# Optical Quasi-periodic Oscillations in the TESS light curves of three
blazars

Ashutosh Tripathi,1,2 Krista Lynne Smith,1,2 Paul J. Wiita,3 and Robert V.
Wagoner4
1Department of Physics, Southern Methodist University, 3215 Daniel Avenue,
Dallas, Texas 75205, USA
2George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and
Astronomy, Texas A&M University, College Station, TX 77843-4242, USA
3Department of Physics, The College of New Jersey, 2000 Pennington Rd., Ewing,
New Jersey 08628-0718, USA
4Department of Physics and KIPAC, Stanford University, Stanford, California
94305, USA E-mail<EMAIL_ADDRESS>
(Last updated 2023 September 5; in original form 2023 September 5)

###### Abstract

We report the time series analysis of TESS light curves of three blazars, BL
Lacertae, 1RXS J111741.0+254858, and 1RXS J004519.6+212735, obtained using a
customized approach for extracting AGN light curves. We find tentative
evidence for quasi-periodic oscillations (QPOs) in these light curves that
range from 2 to 6 days. Two methods of analysis are used for assessing their
significance: generalized Lomb-Scargle periodograms and weighted wavelet
Z-transforms. The different approaches of these methods together ensure a
robust measurement of the significance of the claimed periodicities. We can
attribute the apparent QPOs to the kink instability model which postulates
that the observed QPOs are related to the temporal growth of kinks in the
magnetized relativistic jet. We confirm the application of this model to BL
Lacertae and extend the kink instability model to the other two BL Lac
objects.

###### keywords:

(galaxies:) BL Lacertae objects: general - (galaxies:) BL Lacertae objects:
individual: BL Lacertae - black hole physics - galaxies: general - methods:
data analysis - relativistic processes

††pubyear: 2023††pagerange: Optical Quasi-periodic Oscillations in the TESS
light curves of three blazars–LABEL:lastpage

##  1 Introduction

Active Galactic Nuclei (AGNs) are among the brightest non-transient objects in
the Universe and are understood to possess supermassive black holes (SMBHs) at
their centers. Objects of such high luminosity ($\sim 10^{47}$ erg s-1) are
thought to be powered by the accretion from their surroundings onto the SMBHs.
AGNs are often classified based on their optical spectroscopic properties,
which depend on their inclination to our line of sight, and on the basis of
their radio emission, as radio-loud and radio-quiet objects (e.g. Kellermann
et al., 1989). Radio-quiet objects are believed to have either weak radio jets
emerging close to the central object and not propagating very far or a lack of
radio jet emission. By contrast, blazars are radio-loud AGNs with low
inclinations, and thus with jets aligned close to the line of sight which make
relativistic effects significant (e.g. Urry & Padovani, 1995). Blazars are
usually considered to be the union of BL Lacertae (BL Lac) objects, which have
very weak optical and UV emission lines, and flat spectrum radio quasars
(FSRQs), for which the broad emission lines that are a defining characteristic
of all quasars can be seen. Blazars have a rapidly varying continuum spanning
the radio to $\gamma$-ray wavebands and exhibit strong polarization. The
Doppler-boosted emission enhances the variability throughout the
electromagnetic spectrum and blazars can be distinguished by these and other
extreme properties (Scheuer & Readhead, 1979; Blandford & Königl, 1979). Much
of the radio and optical variability is thought to be due to instabilities
that develop in the jet (e.g. Marscher & Gear, 1985; Marscher et al., 2008).

Quasi-periodic Oscillations (QPOs) have been observed in X-ray light curves of
stellar-mass black hole and neutron star binaries (see Remillard & McClintock,
2006, for a review). The QPOs discovered in these systems are classified into
low- and high-frequency QPOs. Low-frequency QPOs (LFQPOs) are very common and
happen at frequencies ranging from mHz to 30 Hz. The High-frequency QPOs
(HFQPOs) are much rarer and are only found in certain states of hardness and
luminosity, indicating that thermal disk emission is the dominant component.
The associated frequencies often occur in a 3:2 ratio (e.g. Remillard &
McClintock, 2006; Motta, 2016). In neutron star and stellar-mass black hole
systems, these features are believed to originate from the vicinity of the
black hole, probably within a few gravitational radii. Thus QPOs in these
sources can be used to study strong gravity and its effects on the surrounding
plasma. Various models have been proposed to explain the origin of LFQPOs
(Motta, 2016), including Lens-Thirring precession (Ingram et al., 2009) and
unstable spiral density waves (Tagger & Pellat, 1999). For HFQPOs, other
theoretical models could explain the periodicities; these include warped
accretion disks (Kato, 2005), disk-jet coupling (Li & Narayan, 2004),
diskosiesmology (Wagoner et al., 2001), and the relativistic precession model
(Stella et al., 1999). Despite all these efforts, the physical mechanisms for
the origins of QPO are still debatable. Although rare, QPOs have also been
apparently observed in systems possessing SMBHs (Halpern et al., 2003;
Gierliński et al., 2008; Lin et al., 2013; Zhang et al., 2017; Bhatta, 2017;
Smith et al., 2018; Gupta et al., 2018, 2019; Tripathi et al., 2021, and
references therein). The frequencies of the HFQPOs are roughly 40 times
greater than those of the LFQPOs, in both stellar mass and SMBH systems (Zhou
et al., 2015; Smith et al., 2021).

QPOs have been reported in AGNs across the whole electromagnetic spectrum,
from radio to $\gamma$-rays, and claimed timescales vary from a few tens of
minutes to days, to weeks, and even decades. Primarily based on the timescales
of periodicity, different models have been proposed to explain these QPOs.
Valtaoja et al. (1985) reported a periodicity of 15.7 min in the radio light
curve of OJ 287 taken in 1981. Carrasco et al. (1985) claimed the periodicity
of 27 min in the optical observation of the same object taken 2 years later
which they suggested could be explained by the presence of "hotspots" in a
thick accretion disk. Quasi-periods of the order of a few days have been
reported for several AGNs (e.g. Halpern et al., 2003; Smith et al., 2018;
Jorstad et al., 2022; Kishore et al., 2023). Models such as diskoseismic
modes, Lens-Thirring precession, and kink instabilities were employed in those
papers to explain these day-like QPOs. However, the nature of such
periodicities remains uncertain and thus it is important to search for more
such fast putative QPOs.

On longer timescales, Sillanpää et al. (1996) reported the 11.7-year
periodicity in the century-long optical light curve of OJ 287. Periods on the
order of a few years have also been reported in several other (e.g. Raiteri et
al., 2001, 2003; Foschini et al., 2006; Villata et al., 2009; Wang et al.,
2014; Graham et al., 2015; Sandrinelli et al., 2017; Roy et al., 2022; Li et
al., 2023). Those papers suggested various mechanisms that might explain such
year-long periodicities; these include the presence of a binary supermassive
black hole system, a plasma blob moving helically inside the jet, and
precession of the jet.

The frequencies associated with the accretion disk-based oscillations are
inversely proportional to the mass of the central object. This remarkable
relationship has been observed in both stellar mass and SMBHs with masses
spanning many orders of magnitudes from a few solar masses to billions of
solar masses (e.g. Abramowicz et al., 2004; Zhou et al., 2015). This relation
suggests that the origin of such periodicity may be the same for both kinds of
black holes. Moreover, this relation can be used to deduce the mass of an
object if periodicity is detected in its light curve. Although SMBH masses can
be plausibly estimated for some FSRQs (e.g. Foschini et al., 2015; Xiao et
al., 2022), for BL Lacs, there are very few reasonable measurements of SMBH
mass and they are model dependent (e.g. Woo & Urry, 2002; Nguyen et al.,
2021).

Optical QPOs are rarely detected in ground-based surveys due to their uneven
and irregular sampling. If periodicity is searched for in such data, there is
a serious risk that the stochastic red noise characteristic of AGNs can either
mimic or mask a quasi-periodic signal (Vaughan et al., 2016). The Transiting
Exoplanet Survey Satellite (TESS, Ricker, 2015), similar to its predecessor
Kepler (Borucki et al., 2010), has substantially overcome these issues with
improvement in precision over that yielded by the modest aperture ground-based
optical telescopes normally used in these studies and with high-cadence
regular sampling they cannot provide. Thanks to TESS, space-based regularly
sampled light curves with high precision and high cadence can be produced and
used for studying periodicity at higher confidence than ground-based surveys.
In this work, we search for QPOs in the high-cadence and regularly sampled
TESS observations and see that we can interpret the periodicities seen in
three blazars in terms of the kink instability model.

Our team has developed a specialized approach to extracting TESS light curves
of stochastically-varying sources like AGN. Smith et al. (2018) detected a
44-day low-frequency QPO in a _Kepler_ light curve using a preliminary version
of our current approach, which has been significantly improved to avoid over-
fitting of intrinsic AGN variability while handling both background light and
instrumental systematics in a careful and flexible way. Weaver et al. (2020)
analyzed the multi-wavelength observations including the observation from TESS
of BL Lacerate (BL Lac afterward) and suggested a 13-hour timescale. A few
other papers also have examined the variability of individual blazars using
TESS observations (Raiteri et al., 2021a, b; Kishore et al., 2023).

In this paper, we report candidate QPO detections in the TESS light curves of
BL Lacertae itself, as well as in those of two other BL Lac objects: 1RXS
J004519.6+212735 and 1RXS J111741.0+254858 (which has two non-contiguous TESS
observations). In Section 2, we describe the sample selection and our method
of reducing data taken from TESS. In Section 3, we briefly explain the various
methods used for QPO detection. Results of those analysis are reported in
Section 4 and conclusions are given in Section 5.

Figure 1: TESS light curves analyzed in the work. The blue dots represent the
fluxes and the gray curves denote the associated errors, while the red curves
show the running averages. Upper panel: The left plot is of BL Lacertae and
the right one is of 1RXS J004519.6+212735. Lower panel: light curves of 1RXS
J111741.0+254858, where the left and right plots respectively are for the
Cycle 2 and Cycle 4 observations.

##  2 Sample and Data Reduction

###  2.1 Sample Selection

The candidate periodicities presented in this work were discovered as part of
an ongoing large-scale investigation of the TESS light curves of Seyfert
galaxies. We are extracting TESS light curves of 5971 Type 1 Seyferts from the
SDSS-IV SPIDERS catalog: a database of X-ray selected AGN from the ROSAT All-
Sky Survey and the XMM-Newton Slew Survey with either eBOSS or SDSS DR12
spectral coverage (Comparat et al., 2020). Of these objects, the preliminary
periodogram analyses indicated 22 had possible periodicities. Of these, three
objects survived the more rigorous test of periodic significance described
below in Section 3. Two of them, described in this work, were BL Lac objects,
and one is a Seyfert, which will be discussed in terms of the broader Seyfert
sample in an upcoming work.

Because the BL Lac objects were contaminants in the SPIDERS sample of X-ray
selected Seyferts, it is not possible to use these numbers to draw statistical
conclusions about the occurrence rates of periodicities in blazars. In this
large sample, there were only 28 blazars that exhibited detectable variations.
The two presented here, alongside the TESS light curve of BL Lacertae itself,
are the only ones showing possible indications of QPOs and are serendipitous
discoveries that can provide support for the physical interpretation of the
QPO in BL Lacertae reported recently by Jorstad et al. (2022).

###  2.2 The Transiting Exoplanet Survey Satellite (TESS)

The TESS instrument, designed primarily to search for transiting exoplanets
using long-term monitoring at high photometric precision of stars across
nearly the entire sky (Ricker, 2015), provides very high cadence optical
monitoring, with a cadence of 30 minutes in the early cycles (2018 – 2019) and
10 minutes or 2 minutes in later and current cycles (2020 – present). Coverage
is nearly continuous, as there are no gaps due to daylight or seasonal
sunlight. TESS light curves are therefore significantly better-sampled than
ground-based optical light curves, with longer continuous baselines. The
length of nearly continuous monitoring for a TESS source depends upon its
ecliptic latitude, ranging from 27 days nearest the ecliptic plane to
approximately 1 year at the ecliptic poles. After each year-long Cycle, the
spacecraft swaps hemispheres and performs the same survey in the other half of
the sky. Thus, sources will occasionally have two “Sectors” of data, separated
by one year. The mission began in Cycle 1, in the southern hemisphere, then
went to the northern sky in Cycle 2, and returned to the south for Cycle 3,
then the north for Cycle 4. The photometric precision of TESS is dependent
upon the source magnitude; at the magnitudes of our sources, variability at
the $\sim 1-10$% level can be confidently detected.

###  2.3 TESS Data Reduction

Figure 2: Light curves of inactive stars obtained from same postal stamp as
1RXS J004519.6+212735 (left) and 1RXS J111741.0+254858 (right) using our
customized approach.

Because the TESS spacecraft was designed to detect a periodic signal in a
point source, special treatment is required to derive light curves of
stochastic signals that may be in resolved sources like bright host galaxies.
Many of these techniques to study AGN were first developed using very similar
data from the _Kepler_ spacecraft (Smith et al., 2018). We have developed
these techniques significantly, to avoid over-fitting of intrinsic AGN
variability by carefully modeling the background additive light curve effects
and the subtler multiplicative systematics in separate phases using principal
component analysis (PCA), then removing these effects using linear
regression111https://github.com/kristalynnesmith/quaver (Smith & Satori,
2023).

We first download a “postage stamp” of the TESS full-frame images around our
sources, 25$\times$25 pixels on a side. From this image, we select a custom
extraction aperture for each target, allowing us to avoid the inclusion of
nearby sources and to build an aperture appropriate for any spatially resolved
host galaxies.

Next, we define a flux threshold of 1.5 $\sigma$ above the background to
determine what pixels in the downloaded postage stamp are likely to contain
only background sky or very, very faint sources at the background level. These
faint pixels are fit by three principal components. The results of the fit are
stored in a design matrix, representing the additive systematics dominant in
the light curve. This method makes use of the matrix regression methods in the
publicly-available TESSCut (Brasseur et al., 2019) and lightkurve (Lightkurve
Collaboration,, 2018) software packages.

Next, all of the pixels in the field excluded from the previous mask, i.e.,
the “bright” pixels, but excluding the aperture of the source itself, are
corrected by these additive background systematics. Once corrected, the
variability of these bright pixels is again fit by three principal components,
which are stored in a design matrix; this matrix describes the higher-order
and more subtle multiplicative systematics. Armed with these two matrices, we
combine them into a hybrid design matrix that contains principal component
vectors for additive and multiplicative effects.

Finally, the raw light curve of the source is extracted from its aperture and
linearly regressed against the hybrid design matrix. This method was inspired
by the various corrector techniques available from lightkurve and makes heavy
use of its RegressionCorrector
class222https://docs.lightkurve.org/reference/api/lightkurve.correctors.RegressionCorrector.html.

Figure 1 shows the light curves of the sources analyzed in this work. The
upper left panel shows the light curve of BL Lacertae. TESS observed this
source beginning on 12 September 2019 for around 24.6 days in sector (Sec.) 16
of Cycle 2 observations. The gray curves represent the error associated with
the fluxes which are shown in blue. The red curve is the running average of
the light curve taken with the window=20 to highlight the possibly periodic
features present in the data (window defines the number of measurements for
which the running average is calculated). The upper right panel shows the
light curve of 1RXS J004519.6+212735. This BL Lac object was observed
beginning on 8 October 2019 for about 23.7 days in Sec. 17 of Cycle 2
observations. Both lower panels belong to 1RXS J111741.0+254858. The Sec. 22
observation of Cycle 2 began on 21 February 2021 and continued for about 25.6
days; it is shown on the lower left panel of Fig. 1. The lower right panel
shows the light curve of the Sec. 49 observation of Cycle 4 of 1RXS
J111741.0+254858 starting on 28 February 2022 and continuing for about 24.8
days. Figure 2 shows the light curve of two inactive stars from the same
postage stamps as 1RXS J004519.6+212735 and 1RXS J111741.0+254858, reduced
using the same pipeline. The reduced light curves are flat, indicating our
successful removal of systematics.

##  3 Data Analysis

In this section, we discuss the timing analysis techniques used in this work
to search for any periodicities or quasi-periodicities and to assess their
significance in the four light curves shown in Fig. 1. These methods are the
generalized Lomb-Scargle periodogram and weighted wavelet Z-transform
analyses.

###  3.1 Generalized Lomb-Scargle Periodogram

The Lomb-Scargle periodogram (LSP) is a powerful Fourier transform frequently
used to detect periodicity in unevenly sampled data (Lomb, 1976; Scargle,
1982). It uses likelihood minimization to fit sinusoidal components throughout
the data. The Generalized LSP (GLSP) is an advanced version of the ‘classical’
LSP which includes the error of fluxes in the periodogram calculations and
fits the data with both sinusoidal and constant components.

In our work, we employ the GLSP routine provided by pyastronomy333
https://github.com/sczesla/PyAstronomy. It applies the method described in
Zechmeister & Kürster (2009). In the code, the frequency resolution is
governed by the oversampling factor which is needed for the calculation of the
power spectrum and its significance. Here, we took the oversampling factor to
be 2.0, which corresponds to the number of frequencies considered to be equal
to the number of data points in a single observation. We have confirmed that
the result is insensitive to different oversampling factors, by also employing
oversampling factors of 1.0 and 10.0. We found that the values of the
frequencies of the peaks of the signals exceeding 3$\sigma$ significance vary
by no more than 3% for that range of oversampling factors and the value of the
powers at those peaks are within just a few percent of each other.

###  3.2 Weighted wavelet Z-transform analysis

In addition to detecting any possible periodicity in a signal, it is also
imperative to assess its persistence throughout the observation. The wavelet
method decomposes the signal into both frequency and time domains
simultaneously. Instead of the sinusoidal components, this method uses wavelet
functions to fit the observation. In this work, we employed the weighted
wavelet Z-transform (WWZ) method which can be used effectively for unevenly
sampled and sparse data (Witt & Schumann, 2005). It uses the z-statistics for
the calculations (Foster, 1996) and is based on the Morlet wavelet function
which is the function of both frequency and time. In this work, we use the
publicly available WWZ444WWZ software is available at URL:
https://www.aavso.org/software-directory. software that has been used
extensively in the literature (see Bhatta, 2017; Zhang et al., 2017; Tripathi
et al., 2021, and references therein).

The major assumption in wavelet calculations is the type of wavelet function
used. The finite length of the observation can induce the edge effects which
could be observed in the density plots. We also note that the frequency and
temporal resolutions can have important effects on the calculation of the
significance of peaks.

###  3.3 Significance Calculations

Figure 3: Generalized Lomb-Scargle Periodograms for the TESS light curves
analyzed in the work. The black histogram is the power spectral density, the
blue curve shows 3$\sigma$ significance for simple power law and green curves
shows the same confidence for bending power law. Upper panel: BL Lacertae
(left) and 1RXS J004519.6+212735 (right). Lower panel: Cycle 2 (left) and
Cycle 4 (right) observations of 1RXS J111741.0+254858.

It appears that the stochastic processes occurring in both turbulent jets and
accretion disks usually manifest themselves in the light curves of blazars as
is seen in observations taken throughout the electromagnetic spectrum (e.g.
Max-Moerbeck et al., 2014; Nilsson et al., 2018; Bhatta & Dhital, 2020). A
blazar Power Spectrum Density (PSD), $P(f)$, as a function of frequency $f$,
often follows a simple power-law of the form $f^{\alpha}$ where $\alpha$ is
the spectral index for a significant range of frequencies. Any QPO, on the
other hand, is believed to arise from coherent disk or jet processes. So, a
strong QPO feature should project itself in the blazar PSD above the power-
law. It is imperative to estimate the significance of the peak relative to the
red noise of the PSD. As $\alpha$ usually has a negative value, meaning that
the spectrum at lower frequencies fluctuates with larger amplitudes, it is
possible to misinterpret these fluctuations as quasi-periodic signals. In
addition, there may be some fluctuations introduced in the signal by processes
arising from the observational constraints which could be misinterpreted as
periodic peaks. Therefore, it is very important to estimate the significance
of these peaks.

To perform significance calculations, we first simulate light curves having
the same properties as the observation (mean, variance, duration) and power
spectrum. Using the standard discreet Fourier Transform (DFT) for the unevenly
sampled data as used in this work would be problematic as it results in
amplitude fluctuation and unreasonable frequency shifting (e.g. Fan et al.,
2017). Although the TESS data used in this work are far more uniformly spaced
than most observations, they do contain some gaps, so the observations are not
taken at strictly regular intervals. To avoid these issues, we employ the GLSP
method. The light curve analyzed in this work is essentially two segments
separated by a rather wide $\sim$4-day gap (except for BL Lac, where the gap
is only $\sim$1 day). To simulate these light curves realistically, we
essentially treat segments individually. The resultant simulated light curves
consisted of the two $\sim$10-day segments with $\sim$4-day gap in the middle.
We chose the frequency spacing such that the number of frequencies is equal to
the number of data points in the observation. We fit the power spectrum
$P(\nu)$ at a given frequency $\nu$ with a bending power-law plus a constant
representing the instrumental white noise that dominates at high frequencies

$P(\nu)=N\left(1+\left(\frac{\nu}{\nu_{b}}\right)^{-\beta}\right)^{-1.0}+C,$
(1)

where $N$ is the normalization, $\nu_{b}$ is the bending frequency, $\beta$ is
the power-spectrum index, and $C$ is the constant representing the
instrumental white noise. We used the curve fit routine in scipy package of
python to estimate the parameters of the red-noise spectrum. We also fit the
power spectrum with simple power law defined as

$P(\nu)=N\left(\frac{\nu}{\nu_{0}}\right)^{\alpha},$ (2)

where $\nu_{0}$ is the constant. Although a simple power-law and bending
power-law are chosen to model the ‘red-noise’ in the spectra, we note that it
is possible to explain the stochastic process using different mathematical
models; e.g., short memory ARMA models are capable of producing periodic
patterns for AR(p) with p$>$2 (see Schulz & Mudelsee, 2002, and references
therein).

Using the parameters of each power spectrum obtained using GLSP and the other
properties of the source, we simulate 10000 light curves employing the
stingray (Huppenkothen et al., 2019) timing analysis
package555https://github.com/StingraySoftware which implements the algorithm
described in Timmer & Koenig (1995). We calculate GLSP for the simulated light
curves. The significance is estimated from the PSD distribution at each
frequency.

We also perform Monte Carlo Markov Chain (MCMC) simulations to estimate errors
on the best-fit parameters of the best-fit bending power law. Table 1 and
Table 2 show the best-fit parameters and the associated errors for the bending
power-law model and the simple power law, respectively.

To estimate the significance of the signal in the WWZ method, we calculate the
time-averaged WWZ, which is essentially the WWZ power marginalized over time.
This time-averaged WWZ is a periodogram that follows the $\chi^{2}$
distribution with 2 degrees of freedom in the limit of even sampling (Foster,
1996). We calculate time-averaged WWZ for 10,000 simulated light curves. The
significance, as in the case of GLSP, is derived from the distribution of the
simulated time-averaged WWZ at each frequency.

Figure 4: Weighted wavelet Z-transform (WWZ) analysis results for the light
curves analyzed in the work. In each plot, the left sub-panel corresponds to
the color-color diagram of wavelet power (red denoting the maximum power and
decreasing towards violet and black) plotted against time and frequency. The
right sub-panel shows the time-averaged WWZ plotted against frequency in
black; the red dotted lines represent 3$\sigma$ significances. Upper panel: BL
Lac (left) and 1RXS J004519.6+212735 (right). Lower panel: 1RXS
J111741.0+254858 in Cycle 2 (left) and Cycle 4 (right).

##  4 Results

Fig. 1 shows the light curves we obtained from the TESS observations for BL
Lac (upper left panel), 1RXS J004519.6+212735 (upper right panel), and 1RXS
J111741.0+254858 (lower panels). Visually, there seem to be possible quasi-
periodic components that were searched for and confirmed by the GLSP and WWZ
methods. These methods were chosen as they provide distinctive approaches to
the time series data. Explicitly, GLSP gives the Fourier transform of the
data, probing the frequency domain while the wavelet analysis decomposes the
data into frequency and time simultaneously. Only if a peak shows at least 3
$\sigma$ significance in both the GLSP and time-averaged WWZ methods do we
consider it to be a solid QPO signal.

Fig. 3 shows the normalized power (black curve) of the GLSP plotted against
frequency for BL Lac. The blue curves denote the 3$\sigma$ (99.73%)
significance obtained by simulations assuming simple power law as the
underlying model. The green curve corresponds to 3$\sigma$ significance
obtained by simulations assuming bending power law. The most prominent peak,
with a significance of more than 3 $\sigma$, occurs at the period of 4.94
days. The peak around 1.40 days is also prominent ($>$ 3 $\sigma$
significance) for both bending and simple power law.

Note that quoting a period of more than 6 days would correspond to having
fewer than 4 cycles for the timescales of our observations and therefore are
not suitable for claiming a QPO. This limitation is supported by early work
which showed that it is possible to apparently see 3 cycles in many light
curves based purely on flicker noise (Press, 1978). Vaughan et al. (2016)
found that greater numbers of cycles that follow a periodic pattern more
strongly support the presence of periodicity. Even a pure sinusoidal component
could not be distinguished from a simple stochastic process if there are only
$\sim$ 2 cycles. However, the presence of $\sim$ 5 cycles could be easily
distinguished from a stochastic process.

As the time-averaged WWZ is also effectively a periodogram that contains
similar information as the GLSP, showing only one underlying model in the WWZ
plots is sufficient. We note that the claimed frequencies in the Fourier space
lie in the "red-noise" regime where the PSD can be adequately described by
both simple power-law and bending power-law models. So we show the WWZ
significance plots only for bending power laws.

The upper left panel of Fig. 4 shows the result of our WWZ analysis for BL
Lac. The left plot shows the color-color diagram of WWZ power plotted against
time and frequency. The maximum power is concentrated around the period of
5.26 days, and it persists throughout the observation. The second most
prominent concentration, which also persists throughout the observation, is
around 2.94 days. There are other weaker features that seem to be present in
different parts of the observation. For instance, the feature around 1.54 days
becomes stronger in the later half of the observation whereas the feature
around 2.0 days is only visible in the first half. The black curve in the
right sub-panel of Fig. 5 for BL Lac shows the time-averaged WWZ and the red
dotted curve denotes the 3$\sigma$ (99.73%) confidence curve. The feature at
5.26 days, which also appears in the WWZ plot, is found to be above 3$\sigma$.
The signal around 5 days is found to be significant in both methods. However,
as it corresponds to fewer than 5 cycles during the 24.6-day TESS
observations, it cannot be considered to be very convincing.

The upper right plot of Fig. 3 shows the GLSP for Cycle 2 data of 1RXS
J004519.6+212735. The peaks at the period of 5.8 days are marginally
consistent with 3 $\sigma$ significance. The color-color diagram of the WWZ
analysis of this observation (upper right panel of Fig. 4) shows that the peak
around 5.88 days is persistent throughout the observation and is the only peak
having significance more than 3$\sigma$ in the time-averaged WWZ plot on the
right-hand side of the panel. Although the peak at 5.9 days is nominally the
most significant measurement from both methods, it corresponds to only 4.0
cycles during the length of the observations, so cannot be fully trusted.

Table 1: Best fit parameters and associated errors for the bending power-law sampled from MCMC realizations.  Source & Sector | Segment | log $N$ | $\beta$ | $\nu_{b}$ | $C$
---|---|---|---|---|---
BL Lacertae Sec. 16 | - | $-0.95^{+0.09}_{-0.09}$ | $-1.80^{+0.15}_{-0.16}$ | $0.27^{+0.02}_{-0.02}$ | $0.0012^{+0.0001}_{-0.0001}$
1RXS J004519.6+212735 Sec. 17 | 1 | $-0.33^{+0.05}_{-0.05}$ | $-3.21^{+0.28}_{-0.30}$ | $0.24^{+0.02}_{-0.02}$ | $0.01^{+0.001}_{-0.001}$
| 2 | $-0.85^{+0.08}_{-0.07}$ | $-3.33^{+0.29}_{-0.29}$ | $0.32^{+0.03}_{-0.03}$ | $0.009^{+0.0008}_{-0.0008}$
1RXS J111741.0+254858 Sec. 22 | 1 | $-1.44^{+0.13}_{-0.13}$ | $-3.39^{+0.29}_{-0.29}$ | $0.65^{+0.06}_{-0.05}$ | $0.009^{+0.001}_{-0.001}$
| 2 | $-1.68^{+0.14}_{-0.14}$ | $-6.12^{+0.56}_{-0.53}$ | $0.81^{+0.07}_{-0.07}$ | $0.002^{+0.001}_{-0.001}$
1RXS J111741.0+254858 Sec. 49 | 1 | $-1.15^{+0.10}_{-0.11}$ | $-5.61^{+0.51}_{-0.49}$ | $0.81^{+0.07}_{-0.07}$ | $0.003^{+0.0003}_{-0.0003}$
| 2 | $-1.62^{+0.14}_{-0.14}$ | $-8.92^{+0.80}_{-0.81}$ | $0.96^{+0.09}_{-0.08}$ | $0.002^{+0.0002}_{-0.0002}$
Table 2: Best fit parameters and associated errors for the simple power-law sampled from MCMC realizations.  Source & Sector | Segment | log $N$ | $\alpha$ | $\nu_{0}$
---|---|---|---|---
BL Lacertae Sec. 16 | - | $-1.82^{+0.16}_{-0.16}$ | $-1.11^{+0.18}_{-0.19}$ | $0.65^{+0.06}_{-0.06}$
1RXS J004519.6+212735 Sec. 17 | 1 | $-1.41^{+0.13}_{-0.13}$ | $-1.21^{+0.11}_{-0.11}$ | $0.69^{+0.06}_{-0.06}$
| 2 | $-1.40^{+0.12}_{-0.13}$ | $-0.85^{+0.08}_{-0.07}$ | $0.68^{+0.06}_{-0.06}$
1RXS J111741.0+254858 Sec. 22 | 1 | $-1.22^{+0.11}_{-0.10}$ | $-1.24^{+0.11}_{-0.10}$ | $0.38^{+0.03}_{-0.03}$
| 2 | $0.62^{+0.06}_{-0.05}$ | $-2.16^{+0.19}_{-0.19}$ | $0.06^{+0.01}_{-0.01}$
1RXS J111741.0+254858 Sec. 49 | 1 | $-1.32^{+0.11}_{-0.12}$ | $-0.62^{+0.05}_{-0.06}$ | $0.40^{+0.03}_{-0.04}$
| 2 | $0.04^{+0.01}_{-0.01}$ | $-1.14^{+0.11}_{-0.12}$ | $0.06^{+0.01}_{-0.01}$

We now discuss the results obtained for 1RXS J111741.0+254858. There are two
observations of this source, one from sector 22 of Cycle 2 and one from sector
49 of Cycle 4. The lower left panel of Fig. 3 shows the GLSP for Cycle 2 data.
The most prominent period (at less than 6 days) is found at 3.68 days. The
feature around 3.68 days has a significance of $\approx$ 3$\sigma$ and has a
double-horned structure which is confirmed by the other method. Note that we
report the period of the ‘horn’ of this broad peak that has the higher
significance. The double horned structure is also seen in the color-color
diagram of the wavelet of this Cycle 2 observation (lower left panel of Fig. 4
at 3.78 days (higher peak). Interestingly, only one of the peaks is present in
the first half of the observation but as the observation evolves with time,
the second one appears. So, the behavior of this double-horned peak could only
be distinguished when the observation is decomposed into time and frequency
domains simultaneously which is achieved by using WWZ. The feature at the
period of 5 days is relatively stronger at the beginning of the observation
but gets weaker with time and becomes negligible by the end of the
observation. All these methods show the most significance for a possible QPO
at around 3.8 days.

The GLSP for the Cycle 4 data of 1RXSJ111741.0+254858 is plotted in the lower
right panel of Fig. 3. The most prominent peak is found at the period of 6.20
days (although that is over 25% of the length of the observations, so not
convincing) followed by the peaks at 4.38 and 2.86 days. The perseverance of
these peaks can be assessed by the wavelet analysis which is plotted in the
lower right panel of Fig. 4 for Cycle 4 data. The feature at 6.25 days, which
is more than 3$\sigma$ significant, seems to be strongly persistent throughout
the observation, though it corresponds to too few cycles to be strongly
supported. The features at 4.64 days continue to present strongly throughout
the observation which is also the case for the somewhat weaker signal at 2.87
days. So, the most promising possible QPOs indicated by both GLSP and WWZ
methods are around 6.2, 4.5, and 2.9 days, and the latter would correspond to
over 9 cycles.

###  4.1 Caveats

Figure 5: Simulated results demonstrating the effects of caveats for a bending
power law as the underlying model. Left: 1 $\sigma$ confidence region for the
periodogram obtained by simulating the 20-day light curves; with a gap (blue)
and without a gap (red) of 4 days. Middle: 1 $\sigma$ confidence region for
the power spectrum obtained by simulating stationary (blue) and non-stationary
(red) light curves. Right: 1 $\sigma$ confidence region for the periodogram
obtained by simulating light curves having the same (blue) and different (red)
PSD index. For all of these simulations, we used the inputs from the Cycle 2
observation of 1RXS J111741.0+254858.

Figure 6:  Same as Fig. 5 but for a simple power law as the underlying model.

One of the main challenges encountered while analyzing TESS light curves is
the presence of gaps between light curves which are not negligible in
comparison to the duration of the light curves. For instance, a gap of 4 days
is present in the 25-day light curve of Cycle 2 of 1RXS J111741.0+254858. To
assess the effect of gaps on the power spectrum, we have simulated 1000 light
curves each, with and without gaps, and with initial conditions similar to
that of Cycle 2 observation of 1RXS J111741.0+254858. The left panel of Fig. 5
shows the $1\sigma$ confidence bound for the periodogram for the 20-day light
curves both without gaps and with a gap of 4 days, assuming a bending power
law as the underlying model. We could deduce from the plot that the
periodograms in both cases are consistent with each other at higher
frequencies. However, as we go towards lower frequencies
($f~{}\approx~{}0.1$d-1), the periodogram distributions no longer closely
agree with each other. The gap between the light curve segments can affect its
power spectrum at frequencies lower than 0.1 d-1. So, we restrict our analysis
to frequencies above 0.1 d-1 to avoid this issue. We note that the effect of a
gap is more pronounced in the case of simple power law model for the PSD (left
panel of Fig. 6). The confidence intervals for the light curves with gaps are
more stringent as compared to continuous light curves at low frequencies
($f~{}\approx~{}0.07$d-1) whereas at higher frequencies, the confidence bounds
for continuous light curves are less stringent.

A second issue is the possibility of non-stationarity of the data which
affects the power spectrum and the resultant underlying model. To address
this, we performed wavelet analysis to assess not only the significance of the
signal but also its perseverance. So we have simulated 25-day light curves
with a 4-day gap in-between its halves for two cases: stationary, with the
same statistical properties throughout the light curve; and non-stationary,
where the statistical properties are different for the data before and after
the gap. For these simulations, we took the same light curves as before (Cycle
2 observation of 1RXS J111741.0+254858). The middle panels of Figs. 5 and 6
show the 1$\sigma$ error bound for stationary and non-stationary light curves
with gaps for simple power law and bending power law respectively. In both
cases, the bounds are quite similar at frequencies above $f\approx 0.2$d-1 and
only start to significantly diverge below $f\approx 0.1$d-1.

The third challenge is the choice of the underlying model. In this work, we
employed both simple power-laws and bending power-laws which could provide
incomplete descriptions of the data. By limiting the analysis to frequencies
above 0.2 d-1, we assumed that the power spectrum behaves like the simple
power law (or bending power law in some cases) in this regime.

It is, however, also possible that a single model could not explain the
behavior throughout the light curve. So, we fit the segments of the light
curves separately and then simulate the light curves using the best-fit
parameters for each segment. In this case, the confidence intervals with the
same spectral index in the underlying bending power law (right panel of Fig.
5) are somewhat less tightly bound as compared to the different spectral index
cases at higher frequencies; however, at lower frequencies, the confidence
intervals are actually tighter for the same case. In the case of simple power
laws (right panel of Fig. 6), the error bounds agree with each other quite
closely throughout, though the different power-law case is somewhat broader.
It can be concluded from these simulations that if we use the same spectral
index throughout the observation, the confidence bound is underestimated as
compared to the case where different spectral indices are used for different
segments of light curves. So, it is preferable to simulate the segments within
light curves separately if the gap is large in comparison to the resolution of
the data.

The timescales found in this work are in the range of 3–5 days, corresponding
to frequencies of 0.20–0.34 d-1 so the caveats discussed in this section
should have little impact.

###  4.2 Segment-wise analysis

Figure 7: Segment-wise GLSP analysis for the light curves of 1RXS
J004519.6+212735 (left panel) and 1RXS J111741.0+254858 (middle and right
panels). In each plot, the black curve is the GLSP. The green and blue curves
show the 3 $\sigma$ significance level for the simple power law and the
bending power law, respectively.

The stochastic processes occurring in the jets and accretion disks of a blazar
may be reflected in observations taken throughout the whole electromagnetic
spectrum. In general, they may be non-stationary as their statistical
properties can change with time. It is possible that the light curves of
blazars may be non-stationary in nature even on the relatively short
timescales investigated in this work. Indeed, evidence of non-stationarity has
been observed in X-ray light curves of blazars and Seyferts with similar
durations as the TESS light curves explored here (e.g. Alston, 2019;
Bhattacharyya et al., 2020). However, the simulation procedure we have used to
this point to generate significance curves assumes the simple or bending
power-law description of the PSD is constant throughout the observation and
therefore, should be considered cautiously. While the observations of BL Lac
have only a short ($\sim 1$d) gap, the observations of 1RXS J004519.6+212735
and 1RXS J111741.0+254858 have substantial gaps. It is important to examine
whether any QPO signal is present both before and after the gap or if it is
only present in one of the segments. To investigate this issue, we have
divided the light curve into two segments separated by a gap in the middle and
performed the GLSP analysis assuming both simple and bending power laws as the
underlying red noise model. The portion of each light curve before the 4-day
gap is termed as Segment 1 and that after the gap is termed as Segment 2.

The left panels of Fig. 7 show the segment-wise analysis for the Sector 17
observation of 1RXS J004519.6+212735. In Segment 1, a peak at around 6.0 days
is found with at least 3 $\sigma$ significance using the simple power-law but
no peak is found significant employing a bending power-law. For Segment 2, a
similar signal at around 5.9 days is significant at $>3\sigma$ using the
simple power law. While these peaks are not found to be significant using the
bending power law model, their GLSP significance increases to $3\sigma$ when
both segments are considered together (see Fig. 3) and is further supported by
the wavelet analysis (Fig. 4).

We analyzed the Sec. 22 observation of 1RXS J111741.0+254858 by segments and
the GLSP results are shown in the middle panels of Fig. 7. Signals of at least
3 $\sigma$ significance are found at 4.5 days for Segment 1 and 2.8 days for
Segment 2, assuming either simple or bending power laws as the underlying
model. In the combined analysis (Fig. 3), these two peaks blend into a double-
horned peak. This transition from one frequency to another is also evident in
the color-color wavelet analysis of Sec. 22 (Fig. 4).

For the Sec. 49 observation of 1RXS J111741.0+254858, the segment-wise GLSP
results are plotted in the right panels of Fig. 7. In Segment 1, a period of
around 1.8 days is found to be significant at 3 $\sigma$ with the simple power
law, though not with the bending one. This signal is also seen in the combined
wavelet analysis of this observation (Fig. 4) but with less significance, as
it is relatively strong only in the first half of the observation. In Segment
2, a very strong signal with at least 3 $\sigma$ with both simple and bending
power law is found at a period of around 6 days. This evolution of the
possible $\sim$6 day period is shown in the combined wavelet analysis (Fig. 4)
where the power of the signal increases with time.

##  5 Discussion

Figure 8: Corner plot of posterior distributions sampled from 100 walkers and
120000 iterations through MCMC simulations assuming the kink instability model
for BL Lacertae (left) and 1RXS J004519.6+212735 (right).

Figure 9: Same as Fig. 8 but for Cycle 2 (left) and Cycle 4 (right)
observation of 1RXS J111741.0+254858.

Emission from blazars is believed to be dominated by non-thermal processes in
the relativistic jet. As the jet is viewed at a very small inclination angle
($<10^{\circ}$) (Urry & Padovani, 1995), the emission from the jet is
magnified enormously by the relativistic effects and outweighs the cumulative
emission from the host galaxy and accretion disk (e.g. Jorstad et al., 2001).
The emission, extending from radio to $\gamma$-rays, is observed to have
variability on timescales of a few minutes (e.g. Ahnen et al., 2017; Ackermann
et al., 2016; Jorstad et al., 2022) to years (e.g. Villforth et al., 2010;
Acciari et al., 2014). Some variability timescales have also been claimed to
be QPO signatures in blazars (Hayashida et al., 1998; King et al., 2013;
Graham et al., 2015; Tripathi et al., 2021; Jorstad et al., 2022). The short
timescales of much of the variability indicate that the emitting region must
be compact.

###  5.1 Kink Instability

Very recently, Jorstad et al. (2022) reported a quasi-periodic feature with a
0.57 day period in BL Lacertae, one of the objects analyzed in this work,
employing both optical and $\gamma$-ray observations that were taken between
March and December 2020. They considered a kink instability as the reason for
the origin of these QPOs based on the approach given by Dong et al. (2020). In
this model, periodicity is the result of a type of current-driven plasma
instabilities known as kink instabilities. Relativistic magnetohydrodynamic
simulations have shown that these kinds of instabilities can arise in the jet
having a strong toroidal magnetic field and result in both twisting of field
lines and an increase in particle acceleration (Mizuno et al., 2009; Barniol
Duran et al., 2017). This distorted magnetic field grows to produce a quasi-
periodic kink in the jet that manifests into a region of compressed moving
plasma with enhanced emission that could be observed to have a QPO signature.
The timescales of such QPO signatures in blazars can be of the order of days
and are related to the growth of kink instabilities over timescales of weeks
to months.

Jorstad et al. (2022) performed semi-analytical simulations to model the time-
dependent emission from the plasma following Dong et al. (2020). This model
assumes the dominant source of fluctuations arises from a blob in the kink
region and has three components: a constant toroidal magnetic component; a
sinusoidal poloidal component having temporal period $T$ and amplitude
$bp_{amp}$; and a turbulence profile having amplitude $tb_{amp}$ which
averages to $tbsq_{avg}$; $n_{g}$ corresponds to the average emission power
during the observation. They constrained these parameters along with the
normalization using MCMC simulations and found the overall magnetic (and hence
polarization) period to be 1.14 days, which corresponds to the observed flux
variability QPO of 0.57 d. They attributed the 4-day QPO period also detected
using the wavelet method to the growth of this one-day period kink with time.

Quantifying the transverse motion of the kinked region can be used to
calculate its growth rate (Mizuno et al., 2009). Dong et al. (2020) estimated
the kink growth time ($\tau_{KI}$) to be the ratio of the transverse
displacement of the jet from its center ($R_{KI}$) to the average velocity of
propagation ($\left<v_{tr}\right>$). They found the growth time of the kink
instability to be consistent with the period of the QPOs estimated in their
simulations. The period associated with the kink instability ($T_{obs}$) in
the observer’s frame is given by (Dong et al., 2020)

$T_{obs}=\frac{R_{KI}}{\left<v_{tr}\right>\delta}~{},$ (3)

where $\delta$ is the Doppler factor and $R_{KI}$ is the size of the emission
region in the co-moving frame and its value can be $10^{16}$–$10^{17}$ cm for
a typical BL Lac object. In Dong et al. (2020), $\left<v_{tr}\right>$ is
estimated to be $\approx 0.16c$. So, the kink instability is related to the
properties of the jet which include its inclination, velocity, hence the
Doppler factor, and size of the emission region. Calculating $T_{obs}$ for BL
Lacertae using $\delta=15$ yields a time of 1–10 days which is consistent with
the QPOs timescales found in this work. So it is reasonable to suggest that
the apparent quasi-periods of 3–5 days we obtained here for all three sources
using different timing analyses can be attributed to the growth of the quasi-
periodic kinks originating in the inner portions of jets.

The longer QPO periods of around 3–5 days, indicated by both GLSP and wavelet
approaches could be explained by the kink evolving with time. Any quasi-
periods of more than $\sim$ 5 days cannot be firmly detected with the TESS
light curves having a duration of 25–27 days as a period of more than 6 days
corresponds to fewer than 4 cycles. It should also be noted that any QPOs of
periods shorter than a few hours could not be reliably detected in these data
sets as the sampling frequency of these TESS data is 30 minutes so they would
be very close to the Nyquist frequency and lie in the white-noise portion of
the power spectrum.

Still, there are other processes that might be able to explain any QPOs on the
scale of a few days. Possible explanations include the normal modes of
oscillations trapped in the inner part of the accretion geometry due to the
strong gravity of the central compact object (e.g., Perez et al., 1997;
Espaillat et al., 2008). These modes correspond to the internal oscillations
of the accretion disk and could produce the observed QPOs. Magnetorotational
instability (MRI) turbulence in the accretion flow (e.g., Abramowicz et al.,
2004) might also result in such periodicity. The frame-dragging around a
rotating SMBH leads to the Lens-Thirring precession near the inner portion of
the accretion disk which could naturally produce a QPO with a period of the
order of a few weeks (Stella & Vietri, 1998). However, these processes are
more likely to be more important in Seyfert galaxies and quasars where the
emission from the disk dominates over that of the jets, whereas the jet
emission is the most important in these blazars.

###  5.2 Individual Sources

####  5.2.1 BL Lacertae

BL Lacertae is the prototype of the BL Lac objects characterized by very weak
or absent broad emission lines and high variability observed throughout the
electromagnetic spectrum. This object has been observed extensively in all
wavebands, from radio to $\gamma$-rays (e.g. Sandrinelli et al., 2018; Raiteri
et al., 2003; Ren et al., 2023, and references therein). It also has been
claimed to show quasi-periodic behavior in various wavebands with periods
ranging from a few minutes to hours, weeks, and even years. Villata et al.
(2004) claimed a periodicity of 8 years for radio outbursts for this source
which was later supported by Villata et al. (2009). Sandrinelli et al. (2017)
reported a periodicity of around 700 days in both optical and $\gamma$-ray
observations. As mentioned above, Jorstad et al. (2022) analyzed the optical
and $\gamma$-ray observations and claimed a periodicity of around 13 hours in
both. They found a very strong correlation between the optical and
$\gamma$-ray light curves (correlation coefficient: 0.62$\pm$0.04) with a
temporal lag between them consistent with 0. The detection of essentially
identical periodicites implies that both optical and $\gamma$-rays are
produced from the same mechanism which is claimed to be the kink instability
in the jet. They also found a periodicity of the order of a few days which is
explained by the temporal growth of the kink.

In our work, the periodicity of $\sim$5 days is found in the TESS light curves
of BL Lacertae, which we suggest is the optical counterpart of the kink
developed in the jet. BL Lacertae was observed by TESS for 27 days beginning
on 12 September 2019, approximately 5 months prior to the onset of the
observations analyzed by Jorstad et al. (2022). Ren et al. (2023) analyzed the
weekly and monthly binned Fermi-LAT observations of BL Lacertae taken between
August 2008 and April 2021 and found no significant long-lived periodicity.
Although the TESS observation analyzed in this work is subsumed within the
span of these Fermi observations, the cadences are significantly different.
The minimum cadence normally available with Fermi data is 1 day and they
usually need to be binned over longer periods to get adequate statistics.
Therefore, the Fermi observations with a 1 day cadence could not resolve
periodic variability with timescales of just a few days, while the 30 minute
cadence of the Cycle 2 TESS data can.

We have applied the kink instability model to the TESS observation of BL
Lacerate and followed the approach of Jorstad et al. (2022) to fit our 26-day
observation using their publicly available
code666https://zenodo.org/record/6562290. Fig. 8 displays the corner plots
showing the posterior distributions of various parameters using the algorithm
applied in the code. We used the Doppler factor $\delta=15$ which is derived
in Jorstad et al. (2001) for BL Lacertae using Very Large Baseline Array
(VLBA) observations at 43 GHz. We used 100 walkers with 120000 iterations and
fit 5 parameters used in the code. We obtained the period of 1.16 days which
agrees nicely with the results of Jorstad et al. (2022).

####  5.2.2 1RXS J111741.0+254858 and 1RXS J004519.6+212735

1RXS J111741.0+254858 (Véron-Cetty & Véron, 2010; Gaia Collaboration, 2020)
and 1RXS J004519.6+212735 (Mallery et al., 2007; Gaia Collaboration, 2020)
were found serendipitously in the SDSS-IV SPIDERS catalog, which is
essentially a catalog of X-ray-selected Seyferts. For 1RXS J111741.0+254858,
we have analyzed the observations from Sec. 22 (Cycle 2) and Sec. 49 (Cycle 4)
and found probable QPOs of 3.7 and 4.6 days respectively. A QPO signal at 5.8
days is found in the analysis of the Sector 17 observation of 1RXS
J004519.6+212735.

Motivated by the successful application of the kink instability model to BL
Lac’s TESS data, we applied this formalism to these other two sources. For
1RXS J004519.6+212735, the period is found to be 1.20 days. For the Cycle 2
and Cycle 4 observations of 1RXS J111741.0+254858, periods of 1.09 and 1.1
days are found, respectively. The corresponding MCMC corner plots are given in
Figs. 8–9. For this analysis, we used the Doppler factor appropriate for BL
Lac itself, $\delta=15$ for both sources. We also examined whether using
different values of $\delta$ would yield different temporal periods. Hence we
also used the average Doppler factor value of 9.2 derived in Liodakis et al.
(2017) for BL Lac objects. For 1RXS J004519.6+212735, the new period is found
to be 1.26 days, while periods of 1.19 and 1.21 days are found for the Cycle 2
and Cycle 4 observations of 1RXS J111741.0+254858, respectively. So, plausible
variations in $\delta$ have only modest effects on the observed period of the
kink developed in the jet.

##  6 Conclusion

In this work, we have examined densely sampled optical light curves of three
blazars, BL Lacertae, 1RXS J111741.0+254858, and 1RXS J111741.0+254858. These
were extracted from TESS using a customized reduction approach designed to
preserve actual AGN variability. We have discovered tentative evidence for
QPOs in these light curves on the timescale of a few days. For each light
curve, these possible QPOs mainly lie between $\sim 2$ and $\sim 6$ days. We
also ran MCMC simulations to examine the claimed QPOs present in the
observations, assuming they originate with a kink instability in the jet. We
found the periods of the kinks to be consistent with the period reported for
BL Lacertae in Jorstad et al. (2022). Our analysis thus supports the kink
instability hypothesis where the apparently observed QPOs can be produced
through the temporal evolution of a kink developed in the jet.

## Acknowledgements

This work is supported by NASA grant number 80NSSC22K0741. KLS gratefully
acknowledges the staff of the K2 Guest Observer office at NASA Ames,
especially Christina Hedges, for their assistance and advice in adapting the
matrix regression methods.

## Data Availability

The TESS data presented in this paper were obtained from the Mikulski Archive
for Space Telescopes (MAST) at the Space Telescope Science Institute.

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