# Analysis of the intra-night variability of BL Lacertae during its August
2020 flare

A. Agarwal Raman Research Institute, C. V. Raman Avenue, Sadashivanagar,
Bengaluru – 560 080, India  B. Mihov Institute of Astronomy and NAO, Bulgarian
Academy of Sciences, 72 Tsarigradsko Chaussee Blvd., 1784 Sofia, Bulgaria  V.
Agrawal  Twilio, RMZ Ecoworld, Bellandur, Bangalore, India  S. Zola
Astronomical Observatory, Jagiellonian University, ul. Orla 171, 30-244
Krakow, Poland  Aykut Özdönmez  Ataturk University, Faculty of Science,
Department of Astronomy and Space Science, 25240, Yakutiye, Erzurum, Turkey
Ergün Ege  Istanbul University, Faculty of Science, Department of Astronomy
and Space Sciences, 34116, Beyazıt, Istanbul, Turkey  L. Slavcheva-Mihova
Institute of Astronomy and NAO, Bulgarian Academy of Sciences, 72 Tsarigradsko
Chaussee Blvd., 1784 Sofia, Bulgaria  D. E. Reichart  University of North
Carolina at Chapel Hill, Chapel Hill, NC 27599, USA  D. B. Caton  Dark Sky
Observatory, Dept. of Physics and Astronomy, Appalachian State University,
Boone, NC 28608, USA  Avik Kumar Das  Raman Research Institute, C. V. Raman
Avenue, Sadashivanagar, Bengaluru – 560 080, India

###### Abstract

We present an analysis of the $BVRI$ photometry of the blazar BL Lacertae on
diverse timescales from mid-July to mid-September 2020. We have used 11
different optical telescopes around the world and have collected data over 84
observational nights. The observations cover the onset of a new activity phase
of BL Lacertae started in August 2020 (termed as the August 2020 flare by us),
and the analysis is focused on the intra-night variability. On short-term
timescales, (i) flux varied with $\sim$2.2 mag in $R$ band, (ii) the spectral
index was found to be weakly dependent on the flux (i.e., the variations could
be considered mildly chromatic) and (iii) no periodicity was detected. On
intra-night timescales, BL Lacertae was found to show bluer-when-brighter
chromatism predominantly. We also found two cases of significant inter-band
time lags of the order of a few minutes. The duty cycle of the blazar during
the August 2020 flare was estimated to be quite high ($\sim$90% or higher). We
decomposed the intra-night light curves into individual flares and determined
their characteristics. On the basis of our analysis and assuming the turbulent
jet model, we determined some characteristics of the emitting regions: Doppler
factor, magnetic field strength, electron Lorentz factor, and radius. The
radii determined were discussed in the framework of the Kolmogorov theory of
turbulence. We also estimated the weighted mean structure function slope on
intra-night timescales, related it to the slope of the power spectral density,
and discussed it with regard to the origin of intra-night variability.

galaxies: general – galaxies: active – BL Lacertae objects: general – BL
Lacertae objects: individual: BL Lacertae

##  1 Introduction

Blazars are a subclass of radio-loud active galactic nuclei whose relativistic
jets are closely aligned with the line of sight (Urry & Padovani, 1995).
Blazars display peculiar characteristics across the entire electromagnetic
spectrum, including non-thermal continuum emission variables on timescales
ranging from a few minutes to years (e.g. Wagner & Witzel, 1995; Gupta et al.,
2008; Mohan et al., 2015; Bhatta & Dhital, 2020; Agarwal et al., 2021), strong
optical linear polarization, and superluminal motions (Lister et al., 2019).
Blazars are divided into two categories, namely BL Lacertae objects (BL Lacs)
and flat-spectrum radio quasars, based on their optical spectra and compact
radio morphology. Flat-spectrum radio quasars show strong emission lines,
while BL Lacs display very weak or no emission lines in their optical spectra.

The observed spectral energy distribution (SED) of blazars shows two broad
humps: the first one extends from 1012 Hz to 1017 Hz, while the second one is
peaking between 1021 Hz and 1026 Hz (e.g. Abdo et al., 2010a). The low-
frequency hump is attributed to the synchrotron radiation of the relativistic
electrons in the magnetic field of Doppler-boosted jets. On the other hand,
the high-energy hump is generally associated with the inverse Compton
scattering of the infrared/optical/ultraviolet photons by the jet electrons
(Sikora et al., 2009). The seed photons for the inverse Compton scattering
could be originating from the synchrotron emission within the jet, commonly
known as synchrotron self-Compton (Böttcher et al., 2002), or from the
external photon fields such as accretion disk, broad emission line region, and
dusty torus and named as external Compton (Sikora et al., 1994). Blazars are
further classified based on the location of their synchrotron peak as follows
(Abdo et al., 2010a): high synchrotron peaked ($\nu_{\rm peak}\geq
10^{15}\,{\rm Hz}$), intermediate synchrotron peaked ($10^{14}\,{\rm
Hz}\leq\nu_{\rm peak}\leq 10^{15}\,{\rm Hz}$), and low synchrotron peaked
($\nu_{\rm peak}\leq 10^{14}\,{\rm Hz}$).

BL Lacertae is the prototype of the BL Lac class of blazars and has a redshift
of $z=0.0686\pm 0.0004$ (Vermeulen et al., 1995). It is classified as a low-
synchrotron-peaked blazar (Nilsson et al., 2018). BL Lacertae has been of
great interest for numerous intense multi-wavelength (MWL) campaigns (e.g.
Villata et al., 2002, 2003; Böttcher et al., 2003; Raiteri et al., 2010;
Wierzcholska et al., 2015; Agarwal et al., 2017; MAGIC Collaboration et al.,
2019; Weaver et al., 2020; Jorstad et al., 2022; Kalita et al., 2023;
Shablovinskaya et al., 2023); in particular, BL Lacertae is one of the
favorite targets of the campaigns organized by the Whole Earth Blazar
Telescope collaboration. More-than-century-long observations of BL Lacertae
reveal intense variability on diverse timescales ranging from a few minutes
(e.g. Villata et al., 2002; Gaur et al., 2015; Meng et al., 2017; Fang et al.,
2022) to years (Carini et al., 1992; Villata et al., 2004a, b, 2009; Raiteri
et al., 2013). As an example of yearly variability, Carini et al. (1992)
detected an erratic behavior of the source with a $V$ band magnitude ranging
from 14 to 16 over about 17 years of observations. BL Lacertae shows outbursts
of a few magnitudes, which is typical for blazars; for example, Villata et al.
(2004a) reported a brightness excursion of about 3 mag in all bands during the
1997 outburst (see also Bachev, 2018).

BL Lacertae generally shows a bluer-when-brighter (BWB) chromatism, whose
strength was found to be related to the timescale considered: Villata et al.
(2002) reported strongly BWB chromatic, fast flares on intra-night timescales
and mildly chromatic variations on longer timescales (see also Villata et al.,
2004a; Bhatta & Webb, 2018; Gaur et al., 2019). The mildly chromatic component
was explained as arising because of the Doppler factor change, while the
strongly chromatic flares were assumed to be of synchrotron origin.

Previous studies of BL Lacertae in optical bands show both the lack (e.g.
Nesci et al., 1998; Li et al., 2021) and the presence of inter-band time lags,
$\tau$: Papadakis et al. (2003) found a time lag of $\tau=13.8^{+11.4}_{-9.0}$
min between $B$ and $I$ bands ($B$ band leads), Hu et al. (2006) found a lag
of 11.6 min between $e$ and $m$ bands ($e$ band leads), Meng et al. (2017)
found a lag of 11.8 min between $R$ and $V$ bands ($R$ band leads), and Fang
et al. (2022) found a lag of $\sim$16 min between $B$ and $V$ bands ($B$ band
leads) and a lag of $\sim$18 min between $B$ and $R$ bands ($B$ band leads).
Therefore, the so-called soft lag – that is, the lower-frequency/softer energy
emission variations are lagging – dominates the inter-band time lags observed
in BL Lacertae.

The Doppler factor, $\delta$, is an important jet characteristic, and for BL
Lacertae it was determined by a number of authors using various approaches.
Jorstad et al. (2017) used the observed variability timescale and the angular
size of the six moving knots, observed by the Very Long Baseline Array, to get
Doppler factors of $6.2\pm 1.5$, $11.0\pm 5.6$, $5.6\pm 3.3$, $8.4\pm 1.7$,
$8.6\pm 2.6$, and $7.1\pm 4.3$. Liodakis et al. (2017) and Liodakis et al.
(2018) compared observed and intrinsic brightness temperatures and got the
following variability Doppler factors $6.1\pm 0.8$ and
$12.17^{+3.44}_{-2.81}$, respectively, while Chen (2018) used broadband SED to
derive $\delta=3.8$. Zhang et al. (2020) proposed a new method to estimate the
Doppler factor for a source of known $\gamma$-rays and broad emission line
luminosities; the authors got $\delta=8.13$ for BL Lacertae. Ye & Fan (2021)
used the relation between the core and extended radio luminosities to estimate
$\delta=14.22$ for a continuous jet and $\delta=6.66$ for a moving blob; to
get these values, the authors assumed a spectral index $\alpha=0.5$
($F_{\nu}\propto\nu^{-\alpha}$, where $F_{\nu}$ is the monochromatic flux
density). Generally, the different methods result in different Doppler factors
because of the different assumptions made.

During the summer of 2020, a new phase111During this long-lasting activity
phase BL Lacertae reached its historical maximum of $R=11.271\pm 0.003\,\rm
mag$ at JD 2459426.4930 (Jul 30, 2021, Kunkel et al., 2021) of the BL Lacertae
activity began, which continued throughout 2021. The source was reported as
flaring during August 2020 in the optical (Grishina & Larionov, 2020;
Jankowsky & Wagner, 2020; Steineke et al., 2020) and high-energy $\gamma$-rays
(Cheung, 2020; Ojha & Valverd, 2020). The MAGIC system of Cherenkov telescopes
detected very high energy $\gamma$-rays during the night of Aug 19 (Blanch,
2020a); the next peak of the very high energy $\gamma$-rays was detected on
Sep 19 (Blanch, 2020b). A significant optical intra-night variability (INV)
was also observed (Jankowsky & Wagner, 2020).

In this paper, we report the results from our observations of BL Lacertae on
intra-night timescales during its August 2020 flare; the mid-August to mid-
September BL Lacertae activity will be termed by us as an August 2020 flare
throughout the paper. In particular, we focus on the analysis of the
individual intra-night light curves (INLCs) recorded in the course of our
monitoring.

The paper is organized as follows. In Section 2 we describe our observations
and data reduction. In Section 3 the analysis techniques used by us are
described in detail. In Section 4 we present the results obtained, and in
Section 5 we discuss them.

##  2 Observations and Data Reductions

To understand the source behavior in the optical regime, we carried out
optical observations of BL Lacertae from July to September 2020 using 11
different optical telescopes around the globe over 84 observational nights and
gathering $\sim$12 800 frames in $BVRI$ bands. The telescopes used are as
follows: 50 cm OAUJ-CDK500 (Corrected Dall-Kirkham Astrograph, telescope A) of
the Astronomical Observatory operated by the Jagiellonian University, Krakow,
Poland; Kirkham astrograph telescope (KRK, telescope B) of the Jagiellonian
University, Krakow, Poland; 40 cm PROMPT-USASK telescope of Sleaford
Observatory (PSASK, Telescope C); 60 cm Rapid Response Robotic Telescope
(RRRT, telescope D) of the Fan Mountain Observatory, SUH (telescope E); 50/70
cm Schmidt telescope at the Rozhen National Astronomical Observatory, Bulgaria
(telescope F, Kostov, 2010); 2.01 m RC Himalayan Chandra Telescope (HCT,
telescope G) at Indian Astronomical Observatory, Hanle, India; 40 cm telescope
of the Dark Sky Observatory (DSO, telescope H); 40 cm telescope of the Montana
Learning Center (MLC-COS16, telescope I); 60 cm RC robotic telescope, Turkey
(telescope J); and 1.0 m RC telescope, Turkey (telescope K). Telescopes F and
G are described in Agarwal et al. (2019), and telescopes J and K are described
in Agarwal et al. (2021). The technical details about the rest of the
telescopes are given in Table 1. Telescopes A, C, D, H, and I work in the
robotic mode under the Skynet Robotic Telescope Network software (Zola et al.,
2021). The complete log of our observations is presented in Table 2.

The data reduction procedure includes bias/dark subtraction, flat-fielding,
and cosmic-ray treatment which was performed using the standard IRAF222IRAF is
distributed by the National Optical Astronomy Observatories, which are
operated by the Association of Universities for Research in Astronomy, Inc.,
under a cooperative agreement with the National Science Foundation. tasks.
This was followed by the extraction of the instrumental magnitudes of the
source and standard stars in the field using the Dominion Astronomical
Observatory Photometry (DAOPHOT II) software (Stetson, 1987, 1992). To perform
differential photometry, we finally chose stars B and C from the source
field333https://www.lsw.uni-
heidelberg.de/projects/extragalactic/charts/2200+420.html that are in close
proximity to the target and with magnitudes similar to the blazar. A more
detailed data reduction procedure is discussed in Agarwal et al. (2019).

To get the optimum aperture for each night, we performed aperture photometry
for different radii: 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.5, and 3.0 times the full
width at the half-maximum (FWHM) of the field stars. For background
subtraction, we selected the sky annulus to approximately 5$\times$FWHM. We
finally selected the aperture with the best signal-to-noise ratio and minimum
standard deviation of the difference between instrumental magnitudes of
standard stars. The above procedure was applied on all the $BVRI$ frames, and
the calibrated magnitudes of the source were derived.

The calibrated $BVRI$ magnitudes of the blazar were dereddened by subtracting
the Galactic extinction values from the NASA/IPAC Extragalactic Database:
$A_{B}=0.43$ mag, $A_{V}=0.54$ mag, $A_{R}=0.64$ mag, and $A_{I}=0.80$ mag.
The flux from the nucleus of the source is contaminated by its elliptical host
galaxy. Hence, to perform host galaxy subtraction, we converted extinction-
corrected magnitudes to fluxes using the zero point values from Bessell et al.
(1998). Thereafter using the measurements from Nilsson et al. (2007), we
estimated the host galaxy emission in the $R$ band. This $R$ band value is
further used to obtain the corresponding contributions for the $BVI$ bands by
using the galaxy colors (Fukugita et al., 1995) as $B-V=0.96$ mag, $V-R=0.61$
mag, and $R-I=0.70$ mag.

Table 1: Details about the telescopes and instruments used

Telescope | A | B | C | D | E | H | I
---|---|---|---|---|---|---|---
| 50 cm Cassegrain | 50 cm Cassegrain | 40 cm RC | 60 cm RC | 60 cm RC | 40 cm RC | 40 cm RC
CCD model | U47 | Alta U47 | FLI | SBIG STX-16803 | CG47 | U47 | SBIG STXL-11002
| F42 | U42/F42 |  |  |  |  |
Chip size | 1024$\times$1024 px | 1024$\times$1024 px | 2014$\times$2014 px | 4096$\times$4096 px | 1024$\times$1024 px | 1024$\times$1024 px | 4008$\times$2672 px
| 2048$\times$2048 px | 2048$\times$2048 px |  |  |  |  |
Scale | 0$\farcs$81/px | 0$\farcs$399/px | 0$\farcs$12/px | 1$\farcs$13/px | 1$\farcs$12/px | 1$\farcs$23/px | 1$\farcs$53/px
| 0$\farcs$81/px | 0$\farcs$399/px |  |  |  |  |
Field | 13$\farcm$3$\times$13$\farcm$3 | 6$\farcm$8$\times$6$\farcm$8 | 10$\farcm$1$\times$10$\farcm$1 | 25$\farcm$8$\times$25$\farcm$8 | 19$\farcm$1$\times$19$\farcm$1 | 10$\farcm$5$\times$10$\farcm$5 | 34$\farcm$1$\times$22$\farcm$7
| 27$\farcm$7$\times$27$\farcm$7 | 13$\farcm$6$\times$13$\farcm$6 |  |  |  |  |
Gain | 1.41 $\rm e^{-}$/ADU | 1.41 $\rm e^{-}$/ADU | 1.21 $\rm e^{-}$/ADU | 1.27 $\rm e^{-}$/ADU | 1.61 $\rm e^{-}$/ADU | 1.29 $\rm e^{-}$/ADU | 1.74 $\rm e^{-}$/ADU
| 1.69 $\rm e^{-}$/ADU | 1.69 $\rm e^{-}$/ADU |  |  |  |  |
Read-out noise | 8.3 $\rm e^{-}$ rms | 8.3 $\rm e^{-}$ rms | 17.0 $\rm e^{-}$ rms | 16.96 $\rm e^{-}$ rms | 12.81 $\rm e^{-}$ rms | 8.74 $\rm e^{-}$ rms | 11.13 $\rm e^{-}$ rms
| 10.1 $\rm e^{-}$ rms | 10.1 $\rm e^{-}$ rms |  |  |  |  |
Binning used | 1$\times$1 | 2$\times$2 | 1$\times$1 | 2$\times$2 | 3$\times$3 | 1$\times$1 | 1$\times$1
Typical seeing | 2$\farcs$5 to 3$\farcs$5 |  $\sim$3$\farcs$5 | 1$\arcsec$ to 3$\arcsec$ | 1$\arcsec$ to 3$\arcsec$ | 1$\farcs$5 to 2.5$\arcsec$ | 2$\farcs$5 to 3$\farcs$5 | 1$\farcs$5 to 2$\farcs$5

Table 2: Log of photometric observations for the blazar BL Lacertae Date | Telescope | Number of data points | Date | Telescope | Number of data points
---|---|---|---|---|---
(yyyy mm dd) |  | $B$ | $V$ | $R$ | $I$ | (yyyy mm dd) |  | $B$ | $V$ | $R$ | $I$
2020 07 13 | J | 1 | 1 | 1 | 1 | 2020 08 27 | J | 0 | 2 | 1 | 1
2020 07 14 | J | 1 | 1 | 1 | 1 | 2020 08 28 | B | 0 | 0 | 152 | 0
2020 07 15 | J | 1 | 1 | 1 | 1 | 2020 08 28 | C | 0 | 0 | 10 | 0
2020 07 16 | J | 1 | 1 | 1 | 1 | 2020 08 28 | F | 328 | 18 | 18 | 348
2020 07 17 | J | 1 | 1 | 1 | 1 | 2020 08 28 | J | 0 | 2 | 1 | 1
2020 07 19 | J | 1 | 1 | 1 | 1 | 2020 08 29 | D | 0 | 0 | 23 | 0
2020 07 20 | J | 1 | 1 | 1 | 1 | 2020 08 29 | J | 0 | 2 | 1 | 1
2020 07 21 | J | 0 | 1 | 0 | 1 | 2020 08 30 | D | 0 | 0 | 9 | 4
2020 07 22 | J | 0 | 1 | 0 | 1 | 2020 08 30 | J | 0 | 2 | 0 | 1
2020 07 23 | J | 0 | 0 | 1 | 1 | 2020 08 30 | K | 2 | 2 | 518 | 2
2020 07 24 | J | 1 | 0 | 0 | 1 | 2020 08 31 | K | 3 | 77 | 75 | 73
2020 07 25 | J | 1 | 1 | 1 | 1 | 2020 09 01 | J | 0 | 2 | 1 | 1
2020 07 26 | J | 1 | 1 | 1 | 1 | 2020 09 02 | B | 30 | 30 | 1000 | 30
2020 07 28 | J | 0 | 3 | 1 | 1 | 2020 09 02 | C | 0 | 0 | 9 | 0
2020 07 29 | J | 0 | 3 | 0 | 1 | 2020 09 02 | J | 1 | 2 | 1 | 1
2020 07 30 | J | 1 | 3 | 0 | 1 | 2020 09 03 | A | 45 | 39 | 27 | 28
2020 07 31 | K | 3 | 2 | 92 | 3 | 2020 09 03 | B | 13 | 17 | 553 | 15
2020 07 31 | J | 0 | 2 | 0 | 0 | 2020 09 04 | G | 7 | 1 | 7 | 7
2020 08 01 | J | 0 | 1 | 1 | 0 | 2020 09 04 | J | 0 | 0 | 1 | 1
2020 08 02 | J | 0 | 1 | 1 | 1 | 2020 09 05 | D | 25 | 25 | 46 | 0
2020 08 04 | J | 1 | 2 | 0 | 1 | 2020 09 06 | D | 0 | 0 | 87 | 0
2020 08 05 | J | 1 | 2 | 0 | 1 | 2020 09 06 | G | 1 | 1 | 1 | 1
2020 08 06 | J | 1 | 1 | 1 | 1 | 2020 09 06 | J | 1 | 2 | 1 | 1
2020 08 07 | J | 1 | 0 | 1 | 1 | 2020 09 07 | A | 30 | 29 | 24 | 30
2020 08 12 | J | 1 | 2 | 0 | 1 | 2020 09 07 | D | 0 | 0 | 34 | 0
2020 08 13 | J | 1 | 2 | 0 | 1 | 2020 09 07 | J | 1 | 2 | 1 | 1
2020 08 14 | J | 1 | 2 | 1 | 0 | 2020 09 08 | A | 32 | 33 | 32 | 30
2020 08 16 | J | 1 | 2 | 0 | 1 | 2020 09 08 | B | 0 | 0 | 236 | 0
2020 08 18 | J | 1 | 2 | 1 | 1 | 2020 09 08 | J | 0 | 2 | 1 | 1
2020 08 19 | G | 0 | 2 | 0 | 0 | 2020 09 09 | A | 43 | 45 | 44 | 45
2020 08 20 | A | 0 | 0 | 370 | 0 | 2020 09 09 | B | 0 | 0 | 508 | 0
2020 08 20 | B | 95 | 118 | 118 | 108 | 2020 09 09 | J | 1 | 2 | 1 | 1
2020 08 20 | C | 0 | 0 | 6 | 0 | 2020 09 10 | A | 49 | 50 | 217 | 54
2020 08 20 | G | 0 | 1 | 0 | 0 | 2020 09 10 | B | 0 | 0 | 187 | 0
2020 08 21 | A | 0 | 0 | 327 | 0 | 2020 09 10 | J | 1 | 2 | 1 | 1
2020 08 21 | C | 0 | 0 | 6 | 0 | 2020 09 10 | K | 2 | 2 | 206 | 3
2020 08 21 | G | 2 | 1 | 144 | 2 | 2020 09 11 | A | 242 | 59 | 62 | 61
2020 08 22 | C | 0 | 0 | 25 | 0 | 2020 09 11 | B | 0 | 0 | 573 | 0
2020 08 22 | G | 2 | 1 | 2 | 2 | 2020 09 11 | C | 0 | 0 | 9 | 0
2020 08 23 | A | 7 | 7 | 7 | 7 | 2020 09 11 | J | 1 | 2 | 0 | 1
2020 08 23 | C | 0 | 0 | 20 | 0 | 2020 09 12 | A | 41 | 12 | 370 | 11
2020 08 23 | E | 38 | 36 | 37 | 40 | 2020 09 12 | B | 0 | 0 | 56 | 0
2020 08 23 | G | 3 | 0 | 0 | 3 | 2020 09 12 | C | 0 | 0 | 17 | 0
2020 08 23 | J | 0 | 0 | 1 | 0 | 2020 09 12 | I | 0 | 0 | 10 | 0
2020 08 24 | A | 6 | 7 | 6 | 6 | 2020 09 12 | J | 1 | 2 | 1 | 1
2020 08 24 | B | 0 | 0 | 18 | 0 | 2020 09 12 | K | 1 | 2 | 241 | 1
2020 08 24 | C | 0 | 0 | 23 | 0 | 2020 09 13 | A | 14 | 232 | 18 | 221
2020 08 24 | G | 2 | 0 | 0 | 2 | 2020 09 13 | B | 0 | 0 | 376 | 0
2020 08 24 | J | 1 | 2 | 1 | 1 | 2020 09 13 | C | 0 | 0 | 6 | 0
2020 08 25 | B | 54 | 54 | 949 | 54 | 2020 09 13 | I | 0 | 0 | 2 | 0
2020 08 25 | C | 0 | 0 | 12 | 0 | 2020 09 13 | J | 1 | 2 | 1 | 1
2020 08 25 | J | 1 | 2 | 1 | 1 | 2020 09 13 | K | 2 | 2 | 233 | 2
2020 08 26 | C | 0 | 0 | 16 | 0 | 2020 09 14 | A | 18 | 224 | 18 | 223
2020 08 26 | F | 299 | 14 | 15 | 296 | 2020 09 14 | B | 6 | 6 | 710 | 6
2020 08 26 | J | 1 | 2 | 1 | 1 | 2020 09 14 | C | 0 | 0 | 13 | 0
2020 08 27 | C | 0 | 0 | 9 | 0 | 2020 09 14 | D | 0 | 0 | 26 | 0
2020 08 27 | F | 332 | 18 | 18 | 331 | 2020 09 14 | J | 1 | 0 | 1 | 1

##  3 Analysis Techniques

Having obtained the light curves (LCs) in flux units, we

  1. 1.

combined the LCs in the case in which multi-telescope data are available and
cleaned the combined LCs of the outliers if any; and

  2. 2.

corrected the combined LCs for the smooth flux variation in the case in which
the LCs show two variability components.

The corrected LCs were further

  1. 1.

decomposed into individual flares; and

  2. 2.

used to build the structure functions (SFs).

In addition, the corrected MWL LCs were

  1. 1.

used to build the color-magnitude diagrams (CMDs); and

  2. 2.

used to search for inter-band time lags.

Below we shall describe in detail the analysis techniques used in each of the
above steps.

###  3.1 Variability Detection and Amplitude

We quantified the flux variability of BL Lacertae using $C$-, $F$-, and
$\chi^{2}$-tests and the percentage amplitude variation, $A$. A brief
introduction to these methods is given below.

####  3.1.1 C-test

The most frequently used variability detection criterion is the $C$-test
(Romero et al., 1999), which is defined as

$C_{1}={\rm\frac{\sigma(BL-S_{B})}{\sigma(S_{B}-S_{C})}},\quad
C_{2}={\rm\frac{\sigma(BL-S_{C})}{\sigma(S_{B}-S_{C})}},$ (1)

where BL$-$SB, BL$-$SC, and SB$-$SC are the differential instrumental LCs of
the blazar (BL) against the standard star B (SB), BL against the standard star
C (SC), and SB against SC, respectively, while $\sigma$(BL$-$SB),
$\sigma$(BL$-$SC), and $\sigma$(SB$-$SC) are the standard deviations of the
respective LCs. If $C\geq 2.576$, then we marked the LC as a variable at a
confidence level of $99.5$% or greater; otherwise, we call it a non-variable
(here $C$ is a mean over $C_{1}$ and $C_{2}$). As pointed out by Zibecchi et
al. (2017), through their study of INV in active galactic nuclei using various
statistical methods, the $C$-test could be considered a suitable test to
detect variability with more reliable results as compared to the $F$-test.

####  3.1.2 F-test

The $F$-test (Zibecchi et al., 2017) is a powerful tool to quantify
variability at diverse timescales and is defined as

$F_{1}=\frac{\sigma^{2}(\mathrm{BL-
S_{B})}}{\sigma^{2}(\mathrm{S_{B}-S_{C}})},\quad
F_{2}=\frac{\sigma^{2}(\mathrm{BL-S_{C}})}{\sigma^{2}(\mathrm{S_{B}-S_{C}})},$
(2)

where BL$-$SB, BL$-$SC, and SB$-$SC are the differential instrumental LCs of
BL against SB, BL against SC, and SB against SC, respectively, while
$\sigma^{2}$(BL$-$SB), $\sigma^{2}$(BL$-$SC), and $\sigma^{2}$(SB$-$SC) are
the variances of the respective LCs. Averaging $F_{1}$ and $F_{2}$ gives the
mean observational $F$ value, which is then compared with the critical value,
$F_{\rm c}=F^{(\alpha)}_{\nu_{\rm BL},\nu_{\rm S}}$, where $\nu_{\rm BL}$ and
$\nu_{\rm S}$ give the number of degrees of freedom for the blazar and star
LCs, respectively, estimated as the number of measurements, $N_{\rm data}$,
minus 1 ($\nu=N_{\rm data}-1$). The significance level, $\alpha$, is set as
0.1% and 1% (i.e. $3\sigma$ and $2.6\sigma$) for this work. If the mean $F$
value is more than the critical value, the null hypothesis (i.e. no
variability) is rejected and the LC is marked as variable.

####  3.1.3 $\chi^{2}$-test

Further, to detect the genuine variability in our source, we also used the
$\chi^{2}$-test, which is interpreted as:

$\chi^{2}=\sum_{i=1}^{N}\frac{(V_{i}-\overline{V})^{2}}{e_{i}^{2}},$ (3)

where $\overline{V}$ is the mean magnitude and $V_{i}$ the magnitude
corresponding to the $i$-th observation with a respective uncertainty $e_{i}$.
Estimating the exact values of uncertainties is unattainable in the IRAF
package used for data reduction, whereas, the theoretical uncertainties have
been found to be smaller by 1.3–1.75 (Gopal-Krishna et al., 2003). For our
data, the factor is around 1.6, on average. Therefore, for a better
calculation of photometric uncertainties, we should multiply the uncertainties
obtained from data analysis by the above factor. The obtained $\chi^{2}$ value
is then compared with a critical value $\chi_{\alpha,\nu}^{2}$ where $\alpha$
is the significance level and $\nu=N_{\rm data}-1$ is the degree of freedom.
When $\chi^{2}>\chi_{\alpha,\nu}^{2}$, it indicated the presence of
variability.

Depending on the sampling of the individual INLCs and on the monitoring
duration, there could happen the blazar to have both variable and non-variable
status for one and the same night; the nights of Sep 2 and Sep 7 are examples
in this context. In such cases, we adopted the status obtained by testing the
better LCs in terms of sampling and/or duration.

####  3.1.4 Percentage amplitude variation

To estimate the percentage amplitude change in our LCs, we calculated the
variability amplitude parameter $A$ (Heidt & Wagner, 1996):

$\displaystyle
A=100\times\sqrt{{(m_{\mathrm{max}}-m_{\mathrm{min}}})^{2}-2\langle
e^{2}\rangle}\leavevmode\nobreak\ [\%],$ (4)

where $m_{\mathrm{max}}$ and $m_{\mathrm{min}}$ are the maximum and minimum
magnitudes attained by the blazar and $\langle e^{2}\rangle$ the mean squared
uncertainty of the measurements.

###  3.2 Combination of the Light Curves

The INLCs obtained with two or more telescopes were combined in order to get a
single LC for the given night and band. If the individual LCs have overlapping
parts, then, before the combination, the LCs were adjusted such that (i) a
single band LC was adjusted to match the corresponding LC from a MWL data set
(in order to avoid the systematic uncertainties when the LCs are used to build
CMDs) and (ii) a poorly sampled LC was adjusted to match the densely sampled
one (if it does not contradict the first condition). Technically, the
adjustment was made as follows: we interpolated the first LC over the second
one in their overlapping parts, computed the median offset and its standard
uncertainty, and applied the so-obtained offset according to the above
conditions. If the LCs have no overlapping parts, then the LCs were combined
without adjustment. Finally, the observations consisting of a few data points
per band were (adjusted and) combined with the so-built composite INLCs (an
exception were the telescope J data, see below). During the combination of the
LCs, a few outlying measurements were identified and cleaned.

The so-combined INLCs were merged with the rest of the data to build the
short-term444Variability on timescales from days to weeks/months is usually
termed as the short-term variability (STV, Singh & Meintjes, 2020).
variability LCs (STLCs) of BL Lacertae for each band. To these STLCs, the
telescope J STLCs were adjusted (actually, the adjustment was needed only for
the $BV$ bands) and combined.

We are interested in the analysis of the INV, and so the above procedure is
optimized for the accurate combination of the individual INLCs, but not for
the STLCs of the individual telescopes. This would result in increased night-
to-night scatter in the STLCs, but this is not an issue for the presented
research.

###  3.3 Correction for the Smooth Flux Variation

Generally, the INLCs obtained in the course of our study could be described as
flares superimposed onto a smooth flux variation; that is, the LCs show two
variable components. The flare timescales are much shorter than the smooth
component timescale. The latter timescale is usually longer than several
hours, which is longer than the typical duration of a single-telescope intra-
night monitoring session.

We are interested in the analysis of the flaring activity of BL Lacertae and
so a correction has to be done in order to minimize the contribution of the
smooth variability component. For example, to make flares more evident,
Ghisellini et al. (1997) divided their LC by a curve interpolated through the
local minima of the same LC.

The correction of the LCs for the smooth flux variation (or detrending for
short) was done following an approach closely related to that of Villata et
al. (2004a); see also Xiong et al. (2020) and Raiteri et al. (2021). Firstly,
we selected the regions of the LC that are free of flares – they were assumed
to be related to the smooth component we want to correct for. Secondly, we
fitted to these regions a low-degree polynomial. For more complicated LCs, the
fitting was done by splitting the LC into segments and fitting a polynomial to
each segment. The polynomials could be of different degrees for different
segments, or, for some of the segments, the polynomial could be replaced by
another fitting function (e.g. cubic spline or Gaussian). Upon completion of
the fit, care was taken to ensure the individual fitting functions were joined
smoothly. If MWL data are available for a given night, then the fitted regions
and the fitting functions are one and the same for all bands. Finally, we
rescaled each data point of the LC by dividing the corresponding flux value by
the scaling factor $C_{k}(t)=F_{k\rm,fit}(t)/F_{k\rm,min}$ (here $k$
represents the $BVRI$ bands), which is the ratio between the value of the
(composite) fitting function at the corresponding time and the fitting
function minimum value. That minimum value served as the base level in the LC
decompositions.

###  3.4 Decomposition of the INLCs

The INLCs that show flaring activity were decomposed using the following
double exponential function (DE, Abdo et al., 2010b):

$F(\Delta t)=F_{\rm base}+\\\ F_{0}\left[\exp\left(\frac{\Delta t_{0}-\Delta
t}{\mathcal{T}_{\rm r}}\right)+\exp\left(\frac{\Delta t-\Delta
t_{0}}{\mathcal{T}_{\rm d}}\right)\right]^{-1},$ (5)

where $F_{\rm base}$ is the constant base level, $F_{0}$ twice the flare
amplitude (with respect to the base level), $\Delta t_{0}$ the approximate
position in the time of the flare peak, and $\\{\mathcal{T}_{\rm
r},\mathcal{T}_{\rm d}\\}$ the rise and decay timescales. If the LC has been
detrended, then the base level was set to the minimal value of the function,
used to fit the smooth component, and was held fixed during the decomposition.
If no detrending has been done, then the base level is left free (we, however,
have no such LCs). The time variable, $\Delta t=t-t_{0}$, we used represents
the time since the earliest observation (taken at $t_{0}$) among the available
data sets for the given night; the JD of the earliest observation is indicated
in the LC plots.

The characteristics of the DE function can be summarized as follows. The
actual position in the time of the flare maximum is

$\Delta t_{\rm max}=\Delta t_{0}+\frac{\mathcal{T}_{\rm r}\mathcal{T}_{\rm
d}}{\mathcal{T}_{\rm r}+\mathcal{T}_{\rm d}}\ln\left(\frac{\mathcal{T}_{\rm
d}}{\mathcal{T}_{\rm r}}\right)$ (6)

and it is equal to $\Delta t_{0}$ in the case of symmetric flares,
$\mathcal{T}_{\rm r}=\mathcal{T}_{\rm d}$. An estimate of the total duration
of the flare could be found as $\Delta\mathcal{T}\simeq 2\,(\mathcal{T}_{\rm
r}+\mathcal{T}_{\rm d})$. The asymmetry parameter is defined as

$\xi=\frac{\mathcal{T}_{\rm d}-\mathcal{T}_{\rm r}}{\mathcal{T}_{\rm
d}+\mathcal{T}_{\rm r}}\qquad\begin{cases}\xi\in[-1,1];\\\
\xi=0\implies\text{symmetric flare}.\end{cases}$ (7)

Finally, the doubling and halving timescales are equal to
$\ln(2)\mathcal{T}_{\rm r}$ and $\ln(2)\mathcal{T}_{\rm d}$, respectively
(Albert et al., 2007).

###  3.5 Structure Function

The SF was introduced by Simonetti et al. (1985) and is particularly useful
for analyzing unevenly sampled astronomical data (e.g. Bhatta & Webb, 2018).
Various aspects of the SF application are thoroughly discussed by
Emmanoulopoulos et al. (2010) and Kozłowski (2016).

For a time separation $\delta t$ and a bin of size ${\text{d}}t$, we
calculated the first-order SF as

$D^{1}(\delta t,{\text{d}}t)=\frac{1}{N(\delta
t,{\text{d}}t)}\sum_{i>j}[F(t_{i})-F(t_{j})]^{2},$ (8)

where $N(\delta t,{\text{d}}t)$ is the number of pairs $(t_{i},t_{j})$ for
which $\delta t<t_{i}-t_{j}<\delta t+{\rm d}t$. The choice of bin size depends
on the LC sampling. The uncertainties of the SF were calculated simply as the
standard uncertainty of the mean in the bins (see Sergison et al., 2020, for
discussion about the SF uncertainties). The value of $\delta t$ in each bin
was set to the middle of the bin.

Ideally, the SF has two plateaus connected by a curve, whose slope depends on
the nature of the observed flux variation (shot noise, flicker noise, etc.;
see Hughes et al., 1992; Sergison et al., 2020). Let us assume that the LC can
be represented by the sum ($s+n$), where $s$ is the signal and $n$ is the
noise, both having Gaussian distribution. Then, the first plateau (at $\delta
t\rightarrow 0$) equals $2\sigma^{2}_{n}$ and the second one equals to
$2\sigma^{2}_{s}$, where $\sigma^{2}$ represents the corresponding variances.
These plateaus bracket the time separations over which the flux variations are
correlated. The upward-sloping curve between the plateaus is usually
characterized by its logarithmic slope ${\rm d}[\log(D^{1})]/{\rm
d}[\log(\delta t)]$. The time separation at which this curve flattens could be
considered as a robust characteristic variability timescale; if the second
plateau is not reached, then the timescale is longer than the observation
span. Next, the SF could be used to study the time asymmetry of the LCs
(Kawaguchi et al., 1998; Bachev et al., 2017, 2021). Finally, if the LC shows
periodicity, then the SF has a dip at the time separation equal to the
corresponding period.

It is common practice for the measurement uncertainties to be subtracted off
during the SF build, and there are various ways to do that (see Kozłowski,
2016, for discussion on this topic). If the measurement uncertainties, $e$,
are assumed to follow a Gaussian distribution, then $\sigma^{2}_{n}$ could be
approximated as $\sigma^{2}_{n}\simeq\langle e^{2}\rangle$ and, therefore,
$D^{1}(\delta t)-2\langle e^{2}\rangle$ is the noise-free SF estimate we
wanted. The problem here is that any incorrectness in the measurement
uncertainty estimation affects the slope of the SF. Hence, we prefer to add
$2\sigma^{2}_{n}$ as a free parameter during the SF fitting rather than
subtracting $2\langle e^{2}\rangle$ from the SF. In particular, in this way,
we could obtain an independent estimate of the mean measurement uncertainty.

In the case of no noise subtraction, we fitted the SF using a single power-law
(SPL) model plus a noise term to determine the SF slope:

$D^{1}(\delta t)=2\sigma^{2}_{n}+D^{1}_{0}\left(\frac{\delta t}{\delta
t_{0}}\right)^{\varrho},$ (9)

where $D^{1}_{0}$ is the variability amplitude at the fixed timescale $\delta
t_{0}$ (we arbitrarily choose $\delta t_{0}=1$ min), $\varrho$ the power-law
index, and $\sigma^{2}_{n}$ the variance of the measurement noise. The fitting
was done up to the turnover point, $\delta t_{\rm to}$, at which the SF
changes its slope and starts to flatten. After that point, the SPL
overestimates the SF.

It is worth mentioning two issues that affect the SF fitting, namely the lack
of statistical independence and Gaussianity; that is, the individual SF
estimates are not independent of each other and the distribution of the SF
estimates within the individual bins is not Gaussian (Emmanoulopoulos et al.,
2010). The latter problem could be solved particularly by fitting not
$D^{1}(\delta t)$, but $\log[D^{1}(\delta t)]$ as we actually did; see
Emmanoulopoulos et al. (2010) and Kasliwal et al. (2015) for details about
these issues.

There is an approximate relation between the slopes of the power spectral
density (PSD), $\varkappa$, and SF, namely $\varkappa\simeq\varrho+1$ (the
equality is obtained under special conditions, see Emmanoulopoulos et al.,
2010, for details).

###  3.6 Color-magnitude Diagram

Given the BL Lacertae fluxes $F_{\nu}$, we built the following CMDs:
$F_{\nu_{1}}/F_{\nu_{3}}$ vs $F_{\nu_{2}}$ if three- or four-band data are
available ($\nu_{1}>\nu_{2}>\nu_{3}$, where $\nu_{i}$ is the frequency
corresponding to the $i$-th band) and $F_{\nu_{1}}/F_{\nu_{2}}$ vs
$(F_{\nu_{1}}\\!+\\!F_{\nu_{2}})/2$ if two-band data are available. The CMD
forms were chosen to minimize the possibility of introducing spurious effects
if we are correlating the flux ratio with one of the fluxes used to build the
ratio itself (Massaro & Trevese, 1996; Papadakis et al., 2007). The flux
ratios we used are representative for the two-point spectral index,
$\alpha_{\nu_{1}\nu_{2}}\propto-\log(F_{\nu_{1}}/F_{\nu_{2}})$, under the
assumption that $F_{\nu}\propto\nu^{-\alpha}$.

The CMDs were built by selecting the data points from the corresponding LCs
closest to each other. In addition, we required the time intervals among the
data points used to get a single CMD data point to be smaller than a
predefined threshold, which depends on the sampling of the LCs used and was
typically set to a few minutes.

The CMDs were fitted by the power-law model $F_{\nu_{1}}/F_{\nu_{2}}\propto
X^{\varpi}$, where $\varpi$ is the power-law index555The power-law index
corresponds to the slope of the CMD in magnitude units (e.g. Papadakis et al.,
2007). and $X$ either the flux or the mean flux depending on the CMD form
used. Further analysis of the CMDs was done after taking a logarithm of both
sides of the above equation.

To consider a CMD trend significant at 99% confidence level, we required (i)
the linear Pearson correlation coefficient to be $|r|\geq 0.5$ and (ii) the
probability to get such a correlation coefficient by chance to be $p\leq 0.01$
(e.g. Gupta et al., 2016; Agarwal et al., 2021). For the nights for which we
have $BVRI$ band data, we used the following CMD forms: $F_{i}/F_{I}$ vs
$F_{R}$ ($i=B,V$). To assign a significant BWB or redder-when-brighter CMD
trend for these nights, we further required both CMDs to show a significant
correlation.

###  3.7 Cross-correlation Analysis

To search for inter-band time lags, we used a Python implementation
pyDCF666https://github.com/astronomerdamo/pydcf (Robertson et al., 2015) of
the discrete cross-correlation function (DCF, Edelson & Krolik, 1988), which
is suitable to cross-correlate unevenly sampled time series. In our runs (i)
the measurement uncertainties were not taken into account in the build of the
DCF following White & Peterson (1994) and (ii) the Gaussian weighting scheme
was applied in order to assign higher importance to the values closer to the
bin center.

The estimation of the time lag and its uncertainty was done utilizing the flux
randomization/random subset selection method (FR/RSS, Peterson et al., 1998,
2004) based on Monte Carlo simulations. During the RSS process, the data
points counted more than once were rejected. At the end of each FR/RSS run,
the time lag was found as the centroid of the DCF, defined as the DCF-weighted
mean lag. The centroid was calculated using DCF points above a predefined
threshold, which was set to the DCF peak value of less than one to three times
its uncertainty – we varied the threshold value so as to ensure at least ten
data points for the centroid calculation. We ran a total of 2500 cycles, and
the resulting time lags were used to build the cross-correlation centroid
distribution (CCCD). Given the CCCD, the time lag is estimated as the 50th
percentile (or the median) of the CCCD, while the 16th and 84th percentiles
serve as the $1\sigma$ uncertainties.

The significance of the cross-correlation results was estimated by means of
Monte Carlo simulation following the approach of Max-Moerbeck et al. (2014).
To generate the LCs, we used a Python implementation
DELCgen777https://github.com/samconnolly/DELightcurveSimulation (Connolly,
2015) of the method of Emmanoulopoulos et al. (2013), which accounts for the
flux probability density function (PDF) and PSD of the observed LC; the
alternative LC generation method of Timmer & Koenig (1995) produces LCs having
a Gaussian flux PDF. To produce evenly sampled LCs needed for the PSD build,
we used interpolation onto a regular grid having a time interval of 2 min. We
fitted the PSD by a single-slope power law, ${\rm PSD}\propto f^{-\varkappa}$
(here $f$ is the temporal frequency; Vaughan, 2005, 2010; González-Martín &
Vaughan, 2012). The PDF was approximated either with a Gaussian or with a sum
of Gaussians. Each simulated LC has the same statistical properties and
sampling as the observed one. In addition, the noise was added to each
simulated LC according to the mean observational uncertainty. We generated a
total of 2500 LCs for each of the bands involved in the cross-correlation.
Then, we cross-correlated the simulated LCs in the same way as we have done
for the observed ones. Finally, the distribution of the simulated cross-
correlation coefficients for each time lag bin was used to estimate the
significance levels of the observed coefficients.

The LCs produced during a typical intra-night monitoring session are of good
sampling, so it is worth trying the interpolated cross-correlation function
(ICF) for the time lag search. We used a Python implementation
PyCCF888https://bitbucket.org/cgrier/python_ccf_code (Sun et al., 2018) of the
method of Peterson et al. (1998). To estimate the lag and its uncertainty, we
used the cross-correlation peak distribution (CCPD) because there is no need
for additional free parameters, namely the bin size and threshold.

Figure 1: Light curves in $BVRI$ bands from Jul 11 to Sep 14, 2020. The LCs
are ordered as indicated in the top panel; $RI$ band LCs are shifted by the
corresponding offsets for display purposes. The blue dashed lines are the fits
used to determine the shape of the smooth component for the corresponding
nights – see Section 4.2 for the description of the $BRI$ band LCs around JD =
2459088 and of the $R$ band LC around JD = 2459104. Figure 2: Optical ($R$
band, red circles) and the “quick-look” $\gamma$-rays (0.1–300 GeV band, black
stepped curve) LCs from Jul 11 to Sep 14, 2020. The $R$ band flux is in units
of mJy, while the $\gamma$-rays flux is in units of $10^{-7}\,\rm
photons\leavevmode\nobreak\ s^{-1}\,cm^{-2}$. Figure 3: Weighted wavelet
$Z$-transform of the nightly binned and cut $R$ band LC (see text). Left
panel: the colored WWZ power in the time-period plane. Right panel: the time-
averaged WWZ power as a function of the period. The colored dashed curves
represent the corresponding local significance contours.

##  4 Results

###  4.1 Short-term Variability

The LCs from Jul 11 to Sep 14, 2020 (built as described in Section 3.2) are
shown in Figure 1. In Figure 2, we show the $R$ band LC along with the
$\gamma$-rays999The $\gamma$-rays LC is derived at the Large Area Telescope
Instrument Science Operations center in a “quick-look” analysis. These
preliminary flux estimates should be used with caution, so we shall use them
only for illustrative purposes. LC in the 0.1–300 GeV band for inter-band
comparison. The comparison reveals a good correlation between the optical and
$\gamma$-rays LCs. In general, the LCs could be split visually into two parts
– a pre-flare and a flare (see also Shablovinskaya et al., 2023).

The pre-flare LCs (untill the end of July 2020 = JD 2459062, the top panel of
Figure 1) are characterized by a smooth and gradual flux increase. Since the
beginning of August 2020, the flux increase has continued, but it is not as
smooth as in July. During the pre-flare period, we recorded the minimal $R$
band flux of 13.37 mJy (or calibrated magnitude of 14.0545 $\pm$ 0.0016,
telescope J) at JD 2459045.52063. The pre-flare is followed by a period of
flaring activity, namely the August 2020 flare, which starts in the first
decade of August and continues beyond the end of the time interval considered
in this paper. The maximal $R$ band flux of 109.88 mJy (or calibrated
magnitude of 11.8190 $\pm$ 0.0033, telescope A) for the monitoring period was
reached at JD 2459083.45823 – that is, soon after the August 2020 flare onset.
Unfortunately, the period between the flare onset and the flare peak is very
sparsely covered by data points, so we cannot study the shape of the rising
part of the August 2020 flare. According to the preliminary $\gamma$-rays LC,
it seems that the flux rise is steeper than the flux decay; that is, there is
an asymmetry. We also have no information about the optical intra-night
activity of BL Lacertae at that period – we have detected only a non-well-
sampled flare on Jul 31. We cannot, however, rule out the presence of other
flares during the rising phase of the August 2020 flare because of the sparse
sampling and the lack of intra-night monitoring sessions. On the other hand,
the decaying phase of the August 2020 flare shows the high activity of BL
Lacertae on intra-night timescales. That activity will be our focus from now
on: in what follows, we shall not consider the pre-flare, and all analysis
will be related to the August 2020 flare.

####  4.1.1 Searching for Periodicity

To search for periodicity in the STLCs of BL Lacertae, we used the Lomb-
Scargle periodogram (Lomb, 1976; Scargle, 1982) and weighted wavelet
$Z$-transform (WWZ, Foster, 1996) techniques. Before the periodicity search,
we performed nightly binning of our data following the approach of Agarwal et
al. (2021) – in this way, we removed the influence of the different number of
data points per night on the search results. We also cut out the weakly
variable part of the LCs (namely before JD = 2459075). Given our data, we
found no signs of periodicity for all bands using both techniques (Figure 3).
Recently, Jorstad et al. (2022) reported a detection of a transient
periodicity of 0.55 days in the $R$ band LC generated by the Whole Earth
Blazar Telescope; their WWZ time interval encompasses ours.

Figure 4: Spectral energy distribution for the individual nights. Note the
scatter in the $B$ band fluxes (see text). Figure 5: Dependence of the
spectral index on the $R$ band flux. The blue circles are the spectral indices
calculated using the $VRI$ bands, while the red squares are the spectral
indices calculated using the $BVRI$ ones. The error bars reflect the
variability amplitude in the cases when the intra-night monitoring data are
included in the spectral index calculation (see text).

####  4.1.2 Spectral Energy Distribution

For the nights of $BVRI$ observations, we built the SEDs as follows. If a
single measurement is available for the given night, then we use the
corresponding flux directly. If repeating observations were performed during
the given night, then we calculated the weighted mean fluxes for the
corresponding bands. The averaging was done over the same time interval for
the corresponding bands to avoid the influence of the different duration of
the INLCs on the mean value obtained. This time interval was taken to be the
duration of the shortest LC for the given night. The uncertainty of the mean
flux was taken to be the larger between (i) the weighted standard deviation
and (ii) the standard uncertainty of the weighted mean. The effective
wavelengths for the $BVRI$ bands were taken from Bessell et al. (1998). The
so-derived SEDs have been plotted in Figure 4 for all nights jointly.

To estimate the spectral index, we fitted a linear polynomial of the form
$\log(F_{\nu})=-\alpha\log(\nu)+\rm const$ to each SED. We used only $VRI$
bands in the fitting because of the large scatter of the $B$ band fluxes: for
most of the nights the $B$ band flux is below the power-law model expectation.
The similar behavior of the $B$ band measurements was discussed by Weaver et
al. (2020). They attributed this behavior to the combination of the wide $B$
filter band and the spectral shape of BL Lacertae.

We show in Figure 5 the relation between the spectral index and the $R$ band
flux. There is a hint of steepening of the spectral index as the flux
decreases. However, the overall spectral index behavior of BL Lacertae on
short-term timescales could be considered mildly chromatic – the dependence of
$\alpha$ on the flux level is weak. The median spectral index over the August
2020 flare was found to be $\langle\alpha_{VRI}\rangle_{\rm med}=0.885\pm
0.020$ (a standard deviation of 0.096).

For six nights, we were able to calculate the spectral index using the $BVRI$
bands – for these nights, the $B$ band flux behaves not so unusually (see
above). The corresponding median spectral index was calculated to be
$\langle\alpha_{BVRI}\rangle_{\rm med}=1.038\pm 0.025$ (a standard deviation
of 0.061). In any case, the inclusion of the $B$ band leads to slightly
steeper indices (Figure 5).

Figure 6: Intra-night LCs of BL Lacertae. The blue, green, red, and black
colored data points code $BVRI$ bands, respectively; the $B$ band offsets are
indicated. In each plot, the JDs are along the $x$-axis and the BL Lacertae
brightness in magnitudes is along the $y$-axis. The observation date and the
telescope used are indicated in each plot. Figure 6: Continued. Figure 6:
Continued. Figure 6: Continued.

Table 3: Results from the INV tests of BL Lacertae

Date(Telecope) | Band | $N$ |  $C$-test |  $F$-test |  $\chi^{2}$-test | Status | $A$
---|---|---|---|---|---|---|---
(yyyy mm dd) |  |  | $C_{1},C_{2},C$ | $F_{1},F_{2},F,F_{\rm c}(0.99),F_{\rm c}(0.999)$ | $\chi^{2}_{1},\chi^{2}_{2},\chi^{2}_{\rm av},\chi^{2}_{0.99},\chi^{2}_{0.999}$ |  | (%)
2020.07.31(K) | $R$ | 92 | 6.32, 6.62, 6.47 | 40.00, 43.89, 41.94, 1.63, 1.93 | 4955.9, 5553.5, 5254.5, 125.29, 138.44 | V | 18.10
2020.08.20(A) | $R$ | 370 | 3.61, 3.33, 3.47 | 13.14, 11.11, 12.06, 1.27, 1.38 | 18064.0, 4148.8, 11106.4, 435.12, 458.68 | V | 13.39
2020.08.20(B) | $B$ | 95 | 2.48, 2.59, 2.53 | 6.14, 6.72, 6.43, 1.62, 1.90 | 2182.1, 847.15, 1514.62, 128.80, 142.12 | V | 23.15
| $V$ | 108 | 4.24, 4.31, 4.28 | 18.09, 18.62, 18.36, 1.57, 1.83 | 9901.5, 2365.8, 6133.65, 143.94, 157.95 | V | 19.49
| $R$ | 118 | 3.90, 3.78, 3.84 | 15.25, 14.29, 14.77, 1.54, 1.78 | 8487.3, 1791.4, 5139.35, 155.50, 170.01 | V | 19.39
| $I$ | 118 | 4.68, 5.05, 4.87 | 21.92, 25.50, 23.71, 1.54, 1.78 | 18721.0, 31986.0, 25353.5, 155.50, 170.01 | V | 15.30
2020.08.21(A) | $R$ | 327 | 2.70, 3.10, 2.90 | 7.30, 9.60, 8.45, 1.29, 1.41 | 8576.2, 3202.1, 5889.15, 388.32, 410.64 | V | 10.19
2020.08.21(G) | $R$ | 144 | 0.87, 1.17, 1.02 | 0.75, 1.36, 1.06, 1.48, 1.63 | 353.6, 181.7, 267.7, 185.25, 201.00 | NV | –
2020.08.22(C) | $R$ | 25 | 1.99, 2.24, 2.11 | 3.96, 5.01, 4.49, 2.66, 3.73 | 466.93, 159.95, 313.44, 42.98, 51.18 | PV | 6.96
2020.08.23(C) | $R$ | 20 | 1.36, 1.80, 1.58 | 1.85, 3.24, 2.54, 3.03, 4.47 | 206.00, 83.94, 144.97, 36.19, 43.82 | NV | –
2020.08.23(E) | $B$ | 38 | 2.70, 2.60, 2.65 | 7.28, 6.77, 7.03, 2.18, 2.84 | 1363.2, 644.12, 1003.66, 59.89, 69.35 | V | 15.64
| $V$ | 40 | 2.58, 2.69, 2.63 | 6.64, 7.23, 6.94, 2.13, 2.76 | 1991.2, 745.50, 1368.35, 62.43, 72.05 | V | 12.50
| $R$ | 37 | 2.43, 2.70, 2.56 | 5.91, 7.27, 6.59, 2.20, 2.89 | 2093.8, 608.01, 1350.90, 58.62, 67.98 | V | 12.44
| $I$ | 36 | 2.63, 2.70, 2.66 | 6.91, 7.28, 7.10, 2.23, 2.93 | 3843.2, 788.51, 2315.85, 57.34, 66.62 | V | 13.46
2020.08.24(B) | $R$ | 18 | 0.68, 1.09, 0.88 | 0.46, 1.19, 0.83, 3.24, 4.92 | 33.26, 23.78, 28.52, 33.41, 40.79 | NV | –
2020.08.24(C) | $R$ | 23 | 0.81, 0.98, 0.89 | 0.65, 0.95, 0.80, 2.78, 3.98 | 59.27, 19.05, 39.16, 40.29, 48.27 | NV | –
2020.08.25(B) | $B$ | 54 | 8.18, 7.78, 7.98 | 66.88, 60.59, 63.73, 1.91, 2.38 | 8098.8, 4149.8, 6124.3, 79.84, 90.57 | V | 57.98
| $V$ | 54 | 10.57, 10.02, 10.29 | 111.74, 100.36, 106.05, 1.91, 2.38 | 14087.0, 4962.8, 9524.9, 79.84, 90.57 | V | 54.59
| $R$ | 949 | 9.87, 9.99, 9.93 | 97.41, 99.82, 98.62, 1.16, 1.22 | 282320.00, 90054.20, 186187.10, 1052.23, 1088.28 | V | 52.09
| $I$ | 54 | 9.37, 8.60, 8.98 | 87.70, 73.97, 80.84, 1.91, 2.38 | 16613.00, 3338.00, 9975.5 79.84, 90.57 | V | 47.70
2020.08.25(C) | $R$ | 12 | 2.82, 3.33, 3.08 | 8.00, 11.09, 9.53, 4.46, 7.76 | 336.15, 105.48, 220.82, 24.72, 31.26 | V | 9.27
2020.08.26(C) | $R$ | 16 | 2.72, 3.25, 2.99 | 7.40, 10.56, 8.98, 3.52, 5.53 | 335.33, 140.81, 238.07, 30.58, 37.70 | V | 12.68
2020.08.26(F) | $B$ | 299 | 9.34, 9.02, 9.18 | 87.31, 81.33, 84.32, 1.31, 1.43 | 46939.39, 28023.80, 37481.60, 357.72, 379.17 | V | 63.76
| $V$ | 14 |  | 14.13, 13.43, 13.78, 3.91, 6.41 | 6045.40, 2331.0, 4188.2, 27.69, 34.53 | V | 52.09
| $R$ | 15 | 16.92, 16.73, 16.83 | 286.47, 280.05, 283.26, 3.70, 5.93 | 10953.42, 37940.38, 24446.90, 29.14, 36.12 | V | 49.40
| $I$ | 296 | 10.72, 10.69, 10.70 | 114.81, 114.30, 114.56, 1.31, 1.44 | 130575.58, 31436.29, 40502.97, 354.43, 375.79 | V | 47.89
2020.08.27(F) | $B$ | 332 | 4.05, 4.28, 4.17 | 16.43, 18.29, 17.36, 1.29, 1.41 | 10057.95, 7286.32, 8672.13, 393.78, 416.24 | V | 32.35
| $V$ | 18 | 3.96, 4.00, 3.98 | 15.71, 15.96, 15.83, 3.23, 4.92 | 676.71, 269.64, 473.17, 33.41, 40.79 | V | 24.77
| $R$ | 18 | 6.93, 7.23, 7.08 | 47.98, 52.25, 50.11, 3.23, 4.92 | 2306.1, 827.23, 1566.66, 33.41, 40.79 | V | 21.78
| $I$ | 331 | 3.70, 3.83, 3.76 | 13.71, 14.64, 14.17, 1.29, 1.41 | 17955.38, 4469.69, 11212.53, 392.69, 415.12 | V | 22.68
2020.08.28(B) | $R$ | 152 | 1.41, 1.24, 1.32 | 1.99, 1.53, 1.76, 1.46, 1.66 | 227.69, 940.44, 584.06, 194.34, 210.44 | NV | –
2020.08.28(C) | $R$ | 10 | 4.42, 4.27, 4.35 | 19.52, 18.20, 18.86 5.35, 10.11 | 651.35, 144.66, 398.01, 21.67, 27.88 | V | 12.57
2020.08.28(F) | $B$ | 328 | 4.69, 4.41, 4.55 | 21.97, 19.41, 20.69, 1.29, 1.41 | 13419.00, 7752.80, 10585.90, 389.42, 411.76 | V | 40.00
| $V$ | 18 | 8.05, 7.77, 7.91 | 64.76, 60.38, 62.57, 3.24, 4.92 | 25557.00, 10055.00, 17806.00, 33.41, 40.79 | V | 24.97
| $R$ | 18 | 7.57, 7.57, 7.57 | 57.34, 57.28, 57.31, 3.24, 4.92 | 2567.80, 907.91, 1737.86, 33.41, 40.79 | V | 23.39
| $I$ | 348 | 3.63, 3.83, 3.73 | 13.21, 14.64, 13.93, 1.28, 1.39 | 17354.00, 4375.70, 10864.85, 411.21, 434.14 | V | 23.98
2020.08.29(D) | $R$ | 23 | 2.32, 2.90, 2.61 | 5.38, 8.38, 6.88, 2.78, 3.98 | 441.00, 168.93, 304.97, 40.29, 48.27 | V | 12.79
2020.08.30(K) | $R$ | 518 | 1.16, 1.36, 1.26 | 1.35, 1.86, 1.60, 1.23, 1.31 | 5849.1, 935.11, 3392.1, 594.73, 622.09 | PV | 7.59
2020.08.31(K) | $V$ | 77 | 3.00, 3.20, 3.10 | 8.98, 10.27, 9.62, 1.71, 2.05 | 3441.5, 766.02, 2103.8, 107.58, 119.85 | V | 11.59
| $R$ | 75 | 3.53, 3.53, 3.53 | 12.43, 12.43, 12.43, 1.73, 2.07 | 6268.8, 871.79, 3570.0, 105.20, 117.35 | V | 9.59
| $I$ | 73 | 2.38, 2.74, 2.56 | 5.65, 7.51, 6.58, 1.74, 2.09 | 4278.14, 517.97, 2398.05, 102.82, 114.83 | V | 10.09
2020.09.02(B) | $B$ | 30 | 0.42, 0.92, 0.67 | 0.18, 0.86, 0.52, 2.42, 3.29 | 15.61, 26.43, 21.02, 49.59, 58.30 | NV | –
| $V$ | 30 | 0.94, 1.13, 1.04 | 0.89, 1.27, 1.08, 2.42, 3.29 | 89.05, 34.49, 61.77, 49.59, 58.30 | NV | –
| $R$ | 1000 | 2.33, 2.70, 2.52 | 5.45, 7.28, 6.37, 1.16, 1.22 | 20214.00, 7039.00, 13626.50, 1105.92, 1142.85 | V | 11.88
| $I$ | 30 | 0.66, 1.01, 0.84 | 0.43, 1.02, 0.73, 2.42, 3.29 | 58.15, 28.05, 43.10, 49.59, 58.30 | NV | –
2020.09.03(A) | $B$ | 45 | 2.87, 3.11, 2.99 | 8.22, 9.70, 8.96, 2.04, 2.60 | 1150.3, 617.43, 883.86, 68.71, 78.75 | V | 19.56
| $V$ | 39 | 2.67, 2.50, 2.59 | 7.16, 6.27, 6.71, 2.16, 2.80 | 601.69, 197.10, 399.39, 61.16, 70.70 | V | 15.36
| $R$ | 27 | 1.51, 1.49, 1.50 | 2.28, 2.23, 2.25, 2.55, 3.53 | 178.76, 50.45, 114.60, 45.64, 54.05 | PV | 7.86
| $I$ | 28 | 1.52, 1.96, 1.73 | 2.30, 3.82, 3.06, 2.51, 3.44 | 221.04, 82.88, 151.96, 46.96, 55.48 | PV | 6.87
2020.09.03(B) | $B$ | 13 | 2.54, 3.15, 2.85 | 6.43, 9.91, 8.17, 4.15, 7.00 | 159.97, 119.12, 139.55, 26.22, 32.91 | V | 13.63
| $V$ | 17 | 3.75, 4.07, 3.91 | 14.09, 16.55, 15.32, 3.37, 5.21 | 1113.30, 413.97, 763.64, 32.00, 39.25 | V | 10.51
| $R$ | 553 | 2.85, 3.01, 2.93 | 8.13, 9.08, 8.61, 1.22, 1.30 | 1588.70, 5406.10, 3497.40, 632.22, 660.40 | V | 11.68
| $I$ | 15 | 2.40, 2.82, 2.61 | 5.74, 7.95, 6.85, 3.70, 5.93 | 377.34, 114.85, 246.10, 29.14, 36.12 | V | 6.77
2020.09.05(D) | $B$ | 25 | 2.63, 2.73, 2.68 | 6.90, 7.43, 7.17, 2.60, 3.73 | 407.17, 260.16, 333.67, 42.98, 51.18 | V | 24.22
| $V$ | 25 | 4.21, 4.91, 4.56 | 17.76, 24.15, 20.96, 2.66, 3.73 | 1145.60, 599.41, 872.51, 42.98, 51.18 | V | 14.78
| $R$ | 46 | 4.78, 5.22, 5.00 | 22.86, 27.28, 25.07, 2.02, 2.57 | 3506.20, 1310.00, 2408.10, 69.96, 80.08 | V | 14.09
2020.09.06(D) | $R$ | 87 | 3.34, 3.25, 3.30 | 11.18, 10.56, 10.87 1.66, 1.96 | 3189.50, 869.83, 2029.67, 119.41, 132.28 | V | 12.19
2020.09.07(A) | $B$ | 30 | 1.62, 2.04, 1.83 | 2.62, 4.16, 3.39, 2.42, 3.29 | 162.77, 144.31, 153.54, 49.59, 58.30 | PV | 8.60
| $V$ | 29 | 2.18, 2.73, 2.46 | 4.75, 7.43, 6.09, 2.46, 3.36 | 331.61, 207.57, 269.59, 48.28, 56.89 | PV | 9.47
| $R$ | 24 | 2.09, 2.82, 2.46 | 4.35, 7.94, 6.15, 2.72, 3.85 | 290.08, 170.85, 230.47, 41.64, 49.73 | PV | 8.08
| $I$ | 30 | 1.89, 2.33, 2.11 | 3.56, 5.45, 4.51, 2.42, 3.29 | 396.10, 151.15, 273.63, 49.59, 58.30 | PV | 7.28
2020.09.07(D) | $R$ | 34 | 5.22, 5.41, 5.32 | 27.22, 29.25, 28.24, 2.29, 3.04 | 2622.70, 814.59, 1718.65, 54.77, 63.87 | V | 17.09
2020.09.08(A) | $B$ | 32 | 2.85, 3.30, 3.08 | 8.14, 10.91, 9.53, 2.35, 3.15 | 504.54, 381.87, 443.21, 52.19, 61.10 | V | 15.33
| $V$ | 33 | 4.01, 3.47, 3.74 | 16.09, 12.02, 14.06, 2.32, 3.09 | 1074.60, 361.48, 718.04, 53.48, 62.49 | V | 13.67
| $R$ | 32 | 2.88, 2.28, 2.58 | 8.27, 5.20, 6.73, 2.35, 3.15 | 615.64, 143.07, 379.36, 52.19, 61.10 | V | 12.78
| $I$ | 30 | 2.79, 2.30, 2.55 | 7.51, 4.46, 5.99, 2.42, 3.29 | 684.74, 117.63, 401.19, 49.59, 58.30 | V | 9.98
2020.09.08(B) | $R$ | 236 | 3.58, 3.58, 3.58 | 12.78, 12.84, 12.81, 1.36, 1.50 | 6944.60, 2597.40, 4771.00, 288.35, 307.73 | V | 8.98
2020.09.09(A) | $B$ | 43 | 2.05, 1.68, 1.86 | 4.20, 2.78, 3.50, 2.07, 2.66 | 315.49, 127.60, 221.54, 66.21, 76.08 | PV | 8.90
| $V$ | 45 | 1.44, 1.06, 1.25 | 2.07, 1.13, 1.60, 2.04, 2.60 | 177.93, 44.32, 111.12, 68.71, 76.08 | NV | –
| $R$ | 44 | 1.71, 1.25, 1.48 | 2.94, 1.56, 2.25, 2.06, 2.63 | 284.56, 58.14, 171.35, 67.46, 77.42 | NV | –
| $I$ | 45 | 1.55, 1.18, 1.36 | 2.41, 1.38, 1.90, 2.04, 2.60 | 320.68, 53.79, 187.23 68.71, 76.08 | NV | –
2020.09.09(B) | $R$ | 508 | 2.98, 3.02, 3.00 | 8.91, 9.13, 9.02, 1.23, 1.32 | 10540.00, 4086.20, 7313.10, 584.01, 611.13 | V | 7.88
2020.09.10(A) | $B$ | 49 | 5.11, 5.48, 5.29 | 26.09, 29.98, 28.03, 1.98, 2.49 | 2667.50, 1764.70, 2216.1, 73.68, 84.04 | V | 27.86
| $V$ | 50 | 10.12, 9.59, 9.85 | 102.41, 91.96, 97.18, 1.96, 2.46 | 11515.00, 4402.5, 7958.5, 74.92, 85.35 | V | 27.39
| $R$ | 217 | 11.18, 10.85, 11.02 | 125.05, 117.74, 121.40, 1.37, 1.52 | 65010.00, 22353.00, 43681.5, 267.27, 285.96 | V | 26.60
| $I$ | 54 | 6.75, 5.88, 6.31 | 45.57, 34.53, 40.05, 1.91, 2.38 | 7885.6, 1645.50, 4765.55, 79.84, 90.57 | V | 26.60
2020.09.10(B) | $R$ | 187 | 2.00, 1.90, 1.95 | 4.0, 3.62, 3.81, 1.41, 1.58 | 1979.90, 620.30, 1300.1, 251.34, 233.79 | PV | 5.87
2020.09.10(K) | $R$ | 206 | 6.39, 6.54, 6.52 | 40.83, 44.14, 42.49, 1.39, 1.55 | 48841.00, 11330.00, 30086.0, 252.79, 271.00 | V | 21.60
2020.09.11(A) | $B$ | 242 | 7.91, 7.53, 7.72 | 62.57, 56.68, 59.62, 1.35, 1.49 | 31244.00, 16181.00, 23712.5, 294.99, 314.58 | V | 45.28
| $V$ | 59 | 14.87, 15.37, 15.12 | 221.04, 236.16, 228.60, 1.86, 2.18 | 30647.00, 13225.00, 21936, 85.95, 97.04 | V | 34.79
| $R$ | 62 | 9.22, 10.05, 9.64 | 85.02, 101.09, 93.05, 1.83, 2.24 | 14464.00, 5626.00, 10045.0, 89.59, 100.89 | V | 29.80
| $I$ | 61 | 7.93, 8.82, 8.38 | 62.96, 77.73, 70.35, 1.84, 2.25 | 13822.0, 4266.9, 9044.45, 88.38, 99.61 | V | 27.80
2020.09.11(B) | $R$ | 573 | 13.72, 13.58, 13.65 | 188.32, 184.30, 86.31, 1.21, 1.29 | 338540, 100030, 219285.0, 653.61, 682.24 | V | 30.90
2020.09.12(A) | $B$ | 41 | 0.99, 1.26, 1.13 | 0.99, 1.60, 1.29, 2.11, 2.73 | 65.51, 68.46, 66.98, 63.69, 73.40 | NV | –
| $V$ | 12 | 4.48, 5.23, 4.86 | 20.12, 27.41, 23.76, 4.46, 7.76 | 416.35, 265.35, 340.85, 24.72, 31.26 | V | 11.77
| $R$ | 370 | 5.10, 3.37, 5.24 | 26.06, 28.81, 27.44, 1.27, 1.38 | 22405.00, 9697.10, 16051.05, 435.12, 458.68 | V | 14.58
| $I$ | 11 | 3.82, 4.57, 4.20 | 14.59, 20.93, 17.76, 4.85, 8.75 | 421.66, 182.39, 302.02, 23.21, 29.59 | V | 10.19
2020.09.12(B) | $B$ | 56 | 0.64, 1.04, 0.84 | 0.41, 1.09, 0.75, 1.89, 2.34 | 47.20, 47.71, 47.45, 82.29, 93.17 | NV | –
2020.09.12(C) | $R$ | 17 | 2.99, 3.38, 3.18 | 8.92, 11.44, 10.18, 3.37, 5.21 | 432.23, 153.43, 292.83, 32.00, 39.25 | V | 13.68
2020.09.12(K) | $R$ | 241 | 4.00, 3.64, 3.82 | 16.01, 13.24, 14.63, 1.35, 1.50 | 2826.2, 14873.0, 8849.5, 291.68, 311.15 | V | 12.60
2020.09.13(A) | $B$ | 14 | 1.29, 2.23, 1.76 | 1.67, 4.97, 3.32, 3.90, 6.41 | 53.50, 77.75, 65.62, 27.69, 34.53 | NV | –
| $V$ | 232 | 4.63, 4.57, 4.60 | 21.41, 20.86, 21.13, 1.36, 1.50 | 11959.00, 4666.40, 8312.7, 289.92, 303.16 | V | 18.98
| $R$ | 18 | 3.57, 3.36, 3.46 | 12.71, 11.29, 12.00, 3.24, 4.92 | 215.50, 764.95, 490.22, 33.41, 40.79 | V | 13.28
| $I$ | 221 | 3.23, 2.79, 3.01 | 10.45, 7.79, 9.12, 1.37, 1.52 | 8438.00, 1750.70, 5094.35, 271.72, 290.56 | V | 15.19
2020.09.13(B) | $R$ | 376 | 3.39, 3.44, 3.41 | 11.47, 11.83, 11.65, 1.27, 1.38 | 17804.00, 5655.20, 11729.6, 441.63, 465.36 | NV | –
2020.09.13(K) | $R$ | 233 | 3.38, 3.90, 3.64 | 11.45, 15.18, 13.32, 1.36, 1.50 | 14482.00, 3359.80, 8921.0, 285.03, 304.30 | V | 9.00
2020.09.14(A) | $B$ | 18 | 5.91, 6.25, 6.08 | 34.90, 39.05, 36.98, 3.24, 4.92 | 1214.50, 724.54, 969.52, 33.41, 40.79 | V | 24.15
| $V$ | 224 | 9.22, 9.08, 9.15 | 85.03, 82.42, 83.73, 1.24, 1.33 | 44349.00, 16836.00, 30592.5, 275.05, 294.00 | V | 31.89
| $R$ | 18 | 8.67, 8.50, 8.59 | 75.16, 72.30, 73.73, 3.24, 4.92 | 3539.40, 1061.70, 2300.55, 33.41, 40.79 | V | 24.29
| $I$ | 223 | 7.12, 6.86, 7.02 | 51.67, 47.05, 49.36, 1.25, 1.34 | 40396.00, 9137.55, 24766.75, 513.01, 538.50 | V | 25.70
2020.09.14(B) | $R$ | 710 | 9.69, 10.17, 9.93 | 93.87, 103.47, 98.67, 1.19, 1.26 | 192840.0, 63176.0, 128008.0, 799.53, 831.09 | V | 28.60
2020.09.14(C) | $R$ | 13 | 5.93, 5.64, 5.78 | 35.08, 31.82, 33.45, 4.16, 7.00 | 2339.10, 367.92, 1353.51, 26.22, 32.91 | V | 22.27
2020.09.14(D) | $R$ | 26 | 3.44, 3.58, 3.51 | 11.82, 12.84, 12.33, 2.60, 3.63 | 962.49, 289.91, 626.20, 44.31, 52.62 | V | 22.58

Note. — Variability status is abbreviated as follows: V – variable, PV –
probably variable, NV – non-variable.

Figure 7: Combined $BVRI$ band LCs for the nights of intra-night flaring
activity. In each plot, the LCs are ordered from bottom to top as follows: $B$
band – blue, $V$ band – green, $R$ band – red, $I$ band – magenta. The $(V)RI$
band LCs are shifted for display purposes downward by the corresponding
offsets indicated in the plots. The (composite) fitting functions, used to
detrend the corresponding LCs, are overplotted (black solid lines); the
alternative fitting functions are plotted as black dashed lines (see text).
For the sake of clarity, (i) if a few data points are available for a given
band, then its LC is not shown and (ii) only the portion of the LC used in the
decomposition is shown. Figure 7: Continued. Figure 8: Color-magnitude
diagrams built using the non-corrected LCs that show INV. The fitted power-law
models are overplotted. Figure 8: Continued. Figure 9: Structure functions
built using the corrected LCs. For MWL LCs, only $R$ (or $I$) band SFs are
shown. The SPL function fits are overplotted with a red line. Figure 10:
Results from the cross-correlation analysis of the corrected MWL LCs. In each
plot the left panel shows the DCF (black lines) and its uncertainties (green
shaded area). The red solid and dashed lines indicate the significance levels
of 99% and 95%, respectively, while the red dotted line indicates the zero
correlation. The right panel in each plot shows the corresponding CCCD. Figure
11: Decomposition of the corrected MWL LCs. In each plot, we indicate the
evening date, the value of $t_{0}$, the bands plotted, and the corresponding
offsets used for display purposes. The bands are coded as follows: $B$ – blue,
$V$ – green, $R$ – red, $I$ – magenta. The blue dashed lines are the
individual flares to which the LC is decomposed, while the black solid line is
the model LC. The error bars are not shown for the sake of clarity. Figure 12:
Same as in Figure 11, but for the $R$-band-only LCs.

###  4.2 Intra-night Variability

To study the INV of BL Lacertae, we included those nights that have more than
two hours of monitoring. In this way, we got a total of 48 INLCs. They are
shown in Figure 6 and the results from the INV tests are summarized in Table
3.

We tested for variability in the INLCs of each telescope individually for a
total of 25 nights. For 22 of them, BL Lacertae was found to show variable
status, for two of them, probably variable status, and for one of them, non-
variable status. If we define101010A discussion about the duty cycle
definition could be found in Webb et al. (2021). the duty cycle as the number
of nights the blazar shows INV over the total number of nights the blazar
being monitored, then we found a duty cycle of 96% (the probably variable
cases considered variable) or 88% (the probably variable cases considered non-
variable).

After the magnitudes were transformed into fluxes, the multi-telescope data
for the given night and band were combined. In what follows, we shall use the
combined LCs unless otherwise specified. After the combination, we selected a
total of 18 nights of intra-night monitoring suitable to perform an analysis
of the INV of BL Lacertae; the corresponding LCs are of good sampling and show
flaring activity (Figure 7).

The so-combined LCs were then detrended – the (composite) fitting functions
used are shown in Figure 7 along with the LCs. The detrending of the Aug 26
$BI$ and Sep 11 $R$ band LCs deserves special attention. For these LCs, we
were not able to derive the shape of the smooth components that are to be
fitted because of the shape of the LCs themselves (Figure 7). So, we had to
take into account the data for the preceding night to get an idea of what the
smooth component looks like. According to Figure 1, the Aug 26 $R$ band flux
variations are superimposed onto a linearly decaying flux trend marked by a
blue dashed line. We used that fit to determine what regions to fit for the
$BI$ bands. For Sep 11, we also assumed a linear trend, but it is obvious that
alternative functional forms are also possible (Figure 1).

The above considerations show that the main source of uncertainty in the
detrending process is the unknown shape of the underlying, smooth variable
component. In general, the shape, assumed by us for each night, should be
considered as an approximate one; however, the determination of the accurate
shape of the smooth component is beyond the scope of the presented paper. To
test the influence of that shape on the LC decomposition, a few LCs were
detrended using alternative fitting functions (these functions are denoted in
Figure 7 with dashed lines). Another source of uncertainty is the choice of
regions free of flares. However, the choice of these regions is dependent to
some extent on the assumed shape of the underlying component, and so we shall
consider it as an uncertainty source of lower importance.

Generally, the presence of enough data points on the LC that could be
attributed to the smooth component is of utmost importance to estimate its
shape accurately. This requires dense sampling and the large duration of the
LCs that could be achieved performing “around-the-world” observations (e.g.
Bhatta et al., 2013).

####  4.2.1 Color Behaviour

The CMDs of BL Lacertae are shown in Figure 8 and the fitting results are
listed in Table 4; CMDs for the nights at which the MWL LCs are probably
variable or non-variable according to Table 3 were not analyzed. Most of the
non-corrected CMDs show significant BWB trends on intra-night timescales,
already observed by other authors (e.g. Papadakis et al., 2003). We found no
loops in the CMDs.

####  4.2.2 Structure Function

The SFs built using the corrected LCs are presented in Figure 9, and the
results from the SPL fits are listed in Table 5. We found no dependence of the
SF slopes on the bands, and so we weight-averaged all slopes together – their
mean value is $\langle\varrho\rangle_{\rm wt}=1.624\pm 0.007$ (a weighted
standard deviation of 0.275). Regarding the turnover point, its median value
(in the observer’s frame) over all nights and bands is $\langle\delta t_{\rm
to}\rangle_{\rm med}=36.1\pm 3.7$ min (a standard deviation of 19.8 min).

####  4.2.3 Cross-correlation Analysis

For each night of MWL LCs of good sampling, we calculated DCFs using the
original and detrended LCs and ICFs using the detrended LCs. For our further
analysis, we shall consider only the time lags obtained using the DCF, based
on the detrended LCs, while the results from the other two cross-correlation
functions will serve as a check: the consistency among the various values for
a given night and bands supports the reliability of the lag obtained. The DCFs
of BL Lacertae are shown in Figure 10, and the resulting lags are listed in
Table 6.

We have a total of seven nights suitable for cross-correlation analysis. To
consider a given time lag real, we require the lag under consideration to be
larger than (i) the modal sampling of the LCs, (ii) the bin size used to build
the DCF, and (iii) the lag uncertainties obtained by the FR/RSS method; in
addition, the DCF should exceed the 99% confidence limit, and there should be
consistency among the different cross-correlation functions used (see above).
From Figure 10 and Table 6 one can see that the lags satisfying the above
conditions are those for Aug 20 and Aug 26. In both cases, the variability at
shorter wavelengths is leading; that is, we have soft lags. The lag values
themselves are consistent with the previous lag estimates for BL Lacertae.

For Aug 20, the $VI$ band LCs sampling is larger, while the $R$ band LC
sampling is smaller than the lag found (Table 6). To check the reliability of
the lags obtained using such LCs, we performed the following test. We shifted
the detrended $R$ band LC with the measured $V$ vs $R$ time lag (2.2 min); we
choose $V$ band LC for this test because it is of worse sampling compared to
the $I$ band one (Figure 7). Then, the shifted $R$ band LC was interpolated
onto the $V$ band JDs. Finally, the $V$ band LC uncertainties were assigned to
the transformed $R$ band LC. The so-generated fake $V$ band LC was cross-
correlated with the original $R$ band LC – the time lag found is
$2.9^{+6.0}_{-4.8}\,\rm min$; that is, it is consistent with the lag found
using the original detrended $V$ band LC. Hence, we can conclude that the lags
obtained for Aug 20 are reliable and could be used for further analysis.
Regarding Aug 26, we were not able to estimate the significance levels because
of the specific LC shape (Figure 7).

The LCs used for the cross-correlation analysis are a combination of various
numbers of flares, and so the measured time lags are a kind of weight-averaged
lags over the individual flares (Xu et al., 2019). The attempts to measure the
lags using the individual flares, forming the INLCs, lead to inaccurate
results either because of the flare overlapping (mainly) or because of the bad
flare sampling.

Figure 13: Comparison of the timescales obtained after the decomposition of
the LCs detrended using two alternative fitting functions. The symbols denote
the bands as follows: $B$ – blue diamonds, $V$ – green triangles, $R$ – red
circles, $I$ – magenta squares. The timescales along the $x$-axis are those
adopted by us for the further analysis. The dotted line is the line of exact
correspondence. The solid line is the line corresponding to the clipped mean
difference between the timescales of 1.4 min. Figure 14: Plot of the $I$ band
decay timescales against the $B$ band (blue diamonds) and $R$ band (red
circles) ones. The dotted line is the line of exact correspondence. Figure 15:
Distribution of the decay timescales jointly for all bands. Figure 16:
Distribution of the flare durations jointly for all bands.

####  4.2.4 Decomposition of the INLCs

The decomposition of the detrended LCs was done employing a non-linear least-
squares technique implemented into the MPFIT fitter (Markwardt, 2009). If (i)
a flare is not fully recorded, (ii) a flare is of low amplitude, or (iii)
flares overlap to a great extent, then we used a symmetric DE function for
fitting. In addition, if, for a flare, the fitted uncertainties are comparable
to or larger than the fitted values after a general DE fit, then we have
redone the decomposition using the symmetric DE function.

Once we have the flare model at hand, we need to estimate how many flares to
fit. For most of the LCs the number of flares to be fitted, $N_{\rm fla}$,
could easily be obtained; for complex or noisy LCs, however, that tack could
be difficult. Hence, to avoid the overfitting, we used the Bayesian
Information Criterion (BIC, Schwarz, 1978) to get the final estimate of
$N_{\rm fla}$. The BIC penalizes the $\chi^{2}$ of the fit for the newly added
parameters as follows:

${\rm BIC}=\chi^{2}+N_{\rm pars}\,\ln(N_{\rm data}),$ (10)

where $N_{\rm pars}$ is the number of model free parameters and $N_{\rm data}$
the number of the data points of the fitted LC. Using BIC, we could identify
the number of flares beyond which the addition of a new flare does not
significantly improve the fit. To accept the addition of a new flare, we
required BIC to decrease by ten or larger: $\Delta{\rm BIC}={\rm BIC({\it
N_{\rm fla}}+1)}-{\rm BIC({\it N_{\rm fla}})}\geq 10$. The decompositions are
shown in Figures 11 and 12; the fitted parameters are listed in Table 7.

As we mentioned in Section 4.2, the unknown shape of the smooth variability
component is the main source of systematic uncertainties in the timescales. To
make a crude estimate of these uncertainties, we compare in Figure 13 the
timescales obtained using two alternative fitting functions to detrend the
original LC (see Figure 7). The mean difference between the timescales was
found to be 1.4 min with a standard deviation of 3.9 min; these values were
obtained after the most deviant data points were clipped out. These results
give a crude estimate of the systematic uncertainty of the timescales due to
the unknown shape of the underlying smooth component. The difference, however,
is within the scatter of individual data points, and so we shall neglect it in
our further considerations.

Next, we searched for the dependence of the derived decay timescales on the
band. We plot in Figure 14 the $I$ band timescales against the $BR$ band ones:
one can see the lack of significant dependence of ${\mathcal{T}}_{\rm d}$ on
the band; the same applies for the rise timescales as long as all of the flare
fits are done using symmetric DE functions (we have four exceptions of this).
Hence, we plot the distribution of the decay timescales jointly for all bands
(Figure 15) – the clipped modal value is $\langle{\mathcal{T}}_{\rm
d}\rangle_{\rm mode}=11.6^{+10.5}_{-5.1}$ min. The lack of dependence on the
band was found for the flare duration as well, and so we plot in Figure 16 the
distribution of the flare duration altogether for all bands – the clipped
modal value is $\langle\Delta{\mathcal{T}}\rangle_{\rm
mode}=46.6^{+41.0}_{-20.6}$ min. The parameter uncertainties listed above
represent the 16-th and 84-th percentiles of the corresponding distributions.
Finally, using the four asymmetric flares, we calculated a weighted mean
asymmetry parameter $\langle\xi\rangle_{\rm wt}=0.49\pm 0.10$.

Table 4: Results from the power-law fits to the non-corrected CMDs

Date, 2020 | CMD | $\varpi$ | $r$ | $p$ | Trend
---|---|---|---|---|---
Aug 20 |  $F_{B}/F_{I}$ vs $F_{R}$ | 0.265 $\pm$ | 0.020 | 0. | 636 | $<\\!10^{-5}$ | BWB
|  $F_{V}/F_{I}$ vs $F_{R}$ | 0.211 $\pm$ | 0.011 | 0. | 836 | $<\\!10^{-5}$ |
Aug 23 |  $F_{B}/F_{I}$ vs $F_{R}$ | 0.260 $\pm$ | 0.045 | 0. | 555 | $<\\!10^{-3}$ | …
|  $F_{V}/F_{I}$ vs $F_{R}$ | 0.036 $\pm$ | 0.032 | 0. | 058 | 0.739 |
Aug 25 |  $F_{B}/F_{I}$ vs $F_{R}$ | 0.222 $\pm$ | 0.010 | 0. | 909 | $<\\!10^{-5}$ | BWB
|  $F_{V}/F_{I}$ vs $F_{R}$ | 0.147 $\pm$ | 0.007 | 0. | 898 | $<\\!10^{-5}$ |
Aug 26 |  $F_{B}/F_{I}$ vs $\frac{F_{B}+F_{I}}{2}$ | 0.260 $\pm$ | 0.006 | 0. | 916 | $<\\!10^{-5}$ | BWB
Aug 27 |  $F_{B}/F_{I}$ vs $\frac{F_{B}+F_{I}}{2}$ | 0.357 $\pm$ | 0.014 | 0. | 757 | $<\\!10^{-5}$ | BWB
Aug 28 |  $F_{B}/F_{I}$ vs $\frac{F_{B}+F_{I}}{2}$ | 0.525 $\pm$ | 0.012 | 0. | 879 | $<\\!10^{-5}$ | BWB
Aug 31 |  $F_{V}/F_{I}$ vs $F_{R}$ | 0.201 $\pm$ | 0.024 | 0. | 533 | $<\\!10^{-5}$ | BWB
Sep 3 |  $F_{B}/F_{I}$ vs $F_{R}$ | 0.307 $\pm$ | 0.132 | 0. | 548 | 0.004 | BWB
|  $F_{V}/F_{I}$ vs $F_{R}$ | 0.336 $\pm$ | 0.060 | 0. | 751 | $<\\!10^{-5}$ |
Sep 8 |  $F_{B}/F_{I}$ vs $F_{R}$ | 0.289 $\pm$ | 0.051 | 0. | 688 | $<\\!10^{-5}$ | BWB
|  $F_{V}/F_{I}$ vs $F_{R}$ | 0.296 $\pm$ | 0.035 | 0. | 866 | $<\\!10^{-5}$ |
Sep 10 |  $F_{B}/F_{I}$ vs $F_{R}$ |  $-$0.030 $\pm$ | 0.015 |  $-$0. | 173 | 0.210 | …
|  $F_{V}/F_{I}$ vs $F_{R}$ | 0.062 $\pm$ | 0.010 | 0. | 512 | $<\\!10^{-3}$ |
Sep 11 |  $F_{B}/F_{I}$ vs $F_{R}$ | 0.317 $\pm$ | 0.011 | 0. | 923 | $<\\!10^{-5}$ | BWB
|  $F_{V}/F_{I}$ vs $F_{R}$ | 0.189 $\pm$ | 0.007 | 0. | 936 | $<\\!10^{-5}$ |
Sep 13 |  $F_{V}/F_{I}$ vs $F_{R}$ | 0.195 $\pm$ | 0.014 | 0. | 666 | $<\\!10^{-5}$ | BWB
Sep 14 |  $F_{V}/F_{I}$ vs $F_{R}$ | 0.206 $\pm$ | 0.005 | 0. | 901 | $<\\!10^{-5}$ | BWB

Note. — To derive the values of $\varpi$, $r$, and $p$, the CMDs were fitted
in a “log-log” form.

Table 5: Results from the SF fits

Date, 2020 | Band | Bin Size | $\varrho$ | $\delta t_{\rm to}$
---|---|---|---|---
|  | (min) |  | (min)
Aug 20 | $B$ | 2.50 | $0.86\pm 0.11$ | 46.8
| $V$ | 2.50 | $1.72\pm 0.08$ | 44.1
| $R$ | 2.50 | $1.31\pm 0.03$ | 44.1
| $I$ | 2.50 | $1.61\pm 0.06$ | 44.1
Aug 21 | $R$ | 1.50 | $1.34\pm 0.06$ | 28.1
Aug 25 | $R$ | 1.50 | $1.55\pm 0.03$ | 29.7
Aug 26 | $B$ | 2.50 | $2.00\pm 0.02$ | 94.8
| $I$ | 2.50 | $2.00\pm 0.02$ | 94.8
Aug 27 | $B$ | 2.50 | $0.89\pm 0.08$ | 52.1
| $I$ | 2.50 | $1.17\pm 0.09$ | 33.4
Aug 28 | $B$ | 2.50 | $1.62\pm 0.04$ | 49.4
| $I$ | 2.50 | $1.49\pm 0.04$ | 52.1
Aug 30 | $R$ | 1.00 | $1.15\pm 0.09$ | 16.6
Aug 31 | $V$ | 2.00 | $1.30\pm 0.10$ | 26.7
| $R$ | 2.00 | $1.86\pm 0.18$ | 28.9
| $I$ | 2.00 | $1.79\pm 0.20$ | 28.9
Sep 2 | $R$ | 1.50 | $1.07\pm 0.06$ | 21.6
Sep 3 | $R$ | 2.50 | $1.36\pm 0.05$ | 36.1
Sep 8 | $R$ | 1.50 | $1.51\pm 0.05$ | 24.8
Sep 9 | $R$ | 2.50 | $1.58\pm 0.02$ | 49.4
Sep 10 | $R$ | 1.50 | $1.55\pm 0.05$ | 18.4
Sep 11 | $R$ | 2.50 | $1.64\pm 0.02$ | 46.8
Sep 12 | $R$ | 2.00 | $0.93\pm 0.10$ | 31.0
Sep 13 | $V$ | 2.00 | $1.16\pm 0.08$ | 65.2
| $R$ | 1.75 | $1.30\pm 0.06$ | 43.9
| $I$ | 2.00 | $1.36\pm 0.08$ | 52.4
Sep 14 | $V$ | 1.75 | $1.57\pm 0.17$ | 23.4
| $R$ | 1.25 | $1.50\pm 0.02$ | 31.4
| $I$ | 1.75 | $1.97\pm 0.26$ | 29.0

Note. — Column 3: Bin sizes used to build the SFs. Column 5: Position of the
SF turn-off point in the observer’s frame; the SPL is fitted up to this point.

Table 6: Results from the cross-correlation analysis of the LCs

Date, 2020 | DCF | Sampling | $\tau$ | Bin Size | Detrended?
---|---|---|---|---|---
|  | (min) | (min) | (min) |
Aug 20 |  $B$ vs $R$ | 3.24, 0.63 | $+4.9^{+2.1}_{-2.5}$ | 2.00 | Yes
|  |  | $+4.4^{+4.1}_{-2.9}$ | – | Yes
|  |  | $+4.3^{+2.7}_{-2.3}$ | 2.00 | No
|  $V$ vs $R$ | 3.24, 0.63 | $+2.2^{+1.9}_{-1.8}$ | 2.00 | Yes
|  |  | $+1.5^{+1.8}_{-2.9}$ | – | Yes
|  |  | $+1.0^{+2.0}_{-2.0}$ | 2.00 | No
|  $I$ vs $R$ | 3.24, 0.63 | $-2.9^{+2.0}_{-2.0}$ | 2.00 | Yes
|  |  | $-2.6^{+2.3}_{-1.2}$ | – | Yes
|  |  | $-1.0^{+2.0}_{-2.0}$ | 2.00 | No
Aug 26 |  $B$ vs $I$ | 1.40, 1.41 | $+3.8^{+2.5}_{-1.3}$ | 2.50 | Yes
|  |  | $+3.4^{+2.8}_{-2.8}$ | – | Yes
|  |  | $+6.2^{+2.5}_{-2.5}$ | 2.50 | No
Aug 27 |  $B$ vs $I$ | 1.44, 1.44 | $-2.5^{+2.2}_{-2.1}$ | 2.50 | Yes
|  |  | $-0.8^{+2.8}_{-2.8}$ | – | Yes
|  |  | $+1.0^{+2.0}_{-2.1}$ | 2.00 | No
Aug 28 |  $B$ vs $I$ | 1.44, 1.43 | $+0.4^{+2.2}_{-2.2}$ | 1.75 | Yes
|  |  | $+0.6^{+1.4}_{-0.0}$ | – | Yes
|  |  | $+4.4^{+2.3}_{-1.8}$ | 1.75 | No
Aug 31 |  $V$ vs $I$ | 2.01, 2.01 | $+1.8^{+2.3}_{-2.1}$ | 2.00 | Yes
|  |  | $+1.6^{+1.6}_{-1.6}$ | – | Yes
|  |  | $+0.0^{+2.7}_{-2.0}$ | 2.00 | No
|  $R$ vs $I$ | 2.01, 2.01 | $-0.3^{+2.0}_{-2.4}$ | 2.00 | Yes
|  |  | $-2.3^{+2.3}_{-0.0}$ | – | Yes
|  |  | $+0.0^{+2.0}_{-2.1}$ | 2.00 | No
Sep 13 |  $V$ vs $R$ | 1.86, 0.72 | $-4.5^{+4.0}_{-3.0}$ | 1.50 | Yes
|  |  | $+0.7^{+2.7}_{-3.3}$ | – | Yes
|  |  | $+3.1^{+3.6}_{-2.3}$ | 1.50 | No
|  $I$ vs $R$ | 1.87, 0.72 | $-3.6^{+2.4}_{-1.9}$ | 1.50 | Yes
|  |  | $-1.3^{+3.3}_{-3.3}$ | – | Yes
|  |  | $+4.5^{+2.3}_{-2.1}$ | 1.50 | No
Sep 14 |  $V$ vs $R$ | 1.86, 0.68 | $+1.3^{+1.2}_{-1.2}$ | 1.25 | Yes
|  |  | $+1.3^{+1.3}_{-1.3}$ | – | Yes
|  |  | $+1.5^{+1.5}_{-0.8}$ | 1.50 | No
|  $I$ vs $R$ | 1.87, 0.68 | $+0.0^{+1.2}_{-0.9}$ | 1.25 | Yes
|  |  | $+0.7^{+1.3}_{-2.0}$ | – | Yes
|  |  | $+0.0^{+1.5}_{-1.5}$ | 1.50 | No

Note. — Time lags are in the observer’s frame. In our DCF notation, namely
“band1” vs “band2”, the positive lag means that the variability at “band1” is
the leading one (see also Section 5.1). Column 2: Cross-correlated LCs. Column
3: Modal sampling of the cross-correlated LCs. Column 4: Time lag and its
lower and upper uncertainties. Zero lower uncertainties are due to the
strongly asymmetric shape of the lag distribution. Column 5: Bin size used to
build the DCF. The lags with no bin size specified are obtained by means of
the ICF. Column 6: Indication whether the used LCs are detrended or not.

Table 7: Results from the corrected LC decompositions

Date, 2020 | Band | $F_{0}$ | $\Delta t_{0}$ | ${\mathcal{T}}_{\rm r}$ | ${\mathcal{T}}_{\rm d}$ | $\Delta{\mathcal{T}}$ | $\sigma_{\rm fit}$
---|---|---|---|---|---|---|---
|  | (mJy) | (min) | (min) | (min) | (min) | (mJy)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
Aug 20 | $R$ | 16.6 $\pm$ | 0.2 | 240.6 $\pm$ | 0.7 | 20.3 $\pm$ | 0.6 | 20.3 $\pm$ | 0.6 | 81.2 $\pm$ | 1.7 | 0.41
|  | 6.8 $\pm$ | 0.3 | 434.1 $\pm$ | 1.2 | 15.8 $\pm$ | 0.9 | 15.8 $\pm$ | 0.9 | 63.2 $\pm$ | 2.5 |
|  | 4.8 $\pm$ | 0.6 | 472.9 $\pm$ | 1.8 | 12.7 $\pm$ | 2.7 | 12.7 $\pm$ | 2.7 | 50.8 $\pm$ | 7.6 |
|  | 3.4 $\pm$ | 0.8 | 497.5 $\pm$ | 1.8 | 9.9 $\pm$ | 1.5 | 9.9 $\pm$ | 1.5 | 39.6 $\pm$ | 4.2 |
|  | 5.0 $\pm$ | 0.2 | 583.9 $\pm$ | 0.9 | 15.0 $\pm$ | 0.8 | 15.0 $\pm$ | 0.8 | 60.0 $\pm$ | 2.3 |
|  | 4.5 $\pm$ | 0.2 | 620.6 $\pm$ | 0.6 | 10.3 $\pm$ | 0.9 | 10.3 $\pm$ | 0.9 | 41.2 $\pm$ | 2.5 |
| $I$ | 19.8 $\pm$ | 0.3 | 237.7 $\pm$ | 1.1 | 21.8 $\pm$ | 0.8 | 21.8 $\pm$ | 0.8 | 87.2 $\pm$ | 2.3 | 0.36
|  | 1.9 $\pm$ | 0.4 | 324.4 $\pm$ | 2.0 | 6.8 $\pm$ | 2.0 | 6.8 $\pm$ | 2.0 | 27.2 $\pm$ | 5.7 |
|  | 5.9 $\pm$ | 0.4 | 435.0 $\pm$ | 2.1 | 17.4 $\pm$ | 1.7 | 17.4 $\pm$ | 1.7 | 69.6 $\pm$ | 4.8 |
|  | 5.0 $\pm$ | 0.4 | 479.4 $\pm$ | 1.7 | 13.5 $\pm$ | 2.5 | 13.5 $\pm$ | 2.5 | 54.0 $\pm$ | 7.1 |
|  | 2.8 $\pm$ | 0.6 | 508.5 $\pm$ | 1.1 | 3.9 $\pm$ | 1.5 | 3.9 $\pm$ | 1.5 | 15.6 $\pm$ | 4.2 |
|  | 3.4 $\pm$ | 0.3 | 587.6 $\pm$ | 2.0 | 14.1 $\pm$ | 2.3 | 14.1 $\pm$ | 2.3 | 56.4 $\pm$ | 6.5 |
|  | 2.6 $\pm$ | 0.5 | 620.5 $\pm$ | 1.4 | 5.5 $\pm$ | 1.9 | 5.5 $\pm$ | 1.9 | 22.0 $\pm$ | 5.4 |
Aug 21 | $R$ | 2.0 $\pm$ | 0.1 | 56.4 $\pm$ | 0.7 | 5.3 $\pm$ | 0.4 | 16.5 $\pm$ | 1.1 | 43.6 $\pm$ | 2.3 | 0.41
|  | 2.4 $\pm$ | 0.2 | 432.6 $\pm$ | 0.6 | 5.0 $\pm$ | 0.7 | 5.0 $\pm$ | 0.7 | 20.0 $\pm$ | 2.0 |
|  | 3.3 $\pm$ | 0.2 | 459.2 $\pm$ | 0.8 | 9.8 $\pm$ | 0.8 | 9.8 $\pm$ | 0.8 | 39.2 $\pm$ | 2.3 |
|  | 2.8 $\pm$ | 0.2 | 565.6 $\pm$ | 0.6 | 6.5 $\pm$ | 0.6 | 6.5 $\pm$ | 0.6 | 26.0 $\pm$ | 1.7 |
|  | 4.4 $\pm$ | 0.1 | 595.0 $\pm$ | 0.4 | 8.1 $\pm$ | 0.4 | 8.1 $\pm$ | 0.4 | 32.4 $\pm$ | 1.1 |
|  | 9.0 $\pm$ | 0.9 | 805.3 $\pm$ | 2.0 | 8.0 $\pm$ | 1.3 | 27.5 $\pm$ | 4.2 | 71.0 $\pm$ | 8.8 |
Aug 25 | $R$ | 3.3 $\pm$ | 0.1 | 146.4 $\pm$ | 0.7 | 13.4 $\pm$ | 0.7 | 13.4 $\pm$ | 0.7 | 53.6 $\pm$ | 2.0 | 0.35
|  | 5.0 $\pm$ | 0.1 | 190.4 $\pm$ | 0.5 | 15.9 $\pm$ | 0.6 | 15.9 $\pm$ | 0.6 | 63.6 $\pm$ | 1.7 |
|  | 2.0 $\pm$ | 0.1 | 271.1 $\pm$ | 0.9 | 13.1 $\pm$ | 1.0 | 13.1 $\pm$ | 1.0 | 52.4 $\pm$ | 2.8 |
|  | 2.7 $\pm$ | 0.1 | 320.3 $\pm$ | 0.7 | 14.3 $\pm$ | 0.8 | 14.3 $\pm$ | 0.8 | 57.2 $\pm$ | 2.3 |
Aug 26 | $B$ | 13.2 $\pm$ | 5.0 | 4.8 $\pm$ | 8.7 | 47.8 $\pm$ | 34.5 | 47.8 $\pm$ | 34.5 | 191.2 $\pm$ | 97.6 | 0.54
|  | 14.5 $\pm$ | 7.5 | 86.2 $\pm$ | 10.7 | 45.8 $\pm$ | 8.1 | 45.8 $\pm$ | 8.1 | 183.2 $\pm$ | 22.9 |
|  | 3.9 $\pm$ | 0.4 | 212.9 $\pm$ | 4.3 | 37.2 $\pm$ | 3.6 | 37.2 $\pm$ | 3.6 | 148.8 $\pm$ | 10.2 |
| $R$ | 15.3 $\pm$ | 0.5 | 604.8 $\pm$ | 1.2 | 30.4 $\pm$ | 1.0 | 30.4 $\pm$ | 1.0 | 121.6 $\pm$ | 2.8 | 0.58
|  | 13.4 $\pm$ | 0.5 | 716.0 $\pm$ | 1.5 | 30.4 $\pm$ | 1.0 | 30.4 $\pm$ | 1.0 | 121.6 $\pm$ | 2.8 |
| $I$ | 20.0 $\pm$ | 3.3 | 7.5 $\pm$ | 4.0 | 44.3 $\pm$ | 14.2 | 44.3 $\pm$ | 14.2 | 177.2 $\pm$ | 40.2 | 0.76
|  | 23.7 $\pm$ | 4.8 | 88.3 $\pm$ | 4.7 | 43.2 $\pm$ | 4.1 | 43.2 $\pm$ | 4.1 | 172.8 $\pm$ | 11.6 |
|  | 7.3 $\pm$ | 0.5 | 201.8 $\pm$ | 2.6 | 35.8 $\pm$ | 2.2 | 35.8 $\pm$ | 2.2 | 143.2 $\pm$ | 6.2 |
Aug 27 | $B$ | 2.7 $\pm$ | 0.4 | 37.0 $\pm$ | 1.3 | 6.1 $\pm$ | 1.5 | 6.1 $\pm$ | 1.5 | 24.4 $\pm$ | 4.2 | 0.53
|  | 3.2 $\pm$ | 0.3 | 76.1 $\pm$ | 2.2 | 15.8 $\pm$ | 2.6 | 15.8 $\pm$ | 2.6 | 63.2 $\pm$ | 7.4 |
|  | 4.0 $\pm$ | 0.3 | 175.2 $\pm$ | 2.1 | 17.7 $\pm$ | 2.2 | 17.7 $\pm$ | 2.2 | 70.8 $\pm$ | 6.2 |
|  | 4.6 $\pm$ | 0.3 | 235.3 $\pm$ | 2.3 | 21.9 $\pm$ | 2.0 | 21.9 $\pm$ | 2.0 | 87.6 $\pm$ | 5.7 |
|  | 2.3 $\pm$ | 0.6 | 388.2 $\pm$ | 1.0 | 3.1 $\pm$ | 1.0 | 3.1 $\pm$ | 1.0 | 12.4 $\pm$ | 2.8 |
|  | 2.9 $\pm$ | 0.5 | 495.3 $\pm$ | 1.5 | 5.5 $\pm$ | 1.2 | 5.5 $\pm$ | 1.2 | 22.0 $\pm$ | 3.4 |
| $I$ | 4.9 $\pm$ | 0.5 | 37.5 $\pm$ | 1.0 | 6.8 $\pm$ | 1.1 | 6.8 $\pm$ | 1.1 | 27.2 $\pm$ | 3.1 | 0.68
|  | 5.0 $\pm$ | 0.3 | 77.9 $\pm$ | 1.5 | 15.0 $\pm$ | 1.7 | 15.0 $\pm$ | 1.7 | 60.0 $\pm$ | 4.8 |
|  | 7.6 $\pm$ | 0.3 | 179.8 $\pm$ | 1.1 | 16.6 $\pm$ | 1.1 | 16.6 $\pm$ | 1.1 | 66.4 $\pm$ | 3.1 |
|  | 5.1 $\pm$ | 0.3 | 230.5 $\pm$ | 1.5 | 14.4 $\pm$ | 1.5 | 14.4 $\pm$ | 1.5 | 57.6 $\pm$ | 4.2 |
|  | 4.9 $\pm$ | 0.7 | 390.2 $\pm$ | 0.4 | 2.0 $\pm$ | 0.4 | 2.0 $\pm$ | 0.4 | 8.0 $\pm$ | 1.1 |
|  | 6.4 $\pm$ | 0.4 | 494.0 $\pm$ | 0.8 | 8.7 $\pm$ | 0.6 | 8.7 $\pm$ | 0.6 | 34.8 $\pm$ | 1.7 |
Aug 28 | $B$ | 4.2 $\pm$ | 0.2 | 87.0 $\pm$ | 2.8 | 29.4 $\pm$ | 2.6 | 29.4 $\pm$ | 2.6 | 117.6 $\pm$ | 7.4 | 0.57
|  | 9.8 $\pm$ | 0.3 | 171.8 $\pm$ | 0.9 | 22.6 $\pm$ | 1.2 | 22.6 $\pm$ | 1.2 | 90.4 $\pm$ | 3.4 |
|  | 9.0 $\pm$ | 0.2 | 298.3 $\pm$ | 1.2 | 41.3 $\pm$ | 1.2 | 41.3 $\pm$ | 1.2 | 165.2 $\pm$ | 3.4 |
| $I$ | 6.0 $\pm$ | 0.2 | 78.5 $\pm$ | 1.7 | 27.0 $\pm$ | 1.8 | 27.0 $\pm$ | 1.8 | 108.0 $\pm$ | 5.1 | 0.85
|  | 13.5 $\pm$ | 0.2 | 171.5 $\pm$ | 0.7 | 24.8 $\pm$ | 0.9 | 24.8 $\pm$ | 0.9 | 99.2 $\pm$ | 2.5 |
|  | 14.3 $\pm$ | 0.2 | 295.6 $\pm$ | 0.8 | 41.6 $\pm$ | 0.9 | 41.6 $\pm$ | 0.9 | 166.4 $\pm$ | 2.5 |
Aug 30 | $R$ | 2.4 $\pm$ | 0.2 | 30.5 $\pm$ | 0.5 | 8.6 $\pm$ | 0.8 | 8.6 $\pm$ | 0.8 | 34.4 $\pm$ | 2.3 | 0.54
|  | 3.1 $\pm$ | 0.2 | 76.3 $\pm$ | 2.4 | 17.5 $\pm$ | 2.5 | 17.5 $\pm$ | 2.5 | 70.0 $\pm$ | 7.1 |
|  | 3.2 $\pm$ | 0.4 | 101.8 $\pm$ | 0.5 | 7.4 $\pm$ | 1.0 | 7.4 $\pm$ | 1.0 | 29.6 $\pm$ | 2.8 |
|  | 1.3 $\pm$ | 0.3 | 140.8 $\pm$ | 1.0 | 5.5 $\pm$ | 1.7 | 5.5 $\pm$ | 1.7 | 22.0 $\pm$ | 4.8 |
|  | 2.3 $\pm$ | 0.2 | 166.1 $\pm$ | 1.8 | 14.3 $\pm$ | 3.7 | 14.3 $\pm$ | 3.7 | 57.2 $\pm$ | 10.5 |
|  | 5.3 $\pm$ | 0.3 | 199.5 $\pm$ | 1.0 | 14.5 $\pm$ | 0.7 | 14.5 $\pm$ | 0.7 | 58.0 $\pm$ | 2.0 |
Aug 31 | $V$ | 4.5 $\pm$ | 0.4 | 58.5 $\pm$ | 1.0 | 9.0 $\pm$ | 1.0 | 9.0 $\pm$ | 1.0 | 36.0 $\pm$ | 2.8 | 0.36
|  | 3.8 $\pm$ | 0.2 | 87.4 $\pm$ | 1.8 | 13.1 $\pm$ | 1.6 | 13.1 $\pm$ | 1.6 | 52.4 $\pm$ | 4.5 |
|  | 1.5 $\pm$ | 0.3 | 136.6 $\pm$ | 1.3 | 4.2 $\pm$ | 1.3 | 4.2 $\pm$ | 1.3 | 16.8 $\pm$ | 3.7 |
| $R$ | 3.8 $\pm$ | 0.6 | 58.5 $\pm$ | 1.2 | 9.8 $\pm$ | 1.1 | 9.8 $\pm$ | 1.1 | 39.2 $\pm$ | 3.1 | 0.30
|  | 4.4 $\pm$ | 0.4 | 82.8 $\pm$ | 1.7 | 13.2 $\pm$ | 1.8 | 13.2 $\pm$ | 1.8 | 52.8 $\pm$ | 5.1 |
|  | 1.8 $\pm$ | 0.2 | 129.0 $\pm$ | 2.1 | 14.6 $\pm$ | 2.5 | 14.6 $\pm$ | 2.5 | 58.4 $\pm$ | 7.1 |
| $I$ | 5.5 $\pm$ | 0.2 | 61.3 $\pm$ | 0.9 | 10.5 $\pm$ | 0.7 | 10.5 $\pm$ | 0.7 | 42.0 $\pm$ | 2.0 | 0.50
|  | 3.5 $\pm$ | 0.6 | 85.1 $\pm$ | 0.9 | 6.8 $\pm$ | 1.6 | 6.8 $\pm$ | 1.6 | 27.2 $\pm$ | 4.5 |
|  | 2.2 $\pm$ | 0.3 | 109.5 $\pm$ | 4.7 | 15.6 $\pm$ | 3.1 | 15.6 $\pm$ | 3.1 | 62.4 $\pm$ | 8.8 |
Sep 2 | $R$ | 4.8 $\pm$ | 0.2 | 49.9 $\pm$ | 0.8 | 13.5 $\pm$ | 0.6 | 13.5 $\pm$ | 0.6 | 54.0 $\pm$ | 1.7 | 0.48
|  | 5.3 $\pm$ | 0.4 | 86.5 $\pm$ | 1.0 | 10.6 $\pm$ | 1.1 | 10.6 $\pm$ | 1.1 | 42.4 $\pm$ | 3.1 |
|  | 6.1 $\pm$ | 0.7 | 108.7 $\pm$ | 0.6 | 8.1 $\pm$ | 1.1 | 8.1 $\pm$ | 1.1 | 32.4 $\pm$ | 3.1 |
|  | 4.0 $\pm$ | 0.3 | 134.0 $\pm$ | 1.6 | 13.1 $\pm$ | 2.3 | 13.1 $\pm$ | 2.3 | 52.4 $\pm$ | 6.5 |
|  | 3.4 $\pm$ | 0.3 | 165.2 $\pm$ | 0.6 | 7.3 $\pm$ | 1.0 | 7.3 $\pm$ | 1.0 | 29.2 $\pm$ | 2.8 |
|  | 5.6 $\pm$ | 0.3 | 201.7 $\pm$ | 2.0 | 19.0 $\pm$ | 1.8 | 19.0 $\pm$ | 1.8 | 76.0 $\pm$ | 5.1 |
|  | 5.0 $\pm$ | 0.6 | 225.5 $\pm$ | 0.4 | 7.5 $\pm$ | 0.9 | 7.5 $\pm$ | 0.9 | 30.0 $\pm$ | 2.5 |
|  | 3.0 $\pm$ | 0.1 | 292.7 $\pm$ | 0.9 | 9.9 $\pm$ | 1.0 | 9.9 $\pm$ | 1.0 | 39.6 $\pm$ | 2.8 |
|  | 2.3 $\pm$ | 0.2 | 335.2 $\pm$ | 1.1 | 9.9 $\pm$ | 1.5 | 9.9 $\pm$ | 1.5 | 39.6 $\pm$ | 4.2 |
|  | 7.0 $\pm$ | 0.1 | 370.2 $\pm$ | 0.6 | 15.1 $\pm$ | 1.0 | 15.1 $\pm$ | 1.0 | 60.4 $\pm$ | 2.8 |
|  | 6.4 $\pm$ | 0.2 | 409.8 $\pm$ | 0.4 | 8.8 $\pm$ | 0.6 | 8.8 $\pm$ | 0.6 | 35.2 $\pm$ | 1.7 |
|  | 3.8 $\pm$ | 0.2 | 436.9 $\pm$ | 0.6 | 8.4 $\pm$ | 0.7 | 8.4 $\pm$ | 0.7 | 33.6 $\pm$ | 2.0 |
|  | 3.3 $\pm$ | 0.1 | 481.7 $\pm$ | 0.6 | 9.1 $\pm$ | 0.7 | 9.1 $\pm$ | 0.7 | 36.4 $\pm$ | 2.0 |
Sep 3 | $R$ | 1.7 $\pm$ | 0.1 | 58.3 $\pm$ | 1.5 | 24.5 $\pm$ | 2.0 | 24.5 $\pm$ | 2.0 | 98.0 $\pm$ | 5.7 | 0.37
|  | 1.1 $\pm$ | 0.1 | 113.1 $\pm$ | 0.9 | 5.0 $\pm$ | 1.0 | 5.0 $\pm$ | 1.0 | 20.0 $\pm$ | 2.8 |
|  | 3.8 $\pm$ | 0.6 | 174.3 $\pm$ | 1.3 | 14.4 $\pm$ | 1.7 | 14.4 $\pm$ | 1.7 | 57.6 $\pm$ | 4.8 |
|  | 5.1 $\pm$ | 0.8 | 215.9 $\pm$ | 1.9 | 22.6 $\pm$ | 4.6 | 22.6 $\pm$ | 4.6 | 90.4 $\pm$ | 13.0 |
|  | 4.0 $\pm$ | 0.8 | 262.4 $\pm$ | 4.2 | 24.7 $\pm$ | 3.7 | 24.7 $\pm$ | 3.7 | 98.8 $\pm$ | 10.5 |
|  | 4.2 $\pm$ | 0.9 | 317.6 $\pm$ | 1.0 | 6.5 $\pm$ | 1.4 | 6.5 $\pm$ | 1.4 | 26.0 $\pm$ | 4.0 |
|  | 3.6 $\pm$ | 0.3 | 386.0 $\pm$ | 2.1 | 15.5 $\pm$ | 1.9 | 15.5 $\pm$ | 1.9 | 62.0 $\pm$ | 5.4 |
Sep 6 | $R$ | 4.2 $\pm$ | 0.3 | 574.4 $\pm$ | 1.7 | 15.8 $\pm$ | 1.5 | 15.8 $\pm$ | 1.5 | 63.2 $\pm$ | 4.2 | 0.47
|  | 5.5 $\pm$ | 0.4 | 652.3 $\pm$ | 0.9 | 12.2 $\pm$ | 1.2 | 12.2 $\pm$ | 1.2 | 48.8 $\pm$ | 3.4 |
Sep 8 | $R$ | 1.1 $\pm$ | 0.2 | 21.0 $\pm$ | 0.6 | 3.2 $\pm$ | 0.7 | 3.2 $\pm$ | 0.7 | 12.8 $\pm$ | 2.0 | 0.20
|  | 2.1 $\pm$ | 0.1 | 41.7 $\pm$ | 0.6 | 8.4 $\pm$ | 0.6 | 8.4 $\pm$ | 0.6 | 33.6 $\pm$ | 1.7 |
|  | 1.3 $\pm$ | 0.1 | 122.0 $\pm$ | 0.9 | 12.9 $\pm$ | 1.0 | 12.9 $\pm$ | 1.0 | 51.6 $\pm$ | 2.8 |
Sep 9 | $R$ | 1.4 $\pm$ | 0.1 | 115.9 $\pm$ | 1.3 | 16.8 $\pm$ | 1.9 | 16.8 $\pm$ | 1.9 | 67.2 $\pm$ | 5.4 | 0.21
|  | 4.9 $\pm$ | 0.2 | 215.5 $\pm$ | 2.7 | 40.6 $\pm$ | 2.4 | 40.6 $\pm$ | 2.4 | 162.4 $\pm$ | 6.8 |
|  | 1.4 $\pm$ | 0.7 | 286.9 $\pm$ | 8.4 | 22.1 $\pm$ | 9.8 | 22.1 $\pm$ | 9.8 | 88.4 $\pm$ | 27.7 |
|  | 2.3 $\pm$ | 0.7 | 321.9 $\pm$ | 3.6 | 18.7 $\pm$ | 2.3 | 18.7 $\pm$ | 2.3 | 74.8 $\pm$ | 6.5 |
Sep 10 | $R$ | 0.8 $\pm$ | 0.2 | 31.3 $\pm$ | 0.9 | 3.1 $\pm$ | 0.9 | 3.1 $\pm$ | 0.9 | 12.4 $\pm$ | 2.5 | 0.26
|  | 2.1 $\pm$ | 0.1 | 90.1 $\pm$ | 0.9 | 23.4 $\pm$ | 1.1 | 23.4 $\pm$ | 1.1 | 93.6 $\pm$ | 3.1 |
|  | 3.0 $\pm$ | 0.1 | 200.8 $\pm$ | 0.3 | 8.3 $\pm$ | 0.3 | 8.3 $\pm$ | 0.3 | 33.2 $\pm$ | 0.8 |
|  | 2.2 $\pm$ | 0.1 | 261.1 $\pm$ | 0.2 | 4.4 $\pm$ | 0.2 | 4.4 $\pm$ | 0.2 | 17.6 $\pm$ | 0.6 |
|  | 7.5 $\pm$ | 0.0 | 314.4 $\pm$ | 0.3 | 15.2 $\pm$ | 0.2 | 15.2 $\pm$ | 0.2 | 60.8 $\pm$ | 0.6 |
|  | 2.7 $\pm$ | 0.1 | 339.5 $\pm$ | 0.2 | 6.1 $\pm$ | 0.3 | 6.1 $\pm$ | 0.3 | 24.4 $\pm$ | 0.8 |
|  | 2.3 $\pm$ | 0.3 | 390.1 $\pm$ | 0.6 | 5.9 $\pm$ | 0.7 | 5.9 $\pm$ | 0.7 | 23.6 $\pm$ | 2.0 |
|  | 4.8 $\pm$ | 0.3 | 408.8 $\pm$ | 0.6 | 9.2 $\pm$ | 1.4 | 9.2 $\pm$ | 1.4 | 36.8 $\pm$ | 4.0 |
|  | 4.3 $\pm$ | 0.3 | 440.6 $\pm$ | 1.2 | 14.7 $\pm$ | 2.0 | 14.7 $\pm$ | 2.0 | 58.8 $\pm$ | 5.7 |
|  | 3.0 $\pm$ | 0.2 | 481.1 $\pm$ | 1.5 | 16.3 $\pm$ | 1.4 | 16.3 $\pm$ | 1.4 | 65.2 $\pm$ | 4.0 |
Sep 11 | $R$ | 6.5 $\pm$ | 0.3 | 10.3 $\pm$ | 1.4 | 9.7 $\pm$ | 0.9 | 40.4 $\pm$ | 5.0 | 100.2 $\pm$ | 10.2 | 0.26
|  | 1.1 $\pm$ | 0.2 | 63.4 $\pm$ | 0.7 | 4.3 $\pm$ | 1.0 | 4.3 $\pm$ | 1.0 | 17.2 $\pm$ | 2.8 |
|  | 10.6 $\pm$ | 0.5 | 99.0 $\pm$ | 0.8 | 19.2 $\pm$ | 1.0 | 19.2 $\pm$ | 1.0 | 76.8 $\pm$ | 2.8 |
|  | 7.6 $\pm$ | 1.8 | 134.9 $\pm$ | 1.4 | 11.7 $\pm$ | 1.7 | 11.7 $\pm$ | 1.7 | 46.8 $\pm$ | 4.8 |
|  | 9.7 $\pm$ | 2.9 | 157.3 $\pm$ | 1.3 | 11.7 $\pm$ | 2.8 | 11.7 $\pm$ | 2.8 | 46.8 $\pm$ | 7.9 |
|  | 7.8 $\pm$ | 2.6 | 177.0 $\pm$ | 1.4 | 10.6 $\pm$ | 2.5 | 10.6 $\pm$ | 2.5 | 42.4 $\pm$ | 7.1 |
|  | 4.8 $\pm$ | 1.0 | 197.8 $\pm$ | 1.2 | 9.5 $\pm$ | 0.8 | 9.5 $\pm$ | 0.8 | 38.0 $\pm$ | 2.3 |
|  | 1.8 $\pm$ | 0.1 | 264.8 $\pm$ | 1.1 | 12.9 $\pm$ | 1.1 | 12.9 $\pm$ | 1.1 | 51.6 $\pm$ | 3.1 |
|  | 3.6 $\pm$ | 0.1 | 306.1 $\pm$ | 0.6 | 15.3 $\pm$ | 0.7 | 15.3 $\pm$ | 0.7 | 61.2 $\pm$ | 2.0 |
|  | 1.4 $\pm$ | 0.2 | 364.3 $\pm$ | 0.5 | 3.4 $\pm$ | 0.7 | 3.4 $\pm$ | 0.7 | 13.6 $\pm$ | 2.0 |
|  | 1.6 $\pm$ | 0.1 | 400.7 $\pm$ | 2.0 | 15.8 $\pm$ | 2.7 | 15.8 $\pm$ | 2.7 | 63.2 $\pm$ | 7.6 |
|  | 1.5 $\pm$ | 0.1 | 451.1 $\pm$ | 3.3 | 19.7 $\pm$ | 3.4 | 19.7 $\pm$ | 3.4 | 78.8 $\pm$ | 9.6 |
Sep 12 | $R$ | 1.2 $\pm$ | 0.1 | 83.5 $\pm$ | 1.8 | 6.0 $\pm$ | 1.3 | 6.0 $\pm$ | 1.3 | 24.0 $\pm$ | 3.7 | 0.22
|  | 1.2 $\pm$ | 0.3 | 191.5 $\pm$ | 1.5 | 9.7 $\pm$ | 2.9 | 9.7 $\pm$ | 2.9 | 38.8 $\pm$ | 8.2 |
|  | 1.3 $\pm$ | 0.1 | 225.2 $\pm$ | 4.6 | 22.0 $\pm$ | 4.5 | 22.0 $\pm$ | 4.5 | 88.0 $\pm$ | 12.7 |
|  | 1.4 $\pm$ | 0.1 | 276.5 $\pm$ | 0.9 | 14.4 $\pm$ | 1.2 | 14.4 $\pm$ | 1.2 | 57.6 $\pm$ | 3.4 |
|  | 1.7 $\pm$ | 0.1 | 328.9 $\pm$ | 0.4 | 10.7 $\pm$ | 0.4 | 10.7 $\pm$ | 0.4 | 42.8 $\pm$ | 1.1 |
|  | 0.6 $\pm$ | 0.1 | 370.9 $\pm$ | 0.5 | 2.9 $\pm$ | 0.6 | 2.9 $\pm$ | 0.6 | 11.6 $\pm$ | 1.7 |
|  | 0.7 $\pm$ | 0.1 | 389.1 $\pm$ | 0.8 | 7.1 $\pm$ | 0.8 | 7.1 $\pm$ | 0.8 | 28.4 $\pm$ | 2.3 |
|  | 0.5 $\pm$ | 0.1 | 434.3 $\pm$ | 0.6 | 2.5 $\pm$ | 0.6 | 2.5 $\pm$ | 0.6 | 10.0 $\pm$ | 1.7 |
|  | 1.3 $\pm$ | 0.1 | 461.1 $\pm$ | 0.4 | 7.7 $\pm$ | 0.4 | 7.7 $\pm$ | 0.4 | 30.8 $\pm$ | 1.1 |
Sep 13 | $V$ | 2.5 $\pm$ | 0.2 | 119.9 $\pm$ | 2.5 | 21.7 $\pm$ | 2.3 | 21.7 $\pm$ | 2.3 | 86.8 $\pm$ | 6.5 | 0.30
|  | 2.7 $\pm$ | 0.2 | 376.6 $\pm$ | 1.5 | 19.3 $\pm$ | 1.9 | 19.3 $\pm$ | 1.9 | 77.2 $\pm$ | 5.4 |
|  | 1.5 $\pm$ | 0.2 | 443.0 $\pm$ | 2.1 | 11.2 $\pm$ | 2.1 | 11.2 $\pm$ | 2.1 | 44.8 $\pm$ | 5.9 |
| $R$ | 1.5 $\pm$ | 0.3 | 36.9 $\pm$ | 0.7 | 2.6 $\pm$ | 0.7 | 2.6 $\pm$ | 0.7 | 10.4 $\pm$ | 2.0 | 0.35
|  | 3.7 $\pm$ | 0.3 | 91.2 $\pm$ | 1.1 | 11.6 $\pm$ | 1.0 | 11.6 $\pm$ | 1.0 | 46.4 $\pm$ | 2.8 |
|  | 3.9 $\pm$ | 0.3 | 123.6 $\pm$ | 1.4 | 14.2 $\pm$ | 1.7 | 14.2 $\pm$ | 1.7 | 56.8 $\pm$ | 4.8 |
|  | 1.0 $\pm$ | 0.1 | 177.2 $\pm$ | 6.8 | 23.9 $\pm$ | 4.7 | 23.9 $\pm$ | 4.7 | 95.6 $\pm$ | 13.3 |
|  | 1.3 $\pm$ | 0.1 | 260.7 $\pm$ | 0.6 | 5.5 $\pm$ | 0.5 | 5.5 $\pm$ | 0.5 | 22.0 $\pm$ | 1.4 |
|  | 2.0 $\pm$ | 0.0 | 384.3 $\pm$ | 0.4 | 13.7 $\pm$ | 0.6 | 13.7 $\pm$ | 0.6 | 54.8 $\pm$ | 1.7 |
|  | 0.6 $\pm$ | 0.1 | 436.7 $\pm$ | 0.9 | 7.9 $\pm$ | 0.9 | 7.9 $\pm$ | 0.9 | 31.6 $\pm$ | 2.5 |
| $I$ | 2.8 $\pm$ | 0.3 | 103.7 $\pm$ | 2.7 | 11.9 $\pm$ | 2.5 | 11.9 $\pm$ | 2.5 | 47.6 $\pm$ | 7.1 | 0.38
|  | 3.6 $\pm$ | 0.4 | 135.2 $\pm$ | 1.8 | 11.8 $\pm$ | 1.6 | 11.8 $\pm$ | 1.6 | 47.2 $\pm$ | 4.5 |
|  | 3.6 $\pm$ | 0.1 | 374.1 $\pm$ | 1.5 | 32.0 $\pm$ | 1.8 | 32.0 $\pm$ | 1.8 | 128.0 $\pm$ | 5.1 |
|  | 1.6 $\pm$ | 0.2 | 446.9 $\pm$ | 1.6 | 9.4 $\pm$ | 1.8 | 9.4 $\pm$ | 1.8 | 37.6 $\pm$ | 5.1 |
Sep 14 | $V$ | 4.3 $\pm$ | 0.5 | 395.0 $\pm$ | 2.0 | 12.5 $\pm$ | 1.3 | 12.5 $\pm$ | 1.3 | 50.0 $\pm$ | 3.7 | 0.36
|  | 5.5 $\pm$ | 0.4 | 422.0 $\pm$ | 1.3 | 11.5 $\pm$ | 1.2 | 11.5 $\pm$ | 1.2 | 46.0 $\pm$ | 3.4 |
|  | 6.3 $\pm$ | 0.2 | 489.5 $\pm$ | 0.7 | 17.9 $\pm$ | 0.9 | 17.9 $\pm$ | 0.9 | 71.6 $\pm$ | 2.5 |
| $R$ | 1.8 $\pm$ | 0.2 | 45.3 $\pm$ | 0.7 | 5.7 $\pm$ | 1.1 | 5.7 $\pm$ | 1.1 | 22.8 $\pm$ | 3.1 | 0.31
|  | 4.8 $\pm$ | 0.1 | 79.6 $\pm$ | 1.3 | 14.4 $\pm$ | 1.7 | 14.4 $\pm$ | 1.7 | 57.6 $\pm$ | 4.8 |
|  | 2.4 $\pm$ | 0.4 | 100.6 $\pm$ | 0.6 | 4.9 $\pm$ | 1.1 | 4.9 $\pm$ | 1.1 | 19.6 $\pm$ | 3.1 |
|  | 1.2 $\pm$ | 0.2 | 125.3 $\pm$ | 1.7 | 9.2 $\pm$ | 1.7 | 9.2 $\pm$ | 1.7 | 36.8 $\pm$ | 4.8 |
|  | 2.5 $\pm$ | 0.1 | 238.2 $\pm$ | 2.0 | 10.6 $\pm$ | 1.3 | 23.6 $\pm$ | 2.1 | 68.4 $\pm$ | 4.9 |
|  | 5.3 $\pm$ | 0.2 | 402.7 $\pm$ | 1.2 | 15.0 $\pm$ | 0.7 | 15.0 $\pm$ | 0.7 | 60.0 $\pm$ | 2.0 |
|  | 3.6 $\pm$ | 0.3 | 426.9 $\pm$ | 0.7 | 8.4 $\pm$ | 0.8 | 8.4 $\pm$ | 0.8 | 33.6 $\pm$ | 2.3 |
|  | 5.4 $\pm$ | 0.1 | 490.8 $\pm$ | 0.5 | 17.9 $\pm$ | 0.6 | 17.9 $\pm$ | 0.6 | 71.6 $\pm$ | 1.7 |
| $I$ | 4.7 $\pm$ | 0.7 | 398.4 $\pm$ | 1.8 | 12.3 $\pm$ | 1.2 | 12.3 $\pm$ | 1.2 | 49.2 $\pm$ | 3.4 | 0.42
|  | 5.2 $\pm$ | 0.5 | 424.6 $\pm$ | 1.9 | 13.6 $\pm$ | 1.6 | 13.6 $\pm$ | 1.6 | 54.4 $\pm$ | 4.5 |
|  | 7.4 $\pm$ | 0.2 | 490.1 $\pm$ | 0.6 | 18.4 $\pm$ | 0.8 | 18.4 $\pm$ | 0.8 | 73.6 $\pm$ | 2.3 |

Note. — Timescales are in the observer’s frame. Column 3: Twice the flare
amplitude. Column 4: Approximate position of the flare maximum (the actual
position is equal to $\Delta t_{0}$ only for symmetric flares). Column 5:
$e$-folding rise timescale. Column 6: $e$-folding decay timescale. Column 7:
Approximate duration of the flare. Column 8: Standard deviation about the
fitted sum of DE functions.

Table 8: Characteristics of the emitting regions

Parameter | Min | Max | Median | Mode
---|---|---|---|---
${\widetilde{\mathcal{B}}}_{\rm min}$ | 7.5 | 76.3 | $20.5^{+13.3}_{-6.4}$ | $17.7^{+16.1}_{-3.6}$
${\mathcal{B}}_{\rm min}(11.0)$ | 3.4 | 34.3 | $9.2^{+6.0}_{-2.9}$ | $8.0^{+7.2}_{-1.6}$
${\widetilde{\gamma}}_{\rm e,max}$ | 1189 | 5110 | $2540^{+584}_{-548}$ | $2470^{+655}_{-478}$
$\gamma_{\rm e,max}(11.0)$ | 535 | 2298 | $1142^{+263}_{-246}$ | $1111^{+295}_{-215}$
${\widetilde{\mathcal{R}}}_{\rm max}$ | 0.2 | 5.4 | $1.4^{+1.0}_{-0.8}$ | $1.2^{+1.2}_{-0.6}$
${\mathcal{R}}_{\rm max}(11.0)$ | 2.2 | 59.4 | $15.4^{+11.0}_{-8.8}$ | $13.2^{+13.2}_{-6.6}$

Note. — Magnetic field strength is in units of Gauss and the radius is in
astronomical units (AU). The mode is calculated using a clipping technique.
The uncertainties represent the 16-th and 84-th percentiles of the
corresponding distributions.

Figure 17: Time lag of the $BVI$ band variations (with respect to the $R$ band
ones) against the frequency of the corresponding bands (squares). The black
solid curve is fit for this frequency dependence. The black plus signs mark
the randomized lag values, while the green lines are the fits to each set of
randomized time lags (see text). We show the $3\sigma$ error bars for the sake
of comparison with the randomized lag values. Figure 18: Distribution of the
minimal magnetic field strengths. Figure 19: Distribution of the maximal
electron Lorentz factors. Figure 20: Distribution of the maximal radii.

##  5 Discussion

In this paper, we have presented the results from the optical monitoring of
the blazar BL Lacertae for the period Jul 11 – Sep 14, 2020, which encompasses
the August 2020 flare. During this period (more specifically, starting from
the second half of August), we have performed intense intra-night monitoring
of BL Lacertae. The blazar showed very high intra-night activity with a duty
cycle over that period of 96% or 88%, depending on whether the probably
variable cases are considered variable or not. We performed a thorough
analysis of the INV of BL Lacertae during the August 2020 flare, and now we
shall discuss some constraints that the results from our analysis can place on
the blazar jet parameters.

###  5.1 Emitting Region Parameters

First of all, we adopted the turbulent jet model (e.g. Bhatta et al., 2013) in
order to interpret the INV observed. Within this model, a plane shock hits a
turbulent cell and accelerates (energize) the cell electrons, which are then
cooled by synchrotron emission. In this way, a flux pulse is produced, which
manifests itself as a flare on the LC. The combination of the individual
pulses coming from cells of various characteristics leads to the observed INV.
Within this model, the high duty cycle obtained by us means that there is
well-developed turbulence within the jet (e.g. Webb et al., 2021). In a recent
study, Kalita et al. (2023) reported results from the BL Lacertae monitoring
from Oct 1 to Nov 23, 2020 in the optical. According to their Table 2, the
source showed INV during four nights out of ten (the probably variable cases
considered non-variable); see also Shablovinskaya et al. (2023) regarding the
source monitoring in that period. Therefore, the duty cycle could be estimated
as $\sim$40%, which is significantly lower than ours. The probably variable
cases, however, are associated with the intra-night monitoring duration of
$\lesssim$3 h, which could affect the source variability status and, hence,
the duty cycle estimate. In any case, the above-obtained value could be
considered as a lower limit. If, however, we assume that the duty cycle
decrease is real, and not an artifact of the insufficient monitoring duration,
then, following the turbulent jet model, the turbulence within the jet
subsides significantly within about two months since the August 2020 flare
onset.

The details about the processes of particle acceleration taking place in the
jet are not directly relevant to the present scenario, and so we assumed for
simplicity a quasi-instantaneous injection within a time $t^{\prime}_{\rm
inj}\leq{\mathcal{R}}/c$ of a mono-energetic population of high-energy
electrons in a homogeneous region of radius $\mathcal{R}$ (here $c$ stands for
the speed of light); here and below the primed quantities are in the rest
frame. These electrons cool by synchrotron emission and lose half of their
energy within the cooling time, $t_{\rm cool}(\nu)$:

$t_{\rm cool}(\nu)\simeq
4.73\\!\times\\!10^{4}\,{\mathcal{B}}^{-3/2}\,\nu_{15}^{-1/2}\,\left(\frac{\delta}{1+z}\right)^{-1/2}\leavevmode\nobreak\
[\rm s],$ (11)

where $\nu_{15}$ is the observed photon frequency (in units of $10^{15}$ Hz,
$\nu=10^{15}\nu_{15}$ Hz) and ${\mathcal{B}}$ the magnetic field strength (in
units of Gauss). Here we neglected the cooling by the inverse Compton
processes; that is, a zero Compton dominance parameter was assumed. This
assumption is justified because Abdo et al. (2010a) reported a Compton
dominance parameter of 0.2 for BL Lacertae.

In the framework of this scenario, the low-energy electrons result from
initially more energetic ones after their synchrotron cooling, thereby leading
to the soft time lag (e.g. Urry et al., 1997; Tavecchio et al., 1998): the
time lag between two bands corresponding to frequencies $\nu_{1}$ and
$\nu_{2}$ ($\nu_{1}>\nu_{2}$) is equal to $\tau(\nu_{2},\nu_{1})=t_{\rm
cool}(\nu_{2})-t_{\rm cool}(\nu_{1})$. Therefore, if we have estimated the
time lags among the $BVRI$ bands, then we can derive ${\mathcal{B}}$ and
$\delta$ simultaneously. This technique was applied using the Aug 20 lags
(Table 6), and so we have $\tau(\nu_{R},\nu_{k})=t_{\rm cool}(\nu_{R})-t_{\rm
cool}(\nu_{k})$, $k=B,V,I$. In this notation the lags $\tau(\nu_{R},\nu_{B})$
and $\tau(\nu_{R},\nu_{V})$ are positive, while the lag
$\tau(\nu_{R},\nu_{I})$ is negative. The frequency dependence of the observed
lags is shown in Figure 17. Technically, we did randomization of the time lags
within the corresponding asymmetric lag uncertainties to estimate the
parameters and their uncertainties. For each set of randomly drawn lags, we
estimated ${\mathcal{B}}$ and $\delta$ performing an unweighted fit using the
Nelder-Mead fitting method; we ran a total of 2500 cycles. Finally, we built
the parameter distributions and used them to get
${\mathcal{B}}=5.6^{+1.3}_{-0.8}\,\rm G$ and $\delta=11.0^{+0.3}_{-0.3}$; the
weighted Nelder-Mead fit without randomization gave very similar results. The
parameter uncertainties represent the 16-th and 84-th percentiles of the
corresponding distributions, and the fit corresponding to the so-derived
parameters is drawn in Figure 17 with a black line. Using the same approach
and MWL time lags from $\gamma$-rays to optical, Weaver et al. (2020) obtained
a magnetic field strength of $\sim$3.0 G for BL Lacertae.

We have only $B$ vs $I$ time lag for Aug 26, and so we can apply the following
expression to derive the magnetic field strength (e.g. Tavecchio et al., 1998;
Papadakis et al., 2003):

${\mathcal{B}}\,\delta^{1/3}\simeq
1.31\\!\times\\!10^{3}\,\left(\frac{1+z}{\nu_{15,I}}\right)^{1/3}\times\\\
\left[\frac{1-(\nu_{15,I}/\nu_{15,B})^{1/2}}{\tau(\nu_{I},\nu_{B})}\right]^{2/3}\leavevmode\nobreak\
[\rm G],$ (12)

where $\nu_{15,B}$ and $\nu_{15,I}$ are the frequencies corresponding to the
$BI$ bands, respectively (in units of $10^{15}$ Hz) and
$\tau(\nu_{I},\nu_{B})$ the $B$ vs $I$ time lag (in units of seconds). Having
a $B$ vs $I$ time lag of $3.8^{+2.5}_{-1.3}$ min, we got
${\mathcal{B}}\,\delta^{1/3}\simeq 20.3^{+6.4}_{-5.7}\,\rm G$ or
${\mathcal{B}}\simeq 9.1^{+2.9}_{-2.6}\,\rm G$ if we assume a Doppler factor
of $11.0^{+0.3}_{-0.3}$ as estimated above (see also Shablovinskaya et al.,
2023). The uncertainties of $\mathcal{B}$ were derived using the lag and
Doppler factor randomization.

As we mentioned in Section 3.7, the measured time lags are INLC lags rather
than individual flare lags. Therefore, we shall assume the parameters
determined above to be an average over the emitting regions, contributing to
the given INLC. In this regard, our estimate of the Doppler factor is a kind
of local estimate related to the regions, contributing to the Aug 20 INLC.
Nevertheless, it is consistent with the literature values of $\delta$ for BL
Lacertae as mentioned before. We see that the various Doppler factor estimates
for BL Lacertae are consistent with each other irrespective of the band and
method used to get them. This is in contrast with the estimates of $\delta$
for the high-energy synchrotron-peaked blazars, for which dependence on the
band and method used is observed (this is termed as the “Doppler crisis”, e.g.
Piner & Edwards, 2018; Agarwal et al., 2021). An explanation of that
dependence could lie in the more complex internal jet structure in these
sources compared to the other kind of blazars. Hence, our results are in
support of this scenario as far as BL Lacertae is classified as a low-energy
synchrotron-peaked blazar: the lack of discrepancy among the Doppler factor
estimates could mean a simple structure of its jet.

An independent magnetic field strength estimate could be obtained using the
results from the LC decompositions and considering the decay timescale,
${\mathcal{T}}_{\rm d}$, as an upper limit of $t_{\rm cool}$; that is,
${\mathcal{T}}_{\rm d}\geq t_{\rm cool}$ (e.g. Fan et al., 2021). Thus, the
lower limit (or the minimum value) of the magnetic field strength,
${\mathcal{B}}_{\rm min}(\delta)$, inside the emitting region could be derived
by rewriting the Equation (11) as follows:

$\begin{split}{\widetilde{\mathcal{B}}}_{\rm
min}&=1.31\\!\times\\!10^{3}\,{\mathcal{T}}_{\rm
d}^{\,-2/3}\,\nu_{15}^{-1/3}\,(1+z)^{1/3}\leavevmode\nobreak\ [\rm G];\\\
{\mathcal{B}}_{\rm min}(\delta)&={\widetilde{\mathcal{B}}}_{\rm
min}\,\delta^{-1/3};\\\ {\mathcal{B}}&\geq{\mathcal{B}}_{\rm
min}(\delta),\end{split}$ (13)

where ${\mathcal{T}}_{\rm d}$ is in units of seconds. In addition, the results
from the LC decompositions could also be used to set limits on the electron
Lorentz factor in the emitting region and on the radius of the emitting
region.

The electron Lorentz factor, $\gamma_{\rm e}$, which is the electron energy in
units of $m_{\rm e}c^{2}$ can be associated with the observed frequency of the
emitted synchrotron radiation via (Ghisellini et al., 1997)

$\nu=\frac{4}{3}\,\gamma^{2}_{\rm e}\,\nu_{\mathcal{B}}\frac{\delta}{1+z},$
(14)

where $\nu_{\mathcal{B}}=2.80\times 10^{6}\mathcal{B}$ is the cyclotron
frequency. This equation, coupled with Equation (11), yields $\gamma_{\rm
e}\propto t^{1/3}_{\rm cool}(\nu)$. Assuming again that ${\mathcal{T}}_{\rm
d}\geq t_{\rm cool}$, we get an upper limit (or a maximal value) of the
electron Lorentz factor for the corresponding frequency:

$\begin{split}{\widetilde{\gamma}_{\rm
e,max}}&=4.53\\!\times\\!10^{2}\,\nu^{2/3}_{15}\left[{\mathcal{T}}_{\rm
d}\,(1+z)\right]^{1/3};\\\ \gamma_{\rm
e,max}(\delta)&={\widetilde{\gamma}_{\rm e,max}}\,\delta^{-1/3};\\\
\gamma_{\rm e}&\leq\gamma_{\rm e,max}(\delta),\end{split}$ (15)

where ${\mathcal{T}}_{\rm d}$ is in units of seconds.

Accounting for our assumption about the injection time of electrons, the
rising part of the flare LC constrains the light-crossing time, $t_{\rm cros}$
(${\mathcal{T}}_{\rm r}\geq t_{\rm cros}$), thus setting an upper limit (or a
maximal value) on the emitting region radius as follows:

$\begin{split}{\widetilde{\mathcal{R}}_{\rm max}}&=\frac{c\,{\mathcal{T}}_{\rm
r}}{1+z}\leavevmode\nobreak\ [\rm cm];\\\ {\mathcal{R}}_{\rm
max}(\delta)&={\widetilde{\mathcal{R}}_{\rm max}}\,\delta;\\\
{\mathcal{R}}&\leq{\mathcal{R}}_{\rm max}(\delta),\end{split}$ (16)

where ${\mathcal{T}}_{\rm r}$ is in units of seconds. The dominance of the
light-crossing time means also that the rising timescale and, hence, the
emitting region radius are not frequency dependent.

Taking the values of ${\mathcal{T}}_{\rm r}$ and ${\mathcal{T}}_{\rm d}$ from
Table 7, assuming $\delta=11.0$, and using Equations (13), (15), and (16), we
obtained the minimal values of the magnetic field strength, maximal values of
the electron Lorentz factor, and maximal values of the radius that
characterize the emitting regions. The distributions of ${\mathcal{B}}_{\rm
min}(\delta=11.0)\equiv{\mathcal{B}}_{\rm min}(11.0)$, $\gamma_{\rm
e,max}(\delta=11.0)\equiv\gamma_{\rm e,max}(11.0)$, and ${\mathcal{R}}_{\rm
max}(\delta=11.0)\equiv{\mathcal{R}}_{\rm max}(11.0)$ are shown in Figures 18,
19, and 20. Some characteristics of the emitting regions are listed in Table
8.

Using the same approach, Covino et al. (2015) found the following
characteristics for the emitting regions of BL Lacertae assuming $\delta=10.0$
and a Compton dominance parameter of unity: a lower limit for the magnetic
field strength of 6.0 G and an upper limit for the radius of $3\times
10^{-5}\,{\rm pc}=6.2$ AU. In addition, Weaver et al. (2020) obtained a
magnetic field strength of $\sim$3.0 G using a minimal timescale of $\sim$30.0
min, derived on the basis of the BL Lacertae MWL variability.

####  5.1.1 Turbulent Cell Sizes

Following the turbulent jet model, the INLCs are a combination of synchrotron
pulses coming from various turbulent cells within the emitting region in the
jet, hit by a plane shock. The turbulence is a stochastic process, and so each
INLC is a single realization of this process. In the framework of Kolmogorov
theory of turbulence (Kolmogorov, 1941) the Kolmogorov scale is the smallest
spatial scale in a turbulent flow at which scale the turbulence kinetic energy
dissipates. Therefore, the smallest spatial scales, found on the basis of the
minimal timescales of the flux variations, directly probe the Kolmogorov
scale. Following this line of reasoning, the Kolmogorov scale (i.e., the
diameter of the smallest emitting region) derived by us is $\leq$4.4 AU (see
Table 8). Meng et al. (2017) fitted synchrotron pulses to the flares of the BL
Lacertae INLCs and estimated a smallest cell size of $\sim$1.5 AU assuming a
shock speed of $0.1c$ and a Doppler factor of 7.3; the smaller cell size
becomes $\sim$2.3 AU if one assumes $\delta=11.0$.

Figure 20 shows that the limits on the turbulent cell sizes are distributed
continuously up to about 70 AU; the cell size is assumed to be twice
${\mathcal{R}}_{\rm max}(11.0)$. This is in agreement with the results of
Bhatta et al. (2013) but somewhat larger than the cell sizes obtained by Rafle
et al. (2012) and Xu et al. (2019); in all three papers the authors model the
INLCs by means of numerical calculation of the synchrotron pulse profiles,
expected from the energized turbulent cells.

Large cell sizes are rare (Figure 20), which could mean that either (i) the
large cells exist, but they are truly rare because they are unstable, or (ii)
the large cells do not exist and they are actually unresolved groups of
smaller cells. Our INLCs provide examples in support of both possibilities
(see Figures 11 and 12): the INLCs for Aug 28 shows two large, well-pronounced
flares, while the INLC for Sep 11 shows a barely resolved group of four
flares, which in the case of noisy data could be misidentified as a large,
single flare. A few other INLCs show similar, barely resolved groups of flares
(Sep 2, Sep 3, and Sep 14). These groups reflect the synchrotron pulses from
compact, fragmented regions within the jet hit by a shock. A crude estimate of
the upper limit of the radius of such a fragmented region could be made by
summation of the radii of the corresponding individual region that form it –
the upper limits are in the interval $(39-77)$ AU $=(5.8-12)\times
10^{14}\,\rm cm$ for the above-mentioned dates.

The typical jet radii used in the blazar SED modeling are
$(10^{16}\\!-\\!10^{17})\,\rm cm$ (e.g. Chen, 2018); that is, the fragmented
regions we studied are smaller than the jet by a factor of at least $\sim$8.

###  5.2 Power Spectral Density Slopes

Characterization of the PSD on intra-night timescales in the optical has been
performed occasionally in the past. The first systematic study on this topic,
based on a sample of blazars, was published just recently by Goyal (2021). The
PSD characteristics reflect the emission processes in blazars, and so it is
important to deepen our research in that field both by enlarging the samples
and by increasing the number of the single-source INLCs analyzed (as we did).
Here, we used the SF to study the temporal characteristics of the detrended BL
Lacertae INLCs.

We built and fitted a total of 29 SFs of BL Lacertae. The weighted mean (over
the nights and bands) slope was found to be $1.624\pm 0.007$. Using the
approximate relation between the SF and PSD slopes, we found a mean PSD slope
of $\varkappa\simeq 2.6$ (a standard deviation of 0.3, see Section 3.5), that
is our PSD slope is steeper than that of a pure random walk/red-noise process
for which $\varkappa=2$. We should point out, however, that there could be an
additional offset in our estimate because of the PSD slope approximation used
by us – this should be accounted for in the discussion that follows. Papadakis
et al. (2003) estimated the PSD slope for BL Lacertae on intra-night
timescales to be $\varkappa=1.87\pm 0.16$; the individual PSDs were averaged
over nights and bands before the fitting. Carini et al. (2011) found the SF
slopes for the blazar S5 0716+714 to lie mostly between 1 and 2 (corresponding
to the PSD slopes in the range 2–3). Recently, Goyal (2021) found a mean PSD
slope of $3.1\pm 0.3$ for a sample of seven BL Lacs. Our result is consistent
with that of Carini et al. (2011) and Goyal (2021) to within the scatter
quoted and steeper than the PSD slope obtained by Papadakis et al. (2003). The
above groups, however, did not apply any detrending procedure, and so their
results could be affected by the long-term component when present: the results
will be dependent on the number of the INLCs showing a long-term component
(the INLCs without such component could be considered as being detrended
already).

Our assumption about the INLC generation is related to the turbulent jet model
as we mentioned above. In this regard, Calafut & Wiita (2015) and Pollack et
al. (2016) estimated the PSD slopes expected from the turbulence within the
jet flow. Their computations are based on the numerical 2D modeling of
relativistic jet propagation, and both groups found the PSD slopes to average
around $\varkappa=2$ for timescales from a few days to years. Our mean PSD
slope is steeper, but it is derived on the intra-night timescales. However,
the detailed analysis of the PSDs for our data is beyond the scope of the
present paper.

##  6 Summary

The main results of the presented study could be summarized as follows:

  1. 1.

Short-timescale flux variations displayed a total amplitude variation of
$\sim$2.2 mag in $R$ band. In addition, we found that on a short-term basis
the spectral index has a weak dependence on the flux level and the variations
could be mildly chromatic;

  2. 2.

During the August 2020 flare, the median spectral index was calculated to be
$\langle\alpha_{VRI}\rangle_{\rm med}=0.885\pm 0.020$;

  3. 3.

We did not find any significant periodicity;

  4. 4.

The source was found to display BWB chromatism on intra-night timescales;

  5. 5.

The duty cycle was estimated to be $\sim$90% or higher;

  6. 6.

The weighted mean SF slope was found to be $\langle\varrho\rangle_{\rm
wt}=1.624\pm 0.007$;

  7. 7.

The cross-correlation analysis resulted in two cases of significant inter-band
time lags – the lags were of order of a few minutes;

  8. 8.

We obtained an estimate of the Doppler factor, $\delta=11.0^{+0.3}_{-0.3}$,
using the inter-band time lags;

  9. 9.

We derived the values or limits for the magnetic field strength in the
emitting regions using the inter-band time lags or LC decomposition results,
respectively. The typical values/limits for $\mathcal{B}$ were found to be
$\sim$10.0 G if we assume $\delta=11.0$;

  10. 10.

Using the LC decomposition results, we obtained limits for the Lorentz factors
of the emitting electrons and the radii of the emitting regions. In
particular, the smallest upper limit on the radius is 2.2 AU, which we related
to the Kolmogorov scale of the turbulent flow;

  11. 11.

The mean slope of the power spectral density on intra-night timescales,
roughly estimated from the mean SF slope, is steeper than that of a pure
random walk/red-noise process.

We thank the anonymous referee for valuable comments and suggestions, which
helped in improving the paper. The work is partly supported by the NCN grant
No 2018/29/B/ST9/01793. A.A. and A.O. were supported by the Scientific and
Technological Research Council of Turkey (TUBITAK), Project No. 121F427. EE
was supported by the Scientific Research Project Coordination Unit of Istanbul
University, Project No. FDK-2022-19145. We thank TUBITAK National Observatory
for partial support in using T60 and T100 telescopes with project numbers
19BT60-1505 and 19AT100-1486, respectively.

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