11institutetext:  Dipartimento di Fisica, Università degli Studi di Torino,
via Pietro Giuria 1, I-10125, Torino, Italy 22institutetext: Thüringer
Landessternwarte Tautenburg, 07778 Tautenburg, Germany 33institutetext:
Instituto de Astrofísica de Canarias (IAC), Calle Vía Láctea s/n, 38205 La
Laguna, Tenerife, Spain 44institutetext: Departamento de Astrofísica,
Universidad de La Laguna (ULL), 38206 La Laguna, Tenerife, Spain
55institutetext: Department of Space, Earth and Environment, Chalmers
University of Technology, SE-412 96 Gothenburg, Sweden 66institutetext: Max-
Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
77institutetext: Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta de
la Astronomía s/n, 18008 Granada, Spain 88institutetext: INAF – Osservatorio
Astrofisico di Catania, Via S. Sofia 78, 95123 Catania, Italy 99institutetext:
Centro de Astrobiología, (CSIC-INTA), ESAC Campus, Camino bajo del castillo
s/n, 28692 Villanueva de la Cañada, Madrid, Spain 1010institutetext: Steward
Observatory and Department of Astronomy, The University of Arizona, Tucson, AZ
85721, USA 1111institutetext: Hamburger Sternwarte, Gojenbergsweg 112, 21029
Hamburg, Germany 1212institutetext: Astrophysics Group, Keele University,
Staffordshire ST5 5BG, United Kingdom 1313institutetext: Max Planck Institut
für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany
1414institutetext: Departamento de Física de la Tierra y Astrofísica and
IPARCOS-UCM (Instituto de Física de Partículas y del Cosmos de la UCM),
Facultad de Ciencias Físicas, Universidad Complutense de Madrid, 28040 Madrid,
Spain 1515institutetext: Department of Physics, Ariel University, Ariel 40700,
Israel 1616institutetext: Vereniging Voor Sterrenkunde, Oude Bleken 12, 2400
Mol, Belgium 1717institutetext: AstroLAB IRIS, Provinciaal Domein “De
Palingbeek”, Verbrande-molenstraat 5, 8902 Zillebeke, Ieper, Belgium
1818institutetext: Komaba Institute for Science, The University of Tokyo,
3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan 1919institutetext: Department of
Multi-Disciplinary Sciences, Graduate School of Arts and Sciences, The
University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8902, Japan
2020institutetext: Institut d’Estudis Espacials de Catalunya (IEEC), Calle
Gran Capita 2-4, 08034, Barcelona, Spain 2121institutetext: Astrobiology
Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 2222institutetext: Centre
for Mathematical Plasma-Astrophysics, Department of Mathematics, KU Leuven,
Celestijnenlaan 200B, 3001 Heverlee, Belgium 2323institutetext: Centro
Astronómico Hispano en Andalucía (CAHA), Observatorio de Calar Alto, Sierra de
los Filabres, 04550 Gérgal, Spain 2424institutetext: Institut für Astrophysik
und Geophysik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1,
37077 Göttingen, Germany 2525institutetext: Department of Astronomy &
Astrophysics, University of Chicago, Chicago, IL 60637, USA 2626institutetext:
Landessternwarte, Zentrum für Astronomie der Universität Heidelberg,
Königstuhl 12, 69117 Heidelberg, Germany 2727institutetext: Institut de
Ciéncies de l’Espai (CSIC-IEEC), Campus UAB, de Can Magrans s/n, 08193
Bellaterra, Barcelona, Spain 2828institutetext: Centro de Astrobiología (CSIC-
INTA), Carretera de Ajalvir km 4, 28850 Torrejón de Ardoz, Madrid, Spain
2929institutetext: Center for Astroparticles and High Energy Physics (CAPA),
Universidad de Zaragoza, E-50009 Zaragoza, Spain 3030institutetext: School of
Sciences, European University Cyprus, Diogenes street, Engomi, 1516 Nicosia,
Cyprus

# TOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterization

E. Goffo TOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterization P. Chaturvedi TOI-4438 b: a transiting mini-Neptune amenable
to atmospheric characterizationTOI-4438 b: a transiting mini-Neptune amenable
to atmospheric characterization F. Murgas TOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterizationTOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterizationTOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterizationTOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterization G. Morello TOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterization J. Orell-
Miquel TOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterization L. Acuña TOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterizationTOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterization L. Peña-Moñino TOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterizationTOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterization E. Pallé TOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterization A. P. Hatzes
TOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterization S. Geraldía-González TOI-4438 b: a transiting mini-Neptune
amenable to atmospheric characterizationTOI-4438 b: a transiting mini-Neptune
amenable to atmospheric characterizationTOI-4438 b: a transiting mini-Neptune
amenable to atmospheric characterizationTOI-4438 b: a transiting mini-Neptune
amenable to atmospheric characterization F. J. Pozuelos TOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterization A. F. Lanza
TOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterization D. Gandolfi TOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterizationTOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterization J. A. Caballero TOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterizationTOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterization M. Schlecker TOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterization M. Pérez-
Torres TOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterization N. Lodieu TOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterizationTOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterizationTOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterizationTOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterization A. Schweitzer TOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterizationTOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterization C. Hellier TOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterization S. V. Jeffers
TOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterization C. Duque-Arribas TOI-4438 b: a transiting mini-Neptune
amenable to atmospheric characterizationTOI-4438 b: a transiting mini-Neptune
amenable to atmospheric characterization C. Cifuentes TOI-4438 b: a transiting
mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a transiting
mini-Neptune amenable to atmospheric characterization V. J. S. Béjar TOI-4438
b: a transiting mini-Neptune amenable to atmospheric characterizationTOI-4438
b: a transiting mini-Neptune amenable to atmospheric characterizationTOI-4438
b: a transiting mini-Neptune amenable to atmospheric characterizationTOI-4438
b: a transiting mini-Neptune amenable to atmospheric characterization M.
Daspute TOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterization F. Dubois TOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterizationTOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterizationTOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterizationTOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterization S. Dufoer TOI-4438 b: a transiting mini-Neptune
amenable to atmospheric characterizationTOI-4438 b: a transiting mini-Neptune
amenable to atmospheric characterization E. Esparza-Borges TOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterization A. Fukui
TOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterization Y. Hayashi TOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterizationTOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterization E. Herrero TOI-4438 b: a transiting mini-Neptune
amenable to atmospheric characterizationTOI-4438 b: a transiting mini-Neptune
amenable to atmospheric characterization M. Mori TOI-4438 b: a transiting
mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a transiting
mini-Neptune amenable to atmospheric characterization N. Narita TOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterization H. Parviainen
TOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterization L. Tal-Or TOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterizationTOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterization S. Vanaverbeke TOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterizationTOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterizationTOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterizationTOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterizationTOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterizationTOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterization I. Hermelo TOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterization P. J. Amado
TOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterization S. Dreizler TOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterizationTOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterization Th. Henning TOI-4438 b: a transiting mini-Neptune
amenable to atmospheric characterizationTOI-4438 b: a transiting mini-Neptune
amenable to atmospheric characterization J. Lillo-Box TOI-4438 b: a transiting
mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a transiting
mini-Neptune amenable to atmospheric characterization R. Luque TOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterization M. Mallorquín
TOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterization E. Nagel TOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterizationTOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterization A. Quirrenbach TOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterizationTOI-4438 b: a transiting mini-
Neptune amenable to atmospheric characterization S. Reffert TOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterization A. Reiners
TOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterization I. Ribas TOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterizationTOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterizationTOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterizationTOI-4438 b: a transiting mini-Neptune amenable to
atmospheric characterization P. Schöfer TOI-4438 b: a transiting mini-Neptune
amenable to atmospheric characterizationTOI-4438 b: a transiting mini-Neptune
amenable to atmospheric characterization H. M. Tabernero TOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterizationTOI-4438 b: a
transiting mini-Neptune amenable to atmospheric characterization M.
Zechmeister TOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterizationTOI-4438 b: a transiting mini-Neptune amenable to atmospheric
characterization

(Received 30 December 2023 / Accepted dd Month 2024)

We report the confirmation and mass determination of a mini-Neptune transiting
the M3.5 V star TOI-4438 (G 182-34) every 7.44 days. A transit signal was
detected with NASA’s TESS space mission in the sectors 40, 52, and 53. In
order to validate the planet TOI-4438 b and to determine the system
properties, we combined TESS data with high-precision radial velocity
measurements from the CARMENES spectrograph, spanning almost one year, and
ground-based transit photometry. We found that TOI-4438 b has a radius of
$R_{\mathrm{b}}$ = 2.52 $\pm$ 0.13 R⊕ (5% precision), which together with a
mass of $M_{\mathrm{b}}$ = 5.4 $\pm$ 1.1 M⊕ (20% precision), results in a bulk
density of $\rho_{\mathrm{b}}$ = 1.85${}^{+0.51}_{-0.44}$ g cm-3 ($\sim$28%
precision), aligning the discovery with a volatile-rich planet. Our interior
structure retrieval with a pure water envelope yields a minimum water mass
fraction of 46% (1$\sigma$). TOI-4438 b is a volatile-rich mini-Neptune with
likely H/He mixed with molecules, such as water, CO2, and CH4. The primary
star has a $J$-band magnitude of 9.7, and the planet has a high transmission
spectroscopy metric (TSM) of 136 $\pm$ 13\. Taking into account the relatively
warm equilibrium temperature of $T_{\mathrm{eq}}$ = 435 $\pm$ 15 K, and the
low activity level of its host star, TOI-4438 b is one of the most promising
mini-Neptunes around an M dwarf for transmission spectroscopy studies.

###### Key Words.:

planetary systems – planets and satellites: individual: TOI-4438 b – planets
and satellites: atmospheres – methods: radial velocity – techniques:
spectroscopic – stars: low-mass

##  1 Introduction

Precise and accurate planetary mass and radius measurements are necessary to
enable the atmospheric characterization of exoplanets. Current and forthcoming
telescopes such as the James Webb Space Telescope (JWST), the Atmospheric
Remote-sensing Infrared Exoplanet Large-survey (Ariel), and the Extremely
Large Telescope ELT will expand our frontiers of planetary science, allowing
us to explore the composition of the upper gaseous layer of exoplanets.
Selecting the best targets amenable to future atmospheric characterization is
therefore crucial.

Since the beginning of this century, the CoRoT (Baglin et al. 2006), Kepler
(Borucki et al. 2010), K2 (Howell et al. 2014), and TESS (Ricker et al. 2015)
space-based telescopes have opened our eyes to several thousands of transiting
exoplanets in our galaxy. Surprisingly, most of these planets have no
counterparts within our solar system, since their sizes are between the ones
of Earth and Neptune (1.0 $\lesssim\,R_{\mathrm{p}}$ $\lesssim$ 4.0 R⊕;
Batalha et al. 2013). Small planets with radii 1.8$\lesssim\,R_{\mathrm{p}}$
$\lesssim$ 4.0 R⊕, the so-called mini-Neptunes, seem to host hydrogen-
dominated atmospheres. The composition of the outer atmosphere as well as the
internal planetary structure are still not well understood, as they could be
explained by several degenerate models (Zeng et al. 2019). Atmospheric follow-
up observations are needed to solve the discrepancy between various
composition models. With the methodology proposed by Kempton et al. (2018),
one can calculate the expected signal-to-noise ratio (S/N) of transmission
spectra for a transiting planet, based on the strength of spectral features
and the brightness of the host star, assuming cloud-free atmospheres, namely
the transmission spectroscopy metric (TSM). The TSM is used for determining
which transiting exoplanets are the best targets for atmospheric
characterization via transmission spectroscopy, especially with the Near
Infrared Imager and Slitless Spectrograph (NIRISS) on JWST.

Small planets including mini-Neptunes happen to be common around M-type dwarf
stars (Dressing & Charbonneau 2013; Sabotta et al. 2021; Ribas et al. 2023),
as found also in data from the TESS mission (Chaturvedi et al. 2022; Espinoza
et al. 2022; Kawauchi et al. 2022; González-Álvarez et al. 2023; Barkaoui et
al. 2023). Follow-up observations focused on M dwarfs have their own
advantages as these are abundant compared to F, G, and K stars. Their smaller
stellar radii and masses produce deeper transit signals (the transit depth
$\delta$ is inversely proportional to $R_{\star}^{2}$) and larger Doppler
reflex motions (the radial velocity semi-amplitude $K_{\star}$ is inversely
proportional to $M_{\star}^{2/3}$), increasing our capability to detect small
low-mass planets (see, e.g., Ribas et al. 2023).

Here, we present the characterisation of TESS object of interest (TOI)-4438 b,
a transiting mini-Neptune orbiting the M3.5 V star TOI-4438 (G 182-34). We
used CARMENES precise radial velocities (RVs) and ground-based transit
observations to confirm the planet and to derive its fundamental parameters,
such as mass, radius, orbital period, semi-major axis, and equilibrium
temperature. TOI-4438 b with a score of $\sim$136 scales very high on TSM,
thereby, making it a promising target for future atmospheric observations with
JWST. This work is part of the TESS follow-up program within the CARMENES
observations survey (see, e.g., González-Álvarez et al. 2023; Kossakowski et
al. 2023; Mallorquín et al. 2023; Murgas et al. 2023; Palle et al. 2023).

##  2 Observations

###  2.1 TESS photometry

TESS observed TOI-4438 in sector 40, from 25 June to 23 July 2021, with CCD#2
of camera 1 with an integration time of 2 min. TOI-4438 had more observations
taken in sector 52, from 19 May to 12 June 2022, and in sector 53, from 13
June to 8 July 2022. The observations were done again at the same cadence of 2
min with CCD#1 of camera 1 in sector 52, and with CCD#2 of camera 1 in sector
53. We retrieved the TESS data from the Mikulski Archive for Space Telescopes
(MAST)111 https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html. The
data were processed by Science Processing Operations Center (SPOC, Jenkins et
al. 2016) pipeline using the Presearch Data Conditioning Simple Aperture
Photometry (PDCSAP) algorithm. For details, refer Stumpe et al. (2012, 2014);
Smith et al. (2012).

In order to correct for the photometric contamination caused by TESS’s large
pixel size of 21″, a photometric mask is used to extract the TESS Simple
Aperture Photometry (SAP) light curve. We plotted the target pixel files
(TPFs; Fig. 1) of TOI-4438 using the code
tpfplotter222https://github.com/jlillo/tpfplotter. (Aller et al. 2020). The
TPFs include all the Gaia Data Release 3 (DR3, Gaia Collaboration et al. 2023)
sources, down to $\Delta G$ = 10 mag fainter than the target star and
spatially located within the field of view of the TESS aperture. No bright
sources fall inside the photometric aperture of TOI-4438. For the analysis
presented in this paper, we used the TESS contamination corrected PDCSAP light
curves (Fig. 2).

Figure 1: TPFs of TOI-4438 in TESS sectors 40, 52, and 53. The red bordered
squares mark the optimal photometric aperture used to extract the SAP fluxes.
The different sizes of the red circles show the $G$-band magnitudes from
$Gaia$ DR3 for all nearby stars with respect to TOI-4438 (marked with an
”$\times$”) up to $\Delta G$ = 10 mag fainter.

Figure 2: TESS PDCSAP light curve of TOI-4438 from sectors 40, 52, and 53. The
green triangles mark the timings of the transits of TOI-4438 b.

###  2.2 CARMENES radial velocity follow-up

We spectroscopically monitored TOI-4438 with the Calar Alto high-Resolution
search for M dwarfs with Exoearths with Near-infrared and optical Echelle
Spectrographs (CARMENES) installed at the 3.5 m telescope at Calar Alto
Observatory, Almería, Spain. The instrument comprises two highly stable,
fiber-fed spectrographs (Quirrenbach et al. 2014, 2018), one covering the
wavelength range 520–960 nm (VIS) with a resolving power of R $\approx$ 94
600, the other covering the near-infrared range 960–1710 nm (NIR), with a
resolving power of R $\approx$ 80 400.

We collected 34 high-resolution spectra between 21 May 2022 and 2 May 2023.
Depending on the sky condition and observing constraints, each of the spectra
was exposed between 700 and 1800 s. The resultant average S/N was $\sim$47 at
740 nm. We followed the standard caracal pipeline (Zechmeister et al. 2014) to
reduce the CARMENES data.

The precise relative RVs were computed using the
serval333https://github.com/mzechmeister/serval. pipeline (Zechmeister et al.
2018) with the standard corrections for barycentric and secular motions. The
ancillary corrections applied for the nightly drift by incorporating the
nightly zero-points and the atmospheric effects using the telluric lines are
described by Nagel et al. (2023). In our analysis we also used the serval
produced spectral activity indicators, namely, the chromatic index (CRX), the
differential line width (dLW), the H$\alpha$ index, the Ca ii infrared triplet
(IRT) index, and the Na i D doublet index.

We discarded two data points in the CARMENES VIS RV time series due to their
low signal-to-noise ratio (S/N $<$ 21). The final CARMENES dataset includes 32
measurements with a root mean square (RMS) of $\approx$ 2 m s-1. We did not
include the NIR RVs in our further analysis, given their low signal-to-noise
ratio. Table 4 lists the CARMENES VIS RV measurements with the time
information in the Barycentric Julian Date- Barycentric Dynamical Time
(BJDTDB) units.

###  2.3 Ground-based photometry

####  2.3.1 MuSCAT2

TOI-4438 was observed on three nights, namely, on 17 May 2022, 2 May 2023, and
8 July 2023, with the multi-band imager MuSCAT2 (Narita et al. 2019) mounted
on the 1.5 m Telescopio Carlos Sánchez (TCS) at Teide Observatory, Tenerife,
Spain. MUSCAT2 is equipped with four 1k$\times$1k pixel CCDS, with each having
a field of view of 7.4′$\times$7.4′. The cameras can simultaneously observe in
the $g^{\prime}$, $r^{\prime}$, $i^{\prime}$, and $z_{s}$ bands. We observed
two complete transits and one partial transit of TOI-4438 b in each band. The
times for exposure varied between 10 s to 30 s. The data reduction were
carried out using the standard pipeline as described by Parviainen et al.
(2019); Parviainen (2015).

####  2.3.2 SuperWASP

The SuperWASP transit search (Pollacco et al. 2006) observed the field of
TOI-4438 in 2004 and then every year from 2007 to 2010. The eight cameras have
each 2048$\times$2048 CCDs with f/1.8 lenses. A total of 71 000 photometric
data points were obtained over a period of of $\sim$ 120 nights in the visible
passband of 400–700 nm. At $V$ = 13.7 mag, TOI-4438 is relatively faint for
WASP, though it is the brightest star in the 48″ extraction aperture.

####  2.3.3 TJO

We carried out a dedicated photometry monitoring campaign of TOI-4438 on 49
different nights from April to September 2023 with the 0.8 m Telescopi Joan
Oró (TJO; Colomé et al. 2010) at the Montsec Observatory, Lleida, Spain. A
total of 285 data points were observed using the Johnson $R$ filter of the
LAIA imager on a 4k$\times$4k CCD with a 30′ field of view. The standard
reduction process was followed using the AstroImageJ(Collins et al. 2017)
package and the icat pipeline (Colome & Ribas 2006) of the TJO. We used
indigenous pipelines for removal of outliers for observations made during poor
sky conditions. We obtained an effective rms of $\sim$9 ppt in the $R$ filter.

####  2.3.4 LCOGT

We performed photometric monitoring of TOI-4438 in the $V$ band between 2
August and 6 November 2023, using the SBIG camera of one of the 40 cm
telescope of Las Cumbres Observatory Global Telescope (LCOGT; Brown et al.
2013) providing a field of view of 29.2′$\times$19.5′. We took three exposures
of 600 s in each of the 71 different observing epochs during 96 d. The
observing conditions were mostly good and the average seeing varied from 2.0
to 6.0″. Raw data were processed using the BANZAI pipeline (McCully et al.
2018) and differential aperture photometry of TOI-4438 was performed using the
AstroImageJ software. An optimal aperture of 8 pixels ($\sim$ 4.5″) was
adopted to minimize the dispersion of the differential light curves.

We used the 1 m LCOGT telescopes from 9 June to 4 November 2023, to observe
TOI-4438 in the $B$ filter with an exposure time of 120 s until JD = 2460156,
and of 77 s afterwards. This produced 88 images, out of which nine were found
to be bad. The final light curve has an rms of 0.57% $\sim$ with 79 data
points, spanning 148 d.

####  2.3.5 ASAS-SN

TOI-4438 was photometrically monitored by the All-Sky Automated Survey for
Supernovae (ASAS-SN; Shappee et al. 2014; Kochanek et al. 2017). ASAS-SN
consists of a network of 24 robotic telescopes with a diameter of 14 cm,
distributed around the world. We retrieved the ASAS-SN light curve from the
webpage of the project444https://asas-sn.osu.edu/.. There are 246 $V$-band
photometry measurements taken for the star between 3 May 2013 and 25 September
2018 with a rms scatter of $\sim$0.021 mag.

####  2.3.6 e-EYE

TOI-4438 was also observed by e-EYE (Entre Encinas y
Estrellas)555www.e-eye.es/. 16″ ODK corrected-Dall-Kirkham reflector, located
at Fregenal de la Sierra in Badajoz, Spain. $V$ band observations were taken
between 12 September and 13 November 2023 with a Kodak KAF-16803 CCD chip
mounted on ASA DDM85. The images were reduced using the LesvePhotometry
package666www.dppobservatory.net.. The rms of the data was $\sim$0.006 mag.

####  2.3.7 ADONIS

TOI-4438 was observed by the privately owned ADONIS observatory in the $V$
filter from 30 September to 6 December 2023. The telescope used is 0.25 m
skywatcher Quattro Newtonian telescope with a Moravian CCD camera.
LesvePhotometry is used to perform the aperture photometry.

##  3 Stellar properties

The star TOI-4438 was originally discovered as a high-proper motion star (34th
in the plate 182 in Hercules) in the northern hemisphere at the Lowell
Observatory by Giclas (1966). The star has been identified as G 182–34 (Giclas
et al. 1971) and has been tabulated by a number of catalogs on proper motion
(Luyten 1979; Lépine & Shara 2005; Schneider et al. 2016), parallax (Dahn et
al. 1988, 2002; van Altena et al. 1995), or the solar neighborhood (Reid &
Cruz 2002; Lépine & Gaidos 2011; Frith et al. 2013; Gaia Collaboration et al.
2021).

###  3.1 Stellar parameters

We derived the stellar parameters as discussed in Schweitzer et al. (2019).
The total luminosity was computed by summing over the spectral energy
distribution with VOSA (Bayo et al. 2008). Broadband photometry from passbands
Johnson ($B$) to AllWISE ($W4$) and astrometry from Gaia DR3 catalog were
utilized. The stellar atmospheric parameters ($T_{\rm eff}$, $\log{g}$, and
[Fe/H]) were derived with the
SteParSyn777https://github.com/hmtabernero/SteParSyn/. code (Tabernero et al.
2022) using the line list and model grid described by Marfil et al. (2021).
The mass of the star was derived using the the linear mass-radius relation
(Schweitzer et al. 2019) whereas the radius of the star was derived from
Stefan-Boltzmann’s law assuming the bolometric luminosity and the
spectroscopic temperature. The stellar parameters of TOI-4438 are listed in
Table 1. The effective temperature of $T_{\mathrm{eff}}$ = 3422 $\pm$ 81 K and
surface gravity of $\log{g_{\star}}$ = 4.68 $\pm$ 0.04 (cgs) match the $\sim$
M3 V spectral-type from the spectro-photometric relations of Pecaut & Mamajek
(2013) and Cifuentes et al. (2020) and, the M3.5$\pm$0.5 V spectral type from
low-resolution optical spectroscopy by Reid et al. (2003).

Table 1: Stellar parameters of TOI-4438.

Parameter | Value | Reference
---|---|---
Identifiers | G 182-34 | Gic71
| Karmn J18012+355 | Cab16
| TIC 22233480 | TIC
Sp. type | M3.5 V | Reid03
$\alpha$ (ICRS, epoch 2016.0) | 18:01:16.14 |  Gaia DR3
$\delta$ (ICRS, epoch 2016.0) | +35:35:41.6 |  Gaia DR3
$\mu_{a}\,\cos\delta$ (mas yr-1) | +29.212 $\pm$ 0.017 |  Gaia DR3
$\mu_{\delta}$ (mas yr-1) | –517.512 $\pm$ 0.018 |  Gaia DR3
$\varpi$ (mas) | 33.2642 $\pm$ 0.0154 |  Gaia DR3
$d$ (pc) | 30.0623 $\pm$ 0.0139 |  Gaia DR3
$\gamma$ (km s-1) | –35.81 $\pm$ 0.47 |  Gaia DR3
$T_{\text{eff}}$ [K] | 3422 $\pm$ 81 | This work
$\log{g_{\star}}$ [cgs] | 4.68 $\pm$ 0.04 | This work
[Fe/H] [dex] | –0.19 $\pm$ 0.05 | This work
$L_{\star}$ [$L_{\odot}$] | 0.01706 $\pm$ 0.00007 | This work
$R_{\star}$ [$R_{\odot}$] | 0.372 $\pm$ 0.018 | This work
$M_{\star}$ [$M_{\odot}$] | 0.368 $\pm$ 0.021 | This work
$\rho_{\star}$ [g cm-3] | 10.06 $\pm$ 1.62 | This work
$v\sin\,i_{\star}$ [km s-1] |  $<$ 2 | This work
$P_{\text{rot}}$ [d] | 68 $\pm$ 6 | This work
Age [Gyr] | 5.1 $\pm$ 2.8 | This work
$G$ [mag] | 12.50 $\pm$ 0.68 |  Gaia DR3
$T$ [mag] | 11.2695 $\pm$ 0.0073 | TIC
$J$ [mag] | 9.695 $\pm$ 0.021 | 2MASS
$K_{s}$ [mag] | 8.869 $\pm$ 0.018 | 2MASS

Gic71: Giclas et al. (1971); Cab16: Caballero et al. (2016); TIC: Stassun et
al. (2019); Gaia DR3: Gaia Collaboration et al. (2023); 2MASS: Skrutskie et
al. (2006); Reid03: Reid et al. (2003).

###  3.2 Stellar activity

The broad wavelength range coverage offered by the CARMENES VIS and NIR
channels allows us to study a total of sixteen spectral activity indicators.
These are pEW(H$\alpha$) (from which we derived $\log{{\rm H}\alpha/L_{\rm
bol}}$ as Jeffers et al. 2018), HeD3, NaD, Ca IRT-a,-b,-c, He i$\lambda$10830,
Pa $\beta$, CaH2, CaH3, TiO 7050, TiO 8430, TiO 8860, VO 7436, VO 7942, and
FeH Wing-Ford, and they were computed following the methods described by
Schöfer et al. (2019). In addition, the output parameters from serval also
include the CRX and dLW parameters for both channels (Zechmeister et al.
2018), resulting in twenty activity diagnostics.

In all CARMENES spectra we found that H$\alpha$ is in absorption, indicating
that TOI-4438 is a weakly active star with undetectable levels of activity-
induced emission in the H$\alpha$ line. We also did not detect H$\alpha$
emission due to flaring in the CARMENES spectra. Jeffers et al. (2018) showed
that only about one third of M3 dwarfs are H$\alpha$ active. Given the M3.5 V
spectral type, the low activity level of TOI-4438 is therefore not surprising.
We also computed Pearson’s $r$ correlation coefficient between the measured
RVs and the spectral activity indicators as described in Jeffers et al. (2020)
and confirmed the star to be least active.

###  3.3 Rotation period of the star

We inspected the SuperWASP light curves for rotational modulations as
discussed in Maxted et al. (2011). There appears to be significant power at
long periods (see Fig. 3), although the periodograms are hard to interpret. In
each observation season, the amplitude of the modulations is variable, and the
data covered only about two cycles. There are likely to be profile and phase
changes between years. We found a rotational modulation with an amplitude of
$\sim$ 6 mmag, and a period of 68 $\pm$ 6 d. The first harmonic is also
present at $\sim$ 34 d in the years 2004 and 2010.

In addition, we computed the generalized Lomb-Scargle (GLS; Zechmeister &
Kürster 2009) periodograms of the TJO, LCOGT (in $V$\- and $B$\- filters),
ASAS-SN, e-EYE, and ADONIS light curves of TOI-4438, as shown in Fig. 4. The
periodogram of the TJO light curve shows its highest peak at $\sim$34 d, and a
second high peak at $\sim$64 d, consistent with the SuperWASP results. The
same applies to the periodogram of the LCOGT data ($V$-filter), where we found
the highest signal at 68 $\pm$ 10 d, and the ASAS-SN photometry, which shows a
modulation at around 57 d. LCOGT $B$-filter data show a peak at 22 $\pm$ 1 d
with a false alarm probability of $<$0.1%, which could be related to the
second harmonic of the stellar rotation period. Since the e-EYE and ADONIS
data covered a shorter baseline, the periodograms displayed no significant
signals around 68 or 34 d. We concluded that TOI-4438 has a rotation period of
68 $\pm$ 6 d, as derived from SuperWASP photometry.

Figure 3: Periodogram of the SuperWASP data for TOI-4438 from each year and
(top) the periodogram of 2007–2010 combined. The horizontal dashed lines
indicate the estimated 10%- and 1%-likelihood false-alarm levels. The blue
vertical dashed lines mark the stellar rotation period and its first harmonic.
Figure 4: Periodograms of the TJO, LCOGT ($V$ and $B$), ASAS-SN, e-EYE, and
ADONIS data for TOI-4438. The horizontal dashed lines indicate the estimated
10%- and 1%-likelihood false-alarm levels. The blue vertical dashed lines mark
the stellar rotation period and its first harmonic.

###  3.4 Age

With a rotation period of 68 $\pm$ 6 d, TOI-4438 is very likely an old star.
From the relations of Bouma et al. (2023), a maximum rotation period of 25 d
is detected for members of the Praesepe cluster (500–700 Myr; Lodieu et al.
2019). The rotation period of TOI-4438 is at least twice longer, so the age is
expected to be much older, although the relation is not linear. Similarly,
Curtis et al. (2020) showed that field stars with spectral types between K0
and M1–M2 and ages of 6 $\pm$ 2 Gyr have rotation periods around 35–45 d,
again much shorter than TOI-4438. Eventually, we used the rotation–age
relations applicable to M2.5–6.5 dwarfs from the recent study by Engle &
Guinan (2023) and derived an age of 5.1 $\pm$ 2.8 Gyr.

To constrain the age of an M-type dwarf we can also look for wide, higher-
mass, gravitationally bound companions. We searched for possible co-moving
common proper motion companions to TOI-4438 in the $Gaia$ Data Release 3 (DR3)
in a radius of five degrees. The search was based on the proximity in
parallax, using a very conservative interval of 3.25 pc, 200 times more than
the error bar on the parallax reported in DR3. Ten sources were returned but
none of them shows a proper motion in right ascension consistent with the
large motion of TOI-4438 ($\mu_{\alpha}\cos\delta$ $\approx$ $-$517.5 mas
yr-1). Moreover, we checked several parameters that may indicate departure
from a single star astrometric model, such as the renormalized unit weight
error (RUWE), the excess noise of the source and its significance, and the
percent of successful-IPD windows with more than one peak. None of them point
toward the presence of an unseen companion around TOI-4438. This is consistent
with the null results on TOI-4438 of the laser-guided, adaptive-optics, Robo-
AO M-dwarf multiplicity survey of Lamman et al. (2020).

##  4 Analysis

###  4.1 Transit light curve modeling

Prior to performing a joint analysis of the photometric and Doppler time
series (Sect. 4.4), we performed a preliminary analysis of the available
transit light curves to check whether the different passbands provide
consistent transit depths. With this aim, we used the code pyaneti (Barragán
et al. 2017, 2022) to model the TESS light curve from sectors 40, 52, and 53
(Sect. 2.1), and MuSCAT2 ground-based photometry (Sect. 2.3.1). The TESS data
included in the analysis are subsets of 8 h of the PDCSAP light curves,
centered around each transit epoch, and detrended by fitting a second-order
polynomial to the out-of-transit data.

Based on the period and ephemeris reported by SPOC (Porb,b = 7.446303 $\pm$
0.000023 d and T0,b = 2459396.40958 $\pm$ 0.00075 BJDTDB), we allowed the
period to vary between 7.2 and 7.7 d, and the epoch of reference transit
between BJDTDB = 2459396.16 and 2459396.66 d. We adopted the parameterization
proposed by Anderson et al. (2010) for the eccentricity $e$ and the argument
of periastron of the stellar orbit $\omega_{\star}$ (i.e.,
$\sqrt{e}\sin{\omega_{\star}}$ and $\sqrt{e}\cos{\omega_{\star}}$). pyaneti
uses Mandel & Agol (2002)’s limb-darkened quadratic model for the transit
light curve and the limb-darkening parameterization proposed by Kipping
(2013). We adopted Gaussian priors for the linear and quadratic limb-darkening
coefficients ($q_{1}$, $q_{2}$) using the values calculated with the Python
Limb Darkening Toolkit888https://github.com/hpparvi/ldtk. (PyLDTk, Parviainen
& Aigrain 2015) for TESS and MuSCAT2 bands. We set uniform priors for all of
the remaining parameters. A photometric jitter term was added to account for
any instrumental noise not included in the nominal uncertainties. We sampled
the parameter space with 500 walkers using the Markov chain Monte Carlo (MCMC)
ensemble sampler algorithm implemented in pyaneti. We created the posterior
distributions using the last 5000 iterations of the converged chains with a
thin factor of 10, giving a distribution of 250 000 data points for each
sampled parameter. We followed the procedure and the convergence test as
described by Barragán et al. (2017).

In order to check for uniform transit depths in the TESS and MuSCAT2 bands,
and determine the chromaticity of the transits in each available filter, we
performed two analyses: (1) we allowed an independent scaled planetary radius
$R_{\mathrm{b}}/R_{\star}$ for each passband; (2) we modeled a single scaled
planetary radius, namely assuming the same transit depth in all passbands.
Table 2 lists the adopted priors and posterior values of the sampled
parameters. For the parameter estimates and their 1$\sigma$ uncertainties, we
took the median and the 68.3% credibility interval of the posterior
distributions. Fig. 5 shows the phase-folded TESS and MuSCAT2 transit light
curves of TOI-4438 b. We measured an orbital period of $P_{\mathrm{orb,b}}$ =
7.4462800 $\pm$ 0.0000087 d, an epoch of reference transit of
$T_{\mathrm{0,b}}$ = 2459396.4116 $\pm$ 0.0006 d (BJDTDB), and a radius of
$R_{\mathrm{b}}$ = 2.51 $\pm$ 0.12 $R_{\oplus}$ from the unique scaled planet
radius fit (case 2).

Fig. 6 displays the posterior distributions for the sampled scaled planetary
radii $R_{\mathrm{p}}/R_{\star}$ for each passband. We found that the
posterior distributions for the TESS and MuSCAT2 bands overlap, except for the
$g^{\prime}$ and $r^{\prime}$ filters.

We also found a color dependence, given the deeper transits in the
$g^{\prime}$ and $r^{\prime}$ filters. Assuming an astrophysical origin, this
color trend could be caused by contamination from a cooler dwarf (Parviainen
et al. 2019) or unocculted star spots (Ballerini et al. 2012). Potential
inaccuracies due to the stellar limb-darkening parameterization (Morello et
al. 2017) or planetary atmospheric signatures (Chen et al. 2021) should be
negligible compared to the error bars. In order to check the hypothesis of
astrophysical contamination, we fitted the measured planetary radii in
multiple bands by assuming a unique planet-to-star radius ratio
$R_{\mathrm{b}}/R_{\star}$ and different star spots configurations modeled
with ExoTETHyS (Morello et al. 2021). The stellar spectrum was modeled as the
sum of two components, namely the quiet photosphere and the spots,
characterized by different values of $T_{\text{eff}}$ and $\log{g_{\star}}$,
and neglecting stellar limb-darkening effects (Cracchiolo et al. 2021;
Thompson et al. 2023). We adopted the PHOENIX-COND and PHOENIX-DRIFT libraries
of stellar spectra (Claret et al. 2012, 2013; Husser et al. 2013). Technical
details are given in Appendix B. It turned out that a spot filling factor of
$\sim$40$\%$ and temperature contrast of $\sim$500 K are needed to explain the
different measurements between $g^{\prime}$ and TESS filters, but they
struggle to maintain the uniform transit depth observed in $i^{\prime}$,
$z_{s}$, and TESS. Furthermore, these values would denote a very high level of
stellar activity, in contrast with all other indicators (see Sect. 3.2). This
scenario is mathematically equivalent to that of a cooler M-type contaminant
with similar brightness to that of the host star, following the model by
Morello et al. (2023). Such a bright contaminant is also unlikely to be unseen
(see Sect. 3.4). Another possibility is that the transit measurements in
$g^{\prime}$ and $r^{\prime}$ bands are affected by systematic bias, due to
the low S/N of the MuSCAT2 light curves obtained with these filters.
Therefore, we ignored the bluer $g^{\prime}$ and $r^{\prime}$, and used
$i^{\prime}$ and $z_{s}$ only for the following analysis. We discuss an
alternative solution including all bands and potential color contamination in
Appendix B.

Table 2: TOI-4438 b parameters from the transit light curve modeling with
pyaneti.(a)

Parameter Prior Derived value (multi-radii) Derived value (single radius)
Model parameters $P_{\mathrm{orb,b}}$ [d] $\mathcal{U}$[7.2, 7.7] 7.4462801
$\pm$ 0.0000091 7.4462800 $\pm$ 0.0000087 $T_{\mathrm{0,b}}$
[BJD${}_{\mathrm{TDB}}-$2,450,000] $\mathcal{U}$[9396.16, 9396.66] 9396.4116
$\pm$ 0.0007 9396.4116 $\pm$ 0.0006 $R_{\mathrm{b}}/R_{\star}$
$\mathcal{U}$[0.001, 0.09] … 0.0619${}^{+0.0006}_{-0.0005}$
$R_{\mathrm{b}}/R_{\star}\mathrm{TESS}$ $\mathcal{U}$[0.001, 0.09] 0.0606
$\pm$ 0.0012 … $R_{\mathrm{b}}/R_{\star}\mathrm{MuSCAT2\,g^{\prime}}$
$\mathcal{U}$[0.001, 0.09] 0.0678 $\pm$ 0.0026 …
$R_{\mathrm{b}}/R_{\star}\mathrm{MuSCAT2\,r^{\prime}}$ $\mathcal{U}$[0.001,
0.09] 0.0661 $\pm$ 0.0019 …
$R_{\mathrm{b}}/R_{\star}\mathrm{MuSCAT2\,i^{\prime}}$ $\mathcal{U}$[0.001,
0.09] 0.0622 $\pm$ 0.0013 … $R_{\mathrm{b}}/R_{\star}\mathrm{MuSCAT2\,z_{s}}$
$\mathcal{U}$[0.001, 0.09] 0.0624 $\pm$ 0.0010 … $b_{\mathrm{b}}$
$\mathcal{U}$[0, 1] 0.38${}^{+0.16}_{-0.24}$ 0.15${}^{+0.14}_{-0.10}$
$a/R_{\star}$ $\mathcal{U}$[1, 50] 28.6${}^{+4.0}_{-4.9}$ 30 $\pm$ 5
$\sqrt{e_{\mathrm{b}}}\sin{\omega_{\mathrm{\star,b}}}$ $\mathcal{U}$[–1, 1]
–0.13${}^{+0.38}_{-0.25}$ –0.14${}^{+0.40}_{-0.27}$
$\sqrt{e_{\mathrm{b}}}\cos{\omega_{\mathrm{\star,b}}}$ $\mathcal{U}$[–1, 1]
–0.004 $\pm$ 0.447 –0.02 $\pm$ 0.47 Derived parameters $R_{\mathrm{b}}$
[$R_{\oplus}$] … … 2.51 $\pm$ 0.12 $R_{\mathrm{b,TESS}}$ [$R_{\oplus}$] … 2.46
$\pm$ 0.13 … $R_{\mathrm{b,MuSCAT2\,g^{\prime}}}$ [$R_{\oplus}$] … 2.75 $\pm$
0.17 … $R_{\mathrm{b,MuSCAT2\,r^{\prime}}}$ [$R_{\oplus}$] … 2.68 $\pm$ 0.15 …
$R_{\mathrm{b,MuSCAT2\,i^{\prime}}}$ [$R_{\oplus}$] … 2.52 $\pm$ 0.13 …
$R_{\mathrm{b,MuSCAT2\,z_{s}}}$ [$R_{\oplus}$] … 2.53 $\pm$ 0.13 …
$a_{\mathrm{b}}$ [au] … 0.0492${}^{+0.0074}_{-0.0086}$ 0.053 $\pm$ 0.009
$e_{\mathrm{b}}$ … 0.21${}^{+0.24}_{-0.14}$ 0.23${}^{+0.25}_{-0.15}$
$\omega_{\star,\mathrm{b}}$ [deg] … –34${}^{+134}_{-111}$
–37${}^{+139}_{-108}$ $i_{\mathrm{b}}$ [deg] … 89.22${}^{+0.50}_{-0.53}$
89.71${}^{+0.35}_{-0.21}$ $\tau_{14,\mathrm{b}}$ [h] … 2.02 $\pm$ 0.03 2.01
$\pm$ 0.02 $T_{\mathrm{eq,b}}$ [K] (b) … 453${}^{+46}_{-30}$
438${}^{+45}_{-32}$ $S_{\mathrm{b}}$ [S⊕] … 7.04${}^{+3.3}_{-1.7}$
6.1${}^{+2.9}_{-1.6}$ Additional model parameters Parameterized limb-darkening
coefficients $q_{\mathrm{1,TESS}}$ $\mathcal{N}$[0.276, 0.1] 0.30 $\pm$ 0.08
0.30 $\pm$ 0.08 $q_{\mathrm{2,TESS}}$ $\mathcal{N}$[0.261, 0.1] 0.26 $\pm$
0.10 0.26 $\pm$ 0.10 $q_{\mathrm{1,MuSCAT2\,g^{\prime}}}$ $\mathcal{N}$[0.629,
0.1] 0.64 $\pm$ 0.10 0.64 $\pm$ 0.10 $q_{\mathrm{2,MuSCAT2\,g^{\prime}}}$
$\mathcal{N}$[0.312, 0.1] 0.35 $\pm$ 0.10 0.35 $\pm$ 0.10
$q_{\mathrm{1,MuSCAT2\,r^{\prime}}}$ $\mathcal{N}$[0.572, 0.1] 0.55 $\pm$ 0.16
0.55 $\pm$ 0.16 $q_{\mathrm{2,MuSCAT2\,r^{\prime}}}$ $\mathcal{N}$[0.317, 0.1]
0.33 $\pm$ 0.10 0.33 $\pm$ 0.10 $q_{\mathrm{1,MuSCAT2\,i^{\prime}}}$
$\mathcal{N}$[0.334, 0.1] 0.37 $\pm$ 0.08 0.37 $\pm$ 0.08
$q_{\mathrm{2,MuSCAT2\,i^{\prime}}}$ $\mathcal{N}$[0.260, 0.1] 0.31 $\pm$ 0.10
0.31 $\pm$ 0.10 $q_{\mathrm{1,MuSCAT2\,z_{s}}}$ $\mathcal{N}$[0.241, 0.1] 0.16
$\pm$ 0.08 0.16 $\pm$ 0.08 $q_{\mathrm{2,MuSCAT2\,z_{s}}}$
$\mathcal{N}$[0.240, 0.1] 0.19 $\pm$ 0.10 0.19 $\pm$ 0.10 Jitter terms
$\sigma_{\mathrm{TESS}}$ $\mathcal{J}$[0,100] 0.000058 $\pm$ 0.000054 0.000058
$\pm$ 0.000054 $\sigma_{MuSCAT2\,g^{\prime}}$ $\mathcal{J}$[0,100] 0.00021
$\pm$ 0.00020 0.00021 $\pm$ 0.00020 $\sigma_{\mathrm{MuSCAT2\,r^{\prime}}}$
$\mathcal{J}$[0,100] 0.00030 $\pm$ 0.00026 0.00030 $\pm$ 0.00026
$\sigma_{\mathrm{MuSCAT2\,i^{\prime}}}$ $\mathcal{J}$[0,100] 0.00014 $\pm$
0.00012 0.00014 $\pm$ 0.00012 $\sigma_{\mathrm{MuSCAT2\,z_{s}}}$
$\mathcal{J}$[0,100] 0.00018 $\pm$ 0.00026 0.00018 $\pm$ 0.00026 999 a
$\mathcal{U}[a,b]$ refers to uniform priors between $a$ and $b$;
$\mathcal{N}[a,b]$ refers to Gaussian priors with mean $a$ and standard
deviation $b$; $\mathcal{J}[a,b]$ refers to Jeffrey’s priors between $a$ and
$b$. Inferred parameters and uncertainties are defined as the median and the
68.3% credible interval of their posterior distributions. b Assuming zero
albedo and uniform redistribution of heat.

(a)

(b)

(c)

(d)

(e)

Figure 5: Phase-folded transit light curves of TOI-4438 b from the (a) TESS,
and (b) $g^{\prime}$, (c) $r^{\prime}$, (d) $i^{\prime}$, and (e) zs MuSCAT2
bands. Data are shown as filled circles with their nominal uncertainties. The
10-min binned data points are marked with white circles. Figure 6: Posterior
distributions of the sampled scaled planetary radii $R_{\mathrm{b}}/R_{\star}$
for the TESS and MuSCAT2 $g^{\prime}$, $r^{\prime}$, $i^{\prime}$, $z_{s}$
bands. The solid and dotted, vertical, gray lines mark the median and the
68.3% credibility interval of the $R_{p}/R_{\star}$’s posterior distribution,
as derived in case 2 (see Sect. 4.1).

###  4.2 Additional planet search and detection limits

We adopted the box least-square (BLS; (Kovács et al. 2002; Hartman & Bakos
2016) method to look for additional transits in the PDCSAP TESS light curves.
Before applying the BLS we detrended the data using the package
citlalique101010https://github.com/oscaribv/citlalicue, which allows to
detrend light curves using Gaussian processes (GPs) to remove low frequency
signals in the light curves (Barragán et al. 2022).

We used the BoxLeastSquares class of the astropy.timeseries python package.
Whenever the program identified a transit signal, we removed the in-transit
data points and ran the algorithm again to search for additional transiting
planets. The BLS periodograms of the TESS light curves were evaluated over
periods from 0.5 d to half the total length of the observations, namely
$\sim$190 d. The first transit signal was identified at 7.44 d, corresponding
to the orbital period of TOI-4438 b. We found no other signals in the data
corresponding to additional transiting planets.

To identify possible additional companions that might perturb the orbit of
TOI-4438 b, we analysed the available transit light curves for transit timing
variation (TTVs). We used the ten TESS transits from sectors 40, 52, and 53,
and the two full transits observed with MuSCAT2 on 2 May and 8 July 2023. We
co-added the data from MuSCAT2 passbands $z_{s}$ and $i^{\prime}$ prior to the
analysis. We individually modeled the mid-time of each transit with the code
pyaneti (Barragán et al. 2022). We found no significant TTV in the $\sim$770 d
baseline of our observations, as shown in Fig. 7.

Figure 7: Transit-timing variations of TOI-4438 b. Blue points correspond to
TESS transits while the orange points correspond to the full MuSCAT2 transits.

The lack of extra signals attributable to transiting planets in the TESS data
might be due to one of the following scenarios (see, e.g., Wells et al. 2021;
Schanche et al. 2022; Delrez et al. 2022; Pozuelos et al. 2023): (1) no other
planets exist in the system; (2) they do exist, but they do not transit; (3)
they do exist and transit the host star, but have orbital periods longer than
the ones explored in this study; (4) they do exist and transit, but the
photometric precision of the data is not high enough to detect them. Scenarios
(1) and (2) might be further explored by employing RV follow-up data as
discussed in Sect. 2.2. Scenario (3) can be tested when TESS will reobserve
TOI-4438 in sectors 79 and 80 (June-July 2024), extending the available time
baseline111111See the TESS-point Web Tool available at
https://heasarc.gsfc.nasa.gov/wsgi-scripts/TESS/TESS-point_Web_Tool/TESS-
point_Web_Tool/wtv_v2.0.py/.. To evaluate scenario (4), we studied the
detection limits of the current TESS photometry conducting injection-and-
recovery experiments with the MATRIX code121212The MATRIX (Multi-phAse
Transits Recovery from Injected eXoplanets) code is open access on GitHub at
https://github.com/PlanetHunters/tkmatrix. (Dévora-Pajares & Pozuelos 2023).
We used the MATRIX code to inject synthetic planets over the PDCSAP light
curves corresponding to the three TESS sectors used in this study, combining a
unique set of radius, orbital period, and phase for each synthetic planet. We
explored the $R_{\mathrm{planet}}$–$P_{\mathrm{planet}}$ parameter space in
the ranges of 0.5–3.0 R⊕ with steps of 0.25 R⊕, and 0.5–15.0 d with steps of
0.25 d. Moreover, for each combination of
$R_{\mathrm{planet}}$–$P_{\mathrm{planet}}$ MATRIX produces five different
phases, that is, different values of $T_{0}$. In total, we explored 3000
scenarios. For simplicity, the injected planets had impact parameters and
eccentricities equal to zero.

Once the synthetic planets were injected in MATRIX , we detrended the light
curves using a bi-weight filter with a window size of 0.5 d, which was found
to be the optimal value to recover the known planet TOI-4438 b. Moreover,
existing transits in the data were masked out to evaluate the ease of finding
TOI-4438 b-like planets. In MATRIX , a synthetic planet is recovered when its
epoch matches the injected epoch with 1 hour accuracy, and its period is
within 5 % of the injected period. Since we injected the synthetic signals in
the PDCSAP light curve, these signals were not affected by the PDCSAP
systematic corrections; hence, the detection limits that we found represent
the most optimistic scenario (see, e.g., Pozuelos et al. 2020).

The resulting detectability map from this injection-and-retrieval experiment
is shown in Fig. 8. On the one hand, we found that transiting Earth- and sub-
Earth size planets would remain unnoticed for the complete set of periods
explored with recovery rates ranging from 50 to 0$\%$. On the other hand,
mini-Neptunes such as TOI-4438 b, seem to be easily detectable, with recovery
rates higher than 80$\%$ for all the periods, that is, the presence of
additional transiting planets in the system with sizes larger than 1.5 R⊕ and
with orbital periods between 0.5 and 15.0 d seems unlikely.

Figure 8: The resulting detectability map from the injection-and-retrieval
experiment. Each pixel corresponds to ten of the 3000 different scenarios,
that is, ten light curves with injected planets having different
$P_{\mathrm{planet}}$, $R_{\mathrm{planet}}$, and $T_{0}$. Larger recovery
rates are presented in yellow and green colors, while lower recovery rates are
shown in blue and darker hues. The red dot refers to TOI-4438 b, which has a
recovery rate of 100$\%$, highlighting the ease of detecting mini-Neptunes in
this TESS data set with orbital periods shorter than 15 d.

###  4.3 Frequency analysis of the RV data

We performed a frequency analysis to search for periodic signals due to
orbiting planets or stellar activity. We computed the GLS periodograms of the
CARMENES RVs and activity indicators, as shown in Fig. 9. We estimated the
false alarm probability (FAP) using the bootstrap method as described by
Murdoch et al. (1993). We calculated the periodogram of 106 time series
obtained by randomly shuffling the data and their error bars, keeping the time
values fixed. We determined that a peak is significant when it has a FAP $<$
0.1% (Cochran & Hatzes 1996).

The GLS periodogram of the CARMENES RVs in the VIS band (Fig. 9, upper panel)
shows the highest peak at $f_{1}$ = 0.13 d-1, corresponding to a period of
about 7.44 d. This signal is not significant (FAP $<$ 10%) within the
frequency range 0.0–0.3 d-1. However, the peak was found at a known frequency,
namely the orbital frequency of the planet, providing strong evidence that the
signal is related to TOI-4438 b. We estimated the FAP of the peak at orbital
frequency of TOI-4438 b using the ”windowing bootstrap” method described by
Hatzes (2019) and following the procedure done in Osborne et al. (2024). We
found a FAP = 0.002% for $f_{1}$, which confirms the planetary nature of the
signal discovered by TESS.

We also used a pre-whitening technique (Hatzes et al. 2010) to find additional
signals. We subtracted the 7.44 d signal from the VIS RVs. The periodogram of
the RV residuals following the subtraction of the signal at $f_{1}$ (second
panel) shows no other significant signal.

Figure 9: GLS periodograms of the CARMENES VIS RV measurements (upper panel),
and RV residuals after subtracting $f_{1}$ (second panel). The periodograms of
the activity indicators are shown in the remaining panels. The 10%, 1%, and
0.1% FAPs are shown with horizontal blue lines. The red vertical line marks
the orbital frequency of the transiting planet TOI-4438 b. The yellow vertical
lines mark the stellar rotation period at $\sim$ 68 d, and its first and
second harmonics.

Finally, we performed a frequency analysis of the activity indicators obtained
using the serval pipeline. The GLS periodograms show peaks at periods $>$ 30
d, with a FAP $>$ 10% (lower panels of Fig. 9). Although not significant,
peaks at $\sim$68 d and $\sim$34 d are present in the periodograms of the CRX,
the Ca ii IRT2, H$\alpha$, and Na i D indices. The peaks appear at the periods
observed in the ground-based photometry analysis of Sect. 3.3, corroborating
the stellar rotational period, and its first and second harmonic at 68 d, 34
d, and 22 d, respectively.

###  4.4 Joint fit of the transit light curves and RV measurements

We performed a joint modeling combining TESS, MuSCAT2 $i^{\prime}$ and $z_{s}$
transit light curves, and CARMENES VIS RV measurements, following the results
presented in Sect. 4.1 and 4.3. We used the code pyaneti to perform an MCMC
analysis. We followed the procedures described previously in Sect. 4.1,
adopting uniform priors for the orbital period and reference transit epoch of
TOI-4438 b. We used the same parameterizations and prior distributions as in
Sect. 4.1. We performed two analyses: (i) we assumed a circular orbit fixing
$\sqrt{e}\sin{\omega_{\star}}$ and $\sqrt{e}\cos{\omega_{\star}}$ to zero;
(ii) we fitted for an eccentric orbit. In a preliminary analysis we used a
uniform prior for the stellar density $\rho_{\star}$. The modeling of the
transit light curves gave a mean stellar density consistent with the density
of Table. 1, which corroborates the planetary nature of the transit signals.
We then sampled for the stellar density $\rho_{\star}$ using a Gaussian prior
on the stellar mass and radius, derived in Sect. 3.1, and used wide uniform
priors for the remaining model parameters. We fitted for photometric and RV
jitter terms to account for possible signals not captured by our models, or
instrumental noise not included in the nominal error bars.

We explored the parameter space with 500 Markov chains initialized randomly
inside the prior ranges. Once all chains converged, we used the last 5000
iterations of the converged chains with a thin factor of 10 to create the
posterior distributions, leading to a distribution of 250 000 data points for
each sampled parameter. This procedure ensured a homogeneous sampling of the
parameter space. We followed the same procedure and convergence test as
described by Barragán et al. (2017).

Results and prior ranges of the fitted and derived parameters are reported in
Table 3. Since the planet is not close enough to the star to expect tidal
circularization and there are no other reasons to assume so (Sect. 4.5), we
chose the eccentric model as the best-fit, although the resulting eccentricity
is consistent with zero.

Figs. 10 and 11 display the CARMENES VIS time series and the RV curve phase-
folded at the orbital period of TOI-4438 b, respectively, along with the best-
fitting Doppler model. We found an RV semi-amplitude variation of
$K_{\mathrm{b}}$ = 3.50 $\pm$ 0.72 m s-1, and a planetary mass of
$M_{\mathrm{b}}$ = 5.4 $\pm$ 1.1 M⊕ (20% precision). We measured a radius of
$R_{\mathrm{b}}$ = 2.52 $\pm$ 0.13 R⊕ (5% precision). Combining the planetary
mass and radius, we calculated a mean density of $\rho_{\mathrm{b}}$ =
1.85${}^{+0.51}_{-0.44}$ g cm-3 ($\sim$28% precision).

Although the results obtained with pyaneti are robust, we performed a second
joint analysis of the light curves and RVs with juliet (Espinoza et al. 2019)
as a sanity check. The derived planetary parameters are consistent within
errors with those presented in Table 4.4. The juliet planetary radius and mass
are $R_{\mathrm{b}}=2.54\,\pm$ 0.12 R⊕ ($\sim$5 %), and
${M_{\mathrm{b}}}=5.7\,\pm$ 1.1 M⊕ ($\sim$20 %), respectively.

Table 3: TOI-4438 b parameters from the joint RVs and transit modeling with
pyaneti.(a)

Parameter Prior Derived value ($e=0$) Derived value ($e\neq 0$) Model
parameters $P_{\mathrm{orb,b}}$ [d] $\mathcal{U}$[7.2,7.7] 7.44628 $\pm$
0.000009 7.44628 $\pm$ 0.000009 $T_{\mathrm{0,b}}$
[BJD${}_{\mathrm{TDB}}-$2,450,000] $\mathcal{U}$[9396.16, 9396.66] 9396.41178
$\pm$ 0.00066 9396.41168 $\pm$ 0.00065 $R_{\mathrm{b}}/R_{\star}$
$\mathcal{U}$[0.001,0.09] 0.0613 $\pm$ 0.0007 0.062 $\pm$ 0.001
$b_{\mathrm{b}}$ $\mathcal{U}$[0,1] 0.18${}^{+0.15}_{-0.13}$
0.39${}^{+0.16}_{-0.24}$
$\sqrt{e_{\mathrm{b}}}\sin{\omega_{\mathrm{\star,b}}}$ $\mathcal{U}$[–1,1] 0.0
–0.31${}^{+0.20}_{-0.14}$
$\sqrt{e_{\mathrm{b}}}\cos{\omega_{\mathrm{\star,b}}}$ $\mathcal{U}$[–1,1] 0.0
0.11${}^{+0.14}_{-0.19}$ $K_{\mathrm{b}}$ [m s-1] $\mathcal{U}$[0,50] 3.23
$\pm$ 0.69 3.50 $\pm$ 0.72 Derived parameters $M_{\mathrm{b}}$ [$M_{\oplus}$]
… 5.1 $\pm$ 1.1 5.4 $\pm$ 1.1 $R_{\mathrm{b}}$ [$R_{\oplus}$] … 2.48 $\pm$
0.12 2.52$\pm$ 0.13 $\rho_{\mathrm{b}}$ [g cm-3] … 1.80${}^{+0.53}_{-0.44}$
1.85${}^{+0.51}_{-0.44}$ $a_{\mathrm{b}}$ [au] … 0.0509 $\pm$ 0.0028 0.0534
$\pm$ 0.0037 $e_{\mathrm{b}}$ … 0.0 0.14 $\pm$ 0.09
$\omega_{\star,\mathrm{b}}$ [deg] … 0.0 –70${}^{+36}_{-32}$ $i_{\mathrm{b}}$
[deg] … 89.64${}^{+0.25}_{-0.31}$ 89.34${}^{+0.39}_{-0.21}$
$\tau_{14,\mathrm{b}}$ [h] … 2.002 $\pm$ 0.018 2.021${}^{+0.033}_{-0.025}$
$T_{\mathrm{eq,b}}$ [K] ($a$=0) … 446 $\pm$ 13 435 $\pm$ 15
$T_{\mathrm{eq,b}}$ [K] ($a$=0.6) … 354 $\pm$ 7 347 $\pm$ 5 $S_{\mathrm{b}}$
[S⊕] … 6.57${}^{+0.81}_{-0.68}$ 6.00${}^{+0.90}_{-0.77}$ Additional model
parameters $\rho_{\star}$ [g cm-3] $\mathcal{N}$[10.06,1.62]
8.92${}^{+0.49}_{-0.99}$ 10.1${}^{+1.7}_{-1.4}$ $\gamma_{\mathrm{CARMENES}}$
[m s-1] $\mathcal{U}$[–2,2] –0.2 $\pm$ 0.5 –0.3 $\pm$ 0.5 RV jitter term
$\sigma_{\mathrm{RV,CARMENES}}$ [m s-1] $\mathcal{J}$[0,100]
1.48${}^{+0.72}_{-0.79}$ 1.33${}^{+0.71}_{-0.80}$ Phot. jitter term
$\sigma_{\mathrm{TESS}}$ $\mathcal{J}$[0,100] 0.000060 $\pm$ 0.000055 0.000060
$\pm$ 0.000055 Phot. jitter term $\sigma_{\mathrm{MuSCAT2\,i^{\prime}}}$
$\mathcal{J}$[0,100] 0.00014 $\pm$ 0.00011 0.00013 $\pm$ 0.00011 Phot. jitter
term $\sigma_{\mathrm{MuSCAT2\,z_{s}}}$ $\mathcal{J}$[0,100] 0.00018 $\pm$
0.00016 0.00017 $\pm$ 0.00016

Figure 10: CARMENES RV time series of TOI-4438 along with the best-fitting
model. Data are shown as orange filled circles with their nominal
uncertainties and semitransparent error bar extensions accounting for the
jitter term.  Figure 11: Same as Fig. 10 but phase-folded at the orbital
period of TOI-4438 b.

###  4.5 Tidal damping of the eccentricity and its consequences

We estimated the current eccentricity decay timescale of TOI-4438b adopting
the model by Leconte et al. (2010). We assumed that its current value is
$e=0.14$, although its actually measured value is compatible with zero at the
2$\sigma$ level. To apply the tidal model, we expressed the product of the
constant tidal time lag $\Delta t_{\rm p}$, adopted by Leconte et al. (2010),
by the Love number of the planet $k_{2,\rm p}$:

$k_{2,\rm p}\Delta t_{\rm p}=\frac{3}{2Q^{\prime}_{\rm p}n},$ (1)

where $Q^{\prime}_{\rm p}$ is the modified tidal quality factor of the planet
itself, $n=2\pi/P$ is the orbit mean motion, and $P$ the orbital period.

The rheology of mini-Neptune planets is unknown, thus we investigated two
extreme cases. The former assumed $Q^{\prime}_{\rm p}=10^{5}$, which is
comparable to the modified tidal quality factor of Uranus or Neptune as
estimated by for example Tittemore & Wisdom (1990) and Ogilvie (2014), and is
appropriate for a body where the dissipation of the tidal energy occurs in a
fluid interior; the latter assumed that tides are mainly dissipated inside a
rocky core encompassing the whole mass of the planet $5.4$ M⊕, but having a
radius of 1.3 R⊙. This hypotetical core radius corresponds to the peak of the
smallest radius component of the distribution of the transiting planets as
observed by Kepler (Fulton et al. 2017). According to the model by Owen & Wu
(2017) such a radius corresponds to the mean size of the bare core that
remains after the gaseous envelope of a mini-Neptune has been removed by
photoevaporation after the first few 100 Myr after its host star settled on
the main sequence. In the latter case, we adopted $Q^{\prime}_{\rm p}=300$ as
in the case of our Earth (Henning et al. 2009) and neglected the small mass
and tidal dissipation in the fluid outer envelope of the planet in comparison
with that in its rocky core. The present eccentricity tidal decay timescale
was found to be $\tau_{\rm e}\equiv|e/(de/dt)|\sim 24$ Gyr in the former case,
while $\tau_{\rm e}\sim 2$ Gyr in the latter case. We noted that tidal
dissipation inside the star has a negligible effect on the evolution of the
eccentricity because of the relatively large separation of the planet and its
small mass. Our results on the eccentricity decay timescale are critically
dependent on the $Q^{\prime}_{\rm p}$ of the planet, and therefore, it is
difficult to reach a conclusion on the possibility that the current
eccentricity is primordial or is still excited, for example, by the
gravitational perturbation of a distant companion or by other effects (see,
e.g., Correia et al. 2020).

An interesting consequence of the eccentric orbit of TOI-4438b is its expected
pseudosynchronization with the orbital motion, that is, its rotation is
predicted to be faster than the orbital mean motion $n$ because tides tend to
synchronize its rotation with the orbital velocity at periastron where they
are stronger. Using the formalism of Leconte et al. (2010), we predicted a
rotation period of the planet of 6.7 d, that is, 1.12 times shorter than its
orbital period. Such a state of pseudosynchronization is reached after a
timescale of $\sim 0.26$ Myr for $Q^{\prime}_{\rm p}=10^{5}$ or 5.6 kyr for
$Q^{\prime}_{\rm p}=300$, that is, on timescales much shorter than the system
lifetime. We expect tides to dissipate energy inside the planet due to its
eccentric orbit and pseudosynchronous rotation. The maximum dissipated power
is predicted for $Q^{\prime}_{\rm p}=300$ and is $\sim 6.3\times 10^{16}$ W
giving a heat flux of $\sim 20$ W m-2 at the top of its atmosphere with a
radius of $2.5$ R⊕. Such a flux is significantly larger than in the case of
Jupiter, where the heat flux from the interior of the planet is $\sim 5.4$ W
m-2 (Guillot et al. 2004). Therefore, it can play a relevant role in the
atmospheric dynamics of the planet. On the other hand, adopting
$Q^{\prime}_{\rm p}=10^{5}$, we found a dissipated power of $5.2\times
10^{15}$ W and a surface flux of only $\sim 1.6$ W m-2.

##  5 Discussion

TOI-4438 b has a mass of $M_{b}$ = 5.4 $\pm$ 1.1 M⊕, and a radius of $R_{b}$ =
2.52 $\pm$ 0.13 R⊕, with precision of 20% and 5%, respectively. Its bulk
density of $\rho_{b}$ = 1.85${}^{+0.51}_{-0.44}$ g cm-3, is close to Neptune’s
of 1.638 g cm-3 and 32% that of Earth’s density. We calculated an equilibrium
temperature of 435 K, assuming zero albedo. TOI-4438 b belongs to the small
group ($\lesssim$ 50) of planets around M-dwarf stars whose masses and radii
are known with a precision better than 25%. Fig. 12 shows the mass-radius
diagram with different theoretical composition models (Zeng et al. 2016).
TOI-4438 b lies in the mini-Neptunes domain.

The position of TOI-4438 b in the mass-radius diagram is consistent with a
high volatile content; it joins a group of low-density planets abundantly
predicted in planet population syntheses based on the core accretion paradigm
(e.g., Venturini et al. 2020; Burn et al. 2021; Schlecker et al. 2021a, b),
where they correspond to planets with high ($\sim$50 %) water ice mass
fractions that accreted the bulk of their mass outside the water ice line
(e.g., Mordasini 2018; Bitsch et al. 2021). However, the planet’s bulk density
is also compatible with an extended atmosphere.

With an instellation of $S_{\mathrm{b}}$ =
6.00${}^{+0.90}_{-0.77}\,S_{\oplus}$, TOI-4438 b is orbiting significantly
closer to its host star than the predicted inner edge of the habitable zone
(Kopparapu et al. 2014). As such, it may either have undergone an
instellation-induced runaway greenhouse transition (e.g., Kasting 1988;
Nakajima et al. 1992) and lost all its water to space, or currently be in a
post-runaway state and host an extended steam atmosphere. The latter scenario
is consistent with the planet’s low bulk density (Turbet et al. 2020; Dorn &
Lichtenberg 2021) and gains credibility through the expected long durations of
runaway greenhouse phases in planets orbiting M dwarfs (Luger & Barnes 2015).
With a derived instellation of a few times that of Earth, TOI-4438 b is well
within the runaway greenhouse regime but close to the habitable zone inner
edge. As such, and through the precise density estimate presented here,
TOI-4438 b contributes to the currently small sample of planets suited to
probe the habitable zone inner edge discontinuity, a predicted demographic
imprint of the habitable zone inner edge in the radius–density distribution of
small exoplanets (Turbet et al. 2019; Schlecker et al. 2023).

Figure 12: Mass-radius diagram for well-characterized planets (R$<4$ R⊕,
M$<13$ M⊕) around M-dwarf stars (T${}_{\mathrm{eff}}<$4000 K), as retrieved
from the Transiting Extrasolar Planet Catalogue (Southworth 2011). The masses
and radii are known with a precision better than 30% and 10%, respectively.
The theoretical composition models from Zeng et al. (2019) are displayed from
bottom to top, with solid and dashed curves. The location of TOI-4438 b is
marked with a red dot.

###  5.1 Interior modeling

Given the planet mass and radius obtained in our transit and radial velocity
analyses, we derived the interior composition of TOI-4438 b. The forward model
favoured an interior structure model for water-dominated sub-Neptunes (Acuña
et al. 2021; Aguichine et al. 2021). The interior structure of the planet was
assumed to consist of three layers: a Fe-rich core, a silicate mantle, and a
water layer. The phases of water present in the top layer were determined
self-consistently with the irradiation received by the planet from its host
star and an atmospheric model. To compute the temperature in the deep water
layer, we used the $k$-correlated atmospheric model initially introduced in
Marcq et al. (2017) and Pluriel et al. (2019), with up-to-date opacity and
equation of state data from Acuña et al. (2023). The atmospheric model
computed the temperature profile and the radius contribution of the upper
water envelope (from 300 bar to 20 mbar). The interior and the atmospheric
models were coupled self-consistently assuming radiative-convective
equilibrium in the atmosphere, and with an iterative algorithm (Acuña et al.
2021). A water-dominated upper envelope was inferred with 99% H2O and 1% CO2.

TOI-4438 b’s instellation is high enough to have molecules such as water in
gas and supercritical phases. Fig. 13 shows the posterior distribution
functions of the free parameters in our interior composition retrieval. Due to
the low density of the planet and its distance from the star, the refractory
elements probably consist of silicates instead of a Si and Fe mixture
(Aguichine et al. 2020). Therefore, in our analysis we set the Fe mass
fraction to zero. We obtained a water mass fraction of $0.62^{+0.34}_{-0.16}$.
With a composition between 46% and 96% H2O (1$\sigma$), TOI-4438 b is
volatile-rich. Moreover, it is very likely to have H/He atmosphere mixed with
molecules, such as H2O, CO2, and CH4. In the Solar System, comets and other
small bodies with up to 80% of pure water have been found (McKay et al. 2019).
However, low-mass planets with more than 50% water in mass are not a natural
outcome of planetesimal accretion and other planet formation mechanisms
(Kimura & Ikoma 2020; Miguel et al. 2020). Thus, atmospheric characterization
is required to break the degeneracy between the mass and the composition of
the envelope, and to determine whether TOI-4438 b has a massive, high
metallicity envelope (i.e H2O envelope scenario), or a less massive, low
metallicity atmosphere (i.e H/He-dominated envelope scenario).

Figure 13: Posterior probability distributions of the mass, radius and water
mass fraction (WMF) derived from in the interior structure retrieval. The
diagonal panels show the marginalized probability distributions of the
individual parameters. For the observable parameters (mass and radius), the
mean observed value is indicated with a black line. The mean and the 1$\sigma$
interval of the posterior distributions retrieved in the MCMC are shown as
orange dashed lines for all parameters. The PDFs agree well with the mean and
uncertainties of the observed mass and radius.

###  5.2 Prospects for atmospheric characterization

Figure 14: TSM for planets with radii smaller than 4 R⊕ and measured masses,
whose host stars have J-band magnitudes of 8.5 mag $<\,J\,<$ 11.5 mag.
TOI-4438 b is displayed with a star symbol. The color indicates the radius of
the planets.

Figure 15: Synthetic atmospheric transmission spectra of TOI-4438 b. Left:
fiducial models for clear or hazy H/He atmospheres with scaled solar
abundances. Right: model for a steam H2O atmosphere. Simulated measurements
with error bars are shown for the observation of one (left) or two (right)
transits with JWST NIRISS-SOSS, NIRSpec-G395H, and MIRI-LRS configurations.

We used the metric proposed by Kempton et al. (2018) to identify the
transiting planets most amenable for atmospheric characterization via
transmission spectroscopy with the JWST. For TOI-4438 b the derived
transmission spectroscopy metric (TSM) is 136 $\pm$ 13, which is well above
the threshold value of 92 defined for the respective planetary categories
(small mini-Neptunes; 1.5 $R_{\oplus}$ $<$ Rp $<$ 2.75 $R_{\oplus}$) for
follow-up studies (Kempton et al. 2018). In this regard, TOI-4438 b is placed
in the second quartile (rank 25–50%) of JWST targets with the strongest
predicted atmospheric detections, and it is among the first five planets with
the highest TSM, whose host stars have $J$-band magnitudes of 8.5 mag
$<\,J\,<$ 11.5 mag. Therefore, TOI-4438 b is one of the best small mini-
Neptune targets for atmospheric characterization. Fig. 14 shows the TSM value
of TOI-4438 b along with the TSMs of known transiting planets smaller than 4
R⊕ with measured masses, as retrieved from the Transiting Extrasolar Planet
Catalogue (Southworth 2011). We also calculated the emission spectroscopy
metric (ESM) following Kempton et al. (2018) finding an ESM of $\sim$4, which
instead is below the suggested threshold value (i.e., ESM = 7.5). The planet
TOI-4438 b is not a high priority target for emission spectroscopy of
terrestrial planets with JWST.

We explored the potential of TOI-4438 b for transmission spectroscopy with the
JWST through spectral simulations for a set of model atmospheres consistent
with the planetary mass, radius, and equilibrium temperature. We adopted
TauREx 3 (Al-Refaie et al. 2021) to generate the synthetic transmission
spectra. TOI-4438 b could have a rocky core surrounded by an H/He envelope or
be a waterworld (Sect 5.1). We modeled H/He atmospheres with 1$\times$ and
100$\times$ scaled solar abundances using the atmospheric chemical equilibrium
module (Agúndez et al. 2012), including collisionally induced absorption by
H2–H2 and H2–He (Abel et al. 2011, 2012; Fletcher et al. 2018), and Rayleigh
scattering. For each chemical setup, we considered the cases of clear and hazy
atmospheres. The haze was modeled by Mie scattering with the formalism of Lee
et al. (2013), assuming a particle size of $\alpha$ = 0.05 $\mu$m, a mixing
ratio of $\chi_{c}$ = 10-12, and an extinction coefficient of $Q_{0}$ = 40.
While this assumption has some limitations in reproducing the actual behavior
of scattered light due to using spherical particles instead of non-spherical
ones (see, e.g., Mishchenko et al. 1996), this strategy was motivated by its
low computational cost. The same haze parameters were chosen in previous
atmospheric simulation studies (e.g. Orell-Miquel et al. 2023). We note that a
super-solar metallicity is expected from planetary formation theories (e.g.,
Fortney et al. 2013; Thorngren et al. 2016). The equilibrium temperature
within $\sim$400–600 K also points to a high degree of haziness due to
inefficient haze removal (Gao & Zhang 2020; Ohno & Tanaka 2021; Yu et al.
2021). Additionally, we modeled the case of a steam H2O atmosphere.

We used ExoTETHyS (Morello et al. 2021) to simulate the corresponding JWST
spectra, as observed with the NIRISS-SOSS (0.6–2.8 $\mu$m), NIRSpec-G395H
(2.88–5.20 $\mu$m), and MIRI-LRS (5–12 $\mu$m) instrumental modes. The
spectroscopic error bars obtained with ExoTETHyS are consistent with those
calculated using the Exoplanet Characterization Toolkit (ExoCTK, Bourque et
al. 2021) and PandExo (Batalha et al. 2017), as proven in previous studies
(Murgas et al. 2021; Espinoza et al. 2022; Luque et al. 2022a, b; Chaturvedi
et al. 2022; Lillo-Box et al. 2023; Orell-Miquel et al. 2023; Palle et al.
2023). We conservatively increased the error estimates by 20%. We chose the
sizes of the wavelength bins following the recommendations from recent JWST
data synthesis papers that adopted a spectral resolution of $R\sim$100 for
NIRISS and NIRSpec (Carter et al., under review), and a constant bin size of
0.25 $\mu$m for MIRI-LRS observations (Powell et al. 2024).

Fig. 15 shows the synthetic transmission spectra for the atmospheric
configurations above. The H/He model atmospheres exhibit strong H2O and CH4
absorption features of $\gtrsim$100–1000 parts per million (ppm), depending on
metallicity and haze, while the steam H2O atmosphere has absorption features
$\lesssim$100 ppm. The predicted error bars for a single transit observation
are 31–123 ppm (mean error 53 ppm) for NIRISS-SOSS, 38–82 ppm (mean error 56
ppm) for NIRSpec-G395H, and 47–89 ppm (mean error 64 ppm) for MIRI-LRS. Our
simulations suggest that a single transit observation with NIRISS-SOSS or
NIRSpec-G395H is well suited to detect an H/He atmosphere, while at least two
transit observations may be needed to reveal a secondary H2O– dominated
atmosphere.

###  5.3 Expected radio emission from star-planet interaction

Figure 16: Flux density arising from star planet interaction as a function of
stellar mass-loss rate (left and center) and the magnetic field of the planet
(right). The emission expected from Saur–Turnpenney’s model is shown in
orange, while the emission expected from Zarka–Lanza’s model is shown in blue.
The overlap between both models is shown in brown. We show in pale red color
the region when the planet is in the super-Alfvènic regime. The dashed black
line represents an assumed detection threshold of $100\,\rm\mu Jy$. For the
variation with respect to the magnetic field of the planet we used
$\dot{M}=1.92\dot{M}_{\odot}$, which is the maximum value for the mass loss
rate before entering the sub-Alfvènic regime. Left: Expected flux density for
an unmagnetized planet in a closed dipolar geometry, as a function of the
stellar wind mass loss. The pale red colored region indicates the super-
Alfvénic case, when our approach no longer applies. Center: The same as in the
left panel, but for a magnetized planet with $B_{\rm p}$ = 2 G. Right:
Expected flux density for a closed dipolar geometry

Here we assess the feasibility of detecting radio emission arising from
magnetic star-planet interaction (SPI) between TOI-4438b and its host star.
The physical phenomenon responsible for this kind of emission is the electron
cyclotron maser (ECM) instability (Melrose & Dulk 1982), which can generate
auroral radio emission on the star and also the planet itself.

The characteristic frequency associated with ECM is given by the electron
gyrofrequency, $\nu_{G}=2.8\,B$ MHz, where $B$ is the local magnetic field in
Gauss. This non-thermal emission is coherent and circularly polarized
(reaching up to 100% in some instances), and may reach a large bandwidth
($\Delta\,\nu\sim\nu_{G}/2$).

We note, however, that any kind of SPI emission coming from the planet itself
would be undetectable with current instrumentation. The reason is that the
magnetic field of a planet is merely a few Gauss, so the resulting emission
would be blocked by the Earth’s ionosphere. We therefore studied the auroral
emission on the star TOI-4438 induced by the presence of its planet b, since
the global magnetic field of the star is of the order of a few hundreds of
Gauss or larger. As a result, the corresponding electron gyrofrequency would
be of several hundreds or thousands of MHz, a frequency range that can be
studied with current observatories. This phenomenon takes place in the sub-
Alfvénic regime, that is, when the planet is close enough to the star that the
speed of the plasma wind is lower than the Alfvén speed, enabling energy and
momentum to travel back from the planet onto the star in the form of Alfvèn
wings, triggering the radio emission close to the stellar surface.

To estimate the magnetic field of TOI-4438, we used the relation from Reiners
et al. (2022), who estimated the Rossby number as a function of the stellar
mass with the relations from Wright et al. (2018), which yields a value of
$B_{\star}\approx 191.9$ G. We then used the code described in Pérez-Torres et
al. (2021) to estimate the flux density arising from the sub-Alfvénic
interaction between an exoplanet and its host star, for an isothermal wind.
For our calculations, we assumed an isothermal wind with coronal temperature
equal to that of the solar corona, $T=2\,\times\,10^{6}$ K, and which is also
adequate for M dwarf stars. We took the solid angle covered by the ECM
emission to be 1.6 sterradians, in conformity with observations of the
Jupiter-Io decametric radio emission (Ray & Hess 2008). We worked with two
different emission models, Saur–Turnpenney (Saur et al. 2013; Turnpenney et
al. 2018) and Zarka–Lanza (Zarka 2007; Lanza 2009), and in both cases we
considered an efficiency in the conversion from Poynting flux to radio
emission in the range from 0.2 to 1%.

In the left panels of Fig. 16, we show the radio flux density from sub-
Alfvénic star-planet interaction as a function of the mass-loss rate of the
star for an unmagnetized planet and a planet with a magnetic field of 2 G. In
this regime, our approach does not apply and, therefore, no radio emission
from SPI can reach us. In the right panel of Fig. 16 we show the emission as a
function of the magnetic field of the planet. Despite the planet being rather
large in size, the expected radio emission is too low to be detectable in
essentially all scenarios due to the large distance to the system (30 pc).
Only in the case when the planet is highly magnetized and the stellar wind is
relatively powerful there could be marginal chances of detecting a signal from
SPI.

##  6 Conclusions

We have reported the confirmation and characterization of TOI-4438 b, a mini-
Neptune around an M3.5 V star (G 182-34) on a 7.44 d orbit. We performed a
joint modeling of the TESS and MuSCAT2 light curves with CARMENES high-
resolution spectroscopy measurements. We found a radius of 2.52 $\pm$ 0.13 R⊕
and a mass of 5.4 $\pm$ 1.1 M⊕, resulting in a bulk density of 1.85
${}^{+0.51}_{-0.44}$ g cm-3. The equilibrium temperature of the planet is 435
$\pm$ 15 K.

Our interior structure retrieval with a pure water envelope yields a minimum
water mass fraction of 46% (1$\sigma$). The volatile-rich mini-Neptune
TOI-4438 b has likely H/He mixed with molecules, such as water, CO2, and CH4.
TOI-4438 b presents a high transmission spectroscopy metric of 136 $\pm$ 13,
which places the planet among the most suitable targets for atmospheric
observations with JWST. We performed spectral simulations to explore the
potential for transmission spectroscopy with JWST. A single transit
observation of TOI-4438 with NIRISS-SOSS or NIRSpec-G395H should be adequate
to detect an H/He atmosphere, while at least two transits may be needed to
reveal a secondary H2O dominated atmosphere.

###### Acknowledgements.

We acknowledge the use of public TESS data from pipelines at the TESS Science
Office and at the TESS Science Processing Operations Center. TESS data
presented in this paper were obtained from the Milkulski Archive for Space
Telescopes (MAST) at the Space Telescope Science Institute. 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. CARMENES is an
instrument at the Centro Astronómico Hispano en Andalucía (CAHA) at Calar Alto
(Almería, Spain), operated jointly by the Junta de Andalucía and the Instituto
de Astrofísica de Andalucía (CSIC). CARMENES was funded by the Max-Planck-
Gesellschaft (MPG), the Consejo Superior de Investigaciones Científicas
(CSIC), the Ministerio de Economía y Competitividad (MINECO) and the European
Regional Development Fund (ERDF) through projects FICTS-2011-02,
ICTS-2017-07-CAHA-4, and CAHA16-CE-3978, and the members of the CARMENES
Consortium (Max-Planck-Institut für Astronomie, Instituto de Astrofísica de
Andalucía, Landessternwarte Königstuhl, Institut de Ciències de l’Espai,
Institut für Astrophysik Göttingen, Universidad Complutense de Madrid,
Thüringer Landessternwarte Tautenburg, Instituto de Astrofísica de Canarias,
Hamburger Sternwarte, Centro de Astrobiología and Centro Astronómico Hispano-
Alemán), with additional contributions by the MINECO, the Deutsche
Forschungsgemeinschaft (DFG) through the Major Research Instrumentation
Programme and Research Unit FOR2544 ”Blue Planets around Red Stars”, the Klaus
Tschira Stiftung, the states of Baden-Württemberg and Niedersachsen, and by
the Junta de Andalucía. The results reported herein benefitted from
collaborations and/or information exchange within NASA’s Nexus for Exoplanet
System Science (NExSS) research coordination network sponsored by NASA’s
Science Mission Directorate under Agreement No. 80NSSC21K0593 for the program
“Alien Earths”. The Joan Oró Telescope (TJO) of the Montsec Observatory (OdM)
is owned by the Catalan Government and operated by the Institute for Space
Studies of Catalonia (IEEC). G.M. has received fundings from the Ariel
Postdoctoral Fellowship program of the Swedish National Space Agency (SNSA).
S.V.J. acknowledges the support of the DFG priority program SPP 1992
“Exploring the Diversity of Extrasolar Planets (JE 701/5-1). M.P.T. and L.P.M.
acknowledge financial support through grants CEX2021-001131-S and
PID2020-117404GB-C21, funded by MCIU/AEI/ 10.13039/501100011033. L.P.M. also
acknowledges funding through the grant PRE2020-095421, funded by
MCIU/AEI/10.13039/501100011033 and by FSE Investing in your future. This work
is partly supported by JSPS KAKENHI Grant Number JP18H05439, JST CREST Grant
Number JPMJCR1761. This article is based on observations made with the MuSCAT2
instrument, developed by ABC, at Telescopio Carlos Sánchez operated on the
island of Tenerife by the IAC in the Spanish Observatorio del Teide. We
acknowledge financial support from the Agencia Estatal de Investigación
(AEI/10.13039/501100011033) of the Ministerio de Ciencia e Innovación and the
ERDF “A way of making Europe” through projects PID2022-137241NB-C4[1:4],
PID2021-125627OB-C31, PID2019–109522GB–C5[1:4], and the Centre of Excellence
“Severo Ochoa” and “María de Maeztu” awards to the Instituto de Astrofísica de
Canarias (CEX2019-000920-S), Instituto de Astrofísica de Andalucía
(CEX2021-001131-S) and Institut de Ciències de l’Espai (CEX2020-001058-M).
This work was also funded by the Generalitat de Catalunya/CERCA programme, the
DFG through grant HA3279/14-1 and the priority program SPP 1992 “Exploring the
Diversity of Extrasolar Planets” (JE 701/5-1), the JSPS KAK-ENHI through grant
JP18H05439, the JST CREST through grant JPMJCR1761, and the Israel Science
Foundation through grant 1404/22.

## References

  * Abel et al. (2011) Abel, M., Frommhold, L., Li, X., & Hunt, K. L. C. 2011, Journal of Physical Chemistry A, 115, 6805
  * Abel et al. (2012) Abel, M., Frommhold, L., Li, X., & Hunt, K. L. C. 2012, J. Chem. Phys., 136, 044319
  * Acuña et al. (2023) Acuña, L., Deleuil, M., & Mousis, O. 2023, A&A, 677, A14
  * Acuña et al. (2021) Acuña, L., Deleuil, M., Mousis, O., et al. 2021, A&A, 647, A53
  * Aguichine et al. (2021) Aguichine, A., Mousis, O., Deleuil, M., & Marcq, E. 2021, ApJ, 914, 84
  * Aguichine et al. (2020) Aguichine, A., Mousis, O., Devouard, B., & Ronnet, T. 2020, ApJ, 901, 97
  * Agúndez et al. (2012) Agúndez, M., Venot, O., Iro, N., et al. 2012, A&A, 548, A73
  * Al-Refaie et al. (2021) Al-Refaie, A. F., Changeat, Q., Waldmann, I. P., & Tinetti, G. 2021, ApJ, 917, 37
  * Aller et al. (2020) Aller, A., Lillo-Box, J., Jones, D., Miranda, L. F., & Barceló Forteza, S. 2020, A&A, 635, A128
  * Anderson et al. (2010) Anderson, D. R., Cameron, A. C., Hellier, C., et al. 2010, ApJL, 726, L19
  * Baglin et al. (2006) Baglin, A., Auvergne, M., Boisnard, L., et al. 2006, in 36th COSPAR Scientific Assembly, Vol. 36, 3749
  * Ballerini et al. (2012) Ballerini, P., Micela, G., Lanza, A. F., & Pagano, I. 2012, A&A, 539, A140
  * Barkaoui et al. (2023) Barkaoui, K., Timmermans, M., Soubkiou, A., et al. 2023, A&A, 677, A38
  * Barragán et al. (2022) Barragán, O., Aigrain, S., Rajpaul, V. M., & Zicher, N. 2022, MNRAS, 509, 866
  * Barragán et al. (2017) Barragán, O., Gandolfi, D., & Antoniciello, G. 2017, pyaneti: Multi-planet radial velocity and transit fitting, Astrophysics Source Code Library, record ascl:1707.003
  * Batalha et al. (2017) Batalha, N. E., Mandell, A., Pontoppidan, K., et al. 2017, PASP, 129, 064501
  * Batalha et al. (2013) Batalha, N. M., Rowe, J. F., Bryson, S. T., et al. 2013, ApJS, 204, 24
  * Bayo et al. (2008) Bayo, A., Rodrigo, C., Barrado Y Navascués, D., et al. 2008, A&A, 492, 277
  * Bitsch et al. (2021) Bitsch, B., Raymond, S. N., Buchhave, L. A., et al. 2021, 1
  * Borucki et al. (2010) Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977
  * Bouma et al. (2023) Bouma, L. G., Palumbo, E. K., & Hillenbrand, L. A. 2023, ApJ, 947, L3
  * Bourque et al. (2021) Bourque, M., Espinoza, N., Filippazzo, J., et al. 2021, The Exoplanet Characterization Toolkit (ExoCTK), Zenodo
  * Brown et al. (2013) Brown, T. M., Baliber, N., Bianco, F. B., et al. 2013, PASP, 125, 1031
  * Burn et al. (2021) Burn, R., Schlecker, M., Mordasini, C., et al. 2021, Astronomy & Astrophysics, 656, A72
  * Caballero et al. (2016) Caballero, J. A., Guàrdia, J., López del Fresno, M., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9910, Observatory Operations: Strategies, Processes, and Systems VI, ed. A. B. Peck, R. L. Seaman, & C. R. Benn, 99100E
  * Chaturvedi et al. (2022) Chaturvedi, P., Bluhm, P., Nagel, E., et al. 2022, A&A, 666, A155
  * Chen et al. (2021) Chen, G., Pallé, E., Parviainen, H., et al. 2021, MNRAS, 500, 5420
  * Cifuentes et al. (2020) Cifuentes, C., Caballero, J. A., Cortés-Contreras, M., et al. 2020, A&A, 642, A115
  * Claret et al. (2012) Claret, A., Hauschildt, P. H., & Witte, S. 2012, A&A, 546, A14
  * Claret et al. (2013) Claret, A., Hauschildt, P. H., & Witte, S. 2013, A&A, 552, A16
  * Cochran & Hatzes (1996) Cochran, W. D. & Hatzes, A. P. 1996, Ap&SS, 241, 43
  * Collins et al. (2017) Collins, K. A., Kielkopf, J. F., Stassun, K. G., & Hessman, F. V. 2017, AJ, 153, 77
  * Colomé et al. (2010) Colomé, J., Casteels, K., Ribas, I., & Francisco, X. 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7740, Software and Cyberinfrastructure for Astronomy, ed. N. M. Radziwill & A. Bridger, 77403K
  * Colome & Ribas (2006) Colome, J. & Ribas, I. 2006, IAU Special Session, 6, 11
  * Correia et al. (2020) Correia, A. C. M., Bourrier, V., & Delisle, J. B. 2020, A&A, 635, A37
  * Cracchiolo et al. (2021) Cracchiolo, G., Micela, G., Morello, G., & Peres, G. 2021, MNRAS, 507, 6118
  * Curtis et al. (2020) Curtis, J. L., Agüeros, M. A., Matt, S. P., et al. 2020, ApJ, 904, 140
  * Dahn et al. (1988) Dahn, C. C., Harrington, R. S., Kallarakal, V. V., et al. 1988, AJ, 95, 237
  * Dahn et al. (2002) Dahn, C. C., Harris, H. C., Vrba, F. J., et al. 2002, AJ, 124, 1170
  * Delrez et al. (2022) Delrez, L., Murray, C. A., Pozuelos, F. J., et al. 2022, A&A, 667, A59
  * Dévora-Pajares & Pozuelos (2023) Dévora-Pajares, M. & Pozuelos, F. J. 2023, MATRIX: Multi-phAse Transits Recovery from Injected eXoplanets toolkit, Astrophysics Source Code Library, record ascl:2309.007
  * Dorn & Lichtenberg (2021) Dorn, C. & Lichtenberg, T. 2021, The Astrophysical Journal Letters, 922, L4
  * Dressing & Charbonneau (2013) Dressing, C. D. & Charbonneau, D. 2013, ApJ, 767, 95
  * Engle & Guinan (2023) Engle, S. G. & Guinan, E. F. 2023, ApJ, 954, L50
  * Espinoza et al. (2019) Espinoza, N., Kossakowski, D., & Brahm, R. 2019, MNRAS, 490, 2262
  * Espinoza et al. (2022) Espinoza, N., Pallé, E., Kemmer, J., et al. 2022, AJ, 163, 133
  * Fletcher et al. (2018) Fletcher, L. N., Gustafsson, M., & Orton, G. S. 2018, ApJS, 235, 24
  * Foreman-Mackey (2016) Foreman-Mackey, D. 2016, The Journal of Open Source Software, 1, 24
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306
  * Fortney et al. (2013) Fortney, J. J., Mordasini, C., Nettelmann, N., et al. 2013, ApJ, 775, 80
  * Frith et al. (2013) Frith, J., Pinfield, D. J., Jones, H. R. A., et al. 2013, MNRAS, 435, 2161
  * Fulton et al. (2017) Fulton, B. J., Petigura, E. A., Howard, A. W., et al. 2017, AJ, 154, 109
  * Gaia Collaboration et al. (2021) Gaia Collaboration, Smart, R. L., Sarro, L. M., et al. 2021, A&A, 649, A6
  * Gaia Collaboration et al. (2023) Gaia Collaboration, Vallenari, A., Brown, A. G. A., et al. 2023, A&A, 674, A1
  * Gao & Zhang (2020) Gao, P. & Zhang, X. 2020, ApJ, 890, 93
  * Giclas (1966) Giclas, H. 1966, Vistas in Astronomy, 8, 23
  * Giclas et al. (1971) Giclas, H. L., Burnham, R., & Thomas, N. G. 1971, Lowell proper motion survey Northern Hemisphere. The G numbered stars. 8991 stars fainter than magnitude 8 with motions ¿ 0”.26/year
  * González-Álvarez et al. (2023) González-Álvarez, E., Zapatero Osorio, M. R., Caballero, J. A., et al. 2023, A&A, 675, A177
  * Guillot et al. (2004) Guillot, T., Stevenson, D. J., Hubbard, W. B., & Saumon, D. 2004, in Jupiter. The Planet, Satellites and Magnetosphere, ed. F. Bagenal, T. E. Dowling, & W. B. McKinnon, Vol. 1, 35–57
  * Hartman & Bakos (2016) Hartman, J. D. & Bakos, G. Á. 2016, Astronomy and Computing, 17, 1
  * Hatzes (2019) Hatzes, A. P. 2019, The Doppler Method for the Detection of Exoplanets
  * Hatzes et al. (2010) Hatzes, A. P., Dvorak, R., Wuchterl, G., et al. 2010, Astronomy and Astrophysics, 520, A93
  * Henning et al. (2009) Henning, W. G., O’Connell, R. J., & Sasselov, D. D. 2009, ApJ, 707, 1000
  * Howell et al. (2014) Howell, S. B., Sobeck, C., Haas, M., et al. 2014, PASP, 126, 398
  * Husser et al. (2013) Husser, T. O., Wende-von Berg, S., Dreizler, S., et al. 2013, A&A, 553, A6
  * Jeffers et al. (2020) Jeffers, S. V., Dreizler, S., Barnes, J. R., et al. 2020, Science, 368, 1477
  * Jeffers et al. (2018) Jeffers, S. V., Schöfer, P., Lamert, A., et al. 2018, A&A, 614, A76
  * Jenkins et al. (2016) Jenkins, J. M., Twicken, J. D., McCauliff, S., et al. 2016, in Proc. SPIE, Vol. 9913, Software and Cyberinfrastructure for Astronomy IV, 99133E
  * Kasting (1988) Kasting, J. F. 1988, Icarus, 74, 472
  * Kawauchi et al. (2022) Kawauchi, K., Murgas, F., Palle, E., et al. 2022, A&A, 666, A4
  * Kempton et al. (2018) Kempton, E. M. R., Bean, J. L., Louie, D. R., et al. 2018, PASP, 130, 114401
  * Kimura & Ikoma (2020) Kimura, T. & Ikoma, M. 2020, MNRAS, 496, 3755
  * Kipping (2013) Kipping, D. M. 2013, MNRAS, 435, 2152
  * Kochanek et al. (2017) Kochanek, C. S., Shappee, B. J., Stanek, K. Z., et al. 2017, PASP, 129, 104502
  * Kopparapu et al. (2014) Kopparapu, R. K., Ramirez, R. M., SchottelKotte, J., et al. 2014, The Astrophysical Journal Letters, 787, L29
  * Kossakowski et al. (2023) Kossakowski, D., Kürster, M., Trifonov, T., et al. 2023, A&A, 670, A84
  * Kovács et al. (2002) Kovács, G., Zucker, S., & Mazeh, T. 2002, A&A, 391, 369
  * Lamman et al. (2020) Lamman, C., Baranec, C., Berta-Thompson, Z. K., et al. 2020, AJ, 159, 139
  * Lanza (2009) Lanza, A. F. 2009, A&A, 505, 339
  * Leconte et al. (2010) Leconte, J., Chabrier, G., Baraffe, I., & Levrard, B. 2010, A&A, 516, A64
  * Lee et al. (2013) Lee, J.-M., Heng, K., & Irwin, P. G. J. 2013, ApJ, 778, 97
  * Lépine & Gaidos (2011) Lépine, S. & Gaidos, E. 2011, AJ, 142, 138
  * Lépine & Shara (2005) Lépine, S. & Shara, M. M. 2005, AJ, 129, 1483
  * Lillo-Box et al. (2023) Lillo-Box, J., Gandolfi, D., Armstrong, D. J., et al. 2023, A&A, 669, A109
  * Lodieu et al. (2019) Lodieu, N., Pérez-Garrido, A., Smart, R. L., & Silvotti, R. 2019, A&A, 628, A66
  * Luger & Barnes (2015) Luger, R. & Barnes, R. 2015, Astrobiology, 15, 119
  * Luque et al. (2022a) Luque, R., Fulton, B. J., Kunimoto, M., et al. 2022a, A&A, 664, A199
  * Luque et al. (2022b) Luque, R., Nowak, G., Hirano, T., et al. 2022b, A&A, 666, A154
  * Luyten (1979) Luyten, W. J. 1979, LHS catalogue. A catalogue of stars with proper motions exceeding 0”5 annually
  * Mallorquín et al. (2023) Mallorquín, M., Goffo, E., Pallé, E., et al. 2023, A&A, 680, A76
  * Mandel & Agol (2002) Mandel, K. & Agol, E. 2002, ApJL, 580, L171–L175
  * Marcq et al. (2017) Marcq, E., Salvador, A., Massol, H., & Davaille, A. 2017, Journal of Geophysical Research (Planets), 122, 1539
  * Marfil et al. (2021) Marfil, E., Tabernero, H. M., Montes, D., et al. 2021, A&A, 656, A162
  * Maxted et al. (2011) Maxted, P. F. L., Anderson, D. R., Collier Cameron, A., et al. 2011, PASP, 123, 547
  * McCully et al. (2018) McCully, C., Volgenau, N. H., Harbeck, D.-R., et al. 2018, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10707, Proc. SPIE, 107070K
  * McKay et al. (2019) McKay, A. J., DiSanti, M. A., Kelley, M. S. P., et al. 2019, AJ, 158, 128
  * Melrose & Dulk (1982) Melrose, D. B. & Dulk, G. A. 1982, ApJ, 259, 844
  * Miguel et al. (2020) Miguel, Y., Cridland, A., Ormel, C. W., Fortney, J. J., & Ida, S. 2020, MNRAS, 491, 1998
  * Mishchenko et al. (1996) Mishchenko, M. I., Travis, L. D., & Mackowski, D. W. 1996, J. Quant. Spec. Radiat. Transf., 55, 535
  * Mordasini (2018) Mordasini, C. 2018, in Handbook of Exoplanets (Cham: Springer International Publishing), 2425–2474
  * Morello et al. (2023) Morello, G., Parviainen, H., Murgas, F., et al. 2023, A&A, 673, A32
  * Morello et al. (2017) Morello, G., Tsiaras, A., Howarth, I. D., & Homeier, D. 2017, AJ, 154, 111
  * Morello et al. (2021) Morello, G., Zingales, T., Martin-Lagarde, M., Gastaud, R., & Lagage, P.-O. 2021, AJ, 161, 174
  * Murdoch et al. (1993) Murdoch, K. A., Hearnshaw, J. B., & Clark, M. 1993, The Astrophysical Journal, 413, 349
  * Murgas et al. (2021) Murgas, F., Astudillo-Defru, N., Bonfils, X., et al. 2021, A&A, 653, A60
  * Murgas et al. (2023) Murgas, F., Castro-González, A., Pallé, E., et al. 2023, A&A, 677, A182
  * Nagel et al. (2023) Nagel, E., Czesla, S., Kaminski, A., et al. 2023, A&A, 680, A73
  * Nakajima et al. (1992) Nakajima, S., Hayashi, Y.-Y., & Abe, Y. 1992, Journal of the Atmospheric Sciences, 49, 2256
  * Narita et al. (2019) Narita, N., Fukui, A., Kusakabe, N., et al. 2019, Journal of Astronomical Telescopes, Instruments, and Systems, 5, 015001
  * Ogilvie (2014) Ogilvie, G. I. 2014, ARA&A, 52, 171
  * Ohno & Tanaka (2021) Ohno, K. & Tanaka, Y. A. 2021, ApJ, 920, 124
  * Orell-Miquel et al. (2023) Orell-Miquel, J., Nowak, G., Murgas, F., et al. 2023, A&A, 669, A40
  * Osborne et al. (2024) Osborne, H. L. M., Van Eylen, V., Goffo, E., et al. 2024, MNRAS, 527, 11138
  * Owen & Wu (2017) Owen, J. E. & Wu, Y. 2017, ApJ, 847, 29
  * Palle et al. (2023) Palle, E., Orell-Miquel, J., Brady, M., et al. 2023, A&A, 678, A80
  * Parviainen (2015) Parviainen, H. 2015, Monthly Notices of the Royal Astronomical Society, 450, 3233
  * Parviainen & Aigrain (2015) Parviainen, H. & Aigrain, S. 2015, MNRAS, 453, 3821
  * Parviainen et al. (2019) Parviainen, H., Tingley, B., Deeg, H. J., et al. 2019, A&A, 630, A89
  * Pecaut & Mamajek (2013) Pecaut, M. J. & Mamajek, E. E. 2013, ApJS, 208, 9
  * Pérez-Torres et al. (2021) Pérez-Torres, M., Gómez, J. F., Ortiz, J. L., et al. 2021, A&A, 645, A77
  * Pluriel et al. (2019) Pluriel, W., Marcq, E., & Turbet, M. 2019, Icarus, 317, 583
  * Pollacco et al. (2006) Pollacco, D. L., Skillen, I., Collier Cameron, A., et al. 2006, PASP, 118, 1407
  * Powell et al. (2024) Powell, D., Feinstein, A. D., Lee, E. K. H., et al. 2024, Nature
  * Pozuelos et al. (2020) Pozuelos, F. J., Suárez, J. C., de Elía, G. C., et al. 2020, A&A, 641, A23
  * Pozuelos et al. (2023) Pozuelos, F. J., Timmermans, M., Rackham, B. V., et al. 2023, A&A, 672, A70
  * Quirrenbach et al. (2014) Quirrenbach, A., Amado, P. J., Caballero, J. A., et al. 2014, in Proc. SPIE, Vol. 9147, Ground-based and Airborne Instrumentation for Astronomy V, 91471F
  * Quirrenbach et al. (2018) Quirrenbach, A., Amado, P. J., Ribas, I., et al. 2018, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10702, Ground-based and Airborne Instrumentation for Astronomy VII, 107020W
  * Ray & Hess (2008) Ray, L. C. & Hess, S. 2008, Journal of Geophysical Research (Space Physics), 113, A11218
  * Reid & Cruz (2002) Reid, I. N. & Cruz, K. L. 2002, AJ, 123, 2806
  * Reid et al. (2003) Reid, I. N., Cruz, K. L., Allen, P., et al. 2003, AJ, 126, 3007
  * Reiners et al. (2022) Reiners, A., Shulyak, D., Käpylä, P. J., et al. 2022, A&A, 662, A41
  * Ribas et al. (2023) Ribas, I., Reiners, A., Zechmeister, M., et al. 2023, A&A, 670, A139
  * Ricker et al. (2015) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003
  * Sabotta et al. (2021) Sabotta, S., Schlecker, M., Chaturvedi, P., et al. 2021, A&A, 653, A114
  * Saur et al. (2013) Saur, J., Grambusch, T., Duling, S., Neubauer, F. M., & Simon, S. 2013, A&A, 552, A119
  * Schanche et al. (2022) Schanche, N., Pozuelos, F. J., Günther, M. N., et al. 2022, A&A, 657, A45
  * Schlecker et al. (2023) Schlecker, M., Apai, D., Lichtenberg, T., et al. 2023, Bioverse: The Habitable Zone Inner Edge Discontinuity as an Imprint of Runaway Greenhouse Climates on Exoplanet Demographics
  * Schlecker et al. (2021a) Schlecker, M., Mordasini, C., Emsenhuber, A., et al. 2021a, Astronomy & Astrophysics, 656, A71
  * Schlecker et al. (2021b) Schlecker, M., Pham, D., Burn, R., et al. 2021b, Astronomy & Astrophysics, 656, A73
  * Schneider et al. (2016) Schneider, A. C., Greco, J., Cushing, M. C., et al. 2016, ApJ, 817, 112
  * Schöfer et al. (2019) Schöfer, P., Jeffers, S. V., Reiners, A., et al. 2019, A&A, 623, A44
  * Schweitzer et al. (2019) Schweitzer, A., Passegger, V. M., Cifuentes, C., et al. 2019, A&A, 625, A68
  * Shappee et al. (2014) Shappee, B. J., Prieto, J. L., Grupe, D., et al. 2014, ApJ, 788, 48
  * Skrutskie et al. (2006) Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131, 1163
  * Smith et al. (2012) Smith, J. C., Stumpe, M. C., Van Cleve, J. E., et al. 2012, PASP, 124, 1000
  * Southworth (2011) Southworth, J. 2011, MNRAS, 417, 2166
  * Stassun et al. (2019) Stassun, K. G., Oelkers, R. J., Paegert, M., et al. 2019, AJ, 158, 138
  * Stumpe et al. (2014) Stumpe, M. C., Smith, J. C., Catanzarite, J. H., et al. 2014, PASP, 126, 100
  * Stumpe et al. (2012) Stumpe, M. C., Smith, J. C., Van Cleve, J. E., et al. 2012, PASP, 124, 985
  * Tabernero et al. (2022) Tabernero, H. M., Marfil, E., Montes, D., & González Hernández, J. I. 2022, A&A, 657, A66
  * Thompson et al. (2023) Thompson, A., Biagini, A., Cracchiolo, G., et al. 2023, arXiv e-prints, arXiv:2302.04574
  * Thorngren et al. (2016) Thorngren, D. P., Fortney, J. J., Murray-Clay, R. A., & Lopez, E. D. 2016, ApJ, 831, 64
  * Tittemore & Wisdom (1990) Tittemore, W. C. & Wisdom, J. 1990, Icarus, 85, 394
  * Turbet et al. (2020) Turbet, M., Bolmont, E., Ehrenreich, D., et al. 2020, Astronomy & Astrophysics, Volume 638, id.A41, 638, A41
  * Turbet et al. (2019) Turbet, M., Ehrenreich, D., Lovis, C., Bolmont, E., & Fauchez, T. 2019, Astronomy & Astrophysics, 628, A12
  * Turnpenney et al. (2018) Turnpenney, S., Nichols, J. D., Wynn, G. A., & Burleigh, M. R. 2018, ApJ, 854, 72
  * van Altena et al. (1995) van Altena, W. F., Lee, J. T., & Hoffleit, E. D. 1995, The general catalogue of trigonometric [stellar] parallaxes
  * Venturini et al. (2020) Venturini, J., Guilera, O. M., Haldemann, J., Ronco, M. P., & Mordasini, C. 2020, Astronomy & Astrophysics, 643, L1
  * Wells et al. (2021) Wells, R. D., Rackham, B. V., Schanche, N., et al. 2021, A&A, 653, A97
  * Wright et al. (2018) Wright, N. J., Newton, E. R., Williams, P. K. G., Drake, J. J., & Yadav, R. K. 2018, Monthly Notices of the Royal Astronomical Society, 479, 2351
  * Yu et al. (2021) Yu, X., He, C., Zhang, X., et al. 2021, Nature Astronomy, 5, 822
  * Zarka (2007) Zarka, P. 2007, Planet. Space Sci., 55, 598
  * Zechmeister et al. (2014) Zechmeister, M., Anglada-Escudé, G., & Reiners, A. 2014, A&A, 561, A59
  * Zechmeister & Kürster (2009) Zechmeister, M. & Kürster, M. 2009, Astronomy & Astrophysics, 496, 577–584
  * Zechmeister et al. (2018) Zechmeister, M., Reiners, A., Amado, P. J., et al. 2018, A&A, 609, A12
  * Zeng et al. (2019) Zeng, L., Jacobsen, S. B., Sasselov, D. D., et al. 2019, Proceedings of the National Academy of Science, 116, 9723
  * Zeng et al. (2016) Zeng, L., Sasselov, D. D., & Jacobsen, S. B. 2016, ApJ, 819, 127

##  Appendix A Radial velocity data

Table 4: RV data from CARMENES VIS.

Time RV $\sigma$ [BJDTDB] [m s-1] [m s-1] 2459720.6490 1.90 3.99 2459726.5607
–6.93 2.02 2459730.5751 –1.11 2.20 2459736.5664 2.74 1.81 2459738.6403 –3.77
1.97 2459740.6490 –4.64 2.32 2459748.5283 –0.98 3.35 2459768.4734 –1.05 2.57
2459770.4685 –0.97 1.94 2459772.4574 –4.01 2.31 2459776.4443 0.86 2.09
2459778.4424 –3.92 2.31 2459780.6099 1.97 2.25 2459784.6203 0.36 3.21
2459786.5898 –2.26 3.63 2459788.6297 4.97 1.98 2459796.5221 4.49 2.64
2459812.4809 –2.08 2.23 2459812.5016 2.40 1.98 2459813.4895 3.57 3.72
2459813.5105 –2.29 3.12 2459814.4412 –6.17 2.09 2459826.4745 3.32 2.61
2459831.3528 –3.44 2.42 2460007.6296 –0.85 2.37 2460015.6770 –4.80 1.72
2460033.6556 3.51 3.38 2460037.5939 4.58 3.48 2460041.6186 2.86 1.92
2460057.6177 6.53 2.07 2460060.6283 1.38 1.98 2460066.6401 2.64 2.30

##  Appendix B Photometric fit with unocculted star spots

We fitted the color dependent transit depths that appear when considering the
measurements from TESS and all MuSCAT2 filters, including $g^{\prime}$ and
$r^{\prime}$, following the method described in Sect. 4.1. For simplicity, we
fixed the stellar parameters of the quiet photosphere to those of TOI-4438, as
reported in Table 1. The free parameters in our fit were the geometric radius
ratio $R_{b}/R_{\star}$, the star spot photospheric parameters
$T_{\mathrm{spot}}$ and $\log{g_{\mathrm{spot}}}$, and filling factor
$f_{\mathrm{spot}}$. We run emcee (Foreman-Mackey et al. 2013) with 16 walkers
and 30 000 iterations to sample the posterior distribution, then applying a
conservative burn-in of 10 000 iterations, which is more than 20 times the
maximum autocorrelation length. For each parameters set from the chains, we
also computed the apparent transit depth in each passband using the formulae
from Ballerini et al. (2012).

Fig. 17 compares the measured transit depths in each passband with those
derived from the fit with unocculted star spots and the posterior distribution
for the model parameters. While the unocculted star spots may explain the
larger transit depths in $g^{\prime}$ and $r^{\prime}$, they would also cause
a difference between the $i^{\prime}$ and $z_{s}$ that is not seen in the
data. As a compromise, the model tends to underestimate the $g^{\prime}$ and
$r^{\prime}$ transit depths to keep a small difference between $i^{\prime}$
and $z_{s}$. The modelled transit depths are consistent within 1$\sigma$ with
the measured ones, owing to the large error bars in $g^{\prime}$ and
$r^{\prime}$. The spot temperature contrast is -490${}_{-640}^{+380}$ K and
the filling factor is 0.44${}_{-0.17}^{+0.24}$. In this scenario, the planet-
to-star radius ratio is $R_{b}/R_{\star}=0.055_{-0.011}^{+0.004}$, that is
significantly smaller and with order-of-magnitude larger error bars than those
obtained discarding the $g^{\prime}$ and $r^{\prime}$ bands and neglecting
star spots.

If we accepted this solution, the planetary radius would be
$R_{b}=2.24_{-0.55}^{+0.28}\,R_{\oplus}$. This slightly smaller radius would
not significantly change the predicted composition, which is volatile-rich.
With a smaller TSM of 96, TOI-4438 b would remain a top target for atmospheric
characterization with JWST.

Figure 17: Results of the photometric fit with unocculted star spots. Left:
Individual transit depths derived from photometric observations (blue squares)
and best-fit model assuming that the color-dependence is due to unocculted
star spots. Right: corner plot with the posterior distribution of the model
parameters, generated with the corner package (Foreman-Mackey 2016).