# TOI-3884 b: A rare 6-R E planet that transits a low-mass star with a giant and likely polar spot

J M Almenara, X. Bonfils, T. Forveille, N. Astudillo-Defru, D R Ciardi, R P

Schwarz, K A Collins, M. Cointepas, M B Lund, F. Bouchy, et al.

► To cite this version:

J M Almenara, X. Bonfils, T. Forveille, N. Astudillo-Defru, D R Ciardi, et al., TOI-3884 b: A rare 6-R E planet that transits a low-mass star with a giant and likely polar spot. Astronomy & Astrophysics - A&A, 2022, 667, pp.L11. 10.1051/0004-6361/202244791. hal-04975528

HAL Id: hal-04975528

https://hal.science/hal-04975528v1

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LETTER TO THE EDITOR

# TOI-3884 b: A rare 6-$R_E$ planet that transits a low-mass star with a giant and likely polar spot

J. M. Almenara<fnref n="1" />, X. Bonfils<fnref n="1" />, T. Forveille<fnref n="1" />, N. Astudillo-Defru<fnref n="2" />, D. R. Ciardi<fnref n="3" />, R. P. Schwarz<fnref n="4" />, K. A. Collins<fnref n="4" />,
M. Cointepas<fnref n="1" />,<fnref n="5" />, M. B. Lund<fnref n="3" />, F. Bouchy<fnref n="5" />, D. Charbonneau<fnref n="4" />, R. F. Díaz<fnref n="6" />, X. Delfosse<fnref n="1" />, R. C. Kidwell<fnref n="7" />,
M. Kunimoto<fnref n="8" />, D. W. Latham<fnref n="4" />, J. J. Lissauer<fnref n="9" />, F. Murgas<fnref n="10" />,<fnref n="11" />, G. Ricker<fnref n="8" />, S. Seager<fnref n="8" />,<fnref n="12" />,<fnref n="13" />,
M. Vezie<fnref n="8" />, and D. Watanabe<fnref n="14" />

Received 22 August 2022 / Accepted 12 October 2022

ABSTRACT

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-$R_E$ planet. We validate the planet with seeing-limited photometry, high-resolution imaging, and radial velocities. TOI-3884 b, with a radius of $6.00 \pm 0.18 R_E$, 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 – starspots – planetary systems – techniques: photometric – techniques: radial velocities

1. Introduction

Over the past thirty years, more than 3500 transiting plan-
ets 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 $R_E$ exists around stars with $M_\star <$
0.3 $M_\odot$ (Burn et al. 2021). And indeed, no planet larger than
~3.5 $R_E$ has been found so far around any star with $R_\star \le$
0.35 $R_\odot$. One may note an apparent exception to this statis-
tic: the radial-velocity (RV) detection of a giant planet around
the small star GJ 3512 (0.14 $R_\odot$). With a minimum mass of
$M_p \sin i \sim 147 M_E$, GJ 3512 b is presumably larger than ~10 $R_E$
(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, how-
ever, 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$R_E$ 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

<footnote n="1">¹ Univ. Grenoble Alpes, CNRS, IPAG, 414 rue de la Piscine, 38400 St-Martin d'Hères, France
e-mail: Jose-Manuel.Almenara-Villa@univ-grenoble-alpes.fr</footnote>

<footnote n="2">² Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Alonso de Rivera 2850, Concepción, Chile</footnote>

<footnote n="3">³ NASA Exoplanet Science Institute, Caltech/IPAC, 1200 E California Blvd, Pasadena, CA 91125, USA</footnote>

<footnote n="4">⁴ Center for Astrophysics, Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA</footnote>

<footnote n="5">⁵ Observatoire de Genève, Département d'Astronomie, Université de Genève, Chemin Pegasi 51, 1290 Versoix, Switzerland</footnote>

<footnote n="6">⁶ International Center for Advanced Studies (ICAS) and ICIFI (CONICET), ECyT-UNSAM, Campus Miguelete, 25 de Mayo y Francia, 1650 Buenos Aires, Argentina</footnote>

<footnote n="7">⁷ Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA</footnote>

<footnote n="8">⁸ Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA</footnote>

<footnote n="9">⁹ NASA Ames Research Center, Moffett Field, CA 94035, USA</footnote>

<footnote n="10">¹⁰ Instituto de Astrofísica de Canarias (IAC), Vía Láctea, s/n, 38200 La Laguna, Tenerife, Spain</footnote>

<footnote n="11">¹¹ Dept. Astrofísica, Universidad de La Laguna (ULL), Avenida Astrofísico Francisco Sánchez, s/n, 38206 La Laguna, Tenerife, Spain</footnote>

<footnote n="12">¹² Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA</footnote>

<footnote n="13">¹³ Department of Aeronautics and Astronautics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA</footnote>

<footnote n="14">¹⁴ Planetary Discoveries in Fredericksburg, 22 Woodford Dr, Fredericksburg, VA 22405, USA</footnote>
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 et al. 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-min cadence<fnref n="1" />, and in sectors 46 and 49 with a 2-min 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. A.1, Aller et al. 2020; Gaia Collaboration 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. A.2. 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. A.2 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 (0.88–1.55 µm) multi-object
spectrograph fibre-fed by three 60-cm telescopes. Five fibre
positioners intercept the light from one target and four compar-
ison 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 spec-
trograph (R ~ 20), and 60-s exposures. We chose comparison
stars with effective temperatures (Gaia Collaboration 2018) sim-
ilar 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 Program<fnref n="2" /> (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 tele-
scope at Teide observatory in the g' filter, with an exposure
time of 300 s. 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 exo-
planets and assessing the possible contamination of the derived
planetary radii by bound or unbound stars in the TESS aper-
ture (Ciardi et al. 2015), we obtained near-IR adaptive optics
(AO) imaging of TOI-3884 at Palomar Observatory. The Palo-
mar Observatory observations of TOI-3884 were made in the
narrow-band Br-γ filter (λ₀ = 2.1686; Δλ = 0.0326 µm) with the
PHARO instrument (Hayward et al. 2001) behind the P3K nat-
ural guide star AO system (Dekany et al. 2013) on 2022 Febru-
ary 13 in a standard 5-point quincunx dither pattern with steps of
5". Each dither position was observed three times, offset in posi-
tion from each other by 0.5" for a total of 15 frames; the indi-
vidual exposures were 31.1 s long, for a total on-source times of
466 s. PHARO has a pixel scale of 0.025" per pixel for a total
field of view of ~25".

The AO data were processed and analysed with a custom
set of Interactive Data Language tools. The flat fields were gen-
erated 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 func-
tions: 0.105".

The sensitivity of the combined AO image to the compan-
ions was determined by injecting simulated sources around the
primary target at separations of integer multiples of the central
source's FWHM and every 20° in azimuth (Furlan et al. 2017).
The brightness of each injected source was scaled until standard
aperture photometry detected it with 5σ significance, and the
brightness relative to TOI-3884 of the faintest injected sources
that was detected set the contrast limit at that injection loca-
tion. The adopted 5σ limit at each separation is then the aver-
age 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. A.3; no stellar companions were detected.

<footnote n="1">¹ The 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).</footnote>

<footnote n="2">² https://tess.mit.edu/followup</footnote>
## 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 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 ≤1.4 indicate that the Gaia astrometry is consistent with a single star, whereas RUWE values ≥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 s, 2×1 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 $1.9\,\text{m s}^{-1}$ and $1.6\,\text{m s}^{-1}$. The data reduction and RV extraction were done through the ESPRESSO-ESO pipeline<fnref n="3" />. 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 A.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 2016, 2021) to compute the distance and obtain an absolute magnitude of $M_{K_s} = 7.056 \pm 0.017$. We then used the empirical relations of (Mann et al. 2019) and (Mann et al. 2015) to derive a mass of $M_* = 0.2813 \pm 0.0067 M_\odot$ and a radius of $R_* = 0.3043 \pm 0.0090 R_\odot$, which we adopt as our preferred values (Table A.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 A.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 ($T_{\text{eff}} = 3269 \pm 70$ K, which corresponds to an M4 spectral type) and metallicity ([Fe/H] = 0.23 ± 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 A.3. The maximum a posteriori (MAP) model is shown in Fig. A.4. The derived radius ($R_* = 0.3005 \pm 0.0090 R_\odot$), 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 $3270_{-170}^{+270}$ 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 $5.29\,\text{km s}^{-1}$. In comparison, we measured an FWHM = $5.46\,\text{km s}^{-1}$ for TOI-3884. We then followed Hirano et al. (2010) to derive $v \sin i_*$ as $(1.23 \, v \sin i_*)^2 = \text{FWHM}_{\text{TOI-3884}}^2 - \text{FWHM}_{0, \text{LHS 1140}}^2$, and obtained a value of $v \sin i_* = 1.1\,\text{km s}^{-1}$. Given $R_*$ from Table A.2 and $i_*$ from Table A.4, this corresponds to a rotation period of ~18 days. In addition, we measured $\log R'_{HK} = -4.79 \pm 0.13$ in the co-added ESPRESSO spectrum, from which the Astudillo-Defru et al. (2017) activity-rotation relation estimates a rotation period of $P_{\text{rot}} = 30 \pm 6$ 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 $v \sin i_*$ of $350\,\text{km s}^{-1}$. 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

<footnote n="3">³ https://www.eso.org/sci/software/pipelines/</footnote>
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 A.4 lists the parameters, priors, and posteriors. Figures A.5 and A.7 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. A.6). We therefore phase-folded the light curves<fnref n="4" /> to speed up the spot modelling computations. To do this, we used the MAP model of the analysis with *juliet* to normalise the TESS<fnref n="5" />, 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 A.4). We modelled the folded transits in the three bands (*g'*, TESS, and ExTrA) with the *starry* code (Luger et al. 2019, 2021b,a). 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 contrast<fnref n="6" /> is a free parameter for each of the *g'*, TESS, and ExTrA bands, which have a respective $\lambda_{\text{pivot}}$ (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 A.4, and the data and model posterior are shown in Fig. 1. Figure A.9 shows the spot contrast for the three bands. From these three spot contrasts, we estimate a photosphere-spot difference temperature of 187 ± 21 K (Sect. B), which is in the range measured for M dwarfs (Berdyugina 2005; Barnes et al. 2015).

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. A.10). 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. A.11), and reinforce the planetary scenario. The ESPRESSO RVs, the CCF con-

<figure><img src="image_2.png" /><figcaption>**Fig. 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-min 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.</figcaption></figure>

trast, 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 $6.00 \pm 0.18 R_E$ 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 $R_p \sim 6 R_E$ 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; $0.45 M_\odot$, $P_{\text{rot}} \sim 15$ d) is one of

<footnote n="4">⁴ We previously checked, using *juliet*, that there are no significant transit timing variations (Fig. A.8).</footnote>

<footnote n="5">⁵ We did not use the sector 22 transits, observed with a 30-min cadence, which blurs their shape.</footnote>

<footnote n="6">⁶ The spot contrast is defined in *starry* as the fractional change in the intensity at the centre of the spot ($I_{\text{spot}}$) relative to the baseline intensity of an unspotted stellar surface ($I_{\text{photosphere}}$): contrast = $1 - \frac{I_{\text{spot}}}{I_{\text{photosphere}}}$.</footnote>
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** TOI-3884 b in context. Left: planetary vs. 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 vs. 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 $R_E$. 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.</figcaption></figure>

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 axisymmet-
ric. 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⁻¹, 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 (ψ), and we find that it would be misaligned (ψ = 50 ± 12° or 130 ± 12°). This measurement will probe the history of the planet, which almost certainly had to migrate to end up so close to its star.

**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-100. 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 at 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

<table><caption>**Table A.1.** Parameters measured on ESPRESSO spectra of TOI-3884.</caption><thead><tr><th>Time<br>[BJD<sub>TDB</sub>]</th><th>RV<br>[km s<sup>-1</sup>]</th><th>CCF<br>FWHM<br>[km s<sup>-1</sup>]</th><th>CCF<br>contrast<br>[%]</th><th>CCF<br>bisector span<br>[km s<sup>-1</sup>]</th></tr></thead><tbody><tr><td>2459762.484185</td><td>8.3318 ± 0.0019</td><td>5.4367 ± 0.0037</td><td>39.364 ± 0.027</td><td>0.0225 ± 0.0037</td></tr><tr><td>2459773.488742</td><td>8.3589 ± 0.0016</td><td>5.4813 ± 0.0031</td><td>37.378 ± 0.021</td><td>0.0216 ± 0.0031</td></tr></tbody></table>

<table><caption>**Table A.2.** Astrometry, photometry, and stellar parameters for TOI-3884.</caption><thead><tr><th>Parameter</th><th>Value</th><th>Source</th></tr></thead><tbody><tr><td>Designations</td><td>LSPM J1206+1230<br>TIC 86263325</td><td>Lépine & Shara (2005)<br>Stassun et al. (2019)</td></tr><tr><td>RA (ICRS, J2000)</td><td>12<sup>h</sup> 06<sup>m</sup> 17.44<sup>s</sup></td><td>Gaia EDR3</td></tr><tr><td>Dec (ICRS, J2000)</td><td>+12<sup>o</sup> 30' 24.9"</td><td>Gaia EDR3</td></tr><tr><td>μ RA [mas yr<sup>-1</sup>]</td><td>-186.042 ± 0.028</td><td>Gaia EDR3</td></tr><tr><td>μ Dec [mas yr<sup>-1</sup>]</td><td>26.388 ± 0.017</td><td>Gaia EDR3</td></tr><tr><td>Parallax [mas]</td><td>23.074 ± 0.026</td><td>Gaia EDR3<sup>a</sup></td></tr><tr><td>Distance [pc]</td><td>43.338 ± 0.049</td><td>Parallax</td></tr><tr><td>Gaia-BP [mag]</td><td>15.9710 ± 0.0047</td><td>Gaia EDR3</td></tr><tr><td>Gaia-G [mag]</td><td>14.2463 ± 0.0029</td><td>Gaia EDR3</td></tr><tr><td>Gaia-RP [mag]</td><td>12.9897 ± 0.0033</td><td>Gaia EDR3</td></tr><tr><td>2MASS-J [mag]</td><td>11.127 ± 0.021</td><td>2MASS</td></tr><tr><td>2MASS-H [mag]</td><td>10.552 ± 0.020</td><td>2MASS</td></tr><tr><td>2MASS-Ks [mag]</td><td>10.240 ± 0.017</td><td>2MASS</td></tr><tr><td>WISE-W1 [mag]</td><td>10.157 ± 0.023</td><td>WISE</td></tr><tr><td>WISE-W2 [mag]</td><td>9.986 ± 0.021</td><td>WISE</td></tr><tr><td>WISE-W3 [mag]</td><td>9.762 ± 0.048</td><td>WISE</td></tr><tr><td>Mass, M<sub>*</sub> [M<sub>☉</sub>]</td><td>0.2813 ± 0.0067</td><td>Mann et al. (2019)</td></tr><tr><td>Radius, R<sub>*</sub> [R<sub>☉</sub>]</td><td>0.3043 ± 0.0090</td><td>Mann et al. (2015)</td></tr><tr><td>Mean density, ρ<sub>*</sub> [g cm<sup>-3</sup>]</td><td>14.1<sup>+1.4</sup><sub>-1.2</sub></td><td>M<sub>*</sub>, R<sub>*</sub></td></tr><tr><td>Surface gravity, log g [cgs]</td><td>4.921 ± 0.028</td><td>M<sub>*</sub>, R<sub>*</sub></td></tr><tr><td>Effective temperature, T<sub>eff</sub> [K]</td><td>3269 ± 70</td><td>SpecMatch-Emp</td></tr><tr><td>Metallicity, [Fe/H] [dex]</td><td>0.23 ± 0.12</td><td>SpecMatch-Emp</td></tr></tbody></table>

Notes. <sup>a</sup> Corrected with Lindegren et al. (2021). μ is the proper motion.

<table><caption>**Table A.3.** Modelling of the SED.</caption><thead><tr><th>Parameter</th><th>Prior</th><th>Posterior median and 68.3% CI</th></tr></thead><tbody><tr><td>Effective temperature, T<sub>eff</sub> [K]</td><td>N(3269, 70)</td><td>3271±60</td></tr><tr><td>Surface gravity, log g [cgs]</td><td>U(-0.5, 6.0)</td><td>5.24<sup>+0.56</sup><sub>-0.83</sub></td></tr><tr><td>Metallicity, [Fe/H] [dex]</td><td>N(0.23, 0.12)</td><td>0.22±0.12</td></tr><tr><td>Distance [pc]</td><td>N(43.338, 0.049)</td><td>43.338±0.048</td></tr><tr><td>E<sub>(B-V)</sub> [mag]</td><td>U(0, 3)</td><td>0.071<sup>+0.076</sup><sub>-0.049</sub></td></tr><tr><td>Jitter Gaia [mag]</td><td>U(0, 1)</td><td>0.181<sup>+0.080</sup><sub>-0.083</sub></td></tr><tr><td>Jitter 2MASS [mag]</td><td>U(0, 1)</td><td>0.082<sup>+0.099</sup><sub>-0.042</sub></td></tr><tr><td>Jitter WISE [mag]</td><td>U(0, 1)</td><td>0.046<sup>+0.088</sup><sub>-0.034</sub></td></tr><tr><td>Radius, R<sub>*</sub> [R<sub>☉</sub>]</td><td>U(0, 100)</td><td>0.3005±0.0090</td></tr><tr><td>Luminosity [L<sub>☉</sub>]</td><td>U(0, 100)</td><td>0.00930<sup>+0.00055</sup><sub>-0.00038</sub></td></tr></tbody></table>

Notes. N(μ, σ): Normal distribution prior with mean μ, and standard deviation σ. U(l, u): Uniform distribution prior in the range [l, u].
<figure><img src="image_4.png" /></figure>

**Fig. A.1.** TESS target pixel file image of TOI-3884 in sectors 46 and 49 (created with `tfpplotter`, 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 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><img src="image_5.png" /></figure>

**Fig. A.2.** 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><img src="image_6.png" /></figure>

**Fig. A.4.** 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><img src="image_7.png" /></figure>

**Fig. A.3.** Palomar NIR AO imaging and sensitivity curves for TOI-3884 taken in the Br-γ filter. The image reaches a contrast of ~7 magnitudes fainter than the host star within 0.5". Inset: Image of the central portion of the data, centred on the star.
<table><caption>**Table A.4.** Inferred system parameters.</caption><thead><tr><th>Parameter</th><th>Units</th><th>Prior<br> juliet</th><th>Median and 68.3% CI<br> juliet</th><th>Prior<br> starry</th><th>Median and 68.3% CI<br> starry</th><th>Adopted<br> value</th></tr></thead><tbody><tr><td colspan="7" style="text-align: center; font-weight: bold;">Star (TOI-3884)</td></tr><tr><td>Mean density, ρ<sub>★</sub></td><td>[g cm<sup>-3</sup>]</td><td>N(14.1, 1.4)</td><td>14.3 ± 1.1</td><td>N(14.1, 1.4)</td><td>13.83<sup>+0.66</sup><sub>-0.99</sub></td><td> juliet</td></tr><tr><td>Inclination of the spin axis, i<sub>★</sub></td><td>[°]</td><td></td><td></td><td>U(0, 90)</td><td>47.0 ± 8.5</td><td></td></tr><tr><td>Sky-projected spin-orbit angle, λ</td><td>[°]</td><td></td><td></td><td>U(90, 270)</td><td>151 ± 11</td><td></td></tr><tr><td>Polar spot size, α</td><td>[°]</td><td></td><td></td><td>U(0, 90)</td><td>48.6 ± 4.7</td><td></td></tr><tr><td>Polar spot contrast g'</td><td></td><td></td><td></td><td>U(0, 0.9)</td><td>0.664 ± 0.097</td><td></td></tr><tr><td>Polar spot contrast TESS</td><td></td><td></td><td></td><td>U(0, 0.9)</td><td>0.365<sup>+0.064</sup><sub>-0.039</sub></td><td></td></tr><tr><td>Polar spot contrast ExTrA</td><td></td><td></td><td></td><td>U(0, 0.9)</td><td>0.159 ± 0.025</td><td></td></tr><tr><td>q<sub>1</sub> g'</td><td></td><td>U(0, 1)</td><td>0.27<sup>+0.37</sup><sub>-0.20</sub></td><td>U(0, 1)</td><td>0.51 ± 0.25</td><td>starry</td></tr><tr><td>q<sub>2</sub> g'</td><td></td><td>U(0, 1)</td><td>0.36<sup>+0.37</sup><sub>-0.26</sub></td><td>U(0, 1)</td><td>0.29<sup>+0.33</sup><sub>-0.21</sub></td><td>starry</td></tr><tr><td>q<sub>1</sub> TESS</td><td></td><td>U(0, 1)</td><td>0.132<sup>+0.12</sup><sub>-0.077</sub></td><td>U(0, 1)</td><td>0.224 ± 0.13</td><td>starry</td></tr><tr><td>q<sub>2</sub> TESS</td><td></td><td>U(0, 1)</td><td>0.105<sup>+0.081</sup><sub>-0.019</sub></td><td>U(0, 1)</td><td>0.122<sup>+0.19</sup><sub>-0.097</sub></td><td>starry</td></tr><tr><td>q<sub>1</sub> ExTrA</td><td></td><td>U(0, 1)</td><td>0.027<sup>+0.028</sup><sub>-0.014</sub></td><td>U(0, 1)</td><td>0.069<sup>+0.050</sup><sub>-0.027</sub></td><td>starry</td></tr><tr><td>q<sub>2</sub> ExTrA</td><td></td><td>U(0, 1)</td><td>0.54 ± 0.31</td><td>U(0, 1)</td><td>0.61 ± 0.25</td><td>starry</td></tr><tr><td>Systemic velocity, γ</td><td>[km s<sup>-1</sup>]</td><td>U(8.31, 8.37)</td><td>8.3444 ± 0.0015</td><td></td><td></td><td></td></tr><tr><td colspan="7" style="text-align: center; font-weight: bold;">Planet (TOI-3884 b)</td></tr><tr><td>Semi-major axis, a</td><td>[au]</td><td></td><td>0.0354 ± 0.0014</td><td></td><td></td><td></td></tr><tr><td>Eccentricity, e</td><td></td><td></td><td>&lt; 0.32<sup>†</sup>, (0.059<sup>+0.14</sup><sub>-0.044</sub>)</td><td>U(0, 0.9)</td><td>&lt; 0.19<sup>†</sup>, (0.024<sup>+0.063</sup><sub>-0.018</sub>)</td><td> juliet</td></tr><tr><td>Argument of pericentre, ω</td><td>[°]</td><td></td><td>190 ± 150</td><td>U(0, 360)</td><td>72<sup>+150</sup><sub>-62</sub></td><td> juliet</td></tr><tr><td>Inclination, i<sub>p</sub></td><td>[°]</td><td></td><td>89.83<sup>+0.12</sup><sub>-0.16</sub></td><td>U(88, 92)</td><td>90.10 ± 0.29</td><td>starry</td></tr><tr><td>Radius ratio, R<sub>p</sub>/R<sub>★</sub></td><td></td><td>U(0.1, 0.3)</td><td>0.1899 ± 0.0064</td><td>U(0.1, 0.3)</td><td>0.1808 ± 0.0014</td><td>starry</td></tr><tr><td>Scaled semi-major axis, a/R<sub>★</sub></td><td></td><td></td><td>25.01 ± 0.65</td><td></td><td></td><td></td></tr><tr><td>Impact parameter, b</td><td></td><td>U(0, 1)</td><td>0.072 ± 0.072</td><td></td><td>-0.04 ± 0.13</td><td>starry</td></tr><tr><td>Transit duration, T<sub>14</sub></td><td>[h]</td><td></td><td>1.646 ± 0.011</td><td></td><td></td><td></td></tr><tr><td>T<sub>0</sub> - 2 450 000</td><td>[BJD<sub>TDB</sub>]</td><td>U(9642.8624, 9642.8638)</td><td>9642.86314 ± 0.00012</td><td></td><td></td><td></td></tr><tr><td>Orbital period, P</td><td>[d]</td><td>U(4.54447, 4.54468)</td><td>4.5445697 ± 0.0000094</td><td></td><td></td><td></td></tr><tr><td>RV semi-amplitude, K</td><td>[m s<sup>-1</sup>]</td><td></td><td>14.9<sup>+3.4</sup><sub>-1.6</sub></td><td></td><td></td><td></td></tr><tr><td>Radius, R<sub>p</sub></td><td>[R<sup>N</sup><sub>E</sub>]</td><td>U(0, 100)</td><td>6.31 ± 0.28</td><td></td><td>6.00 ± 0.18</td><td>starry</td></tr><tr><td>Mass, M<sub>p</sub></td><td>[M<sub>E</sub>]</td><td></td><td>16.5<sup>+3.5</sup><sub>-1.8</sub></td><td></td><td></td><td></td></tr><tr><td>Mean density, ρ<sub>p</sub></td><td>[g cm<sup>-3</sup>]</td><td></td><td>0.365<sup>+0.095</sup><sub>-0.061</sub></td><td></td><td>0.424<sup>+0.10</sup><sub>-0.061</sub></td><td>starry</td></tr><tr><td>Surface gravity, log g<sub>p</sub></td><td>[cgs]</td><td></td><td>2.612 ± 0.090</td><td></td><td>2.656<sup>+0.088</sup><sub>-0.059</sub></td><td>starry</td></tr><tr><td>Equilibrium temperature, T<sub>eq</sub></td><td>[K]</td><td></td><td>463 ± 12</td><td></td><td></td><td></td></tr><tr><td>Insolation flux</td><td>[F<sub>E</sub>]</td><td></td><td>7.42 ± 0.77</td><td></td><td></td><td></td></tr><tr><td>True spin-orbit angle, ψ</td><td>[°]</td><td></td><td></td><td></td><td>50 ± 12 or 130 ± 12</td><td></td></tr><tr><td>√e cos ω</td><td></td><td>U(-1, 1)</td><td>-0.02 ± 0.33</td><td></td><td>0.07<sup>+0.21</sup><sub>-0.13</sub></td><td> juliet</td></tr><tr><td>√e sin ω</td><td></td><td>U(-1, 1)</td><td>-0.02 ± 0.12</td><td></td><td>0.057 ± 0.090</td><td> juliet</td></tr><tr><td>Dilution g'</td><td></td><td>U(0, 1)</td><td>0.84<sup>+0.11</sup><sub>-0.17</sub></td><td></td><td></td><td></td></tr><tr><td>Dilution TESS sector 22</td><td></td><td>U(0, 1)</td><td>0.683 ± 0.071</td><td></td><td></td><td></td></tr><tr><td>Dilution TESS sector 46</td><td></td><td>U(0, 1)</td><td>0.893 ± 0.060</td><td></td><td></td><td></td></tr><tr><td>Dilution TESS sector 49</td><td></td><td>U(0, 1)</td><td>0.880 ± 0.058</td><td></td><td></td><td></td></tr><tr><td>Dilution ExTrA</td><td></td><td>U(0, 1)</td><td>0.928 ± 0.057</td><td></td><td></td><td></td></tr></tbody></table>

**Notes.** The 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<sub>★</sub> + 180°) and (180° - λ) angles assuming i<sub>p</sub> between 0° and 180°. † Upper limit, 95% confidence. The parameters q<sub>1</sub> and q<sub>2</sub> 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: R<sup>N</sup><sub>☉</sub> = 6.957 × 10<sup>8</sup> m, R<sup>N</sup><sub>eE</sub> = 6.378 1 × 10<sup>6</sup> m, (GM)<sup>N</sup><sub>☉</sub> = 1.327 124 4 × 10<sup>20</sup> m<sup>3</sup> s<sup>-2</sup>, (GM)<sup>N</sup><sub>E</sub> = 3.986 004 × 10<sup>14</sup> m<sup>3</sup> s<sup>-2</sup>. M<sub>☉</sub> = (GM)<sup>N</sup><sub>☉</sub>/G, M<sub>E</sub> = (GM)<sup>N</sup><sub>E</sub>/G. CODATA 2018: G = 6.674 30 × 10<sup>-11</sup> m<sup>3</sup> kg<sup>-1</sup> s<sup>-2</sup>. N(μ, σ): Normal distribution with mean μ, and standard deviation σ. U(a, b): A uniform distribution defined between a lower a and an upper b limit.
<figure><img src="image_8.png" /></figure>

**Fig. A.5.** 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 T<sub>0</sub>, the sector for transits observed with TESS, and the night and the telescope for transits observed with ExTrA.

<figure><img src="image_9.png" /></figure>

**Fig. A.6.** Transit observation (Fig. A.5) 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><img src="image_10.png" /><figcaption>**Fig. A.7.** Phased ESPRESSO RVs (blue error bars, as computed from the spectra), median posterior model (black line), and 68% CI (grey band).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. A.8.** Transit timing variations from TESS sectors 46 and 49 (orange), ExTrA (red), and a transit observed simultaneously by TESS and ExTrA (black).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. A.9.** 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</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Fig. A.10.** Seeing limited images centred on TOI-3884 at epoch 2016.0 (Gaia EDR3 position, Gaia Collaboration 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.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. A.11.** Planet-to-star radius ratio estimated jointly for all bands or for each band individually.</figcaption></figure>
## Appendix B: Spot contrast

We modelled the derived spot contrast in three different bands (Table A.4) to estimate the photosphere-spot difference temperature. The use of a blackbody spectrum fit the data poorly (see Fig. A.9). 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 A.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 B.1. The posterior model is shown in Fig. A.9. The posterior spot contrast in the Ks-band (that was used to derive the stellar parameters in Sect. 3) is 0.159 ± 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 $T_{eff} = 3400^{+460}_{-320}$ K, in agreement with the adopted one (Table A.2), but having a large uncertainty.

<table><caption>**Table B.1.** Modelling of the starspot contrast.</caption><thead><tr><th>Parameter</th><th></th><th>Prior</th><th>Posterior median and 68.3% CI</th><th>Prior</th><th>Posterior median and 68.3% CI</th></tr></thead><tbody><tr><td>Photosphere effective temperature, T<sub>eff</sub></td><td>[K]</td><td>N(3269, 70)</td><td>3249 ± 74</td><td>U(400, 10000)</td><td>3400<sup>+460</sup><sub>-320</sub></td></tr><tr><td>Surface gravity, log g</td><td>[cgs]</td><td>N(4.921, 0.028)</td><td>4.922 ± 0.028</td><td>U(-0.5, 6.0)</td><td>0.8<sup>+2.9</sup><sub>-1.0</sub></td></tr><tr><td>Metallicity, [Fe/H]</td><td>[dex]</td><td>N(0.23, 0.12)</td><td>0.22 ± 0.12</td><td>U(-0.4, 0.5)</td><td>-0.87<sup>+0.92</sup><sub>-1.4</sub></td></tr><tr><td>Photosphere-spot difference temperature</td><td>[K]</td><td>U(0, 500)</td><td>187 ± 21</td><td>U(0, 500)</td><td>214<sup>+95</sup><sub>-51</sub></td></tr></tbody></table>

**Notes.** Modelling with informative and non-informative priors for the effective temperature, surface gravity, and metallicity. N(μ, σ): Normal distribution prior with mean μ, and standard deviation σ. U(l, u): Uniform distribution prior in the range [l, u].