# What Drives the Variability in AGN: Explaining the UV-Xray Disconnect
Through Propagating Fluctuations

Scott Hagen,1 Chris Done,1 Rick Edelson2
1Centre for Extragalactic Astronomy, Department of Physics, University of
Durham, South Road, Durham DH1 3LE, UK
2Eureka Scientific Inc., 2453 Delmer Street, Suite 100, Oakland, CA 94602, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Intensive broadband reverberation mapping (IBRM) campaigns have shown that AGN
variability is significantly more complex than expected from disc
reverberation of the variable X-ray illumination. The UV/optical variability
is highly correlated and lagged, with longer lag at longer wavelength, but the
timescale is longer than expected. More challenging though is that the
UV/optical lightcurves are not well correlated with the X-rays which were
meant to be driving them. Instead, we consider an intrinsically variable
accretion disc, where mass accretion rate fluctuations propagate in through
the flow, modulating the intrinsically faster X-ray variability from the
central regions. We match our model to the parameters of Fairall 9, a well
studied AGN with $L\sim 0.1L_{\mathrm{Edd}}$, where the spectrum is dominated
by the UV/EUV rather than the X-rays, as opposed to NGC4151 and NGC5548 where
$L\sim 0.02L_{\mathrm{Edd}}$. We show that intrinsic variability and
propagation gives X-ray and UV/optical light-curves that are dominated by
variability on two different time-scales, yet are correlated on long time-
scales. We include reprocessing of the X-rays from the disc but this has
negligible impact on the lightcurves for spectra where the EUV dominates the
bolometric power. We also include reverberation of the total (EUV plux X-ray)
variable spectrum off a wind. This results in a bound-free component which
predominantly follows the slow variable EUV, but lagged and smoothed on the
light-travel time. This spectrum is redder than the EUV disc emission, so it
contributes more at longer wavelengths giving the apparent rise in lag time
with wavelength from a constant lag component. We conclude that contrary to
the original motivation for IBRM campaigns, AGN variability is likely driven
by intrinsic fluctuations within the disc, not X-ray reprocessing, and that
the observed lags are produced by the EUV illumination of the wind not the
X-ray illumination of the disc.

###### keywords:

accretion, accretion discs – black hole physics – galaxies: active

††pubyear: 2023††pagerange: What Drives the Variability in AGN: Explaining the
UV-Xray Disconnect Through Propagating Fluctuations–A

##  1 Introduction

Active galactic nuclei (AGN) are powered by accretion onto a supermassive
black hole (SMBH), a process that is generally understood through the
framework of a Shakura & Sunyaev (1973) optically thick and geometrically thin
disc. In this model the energy dissipated through the flow gives rise to a
radial temperature profile, increasing at smaller radii. The full spectral
energy distribution (SED) from this model is the sum of black-body components,
peaking typically in the UV/EUV for bright AGN.

A key feature of the standard Shakura & Sunyaev (1973) disc theory is that the
time-scale for changing the mass-accretion rate through the disc is the
viscous timescale, typically several thousand years for SMBH (see e.g Noda &
Done 2018). However, observations of optical/UV emission from AGN clearly show
stochastic variability on time-scales of months to years (e.g. MacLeod et al.
2010). This is incompatible with standard disc theory (Lawrence, 2018).

However, AGN SEDs are more complex than a simple sum of black-body components.
AGN spectra always include an X-ray tail extending to higher energies, showing
that some energy is dissipated outside of the optically thick disc structure
(Elvis et al., 1994; Lusso & Risaliti, 2016). This X-ray emission is generally
highly variable on short time-scales, showing it is formed in a compact corona
(e.g Lawrence et al. 1987; Ponti et al. 2012). This gives a potential solution
of the variability problems as some fraction of the variable X-ray emission
should illuminate the disc, giving a variable reprocessed component in the
optical/UV (Clavel et al., 1992). This can be directly tested by long, well
sampled lightcurves in the X-ray, UV and optical as it predicts that the
variations in the X-ray emission ’echo’ through the disc, progressively
modulating first the UV from the inner disc then the optical from the outer
disc. The radial temperature profile of a standard disc predicts a relation
for the time lag between variability of the X-ray source and that at any
wavelength $\lambda$ emitted by the disc of $\tau\propto\lambda^{4/3}$
(Collier et al., 1999; Cackett et al., 2007). The recent intensive multi-
wavelength monitoring campaigns of AGN, were designed to use this relation to
map the accretion disc size scale (Edelson et al., 2015, 2017, 2019; McHardy
et al., 2014, 2018; Fausnaugh et al., 2016; Chelouche et al., 2019; Cackett et
al., 2018; Hernández Santisteban et al., 2020; Kara et al., 2021; Vincentelli
et al., 2021).

However, what these campaigns have instead showed is that AGN variability is
significantly more complex than predicted by these models. Generally the
measured lags give a size scale several times larger than expected for a disc,
and often the correlation between the X-ray and UV/optical is poor. The long
lags can be produced if there are additional structures such as e.g, a wind on
the inner edge of the BLR contributing to the re-processed signal (e.g
Dehghanian et al. 2019; Chelouche et al. 2019; Kara et al. 2021). The poor
correlation between the X-ray and UV/optical, however, cannot easily be
explained in the standard reverberation picture. Attempts at directly
modelling the UV/optical light-curves through disc reverberation all predict
light-curves that are highly correlated with the X-ray (Gardner & Done, 2017;
Mahmoud & Done, 2020; Mahmoud et al., 2023; Hagen & Done, 2023a).
Reverberation smooths on timescales similar to the lags, and the optical/UV
lags are of the order a few days, yet these lightcurves have typical
variability timescale of 20-40 days, while the X-rays have typical timescale
of $<1$ day. This amount of smoothing cannot be produced by the same
reverberation material which gives rise to the lags, strongly indicating that
reverberation is not the sole driver of variability in AGN.

Intrinsic variability of the accretion flow is even more directly required in
the ’changing look’ AGN. These show a transition in Balmer line profiles, from
type 1 (broad plus narrow lines) to type 1.8-2 (where the broad component of
the line is strongly suppressed). This change correlates with a drop in the
observed optical/UV continuum flux. Most of these events are not likely due to
obscuration, as posited by unification models, as there is no associated
reddening signature (LaMassa et al., 2015; McElroy et al., 2016; Ruan et al.,
2016; Runnoe et al., 2016). Even more convincingly, the infrared emission
(reprocessed UV from the torus) also follows the optical/UV, showing clearly
that this is an intrinsic change in the AGN accretion flow (Sheng et al.,
2017; Stern et al., 2018; Wang et al., 2018; Ross et al., 2018), likely
marking the transition between some sort of efficient optically thick disc
accretion, to being dominated by a radiatively inefficient optically thin hot
flow (Noda & Done, 2018; Ruan et al., 2019).

In this paper we will consider the scenario where the accretion disc itself is
intrinsically variable on observable time-scales. This obviously departs from
standard Shakura & Sunyaev (1973) disc theory. However, the disc emission
itself also clearly departs from standard Shakura & Sunyaev (1973) theory in
its disc spectral shape. There is a ubiquitous downturn in the far UV which
appears to extrapolate across the unobservable EUV data gap to meet an upturn
below 1 keV relative to the X-ray coronal tail (Laor et al., 1997; Porquet et
al., 2004; Gierliński & Done, 2004) (historically referred to as ’the Big Blue
Bump’ and soft X-ray excess). These two features can be modelled together with
a single warm, optically thick thermal Compton emission component (Mehdipour
et al., 2011, 2015; Done et al., 2012; Kubota & Done, 2018; Petrucci et al.,
2018). Such models can provide successful fits to the optical/UV/X-ray SEDs of
individual AGN (e.g Matzeu et al. 2016; Done & Jin 2016; Czerny et al. 2016;
Hagino et al. 2016, 2017; Porquet et al. 2018, 2019) and also larger samples
of AGN (e.g Jin et al. 2012a, b; Mitchell et al. 2023; Temple et al. 2023).

The warm Comptonisation component generally dominates the bolometric
luminosity, indicating its origin is in the energy generating structure. It
also requires a large optical depth. Both these strongly suggest that this is
emission from the disc itself, but obviously it is not thermalising to a
blackbody as in the Shakura & Sunyaev (1973) models. Instead, one way to
produce the spectrum is to change the vertical structure of the disc such that
the accretion power is dissipated higher in the photosphere (Różańska et al.,
2015; Petrucci et al., 2018; Jiang & Blaes, 2020), rather than concentrated on
the equatorial plane as in the Shakura & Sunyaev (1973) models. However, this
is not yet well understood, so there are no theoretical models to predict the
intrinsic variability of this structure, though this is the subject of recent
numerical studies (Secunda et al., 2023). Instead, we will use the
observations as a guide, and model the intrinsic variability
phenomenologically. The results can then be used to build more physical models
of the disc structure, as well as to give more realistic driving lightcurves
to reverberation map the accretion flow structures.

We take models of the disc and corona variability seen in the stellar mass
black hole binaries as our starting point. Unlike the AGN, these can show
spectra which are dominated by what looks very like a standard Shakura &
Sunyaev (1973) disc (high/soft state see e.g. Done et al. 2007, X-ray tail is
very weak). Again unlike AGN, the variability of this component is well
matched to the viscous timescale of the outer disc of weeks/months (Dubus et
al., 2001; Lasota, 2007, 2008), with no short timescale disc variability.
However, they also show spectra (bright low/hard and intermediate states) with
more X-ray tail where the disc emission is much weaker and distorted from the
Shakura & Sunyaev (1973) models. This disc emission varies on timescales of
seconds (Uttley et al., 2011; De Marco et al., 2017; Kawamura et al., 2022),
much faster than expected from a viscous timescale even from the inner disc.
This may instead indicate a turbulent region on the inner edge of a truncated
disc, where the flow transitions to a hot corona (Kawamura et al., 2022, 2023;
Marcel et al., 2022; Lucchini et al., 2023).

Figure 1: A depiction of what we envisage with the propagating fluctuations.
The disc like structure (green) generates slow variability, giving light-
curves that vary on long time-scales (bottom left). These variations propagate
into the hot Corona (blue), which generates fast variability. The slow
variations from the disc modulate the fast from the corona, giving light-
curves containing both long and short term trends (bottom right). We stress
that this is a sketch, and as such the components and light-curves are not to
scale.

The first models of the variability in stellar mass systems only focused on
the hard X-ray tail, which shows the fastest variability with substantial
power down to $\sim 0.1$ s. The broad band power spectrum, and lags between
different energy bands in the X-ray power law emission can be reproduced in
models based on a propagating fluctuations framework. Larger radii typically
produce longer timescale fluctuations, and these drift inwards, modulating the
shorter timescale fluctuations stirred up at smaller radii (Lyubarskii, 1997;
Arévalo & Uttley, 2006; Ingram & Done, 2011, 2012). These models were then
extended to include the turbulent disc region. The key feature is that there
is a discontinuity in timescale of intrinsic fluctuations at the radius at
which the geometrically thin, cool disc transitions into the geometrically
thick hot flow (Rapisarda et al., 2016; Mahmoud & Done, 2018; Kawamura et al.,
2022, 2023). Since fluctuations propagate inwards, the hard X-ray lightcurve
contains the propagated slow fluctuations from the disc, as well the faster
fluctuations generated in the corona itself, while the intrinsic disc
lightcurve contains only the slow fluctuations generated within the disc, as
shown schematically in Figure 1. This clearly holds out the possibility to
explain the disconnect in variability timescale seen in AGN, where the disc
lightcurve is intrinsically much smoother than the X-ray lightcurve. This also
retains a correlation between disc and X-ray lightcurves on long timescales
due to the propagation of the disc fluctuations, but gives an overall poor
correlation on faster timescales as these are produced only in the X-ray hot
flow so are not intrinsically part of the disc variability.

We will scale up these models from black hole binaries to AGN, using Fairall 9
as a typical example of a moderate luminosity Seyfert 1 galaxy. This AGN was
the subject of a long timescale intensive broadband monitoring campaign,
giving excellent spectral and variability data (Hernández Santisteban et al.,
2020), that show outgoing variations on short time-scales as well as tentative
evidence of propagation on long time-scales where the lag time is 10s of days
(Hernández Santisteban et al., 2020; Neustadt & Kochanek, 2022; Yao et al.,
2023).

We first build a broadband spectral model which is tailored for variability
studies (Section 2) and then parameterise propagating fluctuations in mass
accretion rate through this structure (Section 3). This full spectral-timing
model predicts the intrinsic lightcurves in any energy band, so gives input
for reverberation (Section 4). We allow the X-ray emission to illuminate the
disc to produce a disc reverberation signal, and use the bolometric
(especially extreme UV) flux to reverberate from a larger scale wind on the
inner edge of the BLR. Finally, in Section 5 we generate a set of model light-
curves and explore how the different components within the accretion flow
affect the observables (i.e lags and cross-correlation functions).

##  2 Modelling the SED

Throughout this paper we use $R$ for radius in physical units (measured from
the black hole) and $r$ for dimensionless gravitational radii, where
$R=rR_{G}$ with $R_{G}=GM/c^{2}$. Additionally, we will use $\dot{m}$ for
dimensionless mass accretion rate, scaled by the Eddington rate such that
$\dot{m}=\dot{M}/\dot{M}_{\mathrm{Edd}}$, where $\dot{M}$ is the physical
mass-accretion rate in $g/s$, and $\dot{M}_{\mathrm{Edd}}$ is the Eddington
mass-accretion rate, related to the Eddington luminosity by
$L_{\mathrm{Edd}}=\eta(a)\dot{M}_{\mathrm{Edd}}c^{2}$, where $\eta(a)$ is a
spin-dependent accretion efficiency.

We base our spectral-timing model on the underlying radially stratified SED
agnsed from Kubota & Done (2018). This assumes the radial emissivity profile
for a thin disc, $\epsilon(R)\propto R^{-3}f(R)$ where $f(R)$ describes the
stress free inner boundary condition for the Kerr metric (Novikov & Thorne,
1973; Page & Thorne, 1974), but allows the emission from the flow to
transition from blackbody from a standard outer disc ($r>r_{w}$), to warm
Comptonisation ($r_{w}\geq r>r_{h}$), and then to hot Comptonisation
($r_{h}\geq r>r_{\mathrm{isco}}$). Within our code, the standard outer disc
and warm Comptonisation region are implemented as in Kubota & Done (2018), and
we refer the readers to this paper for details regarding these two regions.
For the hot Comptonisation region, however, we make some small modifications
to better follow the spectral variability.

Figure 2: The mean SED of Fairall 9 during the 1st year of the intensive
monitoring campaign, which will be used throughout this paper to initiate the
variable model. The solid black line shows the total $\tau$agnsed model, with
the components shown separately as dashed lines. The hot Compton flow is in
blue with its neutral reflection in magenta. The whole of the optically thick
disc emission is modelled as warm Compton (green dashed line) rather than
including any standard Shakura & Sunyaev (1973) disc region. This is important
later when we consider the variability time-scales of the system.

The hot Compton emission in agnsed is parameterised by the coronal radius
$r_{h}$, photon index $\Gamma_{h}$, and electron temperature $kT_{e,h}$.
However, the photon index and electron temperature are fundamentally set by
the balance between Compton heating and cooling per electron within the hot
flow (see e.g. Done 2010). Compton heating depends on the power,
$L_{\mathrm{diss}}$, dissipated within $r_{h}$ while cooling depends on the
seed photon luminosity from the disc which is incident on the hot flow,
$L_{\mathrm{seed}}$ (see Hagen & Done 2023b for an update on how
$L_{\mathrm{seed}}$ is calculated). Following Beloborodov (1999) (see also the
qsosed model in Kubota & Done 2018) this gives:

$\Gamma_{h}=\frac{7}{3}\left(\frac{L_{\mathrm{diss}}}{L_{\mathrm{seed}}}\right)^{-0.1}$
(1)

We use this formalism to better capture the time dependent behaviour of the
X-ray spectrum as changes in the seed photons from the disc
$L_{\mathrm{seed}}$ travel at the speed of light, whereas mass accretion rate
fluctuations which modulate $L_{\mathrm{diss}}$ propagate more slowly (see
e.g. Veledina 2016, 2018). This means that the power law spectrum pivots
rather than changing only in normalisation (Mastroserio et al., 2018).

Continuing from Beloborodov (1999), we can then calculate the coronal electron
temperature, $kT_{e,h}$, from $\Gamma_{h}$ and $\tau_{h}$:

$\frac{kT_{e,h}}{m_{e}c^{2}}=\frac{4y}{\tau_{h}(\tau_{h}+1)}\ \ {\rm where}\
y=\left(\frac{4\Gamma_{h}}{9}\right)^{-9/2}$ (2)

where $y$ is the Compton y-parameter.

This gives an SED model where $\Gamma_{h}$ and $kT_{e,h}$ are calculated self-
consistently, and can therefore become time-dependent when we calculate the
variable SED. We call this model $\tau$agnsed.

###  2.1 Fairall 9 fit to $\tau$agnsed.

Component | Parameter (Unit) | Value
---|---|---
phabs |  $N_{H}$ ($10^{20}$ cm-2) | 3.5
$\tau$agnsed |  $M$ ($M_{\odot}$) | $2\times 10^{8}$
| Dist (Mpc) | 200
|  $\log\dot{m}$ ($\dot{M}/\dot{M}_{\mathrm{Edd}}$) | $-1.217^{+0.0196}_{-0.024}$
| $a_{\star}$ | $0.722^{+0.062}_{-0.11}$
| $\cos(i)$ | 0.9
| $\tau_{h}$ | $0.99^{+2.4}_{-0.99}$
|  $kT_{e,w}$ (keV) | $0.394^{+0.037}_{-0.036}$
| $\Gamma_{w}$ | $2.821^{+0.028}_{-0.029}$
| $r_{h}$ | $9.16^{+0.90}_{-0.81}$
| $r_{w}$ | $=r_{\mathrm{out}}$
| $\log r_{\mathrm{out}}$ | $-1$
| $h_{\mathrm{max}}$ | $10$
| Redshift | $0.045$
rdblur | Index | -3
| $r_{\mathrm{in}}$ | $386^{+517}_{-163}$
| $r_{\mathrm{out}}$ | $10^{6}$
| Inc (deg) | 25.8
pexmon | $\Gamma$ | Calculated from $\tau$agnsed
|  $E_{c}$ (keV) | $10^{4}$
| Redshift | $0.045$
| Inc (deg) | 25.8
| Norm ($10^{-3}$) | $4.52^{+0.70}_{-0.60}$
$\chi^{2}$/d.o.f | 229.50/166 = 1.38 |
Table 1: Fit parameters for the mean SED of Fairall 9. Values with no error
were frozen during the fitting process. We note that the inner radius in
rdblur was kept free to fit the Fe-K$\alpha$ line profile which may have
substantial contribution from material further out in the accretion flow
(wind/BLR and torus). The lower limit on $\tau_{h}$ is formally $0$ as this is
degenerate with $kT_{e,h}$ and our data do not have sufficient spectral
coverage to constrain the high energy rollover. This forms our mean SED model
throughout the paper when we calculate the variability.

We refit the Fairall 9 SED from Hagen & Done (2023a) (hereafter HD23a) with
this new model, so as to get the time averaged values of each parameter, using
xspec v.12.13.0c (Arnaud, 1996). As in HD23a, we also include a neutral
reflection component to account for the Fe-K$\alpha$ line and Compton hump,
modelled with pexmon (Nandra et al., 2007; Magdziarz & Zdziarski, 1995)
convolved with rdblur (Fabian et al., 1989) to account for any smearing within
the reflection spectrum. We tie the value of the hard X-ray spectral index in
pexmon to that in $\tau$agnsed. The optical/UV fluxes are dereddened, and host
galaxy subtracted (see Hernández Santisteban et al. 2020 for details), so we
include galactic absorption only on the X-ray data, using phabs with fixed
$N_{H}=0.035\times 10^{22}$ cm2. The final xspec model is then
phabs*($\tau$agnsed + rdblur*pexmon). The resulting intrinsic model SED is
shown compared to the data (unabsorbed, and deconvolved from the instrument
response) in Fig. 2, with the corresponding fit parameters given in Table. 1.

In common with many intensively studied AGN (e.g. Mehdipour et al. 2011, 2016;
Petrucci et al. 2018, there is no need for an outer standard disc component.
The flow is well described using only the warm Comptonised disc from $r\sim
200-10$ and a hot inner flow from $10-r_{\mathrm{isco}}$. We will use these
parameters from the time averaged SED to set the size scale and physical
conditions for our model of variability from propagating fluctuations through
this structure.

##  3 Propagating fluctuations in $\tau$agnsed

###  3.1 Propagating Fluctuations

Figure 3: Example mass-accretion rate time-series and power-spectrum for a two
component flow, consisting of a slowly varying warm disc region (green) and an
intrinsically faster hot corona (blue).
Left: $\sim 1500$ day snapshot from a time-series realisation calculated for
$N=2^{18}$ time-steps and sampling rate $dt=0.1$ days. It is clear that the
time-series from within the hot corona (blue) has both stronger and faster
variability than the disc.
Right: The power-spectrum for the inner part of the warm disc (green) and the
inner part of the hot flow (blue). The coloured lines show the analytic
solution, following Ingram & van der Klis (2013), while the grey line shows
the average power-spectrum from 100 time-series realisations, again using
$N=2^{18}$ and $dt=0.1$ days. As expected, these follow the analytic
treatment. It can be seen that the hot coronal variability closely follows
that of the warm disc at low frequencies, due to the disc variations
propagating into the corona, and hence modulating its variability.

The propagating fluctuations model considers local variations in the mass
accretion propagating down through the accretion flow (Lyubarskii, 1997). In
this work we will consider these propagating mass accretion rate fluctuations
as the driver of intrinsic variability in AGN. The formalism used is based on
Ingram & Done (2011, 2012); Ingram & van der Klis (2013) as used in Kawamura
et al. (2023), and we refer the reader to these papers for a detailed
description of the model. For completeness, however, we give a brief overview
of the key aspects here.

Following Ingram & van der Klis (2013) we split the accretion flow into $N$
annuli centred at $r_{n}$ with equal logarithmic spacing such that
$d\log(r_{n})=dr_{n}/r_{n}=\mathrm{constant}$. The local mass accretion rate
variability is then assumed to have a well defined power-spectrum, given by a
zero-centred Lorentzian peaking at the frequency at which fluctuations are
locally generated, $f_{\mathrm{gen}}$.

$|A(r_{n},f)|=\frac{\sigma^{2}}{\pi
T}\frac{f_{\mathrm{gen}}(r_{n})}{(f_{\mathrm{gen}}(r_{n}))^{2}+f^{2}}$ (3)

where $f$ is the Fourier frequency, and $\sigma$ and $T$ are the variance and
duration of the time-series $a(r_{n},t)$. The fractional root mean square
variability is $\sigma/\mu=F_{\mathrm{var}}/\sqrt{N_{\mathrm{dec}}}$, where
$\mu$ is the mean, fixed at unity, $N_{\mathrm{dec}}$ is the number of radial
bins per decade, and so $F_{\mathrm{var}}$ is the fractional variability
produced per decade in radius in the flow. Using $|A(r_{n},t)|$ we can now
create realisations of the time-series $a(r_{n},t)$ using the method described
in Timmer & Koenig (1995). Since the fluctuations generated within each
annulus propagate down through the flow, the total time-series within each
annulus will be modulated by those from the previous annuli. As this is a
multiplicative process (Ingram & van der Klis, 2013), we can write the mass-
accretion rate time series at each annulus as:

$\dot{m}(r_{n},t)=\dot{m}_{0}\prod_{k=1}^{n}a(r_{k},t-\Delta t_{kn})$ (4)

where $\dot{m}_{0}$ is the mean mass-accretion rate, set to 1 throughout, and
$\Delta t_{kn}$ is the propagation time between the annuli at $r_{k}$ and
$r_{n}$.

In a standard Shakura & Sunyaev (1973) disc, $f_{\mathrm{gen}}$ is given by
the viscous timescale in the flow. However, this is clearly many orders of
magnitude too slow to describe the data (see introduction). Instead, following
Kawamura et al. (2023), we parameterise $f_{\mathrm{gen}}$ as a broken power
law, with a discontinuity at $r_{h}$ to allow for the much faster timescales
expected in the hot flow compared to the warm disc.

$f_{\mathrm{gen}}(r)=\begin{cases}B_{g,h}r^{-m_{g,h}}f_{K}(r)&r<r_{h}\\\
B_{g,w}r^{-m_{g,w}}f_{K}(r)&r>r_{h}\end{cases}$ (5)

where $f_{K}=(1/2\pi)r^{-3/2}$ is the Keplerian frequency at $r$ in units of
$c/R_{G}$.

We tailor our fiducial model for variability to Fairall 9. From the SED fits
above, we see that the warm disc structure extends from $r\sim 200-10$, i.e.
spans more than an order of magnitude. However, the Fairall 9 optical and UV
lightcurves all have similar variability timescales (Hernández Santisteban et
al., 2020), so we fix $m_{g,w}=-3/2$ so as to give constant fluctuation
timescales across the entire warm disc. Conversely, the hot Compton region
barely spans a factor 2 in radius, but there is some evidence for radial
stratification of timescale across the hot flow seen in AGN power spectra,
with more high frequency variability at higher X-ray energies (Ponti et al.,
2012; Ashton & Middleton, 2022; Tortosa et al., 2023). Hence we choose
$m_{g,h}=1$ so that there is a factor $\sim 5$ increase in $f_{\mathrm{gen}}$
with decreasing radius across the hot flow. There are $\sim 3$ years of well
sampled optical/UV lightcurve for Fairall 9 showing variability, so we choose
a generator timescale of $\sim 3$ years for the warm disc (Edelson et al
(sub)). The X-ray shows substantially more fast variability, extending with
large fluctuation power up to around $0.1$ day (Markowitz et al. 2003;
Markowitz & Edelson 2004; Lohfink et al. 2014; HD23a), so we choose a
generator timescale of 0.1 day for the inner edge of the corona at $r\sim 4$.

The propagation timescale between each annuli is characterised in terms of a
propagation frequency $f_{\mathrm{prop}}$ such that $\Delta
t_{kn}=(dr_{n}/r_{n})(1/f_{\mathrm{prop}}(r_{n}))$ (e.g Ingram & van der Klis
2013). Early models had this set to the same as the generator timescale, but
there is clear evidence in the BHBs that propagation of fluctuations happens
on faster timescales than this (see e.g. Kawamura et al. 2023). There are now
some tentative detections of propagation in AGN disc lightcurves, including in
Fairall 9, where the lag time is 10s of days (Hernández Santisteban et al.,
2020; Vincentelli et al., 2022; Neustadt & Kochanek, 2022; Yao et al., 2023).
Hence we set the propagation timescale to be
$f_{\mathrm{prop}}(r)=100f_{\mathrm{gen}}(r)$ i.e. we keep the same power law
indices, but increase the normalization, so that $B_{p,w}=100B_{g,w}$ and
$B_{p,h}=10B_{g,h}$ (see Kawamura et al. 2023). This gives a $10$ day
propagation timescale from $r=100$ to $50$, which are typical radii at which
the disc emission peaks in the V and UVW2 band, respectively, and a 20-40 day
lag for fluctuations in UVW2 to propagate down into the hard X-ray corona.

Parameter | Value
---|---
\- - - - - - - Generator Parameters - - - - - - -
$r_{\mathrm{var,max}}$ | 200
$F_{\mathrm{var},w}$ | 0.6
$F_{\mathrm{var},h}$ | 0.5
$B_{g,w}$ | $5\times 10^{-5}$
$B_{g,h}$ | 1
$m_{g,w}$ | -3/2
$m_{g,h}$ | 1
\- - - - - - - Propagation Parameters - - - - - - -
$B_{p,w}$ | $5\times 10^{-3}$
$B_{p,h}$ | 100
$m_{p,w}$ | -3/2
$m_{p,h}$ | 1
Table 2: Parameter values used for the generator and propagator frequencies
(see Eqn. 5) for all simulation runs in this paper. Parameters denoted with
the subscripts $w$ and $h$ correspond to the warm and hot corona respectively,
while the subscripts $g$ and $p$ correspond to generative and propagation
time-scales respectively.

Fig. 3a shows example time-series for the mass-accretion rate fluctuations
using these parameters (see table 2). We discard the first 200 days to allow
propagation throughout the accretion flow. The lower panel shows the
variability propagating through the inner edge of the warm disc, while the
upper panel shows the total variability in mass accretion rate through the
inner edge of the hot flow. It is clear that the hot flow lightcurve has more
fast variability than the warm disc, but that the two are correlated on long
timescales.

Fig. 3b shows the power spectrum of the mass accretion rate fluctuations
propagating through the inner edge of the hot flow (grey). As the model is
stochastic these data are the power-spectrum averaged over 100 realisations,
but we also show the analytic solution of the propagating fluctuation model
(Ingram & van der Klis, 2013). The blue line shows the result for propagation
through the entire flow, while the green line shows the result for propagation
only through the warm disc. The warm disc has strong, but slow, variability
while the hot corona has this variability propagated into its mass accretion
rate fluctuations, plus much faster variability generated in the hot flow
itself. Plainly this captures some aspects of the observed disconnect in
timescales between the UV and X-ray behaviour seen in Fairall 9, so we use
these parameters in our exploration of how this affects the spectrum as a
function of time in the next section.

###  3.2 Converting $\dot{m}(t,r)$ to Light-Curves via the SED model

Figure 4: Example model output for a run considering the intrinsic variability
only with no suppression of the disc variability seen by the hot corona, using
the $\dot{m}$ realisation from Fig. 3.
Top Left: The output SEDs. The solid black line shows the mean SED of the
system, while the orange lines shows a sample of 200 SEDs randomly selected
from the variable output. The shaded regions show the extraction regions used
to generate light curves in HX (blue), UVW2 (magenta), U (green) and (V)
orange. For UVW2, U, and V the Swift-UVOT effective area curves were used to
calculate the light-curves.
Top Right: Model auto-correlation functions (ACFs) for the extracted light-
curves, where the colours corresponds the the extraction regions in the SED.
These have been calculated by considering 250 day chunks of the light-curves,
and averaging the resulting ACF from each chunk, as current intensive
monitoring data generally covers 200-300 day time-scales. It is clear that
UVW2, U, and V are dominated by long term slow variability. HX on the other
hand has a narrow peak due to the fast variability intrinsic to the hot
corona, but is generally dominated by the slow variability propagating in from
the disc.
Bottom: Model light-curves extracted from the time-dependent SEDs. The colours
correspond to the extraction region in the top left panel. It is exceptionally
clear that the model vastly over-predicts the variability in the X-ray.

Now that we have a description of the variability in mass-accretion rate at
each radial annulus we can calculate the time dependent SED, following the
description in Section 2. Firstly, we generate realisations of $\dot{m}(t,r)$,
following the previous section, for each radial annulus within the flow. This
allows us to calculate the time-dependent emission from each annulus, which we
then sum over to create the time-dependent SEDs. We note that calculating the
SED at any given time is similar to calculating a single mean SED, with the
difference being that we use a local $\dot{m}(t,r)$ at each radial annulus
rather than a single $\dot{m}$ for the entire flow.

For the standard disc region, the spectrum at each radius is directly given by
$\dot{m}(t,r)$ as this sets the effective blackbody temperature. For the warm
Compton region the optical depth and temperature are assumed constant so the
only shape change is from the seed photon temperature which is set by
reprocessing on the underlying passive disc to the effective blackbody disc
temperature.

The hot corona is more complex as the spectral index, $\Gamma_{h}$, and
temperature, $kT_{e,h}$, change with the changing ratio between seed photon
cooling, $L_{\mathrm{seed}}$ and gravitation heating, $L_{\mathrm{diss}}$. We
set the power dissipated in annulii in the corona as
$L_{\mathrm{diss}}(t,r)\propto\dot{m}$, while $L_{\mathrm{seed}}(t,r)$ tracks
the warm Compton power lagged by the light travel time. The difference in time
dependence of heating and cooling lead to changes in $\Gamma_{h}$ and
$kT_{e,h}$ as described by equations 1 and 2.

Fig. 4 shows a resulting model realisation, using the underlying mean SED from
Fig. 2 and the variability parameters used for Fig. 3. For repeatability, we
set the initial random seed to 1113, using the random number generator from
numpy. This seed will remain the same throughout the paper (unless otherwise
stated), such that different model lightcurves can be directly compared, and
also corresponds to the time-series in $\dot{m}$ shown in Fig.3. Additionally,
the radial resolution is set to $N_{\mathrm{dec}}=500$. We discard the first
200 days of our output time-series in order for the propagating fluctuations
to fill the radial grid.

Fig. 4a shows 200 SEDs randomly selected from within the model time-series
(orange lines) compared to the mean SED (black line). It is immediately
obvious that the model has variability which is much larger in the EUV (warm
disc peak) than in the UV/optical, and that there is even more variability
(over an order of magnitude) in the X-rays.

We quantify this energy dependence by extracting lightcurves in Swift UVOT
band UVW2 (pink), U (green) and V (yellow) by multiplying the SEDs with each
filter effective areas. We extract a hard X-ray lightcurve by assuming a
tophat filter from 1.5-10 keV. These are shown in the lower panel of Fig. 4.
The V/U/UVW2 bands show the typical amount of variability seen in Fairall 9 on
timescales of $\sim 1$ year (Hernández Santisteban et al. 2020, Edelson et al.
(submitted)). However, these bands are produced at radii dominated by the
Rayleigh-Jeans tail of the warm Compton disc, where the monochromatic
luminosity $L_{\nu}\propto T\propto\dot{M}^{1/4}$. The unobservable EUV
emission, on the other hand, sees the peak of the warm Compton emission from
each radius rather than the tail, and so its monochromatic luminosity goes as
$L_{\nu}\propto T^{4}\propto\dot{M}$. For a single annulus, a factor 2 change
in optical/UV emission predicts a factor 16 change in the EUV. Any thermally
based spectrum has a Rayleigh-Jeans tail below the peak where the variability
is strongly suppressed such that $L_{\nu}\propto T\propto\dot{M}^{1/4}$
relative to the spectral peak which has $L_{\nu}\propto\dot{M}$.

The X-ray corona is even more variable than the EUV peak as in the model all
the $\dot{m}$ fluctuations from the inner edge of the warm disc are propagated
into the corona, and then the additional fast coronal variability is imprinted
on top of this. There is also additional variabilty from the seed photons
illuminating the hot corona. Thus the model which gives reasonable amplitude
(few 10s of percent over timescales of 6-12 months) variability in the
optical/UV bands, predicts factor of 10 variability in the corresponding X-ray
lightcurve, much larger than is seen in the X-ray data from Fairall 9.

###  3.3 Suppressing the Variability Seen by the Corona

Figure 5: Same as Fig. 3, but now suppressing the disc variability seen by the
hot corona by $f_{h}=0.02$. It is clear in the power-spectrum that the
fluctuatios within the hot corona should now be dominated by rapid
variability, with only a small contribution from the slow. Indeed, it can be
seen in the time-series that the inner hot flow no longer contains a
significant slow trend remesbling the warm; unlike the case in Fig. 3

The overprediction of the X-ray variability is a surprise in the light of the
results from the stellar mass black hole binaries. These strongly require that
fluctuations propagate from the warm/turbulent disc into the corona,
unhindered by strong viscous diffusion which dissipates the fluctuation by
spreading it out on the propagation timescale (compared fits to data with no
diffusive losses: Rapisarda et al. 2016; Mahmoud et al. 2019; Kawamura et al.
2023 with the strong suppression of variability with viscous diffusion:
Mushtukov et al. 2018; Kawamura et al. 2023). The hot corona is likely in the
regime where fluctuations can propagate in a wavelike manner at the sound
crossing time (Ingram, 2016) i.e. have $\alpha\geq H/R$, but the
warm/turbulent disc is at a much lower temperature, so have much lower $H/R$
hence is more likely to be in the regime where viscous diffusion dominates.
The stellar mass black holes likely only generate variability on the inner
edge of the truncated disc, which may be produced by strong turbulence giving
rise to the fluctuations rather than variability in the disc itself. However,
in the supermassive AGN considered here we have considered the warm disc to be
intrinsically variable up to $200\,R_{G}$.

Clearly there is an issue in our assumptions when we calculate the
variability. The two likely culprits are either our SED shape, or the
assumption that all the disc variability is propagated into the X-ray corona
in a lossless fashion. In the case of the SED shape we could imagine a system
where the UVW2 emission does not lie on the Rayleigh-Jeans tail, but is
instead described by a process that makes its luminosity proportional to
$\dot{M}$. This would lead to UVW2 being a $\sim 1:1$ tracer of the intrinsic
disc variability, eliminating the issue where an overly variable disc leads to
extreme X-ray variability. However, for the emission mechanisms generally
associated with accretion discs (multi-colour black body, Comptonised black-
body) this would imply that UVW2 must see the peak emission. The presence of a
soft X-ray excess that appears to point back to the UV down-turn in the
majority of AGN (e.g Laor et al. 1997; Porquet et al. 2004; Gierliński & Done
2004) would suggest an SED shape that links the EUV and soft X-ray emission to
the disc. As UVW2 is generally below the UV down-turn, this feature becomes
incompatible with UVW2 seeing the peak emission. Additionally, timing studies
show that the He II line, often used as a proxy for the ionising EUV (e.g
Mathews & Ferland 1987; Baskin et al. 2013; Ferland et al. 2020, displays
stronger variability than the optical continuum (e.g Homan et al. (2023)).
This could indicate that the UVW2 pass-band cannot be a $\sim 1:1$ traced of
the intrinsic variability, as that would give a similar amount of variability
in He II (and other recombination lines).

The other likely possibility then is that not all of the disc variability
propagates into the hot corona. In our SED modelling we consider the disc/warm
Corona and the hot Corona as separate and distinct regions. These should be
linked, however there is no guarantee that this implies all variations in the
disc must pass directly into the hot corona. For the remainder of the paper we
will consider this case, where the disc/warm coronal variability is suppressed
somehow before propagating into the hot corona. We stress, however, that the
physical mechanisms that could cause this are not well known. Hence, the
following should be treated as a phenomenological approach.

The simplest way to reduce the variability that propagates into the hot corona
is by setting $\sigma_{h}(\dot{m},r_{k})=f_{h}\sigma(\dot{m},r_{k})$, where
$\sigma(\dot{m},r_{k})$ is the variance of $\dot{m}(t)$ at the radial annulus
$r_{k}$; taken here to be the inner annulus of the warm region; $\sigma_{h}$
is the variance of this annulus as seen by the hot corona, and $f_{h}$ is a
scaling fraction, with $0\leq f_{h}\leq 1$ (see e.g. Mahmoud et al. 2019). The
mass-accretion rate time series from the inner annulus of the warm region,
$\dot{m}(r_{k},t)$, as seen by the hot corona, $\dot{m}_{h}(r_{k},t)$, is
then:

$\dot{m}_{h}(r_{k},t)=\big{(}\dot{m}(r_{k},t)-\dot{m}_{0}\big{)}\sqrt{f_{h}}+\dot{m}_{0}$
(6)

This allows fluctuations to propagate without losses through the disc, but
then only allow a fraction $f_{h}$ of these to propagate into the hot corona.

We perform a run of the same model as in Fig. 4, but setting $f_{h}=0.02$.
This will give a X-ray time-series that is dominated by the fast variability,
generated within the corona itself, but with a small contribution from a slow
component propagating in from the disc. This is clear in Fig. 5 where we see
the power-spectrum in $\dot{m}$ for the hot corona (blue line, and grey
points) is now dominated by the fast variability (unlike in Fig. 3), but still
containing a bump at lower frequencies originating from the warm corona, while
the warm corona itself has much more slow variability in $\dot{m}$ (green
line). Importantly, the slow variability present within the hot corona will
still be correlated with the disc/warm corona variability. Any resulting X-ray
light-curve will then have a marginal correlation with the UV/optical, giving
rise to a disconnect driven by the strong fast variability within the hot
corona.

Figure 6: Example model output for a run considering the intrinsic variability
only, using the $\dot{m}$ realisation from Fig. 5.
Top Left: The output SEDs. The solid black line shows the mean SED of the
system, while the orange lines shows a sample of 200 SEDs randomly selected
from the variable output. The shaded regions show the extraction regions used
to generate light curves in HX (blue), UVW2 (magenta), U (green) and (V)
orange. For UVW2, U, and V the Swift-UVOT effective area curves were used to
calculate the light-curves.
Top Right: Model auto-correlation functions (ACFs) for the extracted light-
curves, where the colours corresponds the the extraction regions in the SED,
again calculated using 250 day chunks of the light-curves. It is clear that
UVW2, U, and V are dominated by long term slow variability, whereas HX has a
narrow core due to the rapid variability intrinsic to the hot corona and a
broad base from the slow variability that propagates into the corona from the
disc.
Bottom: Model light-curves extracted from the time-dependent SEDs. The colours
correspond to the extraction region in the top left panel.

We convert this new set of $\dot{m}(r,t)$ fluctuations into a time dependent
SED. The model and resulting lightcurves are shown in Fig. 6. Compared to Fig.
4 we see a significant reduction in the variability of the X-ray portion of
the SED, also highlighted in the hard X-ray light-curve which is now more
clearly dominated by the fast variability. Examining the ACF of the X-ray
light-curve we can see this more clearly. There is a narrow core to the HX
ACF, which originates from the fast hot coronal variability. At lower
correlation coefficients ($\sim 0.3$) we see a broader component to the HX
ACF, showing the longer timescale variability from the fraction of disc/warm
coronal variability that propagates into the hot corona.

##  4 Reverberation

###  4.1 X-ray illumination of the disc

Figure 7: Model power-spectra for HX, UVW, U, and V band light-curves, for
models considering intrinsic variability only (black lines), and intrinsic and
disc reverberation (coloured lines). Due to the stochastic nature of the
model, the power-spectra were created by calculating the averaged power-
spectrum from 1000 $\dot{m}$ realisations of each model. Including the disc
reverberation clearly adds a high frequency contribution to the total power,
but has near to no impact on the slow variability or the total (integrated)
power.

We now have a model that can describe the variability intrinsic to the flow.
However, since a portion of the photons emitted by the corona will be incident
on the disc, giving an additional contribution to the local temperature (e.g
Zycki et al. 1999), we would expect the X-rays to imprint additional
variability on to the UV/optical with an additional lag originating from the
light-travel time (e.g Blandford & McKee 1982; Welsh & Horne 1991; McHardy et
al. 2014; Edelson et al. 2015). As we know the variability within the X-ray
corona, it is possible to directly calculate this re-processed variability
within the disc/warm corona, as first done by Gardner & Done (2017), and later
by Mahmoud & Done (2020); Mahmoud et al. (2023); HD23a.

For details on calculating the re-processed variability we refer the reader to
HD23a (sections 2.2 through to 2.4). In terms of the UV/optical variability
the disc reverberation will imprint variability on a similar time-scale to the
intrinsic X-ray variability, since the strongest response comes from close to
the inner edge of the disc where the illuminating X-ray flux is also the
strongest (HD23a). For the model considered here this time-scale will be of
the order $\sim 0.1$ days (light travel time to the warm disc), whereas the
light travel time to the outer disc edge is of order $\sim 2$ days. This puts
some small fraction of the fast variability into the disc emission, but the
effect of this is small. Fig.7 shows the power spectra of the HX,UVW2, U and V
bands, with the black lines showing the intrinsic variability, and the
coloured lines including the disc reprocessed flux. This only makes a
difference to the very fastest variability, and will predict a change from
negative lags at low frequencies (long time-scales) to positive at high
frequencies (e.g Uttley et al. 2014). The overall effect, however, is small
with a minimal contribution to the total power, as shown directly in HD23a.

Hence we also consider the effect of reprocessing from a larger scale wind in
order to increase both the amplitude of reprocessed variability and the lag
timescale, as required by the data (HD23a).

###  4.2 Including a wind

Figure 8: Left: Model light-curve, using the same $\dot{m}$ realisation as in
Fig. 5, calculated for a system with and without a wind (coloured and black
lines respectively). Additionally, propagation and disc reverberation is
included in both scenarios. We see a clear increase in response in both U and
V bands due to the wind, since these have a stronger contribution from the
free-bound continuum than UVW2, as seen in the right panel.
Right: optical/UV SED. The dashed green line shows the intrinsic emission (i.e
warm Compton component from the disc-like structure), while the dashed red
line shows the free-bound continuum from the wind. The solid black line shows
the total SED. The coloured panels show the extraction regions for UVW2, U,
and V bands, where the colours correspond to the light-curves on the left.
Note that we have subtracted out the line-emission from the free-bound
component, for both clarity and computational efficiency.

There is growing evidence for a large scale height wind on the inner edge of
the BLR. This is seen directly in broad blue-shifted UV absorption lines which
correlate with ’neutral’ time variable X-ray absorption (Kaastra et al., 2014;
Cappi et al., 2016; Mehdipour et al., 2016; Dehghanian et al., 2019; Kara et
al., 2021; Netzer, 2022), as well as in the lag spectra, where there is a
prominent jump in the U band from diffuse Balmer continuum emission (Korista &
Goad, 2001, 2019; Cackett et al., 2018; Lawther et al., 2018; Chelouche et
al., 2019).

We use the same geometric wind model as in HD23a, i.e. a bi-conical outflow,
launched at some radius $r_{l}$ at an angle $\alpha_{l}$ with respect to the
disc. The wind then extends to a maximum radius and height,
$r_{w,\mathrm{max}}$ and $h_{w,\mathrm{max}}$, such that the wind has a total
covering fraction $f_{\mathrm{cov}}$ as seen from the central source (i.e
centred on the black hole). However, we now directly calculate the emitted
spectrum from this wind using the cloudy radiative transfer code (Ferland et
al., 2017) so as to explicitly include the Balmer continuum (unlike HD23a
where the emission was simply modelled as a blackbody).

We divide the the wind surface into a polar grid of $\theta$ and $\phi$, with
spacing $d\cos(\theta)=d\phi=0.01$. Each grid-point is then located at some
distance $r_{w}$ from the central source, with corresponding time-lag
$\tau_{w}$, so sees the SED shape from this time. The central source is
somewhat extended, but here we assume for simplicity and computational
efficiency that this is point like at the centre. We use this as input to
cloudy to calculate the wind emission as seen by the observer for each time-
step. For details on the calculation procedure see Appendix A.

For now we will consider a near face on object, as is likely the case for
Fairall 9; our template AGN. In such an object the observer should be looking
down the bi-cone of the wind, such that they only see the emission reflected
off the wind surface. Hence, we only extract the reflected component from the
cloudy output, not the diffuse and transmitted fluxes which are seen by an
observer looking through the wind.

We take wind parameters from those typically observed, so set the column to
$N_{H}=10^{23}$ cm-2 (Mehdipour et al., 2016; Kara et al., 2021) We assume the
wind is launched from $r_{l}=800$ as this gives typical lags of 5-6 days as
seen in Fairall 9 (Hernández Santisteban et al., 2020), and we set the launch
angle of $\alpha_{l}=60$ deg and total covering fraction of
$f_{\mathrm{cov}}=0.3$ (e.g Baskin & Laor 2018). The ionisation state seen
along the line of sight is low, typically $\log\xi\lesssim 0$ so that moderate
Z elements still have a complete K shell (e.g. CIV). This gives a fairly high
density $n_{H}=10^{13}$ cm-3, as is also required to weight the emission to
Balmer continuum rather than lines.

Fig. 8 shows the resultant lightcurves in UVW2, U and V together with the
UV/optical SED. There is only a slight change to UVW2 as the wind makes little
impact on this wavelength range, while the U and V bands are strongly enhanced
by the Balmer and Pashen continuum respectively.

The resulting variability in the wind light-curves is near identical in shape
and time-scales to the intrinsic variability originating from the disc, as can
be seen both by eye in Fig. 8 and in the cross-correlation functions in the
next section (Fig. 9). Unlike the black-body models of HD23a the cloudy models
are responding to the UV/EUV emission, rather than the X-ray. This is because
for a column-density of $10^{23}$ cm-2 the wind will be optically thin to
X-rays above $\sim 3$ keV, and so most of the X-ray flux is not re-processed.
On the other hand, it will be optically thick to the EUV emission. Hence it
will respond to and re-process the energetically dominant, slowly variable,
EUV component, giving an increase in the overall variability power on long
time-scales.

##  5 Lags from Simulated light-curves

Figure 9: Cross correlation functions for HX, UVW2, U, V band light-curves
(segmented into chuncks of 250 days each), with respect to UVW2, for the same
$\dot{m}$ realisation as in Fig. 5 using only the intrinsic variability
(dashed lines), intrinsic and disc reverberation (solid black line), and the
full model including intrinsic variability, disc reverberation, and wind
reprocessing (coloured solid lines). The dotted vertical line indicates 0-lag,
while the zoomed inset highlights the peak of each CCF; in order to highlight
the effect of each model component. It is clear that the disconnect between
the X-ray and UV/optical is driven by the intrinsic variability, while
including disc and wind reverberation will shift the peak CCF from inward lags
(i.e optical before UV/X-ray) to outward lags.

Our model generates a fully time dependent SED, so we can extract lightcurves
in any energy band, on any sampling, to explore the predicted behaviour, for a
variety of AGN configurations. In this section we explore this in detail,
starting with the cross-correlation function. Note that we always measure the
cross-correlation function with respect to the UVW2 bandpass, as is generally
done in intensive monitoring campaigns (e.g Edelson et al. 2017; Hernández
Santisteban et al. 2020; Vincentelli et al. 2021) since it is the Swift filter
closest to the thermal peak of the SED, although we note that there are
exceptions (e.g Edelson et al. 2015; Fausnaugh et al. 2016) when higher energy
bandpasses are available (through e.g HST). We also note that all cross-
correlations are calculated following Gardner & Done (2017).

We start by generating models considering propagation only, then include disc
reverberation, then finally the full model which includes propagation, disc
reverberation and a wind. These are all run using the same initial seed as
previously (1113 in the numpy random generator), such that our comparisons
show real physical differences rather than random effects unique to each
realisation.

Fig. 9 shows the resulting cross-correlation functions for these model
realisations, for Swift HX (1.5-10 keV), UVW2, U, and V bands, all correlated
against UVW2. The first thing to note is that the correlation between HX and
UVW2 is significantly worse than those for U and V, for all models. This is as
expected, and highlights the result from the previous section that the UV-X-
ray disconnect can be explained through propagating fluctuations. It is the
two distinct regions (disc/warm corona and hot corona) varying intrinsically
on different time scales, but where some variations leak from one to the
other, that give highly correlated UV/optical light-curves (as these should
all originate from the disc/warm corona) with only marginal correlations
between the X-ray and UV/optical. We stress that it is not enough to simply
have two distinct regions varying completely separately from one another, as
although this would give a disconnect between the UV/optical and the X-ray,
the resulting light-curves would appear near incoherent. Data on the other-
hand show correlations between the X-ray and UV/optical that are better than
one would expect for two separate incoherent processes (e.g Edelson et al.
2019).

Continuing with the X-ray to UVW2 correlation, we see a strong peak feature at
short time-scales when we include reverberation. This originates because
X-rays re-processing on the disc will imprint the X-ray variability onto the
UV/optical light-curves. The response in this case will not be very large, as
the SED is strongly dominated by the UV/optical, however it is sufficient that
the cross-correlation will pick up an additional semi-coherent signal on fast
time-scales (as the X-rays are intrinsically dominated by fast variability),
hence the narrow shape of the peak. Additionally, it can also be seen that the
peak of this feature is shifted to $\tau\sim-0.6$ days. This is simply the
light-travel time from the X-ray corona to the inner edge of the disc.

Moving onto the UV/optical cross-correlations (still with respect to UVW2), we
no longer see a disconnect. For all models the UV/optical light-curves are
well correlated with one another, as these all originate from the disc/warm
corona, and as such are driven by the same physical process. One could perhaps
think that because each radial annulus within the flow generates its own
variability, then the observed correlation should not be this good and have a
strong negative lag from the propagations which will be correlated. However,
that would imply that the emission seen from each bandpass does not overlap in
radii. This is of course not the case. Firstly we are considering broad-band
filters, and as such these should receive flux from a range of energies, and
secondly since each radial bin emits broad-band spectra (either black-body or
Comptonised black-body) there should be a dilution effect between each radial
bin, leading to the observed broad-band light-curves sampling a wide range of
effective area. This in turn leads to well correlated UV/optical light-curves,
as there will be overlap between the radii being sampled, as well as a
reduction in the observed lag.

Comparing individual models in the UV/optical correlations we see a systematic
shift from negative to positive lags when going from propagation only to
including disc reverberation and finally including both disc and wind
reverberation. This is not entirely surprising. The propagation only model
will naturally give negative lags, as the mass accretion rate fluctuations
move from the outer (cool) regions to the inner (hot) regions, and as such
will give light-curves from lower energies leading those from higher energy.
Once we include disc reverberation this changes. As seen with the HX CCFs disc
reverberation will imprint an additional feature on short time-scales while
leaving the longer time-scales similar to the propagation case. We see a
similar effect in the UV/optical CCFs, albeit not as pronounced as in HX. The
general shape of the UV/optical CCFs remains the same, but with the peaks
shifted to the right and a slight increase in correlation coefficient on the
shortest time-scales. Unlike the HX CCFs, however, we do not see a strong
spike on short time-scales induced by the disc reverberation. This is due to
the warm corona being the energetically dominant component within the SED (by
$\sim$ an order of magnitude). As such the X-rays make a tiny impact to the
total emission, and hence the variability. If the systems SED was such that
the X-rays dominate, we would obtain UV/optical light-curves dominated by
reverberation and would then see a significantly stronger impact in the CCF
caused by the rapid re-processed variability. Nonetheless, even in our
UV/optical light-curves, which are clearly intrinsically dominated, the
introduction of a weak fast coherent signal is enough to shift the peak
marginally to the right.

When we include a wind the impact on the UV/optical CCFs become more
pronounced, than for just disc reverberation, giving a stronger shift towards
a positive (outgoing) lag. As seen in the previous section, the wind re-
processes the UV/EUV emission, and gives a strong contribution in the optical
part of the SED through the free-bound continuum. As such the re-processed
wind signal is simply a lagged version of the disc/warm corona emission. The
total observed lag will then depend on the relative contribution to the SED
from each component at the relevant energy. For energies/wavelengths where the
wind emission dominates (e.g U/V band and below) the observed lag will be
high, as highlighted in the U and V band CCFs.

It is worth noting that the X-ray CCF is almost unaffected by the inclusion of
a wind. This comes from two parts. Firstly, the wind is optically thin to the
X-ray emission (as discussed previously), and secondly the UVW2 band (which we
are correlating against) is dominated by the intrinsic disc/warm corona
emission. Hence, in this case we are comparing the emission coming directly
from the flow (intrinsic and reverberation), even in the case where a wind is
present.

###  5.1 Lag-Spectra

Figure 10: The data-point show the lags measured for Fairall 9 from Hernández
Santisteban et al. (2020). The lines are lags from the models taken from the
centroid of the CCF of the lightcurve at wavelength $\lambda$ with respect to
1928 Å. The solid lime green line shows results only including intrinsic
propagation model. This gives negative (longer wavelengths lead shorter
wavelengths) lags as the fluctuations start in the outer disc and propagate
inwards. Including X-ray reverberation from the disc gives the solid blue
line. The X-ray reverberation is too small to cancel the negative lag from
propagation, so the lightcurves including both propagation and disc
reverberation still have longer wavelengths leading. The dashed blue line
shows the effect of X-ray reverberation from the disc alone without the
intrinsic propagation, showing how these predicted reverberation lags are
smaller than measured from the data. Finally, the orange solid line shows the
lags from the full model, including UV (and X-ray) reverberation from a wind
(as well as propagation and disc reverberation). Here we see a clear increase
in lag with wavelength due to the increased contribution of the wind at longer
wavelengths, with clear features corresponding to the Balmer and Paschen
continua, Fig. 8.

So far we have only compared light-curves extracted for broad-band filters (in
particular Swift-UVOT filters). However, our model predicts the full variable
SED, and as such allows us to extract light-curves at the spectral resolution
of the model ($d\log(E/\mathrm{keV})=1/125$). This then allows us to predict
the lag-spectrum one would expect from each model, by comparing the model
light-curve in each energy bin to a single reference light-curve. In this case
we pick our reference as $\lambda_{\mathrm{ref}}=1928$ Å, as this is the
centre of the Swift-UVOT UVW2 bandpass, and as such gives the cleanest
comparison to current intensive monitoring campaigns.

For each light-curve extracted from each energy (wavelength) bin in our
variable SED, we calculate the cross-correlation function with respect to the
light-curve in the energy bin that covers $1928$ Å, with the light-curves
chunked into segments of 250 days. We then use these CCFs to extract an
estimate for the model lag as a function of energy (wavelength), defined as
the lag that corresponds to the maximal correlation coefficient. We measure
this centroid lag from the top 20 % of the CCF, following the method used in
the data analysis papers (e.g Peterson et al. 1998; Edelson et al. 2019).

The resulting lag spectrum from each model is shown in Fig. 10. For the
propagation only model (green line) we see a strong negative lag, increasing
to lower energy (higher wavelength), as expected for propagating fluctuations.
An interesting point to note here is the measured lag of ($\sim-2$ days) is
considerably shorter than the model propagation time of $\sim 20$ days between
the inner and outer disc radii (chosen for computational efficiency rather
than physical expectations). This is due to blackbody emission being broad, so
a single wavelength does not just contain emission from a single radius. Hence
the observed time-delay will instead be more representative of the propagation
time between the flux weighted radii for each energy, which will naturally be
smaller than the difference between the outer and inner edge of the disc.

When we include disc reverberation (blue line in Fig. 10) we see a reduction
in the negative lag, not a switch to positive. This is interesting, as it
shows that X-ray reverberation from the disc reverberation does not contribute
enough signal to the light-curves to overcome the negative propagation lag. We
also show (blue dashed line) the reverberation lag alone, without any
underlying propagation through the disc. This clearly shows that X-ray
reverberation in the disc gives a lag which is a few smaller than that
measured, and it is then clear that the propagation plus X-ray disc
reverberation lag prediction is approximately the sum of the negative
propagation lag and the positive disc reverberation lag.

Including reverberation from the wind is the key to matching the data (orange
line in Fig. 10), compared to the data points from Hernández Santisteban et
al. (2020). The lag-wavelength relation calculated for the full model
(propagation, disc reverberation, and wind reverberation) follows the data
remarkably well, especially as this is not a best fit. The model parameters
(generator and propagation timescales, wind size scale) were simply examples
given to roughly match some of the observed properties of the data, yet the
model has roughly both the correct shape and normalisation.

The wind reprocesses the entire SED, but this is dominated by the EUV band, so
the total light-curve in each band is a superposition of the light-curve
emitted by the disc, and that reprocessed by the wind, which will be near
identical to that of the disc but with some lag and additional smoothing on
short time-scales. The measured lag rises with wavelength, not because the
wind lag is intrinsically longer, but because the wind contributes more to the
spectrum at longer wavelengths (see Fig. 8). The spectrum at each wavelength
has a contribution from the intrinsic disc, which is highly correlated with
the small (negative) lag at each wavelength, and the wind reverberation, which
is lagged by a constant (positive) value $\tau_{\mathrm{wind}}$. The measured
lag is then the flux weighted lag from each component, so
$\tau\approx(F_{\mathrm{disk}}(\lambda)\tau_{\mathrm{disk}}(\lambda)+F_{\mathrm{wind}}(\lambda)\tau_{\mathrm{wind}})/F_{\mathrm{tot}}$,
where $\tau_{\mathrm{wind}}$ is constant with wavelength. Only when the wind
is the only component in the spectrum does the lag tend to
$\tau_{\mathrm{wind}}$, otherwise it is diluted (Uttley et al., 2014) by the
intrinsic disc emission which is in both the lightcurves (reference band at
$1928$ which has almost no wind contribution and at $\lambda$). This gives the
characteristic wind lag shape, with lags increasing at longer wavelengths,
with sharp features from the Balmer and Paschen edges superimposed, and our
example model parameters give a fairly good quantatative match to the observed
lags despite not being fit to the data.

##  6 Conclusions

The intensive broadband monitoring campaigns on AGN give simultaneous
lightcurves from optical to X-ray energies. These contain much more
information than simply the lag between any two bands. The amount of lagged
signal is also important as a diagnostic of the geometry, as is the timescale
of variability. A reverberation origin for the UV variability predicts a
lightcurve which is a lagged and smoothed version of the driving X-ray
lightcurve, with lag timescale similar to the smoothing timescale. This is in
clear conflict with the data, where the UV lags behind the X-rays by $\sim 1$
day, while it is smoothed on timescales of $\sim 20$ days. This is the main
issue with X-ray reverberation models, not that the lag timescale is a factor
$\sim 2$ bigger than predicted but that the X-ray and UV lightcurves are so
different.

Instead, we develop a full spectral-timing model to use all of the information
in these intensive broadband monitoring datasets. We use the truncated (warm
Compton) disc/hot inner flow geometry which successfully matches the SED in
Fairall 9 (and other AGN with $L/L_{\mathrm{Edd}}\sim 0.05-0.2$). We assume
that there are intrinsic fluctuations stirred up in the disc, which propagate
down and modulate the much faster fluctuations stirred up in the hot flow.
This produces intrinsic UV variability which is much slower than the intrinsic
X-ray variability, with the slow UV variability correlated with and leading
the slow X-ray variability. The fast X-ray variability has no intrinsic
correlation with the UV as it is stirred up only in the hot flow. Its
reverberation on the truncated disc produces a very small amplitude, fast
variable, lagged signal in the UV, but the UV variability is dominated by the
intrinsic slow fluctuations. Instead, in our model the majority of the lag
seen in the optical/UV arises from the intrinsic slow variable UV/EUV
reverberating off a wind on the inner edge of the BLR. This gives a much
larger amplitude signal than X-ray reverberation from the disc firstly as the
EUV is where the SED peaks, so there is more intrinsic flux than in the
X-rays, and secondly because a vertically extended wind intercepts much more
flux from a central source than a flat disc. The wind reverberation signal is
a lagged and smoothed version of the UV/EUV, but these are already
intrinsically variable only on long timescales (unlike the X-rays), so there
is no longer a mismatch between the lag and smoothing timescale. Perhaps the
most un-intuitive aspect is that the increasing lag as a function of
wavelength is not produced by increasing the scale of the reverberating
structure at longer wavelengths. Instead, in this model it is produced by the
increasing fraction of a fixed size scale reverberation signal at longer
wavelengths, as bound-free continuum has a redder spectrum than the warm disc.

Thus the model succeeds in qualitatively explaining all the puzzling features
of the Fairall 9 dataset, but it does have multiple free parameters even after
the disc/hot flow radius is set by the SED. There is the intrinsic variability
timescale as a function of radius, the propagation timescale as a function of
radius, and the size scale of the wind. These were not derived from a fit to
the data, but simply chosen from order of magnitude arguments about what was
needed.

The model predicts that the warm Compton disc has both SED and variability
peaking in the EUV. There is some evidence for increased variability in the
EUV, e.g from He II photo-ionised line (e.g Homan et al. 2023). However, this
overpredicts the observed X-ray variability if all of the warm disc
fluctuations propagate into the X-ray corona (unlike the BHB, see Kawamura et
al. 2023).

The model can then quantitatively match the observed lags between all the
bands, and the observed variability timescales (auto-correlation function
widths) and amplitudes of variability in each band. These successes highlight
the importance of its fundamental assumption, which is that the (warm Compton)
disc is intrinsically variable. This requires that the disc structure is very
different to the Shakura-Sunyaev prediction. Understanding this would lead to
breakthrough in our understanding of the energy generating structure in AGN. .

## Acknowledgements

We thank Juan Hernández-Santisteban and the entire IBRM collaboration for all
their efforts in organising these campaigns and extracting the data,
motivating this study.

SH acknowledges support from the Science and Technology Facilities Council
(STFC) through the studentship grant ST/V506643/1. CD acknowledges support
from STFC through grant ST/T000244/1.

This work used the DiRAC@Durham facility managed by the Institute for
Computational Cosmology on behalf of the STFC DiRAC HPC Facility
(www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC
capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and ST/S002502/1,
Durham University and STFC operations grant ST/R000832/1. DiRAC is part of the
National e-Infrastructure.

This work made use of the following python modules: scipy (Virtanen et al.,
2020), numpy (Harris et al., 2020), and astropy (Astropy Collaboration et al.,
2013, 2018, 2022). Additionally, all plots were made using matplotlib (Hunter,
2007).

## Data Availability

No new data were produced during this study. The SED in Fig. 2 used archival
data, available from HEASARC (https://heasarc.gsfc.nasa.gov/cgi-
bin/W3Browse/w3browse.pl). The model code developed throughout this study is
available via the corresponding authors GitHub
(https://github.com/scotthgn/AGNvar2).

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##  Appendix A Modelling the wind variability

To model the wind variability we start by considering a bi-conical outflow
launched from radius $r_{l}$ at angle $\alpha_{l}$ with respect to the disc.
The wind subtends a solid angle $\Omega=4\pi f_{\mathrm{cov}}$ as seen from
the central black hole, where $f_{\mathrm{cov}}$ is the covering fraction.
This geometry is identical to that used in HD23a. Here we provide details on
the main calculations of the wind emission, but refer the reader to Appendix A
in HD23a for details on the geometry

Unlike HD23a we use cloudy (v.17.01 Ferland et al. 2017) to calculate the
emission from the wind. As we are mainly interested in the response of the
free-bound emission to the ionising continuum we use a reatively simple cloudy
model, defined by the Hydrogen column desnity, $N_{H}$, Hydrogen number
density, $n_{H}$, covering fraction, $f_{\mathrm{cov}}$, and SED shape. The
strength of the reflected free-bound continuum depends on the intensity of the
ionising continuum at the wind, and so the distance from the illuminating
source. The wind clearly subtends a range of radii from the central region.
However, for simplicity we only calculate a single cloudy run for the wind, at
a distance set in the middle between the launch radius and maximal radial
extent (defined through $f_{\mathrm{cov}}$ \- see HD23a). This will give of
the order correct emission power, and significantly reduces the computational
cost when evaluating the time-dependent version later.

The wind is defined as launching from both sides of the disc, however the
observer will only see the emission from the side facing the observer. Hence,
for the total observed free-bound wind emission, $L_{\mathrm{wind}}$, we only
extract half the total reflected emission given by cloudy, such that:

$L_{\mathrm{wind}}(E)=\epsilon_{\mathrm{wind}}(E)2\pi f_{\mathrm{cov}}$ (7)

where $\epsilon_{\mathrm{wind}}(E)$ is the emissivity of the wind surface
calculated by cloudy.

We stress here that we use the reflected emission, as for our geometry the
observer would be looking down the wind funnel, and therefore only see
emission from the side of the wind facing the illuminating source, not the
transmitted or diffuse components that are emitted from the opposite side (as
defined by cloudy). We also note that this assumes the wind emission is
isotropic (within the lines of sight that will see the wind face), as you can
expect from a diffuse medium.

To calculate the variability of the wind we also need to consider the light-
travel time to different section of the wind. Hence, we start by dividing the
wind into a polar grid in $\cos(\theta)$ and $\phi$, where $\phi$ is the
azimuthal angle in the x-y plane and $\theta$ is the polar angle measured from
the z-axis. The grid is linearly space between $0$ and $2\pi$ with spacing
$d\phi=0.01$, and $0$ and $\cos(\theta_{m})$ with spacing
$d\cos(\theta_{m})=0.01$, for $\phi$ and $\theta$ respectively. Here
$\theta_{m}$ is the maximal polar angle of the wind, defined in terms of the
covering fraction such that $\cos(\theta_{m})=f_{\mathrm{cov}}$. As each grid-
point is defined in terms of $\phi$ and $\theta$, the solid angle of each
grid-point (as seen by the central source) is
$d\Omega_{\mathrm{grid}}=d\cos(\theta)d\phi$. This solid angle is important,
as we use it later to calculate the relative contribution from each grid-point
when calculating the time-dependent emission.

The light-travel time to a grid-point on the wind surface is (HD23a):

$\tau_{w}(\theta,\phi)=\\\
\frac{R_{G}}{c}\left[\sqrt{r_{w}^{2}(\theta)+h_{w}^{2}(\theta)}-h_{w}(\theta)\cos(i)-r_{w}(\theta)\sin(i)\cos(\phi)\right]$
(8)

where $r_{\mathrm{wind}}$ and $h_{\mathrm{wind}}$ are the radius (in the x-y
plane) and the height of the wind grid-point respectively, and are given by:

$r_{\mathrm{wind}}(\theta)=\frac{r_{l}\tan(\alpha_{l})}{\tan(\alpha_{l})-\tan(\pi/2-\theta)}$
(9)

$h_{\mathrm{wind}}(\theta)=r_{\mathrm{wind}}(\theta)\tan(\pi/2-\theta)$ (10)

We note here that Eqn. 8 is the delay assuming the distance from the central
black-hole, whereas the equation in HD23a also includes a term for the height
of the corona above the black hole (assuming a lamppost geometry).

To calculate the wind emission at a given time we now assume the response to
changes in the continuum are linear, bit not necessarily 1:1. This simplifies
the problem significantly, as we now only run two cloudy models. One for the
minimum and another for the maximum luminosity values within the SED;
$L_{\mathrm{min}}(E)$ and $L_{\mathrm{max}}(E)$. The two cloudy models then
give the minimum and maximum emissivity values of the free-bound wind
emission; $\epsilon_{\mathrm{wind},\mathrm{min}}(E)$ and
$\epsilon_{\mathrm{wind},\mathrm{max}}(E)$. The wind emission for any given
intrinsic SED at any given time within our time-series (and hence within
$L_{\mathrm{min}}(E)$ and $L_{\mathrm{max}}(E)$) is then given by:

$\epsilon_{\mathrm{wind}}(E)=f(E)\epsilon_{\mathrm{wind},{min}}(E)+(1-f(E))\epsilon_{\mathrm{wind},{max}}(E)$
(11)

where $f(E)$ is an energy dependent interpolation factor given by:

$f(E)=\frac{L(E)-L_{\mathrm{max}}(E)}{L_{\mathrm{min}}(E)-L_{\mathrm{max}}(E)}$
(12)

where $L(E)$ is the intrinsic SED seen by the wind. We can make this time
dependent by including the time-delay to a point on the wind by writing $L(E)$
as $L(E,t-\tau(\theta,\phi))$, such that $f(E)$ becomes $f(E,t,\theta,\phi)$.
Of course, we need to take into account the size of a grid-point relative to
the total wind area when calculating the time-dependent emission from each
grid. For this we simply note that integrating
$d\Omega_{\mathrm{grid}}=d\cos(\theta)d\phi$ over the wind surface will give
$\Omega_{\mathrm{wind}}/2=2\pi f_{\mathrm{cov}}$. Hence, the luminosity of a
given grid point at time $t$ is simply

$L_{\mathrm{wind},\mathrm{grid}}(E,t,\theta,\phi)=\epsilon_{\mathrm{wind}}(E,t,\theta,\phi)d\cos(\theta)d\phi$
(13)

and hence the total time-dependent wind emission is simply a sum over all grid
points, such that:

$L_{\mathrm{wind}}(E,t)=\sum_{\cos(\theta)=0}^{\cos(\theta_{m})}\sum_{\phi=0}^{2\pi}L_{\mathrm{wind},\mathrm{grid}}(E,t,\theta,\phi)$
(14)

This is of course just for the free-bound wind component. The total variable
SED is then a sum of the intrinsic and wind components, such that
$L_{\mathrm{tot}}(E,t)=L(E,t)+L_{\mathrm{wind}}(E,t)$.