# Four bright eclipsing binaries with $\gamma$ Doradus pulsating components:
CM Lac, MZ Lac, RX Dra and V2077 Cyg

John Southworth 1, Timothy Van Reeth 2
1 Astrophysics Group, Keele University, Staffordshire, ST5 5BG, UK
2 Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven,
Belgium

(Accepted YYYYMMDD. Received YYYYMMDD; in original form YYYYMMDD.)

###### Abstract

The study of pulsating stars in eclipsing binaries holds the promise of
combining two different ways of measuring the physical properties of a star to
obtain improved constraints on stellar theory. Gravity (g) mode pulsations
such as those found in $\gamma$ Doradus stars can be used to probe rotational
profiles, mixing and magnetic fields. Until recently few $\gamma$ Doradus
stars in eclipsing binaries were known. We have discovered g-mode pulsations
in four detached eclipsing binary systems from light curves obtained by the
Transiting Exoplanet Survey Satellite (TESS) and present an analysis of their
eclipses and pulsational characteristics. We find unresolved g-mode pulsations
at frequencies 1–1.5 d-1 in CM Lac, and measure the masses and radii of the
component stars from the TESS data and published radial velocities. MZ Lac
shows a much richer frequency spectrum, including pressure modes and tidally-
excited g-modes. RX Dra is in the northern continuous viewing zone of TESS so
has a light curve covering a full year, but shows relatively few pulsation
frequencies. For V2077 Cyg we formally measure four pulsation frequencies, but
the available data are inadequate to properly resolve the g-mode pulsations.
V2077 Cyg also shows total eclipses, with which we obtain the first
measurement of the surface gravity of the faint secondary star. All four
systems are bright and good candidates for detailed study. Further TESS
observations are scheduled for all four systems, with much improved temporal
baselines in the cases of RX Dra and V2077 Cyg.

###### keywords:

stars: fundamental parameters — stars: binaries: eclipsing — stars:
oscillations

††pagerange: Four bright eclipsing binaries with $\gamma$ Doradus pulsating
components: CM Lac, MZ Lac, RX Dra and V2077 Cyg–References

##  1 Introduction

Eclipsing binary star systems (EBs) are crucial objects for understanding the
physics governing stellar structure and evolution, because they are the only
stars whose masses and radii can be measured to high precision and accuracy
from observational material and geometrical arguments alone. Precise
measurements of the masses and radii of stars in EBs were instrumental in the
development of stellar theory (e.g. Russell, 1914), in the verification of the
first modern generation of theoretical stellar models (e.g. Andersen et al.,
1990; Pols et al., 1997), and continue to be used to guide theoretical
progress (Claret & Torres, 2018; Tkachenko et al., 2020). In the current era
of échelle spectroscopy and space-based light curves it is possible to measure
masses and radii for stars in EB to precisions approaching 0.1% (Maxted et
al., 2020; Graczyk et al., 2021). The impact of photometry from space missions
has been reviewed in detail by Southworth (2021a).

Another class of stars capable of posing high-quality constraints on stellar
theory is that of the pulsating variables (Aerts et al., 2010). Among these,
stars showing gravity-mode (g-mode) pulsations are well-suited to studying the
interiors of stars as g-modes, which have buoyancy as the dominant restoring
force, can travel deep within stars whilst leaving observable signatures on
the stellar surface (Bowman, 2020). $\gamma$ Doradus ($\gamma$ Dor) variables
(Kaye et al., 1999) are stars of spectral types A and F that show g-mode
pulsations with periods ranging from 0.3 d to 4 d and amplitudes up to 0.1 mag
(Henry et al., 2007; Grigahcène et al., 2010).

In recent years, asteroseismic analyses of main-sequence stars with g-mode
pulsations have allowed us to place constraints on multiple phenomena. These
include near-core stellar rotation (e.g. Bouabid et al., 2013; Van Reeth et
al., 2016; Christophe et al., 2018; Takata et al., 2020a; Takata et al.,
2020b; Szewczuk et al., 2021), convective core boundary mixing (e.g.
Michielsen et al., 2019; Wu & Li, 2019; Mombarg et al., 2021), extra envelope
mixing (e.g. Mombarg et al., 2020; Pedersen et al., 2021) and magnetic fields
(e.g. Prat et al., 2019; Van Beeck et al., 2020; Lecoanet et al., 2022).

The advantages of asteroseismology and binarity can be combined in objects
which show both eclipses and pulsations. These systems have the potential to
set exacting constraints on stellar theory (e.g. Johnston et al., 2019b). A
wide variety of pulsating stars are known in EBs, including $\delta$ Scuti
(Maceroni et al., 2014; da Silva et al., 2014; Lee et al., 2020, 2021;
Southworth, 2021d), $\beta$ Cephei (Southworth et al., 2020, 2021; Lee & Hong,
2021), SPB (Clausen, 1996; Southworth & Bowman, 2022) and $\delta$ Cephei
(Pilecki et al., 2018). Over the last decade, multiple $\gamma$ Dor variables
in EBs have been discovered as well. An emblematic case is that of KIC
11285625 (Debosscher et al., 2013). Others were found by, among others,
Hambleton et al. (2013); Lee et al. (2014); Sowicka et al. (2017); Hełminiak
et al. (2017); Guo et al. (2017b); Lampens et al. (2018); Guo et al. (2019)
and Van Reeth et al. (2022).

A sample of 115 $\gamma$ Dor variables in EBs was identified by Gaulme & Guzik
(2019) via a systematic search of the Kepler EB catalogue (Kirk et al., 2016).
Li et al. (2020) performed an asteroseismic evaluation of these targets and
reported the detection of g-mode period-spacing patterns for 34 of them, as
well as for one discovered by Colman et al. (2022). An independent systematic
search and asteroseismic evaluation of the Kepler EB catalogue was conducted
by Sekaran et al. (2020), resulting in a different sample of 93 $\gamma$ Dor
variables in EBs, with detected period-spacing patterns for seven of them. So
far, asteroseismic modelling has only been achieved for a small number of
targets: KIC 10080943 (Schmid & Aerts, 2016; Johnston et al., 2019a), KIC
10486425 (Zhang et al., 2018), KIC 7385478 (Guo & Li, 2019) and KIC 9850387
(Zhang et al., 2020; Sekaran et al., 2021).

The accumulating data from the NASA Transiting Exoplanet Survey Satellite
(TESS; Ricker et al. 2015) is enabling previously unattainable analyses of
many of the bright and well-known variable stars. The catalogue of objects
includes many EBs with a long observational history (e.g. Southworth, 2020).
In the course of trawling this database we have discovered four bright EBs
whose light curves also show $\gamma$ Dor pulsations. In this work we present
these discoveries, studies of their light and radial velocity (RV) curves, and
a first analysis of the nature of their pulsations.

##  2 Observations

Figure 1:  TESS SAP light curves of the four EBs. In each case only one sector
is shown.

The TESS satellite (Ricker et al., 2015) is in the process of observing almost
the entire celestial sphere, split into 69 overlapping sectors. Each sector is
a 24∘$\times$96∘ strip of sky and is photometrically monitored for 27.4 d,
through a filter with a high throughput between 600 nm and 1000 nm, with a
pause near the midpoint for the download of data to Earth. During the nominal
two-year mission, full-frame images (FFI) were taken at a default cadence of
1800 s, while a subset of stars were monitored in short cadence, where
successive exposures were combined to yield light curves with a cadence of 120
s. In the ongoing extended mission, the FFI cadence has been changed to 600 s.

Reduced light curves are obtained from the data by the Science Processing
Operations Center (SPOC; Jenkins et al., 2016) and made available through the
MAST portal111https://mast.stsci.edu/portal/Mashup/Clients/ Mast/Portal.html.
We used the simple aperture photometry (SAP) light curves in the current work.
We visually inspected the TESS light curves of approximately 2000 EBs present
in a bibliography maintained by the first author222A hobby during Covid-19
lockdowns., and identified four objects that show both eclipses and
previously-unrecognised $\gamma$ Dor pulsations: CM Lac, MZ Lac, RX Dra and
V2077 Cyg.

CM Lac was observed only in sector 16 (2019/09/11 to 2019/10/07) and one
further sector of observations is
scheduled333https://heasarc.gsfc.nasa.gov/cgi-bin/tess/ webtess/wtv.py (sector
56, 2022 September). MZ Lac was observed in two consecutive sectors, 16 and 17
(2019/11 to 2019/11/02) and this will recur in sectors 56–57 (2022 Sept-Oct).
By contrast, extensive observations exist for RX Dra because it is sited
within the northern continuous viewing zone (CVZ) of TESS. It was observed in
sectors 14–26 (2019/07/18 to 2020/07/04) and 40–41 (2021/06/08 to 2021/08/20)
and observations are currently being obtained in sectors 47–60 (2021/12/30 to
2023/01/18). V2077 Cyg was observed in sectors 14 (2019/07/18 to 2019/08/15),
26 (2020/06/08 to 2020/07/04) and 41 (2021/07/23 to 2021/08/20), and an
additional four consecutive sectors (53–56) will be observed in 2022. The
light curves for one sector of each of the four targets are shown in Fig. 1.

Following the target selection, we carefully reassessed the quality of the
reduced SAP light curves. First, we inspected the aperture masks used in the
pixel data and evaluated both the level of captured flux and the level of
contamination from nearby stars. For all stars, the standard SPOC aperture
mask was accepted, with the exception of RX Dra. Here, we re-extracted the
light curve from the pixel data using custom aperture masks, to account for
the varying contamination level between sectors by the nearby non-variable
star 2MASS J19024829+5844050. Second, we applied additional detrending to the
light curves by fitting low-order polynomials. To avoid overfitting, these
detrending curves were optimised simultaneously with preliminary binary and
pulsation models, built from a sum of sine waves:

$\displaystyle f(t)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{10}a_{i}\sin\left(2\pi\nu_{i}\left(t-t_{0}\right)+\phi_{i}\right)$
$\displaystyle+\sum_{j=1}^{10}a_{j}\sin\left(2\pi j\nu_{\rm
orb}\left(t-t_{0}\right)+\phi_{j}\right)$

Here, $\nu_{i}$ and $\nu_{\rm orb}$ are estimates for the dominant pulsation
frequencies and the orbital frequency, $a_{i}$ and $a_{j}$ are their
amplitudes, and $\phi_{i}$ and $\phi_{j}$ are their phases, respectively. The
mean timestamp of each light curve was used as the zeropoint $t_{0}$.

Table 1:  Parameters measured from the eclipses using the jktebop code. Parameters with a superscripted N were calculated using the nominal physical constants and solar quantities defined by the IAU (Prša et al., 2016). The $T_{\rm eff}$ values are from Liakos & Niarchos (2012) for CM Lac and from Molenda-Żakowicz et al. (2007) for V2077 Cyg. | CM Lac | MZ Lac | RX Dra | V2077 Cyg
---|---|---|---|---
Fitted parameters:
$r_{\rm A}+r_{\rm B}$ | $0.35383\pm 0.00078$ | $0.2557\pm 0.0024$ | $0.21854\pm 0.000099$ | $0.12303\pm 0.00022$
$k$ | $0.902\pm 0.018$ | $0.893\pm 0.027$ | $0.7144\pm 0.0010$ | $0.4933\pm 0.0063$
$i$ (∘) | $87.573\pm 0.082$ | $88.53\pm 0.41$ | $88.589\pm 0.018$ | $88.668\pm 0.048$
$J$ | $0.5956\pm 0.0074$ | $0.888\pm 0.048$ | $0.8063\pm 0.0055$ | $0.321\pm 0.013$
$L_{3}$ | $-0.0043\pm 0.0070$ | $0.377\pm 0.018$ | $0.0176\pm 0.0013$ | $-0.026\pm 0.018$
$u_{\rm A}$ | $0.237\pm 0.029$ | $0.62\pm 0.13$ | $0.300\pm 0.031$ | $0.248\pm 0.033$
$u_{\rm B}$ | $0.182\pm 0.040$ | $0.673\pm 0.073$ | $0.195\pm 0.057$ | $0.35\pm 0.11$
$v_{\rm A}$ | 0.232 (fixed) | 0.229 (fixed) | $0.081\pm 0.050$ | 0.229 (fixed)
$v_{\rm B}$ | 0.229 (fixed) | 0.229 (fixed) | $0.316\pm 0.096$ | 0.229 (fixed)
$e\cos\omega$ | $0.000289\pm 0.000082$ | $0.00732\pm 0.00025$ | $0.000030\pm 0.000010$ | 0.0 fixed
$e\sin\omega$ | $0.0009\pm 0.0023$ | $0.4116\pm 0.0079$ | $0.00094\pm 0.00062$ | 0.0 fixed
$P$ (d) | $1.6046879\pm 0.0000046$ | $3.158794\pm 0.00009$ | $3.78639491\pm 0.00000009$ | $5.937226\pm 0.000068$
$T_{0}$ (BJD/TDB) | $2458752.67073\pm 0.00032$ | $2458765.08683\pm 0.00014$ | $2458857.410151\pm 0.000002$ | $2458697.52198\pm 0.00028$
$K_{\rm A}$ ( km s-1) | $120.0\pm 3.4$ |  |  | $54.52\pm 0.82$
$K_{\rm B}$ ( km s-1) | $157.0\pm 3.3$ |  |  |
$V_{\rm\gamma,A}$ ( km s-1) | $22.7\pm 1.9$ |  |  | $-0.23\pm 0.40$
$V_{\rm\gamma,B}$ ( km s-1) | $25.0\pm 1.8$ |  |  |
Derived parameters:
$r_{\rm A}$ | $0.1860\pm 0.0015$ | $0.1351\pm 0.0032$ | $0.12747\pm 0.00008$ | $0.08238\pm 0.00038$
$r_{\rm B}$ | $0.1678\pm 0.0021$ | $0.1296\pm 0.0016$ | $0.09107\pm 0.00008$ | $0.04064\pm 0.00034$
$e$ | $0.0009\pm 0.0015$ | $0.4117\pm 0.0079$ | $0.00094\pm 0.00059$ |
$\omega$ (∘) | $71^{+213}_{-14}$ | $88.091\pm 0.041$ | $88.2\pm 1.5$ |
Light ratio | $0.495\pm 0.018$ | $0.691\pm 0.062$ | $0.4088\pm 0.0008$ | $0.0750\pm 0.0022$
Mass of star A ($\mathcal{M}^{\rm N}_{\odot}$) | $2.01\pm 0.10$ |  |  |
Mass of star B ($\mathcal{M}^{\rm N}_{\odot}$) | $1.54\pm 0.09$ |  |  |
Radius of star A ($\mathcal{R}^{\rm N}_{\odot}$) | $1.636\pm 0.032$ |  |  |
Radius of star B ($\mathcal{R}^{\rm N}_{\odot}$) | $1.476\pm 0.031$ |  |  |
Surface gravity of star A | $4.314\pm 0.011$ |  |  |
Surface gravity of star B | $4.286\pm 0.016$ |  |  | $4.607\pm 0.007$
$T_{\rm eff}$ of star A (K) | $8700\pm 300$ |  |  | $7066\pm 201$
$T_{\rm eff}$ of star B (K) | $7034\pm 270$ |  |  |

##  3 Analysis methods

###  3.1 Light curve analysis

All four systems are well-detached EBs so are suitable for analysis with the
jktebop code444http://www.astro.keele.ac.uk/jkt/codes/jktebop.html (Southworth
et al., 2004a; Southworth, 2013). In this code, the fractional radii of the
stars555$r_{\rm A}=\frac{R_{\rm A}}{a}$ and $r_{\rm B}=\frac{R_{\rm B}}{a}$
where $R_{\rm A}$ and $R_{\rm B}$ are the true radii and $a$ is the semimajor
axis of the relative orbit. are parameterised as their sum ($r_{\rm A}+r_{\rm
B}$) and ratio ($k=\frac{r_{\rm B}}{r_{\rm A}}$), and the orbital eccentricity
($e$) and argument of periastron are parameterised using the combination terms
$e\cos\omega$ and $e\sin\omega$. Other fitted parameters include the orbital
period ($P$), midpoint of primary eclipse ($T_{0}$), orbital inclination
($i$), and the central surface brightness ratio ($J$). We used the quadratic
limb darkening (LD) law, fitted for the linear LD coefficients ($u_{\rm A}$
and $u_{\rm B}$), and fixed the quadratic LD coefficients ($v_{\rm A}$ and
$v_{\rm B}$) to theoretical values (Claret, 2018). We define the primary
eclipse to be the deeper of the two types of eclipse, the primary star (the
star eclipsed during primary eclipse) to be star A, and the secondary to be
star B.

The presence of pulsations complicates the light curve analysis. In our
initial analysis we ignored this and fitted the entire light curve with a
jktebop model including one low-order polynomial versus time per TESS half-
sector to remove slow drifts in brightness due to instrumental or
astrophysical effects. Once the residuals of the fit were obtained, these were
subsequently used in the pulsation analysis (see below). For CM Lac and V2077
Cyg we then extracted each eclipse from the data, plus half an eclipse
duration either side, and then fitted these together with a low-order
polynomial for each eclipse representing the pulsation-induced brightness
changes around the time of the eclipse. This procedure was successful in the
case of CM Lac, but led to small systematic biases for V2077 Cyg due to the
amplitude and complexity of the pulsational signature. For RX Dra the surfeit
of data mandated a different approach: an initial fit to the full data
followed by phase-binning down into a more manageable number of observations
(see Section 4.3).

Uncertainties in the fitted parameters were calculated using the Monte Carlo
and residual-permutation simulations implemented in jktebop (Southworth,
2008). The Monte Carlo simulations account for correlations between
photometric parameters, and required the errorbars in the TESS data to be
rescaled to force a reduced $\chi^{2}$ of $\chi^{2}_{\nu}=1$. The residual-
permutation simulations are sensitive to red noise in data, so are useful in
the case of unmodelled pulsations. The two errorbars were checked for every
parameter and the larger of the two retained in each case. From recent
analyses of the TESS light curves of other EBs (Southworth, 2021b, c) we have
found that the Monte Carlo and residual-permutation simulations typically
agree with each other and with errorbars obtained from splitting up light
curves into multiple subsets for analysis in isolation. Additional support for
the reliability of the Monte Carlo and residual-permutation errorbars, and of
the jktebop model, comes from a recent analysis of AI Phe by multiple
researchers working independently and using multiple codes and error
estimation methods (Maxted et al., 2020). The systematics in the modelling of
V2077 Cyg have only a small effect on our results because the system is
totally-eclipsing and thus possesses a higher intrinsic determinacy of the
photometric parameters (Kopal, 1959).

###  3.2 Pulsation analysis

After a good fit to the light curve had been obtained, the residuals of the
fit were calculated and subjected to a frequency analysis. The data points
that were obtained during the eclipses were excluded to minimise the influence
of any remaining binary signal in the residuals. Because the available TESS
data are in most cases limited to one or two sectors per star, which are
generally insufficient to resolve individual g-mode pulsations, this remains a
preliminary analysis. Using a Lomb-Scargle periodogram (Scargle, 1982), we
iteratively prewhitened the light curve (Degroote et al., 2009) to measure the
pulsations for which the signal-to-noise ratio S/N $\geqslant$ 4 (Breger et
al., 1993).

The amplitudes, frequencies and phases of these pulsations were subsequently
optimised with a non-linear least-squares fit (Bowman, 2017; Bowman &
Michielsen, 2021) and the S/N ratios were re-evaluated using these optimised
parameter values. The derived pulsation frequencies and their combinations
were then compared with harmonics of the binary orbital frequency to identify
possible tidally excited (e.g., Fuller, 2017; Guo et al., 2019), tidally
perturbed (e.g., Reyniers & Smeyers, 2003; Bowman et al., 2019; Steindl et
al., 2021) or non-linearly coupled oscillations (e.g., Burkart et al., 2012;
Guo, 2020).

Finally, we searched for combination frequencies

$\nu_{k}=n_{i}\nu_{i}+n_{j}\nu_{j}$

where $\nu_{i}$, $\nu_{j}$ and $\nu_{k}$ are measured frequencies, and $n_{i}$
and $n_{j}$ are integer numbers such that $|n_{i}|+|n_{j}|\leq 2$. Such
combination frequency identifications were accepted if they agreed within
3$\sigma$.

###  3.3 Physical properties

Published spectroscopic orbits exist for both components of the CM Lac system,
allowing us to measure its physical properties directly. To do so we included
the published radial velocities (RVs) in the jktebop analysis in order to
determine the absolute masses and radii of the stars. In the case of V2077 Cyg
RVs are available for the primary but not the secondary component. In this
situation the masses and radii of the stars cannot be measured directly (e.g.
Hilditch, 2001), but it is possible to obtain the surface gravity of the
secondary (Southworth et al., 2004b, 2007) and thus verify its evolutionary
status. The measured properties of the four targets are given in Table 1,
including published effective temperature ($T_{\rm eff}$) values where
available.

Figure 2:  TESS SAP light curves of the four EBs plotted versus orbital phase.
In each case the coloured points give the observations and the black line
shows the best fit. The residuals are plotted below each main panel on an
inflated scale. Black points show the residuals binned according to orbital
phase. The plot for V2077 Cyg has been split into panels concentrating on the
eclipses for clarity. The polynomials versus time have been subtracted from
the observations in all cases to remove any slow trends in brightness that
would otherwise blur the figure in the vertical direction.

##  4 Discussion of individual systems

###  4.1 CM Lacertae

####  4.1.1 Analysis of the binarity

CM Lac was discovered to be an eclipsing binary by Wachmann (1931) and
physical properties were measured by Popper (1968). The most detailed study of
the system was by Liakos & Niarchos (2012), who determined its physical
properties based on light curves observed in the $BVRI$ bands and RVs from a
set of 28 medium-resolution spectra.

The TESS light curve of CM Lac (Fig. 1) contains 12 primary and 14 secondary
eclipses. The pulsations are strong enough to significantly affect the eclipse
depths so we cut out the data around each eclipse from the light curve and
modelled the out-of-eclipse brightness through each as a quadratic function,
as described in Section 3.1. We found that it was possible to constrain one LD
coefficient for each star. We also fitted for the amount of third light
($L_{3}$), obtaining a slightly negative value. This is physically plausible
if the background subtraction in the TESS images is too strong. Although the
orbit appears circular, a slight eccentricity is preferred for the TESS data.
A phased light curve is shown in Fig. 2.

Our final solution includes the RVs from Liakos & Niarchos (2012) in order to
determine the full properties of the system; we fitted for the velocity
amplitudes ($K_{\rm A}$ and $K_{\rm B}$) and systemic velocities
($V_{\rm\gamma,A}$ and $V_{\rm\gamma,B}$) of both stars (Fig. 3). The reduced
$\chi^{2}$ of the fit was forced to unity for each dataset by scaling the data
errors. We tried including historical times of primary eclipse as well
(Wachmann, 1931; Kreiner et al., 2001) but they occurred systematically too
late so we rejected them – this may suggest that the orbital period of CM Lac
is not constant. The uncertainties from residual permutation are larger than
those from Monte Carlo. Our results are in Table 1: they agree well with those
of Liakos & Niarchos (2012). Further RVs are needed to refine the mass
measurements, which are currently precise to only 5%.

Figure 3:  Fitted spectroscopic orbit for CM Lac (blue lines) compared to the
measured RVs from Liakos & Niarchos (2012): red circles for star A and red
squares for star B. The residuals of the fit are shown in the lower panels.

####  4.1.2 Analysis of the pulsations

Our frequency analysis reveals the presence of $\gamma$ Dor pulsations. This
can easily be seen from the Lomb-Scargle periodogram of the TESS light curve
(excluding the eclipses) shown in Fig. 4. There is unresolved variability with
frequencies between 1.0 and 1.5 $\rm d^{-1}$. This is most likely caused by
g-mode pulsations, as the (tentative) dominant frequency that can be obtained
with iterative prewhitening (listed in Table 2), differs significantly from
the harmonics of the binary orbital frequency. However, because the
variability is unresolved, the measured frequency value cannot be assigned to
a single g-mode pulsation. The asymmetric shape of the light curve is also
typical for g-mode pulsations (Kurtz et al., 2015).

We note that the second harmonic of the binary orbital frequency also lies
within the frequency range of the observed variability, as shown in Fig. 4.
However, because the available TESS data only cover 27 d, we cannot
investigate the nature of the variability at this frequency. It may be caused
by a residual signature of the binarity or unresolved nearby g-mode pulsation
frequencies, but more high-precision photometric data are needed to resolve
this issue. The unresolved variability around the fourth and sixth harmonics
of the orbital frequency are likely aliases of the unresolved variability
around the second harmonic, as can be deduced from the spectral window shown
in the top panel of Fig. 4. From a visual inspection of the TESS light curve,
we find that the pulsations remain visible during the primary eclipse, but
vanish during the secondary eclipse. This means that the g-mode pulsations
belong to the secondary star.

Figure 4:  Spectral window (top panel) and part of the Lomb-Scargle periodogram (black) of the TESS light curve of CM Lac, excluding the eclipses. The harmonics of the orbital frequency $\nu_{\rm orb}$ (dotted grey lines) and the extracted pulsation frequency (dashed red line) are also indicated. Table 2:  The parameter values for the significant prewhitened frequency of CM Lac. Bracketed quantities indicate the uncertainty in the final digit of the preceding number. Frequency ($\rm d^{-1}$) | Amplitude (mmag) | Phase (rad) | S/N
---|---|---|---
1.1799 (2) | 12.5 (1) | 0.4276 (12) | 6.0

###  4.2 MZ Lacertae

Figure 5:  Sections of the Lomb-Scargle periodogram (black) of the residual light curve of MZ Lac. The harmonics of the orbital frequency $\nu_{\rm orb}$ (dotted grey lines) and the extracted pulsation frequencies are also indicated. The dashed red lines and dash-dotted blue lines mark non-harmonic and harmonic pulsation frequencies, respectively. Table 3:  An overview of the parameter values for the significant frequencies of MZ Lac. For frequencies that lie close to harmonics of the binary orbital frequency $\nu_{\rm orb}$, the differences between those frequencies and their corresponding harmonics are also given. Possible combination frequencies are listed in the last column. | Frequency ($\rm d^{-1}$) | Amplitude (mmag) | Phase (rad) | S/N | $i$ | $i\nu_{\rm orb}-\nu$ | Comments
---|---|---|---|---|---|---|---
$\nu_{1}$ | 0.31523 (14) | 1.66 (3) |  $0.495$ (2) | 12.3 | 1 |  $-$0.00135 (14) |
$\nu_{2}$ | 0.8729 (4) | 0.70 (2) |  $-0.477$ (5) | 5.9 |  |  |
$\nu_{3}$ | 1.2665 (2) | 1.46 (3) |  $0.131$ (3) | 14.7 | 4 | 0.0002 (2) |
$\nu_{4}$ | 1.5819 (4) | 0.60 (3) |  $-0.103$ (7) | 6.3 | 5 |  $-$0.0010 (4) |
$\nu_{5}$ | 1.6094 (3) | 0.74 (2) |  $-0.326$ (5) | 7.9 |  |  |
$\nu_{6}$ | 1.73674 (14) | 1.86 (2) |  $0.424$ (2) | 20.5 |  |  | $2\nu_{2}+\nu_{6}=11.000(3)\,\nu_{\rm orb}$
$\nu_{7}$ | 1.8991 (4) | 0.74 (3) |  $-0.441$ (5) | 8.2 | 6 |  $-$0.0004 (4) |
$\nu_{8}$ | 2.1884 (2) | 1.13 (2) |  $0.038$ (3) | 14.2 |  |  | $\nu_{5}+\nu_{8}=11.9966(12)\,\nu_{\rm orb}$
$\nu_{9}$ | 2.9357 (3) | 0.85 (2) |  $-0.458$ (4) | 12.8 |  |  | $\nu_{2}+\nu_{9}=12.0303(16)\,\nu_{\rm orb}$
$\nu_{10}$ | 3.79908 (14) | 1.81 (2) |  $0.424$ (2) | 32.5 | 12 | 0.0002 (2) |
$\nu_{11}$ | 46.6367 (7) | 0.33 (2) |  $0.264$ (11) | 13.0 |  |  |
$\nu_{12}$ | 48.362 (2) | 0.14 (2) |  $-0.20$ (3) | 4.4 |  |  |
$\nu_{13}$ | 49.1942 (9) | 0.26 (2) |  $-0.408$ (14) | 8.4 |  |  |
$\nu_{14}$ | 51.2559 (4) | 0.57 (2) |  $-0.334$ (7) | 20.3 |  |  |

####  4.2.1 Analysis of the binarity

MZ Lac was found to be an EB by Miller & Wachmann (1971), where it is labelled
VV 399, but has been the subject of very little study since. It exhibits
apsidal motion (Silhan, 1990) with a period of $424\pm 6$ yr (Bulut et al.,
2016). The TESS light curve shows deep eclipses (Fig. 1) and covers two
sectors. The secondary eclipse is at approximately phase 0.5 but is
significantly longer than the primary, so $e\cos\omega\approx 0$ but
$e\sin\omega$ is large.

The pulsations in MZ Lac have a much lower amplitude than the eclipses so we
treated them as red noise: we fitted the whole dataset with jktebop without
attempting to reject out-of-eclipse data or compensate for the pulsations in
any way. We fitted for the same parameters as for CM Lac, except for those
related to RVs and for the inclusion of only four polynomials for the out-of-
eclipse brightness of the system (one for each TESS half-sector). Due to the
apsidal motion we did not attempt to use published eclipse timings to help
constrain the orbital ephemeris. A substantial amount of third light is
required to obtain a good fit to the data.

We were able to find a good fit to the light curve (Table 1). The residual-
permutation errorbars are significantly larger than the Monte Carlo errorbars,
which suggests it would be possible to improve the light curve analysis in
future by either subtracting the pulsations or fitting for them
simultaneously. The full physical properties of the system cannot be
calculated as no RVs are available. Gaia DR2 (Gaia Collaboration, 2018) gives
a $T_{\rm eff}$ of the system of 6900 K; if this is appropriate for star A
then the $T_{\rm eff}$ of star B is 6700 K based on the value of $J$ we
measure. These values are consistent with the presence of $\delta$ Sct and
$\gamma$ Dor pulsations (Grigahcène et al., 2010; Bowman & Kurtz, 2018; Murphy
et al., 2019).

####  4.2.2 Analysis of the pulsations

From a detailed analysis of the TESS photometry, we find that MZ Lac exhibits
an interesting variety of pulsation behaviour. Following the binary modelling,
we excluded the eclipses from the residual light curve and used this clipped
light curve for the iterative prewhitening. The prewhitened pulsation
frequencies of MZ Lac are listed in Table 3 and shown in Fig. 5.

Ten frequencies below 5 $\rm d^{-1}$ have been measured from the TESS light
curve. Five of these lie significantly close to harmonics of the orbital
frequency666MZ Lac experiences apsidal motion, so its sidereal and anomalistic
periods are not identical. However, they differ by only 0.000035 d so our
analysis is not affected by which we choose to represent the orbital
frequency., and four are well within 3$\sigma$. A visual inspection of the
light curve, phase-folded with the binary orbital period as shown in Fig. 2,
reveals that these frequencies are likely related to tidally excited
pulsations. An imperfect fit of the binary orbit to the light curve might
cause similar residual signal in the Fourier transform at integer multiples of
the orbital frequency. However, the dominant variation seen in the phase-
folded light curve (in Fig. 2) occurs on a timescale much shorter than the
orbital period, which is consistent with tidally excited pulsations, but not
with residual signal of the binary orbit itself.

We also obtain five non-harmonic low frequencies. While these have values of
typical $\gamma$ Dor-type pulsations, three pairs of them form combination
frequencies that (almost) coincide with harmonics of the orbital frequency:
$2\nu_{2}+\nu_{6}=11.000(3)\,\nu_{\rm orb},$
$\nu_{5}+\nu_{8}=11.9966(12)\,\nu_{\rm orb}$ and
$\nu_{2}+\nu_{9}=12.0303(16)\,\nu_{\rm orb}$. The first two combinations match
the closest orbital harmonics within $3\sigma$, and the last one matches the
orbital harmonic within $\nu_{\rm res}$. This can be caused by nonlinear mode
coupling, whereby the orbital harmonic is the parent frequency and the
observed non-harmonic frequencies are the child frequencies (e.g., Burkart et
al., 2012; Weinberg et al., 2013; Guo et al., 2017a). Finally, there are four
pressure-mode (p-mode) pulsations, with frequencies between 46 and 51.5 $\rm
d^{-1}$.

From a visual inspection of the TESS light curve shown in Fig.2, we find that
the pulsations appear to be less visible during _both_ eclipses. Our best
explanation for this apparent physical inconsistency is that the (tidally
excited) pulsations were partially fitted by the binary model in the eclipses.
This could have happened in particular if the pulsations had a similar effect
on the eclipse shape as LD, as some LD coefficients were included as fitted
parameters. The fitted LD coefficients are indeed larger than expected. Whilst
one approach to fixing this problem might be to fix the LD coefficients, this
risks biasing the measured radii because theoretical LD coefficients are not
completely reliable. An alternative approach in future could be to
simultaneously model the eclipses and the pulsations.

###  4.3 RX Draconis

Figure 6:  Sections of the Lomb-Scargle periodogram (black) of the residual light curve of RX Dra. The harmonics of the orbital frequency $\nu_{\rm orb}$ (dotted grey lines) and the extracted pulsation frequencies (dashed red lines) are also indicated. Table 4:  An overview of the parameter values for the significant frequencies of RX Dra. Possible combination frequencies within 3$\sigma$ are also listed. | Frequency | Amplitude | Phase | S/N | Combinations
---|---|---|---|---|---
| ($\rm d^{-1}$) | (mmag) | (rad) |  |
$\nu_{1}$ | 0.40386 (2) | 0.407 (5) |  $0.104$ (2) | 9.4 |
$\nu_{2}$ | 0.48268 (1) | 0.64 (5) |  $-0.486$ (1) | 15.2 |
$\nu_{3}$ | 0.51792 (3) | 0.252 (5) |  $-0.484$ (3) | 6.0 | $\nu_{19}-\nu_{11}$
$\nu_{4}$ | 0.54059 (2) | 0.376 (5) |  $0.296$ (2) | 9.1 |
$\nu_{5}$ | 0.56659 (2) | 0.516 (5) |  $0.492$ (2) | 12.5 |
$\nu_{6}$ | 0.58234 (2) | 0.395 (5) |  $0.371$ (2) | 9.6 |
$\nu_{7}$ | 0.63871 (2) | 0.41 (5) |  $0.235$ (2) | 10.1 |
$\nu_{8}$ | 0.65485 (2) | 0.373 (5) |  $0.491$ (2) | 9.2 | $\nu_{17}-\nu_{2}$
$\nu_{9}$ | 0.69392 (1) | 0.596 (5) |  $0.204$ (1) | 14.8 | $\nu_{18}-\nu_{2}$
$\nu_{10}$ | 0.72459 (4) | 0.224 (5) |  $0.401$ (4) | 5.6 | $\nu_{20}-\nu_{6}$
$\nu_{11}$ | 0.75959 (1) | 0.674 (5) |  $-0.414$ (1) | 17.0 |
$\nu_{12}$ | 0.77293 (2) | 0.397 (5) |  $-0.096$ (2) | 10.0 |
$\nu_{13}$ | 0.81678 (2) | 0.365 (5) |  $0.391$ (2) | 9.3 |
$\nu_{14}$ | 0.91214 (2) | 0.526 (5) |  $-0.26$ (2) | 13.8 |
$\nu_{15}$ | 0.92597 (2) | 0.393 (5) |  $0.115$ (2) | 10.4 |
$\nu_{16}$ | 1.10811 (5) | 0.179 (5) |  $-0.442$ (5) | 5.0 |
$\nu_{17}$ | 1.13749 (2) | 0.466 (5) |  $0.479$ (2) | 13.2 |
$\nu_{18}$ | 1.176625 (6) | 1.405 (5) |  $-0.230$3 (6) | 41.0 |
$\nu_{19}$ | 1.27764 (3) | 0.276 (5) |  $0.265$ (3) | 8.5 |
$\nu_{20}$ | 1.30693 (3) | 0.274 (5) |  $-0.002$ (3) | 8.6 |
$\nu_{21}$ | 1.5985 (6) | 0.132 (5) |  $-0.03$ (6) | 5.1 |
$\nu_{22}$ | 1.71749 (4) | 0.187 (5) |  $0.324$ (4) | 7.7 |
$\nu_{23}$ | 1.74289 (3) | 0.289 (5) |  $-0.44$ (3) | 12.1 |
$\nu_{24}$ | 2.8467 (1) | 0.054 (5) |  $0.37$ (1) | 4.3 |

####  4.3.1 Analysis of the binarity

RX Dra has been known to be eclipsing for over a century. The first
photometric analysis was given by Shapley (1913) but it has received very
little attention subsequently. It came to the authors’ attention by its
inclusion in a list of EBs that are candidates for hosting pulsating stars
(Soydugan et al., 2006), and inspection of its TESS light curve revealed clear
pulsations of the $\gamma$ Dor type. TESS observations are available for 13
consecutive sectors (14–26) followed by two further sectors (40 and 41). At
the time of writing, observations are also ongoing for 14 consecutive sectors
(47–60); see Section 2. We base our results in the current work on the data
from sectors 14–26.

Because of the huge amount of data (230 766 observations sampled at 120 s
cadence for one year) covering a large number of eclipses (approximately 170)
we first obtained a preliminary solution with jktebop to establish the orbital
ephemeris. We then converted the data into orbital phase and binned them into
1000 points equally spaced in phase. The primary eclipse is annular and the
secondary eclipse is total. A good fit to the binned data can be obtained
comparatively easily due to the total eclipses. We included orbital
eccentricity and third light as fitted parameters and both come out at very
small but probably non-zero. It was also possible to fit for _both_ LD
coefficients for both stars, an occurrence the first author has not previously
encountered despite having recently studied the TESS light curves of over 50
EBs. The errorbars from the Monte Carlo algorithm were adopted as they are
significantly larger than those from the residual-permutation approach. Our
results are given in Table 1. The uncertainties in the fractional radii are
below 0.1%, but we recommend that interested users increase them to at least
0.1% as this is the lower limit of reliability established by comparing
independent analyses of the same data for a similar EB (Maxted et al., 2020).

####  4.3.2 Analysis of the pulsations

RX Dra is an outlier in our sample. While we only have one or two TESS sectors
of data available for most of our stars, RX Dra is located in the northern CVZ
and has been observed during 13 consecutive sectors, spanning 351 days in
total. This has allowed us to better resolve the individual stellar pulsation
frequencies and measure them with a higher precision, as listed in Table 4 and
shown in Fig. 6.

Despite the short orbital period, all observed pulsations were found to be
g-mode pulsations. Four of the 24 measured frequencies form combinations
within 3$\sigma$. Additionally, the observed pulsation spectrum is relatively
sparse, given that we have almost a full year of photometry at our disposal.
The inclusion of additional photometric observations, such as those that are
currently being taken during TESS cycle 4, may improve the window functions of
the stellar pulsations sufficiently to allow us to measure additional g-mode
pulsation frequencies.

Similarly to the pulsations observed for MZ Lac, we cannot assign the
pulsations of RX Dra to a specific component. Here, the narrow triangular
shapes of the eclipses do not suffice to evaluate the pulsation amplitudes as
a function of the light contribution of the individual components, as
illustrated in Fig. 2. Additionally, the measured g-mode frequencies are too
sparse to detect clear g-mode period-spacing patterns, which would have
allowed us to perform pulsation mode identification and select a group or
groups of pulsations that originate from the same stellar component in the
binary system. There is, however, clear structure present in the observed
g-mode pulsation spectrum. The inclusion of the forthcoming TESS observations
will be needed to detect period-spacing patterns for RX Dra (e.g. Van Reeth et
al., 2015) if they are present in the data.

###  4.4 V2077 Cygni

####  4.4.1 Analysis of the binarity

V2077 Cyg was identified as an EB using photometry from the Hipparcos
satellite, and given its GCVS designation by Kazarovets et al. (1999) as a
result. The only study of the object so far was performed by Molenda-Żakowicz
et al. (2007), who obtained 29 spectra and measured the spectroscopic orbit of
the primary star. The secondary is much fainter and was not detected in the
spectra. The TESS light curve (Fig. 1) shows distinctive variability
characteristic of $\gamma$ Dor pulsators, plus primary eclipses that are much
deeper than the secondary eclipses. The light curve is strongly reminiscent of
the prototypical system KIC 11285625 (Debosscher et al., 2013), except that
the eclipses are total. TESS has observed the star during three sectors (14,
26 and 41) and four further sectors of observations (53, 54, 55 and 56) are
planned. The four consecutive sectors will be valuable for increasing the
precision of the measured pulsation periods. The following analyses only
includes sectors 14 and 26, as sector 41 was not available at the time they
were performed.

The pulsation amplitudes are comparable to the eclipse depths so we proceeded
as for CM Lac (Sect. 4.1): each eclipse was extracted from the light curve and
normalised to zero differential magnitude using a polynomial function. We
found that a third-order polynomial was necessary in some cases. The light
curve was then fitted as for CM Lac, with the exception that we had RVs for
only the primary star so did not fit for $K_{\rm B}$ or $V_{\rm\gamma,B}$ Fig.
7). The results are given in Table 1. The fractional radii of the stars are
well-measured because of the presence of total eclipses in this system.

Molenda-Żakowicz et al. (2007) measured a $T_{\rm eff}$ of $7066\pm 201$ K for
the primary star, in good agreement with the Gaia DR2 value of 6900 K. Our
value of $J$ then implies a $T_{\rm eff}$ of 5200 K for the secondary star.
Using equation 4 from Southworth et al. (2007) we determined the surface
gravity of the secondary star to be $\log g_{\rm B}=4.607\pm 0.007$, which is
appropriate for a K-dwarf. The $\gamma$ Dor pulsations arise in the primary
component: the secondary is too cool to be a $\gamma$ Dor star and it also
does not contribute enough light to the system to produce pulsations with an
amplitude of 0.1 mag.

Figure 7:  Fitted spectroscopic orbit for V2077 Cyg (blue lines) compared to
the measured RVs from Molenda-Żakowicz et al. (2007) for star A (red circles).
The residuals of the fit are shown in the lower panel.

####  4.4.2 Analysis of the pulsations

Four significant frequencies, listed in Table 5 and illustrated in Fig. 8, can
be obtained from iterative prewhitening of the residual TESS light curve. The
derived parameter values for these frequencies are all consistent with those
expected for $\gamma$ Dor-type pulsations. However, the available TESS data
are again insufficient to ensure that the individual g-mode pulsation
frequencies are properly resolved.

More TESS data are needed to properly identify the individual pulsations in
this system. Fortunately, it is scheduled to be observed in four consecutive
sectors (53–56). If TESS continues to operate as expected, and precise radial
velocities can be measured for both components, V2077 Cyg could become a
useful system for the study of pulsations in a star of known mass and radius.
Fig. 2 shows an increase in residuals for the binary fit during the eclipses.
This arises because the much smaller secondary star is blocking only part of
the surface of the pulsating primary star and thus breaking the spherical
symmetry that attenuates the observed amplitudes of pulsations. V2077 Cyg may
therefore be a good candidate for elipse mapping to spatially resolve the
pulsations on the stellar surface (e.g. Bíró & Nuspl, 2011).

Figure 8:  Part of the Lomb-Scargle periodogram (black) of the residual light curve of V2077 Cyg (after the best-fitting binary model has been subtracted). The harmonics of the orbital frequency $\nu_{\rm orb}$ (dashed grey lines) and the extracted pulsation frequencies (dashed red lines) are also indicated. Table 5:  An overview of the parameter values for the significant frequencies of V2077 Cyg. The identification of possible combination frequencies within 3$\sigma$ is also given. | Frequency | Amplitude | Phase | S/N | Combinations
---|---|---|---|---|---
| ($\rm d^{-1}$) | (mmag) | (rad) |  |
$\nu_{1}$ | 0.48209 (1) | 6.97 (10) |  $-$0.244 (2) | 4.1 |
$\nu_{2}$ | 0.65013 (1) | 8.5 (1) |  $-$0.163 (2) | 5.3 |
$\nu_{3}$ | 0.708696 (3) | 33.4 (1) | 0.4049 (5) | 21.1 |
$\nu_{4}$ | 1.41741 (1) | 8.72 (10) | 0.075 (2) | 6.7 | $\nu_{4}\approx 2\nu_{3}$

##  5 Summary and conclusions

The study of pulsating stars in EBs is a promising opportunity to improve our
understanding of the interior physics of stars. Gravity-mode pulsations are a
high priority because they can probe regions deep inside a star and thus help
constrain stellar rotation, convective core boundary mixing, envelope mixing,
opacity and magnetic fields (see references in Section 1). In pursuit of this
goal we have surveyed the TESS light curves of a sample of known EBs and
detected g-mode pulsations in four of these. In this work we present a
preliminary analysis of all four systems.

CM Lac shows very deep eclipses (0.8 mag for the primary and 0.4 mag for the
secondary) and unresolved g-modes with frequencies between $1\,\rm d^{-1}$ and
$1.5\,\rm d^{-1}$ in the one sector of available TESS data. The available data
are therefore insufficient for asteroseismology, but remain useful for the
determination of the physical properties of stars. We measured the masses and
radii of the component stars using the TESS data and the results of a
published RV study, but additional RVs are needed to measure their masses to
the canonical 2% precision (Andersen, 1991).

MZ Lac is a more interesting system due to its large orbital eccentricity
(Table 1) and richer pulsation spectrum. Based on the two sectors of TESS data
for this object we found evidence for p-modes and for tidally excited or
tidally perturbed g-mode pulsations. No RVs are currently available for this
object so we are not yet able to determine its physical properties.

RX Dra has been observed for 13 consecutive sectors by TESS but this abundance
of data yielded only relatively few identified frequencies. The presence of
total eclipses allowed the fractional radii of the stars to be measured to
exceptional precision so the system is a good candidate for precise
measurement of its physical properties. RX Dra is in the process of being
observed for a further 14 consecutive sectors by TESS and these additional
data may allow the detection of further pulsation frequencies.

V2077 Cyg shows beautiful $\gamma$ Dor pulsations but the few currently-
available TESS light curves are insufficient to perform mode identification.
The system is scheduled for observation in three consecutive sectors by TESS
in 2022 and these new data should greatly improve the asteroseismic potential
of this system. V2077 Cyg is not promising from the viewpoint of determining
its physical properties for two reasons. The eclipses are perturbed by the
high-amplitude g-mode pulsations, but are total so precise fractional radii
are still measurable. The secondary star is much fainter than the primary
(light ratio of 0.075 in the TESS passband) so measuring its RVs will require
high-quality spectroscopy. A single-lined spectroscopic orbit exists for this
system, allowing us to measure the surface gravity of the secondary component
and verify that it is a normal low-mass main-sequence star.

All four systems need detailed spectroscopic study to measure their physical
properties precisely, which in turn provides context for the pulsation
analysis. This work has begun.

## Data availability

All data underlying this article are available in the MAST archive
(https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html).

## Acknowledgements

TVR gratefully acknowledges support from the Research Foundation Flanders
(FWO) under grant agreement N°12ZB620N. We thank Johanna Molenda-Żakowicz for
sending us her RV measurements for V2077 Cyg, and Dominic Bowman and the
anonymous referee for discussions and comments. The TESS data presented in
this paper were obtained from the Mikulski Archive for Space Telescopes (MAST)
at the Space Telescope Science Institute (STScI). STScI is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS5-26555. Support to MAST for these data is provided by the NASA
Office of Space Science via grant NAG5-7584 and by other grants and contracts.
Funding for the TESS mission is provided by the NASA Explorer Program. This
research has made use of the SIMBAD database, operated at CDS, Strasbourg,
France; the SAO/NASA Astrophysics Data System; and the VizieR catalogue access
tool, CDS, Strasbourg, France.

## References

  * Aerts et al. (2010) Aerts C., Christensen-Dalsgaard J., Kurtz D. W., 2010, Asteroseismology. Astron. and Astroph. Library, Springer Netherlands, Amsterdam
  * Andersen (1991) Andersen J., 1991, A&ARv, 3, 91
  * Andersen et al. (1990) Andersen J., Clausen J. V., Nordström B., 1990, ApJ, 363, L33
  * Bíró & Nuspl (2011) Bíró I. B., Nuspl J., 2011, MNRAS, 416, 1601
  * Bouabid et al. (2013) Bouabid M. P., Dupret M. A., Salmon S., Montalbán J., Miglio A., Noels A., 2013, MNRAS, 429, 2500
  * Bowman (2017) Bowman D. M., 2017, Amplitude modulation of pulsation modes in delta Scuti stars. Springer Theses series, Springer, US
  * Bowman (2020) Bowman D. M., 2020, Frontiers in Astronomy and Space Sciences, 7, 70
  * Bowman & Kurtz (2018) Bowman D. M., Kurtz D. W., 2018, MNRAS, 476, 3169
  * Bowman & Michielsen (2021) Bowman D. M., Michielsen M., 2021, A&A, 656, A158
  * Bowman et al. (2019) Bowman D. M., Johnston C., Tkachenko A., Mkrtichian D. E., Gunsriwiwat K., Aerts C., 2019, ApJ, 883, L26
  * Breger et al. (1993) Breger M., et al., 1993, A&A, 271, 482
  * Bulut et al. (2016) Bulut A., Bulut I., Ćiçek C., Erdem A., 2016, in American Institute of Physics Conference Series. p. 040002
  * Burkart et al. (2012) Burkart J., Quataert E., Arras P., Weinberg N. N., 2012, MNRAS, 421, 983
  * Christophe et al. (2018) Christophe S., Ballot J., Ouazzani R. M., Antoci V., Salmon S. J. A. J., 2018, A&A, 618, A47
  * Claret (2018) Claret A., 2018, A&A, 618, A20
  * Claret & Torres (2018) Claret A., Torres G., 2018, ApJ, 859, 100
  * Clausen (1996) Clausen J. V., 1996, A&A, 308, 151
  * Colman et al. (2022) Colman I. L., Bedding T. R., Huber D., Kjeldsen H., 2022, ApJS, 258, 39
  * da Silva et al. (2014) da Silva R., Maceroni C., Gandolfi D., Lehmann H., Hatzes A. P., 2014, A&A, 565, A55
  * Debosscher et al. (2013) Debosscher J., et al., 2013, A&A, 556, A56
  * Degroote et al. (2009) Degroote P., et al., 2009, A&A, 506, 111
  * Fuller (2017) Fuller J., 2017, MNRAS, 472, 1538
  * Gaia Collaboration (2018) Gaia Collaboration 2018, A&A, 616, A1
  * Gaulme & Guzik (2019) Gaulme P., Guzik J. A., 2019, A&A, 630, A106
  * Graczyk et al. (2021) Graczyk D., et al., 2021, A&A, 649, A109
  * Grigahcène et al. (2010) Grigahcène A., et al., 2010, ApJ, 713, L192
  * Guo (2020) Guo Z., 2020, ApJ, 896, 161
  * Guo & Li (2019) Guo Z., Li G., 2019, ApJ, 882, L5
  * Guo et al. (2017a) Guo Z., Gies D. R., Fuller J., 2017a, ApJ, 834, 59
  * Guo et al. (2017b) Guo Z., Gies D. R., Matson R. A., 2017b, ApJ, 851, 39
  * Guo et al. (2019) Guo Z., Fuller J., Shporer A., Li G., Hambleton K., Manuel J., Murphy S., Isaacson H., 2019, ApJ, 885, 46
  * Hambleton et al. (2013) Hambleton K. M., et al., 2013, MNRAS, 434, 925
  * Hełminiak et al. (2017) Hełminiak K. G., Ukita N., Kambe E., Kozłowski S. K., Pawłaszek R., Maehara H., Baranec C., Konacki M., 2017, A&A, 602, A30
  * Henry et al. (2007) Henry G. W., Fekel F. C., Henry S. M., 2007, AJ, 133, 1421
  * Hilditch (2001) Hilditch R. W., 2001, An Introduction to Close Binary Stars. Cambridge University Press, Cambridge, UK
  * Jenkins et al. (2016) Jenkins J. M., et al., 2016, in Proc. SPIE. p. 99133E
  * Johnston et al. (2019a) Johnston C., Tkachenko A., Aerts C., Molenberghs G., Bowman D. M., Pedersen M. G., Buysschaert B., Pápics P. I., 2019a, MNRAS, 482, 1231
  * Johnston et al. (2019b) Johnston C., Pavlovski K., Tkachenko A., 2019b, A&A, 628, A25
  * Kaye et al. (1999) Kaye A. B., Handler G., Krisciunas K., Poretti E., Zerbi F. M., 1999, PASP, 111, 840
  * Kazarovets et al. (1999) Kazarovets E. V., Samus N. N., Durlevich O. V., Frolov M. S., Antipin S. V., Kireeva N. N., Pastukhova E. N., 1999, IBVS, 4659, 1
  * Kirk et al. (2016) Kirk B., et al., 2016, AJ, 151, 68
  * Kopal (1959) Kopal Z., 1959, Close Binary Systems. The International Astrophysics Series, London: Chapman & Hall
  * Kreiner et al. (2001) Kreiner J. M., Kim C.-H., Nha I.-S., 2001, An atlas of O-C diagrams of eclipsing binary stars
  * Kurtz et al. (2015) Kurtz D. W., Shibahashi H., Murphy S. J., Bedding T. R., Bowman D. M., 2015, MNRAS, 450, 3015
  * Lampens et al. (2018) Lampens P., et al., 2018, A&A, 610, A17
  * Lecoanet et al. (2022) Lecoanet D., Bowman D. M., Van Reeth T., 2022, MNRAS, 512, L16
  * Lee & Hong (2021) Lee J. W., Hong K., 2021, AJ, 161, 32
  * Lee et al. (2014) Lee J. W., Kim S.-L., Hong K., Lee C.-U., Koo J.-R., 2014, AJ, 148, 37
  * Lee et al. (2020) Lee J. W., Hong K., Kristiansen M. H., 2020, PASJ, 72, 37
  * Lee et al. (2021) Lee J. W., Hong K., Kim H.-Y., 2021, AJ, 161, 129
  * Li et al. (2020) Li G., Guo Z., Fuller J., Bedding T. R., Murphy S. J., Colman I. L., Hey D. R., 2020, MNRAS, 497, 4363
  * Liakos & Niarchos (2012) Liakos A., Niarchos P., 2012, Ap&SS, 340, 281
  * Maceroni et al. (2014) Maceroni C., et al., 2014, A&A, 563, A59
  * Maxted et al. (2020) Maxted P. F. L., et al., 2020, MNRAS, 498, 332
  * Michielsen et al. (2019) Michielsen M., Pedersen M. G., Augustson K. C., Mathis S., Aerts C., 2019, A&A, 628, A76
  * Miller & Wachmann (1971) Miller W. J., Wachmann A. A., 1971, Ricerche Astronomiche, 8, 12
  * Molenda-Żakowicz et al. (2007) Molenda-Żakowicz J., Frasca A., Latham D. W., Jerzykiewicz M., 2007, AcA, 57, 301
  * Mombarg et al. (2020) Mombarg J. S. G., Dotter A., Van Reeth T., Tkachenko A., Gebruers S., Aerts C., 2020, ApJ, 895, 51
  * Mombarg et al. (2021) Mombarg J. S. G., Van Reeth T., Aerts C., 2021, A&A, 650, A58
  * Murphy et al. (2019) Murphy S. J., Hey D., Van Reeth T., Bedding T. R., 2019, MNRAS, 485, 2380
  * Pedersen et al. (2021) Pedersen M. G., et al., 2021, Nature Astronomy, 5, 715
  * Pilecki et al. (2018) Pilecki B., et al., 2018, ApJ, 862, 43
  * Pols et al. (1997) Pols O. R., Tout C. A., Schroder K.-P., Eggleton P. P., Manners J., 1997, MNRAS, 289, 869
  * Popper (1968) Popper D. M., 1968, ApJ, 154, 191
  * Prat et al. (2019) Prat V., Mathis S., Buysschaert B., Van Beeck J., Bowman D. M., Aerts C., Neiner C., 2019, A&A, 627, A64
  * Prša et al. (2016) Prša A., et al., 2016, AJ, 152, 41
  * Reyniers & Smeyers (2003) Reyniers K., Smeyers P., 2003, A&A, 404, 1051
  * Ricker et al. (2015) Ricker G. R., et al., 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003
  * Russell (1914) Russell H. N., 1914, The Observatory, 37, 165
  * Scargle (1982) Scargle J. D., 1982, ApJ, 263, 835
  * Schmid & Aerts (2016) Schmid V. S., Aerts C., 2016, A&A, 592, A116
  * Sekaran et al. (2020) Sekaran S., et al., 2020, A&A, 643, A162
  * Sekaran et al. (2021) Sekaran S., Tkachenko A., Johnston C., Aerts C., 2021, A&A, 648, A91
  * Shapley (1913) Shapley H., 1913, ApJ, 38, 158
  * Silhan (1990) Silhan J., 1990, JAAVSO, 19, 12
  * Southworth (2008) Southworth J., 2008, MNRAS, 386, 1644
  * Southworth (2013) Southworth J., 2013, A&A, 557, A119
  * Southworth (2020) Southworth J., 2020, The Observatory, 140, 247
  * Southworth (2021a) Southworth J., 2021a, Universe, 7, 369
  * Southworth (2021b) Southworth J., 2021b, The Observatory, 141, 122
  * Southworth (2021c) Southworth J., 2021c, The Observatory, 141, 190
  * Southworth (2021d) Southworth J., 2021d, The Observatory, 141, 282
  * Southworth & Bowman (2022) Southworth J., Bowman D. M., 2022, MNRAS, 513, 3191
  * Southworth et al. (2004a) Southworth J., Maxted P. F. L., Smalley B., 2004a, MNRAS, 351, 1277
  * Southworth et al. (2004b) Southworth J., Zucker S., Maxted P. F. L., Smalley B., 2004b, MNRAS, 355, 986
  * Southworth et al. (2007) Southworth J., Wheatley P. J., Sams G., 2007, MNRAS, 379, L11
  * Southworth et al. (2020) Southworth J., Bowman D., Tkachenko A., Pavlovski K., 2020, MNRAS, 497, L19
  * Southworth et al. (2021) Southworth J., Bowman D. M., Pavlovski K., 2021, MNRAS, 501, L65
  * Sowicka et al. (2017) Sowicka P., Handler G., Dębski B., Jones D., Van de Sande M., Pápics P. I., 2017, MNRAS, 467, 4663
  * Soydugan et al. (2006) Soydugan E., Soydugan F., Demircan O., İbanoǧlu C., 2006, MNRAS, 370, 2013
  * Steindl et al. (2021) Steindl T., Zwintz K., Bowman D. M., 2021, A&A, 645, A119
  * Szewczuk et al. (2021) Szewczuk W., Walczak P., Daszyńska-Daszkiewicz J., 2021, MNRAS, 503, 5894
  * Takata et al. (2020a) Takata M., Ouazzani R. M., Saio H., Christophe S., Ballot J., Antoci V., Salmon S. J. A. J., Hijikawa K., 2020a, A&A, 635, A106
  * Takata et al. (2020b) Takata M., Ouazzani R. M., Saio H., Christophe S., Ballot J., Antoci V., Salmon S. J. A. J., 2020b, A&A, 644, A138
  * Tkachenko et al. (2020) Tkachenko A., et al., 2020, A&A, 637, A60
  * Van Beeck et al. (2020) Van Beeck J., Prat V., Van Reeth T., Mathis S., Bowman D. M., Neiner C., Aerts C., 2020, A&A, 638, A149
  * Van Reeth et al. (2015) Van Reeth T., et al., 2015, A&A, 574, A17
  * Van Reeth et al. (2016) Van Reeth T., Tkachenko A., Aerts C., 2016, A&A, 593, A120
  * Van Reeth et al. (2022) Van Reeth T., Southworth J., Van Beeck J., Bowman D. M., 2022, A&A, 659, A177
  * Wachmann (1931) Wachmann A. A., 1931, AN, 244, 303
  * Weinberg et al. (2013) Weinberg N. N., Arras P., Burkart J., 2013, ApJ, 769, 121
  * Wu & Li (2019) Wu T., Li Y., 2019, ApJ, 881, 86
  * Zhang et al. (2018) Zhang X. B., Fu J. N., Luo C. Q., Ren A. B., Yan Z. Z., 2018, ApJ, 865, 115
  * Zhang et al. (2020) Zhang X., Chen X., Zhang H., Fu J., Li Y., 2020, ApJ, 895, 124