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The Ensemble Photometric Variability of ∼ 25000 Quasars in the

Sloan Digital Sky Survey

Daniel E. Vanden Berk1, Brian C. Wilhite2, Richard G. Kron2,3, Scott F. Anderson4,

Robert J. Brunner5, Patrick B. Hall6,7, Zeljko Ivezic6, Gordon T. Richards6, Donald P.

Schneider8, Donald G. York2,9, Jonathan V. Brinkmann10, Don Q. Lamb2, Robert C.

Nichol11, David J. Schlegel6

ABSTRACT

Using a sample of over 25000 spectroscopically confirmed quasars from the Sloan Digital SkySurvey, we show how quasar variability in the rest frame optical/UV regime depends upon restframe time lag, luminosity, rest wavelength, redshift, the presence of radio and X-ray emission,and the presence of broad absorption line systems. Imaging photometry is compared with three-band spectrophotometry obtained at later epochs spanning time lags up to about two years. Thelarge sample size and wide range of parameter values allow the dependence of variability to beisolated as a function of many independent parameters. The time dependence of variability (thestructure function) is well-fit by a single power law with an index γ = 0.246±0.008, on timescalesfrom days to years. There is an anti-correlation of variability amplitude with rest wavelength – e.g.quasars are about twice as variable at 1000A as 6000A – and quasars are systematically bluer whenbrighter at all redshifts. There is a strong anti-correlation of variability with quasar luminosity– variability amplitude decreases by a factor of about four when luminosity increases by a factorof 100. There is also a significant positive correlation of variability amplitude with redshift,indicating evolution of the quasar population or the variability mechanism. We parameterizeall of these relationships. Quasars with RASS X-ray detections are significantly more variable(at optical/UV wavelengths) than those without, and radio loud quasars are marginally morevariable than their radio weak counterparts. We find no significant difference in the variabilityof quasars with and without broad absorption line troughs. Currently, no models of quasarvariability address more than a few of these relationships. Models involving multiple discreteevents or gravitational microlensing are unlikely by themselves to account for the data. So-calledaccretion disk instability models are promising, but more quantitative predictions are needed.

Subject headings: galaxies: active – quasars: general – techniques: photometric

1Department of Physics and Astronomy, University ofPittsburgh, 3941 O’Hara Street, Pittsburgh, PA 15260.

2Department of Astronomy and Astrophysics, The Uni-versity of Chicago, 5640 South Ellis Avenue, Chicago, IL60637.

3Fermi National Accelerator Laboratory, P.O. Box 500,Batavia, IL 60510.

4Department of Astronomy, University of Washington,Box 351580, Seattle, WA 98195.

5Department of Astronomy/NCSA, University of Illi-nois, 1002 W. Green Street, Urbana, IL 61801.

6Princeton University Observatory, Peyton Hall, Prince-

ton, NJ 08544.7Departamento de Astronomıa y Astrofısica, Pontificia

Universidad Catolica de Chile, Casilla 306, Santiago 22,Chile.

8Department of Astronomy and Astrophysics, ThePennsylvania State University, 525 Davey Laboratory, Uni-versity Park, PA 16802.

9Enrico Fermi Institute, The University of Chicago,5640 South Ellis Avenue, Chicago, IL 60637.

10Apache Point Observatory, P.O. Box 59, Sunspot, NM88349.

11Department of Physics, Carnegie Mellon University,

1

1. Introduction

The luminosities of quasars and other activegalactic nuclei (AGNs) have been observed tovary from X-ray to radio wavelengths, and ontime scales from several hours to many years.The majority of quasars exhibit continuum vari-ability on the order of 10% on timescales ofmonths to years. A minority of AGNs, broadlyclassified as blazars, vary much more dramati-cally on much shorter timescales. The mecha-nisms behind quasar variability are not known, al-though in principle variability is a powerful meansof constraining models for the energy source ofAGNs. The most promising models (for non-blazar variability) include accretion disk insta-bilities (e.g. Rees 1984; Kawaguchi, Mineshige,Umemura, & Turner 1998), so-called Poissonianprocesses such as multiple supernovae (e.g. Ter-levich, Tenorio-Tagle, Franco, & Melnick 1992)or star collisions (Courvoisier, Paltani, & Walter1996; Torricelli-Ciamponi, Foellmi, Courvoisier, &Paltani 2000), and gravitational microlensing (e.g.Hawkins 1993). Only recently have the variousmodels become quantitative enough for meaning-ful comparison with observations. A consensus onthe observational trends with variability is emerg-ing, but disagreements remain and even the mostfundamental relationships need better characteri-zation.

Several dozen studies of quasar optical broad-band variability have appeared in the literature. Anumber of the more important studies are summa-rized in tabular form by Helfand et al. (2001) andGiveon et al. (1999). Most ensemble studies havefocused on establishing correlations between vari-ability (defined in various ways as a measure of thesource brightness change) and a number of param-eters, most importantly time lag, quasar luminos-ity, rest frame wavelength, and redshift. Charac-teristic timescales of variability range from monthsto years (e.g. Collier & Peterson 2001; Cristianiet al. 1996; di Clemente et al. 1996; Smith &Nair 1995; Hook, McMahon, Boyle, & Irwin 1994;Trevese et al. 1994). The amplitude of variabilityrises quickly on those timescales, but may slow oreven level off on longer timescales.

An anti-correlation between quasar variabil-

5000 Forbes Avenue, Pittsburgh, PA 15232.

ity and luminosity was reported by Angione &Smith (1972), and confirmed in numerous sub-sequent studies (Uomoto, Wills, & Wills 1976;Pica & Smith 1983; Lloyd 1984; O’brien, Gond-halekar, & Wilson 1988; Hook, McMahon, Boyle,& Irwin 1994; Trevese et al. 1994; Cid Fernan-des, Aretxaga, & Terlevich 1996; Cristiani et al.1996; Cristiani, Trentini, La Franca, & Andreani1997; Paltani & Courvoisier 1997; Giveon et al.1999; Garcia, Sodre, Jablonski, & Terlevich 1999;Hawkins 2000; Webb & Malkan 2000). Such ananti-correlation is expected in Poissonian models,although complex versions are necessary to explainthe diversity of the relationship among quasars(Cid Fernandes, Sodre, & Vieira da Silva 2000).

There is strong evidence from multiwavelengthobservations of quasars that variability increaseswith decreasing rest wavelength, which holds overa wavelength range spanning at least the ultravi-olet to near infrared (Cutri, Wisniewski, Rieke, &Lebofsky 1985; Neugebauer, Soifer, Matthews, &Elias 1989; Kinney, Bohlin, Blades, & York 1991;Paltani & Courvoisier 1994; di Clemente et al.1996; Cristiani, Trentini, La Franca, & Andreani1997; Giveon et al. 1999; Cid Fernandes, Sodre, &Vieira da Silva 2000; Helfand et al. 2001; Trevese& Vagnetti 2002). The wavelength dependenceis related to the observed tendency for quasarspectra to become harder (bluer) in bright phases(Cutri, Wisniewski, Rieke, & Lebofsky 1985; Edel-son, Pike, & Krolik 1990; Giveon et al. 1999; CidFernandes, Sodre, & Vieira da Silva 2000; Trevese,Kron, & Bunone 2001). The chromatic natureof quasar variability is often taken as evidenceagainst gravitational microlensing as the primarycause of variability (e.g. Cristiani, Trentini, LaFranca, & Andreani (1997); except see Hawkins(1996)), although this may be accounted for if re-gions closer to the center are both brighter andbluer.

A correlation of variability with redshift is oftenreported (Cristiani, Vio, & Andreani 1990; Gial-longo, Trevese, & Vagnetti 1991; Hook, McMahon,Boyle, & Irwin 1994; Trevese et al. 1994; Cid Fer-nandes, Aretxaga, & Terlevich 1996; Cristiani etal. 1996; Trevese & Vagnetti 2002) if wavelengthand luminosity dependencies are not taken into ac-count. For a fixed observer timescale, the increaseof variability with increasing redshift would con-tradict the expected 1 + z effect of time dilation.

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However, it has been shown by Giallongo, Trevese,& Vagnetti (1991), Cristiani et al. (1996), and CidFernandes, Aretxaga, & Terlevich (1996) that theinverse wavelength dependence can easily accountfor the uncorrected redshift correlation, since fora fixed passband in the observer frame, quasarswith higher redshifts are detected at shorter wave-lengths, which systematically vary at a greateramplitude. It is still not clear whether any red-shift dependence remains after accounting for restwavelength and luminosity (which is strongly cor-related with redshift in flux-limited quasar sam-ples). Some studies which have leverage in bothredshift and luminosity suggest a weak correla-tion of redshift and variability (Hook, McMahon,Boyle, & Irwin 1994; Cristiani et al. 1996), butothers show no such effect (Cimatti, Zamorani, &Marano 1993; Paltani & Courvoisier 1994; Netzeret al. 1996; Cristiani, Trentini, La Franca, & An-dreani 1997; Helfand et al. 2001).

Variability is sometimes found to be correlatedwith radio loudness (Pica & Smith 1983; Smith &Nair 1995; Garcia, Sodre, Jablonski, & Terlevich1999; Eggers, Shaffer, & Weistrop 2000; Helfand etal. 2001; Enya et al. 2002), the equivalent widthof the Hβ line (Giveon et al. 1999; Cid Fernandes,Sodre, & Vieira da Silva 2000), and the presence ofbroad absorption line troughs (Sirola et al. 1998),although the results are not conclusive. No largeX-ray detected quasar sample has been systemati-cally studied for optical variability, but since mostblazars are X-ray bright, a greater degree of vari-ability may be expected from such a sample (e.g.Ulrich, Maraschi, & Urry 1997).

In this paper we present results on a quasarvariability program using data from the Sloan Dig-ital Sky Survey (SDSS, York et al. 2000). A com-plementary variability study by de Vries, Becker,&White (2003) presents a comparison of the SDSSEarly Data Release (Stoughton et al. 2002) imag-ing photometry with archival photographic platedata. One of the goals of the present work is tocharacterize the spectroscopic calibrations of theSDSS, in order to examine the spectroscopic vari-ability properties of quasars and other objects ob-served in the spectroscopic survey. The presentwork uses the broad band fluxes of the spectraconvolved with the SDSS filter transmission func-tions in direct comparison with the imaging pho-tometry. This provides photometric data at two

epochs in three bands for every spectroscopicallyconfirmed quasar in the survey – a sample sizecurrently of over 25000 quasars. This is by farthe largest quasar UV/optical variability study todate, and it also includes the largest samples of ra-dio selected, X-ray selected, and broad absorptionline quasars ever examined for variability. Ourgoal is to characterize the ensemble dependence ofvariability on many quasar parameters and types,on timescales from weeks up to several years.

We describe the quasar sample drawn from theSDSS in § 2. Ensemble measurements of the vari-ability are given in § 3. We disentangle the de-pendence of variability upon time lag, luminos-ity, wavelength, and redshift in § 4, show howquasar colors change with variability in § 5, andlook at variability in various quasar subclassesin § 6. The implications of the results are dis-cussed in § 7, and we conclude in § 8. Through-out the paper we assume a flat, cosmological con-stant dominated cosmology with parameter valuesΩΛ = 0.7,ΩM = 0.3, and H0 = 65km/s/Mpc.

2. The Quasar Dataset

2.1. The Sloan Digital Sky Survey

The Sloan Digital Sky Survey (SDSS) is aproject to image 104 deg2 of sky mainly in thenorthern Galactic cap, in five broad photometricbands (u, g, r, i, z) to a depth of r ∼ 23, and toobtain spectra of 106 galaxies and 105 quasars ob-served in the imaging survey (York et al. 2000).All observations are made with a dedicated 2.5mtelescope at Apache Point Observatory in NewMexico. Images are taken with a large mosaicCCD camera (Gunn et al. 1998) in a drift-scanningmode. Absolute astrometry for point sources isaccurate to better than 100 milliarcseconds (Pieret al. 2003). Site photometricity and extinctionmonitoring are carried out simultaneously witha dedicated 20in telescope at the observing site(Hogg, Finkbeiner, Schlegel, & Gunn 2001). Theimaging data are reduced and calibrated using thePHOTO software pipeline (Lupton et al. 2001). Inthis study we use the point-spread function (PSF)magnitudes, which are determined by convolvingthe reduced imaging data with a model of thespatial point-spread function. The PSF magni-tudes are more stable than aperture magnitudesfor point sources, since they are less dependent

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upon seeing variations, and because the PSF back-ground noise is less within the survey seeing limit(which is 1.7 arcseconds). The SDSS photometricsystem is normalized so that the u, g, r, i, z mag-nitudes are on the AB system (Smith et al. 2002;Fukugita et al. 1996; Oke & Gunn 1983). The pho-tometric zeropoint calibration is accurate to bet-ter than 1% (root-mean-squared) in the g, r, andi, bands, and to better than 2% in the u and zbands, measured by comparing the photometry ofobjects in scan overlap regions. The SDSS imagereduction and calibration routines have evolvedthroughout the course of the survey and the imag-ing runs have been reprocessed accordingly. Thusthe object imaging magnitudes deemed “best”,and which we use in this study, may be slightly dif-ferent than those used for the spectroscopic targetselection, although any differences are insignificantto the results of this study. Throughout this paperwe use magnitudes corrected for Galactic extinc-tion according to Schlegel, Finkbeiner, & Davis(1998).

Objects are selected for spectroscopic follow-upas candidate galaxies (Strauss et al. 2002; Eisen-stein et al. 2001), quasars (Richards et al. 2002),and stars (Stoughton et al. 2002). The spectro-scopic targets are grouped by 3 degree diame-ter areas or “tiles” (Blanton et al. 2003). Foreach tile, an aluminum plate is drilled with holescorresponding to the sky locations of the targetsalong with holes for blank sky, calibration stars,and guide stars. The plates are placed at the fo-cal plane of the telescope, and optical fibers runfrom the hole positions to two spectrographs, eachof which accepts 320 fibers allowing for the si-multaneous observation of 640 objects. For eachplate, approximately 500 galaxies, 50 quasars, and50 stars are observed. Spectroscopic observationsgenerally occur up to a few months, but occasion-ally years, after the corresponding imaging obser-vations, depending upon scheduling constraints.The spectroscopic data for this study come from479 spectroscopic plates observed and processedthrough September 2002; 284 of the plates are partof the SDSS First Data Release (DR1, Abazajianet al. 2003), publicly available since April 2003.Seven of the (291) DR1 plates are not included inthis study, since the DR1 plate list was not final-ized until after the sample for this study had beengathered.

2.2. Quasar Target Selection and Sample

Definition

Quasar candidates are selected from the imag-ing sample by their non-stellar colors from thefive-band photometry as well as by matchingSDSS point sources with FIRST radio sources(Stoughton et al. 2002). The selection is similarto that described by Richards et al. (2002), butthe formal implementation of this algorithm wasimposed after the cutoff date for the DR1 quasarsample, and much of the post-DR1 data used inthis study. About two-thirds of the candidates areconfirmed to be quasars from the spectroscopicsurvey. Ultraviolet excess quasars are targeted toa limit of i = 19.1 and higher redshift quasarsare targeted to i = 20.2. These criteria give asample that is estimated to be over 90% complete(Richards et al. 2002). Additional quasars are tar-geted as part of the SERENDIPITY and ROSATclasses (Stoughton et al. 2002) or (incorrectly) asstars.

Quasars are identified from their spectra us-ing a combination of both automated classifica-tion (about 94%) and manual inspection (about6%) of those objects flagged by the spectroscopicpipeline as being less reliably identified. For thepurposes of this study, we define “quasar” to meanany extragalactic object with broad emission lines(pipeline measured full width at half maximum ve-locity width of & 1000kms−1) regardless of lumi-nosity. The definition thus includes objects whichare often classified as less luminous types of ac-tive galactic nuclei (AGNs) rather than quasars,and excludes AGNs without strong broad emis-sion lines such as BL Lacs and some extreme broadabsorption line quasars. To assemble our sample,we extract relevant data for all point sources fromthe SDSS database. Data from only one imagingand one spectroscopic epoch are used per objectto avoid giving extra weight to any object. Ex-tended sources are rejected because they compli-cate the spectrophotometric recalibration (§ 2.4),and their spectra are likely to be seriously con-taminated by host galaxy light. Objects with badspectra (defined to be those with significantly longunprocessable portions of spectrum) are rejected.Those remaining objects spectroscopically identi-fied as stars are used to refine the spectroscopiccalibration (see § 2.4). The remaining 25710 ob-jects identified to be quasars become part of the

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quasar variability sample.

A catalog of quasars found in the SDSS DR1dataset is described by Schneider et al. (2003). Ofthe 16713 listed DR1 quasars, 14705 are includedin this study. The difference can be attributed totwo factors. First, the definitions of “quasar” areslightly different – Schneider et al. (2003) imposean absolute magnitude limit of Mi ≤ −22.0, andthey make no distinction between point-like andextended sources. Second, data from only 284 ofthe 291 DR1 spectroscopic plates are included inthis study. Over half (57%) of the quasars in thisstudy are contained in the DR1 sample, and we ex-pect that our results on quasar variability wouldbe similar (albeit more noisy) for the full DR1sample, except possibly for the extended sources.

Absolute magnitudes in the rest frame i bandMi, are calculated for each quasar using dered-dened observed i band PSF magnitudes and as-suming a power law spectral energy distributionfλ ∝ λαλ , with a wavelength slope of αλ = −1.5.Detailed K-corrections are not yet available forquasars in the SDSS photometric system. How-ever, estimated K-corrections using the compos-ite spectrum from Vanden Berk et al. (2001) areconsistent with the simple power law assumption,and the differences are usually no greater than 0.1magnitude at any redshift. The differences are notsignificant here since the data are coarsely binned(by statistical necessity) when we examine abso-lute magnitude trends.

2.3. SDSS Spectroscopy and Its Calibra-

tion

Spectra are obtained in three to four consec-utive 15-minute observations. There are 32 skyfibers, 8 spectrophotometric standard stars and 8reddening standards stars observed on each plateto help with calibration of the remaining 592 sci-ence targets. Spectral reductions and calibrationsare done using the SDSS SPECTRO 2D pipeline(Stoughton et al. 2002). The 8 spectrophotomet-ric calibration stars are chosen to approximate thestandard F0 subdwarf star BD+174708, and areused by the 2D pipeline for absolute spectral fluxcalibration and dereddening due to Galactic ex-tinction. The 2D pipeline also calculates syntheticspectroscopic magnitudes by convolving the cali-brated spectra with SDSS g, r, and i filter trans-mission curves, assuming 1.2 airmasses of extinc-

tion (the spectra do not cover the entire wave-length ranges of the u and z bands). By usingthese synthetic spectral magnitudes, we obtain asecond photometric data point for every spectro-scopically observed quasar. Additionally, a signal-to-noise (S/N) parameter is calculated for eachof the three bands by convolving the spectral er-ror with the transmission curves and dividing thatinto the corresponding convolved flux. The spec-tral magnitude signal-to-noise is essential for char-acterizing magnitude difference uncertainties (see§ 2.4).

2.4. Refinement of Spectroscopic Calibra-

tion

The differences between the spectroscopic andimaging PSF magnitudes ∆m = ms − mp for allof the spectroscopically confirmed stars in each ofthe g, r, and i bands are shown in Fig. 1. Wellcalibrated data should center around zero mag-nitude difference, and there should be no trendwith magnitude, except for larger uncertaintiesat fainter magnitudes. There are clearly system-atic differences in the magnitudes derived from thespectra and from the images. While the initialspectroscopic calibration is more than adequatefor the primary purposes of the SDSS, namely ob-ject identification and reliable redshift measure-ment, variability studies require more careful cal-ibrations. Fortunately, the magnitude differencetrends with PSF magnitude can be easily under-stood and are almost entirely correctable.

There are three primary sources for the mag-nitude difference discrepancy: the inclusion of ob-jects with bad PSF magnitudes, an aperture ef-fect relating the finite fiber diameters to the PSFmagnitudes, and what may be a very small butsignificant sky under-subtraction in the spectro-scopic data. Occasionally, point sources in the im-ages can have poor photometry if they are closelyblended with other objects, occur where the seeinghas changed very rapidly, or lie where there may beother problems in the imaging data. The long tailsin the histograms of Fig. 1 are populated mainlyby the measurements of these objects. Because theobjects will have unusual measured colors they aresometimes selected as high-redshift quasar candi-dates for spectroscopic follow-up, which turn outto be normal stars upon examination of the spec-tra. Therefore, spectroscopically confirmed stars

5

which were selected as high-redshift quasar candi-dates are removed from the stellar data set forspectrophotometric refinement. Late-type starsidentified by the spectroscopic pipeline are also re-jected because they are often variable.

The median ∆m offset from zero is simply anaperture effect wherein the 3′′ spectroscopic fiberssubtend a smaller fraction of the total object im-age than the point-spread function used to mea-sure the PSF magnitudes in the imaging data. Thespectroscopic fiber flux density to PSF flux densityratio is nearly a constant (but somewhat depen-dent upon seeing at the spectroscopic epoch, seebelow), so the magnitude difference will also be anearly (non-zero) constant.

The downward trend of ∆m with PSF magni-tude seen for each band is most easily accountedfor by a small overestimation of the flux densityin the spectroscopic data, possibly caused by aslight under-subtraction of the sky level. Testsof the imaging photometric calibration show thatthe effect is not likely to be caused by sky over-subtraction in the imaging data. Further tests willhave to be done to determine the cause with cer-tainty. The correction for a flux density overesti-mation combined with a fiber aperture correctiongive

∆m = ms −mp = −2.5 log(fs/fp)

= −2.5 log(a(fp + b)/fp)

= −2.5 log(a+ ab10(mp−C)/2.5) , (1)

where a is the aperture correction, b is the cor-rection for the flux density offset, and C is thezeropoint constant used in converting flux densityto magnitude. Assuming a and b are constants,the function has two adjustable parameters, andfits to the data provide reasonably good descrip-tions of the ∆m vs. mp trend. However, in or-der to account for any other effects, expected orunexpected, we use the following more flexible 3parameter function

∆m = ∆m0 − exp((mp −m0)/m∗) , (2)

where ∆m0, m0, andm∗ are constants to be deter-mined from the fits to the data. For example, theMalmquist bias (Malmquist 1924) will add to themagnitude difference approximately as the squareof the PSF magnitude uncertainty, σp

2. Since σp

ranges from about 0.01 to 0.05, the Malmquist bias

is expected to affect the magnitude difference byat most a few percent. In the absence of this orother higher order effects, the 3-parameter func-tion would almost exactly reproduce the 2 param-eter logarithmic fit. The 3-parameter functionsare fit to the data in each band separately, thensubtracted from the magnitude differences.

After this correction, offsets from zero remainfor the mean ∆m values for stars on the sameplate, in excess of those expected from statisticaluncertainty. These plate-to-plate offsets are dueto differences in the spectral energy distributionsof the stars used as initial spectrophotometric cal-ibrators relative to BD + 174708 and due to dif-ferences in the seeing at the epochs of the plateobservations. The former effect applies to “halfplates” corresponding to the two sets of 320 fibersrunning separately to each spectrograph. To cor-rect for these effects, we reject stars outside of the99% confidence envelope resulting from the ∆mcorrections described above, and work only withhalf plates which have at least 5 remaining stars(the average number is about 20). The median∆m offsets are calculated for each half plate andsubtracted from the values for each of the starsobserved with that half plate.

The final corrected ∆m distributions are shownin Fig. 2 as a function of spectral S/N . The widthof the stellar ∆m distribution is correlated withmagnitude, but the better correlation is with spec-tral S/N . The reason for this is that while magni-tude and S/N are correlated, it is the S/N which isdirectly related to the quality of a spectrum. Asa whole, the 68.3% confidence half-width (nomi-nally 1 standard deviation) is ≈ 0.08 in each bandat a spectroscopic S/N of 10, which is a substan-tial improvement over the initial widths of ≈ 0.13.Fits to the 68.3% confidence half-width as a func-tion of spectral S/N are shown in Fig. 2, for whichwe used a function of the form

σS/N = a0 + a1 exp(a2S/N) , (3)

where a0, a1, and a2 are constants. These fits areused as statistical measurement uncertainties forthe quasars (see § 3).

The same spectrophotometric corrections ap-plied to the stars are also applied to the quasars.The resulting distribution of quasar magnitudedifferences as a function of spectral S/N , and thehistograms of magnitude differences are shown in

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Fig. 3. The mean corrected magnitude differencesfor the quasars are 0.002, -0.004, and -0.011 forthe g, r, and i bands respectively. These valuesare small compared with the measurement uncer-tainties derived from the stars. It is possible thatsmall differences in the spectral energy distribu-tions (SEDs) of stars and quasars affect the re-calibration of the quasar photometry. However,any effect is expected to be quite small since thesame filter transmission curves are used for boththe imaging and spectroscopic photometry, andthe majority of the stars used for the recalibrationwere selected as quasar candidates in the color-selected survey, which guarantees that the SEDsare very similar. The 68.3% confidence limit half-widths of the g, r, and i magnitude difference dis-tributions are 0.134, 0.119, and 0.114 respectivelyat a S/N of 10, substantially larger than thoseof the stars. The stars and quasars were selectedto be point sources, observed simultaneously withthe same instrument, and often were selected withthe same algorithm. The larger magnitude differ-ences among the quasars, therefore, demonstratethe variable nature of the quasars in the sample.The following sections quantify the variability andits dependence upon many quasar parameters.

3. Ensemble Variability Measurement –

The Structure Function

The magnitude difference histograms from theprevious section show that the quasars are signif-icantly more variable as a class than the stars.Assuming no stellar variability, we can use thedistribution of the stellar magnitude differencesto quantify the statistical measurement uncertain-ties. Removing the width of the stellar magnitudedifference distribution in quadrature, the averagequasar magnitude differences (at a spectral S/Nof 10) due to variability in the sample are 0.103,0.086, and 0.080 in the g, r, and i bands respec-tively. Measurement uncertainties must be takeninto account because they are comparable to thevalues of the variability itself. The large sizes ofthe samples (both the quasars and the compari-son stars) allow the measurement uncertainty tobe effectively removed.

We first show the absolute values of the mea-sured quasar magnitude variations, uncorrectedfor measurement uncertainty, in Figures 4 through

7, as a function of quasar rest frame time lag(Fig. 4), absolute magnitude in the rest frame iband (Fig. 5), rest frame wavelength (Fig. 6), andredshift (Fig. 7). Data in each of the three photo-metric bands are shown separately. Average val-ues in a set of bins are also shown. Because fluxdensities in the Lyα forest region are not repre-sentative of the true quasar flux, we have omitteddata in each band at redshifts beyond which theLyα forest covers the band: z = 2.5, 4.75, and6.0 for the g, r, and i bands respectively. Thenumber of measurements rejected for each banddue to the Lyα forest are 742, 45, and 0 for theg, r, and i band respectively. The figures showthat there are several apparent correlations evenin the uncorrected data. In particular, the aver-age magnitude difference increases with time lagin all three bands, and the magnitude differencedecreases with more negative absolute magnitude(decreases with luminosity). No trends are ap-parent at this stage between magnitude differenceand rest wavelength or redshift. Again, it is im-portant to account for measurement uncertaintybefore making any claims about the dependenceof variability on any parameter.

The definition of variability used here is a sta-tistical measure of the magnitude difference, tak-ing into account measurement uncertainty. Thefirst application is to the dependence of variabil-ity on rest frame time lag – the so-called structurefunction. Historically, the structure function hasbeen the primary measure of variability for stud-ies of both individual quasars and quasar ensem-bles. For individual quasars with multiple sam-pling epochs, the structure function is comprisedof the values of the magnitude differences for eachpair of time lags in the data set, and it is closely re-lated to the autocorrelation function (e.g. Simon-etti, Cordes, & Heeschen 1985). In the ensem-ble case, here with only two sampling epochs, thestructure function is simply the average value ofthe magnitude difference for all objects with thesame (or nearly the same) time lags. The erroranalysis is simpler in the ensemble case since allof the data points are independent.

We define the ensemble variability, V , of a setof quasars as

V =(π

2〈|∆m|〉

2− 〈σ2

S/N 〉)

1

2

, (4)

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where ∆m is the measured magnitude difference,σS/N is the statistical measurement uncertainty of∆m (as a function of spectral S/N) derived fromthe fits to the star measurements in §2.4, andthe brackets denote average quantities. The av-erage absolute value of the magnitude difference,along with the scaling factor of π/2, is more robustagainst the presence of outliers in the data thanthe average of the square of the differences. Thevalues V as a function of rest frame time lag ∆τ ,define the structure function, V (∆τ). The samerelation has been used for the structure functionin previous variability studies (e.g. di Clemente etal. 1996).

The binned structure function for all of thequasars in the sample for each of the three photo-metric bands is shown in Fig. 8 with logarithmicaxes. The bins were chosen to have equal intervalsin logarithmic rest frame time lag, and to havereasonably large numbers of objects. The numberof objects per bin range from 241 for the shortesttime lag bin covering 7 to 11 days in the g band, to7919 for the i band bin covering time lags from 111to 176 days. Quasars with magnitude differenceslarger than 0.75 – just over 5 times the 1σ width ofthe distribution for quasars – were rejected fromthe analysis in order to remove outliers. This stepremoves about 1% of the quasars, which is morethan would be expected for a truly normal dis-tribution. In a related paper (Ivezic et al. 2003),the distribution of ultraviolet-excess quasar mag-nitude differences, including very large differences,is discussed in more detail. The apparently highlyvariable quasars may be optically violent variablesand are valuable for follow-up studies, but the fo-cus here is on “typical” quasar variability. Theerror bars were determined by propagating theroot-mean-square errors σ2, in the average mag-nitude difference and measurement uncertainty inquadrature

σ(V ) =1

2V −1

√

π2〈|∆m|〉2σ2(〈|∆m|〉) + σ2(〈σ2

S/N 〉) . (5)

Two trends are obvious: first the structure func-tion increases as a function of time lag – the mag-nitude differences are greater the longer the timebetween measurements. Second, the amplitude ofvariability is greater in the g (bluest) band than inthe others, and the r band amplitude is generallygreater than in the i (reddest) band. This is thevariability anticorrelation with wavelength found

in a several previous studies. We will quantifythe wavelength dependence explicitly in § 4.4, ac-counting for the dependence on other parameters.For the purposes of the remainder of this section,the clear wavelength dependence means that theanalysis of the structure function will address thethree bands individually.

The correlation of variability with time lag hasbeen found in numerous previous studies, howeverthe form of the structure function has remained atopic of debate. We fit the binned structure func-tions with the two most common parameteriza-tions. The first is a power law (e.g. Hook, McMa-hon, Boyle, & Irwin 1994)

V (∆τ) =

(

∆τ

∆τ0

)γ

, (6)

where ∆τ0 and γ are constant parameters to bedetermined. This will appear as a straight linein a log-log plot such as Fig. 8. The second is anasymptotic function – a constant minus an expo-nential – which is the most common parameteri-zation of the structure function (e.g. Bonoli et al.1979; Trevese et al. 1994; Hook, McMahon, Boyle,& Irwin 1994; Enya et al. 2002)

V (∆τ) = V0

(

1− e−∆τ∆τ0

)

, (7)

where again V0 and ∆τ0 are to be determined.The “time scales”, ∆τ0, whatever their values, aresimply parameters of the functions to be fit to thedata, and can not necessarily be directly comparedwith physical characteristic time scales, such asthose associated with accretion disks, star bursts,or gravitational lens dynamics.

Parameter values, uncertainties, and χ2 valuesfor each of the functions in each of the photo-metric bands are given in Table 1. Based on theχ2 values, the functional form that best fits thestructure function in each band is a power law.The power law slopes in each of the three bands– 0.293 ± 0.030, 0.336 ± 0.033, and 0.303 ± 0.035for g, r, and i respectively – are consistent witheach other within one standard deviation of thedifference. The power law scale factors (where thestructure functions would have a value of one) arenot well constrained, mainly because the observedtime lag only extends to about 700 days. Theshape of the structure function at much longertime lags is sometimes observed to “flatten” some-what (Cristiani, Trentini, La Franca, & Andreani

8

(1997); Hook, McMahon, Boyle, & Irwin (1994);Trevese et al. (1994); see Hawkins (2002) for acounter-example), but the data at long time lagsdo not yet favor one parameterization over an-other. In any case, what we can say from thisstudy is that a 2-parameter power law is a gooddescription of the data – and a better descriptionthan a 2-parameter exponential – up to time lagsof about 2 years. As the SDSS proceeds, the rangeof time lags will eventually reach up to about 5years. The rest frame time sampling will continueto improve on all scales, and the power law formcan be even more stringently tested on longer timescales.

The wavelength dependence of the structurefunction becomes clearer at longer time lags (andshould become even clearer by survey end), aswould be expected from either a power law or ex-ponential fit. The distributions of quasar proper-ties – e.g. luminosity and rest frame wavelength– vary with rest frame time lag because of surveyselection effects and the dependence of these prop-erties on redshift. In the next section we disentan-gle the dependence of variability on four primaryquasar parameters.

4. Variability Dependence on Time Lag,

Luminosity, Wavelength, and Redshift

4.1. Selection Function

The structure function calculated in the pre-vious section describes the variability of the fulldata set with respect to time lag. However, vari-ability is almost certainly also a function of quasarluminosity and rest frame wavelength, and possi-bly redshift. In order to separate the dependenceof variability on multiple parameters, the selec-tion biases must be taken into account. Even thestructure function may not give the true depen-dence of variability on rest frame time lag. Theset of quasars within a narrow range of time lagswill be populated with objects with wide rangesof the other parameters. In addition, since restframe time lag is dependent upon redshift, as arethe other parameters, the distributions of quasarparameter values will be correlated with the timelag. For example, high redshift quasars will gener-ally have shorter time lags in the rest frame thanlower redshift quasars.

The selection function – that is, the region of

parameter space occupied by the data set – isshown in projected planes in Fig. 9. A numberof artificial correlations are evident and are dueboth to the survey selection criteria, and to thedependence of luminosity, rest frame time lag, andrest wavelength on redshift. Variability informa-tion can obviously only be obtained in the regionsof the four dimensional parameter space contain-ing a statistically sufficient number of objects.

In order to determine how variability is relatedto a single parameter, the space was divided intosmall regions in three dimensions, and the vari-ability calculated as a function of the remainingparameter in each of the slices. The conditionthat there be enough quasars to reliably measurevariability was the primary limiting factor for thebin sizes. For each parameter there is then a setof variability relations, each set representing theresults of restricting the ranges of the other 3-parameters. As shown in the remainder of thissection, in most cases variability trends are cleareven in independent restricted data sets.

If we make the assumption that the equa-tions describing the multi-parameter dependenceof variability are separable, the results from eachof the slices may be scaled in the single parameterranges where they overlap, in order to find thevariability dependence upon a single parameter.That is, the form of the variability dependence isassumed to be

V (∆τ,Mi, λ, z) = v(∆τ) × v(Mi)× v(λ)× v(z) , (8)

where V is calculated as in the previous sec-tion. While this form is not necessarily correct, itgreatly simplifies the analysis, and the relativelysimple relations found for each parameter suggestthat it is not far from reality. In the followingsubsections, we show the unscaled variability rela-tions for single parameters in the independent re-stricted data sets, then show the results after scal-ing the independent sets together assuming equa-tion 8. The scaled relations are fit with relativelysimple descriptive functions for each parameter.

4.2. Time Lag

We focus first on the dependence of variabilityon rest frame time lag, independent of the other3 parameters. The full quasar sample was firstseparated into 6 redshift bins, each with an equal

9

spacing in logarithmic 1 + z. The redshift binsides are: 0.185, 0.499, 0.895, 1.395, 2.028, 2.829,and 3.840. The quasar sample in each redshift binwas then divided into two halves separated at themedian absolute magnitude of the quasars in theredshift bin: Mi,median = -22.96, -24.07, -25.24,-26.07, -26.73, and -27.09. Taking each photo-metric band separately for the quasars in each ofthe 12 redshift/absolute magnitude bins restrictedthe quasar rest wavelengths to small ranges. Theprocedure produced 36 independent data sets con-fined to small ranges of redshift, absolute magni-tude, and rest wavelength, but unrestricted withrespect to rest frame time lag. The variability am-plitude and uncertainty as a function of time lagwere determined as in § 3, for the quasars in eachof the 36 data sets independently. The time lagbins were set to have a constant logarithmic timewidth, as in § 3, but with twice the width to ac-commodate the smaller number of objects per bin.The rest frame time lag bin sides, in days, are:7.0, 17.6, 44.3, 111.1, 278.7, and 699.2. Unphysical(imaginary) values of the variability amplitude oc-curred in a small number (3) of cases in which thenumber of quasars was relatively small. In mostcases when this occurs the number of quasars is 5or fewer. For all further analysis, binned data setscontaining fewer than 10 quasars or which produceimaginary values of the variability amplitude arerejected. For each of the 36 data sets, the vari-ability with respect to rest frame time lag is anindependent structure function over which the ab-solute magnitude, rest wavelength, and redshift donot vary greatly. The results are shown in Fig. 10.Results for the three photometric bands, corre-sponding to restricted ranges of rest wavelength,are shown in separate panels. The average red-shift of the quasars contributing to each structurefunction is indicated by color, with redder colorsrepresenting progressively higher redshifts. Thestructure functions containing the more luminoushalves of the quasar data sets are shown with solidpoints, and the less luminous halves with openpoints.

It is clear from Fig. 10 that variability is an in-creasing function of rest frame time lag at all red-shifts, absolute magnitudes, and rest wavelengths.Two other trends can also be seen: the less lu-minous quasars vary more than their more lumi-nous counterparts (nearly all of the open points

lie above the solid points within the same red-shift bin); and quasars vary more at shorter wave-lengths, confirming what was shown by the unre-stricted structure functions in § 3.

Under the assumption that the variability as afunction of time lag is separable from the other de-pendencies, the individual structure functions canbe scaled together in the time lag regions wherethey overlap. Using as a reference a structurefunction near the middle of the redshift, luminos-ity, and wavelength distributions, all 36 structurefunctions (excluding bins with too few objects)were scaled such that the sum of the products ofthe amplitudes and the time lag bin widths (theareas under the curves) were equal. The scaledpoints are shown in the last panel of Fig. 10 alongwith the best fit power law. The parameter fitsto a power law and exponential are given in Ta-ble 2. A 2-parameter power law provides a verygood fit and is better than the asymptotic (ex-ponential) form. The power law fit has a slopeof γ = 0.246 ± 0.008, which is comparable tobut shallower than the values found for the un-restricted structure functions in § 3. Scaling thedata points will change the characteristic timescale of the function (the time lag at which thepower law amplitude would be unity), but notthe power law slope. That the slope is relativelyclose to those found for the unrestricted structurefunctions is due to the offsetting variability depen-dencies on luminosity and rest wavelength. FromFig. 9, longer time lags are statistically populatedby more luminous objects, which vary less, butat shorter wavelengths, where the variability isgreater. The significance of the power law slopein relation to variability models is discussed in § 7.

4.3. Absolute Magnitude

The luminosity (absolute magnitude) depen-dence of variability is separated from the otherparameters in a similar way to the time lag in theprevious subsection. The full quasar sample wasseparated into the same 6 redshift bins, and eachseparate quasar sample was further divided intotwo halves at the median rest frame time lag of thequasars in the redshift bin. The median rest frametime lags, in days, for each redshift bin are: 249.9,192.0, 157.1, 122.9, 100.5, and 73.6. Again con-sidering the three photometric bands separatelyrestricted the rest wavelengths of each data set

10

to small ranges. This produced 36 independentquasar subsamples unrestricted with respect to ab-solute magnitude. The variability amplitude as afunction of absolute magnitude, in bins one magni-tude in width, is shown for each quasar subsamplein Fig. 11. The variability amplitude is an increas-ing function of absolute magnitude (brighter ob-jects vary less) for nearly every subsample. Alsoseen in Fig. 11 are the time lag dependence – vari-ability amplitudes are greater at longer time lags(filled symbols) than short time lags (open sym-bols) – and a wavelength dependence, seen mosteasily in the amplitude differences between the gand i bands. This is a clear demonstration thatthe well-known luminosity-variability anticorrela-tion is not simply due to time lag or rest wave-length selection effects.

The data sets were again scaled such that thatthe areas under the curves were equal. In eachredshift bin, the six individual sets (3 wavelengthand 2 time lag bins) were scaled to have identicalsums of the product of the absolute magnitude binwidth and variability amplitude. Then proceed-ing from the lowest to highest redshifts, the scaledredshift sets were rescaled to the adjacent redshiftset so that the areas under the curves were equalin regions where the absolute magnitude coverageoverlapped. The scaled data points are shown inthe last panel of Fig. 11. A straight line can be fitto the data points, but such a description is un-physical since at large luminosities (large negativeabsolute magnitudes) the function becomes neg-ative. In so called Poissonian or discrete-eventsmodels the relative luminosity variability, δL/L isexpected to vary with luminosity as δL/L ∝ L−β,where β = 1

2 in general (e.g. Cid Fernandes, Sodre,& Vieira da Silva 2000). This relationship trans-lates into the absolute magnitude form

v(Mi) ∝ 10βMi/2.5 . (9)

This function and one in which β is held fixed at0.5 were fit to the data. Both fits are shown inFig. 11. The logarithmic equation, with a best fitof β = 0.246 ± 0.005, fits the data as well as astraight line, and avoids the problem of negativevalues. The Poissonian prediction of β = 0.5 givesa poor fit and is clearly inconsistent with the data.Scaling the individual data sets, which accountsfor arbitrary contributions from variability depen-dencies on time lag, rest wavelength, and redshift,

does not change the value of β. We will discussthe implications for Poissonian models further in§7.

4.4. Rest Wavelength

The rest wavelength dependence of variabilitywas isolated for subsamples selected to cover smallranges in redshift, time lag, and absolute magni-tude. The redshift and absolute magnitude binsare the same ones used to isolate the time lag de-pendence (12 separate bins, see §4.2). The datain each of these bins were divided into three sepa-rate samples in the plane of absolute magnitudeand time lag, according to the following lines:Mi = 0.085∆τ−31.25 and Mi = 0.024∆τ−31.42.This gives 36 independent data sets covering smallranges of redshift, absolute magnitude, and timelag. All but one of the data sets contain enoughobjects to compute reliable variability measure-ments. Each data set samples three separate restwavelength points given by the effective rest wave-lengths of the three photometric bands. The vari-ability amplitude as a function of the rest framewavelength for each set is shown in Fig. 12. Theaverage rest frame time lag in each set is colorcoded (longer time lags are redder), the more lumi-nous half of a redshift/time lag bin is shown withsolid points, and each redshift subset is shown ina separate panel.

In most cases, the variability amplitude de-creases with wavelength as expected from previousanalysis. The cases in which the opposite happensoccur at short time lags and very low or very highredshifts, but there are too few cases to make anyclaims about deviations from the general trend.The time lag and luminosity dependencies are alsoevident from the figure.

The data points were scaled in a manner similarto that in § 4.3. For each redshift bin, the six setsof points (3 time lag bins and 2 absolute magni-tude bins) were scaled to the same area under thecurves and then, moving from low to high redshift,the points were rescaled to match the appropriatearea in the adjacent redshift bin. In this case,since the rest wavelength bin limits are not equalfor the separate redshift bins, the three wavelengthpoints in each set were connected by two straightlines, and the area under the lines was calculatedin the regions where they overlapped the wave-lengths of the adjacent redshift bin points. The

11

scaled points as a function of rest wavelength areshown in Fig. 13. If the variability is due to a sim-ple change in the index of a single power law, wewould expect the wavelength dependence to be

v(λ) = −2.5∆αλ log(λ) + C , (10)

where ∆αλ is the difference in the wavelengthpower law index, and C is a constant related tothe pivot wavelength (where the two power lawsintersect) presumed to be much longer than theobserved wavelengths. Figure 13 is plotted witha logarithmic wavelength axis, and so the relationwould be seen as a straight line. A single straightline is an adequate fit from the shortest wave-lengths up to about 4000A, but does not accountwell for the longer wavelength end. Contamina-tion from host galaxy light at longer wavelengthswould have the opposite effect – cause the vari-ability to decrease even faster with wavelength. A3-parameter exponential function, although phys-ically unmotivated, fits the data well

v(λ) = a0 exp(−λ/λ0) + a1 , (11)

with parameter values a0 = 0.616 ± 0.056, λ0 =988± 60A, and a1 = 0.164± 0.003.

4.5. Redshift

The redshift dependence of variability is moredifficult to isolate from the time lag, absolute mag-nitude, and wavelength dependencies. The rea-son for this can be seen from inspection of Fig. 9.For example, samples restricted to a narrow rangeof rest wavelengths will have 3 independent red-shift intervals (taking the three bands separately),but the corresponding absolute magnitude rangesfor the redshift intervals may not overlap appre-ciably. So to isolate the redshift dependence, wefirst found a region of wavelength-absolute mag-nitude space which is covered by quasar data inall three bands. Fig. 14 shows the wavelength-absolute magnitude plane and the selected re-gion which is bounded by a triangle with cor-ners (λ,Mi) : (1250,−27.0), (1250,−29.4), and(3491,−23.8). Outside of this region quasar dataare generally available for only 2 or fewer of thebands. The data contained in this region werethen separated into 16 wavelength bins, shown inFig. 14, at intervals of 100, 200, 400, or 800A, de-pending on the number of objects contained in

the bin. For each bin of wavelength separateddata, objects were selected from a single range oftime lags chosen so that the average absolute mag-nitudes, time lags, and wavelengths were aboutequal for data in each of the three photometricbands. Taking each of the photometric bands sep-arately for a restricted data set gives a wide rangeof redshifts, while keeping the ranges of time lag,absolute magnitude, and rest wavelength nearlyconstant.

The variability amplitudes for all 16 data sets(each with three redshift points) are shown inFig. 15. The number of objects contributing toeach point ranges from 65 to 1150, with a meannumber of 239. Lines connect the points belongingto data sets with nearly the same parameter valuesbut at different redshifts. The color of the pointsand lines corresponds to the average absolute mag-nitude of the data set, with bluer colors represent-ing brighter absolute magnitudes. The point sizescorrespond to the average time lag of the objectscontributing to the points. The absolute mag-nitude correlation, discussed above, is evident inFig. 15, but it is partly counteracted by the nearlymonotonically increasing rest wavelength with av-erage absolute magnitude (seen from Fig. 14), andthe generally longer average time lags associatedwith fainter average absolute magnitudes.

What is of interest here is the dependence of thevariability on redshift. The results are fairly noisyand it is difficult to detect any clear trend with red-shift. The sets of points were scaled by matchingthe areas under the curves of adjacent data sets,as in the previous subsections, starting with thesample with the shortest rest wavelengths. Thescaled data points are shown in Fig. 16. There isa correlation between the scaled variability andredshift – quasars appear to be more variable athigher redshifts. The Spearman rank correlationprobability that the points are uncorrelated is lessthan 10−4, even after accounting for the reductionof the number degrees of freedom by the numberof restricted data sets (16). A straight line fit tothe data points (linear in redshift and variabilityamplitude) gives

v(z) = (0.019± 0.002)z + (0.037± 0.005) . (12)

The correlation, although significant, is weakenough that it could easily have gone unnoticed inprevious variability studies, especially since most

12

of them suffer from a lack of sufficient sample over-lap needed to test the redshift relationship inde-pendently of other parameters. The redshift evolu-tion of variability would have serious consequencesfor a number of currently proposed models. Ifthe effect is intrinsic, the quasar population orthe variability mechanism is changing over time.External causes are also possible, such as gravita-tional microlensing which may increase with red-shift since more potential lenses would be avail-able. The variability correlation with redshift isdiscussed further in §7.

5. Color Dependence

Evidence from previous ensemble studies (Trevese,Kron, & Bunone 2001; Giveon et al. 1999; Edel-son, Pike, & Krolik 1990) suggests that the spec-tral energy distribution of quasars becomes harder(bluer) in bright phases. Indirect evidence alsocomes from the fact that there is a strong wave-length dependence upon variability (§ 4.4). Thiscould happen for example if the index of a powerlaw component of the continuum changes withluminosity (Trevese & Vagnetti 2002).

Quasar colors are a strong function of redshift(Richards et al. 2001), since various spectral fea-tures move into and out of the photometric band-passes with redshift changes. A pure power lawspectrum would have a single set of colors indepen-dent of redshift. The observed quasar color struc-ture is mainly due to the presence of strong emis-sion features, especially broad Fe ii complexes, aswell as the Lyα forest. Figure 17 shows the av-erage imaging photometric colors of quasars as afunction of redshift in two samples selected to beeither brighter or fainter by at least 3 standard de-viations in at least one of the g, r, i bands relativeto the spectrophotometry. We use the imagingphotometric colors rather than the spectroscopicsince they are more precise, and two more colorsare available. Although the color differences aresmall (∼ 0.03), at most redshifts up to at least2.5 and for each color the bright phase sample isbluer than the faint phase sample. Also shownin Fig. 17 are the color differences of the brightphase minus the faint phase samples as a functionof redshift. Both the binned and average colordifferences are shown. The color differences areincreasingly larger for shorter wavelength bands,

i.e. quasars in bright phases are bluer than thosein faint phases, and they are even bluer at shorterwavelengths.

That the color change persists at high red-shift also indicates that it cannot be accounted forsolely by a non-variable red spectral component,such as the quasar host galaxy. Such a compo-nent would contribute a higher fraction of the to-tal quasar light in the faint phases, making quasarsappear redder than in the bright phases. Any rea-sonable host galaxy spectral energy distributionand luminosity would contribute very little lightto the bluest bands, and would quickly be red-shifted out of the other bands. By a redshift of0.5, there should be almost no significant contam-ination from the host galaxies in any of the pass-bands. A host galaxy component cannot accountfor the wide range of redshifts over which the colordifference is significant.

6. Variability of Radio, X-ray, and Broad

Absorption Line Quasars

There is evidence from previous studies thatthe variability amplitude of quasars varies amongdifferent subclasses, such as those with radioemission (Helfand et al. 2001; Eggers, Shaffer, &Weistrop 2000; Garcia, Sodre, Jablonski, & Ter-levich 1999; Pica & Smith 1983) or broad absorp-tion line systems (Sirola et al. 1998). The entireclass of highly variable blazars, for example, isdefined in part by X-ray and radio emission (e.g.Ulrich, Maraschi, & Urry 1997). Here we examinethe variability of broad subclasses of quasars incomparison to carefully selected control samples.

6.1. Radio Detected Quasars

Some SDSS quasar candidates are selected asoptical matches to radio sources in the FIRST sur-vey (White, Becker, Helfand, & Gregg 1997; Ivezicet al. 2002). About 10% of the verified quasars inthe sample have counterparts in the FIRST sur-vey. In the areas of the SDSS sample covered bythe FIRST survey at the time the quasar candi-dates were selected, there are 1553 verified quasarswith FIRST catalog matches. To test whetherquasars with detected radio emission are more orless variable than those without, we have extracteda control sample of quasars without matches in theFIRST catalog. The non-radio-detected control

13

sample was constructed to have the same redshift,luminosity, and time lag distribution as the radio-detected sample, by matching each radio quasarwith a non-radio quasar with nearly the same red-shift, magnitude, and time lag. The standarddeviations of the redshift, magnitude, and timelag differences are σ(∆z) = 0.03, σ(∆m) = 0.04mag., and σ(∆(∆τ)) = 3.6 days, and in no casewere the differences allowed to be greater than0.5, 0.5 mag., and 40 days respectively. Of theFIRST matched quasars, 1376, 1389, and 1388were able to be matched with counterparts in thenon-FIRST sample in the g, r, and i bands respec-tively. Kolmogorov-Smirnov tests comparing theredshift, magnitude, and time lag distributions ofthe radio and control samples show that they arestatistically indistinguishable. This also guaran-tees that the wavelength coverages of the samplesare nearly identical.

A radio-loud (not simply radio-detected) sub-sample and its corresponding non-radio controlsample were also generated. Radio loudness isdefined here as the ratio of the rest frame 5GHzto 4500A flux-densities (e.g. Sramek & Weedman1980), and a quasar is deemed radio-loud if theratio is at least 100. The sample sizes of the radio-loud quasars and the matched radio-quiet controlquasars are 492, 530, and 546 objects for the g, r,and i bands respectively.

The matched time lag structure functions forthe full radio and non-radio samples are shown inFig. 18. There is no significant difference in thebinned structure functions of the radio-detectedand undetected quasars. On the other hand, thematched structure functions for the radio-loudsub-sample, shown in Fig. 19, are about 1.3 timeshigher than the non-radio sample. However, onlythe difference in the i band is significant (matchedpair t-test probability of 1% if the samples werenot truly different). Thus, there is marginal evi-dence that radio-loud quasars are more opticallyvariable than radio-quiet quasars, but a largersample will be needed to confirm this.

The qualitative result that the radio-loudquasars are more variable agrees with most ofthe past suggestions (Helfand et al. 2001; Eggers,Shaffer, & Weistrop 2000; Garcia, Sodre, Jablon-ski, & Terlevich 1999; Pica & Smith 1983). Sincemost blazars are radio-loud (e.g. Ulrich, Maraschi,& Urry 1997), the higher variability amplitude

of radio-loud quasars may reflect a higher frac-tion of blazars. There is not enough informationfrom this survey to reliably classify individual ob-jects as blazars (at the very least, more detailedlightcurves are needed). Further subdivision of thesample by finer radio loudness currently yields toofew quasars for meaningful comparisons. In anycase, the evidence for a correlation between radioloudness and UV/optical variability amplitude issuggestive, but not conclusive.

6.2. X-Ray Detected Quasars

As with radio quasars, some of the SDSSquasars are selected for spectroscopic follow-upas matches to sources in the Rosat All Sky Sur-vey (RASS, Voges et al. 1999, 2000). A detailedanalysis of RASS source matches to the SDSSdata is given by Anderson et al. (2003). About5% of the verified quasars can be matched withRASS sources, giving about 1300 X-ray quasarsin our sample. We constructed the X-ray andcontrol samples in the same way as for the radiosample and its control. The numbers of matchedobjects in each of the g, r, and i bands are 1010,1008, and 1009 respectively. Again, Kolmogorov-Smirnov tests show that with respect to redshift,luminosity, and time lag, the two sets of samplesare indistinguishable.

The time lag structure functions for the X-ray detected and X-ray-non-detected sample areshown in Fig. 20. The X-ray sample is more vari-able than the non-X-ray control sample at timelags up to about 250 days, after which the differ-ences of variability amplitudes are much smaller.Overall the X-ray sample amplitudes are larger bya factor of ≈ 10%. The matched pair t-test prob-abilities of the differences occurring by chance are9.1%, 0.8%, and 0.4% for the g, r, and i bands re-spectively. The data therefore show that X-ray se-lected quasars are significantly more variable thantheir X-ray-non-detected counterparts. However,the difference may become less significant at longertime lags and at longer wavelengths.

The higher X-ray variability amplitude is prob-ably not surprising given the high fraction of X-raydetected objects among blazars. As for the radiosample, the X-ray sample was selected purely byoptical matches to catalog sources; no informationabout the variability of the objects in the samplewas used beforehand. This is the first time that a

14

large X-ray selected sample of quasars has been ex-amined for UV/optical variability. The SDSS sam-ple will soon be large enough to subdivide it fur-ther by X-ray brightness. In the meantime, a cor-relation between X-ray emission and UV/opticalvariability amplitude can be claimed. We discussthis further in § 7.

6.3. Broad Absorption Line Quasars

Broad absorption line quasars (BALQSOs) aredefined by the presence of very strong, blue-shiftedabsorption troughs in their spectra. About 5% ofthe quasars in the SDSS sample can be classifiedas BALQSOs, a fraction which is heavily redshiftand color dependent. The largest systematicallyselected samples of BALQSOs are those of Re-ichard et al. (2003) and Tolea, Krolik, & Tsve-tanov (2002), each of which contain close to thesame sets of objects drawn (in somewhat differentways) from the SDSS Early Data Release quasarsample. Both sets contain about 200 objects, andfor the present purposes the samples are indistin-guishable; we have used the Reichard et al. (2003)sample since the selection process is more auto-mated and is likely to be used for future BALQSOcatalogs. A control sample of non-BAL quasarswas designed to have consistent redshift, luminos-ity, and time lag distributions, in the manner de-scribed above. The matched sample of BALQSOscontains 178, 189, and 190 objects in the g, r, andi bands respectively.

The matched structure functions are shown inFig. 21. The time lag binning is necessarily coarsedue to the relatively small sample sizes. At thelevel of sensitivity of this sample, there is no dif-ference in the variability amplitudes of BAL andnon-BAL quasars. Currently favored models ofthe BAL phenomena attribute the absorption tohigh opacity gas, either as clouds or flows, usu-ally viewed near the plane of an accretion disk.If the presence of BALs is purely a viewing an-gle effect, then continuum variability, if due to thecentral quasar engine is unlikely to be correlated.However, if variability is due to the presence ofobscuring dust of varying attenuation crossing thesightline to a quasar, BAL quasars may be ex-pected to be more highly variable at optical andUV wavelengths. The issue will need to be settledwith a larger sample, but the current results donot support any correlation between the presence

of BAL features and UV/optical variability.

7. Discussion

To summarize our results, we have separatedthe dependence of variability on a number of pa-rameters, and found a power law dependence ontime lag, anticorrelations with wavelength and lu-minosity, and a correlation with redshift. All ofthese relationships have been parameterized. Ra-dio loud and X-ray quasars also appear to bemore variable than their quiet counterparts. Thereis currently no model of quasar continuum vari-ability at optical and UV wavelengths that ad-dresses all of the relationships described here, anduntil recently, there were virtually no quantita-tive predictions. Current models can be classifiedbroadly into three groups: accretion disk instabil-ities, discrete-event or Poissonian processes, andgravitational microlensing. We ignore other evi-dence for or against the theories and describe howvariability as an independent phenomenon mayconstrain the models.

The Poissonian models postulate that quasarluminosity, or at least a significant fraction of it,is generated by some type of multiple discrete andindependent energetic events, such as supernovaeor star collisions (e.g. Terlevich, Tenorio-Tagle,Franco, & Melnick 1992; Torricelli-Ciamponi,Foellmi, Courvoisier, & Paltani 2000). The sta-tistical superposition of the light curves of therandomly occurring events determines the lumi-nosity at any given time. As discussed in § 4.3, thesimplest Poissonian models predict a luminositydependent power law slope of β = 1/2 which isinconsistent with our results. More detailed mod-els in which the event rate, energy, timescale, andbackground contribution are adjustable parame-ters can produce a wide range of slopes (Cid Fer-nandes, Sodre, & Vieira da Silva 2000), but a valueof β = 1/2 is still difficult to avoid in models invok-ing supernovae and their remnants as the events(Paltani & Courvoisier 1997; Aretxaga, Cid Fer-nandes, & Terlevich 1997). Another apparentlyunavoidable consequence of the Poissonian modelsis that the variable luminosity component is notwavelength dependent (Cid Fernandes, Sodre, &Vieira da Silva 2000), and any color changes mustbe the result of a non-variable component (such asa host galaxy), which must be red to qualitatively

15

account for the wavelength correlation found hereand in other studies. We have shown that the vari-able source itself is wavelength dependent, and ahost galaxy component alone cannot account forthe color changes. Quantitative predictions forthe power law slope of the structure function inthe starburst (supernova) model (Kawaguchi, Mi-neshige, Umemura, & Turner 1998) range fromγ ≈ 0.7 − 0.9, which are also quite inconsistentwith the value we find (γ = 0.246). Thus, basedon predictions for the time lag, wavelength, andluminosity dependence of variability, current Pois-sonian models are inconsistent with the observa-tional results of this study. It remains to be seenif non-Poissonian processes, for example in whichthe events are not independent or random, canaccount for the observations.

The idea that the motions of intervening matteralong the geodesics to quasars may cause flux vari-ations (microlensing) was discussed as early as thelate 1970s (Chang & Refsdal 1979), but few quan-titative predictions have been worked out with re-gard to typical quasar variability. Using the simu-lated microlensing light curves of Lewis, Miralda-Escude, Richardson, & Wambsganss (1993) andSchneider & Weiss (1987), Hawkins (2002) gen-erated structure functions which have a powerlaw form with slopes ranging from γ ≈ 0.23 toγ ≈ 0.31 (Hawkins 2002), which is consistentwith what we find. However, the slopes dependupon the unknown lens mass distribution func-tion, velocity distribution, and source size (Wyithe& Turner 2001), so that a wide range of val-ues is possible. There is little doubt that mi-crolensing of quasar images does happen, and ithas likely been detected in at least two cases,Q2237+0305 (Schmidt et al. 2002, and referencestherein) and Q0957+561 (Refsdal, Stabell, Pelt, &Schild 2000). However in each case, the quasar ismacrolensed by a foreground galaxy, which meansthat the geodesics selectively pass through re-gions of relatively high density. Wyithe & Turner(2002) showed that the probability of microlens-ing by stars among single image (not macrolensed)sources is very small; dark matter composed ofcompact objects can improve the probability, butat most only about 10% of sources are expectedto be microlensed at any given time. In addition,since unresolved macrolensed quasars – which ap-pear more luminous than they really are – are

more likely to be microlensed, the anticorrelationof variability amplitude with luminosity is oppo-site to the trend that would be expected from mi-crolensing. While the deflection of light by gravityis achromatic, the wavelength dependence of vari-ability is not necessarily evidence against the mi-crolensing hypothesis, as long as quasar emittingregions are smaller and brighter with decreasingwavelength. It is difficult, however, to see whythere would be any dependence on the radio orX-ray properties of the quasar sample. The am-plitude of quasar variability changes with redshift,but it is nearly as strong at low redshifts as it is athigh redshifts, which also shows that microlensingcan not be the primary cause of variability, sincemicrolensing events should be extremely rare atlow redshift. Finally, reverberation mapping stud-ies (Peterson 2001) show that the quasar broadline region varies in response to changes in the con-tinuum luminosity, showing that a large fractionof variability must be intrinsic to quasars. If somequasar variability is due to microlensing, it will beimportant to isolate it from the other sources sinceit has the potential to constrain the componentsof dark matter and the structure of quasars.

It is widely accepted that quasar luminosity isgenerated through some set of processes relatedto the accretion of matter from a disk onto asupermassive black hole (e.g. Rees 1984). It istherefore natural to consider mechanisms associ-ated with changes in these processes as a sourceof quasar variability. Qualitatively, most schemeswould tend to follow the trends we find here, inparticular the anticorrelations of variability withwavelength and luminosity. For example, thedisk emission spectrum of the standard opticallythick geometrically thin accretion disk model (e.g.Shakura & Sunyaev 1973) is more luminous andbluer when the accretion rate is higher, and therelative luminosity change would be lower in moreluminous objects for a given change in accretionrate. However, it is not known how the accre-tion rate would change nor how the resultant lumi-nosity changes would propagate through the disk.While there has been much theoretical discussionof the various possibilities for the emission mecha-nisms and their instabilities (e.g. Wallinder, Kato,& Abramowicz 1992; Schramkowski & Torkels-son 1996), there are currently few quantitativepredictions which can be compared to observa-

16

tional results. Kawaguchi, Mineshige, Umemura,& Turner (1998) generated structure functions forthe cellular-automaton model for disk instability,and found power law forms with slopes rangingfrom γ = 0.41 to 0.49. While this range is incon-sistent with our results, the model is necessarilysimplified and a number of assumptions need tobe made. The complexity of possible accretiondisk (or jet) instability models is likely what hasprevented more quantitative predictions. Disk in-stability models are clearly promising, but as yetit is difficult to compare them to the observations.

In summary, the weight of the observational ev-idence seems to disfavor gravitational microlensingand generic Poissonian processes as the primarysource of quasar variability. Accretion disk insta-bility models have yet to be adequately developedquantitatively for direct comparison with our re-sults. It is also plausible that a combination ofsources produce variations in quasar lightcurves,and no single model can be completely eliminatedat this time.

8. Conclusions

We have examined the ensemble broadbandphotometric variability of a very large and homo-geneous sample of quasars from the SDSS – thelargest sample ever used to study variability. Thethree-band spectrophotometry of each object wascompared directly to the imaging photometry ob-tained at an earlier epoch. Because of the largenumber of objects and wide coverage of param-eter space, the dependences of variability ampli-tude on time lag, luminosity, wavelength, and red-shift were able to be disentangled for the first time.The variability amplitude increases with time lag(up to about two years) as a power-law with aslope of γ = 0.25. In terms of the variabilityamplitude, more luminous quasars are less vari-able, shorter wavelengths are more variable, andmore distant quasars are somewhat more variable;all of these relationships are parameterized. Ra-dio loud quasars appear to be more variable thantheir radio quiet counterparts, and quasars withdetectable X-ray emission (in the ROSAT survey)are more variable than those without. It is diffi-cult to explain the results in the context of modelsinvolving discrete events (Poissonian models) andgravitational microlensing. Accretion disk insta-

bility models are promising, but more quantita-tive predictions are needed to test them againstthe observational results.

Funding for the creation and distribution of theSDSS Archive has been provided by the Alfred P.Sloan Foundation, the Participating Institutions,the National Aeronautics and Space Administra-tion, the National Science Foundation, the U.S.Department of Energy, the Japanese Monbuka-gakusho, and the Max Planck Society. The SDSSWeb site is http://www.sdss.org/.

The SDSS is managed by the Astrophysical Re-search Consortium (ARC) for the Participating In-stitutions. The Participating Institutions are TheUniversity of Chicago, Fermilab, the Institute forAdvanced Study, the Japan Participation Group,The Johns Hopkins University, Los Alamos Na-tional Laboratory, the Max-Planck-Institute forAstronomy (MPIA), the Max-Planck-Institute forAstrophysics (MPA), New Mexico State Univer-sity, University of Pittsburgh, Princeton Univer-sity, the United States Naval Observatory, and theUniversity of Washington.

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Fig. 1.— The uncorrected spectroscopic minus photometric magnitudes vs. imaging PSF magnitudes forstars observed on the same spectral plates as the quasars (left). Results for all three bandpasses are shown.The curves show the binned median trends and the upper and lower 68.3% confidence envelopes. The rightside panels show the uncorrected magnitude difference histograms. The 68.3% confidence half-widths, σ, aregiven for a spectroscopic S/N of 10.

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Fig. 2.— The corrected spectroscopic minus photometric magnitudes vs. spectral signal-to-noise ratiofor stars observed on the same spectral plates as the quasars (left). Results for all three bandpasses areshown. The curves are the fits to the 68.3% confidence half-width envelopes. The right side panels showthe magnitude difference histograms for each band. The 68.3% confidence half-widths, σ are given for aspectroscopic S/N of 10.

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Fig. 3.— The corrected spectroscopic minus photometric magnitudes vs. spectral signal-to-noise ratio forthe quasars (left). Results for all three bandpasses are shown. The curves are the binned 68.3% confidencehalf-width envelopes. The right side panels show the magnitude difference histograms for each band. The68.3% confidence half-widths, σ, are given for a spectroscopic S/N of 10.

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Fig. 4.— Magnitude difference (uncorrected for measurement uncertainties) vs. rest frame time delay in eachof the three photometric pass bands. The binned points show the mean values while the error bars show theroot-mean-square deviations divided by the square root of the number of objects in a bin.

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Fig. 5.— Magnitude difference (uncorrected for measurement uncertainties) vs. absolute i band magnitudein each of the three photometric pass bands. The binned points show the mean values while the error barsshow the root-mean-square deviations divided by the square root of the number of objects in a bin.

24

Fig. 6.— Magnitude difference (uncorrected for measurement uncertainties) vs. rest wavelength in each ofthe three photometric pass bands. The binned points show the mean values while the error bars show theroot-mean-square deviations divided by the square root of the number of objects in a bin.

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Fig. 7.— Magnitude difference (uncorrected for measurement uncertainties) vs. redshift in each of the threephotometric pass bands. The binned points show the mean values while the error bars show the root-mean-square deviations divided by the square root of the number of objects in a bin.

26

Fig. 8.— Quasar structure functions for each of the three pass bands, color coded by band. No accountinghas been made for any other variability dependencies, such as luminosity, wavelength, or redshift. Singlepower law fits to the data are also shown.

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Fig. 9.— Projected parameter values for all of the quasars. This effectively shows the selection function inthe parameter space given by rest frame time lag, ∆τ , redshift, z, absolute i-band magnitude, Mi, and restwavelength, λrest. The passbands are indicated when necessary. For clarity, only the r band data are shownfor the middle two plots.

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Fig. 10.— Variability as a function of rest frame time lag for independent slices of data (upper left, upperright, and lower left panels). Colors indicate redshift, with redder colors showing data from higher redshiftranges, according to the color key. Solid and open points show results from slices with higher and lowerluminosities respectively, within the same redshift range. Results from the three photometric bands are givenseparately, which effectively restricts the rest wavelength ranges to small values in each slice. The lower rightpanel shows all of the scaled data points, along with the best fit power law for variability vs. ∆τ .

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Fig. 11.— Variability as a function of absolute magnitude for independent slices of data (upper left, upperright, and lower left panels). Colors indicate redshift, with redder colors showing data from higher redshiftranges, according to the color key. Solid and open points show results from slices with longer and shortertime lags respectively, within the same redshift range. Results from the three photometric bands are givenseparately, which effectively restricts the rest wavelength ranges to small values in each slice. The lower rightpanel shows all of the scaled data points, along with a best fit generalized Poissonian function (solid) andPoissonian function with the power law index fixed at 1/2 (dashed).

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Fig. 12.— Variability as a function of rest frame wavelength for independent slices of data. Colors indicateaverage rest frame time lag, with redder colors showing longer time lags, according to the color key. Solidand open points show results from slices with more luminous and less luminous quasars respectively, withinthe same redshift and time lag ranges. Results from the six redshift slices are shown in separate panels.

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Fig. 13.— Scaled variability amplitude points from Fig. 12 as a function of rest frame wavelength. The lineshows the best fit exponential function as described in the text.

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Fig. 14.— The triangle shows the region of the wavelength-absolute magnitude plane that is simultaneouslywell covered by quasar observations in all three photometric bands. The vertical lines – separated by 100,200, 400, or 800A – show the bin sides used to separate the quasar sample by wavelength.

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Fig. 15.— Variability as a function of redshift for independent sets of data. Sets of three connected pointsshow the variability amplitude for data sets with similar distributions of rest wavelength, absolute magnitude,and time lag, but different average redshifts. Colors correspond to the average absolute magnitude for eachset, with bluer colors indicating brighter absolute magnitudes, according to the color key. The point sizescorrespond to average time lags, according to the point size scale.

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Fig. 16.— Scaled variability amplitude as a function of redshift. The best fit line (to linear redshift andvariability amplitude) is also shown.

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Fig. 17.— Average faint phase vs. bright phase colors of quasars as a function of redshift (left). Thestructure in the plots is mainly due to spectral features redshifting into and out of the various passbands.The plots on the right side show the color difference between the bright and faint phases as a function ofredshift, along with the uncertainties in each bin. A more negative value implies a bluer spectrum in thebright phase.

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Fig. 18.— Radio-detected (solid lines and points) vs. undetected (dashed lines and open points) structurefunctions.

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Fig. 19.— Radio-loud (solid lines and points) vs. radio-quiet (dashed lines and open points) structurefunctions.

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Fig. 20.— X-ray-detected (solid lines and points) vs. undetected (dashed lines and open points) structurefunctions.

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Fig. 21.— BALQSO (solid lines and points) vs. non-BALQSO (dashed lines and open points) structurefunctions.

40

Table 1

Parameter values for fits to the binned structure functions.

Power Law

Band ∆τ0 (days) γ χ2

g 9.90± 6.49× 104 0.293± 0.030 1.51r 7.05± 4.23× 104 0.336± 0.033 1.20i 1.66± 1.32× 105 0.303± 0.035 4.20

Exponential

Band V0 (mag) ∆τ0 (days) χ2

g 0.168± 0.005 51.9± 6.0 20.5r 0.155± 0.006 74.7± 8.9 24.8i 0.139± 0.005 62.6± 8.3 39.3

Table 2

Parameter values for fits to the scaled variability time lag dependence.

Power Law

∆τ0 (days) γ χ2

5.36± 1.46× 105 0.246± 0.008 299.0

Exponential

V0 ∆τ0 (days) χ2

0.144± 0.001 40.4± 1.4 443.4

41