswas possiblein358cases. Theslopesvariedfrom0to 2.2 while the average of all the frequency bands was near 1,which is expected if the light curve can be modelled as a 1
/
f
2
-type shot noise. At 37 GHz the average slope was 0.72 and at22 GHz 0.83. We ran the Kruskal-Wallis analysis to see if therewere di
ﬀ
erences between the slopes of the source classes. (AllKruskal-Wallis analyses in this paper have been performed withthe Unistat software, version 5.0.) At 4.8 GHz we found that theBLOs di
ﬀ
ered from other classes signiﬁcantly at
P
>
95% level.At 22 GHz the HPQs di
ﬀ
ered from the BLOs signiﬁcantly. Atother frequencies we could not ﬁnd any statistically signiﬁcantdi
ﬀ
erences between the groups.
4.2. Results of the DCF analysis
We used the Discrete Correlation Function to calculate the auto-correlationfor80sourcesandobtained411autocorrelations.Formany sources, more than one time scale was present in the DCF.For each we determined two time scales wherever possible. Weidentify as the most signiﬁcant time scale the onethat shows sig-niﬁcant positive correlation after the DCF has been on the neg-ative side. We were able to determine 273 such time scales. Wealso identiﬁed a time scale that we call the shortest time scale.This is the ﬁrst peak in the correlation function before it hasgained negative values. It did not occur in all cases and usuallythey appeared as small bumps in the DCF. We could determinethe shortest time scale in 175 cases. Furthermore we calculateda redshift corrected time scale of the most signiﬁcant time scale.Figure 3 shows the distribution of time scales at di
ﬀ
erent fre-quency bands and source classes.Table 2 shows the averagetime scales fromthe DCF analysisfor the di
ﬀ
erent frequency bands, and also for BLOs, HPQs andLPQs separately at each frequency band.
4.3. Results of the Lomb-Scargle periodogram analysis
We also calculated the Lomb-Scargle periodogram for oursource sample. Altogether we obtained 411 periodograms. Onaverage we have data at 5 di
ﬀ
erent frequency bands for eachsource. From these periodograms we found 140 time scales. Inother cases there were no signiﬁcant spikes in the periodogramor the spikes were at time scales over half of the total length of the time series. We did not take these into account because weare interestedin time scales that haveoccurredat least twice dur-ing the observing period. In Fig. 4 we have plotted histogramsof the distribution of time scales at all the frequency bands andfor all the source classes.We have calculated the average time scales at all the fre-quency bands. These are shown in Table 3. We present the re-sults for the whole sample and also separately for BLOs, HPQsand LPQs. Foreach sourceclass also the numberofsources usedto calculate the average are given. Each frequency band is listedseparately, and the averages of the most signiﬁcant time scalesare shown as well as the redshift corrected time scales.

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