Tests for Standard Accretion Disk Models by Variability in A(7)

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

4.VARIABILITYTIMESCALEANDTIMELAG

Twomethodsareappliedtoanalysisofvariabilitytimescale.Oneisthemostcommonde nitionofthevariabilitytimescale(e.g.,Wagner&Witzel1995).Anotherisawellde nedquantity,thezero-crossingtimeoftheautocorrelationfunctionoflightcurves.Timelagsareanalyzedbythez-transformeddiscretecorrelationfunction(ZDCF;Alexander1997).Thentheanalysisresultsarecomparedtopredicationsofaccretiondiskmodels.

4.1.AnalysisofVariabilityTimeScale

Thevariabilitytimescaleshavebeende nedindi erentways.Themostcommonde nitionofthevariabilitytimescaleτ=F/| F/ t|andthemoreconservativeapproachofτ=| t/ lnF|havetheadvantageofweighting uctuationsbytheiramplitudes,whereFisthe ux,and Fisthevariabilityamplitudeinthetimescale t(e.g.,Wagner&Witzel1995).Hereweusethemostcommonde nitionofvariabilitytimescaleτ=F/| F/ t|,whereFistakenasthe uxattheminimum.Inthispaper,werefertotheintervalbetweensubsequentlocalminimaandmaximaattheadjacentvalleysandpeaksintheentirelightcurve.First,weselectsubsequentvalleyandpeaksu cientlydensesampledinonelightcurve.Second,variationsof F/F≥30%betweenthesubsequentminimumandmaximumarerequiredwithinthetimescale t.Theestimatedvaluesofτarelistedincolumn

(2)ofTable2.Theuncertaintyonthevaluesofτareestimatedbytherelationστ= t(σFmin| F| Fmin|σFmax σFmin|)/| F|2,where F=Fmax Fmin,σFmaxistheobserved

errorofFmax,andσFministheobservederrorofFmin.

FormostAGNs,itisdi culttode neasinglecharacteristicvariabilitytimescale.OneapproachtoasingletimescaleisdescribedbyGiveonetal.(1999).Theirde nitionisgivenasthezero-crossingtimeoftheautocorrelationfunction(ACF).Ifthereisanunderlyingsignalwithatypicalvariabilitytimescaleinthelightcurve,thewidthoftheACFpeaknearzerotimelagwillbeproportionaltothisvariabilitytimescale(e.g.,Giveonetal.1999;Netzeretal.1996).Thiszero-crossingtimeoftheACF,τ0,isawellde nedquantity,andisusedasacharacteristicvariabilitytimescale(e.g.,Alexander1997;Giveonetal.1999;Netzeretal.1996).Anotherfunctionusedinvariabilitystudiestoestimatethevariabilitytimescaleisthe rst-orderstructurefunction(SF)(e.g.,Treveseetal.1994).ThereisasimplerelationbetweentheACFandtheSF(seeEq.(8)inGiveonetal.1999)parisonofτwithτ0isperformedtotestthereliabilityofthevariabilitytimescaleτlistedincolumn(2)ofTable2.TheACFisestimatedbytheZDCF(Alexander1997).Ithasbeenshownthatthismethodisstatisticallyrobustevenwhenappliedtoverysparselyandirregularlysampledlightcurves(Alexander