Tests for Standard Accretion Disk Models by Variability in A(8)

In this paper, standard accretion disk models of AGNs are tested using light curves of 26 objects well observed for reverberation mapping. Time scales of variations are estimated by the most common definition of the variability time scale and the zero-cros

1997).TheZDCFwascalculatedforallofthelightcurvesusedtoestimateτ.FollowingGiveonetal.(1999),aleast-squaresprocedureisusedto ta fth-orderpolynomialtotheZDCF,andtheZDCF tisusedtoevaluatethezero-crossingtimeintheobserver’sframe.

Theevaluatedresultsarelistedincolumn(3)ofTable2.Foronelightcurve,theZDCFcodeofAlexander(1997)canautomaticallysethowmanybinsaregivenandusedtocalculatetheACF.Thus,thetimelaganditsuncertaintyareimmediatelygivenforeachbinintheACF.However,thiscodecannotestimatetheuncertaintyonthe tvalueofτ0totheACF.Ifthe tτ0ismostnearthetimelagofonebinintheACF,theuncertaintyofthe tτ0maybeapproximatedbytheuncertaintyoftimelaginthisbinintheACF.Thus,theuncertaintyonthevaluesofτ0inTable2isassumedtobetheerrorsoftheACFpointsnearesttothe tvaluesofτ0.

Forcomparison,weplottedτversusτ0inFigure1.ItcanbeseeninFigure1thatthedatapointsarebasicallysharedbytwosidesofthelineτ0=τ.Thelinearregressionanalysisshowsthatthereisacorrelationbetweenτandτ0withPearsoncorrelationcoe cientr=0.766atthechanceprobabilityP=5.1×10 6.Theregressionline ttedbytheordinaryleast-squaresbisectorregressionanalysis(Isobeetal.1990)is

τ0/(1+z)= 96.1(±33.8)+1.5(±0.3)τ/(1+z),(14)

wherezistheredshift,andτandτ0areinunitsofdays.Thissuggeststhattheτandτ0areacceptabletocharacterizethetypicalvariabilitytimescale,andthattheestimatedresultsofτlistedincolumn(2)ofTable2arereliable.

4.2.AnalysisofTimeLag

Cross-correlationfunction(CCF)analysisisastandardtechniqueintimeseriesanalysisto ndtimelagsbetweenlightcurvesatdi erentwavelengths,andthede nitionoftheCCFassumesthatthelightcurvesareuniformlysampled.However,inmostcasesthesamplingisnotuniform.Theinterpolatedcross-correlationfunction(ICCF)methodofGaskell&Peterson(1987)usesalinearinterpolationschemetodeterminethemissingdatainthelightcurves.Ontheotherhand,thediscretecorrelationfunction(DCF;Edelson&Krolik1988)canutilizeabinningschemetoapproximatethemissingdata.ApartfromtheICCFandDCF,thereisanothermethodofestimatingtheCCFinthecaseofnon-uniformlysampledlightcurves,thatis,thez-transformeddiscretecorrelationfunction(Alexander1997).TheZDCFwasusedasanestimationoftheACFin§4.1;hereitisusedasanestimationoftheCCF.TheZDCFisabinningtypeofmethodasanimprovementoftheDCFtechnique,buthasanotablefeaturethatthedataarebinnedbyequalpopulationratherthanequal