11institutetext: Max-Planck-Institut für Sonnensystemforschung, Justus-von-
Liebig-Weg 3, 37077, Göttingen, Germany 11email<EMAIL_ADDRESS>22institutetext: Department of Astronomy, New Mexico State University, P.O.
Box 30001, MSC 4500, Las Cruces, NM 88003-8001, USA 33institutetext: Baja
Astronomical Observatory of University of Szeged, H-6500 Baja, Szegedi út, Kt.
766, Hungary 44institutetext: ELKH-SZTE Stellar Astrophysics Research Group,
H-6500 Baja, Szegedi út, Kt. 766, Hungary 55institutetext: Konkoly
Observatory, Research Centre for Astronomy and Earth Sciences, H-1121
Budapest, Konkoly Thege Miklós út 15-17, Hungary 66institutetext: ELTE Gothard
Astrophysical Observatory, H-9700 Szombathely, Szent Imre h. u. 112, Hungary
77institutetext: MTA-ELTE Exoplanet Research Group, H-9700 Szombathely, Szent
Imre h. u. 112, Hungary 88institutetext: Université Paris-Saclay, Institut
d’Astrophysique Spatiale, UMR 8617, CNRS, Bâtiment 121, 91405 Orsay Cedex,
France 99institutetext: Department of Physics, Faculty of Science, University
of Zagreb, Bijenička cesta 32, 10000 Zagreb, Croatia 1010institutetext:
Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP,
Rua das Estrelas, 4150-762, Porto, Portugal 1111institutetext: Department of
Astronomy, Yale University, P.O. Box 208101, New Haven, CT 06520-8101, USA
1212institutetext: Dipartimento di Fisica e Astronomia Augusto Righi,
Università degli Studi di Bologna, Via Gobetti 93/2, I-40129 Bologna, Italy
1313institutetext: INAF-Osservatorio di Astrofisica e Scienza dello Spazio di
Bologna, via Gobetti 93/3, I-40129 Bologna, Italy 1414institutetext: School of
Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK
1515institutetext: Institute of Astronomy, KU Leuven, Celestijnenlaan 200D,
B-3001 Leuven, Belgium 1616institutetext: LESIA, Observatoire de Paris,
Université PSL, CNRS, Sorbonne Université, Université de Paris, 92195 Meudon,
France 1717institutetext: Department of Astronomy, The Ohio State University,
140 West 18th Avenue, Columbus OH 43210, USA 1818institutetext: Laboratoire
AIM, DRF/IRFU/SAp, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France
1919institutetext: Department of Physics, Montana State University, P.O. Box
173840, Bozeman, MT 59717-3840, USA 2020institutetext: Monterey Institute for
Research in Astronomy, 200 8th Street, Marina, CA 93933, USA

# KIC 7955301: a hierarchical triple system with eclipse timing variations and
an oscillating red giant

Patrick Gaulme KIC 7955301: a hierarchical triple system with eclipse timing
variations and an oscillating red giantKIC 7955301: a hierarchical triple
system with eclipse timing variations and an oscillating red giantKIC 7955301:
a hierarchical triple system with eclipse timing variations and an oscillating
red giantKIC 7955301: a hierarchical triple system with eclipse timing
variations and an oscillating red giant Tamás Borkovits KIC 7955301: a
hierarchical triple system with eclipse timing variations and an oscillating
red giantKIC 7955301: a hierarchical triple system with eclipse timing
variations and an oscillating red giantKIC 7955301: a hierarchical triple
system with eclipse timing variations and an oscillating red giantKIC 7955301:
a hierarchical triple system with eclipse timing variations and an oscillating
red giantKIC 7955301: a hierarchical triple system with eclipse timing
variations and an oscillating red giantKIC 7955301: a hierarchical triple
system with eclipse timing variations and an oscillating red giantKIC 7955301:
a hierarchical triple system with eclipse timing variations and an oscillating
red giantKIC 7955301: a hierarchical triple system with eclipse timing
variations and an oscillating red giantKIC 7955301: a hierarchical triple
system with eclipse timing variations and an oscillating red giantKIC 7955301:
a hierarchical triple system with eclipse timing variations and an oscillating
red giant Thierry Appourchaux KIC 7955301: a hierarchical triple system with
eclipse timing variations and an oscillating red giantKIC 7955301: a
hierarchical triple system with eclipse timing variations and an oscillating
red giant Krešimir Pavlovski KIC 7955301: a hierarchical triple system with
eclipse timing variations and an oscillating red giantKIC 7955301: a
hierarchical triple system with eclipse timing variations and an oscillating
red giant Federico Spada KIC 7955301: a hierarchical triple system with
eclipse timing variations and an oscillating red giantKIC 7955301: a
hierarchical triple system with eclipse timing variations and an oscillating
red giant Charlotte Gehan KIC 7955301: a hierarchical triple system with
eclipse timing variations and an oscillating red giantKIC 7955301: a
hierarchical triple system with eclipse timing variations and an oscillating
red giantKIC 7955301: a hierarchical triple system with eclipse timing
variations and an oscillating red giantKIC 7955301: a hierarchical triple
system with eclipse timing variations and an oscillating red giant Joel Ong
KIC 7955301: a hierarchical triple system with eclipse timing variations and
an oscillating red giantKIC 7955301: a hierarchical triple system with eclipse
timing variations and an oscillating red giant Andrea Miglio KIC 7955301: a
hierarchical triple system with eclipse timing variations and an oscillating
red giantKIC 7955301: a hierarchical triple system with eclipse timing
variations and an oscillating red giantKIC 7955301: a hierarchical triple
system with eclipse timing variations and an oscillating red giantKIC 7955301:
a hierarchical triple system with eclipse timing variations and an oscillating
red giantKIC 7955301: a hierarchical triple system with eclipse timing
variations and an oscillating red giantKIC 7955301: a hierarchical triple
system with eclipse timing variations and an oscillating red giant Andrew
Tkachenko KIC 7955301: a hierarchical triple system with eclipse timing
variations and an oscillating red giantKIC 7955301: a hierarchical triple
system with eclipse timing variations and an oscillating red giant Benoît
Mosser KIC 7955301: a hierarchical triple system with eclipse timing
variations and an oscillating red giantKIC 7955301: a hierarchical triple
system with eclipse timing variations and an oscillating red giant Mathieu
Vrard KIC 7955301: a hierarchical triple system with eclipse timing variations
and an oscillating red giantKIC 7955301: a hierarchical triple system with
eclipse timing variations and an oscillating red giant Mansour Benbakoura KIC
7955301: a hierarchical triple system with eclipse timing variations and an
oscillating red giantKIC 7955301: a hierarchical triple system with eclipse
timing variations and an oscillating red giant S. Drew Chojnowski KIC 7955301:
a hierarchical triple system with eclipse timing variations and an oscillating
red giantKIC 7955301: a hierarchical triple system with eclipse timing
variations and an oscillating red giant Jean Perkins KIC 7955301: a
hierarchical triple system with eclipse timing variations and an oscillating
red giantKIC 7955301: a hierarchical triple system with eclipse timing
variations and an oscillating red giant Anne Hedlund KIC 7955301: a
hierarchical triple system with eclipse timing variations and an oscillating
red giantKIC 7955301: a hierarchical triple system with eclipse timing
variations and an oscillating red giant Jason Jackiewicz KIC 7955301: a
hierarchical triple system with eclipse timing variations and an oscillating
red giantKIC 7955301: a hierarchical triple system with eclipse timing
variations and an oscillating red giant

KIC 7955301 is a hierarchical triple system with clear eclipse timing and
depth variations that was discovered by the Kepler satellite during its
original mission. It is composed of a non-eclipsing primary star at the bottom
of the red giant branch on a 209-day orbit with a K/G-type main-sequence inner
eclipsing binary, orbiting in 15.3 days. This system was noted for the large
amplitude of its eclipse timing variations (over 4 hours), and the detection
of clear solar-like oscillations of the red-giant component, including p-modes
of degree up to $l=3$ and mixed $l=1$ modes. The system is a single-lined
spectroscopic triple, meaning that only spectral lines from the red giant are
trackable along the orbit. We perform a dynamical model by combining the
4-year-long Kepler photometric data, eclipse timing variations and radial-
velocity data obtained with the high resolution spectrometers ARCES of the
3.5-m ARC telescope at Apache Point observatory and SOPHIE of the 1.93-m
telescope at Haute Provence Observatory. The “dynamical” mass of the red-giant
component is determined with a 2 % precision at
$1.30^{+0.03}_{-0.02}M_{\odot}$. We perform asteroseismic modeling based on
the global seismic parameters and on the individual frequencies. Both methods
provide an estimate of the mass of the red giant that matches the dynamical
mass within the uncertainties. Asteroseismology also reveals the rotation rate
of the core ($\approx 15$ days), the envelope ($\sim 150$ days), and the
inclination ($\sim 75^{\circ}$) of the red giant. Three different approaches
lead to estimating the age to range between 3.3 and 5.8 Gyr, which highlights
the difficulty of determining stellar ages despite the exceptional wealth of
information available for this system. On short timescales, the inner binary
exhibits eclipses with varying depths during a $7.3$-year long interval, and
no eclipses during the consecutive $11.9$ years. This is why Kepler could
detect its eclipses, TESS will not, and the future ESA PLATO mission should.
Over the long term, the system appears to be stable and owes its evolution to
the evolution of its individual components. This triple system could end its
current smooth evolution by merging by the end of the red giant branch of the
primary star because the periastron distance is $\approx 142R_{\odot}$, which
is close to the expected radius of the red giant at the tip of the RG branch.

###### Key Words.:

(Stars:) binaries: spectroscopic - Stars: rotation - Stars: oscillations -
Techniques: spectroscopy - Techniques: radial velocities - Techniques:
photometric - Methods: observational - Methods: data analysis

##  1 Introduction

Among the 5000 stars that are visible with the naked eye, about 2000 are known
to be multiple-star systems. Naked-eye stars are a small fraction of the Milky
Way, but are reasonably representative of the incidence of binarity, which is
estimated to be between 50 and almost 100 % (e.g., Eggleton 2006). Some
systems are close enough to be in contact, others are far apart enough to
evolve almost independently. Binary systems are known with orbital periods as
short as 0.2 days or as long as thousands of years. Studying multiple-star
systems is thus important for understanding the evolution of our Galaxy.

Stars in multiple systems are also precious benchmarks for calibrating
asteroseismology when it is possible to measure the mass of the oscillating
star independently from its oscillation properties. This is the case with
double-lined spectroscopic binary (SB2) where the components eclipse each
other, or optically resolved binary systems. Hitherto, all solar-like
oscillators belonging to eclipsing binary stars (EBs) are red giants (RGs)
detected by the Kepler mission (Hekker et al. 2010; Gaulme et al. 2013, 2014;
Beck et al. 2014, 2015; Kuszlewicz et al. 2019; Benbakoura et al. 2021). So
far, fourteen wide SB2/EBs including an oscillating RG have been fully
characterized with the help of ground-based radial-velocity support (Frandsen
et al. 2013; Rawls et al. 2016; Gaulme et al. 2016; Brogaard et al. 2018;
Themeßl et al. 2018; Benbakoura et al. 2021). Three more bona fide oscillating
RGs in EBs were reported by Gaulme & Guzik (2019), and are currently under
study. Beyond EBs, it is also possible to determine the masses of stars
belonging to visual multiple-systems, where individual components are
spatially resolved, which allows retrieval of their projected orbits, provided
radial velocities are available too. So far, very few of such systems include
solar-like pulsators (Marcadon et al. 2018; Metcalfe et al. 2020). The
drawback of such systems is that the orbits are long and getting accurate
orbital parameters can take decades and rely on heterogeneous datasets.
Finally, hierarchical triple systems are promising types of benchmarks; they
are composed of a close binary with a relatively distant companion (e.g.,
Tokovinin 1997; Ford et al. 2000; Borkovits et al. 2003). Depending on their
orbital configurations, the presence of eclipses and significant eclipse
timing variations (ETVs), it can be possible to determine the masses of their
components (Derekas et al. 2011; Carter et al. 2011; Borkovits et al. 2016).

KIC 7955301 is a hierachical triple system composed of a young RG and a pair
of small main-sequence stars that was observed for nearly four consecutive
years by the original NASA Kepler mission (Borucki et al. 2011) at long
(29.4244-minute cadence). It was pointed out by Gaulme et al. (2013) and
Rappaport et al. (2013) who noticed its outstanding eclipse timing, depth and
duration variations. The pair of main-sequence stars shows partial eclipses,
but the RG does not eclipse the inner binary. The eclipse timings show a
periodicity of about 208.6 days, while the pair of MS stars orbit in 15.3
days. The precession of the orbital plane of the MS stars is quite visible as
well, and has a period longer than the time series. This system was part of
the cohort of systems with ETVs studied by Borkovits et al. (2016). From
eclipse timing only, they estimated the mass of the RG to be $1.5\pm
0.5M_{\odot}$ and the total mass of the MS pair to be $2.2\pm 0.8M_{\odot}$.
From asteroseismic scaling relations and a quick look at the mixed dipole
($l=1$) modes, Gaulme et al. (2013) inferred the RG mass to be $1.2\pm
0.2M_{\odot}$, the radius $5.9\pm 0.2R_{\odot}$, and the core rotation of
about 30 days. By considering that asteroseismic scaling relations tend to
slightly overestimate stellar masses for RGs (Gaulme et al. 2016), the RG is
likely a descendent of an F-type star that had a convective envelope during
the MS and that is expected to rotate slowly.

With this paper, we firstly aim at testing our ability to measure accurate
masses in hierarchical triple systems. We know it is possible to determine the
mass of a star that belongs to an eclipsing binary down to 1 % (e.g., Maxted
et al. 2020). However, no study has specifically been developed for measuring
the mass of an oscillating star in a hierarchical triple that is a single-
lined spectroscopic triple system (ST1). We note that the famous RG in the
hierarchical triple system HD 181068 (Derekas et al. 2011), observed by
Kepler, is triply eclipsing and does not show any oscillations, likely because
of the mode suppression observed in short period multiple systems (Gaulme et
al. 2020). Given the clear ETVs and the clear oscillations of the RG, KIC
7955301 appears to be an ideal case for testing our methods on hierarchical
triple systems with an oscillating component. Subsequently, the approaches
described here should be extended to all known triple with ETVs that include
an oscillating component (Gaulme et al. 2013).

The second objective is a rather unique opportunity to study an RG star in
greater detail, thanks to the multiple approaches used simultaneously to give
a clear picture of the system. Our study makes use of the Kepler photometric
data and high-resolution optical spectra obtained with the 3.5-m telescope at
Apache Point observatory, and with the échelle spectrograph SOPHIE of the
1.93-m telescope at the Haute-Provence Observatory (Sect. 2). The spectra are
used to both determine the radial velocities and the atmospheric parameters
(Sect. 3). The Kepler light curves are used to both analyze the eclipses for
the asteroseismic analysis (Sect. 4) and the dynamical modeling (Sect. 5).
Finally, the deduced parameters – oscillation frequencies, orbital parameters,
etc. – are used to optimize a stellar evolution model (Sect. 6).

Table 1: Archival properties of the KIC 7955301 triple System.

Parameter | Value
---|---
RA (J2000) | $19:20:45$
Dec (J2000) | $+43:43:26$
$K_{\mathrm{p}}$a | $12.672$
$G$b | $12.6283\pm 0.0002$
$G_{\rm BP}^{b}$ | $13.2017\pm 0.0012$
$G_{\rm RP}$b | $11.9340\pm 0.0009$
Bc | $14.082\pm 0.043$
Vd | $12.901\pm 0.008$
g′d | $13.419\pm 0.045$
r′d | $12.553\pm 0.043$
i′d | $12.323\pm 0.026$
Je | $10.992\pm 0.023$
He | $10.487\pm 0.018$
Ke | $10.385\pm 0.011$
W1f | $10.323\pm 0.023$
W2f | $10.409\pm 0.020$
W3f | $10.310\pm 0.056$
W4f | $9.118$
$T_{\rm eff}$ (K)c | $4805\pm 75$
Distance (pc)g | $1375\pm 35$
${\rm[M/H]}^{c}$ | $0.1163\pm 0.0072$
$E(B-V)^{c}$ | $0.044$
$\mu_{\alpha}$ (mas ${\rm yr}^{-1}$)h | $-0.86\pm 0.02$
$\mu_{\delta}$ (mas ${\rm yr}^{-1}$)h | $-8.59\pm 0.02$
111RA and Dec are the system coordinates in the J2000 frame, $K_{\mathrm{p}}$
is the Kepler magnitude, $G$, $G_{\mathrm{BP}}$ and $G_{\mathrm{RP}}$ are the
visible, blue and red passband Gaia magnitudes from DR2, parameters from B to
$W4$ are magnitudes whose origin is detailed in the notes underneath the
Table, $T_{\mathrm{eff}}$ is the effective temperature (Kelvin), [M/H] the
metallicity (dex), $E(B-V)$ the reddening, and $\mu_{\alpha}$ and
$\mu_{\delta}$ the parallaxes along the right ascension and declination
respectively. $a$$a$footnotetext: Kepler Input Catalog (Brown et al. 2011).
$b$$b$footnotetext: Gaia DR2 (Gaia Collaboration et al. 2018).
$c$$c$footnotetext: TESS Input Catalog v8.1 (Stassun et al. 2018)
$d$$d$footnotetext: AAVSO Photometric All Sky Survey (APASS) DR9, (Henden et
al. 2015), http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=II/336/apass9.
$e$$e$footnotetext: 2MASS catalog (Skrutskie et al. 2006). $f$$f$footnotetext:
WISE point source catalog (Cutri et al. 2013). $g$$g$footnotetext: Bailer-
Jones et al. (2021). (h) Gaia EDR3 (Gaia Collaboration et al. 2021). Note
also, that for the SED analysis in Sect. 6.4 the uncertainties of the passband
magnitudes were set to
$\sigma_{\mathrm{mag}}=\mathrm{max}(\sigma_{\mathrm{catalog}},0.030)$ to avoid
the strong overdominance of the extremely accurate Gaia magnitudes over the
other measurements.

##  2 Observations

Figure 1: Kepler light curve of KIC 7955301. Top panel: relative flux as a
function of time, where $t=0$ is the first day of data. The blue curve is the
stitched light curve, whereas the orange curve has the eclipses removed and is
used for asteroseismic analysis. Bottom panel: stitched lightcurve folded over
the inner binary orbital period (15.32 days), where consecutive eclipses are
vertically offset by 0.003 to ease the visibility of the transit depth and
timing variations.

###  2.1 The Kepler light curve

We worked with the Kepler public light curves that are available on the
Mikulski Archive for Space Telescopes
(MAST)222http://archive.stsci.edu/kepler/. Two types of time series are
available: the Simple Aperture Photometry (SAP) and the Pre-search Data
Conditioning Simple Aperture Photometry (PDC-SAP) light curves. The latter
consists of time series that were corrected for discontinuities, systematic
errors and excess flux due to aperture crowding (Twicken et al. 2010). They do
not meet our requirements for monitoring possible rotational modulation or
eclipse depth, which are often altered during the process (e.g., García et al.
2014; Gaulme et al. 2014). We thus made use of the SAP data to preserve any
possible long-term signal. This choice entails our own detrending and
stitching operation on the light curves while ensuring that the rotational
modulation is preserved after each interruption of the time series. The
methods employed to clean the time series are detailed in Gaulme et al.
(2016).

For asteroseismology, light curves with eclipses are an issue because their
signal contaminates the Fourier domain, in which we perform the asteroseismic
analysis. We tried two different approaches to remove the eclipses from the
time series before computing its power density spectrum. The first approach is
better in principle: it consists of modeling each eclipse with a typical
eclipse function, for example that from Mandel & Agol (2002) for fitting
exoplanetary transits, and then subtracting the model from the light curve.
However, this approach, as noted in Gaulme et al. (2016), is less efficient
than simply clipping out the eclipses, followed by gap filling with a second-
order polynomial. Indeed, an eclipse model is always a little imperfect and
when subtracted about 100 times (the number of eclipses during the Kepler run)
they still damage the Fourier transform, unlike eclipse removal and filling.
Figure 1 (top panel) shows the original time series along with the time series
with no eclipses that is used for analyzing the RG oscillations. The bottom
panel highlights the eclipses variations in the form of a folded light curve
where a shift was introduced between consecutive orbits. Eclipse depth varies
from 0.5 to almost 4 %, timing by about 4 hours, and duration from
approximately 4.3 to 6.7 hours.

###  2.2 High-resolution optical spectra

From 2012 to 2019, we were granted time on the ARCES échelle spectrograph of
the 3.5-m telescope of the Astrophysical Research Consortium (ARC) at Apache
Point observatory (APO), which covers the whole visible domain at an average
resolution of $31,000$ (Wang et al. 2003). This allowed us to monitor KIC
7955301 23 times, among observations of other RGs in multiple systems. Even
though the ARC échelle spectrograph was not designed for precise RV
measurements, it has successfully been used for this purpose in earlier work
(e.g., Rawls et al. 2016; Gaulme et al. 2016; Benbakoura et al. 2021). The
measurement error reported in these papers is about 0.5 km s-1 for an RG
spectrum with a signal-to-noise ratio (S/N) between 10 and 20. In practise our
spectra have a S/N ranging from 10 to 25. The ARCES optical spectra were
processed and analyzed in the same way as in Gaulme et al. (2016) and
Benbakoura et al. (2021) and we refer to these papers for details.

In addition to the APO spectrograph, we were granted observing time on the
SOPHIE échelle spectrograph at the 1.93-m telescope at the Haute-Provence
Observatory (OHP). We obtained spectra of KIC 7955301 on June 8, 9 and October
9, 2018. The SOPHIE data processing pipeline directly provides the radial
velocities from the spectra. We refer the reader to Santerne et al. (2011b, a)
for details on the data reduction and radial-velocity measurement based on
SOPHIE spectra.

##  3 Spectroscopic analysis

Figure 2: Broadening function of the 23 spectra taken with the échelle
spectrometer of the 3.5-m ARC telescope at Apache Point observatory. Spectra
were sorted by increasing orbital phase. The position along the $y$ axis
corresponds with the orbital phase. Barycentric velocity corrections are
included in the BF computation. The radial velocity data are indicated with
the red plus symbols and their best-fit Keplerian-orbit model by a dashed
line.

###  3.1 Radial-velocities

From the reduced one-dimensional spectra obtained at APO, we computed the
radial velocities with the Broadening-Function technique (BF, Rucinski 2002).
The fundamental hypothesis of this method is that the observed spectrum is the
theoretical spectrum convolved with a broadening function that accounts for
stellar rotation and instrumental effects. The BF technique deconvolves the
observed spectrum by the theoretical one to extract the BF. For SB2 systems,
the BF shows two peaks, one per component.

We employed the theoretical spectra generated by the PHOENIX BT-Settl code
(Allard et al. 2003), which were computed with the solar abundances derived by
Asplund et al. (2009). As in Gaulme et al. (2016) and Benbakoura et al.
(2021), we used templates of MS stars ($T_{\rm eff}=6200$ K) to maximize the
chance of detecting the signal from the companion star. Such a choice actually
also helps the deconvolution of the RG spectrum because the large number of
absorption lines in RG spectra tends to increase the noise in the BF profiles
when the original spectra do not have an excellent S/N, which is our case. We
computed the BF by making use of the wavelength range 4500 and 5800 Å. Our
final RV data were obtained after correcting the BF profiles from the
barycentric corrections, which account for the rotation and revolution of the
Earth with respect to the target. We computed the barycentric corrections with
the PyAstronomy333https://github.com/sczesla/PyAstronomy routine helcorr,
which is based on the Piskunov & Valenti (2002) algorithm.

All the radial velocities we produced in this work are compiled in Table 5 and
displayed in Fig. 2 together with the series of BF profiles. For illustration
purposes we overplot data points with a fit performed with a simple Keplerian
orbit. This allows us to estimate the standard deviation of the measurements
to be about 0.70 km s-1. We note that employing a Keplerian orbit is a very
simplistic approach as KIC 7955301 is a triple system. In Sect. 5 we show that
the argument of periastron of the outer orbit changed by about 20∘ during the
four years of Kepler observations, meaning that a simple Keplerian orbit is
not sufficient to accurately model the RG orbit.

###  3.2 Disentangling the spectra

A proper estimate of the atmospheric parameters of the RG and possibly of the
inner binary components requires disentangling the spectra. For a triple
system with such a large flux contrast between the evolved primary component
and the inner dwarf binary, it is not an easy task. The spectral lines of the
three components are diluted and it is challenging to disentangle the
individual contributions to the continuum without knowing the light ratio. It
is straightforward to get the ratio from a light curve when all components
eclipse each other. Unfortunately, in our case, the RG is not eclipsing with
the inner pair. The remaining method for measuring the light ratio and
distentangling the spectra consists of scaling the absorption lines from
synthetic spectra to the series of observed spectra (Pavlovski et al. 2009,
2018, and references therein).

According to the dynamical model presented in the following sections, the
contribution of the main-sequence components to the total luminosity is close
to 10 %, which implies that we cannot visualize their signatures neither in
the BF profile, nor in the spectra. Nevertheless, very faint components have
been revealed by spectral disentangling even at a level of only 1-2 % of the
total light, such as the M dwarf in the EB V530 Ori (Torres et al. 2014), the
Roche-lobe filling giant in the inner EB of the Algol triple system (Kolbas et
al. 2015), the main sequence companions of the RGs in EBs (e.g., Gaulme et al.
2016; Hełminiak et al. 2017; Brogaard et al. 2018; Themeßl et al. 2018;
Benbakoura et al. 2021).

In spectral disentangling as originally formulated by Simon & Sturm (1994),
the spectra of the individual components are simultaneously reconstructed with
an optimization of the orbital elements of the multiple-star system. The
problem consists of solving the matrix equation $A\cdot x=b$, where the vector
$x$ represents the unknown individual spectra of the components – which we aim
at extracting – and $b$ all of the observed spectra. The design matrix $A$ is
constructed from Doppler shifts for a given set of the orbital elements and
exposure dates. $A$ could also contain light dilution factors, if known. Since
the set of linear equations are overdetermined, the solution may be calculated
with the linear-algebra technique known as singular value decomposition. In
our particular case, the RVs for the RG component are measured by the BF
(Sect. 3.1, and Table 5). From the dynamical model of the system described in
Sect. 5, we computed the expected values of the RVs of the components of the
inner binary.

The spectral disentangling is performed with the code cres (Ilijic 2004),
which is based on the Simon & Sturm (1994) method for spectral disentangling
in the visible domain. In cres, the RVs, light dilution factors of each
component, and the observed spectra are inputs. The observed spectra should be
given in units of wavelength. The observed spectra do not need to be in an
uniform scale, and the spectra from different spectrographs do not need to be
resampled first. Any portion of the observed spectra could be masked out to
exclude undesired wavelength regions.

Figure 3: A sample region of the mean spectra of the individual components of
the system obtained by disentangling the APO spectra. Top panel: spectrum of
the RG component (blue line) and its best fit obtained with the gssp code
(red) from 5340 to 5450Å. Bottom panel: spectra of the inner binary system Ba
and Bb (blue lines) together with their best-fit synthetic spectra (red
lines). The optimal values representing the best fits are listed in Table 2.

Altogether, we use the 23 high-resolution échelle spectra obtained at APO
(Sect. 2.2). Spectra have a S/N from $\sim 5$ to 60, with an average S/N
$\simeq 27$, as measured in several short line-free windows near 5500 - 5600
Å. The S/N was then used to assign weights to the observed spectra. We note
that the code actually allows an assignment of the weights to individual
pixels, another feature advantageous in the wavelength-domain disentangling,
but we did not use this option, assuming it is sufficient to take the weight
per spectrum, and not per pixel. The selected spectral segments had various
lengths, from about 40 to 90 Å, always taking into account overlapping regions
which serve to double-check the quality of our disentangling.

It is known that without any substantial change in the fractional light of the
components in the course of an orbital cycle, there is no unique solution in
reconstruction of the individual spectra of the components (Pavlovski &
Hensberge 2005). One possibility is to optimize the light dilution factors in
the calculations of the components’ spectra, but with dominant fractional
light contribution of the RG component, which is not eclipsing with the inner
pair of the MS stars, this would be meaningless. The best option is to perform
spectral disentangling in a pure separation mode, and then determine
fractional light contribution for each component from its spectral
characteristics.

The reconstruction of the individual spectra of all three components was
performed in the wavelength range $[5100,5650]$ Å. In addition, the spectral
range of the disentangled spectrum for the RG component was extended to the
range $[4700,6825]$ Å. In the latter case, our intention was to cover the
Balmer lines H$\alpha$, and H$\beta$, as well as the sodium doublet Na i at
$\lambda\lambda$ 5890, and 5896 Å. Some portions of the reconstructed spectra
are shown in Fig. 3. It is evident from a bottom panel of Fig. 3 showing
reconstructed spectra for the stars in the inner (eclipsing) pair that the
signatures of the stellar spectra are successfully revealed, albeit the noise
level is large. We can estimate the gain in the S/N for the spectra of the
disentangled components by simple calculations, since spectral disentangling,
in principle, is working as co-addition of the observed spectra. By assuming
random noise with an average S/N $\simeq 27$, the 23 spectra should lead to
total S/N of about 130. Of this, the RG spectrum benefits most since it
contributes about 91 % of the total light, as is determined later in this
section. The other two components share the remaining $\sim 9$ % – given their
contribution is similar (Sect. 5, Table 4) –, meaning that the S/N of their
reconstructed spectra is about 5-6. Thus, their spectral signatures are
revealed only thanks to the gain of S/N caused by the spectral disentangling.
The successful isolation of the spectra for the stars in the inner system,
which is based on the RVs calculated from the predicted orbit, is an
encouraging confirmation of the correctness of the dynamical model.

The deepest absorption lines in the disentangled spectrum of the RG component
are due to Mg i$b$ triplet lines at 5167, 5172, and 5183 Å, and Na i D doublet
at 5890, and 5896 Å. The depths of these sets of spectral lines constrain the
fractional light contribution of the RG to the total light of the system. A
limit defined by the physical solution is at 89%, which is corroborated with
the optimal fitting (Table 2).

It is well established by observational evidence that lithium is depleted in
the majority of giants (Brown et al. 1989). There are rare exceptions known as
Li-rich giants with abundances $A({\rm Li})>1.5$ (a scale where the number of
hydrogen atoms $\log N({\rm H})=12.0$). Examination of the disentangled
spectrum of the RG component of KIC 7955301 shows almost no trace of the Li i
resonance doublet at 6708 Å. An estimate of an upper limit of the lithium
abundance was made with a comparison to the synthetic line profiles. The
calculations were performed assuming local thermodynamic equilibrium with
model atmospheres calculated with Atlas9 (Castelli & Kurucz 2003), while line
profiles for Li i 6708 Å are calculated with spectrum synthesis code uclsyn
(Smith 1992). The atomic data are from Yan et al. (1998). We were able to
establish only an upper limit in lithium abundance $A({\rm Li})<-1.0$ dex.
This is a common lithium abundance found in modern massive spectroscopic
surveys (Luck & Heiter 2007; Buder et al. 2018; Charbonnel et al. 2020).

###  3.3 Stellar atmospheric parameters

Table 2: Best-fit parameters deduced from the analysis of the atmospheric
parameters.

Red giant (star A)
---
Run A | $T_{\mathrm{eff}}$ | [K] | $4720\pm 105$
$\log g$ | [dex] | 3.0 (fixed)
$v_{\mathrm{micro}}$ | [km s-1] | $1.44\pm 0.35$
$v\sin i$ | [km s-1] | $6.4\pm 1.2$
dilution factor |  | $0.91\pm 0.08$
$[\mathrm{M}/\mathrm{H}]$ | [dex] | $-0.01\pm 0.12$
Run B | $T_{\mathrm{eff}}$ | [K] | $4760\pm 110$
$\log g$ | [dex] | 3.1 (fixed)
$v_{\mathrm{micro}}$ | [km s-1] | $1.36\pm 0.35$
$v\sin i$ | [km s-1] | $6.3\pm 1.2$
dilution factor |  | $0.92\pm 0.07$
$[\mathrm{M}/\mathrm{H}]$ | [dex] | $0.03\pm 0.12$
Run C | $T_{\mathrm{eff}}$ | [K] | $4700\pm 155$
$\log g$ | [dex] | $2.95\pm 0.40$
$v_{\mathrm{micro}}$ | [km s-1] | $1.31\pm 0.35$
$v\sin i$ | [km s-1] | $6.6\pm 1.3$
dilution factor |  | $0.91\pm 0.08$
$[\mathrm{M}/\mathrm{H}]$ | [dex] | $-0.01\pm 0.17$
Inner binary (stars Ba and Bb)
Ba | $T_{\mathrm{eff}}$ | [K] | $5620\pm 580$
$\log g$ | [dex] | 4.5 (fixed)
$v_{\mathrm{micro}}$ | [km s-1] | 2.0 (fixed)
$v\sin i$ | [km s-1] | $19.0\pm 5.5$
dilution factor |  | $0.043\pm 0.013$
$[\mathrm{M}/\mathrm{H}]$ | [dex] | 0.0 (fixed)
Bb | $T_{\mathrm{eff}}$ | [K] | $5330\pm 550$
$\log g$ | [dex] | 4.5 (fixed)
$v_{\mathrm{micro}}$ | [km s-1] | 2.0 (fixed)
$v\sin i$ | [km s-1] | $8.9\pm 7.5$
dilution factor |  | $0.025\pm 0.009$
$[\mathrm{M}/\mathrm{H}]$ | [dex] | 0.0 (fixed)
444Run A: $\log g$ is fixed to 3.0 dex (atmosphere model grid point closest to
$\log g=3.034$ dex reported in Table 4, column ”with SED+PARSEC”); wavelength
range 4700 - 5700 Å. Run B: $\log g$ is fixed to 3.1 dex (atmosphere model
grid point closest to $\log g=3.115$ dex reported in Table 4, column ”without
SED+PARSEC”); wavelength range 4700 - 5700 Å. Run C: $\log g$ is treated as a
free parameter; wavelength range 4700 - 5700 Å. Inner binary, both components;
values of $\log g$ are fixed to 4.5 dex (Table 4); fixing vmicro and [M/H] as
well because spectra are really noisy; wavelength range 5100 - 5630 Å.

For the analysis of the disentangled spectra of all three components of the
KIC 7955301 system, we employed the Grid Search in Stellar
Parameters555https://fys.kuleuven.be/ster/meetings/binary-2015/gssp-software-
package (gssp Tkachenko 2015) software package, specifically its gssp_single
module. It is a grid search-based spectrum analysis algorithm that (on the
fly) generates synthetic spectra in an arbitrary wavelength range based on a
pre-computed grid of model atmospheres, and performs a comparison between the
observed spectrum and each synthetic spectrum from the grid in the $\chi^{2}$
statistical framework. The reported 1-$\sigma$ uncertainties are computed from
the $\chi^{2}$-statistics and taking into account possible correlations
between the free parameters. The gssp algorithm allows for the simultaneous
optimization of the effective temperature, surface gravity, micro- and macro-
turbulent velocities, projected rotational velocity, and metallicity of the
star. Optionally, one can also optimize for the degree of the light dilution
in the spectrum due to the star being a member of a binary and/or higher order
multiple system.

In this study, we employ the grid of LLmodels model atmospheres (Shulyak et
al. 2004) and opt for inclusion of the light dilution factor into the
optimization to account for a priori unknown dilution of the disentangled
spectrum of each of the three stellar components under the analysis. The light
dilution factor is assumed to be wavelength independent, which is a fair
assumption for the RG component given its by far most dominant ($>$ 90%)
contribution to the composite spectrum of the system, as well as for the two
main-sequence components given the limited wavelength interval of some 500 Å
that could be used for the disentangling. We additionally note that the macro-
turbulent velocity parameter was ignored in the analysis of all three stellar
components of the system, and we additionally fixed $\log\,g$ for both main-
sequence components to the values inferred from the light curve solution,
micro-turbulent velocity to 2 km s-1, and assumed solar chemical composition
([M/H] = 0.0 dex) for both of them. The choice to fix so many parameters in
the spectrum analysis of the main-sequence components is dictated by their
small cumulative contribution ($<$ 8-10%) to the total light of the system,
and hence their very noisy disentangled spectra. The consequence of fixing the
macro-turbulent velocity parameter for the RG component is that the $v\,\sin
i$ value reported is in fact representative of the combined spectral line
broadening due to the effects of rotation and macroturbulence. Finally we note
that we explored three different options in the analysis of the RG spectrum,
namely fixing the surface gravity to the two values inferred from the light
curve solution and reported in Table 2, and treating $\log\,g$ as a free
parameter.

##  4 Asteroseismic analysis

Figure 4: Oscillation spectrum of KIC 7955301. Top panel: power spectral
density of the time series after eclipse removal and filling. The red dashed
line represents the stellar background noise: it is the sum of the blue dashed
lines (correlated noise Harvey profiles) and the green dashed line (white
noise). The plain red line represents the fit of the power spectrum including
the stellar background and the Gaussian envelope of the oscillations. Bottom
panel: échelle diagram associated with the frequency spacing
$\Delta\nu=10.48\,\mu$Hz. Colored lines indicate the oscillation universal
pattern, where blue is $l=0$, green $l=1$, and red $l=2$. Modes of degree
$l=3$ are visible half way in between the $l=0$ and $l=1$ ridges. The
horizontal dashed line indicates the location of $\nu_{\mathrm{max}}$.

Figure 5: Oscillation amplitude. Top panel: height of the Gaussian envelope
used to model the contribution of the oscillations to the stellar background.
The red dot highlights the position of KIC 7955301 with respect to a sample of
4500 RGs analyzed by Gaulme et al. (2020) indicated in blue. The sample splits
into three categories: the oscillators with regular amplitude, those with
depleted $l=1$ modes, and the active RGs that show a global mode suppression
($l=0,1,2,3$). KIC 7955301 falls in the regular oscillators. Bottom panel:
amplitude of the largest $l=0$ mode of KIC 7955301 compared with those of the
35 RGs in EBs that were studied by Benbakoura et al. (2021). The figure
displays the relative difference between expected and measured oscillation
amplitudes (%) as a function of orbital period $P_{\mathrm{orb}}$ (days). The
red line indicates 0, i.e., stars whose oscillations are not detected. The two
dashed blue lines represent the region in which relative mode amplitude lies
for systems with orbital periods longer than 180 days within two sigma. The
size of each symbol represents the amplitude of stellar variability (large
means variable; small not variable), and the gray scale indicates the
pulsation mode amplitude (white-no modes; black-large amplitude).

###  4.1 Global oscillation properties

The stellar granulation and accurate values of $\nu_{\mathrm{max}}$ and mode
amplitude $H_{\mathrm{max}}$ are estimated by fitting the power spectrum (Fig.
4) as commonly performed in asteroseismology (Kallinger et al. 2014), and
already used in Gaulme et al. (2016). Following Kallinger et al. (2014), the
power density spectrum is fitted by

$S(\nu)=N(\nu)+\eta(\nu)\left[B(\nu)+G(\nu)\right],$ (1)

where the $N$ is the function describing the noise, $\eta$ is a damping factor
originating from the data sampling, $B$ is the sum of three “Harvey” functions
(super Lorentzian functions centered on 0), and G is the Gaussian function
that accounts for the oscillation excess power

$G(\nu)=H_{\mathrm{max}}\exp{\left[-\frac{(\nu-\nu_{\mathrm{max}})^{2}}{2\sigma^{2}}\right]}.$
(2)

The terms $\nu_{\mathrm{max}}$ and $H_{\mathrm{max}}$ are the central
frequency and height of the Gaussian function. The best-fit values are:
$\nu_{\mathrm{max}}=124.89\pm 0.33\ \mu$Hz, $H=354.2\pm 10.5$ ppm $\mu$Hz-1,
and $\sigma=12.5\pm 0.4\,\mu$Hz.

A first estimate of $\Delta\nu$ is performed with the envelope of the
autocorrelation function (EACF) developed by Mosser & Appourchaux (2009) from
the whitened power spectral density (power spectrum divided by background
function). From the EACF, $\Delta\nu=10.46\pm 0.05$. Then, we used the
universal pattern of RGs introduced by Mosser et al. (2011) to correct it. The
principle of this method is to compare the measured oscillation frequencies to
a theoretical law, the so-called universal pattern, predicting the variations
of these frequencies as a function of $\Delta\nu$ and the radial order. That
way, the value of $\Delta\nu$ is revised to $10.49\pm 0.02\ \mu$Hz.

We then compute a proxy of the stellar masses and radii using the
asteroseismic scaling relations that were originally proposed by Kjeldsen &
Bedding (1995) for SL MS oscillators, and then successfully applied to RGs
(e.g., Mosser et al. 2013). We employ the asteroseismic scaling relations as
proposed by Mosser et al. (2013) for RGs, i.e., where
$\nu_{\mathrm{max,\odot}}=3104\ \mu$Hz, $\Delta\nu_{\odot}=138.8\ \mu$Hz,
$T_{\mathrm{eff,\odot}}=5777$ K, and where the observed $\Delta\nu$ is
converted into an asymptotic one, such as
$\Delta\nu_{\mathrm{as}}=1.038\,\Delta\nu$. By considering the temperature
$T_{\mathrm{eff,A,sce.A}}=4720\pm 105$ K that was found from the disentangled
spectrum by fixing the surface gravity at the value obtained from the most
complete model $\log g=3.0$ (scenario A, Table 2), the stellar parameters are
$R_{\mathrm{A}}=5.91\pm 0.07\ R_{\odot}$, and $M_{\mathrm{A}}=1.27\pm 0.05\
M_{\odot}$.

Finally, we check the amplitude of the oscillations to see whether tidal
interactions in the system have altered their properties, as was observed in
close systems by Gaulme et al. (2014, 2020), and Benbakoura et al. (2021).
From the background fitting, the height of the Gaussian function is $354\pm
11$ ppm $\mu$Hz-1, which is a typical value for an RG with
$\nu_{\mathrm{max}}=125\ \mu$Hz, according to sample of 4500 RGs analyzed by
Gaulme et al. (2020, Fig. 7) with the exact same codes. We note that the
height of the oscillation envelope is a little underestimated because the RG
only contributes approximately 91 % of the photometric flux, whereas most of
the 4500 stars displayed in Gaulme et al. (2020, Fig. 7) are not in multiple
systems and do not suffer from this dilution factor.

An alternative metric on the amplitude of the oscillation modes consists of
measuring the amplitude of the largest $l=0$ mode, as was performed by
Benbakoura et al. (2021). The fitting of the $l=0$ peaks with Lorentzian
functions leads to a maximum $l=0$ amplitude of $7.06\pm 0.08$ ppm, which is
in agreement with the known RGs in EBs that do not show any alteration of
their oscillation properties (Fig. 5). We conclude that the oscillations of
KIC 7955301 have regular amplitudes and do not show any global suppression, as
was observed for RGs in short period binary systems. This result is consistent
with the long-period binary systems observed so far (Benbakoura et al. 2021).

Figure 6: Stretched period échelle diagram for dipole gravity-dominated mixed
modes. Top panel is the work led by co-author Appourchaux with
$\Delta\Pi_{1}=76.3$ s. Fitted ridges with $m=\\{-1,1\\}$ are represented as
red and orange crosses, respectively; while $m=0$ is represented as a
continuous white line. The black stripes show the location of the $l=0,2$ mode
power which has been removed for clarity. Bottom panel summarizes the finding
of co-author Gehan, where peaks with a height-to-background ratio equal to or
above 10 are represented. The symbol size varies as the measured power
spectral density. Ridges with $m=\\{-1,0,1\\}$ are represented in green, light
blue, and red, respectively. Red and green crosses represent the fit of the
ridges with $m=\pm 1$. The location of the $m=0$ ridge is identified
considering that the $m=0$ ridge is median with respect to $m=\pm 1$ ridges.
Figure 7: Peak-bagging results including mixed-mode analysis by co-author
Appourchaux displayed in the form of an échelle diagram of the power spectrum
for $\Delta\nu=$10.44 $\mu$Hz. Fitted frequencies are shown: $l=0$ white
diamonds, $l=2$ orange diamonds, $l=3$ red diamonds, $l=1,m=-1$ white crosses,
$l=1,m=0$ white circles, $l=1,m=+1$ white pluses.  Figure 8: Peak-bagging
results including mixed-mode analysis by co-author Mosser in the form of a
line-by-line échelle diagram, for radial pressure orders 8 to 13. Radial and
quadrupole modes are highlighted in red and green, respectively. The expected
locations of dipole mixed modes are labelled with their mixed radial orders,
and indicated with a color depending on the azimuthal order: dark blue
($m=-1$), light blue ($m=0$), or purple ($m=+1$). $\ell=3$ modes, not
identified in this plot, are located near the abscissa 0.21. The gray dashed
lines correspond to height-to-background ratios of 7 and 10. Table 3: Global
seismic parameters of the RG component.

Parameter |  | Gaulme | Appourchaux | Mosser | Gehan | Vrard
---|---|---|---|---|---|---
$\nu_{\mathrm{max}}$ | [$\mu$Hz] | $124.9\pm 0.4$ |  | $122.7\pm 24.1$ | $122.5\pm 5.6$ |
$\Delta\nu$ | [$\mu$Hz] | $10.49\pm 0.02$ | $10.496\pm 0.011$ | $10.48\pm 0.05$ | $10.45\pm 0.27$ |
$\Delta\Pi_{1}$ | [s] |  | $76.297\pm 0.003$ | $76.27\pm 0.03$ | $76.3\pm 0.1$ | $76.09\pm 0.72$
$q$ |  |  | $0.098\pm 0.002$ | $0.120\pm 0.013$ |  | $0.17\pm 0.05$
$\delta_{01}$ |  |  | $0.027\pm 0.001$ | $0.029\pm 0.006$ |  |
$\epsilon_{p}$ |  |  | 1.233 | $1.194\pm 0.052$ |  |
$\epsilon_{g}$ |  |  | $0.24\pm 0.005$ | $0.246\pm 0.032$ |  |
$\nu_{\mathrm{core}}$ | [nHz] |  | $382\pm 2$ | $382\pm 10$ | $380\pm 6$ |
$\nu_{\mathrm{env}}$ | [nHz] |  | $39\pm 6$ |  |  |
$i$ | [∘] |  | 74.2${}^{+7.7}_{-14.1}$ |  | $75.8^{+14.2}_{-6.6}$ |

666Frequency at maximum amplitude $\nu_{\mathrm{max}}$ ($\mu$Hz), mean large
frequency spacing $\Delta\nu$ ($\mu$Hz), mean dipole mode period spacing
$\Delta\Pi_{1}$ (s), coupling factor $q$, small frequency spacing
$\delta\nu_{01}$, p- and g-mode phase offsets $\epsilon_{p}$ and
$\epsilon_{g}$, respectively, rotational splittings of the core and envelope
$\nu_{\mathrm{core}}$ and $\nu_{\mathrm{env}}$ (nHz), respectively, and the
inclination $i$ (degree).

###  4.2 Mixed modes

####  4.2.1 Formalism

As indicated in the introduction, the RG component of KIC 7955301 shows a rich
spectrum of dipolar mixed pressure and gravity modes. Here we briefly
summarize some basic properties of the mixed modes (see Mosser et al. 2015,
and references therein). The frequency of the mixed modes is an implicit
expression given by

$\tan\theta_{p}=q\tan\theta_{g},$ (3)

where $q$ is the coupling factor between the p and g modes, $\theta_{p}$ is
the phase of the mixed modes with respect to the asymptotic p-mode frequencies
separated by the large separation $\Delta\nu$, while $\theta_{g}$ the phase of
the mixed modes with respect to the asymptotic g-mode periods separated by the
period spacing $\Delta\Pi_{1}$. In other words,

$\theta_{p}=\pi\left(\frac{\nu}{\Delta\nu}-\frac{1}{2}-\epsilon_{p}\right),$
(4)

where $\nu$ is the mixed mode frequency and $\epsilon_{p}$ is a phase offset,
and

$\theta_{g}=\pi\left(\frac{P}{\Delta\Pi_{1}}-\epsilon_{g}\right),$ (5)

where $P$ is the mixed mode period and $\epsilon_{g}$ is a gravity offset.
Solving the implicit relation given by Eq. (3) provides the mixed mode
frequencies.

The period separation $\Delta P$ between two consecutive mixed mode periods is
given by

$\Delta P=\zeta(\nu)\,\ \Pi_{1},$ (6)

where $\zeta$ is related to the phases given above by

$\zeta(\nu)=\left[1+\frac{\nu^{2}\Delta\Pi_{1}}{q\Delta\nu}\frac{\cos^{2}\theta_{g}}{\cos^{2}\theta_{p}}\right]^{-1}.$
(7)

$\zeta$ is also useful for expressing the frequency splitting of these dipole
modes $\delta\nu_{\rm rot}$, since we have

$\delta\nu_{\mathrm{rot}}=\zeta(\nu)\delta\nu_{\mathrm{core}}+(1-\zeta(\nu))\,\delta\nu_{\mathrm{env}},$
(8)

where $\delta\nu_{\mathrm{core}}$ is the rotation mainly sampled by the g
modes, while $\delta\nu_{\mathrm{env}}$ is the rotation mainly sampled by the
p modes. We stress that this expression is not symmetrical in terms of the
azimuthal order $m$ since it depends on the mixed mixed frequency

$\zeta(\nu+\delta\nu_{\mathrm{rot}})\neq\zeta(\nu-\delta\nu_{\mathrm{rot}}),$
(9)

as discussed in Mosser et al. (2012).

Equations (3) and (8) provide a full description of the mode frequency and its
associated rotational splitting. To ensure a robust analysis of the mixed
modes, four independent studies were led by co-authors Appouchaux, Gehan,
Mosser, and Vrard. All followed the approach developed by Mosser et al.
(2015), where the first step consists of stretching the oscillation spectrum
to transform the mixed-mode pattern into a comb-like pattern based on the
gravity period spacing $\Delta\Pi_{1}$. Despite a common approach, each fitter
performed different analysis with independent routines. The following
subsections provide details for all.

####  4.2.2 Analysis adapted from Appourchaux (2020)

The stretched power spectrum is interpolated onto a regular grid for producing
the échelle diagram of Fig. 6 (top panel). The fitted asymptotic frequencies
of the multiplet of the dipole mixed modes are computed using the expressions
of Mosser et al. (2015) where $\zeta$ is replaced by the analytical
formulation given above. We note again that the mode splitting is not
symmetrical due to $\zeta$ depending on frequency. First guesses for
$\Delta\Pi_{1}$, rotation and $q$ are provided by the results of Mosser et al.
(2014, 2015, 2017), respectively. These three parameters are then visually
adjusted to obtain a figure similar to that of Fig. 6. The asymptotic
frequencies obtained by this procedure are very close to the actual peaks,
within $0.1\ \mu$Hz, easing the fitting of the peaks by Maximum Likelihood
Estimation (MLE). Due to the dense forest of dipole mixed modes, the fitting
window for these modes is $0.5\,\mu$Hz, while for $l=0,2$ and 3, it is 3
$\mu$Hz. Fitted frequencies are displayed in Table LABEL:tab:peak_bagging. The
échelle diagram of the power spectrum with the fitted frequencies is shown on
Fig. 7

In a second step, we determine all the parameters of the asymptotic expression
of the dipole mode frequencies, i.e. $\Delta\Pi_{1}$, q, $\epsilon_{g}$,
$\delta_{01}$, $\nu_{\mathrm{core}}$ and $\nu_{\mathrm{env}}$ by doing an
unweighted least squares fit of the fitted frequencies. The optimization is
performed by using the same algorithm as in Appourchaux (2020). Data were
trimmed to exclude three frequencies out of the 62 dipole mode frequencies
(see frequencies in Table LABEL:tab:peak_bagging). The root mean square value
of the optimized difference is $0.00\pm 0.03$ $\mu$Hz. The error bars were
then extracted by doing a Monte Carlo simulation using the error bars on the
dipole mode frequencies extracted from MLE. The optimization was then repeated
100 times. The values obtained are reported in Table 3.

The inclination angle of the star was extracted using a single mixed mode
(n=-94) for which the triplets was clearly fitted. We used the ratio between
heights of modes with different azimuthal orders (Gizon & Solanki 2003). The
ratio between the $m=0$ and the $m=\pm 1$ is 0.08${}^{+0.257}_{-0.060}$
providing an angle of $i=74.2^{+7.7}_{-14.1}$ degrees.

####  4.2.3 Analysis following Gehan et al. (2018)

Before the mixed-mode analysis, the frequency at maximum amplitude
$\nu_{\mathrm{max}}$ was derived by using the FRA pipeline. It computes a
smoothed power spectrum, then fits a Gaussian envelope accounting for
oscillations along with a background contribution to measure
$\nu_{\mathrm{max}}$. It also performs a likelihood ratio test to validate the
measured $\nu_{\mathrm{max}}$ and check whether oscillations are present. The
FRA pipeline was used in Huber et al. (2022) and is explained in detail in
Gehan et al. (2022). The large spacing $\Delta\nu$ was measured by using the
EACF.

The power spectrum was then stretched as in Mosser et al. (2015) and Gehan et
al. (2018). Pressure-dominated mixed modes were located using the RG universal
oscillation pattern (Mosser et al. 2011) and excluded, in order to keep only
gravity-dominated modes that are mostly sensitive to the core rotation rate.
The stretched period échelle diagram reveals that dipole gravity-dominated
mixed modes line up along two ridges, associated with the azimuthal orders
$m=\pm 1$ (Fig. 6, bottom). Fitting these ridges (Fig. 6) allows us to derive
the mean core rotational splitting, which is found to be
$\delta\nu_{\mathrm{rot,core}}=380\pm 6$ nHz (Gehan et al. 2018, for details).
Once the azimuthal order of mixed modes is identified, we measure the
inclination angle of the rotation axis from the ratio between heights of modes
with different azimuthal orders (Gizon & Solanki 2003). Only the ridges with
azimuthal orders $m=\pm 1$ are visible, but we know where to look for the
missing $m=0$ ridge in the background, at the midpoint between the $m=+1$ and
$m=-1$ ridges (Fig. 6, bottom). We obtain an inclination of
$i=75.8^{+14.2}_{-6.6}$ degrees. We refer to Gehan et al. (2021) for the
details of the inclination measurement procedure. We note that the maximal
possible value according to the uncertainties is $i=90^{\circ}$, so that the
star is possibly seen equator-on. This is because the $m=0$ ridge is lost in
the background, thus there is a possibility that the peaks with $m=0$ are
completely absent and that the observed signal is only due to background
noise.

####  4.2.4 Analysis following Mosser et al. (2018)

Stretching the power spectrum first requires a precise fitting of the radial
and quadrupole modes $l=0$ and 2, thanks to the RG universal oscillation
pattern, in order to locate precisely the expected pure pressure dipole modes.
Firstly, the modes with the largest height-to-background ratios (HBR) are used
to determine the parameters of the fit. Secondly, we pick modes with lower
HBRs, still larger than $5$, provided that they stick to the asymptotic
pattern within a frequency range wide six times the spectral resolution. We
applied the method of Mosser et al. (2018) for analyzing the rotational
splittings to overcome the difficulties arising from the overlap of rotational
multiplets larger than period spacings. The global seismic parameters derived
from the fit of the mixed-mode pattern, following the updated approach
described by Pinçon et al. (2019), are reported in Table 3. The frequency
analysis is displayed in Fig. 8.

####  4.2.5 Analysis adapted from Vrard et al. (2016)

The steps that follow the power-spectrum stretching differ from the analysis
led by co-author Mosser. We started by computing the Fourier transform of the
stretched spectrum to obtain a first $\Delta\Pi_{1}$ value following the work
of Vrard et al. (2016). After that, the coupling parameter $q$ was determined
following the method described in Mosser et al. (2017), which involves
searching for the maximum of the Fourier Transform of the stretched
oscillation spectrum. Finally, an iterative process was performed between
$\Delta\Pi_{1}$ and $q$: the $\Delta\Pi_{1}$ measurement was redone with the
measured $q$ value, then the inverse is performed until the $\Delta\Pi_{1}$
and $q$ values converge. The final measurement of $\Delta\Pi_{1}$ and $q$ is
obtained when those values converge. The final fitted parameters lead to
$\Delta\Pi_{1}=76.09\pm 0.72$ for the gravity mode period spacing and
$q=0.17\pm 0.05$ for the coupling parameter.

####  4.2.6 The big picture from the mixed mode analysis

The four independent approaches used to analyse the mixed dipole modes agree
(Table 3). Firstly, the measured mixed-mode period spacing
$\Delta\Pi_{1}\approx 76$ s associated with the large frequency p-mode spacing
$\Delta\nu\approx 10.5\,\mu$Hz, definitely classifies the star as an RGB
(Mosser et al. 2014). Secondly, the fact that the $m=0$ modes are not detected
indicates a large inclination angle $i$ of the RG rotation axis with respect
to the line of sight. The two authors who carefully analysed the ratio of the
amplitudes of the mixed $m=1$ to $m=0$ dipole modes report a very consistent
inclination angle of $\approx 75^{\circ}$, with a rather large uncertainty
ranging from $60^{\circ}$ to $90^{\circ}$. Thirdly, according to Mosser et al.
(2012, Eqs., 22 and 23), the frequency spacing measured on the mixed modes
provides a proxy of the mean core rotation period with the relation
$P_{\mathrm{core}}\approx 1/2\delta\nu_{\mathrm{core}}=15.1\pm 0.2$ days by
assuming an uncertainty of 5 nHz on $\delta\nu_{\mathrm{core}}$. The average
splitting of the envelope determined by co-author Appourchaux is about 10
times smaller, leading to a mean rotation period of $148_{-20}^{+27}$ days. If
we add the fact that no spot-related variability was measured from the Kepler
lightcurve and that the oscillation amplitude meet our expectations for that
type of RG, we can safely conclude that the RG is a regular slow rotator that
does not exhibit any sign of tidally increased spin.

##  5 Dynamical model

The combined photodynamical analysis of the Kepler photometry, the eclipse
timing variations of the inner binary, and the radial-velocity curve of the
outer giant component offer a robust dynamical determination of the mass of
the RG ($M_{\mathrm{A}}$) and of the total mass of the inner binary
($M_{\mathrm{bin}}$). It can be seen as follows. The amplitude of the RV curve
gives the spectroscopic mass function:

$f(M_{\mathrm{bin}})=M_{\mathrm{bin}}\left(\frac{q_{\mathrm{out}}}{1+q_{\mathrm{out}}}\right)^{2}\sin^{3}i_{\mathrm{out}},$
(10)

where $q_{\mathrm{out}}=M_{\mathrm{bin}}/M_{\mathrm{A}}$ and
$i_{\mathrm{out}}$ are the mass ratio and the inclination of the outer (or
wide) binary formed by the RG and the center of mass of the inner eclipsing
pair. In addition, Borkovits et al. (2016) showed that the ETV of the
eclipsing binary is strongly dominated by the gravitational three-body
perturbations against the pure light-travel time effect. Detailed analytical
investigations of third-body perturbed ETVs (Borkovits et al. 2011, 2016) have
shown that such curves tightly constrain almost all of the orbital elements of
both the inner and outer orbits, together with the mutual inclination of the
two orbital planes. Their amplitude is related to the outer mass ratio as

$\mathcal{A}_{\mathrm{dyn}}\propto\frac{1}{1+q_{\mathrm{out}}}.$ (11)

Finally, the variation of the eclipse depth and duration of the inner
eclipsing binary offer a very precise determination of the varying inclination
of both the inner orbit of the eclipsing pair and the outer orbit.

The combined photodynamical analysis was carried out with the software package
Lightcurvefactory (see, e. g. Borkovits et al. 2019, and further references
therein). This code contains a built-in numerical integrator to calculate the
three-body perturbed coordinates and velocities of the three bodies, a multi-
band light curve, ETV and RV curve emulators, and an MCMC-based parameter
search routine for the inverse problem. The latter employs an implementation
of the generic Metropolis-Hastings algorithm (see e. g. Ford 2005).

Figure 9: The long cadence Kepler lightcurve of KIC 7955301 together with the
joint photodynamical model lightcurve solution. Upper left panel: The complete
Q0-Q17 lightcurve (blue) with the bestfitted solution (red). This solution was
obtained with the negligation of the Doppler-boosting effect. The grey curve
show the same model after ‘switching on’ the Doppler-boosting. While this
effect, in theory, should have been observable with the accuracy of Kepler
photometry, the reasons of its negligence is discussed in the text. Upper
right panel: A 1-month-long section of the Kepler observations overplotted
with the photodynamical solution. The pale blue circles represent each
individual observations but, for the analysis only dark blue data (located
within the $\pm 0\aas@@fstack{p}02$ phase environment of each eclipses) were
considered. Lower panels show the residual data.

Figure 10: Left: radial velocity data (red) as a function of time expressed in
modified Julian dates, with their best fit model (blue line). Right: radial-
velocity data (red) folded over the RG orbital period as a function of the
orbital phase. The dispersion of the best-fit curve in blue is caused by the
varying shape of the RV curve as a function of time.

Figure 11: The eclipse timing variations of KIC 7955301 on different
timescales: Observations vs model and predictions. Left: The primary and
secondary ETV curves derived from Kepler-observations (red circles and blue
boxes, respectively), together with the photodynamical model solution
(smaller, pale red and blue symbols, connected with straight lines,
respectively). Right: ETV-like curves derived from the times of the inferior
and superior conjunctions of the secondary of the inner pair (pale red and
blue lines), overplotted with ‘true’, forecasted primary and secondary ETV
points (larger red and blue symbols) during the eclipsing phases of the inner
binary for the first half of the present century. The dynamically forced
short-period apsidal motion of the inner binary is nicely visible. The grey-
shaded area stands for the interval of the Kepler-measurements.  Table 4:
Orbital and astrophysical parameters of KIC 7955301 from the joint
photodynamical lightcurve, RV and ETV solution with and without the
involvement of the stellar energy distribution and PARSEC isochrone fitting.

| without SED+PARSEC | with SED+PARSEC
---|---|---
orbital elements
| subsystem
| Ba–Bb | A–B | Ba–Bb | A–B
$P$ [days] | $15.31831_{-0.00025}^{+0.00024}$ | $209.760_{-0.020}^{+0.020}$ | $15.31825_{-0.00023}^{+0.00023}$ | $209.761_{-0.018}^{+0.018}$
$a$ [R⊙] | $31.68_{-0.11}^{+0.19}$ | $217.0_{-0.9}^{+1.4}$ | $31.68_{-0.08}^{+0.11}$ | $216.9_{-0.5}^{+1.1}$
$e$ | $0.02763_{-0.00015}^{+0.00012}$ | $0.2733_{-0.0014}^{+0.0013}$ | $0.02757_{-0.00007}^{+0.00007}$ | $0.2737_{-0.0014}^{+0.0013}$
$\omega$ [deg] | $295.80_{-0.20}^{+0.22}$ | $117.02_{-0.40}^{+0.39}$ | $295.90_{-0.19}^{+0.18}$ | $116.99_{-0.39}^{+0.39}$
$i$ [deg] | $87.757_{-0.039}^{+0.037}$ | $84.23_{-0.25}^{+0.25}$ | $87.768_{-0.038}^{+0.034}$ | $84.40_{-0.32}^{+0.23}$
$\tau$ [BJD - 2400000] | $54961.5184_{-0.0083}^{+0.0090}$ | $54876.85_{-0.12}^{+0.12}$ | $54961.5225_{-0.0081}^{+0.0076}$ | $54876.84_{-0.11}^{+0.12}$
$\Omega$ [deg] | $0.0$ | $-5.28_{-0.17}^{+0.16}$ | $0.0$ | $-5.19_{-0.17}^{+0.16}$
$i_{\mathrm{m}}$ [deg] | $6.34_{-0.24}^{+0.26}$ | $6.18_{-0.23}^{+0.29}$
$\omega^{\mathrm{dyn}}$ [deg] | $239.7_{-1.4}^{+1.1}$ | $240.6_{-1.4}^{+1.2}$ | $239.2_{-1.4}^{+1.4}$ | $239.9_{-1.4}^{+1.5}$
$i^{\mathrm{dyn}}$ [deg] | $5.36_{-0.21}^{+0.23}$ | $0.97_{-0.04}^{+0.04}$ | $5.23_{-0.20}^{+0.25}$ | $0.95_{-0.04}^{+0.04}$
$\Omega^{\mathrm{dyn}}$ [deg] | $236.4_{-1.1}^{+1.3}$ | $56.4_{-1.1}^{+1.3}$ | $236.9_{-1.4}^{+1.4}$ | $56.9_{-1.4}^{+1.4}$
$i_{\mathrm{inv}}$ [deg] | $84.77_{-0.22}^{+0.21}$ | $84.91_{-0.27}^{+0.20}$
$\Omega_{\mathrm{inv}}$ [deg] | $-4.47_{-0.15}^{+0.14}$ | $-4.39_{-0.14}^{+0.13}$
mass ratio $[q=M_{\mathrm{sec}}/M_{\mathrm{pri}}]$ | $0.929_{-0.009}^{+0.010}$ | $1.399_{-0.012}^{+0.010}$ | $0.933_{-0.003}^{+0.003}$ | $1.406_{-0.013}^{+0.010}$
$K_{\mathrm{pri}}$ [km s-1] | $50.44_{-0.32}^{+0.37}$ | $31.61_{-0.12}^{+0.13}$ | $50.53_{-0.16}^{+0.13}$ | $31.65_{-0.08}^{+0.08}$
$K_{\mathrm{sec}}$ [km s-1] | $54.27_{-0.38}^{+0.46}$ | $22.58_{-0.16}^{+0.23}$ | $54.13_{-0.16}^{+0.24}$ | $22.50_{-0.10}^{+0.23}$
$\gamma$ [km s-1] | $-15.779_{-0.048}^{+0.050}$ | $-15.769_{-0.036}^{+0.036}$
stellar parameters
| Ba | Bb | A | Ba | Bb | A
Relative quantities
fractional radius [$R/a$] | $0.0285_{-0.0005}^{+0.0004}$ | $0.0256_{-0.0007}^{+0.0006}$ | $0.0241_{-0.0005}^{+0.0008}$ | $0.0285_{-0.0002}^{+0.0002}$ | $0.0259_{-0.0001}^{+0.0001}$ | $0.0263_{-0.0006}^{+0.0008}$
fractional flux [in $Kepler$-band] |  |  |  |  |  |
Physical Quantities
$M$ [$M_{\odot}$] | $0.941_{-0.011}^{+0.020}$ | $0.876_{-0.011}^{+0.012}$ | $1.298_{-0.019}^{+0.030}$ | $0.939_{-0.007}^{+0.011}$ | $0.877_{-0.008}^{+0.013}$ | $1.292_{-0.010}^{+0.026}$
$R$ [$R_{\odot}$] | $0.903_{-0.014}^{+0.012}$ | $0.809_{-0.022}^{+0.022}$ | $5.22_{-0.12}^{+0.22}$ | $0.901_{-0.007}^{+0.010}$ | $0.822_{-0.006}^{+0.006}$ | $5.70_{-0.13}^{+0.20}$
$T_{\mathrm{eff}}$ [K] | $5372_{-46}^{+84}$ | $5138_{-48}^{+73}$ | $4800$ | $5580_{-70}^{+44}$ | $5313_{-62}^{+38}$ | $4804_{-48}^{+26}$
$L_{\mathrm{bol}}$ [L⊙] | $0.611_{-0.035}^{+0.043}$ | $0.408_{-0.026}^{+0.034}$ | $13.00_{-0.58}^{+1.12}$ | $0.704_{-0.038}^{+0.038}$ | $0.480_{-0.022}^{+0.022}$ | $15.55_{-0.75}^{+0.87}$
$M_{\mathrm{bol}}$ | $5.27_{-0.07}^{+0.06}$ | $5.71_{-0.09}^{+0.07}$ | $1.95_{-0.09}^{+0.05}$ | $5.15_{-0.06}^{+0.06}$ | $5.57_{-0.05}^{+0.05}$ | $1.79_{-0.06}^{+0.05}$
$M_{V}$ | $5.45_{-0.09}^{+0.08}$ | $5.97_{-0.12}^{+0.08}$ | $2.36_{-0.09}^{+0.05}$ | $5.24_{-0.07}^{+0.07}$ | $5.73_{-0.06}^{+0.07}$ | $2.15_{-0.05}^{+0.06}$
$\log g$ [dex] | $4.503_{-0.014}^{+0.014}$ | $4.565_{-0.019}^{+0.024}$ | $3.115_{-0.026}^{+0.017}$ | $4.500_{-0.005}^{+0.004}$ | $4.551_{-0.003}^{+0.003}$ | $3.034_{-0.023}^{+0.019}$
$\log$(age) [dex] | $-$ | $9.65_{-0.03}^{+0.02}$
$[\mathrm{M/H}]$ [dex] | $-$ | $0.061_{-0.037}^{+0.028}$
$E(B-V)$ [mag] | $-$ | $0.09_{-0.02}^{+0.02}$
$(M_{V})_{\mathrm{tot}}$ | $2.26_{-0.09}^{+0.05}$ | $2.05_{-0.05}^{+0.07}$
distance [pc] | $-$ | $1369_{-29}^{+44}$

777Besides the usual observational system of reference related angular orbital
elements ($\omega$, $i$, $\Omega$), their counterparts in the system’s
invariable plane related dynamical frame of reference are also given
($\omega^{\mathrm{dyn}}$, $i^{\mathrm{dyn}}$, $\Omega^{\mathrm{dyn}}$).
Moreover, $i_{\mathrm{m}}$ denotes the mutual inclination of the two orbital
planes, while $i_{\mathrm{inv}}$ and $\Omega_{\mathrm{inv}}$ give the position
of the invariable plane with respect to the tangential plane of the sky (i.
e., in the observational frame of reference). Note, the instantaneous,
osculating orbital elements, are given for epoch $t_{0}=2454953.0000$ (BJD).

We simultaneously analysed the eclipse photometry from the Kepler light curve,
the primary and secondary ETVs deduced from the same light curve, and the RVs
of the outer RG component. For the Kepler-light curve, we only used the
$\phi\pm 0\aas@@fstack{p}02$ phase sections of each eclipses to reduce
computational costs. Except for preliminary tests, the following parameters
were adjusted with MCMC:

  * (i)

Nine of the twelve orbital parameters. Regarding the inner orbit of the
eclipsing pair, we fit $e_{1}\cos\omega_{1}$, $e_{1}\sin\omega_{1}$, and
$i_{1}$, where $e_{1}$ is the osculating eccentricity, $\omega_{1}$ the
argument of periastron and $i_{1}$ the orbital inclination. About the wide
outer orbit, besides $e_{2}\cos\omega_{2}$, $e_{2}\sin\omega_{2}$, and
$i_{2}$, we fit the anomalistic period $P_{2}$, the periastron passage time
$\tau_{2}$ and the longitude of the node $\Omega_{2}$. Regarding two of the
three “missing” parameters describing the inner orbit, namely, the period
$P_{1}$ and time of a primary eclipse or, with other words, the time of an
inferior conjunction of the secondary component of the inner pair
$\mathcal{T}^{\mathrm{inf}}_{1}$, they were constrained internally at each
trial step via the ETV curves in the manner described in Borkovits et al.
(2019). Finally, the third “missing” orbital parameter, the longitude of the
node of the inner orbit $\Omega_{1}$ was set to zero. It can be done because,
for the complete analysis, only the difference of the nodes, i. e.
$\Delta\Omega=\Omega_{2}-\Omega_{1}$ has relevance. Therefore, in such a
manner, by setting $\Omega_{2}$, one sets $\Delta\Omega$. Note that because
the motion of the three stars are not purely Keplerian, none of the orbital
parameters is constant. Therefore, all the trial values either adjusted,
constrained, or fixed in the initializing phase of any trial steps refer only
a given epoch $t_{0}$.

  * (ii)

Three parameters connected to the masses of the three stars: the RG mass
$M_{\mathrm{A}}$, the spectroscopic mass function $f(M_{\mathrm{B}})$, and the
mass ratio of the eclipsing binary $q_{1}=M_{\mathrm{Bb}}/M_{\mathrm{Ba}}$.

  * (iii)

Four parameters almost exclusively connected to the light curve. The first is
the duration of the primary eclipse around the epoch $t_{0}$ ($\Delta
t_{\mathrm{pri}}$), which is related to the sum of the fractional radii of the
members of the inner eclipsing binary, that is, the radius divided by the semi
major axis. The second is the ratio of the stellar radii of the inner pair
$R_{\mathrm{Bb}}/R_{\mathrm{Ba}}$. The third is the RG radius
$R_{\mathrm{A}}$. The fourth is the ratio of the effective temperatures of the
eclipsing pair $T_{\mathrm{Bb}}/T_{\mathrm{Ba}}$. We note that in case of a
single-band photometric observations, the EB lightcurve only carries
information about the temperature ratio – most strictly speaking, the
passband-dependent surface brightness ratio – of the two components rather
than their true temperatures. Therefore, one of the temperatures should be
taken from some external sources. In a single binary it can be obtained from
spectroscopic analysis or from stellar energy distribution (SED) data (see, e.
g. Miller et al. 2020). In the present situation, most of the total flux comes
from the third RG component which makes the classical approach to be
irrelevant. Therefore, we assume that the two components of the inner binary
are still on the main sequence, by being less massive than the RG, so that at
each trial step the software sets the effective temperature of the primary
component of the eclipsing pair $T_{\mathrm{Ba}}$ accordingly to the mass –
radius and mass – effective temperature relations of Tout et al. (1996) for
main sequence stars. About the RG component, we assume that its contribution
to the light curve is only a constant extra flux888The validity and the limits
of this assumption will be discussed later. Therefore, we fixed its
temperature to $T_{\mathrm{A}}=4\,800$ K for all runs. In such a way,
adjusting the RG radius $R_{\mathrm{A}}$ directly sets its bolometric
luminosity, and indirectly its total flux in the Kepler photometric band.
Hence, the adjustment of $R_{\mathrm{A}}$ was simply used to set the amount of
the extra flux to the eclipsing binary light curve or; in other terms, it
served as a substitute of the usual third light parameter.

Regarding other smaller effects that are mainly light-curve related, we used a
logarithmic limb-darkening law. The corresponding parameters were calculated
internally at each trial step with the use of the publicly available passband-
dependent tables of the Phoebe software (Prša & Zwitter 2005)999These tables
were downloaded from the Phoebe 1.0 Legacy page – http://phoebe-
project.org/1.0/download. For the well detached nature of both the inner and
the outer binaries and, therefore, for the practically spherical shape of the
stars, the gravity darkening coefficients have no detectable effect on the
light curve solution. Therefore, we did not consider the results of the
recent, more sophisticated study of Claret & Bloemen (2011), but simply
adopted the fixed value of $g=0.32$, which is based on the seminal model of
Lucy (1967). Finally, we neglected both the reradiation/illumination and the
Doppler-boosting effects (Loeb & Gaudi 2003; van Kerkwijk et al. 2010). While
the first effect has really no any influence on the light curve of our system,
this is not certainly true for the second one. In this regard, we note that,
as one can see in the left panel of Fig. 9, Doppler-boosting would result in
an $\sim 1000$ ppm amplitude variation in the out-of-eclipse flux level with
the period of the outer orbit, which, in theory, should be detected with
Kepler. Despite this fact, it cannot be seen in the lightcurves, because this
low amplitude variation, having period longer than two quarters was filtered
out during the data processing, which fact justifies its negligence in the
analysis.

The median values of the orbital and physical parameters of the triple system
derived from the MCMC posteriors and their 1-$\sigma$ uncertainties are
tabulated in Table 4. The observed versus model photometric, RV, and ETV
curves are plotted in Figs. 9–11.

##  6 Stellar evolution model

We modeled the system in three independent manners by using parameters that
are deduced from the observations. The first two model the RG component only,
by considering the seismic information and atmospheric parameters. The first
method (PARAM) makes use of the asteroseismic global parameters
$\nu_{\mathrm{max}}$, $\Delta\nu$, and $\Delta\Pi_{1}$ to optimize the model
from a grid of stellar evolution models. The second method makes use of the
individual frequencies of the radial modes and $\Delta\Pi_{1}$. The third
method is a complete model of the triple system, which includes both the RG
and the inner binary, but that does not make use of the asteroseismic
constraints.

###  6.1 Red-giant model based on the global seismic parameters

We used the code PARAM (Rodrigues et al. 2017) to infer the radius, mass, and
age of the RG component by using a combination of seismic and non-seismic
constraints. On the one hand, the average large frequency separation is
computed using the radial-mode frequencies of the models in the grid, not
added as an a-posteriori correction to the scaling relation between
$\Delta\nu$ and the square root of the stellar mean density. On the other
hand, $\nu_{\mathrm{max}}$ in the model grid is computed using a simple
scaling relation (Kjeldsen & Bedding 1995), by considering
$\nu_{\mathrm{max,\odot}}=3090\,\mu Hz$. The period spacing can also be
included in the list of seismic constraints, in which case it is compared with
the asymptotic value of $\Delta\Pi_{1}$ coming out of the models (see
Rodrigues et al. 2017). The grid of stellar evolution models used in PARAM is
the same as the reference grid adopted in Khan et al. (2019) and Miglio et al.
(2021) (G2).

We used the individual radial-mode frequencies in Table LABEL:tab:peak_bagging
to compute an average $\Delta\nu$ that can be directly compared with that from
the model grid, and obtain a value ($\Delta\nu=10.48$) compatible with that
obtained using the EACF (see Sec. 4). With the observational constraints
$\Delta\nu$, $\nu_{\mathrm{max}}$, $T_{\mathrm{eff}}$, and [M/H] we obtain
$R_{\mathrm{A}}=5.84\pm 0.09\,R_{\odot}$, $M_{\mathrm{A}}=1.27\pm
0.05\,M_{\odot}$, and an age of $4.9\pm 0.9$ Gyr. In Fig. 12 we show the
posterior probability density function (PDF) of radius, mass and age, compared
with the same resulting from the asteroseismic modeling based on individual
frequencies that is exposed in the next section (Sect. 6.2).

Figure 12: Mass, radius and age posterior probability distribution functions
derived from seismic constraints. Empty histograms show the distributions
obtained using PARAM, and the dotted lines show their associated $\pm 1\sigma$
credible regions. Solid histograms and shaded regions indicate the same
quantities obtained using the individual mode frequencies.

###  6.2 Red-giant model based on the oscillation frequencies

We generated models using the MESA code (Paxton et al. 2011, 2013, 2015, 2018,
2019) with diffusion and settling of heavy elements, with no core overshoot,
no mass loss, using an Eddington-grey atmosphere, and with chemical abundances
evaluated relative to the solar values of Grevesse & Sauval (1998).

The nonradial mixed modes in red giants can be described as combinations of
pure p- and g-modes — or ”$\pi$” and ”$\gamma$-modes”, in the sense of
Aizenman et al. (1977). We evaluate these pure $\pi$\- and $\gamma$-mode
frequencies using the modified wave operator construction described in Ong &
Basu (2020), which splits the standard wave operator into two complementary p-
and g-mode wave operators, each supporting only one type of wave propagation.
We solve for these mode frequencies numerically with the pulsation code GYRE
(Townsend & Teitler 2013). The observed p-dominated mixed modes of degree
$l\geq 2$ in red giants are very well approximated as pure $\pi$-modes, and as
such we match the model $\pi$-modes against the observed modes of degree
$l\geq 2$. Likewise, dipole modes far from the p-dominated mixed modes have
period spacings close to those of the pure $\gamma$-modes; accordingly, rather
than evaluating $\Delta\Pi_{1}$ using its asymptotic value, we compute it
directly from the pure dipole $\gamma$-modes.

The mode frequencies returned from stellar models are known to be
systematically offset from those which would be obtained from stars of
identical interior structure, owing to systematic errors in the modeling of
stellar surfaces — this is referred to as the asteroseismic ”surface term”. As
such, the model frequencies cannot be used directly to construct likelihood
functions against the observed mode frequencies in the usual fashion. Instead,
we use a nonparametric description — “$\epsilon$-matching”, via the algorithm
of Roxburgh (2016) — to characterize the discrepancy between the model and
observed mode frequencies in a surface-independent fashion. For this purpose
we use only the model $l=0,2,3$ frequencies, since those modes which we
observe are either radial p-modes, or can be approximated as being entirely
p-dominated; here we use the mode frequencies reported by co-author
Appourchaux. For the dipole modes, which are more strongly mixed, we used the
period spacing only, also as reported by co-author Appourchaux. Since the
g-mode cavity is well-localized into the stellar interior, no surface
correction is needed.

To supplement these seismic constraints, we used the spectroscopic constraints
from scenario C ([M/H] and $T_{\mathrm{eff}}$). These were used to compute
log-likelihood functions as

$\chi^{2}=\chi^{2}_{\mathrm{spec}}+\left(\chi^{2}_{\epsilon}+\chi^{2}_{\Delta\Pi_{1}}\right)/f,$
(12)

where $f$ is initially set to 1 when performing the optimization. Optimization
was performed using the differential-evolution algorithm as implemented in the
yabox package (Mier 2017), using the initial mass $M$, initial helium and
metal mass fractions $Y_{i}$ and $Z_{i}$, and the mixing-length efficiency
parameter $\alpha_{\text{MLT}}$ as the independent variables for this
optimization. After the optimization was complete, $f$ is set to the minimum
value of $\chi^{2}_{\mathrm{seis}}$ along the optimization trajectory, in
order to ensure that the reported results are not entirely dominated by the
seismic quantities, which are significantly more precisely constrained than
the spectroscopic ones.

We may consider the optimization trajectory to yield a series of nonuniformly
distributed samples of the likelihood function over the input parameter space.
As such, if we were to report estimates of stellar properties by naively
taking likelihood-weighted averages of the desired quantities over all the
sampled points, or find posterior distributions by constructing likelihood-
weighted histograms, we would be over-representing parts of the parameter
space where the density of samples is high to begin with. Effectively, the
sampling function of the optimization trajectory provides an implicit prior
distribution, which we estimate using a Gaussian kernel density estimator and
divide out to yield a nonuniformly sampled posterior distribution function
assuming a uniform prior. Using this uniform-prior posterior distribution, we
report the posterior median and $\pm 1\sigma$ quantiles of the following of
fundamental stellar properties:

$\displaystyle M_{\mathrm{A}}$ $\displaystyle=$ $\displaystyle
1.281^{+0.015}_{-0.004}\,M_{\odot}$ (13) $\displaystyle R_{\mathrm{A}}$
$\displaystyle=$ $\displaystyle 5.863^{+0.024}_{-0.011}\,R_{\odot}$ (14)
$\displaystyle Y_{i}$ $\displaystyle=$ $\displaystyle 0.291^{+0.010}_{-0.013}$
(15) $\displaystyle Z_{i}$ $\displaystyle=$ $\displaystyle
0.012^{+0.003}_{-0.001}$ (16) $\displaystyle\alpha_{\mathrm{MLT}}$
$\displaystyle=$ $\displaystyle 1.87^{+0.05}_{-0.09}$ (17) $\displaystyle\rho$
$\displaystyle=$ $\displaystyle 0.0089^{+0.00003}_{-0.00003}\,\mbox{g
cm}^{-3}$ (18) $\displaystyle\mathrm{age}$ $\displaystyle=$ $\displaystyle
3.43^{+0.62}_{-0.16}\mbox{Gyr}$ (19)

We compare these results with those obtained with the PARAM code, by plotting
the posterior distributions of the mass, radius and age in Fig. 12. The mass
and radius appear to be in agreement with the values obtained from only
$\Delta\nu$, $\nu_{\mathrm{max}}$, and $\Delta\Pi_{1}$ with the PARAM routine,
while the ages are somewhat different although still within the $1\sigma$
error. We discuss this difference in Sect. 7.2.

###  6.3 Overall system model based on classical parameters

We have compared the classical parameters of the three components of the
system with the isochrones and tracks from the YaPSI database (Spada et al.
2017); the results are summarized in Figure 13. In the two panels on the left
of the Figure, the measured masses and radii are compared with several
isochrones of appropriate metallicity, generated by interpolation from the
YaPSI database. The best fit is obtained for an age of $4.5\pm 0.2$ Gyr for
the primary component (top left panel); the components of the inner binary are
marginally compatible with this isochrone (bottom left). In the right panel of
Figure 13, we show the best-fitting isochrone, together with evolutionary
tracks constructed with the same code and input physics used for the YaPSI
models (see Spada et al. 2017 for details). For the primary star, both the
isochrone and the track are in good agreement with the observed radius and
luminosity (note that the isochrone and track for component A essentially
coincide on the RGB, as a result of the fast-paced evolution typical of this
evolutionary phase). The agreement is less good for the two components of the
inner binary. This could be ascribed to non-standard effects not taken into
account in our models.

Figure 13: Visual fit of observed classical parameters of the system with
theoretical tracks and isochrones from the YaPSI database (Spada et al. 2017).
Left: comparison in the mass radius plane with isochrones of different ages
for the primary (top panel), and the two components of the inner binary
(bottom). Right: comparison in the radius-luminosity plane; a 4.5-Gyr
isochrone and evolutionary tracks of appropriate mass and metallicity are
shown.

Figure 14: The evolutions of some of the orbital elements on a century-long
timescale, obtained via numerical integrations. Left: The variations of the
observable arguments of periastrons ($\omega_{1,2}$) of the inner and outer
orbits (black and red, respectively). The dynamically forced apsidal motions
of both orbits are well visible. The grey and orange shaded regions mark the
intervals of the Kepler and the ground-based spectroscopic follow up
observations, respectively. Right: The inclinations of the inner and outer
binaries (black and red curves, respectively). The green- and lightblue-shaded
horizontal areas denote the inclination domains of the inner and outer orbits
where eclipses can occur. The vertical grey area represents the domain of the
Kepler-observations, while the thin, black, vertical line denotes the interval
of the 27-day-long TESS observations (Sector 14). As one can see, during these
measurements the inner inclination ($i_{1}$) was far below the green line,
therefore, the inner pair did not present any eclipses. (Note, the dark blue
line simply stands for $i_{1,2}=90\degr$ to guide the eye.) Figure 15:
Photodynamical model light curve of KIC 7955301 for the present century. The
$\sim 19.2$ yr-period precession cycle is clearly visible. Within a cycle the
inner pair exhibits regular eclipses during a $\sim 7.3$ yr-long interval. The
last eclipsing session has been finished at 2nd January 2016 with a short,
$\sim 80$ ppm amplitude fading, while the next session is expected to begin at
18 January 2028. The grey vertical area represents the time of Kepler
observations, and TESS Sector 14 observations are also denoted with a thin
vertical line.

###  6.4 Model based on isochrones and SED

We carried out a second spectro-photodynamical study with the software package
Lightcurvefactory that includes the lightcurve, ETVs and RVs, as earlier, but
also the stellar energy distribution (SED), a stellar evolution code (PARSEC,
Bressan et al. 2012), and the Gaia-based accurate trigonometric distance. The
cumulative SED, which consists of pass-band magnitudes, drives the temperature
of sources, or at least of the dominant one. That way, effective temperatures
together with stellar masses and radii determine the locations of the stars on
co-eval PARSEC evolutionary tracks. Besides the lightcurve, RVs, and ETVs, the
publicly available multipassband magnitudes (see Table LABEL:tbl:mags) were
simultaneously fitted against the theoretical passband magnitudes interpolated
from theoretical PARSEC isochrone tables. We refer the reader to Borkovits et
al. (2020) for a detailed description of the method.

With this new model, the free parameters partially differ from Sect. 5.
Firstly, three new free parameters are added: the logarithm of the age; the
metallicity $[M/H]$ of the system; the interstellar extinction $E(B-V)$.
Furthermore, the distance of the system is constrained a posteriori at each
trial step, by minimizing the value of $\chi^{2}_{\mathrm{SED}}$. Secondly,
the stellar radii and temperatures that used to be free are no longer fitted.
However, the code computes them internally by interpolating the appropriate
PARSEC tables according to the trial values of the [mass, age, metallicity]
triplets at each trial runs. The results are tabulated in the last columns of
Table 4.

The comparison of the results from the purely dynamical analysis, which is
independent from stellar evolution models, and the analysis including
dynamical analysis, SED, and isochrones, produce output parameters that lie
within their 1-$\sigma$ uncertainties. This means that the inclusion of
stellar evolution models into the complex analysis did not lead to any
unavoided bias in the values of the physical properties of the system. It also
means that our model-independent lightcurve solution has led to dimensionless
quantities, like fractional radii and temperature ratios, that are fully
consistent with stellar evolutionary models.

The mass of the RG component $M_{\mathrm{A}}=1.30^{+0.03}_{-0.02}\,M_{\odot}$
is in excellent agreement with the asteroseismic analysis. The inner binary
components appear to be on the main sequence with masses
$M_{\mathrm{Ba}}=0.94_{-0.01}^{+0.02}\,\mathrm{M}_{\odot}$,
$M_{\mathrm{Bb}}=0.88\pm 0.01\,\mathrm{M}_{\odot}$, and radii of
$R_{\mathrm{Ba}}=0.90\pm 0.01\,\mathrm{R}_{\odot}$, $R_{\mathrm{Bb}}=0.81\pm
0.02\,\mathrm{R}_{\odot}$ in both solutions.

In contrast, the radius of the RG component $R_{\mathrm{A}}$ and the
temperatures $T_{\mathrm{Ba}}$, $T_{\mathrm{Bb}}$ of the inner stars differ by
about $5-10\%$ between the two models. This is expected because these
quantities are not functions of the lightcurve properties and of the dynamics
of the system. Only the ratio of the effective temperatures of the inner
binary $T_{\mathrm{Bb}}/T_{\mathrm{Ba}}$ is model-independent; this one is in
good agreement (within $1\%$) between the two models. In the purely dynamical
model (Sect. 5), the RG temperature was fixed to the available catalog value
$T_{\mathrm{A}}=4800$ K. Here, the combined dynamical, SED and isochrone model
confirms this value by finding $T_{\mathrm{A}}=4804_{-48}^{+26}$ K, which
within the uncertainty of the value we measured from the disentangled optical
spectra (Table 2). In the dynamical model, the temperature $T_{\mathrm{Ba}}$
of the primary star of the EB was internally set with the use of the zero age
main sequence mass-radius and mass-luminosity relations of Tout et al. (1996),
and the temperature ratio $T_{\mathrm{Bb}}/T_{\mathrm{Ba}}$ was an adjustable
parameter. In the combined model they were constrained from the appropriate
PARSEC isochrone. This latter model has resulted in higher temperatures by
$\sim 200$ K in accordance with the fact that KIC 7955301 is a quite old
system, having an age of $4.5\pm 0.2$ Gyr according to the present model.
Finally, the combined model predicts the RG radius to be of
$R_{\mathrm{A}}=5.70_{-0.13}^{+0.20}\,\mathrm{R}_{\odot}$, which is in
agreement with the asteroseismic value.

The stellar metallicity $[M/H]=0.06_{-0.04}^{+0.03}$ and interstellar
extinction $E(B-V)=0.09\pm 0.02$ obtained from the combined model are
consistent with catalog values (see Table LABEL:tbl:mags). Finally, the SED
fitting provides an estimate of the photometric distance of
$d=1369_{-29}^{+44}$ pc, which is in perfect agreement with trigonometric
distance of $d=1375\pm 35$ pc derived from the Gaia EDR3 catalog by Bailer-
Jones et al. (2021). However, we keep in mind that Gaia EDR3 data are not
corrected for binarity and, might suffer from systematic errors for such a
triple system.

About the dynamical evolution of the system, the small outer-to-inner period
ratio $P_{2}/P_{1}\approx 13.8$ puts the system among the category of compact
hierarchical triple systems. That being said, the small mutual inclination of
$i_{\mathrm{m}}=6\aas@@fstack{\circ}2\pm 0\aas@@fstack{\circ}3$ and almost
circular inner orbit ($e_{1}=0.0276\pm 0.0001$), the orbital configuration of
the system is stable in dynamical timescale. Consequently, the dynamical
evolution of the system will be driven by the stellar evolution of the evolved
RG component. Therefore, here we restrict our discussion for some short term
effects, and their observational consequences.

In Fig. 14 we plot the variations of the observable arguments of periastrons
($\omega_{1,2}$) and inclinations ($i_{1,2}$) of the inner and outer orbits
during the present century. The very fast, dynamically forced apsidal motion
can nicely be seen on the ETV curves determined from the 4-yr-long Kepler-
observations (see Fig. 11). An observationally more dramatic effect caused by
precession of the orbital plane of the inner binary is illustrated in Fig. 15.
As the orbital plane precesses with a half amplitude of
$i^{\mathrm{dyn}}_{1}=5\aas@@fstack{\circ}3\pm 0\aas@@fstack{\circ}2$ around
the invariable plane of the triple (having an inclination of
$i_{\mathrm{inv}}=84\aas@@fstack{\circ}8\pm 0\aas@@fstack{\circ}3$) with a
period of $P_{\mathrm{node}}\approx 19.2$ yr, the inner binary exhibits
eclipses (with varying eclipse depths) during a $\approx 7.3$ yr-long
interval, while in the remaining $\approx 11.9$ yr the system no longer
exhibits eclipses. Fortunately, Kepler observations caught the first half of
the latest eclipsing session and it led to the discovery of this very exciting
system. This eclipsing session has ended at the very beginning of 2016, and
KIC 7955301 will no exhibit any eclipses till the first days of 2028. In
accordance with our findings, TESS spacecraft did not observe any remarkable
light variations of this target when the original Kepler-field was revisited
three times from 2019 to 2022 (Fig. 16).

##  7 Concluding discussion

###  7.1 Dynamical versus seismic masses

With this paper, we illustrate the first detailed analysis of a hierarchical
triple system that includes an oscillating RG by combining asteroseismology,
spectroscopy and dynamical measurements. This system is an important benchmark
for calibrating the measurement of RG masses with asteroseismology. Thanks to
the acquisition of 23 high-resolution spectra with the ARCES spectrometer at
APO we were able to first classify the system as an SB1 and monitor the RV
shift of the RG component along its orbit. Then, with the dynamical model
solution, we could disentangle and co-add the optical spectra to derive the
atmospheric parameters of the three components, even though the parameters of
the inner binary have a low S/N. By fixing the RG surface gravity to that
found by the complete Lightcurvefactory model, we were able to determine the
RG temperature and metallicity at $T_{\mathrm{A}}=4720\pm 105$ K and
[M/H]$=-0.01\pm 0.12$ (Table 2).

Figure 16: TESS observations of KIC 7955301 from 2019 to 2022. Top panel: time
series processed with the software package FITSH (Pál 2012). Bottom panel:
power spectral density of the times series for sector 14 only. The vertical
red dot-dashed line indicates $\nu_{\mathrm{max}}$ and the vertical blue
dashed line indicates Kepler’s Nyquist frequency.

We computed the asteroseismic mass of the RG in three different manners. The
first and most basic one, which consists of applying the asteroseismic scaling
law (SL) tuned for RG stars by Mosser et al. (2013) from the values of
$\nu_{\mathrm{max}}$, $\Delta\nu$, and $T_{\mathrm{A}}$, leads to
$M_{\mathrm{A,SL}}=1.27\pm 0.05\ M_{\odot}$. The second makes use of the
(PARAM) approach that fits the global asteroseismic parameters, including
$\Delta\Pi_{1}$, and by including the metallicity on a grid of stellar-
evolution models. The mass appears to be the exact same as with the
asteroseismic scaling relation: $M_{\mathrm{A,\texttt{PARAM}}}=1.27\pm
0.05\,M_{\odot}$. Finally, the most sophisticated approach based on fitting
individual oscillation frequencies leads to a mass of
$M_{\mathrm{A}}=1.281^{+0.015}_{-0.004}\,M_{\odot}$.

The RG mass that we derive from the purely dynamical modeling – independent
from both asteroseismology and stellar evolution models – is
$1.30^{+0.03}_{-0.02}\,M_{\odot}$. The uncertainty is thus about 2 %, which is
better than what can be obtained from asteroseismology or stellar evolution
models based on atmospheric parameters, luminosity and parallaxes. This type
of system, a solar-like oscillator in an SB1 hierarchical triple system,
should definitely be considered for helping to calibrate the asteroseismic
masses. About ten of them are available in Gaulme et al. (2013) and Gaulme &
Guzik (2019), even though none of them exhibit such clear ETVs, which means
that a 2 % accuracy on mass may not be achievable with those.

It is interesting to notice that in this system the asteroseismic mass matches
the dynamical one, contrary to what was observed with most RGs in EBs (Gaulme
et al. 2016; Brogaard et al. 2018; Benbakoura et al. 2021), where
asteroseismology appeared to overestimate masses by about 15 % on average.
This puzzling fact may just be a case that the sample of RG oscillators with
accurate independent mass estimates is small and the dispersion of the
observed overestimation is large, but it could also highlight issues with RGs
in EBs where the companion is a MS dwarf star. For an RG in an EB, the mass of
the RG is driven by the low S/N Doppler shift that is measured by tracking the
absorption lines of the companion star. In most cases, the flux coming from
the companion is less than 5 % of the total flux. We also note that a good
agreement between dynamical and seismic masses was found with the double RG
KIC 9246715 (Rawls et al. 2016), where there were no issues of noisy RVs.

In contrast, KIC 7955301 is a special target with respect to the bulk of RGs
in EBs because it is the least evolved and thus smallest RG ($\approx
5.85\,R_{\odot}$) for which we have an independent mass measurement. The radii
of all of the RGs in EBs listed in Benbakoura et al. (2021), for example,
range from 8 to $14R_{\odot}$. In addition a significant fraction of them are
red clump stars. So far, no significant overestimation of stellar masses and
radii by asteroseismology was observed for MS and subgiant stars. We may have
an intermediate case here, for which asteroseismic measurements are unbiased.
To clarify the question, our recommendation is to take new RV measurements of
the RG/EBs listed in Benbakoura et al. (2021) and Gaulme & Guzik (2019) with
larger S/N ratios, that is longer exposure times, and with spectrometers that
are dedicated to high-precision RV measurements.

###  7.2 Age of the system

One of the goals of the future ESA PLATO mission (Rauer et al. 2014) is to
provide ages of solar-like MS stars hosting exoplanets with an accuracy of 10
%. Many papers dealing with stellar physics report ages arising from stellar
evolution codes with a similar precision, especially when asteroseismic
measurements are available. With this paper, we could have run a single
stellar evolution code and provided an age with a relatively low uncertainty.
We decided to let four different modelers estimate the age in an independent
fashion. We obtain $4.9\pm 0.9$ Gyr with PARAM, $3.43^{+0.62}_{-0.16}$ Gyr
with MESA, $4.5\pm 0.2$ Gyr with YaPSI, and $4.5\pm 0.2$ Gyr with
Lightcurvefactory, that is 18.4, 22.7, 4.4, and 4.4 % respectively. Both the
YaPSI and Lightcurvefactory estimates have a better precision likely because
the three components of the system are included into the optimization process.
Ages determined for the RG only (component A), with the help of asteroseismic
and classical parameters, have uncertainties of about 20 %, despite RGs are
usually considered to be good cases for age estimate thanks to their fast
evolution along the RG branch.

Our age estimate thus ranges from 3.3 to 5.8 Gyr, which is quite broad given
the unusually vast amount of information that we have. By looking more
closely, the age is consistent between the three “standard” approaches that
use the classical parameters (mass, metallicity, effective temperature), and
the global asteroseismic parameters (PARAM only). On the contrary, the model
based on reproducing the individual oscillation frequencies with MESA stands
apart. Even though a deeper understanding of the differences between the age
estimates goes beyond the scope of the paper, we can already draw some
conclusions. A careful inspection of the posterior probability distribution
(PDF) of the mass, radius and age (Fig. 12) shows a good agreement between the
mass and radius obtained with PARAM and MESA, whereas the age PDF appears to
be bimodal with MESA. In fact, the secondary peak of the age distribution
matches the PARAM estimate (at $\approx 4.5$ Gyr). A key aspect of the MESA-
model based optimization is that the helium abundance $Y$ and metallicity $Z$
are two free independent parameters of the model. The resulting $Y$ value of
0.29 is quite large by considering the slightly sub-solar value found for the
metallicity ($Z\approx 0.012$) and assuming a linear $Y=Y(Z)$ relation, and
that the Sun falls on that relation (see Fig A.4 Miglio et al. 2021). A
difference of about 0.02 in $Y$ with respect to what we have in the grid used
in PARAM, would mean a difference of $\sim 15\,\%$ in age which, considering
also the slight difference in mass, could explain the different age estimate.

###  7.3 Formation of the system

Physical properties and orbital configuration carry information about the
formation process of the systems. We refer the reader to Borkovits et al.
(2022) and references therein, where a connection between dynamical and
orbital parameters and formation scenarios is discussed. An outer eccentricity
of 0.27 for a 209-day period outer orbit is far from being exceptional. For
example, Borkovits et al. (2016) reported 16 triple star candidates with
smaller outer period than that of KIC 7955301 and, 6 of them have larger outer
eccentricities. Since then, further highly eccentric systems amongst
substantially more compact triply eclipsing triples were found in TESS data
(Borkovits et al. 2022).

With an outer semi-major axis less than 10 AU, KIC 7955302 is considered to be
a compact triple system. The formation of such type of systems involves either
a fragmentation of the accretion disk, or a disk-mediated capture of the outer
component. In both cases the outer star forms by being less massive than the
inner binary. Then, thanks to its wider and eccentric orbit it sweeps out a
larger orbit and thus accretes most of the infalling material from the
accretion disk, which eventually leads $q_{\mathrm{out}}$ toward larger
values. Although it is expected that the total mass of the inner pair remains
larger than the mass of the wide component by the end of the complete
formation process, the third component may actually become the most massive
star of the system (Tokovinin 2021), which is the case of KIC 7955301.

###  7.4 Evolution of the system

In the short term, we have shown that the inner binary components eclipse each
other over a period of about 7.3 years, and stop eclipsing for the next 11.9
years. The RG will never eclipse the inner binary as seen from the Earth. This
is why we were able to study the eclipses of the inner binary from the Kepler
data and that we are not able to do so with TESS. The eclipses should be
visible again during the ESA PLATO mission, whose nominal operations should
happen between 2027 and 2031. The PLATO observations will certainly help fine-
tune the model of the system.

Over the long term and in general, stars are likely to merge in close
multiple-star systems when the most massive component reaches the tip of the
RG branch. This is especially true for low-mass stars that ignite helium only
at the tip of the red giant branch, after reaching very large radii ($\sim
200\ R_{\odot}$). RGs may swallow planets or stars, converting their orbital
momentum into spin and friction loss. There is indirect observational evidence
for it. For example, about 3 % of the solar-mass stars on the red-giant branch
and 11 % of the helium-burning stars observed by Kepler display unusually fast
rotation and magnetic fields (e.g., Tayar et al. 2015; Ceillier et al. 2017;
Tayar & Pinsonneault 2018; Gaulme et al. 2020). This means that a fraction of
the RGs in that mass range gains angular momentum between the RGB and the RC.
Until alternative explanations are provided, a likely possibility is that
these stars have engulfed a stellar or substellar companion. Recently, Price-
Whelan et al. (2020) reported a notable dearth of close companions around the
red clump – where companions may have been engulfed when the primary star
ascended the upper giant branch – in the color-magnitude diagram produced by
the APO Galactic Evolution Experiment (APOGEE, Majewski et al. 2017), which is
part of the Sloan Digital Sky Survey IV (e.g., Blanton et al. 2017). The
orbital configuration of KIC 7955301 represents a borderline case for this
type of scenario. The current distance at periastron between the giant and the
inner binary is about $\approx 142\,R_{\odot}$ according to the data from
Table 4.

Figure 17: Top panel: radius evolution of the three components according to
the MIST tracks (with masses and metallicity set equal to their nominal values
from Table 4). The semi-major axis of the inner binary is indicated by the
black dotted line, and the semi-major axis of the outer orbit with dashed
lines. Bottom panel: zoom on the RG component around its RGB and AGB phases.

We can formulate some broad predictions for the long-term evolution of the
system, assuming that the dynamical interaction among the components can be
neglected until they come into contact with each other because of post-MS
evolution. Figure 17 shows the evolution of the radii of the three components
according to evolutionary tracks generated with the MIST web interpolator
(Choi et al. 2016), setting the masses to the central values given in Table 4
(1.29, 0.94, and 0.88 $M_{\odot}$ for components A, Ba, and Bb, respectively),
and the metallicity to +0.06. Significant mass loss takes place during the RGB
and AGB phases, and as a result all three stars reach a final mass of about
$0.5\,M_{\odot}$ by the end of their respective AGB phases. Stellar rotation
is not included in these models. At an age of approximately 5 Gyr, component A
(already a red giant at present time) reaches the RGB tip, and shortly after
(relative to the $10^{9}$ yr time scale) the AGB tip and thermal pulses phase.
The peak radius reached at the tip of the RGB ($\approx 160\,R_{\odot}$) is
mostly equal to the distance at periastron, while the peak radius on some of
the AGB pulses is significantly larger than the semi-major axis of the outer
orbit, and thus merging can be expected to occur at one of those times. Should
merging of component A with the inner binary be avoided because of some
process not considered in this crude analysis, at around 15 Gyr component Ba
will evolve onto the RGB and merge with Bb. In the light of this discussion,
it is comforting that none of our age estimates is larger than 5 Gyr.

###### Acknowledgements.

This paper includes data collected by the Kepler mission. Funding for the
Kepler mission is provided by the NASA Science Mission directorate. Some of
the data presented in this paper were obtained from the Mikulski Archive for
Space Telescopes (MAST). STScI is operated by the Association of Universities
for Research in Astronomy, Inc., under NASA contract NAS5-26555. This work is
also based on observations obtained with the Apache Point Observatory
3.5-meter telescope, which is owned and operated by the Astrophysical Research
Consortium. Part of our spectroscopic observations were done at the
Observatoire de Haute-Provence. We acknowledge the technical team for their
onsite support as well as the “Programme National de Physique Stellaire”
(PNPS) of CNRS/INSU for their financial support. P.G. and F.S. were supported
by the German space agency (Deutsches Zentrum für Luft- und Raumfahrt) under
PLATO data grant 50OO1501. P.G. and J.J. acknowledge NASA grant NNX17AF74G for
partial support. The research leading to these results has (partially)
received funding from the KU Leuven Research Council (grant C16/18/005:
PARADISE) and from the BELgian federal Science Policy Office (BELSPO) through
PRODEX grant PLATO. M.B. acknowledges support from the “Deutscher Akademischer
Austauschdienst” (DAAD) and the Université Paris Diderot.

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##  Appendix A Radial velocity data

Table 5 contains the radial velocities obtained at Apache Point and Haute-
Provence observatories.

Table 5: Radial velocity data obtained at Apache Point and Haute-Provence
observatories.

Date | RV
---|---
(KJD) | km s-1
1569.823526 | -28.82
1591.832861 | -12.50
1611.826703 | -1.44
1648.831533 | 9.92
1704.641327 | -18.59
1711.645679 | -30.04
1737.619563 | -53.19
1766.572107 | -36.50
1958.878300 | -48.28
3090.825370 | 4.65
3189.617802 | -50.82
3195.587236 | -53.95
3203.595970 | -52.70
3216.717021 | -45.36
3217.631558 | -43.57
3219.657836 | -42.14
3416.917213 | -49.39
3444.847760 | -28.94
3445.534184⋆ | -27.26
3446.421966⋆ | -26.49
3542.669669 | 8.53
3557.755987 | 3.53
3559.754443 | 3.12
3568.344058⋆ | -2.24
3616.635254 | -54.58
3617.597411 | -54.06
101010Dates are mid-exposure times expressed in Kepler Julian dates (KJD).
Kepler Julian dates KJD are related to barycentric Julian dates BJD by KJD =
BJD $-$ 2,454,833 days. The first spectrum was taken on April 20th, 2013, and
the last on November, 28th, 2018. Measurement errors are estimated to be 0.5
km s-1. Date with an $\star$ sign are data taken at OHP.

##  Appendix B Oscillation frequencies

Table LABEL:tab:peak_bagging displays the oscillation frequencies of the RG
component, as performed by co-authors Appourchaux and Mosser. 1]

Table 6:  Oscillation frequencies of KIC 7955301 by co-authors Appourchaux and Mosser (COR method). The columns report $\nu_{\mathrm{as}}$ as asymptotic fit ($\mu$Hz); $\nu_{\mathrm{obs}}$ are the observed peak ($\mu$Hz) with uncertainties; $nm$ is the proxy of the mixed order ($nm=np-ng$ ; 0 for p-modes); zeta = zeta factor of mixed modes (Mosser+2015, 2018), and HBR is the height-to-background ratio).  | Appourchaux | Mosser
---|---|---
$l$ | $m$ | $n$ | $nm_{\mathrm{TA}}$ | $nm_{\mathrm{BM}}$ | $\nu_{\mathrm{obs,TA}}$ | err | $\nu_{\mathrm{obs,BM}}$ | err | $\nu_{\mathrm{as}}$ | $\zeta$ | HBR
|  |  |  |  |  $\mu$Hz |  $\mu$Hz |  $\mu$Hz |  $\mu$Hz |  $\mu$Hz |  |
0 | 0 | 7 | 8 | 87 | 0.057 | … | … | … | … | … | …
0 | 0 | 8 | 9 | 0 | 96.918 | 0.027 | 96.905 | 0.063 | 96.593 | 0.000 | 17.4
0 | 0 | 9 | 10 | 0 | 106.921 | 0.009 | 106.917 | 0.009 | 106.920 | 0.000 | 47.4
0 | 0 | 10 | 11 | 0 | 117.396 | 0.013 | 117.402 | 0.018 | 117.323 | 0.000 | 110.2
0 | 0 | 11 | 12 | 0 | 127.851 | 0.167 | 127.753 | 0.013 | 127.802 | 0.000 | 127.9
0 | 0 | 12 | 13 | 0 | 138.268 | 0.018 | 138.230 | 0.027 | 138.357 | 0.000 | 50.2
0 | 0 | 13 | 14 | 0 | 149.021 | 0.004 | 148.943 | 0.013 | 148.987 | 0.000 | 21.1
0 | 0 | 14 | 15 | … | 159.718 | 0.059 | … | … | … | … | …
1 | -1 | … | -136 | … | 90.142 | 0.002 | … | … | … | … | …
1 | 1 | … | -132 | … | 92.323 | 0.022 | … | … | … | … | …
1 | -1 | … | -132 | … | 92.626 | 0.052 | … | … | … | … | …
1 | 1 | 8 | … | -126 | … | … | 98.377 | 0.014 | 98.371 | 0.989 | 5.4
1 | 1 | 8 | -120 | … | 101.235 | 0.004 | … | … | … | … | …
1 | -1 | 8 | -120 | -121 | 101.310 | 0.004 | 101.305 | 0.017 | 101.277 | 0.825 | 19.5
1 | -1 | 8 | -119 | -120 | 101.874 | 0.006 | 101.880 | 0.026 | 101.831 | 0.614 | 64.9
1 | 1 | 8 | -119 | … | 101.881 | 0.006 | … | … | … | … | …
1 | 0 | 8 | -119 | … | 102.112 | 0.001 | … | … | … | … | …
1 | 1 | 8 | -118 | … | 102.342 | 0.015 | … | … | … | … | …
1 | -1 | 8 | -118 | -119 | 102.401 | 0.006 | 102.391 | 0.011 | 102.390 | 0.825 | 14.5
1 | -1 | 8 | -117 | … | 103.061 | 0.002 | … | … | … | … | …
1 | 1 | 8 | -117 | … | 103.103 | 0.005 | … | … | … | … | …
1 | 1 | 8 | … | -118 | … | … | 103.840 | 0.014 | 103.853 | 0.973 | 7.3
1 | -1 | … | -113 | … | 106.245 | 0.000 | … | … | … | … | …
1 | 1 | … | -113 | … | 106.384 | 0.009 | … | … | … | … | …
1 | 1 | … | -112 | … | 107.164 | 0.012 | … | … | … | … | …
1 | -1 | 9 | -110 | -111 | 109.059 | 0.001 | 109.051 | 0.012 | 109.066 | 0.985 | 14.1
1 | 1 | 9 | -109 | -111 | 109.807 | 0.002 | 109.806 | 0.010 | 109.816 | 0.977 | 20.5
1 | 1 | 9 | -108 | -110 | 110.698 | 0.001 | 110.696 | 0.010 | 110.705 | 0.949 | 24.2
1 | -1 | 9 | -108 | -109 | 110.863 | 0.002 | 110.837 | 0.021 | 110.868 | 0.938 | 5.6
1 | 1 | 9 | … | -109 | … | … | 111.554 | 0.012 | 111.550 | 0.823 | 10.6
1 | -1 | 9 | -107 | -108 | 111.717 | 0.001 | 111.719 | 0.016 | 111.693 | 0.770 | 34.8
1 | 1 | 9 | -107 | -108 | 112.305 | 0.013 | 112.254 | 0.035 | 112.188 | 0.567 | 59.1
1 | -1 | 9 | -106 | -107 | 112.369 | 0.016 | 112.325 | 0.016 | 112.299 | 0.584 | 59.3
1 | 0 | 9 | -118 | -107 | 102.592 | 0.001 | 112.514 | 0.021 | 112.547 | 0.699 | 9.2
1 | 1 | 9 | -106 | -107 | 112.862 | 0.004 | 112.876 | 0.022 | 112.836 | 0.822 | 34.9
1 | -1 | 9 | -105 | -106 | 113.013 | 0.011 | 113.041 | 0.016 | 113.014 | 0.870 | 26.1
1 | 1 | 9 | -105 | -106 | 113.699 | 0.010 | 113.703 | 0.012 | 113.718 | 0.949 | 20.3
1 | -1 | 9 | -104 | -105 | 113.913 | 0.007 | 113.931 | 0.011 | 113.933 | 0.959 | 9.1
1 | 1 | 9 | -104 | -105 | 114.624 | 0.002 | 114.655 | 0.013 | 114.673 | 0.976 | 15.8
1 | -1 | 9 | … | -103 | … | … | 115.938 | 0.014 | 115.914 | 0.986 | 45.1
1 | 1 | 9 | … | -103 | … | … | 116.647 | 0.017 | 116.668 | 0.988 | 6.1
1 | -1 | 10 | … | -102 | … | … | 116.953 | 0.011 | 116.941 | 0.988 | 14.6
1 | -1 | 10 | … | -101 | … | … | 117.977 | 0.015 | 117.988 | 0.988 | 5.2
1 | 1 | 10 | … | -101 | … | … | 118.717 | 0.018 | 118.741 | 0.986 | 5.4
1 | -1 | 10 | -99 | -100 | 119.028 | 0.002 | 119.032 | 0.013 | 119.051 | 0.984 | 31.0
1 | 1 | 10 | -99 | -100 | 119.782 | 0.002 | 119.779 | 0.015 | 119.801 | 0.978 | 9.1
1 | -1 | 10 | … | -99 | … | … | 120.157 | 0.019 | 120.127 | 0.974 | 9.3
1 | 1 | 10 | -98 | -99 | 120.854 | 0.003 | 120.858 | 0.010 | 120.864 | 0.954 | 21.1
1 | -1 | 10 | -97 | -98 | 121.186 | 0.008 | 121.180 | 0.015 | 121.199 | 0.935 | 7.1
1 | 0 | 10 | … | -98 | … | … | 121.558 | 0.014 | 121.549 | 0.899 | 6.3
1 | 1 | 10 | -97 | -98 | 121.893 | 0.004 | 121.897 | 0.011 | 121.881 | 0.831 | 94.6
1 | -1 | 10 | -96 | -97 | 122.198 | 0.004 | 122.196 | 0.015 | 122.170 | 0.717 | 110.0
1 | 0 | 10 | … | -97 | … | … | 122.385 | 0.020 | 122.417 | 0.586 | 25.2
1 | 1 | 10 | -96 | -97 | 122.695 | 0.007 | 122.708 | 0.044 | 122.622 | 0.523 | 275.3
1 | -1 | 10 | -95 | … | 122.891 | 0.009 | … | … | … | … | …
1 | 0 | 10 | … | -96 | … | … | 123.030 | 0.027 | 123.078 | 0.707 | 15.7
1 | 1 | 10 | -95 | -96 | 123.394 | 0.000 | 123.384 | 0.012 | 123.369 | 0.823 | 45.7
1 | -1 | 10 | -94 | -95 | 123.685 | 0.022 | 123.723 | 0.009 | 123.721 | 0.897 | 35.3
1 | 0 | 10 | -94 | -95 | 124.054 | 0.006 | 124.054 | 0.013 | 124.069 | 0.933 | 14.4
1 | 1 | 10 | -94 | -95 | 124.409 | 0.001 | 124.416 | 0.010 | 124.430 | 0.953 | 72.8
1 | -1 | 10 | -93 | -94 | 124.812 | 0.001 | 124.809 | 0.015 | 124.834 | 0.966 | 28.1
1 | 0 | 10 | … | -94 | … | … | 125.187 | 0.017 | 125.204 | 0.973 | 5.1
1 | 1 | 10 | -93 | -94 | 125.555 | 0.007 | 125.557 | 0.014 | 125.577 | 0.978 | 15.7
1 | -1 | 10 | … | -93 | … | … | 125.982 | 0.018 | 126.011 | 0.981 | 10.8
1 | -1 | 10 | -91 | -92 | 127.197 | 0.002 | 127.194 | 0.016 | 127.221 | 0.986 | 30.5
1 | 0 | 10 | … | -92 | … | … | 127.627 | 0.017 | 127.598 | 0.986 | 40.7
1 | 1 | 11 | … | -91 | … | … | 129.217 | 0.010 | 129.209 | 0.983 | 16.2
1 | -1 | 11 | -89 | -90 | 129.760 | 0.051 | 129.689 | 0.015 | 129.714 | 0.980 | 28.7
1 | 0 | 11 | -89 | … | 129.919 | 0.000 | … | … | … | … | …
1 | 1 | 11 | -89 | -90 | 130.442 | 0.002 | 130.437 | 0.014 | 130.460 | 0.972 | 33.4
1 | -1 | 11 | -88 | -89 | 130.954 | 0.005 | 130.957 | 0.016 | 130.982 | 0.961 | 16.4
1 | 1 | 11 | -88 | -89 | 131.653 | 0.026 | 131.697 | 0.012 | 131.704 | 0.924 | 9.5
1 | -1 | 11 | -87 | -88 | 132.191 | 0.001 | 132.193 | 0.013 | 132.212 | 0.854 | 101.6
1 | 0 | 11 | -87 | -88 | 132.850 | 0.024 | 132.531 | 0.015 | 132.518 | 0.760 | 7.4
1 | 1 | 11 | -87 | -88 | 132.800 | 0.009 | 132.807 | 0.014 | 132.785 | 0.628 | 114.0
1 | -1 | 11 | -86 | -87 | 133.083 | 0.017 | 133.051 | 0.025 | 133.095 | 0.486 | 27.4
1 | 0 | 11 | … | -87 | … | … | 133.137 | 0.075 | 133.286 | 0.509 | 22.9