Swift Monitoring of NGC 4151: Evidence for a Second X-Ray/UV Reprocessing

The target of this experiment, the Seyfert 1.5 galaxy NGC 4151 (redshift z = 0.00332, de Vaucouleurs et al. 1991; distance D ≈ 19 Mpc; Hönig et al. 2014), is typically the brightest Seyfert 1 galaxy in the sky in the X-ray/UV/optical wavelength range accessible to Swift. For instance, the Swift Burst Alert Telescope (BAT) catalog (Krimm et al. 2013) indicates that NGC 4151 is twice as bright in the 15–50 keV band as the next-brightest Type 1 AGN. NGC 4151 is one of the Seyfert 1 galaxies in the original identification paper on these objects (Seyfert 1943) and is often considered to be an archetype of the class (Ulrich 2000). It is well known to be strongly variable across the wavelength range accessible to Swift (Edelson et al. 1996), making it an ideal monitoring target.

The bolometric luminosity of NGC 4151 is L_bol ≈ 5 × 10⁴³ erg s⁻¹ (Woo & Urry 2002). The central black hole mass has been measured by RM to be ${M}_{\mathrm{BH}}\,\approx {43.57}_{-0.37}^{+0.45}\times {10}^{7}\,{\text{}}{M}_{\odot }$ (Bentz et al. 2006, updated with the calibration of Grier et al. 2013), by gas dynamics to be ${3.0}_{-2.2}^{+0.75}\times {10}^{7}\,{\text{}}{M}_{\odot }$ (Hicks & Malkan 2008), and by stellar dynamics to be (3.76 ± 1.11) × 10⁷ M_⊙ (Onken et al. 2014). We adopt a value of M_BH ∼ 4 × 10⁷ M_⊙ in this paper.

NGC 4151 has been particularly well-studied in the X-rays. The soft X-rays are only weakly variable because that band is dominated by extended line emission (e.g., Zdziarski et al. 2002), but at higher energies (above ∼2 keV) the flux is strongly variable and thought to be coming from the corona. NuSTAR/Suzaku spectroscopy is consistent with reflection from the inner disk in NGC 4151 (Keck et al. 2015). X-ray time lags also show Fe Kα reverberation in this object that would require reflection from the inner disk (Zoghbi et al. 2012; Cackett et al. 2014).

During 2016 February 20 through April 29, Swift executed an intensive monitoring campaign on NGC 4151, consisting of 319 separate visits of at least 120 s, an average of nearly five visits per day. These observations are summarized in Table 1. Start and stop times for Swift observations are originally recorded in Mission Elapsed Time (seconds since the start of 2001) and corrected for the drift of the on-board Swift clock and leap-seconds. These times were converted to Modified Julian Date (MJD), the standard for this observing campaign.

Swift observations with the UVOT were made in mode 0x037a, which allows for hardware windowing in the four longest-wavelength bands. This was done because this source is too bright to be observed in a standard, non-windowed mode. Observations with the X-Ray Telescope (XRT; Burrows et al. 2005) were made in Photon Counting (PC) mode, except for the last seven, which were made in Windowed Timing (WT) mode (Hill et al. 2004). The impacts of these observing modes on the data quality and other details are discussed in the following subsections.

These Swift observations were coordinated with intensive monitoring with numerous ground-based telescopes including the Las Cumbres Observatory Global Telescope network and the Liverpool Telescope at La Palma. Those data will be presented in subsequent papers (K. Horne et al. 2017, in preparation; M. Goad et al. 2017, in preparation).

The UVOT data were taken in a six-filter, blue-weighted mode in which the four longest-wavelength filters (uvw1, u, b, and v) are observed using 5'' × 5'' hardware windows. These reduce the frame time from 11 to 3.6 ms, thereby mitigating the effect of pile-up (coincidence losses) in this bright source. The four hardware window observations are preceded by short (10 s) full-field exposures, but these are not used because the coincidence losses were found to be too large to be corrected reliably. This mode also splits the uvm2 data into two exposures when the time exceeds 300 s; such exposure pairs that survive screening are co-added before final analysis.

All UVOT data were reprocessed for uniformity, applying standard FTOOLS utilities (Blackburn 1995; from version 6.19 of HEASOFT ³⁶ ). The astrometry of each field was refined using the AGN and up to 25 isolated field stars drawn from the HST GSC 2.3.2 (Lasker et al. 2008) and Tycho-2 (Høg et al. 2000) catalogs, yielding residual offsets that were typically ∼03. Fluxes were measured using a 5'' radius circular aperture, and concentric 40''–90'' radius annuli were used to measure the sky background level. The final values include corrections for aperture losses, coincidence losses, large-scale variations in the detector sensitivity across the image plane, and declining sensitivity of the instrument over time. After reprocessing, 25 exposures were screened out to eliminate observations affected by tracking errors or with exposure times shorter than 20 s.

We use a non-default setting when accounting for systematic errors in the aperture correction arising from variations in the UVOT point-spread function. The tool UVOTAPERCORR normally adds a filter-dependent uncertainty of 1.85%–2.15% to measurements made with a 5'' aperture. However, when measuring fluxes of NGC 4151 and the field stars with the highest signal-to-noise ratios (S/Ns), we found the resulting error estimates to be inconsistent with Gaussian statistics. This result is not surprising given that the UVOTAPERCORR documentation notes that the appropriate size of the systematic error is not well established. We empirically examined a range of systematic error estimates by adjusting the parameter FWHMSIG and found that halving this parameter (to 7.5) yielded distributions much more consistent with Gaussian. For instance, in the case of the UV-brightest star in the field (BD+40 2507), 88.7% of the uvm2 and 76.8% of the uvw2 measurements fall within ±1σ of the mean when using the default FWHMSIG setting, whereas these percentages are 72.8% and 64.1% when FWHMSIG = 7.5. By adopting this setting, the flux uncertainties reported here include filter-dependent systematic errors of 0.92%–1.08%.

The resulting light curves, shown in Figure 1, exhibited occasional anomalously low points ("dropouts"), especially in the UV. Similar dropouts were seen in an earlier Swift study of NGC 5548 (Edelson et al. 2015) and were found to be clustered in the detector plane. This may be due to localized regions of reduced sensitivity³⁷ (it should be noted that the deviant flux points are universally low, not symmetric about the light curve as would be expected from AGN variability). As such, we filtered discrepant points in the NGC 4151 data in a fashion similar to that in the Appendix of Edelson et al. This filtering consists of four steps: (1) identify dropouts from the light curves; (2) map the data onto the detector plane; (3) define boxes to enclose clusters of bad data; and (4) use the set of boxes as a mask and screen out all data in these regions in the four shortest-wavelength bands. The procedure used here differs from Edelson et al. (2015) in that we model the light curve by fitting a quadratic expression to each light curve in a sliding window of ±2 days.

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We define dropouts as points below the light curve with absolute deviations that exceed those of the highest positive outliers in the entire light curve. The number found in each band is given in Table 2. Dropouts are most prevalent at short wavelengths, accounting for >10% of measurements in the UV bands, nearly 5% in u, and are barely found in b and v.

As in the case for NGC 5548, the dropouts are highly clustered in the detector plane. We define 23 boxes to enclose clusters of three or more dropouts (Figure 2), ranging in size from a single 1'' × 1'' pixel to over 1000 pixels. We note that the data from both NGC 4151 and NGC 5548 sample the detector plane sparsely and do not cover the exact same regions, so the mask defined for one of these AGNs is not well-suited for the other.

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The final step is to remove all data in the four shortest-wavelength bands that fell within any of these boxes. The mask is not applied to b or v data because the effect is small compared to the statistical uncertainties in these bands. Tallies of the points filtered out and of the final data points are given in Table 2. Data that fall within the detector mask are shown as yellow Xs in Figure 1. The final reduced and filtered UVOT data are given in Table 3 and plotted in the lower six panels of Figure 3.

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The Swift/XRT data were gathered in PC mode for all except the last seven visits, which were gathered in WT mode owing to an error in our observing proposal. The data were analyzed using the tools described by Evans et al. (2009)³⁸ to produce light curves that are fully corrected for instrumental effects such as pile-up, dead regions on the CCD, and vignetting. Because the WT data showed a large flux discontinuity with the PC data at soft energies, these seven WT points were discarded. We additionally excluded all visits where the total good integration time was less than 120 s. This resulted in a final light curve having 319 X-ray points over the 71-day monitoring period (see Table 1).

We generated X-ray light curves in four bands: X1 (0.3–1.25 keV), X2 (1.25–2.5 keV), X3 (2.5–5 keV), and X4 (5–10 keV). These are chosen so each band spans one octave of frequency, except for X1, which is larger because it also has the lowest count rate. We utilize "snapshot" binning, which produces one bin for each spacecraft pointing. This is done because these short visits always occur completely within one orbit. These XRT data are presented in Table 4. As discussed in Section 2.1, the X1 band is only weakly variable because it is known to be dominated by extended emission (Zdziarski et al. 2002).

Besides the pointed UVOT and XRT instruments, Swift also has the BAT, a large-sky monitor originally developed to pinpoint new γ-ray bursts (Barthelmy et al. 2005). The BAT is now also being used to monitor X-ray transients in the hard X-ray (15–50 keV) band (e.g., Krimm et al. 2013). Most AGNs are too faint to produce usable high-cadence BAT light curves. However, NGC 4151 is typically the brightest Seyfert 1 in the sky at these wavelengths, so we were able to utilize these data³⁹ to measure a hard X-ray light curve. This provides an important extension of the XRT light curves to higher energies. These data are reproduced in Table 5. See Krimm et al. (2013) for further details of the BAT data gathering and reduction process.

Figure 3 shows the resulting light curves, presented in order of descending frequency with the highest frequency band at the top and the lowest at the bottom. These data are unprecedented in two respects. First, the average sampling interval of 0.22–0.28 day is about a factor of two better than that obtained for the Swift monitoring of NGC 5548, the previous most-intensive AGN monitoring of this type (Edelson et al. 2015). Second, because NGC 4151 is typically the brightest Seyfert 1 in the sky, it was possible to measure five independent X-ray bands, including the hard X-rays with BAT. All 11 of the resulting light curves in Figure 3 are used for time-series analysis.

The visual impression of the UV/optical (uvw2 through v) light curves is that they are so similar that the variations clearly appear to be related. The same is true for the relatively hard X-rays (BAT through X3). The X2 light curve has a lower S/N and may be related to the harder X-rays, while the X1 light curve shows almost no detectable signal so its relation to other bands cannot be assessed. Comparison of the BAT–X3 bands with the uvw2–v bands shows that while many of the largest peaks seen in the X-rays also appear in the UV/optical, the character of the variations is not identical, in the sense that the most rapid variations seen in the X-rays are not seen in the UV/optical. This could be due to smoothing and lagging of the X-ray light curves to produce the UV/optical light curves or it could be that the two wavelength regimes simply have different drivers and the apparent long-term similarities are just a chance coincidence due to the red-noise character of AGN variability (Vaughan et al. 2003). This caveat that the X-rays may not be driving the UV/optical should be kept in mind throughout this analysis. This possibility will be assessed further in subsequent papers (e.g., K. Horne et al. 2017, in preparation).

The focus of this paper is on testing and constraining continuum-emission models primarily through measurement of interband lags. We used the interpolated cross-correlation function (CCF) as implemented by Peterson et al. (2004) to measure and characterize the temporal correlations and interband lags within these data.⁴⁰

We first normalized the data by subtracting the mean and dividing by the standard deviation. These were derived "locally"—only the portions of the light curves that are overlapping for a given lag are used to compute these quantities. We implemented "2-way" interpolation, which means that for each pair of bands we first interpolated in the "reference" band and then measured the correlation function, next interpolated in the "subsidiary" band and measured the correlation, and subsequently averaged the two to produce the final CCF.

We then used the "Flux Randomization/Random Subset Selection" (FR/RSS) method (Peterson et al. 1998) to estimate uncertainties on the measured lags. This is a Monte Carlo technique in which lags are measured from multiple realizations of the CCF. The FR aspect of this technique perturbs in a given realization each flux point consistent with the quoted errors assuming a Gaussian distribution of errors. In addition, for a time series with N data points, the RSS randomly draws with replacement N points from the time series to create a new time series. In that new time series, the data points selected more than once have their error bars decreased by a factor of ${n}_{\mathrm{rep}}^{-1/2}$, where n_rep is the number of repeated points. Typically a fraction of 1/e of data points are not selected for each RSS realization. In this paper, the FR/RSS is applied to both the "driving" and "responding" light curves in each CCF pair. The CCF (r(τ)) is then measured and a lag determined to be the weighted mean of all points with r > 0.8 r_max, where r_max is the maximum value obtained for r, given in Column 2 of Tables 5 and 6. For the data presented herein, simulated lags are determined for 200,000 realizations and then used to derive 68% confidence intervals.

Because of the wealth of Swift data, covering 11 bands, we perform two complete CCF analyses: one with the UVOT band uvw2 as a reference (Table 6) and the other with the XRT band X3 as the reference (Table 7). This allows for a more sensitive search for small lags within the UV/optical and X-rays as well as between the UV/optical and X-ray regimes than would be possible with just a single set of CCFs. These results are presented graphically in Figure 4 and listed in Tables 6 and 7. We first describe the results relative to the X3 X-ray band and then to the uvw2 UV band.

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Because NGC 4151 is so bright, we are able to measure the BAT/X3 CCF. It shows a correlation consistent with zero lag, though the confidence interval is much larger than with the other X-ray bands, owing to the relatively poor sampling and S/N of BAT compared to the XRT bands. Likewise, the X4/X3 correlation is very strong and consistent with zero lag, with much tighter limits on the lag. The X2/X3 correlation is weaker but still apparently significant, with an ∼1.5 day lag in the sense that the harder band leads the softer band. Because the X1 variations are very weak, consistent with an origin in an extended region (e.g., Zdziarski et al. 2002; Keck et al. 2015), the X1/X3 CCF indicates at best a weak correlation, making the delay measurement much less accurate than for the other bands or perhaps not measurable at all. That correlation does not appear significant and no meaningful lag could be measured with that band. All six UVOT bands are strongly correlated with X3, with well-detected lags in the sense that the X-rays lead the UV/optical by ∼3–4 days.

The uvw2-referenced results are more sensitive to lags within the UV/optical than are the X3-referenced results. These show very strong correlations within the UV with no measurable lags down to limits of ∼0.5 day, while the optical bands appear to show small but possibly significant (∼2σ) lags of ∼0.6–1 day behind uvw2. As with the X3-referenced CCFs, these data also show no correlation with the softest X-ray band (X1) and significant correlations with bands X2–X4, with the lag to the X2 band (∼2 days) midway to the lags with X3 and X4 (3–4 days). The BAT/uvw2 correlation is not significant, probably owing in part to the relatively poor sampling and S/N of the BAT light curve.

We now evaluate the significance of the observed CCF peaks by estimating the probability that the observed r_max could arise by chance from independent AGN-like light curves. This is based on the Monte Carlo simulation technique of Breedt et al. (2009; see also Breedt et al. 2010; Cameron et al. 2012), which cross-correlates the higher-energy observed light curve against simulated observations for a large number of independent AGN-like light curves, each of which are constructed to match the mean and rms of the log(flux) of the lower-energy light curve. Initially, the slope of the power spectral density (PSD) function of the "driving" band (always assumed to be the highest energy of the two) is estimated from the data with a single power-law fit to that PSD. The probability density function of the flux of the driving band is also fitted using a lognormal distribution and used to simulate light curves by the method of Emmanoupolous et al. (2013), as implemented by Connolly (2015), with an appropriate level of measurement noise added back in. Then the simulated light curve is cross-correlated with the actual lower-energy light curve and tested to see if the highest observed value of the correlation coefficient (r_max) exceeded that of the actual data. This process is repeated 10,000 times for each CCF band pair, with a new simulated light curve leading to a new CCF in each realization. This ensemble of simulated CCFs should by construction contain no real correlated signal, allowing us to calculate the probability of finding a particular correlation coefficient at a given lag time by chance.

Our initial testing indicated that for low S/N light curves (e.g., X1) the method used a flat (nearly white [measurement] noise) PSD to generate the synthetic light curves, resulting in implausibly high significances. For instance initial application of this technique yielded 99.6% significance for the X1/uvw2 CCF peak (which had r_max ∼ 0.33) and 87% for the X3/uvw2 peak (r_max ∼ 0.68). In many cases (e.g., X1 and BAT), the data are inadequate to measure even a basic PSD slope. We therefore investigated adapting this procedure to use synthetic PSD slopes plus noise instead of attempting to measure them from the data.

We ran simulations assuming PSD slopes of −3, −2.5, and −2 to cover most of the observed range of AGN behavior (e.g., Gonzalez-Martin & Vaughan 2012; Edelson et al. 2014). Only pure power-law PSD slopes were assumed in this initial analysis; no broken or bending power laws were used. Furthermore, this new approach compares the observed value of r_max to that derived from the simulations across a window of ±10 days, whereas we previously only compared the observed value of r_max with the simulated value of r at that same lag. (This is done because we do not have an a priori idea of where the peak will fall.) In this case, because we tested three PSDs with 30,000 runs each, a total of 90,000 runs were used for each CCF band pair. For consistency, we use this revised technique on all CCFs in this paper. These results are shown in Column 6 of Tables 6 and 7. No more than marginal differences were found for simulations with different PSD slopes. Note, in particular, that the X1/uvw2 CCF significance is now much lower (∼69%, consistent with no correlation), but the X3/uvw2 significance has risen to ∼91%. This is all as expected. Still, these results should be considered preliminary because thorough analysis of this significance test (e.g., testing a wider range of PSD slopes, testing bending PSDs, varying the window size) has not been completed. That is beyond the scope of this paper, but will be presented in a future paper (S. Connolly et al. 2017, in preparation).

In this section, we use these CCF results to establish the relation between lag and wavelength, using a methodology similar to that of Edelson et al. (2015). The uvw2-referenced CCFs are used because those are the most sensitive to small lags within the UV/optical. The BAT and X1 lags are excluded from the analysis because of the relatively low significance of the correlation peak (<1σ) and large errors on the lag confidence intervals (∼6–10 days). The u-band lag is also ignored because of possible contamination of diffuse continuum emission from BLR clouds, which must be present at some level and is expected to be stronger in this band (Korista & Goad 2001). The remaining three X-ray and five UVOT lags are shown as a function of the observing band central wavelength in Figure 5. These data were then modeled with a function of the form

$$\begin{eqnarray}&&\tau ={\tau }_{0}[{(\lambda /{\lambda }_{0})}^{4/3}-1],\end{eqnarray} \tag{ 1 }$$

where λ₀ = 1928 Å, the wavelength of the reference uvw2 band and τ₀ is effectively the fitted lag between wavelength zero and λ₀ in days. The uvw2 autocorrelation function lag is identically zero, so this point does not participate in the fit but instead the fit is forced to pass through this point.

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If the X-rays originate near the black hole and drive emission at longer wavelengths, then the X-ray lag represents the light-travel time from the center of the system. In this way, the X-ray data anchor the zero point of the fit and set the physical size scale of the disk. This is the simplest functional form consistent with the standard reprocessing model, yet the data are not well-fitted by this function, with a reduced χ² of 139.6 for 6 degrees of freedom (dof) for an unacceptable p-value <10⁻²⁷. In particular, this shows that it is impossible to simultaneously fit the X-ray and UV/optical data points due to the ∼3 day lag between variations in these regimes.

The results are quite different if the three X-ray points are excluded from the analysis. Modeling the same function to just the five UVOT points yields a reduced χ² of 0.59/3 dof, for an acceptable p-value of 0.90. If this fit is extrapolated to zero at the inner edge of the disk, then the fit parameter τ₀ = 0.34 ± 0.11 day would indicate that emission from an annulus ∼0.34 lt-day in radius peaks at 1928 Å. Note, however, that the X-ray points are inconsistent with this extrapolation, because the X4, X3, and X2 lags undershoot the fit by 7.7σ, 9.9σ, and 2.5σ, respectively. This indicates that the assumption underlying the reprocessing model, that all interband lags are caused by light-travel time effects, cannot be correct.

This analysis of the NGC 4151 Swift data indicates (1) a clear ∼3 day lag between X-ray and UV/optical variations, (2) smaller ∼0.5–1 day lags within the UV/optical, and (3) at lower confidence, a possible ∼1.5 day lag between relatively hard X-ray (2.5–10 keV) and softer X-ray (1.25–2.5 keV) band variations. Before using these results to test models, we examine size scales that would be related to these timescales under a variety of general assumptions.

The most basic size scale is the light-crossing size. We assume that NGC 4151 has a black hole mass of 4 × 10⁷ M_⊙ (Bentz et al. 2006; Onken et al. 2014), so the gravitational radius r_g = 200 lt-s = 0.0023 lt-day. Thus a lag of 1.5–3 lt-day corresponds to a light-crossing size of R ∼ 650–1300 r_g. This is significantly larger than the expected size of the inner accretion disk/corona region, so we conclude that the observed lags do not correspond to light travel sizes within the central engine.

The dynamical timescale, over which a vertical disturbance in a disk returns to hydrostatic equilibrium, is given by t_dyn ∼ (R³/GM)^1/2 (King (2008). Again for a mass of 4 × 10⁷ M_⊙, a delay of 1.5–3 days corresponds to a region of size R = 23–36r_g for re-establishing hydrostatic equilibrium. Furthermore, the thermal (t_th) and viscous (t_visc) timescales are related to the dynamical timescale by t_dyn ∼ αt_th ∼ α (H/R)² t_visc, where α is the dimensionless viscosity parameter (Shakura & Sunyaev 1973), thought to be of the order of 0.1 in AGNs, and H/R is the ratio of height to radius of the disk, also of the order of 0.1. This means that for both thermal and viscous processes, a delay of 1.5–3 days would correspond to a region smaller than the last stable orbit around the black hole (6r_g).

Thus we conclude that the observed lags of 1.5–3 days cannot be associated with any plausible light-crossing time (that is too large) or thermal or viscous processes (those timescales are too small). However, they may be associated with a process governed by the dynamical timescale (t_dyn), which is also the timescale on which the disk responds to loss of hydrostatic equilibrium (Gardner & Done 2017). A possible implication of associating the observed lags with the dynamical timescale is discussed in Section 4.4.

In this subsection, we compare the size of the accretion disk in NGC 4151 derived from RM and the reprocessing model with theoretical predictions. We start with Equation (12) of Fausnaugh et al. (2016), which gives the light-crossing radius r of a flat, geometrically thin, optically thick accretion disk annulus emitting at a characteristic wavelength λ₀:

$$\begin{eqnarray}&&r={\left(X\displaystyle \frac{k{\lambda }_{0}}{{hc}}\right)}^{4/3}{\left[\left(\displaystyle \frac{{GM}}{8\pi \sigma }\right)\left(\displaystyle \frac{{L}_{\mathrm{Edd}}}{\eta {c}^{2}}\right)(3+\kappa ){\dot{m}}_{{\rm{E}}}\right]}^{1/3}\end{eqnarray} \tag{ 2 }$$

where X is a multiplicative scaling factor of the order of unity that accounts for systematic issues in converting the annulus temperature T to wavelength λ at a characteristic radius R, L_Edd is the Eddington luminosity, η is the radiative efficiency in converting mass into energy, κ is the local ratio of external to internal heating, assumed to be constant with radius, and the Eddington ratio ${\dot{m}}_{\mathrm{Edd}}={L}_{\mathrm{bol}}/{L}_{\mathrm{Edd}}$. Under the assumption that at an annulus of radius R the observed wavelength corresponds to the temperature given by Wien's Law, then X = 4.87. If instead the flux-weighted radius $\langle R\rangle$ is used, then X = 2.49. ($\langle R\rangle ={\int }_{{R}_{0}}^{\infty }B(T(R)){R}^{2}\,{dR}/{\int }_{{R}_{0}}^{\infty }B(T(R))R\,{dR}$, where R₀ is the inner edge of the disk, B(T(R)) is the Planck function, and T(R) is the temperature at radius R.) The flux-weighted estimate assumes that the temperature profile of the disk is described by T ∝ R^−3/4 (Shakura & Sunyaev 1973). In both the Wien and flux-weighted cases, the disk is assumed to have a fixed aspect ratio and to be heated internally by viscous dissipation and externally by the coronal X-ray source extending above the disk (the lamp-post model).

Assuming κ = 0 (negligible external heating compared to internal heating) and η = 0.1 and setting the fiducial wavelength λ₀ = 1928 Å yields a more convenient scaling:

$$\begin{eqnarray}&&r={ct}=0.09{\left(X\displaystyle \frac{\lambda }{1928 \mathring{\rm{A}} }\right)}^{4/3}{M}_{8}^{2/3}{\left(\displaystyle \frac{{\dot{m}}_{\mathrm{Edd}}}{0.10}\right)}^{1/3}\,\mathrm{lt} \mbox{-} \mathrm{days}\end{eqnarray} \tag{ 3 }$$

where r is given in units of light days and M₈ = M_BH/10⁸ M_⊙.

We use as the black hole mass of NGC 4151 M₈ = 0.4 as discussed above. The Eddington ratio is difficult to estimate, and NGC 4151 has a wide variety of estimates including 6% (Meyer-Hofmeister & Meyer 2011), 4% (Kraemer et al. 2005), and 0.6% (Edelson et al. 1996). The first two values were estimated assuming M₈ = 0.13 while the last assumed M₈ = 0.4. Correcting the first two values to a mass of M₈ = 0.4 as used herein yields ${\dot{m}}_{\mathrm{Edd}}=2 \%$, 1.3%, and 0.6%, respectively. Here we assume a value of 1%, close to the harmonic mean of these estimates.

Inputting these values into Equation (3) yields r = 0.19 light-day for the Wein's Law case and r = 0.08 light-day for the flux-weighted assumption. Given the fitted value of t = 0.34 ± 0.11 day, the observed size appears larger than predicted by a factor of ∼2–4. A similar discrepancy was found between the RM-derived and theoretically predicted size of the accretion disk of the Seyfert 1 galaxy NGC 5548 (Edelson et al. 2015; Fausnaugh et al. 2016). Gravitational microlensing of much more distant quasars also appears to derive disk sizes that are larger than predicted (e.g., Morgan et al. 2010).

However, we must note that this particular disk size discrepancy should not be seen as highly significant for two reasons. First, there are large systematic uncertainties in many of the input parameters in Equation (2). As mentioned earlier, the historically derived black hole mass of NGC 4151 ranges over a factor of ∼3 and even after correcting the Eddington ratio estimates to the same mass, those also range over an additional factor of ∼3. In addition, the radiative efficiency η and ratio of external to internal heating κ are also not well established in AGNs, in general, or NGC 4151 in particular. Thus the theoretically expected value of r is not very well determined for this object.

Second, the Swift data alone do not provide strong constraints on the observed light-crossing time t because of the limited wavelength range and the poor S/N, and the weak variability in v band in particular. A much stronger test of the UV/optical data's consistency with the thin-disk model will be possible using the simultaneous ground-based data, which go all the way to the z band (∼9000 Å), at much higher S/N. Those data will be presented and this test will be performed in a future paper (K. Horne et al. 2017, in preparation).

The observed τ ∝ λ^4/3 relation and ∼0.5–1 day lags within the UV/optical are consistent with the standard thin-disk picture (Shakura & Sunyaev 1973), albeit with large uncertainties. However, the full lag-wavelength relation shown in Figure 5, including the X-ray lags, indicates that these results are in fundamental disagreement with the standard lamp-post reprocessing model in which the small X-ray emitting corona directly illuminates and drives variations in the extended UV/optical-emitting accretion disk. The ∼3 day lag between the hard X-rays and UV and the ∼0.5–1 day lags within the UV/optical cannot be simultaneously explained in terms of direct illumination of the disk by the corona.

The UV/optical light curves also appear to be considerably smoother on short timescales than the X-ray light curves. This problem was noted by Gardner & Done (2017) with regards to the NGC 5548 data, where the UV/optical light curves similarly lack the high frequency power seen in the X-rays. However, the NGC 5548 X-ray S/N was much worse than for NGC 4151 (which in most wavelength bands is the brightest Type 1 AGN in the sky), which may be why the NGC 5548 data do not require a long X-ray/UV lag. In retrospect, this effect can be seen, to some degree, in earlier Swift monitoring, such as that of MR2251-178 (Arévalo et al. 2008) and NGC 2617 (Shappee et al. 2014). In all of these cases, it appears that there is no simple way to reconcile the relatively small lags of the UV/optical with the large X-ray/UV lags within the context of the direct reprocessing lamp-post model.

Gardner & Done (2017) developed a picture to explain the poor correlation between the X-ray and UV/optical variability in NGC 5548. That same picture also provides a natural explanation for the wavelength-lag structures seen in NGC 4151, in particular, the observed relatively long 3–4 day X-ray/UV lag and the smaller ≲1 day lags within the UV/optical. This is done by invoking an additional component that emits in the EUV and acts as an intermediary between the corona and disk. This putative EUV component offers an explanation for the origin of both the "big blue bump" and "soft X-ray excess" as low- and high-energy tails of a component that peaks in the intrinsically unobservable EUV spectral region.

The key to this picture is that instead of the X-ray corona directly illuminating the disk that then processes and re-emits the energy in the UV/optical, two separate reprocessings occur: first, the corona illuminates and heats the EUV component, which is smaller than the disk (thus much smaller than light days in size), so the first reprocessing must occur on a timescale longer than the light-crossing size of the EUV component. A sketch of how these components may be arranged physically is shown in Figure 6. Hence, the X-ray/UV lag indicates some slower physical process. This would introduce both a lag and smoothing between the X-ray and UV/optical bands, as has been observed. As discussed earlier, a lag time of ∼1.5–3 days would indicate a size of 23–36r_g for the inner and outer radii of the EUV torus if it was associated with the dynamical timescale. In this model, the heating would cause the torus to puff up in the vertical direction, so a dynamical timescale (the time required for the system to return to relax back in the vertical direction) would be naturally associated with this process.

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Then, in the standard thin-disk picture, a second reprocessing would occur when the EUV torus illuminates and heats the accretion disk on the light-crossing time, which then radiates the observed UV/optical radiation. An alternative scenario, envisioned in the Gardner & Done (2017) picture, is that the inner edge of the disk responds to an increase in heating, caused by increased illumination from the outer edge of the EUV torus, by expanding upward on the dynamical timescale. As a result, the inner disk radii are continually transitioning between a standard thin-disk state and a larger scale height state, which is more similar to that of the material in the EUV torus. This inward/outward pulsating of the EUV torus-standard disk boundary, in response to the X-ray heating of EUV torus inner edge, is potentially a cause of the UV/optical lags in the Gardner & Done (2017) picture.

We emphasize that at this early stage other alternatives are certainly possible. For instance, Korista & Goad (2001) find that the diffuse continuum from the BLR is expected to contribute to the measured UV/optical lag-wavelength relation, even broadly mimicking the increasing lag with wavelength behavior. However, the diffuse continuum from the BLR is unlikely to be the sole source of the observed continuum lag spectrum, and it probably cannot explain the mismatch in lags between the UV/optical and the X-ray continuum bands observed in this object and others.