11institutetext: Département d’Astronomie, Université de Genève, Chemin Pegasi
51, 1290 Versoix, Switzerland 22institutetext: School of Physics and
Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK.

# Coralie radial-velocity search for companions
around evolved stars (CASCADES)

III. A new Jupiter host-star: in-depth analysis of HD 29399 using TESS data.
††thanks: Based on observations collected with the CORALIE echelle
spectrograph on the 1.2-m Euler Swiss telescope at La Silla Observatory, ESO,
Chile.

C. Pezzotti 11 G. Ottoni 11 G. Buldgen 11 A. Lyttle 22 P. Eggenberger 11 S.
Udry 11 D. Ségransan 11 M. Mayor 11 C. Lovis 11 M. Marmier 11 A. Miglio 22 Y.
Elsworth 22 G.R. Davies 22 W.H. Ball 22

(Received December 7, 2020; accepted …)

###### Abstract

Context. Increasing the number of detected exoplanets is far from anecdotal,
especially for long-period planets that require a long duration of
observation. More detections imply a better understanding of the statistical
properties of exoplanet populations, and detailed modelling of their host
stars also enables thorough discussions of star–planet interactions and
orbital evolution of planetary systems.

Aims. In the context of the discovery of a new planetary system, we aim to
perform a complete study of HD 29399 and its companion by means of radial-
velocity measurements, seismic characterisation of the host-star, and
modelling of the orbital evolution of the system.

Methods. High-resolution spectra of HD 29399 were acquired with the CORALIE
spectrograph mounted on the 1.2-m Swiss telescope located at La Silla
Observatory (Chile) as part of the CASCADES survey. We used the moments of the
cross-correlation function profile as well as the photometric variability of
the star as diagnostics to distinguish between stellar and planetary-induced
signals. To model the host star we combined forward modelling with global and
local minimisation approaches and inversion techniques. We also studied the
orbital history of the system under the effects of both dynamical and
equilibrium tides.

Results. We present the detection of a long-period giant planet. Combining
these measurements with photometric observations by TESS, we are able to
thoroughly model the host star and study the orbital evolution of the system.
We derive stellar and planetary masses of $\rm 1.17\pm
0.10\leavevmode\nobreak\ M_{\odot}$ and $1.59\pm 0.08\leavevmode\nobreak\
\rm{M_{Jup}}$, respectively, and an age for the system of $6.2$ Gyr. We show
that neither dynamical nor equilibrium tides have been able to affect the
orbital evolution of the planet. Moreover, no engulfment is predicted for the
future evolution of the system.

###### Key Words.:

Techniques: radial velocities – Planets and satellites: detection – Stars:
interiors – Stars: fundamental parameters – Planet–star interactions – Stars:
individual – HD 29399, TIC 38828538

##  1 Introduction

Owing to the discovery of more than 4300 exoplanets, the booming field of
exoplanetology has seen tremendous success and encountered rapid development.
Today, the attention of the field is not only focused on the detection of
distant worlds but also on the characterisation of exoplanetary populations
(Udry & Santos 2007; Udry 2010; Winn & Fabrycky 2015), revelation of their
formation and orbital history (Mulders 2018; Jontof-Hutter 2019), and the
detailed study of their atmospheric signatures (Seager & Deming 2010;
Kaltenegger 2017; Fujii et al. 2018).

The detection and study of planets around giant stars is particularly
interesting, providing information about the architecture and evolution of
systems orbiting around stars more evolved than our Sun. The advantage of
observing more massive stars at later stages of evolution, in this case along
the red giant branch (RGB), is that they have a decreased surface temperature
and slower surface rotation rate compared to what is observed during the pre-
main sequence (PMS) and main-sequence (MS) phases. This has the effect of
increasing the number of absorption lines and their sharpness (because of
rotational broadening) in the spectra, thus enabling the measurement of
precise stellar radial velocities (RV) suitable for exoplanet searches. The
trend in searching for planets around evolved intermediate-mass stars began
with the announcement of a planet orbiting the K2 III giant $\iota$ Draconis
(Frink et al. 2002), and led to the discovery of more than 100 systems.
However, the correct analysis and interpretation of the radial-velocity
variations remains challenging because of the significant and potentially
periodic intrinsic variability of red giants, which can mimic planetary
signals (see Walker et al. 1989; Hatzes & Cochran 1993, 1994, 1999; Frandsen
et al. 2002; De Ridder et al. 2006; Hekker 2006, for more detailed discussion
on the different potential origins of this variability). In this context, the
CORALIE radial-velocity search for companions around evolved stars (Ottoni
2021 submitted, CASCADES) started monitoring a sample of more than 600 G- and
K-type giant stars in the southern hemisphere in 2006 (see Sect. 2.1.1).

Probing more massive stars is also useful for discussions of the main
competing planet formation scenarios. Indeed, stellar mass has a significant
impact on the evolution of the protostellar disc (Ribas et al. 2015), and thus
its lifetime and ability to form giant planets: a more massive star with a
shorter disc lifetime would favour the disc instability scenario, which can
form massive planets on very short timescales of a few thousand years (e.g.
Helled et al. 2014; Raymond et al. 2014).

While the field of exoplanetology was undergoing its rapid evolution, another
area of research started acquiring importance, asteroseismology. Rapidly,
synergies between exoplanetary studies and the seismic characterisation of
stars became a standard example of successful multidisciplinary studies (e.g.
Christensen-Dalsgaard et al. 2010; Batalha et al. 2011; Huber et al. 2013b, a;
Huber 2018; Campante et al. 2018). Indeed, the precise characterisation of
distant worlds required access to reliable and precise stellar parameters,
such as mass, radius, and age. With the development of sophisticated seismic
modelling techniques, such achievements become feasible, especially owing to
the space-based photometry missions such as CoRoT, Kepler, TESS, and, in the
future, PLATO. Adding the recent publication of Gaia DR2 (Evans et al. 2018;
Gaia Collaboration et al. 2018a), the achievable precision and accuracy of
such detailed studies has reached unprecedented levels for a large number of
stars.

We start in Sect. 2 with a brief description of the CASCADES survey and the
method used for determining stellar parameters. We also present the
acquisition and analysis of the spectroscopic measurements, and a study of the
intrinsic variability of the star HD 29399 using both spectroscopic and
photometric data. In Sect. 3 we present the orbital solution of our new
planetary companion. Section 4 details the asteroseismic analysis, from the
peak-bagging procedure to obtain individual oscillation frequencies for the
host star to the thorough modelling that we carry out, combining forward
modelling and local and global minimisation techniques with seismic
inversions. Finally, in Sect. 5, we investigate both the role of dynamical
tides during the PMS and the role of equilibrium tides on the MS and the RGB
using the stellar parameters determined in Section 4.

##  2 Observations, stellar properties, and intrinsic variability

###  2.1 Observations and stellar parameters

####  2.1.1 The CASCADES survey

The CORALIE radial-velocity Search for Companions ArounD Evolved Stars
(CASCADES) is a 14-year survey of a volume-limited sample of evolved stars of
intermediate mass. The main motivation for this survey was to better
understand the formation of planetary systems and their evolution around stars
more massive than the Sun by completing existing studies of giant host stars
and their companions. Observations began at the end of 2006, and have been
conducted since then with the CORALIE spectrograph on the 1.2-m Leonard Euler
Swiss telescope located at La Silla Observatory (Chile). For a detailed
description of the definition of the sample and first results of the survey,
see Ottoni et al. (2021, submitted), hereafter Paper I. Complete information
on instrumental aspects is given for instance in Queloz et al. (2000),
Ségransan et al. (2010), and Ségransan et al. (2020, submitted).

We collected 28 radial-velocity measurements for HD 29399 with the CORALIE
spectrograph over a time-span of more than 13 years. The obtained spectra have
an average signal-to-noise ratio of 85 (at 5’500 Å) for typical exposure times
of $\sim$180 s. Table 6 gives the list of these radial-velocity measurements
and uncertainties.

####  2.1.2 Additional measurements

An additional set of measurements was used for the analysis: four unpublished
data points from the HARPS spectrograph (Mayor et al. 2003) (mounted on the
3.6 m ESO telescope at La Silla Observatory, Chile), and 22 radial-velocity
points published by Wittenmyer et al. (2017) from the UCLES spectrograph
(Diego et al. 1990) on the AAT and the CHIRON spectrograph (Tokovinin et al.
2013) on the 1.5 m telescope at CTIO.

####  2.1.3 Spectroscopic parameters

The spectroscopic parameters of the stars in the CASCADES sample are provided
in Paper I. Following the method described in Alves et al. (2015), we derived
the effective temperature $T_{\rm{eff}}$, surface gravity $\rm{log\,g,}$ and
metallicity $\rm{[Fe/H]}$ using the CORALIE high-resolution spectra. The
values obtained were in agreement with those found for the subsample of stars
in common with Alves et al. (2015), and with the values from the Gaia-DR2
(Brown et al. 2018). The spectroscopic parameters derived for HD 29399, along
with other stellar parameters obtained from the literature, are presented in
Table 1.

Table 1: Observed and inferred stellar parameters.

|  | ref. | HD 29399
---|---|---|---
|  | TIC | 38828538
|  | GAIA DR2 | 4675576135153914368
Sp. Type |  | [1] | K1III
$V$ | [mag] | [2] | 5.79 $\pm$ 0.01
$B-V$ | [mag] | [2] | 1.03 $\pm$ 0.01
$BC$ |  | [3] | -0.317 $\pm$ 0.019
$\pi$ | [mas] | [4] | 22.62 $\pm$ 0.05
$d$ | [pc] | [5] | 44.2 $\begin{subarray}{c}+0.1\\\ -0.1\end{subarray}$
$M_{V}$ | [mag] | [2,4,5] | 2.56 $\pm$ 0.01
$Bp-Rp$ | [mag] | [4] | 1.170 $\pm$ 0.003
$G$ | [mag] | [4] | 2.26 $\pm$ 0.01
$T_{eff}$ | [K] | [4] | 4803 $\begin{subarray}{c}+66\\\ -64\end{subarray}$
|  | [6] | 4845 $\pm$ 52
$log\,g$ | [cm s-2] | [6] | 3.25 $\pm$ 0.13
$[Fe/H]$ | [dex] | [6] | 0.14 $\pm$ 0.03
$M_{*}$ | [M⊙] | [7] | 1.17 $\pm$ 0.10
$L_{*}$ | [L⊙] | [2,3,4] | 10.04 $\pm$ 0.20
$R_{*}$ | [R⊙] | [2,3,4,6] | 4.50 $\pm$ 0.11
Massopt | [M⊙] | [8] | 1.15 $\pm$ 0.04
Radiusopt | [R⊙] | [8] | 4.47 $\pm$ 0.02
X0 |  | [8] | 0.681 $\pm$ 0.01
Z0 |  | [8] | 0.0170 $\pm$ 0.001
$\alpha_{\rm{MLT}}$ | [HP] | [8] | 2.00 $\pm$ 0.05
Age | [Gy] | [8] | 6.20 $\pm$ 0.5

  * •

[1] - HIPPARCOS catalogue (ESA 1997), [2] - TYCHO-2 catalogue (Høg et al.
2000), [3] - Alonso et al. (1999), [4] - Gaia DR2 (Brown et al. 2018), [5] -
Bailer-Jones et al. (2018), [6] - this paper (see Sect. 2), [7] - model-
independent mass from seismic inversion (see Section 4), [8] parameter of the
optimal stellar model (see Section 4).

Figure 1: Periodogram of the radial-velocity data (first panel), of the
residuals of the radial velocities after subtraction of the fitted periodic
signal (second panel), of the contrast (third panel), of the FWHM (fourth
panel), of the bisector inverse span (fifth panel), and of the equivalent
width of H$\alpha$ activity (sixth panel). The red vertical line represents
the fitted period in the radial-velocity at 892.7 days. Horizontal lines, from
bottom to top, are the FAP levels at 10%, 1%, and 0.1% respectively.

###  2.2 Stellar analysis of HD 29399

We first analysed the radial-velocity time series using the radial-velocity
module of the DACE web platform, which provides open access to a wide range of
observational and theoretical exoplanet data with the corresponding data
visualisation and analysis tools111https://dace.unige.ch/radialVelocities/?.
The formalism of the radial-velocity data analysis implemented in DACE is
described in Ségransan et al. (2021, submitted) and is mainly based on
algorithms presented in Díaz et al. (2014) and Delisle et al. (2016, 2018).

Figure 2: Correlation plot of the radial velocities and H$\alpha$ activity
index time-series for HD 29399. A non-significant correlation is observed,
with a weighted Pearson coefficient value of $R_{P}=0.078\pm 0.103$ and a
weighted Spearman’s rank of $R_{S}=0.017\pm 0.106$.

Our standard approach to search for periodic signals in radial-velocity time-
series is to follow an iterative process consisting in looking for successive
significant dominant peaks in the periodogram of the corresponding radial-
velocity residuals. At each step of the iteration, the radial-velocity
residuals are computed by readjusting the full model composed of the N
independent Keplerians, potential linear, quadratic, or cubic drift terms to
fit long-term trends, the different instrumental offsets, and additional white
noise terms222We fit a combination of white-noise terms corresponding to
individual instrumental precision and intrinsic stellar jitter. The
instrumental precision is well known for each version of CORALIE:
$\sigma_{COR98}=5.0\pm 0.5$ m s-1, $\sigma_{COR07}=8.0\pm 0.5$ m s-1,
$\sigma_{COR14}=3.0\pm 0.5$ m s-1. We proceeded with the periodicity search by
computing the periodogram of the data in the range 10$-$10 000 days using the
algorithm implemented on DACE (see Delisle et al. 2020a, b), and using the
false alarm probability (FAP) to assess the significance of the signal
following the formalism of Baluev (2008).

A periodic signal at 896 days clearly stands out in the periodogram of the
radial velocities of HD 29399 (Fig. 1), but the signal could have various
origins. Radial-velocity variations can be linked to several, potentially
periodic effects that can mimic planetary signals. Giant stars exhibit short-
period solar-like radial pulsations (Walker et al. 1989; Hatzes & Cochran
1993, 1994; Frandsen et al. 2002; De Ridder et al. 2006; Hekker 2006), as well
as non-radial oscillations (Hekker et al. 2006a; De Ridder et al. 2009; Hekker
& Aerts 2010) with lifetimes of hundreds of days (Dupret et al. 2009). They
can also produce longer period variations from a combination of magnetic
cycles (Santos et al. 2010; Dumusque et al. 2011), beating of modes, or
rotational modulations of features on the stellar surface (starspots,
granulation, etc.). To test those possible sources of radial-velocity periodic
signal, we also thoroughly checked for variations of the line profile of the
cross correlation function (CCF), and of the classical spectroscopic
chromospheric indicator (in this case the H$\alpha$ index), as well as the
long-term photometric variation of the star. This is described in the
following sections.

####  2.2.1 Spectroscopic indicator variations

Intrinsic stellar variability can be tracked through changes in the shape of
the spectral lines, and consequently of the CCF (the product of convolution of
the spectrum with a template, used for the estimate of the radial velocities).
The profile of the CCF can be monitored by computing its first moments (Aerts
& Eyer 2000). Here, as proxies, we use the contrast, the full width at half-
maximum (FWHM), and the bisector inverse span (BIS) measured on the CCF. In
the case of HD 29399, none of these show any significant periodicity in their
respective periodogram (Fig. 1). We also checked for potential correlations
between these parameters and radial velocities by computing the corresponding
weighted Pearson coefficient. No significant correlation was found.

On the other hand, we have to mention that Wittenmyer et al. (2017) reported a
$765$-day variation signal with a 50 m s-1 semi-amplitude in the radial-
velocity data obtained with the UCLES and CHIRON spectrographs, which they
associated with an intrinsic stellar variation. After checking for several
potential ‘intrinsic’ origins for the variation, inconclusively333Spots were
rejected because of the large surface coverage required for HD 29399. Such
coverage is needed because of the slow rotation and large radius that lead to
a lower estimate of the period of rotation of $\sim$169 days. The presence of
a debris disc, as first hypothesised by Wittenmyer et al. (2017), was also
excluded because to explain the radial-velocity variation it would have
required the debris disc to be heated to 1500 K, which is too close to the
star to produce such a long periodicity, the authors observed correlations
between the radial velocities and photometric data, and the equivalent width
of H$\alpha$ activity (see their Fig. 6), which led them to the conclusion
that the radial-velocity signal was intrinsic to the star, and the ‘planet’
labelled a false positive.

To check this affirmation, we monitored the impact of magnetic activity on the
chromosphere by computing the H$\alpha$ chromospheric index from our CORALIE
spectra, produced the same figure as in Wittenmyer et al. (2017), and computed
the weighted Pearson correlation coefficient RP and the Spearman’s rank RS
using a bootstrap randomisation technique. These estimates lead to non-
significant correlation values: R${}_{P}=0.078\pm 0.103$ and R${}_{S}=0.017\pm
0.106$ (see Fig. 2). We therefore do not confirm the result reported in
Wittenmyer et al. (2017).

####  2.2.2 Photometric variability analysis

Wittenmyer et al. (2017) also checked the All-Sky Automated Survey V-band
photometric data (ASAS-3, Pojmański 2002) for variability due to intrinsic
stellar processes, and a $\sim$765-day periodicity was also found in the data
(see Wittenmyer et al. 2017, Fig. 1). This highly significant signal and the
large photometric variability of the star further supported the interpretation
of the radial-velocity signal as being intrinsic to the star.

Following Wittenmyer et al. (2017) we also checked the ASAS-3 data 444Courtesy
of Grzegorz Pojmanski, who sent us the complete photometric dataset, along
with technical information on the survey and precautions to take when using
the data. for variability due to intrinsic stellar processes or surface
rotational modulation. We only considered the data flagged as GRADE A quality.
It is important to point out that the ASAS-3 photometry has a saturation level
at $\sim$6 mag, depending on the observing conditions and epoch, which clearly
impacts the consistency and quality of the data acquisition for our star. The
focus of the camera was not stable over time because of many instrumental
issues, the observing scheme and data handling pipeline have evolved over the
years although the data have never been reduced again, and there are obvious
consequences of varying weather conditions such as extinction due to tiny
clouds, increased seeing due to wind, fog, and tracking problems.

From 1850 to 2400 [HJD-2 450 000], there was no treatment of saturation,
causing important discontinuities observed in the data, and exposures were 180
s in length. The induced patterns can generate non-negligible signals in the
periodogram of the data. Later, some saturation corrections were included.
After 2950 [HJD-2 450 000] (Nov, 2003), instead of taking one 180 s exposure,
the standard observation consisted of three 60 s exposures, which reduced the
saturation level by over 1 mag. From that point until the end of 2010, the
system was considered as more or less stable. Despite the fact that the
saturation correction procedure improved the data, for bright stars (V ¡
$\sim\leavevmode\nobreak\ $6.5-7) saturation is still present and is a clear
concern for HD 29399. We also observe in several targets of our sample that
the photometric data between 2950 and 3300 [HJD-2 450 000] presents a very low
dispersion clump that does not match the data acquired later. This ‘step’
creates the equivalent of a long period trend that completely disappears when
removing this portion of the data.

Figure 3: Light curve (top) and periodogram (bottom) of HD 29399 from the
ASAS-3 data (Pojmański (2002)). (Top) Data acquired after 3300 [HJD-2 450
000], the ASAS-3300 dataset (see text). (Bottom) Periodogram of the ASAS-3300
dataset. The red vertical line corresponds to the 897.2-day period found in
the radial velocities, and the dashed-vertical line to the 765-day period
found by Wittenmyer et al. (2017). Horizontal lines, from bottom to top, are
the FAP levels at 10%, 1%, and 0.1% respectively. Table 2: Radial-velocity
observation statistics, best-fit solutions of the model with instrumental
offsets, nuisance parameters, Keplerian orbital parameters, and inferred
planetary parameters.

|  | HD 29399b
---|---|---
Observations
$N_{obs}$ |  | 61
$T_{span}$ | $[days]$ | 4811
$rms_{tot}$ | $[m.s^{-1}]$ | 24.71
$rms_{res}$ | $[m.s^{-1}]$ | 11.18
$\chi^{2}_{red}$ |  | 1.46
Offsets (1)
$\gamma_{COR07}$ | $[m/s]$ | 31659.7 $\pm$ 3.0
$\Delta\,RV_{COR14-COR07}$ | $[m/s]$ | 12.2 $\pm$ 3.7
$\Delta\,RV_{CHIRON-COR07}$ | $[m/s]$ | -31641.5 $\pm$ 6.9
$\Delta\,RV_{UCLES-COR07}$ | $[m/s]$ | -31663.0 $\pm$ 3.8
$\Delta\,RV_{HARPS03-COR07}$ | $[m/s]$ | 20.6 $\pm$ 6.8
Instrumental Noises
$\sigma_{COR98}$ | $[m/s]$ | 5.3 $\pm$ 1.0
$\sigma_{COR07}$ | $[m/s]$ | 7.8 $\pm$ 1.4
$\sigma_{COR14}$ | $[m/s]$ | 3.0 $\pm$ 0.5
$\sigma_{CHIRON}$ | $[m/s]$ | 7.6 $\pm$ 7.1
$\sigma_{UCLES}$ | $[m/s]$ | 4.7 $\pm$ 3.3
$\sigma_{HARPS03}$ | $[m/s]$ | 3.4 $\pm$ 2.6
Stellar Jitter
$\sigma_{jit}$ | $[m.s^{-1}]$ | 8.9 $\pm$ 1.5
Drifts
Lin. | $[m/s/yr]$ | 1.9 $\pm$ 0.6
Keplerians
$P$ | $[days]$ | 892.7 $\pm$ 5.9
$K$ | $[m.s^{-1}]$ | 29.9 $\pm$ 2.2
$e$ |  | 0.05 $\pm$ 0.05
$\omega$ | $[deg]$ | -13.1 $\pm$ 85.1
$\lambda_{0}$ (2) | $[deg]$ | 151.9 $\pm$ 5.0
$T_{p}$ (2) | $[rjd]$ | 5965.0 $\pm$ 210.0
$a$ | $[au]$ | 1.913 $\pm$ 0.008
$m_{2}\,sin\,i$ (3) | $[M_{J}]$ | 1.57 $\pm$ 0.11

  * •

1 The reference instrument is COR07.

  * •

2 The mean longitude is given at $BJD=2\,450\,000$ [d] while $2\,450\,000$ has
been subtracted from the date of passage through periastron (TP).

  * •

3 Using the model-independent mass from seismic inversions (see Sect. 4.2)

Figure 4: First panel: Radial velocities of HD 29399 from CORALIE (COR98 in
blue, COR07 in orange and COR14 in green), HARPS (in red), UCLES (in pink),
and CHIRON (in grey). Overplotted are the fitted Keplerian orbit (purple
curve) and the linear trend (dark line). Second panel: Residual radial
velocities after subtracting the Keplerian fit. Third panel: Radial velocities
phased to the fitted period at 890.9 days. We note that the uncertainties of
the HARPS point have been increased to take into account the calibration of
the offset and the intrinsic variability of the star (see text).

For these reasons and to be on the safe side, we decided to only use for our
analysis the data acquired after 3300 [HJD-2 450 000] (which we refer to as
‘ASAS-3300’) and periodograms, as shown in Fig. 3. The ASAS-3300 dataset
contains 466 epochs spanning 4.9 years with a mean value of 6.12 $\pm$ 0.05
mag and 0.35 mag rms. The dataset has a span of twice the periodicity found in
our CORALIE radial-velocity time-series, and is thus still completely adequate
to check for stellar photometric variability. For the analysis, we also
systematically removed outliers from this subset following a sigma-clipping
process. We observe that those outliers were usually fainter, which reinforces
our intuition that they are linked to bad weather conditions.

The periodogram of the complete photometric time-series shown in Fig.1 of
Wittenmyer et al. (2017) clearly exhibits signals around 200 days
(contribution of data acquired between 1850 and 2400 [HJD-2 450 000]) and 765
days (contribution of data acquired between 2400 and 3300 [HJD-2 450 000])
that completely disappear in the periodogram of the more reliable ASAS-3300
data (see Fig. 3). We can therefore conclude that the photometric analysis
from Wittenmyer et al. (2017) suffered from the non-optimal quality of the
early ASAS data. The conclusion of a false-positive planetary signal in the
radial velocities caused by features on the surface of the star is therefore
not robust.

##  3 Keplerian analysis of the radial velocities

In the absence of any significant periodic signal and correlations in the
activity-related products from the high-resolution spectra, we assumed that
the periodic variation observed in the radial-velocity time-series of HD 29399
is due to a substellar companion orbiting the star. Following the procedure
described in Sect. 2.2, we detected a dominant peak in the periodogram at a
period of $\sim$890.91 days corresponding to a variation with a semi-amplitude
of $\sim$30 ms-1. To fit the parameters of the model, we used the MCMC
algorithm implemented in DACE, developed by Díaz et al. (2014, 2016), to probe
the complete parameter space, with $1.6$ million iterations. We used the
following parameters for the Keplerian model: We used the natural logarithm of
the period (log P) and of the semi-amplitude (log K) to better explore ranges
of several orders of magnitude with a uniform prior; $\sqrt{e\cos{\omega}}$
and $\sqrt{e\sin{\omega}}$ (with $e$ the eccentricity of the orbit and
$\omega$ the argument of the periastron) to obtain a uniform prior for the
eccentricity; and finally the mean longitude at epoch of reference (i.e.,
$BJD=2\,455\,500$ [d]) ($\lambda_{0}$), with a uniform prior. We used a
uniform prior for the COR07 offset of reference, and Gaussian priors for the
relative offsets between COR07 and COR98/14: $\Delta\,RV_{COR98-COR07}$:
$\mathcal{N}(0,4)$ m s-1, $\Delta\,RV_{COR14-COR07}$: $\mathcal{N}(14,4)$ m
s-1. We also used Gaussian priors for the instrumental noise:
$\sigma_{COR98}$: $\mathcal{N}(5,1)$ m s-1, $\sigma_{COR07}$:
$\mathcal{N}(8,1.5)$ m s-1 and $\sigma_{COR14}$: $\mathcal{N}(3,0.5)$ m s-1.
Finally, we used a uniform prior for the stellar jitter parameter. We present
in Appendix B the corner plot of the posterior distributions of the fitted
parameters. Adopting the model-independent stellar mass of 1.17 M⊙ (see Sect.
4.2), the single-planet model yields a minimum mass for the companion of 1.57
MJ, on a 892.7-day period orbit with a semi-major axis of 1.91 au.

In addition, HD 29399 exhibits a clear long-term radial-velocity trend,
probably explained by the presence of an additional companion. The observed
slope could correspond to a substellar object of a couple of Jupiter masses,
with a minimum period of $\sim$8000 days, considering a circular orbit. A
longer time-span is necessary to better constrain this trend. It can be
adequately fitted by a linear drift term, however part of the effect could be
related to the effect of the offset between the disjointed CORALIE
instruments. The use of the four HARPS points555The last HARPS point was
obtained after a major intervention on the instrument, changing the instrument
zero point. An offset was then applied (based on Lo Curto et al. (2015)). We
also increased the uncertainties of the HARPS points to take into account the
effect of the calibration and the minimum intrinsic variability of the star.
broadly spread in time, as well as the inclusion of the CHIRON and UCLES data
sets overlapping with the CORALIE measurements, help to consolidate the
estimate of the offsets between the instruments. In this context, we tried
three types of models: a single planet, a planet $+$ a linear drift, and a
planet $+$ a quadratic drift. A Bayesian model comparison using the Bayesian
information criterion clearly indicates a preference for the model with one
planet $+$ linear drift (change in the BIC of 10 and 2, adding a linear drift,
and then a quadratic term, respectively).

The resulting best model overplotted on the radial velocities, as well as the
residuals around the best solution are shown in Fig. 4. Table 2 presents the
statistics of the distributions (i.e., the median and standard deviation) of
the most common set of Keplerian parameters P, K, e, $\omega$, and the date of
passage through periastron (TP), as well as the distributions of the semi-
major axis and minimum masses derived from the MCMC chains of the fitted
parameters. Appendix B shows the corner plot of the posterior distributions of
the fitted parameters. The weighted rms of the residuals around the solution
is comparable to the radial-velocity dispersion of giant stars with similar
$B-V$ (see Fig.3 from Hekker et al. 2006b).

In light of the presented results, we are fairly confident that the 892.7-day
periodic variation in the radial-velocity data of HD 29399 is not due to
chromospheric stellar activity as there are no significant correlations
between the radial velocities and the classical chromospheric indicators.
Rotational modulation of surface features such as spots can also be excluded,
as they would require a very large percentage of the stellar surface to be
covered, and we do not observe any long-term photometric variability. Finally,
we found no trace of long-period non-radial oscillation modes (either matching
periodicities or corresponding to harmonics in the line profile moments). An
orbiting planet therefore remains a valid hypothesis to explain the radial-
velocity periodic signal detected.

Finally, it is worth noting that long-period planets orbiting giant stars are
good candidates for transit search. Indeed, the decrease in transit
probability due to the long period is compensated by the large radius of the
star. The only downside to using such targets is the small depth of the
transit scaling with the square of the star-to-planet ratio. However, this
remains detectable from space with missions such as TESS (Ricker et al. 2015),
CHEOPS (Benz et al. 2020), and PLATO (Rauer et al. 2014). For the planet
orbiting HD 29399, the transit probability is of the order of 1 % and an
expected depth of $\sim 500$ ppm for a Jupiter-size planet.

##  4 Asteroseismic analysis

###  4.1 Peakbagging and frequency determinations

We studied the acoustic oscillation mode frequencies of HD 29399 (also known
as TIC 38828538) using photometric measurements from TESS. Our method
comprises constructing a power spectrum from the observed flux and measuring
the locations of radial $(\ell=0)$ and quadrupolar $(\ell=2)$ oscillation
modes using the automated peakbagging package PBjam666See
https://github.com/grd349/PBjam (Nielsen et al. 2020).

To construct the power spectrum, we used the lightkurve package (Lightkurve
Collaboration et al. 2018) which makes use of the astropy (Astropy
Collaboration et al. 2013, 2018) and astroquery packages (Ginsburg et al.
2019). We downloaded TESS light curves from the Mikulski Archive for Space
Telescopes (MAST) for all sectors from 1 to 12, except sector 3. Subsequently,
we stitched the Pre-search Data Conditioning Simple Aperture Photometry
(PDCSAP, Stumpe et al. 2012; Smith et al. 2012) flux for each sector, removing
both low-frequency trends, using a Savitzky–Golay filter, and $5$-$\sigma$
outliers. We obtained the power spectrum using the Lomb-Scargle method (Lomb
1976; Scargle 1982). To determine the signal-to-noise ratio, we divided the
power spectrum by an estimate of the background obtained by smoothing with a
moving median in steps of $\frac{1}{2}\log(0.01\,\mu\mathrm{Hz})$.

We used the PBjam package to determine the observed radial, $\nu_{n,0}$, and
quadrupolar, $\nu_{n,2}$, oscillation modes of the star. Initial mode
identification was based on the methods of Davies & Miglio (2016) and a prior
probability distribution constructed from thousands of stars already analysed
using PBjam. The means, $\mu$, and uncertainties, $\sigma$, on the input
parameters used to select stars from the prior are given in Table 3. The input
large frequency separation, $\Delta\nu$, and frequency at maximum power,
$\nu_{\mathrm{max}}$, were obtained using the methods described in Hekker et
al. (2012). We adopted the input effective temperature, $T_{\mathrm{eff}}$,
and colour, $G_{\mathrm{BP}}-G_{\mathrm{RP}}$, from Gaia DR2 (Gaia
Collaboration et al. 2016, 2018b). The inputs primarily determined the window
in which we selected stars from the prior for subsequent mode identification;
they had no influence on the final peakbagging step.

We performed initial mode identification by fitting the asymptotic relation
(Mosser et al. 2013) and used the emcee package (Foreman-Mackey et al. 2013)
to build the final posterior distributions of the modes and provide their
identification. This resulted in values for the large frequency separation,
$\Delta\nu=14.93\pm 0.02\,\mu\mathrm{Hz}$, and the frequency at maximum power,
$\nu_{\mathrm{max}}=196.5\pm 2.5\,\mu\mathrm{Hz}$, both within $1$\- and
$2$-$\sigma$ of their input values, respectively.

Table 3: Global stellar property estimates for TIC 38828538 used as inputs for the PBjam peakbagging pipeline. Input | $\mu$ | $\sigma$
---|---|---
$\Delta\nu\,(\mu\mathrm{Hz})$ | 14.88 | 0.05
$\nu_{\mathrm{max}}\,(\mu\mathrm{Hz})$ | 193.34 | 1.47
$T_{\mathrm{eff}}\,(\mathrm{K})$ | 4803 | 70
$G_{\mathrm{BP}}-G_{\mathrm{RP}}\,(\mathrm{dex})$ | 1.17 | 0.01
Table 4: Individual observed asteroseismic frequencies for TIC 38828538. The radial and quadrupolar oscillation modes, with their statistical uncertainties, are given in the left and right columns respectively. $\nu_{n,0}\,(\mu\mathrm{Hz})$ | $\nu_{n,2}\,(\mu\mathrm{Hz})$
---|---
$139.619\pm 0.070$ | $137.717\pm 0.257$
$154.063\pm 0.051$ | $152.152\pm 0.061$
$168.590\pm 0.027$ | $166.727\pm 0.043$
$183.575\pm 0.024$ | $181.751\pm 0.043$
$198.352\pm 0.017$ | $196.522\pm 0.035$
$213.258\pm 0.040$ | $211.439\pm 0.031$
$228.564\pm 0.039$ | $226.817\pm 0.032$
$244.017\pm 0.224$ | $241.788\pm 0.240$

After the mode identification, we fit a Lorentzian profile to each mode by
sampling its posterior distribution using the Bayesian package PyMC3
(Salvatier et al. 2016). Each mode location, $\nu_{n,l}^{\,\prime}$, from the
previous step was used as the mean of a prior normal distribution,
$\mathcal{N}(\mu,\sigma)$, on the Lorentzian centre given by,

$\nu_{n,l}\sim\mathcal{N}\left(\nu_{n,l}^{\,\prime},0.03\Delta\nu\right),$ (1)

where $\mu$ and $\sigma$ are the mean and standard deviation, respectively.
All other parameters from the previous steps were relaxed.

The oscillation mode locations are given in Table 4, and plotted on the
periodogram in Fig. 5 together with their $68\%$ credible regions. These are
the observed frequencies, uncorrected for any shifts due to the radial-
velocity of the star. The echelle diagram in Fig. 6 shows the locations of the
radial and quadrupolar oscillation modes phase-folded by the large frequency
separation determined above. We give the uncertainties as the standard
deviation of the posterior samples for each mode location.

Figure 5: Asteroseimic signal-to-noise ratio power spectrum for HD 29399 (TIC
38828538) in light grey and the smoothed spectrum in black. The locations of
the radial, $l=0,$ and quadrupolar, $l=2,$ oscillation modes are shown with
dashed lines, and their 68% credible regions are shaded in blue and orange,
respectively.

Figure 6: Echelle diagram for HD 29399 (TIC 38828538). The signal-to-noise
ratio in frequency, $\nu,$ modulo the large frequency separation,
$\Delta\nu=14.93\,\mu\mathrm{Hz}$. The locations of radial, $l=0$ (blue), and
quadrupolar, $l=2$ (orange), oscillation modes are shown with error bars given
by their 68% credible regions.

###  4.2 Seismic modelling

In this section, we carry out a detailed modelling of HD 29399, combining
seismic and classical constraints from both ground-based spectroscopic surveys
and Gaia parallax values. We use the Liège stellar evolution code (Scuflaire
et al. 2008b) combined with the Liège stellar oscillation code (Scuflaire et
al. 2008a) to compute adiabatic oscillations. The available constraints are
summarised in Table 1 with their corresponding references. Following Buldgen
et al. (2019a), the modelling is divided into three steps:

  1. 1.

Forward modelling with the Asteroseismic Inference on a Massive Scale (AIMS,
Rendle et al. 2019; Montalbán et al. 2020) software using radial oscillations
and classical constraints.

  2. 2.

Inversion of the stellar mean density to determine a model-independent mass
range.

  3. 3.

Forward modelling combining the inversion results, classical constraints, and
frequency separations of the radial and quadrupolar modes.

The first step uses a global minimisation technique to determine first
estimates of the global stellar properties and carry out a thorough
exploration of the parameter space. This approach ensures a reliable and
accurate inversion procedure in step 2, which is used to determine the model-
independent mass interval from which the planetary properties are deduced.
Using these results, we carry out a third step using a local minimisation
technique to find a solution in better agreement with the observations,
delivering a robust age estimate for the system better accounting for seismic
and non-seismic constraints.

In the first step, we used two different grids of stellar evolutionary models
described in Rendle et al. (2019) and used in Buldgen et al. (2019b) to study
a sample of Kepler eclipsing binaries. This grid uses the GN93 solar
abundances (Grevesse & Noels 1993) and the corresponding metallicity scale. We
also recomputed a second grid using the AGSS09 revision of the solar
abundances (Asplund et al. 2009) to explore their impact on the determined
stellar parameters. Both grids are evolved up to a cut-off lower
$\nu_{\rm{Max}}$ value determined in Section 4.1. Moreover, Rendle et al.
(2019) and Buldgen et al. (2019b) used an Eddington atmosphere for the outer
boundary layers, which is unsuitable for fitting classical parameters on the
RGB with a solar-calibrated mixing-length parameter value. Therefore, in the
second grid we used a $T(\tau)$ relation from Model C of Vernazza et al.
(1981), which is more suitable for our needs (Sonoi et al. 2019). None of the
grids include microscopic diffusion. The observed frequencies have been
corrected from the line-of-sight Doppler velocity shifts following the
recommendations of Davies et al. (2014). Both grids assume a solar calibrated
mixing-length parameter, use the FreeEOS equation of state (Irwin 2012), and
OPAL opacity tables (Iglesias & Rogers 1996). We summarise the properties of
the AGSS09 grid in Table 5 and refer the reader to Rendle et al. (2019) for
the properties of the GN93 grid.

Table 5: Properties of the AIMS stellar evolution model grids. Parameters | AGSS09 grid values
---|---
Mass $\left(M_{\odot}\right)$ |  $1.00-1.70$ $(0.02\;\rm{step})$
$X_{0}$ |  $\left[0.68,0.72\right]$ $(0.01\;\rm{step})$
$Z_{0}$ |  $\left[0.010,0.050\right]$ $(0.001\;\rm{step})$
$\alpha_{\rm{MLT}}$ $\left(H_{P}\right)$ | $2.03$
$\nu_{\rm{Max}}$ cutoff $\left(\mu\rm{Hz}\right)$ | $40$

The results are illustrated for the AGSS09 grid in Fig. 7. As we can see, the
individual radial frequencies are relatively well fitted by AIMS. A similar
conclusion is reached for the final $\left[\rm{Fe}/\rm{H}\right]$ of the
model, which is $0.095\leavevmode\nobreak\ \rm{dex,}$ and the effective
temperature of $4880\leavevmode\nobreak\ K$. However, the radius of the model
slightly disagrees with the value determined from Gaia and the spectroscopic
constraints reported in Table 1. This small discrepancy is also seen in the
luminosity of the model, found around $10.78\leavevmode\nobreak\ L_{\odot}$
and results from the higher weight of the individual frequencies compared to
the classical constraints. Nevertheless, these discrepancies remain very small
and within $1\sigma$ of the observations. The age of the star is found to be
around $\rm 6.33\leavevmode\nobreak\ Gyr$. The results for the GN93 grid are
similar, with good agreement found in frequencies, but, unsurprisingly given
the atmosphere used, significant mismatches are found for the classical
parameters. Nevertheless, given that they fit the individual frequencies (as
shown in Fig. 8) for the AGSS09 model, both models are suitable for seismic
inversion.

Figure 7: Probability distribution functions for the mass (left) and radius
(right) for HD 29399 (TIC38828538) obtained using AIMS. The vertical lines
indicate the position of the best model obtained from a simple scan of the
grid.

Figure 8: Echelle diagram illustrating the agreement between theoretical and
observed radial $(\ell=0)$ frequencies for the AIMS AGSS09 solution.

Following Reese et al. (2012) and Buldgen et al. (2019b), we carried out an
inversion of the mean density using only radial modes to avoid non-linear
behaviours. The inversion procedure is based on the integral relations linking
relative frequency differences to corrections of thermodynamic variables of
the stellar structure (Dziembowski et al. 1990). From these relations, we can
determine the coefficients, $c_{i}$, such that the mean density of the
reference model is computed in a model-independent way from a recombination of
the individual frequencies:

$\displaystyle\frac{\delta\bar{\rho}}{\bar{\rho}}=\sum_{i}c_{i}\frac{\delta\nu_{i}}{\nu_{i}}.$
(2)

We used the SOLA inversion technique (Pijpers & Thompson 1994), following the
guidelines of Reese et al. (2012) and Buldgen et al. (2019b), applying
different surface effect corrections (namely those of Kjeldsen et al. 2008;
Sonoi et al. 2015; Ball & Gizon 2014; Ball et al. 2016) as well as different
reference models. The final inverted value is $0.01815\,\pm\,1\times 10^{-4}$
g cm-3. Combining it with the radius determined from spectroscopic and
astrometric data, we determine a model-independent mass interval of
$1.17\,\pm\,0.11$ M⊙ for HD 29399777While this mass interval is determined
independently from any stellar model, it is however dependent on the accuracy
of the radii values, hence on the accuracy of bolometric corrections,
extinction laws, and so on..

Figure 9: Left panel: HR diagram showing the CLES evolutionary track and the
position of the optimal seismic solution. Right panel: Agreement of the
$d_{0,2}$ for the optimal solution with the values determined from TESS data.

The final modelling step is carried out using a Levenberg-Marquardt algorithm
(used e.g. in Buldgen et al. 2016; Farnir et al. 2019; Buldgen et al. 2019a)
with the following set of constraints: $\bar{\rho}_{\rm{Inv}}$, the inverted
mean density, $L$, $T_{\rm{eff}}$, $\left[\rm{Fe}/\rm{H}\right]$, and the
individual small frequency separations, denoted
$d_{0,2}=\nu_{n,0}-\nu_{n-1,2}$, using the mass, $M$, the age, $t$, the
mixing-length parameter, $\alpha_{\rm{MLT}}$ , and the initial hydrogen,
$X_{0}$, and metal content, $Z_{0}$, as free parameters. The stellar
evolutionary models and their oscillation spectra are computed on the fly to
avoid any interpolation issues in between and alongside the tracks. Departing
from a solar-calibrated $\alpha_{\rm{MLT}}$ value implies additional
degeneracies in the modelling. However, as we already have a good grasp on the
expected mass of the star, we can ascertain that there will be a gain in
accuracy, as departures from a solar-calibrated value are expected from the
analysis of averaged $3D$ hydrodynamical simulations (Trampedach et al. 2014;
Magic et al. 2015; Sonoi et al. 2019).

Using the small separations is justified as they allow mitigation of the
surface effects, providing more robust estimates than those determined from
individual frequency fitting. Moreover, the small separations have been shown
by Montalbán et al. (2010) to be very sensitive to the mass of the star as a
function of its mean density, which serves our purpose of refining the stellar
mass determination.

We illustrate the agreement of our final model with the constraints used in
Fig. 9, where we present on the left panel its evolutionary track showing its
position on the lower RGB and on the right panel the agreement with the
observed values of the individual $d_{0,2}$. The final parameters for the star
are given in the second part of Table 1, where the reported uncertainties are
determined from the analysis using the Levenberg-Marquardt minimisation. The
precision of the stellar parameters is quite good, as the respective
precisions of $d_{0,2}$, and the mean density are very high. However, it would
be unrealistic to assume that this is the true precision of the determined
stellar parameters. Varying the physical ingredients of the stellar evolution
model may well lead to variations at a level comparable to the precision
reported here. However, we note that this solution is well within the model-
independent mass interval given by our inversion procedure. Therefore, below,
we use the mass value from the second part of Table 1 but consider the
$1\leavevmode\nobreak\ \sigma$ uncertainty to be given by the combination of
the inverted mean density and radius values from Gaia parallaxes and
spectroscopic parameters when determining the planetary parameters.

##  5 Orbital history

In this section, we describe our study of the orbital evolution of the system.
In addition to the non-rotating models of HD 29399 computed with the CLES
stellar evolution code, detailed models that include a comprehensive treatment
of rotational effects and magnetic fields are computed with the Geneva stellar
evolution code (Eggenberger et al. 2008) from the PMS to the RGB phase.

The stellar parameters determined through the seismic modelling described in
the second part of Table 1 are used as a starting point to compute these
models. The assumption of shellular rotation (Zahn 1992) is used and the
internal transport of angular momentum (AM) is then solved simultaneously to
the evolution of the star by accounting for meridional currents, transport by
the shear instability, and transport by magnetic fields in the framework of
the Tayler-Spruit dynamo (Spruit 2002). Consequently, the following equation
for internal AM transport is solved:

$\rho\frac{{\rm d}}{{\rm
d}t}\left(r^{2}\Omega\right)_{M_{r}}=\frac{1}{5r^{2}}\frac{\partial}{\partial
r}\left(\rho r^{4}\Omega U(r)\right)+\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(\rho(D_{\rm shear}+\nu_{\rm TS})r^{4}\frac{\partial\Omega}{\partial
r}\right)\,,$ (3)

with $r$, $\rho(r)$, and $\Omega(r)$ being the radius, the mean density, and
the mean angular velocity on an isobar, respectively. The radial dependence of
the meridional circulation velocity in the radial direction is denoted $U(r)$,
while $D_{\rm shear}$ corresponds to the diffusion coefficient for AM
transport by the shear instability (see Eggenberger et al. 2010, for more
details). The transport of AM by magnetic fields is taken into account through
the viscosity $\nu_{\rm TS}$; this magnetic process is able to operate only
when the shear parameter $q=-\frac{\partial\ln\Omega}{\partial\ln r}$ is
larger than a minimum threshold given by $q_{\min}$ (see Eggenberger et al.
2019a, for mode details). These models that account for rotational and
magnetic effects are able to correctly reproduce the internal rotation of the
Sun together with the observations of surface velocities of stars in open
clusters (Eggenberger et al. 2005, 2019a). However, we note that the same
models do not provide sufficient coupling to correctly reproduce the
asteroseismic core rotation rates of red giants (Cantiello et al. 2014; den
Hartogh et al. 2019), which indicates that an unknown efficient additional AM
transport process is needed for subgiant (Eggenberger et al. 2019b) and red
giant stars (Eggenberger et al. 2012, 2017).

Figure 10: Evolution of the orbital distance after the MS (to the RGB branch).
The solid red line represents the orbit of the planet. The solid black line
shows the value for the orbital distance below which the planet will be
engulfed. The magenta area represents the stellar convective envelope, while
the cyan area shows the extension of the radiative interior.

To follow the evolution of a planetary system, these stellar models are
coupled to our orbital evolution code by taking into account the exchange of
angular momentum between the star and the orbit (Privitera et al. 2016c, a;
Rao et al. 2018). This enables us to test whether an eventual impact due to
dynamical tides (mainly during the PMS) and/or equilibrium tides (at later
evolutionary stages) could have significantly changed the orbit of the planet,
leading to the current architecture of the system, or whether the system has
retained the same architecture since its formation.

The physics included in the orbital evolution code is described in Rao et al.
(2018). We account for the planetary atmospheric evaporation occurring in
Jeans escape or hydrodynamic escape regime conditions, depending on the
properties of the system considered. For the computation of the emitted
stellar X-ray luminosity, following the work by Tu et al. (2015), we use the
formula of Wright et al. (2011), while for the EUV luminosity we refer to the
work of Sanz-Forcada et al. (2011).

We started our study considering the planet at its current orbital distance
from the host star at the beginning of the evolution, namely $\rm
a_{in}=1.910$ au, with a minimum mass $\rm M_{pl}=1.59$ $\rm{M_{J}}$. The
rotational history of the host star HD 29399 being unknown, we considered
three different initial surface angular velocities ($\rm 3.2$, $\rm 5$ and
$\rm 18\,\Omega_{\odot}$), covering the range for slow, medium, and fast
rotators as deduced from surface rotation rates of solar-type stars observed
in open clusters at different ages (Eggenberger et al. 2019a). A disc lifetime
of 6 Myr is used for $\rm\Omega_{in}=3.2,5.0\leavevmode\nobreak\
\Omega_{\odot}$ and 2 Myr for $\rm\Omega_{in}=18\leavevmode\nobreak\
\Omega_{\odot}$. During the disc-locking phase, the surface angular velocity
of the star is simply assumed to remain constant. After remaining constant
during the disc-locking phase, the surface velocity rapidly increases due to
the PMS contraction and reaches a peak at an age of about 25 Myr. Then the
surface rotation rate exhibits a decrease along the evolution during the MS,
indicating the braking of the stellar surface by magnetised winds. After the
end of the MS, the surface rotation decreases rapidly due to the expansion of
the envelope during the subgiant and the red giant phases.

Using the initial setup described above, we did not find any appreciable
change in the orbit of the planet along the evolution. Given the mass of the
planet and that of the star, the planet is found to be at too great an initial
distance for its motion to be significantly affected by tides. Even during the
RGB phase, the limited increase in the stellar radius (a value of only about
$\rm 12.9\,R_{\odot}$ is found for HD 29399, corresponding to $\rm 0.0586$ au)
does not give rise to efficient equilibrium tides. These results are in
perfect agreement with previous results reported in Privitera et al. (2016b).

For the sake of completeness, we computed the orbital evolution of the planet
when the host star climbs the RGB, exploring a range of lower initial orbital
distances to determine the maximal value below which the planet would be
engulfed, denoted here $\rm{a_{Max}}$. As shown in Fig. 10, we find a value of
$\rm{a_{Max}}\approx$ 0.12 au. In the case of HD 29399, this is more than 15
times lower than the observed orbital distance.

##  6 Conclusion

In this paper, we illustrate the advantages of synergy between both
exoplanetary studies and seismic characterisation of stars with the detailed
characterisation of a long-period giant planet orbiting the evolved star HD
29399 detected within the 14-year CASCADES survey (Ottoni et al. 2021,
submitted), and observed by TESS during 11 continuous sectors of 27 days. The
detailed analysis allows us to provide very precise stellar parameters, far
beyond what is achievable from stellar evolutionary tracks alone, paving the
way for an analysis of the orbital evolution of the planetary system.

The newly discovered giant planet is a 1.57 MJ companion at an orbital
distance of 1.913 au, with a period of 892.7 days. The radial-velocity time-
series also present a long-term linear trend that may reveal a substellar
companion. We consulted the corresponding ASAS-3 photometry time-series
(Pojmański 2002) to address the announcement of a false-positive by Wittenmyer
et al. (2017) and demonstrate that the photometric time-series must be
considered carefully, and cannot not rule out the planet hypothesis. We also
checked for any correlation with chromospheric activity and spectroscopic
line-profile variations, and find no significant signals.

In addition to the exquisite coverage of radial velocities, HD 29399 has also
been followed for almost a year by TESS, allowing for the precise
determination of its pulsational properties. This allowed us to determine
individual radial and quadrupolar oscillation modes with high precision and
use them for a detailed modelling of the host star. This allowed us to
determine the mass of HD 29399 to be $1.17\pm 0.10$ M⊙, the radius to be
$4.47\pm 0.02\,\rm{R}_{\odot}$, and the age of the system to be around $6.2\pm
0.5\,\rm{Gyr}$.

The determination of these detailed properties enabled us to study the orbital
evolution of the system using non-standard stellar evolution models computed
with the Geneva stellar evolution code. Our study shows that neither dynamical
nor equilibrium tides have been able to affect the orbital evolution of the
planet, whatever the initial rotational velocity considered for the host star.
Our results are in agreement with those of Privitera et al. (2016c), showing
that a Jupiter-mass planet orbiting at such long periods should remain
unaffected. We carried out our computations up to the tip of the RGB and
predict no engulfment in the future evolution of the system.

Overall, our study demonstrates a perfect example of a multidisciplinary,
multi-instrument approach, where a long-duration ground-based survey combined
with high-cadence space-based observations has led to the discovery of a long-
period planet and precise asteroseismic characterisation of its host star. We
combined precise stellar and planetary characterisations in our study of the
orbital evolution of the system. Such multidisciplinary approaches are crucial
to understanding the statistical properties of planetary systems, their
formation history, and the evolution of their properties as a result of
interactions with their host star.

###### Acknowledgements.

We thank the referee for the useful comments that helped to improve the
quality of the manuscript. This work has been carried out within the framework
of the National Centre for Competence in Research PlanetS supported by the
Swiss National Science Foundation. The authors acknowledge the financial
support of the SNSF.
This publication makes use of the The Data & Analysis Center for Exoplanets
(DACE), which is a facility based at the University of Geneva (CH) dedicated
to extrasolar planet data visualisation, exchange and analysis. DACE is a
platform of the Swiss National Centre of Competence in Research (NCCR)
PlanetS, federating the Swiss expertise in Exoplanet research. The DACE
platform is available at https://dace.unige.ch.
We thank all observers at La Silla Observatory from the past fourteen years
for their quality work. We acknowledge financial support from the Swiss
National Science Foundation (SNSF). This work has, in part, been carried out
within the framework of the National Centre for Competence in Research PlanetS
supported by SNSF.
C.P. acknowledges funding from the Swiss National Science Foundation (project
Interacting Stars, number 200020-172505). G.B. acknowledges funding from the
SNF AMBIZIONE grant No 185805 (Seismic inversions and modelling of transport
processes in stars). P.E. has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 833925, project STAREX). This article used an
adapted version of InversionKit, a software developed within the HELAS and
SPACEINN networks, funded by the European Commission’s Sixth and Seventh
Framework Programmes. AM acknowledges support from the ERC Consolidator Grant
funding scheme (project ASTEROCHRONOMETRY, https://www.asterochronometry.eu,
G.A. n. 772293).

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##  Appendix A Radial-velocity data

Table 6: Radial-velocity measurements and uncertainties for HD 29399 obtained
with the CORALIE and HARPS spectrographs.

JD-2 400 000 | RV | e_RV | Instrument
---|---|---|---
| $[m\,s^{-1}]$ | $[m\,s^{-1}]$ |
54035.785459 | 31627.21 | 3.25 | CORALIE98
54371.782019 | 31694.66 | 5.69 | HARPS
54472.673890 | 31643.86 | 5.06 | CORALIE07
54689.929529 | 31619.75 | 2.72 | CORALIE07
54764.767582 | 31637.14 | 4.96 | CORALIE07
54772.828843 | 31650.47 | 3.67 | CORALIE07
54777.770615 | 31635.50 | 3.91 | CORALIE07
54817.667858 | 31644.19 | 5.27 | CORALIE07
54830.700228 | 31646.48 | 2.98 | CORALIE07
54892.579440 | 31677.34 | 5.74 | HARPS
54894.550719 | 31686.96 | 5.69 | HARPS
55176.740700 | 31676.88 | 2.48 | CORALIE07
55303.503222 | 31696.39 | 5.20 | CORALIE07
55446.907968 | 31642.22 | 2.45 | CORALIE07
55619.560825 | 31625.53 | 2.54 | CORALIE07
55858.875008 | 31680.45 | 2.86 | CORALIE07
55929.675679 | 31692.51 | 2.31 | CORALIE07
56952.825823 | 31709.20 | 8.81 | CORALIE07
57346.699387 | 31648.69 | 2.29 | CORALIE14
57763.623634 | 31714.20 | 2.90 | CORALIE14
57829.493845 | 31725.52 | 2.64 | CORALIE14
57971.892631 | 31697.45 | 2.73 | CORALIE14
58031.714400 | 31675.10 | 2.96 | CORALIE14
58094.561422 | 31674.10 | 2.38 | CORALIE14
58150.558636 | 31645.80 | 8.71 | HARPS
58172.514760 | 31657.57 | 2.59 | CORALIE14
58178.551656 | 31673.40 | 2.75 | CORALIE14
58323.934316 | 31652.97 | 2.58 | CORALIE14
58422.854271 | 31673.54 | 3.63 | CORALIE14
58482.776199 | 31691.53 | 2.73 | CORALIE14
58775.738473 | 31718.29 | 3.07 | CORALIE14
58846.731357 | 31697.70 | 2.68 | CORALIE14

  * •

Note that small radial-velocity offsets between the different versions of each
instruments have to be considered. In this case, the offset between COR98 and
COR07 has been fixed at 0 m s-1, and the offset between COR14 and COR07 has
been considered as a free parameters in the model (see Table 2).

##  Appendix B MCMC - corner plot distributions of fitted parameters

Figure 11: Posterior distributions of the fitted parameters of HD29399. Each
panel contains the two-dimensional histograms of the 1 200 000 samples (after
removal of the burn-in, the first 25% of the chains). Contours are drawn to
improve the visualisation of the 1$\sigma$ and 2$\sigma$ confidence interval
levels. The upper panels of the corner plot show the probability density
distributions of each orbital parameter of the final MCMC sample. The vertical
dashed lines mark the 16th, 50th and the 84th percentiles of the overall MCMC
samples, delimiting the 1$\sigma$ confidence interval.