11institutetext:  Max-Planck-Institut für Astronomie, Königstuhl 17, 69117
Heidelberg, Germany
11email<EMAIL_ADDRESS>22institutetext: Department of Astronomy, Sofia
University “St Kliment Ohridski”, 5 James Bourchier Blvd, BG-1164 Sofia,
Bulgaria 33institutetext: Centro de Astrobiología (CSIC-INTA), ESAC, Camino
bajo del castillo s/n, 28692 Villanueva de la Cañada, Madrid, Spain
44institutetext: Department of Physics, University of Warwick, Gibbet Hill
Road, Coventry CV4 7AL, United Kingdom 55institutetext: Institut de Ciències
de l’Espai (ICE, CSIC), Campus UAB, C/ de Can Magrans s/n, 08193 Cerdanyola
del Vallès, Spain 66institutetext: Institut d’Estudis Espacials de Catalunya
(IEEC), C/ Gran Capità 2-4, 08034 Barcelona, Spain 77institutetext: Instituto
de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía s/n, 18008
Granada, Spain 88institutetext: Landessternwarte, Zentrum für Astronomie der
Universität Heidelberg, Königstuhl 12, 69117 Heidelberg, Germany
99institutetext: Max-Planck-Institut für Sonnensystemforschung, Justus-von-
Liebig Weg 3, 37077 Göttingen, Germany 1010institutetext: Instituto de
Astrofísica de Canarias (IAC), 38205 La Laguna, Tenerife, Spain
1111institutetext: Departamento de Astrofísica, Universidad de La Laguna,
38206 La Laguna, Tenerife, Spain 1212institutetext: Institut für Astrophysik,
Georg-August-Universität, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
1313institutetext: Thüringer Landessternwarte Tautenburg, Sternwarte 5, 07778
Tautenburg, Germany 1414institutetext: Departamento de Física de la Tierra y
Astrofísica & IPARCOS-UCM (Instituto de Física de Partículas y del Cosmos de
la UCM), Facultad de Ciencias Físicas, Universidad Complutense de Madrid,
E-28040 Madrid, Spain 1515institutetext: Department of Physics, Ariel
University, Ariel 40700, Israel 1616institutetext: Hamburger Sternwarte,
Gojenbergsweg 112, 21029 Hamburg, Germany

# The CARMENES search for exoplanets around M dwarfs

Stable radial-velocity variations at the rotation period of AD Leonis - A test
case study of current limitations to treating stellar activity††thanks: Tables
5, 6, 7, 8, and additional data (i.e., stellar activity indicators as shown in
Figs. 9, 10, and 11) are available in electronic form at the CDS via anonymous
ftp to cdsarc.u-strasbg.fr (130.79.128.5)

D. Kossakowski 11 M. Kürster 11 Th. Henning 11 T. Trifonov 1122 J. A.
Caballero 33 M. Lafarga 445566 F. F. Bauer 77 S. Stock 88 J. Kemmer 88 S. V.
Jeffers 99 P. J. Amado 77 M. Pérez-Torres 77 V. J. S. Béjar 10101111 M.
Cortés-Contreras 33 I. Ribas 5566 A. Reiners 1212 A. Quirrenbach 88 J.
Aceituno 77 D. Baroch 5566 C. Cifuentes 33 S. Dreizler 1212 A. Hatzes 1313 A.
Kaminski 88 D. Montes 1414 J. C. Morales 5566 A. Pavlov 11 L. Peña 77 V.
Perdelwitz 15151616 S. Reffert 88 D. Revilla 77 C. Rodríguez López 77 A.
Rosich 5566 S. Sadegi 8811 J. Sanz-Forcada 33 P. Schöfer 771212 A. Schweitzer
1616 M. Zechmeister 1212

(Received 15 April 2022 / Accepted 31 August 2022)

###### Abstract

Context. A challenge with radial-velocity (RV) data is disentangling the
origin of signals either due to a planetary companion or to stellar activity.
In fact, the existence of a planetary companion has been proposed, as well as
contested, around the relatively bright, nearby M3.0 V star AD Leo at the same
period as the stellar rotation of 2.23 d.

Aims. We further investigate the nature of this signal. We introduce new
CARMENES optical and near-IR RV data and an analysis in combination with
archival data taken by HIRES and HARPS, along with more recent data from
HARPS-N, GIANO-B, and HPF. Additionally, we address the confusion concerning
the binarity of AD Leo.

Methods. We consider possible correlations between the RVs and various stellar
activity indicators accessible with CARMENES. We additionally applied models
within a Bayesian framework to determine whether a Keplerian model, a red-
noise quasi-periodic model using a Gaussian process, or a mixed model would
explain the observed data best. We also exclusively focus on spectral lines
potentially associated with stellar activity.

Results. The CARMENES RV data agree with the previously reported periodicity
of 2.23 d, correlate with some activity indicators, and exhibit chromaticity.
However, when considering the entire RV data set, we find that a mixed model
composed of a stable and a variable component performs best. Moreover, when
recomputing the RVs using only spectral lines insensitive to activity, there
appears to be some residual power at the period of interest. We therefore
conclude that it is not possible to determinedly prove that there is no planet
orbiting in synchronization with the stellar rotation given our data, current
tools, machinery, and knowledge of how stellar activity affects RVs. We do
rule out planets more massive than 27 M⊕ (= 0.084 M${}_{\textnormal{Jup}}$).
Likewise, we exclude any binary companion around AD Leo with M$\sin{i}$
greater than 3–6 M${}_{\textnormal{Jup}}$ on orbital periods $<14$ yr.

###### Key Words.:

techniques: radial velocities – stars: late-type – stars: individual: AD Leo –
stars: activity

##  1 Introduction

In the pursuit for Earth-like planets, modern spectrographs are pushing the
limits by reaching m s-1 level precision or even tens of cm s-1 (e.g.,
ESPRESSO; Pepe et al., 2021), which is needed. However, the measurements begin
to succumb to unwanted signals for planet searches. Intrinsic stellar
variability in the form of dark spots, bright plages, and flares can produce
radial-velocity (RV) variations that can conceal true planetary signals, or
even masquerade as a fake planet which then can be effectively modeled with a
Keplerian orbit. To help mitigate these stellar-activity-induced RV signals, a
number of procedures are commonly put in place, such as different modeling
approaches, smart data collection strategies, and extraction of particular
spectral lines.

Statistical techniques such as Gaussian process (GP) regression have been used
by treating stellar activity behavior as a quasi-periodic signal (e.g.,
Haywood et al., 2014; Rajpaul et al., 2015; Jones et al., 2017). This approach
may sometimes lead to better precision and accuracy of planetary parameters,
especially those in the lower-mass regime (e.g., Stock et al., 2020; Amado et
al., 2021). Likewise, these signals can be wavelength-dependent, usually with
the amplitude decreasing in the redder regime of the spectrum, but still
containing some residual effect depending on the star-spot configuration and
temperature difference (Reiners et al., 2010, and references therein). For
this reason, there is a push for higher-precision instruments covering the red
and near-IR wavelength range such as CARMENES111Calar Alto high-Resolution
search for M dwarfs with Exo-earths with Near-infrared and optical Échelle
Spectrographs, http://carmenes.caha.es (Quirrenbach et al., 2014, 2018),
GIARPS222GIAno & haRPS-n, https://www.tng.inaf.it/news/2017/04/04/giarps/
(Claudi et al., 2017), HPF333The Habitable Zone Planet Finder,
https://hpf.psu.edu/ (Mahadevan et al., 2012, 2014), IRD444InfraRed Doppler
instrument (Tamura et al., 2012; Kotani et al., 2014, 2018), MAROON-X555Red-
optical, high-resolution spectrograph with focus on mid- to late-type M
dwarfs, https://www.gemini.edu/instrumentation/maroon-x/ (Seifahrt et al.,
2018, 2020), and SPIRou666A near-infrared spectropolarimeter/velocimeter,
https://spirou.omp.eu/ (Donati et al., 2020).

Focusing on certain spectral lines as activity indicators sensitive to
chromospheric (e.g., H$\alpha$, Ca ii infrared triplet) or photospheric (e.g.,
TiO) effects on active M dwarfs can oftentimes be successful in determining
the star’s rotational period (see Fig. 11 in Schöfer et al., 2019). Even then,
there is no universal approach because it is still not unique as to which
activity indices do peak at the rotational period and under what conditions,
as pointed out by Lafarga et al. (2021). Moreover, efforts for identifying
which spectral lines, in general (i.e., not focusing on already-known specific
lines), seem to be more activity-sensitive than others have been fruitful for
a selection of G–K dwarfs (e.g., Wise et al., 2018; Dumusque, 2018;
Lisogorskyi et al., 2019; Ning et al., 2019; Cretignier et al., 2020; Thompson
et al., 2020). Such an approach proves to be rather challenging for M dwarfs,
where the spectra contain a forest of blended lines, making it almost
impossible to find the continuum (Merrill et al., 1962; Boeshaar, 1976;
Kirkpatrick et al., 1991; Alonso-Floriano et al., 2015). This approach,
however, seems to have been successful for stars that show a clear stellar
activity impact on the RVs (e.g., EV~Lac, Lafarga priv. comm.).

We turn our attention specifically to the active mid-type M dwarf AD Leo, a
star whose stellar rotation period of 2.23 d presents itself both in
photometry and RVs (Morin et al., 2008; Tuomi et al., 2018; Carleo et al.,
2020; Robertson et al., 2020). Despite its strong flaring activity manifesting
itself at many wavelengths (Buccino et al., 2007; Rauer et al., 2011;
Tofflemire et al., 2012; Vidotto et al., 2013), AD Leo has been included in a
number of studies addressing the existence of planets orbiting this star.
Tuomi et al. (2018), referred to as T18 hereinafter, first suggested that a
planet may be orbiting AD Leo in a 1:1 spin-orbit resonance since it proved to
be difficult to simultaneously explain both the photometry and RV measurements
using a variety of star-spot scenarios. Furthermore, they claimed that the RV
measurements were time- and wavelength-independent, and the putative planet
exhibited a semi-amplitude of $\sim$19 m s-1. Despite some evidence for solely
stellar activity behavior, T18 concluded that AD Leo is an active M dwarf
hosting a hot Jupiter ($\geq 0.2M_{\textnormal{Jup}}$) in a 1:1 spin-orbit
resonance.

However, the existence of this hot Jupiter around AD Leo was challenged.
Carleo et al. (2020, referred to as C20 hereinafter) investigated the 2.23 d
signal and obtained observations with GIARPS at the 3.56 m Telescopio
Nazionale Galileo (Claudi et al., 2017). Even though the 2.23 d signal was
persistent, the amplitude heavily diminished as a function of wavelength and
of time. Simultaneous photometric data from STELLA showed a shift of
$\sim$0.25 in phase ($\sim$0.6 d) in comparison to the HARPS-N RV curves.
Therefore, C20 disputed the argument posed by T18, concluding that the RV
modulation is not compatible with a planetary companion. Shortly after, the
conclusions by Robertson et al. (2020), referred to as R20 hereinafter, were
also in line with C20, as they likewise observed a decrease in amplitude
between the two observing seasons using HPF spectroscopic data. At the time of
submission, additional data from SPIRou and SOPHIE777Optical échelle
spectrograph, http://www.obs-hp.fr/guide/sophie/sophie-eng.shtml also
indicated no evidence for a corotating planet (Carmona et al. in prep.).

This concept of such a hot Jupiter around an M dwarf in a synchronized
rotation is unique. Many studies addressing the correlation between the
orbital period ($P_{\textnormal{orb}}$) and the stellar rotation period
($P_{\textnormal{rot}}$) have thus far been focused on more solar-like
transiting host stars rather than M dwarfs, using Kepler data (Borucki et al.,
2010). Findings from McQuillan et al. (2013) and Walkowicz & Basri (2013)
suggest a clear absence of close-in planets ($P_{\textnormal{orb}}\lesssim$
2–3 days) around rapidly rotating stars ($P_{\textnormal{rot}}\lesssim$ 5–10
days), where planets with shorter periods were nearly synchronous
($P_{\textnormal{orb}}\sim P_{\textnormal{rot}}$). Teitler & Königl (2014)
proposed that the cause was orbital decay of close-in planets by their host
stars. Besides, with regards to M dwarfs, Newton et al. (2016) showed that
typical stellar rotation periods would often match the periods of planets in
their habitable zone. Regardless, there is a dearth of close-in massive
planets around M dwarfs where only a handful of such are known, such as NGST-1
b (Bayliss et al., 2018) or TOI-519 b ($M\lesssim
14\,\textnormal{M}_{\textnormal{Jup}}$; Parviainen et al., 2021). This is not
a result of observational bias as these types of planets would be simple to
detect through transits and precise RVs. AD Leo was even used as a test case
to check the effects of XUV irradiation on planet atmospheres (Chadney et al.,
2016), considering that the ionsphere can play an influential role in
regulating the stability of upper atmospheres on giant planets. Therefore,
understanding the case of AD Leo could aid in finding the key methods for
other potential planet detections that suffer from ambiguities due to 1:1
spin-orbit resonances.

In our focused study using optical and near-IR RV measurements, we claim that
one cannot prove (or disprove) the suggested planetary signal in the AD Leo
system, as the problem is completely degenerate and we are limited by our
current state-of-the-art machinery and measurements. We organize the paper as
follows. In Sect. 2, we first introduce the active M dwarf AD Leo and
investigate its presumed binarity for the first time. All available RV data
used for the analysis are presented in Sect. 3. We turn our focus on the
chromaticity of the RV signal for the CARMENES data in Sect. 4. Still
concentrating on the CARMENES data, in Sect. 5 we perform a spectral line
analysis for the target. Then combining all available spectroscopic data for
the first time, we explore various modeling techniques within the Bayesian
framework in Sect. 6. Finally, in Sect. 7, we present a discussion on our
results as well as suggestions to break the degeneracy of this situation. We
summarize our conclusions in Sect. 8.

##  2 AD Leo

###  2.1 Stellar parameters

Table 1: Stellar parameters of AD Leo.

Parameter | Value | Reference
---|---|---
Identifiers
BD | +20 2465 | Arg1861
Ci | 18 1244 | Por1915
GJ | 388 | Gli1957
Karmn | J10196+198 | Cab2016
Astrometry and kinematics
$\alpha$ (epoch J2016.0) | 10:19:35.7 |  Gaia EDR3
$\delta$ (epoch J2016.0) | 19:52:11.3 |  Gaia EDR3
$\mu_{\alpha}\cos\delta$ (mas yr-1) | $-498.62\pm 0.03$ |  Gaia EDR3
$\mu_{\delta}$ (mas yr-1) | $-43.43\pm 0.03$ |  Gaia EDR3
$\pi$ (mas) | $201.41\pm 0.03$ |  Gaia EDR3
$d$ (pc) | $4.964\pm 0.001$ |  Gaia EDR3
$\gamma$ (km s-1) | $12.286\pm 0.021$ | Laf2020
$U$ (km s-1) | $-14.929\pm 0.010$ | This work
$V$ (km s-1) | $-7.444\pm 0.007$ | This work
$W$ (km s-1) | $+3.391\pm 0.017$ | This work
Galactic population | Young disk | Mon2001
Stellar kinematic group | Castor | LS2010
Key photometry
$G$ (mag) | $8.2041\pm 0.0015$ |  Gaia EDR3
$J$ (mag) | $5.449\pm 0.027$ | 2MASS
Photospheric parameters
Spectral type | M3.0 V | AF2015
$T_{\textnormal{eff}}$ (K) | $3477\pm 23$ | Mar2021
$\log{g}$ (cgs) | $5.12\pm 0.12$ | Mar2021
[Fe/H] (dex) | $-0.19\pm 0.12$ | Mar2021
¡Bf¿ (G) | $3357\pm 172$ | Rein2022
Activity
$v\sin{i}$ (km s-1) | $2.4\pm 1.5$ |  This workaa$a$This value was individually produced for AD Leo using a chosen order selection and is consistent, but supersedes previous measurement (e.g., Marfil et al., 2021).
$P_{\textnormal{rot}}$ (d) | $2.2270^{+0.0010}_{-0.0011}$ |  This workbb$b$See Sect. 6.3 for the $P_{\textnormal{rot}}$ determination and Table $a$ for other rotational periods found in the literature.
pEW(H$\alpha$) (Å) | $-3.73\pm 0.41$ | Fuhr2022
$\log{\textnormal{H}\alpha/L_{\textnormal{bol}}}$ | $-3.614\pm 0.003$ | Sch2019
$\log{R^{\prime}_{\textnormal{HK}}}$ | $-3.97\pm 0.05$ | This work
$\log{L_{\textnormal{X}}/L_{\textnormal{bol}}}$ | –3.3 | Fav2000
Physical parameters
$L_{\star}$ ($10^{-5}\,L_{\odot}$) | $2359\pm 11$ | This work
$R_{\star}$ ($R_{\odot}$) | $0.4233\pm 0.0057$ | This work
$\textnormal{M}_{\star}$ ($\textnormal{M}_{\odot}$) | $0.423\pm 0.012$ | This work
$i$ (deg) | $12.9^{+8.4}_{-8.1}$ | This work
888

2MASS: Skrutskie et al. (2006); AF2015: Alonso-Floriano et al. (2015);
Arg1861: Argelander (1861); Cab2016: Caballero et al. (2016); Fav2000: Favata
et al. (2000); Fuhr2022: Fuhrmeister et al. (2022); Gaia EDR3: Gaia
Collaboration et al. (2021); Gli1957: Gliese (1957); Hoj2019: Hojjatpanah et
al. (2019); Laf2020: Lafarga et al. (2020); LS2010: López-Santiago et al.
(2010); Mar2021: Marfil et al. (2021); Mon2001: Montes et al. (2001);
Rein2022: Reiners et al. (2022); Pass2019: Passegger et al. (2019); Por1915:
Porter et al. (1915); Sch2019: Schöfer et al. (2019).

AD Leo (GJ 388), an M3.0 V star at a distance of slightly less than 5 pc and
with $V\sim$ 9.5 mag, is one of the closest and brightest M dwarfs. Already
tabulated in the Bonner Sternverzeichnis by Argelander (1861), AD Leo has been
the subject of numerous investigations in the last century (e.g., Abell, 1959;
Engelkemeir, 1959; Lang et al., 1983; Saar & Linsky, 1985; Hawley & Pettersen,
1991; Hawley et al., 2003; Osten & Bastian, 2008; Hunt-Walker et al., 2012).

In Table 1, we list the stellar properties of AD Leo. In particular, we
tabulate equatorial coordinates, proper motions, and parallax from the Gaia
Early Data Release 3 (EDR3; Gaia Collaboration et al., 2021) and absolute RV
from Lafarga et al. (2020, uncorrected of gravitational redshift for
consistency with the previous literature), from which we recompute
Galactocentric space velocities as in Cortés-Contreras (2016). The spectral
type of Alonso-Floriano et al. (2015) superseded previous determinations
(e.g., Johnson & Morgan, 1953; Bidelman, 1985; Stephenson, 1986; Keenan &
McNeil, 1989), while the photosphere parameters of Passegger et al. (2019)
match previous CARMENES publications and are similar, but not identical, to
those of Rojas-Ayala et al. (2012), Lépine et al. (2013), Gaidos et al.
(2014), or Mann et al. (2015). With the effective temperature of Passegger et
al. (2019), the bolometric luminosity of Cifuentes et al. (2020), and the
Stefan-Boltzmann law we derived the stellar radius and, with the radius-mass
relation of Schweitzer et al. (2019), the stellar mass. For compiling the most
precise parameters of the activity indicators, we used the SVO Discovery
Tool999http://sdc.cab.inta-csic.es/SVODiscoveryTool/. The projected rotational
velocity $v\sin{i}$ was computed by us exactly as in Reiners et al. (2018),
but on the newest CARMENES template spectra (Sect. 3).

As first reported by Montes et al. (2001), the Galactocentric space velocity
of AD Leo is consistent with it belonging to the Galactic young disk (Leggett,
1992). Later, López-Santiago et al. (2010) and Klutsch et al. (2014) proposed
AD Leo as a candidate member of the Castor moving group, in agreement with our
latest kinematic data. The age of the Castor moving group, of about 300–500
Myr (Barrado y Navascués, 1998; Mamajek et al., 2013), is consistent with age
determinations for AD Leo by Shkolnik et al. (2009), Brandt et al. (2014), and
Meshkat et al. (2017).

Such a young age partly explains the flares frequently observed in AD Leo. The
star has been known to exhibit activity ever since the first observed optical
flare event in 1949 (Gordon & Kron, 1949), followed by many others (e.g.,
Liller, 1952; MacConnell, 1968; Pettersen et al., 1984; Crespo-Chacón et al.,
2006)101010Of the 70 reports of the IAU Information Bulletin on Variable Stars
citing AD Leo, only 18 did not have the star name in the title.. Frequent
flaring activity was further observed during an extreme ultraviolet (EUV)
1-month monitoring of AD Leo (Sanz-Forcada & Micela, 2002). Later on, emission
in the X-ray and radio regimes has also been observed (Gurzadyan, 1971;
Robinson et al., 1976), and which has shown bursting radio emission at GHz
frequencies and below (Osten & Bastian, 2008; Villadsen & Hallinan, 2019).
Muheki et al. (2020) and Namekata et al. (2020) have presented the most recent
analyses on high-resolution optical spectroscopy and X-ray observations of
flares on AD Leo.

The young age of AD Leo also explains the moderately large rotational velocity
and short rotational period, of 2.23 d, as well as X-ray, Ca ii H&K, and
H$\alpha$ emission (see references in Table 1). In addition, this star
presents large RV variations: it shows a standard deviation larger than 20 m
s-1, ($\sim 1.5\times$ median absolute deviation RV as in Tal-Or et al. 2019
and Grandjean et al. 2020), which can be connected to stellar activity, the
presence of a planetary companion, or both. For the derivation of
$\log{R^{\prime}_{\textnormal{HK}}}$, we used the method described by
Perdelwitz et al. (2021), albeit with a slight modification of the $k1$
passband used for normalization. A total of 316 archival spectra from ESPaDOnS
(232 spectra), HARPS (58 spectra), HIRES (24 spectra) and NARVAL (2 spectra)
were analyzed, yielding a mean value of
$\log{R^{\prime}_{\textnormal{HK}}}=-3.97\pm 0.05$, which is in agreement with
the result published by Astudillo-Defru et al. (2017) based on the HARPS
spectra alone. We compute the stellar inclination, $i$, to be $\sim
13^{\circ}$ from $10^{6}$ MCMC realizations using $v\sin{i}$, $R$, and
$P_{\textnormal{rot}}$ as provided in the stellar parameters table. Previous
papers quoted values of $\sim 20^{\circ}$ (Morin et al., 2008) and $\sim
15^{\circ}$ (Tuomi et al., 2018).

###  2.2 Photometric rotational period

T18 took a closer look at the available photometry from ASAS-North and ASAS-
South, as well as from short-cadence MOST observations (see T18 for photometry
references). To summarize, the ASAS photometry shows a long-term periodicity
of around 4070 d, most likely due to a stellar activity cycle. During the
brightness minimum of this cycle, the signal at the rotation period of 2.23 d
seems to disappear turning up only during the brightness maximum, in which the
MOST observations were taken. These short-cadence MOST data, taken over the
course of 9 d (Hunt-Walker et al., 2012), showed fluctuations with periodicity
2.23 d, but also demonstrated behavior of slight phase shifts and amplitude
and period variations, indicating that the stellar surface should be
experiencing rapid evolution. Other observations taken with different
instruments are in agreement with the 2.23 d period (see Table $a$).

AD Leo was additionally observed in three of the TESS sectors111111TESS Web
Viewing Tool: https://heasarc.gsfc.nasa.gov/cgi-bin/tess/webtess/wtv.py
Accessed 6 April 2022., particularly in sector 45 (camera #3 CCD #4; 6
November 2021 to 02 December 2021), sector 46 (camera #2 CCD #2; 02 December
2021 to 30 December 2021), and in sector 48 (camera #1 CCD #4; 28 January 2022
to 26 February 2022). All sectors constitute short cadence 20-second and
2-minute integrations. Unfortunately, the pixel column was saturated for
sectors 45 and 46 leading to unusable data. The sector 48 data were still
valuable for determining a periodic variation. In doing so, we removed strong
flare events (up to 275 ppt) and any outliers on the 20-second integration
data which were binned every 10 points for computational reasons. After
applying a sinusoid model, the resulting amplitude and $P_{\textnormal{rot}}$
are $\sim$800 ppm and $P_{\textnormal{rot}}$ of $2.2304\pm 0.0014$ d,
respectively.

###  2.3 Hypothetical binary

There is some confusion in the literature regarding the hypothetical
multiplicity of AD Leo. First of all, in spite of being listed as “$\gamma$
Leo C” in some astronomical databases (e.g., SIMBAD), AD Leo is not a wide
($\rho\sim$ 5.7 arcmin) physical companion to Algieba, which is a binary
system of two G- and K-type giant stars visible to the naked-eye located eight
times further ($γ^01$~Leo \+ $γ^02$~Leo; van Leeuwen, 2007; Gaia Collaboration
et al., 2021).

Next, the Washington Double Star (WDS; Mason et al., 2001) catalog tabulates
AD Leo as a close binary candidate. The presence of a hypothetical companion
around AD Leo was first inferred by Reuyl (1943) from measurements of
photographic plate images. He suggested an eccentric ($e=0.6$) and close-in
($a\sim 0.54$ au, projected angular separation of $\rho\sim 0.11$ arcsec)
orbit with an orbital period of 26.5 years, making the companion a brown dwarf
($\textnormal{M}\approx 0.032\textnormal{M}_{\odot}$). Later, van de Kamp &
Lippincott (1949) found indications of an astrometric trend from photographic
plates that did not fit those orbital elements. Following up two decades
later, Lippincott (1969) found an ambiguous deviation from linear proper
motion, thus deciding that it was inconclusive to determine whether there
could be a variable proper motion due to a companion.

With speckle imaging at 7800 Å at the 3.6 m Canada-France-Hawai’i Telescope,
Balega et al. (1984) resolved a companion candidate to AD Leo at $\rho$ =
0.078 $\pm$ 0.010 arcsec ($r\approx$ 0.39 au) and position angle of $\theta$ =
39 $\pm$ 4 deg in 1981, which received the WDS name 10200+1950 and discoverer
code BAG 32. Two additional measurements in 1983 were provided by Balega &
Balega (1985) at $\rho\sim$ 0.11 arcsec, but with $\theta\sim$ 330 deg.
Afterwards, with lucky imaging in the $I$ band at the 1.5 m Telescopio Carlos
Sánchez, Cortés-Contreras et al. (2017) resolved a candidate source, about 2.0
mag fainter than AD Leo, at $\rho$ = 0.195 $\pm$ 0.061 arcsec ($r\approx$ 0.97
au) and $\theta$ = 23.8 $\pm$ 3.7 deg in 2012. However, it fell on the first
Airy disk, and they were not able to detect it again in observations in 2015
with the same instrument setup. From $\Delta I\approx 2.0$ mag and the
projected physical separation of about 0.97 au, Cortés-Contreras et al. (2017)
estimated an M5 V spectral type for the companion candidate and an orbital
period of $P\sim$ 1.5 yr, which might be the source discovered by Balega et
al. (1984). Because of this hypothetical companion, AD Leo was initially
discarded from the CARMENES guaranteed time observation target list (Caballero
et al., 2016; Reiners et al., 2018).

Further attempts at resolving the companion proved to be unsuccessful, such as
additional speckle imaging by Docobo et al. (2006). With adaptive optics at 8
m-class telescopes, Daemgen et al. (2007) and Brandt et al. (2014) imposed
very strict upper limits to the presence of companions, of $\Delta K_{s}\sim$
7.8 mag at $\rho\geq$ 0.5 arcsec and $\Delta H\sim$ 10.9 mag at $\rho\geq$ 1.5
arcsec, respectively. From Fig. 7 in Daemgen et al. (2007), their
Altair/Gemini North observations discarded any companion of $\Delta K_{s}\sim$
2.0 mag at $\rho\geq$ 0.1 arcsec. No trace of companions is seen either in
archival images obtained in 2005 with the Hubble Space Telescope (ACS/HRC in
F330W) and in 2018 with the Very Large Telescope (SPHERE in $H$)121212Near-
infrared archival images obtained in 1999 with Hubble NICMOS/NIC2 and in 2009
with VLT NAOS+CONICA saturate and are not useful at very close separations..
Therefore, the findings of Balega et al. (1984), Balega & Balega (1985), and
Cortés-Contreras et al. (2017) are most probably not related to a real
physical companion, unless the orbital motion can explain the later
nondetections. Furthermore, the Gaia EDR3 renomalized unit weight error (RUWE)
for AD Leo is 1.15, below the critical threshold of 1.40, the along-scan
observations (astrometric_gof_al) is 3.22, and the excess noise of the source
(astrometric_excess_noise) is 0.16, all of which indicate that it is most
likely a single source (Lindegren et al., 2018). To rule out any wide
companions, we searched for objects with common parallax and proper motions up
to a projected physical separation of 100 000 au, as in Caballero et al.
(2022), and found no hints of any wide potential companion. Figure 1
illustrates all the mentioned findings from the literature. The binary issue
was not addressed by T18, C20, or R20. The absence of any close companion to
AD Leo is further investigated with our spectroscopic data in Sect. 2.4.

###  2.4 Evidence for single star in spectroscopic data

As a first approach to search to address the potential stellar binarity and
look for traces of it, we combined all the RV data (presented in Sect. 3) to
look for long-term trends. However, this method presents its challenges as
there is no temporal overlap between the older and newer data sets and each
instrument has its own, unknown zero-point offset. We nonetheless attempted a
grid search for a Keplerian signal for which we stepped through the period,
amplitude, eccentricity and argument of periastron parameter space, where each
instrument also had its own offset and jitter term131313Jitter terms are added
in quadrature to the given error bars for each respective instrument. (e.g.,
Baluev, 2009). One then adapts the values with the best log likelihood.
Unfortunately, the lack of temporal overlap led to strong ambiguities and
degeneracies. We therefore did not find any conclusive periodicity or
indicative linear trend in this RV analysis.

We continued to investigate whether there is evidence for the presence of a
companion within our CARMENES spectra that could affect the analysis presented
later in this work. We computed a 1D cross-correlation function (CCF) with a
binary mask over a large RV range and did not find any hint of a clear
secondary peak, neither in the VIS nor the NIR data. To be certain, we also
ran todcor (Zucker & Mazeh, 1994), which computes a 2D CCF to get the RVs of
the two components simultaneously, and the results showed no evidence for a
companion as well. The 2D CCF method with todcor is well suited in the case of
double-lined binaries, but since the secondary signal seems to be either too
weak if expected to be $\sim$10 km s-1 away from the primary, or too hidden if
expected to be very close to the primary, we concluded that no secondary
heavily distorts the CCF profile and, therefore, would not cause any
noticeable effects on the CCF parameter values. Therefore, we first discard
the presence of a nearly equal brightness double-lined spectroscopic binary
from the todcor analysis. Secondly, based on our RV time series analysis, we
rule out companions with minimum masses greater than 3–6
$\textnormal{M}_{\textnormal{Jup}}$ using orbital periods of less than
$\sim$2–14 yr and amplitudes 3–5 greater than the rms of the earlier data set
instruments (i.e., HARPS, HIRES), while assuming $\sin{i}\neq 0$. We therefore
conclude that there is no evidence for a stellar binary or brown dwarf
companion of AD Leo within this parameter regime.

Figure 1: Magnitude difference versus projected angular separation diagram for
observations of potential companions around AD Leo. Detections should be above
and to the right of the lines (gray shaded region). Inaccurate detection
claims are indicated with open symbols. Measurements from Gaia EDR3 are
indicated with an extended arrow to the right to demonstrate it continues for
many more arcseconds. The upwards green arrow represents a rough difference in
magnitude interval from the astrometric masses indicated by Reuyl (1943),
which are difficult to pinpoint as they are subject to huge uncertainties. The
bright binary $\gamma$ Leo in the background of AD Leo is discussed in Sect.
2.3. References – CC17: Cortés-Contreras et al. (2017); Bra14: Brandt et al.
(2014); Dae07: Daemgen et al. (2007); BB85: Balega & Balega (1985); Bal84:
Balega et al. (1984); Reu43: Reuyl (1943), astrometry; Gaia EDR3: Gaia
Collaboration et al. (2021), astrometry.

###  2.5 Bursting radio emission from the AD Leo system

Radio bursts from AD Leo have been known for quite a long time, and have been
ascribed to (coherent) plasma emission (Stepanov et al., 2001; Osten &
Bastian, 2008). More recently, Villadsen & Hallinan (2019) detected both
short-duration (seconds to minutes) and long-duration (30 minutes or more)
bursts of AD Leo, using ultrawide-band VLA observations (in the ranges 0.2–0.5
GHz and continuous frequency coverage between 1 and 6 GHz) in several epochs
between 2013 and 2015. The timescale of the short-duration events was
consistent with the duration of “space-weather” events, such as the impulsive
phase of a flare or a coronal mass ejection (CME) crossing the corona. The
long-duration bursts, which lasted hours, and possibly extended for up to days
or even longer times, between observing epochs (similarly to the case of
Proxima Centauri; see, e.g., Pérez-Torres et al. 2021 and references therein)
require an ongoing electron acceleration mechanism during the burst. Candidate
acceleration processes are found within the paradigms of solar radio bursts
and periodic radio aurorae produced by ultracool dwarfs and planets.

The emission mechanism responsible for the aurorae from stars and planets
alike is the electron cyclotron maser (ECM) instability (Melrose & Dulk,
1982), whereby plasma processes within the star (or planet) magnetosphere
generate a population of unstable electrons that amplifies the emission. The
characteristic frequency of the ECM emission is given by the electron
gyrofrequency, $\nu_{G}=2.8\,B$ MHz, where $B$ is the local magnetic field in
the source region, in Gauss. ECM emission is a coherent mechanism that yields
broadband ($\Delta\,\nu\sim\nu_{G}/2$), highly polarized (sometimes reaching
100%), amplified nonthermal radiation. The long-duration bursts seen for AD
Leo by (Villadsen & Hallinan, 2019) between 1 and 6 GHz are characteristic of
ECM emission, implying electron densities $\lesssim 10^{9}$ cm-3, and
originate in regions with field strengths of 0.36–2.1 kG (in fundamental
emission) or 0.18–1.1 kG (if second harmonic). Since the surface magnetic
field of AD Leo is estimated to be of $\simeq 3.36$ kG (Reiners et al., 2022),
the above field strengths are likely to occur relatively close to the star, at
a height less than about one stellar radius.

Auroral cyclotron maser emission is powered by the acceleration of confined
electrons with energies between $\sim$10 keV and up to 1 MeV energies. For
substellar objects, the currents could be driven by the breakdown of rigid
corotation of magnetospheric plasma with the object’s magnetic field, for
example, due to the interaction between a rotating magnetosphere and the
interstellar medium. Corotation breakdown observed in a number of ultracool
(UCD) dwarfs with periods less than about 3 hr seem to be rotation powered,
with radio powers of up to a few times $10^{13}$ erg s-1 Hz-1 (e.g.,
Turnpenney et al., 2017). AD Leo has a rotation period of $\sim$54 hr.
Assuming coronal parameters similar to radio-loud UCDs, any corotation
breakdown in AD Leo will generate a radio power that is $\sim 6\times 10^{11}$
erg s-1 Hz-1, or less than about 100 $\mu$ Jy. Therefore, corotation
breakdown, while is likely to contribute to the overall radio power observed
from AD Leo, cannot account for the observed radio bursts, which reach peaks
of tens of mJy at GHz frequencies (Villadsen & Hallinan, 2019). A discussion
of further potential mechanisms, such as star-planet interaction, to explain
the measurements is continued in Sect. 7.2.3.

##  3 Spectroscopic data

The Doppler data that are used in our analysis are described below. All data
and spectrograph descriptions are summarized in Table 2. Data used for the
analysis are displayed in Fig. 2, along with the generalized Lomb-Scargle
(GLS) periodograms (Zechmeister & Kürster, 2009) of the data in Fig. 3.

Figure 2: Time series of all RVs with instrumental offsets accounted for. The
RV uncertainties are included, though many are too small to be seen in the
plots. The HIRES and HARPS data span a large time range and the rest of the
data come in $\sim$12 years later after the time when the majority of the
previous data were taken. The time axis is interrupted in several places, and
stretched differently between the individual sections. Additionally, the
majority of HIRES and HARPS time series are overlapping each other. Four of
the HARPS-N data overlap with CARMENES data; and the GIANO-B data are taken
all within the first observing run for HARPS-N. The first season of HPF data
overlaps with the HARPS-N and GIANO-B data sets whereas the HPF second season
overlaps with the HARPS-N and CARMENES second season. Figure 3: GLS
periodograms (black) and window functions (gray) for the CARMENES VIS1 and
NIR1 RV data sets, as well as for the combined optical (i.e., VIS1, VIS2,
HARPS, HT1, and HT2) and near-infrared (i.e., NIR1, NIR2, HPF1, HPF2)
instruments, with offsets accounted for. The horizontal lines correspond to
FAP levels of 10% (dashed), 1% (dash-dotted), and 0.1% (dotted). These are
computed by using the randomization technique for 10 000 samples. The orange
solid line corresponds to the rotational period of 2.23 d, whereas the orange
dashed line represents the 1.81 d alias signal due to daily sampling. Table 2:
Summary of the extensive spectroscopic data set for AD Leo. All instruments
used for the analysis along with their wavelength range, spectral resolution,
and number of spectra are listed.

Instrument | CARMENES VIS | CARMENES NIR | HIRES | HARPS | HARPS-N | GIANO-B | HPF
---|---|---|---|---|---|---|---
$\Delta\lambda$ (nm) | 520–960 | 960–1710 | 500-620 | 380–690 | 390–680 | 970–2450 | 810–1280
$R$ | 94 600 | 80 400 | 60 000 | 115 000 | 115 000 | 50 000 | 55 000
Subset aa$a$A subset is treated effectively as an independent data set. | VIS1 | VIS2 | NIR1 | NIR2 |  |  | HT1 | HT2 | G1 | G2 | HPF1 | HPF2
# of spectra | 26 | 20 | 26 | 20 | 43 | 47 | 42 | 21 | 12 | 13 | 5 | 30
rms (m s-1) | 18.54 | 15.63 | 16.44 | 13.61 | 20.74 | 24.23 | 21.97 | 10.51 | 17.92 | 18.41 | 19.01 | 6.68
Start date | Mar 2018 | Feb 2020 | Mar 2018 | Feb 2020 | Jun 2001 | Dec 2003 | Apr 2018 | Nov 2018 | Apr 2018 | May 2008 | Apr 2018 | Dec 2018
End date | Apr 2018 | Feb 2020 | Apr 2018 | Feb 2020 | Dec 2013 | May 2014 | Jun 2018 | Jan 2019 | May 2018 | Jun 2018 | Apr 2018 | Feb 2020

141414

###  3.1 CARMENES

The CARMENES instrument is located at the 3.5 m telescope at the Calar Alto
Observatory in Spain. It is a dual arm instrument to produce RV measurements
at optical (VIS) and near-infrared (NIR) wavelengths (Quirrenbach et al.,
2014, 2018, see Table 2 for the instrument specifications). We obtained 26
spectra for AD Leo (Karmn J10196+198) over the time span of 25 days in 2018,
notably 12 years after the majority of the HIRES and HARPS data were taken. On
some nights, we observed AD Leo multiple times to better sample the short
period of 2.23 d. We call this subset of data “VIS1” and “NIR1” given that AD
Leo was observed again later in 2020 as part of a DDT151515Director’s
Discretionary Time program, for which 20 data points were obtained over
$\sim$3 d, which we call subset “VIS2” and “NIR2”. The RVs and several
activity indicators (e.g., CRX – chromatic index, dLW – differential line
width) from both subsets were first extracted with serval (Zechmeister et al.,
2018), so that they are corrected for barycentric motion, secular
acceleration, and instrumental drift. The final RVs we use had nightly zero-
points applied (Tal-Or et al., 2019; Trifonov et al., 2020). Figure 4 displays
the CARMENES data and the best respective models (detailed in Sect. 6). For
calculating the CCF parameters within CARMENES, weighted binary masks, which
depend on spectral type and $v\sin{i}$, are produced by coadding spectra
corrected for tellurics & RV shifts and then selecting pronounced minima
(Lafarga et al., 2020). From there, the CCF parameters, namely, the bisector
span (BIS), contrast, and full-width half-maximum (FWHM), are obtained for
both the VIS and NIR channel.

Figure 4: Radial-velocity times series and phase-folded plots for the CARMENES
VIS1, NIR1, VIS2, and NIR2 data (top to bottom) with residuals from the best-
fit model. In some cases, especially in the panels with the VIS1 and VIS2
data, the uncertainties are too small to be seen in the plots. The best-fit
models are overplotted (gray line) along with the 68% posterior bands gray
shaded regions).

###  3.2 Archival RV data

##### HARPS-South.

Spectroscopic data for AD Leo are in the public archive from the High-Accuracy
Radial velocity Planet Searcher (HARPS, Mayor et al., 2003), a high-resolution
échelle spectrograph located at the ESO 3.6 m telescope at La Silla
Observatory in Chile. We retrieved a total of 52 spectra that were taken over
a span of $\sim$4500 d. Due to a fiber-upgrade intervention (Lo Curto et al.,
2015), we considered the HARPS data as coming from two separate instruments,
before and after this upgrade. After the intervention, only five data points
were taken close to one another (within 2 d) and much later than the earlier
data. These data also have a very low root-mean-square (RMS) deviation from
their mean. We did not consider them for the further analysis since they could
fit anywhere in a model with a large offset and therefore do not provide new
insight. Thus, for the rest of the analysis, we used the 47 HARPS spectra from
before the intervention. Additionally, the majority of data (i.e., 33 points)
were taken over the course of $\sim$115 d (January–May 2006). The spectra were
first processed using serval (Zechmeister et al., 2018) to obtain the RVs and
helpful stellar activity indicators (e.g., CRX, dLW, H$\alpha$, see
Zechmeister et al., 2018, for more explanation). Nightly zero point
corrections provided by Trifonov et al. (2020) were applied to the
RVs161616https://www2.mpia-hd.mpg.de/homes/trifonov/HARPS_RVBank.html and are
used in our analysis (Table 7).

##### HIRES.

We collected 43 spectra from the high-resolution spectrograph HIRES (Vogt et
al., 1994) mounted on the 10 m Keck-I telescope located at the Mauna Kea
Observatory in Hawai’i, which has been in service since 1994. The data were
taken over the course of 4500 days. A majority of them (i.e., 22 points) were
taken in 2005/2006 over 120 d and, moreover, overlap with the higher cadence
HARPS data (see Fig. 2). Butler et al. (2017) released a large RV database of
64 480 observations for a sample of 1699 stars, which was later reanalyzed by
Tal-Or et al. (2019) for minor, though significant systematic effects, such as
an RV offset due to the CCD upgrade in 2004, long-term drifts, and slight
intra-night drifts. Therefore, we continued with the corrected HIRES RVs
provided by Tal-Or et al. (2019), presented in Table 8. Data from HIRES are
also specifically addressed further under Sect. 5.2.

###  3.3 Additional RV data from the literature

Radial velocities are taken directly from C20 and R20, as described in the
following paragraphs, and are used for the analysis as well.

##### GIARPS.

We include spectroscopic data taken in GIARPS mode (Claudi et al., 2017),
where high resolution spectroscopic measurements are obtained simultaneously
with HARPS-N (Cosentino et al., 2012, extracted with the TERRA pipeline) and
with GIANO-B (Oliva et al., 2006) reduced with the on-line DRS pipeline and
the off-line GOFIO pipeline. Both instruments are located at the 3.58 m
Telescopio Nazionale Galileo (TNG) at the Roque de los Muchachos Observatory
in La Palma, Spain. The GIARPS mode is similar to CARMENES in the sense it
also addresses potential variations of a signal amplitude over a wide
wavelength range. There were two runs of HARPS-N, with only the first run
having simultaneous GIANO-B data; four of the HARPS-N data overlap with the
CARMENES data. To stay consistent with C20, we also consider two separate runs
for HARPS-N and GIANO-B, and designate them as “HT1”, “HT2”, “G1”, and “G2”
for our analysis. Due to the high uncertainty on the data points from the
GIANO-B instrument, we do not include them for our analysis.

##### HPF.

Our last data set comes from the Habitable-zone Planet Finder (HPF), a stable
NIR Doppler spectrograph that is designed to reach 1–3 m s-1 RV precision for
M dwarfs with the help of wavelength calibration via a custom NIR laser
frequency comb (Mahadevan et al., 2012, 2014). The spectrograph is installed
at the 10 m Hobby-Eberly Telescope at McDonald Observatory in Texas. The HPF
data are of high quality providing an RV precision of 1.5 m s-1 on AD Leo. A
total of 35 HPF RVs were obtained, five of which during HPF commissioning and
30 afterwards, namely “HPF1” and “HPF2”. The HPF data in this paper are
tabulated in R20. These data overlap with optical RV data from HARPS-N and
show also an amplitude decrease between observing seasons.

##  4 Wavelength dependence of RV signal in CARMENES data

Signals in RV measurements induced by dark spots corotating on the stellar
surface are expected to be more pronounced in the bluer wavelength regime for
M-dwarf stars (e.g., Desort et al., 2007; Reiners et al., 2010; Mahmud et al.,
2011; Sarkis et al., 2018). In contrast, a true planetary signal should have
an amplitude and phase that are consistent both in time and as a function of
wavelength.

###  4.1 Chromatic index

The chromatic index is a photospheric activity indicator that measures the RV-
log $\lambda$ correlation where a straight line is fit to the RV values
computed from individual échelle orders as a function of log $\lambda$
(Zechmeister et al., 2018; Tal-Or et al., 2018). This acts as a measure of
wavelength dependence as caused by cool spots on the surface of M dwarfs.
However, without further modeling, it does not provide any insight as to how
large the spot coverage fraction is or what the star-spot temperature contrast
may be. The RV-CRX correlations are shown for CARMENES VIS, CARMENES NIR, and
HARPS, in Figs. 9, 10, and 11, respectively. The CARMENES VIS data show a
clear anticorrelation ($r$ = –0.82) indicating chromatic dependency and HARPS
surprisingly demonstrates a strong, positive correlation ($r$ = +0.80),
whereas the CARMENES NIR data present a moderate, negative correlation ($r$ =
–0.30).

Different slopes at different wavelengths are not necessarily unexpected. In
principle, this depends on which mechanism is dominating, namely either the
star-spot temperature contrast or Zeeman broadening, where a more negative RV-
CRX correlation suggests the former to be predominant. This is also equivalent
to the amplitude of the RVs due to stellar features decreasing with longer
wavelengths (Sect. 4.2). The fact that the slopes of CARMENES and HARPS are in
contradiction could simply indicate that different mechanisms are prevalent.
Likewise, taking into consideration that we are probing very different
spectral lines at different wavelengths may introduce a trend with wavelength
that is not yet well understood. When considering only the orders of CARMENES
VIS that overlap with the HARPS wavelength range, the correlation would still
stay the same. Here, the wavelength range considered is now equivalent,
nonetheless the signs of the slopes are inconsistent. No firm conclusion can
be drawn, though this could be a result of different mechanisms dominating
during different time periods of long-term stellar activity.

Furthermore, the correlation plot for the CARMENES VIS demonstrates a “closed-
loop” (circular) behavior (see Sect. 7.2.5 for a discussion). Removing the CRX
trend on just the CARMENES data results in two-fold decrease in the rms of the
corrected RVs (from 18.5 m s-1 to 9.2 m s-1 for first season and from 15.6 m
s-1 to 9.3 m s-1 for the second). This subtraction was not performed on the
CARMENES NIR data since the Pearson-$r$ coefficient was not large enough
(Jeffers et al., 2022).

###  4.2 Wavelength segments

The CARMENES VIS channel records 55 échelle orders, 42 of which are considered
when computing the RV measurement via a weighted mean (Zechmeister et al.,
2018). We chose to combine these orders into four wavelength segments where
each segment consists of ten (or 11) échelle orders in order to preserve some
precision, and the RVs are then recomputed for each respective wavelength
coverage. Similarly, the CARMENES NIR channel has 28 RV orders over the $Y,J,$
and $H$ photometric bands. Due to telluric contamination, especially in the
$J$ band (Reiners et al., 2018), only a selected few orders are considered
(Bauer et al., 2020). We then use two wavelength segments, one for the $Y$ and
another for the $H$ band, consisting of 12 and seven individual orders,
respectively. For this wavelength segment analysis, we consider just the
season one CARMENES data because the data from season two exhibit an amplitude
decrease (see Sect. 6.3), and likewise, breaking up the orders would introduce
noise given the few data points over a short time span.

Each wavelength segment is treated as an individual data set. We fit a simple
sinusoid to obtain $K$, the semi-amplitude of the signal and 1$\sigma$ errors.
When doing this, the semi-amplitude clearly decreases with increasing
wavelength in the optical regime, but then reaches a plateau when continuing
in the near-IR (Fig. 5). This behavior of decreasing but then constant RV
semi-amplitude is in agreement with Reiners et al. (2010) for a dark spot on
the surface of an M dwarf; specifically, a spot covering 1.5% of the projected
surface, with a temperature 200 K cooler than the star (assumed
$T_{\textnormal{eff}}=3700$ K), and a stellar $v\sin{i}$ of 5 km s-1, in line
with AD Leo’s stellar parameters171717We adopt the higher value of $v\sin{i}$
(i.e., 5 km s-1) rather than 2 km s-1 (as also computed in Reiners et al.,
2010) because AD Leo is relatively active for its $v\sin{i}$. (Table 1).
Simulations show a linear relation between spot coverage and RV semi-amplitude
for low spot coverage values. A large spot-star temperature difference
($T_{\textnormal{spot}}=\frac{2}{3}T_{\textnormal{eff}}$), in comparison,
would not cause a notable semi-amplitude dependency as a function of
wavelength; however, it is not likely for cooler stars, such as AD Leo, to
have large spot-star temperature differences (Bauer et al., 2018).

Likewise, the HARPS instrument covers 72 spectral orders, of which 61 produce
reliable RVs after being processed by serval (signal-to-noise is too low for
the others). Similarly, we computed six wavelength chunks with ten spectral
orders each (11 for the reddest chunk) and followed the same methodology as
for the CARMENES wavelength chunks. Reiners et al. (2013) and T18 performed
similar analyses but differed in interpretation. T18 suggested there is no
dependence on wavelength, whereas Reiners et al. (2013) claimed otherwise and
mentioned that this is the first case of a known active star with increasing
amplitude with wavelength. Here, we find a slight positive incline, which is
plotted for comparison to the CARMENES data in Fig. 5. The positive slope can
be interpreted as being due to the Zeeman effect, which has the opposite
effect compared to a spot-temperature difference where the RV amplitude is
predicted to increase for redder wavelengths (Reiners et al., 2013). For the
overlapping wavelengths, the amplitudes of the wavelength chunks do not
particularly agree which can simply be an artifact that HARPS data were taken
at a time where AD Leo exhibited less activity in the RVs, as is likewise
present behavior in the photometry (Sect. 2.2).

Figure 5: Radial-velocity semi-amplitudes as a function of wavelength for the
wavelength chunks from HARPS and the CARMENES VIS and NIR spectographs. The
gray horizontal lines for each data point correspond to the wavelength
coverage considered when recomputing the RV for the wavelength chunk. The
green dots connected by a dashed line represent the theoretical values of a
1.5% spot coverage on a 3700 K, $v\sin{i}$ of 5 km s-1 star with a temperature
difference of 200 K taken from Reiners et al. (2010). The theoretical values
are binned in 2 to serve as a better comparison to the wavelength bins
provided by the real data.

##  5 Identifying individual spectral lines affected by stellar activity

We further investigate the possible effect of stellar activity on the spectral
lines themselves. We follow an approach similar to Dumusque (2018), which has
proven to be fruitful for a selection of G-K dwarfs whose RVs are dominated by
activity. Essentially, we use the CCF technique, where we obtain a binary mask
that contains all available, reliable lines for a spectrum (following Lafarga
et al., 2020). We then compute individual RVs of the few thousands of spectral
lines we have identified, and obtain an RV time series for each line. To
classify the lines according to their sensitivity to activity, we correlate
their RVs to an activity indicator obtained from the same spectra (such as the
CRX, BIS, or the total RV). We then select a subsample of spectral lines that
are least affected by stellar activity (those that do not show a strong
correlation) and recompute the RVs using this subsample to mitigate stellar
activity. The recomputed RVs then have a smaller scatter and the modulation
due to stellar rotation decreases.

This technique has been tested for other M dwarfs similar to AD Leo (spectral
types 3.0 to 4.5 V, relatively low rotational velocities and high activity
levels) that are well-known to exhibit strong stellar activity signals (e.g.,
YZ CMi, EV Lac) and appears to perform well and as expected (Lafarga priv.
comm.). We found similar results regardless of the activity indicator (total
RV, CRX or BIS) used to compute the correlations with the individual line RVs,
with the total RV yielding slightly smaller RV scatters.

###  5.1 Individual spectral lines using the CARMENES data

Specifically for AD Leo, we compare the computed individual line RVs to the
CRX or the BIS and estimate the strength of stellar activity based on these
correlations. We focus on just the first season in 2018, namely, VIS1, since
the second season only covers a bit more than one rotation cycle and this then
introduces too much scatter due to the photon noise being too high. To
quantify the correlation strength, we used the Pearson correlation coefficient
$r$. We considered lines as ‘inactive’ with $r$ close to 0. We also discarded
lines that had time series scatter larger than 400 m s-1, as measured from
their weighted standard deviation (WSTD) RV; these are weak lines that mostly
add noise to the recomputed RVs. Our results for the correlation with the CRX
are shown in Fig. 6, where we generated three line subsamples: (1) all
spectral lines – black, (2) where $|r|\leq 0.30$ – orange , and (3) where
$|r|\leq 0.20$ – blue. We obtained similar values using the correlation with
the BIS. We recomputed the RVs from both the orange and blue subsamples, thus
using the least activity-affected lines. Assuming the scatter of the RV data
to be dominated by the stellar activity, we then would expect a decrease in
the time series RV scatter, which we do not observe. Regarding the modulation,
the periodograms of these two data sets show a decrease in the power of the
2.23 d peak compared to the all-lines data set, but there is still some power
left (false alarm probability (FAP) ¿ 10 % and almost 10 % for the orange and
blue subsets respectively).

This would be clear if the RV scatter decreased for the orange or blue data
sets. However, since we do not observe such a decrease, it is difficult to
discern what is causing the periodogram behavior. The increase in the RV
scatter could be caused by increasing photon noise in the RVs of the orange
and blue data sets (because we are using a smaller number of lines to compute
the RVs, hence less signal). Then, the decrease in the periodogram power could
be due to this increasing noise, and not to a decrease in the activity signal.

Compared to the results from other stars (e.g., EV Lac, YZ CMi), the
correlations found for AD Leo are much weaker (the mean correlation
coefficient $r$ is $\sim$ –0.2 and some of them show $|r|$ $\geq$ 0.8 for the
correlation with the CRX, while for AD Leo, the mean is at 0 and very few
lines have $|r|$ $\geq$ 0.6). This lack of clear correlations indicates that
the correlations that we find for AD Leo do not have much information related
to the activity of the star, and this could be why we are not able to
effectively mitigate the stellar activity signal in the RVs.

This difference in the correlation strength could be due to the different RV
amplitudes of the stars. AD Leo shows a small RV amplitude compared to the
other considered stars: $K$ $\sim$ 25 m s-1 in comparison to EV Lac with
$\sim$100 m s-1. Both stars have similar spectral types and activity levels
(for EV Lac, pEW(H$\alpha$) = –4.983 $\pm$ 0.021, as computed from the
CARMENES observations, Schöfer et al., 2019), so the difference in RV
amplitudes seems to be caused by different spot configurations. AD Leo has a
relatively low inclination ($i\sim 13^{\circ}$) in comparison to EV Lac or the
other considered stars ($\geq 60^{\circ}$, see e.g., Morin et al., 2008). This
close to pole-on inclination could cause any visible corotating spots to
induce a smaller modulation in the RVs simply because $v\sin{i}$ is smaller.
Also, the photosphere of AD Leo could be more homogeneously spotted, also
inducing smaller RV modulations.

To summarize, we recomputed the RVs using only the lines least affected by
activity in the AD Leo spectra, and observed a decrease in the periodogram
peak at 2.23 d (0.1 % to 10 %); however, there was still some significant
residual power and we did not observe a significant decrease in the RV
scatter. This could be due to AD Leo having a different activity signal in the
RVs than other stars with similar characteristics, for which we observe a
clear mitigation of activity.

Figure 6: Three various subsamples of the spectral lines: black – all, orange
– $|r|\leq 0.30$, and blue – $|r|\leq 0.20$. Top: Absolute RVs versus time for
the first CARMENES VIS season. Bottom left: Histogram of the Pearson-$r$
correlation coefficient when comparing the RVs from individual lines to the
CRX. The criteria for the subsamples are illustrated here. Bottom right:
Zoomed-in GLS periodograms around the signal of interest of the RVs of the
subsamples. The orange subsample still shows power at 2.23 d, whereas the blue
subsample still has a nonsignificant bump but it is noise limited.

###  5.2 Other spectroscopic activity indicators

To aid in disentangling the origin of a signal, a variety of stellar activity
indicators, such as photometric variability, bisector inverse slope (BIS, the
measurement of asymmetry of the CCF), Ca ii infrared triplet (IRT), H$\alpha$,
CRX, the dLW, or the TiO bands, has been shown to be successful in identifying
and indicating a nonplanetary signal. In the literature, periodic RV signals
are often considered planetary in nature if the same signal is not found in
activity indicators. Therefore, it is important to examine as many activity
indicators as possible.

Correlations between the mentioned stellar activity indicators (when
available) and the RVs along with the GLS periodograms for these indicators
can be found in Appendix C for both the CARMENES VIS and NIR channel as well
as for the HARPS instrument. The strong correlations found with the CRX was
already discussed in Sect. 4.1. The BIS-RV anticorrelation for the VIS channel
($r$ = –0.79) is inline with the same anticorrelation found by C20 in the
HARPS-N data ($r$ = –0.74). Additionally, we look at the dLW However, in our
case, there seems to be no correlation. Along with the dLW, other expected
stellar activity indicators such as the emission of H$\alpha$ (Kürster et al.,
2003; Hatzes et al., 2015; Hatzes, 2016; Jeffers et al., 2018; Barnes et al.,
2014) and the Ca ii IRT (Gomes da Silva et al., 2011; Martin et al., 2017;
Robertson et al., 2015, 2016) seem to not show peaks at 2.23 d in periodograms
and exhibit no strong or moderate correlations. These findings are in
agreement with Lafarga et al. (2021) for a high-mass and H$\alpha$-active star
as AD Leo. In addition, we also consider the pseudo-equivalent widths (pEW’)
of various potential stellar activity lines as described in Schöfer et al.
(2019), though find no correlations or significant peaks other than suggestive
bumps at 1.8 d (i.e., the daily alias of 2.23 d).

##  6 Modeling the time dependence of the RV signal

Discontinuous sampling over the course of a long time baseline impedes
obtaining key information as to what occurs in large data gaps — as is the
case between the majority of the HIRES, HARPS data and CARMENES, GIARPS, HPF
data for AD Leo being 12 years apart. Remarkably, the 2.23 d signal persisted
over all this time, however, it is evident that amplitude changes and
phaseshifts occurred. Therefore, given such a rich RV data set, we test out
the current approaches for modeling stellar activity on a real-life example
and present our results below.

###  6.1 Model setup

We model the RV data using two components: a stable component (in amplitude,
period, and phase) and a variable component. The former is described by a
deterministic model, typically a circular or eccentric Keplerian orbit. We
would like to emphasize that when we use a Keplerian model, we consider it
more as a “stable model” which may include both a planetary companion and a
stable aspect of stellar activity, or only one of the two. Therefore,
parameters that are typically called $P_{\textnormal{planet}}$ and
$K_{\textnormal{planet}}$ to represent the period and semi-amplitude of the
Keplerian signal are rather referred as $P_{\textnormal{stable}}$ and
$K_{\textnormal{stable}}$ to indicate that we are considering a “stable mode”
that appears persistent. From here on, we use the terms Keplerian model and
stable model interchangeably, though we are not claiming that this component
is solely due to a planetary companion. As for the variable component,
typically comprising the quasi-periodic behavior of stellar activity, we do
not currently possess a deterministic model, therefore, we use a nonparametric
GP model to describe these modulations. We employ two different kernels.

The first is an exponential-squared-sine-squared kernel, that is, quasi-
periodic (QP-GP) kernel, provided by george (Ambikasaran et al., 2015),

$k_{i,j}(\tau)=\sigma^{2}_{\textnormal{GP}}\exp\left[-\alpha_{\textnormal{GP}}\tau^{2}-\Gamma_{\textnormal{GP}}\sin^{2}\left(\frac{\pi\tau}{P_{\textnormal{GP,\
rot}}}\right)\right]$ (1)

where $\tau=|t_{i}-t_{j}|$ is temporal distance between two points,
$\sigma_{\textnormal{GP}}$ is the amplitude of the GP modulation,
$\alpha_{\textnormal{GP}}$ is the inverse length-scale181818The parameter
$\alpha_{\textnormal{GP}}$ is defined as $1/l^{2}$, where $l$ is the timescale
of variations. The original relation in Espinoza et al. (2019) incorrectly
defined $\alpha=1/2l^{2}$. This has since been corrected. of the GP
exponential component, $P_{\textnormal{GP,\,rot}}$ corresponds to the
recurrence timescale, and $\Gamma_{\textnormal{GP}}$ is the smoothing
parameter. The former term is an exponential that can model the decorrelation
due to the changes in phase and amplitude as active regions grow and decay
over time, whereas the latter term accounts for the reoccuring periodicity.

The second kernel is a sum of two stochastically driven, damped harmonic
oscillator (SHO) terms, or a double SHO GP (dSHO-GP), where the power spectrum
of one SHO term is given by Anderson et al. (1990),

$\displaystyle\textnormal{SHO}_{1}(\omega_{\textnormal{GP}})=\sqrt{\frac{2}{\pi}}\frac{S_{0}\omega_{1}^{4}}{(\omega^{2}_{\textnormal{GP}}-\omega^{2}_{1})^{2}+\omega^{2}_{1}\omega_{\textnormal{GP}}^{2}/Q^{2}_{1}}$
(2a) and
$\displaystyle\textnormal{SHO}_{2}(\omega_{\textnormal{GP}})=\sqrt{\frac{2}{\pi}}\frac{S_{0}\omega_{2}^{4}}{(\omega^{2}_{\textnormal{GP}}-\omega^{2}_{2})^{2}+\omega^{2}_{2}\omega_{\textnormal{GP}}^{2}/Q^{2}_{2}},$
(2b) for which we applied a reparametrization using the hyperparameters
$\displaystyle Q_{1}$ $\displaystyle=0.5+Q_{0}+\delta Q$ (2c)
$\displaystyle\omega_{1}$ $\displaystyle=\frac{4\pi
Q_{1}}{P_{\textnormal{GP,\,rot}}\sqrt{4Q_{1}^{2}-1}}$ (2d) $\displaystyle
S_{1}$
$\displaystyle=\frac{\sigma_{\textnormal{GP}}^{2}}{(1+f)\omega_{1}Q_{1}}$ (2e)
$\displaystyle Q_{2}$ $\displaystyle=0.5+Q_{0}$ (2f) $\displaystyle\omega_{2}$
$\displaystyle=2\omega_{1}=\frac{8\pi
Q_{1}}{P_{\textnormal{GP,\,rot}}\sqrt{4Q_{1}^{2}-1}}$ (2g) $\displaystyle
S_{2}$
$\displaystyle=\frac{f\sigma_{\textnormal{GP}}^{2}}{(1+f)\omega_{2}Q_{2}},$
(2h)

and where $\sigma_{\textnormal{GP}}$ is again the amplitude of the GP kernel,
$P_{\textnormal{GP,\,rot}}$ is the primary period of the variability, $Q_{0}$
is the quality factor for the secondary oscillation, $\delta Q$ is the
difference between the quality factors of the first and second oscillations,
and $f$ represents the fractional amplitude of the secondary oscillation with
respect to the primary one.

We investigate multiple models to see which one is preferred for each
individual data set as well as for the combined data set comprising all
available RVs. The models being tested are:

  1. 1.

Keplerian-only models, i.e, both circular and eccentric

  2. 2.

GP-only model, either with a QP-GP or a dSHO-GP, as a proxy to describe the
quasi-periodic nature of the stellar activity

  3. 3.

Mixed (Keplerian + GP) models, to describe stable components with variable
ones

We use the model-fitting python package, juliet (Espinoza et al., 2019) in
order to compare these models using a Bayesian framework. For our purposes,
the RVs are modeled by radvel191919https://radvel.readthedocs.io/en/latest/
(Fulton et al., 2018), and the GP models with the help of
george202020https://george.readthedocs.io/en/latest/ (Ambikasaran et al.,
2015) and celerite212121https://celerite.readthedocs.io/en/stable/ (Foreman-
Mackey et al., 2017), where we use the
dynesty222222https://github.com/joshspeagle/dynesty package (Speagle &
Barbary, 2018; Speagle, 2020) to execute the Nested Sampling algorithm in
order to efficiently compute the Bayesian model log evidence,
$\ln\mathcal{Z}$. The main motivation for calculating the Bayesian log
evidence ($\ln\mathcal{Z}$) is to perform model comparisons. Outlined by
Trotta (2008), we follow the rule of thumb that a $\Delta\ln\mathcal{Z}$
greater than 5 between two models indicates strong evidence in favor for the
model with the larger Bayesian log evidence (odds are $\sim$150 to 1), whereas
a $\Delta\ln\mathcal{Z}$ more than 2.5 indicates moderate evidence for the
winning model, and anything less we consider to be inconclusive.

###  6.2 Prior setup

The RV periodograms (Fig. 3) indicate the interesting signal to be around 2.23
d. Thus, for all models with a Keplerian, the prior of the period of the
stable component, $P_{\textnormal{stable}}$, was kept uniform, but relatively
narrow, $\mathcal{U}$(2.0 d, 2.5 d), in order to avoid picking up aliases
(i.e., 1.8 d due to the daily sampling), and similarly for the time of transit
center232323The time-of-transit center is used within the juliet framework in
regards to the phase of the orbit., $t_{0}$, $\mathcal{U}$(2458200 d, 2458202
d), in order to cover solely one cycle and to avoid picking up other potential
modes. The semi-amplitude prior was simply uniform as well, $\mathcal{U}$(0 m
s-1, 50 m s-1).

As for the GP models, the rotational period, $P_{\textnormal{GP,\ rot}}$
follows the same prior as $P_{\textnormal{stable}}$ for consistency. Then, we
consider wide log-uniform priors for the other QP-GP hyperparameters:
$\Gamma_{\textnormal{GP}}$ (between $10^{-2}$ and $10^{2}$) and
$\alpha_{\textnormal{GP}}$ (between $10^{-10}$ d-2 and $10^{-1}$ d-2,
corresponding to timescales of $\sim$3 d – $\sim$70 000 d). For the dSHO-GP
hyperparameters: $Q_{0}$ was log-uniform (from $10^{2}$ to $10^{5}$), as well
as $\delta Q$ (from $10^{-1}$ to $10^{5}$), and $f$ was kept uniform (between
$0$ and $1$). For both GP kernels, the sigma of the GP was kept log-uniform
(between 0.1 and $70$ m s-1) where each instrument had its own, noted as
$\sigma_{\textnormal{GP,\ inst}}$, because each instrument has its own
characteristics, such as noise level, zero point offset, wavelength range, or
intrinsic stellar jitter. The other GP hyperparameters are shared. Additional
instrumental jitter terms (log-uniform from $10^{-2}$ to $30$ m s-1) and
offsets for each individual instrument were considered as well.

###  6.3 Results

We applied this recipe to the three following data sets: CARMENES VIS only,
CARMENES NIR only, and the entire RV data set. The resulting posteriors on the
$P_{\textnormal{stable}}$ ($P_{\textnormal{GP,\ rot}}$ if using a GP), semi-
amplitude $K_{\textnormal{stable}}$ ($\sigma_{\textnormal{GP,\ inst}}$ if GP),
added instrumental jitter terms ($\sigma_{\textnormal{inst}}$) and
eccentricity along with the differences in Bayesian log evidence can be found
in Table 3.

####  6.3.1 CARMENES only

Focusing on just the CARMENES season one data, an eccentric ($e\sim 0.19$)
Keplerian model is preferred over a circular model for the VIS channel
($\Delta\ln\mathcal{Z}\sim 5$). This is in agreement with the optical HARPS-N
data where C20 also found a similar eccentricity when applying a Keplerian-
only model. As for the NIR channel, a circular model performs best. Moreover,
the NIR data does not actually have a clear model preference, simply
attributed to the lower precision. The amplitude decreased from
$28.43^{+0.94}_{-0.98}$ m s-1 in the VIS1 data to $18.0^{+2.6}_{-2.7}$ m s-1
in the NIR1 data. When introducing combined stable plus GP models, the stable
component dominates and the GP component is not needed, in other words,
$\sigma_{\textnormal{GP}}$ becomes consistent with zero. This finding is
anticipated given that the time span of the data is roughly $\sim$10 cycles of
the periodicity, which would be too short for any noticeable changes of the
stellar spot pattern assuming a long spot lifetime.

For the season two data, we slightly readjusted the priors as introduced in
Sect. 6.2. The time of transit center was moved to $\mathcal{U}$(2458893 d,
2458895 d) to comply with the pertinent time stamps. Likewise, the period was
narrowly constrained, $\mathcal{U}$(2.22 d, 2.24 d), to ensure the correct
periodicity is recognized given the short time baseline of less than two
periodic cycles. Likewise for this reason, just the circular Keplerian-only
models were performed, and as a result, the posterior values are not included
in Table 3. Similarly to the season one data, there was also an amplitude
decrease between the instruments, namely from $17.02^{+0.63}_{-0.62}$ m s-1 to
$9.2^{+3.0}_{-2.6}$ m s-1 between VIS2 and NIR2, respectively, both of which
are lower with respect to their season one counterparts. Figure 4 shows how
the time series and phase-folded plots using the best model fits for the four
CARMENES data sets appear reasonable, showing a uniformly distributed scatter
in the model residuals. We additionally tested out combining the two seasons
with each respective instrument (i.e., VIS1+VIS2, NIR1+NIR2) and applied all
the models (including those with GPs), finding that the circular Keplerian was
favored for both.

####  6.3.2 Whole data set

Next we considered the whole AD Leo RV data set. We expect the Keplerian-only
models, where the amplitude for all given instruments is shared, to perform
poorly given that the amplitude is clearly decreasing as a function of
wavelength (shown in Sect. 4). The idea of combining all available data sets
covering a wide wavelength range is that the data set with the smallest
amplitude, specifically the NIR data, will act as an upper limit for any
stable signal, whereas the GP component will adapt the difference. Within the
framework of juliet, the data is first attempted to fit the deterministic
model as best as possible, where the GP is then applied to the residuals.
Likewise, we expect that the added jitter terms and GP amplitude parameters
for the redder instruments will be consistent with zero in the mixed model
because the stable model will most likely saturate for the redder instruments.
The role of the GP is to account for the excess amplitude observed with the
bluer instruments.

The preferred model among all is the mixed circular Keplerian + dSHO-GP model.
When compared to a Keplerian-only model, it exceeds in Bayesian log evidence
tremendously ($\Delta\ln\mathcal{Z}\sim 180$), but it does not prevail against
a dSHO-GP-only model that enormously ($\Delta\ln\mathcal{Z}\sim 3.5$). As
anticipated, the Keplerian-only models fail to explain the data in the sense
that finding a common amplitude is not feasible such that the jitter values
become too high, and likewise, the phase shifts are quite strong that a simple
sinusoid can not describe this behavior. Surprisingly, even though the dSHO-
GP-only model is flexible and could model the data well, the evidence suggests
that a model containing an additional stable periodic component is favored.
Likewise, the same concept applies for the models including the QP-GP kernel,
though including the additional stable component produced a
$\Delta\ln\mathcal{Z}\sim 10$ rather than $\sim 3.5$ for the dSHO-GP models,
that is, very strong evidence for one model versus bordering just moderate
evidence. This delicate boundary could change the interpretation. Tests on
simulated RV data based on StarSim (Herrero et al., 2016) indicate that a
Keplerian signal is in rare cases more efficient than the QP-GP in modeling a
coherent activity signal (Stock et al., 2022).

We find that the stable signal amplitude $K_{\textnormal{stable}}$ is $16.6\pm
2.2$ m s-1, where both the $P_{\textnormal{stable}}$ and $P_{\textnormal{GP}}$
is rounded to 2.23 d. All posteriors for the different models can be found in
Table 3 and the phase-folded plots for the best model, namely the mixed model,
are shown in Fig. 7.

To summarize our findings, we have the following takeaway points:

  * •

The CARMENES-VIS1 data alone prefer an eccentric Keplerian ($e\sim 0.19$)
model. This is in agreement with the HARPS-N data from C20. Whereas, the
CARMENES-NIR1 data alone prefer a circular Keplerian, that is, a sinusoid.

  * •

The amplitude between the two CARMENES seasons decreased in both instruments.
Namely, from $28.43^{+0.94}_{-0.98}$ m s-1 to $17.02^{+0.63}_{-0.62}$ m s-1 in
the VIS channel, and from $18.0^{+2.6}_{-2.7}$ m s-1 to $9.2^{+3.0}_{-2.6}$ m
s-1 in the NIR channel.

  * •

Combining both seasons of the CARMENES data (i.e., VIS1+VIS2 and NIR1+NIR2), a
circular Keplerian is preferred.

  * •

Combining all data sets, the mixed circular Keplerian + GP model provides the
best fit, for either GP used (QP-GP or dSHO-GP). This signifies that there is
a stable and variable component in the data.

  * •

The $K_{\textnormal{stable}}$ of the preferred model (mixed circular Keplerian
+ dSHO-GP model) is $16.6\pm 2.2$ m s-1. The amplitude of the GP component
($\sigma_{\textnormal{GP,\ inst}}$) is consistent with 0 for the near-IR
instruments, except for HPF2 ($\sigma_{\textnormal{GP,\ HPF2}}=7.8\pm 2.6$ m
s-1). For the optical instruments, the earlier data, namely with HARPS, HIRES,
have similar sigmas ($\sigma_{\textnormal{GP,\ inst}}\sim 20$ m s-1) as well
as the overlapping data, with VIS1 and HT1 ($\sigma_{\textnormal{GP,\
inst}}\sim 13$ m s-1). For VIS2, it was consistent with zero and for HT2, it
was higher with $\sigma_{\textnormal{GP,\ inst}}\sim 22$ m s-1.

  * •

The difference in evidence between a mixed model and a GP-only model depended
on the GP kernel choice (i.e., $\Delta\ln\mathcal{Z}\sim 3.5$ and
$\Delta\ln\mathcal{Z}\sim 10$ when considering a dSHO-GP and QP-GP kernel,
repspectively.)

Figure 7: Phase-folded plots for the optical (left) and near-infrared (right)
instruments using the circular Keplerian + dSHO-GP model after subtracting the
GP component out. The first subset of any given instrument is represented with
a circle, whereas the second subset by a triangle, when applicable. The y-axis
ranges are consistent between the two plots, except for the residual plots to
better visualize the scatter around the fit. Table 3: Model comparison of RV
fits done with juliet comparing a Keplerian model, a red-noise model, and a
mixture of both for the various instrument data sets: CARMENES VIS1, CARMENES
NIR1, and the combined data set. The selected best model is boldfaced (refer
to Sect. 6 for details on the priors used, model choice, and discussion of
results). The columns refer to which model was being used, whereas the rows
correspond to the model parameter. A larger, positive $\Delta\ln\mathcal{Z}$
value indicates a better model.

|  |  | QP-GP | dSHO-GP
---|---|---|---|---
| circ. | ecc. | GP | circ. + GP | ecc. + GP | GP | circ. + GP | ecc. + GP
CARMENES VIS1 |  |  |  |  |  |  |  |
$P_{\textnormal{stable}}$ | $2.2362^{+0.0054}_{-0.0056}$ | $2.2344^{+0.0029}_{-0.0030}$ | … | $2.238^{+0.150}_{-0.014}$ | $2.2346^{+0.0033}_{-0.0035}$ | … | $2.237^{+0.015}_{-0.008}$ | $2.2348^{+0.0036}_{-0.0036}$
$K_{\textnormal{stable}}$ | $25.9^{+1.4}_{-1.5}$ | $28.43^{+0.94}_{-0.98}$ | … | $19^{+11}_{-16}$ | $28.4^{+1.1}_{-1.2}$ | … | $22.6^{+7.2}_{-16.0}$ | $28.3^{+1.2}_{-1.5}$
$P_{\textnormal{GP}}$ | … | … | $2.2332^{+0.0035}_{-0.0038}$ | $2.226^{+0.011}_{-0.019}$ | $2.28^{+0.16}_{-0.19}$ | $2.2329^{+0.0048}_{-0.0049}$ | $2.224^{+0.013}_{-0.017}$ | $2.24^{+0.19}_{-0.14}$
ecc. | 0.0 (fixed) | $0.192^{+0.030}_{-0.031}$ | … | 0.0 (fixed) | $0.186^{+0.032}_{-0.033}$ | … | 0.0 (fixed) | $0.187^{+0.034}_{-0.038}$
$\sigma_{\textnormal{GP, CARMENES\ VIS1}}$ | … | … | $38^{+18}_{-14}$ | $23^{+24}_{-15}$ | $1.8^{+7.6}_{-1.7}$ | $18.3^{+8.2}_{-5.1}$ | $9.7^{+7.7}_{-4.2}$ | $0.79^{+3.00}_{-0.74}$
$\sigma_{\textnormal{CARMENES\ VIS1}}$ | $4.31^{+1.00}_{-0.83}$ | $1.2^{+1.1}_{-1.1}$ | $1.2^{+1.1}_{-1.1}$ | $0.45^{+1.40}_{-0.42}$ | $0.6^{+1.4}_{-0.6}$ | $0.26^{+1.10}_{-0.23}$ | $0.27^{+1.20}_{-0.24}$ | $0.61^{+1.40}_{-0.57}$
$\ln\mathcal{Z}$ | –94.06 | –87.71 | –90.06 | –90.39 | –87.08 | –87.86 | –88.18 | –87.05
$\Delta\ln\mathcal{Z}$ | –6.35 | 0.0 | –2.35 | –2.68 | 0.64 | –0.14 | –0.46 | 0.66
CARMENES NIR1 |  |  |  |  |  |  |  |
$P_{\textnormal{stable}}$ | $2.269^{+0.016}_{-0.017}$ | $2.286^{+0.016}_{-0.014}$ | … | $2.264^{+0.049}_{-0.110}$ | $2.287^{+0.023}_{-0.019}$ | … | $2.270^{+0.051}_{-0.110}$ | $2.288^{+0.025}_{-0.019}$
$K_{\textnormal{stable}}$ | $18.0^{+2.6}_{-2.7}$ | $21.8^{+3.9}_{-3.5}$ | … | $16.3^{+4.1}_{-5.2}$ | $20.7^{+4.3}_{-5.0}$ | … | $16.6^{+4.0}_{-4.7}$ | $20.9^{+4.2}_{-4.7}$
ecc. | 0.0 (fixed) | $0.488^{+0.078}_{-0.130}$ | … | 0.0 (fixed) | $0.470^{+0.092}_{-0.200}$ | … | 0.0 (fixed) | $0.471^{+0.091}_{-0.190}$
$P_{\textnormal{GP}}$ | … | … | $2.276^{+0.023}_{-0.042}$ | $2.19^{+0.19}_{-0.11}$ | $2.22^{+0.20}_{-0.15}$ | $2.286^{+0.031}_{-0.031}$ | $2.174^{+0.200}_{-0.097}$ | $2.22^{+0.21}_{-0.15}$
$\sigma_{\textnormal{GP, CARMENES\ NIR1}}$ | … |  …$20.9^{+16.0}_{-7.4}$ | $16^{+21}_{-11}$ | $4.4^{+17.0}_{-4.3}$ | $17.1^{+12.0}_{-6.8}$ | $7.5^{+7.8}_{-6.2}$ | $2.6^{+6.9}_{-2.5}$ |
$\sigma_{\textnormal{CARMENES\ NIR1}}$ | $0.97^{+5.30}_{-0.93}$ | $0.34^{+2.80}_{-0.31}$ | $0.28^{+2.30}_{-0.25}$ | $0.30^{+2.70}_{-0.26}$ | $0.25^{+2.10}_{-0.23}$ | $0.43^{+4.10}_{-0.39}$ | $0.31^{+2.90}_{-0.28}$ | $0.26^{+2.20}_{-0.23}$
$\ln\mathcal{Z}$ | –109.92 | –108.95 | –109.88 | –108.72 | –108.30 | –111.07 | –108.98 | –108.32
$\Delta\ln\mathcal{Z}$ | 0.0 | 0.97 | 0.04 | 1.19 | 1.62 | –1.15 | 0.94 | 1.60
HIRES + HARPS + CARMENES VIS + CARMENES NIR + HARPS-N + HPF |  |  |
$P_{\textnormal{stable}}$ | $2.226228^{+0.000028}_{-0.000026}$ | $2.227311^{+0.000018}_{-0.000019}$ | … | $2.226329^{+0.000078}_{-0.000066}$ | $2.226364^{+0.000900}_{-0.000088}$ | … | $2.226345^{+0.000098}_{-0.000084}$ | $2.22638^{+0.00095}_{-0.00010}$
$K_{\textnormal{stable}}$ | $17.13^{+0.59}_{-0.58}$ | $28.3^{+0.9}_{-1.0}$ | … | $15.7^{+1.3}_{-4.5}$ | $12.9^{+3.7}_{-2.7}$ | … | $16.57^{+0.85}_{-2.20}$ | $14.6^{+2.5}_{-4.0}$
$P_{\textnormal{GP}}$ | … | … | $2.2245^{+0.0021}_{-0.0021}$ | $2.2250^{+0.0022}_{-0.0021}$ | $2.2252^{+0.0023}_{-0.0021}$ | $2.22775^{+0.00054}_{-0.00067}$ | $2.2270^{+0.0010}_{-0.0011}$ | $2.22735^{+0.00095}_{-0.00120}$
ecc. | … | $0.183^{+0.022}_{-0.023}$ | … | … | $0.072^{+0.100}_{-0.051}$ | … | … | $0.072^{+0.099}_{-0.052}$
$\sigma_{\textnormal{GP, HIRES}}$ | … | … | $25.6^{+4.6}_{-3.5}$ | $20.8^{+3.6}_{-2.8}$ | $21.1^{+3.7}_{-2.9}$ | $23.9^{+5.0}_{-3.7}$ | $19.6^{+3.5}_{-2.7}$ | $19.9^{+3.8}_{-2.9}$
$\sigma_{\textnormal{GP, HARPS}}$ | … | … | $33.2^{+6.9}_{-5.4}$ | $25.5^{+5.6}_{-4.2}$ | $26.0^{+5.8}_{-4.3}$ | $25.9^{+5.9}_{-4.3}$ | $20.6^{+4.7}_{-3.4}$ | $21.2^{+4.7}_{-3.6}$
$\sigma_{\textnormal{GP, CARMENES\ VIS1}}$ | … | … | $26.6^{+9.8}_{-6.6}$ | $15.8^{+7.2}_{-4.8}$ | $16.9^{+7.4}_{-5.1}$ | $19.4^{+6.9}_{-4.8}$ | $13.1^{+5.4}_{-3.6}$ | $13.5^{+5.3}_{-3.8}$
$\sigma_{\textnormal{GP, CARMENES\ NIR1}}$ | … | … | $32^{+17}_{-13}$ | $0.78^{+13.00}_{-0.74}$ | $4.2^{+25.0}_{-4.1}$ | $18.5^{+23.0}_{-9.6}$ | $0.26^{+2.40}_{-0.23}$ | $1.4^{+9.4}_{-1.3}$
$\sigma_{\textnormal{GP, HARPS-N1}}$ | … | … | $26.0^{+8.7}_{-5.6}$ | $16.2^{+6.3}_{-4.1}$ | $17.3^{+6.3}_{-4.5}$ | $21.0^{+5.8}_{-4.3}$ | $14.6^{+4.4}_{-3.3}$ | $15.1^{+4.8}_{-3.4}$
$\sigma_{\textnormal{GP, HARPS-N2}}$ | … | … | $15.8^{+7.7}_{-5.0}$ | $10.3^{+7.3}_{-9.7}$ | $7.5^{+8.3}_{-7.3}$ | $30.2^{+10.0}_{-8.1}$ | $21.6^{+7.9}_{-7.9}$ | $20.7^{+8.5}_{-19.0}$
$\sigma_{\textnormal{GP, CARMENES\ VIS2}}$ | … | … | $17.8^{+8.5}_{-5.1}$ | $1.9^{+6.5}_{-1.9}$ | $5.7^{+5.8}_{-5.4}$ | $11.0^{+5.8}_{-3.3}$ | $0.38^{+2.90}_{-0.35}$ | $3.2^{+4.6}_{-3.0}$
$\sigma_{\textnormal{GP, CARMENES\ NIR2}}$ | … | … | $15.7^{+14.0}_{-8.3}$ | $7.1^{+11.0}_{-6.9}$ | $2.3^{+12.0}_{-2.3}$ | $9.3^{+12.0}_{-5.2}$ | $6.8^{+9.7}_{-5.7}$ | $2.4^{+9.2}_{-2.3}$
$\sigma_{\textnormal{GP, HPF1}}$ | … | … | $30^{+15}_{-12}$ | $17^{+14}_{-16}$ | $19^{+13}_{-18}$ | $23^{+16}_{-14}$ | $4.8^{+16.0}_{-4.7}$ | $13^{+12}_{-12}$
$\sigma_{\textnormal{GP, HPF2}}$ | … | … | $8.4^{+2.6}_{-2.0}$ | $9.2^{+2.7}_{-2.5}$ | $8.2^{+3.0}_{-2.7}$ | $6.1^{+2.7}_{-1.8}$ | $7.8^{+2.6}_{-1.8}$ | $7.2^{+2.5}_{-1.9}$
$\sigma_{\textnormal{HIRES}}$ | $17.0^{+2.0}_{-1.7}$ | $19.5^{+2.5}_{-2.1}$ | $1.4^{+1.2}_{-1.3}$ | $1.3^{+1.2}_{-1.2}$ | $1.3^{+1.1}_{-1.2}$ | $0.78^{+1.20}_{-0.73}$ | $0.69^{+1.20}_{-0.64}$ | $0.59^{+1.30}_{-0.55}$
$\sigma_{\textnormal{HARPS}}$ | $18.0^{+2.1}_{-1.7}$ | $12.2^{+1.5}_{-1.2}$ | $1.05^{+0.44}_{-0.46}$ | $0.93^{+0.48}_{-0.67}$ | $0.92^{+0.47}_{-0.66}$ | $0.12^{+0.48}_{-0.10}$ | $0.108^{+0.410}_{-0.086}$ | $0.094^{+0.370}_{-0.073}$
$\sigma_{\textnormal{CARMENES\ VIS1}}$ | $7.4^{+1.4}_{-1.1}$ | $2.33^{+0.84}_{-0.82}$ | $0.48^{+1.40}_{-0.44}$ | $0.59^{+1.50}_{-0.55}$ | $0.61^{+1.40}_{-0.57}$ | $0.24^{+1.10}_{-0.21}$ | $0.26^{+1.10}_{-0.23}$ | $0.35^{+1.10}_{-0.31}$
$\sigma_{\textnormal{CARMENES\ NIR1}}$ | $1.2^{+5.3}_{-1.2}$ | $11.6^{+3.2}_{-2.7}$ | $0.60^{+4.60}_{-0.56}$ | $1.4^{+5.8}_{-1.4}$ | $1.4^{+6.1}_{-1.4}$ | $0.89^{+6.00}_{-0.85}$ | $1.8^{+5.5}_{-1.7}$ | $2.0^{+5.9}_{-1.9}$
$\sigma_{\textnormal{HARPS-N1}}$ | $11.1^{+1.4}_{-1.2}$ | $4.05^{+0.81}_{-0.62}$ | $1.58^{+0.37}_{-0.32}$ | $1.62^{+0.37}_{-0.33}$ | $1.62^{+0.37}_{-0.32}$ | $0.40^{+0.62}_{-0.36}$ | $0.46^{+0.62}_{-0.42}$ | $0.51^{+0.58}_{-0.45}$
$\sigma_{\textnormal{HARPS-N2}}$ | $9.1^{+1.7}_{-1.3}$ | $16.6^{+3.1}_{-2.4}$ | $4.2^{+1.4}_{-1.1}$ | $4.9^{+1.9}_{-1.5}$ | $5.1^{+1.6}_{-1.6}$ | $0.18^{+1.20}_{-0.15}$ | $0.25^{+2.90}_{-0.22}$ | $0.42^{+5.00}_{-0.38}$
$\sigma_{\textnormal{CARMENES\ VIS2}}$ | $0.098^{+0.410}_{-0.077}$ | $8.9^{+1.9}_{-1.5}$ | $0.095^{+0.380}_{-0.075}$ | $0.098^{+0.380}_{-0.077}$ | $0.10^{+0.40}_{-0.08}$ | $0.097^{+0.380}_{-0.076}$ | $0.092^{+0.370}_{-0.071}$ | $0.110^{+0.360}_{-0.086}$
$\sigma_{\textnormal{CARMENES\ NIR2}}$ | $5.6^{+4.5}_{-5.3}$ | $6.1^{+4.0}_{-5.8}$ | $0.3^{+2.8}_{-0.3}$ | $0.28^{+2.50}_{-0.25}$ | $0.27^{+2.30}_{-0.24}$ | $0.3^{+3.0}_{-0.3}$ | $0.29^{+2.60}_{-0.26}$ | $0.36^{+2.80}_{-0.32}$
$\sigma_{\textnormal{HPF1}}$ | $10.2^{+5.1}_{-2.9}$ | $7.6^{+4.1}_{-2.2}$ | $0.89^{+12.00}_{-0.84}$ | $2.2^{+10.0}_{-2.1}$ | $1.4^{+10.0}_{-1.3}$ | $1.7^{+14.0}_{-1.6}$ | $7.9^{+6.0}_{-7.5}$ | $3.9^{+8.4}_{-3.7}$
$\sigma_{\textnormal{HPF2}}$ | $12.1^{+1.9}_{-1.5}$ | $20.7^{+3.0}_{-2.5}$ | $2.98^{+0.82}_{-0.65}$ | $2.50^{+0.71}_{-0.62}$ | $2.61^{+0.75}_{-0.62}$ | $2.90^{+0.72}_{-0.63}$ | $2.39^{+0.72}_{-0.74}$ | $2.47^{+0.73}_{-0.69}$
$\ln\mathcal{Z}$ | –1143.31 | –1143.38 | –1002.17 | –992.50 | –993.37 | –966.77 | -963.40 | –965.73
$\Delta\ln\mathcal{Z}$ | –179.91 | –179.99 | –38.77 | –29.10 | –29.98 | –3.37 | 0.0 | –2.33

242424

##  7 Discussion and future outlook

In this analysis, we presented new CARMENES optical and near-IR data that
cover a wide wavelength range well into the red part of the spectrum to test
the origin of the 2.23 d signal found in the RVs for AD Leo. The presence of
stellar activity is supported with CARMENES through the proof of wavelength-
and time-dependence. With the CARMENES data alone, the shape of the wavelength
dependence of the RV amplitude can be attributed to a star-spot configuration
following Reiners et al. (2010). Between the two CARMENES seasons, the
amplitude decreased in both instruments. Specifically, from
$28.43^{+0.94}_{-0.98}$ m s-1 to $17.02^{+0.63}_{-0.62}$ m s-1 in the VIS
channel, and from $18.0^{+2.6}_{-2.7}$ m s-1 to $9.2^{+3.0}_{-2.6}$ m s-1 in
the near-IR channel when modeling the signals as pure sinsuoidal variations.
C20 also found an amplitude decrease between the HARPS-N and GIANO-B data,
that is, from 33 m s-1 to less than 23 m s-1 between the instruments and from
33 m s-1 to 13 m s-1 between seasons. However, given the large errorbars of
the GIANO-B data ($\sim$20 m s-1), it was only possible to identify an
amplitude decrease, but not to constrain whether there even is a signal
present in these data at 2.23 d. Likewise, the high-precision near-IR data
from HPF show a much smaller RV amplitude (R20), inconsistent with the higher
RV amplitude in the optical, and there is also a discrepancy even between HPF
seasons, namely, the RMS scatter dropped from 23 m s-1 to 6.4 m s-1. This
decrease in amplitude is also seen in the overlapping, optical HARPS-N data
and second season of CARMENES data (see again Fig. 2). Nonetheless, both our
results and those in C20 and R20 agree that the signal’s amplitude does indeed
decrease rather than increase, indicating that AD Leo is entering a lower-
activity phase. We conclude that the effect of the spots on the RVs is
dominated by temperature differences rather than the Zeeman effect (see
Reiners et al., 2013), in accordance with the other evidence at hand such as
photometric variability (presented in T18).

###  7.1 Possibility of stable stellar-spot signal

It is common practice to model activity-induced RVs with GPs (e.g., Rajpaul et
al., 2015, and further citations), as they are nondeterministic models that
can sufficiently fit stochastic modulations. Provided that the evidence
against a planet outweighs that in favor of a planet, our understanding was to
expect that the GP-only model should have been adequate enough in accounting
for the stellar activity, with the assumption that the signal in AD Leo is
purely stellar activity induced. The modeling instead showed that neither a
stable-only (i.e., sinusoid) model nor red-noise-only (i.e., GP-only) model
can correctly describe the whole data set (Table 3). A mix of a stable and
variable component is rather the preferred model, where the stable model has
an amplitude of ${K}_{\textnormal{stable}}=16.6\pm 2.2$ m s-1, lower than the
value of 19 m s-1 proposed by T18. We do not claim that the stable component
is solely due to a planetary companion as it could perhaps be also due to the
persistent presence of a circumpolar stellar spot imposing a constant
behavior, or even a mixture of both. Thus, we set an 3$\sigma$ upper limit
(i.e., from ${K}_{\textnormal{stable}}=16.6\pm 3\times 2.2$ m s-1) of 27
$\textnormal{M}_{\oplus}$ (or 0.084 $\textnormal{M}_{\textnormal{Jup}}$) for a
putative planetary companion, in comparison to the proposed mass of $\sim$0.2
$\textnormal{M}_{\textnormal{Jup}}$ by T18. Focusing on the amplitudes of the
GP kernel of the mixed model, we found that those of the earlier optical RV
data, specifically HARPS and HIRES ($\sigma_{\textnormal{GP,\ inst}}\sim 20$ m
s-1), were higher than most of the more recent optical data
($\sigma_{\textnormal{GP,\ inst}}\lesssim 13$ m s-1). These values support the
assumption that the star was more active earlier and is entering a less active
phase (Sect. 2.2). The only exception was with HT2, with
$\sigma_{\textnormal{GP,\ HT2}}\sim 22$ m s-1, an even higher value. The GP
amplitudes of near-IR spectrographs were consistent with
$\sigma_{\textnormal{GP,\ inst}}\sim$ 0 m s-1, except for HPF2
($\sigma_{\textnormal{GP,\ HPF2}}\sim 8$ m s-1). The data from the two
instruments that stand out (i.e., HT2 and HPF2) were contemporaneously taken,
indicating that there was some phase shift between the stable and variable
component during this time period. A better understanding of how model
comparison can behave given a purely stellar-activity induced signal would be
necessary to determine if such a result is even expected (see also Sect. 7.2
for future suggestions). Additionally, our analysis of recomputing the RVs,
considering only those spectral lines unaffected by stellar activity, still
showed some strong significant periodicity at 2.23 d (Fig. 6), unlike for
other known active stars where the signals had disappeared (Lafarga priv.
comm.).

Nonetheless, this should raise the question of whether a signal with an
amplitude of $\sim$17 m s-1 that is fully stable over all these observations
($\sim$19 yr) in such an M dwarf is possible. In fact, it is not so surprising
that spot-induced RV fluctuations for M dwarfs are long-lived (e.g., Günther
et al., 2020; Quirrenbach et al., 2022). Though evidence of stable stellar
activity behavior had previously been shown in photometry (e.g., GJ 1243,
Davenport et al., 2020) and RVs (e.g., $\alpha$ Tau, Hatzes et al., 2015) over
time, our paper demonstrates the first case for modeling RVs over time and
wavelength. Future studies performing simulations with software packages such
as StarSim 2.0 (Herrero et al., 2016) or SOAP 2.0 (Dumusque et al., 2014)
using various star-spot configurations may also shed light on why a signal can
stay persistent over many years (Herrero et al., 2016; Rosich et al., 2020).
Utilizing StarSim 2.0 to compare RV-CRX correlations between simulated and
real data as performed by Baroch et al. (2020), with YZ CMi as a case study,
could be beneficial in determining the star-spot temperature difference, the
star-spot filling factor, and the location of the spot. A first test with a
simple assumption (i.e., one big polar spot) could reproduce well the RVs and
CRX values of AD Leo for the CARMENES VIS data (Baroch priv. comm.). But such
an approach can become degenerate when considering so many instruments and
various star-spot configurations, and a planetary signal would act as an
achromatic offset to the RVs and CRX.

###  7.2 Current limitations

AD Leo is a prime case study for M dwarfs to explore the possibility of a
planetary signal with an orbital period indistinguishably close to its stellar
host’s rotation period. Given its rich multi-wavelength spectroscopic
monitoring over a large time baseline, there are no other targets with such an
extensive data set coverage. Even then, we showed that there are apparent
holes in our current state of research in this field in which studies to
address them would require proposals, dedicated surveys and telescope time. In
essence, we are either limited by our modeling approaches, by our
astrophysical knowledge of star-spot configurations, or a combination of both.
In order to progress and disentangle this problem, we propose the following,
in no particular order.

####  7.2.1 Simultaneous multiband photometry

The shape of the curve in Fig. 5 is determined by stellar activity behavior,
whereas the presence of a planet would simply act as an offset across the
wavelength space. Even when doubling (or halving) the number of available
spectroscopic data points, disentangling the contribution of a planetary
signal within this curve still persists as a degenerate problem. Obtaining
continuous and simultaneous multiband photometry would help in painting a
better picture of the stellar-spot map distribution. In doing so, this can
then be translated by forward modeling to determine the stellar-activity
induced RVs (e.g., via Starsim 2.0 or SOAP 2.0; Herrero et al., 2016; Dumusque
et al., 2014). The residual between these computed RVs and those obtained
would indicate the contribution of the Keplerian component. However, the
caveat is, that spot configurations on the stellar surface can vary with time.
Thus, continuous photometry would only apply to simultaneously taken RVs.

####  7.2.2 A better-suited RV data set for spectral line analyses

In the absence of available photometric facilities, the approach of selecting
spectral lines not affected by activity seems promising. This line-by-line
approach has been successfully applied to other stars (e.g., Dumusque, 2018;
Wise et al., 2018; Cretignier et al., 2020). In the case for AD Leo, it is
difficult to decipher whether it was the target itself or the relatively new
method that resulted in an inconclusive outcome. Perhaps the RVs were not as
affected by the stellar spots (i.e., due to the pole-on inclination), the RV
amplitude was not as large as the others, or there were not enough data points
to strengthen the correlation plots by establishing a statistically stronger
correlation. Definitely, if overall activity decreases, then it would be more
difficult to pick up the activity within certain spectral lines. Hence,
observing a star when it is in its most active phase would constitute the
ideal data set. Additionally, our spectral line analysis should be applied to
a wider wavelength range (e.g., HARPS) as well to determine if the signal
disappears. Another possible explanation could be that all lines are
inherently affected by activity to some degree. Nonetheless, determining the
ideal targets for this technique as well as developing the method further will
be paramount in suppressing activity for future M-dwarf studies.

####  7.2.3 Magnetic star-planet interaction

The radio bursts thus far observed for the AD Leo system (outlined in Sect.
2.5) suggest an excess that is unaccounted for. A possible scenario to justify
the discrepancy could be due to sub-Alfvénic interaction of the stellar
magnetosphere with an orbiting body, in this case with a putative planet
around AD Leo, which has $P_{\rm orb}=P_{\rm rot}$. This magnetic interaction
between the star and a planet (Magnetic Star-Planet Interaction; MSPI) would
be responsible for producing: (1) an enhancement of the magnetic activity of
the star, and (2) the long-duration radio emission as that detailed in Sect.
2.5, both effects of which are modulated in phase with the orbital period of
the planet. In a star like AD Leo, the difficulties for detecting this
modulation are that this target is seen nearly pole-on ($i\sim$ 13 deg) and
that a putative planet orbit is synchronized with the stellar rotation. This
produces a weaker MSPI, as it depends on the relative velocity of the planet
with respect to the magnetosphere of the star (Lanza, 2009, 2012) and makes it
difficult to disentangle it from stellar intrinsic activity. However, the
presence of such a planet could explain the need for the stable component and
part of its semi-amplitude ($K_{\textnormal{stable}}$ ¡ 16.6 m s-1) seen in
the RVs in this work. MSPI would show as a hot spot in the chromosphere that
would mainly affect the chromospheric activity indicators such as
$\log{R^{\prime}_{\textnormal{HK}}}$ from the Ca ii H&K lines, not available
in the CARMENES data set but covered by HARPS.

Within this scenario, we followed the prescriptions in Appendix B of Pérez-
Torres et al. (2021) to estimate the flux density expected to arise from the
interaction between a putative planet and its host star. We assume for
simplicity an isothermal Parker wind with a temperature of (2–3)$\times
10^{6}$ K, and a density of 107–8 cm-3 at the base of the corona. Since the
hypothetical planet would be very close to its host star, we find that the
planet is in the sub-Alfvénic regime, which makes it possible that energy and
momentum can be carried upstream of the flow along Alfvén wings. We estimate
that the radio flux in the few GHz band, arising from star-planet interaction,
is in the range from less than about 0.1 mJy and up to a few mJy, or even
higher, depending on the stellar wind parameters at both the corona and at the
orbital position of the planet. We therefore encourage a campaign for
simultaneous monitoring of AD Leo in both optical (i.e., spectroscopy,
including Ca ii H&K lines, and photometry) and in radio wavelengths to test
the star-planet (sub-Alfvénic) interaction scenario. The detection of a
correlation in the radio signal with the orbital period would be the smoking
gun of sub-Alfvénic interaction, serving as an argument in support of the
presence of a planet in the system.

####  7.2.4 Application of GP models for red-noise correlated signals

Additional data may not be what is necessary as much as is a better
understanding of the astrophysical effects stellar spots impose on the star
and how we choose to model them through nonphysically motivated means. The
usage of GPs is currently the most popular approach in the RV field for
modeling stellar activity. That being said, the most commonly used kernels
(i.e., QP-GP, dSHO-GP) are not completely physically motivated, but
nonetheless serve as a very good approximation to our aim of describing the
quasi-periodic behavior. They can play a vital role, especially when trying to
search for low-mass planets with low amplitudes among noisy data (e.g., Stock
et al., 2020; Demangeon et al., 2021; Faria et al., 2022).

The problem is, however, that we do not know just how effective the GPs are in
serving their purpose and what limitations and consequences on (planetary)
parameters they particularly have. Perhaps, the choice of the GP for different
time baselines could be also of importance, as is the case for AD Leo (see
Sect. 6), considering there are, for the time being, too many unknowns that
can lead to misinterpretations. Newer implementations of GP kernels with a
potentially better physical interpretation of stellar processes could become
essential (e.g., Perger et al., 2021; Luger et al., 2021). Applying these
models to real data sets turns out to not be the solution, as it becomes
complicated to correctly interpret the results. For this reason, these models
should be compared against each other on controlled, simulated RV time series.
A more in-depth study could include simulating RV time series by injecting
solely stellar-activity-induced signals and a mixture of stellar activity and
planetary components for a general grasp (Stock et al., 2022). One step
further would be to specifically simulate the stellar characteristics of AD
Leo, together with the time stamps and various wavelengths. Even here, the
limitation of our knowledge of spot distribution maps and correctly choosing
the correct one persists.

In future potential cases where there is a presumed planetary and stellar
activity signal with two instruments covering a wide wavelength range, for
instance, with CARMENES, it is crucial to adequately model the wavelength
dependence of the stellar activity to ensure the most precise planetary
parameters. More so for M-dwarf stars considering that their stellar rotation
periods would often correlate to the orbital periods of planets residing in
the habitable zone (Newton et al., 2016). AD Leo in this regard is an
interesting case study to perform such fits when considering real data,
however, simulated data sets may offer a more controlled environment to test
the effectiveness of our current modeling techniques. Either way, AD Leo
certainly serves as a particularly intriguing system for studying the impact
of a stable spot periodicity on the search for planetary signals.

####  7.2.5 Using auxiliary data to correct for stellar activity

Another promising procedure is to include the stellar activity indicators as
auxiliary data to try to detrend the RVs, commonly done so as linear
components (e.g., Anglada-Escudé et al., 2016) or within the GP kernel (e.g.,
Suárez Mascareño et al., 2020; Cale et al., 2021; Barragán et al., 2022) in
regards to planet detection. Likewise, this technique is being carried out in
a sole stellar activity context (Jeffers et al., 2022; Cardona Guillén et al.,
2022), with particular attention to not only detrending with various
indicators, but also to characterizing the behavior of the correlations.
Specifically, this “closed-loop” relation is becoming more prevalent, as
already exhibited with known active M-dwarf stars such as YZ~CMi (Zechmeister
et al., 2018; Baroch et al., 2020), EV Lac (Jeffers et al., 2022), and GJ~674
(Bonfils et al., 2007), and now for AD Leo as displayed in the RV-CRX and RV-
BIS correlations (Fig. 9). Subtracting out the CRX dependency can improve the
rms of the RVs, though it is nearly impossible to obtain a straight line from
the correction, that is, completely removing all of the stellar activity, even
in cases with very high signal-to-noise ratios and very dense sampling (e.g.,
factor of three improvement for EV Lac; Jeffers et al., 2022). For our case of
AD Leo (Sect. 4.1), the CRX detrending improved the rms roughly by a factor of
two for VIS1 (from 18.5 m s-1to 9.2 m s-1).

In this line, a relatively recent technique is that of using
spectropolarimetric measurements to correct the RVs of the variability
introduced by the activity. The data are used to derived precise RVs and to
obtain maps of the magnetic field and the distribution of surface
inhomogeneities. These maps are then used to obtain a model RV curve which are
the subtracted from the observed RVs. The removal of activity variations with
this technique decreased the amplitude of the RV curve in V830 Tau by a factor
10, which allowed the detection of a hot Jupiter orbiting this very young T
Tauri star (Donati et al., 2016).

While these approaches are encouraging, they still poses limitations when
considering a putative planetary signal whose orbital period is equal to the
rotational period because the same issue, as mentioned already above, of
degeneracy still holds true. Again, a systematic simulated analysis of forward
modeling RVs and stellar activity indicators could be key in understanding
what correlations to expect and how to properly detrend them.

##  8 Conclusions

In this paper we presented new CARMENES optical and near-infrared
spectroscopic data for a known nearby, active M dwarf, AD Leo. The stellar
rotation period of 2.23 d is clearly present in the RVs, and we address the
question of whether there could be a planet orbiting with the same
periodicity, $P_{\rm orb}=P_{\rm rot}$. Taking advantage of the wide
wavelength range of the CARMENES instrument, we demonstrated the shape of the
wavelength dependency of the RV semi-amplitudes to be in agreement with what
is expected for a star-spot temperature difference configuration. The strong
anticorrelation found between CRX-RV and BIS-RV for the CARMENES VIS data were
additional signs of stellar activity. In addition, we recomputed RVs using
spectral lines unaffected by stellar activity, which has not been done before
for AD Leo, and found that there was still some significant residual power at
the rotational period.

When incorporating all available RVs (HARPS, HIRES, CARMENES, and the more
recent HARPS-N, GIANO-B, and HPF data), it became evident that the signal has
undergone various amplitude fluctuations as well as phase shifts, and this is
behavior that cannot be attributed to the presence of a planetary companion.
However, a closer look into the model comparison showed that a mixed model of
a stable plus a quasi-periodic red-noise model best explained the data, where
the stable component had a semi-amplitude of $K_{\textnormal{stable}}$ =
$16.6\pm 2.2$ m s-1, setting a 3$\sigma$ upper limit on the mass of a putative
planet, $27\,\textnormal{M}_{\oplus}$
($=0.084\,\textnormal{M}_{\textnormal{Jup}}$). Based on all this evidence, our
current machinery, and the given data at hand, we conclude we cannot
unequivocally prove, or even disprove, the 2.23 d periodic signal found in AD
Leo to be solely due to stellar activity. We therefore suggest to obtain
simultaneous photometry and spectroscopy in various wavelengths accompanied by
radio observations, and complemented with a clearer insight of stellar
activity behavior on measurements via simulated time series to break the
degeneracy.

###### Acknowledgements.

CARMENES is an instrument for the Centro Astronómico Hispano-Alemán de Calar
Alto (CAHA, Almería, Spain). CARMENES was funded by the Max-Planck-
Gesellschaft (MPG), the Consejo Superior de Investigaciones Científicas
(CSIC), the Ministerio de Economía y Competitividad (MINECO) and the European
Regional Development Fund (ERDF) through projects FICTS-2011-02,
ICTS-2017-07-CAHA-4, and CAHA16-CE-3978, and the members of the CARMENES
Consortium (Max-Planck-Institut für Astronomie, Instituto de Astrofísica de
Andalucía, Landessternwarte Königstuhl, Institut de Ciències de l’Espai,
Institut für Astrophysik Göttingen, Universidad Complutense de Madrid,
Thüringer Landessternwarte Tautenburg, Instituto de Astrofísica de Canarias,
Hamburger Sternwarte, Centro de Astrobiología and Centro Astronómico Hispano-
Alemán), with additional contributions by the MINECO, the Deutsche
Forschungsgemeinschaft (DFG) through the Major Research Instrumentation
Programme and Research Unit FOR2544 “Blue Planets around Red Stars”, the Klaus
Tschira Stiftung, the states of Baden-Württemberg and Niedersachsen, and by
the Junta de Andalucía. We acknowledge financial support from the Agencia
Estatal de Investigación of the Ministerio de Ciencia, Innovación y
Universidades and the ERDF through projects PID2019-109522GB-C51/2/3/4
PGC2018-098153-B-C33 AYA2017-84089, AYA2016-79425-C3-1/2/3-P,
ESP2016-80435-C2-1-R, and the Centre of Excellence “Severo Ochoa” and “María
de Maeztu” awards to the Instituto de Astrofísica de Canarias (SEV-2015-0548),
Instituto de Astrofísica de Andalucía (SEV-2017-0709), and Centro de
Astrobiología (MDM-2017-0737), the Generalitat de Catalunya/CERCA programme,
and the DFG under Research Unit FOR2544. T.T. acknowledges support by the DFG
Research Unit FOR 2544 ”Blue Planets around Red Stars” project No. KU
3625/2-1. T.T. further acknowledges support by the BNSF program “VIHREN-2021”
project No. KP-06-DV/5. This research has made use of the ESO Science Archive
Facility and the CARMENES data archive at CAB (CSIC-INTA). Based on
observations collected at the European Southern Observatory under ESO programs
072.C-0488, 191.C-0505, and 192.C-0224.

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##  Appendix A RV data and short tables

Table 4: Published rotational periods values of AD Leoaa$a$The
$P_{\textnormal{rot}}$ determined by us from the GP component of the best-fit
model on the spectroscopic data (Sect 6.3, Table 3) is
$2.2270^{+0.0010}_{-0.0011}$ d..

Data | $P_{\textnormal{rot}}$ | Reference
---|---|---
| (d) |
Photometry
TESS-s48 | $2.2304\pm 0.0014$ |  This workbb$b$Priors for the NIR arm are slightly different because of the lower data quality. The period priors are wider, $\mathcal{U}$(2.2 d, 2.5 d).
STELLA | $2.237\pm 0.035$ | C20
ASAS-N | $2.22791^{+0.00066}_{-0.00055}$ |  T18cc$c$After subtraction of a longer periodicity in the data, see reference for details.
MOST-1 | $2.289\pm 0.019$ |  T18dd$d$The MOST baseline of 8.9 d was separated into two independent data sets, see reference for details.
MOST-2 | $2.145\pm 0.011$ |  T18dd$d$The MOST baseline of 8.9 d was separated into two independent data sets, see reference for details.
MOST | $2.23^{+0.36}_{-0.27}$ | Hun2012
FCAPT | 2.23 | Eng2009
Spectroscopy