11institutetext:  Landessternwarte, Zentrum für Astronomie der Universität
Heidelberg, Königstuhl 12, 69117 Heidelberg, Germany
11email<EMAIL_ADDRESS>
# Non-radial oscillations mimicking a brown dwarf orbiting the cluster giant
NGC 4349 No. 127

Dane Spaeth Fellow of the International Max Planck Research School for
Astronomy and Cosmic Physics at the University of Heidelberg (IMPRS-HD) Sabine
Reffert  Emily L. Hunt  Adrian Kaminski  Andreas Quirrenbach

(Received 28 March 2024 / Accepted 09 July 2024)

###### Abstract

Context. Several evolved stars have been found to exhibit long-period radial
velocity variations that cannot be explained by planetary or brown dwarf
companions. Non-radial oscillations caused by oscillatory convective modes
have been put forth as an alternative explanation, but no modeling attempt has
yet been undertaken.

Aims. We provide a model of a non-radial oscillation, aiming to explain the
observed variations of the cluster giant NGC 4349 No. 127. The star was
previously reported to host a brown dwarf companion, but whose existence was
later refuted in the literature.

Methods. We reanalyzed 58 archival HARPS spectra of the intermediate-mass
giant NGC 4349 No. 127. We reduced the spectra using the SERVAL and RACCOON
pipelines, acquiring additional activity indicators. We searched for
periodicity in the indicators and correlations between the indicators and
radial velocities. We further present a simulation code able to produce
synthetic HARPS spectra, incorporating the effect of non-radial oscillations,
and compare the simulated results to the observed variations. We discuss the
possibility that non-radial oscillations cause the observed variations.

Results.  We find a positive correlation between chromatic index and radial
velocity, along with closed-loop Lissajous-like correlations between radial
velocity and each of the spectral line shape indicators (full width at half
maximum, and contrast of the cross-correlation function and differential line
width). Simulations of a low-amplitude, retrograde, dipole ($l=1,m=1$), non-
radial oscillation can reproduce the observed behavior and explain the
observables. Photometric variations below the detection threshold of the
available ASAS-3 photometry are predicted. The oscillation and stellar
parameters are largely in agreement with the prediction of oscillatory
convective modes.

Conclusions. The periodic variations of the radial velocities and activity
indicators, along with the respective phase shifts, measured for the
intermediate-mass cluster giant NGC 4349 No. 127, can be explained by a non-
radial oscillation.

###### Key Words.:

stars: individual: NGC 4349 No. 127 – stars: oscillations – stars: evolution –
planets and satellites: detection – techniques: radial velocities

##  1 Introduction

With the advent of large-scale transit surveys such as Kepler (Borucki et al.
2010), K2 (Howell et al. 2014), and TESS (Ricker et al. 2015), the number of
known exoplanets has increased drastically to over 5600 confirmed planets to
date.111https://exoplanetarchive.ipac.caltech.edu/ (June 2024) Despite this
immense wealth of newly discovered planets, less than 200 of these have been
found orbiting evolved stars.222https://www.lsw.uni-
heidelberg.de/users/sreffert/giantplanets/giantplanets.php Due to intrinsic
biases of the transit surveys, the majority of planets orbiting giant stars
have been found using the radial velocity (RV) method.

Targeting giant stars in RV surveys offers important additions to surveys
targeting main-sequence stars. Specifically, it allows the detection of
planets around the evolved counterparts of stars more massive than about
$1.5\,\mathrm{M_{\sun}}$ (Reffert et al. 2015), which are increasingly
inaccessible to RV measurements during their main-sequence phase, due to high
effective temperatures and rotation rates (Sato et al. 2003; Galland et al.
2005; Johnson et al. 2007; Lagrange et al. 2009; Assef et al. 2009).
Furthermore, analyzing the planet population in later evolutionary stages
allows studies of the effect of stellar evolution on planetary systems.
Therefore, detecting a statistically meaningful sample of planets around giant
stars is crucial in order to capture a full picture of the general planet
population. For simplicity, we also refer to low-mass brown dwarfs ($M\lesssim
30M_{\mathrm{Jup}}$) as planets within this work.

However, detecting planets orbiting giants using the RV method has its own
challenges. First, most giant stars are known to undergo short-term,
stochastically driven p-mode oscillations, called solar-like oscillations. For
giants, these typically occur on timescales of hours to days and lead to RV
jitter on the order of $\sim 10\,\mathrm{m\,s^{-1}}$ to $\sim
20\,\mathrm{m\,s^{-1}}$, even for giant stars considered to be relatively
stable (Kjeldsen & Bedding 1995; Hekker et al. 2006, 2008). This intrinsic
noise can often be overcome with sufficient statistics, but limits the
detectable orbital companions to those with rather high masses (typically
Jovian or higher).

Less understood, and therefore more challenging, is the recent discovery of
several giant stars showing long-term, periodic RV variations, some of which
were already attributed to planets or brown dwarfs, yet were later shown to be
incompatible with orbital companions. These include $γ$ Dra (Hatzes et al.
2018), Aldebaran (Reichert et al. 2019), $ϵ$ Cyg (Heeren et al. 2021), 42 Dra
(Döllinger & Hartmann 2021), Sanders 364333SIMBAD identifier: BD+12 1917 (Zhou
et al. 2023), 41 Lyn, 14 And (both Teng et al. 2023), and the cluster giants
NGC 2423 No. 3, NGC 2345 No. 50, NGC 3532 No. 670, and NGC 4349 No.
127444SIMBAD identifiers: NGC 2423 3, NGC 2345 50, HD 96789, and NGC 4349 127
(Delgado Mena et al. 2018, 2023). Döllinger & Hartmann (2021) show that
doubtful planet detections around giant stars mostly accumulate at periods
between 300 d and 800 d and around stars with radii $R>21\,R_{\sun}$, pointing
at a common origin of these RV variations.

Wolthoff et al. (2022) present the planet occurrence rate around evolved stars
as a function of the orbital period, which follows a broken power law relation
peaking at $P\sim 720\,\mathrm{d}$, very close to the periods of the
identified false positive detections. If a significant number of as-yet-
unidentified false positives contaminate the comparably small sample of
planets around giants, significant implications on the planet occurrence rate
and its interpretation are to be expected (Wolthoff et al. 2022).

Moreover, as the phenomenon is still poorly understood and can mimic planets
quite convincingly in RV data, it is often impracticable to unambiguously
confirm new planets around giant stars (see, e.g., Tala Pinto et al. 2020;
Niedzielski et al. 2021; Jeong et al. 2022; Teng et al. 2022; Zhou et al.
2023). This is due to the fact that it is not yet clear which observables are
the most decisive diagnostics to separate intrinsic signals from orbital
companions for giants. This problem is made even worse by the varying
availability, in data sets from different spectrographs, of spectral
diagnostics targeting such intrinsic variations. These are commonly referred
to as activity indicators, and we adopt this term even though we do not
(mainly) target magnetic activity in this work.

So far, several intrinsic origins have been discussed. Radial oscillations can
usually be ruled out due to the long-period nature of the suspected signals,
which appear at periods much longer than the fundamental radial mode (Hatzes &
Cochran 1993; Cox et al. 1972; Hatzes et al. 2018; Reichert et al. 2019). In
the binary system $\epsilon$ Cyg, the heartbeat phenomenon was considered
(Heeren et al. 2021); however, it is not applicable to single stars. Magnetic
surface structures, such as cool spots, can in most cases equally be ruled
out, as they should manifest themselves in much larger photometric variations
than were reported for the above stars (Reichert et al. 2019; Heeren et al.
2021). Delgado Mena et al. (2023) (and references therein) propose magnetic
plages or other magnetic structures that locally reduce convection and could
cause RV variations without associated photometric variations. However, the
magnetic fields on the surface of most giant stars are still not well
understood (see, e.g., Aurière et al. 2011, 2013, 2015; Konstantinova-Antova
et al. 2024, for notable exceptions).

Non-radial oscillations, on the other hand, can cause large amplitude RV
variations at long periods (Hatzes 1996; Hatzes & Cochran 1999). Hatzes et al.
(2018) link the aforementioned phenomenon of false positive planet detections
to oscillatory convective modes presented by Saio et al. (2015). These modes
were proposed as the origin for the sequence D of the long secondary periods
(LSPs) observed for variable bright giant stars (Wood et al. 1999; Wood 2000).
These are dipole ($l=1$) $g^{-}$-modes that become oscillatory in the non-
adiabatic conditions present in the envelopes of luminous ($\log
L/L_{\sun}\gtrsim 3$) giant stars (Takayama & Ita 2020). Reichert et al.
(2019) build on this argument, showing in their Fig. 8 that both $\gamma$ Dra
and Aldebaran fall into a region in the period-luminosity diagram in which an
extrapolation of the models by Saio et al. (2015) could potentially explain
the observed variations. Nevertheless, no satisfying attempt to apply models
of non-radial oscillations to the observables of a known false positive
evolved planet host has been published to date.

In this work we reexamine the RV variations of the cluster giant NGC 4349 No.
127, previously thought to host a brown dwarf companion in a
$P=677.8\,\mathrm{d}$ orbit originally published by Lovis & Mayor (2007).
However, Delgado Mena et al. (2018) show that, while the RV signal is stable
in HARPS (High Accuracy Radial velocity Planet Searcher) spectra, variations
of the full width at half maximum (FWHM) of the cross-correlation function
(CCF) and the $\mathrm{H\alpha}$ index at the orbital period are present. The
authors therefore refute the companion’s existence, favoring rotational
modulation of magnetic activity as the most likely alternative, while not
ruling out non-radial oscillations. Delgado Mena et al. (2023) affirm these
findings by presenting 11 additional RV measurements.

Here, we present a reanalyis of the HARPS data set adding measurements of the
chromatic index (CRX) and differential line width (dLW) variations by using
the SERVAL (Zechmeister et al. 2018) pipeline, as well as the contrast of the
CCF by using the RACCOON (Lafarga et al. 2020) reduction software. We reveal a
linear correlation between the RV, on the one hand, and the CRX and the
$\mathrm{H\alpha}$ indicator, on the other hand, respectively. We further show
that the dLW, the FWHM of the CCF, and the contrast of the CCF each correlate
with the RV in a “closed-loop” relation. We additionally present simulations
of the observational effects of non-radial oscillations that can closely
reproduce the periods and amplitudes of the activity indicator variations, as
well as the phase differences with the RV.

This paper is structured as follows. In Sect. 2 we give an overview of the
stellar parameters, observations, and data reduction. In Sect. 3 we present
the pyoscillot simulation suite developed to simulate the effect of non-radial
oscillations on HARPS spectra. We present newly detected correlations between
the activity indicators and the RVs in Sect. 4, and show that they are
consistent with a retrograde, dipole, non-radial oscillation model. In Sect. 5
we present general observational properties of different oscillation modes. We
discuss our non-radial oscillation model in the context of oscillatory
convective modes in Sect. 6, before summarizing our findings in Sect. 7.

##  2 Observations and stellar parameters

###  2.1 Stellar parameters

NGC 4349 No. 127 is the most evolved star of the open cluster NGC 4349
(Delgado Mena et al. 2018), located at a distance $d=1788.9\pm
2.9\,\mathrm{pc}$ (Hunt & Reffert 2023), with an estimated age of around
$300\,\mathrm{Myr}$ (Holanda et al. 2022; Tsantaki et al. 2023; Hunt & Reffert
2023). The stellar parameters vary considerably between different studies and
are summarized in Table 1.

In Fig. 1, we plot a color-magnitude diagram based on Gaia DR3 photometry
(Gaia Collaboration et al. 2016, 2023), highlighting the position of NGC 4349
No. 127 as the orange star marker. The membership list and isochrone fit were
taken from Hunt & Reffert (2023). The star’s exact evolutionary state is
ambiguous between the first or second ascent on the red giant branch (RGB), as
was discussed in the literature (Delgado Mena et al. 2016; Tsantaki et al.
2023). One reason for this ambiguity is the differential extinction
$0.57^{+0.30}_{-0.31}\,\mathrm{mag}$ in the V band present in the cluster
(Hunt & Reffert 2023), which is also evident from the spread at the turn-off
of the main sequence in Fig. 1. This complicates the determination of a
specific extinction value for NGC 4349 No. 127, adding uncertainty to its
intrinsic photometry. The difficulty in determining an accurate extinction
value could also (partially) explain the variability of the stellar parameters
summarized in Table 1.

We also tested the stellar parameters using Bayesian inference based on Gaia
DR3 parallaxes and photometry using
SPOG+555https://github.com/StephanStock/SPOG (Stock et al. 2018). We use
extinction values taken from the Starhorse catalog (Anders et al. 2022). The
metallicity was taken from Tsantaki et al. (2023). The tool yields a
probability of $99.1\%$ for the star being on the horizontal branch (HB) or
second ascent on the RGB. The stellar parameters derived by SPOG+ are listed
in Table 1.

While the results for $T_{\mathrm{eff}}$ are roughly consistent with other
determinations in the literature, SPOG+ estimates a significantly lower mass,
radius, and luminosity. The age determination of the star
$\tau=930^{+300}_{-330}\,\mathrm{Myr}$ also significantly deviates from the
estimated age for the whole cluster from other studies. We assume that stellar
parameters derived from spectroscopy or ones that take the whole cluster into
account should generally be regarded as more accurate. SPOG+ is also quite
sensitive to the choice of extinction values, which are difficult to
determine. However, as masses of giant stars derived from evolutionary models
are suspected to be overestimated (see, e.g., Lloyd 2011), we note this
somewhat lower mass estimate, as well as the lower mass estimates by Anders et
al. (2022) and Mortier et al. (2013), and the general variability of the
parameters in Table 1.

The star has furthermore been found to show enhanced Li abundance compared to
other giant stars in the cluster and compared to field stars (Carlberg et al.
2016; Delgado Mena et al. 2016; Tsantaki et al. 2023), which was proposed to
be caused by planet engulfment. However, Holanda et al. (2022) notes that the
determined Li abundance is still below the traditional limit for Li-rich giant
stars $\log\epsilon(\textrm{Li})\geq 1.50$.

Given the precision of the results and favoring parameters derived by
spectroscopy, we base our analysis and discussion on the stellar parameters
derived by Tsantaki et al. (2023) and Delgado Mena et al. (2023). For the
simulations presented in Sect. 4, we chose
$(T_{\mathrm{eff}}=4500\,\mathrm{K},\log g=2.0,[\mathrm{Fe/H}]=0.0)$ as the
closest grid point of the PHOENIX spectral library. We refrain from
interpolating the base spectrum of the simulations to the exact parameters due
to the differences shown in the literature (Table 1) and to avoid
interpolation errors. Using the relation by Hekker & Meléndez (2007), we
determine the value of the macroturbulence to be $\zeta=4822\pm
39\,\mathrm{m\,s^{-1}}$ at this effective temperature.

Table 1: Overview of the stellar parameters determined for NGC 4349 No. 127 in
the literature.

$M$ | $R$ | $L$ | $T_{\mathrm{eff}}$ | $\log g[\mathrm{cm\,s^{-2}}]$ | $[\mathrm{Fe/H}]$ | $\xi$ | $v\sin i$ | $\zeta$ | $A_{V}$ | Ref.
---|---|---|---|---|---|---|---|---|---|---
$M_{\sun}$ | $R_{\sun}$ | $L_{\sun}$ | K |  | dex | $\mathrm{km\,s^{-1}}$ | $\mathrm{km\,s^{-1}}$ | $\mathrm{km\,s^{-1}}$ | mag |
$2.20^{+0.38}_{-0.20}$ | $23.3^{+0.9}_{-1.4}$ | $211^{+3}_{-4}$ | $4566^{+123}_{-70}$ | $2.04^{+0.13}_{-0.06}$ | - | - | - | - | $0.67^{+0.06}_{-0.07}$ |  SPOG+, 1 (phot)
$3.29^{+0.28}_{-0.56}$ | - | - | - | - | - | - | - | - | $0.86^{+0.19}_{-0.15}$ | 2 (phot)
$3.01\pm 0.24$ | - | - | $4417\pm 12$ | $1.78\pm 0.05$ | $0.17\pm 0.02$ | - | - | - | 1.16 | 3 (spec)
$3.41\pm 0.27$ | - | - | $4567\pm 31$ | $1.83\pm 0.08$ | - | - | - | - | 1.16 | 3 (phot)
- | $38.0\pm 2.6$ | $575.4$ | - | - | - | - | 4.81 | 4.66 | 1.16 | 4, 3 (spec+phot)
- | - | - | $4420\pm 70$ | $1.70\pm 0.20$ | $-0.13\pm 0.13$ | $1.51\pm 0.07$ | $4.20\pm 0.60$ | $3.0$ | $1.24\pm 0.06$ | 5 (spec)
- | - | - | $4597\pm 39$ | $1.78\pm 0.09$ | - | - | - | - | $1.24\pm 0.06$ | 5 (phot)
$1.56^{+0.20}_{-0.36}$ | - | - | $4199^{+61}_{-57}$ | $1.43^{+0.01}_{-0.03}$ | $-0.29^{+0.01}_{-0.11}$ | - | - | - | $0.67^{+0.06}_{-0.07}$ | 6 (phot)
$2.4$ | - | - | $4370\pm 78$ | $1.70\pm 0.22$ | $-0.20\pm 0.11$ | $1.70\pm 0.06$ | $<2$ | $\sim 5.0$ | 0.90 | 7 (spec)
$3.81\pm 0.23$ | $37.0\pm 4.9$ | $646^{+205}_{-156}$ | $4503\pm 70$ | $1.99\pm 0.19$ | $-0.13\pm 0.04$ | $1.77\pm 0.07$ | $6.13$ | - | - | 8 (spec+phot)
$1.37\pm 0.37$ | - | - | $4445\pm 87$ | $1.64\pm 0.23$ | $-0.25\pm 0.06$ | $1.84\pm 0.08$ | - | - | - | 9 (spec+phot)
$3.77\pm 0.36$ | $44.7\pm 2.5$ | $774^{+467}_{-291}$ | $4519\pm 100$ | $1.92\pm 0.2$ | $-0.21\pm 0.12$ | $2.08\pm 0.20$ | - | - | $1.08$ | 10 (spec+phot)

666Presented are mass $M$, radius $R$, luminosity $L$, effective temperature
$T_{\mathrm{eff}}$, surface gravity $\log g$, metallicity $[\mathrm{Fe/H}]$,
microturbulent velocity $\xi$, rotational broadening $v\sin i$, macroturbulent
velocity $\zeta$, and the V-band extinction $A_{V}$ used to determine the
parameters. The values for reddening $E(B-V)$ given by Holanda et al. (2022)
and Carlberg et al. (2016) were converted assuming $A_{V}=3.1E(B-V)$.

(1) Stock et al. (2018); (2) Hunt & Reffert (2024); (3) Tsantaki et al.
(2023); (4) Delgado Mena et al. (2023); (5) Holanda et al. (2022); (6) Anders
et al. (2022); (7) Carlberg et al. (2016); (8) Delgado Mena et al. (2016); (9)
Mortier et al. (2013); (10) Ghezzi et al. (2010)

Figure 1: Color-magnitude diagram for the open cluster NGC 4349 based on Gaia
DR3 photometry. The membership list and isochrone fit were taken from Hunt &
Reffert (2023). NGC 4349 No. 127 is the most evolved star of the cluster,
indicated by the orange star marker. Its evolutionary state is ambiguous
between the first or second ascent on the RGB.

###  2.2 Data

We downloaded 58 publicly available, extracted HARPS (Mayor et al. 2003)
spectra from the ESO science
archive777http://archive.eso.org/scienceportal/home for NGC 4349 No. 127. The
spectra were taken between 2005 and 2022 and have signal-to-noise ratios (S/N)
per extracted pixel between 25 and 62 (with one outlier at 14.7) at the peak
of the blaze function of (physical) order 102, which is centered around
$6000\,\textrm{\AA}$. The data were taken as part of a long-term RV survey of
intermediate-mass giant stars in open clusters and were originally published
by Lovis & Mayor (2007) and Delgado Mena et al. (2018, 2023).

We reduced the spectra using the SERVAL pipeline (Zechmeister et al. 2018),
which derives radial velocities via a least-squares optimization of the RV
shift relative to a high signal-to-noise stellar template obtained by coadding
the available observations. SERVAL was shown to produce slightly more precise
RV results than the default HARPS data reduction software (DRS) (Trifonov et
al. 2020). The SERVAL pipeline furthermore computes additional activity
indicators, namely the CRX and the dLW (Zechmeister et al. 2018), which have
proven to be effective indicators to detect activity for main-sequence dwarfs,
especially in context of the CARMENES (Calar Alto high-Resolution search for M
dwarfs with Exoearths with Near-infrared and optical Echelle Spectrographs)
survey (Quirrenbach et al. 2014). It moreover calculates several line indices,
of which the $\mathrm{H}\alpha$ index will be used within this work.

We furthermore utilized the RACCOON reduction software (Lafarga et al. 2020)
to obtain radial velocities using a CCF with a weighted binary mask. The mask
was created from the SERVAL template via the routines provided by the RACCOON
software. It furthermore provides three activity indicators targeting the
contrast and FWHM of the CCF and calculates the Bisector Inverse Slope (BIS)
(Lafarga et al. 2020). We prefer the RACCOON pipeline over the default HARPS
DRS, since the latter is not publicly available and can therefore not be used
to reduce the simulated data presented in Sect. 4. The results of the RACCOON
pipeline are generally in good agreement with the HARPS DRS results, but yield
somewhat higher RV uncertainties compared to HARPS DRS and SERVAL (see Table
3).

We separated the spectra taken before (46 spectra) and after (12 spectra) 2
June 2015, to account for the HARPS fiber
change.888https://www.eso.org/sci/facilities/lasilla/instruments/harps/news.html
We did not separate the spectra further, accounting for the HARPS warm-up on
23 March 2020, as only three spectra were taken after this date. We treated
the two subsets as separate RV time series, allowing for a relative offset.

We restricted the analysis of the activity indicators to the 46 spectra
acquired before the HARPS fiber exchange, as we noted significant offsets of
the activity indicators by analyzing the activity time series of quiet stars
using the HARPS RVBANK (Trifonov et al. 2020) (see also Appendix A of Delgado
Mena et al. 2023). As these offsets are (to our knowledge) poorly understood
and constrained, we decided to focus merely on the spectra acquired prior to
the fiber change. Both the SERVAL and RACCOON RVs and activity indicators are
listed in Table 3.

##  3 Simulations

To study whether non-radial oscillations could explain the variations of the
radial velocities and the activity indicators, we simulate the observational
effect of these on HARPS spectra. The simulations are part of the simulation
suite pyoscillot.999https://github.com/DaneSpaeth/pyoscillot

###  3.1 Description of non-radial oscillations

We start out with a model of the stellar photosphere, using spherical
coordinates ($\theta$, $\phi$), with $\theta$ being the polar angle (or
colatitude), measured from the oscillation pole and ranging from 0 to $\pi$,
and $\phi$ being the azimuthal angle, ranging from 0 to $2\pi$. Each of the
grid points has a local effective temperature $T_{\mathrm{eff}}$ and a local
oscillation velocity vector $\@vec{v_{\mathrm{osc}}}$ defined by the three
components (Kurtz 2006; Hatzes 1996; Schrijvers et al. 1997; Kochukhov 2004)

$\displaystyle v_{r}$
$\displaystyle=\frac{{\partial{\xi_{r}}}}{\partial{t}}=i2\pi\nu\frac{a}{C}Y_{l}^{m}(\theta,\phi)\exp(i2\pi\nu
t),$ (1) $\displaystyle v_{\theta}$
$\displaystyle=\frac{\partial{\xi_{\theta}}}{\partial{t}}=i2\pi\nu\frac{Ka}{C}\frac{\partial
Y_{l}^{m}(\theta,\phi)}{\partial\theta}\exp(i2\pi\nu t),$ (2) $\displaystyle
v_{\phi}$
$\displaystyle=\frac{\partial{\xi_{\phi}}}{\partial{t}}=i2\pi\nu\frac{Ka}{C\sin\theta}\frac{\partial
Y_{l}^{m}(\theta,\phi)}{\partial\phi})\exp(i2\pi\nu t),$ (3)

which are oriented in the directions of the local unit vectors of r, $\theta$,
and $\phi$, respectively. We use the real part of the velocity components to
calculate the final local velocities. Here, $\vec{\xi}$ describes the local
displacement vector, $\nu$ is the oscillation frequency, $a$ is the
displacement amplitude in the radial direction, $K$ is the ratio between the
radial and the horizontal displacement, and $Y_{l}^{m}$ are the spherical
harmonics defined as

$Y_{l}^{m}(\theta,\phi)=\sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}P_{l}^{m}(\cos\theta)\exp(im\phi).$
(4)

Here we introduce the quantum numbers $l$ and $m$. $l$ is the number of line
of nodes on the stellar surface, while $m$ is the azimuthal order quantifying
how many of these line of nodes run through the oscillation pole. It ranges
from $-l\leq m\leq l$. $P_{l}^{m}(\cos\theta)$ are the Legendre Polynomials
defined by

$P_{l}^{m}(\cos\theta)=\frac{(-1)^{m}}{2^{l}l!}(1-\cos^{2}\theta)^{\frac{m}{2}}\frac{\mathrm{d}^{l+m}}{\mathrm{d}\cos^{l+m}\theta}(\cos^{2}\theta-1)^{l}.$
(5)

Thus, the velocity amplitude in the radial direction is given by
$v_{\mathrm{osc}}=2\pi\nu\frac{a}{C}$, and consequently $K\cdot
v_{\mathrm{osc}}$ is the velocity amplitude in the $\theta$ and $\phi$
directions. We introduce the normalization factor
$C=\max(\operatorname{Re}(iY_{l}^{m}(\theta,\phi)))$ for $m\neq 0$, and
$C=\max(\operatorname{Re}(Y_{l}^{m}(\theta,\phi)))$ for $m=0$, such that
$v_{\mathrm{osc}}$ can be regarded as the amplitude of the radial component at
the position of maximum variation on the stellar surface, giving it a
straight-forward interpretation. Both $v_{\mathrm{osc}}$ and $K$ are input
parameters to be defined by the user. In the Cowling approximation, neglecting
perturbations of the gravitational potential, $K$ can be expressed as (Aerts
2021; Cowling 1941)

$K=\frac{GM_{\star}}{\left(2\pi\nu\right)^{2}R_{\star}^{3}}.$ (6)

Associated with the (physical) radial displacement are variations of the local
effective temperature of a scale $\delta T_{\mathrm{eff}}$ and at a phase
$\psi_{T}$. Assuming non-adiabatic conditions, we can express the local
effective temperature as (Dupret et al. 2002; De Ridder et al. 2002)

$T_{\mathrm{eff,local}}=T_{\mathrm{eff}}+\delta
T_{\mathrm{eff}}\operatorname{Re}\left(\frac{Y_{l}^{m}(\theta,\phi)}{\max(\operatorname{Re}(Y_{l}^{m}(\theta,\phi)))}\exp(i(2\pi\nu
t+\psi_{T}))\right),$ (7)

defining $\delta T_{\mathrm{eff}}$ as the amplitude of the temperature
variation at the position of maximum variation on the stellar surface. The
scale of the temperature variation is not influenced by the choice of the
oscillation velocity $v_{\mathrm{osc}}$, such that a reasonable value for
$\delta T_{\mathrm{eff}}$ has to be chosen by the user.

As the velocity components of interest are generally low, we neglect
geometrical distortions of the stellar sphere and keep the stellar radius at
unity at all times. Thus, the simulations can be visualized as mapping the
oscillation’s velocity components onto the unit sphere, neglecting the
displacements of the individual positions. We further do not include surface-
normal variations which were found to play a minor role on the variation of
line widths by De Ridder et al. (2002). However, we note that Townsend (1997)
advocates to include both surface-normal and surface-area variations to
accurately reproduce photometric continuum variations.

###  3.2 Calculation of the synthetic spectrum

After choosing an inclination angle, defined as the angle between the polar
axis of the oscillation and the line of sight, the velocity and temperature
fields are projected onto a flat, uniformly spaced 2D grid of size
$N_{\mathrm{grid}}\times N_{\mathrm{grid}}$, using triangulation and linear
interpolation. The directional velocity components are further projected onto
the line-of-sight vector, and can thereafter be summed up to yield the local
combined oscillation velocity for each element on the grid.

We further add a projected rotational velocity for each position characterized
by the (unprojected) rotational velocity $v_{\mathrm{rot}}$. We fix the axis
of rotation to coincide with the polar axis of the simulated star and thus
with the symmetry axis of the oscillation. The summation of the projected,
combined oscillation velocity and the projected rotational velocity yield the
final, local velocities along the line of sight $v_{i}$ for each position $i$.
This local velocity will subsequently be used to Doppler shift the local
spectra as detailed below.

The projected positions on the grid can be regarded as the center points of
small surface areas with the same projected area. For cells at the limb of the
star, a geometric weight $w_{i}$ is calculated as these are only partially
covering the stellar disk.

For each grid cell, a synthetic stellar spectrum is calculated by performing a
cubic spline interpolation with respect to the local effective temperature
$T_{\mathrm{eff},i}$ within a grid of synthetic spectra taken from the PHOENIX
spectral library101010https://phoenix.astro.physik.uni-goettingen.de/ (Husser
et al. 2013). For computational speed, the cubic spline interpolation is
performed in $0.1\,\mathrm{K}$ steps. For simplicity, we do not interpolate
the spectra with respect to the surface gravity $\log g$ and the metallicity
$[\mathrm{Fe/H}]$. Both are fixed to the closest value for NGC 4349 No. 127,
that is $\log g=2.0$ and $[\mathrm{Fe/H}]=0.0$.

Ideally, one would like to use the specific intensity spectra available from
the PHOENIX library, which were calculated for a model atmosphere under
different observation angles $\mu=\cos{\gamma}$, with $\gamma$ being the angle
between the line of sight and the surface normal. However, these are only
available with a sampling rate of $1.0\,\mathrm{\AA}$, too low to simulate
high-resolution Echelle spectra. We are therefore forced to use the high-
resolution spectra of the PHOENIX spectral library. As these are already disk-
integrated spectra, the final simulated spectrum can only be regarded as a
relatively crude approximation.

####  3.2.1 Implementation of the limb-darkening correction

Since we are interested in the observational effects of non-radial
oscillations, which can be dominated by their horizontal components and thus
strongest toward the limb of the stellar disk, we implement a wavelength-
dependent limb-darkening correction. Hestroffer & Magnan (1998) fit the
intensity profile of the solar disk as a function of $\mu$ and wavelength
$\lambda$ using data provided by Pierce & Slaughter (1977) and Neckel & Labs
(1994) and provide the relation

$I(\mu,\lambda)=1-u(1-\mu^{\alpha(\lambda)}).$ (8)

Hestroffer & Magnan (1998) use $u=1$ and determine
$\alpha\sim-0.023+0.292\lambda^{-1}$ for $\lambda$ in units of $\mathrm{\mu
m}$ and $416\,\mathrm{nm}\lesssim\lambda\lesssim 1099\,\mathrm{nm}$. Since the
wavelength range of HARPS extends down to $378\,\mathrm{nm}$ at the blue end,
we extrapolate the model, despite the discontinuity at $\lambda\sim
390\,\mathrm{nm}$ discussed by Hestroffer & Magnan (1998).

As the underlying PHOENIX spectra are already disk-integrated and therefore
include the effect of limb darkening, we first divide each local spectrum by
the wavelength-dependent intensity profile for the mean angle $\bar{\mu}$.
Next, the limb-darkening effect is added back into the models by multiplying
with the respective intensity profile for each cell according to its local
angle $\mu_{i}$. By this approach, we add an implicit weight based on the
intensity according to the angle $\mu_{i}$ during the spectrum combination
process detailed below. Ideally, one would like to use $\mu$ and wavelength-
dependent models for a star more similar to NGC 4349 No. 127, but such models
are not available in the literature.

We further include the effect of macroturbulent broadening by convolving the
local spectra with a wavelength-dependent Gaussian kernel, specified by a
user-defined macroturbulent velocity $\zeta$. Microturbulent broadening has
already been included during the synthesis of the base PHOENIX spectrum,
albeit at a fixed value for each grid point within the PHOENIX library (Husser
et al. 2013).

####  3.2.2 Implementation of the convective blueshift correction

We also include an estimation for the effect of convective blueshift on the
spectral line bisectors, following the approach by Zhao & Dumusque (2023). As
the PHOENIX spectral library is derived from 1D atmospheric models, they do
not properly reproduce line bisector shapes (Zhao & Dumusque 2023). The
intrinsic bisectors of the PHOENIX spectra therefore have to be removed prior
to adding in a more plausible bisector shape stemming from convective
processes.

First, the PHOENIX spectra are normalized using RASSINE (Cretignier et al.
2020). Using the normalized spectra we first calculate the small wavelength
shifts necessary to remove the intrinsic PHOENIX bisector by measuring the
bisectors of five FeI lines (FeI $5250.2084\,\textrm{\AA}$, FeI
$5250.6453\,\textrm{\AA}$, FeI $5434.5232\,\textrm{\AA}$, FeI
$6173.3344\,\textrm{\AA}$, FeI $6301.5008\,\textrm{\AA}$) and calculating the
average. Next, the wavelength shifts from one of the giant stars presented by
Gray (2005) are calculated using the bisector-removed, normalized spectra.
Both the wavelength shifts are applied to each grid point in the unnormalized
PHOENIX spectra. These are then used during the spectrum combination process.
By testing all available bisectors, we found that the observed CCF bisectors
are well reproduced using the bisector model of the giant star $\beta$ Boo.
While this procedure creates CCF bisectors quite similar to the observed ones,
it has little effect on the observables presented in Sect. 4.

####  3.2.3 Combination of the local spectra

Finally, after including each of the effects on the local spectra (local
temperature variation, limb darkening, convective blueshift), the individual
spectra have to be Doppler-shifted before being summed to yield the disk-
integrated spectrum.

To yield accurate Doppler shifts, we first oversample the wavelength grid at
equidistant $0.001\,\textrm{\AA}$ steps and perform a cubic spline
interpolation for the flux values at the oversampled grid points. After the
spectra are Doppler-shifted according to the summed local velocities of
oscillation and rotation $v_{i}$, the shifted flux values are then linearly
interpolated back onto the original wavelength grid, which is common for all
surface grid points. Finally, the spectra are summed, accounting for the
reduced geometrical weights $w_{i}$ for cells at the edge of the star, and
divided by the sum of all geometrical weights to maintain a reasonable
normalization of the spectra. The final, disk-integrated spectrum can be
expressed as

$I_{\mathrm{integrated}}(\lambda)=\frac{1}{\sum_{i}w_{i}}\sum_{i}I_{i}(T_{\mathrm{eff},i},v_{i},\mu_{i},\lambda)\cdot
w_{i}.$ (9)

Applying the above steps, we calculate synthetic stellar spectra including the
effect of non-radial oscillations. The code is also able to study other
effects such as star spots, which is however beyond the scope of this
publication.

####  3.2.4 Conversion to simulated HARPS spectra

The combined spectrum is next smoothed with a Gaussian kernel to bring the
spectrum to the required resolution for each spectrograph. For HARPS,
$R=115\,000$ (Mayor et al. 2003) is used (CARMENES VIS and NIR channels are
also available). The smoothed spectra are then rebinned onto an extracted
wavelength grid of a real HARPS observation of NGC 4349 No. 127. Conceptually,
converting the combined spectrum from an energy flux to a photon flux would
more closely mimic the acquisition process of a real CCD. However, tests
revealed that the differences are negligible and we thus decided to omit the
conversion for simplicity. Next, a measured blaze function is applied to each
spectral order and the necessary FITS header keywords are altered to mimic a
real observation. Finally, the spectrum is saved in the FITS format and the
RVs can be reduced from the simulated, extracted spectra. The simulation is
performed at user-defined epochs.

The final spectra are then reduced using the RACCOON and SERVAL pipelines –
identical to the processing applied to real observations – yielding the
simulated RVs and activity indicators. For the RACCOON pipeline, we use the
weighted mask created from the real observations. For the SERVAL reduction, a
new template is created for the simulation.

For the simulation results presented below, we performed the simulation with a
grid of size $150\times 150$. This number was chosen for computational speed,
but tests at higher resolutions revealed negligible differences.

##  4 Results

###  4.1 Confirmation of the intrinsic nature of the RV variations

From the 46 spectra acquired prior to the HARPS fiber change in 2015, two (BJD
= 2454323.471811, BJD = 2454349.472032) were found to be strong outliers in
the dLW time series. These have $\mathrm{dLW}=96.6$ and $\mathrm{dLW}=96.4$,
respectively, while the dLW time series for the other spectra varies between
$-21$ and $24$. These two spectra were discarded from all time series. We note
that the two data points do not have particularly low S/N and are not
conspicuous in the RVs or any of the other activity indicators. It remains
unclear, why the two spectra are strong outliers in only one indicator.

Another strong outlier (BJD = 2458849.842858) in the RV time series was
removed from the 12 spectra acquired after 2015. Again, it remains uncertain
why the spectrum deviates strongly in the RVs. It has sufficient S/N, albeit
slightly below average.

The remaining 44 spectra taken before the fiber change and 11 spectra taken
after the fiber change are plotted in Fig. 2 as blue and orange data points,
respectively. We treated the two data sets independently and fitted for an
offset. We plot a fitted sinusoid with period $P=674.0\pm 0.1\,\mathrm{d}$ in
black to illustrate that the RV signal is consistent with a periodic variation
stable over at least fifteen years. This RV variation by itself could easily
be attributed to an orbiting companion, which was however shown not to exist
by Delgado Mena et al. (2018, 2023), based on significant periodicity of the
FWHM of the CCF close to the proposed orbital period. Delgado Mena et al.
(2018) further report periodicity of the $\mathrm{H\alpha}$ index close to the
orbital period but with a false alarm probability (FAP) less significant than
$1.0\%$, which is further reduced in significance when including the spectra
acquired after 2015 (Delgado Mena et al. 2023). Moreover, the authors find a
weak but significant correlation between $\mathrm{H\alpha}$ and RV.

Figure 2: RVs reduced by SERVAL prior to (blue) and after (orange) the HARPS
fiber change in 2015 plotted against time. The error bars are smaller than the
size of the markers. A sinusoidal fit is plotted in black and reveals the RVs
to be consistent with a long-lived, coherent signal that could (when examined
in isolation to other diagnostics) be attributed to a brown dwarf orbiting the
primary.

With the additional activity indicators available through the SERVAL and
RACCOON reductions, we can strengthen these findings. Figure 3 shows a
generalized Lomb-Scargle (GLS) periodogram (Zechmeister & Kürster 2009) of the
44 spectra acquired prior to the HARPS fiber change and reveals that the RV
periodicity in the pre 2015 data set (red dashed line) is accompanied by
further significant peaks of the activity indicators close to the orbital
period. Besides the FWHM, also the contrast of the CCF and the dLW of the
SERVAL reduction show strong peaks more significant than the
$\mathrm{FAP}=0.1\%$ level, which was determined via bootstrapping with 10 000
reshuffles. The peak in the contrast GLS is slightly offset at
$P=645.4\,\mathrm{d}$.

Both the $\mathrm{H\alpha}$\- and the CRX-periodograms furthermore have peaks
at the orbital period, or slightly offset in case of the CRX
($P=645.4\,\mathrm{d}$). While both peaks are formally less significant than
the $\mathrm{FAP}=5\%$ level, the fact that they appear very close to the RV
period and are the largest peak in their respective periodograms is certainly
not a coincidence and indicates a signal at the RV period in H$\alpha$ and
CRX. Only the BIS periodogram is inconspicuous. Taken together, the additional
spectral diagnostics strengthen the findings by Delgado Mena et al. (2018,
2023), rule out a physical companion, and thus confirm the intrinsic origin of
the RV variations.

Figure 3: GLS periodograms of the RVs and activity indicators calculated for
the 44 HARPS spectra acquired prior to the fiber change. The FAP of 5% (dashed
line), 1% (dash-dotted line), and 0.1% (dotted line) were determined using a
bootstrap with 10 000 reshuffles and are plotted for each panel. The strong RV
periodicity at $P=672.7\,\mathrm{d}$ is accompanied by significant periodicity
of the FWHM and contrast of the CCF and the dLW. The $\mathrm{H\alpha}$
indicator of the SERVAL reduction and the CRX show peaks at or very close to
the orbital period that, however, have FAP ¿ 5%. The BIS is inconspicuous.

###  4.2 Correlations between the activity indicators and the RVs

Figure 4: Observed (data points) and simulated (lines) correlations between
the activity indicators and the RVs for 44 spectra taken prior to the HARPS
fiber change. For each panel, the mean (RV and any of the indicators,
respectively) was subtracted. Each data point is color-coded with the phase
according to the best RV period $P=674.0\,\mathrm{d}$. While the CRX (top
left) shows a significant positive correlation with the RVs (r=0.54,
$p(\textit{F}\textrm{-test})=0.002\%$), dLW, FWHM, and contrast of the CCF are
correlated with the RVs in a closed-loop behavior. We plot the linear (CRX)
and elliptical fits (dLW, FWHM, contrast) to the simulated data points for the
best model of a $l=1,m=1$ oscillation mode as solid lines applying the same
color-coding. As a linear relationship is predicted between CRX and RV, we
plot the ascending and descending phase relations on top of the black fit to
the simulations. The simulated ellipses can closely reproduce the observed
behavior including the amplitudes, phases, and directions of correlation.
Figure 5: Observed (data points) and simulated (lines) RVs and activity
indicators of the 44 HARPS spectra acquired prior to 2015 plotted against
time. The same color-coding as in Fig. 4 was applied. The mean RV was
subtracted from both the SERVAL and RACCOON RV time series, as well as the
mean of each indicator time series The color-coded solid lines are sinusoidal
fits to the simulated spectra computed for the best $l=1,m=1$ mode. The
simulation is able to reproduce the amplitudes and phases of all indicators,
although some (real) indicators suffer from large scatter. As the simulations
focus solely on the stellar photosphere, no meaningful variation of the
$\mathrm{H\alpha}$ indicator can be simulated.

To find an alternative explanation for the radial velocity variations, we
searched for correlations between the activity indicators and the RVs. Figure
4 shows the CRX, dLW, contrast, and FWHM of the CCF plotted against the RV for
the 44 HARPS spectra taken prior to the 2015 fiber exchange. For all
indicators, we plot the absolute deviation from the mean value, which was
determined by fitting sinusoids and determining the offset, to allow an easier
comparison with the simulated variations.

All data points in Fig. 4 are color-coded with the RV period
$P=674.0\,\mathrm{d}$ present in the full data set (including the spectra
taken after 2015). It is evident from the top left panel that the CRX is
positively correlated with the RV with a Pearson’s r value $r=0.54$ with
p-value $p(r)=0.015\%$. We fitted a linear relation using orthogonal distance
regression and perform an F-test (Fisher 1925) to validate the relation’s
significance against a constant model, resulting in a p-value
$p(\textit{F}\textrm{-test})=0.002\%$. Following Tal-Or et al. (2018) and
Benjamin et al. (2018), we regard the correlation as significant. As Delgado
Mena et al. (2023), we further note a significant positive correlation of the
$\mathrm{H\alpha}$ index with the RV with $r=0.48$ and
$p(\textit{F}\textrm{-test})=0.2\%$ (shown in Fig. 10).

Furthermore, we observe closed-loop correlations with the RV for the dLW,
contrast, and FWHM of the CCF. That is, the activity indicators correlate
elliptically with the RVs with each data point’s position on the ellipse given
by the phase of the variation, indicated by the color-coding. Such ellipses
are effectively Lissajous curves resulting from sinusoidal variations at the
same period but with different amplitudes and phase shifts close to
$\frac{\pi}{2}$.

The color-coding further reveals the direction of correlation, which is anti-
clockwise for FWHM and dLW but clockwise for the contrast, which is to be
expected as FWHM and contrast should be inversely dependent on each other. We
note that the dLW measures a similar line variation as the FWHM and the
contrast and should therefore not be regarded as an entirely independent
indicator (Zechmeister et al. 2018; Jeffers et al. 2022). However, as the
results stem from different reduction pipelines (RACCOON vs SERVAL) with
different approaches to derive the RVs, it is reassuring that we find a
similar behavior for both reductions.

###  4.3 Simulation of a dipole, retrograde ($l=1,m=1$) oscillation mode

In order to explain these peculiar correlations, we simulated a set of models
using the pyoscillot simulation suite detailed in Sect. 3. Motivated by
oscillatory convective modes as published by Saio et al. (2015), we tested
different configurations for dipole ($l=1$) modes using the stellar parameters
($T_{\mathrm{eff}}=4500\,\mathrm{K},\log
g=2.0,[\mathrm{Fe/H}]=0.0,\zeta=4822\,\mathrm{m\,s^{-1}}$) for the base
spectrum as discussed in Sect. 2.

We include the chromatic effect of limb darkening and the effect of convective
blueshift. For the latter, we calculated the CCF bisectors for all measured
bisectors of giant stars from Gray (2005) and compared them to the observed
CCF bisectors for NGC 4349 No. 127. The best match was found for the bright
giant star $\beta$ Boo. We note, however, that the inclusion of the convective
blueshift plays a minor role in modeling the relative variations of the
activity indicators and mostly provides an offset for the absolute value of
the BIS.

Figure 6: CCFs and bisector profiles for the observed spectra (black) and the
simulated spectra (red). Top: CCFs plotted against the velocity shift for the
observed and simulated spectra. The CCFs were shifted with the RV to lie on
top of each other. The measured bisector for each line is overplotted. Bottom:
Zoomed-in image of the measured bisectors. The simulated bisectors (red) show
much less scatter than the observed ones, but generally follow the same
rightward trend.

Using the RV period and the stellar properties, the ratio between the
horizontal and radial oscillation components can be estimated as $K=1856\pm
404$. Therefore, the oscillation is dominated by the horizontal components,
such that the product $K\cdot v_{\mathrm{osc}}$ is the decisive input
parameter and should be interpreted in conjunction (since changes in
$v_{\mathrm{osc}}$ can be balanced out by changing $K$). This behavior was
confirmed through test simulations. $K$ was fixed to $K=1856$ when searching
for the best fitting parameters.

Initial tests at different inclination angles revealed that retrograde ($m=1$)
modes are the only dipole modes capable of reproducing the observed
correlations for NGC 4349 No. 127, as the other allowed modes would either
lead to much smaller amplitudes in the indicators ($m=0$) or to inverse phase
relations ($m=-1$). We examine this behavior more closely in Sect. 5.1 and
focus on $l=1,m=1$ modes for the current discussion. We examine the amplitudes
for inclination $i=90\degr$ (equator-on), which maximizes the amplitude for
the $|m|=1$ modes. We also restrict the models to only include one isolated
oscillation mode.

We further find that the scale of the temperature variations $\delta
T_{\mathrm{eff}}$ directly influences the amplitudes of all activity
indicators and the RV. Nevertheless, as the CRX captures the wavelength
dependence of the RV, it is the most sensitive indicator to the $\delta
T_{\mathrm{eff}}$ parameter. At the same time, the variations of the CRX are
strongly influenced by the chromatic limb-darkening law. At maximum RV, the
visible and projected $\theta$ and $\phi$ components of the $l=1,m=1$
oscillation are entirely positive. As the RVs are dominated by the horizontal
components, the strongest contribution stems from the cells at the limb of the
star due to the projection onto the line of sight. The limb darkening
suppresses their contribution, such that the overall RV is reduced compared to
a model without limb-darkening correction. However, as the limb-darkening law
is chromatic, this reduction is stronger at shorter wavelengths, leading to a
larger RV at longer wavelength and hence a positive CRX at positive RV. This
accounts for the majority of the positive CRX-RV correlation.

Only small temperature variations are therefore consistent with the data, as
an increase in $\delta T_{\mathrm{eff}}$ at phase shift $\psi_{T}=0$ increases
the CRX-RV slope. However, at such small scales the scatter within the data
complicates an exact determination. We therefore estimated a value $\delta
T_{\mathrm{eff}}=2.5\,\mathrm{K}$ predicted for an oscillatory convective mode
as presented in Fig. 6 of Saio et al. (2015). We determined this value based
on the luminosity of the star and the radial amplitude that minimized the
residuals between the simulations and the real data for the other observables.
We note that the models presented by Saio et al. (2015) at the luminosity of
NGC 4349 No. 127 are only available for lower stellar masses. A higher
temperature variation is therefore possible.

For small temperature variations, the simulations become rather insensitive to
small deviations of the phase shift between the temperature variation and the
radial displacement, given the scatter in the real data. This phase shift is
predicted to be small, $\psi_{T}\lesssim 0.1\cdot\pi$ (Saio et al. 2015). We
therefore fixed this value to $\psi_{T}=0$, such that the temperature is
largest when the radial displacement is at its maximum.

With these initial considerations, we ran a large number of models varying
$v_{\mathrm{rot}}$ and $v_{\mathrm{osc}}$, finally selecting the model (by
eye) that minimizes the residuals between the simulations and the real data
for all observables. We simulated 40 synthetic spectra spread over the time
span of the real observations (prior to the fiber change) and reduced these
using SERVAL and RACCOON. We find that the amplitudes of the activity
indicators and the RVs and their phase relations are reproduced using
$v_{\mathrm{rot}}=1700\,\mathrm{m\,s^{-1}}$ and
$v_{\mathrm{osc}}=0.30\,\mathrm{m\,s^{-1}}$. We summarize the best values for
all simulation parameters in Table 2.

Table 2: Parameters used for the retrograde dipole oscillation model. Parameter | Value
---|---
$T_{\mathrm{eff}}$ | $4500\,\mathrm{K}$
$\log g[\mathrm{cm\,s^{-2}}]$ | $2.0$
$[\mathrm{Fe/H}]$ | $0.0$
$l$ | 1
$m$ | 1
$P$ | $674.0\,\mathrm{d}$
$v_{\mathrm{osc}}$ | $0.30\,\mathrm{m\,s^{-1}}$
$K$ | $1856$
$v_{\mathrm{rot}}$ | $1700\,\mathrm{m\,s^{-1}}$
$\zeta$ | $4822\,\mathrm{m\,s^{-1}}$
$\delta T_{\mathrm{eff}}$ | $2.5\,\mathrm{K}$
$\psi_{T}$ | $0\degr$
$i$ | $90\degr$
$N_{\mathrm{grid}}$ | $150\times 150$

###  4.4 Comparison between the data and the oscillation model

Equipped with the non-radial oscillation model, we compare the observed
correlations with the simulated ones in Fig. 4. The aim is to test whether
such an oscillation is able to explain the observed behavior.

The oscillation model predicts a positive correlation between CRX and RV (top
left panel). We fitted the correlation with a linear relation and determined
the slope $m_{\mathrm{sim}}=0.129\pm 0.001\,\mathrm{Np^{-1}}$ (black line).
The overplotted, colored lines encode the phase information on the linear
relation. The simulated data points deviate only marginally from the linear
relation and were left out for visual clarity. The slope is consistent with
the actual slope measured for the real data $m_{\mathrm{real}}=0.101\pm
0.021\,\mathrm{Np^{-1}}$ with a deviation of $1.3\,\sigma$.

For the dLW, contrast, and FWHM, closed-loop correlations very similar to the
observed correlations are predicted. We plot fitted ellipses to the simulated
indicator-RV correlations that closely follow the synthetic data points (which
were therefore left out for visual clarity) in Fig. 4, applying the same
color-coding. An offset for the RV zero points was fitted and removed between
the real and synthetic data sets. We stress again that we show the absolute
deviation from the respective mean of each time series. The absolute values
between the FWHM and contrast of the CCF of the real and synthetic
observations are slightly offset as not all influences can be modeled
adequately.

We also had to downscale the simulated dLW variations by a multiplicative
factor of 0.098 to match the real dLW variations. As the dLW is a differential
quantity sensitive to the second derivative of the SERVAL templates, which are
separate for the real and simulated data sets, such a multiplicative factor is
expected when comparing separate dLW time series (Zechmeister et al. 2018). We
find that the scaling factor is on the order of unity when using the same
template for both reductions, which however reduces the precision of
individual RV and indicator determinations. The multiplicative factor is
presumably caused by the (slightly) different widths of the real and simulated
spectral lines (see also Sect. 4.5). It is furthermore linked to deviations
from the assumption of Gaussian spectral lines due to the oscillation, which
are fundamental to the definition of the dLW (Zechmeister et al. 2018). We
note that, due to this scaling factor, the amplitude of the dLW holds little
quantitative information. The phase relation with the RV, however, holds
valuable information to validate the simulations.

It can be seen from Fig. 4 that all variations of the activity indicators, the
relative phases, and the directions of correlations are well reproduced. We
tested whether radial, p-mode (solar-like) oscillations can explain the
residual scatter present in the activity indicators. From the residual scatter
in the RVs, we calculated associated temperature variations of $\delta
T_{\mathrm{eff}}\sim 0.9\,\mathrm{K}$ using the scaling relations by Kjeldsen
& Bedding (1995). Fine-tuning $v_{\mathrm{osc}}$ to reproduce the residual
scatter in the RVs, we find that the activity indicators are in principle
sensitive to the p-mode oscillations, but the amplitudes are too small to
explain the observed residuals. We assume that instrumental effects and other
intrinsic, stellar noise sources, such as granulation, dominate the residual
scatter.

The CRX is sensitive to the wavelength dependence of the RVs and thus to
temperature variations on the stellar surface, as well as the limb-darkening
coefficients. We find the interplay of limb darkening and the oscillations to
be the dominating influence. As the limb-darkening contrast decreases toward
redder wavelength and the resulting RV is dominated by the horizontal
components at the limb of the star, the resulting RV amplitude is larger in
the redder part of the spectrum and hence the CRX is positively correlated
with the RV. Its slope is commonly referred to as chromaticity (Zechmeister et
al. 2018).

In comparison, the temperature variations have only minor influence on the
observed CRX-RV slope. If we simulate an (unphysical) oscillation model
setting $\delta T_{\mathrm{eff}}=0\,\mathrm{K}$, the resulting slope of the
CRX-RV correlation is reduced slightly to $m_{\delta T=0K}=0.1213\pm
0.0003\,\mathrm{Np^{-1}}$, even closer to the real correlation. A limb-
darkening model adapted to the stellar parameters of NGC 4349 No. 127 could
therefore be beneficial, but is beyond the scope of this work.

The variations of the line shape indicators (dLW, FWHM, contrast) are mainly
influenced by the interplay between the oscillation at the edge of the stellar
disk (for which the dominating horizontal components are most directly
oriented along the line of sight) and the rotational broadening. This can be
easily understood as the oscillation velocities at the limb directly affect
the flanks of the broadened spectral lines, thus introducing variations of the
line width and depth, or introducing asymmetries.

Figure 5 shows the RV and indicator time series data acquired pre 2015. We
fitted sinusoids to the simulated time series and color-code these in the same
way as the real data. The simulations are able to reproduce the amplitudes and
phases of the RVs and all activity indicators, although large scatter is
present in some of the real indicators. We note that the simulations also
predict small variations of the BIS at the same period as the RVs that are
consistent with the observed data, thus adding another argument in favor of
the oscillation model. Large errors and high scatter in the real data likely
obscure these variations, so that they are not detected in the GLS periodogram
search in Fig. 3.

No meaningful $\mathrm{H\alpha}$ variations can be predicted by the models
since the simulations focus solely on the stellar photosphere, while the
$\mathrm{H\alpha}$ variations are mostly caused by chromospheric processes
(Kürster et al. 2003). Whether such chromospheric (and likely magnetic)
processes could be linked to the oscillations is unknown. We note that similar
connections between stellar oscillations and magnetic fields have been
proposed (see, e.g., Lèbre et al. 2014; Georgiev et al. 2023; Konstantinova-
Antova et al. 2024).

###  4.5 Cross-correlation profiles

In Fig. 6 (top), we plot the real, observed CCF profiles from the RACCOON
reduction for NGC 4349 No. 127 (black) as well as the modeled profiles (red).
While the CCF shapes are reasonably well reproduced, slight differences in the
absolute values of the CCF width (FWHM) and its contrast remain. We therefore
plot the differences from the respective means in Fig. 4 and Fig. 5.

Offsets between the simulated and the real CCF profiles are to be expected as
not all influences can be modeled accurately. These include the real
instrumental profile of the HARPS spectrograph, which we model as a Gaussian,
and the intrinsic line shapes inherent to the PHOENIX models. These are given
with fixed microturbulent broadening values for each model. For the base
PHOENIX model employed in the simulation, microturbulence
$\xi=1.49\,\mathrm{km}\,\mathrm{s^{-1}}$ was used, somewhat lower than
expected for our star (see Table 1). Due to these inherent differences, the
line shapes cannot be reproduced accurately.

We also find that the rotational velocity $v_{\mathrm{rot}}$ plays an
important role in reproducing the amplitudes of the line shape indicators
(dLW, FWHM, contrast), with higher rotation velocities generally producing
larger variations. At the same time, $v_{\mathrm{rot}}$ influences the
absolute width of the CCF profile. We find a smaller rotation velocity
($v_{\mathrm{rot}}=1.7\leavevmode\nobreak\ \mathrm{km}\,\mathrm{s^{-1}}$) to
be generally in good agreement with the observed variations for NGC 4349 No.
127. We note that more recent determinations of the star’s rotation velocity
are larger (see Table 1), but find good agreement with the rotation velocity
determined by Carlberg et al. (2016). However, due to the inherent differences
of the PHOENIX spectra to the real spectrum of the star, we caution against
using this value as a determination of the rotation velocity of NGC 4349 No.
127.

The bottom panel of Fig. 6 portrays a zoomed-in image of the calculated
bisector profiles. We observe that the general bisector shape is well
reproduced by the convective blueshift model based on $\beta$ Boo (Gray 2005).
However, we find the bisector shape to have negligible effect on the
observables presented in Fig. 4 and Fig. 5, apart from providing an offset for
the absolute value of the BIS.

###  4.6 Photometric variations

For many types of intrinsic variations manifested in RV data, such as stellar
spots or radial oscillations, photometric variability is expected and can
provide independent constraints on their properties (see, e.g., Hojjatpanah et
al. 2020). One V-band photometric data set, already discussed by Delgado Mena
et al. (2018), is available for the star from the All Sky Automated Survey
(ASAS) at Las Campanas Observatory (Chile) (Pojmanski & Maciejewski 2004). We
use only grade A and B results as suggested by Delgado Mena et al. (2023).

In Fig. 11 (top), we plot the ASAS-3 data (dots) against time. We overplot
binned data points (black rectangles) with bin size $67.4\,\mathrm{d}$ ($10\%$
of the RV period) to add some visual clarity. There is significant scatter at
the $\sim$ 0.1 mag level, albeit without obvious periodicity. We overplot the
predicted V-band variations from the simulated dipole oscillation caused by
the temperature fluctuations. These were derived by integrating the product of
the simulated spectra and the Bessel V-band filter
curve111111http://spiff.rit.edu/classes/phys440/lectures/filters/bess-v.pass
and rescaling to the median magnitude present in the ASAS-3 data set. The
simulations predict a sinusoidal variation at the RV period with an amplitude
of 0.003 mag. This lies well below the average ASAS-3 error for NGC 4349 No.
127 of 0.051 mag and below the scatter present in the ASAS-3 data set. It
would therefore be plausible that the photometric variation stemming from the
oscillation would not be detected by the ASAS-3 photometry. We thus conclude
that the available photometry does not argue against the oscillation
hypothesis. We discuss the available photometry in more detail and comment on
the findings by Delgado Mena et al. (2018) in Appendix B.

ASAS-3 is reported to achieve a differential accuracy of 0.01 mag for ideal,
bright targets (Pojmanski & Maciejewski 2004), which is still larger by a
factor of 3 than the photometric amplitude predicted by the simulation. The
star was further observed in the TESS full-frame images in sectors 11, 37, 38,
64, and 65. However, the short duration of each individual sector, the long
gaps between the available sectors, and instrumental offsets between them
prevent us from analyzing the TESS photometry on timescales similar to the RV
period. Future Gaia data releases might provide the necessary photometric
precision and timescales to detect such
variability.121212https://www.cosmos.esa.int/web/gaia/science-performance
However, as the star is relatively bright, systematic effects might hinder
this analysis.

The calculation of the photometric variability is based solely on the
temperature variations on the stellar surface, neglecting surface-area and
surface-normal variations. The latter are small at the velocity amplitudes
considered but would violate the assumption of spherical symmetry of the star,
complicating the calculation of the velocity fields. Given the small
amplitudes, we find this simplification justified.

We note, however, that Townsend (1997) argues to include surface-area and
surface-normal variations when predicting photometric variations of non-radial
oscillations. However, the oscillation amplitudes considered are much larger
than in the case of NGC 4349 No. 127 and their $K$ values range between 0 and
1. The differences between simulations considering these geometrical surface
variations and those that do not (see their Fig. 6) seem to become less
pronounced toward larger $K$ (as in our case). The surface-normal variations
mainly act to avoid an unphysical photometric minimum at $K\sim 0.85$ with the
differences being on the order of a factor 2 to 3 otherwise. It can therefore
be assumed that the overall photometric variability would not be affected
enough to be easily detectable by the ASAS-3 photometry.

With the presented results, we show that a retrograde, dipole ($l=1,m=1$)
oscillation mode is fully consistent with the radial velocity variations and
the variations of all studied activity indicators, including the amplitudes
and phase relations, as well as the available photometry. We therefore
conclude that non-radial oscillations are indeed present in the star and cause
the observed periodic patterns.

##  5 Other oscillation modes

Having established that a retrograde $l=1$ mode is able to reproduce the
observables, we further aim to qualitatively address the question whether,
alternatively, other oscillation modes are also capable to reproduce the same
behavior.

###  5.1 l=1 modes

Figure 7: RVs and activity indicators of dipole oscillation modes plotted
against time at inclination angle $i=45\degr$. Sinusoids were fitted to all
time series except for the FWHM, dLW, and contrast in case of $m=0$, for which
the simulated points were interconnected. While all three modes cause similar
RV variations (top left panel), the phase relations between the RVs and the
line shape indicators, as well as their amplitudes, are notably different. The
CRX (bottom left) shows a similar behavior for all three modes. The dLW
variations were rescaled with a common factor of $0.1$. Figure 8: Correlation
plots between the activity indicators and the RVs for modes with $l=1$ (left
column), $m=l$ (center column), and $m=-l$ (right column). Linear correlations
were fitted for the correlation between CRX and RV. For the line shape
diagnostics, ellipses were fitted when appropriate. Arrows indicate the
temporal dependence of the ellipses. The individual modes are specified in the
legend at the top of each column. The simulated dLW variations were rescaled
with a common factor of $0.1$.

Motivated by the models for dipole oscillatory convective modes presented by
Saio et al. (2015), we first study the characteristics of $l=1$ modes, for
which the azimuthal orders $m=(-1,0,1)$ are possible. We chose the same
parameters as presented above, aiming to understand the qualitative
differences for the different modes. We simulated the different oscillation
modes at 20 epochs each.

We first note that different modes have different inclination angles at which
the RV amplitudes are maximized or minimized. In case of the $l=1$ modes, the
RV amplitudes are maximized when viewed equator-on ($i=90\degr$) for $|m|=1$,
while the $m=0$ mode has maximum RV amplitude when viewed pole-on ($i=0\degr$)
(Chadid et al. 2001; De Ridder et al. 2002). We therefore ran the simulations
at different inclination angles and present the intermediate inclination angle
$i=45\degr$, for which the amplitudes for the $|m|=1$ and $m=0$ modes are
comparable.

Figure 7 shows the results for radial velocity (top left) and the spectral
diagnostics for the $l=1,m=-1$ mode (red), $l=1,m=0$ mode (black), and
$l=1,m=1$ mode (blue) plotted against time. We fitted sinusoids to the
variations (except for FWHM, dLW, and contrast in the case of $m=0$) to
highlight the phase relations. The differences between the exact simulated
data points and the sinusoids are negligible.

We first note that all three modes are capable of producing RV variations with
similar amplitudes and would thus, in principle, be able to explain the RV
variations for NGC 4349 No. 127. While the $|m|=1$ modes are in phase, the
$m=0$ mode is phase-shifted. However, we note that an artificial phase shift
can be added (and was used to match the retrograde mode with the real
observations) to match the $m=0$ mode to the real RVs.

What cannot be altered, though, are the phase relations between the RV and the
spectral diagnostics. First, we observe that for the line shape diagnostics
(FWHM, dLW, contrast) the $m=1$ and $m=-1$ modes are roughly in antiphase with
each other. The $m=0$ mode shows much smaller line shape variations that are
not perfectly sinusoidal (and therefore presented here by connecting the
simulated data points) and, most notably, vary at roughly half the RV period.
The behavior can be explained as the azimuthal ($\phi$) component of the
axisymmetric $m=0$ mode is in all cases 0. Thus, the interplay between the
$\phi$ oscillation component and the rotation vector, which causes the
majority of the line shape variations as it influences the flanks of the
rotationally broadened line profiles, is not present in the $m=0$ mode. The
now solely dominating $\theta$ component of the $m=0$ oscillation is symmetric
with respect to the rotation axis and thus leads to much smaller line shape
variations.

We also show the phase relations and their directions of correlations in Fig.
8 (left column). The dLW, contrast, and FWHM are plotted in the bottom three
panels, respectively. We fitted ellipses to the correlations and present the
direction of correlation when appropriate. Figure 8 reveals that both $|m|=1$
modes create closed-loop correlations, while the $m=0$ mode leads to an “arc-
like” correlation, clearly not consistent with the real data for NGC 4349 No.
127.

For the two $|m|=1$ modes, we note that the time dependence of the correlation
is reversed (indicated by the arrows), a consequence of the phase shifts of
the line shape indicators in Fig. 7. While the retrograde $m=1$ mode shows an
anti-clockwise correlation for the dLW and FWHM (and therefore clockwise for
the contrast, as for NGC 4349 No. 127), the prograde $m=-1$ mode correlates
with a reversed time dependence and thus cannot explain the real variations.

It is furthermore evident from Fig. 7 (bottom left panel), that the CRX is in
all cases roughly in phase with the RV with only slightly different
amplitudes. These lead to similar positive correlations between CRX and RV in
Fig. 8 (top left panel). The similarity of the CRX variations stems from the
fact that the wavelength-dependent limb darkening in interplay with the
horizontal components at the limb of the star is dominating the CRX amplitude.
The CRX variation is therefore, for all $l=1$ modes considered, in phase with
the RV variation. This relation holds true as long as the temperature
variation $\delta T_{\mathrm{eff}}$ is small. The CRX-RV correlation thus
seems to be a sensitive indicator for the $l$-mode.

The slight differences in the CRX amplitudes (and therefore the slopes of the
correlation) stem from the interplay between the small temperature variation
$\delta T_{\mathrm{eff}}=2.5\,\mathrm{K}$ and the rotational broadening of the
spectral lines. That is, the temperature variation by itself has a small
effect on the RVs and the CRX, as it slightly increases or decreases the flux
from either side of the rotationally broadened line profile, introducing line
asymmetries that are measured as RV shifts. As the contrast (flux ratio)
between the slightly hotter and cooler halves of the star is less pronounced
at longer wavelengths, this effect is wavelength dependent and thus captured
by the CRX. For the $m=-1$ and $m=1$ modes, for instance, the temperature
variation is phase-shifted by $180\degr$. Thus, the effect of temperature
alone enhances the RV and CRX amplitudes in the $m=1$ case, while it decreases
both amplitudes in the $m=-1$ case. The CRX-RV correlation for the $m=0$ mode
coincides with the effect given by the limb darkening alone.

Figure 7 finally presents the BIS variations (bottom right panel). Sinusoidal
variations at the period of the RV are predicted for all three $m$-modes,
albeit with different amplitudes and phase relations. However, as the
variations for NGC 4349 No. 127 cannot be detected given the noise of the
data, no strong conclusion can be drawn from the BIS indicator in this case.
All of the above findings are valid at all inclination angles that produce
non-zero RV variations for the individual modes.

The fact that $m=0$ modes lead only to small line shape variations is somewhat
discouraging when considering the possibility of applying the same analysis to
other evolved stars, suggested to be false positive planet hosts. If an $m=0$
mode is equally likely, and considering the typical scatter in the real time
series of activity indicators, it seems very challenging to detect the line
shape variations predicted for this mode. Only the CRX (and BIS) variation and
its correlation with the RV are predicted to be similar as for NGC 4349 No.
127 and could provide the most direct hint for non-radial oscillations in the
case of an $m=0$ mode. However, if stemming from non-radial oscillations, the
amplitude of the CRX variations also depend directly on the amplitude of the
oscillation velocity $v_{\mathrm{osc}}$. Therefore, identifying oscillations
based on the CRX at much smaller RV amplitudes could be challenging with the
typical data sets.

In principle, all $m$-modes can be expected to be excited, albeit possibly at
different amplitudes and periods in the presence of rotation. For NGC 4349 No.
127, considering only dipole ($l=1$) modes, it seems most straight-forward to
explain the data with a single $l=1,m=1$ mode. In reality though, a
combination of all three $m$-modes could be the most plausible scenario. Such
a combination, however, could also be difficult to detect.

If all three $m$-modes are excited to the same amplitude, and neglecting at
first the change in oscillation periods due to rotation, cancellation effects
between the $m=1$ and $m=-1$ modes remove the largest part of the variations
present in FWHM, contrast, and dLW (as these are in antiphase for the $m=1$
and $m=-1$ modes), while resulting in a large RV variation. Only the
variations in CRX would likely remain above a detectable threshold, a behavior
that we confirmed with test simulations. The resulting RV curves of a
combination of all $m$-modes can be well approximated as the simple sum of the
individual RV curves for each mode shown in Fig. 7. Therefore, a combination
of all three modes at equal and constant amplitudes leads to a phase shift of
the RV curve.

In the presence of rotation, the frequencies $\nu_{l,m}$ for different
azimuthal quantum numbers $m$ are split (see, e.g., Aerts 2021). This
frequency split can be approximated as

$\nu_{l,m}-\nu_{l,0}=\frac{m}{P_{\mathrm{rot}}}\left(1-\frac{1}{l(l+1)}\right)$
(10)

(Chen & Li 2017; Brickhill 1975). For oscillation periods comparable to the
rotation period, this can lead to large period changes.

A superposition of sinusoidal signals at different periods and amplitudes can
lead to RV curves potentially resembling multi-planetary signals. Moreover, if
such signals are insufficiently sampled in context of exoplanet surveys and
interpreted as single-planet signals, a change in period, amplitude, or phase
of the RV curve might be deduced. This could potentially offer an explanation
for the amplitude changes and phase shifts detected in the cases of $\gamma$
Dra (Hatzes et al. 2018) and Aldebaran (Reichert et al. 2019), although more
thorough modeling would be necessary. We also note that a period modulation
has recently been reported for some long-period variables showing long
secondary periods (Takayama 2023). Of course, in the context of oscillations,
it is also plausible to assume that the oscillation periods themselves are not
constant.

###  5.2 Higher-order l-modes

While Saio et al. (2015) discuss that dipole oscillatory convective modes have
properties that resemble the period-luminosity relations of sequence D
variable giant stars, higher-order $l$-modes could also cause false positive
planet detections. To understand qualitatively how different quantum numbers
$l$ affect the simulation results, we simulated (with the same settings) all
possible modes up to $l=4$.

Motivated by the finding that the $l=1,m=1$ is the only dipole mode consistent
with the data for NGC 4349 No. 127, we plot the correlations between the
indicators and the RV for the $m=l$ modes in the central column of Fig. 8. We
also show the variations plotted against time in Fig. 13. As in the previous
section, we present the results at the inclination angle $i=45\degr$, but note
that the results are qualitatively consistent across all inclination angles
that produce significant amplitudes in the RVs. We fitted linear correlations
for the CRX plotted against RV (top) and elliptical relations for the line
shape indicators. All relations follow the simulated data points closely, such
that the latter were omitted. Only slight deviations from the linear CRX
correlations are present. However, as these are much smaller than the scatter
in the real data, we still approximated these with linear relations for visual
clarity.

We first note that different RV amplitudes are predicted. While naively one
would expect the RV amplitude to decrease for higher-order $l$-modes due to
increasing cancellation effects, the behavior is not as straight-forward for
non-radial oscillations dominated by the horizontal components. This behavior
can be understood as the partial derivatives in Equ. 2 and 3 introduce a
factor $m$ that increases the respective amplitudes as well as (slightly)
different normalization factors that were introduced in Equ. 1. The physical
horizontal velocity components therefore differ for different modes with the
same input velocity $v_{\mathrm{osc}}$. As $v_{\mathrm{osc}}$ is a-priori not
well constrained, differences in the RV amplitudes can be overcome when
attempting to match a mode to the real data.

Furthermore, sinusoidal variations at the period of the RV are predicted for
all indicators. The phase differences between the indicators and the RVs again
provide insights to identify the modes. For the line shape indicators FWHM and
dLW, the phase shift is positive for the $l=1,m=1$ mode, while it is negative
for the higher-degree $m=l$ modes. The behavior of the contrast is reversed.
This leads to an inversion of the temporal direction of the elliptical
correlations in Fig. 8. This already disqualifies the higher-degree $m=l$
modes from being consistent with NGC 4349 No. 127.

However, we find the opposite behavior for the $m=-l$ modes, plotted in the
right column of Fig. 8 and Fig. 14. For $l\geq 2$, modes with $m=-l$ produce
sinusoidal variations with an anti-clockwise correlation for dLW and FWHM with
the RV. The amplitudes of these line shape variations generally depend on the
oscillation amplitude $v_{\mathrm{osc}}$ and the rotation velocity
$v_{\mathrm{rot}}$ and could therefore (to some extent) be adapted in an
attempt to match NGC 4349 No. 127. From this finding alone, these higher-order
modes would therefore be able to explain the observables for NGC 4349 No. 127.

Again, we find the CRX to be the most decisive indicator to identify the mode.
As presented in the top panels of Fig. 8, the CRX correlates linearly with the
RV for all modes. However, the slope of the correlation depends critically on
the mode. While the dipole $l=1$ (red) mode has a positive slope very similar
to the real data, the $l=2$ (orange) mode has a decreased but still positive
slope. Increasing the quantum number $l$ decreases the slope further, making
it progressively negative for $l=3$ (black) and $l=4$ (blue). This behavior is
nearly identical for both $m=l$ and $m=-l$ modes, and is roughly consistent
for all $m$-modes for the same order $l$. The same effect is also evident in
the phase shifts between the different CRX time series and the RVs in Fig. 13.

The behavior can be understood due to the geometry of the projected and
combined oscillation velocities in interplay with the wavelength-dependent
limb darkening. The modes are in all cases dominated by the horizontal
components (due to the high $K$ factor). For the low-order $m=l=1$ mode,
maximum RV was reached when the projected $\phi$ component of the oscillation
was fully positive. The wavelength-dependent limb darkening then decreases the
overall maximum RV to a larger extent in the blue vs. the red part of the
spectrum. This leads to a chromatic RV, captured by a maximum positive CRX at
time of maximum positive RV, and therefore the observed positive chromaticity.

For modes with $l=|m|=3$ and $l=|m|=4$ the dominating $\phi$ component facing
the observer is never fully positive. Maximum RV is then reached when the
central part of the disk is receding from the observer, while at the same time
thin strips at the limb of the star have negative RV values. As these thin
strips of cells are most affected by the limb darkening, its wavelength-
dependent reduction of RV now has the inverse effect. That is, it diminishes
the negative parts and thus increases the overall RV. As this effect is more
prominent in the blue vs the red, the phase of maximum CRX is now phase-
shifted by half a period with respect to the phase of maximum RV. As a
consequence, a negative CRX-RV correlation is observed.

The $l=|m|=2$ mode is intermediate. While the $\phi$ component can still
entirely be positive, the $\theta$ component already presents a similar effect
as detailed for the $l>2$ modes. As the $\theta$ component is generally
smaller in summed projected velocities, this acts only to decrease the CRX-RV
slope. Of course, the exact CRX-RV slopes depend on the choice of limb-
darkening parameters, but the overall trend can be expected to be insensitive
to the exact set of parameters.

To further study this behavior, we simulated even higher $l$-modes, observing
that the slopes start to alternate between uneven (positive slope) and even
(negative slope) $l$-modes, as can be expected from geometrical
considerations. However, due to the ever increasing cancellation effects,
simulated RV and indicator amplitudes quickly drop and would thus require much
larger values of $v_{\mathrm{osc}}$ to be detectable.

Finally, we note that these considerations are only valid for small
temperature variations (here $\delta T_{\mathrm{eff}}=2.5\,\mathrm{K}$), for
which the limb darkening dominates the chromatic behavior. Increased
temperature variations change the CRX-RV slope and thus make mode
identification based on CRX measurements ambiguous.

As the overall behavior of the $l=2,m=-2$ mode resembles the retrograde,
dipole mode discussed in Sect. 4, we tested whether we could reproduce the
data for NGC 4349 No. 127 with this mode. While the CRX-RV correlation can be
reproduced with a slightly increased temperature variation $\delta
T_{\mathrm{eff}}\sim 4\,\mathrm{K}$, the amplitudes of the line shape
indicators, especially the FWHM and contrast of the CCF, were somewhat smaller
than presented for the $l=1,m=1$ simulation for all sets of parameters
considered. However, given the scatter of the real data, they provide a fit
nearly as good as presented in Sect. 4. Due to the increased rotation velocity
($v_{\mathrm{rot}}=4.5\ \mathrm{\frac{km}{s}}$), necessary to increase the
amplitude of the FWHM variations, and the altered oscillation geometry, the
amplitude of the BIS variation is increased by a factor $\sim 9.6$ compared to
Fig. 5, yielding a sinusoidal variation slightly larger but still comparable
to the scatter in the BIS data. It is unclear whether this signal would have
been picked up in the HARPS data set. We therefore cannot exclude the
possibility of an $l=2,m=-2$ mode as the cause of the variations.

In general, we find that most modes with $m\neq 0$ for $l\leq 4$, produce
similar sinusoidal variations of the RVs and indicators which lead to
elliptical correlation plots with reversed directions for the two opposite
$m$-modes. One exception are $|m|=1$ for $l\geq 3$ at specific inclination
angles that resemble the arc-like correlations for $m=0$. However, examining
this behavior more closely is beyond the scope of this work. The CRX-RV slopes
are always very similar for all $m$-modes for one quantum number $l$. The
$m=0$ modes always show a similar behavior to the $l=1,m=0$ mode presented in
the previous section and can thus always be refuted for NGC 4349 No. 127.

##  6 Discussion

The aim of this study was to test whether non-radial oscillations are capable
of explaining the variations of the RVs and the activity indicators of the
cluster giant NGC 4349 No. 127. With the data and model presented in Sect. 4,
we can confirm that the variations of the RVs and indicators are indeed of
intrinsic origin and are consistent with a model of a retrograde, dipole
oscillation mode. Here, we put these findings into the context of previously
reported false positive exoplanets and discuss whether the non-radial
oscillations could be oscillatory convective modes. We further discuss that
magnetic activity, as suggested by Delgado Mena et al. (2018), is incompatible
with the observations.

###  6.1 Oscillatory convective modes

Figure 9: Period-luminosity plot adapted from Reichert et al. (2019) showing
the location of giant stars with RV variations of unknown intrinsic origins
(orange star markers). The green dots are photometrically variable giant stars
identified by the OGLE survey, while the black dots are the subset of these
showing long secondary periods. Overplotted in shades of purple are models of
oscillatory convective modes (mixing length parameter of 1.2), as presented by
Saio et al. (2015). We also overplot confirmed planets orbiting giant stars
from the Lick RV survey (see, e.g., Wolthoff et al. 2022) with their
respective Hipparcos identifiers in red and (mostly unpublished) planet
candidates from the same RV survey in blue. NGC 4349 No. 127 lies in close
vicinity to some stars with established intrinsic RV variations, suggesting a
common origin for the phenomenon. This impression is further enhanced by the
clear separation of the regions in which the confirmed planets and planet
candidates fall.

In Fig. 9, we adapt Fig. 8 of Reichert et al. (2019), showing the location of
NGC 4349 No. 127 along with the locations of other giant stars (orange star
markers) with established intrinsic RV variations (some of which were formerly
thought to host planets or brown dwarfs) in a period-luminosity plot. We do
not include $\epsilon$ Cyg, as the RV variations of the star are likely caused
by the heartbeat phenomenon (Heeren et al. 2021). Luminosities and periods
were taken from the respective publications that refute the planets. For
Sanders 364, the luminosity was taken from the Gaia DR3 determination of
astrophysical parameters (Creevey et al. 2023). We note that Zhou et al.
(2023) find RV periodicity at several periods (224 d, 281 d, 333 d, 530 d, 777
d) longer than the published planet period at 121 d by Brucalassi et al.
(2014, 2017), such that the position of the star could move significantly
parallel to the x-axis of Fig. 9.

It is evident that NGC 4349 No. 127 lies in close vicinity to the refuted
planet hosts $\gamma$ Dra (Hatzes et al. 2018) and Aldebaran (Reichert et al.
2019). We also show the locations of variable giant stars in the Large
Magellanic Cloud from the OGLE survey (Soszyński et al. 2009) in green and
stars which show long secondary periods in black (see Reichert et al. 2019 for
details). We further plot the models taken from Saio et al. (2015) for a
mixing length parameter of $\alpha=1.2$.

Overplotted in red are known planets from the Lick survey (see a recent
overview in Wolthoff et al. 2022), along with their Hipparcos identifiers, and
planet candidates identified in the same survey in blue. The latter group has
(semi-)periodic RV signatures, but either high RV jitter or otherwise
complicated RV curves, which are challenging to model with Keplerian orbits.
The two systems HIP 53261 and HIP 39117 were classified as planet candidates
by Tala Pinto et al. (2020) and are also listed with their Hipparcos
identifiers.

There appears to be a perceptible distinction between the groups of confirmed
planets on the one hand and planet candidates and false positives on the
other. The latter mostly accumulate close to the region with periods between
$500\,\mathrm{d}$ and $1000\,\mathrm{d}$ and luminosities
$2.3\lesssim\log_{10}(L/L_{\sun})\lesssim 3.2$. We find two refuted planetary
systems, 42 Dra and NGC 2423 No. 3, to be quite close to established planet
hosts. We also find a group of three false positive planet hosts (Sanders 364,
41 Lyn, 14 And) to be located at much lower luminosities and much shorter
periods. As Reichert et al. (2019), we note that increased luminosity is also
linked to increased levels of RV jitter, which makes the confirmation of
exoplanets more challenging.

However, increased RV jitter alone cannot explain the individual cases of most
refuted planet systems, which show large-amplitude RV variations, such as the
star discussed in this work. As Hatzes et al. (2018) and Reichert et al.
(2019) argue, if one were to extend the low-mass models of oscillatory
convective modes toward longer periods, they would presumably cross the region
covered by Aldebaran, NGC 4349 No. 127, $\gamma$ Dra, and NGC 3532 No. 670.
Another cluster giant, NGC 2345 No. 50, appears to be close to the high-
luminosity end of the Saio models. 42 Dra, NGC 2423 No. 3, Sanders 364, 41
Lyn, and 14 And seem to have luminosities that are too low to be in good
agreement with the Saio models. As the last three also exhibit shorter
periods, one can suspect that another mechanism could be present in these
stars, such as the rotational modulation of stellar spots as suggested by Teng
et al. (2023).

We also note that many of the stars with longer-period RV variations,
including NGC 4349 No. 127 ($M=3.01\pm 0.24\,M_{\sun}$), are significantly
more massive than predicted by the Saio models. These include $\gamma$ Dra
($M=2.14\pm 0.16\,M_{\sun}$; Hatzes et al. 2018), NGC 3532 No. 670 ($M=3.05\pm
0.23\,M_{\sun}$), NGC 2423 No. 3 ($M=2.03\pm 0.14\,M_{\sun}$), and NGC 2345
No. 50 ($M=5.84\pm 0.61\,M_{\sun}$)(Tsantaki et al. 2023). All of these would
need to have higher luminosities to be consistent with the Saio models. Only
Aldebaran ($M=0.91^{+0.04}_{-0.02}\,M_{\sun}$; Stock et al. 2018) and 42 Dra
($M=0.98\pm 0.05\,M_{\sun}$; Döllinger et al. 2009) have lower masses.
Therefore, only Aldebaran is in good agreement with the low-mass models of
Saio et al. (2015).

For NGC 4349 No. 127, as detailed in Sect. 2, a lower mass
$M=2.20^{+0.38}_{-0.20}\,M_{\sun}$ is also possible, which is however also
linked to a lower luminosity $\log_{10}(L/L_{\sun})\sim 2.3$. We also note
that the models could be altered significantly by a more sophisticated
treatment of convection (Saio et al. 2015).

Overall, we find that the properties (mode, amplitude of temperature
variation) of the presented oscillation model seem to be consistent with the
predictions for oscillatory convective modes. We nevertheless are cautious to
conclude that the Saio models provide a good match given the stellar
properties and period of the variation. More detailed modeling of their
properties would be very valuable.

As discussed by several authors (e.g., Reichert et al. 2019; Döllinger &
Hartmann 2021; Delgado Mena et al. 2023), a suspicious accumulation of false
positive planet detections in the vicinity of NGC 4349 No. 127 is evident. As
the non-radial oscillation model convincingly explains the variations of the
star, a generalization to other stars with similar parameters seems plausible.

On the other hand, an alternative explanation for the long secondary periods
of stars on sequence D is binarity. In this scenario, a brown dwarf or low-
mass stellar companion enveloped by a comet-like dust cloud causes the
photometric variations by obscuring the stellar disk (Soszyński 2007;
Soszyński & Udalski 2014), a proposal supported by the detection of secondary
eclipses in mid-infrared photometry for a subsample of variable stars showing
long secondary periods (Soszyński et al. 2021).

While a brown dwarf is indeed consistent with the RVs, this scenario
nevertheless seems implausible for NGC 4349 No. 127, as it would not be clear
how the line shape and CRX variations could be explained. If we, for the sake
of the argument, assume that the brown dwarf exists, is engulfed in a dusty
cloud, and is transiting the star, then transit durations are on the order of
$T_{\mathrm{transit}}\sim 22\,\mathrm{d}$, even if one were to assume a radius
of $10\,R_{\sun}$ for the dust cloud. In principle, the Rossiter-McLaughlin
effect can explain the line shape variations during the transit itself. But
since we observe a near sinusoidal variation, for instance, for the FWHM, the
variations out of transit would remain unaccounted for. It would be even more
problematic in the cases of Aldebaran or $\gamma$ Dra, which show RV amplitude
changes or phase shifts that are hard to explain with an orbital companion.
This scenario can thus not explain the observables.

It is, of course, possible that the variability on sequence D, mostly detected
via photometry, is simply not related to the intrinsic RV variations of the
discussed giants. Moreover, it is possible that even within the small group of
these false positive planet hosts, different intrinsic mechanism are at work.
For NGC 4349 No. 127, non-radial oscillations provide the most convincing
explanation for all observables to date. Whether these are linked to
oscillatory convective modes or not remains uncertain.

###  6.2 Magnetic activity

Delgado Mena et al. (2018) propose the rotational modulation of magnetic spots
as the origin of the RV variations. However, the available photometry and the
newly presented activity indicators lead us to refute the spot hypothesis.

If magnetic surface spots cause the RV periodicity, significant photometric
variations are to be expected. Following Hatzes (2002), we can roughly
estimate the required spot filling factor to produce RV variations with
amplitude $A_{\mathrm{RV}}=235.6\,\mathrm{\frac{m}{s}}$ (see Fig. 2) as
$f=7.3\%$, using $v\sin i=4.81\,\mathrm{\frac{km}{s}}$ (Tsantaki et al. 2023).
Hatzes (2002) used $T_{\mathrm{eff}}=5800\,\mathrm{K}$ and
$T_{\mathrm{spot}}=4600\,\mathrm{K}$ for the effective temperatures of the
stellar photosphere and the star spot, respectively. Downscaling the spot
temperature to yield the same flux ratio as considered by Hatzes (2002), but
at effective temperature $T_{\mathrm{eff}}=4417\,\mathrm{K}$ (Tsantaki et al.
2023), yields a temperature difference $\Delta
T_{\mathrm{spot}}=914\,\mathrm{K}$ between photosphere and spot. Assuming
black body radiation, we calculate the flux in a Bessel V-band filter for a
photosphere with a star spot of filling factor $f$ to be reduced by $5.7\%$
compared to the flux of a photosphere with no spot present. This corresponds
to a magnitude difference of $\Delta V\sim 0.06\,\mathrm{mag}$.

We searched for the largest amplitude consistent with the ASAS-3 V-band
photometry by fitting sinusoids with fixed periods in a box of size
$100\,\mathrm{d}$ around the best RV period $P=674.0\,\mathrm{d}$. The largest
amplitude present in the data is $A_{\mathrm{V}}=0.006\pm
0.003\,\mathrm{mag}$, an order of magnitude lower than predicted by our rough
estimate. We further show in Appendix B that the apparent periodicity at
roughly half the RV period is caused by the window function of the ASAS-3
observations. We therefore conclude that there are no photometric variations
at periods related to the RV period in the ASAS-3 data set that are consistent
with the prediction.

Another strong argument against the RV variations being caused by star spots
is the positive correlation between CRX and RV in Fig. 4 (top left panel). If
spots were to cause the RV variation, one would expect the CRX to be
anticorrelated with the RV (negative chromaticity), as the contrast (flux
ratio) between the cool spot and the stellar photosphere decreases toward
longer wavelengths (Reiners et al. 2010; Barnes et al. 2011; Tal-Or et al.
2018; Baroch et al. 2020; Lafarga et al. 2021). The same relation holds for
hot active regions (Reiners et al. 2013). Thus, the positive chromaticity
observed for NGC 4349 No. 127 is challenging to explain with a spot model,
while it is a natural consequence of the limb darkening in case of dipole non-
radial oscillations.

We note that Delgado Mena et al. (2023) recently discussed other magnetic
structures that locally reduce convection and lead to RV variations without
photometric variations. However, it is not clear if such structures could
explain any of the correlations present for NGC 4349 No. 127. We therefore
find the oscillation hypothesis more convincing.

##  7 Summary

We presented a reanalysis of archival HARPS spectra of the cluster giant NGC
4349 No. 127, which was originally thought to host a brown dwarf orbital
companion (Lovis & Mayor 2007). Delgado Mena et al. (2018, 2023), however,
refute the brown dwarf’s existence based on variations of the FWHM of the CCF
and the $\mathrm{H}\alpha$ index. This makes NGC 4349 No. 127 another example
of a group of giant stars with RV periodicities that mimic orbital companions,
but are of unknown intrinsic origin.

We first presented additional activity indicators available through the use of
the SERVAL and RACCOON reduction pipelines, revealing that, along with the
FWHM, the contrast of the CCF, the dLW, and the CRX also have significant
periodicity at the proposed orbital period. We therefore affirm the findings
of Delgado Mena et al. (2018, 2023) and unequivocally refute the orbital
companion. Furthermore, we find a significant, positive correlation between
the CRX and the RVs, as well as closed-loop correlations between the RVs and
dLW, FWHM, and contrast, respectively, a result of sinusoidal variations with
phase shifts close to $\pi/2$.

We presented the simulation software pyoscillot, capable of simulating
synthetic HARPS spectra, along with their respective RVs and activity
indicators, based on a model of non-radial oscillations in the photosphere of
a star. Such non-radial oscillations were previously proposed to cause the
intrinsic RV variations that mimic orbital companions.

We showed that an isolated, low-amplitude, retrograde, dipole ($l=1,m=1$)
oscillation reproduces the periodicities and amplitudes of the RVs and all
activity indicators, along with their respective phase relations and thus the
Lissajous figures of their correlations. The oscillation model also predicts
low-amplitude photometric variations at the orbital period, which are well
below the detection threshold of the available ASAS-3 data, and thus cannot be
ruled out.

We also showed that other dipole ($l=1$) modes, while producing similar RV
variations, are inconsistent with the activity indicators measured for NGC
4349 No. 127 and present the expected variations for all modes. We showed that
the CRX is the only indicator yielding variations for all azimuthal orders $m$
allowed for dipole ($l=1$) modes, while the line shape diagnostics are mostly
inconspicuous for the $l=1,m=0$ mode. In the presence of scatter in the CRX
data, such modes and combinations of different $l=1$ modes could therefore be
difficult to detect.

Moreover, we explored higher-order modes, showing that modes with $m=l$ and
$m=-l$ show similar behavior to the $l=1,m=1$ mode. The $l=2,m=-2$ mode cannot
entirely be ruled out as the cause of the observed variations. In general, the
CRX provides the most direct approach to differentiate between modes, but it
is also affected by the scale of the temperature variations employed, and thus
cannot be used to rule out modes unambiguously.

We finally showed that the stellar parameters of NGC 4349 No. 127 are broadly
consistent with the models of oscillatory convective modes (Saio et al. 2015)
put forth to cause non-radial oscillations of bright giant stars. We showed
that reliable planet detections, on the one hand, and more uncertain planet
systems and false positives, on the other, are separated in a period-
luminosity plot. This suggests that the same intrinsic phenomenon might be
present in the latter group. We note, however, that the masses of most giant
stars with intrinsic RV signatures are higher than predicted by the models of
Saio et al. (2015). We further give arguments that the correlations observed
for NGC 4349 No. 127 cannot be explained by magnetic spots.

We conclude that the RV variations of NGC 4349 No. 127 are likely caused by a
retrograde, dipole ($l=1,m=1$) non-radial oscillation. We also provide, for
the first time, a testable model capable of explaining the intrinsic RV
variations discussed in the literature.

The case of NGC 4349 No. 127 shows the importance of acquiring a set of
reliable and precise activity indicators targeting the shape of spectral lines
and the wavelength dependence of the RV. However, as $m=0$ modes do not show
large variations of the shape of spectral lines, and would only be detected by
the CRX, which often suffers from significant scatter, long-term RV monitoring
remains crucial in order to detect any changes in the amplitude, period, or
phase of the RV variation.

Applying the same analysis to other evolved stars with similar luminosities
and RV periods could offer valuable insights to constrain their (potential)
intrinsic nature. This analysis may thus contribute to verifying the
reliability of the planet population orbiting evolved stars.

###### Acknowledgements.

We thank the anonymous referee for their helpful comments that improved the
quality of this work. We further thank M. Zechmeister for fruitful discussions
regarding the SERVAL reduction results. This work was supported by the
Deutsche Forschungsgemeinschaft (DFG) within the Priority Program SPP 1992
“The Diversity of Exoplanets” (RE 2694/7-1). Based on data obtained from the
ESO Science Archive Facility with DOI:
https://doi.eso.org/10.18727/archive/33. This research has made use of the
VizieR catalog access tool and the SIMBAD database, operated at CDS,
Strasbourg, France. This work has made use of data from the European Space
Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the
Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research made use of
Astropy,131313http://www.astropy.org a community-developed core Python package
for Astronomy (Astropy Collaboration et al. 2013, 2018). We further
acknowledge use of the Python packages NumPy (Harris et al. 2020), SciPy
(Virtanen et al. 2020), Matplotlib (Hunter 2007), pandas (The pandas
development team 2020; McKinney 2010),
PyAstronomy141414https://github.com/sczesla/PyAstronomy (Czesla et al. 2019),
lsq-ellipse (Hammel & Sullivan-Molina 2020), the RV fitting tool Exo-Striker
(Trifonov 2019), and the color map “vikO” (Crameri 2023).

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##  Appendix A Linear correlation between H$\alpha$ and RV

Figure 10: Correlation between H$\alpha$ indicator (absolute difference from
the mean) and RV for the 44 spectra acquired prior to the HARPS fiber change.
The same color-coding as in Fig. 4 applies. A linear relation (black line) was
fitted to the data. An F-test against a constant model confirms the
correlation to be significant with a p-value
$p(\textit{F}\textrm{-test})=0.2\%$ and Pearson’s r coefficient $r=0.48$.

##  Appendix B Discussion of the ASAS-3 photometry

Figure 11: Observed and simulated photometry for NGC 4349 No. 127. Top: V
magnitudes extracted from ASAS-3 plotted against time (colored dots). The
color-coding is the same as in Fig. 4 and refers to the RV period. We also
plot binned means (black rectangles) as well as the photometric variability
predicted by the $l=1,m=1$ oscillation model (continuous multi-colored line).
We further overplot a sinusoidal fit with period $2571\,\mathrm{d}$ in black.
Bottom: ASAS-3 V magnitudes phase-folded to the $2571\,\mathrm{d}$ period.
Figure 12: GLS periodogram analysis of the ASAS-3 V-band photometry. Top: GLS
periodogram of the original photometry. The FAP levels of $5\%$ (dashed),
$1\%$ (dash-dotted), and $0.1\%$ (dotted) were computed via bootstrapping with
10 000 reshuffles and are plotted as horizontal lines. The periods of the four
most prominent peaks are given in units of days. Middle: GLS periodogram of
the residual photometry after removing the $2571\,\mathrm{d}$ fit. Bottom: LS
periodogram of the window function.

In Fig. 11 (top), we plot the available ASAS-3 data against time, color-coding
the individual data points with respect to the RV period. For enhanced
visibility, we further show binned data points as black rectangles. The
photometric variation predicted by the $l=1,m=1$ oscillation model discussed
in Sect. 4 is overplotted as the colored line and is much smaller than the
scatter present in the ASAS-3 data. It is therefore reasonable to assume that
the photometric variations would not have been detected in the ASAS-3 data
set.

In Fig. 12 (top), we plot the GLS periodogram of the ASAS-3 data, labeling the
four most significant peaks in units of days. Delgado Mena et al. (2018)
discuss the same data set and argue that the peak at $338\,\mathrm{d}$ is
close to the first harmonic ($P/2$) of the RV period or could be linked to the
rotation period of the star. However, we find that the most significant peak
at $2592\,\mathrm{d}$ provides a much more convincing fit to the ASAS-3 data
set, which we plot in black in the top panel of Fig. 11. The sinusoidal fit
has an amplitude of 0.05 mag and a period of $2571\,\mathrm{d}$. The
discrepancy to the $2592\,\mathrm{d}$ period found in the GLS periodogram
stems from the sampling of the periodogram. We show a phase-folded plot of the
ASAS-3 data in the bottom panel of Fig. 11.

In the middle panel of Fig. 12, we show the GLS periodogram of the residuals
after removing the fit with period $2571\,\mathrm{d}$. The bottom panel
further shows the periodogram of the window function computed as a Lomb-
Scargle (LS) periodogram, as suggested by VanderPlas (2018). The formerly
second strongest peak at $430\,\mathrm{d}$ coincides with the alias of the
$2592\,\mathrm{d}$ signal and the strong yearly period present in the window
function. It is therefore much reduced in significance by removing the long-
period signal.

The interesting $338\,\mathrm{d}$ period also changes significantly by
removing the fit with period $2571\,\mathrm{d}$. It is reduced in significance
and shifted to $361\,\mathrm{d}$, coinciding quite closely with the yearly
period. The two neighboring peaks at $248\,\mathrm{d}$ and $532\,\mathrm{d}$
are furthermore close to the alias period of the yearly period and the
$950\,\mathrm{d}$ period present in the window function. We therefore argue
that the $338\,\mathrm{d}$ signal discussed by Delgado Mena et al. (2018) is
likely not real and instead an artifact of the window function, as well as its
neighboring peaks.

The origin of the fourth peak at $1160\,\mathrm{d}$ present in the original
GLS periodogram is less certain. While it is close to the first harmonic of
the long-period signal, it is not entirely suppressed by removing the
$2571\,\mathrm{d}$ fit. It is, however, shifted closer to the
$950\,\mathrm{d}$ peak present in the window function and could therefore
potentially be explained as another windowing artifact.

For the $2592\,\mathrm{d}$ peak, there is no obvious peak present in the
window function. It is also unlikely to be caused by the typical rise of the
periodogram power of the window function at periods longer than the observing
window. We therefore argue that the $2592\,\mathrm{d}$ signal is the only
signal that cannot be explained by the window function. Whether this signal is
of instrumental or stellar origin is uncertain. If it is real, it seems
unlikely to be related to the non-radial oscillation discussed in Sect. 4.

It could, in principle, be related to the rotation period of the star.
However, given the radius of the star, this would lead to a rotation velocity
of $v_{\mathrm{rot}}\sim 0.75\,\mathrm{km\,s^{-1}}$, considerably lower than
most estimates in the literature (see Table 1), but consistent with the
estimate by Carlberg et al. (2016). We note that the presented simulation of a
$l=1,m=1$ oscillation mode also requires a somewhat lower rotation velocity to
reproduce the variations of the line shape indicators, but caution to use this
value to determine the actual rotation velocity of the star.

##  Appendix C Oscillation modes with $m=l$

Figure 13: RVs and activity indicators for oscillation modes with $m=l$
plotted against time at inclination angle $i=45\degr$. Sinusoids are predicted
for all observables and were fitted to the time series. Different amplitudes
and phase relations are predicted and can be useful for mode identification.
The dLW variations were rescaled with a common factor of $0.1$.

##  Appendix D Oscillation modes with $m=-l$

Figure 14: RVs and activity indicators for oscillation modes with $m=-l$
plotted against time at inclination angle $i=45\degr$. Sinusoids are predicted
for all observables and were fitted to the time series. Different amplitudes
and phase relations are predicted and can be useful for mode identification.
The dLW variations were rescaled with a common factor of $0.1$.

##  Appendix E HARPS data set

Table 3: SERVAL and RACCOON reduction results.

BJD | RV | RV | CRX | dLW | H$\alpha$ | FWHM | Contrast | BIS
---|---|---|---|---|---|---|---|---
SERVAL | SERVAL | RACCOON | SERVAL | SERVAL | SERVAL | RACCOON | RACCOON | RACCOON
| $\frac{\mathrm{m}}{\mathrm{s}}$ | $\frac{\mathrm{m}}{\mathrm{s}}$ | $\frac{\mathrm{m}/\mathrm{s}}{\mathrm{Np}}$ | $1000\frac{\mathrm{m^{2}}}{\mathrm{s^{2}}}$ |  | $\frac{\mathrm{km}}{\mathrm{s}}$ | $\%$ | $\frac{\mathrm{km}}{\mathrm{s}}$
$\phantom{(}2453449.783796\phantom{)}$ | $\phantom{(}\phantom{\mathopen{}-}127.1\pm 1.9\phantom{)}$ | $\phantom{(}\mathopen{}-11528.8\pm\phantom{0}6.8\phantom{)}$ | $\phantom{(}\phantom{0}\mathopen{}-7.5\pm 14.8\phantom{)}$ | $\phantom{(}\phantom{\mathopen{}-}19.9\pm 1.7\phantom{)}$ | $\phantom{(}0.4091\pm 0.0017\phantom{)}$ | $\phantom{(}9.205\pm 0.019\phantom{)}$ | $\phantom{(}49.777\pm 0.077\phantom{)}$ | $\phantom{(}0.076\pm 0.016\phantom{)}$
$\phantom{(}2453460.836163\phantom{)}$ | $\phantom{(}\phantom{0}\phantom{\mathopen{}-}71.1\pm 2.8\phantom{)}$ | $\phantom{(}\mathopen{}-11578.2\pm\phantom{0}8.8\phantom{)}$ | $\phantom{(}\phantom{\mathopen{}-}55.8\pm 20.9\phantom{)}$ | $\phantom{(}\phantom{\mathopen{}-}18.6\pm 2.6\phantom{)}$ | $\phantom{(}0.4169\pm 0.0022\phantom{)}$ | $\phantom{(}9.228\pm 0.022\phantom{)}$ | $\phantom{(}48.757\pm 0.087\phantom{)}$ | $\phantom{(}0.095\pm 0.021\phantom{)}$
$\phantom{(}2453469.790637\phantom{)}$ | $\phantom{(}\phantom{0}\phantom{\mathopen{}-}60.0\pm 2.4\phantom{)}$ | $\phantom{(}\mathopen{}-11598.1\pm\phantom{0}6.0\phantom{)}$ | $\phantom{(}\phantom{\mathopen{}-}15.4\pm 18.1\phantom{)}$ | $\phantom{(}\phantom{0}\phantom{\mathopen{}-}5.0\pm 1.8\phantom{)}$ | $\phantom{(}0.4056\pm 0.0015\phantom{)}$ | $\phantom{(}9.157\pm 0.016\phantom{)}$ | $\phantom{(}49.700\pm 0.065\phantom{)}$ | $\phantom{(}0.059\pm 0.014\phantom{)}$
$\phantom{(}2453499.578838\phantom{)}$ | $\phantom{(}\phantom{0}\mathopen{}-24.5\pm 3.4\phantom{)}$ | $\phantom{(}\mathopen{}-11688.5\pm 10.8\phantom{)}$ | $\phantom{(}\phantom{0}\phantom{\mathopen{}-}8.6\pm 26.9\phantom{)}$ | $\phantom{(}\phantom{\mathopen{}-}14.6\pm 4.1\phantom{)}$ | $\phantom{(}0.4180\pm 0.0027\phantom{)}$ | $\phantom{(}9.168\pm 0.020\phantom{)}$ | $\phantom{(}48.874\pm 0.078\phantom{)}$ | $\phantom{(}0.043\pm 0.025\phantom{)}$
$\phantom{(}2453500.641879\phantom{)}$ | $\phantom{(}\phantom{0}\mathopen{}-11.3\pm 9.1\phantom{)}$ | $\phantom{(}\mathopen{}-11654.9\pm 19.4\phantom{)}$ | $\phantom{(}\phantom{\mathopen{}-}36.6\pm 71.7\phantom{)}$ | $\phantom{(}\phantom{\mathopen{}-}19.5\pm 6.3\phantom{)}$ | $\phantom{(}0.4058\pm 0.0051\phantom{)}$ | $\phantom{(}9.215\pm 0.020\phantom{)}$ | $\phantom{(}48.047\pm 0.076\phantom{)}$ | $\phantom{(}0.066\pm 0.044\phantom{)}$
$\phantom{(}2453521.584883\phantom{)}$ | $\phantom{(}\mathopen{}-101.3\pm 2.7\phantom{)}$ | $\phantom{(}\mathopen{}-11754.8\pm\phantom{0}7.7\phantom{)}$ | $\phantom{(}\mathopen{}-18.4\pm 20.4\phantom{)}$ | $\phantom{(}\phantom{\mathopen{}-}10.9\pm 2.3\phantom{)}$ | $\phantom{(}0.4080\pm 0.0019\phantom{)}$ | $\phantom{(}9.184\pm 0.018\phantom{)}$ | $\phantom{(}48.917\pm 0.070\phantom{)}$ | $\phantom{(}0.069\pm 0.018\phantom{)}$
$\phantom{(}2453787.795108\phantom{)}$ | $\phantom{(}\phantom{0}\mathopen{}-37.8\pm 2.0\phantom{)}$ | $\phantom{(}\mathopen{}-11698.1\pm\phantom{0}6.2\phantom{)}$ | $\phantom{(}\phantom{0}\mathopen{}-6.9\pm 15.5\phantom{)}$ | $\phantom{(}\mathopen{}-20.4\pm 1.9\phantom{)}$ | $\phantom{(}0.4042\pm 0.0016\phantom{)}$ | $\phantom{(}9.015\pm 0.016\phantom{)}$ | $\phantom{(}50.623\pm 0.067\phantom{)}$ | $\phantom{(}0.059\pm 0.014\phantom{)}$
$\phantom{(}2453812.758975\phantom{)}$ | $\phantom{(}\phantom{0}\mathopen{}-28.2\pm 2.3\phantom{)}$ | $\phantom{(}\mathopen{}-11687.3\pm\phantom{0}7.3\phantom{)}$ | $\phantom{(}\mathopen{}-28.3\pm 17.2\phantom{)}$ | $\phantom{(}\mathopen{}-17.7\pm 2.1\phantom{)}$ | $\phantom{(}0.4077\pm 0.0019\phantom{)}$ | $\phantom{(}9.027\pm 0.016\phantom{)}$ | $\phantom{(}50.444\pm 0.066\phantom{)}$ | $\phantom{(}0.052\pm 0.017\phantom{)}$
$\phantom{(}2453833.703149\phantom{)}$ | $\phantom{(}\phantom{0}\phantom{\mathopen{}-}37.9\pm 2.0\phantom{)}$ | $\phantom{(}\mathopen{}-11624.7\pm\phantom{0}5.5\phantom{)}$ | $\phantom{(}\mathopen{}-13.6\pm 15.4\phantom{)}$ | $\phantom{(}\mathopen{}-15.0\pm 1.6\phantom{)}$ | $\phantom{(}0.3778\pm 0.0014\phantom{)}$ | $\phantom{(}9.069\pm 0.014\phantom{)}$ | $\phantom{(}50.372\pm 0.057\phantom{)}$ | $\phantom{(}0.055\pm 0.013\phantom{)}$
$\phantom{(}2453862.612919\phantom{)}$ | $\phantom{(}\phantom{0}\phantom{\mathopen{}-}62.3\pm 2.1\phantom{)}$ | $\phantom{(}\mathopen{}-11594.8\pm\phantom{0}5.7\phantom{)}$ | $\phantom{(}\phantom{\mathopen{}-}38.2\pm 14.9\phantom{)}$ | $\phantom{(}\mathopen{}-14.3\pm 1.7\phantom{)}$ | $\phantom{(}0.4094\pm 0.0015\phantom{)}$ | $\phantom{(}9.062\pm 0.020\phantom{)}$ | $\phantom{(}50.258\pm 0.082\phantom{)}$ | $\phantom{(}0.079\pm 0.013\phantom{)}$
$\phantom{(}2453883.589922\phantom{)}$ | $\phantom{(}\phantom{\mathopen{}-}105.1\pm 1.8\phantom{)}$ | $\phantom{(}\mathopen{}-11550.8\pm\phantom{0}5.0\phantom{)}$ | $\phantom{(}\phantom{\mathopen{}-}15.2\pm 13.7\phantom{)}$ | $\phantom{(}\mathopen{}-14.1\pm 1.4\phantom{)}$ | $\phantom{(}0.4096\pm 0.0013\phantom{)}$ | $\phantom{(}9.060\pm 0.018\phantom{)}$ | $\phantom{(}50.306\pm 0.073\phantom{)}$ | $\phantom{(}0.074\pm 0.012\phantom{)}$
$\phantom{(}2453922.499565\phantom{)}$ | $\phantom{(}\phantom{\mathopen{}-}126.4\pm 4.4\phantom{)}$ | $\phantom{(}\mathopen{}-11532.0\pm 10.3\phantom{)}$ | $\phantom{(}\mathopen{}-42.2\pm 34.2\phantom{)}$ | $\phantom{(}\phantom{0}\phantom{\mathopen{}-}1.5\pm 2.8\phantom{)}$ | $\phantom{(}0.4213\pm 0.0027\phantom{)}$ | $\phantom{(}9.158\pm 0.019\phantom{)}$ | $\phantom{(}49.554\pm 0.075\phantom{)}$ | $\phantom{(}0.076\pm 0.024\phantom{)}$
$\phantom{(}2453950.476290\phantom{)}$ | $\phantom{(}\phantom{\mathopen{}-}173.6\pm 3.3\phantom{)}$ | $\phantom{(}\mathopen{}-11476.0\pm\phantom{0}7.7\phantom{)}$ | $\phantom{(}\phantom{\mathopen{}-}43.5\pm 24.5\phantom{)}$ | $\phantom{(}\phantom{0}\mathopen{}-2.2\pm 2.2\phantom{)}$ | $\phantom{(}0.4101\pm 0.0020\phantom{)}$ | $\phantom{(}9.163\pm 0.019\phantom{)}$ | $\phantom{(}49.765\pm 0.075\phantom{)}$ | $\phantom{(}0.062\pm 0.018\phantom{)}$
$\phantom{(}2454117.846012\phantom{)}$ | $\phantom{(}\phantom{\mathopen{}-}129.3\pm 4.8\phantom{)}$ | $\phantom{(}\mathopen{}-11514.1\pm\phantom{0}9.7\phantom{)}$ | $\phantom{(}\phantom{\mathopen{}-}13.6\pm 37.6\phantom{)}$ | $\phantom{(}\phantom{0}\phantom{\mathopen{}-}4.3\pm 2.9\phantom{)}$ | $\phantom{(}0.4131\pm 0.0026\phantom{)}$ | $\phantom{(}9.149\pm 0.013\phantom{)}$ | $\phantom{(}49.463\pm 0.053\phantom{)}$ | $\phantom{(}0.048\pm 0.023\phantom{)}$
$\phantom{(}2454137.806299\phantom{)}$ | $\phantom{(}\phantom{0}\phantom{\mathopen{}-}73.9\pm 2.0\phantom{)}$ | $\phantom{(}\mathopen{}-11582.4\pm\phantom{0}5.7\phantom{)}$ | $\phantom{(}\mathopen{}-22.1\pm 14.5\phantom{)}$ | $\phantom{(}\phantom{0}\phantom{\mathopen{}-}6.5\pm 1.6\phantom{)}$ | $\phantom{(}0.4061\pm 0.0014\phantom{)}$ | $\phantom{(}9.191\pm 0.018\phantom{)}$ | $\phantom{(}49.416\pm 0.072\phantom{)}$ | $\phantom{(}0.073\pm 0.013\phantom{)}$
$\phantom{(}2454169.723975\phantom{)}$ | $\phantom{(}\phantom{0}\phantom{0}\mathopen{}-0.8\pm 1.5\phantom{)}$ | $\phantom{(}\mathopen{}-11653.4\pm\phantom{0}5.0\phantom{)}$ | $\phantom{(}\phantom{0}\phantom{\mathopen{}-}6.5\pm 11.5\phantom{)}$ | $\phantom{(}\phantom{0}\phantom{\mathopen{}-}0.4\pm 1.3\phantom{)}$ | $\phantom{(}0.4059\pm 0.0013\phantom{)}$ | $\phantom{(}9.178\pm 0.017\phantom{)}$ | $\phantom{(}49.463\pm 0.068\phantom{)}$ | $\phantom{(}0.061\pm 0.012\phantom{)}$
$\phantom{(}2454194.779612\phantom{)}$ | $\phantom{(}\phantom{0}\mathopen{}-33.6\pm 2.1\phantom{)}$ | $\phantom{(}\mathopen{}-11691.2\pm\phantom{0}6.1\phantom{)}$ | $\phantom{(}\mathopen{}-14.2\pm 15.7\phantom{)}$ | $\phantom{(}\phantom{0}\phantom{\mathopen{}-}6.9\pm 1.9\phantom{)}$ | $\phantom{(}0.4022\pm 0.0016\phantom{)}$ | $\phantom{(}9.207\pm 0.016\phantom{)}$ | $\phantom{(}49.253\pm 0.063\phantom{)}$ | $\phantom{(}0.053\pm 0.014\phantom{)}$