11institutetext:  Max Planck Institute for Extraterrestrial Physics (MPE),
Giessenbachstr.1, 85748 Garching, Germany 22institutetext: LESIA, Observatoire
de Paris, Université PSL, CNRS, Sorbonne Université, Univ. Paris Diderot,
Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon, France
33institutetext: I. Institute of Physics, University of Cologne, Zülpicher
Straße 77, 50937 Cologne, Germany 44institutetext: Departments of Physics and
Astronomy, Le Conte Hall, University of California, Berkeley, CA 94720, USA
55institutetext: Department of Physics and Astronomy, University of
Southampton, Southampton, UK 66institutetext: Department of Physics, Kyoto
Sangyo University, Kita-ku, Japan 77institutetext: Université Côte d’Azur,
Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Nice, France
88institutetext: School of Physics and Astronomy, Tel Aviv University, Tel
Aviv 69978, Israel 99institutetext: Univ. Grenoble Alpes, CNRS, IPAG, 38000
Grenoble, France 1010institutetext: Center for Computational Astrophysics,
Flatiron Institute, 162 5th Ave., New York, NY 10010, USA 1111institutetext:
European Southern Observatory, Casilla 19001, Santiago 19, Chile
1212institutetext: European Southern Observatory, Karl-Schwarzschild-Str. 2,
85748 Garching, Germany 1313institutetext: Sterrewacht Leiden, Leiden
University, Postbus 9513, 2300 RA Leiden, The Netherlands 1414institutetext:
Max Planck Institute for Radio Astronomy, Auf dem Hügel 69, 53121 Bonn,
Germany 1515institutetext: Universidade de Lisboa - Faculdade de Ciências,
Campo Grande, 1749-016 Lisboa, Portugal 1616institutetext: Faculdade de
Engenharia, Universidade do Porto, rua Dr. Roberto Frias, 4200-465 Porto,
Portugal 1717institutetext: CENTRA - Centro de Astrofísica e Gravitação, IST,
Universidade de Lisboa, 1049-001 Lisboa, Portugal 1818institutetext: Max
Planck Institute for Astronomy, Königstuhl 17, 69117, Heidelberg, Germany
1919institutetext: Instituto de Astrofísica de Canarias (IAC), E-38200 La
Laguna, Tenerife, Spain 2020institutetext: Department of Astrophysical &
Planetary Sciences, JILA, University of Colorado, Duane Physics Bldg., 2000
Colorado Ave, Boulder, CO 80309, USA 2121institutetext: Research School of
Astronomy and Astrophysics, Australian National University, Canberra, ACT
2611, Australia 2222institutetext: Retired

# Towards measuring supermassive black hole masses with interferometric
observations of the dust continuum

GRAVITY Collaboration: A. Amorim GRAVITY is developed in a collaboration by
the Max Planck Institute for Extraterrestrial Physics, LESIA of Observatoire
de Paris/Université PSL/CNRS/Sorbonne Université/Université de Paris and IPAG
of Université Grenoble Alpes /CNRS, the Max Planck Institute for Astronomy,
the University of Cologne, the CENTRA - Centro de Astrofisica e Gravitação,
and the European Southern Observatory. Corresponding authors: J. Shangguan
<EMAIL_ADDRESS>and Y. Cao (ycao@mpe.mpg.de)15151717 G. Bourdarot 11 W.
Brandner 1818 Y. Cao 11 Y. Clénet 22 R. Davies 11 P. T. de Zeeuw 111313 J.
Dexter 202011 A. Drescher 11 A. Eckart 331414 F. Eisenhauer 11 M. Fabricius 11
N. M. Förster Schreiber 11 P. J. V. Garcia 111116161717 R. Genzel 1144 S.
Gillessen 11 D. Gratadour 222121 S. Hönig 55 M. Kishimoto 66 S. Lacour 221212
D. Lutz 11 F. Millour 77 H. Netzer 88 T. Ott 11 T. Paumard 22 K. Perraut 99 G.
Perrin 22 B. M. Peterson 2222 P. O. Petrucci 99 O. Pfuhl 1212 M. A. Prieto
1919 D. Rouan 22 D. J. D. Santos 11 J. Shangguan 11 T. Shimizu 11 A. Sternberg
881010 C. Straubmeier 33 E. Sturm 11 L. J. Tacconi 11 K. R. W. Tristram 1111
F. Widmann 11 J. Woillez 1212

This work focuses on active galactic nuclei (AGNs), and the relation between
the sizes of the hot dust continuum and the broad-line region (BLR). We find
that the continuum size measured using optical/near-infrared interferometry
(OI) is roughly twice that measured by reverberation mapping (RM). Both OI and
RM continuum sizes show a tight relation with the H$\beta$ BLR size with only
an intrinsic scatter of 0.25 dex. The masses of supermassive black holes (BHs)
can hence be simply derived from a dust size in combination with a broad line
width and virial factor. Since the primary uncertainty of these BH masses
comes from the virial factor, the accuracy of the continuum-based BH masses is
close to those based on the RM measurement of the broad emission line.
Moreover, the necessary continuum measurements can be obtained on a much
shorter timescale than those required monitoring for RM, and are also more
time efficient than those needed to resolve the BLR with OI. The primary goal
of this work is to demonstrate measuring the BH mass based on the dust
continuum size with our first calibration of the
$R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relation. The current limitation and
caveats are discussed in detail. Future GRAVITY observations are expected to
improve the continuum-based method and have the potential to measure BH masses
for a large sample of AGNs in the low-redshift Universe.

###### Key Words.:

galaxies: active – galaxies: nuclei – galaxies: Seyfert – quasars:
supermassive black holes

##  1 Introduction

Measuring the mass of supermassive black holes (BHs) is challenging as this
requires resolving stellar or gas dynamics inside the BH’s sphere of influence
(e.g. Thomas et al., 2004; Onken et al., 2014; Saglia et al., 2016; Hicks &
Malkan, 2008; Davis, 2014; Onishi et al., 2017; Boizelle et al., 2019). In
active galactic nuclei (AGNs) with broad recombination lines, the
reverberation mapping (RM) method has been developed to measure the size of
the broad-line region (BLR) and hence lead to the measurement of the BH mass
(Blandford & McKee, 1982; Peterson, 1993; Peterson et al., 2004). By
monitoring the variability of the UV/optical continuum and an emission line,
typically H$\beta$, the BLR size can be obtained from the RM method. Assuming
that the BLR is a virialized system, one can calculate the BH mass,

${\mbox{$M_{\mathrm{BH}}$}}=f\frac{\mbox{$R_{\mathrm{BLR}}$}(\Delta
V)^{2}}{G},$ (1)

where $R_{\mathrm{BLR}}$ is the BLR radius, $\Delta V$ is the velocity width
of the broad emission line, $f$ is the corresponding virial factor, and $G$ is
the gravitational constant. The FWHM or second moment
($\sigma_{\mathrm{line}}$) of the broad line is usually used as $\Delta V$.
The virial factor $f$ depends on the geometry, kinematics, and inclination of
the BLR clouds and is likely different from object to object. For example, the
virial factor is 0.75 assuming an isotropic velocity distribution of Keplerian
motion (Netzer, 1990). The mean virial factor, $\langle f\rangle$, can be
calibrated using nearby AGNs assuming that the AGN and quiescent galaxies
follow the same $M_{\mathrm{BH}}$–$\sigma_{*}$ relation (e.g. Onken et al.,
2004; Woo et al., 2010; Graham et al., 2011; Grier et al., 2013; Ho & Kim,
2014). This method based on RM measurements has been used successfully for
many years, despite potential biases and caveats in the virial factor
calibration (Shankar et al., 2019). For a particular source, the uncertainty
in $f$ may come from its unknown inclination angle and other effects such as
the radiation pressure (Collin et al., 2006; Mejía-Restrepo et al., 2018).

Moreover, a scaling relation between the BLR radius and AGN luminosity was
discovered from RM measurements (Kaspi et al., 2000; Dalla Bontà et al., 2020,
and references therein). This $R$–$L$ relation enables one to estimate the BH
mass only with the AGN luminosity and the FWHM of a broad emission line from
single-epoch spectra (Shen, 2013). Thanks to its simplicity, the single-epoch
method has been widely used with different broad lines in the UV and optical,
although the uncertainty is around 0.5 dex or above (e.g. Vestergaard &
Peterson, 2006). Other methods have been developed to estimate the BH mass and
tested against the RM measured BH mass. For example, the coronal line [Si
VI]1.963 $\mu$mcan be used to estimate the BH mass with an uncertainty of
about 0.5 dex (Prieto et al., 2022).

Recently, the BLRs of three AGNs have been spatially resolved by GRAVITY, a
second-generation Very Large Telescope Interferometer (VLTI) instrument
(Gravity Collaboration et al., 2018, 2020a, 2021b). GRAVITY has greatly
improved the sensitivity of earlier efforts and has been able to combine all
four of the 8-m Unit Telescope (UT) beams to yield six simultaneous baselines
(Gravity Collaboration et al., 2017). With a few hours of on-source exposure,
GRAVITY measures the differential phase signal of a broad emission line in the
near-infrared (NIR) $K$-band, which reflects the offsets of the photocenters
from the center of the continuum emission in each wavelength channel (Petrov
et al., 2001; Marconi et al., 2003). The BLR size is then inferred from the
differential phase data by fitting a dynamical BLR model such as the widely
used Pancoast et al. (2014a) model. The GRAVITY-measured BLR size and BH mass
are in good agreement with RM measurements (Gravity Collaboration et al.,
2021b, a).

GRAVITY can also resolve the size of the NIR continuum emission of the AGN,
which comes from the thermal radiation of hot dust that is reprocessing the
UV/optical continuum from the accretion disk (e.g. Rees et al., 1969;
Barvainis, 1987). Spatial sizes of the dust continuum emission can be measured
by both continuum RM (e.g. Clavel et al., 1989; Baribaud et al., 1992; Glass,
1992; Sitko et al., 1993; Minezaki et al., 2004; Suganuma et al., 2006;
Koshida et al., 2014; Pozo Nuñez et al., 2014, 2015; Mandal et al., 2018;
Minezaki et al., 2019; Sobrino Figaredo et al., 2020; Mandal et al., 2021a, b)
as well as optical/NIR interferometry (OI; Swain et al., 2003; Wittkowski et
al., 2004; Kishimoto et al., 2009, 2011; Gravity Collaboration et al., 2020b;
Leftley et al., 2021). Similar to the BLR size, the dust continuum size also
scales with the AGN luminosity $\propto L^{0.5}$ (Suganuma et al., 2006;
Kishimoto et al., 2011; Koshida et al., 2014; Minezaki et al., 2019; Gravity
Collaboration et al., 2020b). Such a relation is expected if the dust
temperature and the inner radius of the dust distribution are determined by
radiation equilibrium and dust sublimation, respectively (Barvainis, 1987;
Kishimoto et al., 2007). The dust continuum RM radius is a factor of four or
five larger than the BLR radius (Koshida et al., 2014) and is consistent with
BLR models that place hot dust on the outskirts of the BLR (e.g. Wang et al.,
2017; Baskin & Laor, 2018). Moreover, RM-measured dust continuum sizes are
systematically smaller than those measured from OI, likely because the RM size
is weighted by the time lag over the emitting region while the OI size is
weighted by the intensity of hot dust emission (Koshida et al., 2014;
Kishimoto et al., 2011; Gravity Collaboration et al., 2020b).

GRAVITY can observe the dust continuum independently of the full
spectroastrometric measurements and has demonstrated excellent efficiency
(e.g. $\lesssim 1$ hour per source; Gravity Collaboration et al. 2020b and in
preparation). Establishing a link between the BLR and dust continuum size will
enable BH mass estimations from these more accessible dust continuum
observations. In this work, we investigate the correlation between BLR and
dust continuum size in the context of estimating the BH mass. The four methods
discussed in this paper are:

  1. 1.

Reverberation mapping of the broad emission line, where a sequence of
measurements over months or years, yielding the time delay for variations in
the broad-line emission, leads to an estimate $R_{\mathrm{BLR}}$; and hence,
via Equation (1), the BH mass (Peterson, 2014). This method has enabled
empirical calibration of a sample-average virial factor $\langle f\rangle$.
More recently, the velocity resolved RM data can constrain a BLR dynamical
model and enable the estimation of $f$ for individual sources (e.g. Pancoast
et al., 2014a, b).

  2. 2.

The single-epoch method, where $R_{\mathrm{BLR}}$ is estimated from a single
measurement of the AGN luminosity via the relation between these quantities
(Shen, 2013; Dalla Bontà et al., 2020). This method relies on Equation (1) and
the virial factor calibrated via RM.

  3. 3.

Continuum size measurements, the method introduced in this paper, where
$R_{\mathrm{BLR}}$ is estimated from an interferometric measurement of
$R_{\mathrm{d}}$. Like the single-epoch method, this also relies on Equation
(1) and a pre-calibration of the virial factor.

  4. 4.

Spectrally resolved differential phase measurements of the broad-line
emission. Using the interferometric data as constraints on a dynamical BLR
model allows one to derive BH mass; and hence also infer a value of the virial
factor for individual sources that is independent of RM (Gravity Collaboration
et al., 2020a).

We show that the H$\beta$ BLR size scales tightly with the dust continuum
size, which allows us to estimate the BH mass from the dust continuum size
with an uncertainty similar to the RM BLR method. We discuss the prospects of
this method for BH mass estimations in the low-redshift Universe, especially
with the upgrade of GRAVITY in Section 5. This work adopts the following
parameters for a $\Lambda$CDM cosmology: $\Omega_{m}=0.308$,
$\Omega_{\Lambda}=0.692$, and $H_{0}=67.8$ km s-1 Mpc-1 (Planck Collaboration
et al., 2016).

##  2 Samples

Table 1: Physical properties of AGNs with dust continuum size measurements

Name | Redshift |  $R_{\mathrm{d}}$ (RM) | Ref. |  $R_{\mathrm{d}}$ (OI) | Ref. | $R_{\mathrm{BLR}}$ | FWHM | $\log\,\mbox{$\lambda L_{\lambda}$ (5100 \AA)}$ | Ref. | $\log\,{\mbox{$M_{\mathrm{BH}}$}}$ | $\log\,\mbox{$\lambda_{\mathrm{Edd}}$}$
---|---|---|---|---|---|---|---|---|---|---|---
|  | (ld) |  | (ld) |  | (ld) |  ($\mathrm{km\,s^{-1}}$) |  ($\mathrm{erg\,s^{-1}}$) |  |  ($M_{\odot}$) |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12)
Mrk 335 | 0.0258 |  167.5$\pm$ | 6.0 | 1 |  185$\pm$ | 49 | 2 |  14.0$\pm$ | 4.0 |  1707$\pm$ | 79 | 43.8 | 3 |  6.90$\pm$ | 0.13 | $-0.29$
UGC 545 | 0.0612 | … | … |  707$\pm$ | 77 | 4 |  37.2$\pm$ | 4.7 |  1131$\pm$ | 37 | 44.5 | 5 |  6.97$\pm$ | 0.06 | 0.39
Mrk 590a,b | 0.0264 |  33.5$\pm$ | 4.2 | 1 | … | … |  25.6$\pm$ | 5.9 |  2716$\pm$ | 202 | 43.5 | 3 |  7.57$\pm$ | 0.12 | $-1.21$
3C 120 | 0.0330 |  94.4$\pm$ | 5.5 | 6 |  379$\pm$ | 85 | 2 |  26.2$\pm$ | 7.7 |  2472$\pm$ | 729 | 44.0 | 3 |  7.49$\pm$ | 0.29 | $-0.64$
H0507+164 | 0.0179 |  35$\pm$ | 11 | 7 | … | … |  3.0$\pm$ | 1.2 |  4062$\pm$ | 247 | 42.6 | 8 |  6.99$\pm$ | 0.18 | $-1.57$
Akn 120 | 0.0327 |  138$\pm$ | 18 | 1 |  387$\pm$ | 77 | 4 |  39.5$\pm$ | 8.2 |  6077$\pm$ | 147 | 43.9 | 3 |  8.45$\pm$ | 0.09 | $-1.73$
MCG+08-11-011 | 0.0205 |  72.7$\pm$ | 1.6 | 1 | … | … |  15.7$\pm$ | 0.5 |  4139$\pm$ | 207 | 43.3 | 3 |  7.72$\pm$ | 0.05 | $-1.54$
Mrk 6a | 0.0195 | … | … |  214$\pm$ | 60 | 9 |  18.5$\pm$ | 2.5 |  5457$\pm$ | 16 | 43.6 | 10 |  8.03$\pm$ | 0.06 | $-1.55$
Mrk 79 | 0.0222 |  67.7$\pm$ | 4.8 | 1 | … | … |  15.6$\pm$ | 5.0 |  4793$\pm$ | 145 | 43.7 | 3 |  7.84$\pm$ | 0.14 | $-1.31$
PG 0844+349 | 0.0640 |  99$\pm$ | 11 | 1 | … | … |  32.3$\pm$ | 13.6 |  2694$\pm$ | 58 | 44.2 | 3 |  7.66$\pm$ | 0.18 | $-0.59$
Mrk 110 | 0.0353 |  116.6$\pm$ | 6.3 | 1 | … | … |  25.6$\pm$ | 8.1 |  1634$\pm$ | 83 | 43.7 | 3 |  7.13$\pm$ | 0.14 | $-0.61$
PG 0953+414 | 0.2341 |  566$\pm$ | 44 | 1 | … | … |  150.1$\pm$ | 22.1 |  3071$\pm$ | 27 | 45.2 | 3 |  8.44$\pm$ | 0.06 | $-0.40$
NGC 3227 | 0.0038 |  14.37$\pm$ | 0.70 | 1 |  45.0$\pm$ | 7.2 | 4 |  3.8$\pm$ | 0.8 |  4112$\pm$ | 206 | 42.2 | 3 |  7.10$\pm$ | 0.10 | $-2.00$
NGC 3516a | 0.0088 |  72.7$\pm$ | 4.6 | 1 | … | … |  11.7$\pm$ | 1.3 |  5384$\pm$ | 269 | 42.8 | 3 |  7.82$\pm$ | 0.06 | $-2.18$
NGC 3783 | 0.0097 |  76.3$\pm$ | 14.1 | 11 |  131$\pm$ | 20 | 2 |  9.6$\pm$ | 0.7 |  4486$\pm$ | 35 | 43.0 | 12 |  7.58$\pm$ | 0.03 | $-1.70$
NGC 4051 | 0.0023 |  16.30$\pm$ | 0.57 | 1 |  38.1$\pm$ | 6.0 | 13 |  2.1$\pm$ | 0.8 |  1076$\pm$ | 277 | 41.9 | 3 |  5.68$\pm$ | 0.28 | $-0.92$
NGC 4151a | 0.0033 |  46.11$\pm$ | 0.44 | 1 |  44.1$\pm$ |  8.3c | 9 |  6.6$\pm$ | 1.0 |  6371$\pm$ | 150 | 42.1 | 3 |  7.72$\pm$ | 0.07 | $-2.77$
3C 273 | 0.1583 |  409$\pm$ | 41 | 14 |  675$\pm$ | 126 | 2 |  146.8$\pm$ | 10.2 |  3314$\pm$ | 59 | 45.9 | 3 |  8.50$\pm$ | 0.03 | 0.28
NGC 4593 | 0.0083 |  41.82$\pm$ | 0.90 | 1 |  54.8$\pm$ | 8.8 | 4 |  4.0$\pm$ | 0.8 |  5142$\pm$ | 572 | 42.6 | 3 |  7.31$\pm$ | 0.13 | $-1.84$
MCG-6-30-15 | 0.0078 |  19.6$\pm$ | 4.9 | 15 | … | … |  5.7$\pm$ | 1.8 |  1947$\pm$ | 58 | 41.6 | 3 |  6.63$\pm$ | 0.14 | $-2.13$
NGC 5548 | 0.0172 |  61.21$\pm$ | 0.30 | 1 | … | … |  13.9$\pm$ | 8.7 |  7256$\pm$ | 2203 | 43.3 | 3 |  8.15$\pm$ | 0.38 | $-2.00$
Mrk 817 | 0.0313 |  92.6$\pm$ | 8.9 | 1 | … | … |  19.9$\pm$ | 8.3 |  5348$\pm$ | 536 | 43.7 | 3 |  8.05$\pm$ | 0.20 | $-1.45$
PG 1613+658 | 0.1211 |  334$\pm$ | 40 | 1 | … | … |  40.1$\pm$ | 15.1 |  9074$\pm$ | 103 | 44.8 | 3 |  8.81$\pm$ | 0.16 | $-1.19$
Z 229-15 | 0.0279 |  20.4$\pm$ | 5.8 | 16 | … | … |  3.9$\pm$ | 0.8 |  3350$\pm$ | 72 | 42.9 | 17 |  6.93$\pm$ | 0.09 | $-1.18$
Mrk 509 | 0.0344 |  121.3$\pm$ | 1.6 | 1 |  297$\pm$ | 31 | 2 |  79.6$\pm$ | 5.8 |  3015$\pm$ | 2 | 44.2 | 3 |  8.15$\pm$ | 0.03 | $-1.11$
NGC 7469 | 0.0163 |  85.29$\pm$ | 0.43 | 1 | … | … |  10.8$\pm$ | 2.4 |  4369$\pm$ | 6 | 43.5 | 3 |  7.60$\pm$ | 0.10 | $-1.24$
NGC 1365a | 0.0055 | … | … |  38.1$\pm$ | 4.8 | 2 | … |  1586$\pm$ | 465 | 41.9 |  2f,18 | … | …
IRAS 03450+0055 | 0.0315 |  157.4$\pm$ | 5.9 | 1 | … | … | … |  3098$\pm$ | 55 | 43.9 | 1,19 | … | …
IRAS 09149-6206d | 0.0573 | … | … |  482$\pm$ | 49 | 2 | … |  4281$\pm$ | 121 | 45.0 | 20,21 |  8.06$\pm$ | 0.25 | $-0.26$
Mrk 1239 | 0.0199 | … | … |  189$\pm$ | 30 | 4 | … |  830$\pm$ | 10 | 44.5 |  22f | … | …
WPVS 48 | 0.0370 |  70.8$\pm$ | 4.6 | 23 | … | … | … |  1890$\pm$ | 60 | 43.6 | 24 | … | …
Mrk 744 | 0.0091 |  19.9$\pm$ | 2.2 | 1 | … | … | … |  5616$\pm$ | 129 | 41.8 |  1,21g | … | …
HE 1029$-$1401 | 0.0858 | … | … |  880$\pm$ | 133 | 4 | … |  5684$\pm$ | 284 | 44.6 | 21 | … | …
GQ Com | 0.1650 |  210$\pm$ | 40 | 25 | … | … | … |  5036$\pm$ | 252 | 44.6 | 26 | … | …
Mrk 231 | 0.0422 | … | … |  393$\pm$ | 83 | 13 | … |  3130$\pm$ | 156 | 45.0 |  2f,27 | … | …
ESO 323-G77 | 0.0150 | … | … |  100.0$\pm$ | 4.8 | 28 | … |  2635$\pm$ | 132 | 43.1 | 21 | … | …
IRAS 13349+2438 | 0.1076 | … | … |  1096$\pm$ | 71 | 13 | … |  1796$\pm$ | 90 | 45.0 | 29 | … | …
IC 4329A | 0.0161 | … | … |  178$\pm$ | 10 | 4 | … |  6472$\pm$ | 324 | 43.2 | 21 | … | …
PGC 50427e | 0.0235 |  46.7$\pm$ | 2.2 | 30 | … | … | … |  3036$\pm$ | 74 | 43.1 | 30 |  7.34$\pm$ | 0.04 | $-1.42$
PDS 456 | 0.1840 | … | … |  1599$\pm$ | 213 | 2 | … |  3974$\pm$ | 764 | 46.3 | 31 | … | …
PGC 89171 | 0.0270 | … | … |  303$\pm$ | 36 | 4 | … |  2644$\pm$ | 132 | 43.9 | 21 | … | …
NGC 7603 | 0.0288 | … | … |  332$\pm$ | 66 | 4 | … |  6350$\pm$ | 318 | 44.4 | 21 | … | …

  * a

This target is discussed as a changing-look quasar in the literature.

  * b

Mrk 590 is a changing look AGN displaying strong variability over the time of
BLR and dust continuum observations (Denney et al., 2014). We do not include
it in our statistical analysis.

  * c

The continuum radius of NGC 4151 reported by Kishimoto et al. (2011) is
statistically consistent with the recent measurement from Kishimoto et al.
(2022). We prefer the early measurements because it is close to the RM
measurement.

  * d

The BLR of IRAS 09149$-$6206 was resolved by GRAVITY and the BH mass is
derived by Gravity Collaboration et al. (2020a). The $\lambda L_{\lambda}$
(5100 Å) and FWHM of H$\beta$ are from the BASS catalog (Koss et al., 2017).

  * e

The BH mass of PGC 50427 was measured by RM of the H$\alpha$ line (Pozo Nuñez
et al., 2015). We quote the $\lambda L_{\lambda}$ (5100 Å) from Probst &
Kollatschny (2020) observed close in time to H$\alpha$ RM. We measure the FWHM
of H$\beta$ line from the 6dF spectrum (Jones et al., 2009).

  * f

We converted the bolometric luminosity to $\lambda L_{\lambda}$ (5100 Å) using
Equation (A.2) of Gravity Collaboration et al. (2020b).

  * g

H$\beta$ FWHM is converted from H$\alpha$ FWHM assuming
$\mathrm{FWHM_{H\beta}}/\mathrm{FWHM_{H\alpha}}=1.17$ (Greene & Ho, 2005).

Notes. Col. (1): Target name. Col. (2): Redshift from NASA/IPAC Extragalactic
Database (NED). Col. (3): Dust continuum radius based on RM measurement. Col.
(4): Reference of $R_{\mathrm{d}}$ (RM). Col. (5): Dust continuum radius based
on OI measurement. Col. (6): Reference of $R_{\mathrm{d}}$ (OI). Col. (7): BLR
radius based on H$\beta$ time lag. Col. (8): H$\beta$ FWHM. Col. (9): AGN
optical luminosity at 5100 Å. Col. (10): References of $R_{\mathrm{BLR}}$,
FWHM, and $\lambda L_{\lambda}$ (5100 Å). Col. (11): BH mass derived from
$R_{\mathrm{BLR}}$ and FWHM from Columns (7) and (8) assuming the virial
factor $f=1$. Col. (12): Eddington ratio derived from $\lambda L_{\lambda}$
(5100 Å) and BH mass from Columns (9) and (11) with the bolometric correction
factor 9 (Peterson et al., 2004).
References: (1) Minezaki et al. (2019), (2) Gravity Collaboration et al.
(2020b), (3) Du & Wang (2019), (4) GRAVITY Collaboration (in preparation), (5)
Huang et al. (2019), (6) Ramolla et al. (2018), (7) Mandal et al. (2018), (8)
Stalin et al. (2011), (9) Kishimoto et al. (2011), (10) Du et al. (2018), (11)
Lira et al. (2011), (12) Bentz et al. (2021), (13) Kishimoto et al. (2009),
(14) Sobrino Figaredo et al. (2020), (15) Lira et al. (2015), (16) Mandal et
al. (2021a), (17) Barth et al. (2011), (18) Onori et al. (2017), (19) Rashed
et al. (2015), (20) Gravity Collaboration et al. (2020a), (21) Koss et al.
(2017), (22) Pan et al. (2021), (23) Pozo Nuñez et al. (2014), (24) Probst &
Kollatschny (2020), (25) Sitko et al. (1993), (26) Shangguan et al. (2018),
(27) Zheng et al. (2002), (28) Leftley et al. (2021), (29) Dong et al. (2018),
(30) Pozo Nuñez et al. (2015), (31) Nardini et al. (2015).

###  2.1 Dust continuum measurements

We collect type 1 AGNs with dust continuum sizes measured by RM and/or OI in
Table 1. The dust continuum sizes based on $K$-band RM observations are mainly
measured by Koshida et al. (2014). Minezaki et al. (2019) summarized the
results of Koshida et al. (2014) in their Table 6, using the power index
$\alpha=0.1$ to remove the NIR emission from the accretion disk (i.e.
$f_{\nu}\propto\nu^{\alpha}$) to be consistent with their primary results. The
assumption of the power index may introduce a $\lesssim 10\%$ difference in
the time lag, which is typically not significant compared to the measurement
uncertainty (Koshida et al., 2014; Minezaki et al., 2019). We, therefore,
adopt the time lags from Minezaki et al. (2019) whenever available. The
$K$-band time lag probes the dust emission size at shorter wavelength for
higher redshift sources, so we prefer not to include AGNs much higher than
$z\approx 0.2$ (see also Section 3.3). Minezaki et al. (2019) also reported
$K$-band continuum RM measurements for a sample of quasars at $z\approx$
0.1–0.6, most of which do not have H$\beta$ RM measurements. We only include
three quasars from this sample, PG 0844+349, PG 0953+414, and PG 1613+658,
because they have H$\beta$ RM measurements and are valuable to study the
relation of the dust continuum and BLR sizes. $K$-band RM measurements of the
other targets, e.g. 3C 120 (Ramolla et al., 2018) and H 0507+164 (Mandal et
al., 2018), are collected from individual papers.

We also find 23 AGNs with dust continuum sizes measured by OI, which consists
of Keck (Kishimoto et al., 2009, 2011) and recent VLTI/GRAVITY observations
(Gravity Collaboration et al. 2020b and in preparation). Kishimoto et al.
measured the dust continuum size by fitting squared visibility amplitudes
($V^{2}$) with a thin-ring model. They corrected the influence of the
accretion disk assuming a point source contribution to the visibilities.
Gravity Collaboration et al. (2020b) measured the dust continuum size by
fitting a Gaussian model to the $V^{2}$. They converted the Gaussian FWHM to
the thin-ring radius and corrected the radius assuming a 20% contribution of
the coherent flux from the accretion disk. Recent GRAVITY observations
measured the dust continuum size of seven AGNs with $\lesssim 1$ hour
observation time (GRAVITY Collaboration in preparation). We follow the method
of Gravity Collaboration et al. (2020b) to measure their continuum size. To
estimate the uncertainty, we sum in quadrature the statistical uncertainty of
the size measurements of individual exposures and a 10% systematic uncertainty
(Gravity Collaboration et al. 2020b and in preparation).

###  2.2 BLR measurements

The BLR size can be probed by different broad emission lines. The H$\beta$
line has been the most extensively used in RM campaigns of low-$z$ AGNs (e.g.
Bentz & Katz, 2015). Meanwhile, GRAVITY spectroastrometric observations probe
the BLR with Pa$\alpha$ and Br$\gamma$ (Gravity Collaboration et al., 2018,
2020a, 2021b). Different broad lines of a BLR may show a different size due to
the photoionization conditions and optical depth (Korista & Goad, 2004). In
this work, we only study the relation of the dust continuum radius and
H$\beta$ BLR radius from the RM time lag for simplicity. Meanwhile, future
GRAVITY observations measuring the BLR and dust continuum simultaneously in
$K$-band will be powerful to investigate their relation (see Section 5). We
collect the H$\beta$ BLR radii of most targets, together with their optical
luminosities at 5100 Å and H$\beta$ line FWHMs, from Table 1 of Du & Wang
(2019).111Similar data are also collected in Dalla Bontà et al. (2020). Du &
Wang (2019) averaged the quantities if there is more than one measurement. We
only find a few additional AGNs from the other references (see Table 1). We
adjust the $\lambda L_{\lambda}$ (5100 Å) to our cosmology when necessary
unless the distances of some nearby AGNs are explicitly specified in the
references.

Figure 1: Size–luminosity relations of the dust continuum measured by (a) RM
(blue circles) and (b) OI (red diamonds) are systematically above that of the
BLR (black circles; same in both panels). The open-colored circles indicate
sources without BLR measurements. The dashed lines in both panels are based on
RM-measured $R$–$L$ relations of the dust continuum (blue; Gravity
Collaboration et al., 2020b) and the BLR (black; Bentz et al., 2013). The
typical uncertainties are shown on the lower right of each panel.

###  2.3 AGN variability

We emphasize that the BLR and dust continuum sizes are usually measured in
different epochs. While the BLR RM campaigns of many targets conducted from
2000–2010 coincide with dust continuum RM campaigns (Koshida et al., 2014;
Minezaki et al., 2019), the time difference between BLR and dust measurements
for some targets can be over a decade. For example, we adopt the BLR RM of NGC
3783 measured in 2020 (Bentz et al., 2021), while its dust continuum RM was
measured around 2009 (Lira et al., 2011). We find that the early BLR RM
measurements of NGC 3783 (Onken & Peterson, 2002) yielded a BLR radius and
$\lambda L_{\lambda}$ (5100 Å) quite close to the recent measurements. We,
therefore, prefer to adopt the new measurements for simplicity. The
asynchronous effect will contribute to the scatter of the size correlation
(see discussion in Section 3).

Targets with substantial variability may show a large scatter in the
$R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relation. We identify so-called ‘changing-
look’ AGNs in our sample from the literature and discuss them in the
following. In the end, we are convinced that only Mrk 590 is not suitable to
be included in our analysis. Mrk 590 transformed from a classical type 1 AGN
to type 1.9–2, as its continuum luminosity decreased by 100 over the past four
decades (Denney et al., 2014). The lack of evidence of intrinsic absorption
indicates that changes in continuum and emission lines are due to the decline
of the BH accretion rate instead of obscuration along the line of sight. The
BLR of Mrk 590 was monitored through the H$\beta$ line from 1990–1996
(Peterson et al., 1998) and recently, with H$\alpha$ in 2018 (Mandal et al.,
2021b). The BLR size of Mrk 590, around 25 ld, does not change significantly
at the bright and faint states. The dust continuum RM campaign was conducted
from 2003 to 2007 (Koshida et al., 2014) during the rapid decline of the AGN
luminosity in Mrk 590. Kokubo & Minezaki (2020) found that the dust continuum
size of Mrk 590 is quite small, only $\sim 33$ ld, reflecting the rapid
replenishment of the dust in the innermost region of the dusty interstellar
medium. We exclude Mrk 590 in our statistics to avoid the complicated physics
of this target. There are other changing-look AGNs in our sample, e.g. Mrk 6
and NGC 4151 (flagged in Table 1). They change either from type 1 to type 2 or
vice versa (see Marin et al. 2019; Senarath et al. 2021 and references
therein). However, we find the BLR and continuum were measured when they
stayed the same type, so we include them in our analysis. Moreover, some
changing-look AGNs flagged in our sample are likely caused by temporary
changes in the line-of-sight obscuration (e.g. Goodrich, 1989; Shapovalova et
al., 2019). This mechanism does not relate to any intrinsic change in the AGN
properties, so it will not affect the $R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$
relation that we are interested in for this work. Nevertheless, our results
stay the same if we exclude all the changing-look AGNs. Clavel et al. (1989)
published their $K$-band RM measurement of Fairall 9 while its UV continuum
flux was dropping by a factor of $\sim 30$. However, an H$\beta$ RM
measurement before it changed to the faint state does not exist. The extreme
variability, from type 1 to almost type 2 (Kollatschny & Fricke, 1985; Lub &
de Ruiter, 1992), prevents a simple choice of $\lambda L_{\lambda}$ (5100 Å)
and H$\beta$ FWHM for our analysis. Therefore, we chose to exclude Fairall 9
in this work.

##  3 Scaling relations of BLR and dust continuum size

Figure 1 displays our sample. The dust $R$–$L$ relations measured by both RM
and OI are systematically above the BLR $R$–$L$ relation. Moreover, the
continuum size measured by OI is above that of RM. The $R$–$L$ relations of
dust RM and OI measurements have been discussed in many previous works (e.g.
Suganuma et al., 2006; Kishimoto et al., 2009, 2011; Koshida et al., 2014;
Minezaki et al., 2019; Gravity Collaboration et al., 2020b). One common
explanation is that the RM-measured time lag is weighted by the amplitude of
flux variations, which is expected to originate most strongly from the inner
boundary of the hot dust; on the other hand, OI-measured sizes are mainly
flux-weighted and, therefore, are elevated by contributions of lower
temperature dust at larger radii (Kishimoto et al., 2009, 2011). Another
slightly different explanation assumes that the dust structure is a ‘bowl
shape’ (Kawaguchi & Mori, 2010): we mainly observe the foreground side at a
low inclination angle for type 1 AGNs, and dust at larger radii is also closer
to the observer, so the projected size increases more significantly than their
RM time lag towards larger radii (Pozo Nuñez et al., 2014; Sobrino Figaredo et
al., 2020). A more detailed discussion of BLR and dust structure models is
beyond the scope of this paper. Throughout the paper, we use OI and RM to
refer to the dust continuum measurements from these two different methods
unless otherwise clarified.

Previous works studying both RM and OI observations have hinted that the slope
of the dust continuum $R$–$L$ relation is shallower than $\propto L^{0.5}$.
The slopes in our current RM and OI samples are both about 0.4, consistent
with previous works. We leave a more detailed discussion of the $R$–$L$
relation in a separate paper (GRAVITY Collaboration et al. in preparation). In
summary, previous RM studies provide various explanations of the shallower
slope relating to the dust structure, dust response to the accretion disk
emission, and its putative effect on the observed optical luminosity (e.g.
Minezaki et al., 2019; Sobrino Figaredo et al., 2020). Based on the OI
measurements, Gravity Collaboration et al. (2020b) suspect the continuum
emission of the accretion disk may bias the OI size measurement for the most
luminous sources.

Likewise, recent RM BLR measurements (Du et al., 2015; Grier et al., 2017)
also find targets with $R_{\mathrm{BLR}}$ significantly lower than the
canonical BLR $R$–$L$ relation (e.g. Bentz et al., 2013). Some works find that
the deviation closely correlates with the accretion rate of the BH (Du et al.,
2015; Du & Wang, 2019; Martínez-Aldama et al., 2019; Dalla Bontà et al.,
2020), while the physical driver remains unclear in some other works (Grier et
al., 2017; Fonseca Alvarez et al., 2020). The physical explanation of the
deviation of BLR and dust continuum $R$–$L$ relations is beyond the scope of
the current paper. As discussed in the following sections, we find tight
relations between $R_{\mathrm{BLR}}$ and $R_{\mathrm{d}}$, which are close to
linear, reflecting a simple link between the two physical structures.

Figure 2: Tight relations between the radii of the BLR and the dust continuum.
The blue circles are AGNs measured by RM, and the red diamonds are the targets
measured by OI. The gray dashed lines connect the same sources with both RM
and OI measured $R_{\mathrm{d}}$. The best-fit
$R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relations of RM and OI datasets (formula
in the lower right corner of each panel) are shown as the blue and red lines,
respectively. The black circles enclose the known changing-look AGNs. We only
exclude Mrk 590, denoted as the empty blue circle, in our analysis. The best-
fit results with the slope free (a) or fixed to unity (b) are almost the same,
so we mainly adopt the results of (b) in the discussion. The linear scale is
indicated on the top and right axes.

###  3.1 Statistical analysis

We find strong correlations between BLR and dust continuum sizes, as shown in
Figure 2. We fit the relations for RM- and OI-measured $R_{\mathrm{d}}$
separately as they show systematic differences. We fit the data with a power-
law relation,

$\log\,(\mbox{$R_{\mathrm{BLR}}$}/\mathrm{ld})=\alpha+\beta\log\,(\mbox{$R_{\mathrm{d}}$}/R_{\mathrm{d,0}}),$
(2)

where $\alpha$ and $\beta$ are the intercept and the slope, and
$R_{\mathrm{d,0}}$ is a pivot point fixed close to the median of
$R_{\mathrm{d}}$ of the data to reduce the degeneracy of $\alpha$ and $\beta$.
We adopt the Bayesian Markov Chain Monte Carlo (MCMC) approach to fit the
data. The likelihood is,

$\ln\,\mathcal{L}=-\frac{1}{2}\Sigma_{i}\left(\ln(2\pi\sigma_{i}^{2})+\frac{(y_{i}-m_{i})^{2}}{\sigma_{i}^{2}}\right),$
(3)

where $y_{i}$ is $\log\,\mbox{$R_{\mathrm{BLR}}$}$ data, $m_{i}$ is the model
value based on $\log\,\mbox{$R_{\mathrm{d}}$}$ and Equation (2), and
$\sigma_{i}^{2}=(\beta\,\sigma_{x,i})^{2}+\sigma_{y,i}^{2}+\epsilon^{2}$
includes the measurement uncertainties of the dust continuum ($\sigma_{x,i}$)
and BLR ($\sigma_{y,i}$) radius as well as the intrinsic scatter ($\epsilon$).
We adopt uniform priors of the parameters that are wide enough and sample the
posterior with the widely used Python package of MCMC, emcee (Foreman-Mackey
et al., 2013). We used 32 walkers and 5000 steps, with the first 500 steps
discarded as burn-in steps. The fitting is well converged.

Table 2: Best-fit parameters of Equation (2)

Relation | $\alpha$ | $\beta$ | $\epsilon$ | $\log\,(R_{\mathrm{d,0}}/\mathrm{ld})$
---|---|---|---|---
RM (free) | $1.20^{+0.05}_{-0.05}$ | $1.12^{+0.13}_{-0.13}$ | $0.21^{+0.05}_{-0.04}$ | 1.9
OI (free) | $1.19^{+0.08}_{-0.08}$ | $1.10^{+0.18}_{-0.18}$ | $0.25^{+0.08}_{-0.06}$ | 2.2
RM (fixed) | $1.20^{+0.05}_{-0.05}$ | 1 | $0.21^{+0.04}_{-0.04}$ | 1.9
OI (fixed) | $1.19^{+0.08}_{-0.08}$ | 1 | $0.24^{+0.07}_{-0.05}$ | 2.2

Notes. We fit the $R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relation for AGNs with
$R_{\mathrm{d}}$ measured by RM and OI, respectively. $\alpha$, $\beta$, and
$\epsilon$ are the intercept, slope, and intrinsic scatter of a linear
relation. $R_{\mathrm{d,0}}$ is the pivot point fixed in the fitting. The
first two rows provides the best-fit parameters with $\beta$ free, while the
last two rows are results with $\beta$ fixed to unity. OI-measured
$R_{\mathrm{d}}$ is about 0.3 dex larger than that measured by the RM.

The best-fit relations are very close to linear (throughout the paper,
‘linear’ refers $\beta=1$) when we allow the slope to be free, as shown in
Figure 2a. Therefore, we also fit the data with the slope fixed to unity
(Figure 2b). The best-fit results are listed in Table 2. For simplicity, we
will take the $\beta$-fixed fitting results in the following discussion and
derive the BLR radius and BH mass with $R_{\mathrm{d}}$ in Section 4. Given
current uncertainties, whether we adopt the relations with $\beta$ free or
fixed does not affect these results.

The dust continuum size measured from RM is about 0.7 dex (five times) larger
than the BLR size, which is consistent with previous works (e.g. Koshida et
al., 2014; Kokubo & Minezaki, 2020). The dust continuum size measured by OI is
about 0.3 dex (two times) larger than that measured by RM, again consistent
with previous studies (Kishimoto et al., 2011; Koshida et al., 2014; Gravity
Collaboration et al., 2020b). We do not find the slope of the
$R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relation to significantly deviate from
unity. However, for the relation to be linear, any departure from $\propto
L^{0.5}$ for the $R$–$L$ relations of the dust continuum and the BLR must be
similar. Our sample shows a more significant deviation in
$R_{\mathrm{d}}$–$\lambda L_{\lambda}$ (5100 Å) than that in
$R_{\mathrm{BLR}}$–$\lambda L_{\lambda}$ (5100 Å). This difference may
contribute to the scatter of the $R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relation,
which we will discuss in Section 3.3.

###  3.2 Intrinsic scatter

The intrinsic scatter of the best-fit relations is about 0.25 dex for both RM
and OI relations. The physical difference between the BLR and dust structure
for individual targets may contribute to the scatter. Such a variation was
observed in the mid-IR, where a large scatter of the $R$–$L$ relation was
observed (Burtscher et al., 2013). However, besides the individual BLR and
dust structure difference, for the $R$–$L$ relation, the uncertainty of the
bolometric luminosity of an AGN is another primary source of the scatter.
Studying the BLR and dust structure with the
$R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relation allows us to avoid the uncertain
bolometric luminosity. However, AGN variability still likely introduces
considerable intrinsic scatter because the BLR and dust continuum sizes are
not measured in a state where they reflect the same AGN bolometric luminosity.

As described in Section 2.3, most of the dust continuum RM measurements are
made between 2001 and 2008, while the OI measurements are around 2009–2010
(Keck interferometry) and 2018–2022 (GRAVITY). The BLR measurements are
conducted from the 1980s until recent years. NGC 5548 is one of the best
targets to investigate the variability: Du & Wang (2019) collected 18 epochs
of its BLR RM measurements from 1989 to 2015, and the BLR radius (time lag)
varies from 4.2 ld to 26.5 ld with a standard deviation of 0.24 dex. Mrk 335
and Mrk 817 also have four epochs of BLR measurements across $>10$ years, and
their BLR radii changes are $\gtrsim 0.3$ dex. For the dust continuum, Koshida
et al. (2014) reported 6 epochs of dust continuum RM measurements for NGC 5548
from 2001 to 2007; the RMS of the dust continuum radius is about 0.1 dex.
Other AGNs, NGC 3227, NGC 4051, and NGC 4151, with $\geq 4$ epochs of dust RM
measurements, also show similar $\sim 0.1$ dex RMS variation. NGC 3783 was
observed from 1974–1990 (Glass, 1992) and later in 2006–2009 (Lira et al.,
2011); the measured $K$-band time lags are consistent within their
uncertainties. It is not surprising that the dust continuum size shows less
variability: the dust re-radiation effectively averages the variability of the
central engine on a longer time scale, while the RM technique measures the
averaged size over the monitoring period.

Since the BLR size variation of the AGNs with multiple measurements always
reach $\gtrsim 0.2$ dex, we conclude that the observed 0.25 dex intrinsic
scatter of the $R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relations can be explained
by the time variation of the BLR and dust continuum sizes, while the physical
difference between the BLR and dust structures of individual targets also
plays a role. Future simultaneous measurements of BLR and dust continuum sizes
have the potential to reveal a tighter $R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$
relation, while the related caveats are discussed in Section 4.3.

###  3.3 Higher-order correlations

Figure 3: Ratio of BLR and dust continuum sizes as a function of (a) the AGN
luminosity and (b) the Eddington ratio. The $p$-values of Spearman’s rank
correlation coefficient are reported in the lower right corner of each panel
for RM- and OI-measured samples. The notations are the same as in Figure 2.
The error bars are the quadrature sum of the uncertainties of
$R_{\mathrm{BLR}}$ and $R_{\mathrm{d}}$. Panel (c) displays the histograms of
the ratios. The dashed lines indicate the averaged
$\mbox{$R_{\mathrm{BLR}}$}/\mbox{$R_{\mathrm{d}}$}$ based on the fitting of
the $R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relations with $\beta$ fixed to unity
(see also Equation (4)). $\mbox{$R_{\mathrm{BLR}}$}/\mbox{$R_{\mathrm{d}}$}$
does not show a statistically significant correlation with the $\lambda
L_{\lambda}$ (5100 Å) nor the Eddington ratio. The linear scale of
$\mbox{$R_{\mathrm{BLR}}$}/\mbox{$R_{\mathrm{d}}$}$ is indicated on the right.

We investigate whether the scatter of the $R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$
relation correlates with the physical properties of the AGN. Since the
$R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relations are remarkably close to linear
(Section 3.1), we study the dependence of the ratio,
$\mbox{$R_{\mathrm{BLR}}$}/\mbox{$R_{\mathrm{d}}$}$, on the other physical
parameters of the AGN. We calculate Spearman’s rank correlation coefficients
to test the significance of the correlations. We perturb
$\mbox{$R_{\mathrm{BLR}}$}/\mbox{$R_{\mathrm{d}}$}$ 500 times with the
measurement uncertainties of $R_{\mathrm{BLR}}$ and $R_{\mathrm{d}}$ to
calculate the $p$-value distribution. The resulting $p$-values do not support
any significant correlations (i.e. $p>0.05$). We find that the conclusions do
not change when we study the deviation of the
$R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relations from the best-fit results with
$\beta$ free.

We first investigate $\mbox{$R_{\mathrm{BLR}}$}/\mbox{$R_{\mathrm{d}}$}$
against the AGN luminosity and the Eddington ratio (Figure 3). Although the
$p$-values do not support significant correlations, we notice that, at
$\mbox{$\lambda L_{\lambda}$ (5100 \AA)}>10^{45}$ $\mathrm{erg\,s^{-1}}$, the
three data points of PG 0953+414 and 3C 273 are all above the averaged values
of $\mbox{$R_{\mathrm{BLR}}$}/\mbox{$R_{\mathrm{d}}$}$. This trend may drive
the slope to values slightly larger than 1 when we fit the
$R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relation with the slope free.
Unfortunately, our current sample has too few luminous AGNs to confirm such a
trend. Previous BLR RM studies have found that the Eddington ratio may drive
the deviation of the $R$–$L$ relation such that highly accreting AGNs display
shorter time lags (e.g. Du et al., 2015). We do not find a dependence of
$\mbox{$R_{\mathrm{BLR}}$}/\mbox{$R_{\mathrm{d}}$}$ on the Eddington ratio.
Nevertheless, our targets do not show significant deviation from the Bentz et
al. (2013) relation either (Figure 1). It is worth noting that Mrk 509, which
displays the highest deviation of
$\mbox{$R_{\mathrm{BLR}}$}/\mbox{$R_{\mathrm{d}}$}$ has only $\mbox{$\lambda
L_{\lambda}$ (5100 \AA)}\approx 10^{44.2}$ $\mathrm{erg\,s^{-1}}$ and
intermediate Eddington ratio. More observations are needed to understand the
details of the $R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relation.

We further investigate the dependence of
$\mbox{$R_{\mathrm{BLR}}$}/\mbox{$R_{\mathrm{d}}$}$ on $R_{\mathrm{d}}$,
$R_{\mathrm{BLR}}$, and the FWHM of H$\beta$ (Figure 4). Again, Spearman’s
rank correlation coefficients do not support a significant correlation with
any of the three parameters. We only find a tentative trend that all three
targets, PG 0953+414, 3C 373, and Mrk 509, with
$\mbox{$R_{\mathrm{BLR}}$}\gtrsim 50$ ld in Figure 4b show
$\mbox{$R_{\mathrm{BLR}}$}/\mbox{$R_{\mathrm{d}}$}$ above the averaged values.
This trend is similar to what is discussed above for $\mbox{$\lambda
L_{\lambda}$ (5100 \AA)}>10^{45}$ $\mathrm{erg\,s^{-1}}$ AGNs, although we
caution that the intrinsic scatter of the $R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$
relation will naturally lead to the correlation between
$\mbox{$R_{\mathrm{BLR}}$}/\mbox{$R_{\mathrm{d}}$}$ and $R_{\mathrm{BLR}}$.

Moreover, in our sample, AGNs with the largest radii are at $z\gtrsim 0.1$.
Their $K$-band measurements probe continuum emission at a slightly shorter
wavelength ($\lesssim 2\,\mu$m) than the rest of the sample. One may simply
expect the continuum size to be smaller at shorter wavelengths because of a
larger contribution from higher temperature dust that is closer to the central
engine (e.g., Oknyansky et al., 2015). Indeed, a sharp decrease in the time
lag towards shorter wavelength was observed for NGC 4151 (Oknyanskij et al.,
1999) and GQ Com (Sitko et al., 1993). However, it is more common that the
time lags of dust emission only decrease moderately towards shorter wavelength
(Oknyansky et al., 2015). A bi-conical dust distribution can explain this
because dust with different temperatures is located on similar isodelay
surfaces. Therefore, we suggest that the redshift effect is not likely to
cause the observed deviation. However, it is hard to draw a firm conclusion
with the limited number of measurements. More observations of AGNs with high
luminosity and/or at high redshift would be essential to investigate this
problem further.

Figure 4: Ratio of $R_{\mathrm{BLR}}$ and $R_{\mathrm{d}}$ as a function of
(a) the continuum radius, (b) the BLR radius, and (c) the FWHM of H$\beta$
line. We do not find any statistically significant correlation. However, all
of the AGNs with $\mbox{$R_{\mathrm{BLR}}$}\gtrsim 50$ ld show
$\mbox{$R_{\mathrm{BLR}}$}/\mbox{$R_{\mathrm{d}}$}$ above the averaged values.
We discuss this tentative trend in the text. The notations are the same as in
Figure 3. The linear scale of
$\mbox{$R_{\mathrm{BLR}}$}/\mbox{$R_{\mathrm{d}}$}$ is indicated on the right.

##  4 Estimating BH masses with dust continuum sizes

###  4.1 BLR radius based on dust continuum measurements

We can now use the measured dust continuum radius to estimate the BLR radius
for AGNs lacking BLR measurements. We adopt the best-fit parameters with
$\beta=1$ in Table 2 so that Equation (2) can be written as

$\displaystyle\log\,\mbox{$R_{\mathrm{BLR}}$}=\begin{cases}\log\,\mbox{$R_{\mathrm{d}}$}-0.70&\text{(RM)},\\\
\log\,\mbox{$R_{\mathrm{d}}$}-1.01&\text{(OI)}.\end{cases}$ (4)

We adopt 0.25 dex as the uncertainty of $R_{\mathrm{BLR}}$ based on the
intrinsic scatter of the $R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relation. We
expect the uncertainty of the $R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relations to
be reduced in the future with more observations of the BLR and continuum close
in time and of more luminous AGNs with large BLR sizes (however, see Section
4.3).

In Table 3, we report the $R_{\mathrm{BLR}}$ derived from Equation (4) for
AGNs in the lower part of Table 1. We plot these targets in Figure 5 with
their $R_{\mathrm{BLR}}$ against $\lambda L_{\lambda}$ (5100 Å). The
continuum-based $R_{\mathrm{BLR}}$ follows the $R$–$L$ relation of the direct
BLR measurements. Similar to Du & Wang (2019) and Grier et al. (2017)’s
results, we find AGNs more likely below the Bentz et al. (2013) relation,
especially for the luminous ($\mbox{$\lambda L_{\lambda}$ (5100
\AA)}>10^{44}\,{\mbox{$\mathrm{erg\,s^{-1}}$}}$) objects. Although we are
still limited by the small number of objects, the distribution of our targets
closely resembles the RM measured sample. Four targets, Mrk 1239, Mrk 231,
IRAS 09149$-$6206, and PDS 456, show the most significant deviation from the
Bentz et al. (2013) relation. As discussed in Section 4.2, all of them, except
for IRAS 09149$-$6206, are at or above or above their Eddington luminosity.
The Eddington ratio of IRAS 09149$-$6206 (about 0.4) is also high among the
typical AGNs (e.g. the rest of the sample). Thus the large deviations from the
$R$–$L$ relation could be related to their high accretion rates (Du & Wang,
2019).

Table 3: BH mass and Eddington ratio derived with dust continuum size

Name | $\log\,\mbox{$R_{\mathrm{BLR}}$}$ | $\log\,{\mbox{$M_{\mathrm{BH}}$}}$ | $\log\,\mbox{$\lambda_{\mathrm{Edd}}$}$
---|---|---|---
| (ld) |  ($M_{\odot}$) |
(1) | (2) | (3) | (4)
NGC 1365 |  0.57 $\pm$ | 0.26 | 6.26 | $-1.51$
IRAS 03450+0055 |  1.50 $\pm$ | 0.25 | 7.77 | $-1.02$
IRAS 09149$-$6206 |  1.67 $\pm$ | 0.25 | 8.23 | $-0.37$
Mrk 1239 |  1.27 $\pm$ | 0.26 | 6.40 | 0.96
WPVS 48 |  1.15 $\pm$ | 0.25 | 6.99 | $-0.54$
HE 1029$-$1401 |  1.93 $\pm$ | 0.26 | 8.73 | $-1.28$
Mrk 744 |  0.60 $\pm$ | 0.25 | 7.39 | $-2.73$
GQ Com |  1.62 $\pm$ | 0.26 | 8.32 | $-0.86$
Mrk 231 |  1.58 $\pm$ | 0.27 | 7.87 | $-0.01$
ESO 323-G77 |  0.99 $\pm$ | 0.25 | 7.12 | $-1.17$
IRAS 13349+2438 |  2.03 $\pm$ | 0.25 | 7.83 | $0.02$
IC 4329A |  1.24 $\pm$ | 0.25 | 8.17 | $-2.10$
PGC 50427 |  0.97 $\pm$ | 0.25 | 7.22 | $-1.27$
PDS 456 |  2.19 $\pm$ | 0.26 | 8.68 | 0.47
PGC 89171 |  1.47 $\pm$ | 0.26 | 7.61 | $-0.85$
NGC 7603 |  1.51 $\pm$ | 0.26 | 8.41 | $-1.15$

Notes. Col. (1): Target name. Col. (2): BLR radius derived from
$R_{\mathrm{d}}$ with Equation (4). However, we find the difference between
the results using the best-fit relations with $\beta$ fixed and free is much
smaller than the uncertainties. Col. (3): BH mass derived from
$R_{\mathrm{BLR}}$ and H$\beta$ FWHM assuming the virial factor $f=1$. Col.
(4): Eddington ratio derived from $\lambda L_{\lambda}$ (5100 Å) (Col. (9) of
Table 1) and BH mass from Col. (3).

Figure 5: BLR $R$–$L$ relation of AGNs measured by the continuum-based method.
The blue circles and red diamonds are dust radii measured by RM and OI,
respectively. The black stars are three AGNs whose BLR kinematics are resolved
by GRAVITY (Gravity Collaboration et al., 2018, 2020a, 2021b). The gray
circles and squares are BLR RM measured AGNs from Du & Wang (2019) and Grier
et al. (2017) respectively. The dashed line is the best-fit $R$–$L$ relation
from Bentz et al. (2013). The continuum-based method derives the BLR radii
following a similar trend of the $R$–$L$ relation based on RM and GRAVITY BLR
measurements.

###  4.2 BH mass

Since our continuum-based $R_{\mathrm{BLR}}$ is fully consistent with RM-
measured $R_{\mathrm{BLR}}$, one can simply adopt Equation (1) and the virial
factor previously calibrated for the RM method to derive the BH mass. We adopt
$f=1$ to be consistent with Du & Wang (2019). The FWHM of the H$\beta$ line
can be obtained from a single-epoch spectrum (Table 1). We further calculate
the Eddington ratios of these targets with the bolometric luminosities scaled
from $\lambda L_{\lambda}$ (5100 Å) by a bolometric correction factor of 9
(Peterson et al., 2004). Four AGNs, Mrk 1239, Mrk 231, IRAS 13349+2438, and
PDS 456, are close to or above the Eddington accretion. Their optical spectra
(see their references in Col. (10) of Table 1) commonly show features of
narrow widths of the broad H$\beta$ lines, weak or no [O III]
$\lambda\lambda$4959,5007 lines, and strong Fe II features. These features
strongly indicate that these targets have high accretion rates (Boroson &
Green 1992; Shen & Ho 2014 and references therein), consistent with our
Eddington ratios. The BH masses of IRAS 09149$-$6206 and PGC 50427 are
$10^{8.06}\,{\mbox{$M_{\odot}$}}$ and $10^{7.34}\,{\mbox{$M_{\odot}$}}$, based
on the spectroastrometry of the broad Br$\gamma$ line Gravity Collaboration et
al. (2020a) and the RM of H$\alpha$ line (Pozo Nuñez et al., 2015),
respectively. Our derived BH masses by the continuum-based method are very
close ($<0.2$ dex) to those from the direct BLR measurements.

The primary uncertainty of the RM method to measure the BH mass comes from the
virial factor. The calibration of $f$ typically shows $\sim 0.4$ dex intrinsic
scatter (Woo et al., 2010; Ho & Kim, 2014), which consists of the variation of
the BLR structure of individual targets and the intrinsic scatter of the
$M_{\mathrm{BH}}$–$\sigma_{*}$ relation (Gebhardt et al., 2000; Ferrarese &
Merritt, 2000). The sample selection and regression method may introduce a
factor of 2 systematic difference of the virial factor (Graham et al., 2011;
Park et al., 2012). The virial factor may differ by a factor of 2 depending on
the bulge type (classical bulge and pseudo bulge; Ho & Kim 2014), which might
be correlated to the systematics of the sample selection. For individual AGNs,
one way to measure their virial factors is via dynamically modeling the
velocity-resolved RM data (Pancoast et al., 2014a, b). Based on dynamical
modeling results, Williams et al. (2018) find 0.2–0.5 dex uncertainties on the
predictive distribution of the virial factor corresponding to different
definitions of the line width. One should be cautious that the small scatter
(e.g. 0.2 dex) may be due to the narrow range of parameter space spanned by
their small sample. Altogether, to derive the BH mass, the uncertainty of the
virial factor is likely $\gtrsim 0.3$ dex. Since the continuum-based method
shares the same uncertainty of the virial factor as the RM method, the
$R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relation provides a promising method to
measure the BH mass close, if not equivalent, to the accuracy of the RM
method.

The single-epoch method is generally believed to be much more uncertain than
RM due to the intrinsic scatter and the systematic bias of the $R$–$L$
relation. The systematic deviation of the measured $R_{\mathrm{BLR}}$ from the
canonical $R$–$L$ relation is discussed in Section 3 and Section 4.1. Dalla
Bontà et al. (2020) recently provided the state-of-the-art calibration of the
single-epoch method with the RM database (Bentz & Katz, 2015) and SDSS RM
(Grier et al., 2017) samples. Their calibration empirically includes the
secondary dependence of the Eddington ratio which results in an intrinsic
scatter of of 0.31 dex for the virial product
($\propto\mbox{$R_{\mathrm{BLR}}$}\Delta V^{2}$) when the line dispersion (in
contrast to the FWHM) is used. This scatter is equivalent to the intrinsic
scatter we find for the $R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relation
($\lesssim 0.25$ dex) which will be the main contributor to the continuum-
based virial product. We expect the continuum-based method scatter can be
reduced with future observations of the BLR and the dust continuum size close
in time, but we discuss the caveats in Section 4.3. We believe the direct dust
continuum size measurement, which we have shown is tightly linked to the BLR
size, is promising to provide high-accuracy BH masses in the future (more
discussion in Section 5).

###  4.3 Caveats

The primary goal of this work is to propose the idea of measuring the BH mass
based on the dust continuum size, particularly with time-efficient OI
observations. The current calibration is not ideal because the BLR and dust
continuum are not measured in the same AGN luminosity state. The line width
measurements in the lower part of Table 1 are collected from different epochs
too. This may introduce significant uncertainty on the BH mass (Table 3),
because the latter is $\propto\Delta V^{2}$. Due to this reason, we caution
that the BH mass and Eddington ratio in Table 3 are only for the purpose to
discuss the new method instead of rigorous measurements.

In practice, it may not be easy to measure the BLR and dust continuum size
when they reflect the same luminosity state in order to calibrate the
$R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relation. It is more feasible to measure
the BLR and the dust continuum sizes close in time (see also Section 5). In
this way, the different time lags of the BLR and the hot dust will contribute
to the intrinsic scatter of their size relation. The BLR size and the line
width may show quicker and stronger variations than the dust continuum because
the BLR has 5–10 times smaller size than the dust continuum. We expect a
stronger averaging effect on the dust continuum too. For an extreme example,
the dust continuum size may be correlated with a long-term average of the AGN
luminosity over the previous several years, as discussed by Kishimoto et al.
(2013) for NGC 4151. Future observations are important to quantify how much we
can improve the $R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relation from the first
calibration provided in this work.

The current calibration of the $R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relation is
limited by the sample size. As briefly discussed in Section 3, various
physical mechanisms may lead to the deviation of $R\propto L^{0.5}$ for both
the BLR and the hot dust. Such deviations may reflect variations of BLR and
hot dust structures, which may not necessarily follow each other. For example,
the BLR radius may have a secondary dependence on the BH accretion rate (Du et
al., 2015; Du & Wang, 2019), while the dust sublimation radius does not depend
on it (Barvainis, 1987; Kishimoto et al., 2007). This difference means that
the $R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relation may depend on some secondary
physical parameters, such as the accretion rate. Such a dependence is not
found in the current sample (Section 3.3), but we cannot rule out the
possibility that we are limited by the parameter space of the current sample.
Following Du et al. (2015, Equation 2), we calculate the dimensionless
accretion rate, $\mathcal{\dot{M}}$ and find that our targets span
$10^{-3}\lesssim\mathcal{\dot{M}}\lesssim 200$, with 23% of our targets
falling in their super-Eddington regime ($\mathcal{\dot{M}}>3$). We do not
find a correlation between $\mbox{$R_{\mathrm{BLR}}$}/\mbox{$R_{\mathrm{d}}$}$
and $\mathcal{\dot{M}}$, but this correlation is worth revisiting with future
larger samples including more high accretion rate AGNs.

##  5 Prospects

OI observations have great potential to measure the BLR and continuum of AGNs
in the future. Current GRAVITY observations are limited to the brightest
targets ($K<11$). With the ongoing upgrade to significantly improve its
sensitivity and sky coverage, GRAVITY+ will be capable of observing $K<13$
AGNs in on-axis mode or even fainter AGNs in off-axis mode with a phase
reference source at $\lesssim 30\arcsec$ (GRAVITY+ Collaboration et al.,
2022), enabling observations of thousands of AGNs from $z\lesssim 0.2$ out to
$z\gtrsim 2$.

Dust continuum sizes are a side product of GRAVITY(+) spectroastrometry
observations of the BLR if the AGN itself is bright enough to be the phase
reference. Half of the AGN emission is split into the low-resolution beam
combiner (or the ‘fringe tracker’), which is used as the phase reference of
the long-time exposure in the science channel to measure the BLR
spectroastrometric signal (Gravity Collaboration et al., 2017). The dust
continuum size can be measured from the visibility of the fringe tracker data.
Since the spectroastrometric measurement constrains the BLR geometry and
dynamics and the BH mass for individual AGNs, we can use the simultaneous
measurements of the BLR and continuum to calibrate the virial factor in
Equation (1) directly for $R_{\mathrm{d}}$ instead of $R_{\mathrm{BLR}}$. This
approach will further improve the accuracy of BH masses from the continuum-
based method.

GRAVITY can efficiently measure the dust continuum size, e.g. $\lesssim 1$
hour observation for one source with current sensitivity (GRAVITY
Collaboration et al. in preparation). We can put all AGN light into the fringe
tracker so that the continuum method can measure the BH mass of AGNs a factor
of two (or 0.75 mag) fainter than the spectroastrometry method. GRAVITY+
observations will be able to measure the dust continuum size and derive the
BLR size of a few hundreds of $z\lesssim 0.2$ AGNs, including many sources
with $\mbox{$\lambda L_{\lambda}$ (5100
\AA)}>10^{45}\,{\mbox{$\mathrm{erg\,s^{-1}}$}}$. They will be crucial to
understanding the dependence of BLR properties and the BH accretion on the AGN
properties.

##  6 Summary

In this paper, we collect 42 AGNs with dust continuum size measurements from
RM and/or OI observations. Among them, the BLR size based on H$\beta$ RM
measurements of 26 AGNs are available. We find close linear relations between
the BLR and dust continuum radius with an intrinsic scatter of only 0.25 dex.
The dust continuum radius measured by OI is about twice as large as that
measured by RM. Dust continuum radii measured by RM and OI are about five and
ten times the radius of the BLR, respectively. We provide simple scaling
relations to derive the BLR radius based on the dust continuum radius,
measured with RM and OI separately. For the remaining 16 AGNs, we calculate
their BLR radii, BH masses, and Eddington ratios using the
$R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relations. We find that these AGNs
consistently follow the BLR $R$–$L$ relation of previous RM and GRAVITY
measurements. All targets significantly below the Bentz et al. (2013) relation
show a high Eddington ratio.

The accuracy of the continuum-based BH mass is comparable to that of the
integrated broad emission line RM measurements since the primary uncertainty
comes from the virial factor. The primary goal of this paper is to propose a
new method to measure the BH mass based on the dust continuum size. We discuss
the caveats of the method in detail. More continuum observations close in time
with BLR measurements will be essential to study better the
$R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relation in the future. In particular, it
is important to test whether luminous AGNs with large BLRs show a different
$R_{\mathrm{BLR}}$–$R_{\mathrm{d}}$ relation compared to their low-luminosity
counterparts. With its improved sensitivity, GRAVITY+ will be powerful to
improve the continuum-based method and to efficiently measure the BH mass for
a large sample of AGNs in the low-redshift Universe using this method.

###### Acknowledgements.

We thank the anonymous referees for their careful reading and suggestions that
helped to improve this manuscript. This research has made use of the NASA/IPAC
Extragalactic Database (NED) which is operated by the California Institute of
Technology, under contract with the National Aeronautics and Space
Administration. This research has made use of the SIMBAD database, operated at
CDS, Strasbourg, France.

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