# The outer dusty edge of accretion disks in active galactic nuclei

Hermine Landt 1,∗

###### Abstract

Recent models for the inner structure of active galactic nuclei (AGN) aim at
connecting the outer region of the accretion disk with the broad-line region
and dusty torus through a radiatively accelerated, dusty outflow. Such an
outflow not only requires the outer disk to be dusty and so predicts disk
sizes beyond the self-gravity limit but requires the presence of nuclear dust
with favourable properties. Here we investigate a large sample of type 1 AGN
with near-infrared (near-IR) cross-dispersed spectroscopy with the aim to
constrain the astrochemistry, location and geometry of the nuclear hot dust
region. Assuming thermal equilibrium for optically thin dust, we derive the
luminosity-based dust radius for different grain properties using our
measurement of the temperature. We combine our results with independent dust
radius measurements from reverberation mapping and interferometry and show
that large dust grains that can provide the necessary opacity for the outflow
are ubiquitous in AGN. Using our estimates of the dust covering factor, we
investigate the dust geometry using the effects of the accretion disk
anisotropy. A flared disk-like structure for the hot dust is favoured.
Finally, we discuss the implication of our results for the dust radius-
luminosity plane.

##  1 Keywords:

Active galactic nuclei — Quasars — Dust continuum emission — Dust physics —
Near-infrared astronomy

##  2 Introduction

A prominent feature in the multi-wavelength spectral energy distributions
(SEDs) of active galactic nuclei (AGN) is strong infrared (IR) continuum
emission, which originates from thermal dust radiation. The structure of the
dust producing it is commonly assumed to be an optically thick toroid, which
is aligned with the plane of the accretion disk. The accretion disk then
provides the UV/optical photons that heat the dust. An alternative to the
dusty torus was already proposed early on by Phinney (1989), namely, an
extended and warped dusty disk. Such a structure is attractive since it could
present a natural transition between the accretion disk and the central dust,
without the stability problem of the torus (Krolik, 2007).

The extended dust structure emits over a large IR wavelength range, with the
near-IR believed to be dominated by the hottest dust located closest to the
central supermassive black hole. Therefore, AGN are most suitable to
investigate the chemical composition and grain properties of astrophysical
dust. Their UV/optical luminosities are usually high enough to heat the
central dust to sublimation temperatures, a property which they have in common
with protoplanetary disks around young stars. If we can observe these highest
temperatures, we can in principle constrain the chemistry since different
species condense out of the gas phase in different environmental conditions.
Previously dust temperatures were measured in a handful of AGN by obtaining
simultaneous photometry at several near-IR wavelengths (Clavel et al., 1989;
Glass, 2004; Schnülle et al., 2013, 2015), but has only come of age with the
availability of efficient near-IR cross-dispersed spectrographs. Landt et al.
(2011b) and Landt et al. (2014) derived dust temperatures from such
spectroscopy for the largest sample of type 1 AGN so far ($\sim 30$ sources).
They found a very narrow dust temperature distribution with an average value
of $T\sim 1400$ K. This result, which is similar to what has been found in
protoplanetary disks (Monnier et al., 2005), either indicates dust at
sublimation but composed mainly of silicate grains and so an oxygen-rich
environment from which the dust formed. Or, if carbon dominates the
composition, then the dust is not heated to close to sublimation, since
carbonaceous dust, e.g. graphite, can survive up to $T\sim 2000$ K (Salpeter,
1977).

The first near-IR spectroscopic monitoring campaign of the hot dust in an AGN,
namely, NGC 5548, found that a single component dominated both the mean and
variable emission, with the dust reponse time and luminosity-based dust radius
being consistent with each other if the emissivity of a pure blackbody was
assumed (Landt et al., 2019). Thus, the dust grain size of the hot dust in
this AGN could be constrained to relatively large values (of a few $\mu$m).
From the estimated dust temperature and its variability they concluded that
the dust composition was predominantly carbonaceous and well below the
sublimation threshold. The reverberation signal of the dust was then mainly
due to a heating and cooling process in response to the variable UV/optical
accretion disk flux irradiating it. Most importantly, the dust reverberation
signal showed tentative evidence for a second hot dust component most likely
associated with the accretion disk. The existence of dust in the outer regions
of the accretion disk is a prerequisite for the recent models of the AGN
structure proposed by Czerny et al. (2017) and Baskin and Laor (2018). These
authors explain both the broad emission line region (BLR) and the dusty torus
as part of the same outflow launched from the outer accretion disk by
radiation pressure on dust. Since carbonaceous dust has a higher opacity than
silicate dust, the former is preferred in this scenario. Most recently, Landt
et al. (2023) presented results from a near-IR spectroscopic monitoring
campaign on the high-luminosity AGN Mrk 876. The comparison of the mean and
variable (rms) spectrum clearly showed that at least two hot dust components
are present in AGN, with the second component possibly originating in the
accretion disk. However, contrary to the case of NGC 5548, the independent
measure of the dust radius via reverberation mapping yielded a value a factor
of $\sim 2$ lower than the luminosity-based dust radius estimate, indicating
that the anisotropy effects of the accretion disk illumination and through it
the dust geometry can be detected and studied in this way. For Mrk 876, it was
concluded that the geometry is most likely a flared, dusty disk with an
enlarged ’inner hole’, which is very similar to the paradigm commonly assumed
for protoplanetary disks.

Given the high potential that a comparison between luminosity-based dust radii
and those obtained by independent methods, e.g. by reverberation mapping,
carries for revealing the astrochemistry and geometry of the hot dust in AGN,
this study sets out to apply it to a large sample of AGN. In Section 3, we
discuss the selection of the sample, whereas Section 4 gives details of the
measurements required for the estimation of the luminosity-based dust radius.
In Section 5, we discuss our main results and finally present the conclusions
in Section 6.

##  3 The sample selection

As discussed by Landt et al. (2019, 2023), combining the assumption of
radiative equilibrium of optically thin dust, which encodes a dependence on
the dust chemical species and grain size through the dust emissivity parameter
(see eq. 1), with an independent measurement of the dust radius can constrain
the dust astrochemistry. Alternative dust radius measurements can come from,
e.g., the dust response time estimated by reverberation mapping or the
geometric distance of the dust measured through near-IR interferometry. A
promising new method is also the estimate of the location of the scattering
region through optical spectropolarimetry of the BLR (Shablovinskaya et al.,
2020). Therefore, this study required a sample of type 1 AGN with an
optical/near-IR continuum dominated by the AGN rather than the host galaxy
starlight. Furthermore, the available near-IR spectrum should cover a
sufficiently large wavelength range to allow for both an estimate of the
accretion disk flux level as well as a measurement of the hot dust temperature
in order to be able to calculate the luminosity-based dust radius. This
requirement is usually met only by cross-dispersed near-IR spectra (Landt et
al., 2011b). Finally, the type 1 AGN with available cross-dispersed near-IR
spectra were required to have a published near-IR dust radius based on an
alternative method. The total sample comprises of 39 objects and its
properties are listed in Table 1. The independent dust radius measurement was
mostly from near-IR (photometric) dust reverberation mapping campaigns
covering the wavelength region of up to a few $\mu$m and so sampling the SED
of the hot dust, and in only 4/39 sources the dust radius measurement was
based on near-IR interferometry.

Table 1:  The sample

Object name | $z$ | log $L_{\rm uv}$ | $T_{\rm d}$ | log $L_{\rm d}$ | $R_{\rm d,lum}$ | Ref. | $R_{\rm d,rev}$ | Ref.
---|---|---|---|---|---|---|---|---
|  | (erg s-1) | (K) | (erg s-1) | (lt-days) |  | (lt-days) |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
3C 351 | 0.372 | 46.63 | 1373 | 46.78 | 792 | R06 | 1203$\pm$542 | Lyu19
PG 2233$+$134 | 0.326 | 46.85 | 1413 | 45.65 | 964 | This work | 343$\pm$29 | Lyu19
PG 0953$+$414 | 0.234 | 46.45 | 1391 | 45.75 | 627 | This work | 566${}^{+50}_{-38}$ | M19
PG 0947$+$396 | 0.206 | 45.22 | 1406 | 44.96 | 149 | This work | 1629$\pm$23 | Lyu19
PDS 456 | 0.184 | 47.41 | 1425 | 46.83 | 1806 | L08 | 1599$\pm$213⋆ | G20
3C 273 | 0.158 | 47.59 | 1443 | 46.76 | 2166 | L08 | 1656$\pm$59 | Lyu19
PG 0052$+$251 | 0.155 | 46.07 | 1198 | 45.29 | 546 | L13 | 347$\pm$37 | Lyu19
PG 1307$+$085 | 0.155 | 46.22 | 1307 | 45.25 | 545 | L13 | 310$\pm$40 | Lyu19
PG 0026$+$129 | 0.145 | 46.41 | 1127 | 45.15 | 913 | L13 | 487$\pm$36 | Lyu19
PG 1519$+$226 | 0.137 | 45.85 | 1538 | 45.24 | 257 | R06 | 170$\pm$99 | Lyu19
PG 1612$+$261 | 0.131 | 45.67 | 1549 | 45.25 | 206 | R06 | 555$\pm$35 | Lyu19
Mrk 876 | 0.129 | 46.09 | 1306 | 45.55 | 469 | L23 | 334${}^{+42}_{-37}$ | M19
PG 1415$+$451 | 0.114 | 45.92 | 1461 | 45.07 | 309 | R06 | 269$\pm$208 | Lyu19
PG 0804$+$761 | 0.100 | 45.97 | 1314 | 45.70 | 405 | L13 | 600$\pm$21 | Lyu19
PG 1211$+$143 | 0.081 | 46.22 | 1337 | 45.10 | 521 | L13 | 338$\pm$83 | Lyu19
Mrk 478 | 0.079 | 46.11 | 1547 | 45.34 | 343 | R06 | 237$\pm$37 | Lyu19
PG 1448$+$273 | 0.065 | 45.31 | 1477 | 44.55 | 150 | R06 | 263$\pm$30 | Lyu19
PG 0844$+$349 | 0.064 | 46.18 | 1190 | 44.76 | 628 | L11 | 99${}^{+13}_{-10}$ | M19
Mrk 1513 | 0.063 | 46.33 | 1356 | 45.04 | 575 | L13 | 494$\pm$42 | Lyu19
I Zw 1 | 0.060 | 45.58 | 1412 | 45.21 | 224 | G12 | 274$\pm$41 | Lyu19
PG 1126$-$041 | 0.060 | 45.65 | 1515 | 45.09 | 211 | R06 | 523$\pm$26 | Lyu19
Mrk 734 | 0.050 | 45.70 | 1597 | 44.48 | 201 | This work | 103$\pm$7 | Lyu19
Mrk 231 | 0.041 | 45.67 | 1529 | 45.71 | 212 | This work | 393$\pm$83⋆ | K09
Mrk 841 | 0.036 | 45.22 | 1437 | 44.23 | 143 | P22 | 110$\pm$15 | Lyu19
Mrk 110 | 0.035 | 46.00 | 1452 | 44.56 | 343 | L11 | 117$\pm$6 | K14
Mrk 509 | 0.034 | 45.77 | 1398 | 44.79 | 284 | L08 | 121$\pm$2 | K14
Ark 120 | 0.033 | 45.45 | 1102 | 45.16 | 316 | L08 | 138$\pm$18 | K14
3C 120 | 0.033 | 45.28 | 1389 | 44.59 | 164 | L13 | 94${}^{+4}_{-7}$ | R18
Mrk 817 | 0.031 | 45.65 | 1401 | 44.53 | 246 | L11 | 93$\pm$9 | K14
Mrk 290 | 0.030 | 45.36 | 1353 | 44.26 | 189 | L08 | 124$\pm$3 | Lyu19
H 2106$-$099 | 0.027 | 45.28 | 1342 | 44.27 | 175 | L08 | 303$\pm$36⋆ | G23
Mrk 335 | 0.026 | 45.63 | 1308 | 44.40 | 276 | L08 | 168$\pm$6 | K14
Mrk 79 | 0.022 | 44.82 | 1364 | 44.21 | 100 | L11 | 68$\pm$5 | K14
Mrk 1239 | 0.019 | 44.85 | 1443 | 44.64 | 92 | R06 | 189$\pm$30⋆ | G23
NGC 5548 | 0.017 | 44.50 | 1450 | 44.02 | 61 | L19 | 75$\pm$8 | L19
NGC 7469 | 0.016 | 45.28 | 1551 | 44.19 | 131 | L08 | 85$\pm$1 | K14
NGC 3783 | 0.010 | 45.23 | 1472 | 44.01 | 138 | P22 | 131${}^{+25}_{-50}$ | E23
NGC 4593 | 0.009 | 44.72 | 1380 | 43.71 | 87 | L08 | 42$\pm$1 | K14
NGC 4151 | 0.003 | 43.24 | 1328 | 43.06 | 17 | L08 | 46$\pm$1 | K14

The columns are: (1) object name; (2) redshift; (3) total accretion disk
luminosity; for a blackbody emissivity (4) dust temperature; (5) total dust
luminosity and (6) dust radius; (7) reference for the cross-dispersed near-IR
spectral data used for the continuum fits; (8) near-IR dust reverberation lag
time in the rest-frame taken from reference in (9). References are E23: Esser
et al. (2023), G12: Garcia-Rissmann et al. (2012), G20: Gravity Collaboration
et al. (2020), G23: Gravity Collaboration et al. (2023), K09: Kishimoto et al.
(2009), K14: Koshida et al. (2014), L08: Landt et al. (2008), L11: Landt et
al. (2011b), L13: Landt et al. (2013), L19: Landt et al. (2019), L23: Landt et
al. (2023), Lyu19: Lyu et al. (2019), M19: Minezaki et al. (2019), P22: Prieto
et al. (2022), R06: Riffel et al. (2006) and R18: Ramolla et al. (2018).

⋆ near-IR interferometric dust radius

##  4 The luminosity-based dust radius

The calculation of the luminosity-based dust radius requires a measurement of
the dust temperature and an estimate of the UV/optical (accretion disk)
luminosity that heats the dust. Cross-dispersed near-IR spectra with their
relatively large wavelength range cover about half the hot dust SED in low-
redshift AGN and additionally a considerable part of the accretion disk
spectrum. The latter is expected to be the dominant contributor to the
continuum flux up to $\sim 1~{}\mu$m (Landt et al., 2011a, b). In order to
obtain the necessary measurements, we decomposed the spectral continuum into
these two components, accretion disk and dust emission, following the approach
described in Landt et al. (2019). As already noted in Landt et al. (2019), due
to the relatively small spectral aperture usually used for near-IR
spectroscopy, the contribution of host galaxy light to the total observed
continuum flux is expected to be negligible in most AGN. In short, we have
first approximated the rest-frame wavelength range of $\mathrel{\hbox{\hbox
to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}1~{}\mu$m with the
spectrum of a standard accretion disk, which we have subsequently subtracted
from the total spectrum. For the calculation of the accretion disk spectrum we
adopted the black hole mass listed in Bentz and Katz (2015) for a geometrical
scaling factor of $f=4.3$, where available, and else we have estimated it
based on the relationship between black hole mass and near-IR virial product
presented in Landt et al. (2013). The scaling of the accretion disk spectrum
to the near-IR spectrum then readily gives the accretion rate. Furthermore, we
assumed that the disk is relatively large and extends out to $r_{\rm
out}=10^{4}r_{\rm g}$. We then fitted the resultant hot dust spectrum at
wavelengths $>1~{}\mu$m with a blackbody, representing emission by large dust
grains. Table 1 lists the physical parameters derived from the near-IR
spectral decomposition. As in Landt et al. (2019), we then calculated
luminosity-based dust radii, $R_{\rm d,lum}$, from the best-fit dust
temperatures assuming radiative equilibrium between the luminosity of the
irradiating source and the dust:

$\frac{L_{\rm uv}}{4\pi R_{\rm d,lum}^{2}}=4\sigma T^{4}\langle Q^{\rm
em}\rangle,$ (1)

where $\sigma$ is the Stefan-Boltzmann constant and $\langle Q^{\rm
em}\rangle$ is the Planck-averaged emission efficiency. We have approximated
$L_{\rm uv}$ with the accretion disk luminosity, as given in Table 1, and have
used $\langle Q^{\rm em}\rangle=1$, which is appropriate for the case of a
blackbody. The blackbody case is reached for grain sizes of
$a\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}0.4~{}\mu$m and $\mathrel{\hbox{\hbox
to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}2~{}\mu$m for carbon
and silicate dust, respectively (see Fig. 8 of Landt et al., 2019). As Landt
et al. (2019, 2023) have shown, radii for dust composed of small grains
($a\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}0.1~{}\mu$m) are a factor of $\sim 6$ and
$\sim 10$ larger if the composition is mainly carbonaceous or silicates,
respectively. The error on the luminosity-based dust radius is determined by
the error on the temperature and the error on the total accretion disk
luminosity. Due to the relatively large wavelength coverage of the cross-
dispersed near-IR spectra, which are also of relatively high signal-to-noise,
the error on the temperature is small ($\sim 10-30$ K; Landt et al., 2019,
2023). The accretion disk luminosity is mostly determined by the accretion
rate, which is set by the flux level of the near-IR spectrum. As discussed in
Landt et al. (2019, 2023), since the telluric standard star is observed close
in time with the science target, photometric correction factors are on average
$\sim 10-15\%$. Therefore, errors on the luminosity-based dust radius are
assumed to be $\sim 10\%$.

##  5 Results and discussion

Based on two well-studied sources, namely, NGC 5548 and Mrk 876, Landt et al.
(2023) have recently suggested that there are considerable similarities
between the hot dust in AGN and that present in protoplanetary disks around
young stars. In particular, these similarities are: (i) the prevalance of
large dust grain sizes, which in protoplanetary disks might have formed in the
midplane of the disk by differential settling; (ii) the presence of an ’inner
hole’ beyond the dust-free zone expected due to dust sublimation, which in
protoplanetary disks is thought to be a cavity filled with gaseous disk
material that can change its optical thickness and thus influence the location
of the puffed-up so-called ’wall’ or ’inner rim’; and (iii) a general
temperature-radius relationship consistent with that for a dusty, flared and
passively illuminated protoplanetary disk. Here we will test and expand on
this proposition by using a sample of $\sim 40$ AGN.

###  5.1 The prevalence of large grains in the hot dust of AGN

Figure 1:  The logarithmic near-IR dust reverberation lag time versus the
logarithmic luminosity-based dust radius for large dust grain sizes. The solid
black line indicates the line of equality, which corresponds to the mean ratio
between the two radii, whereas the dashed black lines mark the $1~{}\sigma$
region around it, corresponding to a factor of 2. Luminosity-based dust radii
corresponding to small-grain ($a\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}0.1~{}\mu$m) carbon and silicate dust are
expected to be larger than the values for large dust grains by a factor of
$\sim 6$ (solid green line) and $\sim 10$ (solid blue line), respectively.

As discussed in Landt et al. (2023), the assumption of dust, which is
optically thick to the UV/optical radiation heating it and optically thin to
its own IR radiation, allows us to constrain the astrochemistry of the dust
through the dust emissivity parameter $Q^{\rm em}$ (see eq. 1), if the dust
radius can be measured by an independent method. Fortunately, several such
methods exist, including, e.g., the dust response time measured through
reverberation mapping and a geometric dust radius measurement based on near-IR
interferometry. Here we have compiled dust radius measurements from these two
independent methods for a sample of AGN with available near-IR spectra that
were suitable to derive luminosity-based dust radii. Fig. 1 plots the
comparison between the luminosity-based dust radius and that obtained by an
independent method, which was mostly near-IR photometric reverberation
mapping. We note that the two dust radii are generally not contemporaneous and
so could be affected by the variability effects of the irradiating luminosity,
although Landt et al. (2019) argued that the hot dust radius is not set by
sublimation and is probably luminosity-invariant.

Fig. 1 shows that the large majority of the data points cluster around the
line of equality for a blackbody emissivity, which corresponds to the largest
dust grains. The mean ratio is $R_{\rm d,lum}/R_{\rm rev}=1$, with a
dispersion around the mean of a factor of $\sim 2$. We note that for most
sources the luminosity-based dust radius is larger than the dust radius
measured by the independent method. Assuming instead an emissivity
corresponding to small dust grains of predominantly carbonaceous or silicate
composition would move the line of equality to luminosity-based dust radii
larger by a factor of $\sim 6$ (green solid line) and $\sim 10$ (blue solid
line), respectively. Although in a handful of sources this possibility cannot
be excluded, in general small grains do not seem to dominate the hot dust
composition.

###  5.2 A flared, passively illuminated (hot) dusty disk with an ’inner hole’
in AGN

Figure 2:  The logarithmic total accretion disk luminosity (left panel) and
the dust temperature (right panel) versus the logarithm of the ratio between
the luminosity-based dust radius and the dust radius measured by reverberation
mapping (or near-IR interferometry).

It is in general of high interest to understand the origin of the scatter in
Fig. 1 and if it is caused by AGN physics rather than being mainly due to
variability and/or measurement uncertainties. Such an investigation is
particularly timely, since considerable research is under way to help
establish the hot dust radius as a suitable standard candle for cosmological
studies using AGN (e.g. Hönig et al., 2017). A first indication that a
considerable inconsistency can be present between the luminosity-based hot
dust radius and that obtained by other means was found for the high-luminosity
AGN Mrk 876 by (Landt et al., 2023). The difference between the two values of
a factor of $\sim 2$ could be well explained if the dust was assembled in a
flared, disk-like geometry, which is naturally expected to be carved out by
the anisotropy of the accretion disk illumination (Kawaguchi and Mori, 2010,
2011). Then, the dust is expected to be illuminated by the UV/optical
accretion disk luminosity reduced by a factor $\cos\theta$, with $\theta$ the
angle between the accretion disk rotation axis and the location of the dust.
Since such a pronounced difference was not found for the low-luminosity AGN
NGC 5548 by the study of Landt et al. (2019), there is a strong possibility
that the anisotropy effect depends on AGN luminosity. An indication that the
accretion disk illumination anisotropy considered by Kawaguchi and Mori (2010,
2011) affects low- and high-luminosity AGN differently was also found by
Minezaki et al. (2019). Their sample showed a best-fit slope for the
logarithmic reverberation dust radius versus optical luminosity relationship
of $\sim 0.4$, i.e. shallower than the slope of 0.5 predicted by radiative
equilibrium considerations (see eq. 1).

Fig. 2 (left panel) investigates this conjecture, where we plot the total
accretion disk luminosity estimated from the fits to the near-IR spectroscopy
versus the ratio between the luminosity-based dust radius and the dust radius
measured by reverberation mapping (or near-IR interferometry). There is
clearly a trend for the dust radius ratio to increase with luminosity and so
for the disk structure to be more pronounced in high-luminosity sources. The
significance of a linear correlation is $P=95.5\%$. Fig. 2 (right panel) plots
the dependence of the dust radius ratio on the dust temperature. The dust
temperatures reach values well above the sublimation temperature for silicates
(of $\sim 1400$ K) and, therefore, the hot dust in AGN is most likely
carbonaceous, as already argued previously (e.g. Mor et al., 2009; Landt et
al., 2011a, 2019). A trend is present also in Fig. 2 (right panel) that
indicates that the ’inner hole’ of the disk is more enlarged in high-
luminosity sources, if we assume a single chemical species for the hot dust
and so a unique sublimation temperature (e.g. $\sim 1900$ K, corresponding to
carbonaceous dust). The significance of a linear correlation is $P=93.5\%$.
However, we note that the scatter is considerable in both relationships
displayed in Fig. 2.

###  5.3 The outer dusty edge of accretion disks in AGN

Figure 3:  The logarithmic hot dust covering factor versus the logarithm of
the ratio between the luminosity-based dust radius and the dust radius
measured by reverberation mapping (or near-IR interferometry). The best-fit
relationship is shown as the black solid line. See text for more details.

A considerably tighter relationship is present between the hot dust covering
factor, defined as the ratio between the total dust luminosity and the total
accretion disk luminosity, and the dust radius ratio (see Fig. 3). The
significance of a linear correlation is $P>99.999\%$ for a best-fit
relationship of $\log(L_{\rm d}/L_{\rm UV})=(-0.72\pm 0.05)-(0.84\pm
0.15)\cdot\log(R_{\rm d,lum}/R_{\rm d,rev})$. This relationship could be
interpreted as evidence that the ratio between the luminosity-based dust
radius and that measured by reverberation mapping (or near-IR interferometry)
is a suitable indicator of the dust geometry. Then, the larger this ratio, the
more pronounced the flared, dusty disk structure and so the lower the expected
dust covering factor. In other words, as the ratio increases, the dust
geometry moves from almost circular to a short disky component with a large
flare and eventually to a long disky component with a smaller flare. A sketch
of this possible change in dust geometry is exemplified in Fig. 4. Baskin and
Laor (2018) investigated the geometry of a dusty, inflated accretion disk
structure. Whereas they assumed largely varying grain sizes (spanning about
two orders of magnitude), which resulted in the sublimation process occuring
over a transition zone rather than at a single location, the scale height of
this geometry could be related to the change in dust geometry proposed here.
In their model, the disk height is set by the balance between the effects of
gravity and radiation pressure and depends only on the accretion rate and the
opacity of the material (see their eq. 22). Therefore, it predicts that more
luminous AGN have higher dust covering factors, which is contrary to the
observed relationship presented in Fig. 2.

Recently, Landt et al. (2023) presented the comparison of the mean and
variable (rms) spectrum in the high-luminosity AGN Mrk 876, which showed
clearly that at least two hot dust components are present in AGN. They found
that the total dust emission was dominated (at the $\sim 70\%$ level) by the
non-variable component, which could have its origin in the outer (flared) part
of the dusty disk or the outer dusty edge of the accretion disk or both. How
would such a second hot dust component manifest itself in the relationship
shown in Fig. 3? We would need to know the viewing angle, $\theta$, in order
to estimate the accretion disk luminosity as seen by the dust. Then, the true
dust covering factor should be calculated relative to this (reduced) $L_{\rm
UV}$. In other words, if the main origin of the second dust component is from
the flare, the more correct our already assumed accretion disk luminosity is.
However, if the main origin of the second dust component is instead in the
accretion disk, we should expect to have overestimated the accretion disk
luminosity as seen by the dust and, therefore, underestimated the dust
covering factor. If indeed the dust radius ratio is a suitable indicator of
the dust geometry as described above, as this value increases, the flare
becomes less important and the accretion disk dust more important, thus
apparently reducing the dust covering factor and tightening the relationship
displayed in Fig. 3. We note that although the dust in the accretion disk will
not necessarily be externally illuminated, if it is composed of $\mu$m-sized
grains, it will be well-coupled to the accretion disk gas and so affected by
its (transmitted) total luminosity in a similar way. Stalevski et al. (2016)
investigated in detail the effects of the anisotropy of the accretion disk
luminosity on the covering factor for the torus dust. They found that for type
1 AGN low and high covering factors will be underestimated and overestimated,
respectively, with the effect being more pronounced for the former than for
the latter (see their Fig. 10). Their results could partly explain the steep
slope we find for the relationship in Fig. 3 and also the fact that we find
dust covering factors exceed unity as well as extremely low values (of only a
few per cent).

Figure 4:  Sketch of a possible change in the dust geometry as the ratio
between the luminosity-based dust radius and the dust radius measured by
reverberation mapping (or near-IR interferometry) increases and thus the dust
covering factor decreases.

##  6 Conclusions

This study used a sample of $\sim 40$ AGN with available cross-dispersed near-
IR spectroscopy to perform a comparison between luminosity-based dust radii
and those obtained by independent methods, e.g. by reverberation mapping. The
main aim was to reveal the astrochemistry and geometry of the hot dust in AGN.
The main results can be summarized as follows.

(i) It could be shown that luminosity-based dust radii obtained with the
assumption of a blackbody emissivity were consistent within a factor of $\sim
2$ with dust radii obtained by independent methods. Therefore, large dust
grains (with sizes of a few $\mu$m) appear to be ubiquitous in the hot dust of
AGN. Assuming instead an emissivity corresponding to small dust grains implied
much larger luminosity-based dust radii (by factors of $\sim 6-10$).

(ii) We found a tight relationship between the dust covering factor and the
ratio between the luminosity-based dust radius and the dust radius measured by
reverberation mapping (or near-IR interferometry). This new relationship can
be understood if one assumes that the dust radius ratio is a suitable
indicator of the dust geometry and that hot dust is present in the accretion
disk in addition to that assembled in a flared, dusty disk, as proposed by
Landt et al. (2023).

(iii) The effects of the anisotropy of the accretion disk illumination might
be different for low- and high-luminosity AGN, as suggested by the flattening
of the slope at high luminosities in the dust radius-luminosity plane of
Minezaki et al. (2019). We find a trend that the dust radius ratio increases
with AGN luminosity, indicating a change in dust geometry whereby the disk
structure becomes more pronounced in high-luminosity AGN.

## Conflict of Interest Statement

The author declares that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

The first author has performed all original measurements and analysis of the
data, which were not already reported elsewhere. The first author has written
the entire manuscript.

## Funding

HL acknowledges a Daphne Jackson Fellowship sponsored by the Science and
Technology Facilities Council (STFC), UK, and support from STFC grants
ST/P000541/1, ST/T000244/1 and ST/X001075/1.

## Data Availability Statement

The near-IR spectra used for the first time in this work are available on
request from the first author.

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