11institutetext:  Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France 22institutetext: Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Alonso de Rivera 2850, Concepción, Chile 33institutetext: NASA Exoplanet Science Institute, Caltech/IPAC, Pasadena, CA 91125, USA 44institutetext: Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 55institutetext: Observatoire de Genève, Département d’Astronomie, Université de Genève, Chemin Pegasi 51, 1290 Versoix, Switzerland 66institutetext: International Center for Advanced Studies (ICAS) and ICIFI (CONICET), ECyT-UNSAM, Campus Miguelete, 25 de Mayo y Francia, (1650) Buenos Aires, Argentina 77institutetext: Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD, 21218, USA 88institutetext: Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 99institutetext: NASA Ames Research Center, Moffett Field, CA 94035, USA 1010institutetext: Instituto de Astrofísica de Canarias (IAC), E-38200 La Laguna, Tenerife, Spain 1111institutetext: Dept. Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain 1212institutetext: Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 1313institutetext: Department of Aeronautics and Astronautics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA 1414institutetext: Planetary Discoveries in Fredericksburg, VA 22405, USA

# TOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a low-mass star
with a giant and likely polar spot

J.M. Almenara TOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spotTOI-3884 b: A rare
6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spot X. Bonfils TOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that
transits a low-mass star with a giant and likely polar spotTOI-3884 b: A rare
6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spot T. Forveille TOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet
that transits a low-mass star with a giant and likely polar spotTOI-3884 b: A
rare 6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spot N. Astudillo-Defru TOI-3884 b: A rare 6-$\mathrm{R_{E}}$
planet that transits a low-mass star with a giant and likely polar
spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a low-mass star
with a giant and likely polar spot D.R. Ciardi TOI-3884 b: A rare
6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot R.P. Schwarz TOI-3884 b: A
rare 6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot K.A. Collins TOI-3884 b: A
rare 6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot M. Cointepas TOI-3884 b: A
rare 6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spotTOI-3884 b: A rare
6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot M.B. Lund TOI-3884 b: A rare
6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot F. Bouchy TOI-3884 b: A rare
6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot D. Charbonneau TOI-3884 b: A
rare 6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot R.F. Díaz TOI-3884 b: A rare
6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot X. Delfosse TOI-3884 b: A
rare 6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot R.C. Kidwell TOI-3884 b: A
rare 6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot M. Kunimoto TOI-3884 b: A
rare 6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot D.W. Latham TOI-3884 b: A
rare 6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot J.J. Lissauer TOI-3884 b: A
rare 6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot F. Murgas TOI-3884 b: A rare
6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spotTOI-3884 b: A rare
6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot G. Ricker TOI-3884 b: A rare
6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot S. Seager TOI-3884 b: A rare
6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spotTOI-3884 b: A rare
6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spotTOI-3884 b: A rare
6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot M. Vezie TOI-3884 b: A rare
6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot D. Watanabe TOI-3884 b: A
rare 6-$\mathrm{R_{E}}$ planet that transits a low-mass star with a giant and
likely polar spotTOI-3884 b: A rare 6-$\mathrm{R_{E}}$ planet that transits a
low-mass star with a giant and likely polar spot

The Transiting Exoplanet Survey Satellite mission identified a deep and
asymmetric transit-like signal with a periodicity of 4.5 days orbiting the M4
dwarf star TOI-3884. The signal has been confirmed by follow-up observations
collected by the ExTrA facility and Las Cumbres Observatory Global Telescope,
which reveal that the transit is chromatic. The light curves are well modelled
by a host star having a large polar spot transited by a 6-$\mathrm{R_{E}}$
planet. We validate the planet with seeing-limited photometry, high-resolution
imaging, and radial velocities. TOI-3884 b, with a radius of $6.00\pm 0.18$
$\mathrm{R_{E}}$, is the first sub-Saturn planet transiting a mid-M dwarf.
Owing to the host star’s brightness and small size, it has one of the largest
transmission spectroscopy metrics for this planet size and becomes a top
target for atmospheric characterisation with the James Webb Space Telescope
and ground-based telescopes.

###### Key Words.:

stars: individual: TOI-3884 – stars: low-mass – (stars:) starspots – (stars:)
planetary systems – techniques: photometric – techniques: radial velocities

##  1 Introduction

Over the past thirty years, more than 3,500 transiting planets have been
discovered and important statistical properties have emerged. For example, it
has been found that planets larger than Neptune are uncommon around later type
stars. The Bern population synthesis model, for instance, predicts that no
planet larger than 5 $\mathrm{R_{E}}$ exists around stars with $M_{\star}<0.3$
$M_{\odot}$ (Burn et al. 2021). And indeed, no planet larger than $\sim 3.5$
$\mathrm{R_{E}}$ has been found so far around any star with $R_{\star}\lesssim
0.35$ $R_{\odot}$. One may note an apparent exception to this statistic: the
radial-velocity (RV) detection of a giant planet around the small star GJ 3512
(0.14 $R_{\odot}$). With a minimum mass of $M_{p}\sin i\sim 147$
$\mathrm{M_{E}}$, GJ 3512 b is presumably larger than $\sim 10$
$\mathrm{R_{E}}$ (Morales et al. 2019). GJ 3512 b is, however, found beyond
the ice line ($a=0.34$ au) of its star, and Burn et al. (2021) could, although
with difficulty, form a similar planet by slowing down planet migration in
their models. That model tuning would, however, not work for a giant planet
closer to the star, as its migration could not be turned down.

TOI-3884.01, a planet candidate identified by the Transiting Exoplanet Survey
Satellite (TESS, Ricker et al. 2015) mission, is also at odds with this
statistic, and is another potential challenge to planet formation models.
TOI-3884.01 would be 6 $\mathrm{R_{E}}$ in radius and orbiting at $0.035$ au
from a 0.30 $R_{\odot}$ star. This is above the largest predicted size for
planets around such stars, especially at a small separation.

In addition, the transits of TOI-3884.01 are asymmetric, which we show can be
explained by a giant polar spot on the surface of its fully convective host
star. Polar spots do not exist on the Sun but their existence has been
proposed since the first Doppler image of stars with short rotation periods
(Vogt & Penrod 1983; Strassmeier 1996). Concerning M dwarfs, several studies
using spectropolarimetric observations (Donati et al. 2006; Morin et al. 2008;
Moutou et al. 2017) have demonstrated that mid-M dwarfs with short rotation
periods frequently develop a dipolar and axisymmetric magnetic field. Hébrard
et al. (2016) show that, for this spectral type, spots are frequently
concentrated at the magnetic pole, and this suggests that polar spots may be
frequent.

In this Letter we report the confirmation of the sub-Saturn planet TOI-3884 b
transiting an M4 dwarf. This planet is the best target of its size for
atmospheric characterisation.

##  2 Observations

###  2.1 TESS

The TESS mission observed TOI-3884 (TIC 86263325) in sector 22 with a
30-minute cadence111The photometry is available in the full-frame images
(FFIs) that were calibrated by the SPOC at NASA’s Ames Research Center
(Jenkins et al. 2016)., and in sectors 46 and 49 with a 2-minute cadence, for
a total of 81 days spanning 765 days. A transit signature with a 4.545-day
period and 4% depth was first identified by the Faint Star Search (Kunimoto et
al. 2022) using data products from the Quick-Look Pipeline (QLP, Huang et al.
2020; Kunimoto et al. 2021). Gaia has detected no other star within the TESS
aperture (Fig. 3, Aller et al. 2020; Gaia Collaboration et al. 2018). The TESS
Science Processing Operations Centre (SPOC) Simple Aperture Photometry (SAP,
Twicken et al. 2010; Morris et al. 2020) and the QLP Kepler Spline SAP
(KSPSAP, Vanderburg & Johnson 2014) light curves are shown in Fig. 4. The SPOC
SAP light curve shows no signs of rotational modulation. For the analysis in
Sect. 4, we used the KSPSAP (sector 22) and the Pre-search Data Conditioning
(PDC) SAP (PDCSAP, Stumpe et al. 2012, 2014; Smith et al. 2012, sectors 46 and
49) light curves in which long-term trends have been removed. TESS observed a
total of 13 transits and Fig. 4 shows one composite transit per sector. The
asymmetry of the transit-like signature in sector 46 triggered our follow-up
observations with the Exoplanets in Transits and their Atmospheres (ExTrA)
project.

###  2.2 Near-IR photometry with ExTrA

The ExTrA facility (Bonfils et al. 2015), located at La Silla observatory,
consists of a near-IR (0.88 to 1.55 $\mu$m) multi-object spectrograph fibre-
fed by three 60-cm telescopes. Five fibre positioners intercept the light from
one target and four comparison stars at the focal plane of each telescope. We
observed five transits of TOI-3884 b on nights UTC 2022 February 22, 2022
March 3, 2022 April 4, 2022 April 13, and 2022 May 15. We observed with one,
two, or three telescopes simultaneously. We used the 8″ aperture fibres, the
low-resolution mode of the spectrograph (R$\sim$20), and 60-second exposures.
We chose comparison stars with effective temperatures (Gaia Collaboration et
al. 2018) similar to that of TOI-3884. The ExTrA data were analysed using
custom data reduction software.

###  2.3 Seeing-limited optical photometry

We obtained one additional ground-based photometric follow-up observation of
TOI-3884 with the Las Cumbres Observatory Global Telescope (LCOGT; Brown et
al. 2013) and as part of the TESS Follow-up Observing
Program222https://tess.mit.edu/followup (TFOP; Collins 2019), in an attempt to
rule out or identify nearby eclipsing binaries (NEBs) as potential sources of
the TESS detection, to measure the transit-like event on target to confirm its
depth, and thus the TESS photometric deblending factor, and to refine the TESS
ephemeris. We scheduled our transit observations using the TESS Transit
Finder, which is a customised version of the Tapir software package (Jensen
2013).

The LCOGT observation was performed with a 1-m telescope at Teide observatory
in the g’ filter, with an exposure time of 300 seconds. The data were
calibrated by the standard LCOGT BANZAI pipeline (McCully et al. 2018), and
the photometry was extracted using AstroImageJ (Collins et al. 2017). The
transit of TOI-3884 b was detected on-target within the photometric aperture
of 4.7″.

###  2.4 High-resolution imaging

As part of our standard process for validating transiting exoplanets and
assessing the possible contamination of the derived planetary radii by bound
or unbound stars in the TESS aperture (Ciardi et al. 2015), we obtained near-
IR adaptive optics (AO) imaging of TOI-3884 at Palomar Observatory. The
Palomar Observatory observations of TOI-3884 were made in the narrow-band
Br-$\gamma$ filter
$(\lambda_{o}=2.1686;\Delta\lambda=0.0326\leavevmode\nobreak\ \mu$m) with the
PHARO instrument (Hayward et al. 2001) behind the P3K natural guide star AO
system (Dekany et al. 2013) on 2022 February 13 in a standard 5-point quincunx
dither pattern with steps of 5″. Each dither position was observed three
times, offset in position from each other by 0.5″ for a total of 15 frames;
the individual exposures were 31.1 seconds long, for a total on-source times
of 466 seconds. PHARO has a pixel scale of $0.025\arcsec$ per pixel for a
total field of view of $\sim 25\arcsec$.

The AO data were processed and analysed with a custom set of Interactive Data
Language tools. The flat fields were generated from a median of dark
subtracted exposures of the sky. Those flats were then normalised to a median
value of one. The sky frames were generated as the median of the 15 dithered
science frames; each science image was then sky-subtracted and flat-fielded.
The reduced science frames were combined into a single combined image using an
intra-pixel interpolation that conserves flux, shifts the individual dithered
frames by the appropriate fractional pixels, and median-co-adds the frames.
The resolution of the combined image was determined from the full-width half-
maximum (FWHM) of the point spread functions: 0.105″.

The sensitivity of the combined AO image to the companions was determined by
injecting simulated sources around the primary target at separations of
integer multiples of the central source’s FWHM and every $20^{\circ}$ in
azimuth (Furlan et al. 2017). The brightness of each injected source was
scaled until standard aperture photometry detected it with $5\sigma$
significance, and the brightness relative to TOI-3884 of the faintest injected
sources that was detected set the contrast limit at that injection location.
The adopted $5\sigma$ limit at each separation is then the average of the
limits for the 18 azimuths at that radial distance, and the uncertainty on the
limit is their root mean square dispersion. The resulting sensitivity curve
for the Palomar image is shown in Fig. 5; no stellar companions were detected.

###  2.5 Gaia assessment

We used Gaia to probe for wide stellar companions outside the field of view of
the PHARO image that would be bound to TOI-3884. Such stars are typically
already in the TESS Input Catalog, and their flux dilution to the transit has
therefore already been accounted for in the transit fits and the associated
derived parameters. Based upon parallax and proper motion similarity (e.g.
Mugrauer & Michel 2020, 2021), Gaia identifies no such companions.

The Gaia data release 3 (DR3) astrometry (Gaia Collaboration et al. 2022) also
contains indirect information on potential inner companions that would have
gone undetected by both Gaia and the AO imaging. The Gaia renormalised unit
weight error (RUWE) is a goodness of fit metric, akin to a reduced chi-square.
Values ${\lesssim}\leavevmode\nobreak\ 1.4$ indicate that the Gaia astrometry
is consistent with a single star, whereas RUWE values
${\gtrsim}\leavevmode\nobreak\ 1.4$ indicate excess astrometric noise, which
can be from an unseen companion (e.g. Ziegler et al. 2020). TOI-3884 has a
Gaia DR3 RUWE value of 1.25, and its Gaia astrometry is therefore fully
consistent with a single star.

###  2.6 ESPRESSO

We obtained RV measurements of TOI-3884 with ESPRESSO (Pepe et al. 2021), a
fibre-fed spectrograph installed at the Very Large Telescope at Paranal
Observatory. Under the ESO programme 109.24EP, we collected two reconnaissance
spectra with an exposure time of 1300 seconds, 2x1 binning, using the slow
readout mode, and the single-telescope HR mode (R$\sim$140 000). During the
exposure we elected to maintain the calibration fibre on sky. The signal-to-
noise ratio at 756 nm for the two measurements were 25 and 29, equivalent to
expected RV precisions of 1.9 m/s and 1.6 m/s. The data reduction and RV
extraction were done through the ESPRESSO-ESO
pipeline333https://www.eso.org/sci/software/pipelines/. The cross-correlation
function (CCF) was derived by using the M4 mask available in the pipeline. The
RVs and CCF characteristics are listed in Table 1. Visual inspection of the
CCFs clearly exclude the case of a double-line spectroscopic binary. We do
measure a slight but significant variation in the contrast and FWHM of the
CCFs between the two epochs. This could be due to a change in the contribution
from the starspot to the averaged temperature.

##  3 Stellar parameters

We used empirical relations to derive the mass and radius of TOI-3884 from its
absolute magnitude in the $K_{s}$-band, which is less affected by stellar
spots and other activity phenomena than bluer bands. We used the zero-point
corrected (Lindegren et al. 2021) Gaia parallax (Gaia Collaboration et al.
2016, 2021) to compute the distance and obtain an absolute magnitude of
$M_{K_{s}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 7.056\leavevmode\nobreak\
\pm\leavevmode\nobreak\ 0.017$. We then used the empirical relations of (Mann
et al. 2019) and (Mann et al. 2015) to derive a mass of $M_{\star}=0.2813\pm
0.0067$ $M_{\odot}$ and a radius of $R_{\star}=0.3043\leavevmode\nobreak\
\pm\leavevmode\nobreak\ 0.0090$ $R_{\odot}$, which we adopt as our preferred
values (Table 2). We derived an alternative stellar radius from the spectral
energy distribution (SED), which we constructed using photometry from Gaia
(Riello et al. 2021), the 2-Micron All-Sky Survey (2MASS, Skrutskie et al.
2006; Cutri et al. 2003), and the Wide-field Infrared Survey Explorer (WISE,
Wright et al. 2010; Cutri & et al. 2013). We modelled the photometry (Table 2)
using the Bayesian procedure described in Díaz et al. (2014), with the
PHOENIX/BT-Settl (Allard et al. 2012) stellar atmosphere models. We used
informative priors for the effective temperature ($T_{\mathrm{eff}}=3269\pm
70$ K, which corresponds to an M4 spectral type) and metallicity
($[\rm{Fe/H}]=0.23\pm 0.12$ dex), both derived from the co-added ESPRESSO
spectra (which we analysed with SpecMatch-Emp; Yee et al. 2017), and the
distance from Gaia. We used non-informative priors for the rest of the
parameters. We used one jitter term (Gregory 2005) for each set of photometric
bands (Gaia, 2MASS, and WISE). The parameters, priors, and posterior median,
and the 68% credible interval (CI) are listed in Table 3. The maximum a
posteriori (MAP) model is shown in Fig. 6. The derived radius
($R_{\star}=0.3005\pm 0.0090$ $R_{\odot}$), is compatible with the adopted
one. Evolution models predict that TOI-3884 is fully convective (Chabrier &
Baraffe 1997). We repeated the SED modelling with non-informative priors for
the effective temperature and metallicity, and obtained an effective
temperature of 3270${}^{+270}_{-170}$ K, compatible with the one derived with
the ESPRESSO spectra, but less precise.

Rotational broadening can be measured with the FWHM of the CCF (Santos et al.
2002). This usually requires calibrating the FWHM0 for a set of stars with a
range of temperatures and with unresolved stellar rotations. We did not
perform such a full calibration but, instead, we picked one other M dwarf, LHS
1140, with a similar spectral type (M4.5) to TOI-3884, a slow rotation (131
days), and also observed with ESPRESSO (Lillo-Box et al. 2020). We reduced LHS
1140 spectra taken at three epochs (2019 December 9, 14, and 15) and measured
an average CCF FWHM of 5.29 km/s. In comparison, we measured an FWHM = 5.46
km/s for TOI-3884. We then followed Hirano et al. (2010) to derive $v$ sin
$i_{\star}$ as $(1.23\;\hbox{$v$\,sin\,$i_{\star}$})^{2}={\rm FWHM}_{\rm
TOI-3884}^{2}-{\rm FWHM}_{\rm 0,\;LHS\,1140}^{2}$, and obtained a value of $v$
sin $i_{\star}$ = 1.1 km/s. Given $R_{\star}$ from Table 2 and $i_{\star}$
from Table 4, this corresponds to a rotation period of $\sim$18 days. In
addition, we measured $\log R^{\prime}_{HK}=-4.79\pm 0.13$ in the co-added
ESPRESSO spectrum, from which the Astudillo-Defru et al. (2017) activity-
rotation relation estimates a rotation period of $P_{\rm rot}=30\pm 6$ days;
this measurement must be taken with caution though, because of the poor
signal-to-noise ratio of the spectrum at the H and K Ca II lines.

##  4 Analysis

The shape of the asymmetric transit of TOI-3884 b seems stable over at least
two years, and it changes with observing wavelength (chromatic effect). In
addition, the TESS light curve shows no signs of rotational modulation. We
explored two models that can explain such characteristics: a rotationally
flattened and gravity darkened very rapidly rotating star (Barnes 2009;
Dholakia et al. 2022), and a polar spot. The rapidly rotating star model is
strongly excluded by the ESPRESSO spectra because it needs $v$ sin $i_{\star}$
of 350 km/s. Therefore, the polar spot is our adopted model. It should be
noted that a circumpolar inhomogeneity such as a ring would match the data as
well as a polar spot.

We first modelled the TESS, ExTrA, and LCOGT transits and the ESPRESSO RVs
with juliet (Espinoza et al. 2019), which uses batman (Kreidberg 2015) for its
transit model and radvel (Fulton et al. 2018) to model the RVs. We added a
Gaussian process (GP) regression model, with an approximate Matern kernel
(celerite, Foreman-Mackey et al. 2017) for the model of the error terms of the
photometry (with different kernel hyperparameters for each transit
observation, except for the TESS data, for which we used common kernel
hyperparameters within each sector), to account for the asymmetry of the
transit, and instrument-dependent dilution factors to deal with the
chromaticity. We used a quadratic limb-darkening law (Manduca et al. 1977),
parameterised following Kipping (2013). We used a stellar density prior from
Sect. 3, and adopted non-informative priors for the rest of the parameters. We
sampled from the posterior with dynesty (Speagle 2020). Table 4 lists the
parameters, priors, and posteriors. Figures 7 and 9 show the data and the
model posterior.

We performed a second analysis, using only the transit data, to model the
starspot. Within the uncertainties of the light curves, the shape of the
transit remained constant at each wavelength (Fig. 8). We therefore phase-
folded the light curves444We previously checked, using juliet, that there are
no significant transit timing variations (Fig. 10). to speed up the spot
modelling computations. To do this, we used the MAP model of the analysis with
juliet to normalise the TESS555We did not use the sector 22 transits, observed
with a 30-minute cadence, which blurs their shape., ExTrA, and LCOGT transits
(imposing a straight line between the first and fourth contacts in order to
not remove the transit or the asymmetry), and folded the transits according to
the posterior ephemeris of the analysis with juliet (Table 4). We modelled the
folded transits in the three bands (g’, TESS, and ExTrA) with the starry code
(Luger et al. 2019, 2021b, 2021a). We describe the surface of the star with a
quadratic limb-darkening law (as in the modelling with juliet), and a top hat
function in angular separation (from a longitude and latitude of the stellar
surface) described with spherical harmonic expansion up to degree 30,
representing a circular spot with uniform contrast. As we assume a polar spot,
we fixed its latitude to 90°; thus, the model is independent of the value of
its longitude and the rotation period of the star. The inclination of the spin
axis and the sky-projected spin-orbit angle, which determine the position of
the pole relative to the transit cord, are free parameters of the model. The
spot contrast666The spot contrast is defined in starry as the fractional
change in the intensity at the centre of the spot ($I_{\rm spot}$) relative to
the baseline intensity of an unspotted stellar surface ($I_{\rm
photosphere}$): ${\rm contrast}=1-\frac{I_{\rm spot}}{I_{\rm photosphere}}$.
is a free parameter for each of the g’, TESS, and ExTrA bands, which have a
respective $\lambda_{\rm pivot}$ (Koornneef et al. 1986) of 476, 770, and 1165
nm. We oversampled the model in time and accounted for the integration time of
the observations through binning (Kipping 2010). We used the pymc3 package
(Salvatier et al. 2016; Hoffman & Gelman 2011) to sample from the posterior.
The list of parameters, priors, and posteriors are shown in Table 4, and the
data and model posterior are shown in Fig. 1. Figure 11 shows the spot
contrast for the three bands. From these three spot contrasts, we estimate a
photosphere-spot difference temperature of 187 $\pm$ 21 K (Sect. B), which is
in the range measured for M dwarfs (Berdyugina 2005; Barnes et al. 2015).

Figure 1: Phased transits observed by LCOGT, TESS (only sectors 46 and 49),
and ExTrA, and the posterior model of a planet transiting a star with a polar
spot. For clarity, the figure only shows 5-minute bins in the phased light
curve for TESS and ExTrA. Above the panel: Model of the stellar surface and
the planet for the median posterior values, as seen by TESS at different times
during the transit.

Follow-up transit observations combined with the high-resolution imaging rule
out most nearby eclipsing binary scenarios. Thanks to the significant proper
motion of the target, historical images exclude background eclipsing binaries
(Fig. 12). This leaves as a false positive scenario only a bound triple star,
in which an eclipsing binary is diluted by a third star. The high-resolution
imaging, the Gaia RUWE, and the single-lined ESPRESSO spectra excluded some of
these configurations. As an additional check, we repeated the modelling with
starry, allowing for a different planet-to-star radius ratio in each band. The
three ratios are then compatible (see Fig. 13), and reinforce the planetary
scenario. The ESPRESSO RVs, the CCF contrast, and the CCF bisector span are
compatible with a single star, and confirm the planetary nature of TOI-3884 b.

##  5 Results and discussion

The TOI-3884 system is composed of an M4 dwarf star orbited by a $6.00\pm
0.18$ $\mathrm{R_{E}}$ planet, TOI-3884 b, with a period of 4.5 days. This
planet size, between gas Giants and Neptunians, is especially rare around low-
mass stars (Fig. 2). The very preliminary mass that we obtain for TOI-3884 b,
assuming neither stellar activity nor other planets contribute to the RVs, is
consistent with that of Neptune, but the derived planet density would be one-
fourth that of Neptune. The transmission spectroscopy metric (TSM, Kempton et
al. 2018) of TOI-3884 b is $442\pm 90$. As shown by Fig. 2, this TSM is the
most favourable to date for an $R_{p}\sim 6$ $\mathrm{R_{E}}$ planet, owing to
the small size of the host star and its relative brightness.

Follow-up investigations for atmospheric characterisation will require good
constraints on the mass of the planet, to 20% precision or better. Indeed,
transmission spectroscopy depends on the scale height of the atmosphere, which
itself depends on the planet mass. A firm mass measurement with additional RVs
is therefore required to support further follow-up observations with the James
Webb Space Telescope (JWST, Gardner et al. 2006).

Our preferred model has the planet occulting a giant polar spot of 49° radius,
covering about 17% of the stellar surface. Due to the inclination of the
stellar spin axis, a similar spot on the opposite pole would be hidden from
our viewpoint, and thus would not affect the observed properties of TOI-3884
b. A giant polar spot suggests a large-scale magnetic field that is
axisymmetric and poloidal. This is in agreement with spectropolarimetric
surveys of M dwarfs that have shown that stars with mass ranging from 0.15 to
0.5 $M_{\odot}$, and with short rotation periods, generate strong and long-
lived magnetic fields featuring a significant large-scale poloidal component
(Donati et al. 2006, Morin et al. 2008, and for a general picture, see Fig. 14
of Moutou et al. 2017). This magnetic topology is very common for rotation
periods of less than 10 days, but there are weak constraints on whether this
behaviour extends to longer periods of 20 days. CE Boo (GJ 569A; 0.45
$M_{\odot}$, $P_{\rm rot}\sim 15$ d) is one of the few stars in this period
range, close to that of TOI-3884, with a reconstructed magnetic map (Donati et
al. 2008), and its topology is almost completely poloidal and mostly
axisymmetric. The magnetic activity of the star could be further studied with
spectropolarimetry in the near infrared, for example with SPIRou (Donati et
al. 2020). This could be important because of the influence this large spot
can have on the planet transmission spectroscopy.

The expected amplitude of the Rossiter–McLaughlin effect (Rossiter 1924;
McLaughlin 1924) for TOI-3884 b is 40 m/s, and this observation will provide
an independent measurement of the obliquity that can validate the results from
the polar spot model. A more detailed analysis of spectroscopic observations
during transit can map the stellar surface occulted by the planet (Bourrier et
al. 2021) and will test the presence of the polar spot. If confirmed, the
polar spot allows the measurement of the true spin–orbit angle ($\psi$), and
we find that it would be misaligned ($\psi=50\pm 12\degree$ or $130\pm
12\degree$). This measurement will probe the history of the planet, which
almost certainly had to migrate to end up so close to its star.

Figure 2: TOI-3884 b in context. Left: Planetary versus stellar radius. Black
dots are transiting planets listed in the NASA Exoplanet Archive
(https://exoplanetarchive.ipac.caltech.edu/ with a planetary radius
uncertainty smaller than 20%. The blue circle with error bars marks the
position of TOI-3884 b. The histogram of known planets is shown at the right
of the panel. Centre: Planet radius versus insolation flux. Error bars mark
the position of TOI-3884 b (blue) and grey dots are planets listed in the NASA
Exoplanet Archive with a radius uncertainty smaller than 20%. Right:
Transmission spectroscopy metric for planets with a radius between 1 and 10
$\mathrm{R_{E}}$. Data are taken from the NASA Exoplanet Archive. Grey areas
are below the cutoffs that Kempton et al. (2018) suggested for follow-up
efforts. The symbol with error bars is TOI-3884 b. The colour indicates the
equilibrium temperature of the planet, computed for zero albedo and full day-
night heat redistribution.

###### Acknowledgements.

Funding for the TESS mission is provided by NASA’s Science Mission
Directorate. We acknowledge the use of public TESS data from pipelines at the
TESS Science Office and at the TESS Science Processing Operations Center. This
research has made use of the Exoplanet Follow-up Observation Program website,
which is operated by the California Institute of Technology, under contract
with the National Aeronautics and Space Administration under the Exoplanet
Exploration Program. Resources supporting this work were provided by the NASA
High-End Computing (HEC) Program through the NASA Advanced Supercomputing
(NAS) Division at Ames Research Center for the production of the SPOC data
products. This paper includes data collected by the TESS mission that are
publicly available from the Mikulski Archive for Space Telescopes (MAST). This
research has made use of the Exoplanet Follow-up Observation Program (ExoFOP;
DOI: 10.26134/ExoFOP5) website, which is operated by the California Institute
of Technology, under contract with the National Aeronautics and Space
Administration under the Exoplanet Exploration Program. We thank the ESO
Director General for allocating discretionary observing time with ESPRESSO for
reconnaissance spectroscopy of TOI-3884. We are grateful to the ESO/La Silla
staff for their support of ExTrA. This work has made use of data from the
European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia),
processed by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This work makes use of
observations from the LCOGT network. Part of the LCOGT telescope time was
granted by NOIRLab through the Mid-Scale Innovations Program (MSIP). MSIP is
funded by NSF. This research has made use of the Spanish Virtual Observatory
(https://svo.cab.inta-csic.es) project funded by
MCIN/AEI/10.13039/501100011033/ through grant PID2020-112949GB-I00. The Pan-
STARRS1 Surveys (PS1) and the PS1 public science archive have been made
possible through contributions by the Institute for Astronomy, the University
of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its
participating institutes, the Max Planck Institute for Astronomy, Heidelberg
and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns
Hopkins University, Durham University, the University of Edinburgh, the
Queen’s University Belfast, the Harvard-Smithsonian Center for Astrophysics,
the Las Cumbres Observatory Global Telescope Network Incorporated, the
National Central University of Taiwan, the Space Telescope Science Institute,
the National Aeronautics and Space Administration under Grant No. NNX08AR22G
issued through the Planetary Science Division of the NASA Science Mission
Directorate, the National Science Foundation Grant No. AST-1238877, the
University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos
National Laboratory, and the Gordon and Betty Moore Foundation. Based on
observations collected at the European Southern Observatory under ESO
programme 109.24EP. This work has been supported by a grant from Labex
OSUG$@$2020 (Investissements d’avenir – ANR10 LABX56). Based on data collected
under the ExTrA project at the ESO La Silla Paranal Observatory. ExTrA is a
project of Institut de Planétologie et d’Astrophysique de Grenoble
(IPAG/CNRS/UGA), funded by the European Research Council under the ERC Grant
Agreement n. 337591-ExTrA.

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##  Appendix A Additional figures and tables

Figure 3: TESS target pixel file image of TOI-3884 in sectors 46 and 49
(created with tpfplotter, Aller et al. 2020). The electron counts are colour-
coded. The pixels highlighted in red are used for the simple aperture
photometry. The positions of the stars in the Gaia DR2 (Gaia Collaboration et
al. 2018) are indicated with red circles (and labelled with numbers according
to the distance to the main target in the aperture labelled ‘1’), and their
sizes are proportional to the Gaia DR2 magnitudes.

Figure 4: TESS observations. Top: Light curves from QLP KSPSAP (sector 22) and SPOC SAP (sectors 46 and 49) are shown by blue dots, with the transits of TOI-3884 b highlighted in orange. The SAP light curves do not show significant variability, apart from the one due to the satellite orbit. Bottom: Transits observed by TESS folded for each sector.  Figure 5: Palomar NIR AO imaging and sensitivity curves for TOI-3884 taken in the Br-$\gamma$ filter. The image reaches a contrast of $\sim 7$ magnitudes fainter than the host star within 0.5″. Inset: Image of the central portion of the data, centred on the star.  Table 1: Parameters measured on ESPRESSO spectra of TOI-3884. |  | CCF | CCF | CCF
---|---|---|---|---
Time | RV | FWHM | contrast | bisector span
$[{\rm BJD_{TDB}}]$ |  [km s-1] |  [km s-1] | [%] |  [km s-1]
2459762.484185 | $8.3318\pm 0.0019$ | $5.4367\pm 0.0037$ | $39.364\pm 0.027$ | $0.0225\pm 0.0037$
2459773.488742 | $8.3589\pm 0.0016$ | $5.4813\pm 0.0031$ | $37.378\pm 0.021$ | $0.0216\pm 0.0031$
Table 2: Astrometry, photometry, and stellar parameters for TOI-3884.

Parameter | Value | Source
---|---|---
Designations | LSPM J1206+1230 | Lépine & Shara (2005)
| TIC 86263325 |  Stassun et al. (2019)
RA (ICRS, J2000) |  12h 06m 17.44s | Gaia EDR3
Dec (ICRS, J2000) |  $+12^{\rm o}$ 30’ 24.9” | Gaia EDR3
$\mu$ RA [mas yr-1] |  -186.042 $\pm$ 0.028 | Gaia EDR3
$\mu$ Dec [mas yr-1] |  26.388 $\pm$ 0.017 | Gaia EDR3
Parallax [mas] |  23.074 $\pm$ 0.026 |  Gaia EDR3a
Distance [pc] |  43.338 $\pm$ 0.049 | Parallax
Gaia-BP [mag] |  15.9710 $\pm$ 0.0047 | Gaia EDR3
Gaia-G [mag] |  14.2463 $\pm$ 0.0029 | Gaia EDR3
Gaia-RP [mag] |  12.9897 $\pm$ 0.0033 | Gaia EDR3
2MASS-J [mag] |  11.127 $\pm$ 0.021 | 2MASS
2MASS-H [mag] |  10.552 $\pm$ 0.020 | 2MASS
2MASS-Ks [mag] |  10.240 $\pm$ 0.017 | 2MASS
WISE-W1 [mag] |  10.157 $\pm$ 0.023 | WISE
WISE-W2 [mag] |  9.986 $\pm$ 0.021 | WISE
WISE-W3 [mag] |  9.762 $\pm$ 0.048 | WISE
Mass, $M_{\star}$ [$M_{\odot}$ ] | $0.2813\pm 0.0067$ | Mann et al. (2019)
Radius, $R_{\star}$ [$R_{\odot}$ ] | $0.3043\pm 0.0090$ | Mann et al. (2015)
Mean density, $\rho_{\star}$ [$\mathrm{g\;cm^{-3}}$] | $14.1^{+1.4}_{-1.2}$ |  $M_{\star}$, $R_{\star}$
Surface gravity, $\log g$ [cgs] | $4.921\pm 0.028$ |  $M_{\star}$, $R_{\star}$
Effective temperature, $T_{\rm eff}$ [K] | $3269\pm 70$ | SpecMatch-Emp
Metallicity, [Fe/H] [dex] | $0.23\pm 0.12$ | SpecMatch-Emp
777a Corrected with Lindegren et al. (2021). $\mu$ is the proper motion.

Table 3: Modelling of the SED.

Parameter |  | Prior | Posterior median
---|---|---|---
|  |  | and 68.3% CI
Effective temperature, $T_{\mathrm{eff}}$ | [K] |  $N$(3269, 70) |  3271$\pm$60
Surface gravity, $\log g$ | [cgs] |  $U$(-0.5, 6.0) |  5.24${}^{+0.56}_{-0.83}$
Metallicity, $[\rm{Fe/H}]$ | [dex] |  $N$(0.23, 0.12) |  0.22$\pm$0.12
Distance | [pc] |  $N$(43.338, 0.049) |  43.338$\pm$0.048
$E_{\mathrm{(B-V)}}$ | [mag] |  $U$(0, 3) |  0.071${}^{+0.076}_{-0.049}$
Jitter Gaia | [mag] |  $U$(0, 1) |  0.181${}^{+0.20}_{-0.083}$
Jitter 2MASS | [mag] |  $U$(0, 1) |  0.082${}^{+0.099}_{-0.042}$
Jitter WISE | [mag] |  $U$(0, 1) |  0.046${}^{+0.088}_{-0.034}$
Radius, $R_{\star}$ |  [$R_{\odot}$ ] |  $U$(0, 100) |  0.3005$\pm$0.0090
Luminosity |  [L⊙] |  |  0.00930${}^{+0.00055}_{-0.00038}$
888$N$($\mu$, $\sigma$): Normal distribution prior with mean $\mu$, and
standard deviation $\sigma$. $U$(l, u): Uniform distribution prior in the
range [l, u].

Figure 6: Spectral energy distribution of TOI-3884. Top panel: Maximum a
posteriori PHOENIX/BT-Settl interpolated synthetic spectrum (solid line). The
red circles show photometric observations and the grey open circles are the
result of integrating the synthetic spectrum in the observed bandpasses.
Bottom panel: Residuals of the MAP model (the jitter has been added
quadratically to the error bars of the data). Figure 7: TESS, ExTrA, and LCOGT
transits (offset for clarity) modelled with juliet. The blue symbols with
error bars are the data, the grey line is the juliet posterior median model,
and the black line is the corresponding pure transit model. Each transit is
labelled with the epoch relative to T0, the sector for transits observed with
TESS, and the night and the telescope for transits observed with ExTrA. Figure
8: Transit observation (Fig. 7) residuals (data divided by model) from the
posterior median model of a planet transiting a star with a polar spot from
the analysis with the starry code. Figure 9: Phased ESPRESSO RVs (blue error
bars, as computed from the spectra), median posterior model (black line), and
68% CI (grey band). Figure 10: Transit timing variations from TESS sectors 46
and 49 (orange), ExTrA (red), and a transit observed simultaneously by TESS
and ExTrA (black). Table 4: Inferred system parameters.

Parameter | Units | Prior | Median and 68.3% CI | Prior | Median and 68.3% CI | Adopted
---|---|---|---|---|---|---
|  | juliet | juliet | starry | starry | value
_Star (TOI-3884)_ |  |  |  |  |  |
Mean density, $\rho_{\star}$ |  [$\mathrm{g\;cm^{-3}}$] | $N(14.1,1.4)$ | $14.3\pm 1.1$ | $N(14.1,1.4)$ | $13.83^{+0.66}_{-0.99}$ | juliet
Inclination of the spin axis, $i_{\star}$ | [°] |  |  | $U(0,90)$ | $47.0\pm 8.5$ |
Sky-projected spin-orbit angle, $\lambda$ | [°] |  |  | $U(90,270)$ | $151\pm 11$ |
Polar spot size, $\alpha$ | [°] |  |  | $U(0,90)$ | $48.6\pm 4.7$ |
Polar spot contrast g’ |  |  |  | $U(0,0.9)$ | $0.664\pm 0.097$ |
Polar spot contrast TESS |  |  |  | $U(0,0.9)$ | $0.365^{+0.064}_{-0.039}$ |
Polar spot contrast ExTrA |  |  |  | $U(0,0.9)$ | $0.159\pm 0.025$ |
$q_{1}$ g’ |  | $U(0,1)$ | $0.27^{+0.37}_{-0.20}$ | $U(0,1)$ | $0.51\pm 0.25$ | starry
$q_{2}$ g’ |  | $U(0,1)$ | $0.36^{+0.37}_{-0.26}$ | $U(0,1)$ | $0.29^{+0.33}_{-0.21}$ | starry
$q_{1}$ TESS |  | $U(0,1)$ | $0.132^{+0.12}_{-0.077}$ | $U(0,1)$ | $0.224\pm 0.13$ | starry
$q_{2}$ TESS |  | $U(0,1)$ | $0.105^{+0.19}_{-0.081}$ | $U(0,1)$ | $0.122^{+0.19}_{-0.097}$ | starry
$q_{1}$ ExTrA |  | $U(0,1)$ | $0.027^{+0.028}_{-0.014}$ | $U(0,1)$ | $0.069^{+0.050}_{-0.027}$ | starry
$q_{2}$ ExTrA |  | $U(0,1)$ | $0.54\pm 0.31$ | $U(0,1)$ | $0.61\pm 0.25$ | starry
Systemic velocity, $\gamma$ |  [ km s-1] |  $U$(8.31, 8.37) | $8.3444\pm 0.0015$ |  |  |
_Planet (TOI-3884 b)_ |  |  |  |  |  |
Semi-major axis, $a$ | [au] |  | $0.0354\pm 0.0014$ |  |  |
Eccentricity, $e$ |  |  |  ¡ 0.32†, ($0.059^{+0.14}_{-0.044}$) | $U(0.0,0.9)$ |  ¡ 0.19†, ($0.024^{+0.063}_{-0.018}$) | juliet
Argument of pericentre, $\omega$ | [°] |  | $190\pm 150$ | $U(0,360)$ | $72^{+150}_{-62}$ | juliet
Inclination, $i_{p}$ | [°] |  | $89.83^{+0.12}_{-0.16}$ | $U(88,92)$ | $90.10\pm 0.29$ | starry
Radius ratio, $R_{\mathrm{p}}/R_{\star}$ |  | $U(0.1,0.3)$ | $0.1899\pm 0.0064$ | $U(0.1,0.3)$ | $0.1808\pm 0.0014$ | starry
Scaled semi-major axis, $a/R_{\star}$ |  |  | $25.01\pm 0.65$ |  |  |
Impact parameter, $b$ |  | $U(0,1)$ | $0.072\pm 0.072$ |  | $-0.04\pm 0.13$ | starry
Transit duration, $T_{14}$ | [h] |  | $1.646\pm 0.011$ |  |  |
T0 \- 2 450 000 |  [BJDTDB] | $U(9642.8624,9642.8638)$ | $9642.86314\pm 0.00012$ |  |  |
Orbital period, $P$ | [d] | $U(4.54447,4.54468)$ | $4.5445697\pm 0.0000094$ |  |  |
RV semi-amplitude, K |  [m s-1] |  $U$(0, 100) | $14.9^{+3.4}_{-1.6}$ |  |  |
Radius, $R_{\mathrm{p}}$ |  [$\mathcal{R}^{\rm N}_{e\rm E}$] |  | $6.31\pm 0.28$ |  | $6.00\pm 0.18$ | starry
Mass, $M_{\mathrm{p}}$ |  [$\mathrm{M_{E}}$ ] |  | $16.5^{+3.5}_{-1.8}$ |  |  |
Mean density, $\rho_{\mathrm{p}}$ |  [$\mathrm{g\;cm^{-3}}$] |  | $0.365^{+0.095}_{-0.061}$ |  | $0.424^{+0.10}_{-0.061}$ | starry
Surface gravity, $\log$ $g_{\mathrm{p}}$ | [cgs] |  | $2.612\pm 0.090$ |  | $2.656^{+0.088}_{-0.059}$ | starry
Equilibrium temperature, Teq | [K] |  | $463\pm 12$ |  |  |
Insolation flux |  [FE] |  | $7.42\pm 0.77$ |  |  |
True spin–orbit angle, $\psi$ | [°] |  |  |  |  $50\pm 12$ or $130\pm 12$ |
$\sqrt{e}\cos{\omega}$ |  | $U(-1,1)$ | $-0.02\pm 0.33$ |  | $0.07^{+0.21}_{-0.13}$ | juliet
$\sqrt{e}\sin{\omega}$ |  | $U(-1,1)$ | $-0.02\pm 0.12$ |  | $0.057\pm 0.090$ | juliet
Dilution g’ |  | $U(0,1)$ | $0.84^{+0.11}_{-0.17}$ |  |  |
Dilution TESS sector 22 |  | $U(0,1)$ | $0.683\pm 0.071$ |  |  |
Dilution TESS sector 46 |  | $U(0,1)$ | $0.893\pm 0.060$ |  |  |
Dilution TESS sector 49 |  | $U(0,1)$ | $0.880\pm 0.058$ |  |  |
Dilution ExTrA |  | $U(0,1)$ | $0.928\pm 0.057$ |  |  |
999The table lists: prior, posterior median, and 68.3% CI, for the analysis of
transit photometry and RVs (juliet, in which the asymmetry of the transit is
modelled using GPs, and the chromaticity with dilution factors), and the
analysis of transit photometry including a starspot in the model (starry). If
a parameter is estimated from both modellings, the last column specifies the
adopted value. There is a degeneracy with ($i_{\star}+180\degree$) and
($180\degree-\lambda$) angles assuming $i_{p}$ between 0° and 180°. † Upper
limit, 95% confidence. The parameters $q_{1}$ and $q_{2}$ are the quadratic
limb-darkening coefficients parameterised using Kipping (2013). The planetary
equilibrium temperature is computed for zero albedo and full day-night heat
redistribution. The true spin–orbit angle is given for the prograde and
retrograde configurations, which cannot be distinguished with the available
data. IAU 2012: au = 149 597 870 700 m . IAU 2015: $\mathcal{R}^{\rm
N}_{\odot}$= 6.957$\;\times 10^{8}$ m, $\mathcal{R}^{\rm N}_{e\rm E}$ = 6.378
1$\;\times 10^{6}$ m, $\mathcal{(GM)}^{\rm N}_{\odot}$= 1.327 124 4$\;\times
10^{20}$ $\rm{m^{3}\;s^{-2}}$, $\mathcal{(GM)}^{\rm N}_{\rm E}$= 3.986
004$\;\times 10^{14}$ $\rm{m^{3}\;s^{-2}}$. $M_{\odot}$ = $\mathcal{(GM)}^{\rm
N}_{\odot}$/$\mathcal{G}$, $\mathrm{M_{E}}$ = $\mathcal{(GM)}^{\rm N}_{\rm
E}$/$\mathcal{G}$. CODATA 2018: $\mathcal{G}$ = 6.674 30$\;\times 10^{-11}$
$\rm{m^{3}\;kg^{-1}\;s^{-2}}$. $N(\mu,\sigma)$: Normal distribution with mean
$\mu$, and standard deviation $\sigma$. $U(a,b)$: A uniform distribution
defined between a lower $a$ and an upper $b$ limit.

Figure 11: Polar spot contrast as a function of the photometric band (blue for
g’, orange for TESS, and red for ExTrA) from the analysis of Sect. 4. The g’
and TESS transmission curves were retrieved from the SVO Filter Profile
Service (http://svo2.cab.inta-csic.es/theory/fps/, Rodrigo et al. 2012;
Rodrigo & Solano 2020). The blue, orange, and red bands represent the median
and the 68% CI of the contrast as a function of wavelength predicted by the
corresponding band contrast measurement assuming blackbody emission. The black
error bars and grey curves are the posterior (median and the 68% CI) of the
measured contrast and the contrast as a function of wavelength, respectively,
from the modelling using the PHOENIX/BT-Settl stellar atmosphere models (Sect.
B). Above the panel: Model of the stellar surface as seen at each band for the
median posterior values. Figure 12: Seeing limited images centred on TOI-3884
at epoch 2016.0 (Gaia EDR3 position, Gaia Collaboration et al. 2021, marked
with a red circle) from the Palomar Observatory Sky Survey (POSS, two epochs,
retrieved from https://archive.eso.org/dss/dss) and Pan-STARRS (Chambers et
al. 2016). The proper motion can be appreciated over the 60-year time span. No
background star is detected at the current position of TOI-3884. Figure 13:
Planet-to-star radius ratio estimated jointly for all bands or for each band
individually.

##  Appendix B Spot contrast

We modelled the derived spot contrast in three different bands (Table 4) to
estimate the photosphere-spot difference temperature. The use of a blackbody
spectrum fit the data poorly (see Fig. 11). Therefore, we used the PHOENIX/BT-
Settl (Allard et al. 2012) stellar atmosphere models, which we integrated in
the g’, TESS, and ExTrA bands. We used as input parameters the effective
temperature, surface gravity, and metallicity with normal priors from Sect. 3
(Table 2), and a photosphere-spot difference temperature with a uniform prior.
We sampled from the posterior with emcee (Goodman & Weare 2010; Foreman-Mackey
et al. 2013). The parameters, priors, and posteriors are listed in Table 5.
The posterior model is shown in Fig. 11. The posterior spot contrast in the
$K_{s}$-band (that was used to derive the stellar parameters in Sect. 3) is
$0.159\pm 0.017$.

The measurement of the spot contrast in several bands allows for an
independent determination of the effective temperature of the star, in
addition to the photosphere-spot difference temperature, subject to the
validity of the stellar atmosphere model. We repeated the modelling using a
uniform prior for the photosphere effective temperature, surface gravity, and
metallicity, and obtained $T_{\mathrm{eff}}=3400^{+460}_{-320}$ K, in
agreement with the adopted one (Table 2), but having a large uncertainty.

Table 5: Modelling of the starspot contrast.

Parameter |  | Prior | Posterior median | Prior | Posterior median
---|---|---|---|---|---
|  |  | and 68.3% CI |  | and 68.3% CI
Photosphere effective temperature, $T_{\mathrm{eff}}$ | [K] |  $N$(3269, 70) | $3249\pm 74$ |  $U$(400, 10000) | $3400^{+460}_{-320}$
Surface gravity, $\log g$ | [cgs] |  $N$(4.921, 0.028) | $4.922\pm 0.028$ |  $U$(-0.5, 6.0) | $0.8^{+2.9}_{-1.0}$
Metallicity, $[\rm{Fe/H}]$ | [dex] |  $N$(0.23, 0.12) | $0.22\pm 0.12$ |  $U$(-0.4, 0.5) | $-0.87^{+0.92}_{-1.4}$
Photosphere-spot difference temperature | [K] |  $U$(0, 500) | $187\pm 21$ |  $U$(0, 500) | $214^{+95}_{-51}$
101010Modelling with informative and non-informative priors for the effective
temperature, surface gravity, and metallicity. $N$($\mu$, $\sigma$): Normal
distribution prior with mean $\mu$, and standard deviation $\sigma$. $U$(l,
u): Uniform distribution prior in the range [l, u].