11institutetext:  Dipartimento di Matematica e Fisica, Università Roma Tre,
via della Vasca Navale 84, I-00146 Rome, Italy 11email:
<EMAIL_ADDRESS>22institutetext: IRAP, Université de Toulouse, CNRS,
UPS, CNES 9, Avenue du Colonel Roche, BP 44346, F-31028, Toulouse Cedex 4,
France 33institutetext: INAF – Osservatorio Astrofisico di Arcetri, Largo
Enrico Fermi 5, I-50125 Firenze, Italy 44institutetext: Department of Physics
and Institute of Theoretical and Computational Physics, University of Crete,
71003 Heraklion, Greece 55institutetext: Institute of Astrophysics, FORTH,
GR-71110 Heraklion, Greece 66institutetext: Astronomical Institute of the
Czech Academy of Sciences, Boční II 1401, CZ-14100 Prague, Czech Republic
77institutetext: MIT Kavli Institute for Astrophysics and Space 2 Research,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA

# Broadband X-ray/UV/optical time-resolved spectroscopy of NGC 5548: The
origin of the UV/optical variability in active galactic nuclei

E. Kammoun Broadband X-ray/UV/optical time-resolved spectroscopy of NGC 5548:
The origin of the UV/optical variability in active galactic nucleiBroadband
X-ray/UV/optical time-resolved spectroscopy of NGC 5548: The origin of the
UV/optical variability in active galactic nucleiBroadband X-ray/UV/optical
time-resolved spectroscopy of NGC 5548: The origin of the UV/optical
variability in active galactic nucleiBroadband X-ray/UV/optical time-resolved
spectroscopy of NGC 5548: The origin of the UV/optical variability in active
galactic nucleiBroadband X-ray/UV/optical time-resolved spectroscopy of NGC
5548: The origin of the UV/optical variability in active galactic
nucleiBroadband X-ray/UV/optical time-resolved spectroscopy of NGC 5548: The
origin of the UV/optical variability in active galactic nuclei I. E. Papadakis
Broadband X-ray/UV/optical time-resolved spectroscopy of NGC 5548: The origin
of the UV/optical variability in active galactic nucleiBroadband
X-ray/UV/optical time-resolved spectroscopy of NGC 5548: The origin of the
UV/optical variability in active galactic nucleiBroadband X-ray/UV/optical
time-resolved spectroscopy of NGC 5548: The origin of the UV/optical
variability in active galactic nucleiBroadband X-ray/UV/optical time-resolved
spectroscopy of NGC 5548: The origin of the UV/optical variability in active
galactic nuclei M. Dovčiak Broadband X-ray/UV/optical time-resolved
spectroscopy of NGC 5548: The origin of the UV/optical variability in active
galactic nucleiBroadband X-ray/UV/optical time-resolved spectroscopy of NGC
5548: The origin of the UV/optical variability in active galactic nuclei C.
Panagiotou Broadband X-ray/UV/optical time-resolved spectroscopy of NGC 5548:
The origin of the UV/optical variability in active galactic nucleiBroadband
X-ray/UV/optical time-resolved spectroscopy of NGC 5548: The origin of the
UV/optical variability in active galactic nuclei

(Received ; accepted )

###### Abstract

Context. Recently, nearby bright galaxies have been the subject of long
monitoring surveys in the X-rays, UV, and optical. All of these campaigns
revealed a strong correlation between the various UV and optical bands, with
time lags that increase with wavelength. In a series of papers, we have
demonstrated that a scenario in which an X-ray source located on the rotation
axis of the central black hole illuminating the accretion disc is a viable
explanation for the observed correlations. However, some of the monitored
sources showed low or moderate correlation between the X-rays and the UV,
which could challenge this scenario.

Aims. In this paper, we analyse the time-averaged and the variable broadband
X-ray/UV/optical spectral energy distributions (SEDs) of NGC 5548, one of the
most intensely monitored Seyfert 1 galaxies, using Swift, HST, and ground-
based telescopes. The aim of this paper is to test whether the broadband
spectral behaviour of the source could be explained with the X-ray
illumination hypothesis, despite the apparently moderate correlation between
the X-rays and longer wavelength.

Methods. We modelled the broadband time-averaged SED, from the Space Telescope
and Optical Reverberation Mapping (STORM) monitoring campaign of the source
performed using the KYNSED model, which assumes X-ray illumination of the
disc. We also modelled 15 time-resolved SEDs extracted from the same campaign
to check whether this model can account for the spectral variability seen in
the various wavebands during the monitoring. We assumed in our modelling that
the X-ray corona is powered via the accretion process.

Results. Our results show that the proposed scenario could describe the time-
averaged and the time-resolved SEDs of NGC 5548 perfectly well. In this
scenario, the height of the corona, the X-ray photon index, and the power that
is transferred to the corona all vary. This would explain the variability
behaviour at the various wavelengths. The best-fit model is obtained for a
non-spinning black hole accreting at a constant rate of 5% of its Eddington
limit. Since each of the variable parameters will affect the observed flux in
a particular way, the combined variability of all of these parameters will
then account for the moderate correlation between the X-rays and UV/optical.

Conclusions. We demonstrate in this paper that X-ray illumination of the
accretion disc can actually explain the observed properties of NGC 5548. In
fact, this model not only fits the broadband spectra of the source well, but
it also explains the time-lag behaviour as a function of wavelength as well as
the power spectral distribution, providing a complete description of the
behaviour of this source.

###### Key Words.:

Accretion, accretion discs – Galaxies: active – Galaxies: Seyfert

##  1 Introduction

Active galactic nuclei (AGN) are the most luminous persistent sources of
electromagnetic radiation in the universe. It is commonly accepted that these
objects are powered by accretion onto a supermassive black hole (SMBH) in the
form of an optically thick and geometrically thick accretion disc (Shakura &
Sunyaev, 1973; Novikov & Thorne, 1973). AGN are also known to be strong X-ray
emitters. However, the nature and the geometry of the X-ray source (known as
the X-ray corona) remains unknown. Owing to the difficulties in spatially
resolving the accretion disc and the X-ray corona using current facilities,
various techniques have been proposed to study the innermost regions of the
accretion flow. For example, gravitational microlensing has been used to study
the structure of the accretion disc and the X-ray corona (e.g. Pooley et al.,
2007; Chartas et al., 2009; Mosquera et al., 2013; Blackburne et al., 2014;
Chartas et al., 2016). Alternatively, reverberation mapping techniques have
been widely used to study the different components of AGN. This technique uses
the time lags between light curves observed at different wavelengths to infer
the geometry and kinematics of the regions emitting at those wavelengths (see
Cackett et al., 2021, for a recent review).

Originally, this technique is used to measure the lags between the light curve
in the H$\beta$ emission line and the UV continuum, aiming at studying the
size scale and the dynamics of the broad line region (BLR; e.g. Blandford &
McKee, 1982; Peterson, 1993; Peterson et al., 2004). A variant of this
technique was later used to study time delays between the variations in
various X–ray wavebands. The objective was to constrain the geometry of the
X–ray source with respect to the accretion disc, via the study of time-lags
spectra between bands dominated by the X–ray continuum emission and those
bands where the X–ray reflection features are dominant (e.g. Fabian et al.,
2009; De Marco et al., 2013; Kara et al., 2016; Emmanoulopoulos et al., 2011;
Epitropakis et al., 2016; Caballero-García et al., 2018, 2020).

More recently, a huge observational effort has been invested in long
monitoring campaigns of many Seyfert galaxies using ground-based and space-
based observatories in order to investigate the correlated (or not) variations
of AGN in the X–rays and the UV/optical bands (see e.g. De Rosa et al., 2015;
Fausnaugh et al., 2016; McHardy et al., 2018; Cackett et al., 2018; Edelson et
al., 2019; Cackett et al., 2020; Hernández Santisteban et al., 2020; Pahari et
al., 2020; Kara et al., 2021; Cackett et al., 2023; Miller et al., 2023; Kara
et al., 2023; Kumari et al., 2023, 2024). The main objective is to test
theoretical models for the interplay between the X–ray and the fast UV/optical
variations in AGN. For example, in the case when the central X-ray corona
illuminates the accretion disc, most of the X-rays are absorbed by the disc,
which adds to its heating. Since the X-rays are highly variable in AGN, this
mechanism drives variations in the thermally reprocessed continuum emission of
the disc. These variations are seen with some delay which depends on the
location of the X-rays above the disc as well as the BH mass, the disc
accretion rate, etc.

A standard Shakura & Sunyaev (1973) accretion disc gives rise to wavelength-
dependent lags following $\tau\propto\lambda^{4/3}$ (see Cackett et al., 2007,
and references therein), and the cross-correlation results from the monitoring
campaigns confirmed this prediction. However, it was originally thought that
the amplitude of this relation underestimated the observed lags by a factor of
$\sim 3-4$. Recently, we published the results from a detailed study of X–ray
thermal reverberation in a series of papers. Kammoun et al. (2021a, K21
hereafter) studied in, detail, the response of a Novikov & Thorne (1973) disc
to X-ray illumination by a point-like source, located above the central BH,
taking all relativistic effects into account. Kammoun et al. (2019, 2021b)
showed that, when treated properly, the model time lags in the case of X-ray-
illuminated discs are in perfect agreement with the observations, contrary to
the previous claims. In addition, the K21 model can also explain the observed
power spectra in the optical/UV bands of NGC 5548 (Panagiotou et al., 2020,
2022a).

Recently, Kammoun et al. (2023, K23 hereafter) published a new code that can
be used to fit the observed time-lags when X–rays illuminate the disc, both in
the case when the X–ray corona is powered by the accretion process, and when
it is powered externally, by an unknown source not directly associated with
the accretion power. In the former case, the model assumes that the accretion
power dissipated in the disc below a transition radius, $r_{\rm tr}=R_{\rm
tr}/$$r_{\rm g}$111$r_{\rm g}$$=G$$M_{\rm BH}$$/c^{2}$ is the gravitational
radius of a BH with a mass of $M_{\rm BH}$., is transferred to the X-ray
source (by a yet unspecified physical mechanism). In this way, a direct link
between the accretion disc and the X–ray source is set. They also considered
various colour-correction factors, $f_{col}$, for the disc emission, and they
presented results from fits to the time-lags spectra of four sources. In fact,
electron scattering is known to play an important role and can lead to
deviations from blackbody emission. In general, transfer of energy between
electrons and photons that are Compton scattered enforces a Wien tail at the
high energy end of the spectrum. The resulting spectra can be approximately
modelled by modifying the blackbody spectrum using $f_{\rm col}$ as a
multiplicative factor by which spectral features are shifted to higher
energies, while the normalisation of the spectrum is multiplied by $f_{\rm
col}$-4 to keep the frequency integrated flux fixed. In this case the modified
blackbody spectrum can be expressed as follows,

$I_{\nu}=\frac{2h}{c^{2}f_{\rm col}^{4}}\frac{\nu^{3}}{\exp(\frac{h\nu}{f_{\rm
col}kT})-1},$ (1)

where $I_{\nu}$ is the specific intensity, $k$ is Boltzmann’s constant, $h$ is
Planck’s constant, and $T$ is the disc temperature.

Time-lag spectra (i.e. distribution of time lags as a function of wavelength)
are just one of the many tests one can perform in order to test a theoretical
model for the X–ray/UV/optical emission in AGN. The available data from the
long, multi-wavelength light curves can be used to test models in various
ways. We aim to explore two of them, using data from the original Space
Telescope and Optical Reverberation Mapping (STORM) multi-wavelength
monitoring campaign in 2014 (De Rosa et al., 2015; Fausnaugh et al., 2016).
NGC 5548 is, perhaps, the best target for these tests at the moment due to the
quality of the available light curves (which are long and very well sampled)
but also because the host galaxy emission and the ‘small-blue-bump’ emission
(i.e. the blended FeII with the Balmer continuum components) are well
established in this source.

Firstly, we use the spectral model KYNSED to fit the average SED of the
source. KYNSED was developed by Dovčiak et al. (2022) (D22 hereafter) to
calculate the broadband optical/UV/X-ray spectral energy distribution (SED) of
an AGN, assuming a Novikov & Thorne (1973) accretion disc which is illuminated
by a point-like X-ray corona. The model can provide the theoretical, broad
band SED both in the case when the X–ray corona is powered by an unknown
source as well as in the case when it is powered by the accretion process
itself, similar to the case of the time lag model described above. In the
latter case, a direct link between the accretion disc and the X–ray source is
set. D22 already showed that this model can fit well the mean SED of NGC 5548.
Our aim is to investigate whether the model can fit well the mean SED assuming
the best-fit results of Kammoun et al. (2023). In this way, we will be able to
demonstrate that the model can fit well both the average SED and the average
time-lags spectra in this source.

Secondly, we construct numerous SEDs, sampling X–ray, UV and optical data
throughout the campaign and investigate whether X–ray reverberation can indeed
explain the complex, X–ray/UV/optical spectral variability of the source. The
second objective aims at directly addressing a puzzling result from the cross-
correlation analysis of the X–ray and the UV/optical light curves. The STORM
light curves of NGC 5548 implied a rather low cross-correlation amplitude
between the X–ray and the UV/optical light curves. Even by visual inspection,
the shape of the X-ray light curve did not appear capable of driving the
UV/optical variability (Gardner & Done, 2017; Starkey et al., 2017).
Panagiotou et al. (2022a) already showed that an apparent low cross-
correlation between the X–ray and the UV/optical light curves can be explained
in a physical way within the X–ray reverberation model. This work shows that,
indeed, the multiwavelength light curves of NGC 5548 are fully consistent with
the assumption of X–ray thermal reverberation being the only driver of the
observed variations in the UV/optical bands.

The paper is organised as follows. In Sect. 2, we present the fitting of the
time-averaged SED of NGC 5548. In Sect. 3 we present the analysis of the time-
variable SED of NGC 5548. Finally, we summarise our results and discuss the
main implications of our work in Sect. 4.

##  2 Fitting the average SED

We use the same data set presented in D22 as described in their Section 4.1 to
construct the mean SED. We also adopt the same model used in D22 to it the
data. The overall model, written in XSPEC (Arnaud, 1996) parlance, is as
follows,

$\displaystyle\rm Model$ $\displaystyle=$ $\displaystyle{\rm\tt
extinction_{Czerny}\times{\tt KYNSED}_{UV/opt.}}$ (2) $\displaystyle+$
$\displaystyle{\rm\tt TBabs\times zTBabs\times zxipcf\times{\tt KYNSED}_{\rm
X-rays}}.$

We use KYNSED twice; one for the optical/UV and the other for the X-ray
spectra. Obviously, all the relevant KYNSED parameters were tied in the
UV/optical and X-rays range. The TBabs component (Wilms et al., 2000) accounts
for the effects of the Galactic absorption in the X–ray band (the UV/optical
data points are already corrected for Galactic extinction). We fix its column
density to $1.55\times 10^{20}\leavevmode\nobreak\ \rm cm^{-2}$ (HI4PI
Collaboration et al., 2016).

We also consider the possibility of intrinsic absorption in the host galaxy.
We assume the extinction curve of Czerny et al. (2004) for the absorption in
the optical/UV. The extinctionCzerny component is determined by $E(B-V)_{\rm
host}$ which is left as a free parameter of the fit. The zTBabs component
accounts for the absorption in the X–ray band of the same absorber, and its
column density was kept linked to $E(B-V)_{\rm host}$ through the relation
$N_{\rm H,host}=8.3\times 10^{21}\leavevmode\nobreak\ E(B-V)_{\rm
host}\leavevmode\nobreak\ {\rm cm^{-2}}$ (Liszt, 2021). Finally, we use the
spectral component zxipcf (Reeves et al., 2008), to account for possible warm
absorption. This component accounts for absorption from partially ionised
absorbing material, which covers some fraction of the source, while the
remaining of the spectrum is seen directly. First, we let the covering
fraction of this component free, but it turned out to be consistent with one.
So we fix it at this value.

Figure 1: Model SEDs estimated by choosing randomly 1000 values of $h$,
$\Gamma$, and $L_{\rm transf}/L_{\rm disc}$ (see Sect. 2.1). The individual
realisations are shown as grey dotted lines. The black solid line correspond
to the median of all 1000 SEDs. The red line corresponds to the SED estimated
by considering the average values of $h$, $\Gamma$, and $L_{\rm transf}/L_{\rm
disc}$.

###  2.1 Comments on the physical meaning of the mean SED.

K23 presented the fits to the time-lag spectrum of NGC 5548. These fits were
performed by assuming $\hbox{$M_{\rm BH}$}=7\times 10^{7}\,M_{\odot}$ (Horne
et al., 2021) and an inclination angle of $40\degr$ (Pancoast et al., 2014).
K23 showed that the model explains the observed time lags for a broad range of
parameters. To reduce the degeneracy between these parameters, they considered
a grid of $\hbox{$a^{\ast}$},f_{\rm col},$ and $L_{\rm transf}/L_{\rm disc}$,
and fitted for the accretion rate and the corona height. For NGC 5548, they
considered as ‘best fit’ only the cases which resulted in
$\dot{m}/\dot{m}_{\rm Edd}$ between 0.025 and 0.1, and that could also
reproduce the observed X-ray luminosity.

The chosen range of $\dot{m}/\dot{m}_{\rm Edd}$ is based on the fact that the
accretion rate in this source is assumed to be $\sim 0.05$ of the Eddington
limit (Fausnaugh et al., 2016). This is based on the observed Eddington ratio
$\lambda_{\rm Edd}=L_{\rm bol}/L_{\rm Edd}$, where $L_{\rm bol}$ and $L_{\rm
Edd}$ are the bolometric and Eddington luminosity, respectively. However,
accretion rate estimates which are based on observed $\lambda_{\rm Edd}$
values are rather uncertain, for various reasons. As K23 explained, the
bolometric luminosity is usually computed by measuring the observed luminosity
in a given spectral band and then applying a bolometric correction factor. The
observed luminosity may not be representative of the mean luminosity in this
band, and the correction factor is quite uncertain. Furthermore, computation
of $L_{\rm Edd}$ requires accurate knowledge of the BH mass, while converting
from $L_{\rm bol}/L_{\rm Edd}$ to $\dot{m}/\dot{m}_{\rm Edd}$ requires a
knowledge of the inclination of the system. In addition, the bolometric
luminosity estimates do not take into consideration intrinsic reddening due to
dust that may be present in the host galaxy (see e.g. Gaskell et al., 2023).
For al these reasons, K23 considered a conservative uncertainty on the
observed $\lambda_{\rm Edd}$ by a factor of $\sim 2$. For NGC 5548 this
translates into a range between 0.025 and 0.1, if $\lambda_{\rm Edd}=0.05$
(Fausnaugh et al., 2016).

Based on this, K23 found that the model can fit the time-lag spectrum of NGC
5548 for 5 different model parameter combinations:

  * •

$\hbox{$a^{\ast}$}=0,f_{\rm col}=1.7,L_{\rm transf}/L_{\rm disc}=0.5,h\geq
8.4\,r_{\rm g}$;

  * •

$\hbox{$a^{\ast}$}=0,f_{\rm col}=2.5,L_{\rm transf}/L_{\rm
disc}=0.9,h\in[18.9,28.5]\,r_{\rm g}$;

  * •

$\hbox{$a^{\ast}$}=0.7,f_{\rm col}=1.7,L_{\rm transf}/L_{\rm disc}=0.5,h\geq
64\,r_{\rm g}$;

  * •

$\hbox{$a^{\ast}$}=0.998,f_{\rm col}=2.5,L_{\rm transf}/L_{\rm
disc}=0.5,h\in[5.6,28.5]\,r_{\rm g}$;

  * •

$\hbox{$a^{\ast}$}=0.998,f_{\rm col}=2.5,L_{\rm transf}/L_{\rm
disc}=0.9,h\in[18.9,42.7]\,r_{\rm g}$.

As reported in Table 2 of K23, the model fits for all the above parameter
combinations are statistically acceptable, in the sense that $p_{\rm
null}>0.01$, in all cases. For that reason, we refer to the above five models
as the ‘best-fit’ models, although they are not exactly ‘best model fits’ in
the traditional statistical sense. They provide a good fit to the time lags
(i.e. $p_{\rm null}>0.01$), with accretion rates within the limits we
mentioned above, given the assumption that $a^{\ast}$, $f_{\rm col}$, and
$L_{\rm transf}/L_{\rm disc}$, have the values listed above. Obviously, given
the a priori determination of a few model parameters, the confidence regions
on the corona heights which are reported above are smaller than the case in
which all the parameters are left free in the first place. We suspect that the
model can still fit the time-lags even if we consider BH mass very different
from the mass that is measured for this source and/or accretion rates outside
the range we consider, but we believe that these models cannot be considered
as reliable.

In the following, we explain how we fitted the average SED using Eq. 2, and
the models listed above, in order to investigate whether any or all of them
can explain simultaneously the time lags and the SED. We explain below why
this should be the case, hence why this is a meaningful test for the model.

The flux from a compact X-ray corona can vary due to changes in radius,
temperature, optical depth, height, or the BH accretion rate (see for example
Eq. 12 in Zhang et al., 2023). Similarly to almost all AGN, the X–ray light
curve of NGC 5548 from the STORM campaign is highly variable, at all sampled
time scales. These variations imply that one, or even all of the corona
physical parameters change constantly with time. Even if it is variable, the
accretion rate itself cannot explain the observed X–ray variations in AGN. The
variability amplitude of the X–ray light curves (in NGC 5548 and almost all
other AGN) is of larger than the variability amplitude of the UV/optical light
curves. Consequently, the mechanism that powers the corona should be variable
and, somehow, at least one of the coronal parameters (radius, temperature,
optical depth, and height) should vary with time.

In this case, it is not straightforward to understand what the mean SED
represents in such a variable source. To investigate this issue in the case of
a variable X–ray source which illuminates the disc below, we produce 1000
optical/UV/X-ray spectra using KYNSED, and assuming a BH mass of $7\times
10^{7}\,M_{\odot}$, an inclination angle of $40\degr$, and an accretion rate
of 0.05 the Eddington limit (just like NGC 5548). Furthermore, we consider an
X–ray source whose height changes randomly between 4 $r_{\rm g}$ and 40
$r_{\rm g}$, $L_{\rm transf}/L_{\rm disc}$ changes between 0.5 and 0.9, and
$\Gamma$ changing between 1.3 and 2.1 (the latter being indicative of either
temperature and/or optical depth variations), by drawing these parameters from
uniform distributions. The range of the parameter values we considered
corresponds to the range of the parameters we find later on (see Sect. 3.3),
where we study the SED variability of NGC 5548. These spectra are plotted in
Fig. 1 with grey dotted lines.

The solid black line in the same figure shows the median of all the 1000 SEDs
and the red solid line shows the SED when we assume the mean of the varying
model parameters. This figure shows that, at least for the adopted range of
parameter values (which is appropriate for the source we study), the mean SED
is fully consistent with the SED of the mean of the variable parameters222We
observe a slight difference between the median SED and the SED of the mean
parameter values in the X-ray band, above $\sim 2$ keV. This is due to the
large range of $\Gamma$ variations we consider, and is a problem with all
power-law-like spectra. If the spectrum has a power-law-like shape and changes
in spectral slope, then the mean spectrum will not be equal to the power law
with the mean slope.. The same applies for the case when the parameters are
correlated, as long as they vary within the range that is observed for NGC
5548 (see Sect. 4.1 for more details). Consequently, in the case of NGC 5548,
fitting the mean SED with KYNSED will provide best-fit parameter values which
will be very close to the mean of the variable physical parameters (except,
perhaps, with $\Gamma$). We note that the changes in the parameters of the
system may also affect the time lags computed using the total light curves
from multi-wavelength campaigns, as we discuss below in Sect. 4.2.

###  2.2 The best-fit results

Figure 2: Time-averaged SED of NGC 5548 fitted with Case 1 and Case 3 models, using the best-fit parameters listed in Table 2 (see §2 for details). The blue data points correspond to the UV/optical data, the black, grey, and orange data points correspond to the Swift/XRT spectra extracted from three different time intervals that are consistent with the average X-ray flux. The bottom panels show the corresponding data-to-model ratios. Table 1: Best-fit $\chi^{2}$ values when fitting the average SED. $\dot{m}/\dot{m}_{\rm Edd}$ | 0.025 | 0.05 | 0.1
---|---|---|---
|  $\hbox{$a^{\ast}$}=0$, $f_{\rm col}=1.7$
$\rm\chi^{2}/dof$ | 609/395 | 436/395 | 497/395
$p_{\rm null}$ | $<0.01$ | 0.08 | $<0.01$
|  $\hbox{$a^{\ast}$}=0$, $f_{\rm col}=2.5$
$\rm\chi^{2}/dof$ | 786/395 | 503/395 | 443/395
$p_{\rm null}$ | $<0.01$ | $<0.01$ | 0.05
|  $\hbox{$a^{\ast}$}=0.7$, $f_{\rm col}=1.7$
$\rm\chi^{2}/dof$ | 629/395 | 435/395 | 437/395
$p_{\rm null}$ | $<0.01$ | 0.08 | 0.07
|  $\hbox{$a^{\ast}$}=0.998$, $f_{\rm col}=2.5$
$\rm\chi^{2}/dof$ | 928/395 | 643/395 | 625/395
$p_{\rm null}$ | $<0.01$ | $<0.01$ | $<0.01$

The important physical parameters of KYNSED are: BH mass, $M_{\rm BH}$, spin,
$a^{\ast}$, accretion rate, $\dot{m}/\dot{m}_{\rm Edd}$, the ratio of the
accretion power transferred to the corona over the total power given to the
disc, $L_{\rm transf}/L_{\rm disc}$, the corona height, $h$, the X–ray
spectrum parameters, $\Gamma$ and $E_{\rm cut}$, the disc inclination, and the
colour correction factor, $f_{\rm col}$. Similarly to K23, we fix the BH mass
at $7\times 10^{7}\leavevmode\nobreak\ \rm M_{\odot}$ (Horne et al., 2021),
$E_{\rm cut}$ at 150 keV (in agreement with previous estimates by Nicastro et
al., 2000; Ursini et al., 2015), the inclination at 40° (in agreement with the
estimate by Pancoast et al., 2014). We also fix $\dot{m}/\dot{m}_{\rm Edd}$ at
three different values: 0.025, 0.05, and 0.1 following the reasoning explained
in K23 and in the previous section. We then fit the average SED by considering
the various combinations of $a^{\ast}$ and $f_{\rm col}$ that fit the time
lags as presented above. The free parameters are $L_{\rm transf}/L_{\rm
disc}$, $h$, $\Gamma$, $E(B-V)_{\rm host}$, $N_{\rm H}$, and $\log\xi$.

We note again that fixing $a^{\ast}$, $f_{\rm col}$, and $\dot{m}/\dot{m}_{\rm
Edd}$ will reduce the degeneracy that is present between some of the
parameters, and will naturally lead to a decrease in the uncertainty on the
free parameters (mainly $L_{\rm transf}/L_{\rm disc}$ and $h$). Most probably,
the SED can also be fitted well by different parameter combinations. But our
objective is to investigate whether the same model parameter combinations are
capable of explaining the SED as well as the time lags, since K23 fitted the
time lags following the same approach. Obviously, our results should be
considered as a sub-sample of a wider parameter space within the X-ray
illuminated disc model that could fit the data.

We list in Table 1 the values of $\rm\chi^{2}/dof$ and $p_{\rm null}$ obtained
for each of the combinations. There are four $a^{\ast}$, $f_{\rm col}$, and
$\dot{m}/\dot{m}_{\rm Edd}$ combinations that result in a statistically
acceptable fit, namely: Case 1: $\hbox{$a^{\ast}$}=0,f_{\rm
col}=1.7,\dot{m}/\dot{m}_{\rm Edd}=0.05$; Case 2: $\hbox{$a^{\ast}$}=0,f_{\rm
col}=2.5,\dot{m}/\dot{m}_{\rm Edd}=0.1$; Case 3: $\hbox{$a^{\ast}$}=0.7,f_{\rm
col}=1.7,\dot{m}/\dot{m}_{\rm Edd}=0.05$; and Case 4:
$\hbox{$a^{\ast}$}=0.7,f_{\rm col}=1.7,\dot{m}/\dot{m}_{\rm Edd}=0.1$.
Although the combination of $a^{\ast}$=0.998 and $f_{\rm col}$=2.5 can provide
good fits to the time lags, it cannot fit the average SED well, for any
$\dot{m}/\dot{m}_{\rm Edd}$ value within the range of 0.01–0.1 that we
consider.

Furthermore, two out of the four combinations which fit the SED well cannot be
accepted as the resulting best-fit $h$ and/or $L_{\rm transf}/L_{\rm disc}$
values are not in agreement with the results obtained by fitting the time
lags. For example, the Case 2 best-fit results suggest $h>64\,r_{\rm g}$ and
$L_{\rm transf}/L_{\rm disc}\in[0.2,0.3]$ (these are the 90 per cent
confidence limits for a single parameter). These limits are in contradiction
with the time lags best-fit results (for the same combination of $a^{\ast}$
and $f_{\rm col}$), which implied $h\in[18.9,28.5]\,r_{\rm g}$ and $L_{\rm
transf}/L_{\rm disc}=0.9$. Similarly, according to the Case 4 best-fit
results, $h\in[2.6,13.7]\,r_{\rm g}$, while $h\geq 64\,r_{\rm g}$ according to
the best-fit results to the time-lags. Thus, in the following we will adapt
Cases 1 and 3 as the only acceptable model parameters which can fit well both
the time lags and the average SED.

Table 2 lists the Case 1 and Case 3 best-fit parameters, and Fig. 2 shows the
respective best-fit models to the average SED. We note that, except for the
spin and the height, all the parameters are consistent between the two cases.
The last row in Table 2 list the corona radius ($R_{\rm C}$) for both models.
KYNSED computes a posteriori the corona radius for a spherical geometry, based
on the best-fit model parameters, assuming conservation of energy and photon
number. All the assumptions and the method of calculating $R_{\rm C}$ are
detailed in Dovčiak & Done (2016) and D22. A sanity check of the validity of
the two model fits is that the corona should be outside the event horizon of
the BH. Consequently, $R_{\rm C}$ should be smaller than the difference
between the corona height and the event horizon radius ($R_{\rm H}$). For both
cases, the corona is above the event horizon.

Table 2: Case 1 and Case 3, best-fit parameter values obtained by fitting the average SED of NGC 5548. Parameter | Case 1 | Case 3
---|---|---
$a^{\ast}$ | 0 | 0.7
$f_{\rm col}$ | 1.7 | 1.7
$\dot{m}/\dot{m}_{\rm Edd}$ | 0.05 | 0.05
$E(B-V)_{\rm host}$ | $0.14_{-0.01}^{+0.02}$ | $0.14_{-0.02}^{+0.01}$
$L_{\rm transf}/L_{\rm disc}$ | $0.55_{-0.01}^{+0.02}$ | $0.54_{-0.01}^{+0.02}$
$h\,(r_{\rm g})$ | $31.4_{-9.1}^{+10.4}$ | $79.5_{-10.5}^{+11.5}$
$\Gamma$ | $1.83_{-0.04}^{+0.06}$ | $1.81_{-0.06}^{+0.07}$
$N_{\rm H}\,(10^{22}\rm cm^{-2})$ | $2.54_{-0.25}^{+0.28}$ | $2.49_{-0.24}^{+0.26}$
$\log\left(\xi/\rm erg\,cm\,s^{-1}\right)$ | $1.37_{-0.08}^{+0.10}$ | $1.36_{-0.07}^{+0.09}$
$R_{\rm C}\,(r_{\rm g})$ | 23 | 31

##  3 Modelling the complex X–ray/UV/optical SED variability in NGC 5548

Figure 3: Mulit-wavelength light curves of NGC 5548. Top and bottom panels:
The observed X-ray and UV light curves normalised to their mean (H1, H3, and
UVW2 bands are plotted in red, orange, and blue points, respectively). The
vertical dotted lines indicate the times at which we selected the UV data
points. The grey shaded areas show the corresponding time intervals over which
we integrated the X-ray spectrum (See text for details).

Figure 3 shows the Swift/XRT light-curve of NGC 5548 (taken from Panagiotou et
al., 2020), the HST light curves at 1158 Å and 1479 Å (hereafter H1 and H3)
and the Swift/UVOT light curve in the UVW2 band, centred at 1928 Å(taken from
Edelson et al., 2015; Fausnaugh et al., 2016, for the UVOT and HST data,
respectively). The light curves are normalised to their corresponding mean,
and they show clearly the complex variability of the source. For example, we
can see mini-flares and dips in X–rays which are absent in the UV bands, the
X–ray flux decreases (overall) in the period between $\sim 56760-56810$ MJD,
while the UV flux appears to increase, etc. Even within the UV bands, there
are differences in the variability amplitude, which imply significant UV
spectral variability with the spectrum becoming bluer with increasing flux.
Any physical model should be able to explain these complex spectral
variability patterns.

The vertical dotted lines and the shaded areas (in the bottom and top panels
in Fig. 3) indicate 15 times (in the UV) and 15 X–ray light curve segments
(s1-s15), which sample characteristic features in the X–ray light curve such
as maxima (e.g. s1, s7), minima (e.g. s6, s15), or large amplitude variations
(s8-s12), which do not necessarily correspond to similar features/trends in
the UV light curves. Our objective is to extract the UV/X–ray SEDs during
these times, fit them with KYNSED and investigate whether X–ray reverberation
can explain the complex X–ray/UV variability in this source, the origin of the
X–ray variability as well as the reason for the rather poor X–ray/UV
correlation in this source, that is to say why the X–ray and the UV fluxes do
not appear to vary in phase, at all times.

###  3.1 Constructing the time-resolved SEDs

Figure 4: Zoom-in on the time segment s10 as an example of the data selection.
The top and bottom panels are similar to the ones shown in Fig. 3.

If X–rays illuminate the accretion disc and produce the variable UV/optical
emission, we cannot simply use data from simultaneous UV/X–ray observations to
construct the SED. For example, Fig. 4 shows a small part of the X–ray and UV
light curves around segment s10, as labelled in Fig. 3. The vertical dotted
line in the bottom panel of Fig. 4 indicates the H1, H3 and UVW2 points we use
for the s10 SED. There is an Swift/XRT observation just after the dotted line
but we cannot consider the X–ray spectrum from this observation to construct
the X–ray/UV SED.

In the case of X–ray thermal reverberation, the variable UV emission at any
time does not depend on the X–rays that were emitted at that time. Instead,
this emission should be equal to the convolution of the X–rays with the disc
response function at all times before the UV observations took place (see Eq.
(4) in K21, for example), and it is the width of the response function that
determines how far in the past we should consider this convolution. According
to Fig. 8 in K21a, the disc response function at $\sim 1200$ Å in the case of
M${}_{\rm BH}=5\times 10^{7}$ M⊙ is $\sim 10-15$ time smaller than its peak
$\sim 1$ day after the X–rays were emitted (both for $a^{\ast}$=0 and 1). In
the case of the UVW2 band, the response function is $\sim 10$ time smaller
than its peak at approximately $\sim 2$ days after the X–ray emission.
Consequently, the X–ray spectrum in the s10 SED should be the average X–ray
spectrum over a period of $\sim 1-2$ days before the UV data.

We therefore extract the spectrum from all the X–ray observations that were
taken within $1.3$ days before the UV observations (we use 1.3 days as a
compromise between 1 and 2 days). This time interval is indicated by the grey
area in the upper panel in Fig. 4. In fact, all the grey-shaded areas in the
top panel in Fig. 3 indicate the 1.3 day intervals we consider to extract the
X–ray spectrum for each UV/X-ray SED. There are typically $3-4$ Swift/XRT
observations in such a period for the SED of all segments.

We cannot add longer wavelengths in the SED because the width of the disc
response function increases with increasing wavelength. For example, Fig. 8 in
K21a shows that the width of the response function is at least $\sim 10$ days
in the $g^{\prime}$ band. Consequently, if we wish to include longer
wavelengths in the SED, then we should extract the X-ray spectrum using
observations over a period longer than 1–2 days. But then, we must construct
different X–ray spectra for the UV and the optical SEDs at each time, and the
model fitting of such data sets would have been very challenging.

We extract the X–ray spectra of each interval using the automatic Swift/XRT
data products generator333https://www.swift.ac.uk/user_objects/ (Evans et al.,
2009). We consider the $0.4-8$ keV band in each spectrum, and grouped the
spectra to have at least 25 counts per bin. For the UV/optical data, we use
the FTOOLS (Nasa High Energy Astrophysics Science Archive Research Center
(2014), Heasarc) command
ftflx2xsp444https://www.heasarc.gsfc.nasa.gov/lheasoft/ftools/headas/ftflx2xsp.html
to create spectral files in the UV/optical range in a format that is
compatible with XSPEC, which we use to model the SED as described in the
following section. In the following, all uncertainties are estimated for a
change of fit statistic by $\Delta\chi^{2}=2.706$ for a single parameter.

###  3.2 Spectral fitting of the variable SEDs

Figure 5: Best-fit to all 15 SEDs for the Case 1 model. The corresponding
ratios (data/model) are plotted in the bottom panel.

We fitted all the 15 SEDs with the model defined by Eq. 2 twice. First, we
fixed spin and $f_{\rm col}$, to 0 and 1.7, and then to 0.7 and 1.7. These are
the Case 1 and Case 3 model values, which fit well both the average SED as
well the time-lags spectrum of the source. Both times, we assume the BH mass,
inclination and $E_{cut}$ values that we use when fitting the average SED, and
we fix the covering fraction of the ionised absorber to 1. Each time, we fix
the accretion rate, and the host galaxy extinction to the best-fitting values
listed in Table 1. We then fit separately each of the 15 SEDs letting free the
height, $L_{\rm transf}/L_{\rm disc}$, the photon index, the column density
and the ionisation parameter of the ionised absorber.

The majority of the spectra resulted in statistically unacceptable fits when
assuming the Case 3 model (i.e. $p_{null}$ is smaller than 0.01 in most
cases). On the contrary, all 15 SEDs could be well fitted assuming Case 1,
model. The only exception was the s2 SED, in which case
$\chi^{2}$/dof=32.8/13, $p_{null}=0.002$. Adding a systematic error on the
model for this fit of 3%, using the systematic command in XSPEC, improved the
fit significantly (rendering the fit statistically acceptable). We also
compute the corona radius, $R_{\rm C}$, when fitting each SED. In all cases,
the radius was physically accepted, i.e. it was smaller than the corona height
minus the BH horizon, except for the s8 and s9 SEDs. For these two spectra,
the best-fit $\Gamma$ exceeded 2.3, which in turn results in $R_{\rm C}$
values larger than the height of the corona. For that reason, we set an upper
limit of $\Gamma=1.9$ for these two spectra (this is consistent with the upper
uncertainty we get on $\Gamma$ by fitting the time-averaged SED). By doing so,
the fit remains statistically accepted and the value of $R_{\rm C}$ become
physically accepted as well. Figure 5 shows the 15 SEDs together with the
corresponding best-fitting model.

###  3.3 Best-fit results

Figure 6: Evolution of the best-fit parameters as fuction of time.Top two
panels show the observed and the best-fit model-predicted X-ray and UV light
curves (top and bottom panels, respectively). Grey circles show the mean count
rate from all observations, while the black circles indicate the average count
rate in each of the 15 time segments we computed the SEDs. Blue diamonds
indicate the best-fit flux in each band, during the same segments. All fluxes
and the black circles count rates are normalised to their respective mean. The
bottom six panels show the evolution of the best-fit parameters as a function
of time. The horizontal grey dashed lines and shaded areas correspond to the
best-fit parameters (and the corresponding uncertainty) obtained by fitting
the time-averaged SED in Sect. 2. Blue circles in the bottom panel show the
corona radius, and the empty black squares indicate the difference between the
height of the corona and the event horizon radius (errors on these points take
into account only the error on the height). The red shaded area highlights the
s8-s12 period in which $L_{\rm transf}/L_{\rm disc}$ is the main driver of the
variability (see Sect. 4.1 for more details).

The best-fit parameter values for all the SED model fits are listed in Table
4. Using these values, we compute the model X–ray flux in the $2-7$ keV band,
as well as the UV flux in the bands we consider. Grey circles in the top two
panels of Fig. 6 correspond to the observed mean count rate during each
observation in the X–ray and in the H3 band (first and second panels,
respectively). Black circles in both panels indicate the count rates during
the times chosen to construct the 15 X–ray/UV SEDs, normalised to their mean.
The open diamonds in both panels indicate the best-fit model luminosity in
each band, normalised to the respective mean values. The agreement between the
model and the observed fluxes is very good. Just like the model/data ratios in
Fig. 5, this figure demonstrates clearly the quality of the model fits to the
data.

Figure 6 shows that the model and the observed UV fluxes, plotted in the
second panel in Fig. 6, agree at the $\sim 1.5\%$ level, on average. Since we
fitted the SEDs by keeping $\dot{m}/\dot{m}_{\rm Edd}$ constant, this implies
that X–ray thermal reverberation could account for all the observed UV
variations in NGC 5548. This result is even more impressive given the fact
that the correlation between the X–rays and the UV is rather weak. As we have
already discussed, this fact raised the question of whether the X-rays are the
main driver of the variations seen in the UV/optical or not. Our results show
that, despite the fact that the two light curves appear to be weakly
correlated, it is entirely possible that all the UV variations are caused by
the X–rays which illuminate the disc. The reason for the less than perfect
correlation between the two light curves can be understood, to some extent,
when we investigate the variability of the best-fit model parameters.

Figure 6 shows how the best-fit parameters vary as a function of time. The
horizontal dashed lines and the grey shaded areas around them indicate the
best-fit values we get when we fitted the mean SED, and their 1$\sigma$
confidence limit, respectively. The column density and ionisation parameter of
the ionised absorber do not appear to vary significantly. They remain roughly
constant close to the best-fit values we get when we fitted the mean SED. On
the other hand, the corona parameters, i.e $L_{\rm transf}/L_{\rm disc}$,
$\Gamma$, and height, are highly variable. The height of the corona changes
significantly in the first 8 segments, and then remains roughly constant
between $\sim 10-20$ $r_{\rm g}$ before increasing in the last two segments
above $30\,r_{\rm g}$. In particular, the height appears to determine the
X–ray and UV variability during the period between s1 and s5 SEDs. During this
period, the X–ray and UV flux vary in the same way (as shown by the first 5
open diamonds in the top two panels of Fig. 6). The third panel in Fig. 6
shows that the height varies in exactly the same way. As the height decreases
(from s1 to s2) the X–ray and the UV flux decrease as well. This is because as
the height decreases X–rays are strongly lensed towards the BH, the X-rays
detected by the observer decrease, and so are the X–rays illuminating the
disc, hence producing less of the variable UV flux. As the height increases
(for s3 to s4 and s5). then both the X-rays emitted towards the observer and
towards the disc increase as well.

We also see $L_{\rm transf}/L_{\rm disc}$ variations as well. This parameters
also influences significantly the observed X–ray and UV variations. For
example, in the period between s8 to s12, the X–ray flux systematically
decreases, while the UV flux increases. This apparent contradictory behaviour
can be explained by the fact that $L_{\rm transf}/L_{\rm disc}$ decreases
during the same period (see Fig. 6, in the period between 56760–56780 MJD).
For that reason the X–ray flux decreases as well but, since the power that is
transferred to the corona is taken by the accretion process, as $L_{\rm
transf}/L_{\rm disc}$ decreases, the disc luminosity increases, and hence the
UV flux as well. Therefore, although the two light curves during the period
between the s8 and s12 segments appear to vary in the opposite directions,
this can be explained, and is entirely consistent, with the X–ray, disc
reverberation. Within the context of our model, as long as the power
transferred to the corona is part of the accretion power released in the disc,
and it is variable, then the X–ray and the UV flux variations can be even
anti-correlated.

The last panel in Fig. 6 shows the radius of the corona during the full
period, computed as explained in D22. In almost all cases the radius is
smaller than or equal (within the errors) to the height of the corona (minus
the horizon radius), as it should be. The radius does not vary a lot, and most
of the time sit is $\sim 10$ $r_{\rm g}$, although it reduces to less than 5
$r_{\rm g}$, during s2 and s3 (when the height is small as well), and in a
couple of cases (s4, s5 and s15) it may be up to 20 or even 40 $r_{\rm g}$.

###  3.4 Comparison between the best-fit results to the mean and to the time
variable SEDs

Figure 7: Best-fit model SEDs to the 15 spectra of NGC 5548 (dotted grey
lines). The solid black line shows the average of the 15 SEDs. The solid red
line corresponds to the best-fit model to the time-averaged SED. Blue circles
indicate the average flux of the 15 segments in the UV bands. The orange
shaded areas highlight the energy ranges used to fit the time-averaged SED.

Figure 6 shows that the best-fit log($\xi)$ and $N_{H}$ values when we fit the
time variable SEDs and the mean SED are in agreement. But this is not the case
with the other model parameters. The height of the corona during the 15
segments is systematically smaller than the height we measure when we fit the
mean SED. The opposite is true with $L_{\rm transf}/L_{\rm disc}$. The $L_{\rm
transf}/L_{\rm disc}$ we measure when we fit the average SED is systematically
lower than the best-fit $L_{\rm transf}/L_{\rm disc}$ values we get when we
fit the individual SEDs. The disagreement between the mean of the model-
parameters of the 15 SEDs and the model parameters of the (overall) mean SED
is because the segments we choose to study the spectral variability of the
source were not chosen so that their mean flux would be equal to the overall
mean of the light curves. It turns out that the mean flux of these segments is
smaller than the mean of the whole light curves, and this can explain the
differences we see.

The grey dotted lines in Fig. 7 indicate the best-fit KYNSED models to the 15
variable SEDs. The solid black line shows the mean of these 15 models. Of
course, the mean of the 15 best-fit models should give a good fit to the mean
observed spectrum of the 15 segments. Indeed, the three blue points in Fig. 7
show the mean H1, H3 and UVW2 flux of the UV observations in the 15 segments,
and they are in perfect agreement with the black solid line. The solid red
line in the same figure shows the best KYNSED model-fit to the mean SED. The
mean spectrum of the individual SEDs is lower than the overall mean SED.
Consequently, it is not necessary that the mean of the parameter values from
the best-fits to the 15 SEDs is equal to the respective model parameters we
derive from the model fit to the average SED.

At first, it seems rather strange that the mean $L_{\rm transf}/L_{\rm disc}$
during the 15 segments is larger than the $L_{\rm transf}/L_{\rm disc}$ we
determined when we fitted the average SED, while the mean X–ray flux during
the 15 segments is lower than the overall mean. We would expect the opposite,
but, at the same time, the mean corona height is quite smaller than the height
we measure when we fitted the mean SED. Therefore, although the power
transferred to the corona during the 15 segments is larger than the average
power, a larger fraction of the X–rays are emitted away from the observer,
towards the BH, so that the observed X–ray flux is smaller than average.

##  4 Summary and discussion

This is the seventh paper in which we study the X-ray and UV/optical
variability of NGC 5548, using the data originally collected during the STORM
campaign, assuming X-ray illuminated accretion disc. In Kammoun et al. (2019,
2021b) we fitted the UV/optical continuum time lags assuming that the energy
powering the X–ray source was not part of the power released in the disc
through accretion. Assuming the same scenario, we were also able to explain
the UV/optical power spectral density (PSD) of the source in Panagiotou et al.
(2020, 2022a). Later on, D22 presented KYNSED, which can be used to model the
broadband SED of AGN assuming X-ray illumination and by allowing for two
scenarios for powering the corona: either the power is external to the
accretion disc, or the corona is powered via the accretion process. D22 used
the latter scenario to model the broadband time-averaged SED of NGC 5548. More
recently, K23 modelled the time-lags of the source assuming the same scenario.
In this work, we use the best-fit results obtained by K23 to model both the
time-averaged and the variable SEDs of the source. In the latter case, we
consider 15 segments of the X-ray and UV bands (at wavelengths shorter than
$\sim 2000$ Å) and we model the complex, X–ray/UV spectral variability of the
source.

The scenario of an X-ray illuminated accretion disc can explain the time lags
and the time-averaged SED of NGC 5548 with the same model parameters. In this
case, two values of the BH spin are acceptable (0 and 0.7), assuming
$\dot{m}/\dot{m}_{\rm Edd}=0.05$. Both spin values result in comparable values
of $\Gamma\sim 1.8$, $L_{\rm transf}/L_{\rm disc}\sim 0.55$. However, the
lower spin value resulted in $h\sim 31\,r_{\rm g}$, while the higher spin
value resulted in $h\sim 80\,r_{\rm g}$. To the best of our knowledge, our
model is the only model that can explain both the spectral as well as the
timing properties of NGC 5548. Furthermore, this model can even explain the
time-variable X-ray/UV SEDs of the source, assuming a constant accretion rate.
The fact that the X-rays are the only driver of the UV/optical variability can
resolve problems associated with the fast UV/optical variations observed in
AGN, and could not be explained by the accretion disc (slow) characteristic
time scales.

We find that only the zero BH spin case can explain the variable SED. The
spectral variability can be thus explained by variations in the corona height
(between $\sim 4\,r_{\rm g}$ and $50\,r_{\rm g}$), $L_{\rm transf}/L_{\rm
disc}$ (between $\sim 0.55$ and 0.9), and $\Gamma$ (between $\sim 1.25$ and
2.1). Since these model parameters affect the observed flux in various bands
in different ways, the model can account for the weak correlation that is
observed between the X-rays and the UV/optical light curves (Fausnaugh et al.,
2016).

###  4.1 The origin of the X–ray variability and the weak X–ray/UV correlation

Table 3: Best-fit slopes obtained by fitting a straight line, in linear or logarithmic scale, for various combinations of the best-fit model parameters. Values in parenthesis below the best-fit slopes indicate the significance of the hypothesis that the slope is different than zero. In the last column, we list the Spearman’s rank correlation coefficient and the null hypothesis probability (values in parenthesis). Parameters | Space | Slope | $\rho_{\rm S}$
---|---|---|---
$h\,(r_{\rm g})$ vs $L_{\rm transf}/L_{\rm disc}$ | linear | $56.92\pm 22.94$ | 0.27
|  |  ($2.5\sigma$) | (0.33)
$\Gamma$ vs $L_{\rm transf}/L_{\rm disc}$ | linear | $-1.00\pm 0.35$ | -0.56
|  |  ($2.8\sigma$) | (0.03)
$F_{\rm X}/\langle F_{\rm X}\rangle$ vs $L_{\rm transf}/L_{\rm disc}$ | linear | $0.04\pm 0.46$ | 0.00
|  |  ($0.1\sigma$) | (1.00)
$F_{\rm UV}/\langle F_{\rm UV}\rangle$ vs $L_{\rm transf}/L_{\rm disc}$ | linear | $-1.41\pm 0.30$ | -0.79
|  |  ($4.6\sigma$) | (0.0004)
$\Gamma$ vs $h\,(r_{\rm g})$ | linear | $-0.006\pm 0.004$ | -0.46
|  |  ($1.6\sigma$ ) | (0.08)
$F_{\rm X}/\langle F_{\rm X}\rangle$ vs $h\,(r_{\rm g})$ | log | $0.15\pm 0.06$ | 0.57
|  |  ($2.5\sigma$) | (0.03)
$F_{\rm UV}/\langle F_{\rm UV}\rangle$ vs $h\,(r_{\rm g})$ | log | $0.19\pm 0.08$ | 0.31
|  |  ($2.4\sigma$) | (0.26)
$F_{\rm X}/\langle F_{\rm X}\rangle$ vs $\Gamma$ | log | $0.62\pm 0.21$ | 0.26
|  |  ($3.0\sigma$) | (0.35)
$F_{\rm UV}/\langle F_{\rm UV}\rangle$ vs $\Gamma$ | linear | $0.33\pm 0.16$ | 0.37
|  |  ($2.1\sigma$) | (0.18)
$F_{\rm UV}/\langle F_{\rm UV}\rangle$ vs $F_{\rm X}/\langle F_{\rm X}\rangle$ | linear | $1.13\pm 0.11$ | 0.38
|  |  ($9.9\sigma$) | (0.16)

We show in Fig. 8 correlations between various best-fit corona parameters as
well as the model X-ray and UV fluxes. Such correlations, if real, can help us
investigate the origin of the observed variability. In other terms, these
correlations can help us identify the primary physical parameters that are
responsible for the observed variability in the X-rays and in the UV/optical
bands.

In order to search for significant correlations between the various parameters
that are plotted in Fig. 8, we fitted a line (either in the linear or in the
logarithmic space) and we then check if the best-fit slope is consistent with
zero. If the slope is significantly different than zero, this would imply that
there is a significant correlation between the two parameters. We used the
ordinary least-squares (OLS) regression method (Isobe et al., 1990) of Y
against X, OLS(Y—X), for all the relations except for the relation between the
X-ray and UV flux (Panel j). Since neither the X-ray nor the UV flux can be
consider as the “effect” and/or the “cause” of the other, it is better to
treat these variables symmetrically. For that reason, we follow Isobe et al.
(1990) and we use the bisector when fitting the respective plot in Fig. 8. The
best-fit lines are shown as red solid lines in Fig. 8, and the best-fit slopes
are listed in Table 3.

We also report the significance of the best-fit slopes being non-zero in each
of the panels, $\sigma_{\rm slope}$, which should be indicative of the
significance of the correlation between the various parameters plotted in Fig.
8. Of course this is true only if the best-fit line does indeed provide a good
fit to the data (the Isobe et al. (1990) method already assumes that there is
a linear relation between the two variables). However, the various plots in
Figure 8 show that a simple line (either in the linear or in the logarithmic
space) may not provide a good fit to the data. The scatter between the data
points and the best-fit lines in many plots is quite large. Significant
outliers exist, which makes it difficult to even consider (let alone prove)
whether there is a correlation or not. We suspect that most of the best-fit
parameter values depend on more than one of the other physical parameters, and
it is for this reason that the correlation between most parameters appears to
be rather messy.

For these reasons, we computed the Spearman’s rank correlation coefficient,
$\rho_{\rm S}$, for all the correlations plotted in the same figure (results
are listed in the last column of Table 3), and we also binned the data in the
various panels in Fig. 8 (except Panels (d) and (j) where $\sigma_{\rm slope}$
and/or the correlation coefficient imply a strong correlation between the
parameters). The binned points should indicate the mean correlation between
the various parameters in each plot, and will allow us to investigate if they
are correlated, at least on average (i.e. when the scatter of the points is
taken off, to some extend). The black diamonds in these panels indicate the
resulting mean correlation in each plot. In many cases, the binned data are
consistent with the best fit lines. We consider this as an indication of an
intrinsic correlation between various model parameters.

For example, Panels (b) and (e) show that the binned data are very close to
the best-fit line. In Panel (b), both $\rho_{\rm S}$ and $\sigma_{\rm slope}$
indicate an intrinsic correlation between $\Gamma$ and $L_{\rm transf}/L_{\rm
disc}$, while in Panel (e), $\sigma_{\rm slope}$ is small. This is most likely
due to the fact that, even if there is an intrinsic linear relation between
the two parameters, the slope is small and we cannot detect it with a high
accuracy with the available data. The binned data are close to the best-fit
lines in Panels (f) and (h) as well. In the first case, both $\sigma_{\rm
slope}$ and $\rho_{\rm S}$ indicate that an intrinsic correlation may be real,
while in the second case $\rho_{\rm S}$ is rather small. But this is not
surprising, because the correlation coefficient measures the scatter of the
points around the best-fit line to the data in the linear space, while the
best-fit relation in this panel is non-linear. As for Panel (i), $\rho_{\rm
S}$ implies a weak correlation, although the binned points are close to the
best-fit line, most probably because of the presence of two points which are
outliers in this plot.

We discuss below some of the correlations in the cases when the binned data
are close to the best-fit lines, and $\sigma_{\rm slope}$ and/or $\rho_{\rm
S}$ suggest either significant (at the 3$\sigma$ level) or quite probable
correlations between the parameters. These correlations indicate that the main
driver of the X-ray variability could be $L_{\rm transf}/L_{\rm disc}$ and the
corona height, and can explain why the correlation between the observed X–ray
and UV fluxes may not always be perfect. In fact if the X-ray source is
powered by the accretion process, it is even possible that UV and X-rays will
be anti-correlated.

Figure 8: Correlations between various best-fit parameters, as determined by
fitting the variable SEDs. Solid red lines show the best-fit straight line or
power law model. Black diamonds show the binned data. Numbers in each plot
indicate the significance of each slope not being zero (see Sect. 4.1).

Panel (d) in Fig. 8 shows that the UV flux strongly anti-correlates with
$L_{\rm transf}/L_{\rm disc}$: as the power given to the corona increases, the
available disc luminosity (hence UV flux as well) decreases. At the same time,
as $L_{\rm transf}/L_{\rm disc}$ increases, the X–ray flux should also
increase. This can lead to an anti-correlation between UV and X–rays, if
$L_{\rm transf}/L_{\rm disc}$ were the only or the major parameter to vary.
This is what most likely happens between segments s8-s12 (i.e. between MJD
56760 and 56800, highlighted as red shaded area in Fig. 3). Fig. 6 shows that
$L_{\rm transf}/L_{\rm disc}$ is the main parameter that varies during this
period. $L_{\rm transf}/L_{\rm disc}$ is overall decreasing during this
period, which explains why the UV flux is increasing, while the X-ray flux is
decreasing.

The spectrum photon index appears to negatively correlate with both $L_{\rm
transf}/L_{\rm disc}$ and $h$ (Panels (b) and (e) in Fig. 8). The first
correlation could be due to the fact that part of the power that is
transferred to the corona should be used to heat it up, hence large power
should imply a flatter spectrum (as suggested by Panel (b)). At the same time,
Panel (e) implies weak, but suggestive, correlation between $\Gamma$ and $h$.
The spectrum becomes flatter (i.e. it is most likely that the corona becomes
hotter) as the height increases. In this case, the solid angle of the corona
subtended to the disc decreases (note that, as we commented in the previous
section, the corona radius appears to be constant, irrespective of the
height). Consequently, the number of photons that enter in the corona should
also decrease, hence the corona cooling should decrease, and the temperature
increases (causing the $\Gamma$ flattening).

Interestingly, Panel (c) in Fig. 8 shows that the observed X-ray flux, in the
$2-7$ keV band, is not correlated with $L_{\rm transf}/L_{\rm disc}$. This can
be explained by the $\Gamma$-$L_{\rm transf}/L_{\rm disc}$ anti-correlation in
Panel (b): as the power given to the corona increases (from $L_{\rm
transf}/L_{\rm disc}$$\sim 0.5$ to 0.9), $\Gamma$ flattens from $\sim 2.1$ to
$\sim 1.25$. This $\Gamma$ variability will cause a significant variation in
the observed X-ray flux in the $2-7\,\rm keV$ band, as Panel (h) shows (this
is the ‘steeper when brighter’ X-ray spectral variability pattern commonly
observed in Seyfert galaxies). Therefore, as the power given to the corona
increases, the spectral slope flattens, hence the observed X-ray flux in the
$2-7$ keV band decreases, and this results in an apparent lack of correlation
between $L_{\rm transf}/L_{\rm disc}$ and $F_{\rm X}$.

Our results are in agreement with the findings of Panagiotou et al. (2022b).
The authors demonstrated that the low correlation between the X-rays and the
UV/optical in NGC 5548, obtained by using the cross-correlation function
(CCF), could be caused by the lack of a static source configuration, as is for
example the case of a dynamic corona, accompanied by changes in the photon
index. The authors concluded that the moderate X-ray/UV CCF does not
contradict a scenario in which the UV/optical variability is driven by X-ray
illumination of the accretion disc. In the current work, we show that not only
that changes in height and photon index are required to reproduce the
broadband variability but also a change in the heating power given to the
corona is needed.

We note that, although our results fully support the hypothesis of X-ray
illumination of the disc, we do not have a physical explanation for various
aspects of the model. For example, our results support the idea that $L_{\rm
transf}/L_{\rm disc}$ is the main driver of the X-ray variability. This
implies that the radius below which the power is transferred to the corona
varies. However, the mechanism that could produce these variations remains
unknown. The processes that could transfer the power from the disc to the
corona remain unknown as well. It could be probable that the corona is
consistent with a failed jet as suggested by Ghisellini et al. (2004). The
corona could also be formed by magnetic reconnection which is also how the
power could be transferred to the corona from the accretion disc (see e.g. Di
Matteo, 1998; Merloni & Fabian, 2001a, b). Additional models invoking magnetic
field, such as the jet emitting disc (JED) standard accretion disc (JED-SAD;
Ferreira et al., 2006; Petrucci et al., 2008), can potentially explain the
heating of the X-ray corona and its connection to magnetic field. However, to
the best of our knowledge, this model has not been applied to the correlated
X-ray/UV variability that is observed in AGN.

###  4.2 Implications for the time lags

Figure 9: Model time lags obtained by considering the best-fit parameters from
the 15 SEDs of NGC 5548 (thin blue lines). The thick blue line correspond to
the average of the 15 time-lag spectra. The dashed red line corresponds to the
time lag spectrum obtained by considering the average values of the best-
fitted parameters obtained by modelling the variable SEDs. The black circles
are the observed continuum time lags of NGC 5548. We show as empty black
circles the measurements that are expected to be affected by the continuum
emission from the broad-line region resulting in an excess in the delay.

The observed X-ray and UV/optical flux variations in NGC 5548 are due to
continuous changes of the power that heats the corona ($L_{\rm transf}/L_{\rm
disc}$) as well as its height. These two parameters strongly affect the
UV/optical continuum responses of the accretion disc expected from thermal
reverberation as demonstrated in Kammoun et al. (2019, 2021a) and K23.
Therefore, we expect that the cross-correlation between the X-rays and the
UV/optical bands, as well as the cross-correlation between the various
UV/optical bands to continuously change in response to these variations in
$L_{\rm transf}/L_{\rm disc}$ and $h$. This implies that the time-lags between
the various wavebands should also change.

In Fig. 9 we show the observed time lags during the full STORM campaign as a
function of wavelength (Fausnaugh et al., 2016), hereafter the time-lag
spectrum. We also show the model time-lag spectra expected during each of the
15 segments we analyse in this work, as thin blue lines. These time-lag
spectra are estimated using the code of K23. We use as input parameters the
values of $h,\Gamma$, and $L_{\rm transf}/L_{\rm disc}$ listed in Table 4. The
average of the 15 time-lag spectra is shown as thick blue solid line. The red
dashed line corresponds to the model time-lag spectrum when we assume the mean
of the parameters listed in Table 4, i.e. $h=23.2\,r_{\rm g}$, $\Gamma=1.67$,
and $L_{\rm transf}/L_{\rm disc}=0.73$. The difference between the two lines
is less than 10% on average. This indicates that the time lags usually
computed using long light curves are equivalent to the time lags of the mean
of the parameters that determine the disc/corona system (i.e. $h$, $L_{\rm
transf}/L_{\rm disc}$, $\dot{m}/\dot{m}_{\rm Edd}$, $\Gamma$). Our results are
affected by the fact that we choose only a sub-sample of 15 SEDs. Thus the
changes in the model parameters and consequently in the time lags is
restricted to this sub-sample while in reality this may not be following the
continuous changes in the source. Despite this caveat, the average time-lag
spectrum and the time-lag spectrum obtained from the average parameters are
both in good agreement with the observed time lags within $\sim 1.3\,\sigma$,
$1.2\,\sigma$ on average, respectively. It is also worth noting that despite
the fact that the variable SEDs modelled in this work are limited to
wavelengths below $\sim 2000$ Å, the predicted time-lags describe well the
observations up to much longer wavelength.

An important implication of these results is that, in reality, time lags are
not constant. Thus, the measured time lags should depend on the duration of
the light curves. As the length of observed light curves increases, the mean
of the system parameters changes and so do the time lags. In this case, the
observed time lags using various light curve segments should not be the same,
but should indeed change with the light curve duration.

We have shown that the spectral and timing properties of NGC 5548 during the
STORM campaign could be well explained within the context of X-ray
illumination of the accretion disc. Using this model we are able to explain
the UV/optical continuum time-lag spectrum of this source, its power spectral
density, as well as the time-averaged and the time-dependent SEDs, which are
crucial tests for any model proposed to explain variability in a given source.
We plan to repeat this kind of modelling of all the various available and
suitable data sets of other sources in the future.

##  5 Acknowledgements

###### Acknowledgements.

We would like to thank the anonymous referee for their comments which helped
improve the clarity of the manuscript. EK acknowledges financial support from
the Centre National d’Etudes Spatiales (CNES). EK and IEP acknowledge support
from the European Union’s Horizon 2020 Programme under the AHEAD2020 project
(grant agreement n. 871158). MD acknowledges support from GACR project
21-06825X and thanks for the institutional support from RVO:67985815. This
work made use of data supplied by the UK Swift Science Data Centre at the
University of Leicester. This research is based on observations made with the
NASA/ESA Hubble Space Telescope obtained from the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Inc., under NASA contract NAS 5–26555. This work makes use of
Matplotlib (Hunter, 2007), NumPy (Harris et al., 2020), and SciPy (Virtanen et
al., 2020).

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##  Appendix A Best-fit results to the time-resolved SEDs

Table 4: Best-fit results obtained by fitting the individual spectra as
presented in Sect. 3.2. We also show in the bottom line the average of the
best-fitted parameters.

| Time | $L_{\rm transf}/L_{\rm disc}$ | $h$ | $\Gamma$ | $N_{\rm H}$ | $\log\xi$ | $R_{\rm C}$ | $\rm\chi^{2}/dof$ | $p_{\rm null}$
---|---|---|---|---|---|---|---|---|---
| (MJD) |  | ($r_{\rm g}$) |  | $(10^{22}\,\rm cm^{-2})$ | $(\rm erg\leavevmode\nobreak\ cm\leavevmode\nobreak\ s^{-1})$ | ($r_{\rm g}$) |  |
s1 | $56712.9-56714.2$ | $0.90_{-0.02}^{u}$ | $40.9_{-6.3}^{+4.3}$ | $1.52_{-0.05}^{+0.06}$ | $2.56_{-0.50}^{+0.93}$ | $1.31_{-0.17}^{+0.21}$ | 15.4 | 15.0/20 | 0.77
s2 | $56721.8-56723.2$ | $0.81_{-0.01}^{+0.02}$ | $3.9_{-0.3}^{+1.2}$ | $1.71_{-0.23}^{+0.25}$ | $3.79_{-1.74}^{+1.32}$ | $1.29_{-0.13}^{+0.19}$ | 7.3 | 20.0/13 | 0.09
s3 | $56731.6-56732.9$ | $0.79_{-0.01}^{+0.06}$ | $4.3_{-0.3}^{+2.0}$ | $1.60_{-0.17}^{+0.11}$ | $1.64_{-0.46}^{+2.51}$ | $0.59_{-3.59}^{+0.98}$ | 5.3 | 29.0/15 | 0.02
s4 | $56740.0-56741.3$ | $0.76_{-0.13}^{+0.14}$ | $15.8_{-3.9}^{+12.0}$ | $2.09_{-0.40}^{+0.14}$ | $2.04_{-0.21}^{+0.54}$ | $0.83_{-0.26}^{+0.70}$ | 27.0 | 14.5/13 | 0.33
s5 | $56747.9-56749.2$ | $0.64_{-0.05}^{+0.07}$ | $27.3_{-6.6}^{+13.1}$ | $1.89_{-0.16}^{+0.21}$ | $2.47_{-0.60}^{+0.69}$ | $1.39_{-0.22}^{+0.27}$ | 27.0 | 31.0/32 | 0.50
s6 | $56752.7-56754.0$ | $0.88_{-0.03}^{+0.01}$ | $44.8_{-9.0}^{+13.6}$ | $1.26_{-0.03}^{+0.03}$ | $1.00_{-0.34}^{+0.32}$ | $1.32_{-0.35}^{+0.21}$ | 7.4 | 40.0/33 | 0.19
s7 | $56755.7-56757.0$ | $0.88_{-0.04}^{u}$ | $38.3_{-3.4}^{+16.2}$ | $1.49_{-0.05}^{+0.06}$ | $1.31_{-0.29}^{+1.13}$ | $1.34_{-0.16}^{+0.38}$ | 13.1 | 53.7/36 | 0.03
s8 | $56758.8-56760.0$ | $0.73_{-0.01}^{+0.03}$ | $7.3_{-0.6}^{+0.5}$ | $1.90_{-0.11}^{u}$ | $1.88_{-0.16}^{+0.45}$ | $1.09_{-0.24}^{+0.16}$ | 11.5 | 40.9/33 | 0.16
s9 | $56769.7-56771.0$ | $0.58_{-0.07}^{+0.03}$ | $12.0_{-1.4}^{+3.8}$ | $1.90_{-0.08}^{u}$ | $2.99_{-0.63}^{+0.32}$ | $1.74_{-0.18}^{+0.18}$ | 14.7 | 35.0/22 | 0.04
s10 | $56779.2-56780.0$ | $0.70_{-0.06}^{+0.03}$ | $20.5_{-3.1}^{+8.7}$ | $1.60_{-0.08}^{+0.16}$ | $2.85_{-0.99}^{+0.70}$ | $1.64_{-0.32}^{+0.22}$ | 10.3 | 27.9/33 | 0.72
s11 | $56788.0-56789.0$ | $0.61_{-0.06}^{+0.08}$ | $13.4_{-4.2}^{+1.9}$ | $1.56_{-0.05}^{+0.19}$ | $2.18_{-0.84}^{+1.08}$ | $1.48_{-0.29}^{+0.34}$ | 6.6 | 47.9/29 | 0.02
s12 | $56798.7-56800.0$ | $0.55_{-0.03}^{+0.08}$ | $11.5_{-3.6}^{+2.4}$ | $1.73_{-0.15}^{+0.21}$ | $2.93_{-0.67}^{+0.49}$ | $1.29_{-0.15}^{+0.14}$ | 9.4 | 30.1/29 | 0.40
s13 | $56810.7-56812.0$ | $0.68_{-0.07}^{+0.08}$ | $23.1_{-3.2}^{+4.5}$ | $1.54_{-0.07}^{+0.18}$ | $1.81_{-0.42}^{+1.60}$ | $1.09_{-0.33}^{+0.42}$ | 9.3 | 27.9/21 | 0.14
s14 | $56823.4-56824.7$ | $0.68_{-0.10}^{+0.07}$ | $48.0_{-7.7}^{+14.9}$ | $1.88_{-0.09}^{+0.11}$ | $2.41_{-1.14}^{+1.06}$ | $1.26_{-0.56}^{+0.24}$ | 41.5 | 10.6/13 | 0.65
s15 | $56827.2-56828.5$ | $0.78_{-0.03}^{+0.02}$ | $36.3_{-8.3}^{+12.9}$ | $1.37_{-0.02}^{+0.06}$ | $1.15_{-0.17}^{+0.33}$ | $1.09_{-0.19}^{+0.20}$ | 8.6 | 41.3/34 | 0.18
Mean |  | 0.73 | 23.2 | 1.67 | 2.20 | 1.25 | 14.3 |  |

  * u

Upper limit.


# A Lower Bound for the Area of the Fundamental Region of a Binary Form

Jason Fang and Anton Mosunov  University of Waterloo

###### Abstract

Let

$F(x,y)=\prod\limits_{k=0}^{n-1}(\delta_{k}x-\gamma_{k}y)$

be a binary form of degree $n\geq 1$, with complex coefficients, written as a
product of $n$ linear forms in $\mathbb{C}[x,y]$. Let

$h_{F}=\prod\limits_{k=0}^{n-1}\sqrt{|\gamma_{k}|^{2}+|\delta_{k}|^{2}}$

denote the _height_ of $F$ and let $A_{F}$ denote the area of the _fundamental
region_ $\mathcal{D}_{F}$ of $F$:

$\mathcal{D}_{F}=\left\\{(x,y)\in\mathbb{R}^{2}\colon|F(x,y)|\leq 1\right\\}.$

We prove that $h_{F}^{2/n}A_{F}\geq\left(2^{1+(r/n)}\right)\pi$, where $r$ is
the number of roots of $F$ on the real projective line
$\mathbb{R}\mathbb{P}^{1}$, counting multiplicity.

##  1 Introduction

Let

$F(x,y)=\prod\limits_{k=0}^{n-1}(\delta_{k}x-\gamma_{k}y)$

be a binary form of degree $n\geq 1$, with complex coefficients, written as a
product of $n$ linear forms in $\mathbb{C}[x,y]$. Let

$h_{F}=\prod\limits_{k=0}^{n-1}\sqrt{|\gamma_{k}|^{2}+|\delta_{k}|^{2}}$

denote the _height_ of $F$. We define the _fundamental region_ (also known as
the _fundamental domain_) $\mathcal{D}_{F}$ of $F$ as

$\mathcal{D}_{F}=\left\\{(x,y)\in\mathbb{R}^{2}\colon|F(x,y)|\leq 1\right\\}.$

The area $A_{F}$ of the fundamental region $\mathcal{D}_{F}$ plays an
important role in number theory. Consider the special case when
$F\in\mathbb{Z}[x,y]$ is a binary form of degree $n\geq 3$ and nonzero
discriminant $D_{F}$. In 1933, Mahler [4] proved that the number of solutions
$Z_{F}(m)$ to the _Thue inequality_

$|F(x,y)|\leq m$

can be approximated by $A_{F}m^{2/n}$, provided that $F$ is irreducible over
the rationals (here $m$ is a non-negative integer). More precisely, he proved
the existence of a positive real number $c_{F}$, which depends only on $F$,
such that

$\left|Z_{F}(m)-A_{F}m^{2/n}\right|\leq c_{F}m^{1/(n-1)}.$

In 2019, Stewart and Xiao [6, Theorem 1.1] proved that the number of integers
$R_{F}(m)$ of absolute value at most $m$ which are represented by $F$ is
asymptotic to $W_{F}A_{F}m^{2/n}$, where $W_{F}$ is a positive rational number
that can be computed in terms of the _automorphism group_ of $F$ (see [6] for
the definition of the automorphism group). Thus it is interesting to
investigate how large and how small can $A_{F}$ be. Motivated by this question
we succeeded in proving the following.

######  Theorem 1.1.

Let $F$ be a binary form with complex coefficients of degree $n\geq 1$, with
exactly $r$ roots on the real projective line $\mathbb{R}\mathbb{P}^{1}$,
counting multiplicity. Then

$h_{F}^{2/n}A_{F}\geq\left(2^{1+(r/n)}\right)\pi.$ (1)

For complex numbers $x$ and $y$ with positive real parts, let

$B(x,y)=2\int\limits_{0}^{\pi/2}(\sin\theta)^{2x-1}(\cos\theta)^{2y-1}d\theta$
(2)

denote the _beta function_ , written in its trigonometric form. The lower
bound for $A_{F}$ given in Theorem 1.1 complements the upper bound for $A_{F}$
found by Bean [1, Theorem 1]. Bean’s result asserts that for any binary form
with complex coefficients having degree $n\geq 3$ and nonzero discriminant
$D_{F}$ the following inequality holds:

$|D_{F}|^{1/n(n-1)}A_{F}\leq 3B\left(\frac{1}{3},\frac{1}{3}\right).$

One of the ingredients in our proof is Jensen’s inequality, which states that

$\varphi\left(\frac{1}{b-a}\int\limits_{a}^{b}f(x)dx\right)\geq\frac{1}{b-a}\int\limits_{a}^{b}\varphi(f(x))dx$
(3)

for every concave function $\varphi(x)$ and every real-valued function $f(x)$
that is Lebesgue-integrable on an interval $[a,b]$. Another important
ingredient is the polar formula for the computation of $A_{F}$. Since the
curve $|F(x,y)|=1$ can be expressed in polar form as

$r(\theta)=\frac{1}{\left|F(\cos\theta,\sin\theta)\right|^{1/n}},$

from calculus we know that the area $\mathcal{D}_{F}$ bounded by this curve
can be computed as follows:

$A_{F}=\int\limits_{0}^{2\pi}\frac{1}{2}r^{2}(\theta)d\theta=\frac{1}{2}\int\limits_{0}^{2\pi}\frac{d\theta}{\left|F(\cos\theta,\sin\theta)\right|^{2/n}}.$
(4)

With this formula one can also notice that $A_{cF}=|c|^{-2/n}A_{F}$ for any
nonzero complex number $c$. Since $h_{cF}=|c|h_{F}$, we can see that the
quantity $h_{F}^{2/n}A_{F}$ is invariant under scaling by a nonzero complex
number, i.e.,

$h_{cF}^{2/n}A_{cF}=\left(|c|h_{F}\right)^{2/n}\left(|c|^{-2/n}A_{F}\right)=h_{F}^{2/n}A_{F}.$

When $r=0$ or $r=n$ the right-hand side of the inequality (1) becomes
independent of both $r$ and $n$. In the case $r=n$ there does exist a family
of binary forms $F_{n,n}$ such that

$\lim\limits_{n\rightarrow\infty}\left(h_{F_{n,n}}A_{F_{n,n}}\right)=4\pi.$

This family was studied by Bean and Laugesen [3] and a binary form $F_{n,n}$
is defined by

$F_{n,n}(x,y)=\prod\limits_{k=1}^{n}\left(x\sin\frac{k\pi}{n}-y\cos\frac{k\pi}{n}\right).$

In this case, for $n\geq 3$ we have

$h_{F_{n,n}}=1\quad\text{and}\quad
A_{F_{n,n}}=4^{1-1/n}B\left(\frac{1}{2}-\frac{1}{n},\frac{1}{2}\right).$

In the case $r=0$ the family of binary forms $F_{n,0}$ defined as

$F_{n,0}(x,y)=(x-iy)^{n}$

satisfies $h_{F_{n,0}}^{2/n}A_{F_{n,0}}=2\pi$ for every positive integer $n$.
To see that this is the case, note that $h_{F_{n,0}}=2^{n/2}$, and that

$|F(\cos\theta,\sin\theta)|=|(\cos\theta-i\sin\theta)^{n}|=\left|\exp\left(-\theta
i\right)^{n}\right|=\left|\exp(-n\theta i)\right|=1.$

Thus it follows from (4) that $A_{F_{n,0}}=\pi$. The family $F_{n,0}$ has a
rather special property that the quantity $h_{F}^{2/\deg F}A_{F}$ attains its
smallest value when $F=cF_{n,0}$ for some $n\in\mathbb{N}$ and nonzero
$c\in\mathbb{C}$.111Notice that there are other families that also attain the
minimum. For example, for even $n$ and nonzero $c\in\mathbb{C}$ the minimum is
attained by the form $c(x^{2}+y^{2})^{n/2}$. By combining the above two
examples together, we define the family of binary forms

$F_{n,r}(x,y)=(x-iy)^{n-r}\prod\limits_{k=1}^{r}\left(x\sin\frac{k\pi}{r}-y\cos\frac{k\pi}{r}\right)$
(5)

and conjecture the following.

######  Conjecture 1.2.

Let $F$ be a binary form with complex coefficients of degree $n\geq 1$, with
exactly $r$ roots on the real projective line $\mathbb{R}\mathbb{P}^{1}$,
counting multiplicity. Then

$h_{F}^{2/n}A_{F}\geq h_{F_{n,r}}^{2/n}A_{F_{n,r}}.$

For a $2\times 2$ matrix $M=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)$ define

$F_{M}(x,y)=F(ax+by,cx+dy).$

We say that two binary forms $F$ and $G$ are _equivalent under
$\operatorname{GL}_{2}(\mathbb{R})$_ if and only if there exists an invertible
$2\times 2$ matrix $M$, with real coefficients, such that $F_{M}=G$.
Conjecture 1.2 naturally complements the following conjecture of Bean [2,
Conjecture 1].

######  Conjecture 1.3.

The maximal value $M_{n}$ of the quantity $|D_{F}|^{1/n(n-1)}A_{F}$ over the
class of forms of degree $n$ with complex coefficients and nonzero
discriminant $D_{F}$ is attained precisely when $F$ is a form which, up to
multiplication by a complex number, is equivalent under
$\operatorname{GL}_{2}(\mathbb{R})$ to the form $F_{n,n}$. In particular,

$M_{n}=D_{F_{n,n}}^{1/n(n-1)}A_{F_{n,n}}.$

The paper is organized as follows: in Section 2 we prove two auxiliary lemmas,
in Section 3 we prove Theorem 1.1, and in Section 4 we compute the height and
the area of the fundamental region of a binary form $F_{n,r}$ defined in (5).

##  2 Auxiliary Results

In this section we prove two auxiliary lemmas.

######  Lemma 2.1.

$\displaystyle\int\limits_{0}^{\frac{\pi}{2}}\log(\sin(x))dx=-\frac{\pi}{2}\log
2$.

###### Proof.

Denote the left-hand side by $I$. Then

$I=\int\limits_{0}^{\frac{\pi}{2}}\log(\sin(x))dx=\int\limits_{0}^{\frac{\pi}{2}}\log\left(\cos\left(\frac{\pi}{2}-x\right)\right)dx=\int\limits_{0}^{\frac{\pi}{2}}\log(\cos(x))dx,$

so

$\displaystyle 2I$
$\displaystyle=\int\limits_{0}^{\frac{\pi}{2}}\log(\sin(x))dx+\int\limits_{0}^{\frac{\pi}{2}}\log(\cos(x))dx$
$\displaystyle=\int\limits_{0}^{\frac{\pi}{2}}\log(\sin(x)\cos(x))dx$
$\displaystyle=\int\limits_{0}^{\frac{\pi}{2}}\log\left(\frac{\sin(2x)}{2}\right)dx$
$\displaystyle=\int\limits_{0}^{\frac{\pi}{2}}\log(\sin(2x))dx-\frac{\pi}{2}\log
2$ $\displaystyle=\frac{1}{2}\int\limits_{0}^{\pi}\log(\sin
x)dx-\frac{\pi}{2}\log 2$ $\displaystyle=I-\frac{\pi}{2}\log 2.$

Hence $I=-\frac{\pi}{2}\log 2$. ∎

######  Lemma 2.2.

For every real number $\beta$,

$\int\limits_{0}^{\pi}\log((\sin\beta\cos\theta)^{2}+(\cos\beta\sin\theta)^{2})\
d\theta\leq-\pi\log 2.$

Furthermore, the equality is attained if and only if
$\beta=\frac{\pi}{4}+\frac{\pi}{2}k$ for some $k\in\mathbb{Z}$.

###### Proof.

Let $I(\beta)$ denote the left-hand side. Then it follows from Jensen’s
inequality (3) with $\varphi(x)=\log x$ that

$\displaystyle I(\beta)$
$\displaystyle\leq\pi\log\left(\frac{1}{\pi}\int\limits_{0}^{\pi}\left(\sin^{2}\beta\cos^{2}\theta+\cos^{2}\beta\sin^{2}\theta\right)\
d\theta\right)$
$\displaystyle=\pi\log\left(\frac{1}{\pi}\sin^{2}\beta\int\limits_{0}^{\pi}\cos^{2}\theta\
d\theta+\cos^{2}\beta\int\limits_{0}^{\pi}\sin^{2}\theta\ d\theta\right)$
$\displaystyle=\pi\log\left(\frac{1}{\pi}\sin^{2}\beta\cdot\frac{\pi}{2}+\cos^{2}\beta\cdot\frac{\pi}{2}\right)$
$\displaystyle=\pi\log\left(\frac{1}{2}\right)$ $\displaystyle=-\pi\log 2.$

Since the function $\log x$ is strictly convex, the equality holds if and only
if

$\log((\sin\beta\cos\theta)^{2}+(\cos\beta\sin\theta)^{2})$ (6)

is constant on $(0,\pi)$. This means that for any
$\theta_{1},\theta_{2}\in(0,\pi)$,

$\displaystyle\log((\sin\beta\cos\theta_{1})^{2}+(\cos\beta\sin\theta_{1})^{2})$
$\displaystyle=\log((\sin\beta\cos\theta_{2})^{2}+(\cos\beta\sin\theta_{2})^{2}),$
$\displaystyle(\sin\beta\cos\theta_{1})^{2}+(\cos\beta\sin\theta_{1})^{2}$
$\displaystyle=(\sin\beta\cos\theta_{2})^{2}+(\cos\beta\sin\theta_{2})^{2}.$

We can pick $\theta_{1}=\frac{\pi}{6}$ and $\theta_{2}=\frac{2\pi}{3}$ to
obtain:

$\displaystyle\left(\sin\beta\cdot\frac{\sqrt{3}}{2}\right)^{2}+\left(\cos\beta\cdot\frac{1}{2}\right)^{2}$
$\displaystyle=\left(\sin\beta\cdot-\frac{1}{2}\right)^{2}+\left(\cos\beta\cdot\frac{\sqrt{3}}{2}\right)^{2},$
$\displaystyle\frac{3}{4}\sin^{2}\beta+\frac{1}{4}\cos^{2}\beta$
$\displaystyle=\frac{1}{4}\sin^{2}\beta+\frac{3}{4}\cos^{2}\beta,$
$\displaystyle\frac{1}{2}\sin^{2}\beta$
$\displaystyle=\frac{1}{2}\cos^{2}\beta,$ $\displaystyle\sin^{2}\beta$
$\displaystyle=\cos^{2}\beta.$

This happens if and only if $\beta=\frac{\pi}{4}+\frac{\pi}{2}k$ for some
integer $k$. Plugging this value of $\beta$ into (6), we see that the
resulting function is constant and equal to $-\log 2$. Hence
$I\left(\frac{\pi}{4}+\frac{\pi}{2}k\right)=-\pi\log 2$ for every integer $k$.
∎

##  3 Proof of Theorem 1.1

Suppose that the binary form $F(x,y)$ has exactly $r$ roots
$(\gamma_{j}\colon\delta_{j})$ on $\mathbb{R}\mathbb{P}^{1}$, counting
multiplicity. Write

$F(x,y)=\prod\limits_{j=0}^{r-1}(\delta_{j}x-\gamma_{j}y)\prod\limits_{k=r}^{n-1}(\delta_{k}x-\gamma_{k}y),$

where $\delta_{j},\gamma_{j}\in\mathbb{R}$ and
$\delta_{k},\gamma_{k}\in\mathbb{C}$. Notice that we can rewrite $F$ as
follows:

$F(x,y)=h_{F}\prod\limits_{j=0}^{r-1}(\delta_{j}^{\prime}x-\gamma_{j}^{\prime}y)\prod\limits_{k=r}^{n-1}(\delta_{k}^{\prime}x-\gamma_{k}^{\prime}y),$

where we now have

$(\gamma_{j}^{\prime})^{2}+(\delta_{j}^{\prime})^{2}=1\quad\text{and}\quad\left|\gamma_{k}^{\prime}\right|^{2}+\left|\delta_{k}^{\prime}\right|^{2}=1.$

In particular, notice that for $j=0,1,\ldots,r-1$ each point
$(\gamma_{j}^{\prime},\delta_{j}^{\prime})$ lies on the unit circle, so
$(\gamma_{j}^{\prime},\delta_{j}^{\prime})=(\cos\alpha_{j},\sin\alpha_{j})$
for some $\alpha_{j}\in\mathbb{R}$. Since

$\displaystyle F(\cos\theta,\sin\theta)$
$\displaystyle=h_{F}\prod\limits_{j=0}^{r-1}\sin\left(\theta-\alpha_{j}\right)\prod\limits_{k=r}^{n-1}(\delta_{k}^{\prime}\cos\theta-\gamma_{k}^{\prime}\sin\theta),$

we can use the formula (4) to compute $A_{F}$:

$\displaystyle A_{F}$
$\displaystyle=\frac{1}{2}\int\limits_{0}^{2\pi}\frac{d\theta}{|F(\cos\theta,\sin\theta)|^{2/n}}$
$\displaystyle=\frac{1}{2h_{F}^{2/n}}\int\limits_{0}^{2\pi}\sqrt[n]{\prod\limits_{j=0}^{r-1}\csc^{2}\left(\theta-\alpha_{j}\right)}\cdot\sqrt[n]{\prod\limits_{k=r}^{n-1}|\delta_{k}^{\prime}\cos\theta-\gamma_{k}^{\prime}\sin\theta|^{-2}}\
d\theta.$

Since the integrand has period $\pi$, we conclude that

$h_{F}^{2/n}A_{F}=\int\limits_{0}^{\pi}\sqrt[n]{\prod\limits_{j=0}^{r-1}\csc^{2}\left(\theta-\alpha_{j}\right)}\cdot\sqrt[n]{\prod\limits_{k=r}^{n-1}|\delta_{k}^{\prime}\cos\theta-\gamma_{k}^{\prime}\sin\theta|^{-2}}\
d\theta.$

By Jensen’s inequality (3) with $\varphi(x)=\log x$,

$\displaystyle\log\left(\frac{h_{F}^{2/n}A_{F}}{\pi}\right)$
$\displaystyle\geq\frac{1}{\pi}\int\limits_{0}^{\pi}\log\left(\sqrt[n]{\prod\limits_{j=0}^{r-1}\csc^{2}\left(\theta-\alpha_{j}\right)}\cdot\sqrt[n]{\prod\limits_{k=r}^{n-1}|\delta_{k}^{\prime}\cos\theta-\gamma_{k}^{\prime}\sin\theta|^{-2}}\right)\
d\theta$
$\displaystyle=\frac{1}{n\pi}\sum\limits_{j=0}^{r-1}\int\limits_{0}^{\pi}\log(\csc^{2}(\theta-\alpha_{j}))d\theta+\frac{1}{n\pi}\sum\limits_{k=r}^{n-1}\int\limits_{0}^{\pi}\log\left(|\delta_{k}^{\prime}\cos\theta-\gamma_{k}^{\prime}\sin\theta|^{-2}\right)d\theta$
$\displaystyle=\frac{r}{n\pi}\int\limits_{0}^{\pi}\log(\csc^{2}(\theta))d\theta+\frac{1}{n\pi}\sum\limits_{k=r}^{n-1}\int\limits_{0}^{\pi}\log\left(|\delta_{k}^{\prime}\cos\theta-\gamma_{k}^{\prime}\sin\theta|^{-2}\right)d\theta$
$\displaystyle\text{Lemma \ref{lem:logsin}}\rightarrow$
$\displaystyle=\frac{r}{n}\log 4+\frac{1}{n\pi}I,$

where

$I=\sum\limits_{k=r}^{n-1}\int\limits_{0}^{\pi}\log\left(|\delta_{k}^{\prime}\cos\theta-\gamma_{k}^{\prime}\sin\theta|^{-2}\right)d\theta.$

It remains to prove that $I\geq(n-r)\pi\log 2$. Since
$\left|\gamma_{k}^{\prime}\right|^{2}+\left|\delta_{k}^{\prime}\right|^{2}=1$,
there exist real numbers $\beta_{k}$ such that

$|\gamma_{k}^{\prime}|=\cos\beta_{k}\quad\text{and}\quad|\delta_{k}^{\prime}|=\sin\beta_{k}.$

Notice that

$\displaystyle|\delta_{k}^{\prime}\cos\theta-\gamma_{k}^{\prime}\sin\theta|\cdot|\delta_{k}^{\prime}\cos(\pi-\theta)-\gamma_{k}^{\prime}\sin(\pi-\theta)|$
$\displaystyle=|(\delta_{k}^{\prime}\cos\theta)^{2}-(\gamma_{k}^{\prime}\sin\theta)^{2}|$
$\displaystyle\leq\left|\delta_{k}^{\prime}\cos\theta\right|^{2}+\left|\gamma_{k}^{\prime}\sin\theta\right|^{2}$
$\displaystyle=(\sin\beta_{k}\cos\theta)^{2}+(\cos\beta_{k}\sin\theta)^{2},$

so

$(|\delta_{k}^{\prime}\cos\theta-\gamma_{k}^{\prime}\sin\theta||\delta_{k}^{\prime}\cos(\pi-\theta)-\gamma_{k}^{\prime}\sin(\pi-\theta)|)^{-2}\geq((\sin\beta_{k}\cos\theta)^{2}+(\cos\beta_{k}\sin\theta)^{2})^{-2}.$

Consequently,

$\displaystyle 2I$
$\displaystyle=\sum\limits_{k=r}^{n-1}\int\limits_{0}^{\pi}\log\left(|\delta_{k}^{\prime}\cos\theta-\gamma_{k}^{\prime}\sin\theta|^{-2}\right)d\theta+\sum\limits_{k=r}^{n-1}\int\limits_{0}^{\pi}\log\left(|\delta_{k}^{\prime}\cos(\pi-\theta)-\gamma_{k}^{\prime}\sin(\pi-\theta)|^{-2}\right)d\theta$
$\displaystyle=\sum\limits_{k=r}^{n-1}\int\limits_{0}^{\pi}\log\left(|\delta_{k}^{\prime}\cos\theta-\gamma_{k}^{\prime}\sin\theta|\cdot|\delta_{k}^{\prime}\cos(\pi-\theta)-\gamma_{k}^{\prime}\sin(\pi-\theta)|\right)^{-2}d\theta$
$\displaystyle\geq\sum\limits_{k=r}^{n-1}\int\limits_{0}^{\pi}\log((\sin\beta_{k}\cos\theta)^{2}+(\cos\beta_{k}\sin\theta)^{2})^{-2}d\theta$
$\displaystyle\text{Lemma \ref{lem:bound}}\rightarrow$ $\displaystyle\geq
2(n-r)\pi\log 2.$

In summary, we proved that

$\log\left(\frac{h_{F}^{2/n}A_{F}}{\pi}\right)\geq\frac{r}{n}\log
4+\frac{n-r}{n}\log 2.$

Exponentiation on both sides yields the desired inequality
$h_{F}^{2/n}A_{F}\geq\left(2^{1+(r/n)}\right)\pi$.

##  4 The Family $F_{n,r}$

In this section we derive formulas for the height and the area of the
fundamental region of a binary form $F_{n,r}$ defined in (5). By definition,
the height of $F_{n,r}$ is given by $h_{F_{n,r}}=2^{(n-r)/2}$. To compute
$A_{F_{n,r}}$, notice that for $r\geq 1$ we have

$\displaystyle\left|F_{n,r}(\cos\theta,\sin\theta)\right|$
$\displaystyle=\left|(\cos\theta-i\sin\theta)^{n-r}\prod\limits_{k=1}^{r}\left(\cos\theta\sin\frac{k\pi}{r}-\sin\theta\cos\frac{k\pi}{r}\right)\right|$
$\displaystyle=\left|\prod\limits_{k=1}^{r}\sin\left(\theta-\frac{k\pi}{r}\right)\right|=\left|2^{-(r-1)}\sin(r\theta)\right|,$

where the last equality follows from the identity
$\sin(r\theta)=2^{r-1}\prod_{k=1}^{r}\sin\left(\frac{k\pi}{r}-\theta\right)$
(see, for example, [5, Section 2]). By (4),

$\displaystyle A_{F_{n,r}}$
$\displaystyle=\frac{1}{2}\int\limits_{0}^{2\pi}\sqrt[n]{4^{r-1}\csc^{2}(r\theta)}d\theta$
$\displaystyle=2^{2(r-1)/n-1}\int\limits_{0}^{2\pi}\sqrt[n]{\csc^{2}(r\theta)}d\theta$
$\displaystyle=\frac{2^{2(r-1)/n-1}}{r}\int\limits_{0}^{2\pi
r}\sqrt[n]{\csc^{2}\theta}d\theta$
$\displaystyle=\frac{2^{2(r-1)/n-1}}{r}\cdot
4r\int\limits_{0}^{\pi/2}(\sin\theta)^{-\frac{2}{n}}d\theta.$

Now, for $n=1,2$ the above integral diverges, while for $n\geq 3$ it follows
from (2) that

$A_{F_{n,r}}=2^{2(r-1)/n}\cdot
2\int\limits_{0}^{\pi/2}(\sin\theta)^{-\frac{2}{n}}d\theta=2^{2(r-1)/n}B\left(\frac{1}{2}-\frac{1}{n},\frac{1}{2}\right).$

Thus,

$h_{F_{n,r}}^{2/n}A_{F_{n,r}}=\begin{cases}2\pi&\text{if $r=0$,}\\\
\infty&\text{if $n=1,2$ and $r>0$,}\\\
2^{1+(r-2)/n}B\left(\frac{1}{2}-\frac{1}{n},\frac{1}{2}\right)&\text{if $n>2$
and $r>0$.}\end{cases}$

## References

  * [1] M. A. Bean, _An isoperimetric inequality for the area of plane regions defined by binary forms_ , Compositio Math. 92, pp. 115–131, 1994.
  * [2] M. A. Bean, _Binary forms, hypergeometric functions and the Schwarz-Christoffel mapping formula_ , Trans. of the Amer. Math. Soc. 347 (12), pp. 4959–4983, 1995.
  * [3] M. A. Bean and R. S. Laugesen, _Binary forms, equiangular polygons and harmonic measure_ , Rocky Mt. J. Math. 30, pp. 15–62, 2000.
  * [4] K. Mahler, _Zur Approximation algebraischer Zahlen III_ , Acta Math. 62, pp. 91–166, 1933.
  * [5] A. Mosunov, _On the area bounded by the curve $\prod_{k=1}^{n}\left|x\sin\frac{k\pi}{n}-y\cos\frac{k\pi}{n}\right|=1$_, Rocky Mountain J. Math. 50 (5), pp. 1773–1777, 2020.
  * [6] C. L. Stewart and S. Y. Xiao, _On the representation of integers by binary forms_ , Math. Ann. 375, pp. 133–163, 2019.