11institutetext:  Leiden Observatory, Leiden University, PO Box 9513, 2300 RA
Leiden, The Netherlands
11email<EMAIL_ADDRESS>22institutetext: Institut de Recherche en
Astrophysique et Planétologie, Université de Toulouse, CNRS, IRAP/UMR 5277, 14
avenue Edouard Belin, F-31400, Toulouse, France 33institutetext: Science
Division, Directorate of Science, European Space Research and Technology
Centre (ESA/ESTEC), Keplerlaan 1, 2201 AZ, Noordwijk, The Netherlands
44institutetext: Laboratoire Univers et Particules de Montpellier, Université
de Montpellier, CNRS, F-34095, Montpellier, France 55institutetext: University
of Vienna, Department of Astrophysics, Türkenschanzstrasse 17, A-1180 Vienna,
Austria 66institutetext: INAF - Osservatorio Astrofisico di Arcetri, Largo E.
Fermi 5, 50125, Firenze, Italy 77institutetext: INAF - Osservatorio
Astronomico di Palermo, Piazza del Parlamento 1, 90134, Palermo, Italy

# Spectropolarimetric characterisation of exoplanet host stars in preparation
of the $Ariel$ mission

Magnetic environment of HD 63433

S. Bellotti 1122 0000-0002-2558-6920 D. Evensberget 11 0000-0001-7810-8028 A.
A. Vidotto 11 0000-0001-5371-2675 A. Lavail 22 0000-0001-8477-5265 T.
Lüftinger 33 0009-0000-4946-6942 G. A. J. Hussain 33 0000-0003-3547-3783 J.
Morin 44 0000-0002-4996-6901 P. Petit 22 0000-0001-7624-9222 S. Boro Saikia 55
0000-0002-3673-3746 C. Danielski 66 0000-0002-3729-2663 G. Micela 77
0000-0002-9900-4751

(Received ; accepted )

###### Abstract

Context. The accurate characterisation of the stellar magnetism of planetary
host stars has been gaining momentum, especially in the context of
transmission spectroscopy investigations of exoplanets. Indeed, the magnetic
field regulates the amount of energetic radiation and stellar wind impinging
on planets, as well as the presence of inhomogeneities on the stellar surface
that hinder the precise extraction of the planetary atmospheric absorption
signal.

Aims. We initiated a spectropolarimetric campaign to unveil the magnetic field
properties of known exoplanet hosting stars included in the current list of
potential $Ariel$ targets. In this work, we focus on HD 63433, a young solar-
like star hosting two sub-Neptunes and an Earth-sized planet. These exoplanets
orbit within 0.15 au from the host star and have likely experienced different
atmospheric evolutionary paths.

Methods. We analysed optical spectropolarimetric data collected with ESPaDOnS,
HARPSpol, and Neo-Narval to compute the magnetic activity indices ($\log
R^{\prime}_{\mathrm{HK}}$, H$\alpha$, and Ca ii infrared triplet), measure the
longitudinal magnetic field, and reconstruct the large-scale magnetic topology
via Zeeman-Doppler imaging (ZDI). The magnetic field map was then employed to
simulate the space environment in which the exoplanets orbit.

Results. The reconstructed stellar magnetic field has an average strength of
24 G and it features a complex topology with a dominant toroidal component, in
agreement with other stars of a similar spectral type and age. Our simulations
of the stellar environment locate 10% of the innermost planetary orbit inside
the Alfvén surface and, thus, brief magnetic connections between the planet
and the star can occur. The outer planets are outside the Alfvén surface and a
bow shock between the stellar wind and the planetary magnetosphere could
potentially form.

###### Key Words.:

Stars: magnetic field – Stars: individual: HD 63433 – Stars: activity –
Techniques: polarimetric

##  1 Introduction

The field of exoplanetology has flourished significantly over the past three
decades. With more than $5,500$ confirmed detections111www.exoplanet.eu,
December 2023, the main goal is to progressively focus on a comprehensive
characterisation of the planetary population. Our knowledge of planet
formation and evolution will indeed progress by inspecting the chemical
variety of exoplanet atmospheres and, in turn, it will allow us to constrain
planet formation theories and refine the assessment framework of potential
biomarkers. These considerations motivated the deployment of space-based
missions currently in operation such as $JWST$ (Gardner et al., 2006), as well
as future ones such as $Ariel$ (Tinetti et al., 2021), which is a medium-class
science mission planned for launch by the European Space Agency in 2029. It
will target a comprehensive and diverse sample of telluric, ocean, and gas
giant planets discovered primarily via radial velocities and transit
photometry, orbiting around stars of all accessible spectral types, from early
A to late M (Zingales et al., 2018; Edwards et al., 2019; Edwards & Tinetti,
2022) types.

When it comes to atmospheric characterisation surveys of exoplanets, stellar
magnetic activity is a crucial aspect that ought to be taken into account to
avoid wrongful or biased interpretations of the planetary nature. In fact,
stellar activity is responsible for photospheric inhomogeneities such as spots
and faculae, to which the optical and near-infrared (NIR) are sensitive within
the spectral ranges commonly examined for exoplanet detection and
characterisation. These inhomogeneities contaminate radial velocity signals
(e.g. Huélamo et al., 2008; Dumusque et al., 2011), either mimicking a
planetary signal (e.g. Carmona et al., 2023) or preventing an accurate
estimation of the planetary mass, ultimately affecting a correct atmospheric
retrieval (Changeat et al., 2020; Di Maio et al., 2023). In the case of
transmission spectroscopy, stellar activity can bias the observed spectrum,
causing the transit chord to differ from the disk-integrated stellar spectrum,
generating ambiguity in the extraction of the planetary atmospheric absorption
signal (e.g. Rackham et al., 2018, 2019; Salz et al., 2018). For instance,
spots affect the transit depth with at least the same order of magnitude as
transiting exoplanets, potentially emulating the presence of specific chemical
species like H2O (Rackham et al., 2018). Therefore, designing activity-
filtering techniques for transmission spectroscopy is necessary (Cracchiolo et
al., 2021; Thompson et al., 2024).

In addition, magnetic activity shapes the environment in which exoplanets
orbit, and it affects their atmospheres, especially during the early stages of
their formation (e.g. Ribas et al., 2005; Allan & Vidotto, 2019; Ketzer &
Poppenhaeger, 2023). Strong activity can endanger habitability (e.g.,
Airapetian et al., 2017; Tilley et al., 2019), as it indeed correlates with
energetic phenomena, such as flares, ultimately modifying the atmospheric
chemical composition (Segura et al., 2010; Guenther & Kislyakova, 2020;
Konings et al., 2022; Louca et al., 2023). Ionising radiation (e.g. in the
extreme-ultraviolet and X-rays) and particles from strong stellar winds induce
and shape hydrodynamic escape of the atmosphere, stripping it away over time
(Lammer et al., 2003; Ribas et al., 2016; Carolan et al., 2021; Hazra et al.,
2020). XUV radiation also impacts secondary atmospheres, such as the case of
Trappist-1 (Van Looveren et al., 2024). Moreover, Chavez et al. (2023) show
that the cloud coverage on Neptune is temporally correlated with the solar
magnetic cycle. For these reasons, there is increased attention towards a
comprehensive and homogeneous characterisation of exoplanet hosts (e.g.
Danielski et al., 2022; Magrini et al., 2022) and their magnetic activity
(Fares et al., 2013; Mengel et al., 2016; Fares et al., 2017; Folsom et al.,
2020). A recent example is given by our work on GJ 436 and its orbiting hot
Neptune (Bellotti et al., 2023a; Vidotto et al., 2023).

To obtain a complete picture of the magnetic activity of stars in the current
list of potential $Ariel$ targets (Edwards et al., 2019; Edwards & Tinetti,
2022), we initiated a dedicated spectropolarimetric programme divided into
three steps: 1) a snapshot campaign to assess the detectability of the large-
scale magnetic field and optimise further observations; 2) an observing
campaign to reconstruct the topology of the surface stellar magnetic field;
and 3) a long-term monitoring to constrain the evolution and variability of
the field. Spectropolarimetry enables us to achieve these goals effectively,
as we detect the polarisation properties of the Zeeman-split components of
spectral lines (Zeeman, 1897). From the time series of polarised spectra, we
mapped the large-scale magnetic field with Zeeman-Doppler imaging (ZDI; Semel
1989; Donati et al. 1997), constraining its configuration and main features.
This technique has been applied extensively in spectropolarimetric studies,
revealing a variety of field geometries for cool stars (e.g. Petit et al.,
2005; Hussain et al., 2016; Morin et al., 2008b, 2010) and long-term evolution
(Boro Saikia et al., 2016, 2018; Bellotti et al., 2023b).

This paper focuses on HD 63433, a young, active solar-like star which is part
of the current list of potential $Ariel$ targets and hosts two transiting mini
Neptunes (planets b and c orbiting at 0.07 and 0.15 AU; Mann et al. 2020;
Mallorquín et al. 2023; Damasso et al. 2023). As reported by transmission
spectroscopy studies of their atmospheres (Zhang et al., 2022), both planets
do not exhibit helium absorption, whereas only planet c shows Ly$\alpha$
absorption. Considering the architecture of the system, along with the age and
activity of the star, this suggests that HD 63433 b has probably lost most of
its primordial H/He atmosphere, while HD 63433 c is still experiencing
atmospheric evaporation. These results make the HD 63433 system an attractive
benchmark for comparative studies of distinct atmospheric evolution tracks and
their timescales. Recently, a third planet with an orbital period of 4.2 d was
announced (Capistrant et al., 2024) and its proximity to the star makes it a
potentially interesting target for star-planet interaction studies.

The paper is structured as follows. In Sect. 2, we describe the observations
obtained with ESPaDOnS and HARPS-Pol, and the subsequent Neo-Narval
monitoring. In Sect. 3, we outline the computation of the activity proxies,
longitudinal magnetic field, and their correlation analysis. We present in
Sect. 4 the large-scale magnetic field reconstruction by means of ZDI, and in
Sect. 5 the ZDI-driven stellar wind models. Finally, we draw our conclusions
in Sect. 6.

##  2 Observations

HD 63433 is a G5 dwarf at a distance of 22.34$\pm$0.04 pc (Gaia Collaboration,
2020). It is a bright ($V=6.9$) star belonging to the Ursa Major moving group,
with an estimated age of $414\pm 23$ Myr (Gagné et al., 2018; Mann et al.,
2020). The star has a rotation period of 6.45 d, and is magnetically active,
with a reported chromospheric activity index $\log
R^{\prime}_{\mathrm{HK}}=-4.39\pm 0.05$ and an X-ray-to-bolometric luminosity
index $\log(L_{X}/L_{\mathrm{bol}})=-4.04\pm 0.13$ (Mann et al., 2020). The
$L_{X}/L_{\mathrm{bol}}$ is two orders of magnitude larger than the solar
value (Wright et al., 2011), while the $\log R^{\prime}_{\mathrm{HK}}$ is 0.5
dex larger (Egeland et al., 2017). A summary of the fundamental parameters of
HD 63433 is given in Table 1. Two sub-Neptunes were discovered to orbit the
star via photometric transits (Mann et al., 2020): HD 63433 b with a radius
and orbital period of 2.02 R⊕ and 7.11 d, and HD 63433 c, with a radius of
2.44 R⊕ and on a 20.55 d orbit. Radial velocity follow-up of this system by
Damasso et al. (2023) resulted in planetary mass upper limits of 11 M⊕ for
planet b and 31 M⊕ for planet c. The work of Mallorquín et al. (2023) set a
mass of 15.5 M⊕ for planet c and an upper limit of 22 M⊕ for planet b. A third
planet was also reported recently via photometric transit, with a radius of
1.073 R⊕ and orbital period of 4.2 d (Capistrant et al., 2024).

###  2.1 ESPaDOnS

We obtained one observation with
ESPaDOnS222https://www.cfht.hawaii.edu/Instruments/Spectroscopy/Espadons/https://www.cfht.hawaii.edu/Instruments/Spectroscopy/Espadons/
on 12 September 2022, within the snapshot campaign (22BF23, P.I. Bellotti S.)
of the spectropolarimetric characterisation programme (see Table 2). ESPaDOnS
is the optical spectropolarimeter on the 3.6 m CFHT located atop Mauna Kea in
Hawaii, operating between 370 and 1050 nm with a spectral resolving power of
$R=65,000$ (Donati, 2003). A polarimetric sequence is obtained from four
consecutive sub-exposures. Each sub-exposure is taken with a different
rotation of the retarder waveplate of the polarimeter relative to the optical
axis. In circular polarisation mode, the output is given by the total
intensity spectrum (Stokes $I$) and the circularly polarised spectrum (Stokes
$V$). The Stokes $I$ spectrum is computed by summing the four sub-exposures,
while the Stokes $V$ spectrum from the ratio of sub-exposures with orthogonal
polarisation states. Circular polarisation mode is more appropriate for
snapshot campaigns of main sequence cool dwarf stars, since the amplitude of a
Zeeman signature is larger than in linear polarisation (Landi Degl’Innocenti,
1992). The data reduction was performed with the LIBRE-ESPRIT pipeline (Donati
et al., 1997), and the maximum signal-to-noise ratio (S/N) per CCD pixel of
the circularly polarised spectrum is 735.

We applied least-squares deconvolution (LSD) (Donati et al., 1997; Kochukhov
et al., 2010) to compute the Stokes $I$ and $V$ mean-line profiles. Both the
unpolarised and polarised spectra are deconvolved with a line mask, which is a
list of photospheric absorption lines for the specific spectral type examined.
Least-squares deconvolution results in high-S/N line profiles encapsulating
the average information of thousands of lines. We used a line mask synthesised
using the Vienna Atomic Line Database333http://vald.astro.uu.se/ (VALD,
Ryabchikova et al., 2015), characterised by $\log g=$ 4.5 [cm s-2],
$v_{\mathrm{micro}}=$ 1 km s-1, and $T_{\mathrm{eff}}=5500$ K. The mask
includes 4620 atomic lines between 360–1080 nm, with known sensitivity to the
Zeeman effect (also known as Landé factor and indicated as $g_{\mathrm{eff}}$)
and an absorption depth greater than 40 % of the continuum level. This value
is chosen to obtain a larger effective S/N of the LSD profiles (Moutou et al.,
2007). The normalisation wavelength and the Landé factor for LSD are 700 nm
and 1.0, respectively.

The Stokes LSD profiles for the snapshot observation are shown in Fig. 1.
Owing to an LSD multiplex gain of about 30, the S/N per 1.8 km s-1 velocity
bin of the Stokes $V$ profile is $\sim$20,000. We note an evident Zeeman
signature in Stokes $V$ within $\pm$17 km s-1 from line centre at $-15.9$ km
s-1, and with an associated false-alarm probability lower than $10^{-5}$,
indicating a reliable magnetic detection (Donati et al., 1997). The
polarimetric pipeline provides the null spectrum (Stokes $N$) as well,
obtained by dividing sub-exposures with the same polarisation state. We
computed the associated LSD profile, with which we can assess the presence of
spurious polarisation signatures and the overall noise level in the LSD output
(Donati et al., 1997; Bagnulo et al., 2009). Within $\pm$17 km s-1 from line
centre, the Stokes $N$ profile has a mean of 4.7e-6, a standard deviation of
3.6e-5, and a false-alarm probability of 7.9e-1, confirming the absence of
spurious polarimetric signatures. The success of this preliminary
spectropolarimetric snapshot of HD 63433 represents the gateway for a
subsequent monitoring dedicated to the characterisation of the large-scale
magnetic field.

Our observations were carried out in circular polarisation mode only,
considering the prohibitive exposure time required for linear polarisation
mode for main sequence stars such as HD 63433. The inclusion of linear
polarisation would allow us to reduce the cross-talk between the radial and
meridional field components and obtain a better constrain of smaller magnetic
field features (e.g. Rosén et al., 2015).

Table 1: Properties of HD 63433 that are relevant for our study.

Parameter | Value | Reference
---|---|---
B [mag] | $7.59\pm 0.02$ | [1]
V [mag] | $6.91\pm 0.01$ | [1]
Distance [pc] | 22.34$\pm$0.04 | [2]
Teff [K] | $5700\pm 75$ | [3]
$\log g$ [dex] | $4.54\pm 0.05$ | [3]
Mass [M⊙] | $0.99\pm 0.03$ | [3]
Radius [R⊙] | $0.897\pm 0.019$ | [3]
Age [Myr] | $414\pm 23$ | [4]
Prot [d] | $6.45\pm 0.05$ | [4]
$v_{\mathrm{eq}}\sin i$ [km s-1] | $7.3\pm 0.3$ | [4]
Inclination [∘] | $>74$ | [4]
$S$-index | $0.433\pm 0.041$ | [5]
$\log R^{\prime}_{\mathrm{HK}}$ [dex] | $-4.345\pm 0.055$ | [5]
444References: (1) Høg et al. (2000), (2) Gaia Collaboration (2020), (3)
Damasso et al. (2023), (4) Mann et al. (2020), (5) This work.

Figure 1: Least-squares deconvolution profiles for the snapshot observations
of HD 63433 with ESPaDOnS. The panels show the Stokes $I$ (top), $V$ (middle)
and $N$ (bottom) profiles, obtained from the combination of 4620 atomic
spectral lines. We note the presence of a clear Zeeman signature in Stokes
$V$, indicating that the large-scale magnetic field is amenable to
characterisation.  Table 2: Our observations of HD 63433.

Date | UT | HJD | $n_{\mathrm{cyc}}$ | Instrument | $t_{\mathrm{exp}}$ | S/N | $\sigma_{\mathrm{LSD}}$
---|---|---|---|---|---|---|---
[dd-mm-yyyy] | [hh:mm:ss] | [$-2450000$] |  |  | [s] |  | [$10^{-5}I_{c}$]
12-09-2022 | 15:01:01.380 | 9835.12079 |  $-$27.17 | ESPaDOnS | 4x340 | 743 | 4.83
23-12-2022 | 05:32:01.248 | 9936.73575 |  $-$11.42 | HARPSpol | 4x900 | 394 | 13.70
25-12-2022 | 05:23:12.480 | 9938.72971 |  $-$11.11 | HARPSpol | 4x900 | 327 | 201.53
06-02-2023 | 04:22:00.480 | 9981.68718 |  $-$4.45 | HARPSpol | 4x900 | 245 | 11.42
06-03-2023 | 20:55:29.485 | 10010.37541 | 0.00 | Neo-Narval | 4x900 | 520 | 9.60
07-04-2023 | 19:46:09.397 | 10042.32441 | 4.95 | Neo-Narval | 4x900 | 556 | 11.70
08-04-2023 | 20:17:03.480 | 10043.34576 | 5.11 | Neo-Narval | 4x900 | 465 | 8.71
16-04-2023 | 19:25:34.170 | 10051.30923 | 6.35 | Neo-Narval | 4x900 | 494 | 9.83
02-05-2023 | 19:49:12.432 | 10067.32409 | 8.83 | Neo-Narval | 4x900 | 381 | 13.60
05-05-2023 | 20:45:55.940 | 10070.36319 | 9.30 | Neo-Narval | 4x900 | 417 | 9.44
555The following quantities are listed: 1-2) date and time of the observation;
3) Heliocentric Julian date; 4) cycle number based on the ephemeris in Eq. 1;
5) instrument used; 6) exposure time per polarimetric sequence; 7) S/N per
polarimetric sequence at 650 nm per CCD pixel of unpolarised spectrum; and 8)
RMS noise level of Stokes $V$ signal in units of unpolarised continuum.

###  2.2 HARPSpol

HD 63433 was observed with
HARPSpol666https://www.eso.org/sci/facilities/lasilla/instruments/harps.htmlhttps://www.eso.org/sci/facilities/lasilla/instruments/harps.html
(Snik et al., 2011; Piskunov et al., 2011), the spectropolarimeter for the
HARPS spectrograph (Mayor et al., 2003) at the ESO 3.6 m telescope at La Silla
observatory, Chile. The HARPSpol instrument is mounted at the Cassegrain focus
of the telescope and has two polarimeters: for circular (Stokes $V$) and
linear (Stokes $QU$) polarisation, respectively. Each polarimeter consists of
a rotating retarder wave plate (a quarter-wave plate for the circular
polarimeter and a half-wave plate for the linear polarimeter) and a beam-
splitter – in this case a Foster prism – which splits the light into two beams
with opposite polarisation states. These two beams are then fed via fibers to
the HARPS spectrograph. Our HARPSpol observations cover the wavelength range
between 380 and and 691 nm with a 8 nm gap at 529 nm separating the red and
blue detectors. The spectral resolution is $R=\delta\lambda/\lambda\approx
110000$. We obtained circular polarisation observations on three separate
nights: 22 December 2022, 24 December 2022, and 06 February 2023.

To reduce the HARPSpol data, we used the PyReduce
package777https://github.com/AWehrhahn/PyReduce (Piskunov et al., 2021), the
updated implementation in python of the versatile REDUCE package (Piskunov &
Valenti, 2002). The reduction with PyReduce is run with a series of standard
steps, similar to REDUCE as described, for instance, Rusomarov et al. (2013).
First, the mean of both bias and flat frames is computed, followed by the
substraction of the mean bias from the mean flat. The final mean flat frame is
then used to compute the normalized flat field, and to determine the location
and shape of the spectral orders on the detectors. Science frames first get
the bias subtracted and are divided by the normalized flat field to correct
pixel-to-pixel variation in sensitivity. Scattered light is then subtracted
from the science frames. Finally, the spectra from the two polarisation
channels are extracted for each science frame. At this point, the extracted
science spectra are in pixel space. To convert from pixel to wavelength space,
we constructed a 2D wavelength solution from the wavelength calibration frames
taken with the Thorium-Argon (ThAr) lamp as part of the regular daily
calibrations. In this reduction, the blue and red detectors of HARPS are
reduced independently.

We then applied LSD to the collected HARPSpol spectra using the same mask as
ESPaDOnS. Compared to ESPaDOnS, the final LSD profiles are expected to be
penalised due to the narrower spectral coverage of HARPSpol. For the night of
22 December 2022, we detected a Zeeman signature with FAP 8.7e-9, and the
Stokes $V$ S/N is 7900. The two other nights do not show detections (FAP is
9.4e-1 and 3.8e-2) and they correspond to the observations with the lowest S/N
(see Table 2). Therefore, while we computed the longitudinal field only for
the first observation, we measured the chromospheric activity indices for all
three observations. Given that the spectral coverage of HARPSpol stops at 691
nm, we did not compute the Ca ii infrared triplet activity diagnostics (see
Sect. 3).

###  2.3 Neo-Narval

HD 63433 was observed by Neo-Narval888https://tbl.omp.eu/instruments/neo-
narval/https://tbl.omp.eu/instruments/neo-narval/ between March and May 2023.
We obtained six observations and although the phase coverage is not ideal, it
is sufficient to reconstruct the topology of the large-scale magnetic field.
The complete time series is summarised in Table 2. Neo-Narval is an upgrade to
the Narval instrument at Télescope Bernard Lyot (Pic du Midi Observatory,
Aurière 2003), operating between 370 and 1000 nm with a spectral resolving
power of $R=65,000$.

The Neo-Narval upgrade, implemented in 2019, included the installation of a
new detector and improved velocimetric capabilities (López Ariste et al.,
2022). Issues with the extraction of blue spectral orders occurred during the
period of our observations (López Ariste et al., 2022), affecting especially
those with very low S/N (below approximately 400 nm). As explained in the next
section, this prevented us from studying Ca II H&K lines. In contrast, the
computation of LSD profiles was less affected because LSD uses a S/N-weighting
scheme. We adopted the same synthetic mask of 4620 lines as for ESPaDOnS and
normalisation parameters, obtaining Stokes $V$ profiles with the maximum S/N
spanning between 7300 and 11,480. In the next sections, all the observations
will be phased with the following ephemeris

$\mathrm{JD}=2460010.37186+\mathrm{P}_{\mathrm{rot}}\cdot n_{\mathrm{cyc}},\\\
$ (1)

where we used the first Neo-Narval observation as JD reference, Prot is the
stellar rotation period, and $n_{\mathrm{cyc}}$ represents the rotation cycle
(see Table 2).

##  3 Magnetic activity proxies

###  3.1 Activity indices

The spectral coverage of both ESPaDOnS, HARPSpol, and Neo-Narval encompasses
chromospheric lines that are typically used to gauge chromospheric flux and
thus assess stellar activity. As described in the next sections, we computed
canonical activity indices using the Ca ii H&K lines, the H$\alpha$ line and
Ca ii infrared triplet lines. In the absence of star-planet interactions,
these indices should vary at the stellar rotation period timescale (e.g.
Mittag et al., 2017; Kumar & Fares, 2023), although it is known that
chromospheric complexity hampers rotation period measurements. For this
reason, and considering the low number of observations, we did not perform a
temporal analysis.

####  3.1.1 Ca II H&K index

Stellar activity has been quantified with the $S$ index (Vaughan et al., 1978)
and it was firstly adopted to search for activity cycles within the Mount
Wilson project (Wilson, 1968; Duncan et al., 1991). It is defined as the ratio
between the flux of the Ca ii H&K lines and the flux of the nearby continuum.
Formally,

$S=\frac{aF_{H}+bF_{K}}{cF_{R}+dF_{V}}+e,$ (2)

where $F_{H}$ and $F_{K}$ are the fluxes in two triangular band passes with
FWHM = 1.09 Å centred on the cores of the H line (3968.470 Å) and K line
(3933.661 Å), whereas $F_{R}$ and $F_{V}$ are the fluxes within two 20-Å
rectangular band passes centred at 3901 and 4001 Å, respectively. The set of
coefficients $\\{a,b,c,d,e\\}$ is used to convert the $S$-index from a
specific instrument scale to the Mount Wilson scale. The coefficients for the
ESPaDOnS instrument were estimated by Marsden et al. (2014), while for
HARPSpol we used the optimisation of Boro Saikia et al. (2018). Given the Neo-
Narval pipeline issue for the spectral region below 400 nm, we did not compute
the activity index for the corresponding observations.

We determined the $S$-index from the intensity spectra of all four sub-
exposures composing the ESPaDOnS snapshot observation (see Sect. 2), which we
Doppler-shifted according to the radial velocity of the star ($-15.9$ km s-1).
We found a mean value of $0.433\pm 0.021$, where the error bar is obtained as
standard deviation between the four sub-exposures divided by two (i.e. the
square root of the number of sub-exposures). We took this approach since the
measurement of an activity indicator for a particular night has the benefit of
an increased S/N thanks to the combination of sub-exposures. The value of
$S$-index remains consistent if computed directly from the intensity spectrum
obtained as combination of the four sub-exposures. The S/N in the spectral
region where the Ca ii H&K lines lie is too low in the HARPSpol observations
for a reliable measurement.

The $S$-index contains a colour dependence that biases the comparison of
activity levels between stars, as well as a photospheric contribution in the
line wings due to magnetic heating (Noyes et al., 1984). For this reason, the
chromospheric activity index $\log R^{\prime}_{\mathrm{HK}}$ is typically used
(Middelkoop, 1982; Noyes et al., 1984; Rutten, 1984). The conversion yielded a
value of $-4.345\pm 0.027$ for the ESPaDOnS observation.

The index values we estimated from the ESPaDOnS spectrum are consistent with
the range for which large-scale magnetic fields of active cool dwarfs are
detectable (Marsden et al., 2014). They are compatible within uncertainties
with the range reported by Pace (2013) and Boro Saikia et al. (2018), but they
are larger by 0.06 dex (3$\sigma$) than the individual measurement in 2010
computed by Marsden et al. (2014).

####  3.1.2 H$\alpha$ index

We measured the flux of the H$\alpha$ line normalised by the nearby continuum
following Gizis et al. (2002). This line is formed in the upper chromosphere,
thus providing insights into this particular region of the stellar atmosphere.
Formally,

$\mathrm{H}\alpha=\frac{F_{\mathrm{H}\alpha}}{H_{R}+H_{V}},$ (3)

where $F_{\mathrm{H}\alpha}$ is the flux within a rectangular band pass of
3.60 Å centred on the H$\alpha$ line at 6562.85 Å, and $H_{V}$ and $H_{R}$ are
the fluxes within two rectangular band passes of 2.2 Å centred on 6558.85 Å
and 6567.30 Å.

The temporal evolution of the values computed from Neo-Narval spectra is shown
in Fig. 3 and the values are reported in Table 3. From the ESPaDOnS snapshot
observation, we obtained a value of $0.341\pm 0.002$, from HARPSpol we
measured $0.339\pm 0.001$, $0.339\pm 0.001$, and $0.331\pm 0.002$, and from
Neo-Narval, we estimated values between 0.336 and 0.342 with a mean of
$0.339\pm 0.002$. Even though all observations spanned eight months in total,
the activity level, as quantified by the H$\alpha$ index, did not indicate a
substantial evolution.

Figure 2: Normalised fluxes around the H$\alpha$ line for the three
instruments used in this study. From the top: ESPaDOnS, HARPSpol, and Neo-
Narval. The black rectangles indicate the regions used in the definition of
the corresponding activity index. The observations were taken between
September 2022 and May 2023.

####  3.1.3 Ca II infrared triplet index

Another chromospheric activity indicator is the Ca ii infrared-triplet (IRT)
index, which is a diagnostic for the lower chromosphere (Montes et al., 2000;
Hintz et al., 2019). We followed Petit et al. (2013) and Marsden et al.
(2014), computing:

$\mathrm{CaIIIRT}=\frac{\mathrm{IR1}+\mathrm{IR2}+\mathrm{IR3}}{\mathrm{IR}_{R}+\mathrm{IR}_{V}},$
(4)

where IR1, IR2, and IR3 are the fluxes within a rectangular band pass of 2 Å
centred on the Ca ii lines at 8498.023, 8542.091, and 8662.410 Å,
respectively, and IRR and IRV are the fluxes within two rectangular band
passes of 5 Å centred on 8704.9 Å and 8475.8 Å. The spectral coverage of
HARPSpol does not encompass these calcium lines; hence, we did not compute the
index for the corresponding observations.

The temporal evolution is shown in Fig. 3 and the values reported in Table 3.
From the ESPaDOnS snapshot observation, we obtained a value of $0.899\pm
0.002$, whereas from the Neo-Narval observations, we estimated values between
0.880 and 0.910 with a mean of $0.891\pm 0.002$. In a similar manner to
H$\alpha$ index, the magnetic activity level seems to have remained reasonably
stable, with the oscillations due to intrinsic variability of active regions.
In general, we caution that robust conclusions on the evolution of the
activity level based on the activity proxies computed here cannot be drawn,
considering the large gaps between observations as well as the low number of
observations.

Figure 3: Time series of longitudinal magnetic field and activity indices
measurements for HD 63433 from the Neo-Narval observations. The longitudinal
field (top panel) exhibits intermittency in the sign of the magnetic polarity,
which is symbolic of a likely complex or non-axisymmetric topology. The
H$\alpha$ index (middle panel) does not exhibit strong variations, while the
Ca ii IRT index (bottom panel) has an increased scatter towards the latest
observations. The H$\alpha$ and Ca ii IRT indices show correlated temporal
evolution. Figure 4: Correlation plots for the activity proxies computed from
the six Neo-Narval observations. From the left: H$\alpha$ vs $|$B${}_{l}|$, Ca
ii IRT vs $|$B${}_{l}|$, and H$\alpha$ vs Ca ii IRT. Each data point is
colour-coded based on the rotational cycle as computed from Eq.1. Each panel
displays the Pearson correlation coefficient ($\rho$) at the top and a linear
fit as black dashed line. The green cross indicates the values of the ESPaDOnS
snapshot observation, but it was not included in the computation of the
Pearson correlation coefficient. We included the first HARPSpol observation in
the first panel as a blue cross. Table 3: Activity indices and longitudinal
magnetic field measurements.

Date | Instrument |  $S$-index | $\log R^{\prime}_{\mathrm{HK}}$ | H$\alpha$ index | Ca ii IRT | Bl
---|---|---|---|---|---|---
[dd-mm-yyyy] |  |  |  |  |  | [G]
12-09-2022 | ESPaDOnS | $0.433\pm 0.041$ | $-4.345\pm 0.055$ | $0.341\pm 0.002$ | $0.899\pm 0.002$ | $3.2\pm 0.9$
22-12-2022 | HARPSpol | … | … | $0.339\pm 0.002$ | $\ldots$ | $6.4\pm 2.0$
24-12-2022 | HARPSpol | … | … | $0.339\pm 0.002$ | $\ldots$ | $\ldots$
05-02-2023 | HARPSpol | … | … | $0.331\pm 0.003$ | $\ldots$ | $\ldots$
06-03-2023 | Neo-Narval | $\ldots$ | $\ldots$ | $0.339\pm 0.002$ | $0.880\pm 0.003$ | $-5.1\pm 1.9$
07-04-2023 | Neo-Narval | $\ldots$ | $\ldots$ | $0.337\pm 0.002$ | $0.885\pm 0.003$ | $-0.6\pm 1.9$
08-04-2023 | Neo-Narval | $\ldots$ | $\ldots$ | $0.338\pm 0.002$ | $0.889\pm 0.003$ | $-3.3\pm 2.0$
16-04-2023 | Neo-Narval | $\ldots$ | $\ldots$ | $0.340\pm 0.003$ | $0.898\pm 0.005$ | $3.6\pm 1.9$
02-05-2023 | Neo-Narval | $\ldots$ | $\ldots$ | $0.336\pm 0.006$ | $0.882\pm 0.006$ | $-0.2\pm 2.8$
05-05-2023 | Neo-Narval | $\ldots$ | $\ldots$ | $0.342\pm 0.007$ | $0.910\pm 0.006$ | $-9.4\pm 2.5$

999The error bar for $S$-index and $\log R^{\prime}_{\mathrm{HK}}$ is the
standard deviation of the measurements from the four sub-exposures. For the
H$\alpha$ index, the Ca ii IRT index and longitudinal magnetic field, they are
obtained from formal propagation.

###  3.2 Longitudinal magnetic field

We computed the disk-integrated, line-of-sight-projected component of the
large-scale magnetic field following Donati et al. (1997) for the snapshot
ESPaDOnS observation, the first HARPSpol observation, and the six Neo-Narval
observations. We used the general formula as in Cotton et al. (2019),

$\mathrm{B}_{l}=\frac{h}{\mu_{B}\lambda_{0}\mathrm{g}_{\mathrm{eff}}}\frac{\int
vV(v)dv}{\int(I_{c}-I(v))dv}\,,$ (5)

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, $h$ is the Planck’s constant, and $\mu_{B}$
is the Bohr magneton. To express it as in Rees & Semel (1979) and Donati et
al. (1997), it is possible to use $hc/\mu_{B}=0.0214$ Tm, where $c$ is the
speed of light in m s-1.

The computation was carried out within $\pm 20$ km s-1 from line centre at
$-15.9$ km s-1 for both Stokes $I$ and $V$. We found a value of $3.2\pm 0.9$ G
for the ESPaDOnS snapshot observation, $6.4\pm 2.0$ G for the HARPSpol
observation and a range between $-9.4$ and $3.6$ G for the Neo-Narval
observations. The median formal error bar is 2.0 G. In comparison, the value
reported in 2010 by Marsden et al. (2014) is $2.5\pm 0.5$ G. We illustrate the
time series of Bl measurements in Fig. 3. The fast-changing sign is symbolic
of a complex magnetic field topology, which is expected for stars of similar
properties to HD 63433 (e.g. Morgenthaler et al., 2012; Strassmeier et al.,
2023).

Figure 5: Time series of circularly polarised Stokes profiles obtained with
Neo-Narval. Observations are shown as black dots and ZDI models as red lines
and they are offset vertically for better visualisation. The number on the
right is the rotational cycle (see Eq. 1).

###  3.3 Correlations of activity proxies

We inspected the correlations between $|$B${}_{l}|$, the H$\alpha$ index and
the Ca ii IRT index for the six Neo-Narval observations. The results are given
in Fig. 4. In general, we caution that these correlations are found with a
statistically low number of observations. Therefore, our main aim is to check
that we see the expected trends, even for a low number of observations.

The activity indices are computed from unpolarised spectra, which are
sensitive to both small and large spatial scales and, thus, to most magnetic
features. Instead, the longitudinal field is derived from circularly polarised
spectra, which are known to be sensitive to large-scales only, owing to
polarity cancellation effects. For this reason, moderate correlations or more
complex behaviours between these quantities have been reported for cool dwarfs
(Petit et al., 2008; Morgenthaler et al., 2012; Marsden et al., 2014; Boro
Saikia et al., 2016; Brown et al., 2022).

Here, we obtained a strong positive correlation between H$\alpha$ index and
$|$B${}_{l}|$ of Pearson correlation coefficient $\rho=0.95$. The ESPaDOnS and
HARPSpol values are not included in the computation of $\rho$, because the
corresponding observations were collected months before (see Table 3), hence
intrinsic variability may have occurred (the inclusion of these two points
yields $\rho=0.79$, as expected from a loss of coherency of activity over long
timescales). For the Ca ii IRT index versus $|$B${}_{l}|$ case and the
H$\alpha$ versus Ca ii IRT index case, we obtained $\rho=0.75$ and 0.77,
respectively. While still strong, the lower level of correlation is likely due
to the different chromospheric regions that H$\alpha$ index and Ca ii IRT
index probe (see e.g. Hintz et al., 2019). Figure 4 also demonstrates the high
variability of magnetic activity for this star, since the last two
measurements (yellow circles beyond rotational cycle 8.0) are at the extremes
of the measurements range of the indices (see also Table 3).

##  4 Zeeman-Doppler imaging

We applied ZDI on the six Neo-Narval observations to map the large-scale
magnetic field at the surface of HD 63433. Formally, the field is expressed as
the sum of a poloidal and toroidal component, and both components are
described by means of spherical harmonic decomposition (Donati et al., 2006;
Lehmann & Donati, 2022). The ZDI algorithm generates a time series of Stokes
$V$ profiles and adjusts them to the observations iteratively until a target
$\chi^{2}_{r}$ is reached. The goal is to fit the spherical harmonics
coefficients $\alpha_{\ell,m}$, $\beta_{\ell,m}$, and $\gamma_{\ell,m}$ (with
$\ell$ and $m$ being the degree and order of the mode, respectively). The
algorithm adopts a maximum-entropy regularisation scheme to reconstruct the
magnetic field configuration compatible with the data and with the lowest
information content (for more information see Skilling & Bryan, 1984; Donati
et al., 1992, 1997). In practice, we employed the python zdipy code described
in Folsom et al. (2018) 101010Available at
https://github.com/folsomcp/ZDIpyhttps://github.com/folsomcp/ZDIpy, assuming
weak-field approximation (see e.g. Landi Degl’Innocenti, 1992).

We adopted the following stellar input parameters (Mann et al., 2020):
inclination of 70∘, projected equatorial velocity ($v_{\mathrm{eq}}\sin i$) of
7.3 km s-1, rotation period of 6.45 d, and solid body rotation. The value of
$v_{\mathrm{eq}}\sin(i)$ is consistent with other dedicated studies (Rainer et
al., 2023). We set a linear limb darkening coefficient of 0.7 (Claret &
Bloemen, 2011) and the maximum degree of harmonic expansion
$\ell_{\mathrm{max}}=5$, consistently with the spatial resolution allowed by
$v_{\mathrm{eq}}\sin(i)$. Using larger values of $\ell_{\mathrm{max}}$ did not
improve the fit since the magnetic energy is stored only in the modes up to
$\ell=5$.

The time series of Neo-Narval Stokes $V$ profiles is shown in Fig. 5. The
shape of the LSD profiles suggests that the field topology includes a
significant contribution from the toroidal component. The last observation at
cycle 9.30 seems to have a more antisymmetric shape, which is typical of a
more dipolar configuration instead (Morin et al., 2008a, b; Bellotti et al.,
2023a, b). Such variation in profile shape is likely due to stellar rotation
rather than intrinsic evolution, considering the short phase separation with
the previous observations. As a result, we expect a magnetic feature of
negative polarity in the radial field at that phase.

A qualitative assessment of the axisymmetric component of the magnetic field
can be obtained by decomposing the median Stokes $V$ profile into its
symmetric and asymmetric components (Lehmann & Donati, 2022). We show this
exercise in Fig. 6, where we see that the median Stokes $V$ shape is mostly
symmetric, suggesting a significant contribution from the toroidal component
of the field. In parallel, the antisymmetric shape exhibits a strong signal,
meaning that the poloidal component is also contributing substantially to the
profile shape. From this, one can expect the axisymmetric component of the
field to be mostly toroidal (in agreement with Petit et al., 2008), but with a
substantial poloidal contribution as well. This confirms the capability of the
symmetric-antisymmetric decomposition to qualitatively determine the main
magnetic field component. This could be particularly useful when a large
number of stars have to be analysed before a detailed ZDI analysis is carried
out.

Figure 6: Decomposition of the median Stokes $V$ LSD profile into its
symmetric and antisymmetric components. We note that the median Stokes $V$
(red line) has mostly a symmetric shape (yellow line) which indicates a strong
toroidal component. At the same time, there is still a significant
contribution from the antisymmetric shape (blue line), namely, the poloidal
component. Table 4: Properties of the magnetic map.

Bmean | Bmax | Epol/Etot | Etor/Etot | Edip/Epol | Equad/Epol | Eoct/Epol | Eaxi/Etot | Eaxi,pol/Epol | Eaxi,tor/Etor
---|---|---|---|---|---|---|---|---|---
[G] | [G] | [%] | [%] | [%] | [%] | [%] | [%] | [%] | [%]
23.8${}^{+6.2}_{-4.4}$ | 54.4${}^{+19.1}_{-15.3}$ | 46.3${}^{+15.3}_{-9.4}$ | 53.7${}^{+9.4}_{-15.3}$ | 30.2${}^{+25.2}_{-4.7}$ | 25.6${}^{+8.9}_{-3.8}$ | 15.2${}^{+0.7}_{-10.6}$ | 65.7${}^{+24.6}_{-17.3}$ | 37.7${}^{+39.1}_{-14.8}$ | 89.9${}^{+8.3}_{-5.9}$

111111The following quantities are listed: mean absolute magnetic strength,
maximum value of the magnetic field modulus, 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, poloidal axisymmetric energy as a
fraction of the poloidal energy, and toroidal axisymmetric energy as a
fraction of the toroidal energy. The variation bars were computed by
reconstructing ZDI maps and including the uncertainties on the input stellar
parameters.

Figure 7: ZDI reconstruction in flattened polar view of the large-scale field
of HD 63433. From the left, the radial, azimuthal, and meridional components
of the magnetic field vector are illustrated. 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), as well as the equator (solid line). The geometry is
complex, with a significant toroidal fraction and the poloidal component
distributed mainly in the dipolar, quadrupolar, and octupolar modes. The
colour bar encapsulates the magnetic field strength, up to a maximum of 55 G.

The Stokes $V$ profiles were fit down to $\chi^{2}_{r}=1.2$, from an initial
value of 6.5 (see Fig. 5). The value was chosen to prevent artificial
injection of magnetic energy in high order modes or reconstruction of
substantially weak fields, which would be symbolic of overfitting and
underfitting of the Stokes $V$ profiles, respectively (see e.g. Alvarado-Gómez
et al., 2015). Lowering the $\chi^{2}_{r}$ further would likely require the
inclusion of differential rotation as intrinsic variability mechanism, but
with our sparse observations it cannot be constrained.

The properties of the magnetic map are given in Table 4 and the map is shown
in Fig. 7. The mean magnetic field strength is B${}_{\mathrm{mean}}=24$ G,
with the toroidal component accounting for 54% of the magnetic energy and the
poloidal component for 46%. The dipolar mode accounts for 30% of the poloidal
energy and both the quadrupolar and quadrupolar modes store a significant
fraction, namely, 26% and 15%.

To set the uncertainties on the magnetic field properties, we estimated
variation bars following Mengel et al. (2016), Fares et al. (2017), and
Bellotti et al. (2023a). The idea is to vary the input parameters, namely
inclination, $v_{\mathrm{eq}}\sin(i)$, and rotation period to plausible values
consistent with the literature. One at a time, we set these input parameters
to the extremes of the range of values reported in the literature, and
reconstructed the corresponding map of the magnetic field. The variation bars
on the field characteristics are then the maximum deviations of these new maps
from the reconstruction with the optimised parameters. These variation bars do
not capture the actual error bars, but, rather, the sensitivity of the
reconstructed properties to input parameters.

Quantitatively, Mann et al. (2020) constrained the stellar inclination to be
larger than $74^{\circ}$; hence, we reconstructed maps with $i=60^{\circ}$ and
$i=80^{\circ}$. For $v_{\mathrm{eq}}\sin i$, we used 6.5 km s-1 and 7.9 km s-1
(Damasso et al., 2023) and for Prot we used 6.0 d (Mallorquín et al., 2023)
and 7.6 d (Marsden et al., 2014). We adopted a target $\chi^{2}_{r}$ of 1.2 in
every case, except for P${}_{\mathrm{rot}}=6.0$ d for which we used
$\chi^{2}_{r}=2.4$. This already indicates that using such value of rotation
period degrades significantly the magnetic field model. The variation bars of
the magnetic characteristics are driven predominantly by the uncertainties on
the rotation period, since the variations in inclination and
$v_{\mathrm{eq}}\sin i$ have led to magnetic maps that are reasonably
consistent with the reconstruction when using the optimised parameters.

To check the robustness of our results, we reconstructed a map excluding the
last observation. The only change is in the energy fraction in the poloidal
component, as it decreases to 25%. This is not surprising considering that the
shape of Stokes $V$ in the last spectropolarimetric observation is the most
similar to a typical poloidal-dipolar field (e.g. Morin et al., 2008b;
Bellotti et al., 2023a). The average field strength changes only by 1 G, while
the dipolar, quadrupolar and octupolar energy fraction by less than 2% and the
axisymmetric fraction by less than 15%. Such changes are encompassed by the
variation bars we estimated in this work.

##  5 ZDI-driven stellar wind models

We created the stellar wind models using the Alfvén wave solar model (AWSoM,
Sokolov et al., 2013; van der Holst et al., 2014), which is a component in the
Space Weather Modelling Framework (SWMF, Tóth et al., 2005, 2012). This model
solves the ideal two-temperature magnetohydrodynamic equations, supplemented
by a pair of equations describing the propagation of Alfvén wave energy along
magnetic field lines. The model extends from the chromosphere, where the
temperature is 5e4 and the density is 2e17, through the transition region to
the stellar corona. The model is driven by the energy flux of Alfvén waves,
which crosses the inner boundary at a rate proportional to the local magnetic
field strength at the stellar surface. The Poynting flux-to-field ratio is set
to $1.1\times 10^{6}$ W m-2 T-1 and the turbulence correlation length is
$1.5\times 10^{5}$ m T1/2. The radial component of the surface magnetic field
is held to the $B_{r}$ values in Fig. 7, while the transverse components are
left to evolve with the numerical solution. The AWSoM model was configured
using solar values that have been shown to reproduce solar conditions (Meng et
al., 2015; van der Holst et al., 2019; Sachdeva et al., 2019). The coronal
heating due to Alfvén waves is likely to resemble that of the Sun, according
to the non-thermal velocity measurements of sun-like stars of similar rotation
periods in Boro Saikia et al. (2023). For a more detailed description of the
methodology behind the wind models, we refer to Evensberget et al. (2021,
2022, 2023).

The top panel of Fig. 8 shows the stellar magnetic field embeded in the
stellar wind equator-on. The large scale coronal magnetic field resembles a
dipole with closed regions pushed towards the positive $z$ axis. We also see
arch-like structures where the magnetic field lines connect regions of
opposite polarity, which are the result of quadrupolar and octupolar
components of the magnetic field as may be seen also in the radial field plot
of Fig. 7. In our model, the wind mass loss rate is $1.1\times
10^{-13}\,\text{M}_{\odot}/\text{yr}$, which is about five times the solar
wind mass loss rate.

The bottom panel of Fig 8 shows the Alfvén surface, where the stellar wind
speed equals the Alfvén wave velocity. This quantity is crucial to understand
the regime in which magnetic or inertial forces dominate, within or outside
the surface, respectively (e.g. Vidotto, 2021). A practical quantity in this
context is the Alfvén Mach number, which is defined as the ratio of the wind
speed, $v_{\textsc{sw}}$, to the Alfvén wave speed of
$v_{\textsc{a}}=B/\sqrt{4\pi\rho_{\textsc{sw}}}$, where $B$ is the local
magnetic field strength and $\rho_{\textsc{sw}}$ is the local wind density.
The Alfvén Mach number quantifies whether the planetary motion is
super-(M${}_{\mathrm{A}}>1$) or sub-Alfvénic (M${}_{\mathrm{A}}<1$). The
Alfvén surface corresponds to M${}_{\mathrm{A}}=1$.

The orbits of planet d, b, and c are shown in the bottom panel of Fig. 8 as
concentric curves at 9.9, 16.8, and 36.1 R⋆ (Damasso et al., 2023; Mallorquín
et al., 2023; Capistrant et al., 2024). For the three planets, we assume that
the planets have circular orbits in the equatorial plane of the star,
consistently with the literature estimates within error bars (Mann et al.,
2020; Damasso et al., 2023; Capistrant et al., 2024). The orbit of the
innermost planet in our model crosses briefly the Alfvén surface into the sub-
Alfvénic regime; we return to this in Fig. 10 and discuss it further in Sect.
6.

Figure 8: Simulated environment of HD 63433. Top: Steady-state coronal
magnetic field structure obtained from our ZDI-driven stellar wind model. Open
magnetic field lines are truncated at two stellar radii, while the closed
field lines are not truncated. The star is seen equator on at the rotational
phase 0.35. The large-scale structure exhibits a mostly dipolar topology. The
colours are the same as Fig. 7. Bottom: Alfvén surface and wind radial speed.
The two-lobed Alfvén surface is characteristic for a dipole-dominated coronal
magnetic field. The intersection of the Alfvén surface and the stellar
equatorial plane is shown as a white curve, while the the planetary orbits are
shown in orange, red, and pink. The orbital plane is colour-coded by the wind
radial speed.

Figure 9 shows the wind speed $v_{\textsc{sw}}$, density,
$\rho_{\textsc{sw}}$, and wind ram pressure,
$\rho_{\textsc{sw}}v_{\textsc{sw}}^{2}$, in the planetary frame, as a function
of stellar rotation phase. We note the presence of three dips in the wind
speed curves. These are correlated to the three arms of the wind spiral shape
which, in turn, are due to stellar rotation (see Fig. 8). Such spiral features
were already reported, and tend to build more arms if the field complexity is
larger (Evensberget et al., 2021). The wind density experienced by each planet
can vary by a factor of $\sim 5$ along planetary orbits. For the ram pressure,
the variation is somewhat smaller than that of the density.

Figure 10 shows the Alfvén Mach number (MA) computed in the planetary frame as
a function of stellar rotational phase. In our model, we see that planet d
experiences sub-Alfvénic winds at stellar rotation phases 0.38–0.50, whereas
planets b and c orbit in super-Alfvénic motions. In the next section, we
discuss what these locations translate to in terms of star-planet interactions
and their observable signatures.

Figure 9:  Stellar wind conditions in the planetary frame. From the top: Wind
speed, density, and ram pressure are shown as a function of stellar rotation
phase. Planets b, c, and d are colour-coded in red, pink, and orange.  Figure
10:  Alfvén Mach number computed in the planetary frame as a function of
stellar rotation phase. The colours and labels are the same as in Fig. 9. The
phase interval where our model’s planet d crosses into the sub-Alfvénic regime
is shown as a green shaded region.

##  6 Summary and discussion

We have characterised the magnetic field and environment of the active, young
G5 dwarf star HD 63433, which is known to host two sub-Neptunes and an Earth-
sized planet. We computed magnetic activity diagnostics such as the
chromospheric $S$ index, $\log R^{\prime}_{\mathrm{HK}}$, H$\alpha$, Ca ii
infrared triplet, and longitudinal magnetic field, using observations from
ESPaDOnS, HARPSpol, and Neo-Narval. We then reconstructed the large-scale
magnetic field by means of ZDI and used it as boundary condition for the
simulation of the system’s stellar wind from the corona to the orbital
distances of the planets b, c, and d, following the method presented in
Evensberget et al. (2021). In general, the observations and analyses were
conducted as part of a survey dedicated to the spectropolarimetric
characterisation of bright stars in the current list of potential $Ariel$
targets, in order to efficiently construct the final target list before the
launch and, consequently, optimize the science outcome of the mission.

Figure 11: Properties of the magnetic topologies for cool, main sequence stars
obtained via ZDI. The location of HD 63433 is highlighted. The $y$\- and
$x$-axes represent the mass and rotation period of the star, and iso-Rossby
number curves are overplotted using the empirical relations of Wright et al.
(2018). The symbol size, colour and shape encodes the ZDI average field
strength, the poloidal and toroidal energy fraction, and axisymmetry. Data
entering the plot are taken from Petit et al. 2008; Morgenthaler et al. 2012;
Fares et al. 2013; Boro Saikia et al. 2015; do Nascimento et al. 2016; Boro
Saikia et al. 2016; Hussain et al. 2016; Alvarado-Gómez et al. 2015; Folsom et
al. 2016, 2018, 2020; Klein et al. 2021; Willamo et al. 2022; Martioli et al.
2022.

We estimated $S=0.433\pm 0.041$ and $\log R^{\prime}_{\mathrm{HK}}=-4.345\pm
0.055$, which are compatible with the measurements of other Sun-like stars
with analogous spectral type, mass, rotation period, and age ($\xi$ Boo A,
Morgenthaler et al. 2012; Strassmeier et al. 2023; HD 1237, Alvarado-Gómez et
al. 2015; HD 147513, Hussain et al. 2016; $\kappa$ Cet, do Nascimento et al.
2016). In comparison, the average values for the Sun at cycle maxima and
minima are $\log R^{\prime}_{\mathrm{HK}}=-4.905$ and $-4.984$ (Egeland et
al., 2017).

The longitudinal magnetic field of HD 63433 varies between $-11$ and 5 G,
which is typical for solar-like stars. For instance, Morgenthaler et al.
(2012) found variations between $-2.3$ and 22 G for $\xi$ Boo A, Boro Saikia
et al. (2016) reported values spanning $-20$ and 10 G across different epochs
of HN Peg, and Alvarado-Gómez et al. (2015) reported a range between $-10$ and
10 G for HD 1237. The large-scale field is three times stronger (in absolute
value) than the Sun, whose large-scale field values can reach 3-4 G (Kotov et
al., 1998).

The reconstructed magnetic field topology of HD 63433 is complex, with a
significant amount of energy stored in the toroidal and poloidal components
(see Sect. 3). The average field strength in absolute value is around 23 G.
This is of the same order of magnitude as estimated by Zhang et al. (2022) who
employed the age-magnetic field strength law of Vidotto et al. (2014) to
include the effects of stellar magnetic fields in the outflow models of planet
b and c. In our model, the derived stellar wind mass loss rate is $1.1\times
10^{-13}\,\text{M}_{\odot}/\text{yr}$, which is 2.9 times smaller than the
value derived from the simulations of Zhang et al. (2022). The stellar wind
can affect atmospheric escape in exoplanets in several ways, from potentially
reducing escape rates (Vidotto & Cleary, 2020) and altering the signatures of
atmospheric evaporation through spectroscopic transits (Carolan et al., 2021),
even to possibly creating a tenuous atmosphere through sputtering processes
(Vidotto et al., 2018). Recently, Zhang et al. (2022) modelled the escaping
atmospheres of HD 63433 b and c, including the effects of the interaction with
the stellar wind. So far, no model exists for the escaping atmosphere of the
recently discovered innermost planet HD 63433 d. With the optimally
constrained stellar wind properties presented in our work, future atmospheric
models of HD 63433 d will be able to better pin-point the effects of the wind
of the host star on the upper atmosphere of HD 63433 d, as well as potential
signatures in Ly-$\alpha$ and He i transits. Such models are important to
assess the presence and evolution of the atmosphere of HD 63433 d (Kubyshkina
et al., 2022).

Finally, we illustrate the properties of the field geometry of HD 63433
compared to other stars in Fig. 11. Although the number of observations used
for tomographic inversion is limited, the magnetic field exhibits similarities
with other Sun-like stars of similar properties and remarkably with the field
map of $\xi$ Boo A from 2008 (Morgenthaler et al., 2012) because it also
exhibits a dominating polarity at the north pole, as well as lower latitude
magnetic spots.

In terms of evolution, Maldonado et al. (2022) analysed activity indices and
photometric light curves of HD 63433, and estimated the lifetime of active
regions to be approximately 60 d. Additionally, Lehtinen et al. (2016)
reported cyclic photometric variability on a 2.7 yr timescale and a less
significant variation of approximately 8.0 yr. If we maintain the analogy with
$\xi$ Boo A, Morgenthaler et al. (2012) reported fast variations over
timescales of six months of the field topology and complexity, from a
poloidal-dominated to a toroidal-dominated regime. Overall, this shows that HD
63433 could manifest evident and important variability, which motivates
additional spectropolarimetric monitoring.

From the stellar environment simulations, we obtained a largely dipolar
coronal magnetic field, aligned with the stellar axis of rotation. We also see
arch-like structures where the magnetic field lines connect regions of
opposite polarity, resulting from the quadrupolar and octupolar components of
the magnetic field. Using our wind model, we computed the location of the
Alfvén surface where the wind speed equals the Alfvén wave velocity. The
Alfvén surface is two-lobed, which is characteristic for a dipole-dominated
coronal magnetic field.

We found that the outermost two planets orbit in the super-Alfvénic region. In
this regime, a bow shock between the stellar wind and a planetary
magnetosphere (if present) can be generated and the stellar wind is deflected
around the planet (Chapman & Ferraro, 1931; Vidotto et al., 2009, 2010). Also,
the formation of a tail-like structure of evaporated planetary material can
occur (e.g. Schneiter et al., 2007; Villarreal D’Angelo et al., 2018; Zhang et
al., 2022). Star-planet interactions in super-Alfvénic regime are expected to
produce auroral radio emission from the planetary magnetosphere, similarly to
Solar System planets (Zarka et al., 2001).

The innermost Earth-sized planet is located in the trans-Alfvénic region, as
our modelling indicates that the stellar wind conditions along its orbit may
switch between sub- and super-Alfvénic. Within the sub-Alfvénic regime, the
planet can directly link to the stellar magnetic field, and its motion can
perturb the field lines producing Alfvén waves travelling starward;
alternatively, it can induce magnetic reconnection events (if magnetised) or a
combination of both (e.g. Neubauer, 1998; Ip et al., 2004; Strugarek et al.,
2015; Kavanagh et al., 2022). In turn, these phenomena generate detectable
signatures such as coronal radio emission on the star (Saur et al., 2013;
Kavanagh et al., 2021) and local chromospheric flux enhancement (Shkolnik et
al., 2008; Lanza, 2009).

We note that the magnetic field strength recovered from ZDI could be
underestimated (Lehmann et al., 2019). On one side, the latitudinal coverage
of the star is limited by definition, as we can only model the disk
intersected by the line of sight. On the other side, the low number of
observations combined with the poor longitudinal coverage of the stellar
rotation may prevent us from reconstructing additional large-scale magnetic
features. For these reasons, a more robust wind model will be provided in the
future with an updated map built from a denser spectropolarimetric monitoring.
Although spectropolarimetric time series with a scarce sampling may limit the
reconstruction of the large-scale magnetic topology, the first-order
information that we can derive from them is still relevant feedback for
$Ariel$ target planning. They allow us to assess the magnetic activity level
of the star and its variation over distinct timescales: from rotational
modulation to long-term evolution, potentially in the form of a magnetic
cycle, which could be seen as a magnetic polarity reversal even from multiple
epochs in which the rotational phase coverage is not optimal (see Fig. 1 Boro
Saikia et al., 2016, for example).

This work has been developed within the science development of the ESA $Ariel$
space mission, for which more magnetic activity characterisation studies will
be presented in the upcoming future. It represents the start of a campaign
dedicated to the magnetic characterisation of stars in the current list of
$Ariel$ targets (Edwards & Tinetti, 2022). In this regard, additional
observations of HD 63433 are necessary to understand the long-term evolution
of its large scale magnetic field. This will be also valuable to our
understanding of the observational signatures of star-planet interactions due
to the correlated long-term evolution of the Alfven surface. Indeed, it would
be interesting to observe the star along its likely magnetic cycle to
characterize how the Alfvén surface moves with respect to the planetary
orbits. Observations in the UV range will also be essential, as X-ray
observations show a difference of around two orders of magnitude between the
Sun and HD 63433. This raises the question of how the UV wavelength range
differs and the consequences on the atmospheric photochemistry, heating, and
loss. Ultimately, this information can provide practical constraints for the
observing strategies of $Ariel$, for such targets as the HD 63433 system. Once
the mission is launched, the goal of our spectropolarimetric campaign will
become more of a follow-up work, aimed at complementing the observations
already performed and providing new insights for the interpretation of $Ariel$
data.

###### Acknowledgements.

This publication is part of the project ”Exo-space weather and contemporaneous
signatures of star-planet interactions” (with project number OCENW.M.22.215 of
the research programme ”Open Competition Domain Science- M”), which is
financed by the Dutch Research Council (NWO). This work used the Dutch
national e-infrastructure with the support of the SURF Cooperative using grant
nos. EINF-2218 and EINF-5173. SB acknowledges funding from the SCI-S
department of the European Space Agency (ESA), under the Science Faculty
Research fund E/0429-03. AAV and DE acknowledge funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No 817540, ASTROFLOW). SBS was supported
by the Austrian Science Fund (FWF) Lise Meitner grant M2829-N. CD acknowledges
financial support from the INAF initiative “IAF Astronomy Fellowships in
Italy”, grant name GExoLife. GM acknowledges the support of the ASI- INAF
agreement 2021-5-HH.0 and PRIN MUR 2022 (project No. 2022J7ZFRA, EXO-CASH).
Based on observations obtained at the Canada-France-Hawaii Telescope (CFHT)
which is operated by the National Research Council of Canada, the Institut
National des Sciences de l’Univers of the Centre National de la Recherche
Scientique of France, and the University of Hawaii. Based on observations
collected at the European Southern Observatory under ESO programme
110.24C8.002 (P.I. S. Bellotti). We thank the TBL team for providing service
observing with Neo-Narval. 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) and PyAstronomy (Czesla et al., 2019).

## References

  * Airapetian et al. (2017) Airapetian, V. S., Glocer, A., Khazanov, G. V., et al. 2017, ApJ, 836, L3
  * Allan & Vidotto (2019) Allan, A. & Vidotto, A. A. 2019, MNRAS, 490, 3760
  * Alvarado-Gómez et al. (2015) Alvarado-Gómez, J. D., Hussain, G. A. J., Grunhut, J., et al. 2015, A&A, 582, A38
  * 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
  * Aurière (2003) Aurière, M. 2003, in EAS Publications Series, Vol. 9, EAS Publications Series, ed. J. Arnaud & N. Meunier, 105
  * Bagnulo et al. (2009) Bagnulo, S., Landolfi, M., Landstreet, J. D., et al. 2009, PASP, 121, 993
  * Bellotti et al. (2023a) Bellotti, S., Fares, R., Vidotto, A. A., et al. 2023a, A&A, 676, A139
  * Bellotti et al. (2023b) Bellotti, S., Morin, J., Lehmann, L. T., et al. 2023b, A&A, 676, A56
  * Boro Saikia et al. (2016) Boro Saikia, S., Jeffers, S. V., Morin, J., et al. 2016, A&A, 594, A29
  * Boro Saikia et al. (2015) Boro Saikia, S., Jeffers, S. V., Petit, P., et al. 2015, A&A, 573, A17
  * Boro Saikia et al. (2023) Boro Saikia, S., Lueftinger, T., Airapetian, V. S., et al. 2023, ApJ, 950, 124
  * Boro Saikia et al. (2018) Boro Saikia, S., Lueftinger, T., Jeffers, S. V., et al. 2018, A&A, 620, L11
  * Brown et al. (2022) Brown, E. L., Jeffers, S. V., Marsden, S. C., et al. 2022, MNRAS, 514, 4300
  * Capistrant et al. (2024) Capistrant, B. K., Soares-Furtado, M., Vanderburg, A., et al. 2024, AJ, 167, 54
  * Carmona et al. (2023) Carmona, A., Delfosse, X., Bellotti, S., et al. 2023, A&A, 674, A110
  * Carolan et al. (2021) Carolan, S., Vidotto, A. A., Villarreal D’Angelo, C., & Hazra, G. 2021, MNRAS, 500, 3382
  * Changeat et al. (2020) Changeat, Q., Keyte, L., Waldmann, I. P., & Tinetti, G. 2020, ApJ, 896, 107
  * Chapman & Ferraro (1931) Chapman, S. & Ferraro, V. C. A. 1931, Terrestrial Magnetism and Atmospheric Electricity (Journal of Geophysical Research), 36, 77
  * Chavez et al. (2023) Chavez, E., de Pater, I., Redwing, E., et al. 2023, Icarus, 404, 115667
  * Claret & Bloemen (2011) Claret, A. & Bloemen, S. 2011, A&A, 529, A75
  * Cotton et al. (2019) Cotton, D. V., Evensberget, D., Marsden, S. C., et al. 2019, MNRAS, 483, 1574
  * Cracchiolo et al. (2021) Cracchiolo, G., Micela, G., & Peres, G. 2021, MNRAS, 501, 1733
  * Czesla et al. (2019) Czesla, S., Schröter, S., Schneider, C. P., et al. 2019, PyA: Python astronomy-related packages
  * Damasso et al. (2023) Damasso, M., Locci, D., Benatti, S., et al. 2023, A&A, 672, A126
  * Danielski et al. (2022) Danielski, C., Brucalassi, A., Benatti, S., et al. 2022, Experimental Astronomy, 53, 473
  * Di Maio et al. (2023) Di Maio, C., Changeat, Q., Benatti, S., & Micela, G. 2023, A&A, 669, A150
  * do Nascimento et al. (2016) do Nascimento, J. D., J., Vidotto, A. A., Petit, P., et al. 2016, ApJ, 820, L15
  * 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 et al. (1992) Donati, J. F., Brown, S. F., Semel, M., et al. 1992, A&A, 265, 682
  * Donati et al. (2006) Donati, J.-F., Forveille, T., Collier Cameron, A., et al. 2006, Science, 311, 633
  * Donati et al. (1997) Donati, J. F., Semel, M., Carter, B. D., Rees, D. E., & Collier Cameron, A. 1997, MNRAS, 291, 658
  * Dumusque et al. (2011) Dumusque, X., Udry, S., Lovis, C., Santos, N. C., & Monteiro, M. J. P. F. G. 2011, A&A, 525, A140
  * Duncan et al. (1991) Duncan, D. K., Vaughan, A. H., Wilson, O. C., et al. 1991, ApJS, 76, 383
  * Edwards et al. (2019) Edwards, B., Mugnai, L., Tinetti, G., Pascale, E., & Sarkar, S. 2019, AJ, 157, 242
  * Edwards & Tinetti (2022) Edwards, B. & Tinetti, G. 2022, AJ, 164, 15
  * Egeland et al. (2017) Egeland, R., Soon, W., Baliunas, S., et al. 2017, ApJ, 835, 25
  * Evensberget et al. (2021) Evensberget, D., Carter, B. D., Marsden, S. C., Brookshaw, L., & Folsom, C. P. 2021, MNRAS, 506, 2309
  * Evensberget et al. (2022) Evensberget, D., Carter, B. D., Marsden, S. C., et al. 2022, MNRAS, 510, 5226
  * Evensberget et al. (2023) Evensberget, D., Marsden, S. C., Carter, B. D., et al. 2023, MNRAS, 524, 2042
  * Fares et al. (2017) Fares, R., Bourrier, V., Vidotto, A. A., et al. 2017, MNRAS, 471, 1246
  * Fares et al. (2013) Fares, R., Moutou, C., Donati, J. F., et al. 2013, MNRAS, 435, 1451
  * Folsom et al. (2018) Folsom, C. P., Bouvier, J., Petit, P., et al. 2018, MNRAS, 474, 4956
  * Folsom et al. (2020) Folsom, C. P., Ó Fionnagáin, D., Fossati, L., et al. 2020, A&A, 633, A48
  * Folsom et al. (2016) Folsom, C. P., Petit, P., Bouvier, J., et al. 2016, MNRAS, 457, 580
  * Gagné et al. (2018) Gagné, J., Mamajek, E. E., Malo, L., et al. 2018, ApJ, 856, 23
  * Gaia Collaboration (2020) Gaia Collaboration. 2020, VizieR Online Data Catalog, I/350
  * Gardner et al. (2006) Gardner, J. P., Mather, J. C., Clampin, M., et al. 2006, Space Science Reviews, 123, 485
  * Gizis et al. (2002) Gizis, J. E., Reid, I. N., & Hawley, S. L. 2002, AJ, 123, 3356
  * Guenther & Kislyakova (2020) Guenther, E. W. & Kislyakova, K. G. 2020, MNRAS, 491, 3974
  * Hazra et al. (2020) Hazra, G., Vidotto, A. A., & D’Angelo, C. V. 2020, MNRAS, 496, 4017
  * Hintz et al. (2019) Hintz, D., Fuhrmeister, B., Czesla, S., et al. 2019, A&A, 623, A136
  * Høg et al. (2000) Høg, E., Fabricius, C., Makarov, V. V., et al. 2000, A&A, 355, L27
  * Huélamo et al. (2008) Huélamo, N., Figueira, P., Bonfils, X., et al. 2008, A$A, 489, L9
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90
  * Hussain et al. (2016) Hussain, G. A. J., Alvarado-Gómez, J. D., Grunhut, J., et al. 2016, A&A, 585, A77
  * Ip et al. (2004) Ip, W.-H., Kopp, A., & Hu, J.-H. 2004, ApJ, 602, L53
  * 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
  * Ketzer & Poppenhaeger (2023) Ketzer, L. & Poppenhaeger, K. 2023, MNRAS, 518, 1683
  * Klein et al. (2021) Klein, B., Donati, J.-F., Moutou, C., et al. 2021, MNRAS, 502, 188
  * 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
  * Kotov et al. (1998) Kotov, V. A., Scherrer, P. H., Howard, R. F., & Haneychuk, V. I. 1998, ApJS, 116, 103
  * Kubyshkina et al. (2022) Kubyshkina, D., Vidotto, A. A., Villarreal D’Angelo, C., et al. 2022, MNRAS, 510, 2111
  * Kumar & Fares (2023) Kumar, M. & Fares, R. 2023, MNRAS, 518, 3147
  * Lammer et al. (2003) Lammer, H., Selsis, F., Ribas, I., et al. 2003, ApJl, 598, L121
  * Landi Degl’Innocenti (1992) Landi Degl’Innocenti, E. 1992, Magnetic field measurements., ed. F. Sanchez, M. Collados, & M. Vazquez, 71
  * Lanza (2009) Lanza, A. F. 2009, A&A, 505, 339
  * Lehmann & Donati (2022) Lehmann, L. T. & Donati, J. F. 2022, MNRAS, 514, 2333
  * Lehmann et al. (2019) Lehmann, L. T., Hussain, G. A. J., Jardine, M. M., Mackay, D. H., & Vidotto, A. A. 2019, MNRAS, 483, 5246
  * Lehtinen et al. (2016) Lehtinen, J., Jetsu, L., Hackman, T., Kajatkari, P., & Henry, G. W. 2016, A&A, 588, A38
  * López Ariste et al. (2022) López Ariste, A., Georgiev, S., Mathias, P., et al. 2022, A&A, 661, A91
  * Louca et al. (2023) Louca, A. J., Miguel, Y., Tsai, S.-M., et al. 2023, MNRAS, 521, 3333
  * Magrini et al. (2022) Magrini, L., Danielski, C., Bossini, D., et al. 2022, A&A, 663, A161
  * Maldonado et al. (2022) Maldonado, J., Colombo, S., Petralia, A., et al. 2022, A&A, 663, A142
  * Mallorquín et al. (2023) Mallorquín, M., Béjar, V. J. S., Lodieu, N., et al. 2023, A&A, 671, A163
  * Mann et al. (2020) Mann, A. W., Johnson, M. C., Vanderburg, A., et al. 2020, AJ, 160, 179
  * Marsden et al. (2014) Marsden, S. C., Petit, P., Jeffers, S. V., et al. 2014, MNRAS, 444, 3517
  * Martioli et al. (2022) Martioli, E., Hébrard, G., Fouqué, P., et al. 2022, A&A, 660, A86
  * Mayor et al. (2003) Mayor, M., Pepe, F., Queloz, D., et al. 2003, The Messenger, 114, 20
  * Meng et al. (2015) Meng, X., van der Holst, B., Tóth, G., & Gombosi, T. I. 2015, MNRAS, 454, 3697
  * Mengel et al. (2016) Mengel, M. W., Fares, R., Marsden, S. C., et al. 2016, MNRAS, 459, 4325
  * Middelkoop (1982) Middelkoop, F. 1982, A&A, 107, 31
  * Mittag et al. (2017) Mittag, M., Robrade, J., Schmitt, J. H. M. M., et al. 2017, A&A, 600, A119
  * Montes et al. (2000) Montes, D., Fernández-Figueroa, M. J., De Castro, E., et al. 2000, A&AS, 146, 103
  * Morgenthaler et al. (2012) Morgenthaler, A., Petit, P., Saar, S., et al. 2012, A&A, 540, A138
  * 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
  * Moutou et al. (2007) Moutou, C., Donati, J. F., Savalle, R., et al. 2007, A&A, 473, 651
  * Neubauer (1998) Neubauer, F. M. 1998, J. Geophys. Res., 103, 19843
  * Noyes et al. (1984) Noyes, R. W., Hartmann, L. W., Baliunas, S. L., Duncan, D. K., & Vaughan, A. H. 1984, ApJ, 279, 763
  * Pace (2013) Pace, G. 2013, A&A, 551, L8
  * Petit et al. (2013) Petit, P., Aurière, M., Konstantinova-Antova, R., et al. 2013, in Lecture Notes in Physics, Berlin Springer Verlag, ed. J.-P. Rozelot & C. E. . Neiner, Vol. 857, 231
  * Petit et al. (2008) Petit, P., Dintrans, B., Solanki, S. K., et al. 2008, MNRAS, 388, 80
  * Petit et al. (2005) Petit, P., Donati, J. F., Aurière, M., et al. 2005, MNRAS, 361, 837
  * Piskunov et al. (2011) Piskunov, N., Snik, F., Dolgopolov, A., et al. 2011, The Messenger, 143, 7
  * Piskunov et al. (2021) Piskunov, N., Wehrhahn, A., & Marquart, T. 2021, A&A, 646, A32
  * Piskunov & Valenti (2002) Piskunov, N. E. & Valenti, J. A. 2002, A&A, 385, 1095
  * Rackham et al. (2018) Rackham, B. V., Apai, D., & Giampapa, M. S. 2018, ApJ, 853, 122
  * Rackham et al. (2019) Rackham, B. V., Apai, D., & Giampapa, M. S. 2019, AJ, 157, 96
  * Rainer et al. (2023) Rainer, M., Desidera, S., Borsa, F., et al. 2023, A&A, 676, A90
  * Rees & Semel (1979) Rees, D. E. & Semel, M. D. 1979, A&A, 74, 1
  * Ribas et al. (2016) Ribas, I., Bolmont, E., Selsis, F., et al. 2016, A&A, 596, A111
  * Ribas et al. (2005) Ribas, I., Guinan, E. F., Güdel, M., & Audard, M. 2005, ApJ, 622, 680
  * Rosén et al. (2015) Rosén, L., Kochukhov, O., & Wade, G. A. 2015, ApJ, 805, 169
  * Rusomarov et al. (2013) Rusomarov, N., Kochukhov, O., Piskunov, N., et al. 2013, A&A, 558, A8
  * Rutten (1984) Rutten, R. G. M. 1984, A&A, 130, 353
  * Ryabchikova et al. (2015) Ryabchikova, T., Piskunov, N., Kurucz, R. L., et al. 2015, Phys. Scr., 90, 054005
  * Sachdeva et al. (2019) Sachdeva, N., van der Holst, B., Manchester, W. B., et al. 2019, ApJ, 887, 83
  * Salz et al. (2018) Salz, M., Czesla, S., Schneider, P. C., et al. 2018, A&A, 620, A97
  * Saur et al. (2013) Saur, J., Grambusch, T., Duling, S., Neubauer, F. M., & Simon, S. 2013, A&A, 552, A119
  * Schneiter et al. (2007) Schneiter, E. M., Velázquez, P. F., Esquivel, A., Raga, A. C., & Blanco-Cano, X. 2007, ApJ, 671, L57
  * Segura et al. (2010) Segura, A., Walkowicz, L. M., Meadows, V., Kasting, J., & Hawley, S. 2010, Astrobiology, 10, 751
  * Semel (1989) Semel, M. 1989, A&A, 225, 456
  * Shkolnik et al. (2008) Shkolnik, E., Bohlender, D. A., Walker, G. A. H., & Collier Cameron, A. 2008, ApJ, 676, 628
  * Skilling & Bryan (1984) Skilling, J. & Bryan, R. K. 1984, MNRAS, 211, 111
  * Snik et al. (2011) Snik, F., Kochukhov, O., Piskunov, N., et al. 2011, in Astronomical Society of the Pacific Conference Series, Vol. 437, Solar Polarization 6, ed. J. R. Kuhn, D. M. Harrington, H. Lin, S. V. Berdyugina, J. Trujillo-Bueno, S. L. Keil, & T. Rimmele, 237
  * Sokolov et al. (2013) Sokolov, I. V., van der Holst, B., Oran, R., et al. 2013, ApJ, 764, 23
  * Strassmeier et al. (2023) Strassmeier, K. G., Carroll, T. A., & Ilyin, I. V. 2023, A&A, 674, A118
  * Strugarek et al. (2015) Strugarek, A., Brun, A. S., Matt, S. P., & Réville, V. 2015, ApJ, 815, 111
  * Thompson et al. (2024) Thompson, A., Biagini, A., Cracchiolo, G., et al. 2024, ApJ, 960, 107
  * Tilley et al. (2019) Tilley, M. A., Segura, A., Meadows, V., Hawley, S., & Davenport, J. 2019, Astrobiology, 19, 64
  * Tinetti et al. (2021) Tinetti, G., Eccleston, P., Haswell, C., et al. 2021, arXiv e-prints, arXiv:2104.04824
  * Tóth et al. (2005) Tóth, G., Sokolov, I. V., Gombosi, T. I., et al. 2005, Journal of Geophysical Research (Space Physics), 110, A12226
  * Tóth et al. (2012) Tóth, G., van der Holst, B., Sokolov, I. V., et al. 2012, Journal of Computational Physics, 231, 870
  * van der Holst et al. (2019) van der Holst, B., Manchester, W. B., I., Klein, K. G., & Kasper, J. C. 2019, ApJ, 872, L18
  * van der Holst et al. (2014) van der Holst, B., Sokolov, I. V., Meng, X., et al. 2014, The Astrophysical Journal, 782, 81
  * van der Walt et al. (2011) van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science and Engineering, 13, 22
  * Van Looveren et al. (2024) Van Looveren, G., Güdel, M., Boro Saikia, S., & Kislyakova, K. 2024, A&A, 683, A153
  * Vaughan et al. (1978) Vaughan, A. H., Preston, G. W., & Wilson, O. C. 1978, PASP, 90, 267
  * Vidotto (2021) Vidotto, A. A. 2021, Living Reviews in Solar Physics, 18, 3
  * Vidotto et al. (2023) Vidotto, A. A., Bourrier, V., Fares, R., et al. 2023, A&A, 678, A152
  * Vidotto & Cleary (2020) Vidotto, A. A. & Cleary, A. 2020, MNRAS, 494, 2417
  * Vidotto et al. (2010) Vidotto, A. A., Jardine, M., & Helling, C. 2010, ApJ, 722, L168
  * Vidotto et al. (2014) Vidotto, A. A., Jardine, M., Morin, J., et al. 2014, MNRAS, 438, 1162
  * Vidotto et al. (2018) Vidotto, A. A., Lichtenegger, H., Fossati, L., et al. 2018, MNRAS, 481, 5296
  * Vidotto et al. (2009) Vidotto, A. A., Opher, M., Jatenco-Pereira, V., & Gombosi, T. I. 2009, ApJ, 703, 1734
  * Villarreal D’Angelo et al. (2018) Villarreal D’Angelo, C., Esquivel, A., Schneiter, M., & Sgró, M. A. 2018, MNRAS, 479, 3115
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Burovski, E., et al. 2020, scipy/scipy: SciPy 1.5.3
  * Willamo et al. (2022) Willamo, T., Lehtinen, J. J., Hackman, T., et al. 2022, A&A, 659, A71
  * 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
  * Wright et al. (2018) Wright, N. J., Newton, E. R., Williams, P. K. G., Drake, J. J., & Yadav, R. K. 2018, MNRAS, 479, 2351
  * Zarka et al. (2001) Zarka, P., Treumann, R. A., Ryabov, B. P., & Ryabov, V. B. 2001, Ap&SS, 277, 293
  * Zeeman (1897) Zeeman, P. 1897, Nature, 55, 347
  * Zhang et al. (2022) Zhang, M., Knutson, H. A., Wang, L., et al. 2022, AJ, 163, 68
  * Zingales et al. (2018) Zingales, T., Tinetti, G., Pillitteri, I., et al. 2018, Experimental Astronomy, 46, 67