MNRAS000,1–29(2016) Preprint18February2016 CompiledusingMNRASLATEXstylefilev3.0 Towards modelling X-ray reverberation in AGN: Piecing together the extended corona D. R. Wilkins1(cid:63), E. M. Cackett2, A. C. Fabian3 and C. S. Reynolds4,5 1Department of Astronomy & Physics, Saint Mary’s University, Halifax, NS. B3H 3C3, Canada 2Department of Physics & Astronomy, Wayne State University, 666 W. Hancock St., Detroit, MI 48201, USA 3Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge. CB3 0HA, UK 4Department of Astronomy, University of Maryland, College Park, MD 20742, USA 5Joint Space-Science Institute (JSI), College Park, MD 20742, USA Accepted2016January29.Received2016January7;inoriginalform2015November5 ABSTRACT Models of X-ray reverberation from extended coronæ are developed from general rel- ativistic ray tracing simulations. Reverberation lags between correlated variability in the directly observed continuum emission and that reflected from the accretion disc arise due to the additional light travel time between the corona and reflecting disc. X-ray reverberation is detected from an increasing sample of Seyfert galaxies and a numberofcommonpropertiesareobserved,includingatransitionfromthecharacter- istic reverberation signature at high frequencies to a hard lag within the continuum component at low frequencies, as well a pronounced dip in the reverberation lag at 3keV. These features are not trivially explained by the reverberation of X-rays orig- inating from simple point sources. We therefore model reverberation from coronæ extended both over the surface of the disc and vertically. Causal propagation through its extent for both the simple case of constant velocity propagation and propagation linked to the viscous timescale in the underlying accretion disc is included as well as stochastic variability arising due to turbulence locally on the disc. We find that the observed features of X-ray reverberation in Seyfert galaxies can be explained if the long timescale variability is dominated by the viscous propagation of fluctuations through the corona. The corona extends radially at low height over the surface of the discbutwithabrightcentralregioninwhichfluctuationspropagateuptheblackhole rotation axis driven by more rapid variability arising from the innermost regions of the accretion flow. Keywords: accretion,accretiondiscs–blackholephysics–galaxies:active–X-rays: galaxies. 1 INTRODUCTION it is formed and how it evolves to give rise to the extreme variabilitythatisseenintheX-rayemission.Adeepunder- The accretion of material onto supermassive black holes in standingofthecoronaisavitalcomponenttounderstanding activegalacticnuclei(AGN)powerssomeofthemostlumi- how supermassive black holes can power some of the most nous objects in the Universe. An intense X-ray continuum luminous objects in the Universe. is seen, originating from a corona of energetic particles as- sociated with the innermost regions of the accretion flow. In addition to being observed directly, the X-ray con- X-raysareproducedwhenthermalultravioletphotonsfrom tinuum illuminates the accretion disc, from which it is ‘re- the accretion disc are Compton up-scattered by the parti- flected.’Moreprecisely,itisback-scatteredthroughthepro- cles of the corona (Sunyaev & Tru¨mper 1979), thought to cesses of Compton scattering, photoelectric absorption and be accelerated and confined by magnetic fields associated subsequent fluorescent line emission, and bremsstrahlung with the disc (Galeev et al. 1979; Haardt & Maraschi 1991; (Ross&Fabian2005).ThespectrumofthereflectedX-rays Merloni & Fabian 2001). Much remains unknown, however, is shifted inenergy andblurredby Doppler shiftingandas- abouttheexactstructureofthecorona,theprocessbywhich sociatedrelativisticbeamingduetotheorbitalmotionofthe reflectingmaterialinthediscaswellasthegravitationalred- shift associated with the spacetime curvature in such close (cid:63) E-mail:[email protected] proximitytotheblackhole(Fabianetal.1989;Laor1991). (cid:13)c 2016TheAuthors 2 D. R. Wilkins et al. In particular, the relativistically blurred reflection from the of objects in terms of the characteristic length scale in the accretiondiscresultsina‘softexcess’inemissionabovethe gravitational field, represented by the gravitational radius power law continuum between around 0.3 and 1keV com- (1r =GM/c2).Reverberationlagshaveevenbeendetected g posed of a number of emission lines from the ionised disc inGalacticstellarmassX-raybinarieswheretheequivalent material. These become blurred and smoothed. The promi- lags are on the order milliseconds (Uttley et al. 2011). The nent Kα fluorescence line of iron is seen at 6.4keV with a measured lags correspond to just a few gravitational radii broad wing extending as low as 3keV where photons emit- showing that reverberation measurements are probing the tedinthislinefromtheinnerregionsofthediscareshifted innermost regions of the accretion flow in the immediate to lower energy. vicinity of the black hole event horizon. The X-ray emission from accreting black holes, in par- Whilethesignatureofreverberationfromthedisc(the ticular the narrow line Seyfert 1 (NLS1) galaxies, is seen to ‘softlag’or‘reverberationlag’)isseeninthehighfrequency be highly variable (Leighly 1999; Turner et al. 1999; Ponti components of the X-ray variability, at lower frequencies, et al. 2012) and in recent years studies of this variability the measured time lags undergo a transition to a different haveaddedafurtherdimensiontoourunderstandingofthe behaviour. At low frequencies, variability in harder X-rays corona and the inner regions of the accretion flow. The re- are seen to systematically lag behind that in lower energy flected X-rays vary in luminosity according to variations in ‘(softer)X-rays.Measuringthefrequencydependenceofthe the flux received from the corona.Where there are energy lag between the soft 0.3-1keV reflection-dominated band bands in which the emission is predominantly composed of (thesoftexcess)andthehard1-4keVcontinuum-dominated reflection from the accretion disc (i.e. in the 0.3-1keV soft band, a clear transition is seen with the hard band leading excessandthebroadenedironKαlinebetweenaround3∼4 the soft at high frequencies (i.e. reverberation) but the soft and 7keV), the variability is expected to lag behind corre- bandleadingthehard.Thelagtimeincreasessmoothly,fol- lated variations in energy bands dominated by the directly lowing an approximately log-linear dependence on energy. observedcontinuumemission(1-4keV).Thetimedelaycor- BeforeitsdiscoveryinAGN,this‘hardlag’wasawellknown respondstotheadditionallighttraveltimefromthecoronal featureinX-raybinaries(Miyamotoetal.1988;Miyamoto& source to the reflecting accretion disc (Uttley et al. 2014). Kitamoto 1989; Nowak et al. 1999) and is often attributed Since their initial detection (Fabian et al. 2009), X-ray to the propagation of fluctuations in mass accretion rate reverberation lags between the continuum and soft excess throughthedisc,energisingthelessenergeticouterpartsof have been detected in an ever-growing sample of Seyfert the corona before reaching the more energetic inner parts galaxies (Emmanoulopoulos et al. 2011; De Marco et al. (Kotov et al. 2001; Ar´evalo & Uttley 2006). 2011; Zoghbi & Fabian 2011; De Marco et al. 2013; Fabian Following the almost ubiquitous detection of X-ray re- etal.2013;Cackettetal.2013;Alstonetal.2013)andalso verberation across the AGN in Seyfert galaxies, attention betweenthecontinuumandbroadironKline(Zoghbietal. has turned to modelling the phenomenon in order to trans- 2012,2013;Karaetal.2013a,b).TheadventofthehardX- late the frequency and energy dependence of reverberation ray imaging telescope, NuSTAR, has enabled the detection lagsintoaccuratemeasurementsofthegeometryoftheinner of reverberation lags between the continuum and the broad accretion flow and the processes forming and injecting en- Comptonhumpthatappearsaround20keVinthereflection ergy into the corona. The modelling of X-ray reverberation spectrum (Zoghbi et al. 2014; Kara et al. 2015). has been largely motivated by Reynolds et al. (1999) who X-ray timing analysis is typically conducted as a func- computed‘2Dtransferfunctions’givingtheobservedfluxas tion of Fourier frequency; that is the different frequency afunctionofobservedphotonenergyandtimefromasingle, components that make up the X-ray light curves. X-ray localised, X-ray flare withYoung & Reynolds (2000) relat- reverberation is detected in the higher frequency compo- ingthisworktoobservablesignaturesthatmaybedetected nents representing the variability occurring on the shortest byfuturegenerationX-rayobservatories,comparingtheob- timescales. Measuring the time lag (with respect to some served reverberation of X-ray flares with a library of com- referenceband)ofsuccessiveX-rayenergybandsoverthese putedtransferfunctionstodeterminethemassoftheblack high frequency components reveals a profile reminiscent of hole as well as the location of the flare. Wilkins & Fabian thereflectionspectrum.Emissioninthesoftexcessandthe (2013) develop general relativistic ray tracing simulations ironKαlineisseentoarrivelaterthanthecontinuumemis- of the propagation of X-rays from coronæ, both point-like sion (Kara et al. 2013a, 2014) while the redshifted wing of andextended,tomodelthefrequencydependenceofthelag the iron K line responds more rapidly than the core of the and show how the reverberation lag provides a measure of line as might be expected if the redshifted wing is originat- the height of a point like corona or average vertical extent ing from the inner parts of the disc, closer to a relatively of a corona extended above the accretion disc, while being compactcoronalX-raysource(Zoghbietal.2012).Remark- relatively insensitive to the radial extent. ably,thehighfrequencyenergydependenceovertheironK Cackett et al. (2014) extend this analysis to the fre- lineofthelagshowsacommonprofileacrossmanyobjects, quency and energy dependence of lags arising from the re- oncescaledfortheabsolutemagnitudeofthelag,suggesting verberation of X-rays originating from a point source. As thatthereisacommonreverberationprocessproducingthe wellasconsideringthesourceheight,thedependenceofthe lag in these objects (Kara et al. 2013b). lags on a number of parameters commonly associated with The magnitude of the reverberation lag between both spectral modelling of X-ray reflection is computed and the thesoftexcess(DeMarcoetal.2013)andironKline(Kara frequency and energy dependence of the lags in the Seyfert et al. 2013b) is found to scale approximately linearly with galaxy NGC4151 are fitted with this model. Further at- the black hole mass suggesting that the process inducing tempts to fit point source models of X-ray reverberation to thelagisoccurringonasimilarsizescaleacrossthesample lag measurements have reinforced the requirement for com- MNRAS000,1–29(2016) Modelling X-ray reverberation 3 pactcoronæconfinedtoasmallregionaroundtheblackhole isgivenbytheFouriertransformofthelightcurveandcan (cid:12) (cid:12) and inner regions of the accretion flow (Kara et al. 2014; be written F˜(ω)=(cid:12)F˜(ω)(cid:12)eiϕ and is computed by (cid:12) (cid:12) Emmanoulopoulos et al. 2014; Chainakun & Young 2015). PointsourcemodelsoftheX-rayemissionhavebeenfound, 1 (cid:90) F˜(ω)= √ F(t)eiωtdt however,tohavelimitedapplicabilitytoreproducingthefull 2π detailsofX-raytimingmeasurements(Chainakun&Young The phase lag between two time series, say the hard and 2012). In particular the self-consistent combination of hard soft light curves, H(t) and S(t) respectively, can be found lagsproducedbypropagationthroughanextendedemitting byconsideringthecomplexformoftheirFouriertransforms region and reverberation from a compact, point-like corona (cid:12) (cid:12) (cid:12) (cid:12) H˜(ω)=(cid:12)H˜(ω)(cid:12)e−iϕ andS˜(ω)=(cid:12)S˜(ω)(cid:12)e−iθ andcomputing as well as the reproduction of the fine features of the en- (cid:12) (cid:12) (cid:12) (cid:12) ergy dependence is challenging. Moreover, measurements of the cross spectrum theilluminationpatternoftheaccretiondiscbythecoronal (cid:12) (cid:12)(cid:12) (cid:12) C˜(ω)=S˜∗(ω)H˜(ω)=(cid:12)S˜(ω)(cid:12)(cid:12)H˜(ω)(cid:12)ei(θ−ϕ) (1) X-ray source, exploiting the energy shifts imparted on pho- (cid:12) (cid:12)(cid:12) (cid:12) tonsreflectedatdifferentradii,hassuggestedthataccretion The argument of which gives the time lag, τ, since ϕ = ωt discsinanumberofNLS1galaxiesareilluminatedbyanex- (and converting from angular to linear frequency, f) tended corona (Wilkins & Fabian 2011, 2012; Fabian et al. 1 (cid:16) (cid:17) 2013; Wilkins & Gallo 2015). τ(f)= arg C˜(f) (2) We here develop a self-consistent model of the rever- 2πf beration of X-rays illuminating the disc from an extended Followingthis signconvention, apositive time lagindicates corona, accounting for the finite speed at which luminos- thatthevariabilityinthehardband,H(t),islaggingbehind ityfluctuationscanpropagatethroughitsextent.Thetech- that in the soft band, S(t). niques for X-ray timing and measuring X-ray reverberation ThelagtimeisafunctionofboththeFourierfrequency, as well as the common results are reviewed in Section 2 be- that is whether the lag is seen in rapid components of the fore techniques for simulating X-ray reverberation through variability or in more slowly varying components, and the general relativistic ray tracing simulations are discussed in photon energy, via the energy channels that are included Section 3. The features of reverberation from point-like X- when accumulating the light curves from which the cross ray sources are revisited in Section 4 before models of ex- spectrumiscomputed.Inordertomaximisesignaltonoise, tendedcoronæaredevelopedinSection5.Thesebeginwith the lag is measured as a function of Fourier frequency be- simplified prescriptions for the propagation of fluctuations tweenlightcurvesintwobroadenergybandshenceforthre- beforeintroducingamodelforthehardlagandlinkingprop- ferred to as the (broadband) lag-frequency spectrum. When agationtotheunderlyingaccretiondisc.Theresultsofthese investigating reverberation from the accretion disc, energy simulations are discussed in the context of explaining the bandswillbeselectedsuchthatoneisexpectedtobedom- lags measured in the NLS1 galaxy 1H0707−495, display- inatedbythedirectlyobservedcontinuumemissionandthe ingX-raytimingpropertiesrepresentativeofthefindingsin other is expected to be dominated by X-rays reflected from Seyfertgalaxiesandonwhichhighqualitydataareavailable the disc. The cross spectrum is averaged over a range of obtained from more than 1Ms of observations with XMM- Fourierfrequenciestoproducehighersignaltonoiseinalag Newton. spectrum binned by frequency. Alternatively, the lag can be computed as a function of photon energy to produce a lag-energy spectrum. In this case,crossspectraarecomputedbetweeneach(narrow)en- 2 MEASUREMENT OF X-RAY ergy band of interest and a common reference band. To re- REVERBERATION duceuncertaintyduetocountingstatistics,areferenceband SimulationsofX-rayreverberationaredevelopedinthecon- spanningabroadenergyrangeistaken,commonlyoverthe text of the commonly employed techniques for its measure- full instrument bandpass (0.3-10keV in the case of XMM- ment from light curves in distinct energy bands obtained Newton)oroverthereflectedsoftexcess(0.3-0.8keV)where from X-ray telescopes, reviewed by Uttley et al. (2014). In the effective area of the detector is high. The energy chan- thissection,thesetechniquesareoutlinedasrelevanttothe nel of interest is subtracted from the reference band dur- simulation of reverberation and developing models of the ing the calculation of each cross spectrum such that errors process and typical features of the measurements are re- are not correlated between the bands (Zoghbi et al. 2012). viewed. Finally, the lag of each energy band with respect to the FollowingNowaketal.(1999),timelagsbetweenX-ray reference band is computed from the cross spectrum av- energy bands, used as proxies for different spectral compo- eraged over a broad range of frequencies (to minimise the nents,arecomputedfromtheFouriertransformsofthelight errors). Frequency ranges are selected from the broadband curves;thevariationovertimeofthearrivalrateofphotons lag-frequency spectrum to encompass a single region that falling within those specific energy bands. The light curve, appears to be dominated by the same process (for instance F(t), is considered to be the sum over Fourier components whether the lag is positive or negative). ofallfrequencies,ω,describingthecontributionsoflongand Reverberation measurements are illustrated in Fig. 1, short time scale variations to the overall variability: showing the broadband lag-frequency spectrum as well as 1 (cid:90) thelag-energyspectrumatbothlowandhighfrequenciesfor F(t)= √ F˜(ω)e−iωtdω the NLS1 galaxy 1H0707−495. The features displayed are 2π common to many NLS1 galaxies. The lag-frequency spec- Theamplitudeandphaseofeachfrequencycomponent trumiscomputedbetweenthe0.3to1keVenergyband,ex- MNRAS000,1–29(2016) 4 D. R. Wilkins et al. (a) Lag-frequencyspectrum (b) Lag-energy,lowfrequency (c) Lag-energy,highfrequency Figure1.(a)Thebroadbandlag-frequencyspectrumcalculatedfrom1.3MsofobservationsoftheNLS1galaxy1H0707−495withXMM- Newton asinKaraetal.(2013a).ThelagisshownasafunctionofFourierfrequency(i.e.forlongandshorttimescalecomponentsof theobservedvariability)betweenthereflection-dominated0.3-1keVenergybandandthecontinuum-dominated1-4keVband.Apositive lag indicates that the harder, 1-4keV band is lagging behind the softer, thus X-ray reverberation is seen at higher frequencies where the lag is negative and the reflection-dominated band is lagging behind the continuum. Lag-energy spectra are computed, averaged overfrequencyrangesthatshowpositiveandnegativelags.(b)Thelag-energyspectrumof1H0707−495overthelowfrequencyrange (1.65−5.44)×10−4Hzshowingthehardlag,wherethelagtieincreasessystematicallyforhigherphotonenergies.(c)Thehighfrequency lag-energyspectrum,averagedoverthehighfrequencyrange(0.96−2.98)×10−3Hzwherereverberationisseen.Variabilityinthesoft excessbetween0.3and1keVandintheironKαfluorescencelinefromthediscbetween4and7keVisseentolagbehindthecontinuum, whilethereisacharacteristicdipinthelag-energyspectrumat3keV.ManySeyfertgalaxiesshowacommonlag-energyspectrumshape between2and10keV. pected from the time-averaged X-ray spectrum to be dom- extended light curves. In this formulation, the underlying inated by reflection from the accretion disc (Zoghbi et al. variabilityintheluminosityoftheX-raysourceisdescribed 2010), and the 1 to 4keV band, dominated by directly ob- bysometimeseriesL(t).Thesefluctuationswillthenprop- servedcontinuumemission.Apositivelagindicatesthatthe agatethroughthesourcecausingvariationintheX-rayflux harder band is lagging behind the softer, hence reverber- from different regions. ation can be seen at higher frequencies (between around Once the fluctuation reaches a given region of the 5×10−4 and3×10−3Hz).Lowerfrequenciesshowthehard source, the emanating rays will propagate in the curved lag, thought to arise within the continuum and character- spacetime around the black hole to reach the observer di- istic of both AGN and accreting X-ray binaries. Frequency rectly to be observed as part of the continuum. The prop- ranges for the lag-energy spectrum are selected from this agation of the fluctuation through the source region and plot, taking the hard-lag dominated range showing a posi- subsequent propagation of the emitted rays to the observer tive lag for the low frequency lag-energy spectrum and the from each location within the source is encoded in the im- reverberation-dominatedrangewherethelagisnegativefor pulseresponsefunctionT (E,t).Thisdescribestheresponse C the high frequency lag-energy spectrum. Zero time lag cor- seen by the observer; the number of photons received as a responds to the average arrival time of the reference band, function of photon energy (i.e. the shape of the continuum here taken to be 0.3-0.8keV, with more negative times in- spectrum)andtimeafteraninstantaneous(δ-function)flash dicating an earlier response. The low frequency lag-energy propagatingthroughthecoronafromthesiteofenergyinjec- spectrum shows a steady rise in lag as a function of photon tion.Thelightcurveinagivenenergybandisgivensimply energy, while the high frequency spectrum is reminiscent of by the convolution of this response function with the time theX-rayreflectionspectrum.Thesoftexcess(below1keV) series describing the underlying variability. and iron Kα line (5-7keV) are seen to respond later than theX-raycontinuum(1-2keV).Theearliestresponseisseen L (t)=L(t)⊗T (E,t) (3) E C fromphotonsaround3keVproducingadipinthelag-energy spectrum.Theoriginofthisdiphasnotpreviouslybeenwell X-rays emitted from the coronal X-ray source will also understood but it is a common feature seen across Seyfert reachtheaccretiondiscwhereupontheywillproducethere- galaxies. Once scaled for the absolute value of the lag, the flection spectrum. The reflected X-rays will then propagate shape of the 2-10keV high frequency lag-energy spectrum fromthelocationoftheiremissionupontheaccretiondiscto is almost identical across the sampled objects (Kara et al. theobserver.Theresponseseenbytheobserverinthereflec- 2013b). tion from the accretion disc of the fluctuation in the X-ray continuum is again described by an impulse response func- tion, T (E,t). The photon count is expressed as a function R of energy (the spectrum of the reflected X-rays in the rest 3 SIMULATION OF X-RAY REVERBERATION frameofthematerialintheaccretiondiscshiftedaccording Thereverberationoffluctuationsinluminositypropagating to the gravitational redshift and Doppler shift determined through an extended source region or corona is considered bythelocationandorbitalvelocityofthematerial)andto- in the above formalism of a cross spectrum computed from tal propagation and ray travel time from the patch of the MNRAS000,1–29(2016) Modelling X-ray reverberation 5 X-ray source to different sites of reflection on the disc and Rays are propagated through numerical integration of then to the observer. the geodesic equations. In the Kerr geometry, the null Thetotalresponseseenbytheobserverisobtainedsim- geodesic equations describing the propagation of photons ply by summing the responses of the directly observed con- canbewrittenintermsofthefirstderivativesoftheBoyer- tinuum and the reflection from the accretion disc. Lindquist co-ordinates with respect to some affine parame- ter. T(E,t)=T (E,t)+T (E,t) (4) C R (cid:2)(r2+a2cos2θ)(r2+a2)+2a2rsin2θ(cid:3)k−2arh Itisimportanttonotethatinthisformalism,asisnec- t˙= (6) (cid:16) (cid:17) essarywhenX-rayreverberationismeasuredfromthecross r2 1+ a2cos2θ − 2 (r2+a2)+2a2rsin2θ r2 r spectra computed from extended light curves, the propaga- 2arksin2θ+(r2+a2cos2θ−2r)h tionandspectralresponseisassumedtobeconstantacross ϕ˙ = (7) (r2+a2)(r2+a2cos2θ−2r)sin2θ+2a2rsin4θ allfluctuations.Ifthereisanyvariation(e.g.intheoriginof thefluctuation,howitpropagatesanditsspectralresponse), θ˙2 = Q+(kacosθ−hcotθ)(kacosθ+hcotθ) (8) thiswillbeimplicitlyaveragedbytheFouriertransformsof ρ4 the measured time series. ∆ (cid:104) (cid:105) r˙2 = kt˙−hϕ˙ −ρ2θ˙2 (9) Given this definition of the impulse response function ρ2 and noting that the convolution theorem allows the Fourier With the starting co-ordinates of the ray set and the transformofthelightcurveinagivenenergybandtobeex- constants of motion h and Q determined from the initial pressed as the product of the Fourier transforms of the un- direction of propagation (k is set such that each photon as derlyingvariabilityandoftheresponsefunction,L˜ =L˜T˜ , E E unitenergy;thisdoesnotaffectthepropagation),theaffine the cross spectrum of one energy band with some reference parameter is advanced to propagate the ray. Clearly, the band can be written equations in r˙ and θ˙ allow for both positive and negative (cid:12) (cid:12)2 C˜ =L˜∗L˜ =(cid:12)L˜(cid:12) T˜∗T˜ (5) solutions, corresponding to propagation in the direction of E ref (cid:12) (cid:12) E ref increasing and decreasing values of these co-ordinates. The The time lag is again calculated from the argument of this initial signs of these steps are set to produce the correct (cid:12) (cid:12)2 and is entirely encoded in the term T˜∗T˜ . (cid:12)L˜(cid:12) is the initial direction of motion, then the sign is switched during E ref (cid:12) (cid:12) (frequency-dependent) power spectral density of the time propagationwheneverr˙2 orθ˙2 passesthroughzerobetween (cid:12) (cid:12)2 integration steps. seriesandistypicallyapowerlawinfrequency,(cid:12)L˜(cid:12) ∝f−α (cid:12) (cid:12) The geodesic equations are integrated until the ray withα∼2.Beingarealquantity,thisdoesnotaffectthelag reacheseithertheequatorialplane,whereitistakentohave in narrow frequency bins. It becomes important, however, hittheaccretiondiscsolongasithitsoutsidetheinnermost when the cross spectrum is averaged over broad frequency stable orbit for the value of the spin parameter in question, bins since this term acts as a low-pass filter, increasing the untilitreachesalimitingouterradiusof1000r oritislost g influence of the lower frequency components where the lag through the black hole event horizon. The rays that hit the varies significantly over a bin. It is also important in lag disc are binned in r and ϕ and their redshifts, which repre- measurements of real data since it will decrease the signal sentboththeshiftinphotonenergyandphotonarrivalrate to noise at high frequency. are calculated (for a source emitting photons at a given en- ergy and constant rate along each ray in its own rest frame asmeasuredbysomeobserveronthedisc).Foranobserver 3.1 The Reverberation Response Function with4-velocityv receivingphotons(4-momentump)from O Impulse response functions are calculated through general an emitter with 4-velocity v , the total redshift (which in- E relativisticraytracingsimulationsofthepropagationofrays cludes both the gravitational redshift and Doppler shift) is intheKerrspacetimearoundarotatingblackholewithspin given by parameter a=J/Mc following the procedure of Wilkins & ν v ·p(O) g vµpν(O) Fabian (2012, 2013). g−1 ≡ O = O = µν O (10) ν v ·p(E) g vρpσ(E) In order to simulate X-rays originating from an ex- E E ρσ E tendedcorona,raysarestartedatrandomlocationswithina Oncetheilluminationofthediscbythesourcehasbeen cylindricalsourceregion,definedbyanouterradius(R)and computed,itisnecessarytocalculatetheappearanceofthis lowerandupperverticalheightabovetheequatorialplanein disc to the distant observer carrying out the reverberation whichtheaccretiondiscistakentolie.Theprobabilitydis- measurements.Inordertodothis,theobserverwillmeasure tributionsareuniforminverticalpositionz,azimuthalangle the rays travelling parallel to one another arriving at the ϕ, and radial distance ρ from the black hole rotation axis. telescope. Hence to visualise the accretion disc, an ‘image Each ray is weighted by the volume element 2πρdρdz to plane’ is constructed a large distance (in this case 10,000r g produce uniform luminosity per unit volume of the corona. tobefreefromthegravitationalinfluenceoftheblackhole) Rays propagate in random directions, with uniform proba- centred on a line of sight inclined at some angle i to the bilitydistributionsincosαandβ whicharerespectivelythe rotation axis. The image plane is a regular grid of parallel polar and azimuthal angle of the rays measured in the rest rays (travelling perpendicular to its face and parallel to the frame of the emitting material. line of sight to the black hole from the centre of the image Propagation of the luminosity fluctuation through the plane) whose dimensions are taken to be 500×500r . g corona is accounted for by setting the starting time co- The Kerr metric is time-reversible under the transfor- ordinate, t of the ray at its origin. The ray is emitted once mation a→−a, hence rather than tracing all possible rays the propagation reaches that part of the corona. fromtheilluminatedaccretiondiscandfindingthosewhich MNRAS000,1–29(2016) 6 D. R. Wilkins et al. reachtheplane,thegridofparallelrayscanbetracedback- Table1.Parametersofthebest-fittingmodelreflectionspectrum wards in time to find where they originated on the disc. As to observations of the NLS1 galaxy, 1H0707−495, used as the for the illumination of the disc by the corona, the rays are basisforsimulationsofX-rayreverberation. traced from the plane until they reach the equatorial plane where they are again binned and the average travel time Component Parameter Value (the time taken for a ray from part of the disc to reach the Continuum Photonindex,Γ 3.0 observer)andredshiftarecalculatedforeachbin.Thenum- ber of rays reaching each bin, divided by the proper area Discreflection Inclination,i/deg 53 of the bin, represents the projected area of that bin as seen Spinparameter,a/GMc−2 0.998 by the observer and accounts for the magnification of parts Innerradius/ rg 1.235 of the disc by gravitational lensing. It is implicitly assumed Outerradius/ rg 1000 that each ray hitting the disc illuminates equal proper area Ironabundance/Solar 8 Ionisationparameter,ξ 50 of the disc surface. Hence, for each ray that hits the disc from the source, Reflectionfraction(0.1-10keV) 2.5 it is possible to look up from a pre-calculated table the to- tal travel time from the source to the disc to the observer and the perceived brightness of reflection from that part of the disc. When a ray hits the disc, it is assumed to pro- ducetherestframereflectionspectrumaccordingtothere- flionx model of Ross & Fabian (2005), which for each ray, is shifted by the appropriate redshift from the disc to the observer. The numbers of photons arriving at the observer (withthearrivalratealongeachrayappropriatelyshiftedby theredshiftsfromthesourcetothediscandthedisctothe observer) are then binned by energy and time, representing the impulse response function, T (E,t). R 3.2 The Continuum Response Function When considering a spatially extended X-ray source, it is necessary to consider the response to the luminosity fluctu- ation seen in the continuum as well as in its reverberation from the accretion disc, owing to the range of ray paths fromanextendedsourcetotheobserver,nottomentionthe additional delay caused by propagation of the fluctuation through the source on a finite time scale. On the contrary, Figure 2.Thetime-averagespectrumofthemodelshowingthe from a point source, there is precisely one ray path that contributionsfromthedirectlyobservedcontinuumemissionand reaches the observer. that reflected from the accretion disc, along with the fractional Theimpulseresponsefunctioniscalculatedinthesame contributionofthereflectionfromtheaccretiondiscvs.thecon- wayastheobservationoftheaccretiondiscbytheobserver. tinuumateachenergy. For a corona extended radially over the accretion disc, all rays are taken to originate from the same height above the disc (i.e. dz (cid:28) R). Rays are again traced from an image We adopt the best-fitting spectral parameters of the plane, a regular grid of parallel rays 10,000r centred on a NLS1 galaxy, 1H0707−495, shown in Table 1 for the pho- g line of sight at inclination i to the black hole rotation axis, tonindexofthepowerlawcontinuumemittedbyeachpoint backward in time, until they intercept the plane taken to in the corona and for the parameters of the black hole, ac- represent the X-ray source. The ray travel times, redshifts cretion disc and rest frame reflection spectrum. The time- and numbers of rays (representing the projected area and average spectrum of the model is shown in Fig. 2. gravitational lensing) are recorded for bins on the source Raytracingisconductedinspatialunitsofgravitational plane. Propagation of the source fluctuation is accounted radii(1r =GM/c2)andtemporalunitsofGM/c3 inorder g for by incrementing the travel time of the ray according to toexploitthescaleinvarianceofthegravitationalfieldwith itsradiusoforigininthesource(i.e.theradiusatwhichthe respecttotheblackholemass,M.Whenexpressedingrav- backward-traced ray lands on the source plane) and, again, itational radii, the effect of light bending at the equivalent the impulse response function, T (E,t) is computed as the distance from the singularity is identical regardless of the C numberofraysarrivingattheobserverfromthecontinuum blackholemass.Sotooisthe(relativistic)Keplerianorbital source as a function of photon energy and time. velocity and the associated Doppler shift and gravitational The reflection and continuum response functions are redshift.Theresultsofraytracingcalculationscantrivially summedwithcoefficientsappropriatetoproducethedesired be applied to the reverberation of X-rays around any black reflection fraction, here taken to be a ratio of 2.5 between hole simply by converting the natural gravitational units the reflection and continuum spectrum photon count rates into physical units. The reverberation lags scale simply as overthe0.1−10keVenergybandinlinewiththebest-fitting the black hole mass, M, and the frequencies at which they model to the X-ray spectrum of 1H0707−495. areobserved,measuredintheequivalentunitsofc3(GM)−1, MNRAS000,1–29(2016) Modelling X-ray reverberation 7 lagcorrespondstotheaveragelighttraveltimeoverallpaths from the source to the disc in the curved spacetime around the black hole (with light travelling more slowly the closer it passes to a massive object, Shapiro 1964). The time lag h isthendilutedaccordingtothereflectionfractionineachof theenergybandsasboththecontinuumandreflectedcom- ponentswillcontributeinvaryingproportionstoeachband, delaying the arrival of a ‘continuum-dominated’ band and advancingthearrivalofone‘reflectiondominated.’Varying theoverallreflectionfractionsimplyintroducesalinearscal- ingofthetimelag,whilepreservingtheoverallshapeofthe lag spectrum (Wilkins & Fabian 2013; Cackett et al. 2014). Adeeperunderstandingofthelighttraveltimesassoci- Figure 3.Inthesimplestreverberationmodel,X-raysareemit- atedwithX-rayreverberationcanbegleanedfromtheaver- tedfromanisotropicpointsourcethatisstationaryontheblack age arrival time of photons in just the reflected component holerotationaxisabovetheplaneoftheaccretiondisc. showninFig.5(a)andalsointhetime-andenergy-resolved response function (Fig. 4), which will be instructive when scaleinverselywiththeblackholemass.Thepredictedpro- interpreting the lag-energy spectra obtained from luminos- files of the lag-frequency and lag-energy spectra are un- ity fluctuations that propagate through extended coronæ. changed.Takingthemassoftheblackholein1H0707−496 Focusing particularly on the iron Kα fluorescence line be- to be 2×106M (Zhou & Wang 2005), GM/c3 ∼10 tween 3 and 10keV in which photons are shifted from their (cid:12) s and c3(GM)−1 ∼0.1Hz. rest frame energy of 6.4keV, the earliest photons to be ob- served are those at 5keV. This is expected considering the light travel time of photons in flat, Euclidean space from a point source, to the disc and then to the observer, neglect- 4 LAG SPECTRA FOR A POINT SOURCE ingtheeffectofspacetimecurvature.Theclassicalreflected Before considering luminosity fluctuations propagating path length from a point source 5rg above the disc plane throughextendedcoronae,welookfirstatthesimplecaseof to an observer at an inclination of 53deg is minimised for a point source shown in Fig. 3 in order to illustrate the ba- reflections approximately 8rg from the centre and this an- sicfeaturesoflagspectra.X-rayreverberationandlagspec- nulus on a relativistic accretion disc around a black hole tra from point sources is considered at length by Wilkins willshiftthe6.4keVlineemissiontobetween4and7.5keV, &Fabian(2013)andCackettetal.(2014).Fig.4showsthe with a weighted average at 5keV. The light travel time is timeandenergyresolvedresponsefunctionsofraysreflected longer to the outer parts of the disc, producing the delayed from the accretion disc. Shading corresponds to the photon double-peakedlagprofilecentredontherestframeenergyof flux received by an observer as a function of time and pho- 6.4keV. The double peaks correspond to the Doppler shift- ton energy after the continuum rays from the point source ing of reflection from the more slowly approaching and re- are received. ceding material orbiting in the outer disc. The simplest model of the lag-energy spectrum is the Whatisnotexpectedclassically,however,isthelongde- average arrival time of photons at a given energy in the layexperiencedbylinephotonsbetween2and4keV.These impulse response function, calculated as the mean of the photons are delayed from their classically predicted arrival arrival time from the response function; timetoagreaterextentbytheShapirodelaytheyexperience 1 (cid:90) passingclosetotheblackhole(bothtowardsandawayfrom τ¯(E)= (cid:82) tN(E,t)dt (11) the disc). This results in the leading edge of the response N(E,t)dt function curving to later times for the most redshifted and Theaveragearrivaltimeofphotonsoriginatingfroman blueshiftedemissionreceivedfromtheinneredgeofthedisc isotropic point source located at a height of 5r above the on the nearside to the observer. g singularity is shown in Fig. 5(a). The average arrival time A distinctive ‘loop back’ pattern is produced in the of all photons is shown as well as that for photons in just emission line response, visible between around 2 and 7keV the directly-observed continuum and reflected components. where the line tracing the delayed reception of the most The form of the lag-energy spectrum can be simply inter- blueshifted photons appears to curve back in energy form- preted,withtheearliestarrivingphotonsinthe1-2keVen- ing a second peak at later times in the redshifted wing of ergyband,whichisdominatedbythedirectly-observedcon- the line. This consists of photons reflected from the back tinuum photons, as can be seen from the reflection fraction side of the disc (that would classically be hidden behind shown in Fig. 2. These continuum photons travel directly the blackhole’s shadow).These photons are gravitationally from the point source to the observer and, therefore, will lensed into the observer’s line of sight, starting with those always travel the shortest path. blueshiftedfromtheapproachingsideofthebackofthedisc Theenergybands0.5-1keVand4-7keVaredominated (justoutofsightoftheobserver)andshiftingtothosethat by the reflection from the accretion disc, corresponding to have travelled round the black hole from the receding side thesoftexcessandtheironKfluorescenceline,respectively. ofthedisc,leadingtoare-emergenceoftheredwingofthe Due to the extra distance travelled by these photons from line. An intense response is seen from the part of the disc the primary source to the disc, they are delayed with re- immediately behind the black hole shadow. This produces spect to the continuum-dominated 1-2keV band. The time theflashrepresentedbythedarkbandandthefollowingtri- MNRAS000,1–29(2016) 8 D. R. Wilkins et al. 10 10 9 9 8 8 7 7 V V e e y / k6 y / k6 g g er5 er5 n n E E 4 4 3 3 2 2 1 1 Time Time (a) h=5rg (b) h=1.5rg Figure 4.Timeandenergyresolvedresponsefunctionsforanaccretiondiscilluminatedbyaninstantaneousflashfromapointsource at a height of (a) 5rg and (b) 2rg above the singularity located on the spin axis of the black hole. The rate of photon arrival from the accretion disc at a distant observer is shown as a function of time and for distinct photon energies (as measured by the observer, accountingforDopplershiftsandgravitationalredshift)withdarkershadingindicatinggreaterphotonflux. (a) (b) Figure 5.(a)Theaveragearrivaltimes(relativetotheearliestarrivalofphotonsineachcase)ofphotonsasafunctionofenergyafter aninstantaneousflashofemissionfromapointsourcelocated5rg abovetheplaneoftheaccretiondiscontheblackhole’srotationaxis. Alsoshownarethearrivaltimesofthephotonsthatmakeupthecontinuumandreflectedcomponents.(b)Thevariationintheoverall averagephotonarrivaltimesasthepointsourceismovedinheightabovethesingularity. angular structure between 4 and 5keV (redshifted from the forebeingturnedinthedirectionoftheobserverandbeing rest frame energy of the line as these photons are reflected observed at a much later time. The combination of these from the innermost disc radii). The region of the accretion effectsleadstotheextendedlow-energytailoftheresponse discimmediatelybehindthesingularityismostmagnifiedas function and the 2-4keV hump in the average arrival time its emission is lensed into the line of sight (Reynolds et al. of the reflected photons, however very few of the line pho- 1999). Since the full reflection spectrum is considered here, tonsarereflecteddowntothisenergybandsotheybecome includingnotjustlineemissionbutthereflectedcontinuum, lostinthecontinuum(thatarrivesmuchearlier)andarenot this re-emergence results in a flash of emission across all detected in the overall lag-energy spectrum. X-ray energies, not just the redshifted energies in the line, If the radius at which photons were reflected could be represented by the vertical band seen to coincide with the determined by the distant observer, in addition to the out- re-emergence of lien photons down the response function. ward propagating ellipsoid representing the response from Photons reflected from the innermost parts of the disc successively larger radii in the disc at later times, the re- become‘trapped’closetothephotonorbitat2r arounda sponse from the inner disc would appear to move inwards g maximally spinning black hole. These extremely redshifted at later times. The two components to the response would photons can spiral around the black hole several times be- split at the radius from which the earliest response is seen MNRAS000,1–29(2016) Modelling X-ray reverberation 9 and the delay experienced by photons passing close to the ning (and for retrograde spin) also places the inner edge of black hole would lead to the inner disc response appearing thediscfurtherfromthepointsourceabovetheblackhole, asaninwardpropagatingringtowardstheinnermoststable increasing the average lag. These dependences will remain orbit. in lag spectra for extended coronæ and the propagation of A similar effect is seen in the soft X-ray excess. Once fluctuationsthroughtheseandtheseparameterswillnotbe shiftedbyDopplerandgravitationalredshifts,theresponses considered further in the present work. of the component emission lines sit atop one another in en- ergy so the specific features of each line cannot be distin- guished. In the overall lag-energy spectrum, a broad hump 4.1 Frequency dependence of the lag is seen from the delayed arrival of these reflected photons Thelag-frequencyspectrum,showingthetimelagasafunc- with respect to the continuum and in the average arrival tionofFourierfrequency(ofthetemporalcomponentsmak- of just the reflected photons, slightly earlier arrival is seen ingupthelightcurve)ofthecontinuum-dominated1-4keV above0.5keV withlaterarrivalbelowthisoftheredshifted band with respect to the reflection-dominated 0.3-1keV wings of these lines. band is shown in Fig. 6(a). Following the usual convention, Fig.5(b)showsthevariationinthelag-energyspectrum apositivelagindicatesthattheharderbandlagsbehindthe asafunctionofthesourceheightabovethesingularity.Com- soft, hence the negative lag apparent here indicates that at paringtheaveragearrivaltimesofphotonsoriginatingfrom all frequencies, variability in the reflection-dominated soft sources at 5 and 10r above the black hole, it is clear that g band is lagging behind that in the continuum-dominated the most significant effect of increasing the source height is hard band, as expected due to the extra light travel time increasing the average lag time between the arrival of con- from the source to the disc. tinuum photons, dominating the 1-4keV energy band and AsdiscussedbyWilkins&Fabian(2013),thelagforthe both the reflection seen in the soft excess (0.3-1keV) and lowestfrequencycomponentsisequaltothemeanofthelag the iron K line (4-7keV). Indeed, Wilkins & Fabian (2013) fromtheresponsefunction.Atimelagτ greaterthan 1 can- showthatthesourceheightisthemostsignificantfactorin 2f not be detected since at this point the (sinusoidal) Fourier determiningthereverberationlagtime,thisdefiningthead- component has been shifted by half of its wavelength. It is ditional light travel time between source and reflector. The thereforeimpossibletotell(giventhedefinitionofthephase lag-energy profile remains similar between the sources at 5 angle between the two components in the range −π to π) and 10r , however changes significantly when the source is g whetherthecomponenthasbeenshiftedforwardorbackby extremely close to the black hole, as can be seen in the lag- half a wavelength. At f = 1 the phase wraps around from energy spectrum for a source at 2r above the singularity. 2τ g −π to π and the sign of the measured lag flips. Moving to Whenthesourceissoclosetotheblackhole,themajorityof higher and higher frequency, the sign continues to flip ev- X-rays that are emitted are focused towards the black hole ery 2π shift in phase and the lag, averaged across the time and hence onto the inner regions of the accretion disc. This series, decays to zero. results in a significant response from the redshifted wing of In the lag-frequency spectrum for smoothly decaying thelinethatisdelayedasthepassageofphotonspropagat- response functions (as is expected for reverberation from ing in the strong gravitational field is slowed. By contrast, the disc), the time lag contributed by the longest photon few photons are able to reach the outer parts of the accre- paths for which τ > 1 are averaged to zero, leaving only tion disc, resulting in little response from the line core at 2f the shorter paths to contribute to the measured lag. This late times. Moreover, few photons are able to escape to be results in the gradual decrease in the measured lag towards observed directly as part of the continuum, explaining the high frequencies. skewedshapeoftheprofilewithrespecttothoseforsources Whenmeasuringthelag-energyspectrumatagivenfre- at greater heights. quency, as for real data, the overall shape of the spectrum Cackett et al. (2014) consider the dependence of rever- is retained from the profile of the average photon arrival beration time lags on the parameters generally associated time as a function of energy when only the reverberation withthespectralmodellingofrelativisticallyblurredreflec- processisconsidered. The lagtimes are scaledtolower val- tion from accretion discs. In addition to the height of the ueswhenthemeasurementistakenathigherfrequencyand X-ray source above the disc, they find that the lag spectra thesharpfeaturesinthespectrumbecomesmoothedout,as dependupontheinclinationofthelineofsighttotheaccre- demonstratedinFig.6(b).Wethereforeconsidertheaverage tion disc. Observing the disc at a higher inclination (closer photonarrivaltimesasaproxyforthelag-energyspectrum to edge-on) increases the component of the orbital velocity tounderstandtheformanddependenceondifferentparam- projectedalongthelineofsight,thusincreasestherangeof eters such that observed features can be identified but note Doppler shifts. For greater inclination, the lagged emission that if these models were to be used to fit real data, the (relative to the continuum) extends up to greater energy lag-energyspectrumshouldbecomputedfromthemodelled with a broader line profile observed in the lag-energy spec- response function for the frequency range considered. trum, while there is little effect on the lag-frequency spec- trum between two broad energy bands. The black hole spin was found to influence lag measurements through the loca- 4.2 The requirement for more complex models tionoftheinnermoststableorbit.Formorerapidlyspinning blackholes,theaccretiondiscextendscloserinandenables It is apparent, comparing the frequency and energy depen- the reflection of more highly redshifted photons in a more denceofthelagmeasuredin1H0707−495(showninFig.1) extended wing of the line, lagging behind the continuum. andotherSeyfertgalaxieswiththepredictionsofmodelsin Truncationofthediscatlargerradiusformoreslowlyspin- which X-rays originate from a point source (Figs. 5 and 6), MNRAS000,1–29(2016) 10 D. R. Wilkins et al. (a) Lag-frequencyspectrum (b) Frequencydependenceoflag-energyspectrum Figure 6.(a)Thelagbetweenthe0.3-1keVreflection-dominatedbandand1-4keVcontinuum-dominatedbandforreverberationfrom an accretion disc illuminated by point source 5rg above the singularity. A positive lag indicates the harder band lags behind the soft. A negative lag hence represents X-ray reverberation where the softer reflection-dominated band is lagging behind the continuum. (b) Thevariationinthelag-energyspectrum(thelagbetweenvariabilityinthe0.1-10keVreferencebandandthatinnarrowenergybands) averagedoverhighandlowfrequencyintervals,illustratingtheeffectofmeasuringthelag-energyspectrumatdifferentfrequencies. that this simplified model of X-ray reverberation is inade- of emission located above the black hole (Wilkins & Fabian quatetoexplaintheobservedfeatures.Thedipobservedin 2011, 2012; Fabian et al. 2013; Wilkins & Gallo 2015). It many objects in the high frequency lag-energy spectrum at is therefore a natural extension to point source models to 3keV, apparent in Fig. 1(c), is not explained by X-ray re- consider the lag spectra that would be expected from these verberation from a point source. There is precisely one ray extended coronae to understand how such models can be path from an infinitesimal point source to the observer and further constrained by X-ray timing analyses and to test this path will always be shorter than those passing via the whetherthesecanexplaintheobservedfeaturesofreverber- accretiondisc.Hencetheearliestresponseshouldalwaysbe ation. seen in the continuum-dominated 1-2keV band rather than a sharp dip at 3keV. In this section, models of X-ray reverberation from ex- Point source models do not account for the transition tended coronæ are developed, both for the inward and out- to the ‘hard lag,’ denoted by not only the softer reflection- wardpropagationoffluctuationsthroughcoronæextending dominated band leading the harder continuum-dominated over the accretion disc at a low height and upward through band in the broadband lag-frequency spectrum but by the vertically collimated coronæ. We begin with simplified pre- change in energy dependence from the characteristic profile scriptionsofpropagationataconstantvelocitythroughthe of X-ray reflection to the systematic increase in lag time corona in Sections 5.1 and 5.2, before considering propa- with X-ray energy. Models in which luminosity fluctuations gation linked to viscous propagation through the underly- propagate inwards from the less energetic outer regions to ing accretion disc in Section 5.4. Once the propagation is more energetic inner regions of the corona or from outer to established, the hard lag is self-consistently included as a innerregionsoftheaccretiondisctoexplainthisbehaviour gradientinthephotonindexofthecontinuumproducedby implicitly assume a finite spatial extent to the corona, thus the corona in Section 5.3 and the seeding of the corona by cannot trivially be reconciled with a point source origin of stochasticvariationsacrossitsextentareconsideredinSec- the X-ray continuum. tion 5.5. CylindricalcoronæareconsideredwiththeX-rayemit- ting region in each case defined by a maximum radius over the disc and by a lower and upper vertical bound above 5 EXTENDED CORONAE AND the plane of the disc and the coronæ are taken to have uni- PROPAGATING SOURCE FLUCTUATIONS form luminosity throughout their extent. While this is not Measurements of the emissivity profiles of AGN accretion a completely realistic model (for instance the corona may discs, that is the pattern of illumination of the disc by the be closer to spheroidal and brighter in the central regions X-rays emitted from the corona, suggest that in a growing wheremoreenergyisinjectedfromtheaccretionflow)these number of cases, the coronæ extend over the surface of the simplifiedmodelswillallowthebasicpropertiesofextended accretiondisc.Theyhavebeenfoundtoextenduptoafew coronæ to be explored through X-ray timing. Where good tens of gravitational radii and around 30r in the case of agreementisfoundtoobserveddata,theuniformcylindrical g 1H0707−495 rather than being compact, point-like sources model will represent the approximate geometry and extent MNRAS000,1–29(2016)