# TESS Asteroseismic Analysis of HD 76920: The Giant Star Hosting An Extremely
Eccentric Exoplanet

Chen Jiang(姜晨) Max-Planck-Institut für Sonnensystemforschung, Justus-von-
Liebig-Weg 3, 37077 Göttingen, Germany  Tao Wu (吴涛) Yunnan Observatories,
Chinese Academy of Sciences, 396 Yangfangwang, Guandu District, Kunming,
650216, People’s Republic of China  Key Laboratory for the Structure and
Evolution of Celestial Objects, Chinese Academy of Sciences, 396 Yangfangwang,
Guandu District, Kunming, 650216, People’s Republic of China  Center for
Astronomical Mega-Science, Chinese Academy of Sciences, 20A Datun Road,
Chaoyang District, Beijing, 100012, People’s Republic of China  University of
Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China
Adina D. Feinstein NSF Graduate Research Fellow  Department of Astronomy and
Astrophysics, University of Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA
Keivan G. Stassun Department of Physics and Astronomy, Vanderbilt University,
Nashville, TN 37235, USA  Timothy R. Bedding Sydney Institute for Astronomy
(SIfA), School of Physics, University of Sydney, NSW 2006, Australia  Stellar
Astrophysics Centre (SAC), Department of Physics and Astronomy, Aarhus
University, Ny Munkegade 120, 8000 Aarhus C, Denmark  Dimitri Veras Centre for
Exoplanets and Habitability, University of Warwick, Coventry CV4 7AL, UK
Centre for Space Domain Awareness, University of Warwick, Coventry CV4 7AL, UK
Department of Physics, University of Warwick, Coventry CV4 7AL, UK  Enrico
Corsaro INAF — Osservatorio Astrofisico di Catania, via S. Sofia 78, 95123
Catania, Italy  Derek L. Buzasi Department of Chemistry & Physics, Florida
Gulf Coast University, 10501 FGCU Boulevard S., Fort Myers, FL 33965, USA
Dennis Stello School of Physics, University of New South Wales, NSW 2052,
Australia  Sydney Institute for Astronomy (SIfA), School of Physics,
University of Sydney, NSW 2006, Australia  Stellar Astrophysics Centre (SAC),
Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000
Aarhus C, Denmark  Yaguang Li(李亚光) Sydney Institute for Astronomy (SIfA),
School of Physics, University of Sydney, NSW 2006, Australia  Stellar
Astrophysics Centre (SAC), Department of Physics and Astronomy, Aarhus
University, Ny Munkegade 120, 8000 Aarhus C, Denmark  Savita Mathur Instituto
de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain
Universidad de La Laguna (ULL), Departamento de Astrofísica, E-38206 La
Laguna, Tenerife, Spain  Rafael A. García Université Paris-Saclay, Université
Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France  Sylvain N. Breton
Département d’Astrophysique/AIM, CEA/IRFU, CNRS/INSU, Univ. Paris-Saclay &
Univ. de Paris, 91191 Gif-sur-Yvette, France  Mia S. Lundkvist Stellar
Astrophysics Centre (SAC), Department of Physics and Astronomy, Aarhus
University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark  Przemysław J.
Mikołajczyk Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4,
00-478 Warsaw, Poland  Astronomical Institute, University of Wrocław, Mikołaja
Kopernika 11, 51-622 Wrocław, Poland  Charlotte Gehan Max-Planck-Institut für
Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany
Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, Rua das
Estrelas, 4150-762 Porto, Portugal  Tiago L. Campante Instituto de Astrofísica
e Ciências do Espaço, Universidade do Porto, Rua das Estrelas, 4150-762 Porto,
Portugal  Departamento de Física e Astronomia, Faculdade de Ciências da
Universidade do Porto, Rua do Campo Alegre, s/n, 4169-007 Porto, Portugal
Diego Bossini Instituto de Astrofísica e Ciências do Espaço, Universidade do
Porto, Rua das Estrelas, 4150-762 Porto, Portugal  Stephen R. Kane Department
of Earth and Planetary Sciences, University of California, Riverside, CA
92521, USA  Jia Mian Joel Ong (王加冕) Department of Astronomy, Yale University,
P.O. Box 208101, New Haven, CT 06520-8101, USA  Mutlu Yıldız Department of
Astronomy and Space Sciences, Science Faculty, Ege University, 35100, Bornova,
İzmir, Türkiye  Cenk Kayhan Department of Astronomy and Space Sciences,
Science Faculty, Erciyes University, 38030, Melikgazi, Kayseri, Türkiye
Zeynep Çeli̇k Orhan Department of Astronomy and Space Sciences, Science
Faculty, Ege University, 35100, Bornova, İzmir, Türkiye  Si̇bel Örtel
Department of Astronomy and Space Sciences, Science Faculty, Ege University,
35100, Bornova, İzmir, Türkiye  Xinyi Zhang (张昕旖) State Key Laboratory of
Lunar and Planetary Sciences, Macau University of Science and Technology,
Macau, People’s Republic of China  Margarida S. Cunha Instituto de Astrofísica
e Ciências do Espaço, Universidade do Porto, Rua das Estrelas, 4150-762 Porto,
Portugal  Departamento de Física e Astronomia, Faculdade de Ciências da
Universidade do Porto, Rua do Campo Alegre, s/n, 4169-007 Porto, Portugal
Bruno Lustosa de Moura Departamento de Fisica, Universidade Federal do Rio
Grande do Norte, 59072-970 Natal, RN, Brazil  Instituto Federal do Rio Grande
do Norte—IFRN, Brazil  Jie Yu (余杰) Max-Planck-Institut für
Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany
Daniel Huber Institute for Astronomy, University of Hawai‘i, 2680 Woodlawn
Drive, Honolulu, HI 96822, USA  Jian-wen, Ou (欧建文) School of Physics and
Electromechanical Engineering, Shaoguan University, 512005 Shaoguan, Guangdong
Province, China  Robert A. Wittenmyer University of Southern Queensland,
Centre for Astrophysics, USQ Toowoomba, QLD 4350, Australia  Laurent Gizon
Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077
Göttingen, Germany  Institut für Astrophysik, Georg-August-Universität
Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany  Center for Space
Science, NYUAD Institute, New York University Abu Dhabi, Abu Dhabi, UAE
William J. Chaplin Stellar Astrophysics Centre (SAC), Department of Physics
and Astronomy, Aarhus University, Ny Munkegade 120, 8000 Aarhus C, Denmark
School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT,
UK

###### Abstract

The Transiting Exoplanet Survey Satellite (TESS) mission searches for new
exoplanets. The observing strategy of TESS results in high-precision
photometry of millions of stars across the sky, allowing for detailed
asteroseismic studies of individual systems. In this work, we present a
detailed asteroseismic analysis of the giant star HD 76920 hosting a highly
eccentric giant planet ($e=0.878$) with an orbital period of 415 days, using 5
sectors of TESS light curve that cover around 140 days of data. Solar-like
oscillations in HD 76920 are detected around $52\,\mbox{$\mu$Hz}$ by TESS for
the first time. By utilizing asteroseismic modeling that takes classical
observational parameters and stellar oscillation frequencies as constraints,
we determine improved measurements of the stellar mass ($1.22\pm
0.11\,\\!{M_{\sun}}$), radius ($8.68\pm 0.34\,R_{\sun}$), and age ($5.2\pm
1.4\,$Gyr). With the updated parameters of the host star, we update the semi-
major axis and mass of the planet as $a=1.165\pm 0.035$ au and $M_{\rm
p}\sin{i}=3.57\pm 0.22\,M_{\rm Jup}$. With an orbital pericenter of $0.142\pm
0.005$ au, we confirm that the planet is currently far away enough from the
star to experience negligible tidal decay until being engulfed in the stellar
envelope. We also confirm that this event will occur within about 100 Myr,
depending on the stellar model used.

asteroseismology — stars: individual (HD 76920) — planets and satellites:
dynamical evolution and stability

††facilities: TESS, Gaia††software: eleanor (Feinstein et al., 2019),
lightkurve (Lightkurve Collaboration et al., 2018), TessCut (Brasseur et al.,
2019)

##  1 Introduction

The excellent quality of photometric data from space observation missions,
such as CoRoT (Baglin et al., 2006) and Kepler (Borucki et al., 2010), allow
for major advancements in the understanding of stellar interior physics and
evolution using asteroseismology. Asteroseismology is the study of internal
structure of stars by the interpretation of their oscillation frequencies. In
particular, the detection of oscillations in solar-type and red giant stars
has led to breakthroughs such as the discovery of fast core rotation (Beck et
al., 2012) and a way to distinguish between hydrogen-shell-burning stars and
stars that are also burning helium in their cores (Bedding et al., 2011).

The advent of space photometry has also brought in advancements of data
analysis techniques (e.g., Davies & Miglio, 2016; Lund et al., 2017; Corsaro &
De Ridder, 2014; Corsaro et al., 2015) and stellar modeling strategies (e.g.,
Wu & Li, 2016; Serenelli et al., 2017; Silva Aguirre et al., 2017; Wu & Li,
2017). Furthermore, by using individual oscillation frequencies as constraints
in the model optimization process (Metcalfe et al., 2010; Jiang et al., 2011;
Mathur et al., 2012; Paxton et al., 2013; Rendle et al., 2019), asteroseismic
modeling has proven to be a robust tool to determine fundamental stellar
properties, including stellar distances (Silva Aguirre et al., 2012; Rodrigues
et al., 2014), radii and masses (Casagrande et al., 2014; Pinsonneault et al.,
2014; Sharma et al., 2016), and most importantly ages and core size for red
giants and clump stars (Casagrande et al., 2016; Anders et al., 2017;
Pinsonneault et al., 2018; Zhang et al., 2018, 2020). Consequently, this
enables us to characterize systematically the properties of the exoplanet-host
stars through asteroseismology, which in turn provides an unprecedented level
of precision in the parameters estimated for the planets (Ballard et al.,
2014; Campante et al., 2015; Lundkvist et al., 2016; Kayhan et al., 2019).
Furthermore, the synergy between asteroseismology and exoplanetary research
also enables us to set constraints on the spin-orbit alignment of exoplanet
systems (Huber et al., 2013; Benomar et al., 2014; Chaplin et al., 2014; Lund
et al., 2014; Campante et al., 2016a; Kamiaka et al., 2019) and to perform
statistical inferences on the planetary orbital eccentricities, by making use
of asterodensity profiling (Kane et al., 2012; Sliski & Kipping, 2014; Van
Eylen & Albrecht, 2015; Van Eylen et al., 2019).

The Transiting Exoplanet Survey Satellite (TESS) Mission (Ricker et al., 2015)
is NASA’s near all-sky survey for exoplanets, which launched in 2018. The
large sample of monitored systems guarantees the synergy between
asterosismology and exoplanetary science to continue to expand (Campante et
al., 2018; Huber, 2018; Hatt et al., 2022). TESS searches for exoplanets using
the transit method in an area 400 times larger than that covered by the Kepler
mission. Although the exploration of new exoplanetary systems is the main
scientific goal of the mission, TESS has also provided aids toward the
characterization of previously known systems (Kane et al., 2021). Thanks to
the high-quality of TESS photometry and large sky coverage, oscillations are
expected to be detected in hundreds of thousands of solar-like oscillators
(Campante et al., 2018; Huber, 2018; Schofield et al., 2019), including
several hundred asteroseismic exoplanet hosts (Campante et al., 2016b).
Detections of oscillations by TESS in previously known exoplanet-host stars
were reported by several works (e.g. Campante et al., 2019; Jiang et al.,
2020b; Nielsen et al., 2020; Hill et al., 2021; Huber et al., 2022), following
on the discovery of the first planet transiting a star (TOI-197) in which
oscillations could be measured by TESS (Huber et al., 2019). These
extraordinary synergies between asteroseismology and exoplanetary science
significantly improve our understanding of planet systems outside of the solar
system and provide insight into the occurrence rates of exoplanets as a
function of their host stars’ property and evolutionary state, as well as the
planets’ mass, size and orbital architecture. One of the most interesting
architectures is the planets having very eccentric orbits that are considered
as a possible origin of hot Jupiter (see Dawson & Johnson, 2018, for a
review). Planets in very eccentric orbits are a prime example and testbed of
how planetary systems form and evolve.

The first planet discovered to orbit around a evolved giant star, $\iota$
Draconis b, is on a highly eccentric orbit with $e=0.71$ (Frink et al., 2002;
Kane et al., 2010). Even larger orbital eccentricity is found in a planet
around the giant star HIP 126844 with $e=0.76$ (Adamów et al., 2012, 2018). In
this work, using TESS data, we present an asteroseismic analysis of the K
giant star HD 76920 (TIC 302372658) known to host a planet. At the time that
HD 76920 b was first detected through the radial-velocity (RV) survey of Pan-
Pacific Planet Search (Wittenmyer et al., 2017), it was the most eccentric
exoplanet known to orbit an evolved star, with an orbital eccentricity of
$0.856\pm 0.009$. Later, with the help of new multi-site RV measurements,
Bergmann et al. (2021) refined the planetary properties, finding an even
higher eccentricity of $0.8783\pm 0.0025$ and an orbital period of
$415.891^{+0.043}_{-0.039}$ days, a minimum planet mass of
$3.13^{+0.41}_{-0.43}\,M_{\rm Jup}$, and a semi-major axis of
$1.091^{+0.068}_{-0.077}$ au. There is no evidence of any unseen binary
companion, suggesting a scattering event rather than Kozai oscillations as the
probable explanation for the observed eccentricity, and making the system
valuable to the study the evolution and occurrence of planets around evolved
stars. TESS detected solar-like oscillations in HD 76920 for the first time.
Bergmann et al. (2021) analyzed three sectors (9, 10 and 11) of TESS data and
estimated the stellar mass and radius through the scaling relations (Brown et
al., 1991; Kjeldsen & Bedding, 1995; Stello et al., 2008; Kallinger et al.,
2010) using the measurements of global seismic parameters. While the scaling
relations provide decent estimates of the stellar mass and radius for stars
showing solar-like oscillations, improvements, in terms of accuracy and
precision, of the estimates can be achieved by using asteroseismic modeling
and individual oscillation modes.

In this paper, we aim to analyze the solar-like oscillations from 5 sectors of
TESS photometric light curve. Including these oscillations in detailed stellar
modeling can help us derive precise fundamental stellar properties of the host
star. The improved stellar parameters can further be used to update the
properties of the orbiting planet, which is of great importance in
characterizing the planetary system. The determined stellar age from
asteroseismology and modeling provide information about the evolution of the
system, which assists at predicting the final fate of the planet.

Table 1: Stellar Parameters for HD 76920 Parameter | Value | References
---|---|---
Basic Properties
TIC | 302372658 | 1
Hipparcos ID | 43803 | 2
TESS Mag. | 6.87 | 1
Sp. Type | K1 III | 3
Spectroscopy
$T_{\rm eff}$ (K) | $4698\pm 100$ | 4
| $4664\pm 53$ | 5
$[{\rm Fe}/{\rm H}]$ (dex) | $-0.11\pm 0.10$ | 4
| $-0.19\pm 0.06$ | 5
$\log g$ (cgs) | $2.94\pm 0.15$ | 4
| $2.71\pm 0.04$ | 5
SED & Gaia DR3 ParallaxaaTwo SED pipelines are used to obtain $A_{V}$, $F_{\rm
bol}$, $R_{\star}$ and $L_{\star}$ (Section 2.2). The second value of each
parameter is from the SEDEX pipeline (Yu et al., 2021, 2022).
$\pi$ (mas) | $5.4618\pm 0.0187$ | 6
$A_{V}$ | $0.20\pm 0.05$ | 7
| $0.15\pm 0.07$ | 7
$F_{\rm bol}$ (${\rm erg\,s^{-1}\,cm^{-2}}$) | $(3.12\pm 0.11)\times 10^{-8}$ | 7
| $(3.08\pm 0.11)\times 10^{-8}$ | 7
$R_{\star}$ ($R_{\sun}$) | $8.64\pm 0.37$ | 7
| $8.70\pm 0.25$ | 7
$L_{\star}$ ($L_{\sun}$) | $32.54\pm 1.17$ | 7
| $32.06\pm 1.16$ | 7
$M_{\star}$ ($\\!{M_{\sun}}$) |  $1.68\pm 0.10$bbBased on extrapolated relations of Torres et al. (2010). | 7
Asteroseismology
$\Delta\nu$ ($\mu$Hz) | $5.53\pm 0.15$ | 7
$\nu_{\rm max}$ ($\mu$Hz) | $53.2\pm 2.3$ | 7
$M_{\star,\rm seis}$ ($\\!{M_{\sun}}$) | $1.22\pm 0.11$ | 7
$R_{\star,\rm seis}$ ($R_{\sun}$) | $8.68\pm 0.34$ | 7
$\rho$ (gcc) | $0.0026\pm 0.0004$ | 7
$\log g$ (cgs) | $2.648\pm 0.037$ | 7
$t$ (Gyr) | $5.2\pm 1.4$ | 7

##  2 Observations

###  2.1 TESS Observations

HD 76920 has been observed by TESS in Sectors 9, 10, 11 at 30-min cadence
during its Cycle 1 observations, and Sectors 36 and 37 at 10-min cadence111HD
76920 has also been observed in Sector 38 at 10-min cadence, but the data was
not released at the time of this analysis. during its Cycle 3 observations. To
extract these light curves, we used the open-source Python package eleanor
(v2.0.3; Feinstein et al., 2019)222For sectors 36 and 37, eleanor uses the
TessCut tool (Brasseur et al., 2019).. eleanor performs background
subtraction, systematics corrections, and aperture selection per each sector
of data. We extracted $13\times 13$ pixels postage stamps and applied the
eleanor default apertures for light curve extraction (top panel of Figure 1).
Although the aperture selection process is optimized for exoplanet searches,
we found also these default apertures to perform well for asteroseismic
measurements.

Figure 1: eleanor best-fit apertures (top) overlaid on the TESS target pixel
files (TPFs) extracted per each sector. The TPFs are all scaled from 0 to
30000 e-1 s-1. We use these apertures to extract the flux within eleanor,
which are then corrected via the default eleanor corrected flux routine
(bottom). Apertures and light curves are colored by TESS sector.

We used the eleanor corrected flux; this flux option corrects for systematic
issues by regressing against a linear model of time, background, and position
and removing trends by the co-trending basis vectors provided by the Science
Process Operations Center pipeline (Jenkins et al., 2016). We additionally
applied quality masks, provided by eleanor, to remove any bad cadences and
removed outliers $\geq 7\sigma$ from the median of each light curve. The
resulting light curves are shown in the bottom panel of Figure 1. No transit
of the planet was detected in existing TESS data due to the short observation
coverage compared to its long orbital period. TESS is expected to observe HD
76920 again for 5 months in Cycle 5.

###  2.2 Broadband Photometry and Spectral Energy Distribution

As an independent determination of the basic stellar parameters, we performed
an analysis of the broadband spectral energy distribution (SED) of the star
together with the Gaia DR3 parallax (with no systematic offset applied; see,
e.g., Stassun & Torres, 2021), in order to determine an empirical measurement
of the stellar radius, following the procedures described in Stassun & Torres
(2016) and Stassun et al. (2017, 2018). We pulled the $B_{T}V_{T}$ magnitudes
from Tycho-2, the $JHK_{S}$ magnitudes from the Two Micron All Sky Survey, the
W1–W4 magnitudes from the Wide-field Infrared Survey Explorer, and the $G_{\rm
BP}G_{\rm RP}$ magnitudes from Gaia DR3. Together, the available photometry
spans the full stellar SED over the wavelength range 0.4–22 $\mu$m (see Figure
2).

We performed a fit using Kurucz stellar atmosphere models, with the effective
temperature ($T_{\rm eff}$), surface gravity ($\log g$), and metallicity
([Fe/H]) adopted from the spectroscopic analysis of Wittenmyer et al. (2017).
The remaining free parameter is the extinction $A_{V}$, which we limited to
the maximum line-of-sight extinction from the Galactic dust maps of Schlegel
et al. (1998). The resulting fit (Figure 2) has a reduced $\chi^{2}$ of 2.2,
and best-fit $A_{V}=0.20\pm 0.05$. Integrating the (unreddened) model SED
gives the bolometric flux at Earth, $F_{\rm bol}=3.12\pm 0.11\times 10^{-8}$
erg s-1 cm-2. Taking the $F_{\rm bol}$ and $T_{\rm eff}$ together with the
Gaia parallax, gives the stellar radius, $R_{\star}=8.64\pm 0.37\,R_{\sun}$,
revealing the star to be clearly evolved. In addition, we can estimate the
stellar mass directly from $R_{\star}$ together with the spectroscopic $\log
g$ from (Wittenmyer et al., 2017, see Table 1), which gives $M_{\star}=2.37\pm
0.41$ M⊙; this is higher than the value obtained from the empirical relations
of Torres et al. (2010), giving $M_{\star}=1.68\pm 0.10$ M⊙, indicating an
overestimate of spectroscopic $\log g$ (see Section 4). Combining $F_{\rm
bol}$ and Gaia DR3 parallax allows us to derive a luminosity
$L_{\star}=32.54\pm 1.17\,L_{\sun}$. We also performed an independent SED
fitting using the SEDEX pipeline (Yu et al., 2021, 2022) with MARCS atmosphere
models (Gustafsson et al., 2008) and same input atmospheric parameters. The
resulting parameters from both SED fittings are well consistent, as listed in
Table 1.

Figure 2: Spectral energy distribution of HD 76920. Red symbols represent the
observed photometric measurements, where the horizontal bars represent the
effective width of the passband. Blue symbols are the model fluxes from the
best-fit Kurucz atmosphere model (black).

##  3 Asteroseismic Analysis

###  3.1 Global Oscillation Parameters

For the seismic analysis, the TESS light curve prepared by eleanor was
distributed to several groups using different pipelines (e.g., Huber et al.,
2009; Mathur et al., 2010; Jiang et al., 2011; Lundkvist, 2015; Campante et
al., 2017; Yu et al., 2018; Corsaro et al., 2020; De Moura et al., 2020; Li et
al., 2020) to extract the seismic parameters. Figure 3 shows the power density
spectrum of HD 76920 computed based on the eleanor light curve, combining both
Cycle 1 and Cycle 3 data. The 10-min cadence sectors were rebinned to 30-min
cadence, with a simple average over three measurements. The power spectrum
shows a frequency-dependent background signal due to stellar activity,
granulation, and faculae that can be modeled by a superposition of several
Lorentzian-like functions (i.e., Harvey-like models, Harvey, 1985; Karoff,
2008; Jiang et al., 2011; Kallinger et al., 2014; Corsaro et al., 2017), and a
white noise term. The background shown as the cyan dotted curve in Figure 3
was obtained by fitting the background model with two Harvey-like components
and one white noise to the smoothed power spectrum. The target pixel time
series of the Cycle 3 sectors have been detrended with a moving median to
remove long timescale stellar variations. As a result, the background noise at
low frequency is greatly suppressed during this process. Therefore, the
background fit displayed in Figure 3 disregarded frequencies below $\sim$10
$\mu$Hz. The global seismic parameters such as the frequency of maximum power
($\nu_{\rm max}$) and the mean large frequency separation ($\Delta\nu$) were
based on the analysis of the background-corrected power spectrum generated by
each group.

In general, $\nu_{\rm max}$ was measured by fitting a Gaussian distribution
profile to the power excess hump of the smoothed power spectrum (e.g., Hekker
et al., 2010) or through the 2D autocorrelation function (e.g., Viani et al.,
2019). To measure $\Delta\nu$, techniques like autocorrelation of the
amplitude spectrum (e.g., Huber et al., 2009; Mosser & Appourchaux, 2009),
power spectrum of the power spectrum (e.g., Kjeldsen & Bedding, 1995; Mathur
et al., 2010; Jiang, 2015), matched filter response function (e.g., Gilliland
et al., 2011), and asymptotic or linear fit to the frequencies of the radial
modes (individual mode extraction given in Section 3.2) were used. However, a
clear shift of the oscillation power excess region to lower frequencies is
observed in the power spectrum generated with Cycle 3 data (Sectors 36 and
37), compared with the Cycle 1 one (Sectors 9 to 11), as illustrated in the
upper panel of Figure 4. This is principally due to the stochastic nature of
the oscillations so that the change in mode amplitudes impacts the value of
$\nu_{\rm max}$. This shift of the power excess region may lead to a larger
uncertainty in $\nu_{\rm max}$ when compared to the corresponding formal
uncertainty originating from different pipelines. With this in mind, we
computed consolidated results from the 8 independent determinations of the
global seismic parameters. In particular, we adopted the mean values of the
parameter estimates returned by all methods and recalculated the uncertainties
by adding in quadrature the corresponding formal uncertainty and the standard
deviation. The consolidated results are $\mbox{$\nu_{\rm max}$}=53.2\pm
2.3\,\mbox{$\mu$Hz}$ and $\mbox{$\Delta\nu$}=5.53\pm 0.15\,\mbox{$\mu$Hz}$. We
note that due to the intrinsic change of the distribution of mode amplitudes
observed between Cycle 1 and Cycle 3 data, this uncertainty of $\Delta\nu$
resulting from our statistical consolidation approach, can also be
overestimated and can be decreased by examining the different $\Delta\nu$
values against the échelle diagrams (Stello et al., 2011). As an alternative,
instead of taking into account the scatter across results, we also quote here
the pipeline results that are of the smallest deviation from the mean values
of all pipeline results as $\mbox{$\nu_{\rm max}$}=52.4\pm
0.3\,\mbox{$\mu$Hz}$ and $\mbox{$\Delta\nu$}=5.52\pm 0.02\,\mbox{$\mu$Hz}$,
considering both parameters simultaneously (i.e. both parameters returned by
the same pipeline). This level of uncertainty of $\Delta\nu$ is of comparable
magnitude to those extracted from 2 sectors of TESS observations of red giants
(Silva Aguirre et al., 2020). The combined power spectrum corrected from the
background model is shown in the bottom panel of Figure 4, where signals with
$\nu<50\,\mbox{$\mu$Hz}$ are largely enhanced by the Cycle 3 data, and those
with $\nu>50\,\mbox{$\mu$Hz}$ are due to the Cycle 1 data. The asteroseismic
analysis discussed in Section 3 were performed based on the combined power
spectrum (lower panel of Figure 4).

Figure 3: Power spectral density (PSD) of HD 76920 and corresponding global
background model fit (cyan dotted curve). The background model consist of two
Harvey-like profiles (green and blue dashed curves) and white noise (orange
dashed line). The solid red curve depicts the summation of the background and
a Gaussian fit (purple dashed) to the oscillation power excess envelope. The
PSD is generated using 5 sectors of TESS data (Figure 1). The 10-min cadence
sectors are rebinned to 30-min cadence, with simple average over three
measurements. The original PSD is shown in gray and a heavily smoothed
(Gaussian with an FWHM of $\Delta\nu$) version in black. The fit displayed
here disregarded frequencies below $\sim$10 $\mu$Hz (Section 3.1).

Figure 4: Background corrected power spectra of HD 76920 depicted in the power
excess region. Top panel: power spectra of the Cycle 1 (Sectors 9 to 11) and
Cycle 3 (Sectors 36 and 37) light curve. The Cycle 1 data shows a larger value
of $\nu_{\rm max}$, compared with the Cycle 3 data. Bottom panel: power
spectra of the combined light curve that is used for the asteroseismic
analysis.

###  3.2 Individual Mode Frequencies

In the bottom panel of Figure 4 the combined power spectrum shows a regular
series of peaks corresponding to solar-like oscillations within the frequency
range between 30 to 80 $\mu$Hz. To extract individual oscillation modes from
the power spectrum several independent methods ranging from traditional
iterative fitting of sine waves, i.e., pre-whitening (e.g. Kjeldsen et al.,
2005; Lenz & Breger, 2005; Bedding et al., 2007; Jiang et al., 2011), to
fitting of Lorentzian mode profiles individually or in a global power spectrum
model (e.g. Handberg & Campante, 2011; Appourchaux et al., 2012; Mosser et
al., 2012; Corsaro et al., 2015; Vrard et al., 2015; Davies & Miglio, 2016;
Handberg et al., 2017; Roxburgh, 2017; Kallinger et al., 2018; Corsaro et al.,
2020; Li et al., 2020; Breton et al., 2022) were used by different pipelines.
The extracted mode frequencies and corresponding uncertainties are listed in
Table 2. The extracted radial modes also allowed us to measure $\Delta\nu$ by
fitting a straight line to the radial-mode frequencies. Thus, the slope of the
line is $\Delta\nu$ that is $5.62\pm 0.03\,\mbox{$\mu$Hz}$ using this method.
The échelle diagram (Bedding & Kjeldsen, 2010) generated using this
$\Delta\nu$ is depicted in Figure 5. The two vertical ridges located near the
right edge of the figure correspond to $\ell=0$ (red circles) and 2 (blue
triangles) modes. However, due to the relatively short duration of our TESS
data for HD 76920, clear mixed-mode pattern is not visible in the power
spectrum or in the échelle diagram, though there are a few peaks corresponding
to $\ell=1$ mixed modes (green diamonds in Figure 5) appearing in the
spectrum. In Table 2 we also list the $\ell=1$ modes that are extracted by at
least two independent sources, including a pair of mixed modes around
$53\,\mbox{$\mu$Hz}$ that were simultaneously extracted by 3 independent
sources.

Table 2: Extracted Oscillation Frequencies from TESS light curve and Mode
Identification based on the best-fitting model for HD 76920

$\ell$ |  $n$($n_{\rm p}$) | $\nu$ | $\sigma_{\nu}$ | SNR
---|---|---|---|---
|  | ($\mu$Hz) | ($\mu$Hz) |
0 | 7 | 44.80 | 0.13 | 6.63
0 | 8 | 50.32 | 0.10 | 8.08
0 | 9 | 55.85 | 0.08 | 12.50
0 | 10 | 61.48 | 0.08 | 6.71
0 | 11 | 67.18 | 0.13 | 3.83
0 | 12 | 72.73 | 0.15 | 4.0
1 | 6 | 42.41 | 0.08 | 12.05
1 | 8 | 53.12 | 0.03 | 9.54
1 | 8 | 53.27 | 0.03 | 6.73
1 | 9 | 58.90 | 0.07 | 6.95
1 | 10 | 64.13 | 0.08 | 3.61
2 | 5 | 38.61 | 0.29 | 3.71
2 | 6 | 43.93 | 0.24 | 5.88
2 | 7 | 49.53 | 0.12 | 7.13
2 | 8 | 55.10 | 0.07 | 5.42
2 | 9 | 60.59 | 0.20 | 4.48
2 | 10 | 66.32 | 0.07 | 3.89

.

Figure 5: Échelle diagram of the background-corrected PSD, folded on an large frequency separation of $\mbox{$\Delta\nu$}=5.62\,\mbox{$\mu$Hz}$ estimated by fitting the radial-mode frequencies (c.f. Section 3.2). Identified individual mode (Table 2) frequencies are marked with red circles ($\ell=0$), green diamonds ($\ell=1$) and blue triangles ($\ell=2$). Table 3: Modeling configurations from different sources. One entry is used where all five teams used the same input physics. Team | Ong | Kayhan | Izmir | Zhang | BESTP
---|---|---|---|---|---
Evolution code |  MESA (r12778) |  MESA (r12778) |  MESA (r15140) |  MESA (r10398) | ASTEC
Oscillation code | GYRE | ADIPLS | ADIPLS | ADIPLS | ADIPLS
EoSaaThe MESA/OPAL tables are based on the 2005 update of the OPAL EoS tables (Rogers & Nayfonov, 2002). |  MESA/OPAL |  MESA/OPAL |  MESA/OPAL |  MESA/OPAL | OPAL
Surface CorrectionbbThe adopted methods for surface correction are Kjeldsen et al. (2008) (KB) and Ball & Gizon (2014) two-term correction (BG-2term) | BG-2term | KB | KB | None | BG-2term
Nuclear reactions | NACRE (Angulo et al., 1999)
High-$T$ opacities | OPAL (Iglesias & Rogers, 1993, 1996)
Low-$T$ opacities | Ferguson et al. (2005)
Solar mixtureccSolar composition given in Grevesse & Sauval (1998) (GS98) and Asplund et al. (2009) (AGSS09) are used for initial chemical composition. | GS98 | AGSS09 | AGSS09 | GS98 | AGSS09
AtmosphereddThe atmosphere choices used in MESA are introduced in Paxton et al. (2011), and ASTEC uses the Kurucz model (Kurucz, 1991) for atmosphere. | Eddington gray | simple photosphere | simple photosphere | Eddington gray | Kurucz
$\alpha_{\mathrm{MLT}}$ | 1.83 | 2.175 | 1.828 | 2.0 | 1.7 – 2.1
Overshoot | Step overshooteeThe overshoot parameters $f_{\rm ov}$ and $f_{\rm 0,ov}$ are set as 0.006 and 0.003. The overshooting is applied at all convective boundaries. | None | None | None | None
Diffusion | Thoul et al. (1994) | None | None | None | None
Table 4: Modeling results from different sources. Team | Ong | Kayhan | Izmir | Zhang | BESTP
---|---|---|---|---|---
$M_{\star}$ ($\\!{M_{\sun}}$) | $1.31\pm 0.06$ | $1.28\pm 0.16$ | $1.15\pm 0.14$ | $1.17\pm 0.01$ | $1.20\pm 0.10$
$R_{\star}$ ($R_{\sun}$) | $8.96\pm 0.14$ | $8.29\pm 0.14$ | $8.62\pm 0.39$ | $8.67\pm 0.02$ | $8.87\pm 0.26$
$\log g$ (cgs) | $2.650\pm 0.008$ | $2.709\pm 0.002$ | $2.630\pm 0.019$ | $2.630\pm 0.001$ | $2.620\pm 0.010$
$t$ (Gyr) | $4.1\pm 0.7$ | $4.3\pm 0.5$ | $5.8\pm 2.1$ | $6.3\pm 0.06$ | $5.6\pm 1.4$

Figure 6: Evolutionary tracks for a series of ASTEC models with different
initial masses but same chemical abundance ($X=0.714$ and $Z=0.0142$,
corresponding to [Fe/H] = -0.09) and mixing length parameter ($\alpha=1.927$)
that are the optimization outputs of the BESTP pipeline. The initial masses of
the models increase from 0.8 to 2.0 $\\!{M_{\sun}}$ with a step of 0.1
$\\!{M_{\sun}}$. The evolutionary track indicated by orange is with an initial
mass of 1.2 $\\!{M_{\sun}}$ that is closest to the adopted value (Table 1).
Sampling points by BESTP are drawn in the diagram and also in the inset, with
color-coded normalized likelihood values. The green star marks the location of
our best-fitting model ($M=1.21\,\\!{M_{\sun}}$, $X=0.713$ and $Z=0.0145$)
from BESTP.

Figure 7: Background-corrected PSD (dark grey) plotted in the échelle diagram
format that divides the spectrum into bins each $\Delta\nu$ wide. The blue and
red peaks are the $\ell=0$ and 2 modes, respectively, of the best-fitting
model returned by BESTP (Figure 6). The line width and amplitude of the
theoretical modes are derived using the formulae introduced in Lund et al.
(2017) and Ball et al. (2018). The corresponding observed frequencies and
uncertainties (Table 2) are indicated by the horizontal spans, with additional
$\ell=1$ modes in green.

##  4 Asteroseismic Modelling

Asteroseismic modelling is a powerful tool to estimate fundamental stellar
properties. Five independent teams performed modeling efforts to search for
the stellar models that best match the classical and asteroseismic constraints
from observations for HD 76920. These teams employ different stellar evolution
codes (ASTEC, MESA; Christensen-Dalsgaard, 2008a; Paxton et al., 2011, 2013,
2015), oscillation codes (ADIPLS, GYRE; Christensen-Dalsgaard, 2008b; Townsend
& Teitler, 2013), and optimization methods (Mier, 2017; Kayhan et al., 2019;
Wu & Li, 2019; Yıldız et al., 2019; Jiang & Gizon, 2021; Zhang et al., 2022).
The input physics adopted by each team is detailed in Table 3. Diffusion and
overshoot were turned off except for one team. For the modeling, all five
teams used the luminosity $L_{\star}$ of $32.54\pm 1.17\,L_{\sun}$ derived
with Gaia DR3 parallax (Section 2.2) and the spectroscopic measurements, i.e.,
the effective temperature $T_{\rm eff}$ and metallicity [Fe/H] from Wittenmyer
et al. (2017), as constraints. As for the seismic constraints, the large
frequency separation $\mbox{$\Delta\nu$}=5.52\pm 0.02\,\mbox{$\mu$Hz}$ and
$\ell=0$ and 2 mode frequencies in Table 2 were used333The identified five
$\ell=1$ modes listed in Table 2 do not show clear pattern of mixed modes,
thus, provide very limited constraints on the models, as one can always find a
match to the observed mode among the dense theoretical dipolar modes..
Generally, the most p-dominated $\ell=2$ mixed modes of each model are
selected to be compared with the observations, except for team Ong who
computed the pure quadrupole p-modes of stellar models using the $\pi$-mode
isolation condition of Ong & Basu (2020) as implemented in GYRE444These pure
quadrupole p-modes are matched to the frequencies of the observed p-dominated
quadrupole mixed modes, under the assumption that interference from mode
mixing in these high-luminosity red giants may be neglected (cf. discussion in
Ong et al., 2021).. The constraint on $\nu_{\rm max}$ were not considered in
the modeling due to the discrepancy of the oscillation power excess region
between the sectors (Section 3.1). We measured the phase shift $\epsilon_{\rm
c}$ of the central radial mode, which is the linear offset in the asymptotic
fit to the acoustic modes, as $\sim 0.94$, indicating that the star is still
on the red giant branch (RGB) burning only hydrogen in a shell, because the
more evolved horizontal branch stars (e.g. red clump and secondary clump
stars) that have already ignited helium in the core would have a
systematically smaller $\epsilon_{\rm c}$ than their counterparts on the RGB
(Kallinger et al., 2012). Moreover, horizontal branch tracks hardly cross the
observational constraints in the HR diagram, thus, we limited the modeling
only to the RGB.

The outputs from the five teams are generally in agreement with each other
(Table 4), though differences are inevitably seen due to the diversity of
modeling codes, procedures and input physics adopted by different teams. We
note that the amount of convective overmixing ($f_{\rm ov}=0.006$) adopted by
team Ong is mainly for numerical softening and is too small to induce an
appreciable physical difference to the spectroscopic parameters (see
discussions in, e.g., Claret & Torres, 2017, 2018, 2019; Guo & Li, 2019; Zhang
et al., 2022). Furthermore, since we have only used even-degree ($\ell=2$)
p-dominated modes in this work, the effects of overshoot on the g-mode cavity
(e.g. Lindsay et al., 2022) are not relevant. For the even-degree p-modes,
envelope overshoot will reduce the amplitude of the convective-glitch
signature induced into the p-modes, which is already very small on the main
sequence, and decreases in amplitude as the convective boundary retreats
closer to the core as the star ascends the RGB. The consolidated values (the
mean from all sources) for stellar mass $M_{\star,\rm seis}$, radius
$R_{\star,\rm seis}$, age $t$, surface gravity log $g$ and density $\rho$ are
summarized in Table 1, constraining the corresponding parameters to a
precision level of 9%, 4%, 27%, 1%, 15%, respectively, which are the most
precise results for HD 76920 so far. The final uncertainties on these stellar
parameters were recalculated by adding in quadrature the corresponding formal
uncertainty for a given parameter to the standard deviation of the parameter
estimates returned by all teams. Therefore, both random and systematic errors
arising from the diversity of modeling methods from different teams were taken
into account. The estimated log $g$ from modeling is distinctly smaller than
the spectroscopic results reported by Wittenmyer et al. (2017), but matches
the one measured by Bergmann et al. (2021) (who used updated spectroscopy
data) within $1\sigma$, confirming the SED results (see Section 2.2).

Figure 6 shows the locations of the sampling points by BESTP in the HR
diagram, along with a series of evolutionary tracks with different initial
masses generated by ASTEC. The sampling points are color-coded according to
the normalized likelihood, thus, redder samples have higher possibilities as
the representation of the real star. According to the figure, HD 76920 is most
likely approaching closely to the RGB luminosity bump, where the properties of
mixed-modes are significantly impacted by the buoyancy glitch (Cunha et al.,
2015, 2019; Jiang et al., 2020a), a signature that can help us inspect the
stellar mid-layer structures (Pinçon et al., 2020; Jiang et al., 2022).
However, the short observation duration for HD 76920 limits the detection of
mixed-modes from the TESS power spectrum, thus, such investigation of stellar
interior structure with the help of mixed modes is not feasible with current
TESS data. Nevertheless, by matching the theoretical oscillation frequencies
calculated for the best-fitting model with the observed TESS power spectrum
(Figure 7), we could confirm the mode degree and identify the mode order for
the oscillation modes that were extracted by different groups (Table 2). The
theoretical modes used to identify observed modes are corrected for the
surface effect (e.g., Houdek et al., 2017) that yields a systematic offset
between the calculated and the observed oscillation frequencies. For
theoretical dipolar mixed modes, correction for the surface effect was not
performed due to the lack of enough observed dipolar frequencies, thus, we did
not show them in Figure 7.

##  5 Characterization of HD 76920 b

According to the prediction for the transit ephemeris made by Bergmann et al.
(2021), if HD 76920 b were to transit, it would have done so during TESS
Sector 9. However, no clear transit signal was found by Bergmann et al. (2021)
or in our data. Thus, we could not measure the planetary orbit or eccentricity
through the TESS photometric data.

However, we can combine our newly computed asteroseismic stellar mass
($M_{\star,\rm seis}=1.22\pm 0.11\,\\!{M_{\sun}}$) with the orbital period, RV
semi-amplitude and eccentricity values found by Bergmann et al. (2021) to
compute updated values for the planet’s semi-major axis and minimum mass.
Using values from Bergmann et al. (2021), we find $a=1.165\pm 0.035$ au and
$M_{\rm p}\sin{i}=3.57\pm 0.22\,M_{\rm Jup}$. These updated parameters place
the orbital pericenter of the planet at $0.142\pm 0.005$ au, revealing that
the planet’s orbit is sufficiently far from the star such that tidal decay is
currently negligible until the star expands its radius by at least 50-100%
(Villaver & Livio, 2009) and subsequently engulfs the planet, in line with the
findings of Bergmann et al. (2021).

The time at which the star will engulf the planet will depend on when the star
began ascending the RGB. During this phase of stellar evolution, the star’s
radius will expand to a distance approximately one order of magnitude greater
than the planet’s orbital pericenter, allowing the star to easily and quickly
engulf the planet. For instance, further evolving the best-fitting model
(Figure 6) along the evolutionary track, we predict that the expanding star
will have a radius one order of magnitude greater than the planet’s orbital
pericenter and engulf the planet in about 130 Myr.

As an independent estimation of the planetary engulfment time, we also
calculated models using the SSE prescription from Hurley et al. (2000).
According to this prescription, a $1.22\,M_{\odot}$ star with a metallicity of
$Z=0.02$ will begin the red giant phase at about 5.58 Gyr and leave it at 6.01
Gyr. During this interval, the planet will be engulfed. Asteroseismology has
allowed us to estimate the current age of the star to be $5.2\pm 1.4$ Gyr.
This value clearly demonstrates that the star is on the verge of engulfing the
planet. However, the uncertainty on the age is about triple the value of the
duration of the red giant phase. Nevertheless, asteroseismology has also
allowed us to constrain the stellar radius to a value of $8.68\pm
0.34\,R_{\odot}$. This value corresponds to an age from the SSE model of
$5.939\pm 0.004$ Gyr, meaning that the planet will be engulfed in less than
about 150 Myr. However, this approximate upper bound is likely an overestimate
given the fact that the star’s metallicity is sub-Solar.

Other stellar models produce similar results, just renormalized to other
absolute ages within the measured asteroseismic uncertainty. For example, a
$1.22\,M_{\odot}$ star with a sub-Solar metallicity of $Z=0.01$ will begin the
red giant phase at about 4.72 Gyr and leave it at 5.08 Gyr. In this case, the
asteroseismically measured radius corresponds to an age from the SSE model of
$5.007\pm 0.004$ Gyr. This value illustrates that the planet will be engulfed
within about 70 Myr. This result is in agreement with that of Bergmann et al.
(2021).

##  6 Conclusions

In this work, we have analyzed the TESS photometric data for HD 76920 to
determine the star’s fundamental parameters using asteroseismology, and to
characterize the exoplanet system consisting of a planet with an extremely
large orbital eccentricity. In total, 5 sectors of TESS light curves are used
for the extraction of asteroseismic parameters for the host star, including 17
individual oscillation frequencies. Modeling by various pipelines that utilize
the extracted asteroseismic parameters, classical spectroscopic observables,
as well as luminosity from Gaia DR3 parallax, places strong constraints on the
stellar parameters. Through the asteroseismic analysis, we obtain a value for
the stellar mass of $1.22\pm 0.11\,\\!{M_{\sun}}$, a stellar radius of
$8.68\pm 0.34\,R_{\sun}$ and an age of $5.2\pm 1.4$ Gyr, which provide the
most precise estimations for HD 76920 to date. The stellar models reveal that
the star is ascending the red giant branch and most likely approaching closely
to the luminosity bump where the properties of mixed-modes are significantly
impacted by the buoyancy glitch. However, the current power spectrum of HD
76920 from TESS does not allow the extraction of a sufficient number of mixed
modes as would be required for the investigation of the buoyancy glitch.

The updated stellar parameters of the host star from our asteroseismic
analysis have enabled improved estimations for the semi-major axis and mass of
the planet as $a=1.165\pm 0.035$ au and $M_{\rm p}\sin{i}=3.57\pm 0.22\,M_{\rm
Jup}$. With an orbital pericenter of $0.142\pm 0.005$ au, we confirm that the
planet is currently far away enough from the star to experience negligible
tidal decay before being engulfed in the stellar envelope. However, we predict
that this event will occur within about 100 Myr, depending on the stellar
model used.

HD 76920 will be observed in 5 more sectors by TESS in Cycle 5. The prolonged
data will possibly enable the detection of mixed-modes, and thus, the
investigation of stellar interior through these modes. Moreover, our
asteroseismic analysis emphasizes the potential of TESS for characterizing
exoplanet systems through the synergy between exoplanet research and
asteroseismology.

##  7 acknowledgments

The project leading to this publication has received funding from the B-type
Strategic Priority Program of the Chinese Academy of Sciences (Grant No.
XDB41000000). ADF acknowledges the support from the National Science
Foundation Graduate Research Fellowship Program under Grant No. (DGE-1746045).
M.S.L would like to acknowledge the support from VILLUM FONDEN (research grant
42101) and The Independent Research Fund Denmark’s Inge Lehmann program (grant
agreement no.: 1131-00014B). CK is supported by Erciyes University Scientific
Research Projects Coordination Unit under grant number DOSAP MAP-2020-9749. T.
Wu thanks the supports from the National Key Research and Development Program
of China (Grant No. 2021YFA1600402), from the National Natural Science
Foundation of China (Grant Nos. 11873084, 12133011, and 12273104), from Youth
Innovation Promotion Association of Chinese Academy of Sciences, and from Ten
Thousand Talents Program of Yunnan for Top-notch Young Talents. X.-Y. Zhang
thanks the support from the National Natural Science Foundation of China
(Grant No. 12173105). S.M. acknowledges support from the Spanish Ministry of
Science and Innovation (MICINN) with the Ramón y Cajal fellowship no.
RYC-2015-17697, grant no. PID2019-107187GB-I00 and PID2019-107061GB-C66, and
through AEI under the Severo Ochoa Centres of Excellence Program 2020–2023
(CEX2019-000920-S). R.A.G. acknowledges the support from PLATO and GOLF CNES
grants. This work was also supported by Fundação para a Ciência e a Tecnologia
(FCT) through research grants UIDB/04434/2020 and UIDP/04434/2020. TLC and MSC
are supported by FCT in the form of a work contracts (CEECIND/00476/2018,
CEECIND/02619/2017). D.H. acknowledges support from the Alfred P. Sloan
Foundation and the National Aeronautics and Space Administration
(80NSSC21K0652, 80NSSC20K0593).

, LAURA (De Moura et al., 2020), FAMED (Corsaro et al., 2020), echelle,
SolarlikePeakbagging (Li et al., 2020), apollinaire (Breton et al., 2022),
fnpeaks, SYD (Huber et al., 2009), A2Z (Mathur et al., 2010), Period04 (Lenz &
Breger, 2005), Yabox (Mier, 2017), SEDEX (Yu et al., 2021, 2022)

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