AGN STORM 2. VII. A Frequency-resolved Map of the Accretion Disk in Mrk 817: Simultaneous X-Ray Reverberation and UVOIR Disk Reprocessing Time Lags

The Neil Gehrels Swift Observatory (Gehrels et al. 2004) performed daily monitoring of Mrk 817 for 15 months (2020 November 22 through 2022 February 24), and these Swift AGN STORM 2 results were initially presented in Paper IV (Cackett et al. 2023). Our analysis focuses on data collected simultaneously by Swift and ground-based telescopes in the 420 day time window THJD = 9177–9597 shown in Figure 1, where the Truncated HJD (THJD) is defined as THJD = HJD–2450000. This time range is roughly 90% of that used by Cackett et al. (2023), since we exclude the last ∼40 days of the campaign to remove a large gap in the Swift data resulting from the observatory entering safe mode due to the failure of a reaction wheel. Interpolating over this gap using GP regression results in larger uncertainties on the lags, although the results are consistent with those shown here. The Swift X-ray light curves were produced using the Swift-XRT data products generator (Evans et al. 2007, 2009). ³² The background-subtracted count rates were produced with per-observation binning in a soft band (0.3–1.5 keV) and a hard band (1.5–10 keV). We refer the reader to Cackett et al. (2023) for details on the Swift UVOT data reduction.

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The ground-based observations were carried out by the following observatories: Las Cumbres Observatory Global Telescope network (Brown et al. 2013), Liverpool Telescope (Steele et al. 2004), the Calar Alto Observatory, Wise Observatory (Brosch et al. 2008), the Yunnan Observatory, and the Zowada Observatory (Carr et al. 2022). The intercalibrated ground-based photometry is presented in J. Montano (2024, in preparation). The SDSS and Pan-STARRS z_s filters were used, with the measurements in both the SDSS and Pan-STARRS z-labeled filters combined. We hereafter refer to the filters as the ugriz bands. The ground-based Johnson/Bessel B, V filters are denoted B_g , V_g to avoid confusion with the Swift filters. We refer the reader to Kara et al. (2021) for details on the ground-based data reduction, and a full description of the ground-based campaign and photometry will be presented by J. Montano (2024, in preparation).

XMM-Newton carried out a total of four observations throughout the AGN STORM 2 campaign, which were presented in Paper IX (Zaidouni et al. 2024). For all but one observation, the X-ray flux was an order of magnitude fainter than historical observations, due to obscuring gas along the line of sight. It was serendipitous that the 2021 April observation (obsid: 0882340601) coincided with the only prominent X-ray flare observed in the campaign on THJD = 9329, as shown in Figure 1. The high-resolution grating spectra of this observation showed strong narrow absorption lines, from which Zaidouni et al. (2024) revealed that the obscuring gas is a multi-phase, ionized wind. The observation also provides a fortuitous opportunity for timing analysis, with high-amplitude variability exhibited by the light curves in Figure 2.

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We performed the same standard data reduction procedure as Zaidouni et al. (2024): the EPIC-pn data were reduced using the XMM-Newton Science Analysis System (sas v. 19.0.0). The light curves were extracted using circular source and background regions, both with 35 arcsec radii, and then binned to 30 s bins. Both regions are located on the same chip and not near the edges of the chip. We avoided background flares by constructing a good time interval filter with a background count rate cutoff of 0.4 cts s⁻¹, in addition to avoiding spurious detections using the conditions “PATTERN≤4” and “FLAG== 0.” Soft photon background flares were also cleaned from both ends of the observation, resulting in a final exposure of ∼120 ks.

We compute the lags as a function of frequency using a standard procedure involving Fourier techniques (for a review, see Uttley et al. 2014). For the regularly sampled XMM-Newton observation, we can perform the procedure immediately: the cross-spectrum is computed between a light curve dominated by reflection (0.3–1 keV) and another dominated by the direct continuum of the corona (1–4 keV). These energy bands were selected based on the spectral features shown in Figure 2, namely the soft excess commonly attributed to reprocessing/reflection below 1 keV (e.g., Ross & Fabian 2005; García et al. 2014) and a featureless power law from 1 to 4 keV. This selection is in agreement with modeling of the spectral energy distribution (SED) in Paper IX (Zaidouni et al. 2024). The cross-spectrum is then binned into coarser frequency bins, each centered at frequency ν_i . The phase of the cross-spectrum at each frequency is converted to a final time lag by dividing by 2π ν_i , producing a lag–frequency spectrum.

We also measure how nine finer energy bands lag or lead a common reference band—in this case, the 0.3–10 keV broadband—in a given frequency range (i.e., a lag–energy spectrum). In this case, the light curve in each energy band of interest is subtracted from that of the reference band to remove Poisson noise that would otherwise be correlated between light curves. The lag–energy spectrum reveals how each energy band contributes to the lags, and thus is useful for studying the causal relationship between different spectral components (Uttley et al. 2014).

In order to compute the frequency-resolved lags for the irregularly sampled Swift and ground-based observations, we instead apply the aforementioned procedure to GP realizations of the light curve. As introduced in Section 3.2, these realizations have been informed by the empirical variability at all timescales present in the data. A lag–frequency spectrum is then computed for each of the 1000 realization pairs (1000 realizations in each wave band of interest and in the UVW2 reference band). The final lag–frequency spectrum and 1σ uncertainties are produced by computing the mean and standard deviation of this distribution of 1000 lag–frequency spectra. This number of realizations was selected based on convergence of the lag distribution: a two-sample C-vM test between the empirical cumulative distribution function (ECDF) of the lags produced from the first 1000 samples to that from 5000 samples results in a p-value of ∼0.3, and we are thus unable to reject the null hypothesis that both distributions are the same.

The minimum frequency accessible using this approach is limited by the length of the light curve ( for length L), whereas the maximum frequency is set by the sampling rate ( for sampling rate Δt). The XMM-Newton observation allows us to probe frequencies on the order of 10⁻⁵–10⁻³ Hz, equivalent to studying radii from ∼1 to 100 R_g , i.e., the innermost accretion disk. The Swift and ground-based data allow us to study variability operating on much longer timescales, down to frequencies on the order of 10⁻³ day⁻¹, equivalent to radii from ∼100 to 10⁵ R_g , thus probing the outermost parts of the accretion disk and the BLR.

We include a brief overview of GP regression here, but refer the reader to more detailed introductions by Rasmussen & Williams (2006), Wilkins (2019), and Griffiths et al. (2021).

We can consider our light curve data to be a vector of fluxes d observed at times t . The “Gaussian” in “Gaussian process” arises in that we assume the data have been drawn from a multivariate Gaussian distribution; or, in other words, that the observed data are a realization drawn from a GP posterior. This procedure assumes the data are normally (or log-normally) distributed, the validity of which we explore in Appendix A, where we conclude that the data in all wave bands agree better with a log-normal distribution and thus train the GPs on the log-transformed flux values. The properties of this multivariate Gaussian is informed by the data, including the mean function ( denotes the expected value of x and f(t) the function of fluxes observed at time t_i ), and the covariance function, hereafter referred to as the kernel function, .

The mean function is taken to be m = 0, as it is common practice to first standardize the data (subtract the mean of the light curve before dividing by the standard deviation). The kernel function encodes how the light curve deviates from the mean and thus models the observed variability. Functional forms for the kernel function commonly depend on the separation in time between points (i.e., ), thus modeling the empirical variability on every timescale present. The choice of kernel function form has been found to impact the significance of lag recovery, depending on the data quality (Griffiths et al. 2021; Lewin et al. 2022). We detail the selection of the functional form of the kernel function in Appendix B, choosing between kernel functions that have shown success in modeling AGN variability for a wide range of data quality (e.g., Wilkins 2019; Griffiths et al. 2021; Lewin et al. 2022, 2023). In summary, the rational quadratic kernel performs the best statistically in all bands, although the final lags are consistent across the three kernel forms. At all frequencies, the lags agree within 5% of each other on average, with the lag uncertainties agreeing within 10%. Every functional form comes with its own hyperparameters (θ), each encoding a property of the variability, for instance timescales and amplitudes. These hyperparameters are determined by maximizing the likelihood of the model given the observed data; or, more commonly, by minimizing the negative log marginal likelihood (NLML; Equation (17) in Griffiths et al. 2021).

Equally sampled realizations with flux values d _* can then be generated by taking random draws from the conditional distribution ( d _*∣ d ) (Equation (5) in Wilkins 2019), which is defined by the optimized Gaussian and the observed flux vector d .

The architecture used for GP model training and subsequent regression was created by using Scikit-learn ³³ and the X-ray timing analysis package pyLag ³⁴ (Wilkins 2019).

Due to heavy obscuration studied throughout the campaign in Paper III (Partington et al. 2023), the average X-ray flux is an order of magnitude lower than that observed at the time of the flare simultaneous with the XMM-Newton observation. The flare revealed prominent X-ray variability, as shown by the light curves in Figure 2, providing a momentary glimpse of the innermost accretion disk that we aimed to study with X-ray timing analysis.

Here, we present the frequency-resolved X-ray time lags of Mrk 817 computed from the ∼120 ks XMM-Newton observation using the procedure described in the previous section. The lags span temporal frequencies of 10⁻⁵–10⁻³ Hz and were computed between an energy band dominated by the coronal continuum (1–4 keV, hereafter the hard band) and another band dominated by the reprocessed emission (0.3–1 keV, hereafter the soft band).

The resulting lag–frequency spectrum and associated (bias-subtracted) coherence values are shown on the left side of Figure 3. The lag–frequency spectrum exhibits a shape typical for AGN (e.g., Zoghbi et al. 2010; De Marco et al. 2013; Kara et al. 2013): at low frequencies, the hard band lags the soft band, whereas at slightly higher frequencies, the soft band instead lags the hard band (i.e., a soft lag, 150 ± 68 s in the (1–3) × 10⁻⁴ Hz bin). This is the first detection of a soft lag in this source, as the only archival XMM-Newton observation of Mrk 817 (previous to the STORM 2 campaign) was 11 ks long (Winter et al. 2011). Both the measured lag amplitude and frequency of this soft lag are consistent with the lag–mass scaling relations found by De Marco et al. (2013), which predict a lag amplitude of 210 ± 80 s at (1.6 ± 0.3) × 10⁻⁴ Hz for a black hole mass of M_BH = 3.85 × 10⁷ M_⊙. The hard and soft lags have high coherence (>0.90 for the hard lag and >0.75 for the soft lag), indicating a high degree of linear correlation between light curves.

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In both frequency ranges at which we detect a hard or soft lag, we computed a lag–energy spectrum by measuring how the variability in finer energy bands lags or leads a common (0.3–10 keV) reference band. The lag–energy spectrum at low frequencies (the top right panel of Figure 3) increases roughly monotonically with energy. This common shape in the lag–energy spectrum (e.g., Kara et al. 2016) is often attributed to fluctuations in the mass-accretion rate that propagate inward from colder to hotter regions, thus resulting in soft photons that respond before the hard photons (Lyubarskii 1997; Kotov et al. 2001; Arévalo & Uttley 2006). The lag–energy spectrum when isolating the higher frequencies where we detected a soft lag (the bottom right panel of Figure 3) instead shows an anticorrelation between lag and energy, except for the 6–10 keV bin. The lag–energy spectrum thus appears similar to previous AGN lag spectra exhibiting reverberation signatures, including peaks in the lags below 1 keV where we expect the so-called soft excess, and in the 6–10 keV bin where we expect the Fe K. We measure a ∼500 s difference in the lag between the 6–10 keV and 3–6 keV bins, albeit with large uncertainties due to low signal-to-noise ratio. This difference is consistent with the expected amplitude of the Fe K lag based on the lag–mass scaling relation confirmed by Kara et al. (2016).

Here, we present the frequency-resolved UVOIR time lags of Mrk 817 computed from the AGN STORM 2 Swift and ground-based campaign light curves shown in Figure 1. Given the length of the campaign, we computed the lags using two treatments of the light curves: (1) using the full 420 day light curves and (2) when splitting the light curves into three epochs of equal length (140 days). These epochs fortuitously coincide with periods of high (Epochs 1 and 3) and low (Epoch 2) average column density in the obscurer, according to the NICER spectral analysis in Paper III (Partington et al. 2023). The Swift X-ray–UV lags were also measured, but a tentative lag (with coherence less than 0.2) is measured in only the second epoch as a result of the X-ray flare. As such, the X-ray–UV lags are only discussed when presenting the epoch-resolved lags (Section 4.2.2).

4.2.1. Full Campaign Lags

The lags and bias-corrected coherence were computed with respect to the UVW2 band centered at 1928 Å for the full 420 day Swift and ground-based light curves in six bins spanning 7 × 10⁻³–10⁰ day⁻¹. This frequency range translates to light travel times from the black hole out to roughly 10⁵ R_g , thus probing reprocessing by the outermost accretion disk and the BLR (Cackett et al. 2021). The lowest frequency we probe is set by the length of the observation, but we expect to see contributions from emission by the dusty torus on these long timescales Netzer (2022), which may complicate the analysis.

The lags as a function of frequency for all of the wave bands are shown in Figure 4, alongside lag models detailed in the following section. These lags are also provided in the Appendix (Table 1). Both lags and their coherence values decrease with frequency, similar to those reported for NGC 5548 (Cackett et al. 2022) and Mrk 335 (Lewin et al. 2023). The lags continue to increase at lower frequencies (i.e., the lags do not level out), indicating additional reprocessing beyond timescales of 20 days. This is consistent with reprocessing by the BLR based on the 23 day Hβ lag measured in Paper I (Kara et al. 2021).

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The average coherence measured in the two lowest frequency bins is high at >0.9 and >0.75, respectively. The coherence is notably lower—producing larger uncertainties on the lags—in the ground-based u, B_g , and z bands, presumably due to coarse data sampling near the end of the campaign. We nonetheless verified that simulated lags are faithfully recovered in these cases where GP regression over substantial gaps produces substantive drops in coherence, as detailed in the Appendix. The drop in coherence on short timescales (even without the use of GPs) suggests that white noise dominating the variability at higher frequencies may be breaking the correlation between light curves, although the washing out of variability in the disk may also be playing a role.

The lags in the lowest three frequency bins are presented as a function of wavelength in Figure 5. Each set of lags is generally well described by a τ ∝ λ^4/3 relation, as expected from reprocessing by a standard accretion disk (Cackett et al. 2007). In detail, the lags in each frequency range were independently fit with the function , fitting for the normalization (τ₀) given the reference band rest-frame wavelength for the UVW2 band (λ₀ = 1869 Å). This results in the following best-fit normalization values for the three lowest frequency bins (in order of increasing frequency): τ₀ = 1.00 ± 0.05, 0.58 ± 0.03, and 0.23 ± 0.01 days. We also note a discrepancy between the lags in the U and u bands: in the lowest frequency bin, the ground-based u-band lag is >1σ larger than that in the Swift U band. A very similar discrepancy was found in NGC 5548 (Cackett et al. 2022).

As a point of comparison, we modeled the frequency-resolved lags expected from disk reprocessing given the mass and accretion rate of Mrk 817 by generating impulse-response functions, as detailed in Section 5. The lags at low frequencies (<0.05 day⁻¹, corresponding to timescales >20 days; see the two leftmost panels in Figure 5) are significantly longer than those predicted by the disk reprocessing model–on average, by over a factor of 3 in the lowest frequency bin and a factor of 2 in the second-lowest frequency bin. This discrepancy is the most significant in the u band (3465 Å), where the lag exceeds the τ ∝ λ^4/3 fit by a factor of 2 in the lowest frequency bin, as also found in the CCF lags in Paper IV (Cackett et al. 2023). Similar levels of disagreement from disk reprocessing have been reported by previous continuum reverberation mapping campaigns, including especially long lags in the U band (e.g., Cackett et al. 2018; Edelson et al. 2019; Hernández Santisteban et al. 2020; Vincentelli et al. 2021; Kara et al. 2023).

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At higher frequencies (>0.05 day⁻¹, corresponding to timescales <20 days; see the rightmost panel in Figure 5), the lags have decreased in size and show general agreement with the disk reprocessing model. The U-band excess is also absent at these frequencies, with the lag now within 1σ from the τ ∝ λ^4/3 fit. Given the Hβ lag measured in Paper I (τ_Hβ = 23.2 ± 1.6 days; Kara et al. 2021), these findings are similar those in Mrk 335 and NGC 5548: the lags are consistent with disk reprocessing (including the U-band excess) when homing in on timescales shorter than the Hβ lag indicative of the BLR size scale. These results thus further support that reprocessing by the BLR is producing the longer-than-predicted lags reported by previous campaigns, as discussed in Section 6.

4.2.2. Epoch-resolved Lags

To study how the UVOIR lags evolve over the course of the 420 day Swift and ground-based campaign, we evenly split the campaign into thirds and computed the lags in the three resulting 140 days epochs delineated at THJD = 9317, 9457. For convenience, the epochs are labeled in the UVW1 light curve in the top panel of Figure 6. This number of equal segments was selected to probe the same low-frequency range, given that the lowest frequency probed is set by the length of the observation. We also noticed that the epochs correspond approximately to periods of high and low column density of the obscurer reported from the NICER spectral analysis in Paper III (Partington et al. 2023). As shown in the top panel of Figure 6, Epochs 1 and 3 coincide with periods of similarly high average column density ( and 12.9, respectively), whereas Epoch 2 instead overlaps with a period of lower average column density (lower by over a factor of 2 than the other epochs; ). The data quality does not vary dramatically across the three epochs, especially between Epochs 1 and 2, although notable gaps are present in the u, B_g , and z bands in Epoch 3. Given these large u and B_g data gaps, we instead rely on the Swift U and B bands in this epoch. We nonetheless demonstrate that simulated lags are successfully recovered in all epochs and wave bands (see Appendix C).

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The low-frequency lags (at frequencies 0.014–0.048 day⁻¹, equivalent to timescales of 20–70 days) computed for each epoch are presented in the bottom panel of Figure 6. ³⁵ These lags are also provided in the Appendix (Table 2). The epoch-resolved lags show an expected anticorrelation between amplitude and frequency, but with notably larger uncertainties than the full campaign lags, with data gaps taking up a larger fraction of the data.

We perform the same procedure as the lags computed using the full campaign: the low-frequency lags in each epoch are fit with a τ ∝ λ^4/3 relation (i.e., fitting for the normalization τ₀), as expected from standard accretion disk reprocessing (Cackett et al. 2007). We also compare the lags to those predicted by the disk reprocessing model, as detailed in Section 5.

The lags in higher-column-density Epochs 1 and 3 (left and right panels in Figure 6) are generally consistent, producing best-fit normalizations of τ₀ = 0.69 ± 0.03 and 0.66 ± 0.05 days, respectively. These lags are longer than those predicted by the disk reprocessing model by a factor of 2–3. Similar to the lags computed from the full campaign, we also observe a U-band lag excess in both the first and third epochs, where the lag surpasses the τ ∝ λ^4/3 best fit by roughly 110% in Epoch 1 and 80% in Epoch 3.

The lags in lower-column-density Epoch 2 (middle panel in Figure 6) are systematically shorter than those measured in the other two epochs: the best-fit normalization of τ₀ = 0.35 ±0.02 days is smaller by over a factor of two than the normalizations of the other epochs, and >10σ from that measured in Epoch 1. Again, the data quality in Epochs 1 and 2 is consistent in general, and we do not find systematically poorer recovery of simulated lags in this epoch. Unlike Epochs 1 and 3, however, the lags in Epoch 2 are nearly consistent with those expected from disk reprocessing. This includes a lack of the U-band excess observed in the other epochs (the τ ∝ λ^4/3 fit lies within 1σ from the U-band lag in Epoch 2). We find in Section 6 that, if we extend the time range of Epoch 2 to include the first half of Epoch 3 where the column density is still low, the lags become fully consistent with the disk reprocessing model. The measured bias-subtracted coherences are high at these low frequencies, with an average coherence of 0.93 in Epoch 2 and 0.79 in Epochs 1 and 3.

In order to evaluate how the variability changes across epochs, we computed the PSD in each UVOIR band per epoch using the GP realizations. The PSDs are then fit with a power law P(ν) = A ν^β at frequencies <0.1 day⁻¹. These frequencies are low enough to avoid white noise while including timescales where we expect to see reprocessing by the disk. The best-fit PSD slopes (β) range from −3.5 to −2.4 depending on the band and epoch, which are similar to those found for other AGN (Edelson et al. 2014; Panagiotou et al. 2022b). The respective average PSD slopes of each epoch are −3.3 ± 0.2 (Epoch 1), −2.7 ± 0.3 (Epoch 2), and −2.9 ± 0.4 (Epoch 3). This indicates there is more power at high frequencies relative to low frequencies that in Epochs 2 and 3 than in the first epoch. It may be that the first half of Epoch 3, which we found contributes to shorter lags similar to those in Epoch 2 (see Figure 7), is contributing to a shallower slope of the epoch’s PSD. Splitting Epoch 3 to test this, however, renders us unable to meaningfully constrain the slope of the Epoch 3 PSD. An advantage of a frequency-resolved analysis is that the underlying slope of the PSD does not systematically skew the measured lags and thus the inferred size scale, as is the case for the CCF approach, due to the variability amplitude being strongest on the longest timescales.

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These significant changes in the lags from epoch to epoch provide a unique opportunity, especially to explore the origin of the perplexingly long lags reported from previous continuum reverberation mapping campaigns. In summary, we measure longer UVOIR lags at times of higher column density, similar to the UV emission-line lags presented in Paper V (Homayouni et al. 2024). Potential explanations are discussed in Section 6.

We measure a (Swift) X-ray–UV lag in only the second epoch, which is unsurprising given the low X-ray flux throughout the rest of the campaign. Figure 8 presents the X-ray–UV lags in both the soft (0.3–1.5 keV) and hard (1.5–10 keV) bands. Similar to the frequency-resolved lags in Mrk 335 (Lewin et al. 2023), the X-ray lags the UV at low frequencies; in this case, the soft and hard bands lag the UV by 3 and 4 days, respectively, in the 0.014–0.048 day⁻¹ frequency range. This lag is visualized with the X-ray and UVW2 light curves shown in the same figure. While there are some notable features that could explain the lag, it is also possible that the lags are being produced by the overall (long-timescale) negative slope present in both bands.

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These lags appear contrary to the central reprocessing picture, in which the coronal X-rays are reprocessed by circumnuclear material to generate delayed variability at longer wavelengths. In Mrk 335, the positive X-ray lag was explained by reprocessing by photoionized gas near the BLR that dominates the X-ray spectrum below 2 keV (Lewin et al. 2023). We are similarly able to reproduce the observed X-ray–UV lags with additional reprocessing from beyond the disk at the same radius consistent with the BLR used to describe the UVOIR lags. Nonetheless, the lags may be the result of propagating fluctuations in the mass-accretion rate (Arévalo & Uttley 2006) or reprocessing by the obscuring disk wind positioned at ∼4 lt-days from the black hole (Zaidouni et al. 2024).

The disk model is consistent with the lags at high frequencies (>0.05 day⁻¹), but undershoots the lags at low frequencies by a factor of 3–4 on average. The reduced chi-squared of the disk model is , which does not include the X-ray bands (we observe a significant lag only in the epoch-resolved lags). The model is qualitatively different from the observed lags at frequencies below 0.05 day⁻¹: the disk model predicts a constant lag, whereas the lags continue to increase at lower frequencies. This indicates additional reprocessing at larger radii not being accounted for by the disk model.

These results are similar to those found from modeling the frequency-resolved lags in NGC 5548 (Cackett et al. 2022) and Mrk 335 (Lewin et al. 2023): the low-frequency lags cannot be reproduced with disk models alone, even with higher temperatures producing larger effective radii of reprocessing. Instead, the lags are successfully modeled by including an additional impulse-response function, corresponding to reprocessing from a more distant reprocessor. In both sources, the best-fit radius of this distant reprocessor is consistent with the BLR size scale inferred from previous Hβ measurements.

We apply this procedure in an attempt to model our lags using a phenomenological log-normal impulse-response function as in Cackett et al. (2022) and Lewin et al. (2023):

where the median (e^M ) and standard deviation (S) are determined by minimizing the total chi-squared. While this is a phenomenological model, it can be interpreted as reprocessing from beyond the disk, and gauges the geometry (radius and width) of the distant reprocessor required to reproduce the observed lags. While the standard deviation of the component is a proxy for the width of the reprocessor, the two are not interchangeable. For context, this log-normal impulse response is a smoother alternative to the popular top-hat response function used to model reprocessing by a spherical shell (Uttley et al. 2014). The log-normal response function is also asymmetric, with a tail to long lags.

The final impulse-response function is a linear combination of the disk model (ψ_disk) and the distant reprocessor model (ψ_BLR):

where f is the fractional contribution of the distant reprocessor component to the final model. We fit for this fraction in each band independently, akin to the BLR diffuse line and continuum varying across wave bands (Korista & Goad 2001; Lawther et al. 2018; Korista & Goad 2019; Netzer 2022). For clarity, the BLR fraction does not represent the fraction of the observed flux originating from the BLR. Instead, the model parameter has a nonlinear relationship with the amount that the observed lags exceed the disk model. Modeling the lags in NGC 5548 and Mrk 335 in this way produced f values as a function of wavelength that are qualitatively similar to the BLR continuum (Figures 7 and 8 in Cackett et al. 2022; Lewin et al. 2023, respectively). This includes higher f values (i.e., the distant reprocessor contributed more to the total model) required in the U band versus neighboring bands.

The final disk+BLR model (shown in Figure 4) is a significant improvement in reproducing the lags at low frequencies compared to the disk model: the reduced chi-squared is improved to from in the case of the disk model. The median of the BLR model—a proxy for the size scale of additional reprocessing needed to reproduce the lags—is constrained to e^M = 22.5 ± 3.6 days. This radius is consistent with that of the BLR based on Hβ lag measurement from Paper I (τ_Hβ = 23.2 ± 1.6 days; Kara et al. 2021). The standard deviation of the component is constrained to S = 4.3 ± 0.8 days. This value is larger by over a factor of 4 than that found when the same modeling treatment was applied to the frequency-resolved lags in NGC 5548 and Mrk 335 (Cackett et al. 2022; Lewin et al. 2023). In Section 6, we demonstrate that this larger standard deviation can be produced as a result of averaging over lags produced at different radii in different epochs.

The best-fit BLR fractions are presented in Figure 9. Like NGC 5548 and Mrk 335, the shape of the BLR fraction as a function of wavelength shows several similarities to that of the BLR diffuse line and continuum (overlaid in the bottom of the same figure for reference). Most notable are a local maximum in the U band, a change in slope in before and after the Balmer jump, and larger BLR fractions at longer wavelengths (Korista & Goad 2001, 2019).

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The UVOIR lags show systematic changes between the three epochs defined by splitting the 420 day light curves evenly into three segments. Epochs 1 and 3 exhibit similar (low-frequency) lags (see Figure 6), producing lag normalizations within 1σ of each other. The lags in Epoch 2, however, are shorter than those in the other epochs by over a factor of 2 on average.

We apply the same procedure as the full campaign lags to the lags in each epoch: We evaluate the performance of the disk model, and then include the distant reprocessor component to constrain the geometry (namely, the median and standard deviation in radius) of the reprocessor needed to describe the low-frequency lags. As shown in Figure 6, the shorter lags in Epoch 2 are nearly consistent with the disk model, whereas the lags in Epochs 1 and 3 exceed the disk model by a factor of ∼3 on average. This is reflected by the reduced chi-squared values totaled across all UVOIR bands and frequencies: in Epoch 2, versus and in Epochs 1 and 3, respectively (fewer degrees of freedom as the u, B_g bands are not used in Epoch 3).

We then include the log-impulse distant reprocessor model component, fitting for the median and standard deviation in each epoch independently. The reduced chi-squared significantly improves in Epochs 1 and 3: (Epoch 1) and (Epoch 3). The fit statistic for Epoch 2 improves slightly to , as expected given the low contribution of the BLR component needed to describe the lags. The best-fit model parameters agree within 1σ across epochs, suggesting that the same distant reprocessor is contributing to the low-frequency lags in all epochs. The best-fit medians are all consistent with the BLR size scale inferred from the 23 day Hβ lag (Kara et al. 2021), with e^M = 22.3 ± 6.3 days in Epoch 1, e^M = 24.3 ± 5.8 days in Epoch 2, and e^M = 24.0 ± 7.4 days in Epoch 3. The best-fit standard deviations of the distant reprocessor are S = 0.9 ± 0.7 in Epoch 1, S = 1.7 ± 0.9 in Epoch 2, and S = 1.9 ± 1.2 in Epoch 3. These values are over a factor of 2 smaller than that required to describe the full campaign lags (S = 4.3 ± 0.8 days), and are similar to previous values to model the lags in NGC 5548 (S = 1.1 ± 0.2; Cackett et al. 2022) and Mrk 335 ( Lewin et al. 2023). These statistics are used in the next section to demonstrate that the larger standard deviation of the BLR component required to describe the full campaign lags can be reproduced by averaging over lags produced at different effective radii in different epochs.

The similar lags in Epochs 1 and 3 require the BLR model to compose a high fraction of the total model, with an average BLR fraction of f = 0.74 ± 0.09, 0.70 ± 0.14 in Epochs 1 and 3, respectively. The BLR fraction in Epoch 2 is lower by almost a factor of 2 (f = 0.40 ± 0.11), as expected given that the low-frequency lags in this epoch are nearly consistent with disk reprocessing. In Section 6, we discuss that line-of-sight obscuration by a disk wind may be obscuring our view of reprocessing by the disk more significantly in Epochs 1 and 3, producing longer lags from instead viewing mostly reprocessing at larger radii, including the BLR. In this scenario, we expect higher values for the BLR fraction in Epochs 1 and 3 given that the median response radius of the reprocessor is not changing significantly, which is consistent with the results.

The hard and soft (Swift) X-ray bands were found to lag the UVW2 band by 4.1 ± 0.5 and 3.0 ± 0.4 days, respectively, in the lowest frequency bin in Epoch 2 (see Figure 8). This indicates that the hard X-rays lag the soft by ∼1 day. A simple order-of-magnitude comparison to the XMM-Newton hard lag reveals that the lags are consistent after adjusting for the difference in frequency: τ = 10⁵ s, ν = 10⁻⁸ Hz for Swift versus τ = 10² s, ν = 10⁻⁵ Hz for XMM-Newton. This assumption is motivated by theoretical work for the frequency-amplitude scaling of inward-propagating fluctuations in the mass-accretion rate (Lyubarskii 1997; Kotov et al. 2001; Arévalo & Uttley 2006), and suggests that the lags may originate from this same physical process.

We also modeled the X-ray–UV lags on their own to gauge the region of reprocessing capable of reproducing these positive lags. The best-fit parameters of the BLR component are consistent with those constrained with the UVOIR lags: the median/effective radius is e^M = 25.0 ± 8.7 days, with a standard deviation of S = 2.5 ± 1.5 days. Nonetheless, the coherence is moderate (∼0.5–0.7), and more notably that the signal-to-noise ratio is low in these bands. As such, these results are tentative, albeit similar to Mrk 335, where the soft X-rays were also measured to lag the UV (Lewin et al. 2023). It is nonetheless worth mentioning that the variability we see in this low-column-density epoch more directly reflects variations in the intrinsic X-ray continuum rather than in the transparency of the obscuration.

The lag measurements from recent, high-cadence monitoring campaigns of AGN using Swift and ground-based telescopes have consistently deviated from theoretical predictions. First, the normalizations of the expected τ ∝ λ^4/3 relation are typically a factor of 2–3 larger than predicted given the black hole mass and accretion rate. The lags have been especially long in the U band (3465 Å), exceeding the best-fit τ ∝ λ^4/3 relation even with the large normalizations by a factor of ∼2 (Cackett et al. 2018; Edelson et al. 2019; Hernández Santisteban et al. 2020; Vincentelli et al. 2021; Kara et al. 2023). These U-band excesses are commonly attributed to the diffuse line and continuum of the BLR (Korista & Goad 2001; Lawther et al. 2018; Korista & Goad 2019; Netzer 2022).

The lags at frequencies below 0.05 day⁻¹ are longer than those expected from disk reprocessing given the mass and accretion rate of Mrk 817. The lags at frequencies above 0.05 day⁻¹ are instead well described by the disk reprocessing model, including the lag in the U band. If the discrepancies from disk reprocessing at low frequencies are due to contamination from a distant reprocessor, then the reprocessing is occurring on timescales longer than 1/0.05 = 20 days. This timescale is consistent with the outer parts of the BLR according to the 23 day Hβ lag in Paper I (Kara et al. 2021). Nonetheless, a small lag excess remains above this frequency beyond 6000 Å that resembles the Paschen continuum, suggesting that BLR reprocessing continues to smaller radii. This is consistent with the extent of the BLR inferred from the lags in Paper I, which range from the UV to Hβ and span from a few light days to beyond 23 light days.

If we instead fit the disk model temperatures to the measured lags in the lowest frequency bin, an Eddington ratio an order of magnitude higher than observed is required (L/L_Edd = 6.7 versus 0.2; Kara et al. 2021). The assumptions of the thin-disk model are not expected to hold at this accretion rate (Netzer & Trakhtenbrot 2014; Wang et al. 2014; Du et al. 2018). To avoid this unlikely conclusion, we included a simple log-normal impulse-response function to account for additional reprocessing from beyond the disk. The inclusion of this component significantly improves the description of the lags ( versus from the disk model alone).

Similar to the modeling results in Mrk 335 and NGC 5548, the required median radius of the component (22.5 ± 3.6 days) is consistent with the 23 days Hβ lag (Kara et al. 2021). Photoionization models have shown that the mean emissivity radius of the diffuse continuum is smaller than that of Hβ by a factor of ∼2–3 (Korista & Goad 2019; Netzer 2020, 2022). Given that the log-normal distribution is right-skewed, the distribution peaks at a value smaller than the median; in other words, the radius of maximum reprocessing is smaller than the median value. A median of 23 days and a standard deviation of 1 day found from fitting the individual epochs places the radius of maximum reprocessing at 8.2 days, with significant reprocessing down to a few days. This radii range is consistent with the lags from Paper I indicative of the size scale of the BLR and the aforementioned modeling by Korista & Goad (2019).

The standard deviation of this additional component, which is related to the width of the extended reprocessor, is constrained to 4.3 ± 0.8 days. This value is over a factor of 4 larger than that required to describe the lags in NGC 5548 and Mrk 335 (using the same log-normal and disk model; Cackett et al. 2022; Lewin et al. 2023). Modeling the lags in each epoch results in a smaller standard deviation (∼1 day). This suggests that the larger standard deviation may be the result of combining epochs across which the lags differ notably, for instance due to reprocessing at different radii in different epochs. This idea agrees with the full campaign lags being roughly an average of the shorter lags in Epoch 2 and longer lags in Epochs 1 and 3. To test this idea, we compute an effective radius of reprocessing for each epoch (in each wave band) by using the best-fit BLR fraction (f) and radius (R_BLR): R_eff = (1 − f)R_disk + fR_BLR. We then compute the standard deviation of the effective radii across the three epochs to compare to the large standard deviation required to model the full campaign lags. The radius of disk reprocessing (R_disk) depends on wavelength and is computed using Equation (12) in Fausnaugh et al. (2016). We set R_BLR = 23 days to match the Hβ lag, which is consistent with the best-fit median in all epochs. The resulting standard deviation of effective radii across the three epochs ranges from 2.5 to 6 days, depending on the band, or 4.1 ± 1.0 days on average. This is consistent with the larger standard deviation needed to reproduce the full campaign lags, thus indicating this value could be the result of averaging over lags produced at different effective radii in different epochs. Specifically, the effective radii of reprocessing averaged across bands are 17.5 days (Epoch 1), 9.1 days (Epoch 2), and 13.0 days (Epoch 3).

We caution that our modeling of the lags is limited: we apply a single model for disk reprocessing and a simple analytic treatment for the extended reprocessor/BLR. As such, additional modeling with more physical models is warranted, such as those that include a more realistic treatment of the coronal geometry (Kammoun et al. 2021a, 2021b), the thickness of disk (Starkey et al. 2023), and models for the BLR (Korista & Goad 2019; Netzer 2020). Modeling of the lags from inwardly propagating fluctuations in the mass-accretion rate (e.g., Lyubarskii 1997; Neustadt et al. 2024) is also of importance.

There are several key observables presented here and in previous AGN STORM 2 papers that need to be explained:

• 1.  
  Paper III (Partington et al. 2023) found the line-of-sight X-ray column density and the equivalent width of the blueshifted broad UV absorption troughs vary together on timescales of ∼100 days. Here, we separated the observations into three epochs corresponding to these periods of high (Epochs 1 and 3) and low (Epoch 2) column density.
• 2.  
  Paper IX (Zaidouni et al. 2024) found concurrent X-ray and UV absorption lines with the same velocity profile, i.e., from the same outflow. It is estimated that the launching radius of the outflow is roughly 1000 R_g (2–3 light days) and that the time it takes the wind to travel from the disk into our line of sight is 100–200 days.
• 3.  
  Paper IV (Cackett et al. 2023) found that, while the UVW2 and HST emission at shorter wavelengths is usually highly correlated, there is a relative excess of the UVW2 emission compared to all HST bands when the column density is high during the campaign corresponding to our Epoch 1.
• 4.  
  Paper V (Homayouni et al. 2024) found that the C iv lag changes dramatically over time, with the longest lag occurring when the UVW2/HST correlation weakens and the column density is relatively high in Epoch 1.
• 5.  
  Here, we find the continuum lags vary significantly from epoch to epoch. The lags are over a factor of 2 larger when the column density is high (Epochs 1 and 3) than when the column density is low (Epoch 2).

The results presented in this paper demonstrate that the continuum lags can be strongly affected by the presence of a variable obscurer. While a holistic analysis that includes modeling changes in the emission and obscuration is required to fully characterize the lags, here we present one possible explanation, illustrated in Figure 10. In this physical picture, the epoch-to-epoch changes in the lags are due to changing obscuration by an accretion disk wind, launched at ∼1000 R_g . We can think of this as a limit cycle, where the wind is first launched from the disk, and has a high column density. The wind is lifted from the disk, becoming more clumpy and of lower column density, as it reaches our line of sight. This wind moving from the disk surface and up into our line of sight explains the changes in X-ray column density and UV absorption troughs on timescales of 100–200 days (Papers III and IX).

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In the context of our delineated “epochs,” the disk limit cycle begins with the high column density wind being launched from the disk in Epoch 2. This wind intercepts the radiation to the BLR, thus leading to relatively short lags. ³⁷ As the wind is lifted from the disk's surface and into our line of sight (Epochs 1 and 3), it becomes more clumpy and lower column density, and the BLR is exposed to the ionizing source, thus producing longer lags that are indeed dominated by the diffuse BLR continuum. It is also in these epochs where the longest C iv lags are observed (Paper V).

This model is appealing in that it fits within the larger context of UVOIR campaigns, where in all the unobscured AGN, we observe lags that are ubiquitously longer than predicted by disk reprocessing by a factor of 2 or more. It is for this reason that studies have invoked more distant reprocessing from the diffuse BLR (e.g., Korista & Goad 2001; Lawther et al. 2018; Korista & Goad 2019; Netzer 2022). However, here in Epoch 2 in the campaign of Mrk 817, we observe (to our knowledge, for the first time), lags that are fully consistent with expectations from disk reprocessing alone. Perhaps it is precisely because this source has a time-variable obscurer that, at some unobscured epochs, we see short lags more consistent with just the disk.

There are important details to modeling that will be discussed in H. Netzer et al. (2024, in preparation). For instance, because the wind base is a high column density, photoionized plasma, it is expected to radiate its own reprocessed emission at a radius of a few light days from the black hole. In this model, in Epoch 2, the lags are dominated by the reradiation from the wind base, which is still effectively located at a smaller radius than the diffuse BLR.

While this model explains many of the observations from Papers I–IX, there are some caveats that require future work. For instance, it remains unclear how the emission-line fluxes fit into the picture. Paper V shows the C iv flux has its strongest response relative to the continuum when the C iv and continuum lags are shortest (Epoch 2, Phase C). Yet this result implies a strong contribution to continuum lag measurements from the BLR, contrary to our observation of short continuum lags consistent with primarily disk emission. It is difficult to imagine how this picture could produce the short lags that are fully consistent with reprocessing by just the disk at times of low column density, given that BLR reprocessing is present at all times in the Paper V scenario. Moreover, one might expect that in Epoch 2, where, in our model, the obscurer does not affect the continuum as much, and the lags do not require contributions from the BLR, we would expect a larger correlation between X-rays and UVOIR, but this is not observed. Some of this is likely because the obscuration is still present (albeit at a lower level), but may also require examination of effects of dynamic variability of the X-ray corona (Panagiotou et al. 2022a) and/or fluctuations in the disk properties in response to FUV or X-ray heating (Gardner & Done 2017; Sun et al. 2020; Chen et al. 2024; Secunda et al.2024).

While this physical model does broadly explain many of the main results of the AGN STORM 2 campaign, a more holistic approach to modeling the changes in emission and obscuration will be needed in order to disentangle contributions from the disk, obscurer, and BLR. The specifics of a more complete model predicting the time lags together with the emission/absorption line kinematics will be examined in detail in forthcoming works.