# Optical Variability of Blazars in the Tomo-e Gozen Northern Sky Transient
Survey

TianFang Zhang  Department of Astronomy, Graduate School of Science, The
University of Tokyo, 7-3-1, Hongō, Bunkyo City, Tokyo, 113-8654, Japan
Institute of Astronomy, Graduate School of Science, The University of Tokyo,
2-21-1 Osawa, Mitaka, Tokyo, 181-0015, Japan  National Astronomical
Observatory of Japan, National Institutes of Natural Science, 2-21-1 Osawa,
Mitaka, Tokyo, 181-8588, Japan  Mamoru Doi  Institute of Astronomy, Graduate
School of Science, The University of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo,
181-0015, Japan  National Astronomical Observatory of Japan, National
Institutes of Natural Science, 2-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan
Research center for the early universe School of Science Bldg. No.4, The
University of Tokyo, 7-3-1, Hongō, Bunkyo City, Tokyo, 113-0033, Japan
Mitsuru Kokubo  National Astronomical Observatory of Japan, National
Institutes of Natural Science, 2-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan
Shigeyuki Sako  Institute of Astronomy, Graduate School of Science, The
University of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo, 181-0015, Japan  UTokyo
Organization for Planetary Space Science, The University of Tokyo, 7-3-1,
Hongō, Bunkyo City, Tokyo, 113-8654, Japan  Collaborative Research
Organization for Space Science and Technology, The University of Tokyo, 7-3-1,
Hongō, Bunkyo City, Tokyo, 113-8654, Japan  Ryou Ohsawa  National Astronomical
Observatory of Japan, National Institutes of Natural Science, 2-21-1 Osawa,
Mitaka, Tokyo, 181-8588, Japan  Nozomu Tominaga  National Astronomical
Observatory of Japan, National Institutes of Natural Science, 2-21-1 Osawa,
Mitaka, Tokyo, 181-8588, Japan  Astronomical Science Program, Graduate
Institute for Advanced Studies, SOKENDAI, 2-21-1 Osawa, Mitaka, Tokyo,
181-8588, Japan  Department of Physics, Faculty of Science and Engineering,
Konan University, 8-9-1 Okamoto, Kobe, Hyogo, 658-8501, Japan  Masaomi Tanaka
Astronomical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan
Division for the Establishment of Frontier Sciences, Organization for Advanced
Studies, Tohoku University, Sendai, 980-8577, Japan  Yasushi Fukazawa
Department of Physics, Hiroshima University, 1-3-1 Kagamiyama, Higashi-
Hiroshima, 739-8526, Japan  Hidenori Takahashi  Kiso Observatory, Institute of
Astronomy, Graduate School of Science, The University of Tokyo, Mitake
10762-30, Kiso-machi, Kiso-gun, Nagano, 397-0101, Japan  Noriaki Arima
Institute of Astronomy, Graduate School of Science, The University of Tokyo,
2-21-1 Osawa, Mitaka, Tokyo, 181-0015, Japan  Naoto Kobayashi  Kiso
Observatory, Institute of Astronomy, Graduate School of Science, The
University of Tokyo, Mitake 10762-30, Kiso-machi, Kiso-gun, Nagano, 397-0101,
Japan  Ko Arimatsu  Hakubi Center / Astronomical Observatory, Graduate School
of Science, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo-ku, Kyoto,
606-8502, Japan  Shin-ichiro Okumura  Japan Spaceguard Association, Bisei
Spaceguard Center, 1716-3 Okura, Bisei, Ibara, Okayama, 714-1411, Japan  Sohei
Kondo  Kiso Observatory, Institute of Astronomy, Graduate School of Science,
The University of Tokyo, Mitake 10762-30, Kiso-machi, Kiso-gun, Nagano,
397-0101, Japan  Toshihiro Kasuga  National Astronomical Observatory of Japan,
National Institutes of Natural Science, 2-21-1 Osawa, Mitaka, Tokyo, 181-8588,
Japan  Yuki Mori  Kiso Observatory, Institute of Astronomy, Graduate School of
Science, The University of Tokyo, Mitake 10762-30, Kiso-machi, Kiso-gun,
Nagano, 397-0101, Japan  Yuu Niino  Kiso Observatory, Institute of Astronomy,
Graduate School of Science, The University of Tokyo, Mitake 10762-30, Kiso-
machi, Kiso-gun, Nagano, 397-0101, Japan

###### Abstract

We studied the optical variability of 241 BL Lacs and 83 flat-spectrum radio
quasars (FSRQ) from the 4LAC catalog using data from the Tomo-e Gozen Northern
Sky Transient Survey, with $\sim$ 50 epochs per blazar on average. We excluded
blazars whose optical variability may be underestimated due to the influence
of their host galaxy, based on their optical luminosity ($L_{O}$). FSRQs with
$\gamma$-ray photon index greater than 2.6 exhibit very low optical
variability, and their distribution of standard deviation of repeated
photometry is significantly different from that of the other FSRQs (KS test P
value equal to $5\times 10^{-6}$ ). Among a sample of blazars at any
particular cosmological epoch, those with lower $\gamma$-ray luminosity
($L_{\gamma}$) tend to have lower optical variability, and those FSRQs with
$\gamma$-ray photon index greater than 2.6 tend to have low $L_{\gamma}$. We
also measured the structure function of optical variability and found that the
amplitude of the structure function for FSRQs is higher than previously
measured and higher than that of BL Lacs at multiple time lags. Additionally,
the amplitude of the structure function of FSRQs with high $\gamma$-ray photon
index is significantly lower than that of FSRQs with low $\gamma$-ray photon
index. The structure function of FSRQs of high $\gamma$-ray photon index shows
a characteristic timescale of more than 10 days, which may be the variability
timescale of the accretion disk. In summary, we infer that the optical
component of FSRQs with high $\gamma$-ray photon index may be dominated by the
accretion disk.

Blazars — BL Lac — FSRQ — Optical Variability — Structure Function

##  1 Introduction

Blazars are a type of active galactic nucleus (AGN) characterized by a
relativistic jet pointed directly at the observer(Blandford & Königl, 1979).
As one of the most powerful phenomena in the universe, blazars have garnered
significant attention in the field of extra-galactic astronomy. Jets of
blazars are responsible for producing a wide range of observable energies, the
majority of which are in the form of non-thermal radiation. The non-thermal
radiation includes synchrotron radiation, which occurs across radio to X-ray
frequencies, as well as inverse Compton scattering, which occurs from X-ray to
$\gamma$-ray frequencies (Konigl, 1981).

Blazars can be classified into two primary intrinsic types, namely low-
ionization (BL Lacs) and high-ionization (FSRQs, predominantly beamed
Fanaroff–Riley class II sources) ((Giommi et al., 2012); (Fanaroff & Riley,
1974)). The distinction between BL Lacs and FSRQs is primarily determined by
the presence or absence of emission lines in their optical-IR spectra.
Specifically, BL Lacs do not display prominent emission lines in their
spectra, whereas FSRQs do display prominent emission lines in their spectra.

In a recent study, (Ghisellini et al., 2017) examined the dependence of the
Spectral Energy Distribution (SED) on $\gamma$-ray luminosity ($L_{\gamma}$)
using Fermi/LAT and other data. The study found that FSRQs exhibit a similar
SED across a wide range of $L_{\gamma}$, while BL Lacs display decreasing
synchrotron peak frequencies with increasing $L_{\gamma}$. Furthermore, the
nature of blazars at various wavelengths has been discussed by several studies
((Richards et al., 2011), (Landi et al., 2015), (Acciari et al., 2021), (Kaur
et al., 2021), (Kerby et al., 2021)).

Studying the optical variability of blazars is essential for understanding
their nature. (Clements et al., 2003) monitored the intra-night optical
variability of FSRQ PKS 0736+017 and observed a dramatic flare, with the
source brightening by 1.3 magnitudes ($R$ band) in just 2 hours. In a study
comparing the intra-night optical variability rates of blazars, (Bachev et
al., 2012) found indications for the presence of quasi-periodic micro-
oscillations with periods of about an hour. Similarly, (Rani et al., 2013)
monitored the intra-night multi-wavelength luminosity of BL Lac object
S50716+714 over a period of four years and identified a long-term variability
trend (about 350 days) and a shorter time scales (about 60 days). (Pandey et
al., 2020) studied the optical variability of BL Lac 1ES 0806+524 in three
bands ($VRI$) using high cadence observations for 153 nights during
2011$\sim$2019, and found a small but significant variation in both $V$ band
and $R$ band light curves in only one night. In a recent study, (Goyal, 2021)
examined the intra-night optical variability of 9 BL Lacs and 5 FSRQs,
carrying out a Fourier transform on their light curves, and obtained their
power spectral densities. They found that most of the power spectral densities
were well fit by simple power-laws with slopes ranging from 1.4 to 4.0, with
no significant difference between BL Lacs and FSRQs. These studies highlight
the diversity of the optical variability of blazars which may give insights
into the physical processes driving their emission.

The optical variability of FSRQs is a topic of much debate among researchers.
One viewpoint is that FSRQs, dominated by accretion disks, should exhibit
smaller optical variability than jet-dominated BL Lacs (Wiita, 2005).
Conversely, others suggest that FSRQs are a mixture of various types,
including optically violently variable quasars, highly polarized quasars,
core-dominated quasars, and superluminal motion sources, with some objects
exhibiting optical variation of up to 50$\%$ in one day ((Penston & Cannon,
1970); (Fan, 2005)). (Bonning et al., 2012) conducted observations on 9 FSRQs
and revealed that in multiple bands ($BVRJ$), the fractional variability
amplitude of 3C 273 was one magnitude lower than that of other FSRQs,
providing support for the aforementioned bipolarity.

In a study by (Bauer et al., 2009a), the statistical variability of blazars,
which included 276 FSRQs and 86 BL Lacs, and nearly 23000 quasars were
analyzed using data from the Palomar-QUEST Survey (Djorgovski et al., 2008).
The authors investigated the overall variability of BL Lacs and FSRQs using
the structure function and found that different objects exhibit varying
degrees of variability details on timescales up to a few years.

Utilizing the Tomo-e Gozen Northern Sky Transient Survey data, we have
investigated the optical variability of blazars in this work. Compared to the
previous work of (Bauer et al., 2009a), our study has a larger ample size
(with redshift) and a larger number of measurements per object, with an
average of $\sim$50 epochs per blazar compared to an average of 6 epochs in
(Bauer et al., 2009a). Moreover, we have used the latest catalog, the 4LAC
catalog (Ajello et al., 2020), to classify blazars.

In Section 2, we provide a concise description of the data used in this study.
Our investigation into the variability of blazars, utilizing the standard
deviation of repeated photometry and the structure function, is presented in
Section 3. In Section 4, we discuss the differences between this study and
previous studies and outline its limitations. Our conclusions are summarized
in Section 5. The calculations and interpretations presented in this work are
based on a cosmological model defined as a Flat Lambda Cold Dark Matter
($\Lambda$CDM) cosmology. Specifically, we adopt a flat universe assumption
with Hubble constant $H_{0}=70\,\text{kms}^{-1}\text{Mpc}^{-1}$ and a matter
density parameter $\Omega_{m}=0.3$.

##  2 Method

###  2.1 Tomo-e Gozen observation of Fermi Blazar

The Tomo-e Gozen Camera (Sako et al., 2018) is a wide-field camera which
deploys 84 CMOS sensors and covers about 30$\%$ of the entire focal plane (9
degree in diameter) of the 1.05-m Kiso Schmidt telescope (Takase et al.,
1977).The camera has a peak photoelectric conversion efficiency of 0.72 at a
wavelength of 500 nm; the efficiency falls to 0.36 at wavelengths of 380 and
710 nm.(Figure 1(e)). Since the CMOS sensor can read out the signal pixel by
pixel, the read out speed is fast — much more rapid than that of traditional
astronomical CCDs. We typically operate the camera in a high-speed mode,
reading out all 84 sensors at a rate of 2 frames per second.

In the Tomo-e Gozen project, most of the observation time is used for Northern
Sky Transient Survey (Tominaga et al. in preparation). The integration time
for a single image is 0.5 seconds, but we combine frames in groups of 12 (or
18 since August 14, 2020): each frame is stacked with 11 (or 17) other images
in its group after we have removed the maximum value at each pixel position to
avoid cosmic rays. The effective exposure time for our measurements is
therefore 6 seconds. In our all-sky survey mode, the area covered each night
is about 12,000 square degrees, and a sub-region with low atmospheric
extinction (above 40 degrees in elevation) is observed 2-3 times during the
night111Since December 5, 2019, we sometimes operate in a high-cadence survey
mode, covering $3000\,\mathrm{deg}^{2}$ more than 10 times every night at
intervals shorter than 1 hour, rather than the all-sky survey mode. By making
multiple observations of targets each night in this way, we are exploring
short-timescale transients such as supernovae and blazars.

To ensure the accuracy of photometric measurements in the Northern Sky
Transient Survey, we adopted SDSS standard stars given by (Ivezić et al.,
2007) and observed with the Tomo-e Gozen camera. Following the standard data
reduction procedures such as dark and bias subtraction and flat-field
correction, we performed aperture photometry using an aperture diameter of
16.6 arcseconds, compared to a typical Full Width at Half Maximum (FWHM) of 7
arcseconds. The magnitude zero point in our unfiltered images was calculated
using G-band data from the Gaia satellite DR2 (Gaia et al., 2018). We
evaluated the photometric uncertainty of the survey by determining the
standard deviation of repeated photometry across all observations for stars
measured at least 10 times, in the following manner. We initially select all
objects that are suitable for determining the magnitude zero point (zeromag),
based on various factors, such as their magnitude, parallax, and the absence
of other surrounding objects in the Pan-STARRS1 (Kaiser et al., 2002) and Gaia
catalogs. Subsequently, we calculate the zeromag for each object in a frame
and then use the median value as the overall zeromag of the frame. During this
process, the standard deviation in zeromag calculated for different objects is
approximately 0.03 mag. The zeromag error along with other errors (such as
flat error, etc.) results in a systematic error of 0.01 to 0.02 mag, and the
overall photometric uncertainty of the Northern Sky Transient Survey is
presented in Figure 1(a).

We chose as our initial source for candidate blazars the Fourth LAT AGN
Catalog (4LAC), derived from the first eight years of data collected by the
Fermi Gamma-ray Space Telescope within the energy range from 50 MeV to 1 TeV
(Ajello et al., 2020). The observational footprint of the Tomo-e Gozen camera
covers nearly 60$\%$ of the 4FGL survey’s area, and its detection capability
down to mag 18.5 is able to identify 44.6$\%$ of the blazar sources listed in
the 4LAC. Figure 1(b) shows the number of blazars observed and the histogram
of the number of detections of each blazar observed by the Northern Sky
Transient Survey from 08-31-2019 to 11-04-2022. Figure 1(c) depicts a
histogram of the time lag $\tau$ (days) in the rest frame between two
observations of the same blazar. The sparsity of observational data in the
$\tau\sim 100-250$ range can be attributed to the seasonal visibility of
objects, the climate of the Kiso Observatory, and instrument adjustments of
Tomo-e Gozen. Similarly, the scarcity of observational data in the $\tau>400$
range can be attributed to the limitation of the survey period. Figure 1(d),
which displays a histogram of the mean Tomo-e Gozen magnitude in repeated
photometry for BL Lac and FSRQ blazars. The distribution of magnitudes for BL
Lac blazars in this study ranges from magnitude 14-18, with a peak value of
approximately magnitude 17.3. Similarly, FSRQ blazars have magnitudes
distributed in the range of magnitude 15-18, with a peak value of
approximately magnitude 17.3.

Figure 1: (a): Photometric uncertainties of Tomo-e Gozen’s photometry for SDSS
standard stars. We obtained a total of 8,057,340 epochs of 283,605 SDSS
standard stars between MJD 58726 and 59869. To determine the photometric
uncertainty of Tomo-e Gozen at a given magnitude, we calculated the median of
the standard deviation of repeated photometry for each star with the same
r-band bin (0.01 mag) in the SDSS catalog; (b): The relationship between
number of measurements and number of objects. In this study we only used
blazars with more than 5 epochs; (c): Number of measurements pairs with time
lag $\tau$ (day) of FSRQs and BL Lacs. (Due to the difference in redshift
between BL Lacs and FSRQs, their $\tau$ distributions in the rest frame are
different.); (d): Mean Tomo-e Gozen magnitude histogram of FSRQs and BL Lacs.
(e): Bandpass of Tomo-e Gozen transparent window and Gaia G-band (Gaia et al.,
2018).

To facilitate the visual understanding of our analyzed dataset, we display
example light curves of four well-observed FSRQ objects in Figure 2.

Figure 2: Light curves of well observed 4 FSRQs

###  2.2 Sample selection $\&$ variability measurement

The luminosity relationship between blazars and their host galaxies has a
significant impact on our study. When the luminosity of the blazar is not
greater than that of the host galaxy, the continuous spectrum emitted by the
galaxy can overwhelm the emission lines from the AGN core of the blazars. This
phenomenon can also lead to an increase in the radiation photon index from the
jet, resulting in a redder spectrum. Consequently, this overlap in spectral
features can lead to the misclassification of certain FSRQs as members of the
BL Lac class, thereby introducing ambiguity and potential errors.

The assessment of variability of blazars is also complicated by the presence
of their host galaxies. First, the optical emission from the host galaxy
merges with that from the blazar, making it challenging to isolate the
blazar’s variability. This is also particularly notable when the blazar’s
luminosity is not significantly higher than that of the host galaxy, leading
to potential underestimation of the blazar’s relative variability. Second,
since we used a fixed photometric aperture size (16.6 arcseconds in diameter),
variations in seeing can cause variations of the galaxy flux within the
aperture, possibly resulting in an overestimation of the blazar’s absolute
variability.

In order to minimize the potential for errors in our analysis, we attempt to
select objects in which the luminosity contributed by the central engines
(blazars) in our sample will be greater than the luminosity contributed by
their host galaxies. A previous study (Urry et al., 2000) employing the Hubble
Space Telescope to examine the host galaxies of BL Lac objects, spanning
redshifts from 0.1 to 1.3, revealed an absolute magnitude for these galaxies
of $M_{B}=-21.4\pm 0.6$ AB mag (after adjustments for bandpass, Hubble
constant, and conversion from Vega to AB system). Hence the bright end of
blazar host galaxies within a 2$\sigma$ range is $M_{B}=-22.6$ AB mag. This
finding is consistent with bright end of the luminosity function of galaxies,
as reported by the SDSS survey (Blanton et al., 2003), which is around -22.8
AB mag (in the g-band). Consequently, we will exclude candidates whose optical
luminosity $L_{O}$ (including blazar and host galaxy) is lower than the
K-corrected optical luminosities of any hypothetical host galaxy type
(including elliptical, Sc, Sa, and starburst galaxies) corresponding to an
absolute magnitude of $M_{B}=$ -23 AB magnitudes. We present in Figure 3 the
K-corrected optical luminosity,$L_{g}$, of different types of hypothetical
galaxies as a reference for galaxy evolution at various redshifts; we define
$L_{g}$ as

$L_{g}=\frac{L_{g}^{\prime}}{k_{g}(z)}$ (1)

where $L_{g}^{\prime}$ is the optical luminosity of galaxies with absolute
magnitude $M_{B}=$-23 AB mag before K-correction and $k_{g}(z)$ is the
K-correction factor for galaxies with

$k_{g}(z)=(1+z)\frac{\int_{\nu_{1g}}^{\nu_{2g}}\frac{f_{\nu}(\nu)}{\nu}R_{g}(\nu)d\nu}{\int_{(1+z)\nu_{1B}}^{(1+z)\nu_{2B}}\frac{f_{\nu}(\nu)}{\nu}R_{B}(\nu)d\nu}$
(2)

where $\nu_{1g}$ and $\nu_{2g}$ are the starting and ending frequencies,
respectively, for the Gaia G-band, and $R_{g}(\nu)$ represents its quantum
efficiency. Similarly, $\nu_{1B}$ and $\nu_{2B}$ are designated as the initial
and terminal frequencies for the Gaia B-band, with $R_{B}(\nu)$ denoting its
quantum efficiency. $f_{\nu}(\nu)$ is the spectral density of flux of each
galaxy template provided by Kinney et al. (1996).

###  2.3 Optical luminosity of Blazers

The optical luminosity of Blazars without K correction, $L_{O}^{\prime}$, was
calculated using redshift data from the 4LAC database, average magnitude data
from the Tomo-e Gozen survey. and the
$astropy.cosmology.luminosity\\_distance$ method (Robitaille et al., 2013).
Power-law fitting was performed on the flux of each object in the Pan-STARRS
g, r, i bands (Kaiser et al., 2002) to determine its photon index $\Gamma_{O}$
in the optical band, which was then used to perform K-correction and yield the
optical luminosity $L_{O}$ in the following manner:

$L_{O}=\frac{L_{O}^{\prime}}{k(z)}$ (3)

where

$k(z)=(1+z)\frac{\int_{\nu_{1t}}^{\nu_{2t}}\nu^{-\Gamma_{O}}R_{t}(\nu)d\nu}{\int_{(1+z)\nu_{1t}}^{(1+z)\nu_{2t}}\nu^{-\Gamma_{O}}R_{t}(\nu)d\nu}$
(4)

Here, $\nu_{1t}$ and $\nu_{2t}$ are the starting and ending frequencies of the
Tomo-e Gozen band and $R_{t}(\nu)$ represents Tomo-e Gozen’s quantum
efficiency (see Figure 1(e)).

In order to assess the variability of blazars, we first calculated the
standard deviation in repeated photometry (light curve). To obtain the
intrinsic standard deviation $\sigma$ of blazars, we subtracted the
photometric uncertainty, which is shown in Figure 1(a), from this initial
standard deviation $\sigma^{\prime}$.

$\sigma=\sqrt{\sigma^{\prime 2}-magerr(mag)^{2}}$ (5)

Figure 3: The distribution of K-corrected optical luminosity $L_{O}$, plotted
against redshift for FSRQs and BL Lacs. The color of each symbol corresponds
to the standard deviation of the repeated photometry (light curve). We have
excluded blazars with $L_{O}$ below any of the $L_{g}$ lines, which represent
the K-corrected optical luminosity of differents types of hypothetical
galaxies with absolute magnitude of $M_{B}=$-23 AB mag. The symbols of the
excluded blazers were blurred.

A total of 241 BL Lacs and 83 FSRQs were included in the original sample. To
ensure accurate observations of optical luminosity and avoid interference from
the stellar component of host galaxies, our study excluded blazars located
below any of the lines in Figure 3 and made the final sample of 120 BL Lacs
and 78 FSRQs for more detailed investigation.

##  3 Result & Discussion

###  3.1 Standard deviation of repeated photometry

The distribution of photon indices of $\gamma$-ray222The $\gamma$-ray photon
index $\Gamma_{\gamma}$ taken from 4LAC is evaluated by fitting a sub-
exponentially cutoff power law $dN/dE\propto
E^{-\Gamma}\exp(a(E_{0}^{b}-E^{b}))$, where $a$, $b$ and $E_{0}$ are fitting
parameters. and optical wavelengths333The optical photon index $\Gamma_{o}$ is
defined as $dN/dE\propto E^{-\Gamma}$ or $f_{\nu}\propto\nu^{1-\Gamma}$, where
$N$ is the number of photons of the Pan-STARRS g, r, i bands (Kaiser et al.,
2002), $E$ is the photon energy, and $f_{\nu}$ is the flux density at the
frequency $\nu$., and their relationship to the standard deviation of repeated
photometry, are shown in Figure 4.

Figure 4: The relationship between the $\gamma$-ray photon indices (x-axis)
and the optical photon indices (y-axis) is depicted for FSRQs (left panel) and
BL Lacs (right panel). The top inset shows the histogram of the $\gamma$-ray
photon indices, and the right inset displays the optical photon indices, with
FSRQs indicated in blue and BL Lacs in red. The color intensity of each symbol
reflects the standard deviation of the optical repeated photometry (light
curve) for the respective sources.

The distribution of optical photon indices for BL Lacs and FSRQs, while
overlapping in range, demonstrates a significant difference with a KS-test
p-value of 2.6e-5, possibly indicating multiple emission mechanisms. The peak
of the optical photon index distribution for FSRQs is notably lower, which may
be attributed to the bluer color of their accretion disks. Conversely, the
$\gamma$-ray photon index distribution shows a pronounced difference between
the two classes, with FSRQs typically exhibiting higher values than BL Lacs,
as indicated by a KS-test p-value of 6.6e-31. This disparity is likely due to
the differences in the peak of the inverse Compton scattering between the two.
The photon index of BL Lacs at $\gamma$-ray and optical wavelengths is
positively correlated, whereas those of FSRQs display an anti-correlation.
Furthermore, FSRQs with a $\gamma$-ray photon index greater than $\sim$2.6
(highlighted by the yellow region in the left panel of Figure 4) present a
smaller standard deviation in repeated photometry measurements compared to
those with a lower $\gamma$-ray photon index. Further discussion of this trend
is presented in section 3.2.

In Figure 5, we present a histogram that shows the standard deviation observed
in repeated photometric measurements of blazars, specifically focusing on
those observed more than five times. Our analysis reveals a notable
distinction in the variability patterns of BL Lacs and FSRQs. The standard
deviation for BL Lacs predominantly clusters within the 0.0 to 0.2 magnitude
range. Conversely, the standard deviation for FSRQs demonstrates a more
uniform distribution across different magnitude bins.

Further scrutiny, employing the Kolmogorov-Smirnov (KS) test, accentuates the
disparity in the distribution patterns of FSRQs based on their gamma-ray
photon indices. FSRQs characterized by high gamma-ray photon indices
(exceeding 2.6) manifest a statistically significant divergence in their
distribution when compared to two groups: FSRQs with lower gamma-ray photon
indices (resulting in a p-value of 3.8E-6 ) and the general population of
FSRQs (yielding a p-value of 2.3E-3 ).

Figure 5: Distribution of standard deviation of repeated photometry of blazars
which were observed more than 5 times. The blue line represents the overall
FSRQs, the red line represents the FSRQs with high $\gamma$-ray photon indices
(greater than $>2.6$), and the green line represents BL Lacs.

Figure 2 provides detailed illustrations of these relationships for several
well-observed objects. The FSRQ 3C232, which has a high $\gamma$-ray photon
index, displays notable stability throughout our nearly three-year observation
period. Another largely quiescent object is 4C+21.35, which has a relatively
low $\gamma$-ray photon index; it exhibits slight fluctuations during the
initial stage of our observations (before MJD 59000) before becoming quiet in
subsequent observations. On the other hand, OI275 and Ton599, both of which
have relatively low $\gamma$-ray photon indices, display dramatic variations
during our observation period. The maximum optical luminosity of these two
objects can exceed the minimum optical luminosity by more than ten times.

Figure 6 presents the relationship between redshift and K-corrected444Similar
to the K-correction of galaxies and optical band but the spectral density of
flux $f_{\nu}(\nu)$ will be $\nu^{1-\Gamma_{\gamma}}$ where $\Gamma_{\gamma}$
is the $\gamma$-ray photon indices from 4LAC catalog $\gamma$-ray luminosity
($L_{\gamma}$) for FSRQs and BL Lacs. Despite the inevitable selection bias in
the data due to sensitivity limitations and volume effects, the limited
dataset still reveals a noticeable trend: blazars with lower optical
variability (reddish symbols) tend to cluster in regions of lower $\gamma$-ray
luminosity ($L_{\gamma}$) across different cosmological epochs.

Figure 6: The distribution of K-corrected $L_{\gamma}$ versus redshift of
FSRQs and BL Lacs. The color of the symbol stands for the standard deviation
of the repeated photometry (light curve).

The analysis of blazars by (Ghisellini et al., 2017) primarily utilized
$L_{\gamma}$ as the distinguishing characteristic. While this approach offers
a straightforward method of categorization, it could overlook a critical
aspect: the potential diversity in physical properties of blazars at different
redshifts, despite having similar $L_{\gamma}$. This oversight is significant,
considering that the properties and surrounding environment of an AGN are
intricately linked to its redshift. Consequently, relying solely on
$L_{\gamma}$ for classification, without considering the redshift, might lead
to a skewed understanding of blazar properties.

To quantitatively analyze the relationship between luminosity and redshift, we
introduce a new metric: the relative luminosity index ($R_{L}$).

$R_{L}(L_{\gamma},z)=log(\frac{L_{\gamma}}{4\pi k(z)D_{L}^{2}})+12$ (6)

where the term $k(z)$ represents the unified K-correction
coefficient555Similar to the K-correction of galaxies and optical band but the
spectral density of flux $f_{\nu}(\nu)$ will be $\nu^{1-\Gamma}$ where
$\Gamma$ is the average photon index specific to each blazar type; this
equates to 2.48 for FSRQs and 2.00 for BL Lacs. and $D_{L}$ denotes the
luminosity distance, which is also dependent on the redshift.

Figure 7 delineates the correlation between the optical variability of FSRQs
and BL Lacs and their respective $R_{L}$ values. The red solid line in the
figure indicates a linear regression fit to the data. The optical variability
of both FSRQs and BL Lacs shows a positive correlation with their relative
luminosity index ($R_{L}$). For FSRQs, this correlation is weak and not
statistically significant, with a Pearson correlation coefficient of r=0.19
and a confidence level of p=0.09. In contrast, for BL Lacs, the correlation is
moderate and statistically significant, with a Pearson correlation coefficient
of r=0.46 and a confidence level of p=1.6e-7.

Figure 7: The distribution of relative luminosity index ($R_{L}$) versus the
optical variability of FSRQs and BL Lacs. The red solid line in the figure
indicates the linear regression fit to the data. The color of the symbol
stands for the $\gamma$-ray photon indices.

This evidence for a difference between FSRQs and BL Lacs led us to look
further into the relationship between the relative luminosity index ($R_{L}$)
and the dual-band spectral index. For a balanced and methodical analysis, we
categorized FSRQs into three distinct groups according to their $R_{L}$
values, with the group boundaries set at log7 and log20. This stratification
was designed to ensure an approximately equal number of FSRQs in each
category. The same criteria for group classification were applied to BL Lacs,
thereby maintaining consistency and comparability in our analytical approach.
The FSRQs shown in Figure 2 are located in different zones. Specifically,
3C232 is located in zone 1, while OI275 is in zone 2. On the other hand,
4C+21.35 and Ton599 are both in zone 3.

Figure 8: The distribution of the $\gamma$-ray photon index and the optical
photon index of FSRQs (left) and BL Lacs (right) for each zone. (The kernel
density estimate plot was generated using the sns.kdeplot function from the
Seaborn data visualization library (version 0.12) in Python.)

Figure 8 illustrates the distribution of $\gamma$-ray photon index and optical
photon index for FSRQs and BL Lacs in each zone. Zone 1 FSRQs dominate the
high $\gamma$-ray photon index group, suggesting that these FSRQs tend to have
lower $L_{\gamma}$ compared to other FSRQs. Conversely, the low $\gamma$-ray
photon index group does not appear to be dominated by any specific zones of
FSRQ, as all three zones exhibit similar tendencies. The optical photon index
of the high $\gamma$-ray photon index FSRQs is concentrated around 1.5, which
is the typical color of a quasar accretion disk (Kishimoto et al., 2008). This
suggests that FSRQs dominated by accretion disks differ significantly from
those dominated by jets in terms of optical variability and $\gamma$-ray
photon indices. Notably, zone 2 FSRQs also have optical photon indices
concentrated around 1.5, despite their relatively scattered $\gamma$-ray
photon indices, while zone 3 FSRQs have a relatively concentrated $\gamma$-ray
photon indices ($\sim$ 2.3) but have relatively scattered optical photon
indices.

###  3.2 Structure function

In addition to the standard deviation of repeated photometry, which does not
involve the rest frame time interval $\tau$, we also studied the structure
function, following the method by (Berk et al., 2004). The structure function
can be defined as

$\mathrm{SF}(\tau)=\sqrt{\left\langle[m(t)-m(t-\tau)]^{2}\right\rangle-\left\langle\sigma_{SF}^{2}\right\rangle}$
(7)

where $m(t)$ and $m(t-\tau)$ are magnitudes of two measurements of the same
object, separated by (rest frame) time $\tau$, and $\sigma_{SF}$ is
uncertainty of the measurements given by (8), where $magerr(t)$ represents the
blue line in Figure 1(a).

$\sigma_{SF}=\sqrt{magerr(t)^{2}+magerr(t-\tau)^{2}}$ (8)

The optical variability of an object can be quantified using the structure
function, which depends on the rest frame time interval $\tau$. This function
provides insight into the temporal characteristics of the object’s
variability. A change in the slope of the structure function when plotted
again $\tau$ may signify a significant timescale relevant to understanding the
underlying mechanism of optical luminosity variation. To construct the
structure function for a single object, a large number of observation epochs
that are widely distributed in time are required. However, due to limited data
availability, we constructed structure functions for a class of blazars as a
group, combining measurements from objects with similar properties.

Figure 9 illustrates the structure functions of FSRQs and BL Lacs. In the
process of constructing the structure function for a specific class of
blazars, either BL Lacs or FSRQs, a comprehensive approach was adopted to
extract and analyze the observational data. The initial step involved the
extraction of all available data pairs (magnitudes of one object at two
epochs) from the selected blazar class, recording their respective rest frame
time lag $\tau$ and structure function values. This extensive data set was
then combined into a unified set, encompassing observations from various
sources.

To facilitate a detailed analysis, the combined data pairs were organized in
ascending order based on their $\tau$ values. The ordered data set was then
equally divided into 20 groups to ensure a balanced representation across
varying lag intervals. For each of these groups, a representative data point
was generated to populate the structure function graph. The x-coordinate of
these data points, indicating the lag value, was calculated as the mean of all
lag values within the respective group. Within each group, we computed two
y-coordinates: one for the 50th percentile of the SF values, the other for the
75th percentile of the SF values.

Figure 9: Structure function of FSRQs (brown; 10966 data pairs for each point)
and BL Lacs (green; 34015 data pairs for each point), and previous results
from (Bauer et al., 2009a) and (Bauer et al., 2009b). The rest frame interval
$\tau$ is in unit of day.

In the rest-time time interval $\tau$ spanning 10 to 100 days, it is evident
that the median (50th percentile) variability of Flat Spectrum Radio Quasars
(FSRQs) is lower than that of BL Lacertae objects (BL Lacs), while their 75th
percentile variability exceeds that of BL Lacs. This observation suggests that
FSRQs exhibit greater internal heterogeneity than BL Lacs. Despite this
difference, it’s important to note that both BL Lacs and FSRQs have a similar
smooth pattern in the intraday timescale. Additionally, contrary to the
findings of Emmanoulopoulos and Bauer (2009), no distinct trough in
variability was observed in the 70-250 day range. We attribute this
discrepancy to the expanded dataset now available and the refined criteria for
sample selection in our study.

We also performed further analysis on FSRQs grouped by their $\gamma$-ray
photon index, as mentioned in Section 3.1; see Figure 10.

Figure 10: Structure function of low $\gamma$-ray photon index FSRQs (blue;
7689 data pairs for each point), BL Lacs (green; 34015 data pairs for each
point) and high $\gamma$-ray photon index FSRQs (red; 3278 data pairs for each
point).

In this comparative analysis, the amplitude of the structure function for high
$\gamma$-ray photon index FSRQs stands out for its significantly lower value
when compared with both BL Lacs and the low $\gamma$-ray photon index FSRQs.
Notably, the structure function of high $\gamma$-ray photon index FSRQs shows
substantial variability only beyond a 200-day timescale. This observation
aligns with the hypothesis that the variability in these FSRQs resembles that
of accretion disks, which are characterized by timescales over 100 days (Burke
et al., 2021). In contrast, for low $\gamma$-ray photon index FSRQs, optical
variability within a shorter span, particularly under 10 days, is likely
attributed to jet components ((Penston & Cannon, 1970); (Fan, 2005)).
Furthermore, figure 10 reveals that low $\gamma$-ray photon index FSRQs
(sometimes thought to be dominated by an accretion disk) exhibit more
significant variability than the jet-dominated BL Lacs, particularly within
the 10- to 100-day range (KS-test p$<$1E-16). This finding suggests a complex
role of accretion disks and jets in these sources, pointing towards the
necessity of further spectroscopic observations to determine the dominant
factor driving these changes.

##  4 Discussion

Based on our analysis, we infer that the optical component of FSRQs with high
$\gamma$-ray photon index is dominated by the accretion disk. This hypothesis
is based on several observational results: their low optical variability,
their lower $L_{\gamma}$ compared with other FSRQs in the same cosmic period,
and their accretion-disk-like characteristic timescale. These findings are
consistent with the results of (Shaw et al., 2012), who reported a weak
correlation between non-thermal dominance of FSRQs and low photon indices in
Fermi observations. Unfortunately, due to the limited sample size, our study
only separated FSRQs by $\gamma$-ray photon index into two groups. Our result
is different from the result of previous studies (Ghisellini et al., 2017)
which only detected accretion disc components in spectra (from ASI
Astrophysical Data Center) of FSRQs at high redshift, while FSRQs at low
redshift rarely showed accretion disc components in the optical band. Their
result may be attributed to the accretion disc component becoming more
prominent at larger redshifts, thus at higher luminosities. However our study
provides the first confirmation of accretion disk-dominated FSRQs at low
redshifts through their low optical variability and long characteristic
timescale. It remains unclear whether there is a continuous relationship
between the $\gamma$-ray photon index and the dominance of the accretion disk,
or if the high accretion-disk dominance ratio only appears in the high
$\gamma$-ray photon index region.

(Ghisellini, 2016) argued that the IR-optical band SED of FSRQs with low
$L_{\gamma}$ contains smaller accretion-disk components than other FSRQs, and
the flux at optical wavelengths is dominated by non-thermal radiation of the
jet, which differs from the results of this study. The main reason for this
difference may be our different blazar selection and grouping approach.
(Ghisellini, 2016) grouped the FSRQs only according to their $L_{\gamma}$,
while in this study, we considered both the $L_{\gamma}$ and redshift of the
blazar simultaneously. The reason for this approach is that we believe the
nature of blazars with the same $L_{\gamma}$ is not precisely identical but
closely linked to their cosmic epoch.

It is important to acknowledge that the $\gamma$-ray photon index utilized in
our study are based on the Fermi/LAT eight-year integration, which represents
an average of the blazar $\gamma$-ray spectrum over an eight-year period.
Conversely, the optical spectrum index utilized in our study is based on the
result of short-term observations by Pan-STARRS1, indicating that our two
photon indices are not synchronized. Consequently, caution must be exercised
in interpreting the results of our study, particularly with regards to the
correlation between the $\gamma$-ray and optical photon index properties.

Also, according to the blazar sequence given by (Ghisellini et al., 2017), for
FSRQs and BL Lacs at high redshift, the frequency observed by Tomo-e Gozen
(about $10^{14}$ Hz) is higher than their synchrotron radiation peak $\nu_{s}$
(about $10^{12}$ Hz), while for BL Lacs at low redshift, the frequency
observed by Tomo-e Gozen is higher than their synchrotron radiation peak
$\nu_{s}$ (about $10^{16}$ Hz). This redshift dependence may lead to
differences in observed optical variability. However, considering that the BL
Lacs in the sample have roughly the same redshift distribution as the FSRQs,
and the standard deviation of repeated photometry of both FSRQs and BL Lacs
did not appear to be redshift-dependent, we conclude that this phenomenon does
not significantly affect our results.

##  5 Conclusion

We conducted a study of the optical variability of blazars selected from the
4LAC catalog, using observations from the Tomo-e Gozen Northern Sky Transient
Survey, covering the period from MJD 58726 to 59887. Our high-speed sky survey
allowed us to observe 324 blazars, with an average of 50 epochs per blazar,
representing a significant increase compared to the previous study by (Bauer
et al., 2009a), which had an average of only six epochs per blazar.
Furthermore, the catalog used in our study was updated, which provided
additional information not present in the previous study.

We focused exclusively on blazars with an optical luminosity $L_{O}$ higher
than that of hypothetical galaxies with absolute magnitudes of $M_{B}$=-23 (AB
magnitude). This criterion was applied in order to enhance the probability of
the correct spectroscopic classification of low-luminosity blazars and
accurately calculate their optical variability.

Our study shows that there is no significant difference in the optical
variability of blazars at different cosmological epochs, despite the orders of
magnitude difference in their luminosity. This finding indicates that the
optical variability of blazars is not directly dependent on their luminosity,
nor on their redshift. As we considered the redshift dependence and
$L_{\gamma}$ dependence simultaneously in our analysis, we found that blazars
with low $L_{\gamma}$ at a given cosmic epoch tended to have lower optical
variability.

The $\gamma$-ray photon index of FSRQs is typically higher than that of BL
Lacs. We found that the photon index of BL Lacs is positively correlated at
$\gamma$-ray and optical wavelengths, while the that of FSRQs is anti-
correlated. Also, FSRQs with high $\gamma$-ray photon index tend to have lower
optical variability, especially those with $\gamma$-ray photon index exceeding
2.6. Moreover, the distribution of standard deviation in repeated photometry
of FSRQs with $\gamma$-ray photon index greater than 2.6 is significantly
different from that of FSRQs with $\gamma$-ray photon index less than 2.6. The
difference between these two distributions is statistically significant with a
KS test P-value equal to 2.6e-6.

FSRQs with high $\gamma$-ray photon index (greater than 2.6) typically exhibit
low optical variability. From an evolutionary perspective, these FSRQs are
often found in regions of low $L_{\gamma}$ at a cosmic epoch, a pattern that
aligns with what is observed in disk-dominated blazars. While their
characteristic timescales are not immediately apparent from our observation
alone, further observations and analysis will, we hope provide more insight.

When examining the structure functions of these FSRQs in comparison to those
of BL Lacs, a clear distinction emerges: FSRQs generally show higher
amplitudes in their structure functions. Specifically, FSRQs with high
$\gamma$-ray photon index have significantly lower amplitudes compared to
their low-index counterparts. This finding is intriguing as these high-index
FSRQs also display noticeable variability, predominantly on timescales
extending beyond 100 days. This pattern of variability, emerging from the
analysis of structure functions, suggests a resemblance to the longer
timescales typically associated with accretion disks. In contrast, FSRQs with
low $\gamma$-ray photon index exhibit variability on much shorter timescales,
often less than ten days, likely indicative of jet activity. Moreover, these
low $\gamma$-ray photon index FSRQs demonstrate even greater variability than
jet-dominated BL Lacs, not only on timescales less than ten days but also
particularly within the 10 to 100-day range (KS-test p$<$1E-16), pointing
towards an area worth further exploration to determine the underlying causes.
Despite these differences, it’s important to note that the structure functions
of both BL Lac and FSRQs consistently exhibit smooth characteristics at the
intraday scale.

We are initiating spectral studies of FSRQs, particularly focusing on emission
line equivalent widths, to decipher the roles of accretion disks and jets. In
parallel, we will continue our engagement with the Tomo-e Gozen Northern Sky
Transient Survey to enrich our dataset. This sustained effort is expected to
enhance our understanding of blazar luminosity variability and further
illuminate the complex mechanisms driving these FSRQs.

This work has been supported by the Japan Society for the Promotion of Science
(JSPS) KAKENHI grants, 24K17097, 23H04894, 21H04491, 18H05223, 18H01272,
18H01261, 18K13599, 17H06363, 16H06341, 16H02158, 26247074, and 25103502. This
work is supported in part by the Optical and Near-Infrared Astronomy Inter-
University Cooperation Program (OISTER), the Ministry of Education, Culture,
Sports, Science and Technology (MEXT) of Japan. We acknowledge the use of data
from the Fermi Gamma-ray Space Telescope/Large Area Telescope (Fermi/LAT) the
Sloan Digital Sky Survey (SDSS), the Panoramic Survey Telescope and Rapid
Response System (Pan-STARRS), and the Gaia mission. Fermi/LAT is a
collaboration between NASA and the Department of Energy in the United States,
and is supported by institutions in France, Italy, Japan, and Sweden. This
work was supported in part by Japan Foundation for Promotion of Astronomy.
SDSS is managed by the Astrophysical Research Consortium for the Participating
Institutions, funding for the SDSS and SDSS-II has been provided by the Alfred
P. Sloan Foundation, the Participating Institutions, the National Science
Foundation, the US Department of Energy, the National Aeronautics and Space
Administration, the MEXT, the Max Planck Society, and the Higher Education
Funding Council for England. Pan-STARRS is supported by the National
Aeronautics and Space Administration (NASA) under Grant No. NNX12AR65G and
Grant No. NNX14AM74G, and the National Science Foundation (NSF) under Grant
No. AST-1238877. The Gaia mission is a project of the European Space Agency
(ESA) and funded by ESA member states and the European Union. Finally I would
like to express my sincere gratitude to Professor Michael Richmond from the
Rochester Institute of Technology for his invaluable guidance and suggestions
during the final stages of this work. His expertise and insightful feedback
greatly contributed to the overall clarity of this paper. We are truly
grateful for his time, patience, and dedication in reviewing this work and
providing valuable recommendations. His mentorship and support have been
instrumental in shaping the final outcome of this paper.

## References

  * Acciari et al. (2021) Acciari, V., Ansoldi, S., Antonelli, L., et al. 2021, Monthly Notices of the Royal Astronomical Society, 504, 1427
  * Ajello et al. (2020) Ajello, M., Angioni, R., Axelsson, M., et al. 2020, The Astrophysical Journal, 892, 105
  * Bachev et al. (2012) Bachev, R., Semkov, E., Strigachev, A., et al. 2012, Monthly Notices of the Royal Astronomical Society, 424, 2625
  * Bauer et al. (2009a) Bauer, A., Baltay, C., Coppi, P., et al. 2009a, The Astrophysical Journal, 699, 1732
  * Bauer et al. (2009b) —. 2009b, The Astrophysical Journal, 696, 1241
  * Berk et al. (2004) Berk, D. E. V., Wilhite, B. C., Kron, R. G., et al. 2004, The Astrophysical Journal, 601, 692
  * Blandford & Königl (1979) Blandford, R., & Königl, A. 1979, The Astrophysical Journal, 232, 34
  * Blanton et al. (2003) Blanton, M. R., Hogg, D. W., Bahcall, N. A., et al. 2003, The Astrophysical Journal, 592, 819
  * Bonning et al. (2012) Bonning, E., Urry, C. M., Bailyn, C., et al. 2012, The Astrophysical Journal, 756, 13
  * Burke et al. (2021) Burke, C. J., Shen, Y., Blaes, O., et al. 2021, Science, 373, 789
  * Clements et al. (2003) Clements, S. D., Jenks, A., & Torres, Y. 2003, The Astronomical Journal, 126, 37
  * Djorgovski et al. (2008) Djorgovski, S., Baltay, C., Mahabal, A., et al. 2008, Astronomische Nachrichten: Astronomical Notes, 329, 263
  * Fan (2005) Fan, J.-H. 2005, Chinese Journal of Astronomy and Astrophysics, 5, 213
  * Fanaroff & Riley (1974) Fanaroff, B. L., & Riley, J. M. 1974, Monthly Notices of the Royal Astronomical Society, 167, 31P
  * Gaia et al. (2018) Gaia, C., Brown, A., Vallenari, A., et al. 2018, Astronomy & Astrophysics, 616
  * Ghisellini (2016) Ghisellini, G. 2016, Galaxies, 4, 36
  * Ghisellini et al. (2017) Ghisellini, G., Righi, C., Costamante, L., & Tavecchio, F. 2017, Monthly Notices of the Royal Astronomical Society, 469, 255
  * Giommi et al. (2012) Giommi, P., Padovani, P., Polenta, G., et al. 2012, Monthly Notices of the Royal Astronomical Society, 420, 2899
  * Goyal (2021) Goyal, A. 2021, The Astrophysical Journal, 909, 39
  * Ivezić et al. (2007) Ivezić, Ž., Smith, J. A., Miknaitis, G., et al. 2007, The Astronomical Journal, 134, 973
  * Kaiser et al. (2002) Kaiser, N., Aussel, H., Burke, B. E., et al. 2002, in Survey and Other Telescope Technologies and Discoveries, Vol. 4836, International Society for Optics and Photonics, 154–164
  * Kaur et al. (2021) Kaur, A., Falcone, A. D., & Stroh, M. C. 2021, The Astrophysical Journal, 908, 177
  * Kerby et al. (2021) Kerby, S., Kaur, A., Falcone, A. D., et al. 2021, The Astronomical Journal, 161, 154
  * Kinney et al. (1996) Kinney, A. L., Calzetti, D., Bohlin, R. C., et al. 1996, The astrophysical journal. Chicago. Vol. 467, no. 1, pt. 1 (Aug. 1996), p. 38-60
  * Kishimoto et al. (2008) Kishimoto, M., Antonucci, R., Blaes, O., et al. 2008, Nature, 454, 492
  * Konigl (1981) Konigl, A. 1981, The Astrophysical Journal, 243, 700
  * Landi et al. (2015) Landi, R., Bassani, L., Stephen, J. B., et al. 2015, Astronomy & Astrophysics, 581, A57
  * Pandey et al. (2020) Pandey, A., Gupta, A. C., Kurtanidze, S. O., et al. 2020, The Astrophysical Journal, 890, 72
  * Penston & Cannon (1970) Penston, M., & Cannon, R. 1970, Royal Greenwich Observatory Bulletins, 159, 83
  * Rani et al. (2013) Rani, B., Krichbaum, T., Fuhrmann, L., et al. 2013, Astronomy & Astrophysics, 552, A11
  * Richards et al. (2011) Richards, J. L., Max-Moerbeck, W., Pavlidou, V., et al. 2011, The Astrophysical Journal Supplement Series, 194, 29
  * Robitaille et al. (2013) Robitaille, T. P., Tollerud, E. J., Greenfield, P., et al. 2013, Astronomy & Astrophysics, 558, A33
  * Sako et al. (2018) Sako, S., Ohsawa, R., Takahashi, H., et al. 2018, in Ground-based and Airborne Instrumentation for Astronomy VII, Vol. 10702, International Society for Optics and Photonics, 107020J
  * Shaw et al. (2012) Shaw, M. S., Romani, R. W., Cotter, G., et al. 2012, The Astrophysical Journal, 748, 49
  * Takase et al. (1977) Takase, B., Ishida, K., Shimizu, M., et al. 1977, Annals of the Tokyo Astronomical Observatory, 16, 74
  * Urry et al. (2000) Urry, C. M., Scarpa, R., O’Dowd, M., et al. 2000, The Astrophysical Journal, 532, 816
  * Wiita (2005) Wiita, P. J. 2005, arXiv preprint astro-ph/0507141

##  Appendix A The biases from the host galaxy subtraction

Figure 11 shows the selection bias that may be introduced after the Data
selection of section 2.2

Figure 11: Correlation between the $\gamma$-ray and optical photon indices for
FSRQs and BL Lacs. The semi-transparent circular markers denote the FSRQs and
BL Lacs included in this analysis, whereas the opaque star-shaped markers
indicate those FSRQs and BL Lacs excluded due to potential contamination from
host galaxy luminosity, which could lead to underestimation of variability
(see section 2.2). The color coding of each symbol corresponds to the standard
deviation of the optical photometry (light curve) measurements.

It is evident from the figure that the photon index distribution of the
excluded BL Lacs and FSRQs generally falls within the range of those utilized
in this study. However, the mean optical photon index of these excluded BL
Lacs and FSRQs is higher than that of the BL Lacs and FSRQs included at the
same $\gamma$-ray photon indices. This provides substantial evidence that the
optical bands of these blazars are likely contaminated by their host galaxies,
resulting in a redder appearance compared to uncontaminated blazars. Several
of the excluded BL Lacs exhibit $\gamma$-ray photon indices higher than those
of the BL Lacs included in this study. Given that the excluded BL Lacs tend to
have lower redshifts, and considering the pronounced evolution of blazars (see
Fig. 6), it is entirely plausible that low-redshift BL Lacs have weaker jets
and redder $\gamma$-ray spectra. Excluding these BL Lacs could potentially
introduce a selection bias; however, as discussed in Section 2.2,
distinguishing between the blazar component and the host galaxy contribution
in optical band, as well as attributing their variability to either the blazar
or the host galaxy, is not feasible. Given that these BL Lacs constitute less
than three percent of the total BL Lac population, we believe their exclusion
does not introduce a significant selection bias to the final conclusions.

##  Appendix B The biases from the standard deviation subtraction

Figure 12: a: Correlation between the mean mag and standard deviation for
FSRQs and BL Lacs. b: Histogram of the mean mag of FSRQs with different
$\gamma$-ray photon indices.

The limitations imposed by the systematic error in standard deviation
subtraction are acknowledged, particularly in the context of darker objects
where this method may lead to an underestimation of standard deviation.
Despite this potential bias, our analysis indicates that this selection error
remains consistent across various FSRQs with different $\gamma$-ray photon
indices. This uniformity is evidenced by the similarity in the rank
distribution of FSRQs with high $\gamma$-ray photon indices compared to those
with standard $\gamma$-ray photon indices, as demonstrated by a KS-test
yielding a p-value of 0.21. Consequently, we infer that the observed low
variability in FSRQs with high $\gamma$-ray photon indices is not a result of
the standard deviation subtraction process.

Also in our analysis, we have specifically plotted the structure function for
blazars exhibiting an average Tomo-e Gozen magnitude brighter than 17.5, as
depicted in Figure 13. The observed behavior of the structure function in this
figure closely mirrors that presented in Figure 10. Notably, the similarity in
the patterns between these two figures substantiates the assertion that the
biases potentially introduced by the standard deviation subtraction do not
significantly impact our results.

Figure 13: Structure function of low $\gamma$-ray photon indices FSRQs with
average Tomo-e Gozen magnitude brighter than 17.5 (blue; 7504 data pairs for
each point), BL Lacs with average Tomo-e Gozen magnitude brighter than 17.5
(green; 32442 data pairs for each point) and high $\gamma$-ray photon indices
FSRQs with average Tomo-e Gozen magnitude brighter than 17.5 (red; 2953 data
pairs for each point).