Tests for Standard Accretion Disk Models by Variability in Active Galactic Nuclei

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

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aTestsforStandardAccretionDiskModelsbyVariabilityinActiveGalacticNucleiH.T.Liu1,J.M.Bai1,X.H.Zhao1,andL.Ma2ABSTRACTInthispaper,standardaccretiondiskmodelsofAGNsaretestedusinglightcurvesof26objectswellobservedforreverberationmapping.Timescalesofvariationsareestimatedbythemostcommonde nitionofthevariabilitytimescaleandthezero-crossingtimeoftheautocorrelationfunctionoftheopticallightcurvesforeachsource.Themeasuredtimescalesofvariationsbythetwomethodsareconsistentwitheachother.Ifthetypicalvalueoftheviscosityparameterα～0.1isadopted,themeasuredopticalvariabilitytimescalesaremostclosetothethermaltimescalesofthestandarddisks.Ifαisallowedtorangefrom～0.03to～standard0.2,thedisks.measuredTheretimeisascaleslineararerelationconsistentbetweenwiththethemeasuredthermaltimevariabilityscalesoftimethescalesandblackholemasses,andthislinearrelationisqualitativelyconsistentwithexpectationofthestandardaccretiondiskmodels.ThetimelagsmeasuredbytheZDCFbetweendi erentbandsareontheorderofdays.ThemeasuredtimelagsofNGC4151andNGC7469aremarginallyconsistentwiththetimelagsestimatedinthecaseofcontinuumthermalreprocessingforthestandardaccretiondiskmodels.However,themeasuredtimelagsofNGC5548andFairall9areunlikelytobethecaseofcontinuumthermalreprocessing.OurresultsareunlikelytobeinconsistentwithorarelikelytobeconditionallyinfavorofthestandardaccretiondiskmodelsofAGNs.

Subjectheadings:accretion,accretiondisks—blackholephysics—galaxy:active—galaxy:quasars—galaxy:Seyfert

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

1.INTRODUCTION

Large uxvariationsontimescalesfromyearstohoursarecommoninactivegalacticnuclei(AGNs),andlongertimescalevariationsoftheorderofmonthstoyearsmayberelatedtothepropagationoftheshortertimescalevariations(e.g.,Ulrichetal.1997).Thecombinationofhigh uxvariabilityandshortvariabilitytimescalesimpliesthattheenergyconversioninAGNsismoree cientthantheordinarystellarprocesses.Accretionofmatterontoablackholecanhavehighenergyreleasee ciency(Reesetal.1982;Rees1984).TheevidencethatAGNssuchasquasarsandSeyfertgalaxiesarepoweredbygravitationalaccretionofmatterontosupermassiveblackholesisnowquiteconvincing.Certainlytherehasbeennode nitivedetectionoftherelativistice ectsthatwouldberequiredforunam-biguousidenti cationofasingularity,althoughstudiesoftheironKαemissionlineintheX-rayspectraofAGNscurrentlyprovidessomepromise(e.g.,Reynolds&Nowak2003).

Ingeneraloptical–UVradiationsofmostnon-blazartypesourcesarewithintheso-calledBigBlueBump.Theopticalvariabilityischaracterizedaspoorlyunderstood,butisneverthelessrecognizedasameansofprobingphysicalscalesthatcannotberesolvedspatiallybyanytelescopeorinstrument(e.g.,Netzer&Peterson1997;Petersonetal.2004;Woldetal.2007).Anumberofmodelshavebeenproposedtoexplainoptical–UVquasarvariability.Onewayofattemptingtohelpconstrainingtheproposedmodelsisto ndrelationshipsbetweenvariabilityandotherparametersofAGNs,suchasblackholemass.TheblackholemassisafundamentalparameterofAGNs,andthediscoveryofsucharelationship–orlackthereof–mayprovideusefulcluestothephysicalmechanismsbehindthevariability(e.g.,Woldetal.2007).Processesintrinsictothecentralengineitselfcoulddominate.Woldetal.(2007)investigatethedependenceofquasarvariabilityonblackholemass,and ndthatblackholemasscorrelateswiththemeasuredvariabilityamplitude.Anumberofmodelsforquasaropticalvariabilityexistbuttherearenoclearpredictionsrelatingblackholemassandvariabilityamplitude.Di erentsourcesofopticalvariationscanbeassociatedwithdi erentcharacteristictimescales,andmanyofthesetimescalesdependonblackholemass.Collier&Peterson(2001)attempttode nearelationshipbetweenblackholemassandcharacteristicvariabilitytimescale.Studying10well-monitoredAGNs,theyreportevidenceofblackholemasscorrelatingwithcharacteristicopticalvariabilitytimescalesthatareroughlyconsistentwithaccretiondiskthermaltimescales.

Thestandardaccretiondiskisthebasicmodelforaradiativelye cient,geometricallythin,opticallythickdisk(Shakura&Sunyaev1973).Inthestandardpicturethisaccretiondiskradiatesthermallymainlyintheoptical/UVbandsforAGNswithblackholemassesof～106–109M⊙.AGNswithblackholemassesof～107–109M⊙wouldbeexpectedtohaveaccretiondiskthermalcharacteristictimescalesoftheorderofmonthstoyears.Many

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

investigationsbasedoncentralradiationsfromthinaccretiondiskshavebeendone(e.g.,Ebisawaetal.1991;Hanawa1989;Lietal.2005;Pereyraetal.2006;Zimmermanetal.2005).Connectionsofjetsanddisks,averyimportantaspectofAGNresearches,havebeeninvestigatedonthebasisofstandardaccretiondisks(e.g.,Meier2001,2002).Thoughmanyinvestigationsareonthebasisofstandarddisks,onlyafewinvestigationsaimattestingstandardaccretiondiskmodelsbyobservations.Collieretal.(1998)brie ydiscussedtherelationoftimedelaysbetweentheUVandopticalcontinuumvariationswithaccretiondiskinNGC7469.Itisbelievedfornon-blazartypeAGNsthattheoptical/UVemissionsareproducedthermallyfromaccretiondisks.Theradiationenergiesofthermalemissionsemittedinaccretiondisksarefromtwopossiblecontributions.Onewell-knownoriginisthelocalviscousdissipationinaccretiondisks.Thislocalviscousdissipationcanproducethelocalthermalequilibrium,andthenthelocalblackbodyemissions(e.g.,Krolik1999).AnotheroriginisthereprocessedX-rays.TheX-raysarecommonlyattributedtoComptonup-scatteringofthethermalUVphotonsproducedbytheviscousdissipation(e.g.,Sunyaev&Titarchuck1980;Haardt&Maraschi1991).Inthecaseofthermalemissionsfromviscousdissipation,theaccretion ow uctuationstravellinginwardsacrosstheemittingregionsa ect rsttheopticalemittingregionatouterradii,andthentheUVemittingregionatinnerradii.Thenthelongerwavelengthvariationsarelikelytoleadtheshorterwavelengthones.Iftheradialtemperaturepro lesofaccretiondisksarenotsetprimarilybyviscouse ects,butbyirradiationfromthecentralX-raysources,thelongerwavelengthvariationsarelikelytolagtheshorterwavelengthonesforthermalemissionsfromcontinuumthermalreprocessing.The uxvariabilitymustoccuronaphysicaltimescalethatisconsistentwiththechosenmodel.Thetimescalesofinterestarethelightcrossing,dynamical,thermal,andsoundcrossingtimescalesthataresetbytheblackholemass(Frank,King,&Raine2002),andtheorder-of-magnitudescalesare

τl=6M8ξ3days,

τdyn=6M8ξ33/2(1)(2)

(3)

(4)months,3/2τth=τdyn/α=5M8ξ3τs=70M8ξ3T5 1/2yrs,yrs,

whereM8=MBH/108M⊙,α(～0.1)istheShakura-Sunyaevviscosityparameter(Shakura&Sunyaev1973),T5=T/105K,andξ3=rd/103rg(rdisthediskradius,andrg=GMBH/c2thegravitationalradius).Inordertodeterminewhichphysicalmechanismisresponsibleforthevariabilityandtoteststandardaccretiondiskmodels,itisnecessarytoconnecttheobservedvariabilitytimescaleswithoneoftheabovephysicaltimescalesandtosearchthecorrelationofblackholemasseswithcharacteristicopticalvariabilitytimescales,anditis

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

importanttocomparetheobservedtimelagsbetweendi erentbandswiththetheoreticalvaluespredictedbythestandardaccretiondiskmodels.

Thestructureofthispaperisasfollows.Thesampleanddataarein§2.Thecal-culationsoftemperaturepro lesaredescribedin§3.§4presentsvariabilitytimescalesandtimelags.§4.1isanalysisofvariabilitytimescale,§4.2analysisoftimelag,and§

4.3comparisontomodels.Thelastsectionisdiscussionsandconclusions.Throughoutthispaper,weusea atcosmologywithadecelerationfactorq0=0.5andaHubbleconstantH0=75kms 1Mpc 1.

2.THESAMPLEANDDATA

TheobjectslistedinTable1arebasedonthesamplesanalyzedbyKaspietal.(2000)andPetersonetal.(2004),butthelightcurvedatacomesfromavarietyofsources.Therestframewavelengthsandreferencesoflightcurvesarelistedincolumns(3)and(4)ofTable1,respectively.Theopticalvariabilitytimescalesareestimatedbythelightcurvesaround5100 A.Therearefourobjects,Fairall9,NGC4151,NGC5548,andNGC7469,thathavemulti-wavelengthlightcurveswellobservedattheoptical/UVbands.Themulti-wavelengthlightcurvesareusedtoestimatetimelagsforthefourobjects.

TheblackholemassesforAGNshavebeenwellestimatedbythereverberationmappingtechnique(e.g.,Kaspietal.2000,2005;Petersonetal.2004,2005;Vestergaard&Peterson2006).Themassesofthecentralblackholesofquasarsspanalargerangeof107M⊙ MBH 3×109M⊙,andhaveanupperlimitofMBH<1010M⊙(McLure&Dunlop2004;Vestergaard2004).TheblackholemassesusedinthispaperaretakenfromPetersonetal.(2004),andarelistedincolumn(5)ofTable1.ThebolometricluminosityLbolofobjectsexceptforMrk279aretakefromWoo&Urry(2002),andarelistedincolumn(6)ofTable

1.ThebolometricluminosityofMrk279isestimatedbyLbol≈9λLλ(5100 A)(Kaspietal.2000),withλLλ(5100 A)takenfromPetersonetal.(2004).

3.CALCULATIONSOFTEMPERATUREPROFILES

Thelocale ectivetemperaturesofaccretiondisksarefunctionsofradiird,blackhole˙(e.g.,Ebisawaetal.1991;Hanawa1989;massMBH,spina ,andmassaccretionrateM

Kubotaetal.2005;Lietal.2005;Pereyraetal.2006;Shakura&Sunyaev1973;Zim-mermanetal.2005).Thestandardaccretiondiskisthebasicmodelforaradiativelye cient,geometricallythindisk.IfthecentralblackholesareKerrones,thelocale ective

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

temperatureofthestandarddiskisgivenintheKerrmetricas(Krolik1999)

Te (Xd)= ˙3GMBHM

B(Xd)

3,(6)wherethefunctionsB(Xd)andC(Xd)are,respectively,(Krolik1999)B(Xd)=1

and

C(Xd)=1 yms

2yln3/2Xd,(7) y

ln

ln

ln y y1y y2y y3yy1(y1 y2)(y1 y3)yy2(y2 y1)(y2 y3)

yy3(y3 y1)(y3 y2)

2Xd,a =cJ/GMBHisthedimensionlessspinparameterofthecentralblack√holewiththespinangularmomentumJ,yms=

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

Themarginallystableorbitsintheequatorialplanecorrespondtothemaximume ciencyofenergyreleaseasaresultofaccretion,assumingprogradeorbits(Kembhavi&Narlika1999)

1/2Xms 2+a Xmsηmax=1 (12) .

1/2XmsXms 3+2a Xms

Accordingtothede nationofthee ciencyηwithwhichvarioustypesofblackholesconvertrestmass-energyintooutgoingradiation(Thorne1974),themassaccretionrateofthecentralblackholecanbeestimatedbytheformula

˙=LbolM

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

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

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

binwidth τasintheDCF.IthasbeenshowninpracticethatthecalculationoftheZDCFismorerobustthantheICCFandtheDCFwhenappliedtosparselyandunequallysampledlightcurves(e.g.,Edelsonetal.1996;Giveon1999;Royetal.2000).TheZDCFiscalculatedinthispaper.

Ingeneral,itseemstobetruethatthetimelagisbettercharacterizedbythecentroidτcentoftheDCFandtheICCFratherthanbythepeakτpeak,namely,thetimelagwherethelinearcorrelationcoe cienthasitsmaximumvaluermax(e.g.,Petersonetal.2004,2005).τpeakismuchlessstablethanτcentinboththeDCFandtheICCF,andτpeakismuchlessstableintheDCFthanintheICCF(Petersonetal.2005).ThenwepreferthatthetimelagestimatedfromtheZDCFmethodischaracterizedbythecentroidτcentoftheZDCF,fortheZDCFisanimprovementoftheDCFmethod.Thecentroidtimelagsτcentarecomputedusingallpointswithcorrelationcoe cientsr≥0.8rmax,andtheuncertaintiesfortimelagsofdatapointsintheZDCFarecomputedwithalargenumber(1,000)ofMonteCarlorealizations.TheZDCFsoffourobjectsarepresentedinFigures2–5,andthemeasuredtimelagsarelistedincolumn(4)ofTable3.

parisontoModels

ThereisacorrelationbetweentheblackholemassM8andthemeasuredcharacteristicvariabilitytimescaleτwithPearsoncorrelationcoe cientr=0.760atthechanceprob-abilityP=6.6×10 6(seeFig.6).Theregressionlines ttedbythebisectorregressionanalysisare

τ/(1+z)=0.27(±0.04)+0.12(±0.02)M8yrs.(15)

Ifrd～100rginequation(3)withviscosityparameterα=0.1,thereisarelationofτth～0.15M8.Thoughtheinterceptinequation(15)di ersfromtheinterceptpredictedbyequation(3),thispredictedslopeof～0.15isconsistentwiththeoneinequation(15).Thisindicatesthatthelinearcorrelationbetweenblackholemassandcharacteristicvariabilitytimesaleisqualitativelyconsistentwiththeexpectationofequation(3)thatthethermaltimescaleisessentiallylinearlyrelatedwiththeblackholemass.Thus,equation(15)isqualitativelyconsistentwithexpectationsofthestandardaccretiondiskmodels.

Accordingtothestandardaccretiondiskmodels,theoptical/UVemissionsareproducedthermallyinaccretiondisks.Thestandardaccretiondiskmodelsareusedtoestimatetheradiiofmaximumoptical/UVemissions.Weconsideraccretiondisktobecomposedofringswithapproximatelyuniformtemperatureradiatinglocallyasblackbody,andestimatetheradiiofmaximum uxemissionatdi erentwavelengthsusingadiskradialtemperaturepro legivenbyequation(5).Thenthelightcrossing,dynamical,thermal,andsoundcrossing

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

timescalesareestimatedbyequations(1)–(4),respectively,assumingviscosityparameterα=0.1.Thecalculatedresultsarepresentedincolumns(4)–(7)ofTable2,respectively.Itcanbeseenfromcolumns(2)–(7)ofTable2thatthethermaltimescalesaremostclosetotheopticalvariabilitytimescalesamongthefourphysicaltimescales,butthelightcrossinganddynamicaltimescalesaremuchsmallerthanthemeasuredtimescalesofopticalvariations,andthesoundcrossingtimescalesaremuchlargerthanthemeasuredtimescales.Thismightindicatethattheopticalvariationsresultfromthethermalinstabilityinaccretiondisksoronemechanismrelatedtoit.Though,itcannotbea rmedthattheopticalvariationsresultfromtheaccretiondiskthermalinstability,thelinearrelationpresentedinequation

(15)isqualitativelyconsistentwithexpectationofequation(3)thatthethermaltimescaleisessentiallylinearlyrelatedwiththeblackholemass.Theseaboveresultsareobtainedbyadoptingtheviscosityparameterα=0.1foreachsourceinoursample.Inpractice,variousvaluesofαaresuggestedandusedininvestigations(e.g.,Afshordi&Paczynski2003;Khajenabi&Shadmehri2007;Merloni2003;Merloni&Nayakshin2006;Parievetal.2003).Iftheviscosityparameterαisallowedtorangefromα～0.03toα～0.2(e.g.,Afshordi&Paczynski2003),calculationsshowthatthecombinationsofα～0.03–0.2anda =0.5–0.998canresultinthethermaltimescalesthatareinwellagreementwiththeopticalvariabilitytimescalespresentedinTable2.Thenitislikelythattheopticalvariationsresultfromtheaccretiondiskthermalinstability.

Theradiationenergiesemittedinaccretiondisksareprobablyfromthecontinuumthermalreprocessingand/orthelocalviscositydissipation(e.g.,Ulrichetal.1997).IftheX-raysilluminatingopticallythickmaterialinthindiskproducetheoptical–UVemissionsthroughthermalreprocessing,theopticalandUVvariationsfollowingtheX-rayvariationsareprobablycorrelatedwiththeUVvariationsleadingtheopticalones.Thetimelagsinthecaseofcontinuumthermalreprocessingareestimatedforthestandardaccretiondiskswithblackholespinparametera =0.5,0.8,0.9,and0.998.Therelevanttimelagsarelistedincolumn(5)ofTable3.Theplussignsofvaluesincolumn(5)meanthatthevariationsatlongerwavelengthslagthevariationsatshorterwavelengths.Itcanbeseenfromcolumns

(4)and(5)ofTable3thatthemeasuredtimelagsaremarginallyconsistentwiththosepredictedbythestandardaccretiondisksforNGC4151andNGC7469.ThisimpliesthattheopticalandUVemissionsarelikelytobethereprocessedX-raysforNGC4151andNGC7469.Inaddition,thetimelagsdecreaseslightlyasspinparametera increases.ForFairall9andNGC5548,thesignsofthecontinuumthermalreprocessingtimelagsarecontrarytothoseofthemeasuredtimelags.Thisindicatesthattheoptical/UVemissionsareunlikelytobethereprocessedX-raysforFairall9andNGC5548.Ifvariationsintheaccretion owa ect rstthe uxatouterradii,andthenintheinnerregion,thismayberesultincorrelatedoptical/UVlightcurveswithlongerwavelengthvariationsleadingshorter

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

wavelengthvariations.Asareferencetimescale,thesoundcrossingtimeinastandardaccretiondiskbetweentheseradiiareestimatedforFairall9andNGC5548byadoptinga =0.998.Theestimatedresultsarelistedincolumn(6)ofTable3.Theminussignsofvaluesincolumn(6)meanthatthevariationsatouterradiileadthevariationsatinnerradii.Itcanbeseenfromcolumns(4)and(6)ofTable3thatthemeasuredtimelagsaremuchsmallerthanthosepredictedbythestandardaccretiondisksinthecaseofaccretion ow uctuationstravellinginwards.

5.DISCUSSIONSANDCONCLUSIONS

Onewayofattemptingtohelptestingthestandardaccretiondiskmodelsisto ndrelationshipsbetweenvariabilityandfundamentalparametersofAGNs,suchasblackholemass.Thediscoveryofsucharelationship–orlackthereof–mayprovideusefulcluestothephysicalmechanismsbehindthevariability.Di erentsourcesofopticalvariationscanbeassociatedwithdi erentcharacteristictimescales,andmanyofthesetimescalesdependonblackholemass.Woldetal.(2007)investigatethedependenceofquasarvariabilityonblackholemass,and ndthatthemeasuredvariabilityamplitudecorrelateswithblackholemass.Collier&Peterson(2001)attemptedtode nearelationshipbetweenblackholemassandcharacteristicvariabilitytimescale.Theyreportedevidenceofblackholemassescorrelatingwithcharacteristicopticalvariabilitytimescalesforasampleof10well-monitoredAGNs.Inthispaper,alinearcorrelationbetweenthemeasuredtimescalesofopticalvariationsandtheblackholemassesisfoundforasampleof26well-monitoredAGNsbyreverberationmapping.ThislinearcorrelationsupportssuggestionofCollier&Peterson(2001).Theslopeofthiscorrelationinequation(15)is～0.12,andthisslopeisconsistentwiththeoneof～0.15predictedbyequation(3)withtheviscosityparameterα=0.1andtheemittingradiusrd=100rg.Theslopesbetweenthethermaltimescaleandtheblackholemassareestimatedforanothertwoemittingradiiofrd=50rgandrd=200rginequation(3)withα=0.1.ThethreetheoreticallinesbetweentimescalesandblackholemassesarepresentedinFigure6b.ItcanbeseeninFigure6bthatthetheoreticallineofrd=100rgmatchestheobserveddatapointsandthebest tlinebetterthantheothertwolinesdo.Thismeansthatthemeasuredcharacteristictimescalesofopticalvariationsarelikelytobefromtheaccretiondiskthermalinstability.Thenthestandardaccretiondiskmodelsarelikelytobeconditionallyfavoredbyobservations.

Anotherwayofattemptingtohelptestingthestandardaccretiondiskmodelsisconnecttheobservedvariabilitytimescalewithoneofthephysicaltimescalesinequations(1)–(4).Amongthefourphysicaltimescales,thethermaltimescaleismostclosetothemeasured

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

opticalvariabilitytimescaleasα=0.1.Theviscosityparameterαhasthetypicalvalueof～0.1forthestandardaccretiondisks(Shakura&Sunyaev1973).Avalueofα 1/2isimpliedbytheconditionthattheturbulenceshouldbesubsonicinthestandarddisks(Merloni2003).Alowervalueofα<0.14issuggestedbynumericalinvestigationsforthinaccretiondiskswithaconstante ectivespeedofsound(Afshordi&Paczynski2003).Merloni&Nayakshin(2006)alsolimitedasimilarrangeofα 0.15onthebasisofstudyingthelimit-cycleinstabilityinmagnetizedaccretiondisks.Khajenabi&Shadmehri(2007)adoptedα～0.03–0.3tostudythedynamicalstructureofaself-gravitatingdisk.Thentheviscosityparamterαinthestandarddiskspossiblyhasawiderangeincludingthetypicalvalueofα～0.1.Iftheviscosityparameterαisallowedtorangefrom～0.03to～0.2,thetimescalesofopticalvariationsareconsistentwiththethermaltimescalespredictedbythestandardaccretiondiskmodels.Thisimpliesthatthemeasuredcharacteristictimescalesofopticalvariationsarelikelytobeproducedbytheaccretiondiskthermalinstability.

Theanalysisshowsthatthewavelengthdi erences λarecorrelatedwiththerelevanttimelagsbetweendi erentbandsforNGC7469,butthereisnocorrelationbetweenthetwoquantitiesforNGC5548.Ifthe uxvariationsarecausedbytheaccretion ow uctuationstravellinginwardsacrosstheemittingregions,itislikelythattheshorterwavelengthvari-ationslagthelongerwavelengthvariations.However,thelongerwavelengthvariationslagtheshorterwavelengthvariationsforNGC4151andNGC7469.TheshorterwavelengthvariationslagthelongerwavelengthvariationsforNGC5548exceptthatthevariationsat2787 Alagthoseat1841 A.Thevariationsat1390 Alagthoseat1880 AforFairall9.Iftheoptical/UV uxesarethereprocessedcontinuumwithharderphotonsfromthecenterofaccretiondiskandsofteronesatradiifartherout,itisexpectedthatthelongerwavelengthvariationslagtheshorterwavelengthvariations.However,thelongerwavelengthvariationsleadtheshorterwavelengthvariationsforNGC5548andFairall9,andthisisinconsistentwiththeexpectationinthecaseofcontinuumreprocessing.ThecalculationsforNGC7469andNGC4151showthatthetimelagsestimatedinthecaseofcontinuumreprocessingaremarginallyconsistentwiththemeasuredtimelags.Inaddition,Fairall9andNGC5548havetheblackholemassMBH>5×107M⊙withthelongerwavelengthvariationsleadingtheshorterwavelengthvariations,butNGC4151andNGC7469haveMBH<5×107M⊙withtheshorterwavelengthvariationsleadingthelongerwavelengthvariations(seeTable

3).Thereseemstobeatrendbetweenblackholemassandtimelag.Asblackholemassisabovesomevalue,thelongerwavelengthvariationsmightleadtheshorterwavelengthvariations,butblackholemassisbelowthisvalue,theshorterwavelengthvariationsmightleadthelongerwavelengthvariations.

Theoriginoftheradiationenergiesemittedinaccretiondiskisakeytotheissuethattheharderphotonsleadorlagthesofterones.Fornon-blazartypeobjects,iftheoptical/UVra-

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

diationsarethereprocessedX-raysthatarecommonlyattributedtoComptonup-scatteringofthermalUVseedphotonsbyhotelectronsinacorona(e.g.,Sunyaev&Titarchuck1980;Haardt&Maraschi1991),theopical/UVandX-raylightcurvesareexpectedtobecorrelatedwiththeX-raysleadingtheopitcal/UVradiation,andthentheharderandsofterphotonsinoptical–UVregimearecorrelatedwiththeharderphotonsleadingthesofterones.Theopitcal/UVemissionsinNGC7469andNGC4151probablybelongtothiscase.Ifthebulkoftheobservedoptical/UVcontinuumarisesfromtheviscousdissipationinaccretiondisk,theresultinglightcurveswouldbecorrelatedbuttheUVradiationsshouldleadtheX-rays.ThisscenarioissupportedbyobservationsoftheSeyfertgalaxyMCG–6-30-15(Ar´evaloetal.2005).Inthisscenario,theobservedUVandtheseed-photon-emittingregionsarecon-nectedbyperturbationsoftheaccretion owtravellinginwardsthroughtheaccretiondisk,a ecting rstthemainUVemittingradiiandthentheinnermostregionwherethebulkoftheseedphotonsisexpectedtobeproduced(e.g.,Ar´evaloetal.2005).Weanalyzedthe uxvariationsin1–2KeV(Leighlyetal.1997)and1855 A(O’Brienetal.1998)for3C390.3,andasimilarbehaviortoMCG–6-30-15isfound.ThetimelagestimatedbytheZDCF

1.28obcentroidfor3C390.3isτcent= 4.01

+0.77dayswiththeX-rayslaggingtheUVradiation.

TheUVradiationemittedbyNGC5548andFairall9mightbelongtothethermalradia-tionfromtheviscousdissipation,andperturbationsoftheaccretion owtravellinginwardsthroughtheaccretiondiskresultinthesofterphotonsleadingtheharderones.Ourresultsmaysupportthatthesignsoftimelagsdi erfromcasetocase(e.g.,Maozetal.2002).Theexistencesofnegativeaswellaspositivetimelagsimplythatdi erentprocessescouldbedominatingtheemissionsatdi erentcases,andgenerallydon’tindicateanysimplerelationbetweentheenergybands.

Inthispaper,asampleof26objectswellobservedforreverberationmappingisusedtotestthestandardaccretiondiskmodelsacceptedwidelybycomparingthetheoreticalexpectationstothemeasuredtimescalesofopticalvariations,theobservedrelationoftheblackholemasseswiththemeasuredtimescales,andthemeasuredtimelagsbetweentheoptical/UVbands.Thetimescalesmeasuredbyboththemostcommonde nitionofthevariabilitytimescaleandthezero-crossingtimeoftheACFareconsistentwitheachother(seeFig.1).Theobservedvariabilitytimescalesarelinearlycorrelatedwiththeblackholemasses(seeFig.6),andthislinearrelationisconditionallyconsistentwithexpectationforthethermaltimescalesandtheblackholemassesinequation(3).Adoptingtheviscosityparametertypicallyofα～0.1(Shakura&Sunyaev1973),thethermaltimescalesaremostclosetothemeasuredtimescalesofopticalvariations.Thecombinationsofα～0.03–0.2anda =0.5–0.998couldresultinthethermaltimescalesthatareinwellagreementwiththeopticalvariabilitytimescalespresentedinTable2.Thenitislikelythattheopticalvariationsresultfromtheaccretiondiskthermalinstability.Thetimelagsaremeasuredby

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

theZDCFmethodforfouronesoutofthese26objects.Theanalyzedresultsshowthattheharderandsofterphotonsattheoptical/UVbandsarecorrelatedwiththeharderphotonsleadingthesofteronesforNGC4151andNGC7469,andwiththeharderphotonslaggingthesofteronesforNGC5548andFairall9(seeTable3).ForNGC7469andNGC4151,themeasuredtimelagsaremarginallyconsistentwiththetimelagsestimatedinthecaseofcontinuumthermalreprocessing.Itispossiblethattheoptical/UVemissionsofNGC4151andNGC7469arethereprocessedX-raysthatarecommonlyattributedtoComptonup-scatteringofthermalUVseedphotonsbyhotelectrons.ForNGC5548andFairall9,theUVphotonsareunlikelytobefromthecontinuumthermalreprocessingintheaccretiondisk.Ourinvestigationsonthevariabilitytimescales,therelationofthevariabilitytimescaleswiththeblackholemasses,andthetimelagsbetweendi erentbandsareunlikelytobeinconsistentwithorarelikelytobeconditionallyinfavorofthestandardaccretiondiskmodelsofAGNs.

Wearegratefultotheanonymousrefereeforhis/herconstructivecommentsandsug-gestionsleadingtosigni cantimprovementofthispaper.WearealsogratefultoProf.S.Kaspiforhisconstructivecommentsandsuggestionsthathelpedtosigni cantlyimprovethispaper.Prof.T.AlexanderisthankedforkindlyprovidinghisZDCFcode.HTLthanksfor nancialsupportbyNationalNaturalScienceFoundationofChina(Grant10778702).JMBissupportedbyNSFC(Grant10573030)and(Grant10778726).

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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

Table1.Sampleanddata

Objects

(1)z(2)λ( A)(3)Refs.(4)MBHergss 1

(5)(6)

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

Table1—Continued

Objects

(1)z(2)λ( A)(3)Refs.(4)MBHergss 1

(5)(6)

Note.—Col:(1)name.Col:(2)redshift.Col:(3)

therestframewavelengthsoflightcurves.Col:(4)the

referencesoflightcurves.Col:(5)blackholemass.Col:

(6)logofthebolometricluminosity.

References.—(1)Kaspietal.2000;(2)Kaspietal.

2005;(3)Santos-Lleoetal.1997;(4)Rodriguez-Pascual

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etal.1998;(15)Krissetal.2000.

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

Table2.Calculatedresults

Objects

(1)τ/(1+z)daysτldaysτthdays

(2)(3)(4)(5)(6)(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

Table2—Continued

Objects

(1)τ/(1+z)daysτldaysτthdays

(2)(3)(4)(5)(6)(7)

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