11institutetext:  Institut de Recherche en Astrophysique et Planétologie,
Université de Toulouse, CNRS, IRAP/UMR 5277, 14 avenue Edouard Belin, F-31400,
Toulouse, France
11email<EMAIL_ADDRESS>22institutetext: Science Division,
Directorate of Science, European Space Research and Technology Centre
(ESA/ESTEC), Keplerlaan 1, 2201 AZ, Noordwijk, The Netherlands
33institutetext: Laboratoire Univers et Particules de Montpellier, Université
de Montpellier, CNRS, F-34095, Montpellier, France 44institutetext: Tartu
Observatory, University of Tartu, Observatooriumi 1, Tõravere, 61602 Tartumaa,
Estonia 55institutetext: Department of Physics and Astronomy, Uppsala
University, Box 516, SE-75120 Uppsala, Sweden 66institutetext: Univ. Grenoble
Alpes, CNRS, IPAG, 38000 Grenoble, France 77institutetext: Laboratório
Nacional de Astrofísica, Rua Estados Unidos 154, 37504-364, Itajubá - MG,
Brazil 88institutetext: Institut d’Astrophysique de Paris, CNRS, UMR 7095,
Sorbonne Université, 98 bis bd Arago, 75014 Paris, France 99institutetext:
Universidade Federal de Minas Gerais, Belo Horizonte, MG, 31270-901, Brazil
1010institutetext: Université de Montréal, Département de Physique, IREX,
Montréal, QC H3C 3J7, Canada 1111institutetext: Aix Marseille Univ, CNRS,
CNES, LAM, Marseille, France 1212institutetext: Observatoire de Genève,
Université de Genève, Chemin Pegasi, 51, 1290 Sauverny, Switzerland
1313institutetext: Observatoire de Haute Provence, St Michel l’Observatoire,
France 1414institutetext: Department of Physics & Space Science, Royal
Military College of Canada, PO Box 17000 Station Forces, Kingston, ON, Canada
K7K 0C6 1515institutetext: Department of Physics & Astronomy, McMaster
University, 1280 Main St West, Hamilton, ON, L8S 4L8, Canada

# Monitoring the large-scale magnetic field of AD Leo with SPIRou, ESPaDOnS,
and Narval

Towards a magnetic polarity reversal?

S. Bellotti 1122 0000-0002-2558-6920 J. Morin 33 0000-0002-4996-6901 L. T.
Lehmann 11 0000-0001-5674-2116 C. P. Folsom 44 0000-0002-9023-7890 G. A. J.
Hussain 22 0000-0003-3547-3783 P. Petit 11 0000-0001-7624-9222 J-F. Donati 11
0000-0001-5541-2887 A. Lavail 1155 0000-0001-8477-5265 A. Carmona 66
0000-0003-2471-1299 E. Martioli 7788 0000-0002-5084-168X B. Romano Zaire 99
0000-0002-9328-9530 E. Alecian 66 0000-0001-5260-7179 C. Moutou 11
0000-0002-2842-3924 P. Fouqué 11 0000-0002-1436-7351 S. Alencar 99 E. Artigau
1010 0000-0003-3506-5667 I. Boisse 1111 0000-0002-1024-9841 F. Bouchy 1212
0000-0002-7613-393X C. Cadieux 1010 0000-0001-9291-5555 R. Cloutier 1515
0000-0001-5383-9393 N. J. Cook 1010 0000-0003-4166-4121 X. Delfosse 66
0000-0001-5099-7978 R. Doyon 1010 0000-0001-5485-4675 G. Hébrard 881313
0000-0001-5450-7067 O. Kochukhov 55 0000-0003-3061-4591 G. A. Wade 1414

(Received ; accepted )

###### Abstract

Context. One clear manifestation of dynamo action on the Sun is the 22-yr
magnetic cycle, exhibiting a polarity reversal and a periodic conversion
between poloidal and toroidal fields. For M dwarfs, several authors claim
evidence of activity cycles from photometry and analyses of spectroscopic
indices, but no clear polarity reversal has been identified from
spectropolarimetric observations. These stars are excellent laboratories to
investigate dynamo-powered magnetic fields under different stellar interior
conditions, that is partly or fully convective.

Aims. Our aim is to monitor the evolution of the large-scale field of AD Leo,
which has shown hints of a secular evolution from past dedicated
spectropolarimetric campaigns. This is of central interest to inform distinct
dynamo theories, contextualise the evolution of the solar magnetic field, and
explain the variety of magnetic field geometries observed in the past.

Methods. We analysed near-infrared spectropolarimetric observations of the
active M dwarf AD Leo taken with SPIRou between 2019 and 2020 and archival
optical data collected with ESPaDOnS and Narval between 2006 and 2019. We
searched for long-term variability in the longitudinal field, the width of
unpolarised Stokes profiles, the unsigned magnetic flux derived from Zeeman
broadening, and the geometry of the large-scale magnetic field using both
Zeeman-Doppler imaging and principal component analysis.

Results. We found evidence of a long-term evolution of the magnetic field,
featuring a decrease in axisymmetry (from 99% to 60%). This is accompanied by
a weakening of the longitudinal field ($-$300 to $-$50 G) and a correlated
increase in the unsigned magnetic flux (2.8 to 3.6 kG). Likewise, the width of
the mean profile computed with selected near-infrared lines manifests a long-
term evolution corresponding to field strength changes over the full time
series, but does not exhibit modulation with the stellar rotation of AD Leo in
individual epochs.

Conclusions. The large-scale magnetic field of AD Leo manifested first hints
of a polarity reversal in late 2020 in the form of a substantially increased
dipole obliquity, while the topology remained predominantly poloidal and
dipolar for 14 yr. This suggests that low-mass M dwarfs with a dipole-
dominated magnetic field can undergo magnetic cycles.

###### Key Words.:

Stars: individual: AD Leo, Stars: magnetic field – Stars: activity –
Techniques: polarimetric

##  1 Introduction

Studying stellar surface magnetic fields yields relevant insights into the
internal structure of stars, as well as their essential role in stellar
formation, evolution, and activity (Donati & Landstreet, 2009). For cool
stars, monitoring secular changes of the field’s configuration provides useful
feedback on the dynamo processes operating in the stellar interior and
constraints on stellar wind models. The latter is fundamental to understanding
atmospheric hydrodynamic escape of embedded planets since magnetic cycles
modulate the star’s activity level and thus its radiation output (Vidotto &
Cleary, 2020; Hazra et al., 2020).

The Sun is an important benchmark in this context: its long-term monitoring
revealed a periodic variation in sunspot number, size, and latitude (Schwabe,
1844; Maunder, 1904; Hathaway, 2010), and a polarity reversal of the large-
scale magnetic field over a timescale of 11 yr (Hale et al., 1919). The
proposed mechanism to reproduce these phenomena theoretically is the
$\alpha\Omega$ dynamo (Parker, 1955; Charbonneau, 2010), namely the
combination of differential rotation and cyclonic turbulence at the interface
between the radiative and convective zones, known as tachocline. A different
model is the Babcock-Leighton mechanism, which describes the conversion from a
toroidal to poloidal field via a poleward migration of bipolar magnetic
regions (Babcock, 1961; Leighton, 1969). However, there is still no model that
can account for all the solar magnetic processes (Petrovay, 2020).

For other stars, magnetic field measurements can be performed with two
complementary approaches (Morin, 2012; Reiners, 2012). One is to model the
Zeeman splitting in individual unpolarised spectral lines and estimate the
total unsigned magnetic field, which is insensitive to polarity cancellation.
The other is to apply tomographic techniques that use the polarisation
properties of the Zeeman-split components to recover the orientation of the
local field. In addition to these well-established methods, Lehmann & Donati
(2022) show that fundamental properties of the large-scale field topology can
be derived directly from the circularly polarised Stokes $V$ time series using
principal component analysis (PCA), without prior assumptions. This method
allows us to qualitatively infer the predominant component of the field
topology, as well as its complexity, axisymmetry, and evolution. Altogether,
these observational constraints guide dynamo theories to a comprehensive
description of the magnetic field generation and dynamic nature in the form of
magnetic cycles (Reiners et al., 2010; Gregory et al., 2012; See et al.,
2016).

Over the last three decades, Zeeman-Doppler imaging (ZDI, Semel 1989; Donati &
Brown 1997) has been applied to reconstruct the poloidal and toroidal
components of stellar magnetic fields, providing evidence of a wide variety of
the large-scale magnetic topologies (e.g., Morin et al., 2016). Among rapidly
rotating cool stars, the partly convective ones with masses above 0.5 M⊙ tend
to have moderate, predominantly toroidal large-scale fields generally
featuring a non-axisymmetric poloidal component (Petit et al., 2008; Donati et
al., 2008a; See et al., 2015). Those with masses between 0.2 M⊙ and 0.5 M⊙ –
close to the fully convective boundary at 0.35 M⊙ (Chabrier & Baraffe, 1997) –
generate stronger large-scale magnetic fields, dominated by a poloidal and
axisymmetric component. For fully convective stars with M$<$0.2 M⊙,
spectropolarimetric analyses have revealed a dichotomy of field geometries:
either strong, mostly axisymmetric dipole-dominated or weak, non-axisymmetric
multipole-dominated large-scale fields are observed (Morin et al., 2010). The
latter findings could be understood either as a manifestation of dynamo
bistability (Morin et al., 2011; Gastine et al., 2013), that is two dynamo
branches that coexist over a range of stellar rotation periods and masses, or
of long magnetic cycles, implying that different topologies correspond to
different phases of the cycle (Kitchatinov et al., 2014). Yet, no firm
conclusion has been reached. In parallel, studies relying on the analysis of
unpolarised spectra have shown that the average (unsigned) surface magnetic
field of cool stars follows a classical rotation-activity relation including a
non-saturated and a saturated (or quasi-saturated) regime, without a simple
relation with the large-scale magnetic geometry (Reiners & Basri, 2009;
Shulyak et al., 2019; Kochukhov, 2021; Reiners et al., 2022). Similarly,
recent dynamo simulations conducted by, for instance, Zaire et al. (2022)
confirm that the influence of rotation on convective motions alone could not
explain the observed variety of magnetic geometry. Only in the case of fully
convective very fast rotators, Shulyak et al. (2017) found that the strongest
average fields were measured for stars with large-scale dipole-dominated
fields. Kochukhov (2021) show that the fraction of magnetic energy contained
in the large-scale field component is also the highest for these stars.

Cyclic trends for Sun-like stars were found via photometric and chromospheric
activity (i.e. Ca II H&K lines) monitoring, and timescales shorter (e.g., 120
d for $\tau$ Boo, Mittag et al. 2017) or longer ($\simeq$ 20 yr for HD 1835;
Boro Saikia et al. 2018) than the solar magnetic cycle were reported (Wilson,
1968; Baliunas et al., 1995; Boro Saikia et al., 2018). Moreover, polarity
flips of the large-scale field were detected for a handful of stars based on
optical spectropolarimetric observations (Donati et al., 2008b; Petit et al.,
2009; Fares et al., 2009; Morgenthaler et al., 2011; Boro Saikia et al., 2016;
Rosén et al., 2016; Jeffers et al., 2018; Boro Saikia et al., 2022; Jeffers et
al., 2022). For M dwarfs, numerous studies relying on photometry and
spectroscopic indices claimed evidence of activity cycles (e.g., Gomes da
Silva et al., 2012; Robertson et al., 2013; Mignon et al., 2023), and radio
observations suggest the occurrence of polarity reversal at the end of the
main sequence (Route, 2016), but no polarity reversal has been directly
observed with spectropolarimetry so far. This motivates long-term
spectropolarimetric surveys, to reveal secular changes in the field topology
and shed more light on the dynamo processes in action.

A well-known active M dwarf is AD Leo (GJ 388), whose mass (0.42 M⊙) falls at
the boundary between the domains where toroidal- and dipole-dominated magnetic
topologies have previously been identified, and thus represents an interesting
laboratory to study stellar dynamos. Morin et al. (2008b) analysed the large-
scale magnetic field from spectropolarimetric data sets collected with Narval
at Télescope Bernard-Lyot in 2007 and 2008 and reported a stable,
axisymmetric, dipole-dominated geometry. Later, Lavail et al. (2018) examined
data collected with ESPaDOnS at Canada-France-Hawaii Telescope (CFHT) from
2012 and 2016, and showed an evolution of the field in the form of a global
weakening (about 20%) and small-scale enhancement. The latter was
quantitatively expressed by a decrease in the magnetic filling factor (from
13% to 7%), meaning that the field was more intense on local scales. No
polarity reversal was reported on AD Leo (Lavail et al., 2018). The large-
scale magnetic topology has remained stable since spectropolarimetric
observations of AD Leo have been initiated (2007–2016): dominated by a strong
axial dipole, the visible pole corresponding to negative radial field
(magnetic field vector directed towards the star).

Here, we extend the magnetic analysis of AD Leo using both new optical
ESPaDOnS observations collected in 2019 and near-infrared spectropolarimetric
time series collected with SPIRou at CFHT in 2019 and 2020 under the SPIRou
Legacy Survey (SLS), which adds to the previous optical data sets collected
with ESPaDOnS and Narval between 2006 and 2016. The aim is to apply distinct
techniques to search for long-term variations that may or may not resemble the
solar behaviour.

The paper is structured as follows: in Sec. 2 we describe the observations
performed in the near-infrared and optical domains, in Sec. 3 we outline the
temporal analysis of the longitudinal magnetic field, the Full-Width at Half
Maximum (FWHM) of the Stokes $I$ profile, and the total magnetic flux inferred
from Zeeman broadening modelling. Then, we describe the magnetic geometry
reconstructions by means of ZDI and PCA. In Sec. 4 we discuss the wavelength
dependence of magnetic field measurements and in Sec. 5 we present our
conclusions.

##  2 Observations

AD Leo is an M3.5 dwarf with a $V$ and $H$ band magnitude of 9.52 and 4.84,
respectively (Zacharias et al., 2013), at a distance of 4.9651$\pm$0.0007 pc
(Gaia Collaboration et al., 2021). Its age was estimated to be within 25 and
300 Myr by Shkolnik et al. (2009). AD Leo has a rotation period of 2.23 days
(Morin et al., 2008b; Carmona et al., 2023) and an inclination $i=20^{\circ}$,
implying an almost pole-on view (Morin et al., 2008b). Its high activity level
is seen in frequent flares (Muheki et al., 2020; Namekata et al., 2020) and
quantified by an X-ray-to-bolometric luminosity ratio
($\log(\mathrm{L}_{\mathrm{X}}/\mathrm{L}_{\mathrm{bol}})$) of -3.62 (Wright
et al., 2011) and a mean CaII H&K index
($\mathrm{logR^{\prime}}_{\mathrm{HK}}$) of -4.00 (Boro Saikia et al., 2018).

AD Leo’s mass is 0.42 $M_{\odot}$ (Mann et al., 2015; Cristofari et al.,
2023), which places it above the theoretical fully convective boundary at 0.35
$M_{\odot}$ (Chabrier & Baraffe, 1997). The latter value is in agreement with
observations, as it has been invoked to explain the dearth of stars with
$M_{G}\sim$10.2, known as Gaia magnitude gap (Feiden et al., 2021). However,
it is not an absolute limit: age (Maeder & Meynet, 2000) and metallicity
affect the depth of the convective envelope (van Saders & Pinsonneault, 2012;
Tanner et al., 2013), and the presence of strong magnetic fields quenches
convection and could push the theoretical boundary towards later spectral type
(Mullan & MacDonald, 2001).

Figure 1: Example Stokes profiles for AD Leo for the 9th December 2019
observation collected with SPIRou. From the top: Stokes $I$ profile
(unpolarised); Stokes $V$ profile (circularly polarised) with a clear Zeeman
signature of S/N = 6 500, and Null $N$ profile, used for quality check of the
Stokes profiles (Donati et al., 1997; Bagnulo et al., 2009). The LSD profiles
were obtained using a mask containing 838 lines. In all panels, the units are
relative to the unpolarised continuum.

###  2.1 Near-infrared

A total of 77 spectropolarimetric observations in the near-infrared were
collected with the SpectroPolarimètre InfraRouge (SPIRou) within the SLS.
SPIRou is a stabilised high-resolution near-infrared spectropolarimeter
(Donati et al., 2020) mounted on the 3.6 m CFHT atop Maunakea, Hawaii. It
provides a full coverage of the near-infrared spectrum from 0.96 to $2\mu$m at
a spectral resolving power of $R\sim 70,000$. Optimal extraction of SPIRou
spectra was carried out with A PipelinE to Reduce Observations (APERO
v0.6.132), a fully automatic reduction package installed at CFHT (Cook et al.,
2022). The same data set was used in Carmona et al. (2023) to perform a
velocimetric study and reject the hypothesis of a planetary companion by Tuomi
et al. (2018) in favour of activity-induced variations, in agreement with
Carleo et al. (2020).

Observations were performed in circular polarisation mode between February
2019 and June 2020, spanning 482 days in total; the journal of observations is
available in Table LABEL:tab:log. The mean airmass is 1.32 and the signal-to-
noise ratio (S/N) at $1,650$ nm per spectral element ranges from 68 to 218,
with an average of 168. We applied least-squares deconvolution (LSD) to atomic
spectral lines to derive averaged-line Stokes $I$ (unpolarised) and $V$
(circularly polarised) profiles (Donati et al., 1997; Kochukhov et al., 2010).
This numerical technique assumes the spectrum to be the convolution between a
mean line profile and a line mask, that is to say a series of Dirac delta
functions centred at each absorption line in the stellar spectrum, with
corresponding depths and Landé factors (i.e. sensitivities to the Zeeman
effect at a given wavelength). The output mean line profile gathers the
information of thousands of spectral lines and, because of the consequent high
S/N, enables the extraction of polarimetric information from the spectrum. The
adopted line mask was generated using the Vienna Atomic Line
Database111http://vald.astro.uu.se/ (VALD, Ryabchikova et al., 2015) and a
MARCS atmosphere model (Gustafsson et al., 2008) with $T_{\mathrm{eff}}=3,500$
K, $\log g=$ 5.0 [cm s-2] and $v_{\mathrm{micro}}=$ 1 km s-1. It contains
$1,400$ atomic lines between 950–$2,600$ nm and with known Landé factor
(ranging from 0 to 3) and with depth larger than 3 % of the continuum level.

We discarded six observations in February 2019 since one optical component of
the instrument was not working nominally, one observation in November 2019
because likely affected by a flare (the corresponding radial velocity is $>$8
sigma lower than the bulk of the measurements) and two observations in 2020 as
they led to noisier (by a factor of 10) LSD profiles. Therefore, the data set
analysed in this work comprises 68 polarimetric sequences, whose
characteristics are reported in Table LABEL:tab:log.

The near-infrared observations were performed monthly between 2019 and 2020,
except for two gaps of approximately two and three months. There is also a gap
of 1.5 month between the end of 2019 and beginning of 2020. We thus split the
time series in four epochs to maintain coherency of magnetic activity over
short time scales and for clearer visualisation: 2019a (15th April 2019 to
21st June 2019, i.e. 2019.29 to 2019.47), 2019b (16th October 2019 to 12th
December 2019, i.e. 2019.79 to 2019.95) 2020a (26th January 2020 to 12th March
2020, i.e. 2020.07 to 2020.19), and 2020b (8th May 2020 to 10th June 2020,
i.e. 2020.35 to 2020.44).

The near-infrared domain covered by SPIRou is polluted by strong and wide
telluric bands due to Earth’s atmospheric absorption. Their contribution to
the stellar spectra is corrected using a telluric transmission model (which is
built from observations of standard stars since the start of SPIRou
operations, and using Transmissions of the AtmosPhere for AStromomical data
(TAPAS) atmospheric model Bertaux et al. 2014) and a PCA method implemented in
the APERO pipeline (Artigau et al., 2014). To account for potential residuals
in the telluric correction, we ignored the following intervals of the spectrum
when computing the LSD profiles: [950, 979], [1116, 1163], [1331, 1490],
[1790, 1985], [1995,2029], [2250, 2500] nm. These intervals correspond to H2O
absorption regions, with transmission typically smaller than 40%. We assessed
whether removing these telluric intervals optimises the quality of the Stokes
$V$ profiles. In a first test, we searched for stellar absorption lines deeper
than 75 % of the continuum level and within $\pm 100$ km s-1 from telluric
lines included in the transmission model of APERO. This approach allowed us to
identify stellar lines that are contaminated by the telluric lines throughout
the year. When the telluric-affected spectral lines were removed, no
significant improvement was reported in the final LSD profiles, indicating a
robust telluric correction as already reported in (Carmona et al., 2023). In a
second test, we extended the intervals by 25 and 50 nm or reduced them by 10
nm and noticed an increment of the noise level in LSD profiles up to 20 %, so
we proceeded with the previous intervals.

Accounting for the ignored telluric intervals, the number of spectral lines
used in LSD is 838. We show the LSD Stokes profiles for one example
observation in Fig. 1. The average noise level in Stokes $V$ for the entire
time series is $1.6\cdot 10^{-4}$ relative to the unpolarised continuum,
similar to the optical domain (Morin et al., 2008b). We also note that the
profiles are broader than in the optical by more than $10$ km s-1, owing to a
stronger Zeeman effect in the near-infrared domain (Zeeman, 1897).

###  2.2 Optical

For most of the analyses presented here, we considered all archival
observations collected with ESPaDOnS and Narval, and studied previously in
Morin et al. (2008b) and Lavail et al. (2018). We also included six new
observations taken in November 2019 (from 2019.87 to 2019.89) with ESPaDOnS
for CFHT programme 19BC06, PI A. Lavail (reported in Table LABEL:tab:log_opt).
They are contemporaneous to the SPIRou ones for the same period, hence
enabling us to study the dependence of the measured magnetic field strength on
the wavelength domain employed (see Sec. 4).

ESPaDOnS is the optical spectropolarimeter on the 3.6 m CFHT located atop
Mauna Kea in Hawaii, and Narval is the twin instrument on the 2 m TBL at the
Pic du Midi Observatory in France (Donati, 2003). The data reduction was
performed with the LIBRE-ESPRIT pipeline (Donati et al., 1997), and the
reduced spectra were retrieved from the PolarBase archive (Petit et al.,
2014).

The LSD profiles were computed similar to the near-infrared, but using an
optical VALD mask containing 3330 lines in range 350-1080 nm and with depths
larger than 40% the continuum level, similar to Morin et al. (2008b); Bellotti
et al. (2022). The number of lines used is 3240 and accounts for the removal
of the following wavelength intervals, which are affected by telluric lines or
in the vicinity of H$\alpha$: [627,632], [655.5,657], [686,697], [716,734],
[759,770], [813,835], and [895,986] nm. For the 2019 observations, the average
noise in Stokes $V$ is $3\cdot 10^{-4}$ relative to the unpolarised continuum.

In the next sections, the near-infrared and optical observations will be
phased with the following ephemeris:

$\mathrm{HJD}=2458588.7573+\mathrm{P}_{\mathrm{rot}}\cdot
n_{\mathrm{cyc}}\text{ , }$ (1)

where we used the first SPIRou observation taken in April 2019 as reference,
P${}_{\mathrm{rot}}=$2.23 days is the stellar rotation period (Morin et al.,
2008b), and $n_{\mathrm{cyc}}$ corresponds to the rotation cycle (see Table
LABEL:tab:log).

##  3 Magnetic analysis

###  3.1 Longitudinal magnetic field

Figure 2: Temporal evolution of the longitudinal magnetic field (Bl). Top: full time series of measurements between 2006 and 2020 with ESPaDOnS, Narval and SPIRou. The shape of the data points corresponds to the wavelength domain, optical (circles) or near-infrared (pluses), and the colour to the epoch in which the observations were performed. Bottom: SPIRou time series split in four epochs phase-folded according to Eq. 1. The least-square sine fit corresponding to Eq. 3 (Stibbs, 1950), to assess the change in magnetic obliquity, is shown as dashed lines. The six ESPaDOnS observations taken in 2019 are plotted as pink circles.  Table 1: Best fit values of the polar field and obliquity of Eq. 3 obtained for optical and near-infrared epochs. | 2006 | 2007 | 2008 | 2012 | 2016 | 2019a | 2019b | 2020a | 2020b
---|---|---|---|---|---|---|---|---|---
$B_{p}$ [G] | $-868\pm 20$ | $-842\pm 21$ | $-873\pm 15$ | $-736\pm 19$ | $-555\pm 10$ | $-814\pm 17$ | $-882\pm 22$ | $-828\pm 55$ | $-923\pm 70$
$\beta$ [∘] | $13\pm 6$ | $12\pm 4$ | $26\pm 1$ | $7\pm 4$ | $12\pm 1$ | $3\pm 4$ | $23\pm 2$ | $37\pm 4$ | $59\pm 2$
Table 2: Comparison of a constant line vs a sine fit for the FWHM phase
variations.

Epoch | Mask | Mean | Mean Error | STD | RMSconst | $\chi^{2}_{r,\mathrm{const}}$ | RMSsine | $\chi^{2}_{r,\mathrm{sine}}$
---|---|---|---|---|---|---|---|---
|  | [km s-1] | [km s-1] | [km s-1] | [km s-1] |  | [km s-1] |
2019a | default | 19 | 0.29 | 0.40 | 0.40 | 1.8 | 0.39 | 2.0
| g${}_{\mathrm{eff}}>1.2$ | 23 | 0.46 | 1.16 | 1.16 | 6.9 | 1.13 | 8.0
2019b | default | 21 | 0.41 | 0.63 | 0.63 | 2.3 | 0.62 | 2.5
| g${}_{\mathrm{eff}}>1.2$ | 27 | 0.84 | 1.77 | 1.77 | 6.7 | 1.71 | 6.9
2020a | default | 21 | 0.41 | 0.68 | 0.68 | 3.2 | 0.61 | 3.4
| g${}_{\mathrm{eff}}>1.2$ | 27 | 0.82 | 2.47 | 2.47 | 12.2 | 2.29 | 13.3
2020b | default | 19 | 0.30 | 0.47 | 0.47 | 3.0 | 0.47 | 3.4
| g${}_{\mathrm{eff}}>1.2$ | 22 | 0.46 | 0.83 | 0.83 | 5.3 | 0.79 | 7.3
222The columns are: 1) subset of the time series, 2) line list used in LSD
computation, 3) mean value of FWHM, 4) mean error bar on FWHM, 5) dispersion
of the data set, 6) RMS (Root Mean Square) residual of a constant line fit
equal to the average of the data set, 7) reduced $\chi^{2}$ of the constant
fit, 8) RMS residual of a sine fit at the stellar rotation period, 9) reduced
$\chi^{2}$ of the sine fit.

We measured the line-of-sight component of the magnetic field integrated over
the stellar disk (Bl) for all the available observations, in optical
(2006–2019) and near-infrared (2019–2020). Since Bl traces magnetic features
present on the visible hemisphere, its temporal variations are modulated at
the stellar rotation period and can be therefore used as a robust magnetic
activity proxy (Folsom et al., 2016; Hébrard et al., 2016; Fouqué et al.,
2023). Formally, it is computed as the first-order moment of Stokes V (Donati
et al., 1997):

$\mathrm{B}_{l}\;[G]=\frac{-2.14\cdot
10^{11}}{\lambda_{0}\mathrm{g}_{\mathrm{eff}}c}\frac{\int
vV(v)dv}{\int(I_{c}-I)dv}\,,$ (2)

where $\lambda_{0}$ and $\mathrm{g}_{\mathrm{eff}}$ are the normalisation
wavelength and Landé factor of the LSD profiles, $I_{c}$ is the continuum
level, $v$ is the radial velocity associated to a point in the spectral line
profile in the star’s rest frame and $c$ the speed of light in vacuum. For the
near-infrared and optical Stokes profiles, the normalisation wavelength and
Landé factor are 1700 nm and 1.2144, and 700 nm and 1.1420, respectively. In
accordance with the fact that near-infrared lines are broader than optical
ones, the integration was carried out within $\pm$ 50 km s-1 from line centre
in the former case and $\pm$ 30 km s-1 in the latter case, to include the
absorption ranges of both Stokes $I$ and $V$ profiles.

The list of measurements is reported in Table LABEL:tab:Bl_log. The values are
of constant sign (negative), which is expected when observing one polarity of
a dipole almost aligned with the stellar rotation axis over the entire stellar
rotation, especially for a star observed nearly pole-on as AD Leo (Morin et
al., 2008b; Lavail et al., 2018). The near-infrared measurements range between
$-263$ and $-46$ G, with an average of $-179$ G and a median error bar of 15
G. The optical measurements range between $-297$ and $-155$ G, with an average
of $-233$ G and a median error bar of 10 G. The lower error bar is likely due
to the narrower velocity range over which the optical measurements are
performed, since less noise is introduced in Eq. 2. A discussion about
chromatic differences in the longitudinal field measurements is presented in
Sec. 4.

We plot the temporal evolution of Bl in Fig. 2. In general, we note a secular
weakening of the field strength over 14 yr, with an oscillation between 2016
and 2019 followed by a rapid decrease in strength (in absolute value). We also
note that the intra-epoch dispersion increases for the last two epochs.

By phase-folding the near-infrared data at Prot, we observe a systematic
increase in the rotational modulation towards 2020b, meaning that the
axisymmetry level of the field has likely decreased (see Fig. 2). For a first
quantitative evaluation, we followed Stibbs (1950) and Preston (1967) to model
the phase variations of the longitudinal field for a predominantly-dipolar
magnetic configuration. Formally,

$\mathrm{B}_{l}\;[G]=\frac{1}{20}\frac{15+\varepsilon}{3-\varepsilon}\mathrm{B}_{p}\left(\cos\beta\cos
i+\sin\beta\sin i\cos(2\pi p)\right)\,,$ (3)

with $\varepsilon$ the limb darkening coefficient (set to 0.3; Claret &
Bloemen 2011), $p$ the rotational phase, Bp the longitudinal field of the
dipole, $i$ the stellar inclination and $\beta$ the obliquity between magnetic
and rotation axes. The results are listed in Table 1, for both near-infrared
and optical time series for completeness. The six optical observations in
November 2019 have poor coverage (three of them are clustered around phase
0.9) and lead to a less reliable sine fit. Nevertheless, they are compatible
with the 2019b fit curve. These clues clearly indicate that the magnetic field
of AD Leo is evolving, in agreement with Lavail et al. (2018), and demonstrate
the interest of long-term spectropolarimetric monitoring of active M dwarf
stars.

###  3.2 The mean line width

Figure 3: Phase variations of the FWHM, folded at a stellar rotation period of
2.23 days. The data sets are computed with the default (black), high-geff
(yellow), and low-geff (purple) mask for three epochs: 2019a (top left), 2019b
(top right), 2020a (bottom left), 2020b (bottom right). Although no evident
rotational modulation is observed, we notice a long-term oscillation of the
mean value in a similar way to the longitudinal field (absolute) values.

The width of near-infrared spectral lines of stars with intense fields and low
equatorial velocity such as AD Leo ($v_{e}\sin(i)=$ 3 km s-1; Morin et al.
2008b) are sensitive to the Zeeman effect, given its proportionality to
wavelength, field strength, and Landé factor. The rotationally-modulated line
broadening correlates with the azimuthal distribution of the unsigned small-
scale magnetic flux, a useful diagnostic for stellar activity radial velocity
contamination, as shown for the Sun by Haywood et al. (2022). In this context,
Klein et al. (2021) adopted a selection of magnetically sensitive lines for
the young star AU Mic and saw a correspondence in the variations of RV and
FWHM of the Stokes $I$ profiles at the stellar rotation period. This confirmed
the sensitivity of the FWHM to the distortions induced by magnetic regions on
the stellar surface.

Here, we proceeded analogously in an attempt to connect modulations of the
FWHM with variations of the large-scale field. We applied LSD on the near-
infrared data using a mask of 417 lines characterised by
g${}_{\mathrm{eff}}>1.2$, following Bellotti et al. (2022). The near-infrared
time series was divided in four epochs as in Sec. 3.1 for consistency.

In Table 2, we compare the phase variations of the FWHM when adopting the
default and high-geff masks, and we inspect whether they are more compatible
with a sine fit or a constant fit equal to the mean of the data set. In all
cases, there is no clear rotational modulation of the data points, as the sine
fit does not provide a better description (i.e. lower $\chi^{2}_{r}$) of the
variations than the constant fit. This is confirmed by a quick inspection of
the periodogram applied to the FWHM data for each individual epoch. The
$\chi^{2}_{r}$ increase when using a sine model rather than a constant is not
statistically significant. The observed variations are attributable to
dispersion, as illustrated in Fig. 3. We observed that the FWHM is
systematically larger in all epochs for the high-geff mask, as expected given
the linear dependence of Zeeman effect to geff, and the dispersion is between
1.8 and 3.0 times larger. The lack of rotational modulation prevents us from
searching for correlations with other quantities such as RV and Bl as done in
Klein et al. (2021).

From Fig. 3, we also noticed an evident long-term evolution of the mean FWHM.
Such evolution has a moderate correlation (Pearson $R$ coefficient of 0.5)
with the variations of the mean Bl for the same epochs, meaning that the FWHM
is a reasonable proxy to trace long-term evolution of the field. This is
consistent with the recent Donati et al. (2023) analysis analysis of AU Mic.
When using the default mask, the mean FWHM oscillated from 19 km s-1 in 2019a
to 21 km s-1 in 2019b and 2020a, and back to 19 km s-1 in 2020b. As expected,
such oscillation is enhanced when considering the magnetically sensitive lines
and goes from 23 km s-1 in 2019a to 27 km s-1 in 2019b and 2020a, and back to
22 km s-1 in 2020b. We performed the same analysis with low-Landé factor lines
(i.e. g${}_{\mathrm{eff}}<1.2$ and 406 lines) and noticed no appreciable
variation of the mean FWHM, since it remained stable at $\sim$15 km s-1. A
view of the Stokes $I$ profiles computed with the three different line lists
can be found in Appendix A.

The FWHM analysis was also carried out on the ESPaDOnS and Narval data between
2006 and 2019. When using low-geff lines, the mean width of Stokes $I$ is
reasonably stable around 9.7 km s-1, stressing their potential for precise
radial velocity measurements. The full (high-geff) mask yields a mean value at
10 km s-1 (12 km s-1) between 2006 and 2012, which then increases to 11 km s-1
(13 km s-1) in 2016 and 2019. Such long-term evolution is only moderate
compared to the one seen in the near-infrared time series. The entire
evolution is illustrated in Fig. 4.

Figure 4: Secular evolution of the epoch-averaged FWHM of Stokes $I$. The
scale on the left refers to optical data, while on the right to the near-
infrared, and the colours indicate the use of different masks similar to Fig.
3. Both optical (circles) and near-infrared (pluses) observations feature a
long-term variation, which is enhanced (quenched) when using magnetically
sensitive (insensitive) lines. We note that the optical data point of 2019
falls behind the near-infrared ones when using magnetically sensitive lines.

The difference between the mean FWHM of low-geff lines in optical ($\sim$9.5
km s-1) and near-infrared ($\sim$16 km s-1) can be attributed to lines that
have non-zero Landé factor. Indeed, the quadratic differential broadening
between the two domains is $11.4$ km s-1, corresponding to a total magnetic
field of 2.5 kG for a line at 1700 nm with geff=0.96 (the normalisation values
of the low-geff mask). Although we assumed that the Zeeman effect for low-geff
lines is negligible in the optical with this exercise, the inferred value of
total magnetic field is reasonably consistent with what is reported in the
literature (Saar, 1994; Shulyak et al., 2017, 2019), indicating that the
magnetic field accounts mostly for the difference in width between optical and
near-infrared low-geff lines.

Our analysis confirms that the FWHM is capable of tracing secular changes in
the total, unsigned magnetic field, which could be used to better understand
stellar activity jitter. Activity-mitigating techniques would benefit from
this information even for low-inclination stars such as AD Leo, for which the
phase modulation of the radial velocity jitter is more difficult to constrain.
At the same time, the analysis highlights the presence of short-term
variability producing scatter and that is not rotationally modulated.

Figure 5: Example fits including Zeeman broadening resulting from the MCMC-
based approach of radiative transfer modelling. Top: ESPaDOnS observation from
24 Febuary 2016. Bottom: SPIRou observation from 3 November 2020. The panels
have the same wavelength scale with minor ticks at 0.1 nm.

###  3.3 Modelling Zeeman broadening

To further investigate the small-scale magnetic field of AD Leo we conducted a
Zeeman broadening analysis. For this analysis we used the full set of new and
archival data, both in the near-infrared from SPIRou and in the optical from
ESPaDOnS and Narval. All the data sets require a telluric correction, since
telluric lines are present in much of the SPIRou wavelength range, and the red
end of the ESPaDOnS and Narval range. For the SPIRou data we relied on the
telluric correction from the APERO pipeline (Sec 2.1, Cook et al. 2022 for
more detail). For the ESPaDOnS and Narval data, we made a telluric correction
using the molecfit333https://www.eso.org/sci/software/pipelines/ pipeline,
originally designed for handling spectra from ESO instruments (Smette et al.,
2015; Kausch et al., 2015). molecfit retrieves weather conditions and other
relevant information at the time of observation and models the atmosphere in
the line of sight. It performs radiative transfer and iteratively models the
telluric component in the input spectrum while also fitting the continuum and
the wavelength scale of the spectrum. It finally corrects telluric lines and
provides a telluric-corrected output spectrum. After telluric correction the
spectra were re-normalised in the regions of interest using a low order
polynomial fit through carefully selected continuum regions. A few ESPaDOnS
and Narval spectra were affected by fringing effects, hence we adopted a
higher-order polynomial fit to normalise to a flatter continuum. Finally, we
discarded any observations where the telluric correction left a noticeable
residual feature that was blended with the stellar lines of interest.

To characterise the magnetic field, we fitted synthetic spectra to the
observed Stokes $I$ spectra, incorporating both the Zeeman broadening and
intensification effects. Synthetic spectra were calculated with Zeeman
(Landstreet, 1988; Wade et al., 2001; Folsom et al., 2016), using model
atmospheres from marcs (Gustafsson et al., 2008). Zeeman performs polarised
radiative transfer including the Zeeman effect. However, a major limitation
for M-dwarfs is that the programme does not currently include molecular lines,
which are not typically used in Zeeman broadening analyses. Weak molecular
lines are blended with many atomic lines in the spectra of M-dwarfs. With
careful attention we identified a set of atomic lines suitable for AD Leo,
with no evident distortion in the line shape by molecular blends. Thus the
systematic error from this limitation is expected to be negligible, but the
inclusion of molecular lines in the future would substantially simplify the
selection of lines for Zeeman broadening analyses, as shown in the recent work
of Cristofari et al. (2023) and previously applied to AD Leo by Shulyak et al.
(2014, 2017). To check the validity of the analysis presented here, a second
analysis of the ESPaDOnS and Narval spectra was carried out with the SYNMAST
code (Kochukhov et al., 2010). The analyses used nearly the same set of Ti i
lines, and the results we obtained were consistent within uncertainty.

For the ESPaDOnS and Narval observations, we used the Ti i lines at 9675.54 Å
(g${}_{\mathrm{eff}}=1.35$), 9688.87 Å (g${}_{\mathrm{eff}}=1.50$), 9705.66 Å
(g${}_{\mathrm{eff}}=1.26$), 9728.40 Å (g${}_{\mathrm{eff}}=1.00$), 9743.61 Å
(g${}_{\mathrm{eff}}=0.00$), and 9770.30 Å (g${}_{\mathrm{eff}}=1.55$). These
lines have been used extensively for Zeeman broadening analysis (e.g.,
Kochukhov & Lavail, 2017; Hill et al., 2019; Kochukhov, 2021, and references
therein) and have reliable oscillator strengths and Landé factors in VALD.
These lines have relatively weak telluric blending, very little molecular
blending, and a wide range of effective Landé factors.

For the SPIRou observations, we selected a set of lines using similar
criteria, but also avoided lines with large pressure broadened wings, since
small errors in the pressure broadening could cause larger errors in the
Zeeman broadening estimation. In order to maximise the range of available
effective Landé factors we used the Fe i lines at 11422.32 Å
(g${}_{\mathrm{eff}}=1.98$), 11593.59 Å (g${}_{\mathrm{eff}}=2.50$), 11607.57
Å (g${}_{\mathrm{eff}}=1.66$), 11638.26 Å (g${}_{\mathrm{eff}}=1.58$), and
11783.26 Å (g${}_{\mathrm{eff}}=1.14$) and the Ti i lines at 11892.88 Å
(g${}_{\mathrm{eff}}=0.75$), 12821.67 Å (g${}_{\mathrm{eff}}=1.26$), 12831.44
Å (g${}_{\mathrm{eff}}=0.67$), 12847.03 Å (g${}_{\mathrm{eff}}=1.08$),
22232.84 Å (g${}_{\mathrm{eff}}=1.66$), and 22310.61 Å
(g${}_{\mathrm{eff}}=2.50$). This provides multiple lines with both high and
low effective Landé factors, but uses lines from two different ions, which we
compensated for by using the Ti and Fe abundances as independent free
parameters in our analysis. There are a few other Ti i lines near 22000 Å with
large effective Landé factors, but there is a relatively severe blending by
many weak molecular lines in this region, hence we did not include these
lines. Line data were extracted from VALD. In these line lists, experimental
oscillator strengths for Ti i lines were from Lawler et al. (2013) (consistent
with Blackwell-Whitehead et al., 2006), except for 22232.84 Å from Blackwell-
Whitehead et al. (2006), and a theoretical value for 22310.61 Å from the
compilation of R. L. Kurucz444http://kurucz.harvard.edu. Oscillator strengths
for the Fe i lines were taken from O’Brian et al. (1991).

The total magnetic field was modelled with a grid of field strengths and
filling factors for the fraction of the surface area with the corresponding
field strength (e.g. Johns-Krull et al., 1999). A uniform radial orientation
was assumed for the magnetic field, since Stokes $I$ spectra have little
sensitivity to magnetic field orientation. This is also a reasonable
assumption given the magnetic field maps reconstructed in Sec. 3.4. For the
optical spectra, we adopted magnetic fields of 0, 2, 4, 6, 8, and 10 kG, and
derived their filling factors. For the SPIRou spectra, we used a finer grid of
1 kG from 0 to 10 kG, since the sensitivity to Zeeman effect is larger at
longer wavelengths, and a finer grid is needed to produce smooth line
profiles.

To derive the magnetic filling factors we applied an MCMC-based approach,
using the emcee package (Foreman-Mackey et al., 2013) integrated with Zeeman.
The filling factors for $B>0$ were treated as free parameters, with the
filling factor for $B=0$ ($f_{B=0}$) calculated from $1-\sum_{B>0}f_{B}$.
Proposed steps in the chain where $\sum_{B>0}f_{B}>1$ were rejected to ensure
that the filling factors sum to unity. The projected rotational velocity
$v_{e}\sin(i)$ and the abundance of Ti (and Fe for SPIRou) were included as
free parameters in the MCMC process. The modelling used $T_{\rm eff}=3500$ K,
$\log g=5.0$, a microturbulence of 1 km s-1. The chemical abundances may be
unreliable since they do not account for elements bound in molecules, making
them effectively nuisance parameters in this study. However, this provides the
code with flexibility for fitting line strength and width in the absence of a
magnetic field, reducing the sensitivity of the results to small errors in
non-magnetic parameters.

Example fits resulting from the MCMC-based approach are show in Figs. 5 for
ESPaDOnS and SPIRou. The shapes of the posterior distributions are generally
similar for all observations using the same sets of lines, and are illustrated
in Appendix C. There are important anti-correlations between filling factors
with adjacent magnetic field strengths, and weak correlations between filling
factors spaced by two bins in field strength. Therefore, some caution should
be taken in interpreting the uncertainties from this and similar analyses. The
filling factor for $B=0$ and the quantity $\sum_{i}B_{i}f_{i}$ summed over
magnetic field bins (abbreviated to $\sum Bf$), were calculated from samples
in the MCMC chain. The resulting distribution was used to provide the median
value from the 50th percentile, with uncertainties from the 16th and 84th
percentile.

The results for all observations, and averages for each epoch, are presented
in Fig. 6, and values for each epoch are provided in Tables 4 and 5. The
quantity $\sum Bf$ (sometimes called the magnetic flux, and analogous to a
magnetic flux density) ranges between 2.6 kG and 3.7 kG, which is consistent
with previous measurements (e.g., Saar, 1994; Kochukhov et al., 2009; Shulyak
et al., 2014, 2017, 2019; Cristofari et al., 2023). We observe a long-term
increase of the average $\Sigma\,Bf$ from 2.8 kG in 2007 to 3.6 kG in 2016,
followed by a weakening towards 3.4 kG with the latest SPIRou observations.
Such behaviour correlates with the long-term decrease (in absolute value) of
the longitudinal field (Pearson coefficient $R$=0.6, excluding the 2006 data
point). Likewise, the average $\Sigma\,Bf$ time series correlates with the
average FWHM of Stokes $I$, demonstrating its capability at tracing the
evolution of the total, unsigned magnetic field (Donati et al., 2023).

The $\Sigma\,Bf$ values for the ESPaDOnS optical data acquired in 2006 fall
out from this trend. This could stem from residuals of the telluric correction
blending with the lines used in the modelling and/or instrumental effects such
as fringing, for which the results are sensitive to the choice of continuum
normalisation. Attempts were made to correct for these potential systematic
errors: rejecting observations where the telluric correction left residual
features in the used portion of the spectrum, and careful continuum
normalisation to remove any weak fringing. However, it is possible these
attempts were not fully successful, and thus the departure from the general
trend of the 2006 result should be treated with caution.

Figure 6: Long-term evolution of the magnetic flux density measured from
Zeeman broadening modelling. The data point format is the same as Fig. 2, and
for each epoch we overplotted the mean value and standard deviation as error
bar. There is an overall trend that reflects the variations of Bl across both
optical and near-infrared data. In addition, for the near-infrared SPIRou time
series, we also observe the same oscillating behaviour found for the FWHM of
Stokes $I$.

###  3.4 Magnetic imaging

Figure 7: Full SPIRou and ESPaDOnS 2019 time series of Stokes $V$ profiles of
AD Leo normalised by the unpolarised continuum intensity. Two panels are
present for 2019a (top left), 2019b (top right), one panel for 2019 in optical
(bottom left) and 2020a (bottom middle) and two panels for 2020b (bottom
right). In all panels, both the observed profiles (black) and Unno-Rachkovsky
models (red) are shown. They are shifted vertically for visualisation
purposes, and their associated rotational cycle is reported on the right (see
Eq. 1). A remarkable feature is the increased intermittency of the profiles
amplitude for the most recent observations, for example at cycle 174.44 and
174.95 of 2020b against cycle 1.40 and 1.88 in 2019a, suggesting an evolution
of the magnetic field obliquity.

We applied ZDI to the SPIRou and 2019 ESPaDOnS time series of Stokes $V$
profiles to recover the large-scale magnetic field at the surface of AD Leo.
The magnetic geometry is modelled as the sum of a poloidal and a toroidal
component, which are both expressed through spherical harmonics decomposition
(Donati et al., 2006; Lehmann & Donati, 2022). The algorithm compares observed
and synthetic Stokes $V$ profiles iteratively, fitting the spherical harmonics
coefficients $\alpha_{\ell,m}$, $\beta_{\ell,m}$, and $\gamma_{\ell,m}$ (with
$\ell$ and $m$ the degree and order of the mode, respectively), until they
match within a target reduced $\chi^{2}$. Because the inversion problem is
ill-posed, a maximum-entropy regularisation scheme is applied to obtain the
field map compatible with the data and with the lowest information content
(for more details see Skilling & Bryan 1984; Donati & Brown 1997; Folsom et
al. 2018).

In practice, we used the zdipy code described in Folsom et al. (2018). In its
initial version, the code performed tomographic inversion under weak-field
approximation, for which Stokes $V$ is proportional to the first derivative of
Stokes $I$ over velocity (Landi Degl’Innocenti & Landolfi, 2004). For the
present study, we have implemented the Unno-Rachkovsky’s solutions to
polarised radiative transfer equations in a Milne-Eddington atmosphere (Unno,
1956; Rachkovsky, 1967; Landi Degl’Innocenti & Landolfi, 2004) and
incorporated the filling factor formalism outlined in Morin et al. (2008b) and
Donati et al. (2023). The implementation of Unno-Rachkovsky’s solutions was
motivated by the need of a more general model for the observed Stokes V
profiles. Near-infrared observations of stars with intense magnetic fields are
indeed more susceptible to distortions and broadening due to an enhanced
Zeeman effect.

As input parameters for ZDI, we assumed $i=$ 20∘, $v_{e}\sin(i)=$ 3 km s-1,
P${}_{\mathrm{rot}}=$ 2.23 days, and solid body rotation. We adopted a linear
limb darkening coefficient in H band of 0.3 and V band of 0.7 (Claret &
Bloemen, 2011). We set the maximum degree of the harmonic expansion
$\ell_{\mathrm{max}}=$ 8 (considering the low $v_{e}\sin(i)$) and allowed an
entropy weighting scheme proportional to $\ell$ during ZDI inversion, to
favour simple geometries as in Lavail et al. (2018). The SPIRou near-infrared
time series was split similarly to Sec. 3.1: 2019a (21 observations over 30
cycles), 2019b (21 observations over 26 cycles) 2020a (30 observations over 20
cycles), and 2020b (18 observations over 15 cycles). The Stokes V time series
of SPIRou and 2019 ESPaDOnS data are shown in Fig. 7.

Figure 8: Reconstructed ZDI maps in flattened polar view of AD Leo for the
SPIRou and 2019 ESPaDOnS time series. The columns correspond to the epochs
examined, from the left: 2019a, 2019b, 2019 optical, 2020a, and 2020b. In each
column, the radial (top), azimuthal (middle) and meridional (bottom)
components of the magnetic field vector are displayed. The radial ticks are
located at the rotational phases when the observations were collected, while
the concentric circles represent different stellar latitudes: -30 ∘, +30 ∘ and
+60 ∘ (dashed lines) and equator (solid line). The colour bar range is set by
the maximum (in absolute value) of the magnetic field and illustrates the
positive (red) and negative (blue) polarity for each epoch.  Figure 9:
Evolution of the magnetic field obliquity with respect to the stellar rotation
axis. Values estimated from Eq. 3 (Stibbs, 1950) using optical and near-
infrared data are illustrated as circles and pluses, colour-coded by epoch,
whereas values from ZDI are shown as diamonds and crosses. The y axis is
expressed in degrees of colatitude.

For the 2019a, 2019b, 2020a, and 2020b epochs, we fitted the Stokes $V$
profiles to a $\chi^{2}_{r}$ level of 1.2, 1.0, 1.1, and 1.1 from an initial
value of 10.3, 15.8, 14.7, and 8.5, respectively. For the ESPaDOnS 2019 time
series, we fitted down to $\chi^{2}_{r}=2.5$ from an initial value of 156.3.
We attempted to merge the 2019b and 2020a epochs and reconstruct a single map,
since they are separated by the shortest time gap. The quality of the final
model is deteriorated ($\chi^{2}_{r}$=1.3) with respect to the two epochs
separately, but the corresponding map and magnetic energies are consistently
recovered. We therefore kept these two epochs separate.

From Fig. 7, it is evident that the near-infrared Stokes $V$ profiles manifest
structures and stochastic variability in both lobes. This is not extended in
the continuum, since the residuals with respect to the mean profile are
compatible with the noise level, it is not rotationally-modulated, and it is
not exhibited by Stokes $N$. The presence of such variability was already
suggested by the phase-folded variations in Bl (Fig. 2), as some data points
featured a departure from a pure rotational modulation. Likewise, the
residuals of the Stokes $I$ profiles show clear variability, but the
application of a 2D periodogram does not reveal any significant periodicity.
While our ZDI model is capable of describing the general shape of Stokes $V$
profiles, it is limited at reproducing the structures and at capturing all the
information present. These considerations are also valid for optical
observations in 2019, as the amplitude of Stokes $V$ is not matched exactly by
our ZDI model, and overall translate into an underestimate of the field
strength. This motivates further the use of the PCA method described by
Lehmann & Donati (2022), which is a data-driven approach offering a
complementary view on the magnetic field evolution, as outlined in the next
section.

We are able to constrain the filling factor $f_{V}$ following a $\chi^{2}$
minimisation prescription similar to (Petit et al., 2002). We found $f_{V}$
values oscillating between 9% in 2019a, 16% in 2019b and back to $\sim$11% in
the remaining epochs, compatible with Morin et al. (2008b), and larger by a
factor of 1.7 than Lavail et al. (2018). This would indicate a weakening of
the local small-scale field since 2016, on top of a decrease in large-scale
field intensity as seen in the reconstructions (Fig. 8).

The filling factor $f_{I}$ was inspected by considering a grid of values
between 0% and 100%; for each $f_{I}$ value, we synthesised a time series of
model Stokes $I$ profiles, computed the corresponding time series of
$\chi^{2}_{r}$ with the observations, and phase-folded the $\chi^{2}_{r}$
curve at Prot. We then assessed at what value of $f_{I}$ the $\chi^{2}_{r}$
curve would start manifesting rotational modulation, because it would indicate
that certain model profiles deviate from the observations. We noticed that
values above 30% deteriorate the fit of the profile core progressively,
yielding variability and rotational modulation of the Stokes $I$ profiles,
which is not observed otherwise (see Sec. 3.2). Values of $f_{I}=30\%$ are
three times larger than $f_{V}$, in agreement with (Morin et al., 2008b).
Since the plausible $f_{I}$ values are consistent with 0%, we adopted
$f_{I}=0\%$ in the ZDI modelling.

The five maps of surface magnetic flux (one for each SPIRou epoch, and one for
the ESPaDOnS 2019 epoch) are shown in Figure 8 and their properties are
reported in Table 3. In all cases, the configuration is predominantly
poloidal, storing $>$95% of the magnetic energy. The main modes are dipolar
and quadrupolar, as they account for 70-90% and 15-20% of the magnetic energy.
We report a weakening of the mean field strength ($\langle B\rangle$) of
factor of 1.5 and 2.4 relative to the optical maps reconstructed by Morin et
al. (2008b) and Lavail et al. (2018), respectively. The most remarkable
feature is the reduction of magnetic energy contained in the axisymmetric
mode, going from $>$ 99% in 2019a to 60% in 2020b, translating into an
increase of the dipole obliquity relative to the rotation axis, from
$3^{\circ}$ to $38^{\circ}$.

We note that the maximum field strength reconstructed with ZDI is between 1.2
and 2.4 times smaller than obtained via Eq. 3 (Stibbs, 1950). Likewise, the
magnetic field obliquity is underestimated, as illustrated in Fig. 9. On one
side, this difference stems from the limitation of the Stokes $V$ ZDI model,
since it does not encompass the full amplitude of the two lobes for some
observations, and on the other side Eq. 3 assumes a purely dipolar field,
contrarily to our reconstructions (the dipole accounts for 70-90% of the
energy). Nevertheless, both approaches allow us to observe an evident
evolution of the obliquity, featuring a rapid increase in the most recent
epochs.

Finally, we merged the 2019b, 2020a and 2020b data sets and attempted a joint
rotation period and differential rotation search following Petit et al.
(2002). The results were inconclusive, likely due to the significant evolution
of the surface magnetic field between each epoch.

The summary of the magnetic field’s evolution is illustrated in Fig. 10. We
performed ZDI reconstructions also for the archival ESPaDOnS and Narval data
for consistency, finding reasonably compatible results with previous studies
(Morin et al., 2008b; Lavail et al., 2018). We observe a globally simple
geometry (i.e. predominantly poloidal and dipolar) over 14 yr, with a
decreasing strength. Our latest SPIRou observations revealed a clear evolution
of the dipole obliquity in the form of a reduced axisymmetry, suggesting a
potential dynamo magnetic cycle. These features are indeed compatible with the
variations observed by Sanderson et al. (2003) and Lehmann et al. (2021) for
the solar cycle.

Figure 10: Evolution of the magnetic topology of AD Leo over 14 yr. The time
series includes optical ESPaDOnS and Narval data between 2007 and 2019, and
near-infrared SPIRou data collected between 2019 and 2020. Field strength,
level of axisymmetry and dominant field component (poloidal or toroidal) are
encoded with the symbol size, shape, and colour, respectively, and were
computed using the Unno-Rachkovsky implementation described in this work for
consistency. The topology remained predominantly poloidal and dipolar, while
the field strength has decreased over time. The most striking change is a
decrease in axisymmetry in the most recent epoch. Table 3: Properties of the
magnetic maps for the 2019a, 2019b, 2020a, and 2020b SPIRou epochs and the
2019 ESPaDOnS data set.

| 2019a | 2019b | 2019 | 2020a | 2020b
---|---|---|---|---|---
| NIR | NIR | VIS | NIR | NIR
fV | 9% | 16% | 12% | 12% | 11%
$\langle B\rangle$ [G] | 111.2 | 132.3 | 158.0 | 115.3 | 93.4
Bmax [G] | 481.2 | 764.0 | 577.2 | 555.1 | 434.3
Bpol [%] | 100.0 | 99.9 | 95.0 | 99.3 | 98.7
Btor [%] | 0.0 | 0.1 | 5.0 | 0.7 | 1.3
Bdip [%] | 81.3 | 71.1 | 81.7 | 75.7 | 70.1
Bquad [%] | 14.9 | 19.0 | 14.6 | 17.7 | 21.2
Boct [%] | 2.8 | 6.2 | 2.7 | 4.4 | 5.9
Baxisym [%] | 99.8 | 94.5 | 77.0 | 85.8 | 58.3
Obliquity [∘] | 2.5 | 12.5 | 21.5 | 19.5 | 38.0
555The following quantities are listed: filling factor, mean magnetic
strength, maximum magnetic strength, poloidal and toroidal magnetic energy as
a fraction of the total energy, dipolar, quadrupolar and octupolar magnetic
energy as a fraction of the poloidal energy, axisymmetric magnetic energy as a
fraction of the total energy, and tilt of the magnetic axis relative to the
rotation axis. The time span of the 2019 optical epoch is encompassed in the
2019b near-infrared epoch.

###  3.5 Diagnosing the large-scale field using PCA

AD Leo is an ideal target for analysing large-scale field evolution with the
data-driven PCA method recently presented by Lehmann & Donati (2022), given
its magnetic field strength and $v_{e}\sin(i)$. Principal component analysis
allows us to uncover details about the stellar large-scale field directly from
the LSD Stokes $V$ profiles and to trace its magnetic field evolution across
the observation run, without prior assumptions. Here, we analyse only the
near-infrared time series, because the number of optical 2019 observations is
not sufficient.

First, we can get insights about the star’s axisymmetric large-scale field by
analysing the mean Stokes $V$ profile determined over all Stokes $V$ LSD
profiles (see Lehmann & Donati (2022) for further details). Fig. 11 displays
the mean profile and the decomposition into its antisymmetric and symmetric
parts, denoting the poloidal and toroidal axisymmetric components,
respectively. We clearly see that the mean profile is antisymmetric, which
indicates a poloidal-dominated axisymmetric large-scale field. The amplitude
of the symmetric part is comparable to the noise, and likely due to an
artefact of uneven phase coverage rather than a true toroidal field signal
(Lehmann & Donati, 2022). Compared to the mean-subtracted Stokes $V$ profiles,
the amplitude of the mean profile is generally strong, marking a dominant
axisymmetric field. However, we observe an increase in the amplitude of the
mean-subtracted Stokes $V$ in the last two epochs 2020a and 2020b, which
provides a first hint towards a less axisymmetric configuration.

Second, the application of PCA to the mean-subtracted Stokes $V$ profiles
yields insights on the non-axisymmetric field, (Lehmann & Donati, 2022). For
the mean-subtracted Stokes $V$ profiles, we applied the mean profile computed
across all epochs, which allows a direct reflection of the epoch-to-epoch
variations in PCA coefficients (e.g. in amplitude and mean value). If the mean
Stokes $V$ profile were computed per epoch, we would miss such information,
that is to say the mean value of the coefficients would be centred for each
epoch, and the amplitudes could no longer be compared to each other. Fig. 12
presents the first three eigenvectors and their corresponding coefficients for
the mean-subtracted Stokes $V$ profiles separated by epoch and colour-coded by
rotation cycle. The first eigenvector displays an antisymmetric shape
proportional to the first derivative of the Stokes $I$ profile and, together
with the associated coefficient, scales mainly with the longitudinal magnetic
field (Landi Degl’Innocenti & Landolfi, 2004). The second eigenvector shows a
more symmetric shape, more closely related to the second derivative of the
Stokes $I$ profile, and describes the temporal evolution of the Stokes $V$
profiles between the maxima of the longitudinal field. According to Lehmann &
Donati (2022), a strongly antisymmetric eigenvector traces the radial
component and a symmetric eigenvector the azimuthal component for a dipole
dominated field that is strongly poloidal, which is the case for AD Leo. The
third eigenvector features a signal as well (antisymmetric, and related to the
third derivative of the Stokes $I$ profile), which is detectable due to the
high S/N of the data set, while the further eigenvectors are dominated by
noise. Seeing three eigenvectors indicates that even if the axisymmetric field
is likely to be dominant, we are able to detect and to analyse the non-
axisymmetric field in great detail.

The coefficients of the eigenvectors suggest an evolving large-scale field as
their trend changes for every epoch, see Fig. 12 2nd-5th row. In 2019a, the
coefficients related to the first eigenvector show only a flat distribution
around zero, implying a predominantly axisymmetric field. For the following
epochs, 2019b, 2020a and 2020b, we see sine-like trends of the first two
coefficients with rotational phase. The amplitude increases from epoch to
epoch, indicating a growing obliquity of the dipole-dominated large-scale
field. For the 2020b epoch, the obliquity becomes so large that the
coefficients of the third eigenvector start to show a sine-like trend as well,
which translates into a significant non-axisymmetric field.

Furthermore, the extremes of the coefficients associated to the antisymmetric
and symmetric profile (first and second eigenvector) for the same epoch
feature an apparent phase shift of $\approx 0.25$, which demonstrates that the
dipolar component is poloidal dominated with little to no toroidal
contribution (Lehmann & Donati, 2022). The extremes of the coefficients
related to the antisymmetric eigenvector locate the pointing phase of the
dipole (Lehmann & Donati, 2022). For the last three epochs, the maximum of
this coefficient occurs at a pointing phase of $\approx 0.3$ for the northern
pole of the dipole, and the sign of the eigenvector implies a negative
polarity. The extremes of the coefficients occur at the same rotational phase
throughout the whole observation run, designating a stable pointing phase of
the dipole, in agreement with the $B_{\ell}$ measurements (see middle panel of
Fig. 2).

By applying the PCA method on the time series of Stokes $V$ (Lehmann & Donati,
2022), we confirm that AD Leo features a dipolar large-scale field, whose
obliquity increased during the latest epochs (2020a and 2020b). As the large-
scale field became more non-axisymmetric, the pointing phase of the dipole
remained stable.

Figure 11: Decomposition of the Stokes $V$ mean profile of AD Leo SPIRou
observations. The mean profile (solid red line) and its antisymmetric (dash
dotted blue line) and symmetric (dashed yellow line) parts, related to the
poloidal and toroidal components of the axisymmetric large-scale field, are
shown.

Figure 12: Principal component analysis applied to the mean-subtracted Stokes
$V$ profiles. The first three eigenvectors (top row) and their corresponding
coefficients (2nd-4th row) are shown. The vertical line in the eigenvector
panels indicates the radial velocity offset for AD Leo, of approximately 12.4
km s-1, while the horizontal dotted line in all panels represents the null
line. The coefficients are displayed by epoch: 2019a (2nd row), 2019b (3rd
row), 2020a (4th row) and 2020b (5th row). They are phase folded using Eq. 1
and colour-coded by rotation cycle.

##  4 Achromaticity of the magnetic field

The impact of stellar magnetic activity on radial velocity measurements
features a chromatic dependence stemming from a combination of magnetic field
and spot temperature contrast (Reiners et al., 2013; Baroch et al., 2020).
Indeed, at near-infrared wavelengths the Zeeman broadening is expected to be
stronger, while starspots contribute less owing to a lower contrast with the
photosphere. The situation is reversed in the optical domain. For AD Leo,
recent work by Carmona et al. (2023) demonstrated the strong chromatic
behaviour of radial velocity jitter, the latter being significantly weaker in
the near-infrared domain than in optical. The combination of these effects
becomes increasingly important with the activity level of the star, since the
number of spots would be correspondingly larger (Reinhold et al., 2019), and
it could possibly result in distinct contributions to the magnetic field
strength, which can then be used to facilitate the modelling of stellar
activity.

Fast-rotating stars are expected to feature high active latitudes and large
polar spots (e.g., Cang et al., 2020), because the Coriolis force would
overcome the buoyancy force, making the flux tubes ascend parallel to the
stellar rotation axis (Schuessler & Solanki, 1992; Granzer et al., 2000).
There are some cases, however, in which fast rotation does not correlate with
the presence of a polar spot (Barnes et al., 2004; Morin et al., 2008a). The
fact that AD Leo is a moderate rotator observed nearly pole-on makes it an
interesting case to investigate whether longitudinal field measurements are
chromatic, reflecting the behaviour of an underlying spot.

Previous studies dedicated to the Sun have shown that the magnetic field
strength measured in individual lines varies significantly (Demidov et al.,
2008; Demidov & Balthasar, 2012), and differences between optical and near-
infrared domains have unveiled a dependence of the field strength on
atmospheric height: the field increases while going towards deeper internal
layers (Zayer et al., 1989; Solanki, 1993). For other stars, Valenti et al.
(1995) reported a chromatic difference in magnetic field strength for the
moderatively active K dwarf $\varepsilon$ Eri, but attributed its origin to
incomplete modelling of the spectral lines used for the Zeeman broadening
analysis. No wavelength dependence of the field strength was reported more
recently, neither for $\varepsilon$ Eri (Petit et al., 2021) nor for T Tauri
stars (Finociety et al., 2021). The same conclusion was reached by Bellotti et
al. (2022) when computing longitudinal field values for the active M dwarf EV
Lac using blue ($<550$ nm) and red ($>550$ nm) lines of an optical line list.

To investigate the longitudinal field chromaticity, we analyse the
contemporaneous observations taken with SPIRou and ESPaDOnS in November 2019.
We limit the LSD computation within successive wavelength bins of the line
mask, and evaluate the longitudinal field for each case. Including both
optical and near-infrared domains, we considered 11 subsets of lines in the
following ranges: [350,390], [390,430], [430,480], [480,550], [550,650],
[650,1100], [950,1100], [1100,1400], [1400,1600], [1600,1800], [1800,2500] nm.
The [650,1100] and [950, 1100] nm ranges represent the red end of ESPaDOnS
spectra and the blue end of SPIRou spectra, respectively. We adopt more
wavelength regions than those presented in Bellotti et al. (2022), allowing a
finer search of chromatic trends. The number of lines used varies between 100
and 1000 in the optical, and between 120 and 300 in the near-infrared (see
Fig. 16). In addition, we compute LSD using a 50-lines mask in the overlapping
wavelength region of ESPaDOnS and SPIRou spectra ([950,1050] nm).

Stokes $I$ and $V$ profiles were computed for the simultaneous SPIRou and
ESPaDOnS epochs, namely 2019b and 2019, respectively. To increase the S/N and
allow a more precise estimate of Bl, the profiles obtained with a specific
line list subset and belonging to the same epoch were co-added. This is
reasonable considering the marginal amplitude variation over the epochs
examined and the unchanged polarity of Stokes $V$. The longitudinal field was
then computed with Eq. 2 using the specific normalisation wavelength and Landé
factor of each line subset, and adapting the velocity integration range
according to the width of the co-added Stokes $V$ profile.

From Fig. 13, we observe no clear chromaticity of Bl. The distribution of
field strength is flat around $-$200 G with a total scatter of 20 G. Such
dispersion is mainly due to LSD computations with a low number of lines,
implying Stokes shapes more sensitive to variations in individual lines,
blends and residuals of telluric correction. For the same reason, some
profiles appear deformed and lead to evident outliers (see Fig.16). For
instance, the Bl value obtained from ESPaDOnS data in the spectral region
overlapping with SPIRou is 100 G weaker (in absolute value) than the Bl value
obtained from SPIRou data in the same wavelength region. This could be due to
the low S/N at the very red edge of ESPaDOnS.

The case of [390,430] nm leads to a field value of $-$750 G, despite the
Stokes profiles do not show a particular deformation. We attribute this
behaviour to an imprecise continuum normalisation of the spectra, likely due
to a challenging identification of the continuum level in the blue part of the
spectrum, where M dwarfs feature forests of spectral lines. The effect is a
smaller depth (and equivalent width) of the Stokes $I$ profile relative to the
other cases, which artificially increases the value of the field (in absolute
value). Overall, although the [350,390] and [390,430] nm bins contain more
than half of the lines in the optical mask, their weight in the LSD
computation is small (Kochukhov et al., 2010) making their effect in the
computation of Bl with the full mask negligible.

Figure 13: Test on Bl measurements chromaticity. Shown are Bl measurements
obtained with line mask subsets based on wavelength of the optical (circles)
and near-infrared (pluses) domain. The horizontal black lines indicate the
values of Bl computed with the full masks and the vertical dashed line
separates arbitrarily optical from near-infrared measurements. The error bars
are smaller than the symbol size. For visualisation purposes, the [350, 390]
nm wavelength bin is not shown since it leads to an outlier data point at
around $-$1 kG, and the [390, 430] nm bin yields a value at $-$750 G so it is
indicated with a downward arrow. Overall, no chromatic trend emerges from the
data. Figure 14: Dependence of the magnetic field strength on limb darkening.
A monotonic decreasing (in absolute value) trend is observed from large
(optical) to small (near-infrared) coefficient values. This effect is small,
and negligible compared to noise in real observations.

We repeated the same exercise for the other SPIRou epochs and found a similar
behaviour, the only difference being 2020b data points shifting upwards
because of the field global weakening. A possible implication of the lack of a
chromatic trend may be the absence of a polar spot for AD Leo. This would be
justified considering that other faster-rotating M dwarfs like V374 Peg (Morin
et al., 2008a) and HK Aqr (Barnes et al., 2004) do not show polar spots.

A potential source of chromaticity for Bl values may come from limb darkening.
This radial gradient in stellar brightness over the visible disk can be
expressed as a linear function of the angle between the line of sight and the
normal to a surface element ($\theta$)

$\frac{I}{I_{0}}=1-\varepsilon(1-\cos\theta),$ (4)

where $I_{0}$ is the brightness at disk centre ($\theta=0^{\circ}$) and
$\varepsilon$ is the limb darkening coefficient. Claret & Bloemen (2011) show
that $\varepsilon$ decreases with wavelength, being 0.7 in V band and 0.3 in H
band. The linear limb darkening law in Eq.4 is the one implemented in the ZDI
reconstruction (Folsom et al., 2018).

Owing to the stronger limb darkening in optical than in near-infrared, there
is the possibility of additional polarity cancellation in the latter domain,
which would lead to weaker field measurements. For the specific case of AD
Leo, the low stellar inclination makes the equator appear at the limb and
near-infrared observations would be more sensitive to this region. In
particular, the sign of large-scale dipolar magnetic field lines exiting the
pole would cancel out more with those at equator, compared to optical
observations.

To verify this, we 1) linearly interpolated the limb darkening coefficients in
Claret & Bloemen (2011) at the wavelengths examined for the thermal contrast
test (see Fig. 13), 2) synthesised Stokes profiles for the same coefficients
assuming an axisymmetric dipole of 1 kG seen pole-on (akin to AD Leo in 2019a)
and infinite S/N, and 3) computed the associated field values with Eq. 2. The
results are illustrated in Fig. 14. We observe a small (7%) weakening of the
field from optical to near-infrared, which is overwhelmed by noise in real
observations.

##  5 Discussion and conclusions

In this paper, we presented the results of an extended spectropolarimetric
monitoring of the active M dwarf AD Leo, using near-infrared observations
collected with SPIRou between 2019 and 2020 as part of the SLS survey. They
add to the previous optical data obtained with ESPaDOnS and Narval between
2006 and 2019, making the entire time series encompass approximately 14 yr. To
carry out our magnetic analysis, we computed the longitudinal magnetic field,
tracked the variations of the Stokes $I$ FWHM, modelled Zeeman broadening on
individual selected lines, reconstructed the large-scale field topology via
ZDI, and assessed axisymmetry variations by means of a novel PCA method.

Initially, Morin et al. (2008b) reported an axisymmetric, dipole-dominated
structure that was stable over one year; later, Lavail et al. (2018) pointed
out a large-scale weakening and small-scale enhancement of the field but no
variation in the geometry. We found strong evidence of a large-scale field
evolution, that is summarised as follows:

  1. 1.

The longitudinal magnetic field has weakened between 2006 and 2020, from
$-300$ to $-50$ G, with a rapid decrease of 100 G in the 2020b. The dipolar
longitudinal magnetic field evolved in the same time frame, starting from -850
G in 2006, reaching -560 G in 2016 and restoring back to -900 G in 2020.

  2. 2.

The FWHM of Stokes $I$ profiles does not show rotational modulation, but a
dispersion that may partly be due to short-term variability. The epoch-
averaged FWHM manifests a long-term variation both in optical and near-
infrared, being wider in 2019b and 2020a, and narrower in 2019a and 2020b. The
variations are enhanced when the Stokes profiles are computed with
magnetically-sensitive lines, as opposed to the insensitive ones. The near-
infrared data in particular feature a trend moderately correlated with Bl (in
absolute value).

  3. 3.

The magnetic flux estimated from the modelling of Zeeman broadening exhibits a
global increase over time, which is also correlated to the long-term trend of
the longitudinal magnetic field (in absolute value). Moreover the epoch-
averaged magnetic flux obtained for the near-infrared SPIRou time series
oscillates in a similar manner to the FWHM of Stokes $I$, demonstrating that
the latter is capable of tracing secular evolution of the total, unsigned
magnetic field.

  4. 4.

Zeeman-Doppler imaging reconstructions confirmed the same kind of topological
evolution, with the axisymmetric level decreasing to $60\%$ and the obliquity
between magnetic and rotation axis increasing to $38^{\circ}$. This already
found support by the enhanced intermittency of the amplitude of Stokes $V$
profiles in late 2020.

  5. 5.

The PCA method confirmed the predominantly poloidal and dipolar geometry of
the large-scale field, as well as a lower axisymmetry in 2020a and 2020b. In
addition, the pointing phase of the dipole remained stable during the
evolution.

  6. 6.

Measurements of the magnetic field strength are overall achromatic, since they
manifest only a marginal wavelength dependence due to limb darkening.

Our results altogether suggest that AD Leo may be entering a polarity reversal
phase of a long-term magnetic cycle, analogous to the solar one. The
combination of chromospheric activity studies and spectropolarimetric
campaigns show that some Sun-like stars may manifest magnetic cycles and
polarity reversals in phase with chromospheric cycles (Boro Saikia et al.,
2016; Jeffers et al., 2017), while others have a more complex behaviour where
very regular chromospheric oscillations have no straightforward polarimetric
counterpart (Boro Saikia et al., 2022).

Predicting when the polarity reversal may occur for AD Leo is not a trivial
task, as the Bl data set does not feature a clear minimum or maximum.
Recently, Fuhrmeister et al. (2023) did not report any evident trends from a
long-term campaign of chromospheric indexes, whereas previous studies based on
photometric observations reported either two co-existing timescales for
cycles, namely 7 yr and 2 yr (Buccino et al., 2014), or an individual one of
about 11 yr (Tuomi et al., 2018). However, these time scales are not
compatible with the variations in Bl observed over 14 yr. The axisymmetric
level of the large-scale topology is a more suitable proxy to track the cycle
(Lehmann et al., 2021), but we recorded its change only in the most recent
observations.

A comparison between the magnetic field evolution described here and that of
the radial velocity jitter obtained in Carmona et al. (2023) leads to a
puzzling situation. Carmona et al. (2023) show that radial velocity variations
in optical are essentially due to the presence of a spot and that this signal
has changed only slightly (in phase and amplitude, the latter varies from
25.6$\pm$0.3 m s-1 to 23.6$\pm$0.5 m s-1) between 2005 and 2021. Such radial
velocity signal is not detected in infrared with SPIRou, corroborating its
strong chromaticity and therefore its origin due to stellar activity Carmona
et al. (2023). The fact that the dipolar field evolution is disjointed from a
surface brightness evolution is not a surprise: Morin et al. (2008a) show that
the mainly-dipolar topology of V374 Peg did not correlate with the complex
brightness map reconstructed via Doppler imaging.

These considerations motivate long-term spectropolarimetric and velocimetric
campaigns of active M dwarfs. For AD Leo in particular, additional monitoring
is required to observe the polarity reversal and the cycle’s extremes, to
constrain a precise time scale. An extended temporal baseline could also give
more insight on the link between topological variations and high-energy
flaring events (e.g., Stelzer et al., 2022). At the same time, we could shed
more light on the relation between the evolution of the large-scale magnetic
field topology and the stability of the radial velocity jitter.

An additional detail we could infer about AD Leo’s magnetic field is the
helicity, which quantifies the linkage between poloidal and toroidal field
lines and thus describes the complexity of the magnetic topology (Lund et al.,
2020, 2021). For the Sun, Pipin et al. (2019) reported a temporal variation of
the value correlated to the magnetic cycle. Indeed, helicity maxima and minima
occur when the axis of symmetry of the poloidal and toroidal field components
are aligned and orthogonal, respectively.

For AD Leo, the fraction of toroidal energy is only a negligible fraction of
the total one, hence we should exert caution when deriving quantities from it.
Over time, we observe that the poloidal axisymmetric ($m=0$) mode maintains
$>$80% of the magnetic energy and features a drop to 45% in 2020b, while the
energy in the toroidal axisymmetric mode decreases from 30% to 6%. As a
result, the two components maintained an overall misaligned configuration, but
in the most recent epoch, the poloidal component became more aligned with the
toroidal one due to the axisymmetry decrease. Following the practical
visualisation of Lund et al. (2021), this evolution would correspond to an
increase in field helicity.

The existence of a magnetic cycle for AD Leo is in agreement with the
observational evidence of such phenomena for M dwarfs from radial velocity
exoplanet searches (Gomes da Silva et al., 2012; Lopez-Santiago et al., 2020).
In general, studies have shown that magnetic cycles introduce long-term
signals in radial velocity data sets that can dominate over planetary
signatures (Meunier et al., 2010; Meunier & Lagrange, 2019), as they modulate
the appearance and number of heterogeneities on the stellar surface. It is
therefore necessary to have an accurate constraint on the temporal variations
of the cycle, in order to remove its contamination and allow a more reliable
planetary detection and characterisation (Lovis et al., 2011; Costes et al.,
2021; Sairam & Triaud, 2022).

Furthermore, activity cycles modulate the stellar radiation output and winds
in which close-in planets are immersed (Yeo et al., 2014; Hazra et al., 2020).
This leads to a temporal variation in the planetary atmospheric stripping with
consequent alteration of the chemical properties and habitability (Lanza,
2013; McCann et al., 2019; Louca et al., 2023; Konings et al., 2022). Details
on the occurrence of the cycle extremes can thus inform the most suitable
interpretation framework and observing plans for missions dedicated to
transmission spectroscopy like $Ariel$ (Tinetti et al., 2021). At the same
time, periodic variations in the large-scale field geometry need to be
considered for an accurate and updated modelling of the low-frequency radio
emission discovered for M dwarfs (Callingham et al., 2021), which has been
recently proposed to potentially reveal the presence of close-in magnetic
planets (Vedantham et al., 2020; Kavanagh et al., 2021, 2022).

Finally, AD Leo may not be an isolated case. To verify this, it is essential
to explore the possibility for such cycles over a wider area of the stellar
parameter space, namely mass and rotation period.

###### Acknowledgements.

We acknowledge funding from the French National Research Agency (ANR) under
contract number ANR-18-CE31-0019 (SPlaSH). SB acknowledges funding from the
European Space Agency (ESA), under the visiting researcher programme. LTL
acknowledges funding from the European Research Council under the H2020
research & innovation programme (grant #740651 NewWorlds). XD and AC
acknoweldge for funding in the framework of the Investissements dAvenir
programme (ANR-15-IDEX-02), through the funding of the ‘Origin of Life’
project of the Univ. Grenoble-Alpes. OK acknowledges support by the Swedish
Research Council (grant agreement no. 2019-03548), the Swedish National Space
Agency, and the Royal Swedish Academy of Sciences. Based on observations
obtained at the Canada-France-Hawaii Telescope (CFHT) which is operated by the
National Research Council (NRC) of Canada, the Institut National des Sciences
de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of
France, and the University of Hawaii. The observations at the CFHT were
performed with care and respect from the summit of Maunakea which is a
significant cultural and historic site. We gratefully acknowledge the CFHT QSO
observers who made this project possible. This work has made use of the VALD
database, operated at Uppsala University, the Institute of Astronomy RAS in
Moscow, and the University of Vienna; Astropy, 12 a community-developed core
Python package for Astronomy (Astropy Collaboration et al., 2013, 2018); NumPy
(van der Walt et al., 2011); Matplotlib: Visualization with Python (Hunter,
2007); SciPy (Virtanen et al., 2020).

## References

  * Artigau et al. (2014) Artigau, É., Astudillo-Defru, N., Delfosse, X., et al. 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9149, Observatory Operations: Strategies, Processes, and Systems V, ed. A. B. Peck, C. R. Benn, & R. L. Seaman, 914905
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33
  * Babcock (1961) Babcock, H. W. 1961, ApJ, 133, 572
  * Bagnulo et al. (2009) Bagnulo, S., Landolfi, M., Landstreet, J. D., et al. 2009, PASP, 121, 993
  * Baliunas et al. (1995) Baliunas, S. L., Donahue, R. A., Soon, W. H., et al. 1995, ApJ, 438, 269
  * Barnes et al. (2004) Barnes, J. R., James, D. J., & Collier Cameron, A. 2004, MNRAS, 352, 589
  * Baroch et al. (2020) Baroch, D., Morales, J. C., Ribas, I., et al. 2020, A&A, 641, A69
  * Bellotti et al. (2022) Bellotti, S., Petit, P., Morin, J., et al. 2022, A&A, 657, A107
  * Bertaux et al. (2014) Bertaux, J. L., Lallement, R., Ferron, S., Boonne, C., & Bodichon, R. 2014, A&A, 564, A46
  * Blackwell-Whitehead et al. (2006) Blackwell-Whitehead, R. J., Lundberg, H., Nave, G., et al. 2006, MNRAS, 373, 1603
  * Boro Saikia et al. (2016) Boro Saikia, S., Jeffers, S. V., Morin, J., et al. 2016, A&A, 594, A29
  * Boro Saikia et al. (2022) Boro Saikia, S., Lüftinger, T., Folsom, C. P., et al. 2022, A&A, 658, A16
  * Boro Saikia et al. (2018) Boro Saikia, S., Marvin, C. J., Jeffers, S. V., et al. 2018, A&A, 616, A108
  * Buccino et al. (2014) Buccino, A. P., Petrucci, R., Jofré, E., & Mauas, P. J. D. 2014, ApJ, 781, L9
  * Callingham et al. (2021) Callingham, J. R., Vedantham, H. K., Shimwell, T. W., et al. 2021, Nature Astronomy, 5, 1233
  * Cang et al. (2020) Cang, T. Q., Petit, P., Donati, J. F., et al. 2020, A&A, 643, A39
  * Carleo et al. (2020) Carleo, I., Malavolta, L., Lanza, A. F., et al. 2020, A&A, 638, A5
  * Carmona et al. (2023) Carmona, A., Delfosse, X., Bellotti, S., et al. 2023, A&A, 674, A110
  * Chabrier & Baraffe (1997) Chabrier, G. & Baraffe, I. 1997, A&A, 327, 1039
  * Charbonneau (2010) Charbonneau, P. 2010, Living Reviews in Solar Physics, 7, 3
  * Claret & Bloemen (2011) Claret, A. & Bloemen, S. 2011, A&A, 529, A75
  * Cook et al. (2022) Cook, N. J., Artigau, É., Doyon, R., et al. 2022, PASP, 134, 114509
  * Costes et al. (2021) Costes, J. C., Watson, C. A., de Mooij, E., et al. 2021, MNRAS, 505, 830
  * Cristofari et al. (2023) Cristofari, P. I., Donati, J. F., Folsom, C. P., et al. 2023, MNRAS, 522, 1342
  * Demidov & Balthasar (2012) Demidov, M. L. & Balthasar, H. 2012, Sol. Phys., 276, 43
  * Demidov et al. (2008) Demidov, M. L., Golubeva, E. M., Balthasar, H., Staude, J., & Grigoryev, V. M. 2008, Sol. Phys., 250, 279
  * Donati (2003) Donati, J. F. 2003, in Astronomical Society of the Pacific Conference Series, Vol. 307, Solar Polarization, ed. J. Trujillo-Bueno & J. Sanchez Almeida, 41
  * Donati & Brown (1997) Donati, J. F. & Brown, S. F. 1997, A&A, 326, 1135
  * Donati et al. (2023) Donati, J. F., Cristofari, P. I., Finociety, B., et al. 2023, MNRAS[arXiv:2304.09642]
  * Donati et al. (2006) Donati, J. F., Howarth, I. D., Jardine, M. M., et al. 2006, MNRAS, 370, 629
  * Donati et al. (2020) Donati, J. F., Kouach, D., Moutou, C., et al. 2020, MNRAS, 498, 5684
  * Donati & Landstreet (2009) Donati, J. F. & Landstreet, J. D. 2009, ARA&A, 47, 333
  * Donati et al. (2008a) Donati, J. F., Morin, J., Petit, P., et al. 2008a, MNRAS, 390, 545
  * Donati et al. (2008b) Donati, J. F., Moutou, C., Farès, R., et al. 2008b, MNRAS, 385, 1179
  * Donati et al. (1997) Donati, J. F., Semel, M., Carter, B. D., Rees, D. E., & Collier Cameron, A. 1997, MNRAS, 291, 658
  * Fares et al. (2009) Fares, R., Donati, J. F., Moutou, C., et al. 2009, MNRAS, 398, 1383
  * Feiden et al. (2021) Feiden, G. A., Skidmore, K., & Jao, W.-C. 2021, ApJ, 907, 53
  * Finociety et al. (2021) Finociety, B., Donati, J. F., Klein, B., et al. 2021, MNRAS, 508, 3427
  * Folsom et al. (2018) Folsom, C. P., Bouvier, J., Petit, P., et al. 2018, MNRAS, 474, 4956
  * Folsom et al. (2016) Folsom, C. P., Petit, P., Bouvier, J., et al. 2016, MNRAS, 457, 580
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306
  * Fouqué et al. (2023) Fouqué, P., Martioli, E., Donati, J. F., et al. 2023, A&A, 672, A52
  * Fuhrmeister et al. (2023) Fuhrmeister, B., Czesla, S., Perdelwitz, V., et al. 2023, A&A, 670, A71
  * Gaia Collaboration et al. (2021) Gaia Collaboration, Smart, R. L., Sarro, L. M., et al. 2021, A&A, 649, A6
  * Gastine et al. (2013) Gastine, T., Morin, J., Duarte, L., et al. 2013, A&A, 549, L5
  * Gomes da Silva et al. (2012) Gomes da Silva, J., Santos, N. C., Bonfils, X., et al. 2012, A&A, 541, A9
  * Granzer et al. (2000) Granzer, T., Schüssler, M., Caligari, P., & Strassmeier, K. G. 2000, A&A, 355, 1087
  * Gregory et al. (2012) Gregory, S. G., Donati, J. F., Morin, J., et al. 2012, ApJ, 755, 97
  * Gustafsson et al. (2008) Gustafsson, B., Edvardsson, B., Eriksson, K., et al. 2008, A&A, 486, 951
  * Hale et al. (1919) Hale, G. E., Ellerman, F., Nicholson, S. B., & Joy, A. H. 1919, ApJ, 49, 153
  * Hathaway (2010) Hathaway, D. H. 2010, Living Reviews in Solar Physics, 7, 1
  * Haywood et al. (2022) Haywood, R. D., Milbourne, T. W., Saar, S. H., et al. 2022, ApJ, 935, 6
  * Hazra et al. (2020) Hazra, G., Vidotto, A. A., & D’Angelo, C. V. 2020, MNRAS, 496, 4017
  * Hébrard et al. (2016) Hébrard, É. M., Donati, J. F., Delfosse, X., et al. 2016, MNRAS, 461, 1465
  * Hill et al. (2019) Hill, C. A., Folsom, C. P., Donati, J. F., et al. 2019, MNRAS, 484, 5810
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90
  * Jeffers et al. (2017) Jeffers, S. V., Boro Saikia, S., Barnes, J. R., et al. 2017, MNRAS, 471, L96
  * Jeffers et al. (2022) Jeffers, S. V., Cameron, R. H., Marsden, S. C., et al. 2022, A&A, 661, A152
  * Jeffers et al. (2018) Jeffers, S. V., Mengel, M., Moutou, C., et al. 2018, MNRAS, 479, 5266
  * Johns-Krull et al. (1999) Johns-Krull, C. M., Valenti, J. A., & Koresko, C. 1999, ApJ, 516, 900
  * Kausch et al. (2015) Kausch, W., Noll, S., Smette, A., et al. 2015, A&A, 576, A78
  * Kavanagh et al. (2021) Kavanagh, R. D., Vidotto, A. A., Klein, B., et al. 2021, MNRAS, 504, 1511
  * Kavanagh et al. (2022) Kavanagh, R. D., Vidotto, A. A., Vedantham, H. K., et al. 2022, MNRAS, 514, 675
  * Kitchatinov et al. (2014) Kitchatinov, L. L., Moss, D., & Sokoloff, D. 2014, MNRAS, 442, L1
  * Klein et al. (2021) Klein, B., Donati, J.-F., Moutou, C., et al. 2021, MNRAS, 502, 188
  * Kochukhov (2021) Kochukhov, O. 2021, A&A Rev., 29, 1
  * Kochukhov et al. (2009) Kochukhov, O., Heiter, U., Piskunov, N., et al. 2009, in American Institute of Physics Conference Series, Vol. 1094, 15th Cambridge Workshop on Cool Stars, Stellar Systems, and the Sun, ed. E. Stempels, 124–129
  * Kochukhov & Lavail (2017) Kochukhov, O. & Lavail, A. 2017, ApJ, 835, L4
  * Kochukhov et al. (2010) Kochukhov, O., Makaganiuk, V., & Piskunov, N. 2010, A&A, 524, A5
  * Konings et al. (2022) Konings, T., Baeyens, R., & Decin, L. 2022, A&A, 667, A15
  * Landi Degl’Innocenti & Landolfi (2004) Landi Degl’Innocenti, E. & Landolfi, M. 2004, Polarization in Spectral Lines, Vol. 307
  * Landstreet (1988) Landstreet, J. D. 1988, ApJ, 326, 967
  * Lanza (2013) Lanza, A. F. 2013, A&A, 557, A31
  * Lavail et al. (2018) Lavail, A., Kochukhov, O., & Wade, G. A. 2018, MNRAS, 479, 4836
  * Lawler et al. (2013) Lawler, J. E., Guzman, A., Wood, M. P., Sneden, C., & Cowan, J. J. 2013, ApJS, 205, 11
  * Lehmann & Donati (2022) Lehmann, L. T. & Donati, J. F. 2022, MNRAS, 514, 2333
  * Lehmann et al. (2021) Lehmann, L. T., Hussain, G. A. J., Vidotto, A. A., Jardine, M. M., & Mackay, D. H. 2021, MNRAS, 500, 1243
  * Leighton (1969) Leighton, R. B. 1969, ApJ, 156, 1
  * Lopez-Santiago et al. (2020) Lopez-Santiago, J., Martino, L., Míguez, J., & Vázquez, M. A. 2020, AJ, 160, 273
  * Louca et al. (2023) Louca, A. J., Miguel, Y., Tsai, S.-M., et al. 2023, MNRAS, 521, 3333
  * Lovis et al. (2011) Lovis, C., Dumusque, X., Santos, N. C., et al. 2011, arXiv e-prints, arXiv:1107.5325
  * Lund et al. (2020) Lund, K., Jardine, M., Lehmann, L. T., et al. 2020, MNRAS, 493, 1003
  * Lund et al. (2021) Lund, K., Jardine, M., Russell, A. J. B., et al. 2021, MNRAS, 502, 4903
  * Maeder & Meynet (2000) Maeder, A. & Meynet, G. 2000, ARA&A, 38, 143
  * Mann et al. (2015) Mann, A. W., Feiden, G. A., Gaidos, E., Boyajian, T., & von Braun, K. 2015, ApJ, 804, 64
  * Maunder (1904) Maunder, E. W. 1904, MNRAS, 64, 747
  * McCann et al. (2019) McCann, J., Murray-Clay, R. A., Kratter, K., & Krumholz, M. R. 2019, ApJ, 873, 89
  * Meunier et al. (2010) Meunier, N., Desort, M., & Lagrange, A. M. 2010, A&A, 512, A39
  * Meunier & Lagrange (2019) Meunier, N. & Lagrange, A. M. 2019, A&A, 628, A125
  * Mignon et al. (2023) Mignon, L., Meunier, N., Delfosse, X., et al. 2023, arXiv e-prints, arXiv:2303.03998
  * Mittag et al. (2017) Mittag, M., Robrade, J., Schmitt, J. H. M. M., et al. 2017, A&A, 600, A119
  * Morgenthaler et al. (2011) Morgenthaler, A., Petit, P., Morin, J., et al. 2011, Astronomische Nachrichten, 332, 866
  * Morin (2012) Morin, J. 2012, in EAS Publications Series, Vol. 57, EAS Publications Series, ed. C. Reylé, C. Charbonnel, & M. Schultheis, 165–191
  * Morin et al. (2011) Morin, J., Delfosse, X., Donati, J. F., et al. 2011, in SF2A-2011: Proceedings of the Annual meeting of the French Society of Astronomy and Astrophysics, ed. G. Alecian, K. Belkacem, R. Samadi, & D. Valls-Gabaud, 503–508
  * Morin et al. (2008a) Morin, J., Donati, J. F., Forveille, T., et al. 2008a, MNRAS, 384, 77
  * Morin et al. (2008b) Morin, J., Donati, J. F., Petit, P., et al. 2008b, MNRAS, 390, 567
  * Morin et al. (2010) Morin, J., Donati, J. F., Petit, P., et al. 2010, MNRAS, 407, 2269
  * Morin et al. (2016) Morin, J., Hill, C. A., & Watson, C. A. 2016, in Astrophysics and Space Science Library, Vol. 439, Astronomy at High Angular Resolution, ed. H. M. J. Boffin, G. Hussain, J.-P. Berger, & L. Schmidtobreick, 223
  * Muheki et al. (2020) Muheki, P., Guenther, E. W., Mutabazi, T., & Jurua, E. 2020, A&A, 637, A13
  * Mullan & MacDonald (2001) Mullan, D. J. & MacDonald, J. 2001, ApJ, 559, 353
  * Namekata et al. (2020) Namekata, K., Maehara, H., Sasaki, R., et al. 2020, PASJ, 72, 68
  * O’Brian et al. (1991) O’Brian, T. R., Wickliffe, M. E., Lawler, J. E., Whaling, W., & Brault, J. W. 1991, Journal of the Optical Society of America B Optical Physics, 8, 1185
  * Parker (1955) Parker, E. N. 1955, ApJ, 122, 293
  * Petit et al. (2009) Petit, P., Dintrans, B., Morgenthaler, A., et al. 2009, A&A, 508, L9
  * Petit et al. (2008) Petit, P., Dintrans, B., Solanki, S. K., et al. 2008, MNRAS, 388, 80
  * Petit et al. (2002) Petit, P., Donati, J. F., & Collier Cameron, A. 2002, MNRAS, 334, 374
  * Petit et al. (2021) Petit, P., Folsom, C. P., Donati, J. F., et al. 2021, A&A, 648, A55
  * Petit et al. (2014) Petit, P., Louge, T., Théado, S., et al. 2014, PASP, 126, 469
  * Petrovay (2020) Petrovay, K. 2020, Living Reviews in Solar Physics, 17, 2
  * Pipin et al. (2019) Pipin, V. V., Pevtsov, A. A., Liu, Y., & Kosovichev, A. G. 2019, ApJ, 877, L36
  * Preston (1967) Preston, G. W. 1967, ApJ, 150, 547
  * Rachkovsky (1967) Rachkovsky, D. N. 1967, Izvestiya Ordena Trudovogo Krasnogo Znameni Krymskoj Astrofizicheskoj Observatorii, 37, 56
  * Reiners (2012) Reiners, A. 2012, Living Reviews in Solar Physics, 9, 1
  * Reiners & Basri (2009) Reiners, A. & Basri, G. 2009, A&A, 496, 787
  * Reiners et al. (2010) Reiners, A., Bean, J. L., Huber, K. F., et al. 2010, ApJ, 710, 432
  * Reiners et al. (2013) Reiners, A., Shulyak, D., Anglada-Escudé, G., et al. 2013, A&A, 552, A103
  * Reiners et al. (2022) Reiners, A., Shulyak, D., Käpylä, P. J., et al. 2022, A&A, 662, A41
  * Reinhold et al. (2019) Reinhold, T., Bell, K. J., Kuszlewicz, J., Hekker, S., & Shapiro, A. I. 2019, A&A, 621, A21
  * Robertson et al. (2013) Robertson, P., Endl, M., Cochran, W. D., & Dodson-Robinson, S. E. 2013, in American Astronomical Society Meeting Abstracts, Vol. 221, American Astronomical Society Meeting Abstracts #221, 423.05
  * Rosén et al. (2016) Rosén, L., Kochukhov, O., Hackman, T., & Lehtinen, J. 2016, A&A, 593, A35
  * Route (2016) Route, M. 2016, ApJ, 830, L27
  * Ryabchikova et al. (2015) Ryabchikova, T., Piskunov, N., Kurucz, R. L., et al. 2015, Phys. Scr, 90, 054005
  * Saar (1994) Saar, S. H. 1994, in Infrared Solar Physics, ed. D. M. Rabin, J. T. Jefferies, & C. Lindsey, Vol. 154, 493
  * Sairam & Triaud (2022) Sairam, L. & Triaud, A. H. M. J. 2022, MNRAS, 514, 2259
  * Sanderson et al. (2003) Sanderson, T. R., Appourchaux, T., Hoeksema, J. T., & Harvey, K. L. 2003, Journal of Geophysical Research (Space Physics), 108, 1035
  * Schuessler & Solanki (1992) Schuessler, M. & Solanki, S. K. 1992, A&A, 264, L13
  * Schwabe (1844) Schwabe, H. 1844, Astronomische Nachrichten, 21, 233
  * See et al. (2016) See, V., Jardine, M., Vidotto, A. A., et al. 2016, MNRAS, 462, 4442
  * See et al. (2015) See, V., Jardine, M., Vidotto, A. A., et al. 2015, MNRAS, 453, 4301
  * Semel (1989) Semel, M. 1989, A&A, 225, 456
  * Shkolnik et al. (2009) Shkolnik, E., Liu, M. C., & Reid, I. N. 2009, ApJ, 699, 649
  * Shulyak et al. (2017) Shulyak, D., Reiners, A., Engeln, A., et al. 2017, Nature Astronomy, 1, 0184
  * Shulyak et al. (2019) Shulyak, D., Reiners, A., Nagel, E., et al. 2019, A&A, 626, A86
  * Shulyak et al. (2014) Shulyak, D., Reiners, A., Seemann, U., Kochukhov, O., & Piskunov, N. 2014, A&A, 563, A35
  * Skilling & Bryan (1984) Skilling, J. & Bryan, R. K. 1984, MNRAS, 211, 111
  * Smette et al. (2015) Smette, A., Sana, H., Noll, S., et al. 2015, A&A, 576, A77
  * Solanki (1993) Solanki, S. K. 1993, Space Sci. Rev., 63, 1
  * Stelzer et al. (2022) Stelzer, B., Caramazza, M., Raetz, S., Argiroffi, C., & Coffaro, M. 2022, A&A, 667, L9
  * Stibbs (1950) Stibbs, D. W. N. 1950, MNRAS, 110, 395
  * Tanner et al. (2013) Tanner, J. D., Basu, S., & Demarque, P. 2013, ApJ, 767, 78
  * Tinetti et al. (2021) Tinetti, G., Eccleston, P., Haswell, C., et al. 2021, arXiv e-prints, arXiv:2104.04824
  * Tuomi et al. (2018) Tuomi, M., Jones, H. R. A., Barnes, J. R., et al. 2018, AJ, 155, 192
  * Unno (1956) Unno, W. 1956, PASJ, 8, 108
  * Valenti et al. (1995) Valenti, J. A., Marcy, G. W., & Basri, G. 1995, ApJ, 439, 939
  * van der Walt et al. (2011) van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science and Engineering, 13, 22
  * van Saders & Pinsonneault (2012) van Saders, J. L. & Pinsonneault, M. H. 2012, ApJ, 746, 16
  * Vedantham et al. (2020) Vedantham, H. K., Callingham, J. R., Shimwell, T. W., et al. 2020, Nature Astronomy, 4, 577
  * Vidotto & Cleary (2020) Vidotto, A. A. & Cleary, A. 2020, MNRAS, 494, 2417
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Burovski, E., et al. 2020, scipy/scipy: SciPy 1.5.3
  * Wade et al. (2001) Wade, G. A., Bagnulo, S., Kochukhov, O., et al. 2001, A&A, 374, 265
  * Wilson (1968) Wilson, O. C. 1968, ApJ, 153, 221
  * Wright et al. (2011) Wright, N. J., Drake, J. J., Mamajek, E. E., & Henry, G. W. 2011, ApJ, 743, 48
  * Yeo et al. (2014) Yeo, K. L., Krivova, N. A., & Solanki, S. K. 2014, Space Sci. Rev., 186, 137
  * Zacharias et al. (2013) Zacharias, N., Finch, C. T., Girard, T. M., et al. 2013, AJ, 145, 44
  * Zaire et al. (2022) Zaire, B., Jouve, L., Gastine, T., et al. 2022, MNRAS, 517, 3392
  * Zayer et al. (1989) Zayer, I., Solanki, S. K., & Stenflo, J. O. 1989, A&A, 211, 463
  * Zeeman (1897) Zeeman, P. 1897, Nature, 55, 347

##  Appendix A FWHM of Stokes I

In this appendix, the Stokes $I$ profiles computed using different line
selections are shown. We compare the width of the profiles obtained with the
full atomic mask, high-Landé factor (g${}_{\mathrm{eff}}>1.2$) lines and low-
Landé factor (g${}_{\mathrm{eff}}<1.2$) lines.

Figure 15: Median Stokes $I$ profiles computed for the four near-infrared
epochs: 2019a, 2019b, 2020a, and 2020b. The profiles are obtained using the
full mask (top) magnetically sensitive (middle) and insensitive (bottom) line
lists in LSD. Following the linear dependence of the Zeeman effect on the
Landé factor, high-geff lines are broader than low-geff lines on average. The
width of the profiles changes over time in the high-geff case, while it
remains reasonably constant in the low-geff case.

##  Appendix B Chromatic Stokes profiles

The various Stokes profiles computed with different wavelength-based line
lists for LSD are reported. The wavelength intervals of the line subsets are:
[350,390], [390,430], [430,480], [480,550], [550,650], [650,1100], [950,1100],
[1100,1400], [1400,1600], [1600,1800], [1800,2500] nm.

Figure 16: Series of Stokes profiles computed with subsets of the line list
used for LSD, as described in Sec.4. Every panel displays Stokes $I$ (top),
$V$ (middle) and $N$ (bottom) for each line subset and vertical dashed lines
indicate the adjusted velocity range over which Bl was estimated. The optical
profiles are obtained by stacking the ESPaDOnS observations from 2019, whereas
the near-infrared ones from SPIRou 2019b time series, for each panel. The
number of lines used in the LSD computation as well as the mean wavelength are
displayed.

##  Appendix C Zeeman broadening examples

Example plots of the posterior distributions from the Zeeman broadening MCMC
analysis are shown in Fig. 17 for a ESPaDOnS observation and in Fig. 18 for a
SPIRou observation. Summaries of the results of the MCMC analysis for each
epoch are provided in Tables 4 and 5.

Figure 17: Posterior distribution for parameters from the Zeeman broadening analysis of the ESPaDOnS observation on 24 Febuary 2016. The filling factor for a given magnetic field strength is $f$, and abundances are in $\log n_{X}/n_{\rm H}$ units. Figure 18: Posterior distribution for parameters from the Zeeman broadening analysis of the SPIRou observation on 3 November 2020.  Table 4: Parameters from the Zeeman broadening analysis, averaged for each epoch, for ESPaDOnS and Narval optical observations. Uncertainties are the standard deviations for each epoch. epoch | 2006 | 2007 | 2008 | 2011 | 2012 | 2016 | 2019
---|---|---|---|---|---|---|---
$v_{e}\sin(i)$ [km s-1] | 4.27$\pm$0.38 | 4.25$\pm$0.17 | 4.16$\pm$0.29 | 3.46$\pm$0.50 | 2.65$\pm$0.48 | 3.57$\pm$0.21 | 3.39$\pm$0.45
$[$Ti/H$]$ |  $-$6.97$\pm$0.01 |  $-$6.90$\pm$0.01 |  $-$6.93$\pm$0.02 |  $-$6.99$\pm$0.01 |  $-$7.00$\pm$0.01 |  $-$6.96$\pm$0.01 |  $-$6.95$\pm$0.01
$f_{0\rm kG}$ | 0.06$\pm$0.02 | 0.24$\pm$0.06 | 0.18$\pm$0.06 | 0.05$\pm$0.06 | 0.02$\pm$0.01 | 0.05$\pm$0.02 | 0.06$\pm$0.01
$f_{2\rm kG}$ | 0.46$\pm$0.01 | 0.32$\pm$0.07 | 0.39$\pm$0.07 | 0.60$\pm$0.06 | 0.52$\pm$0.02 | 0.44$\pm$0.01 | 0.41$\pm$0.01
$f_{4\rm kG}$ | 0.33$\pm$0.02 | 0.31$\pm$0.02 | 0.28$\pm$0.04 | 0.19$\pm$0.07 | 0.29$\pm$0.01 | 0.33$\pm$0.01 | 0.36$\pm$0.01
$f_{6\rm kG}$ | 0.09$\pm$0.01 | 0.06$\pm$0.01 | 0.08$\pm$0.01 | 0.11$\pm$0.05 | 0.09$\pm$0.01 | 0.11$\pm$0.01 | 0.09$\pm$0.01
$f_{8\rm kG}$ | 0.05$\pm$0.01 | 0.03$\pm$0.01 | 0.04$\pm$0.01 | 0.04$\pm$0.03 | 0.05$\pm$0.01 | 0.05$\pm$0.01 | 0.05$\pm$0.01
$f_{10\rm kG}$ | 0.04$\pm$0.01 | 0.02$\pm$0.01 | 0.02$\pm$0.01 | 0.03$\pm$0.03 | 0.02$\pm$0.01 | 0.04$\pm$0.01 | 0.04$\pm$0.01
$\sum Bf$ [kG] | 3.59$\pm$0.04 | 2.82$\pm$0.13 | 2.97$\pm$0.13 | 3.22$\pm$0.15 | 3.37$\pm$0.06 | 3.65$\pm$0.04 | 3.62$\pm$0.03
Table 5: Parameters from the Zeeman broadening analysis, averaged for each epoch, for SPIRou observations. Uncertainties are the standard deviations for each epoch. epoch | 2019a | 2019b | 2020a | 2020b
---|---|---|---|---
$v_{\mathrm{eq}}\sin(i)$ [km s-1] | 3.42$\pm$0.23 | 3.34$\pm$0.18 | 3.48$\pm$0.17 | 3.48$\pm$0.21
$[$Ti/H$]$ |  $-$6.78$\pm$0.02 |  $-$6.78$\pm$0.02 |  $-$6.79$\pm$0.02 |  $-$6.78$\pm$0.01
$[$Fe/H$]$ |  $-$4.20$\pm$0.01 |  $-$4.22$\pm$0.01 |  $-$4.20$\pm$0.03 |  $-$4.21$\pm$0.02
$f_{0\rm kG}$ | 0.06$\pm$0.02 | 0.05$\pm$0.02 | 0.05$\pm$0.02 | 0.06$\pm$0.01
$f_{1\rm kG}$ | 0.15$\pm$0.02 | 0.12$\pm$0.03 | 0.15$\pm$0.02 | 0.17$\pm$0.02
$f_{2\rm kG}$ | 0.20$\pm$0.04 | 0.23$\pm$0.05 | 0.19$\pm$0.04 | 0.19$\pm$0.04
$f_{3\rm kG}$ | 0.24$\pm$0.04 | 0.19$\pm$0.05 | 0.23$\pm$0.05 | 0.24$\pm$0.04
$f_{4\rm kG}$ | 0.12$\pm$0.04 | 0.16$\pm$0.04 | 0.14$\pm$0.03 | 0.13$\pm$0.03
$f_{5\rm kG}$ | 0.06$\pm$0.02 | 0.08$\pm$0.02 | 0.06$\pm$0.02 | 0.06$\pm$0.01
$f_{6\rm kG}$ | 0.05$\pm$0.01 | 0.05$\pm$0.02 | 0.05$\pm$0.01 | 0.04$\pm$0.01
$f_{7\rm kG}$ | 0.03$\pm$0.01 | 0.02$\pm$0.01 | 0.03$\pm$0.01 | 0.03$\pm$0.01
$f_{8\rm kG}$ | 0.02$\pm$0.01 | 0.02$\pm$0.01 | 0.02$\pm$0.01 | 0.02$\pm$0.01
$f_{9\rm kG}$ | 0.03$\pm$0.01 | 0.03$\pm$0.01 | 0.03$\pm$0.01 | 0.03$\pm$0.01
$f_{10\rm kG}$ | 0.03$\pm$0.01 | 0.03$\pm$0.01 | 0.04$\pm$0.01 | 0.03$\pm$0.01
$\sum Bf$ [kG] | 3.40$\pm$0.09 | 3.49$\pm$0.09 | 3.44$\pm$0.04 | 3.29$\pm$0.07

##  Appendix D Observing log

This appendix contains the journal of observations of AD Leo, for both optical
and near-infrared observations. It also includes all measurements of
longitduinal magnetic field and $\Sigma Bf$.

Table 6:  List of AD Leo observations collected with SPIRou. The columns are: (1 and 2) date and universal time of the observations, (3) rotational cycle of the observations found using Eq. 1, (4) exposure time of a polarimetric sequence, (5) signal-to-noise ratio at 1650 nm per spectral element, (6) RMS noise level of Stokes $V$ signal in units of unpolarised continuum. Date | UT | $n_{\mathrm{cyc}}$ | $t_{exp}$ | S/N | $\sigma_{\mathrm{LSD}}$
---|---|---|---|---|---
| [hh:mm:ss] |  | [s] |  | [$10^{-4}I_{c}$]
2019
April 15 | 06:11:02.35 | 0.00 | 4x61 | 151 | 1.9
April 16 | 12:03:07.61 | 0.56 | 4x61 | 130 | 1.8
April 18 | 09:00:59.77 | 1.40 | 4x61 | 138 | 1.9
April 19 | 10:36:14.55 | 1.88 | 4x61 | 143 | 1.8
April 20 | 08:56:19.44 | 2.29 | 4x61 | 143 | 1.8
April 21 | 05:42:43.43 | 2.68 | 4x61 | 165 | 1.5
April 22 | 08:58:10.67 | 3.19 | 4x61 | 147 | 1.9
April 23 | 07:01:28.73 | 3.60 | 4x61 | 154 | 1.9
April 24 | 06:04:06.74 | 4.03 | 4x61 | 147 | 1.9
April 25 | 09:41:45.90 | 4.55 | 4x61 | 152 | 1.9
April 26 | 08:13:21.75 | 4.97 | 4x61 | 162 | 1.9
April 27 | 08:17:48.22 | 5.42 | 4x61 | 139 | 2.1
May 01 | 08:59:11.25 | 7.23 | 4x61 | 153 | 1.8
May 15 | 06:11:38.01 | 13.45 | 4x61 | 165 | 1.6
June 13 | 06:30.59.49 | 26.46 | 4x61 | 186 | 2.1
June 14 | 05:44:29.79 | 26.89 | 4x61 | 193 | 2.0
June 15 | 06:04:31.16 | 27.35 | 4x61 | 192 | 2.1
June 16 | 05:44:52.21 | 27.79 | 4x61 | 173 | 1.8
June 17 | 06:08:46.48 | 28.25 | 4x61 | 150 | 1.8
June 19 | 05:47:00.42 | 29.14 | 4x61 | 175 | 1.5
June 21 | 06:18:29.14 | 30.05 | 4x61 | 169 | 1.7
October 16 | 15:31:29.14 | 82.69 | 4x61 | 180 | 1.4
October 31 | 15:32:56.51 | 89.41 | 4x61 | 169 | 1.6
November 01 | 15:22:10.08 | 89.86 | 4x61 | 159 | 1.5
November 02 | 15:36:52.90 | 90.31 | 4x61 | 170 | 1.4
November 03 | 14:53:22.07 | 90.75 | 4x61 | 151 | 1.5
November 04 | 15:43:18.06 | 91.21 | 4x61 | 137 | 1.9
November 05 | 15:33:34.93 | 91.66 | 4x61 | 164 | 1.3
November 06 | 15:37:27.47 | 92.10 | 4x61 | 190 | 1.5
November 07 | 15:01:11.73 | 92.54 | 4x61 | 165 | 1.5
November 09 | 14:03:02.38 | 93.42 | 4x61 | 116 | 1.6
November 10 | 15:51:32.68 | 93.90 | 4x61 | 151 | 1.5
November 13 | 14:47:09.89 | 95.22 | 4x61 | 201 | 1.8
November 14 | 14:13:58.20 | 95.66 | 4x61 | 197 | 1.4
December 05 | 15:12:55.68 | 105.10 | 4x61 | 116 | 1.4
December 05 | 15:35:03.10 | 105.10 | 4x61 | 68 | 1.5
December 07 | 15:08:39.83 | 105.99 | 4x61 | 133 | 2.2
December 08 | 14:29:19.93 | 106.43 | 4x61 | 181 | 1.8
December 09 | 14:21:48.74 | 106.88 | 4x61 | 193 | 1.5
December 10 | 13:15:39.48 | 107.31 | 4x61 | 194 | 1.5
December 11 | 14:58:07.44 | 107.79 | 4x61 | 189 | 3.0
December 12 | 14:33:27.45 | 108.23 | 4x61 | 190 | 3.6
2020
January 26 | 12:11:48.90 | 128.36 | 4x61 | 218 | 2.0
January 27 | 12:07:16.45 | 128.81 | 4x61 | 166 | 1.5
January 28 | 12:06:25.46 | 129.26 | 4x61 | 193 | 1.5
February 05 | 08:18:51.61 | 132.78 | 4x61 | 174 | 1.4
February 16 | 07:39:02.87 | 137.70 | 4x61 | 193 | 1.8
February 17 | 09:20:45.46 | 138.18 | 4x61 | 171 | 1.9
February 18 | 07:39:13.56 | 138.59 | 4x61 | 191 | 1.2
February 19 | 08:37:09.84 | 139.06 | 4x61 | 210 | 1.5
March 12 | 09:16:56.32 | 148.94 | 4x61 | 217 | 2.0
May 08 | 05:59:01.58 | 174.43 | 4x61 | 194 | 1.4
May 09 | 09:42:30.22 | 174.95 | 4x61 | 206 | 1.5
May 12 | 09:39:45.19 | 176.30 | 4x61 | 177 | 1.4
May 13 | 09:43:17.10 | 176.75 | 4x61 | 204 | 1.4
May 14 | 07:46:35.66 | 177.16 | 4x61 | 208 | 1.5
May 15 | 09:50:31.27 | 177.65 | 4x61 | 164 | 1.6
May 31 | 06:22:44.29 | 184.76 | 4x61 | 215 | 1.2
June 01 | 07:29:57.32 | 185.23 | 4x61 | 189 | 1.3
June 02 | 06:23:31.70 | 185.65 | 4x61 | 202 | 1.4
June 03 | 07:38:43.10 | 186.13 | 4x61 | 196 | 1.3
June 04 | 06:32:43.13 | 186.55 | 4x61 | 197 | 1.3
June 05 | 06:42:42.88 | 187.01 | 4x61 | 139 | 1.8
June 06 | 07:53:54.53 | 187.48 | 4x61 | 167 | 1.2
June 07 | 06:58:07.88 | 187.91 | 4x61 | 154 | 1.6
June 08 | 06:03:47.85 | 188.34 | 4x61 | 135 | 1.3
June 08 | 06:10:09.48 | 188.34 | 4x61 | 143 | 1.4
June 09 | 06:31:50.17 | 188.79 | 4x61 | 120 | 1.3
June 10 | 06:57:13.44 | 189.25 | 4x61 | 200 | 1.8
October 31 | 15:06:44.49 | 253.75 | 4x61 | 205 | 1.5
November 03 | 15:29:52.78 | 255.11 | 4x61 | 220 | 1.5
Table 6: continued. Table 7:  List of AD Leo observations collected with ESPaDOnS in 2019. The columns are: (1 and 2) date and universal time of the observations, (3) rotational cycle of the observations found using Eq. 1, (4) exposure time of a polarimetric sequence, (5) signal-to-noise ratio at 650 nm per spectral element, (6) RMS noise level of Stokes $V$ signal in units of unpolarised continuum. Date | UT | $n_{\mathrm{cyc}}$ | $t_{exp}$ | S/N | $\sigma_{\mathrm{LSD}}$
---|---|---|---|---|---
| [hh:mm:ss] |  | [s] |  | [$10^{-4}I_{c}$]
November 15 | 13:24:01.40 | 96.32 | 4x300 | 234 | 1.7
November 16 | 14:20:01.30 | 96.79 | 4x300 | 230 | 1.9
November 19 | 14:08:04.00 | 98.13 | 4x300 | 151 | 3.6
November 19 | 14:33:03.00 | 98.14 | 4x300 | 159 | 3.2
November 19 | 14:56:04.00 | 98.15 | 4x300 | 180 | 2.8
November 21 | 15:43:01.00 | 99.06 | 4x300 | 265 | 1.8
Table 8:  List of optical and near-infrared measurements of longitudinal magnetic field and magnetic flux. The columns are: (1) Heliocentric Julian date of the observation, (2) Bl with formal error bar (see Eq. 2), and (3) magnetic flux from Zeeman broadening modelling, when a reliable measurement was possible, and (4) the instrument used. HJD | Bl | $Bf$ | Instrument
---|---|---|---
$[-2450000]$ | [G] | [kG] |
3747.0876 |  $-$269.4 $\pm$ 26.4 | $3.65^{+0.07}_{-0.08}$ | ESPaDOnS
3748.8868 |  $-$272.4 $\pm$ 15.1 | $3.52^{+0.09}_{-0.10}$ | ESPaDOnS
3780.0705 |  $-$280.7 $\pm$ 13.1 | $3.55^{+0.10}_{-0.10}$ | ESPaDOnS
3895.8047 |  $-$291.2 $\pm$ 10.7 | $3.62^{+0.08}_{-0.09}$ | ESPaDOnS
3896.8124 |  $-$260.1 $\pm$ 7.6 | $3.61^{+0.08}_{-0.09}$ | ESPaDOnS
3897.8005 |  $-$266.7 $\pm$ 7.4 | $3.63^{+0.08}_{-0.09}$ | ESPaDOnS
3898.7785 |  $-$294.2 $\pm$ 8.1 | $3.58^{+0.08}_{-0.10}$ | ESPaDOnS
4127.5975 |  $-$294.6 $\pm$ 16.5 | $\ldots$ | Narval
4128.6088 |  $-$248.6 $\pm$ 10.8 | $2.95^{+0.16}_{-0.16}$ | Narval
4129.5717 |  $-$296.2 $\pm$ 11.8 | $2.85^{+0.21}_{-0.22}$ | Narval
4130.6084 |  $-$253.8 $\pm$ 9.4 | $2.65^{+0.21}_{-0.20}$ | Narval
4133.6312 |  $-$274.8 $\pm$ 10.4 | $\ldots$ | Narval
4134.6112 |  $-$271.7 $\pm$ 11.4 | $\ldots$ | Narval
4135.6217 |  $-$231.4 $\pm$ 10.4 | $\ldots$ | Narval
4136.5925 |  $-$294.9 $\pm$ 13.0 | $\ldots$ | Narval
4276.7715 |  $-$261.0 $\pm$ 8.2 | $\ldots$ | ESPaDOns
4485.5177 |  $-$290.1 $\pm$ 13.2 | $3.18^{+0.17}_{-0.20}$ | Narval
4489.5683 |  $-$248.6 $\pm$ 10.3 | $3.10^{+0.17}_{-0.18}$ | Narval
4492.5379 |  $-$285.3 $\pm$ 10.7 | $\ldots$ | Narval
4493.5486 |  $-$204.6 $\pm$ 10.6 | $3.03^{+0.18}_{-0.19}$ | Narval
4495.5611 |  $-$227.2 $\pm$ 12.6 | $2.81^{+0.19}_{-0.19}$ | Narval
4499.5675 |  $-$256.8 $\pm$ 11.1 | $\ldots$ | Narval
4501.5473 |  $-$288.6 $\pm$ 12.1 | $\ldots$ | Narval
4502.5475 |  $-$202.8 $\pm$ 9.7 | $\ldots$ | Narval
4506.5576 |  $-$218.9 $\pm$ 10.0 | $2.84^{+0.19}_{-0.20}$ | Narval
4508.5516 |  $-$265.5 $\pm$ 11.0 | $\ldots$ | Narval
4509.5564 |  $-$219.6 $\pm$ 12.2 | $2.90^{+0.18}_{-0.20}$ | Narval
4510.5523 |  $-$286.6 $\pm$ 15.3 | $\ldots$ | Narval
4511.5694 |  $-$200.2 $\pm$ 10.5 | $2.86^{+0.17}_{-0.17}$ | Narval
4512.5537 |  $-$296.7 $\pm$ 10.8 | $2.99^{+0.16}_{-0.16}$ | Narval
5896.7560 |  $-$249.3 $\pm$ 11.5 | $3.22^{+0.15}_{-0.16}$ | Narval
5934.6407 |  $-$247.0 $\pm$ 9.5 | $3.39^{+0.11}_{-0.12}$ | Narval
5935.6765 |  $-$225.9 $\pm$ 9.3 | $3.50^{+0.09}_{-0.11}$ | Narval
5936.6050 |  $-$241.7 $\pm$ 10.9 | $3.38^{+0.09}_{-0.10}$ | Narval
5937.7575 |  $-$254.6 $\pm$ 11.2 | $3.44^{+0.08}_{-0.10}$ | Narval
5938.6659 |  $-$243.9 $\pm$ 8.8 | $3.31^{+0.08}_{-0.10}$ | Narval
5939.6031 |  $-$234.2 $\pm$ 10.6 | $3.34^{+0.08}_{-0.10}$ | Narval
5940.6416 |  $-$203.8 $\pm$ 9.1 | $3.31^{+0.09}_{-0.09}$ | Narval
5941.6411 |  $-$232.7 $\pm$ 9.7 | $3.36^{+0.11}_{-0.13}$ | Narval
5942.6256 |  $-$220.5 $\pm$ 10.2 | $3.30^{+0.09}_{-0.10}$ | Narval
7435.7957 |  $-$177.0 $\pm$ 5.5 | $3.68^{+0.07}_{-0.08}$ | ESPaDOnS
7436.8831 |  $-$165.8 $\pm$ 5.3 | $3.67^{+0.08}_{-0.09}$ | ESPaDOnS
7441.8954 |  $-$181.4 $\pm$ 5.3 | $3.57^{+0.06}_{-0.07}$ | ESPaDOnS
7443.0074 |  $-$166.1 $\pm$ 5.3 | $3.60^{+0.09}_{-0.10}$ | ESPaDOnS
7447.9103 |  $-$160.7 $\pm$ 6.4 | $3.65^{+0.14}_{-0.16}$ | ESPaDOnS
7449.0011 |  $-$182.9 $\pm$ 7.7 | $3.66^{+0.11}_{-0.13}$ | ESPaDOnS
7449.9154 |  $-$154.6 $\pm$ 6.5 | $3.66^{+0.12}_{-0.15}$ | ESPaDOnS
7450.8328 |  $-$190.5 $\pm$ 6.9 | $3.70^{+0.09}_{-0.11}$ | ESPaDOnS
7495.8253 |  $-$192.9 $\pm$ 8.8 | $3.66^{+0.12}_{-0.15}$ | ESPaDOnS
7498.7442 |  $-$170.6 $\pm$ 5.8 | $3.68^{+0.09}_{-0.09}$ | ESPaDOnS
8588.7573 |  $-$194.7$\pm$17.2 | $3.40^{+0.08}_{-0.09}$ | SPIRou
8590.0016 |  $-$212.3$\pm$19.9 | $\ldots$ | SPIRou
8592.9410 |  $-$206.5$\pm$17.4 | $3.33^{+0.09}_{-0.11}$ | SPIRou
8593.8715 |  $-$202.5$\pm$16.5 | $3.37^{+0.11}_{-0.11}$ | SPIRou
8594.7370 |  $-$234.3$\pm$16.8 | $3.42^{+0.10}_{-0.11}$ | SPIRou
8595.8726 |  $-$211.7$\pm$16.4 | $3.47^{+0.10}_{-0.13}$ | SPIRou
8596.7915 |  $-$217.9$\pm$16.0 | $3.37^{+0.09}_{-0.09}$ | SPIRou
8597.7516 |  $-$219.1$\pm$18.6 | $3.33^{+0.10}_{-0.09}$ | SPIRou
8598.9026 |  $-$240.0$\pm$16.9 | $3.49^{+0.09}_{-0.09}$ | SPIRou
8599.8411 |  $-$263.6$\pm$17.6 | $3.36^{+0.09}_{-0.10}$ | SPIRou
8600.8441 |  $-$182.9$\pm$16.7 | $3.36^{+0.10}_{-0.11}$ | SPIRou
8604.8725 |  $-$190.5$\pm$16.3 | $3.27^{+0.11}_{-0.12}$ | SPIRou
8618.7549 |  $-$216.5$\pm$15.1 | $3.27^{+0.08}_{-0.09}$ | SPIRou
8647.7665 |  $-$217.0$\pm$12.9 | $3.26^{+0.09}_{-0.09}$ | SPIRou
8648.7341 |  $-$221.3$\pm$13.1 | $3.38^{+0.09}_{-0.09}$ | SPIRou
8649.7480 |  $-$230.0$\pm$15.3 | $3.23^{+0.10}_{-0.10}$ | SPIRou
8650.7343 |  $-$194.7$\pm$13.6 | $3.49^{+0.09}_{-0.10}$ | SPIRou
8651.7509 |  $-$209.4$\pm$16.0 | $3.49^{+0.08}_{-0.10}$ | SPIRou
8653.7357 |  $-$227.2$\pm$13.6 | $3.38^{+0.09}_{-0.09}$ | SPIRou
8655.7575 |  $-$250.3$\pm$20.3 | $3.42^{+0.09}_{-0.10}$ | SPIRou
8773.1472 |  $-$223.6$\pm$14.4 | $3.39^{+0.10}_{-0.10}$ | SPIRou
8788.1497 |  $-$223.7$\pm$13.4 | $3.32^{+0.09}_{-0.10}$ | SPIRou
8789.1423 |  $-$184.8$\pm$13.7 | $3.43^{+0.10}_{-0.09}$ | SPIRou
8790.1526 |  $-$245.2$\pm$15.4 | $3.50^{+0.09}_{-0.10}$ | SPIRou
8791.1225 |  $-$192.7$\pm$13.8 | $3.50^{+0.12}_{-0.12}$ | SPIRou
8792.1572 |  $-$256.3$\pm$16.3 | $3.46^{+0.09}_{-0.10}$ | SPIRou
8793.1506 |  $-$204.8$\pm$13.6 | $3.50^{+0.09}_{-0.10}$ | SPIRou
8794.1534 |  $-$246.5$\pm$13.5 | $3.53^{+0.09}_{-0.09}$ | SPIRou
8795.1283 |  $-$209.1$\pm$15.4 | $3.48^{+0.09}_{-0.09}$ | SPIRou
8797.0881 |  $-$189.3$\pm$25.7 | $3.51^{+0.10}_{-0.09}$ | SPIRou
8798.1635 |  $-$202.8$\pm$17.8 | $3.63^{+0.10}_{-0.10}$ | SPIRou
8801.1190 |  $-$254.3$\pm$14.6 | $3.56^{+0.11}_{-0.12}$ | SPIRou
8802.0961 |  $-$186.1$\pm$14.1 | $3.58^{+0.08}_{-0.09}$ | SPIRou
8803.0577 |  $-$208.3$\pm$7.0 | $3.67^{+0.13}_{-0.14}$ | ESPaDOnS
8804.0967 |  $-$189.1$\pm$7.2 | $3.58^{+0.10}_{-0.11}$ | ESPaDOnS
8807.0890 |  $-$178.9$\pm$12.9 | $3.65^{+0.07}_{-0.09}$ | ESPaDOnS
8807.1062 |  $-$198.8$\pm$11.5 | $3.61^{+0.08}_{-0.09}$ | ESPaDOnS
8807.1223 |  $-$169.3$\pm$10.1 | $3.63^{+0.07}_{-0.07}$ | ESPaDOnS
8809.1547 |  $-$161.1$\pm$6.4 | $3.57^{+0.08}_{-0.09}$ | ESPaDOnS
8823.1385 |  $-$254.6$\pm$30.7 | $3.48^{+0.08}_{-0.09}$ | SPIRou
8823.1539 |  $-$242.9$\pm$42.2 | $3.69^{+0.10}_{-0.09}$ | SPIRou
8825.1357 |  $-$193.2$\pm$22.1 | $3.60^{+0.08}_{-0.08}$ | SPIRou
8826.1084 |  $-$236.2$\pm$14.4 | $3.41^{+0.09}_{-0.09}$ | SPIRou
8827.1033 |  $-$178.9$\pm$13.7 | $3.42^{+0.09}_{-0.09}$ | SPIRou
8828.0574 |  $-$249.2$\pm$13.9 | $3.40^{+0.11}_{-0.11}$ | SPIRou
8829.1286 |  $-$152.6$\pm$14.4 | $3.44^{+0.10}_{-0.10}$ | SPIRou
8830.1115 |  $-$248.1$\pm$14.8 | $3.35^{+0.10}_{-0.09}$ | SPIRou
8875.0136 |  $-$176.0$\pm$12.2 | $3.43^{+0.09}_{-0.10}$ | SPIRou
8876.0104 |  $-$118.6$\pm$13.3 | $3.54^{+0.08}_{-0.09}$ | SPIRou
8877.0098 |  $-$238.7$\pm$14.5 | $3.40^{+0.10}_{-0.11}$ | SPIRou
8884.8515 |  $-$122.9$\pm$12.6 | $3.42^{+0.10}_{-0.11}$ | SPIRou
8895.8233 |  $-$132.6$\pm$11.9 | $3.40^{+0.10}_{-0.10}$ | SPIRou
8896.8939 |  $-$219.3$\pm$13.7 | $3.44^{+0.11}_{-0.10}$ | SPIRou
8897.8233 |  $-$157.3$\pm$12.7 | $3.43^{+0.09}_{-0.11}$ | SPIRou
8898.8635 |  $-$216.3$\pm$13.3 | $3.44^{+0.10}_{-0.11}$ | SPIRou
8920.8895 |  $-$176.4$\pm$12.6 | $3.42^{+0.09}_{-0.10}$ | SPIRou
8977.7467 |  $-$177.7$\pm$11.8 | $3.27^{+0.09}_{-0.10}$ | SPIRou
8978.9018 |  $-$114.9$\pm$11.5 | $3.33^{+0.09}_{-0.08}$ | SPIRou
8981.8996 |  $-$214.6$\pm$14.0 | $3.28^{+0.10}_{-0.11}$ | SPIRou
8982.9020 |  $-$81.2$\pm$11.5 | $3.43^{+0.11}_{-0.10}$ | SPIRou
8983.8209 |  $-$190.0$\pm$11.2 | $3.27^{+0.08}_{-0.09}$ | SPIRou
8984.9069 |  $-$100.5$\pm$16.8 | $3.44^{+0.11}_{-0.11}$ | SPIRou
9000.7614 |  $-$50.4$\pm$10.5 | $3.26^{+0.12}_{-0.11}$ | SPIRou
9001.8080 |  $-$217.7$\pm$17.7 | $3.32^{+0.09}_{-0.10}$ | SPIRou
9002.7618 |  $-$61.7$\pm$11.9 | $3.20^{+0.09}_{-0.11}$ | SPIRou
9003.8140 |  $-$199.2$\pm$12.7 | $3.18^{+0.10}_{-0.11}$ | SPIRou
9004.7681 |  $-$110.4$\pm$11.8 | $3.26^{+0.09}_{-0.09}$ | SPIRou
9005.7750 |  $-$96.4$\pm$15.5 | $3.30^{+0.09}_{-0.09}$ | SPIRou
9006.8244 |  $-$119.8$\pm$14.9 | $3.30^{+0.09}_{-0.10}$ | SPIRou
9007.7856 |  $-$70.0$\pm$15.1 | $3.30^{+0.10}_{-0.10}$ | SPIRou
9008.7478 |  $-$178.4$\pm$16.2 | $3.28^{+0.10}_{-0.10}$ | SPIRou
9008.7522 |  $-$185.2$\pm$16.6 | $3.30^{+0.09}_{-0.10}$ | SPIRou
9009.7672 |  $-$64.5$\pm$18.5 | $3.35^{+0.10}_{-0.11}$ | SPIRou
9010.7848 |  $-$194.7$\pm$13.2 | $3.17^{+0.10}_{-0.09}$ | SPIRou
9154.1315 |  $-$62.2$\pm$10.1 | $3.31^{+0.07}_{-0.07}$ | SPIRou
9157.1479 |  $-$46.1$\pm$9.9 | $3.28^{+0.08}_{-0.09}$ | SPIRou
Table 8: continued.