# Intensive broadband reverberation mapping of Fairall 9 with 1.8 years of
daily Swift monitoring

R. Edelson Eureka Scientific, Inc., 2452 Delmer Street, Suite 100, Oakland, CA
94602, USA  email<EMAIL_ADDRESS>B. M. Peterson Retired  J. Gelbord
Spectral Sciences Inc., 30 Fourth Ave. Suite 2, Burlington, MA 01803, USA
Keith Horne SUPA Physics and Astronomy, University of St. Andrews, Fife, KY16
9SS, Scotland, UK  M. Goad  University of Leicester, Department of Physics and
Astronomy, Leicester, LE1 7RH, UK  I. McHardy School of Physics and Astronomy,
University of Southampton, Highfield, Southampton, SO17 1BJ, UK  S. Vaughan
University of Leicester, Department of Physics and Astronomy, Leicester, LE1
7RH, UK  M. Vestergaard Niels Bohr Institute, University of Copenhagen,
Jagtvej 155, 2200 Copenhagen N, Denmark  Steward Observatory, Dept. of
Astronomy, University of Arizona, 933 N. Cherry Ave, Tucson AZ 85721, USA

###### Abstract

We present 1.8 years of near-daily Swift monitoring of the bright, strongly
variable Type 1 AGN Fairall 9. Totaling 575 successful visits, this is the
largest such campaign reported to date. Variations within the UV/optical are
well-correlated, with longer wavelengths lagging shorter wavelengths in the
direction predicted by thin disk/lamp-post models. The correlations are
improved by “detrending;” subtracting a second-order polynomial fit to the
UV/optical light curves to remove long-term trends that are not of interest to
this study. Extensive testing indicates detrending with higher-order
polynomials removes too much intrinsic variability signal on reverberation
timescales. These data provide the clearest detection to date of interband
lags within the UV, indicating that neither emission from a large disk nor
diffuse continuum emission from the broad-line region can independently
explain the full observed lag spectrum. The observed X-ray flux variations are
poorly correlated with those in the UV/optical. Further, subdivision of the
data into four $\sim$160 day light curves shows that the UV/optical lag
spectrum is highly stable throughout the four periods, but the X-ray to UV
lags are unstable, significantly changing magnitude and even direction from
one period to the next. This indicates the X-ray to UV relationship is more
complex than predicted by the simple reprocessing model often adopted for AGN.
A “bowl” model (lamp-post irradiation and blackbody reprocessing on a disk
with a steep rim) fit suggests the disk thickens at a distance ($\sim$10 lt-
day) and temperature ($\sim 8000$K) consistent with the inner edge of the BLR.

galaxies: active – galaxies: nuclei – galaxies: Seyfert – galaxies: Individual
(Fairall 9)

††software: HEAsoft (v6.22.1; Arnaud 1996), FTOOLS (Blackburn 1995), sour
(Edelson et al. 2017)

##  1 Introduction

It is generally thought that essentially all massive galaxies contain a
supermassive black hole (SMBH) in their centers, that is either “active” or
“quiescent.” Active galactic nuclei (AGN) – which are the most powerful,
discrete, quasi-continuous sources of luminosity in the known universe – are
seen in a few percent of bright galaxies, while “quiescent” SMBH occur in the
vast majority of massive galaxies (Ferrarese & Merritt, 2000; Gebhardt et al.,
2000) including our own (Sgr A∗; Genzel et al. 1996; Ghez et al. 1998). AGN
show a highly luminous, hard ionizing source, with strong excesses in the UV
and X-rays, that is often more luminous than the entire underlying galaxy. AGN
also typically exhibit broad emission lines from gas excited by the hard
radiation field. Support for this dichotomy between quiescent and active
galactic nuclei comes from the identification of “changing look” AGN, cases in
which a nuclear continuum and broad emission lines emerge or disappear on
short timescales of months or years (e.g., Penston & Perez 1984; Denney et al.
2014; Shappee et al. 2014; Runnoe et al. 2016; LaMassa et al. 2017; MacLeod et
al. 2019; Raimundo et al. 2019). For this reason, it is crucial to understand
the physical conditions and processes that give rise to AGN activity driven by
SMBH.

A key feature of the active state of SMBH is that AGN are thought to be
accreting matter onto the SMBH at high rates. In the current paradigm of AGN,
the SMBH is immediately surrounded by a hot, relatively spherically symmetric
X-ray-emitting corona, with a larger, flattened disk of optically thick matter
accreting inward. This disk/corona system is thought to be where the bulk of
an AGN’s large luminosity is produced, ultimately by the liberation of
gravitational potential energy as matter heats up as it spirals in towards the
SMBH. The corona and inner disk, which are thought to produce the bulk of the
ionizing photons, are thought to have size scales of order 10 and 100
gravitational radii, respectively. Thus, for a $\sim 10^{7}M_{\odot}$ black
hole these sizes would be of order $\sim 0.01$ and 0.1 lt-day (Frank et al.,
2002). Outflows from the disk or blobs of accreting matter produce larger
regions of ionized gas: the broad-line region (BLR; generally light days to
light weeks in size) and narrow-line region (NLR; up to lt-yrs in size).

Unfortunately, with a few exceptions (e.g., 3C 273 [Gravity Collaboration et
al. 2018]; M 87 [Event Horizon Telescope Collaboration 2019]; NGC 1068
[Gravity Collaboration et al. 2020a], IRAS 09149$-$6206 [Gravity Collaboration
et al. 2020b], NGC 3783 [Gravity Collaboration et al. 2021]), the small sizes
of AGNs and their large physical distances from us mean that the angular sizes
of the central engines are too small to be imaged directly. Thus indirect
probes, such as source variability, provide some of the strongest constraints
on the compact central regions. In particular, reverberation mapping has
proven to be a particularly powerful technique (Blandford & McKee, 1982;
Peterson, 1993). Measured lags (or “light echoes”) of the variable driving
continuum in the spatially extended, variable BLR emission line gas have been
used to constrain the BLR size, structure and orientation, and to estimate the
SMBH mass of over 100 AGN (for recent compilations, see Bentz & Katz 2015 and
Dalla Bontà et al. 2020). Gravitational lensing (e.g., Morgan et al. 2010) can
also be used to study the structure of the central engines of more distant
AGN.

These basic reverberation mapping principles are now also being used to probe
the smaller but more physically and energetically important accretion
disk/corona region. The “reprocessing” model predicts strong correlations
among the continuum bands, with X-rays leading the UV and the UV leading the
optical. Although many early campaigns found good interband correlations
(e.g., Cackett et al. 2007), for over a decade, none had unambiguously
detected ($>3\sigma$) the expected interband lags. The breakthrough came with
the first AGN intensive broadband reverberation mapping (IBRM)111The previous
papers listed in this paragraph used a slightly different acronym, IDRM, for
Intensive Disk Reverberation Mapping. However this and other recent campaigns
show that the accretion disk is not the only, and in fact may not even be the
dominant cause of the observed broadband interband lags. Thus we believe this
name more aptly describes this technique. monitoring campaign in 2014, of the
target NGC 5548, combining Swift (Edelson et al., 2015), ground-based optical
(Fausnaugh et al., 2016) and HST (De Rosa et al., 2015) to blanket the
accessible X-ray, UV and optical wavelengths with unprecedented coverage in
both temporal and photon frequencies. Subsequent IBRM campaigns were done on
several targets (e.g., NGC 4151 [Edelson et al. 2017], NGC 4593 [McHardy et
al. 2018], Mrk 142 [Cackett et al. 2018], Mrk 509 [Edelson et al. 2019],
Fairall 9, [F9 hereafter; Hernandez Santisteban et al. 2020], Mrk 110
[Vincentelli et al. 2021], Mrk 817 [Kara et al. 2021], and Mrk 335 [Kara et
al. 2023]).

The common feature of all these IBRM campaigns is intensive (daily or sub-
daily) monitoring for a total of 200+ visits with Swift in the 0.3–10 keV
X-rays and six broadband UV/optical filters spanning $\sim 1900-5500$ Å,
simultaneous with ground-based robotic monitoring and often additional
observatories (e.g., HST for NGC 4593, Cackett et al. 2020). These campaigns
yielded three main observational results, which appear to be generally
consistent from object to object:

  1. 1.

The UV/optical variations are strongly correlated with lags $\tau$ increasing
with wavelength $\lambda$ consistent with $\tau\propto\lambda^{4/3}$, as
predicted by the standard thin disk model (Shakura & Sunyaev, 1973), although
the disk size appear to be larger than predicted by a factor of $\sim$2–3.

  2. 2.

The $U$ band (which contains the Balmer jump) shows an excess lag relative to
this trend, apparently due to diffuse continuum emission (DCE) from the broad-
line region.

  3. 3.

The correlation between X-ray and UV variations is much weaker than the
correlations among the UV/optical, casting doubt on the standard
“reprocessing” model that links the observed X-ray emission (thought to arise
in a hot corona immediately surrounding the SMBH) and the UV-emitting
accretion disk.

These three observational results challenge various aspects of the current
standard model of AGN central engines, and thus have spurred a tremendous
amount of theoretical work. Perhaps the first such theoretical interpretation
of IBRM data was Gardner & Done (2017), which utilized a far-UV(FUV)-emitting
Comptonized reprocessor that lies between the X-ray emitting corona and an
accretion disk with a truncated inner edge (at $\sim 70R_{g}$, where
$R_{g}=G\,M/c^{2}$ is the gravitational radius) to explain the observed weak
X-ray to UV correlation in the original NGC 5548 campaign. A similar approach
by Mahmoud & Done (2020) found that the NGC 4151 IBRM data again required a
Comptonized reprocessor, with a highly truncated accretion disk or possibly no
disk at all. This methodology was also applied by Hagen & Done (2023) and
Hagen et al. (2024) to the initial F9 dataset of Hernandez Santisteban et al.
(2020), finding further evidence of a second FUV component and a truncated
disk.

Alternatively, Kammoun et al. (2019, 2021a, 2021b) explained these and other
IBRM data with a more conventional geometry, albeit with a hotter disk, and
larger lamp-post height . Panagiotou et al. (2022a, b) expanded on these
results by performing the analysis in the temporal frequency domain and
simultaneously measuring the power spectral density functions, yielding
similar results.

Additionally some theoretical studies focused only on the first two points,
based on UV/optical data only. Neustadt & Kochanek (2022) models the
UV/optical light curves of seven AGN IRBM campaigns by assuming a standard
Shakura & Sunyaev (1973) disk and a lamp-post illuminator. They found that the
fluctuations consisted of both ingoing waves, along with outgoing waves as
would be expected from the standard reprocessing model. Netzer (2022) analyzed
UV/optical IBRM light curves of six AGN, concluding that they could be
explained by a combination of BLR emission and a standard-sized Shakura &
Sunyaev (1973) disk. Clearly the IBRM technique, in use for less than a
decade, has motivated a wide variety of theoretical models. This in turn
underlines the need for even more constraining IBRM data.

As noted above, Hernandez Santisteban et al. (2020) reported the results of
IBRM monitoring of F9 ($z=0.047$, Neeleman et al. 2016) over a $\sim$9 month
period. It is worth explicitly noting that a number of studies have already
successfully modeled this initial dataset (e.g. Netzer, 2022; Kammoun et al.,
2023; Hagen & Done, 2023; Hagen et al., 2024). In fact this target was
observed by Swift on an approximately daily basis during the period MJD
58251.50–58900.50, a total of $\sim$1.8 yr. This has yielded the longest Swift
AGN IBRM dataset that has been gathered to date, with at least twice the time
span and nearly twice the total visits of any AGN. This longer dataset is
reported in the current paper. Additional monitoring was obtained, i.e.,
ground-based robotic optical photometry and spectroscopy and NICER X-ray
spectroscopy; those results will be reported elsewhere.

The intensive F9 monitoring presented herein is by far the largest Swift AGN
IBRM dataset ever gathered (although a current campaign, on Mrk 817 (Cackett
et al., 2023), is poised to eventually surpass it). This allows analyses that
were not previously feasible. With such a large, homogeneous dataset, we can
divide the full 1.8 year light curve into subsets that are equivalent in size
to individual past IBRM datasets and compare the interband lag spectra at
different times and when the target is in different flux states. An analogous
situation exists in emission-line reverberation studies: NGC 5548 has been
subject to nearly 20 reverberation campaigns, enabling study of the line
response at different times and under a variety of conditions (see Peterson et
al. 2013; Pei et al. 2017; De Rosa et al. 2018. and associated references and
citations).

As the main goal of this paper is to present this new dataset to the
community, only a small amount of model testing is done herein. Instead it is
hoped that the aforementioned authors (and others) will utilize these data to
test and refine their models, and perhaps to develop entirely new ones. This
paper is organized as follows: Section 2 summarizes the observations and data
reduction, Section 3 presents the timing analysis, Section 4 discusses
theoretical implications of these results, and Section 5 gives some brief
concluding remarks.

##  2 Observations and Data Reduction

###  2.1 Observations

F9 was the subject of daily IBRM monitoring for 1.8 years, covering the period
MJD 58251–58901. Data from two of Swift’s three instruments are used in this
paper: the X-Ray Telescope (XRT, Burrows et al., 2005) and the
UltraViolet/Optical Telescope (UVOT, Roming et al., 2005). All XRT
observations in this campaign were made in photon counting (PC) mode. All UVOT
observations were made using 6-filter modes. Initially the workhorse mode
0x30ed was employed, but observations were switched to 0x224c once this mode
became available (from MJD 58594 onwards). This new UVOT mode is specifically
designed for IBRM campaigns, with extra weight given to exposures in the
extremal filters $W2$ and $V$. In particular the fraction of time on target
assigned to $V$ band is increased by a factor of $\sim$3, contributing to
clear improvements in the signal/noise ratio of the $V$ light curve (see
Section 2.2) and $W2$–$V$ lag constraints in the latter half of the campaign.

Additional multi-wavelength monitoring was obtained with the NICER X-ray
mission and the ground-based robotic Las Cumbres Observatory global telescope
network, operating at optical wavelengths. Furthermore Swift collected
additional data at lower cadence (once every $\sim$4 days) during the period
MJD 58902–59230. None of these data are analyzed in this paper because here we
focus on the analysis of a homogeneous, relatively uninterrupted set of Swift
light curves. These other datasets will be the subject of future papers.

###  2.2 UVOT data reduction

This paper’s UVOT data reduction follows the general procedure described by
Hernandez Santisteban et al. (2020). This process has four steps: definition
of the observing wavebands, flux measurement, removal of data points that fail
quality checks, and identification and masking of low sensitivity regions of
the detector. Each step is described in turn below.

In this work the band centers are defined (Koornneef et al., 1986) as the
“pivot wavelength,”

$\lambda_{p}(P)=\sqrt{{\int P(\lambda)\,\lambda\,d\lambda}\over{\int
P(\lambda)\,d\lambda/\lambda}}\ ,$ (1)

where $P(\lambda)$ is the passband transmission curve. The colored dashed
curves in Fig. 1 show $P(\lambda)$ for each band. The associated rms bandwidth
is

$\lambda_{rms}=\sqrt{{\int(\lambda-\lambda_{p})^{2}\,P(\lambda)\,d\lambda}\over{\int
P(\lambda)\,d\lambda}}\ .$ (2)

This source-independent parameterization has the advantage that it allows an
exact conversion between broadband flux densities $f_{\nu}$ and $f_{\lambda}$.
Indeed using the pivot point as reference and normalization makes the
parameters of any 2-parameter linear regression independent, and thus gives a
diagonal Hessian matrix which is trivial to invert. These quantities are
tabulated in Table 1 and the UVOT passbands and a spectrum of F9 are shown
Fig. 1.

Figure 1: UV/optical spectrum of F9 created by combining HST FOS spectra
obtained on 1993 January 21 with a spectrum from the ESO 1.5-m telescope taken
on 1994 July 14 (Santos-Lleó et al., 1997). Spectra are scaled to match in
overlapping wavelength regions as the ground-based data were obtained under
non-photometric conditions. The colored, dashed lines show the passband
transmission curves of each Swift UVOT band. Note that the names “W2,” “M2,”
and “W1” are shorthand for “UVW2,” “UVM2,” and “UVW1”, as used throughout this
paper. Table 1: Swift Observing bands and basic variability information

(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
---|---|---|---|---|---|---|---|---
Band/ | Band | Band |  |  | Mean Sampling | Mean |  |
Filter | Center | Range | Unit | $N$ | Rate (days) | Flux | $F_{var}$ | Instrument
$HX$ | 3.9 | 1.5–10 | kev | 575 | 1.13 | 0.98 | 0.20 | XRT
$SX$ | 0.7 | 0.3–1.5 | kev | 575 | 1.13 | 0.49 | 0.15 | XRT
$W2$ | 2055 | 1596–2261 | Å | 479 | 1.35 | 4.89 | 0.25 | UVOT
$M2$ | 2246 | 2018–2474 | Å | 443 | 1.47 | 4.20 | 0.24 | UVOT
$W1$ | 2580 | 2134–3066 | Å | 442 | 1.47 | 3.47 | 0.22 | UVOT
$U$ | 3463 | 3192–3739 | Å | 459 | 1.41 | 2.41 | 0.20 | UVOT
$B$ | 4350 | 4055–4729 | Å | 512 | 1.27 | 1.40 | 0.17 | UVOT
$V$ | 5425 | 5202–5734 | Å | 534 | 1.22 | 1.11 | 0.13 | UVOT

Column 9: Swift instrument (XRT or UVOT).

Note. — Column 1: Filter/band name used in this paper. Column 2: Band center.
Column 3: Band range. Column 4: The physical unit for the quantity in Columns
2 and 3. Column 5: Number of good data points in each band ($N$), after
removal of dropouts and other censored data. Column 6: Mean sampling rate for
each band, after removal of bad data. Column 7: Mean flux, in units of
$10^{-14}$ erg cm-2 s-1 Å-1 for UVOT data and ct s-1 for XRT data. Column 8:
Fractional variability ($F_{\rm var}$) Vaughan et al. (2003).

In the next step, we reprocess all data for uniformity (using version 6.28 of
HEASOFT) and refined their astrometry (following the procedures of Edelson et
al., 2015) before measuring fluxes using UVOTSOURCE from the
FTOOLS222http://heasarc.gsfc.nasa.gov/ftools/ package (Blackburn, 1995).
Details of this instrument are given e.g., by Poole et al. (2008). The F9
nucleus and numerous selected field stars are measured, the latter in order to
allow various checks. Source photometry is measured in circular extraction
regions of 5″ radius, while background fluxes were taken from concentric
40″–90″ annuli. In the $V$ band in particular, the galaxy contributes
significantly to the reported source flux (the background annulus lies beyond
the host galaxy). The final flux values are corrected for aperture losses,
coincidence losses, large-scale variations in the detector sensitivity across
the image plane, and declining sensitivity of the instrument over time. These
corrections include an updated model for the time dependence of UVOT
sensitivity, first introduced in Sept. 2020, which causes the fluxes reported
here to differ systematically from those of Hernandez Santisteban et al.
(2020) and earlier papers. For details, refer to the calibration document
SWIFT-UVOT-CALDB-15-06
(https://heasarc.gsfc.nasa.gov/docs/heasarc/caldb/swift/docs/uvot/uvotcaldb_throughput_06.pdf).
In general the fluxes reported herein tend to be lower, differing by a few
percent from those in Hernandez Santisteban et al. (2020), with larger
disparities of $\sim$5–8% in the M2 and W1 filters. These calibration changes
should not strongly affect the conclusions of Hernandez Santisteban et al.
(2020) because the systematic light curve shifts are negated by the detrending
process.

In the third step, the resulting measurements are used for both automated
quality checks and to flag individual observations for visual inspection.
These automated checks include aperture ratio screenings to catch instances of
extended point-spread functions (PSFs) or when the astrometric solution is
off, and that the exposure time was a minimum of 20 s. Data are flagged for
inspection when the fitted PSFs of either the AGN or several field stars are
found to be unusually large or asymmetric, or if fewer than 10 field stars
with robust centroid positions are available for astrometric refinement. Upon
inspection, observations are rejected if there are obvious astrometric errors,
doubled or distorted PSFs, or prominent image artifacts (e.g., readout streaks
or scattered light) that could affect the AGN measurement. Note that we adopt
a non-standard setting of 7.5% for the UVOTSOURCE parameter FWHMSIG because
this yields flux uncertainties more consistent with the rms point-to-point
scatter in the data (Edelson et al., 2017).

Edelson et al. (2015) first noted that $\sim 10-15$% of Swift UVOT data
suffered from “dropouts,” anomalously low fluxes, especially in the three UV
bands, which occurs when the target falls on low-sensitivity regions in the
UVOT detector. That work, and subsequent IBRM campaigns, used a bootstrap
method to identify and filter out such anomalous data. Hernandez Santisteban
et al. (2020) instead utilized the daily Swift monitoring of SgrA* to derive a
much more comprehensive and stable map of the entire UVOT detector for this
purpose, and applied it to identification and filtering of UVOT dropouts. That
improved technique was applied to filter out UVOT dropouts in the current
dataset as well.

###  2.3 XRT data reduction

The XRT data reduction is essentially the same as described by Edelson et al.
(2017) so it will only be briefly summarized here. We utilize the standard
Swift analysis tools333http://www.swift.ac.uk/user_objects described by Evans
et al. (2009) to produce light curves that are fully corrected for
instrumental effects such as pile up, dead regions on the CCD, and vignetting.
The source aperture varies dynamically according to the source brightness and
position on the detector.

We extract the observation times (the midpoint between the start and end
times) in MJD for ease of comparison with the UVOT data. We utilize “snapshot”
binning, which produces one temporal bin for each continuous spacecraft
pointing. This is done because these short visits always occur completely
within one orbit with one set of corresponding exposures in the UVOT filters.
We generate X-ray light curves covering two bands, the standard soft (0.3–1.5
keV) and hard ( 1.5–10 keV) bands, referred to as $SX$ and $HX$, respectively.
For a detailed discussion of this tool and the default parameter values, see
Evans et al. (2009).

###  2.4 Light Curves

The resulting light curves are shown in Fig. 2 and tabulated in Table 2. The
XRT and UVOT observing bands are tabulated in Table 1.

Table 2: Swift F9 IBRM Campaign Data

(1) |  (2) |  (3) |  (4)
---|---|---|---
Filter | MJD | Flux | Error
$W1$ | 58251.6563 | 2.970 | 0.062
$U$ | 58251.6570 | 2.096 | 0.052
$B$ | 58251.6574 | 1.175 | 0.031

Note. — Column 1: Filter/band used to measure the broadband flux and error.
Column 2: Modified Julian Day at the midpoint of the exposure. Column 3: Mean
flux of the data point. UVOT fluxes are given in units of $10^{-14}$ erg cm-2
s-1 Å-1 and X-ray fluxes in units of ct s-1. Column 4: Uncertainty on the
flux, in the same units as Column 3. Only a portion of this table is shown
here to demonstrate its form and content. A machine-readable version of the
full table is available online.

Figure 2: Left: light curves for the intensive Swift monitoring campaign.
Units are the same as in Table 2. The vertical grey line in the bottom
($V$-band) plot at MJD 58594 denotes the switch to UVOT mode 0x224c, resulting
in a considerable improvement in $S/N$ in this band. The time intervals used
in this paper are shown at the top: the “Full” analysis period incorporates
all data shown and the four quarterly periods (Q1…Q4) are delimited by dashed
lines. As discussed in Section 3.3, the X-ray data were not detrended and
UV/optical were detrended by subtracting a second-order polynomial, shown as a
black line. Right: CCF (black lines) and probability density functions
(colored histograms) as a function of time delay (lag) for the full period.
The observing bands are shown in the same colors and order as the light
curves. Grey vertical lines mark a zero-day lag. All CCFs are measured
relative to the ultraviolet $W2$ band. The median lag estimates are shown as a
solid vertical black line and the 68% confidence interval as dashed vertical
black lines.

##  3 Analysis

###  3.1 CCF Methodology

Cross-correlation functions (CCFs) were constructed using the interpolated
cross-correlation function method (Gaskell & Peterson, 1987). We used the sour
code444This code is available at https://github.com/svdataman/sour, which is
based on the specific implementation presented by Peterson et al. (1998) and
modified by Peterson et al. (2004). In computing the cross-correlation
functions, the mean and standard deviations for each lag value were derived
“locally,” i.e., using only the portions of the light curves that overlap in
time for a given test lag. Because this time window is fixed the number of
overlapping points will change by a small amount between different lags. As
noted in point 2 below, we limit the half-widths of the lag searches to
$\sim$1/3 of the total light curve duration, meaning that at least $\sim$1/3
of the data points do not participate in the measurement of $r$ at any $\tau$.
As discussed by White & Peterson (1994), local rather than global
normalization is appropriate for non-stationary AGN light curves.

Due to the uneven sampling of the light curves, we use interpolation to
perform the CCF analysis. The CCF is computed in both directions, once with
the reference light curve leading and once with the subsidiary light curve
leading. The $W2$ light curve is always the reference and the other bands are
considered to be the subsidiary bands in this analysis. W2 band is chosen
because it has the shortest UV wavelength and thus is closest to the thermal
peak of the accretion disk, and because it shows the strongest fractional
variability of all the UV/optical bands (see Table 1).

We then use the “flux randomization/random subset selection” (FR/RSS) method
(Peterson et al., 1998, 2004) to estimate uncertainties on the measured lags.
This is a Monte Carlo technique in which lags are measured from multiple
realisations of the CCF. The FR aspect of this technique perturbs in a given
realization each flux point by a random Gaussian deviate whose width is based
on the quoted uncertainty associated with that data point. 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 $n_{\mathrm{rep}}$
is the number of repeated points. Typically a fraction of
$(1-[1/N])^{N}\rightarrow 1/e$ of data points are not selected for each RSS
realization. In this paper, the FR/RSS is applied to both the reference and
subsidiary light curves in each CCF pair. The CCF is then measured and a lag
determined to be the weighted mean of all points with
$r>0.8\,r_{\mathrm{max}}$, where $r_{\mathrm{max}}$ is the maximum value
obtained for the correlation coefficient $r$ (e.g., Edelson et al., 2019).
Implementation of the FR/RSS version of the interpolation-based cross-
correlation method requires specifying a number of parameters:

  1. 1.

The interpolation step size. This should be smaller than the typical spacing
between data points in the light curves. However, using too fine a step size
results in artifacts on the shortest timescales because points midway between
real data points are highly correlated with the adjacent interpolated points,
thus producing artificially high local maxima in the CCF on timescales of half
the step size. We find that a step size of 0.2 days works well for these data.

  2. 2.

The range of lags to be explored. As the light curves are shifted away from
one another, an increasing number of points at the ends of a light curve has
no counterpart in the other time-shifted spectrum. Thus the number of points
that contribute to the CCF computation decreases with increasing lag. To avoid
edge effects resulting from a smaller data sample, we set the lag range to a
value around 1/3 the length of the light curve being analyzed. This yields a
lag range of 216 days for the “full” sample analyzed in Section 3.2, and 54
days for the “quarterly” dataset analyzed in Section 3.4.

  3. 3.

The number of Monte Carlo realizations should be large enough that the
resulting probability density functions, or cross-correlation centroid
distributions (CCCDs; e.g., the shaded regions on the right side of Fig. 2)
should be smooth, with all features due to the data/method themselves, and not
to small-number statistics. For these data, we find that 10,000 realizations
met this criterion.

###  3.2 CCF Analysis of the Full Dataset

We first consider a cross-correlation analysis of light curves from the full-
period data, from MJD 58251 through MJD 58900. The CCFs and CCCDs, measured
relative to the Swift $W2$ band, are shown for all the bands in the right-hand
column of Fig. 2, with the corresponding light curves in the left-hand panel.
The CCFs are shown as solid curves and the CCCDs, from the FR/RSS analysis are
shown as colored histograms. Correlated variations are evident in the light
curves, and the response of each band is highly localized to a several-day
window near zero. Table 3 presents the measured interband lags and 68%
confidence limits from the CCCDs.

The UV/optical bands are all highly correlated, with $r_{max}\geq 0.95$, and
the UV/optical lags are well-enough measured to reveal a clear increase with
wavelength. In contrast, the X-ray bands are only weakly correlated with $W2$,
with $r_{max}<0.7$, and the X-ray lags have much larger uncertainties. While
the general shape of the UV/optical variations is evident in the X-ray light
curves, the faster X-ray variations are absent or much weaker in the
UV/optical light curves. The flare near MJD 58440 is stronger in the X-rays
than in the UV/optical. Similar differences in the X-ray vs UV/optical
behavior have been seen in all previous AGN IBRM campaigns, suggesting that
the physical process(es) dominating the UV/optical variations are not simply
related to (or driven by) the X-ray variability.

Table 3: Full Period Interband Lags for the Zeroth and Second Orders

| No detrending (Order = 0) |  | Parabolic detrending (Order = 2)
---|---|---|---
Band | $r_{\rm max}$ | $\tau$ | $\langle\sigma\rangle$ | $S$ |  | $r_{\rm max}$ | $\tau$ | $\langle\sigma\rangle$ | $S$
(1) | (2) | (3) | (4) | (5) |  | (6) | (7) | (8) | (9)
$HX$ | 0.67 | $1.80^{+1.91}_{-2.07}$ | $1.99$ | $0.90$ |  | 0.62 | $-4.20^{+1.96}_{-1.84}$ | $1.90$ | $-2.21$
$SX$ | 0.42 | $12.74^{+5.37}_{-3.84}$ | $4.61$ | $2.76$ |  | 0.42 | $0.63^{+3.79}_{-3.51}$ | $3.65$ | $0.17$
$W2$ | 1.00 | $0.01^{+0.75}_{-0.78}$ | $0.76$ | $0.01$ |  | 1.00 | $0.00^{+0.46}_{-0.45}$ | $0.46$ | $-0.01$
$M2$ | 0.99 | $1.60^{+0.86}_{-0.92}$ | $0.89$ | $1.81$ |  | 0.99 | $0.98^{+0.54}_{-0.50}$ | $0.52$ | $1.89$
$W1$ | 0.99 | $1.87^{+1.06}_{-0.97}$ | $1.01$ | $1.84$ |  | 0.99 | $1.57^{+0.55}_{-0.56}$ | $0.55$ | $2.83$
$U$ | 0.97 | $4.82^{+1.48}_{-1.48}$ | $1.48$ | $3.26$ |  | 0.98 | $3.18^{+0.62}_{-0.61}$ | $0.61$ | $5.20$
$B$ | 0.97 | $3.10^{+1.54}_{-1.52}$ | $1.53$ | $2.02$ |  | 0.97 | $3.55^{+0.63}_{-0.65}$ | $0.64$ | $5.52$
$V$ | 0.95 | $1.78^{+2.28}_{-2.43}$ | $2.35$ | $0.76$ |  | 0.95 | $4.01^{+1.01}_{-0.93}$ | $0.97$ | $4.14$

Note. — Column 1: band/filter name. Column 2: the maximum value ($r_{\rm
max}$) of the cross-correlation coefficient $r(\tau)$. Column 3: the median
lag ($\tau$) with the asymmetric 68% confidence interval ($\sigma_{\rm upper}$
and $\sigma_{\rm lower}$) from the CCCDs shown in Fig. 2. Column 4: average
uncertainty on the median lag, $\langle\sigma\rangle=\left[\left(\sigma_{\rm
upper}^{2}+\sigma_{\rm lower}^{2}\right)/2\right]^{1/2}$. Column 5: the
significance, $S\equiv\tau/\langle\sigma\rangle$. Columns 6–9 are the same as
Columns 2–5, but for the Order 2 detrended light curve. The first two rows
give results for hard and soft X-ray (XRT) and the last six for UV/optical
(UVOT) filters. The third row ($W2$) is for the autocorrelation instead of
cross-correlation.

###  3.3 Detrending the UV/optical light curves

Examination of Fig. 2 suggests that while the UV/optical light curves are
dominated by variations on long timescales, the X-rays are dominated by more
rapid variations. To quantify this, we construct a simplified power spectral
density (PSD) function by first resampling these unevenly sampled data to a
regular grid, computing the periodogram by discrete Fourier transform (DFT),
and then fitting a simple power law ($P(f)\propto f^{\alpha}$) plus noise
model. This confirms that the UV/optical power spectra are quite steep, with
power-law indices in the range $-2.0\leq\alpha\leq-2.6$. Similar behavior is
seen in other AGN, especially in observations made with Kepler, which has
orders of magnitude faster time sampling. In the UV/optical, AGN have steep
“red” power spectra with values of the spectral index in the range $-2$ to
$-4$ (e.g., Edelson et al. 2014; Vaughan et al. 2016; Aranzana et al. 2018).
The X-ray power spectra, however, are much flatter, with
$-1.0\leq\alpha\leq-1.2$. Note that a more detailed analysis that operates in
the time domain would avoid the resampling and DFT steps; that will follow in
a future contribution.

In the case of the UV/optical light curves, the fact that so much of the
variability power is on timescales much longer than the reverberation
timescales we are investigating here could compromise accurate lag
measurement. In order to reduce the risk of bias, many previous investigations
sought to reduce the low-frequency power by “detrending” the light curves
(Welsh, 1999), i.e., subtracting relatively slow variability signals in order
to isolate or enhance the rapid variability that is most suited to measuring
short lags (e.g., Denney et al. 2010; Li et al. 2013; Peterson et al. 2014;
McHardy et al. 2014; Li et al. 2018; McHardy et al. 2018; Zhang et al. 2019;
Raiteri et al. 2021; Rakshit 2020; Hernandez Santisteban et al. 2020; Li et
al. 2022; Beard et al. 2023).

There is, however, no standard prescription for how to detrend light curves;
detrending is generally performed on an ad hoc basis, which may sometimes lead
to controversial, or in the worst case, contradictory results. As an example,
consider the case of NGC 7469 which was observed with the Rossi X-Ray Timing
Explorer (RXTE) in the 2–10 keV range and with the International Ultraviolet
Explorer (IUE) in the 1150–1975 Å range in 1996. This was the most intensive
AGN monitoring program that predates Swift, with nearly continuous monitoring
for a month. Initial CCF analysis with no detrending found that the X-rays
lagged the 1315 Å continuum by $\sim$4 days (Nandra et al., 1998, 2000), while
reanalysis by Pahari et al. (2020) showed that the heavily detrended X-rays
led the UV by $\sim$0.3 days. It is not desirable that contradictory results
be obtained from the same dataset based solely on details of how a user
prepares the data for analysis. Thus we have constructed a methodology to
perform an “optimal” detrending without allowing the user to set the input
parameters (and thus to determine the resultant interband lags).

The high quality of the F9 light curves allows a systematic investigation of
how to optimally set detrending parameters, although once the methodology is
constructed it can be applied to any IBRM dataset. We first explore detrending
the UV/optical data by subtracting a polynomial of the form

$F(t)=\sum_{i=0}^{N}a_{i}\,(t-t_{0})^{i},$ (3)

where $N$ is the order of the polynomial, $t_{0}$ is the approximate midpoint
of the observations (MJD 58576) and $F(t)$ is treated as the “background” flux
at time $t$, and $a_{i}$ are the best-fit coefficients. Equation (3) was
fitted to each of the UV/optical light curves by linear regression, minimizing
the unweighted sum of the squared residuals. The resulting fits are subtracted
from the original light curves and the cross-correlation analysis is
performed. A wide range of polynomial orders is used: 0–8, 10, 12, 14, and 16.
(Order 0 is the case where no detrending is performed.) Note that detrending
is performed only on the steep PSD UV/optical light curves.

The resulting CCFs, FR/RSS histograms and 68% confidence intervals are shown
in Fig. 3. We define a figure of merit to be the “significance” of the lag
measurement: $S\equiv\tau/\sigma$, e.g., the ratio of Columns (3) to (4), (6)
to (7), and (11) to (12) in Table 3. The significance $S$ is tabulated in
Table 4 for all bands and for light curves detrended with polynomials of order
0 through 16. The last column of this table shows the average of the five
previous columns, that is, the UV/optical CCFs, excluding the $W2$ ACF (which
will always be close to zero) and the X-ray CCFs, which show much weaker
correlations. Note that the average value of the figure of merit $S$ is
highest for Orders 1–3, with the highest value for Order 2.

The results summarised in Table 4 indicate that some detrending does clearly
improve the significance of the lag detections, as the Order 0 (no detrending)
values are quite low. However, note also that the significance then declines
for Order 3 and higher. Examination of Fig. 3 shows why: the lag uncertainties
($\sigma$) become marginally smaller at higher orders, but the lags ($\tau$)
get smaller even faster. This is because overly aggressive detrending removes
not only an unrelated confusing signal but also part of the actual lag signal
we are trying to measure. This underscores the importance of being very
conservative about performing detrending with anything higher order than, say,
a second-order (parabolic) function, or similarly, using a smoothing function
with a kernal significantly smaller than the total duration of the experiment.
Based on these results, we utilize a detrending polynomial of order 2 for the
UVOT data throughout the rest of this paper.

Note that Hernandez Santisteban et al. (2020) studied the first $\sim$10
months of these dataset, including ground-based LCO uBgVriz data, through MJD
58550 (see Fig. 1 of that paper). Those data are dominated by a single low
temporal frequency parabolic feature covering the full duration studied in
that paper. They detrended this feature and measured interband correlations
within it, finding the opposite lag dependence than in the rapid variations
(longer wavelength leading shorter wavelengths). Unsurprisingly, this parabola
did not persist into the current analysis, as it has often been noted that the
low temporal frequency features seen in red-noise AGN light curves should be
viewed with skepticism (see e.g. Press 1978 and subsequent works).

Table 4: FR/RSS significance for each band/polynomial order

(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
---|---|---|---|---|---|---|---|---|---
Order | $HX$ | $SX$ | $W2$ | $M2$ | $W1$ | $U$ | $B$ | $V$ | Mean
0 | 0.9 | 2.76 | 0.01 | 1.81 | 1.84 | 3.26 | 2.02 | 0.76 | 1.62
1 | $-0.46$ | 1.44 | 0.00 | 1.75 | 2.78 | 4.51 | 4.95 | 3.07 | 2.84
2 | $-2.21$ | 0.17 | $-0.01$ | 1.89 | 2.83 | 5.2 | 5.52 | 4.14 | 3.26
3 | $-0.01$ | 1.06 | 0.00 | 1.58 | 2.85 | 3.83 | 4.84 | 3.92 | 2.84
4 | $-1.53$ | 1.42 | 0.00 | 1.35 | 1.8 | 3.52 | 4.34 | 3.64 | 2.44
5 | $-1.41$ | 1.74 | 0.00 | 0.82 | 0.87 | 2.78 | 3.56 | 3.57 | 1.93
6 | $-0.66$ | 0.71 | 0.00 | $-0.03$ | 0.99 | 3.01 | 3.08 | 2.89 | 1.66
7 | $-0.72$ | 0.26 | 0.00 | $-0.12$ | 0.86 | 3.14 | 3.06 | 2.91 | 1.64
8 | $-1.14$ | 0.14 | 0.01 | $-0.3$ | 0.85 | 3.31 | 3.3 | 2.95 | 1.68
10 | $-0.66$ | 0.11 | 0.00 | $-0.4$ | 0.82 | 2.55 | 3.43 | 2.82 | 1.54
12 | $-0.49$ | 0.04 | 0.00 | $-0.64$ | 0.52 | 1.98 | 3.14 | 2.7 | 1.28
14 | $-0.72$ | 0.04 | 0.00 | $-0.6$ | 0.54 | 2.08 | 3.26 | 2.68 | 1.33
16 | $-0.73$ | 0.35 | 0.00 | $-0.31$ | 0.32 | 1.5 | 2.1 | 2.19 | 0.97

Note. — Column 1: order of the polynomal fit used for detrending. Order zero
means no detrending was performed Columns 2–9: The figure of merit
($\tau/\sigma)$ for each CCF between the listed band and $W2$. Column 4 refers
to the $W2$ band autocorrelation function. Column 10: average of Columns 5–9,
that is, all of the UVOT data except the $W2/W2$ ACF.

Figure 3: Cross-correlation functions (CCFs, black curves) and cross-
correlation centroid distributions (CCCDs, colored histograms) as a function
of lag for the full period, as in the right-hand column of Fig. 2. All CCFs
are measured relative to the Swift $W2$ band. The median lag estimates are
shown as a solid vertical black line and the 68% confidence interval as dashed
vertical black lines. Each row shows the CCFs and CCCDs for all bands, with
the UV/optical light curves detrended by a polynomial of order $N$ (Equation
3) indicated in the left-hand legend. Note the different x-axis scales:
$\pm$30 days for the X-ray bands on the left and $\pm$7 days for the
UV/optical bands on the right.

###  3.4 Quarterly Analysis

In this section, we perform a novel analysis to compare the stability of lags
measured between the UV and X-rays with those measured within the UV/optical.
The $\sim$650 day campaign is subdivided into four equal $\sim$162 day
periods, denoted as Q1 through Q4, as listed in Table 5. Each of these show
sufficient variability to measure interband correlations and lags (see Fig.
2). For the light curves in each quarter, interband lags are measured as
described in Section 3.1. The resulting CCFs are shown in Fig. 4 and listed in
Table 5. The results are stacked by quarter to facilitate comparison. The
right-hand part of Fig. 4 shows that the lags within the UV/optical are all
well-behaved: for each band the medians measured in all four quarters agree to
within 1$\sigma$, typically with $r_{max}>0.9$. This indicates that it is safe
to detrend the light curves when measuring lags within the UV/optical.

The left-hand side of Fig. 4 shows that the lags between the reference $W2$
band and the X-rays are not consistent; from quarter to quarter they do not
agree and they are also generally inconsistent with the UV/optical lags. It
has often been noted (e.g. McHardy et al., 2018; Edelson et al., 2019) that
the UV/optical lag spectra of IBRM AGN generally follow the same
($\tau\propto\lambda^{4/3}$) relation but that the UV/X-rays correlations and
lags are inconsistent, but this is the first time this relation has been
documented in the same object. This result also supports the decision in the
previous section not to detrend the X-ray light curves.

Figure 4:  Plot of CCF (black lines) and probability density functions
(colored histograms, corresponding to the colors of the light curves in Fig.
2) as a function of lag for the four quarterly periods. The first quarter is
on top and the last on the bottom. Each row shows the CCFs for a different
quarter, measured relative to the ultraviolet $W2$ band. The median lag
estimates are shown as a solid vertical black line and the 68% confidence
interval as dotted vertical black lines. Note the different x-axis scales:
$\pm$30 days for the HX/SX bands and $\pm$7 days for the six UVOT bands.
Table 5: Interband Lags for Quarterly Periods.

(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13)
---|---|---|---|---|---|---|---|---|---|---|---|---
Band/ | Q1 Lags (days) | Q2 Lags (days) | Q3 Lags (days) | Q4 Lags (days)
Filter | $r_{\rm max}$ | Median | Conf. Int. | $r_{\rm max}$ | Median | Conf. Int. | $r_{\rm max}$ | Median | Conf. Int. | $r_{\rm max}$ | Median | Conf. Int.
$HX$ | 0.49 | $-10.8$ | $(-13.5,-3.3)$ | 0.47 | $7.8$ | $(2.9,13.4)$ | 0.42 | $0.1$ | $(-1.5,0.9)$ | 0.43 | $-2.2$ | $(-3.6,-1.2)$
$SX$ | 0.49 | $-14.3$ | $(-16.3,-12.5)$ | 0.46 | $22.8$ | $(15.6,27.6)$ | 0.27 | $2.6$ | $(-6.5,9.3)$ | 0.25 | $0.4$ | $(-29.3,17.0)$
$W2$ | 1.00 | $0.0$ | $(-0.4,0.4)$ | 1.00 | $0.0$ | $(-0.5,0.5)$ | 1.00 | $0.0$ | $(-0.4,0.4)$ | 1.00 | $0.0$ | $(-0.4,0.3)$
$M2$ | 0.96 | $0.3$ | $(-0.1,0.7)$ | 0.97 | $-0.3$ | $(-0.8,0.2)$ | 0.96 | $-0.2$ | $(-0.7,0.4)$ | 0.87 | $-0.3$ | $(-0.9,0.4)$
$W1$ | 0.94 | $0.0$ | $(-0.5,0.5)$ | 0.96 | $0.0$ | $(-0.5,0.6)$ | 0.96 | $0.4$ | $(-0.3,1.0)$ | 0.89 | $0.1$ | $(-0.4,0.6)$
$U$ | 0.89 | $2.1$ | $(1.0,3.1)$ | 0.89 | $1.4$ | $(0.7,2.3)$ | 0.89 | $0.1$ | $(-0.7,0.9)$ | 0.78 | $0.5$ | $(-0.1,1.1)$
$B$ | 0.87 | $0.9$ | $(0.1,1.7)$ | 0.84 | $3.5$ | $(2.3,4.7)$ | 0.89 | $0.3$ | $(-0.4,1.0)$ | 0.73 | $0.8$ | $(-0.1,1.9)$
$V$ | 0.69 | $0.9$ | $(-0.5,2.9)$ | 0.69 | $2.8$ | $(1.0,4.8)$ | 0.83 | $2.4$ | $(1.3,3.4)$ | 0.68 | $1.0$ | $(-0.4,3.0)$

Note. — Measured interband lag analysis results measured in the four time
periods, with all CCFs performed relative to the hardest Swift UV band ($W2$).
Column 1: band/filter name. Columns 2-4: for the quarter 1 (Q1) analysis,
these give the maximum value of $r$, followed by the median lag and 68%
confidence interval measured with the FR/RSS technique. Columns 5-7, 8-10, and
11-13 give the same information as Columns 2-4, but for Q2..Q4, respectively.
The first two rows give the hard and soft X-ray (XRT) results and the last six
give those for the six UV/optical (UVOT) filters, so the third row ($W2$) is
for the autocorrelation instead of cross-correlation.

Another interesting feature of this analysis is the relation between the
duration of light curves examined and the derived interband lags. The largest
UV/optical wavelength interval, and thus the largest expected interband lag,
is between the UVOT bands W2 and V. For the full 650-day campaign,
$\tau_{V/W2}=4.0$ days (Table 3), and the mean of the quarterly 162.5-day
segments is $\tau_{V/W2}=1.8$ days (Table 5). For the initial 300-day
campaign, Hernandez Santisteban et al. (2020) finds $\tau_{V/W2}=2.2$ days.
That is, $\tau_{V/W2}$ increases with duration, albeit with substantial errors
and dispersion.

Such behavior is not a surprise given that AGN light curves show “red-noise”
variability over a wide range of timescales dominated by the longest
timescales sampled, combined with the fact that the expected transfer
functions are not delta functions but rather quite broad and asymmetric. This
means that duration light curves will sample preferentially longer lags in the
transfer function, or more distant parts of the transfer function. Thus the
“lag” is not merely a function of the physical reprocessor, but also of the
driving light curve and the sampling pattern, and we urge caution in
interpreting lags as completely well-determined values, independent of the
details of the campaigns in which they are measured.

###  3.5 Flux-Flux Analysis

The aim of a flux-flux analysis is to use the observed variations to decompose
the AGN light into constant (host galaxy) and variable (accretion disk and/or
BLR) components. This method was developed following the Winkler et al. (1992)
finding that linear relationships hold among observed fluxes in different
photometric bands as the AGN brightens and fades. Thus AGN variations are well
described by one spectral energy distribution (SED) for the variable
component, and another for the constant component, which includes, for
example, starlight from the host galaxy (Winkler et al., 1992). We model the
observed fluxes $F(\lambda,t)$ versus a dimensionless light curve $X(t)$ that
describes the shape of the AGN brightness variations. This method has been
used in the analysis of data from several previous IBRM campaigns, including
the AGN STORM campaign on NGC 5548 (Starkey et al., 2017), and the first year
of F9 monitoring (Hernandez Santisteban et al., 2020). The results demonstrate
the adequacy of the linear model to describe the UV and optical variations, by
up to factors of 5 in the brightness in the case of NGC 5548.

Fig. 5 illustrates our flux-flux analysis of the six-band Swift data on F9. In
the left panel, the observed fluxes in the light curves shown in Fig. 2 are
plotted against a dimensionless brightness level $X$, defined to be zero at
minimum and unity at maximum brightness. the dimensionless lightcurve shape
$X(t)$, normalised to zero mean $\left<X\right>=0$ and unit variance
$\left<X^{2}\right>=1$. The flux-flux analyses assumes a linear model,

$\displaystyle F(\lambda,X(t))$ $\displaystyle=$ $\displaystyle
F_{0}(\lambda)+F_{1}(\lambda)\,X(t)$ (4) $\displaystyle=$ $\displaystyle
F_{\rm gal}(\lambda)+F_{\rm disk}(\lambda)\,\left(X(t)-X_{\rm gal}\right)\ ,$
(5)

representing the total (disk+galaxy) light at different brightness levels. The
intercept $\bar{F}(\lambda)$ at $X=0$, and the slope $dF/dX=\Delta
F(\lambda)$, are the mean and rms of the variations at each wavelength. the
minimum flux $F_{0}(\lambda)$ at $X=0$, and the increase in flux by
$F_{1}(\lambda)$ at $X=1$. Note that $X(t)$ is also modelled; it is not simply
the observed X-ray light curve.) The slopes in Fig. 5 are evidently steeper in
the UV and become flatter at longer wavelengths, indicating that the variable
component has a relatively blue SED, with $F_{\nu}$ decreasing with
wavelength. Bucking this trend, the $U$-band slope is steeper than the
adjacent bands. Extrapolating the linear model to fainter levels, $X<0$
$X<X_{\rm min}$, predicts the flux with the variable disk component at fainter
levels than observed during the campaign. Because fluxes must be positive, we
define $X_{\rm gal}=-3.79$ at the point where the $W2$ flux is $1\sigma$ above
zero. Evaluating the model fluxes at $X_{\rm gal}$ then gives an estimate of
the constant (host galaxy) SED $F_{\rm gal}(\lambda)$, and subtracting this
gives the variable (disk) SED $F_{\rm disk}(\lambda,X)$.

Figure 5:  Flux-flux analysis and resulting spectral energy distributions
(SEDs) for the variable (disk) and constant (host galaxy) components of the
AGN light. Left: Observed fluxes in the 6 Swift bands are plotted against a
dimensionless measure $X$ of the brightness, where $X=0$ corresponds to the
mean and $X=\pm 1$ the mean $\pm$ rms brightness. Best linear fits and $\pm
1\sigma$ uncertainties are shown for each band. Extrapolating to fainter
levels, define $X_{\rm gal}=-3.79$ where the UVW2 flux is 1$\sigma$ above
zero. Evaluating the extrapolated fluxes at $X_{\rm gal}$ defines the constant
galaxy SED $F_{\rm gal}(\lambda)$, and subtracting that leaves the variable
disk flux $F_{\rm disk}(\lambda,X)$. Right: SEDs corrected for Milky Way dust
extinction (black), $F_{\rm gal}(\lambda)$ (red) for the host galaxy and
$F_{\rm disk}(\lambda,X)$ (blue) for the variable disk at minimum and maximum
brightness. Power-law SEDs with $F_{\nu}\propto\nu^{1/3}$ (blue dashed) are
shown for reference. Horizontal error bars on the galaxy SED indicate the
Swift filter bandwidths.  Table 6: Galaxy and Disk SEDs from the Flux-Flux
Analysis in Fig. 5.

(1) | (2) | (3) | (4) | (5)
---|---|---|---|---
band | $\lambda_{p}$ | $F_{\rm gal}$ | $F_{\rm disk,min}$ | $F_{\rm disk,max}$
| (Å) | (mJy) | (mJy) | (mJy)
$W2$ | 1965 | $0.32\pm 0.32$ | $4.52\pm 0.45$ | $12.18\pm 0.45$
$M1$ | 2147 | $0.61\pm 0.18$ | $4.49\pm 0.24$ | $12.09\pm 0.24$
$W1$ | 2467 | $1.57\pm 0.05$ | $4.10\pm 0.06$ | $11.03\pm 0.06$
$U$ | 3310 | $2.86\pm 0.15$ | $4.38\pm 0.21$ | $11.80\pm 0.21$
$B$ | 4158 | $3.50\pm 0.14$ | $3.44\pm 0.19$ | $9.27\pm 0.19$
$V$ | 5186 | $6.04\pm 0.13$ | $3.18\pm 0.17$ | $8.57\pm 0.17$

Note. — Column 1: Swift band, Column 2: Rest-frame pivot wavelength, Columns
3-5: Flux densities, corrected for MW dust extinction E(B-V)=0.026 mag, for
the constant host galaxy and for the variable accretion disk at the minimum
and maximum brightness levels during the Swift campaign.

The resulting galaxy and disk SEDs are detailed in Table 6 and shown in the
right-hand panel of Fig. 5. The observed flux densities are converted from
$F_{\lambda}$ in Fig. 2 to $F_{\nu}$ in Fig. 5 using pivot wavelengths
computed from the sensitivity curves of the Swift passbands. The rest-frame
pivot wavelengths are given in Table 6. This is advantageous because the disk
SED is relatively flat in $F_{\nu}$. The SEDs are also corrected for Milky Way
dust extinction for $E(B-V)=0.026\,{\rm mag}$ (Schlafly & Finkbeiner, 2011).

Note that the disk SED increases by a factor of 2.7 between the minimum and
maximum levels observed during the campaign. It is remarkable that this
relatively large increase occurs with little change in the SED of the variable
component. The rms residuals to the fit in Fig. 5 are below 0.35 mJy in all
six bands. This linearity in the response across a wide range of UV and
optical wavelengths is well documented in other Type 1 AGN, for example in NGC
5548 (Starkey et al., 2017) the disk SED remains unchanged over a factor 5
change in brightness.

While the observed disk+galaxy SED appears to be “bluer when brighter,” the
linear model attributes this to a relatively red constant galaxy SED plus a
relatively blue variable disk SED. Alternatively, Korista & Goad (2001)
suggest that as much as 1/3 of this effect may arise from the variable DCE
contribution. Note in Fig. 5 that the disk SED is close to the
$F_{\nu}\propto\nu^{1/3}$ SED predicted for reprocessing on a steady-state
thin blackbody accretion disk. Note also a significant excess flux in the $U$
band of about 16% relative to the adjacent bands, indicating the size of the
Balmer jump and thus the level of Balmer continuum emission in the correlated
variations.

##  4 Discussion

###  4.1 Accretion disk + BLR model

We use the available observations of F9 to construct a combined disk-BLR model
for the source. We assume that the UV/optical continuum SED is produced by an
optically thick, geometrically thin accretion disk as calculated by Slone &
Netzer (2012). The disk properties and BH mass are adjusted to agree with
earlier spectroscopic and X-ray measurements and with the observed flux in the
$W2$ Swift band (Hernandez Santisteban et al., 2020). This gives $M_{\rm
BH}=2.6\times 10^{8}M_{\odot}$ $L/L_{\rm Edd}=0.035$, $a=0.7$, and
$\alpha_{ox}=1.4$, where a is the spin parameter. Assuming face-on inclination
results in $\lambda L_{\lambda}(5100\,{\text{\AA}})=10^{44}$ erg s-1, but the
luminosity seen by the BLR can differ by a factor of 1.5–2 due to the unknown
disk inclination and the large amplitude continuum variations observed along
our line of sight during the campaign.

To further investigate the UV/optical variability, Fig. 6 shows the UV/optical
lags $\tau$ and uncertainties as a function of the central wavelength
$\lambda$ of each band. The lag data measured relative to the $W2$ band are
fitted with a power-law function of the form
$\tau=\tau_{0}\,\left((\lambda/\lambda_{0})^{4/3}-1\right)$, where $\tau_{0}$
is the lag at the reference wavelength $\lambda_{0}=2050$ Å (the center of the
$W2$ band), and where $\tau_{0}$ is subtracted to force the lag to be zero at
$\lambda_{0}$. The parameter $\tau_{0}$, interpreted as light travel time,
measures the mean radius of the disk, weighted by its response at
$\lambda_{0}$ to changes in the irradiation. This fit, shown as a dashed line
in Fig. 6, formally yields a fitted value of $\tau_{0}=1.81\pm 0.20$ lt-day,
and although the lags generally do increase with wavelength, the overall fit
is rather poor, with $\chi_{\nu}^{2}=7.9/4$ dof.

Figure 6:  Plot of lag vs. wavelength for the UVOT data in Table 3. The dashed
line is a fit to $\tau\propto\lambda^{4/3}$ for the five longer wavelength
UVOT CCF points, shown as filled circles. (The $W2$ ACF, shown as an unfilled
circle at 1928 Å, does not participate in the fit because the fit is forced
through $\tau=0$ for the ACF.) These data are in the observed (not rest) frame
with no correction for the small effects of time dilation. In addition the
blue shaded area is generated from the estimated black hole mass and Eddington
ratio parameters, under the Wien assumption (upper bound) and flux-weighted
assumption (lower bound). The red triangles show interband lag predictions
based on the Netzer (2022) BLR plus zero-lag disk assuming the same source
parameters as in that paper (H. Netzer, priv. comm.). See text for further
details.

Next we use Eqn. 1 of Edelson et al. (2019) to estimate the light-crossing
radius $R$ of an annulus emitting at a characteristic wavelength $\lambda$,
which yields:

$R=0.09\left(X\frac{\lambda}{1928{\text{\AA}}}\right)^{4/3}M_{8}^{2/3}\left(\frac{\dot{m}_{\rm
Edd}}{0.10}\right)^{1/3}\textrm{lt-dy}$

where $X$ is a multiplicative scaling factor of order unity that accounts for
systematic issues in converting the annulus temperature $T$ to wavelength
$\lambda$ at a characteristic radius $R$, $M_{8}$ is the black hole mass in
units of $10^{8}\,{\rm M}_{\odot}$ and $\dot{m}_{\mathrm{Edd}}$ is the
Eddington ratio $L_{\mathrm{bol}}/L_{\mathrm{Edd}}$. Note that this assumes a
face-on disk. If we assume an annulus of radius $R$ emits at an observed
wavelength corresponding to the temperature given by Wien’s Law, then
$X=4.97$. If instead the flux-weighted radius is used, then $X=2.49$. (The
flux-weighted estimate assumes that the temperature profile of the disk is
described by $T\propto R^{-3/4}$; Shakura & Sunyaev 1973.) In both the Wien
and flux-weighted cases, the disk is assumed to have a fixed aspect ratio
$H/R$ and to be heated internally by viscous dissipation and externally by the
coronal X-ray source extending above the disk. These two assumptions are used
to generate the blue shaded area in Fig. 6, where the Wien assumption gives
the upper bound and the flux-weighted assumption yields the lower bound. The
estimated disk size overshoots that derived under the Wien assumption by a
factor of 1.8 and the flux-weighted assumption by 4.5.

In summary, the lag spectrum points to an accretion disk that is somewhat
larger than expected, even discounting the $U$ band, where DCE from the BLR
clearly contributes to the large net lag. We note again, however, that the
systematic uncertainties are quite large. We therefore consider below the
possibility that all of the interband lags are attributable to DCE. Under this
hypothesis, the interband lags from the accretion disk are too short for us to
measure and the continuum light curves for the various bands are, for all
practical purposes, assumed to be identical. In the next section, we model DCE
from the BLR to test this hypothesis.

###  4.2 BLR diffuse continuum and emission line emission

Recently Netzer (2022) analyzed IBRM data from F9 as well as other AGN and
finds that the entire lag spectrum can be explained as emission from the BLR
(DCE plus line emission) and a “zero-lag” disk (one that is so small that the
interband lags are not measurable with current data). In Fig. 6 the red
triangles show the result of using the Netzer (2022) model to estimate the
emission-weighted mean distance from the central source to the emitting
region, as a function of wavelength and then convolved with the Swift bands,
for comparison with the observed lags (black dots with error bars). Note that
the $M2$ lag is predicted to be strongly negative relative to $W2$ in this
model. The $M2$ lag is smaller than the $W2$ lag in all realizations using the
original input source parameters of the Netzer (2022) paper. Fig. 1 shows why:
a complex of strong BLR emission lines (e.g., CIV 1550Å, CIII] 1909Å) falls in
the observed $W2$ band, but the $M2$ band contains much less line emission.
This substantial BLR emission increases the mean emissivity distance and thus
the expected W2/M2 lag, in the same fashion as Balmer jump (continuum)
emission from the BLR produces excess U band lags. Thus the model lags due to
BLR line plus DCE emission alone are always larger for $W2$ than for $M2$. The
addition of a zero-lag (or near zero-lag) disk, no matter how strong its
variability amplitude is compared to that of emission from the BLR, may affect
the magnitude of this interband lag relationship, but cannot change its
direction.

We also note that similar behavior has been observed in two earlier IBRM
campaigns: early IUE monitoring of NGC 7469 finds that 1350 Å continuum leads
1825 Å by $\sim$0.25 day (Wanders, 1997) and the 2014 Swift/HST campaign on
NGC 5548 shows small but significantly increasing lags throughout the UV
(Fausnaugh et al., 2016). Most other IBRM campaigns have lacked the
sensitivity and/or temporal resolution to characterize significant lags within
the UV, but it is worth noting that of the three for which such lags have been
detected, all have been positive (in the sense that the shorter wavelengths
lead the longer wavelengths).

Thus we conclude that the observed Swift lag spectrum of F9 and likely the
other two AGN cannot be completely explained by the BLR plus zero-lag disk
model of Netzer (2022). We note however that for this particular model of the
BLR there is a great deal of freedom, so for instance if the Fe ii emission (a
notoriously difficult feature to characterize) is much stronger than used in
the original model, the $M2$ and $W1$ lags would become larger relative to
$W2$, and could even be positive as seen in the data. Further modeling is
planned to investigate this and other possibilities.

Figure 7: Bowl model predictions fitted to the six-band CCF lags (top right)
and two six-band AGN SEDs (top left) at the minimum and maximum levels
observed during the campaign. Blue and red curves give the model predictions
with the lamp-post irradiation on and off, respectively. The model thickness
profile $H(R)$ (lower right) has a high power-law index, giving a flat disk
with a steep rim near $R_{\rm out}/c\sim 10$ days. Here colored lines depict
light rays from the lamp-post down to the disk and up to the observer at
$i=45^{\circ}$. Note the aspect ratio is $H/R\sim 5$% at the crest of the rim.
The temperature profile $T(R)$ (lower left) balances blackbody cooling and the
sum of internal viscous and external irradiative heating. The flat disk has
$T\propto R^{-3/4}$, falling from $10^{5}$ K to $2000$ K and then rising to
$8000$ K at the crest of the steep rim. Green curves in the upper panels show
SEDs and lags for the flat disk and steep rim components separately. Yellow
and cyan curves for 30 random MCMC samples indicate the fit uncertainty.
Histograms on the lower and right axes of the lower panels give posterior
distributions for 5 fiducial temperatures, inner and outer radii, and
thickness at the rim.

###  4.3 The accretion disk “bowl” picture

The lag spectrum in Fig. 6 clearly rises in the UV and then appears to flatten
at longer wavelengths, albeit with some uncertainty. This convex lag spectrum
shape is commonly seen in the IBRM lag spectra of other AGN as well (e.g.
Edelson et al., 2015; Hernandez Santisteban et al., 2020). In a picture built
around a standard (Shakura & Sunyaev, 1973) flat accretion disk, one obvious
way to produce such a lag spectrum is to introduce an additional component of
“piled-up” material at an outer edge of the disk.

Thus in this section we fit these data with a “bowl” model (Starkey et al.,
2023) that assumes lamp-post irradiation and 100% blackbody reprocessing on
the disk surface, but relaxes the assumption that the disk thickness profile
$H(R)$ is negligible and constant for all radii. This allows us to introduce a
“wall” of height $H_{\rm out}$ at radius $R_{\rm out}$, which is associated
with the opening angle $H_{\rm out}/R_{\rm out}$. This wall could represent
material in the BLR, a wind, a dusty torus, or (most likely) a combination of
these three physical components. Additionally we assume a lamp-post irradiator
at a height $H_{\rm x}$ above the disk and that the disk itself is truncated
at the inner radius $R_{\rm in}$. We also assume a fixed inclination angle
$i=45^{\circ}$, luminosity distance $D_{\rm L}=205$ Mpc, from the redshift
$z=0.046145$ and $\Lambda$CDM cosmology
$(h_{0},\Omega_{\Lambda},\Omega_{M})=(0.7,0.7,0.3)$, and black hole mass
$M=2.0\times 10^{8}$ M⊙.

This yields an initial model with seven parameters: $\dot{M}$, $R_{\rm in}$,
$R_{\rm out}$, $H/R$, $\Delta L_{\rm x}$, $H_{\rm x}$, and the model SED
uncertainty $\sigma_{0}$. Note that because the current dataset has
approximately daily sampling, it has very little power to constrain the
parameters $R_{\rm in}$ and $H_{\rm x}$, as these distances correspond to
light-travel times of less than an hour. Thus we fix these at $R_{\rm
in}=R_{\rm g}$ $H_{x}=6\,R_{\rm g}$, corresponding to a high prograde black
hole spin and a moderate X-ray corona height. Our fits prefer such small
values, but we emphasise that these are unreliable parameters for this
analysis. (These inner-disk parameters are best probed with XMM long-looks,
e.g. Fabian et al. (2016).)

Fig. 7 presents a bowl model fit of the five remaining free parameters
($\dot{M}$, $R_{\rm out}$, $H/R$, $\Delta L_{\rm x}$, and $\sigma_{0}$) to the
six lags measured in the cross-correlation analysis (Table 3), and
simultaneously to the minimum and maximum AGN SEDs from the flux-flux analysis
(Table 6), a total of $3\times 6=18$ data constraints. The bowl model shows
rms residuals of $\sigma_{0}\sim 10$ % when fitted to the observed bright and
faint SED. An upper limit to the accretion rate, $\dot{M}<0.053\,M_{\odot}$
yr-1, is set by the faint disk SED, since irradiation also contributes to
heating, and viscous timescales are too long to account for the observed
variations. With the disk SED being $F_{\nu}\propto\nu^{1/3}\propto T^{1/3}$,
the observed bright/faint flux ratio $\approx 2.7$ requires a rise in
$\log{T}$ of $\log{(2.7)}/3=0.14$ dex due to increased irradiation. The wall
at $R_{\rm out}/c\sim 10$ d, has a lag $(R_{\rm out}/c)(1+(2/3)\cos{i})\approx
14$ d. The wall is implemented as $H\propto R^{100}$. Irradiation heats the
wall from $\sim 2000$ K at its base to $\sim 8000$ K at its crest. The
resulting wavelength-dependent mix of short disk lags and longer wall lags
matches the size and convex shape of the observed lag spectrum. The wall is
not tall, $H/R\sim 5$%, and it sits near to where the disk temperature,
falling as $T\propto r^{-3/4}$ , crosses dust sublimation temperatures. The
disk likely extends to larger radii where dust grains can survive in the
cooler disk atmosphere. Here we can expect an increase in thickness (Baskin &
Laor, 2018) and launching of a failed radiatively accelerated dusty outflow
(FRADO; Czerny & Hryniewicz, 2011; Naddaf et al., 2021). The bowl model’s wall
may thus correspond to the transition from a dusty disk atmosphere launching
BLR clouds, to a dust-free disk atmosphere inside the inner edge of the BLR. A
similar picture, with dust-free gas being launched from the outer disk to form
the inner edge of the BLR may also be seen in NGC 4395 (McHardy et al., 2023).

###  4.4 X-ray/UV relationship

Probably the most surprising result of the early Swift IBRM experiments was
that, while UV and optical variations were always highly correlated with lag
spectra that followed the predicted $\tau\propto\lambda^{4/3}$ relation, the
X-ray to UV correlations were much weaker and the interband relationship was
unclear. In some cases the X-rays and UV appear strongly related, especially
after detrending of the UV (e.g., Hernandez Santisteban et al., 2020; Lawther
et al., 2023) while in others the relationship is weaker or non-existent
(e.g., Edelson et al., 2017).

The sheer size of this F9 dataset allows deeper investigation of this complex
relationship. The quarterly analysis shown in Fig. 4 indicates that the lags
within the UV/optical are highly stable, but the X-ray to UV lags vary in
magnitude and in some cases even sign from one quarter to the next, in the
same object. This strengthens the view that the observed 0.3-10 keV X-ray
variability is not the sole driver of the observed UV/optical variability.
Instead, the driver must either be obscured from our line of sight, or it must
be in a band that is either higher or lower in energy than observed by Swift,
or (the most extreme possibility) the entire reprocessing model could be
invalid.

##  5 Conclusions

This F9 campaign is the most extensive multiband AGN monitoring campaign
undertaken up to this point in time (MJD 58900), although an ongoing campaign
on Mrk 817 should eventually exceed this Cackett et al. (2023). We perform a
systematic analysis of the impact of detrending on the standard cross-
correlation analysis and find that in this case a second order polynomial
function provides the most accurate measurement of interband lags. This
methodology is easily applicable to other IBRM datasets, and because it offers
and objective criterion for selecting the detrending scheme without user
intervention, it could lessen the likelihood of controversial results, e.g.
the widely disparate X-ray/optical lags claimed in the same IUE/ASCA
observations of NGC 7469 (Nandra et al. 2000, Pahari et al. 2020). Future
papers on these data will systematically investigate detrending in even
greater detail, for instance studying Gaussian and boxcar smoothing and
comparing the results with JAVELIN (Peterson et al., in prep.).

This large dataset provided the clearest detection to date of positive
interband lags within the UV, with significant detections between the Swift
$W2$ and both $M2$ and $W1$ bands. The presence of strong BLR line emission in
$W2$ means that this lag spectrum cannot be explained by a combination of
emission from the BLR and a small disk. Likewise the excess $U$-band lag seen
in this campaign as well as almost all previous IBRM campaigns indicates that
the full lag spectrum cannot be the result of a pure thin disk of any size.
Thus we conclude that neither BLR DCE, a thin accretion disk, nor a
combination of both can fully account for the observed lag spectrum. At this
time the most promising theoretical approaches to solving this problem involve
truncated disks (e.g., Hagen & Done, 2023; McHardy et al., 2023) or
reprocessing without reference to an X-ray corona (e.g., Neustadt & Kochanek,
2022).

Along these lines, Fig. 7 shows the result of fitting the F9 lags and
UV/optical SEDs with a ‘bowl’ model that invokes lamp-post irradiation and
blackbody reprocessing on a power-law disk thickness profile $H\propto
R^{\beta}$, as introduced in the context of fitting the UV/optical variations
of NGC 5548 (Starkey et al., 2023). We find that the observed lags imply that
this model’s steep rim occurs at a distance ($\sim$10 lt-dy) where the flat
disk temperature profile crosses the dust sublimation temperature, in
agreement with the predictions of models that invoke dust opacity to increase
the disk thickness (Baskin & Laor, 2018) and/or to launch a failed radiatively
accelerated dusty outflow (Czerny & Hryniewicz, 2011; Naddaf et al., 2021)
that has been proposed as the origin of the low-ionization component of the
BLR.

Finally we note that extensive ground-based photometry and spectroscopy and
high throughput NICER X-ray monitoring was also gathered for F9 throughout
this period. The analysis of those datasets should further elucidate the role
of the disk and BLR in producing the observed relations between variations in
different bands. Those data and results will be reported in future papers.

The authors thank the Neil Gehrels Swift Observatory team for their tireless
and successful efforts to execute this program. We thank the anonymous referee
for constructive criticism of an earlier version of this paper. RE and JMG
gratefully acknowledge support from NASA under the ADAP award 80NSSC17K0126.
JMG also gratefully acknowledges support from NASA under the Swift awards
80NSSC22K1492 and 80NSSC24K0513. IMcH thanks the UK Science and Technology
Facilities Council for support under grant ST/J001600/1. MV gratefully
acknowledges financial support from the Independent Research Fund Denmark via
grant number DFF 8021-00130.

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