# A variable star population in the open cluster NGC 6791 observed by the
Kepler spacecraft

S. Sanjayan1,2 A.S. Baran1,3,4 P. Németh1,5,6 K. Kinemuchi7,8 J. Ostrowski1
and S.K. Sahoo1,2
1ARDASTELLA Research Group  Institute of Physics  Pedagogical University of
Cracow
ul. Podchora̧żych 2  30-084 Kraków  Poland
2Centrum Astronomiczne im. Mikołaja Kopernika  Polskiej Akademii Nauk  ul.
Bartycka 18  00-716 Warszawa  Polska
3Embry-Riddle Aeronautical University  Department of Physical Sciences
Daytona Beach  FL 32114  USA
4Department of Physics  Astronomy  and Materials Science  Missouri State
University  Springfield  MO 65897  USA
5Astroserver.org  Fő tér 1  8533 Malomsok  Hungary
6Astronomical Institute of the Czech Academy of Sciences  Fričova 298  CZ-251
65 Ondřejov  Czech Republic
7Department of Astronomy  New Mexico State University  Box 30001  MSC 4500
Las Cruces  NM 88003  USA
8Apache Point Observatory  2001 Apache Point Road  P.O. Box 59  Sunspot  NM
88349-0059

(April 15, 2022)

###### Abstract

We present the list of variable stars we found in the Kepler superstamp data
covering approximately 9 arcminutes from the central region of NGC 6791. We
classified the variable stars based on the variability type and we established
their cluster membership based on the available Gaia Early Data Release 3
astrometry, by means of the Bayesian Gaussian mixture models. In total we
found 278 variable objects, among which 17 binaries, 45 pulsators, 62
rotational and five unclassified variables are cluster members. The remaining
28 binaries, 25 pulsators, 83 rotational, four unclassified and nine
unidentified variables are either not members or their membership is not
established. In the case of eclipsing binaries we calculated the mid-times of
eclipses and derived ephemerides. We searched for eclipse timing variation by
means of the observed minus calculated diagrams. Only three objects show
significant orbital period variation. Independently of a report published just
recently by Colman et al. (2022) we found 119 new variables. We used
isochrones calculated within the MIST project and derived the age (8.91 Gyr),
average distance (4134 pc) and iron content [Fe/H] (0.26-0.28), of NGC 6791.
Using the cluster members with membership probabilities greater than 0.9, we
calculated the distance to the cluster of 4123(31) pc, which agrees with the
result from our isochrone fitting.
Open clusters and associations : individual: NGC 6791 – binaries: general –
Stars oscillations – Stars : rotation

##  1\. Introduction

NGC 6791 has been first described as a metal rich cluster by Baade (1931) and
listed as an old open cluster by King (1964). The authors provided no age
estimations. Kinman (1965) presented a detailed comparative study of color –
magnitude diagrams (CMD) of NGC 6791 along with other two open clusters, M 67
(4 Gyr) and NGC 188 (6.8 Gyr). From the first photometric observations in the
B-V color, Harris and Canterna (1981) determined a reddening of E(B-V) = 0.13
mag. According to the recent studies NGC 6791 is 7 – 9 Gyr old (Chaboyer et
al. 1999, Carraro et al. 2006, Basu et al. 2011), and it has a mass of around
4 000 M⊙ (Kaluzny and Udalski 1992, Carraro et al. 2006, Platais et al. 2011,
Tofflemire et al. 2014). The cluster is located $\sim$8000 pc from the
Galactic center and 1000 pc above the Galactic plane. According to some
hypotheses the cluster may have formed in the bulge of the Galaxy and radially
migrated to its current location (Jilkova et al. 2012, Villanova et al. 2018).
The distance to the cluster is approximately 3 614 pc, which was estimated for
the first time by Stetson et al. (2003) from the de-reddened distance modulus
of (m-M)${}_{0}\approx$12.79 mag. The authors derived E(B-V) = 0.09 mag.
According to Villanova et al. (2018) NGC 6791 is a super metal-rich cluster
with [Fe/H] = +0.3 – +0.4. Geisler et al. (2012) showed that NGC 6791 has
multiple stellar populations, which makes the cluster chemically peculiar. NGC
6791 has an anomalous horizontal branch with a red clump (RC) region. Liebert
et al. (1994) found a group of extreme horizontal branch members using
spectrophotometry of blue targets observed by Kaluzny and Udalski (1992). The
age of the cluster predicts that it should have a rich population of cooling
white dwarfs, hence Bedin et al. (2005) observed the cluster using the Hubble
Space Telescope up to m${}_{\mathrm{F606W}}\approx$ 28.5 mag. They found the
white dwarf luminosity function to give a peak at 27.4 mag, which corresponds
to an age of 2.5 Gyr. Such an estimate does not agree with the age derived
from the main sequence (MS) or red giant branch (RGB) population (Chaboyer et
al. 1999, Carraro et al. 2006). Thus far, these studies show the cluster is
very unusual. A more detailed study is required to constrain the age and metal
abundances for understanding the formation and evolution of NGC 6791. A clear
picture of the cluster could be achieved by deriving an entire population of
variable stars and analysis of components of the stars to find their ages and
chemical abundances.

NGC 6791 has been a subject of extensive search for variable stars. Kaluzny
and Udalski (1992) and Kaluzny and Rucinski (1993) did an extensive
photometric survey finding 17 variable stars which includes 8 contact
binaries, two blue stragglers and one binary consisting of a hot subdwarf B
star. Rucinski et al. (1996) found three detached binaries and one cataclysmic
variable (CV) star exhibiting a three day outburst. As a part of search for
planets in stellar clusters, Mochejska et al. (2002) found 47 new low
amplitude variable stars. The authors reported several BY Dra type and two
outbursting CV stars, confirming the CV found by Rucinski et al. (1996).
Mochejska et al. (2003) reported seven new variable stars with a long and
periodic flux variation. Kaluzny (2003) found four new variable stars by
reanalyzing archived data from Kaluzny and Rucinski (1993). A search for
transiting events by giant planets reported by Bruntt et al. (2003) yielded 22
new low amplitude objects along with 20 previously known variable stars. Using
a high precision time-series photometry, Hartman et al. (2005) detected 10 new
variable stars including one $\delta$-Scuti type star and 8 contact binaries.
Mochejska et al. (2005) detected 14 more variable stars and reported 9
eclipsing binaries. Using a high precision photometry in the Johnson V band,
de Marchi et al. (2007) detected 260 variables in the cluster area, although
not all stars are members of the cluster.

From the launch in 2009, for almost 10 years the Kepler spacecraft has served
mankind by providing very precise and almost continuous photometric
measurements (Koch et al. 2010). The Kepler has observed more than five
hundred thousand stars during its entire mission time. The Kepler mission was
completed in two phases. During the first mission, Kepler observed 0.25% of
the sky in the direction of Cygnus and Lyra constellation for 1460 days. The
mission was reborn as K2 (second mission) after the second reaction wheel
failed. K2 mission made 80 day observing campaigns along the ecliptic equator,
which lasted 1695 days (Howell et al. 2014). During both missions, the
observations were obtained using two different exposures, 30 minutes for the
long cadence (LC) and 1 minute for the short cadence (SC) mode (Koch et al.
2010, Borucki et al. 2010, Caldwell et al. 2010, Thompson et al. 2016). During
the first mission, four open clusters were inside the Kepler field of view,
NGC 6791, NGC 6819, NGC 6811 and NGC 6866. Two of the open clusters, NGC 6791
and NGC 6819, were observed by using the so-called LC superstamps.

Recently, Colman et al. (2022) presented light curves of KIC stars obtained
from the Kepler superstamp data. The authors used an image subtraction method
to derive light curves of all Kepler cataloged targets. They identified
variability in 239 out of 5342 stars they extracted light curves of. The
number of new variables is not given. We stress that our work has been
performed simultaneously to, yet independently from, Colman et al. (2022) and
contains additional analysis. By comparing our results with results of Colman
et al. (2022), we have noticed that the authors applied a very strong
detrending policy removing either eclipses or out-of-eclipse variations in
binaries or variations in other objects that we claim to be variables.

In Section 2 of this paper, we present a brief description of the Kepler data
and method used for obtaining the light curves of variable objects. In Section
3, we present a spectroscopic study of the variable stars found in this
project using either archived spectra from public surveys or our own data. In
Section 4, we describe the method of deriving the membership probabilities of
our new variable star findings. In Section 5, we report individual variable
star cluster members divided into variability classes. The field variable
counterparts are listed in the Tables 5–8. In Section 6, we present the result
of isochrone fitting.

##  2\. Kepler Photometry

We downloaded the Kepler superstamp data of NGC 6791 from the Mikulski Archive
for Space Telescopes (MAST111https://archive.stsci.edu/). The data are 20 x
100 pixel boxes piled up in two contiguous 10 box stacks. The field of view of
all pixels is 800 x 800 arcseconds and covers the most central part of the
cluster. The superstamps data are collected in the LC mode. The pixel scale of
an individual square pixel is 4 arcsec. The data have been collected over 1460
days and are split into 18 quarters.

We searched for a flux variation by extracting fluxes for all time stamps in
individual pixels for each of the quarters Q 2 – 5. Then, a Fourier transform
of the time-series data was performed in each pixel and each quarter
separately. The pixels showing peaks (representing signal) in the amplitude
spectra were selected. Signals that were identified with artifacts, either
reported by Baran (2013) or those found in this project, were discarded. We
combined all contiguous pixels showing the same signal and defined an optimal
aperture of pixels. To keep the solar cells exposed to the sunlight, every
quarter the spacecraft rolled 90 degrees, hence, with each quarterly rotation
of the spacecraft, our targets landed on different CCD chips. This positioning
caused different target images, and consequently, different optimal apertures
(Bryson et al. 2010). Fortunately, every four quarters the images and
apertures were the same, so we have defined the apertures only in four
quarters, i.e. Q 2 – 5, and propagate them to the corresponding quarters (e.g.
Q 2, 6, 10, 14). Next, using the optimal apertures for all targets showing
flux variation we used PyKE software (Kinemuchi et al. 2012) to pull out the
fluxes and correct them for instrumental artifacts by means of Co-trending
Basis Vectors. Finally, using our custom scripts, we clipped the data at 4.5
sigma, detrended using spline fits, and normalized them to parts per thousand
(ppt). The variable stars discovered in our work will be presented in Section
5.

##  3\. Spectroscopy

We searched for spectra in the literature of all variables we detected. We
found optical or infrared spectra for 111 objects in the archives of APOGEE
(Ahn et al. 2014, Majewski et al. 2017), SDSS (Blanton et al. 2017), LAMOST
(Zhao et al. 2012), ESO (Gilmore et al. 2012, Randich et al. 2013), and the
HECTOSPEC (Fabricant et al. 2005) surveys. All spectra with T${}_{\rm
eff}<15\,000$ K were modeled with interpolated local thermal equilibrium (LTE)
synthetic spectra drawn from the BOSZ (Bohlin et al. 2017) spectral library to
determine the fundamental atmospheric parameters. The BOSZ library was
calculated for scaled solar metallicity with carbon and $\alpha$-element
enhancement; therefore, individual abundance patterns cannot be investigated
with our method.

Our fitting procedure (XTgrid; Németh et al. 2012) is based on a steepest-
gradient chi-square minimizing method, which was originally developed to model
hot stars. To improve its performance for cool stars, we added a grid-search
preconditioning to the procedure. We step through a set of models to search
for the best starting model for the steepest-descent part. Next, the descent
part takes over in driving the fit and converges on the best solution. Once a
convergence is achieved, the procedure explores the parameter errors by
stepping through a set of points around the best solution. If a better
solution is found during error calculations, then the procedure returns to the
descent part, and hence pushing the solution towards the global minimum.
XTgrid fits the radial velocity and projected rotation velocity of each
spectra along with the stellar surface parameters, such as the effective
temperature (Teff), surface gravity ($\log$g) and abundances.

In addition, the procedure accesses photometric data from the VizieR
Photometry Viewer222http://vizier.u-strasbg.fr/vizier/sed/, distance data from
the Gaia EDR3 database, and extinction values from the NED online services.
The spectroscopic surface parameters combined with these measurements allow us
to reduce systematics and derive absolute stellar parameters, such as mass,
radius, and luminosity. An anti-correlation is observed between Teff and
[Fe/H]. Fortunately, the spectral energy distribution (SED) helps in resolving
this bias by restricting the Teff. Another bias is observed in surface
gravity, in particular below Teff = 4 000 K. At such low temperatures, the
spectrum is insensitive to the surface gravity. When the spectral coverage is
very limited, we could not determine an accurate value for $\log$g. We do not
report atmospheric parameters for such stars.

The archival spectroscopic data are very inhomogeneous. Consequently, high
resolution spectra (e.g. obtained with ESO instruments) with a short
wavelength coverage are more suitable for radial velocity measurements, while
low resolution spectra (e.g. from the SDSS and LAMOST surveys) can provide
more consistent atmospheric parameters, but less precise velocities. Some ESO
spectra cover only 5 300-5 600 Å range at a resolution of R=20 000, and only
weak spectral features are visible. For such spectra, at a relatively low
signal-to-noise ratio (SNR), the fitting procedure increases the projected
rotation above 100 km s-1, which decreases the radial velocity accuracy. In
general, low SNR spectra limit our analysis the most, while crowding in dense
stellar fields and a limited spectral coverage affects the parameter
determination.

##  4\. Cluster membership

We used Gaia astrometry to determine the membership probabilities of all
variable stars we found. We used five parameters, i.e. equatorial coordinates
$\alpha$ and $\delta$, proper motions $\mu_{\alpha}$ and $\mu_{\delta}$, and
parallax $\pi$ (further called five astrometric parameters). First, we
adopted/estimated mean values of these parameters. The cluster center has been
taken from Kamann et al. (2019) to be at $\alpha_{\rm 2000}$ = 19:20:51.3 and
$\delta_{\rm 2000}$ = +37:46:26. Next, we downloaded Gaia Early Data Release 3
(EDR3) (Gaia collaboration et al. 2016, 2021) data for all stars within a
tidal radius of 23 arcmin (Platais et al. 2011). The area contains 36 647
targets accessible to our analysis; however, we filtered out dubious targets
with parallaxes to be negative or greater than 1 arcsec or relative
uncertainties for any of the proper motion or parallax values to be greater
than 50%. A cluster environment, particularly toward the center, is very
dense, which can lead to unrealistic or imprecise estimates of these three
parameters ($\mu_{\alpha}$, $\mu_{\delta}$, $\pi$). In addition, we limited
our sample to targets for which the zero point offset corrections of parallax
have been applied (Lindegren et al. 2021). After filtering and correcting for
the parallax zero offset, we ended up with 11 466 targets.

To determine the membership probabilities, we used the Bayesian Gaussian
Mixture Models (GMM) using scikit–learn python toolkit (Pedregosa et al.
2011). The GMM assumes each data point to be a combination of finite Gaussian
functions, in which the number of these functions is determined using a
variational Bayesian inference model with Dirichlet process prior (Ferguson
1973). We performed 10 000 iterations using the Expectation – Maximization
algorithm (Dempster et al. 1977), and we derived membership probabilities for
each target in our sample based on all five astrometric parameters. In the
case of targets that we found variables in the superstamp area, we estimated
their membership probabilities regardless of precision of their five
astrometric parameters. If the uncertainties were larger than 50%, we
considered corresponding parameters to be error-free, while the negative
parallaxes were ignored and only four astrometric parameters were used.

To strengthen the probability, the radial velocity of individual stars can
also be used, however they need to be corrected for the effects of binarity,
rotation, and pulsations. Different instruments differ in instrumental
calibration which often bias the radial velocity (RV) estimates. Since we did
not conduct one single survey that could provide us with consistent RV
estimations, we decided not to use RVs for the membership analysis. Since
binarity and rotation affects the measurement of intrinsic motion, we expect
the RVs will be random values and, as it will be seen in Section 5, the values
in Tables 1–3, confirm our suspicion. We expected the most consistent
estimates for single solar-like pulsators, since their oscillation motion on
the surface is of a very small amplitude. In fact, in only three cases the RVs
are far from the average cluster value (-47.46$\pm$1.08 km/s, Carrera et al.
2019), since the stars may belong to binary systems. On the other hand, RVs
that are consistent indicate that the stars are likely single or the orbital
motion (if any) is very slow or the spectra have been taken when both stars
were aligned with the observer’s line of sight. The RVs of the solar-like
stars that are unlikely to be members (Table 6) are not close to the cluster
average and seem to confirm their field membership, unless they are in
binaries.

##  5\. A zoo of variable stars

In total, we found 278 variable objects in the superstamp area. Our sample
contains cluster members as well as foreground and background stars. In
Section 4, we provided details on a membership analysis. Our prime focus is on
the members of NGC 6791. The non-members and objects with unknown membership
status, as a consequence of a lack of the Gaia astrometry, are listed in the
Table 5-8. Their variability is classified the same way as for the cluster
members.

Figure 1: Examples of light curves and corresponding amplitude spectra of a
zoo of variable stars in the open cluster NGC 6791.

Based on flux variations, we classified the stars into three main variability
types, i.e. eclipsing, pulsating, and rotating stars. The first two types are
further split into specific classes. Five stars remained unclassified. Their
light curves show variations that we are unable to unambiguously identify as
one of the three types listed. These objects show flux variations which can
origin in e.g. a reflection effect, ellipsoidal variation or a rotation of a
spotted star. These stars have typically low amplitude flux variations. In
Fig. 1 we present examples of light curves and their corresponding amplitude
spectra for each type and a selection of classes of variable stars we found.

###  5.1 Binary systems

We selected binary stars with sharp eclipses typical for semi- and detached
systems. Some eclipsing systems show additional out-of-eclipse variation,
which can be caused by a chromospheric activity, and we call them ”active”
eclipsing. We identified contact systems, which are characterized by a
continuous flux change and typical for W UMa stars. Another class contains
outbursting stars, which we associate with binaries experiencing a rapid mass
transfer causing sudden eruptions, e.g. novae, dwarf novae, nova-like
variables. We stress that our classification is not based on radial
velocities. Some of the stars may not be classified correctly, e.g. a smoothly
continuous and small amplitude flux changes may be misidentified with
rotational variables. However, the flux change over the course of observations
is not modulated (see explanation in Section 5.3) or they can be long-period
pulsating stars. In Fig. 2 we present the phased light curves of three stars
that we consider new discoveries. The sample includes all the classes of
binary stars we identified in the superstamp data. We found 17 binaries to be
cluster members (Table 1), 28 binaries are field objects, including two
binaries for which we could not establish membership due to the lack of Gaia
astrometry data (Table 5). For binary systems their membership has been
derived based on all five astrometric parameters. Majority of binaries in the
cluster are main sequence (MS) stars with just two exceptions, assuming the
position of the latter in the CMD is correct. Gaia EDR3 2051105720053889536 is
a post-MS star on its early ascent of the red giant branch (RGB), while Gaia
EDR3 2051293186783992320 is located below the RGB, which can be explained by
an incorrect color index or pre-MS evolutionary status. Among the member
counterparts, five are eclipsing, six active eclipsing, five contact, and one
outbursting stars.

Table 1: List of binary stars that are cluster members.

Gaia EDR3 | KIC | Porb | T0 | G | CMD | HRD | Teff | $\log$g | RV | [Fe/H] | Ref
---|---|---|---|---|---|---|---|---|---|---|---
[days] | [BJD] | [mag] | [K] | [km/s]
eclipsing
2051287203888386688 | 2436378 10L | 8.531633(8) | 2 454 969.2757(8) | 19.031 | MS+ | – | – | – | – | – | –
2051288372129693440 | 2569175 X | 43.499395(21) | 2 455 041.61331(38) | 18.383 | MS | RGB | 5 620(40) | 3.517(18) | -65.8(4) | -0.67(9) | 2∗
2051292980625561216 | 2437149 2S,6L | 18.7986285(37) | 2 454 971.02807(16) | 17.480 | MS | MS+MS | 5 100(900) | 4.5(4) | -80.9(9) | -0.25(25) | 2∗
2051293083698347008 | 2437482 X | 17.551175(14) | 2 454 966.9452(7) | 18.210 | MS | RGB | 4 860(50) | 3.042(20) | -67.5(5) | -0.48(15) | 1∗
2051293186777509376 | 2437041 X | 34.859569(14) | 2 454 979.90568(33) | 18.055 | MS | MS | 6 360(40) | 4.58(7) | -57.0(8) | -0.42(11) | 2∗
active eclipsing
2051105342091761536 | 2437783 4L | 7.453112(10) | 2 454 964.7652(11) | 18.391 | MS | BS | 7 160(50) | 3.60(8) | -20.1(6) | -0.82(9) | 2∗
2051105548255223680 | 2438490 4L | 3.3157657(10) | 2 454 967.85556(27) | 17.887 | MS+ | HB | 5 410(90) | 2.87(7) | -62.2(12) | -0.68(15) | 2∗
2051105784476572032 | 2438061 1S,12L | 4.8858826(16) | 2 454 965.01144(28) | 17.721 | MS | HB | 5 220(30) | 2.197(10) | -46.5(45) | 0.42(22) | 1∗
2051293186783992320 | 2437060 6L | 3.1871038(8) | 2 454 969.04204(21) | 16.989 | RGB | MS | 5 270(20) | 4.253(45) | -59.6(7) | -0.445(47) | 1∗
2051294423734919552 | 2569494 1S,5L | 1.5232747(12) | 2 454 965.3339(7) | 17.288 | MS | MS | 5 580(50) | 4.06(6) | -116(8) | -0.36(11) | 1∗
2051295351447975168 | 2570480 X | 0.7349394(16) | 2 454 964.9628(18) | 19.340 | MS+ | HB | 5 870(160) | 2.34(11) | 41.2(30) | -0.14(7) | 2∗
contact
2051105543955885696 | 2438413 X | 0.31754621(10) | 2 454 964.63952(27) | 18.281 | MS | RGB | 5 380(100) | 3.347(38) | -34.70(13) | -0.631(46) | 2∗
2051105720053889536 | 2438148 X | 0.29468323(11) | 2 454 964.79285(30) | 16.612 | RGB | – | – | – | – | – | –
2051291361416937856 | 2568950 X | 0.3917614(15) | 2 454 964.91341(33) | 17.555 | MS | MS/RGB | 5 300(20) | 3.926(20) | -39.50(10) | -1.131(26) | 1∗
2051293324222974976 | 2569965 X | 0.325587259(49) | 2 454 964.71162(13) | 17.510 | MS+ | MS | 5 380(20) | 4.46(8) | -63(6) | -0.36(8) | 1∗
2051294114497255936 | 2569630 X | 0.31265938(26) | 2 454 964.7815(7) | 16.964 | MS | MS | 5 620(50) | 4,030(56) | -41(2) | -0.14(9) | 1∗
outbursting
2051105788768615040 | 2438114 4L | 32.27 | – | 18.192 | blue of MS+ | blue of MS | 27 000(5 000) | 7.3(6) | -34(25) | -1(1) | 1∗

Figure 2: Phased light curves of three newly discovered binary star members of
NGC 6791.

Since the objects considered in this section are binaries, magnitudes and
color indices used in the top panel of Fig. 3 and Teff and $\log$g in the top
panel of Fig. 6 (this figure will be explained in Section 6) may be averages
of all components in a specific binary system and may not indicate a proper
location of individual stars in the CMD. If the average location in the CMD
diagram is in the well defined regions, e.g. MS or RGB, we can expect the
components are of similar properties (or a secondary component is not
detectable in our data), and the average is not far from single star values
(unless the binary components are much different but the shift is only along a
given evolutionary stage, e.g. along the RGB branch). If the location is
outside those regions, e.g. below MS or RGB, then the individual components in
a binary system are not alike, e.g. WD + MS, and the average will be somewhere
between a WD track and the MS (as is the case for Gaia EDR3
2051105788768615040). For this same reason, the RV of binaries is typically
different from the average RV of a cluster. We expect the orbital RV to be
dominating. There are only one exception with the RV being close to the
average cluster value. In the case of Gaia EDR3 2051105784476572032 the
spectrum could have been taken during the eclipse or it is a spectroscopic
single-line binary.

Figure 3: The color-magnitude diagram for NGC 6791. We listed variable stars
such as binaries, pulsators and rotational cluster members in the top, middle
and bottom panels, respectively. Unclassified stars are plotted in the bottom
panel.

We estimated the mid-times of eclipses or, in the case of non-eclipsing
systems, a minimum of a light variation by means of the method described in
Kwee & van Woerden (1956). We used the mid-times to derive the ephemeris, i.e.
a reference epoch, defined as the first mid-time in a given dataset, and an
orbital period for each binary system we found. We provided estimates of these
two parameters in Table 1 and Table 5. The exceptions are systems for which
the data are very noisy, not allowing us to derive precise mid-times, and
outbursting stars. For these systems we reported only rough estimates
(arbitrarily adopted two decimal places for precision) of their orbital
periods, derived from the Fourier amplitude spectra. The ephemerides were used
to calculate the Observed - Calculated (O-C) diagrams to check on the orbital
period variation. For most of the stars the O-C diagrams do not show any
significant period variation. The exceptions are shown in Fig. 4. Gaia EDR3
2051294114497255936 (the top panel of Fig. 4) shows quite a large amplitude
period variation with only one full cycle covered. These two O-C diagrams are
constructed based on an average primary and secondary eclipses. Gaia EDR3
2051105342091761536 (the bottom panel of Fig. 4) shows cyclic variations of
the orbital period derived solely from primary eclipses. To explain a periodic
change an additional body in the system can be invoked. To test this
hypothesis and, if confirmed, to constrain the parameters of those bodies, RV
time-series data are required.

Figure 4: O-C diagrams for three binary systems that show variation of the
orbital period.

###  5.2 Pulsators

We found 70 objects that show an intrinsic variability caused by stellar
oscillations. We identified the following types of pulsators, i.e. $\delta$
Scuti and $\gamma$-Dor, solar-like along the RGB, RR Lyr, pulsating hot
subdwarf B stars (sdBV) on the EHB, and semi-regular along the RGB and AGB. In
Table 2 we show only the cluster member counterpart, while the field pulsators
are listed in Table 6.

Among the cluster members we found one blue straggler, 36 solar-like pulsators
(10 being RC objects, while 26 are still on the RGB), three sdBVs and five
semi-regular giants. We analyzed the pulsation component in the solar-like
counterpart, and the results will be published elsewhere. The analysis of the
only three sdBVs we found in NGC6791 has been already reported by Sanjayan et
al. (2022). Among sdBVs only Gaia EDR3 2051105509596144768 is a known binary
star. The RV is likely influenced by orbital motion. The other two sdBVs also
show different RVs from the cluster average, which may be an indication of
their binary nature. Sanjayan et al. (2022) reported on the time-series
spectroscopy (their Figure 4), which indicates a RV variation. The RVs of
other pulsators we found to be members of NGC 6791 are close to the average
value, with only a few exceptions. For instance, Gaia EDR3 2051293255503478528
shows a much faster motion, but the quality flag of the spectra suggests a
possible contamination. We do not have resources to sort out these
inconsistency cases. Perhaps the explanation of these cases is an additional
RV component either of a binary or a rotational nature. In the case of other
objects the RV confirms their membership and a single nature. We are aware
that their spectra could have been taken while their orbital RV is negligible,
but we consider it to be a less likely case. As in the case of binaries, the
location of pulsators in the CMD (the middle panels of Fig. 3 and 6) may still
be influenced by binary components.

Table 2: List of cluster members showing pulsations. See the caption of Table
1 for explanation. Ref 6 refers to Sanjayan et al. (2022).

Gaia EDR3 | KIC | G | CMD | HRD | Teff | $\log$g | RV | [Fe/H] | Ref
---|---|---|---|---|---|---|---|---|---
[mag] | [K] | [km/s]
$\gamma$-Dor
2051105479535706752 | 2438249 17L | 15.573 | BS | BS | 6 980(90) | 3.636(84) | -56.7(9) | -0.15(13) | 2∗
solar-like
2051098710667044608 | 2297384 17L | 14.204 | RC | RC | 4 512(91) | 2.460(28) | -45.71(17) | 0.400(7) | 5
2051099260422933632 | 2437539 17L | 14.545 | RGB | RGB | 4230(30) | 2.353(10) | -39.9(4) | -0.57(7) | 2∗
2051099329142406400 | 2437240 17L | 14.648 | RGB | RGB | 4 380(70) | 3.116(17) | -45.5(3) | -0.12(13) | 2∗
2051104586182496000 | 2438462 12L | 14.586 | RGB | RGB | 4 240(80) | 2.020(23) | -45.71(37) | -0.39(7) | 2∗
2051105135938204160 | 2437507 17L | 14.032 | RGB | RGB | 4 262(80) | 2.160(46) | -47.504(21) | 0.400(7) | 5
2051105135938218624 | 2437698 17L | 14.191 | RC | RC | 4 521(93) | 2.370(28) | -45.25(6) | 0.400(8) | 4
2051105239017444992 | 2437564 17L | 14.226 | RC | RC | 4 515(99) | 2.400(29) | -48.53(11) | 0.400(9) | 5
2051105342096702080 | 2437804 17L | 14.110 | RC | RC | 4 459(97) | 2.350(27) | -45.367(23) | 0.400(9) | 5
2051286795872609920 | 2436209 17L | 14.802 | RGB | RGB | 4 498(103) | 2.650(45) | -47.91(45) | 0.400(9) | 5
2051287002031072768 | 2437164 12L | 14.157 | RC | RC | 4 300(50) | 2.515(10) | -48.26(37) | -0.58(13) | 2∗
2051287070750544128 | 2436900 17L | 14.456 | RGB | RGB | 4 428(101) | 2.500(45) | -47.92(8) | 0.400(9) | 5
2051287242549228032 | 2436608 10L | 14.446 | RGB | RGB | 4 500(50) | 2.254(35) | -45.5(15) | -0.07(7) | 2∗
2051287311268703232 | 2436543 12L | 14.120 | RC | – | – | – | – | – | –
2051287311268706432 | 2436540 17L | 14.856 | RGB | RGB | 4 473(104) | 2.560(46) | -48.13(14) | 0.400(10) | 4
2051288067183227008 | 2436417 17L | 14.104 | RC | – | – | – | – | – | –
2051288135902691840 | 2436332 17L | 14.302 | RGB | RGB | 4 360(60) | 3.445(20) | -47.54(30) | -0.07(12) | 2∗
2051288170262450816 | 2436458 17L | 14.532 | RGB | RGB | 4 250(30) | 2.602(10) | -46.36(30) | -0.24(11) | 2∗
2051291228279110912 | 2435987 17L | 14.529 | RGB | RGB | 4 428(98) | 2.490(46) | -44(4) | 0.400(9) | 5
2051292877546340864 | 2437340 17L | 13.462 | RGB | RGB | 4 058(82) | 1.660(43) | -45.0(8) | 0.300(9) | 5
2051293083704792960 | 2437444 17L | 13.998 | RGB | RGB | 4 105(66) | 2.35(11) | -56(4) | 0.12(6) | 3
2051293118064539520 | 2437496 17L | 13.069 | RGB | RGB | 3 920(67) | 1.350(45) | -47.8(7) | 0.300(7) | 5
2051293118064542976 | 2437653 17L | 15.090 | RGB | RGB | 4 559(97) | 2.710(48) | -47(1) | 0.400(8) | 5
2051293152424246912 | 2436884 10L | 13.476 | RGB | RGB | 4 061(83) | 1.670(43) | -44.482(34) | 0.300(9) | 4
2051293186784001152 | 2437103 17L | 14.378 | RGB | RGB | 4 500(50) | 3.356(41) | -48.6(15) | 0.002(92) | 2∗
2051293221143724032 | 2436824 17L | 14.478 | RGB | – | – | – | – | – | –
2051293221143725952 | 2436814 17L | 14.209 | RGB | – | – | – | – | – | –
2051293255503469184 | 2436912 17L | 14.181 | RC | RC | 4 330(90) | 2.140(46) | -45.75(42) | 0.300(8) | 4
2051293255503478528 | 2437040 17L | 14.158 | RGB | RGB | 3 760(50) | 2.955(27) | -81.7(8) | 0.003(10) | 5 ∗
2051293319920668160 | 2569935 17L | 13.065 | RGB | RGB | 4 032(65) | 1.500(46) | -47.90(8) | 0.400(6) | 5
2051293599100902016 | 2570518 17L | 14.683 | RGB | RGB | 4 509(89) | 2.600(49) | -48.68(8) | 0.300(7) | 5
2051293977058270592 | 2436732 17L | 14.135 | RC | RGB | 4 300(100) | 2.60(7) | -44.9(2) | -0.36(8) | 2∗
2051294045777739520 | 2569360 17L | 14.093 | RGB | RGB | 4 287(86) | 2.180(46) | -46.44(47) | 0.300(8) | 5
2051294114497261952 | 2569618 17L | 14.797 | RGB | RGB | 4 486(103) | 2.650(46) | -45.75(9) | 0.400(9) | 5
2051294213273828224 | 2569624 X | 12.965 | RGB | RGB | 3 896(62) | 1.300(45) | -44.98(7) | 0.300(6) | 5
2051294251936167424 | 2569204 11L | 13.293 | RGB | RGB | 3 967(77) | 1.430(43) | -47.909(11) | 0.300(8) | 5
2051297275593144192 | 2569055 17L | 14.173 | RC | RC | 4 543(97) | 2.450(29) | -46.881(12) | 0.400(8) | 5
sdBV
2051105509596144768 | 2438324 35S,12L | 17.859 | EHB | EHB | 25 290(300) | 5.510(43) | -90(3) | -2.62(11) | 6
2051105754408746880 | 2437937 1S,1L | 17.878 | EHB | EHB | 24 860(270) | 5.348(52) | -75(6) | -2.46(12) | 6
2051294183216739584 | 2569576 12S,5L | 17.752 | EHB | EHB | 23 540(210) | 5.311(35) | -28(4) | -2.73(23) | 6
semi-regular
2051105101578558464 | 2438151 17L | 15.655 | RGB | MS | 5 420(60) | 3.912(19) | -50.0(15) | -0.10(9) | 1∗
2051105616974709504 | 2438421 17L | 12.246 | AGB | AGB | 3 555(57) | 0.510(43) | -46.540(46) | 0.200(8) | 5
2051287002031070208 | 2437171 17L | 12.394 | AGB | AGB | 3 540(56) | 0.610(42) | -47.11(16) | 0.300(8) | 5
2051287173829740032 | 2436324 8L | 12.373 | AGB | AGB | 3 332(53) | 0.270(41) | -49.315(19) | 0.200(10) | 5
2051292976324527232 | 2437317 17L | 14.939 | RGB | – | – | – | – | – | –

###  5.3 Rotational Variables

We defined rotational type as stars showing modulated flux variation. Such
variation is caused by the presence of migrating spots on the surface, which
contribute to a flux modulation as a star rotates. The rotational variables
can mimic other types of variability, e.g. ellipsoidal or contact binaries,
and high-amplitude radially pulsating stars. We stress that our identification
may not be ideal for rotational variable stars. We assumed that binaries show
none or negligible modulation of a flux variation. This modulated variation
can be verified by either a light curve shape or a profile of peaks in
amplitude spectra. A complex profile indicates either period or amplitude
change. Objects showing light curves with modulated flux variation have been
classified as rotational variables.

We found a total of 145 rotational variables out of which 62 are cluster
members and are listed in Table 3. The field counterpart is listed in Table 7.
Likewise among binaries, there is no Gaia astrometry for four rotational
variables and they are also listed in Table 7. We show the CMD location of
members in the bottom panel of Fig. 3. The members occupy mostly the MS
region, with three exceptions being the RGB objects and three exceptions being
BS objects. Likewise in the case of binaries and pulsators, their true
location may be influenced by binarity.

Table 3: List of cluster members showing rotational variability. See the
caption of Table 1 for explanation.

Gaia EDR3 | KIC | Period | G | CMD | HRD | Teff | $\log$g | RV | [Fe/H] | Ref
---|---|---|---|---|---|---|---|---|---|---
[days] | [mag] | [K] | [km/s]
2051099015605500672 | 2297584 X | 18.02 | 18.299 | MS+ | – | – | – | – | – | –
2051104757981275264 | 2438685 X | 7.78 | 17.256 | RGB | MS | 5 500(50) | 4.953(20) | -44.8(44) | 0.47(15) | 1 ∗
2051104891118311040 | 2437994 X | 1.85 | 19.228 | MS+ | – | – | – | – | – |
2051104891120796672 | 2437990 X | 13.15 | 19.148 | MS+ | – | – | – | – | – | –
2051104959840285696 | 2437791 X | 15.28 | 18.067 | MS+ | – | – | – | – | – | –
2051104964139549824 | 2437941 X | 10.12 | 17.944 | MS | – | – | – | – | – | –
2051104994200024448 | 2438113 X | 5.80 | 17.288 | MS | – | – | – | – | – | –
2051105097281783040 | 2438031 X | 8.28 | 17.591 | MS | MS | 5 810(40) | 4.216(34) | -62.3(23) | -0.158(47) | 1 ∗
2051105131636983808 | 2437646 X | 13.23 | 17.558 | MS | – | – | – | – | – | –
2051105234715895552 | 2437707 X | 11.00 | 17.867 | MS | MS | 6 080(30) | 4.641(35) | -47.8(48) | 0.60(14) | 1 ∗
2051105273377206784 | 2437801 X | 7.19 | 18.508 | MS+ | MS | 5 380(70) | 4.111(26) | -65.30(10) | -0.40(10) | 1 ∗
2051105342091728384 | 2437761 X | 11.81 | 19.001 | MS | – | – | – | – | – | –
2051105372154852736 | 2437849 4L | 11.02 | 16.863 | MS | MS | 5 260(50) | 3.923(16) | -51.7(14) | -0.31(11) | 1 ∗
2051105372154855552 | 2437944 X | 1.96 | 18.543 | MS+ | – | – | – | – | – | –
2051105372157195264 | – | 2.45 | 19.556 | MS+ | MS | 5480(50) | 4.835(21) | -56(4) | 0.45(17) | 1 ∗
2051105578318626304 | 2438569 X | 7.00 | 17.279 | MS | MS | 5 970(50) | 4.295(20) | -69.2(15) | 0.32(14) | 1 ∗
2051105612675379968 | 2438390 X | 3.80 | 17.987 | MS | – | – | – | – | – | –
2051105685689315968 | 2438129 X | 16.58 | 17.853 | MS | – | – | – | – | – | –
2051105784484742144 | 2570443 X | 2.20 | 17.643 | MS | MS | 5 610(50) | 4.298(20) | -133.2(45) | -0.62(30) | 1 ∗
2051105818833811584 | 2438344 X | 7.74 | 18.616 | MS+ | – | – | – | – | – | –
2051105857492894592 | 2570649 X | 13.36 | 17.293 | MS | – | – | – | – | – | –
2051105921913036160 | 2570622 X | 9.57 | 19.026 | MS+ | – | – | – | – | – | –
2051105926212896512 | 2570559 X | 1.32 | 20.343 | MS+ | – | – | – | – | – | –
2051107055787471104 | 2438631 X | 7.36 | 19.729 | MS+ | – | – | – | – | – | –
2051287002024520064 | – | 10.86 | 19.640 | MS+ | – | – | – | – | – |
2051287066449451136 | 2436942 X | 9.26 | 18.098 | MS | – | – | – | – | – | –
2051287066449464064 | 2436959 X | 2.65 | 18.852 | MS+ | – | – | – | – | – | –
2051287070744003456 | 2436969 X | 7.92 | 19.132 | MS+ | RGB | 4 500(50) | 3.09(8) | -45.0(8) | -0.155(20) | 2 ∗
2051287242549232384 | 2436767 X | 12.32 | 17.309 | MS | MS | 5 690(50) | 4.163(20) | -43.2(57) | 0.148(33) | 1 ∗
2051287895379273088 | 2436011 X | 10.00 | 19.632 | MS+ | – | – | – | – | – | –
2051288372119657728 | 2569185 X | 13.43 | 17.258 | MS | MS | 5 660(50) | 4.117(34) | -97.9(35) | -0.08(10) | 1 ∗
2051288376420896384 | 2569162 X | 5.90 | 18.088 | MS | – | – | – | – | – | –
2051291262638872192 | 2568864 X | 14.42 | 17.618 | MS | MS | 5 370(30) | 4.78(10) | -61.5(42) | 0.58(27) | 1 ∗
2051291503157017472 | 2568685 X | 19.87 | 17.342 | MS | – | – | – | – | – | –
2051292873245240192 | 2437350 X | 16.82 | 17.885 | MS | MS | 5 790(50) | 4.913(20) | -74.3(15) | 0.13(8) | 1 ∗
2051292911906089728 | 2437521 X | 11.27 | 17.987 | MS+ | – | – | – | – | – | –
2051292946259301504 | 2437092 2L | 3.20 | 18.145 | MS+ | MS | 4 700(100) | 4.619(50) | -78(1) | -1.87(6) | 1 ∗
2051293014978876672 | 2437584 X | 5.90 | 18.486 | –+ | – | – | – | – | – | –
2051293083704791040 | 2437354 8L | 7.21 | 16.465 | MS | MS | 5 580(50) | 4.391(20) | -55.6(8) | -0.413(46) | 2 ∗
2051293113776412416 | 2437613 X | 11.05 | 17.787 | MS | – | – | – | – | – | –
2051293148133456896 | 2437062 X | 8.82 | 17.133 | RGB | – | – | – | – | – | –
2051293152424243968 | 2436909 X | 12.00 | 16.778 | MS | MS | 5 640(40) | 4.654(21) | -69.3(26) | -0.203(17) | 1 ∗
2051293186783997056 | – | 9.63 | 18.232 | MS+ | – | – | – | – | – | –
2051293186784004096 | 2437238 X | 4.00 | 16.079 | BS | BS | 5 880(30) | 3.546(23) | -48.0(4) | -0.34(8) | 2 ∗
2051293255503480320 | 2437079 X | 5.81 | 16.794 | BS | – | – | – | – | – | –
2051293289863230720 | 2437338 X | 1.24 | 16.655 | BS | – | – | – | – | – | –
2051293319934813184 | 2569984 X | 12.50 | 17.704 | MS | – | – | – | – | – | –
2051293358583014528 | 2569763 4L | 8.29 | 16.648 | MS | MS/RGB | 5 510(20) | 3.788(49) | -38.6(5) | -0.397(48) | 2 ∗
2051293392942762240 | 2569825 X | 0.48 | 18.754 | MS+ | MS | 5 000(50) | 4.86(15) | -86.3(7) | 0.02(14) | 1 ∗
2051293427302195584 | 2437884 X | 3.23 | 16.871 | MS+ | MS | 5 930(20) | 4.048(26) | -43.9(3) | -0.44(10) | 2 ∗
2051293530375009920 | 2570217 X | 5.55 | 17.891 | MS | MS | 5 760(20) | 4.49(7) | -62.4(23) | 0.07(8) | 1 ∗
2051293663519464192 | 2570420 X | 19.96 | 17.951 | MS | – | – | – | – | – | –
2051293977053182592 | 2436790 X | 9.61 | 18.861 | MS | MS | 4 990(80) | 4.832(25) | -169(3) | -0.356(15) | 1 ∗
2051294007116759808 | 2569597 X | 7.17 | 18.091 | MS+ | – | – | – | – | – | –
2051294148851985536 | – | 9.66 | 20.397 | MS+ | – | – | – | – | – | –
2051294148857020160 | 2569767 X | 2.40 | 17.323 | MS+ | – | – | – | – | – | –
2051294217574157440 | – | 4.92 | 19.300 | MS+ | – | – | – | – | – | –
2051294251931077888 | 2569279 X | 1.14 | 18.883 | MS+ | – | – | – | – | – | –
2051294281994746880 | 2569324 X | 10.90 | 17.983 | MS | – | – | – | – | – | –
2051294286296294528 | 2569334 X | 6.29 | 17.447 | MS | MS | 5 330(20) | 4.220(25) | -47.4(23) | 0.15(8) | 1 ∗
2051294797390943872 | 2569591 X | 14.89 | 17.679 | MS | – | – | – | – | – | –
2051296176081696768 | 2570281 12L | 14.42 | 14.785 | RGB | – | – | – | – | – | –

###  5.4 Unclassified and unidentified variables

In the case of nine objects, we were unable to unambiguously classify them
according to the three types described above. The amplitude of a flux
variation is low and it is not clear if the variation remains stable over
time. The latter indicates that it is not unlikely these stars may be
rotational variables. We leave a definite classification for further analysis,
preferentially if based on better quality data. Five members of the cluster
are listed in Table 4 and they are plotted in the bottom panel of Fig. 3,
while four non-members are listed in Table 8.

While checking the pixel content we found signals in the amplitude spectra,
associated with optical detections found in Pan-STARRS data (Chambers et al.
2016, Flewelling et al. 2020) survey that do not have any designations. We
classified the signal to a proper variability type and estimated its period.
We show the list of these unidentified objects in Table 9. Since the stars are
not listed in the Gaia catalog, we are unable to estimate their membership. At
three locations on the silicons (last three coordinates listed), we detected a
signal in the amplitude spectra, however, we are unable to associate these
coordinates with any optical objects in the Pan-STARRS survey. In addition,
the last two signals listed may be of the same origin, though the coordinates
are different. This result indicates that these three signals listed are
residual signals of some other variable stars in the Kepler field of view that
were spread over the silicons. We consider the signals detected at those three
specific locations on the detector to be artifacts and not real sources.

Table 4: List of cluster members with unclassified variability. See the
caption of Table 1 for explanation.

Gaia EDR3 | KIC | Period | G | CMD | HRD
---|---|---|---|---|---
[days] | [mag]
2051105269075634176 | 2437760 X | 0.44 | 18.064 | MS | –
2051288067182949760 | 2436421 8L | 1.44 | 14.937 | BS | –
2051291498855854208 | 2568724 X | 0.63 | 20.141 | MS+ | –
2051293049345064832 | 2437745 X | 1.43 | 15.968 | BS | –
2051294148857016192 | 2569676 X | 0.27 | 18.037 | MS+ | –

##  6\. The distance and age estimation

We downloaded a grid of isochrones given in the Gaia photometric system from
the MESA Isochrones and Stellar Tracks (MIST) project (Choi et al. 2016,
Dotter 2016). The current version of MIST is 1.2. The MESA version 7503 was
employed to calculate isochrones. We selected V/Vcrit = 0. The grid covers age
in a logarithmic scale between 9.8 and 10.3 with a step of 0.01, while [Fe/H]
was between 0.20 and 0.45 with a step of 0.01.

At least 50$\%$ of the stars are expected to be in binary systems. The
observed magnitude of a given target may include the flux contribution from
all companions and not a single star, which shifts the position of the target
in the CMD. Therefore, we excluded outlying stars where their positions in the
CMD are uncertain. For the fit, we only kept the MS, RGB and RC stars. We
included magnitude uncertainties as weights in the fit, which prevented the MS
targets from over-fitting. The RC and RGB targets, even though less numerous,
are brighter, and hence also remains significant in the fit.

The MIST synthetic isochrones are given in absolute magnitudes, and we
selected no extinction. To account for a distance and a non-vacuum
environment, in our fit, we included a shift (m-M) in the Gaia G magnitude and
in Bp-Rp color. The best-fit isochrones point to a narrow range of age and
[Fe/H]. The age is 8.91 Gyr, while [Fe/H] is between 0.26 and 0.28. The
apparent distance modulus (m-M) equals 13.424, while E(Bp-Rp) from 0.165 to
0.176. We show the fits in Fig. 5. Based on the extinction curve from Cardelli
et al. (1989), Bressan et al. (2012) showed a rough relation between
extinction in the Gaia G band and E(Bp-Rp), which is A${}_{\rm
G}\approx$2$\cdot$E(Bp-Rp). Taking the average of E(Bp-Rp) = 0.171 and using
this relation, we derived AG = 0.342 mag. Subtracting AG from (m-M) we find
the true distance modulus of 13.082, which gives the distance to the cluster
of 4134 pc. We also derived distance from the parallaxes of the cluster
members with probability membership at least 90% and the relative parallax
uncertainty of smaller than 10%, which equals 4123(31) pc. The distance
estimated from the isochrone fit lies well within the uncertainty of the
distance estimations from the cluster parallax. The cluster age we derived
from the isochrone fit is also comparable to the age reported by Choi et al.
(2018).

To verify the correctness of the spectroscopic fits we also engaged isochrones
in Teff and $\log$g plane (HRD). We used the isochrones for the age and the
average metallicity, which we derived from the isochrone fitting in the CMD.
Then, we overplotted the isochrones with our variable stars from Tables 1-3,
for which Teff and $\log$g are listed. The column ’HRD’ in Tables 1-3
describes the location of a given star in the HRD. If the location agrees with
the one in CMD, we can expect the spectroscopic fit is likely correct and we
obtained an agreement for the majority of stars. It should be noted that the
scatter in the HRD is larger, hence some objects may have not been allocated
properly. There are clear exceptions, eight among binaries, two among
pulsators and two among rotational variables. These cases should be treated
with caution.

Figure 5: The color-magnitude diagram of NGC 6791, showing the best MIST
isochrone fits. The top panel shows the overall diagram, while the middle
panel shows the red clump region and the bottom panel shows the main-sequence
region.

Figure 6: The Teff–$\log$g diagram showing the cluster members with
atmospheric parameters derived in this work. The top, middle and bottom panels
show binary, pulsating and rotational stars, respectively. The isochrones
represent the best MIST fits in Teff and $\log$g coordinates. Table 5: List of
binary stars that are not cluster members. See the caption of Table 1 for
explanation. Targets with no astrometry available, hence with no membership
established, are denoted in italic.

Gaia EDR3 | KIC | Porb | T0 | G | Teff | $\log$g | RV | [Fe/H] | Ref
---|---|---|---|---|---|---|---|---|---
[days] | [BJD] | [mag] | [K] | [km/s]
eclipsing
2051104826700758016 | 2438661 X | 196.34051(1) | 2 455 037.21466(43) | 16.449 | – | – | – | – | –
2051105131638986752 | – | 23.878352(36) | 2 454 967.0873(12) | 19.782 | – | – | – | – | –
2051107128803742336 | 2438562 X | 10.406512(6) | 2 454 965.58747(47) | 19.369 | 6 220(130) | 2.864(51) | -49(4) | -1.29(17) | 2∗
2051287311268711168 | 2436579 X | 9.287153(22) | 2 454 971.1134(18) | 19.613 | – | – | – | – | –
2051291739386261248 | 2568780 X | 3.5316020(11) | 2 454 964.71245(25) | 19.913 | – | – | – | – | –
2051292907615354240 | 2437452 1S,17L | 14.4699358(24) | 2 454 974.81692(14) | 17.286 | 6 170(50) | 4.26(9) | -92.1(6) | -0.86(9) | 2∗
2051293079414230912 | 2437505 10L | 21.476420(11) | 2 454 982.26244(40) | 18.324 | 4 920(50) | 3.25(6) | -19.6(30) | -0.25(19) | 1∗
2051293118063233408 | 2437675 X | 1.75 | – | 21.175 | – | – | – | – | –
2051294629893394176 | 2569880 X | 0.51323254(22) | 2 454 964.60382(35) | 17.866 | – | – | – | – | –
active eclipsing
2051104586182512128 | 2438502 6S,16L | 8.35905(26) | 2 455 005.2748(14) | 16.216 | 5 563(234) | 4.17(37) | -14(5) | 0.37(22) | 3
2051104684959920896 | 2438464 X | 0.43657907(7) | 2 454 964.92853(13) | 19.443 | 4 310(50) | 4.169(20) | 41(16) | -0.91(6) | 1∗
2051286761512868992 | 2436203 X | 2.06 | – | 14.328 | – | – | – | – | –
2051288170257319808 | – | 3.7024382(14) | 2 454 968.16622(31) | 20.135 | – | – | – | – | –
2051288372119654528 | 2569138 X | 0.88305272(12) | 2 454 964.71084(11) | 19.205 | – | – | – | – | –
2051293599094508032 | – | 2.8516350(19) | 2 454 967.1274(6) | 20.335 | – | – | – | – | –
contact
2051105543955885312 | 2438471 X | 0.27345629(34) | 2 454 964.8195(11) | 19.527 | 4 300(50) | 3.92(7) | 3.9(9) | -1.354(25) | 1∗
2051288269040355584 | 2436044 X | 0.24533697(21) | 2 454 964.6646(7) | 19.452 | – | – | – | – | –
2051286417915483392 | 2297170 X | 0.3664306(6) | 2 455 002.6182(13) | 17.797 | – | – | – | – | –
2051287070750548736 | 2437038 17L | 0.267678275(46) | 2 454 964.58218(14) | 16.149 | – | – | – | – | –
2051291228279113216 | 2435971 17L | 0.27182769(8) | 2 454 964.81810(25) | 16.276 | – | – | – | – | –
2051291468797350528 | 2568971 17L | 5.088522(10) | 2454966.2350(17) | 13.081 | – | – | – | – | –
2051293530381417856 | 2570289 17L | 0.279027932(31) | 2 454 964.57880(1) | 15.781 | – | – | – | – | –
2051295076563264000 | 2570552 X | 0.28417822(44) | 2 454 964.5913(13) | 20.279 | – | – | – | – | –
2051295454527201792 | 2570460 X | 0.2659879(15) | 2 454 964.61531(47) | 18.689 | – | – | – | – | –
2051296313520692864 | 2708123 X | 0.302184513(81) | 2 454 964.69239(22) | 17.742 | – | – | – | – | –
2051297374371300992 | 2569082 X | 0.33401898(23) | 2 454 964.8252(6) | 17.579 | 5 750(50) | 4.052(15) | -38.30(54) | -1.069(20) | 1∗
2051297477460338304 | – | 0.2690446(1) | 2 454 964.7175(30) | 20.838 | – | – | – | – | –
outbursting
2051287203888388736 | 2436450 15S,12L | 35.71 | – | 19.884 | – | – | – | – | –

Table 6: List of pulsators that are not cluster members. See the caption of
Table 1 for explanation.

Gaia EDR3 | KIC | G | Teff | $\log$g | RV | [Fe/H] | Ref
---|---|---|---|---|---|---|---
[mag] | [K] | [km/s]
$\delta$-scuti
2051098981244205056 | 2297728 1S,18L | 9.929 | 6 888(25) | 3.680(40) | -37(7) | 0.272(22) | 3
2051107369321426432 | 2570760 18L | 13.459 | – | – | – | – | –
solar-like
2051098710667526912 | 2297357 1L | 12.115 | – | – | – | – | –
2051099054264914048 | 2437622 12L | 15.583 | – | – | – | – | –
2051099329142404480 | 2437207 X | 18.731 | – | – | – | – | –
2051104105146054784 | 2297793 17L | 13.626 | 4 085(81) | 1.430(52) | -83.615(30) | -0.300(10) | 5
2051104139506469760 | 2438094 17L | 15.322 | 4 060(50) | 1.11(15) | -84.7(6) | -1.14(8) | 2∗
2051104861060304640 | 2437816 17L | 13.958 | 4 229(77) | 2.090(47) | 0.400(7) | -47.27(9) | 5
2051105720053903232 | 2438289 10L | 14.384 | – | – | – | – | –
2051105857492905600 | 2570715 7L | 12.678 | 4 713(84) | 2.330(36) | -3.30(7) | -0.200(7) | 5
2051107369321441920 | 2570794 10L | 15.158 | – | – | – | – | –
2051286520994706432 | 2436457 18L | 13.245 | 4 969(24) | 3.015(39) | -19.14(4) | -0.422(22) | 3
2051286933311578112 | 2436680 10L | 15.187 | – | – | – | – | –
2051288445140366464 | 2569078 10L | 13.650 | 4 418(90) | 2.290(49) | -5.428(56) | 0.200(8) | 5
2051291434437586304 | 2568912 8L | 12.859 | 4 701(95) | 3.01(6) | -7.122(55) | -0.100(8) | 5
2051291567575397120 | 2568656 X | 15.618 | – | – | – | – | –
2051291571876502912 | 2568654 11L | 13.140 | 4 301(83) | 2.07(5) | -73.155(12) | 0.100(8) | 5
2051291674955780992 | 2568888 10L | 14.131 | 4 387(100) | 2.070(51) | -58.47(23) | -0.100(11) | 5
2051291949833615104 | 2568575 5L | 13.407 | 4 501(50) | 2.55(8) | -2.26(4) | 0.181(48) | 3
2051293118064546944 | 2437692 15L | 11.203 | 4 817(86) | 2.610(38) | -12.780(24) | 0.200(6) | 5
2051293702180102144 | 2570002 X | 13.628 | – | – | – | – | –
2051294320655654912 | 2569137 10L | 13.630 | 4 298(102) | 1.71(6) | -194.10(44) | -0.800(16) | 5
2051295282728537216 | 2570696 2L | 13.620 | 4 461(80) | 2.13(6) | -11.38(9) | -0.300(8) | 5
RRLyr/BLBoo
2051294934839887872 | 2569850 X | 20.426 | – | – | – | – | –
semi-regular
2051105032859063424 | 2438242 X | 16.817 | 4 511(99) | 2.660(47) | -47.85(8) | 0.400(9) | 4

Table 7: List of rotational variables that are not cluster members. See the
caption of Table 1 for explanation.

Gaia EDR3 | KIC | Period | G | Teff | $\log$g | RV | [Fe/H] | Ref
---|---|---|---|---|---|---|---|---
[days] | [mag] | [K] | [km/s]
2051098951180105088 | – | 1.33 | 20.929 | – | – | – | – | –
2051099088624172928 | 2297416 X | 2.66 | 20.201 | – | – | – | – | –
2051099187404218752 | 2436945 X | 4.58 | 16.735 | – | – | – | – | –
2051104139500783744 | – | 0.64 | 20.943 | – | – | – | – | –
2051104139505811328 | 2297846 X | 13.40 | 14.901 | – | – | – | – | –
2051104272645525760 | 2438284 X | 2.63 | 19.629 | – | – | – | – | –
2051104272645529472 | – | 1.53 | 20.277 | – | – | – | – | –
2051104276944815744 | 2438272 18L | 11.48 | 13.173 | – | – | – | – | –
2051104586182508928 | 2438513 15S,17L | 13.27 | 13.939 | 5 652(125) | 4.51(8) | -35.74(33) | 0.000(10) | 5
2051104753679401728 | 2438740 X | 0.42 | 19.653 | – | – | – | – | –
2051104826696024832 | – | 1.06 | 20.370 | – | – | – | – | –
2051104925480543616 | 2437773 X | 0.51 | 19.083 | 6 070(110) | 2.9(1) | 8(4) | -0.54(15) | 2∗
2051105062917626624 | 2437959 X | 15.27 | 17.987 | – | – | – | – | –
2051105067218772096 | 2437888 8L | 10.34 | 14.730 | 6 120(80) | 4.93(5) | 16.42(37) | -0.39(12) | 2∗
2051105067218785280 | 2438003 18L | 0.70 | 13.025 | 7 132(30) | 4.102(50) | 3(10) | -0.22(3) | 3
2051105135933158144 | – | 3.99 | 20.627 | – | – | – | – | –
2051105239017448448 | 2437649 X | 7.62 | 16.377 | – | – | – | – | –
2051105273377200384 | 2437789 X | 16.61 | 17.023 | – | – | – | – | –
2051105303437700224 | 2437984 X | 4.70 | 19.703 | – | – | – | – | –
2051105307736986496 | 2437996 17L | 14.61 | 13.887 | 5 753(37) | 4.60(6) | -38(5) | 0.104(35) | 3
2051105410811376640 | 2438305 X | 7.01 | 19.692 | – | – | – | – | –
2051105823133141888 | 2438376 X | 11.44 | 17.788 | – | – | – | – | –
2051105926207653504 | – | 5.11 | 20.540 | – | – | – | – | –
2051106063646757888 | 2438861 X | 1.48 | 19.672 | – | – | – | – | –
2051107124503872256 | 2438516 X | 5.13 | 18.313 | 5 150(50) | 4.773(27) | -36.9(15) | 0.64(30) | 1∗
2051107472400670720 | 2570846 17L | 10.82 | 15.313 | – | – | – | – | –
2051107541120138752 | 2570736 X | 1.61 | 18.428 | – | – | – | – | –
2051286997729896832 | – | 0.86 | 19.542 | – | – | – | – | –
2051287066448488192 | – | 1.62 | 19.279 | 4 850(50) | 4.871(40) | -112.4(1) | -0.15(12) | 1∗
2051287139470009088 | 2436635 17L | 1.17 | 15.989 | 4 445(291) | 4.09(45) | -76(6) | -0.32(27) | 3
2051287547485746432 | 2436206 X | 8.25 | 17.626 | – | – | – | – | –
2051288032823472128 | 2436274 X | 4.58 | 18.220 | – | – | – | – | –
2051288101537860992 | – | 3.39 | 20.539 | – | – | – | – | –
2051288165961197952 | 2436416 X | 3.99 | 19.859 | – | – | – | – | –
2051288445135225472 | 2569015 X | 1.32 | 20.103 | – | – | – | – | –
2051290919041461632 | 2435889 X | 14.47 | 16.670 | – | – | – | – | –
2051291434437576320 | 2568884 X | 10.50 | 18.224 | – | – | – | – | –
2051291468792223104 | – | 7.37 | 20.190 | – | – | – | – | –
2051291571876503424 | 2568672 X | 18.09 | 16.477 | – | – | – | – | –
2051292976335015296 | 2437180 X | 4.07 | 17.907 | – | – | – | – | –
2051292980625560448 | 2437292 X | 5.51 | 18.356 | – | – | – | – | –
2051292980625569792 | 2437234 17L | 24.10 | 13.153 | – | – | – | – | –
2051293014978879744 | 2437723 X | 1.17 | 18.608 | – | – | – | – | –
2051293014985313152 | 2437469 X | 9.85 | 19.161 | – | – | – | – | –
2051293014985313792 | 2437574 X | 11.683 | 17.183 | 5 650(50) | 4.5(8) | -45.0(3) | 0.05(7) | 1∗
2051293049338645504 | – | 1.97 | 20.310 | – | – | – | – | –
2051293083704793856 | 2437359 17L | 16.91 | 14.598 | – | – | – | – | –
2051293152424251648 | 2436988 X | 4.69 | 20.213 | – | – | – | – | –
2051293221143720832 | 2436808 X | 4.94 | 16.764 | – | – | – | – | –
2051293255503475200 | 2436958 X | 4.50 | 20.342 | – | – | – | – | –
2051293289863227520 | 2437329 16L | 11.67 | 15.925 | – | – | – | – | –
2051293358577984768 | – | 1.55 | 20.049 | – | – | – | – | –
2051293388640138880 | – | 8.48 | 20.028 | – | – | – | – | –
2051293496015260416 | 2570182 X | 1.99 | 18.719 | – | – | – | – | –
2051293530375018112 | – | 4.27 | 20.832 | – | – | – | – | –
2051293599094510848 | 2570536 17L | 10.03 | 14.507 | – | – | – | – | –
2051293839619440640 | 2570259 1S,1L | 3.84 | 16.128 | – | – | – | – | –
2051293908333956352 | 2570154 X | 6.39 | 18.171 | 5 380(50) | 4.004(20) | -68.6(36) | 0.12(8) | 1∗
2051293942698672000 | 2570258 X | 7.40 | 16.852 | – | – | – | – | –
2051293942698685056 | 2570313 X | 1.39 | 19.449 | – | – | – | – | –
2051293972756992384 | 2436621 X | 7.58 | 18.758 | – | – | – | – | –
2051293977053154688 | 2436734 X | 1.59 | 20.187 | – | – | – | – | –
2051294011418019072 | 2436852 X | 3.31 | 18.871 | – | – | – | – | –
2051294080137491968 | 2569421 X | 13.51 | 18.689 | – | – | – | – | –
2051294114495072896 | – | 3.54 | 20.960 | – | – | – | – | –
2051294148851973888 | 2569675 X | 5.24 | 20.253 | – | – | – | – | –
2051294148857011712 | 2569737 14L | 13.54 | 16.109 | – | – | – | – | –
2051294492454406272 | 2569431 X | 2.49 | 17.007 | – | – | – | – | –
2051294664251895808 | – | 4.90 | 21.163 | – | – | – | – | –
2051294728671468672 | 2569908 X | 6.32 | 16.820 | – | – | – | – | –
2051294831750674944 | 2569761 X | 1.99 | 20.008 | – | – | – | – | –
2051294870413630208 | 2569467 X | 9.78 | 15.965 | – | – | – | – | –
2051295007845559552 | – | 6.08 | 20.344 | – | – | – | – | –
2051295007845583744 | – | 1.80 | 21.103 | – | – | – | – | –
2051295488886958592 | 2570555 X | 6.35 | 16.869 | – | – | – | – | –
2051295557606451456 | 2570581 X | 16.52 | 16.997 | – | – | – | – | –
2051296652817037056 | 2707961 X | 8.00 | 18.949 | – | – | – | – | –
2051297516106367232 | – | 2.65 | 21.041 | – | – | – | – | –
2051297962788050816 | 2707771 X | 0.73 | 18.021 | – | – | – | – | –
2051298031507519616 | 2707692 X | 2.50 | 19.376 | – | – | – | – | –
2051299474616582912 | 2707858 X | 0.91 | 20.076 | – | – | – | – | –
2051297275593155328 | 2569073 X | 14.66 | 14.236 | – | – | – | – | –
2051293255496998912 | – | 9.00 | 19.651 | – | – | – | – | –

Table 8: List of unclassified variables that are not cluster members. See the
caption of Table 1 for explanation.

Gaia EDR3 | KIC | Period | G | Teff | $\log$g | RV | [Fe/H] | Ref
---|---|---|---|---|---|---|---|---
[days] | [mag] | [K] | [km/s]
2051104654901977984 | 2438433 X | 0.93 | 20.356 | – | – | – | – | –
2051105307732052864 | 2438028 2L | 0.06 | 20.639 | 3 470(50) | 3.842(47) | -48.80(30) | 0.09(10) | 1∗
2051288337759883648 | 2436293 X | 0.20 | 18.578 | – | – | – | – | –
2051293736533447424 | 2570195 X | 0.77 | 20.856 | – | – | – | – | –

Table 9: A list of equatorial coordinates that are associated with superstamp
pixels showing signal in their amplitude spectra. In the case of the last
three positions listed, we found no optical objects that could be a source of
the signal.

$\alpha_{\rm 2000}$ | $\delta_{\rm 2000}$ | Period | Type
---|---|---|---
[hh mm ss.ss] | [dd mm ss.s] | [days]
19 20 51.72 | +37 47 45.4 | 1.10 | rotational
19 20 53.56 | +37 47 05.4 | 1.20 | eclipsing
19 20 56.85 | +37 47 43.2 | 2.40 | rotational
19 21 13.06 | +37 42 36.3 | 0.25 | binary
19 20 31.99 | +37 47 42.4 | 0.41 | binary
19 20 45.84 | +37 47 04.1 | 3.07 | rotational
19 20 47.82 | +37 42 18.2 | 0.68 | rotational
19 21 03.71 | +37 44 17.6 | 11.68 | rotational
19 20 51.85 | +37 39 51.5 | 9.21 | solar-like
19 20 32.77 | +37 43 23.8 | 0.15 | binary
19 20 44.47 | +37 40 51.5 | 5.73 | rotational
19 20 46.46 | +37 40 18.9 | 5.73 | rotational

##  7\. Summary

We presented a search for variable stars in the Kepler superstamp data. All
available pixels were searched, by means of a Fourier amplitude spectrum, and
a contiguous optimal aperture for each object that shows a significant flux
variation were defined. The coordinates of these optimal apertures were
matched with optical counterparts using Pan-STARRS. In total, we detected 278
variable stars. We cross-matched our variable star sample with those reported
in the literature and found 16 variable stars reported by Kaluzny and Rucinski
(1993), four by Rucinski et al. (1996), 23 by Mochejska et al. (2002), six by
Mochejska et al. (2003), four by Kaluzny (2003), 15 by Bruntt et al. (2003),
seven by Mochejska et al. (2005), one by Hartman et al. (2005) and 33 by de
Marchi et al. (2007). We found 240 stars having KIC designations, out of which
140 stars do not have data delivered to the MAST. A variability of 119 stars
was not known prior to our analysis. These stars are marked in bold in Tables
1-8, including the first nine objects listed in Table 9. Just recently, and
independently of our work, Colman et al. (2022) reported light curves of stars
with a KIC designation, however the authors did not specify which stars were
found variables, only reporting a total number of 239 variables. Even
accounting for this report, which is limited to the KIC stars only, we can
consider 26 variable stars having no KIC designation (including seven objects
marked in bold), to be new detections. No ground-based work on the detection
of the variability of these 26 stars was reported, either.

Using Gaia EDR3 astrometry, we calculated the membership probabilities for all
variable stars in our sample by applying Bayesian Gaussian mixture models. A
star is considered to be a cluster member if the probability is higher than
50%. In total, we found 129 variable stars that are cluster members, 17 binary
systems, 45 pulsators, 62 rotationally and five unclassified variables. The
locations of these cluster variable stars in the CMD diagram were estimated
and indicate the evolutionary status of cluster members. In the CMD, a
majority of our variable stars are located in the MS. Solar-like pulsators are
mostly located in the RGB and RC, while semi-regular variables are located in
the RGB and AGB. In addition there are five BS and three EHB stars in the CMD.

In the case of binary systems, we estimated mid-times of eclipses and derived
ephemerides. We calculated the O-C diagrams and checked for any orbital period
variation. Only three binary systems show significant period variation,
however its nature is not confirmed. The solar-like counterpart has been a
subject of a detailed analysis of its pulsation content, and the results will
be published elsewhere. The analysis of three sdBVs is already reported by
Sanjayan et al. (2022). The rotational variables are not subject to our
detailed analysis. This type of variables can be very useful toward
gyrochronology.

We utilized public and archived spectra for 111 variable stars. Spectra for 64
stars were fitted with XTgrid, while for the remainder of the sample, we
adopted the fit values from the surveys. We derived Teff, $\log$g, [Fe/H] and
RVs. Our spectral analysis was able to recover consistent stellar parameters
from very diverse spectroscopic data. This consistency is reflected by the
similar distribution of stars in the CMD and HRD. The most significant
limiting factor on the parameter determination was the low SNR of some
spectra, while the spectral coverage and crowding in dense fields played less
significant roles. We found that the metallicity is not consistent among the
cluster members and it is still unclear what causes it. If the inconsistency
is real a possible explanation would be the presence of multiple stellar
populations within the cluster as mentioned by Geisler et al. (2012). To
confirm this hypothesis a uniform spectroscopic survey is required.

MIST isochrones were fit to our CMD comprised of cluster members, including
our variable star population. From our best three fits, we derived a
metallicity range of 0.26 to 0.28 and the age of NGC 6791 to be 8.91 Gyr. Our
age estimate agrees with the values reported by e.g. Choi et al. 2018\. The
average distance estimate from the distance modulus is 4134 pc which overlaps
with our independent estimate of 4123(31) pc we derived from the Gaia
astrometry.

## Acknowledgement

Financial support from the National Science Centre under projects No.
UMO-2017/26/E/ST9/00703 and UMO-2017/25/B/ST9/02218 is acknowledged. PN
acknowledges support from the Grant Agency of the Czech Republic (GAČR
22-34467S). The Astronomical Institute in Ondřejov is supported by the project
RVO:67985815. This paper includes data collected by the Kepler mission and
obtained from the MAST data archive at the Space Telescope Science Institute
(STScI). Funding for the Kepler mission is provided by the NASA Science
Mission Directorate. STScI is operated by the Association of Universities for
Research in Astronomy, Inc., under NASA contract NAS 5-26555. This work has
made use of data from the European Space Agency (ESA) mission. Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (https://www.cosmos.esa.int/web/gaia/dpac/consortium).
Funding for the DPAC has been provided by national institutions, in particular
the institutions participating in the Gaia Multilateral Agreement. This
research has made use of the NASA/IPAC Extragalactic Database (NED), which is
operated by the Jet Propulsion Laboratory, California Institute of Technology,
under contract with the National Aeronautics and Space Administration. This
research has used the services of www.Astroserver.org under reference XZR329.
We thank the anonymous referee for valuable comments, which have significantly
improved the quality of the manuscript.

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