$i$ (deg) | 89.21${}^{+0.31}_{-0.21}$ |  $\geqslant$ 88.4bb$b$16th percentile.
$T_{14}$ (hour) | 4.258${}^{+0.149}_{-0.103}$ | 3.127${}^{+0.108}_{-0.127}$
$e$ |  $\leqslant$ 0.13aa$a$84th percentile. |  $\leqslant$ 0.14aa$a$84th percentile.
$K$ (m s-1) | 15.7$\pm$1.5 | 4.1${}^{+1.7}_{-1.6}$
$M_{\rm p}$ ($M_{\oplus}$) | 58.7${}^{+5.7}_{-5.6}$ | 11.8${}^{+4.8}_{-4.7}$
$\rho_{\rm p}$ (g cm-3) | 0.29$\pm$0.05 | 1.50$\pm$0.68
$g$ (m s-2) | 5.36$\pm$0.86 | 8.99$\pm$3.76
$T_{\rm eq}$ (K) | 947$\pm$28 | 1242$\pm$37
141414

###  5.1 Peculiar architecture

TOI-5398 is a compact multi-planet system composed of a warm giant (TOI-5398
b, $P_{\rm orb}$ $\sim$ 10.59 d) and a hot sub-Neptune planet (TOI-5398 c,
$P_{\rm orb}$ $\sim$ 4.77 d) orbiting a moderately young solar-analogue star.
The peculiarity of this system comes from its compactness and planetary
architecture, which is uncommon among known multi-planet systems with short-
period giant planets. In fact, the sub-Neptune is the closest planet to the
host star. There are currently very few notable examples of compact systems
consisting of an inner orbit small-size planet and an outer short-period giant
companion (e.g., Hord et al. 2022). The most famous multi-planet system with
similar characteristics is WASP-47 (Hellier et al., 2012; Becker et al., 2015;
Nascimbeni et al., 2023). Within this family of peculiar planetary systems, we
also have Kepler-730 (Zhu et al., 2018; Cañas et al., 2019), TOI-1130 (Huang
et al., 2020; Korth et al., 2023), TOI-2000 (Sha et al., 2022), and WASP-132
(Hord et al., 2022). We list them in Table 7, together with a list of
available datasets and a few notable parameters that we want to highlight.

Among these compact systems, TOI-5398 stands out, along with WASP-47 and
TOI-2000, due to its precise transit photometry and radial velocity
measurements. These observations enable the precise measurement of planetary
bulk densities and make it an extremely appealing target for continued
monitoring with follow-up observations and surveys, including PLATO and Ariel,
along with telescopes like ELTs.

Moreover, TOI-5398 is the youngest compact system with a gas giant ever
confirmed with a quite good age estimation.

Table 7: Confirmed compact multi-planet systems, sorted by age.

System | n∘ known | Transiting | PRVaa$a$Precise Radial Velocity. | TTVbb$b$Transit Time Variations. | TSMcc$c$Transmission Spectroscopy Metric of the giant planet in the system. | Age | Reference
---|---|---|---|---|---|---|---
| planets | planets |  |  |  | (Gyr) |
TOI-5398 | 2 | True, 2/2 | True | Potential | 288 | 0.65$\pm$0.15 | This paper (Sect. 3.7)
WASP-132 | 2 | True, 2/2 | True, 1/2 | False | 106 | 3.2$\pm$0.5 |  Hord et al. (2022)
TOI-2000 | 2 | True, 2/2 | True | Potential | 68 | 5.3$\pm$2.7 |  Sha et al. (2022)
WASP-47 | 4 | True, 4/4 | True | True | 47 | 6.5${}^{+2.6}_{-1.2}$ |  Hellier et al. (2012)
TOI-1130 | 2 | True, 2/2 | True | True | 345 | 8.2${}^{+3.8}_{-4.9}$ |  Huang et al. (2020)
Kepler-730 | 2 | True, 2/2 | False | False | 25 | 9.5${}^{+2.5}_{-2.7}$ |  Zhu et al. (2018)

151515

###  5.2 Variety among compact multi-planet systems

In Fig. 15, we show the architecture of compact multi-planet systems with
small-size planets orbiting interior to short-period giant planets ($P$
$\lesssim$ 10 d). For comparison, we show the next three systems with giants
whose orbital periods are $P$ ¡ 25 d (Butler et al., 1997; Bourrier et al.,
2018; Weiss et al., 2013; Nesvorný et al., 2013). The semicircular dots
represent the host stars, colour-coded by their age, while the size represents
their radii. Planets, on the other hand, are colour-coded by their equilibrium
temperature $T_{\rm eq}$, and their size reflects their planetary masses. The
inner planet of WASP-132 and the Kepler-730 planets do not yet have mass
measurements, so we extracted them following Wolfgang et al. (2016) or we
showed upper limits (Hord et al., 2022).

Figure 15: Architecture of compact multi-planet systems hosting small-size
planets orbiting inner to short-period gas giants ($P$ $\lesssim$ 10 d). Each
row represents one planetary system ($y$-axis) and the planetary orbital
periods ($x$-axis). The sizes of the dots correspond to the planet masses, and
the colours of the points to the equilibrium temperatures (see colorbar to the
right). From top to bottom, the systems are sorted in ascending order of the
period of the giant. Shaded dots represent the next three systems with gas
giants on $P$ ¡ 25 d orbits. Dots with a vertical line represent planets whose
mass is multiplied by $\sin(i)$. The semi-circular dots filled with a star
shape stand for the host stars, colour-coded by their age (see colorbar to the
top), while the size represents their radii.

TOI-5398 hosts the youngest gas giant planet with $P$ ¡ 25 d and $M_{p}$ ¿ 1/2
$M_{\rm Saturn}$ that is known to have an inner companion. The gas giant
TOI-5398 b has a radius similar to Jupiter and a mass close to 2/3 of Saturn,
which is the smallest mass among giants in compact systems. As a result, its
bulk density is around half that of Saturn and roughly equal to that of
TOI-1130 c (0.38 g $cm^{-3}$, Huang et al. 2020). These properties, as well as
its orbital period, make TOI-5398 b quite similar to the hot-Saturn planet
TOI-2000 c, while they are in contrast with the properties of the larger
WASP-47 b, Kepler-730 b, and TOI-1130 c, which all have radii $\sim$ 1.1
$R_{J}$ and masses $\sim$ 1 $M_{J}$ (apart from Kepler-730 b that does not
have a mass measurement yet). However, the larger radius and smaller mass of
TOI-5398 b compared to those of TOI-2000 c – which results in having a
relatively low density compared to giant planets of similar mass (see Fig. 16
and cf., for example, Yee et al. 2022) – make the giant planet under study
more like a puffy Saturn (Naponiello et al., 2022). Moreover, we calculated
the equilibrium temperature $T_{\rm eq}$ (cf. Eq. 4 from Cowan & Agol 2011) of
TOI-5398 b, assuming zero albedo and full day-night heat redistribution
following

$T_{\rm eq}=T_{\rm
eff}\sqrt{\frac{R_{\star}}{a}}\left(\frac{1}{4}\right)^{1/4},$ (3)

where $a$ is the orbital semi-major axis given in the same units as
$R_{\star}$. We obtained a value of 947$\pm$28 K, which indicates that
TOI-5398 b is unlikely to be affected by the hot-Jupiter anomalous radius
inflation mechanism (Thorngren et al., 2016). By contrast, the $T_{\rm eq}$ of
the hot-Saturn TOI-2000 c is slightly above 1000 K (Sha et al., 2022);
therefore, considering the $T_{\rm eq}$ = 1000 K threshold and its orbital
period exceeding 10 days, we describe TOI-5398 b as a warm-Saturn planet.

In addition, TOI-5398 b is located in the Neptunian “savanna” (Bourrier et
al., 2023), a light deficit of planets close to the Hot-Neptune desert
(Lecavelier Des Etangs, 2007; Mazeh et al., 2016).

Figure 16: Mass-density distribution of all confirmed planets from the NASA
Exoplanet Archive with mass and radius determination better than 20%. The red
dots represent TOI-5398 b and c, while the remaining planets mentioned in
Table 7 are represented by dots of different colours. We also plot theoretical
mass-radius curves for planets of various pure compositions from Zeng et al.
(2019, online at https://lweb.cfa.harvard.edu/~lzeng/planetmodels.html): solid
red indicates a pure-Iron core, brown an Earth-like rocky core (32.5% Fe and
67.5% MgSiO3), light-blue a 100% Water world at 1000K, and blue a 100% cold-
Hydrogen world.

The sub-Neptune TOI-5398 c is one of the very few inner companions to short-
period gas giants with precise values of both mass and radius. In fact, only
four small planets with $R_{\rm p}<$ 4 $R_{\oplus}$ in compact systems (P
$\lesssim$ 15 d) have mass measurements: WASP-47 d, WASP-47 e (Vanderburg et
al., 2017), TOI-1130 c (Korth et al., 2023), and TOI-2000 b. It is worth
noting that they have quite different bulk densities (see Fig. 17), ranging
from being composed of rocky cores to having masses and radii similar to those
of Neptune and Uranus. The latter composition is true for TOI-5398 c, which
shares a bulk density similar to that of the inner companion planet WASP-47 d.
As a result, we provide additional evidence that inner companions to
transiting giant planets tend to have the same density diversity as other
small planets (Sha et al., 2022).

Figure 17: Mass-radius distribution of all confirmed planets with Rp $<$ 4 R⊕
from the NASA Exoplanet Archive with mass and radius determination better than
20%. The red dot represents TOI-5398 c, the beige one TOI-2000 b, the green
one TOI-1130 c, and the violet dots are respectively WASP-47 d and e. We
include the Solar System planets Earth, Venus, Uranus, and Neptune. We add
theoretical mass-radius curves from Zeng et al. (2019): solid brown indicates
an Earth-like rocky core (32.5% Fe and 67.5% MgSiO3), beige a 100% water world
at 1000K, and dotted beige a 1% hydrogen envelope and 99% Earth-like rocky
core at the same temperature. Grey dashed curves represent densities $\rho$ =
0.5, 1, and 5 g/cm3, respectively.

###  5.3 Ephemeris improvements

An important step of our analysis is the derivation of new and updated mean
ephemeris for TOI-5398 b and c. Our best-fit relation for the warm Saturn and
the sub-Neptune are:

$\begin{split}T_{0,\rm b}&=2459616.49232\pm 0.00022\,\mathrm{BJD_{TDB}}\\\
&\quad+N\times(10.590547\pm 0.000012),\end{split}$ (4) $\begin{split}T_{0,\rm
c}&=2459628.6178\pm 0.0009\,\mathrm{BJD_{TDB}}\\\ &\quad+N\times(4.77271\pm
0.00016),\end{split}$ (5)

where the variable $N$ is an integer number commonly referred to as the
“epoch” and arbitrarily set to zero at our reference transit time $T_{\rm
ref}$. We emphasise that if we propagate the new ephemeris at 2030-01-01 (see
Fig. 18, and 19), the level of uncertainty is significantly reduced to $\sim$
5 minutes compared to the previous $\sim$ 197 minutes for TOI-5398 b when only
TESS photometry was available. This means that when also the ground-based
photometry is taken into account, the error bar for TOI-5398 b is 98% smaller
compared to using TESS data alone. For TOI-5398 c, the error bar is 60%
smaller than when we rely solely on TESS data. Accurately identifying the
transit windows is crucial for upcoming space-based observations, given the
significant investment in observing time and the time-critical nature of such
observations. It is crucial to note that no further observations of TOI-5398
are planned in the current TESS Extended Mission161616As it results from the
Web TESS Viewing Tool https://heasarc.gsfc.nasa.gov/cgi-
bin/tess/webtess/wtv.py.

Figure 18: Ephemeris uncertainty of TOI-5398 b propagated until the start of
2030. On the $x$-axis, we show the date, while on the $y$-axis the error bars
in minutes. In grey, we show the uncertainty propagation considering TESS data
alone. In red, we show the improvement considering also Ground-based
photometry data.  Figure 19: As in Fig. 18, but for TOI-5398 c.

###  5.4 Planetary system formation and evolution

The obliquity between the planetary orbital plane and the stellar rotation
axis is a key diagnostic for the mechanisms of formation and orbital migration
of exoplanets (e.g., Naoz et al. 2011). This can be detected with in-transit
RVs through the Rossiter-McLaughlin effect (RM, Ohta et al. 2005; Rossiter
1924; McLaughlin 1924; Queloz et al. 2000). Short-period giant planets are
thought to form in situ close to the final orbit, or in the outer regions and
migrate inward (Dawson & Johnson, 2018). Different mechanisms, such as
dynamical interactions (high-eccentricity migration) through planet-planet
scattering (Marzari et al., 2006) or the Kozai mechanism (Wu & Murray, 2003),
and disc-planet interaction (Lin et al., 1996), can shrink their orbits. These
mechanisms are expected to imprint different signatures in the planets’
obliquity. Scattering encounters should randomise the alignments of the
orbital planes, while migration through disc-planet interactions should keep
the planetary orbits roughly co-planar throughout the entire process.

TOI-5398’s uncommon architecture and moderately young age make it particularly
promising for measuring the obliquity between the orbital plane of the giant
and the spin axis of the star. First, we can access the original configuration
when observing systems young enough to have avoided tidal alterations of the
obliquity. Then, unlike ordinary short-period giants, we can rule out the
high-eccentricity migration scenario (Mustill et al., 2015) for compact
systems such as TOI-5398 and test the other formation models through detailed
atmospheric characterisation.

Following Eq. 40 from Winn (2010), we determined the expected amplitude of the
radial velocity variation produced by the RM effect when planet b transits (58
m s-1) or when planet c transits (7.0 m s-1). Given the activity level of our
target (typical RV dispersion: 27 m s-1) and considering a planetary transit
time scale, that is, when the activity level is significantly lower than its
typical amplitude, we predict a perfectly suitable detection of the RM effect
caused by TOI-5398 b and a possible detection for TOI-5398 c.

####  5.4.1 Circularisation time scale and $\alpha$ parameter

We calculated the circularisation time scale, denoted as $\tau_{\rm circ}$,
for TOI-5398 b, using Eq. 6 from Matsumura et al. (2008). Assuming a circular
orbit and a modified tidal quality factor $Q$ of $10^{5}$ (Ogilvie, 2014), the
calculated value for $\tau_{\rm circ}$ is 2.57 $\pm$ 0.88 Gyr, which
significantly exceeds the age of the system. This suggests that the near-
circular orbit of TOI-5398 b may well be primordial, indicating favourable
conditions for preserving its close planetary companion.

Following the methodology described in Bonomo et al. (2017), we computed the
parameter $\alpha$, which represents the ratio of the planet’s semi-major axis
to its Roche limit. Bonomo et al. (2017) conclude that planets with $\alpha$ ¿
5 and circular orbits are unlikely to undergo high eccentricity migration.
With an $\alpha$ value of 5.6, TOI-5398 b falls in the middle $\alpha$ range
(4.55 - 7.44) for short-period giants with close companions. This finding
favours the notion that short-period giants with close companions should not
be distinguished between hot (P ¡ 10 days) and warm (P ¿ 10 days) planets,
that is, they belong to the same population of exoplanets, distinct in turn
from the typical giants that experience the high-eccentricity migration
scenario.

####  5.4.2 Atmospheric mass-loss

We investigated how the planetary masses, radii, and atmospheric mass-loss
rates change with time due to photoevaporation and internal heating.

We evaluated the mass-loss rate of the two planets’ atmospheres using the
hydro-based approximation developed by Kubyshkina et al. (2018a, b), coupled
with the planetary core-envelope model by Lopez & Fortney (2014) and the MESA
Stellar Tracks (MIST; Choi et al. 2016). For the stellar X-ray emission at
different ages, we adopted the analytic description by Penz et al. (2008),
with the current X-ray luminosity, $L_{\rm x}=10^{29}$ erg s-1 in the 5–100
$\AA$ band. This value was derived from the rotation-activity relationships by
Pizzolato et al. (2003), and closely resembles the median X-ray luminosity of
Hyades stars. The stellar EUV luminosity (100–920 Å) was computed at any given
time using the scaling law by Sanz-Forcada et al. (2022). The subsequent
paragraphs present the outcomes of our forward-in-time simulations, which
include the evolution of the XUV irradiation and the planetary structure in
response to stellar behaviour. More details on our modelling of atmospheric
evaporation are provided in Maggio et al. (2022) and Damasso et al. (2023).

We performed several simulations assuming different possible values for the
planetary core radius at the current age. The test cases were selected by
comparing them with the grids of planetary internal structures by Fortney et
al. (2007), assuming cores composed of 50% rocks and 50% ices. For planet b,
we explored core radii ranging from 2.4 to 7 $R_{\rm\oplus}$, corresponding to
core masses of 10 to 25 $M_{\oplus}$. As for planet c, due to its smaller
size, we limited our analysis to core radii between 1 and 2.5 $R_{\rm\oplus}$,
with core masses of $\sim 10M_{\oplus}$. In Table 8, we show the results of
the simulations.

Table 8: Atmospheric mass-loss simulations

Core Radius | Current age | Evaporation |  $t=5$ Gyr
---|---|---|---
($R_{\oplus}$) |  $f_{\rm atm}$(%) | Mass loss rate (g/s) | time scalea | Mass ($M_{\oplus}$) | Radius ($R_{\oplus}$) |  $f_{\rm atm}$(%) | Mass loss rate (g/s)
Planet b
2.4 | 98.9 | 5.3$\times 10^{11}$ |  $>5$ Gyr | 56.7 | 8.9 | 98.8 | 1.6 $\times 1.6^{10}$
5 | 50.4 | 5.3$\times 10^{11}$ |  $>5$ Gyr | 56.5 | 9.3 | 48.5 | 2.0 $\times 1.6^{10}$
7 | 22.7 | 5.3$\times 10^{11}$ |  $>5$ Gyr | 56.4 | 9.5 | 19.6 | 2.2 $\times 1.6^{10}$
Planet c
1 | 7.0 | 3.5$\times 10^{12}$ | 96 Myr | 9.7 | 1.4 | 0.4 | 5.2$\times 10^{8}$
1.8 | 3.5 | 3.5$\times 10^{12}$ | 27 Myr | 10.3 | 1.8 | 0.0 | 0.0
2.5 | 1.5 | 3.5$\times 10^{12}$ | 8 Myr | 10.2 | 2.5 | 0.0 | 0.0

171717$a$$a$footnotetext: The evaporation time scale is equal to the time
required for losing half of the atmospheric mass.

In our reference model for planet b, the core has a radius of $R_{\rm
core}=5R_{\rm\oplus}$ and a core mass of $M_{\rm core}\sim 29M_{\oplus}$,
resulting in an atmospheric mass fraction $f_{\rm env}$ of $\sim 50\%$. The
current photo-evaporation rate is $\sim 5.3\times 10^{11}$ g s-1, and the
planet will maintain a large envelope mass fraction throughout its main-
sequence lifetime, with $f_{\rm env}\sim 48\%M_{\rm p}$ at time $t\sim 5$ Gyr.
Its radius will only be reduced by $\sim 10$ %. In the range explored, these
results depend little on the assumed characteristics of the core.

Conversely, the evolution of the inner planet is very different due to the
smaller distance from the host star, higher equilibrium temperature, and
higher high-energy irradiation. Our reference model has a core radius $R_{\rm
core}=1.8R_{\rm\oplus}$ and a core mass $M_{\rm core}\sim 10M_{\oplus}$,
resulting in an atmospheric mass fraction $f_{\rm env}$ of $\sim 3.5\%M_{\rm
p}$. The current photo-evaporation rate is $\sim 3.5\times 10^{12}$ g s-1, and
the planet will lose its entire envelope in $\lesssim 200$ Myr from now. The
planetary size will decrease to match that of the core. However, a larger core
radius implies a smaller atmospheric mass fraction and shorter evaporation
time scales. For example, for a core radius $R_{\rm core}=1R_{\rm\oplus}$, the
planet would keep a residual atmospheric envelope even at $t=5$ Gyr.

####  5.4.3 TOI-5398’s global formation history

The different masses of the two planets could be either primordial or, as
suggested by their different evaporation rates discussed in Sect. 5.4.2, the
result of distinct photoevaporation histories. However, these two scenarios
have different implications regarding formation regions and bulk compositions.
Our preliminary exploration of the formation tracks of the two planets using
the methodology from Johansen et al. (2019) highlights the following
possibilities.

For the mass of TOI-5398 c to be primordial or close to the original one, the
planet should not have captured significant quantities of disk gas after
reaching its pebble isolation mass. This condition is satisfied if the planet
had started its formation beyond about ten au and comparatively late ($\sim$ 1
Myr) in the life of its native disk. However, in this scenario, TOI-5398 b
should still be able to capture its present gaseous envelope. For this to
occur, planet b should have started its formation at an earlier time ($\sim$
0.1 Myr) than planet c: as a result, planet b would already be close to its
current orbit while planet c is forming and migrating, and the two planets
would have to cross orbits to reach their current architecture. Such an
encounter would likely result in a planet-planet scattering event: this
scenario would result in higher eccentricities and inclinations than the
currently observed ones. Even in the case that the dynamical excitation
created by the planet-planet scattering event is removed by the interactions
of the two planets with the disk gas, the density of planet c appears too low
for a realistic mixture of rock and ice resulting from its growth track (as a
comparison, the density of the ice-rich dwarf planet Pluto is 2 g cm-3).

The other possible scenario results from the two planets having started
forming early in the lifetime of their native disk at a few au from their host
star. In this case, the simulated growth tracks favour planet c to have
possessed an extended primordial atmosphere of comparable mass to that of
planet b. In this scenario, the present-day mass disparity between the two
planets would result from their different photoevaporation histories, with
planet c experiencing a much higher mass loss than its outer counterpart.
Their gas accretion phases would have occurred close to their final orbits, in
the innermost and hottest regions of the native disk, which suggests that
their atmospheres could exhibit stellar composition.

###  5.5 Planetary bulk composition prediction

Accurate data on mass, eccentricity, and radius allow for measuring precise
inner bulk densities and exploring the differences in planetary structure and
evolution, from inflated Hot Jupiters (HJ) to “over-dense” Warm Jupiters
(WJs,Fortney et al. 2021). While the prediction of hotter interiors and larger
radii for HJs (Guillot et al., 1996), compared to Jupiter, has been proven
true, the mechanism(s) behind some HJs having anomalously large radii remain a
challenge to explain (Thorngren & Fortney, 2018). For non-inflated giant
planets ($F_{\star}$ ¡ 2 $\times$ 108 erg s-1 cm-2), Thorngren et al. (2016)
found a relation between planet mass and bulk metallicity, which confirms a
key prediction of the core-accretion planet formation model (Mordasini et al.,
2014). A recent study (Müller & Helled, 2023b) presents the current knowledge
of mass-metallicity trends for warm giant exoplanets (Teske et al., 2019;
Müller et al., 2020; Müller & Helled, 2023a), and raises some doubts about its
extent and existence. Müller & Helled (2023b) link this ambiguity to
theoretical uncertainties on the assumed models and the need for accurate
stellar age and atmospheric measurements.

Understanding this relationship and open questions regarding giants require
characterising planets and host stars, focusing on the metal enrichment of
planetary atmospheres (Miller & Fortney, 2011). It is particularly true for
warm giants – scarce among confirmed planets181818Only 20 warm giants have
precise bulk densities (density determination better than 20%) in the NASA
Exoplanet Archive – unaffected by the radius inflation mechanism, as we can
reasonably constrain their bulk metal enrichment and interpret atmospheric
features more safely (Thorngren et al., 2016).

TOI-5398 b is a perfect case study if we consider both its low insolation
$F_{\star}$ and its good age estimation. Therefore, we estimated its planetary
bulk heavy-element mass fraction (or bulk metallicity), using evolution
models191919https://github.com/tiny-hippo/planetsynth/blob/main from Müller &
Helled (2021). Figure 20 shows the radius evolution for various bulk
metallicities (coloured lines) for three different atmospheric heavy-element
mass fractions. The bulk heavy-element mass fraction of TOI-5398 b, depending
on the adopted atmospheric metallicity, varies between 20 and 30 per cent of
its total mass. Our result follows the mass-metallicity trend from Thorngren
et al. (2016), but the heavy-element mass appears to be slightly lower than
expected. Therefore, we may infer that the trend from Müller & Helled (2023a),
which predicts a lower heavy-element mass for a given planetary mass compared
to Thorngren et al. (2016), might offer a more plausible explanation of our
result.

Figure 20: Radius evolution in time ($x$-axis) for various bulk metallicities
(coloured lines, in units of planet b masses). The three plots adopt three
different atmospheric heavy-element mass fractions. The light-blue dot
represents TOI-5398 b.

###  5.6 Future atmospheric characterisation

To break the degeneracy in determining the planetary bulk composition, it is
crucial to perform atmospheric measurements and to get information on metal
enrichment. Müller & Helled (2023b) show that atmospheric measurements by JWST
and Ariel can significantly reduce this degeneracy and that this is
particularly promising for warm giant planets (Müller & Helled, 2023a). The
precise characterisation of the TOI-5398 b atmosphere will be crucial to
validate or disprove formation and evolution theories.

TOI-5398 is a fascinating compact system, as planet b has the highest
Transmission Spectroscopy Metric (TSM, Kempton et al. 2018) value ($\sim$ 300)
among warm giant planets (10 ¡ $P$ ¡ 100, $M_{\rm p}$ ¿ 0.1 $M_{J}$) currently
known, making it ideal for atmospheric characterisation by JWST. Indeed, the
TSM parameter quantifies the expected signal-to-noise in transmission
spectroscopy for a given planet, and according to Kempton et al. (2018), a
giant planet’s atmosphere is considered amenable to JWST observations when its
TSM value is greater than 90. In Fig. 21, we include all confirmed planets,
with colour-coding only for those with a TSM ¿ 90, and $M_{\rm p}$ ¿ 0.1
$M_{\rm J}$. In this plot, we show the planetary masses versus radii, where
planets are colour-coded by their orbital period. We colour-code planets with
periods longer than 10 days in orange to highlight the warm-giant planets,
defined as giant planets with periods exceeding 10 days (e.g., Yee et al.
2021; Gan et al. 2023 and references therein). Instead, the size of the dots
tracks the TSM. For comparison, WASP-47 b has only a modest TSM value of
$\sim$ 47 (Bryant & Bayliss, 2022).

Figure 21: Mass radius distribution of all confirmed planets that are present
in the TEPCat catalogue (Southworth, 2011). The orbital period of a planet is
colour-coded when it has TSM ¿ 90, $M_{\rm p}$ ¿ 0.1 $M_{\rm J}$, and $R_{\rm
p}$ ¿ 8 $R_{\oplus}$. Planets without these characteristics are coloured grey.
The dot size tracks the TSM. The planetary mass is shown on a logarithmic
scale.

###  5.7 Characterisation with JWST/NIRSpec

TOI-5398 b is an ideal candidate for precise transmission spectroscopy and
atmospheric characterisation, with a focus on the carbon-to-oxygen (C/O)
ratio. This ratio is essential for understanding planetary formation
mechanisms and grasping planetary atmosphere composition, shedding light on
volatile content and atmospheric chemistry. The availability of carbon and
oxygen determines different chemical reactions as well as the stability of
molecules of planetary atmospheres. Understanding the C/O ratio helps in
predicting the composition and behaviour of atmospheric constituents like
carbon dioxide (CO2), carbon monoxide (CO), methane (CH4), and water (H2O)
(Keyte et al., 2023).

To test the feasibility of atmospheric characterisation using JWST, we
investigated three different atmospheric scenarios for TOI-5398 b. We assumed
equilibrium chemistry as a function of temperature and pressure using FastChem
(Stock et al., 2018) and three different C/O ratios: 0.5, 1.0 and 1.5. We used
FastChem within TauREx3 (Al-Refaie et al., 2021) using the taurex-
fastchem202020https://pypi.org/project/taurex-fastchem plugin. TauREx is a
retrieval code that uses a Bayesian approach to infer atmospheric properties
from observed data, utilising a forward model to generate synthetic spectra by
solving the radiative transfer equation throughout the atmosphere. We used all
the possible gases contributions within FastChem and the active absorption
contribution given by K, Na, HCN, H2CO, CH4, CO, CO2, C2H2, C2H4, H2O, NH3,
SiO, TiO, VO and SO2 opacities.

After generating the transmission spectra using TauREX+FastChem, we simulated
a JWST observation using Pandexo (Batalha et al., 2017), a software tool
specifically developed for the JWST mission. The software allows users to
model and simulate various atmospheric scenarios, incorporating factors such
as atmospheric composition, temperature profiles, and molecular opacities. We
simulated a NIRSpec observation in bots mode, using the s1600a1 aperture with
g235h disperser, sub2048 subarray, nrsrapid read mode and f170lp filter. We
simulated one single transit and an observation 1.75 $T_{14}$ long to ensure a
robust baseline coverage. We fixed this instrumental configuration for all
three scenarios. In Fig 22, we show the resulting spectra for the different
C/O ratios and their best-fit models.

We performed three atmospheric retrievals on the NIRSpec/JWST simulations
using a Nested Sampling algorithm with the
nestle212121https://github.com/kbarbary/nestle library with 1000 live points.
We used the cuda transmission model with the taurex-
cuda222222https://github.com/ucl-exoplanets/taurex_cuda/ TauREx plugin. We
fitted three parameters: the radius of the planet $R_{p}$, the equilibrium
temperature of the atmosphere $T_{\rm eq}$, and the C/O ratios.

Using NIRspec with the g235h disperser wavelength range (1.66 $\mu$m - 3.07
$\mu$m), we can assess the C/O ratio under the three assumptions (see Tab 9).
In particular, when assuming C/O ratios of 0.5 and 1.0, we can retrieve the
correct value within 1$\sigma$ error bar, while under the C/O = 1.5
assumption, we retrieved a value of $1.87\pm 0.15$, within 2.5$\sigma$ error
bar.

The three atmospheres can be explained with three distinct sets of parameters
(Fig 23). The results of atmospheric retrievals confirm and quantify the
feasibility of atmospheric characterisation using NIRSpec@JWST. Furthermore,
they demonstrate that TOI-5398 b is an excellent candidate for comprehensive
atmospheric analysis, to measure the C/O ratio and, therefore, to constrain
planet formation theories for this system.

Table 9: Retrieval results for the three different scenarios. Parameter |  C/O = 0.5 |  C/O = 1.0 |  C/O = 1.5
---|---|---|---
$R_{p}$ $(R_{J})$ |  $0.9214_{-0.0005}^{+0.0006}$ |  $0.9211\pm 0.0004$ |  $0.9226\pm 0.0005$
$T_{\rm eq}$ (K) |  $929\pm 13$ |  $932_{-17}^{+19}$ |  $887\pm 9$
C/O |  $0.47_{-0.07}^{+0.06}$ |  $1.005_{-0.008}^{+0.009}$ |  $1.87\pm 0.15$
$\mu$ (derived) |  $2.3298\pm 0.0008$ |  $2.3361\pm 0.0001$ |  $2.3467\pm 0.0019$
Figure 22: NIRSpec observation simulation using the g235h disperser with
f170lp filter (scatter points) and best-fit models from TauREx (lines). The
three colours indicate three scenarios: C/O = 0.5 in red, C/O = 1.0 in yellow,
and C/O = 1.5 in blue. Figure 23: Posterior distributions for the three
different scenarios. We show C/O = 0.5 in red, C/O = 1.0 in yellow, and C/O =
1.5 in blue.

##  6 Conclusions

In this study, we presented the discovery of the youngest transiting planetary
system containing a sub-Neptune planet orbiting interior to a Saturn-mass
planet with $P<$ 15 days. Using HARPS-N radial velocity measurements of the
host star TOI-5398 and multidimensional Gaussian processes, we modelled the
stellar activity and confirmed the planetary nature of both candidates
identified in the TESS light curve measuring their masses. Furthermore, our
methodology allowed us to accurately determine the stellar parameters.

With a Transmission Spectroscopy Metric value of around 300, the warm Saturn
TOI-5398 b is the most suitable warm giant planet for atmospheric
characterisation using JWST. Such investigations are crucial to validate or
disprove existing formation and evolution theories. By measuring atmospheric
chemistry, we can gain information on metal enrichment and effectively break
the degeneracy in determining the planetary bulk composition.

The presence of two planets in this relatively young system offers the
opportunity to examine distinct evolutionary paths over the initial hundreds
of millions of years following system formation, under the influence of the
same host star. In this study, we provided a characterisation of the system,
with a special focus on the future evolution of the planetary atmospheres.
Future works will focus on investigating the past evolution of the system. We
explored the evolution of the atmospheres of both planets, considering the
decay of stellar activity and XUV irradiation with time. We estimated that
given reasonable assumptions regarding core radius and mass, planet b probably
retains a substantial atmosphere, with a mass fraction $\sim 0.5M_{\rm p}$,
making it amenable to be probed with transmission spectroscopy. Conversely,
planet c is expected to possess a tiny atmospheric envelope. At the current
age, the mass loss rate of planet c exceeds that of planet b by a factor of 7,
implying that planet c will completely lose its residual atmosphere within a
few hundred million years, while planet b will retain a thick atmosphere even
at the solar age.

Notably, TESS observed TOI-5398 during sector 48 and no further observations
are planned in the current Extended Mission. Consequently, our ground-based
light curves play a pivotal role in refining the ephemeris of both planets.
The improved ephemeris we calculated are vital for future follow-up
observations and surveys, including those conducted by CHEOPS, JWST, and
upcoming missions such as PLATO and Ariel, along with telescopes like ELTs.

###### Acknowledgements.

We are extremely grateful to the anonymous referee for their meticulous
feedback, which undoubtedly improved the quality of this manuscript. Based on
observations collected at Copernico 1.82m telescope (Asiago Mount Ekar, Italy)
INAF - Osservatorio Astronomico di Padova. This work has made use of data from
the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular, the institutions
participating in the Gaia Multilateral Agreement. This 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. This work has been supported by the PRIN-INAF 2019
“Planetary systems at young ages (PLATEA)”. This work is partly financed by
the Spanish Ministry of Economics and Competitiveness through grants
PGC2018-098153-B-C31. This work is partly supported by JSPS KAKENHI Grant
Numbers JP17H04574, JP18H01265, and JP18H05439, Grant-in-Aid for JSPS Fellows
Grant Number JP20J21872, JST PRESTO Grant Number JPMJPR1775, and a University
Research Support Grant from the National Astronomical Observatory of Japan
(NAOJ). This work makes use of observations from the LCOGT network. Part of
the LCOGT telescope time was granted by NOIRLab through the Mid-Scale
Innovations Program (MSIP). MSIP is funded by NSF. This research has made use
of the Exoplanet Follow-up Observation Program (ExoFOP; DOI: 10.26134/ExoFOP5)
website, which is operated by the California Institute of Technology, under
contract with the National Aeronautics and Space Administration under the
Exoplanet Exploration Program. Funding for the TESS mission is provided by
NASA’s Science Mission Directorate. KAC and CNW acknowledge support from the
TESS mission via subaward s3449 from MIT. TZi acknowledges NVIDIA Academic
Hardware Grant Program for the use of the Titan V GPU card. This work is
partly supported by JST CREST Grant Number JPMJCR1761. The postdoctoral
fellowship of KB is funded by F.R.S.-FNRS grant T.0109.20 and by the Francqui
Foundation. D.V.C. and I.A.S. acknowledge the support of M.V. Lomonosov Moscow
State University Program of Development. R.L. acknowledges funding from
University of La Laguna through the Margarita Salas Fellowship from the
Spanish Ministry of Universities ref. UNI/551/2021-May 26, and under the EU
Next Generation funds. This work is partly supported by the CHEOPS ASI-INAF
agreement n. 2019-29-HH.0. G.N. thanks for the research funding from the
Ministry of Education and Science programme the “Excellence Initiative -
Research University” conducted at the Centre of Excellence in Astrophysics and
Astrochemistry of the Nicolaus Copernicus University in Toruń, Poland. We
acknowledge the use of public TESS data from pipelines at the TESS Science
Office and at the TESS Science Processing Operations Center. 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. AM, DL, DT and
DP acknowledge partial financial support from the ARIEL ASI-INAF agreement
n.2021-5-HH.0. AM also acknowledges “The HOT-ATMOS Project” (PRIN-INAF 2019).
DP acknowledges the support from the Istituto Nazionale di Oceanografia e
Geofisica Sperimentale (OGS) and CINECA through the program “HPC-TRES (High
Performance Computing Training and Research for Earth Sciences)” award number
2022-05. P.K. acknowledges the funding from a grant LTT-20015.

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##  Appendix A Statistical validation

In the first step of our vetting and validation study, we used Gaia EDR3 data
to identify nearby contaminating stars that might be blended eclipsing
binaries (BEBs). This analysis shows that besides TOI-5398, there is a star
(Gaia DR2 750888586899153536) at 45.3$\arcsec$ separation from TOI-5398 that
might reproduce the transit signal of TOI-5398.02, see Fig. 24. To ensure that
the transits of TOI-5398.02 are genuine and not caused by contaminant
neighbours, we carried out the in- / out-of-transit difference centroid check
outlined by Nardiello et al. (2020) and Nardiello (2020). We show the outcomes
in Fig. 25. The in- / out-of-transit mean difference centroids are consistent
with TOI-5398’s position and far from any potential contaminants, further
proving the planetary nature of TOI-5398.02.

Figure 24: Gaia stars identified in the TESS field. The letter “t” denotes the
position of TOI-5398, while the position of the potential contaminant star is
enclosed in a green circle.  Figure 25: Computation of the in- / out-of-
transit difference centroid test for Sector 48 (violet dot and error bars).
The potential contaminant star’s location is instead surrounded by a green
circle.

To corroborate that TOI-5398.02 is not an FP, we used the
VESPA232323https://github.com/timothydmorton/VESPA software (Morton 2012,
2015) as a final check. We have followed the procedure adopted in Mantovan et
al. (2022), which takes into account the major issues reported in Morton et
al. (2023) and allows us to get reliable results while using VESPA. We used
our detrended light curve (see Sect. 2.1), which we normalised using wotan,
and we phase-folded after removing TOI-5398 b’s signal identified through the
TLS analysis (see Sect. 4.1.1). We find a 100% probability of having a
keplerian transiting companion around TOI-5398, while the probability of an FP
is of the order of $\sim$ 9 x 10-4. It is important to note that candidates in
multi-planet systems have a higher probability of being genuine planets
(Latham et al. 2011; Lissauer et al. 2012). Therefore, our FPP (False Positive
Probability) should be even lower. All these analyses show that TOI-5398.02 is
completely statistically vetted and only needs to go through reconnaissance
spectroscopy to be promoted to the status of a statistically validated
planetary companion (Mantovan et al. 2022). As a result of the extensive
HARPS-N spectroscopy examination of the planetary system TOI-5398, the sub-
Neptune exoplanet is statistically validated and will henceforth be labelled
as TOI-5398 c.

Figure 26: SAP-corrected light curve of TOI-5398 c that has been detrended,
normalised, and phase-folded. The gold line is a trapezoidal fitting model
produced by VESPA.

High angular resolution observations described in Sect. 2.6 rule out – down to
the sensitivity of the data – the presence of stellar companion which could
have been missed by Gaia photometry. The detection limits for companions
($\Delta I_{c}$ mag vs separation) were translated into physical properties of
companions of the star, using the expected V-I color from the $T_{\rm eff}$ of
the target and the stellar models by Baraffe et al. (2015) for the suitable
age of the system. The mass detection limits span from 0.65 $M_{\odot}$ at 13
au (projected separation) to 0.28 $M_{\odot}$ at 50 au, and 0.17 $M_{\odot}$
at 100 au.

##  Appendix B Combined analysis of HARPS-N, TRES, and McDonald datasets

###  B.1 Observations and data reduction of TRES data

Spectra of TOI-5398 were obtained between UT March 28, 2022 - April 19, 2022
using the Tillinghast Reflector Echelle Spectrograph (TRES; Fűrész 2008)
mounted on the 1.5m Tillinghast Reflector telescope at the Fred Lawrence
Whipple Observatory (FLWO) atop Mount Hopkins, Arizona. TRES is an optical,
fibre-fed echelle spectrograph with a wavelength range of 390-910nm and a
resolving power of R 44,000. The TRES spectra were extracted as described in
Buchhave et al. (2010). The spectra were visually inspected to check that
there were no additional stellar companions contaminating the radial
velocities. Then, a multi-order relative velocity analysis was performed by
cross-correlating the strongest signal-to-noise observed spectrum as a
template, order by order, against the remaining spectra to produce an orbit.

###  B.2 Modelling of HARPS-N, TRES, and McDonald RVs

We conducted a modelling analysis using PyORBIT on the HARPS-N, TRES, and
McDonald data, by using the same configuration and priors as described in
Sect. 4.3 (Case 3). The results of this modelling are reported in Tables 10
and 11, in Fig. 27 and 28. Due to the considerable errors associated (see
their associated jitters) with these measurements and less precise activity
indexes, the TRES and McDonald RVs were not used in determining the planet
parameters, as we cannot reliably determined their GP parameters (see their
$V_{c}$, $V_{r}$ coefficients and error bars). Consequently, we decided to
exclude them from the analysis conducted in our study.

Figure 27: Phase-folded RV fit of TOI-5398 b planetary signal. The three
different colours represent HARPS-N (violet), TRES (green), and McDonald 2.7m
(red) data. The reported error bars include the jitter term, added in
quadrature. The bottom panel displays the residuals of the fit.  Figure 28: As
in Fig. 27, but for planet c. Table 10:  Priors and outcomes of the model of
planet b and c from the analysis of combined HARPS-N, TRES, and McDonald
spectroscopic series. GP framework parameters
---
Parameter | Unit | Prior | Value
Uncorrelated RV jitter HARPS-N ($\sigma^{\rm RV}_{\rm jitter,0}$) | m s-1 | … | 1.28${}^{+1.4}_{-0.89}$
RV offset HARPS-N ($\gamma^{\rm RV}_{0}$) | m s-1 | … | -9435.1${}^{+3.7}_{-3.9}$
Uncorrelated RV jitter McDonald ($\sigma^{\rm RV}_{\rm jitter,0}$) | m s-1 | … | 26${}^{+29}_{-18}$
RV offset McDonald ($\gamma^{\rm RV}_{0}$) | m s-1 | … | 27778 $\pm$ 20
Uncorrelated RV jitter TRES ($\sigma^{\rm RV}_{\rm jitter,0}$) | m s-1 | … | 11.3${}^{+13}_{-7.8}$
RV offset TRES ($\gamma^{\rm RV}_{0}$) | m s-1 | … | -6${}^{+23}_{-20}$
Uncorrelated BIS jitter ($\sigma^{\rm BIS}_{\rm jitter,0}$) | m s-1 | … | 17.3${}^{+1.8}_{-1.6}$
BIS offset ($\gamma^{\rm BIS}_{0}$) | m s-1 | … | 10.9${}^{+3.1}_{-3.0}$
Uncorrelated log$R^{{}^{\prime}}_{HK}$ jitter ($\sigma^{\rm logR^{{}^{\prime}}_{HK}}_{\rm jitter,0}$) |  | … | 0.0132${}^{+0.0015}_{-0.0014}$
log$R^{{}^{\prime}}_{HK}$ offset ($\gamma^{\rm logR^{{}^{\prime}}_{HK}}_{0}$) |  | … | -4.4165$\pm$0.0035
Uncorrelated H${\alpha}$ jitter ($\sigma^{\rm H{\alpha}}_{\rm jitter,0}$) |  | … | 0.0018$\pm$0.0002
H${\alpha}$ offset ($\gamma^{\rm H\alpha}_{0}$) |  | … | 0.1513$\pm$0.0006
Uncorrelated FWHM jitter ($\sigma^{\rm FWHM}_{\rm jitter,0}$) | km s-1 | … | 0.043$\pm$0.004
FWHM offset ($\gamma^{\rm FWHM}_{0}$) | km s-1 | … | 11.36$\pm$0.01
Uncorrelated $W_{\rm CCF}$ jitter ($\sigma^{W_{\rm CCF}}_{\rm jitter,0}$) | km s-1 | … | 0.0014${}^{+0.0014}_{-0.0009}$
$W_{\rm CCF}$ offset ($\gamma^{W_{\rm CCF}}_{0}$) | km s-1 | … | 3.503$\pm$0.004
$V_{c}$ HARPS-N | m s-1 |  $\mathcal{U}$(-100.0, 100.0) | -13.4${}^{+3.1}_{-3.7}$
$V_{r}$ HARPS-N | m s-1 |  $\mathcal{U}$(0.0, 100.0) | 31.3${}^{+6.7}_{-4.6}$
$V_{c}$ McDonald | m s-1 |  $\mathcal{U}$(-100.0, 100.0) | 2${}^{+30}_{-31}$
$V_{r}$ McDonald | m s-1 |  $\mathcal{U}$(0.0, 100.0) | 56${}^{+27}_{-28}$
$V_{c}$ TRES | m s-1 |  $\mathcal{U}$(-100.0, 100.0) | -13${}^{+22}_{-24}$
$V_{r}$ TRES | m s-1 |  $\mathcal{U}$(0.0, 100.0) | 18${}^{+21}_{-12}$
$B_{c}$ | m s-1 |  $\mathcal{U}$(-100.0, 100.0) | 6.1${}^{+4.2}_{-3.8}$
$B_{r}$ | m s-1 |  $\mathcal{U}$(-100.0, 100.0) | -40.7${}^{+6.2}_{-8.0}$
$L_{c}$ (log$R^{{}^{\prime}}_{HK}$) |  |  $\mathcal{U}$(-0.1, 0.1) | -0.012$\pm$0.003
$L2_{c}$ (H${\alpha}$) |  |  $\mathcal{U}$(-0.1, 0.1) | -0.0021${}^{+0.0004}_{-0.0005}$
$L3_{c}$ (FWHM) | km s-1 |  $\mathcal{U}$(-0.5, 0.5) | -0.046${}^{+0.009}_{-0.011}$
$L4_{c}$ ($W_{\rm CCF}$) | km s-1 |  $\mathcal{U}$(-0.02, 0.02) | -0.015${}^{+0.002}_{-0.003}$
Stellar activity
Parameter | Unit | Prior | Value
Rotational period ($P_{\rm rot}$) | days |  $\mathcal{N}$(7.18, 0.21) | 7.42$\pm$0.04
Decay Timescale of activity ($P_{\rm dec}$) | days |  $\mathcal{U}$(10.0, 2000.0) | 26.8${}^{+3.3}_{-3.1}$
Coherence scale ($w$) |  |  $\mathcal{U}$(0.01, 0.60) | 0.40${}^{+0.04}_{-0.03}$
Table 11: Continuation of Table 10. Planet b
---
Parameter | Unit | Prior | Value
Orbital period ($P_{\rm b}$) | days |  $\mathcal{N}$(10.59049, 0.00002) | 10.59049$\pm$0.00002
Central time of the first transit ($T_{\rm 0,b}$) | BTJD |  $\mathcal{N}$(2616.4921, 0.0003) | 2616.4921$\pm$0.0003
Orbital eccentricity ($e_{\rm b}$) |  |  $\mathcal{N}$(0.00, 0.098) | 0.05${}^{+0.06}_{-0.04}$
Argument of periastron ($\omega_{\rm b}$) | deg | … | 79${}^{+64}_{-105}$
Semi-major axis to stellar radius ratio ($a_{\rm b}/R_{\star}$) |  | … | 19.9$\pm$1.2
Orbital semi-major axis ($a_{\rm b}$) | au | … | 0.098${}^{+0.004}_{-0.004}$
RV semi-amplitude ($K_{\rm b}$) | m s-1 |  $\mathcal{U}$(0.01, 100.0) | 16.1${}^{+1.5}_{-1.5}$
Planetary mass ($M_{\rm p,b}$) | $M_{\oplus}$ | … | 58.8${}^{+7.7}_{-7.3}$
Planet c
Parameter | Unit | Prior | Value
Orbital period ($P_{\rm c}$) | days |  $\mathcal{N}$(4.7734, 0.0004) | 4.7734$\pm$0.0004
Central time of the first transit ($T_{\rm 0,c}$) | BTJD |  $\mathcal{N}$(2628.6188, 0.0010) | 2628.6188$\pm$0.0010
Orbital eccentricity ($e_{\rm c}$) |  |  $\mathcal{N}$(0.00, 0.098) | 0.07${}^{+0.07}_{-0.05}$
Argument of periastron ($\omega_{\rm c}$) | deg | … | 132${}^{+93}_{-142}$
Semi-major axis to stellar radius ratio ($a_{\rm c}/R_{\star}$) |  | … | 11.7$\pm$0.7
Orbital semi-major axis ($a_{\rm c}$) | au | … | 0.058${}^{+0.002}_{-0.003}$
RV semi-amplitude ($K_{\rm c}$) | m s-1 |  $\mathcal{U}$(0.01, 100.0) | 4.2$\pm 1.6$
Planetary mass ($M_{\rm p,c}$) | $M_{\oplus}$ | … | 11.6${}^{+4.7}_{-4.6}$

##  Appendix C Stellar parameters using TRES spectra

The TRES spectra were also used to derive stellar parameters using the Stellar
Parameter Classification tool (SPC; Buchhave et al. 2012). SPC cross
correlates an observed spectrum against a grid of synthetic spectra based on
the Kurucz atmospheric model (Kurucz 1992) which derives the effective
temperature $T_{\rm eff}=5933\pm 50$ K, surface gravity $\log g=4.42\pm 0.10$,
metallicity $[m/H]=0.03\pm 0.08$, and rotational velocity of the star
$v\sin{i_{\star}}=8.4\pm 0.5$ km s-1. The stellar parameters align with the
estimates that we presented in Sect. 3 and Table 2.

##  Appendix D Independent RV analysis using tweaks

The HARPS-N CCFs were independently analysed for planetary signals using
tweaks (Time and Wavelength domain stEllar Activity mitigation using kima and
scalpels). This pipeline was designed to achieve a sub-m/s detection threshold
at extended orbital periods (Anna John et al. 2023 submitted). This is
achieved by obtaining a set of time-domain stellar activity-decorrelation
vectors using scalpels (Collier Cameron et al. 2021), by doing principal-
component analysis on the autocorrelation function of the CCF. These basis
vectors were then used for the spectral line-shape decorrelation (Collier
Cameron et al. 2021) in kima (Faria et al. 2018), as (John et al. 2022)
reported that using the scalpels basis vectors to de-trend the RVs for line
shape variations results in a model that is significantly better than a model
that does not take these stellar activity signatures into account.

A model with up to five unidentified Keplerian signals was used in our first
blind search of the RVs using the kima nested-sampling package (Faria et al.
2018). As (John et al. 2022) have found, some planet-like signals elude
scalpels analysis, so any remaining rotationally-modulated signals were
modelled with GP regression applied to the RVs, using a quasi-periodic GP
kernel. The joint posteriors showed clear detection of planet b, at orbital
period 10.5905 $\pm$ 0.0002 days with an RV semi-amplitude of 14.51 $\pm$ 3.74
ms-1. Additionally, the GP strongly constrained the stellar rotation to
$P_{\rm rot}=7.36_{-0.08}^{+0.11}$ days. This independent analysis aligns with
the RV reference model solution and the stellar parameters discussed in Sect.
3.5.

##  Appendix E Extra Tables

Table 12: Second part of Table 3. Photometric time-series fit
---
Parameter | Unit | Prior | Value
TESS dilution factor |  |  $\mathcal{N}$(0.00735, 0.00005) | 0.007348$\pm$0.000050
TESS jitter ($\sigma^{\rm TESS}_{\rm jitter}$) | e- s-1 | … | 5.0${}^{+1.1}_{-1.4}$
Asiago jitter ($\sigma^{\rm Asiago}_{\rm jitter}$) |  | … | 0.00074${}^{+0.00006}_{-0.00007}$
LCO_McD_c jitter ($\sigma^{\rm McD\\_c}_{\rm jitter}$) |  | … | 0.00145${}^{+0.00011}_{-0.00010}$
LCO_McD_b jitter ($\sigma^{\rm McD\\_b}_{\rm jitter}$) |  | … | 0.00073${}^{+0.00009}_{-0.00008}$
MuSCAT2 jitter ($\sigma^{\rm MuSCAT2}_{\rm jitter}$) |  | … | 0.00014${}^{+0.00012}_{-0.00008}$
KeplerCam jitter ($\sigma^{\rm KeplerCam}_{\rm jitter}$) |  | … | 0.00279${}^{+0.00011}_{-0.00010}$
Acton jitter ($\sigma^{\rm Acton}_{\rm jitter}$) |  | … | 0.00349$\pm$0.00010
LCO_McD_bc jitter ($\sigma^{\rm McD\\_bc}_{\rm jitter}$) |  | … | 0.00095${}^{+0.00008}_{-0.00008}$
LCO_Teid jitter ($\sigma^{\rm Teid}_{\rm jitter}$) |  | … | 0.00067${}^{+0.00012}_{-0.00013}$
TESS quadratic LD coefficient ($u^{TESS}_{1}$) |  |  $\mathcal{U}$(0, 1) | 0.089${}^{+0.089}_{-0.061}$
TESS quadratic LD coefficient ($u^{TESS}_{2}$) |  |  $\mathcal{U}$(0, 1) | 0.52${}^{+0.11}_{-0.14}$
Asiago quadratic LD coefficient ($u^{Asiago}_{1}$) |  |  $\mathcal{N}$(0.489, 0.014) | 0.455$\pm$0.013
Asiago quadratic LD coefficient ($u^{Asiago}_{2}$) |  |  $\mathcal{N}$(0.151, 0.037) | 0.007${}^{+0.033}_{-0.032}$
McD_c quadratic LD coefficient ($u^{McD\\_c}_{1}$) |  |  $\mathcal{N}$(0.34, 0.01) | 0.340$\pm$0.010
McD_c quadratic LD coefficient ($u^{McD\\_c}_{2}$) |  |  $\mathcal{N}$(0.14, 0.03) | 0.14$\pm$0.03
McD_b quadratic LD coefficient ($u^{McD\\_b}_{1}$) |  |  $\mathcal{N}$(0.34, 0.01) | 0.343$\pm$0.010
McD_b quadratic LD coefficient ($u^{McD\\_b}_{2}$) |  |  $\mathcal{N}$(0.14, 0.03) | 0.17$\pm$0.03
MuSCAT2 quadratic LD coefficient ($u^{MuSCAT2}_{1}$) |  |  $\mathcal{N}$(0.34, 0.01) | 0.337$\pm$0.010
MuSCAT2 quadratic LD coefficient ($u^{MuSCAT2}_{2}$) |  |  $\mathcal{N}$(0.14, 0.03) | 0.13$\pm$0.03
KeplerCam quadratic LD coefficient ($u^{KeplerCam}_{1}$) |  |  $\mathcal{N}$(0.34, 0.01) | 0.334$\pm$0.010
KeplerCam quadratic LD coefficient ($u^{KeplerCam}_{2}$) |  |  $\mathcal{N}$(0.14, 0.03) | 0.10$\pm$0.03
Acton quadratic LD coefficient ($u^{Acton}_{1}$) |  |  $\mathcal{N}$(0.40, 0.01) | 0.4013$\pm$0.0099
Acton quadratic LD coefficient ($u^{Acton}_{2}$) |  |  $\mathcal{N}$(0.14, 0.03) | 0.15$\pm$0.03
McD_bc quadratic LD coefficient ($u^{McD\\_bc}_{1}$) |  |  $\mathcal{N}$(0.34, 0.01) | 0.340$\pm$0.010
McD_bc quadratic LD coefficient ($u^{McD\\_bc}_{2}$) |  |  $\mathcal{N}$(0.14, 0.03) | 0.14$\pm$0.03
Teid quadratic LD coefficient ($u^{Teid}_{1}$) |  |  $\mathcal{N}$(0.34, 0.01) | 0.339$\pm$0.010
Teid quadratic LD coefficient ($u^{Teid}_{2}$) |  |  $\mathcal{N}$(0.14, 0.03) | 0.14$\pm$0.03