General relativistic modelling of the negative reverberation X-ray time delays in AGN

20 Pages·2014·1.75 MB·English

MNRAS439,3931–3950(2014) doi:10.1093/mnras/stu249
AdvanceAccesspublication2014March3
General relativistic modelling of the negative reverberation X-ray time
(cid:2)
delays in AGN
D. Emmanoulopoulos,1† I. E. Papadakis,2,3 M. Dovcˇiak4 and I. M. McHardy1
1PhysicsandAstronomy,UniversityofSouthampton,SouthamptonSO171BJ,UK
2DepartmentofPhysicsandInstituteofTheoreticalandComputationalPhysics,UniversityofCrete,GR-71003Heraklion,Greece
3IESL,FoundationforResearchandTechnology,GR-71110Heraklion,Greece
4AstronomicalInstituteoftheAcademyofSciences,Bocˇn´ıII1401,CZ-14100Praha4,CzechRepublic
D
o
w
Accepted2014February4.Received2014January28;inoriginalform2013December17 n
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e
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ABSTRACT fro
m
We present the first systematic physical modelling of the time-lag spectra between the soft h
(0.3–1keV)andthehard(1.5–4keV)X-rayenergybands,asafunctionofFourierfrequency, ttp
s
in a sample of 12 active galactic nuclei which have been observed by XMM–Newton. We ://a
c
concentrate particularly on the negative X-ray time-lags (typically seen above 10−4Hz), i.e. ad
e
soft-band variations lag the hard-band variations, and we assume that they are produced by m
ic
reprocessingandreflectionbytheaccretiondiscwithinalamp-postX-raysourcegeometry. .o
u
p
Wealsoassumethattheresponseoftheaccretiondisc,inthesoftX-raybands,isadequately .c
describedbytheresponseintheneutralFeKαlineat6.4keVforwhichweusefullygeneral om
/m
relativisticray-tracingsimulationstodetermineitstimeevolution.Theseresponsefunctions, n
andthusthecorrespondingtime-lagspectra,yieldmuchmorerealisticresultsthanthecom- ras
/a
monlyused,buterroneous,top-hatmodels.Additionally,weparametrizethepositivepartof rtic
the time-lag spectra (typically seen below 10−4Hz) by a power law. We find that the best- le
-a
fitting black hole (BH) masses, M, agree quite well with those derived by other methods, b
s
thus providing us with a new tool for BH mass determination. We find no evidence for any tra
c
correlation between M and the BH spin parameter, α, the viewing angle, θ, or the height of t/4
3
9
the X-ray source above the disc, h. Also on average, the X-ray source lies only around 3.7 /4
gravitational radii above the accretion disc and θ is distributed uniformly between 20◦ and /39
3
60◦.Finally,thereisatentativeindicationthatthedistributionofαmaybebimodalaboveand 1/1
below0.62. 17
2
4
Keywords: accretion,accretiondiscs–blackholephysics–relativisticprocesses–galaxies: 31
b
active–galaxies:nuclei–X-rays:galaxies. y
g
u
e
s
t o
n
thecentroidoftheX-rayemittingsource),lyingabovethecentral 0
1 INTRODUCTION BH,ontheaxisofsymmetryofthesystem(i.e.BHspinaxis).This 9 A
p
Inthecurrentparadigm,activegalacticnuclei(AGN)containacen- arrangementisknownasthe‘lamp-postgeometry’.Thephotons, ril 2
tralblackhole(BH)whichisfed,inmostcases,byanopticallythick fromtheX-raysource,formapower-lawspectrum,anddepending 0
1
andgeometricallythinaccretiondiscasaresultofmattertransporta- onthelocationoftheX-raysource,asubstantialpartofthemmay 9
tioninwardsandangularmomentumoutwards.Thisdiscradiatesas illuminatetheaccretiondisc.Inthiscase,theyareeitherCompton
aseriesofblackbodycomponents(Shakura&Sunyaev1973)with scatteredbyfreeorboundelectrons,orphotoelectricallyabsorbed
peak emission at optical–ultraviolet wavelengths (Malkan 1983). followedbyfluorescentlineemission,orbyAugerde-excitation.
PartofthisradiationisassumedtobeComptonup-scatteredwithina Thisyieldstheso-calledreflectionspectrumconsistingofanumber
mildlyrelativistichotelectroncloud,oftencalledtheX-raycorona. of emission and absorption lines (mainly below 1keV), together
Thismediumisoftenapproximatedasapointsource(representing withthe6.4keVFeKαemissionlinefromneutralmaterial,which
is the strongest X-ray spectral feature (George & Fabian 1991).
Depending on the proximity of the reflection process to the BH,
(cid:4) thevariouslinescanundergorelativisticbroadening(Fabianetal.
BasedonobservationsobtainedwithXMM–Newton,anESAsciencemis-
sionwithinstrumentsandcontributionsdirectlyfundedbyESAMember 1989; Laor 1991). These relativistically broadened lines may ac-
StatesandNASA. count for the observed soft X-ray excess (Crummy et al. 2006),
†E-mail:[email protected] where the X-ray spectrum below 1keV (soft band) lie above the
(cid:3)C 2014TheAuthors
PublishedbyOxfordUniversityPressonbehalfoftheRoyalAstronomicalSociety
3932 D.Emmanoulopoulos etal.
power-lawextrapolationofthecontinuum,usuallymeasuredinthe Inordertoproperlymodeltheimpulseresponsefunction,onehas
1.5–4keV(hardband).However,alternativeinterpretationsforthe totakeintoaccountindetailthevariousgeneralrelativistic(GR)
softexcessarealsopossiblee.g.Comptonizationofdiscphotons effects which affect the geometric path of both the hard and re-
fromtheX-raysource(Pageetal.2004),Comptonizationofdisc flectedX-rays.AfirstattemptatsuchamodellingofNXTLswas
photonswithinahotlayerontheaccretiondisc(Wang&Netzer performedbyChainakun&Young(2012)fortheAGN1H0707–
2003;Doneetal.2012).Inadditiontotheprimaryandthereflected 495 using a GR light-bending model and a moving X-ray source
components,severalAGNexhibitabsorptionfeaturesintheirX-ray in order to take account of the observed X-ray variability. Their
spectraaswell.Thesefeaturecanariseby(i)outflowswhichare conclusion was that a more complex physical model is required
eitherthermallyorradiativelyormagneticallydrivenwinds(Cappi in order to explain both the source’s geometry and intrinsic vari-
2006)andcanreachmildlyrelativisticspeeds(Tombesietal.2010), ability.Morerecently,Wilkins&Fabian(2013)usingthefullGR
(ii)inflows(Krug,Rupke&Veilleux2010)and(iii)X-rayabsorp- treatmentwithavarietyofdifferentgeometriesofcorona,forthe
tionclouds(Risaliti,Elvis&Nicastro2002).Thepositionofthese sameAGN1H0707–495,foundtheX-raysourceextendsradially
structuresvariesfromtensuptothousandsofgravitationalradii,1 outwards to around 35rg and at a height of around 2rg above the D
rg = GMBH/c2. In this framework, and taking advantage of the planeoftheaccretiondiscandthatpropagatingfluctuationsmight ow
n
highly variable behaviour of AGN, detection of time delays, as a accountforthepositivelow-frequencytimedelays.Inthispaper, lo
a
function of the Fourier frequency, between the soft-band and the weperformthefirstsystematicanalysisofthetime-lagspectraofa d
e
hnaisrmd-baannddthpehogteoonms,ectarynoshfetdhelsieghstyostnemthse. CXu-rrareynetlmy,istshieonobmseercvhead- sfuamncptiloenosf(1G2RAIGRNFsu,shienrgeafuftlelyr)g.eTnheersaelrfeulnatcitviiosntiscaimrepguelsneerraetsepdonusse- d from
negativeX-raytime-lags(i.e.soft-bandvariationslaggingthehard- ingalamp-postmodelwithvariableBHmass,BHspinparameter, h
band variations; NXTL hereafter) have triggered a great deal of viewingangleandheightoftheX-raysourceabovethedisc.The ttp
s
scientificinterestoninterpretingtheirnature.Aftertheirfirsttenta- paperisorganizedasfollows:Initially,inSection2,wepresentthe ://a
c
tivedetectionintheAGNArk564(McHardyetal.2007),wherean observationsanddatareductionprocedures.Then,inSection3,we a
d
origininreflectionfromtheaccretiondiscwasfirstproposed,the describethelamp-postmodelandthemethodthatweusetoderive em
firststatisticalsignificantdetectioncamefromFabianetal.(2009) theGRIRFs.Inthenextsection,weoutlinetheprocedureforthe ic
.o
fortheAGN1H0707–495.Then,Emmanoulopoulos,McHardy& estimationofthetime-lagspectramodelsfromthecorresponding u
p
Papadakis(2011)foundthatthistime-delayedX-raybehaviouris GRIRFs, and in Section 5, we describe the fitting methodology. .c
o
much more common than was initially thought by analysing the Section 6 contains the results of the best-fitting time-lag spectral m
/m
datasetsfromtwowidelystudiedbrightAGN,MCG–6-30-15and models and finally, in the last section we give a summary of our n
Mrk766.Subsequently,DeMarcoetal.(2013)inasystematicanal- results,andwediscussourconclusions.Throughoutthepaper,the ras
yhsigishosfta3t2istAicGalNc,ofnofiunddenaceto.tTalheofo1p5poAsGiteNtiemxhei-bdietlianygeNdXbeThLasviwoiutrh, ethrreo6r8e.3stpimeracteesntfocronthfiedevnacreioiunsteprvhaylssicuanllepsasraomtheetrewrsisceosrtraetsepdo.nSdimto- /article
i.e. positive X-ray time delays (hard-band variations lag the soft- ilarly,theerrorbarsoftheplotpointsinallthefiguresindicatethe -a
b
bandvariations),hasbeenknownforquitesometimeinbothAGN 68.3percentconfidenceintervals. stra
(e.g. Papadakis, Nandra & Kazanas 2001; McHardy et al. 2004; c
Are´valoetal.2006;Sriram,Agrawal&Rao2009)andX-raybi- t/43
2 OBSERVATIONS AND DATA REDUCTION 9
naries(e.g.Miyamoto&Kitamoto1989;Nowak&Vaughan1996; /4
Nowaketal.1999).Althoughpositivetime-lagsareexpectedinthe In Table 1, we list the details of the XMM–Newton observations /39
3
standardComptonizationprocesswithintheX-raysource(Nowak whichweusedinthiswork.Inthesecondcolumn,welisttheob- 1
/1
etal.1999),theycanalsobeproducedbydiffusivepropagationof servationIDforeachobject.Thelettersinparenthesesrefertothe 1
7
perturbations in the accretion flow (Kotov, Churazov & Gilfanov pnobservationmode;fw,lwandswrefertothePrimeFullWindow, 24
3
2001).TheoriginofNXTLstillremainsunclear.Assumingalamp- PrimeLargeWindow and PrimeSmallWindow modes, respec- 1
b
post geometry, in which the reflection of the hard X-rays occurs tively. The XMM–Newton data were processed using Scientific y
g
very close to the BH (few rg), the NXTL arise from the differ- AnalysisSystem(SAS;Gabrieletal.2004)version12.0.1.Wecon- ue
ence in the path-lengths between the soft (reflected photons) and sider only the EPIC pn data (Stru¨der et al. 2001) as they have a st o
thehardX-rayphotons(comingdirectlyfromtheX-raysource)to highercountrateandlowerpile-updistortionthantheMOSdata. n
0
theobserver(Fabianetal.2009,2013;Zoghbietal.2010;Zoghbi, ThesourcecountsforeachAGNwereaccumulatedfromacircular 9
A
Uttley&Fabian2011;Cackettetal.2013).Ontheotherhandbased apertureofradius40arcseccentredonthesource.Thebackground p
on the large-scale distant scattering scenario (Miller et al. 2010; countswereaccumulatedfromasource-freecircularregiononthe ril 2
Leggetal.2012),NXTLsmightbecausedbyscatteringofX-rays sameCCDchipasthesource.Forthetypeofevents,weselected 01
as they pass through, or are scattered from, a distant (tens up to onlysingleanddoublepixelevents,i.e.PATTERN==0–4,andwere- 9
hundredr )absorbingclumpymedium(e.g.wind,outflow,cloud), jectedbadpixelsandeventstooclosetotheedgesoftheCCDbyus-
g
thatpartiallycoverstheX-raysource,andwhoseopacitydecreases ingthestandardqualitycriterionFLAG==0.Sourceandbackground
withincreasingenergy.Currently,modellingofNXTLsisusually lightcurveswereextractedusingevselectin10stimebins.We
doneintermsofsimpletop-hatimpulseresponsefunctions(THIRF, checkedalllightcurvesforpile-upusingthetaskepatplot.We
hereafter;Milleretal.2010;Emmanoulopoulosetal.2011;Zoghbi found that only three observations of Ark 564 were affected by
etal.2011).Thisapproachis,however,justaparametrizationofthe moderate pile-up with Obs. IDs: 0670130201, 0670130501 and
NXTLspectraanddoesnotcarryanyphysicalinformationabout 0670130901.Inthesecases,weusedanannularregiontoexclude
thegeometryandthephysicalpropertiesoftheBH(i.e.mass,spin). theinnermostsourceemissionhavingaradius,inpixels,of:280,
200and250,respectively.Thesourcelightcurveswerescreened
forhighbackgroundandflaringactivity.Thelight-curvepartswith
1ForaBHmassof2×106M(cid:4)photonsneedtg=rg/c=9.8stotravela highbackgroundactivity,usuallyatthebeginningand/orendofan
distanceof1rg. observation,wereremovedfromthefinaldataproducts.Thetotal
MNRAS439,3931–3950(2014)
GRmodelling ofNXTL 3933
Table1. XMM–Newtonobservations.Thefirstcolumn,(1),givesthenamesoftheAGNandtheirBHmasses.EachBH
massestimatecomeswithafootnote(attheendofthetable)indicatingthecorrespondingliteraturereference.ThoseBH
massesthathavebeenestimatedusingthereverberationmappingtechniquearefollowedbytheindication(r).Thesecond
column,(2),givestheXMM–NewtonobservationIDstogetherwiththecorrespondingobservingmodeoftheEPIC-pncamera:
smallwindow(sw),largewindow(lw)andfullwindow(fw)mode.Thethirdcolumn,(3),givesthenetexposuretimeofthe
observationsi.e.durationafterbackgroundsubtractionanddatascreening.
(1) (2) (3) (1) (2) (3)
AGNname Obs.ID(PNmode) Netexp. Name Obs.ID(PNmode) Netexp.
BHMass(×106M(cid:4)) (ks) BHMass(×106M(cid:4)) (ks)
NGC4395 1H0707−495
0.36±0.11a(r) 0142830101(fw) 108.7 2.34±0.71f 0110890201(fw) 40.7
NGC4051 0148010301(fw) 78
D
1.73+−00..5552b(r) 0109141401(sw) 117 0506200201(lw) 38.7 ow
0157560101(lw) 50 0506200301(lw) 38.7 n
lo
0606320101(sw) 45.3 0506200401(lw) 40.6 a
d
0606320201(sw) 44.4 0506200501(lw) 40.9 e
d
0606320301(sw) 31.3 0511580101(lw) 121.6 fro
0606320401(sw) 28.8 0511580201(lw) 102.1 m
0606321301(sw) 30.1 0511580301(lw) 104.2 h
0606321401(sw) 39.2 0511580401(lw) 101.8 ttps
0606321501(sw) 38.8 0554710801(lw) 96.1 ://a
0606321601(sw) 41.5 0653510301(lw) 113.9 ca
d
0606321701(sw) 38.3 0653510401(lw) 125.8 e
m
0606321801(sw) 39.9 0653510501(lw) 117 ic
0606321901(sw) 6.2 IRAS13224−3809 .o
u
0606322001(sw) 36.9 5.75±0.82f 0110890101(fw) 60.9 p.c
0606322101(sw) 37.7 0673580101(lw) 126.1 o
m
0606322201(sw) 41.1 0673580201(lw) 125.1 /m
0606322301(sw) 42.3 0673580301(lw) 125 n
ra
Mrk766 0673580401(lw) 127.5 s
1.76+−11..5460c(r) 00130094104310310011((ssww)) 19258.1.6 ES6O.9611±3−0.G240g10 0301890101(fw) 92.6 /article
0304030301(sw) 98.5 NGC7469 -a
b
0304030401(sw) 98.5 12.2±1.4h(r) 0112170101(sw) 17.6 s
0304030501(sw) 95.1 0112170301(sw) 23.1 trac
0304030601(sw) 98.5 0207090101(sw) 84.6 t/4
3
0304030701(sw) 34.6 0207090201(sw) 78.7 9
MCG-6−30−15 Mrk335 /4/3
2.14±0.36d 0111570101(sw) 43.2 26±8i 0306870101(sw) 132.8 93
1
0111570201(sw) 55 0600540501(fw) 80.7 /1
0029740101(sw) 83.5 0600540601(fw) 130.3 17
2
0029740701(sw) 127.4 NGC3516 4
0029740801(sw) 125 31.7+−24..82b(r) 0107460601(sw) 79.3 31 b
Ark564 0107460701(sw) 120.5 y
2.32±0.41e 0006810101(sw) 10.6 0401210401(sw) 51.7 gu
e
0206400101(sw) 98.9 0401210501(sw) 62.6 s
0670130201(sw) 59.1 0401210601(sw) 61.6 t o
n
0670130301(sw) 55.5 0401211001(sw) 42.2 0
9
0670130401(sw) 62.5 NGC5548 A
00667700113300560011((ssww)) 6660..95 44.2+−91.39.8b(r) 0089960301(sw) 85.2 pril 2
0
0670130701(sw) 55.3 19
0670130801(sw) 57.8
0670130901(sw) 55.5
aPetersonetal.(2005).
bDenneyetal.(2010).
cBentzetal.(2009).
dEstimatedfromequation3inGu¨ltekinetal.(2009)usingthestellarvelocitydispersionvalueofMcHardyetal.(2005).
eEstimatedfromequation5inVestergaard&Peterson(2006)usingthemeanvaluesofFWHM(Hβ)andλLλ(5100Å)of
Romanoetal.(2004).
fZhou&Wang(2005).
gEstimatedfromequation5inVestergaard&Peterson(2006)usingthemeanvaluesofFWHM(Hβ)andtheλLλ(5100Å)
estimatedbyCackettetal.(2013)usingdataofPietschetal.(1998).
hPetersonetal.(2004).
iGrieretal.(2012).
MNRAS439,3931–3950(2014)
3934 D.Emmanoulopoulos etal.
netpnexposuretime,i.e.aftertheexclusionofthehighbackground
activityperiods,islistedinthethirdcolumnofTable1.Sincewe
areinterestedinstudyingthetime-lagsbetweenthesoftexcessand
theX-raycontinuum,thesourceandbackgroundlightcurveswere
extractedinthoseenergybandswherethesoftexcesscontribution
is maximized and the continuum emission is dominating, respec-
tively.Thus,forallthesources,weuseforthesoftexcessandthe
continuumtheenergybandsof0.3–1and1.5–4keV,respectively.
The background subtracted light curves were produced using the
(SAS)taskepiclccorr.NotethatinSection4.2.1wediscusswhy
for the purposes of our paper detailed fine-tuning of the selected
energybands isnotnecessary. Theresultingdatasetsarecontin-
uouslysampledsufferingonlyfromveryfewdatagapsandcount D
o
ratedrops(asaconsequenceoftelemetrydrop-outs).Withrespect w
n
tothedatagaps,onaveragethereareatmost3–6missingobserva- lo
a
tionswhichwefilledupbylinearinterpolation.Forthecountrate d
e
dwreoprse,swcahleichthaerecoaulsnotvreartye,fewwith(oinnaevaecrhagtiembeetbwine,enac2caonrddi5ngpotointtsh)e, Figure1. Geometricallayoutofthelamp-postmodel.Theaccretiondisc d from
correspondingfractionalexposuretime. isanopticallythick,geometricallythin,Kepleriancold(i.e.neutral)disc http
3 THE LAMP-POST MODEL tαh,attoex1t0e0n0drsgf.roTmhethXe-IrSayCOso,urricne,wishidcehpiisctdeedfibnyedthbeyothraenBgHesspphinerpeaaranmdeitteirs, s://ac
situatedataheighthabovetheBH.Thebluesolidlinescorrespondtothe a
d
In this section, we describe the basic physical and geometrical trajectoriesoftwoexamplehardX-rayphotonsleavingtheX-raysource, em
propertiesofthelamp-postmodel.Then,weoutlinetheprocessthat oneheadingtowardstheobserverandanotheronetowardstheaccretion ic
weusetoestimatetheGRIRFs.Allthetime-scalesareestimated disc.Thelatterfollowsacurvedtrajectory,duetotheGReffects,andthe .ou
innormalizedtimeunits,oft ,andthustheyscalelinearlywiththe trajectoryofthecorrespondingsoftX-rayphotonfromtheaccretiondisc p.c
BHmass,M,viatherelationg (reflectedproduct)isshownwiththereddottedline.Thewholesystemis om
viewedfromanangleθ.Acolourversionofthisfigureisavailableinthe /m
tg,M =4.9255Ms (1) onlineversionofthejournal. nra
s
whereMisgiveninunitsof106M(cid:4). athnedpfrheoqtuoennscwieisllsfcoallleowlindeiafrfleyrewnitthtriatj(eecqtouraiteisonfr1o)m.ItnheeaXch-rgaeyosmoeutrrcye, /article
tothediscandfromthedisctotheobserver,andthus,theresponse -a
3.1 Thegeometricallayoutandparameterspace b
ofthesystemwillbedifferent. stra
Thelamp-postmodelthatweconsiderinthispaperconsistsofthe c
following three physical components: a central supermassive BH t/43
3.2 EstimationoftheGRIRFs 9
withanaccretiondiscilluminatedbyapoint-likeX-raysourcelo- /4
catedontheaxisofthesystem(Fig.1).Thissystemischaracterized Inordertocomputetheresponseoftheaccretiondisctotheprimary /39
3
bythefollowingparameters:spinparameterandmassofthecentral illuminationfromtheX-raysource(describedbythepowerlaw), 1
/1
BH,heightoftheX-raysource,andviewingangle.Theaccretion weuseaflarewithastepfunctionprofilethathasveryshortduration 1
7
discextendsfromtheinnermoststablecircularorbit(ISCO),rin,to of1tg.Theprimaryintrinsicspectrumhasanormalizationofunity. 24
3
1co0l0d0(rig.e,.annedutirtails)dainsco.pTtihceaXlly-rathyicsko,urgceeolmieestaribcoavlleyththeinB,HKaetphleeirgiahnt Tonhleyn,ofwteheesntiemuatrtaeltflheuorreesspcoennsteFoefKthαedliinsec,bayt 6m.4eakseuVrinigntthheeflreusxt 1 by
g
handitcomprisestheprimarysourceofX-rays,whichweassume frameoftheaccretiondisc. u
e
ttsropuibmneposatfraaptmihcoeattonenrd,iαtno,dabenexde(cid:7)imts=itmti2an.sgTs,ihsMeot.crFeoonprtirctahallelByfoHwrmiistehcrh,awaprhoaiwcctheerrd-izleaefiwdnebsspyteihctes- ttiaskcTeaosttcienortimongpaucctcoeodutehneNtObFAoeRthK(Dαthuelmindoeinreflt,cutAxa,bnrwdaesisnuavrseter&steheCCoMolmloinnptt2eo0nC0s0ac)ra,lotwtemhriiunclgh- st on 09
I0S.6C7O6,(iwnetecrmonesdiidaetretshprieneBvHal,ureins:=03(.S5crhg)waanrdzs1ch(mildaxBimHa,lrlyinr=ota6tring)g, pdaroncceeseseqsu.aWltoetahsesuSmolearavnaeluuetr.aTlhaeccrreestuioltnandtiflscuxanadtthaneoirbosneravbeurnis- April 2
KerrBH,rin = 1rg). We have chosen a variety ofheights forthe computedusingalltheGReffects(Dovcˇiak,Karas&Matt2004), 019
X-raysource,dependingonthespinoftheBH.Forthecaseofa withouttakingintoaccounthigherorderimagesofthedisc.Thus,
SchwarzschildBH,weselectanensembleof18heights:{2.3,2.9, we exclude photons emitted from the accretion disc (either from
3.6,4.5,5.7,7,8.8,11,13.7,17.1,21.3,26.5,33.1,41.3,51.5,64.3, the top or the bottom surface) that go around the BH (any num-
80.2,100}r .Fortheintermediatecase,weaddtotheensemble ber of times), which reach the observer by travelling through the
g
a lower height of 1.9r , and for the Kerr BH we add yet another gapbetweentheISCOandtheBH.Higherorderimagesaremore
g
heightof 1.5r , respectively. Finally, thesystemis observed bya prominentforaSchwarzshildBHwheretheISCOisthefurthest
g
distantobserverataviewingangleofθ,i.e.θ=0or90◦ifthedisc outandthegapbetweentheinneredgeofthediscandtheBHis
isfaceonoredgeon,respectively.ForeachoneofthethreeBH the largest. Nevertheless, even in this case these photons do not
spinparametersandeachoneheight,weconsiderthreeangles:20◦, contributeverymuchtothetotalflux(Beckwith&Done2004).An
40◦and60◦.Thiswideparameterspace,consistingofthevariables interestingpointisthatinthecaseofthelowerspinBHs,matter
α,handθ,yieldsatotalof(20+19+18)×3=171different fromtheaccretiondiscfallsdownontotheBHratherquickly(on
geometricallayoutsofthelamp-postmodel.TheBHmassisnotan theorbitaldynamicaltime-scaleattheISCO).Thus,itisexpected
additionalvariableinourestimationofGRIRFsasalltime-scales thatthisfree-fallingmaterial,fillingthegapbetweentheISCOand
MNRAS439,3931–3950(2014)
GRmodelling ofNXTL 3935
D
o
w
n
lo
a
d
e
d
fro
m
h
ttp
s
Figure2. EstimationoftheGRIRFforthelamp-postmodelwithα=0.676,θ=40◦andh=3.6rg.Left-handpanel:thedynamicreflectionspectrumwitha ://aca
timeresolutionof0.1tg.Right-handpanel:theGRIRFderivedbyaddinguptheFeKαlineflux,atagiventime(blackpoints),andthedottedlineistheresult de
ofthecubicinterpolation.Acolourversionofthisfigureisavailableintheonlineversionofthejournal. m
ic
.o
u
theBH,willhavemuchlowerdensity,andconsequently,itcould timealltheFeKα linefluxesintheenergyband0–8keV.Inthe p.c
be heavily ionized contributing to the overall reflection spectrum right-handpanelofFig.2,weshowtheGRIRFwhichcorresponds om
(Reynolds 1997). In this case, higher order images would not be tothedynamicreflectionspectrumoftheleft-handpanel.Thisplot /m
produced. The soft excess, of the contributed re-processed emis- ineffectshowshowtheFeKαlineflux(in0–8keV)evolveswith nra
s
sion,ismoreprominentinthestateswithhigherionization,butdue time,asseenbyadistantobserver.Theactualsumsaredepictedby /a
tothemuchlowerdensityofthematterintheplungingregion,we theblackpointsandthedottedlineisacubicinterpolation.Note rtic
donotexpectaverylargecontributiontotheoverallemissionand thatthecubicinterpolationisneededbecauseinsomecasesthere le-a
thustotheresponsefunctionsofthesystem.Therefore,inourcom- are very small numerical errors during the estimation of the line b
s
putationswedonottakeintoaccounttheionizedemissionbelow fluxinthedynamicspectra(oftheorderof10−5)whichcancreate tra
c
theISCOinthecaseoflowerspinBHs.TheGReffectsincludethe oscillatory artefacts (Fig. 3) in the corresponding GRIRF (during t/4
3
light bending that causes the gravitational lensing, energy shift – the summation process). By performing a cubic interpolation (as 9
/4
Dopplerandgravitational–andtherelativistictimedelaysforboth opposedtoalinearorquadratic)theseartefactsaresuppressedand /3
9
thephotonstravellingfromtheprimarysourcetothedisc,aswellas theintegrationprocesswhichisusedtoderivethetime-lagspectra 3
1
there-processedphotonsemittedfromthedisctravellingtowards (Section4.1)ismuchfaster,i.e.thereisnoneedtotakeglobally /1
1
the observer. Due to the lamp-post geometry and relativistic ef- a very small integration step. As we can see from the right-hand 7
2
fects,theilluminationoftheaccretiondisc,fromtheprimaryX-ray panelofFig.2,thegeneralshapeoftheIRFconsistsofanabrupt 43
1
source,isuneven(i.e.dependingontheradius).Thereflectedflux risefollowedbytwopeaksandadecay.Theonsetofthefirstpeak b
y
fromthediscisproportionaltotheincidentfluxonthediscwhich corresponds to the time that the first hard X-ray photons hit the g
u
isapowerlawwithitsnormalizationbeingafunctionofα,h,θand disc and from that point, a ring of an expanding echo is created e
s
(cid:7) (seeequation3inDovcˇiaketal.2011).DuetotheGReffects, onthediscaroundtheBH.Asthepartoftheechoapproachesthe t o
n
thefluxoftheFeKαlineisspreadoverawiderangeofenergies, BH, it is deformed in such a way that eventually two echoes are 0
9
asafunctionoftime,yieldingforeachlamp-postconfigurationa created, inner and outer, which correspond to the second peak in A
0d.y1ntag.mIinctrheeflleecftti-ohnansdpepcatnreulmo,f2Feisgti.m2,awteedswhoitwhaantiemxeamrepsloelouftiaonnFoef tohneeIcRoFn.tiTnhueesinwneitrhriintsgriasdtihaelnexmpoavnisniogntoawwaarydsfrtohmeBthHe,ctheentoreutoefr pril 20
Kαdynamicreflectionspectrumforthecaseofα=0.676,θ=40◦ the disc and the IRF is gradually fading out. For low spin values 19
andh=3.6r .AswecanseethefluxoftheFeKα lineisspread (e.g.fortheSchwarzschildBH),theinnerringisnotcreateddueto
g
overawiderangeofenergiesbetween2.5and7keVandfades-out theexistenceofthegapintheaccretiondiscbelowtheISCO.The
astimegoeson,indicatingthedecayofthe‘echo’asitismoving secondpeakinthetransferfunctioninthiscasecorrespondstothe
awayfromthecentreoftheaccretiondisc.Inordertoderivethe re-appearanceofthepartoftheouterring,behindtheBH,afterit
GRIRF, (cid:8)(t), of the accretion disc for the Fe Kα emission line, was‘hidden’belowtheISCO.
weusethedynamicreflectionspectrumandweaddupforagiven
4 TIME-LAG SPECTRA ESTIMATION
2Sometimesthedynamicreflectionspectrumiscalleda‘2Dtransferfunc-
tion’(Campana&Stella1995;Reynoldsetal.1999)despitethefactthat 4.1 Modeltime-lagspectra
strictlyspeakinginsignalprocessingtheterm‘transferfunction’refersto
the Fourier domain, i.e. ratio of the Fourier transformed input to output WecannowusetheGRIRFs,whichwehavecomputedfortheFe
signals. Kαlinephotons,toconstructthemodeltime-lagspectra.Assume
MNRAS439,3931–3950(2014)
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Figure3. Small-scaleoscillatoryartefacts.TheplotshowsthelinearlyinterpolatedGRIRFforthelamp-postmodelwithα=0,θ=60◦andh=51.5rg.The de
dashedrectangleszoomintworegionsoftheGRIRFwheretheoscillatoryartefactsbecomeprominent. m
ic
.o
u
thattheX-rayspectrumofAGNcanbedescribedbyapower-law whereEistheexpectationoperator,andthevaluesinanglebrackets p
.c
form a(t)E−(cid:7), where (cid:7) is constant as a function of time and all denotemeanvalues.Itcanbeshownthatinourcase, om
(cid:3)
theobservedfluxvariabilityisduetothevariationsofthenormal- ∞ /m
ization a(t) (primary continuum). Then, the source’s hard X-ray rs,h(τ)=bkra,a(τ)+fb (cid:8)(t(cid:6))ra,a(τ +t(cid:6))dt(cid:6), (5) nra
emission in the 1.5–4keV band, h(t), which is dominated by the 0 s/a
X-raycontinuumsourcecanbewrittenas wherera,a(τ)istheautocovariancefunctionofthecontinuumvari- rtic
abilityprocess, le
-a
h(t)=ba(t), (2) ra,a =E[(a(t)−(cid:7)a(t)(cid:8))(a(t+τ)−(cid:7)a(t)(cid:8))]. (6) bstra
twamheirneabtio=n(cid:2)o1f4.5ktkehVeeVcEo−n(cid:7)tidnEuu(minbSyecthtieonre7flwecetidoinsccuosmsfpuorntheenrt)t.hLeectouns- IwfewoebttaakinetheFourier(cid:4)transfofrm(cid:3)o∞fbothsidesofthe(cid:5)aboveequation, ct/439/4
then assume that s(t) is the variable source’s soft X-ray emission Ps,h(ν)=bkPa,a(ν) 1+ k (cid:8)(t(cid:6))e−i2πνt(cid:6)dt(cid:6) , (7) /393
inthe0.3–1keVenergyrangewhere,inadditiontothecontinuum, 0 1
/1
wealsodetectemissionfromthereprocessingcomponentsothat where Ps,h(ν) is the cross-spectral density function between h(t) 17
s(t)=ka(t)+f (cid:3)0∞(cid:8)(t(cid:6))a(t−t(cid:6))dt(cid:6), (3) t(rriea.flel.edctehtenedsiietnympufisutsnhicoatnriod)naXto-ffrraetyqhueeemcnocinsystiiνno,unau)nmadn.PdTahs,e(at()fνu()ni.icest.itohthneepPoosuw,htep(rνut)spsiesocfa-t 2431 by g
complex function, and its complex argument defines its phase, at u
e
wKαherlienekp=ho(cid:2)t0o1.3nkkesVeVoEve−r(cid:7)tdhEe 0a–n8dke(cid:8)V(t(cid:6)b)aanrdeththaet wGeRIdRisFcsusfsoerdthinetFhee fhtir(metq)e.u-Selainngcc,yeτ(νPν,)a,i,.abe(.eνt)twhieeseapnhrteahaseel-tfwluaongcttbiimeotnew,seeeeqrniueastthiaoetnftir(me7q)euiemsnepcrilyeiesνssits(hta)gtiavtnehnde st on 09
previous section (Section 3.2). In this way, we basically assume A
by p
tohbastetrhveedph0o.t3o–n1skoefVtheenreerflgeyctbeadncdo,mapnodntehnet,FwehKicαhclionnetrpibhuotteotnos,thaet arg(cid:6)1+(f/k)(cid:2)∞(cid:8)(t(cid:6))e−i2πνt(cid:6)dt(cid:6)(cid:7) ril 20
6.4keV,areproducedinthesamepartsoftheaccretiondisc;hence, τν =− 20πν . (8) 19
theshapeofthesoft-bandGRIRFissimilartotheGRIRFsthatwe
Weuseequation(8)toestimatethemodeltime-lagspectrafor
haveestimatedfortheFeKα linephotons.However,evenifthis
eachoneofthemodelGRIRF(intotal171,Section3).Foreach
is the case, we do not expect the flux of the soft-band reflection (cid:8)(t(cid:6)), the integral in this equation was estimated at 491 frequen-
spectrumtobethesameastheFeKα lineflux(i.e.differencein cies: ν = 10kHz with k = −5, −4.99, ..., −0.11, −0.1. These
the normalization), hence the use of the constant f in the above frequenciescoverthetypicalfrequencyrangeobservedbyXMM–
equation.AsweexplainindetailinSections5and7,aswellasin Newton. Note that for the integration procedure we use adaptive
AppendixA,duringouranalysiswesetthisvalueto0.3.Inorder
integrationmethodwhichidentifiestheproblematicintegrationar-
to derive the time-lag spectrum between the s(t) and h(t) bands,
eas, which in our case are usually the regions of the two peaks,
weneedtoestimatethecross-covariancefunctionbetweenthetwo
andconcentratethecomputationaleffort(i.e.samplingpoints)on
timeseries.Thisfunctionatatime-lagτ isdefinedas
them(Malcolm&Simpson1975;Krommer&Ueberhuber1998).
The resulting time-lag model spectra have an almost continuous
rs,h(τ)=E[(s(t)−(cid:7)s(t)(cid:8))(h(t+τ)−(cid:7)h(t)(cid:8))], (4) profile,duetotheveryfinefrequencyresolutionwehaveadopted.
MNRAS439,3931–3950(2014)
GRmodelling ofNXTL 3937
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Figure4. Estimationofthetime-lagspectrumforthelamp-postmodelwithα=0.676,θ=40◦andh=3.6rg,forM=5×106M(cid:4).Left-handpanel:the mic
GRIRFmodelinphysicalunits(ascaledversionofFig.2,right-handpanel).Right-handpanel:thecorrespondingtime-lagspectrum. .o
u
p
.c
o
Therefore,thisallowsustointerpolatelinearlyamongthevarious model. We employthe caseofthelamp-postmodel withthe fol- m
frequencies without adding additional structure into the resulting lowingparameters:α=1,θ =40◦andh=26.5r ,foraforaBH /m
g n
time-lag spectra. These time-lag spectra (equation 8) contain the massM=2×106M(cid:4).Forthiscase,theGRIRFisshowninthe ra
Tinhteugsr,ailfoofnethweaGntRsItRoFw,o(cid:8)rk(τi(cid:6)n),cionnjnuonrcmtiaolnizwedithtimreealudnaittas ((ei..eg..tfgo)r. ltewfot-htoapn-dhpaatnpealraomfFeitgri.z5atwiointhstcheenbalraiockss(boolitdhloinfet.hTehmenn,owrmeacloinzesiddetor s/artic
le
fitting purposes, as we are doing in Section 6) then one has to unity)torepresentthegivenGRIRF:onewithwidthdefinedtobe -a
b
transformthetime-lagspectraintorealphysicalunits(i.e.seconds) equaltotheseparationofthetwopeaksintherealGRIRF(Top-hat s
bydividingandmultiplyingtheabscissas(frequencies,ν)andthe 1,Fig.5,left-handpanel,dottedthickgreyline)andtheotherone tra
c
ordinates (time-lag estimates, τν), respectively, by tg,M (equation startingatthesametimeastherealGRIRFandextendingoverthe t/43
1).Fordemonstrativepurposesintheleft-handpanelofFig.4,we full time range covered by it (Top-hat 2, Fig. 5, left-hand panel, 9
/4
showtheGRIRFinphysicaltimeunitsforaBHmassof5×106 solid thick grey line). As we can see from the right-hand panel /3
9
M(cid:4)correspondingtotheGRIRFshowninnormalizedunitsinthe of Fig. 5, the corresponding time-lag spectra of the two THIRFs 3
1
right-handpanelofFig.2(α=0.676,θ =40◦andh=3.6r ).The differgenuinelyfromthatoftheGRIRF.Thepositionoftheneg- /1
g 1
correspondingtime-lagspectrumforthisGRreflectionscenariois ative constant plateau, the position and the amplitude of the first 72
4
ingeneralanegativefunction(Fig.4,right-handpanel).Thefirst positivepeakaswellasthebehaviourishighfrequencies(above 3
1
morphological characteristic of this time-lag spectrum is that at 10−3Hz)differsignificantlyfromthoseinthetime-lagspectrumof b
y
lowfrequencies,10−5−10−4Hz(i.e.longtime-scales)itexhibitsa theGRIRF.Notethatsimilardiscrepanciesoccurforlowerheights g
u
negativetailthatformsaconstantplateauat−215s.Thisplateau oftheX-raysource.Thetop-hatparametrizationyieldsalwaystwo es
definesthemostnegativetime-delayedreflectedemissionfromthe numbers:thestart-andtheend-timeoftherectangularpulse.Inboth t o
n
disc(i.e.itsordinate)andthecorrespondingtime-scalesthatthese scenariosthatwehaveconsidered,thestart-timeisassociatedwith 0
9
delaysoccur(i.e.10–100ks).Then,thevaluesofthetime-lagspec- the beginning of the reverberation phenomenon on the accretion A
p
tartum1.5in×cre1a0s−e3aHsza,fpuenacktiionngoafrofruenqdue(n2cy×an1d0−th3eHyzb,e3c0oms)e.pFoinsiatlilvye, dphisocto(finrssthpiteathkeodfitshce,GwRhIicRhF)i,si.ae.pthhyestiicmalelythraetatlhepafirrasmthetaerrdoXf-rthaye ril 20
1
thetime-lagspectraexhibitadampedoscillatingbehaviour(Fig.4, system.Theend-timecouldcorrespondeithertothesecondpeak 9
inset)aroundzero.Ingeneral,allthetime-lagspectraexhibitthese oftheGRIRFwheretheiso-delayedarcs(travellingoppositeand
featuresbuttheyappearatdifferentfrequenciesdependingonthe aroundtheBH)meetforthefirsttime,(Top-hat1)ortotheoverall
BHmass,theheightoftheX-raysourceandr .Notethatwhether timeofthereverberationphenomenon(Top-hat2)(seeSection3.2
in
thetime-lagvaluesbecomepositiveorremainalwaysonanegative for the various peaks and times of the GRIRF). However, none
leveldependsonlyontheshapeofcorrespondingGRIRF(asshown of these scenarios yield a time-lag that matches that of GRIRF.
inequation8).InAppendixB1,weexplorethemodelparameter Finally, in order to retrieve the best-fitting THIRF model, which
spacefordifferentconfigurationsofthelamp-postmodel. couldcorrespondtothetime-lagspectrumofthegivenlamp-post
geometry,wefittothetime-lagspectralestimates(showninFig.5,
right-handpanel,blackline)equation(8),using(cid:8)(τ)asatop-hat
4.1.1 GeneralrelativisticIRFversustop-hatIRF
function leaving τ(cid:6) and τ(cid:6) as free parameters (i.e. start and end-
1 2
Inthissection,wecomparethephysicallyjustifiedGRIRFswiththe times) under the condition τ(cid:6) <τ(cid:6) (e.g. Emmanoulopoulos et al.
1 2
widelyused(seeforreferencesSection1)THIRFparametrization 2011;Zoghbietal.2011).Thisisdonebyfindingthepair{τ(cid:6),τ(cid:6)}
1 2
MNRAS439,3931–3950(2014)
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Figure5. GRIRFversusTHIRFforthelamp-postmodelwithα=1,θ =40◦andh=26.5rg,forM=2×106M(cid:4).Left-handpanel:theGRIRF(black mic
solidline)andthetwotop-hatparametrizationmodelseachonedepictingthetwopeaks(dottedthickgreyline)andtheoverallshape(solidthickgreyline)of .o
u
theGRIRF,respectively.Right-handpanel:thecorrespondingtime-lagspectra. p
.c
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3
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Figure6. Fittingthetime-lagspectrumofthelamp-postmodelwithα=1,θ=40◦andh=26.5rg,forM=2×106M(cid:4)withatime-lagspectrumcoming 9
fromaTHIRF.Left-handpanel:thebest-fittingtime-lagspectrum(thickgreyline),for{τ(cid:6),τ(cid:6)}={−4545,3108}s,andthelamp-posttime-lagspectrum(black
1 2
line,alsoshownintheright-handpanelofFig.5).Right-handpanel:thecorrespondingTHIRF,derivedfromthebest-fittingparameters(thickgreyline),
togetherwiththeGRIRFofthelamp-postmodel(blackline,alsoshownintheleft-handpanelofFig.5).
thatminimizesthesquaredsumofthedistanceofbetweenthetwo cannotbeassociatedwithanytime-scaleofthesystem.3Aswecan
functionsestimatedoverthe491frequenciespointsofthetime-lag seefromtheright-handpanelofFig.6,thetop-hatmodel,which
spectrum(Section4.2).Thebest-fittingTHIRFmodelisshownin
theleft-handpanelofFig.6withthethickgreyline,togetherwith
thephysicallyrealisticGRIRFmodel(blackline).Thequalityof 3Constraining the problem only to the positive domain, i.e. 0<τ(cid:6) <
thefitispoor(squaredsumequalsto188288s2 for489degrees τ(cid:6), yields practically a zero-width best-fitting THIRF with {τ(cid:6),τ(cid:6)1}=
2 1 2
of freedom, d.o.f.), but more importantly, the best-fitting THIRF {1.1,1.3}×10−5s,havingaverypoorfitcharacterizedbyasquaredsum
yieldsphysicallyunrealistictimes{τ(cid:6),τ(cid:6)}={−4545,3108}sthat of2.12×107s2for489d.o.f.
1 2
MNRAS439,3931–3950(2014)
GRmodelling ofNXTL 3939
correspondstothebest-fittingtimes(thickgreyline),coverscom- due to the large number of averaging points, appearing statistical
pletely different time-scales from the GRIRF model (black line) meaningfulevenifthereisnotarealcorrelationbetweenthephases
andthuscannotbeassociatedwithanygenuinephysicalproperty and hence nomeaningful timedelay. In allouranalysis, we esti-
ofthesystem. mate the time-lag spectra down to (3–5) × 10−3Hz, but for the
fittingprocedureweconsideronlythetime-lagestimatesforwhich
the coherence is greater than 0.15 corresponding to a physically
4.2 Observedtime-lagspectra
meaningfulphasecorrelation.
Inordertoestimatethetime-lagspectrabetweentwolightcurves,
we use we use the standard analysis method outlined in Bendat
&Piersol(1986)andNowaketal.(1999).Inbrief,considerfora 4.2.1 Selectionofenergybands
given source a soft and a hard light curve, s(t) and h(t), obtained
AswediscussedinSection2,forallthesourcesweextractedthe
simultaneously, consisting of the same number of N equidistant
lightcurvesbetween0.3–1(softband)and1.5–4(hardband)keV
observationswithasamplingperiodtbin (thesearediscretizedand energybandsandthesearetheonesthatweusefortheextractionof Do
finite-lengthversionsofequations3and2,respectively).Foragiven w
thetime-lagspectra.Thesebandsdepictquiteaccuratelythegeneral n
Fourierfrequency,fj=j/(Ntbin)forj =0,1,...,[N/2−1or(N− behaviourofthereflectioncomponent(i.e.softexcess)andthatof loa
1)/2](forevenoroddN),weestimatethecross-spectrum,C(fj),4 theX-raysource(i.e.continuum)despitethefactthatforeachsource ded
(aes.g.Priestley1981)betweenthetwolightcurvesinaphasorform separatelytheexactlimitscouldbeshiftedslightlytowardshigher from
orlowerenergies.Inthispaper,weactuallymodeltheresponseof
h
Cs,h(fj)=S∗(fj)H(fj)=|S(fj)||H(fj)|ei(φH(fj)−φS(fj)) (9) the6.4keVFeKαlineasaproxyforthesoftreprocessedemission ttp
s
in which S(fj) and H(fj) are the discrete Fourier transforms5 of (oSfethcteiolonw3-.2en).eGrgiyvernepthroecpeosssseidbleemuinscsieorntaainntdietshbateotwfetheen6th.4ekreesVpolinnsee, ://ac
stu(td)easn|dS(hfj()t|),arnedsp|Hec(tfijv)|e,lyre,swpeitchtipvhealys.eTshφeS(afsj)tearnisdkφdHen(foj)teasncdoammpplelix- afrnodmalrseoprthoecefsascetdtheamtaisnsyiohna,rtdhebraenidswlititlllecpoonitnatininawsmoraryllicnognttoroibmutuiocnh adem
conjugation. about the exact choices of soft and hard energy bands. Thus, the ic.o
froTmheanl,lwtheeavoebrsaegrevathtieoncos,mopvleerxcaronsusm-sbpeerctorufmatesletiamstat1e0s,ccoonmsiencg- soeflbecottehdtheenesorgftyabnadntdhseohfafredrbueshaavsiiomuprlfeohrothmeoegnesneemobulsedoefsscoruiprtcieosn. up.com
utive frequency bins, yielding m average cross-spectra estimates, /m
(cid:7)Cs,h(fbin,i)(cid:8) at m new (averaged) frequency bins fbin,i for i = nra
1,2,...,m.Finally,foreachaveragecrossspectrumwederiveits 5 THE FITTING PROCEDURE s
/a
caolsmopalsexphaarsgeu,mφe(nfbti,n,ii.)e.aintdsawneglceonwviethrttihtetopopshiytisviecarletailmaexiusn,iktsnown mInforredqeuretnocyfitbtihnes (oSbescetrivoend4t.i2m),e-wlaegrsepqeucitrrea,twτo(fbtiinm,i)e,-lbaignnsepdecitnrtaol rticle-a
τ(fbin,i)= φ2π(ffbbinin,i,i). (11) mcoomdpelonceonmt,peosnteimntast.eTdhferofimrsteqounaeticoonrr(e8s)pofonrdsatgoivtehneGGRRIRreFfl.eTctheids bstrac
component carries all the physical and geometrical information t/4
Foreachtime-lagestimate,wecalculatethecorrespondingstandard 3
deviation, std{τ(fbin,i)} via equations 16 and 17 in Nowak et al. alabgosu,tptahretilcaumlapr-lpyoastttmheodloewla-nfrdeqguiveenscryisreanpgreed.oInmtiontaanlt,lwyetohnaevgea1ti7v1e 9/4/3
(1999).Atthesametimefromthecross-spectrum,weestimatethe 9
coherencebetweens(t)andh(t)asafunctionofFourierfrequency of these negative time-lag spectral models, τν(M, α, θ, h), (each 31
(Vaughan&Nowak1997) onecorrespondingtoadifferentsetof{α,θ,h}lamp-postmodel /11
7
parametersforagivenM)estimatedfortheensembleofGRIRFs 2
γs2,h(fj)= (cid:7)|S(|f(cid:7)jC)|s2,h(cid:8)((cid:7)f|Hj)((cid:8)f|2j)|2(cid:8). (12) (8S)eschtioounld4.b1e).leInftparsinacifprelee,mthoedceolnpsatraanmtefte(ar.ppHeoawrienvgeri,ngeivqeunatitohne 431 by
Thisquantitytakesvaluesbetween0and1anditisameasureofthe complexity of the model fitting (as explained below), this would gue
resultinaprohibitivelylargenumberofmodelspectratocompareto s
wlfeirnnehceqoeausreevncadocluiyrfre.feesAlracetonviorecrnreeyssbpiaemortewnpdoeaecrttntoaunatuhtlnlecycaotduwretroiepolilnaciatgteerhdydtppcbhuoyarinvtshteeessi,sφaφtt(hSfaa(tfgj)sivm)aen(anadlvlFeφcoroHauhg(rfeiejer)d-r, awallleiemfinxieterfgdtiones0u,.m3is,b3seo0ropthfearotcbteshneertvttohetadatlpoeofmitnhitsessifinooprnuefltauXcxh-rooafbyjtehcceotn.FtFeinoKuruαthmilsinsrpeee,acostvorenarl, t on 09 Ap
hovaseraararathnegreuonfifforermqudeinsctrieibsu).tiTohnuisn,tfhoerruanncgoerr(e−laπte,dπp]bih(nd,aiuseest,oφp(hfbains,ei-) flthuex.reTphriosciesssininrgoucgohmapgorneeenmteflnutxtootvheerotbhseercvoendtiflnuuxumratfliouxbeattwseoefnt ril 201
energies(e.g.Crummyetal.2006).Theeffectoff,intheresulting 9
wrapping)thataveragesalwaystozero.Thatmeansthatforsmall
time-lagspectra,isdiscussedinSection7andinAppendixA.The
coherencevalueswegetatime-lagof0thathassmalluncertainties,
secondcomponentconsistsofasimplepowerlaw,PLν(A,s)=Aν−s,
providing us with positive time-lags. Thus, we deal with a five-
4Thisisanaturalestimatorofthecontinuouscross-spectrumequation(7). dimensionalmodelparameterspaceconsistingofmodelparameter
5The discrete Fourier transform for the soft light curve, s(t), at a given vectorsoftheform,v={M,α,θ,h,A,s}.Inthisframework,the
Fourierfrequency,fj,isdefinedasfollows: overalltime-lagspectralmodelatagivenfrequency,ν,isgivenby
thesumofthetwocomponents
(cid:8)N
DFTs(j)= s(tk)e−2πi(k−1)j/N (10) TLν(v)=τν(M,α,θ,h)+PLν(A,s). (13)
k=1
forj=0,1,...,[N/2−1or(N−1)/2](forevenoroddN).Notethatthe Then, we transform each one of the 171 GR reflected model
exponentialfunctioncontainsasarunningindexk−1insteadofk,since components into physical time units (using equation 1) for an
thedatastartfork=1andnotk=0. ensemble of 12 BH masses: (0.01, 0.05, 0.1, 0.5, 1, 2, 5, 10,
MNRAS439,3931–3950(2014)
3940 D.Emmanoulopoulos etal.
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Fpaigraumreet7e.rsDAi=scr0e.t2izaantidons=an0d.8a.vTerhaegcinognteinffueocutssfmoordtheelslaarmepsh-poowsntcwaistehmthoeddeolttwedithlinαes=an0d.6t7h6e,cθor=re2sp0o◦n,hdi=ng3d3is.1crregt,efoesrtiMma=tes2(×ave1r0a6geMd(cid:4)overanthdepforewqeure-lnacwy d from
rangesindicatedbythehorizontallines)withthefilledcircles.Left-handpanel:theGRreflectedcomponent.Middlepanel:Thepower-lawcomponent. h
Left-handpanel:theoveralltime-lagmodel. ttp
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://a
50, 100, 200, 500) × 106 M(cid:4), yielding a grid of 2052 cells, minimization, we use the classical Levenberg–Marquardt method ca
d
each one corresponding to a given set of parameter values {M, (Bevington & Robinson 1992). After localizing the vgbf, we esti- em
α, θ, h}. For each grid-cell, we treat the model (i.e. equation matethe68.3percentconfidencebandsforeachbest-fittingmodel ic
.o
13) in exactly the same way as the observed time-lag spectra, parameter intheusualway, i.e.byvaryingitsvalue overagiven u
p
i.e. we discretize it and average it over exactly the same number range and deriving each time for the rest model parameters their .c
o
oe1rf,a2gf,ree.q.vu.a,elunmec.yFoofbritnhdsee.mToovhneissratrylaliteitvlidmesep-fulorarpgoesmaecsoh,dienflr,eFq(cid:7)igTu.eLn7fcwbyin,e,i(fsbvhin)o,(cid:8)iw,foathrneiaev=f-- bDχek2us(rtvi-nfi)gtstpitnahgceeveaarlrruooeurntehdsatititmysiaBetliFdonbae(cid:14)psrtχ-ofic2tet=sins,g1wfgrreoidmcuptbahirecaammllieynteiimrntvueamrlpuoχelks2a(tivengbtthhf)ee. m/mnras
fmecotdseolfTthLeνd(2is×cre1t0iz6aMtio(cid:4)n,a0n.d67a6v,er2a0g◦i,n3g3o.1nrtgh,e0c.o2nst,in0u.8o)uasntdimoen-laitgs hfoelilgohwticnegllswaabyo:vtewaondBHbelmowasshgcbeflalsndabaollvteheansdpinbeplaorwamMetgebrf,stawnod /article
twocomponents.Forthediscretization,weassumeanobservational viewing angles (each one consisting of three cells). This yields a -a
b
TdatahtFeanos,uerttiheoersffer8e0bqiunkensnebcdiienmsneobddeteiwnlee2es0ntims(1,a.yt2ei5esl×dairn1eg0−a2v50ea0rna0dgme2d.o5do×evle1re0s1t−i6m2)afHtreezs-. cmeorronodtrieneluspotaiumrasamvtieeortnesirwosnceoouvfseeχritkn2h(gev5χ)ka×2(rvo5u)×nspd3avc×gebfa3,ro=χun22(d2v5t)h.geTribhdeu-scste,-lfidlstut,rilniengagvgitrnhigde stract/43
quencybinswhoseoverallrangeisshownwiththehorizontalline. thepower-lawmodelparameters{A,s}tovaryfreely.Sincevgbf 9/4
Theleft-handpanelofFig.7showstheGRreflectedcomponent, hasbeenderivedfromthegrid-cells(i.e.fixedvalues),duringthe /39
vcτieνer(ss2i(o×4n×a10g16r0eM−a3t(cid:4)−de,2a0×l.o6f17‘06o−,s22c)0ilH◦la,zt3,o3wry.h1-irscgtrh)u,acertxeuhrseiubspi’tptionrewgsasinreddtshdheuigrcihonngfrtetihnqeuuoeaunvs-- tfiehrtretoinirngteesprtamimraeamdtiieaottneergprrreiodgc-iecosensl.lnTveahwleuevqsau,lauχne2tsi(tvyobffχ)χ,2w2(v(ivtbhf))inairntehdeeicmsaeteelresgcittnehgdebffireonsmat-l 31/11724
eragingprocess.ThemiddlepanelofFig.7showsthepower-law best-fittingparametervaluesthatwekeepfromourfittingprocess, 31
mcreotdeeelsctoimmaptoensednift,fePrLfrνo(m0.2thse,a0c.8tu),alincownhtiincuho,uosncpeowagera-ilna,wthmeoddies-l vvibsfu=ali{zMebthf,eαibnft,eθrbpfo,lhabtef,dAχb2f,(vsb)fs}.paDceesdpuiteettoheitsfahcitgthhadtimweensciaonnnaol-t by gu
e
epsrtoicmeasste.sT,haetmthoestcporroremsipnoenndtidnigffefrreeqnuceenscaireesa,tdtuheeltoowtehsetfarveeqruaegnincgy iintyte(rspioxl-adtiemdevnesrisoinosn)sionfAthpepernefldiexctBed2c(oFmigpsoBne4n–tBs,6τ),νw(Me,sαh,owθ,thh)e, st on
bins, particularly the first one at 10−4Hz. Finally, the right-hand which are the actual discrete components that we interpolate in 09
pthaeneplreovfioFuisg.tw7osmhoowdseltchoemopvoenraelnlttsi.mTeh-eladgismcroedteela,vie.era.gthedevsuermsioonf eimqupaotritoannt(p1o4i)ntwshoifchtheapopveeararllvpiaroTceLdfubrine,i(avre) a(esqfuoalltoiowns.13). Some April 2
oftheoveralltime-lagmodelcarriesallthepreviouslymentioned 01
9
differences with respect to its continuous version. Then, for each
averaged discretized overall time-lag spectral model, within each (i) Thebest-fittingvaluesderivedfromthegridparameterspace,
grid-cell,k,weestimatethesquareddifferencesbetweenthemodel v ,are,infact,veryclosetothebest-fittingvaluesderivedfrom
gbf
andthedatausingthefollowingχ2indicator the interpolated parameter space, v . All the v lie within the
bf gbf
(cid:9) (cid:10) 86.6percent confidence intervals (1.5 standard deviations) of the
χ2(v)=(cid:8)m (cid:7)TLfbin,i(v)(cid:8)−τ(fbin,i) 2, (14) best-fitting values of vbf. Note that the only reason that we per-
k i=1 std{τ(fbin,i)}2 f6o8r.m3ptehreciennttercpoonlfiatdieonncienbtahnedfisrsatnpdlathceevisfoisrtahenadteurriavlaptiroonduocftthoef
bf
andforeachgrid-cellweminimizethisquantitywithrespecttothe thisprocess.
twopower-lawmodelparameters,{A,s}.Finally,weendupwith (ii) Weleavethepower-lawparametersfreesincetheirvaluesfix
anensembleof2052globalminimumvaluesofχ2(v)fromwhich thenormalizationoftheoverallmodel.Notethatforagivensetof
k
thesmallestonecorrespondstothesetofthebest-fittinggridmodel physicallamp-postmodelparameterthenegativereflectedcompo-
parameter values vgbf ={Mgbf,αgbf,θgbf,hgb,Agbf,sgbf}. For the nent,τν(M,α,θ,h)isabsolutelyfixed.Thus,boththenormalization
MNRAS439,3931–3950(2014)