# Continuum reverberation mapping based accretion disk sizes for AGN selected
from the ZTF survey

Vivek Kumar Jha1,2, Ravi Joshi3,4, Hum Chand5, Xue-Bing Wu4,6, Luis C. Ho4,6,
Shantanu Rastogi2, and Qinchun Ma4,6

1Aryabhatta Research Institute of observational sciencES, Nainital, 263002;
India
2Deen Dayal Upadhyaya Gorakhpur University, Gorakhpur, 273009; India
3Indian Institute of Astrophysics, Koramangla, Bangalore, 560034; India
4Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing
100871, China
5Central University of Himachal Pradesh, Dharamshala, 176215; India
6Department of Astronomy, School of Physics, Peking University, Beijing
100871, China E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present the accretion disk size estimates for a sample of 19 active
galactic nuclei (AGN) using the optical $g$, $r$, and $i$ band light curves
obtained from the Zwicky Transient Facility (ZTF) survey. All the AGN have
reliable supermassive black hole (SMBH) mass estimates based on previous
reverberation mapping measurements. The multi-band light curves are cross-
correlated, and the reverberation lag is estimated using the Interpolated
Cross-Correlation Function (ICCF) method and the Bayesian method using the
javelin code. As expected from the disk reprocessing arguments, the $g-r$ band
lags are shorter than the $g-i$ band lags for this sample. The interband lags
for all, but 5 sources, are larger than the sizes predicted from the standard
Shakura Sunyaev (SS) analytical model. We fit the light curves directly using
a thin disk model implemented through the javelin code to get the accretion
disk sizes. The disk sizes obtained using this model are on an average 3.9
times larger than the prediction based on the SS disk model. We find a weak
correlation between the disk sizes and the known physical parameters, namely,
the luminosity and the SMBH mass. In the near future, a large sample of AGN
covering a range of luminosity and SMBH mass from large photometric surveys
would be helpful in a better understanding of the structure and physics of the
accretion disk.

###### keywords:

accretion, accretion discs – galaxies: active – galaxies: Seyfert – galaxies:
nuclei – quasars: supermassive black holes.

††pubyear: 2021††pagerange: Continuum reverberation mapping based accretion
disk sizes for AGN selected from the ZTF survey–Continuum reverberation
mapping based accretion disk sizes for AGN selected from the ZTF survey

##  1 Introduction

Accretion disks in AGN form as a result of inflowing matter onto the central
Supermassive black hole [SMBH] (Salpeter, 1964; Lynden-Bell, 1969). Various
accretion disk models have been proposed ranging from the standard Shakura
Sunyaev (SS) disk (Shakura & Sunyaev, 1973), the slim disk (Abramowicz et al.,
1988) to more complicated models like the advection dominated accretion disk
presented by Narayan & Yi (1995) and the relativistic accretion disk around a
Kerr black hole presented by Li et al. (2005). Based on these models, the
accretion disk has a temperature profile, with higher temperatures in the
inner regions and decreasing temperatures in the outer regions. The growing
observational shreds of evidence suggest that the X-ray emission in the
innermost regions of the AGN accretion disk are reprocessed as the UV/optical
continuum flux. Further, the rapid X-ray variability and relative strength of
the hard X-ray component point towards the presence of a corona in the
proximity of the SMBH, which irradiates the accretion disk. However, the exact
size and structure of this corona are not known precisely (e.g., see Meyer-
Hofmeister et al., 2017; Arcodia et al., 2019; Sun et al., 2020).

According to the standard SS disk model, the effective temperature $T$ at a
particular location in the accretion disk $R_{disk}$ is related to the disk
size and the SMBH mass as follows:

$T(R_{disk})=\left(\frac{3GM_{BH}\dot{M}}{8\pi\sigma(R_{disk})^{3}}\right)^{1/4},$
(1)

where $\sigma$ denotes the the Stefan-Boltzmann constant and $G$ and $\dot{M}$
represent the gravitational constant and the mass accretion rate respectively.
As is evident from this equation, the accretion disk size (R) scales inversely
with the temperature (T) by a power law of index 4/3.

The photon wavelength ($\lambda_{0}$) at a particular disk radius is related
to the temperature by the Wein’s law. The size of the accretion disk at the
wavelength $\lambda_{0}$ can be determined by two observable parameters, the
SMBH mass, and the mass accretion rate. The radiation from the accretion disk
is assumed to be composed of multiple black bodies, and the disk sizes at
various wavelengths can be calculated using the equation below:

$\centering
R_{\lambda}=R_{\lambda_{0}}\left[\left(\frac{\lambda}{\lambda_{0}}\right)^{\beta}\right]\@add@centering$
(2)

here, $R_{\lambda_{0}}$ is the disk size at a reference wavelength
$\lambda_{0}$ and $\beta$ is 4/3=1.33 for the standard SS disk. The
temperature profile of the disk generated using this formalism can be used to
test the application of the SS disk to the AGN population.

Direct observation of these regions is difficult as they project sub-micro arc
second angular resolutions, which cannot be resolved by any current or near-
future observational facilities. The disk sizes for a few AGN have been
estimated using microlensing studies in (see Pooley et al., 2007; Dai et al.,
2010; Morgan et al., 2010; Blackburne et al., 2011; Mosquera et al., 2013).
Another technique that remains promising is Reverberation Mapping (RM).
Utilizing the variable nature of AGN at most time scales, RM substitutes
spatial resolution with time resolution to measure the lag between the flux
emitted from different regions, thereby enabling us to reconstruct the echo
images that encode the information about the structure and kinematics of AGN
innermost region (see Bahcall et al., 1972; Blandford & McKee, 1982; Peterson,
2006; Bentz et al., 2009). Traditionally the RM technique was used to estimate
the size of the Broad Line Region (BLR) using the cross-correlation between
the accretion disk continuum and the H$\beta$ emission line, which has yielded
a significant correlation between the size of the BLR and the AGN luminosity
(Kaspi et al., 2000; Bentz et al., 2009; Du et al., 2016, 2018). Resolving the
accretion disk structure using RM through multi-band observations is possible
assuming the lamppost model, which implies a disk being irradiated by X-rays
originated in the innermost central regions, e.g., corona above the disk, and
the photons being reprocessed in the form of UV and optical wavelength
emissions (Cackett et al., 2007; Fausnaugh et al., 2017). As the emission from
the disk is expected to be of the black body type peaking at different
wavelengths, depending on the temperature of the disk, continuous,
simultaneous observations in multiple wavelengths ranges to cover hotter inner
regions and cooler outer regions are expected to yield the structure and
temperature profile of the accretion disk itself.

However, due to the complexity involved in executing such multi-band
monitoring campaigns, accretion disk structure has been mapped for only a
handful of AGN so far (see Rosa et al., 2015; Fausnaugh et al., 2016; Starkey
et al., 2016b; Pei et al., 2017; Edelson et al., 2019; Hernández Santisteban
et al., 2020). Alternatively, with the availability of large all-sky surveys,
it has been possible to constrain the accretion disk sizes based on interband
lags for a sample of AGN using observations in the optical regime only. In
Sergeev et al. (2005) the interband lags were detected for a sample of 14 AGN
in the optical wavelengths. In Jiang et al. (2017), inter-band lags based on
the PanStarrs light curves were presented while Mudd et al. (2018) presented
results for 15 quasars from the OzDES survey and Yu et al. (2020) performed
the analysis with around 23 quasars from the same survey but a different
field. In Homayouni et al. (2019), inter-band lags for around 33 quasars were
calculated from the sample of SDSS RM quasars. These studies used the optical
wavelength lags only, and the accretion disk profile from the X-ray (innermost
regions) to optical wavelengths (outer accretion disk) could not be
appropriately constrained. However, the presence of lags itself indicates
evidence of the reprocessing of photons in the accretion disk. These results
put an upper limit on the accretion disk sizes; nevertheless, the correlated
reverberation disk signals are detected. Interestingly, although the interband
lag scales with wavelength as a power law of index 4/3, the size of the disk
has been observed to be 3-4 times larger than expected from the standard
Shakura Sunyaev disk (Fausnaugh et al., 2018).

Figure 1:  The $g$, $r$, and $i$-band light curves for Mrk 493. There are
seasonal gaps between the ZTF light curves after MJD 58400 and MJD 58800,
based on which the light curves are divided in season 1 (MJD ranging from
58200 to 58400) and season 2 (MJD ranging from 58450 to MJD 58800),
respectively. The dashed lines denote the beginning of a season, followed with
a solid line denoting the end of a season. The $i$-band observations are not
available beyond MJD 58850.

Furthermore, Edelson et al. (2019) have shown that the optical wavelength flux
variations lead the UV band flux variations, which is challenging to the disk
reprocessing models. Also, the scaling of the physical parameters such as the
SMBH mass, luminosity, and accretion rate with the size of the accretion disk
has eluded us for the time being. Only a handful of AGN has both the accretion
disk size measurements and SMBH mass estimation, making multi-band monitoring
very important. In this work, using the publicly available Zwicky Transient
Facility (ZTF)111https://www.ztf.caltech.edu/, we aim to constrain the
accretion disk sizes and their correlations with the physical parameters such
as the SMBH mass and bolometric luminosities of AGNs with well-constrained
SMBH masses through prior RM studies.

This paper is structured as follows: In Section 2 we present the sample of AGN
studied in this work along with the details of the observations and data. In
Section 3 we present the analysis based on various lag estimation methods, and
the results of our comparison of disk sizes with the standard SS disk,
followed by the correlations with various physical parameters, are presented
in Section 4. The discussion is presented in Section 5 and we conclude our
results in Section 6.

Figure 2: The $g-$band (green), $r-$band (blue), and $i-$band (black) light
curves for Mrk 335 and RMID101 are shown in the top panel, for Mrk 817 and Mrk
290 are shown in the middle panel, and for PG0052+251 and Mrk 486 are shown in
the bottom panel. The first four light curves are from season 1, while for the
sources in the bottom panel, light curves are from season 2. The units for
magnitude are calibrated magnitudes as available from the ZTF database. The
light curves include the error bars, however the error bars are smaller than
the markers used here for some of the sources. The statistics for these light
curves are available in Table 2.

(a) Mrk 486 (b) Mrk 335 (c) 1RXS J1858+4850

Figure 3:  The $g-r$ lag distribution obtained using javelin (black) and ICCF
(red) for 3 AGN: Mrk 486 (left), Mrk 335 (middle) and 1RXSJ1858+4850 (right)
are shown in top panel. The $g-i$ lag distribution is shown in the bottom
panel with the javelin distribution denoted as blue color and the ICCF
distribution denoted as orange color. The CCF obtained using the ICCF method
is shown in grey color on the plots. The maximum correlation coefficient
obtained using the CCF method is denoted as rmax. Table 3 contains the results
obtained from the ICCF and the javelin methods.

##  2 The sample and data

To measure the accretion disk sizes and thereby test the accretion disk
models, we assembled a set of 122 reverberation mapped AGN selected from
various RM campaigns, including 61 AGNs from the AGN Black hole mass
database222http://www.astro.gsu.edu/AGNmass/(see Bentz & Katz, 2015), 44 AGNs
from Sloan Digital Sky Survey - Reverberation Mapping (SDSS RM) project (Grier
et al., 2019), and 17 AGNs from the Super Eddington Accreting Massive Black
Holes (SEAMBH) sample available in Du et al. (2018). Having an advantage of
accurate BH mass estimation, an essential input for the disk size measurement,
the sample spans a significant range of luminosities (42.57$\leq$ log(L5100Å)
$\leq$ 44.97) and redshifts (0.013 $\leq$ z $\leq$ 0.646) providing a
homogeneous set to study the AGN population.

For the multi-band continuum light curves of these AGNs, we explore the Zwicky
Transient Facility (ZTF) time-domain survey (Graham et al., 2019). The ZTF
uses the 48 inch Samuel Oschin Schmidt telescope with a field of view of 47
deg2 to map the sky in $g$, $r$, and $i$ optical bands with a typical exposure
time of 30 seconds, reaching a magnitude $\sim$20.5 in r$-$band (Bellm et al.,
2019). The processing of the data is done using the Infrared Processing and
Analysis Centre (IPAC) pipeline (Masci et al., 2019). In addition, the average
cadence of 3 days makes it a comparatively better survey for continuum RM
observations which has otherwise been not possible in some recent works using
all-sky surveys (see Jiang et al., 2017; Mudd et al., 2018; Yu et al., 2020).
We obtained the light curves using the ZTF-API
333https://irsa.ipac.caltech.edu/docs/program_interface/ztf_lightcurve_api.html
by providing the positions of individual objects in terms of their RA, DEC sky
positions. Observations for all but 5 AGNs were available for at least one
epoch in each band. We note that the $g$ and $r$ bands are well sampled in ZTF
while the $i-$ band has the least number of observations (see Figure 1).
Keeping this into account, we set a criterion of at least 15 observation
epochs in each band. This limit was kept in order to obtain enough
reverberation signal in order to get the interband lags. Since there are gaps
between the observations in all the three bands, we divide the light curves in
two seasons, the first season (season 1) running between MJD 58200 to 58400
broadly and the second season (season 2) running between 58450 and 58800.
Since the light curves are sampled well, and the lags we expect are of the
order of a few days, this division will not affect the lag estimates, and
dividing the light curves has an additional advantage of getting two disk size
measurements between these intervals.

This criterion resulted in the final sample of 57 reverberation mapped AGNs
for season 1, including 34 sources from the SDSS-RM project, 23 AGNs from the
AGN-Mass sample, and 2 AGNs from the SEAMBH sample, which are part of the AGN-
Mass sample. In season 2, our sample size was limited to 22 AGN primarily
because of the poor sampling in the $i$\- band. Light curves for some of the
sources are presented in Figure 2. We moved forward with the final set of 57
AGNs from season 1 and 22 AGN from season 2 for inter-band lag estimation. All
the 22 sources available in season 2 also satisfied the criterion in season 1
and thus, were part of the 57 sources selected from season 1. This provides us
with the opportunity to measure the disk sizes for 22 AGN in two different
time ranges.

##  3 Measurement of inter band lags

Before measuring the lags between the multi-band continuum light curves, we
first exclude any possible outliers in the light curves by applying a
3$\sigma$ clipping. We employ two most commonly used lag estimation methods,
namely the Interpolated Cross-Correlation Function (ICCF) method (Peterson et
al., 1998) and the javelin method (Zu et al., 2011) which employs a Bayesian
approach to estimate the lags. Given that the expected lags between the $g$
versus $r$ band are expected to be shorter than the $g$ versus $i$ band, we
used the $g$-band light curves as a reference for estimating the interband
lags.

###  3.1 javelin based lags

We use the publicly available code javelin 444https://github.com/nye17/JAVELIN
to estimate the interband reverberation lags. It models the variability of the
AGN as a Damped Random Walk (DRW) or, in other terms, an Ornstein Uhlenbeck
(OU) process for timescales longer than a few days (Zu et al., 2011, 2013).
This approach has been demonstrated by Kelly et al. (2009) for modeling 100
quasar light curves from the OGLE database and further by MacLeod et al.
(2010) for a sample of around 9000 quasars from the SDSS Stripe 82 region. The
covariance function for the DRW process takes the following form:

$S(\triangle t)=\sigma_{\tiny d}^{2}e^{(-|\triangle t/\tau_{d}|)}$ (3)

Where $\triangle t$ is the time interval between the two epochs and
$\sigma_{d}$ and $\tau_{d}$ are the amplitude of variability and the damping
time scale, respectively. Lag estimation in javelin is based on the assumption
that the responding light curve variability is a scaled, smoothed, and
displaced version of the driving light curve. Here, we took the $g$-band light
curve as the driving light curve, and the $r$ and $i$ band light curves were
taken as the responding light curves while estimating the lags. Initially,
using the driving light curves, we build the model to determine the parameters
$\sigma_{d}$ and $\tau_{d}$. Then we fit the other two band light curves using
this model to build the distribution for the time lag, the top hat smoothing
factor, and the flux scaling factor.

We derive the lag, tophat width, and scale factor distribution by shifting,
smoothing, and scaling the light curves. We use 20,000 Markov Chain Monte
Carlo (MCMC) chains to get the best fit parameter distribution. We set the lag
limits to [-50, 50] days for the initial run. Given that the observed inter-
band lags for various AGNs in previous studies were typically smaller than 10
days (see, Jiang et al., 2017; Mudd et al., 2018; Yu et al., 2020; Homayouni
et al., 2019), we rejected the sources for which the lags were obtained
outside the limits of [$-$10,10] days as these estimates could be unphysical.

Recall that in season 1, we had 57 sources with sufficient sampling (no. of
observations in $i$ band $\geq$ 15), and in season 2, we had 22 sources with
sufficient sampling. Out of 57 sources in season 1, we could constrain the lag
within the range of [$-$10, 10] days using javelin for 25 sources, while in
season 2, out of 22 sources, we could constrain the lags for 19 sources within
the same limits. Comparing the lag estimates between the two seasons, we find
that lags for 14 sources were available in both season 1 and season 2;hence we
have two measurements for this sample of sources while we have 11 individual
measurements from season 1 and 5 individual measurements from season 2.

###  3.2 ICCF based lags

The Interpolated Cross-Correlation Function (ICCF) method (Gaskell & Peterson,
1987; Peterson et al., 1998; Peterson, 2006) is quite frequently used to
estimate inter-band reverberation lags in reverberation mapping studies. We
used the $g$-band light curve as the model light curve, which was assumed to
drive the variations in the other two light curves. The observational gaps
were interpolated linearly to create a uniformly sampled dataset, and the
driving and responding light curves were sampled together, taking the effect
of uncertainties into account. These interpolated light curves were cross-
correlated to calculate the Cross-Correlation Function (CCF). This step was
repeated multiple times to build the Cross-correlation Centroid distribution
(CCCD) and Cross-correlation Peak distribution (CCPD). The CCCD has been used
frequently as indicative of the interband lag (e.g., see, Peterson, 2006;
Homayouni et al., 2019). The interpolation frequency was set to 0.5 days as
the ZTF light curves are quite well sampled in the $g$ and $r$ bands. The lag
uncertainties were measured by employing the flux randomization (FR) and the
random subset selection (RSS) sampling method over 5000 iterations with a
significance of $r_{max}\leq 0.5$. We used a python implementation of this
method, known as PyCCF developed by Sun et al. (2018).

The $g-r$ lags were shorter than the $g-i$ lags for most of the cases. For
instance, in the case of Mrk 335, we obtained a lag of $1.5^{+1.2}_{-1.3}$
days between the $g$ and $r$ bands while we obtained a lag of
$2.4^{+1.6}_{-2.0}$ days between the $g$ and $i$ bands. We note that ICCF lags
suffer significant uncertainties compared to other methods Fausnaugh et al.
(2017); Li et al. (2019); Yu et al. (2020); Homayouni et al. (2019) which is
likely because the ICCF method linearly interpolates between the data points,
rather than assuming an underlying model when there are observational gaps. We
estimated the lags using the ICCF method for only the sources for which
javelin lags were reasonable. The reason was that since javelin directly
models the light curves as compared to the linear interpolation applied in the
ICCF method, we expect the lag estimates to be more robust in the case of
javelin as has been observed using simulations (Li et al., 2019). However, the
lags estimated using the ICCF method serve as a check for the lags estimated
through javelin. Figure 3 shows the distribution of lags obtained between the
$g-r$ and $g-i$ bands for a few of the sources in our sample using both
methods. The estimated lags are listed in columns B, C, D, and E of Table 3.
Out of the sources with lag estimates in the reasonable range, we picked up
the sources where at least one method showed positive lags that increase with
wavelength in any of the seasons. This criterion resulted in the final sample
of 19 sources which are the sources we use for further analysis (information
available in Tables 1, 2, 3). The lags obtained were larger than that
predicted by the Standard SS disk model for 14 sources in this sample, while
for 5 sources, the lags were within the expectations of the SS disk models.

We note that for 4 AGN in our sample, previous interband continuum lags are
available in the literature, which can be directly compared with our lag
measurements. For Mrk 335, in Sergeev et al. (2005), the B-R lag is reported
to be $2.36^{+0.74}_{-0.85}$ days, and the B-I lag is reported to be
$2.33^{+0.30}_{-1.86}$ days. We obtained a $g-r$ band lag of
$1.5^{+1.2}_{-1.3}$ days and a $g-i$ band lag of $2.4^{+1.6}_{-2.0}$ days.
Even though the filter set in ZTF is a bit different than the filters used by
Sergeev et al. (2005), the interband lags between the two can be compared as
the two sets of filters cover similar wavelength ranges. Mrk 817 is being
monitored as part of the AGN STORM-II campaign (Kara et al., 2021) and based
on the initial results, the $g-r$ band lag is $1.50^{+0.86}_{-0.76}$ days and
the $g-i$ band lag is $2.31^{+0.70}_{-0.86}$ using the ICCF method, while we
obtain a lag of $2.5^{+1.5}_{-1.5}$ days between the $g-r$ band and a lag of
$3.6^{+1.1}_{-1.4}$ days between the $g-i$ band using the same methods. These
results are in agreement within the limits of the uncertainties. Furthermore,
the interband lags between the $g$ and $i$ bands were obtained for RMID 101
and RMID 300 in Homayouni et al. (2019) as $1.54^{+2.06}_{-3.08}$ and
$2.93^{+1.06}_{-4.24}$ days respectively using the ICCF method. We obtained a
lag of $1.5^{+5.5}_{-5}$ and $2.0^{+6.5}_{-4.5}$ days respectively for these
sources using the ICCF method.

Our $g-r$ band lags have relatively smaller uncertainties than $g-i$ band
lags, likely due to inadequate sampling in the $i$ band in the ZTF survey.
Another concern is that emission from the BLR emission lines may contribute to
the accretion disk continuum. However, the variability timescale of BLR is
known to be higher than the timescales for disk emission based on the
reverberation mapping measurements of the $H\beta$ emission line. Further,
recent works (e.g. Jiang et al., 2017) have concluded that the accretion disk
continuum contributes more than 90 % flux to the total flux, and thus emission
line contamination may not be a significant contribution to broadband
variations. In other recent works, the contribution for the BLR emission has
not been found out to be affecting the lag results (Homayouni et al., 2019).
Keeping this into account, we did not attempt to remove the emission line
contamination from the light curves before estimating the lags.

Table 1: Physical properties of the sample of reverberation mapped AGNs used
in this study.

Name | RA | Dec. | z | log(M${}_{BH}/M_{\odot}$) | log(L5100)
---|---|---|---|---|---
(A) | (B) | (C) | (D) | (E) | (F)
Mrk 335 | 00 06 19.44 | +20 12 07.2 | 0.026 | 7.23 | 43.76
PG 0026+129 | 00 29 13.44 | +13 16 01.2 | 0.142 | 8.48 | 44.97
PG 0052+251 | 00 54 52.08 | +25 25 37.2 | 0.154 | 8.46 | 44.81
NGC 4253 | 12 18 26.40 | +29 48 43.2 | 0.013 | 6.82 | 42.57
PG 1307+085 | 13 09 46.80 | +08 19 48.0 | 0.155 | 8.53 | 44.85
RMID733∗ | 14 07 59.04 | +53 47 56.4 | 0.455 | 8.20 | 43.40
RMID399∗ | 14 10 31.20 | +52 15 32.4 | 0.608 | 8.10 | 44.10
RMID101∗ | 14 12 14.16 | +53 25 44.4 | 0.458 | 7.90 | 43.40
PG 1411+442 | 14 13 48.24 | +44 00 10.8 | 0.089 | 8.54 | 44.56
RMID779∗ | 14 19 23.28 | +54 22 01.2 | 0.152 | 7.40 | 42.60
RMID300∗ | 14 19 41.04 | +53 36 46.8 | 0.646 | 8.20 | 44.00
Mrk 817 | 14 36 22.08 | +58 47 38.4 | 0.031 | 7.58 | 43.74
Mrk 290 | 15 35 52.08 | +57 54 07.2 | 0.029 | 7.27 | 43.17
Mrk 486 | 15 36 38.16 | +54 33 32.4 | 0.039 | 7.63 | 43.69
Mrk 493 | 15 59 09.36 | +35 01 44.4 | 0.031 | 6.19 | 43.11
PG 1613+658 | 16 13 57.12 | +65 43 08.4 | 0.129 | 8.33 | 44.77
PG 1617+175 | 16 20 11.28 | +17 24 25.2 | 0.112 | 8.66 | 44.39
1RXS J1858+4850 | 18 58 00.96 | +48 50 20.4 | 0.079 | 6.70 | 43.65
PG 2130+099 | 21 32 27.60 | +10 08 16.8 | 0.063 | 7.43 | 44.20

Note: Column (A) denotes the common name of the sources, Column (B) and (C)
present the right ascension (RA) and declination (Dec.) of the sources as
obtained from the literature. Column (D) is the redshift value. Column (E)
denotes the SMBH mass in the units of $M_{\odot}$, and Column (F) is the
luminosity at 5100Å. The information for all the parameters is obtained from
the AGN Black Hole Mass database (Bentz & Katz, 2015) except for the sources
marked with ${*}$ which are taken from Grier et al. (2017).

Figure 4: Comparison of the $g-i$ band lags obtained using the javelin and
ICCF methods for the 19 sources used in this study. The red dashed line
indicates a one-to-one correlation between the lags obtained using both the
methods.  Figure 5: Comparison of the disk sizes ($R_{\lambda_{0}}$) at
$g$-band rest wavelength. The estimation for $R_{\lambda_{0}}$ has been made
using the javelin thin disk model, while the analytical disk size has been
calculated using the SS disk model dependent on the SMBH mass and accretion
rates. The red dashed line indicates a one-to-one correlation between the two
estimates.

Figure 6: Distribution for the disk size at $g$-band rest wavelength
($R_{\lambda_{0}}$) for 6 sources from our sample are shown here. This
Distribution for $R_{\lambda_{0}}$ is obtained using the javelin thin disk
model assuming disk size scaling with temperature as a power-law index of 4/3.
For most of the sources, the estimated $R_{\lambda_{0}}$ is larger than the
predictions of the SS disk model (see columns F and G of Table 3). Table 2:
Light curve statistics for the sample of 19 AGN with reasonable lag estimates.

| $g$-band | r-band | i-band
---|---|---|---
Name | nobs | Mag | nobs | Mag | nobs | Mag
(A) | (B) | (C) | (D) | (E) | (F) | (G)
Mrk 335 (S1) | 48 | 14.82 $\pm 0.01$ | 52 | 14.28 $\pm 0.01$ | 27 | 14.34 $\pm 0.01$
Mrk 335 (S2) | 72 | 14.80 $\pm 0.01$ | 67 | 14.25 $\pm 0.01$ | 24 | 14.27 $\pm 0.01$
PG0026+129 (S2) | 70 | 15.31 $\pm 0.01$ | 120 | 15.11 $\pm 0.01$ | 19 | 14.68 $\pm 0.01$
PG0052+251 (S1) | 43 | 16.21 $\pm 0.01$ | 52 | 16.02 $\pm 0.01$ | 24 | 15.36 $\pm 0.01$
PG0052+251 (S2) | 68 | 16.09 $\pm 0.01$ | 67 | 15.93 $\pm 0.01$ | 18 | 15.31 $\pm 0.01$
NGC4253 (S1) | 304 | 15.07 $\pm 0.01$ | 337 | 14.46 $\pm 0.01$ | 45 | 14.32 $\pm 0.02$
NGC4253 (S2) | 168 | 15.17 $\pm 0.01$ | 143 | 14.51 $\pm 0.01$ | 29 | 14.35 $\pm 0.02$
PG1307+085 (S1) | 49 | 16.00 $\pm 0.01$ | 102 | 15.83 $\pm 0.01$ | 20 | 15.35 $\pm 0.01$
PG1307+085 (S2) | 63 | 15.70 $\pm 0.01$ | 72 | 15.59 $\pm 0.01$ | 19 | 15.21 $\pm 0.01$
RMID733 (S1) | 231 | 18.34 $\pm 0.04$ | 331 | 17.85 $\pm 0.03$ | 25 | 17.46 $\pm 0.03$
RMID399 (S1) | 113 | 20.77 $\pm 0.19$ | 237 | 20.37 $\pm 0.15$ | 19 | 19.76 $\pm 0.13$
RMID101 (S1) | 227 | 18.84 $\pm 0.06$ | 325 | 18.81 $\pm 0.06$ | 23 | 18.72 $\pm 0.06$
PG1411+442 (S1) | 452 | 14.98 $\pm 0.01$ | 447 | 14.75 $\pm 0.01$ | 41 | 14.66 $\pm 0.01$
PG1411+442 (S2) | 196 | 14.97 $\pm 0.02$ | 211 | 14.75 $\pm 0.01$ | 32 | 14.65 $\pm 0.02$
RMID779 (S1) | 340 | 19.69 $\pm 0.14$ | 321 | 19.26 $\pm 0.09$ | 24 | 18.83 $\pm 0.07$
RMID300 (S1) | 392 | 19.24 $\pm 0.09$ | 318 | 19.20 $\pm 0.07$ | 21 | 19.23 $\pm 0.09$
Mrk 817 (S1) | 227 | 14.31 $\pm 0.01$ | 246 | 14.08 $\pm 0.01$ | 25 | 14.15 $\pm 0.01$
Mrk 817 (S2) | 93 | 14.36 $\pm 0.01$ | 102 | 14.11 $\pm 0.01$ | 16 | 14.16 $\pm 0.01$
Mrk 290 (S1) | 375 | 15.10 $\pm 0.01$ | 412 | 14.72 $\pm 0.01$ | 28 | 14.78 $\pm 0.01$
Mrk 486 (S1) | 377 | 14.78 $\pm 0.01$ | 432 | 14.28 $\pm 0.01$ | 27 | 14.32 $\pm 0.01$
Mrk 486 (S2) | 144 | 14.80 $\pm 0.01$ | 188 | 14.28 $\pm 0.01$ | 21 | 14.31 $\pm 0.01$
Mrk 493 (S1) | 681 | 15.77 $\pm 0.01$ | 623 | 15.33 $\pm 0.01$ | 47 | 15.22 $\pm 0.01$
Mrk 493 (S2) | 297 | 15.77 $\pm 0.01$ | 236 | 15.31 $\pm 0.01$ | 49 | 15.18 $\pm 0.01$
PG1613+658 (S1) | 644 | 15.08 $\pm 0.02$ | 615 | 15.07 $\pm 0.01$ | 48 | 14.51 $\pm 0.01$
PG1613+658 (S2) | 335 | 15.45 $\pm 0.02$ | 318 | 15.38 $\pm 0.01$ | 53 | 14.74 $\pm 0.01$
PG 1617+175 (S2) | 40 | 15.49 $\pm 0.01$ | 78 | 15.49 $\pm 0.01$ | 25 | 15.01 $\pm 0.01$
1RXS J1858+4850 (S1) | 452 | 16.57 $\pm 0.01$ | 728 | 16.22 $\pm 0.01$ | 28 | 16.27 $\pm 0.01$
1RXSJ1858+4850 (S2) | 288 | 16.66 $\pm 0.01$ | 276 | 16.32 $\pm 0.01$ | 28 | 16.36 $\pm 0.01$
PG 2130+099 (S1) | 63 | 14.54 $\pm 0.01$ | 68 | 14.27 $\pm 0.01$ | 26 | 14.38 $\pm 0.01$
PG2130+099 (S2) | 41 | 14.65 $\pm 0.01$ | 37 | 14.37 $\pm 0.01$ | 25 | 14.51 $\pm 0.01$

Note: Column (A) denotes the common name of the sources as available in Bentz
& Katz (2015) and Grier et al. (2017). S1 and S2 denote the light curves for
the two seasons used in this study. Columns (B), (D), and (F) are the number
of observations (nobs) in the $g$, $r$, and $i$ filters, respectively, as
available in ZTF-DR6. The mean magnitudes and their uncertainties in the $g$,
$r$, and $i$ filters (Mag) throughout the observations are presented in
columns (C), (E), and (G).

Table 3:  The lags and the disk size estimates along with the SS disk
prediction for 19 sources in our sample.

Common name |  $\tau_{r}$ (javelin ) |  $\tau_{i}$ (javelin ) |  $\tau_{r}$ (ICCF) |  $\tau_{i}$ (ICCF) |  $R_{0}$ ( $g$-band) |  $R_{0}$ (SS disk) |
---|---|---|---|---|---|---|---
| (days) | (days) | (days) | (days) | (light-days) | (light-days) |
(A) | (B) | (C) | (D) | (E) | (F) | (G) |
Mrk 335 (S1) | $0.94^{+0.45}_{-1.60}$ | $1.32^{+0.76}_{-0.68}$ | $1.5^{+1.2}_{-1.3}$ | $2.4^{+1.6}_{-2.0}$ | $1.39^{+0.46}_{-0.44}$ | 1.33 |
PG 0026+129 (S2) | $3.30^{+3.70}_{-7.62}$ | $2.48^{+2.82}_{-2.95}$ | $0.6^{+0.9}_{-0.2}$ | $2.0^{+2.5}_{-2.6}$ | $6.29^{+1.24}_{-0.87}$ | 2.81 |
PG 0052+251 (S2) | $2.88^{+1.75}_{-2.37}$ | $-1.79^{+6.24}_{-6.40}$ | $2.5^{+2.5}_{-0.0}$ | $3.5^{+4.0}_{-7.0}$ | $2.79^{+1.98}_{-3.26}$ | 2.44 |
NGC 4253 (S1) | $-0.01^{+2.06}_{-1.98}$ | $-0.22^{+3.98}_{-2.79}$ | $0.5^{+3.9}_{-2.0}$ | $-2.0^{+3.0}_{-6.5}$ | $5.98^{+5.89}_{-2.09}$ | 0.55 |
PG 1307+085 (S2) | $0.10^{+7.27}_{-0.46}$ | $-0.05^{+3.07}_{-5.12}$ | $3.0^{+2.6}_{-2.6}$ | $1.5^{+4.5}_{-7.5}$ | $9.16^{+4.96}_{-0.62}$ | 2.52 |
RMID733 (S1) | $0.92^{+0.47}_{-2.47}$ | $3.37^{+1.00}_{-5.05}$ | $-0.5^{+1.9}_{-2.5}$ | $-2.0^{+6.0}_{-4.5}$ | $3.42^{+0.32}_{-1.36}$ | 0.56 |
RMID399 (S1) | $2.62^{+5.12}_{-5.59}$ | $4.64^{+4.03}_{-8.97}$ | $6.8^{+2.2}_{-10.3}$ | $5.6^{+1.5}_{-3.6}$ | $5.91^{+3.51}_{-3.11}$ | 0.80 |
RMID101 (S1) | $-2.95^{+5.13}_{-5.28}$ | $0.78^{+5.39}_{-5.52}$ | $3.3^{+4.1}_{-6.2}$ | $1.5^{+5.5}_{-5.0}$ | $3.20^{+1.93}_{-3.34}$ | 0.56 |
PG 1411+442 (S2) | $-3.67^{+0.67}_{-0.82}$ | $-0.46^{+0.70}_{-2.00}$ | $1.5^{+7.0}_{-6.4}$ | $2.5^{+2.5}_{-3.5}$ | $3.79^{+0.31}_{-3.65}$ | 2.22 |
RMID779 (S1) | $4.56^{+3.90}_{-3.04}$ | $6.34^{+2.83}_{-7.21}$ | $9.0^{+0.0}_{-2.8}$ | $5.1^{+3.4}_{-9.1}$ | $6.21^{+2.91}_{-2.82}$ | 0.46 |
RMID300 (S1) | $3.77^{+1.62}_{-6.41}$ | $0.75^{+7.00}_{-4.43}$ | $2.0^{+2.6}_{-3.0}$ | $2.0^{+6.5}_{-4.5}$ | $7.84^{+3.39}_{-1.56}$ | 0.71 |
Mrk 817 (S1) | $4.31^{+0.21}_{-1.99}$ | $4.98^{+0.82}_{-2.89}$ | $2.5^{+1.5}_{-1.5}$ | $3.6^{+1.1}_{-1.4}$ | $4.70^{+0.18}_{-3.83}$ | 1.30 |
Mrk 290 (S1) | $0.84^{+3.28}_{-1.64}$ | $0.27^{+2.06}_{-1.62}$ | $4.1^{+1.9}_{-5.9}$ | $2.0^{+2.5}_{-2.1}$ | $5.14^{+2.05}_{-2.62}$ | 0.84 |
Mrk 486 (S2) | $0.38^{+0.21}_{-0.36}$ | $4.35^{+2.41}_{-0.89}$ | $0.5^{+1.0}_{-1.0}$ | $4.5^{+1.1}_{-1.5}$ | $0.93^{+0.11}_{-2.05}$ | 1.23 |
Mrk 493 (S2) | $0.68^{+1.68}_{-0.29}$ | $0.70^{+1.53}_{-1.43}$ | $2.0^{+0.6}_{-0.5}$ | $4.9^{+3.2}_{-3.2}$ | $5.52^{+1.10}_{-0.98}$ | 0.81 |
PG 1613+658 (S1) | $-0.38^{+5.86}_{-0.46}$ | $-1.26^{+5.51}_{-5.39}$ | $2.0^{+1.9}_{-2.6}$ | $5.1^{+3.4}_{-6.1}$ | $5.16^{+0.52}_{-0.32}$ | 2.46 |
PG 1617+175 (S2) | $3.41^{+2.04}_{-1.98}$ | $8.28^{+1.29}_{-2.51}$ | $5.2^{+2.9}_{-1.4}$ | $6.2^{+1.9}_{-4.2}$ | $8.14^{+1.75}_{-1.31}$ | 1.89 |
1RXS J1858+4850 (S2) | $0.37^{+0.13}_{-0.91}$ | $1.40^{+3.68}_{-0.81}$ | $1.0^{+1.1}_{-1.5}$ | $3.1^{+1.9}_{-2.6}$ | $2.74^{+1.90}_{-0.25}$ | 1.13 |
PG 2130+099 (S2) | $2.67^{+0.73}_{-3.05}$ | $3.24^{+3.06}_{-0.90}$ | $1.2^{+0.6}_{-0.5}$ | $1.3^{+1.3}_{-0.5}$ | $4.42^{+2.30}_{-0.54}$ | 1.76 |

Note: Column (A) denotes the common name of the sources as available in Bentz
& Katz (2015) and Grier et al. (2017). S1 and S2 denote the respective season
in which the interband lags increasing with wavelength were obtained using
either of the methods. The $g-r$ and $g-i$ reverberation lags obtained using
javelin are shown in Columns (B) and (C), while the lags obtained using ICCF
method for $g-r$ and $g-i$ are presented in columns (D) and (E) respectively.
The disk sizes obtained using the javelin thin disk model are presented in
column (F) and the expected theoretical disk sizes at $g-$band effective
wavelength assuming the SS disk model are presented in column (G).

##  4 Accretion disk sizes

###  4.1 Disk sizes using javelin thin disk model

A recent extension in the javelin code known as the thin disk model introduced
by Mudd et al. (2018) allows us to directly model and estimate the size of the
accretion disk at a particular wavelength based on the multi-band light
curves. It fits the light curves assuming the SS disk approximation, and the
accretion disk size is given according to equation 2.

This method was used by Mudd et al. (2018) to obtain the accretion disk size
at 2500Å for a set of 15 quasars from the DES survey. Further, Kokubo (2018)
and Yu et al. (2020) have also used this model to derive the accretion disk
sizes for different quasars. We used the thin disk model for the light curves
to obtain the disk sizes at $g$-band rest wavelength and compare it to the
predicted SS disk model based on individual SMBH mass and accretion rates. It
is noticeable that the disk sizes predicted for javelin assume the SS disk as
we have fixed $\beta$ to 1.33 (4/3). Leaving the parameter $\beta$ free does
not converge to any particular value, as noticed in the recent works (Kokubo,
2018; Yu et al., 2020).However, a possible concern is that this model could be
unable to recover the accurate parameters for AGN accreting at a faster rate
which may require additional physical processes or special geometry to be
explained (Castelló-Mor et al., 2017).

The distribution for $R_{\lambda_{0}}$ gives the size of the accretion disk at
the respective wavelength $\lambda_{0}$. Since the sources in our sample were
located at varying redshifts, the corresponding wavelength ($\lambda_{0}$) for
the disk size varied from 2717 Å to 4677 Å according to the $g$-band rest-
frame wavelength. The values for $R_{\lambda_{0}}$ ranged from 0.93 light days
to 9.16 light days with a mean value of 4.88 light days. Since the sources for
this analysis have large SMBH mass and luminosity ranges, it is plausible that
the estimated disk sizes vary from source to source. The distribution for
$R_{\lambda_{0}}$ for a few sources is shown in Figure 6.

###  4.2  Comparison with analytical SS disk model

The standard SS disk model predicts the disk sizes based on the SMBH mass, and
the mass accretion rate. We compare the disk sizes obtained using this
analytical model with the disk sizes obtained by fitting the light curves
through the javelin thin disk model. Following Jiang et al. (2017), the light
travel time across two different radii where photons with wavelengths
$\lambda_{g}$ and $\lambda_{x}$ are emitted, can be put as:

$\centering\Delta t_{g-x}=\left(X\frac{k_{\rm
B}\lambda_{g}}{hc}\right)^{4/3}\left(f_{i}\frac{3GM_{BH}\dot{M}}{8\pi\sigma}\right)^{1/3}\left[\left(\frac{\lambda_{x}}{\lambda_{g}}\right)^{4/3}-1\right]\@add@centering$
(4)

where, $k_{B}$, h, c, G and $\sigma$ are the Boltzmann constant, Planck
constant, speed of light, gravitational constant and Stefan’s constant
respectively, $M_{BH}$ is the SMBH mass, $\dot{M}$ is the mass accretion rate,
and X is a scaling factor. The value of X has been chosen to be 2.49 as per
the recent works (see Fausnaugh et al., 2016; Edelson et al., 2019).

The disk size estimated by modeling the light curves through the javelin thin
disk model should agree with the lags if the SS disk assumption is held valid.
Figure 5 shows the comparison of the disk sizes obtained using the javelin
thin disk model and the prediction of the SS disk model. For 16 sources in our
sample, the obtained disk sizes are larger than the theoretical disk size
estimates, which has been observed in previous works as well (see Shappee et
al., 2014; Fausnaugh et al., 2016; Starkey et al., 2016a; Edelson et al.,
2019, and references therein). However, for 3 sources, the disk sizes are in
agreement with the SS disk predictions. For instance, our estimated disk size
of $1.39^{+0.46}_{-0.44}$ days at 4400 Å for Mrk 335 is in close agreement
with the disk size of 1.33 light days predicted based on standard SS disk at
the same wavelength.

We calculated the lag spectrum based on the predicted SS disk model, the
obtained thin disk sizes by modeling the light curves using the javelin thin
disk model, and the $g-r$ and $g-i$ lags obtained using the javelin and ICCF
methods. We had only 2 interband lag measurements, while we used the
combination of $R_{\lambda_{0}}$ obtained using the javelin thin disk model
and the equation 2 to compare the accretion disk sizes at different
wavelengths. We find out that most sources have disk sizes and interband lags
larger than the SS disk prediction. The results are shown in Figure 7.

Figure 7: The SS disk prediction (black line) and the disk sizes as obtained
through the javelin thin disk model (green dashed line) for 6 sources in our
sample are shown here. Also, the $g-r$ and $g-i$ lags estimated using javelin
and ICCF are shown in red and cyan colors, respectively. The results for all
the sources with lag estimates constrained between [$-$10, 10] days are made
available as supplementary material online. Figure 8:  The scaling of the disk
sizes with the SMBH masses are shown here. The sample used in this study is
denoted as blue circles, the sources from Yu et al. (2020) are shown as red
inverted triangles, sources from Mudd et al. (2018) are shown as green
triangles, and the sources taken from Morgan et al. (2010) are shown as green
squares. The uncertainty in SMBH mass estimate is approximately 0.4 dex, which
is shown as a black line on the bottom left. The predictions based on the
dimensionless mass accretion rates of 0.01 (blue dashed line), 0.1 (green
dotted line), and 1 (solid red line) are also shown in the figure.

###  4.3  Scaling of Disk size with physical parameters

We used the disk sizes obtained using the javelin thin disk model to study the
relation of disk sizes with the physical parameters, namely the SMBH mass and
the luminosity at 5100 Å. According to predictions of the SS disk model, the
disk size scales with the SMBH mass as $M^{2/3}$ (see Morgan et al., 2010).
The RM-based SMBH mass estimates have been the most robust so far in the
absence of any other reliable techniques for SMBH, covering a significant
range of redshifts and luminosities. In Figure 8, we show the relation between
the SMBH masses and accretion disk sizes obtained using the javelin thin disk
model. We include the previous microlensing-based results from Morgan et al.
(2010). We converted the units of the accretion disk size to light days and
converted the reference wavelength at 2500Å. We also included the results from
Mudd et al. (2018) and Yu et al. (2020) in this relation. We also plot the
expected relation between the SMBH mass and the accretion disk size based on
the dmensionless accretion rates of 0.01, 0.1 and 1. The dimensionless
acccretion rate is $\dot{M}/\dot{M_{\textsc{edd}}}$, where $\dot{M}$ is the
mass accretion rate and $\dot{M_{\textsc{edd}}}$ is the Eddington mass
accretion rate defined as $L_{\textsc{edd}}/c^{2}$ and
$L_{\textsc{edd}}=1.45\times 10^{38}\times\frac{M_{BH}}{M\odot}erg/s.$

We find a weak correlation between the obtained disk sizes and the SMBH
masses, with the Spearman rank correlation coefficient being 0.38. Although
Morgan et al. (2010) found a relation linking the SMBH mass and disk sizes
obtained using microlensing, their sample covered the higher end of SMBH
masses only, while our sample covers a range of log (M${}_{BH}/M_{\odot}$)
ranging from 6.19 to 8.66 which is larger than the range covered in that work.
Although the correlation is not very strong, the general trend of AGN with
higher SMBH mass tends to have larger accretion disk sizes. Both the disk size
and SMBH mass estimates suffer large uncertainties, and thus, proper
monitoring and the robust estimates of the accretion disk size will be very
helpful in establishing this relation further.

Further, in Figure 9, we show the relation between the accretion disk sizes
and the luminosity at 5100Å. We expect that the disk sizes should be related
with intrinsic luminosity as more luminous quasars are expected to have large
accretion disks and hence larges inter-band lags, based on the light travel
times (Jiang et al., 2017). In Sergeev et al. (2005), the interband lags are
found to scale with the luminosity as $\tau=L^{0.4-0.5}$. However, we find a
very weak correlation between the luminosities and the disk size, with a
Spearman rank correlation coefficient of 0.18 and a null hypothesis
($p_{null}$) value of 0.31.

##  5 Discussion

To constrain the size of the accretion disk for a sample of quasars with known
SMBH masses, we estimated the inter-band reverberation lags between the
optical $g$, $r$, and $i$ bands observations from the ZTF survey. The
reverberation lags represent the light travel time across the two different
regions of the disk. We found that the interband light curves are correlated,
and for 19 sources, the lags increasing with wavelength were recovered
successfully using the javelin and ICCF methods. The $g-r$ inter-band lags
were shorter than the $g-i$ inter-band lags for these sources, which is
consistent with the disk reprocessing lamppost model, according to which the
photons arising from the innermost regions are reprocessed in the form of
emission from the outer regions resulting in a lag. In multi-wavelength
monitoring campaigns involving X-ray, UV, and optical wavebands, such as the
AGN STORM campaign (Starkey et al., 2016a), inter-band lags have been reported
in all the wavelengths, with longer wavelengths lagging the short wavelengths.
Further, even using only optical band observations, inter-band lags have been
successfully recovered for a large sample of AGN (see Jiang et al., 2017; Mudd
et al., 2018; Homayouni et al., 2019; Yu et al., 2020). Our results complement
these important observations.

We find out that the size of the accretion disk obtained at a reference
wavelength is larger than predicted by the Shakura Sunyaev accretion disk
model for a majority of the sources see Figure 5. We note that the interband
lags for 5 sources, namely PG0026+129, NGC 4253, PG1411+442, RMID300, and
PG2130+099, follow the predictions of the SS disk model, while 14 sources have
lags of about 3 to 4 times larger than the expectations from the SS disk
model. However, we could not zero in on a parameter which distinguished these
sources from the sources where the disk sizes are much larger than the SS disk
assumption. In previous works, the disk size has been known to scale with the
wavelength as a power law of index 4/3, but the sizes have been reported to be
larger than predicted by the SS disk (Fausnaugh et al., 2016; Jiang et al.,
2017; Fausnaugh et al., 2018, and references therein). Contrarily, Mudd et al.
(2018) reported that the disk sizes for their sample of 15 sources did not
differ much from the SS disk analytical model. Interestingly the disk sizes
obtained through microlensing have also reached similar conclusions (Pooley et
al., 2007; Blackburne et al., 2011; Mosquera et al., 2013). Larger accretion
disk sizes obtained through both the continuum reverberation mapping and the
microlensing methods for a majority of the sources may indicate that the
standard SS disk assumption does not hold for the AGN in general and
additional components may be needed while modeling the accretion disks in AGN.
We note that some of the sources in our sample have higher accretion rates (Du
et al., 2015). It is possible that the accretion mechanism in the AGN with
higher accretion rates is different and thus, the disk sizes obtained assuming
the SS disk model may not hold true for these kinds of AGN. Evidence of non
disk component in the optical continuum for Mrk 279 has been reported by
Chelouche et al. (2019). This might be a possibility for the larger than
expected lags for some of the sources in this sample too. However, the exact
contribution from such a component can be achieved through either a
combination of narrow band filters or simultaneous spectra which has not been
possible in our case. Another possible explanation for the larger disk sizes
has been proposed by Gaskell (2017) where, the reddening effects could lead to
larger than expected accretion disk sizes. However, Nuñez et al. (2019) find
out that for Mrk 509, this effect is not able to explain the larger disk
sizes.

Continuum reverberation mapping gives more robust estimates with multi-
wavelength campaigns, as we see in the case of NGC5548 from the AGN STORM
campaign. Nevertheless, with optical data points only, we have been able to
constrain the disk sizes for a large number of objects. While the exact
structure of the AGN accretion disks remains unknown so far, there is
overwhelming evidence in favor of disk reprocessing based on confirmed inter-
band lags, increasing with the wavelength (McHardy et al., 2014; Fausnaugh et
al., 2018). Our results for a set of reverberation mapped AGN differing in
SMBH mass and luminosities provide another set of observational evidence for
testing the accretion mechanism in a wide range of such objects. While we find
a weak correlation between the obtained disk size, the SMBH mass, and the
luminosity, this might be due to the limit of observations as we cannot cover
a significant wavelength range using the ZTF dataset. Another reason behind
the weak correlation could be the wide range of SMBH masses and luminosities
covered in the sample. In the future, with the availability of surveys such as
the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST) (Ivezić
et al., 2019), it will be possible to cover a larger sample with much better
cadence and hence will be helpful in better constraining these correlations.

Figure 9:  The scaling of the disk sizes corresponding to the $g$-band rest
wavelength with the luminosity at 5100 Å for 19 AGN from this sample. The red
line denotes the fitted straight line.

##  6 Conclusions

We obtained the accretion disk size measurements of AGN with previous SMBH
mass estimates through reverberation mapping, with a wide range of
luminosities, redshifts, and SMBH masses, based on $g$, $r$, and $i$ band
observations from the ZTF survey. The high cadence observations from ZTF
provide an excellent dataset to constrain interband reverberation lags
efficiently. The primary conclusions from this work are as follows:

  1. 1.

The $g$, $r$, and $i$ band light curves are correlated, and the $g-r$ inter-
band lags are shorter than $g-i$ inter-band lags for 19 sources in our sample,
which provide strong evidence in favor of the disk reprocessing.

  2. 2.

The interband lags and the disk sizes obtained using the javelin thin disk
model are larger than the ones predicted by the standard SS thin disk model
for a majority of sources, which is consistent with the recent findings and
raises the question of the usage of the simple SS disk model for AGN accretion
disks.

  3. 3.

For 5 sources, we obtained the interband lags in agreement with the SS disk
predictions. However, no significant parameter distinguishes these AGN from
the ones where we found larger disk sizes.

  4. 4.

There is a weak correlation found in the SMBH mass versus disk size and the
luminosity versus disk size, which may be due to the uncertainties in both the
accretion disk size measurements and the SMBH mass measurements, which have
been known to suffer uncertainties up to 0.4 dex.

Although mapping the entire disk profile is only possible with multi-
wavelength campaigns such as the AGN-STORM campaign of NGC 5548 (Edelson et
al., 2015), public surveys as the ZTF provide us the opportunity to use light
curves for a large number of sources and obtain inter-band lags to understand
the accretion mechanisms responsible for the interband lags, albeit with a
smaller wavelength coverage. This work can be extended with larger AGN samples
and derive the accretion mechanisms powering them to understand the AGN
accretion disks better.

## Acknowledgements

LCH and XBW were supported by the National Science Foundation of China
(11721303, 11991052) and the National Key R&D Program of China
(2016YFA0400702, 2016YFA0400703). This research is part of the DST-SERB
project under grant no. EMR/2016/001723. VKJ and HC acknowledge the financial
support provided by DST-SERB for this work. VKJ is thankful to Suvendu Rakshit
(ARIES, Nainital) for useful discussions and help regarding the javelin code.
Based on observations obtained with the Samuel Oschin 48-inch Telescope at the
Palomar Observatory as part of the Zwicky Transient Facility project. ZTF is
supported by the National Science Foundation under Grant No. AST-1440341 and
collaboration including Caltech, IPAC, the Weizmann Institute for Science, the
Oskar Klein Center at Stockholm University, the University of Maryland, the
University of Washington, Deutsches Elektronen-Synchrotron and Humboldt
University, Los Alamos National Laboratories, the TANGO Consortium of Taiwan,
the University of Wisconsin at Milwaukee, and Lawrence Berkeley National
Laboratories. Operations are conducted by COO, IPAC, and UW.

## Data Availability

The data being used for this study is publicly available in the ZTF DR6
database.

## References

  * Abramowicz et al. (1988) Abramowicz M. A., Czerny B., Lasota J. P., Szuszkiewicz E., 1988, ApJ, 332, 646
  * Arcodia et al. (2019) Arcodia R., Merloni A., Nandra K., Ponti G., 2019, Astronomy & Astrophysics, 628, A135
  * Bahcall et al. (1972) Bahcall J. N., Kozlovsky B.-Z., Salpeter E. E., 1972, The Astrophysical Journal, 171, 467
  * Bellm et al. (2019) Bellm E. C., et al., 2019, PASP, 131, 018002
  * Bentz & Katz (2015) Bentz M. C., Katz S., 2015, Publications of the Astronomical Society of the Pacific, 127, 67
  * Bentz et al. (2009) Bentz M. C., et al., 2009, The Astrophysical Journal, 705, 199
  * Blackburne et al. (2011) Blackburne J. A., Pooley D., Rappaport S., Schechter P. L., 2011, ApJ, 729, 34
  * Blandford & McKee (1982) Blandford R. D., McKee C. F., 1982, The Astrophysical Journal, 255, 419
  * Cackett et al. (2007) Cackett E. M., Horne K., Winkler H., 2007, MNRAS, 380, 669
  * Castelló-Mor et al. (2017) Castelló-Mor N., et al., 2017, MNRAS, 467, 1209
  * Chelouche et al. (2019) Chelouche D., Pozo Nuñez F., Kaspi S., 2019, Nature Astronomy, 3, 251
  * Dai et al. (2010) Dai X., Kochanek C. S., Chartas G., Kozłowski S., Morgan C. W., Garmire G., Agol E., 2010, ApJ, 709, 278
  * Du et al. (2015) Du P., et al., 2015, The Astrophysical Journal, 806, 22
  * Du et al. (2016) Du P., et al., 2016, The Astrophysical Journal, 825, 126
  * Du et al. (2018) Du P., et al., 2018, ApJ, 856, 6
  * Edelson et al. (2015) Edelson R., et al., 2015, The Astrophysical Journal, 806, 129
  * Edelson et al. (2019) Edelson R., et al., 2019, The Astrophysical Journal, 870, 123
  * Fausnaugh et al. (2016) Fausnaugh M. M., et al., 2016, The Astrophysical Journal, 821, 56
  * Fausnaugh et al. (2017) Fausnaugh M. M., Peterson B. M., Starkey D. A., Horne K., 2017, Frontiers in Astronomy and Space Sciences, 4, 55
  * Fausnaugh et al. (2018) Fausnaugh M. M., et al., 2018, The Astrophysical Journal, 854, 107
  * Gaskell (2017) Gaskell C. M., 2017, MNRAS, 467, 226
  * Gaskell & Peterson (1987) Gaskell C. M., Peterson B. M., 1987, ApJS, 65, 1
  * Graham et al. (2019) Graham M. J., et al., 2019, PASP, 131, 078001
  * Grier et al. (2017) Grier C. J., et al., 2017, The Astrophysical Journal, 851, 21
  * Grier et al. (2019) Grier C. J., et al., 2019, The Astrophysical Journal, 887, 38
  * Hernández Santisteban et al. (2020) Hernández Santisteban J. V., et al., 2020, Monthly Notices of the Royal Astronomical Society, 498, 5399
  * Homayouni et al. (2019) Homayouni Y., et al., 2019, The Astrophysical Journal, 880, 126
  * Ivezić et al. (2019) Ivezić Ž., et al., 2019, ApJ, 873, 111
  * Jiang et al. (2017) Jiang Y.-F., et al., 2017, ApJ, 836, 186
  * Kara et al. (2021) Kara E., et al., 2021, arXiv e-prints, p. arXiv:2105.05840
  * Kaspi et al. (2000) Kaspi S., Smith P. S., Netzer H., Maoz D., Jannuzi B. T., Giveon U., 2000, The Astrophysical Journal, 533, 631
  * Kelly et al. (2009) Kelly B. C., Bechtold J., Siemiginowska A., 2009, The Astrophysical Journal, 698, 895
  * Kokubo (2018) Kokubo M., 2018, Publications of the Astronomical Society of Japan, 70, 97
  * Li et al. (2005) Li L.-X., Zimmerman E. R., Narayan R., McClintock J. E., 2005, ApJS, 157, 335
  * Li et al. (2019) Li J. I.-H., et al., 2019, The Astrophysical Journal, 884, 119
  * Lynden-Bell (1969) Lynden-Bell D., 1969, Nature, 223, 690
  * MacLeod et al. (2010) MacLeod C. L., et al., 2010, The Astrophysical Journal, 721, 1014
  * Masci et al. (2019) Masci F. J., et al., 2019, PASP, 131, 018003
  * McHardy et al. (2014) McHardy I. M., et al., 2014, Monthly Notices of the Royal Astronomical Society, 444, 1469
  * Meyer-Hofmeister et al. (2017) Meyer-Hofmeister E., Liu B. F., Qiao E., 2017, Astronomy & Astrophysics, 607, A94
  * Morgan et al. (2010) Morgan C. W., Kochanek C. S., Morgan N. D., Falco E. E., 2010, The Astrophysical Journal, 712, 1129
  * Mosquera et al. (2013) Mosquera A. M., Kochanek C. S., Chen B., Dai X., Blackburne J. A., Chartas G., 2013, ApJ, 769, 53
  * Mudd et al. (2018) Mudd D., et al., 2018, The Astrophysical Journal, 862, 123
  * Narayan & Yi (1995) Narayan R., Yi I., 1995, ApJ, 452, 710
  * Nuñez et al. (2019) Nuñez F. P., et al., 2019, Monthly Notices of the Royal Astronomical Society, 000, 1
  * Pei et al. (2017) Pei L., et al., 2017, The Astrophysical Journal, 837, 131
  * Peterson (2006) Peterson B. M., 2006, The Broad-Line Region in Active Galactic Nuclei, Lect. Notes Phys., 693, 77
  * Peterson et al. (1998) Peterson B., Wanders I., Horne K., Collier S., Alexander T., Kaspi S., Maoz D., 1998, Publications of the Astronomical Society of the Pacific, 110, 660
  * Pooley et al. (2007) Pooley D., Blackburne J. A., Rappaport S., Schechter P. L., 2007, ApJ, 661, 19
  * Rosa et al. (2015) Rosa G. D., et al., 2015, The Astrophysical Journal, 806, 128
  * Salpeter (1964) Salpeter E. E., 1964, ApJ, 140, 796
  * Sergeev et al. (2005) Sergeev S. G., Doroshenko V. T., Golubinskiy Y. V., Merkulova N. I., Sergeeva E. A., 2005, The Astrophysical Journal, 622, 129
  * Shakura & Sunyaev (1973) Shakura N. I., Sunyaev R. A., 1973, A&A, 500, 33
  * Shappee et al. (2014) Shappee B. J., et al., 2014, ApJ, 788, 48
  * Starkey et al. (2016a) Starkey D. A., Horne K., Villforth C., 2016a, Monthly Notices of the Royal Astronomical Society, 456, 1960
  * Starkey et al. (2016b) Starkey D., et al., 2016b, The Astrophysical Journal, 835, 65
  * Sun et al. (2018) Sun M., Grier C. J., Peterson B. M., 2018, PyCCF: Python Cross Correlation Function for reverberation mapping studies (ascl:1805.032)
  * Sun et al. (2020) Sun M., et al., 2020, ApJ, 891, 178
  * Yu et al. (2020) Yu Z., et al., 2020, The Astrophysical Journal Supplement Series, 246, 16
  * Zu et al. (2011) Zu Y., Kochanek C. S., Peterson B. M., 2011, The Astrophysical Journal, 735, 80
  * Zu et al. (2013) Zu Y., Kochanek C. S., Kozłowski S., Udalski A., 2013, The Astrophysical Journal, 765, 106