Issue 
A&A
Volume 594, October 2016



Article Number  A71  
Number of page(s)  17  
Section  Extragalactic astronomy  
DOI  https://doi.org/10.1051/00046361/201527748  
Published online  13 October 2016 
Theoretical modelling of the AGN iron line vs. continuum timelags in the lamppost geometry
^{1} Department of Physics and Institute of Theoretical and Computational Physics, University of Crete, 71003 Heraklion, Greece
email: epitrop@physics.uoc.gr
^{2} IESL, Foundation for Research and TechnologyHellas, 71110 Heraklion, Crete, Greece
^{3} Astronomical Institute, Academy of Sciences, Boční II 1401, 14131 Prague, Czech Republic
^{4} Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK
Received: 15 November 2015
Accepted: 7 July 2016
Context. Theoretical modelling of timelags between variations in the Fe Kα emission and the Xray continuum might shed light on the physics and geometry of the Xray emitting region in active galaxies (AGN) and Xray binaries. We here present the results from a systematic analysis of timelags between variations in two energy bands (5−7 vs. 2−4 keV) for seven Xray bright and variable AGN.
Aims. We estimate timelags as accurately as possible and fit them with theoretical models in the context of the lamppost geometry. We also constrain the geometry of the Xray emitting region in AGN.
Methods. We used all available archival XMMNewton data for the sources in our sample and extracted light curves in the 5−7 and 2−4 keV energy bands. We used these light curves and applied a thoroughly tested (through extensive numerical simulations) recipe to estimate timelags that have minimal bias, approximately follow a Gaussian distribution, and have known errors. Using traditional χ^{2} minimisation techniques, we then fitted the observed timelags with two different models: a phenomenological model where the timelags have a powerlaw dependence on frequency, and a physical model, using the reverberation timelags expected in the lamppost geometry. The latter were computed assuming a pointlike primary Xray source above a black hole surrounded by a neutral and prograde accretion disc with solar iron abundance. We took all relativistic effects into account for various Xray source heights, inclination angles, and black hole spin values.
Results. Given the available data, timelags between the two energy bands can only be reliably measured at frequencies between ~5 × 10^{5} Hz and ~10^{3} Hz. The powerlaw and reverberation timelag models can both fit the data well in terms of formal statistical characteristics. When fitting the observed timelags to the lamppost reverberation scenario, we can only constrain the height of the Xray source. The data require, or are consistent with, a small (≲ 10 gravitational radii) Xray source height.
Conclusions. In principle, the 5−7 keV band, which contains most of the Fe Kα line emission, could be an ideal band for studying reverberation effects, as it is expected to be dominated by the Xray reflection component. We here carried out the best possible analysis with XMMNewton data. Timelags can be reliably estimated over a relatively narrow frequency range, and their errors are rather large. Nevertheless, our results are consistent with the hypothesis of Xray reflection from the inner accretion disc.
Key words: galaxies: active / Xrays: galaxies / accretion, accretion disks / galaxies: Seyfert / relativistic processes
© ESO, 2016
1. Introduction
According to the currently accepted paradigm, active galactic nuclei (AGN) contain a central, supermassive (M_{BH} ~ 10^{6−9}M_{⊙}) black hole (BH), onto which matter accretes in a disclike configuration. In the standard αdisc model (Shakura & Sunyaev 1973), this accretion disc is optically thick and releases part of its gravitational energy in the form of blackbody radiation, which peaks at optical to ultraviolet wavelengths. A fraction of these lowenergy thermal photons is assumed to be Compton upscattered by a population of highenergy (~ 100 keV) electrons, which is often referred to as the corona. The Compton upscattered disc photons form a powerlaw spectrum that is observed in the Xray spectra of AGN at energies ~ 2−10 keV (e.g. Haardt & Maraschi 1991). We here refer to this source as the Xray source and to its spectrum as continuum emission. Depending on the Xray source and disc geometry, a significant amount of continuum emission may illuminate disc and be reflected towards a distant observer.
The strongest observable features of such a reflection spectrum from neutral material are the fluorescent Fe Kα emission line at ~ 6.4 keV and the socalled Compton hump, which is an excess of emission at energies ~ 10−30 keV (e.g. George & Fabian 1991). Additionally, if the disc is mildly ionised, an excess of emission at energies ~ 0.3−1 keV can be observed (e.g. Ross & Fabian 2005). In addition to these spectral features, the Xray reflection scenario also predicts unique timing signatures. For example, Papadakis et al. (2016) showed that Xray reflection should leave its imprint in the Xray power spectra. Owing to Xray illumination, the observed power spectra should show a prominent dip at high frequencies, and an oscillatory behaviour, with decreasing amplitude, at higher frequencies. These reverberation echo features should be more prominent in energy bands where the reflection component is more pronounced. Furthermore, as a result of the different light travel paths between photons arriving directly at a distant observer and those reflected off the surface of the disc, variations in the reprocessed disc emission are expected to be delayed with respect to continuum variations. The magnitude of these delays will depend on the size and location (with respect to the disc) of the Xray source, the viewing angle, the mass, and spin of the BH.
Hints for such reverberation delays were first reported by McHardy et al. (2007) in Ark 564. The first statistically robust detection was later reported by Fabian et al. (2009) in 1H 0707–495, where variations in the 0.3−1 keV band (henceforth, the soft band) were found to lag behind variations in the 1−4 keV band by ~ 30 s on timescales shorter than ~ 30 min. The discovery of these timelags, commonly referred to as soft lags in the literature, has triggered a significant amount of research over the past few years. Soft lags have been discovered in ~ 20 AGN so far (see e.g. Uttley et al. 2014, for a review).
A growing number of AGN show evidence of reverberation timelags between the Fe Kα emission line and the continuum (e.g. Zoghbi et al. 2012, 2013; Kara et al. 2013b,a,c; Marinucci et al. 2014a), and between the Compton hump and the continuum (e.g. Zoghbi et al. 2014; Kara et al. 2015). Detecting them is a particularly difficult task because of the low sensitivity of most current detectors and the intrinsically low brightness of AGN at Fe Kα line and Compton hump energies.
Theoretical modelling of Xray timelags can elucidate the physical and geometrical nature of the Xray emitting region in AGN. This requires knowledge of how the disc responds to the continuum emission, and the construction of theoretical timelag spectra, which can then be fitted to the observed ones. Initial modelling attempts were based on the assumption that this response is a simple tophat function (e.g. Zoghbi et al. 2011; Emmanoulopoulos et al. 2011). Chainakun & Young (2012) were the first to consider a more realistic scenario, in which relativistic effects and a moving Xray source were considered to quantify the response of the disc. They deduced that, for 1H 0707–495, a more complex physical model is required to explain both the source geometry and intrinsic variability. More recently, Wilkins & Fabian (2013) considered a variety of different geometries for the primary Xray source and deduced that, in 1H 0707–495, it has a radial extent of ~ 35r_{g} (where r_{g} ≡ GM_{BH}/c^{2} is the gravitational radius) and is located at a height of ~ 2r_{g} above the disc plane.
Emmanoulopoulos et al. (2014, ; E14 hereafter) were the first to perform systematic model fitting of the timelags between the 0.3−1 and 1.5−4 keV bands (henceforth, the soft excess vs. continuum timelags) for 12 AGN. They assumed the Xray source to be pointlike and located above the BH (the socalled lamppost geometry; e.g. Matt et al. 1991), and calculated the response of the disc taking all relativistic effects into account. They deduced that the average Xray source height is ~ 4r_{g} with little scatter. Cackett et al. (2014) were the first to model the timelags between the 5−6 keV (which contains most of the photons from the red wing of a relativistically broadened Fe Kα line) and 2−3 keV bands in the AGN NGC 4151. They used a similar procedure to E14, and deduced that the Xray source height is ~ 7r_{g}, while the viewing angle of the system is < 30°. More recently, Chainakun & Young (2015; CY15, hereafter) simultaneously fitted, for the first time, the 4−6.5 vs. 2.5−4 keV timelags and the 2−10 keV spectrum of Mrk 335. They found that the Xray source is located very close to the central BH, at a height of ~ 2r_{g}.
Our main aim is to study the iron line vs. continuum timelag spectra (hereafter, the iron line vs. continuum timelags), within the context of the lamppost geometry, similarly to E14, C14, and CY15. To this end, we chose the 5−7 keV band as representative of the energy band where most of the iron line photons will be (henceforth, the iron line band), and the 2−4 keV band as the energy band where the primary Xray continuum dominates (henceforth, the continuum band). In our case, the exact choice of these two energy bands is relatively unimportant since, contrary to previous works (with the exception of CY15), we take into account the full disc reflection spectrum in both the iron line and continuum bands when constructing the theoretical lamppost timelag models, which we subsequently fitted to the observed iron line vs. continuum timelag spectra.
Our sample consists of seven AGN. We chose these objects because they are Xray bright and have been observed many times by XMMNewton. We used all the existing XMMNewton archival data for these objects to estimate their iron line vs. continuum timelags. Our work improves significantly on the estimation of timelags. We relied on Epitropakis & Papadakis (2016; EP16, hereafter) to calculate timelag estimates that are minimally biased, have known errors, and are approximately distributed as Gaussian variables. These properties render them appropriate for model fitting using traditional χ^{2} minimisation techniques.
Our results indicate that the data are consistent with a reverberation scenario, although the quality of the data is not high enough to estimate the various model parameters with high accuracy, except for the Xray source height.
2. Observations and data reduction
Table 1 lists the details of the XMMNewton observations we used. Columns 1−4 show the source name, mass of the central BH in units of 10^{6}M_{⊙}, identification number (ID) of each observation, and net exposure in units of ks, respectively.
XMMNewton observations log.
We processed data from the XMMNewton satellite using the Scientific Analysis System (SAS, v. 14.0.0; Gabriel et al. 2004). We only used EPICpn (Strüder et al. 2001) data. Source and background light curves were extracted from circular regions on the CCD, with the former having a fixed radius of 800 pixels (40′′) centred on the source coordinates listed on the NASA/IPAC Extragalactic Database. The positions and radii of the background regions were determined by placing them sufficiently far from the location of the source, while remaining within the boundaries of the same CCD chip.
The source and background light curves were extracted in the iron line and continuum bands with a bin size of 100 s, using the SAS command evselect. We included the criteria PATTERN==0–4 and FLAG==0 in the extraction process, which select only single and doublepixel events and reject bad pixels from the edges of the detector CCD chips. Periods of high solar flaring background activity were determined by observing the 10−12 keV light curves (which contain very few source photons) extracted from the whole surface of the detector, and subsequently excluded during the source and background light curve extraction process.
We checked all source light curves for pileup using the SAS task epatplot and found that only observations 670130201, 670130501, and 670130901 of Ark 564 are affected. For those observations we used annular instead of circular source regions with inner radii of 280, 200, and 250 pixels (the outer radii were held at 800 pixels), respectively, which we found to adequately reduce the effects of pileup.
The background light curves were then subtracted from the corresponding source light curves using the SAS command epiclccorr. Most of the resulting light curves were continuously sampled, except for a few cases that contained a small (≲ 5% of the total number of points in the light curve) number of missing points. These were either randomly distributed throughout the duration of an observation, or appeared in groups of ≲ 10 points. We replaced the missing points by linear interpolation, with the addition of the appropriate Poisson noise.
3. Timelag estimation
XMMNewton light curve characteristics relevant to the estimation of timelags.
We used standard Fourier techniques (see, e.g., Nowak et al. 1999; Uttley et al. 2014) to estimate timelags between light curves in the iron line and continuum bands for our sample.
We denote by { x(t_{r}),y(t_{r}) } a pair of light curves in two energy bands, where t_{r} = Δt,2Δt,...,NΔt, N is the number of points and Δt = 100 s is the time bin size. The discrete Fourier transforms (DFTs), { ζ_{x}(ν_{p}),ζ_{y}(ν_{p}) }, of the light curves are where and are the lightcurve sample means, and ν_{p} = p/NΔt (p = 1,2,...,N/ 2). The crossperiodogram, I_{xy}(ν_{p}), of the lightcurve pair is defined as (Priestley 1981, henceforth P81) (3)The crossperiodogram is an estimator of the intrinsic crossspectrum (CS), C_{xy}(ν_{p}), which is a measure of the crosscorrelation between two random signals in Fourier space. The crossperiodogram is generally biased, in the sense that the mean of I_{xy}(ν_{p}) is not equal to C_{xy}(ν_{p}). The traditional timelag estimator, which we define below, is based on the crossperiodogram. Therefore, the statistical properties of the two estimators are closely dependent. As shown by EP16, there are two main factors that contribute to the bias of the crossperiodogram: the finite duration of the light curves, and their sampling rate and time bin size (in our work, the sampling rate is equal to the time bin size).
Discrete sampling of a continuous process introduces aliasing effects to the CS of the resulting discrete process, which is only defined in the frequency range [− 1 / 2Δt,1 / 2Δt] and is equal to the superposition of the intrinsic CS at frequencies ν,ν ± 1 / Δt,ν ± 2 / Δt, etc. Aliasing effects are reduced when the light curves are binned. They are similar to the aliasing effects in the powerspectral density (PSD) of a light curve, although while PSDs are always positive, this is generally not the case with CS. As a result, aliasing effects are more complex in this case. EP16 found that lightcurve binning generally causes the measured timelags to converge to zero at frequencies ≳ ν_{Nyq}/ 2, where ν_{Nyq} = 1 / 2Δt is the Nyquist frequency. In this work ν_{Nyq} = 5 × 10^{3} Hz, and hence we only computed crossperiodograms at frequencies ≤ 2.5 × 10^{3} Hz.
Owing to the finite lightcurve duration, the mean of the crossperiodogram is equal to the convolution of the intrinsic CS (as modified by aliasing effects) with a particular window function, just like the case of the periodogram (i.e. the traditional PSD estimator; see e.g. Papadakis & Lawrence 1993). However, the effects of this convolution on the timelag estimates cannot be predicted a priori, since they depend on the shape of the (unknown) intrinsic CS (and not just on the intrinsic timelag spectrum). They were quantitatively investigated by EP16, who considered three different types of timelag spectra that are typically observed between Xray light curves of accreting systems: constant timelags, timelags with a powerlaw dependence on frequency, and timelags that have a characteristic oscillatory behaviour with frequency, similar to what is expected in a reverberation scenario. For the model CS they considered, they concluded that timelag estimates based on the crossperiodogram will not be significantly biased, in the sense that their mean will be within ~ 15% (in absolute value) of their corresponding intrinsic values when the lightcurve duration is ≳ 20 ks.
The crossperiodogram has a large and unknown variance. As a result, this feature will be shared by the timelag estimates computed from it. This problem is ameliorated in practice by either binning together m neighbouring frequency ordinates of the crossperiodogram (a process called smoothing), and/or binning different crossperiodogram ordinates at a given frequency obtained from m distinct lightcurve pairs. If, as is often the case in practice, the real and imaginary parts of the intrinsic CS vary in a nonlinear fashion over the smoothed frequency range, then smoothing will introduce an additional source of bias to the crossperiodogram. This bias can only be taken into account a posteriori when fitting observed timelags by prescribing a model CS (and not just a model timelag spectrum), as it affects the crossperiodogram itself.
Since this is a complicated modeldependent procedure, we did not perform any smoothing on the crossperiodograms. We instead divided the available XMMNewton observations of each source into shorter segments of duration 20−40 ks. The segment duration for each source (listed in Col. 2 of Table 2) was determined in such a way as to maximise their number, m, for the total available light curves (m is listed in Col. 3 of Table 2).
For each segment we calculated the crossperiodogram according to Eq. (3), and adopted (4)and (5)as our estimates of the CS and timelag spectrum, respectively ( is the crossperiodogram of the kth segment at frequency ν_{p}). We adopted the standard convention of defining arg [Ĉ_{xy}(ν_{p})] on the interval (− π,π]. The analytic error estimate of is given by (e.g. P81; Nowak et al. 1999) (6)where (e.g. P81; Vaughan & Nowak 1997) (7) and are the traditional periodograms of the two light curves, which are also calculated by binning over m segments. Equation (7) defines an estimator of the socalled coherence function. This function is defined on the interval [0,1] and quantifies the degree of linear correlation between sinusoidal components of two light curves at each frequency.
Figure 1 shows the sample iron line vs. continuum coherence and timelag spectrum of MCG–63015 (top and bottom panel, respectively), which were calculated using Eqs. (7) and (5). The sample coherence decreases to zero with increasing frequency. This loss of coherence is mostly caused by Poisson noise. In the presence of measurement errors, even if the intrinsic coherence is unity at all frequencies, the resulting coherence will decrease towards zero at frequencies where the amplitude of experimental noise variations dominates the amplitude of the intrinsic variations. The sample coherence will, however, always converge to a constant value 1 /m at these frequencies. EP16 found that this decrease can be reasonably approximated by an exponential function of the form (8)where ν_{0} and q are parameters that are determined by fitting this function to the coherence estimates. This was empirically found by EP16 to fit the sample coherence well, using many simulations of light curves in the case of various model CS and light curve signaltonoise ratios (S/N). An example of such a fit to the coherence estimates of MCG–63015 is shown in the top panel of Fig. 1 (brown dashed line). The fit describes the sample coherence function well (this was the case for all sources).
Fig. 1 Sample iron line vs. continuum coherence function (top panel) and timelag spectrum (bottom panel) of MGG–603015, estimated using the data listed in Table 2. The dashed brown line in the top panel shows the bestfit model to the sample coherence. The continuous red vertical line indicates the highest frequency up to which timelags should be estimated, and the horizontal blue dotteddashed line indicates the coherence value at this frequency (see Sect. 3). 

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According to Eq. (6), the error of the timelag estimates increases as the coherence decreases. Therefore, above a certain maximum frequency, ν_{max}, when the coherence is sufficiently small (i.e. ~ 0), we expect that Poisson noise will severely affect the reliability of the timelag estimates. The effects of Poisson noise on the bias and distributions of the timelag estimates were quantitatively investigated by EP16. They found that ν_{max} decreases as the S/N of the light curves decreases. In addition, ν_{max} is mainly affected by the energy band with the lowest mean count rate, which in our case corresponds to the iron line band. In Cols. 4 and 5 of Table 2 we list the mean count rate in the iron line and continuum band, respectively. According to EP16, ν_{max} corresponds approximately to the frequency at which the sample coherence function becomes equal to 1.2 / (1 + 0.2m). Above ν_{max}, EP16 found that Poisson noise has the following effects on the timelag estimates: (a) The analytic error estimate given by Eq. (6) increasingly underestimates their true scatter; and (b) their distribution becomes uniform and symmetrical about a zero timelag value. As a result, the timelag estimates become biased, in the sense that their mean converges to zero, independent of the intrinsic timelag spectrum. Below ν_{max}, and as long as m ≳ 10, the mean of the timelag estimates is not affected, Eq. (6) provides a reliable estimate of their true scatter, and their distribution is approximately Gaussian.
We therefore fitted the coherence estimates of each source to the exponential function given by Eq. (8) (as we did for MCG–63015), and equated this function to the constant 1.2 / (1 + 0.2m) to estimate ν_{max} in each case. The values of ν_{max} calculated thus are listed in Col. 6 of Table 2. We did not estimate timelags above this frequency. Instead of using the values of the sample coherence function to determine the errors of the timelag estimates according to Eq. (6), we used the values of the bestfit exponential model. We found that the resulting errors are more representative of the observed scatter of the timelag estimates, although the differences are small (≲ 20%). The iron line vs. continuum timelag estimates for each source, along with their errors, obtained by the above procedure are shown in Figs. 2 and 3.
4. Theoretical modelling of the timelag spectra
In this section we describe the basic physical and geometrical properties of the lamppost model and show how we determined the corresponding theoretical iron line vs. continuum timelag spectra. All physical quantities in the lamppost model are estimated in geometrised units (G = c = 1) and scale with M_{BH}. Thus, for instance, timescales have to be multiplied by a factor t_{g} ≡ GM_{BH}/c^{3} ~ 5(M_{BH}/ 10^{6}M_{⊙}) s to be converted into units of seconds.
Fig. 2 Observed iron line vs. continuum timelag spectra for 1H 0707−495 (first row), MCG–63015 (second row), and Mrk 766 (third row). The solid brown and dashed red lines indicate the bestfit models A and B, respectively, to each timelag spectrum (see Sect. 5 for details on these models). 

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Fig. 3 Same as in Fig. 2 for NGC 4051 (first row), Ark 564 (second row), NGC 7314 (third row), and Mrk 335 (fourth row). 

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4.1. Geometrical layout of the model
The lamppost model consists of a BH, surrounded by an equatorial accretion disc, that is illuminated by an Xray source located on the disc symmetry axis. The parameters of the model are the mass and spin (a) of the BH, the height (h) of the Xray source, and the viewing angle (θ) of a distant observer with respect to the disc axis.
The disc is assumed to be geometrically thin and Keplerian, corotating with the BH, with a radial extent ranging from the innermost stable circular orbit (ISCO), r_{ISCO}, up to an outer radius r_{out}. The BH spin uniquely defines r_{ISCO}. When measured in geometrised units, the spin can attain any value between zero and unity, with a = 0 (r_{ISCO} = 6r_{g}) and a = 1 (r_{ISCO} = 1r_{g}) indicating a nonspinning (i.e. Schwarzschild) and maximally spinning (i.e. extreme Kerr) BH, respectively. The Xray source is assumed to be pointlike and located at a fixed position above the BH. It emits isotropically with an intrinsic (i.e. restframe) spectrum of N(t)E^{2}exp(−E/ 300 keV). We assumed it to be variable in amplitude only, and that N(t) is a stationary random process (i.e. that it has a finite and timeindependent mean and variance).
4.2. Observed fluxes at infinity
We assumed that the total flux recorded by an observer at a very large distance in a given energy band ℰ = [E_{1}(keV),E_{2}(keV)] is F_{ℰ}(t;a,θ,h,r_{out}). This flux is equal to the sum of the continuum and reprocessed flux from the disc, and , respectively. In other words, (9)where Ψ_{ℰ}(t′;a,θ,h,r_{out}) is the socalled response function, which quantifies the response of the disc to an instantaneous flare of continuum emission. We define the normalisation of the response function such that its timeintegrated value is equal to the observed ratio of reprocessedtocontinuum photons.
The observed continuum spectrum differs in amplitude from its restframe value as a result of relativistic effects (Dovčiak et al. 2011). This is quantified by the factor G(a,θ,h), such that . The dependence of the various terms on the righthand side of Eq. (9) on the parameters of the lamppost model were explicitly included and are henceforth be omitted for reasons of brevity.
4.3. Timelag spectra
We assumed that the total photon fluxes observed in the iron line and continuum bands are F_{5−7}(t) and F_{2−4}(t), respectively. According to Eq. (9), The crosscorrelation function (CCF) between the iron line and continuum bands, R_{5−7,2−4}(τ), is then (12)where E is the expectation operator, and μ_{5−7} (μ_{2−4}) is the mean flux in the iron line (continuum) band. The CS, C_{5−7,2−4}(ν), between the two energy bands is, by definition, the Fourier transform of the CCF. Hence (see Appendix A for a more detailed derivation) (13)where the asterisk denotes complex conjugation, is the Fourier transform of the response function, and is the CS of the continuum emission.
The iron line vs. continuum timelag spectrum, τ_{5−7,2−4}(ν), is defined as τ_{5−7,2−4}(ν) ≡ (2πν)^{1}arg [C_{5−7,2−4}(ν)]. Given our adopted convention, a positive value of τ_{5−7,2−4}(ν) indicates that variations in the iron line band lead variations in the continuum band (and vice versa). According to Eq. (13), (14)The equation above shows that the timelags between the observed variations in the two energy bands equal the sum of two terms; timelags between variations in the Xray continuum, , and timelags due to reprocessed Xray emission from the disc, (henceforth, the continuum and reverberation timelags, respectively). The continuum timelags are given by , while the reverberation timelags are given by (15)This function is uniquely determined by the disc response functions in the iron line and continuum bands.
Model fit results.
4.4. Response functions in the lamppost geometry
To determine the response function of the disc, we assumed that the primary Xray source isotropically emits a flare of duration equal to 1t_{g}. Upon being illuminated, each area element of the disc responds to this flare by isotropically and instantaneously emitting a reflection spectrum in its restframe. We assumed that the reprocessed flux is proportional to the incident flux and that the disc material is neutral, with an iron abundance equal to the solar value. We then used the restframe reflection spectrum computed with the multiscattering code NOAR (Dumont et al. 2000). We determined the timevarying 0.1−8 keV disc reflection spectrum at infinity, with a time resolution of 0.1t_{g} and energy resolution of 20 eV, taking all relativistic effects into account (e.g. gravitational and Doppler energy shifts, light bending, and time delays; Karas 2006).
Finally, we calculated the disc response function in the iron line and continuum bands by integrating the observed disc reflection spectrum in the appropriate energy ranges. In Appendix B we show how the disc response functions depend on the parameters of the lamppost model, while in Appendix C we show how we computed the model timelag spectra given by Eq. (15) from the numerically computed disc response functions. In Appendix D we discuss how these model timelag spectra depend on the parameters of the lamppost model.
The response functions we computed are similar to those presented by Reynolds et al. (1999) and E14, although our approach is different. They calculated the response function considering only the Fe Kα photons emitted in the disc restframe. In contrast, we counted all the photons from the reflection component that an observer will detect in an energy band (within 0.1−8 keV) at each time step. We therefore considered the total reflection spectrum, as emitted by the disc rest frame, hence we computed the total reflection response function and selfconsistently included the reflection component in both the continuum and iron line bands along with the Xray continuum emission. This approach is be more appropriate for comparing our predictions with data. Our approach is similar to the one adopted by CY15, although we did not consider the effects of disc ionisation.
5. Fitting procedure
As explained in Sect. 3, our timelag estimates should be approximately distributed as Gaussian random variables. Fitting the observed iron line vs. continuum timelags was therefore based on minimising the χ^{2} function, which is defined as (16)where { a_{1},a_{2},...,a_{q} } are the parameters of the model, is the timelag estimate with error , τ(ν_{p};a_{1},a_{2},...,a_{q}) is the model timelag spectrum, and n is the number of timelag estimates. The location of the χ^{2}(a_{1},a_{2},...,a_{q}) minimum, say , determines the bestfit parameter values. Their corresponding 68% (95%) confidence intervals are determined by the standard Δχ^{2} = 1 (Δχ^{2} = 4) method for one independent parameter. Unless otherwise mentioned, confidence intervals of bestfit parameters are henceforth quoted at the 68% level.
As we showed in Sect. 4.3, the observed timelags should depend on both the continuum and reverberation timelags. We thus considered two different model timelag spectra, one for the continuum and the other for the reverberation timelags. We describe them in more detail below.
5.1. Model A: continuum timelags model
In AGN and Xray binaries, timelag spectra between Xray light curves are typically observed to have a powerlaw dependence on frequency. Highenergy bands are delayed with respect to lower energy bands, and the magnitude of the timelags decreases with increasing frequency, typically following a powerlaw like form (e.g. Miyamoto & Kitamoto 1989; Nowak & Vaughan 1996; Nowak et al. 1999; Papadakis et al. 2001; McHardy et al. 2004; Arévalo et al. 2006, 2008; Sriram et al. 2009). In addition, the magnitude of these timelags is observed to increase with increasing energy separation between the two energy bands. We therefore considered a powerlaw model of the form (17)where A and s are positive, to account for the continuum timelags. These continuum timelags are expected to be negative in our case, meaning that variations in the iron line band should be delayed with respect to variations in the continuum band.
5.2. Model B: Reverberation timelags model
The model B timelag spectrum corresponds to the function given by Eq. (15), that is to say, it accounts for the reverberation timelags. This function is uniquely determined by the Fourier transforms of the response functions in the iron line and continuum bands. Since these response functions are not given by an analytical formula, we had to numerically compute them (following the procedure outlined in Sect. 4.4) on a grid of points corresponding to different combinations of { a,θ,h,M_{BH} } values (we set r_{out} = 10^{3}r_{g} in all cases).
6. Results
According to Eq. (14), we should fit the observed timelags with the sum of models A and B. However, we discovered that due to the limited frequency range of the observed timelag spectra and the relatively large errors of the timelag estimates, it was not possible to simultaneously constrain the parameters of both models in a meaningful way. We therefore decided to fit the two models separately to the data and then investigate whether they provided a good fit or not. The only exceptions were Ark 564 and NGC 7314, whose observed timelag spectra we also fitted to a combined model A+B as well for reasons we discuss in Sect. 6.2 below.
The continuum timelags model (i.e. model A) is defined by Eq. (17). The model has two free parameters (A and s). For each observed timelag spectrum shown in Figs. 2 and 3, we calculated χ^{2}(A,s) using Eq. (16). We then minimised this function numerically using the LevenbergMarquardt method, and determined the bestfit values and confidence intervals of the model A parameters.
For the reverberation timelags (i.e. model B), the parameter space we considered for the model parameters { a,θ,h,M_{BH} } is similar to the one used by E14. First, we considered three spin values, a = { 0,0.676,1 }. For each spin value we considered an ensemble of 18 heights ranging from 2.3 to 100r_{g}. For every such combination we finally considered three values for the viewing angle, θ = { 20°,40°,60° }, and 1000 values for M_{BH} ranging from 0.1 × 10^{6}M_{⊙} to 100 × 10^{6}M_{⊙} with a step of 0.1 × 10^{6}M_{⊙}. The parameter space thus consists of a grid of 3 × 3 × 18 × 1000 = 162 000 points.
We computed the response functions in the iron line and continuum bands for each point in the parameter space, and used Eq. (15) to estimate the corresponding model B timelag spectrum. We then calculated χ^{2}(a,θ,h,M_{BH}) on the parameter space, based on the observed timelag spectra of each source, according to Eq. (16). The resulting grid of χ^{2} points was subsequently interpolated quadratically in the parameters { a,θ }, and cubically in { h,M_{BH} }. We finally used the continuous, interpolated χ^{2}(a,θ,h,M_{BH}) space to obtain , along with the corresponding bestfit values and confidence intervals of the model B parameters.
6.1. Model A bestfit results
Model A fits the observed timelag spectra well for all sources. Our bestfit results are listed in Cols. 2−4 of Table 3. The bestfit models are shown as continuous brown lines in Figs. 2 and 3. The observed timelag spectra of 1H 0707–495, MCG–63015, and Mrk 766 are flat. The fit is thus contrived for these sources, in the sense that the bestfit s value is ~ 0. In Col. 3 of Table 3 we therefore list only the upper limit on s. We refitted the observed timelag spectra of these sources to a constant delay (i.e. we set s = 0). The resulting fit is statistically acceptable (, 19.0/24, and 19.5/14 for 1H 0707–495, MCG–63015, and Mrk 766, respectively), and the bestfit normalization (i.e. the bestfit constant delay in this case) is , − 21 ± 9 s and − 71 ± 19 s for 1H 0707–495, MCG–63015, and Mrk 766, respectively. When we assumed a Gaussian distribution for the bestfit A values, the bestfit errors can be used for s = 0 to estimate the probability of A = 0 (i.e. the probability that the observed timelag spectrum is identically zero). We find a probability of 16%, 2%, and 0.02% for 1H 0707–495, MCG–63015, and Mrk 766, respectively (this is a rough estimate and should thus only be considered as indicative).
The timelag spectra for the remaining sources show evidence of curvature at low frequencies (≲ 2 × 10^{4} Hz), in the sense that model A requires a nonzero bestfit s value.
6.2. Model B bestfit results
Model B fits the observed timelag spectra of all sources well. When allowing for all four model B parameters to be free during the fitting procedure, we found that a and θ are unconstrained, in the sense that even their 68% confidence interval is larger than the broadest allowed range for the parameter value (0−1 and 20°−60° for a and θ, respectively). The reason is the large errors of the observed timelags and, as discussed in Appendix D, the weak dependence of the model B timelag spectra on these parameters. Furthermore, for most sources there is a degeneracy between h and M_{BH}, which is caused by the similar dependence of the model B timelag spectra on these parameters (see Appendix D).
To constrain a and h in the best possible way, we set θ = 40° (the mean value found for a similar sample of sources studied by E14) for all sources, and M_{BH} to the values listed in Col. 2 of Table 1. We then repeated the fitting procedure to obtain the bestfit a and h values. The bestfit models are shown as dashed red lines in Figs. 2 and 3.
Even by fixing θ and M_{BH}, we found that a cannot be constrained. In the last column of Table 3 we list the value when we allowed a to be free during the fitting procedure, while in parentheses we list the corresponding values when we froze a to the value of 0 and 1, respectively. They are very similar for almost all sources, indicating that we are unable to constrain a. MCG–63015 stands as an exception, since for this source we obtained a bestfit a value of . The upper 95% level is 0.8, which is somewhat inconsistent with the results obtained by modelling the Xray spectrum of this source, which requires a ~ 1 (e.g. Marinucci et al. 2014b). This is due to our choice of M_{BH} during the fitting procedure. For example, when we set M_{BH} = 3 × 10^{6}M_{⊙} (which is consistent, within the errors, with the value listed in Col. 2 of Table 1), the bestfit value of a is 0.4, while the 95% confidence level ranges from 0 to 1.
Column 5 of Table 3 lists our bestfit results for h. The Xray source height is well defined only for Mrk 766 and Mrk 335. For 1H 0707–495 and MCG–63015 the bestfit h values are 4r_{g} and 2.3r_{g} (which is the lowest allowed fitting value for h), respectively. The lower 68% limit is 2.3r_{g} for 1H 0707–495. The upper limit is 20r_{g} and 3r_{g} for 1H 0707–495 and MCG–63015, respectively. For NGC 4051 we obtained a bestfit h value of 17r_{g}, with a lower and upper 68% limit of 2.3r_{g} and 30r_{g}, respectively. Given that the lower limit is equal to the lowest value we considered for h, we list only the upper limit on h for these three sources.
The bestfit h values for Ark 564 and NGC 7314 are 83r_{g} and 100r_{g} (which is the highest allowed fitting value for h), respectively. The Xray source height is consistent with the value of 100r_{g} for Ark 564. The lower limit on the bestfit h value is 28r_{g} and 82r_{g} for Ark 564 and NGC 7314, respectively. As a result, we list the lower limit of this parameter for these two sources in Table 3. The high h values arise because the observed timelag spectra of these sources increase (in magnitude) with decreasing frequency. The lower limit of h for NGC 7314 is higher than for Ark 564 because M_{BH} in lower in the former source.
However, it is not certain that these two sources have a large Xray source height. To investigate this further, we fitted their observed timelag spectra with a model A+B combination. We kept the Xray source fixed at h = 3.7r_{g} (the mean height found by E14), set θ = 40°, fixed M_{BH} to the respective values listed in Col. 2 of Table 1, and let a = 1. In effect, we kept all the model B parameters fixed to a given value during the fit (as we explained above, we cannot reach a meaningful fit when we let all the model A and B parameters free during the fit) so that the number of degrees of freedom is the same as when we fit the data with model A. Our bestfit results in this case are , , , and . As expected, the reverberation timelag component causes the resulting bestfit A and s values to be lower and steeper, respectively, than the respective bestfit model A values listed Table 3. The quality of the combined model A+B fit is similar to that of model A: , and in the case of Ark 564 and NGC 7314, respectively. This result shows that the observed iron line vs. continuum timelags of Ark 564 and NGC 7314 can be fitted well by a combination of a continuum plus reverberation component, the latter of which corresponds to a low h and high a value.
7. Discussion and conclusions
We performed a systematic analysis of the iron line vs. continuum (5−7 vs. 2−4 keV) timelags in seven AGN. The AGN we studied are Xray bright and highly variable. The BH mass estimates for these sources are ≲ 5 × 10^{6}M_{⊙}, except for Mrk 335, which has a corresponding estimate of ~ 3 × 10^{7}M_{⊙} (note that these mass estimates are determined from optical techniques like reverberation mapping, and are not derived here).
Our measurements are among the best that can currently be achieved and are able to be obtained for many years to come (with current Xray satellites). Our choice of focusing on the iron line band was motivated by the simple fact that its existence indicates the presence of an Xray reflection component (either from the disc or from distant material) in this band. It is thus is a clean band, ideal for investigating whether Xray reflection operates in the inner part of the putative accretion disc. However, the low number of photons in this band undermines this advantage. Nevertheless, we found that the iron line vs. continuum timelags are consistent with the simplest Xray reflection scenario. They also imply Xray source heights that are close to those derived using data from lower energy bands. This result supports the hypothesis that the Xray soft excess in these sources is a reflection component (see the relevant discussion in Sect. 7.3).
7.1. Estimation of timelag spectra
We used all the available archival XMMNewton data for seven Xray bright and highly variable Seyfert galaxies and employed standard Fourier techniques to estimate the iron line vs. continuum timelag spectrum of each source. These sources have a large (≳ 0.3 Ms) amount of archival XMMNewton data. We also took the results obtained from extensive numerical simulations performed by EP16 into account, who studied the effects of the light curve characteristics (duration, time bin size, and Poisson noise) on the statistical properties of the traditional timelag estimators assuming various intrinsic timelag spectra commonly observed between Xray light curves of accreting systems. EP16 found the following:

a)
Timelag estimates should be computed at frequencies lowerthan half the Nyquist frequency. This minimises the effects oflightcurve binning on their mean values.

b)
The crossperiodogram should not be binned over neighbouring frequencies, as this may introduce significant bias that can only be taken into account when a model CS (and not just a model timelag spectrum) is assumed.

c)
Timelags should be estimated from a crossperiodogram that is averaged over pairs of continuous lightcurve segments with the same duration.

d)
If the number of segments, m, is ≳ 10, the timelag estimates will have known errors and approximately follow a Gaussian distribution, provided they are estimated at frequencies at which the sample coherence is ≳ 1.2 / (1 + 0.2m). This minimises the effects of Poisson noise on their mean values.
Following these results, we chose the segment duration to be ~ 20 ks. This limits the minimum frequency that can be reliably probed to be ~ 5 × 10^{5} Hz. A longer segment duration would allow us to probe even lower frequencies, but at the same time it would decrease the number of the available segments, and, consequently, increase the error of the resulting timelag estimates. According to EP16, if the segment duration is ≳ 20 ks, then the timelag bias should be ≲ 15% compared to their intrinsic values for the model CS they considered. In Appendix E, we demonstrate that we do not expect the timelag bias to be a problem in our study.
The maximum frequency that can be reliably probed by the current data is set by point (d) above. The frequency at which the coherence becomes lower than the critical value of ~ 1.2 / (1 + 0.2m) depends on the number of segments and is mainly determined by the energy band with the lowest average count rate. This is the iron line band in all cases; the mean count rate of all light curves in our sample is 0.38 ± 0.27 cts / sec and 1.49 ± 0.98 cts / s for the iron line and continuum band, respectively. We found that the maximum frequency is ≲ 10^{3} Hz for all sources. Given that the sources in our sample are Xray bright and have a large amount of archival data, the available XMMNewton data allow for the reliable estimation of iron line vs. continuum timelags at frequencies between ~ 5 × 10^{5} Hz and ~ 10^{3} Hz.
A direct comparison with published iron line vs. continuum timelags for the sources in our sample is complicated by three factors: the choice of energy bands, the XMMNewton observations used to estimate them, and the crossperiodogram smoothing and/or averaging scheme employed to estimate the timelags.
Similar energy bands to ours have been used for Mrk 335, NGC 7314, NGC 4151, and MCG–52316. For Mrk 335, the timelag magnitudes and errors we find are consisted with those reported by CY15, although they only used data from a single XMMNewton observation, which corresponds to ~ 40% of the data we used. The iron line vs. continuum timelags reported by Zoghbi et al. (2013) for NGC 7314 are also roughly consistent in magnitude with our findings. They used data from only two XMMNewton observations, which corresponds to ~ 30% of the data we used. Their timelags are larger (in magnitude) than ours at low frequencies. They provide timelag estimates at frequencies lower than ours. Owing to the limited length of the data sets they used, their lowfrequency estimates must have been obtained from averaging a small number of crossspectral estimates at neighbouring frequencies. As a result, according to EP16, these estimates should be far from being Gaussiandistributed, and the frequently used timelag error prescription of Nowak et al. (1999) should severely underestimate the true scatter of these estimates around their mean. We did not estimate timelags for NGC 4151 and MCG–52316, since the available XMMNewton archival light curves at the time we were analysing the data were not long enough to obtain reliable (in the sense explained in Sect. 3) timelag estimates.
7.2. Modelling the observed timelag spectra
We considered two different model timelag spectra: (a) a powerlaw timelag spectrum that describes delays between Xray continuum variations in different energy bands (model A); and (b) a reverberation timelag spectrum that describes delays between the Xray continuum and reprocessed disc emission in a lamppost geometry (model B). The first is a phenomenological model, while the second is a physical model that depends on the central source geometry. We calculated the model B timelag spectra by determining accurate disc response functions in the iron line and continuum bands. We fixed the photon index of the Xray source at a value of 2 and assumed a neutral, prograde disc with an iron abundance equal to the solar value, around a spinning BH. The inner disc radius was set to the location of the ISCO, and the outer radius was fixed at 10^{3}r_{g}. We took all relativistic effects into account and considered the total reprocessed disc emission (and not just the photons initially emitted by the disc at 6.4 keV) in both the iron line and continuum bands. In this respect, our modelling is more accurate than previous attempts (e.g. E14 and C14).
We found that the model B timelag spectra have a weak dependence on the BH spin and viewing angle. On the other hand, they depend strongly on the BH mass and Xray source height. These parameters affect the model B timelag spectra in a similar way. As the height increases, the model B timelag spectra flatten at lower frequencies, and to a lower level; the same effect can also be produced by a higher BH mass for the same height (in units of r_{g}). In addition, the characteristic flattening of the reverberation timelag spectra to a constant value at low frequencies also depends on the outer disc radius. Therefore, the magnitude of this constant level cannot be used in a straightforward way to determine either the Xray source height or the outer disc radius, even when the BH mass is known.
Our modelling can be improved in many ways. For example, we could let the slope of the Xray continuum spectrum, as well as the iron abundance, be free parameters. These parameters mainly influence the amplitude of the disc response function (as they affect the reflection fraction in each energy band). In this case, these parameters should affect the response functions similarly to the BH spin (at small heights). Consequently, we do not expect the difference in the resulting model timelag spectra to be significant (see the bottom left panel in Fig. B.1). As shown by CY15, for instance, disc ionisation also affects the model timelag spectra and should be included in the determination of the response functions. More importantly, however, the main limitation of our modelling is the adopted geometry. The lamppost geometry is a simplification of the AGN Xray emitting region. A different geometry can significantly affect the shape and amplitude of the disc response function, and as a result, it can significantly alter the resulting model timelag spectrum (see the discussion in Appendices B and D). We adopted it (as has been done by many authors in the past) because the estimation of the disc response is relatively straightforward in this case. Furthermore, our intention was to investigate whether the observed iron line vs. continuum timelag spectra are consistent with the simplest theoretical reverberation model, and to see which constraints they can impose on the Xray source and disc geometry. In retrospect, given the results of our study (see the discussion below), the current data sets fail to distinguish between the predictions of the lamppost model and those from a more detailed approach.
7.3. Modelfit results
We fitted models A and B separately to the observed timelag spectra because given their quality (limited frequency range and large errors), we would not have been able to constrain the lamppost parameters by fitting a combined model A+B to the data. Both models provide statistically acceptable fits. We therefore cannot prefer one model based on the quality of the model fits.
However, our bestfit results do provide useful hints. For example, the bestfit model A powerlaw index values for 1H 0707–495, MCG–63015, and Mrk 766 are consistent with zero. The observed timelags in these sources are flat, and the bestfit model A reduces to just a constant. This result (i.e. that the bestfit powerlaw model to the data is a horizontal line) leads us to believe that the case for Xray reverberation timelags is strong, at least in these three sources. If the observed timelags were indeed representative of continuum timelags, we would expect a nonzero bestfit slope.
As we showed in Sect. 4.3, the observed timelags should have both a continuum and a reverberation component. The lack of a significant detection of the expected continuum component for these three sources (at least) is not surprising and can be explained physically. The continuum timelags depend on the energy separation between the chosen energy bands, which is small in our case. Our bestfit model A amplitude values are systematically lower than the respective bestfit values found by E14. This is what we should expect for continuum timelags, as the energy separation between the iron line and continuum bands is smaller than the separation between the 1.5−4 and 0.3−1 keV bands used by E14.
When fitting the observed timelags to the model B timelag spectrum, we found that the BH spin and inclination cannot be constrained. This is due to the large errors of the timelag estimates and the weak dependence of the model B timelag spectra on these parameters. Furthermore, there is a degeneracy between the Xray source height and the BH mass that is due to the similar dependence of the model B timelag spectrum on these parameters. We thus froze the BH mass value for each source to the most accurate and reliable values we could find in the literature and managed to constrain the Xray source height. The observed iron line vs. continuum timelag spectra either require, or are consistent with, small Xray source heights. For example, the bestfit height estimates are ≲ 10r_{g} in three sources. The bestfit height for NGC 4051 is also consistent with such a low value. Even for Ark 564 and NGC 7314, the data are consistent with an Xray source height as small as ~ 4r_{g} when we considered a combined model A+B.
Figure 4 shows the our bestfit h values versus the E14 bestfit results. The red dashed line indicates the onetoone relation. Although most of the points are located above this line, given the large uncertainties, the plot suggests a broad agreement with the results of E14. The direct comparison is complicated because we considered more data sets than E14 for some sources. Alston et al. (2013) showed that the soft lags of NGC 4051 vary significantly and systematically with source flux. In our case, we cannot fit model B to timelag spectra estimated from low and highflux segments, as the uncertainty on the model parameters will be significantly larger than what we obtain when we fit the overall timelags. Nevertheless, if this trend is present in all AGN and in the iron line vs. continuum timelags as well, then when we average over data with a wide flux range, segments with the highest flux may dominate the crossperiodogram, as they may be associated with higher amplitude variations (due to the rmsflux relation; Uttley & McHardy 2001). If the data sets we considered exhibit a wider flux variability range than the one in the E14 data sets, differences in the bestfit results may be easier to explain.
In conclusion, the soft excess vs. continuum timelags are consistent with the iron line vs. continuum timelags we presented here, in that they both support the hypothesis of disc reflection from an Xray source that is located very close to the disc and the central BH.
Fig. 4 Comparison between the bestfit Xray source height obtained by fitting the iron line vs. continuum timelags (vertical axis; this work) with those obtained by fitting the soft excess vs. continuum timelags (horizontal axis; E14). 

Open with DEXTER 
7.4. Implications for the Xray reflection scenario
Except for the source height, we are unable to constrain additional reverberation model parameters such as the BH mass and spin, viewing angle, and the outer disc radius. Accurate determination of these parameters would require a significant reduction in the errors of the timelag estimates and/or an increase in the frequency range that can be reliable probed. However, this requires a substantial increase in the number of Xray observations of AGN.
For example, to probe frequencies lower by a factor of ~ 5 (i.e. to reach a low limit of ~ 10^{5} Hz), segments with a duration of ~ 100 ks are required. Assuming the number of segments used for the timelag estimation remains the same as in the present work, this would require the net XMMNewton exposure times to increase by a factor of ~ 5 for each source (on average). This will, however, neither decrease the error of the timelag estimates nor allow allow us to probe higher frequencies, since both require an increase in the number of segments. Extending the highfrequency limit requires an increase of ν_{max}, which can only be achieved by increasing the number of segments. For example, to probe frequencies ~ 2 × 10^{3} Hz for MCG–63015, the critical coherence value has to decrease from its present value of ~ 0.18 to ~ 0.05 (see Fig. 1). This requires the number of segments to increase from 28 to 115, which corresponds to an increase in the net XMMNewton exposure times by a factor of ~ 4. This would, in turn, reduce the errors of the timelag estimates by a factor of ~ 2. In this case, however, we would be unable to probe lower frequencies, since this requires segments of longer duration.
One possibility to extend the frequency range of the observed timelag spectra would be to use the large volume of available archival data from past and current lowEarth orbit satellites (e.g. ASCA, Chandra, and Suzaku). The idea would be to bin the respective light curves at one orbital period (~ 96 min) to probe low frequencies, although this requires a large number of long observations. For instance, estimating timelags at frequencies lower than ~ 10^{5} Hz requires an ensemble of at least ten observations, which will be longer than at least a few days. We are currently investigating this possibility to estimate timelag spectra over a wider frequency range.
Given the quality of the present data sets in the iron line band and the resulting iron line vs. continuum timelag spectra, the need for constructing more sophisticated theoretical disc response functions is questionable. It seems that the best way to test the Xray reverberation scenario and significantly constrain the model parameters is to focus on the soft excess vs. continuum timelag modelling, where the S/N of the existing light curves in the soft band is much higher than those in the iron line band. This would require considering the ionisation structure of the disc in the construction of appropriate disc response functions.
Modelling the energy dependence of the timelag spectra is another possibility. However, we note that the errors of the resulting timelag estimates are dictated by the energy band with the lower average count rate. As such, the use of light curves over a broad energy band as a reference should not significantly lower the errors of the resulting timelag estimates, even at the lowest possible frequencies. We plan to model the energy dependence of the observed timelag spectra in a future work, where we will also consider NuSTAR data to study timelags between the Compton hump and the Xray continuum.
Acknowledgments
We thank the referee for his/her suggestions, which significantly improved the quality and clarity of the manuscript. This work was supported by the AGNQUEST project, which is implemented under the Aristeia II Action of the Education and Lifelong Learning operational programme of the GSRT, Greece. The research leading to these results has also received funding from the European Union Seventh Framework Programme (FP7/2007−2013) under grant agreement No. 312789, and by the grant PIRSESGA201231578 EuroCal.
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Appendix A: Model CS
Substituting Eqs. (10) and (11) into Eq. (12), we can compute the CCF between the iron line and continuum bands as follows: (A.1)where is the CCF of the continuum emission. Consequently, the intrinsic CS is (A.2)Setting and applying the convolution theorem, Eq. (A.2) becomes (A.3)Subsequently, the model timelag spectrum is given by (A.4)where we used the property arg [z_{1}z_{2}] = arg [z_{1}] + arg [z_{2}], for the complex numbers z_{1}, z_{2}. The functions appearing on the righthand side of Eq. (A.4) are defined as , and . The total timelag spectrum is therefore equal to the sum of two timelag spectra; one due to delays between variations of different energy bands in the continuum, , and one due to delays between the reprocessed disc emission and the continuum, . The function can be written as follows: (A.5)where (A.6)(A.7), and . In other words, the model timelag spectrum depends in a nontrivial way on the Fourier transforms of the iron line and continuum response functions.
Appendix B: Disc response functions
Fig. B.1 Dependence of the 5−7 keV (top panels; continuous lines) and 2−4 keV (top panels; dashed lines) disc response functions on a (top left panel), θ (top middle panel) and h (top right panel), along with the corresponding timelag spectra (bottom panels). The response functions have been normalised such that their area is equal to the observed ratio of reprocessedtocontinuum photons. 

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The top panels in Fig. B.1 show various disc response functions in the energy bands 5−7 keV, Ψ_{5−7}(t) (solid lines), and 2−4 keV, Ψ_{2−4}(t) (dashed lines), in the case when the disc extends to 10^{3}r_{g}, and M_{BH} = 10^{6}M_{⊙}. The horizontal axes show time in the observer frame, with the origin (t = 0) corresponding to the beginning of the primary Xray flare (which, as we mentioned in Sect. 4.4, ends abruptly at t = 1t_{g}). The response functions plotted in these panels are defined such that is equal to the observed ratio of reprocessedtocontinuum photons.
The parameters a, θ, h, and r_{out} affect both the shape and amplitude of the response function. The parameter M_{BH} changes its shape (by either uniformly stretching or contracting it in the horizontal direction, depending on whether M_{BH} is increased or decreased), but not their amplitude. In all cases the response function shows an initial sharp rise at a time t_{rise}> 0, followed by a second peak at a later times, and a final gradual decline up to a maximum time t_{max}. The initial rise time corresponds to the instant the observer detects the first reflected emission from the near side of the disc. The second peak appears when the observer detects emission from the far side of the disc. At longer timescales we detect emission from the outer disc radii, where the reflection amplitude is reduced, and hence Ψ_{ℰ}(t) gradually decreases up until t_{max}, when we detect the last emission from the edge of the far side of the disc, and Ψ_{ℰ}(t) abruptly drops to zero. t_{max} depends mainly on h and r_{out}. If r_{out} → ∞, then t_{max} → ∞ and the response function has a ~ t^{2} behaviour at long times (e.g. Wilkins & Fabian 2013).
Given our adopted convention, the amplitude of the response function in a given energy band depends on the strength of the reprocessed disc emission relative to the continuum emission in that energy band. Therefore, in almost all cases shown in Fig. B.1, the amplitude of Ψ_{5−7}(t) is larger than the amplitude of Ψ_{2−4}(t) by a factor of ~ 10. However, for high spin values and small heights, this difference is as small as ~ 4 (top left panel in Fig. B.1). Since the amplitude of both Ψ_{5−7}(t) and Ψ_{2−4}(t) affect the model timelag spectra (see Appendix A), Ψ_{2−4}(t) should not be neglected when estimating the theoretical timelag spectra.
The BH spin affects neither the width of the response function, nor t_{rise} (top left panel in Fig. B.1). This may seem counterintuitive at first, as r_{ISCO} decreases from 6r_{g} for a nonspinning BH, to almost 1r_{g} for a maximally spinning BH, reducing the distance between the Xray source and the inner disc. However, for the θ and h values we considered, the reprocessed emission that is initially detected by the distant observer is emitted from a part of the disc that is located towards the observer’s direction at a radius larger than 6r_{g}, independent of a. Nevertheless, a does affect the amplitude of the response function, especially when h is small. When the Xray source is located close to the BH, lightbending effects are strong, and most of the continuum emission illuminates the region of the disc close to the BH. For a maximally spinning BH the disc extends to very small radii, and hence the amplitude of the response function when a = 1 is significantly larger than when a = 0 (brown and black lines, respectively, in the top left panel of Fig. B.1).
The rise time decreases and the width of the response function increases with increasing inclination (top middle panel in Fig. B.1). The rise time decrease is due to the decreased light path difference between the continuum and the disc emission with increasing θ. The increase in the width appears because the difference between the light travel time from the nearside of the disc and from the Xray source increases with increasing θ. The amplitude of the response function decreases with increasing θ, since the projected area of the disc (and hence the observed amount of reflected emission) is proportional to cosθ.
The top right panel in Fig. B.1 shows that t_{rise} depends mainly on the height of the Xray source, as it strongly increases with increasing h. The reason is the increase in the light travel time between the Xray source and the disc emission. The width of the response function also increases with increasing h. As h increases, the difference between t_{rise} and the time when the second peak appears increases as well because the light travel time difference between the near and farside of the disc increases.
Appendix C: Effects of finite continuum flare duration on the response function estimation
As mentioned in Sect. 4.4, the disc response functions were determined assuming a flare of constant flux continuum emission that lasted for 1t_{g}. Our response functions thus formally describe the response of the disc to a flare with a finite duration and a boxlike light curve (instead of a deltafunction), and we henceforth designate them as . In this appendix we investigate the relation between the Fourier transforms of Ψ_{ℰ}(t) (i.e. the disc response to an instantaneous continuum flare) and . This is necessary, because the model timelag spectra given by Eq. (15) depend on the Fourier transforms of Ψ_{ℰ}(t) in the iron line and continuum bands.
We assumed that , as it appears on the righthand side of Eq. (10), is equal to , where H(x) is the Heaviside step function (H(x) is defined as being equal to unity when x ≥ 0, and zero otherwise) and F_{0} is a constant. According to Eq. (10), the observed flux is then (C.1)where . When t_{g} → 0 (i.e. when the flare becomes instantaneous), , as expected.
According to Eq. (C.1), is equal to the convolution of Ψ_{ℰ}(t) with a constant kernel that is nonzero for 0 ≤ t ≤ t_{g}. The relation between the Fourier transforms of the two functions is therefore (C.2)In other words, the Fourier transform of has to be divided by the factor e^{iπνtg}sinc(πνt_{g}) to account for the finite width of the continuum flare. This correction term becomes important only at frequencies , which corresponds to ≳ 0.2 Hz for M_{BH} = 10^{6}M_{⊙}. As explained in Sect. 2, timelags cannot be reliably estimated at such high frequencies with current data, hence the correction term has a negligible effect on the model timelag spectra in our work. Nevertheless, we applied Eq. (C.2) to determine and , which were subsequently used to calculate the corresponding model iron line vs. continuum timelag spectrum according to Eq. (15).
Appendix D: Lamppost model timelag spectra
The bottom panels in Fig. B.1 (and both panels in Fig. D.1) show various iron line vs. continuum model timelag spectra, , calculated using the parameters that we used to estimate the response functions appearing in the same figure. The timelags are predominantly negative, meaning that variations in the iron line band are delayed with respect to variations in the continuum band. They all share similar characteristics. They all flatten to a constant, negative plateau, τ_{plateau}, below a frequency ν_{plateau}. At higher frequencies they rise to a maximum positive bump, τ_{bump}, at a frequency ν_{bump}, followed by sinusoidal behaviour with decreasing amplitude around a zero timelag value.
Fig. D.1 Top panel: model timelag spectra between the 5−7 and 2−4 keV bands for various M_{BH} values. The remaining model parameters are fixed at a = 0.676, θ = 40°, h = 11r_{g}, and r_{out} = 10^{3}r_{g}. Bottom panel: model timelag spectra between the 5−7 keV and 2−4 keV bands for various values of r_{out}. The remaining model parameters are fixed at a = 0.676, θ = 40°, h = 11r_{g}, and M_{BH} = 10^{6}M_{⊙}. 

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As in the case of the response functions, the parameters a, θ, h, M_{BH}, and r_{out} affect both the shape and magnitude of the timelags. The bottom left panel in Fig. B.1 shows that the BH spin has a weak effect on the model timelag spectra. The timelags increase slightly in (absolute) magnitude with increasing a. The frequency ν_{plateau} does not depend on a, while ν_{bump} and τ_{bump} depend weakly on a. The dependence of the model timelag spectra on a is in contrast to the strong dependence of the response function amplitude on the same parameter (top left panel in Fig. B.1). This result shows that the response function amplitude does not significantly influence the timelag characteristics. It also highlights the importance of including the reprocessed emission in the continuum band as well (as we did here), since response functions of comparable magnitudes in two bands will lead to a model timelag spectrum that is close to zero.
Fig. E.1 Expected mean sample timelag spectra computed from 20 ks segments for various model CS parameters (see the text for more details). The continuous blue curve indicates the model timelag spectrum in each case. 

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In general, when Ψ_{5−7}(t) = Ψ_{2−4}(t) ≠ 0 (i.e. when the observed ratio of reprocessedtocontinuum photons is equal in the two energy bands), Eq. (A.5) reduces to . As we mentioned in Appendix B, although Ψ_{5−7}(t) increases with increasing a, so does the amplitude of Ψ_{2−4}(t), and in fact even more so. For a rapidly spinning BH and Xray reflection from the innermost part of the disc, the red wing of the iron line is well into the 2−4 keV band due to strong gravitational effects. Including reprocessed emission in the continuum band has the wellknown effect of decreasing, or diluting, the magnitude of the model timelag spectrum (e.g. Wilkins & Fabian 2013). This can be seen by taking the limit Ψ_{2−4}(t) → 0 in Eqs. (A.6) and (A.7), in which case is minimised, is maximised, and hence the is maximised according to Eq. (A.5).
The bottom middle panel in Fig. B.1 shows that the timelags increase in magnitude with decreasing inclination. The reason is that the rise time of the response functions decreases as θ increases. Similarly to the BH spin, ν_{plateau} and ν_{bump} have a weak dependence on θ.
The (absolute) magnitude of τ_{plateau} in the middle bottom panel of Fig. B.1 is larger than the corresponding magnitude in the bottom left panel because we considered a larger Xray source height in the former case. In a lamppost geometry, the timelag spectra are mainly affected by Xray source height (bottom right panel in the same figure). The (absolute) magnitude of the timelags and τ_{plateau} strongly increase and ν_{plateau} and ν_{bump} decrease with increasing h. Both effects are caused mainly by the fact that t_{rise} increases substantially with increasing source height. The (absolute) magnitude of τ_{bump} also increases with increasing h, although this increase is not as pronounced as in the case of τ_{plateau}.
The BH mass likewise strongly affects the magnitude and shape of the timelag spectrum, as seen in the top panel of Fig. D.1. At a given Xray source height, the magnitude of the timelags increases while ν_{plateau} and ν_{bump} decrease with increasing M_{BH}. This is due to the increased light travel time between the Xray source and the disc, since the physical size of the Xray source/disc system scales proportionally with M_{BH}. In other words, since timelags scale with t_{g} and frequencies scale with , when M_{BH} is increased the timelag spectrum is uniformly stretched and squeezed in the horizontal and vertical direction, respectively. The dependence of the timelags on M_{BH} is thus very similar to their dependence on h.
The bottom panel in Fig. D.1 shows the timelag spectrum for various values of the outer disc radius. As r_{out} increases, τ_{plateau} increases in magnitude and ν_{plateau} decreases. The timelag spectrum remains unaffected by r_{out} at frequencies higher than ν_{plateau}. In other words, r_{out} sets the level of the constant plateau at low frequencies, while this plateau occurs at increasingly lower frequencies as r_{out} increases. Our results are in agreement with CY15, who reported a similar dependence of the magnitude and shape of reverberation timelag spectra on r_{out}. The frequency ν_{plateau} is proportional to , and thus depends on h and r_{out}.
Given our discussion above, it is clear that ν_{plateau} and τ_{plateau}, which are the most pronounced features in the theoretical timelag spectra, depend on h, M_{BH} and r_{out}. Observationally, τ_{plateau} appears to depend mainly on M_{BH} (e.g. De Marco et al. 2013), which implies that h and r_{out} should be approximately the same in all AGN. Even so, the normalisation of this relation cannot directly indicate the height of the Xray source, as τ_{plateau} also depends on r_{out}. The dependence is not as strong as that on h, but is present nevertheless. A more detailed theoretical study of the M_{BH}−τ_{plateau} relation is necessary before reaching conclusions based on the observed normalisation of this relation. This conclusion is strengthened by the fact that the discussion above is based on response functions estimated for the lamppost geometry . If the Xray source has a finite size, we expect the response function rise time to be altered. Since this directly affects τ_{plateau}, a study of the response functions from more complicated geometries is necessary to interpret the observations.
Appendix E: Expected timelag bias
The bestfit model B parameters listed in Table 3 could significantly differ from their intrinsic values if the bias of the timelag estimates has a magnitude comparable to, or larger than, their error. To investigate this possibility, we estimated the timelag bias for { a,h } = { 1,2.3r_{g} } and { 0,100r_{g} } (we assumed θ = 40°, M_{BH} = 2 × 10^{6}M_{⊙} and r_{out} = 10^{3}r_{g} in both cases). These are two extreme cases in the parameter space we considered.
Alston et al. (2013) were the first to quantify the effects of windowing on the timelag bias. They showed that the timelag bias can be up to ~ 30% of the intrinsic value. EP16 also studied these effects in detail by exploring a wider parameter space. They showed that a model CS (not just a model timelag spectrum) needs to be prescribed to estimate the timelag bias, as the bias is introduced to the crossperiodogram itself. We assumed a model CS given by Eq. (A.3) and that there are no delays between variations of different energy bands in the continuum. To determine the amplitude of the model CS, we assumed that a) the continuum PSD in both energy bands is equal to the characteristic bending powerlaw shape observed in the Xray light curves of many AGN (e.g. GonzálezMartín & Vaughan 2012), and b) the intrinsic coherence between the two energy bands is unity at all frequencies. This uniquely determines the intrinsic continuum CS, , appearing in Eq. (A.3), which is then given by (E.1)where A is the amplitude, α the highfrequency slope, and ν_{b} the socalled bendfrequency. The typical values for these parameters are A~ 0.01 (in socalled rootmeansquare units), 2 ≲ α ≲ 3, and ν_{b} ~ 10^{5}−10^{4} Hz for M_{BH} ~ 10^{6}−10^{7}M_{⊙}. We therefore considered the cases {α,ν_{b}} = {2,2 × 10^{4} Hz}, {3,2 × 10^{4} Hz} and {2,2 × 10^{5} Hz} that are appropriate for our sample. We furthermore set A= 0.01, as the PSD amplitude was found by EP16 to not affect the timelag bias.
Given our model CS, we finally determined the expected mean of the timelag estimates computed from 20 ks segments using Eq. (13) in EP16. The results are shown in Fig. E.1. The continuous blue line in the two panels indicates the model timelag spectrum. Filled black circles, open red squares, and green stars correspond to the expected mean sample timelag spectrum for different { α,ν_{b} } values (as noted in the figure).
The horizontal axis indicates the widest frequency range for which we were able to obtain reliable timelag estimates using real data. The range of values in the vertical axis is identical to the corresponding range in Figs. 2 and 3 (with the exception
of Ark 564). The difference between each point and the corresponding model at a given frequency in Fig. E.1 is equal to the expected timelag bias, whose magnitude needs to be compared to the error bars in Figs. 2 and 3. As noted by EP16, the mean of the timelag estimates is always smaller (in magnitude) than their intrinsic values at each frequency.
For low Xray source height values, it is clear that the magnitude of the bias is entirely negligible compared to the timelag errors and hence should not affect our results. For high Xray source height values the bias is more significant at low (≲ 10^{4} Hz) frequencies, although still smaller than the timelag errors for all the sources in our sample. Perhaps in this case the bestfit height values may slightly underestimate their intrinsic values, although this effect should not be significant.
All Tables
All Figures
Fig. 1 Sample iron line vs. continuum coherence function (top panel) and timelag spectrum (bottom panel) of MGG–603015, estimated using the data listed in Table 2. The dashed brown line in the top panel shows the bestfit model to the sample coherence. The continuous red vertical line indicates the highest frequency up to which timelags should be estimated, and the horizontal blue dotteddashed line indicates the coherence value at this frequency (see Sect. 3). 

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In the text 
Fig. 2 Observed iron line vs. continuum timelag spectra for 1H 0707−495 (first row), MCG–63015 (second row), and Mrk 766 (third row). The solid brown and dashed red lines indicate the bestfit models A and B, respectively, to each timelag spectrum (see Sect. 5 for details on these models). 

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In the text 
Fig. 3 Same as in Fig. 2 for NGC 4051 (first row), Ark 564 (second row), NGC 7314 (third row), and Mrk 335 (fourth row). 

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In the text 
Fig. 4 Comparison between the bestfit Xray source height obtained by fitting the iron line vs. continuum timelags (vertical axis; this work) with those obtained by fitting the soft excess vs. continuum timelags (horizontal axis; E14). 

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In the text 
Fig. B.1 Dependence of the 5−7 keV (top panels; continuous lines) and 2−4 keV (top panels; dashed lines) disc response functions on a (top left panel), θ (top middle panel) and h (top right panel), along with the corresponding timelag spectra (bottom panels). The response functions have been normalised such that their area is equal to the observed ratio of reprocessedtocontinuum photons. 

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In the text 
Fig. D.1 Top panel: model timelag spectra between the 5−7 and 2−4 keV bands for various M_{BH} values. The remaining model parameters are fixed at a = 0.676, θ = 40°, h = 11r_{g}, and r_{out} = 10^{3}r_{g}. Bottom panel: model timelag spectra between the 5−7 keV and 2−4 keV bands for various values of r_{out}. The remaining model parameters are fixed at a = 0.676, θ = 40°, h = 11r_{g}, and M_{BH} = 10^{6}M_{⊙}. 

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In the text 
Fig. E.1 Expected mean sample timelag spectra computed from 20 ks segments for various model CS parameters (see the text for more details). The continuous blue curve indicates the model timelag spectrum in each case. 

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In the text 
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