11institutetext:  Observatoire de Genève, Université de Genève, Chemin Pegasi
51, 1290 Versoix, Switzerland. 22institutetext: Institute for Particle Physics
and Astrophysics, ETH Zürich, Otto-Stern-Weg 5, 8093 Zürich, Switzerland.
33institutetext: Center for Theoretical Astrophysics and Cosmology, Institute
for Computational Science, University of Zürich, Winterthurerstrasse 190, 8057
Zürich, Switzerland.
33email<EMAIL_ADDRESS>
# From Super-Earths to Sub-Neptunes: observational constraints and connections
to theoretical models

Léna Parc From Super-Earths to Sub-Neptunes: observational constraints and
connections to theoretical modelsFrom Super-Earths to Sub-Neptunes:
observational constraints and connections to theoretical models
0000-0002-7382-1913 François Bouchy From Super-Earths to Sub-Neptunes:
observational constraints and connections to theoretical modelsFrom Super-
Earths to Sub-Neptunes: observational constraints and connections to
theoretical models 0000-0002-7613-393X Julia Venturini From Super-Earths to
Sub-Neptunes: observational constraints and connections to theoretical
modelsFrom Super-Earths to Sub-Neptunes: observational constraints and
connections to theoretical models 0000-0001-9527-2903 Caroline Dorn From
Super-Earths to Sub-Neptunes: observational constraints and connections to
theoretical modelsFrom Super-Earths to Sub-Neptunes: observational constraints
and connections to theoretical models 0000-0001-6110-4610 Ravit Helled From
Super-Earths to Sub-Neptunes: observational constraints and connections to
theoretical modelsFrom Super-Earths to Sub-Neptunes: observational constraints
and connections to theoretical models 0000-0001-5555-2652

(Received March 9, 2024; revised May 28, 2024; accepted June 12, 2024)

The growing number of well-characterized exoplanets smaller than Neptune
enables us to conduct more detailed population studies. We update the PlanetS
catalog of transiting planets with precise and robust mass and radius
measurements and use this comprehensive catalog to explore Mass-Radius (M-R)
diagrams. On one hand, we propose new M-R relationships to separate exoplanets
into three populations: rocky planets, volatile-rich planets and giant
planets. On the other hand, we explore the transition in radius and density
between Super-Earths and Sub-Neptunes around M-dwarfs and compare it with
those orbiting K- and FG-dwarfs. Using Kernel Density Estimation method with a
re-sampling technique, we estimate the normalized density and radius
distributions, revealing connections between observations and theories on
composition, internal structure, formation, and evolution of these exoplanets
orbiting different spectral types. Firstly, the substantial 30% increase in
the number of well-characterized exoplanets orbiting M-dwarfs compared with
previous studies shows us that there is no clear gap in either composition or
radius between Super-Earths and Sub-Neptunes. The ”water-worlds” around
M-dwarfs cannot correspond to a distinct population, their bulk density and
equilibrium temperature can be interpreted by several different internal
structures and compositions. The continuity in the fraction of volatiles in
these planets suggests a formation scenario involving planetesimal or hybrid
pebble-planetesimal accretion. Moreover, we find that the transition between
Super-Earths and Sub-Neptunes appears to happen at different mass (and radii)
depending on the spectral type of the star. The maximum mass of Super-Earths
seems to be close to 10 M⊕ for all spectral types, but the minimum mass of
Sub-Neptunes increases with the star’s mass, and is around 1.9 M⊕, 3.4 M⊕ and
4.3 M⊕, for M-dwarfs, K-dwarfs, and FG-dwarfs, respectively. The precise value
of this minimum mass may be affected by observational bias, but the trend
appears to be reliable. This effect, attributed to planet migration, also
contributes to the fading of the radius valley for M-planets compared to FGK-
planets. While Sub-Neptunes are less common around M-dwarfs, smaller ones (1.8
R⊕ ¡ Rp ¡ 2.8 R⊕) exhibit lower density than their equivalents around FGK-
dwarfs. Nonetheless, the sample of well-characterized small exoplanets remains
limited, and each new discovery has the potential to reshape our understanding
and interpretations of this population in the context of internal structure,
composition, formation, and evolution models. Broader consensus is also needed
for internal structure models and atmospheric compositions to enhance density
interpretation and observable predictions for the atmospheres of these
exoplanets.

###### Key Words.:

Planets and satellites: composition – Planets and satellites: formation –
Methods: statistical

##  1 Introduction

Over the last three decades, planetary sciences have undergone a revolutionary
transformation, with the identification of over 5 500 exoplanets orbiting
stars within our galaxy. The majority, over 4 100, have been discovered using
the transit method, thanks to the efforts of space telescopes conducting
large-scale surveys like CoRoT (Baglin et al., 2006), Kepler/K2 (Borucki et
al., 2010a; Howell et al., 2014), and TESS (Ricker et al., 2015). Notably,
planets with sizes smaller than Neptune ($<$ 4 R⊕), are prevalent.
Approximately every star is known to host such planets, detected at an orbital
period of less than 100 days (Borucki et al., 2010b; Batalha et al., 2011).

M-type stars, being the most abundant stellar type in our galaxy (Winters et
al., 2015), appear to have a higher occurrence rate of exoplanets (Dressing &
Charbonneau, 2013; Bonfils et al., 2013; Mulders et al., 2015; Gaidos et al.,
2016), especially small ones (Bonfils et al., 2013; Dressing & Charbonneau,
2015). Their planets are easier to detect and study due to the smaller size
and lower mass of their host star. The full sky coverage of TESS compared to
Kepler has significantly increased the detection of planets around these
stars. This was further enhanced by ground-based instruments and telescopes
dedicated to infrared photometry such as TRAPPIST (Jehin et al., 2011),
SPECULOOS (Delrez et al., 2018) or ExTrA (Bonfils et al., 2015) that not only
validated TESS/Kepler candidates but also discovered new M-planets.
Additionally, the development of high-resolution near-infrared spectrographs
like CARMENES (Quirrenbach et al., 2020), MAROON-X (Seifahrt et al., 2016),
SPIRou (Artigau et al., 2014) or NIRPS (Bouchy et al., 2017) enabled efficient
follow-up of these transiting exoplanets to obtain mass measurements. These
surveys have led to a substantial increase in the number of planets around
low-mass stars with precise measured mass and radius. This, in turn,
facilitates statistical studies on larger populations, allowing for more
reliable conclusions.

The Mass-Radius (M-R) diagram stands out as a powerful tool for investigating
the demographics of exoplanets. It serves various purposes, such as deriving
mass-radius relationships for predicting the estimated masses of transiting
objects in radial velocity follow-up campaigns (Weiss & Marcy, 2014; Wolfgang
et al., 2016; Chen & Kipping, 2017; Bashi et al., 2017; Kanodia et al., 2019;
Otegi et al., 2020; Edmondson et al., 2023; Parviainen et al., 2024). Despite
the degenerate nature of the problem, the M-R diagram plays a crucial role in
exploring the composition and internal structure of exoplanets. They enable us
to study the radius and mass distributions of exoplanets, offering insights
into their bulk density, and facilitates comparisons with theoretical models.

One of the best-known features of the radius distribution of small planets is
the ”radius valley” (Fulton et al., 2017). It separates two sub-populations of
Super-Earths and Sub-Neptunes by a gap at $\sim$ 1.5-2 R⊕, with the exact
location depending on the mass of the host star (Fulton & Petigura, 2018;
Berger et al., 2020; Ho & Van Eylen, 2023). Different theories contribute to
the understanding of the origin of the radius valley. Some of them focus
exclusively on processes happening after the disc dissipation, such as
photoevaporation (Owen & Wu, 2017; Jin & Mordasini, 2018) and core-powered
mass-loss (Ginzburg et al., 2018; Gupta & Schlichting, 2019). Other theories
combine formation and evolution processes into a single framework to explain
the radius valley (Venturini et al., 2020, 2024; Burn et al., 2024).

In the view of the pure evolution models (like photoevaporation or core-
powered mass-loss), some of the planets lose their atmospheres, evolving into
Super-Earths, while those retaining their atmospheres are classified as
today’s Sub-Neptunes. Based on the assumption of a unique core composition and
H/He atmospheres, such models find that the correct location of the valley can
only be reproduced if the underlying cores are rocky, implying a formation
inside the water ice line for both super-Earths and sub-Neptunes (Owen & Wu,
2017; Gupta & Schlichting, 2019). On the contrary, combined formation and
evolution models predict that Sub-Neptunes are typically water-worlds
assembled beyond the water ice line (Venturini et al., 2020, 2024; Burn et
al., 2024). In particular, Venturini et al. (2020) showed that larger icy
versus smaller rocky cores form naturally by pebble accretion, due to the
difference of sticking properties between dry and icy pebbles. The separation
between the two populations occurs exactly at the location of the observed
gap. However, this bi-modality in mass and size distribution is hidden at the
end of the formation phase due to the presence of gaseous atmospheres.
Photoevaporation is needed to remove the gas and reveal the radius valley that
was set by core formation in the first place. Venturini et al. (2024) expand
these models for a range of stellar masses, pointing out the role of migration
in the fading of the radius valley towards M-dwarfs, as well as the effect of
the phases of water in the enhancement of the radius valley. Burn et al.
(2024) perform similar calculations as Venturini et al. (2020) but for
planetesimal accretion, highlighting the role of steam atmospheres in the
emergence of the radius valley. However, these previous studies neglect any
compositional coupling of water between the deep interior and the surface (Luo
et al., 2024).

The characterization of sub-Neptune-sized planets is an active area of
studies. These planets, unlikely to have a pure rocky composition (Rogers,
2015; Fulton et al., 2017)), inhabit a degenerate region in the M-R parameter
space. Their bulk densities can be equally well explained by solid rocky/iron
cores with primordial gaseous hydrogen-rich atmospheres (”gas dwarfs”, Lopez &
Fortney 2014; Rogers et al. 2023) or by a water-rich interior and/or
atmosphere (”water worlds”, Léger et al. 2004; Dorn & Lichtenberg 2021;
Aguichine et al. 2021; Luque & Pallé 2022).

Around M-dwarfs, the existence of this radius valley remains unclear (Cloutier
& Menou, 2020; Luque & Pallé, 2022; Ho et al., 2024) and the existence of a
water-world population is still debated (Luque & Pallé, 2022; Parviainen et
al., 2024). Luque & Pallé (2022) propose the existence of a compositional gap
separating rocky from water-rich planets. They claim that the small transiting
planets orbiting M-dwarfs can be categorized into three distinct groups
(”rocky planets”, ”water-worlds” and ”puffy sub-Neptunes”), rather than
constituting a seamless continuum. Among these groups, two are positioned
along specific compositional lines in the M-R diagram: one representing Earth-
like composition (”rocky planets”, noted $\rho_{\oplus\text{,s}}$), and the
other representing a composition of 50% condensed water and 50% silicates
(”water-worlds”). An examination of the density and radius distributions
within these three sub-categories indicates an absence of a radius valley.
However, they find a compositional gap at a normalized density value of 0.65
$\rho_{\oplus\text{,s}}$, where there is an absence of planets and so no
overlap observed between the distribution of ”rocky planets” and that of
”water-worlds”. On the contrary, Parviainen et al. (2024) shows strong support
against the existence of a water-world population around M-dwarfs using the
python module spright 111https://github.com/hpparvi/spright and an updated
sample from Luque & Pallé (2022). Indeed, they present spright, a novel
probabilistic mass-density-radius relationship for small planets. It employs a
Bayesian model comparison to represent the joint planetary radius and bulk
density probability distribution as a mean posterior predictive distribution
of an analytical three-component mixture model. The three components represent
rocky planets, water-rich planets, and sub-Neptunes, and the final numerical
probability model is obtained by marginalizing over all analytical model
solutions allowed by observations.

From a formation point of view, to explain this population of ”water-worlds”
around M-stars, theoretical studies by Alibert (2017), Miguel et al. (2020)
and Burn et al. (2021) suggest that low-mass water-rich planets should be
prevalent in close-in orbits around M-dwarfs. This is attributed to the
proximity of water ice lines to the stars and the increased efficiency of
inward migration at lower planetary masses around lower-mass stars.
Alternatively, formation models that do not consider planetary migration show
that these low-density planets orbiting M-dwarfs could consist of rocky cores
surrounded by H/He envelopes (Lee et al., 2022).

In this work, we update the PlanetS catalog initially publish by Otegi et al.
(2020) of transiting planets with precise mass and radius measurements and the
associated Mass-Radius relationships in Section 2. Subsequently, we conduct a
comprehensive examination of the M-R diagram for small planets around M-dwarfs
in Section 3. This includes deriving distributions of radius and density
(Sect. 3.2) for analysis and discussing observed properties with theoretical
models of internal structure (Sect. 3.3) and formation and evolution (Sect.
3.4). We further broaden our investigation to include planets orbiting FGK-
type stars, allowing for a comparative analysis of the distinctions between
the population of planets orbiting M-dwarfs and those around earlier spectral
types in Section 4. The conclusions drawn from our study are presented in
Section 5.

##  2 The PlanetS catalog: precise and reliable mass and radius for transiting
planets

Figure 1: Mass-Radius Diagram comparing the data from the PlanetS catalog and
the data from NASA Exoplanet Archive to highlight the effects of our selection
criteria and our detailed study of references. The grey dots represented the
default parameters from NASA, the light blue triangles the same but filtered
with our criteria on uncertainties and the blue dots correspond to the PlanetS
catalog. The resulting M-R relationships from Sect.2.2 are in solid red lines
with the confidence intervals represented by the shaded red areas. The
transitions between the three populations are represented by the composition
line of 20% water at 650 K from Luo et al. (2024) in dashed orange and the
dotted black line at 138 M⊕.

###  2.1 Update and redesign of the catalog

We present here the PlanetS catalog222https://dace.unige.ch/exoplanets/ of
transiting planets with reliable parameters. Initially published in Otegi et
al. (2020), where it was limited to 120 M⊕, the catalog has since been
extended to all planetary masses, then updated with new discoveries and
revisited parameters of old systems, and modified to include more planetary
and stellar parameters.

The overall aim of this catalog is to bring together the orbital, stellar and
planetary parameters of transiting exoplanets with precise and reliable mass
and radius measurements, and to keep it up to date. To achieve this, as in
Otegi et al. (2020), the catalog update is based on the NASA Exoplanet
Archive333https://exoplanetarchive.ipac.caltech.edu, as this is the most up-
to-date and complete catalog in terms of parameters and references for each
exoplanet detected. Selection criteria are also identical to Otegi et al.
(2020):

  * •

We consider planets with relative measurement uncertainties smaller than 25%
in mass ($\sigma_{M}{{}_{p}}/M_{p}\leq$ 25%) and 8% in radius
($\sigma_{R}{{}_{p}}/R_{p}\leq$ 8%).

  * •

Reference is checked before a planet is added to the catalog to ensure the
reliability and robustness of the analysis of the photometric and
spectroscopic data.

  * •

Among several possible references, the most precise (without underestimated
uncertainties) and the most recent is chosen. A single reference is chosen for
a given system, even if the precision for one of the planets is better in
another study, in order to maintain consistency.

A special case here are planets whose mass is determined by Timing Transit
Variations (TTVs). The robustness of these determinations is more difficult to
assess, as TTVs models often give several solutions. Like Otegi et al. (2020),
we only consider planets whose mass estimations have been shown to be robust
against mass/eccentricity degeneracy (Lithwick et al., 2012; Hadden &
Lithwick, 2017; Leleu et al., 2023). Indeed, inferred planet masses are
considered ”robust” if the 1$\sigma$ credible region of the default posterior
(logarithmic mass and uniform eccentricity priors) excludes zero and includes
the peak of the high mass posterior (uniform mass and logarithmic eccentricity
priors). This criterion is important, because if the mass posterior is
strongly dependent on the prior, then the data are considered insufficient to
lift this degeneracy. We therefore ignore references giving only one solution
of TTVs or two solutions that disagree.

Thresholds for uncertainties have been maintained since Otegi et al. (2020)
and correspond to median uncertainties at that time, with the same impact for
mass and radius on density uncertainty. Nevertheless, advancements in
detection and analysis techniques suggest that these medians would now hover
around 15% for relative mass error and 5% for relative radius error (retaining
this factor of 3 for the impact on density error). However, adopting this new
criterion would result in a 40% reduction in the total sample size, and sub-
samples would be affected to an even greater extent, potentially compromising
the statistical analysis’s robustness. We have therefore chosen to keep the
original thresholds, leaving open the possibility of reducing them for a given
study.

Finally, on the 11th of December 2023, among the 5 557 known exoplanets, the
PlanetS catalog contains 715 transiting exoplanets. Figure 1 illustrates the
M-R diagram derived from our catalog, comparing it to the NASA Exoplanet
Archive database. The comparison is made using both the default parameter of
the NASA Exoplanet Archive and this default parameter but filtered with the
same uncertainty criteria as ours. This provides insight into the effects of
our selection process: first, our uncertainty thresholds, and then our
detailed study of references, enabling the removal of less reliable data
points. Exoplanets identified using the default parameter from the NASA
Exoplanet Archive often lack mass uncertainty information or provide only
upper limit masses, leading to a wide dispersion in the data. While the
uncertainty criterion of the PlanetS catalog can help address this problem,
challenges persist with masses determined through TTVs, where verifying
robustness is difficult. Additionally, planets analyzed spectroscopically may
exhibit underestimated uncertainties in their masses, especially when radial
velocity curves include only few data points. This underscores the importance
of our comprehensive study of references in addressing these issues.

Another important goal of the catalog modification was to offer the community
a complete catalog in terms of parameters. To accomplish this, we sourced the
primary orbital, planetary and stellar parameters from the NASA Exoplanet
Archive, with the reference paper selected. Any missing parameters were
supplemented by cross-checking with the study due to potential import issues
between the paper and the database. In addition, a cross-match between
catalogs was performed to obtain stellar parameters from Gaia
DR3444https://gea.esac.esa.int/archive/ (Gaia Collaboration et al. 2016; Gaia
Collaboration et al. 2023), as well as orbital obliquities from
TEPCat555https://www.astro.keele.ac.uk/jkt/tepcat/ (Southworth 2011). We
assigned each star’s spectral type using the table in Pecaut & Mamajek (2013),
taking the star’s effective temperature from the reference study.

We have also included classical calculations in the database - planet bulk
density, insolation flux, equilibrium temperature and expected radial velocity
semi-amplitude - in order to have these values for all planets in a more
homogeneous way (see appendix A).

In December 2023, the PlanetS catalog includes 715 transiting exoplanets along
with over 40 parameters and their respective uncertainties. This comprehensive
catalog is accessible through the Data & Analysis Center for Exoplanet
(DACE)666https://dace.unige.ch/.

###  2.2 Updated Mass-Radius relationships

The PlanetS catalog created by Otegi et al. (2020) aimed to investigate Mass-
Radius (M-R) relationships and the transitional boundaries between rocky and
volatile-rich planets. The present goal is to update these M-R relationships
with the updated PlanetS catalog that includes a broader range of masses and
new discoveries.

Numerous studies propose M-R relationships for the observed planetary
population, each with unique characteristics in terms of sample selection,
laws employed, or methods utilized to identify transitions between different
planetary populations. For instance, Chen & Kipping (2017) suggest a mass-
based separation for Super-Earths and Sub-Neptunes at 2 M⊕, while Weiss &
Marcy (2014) propose a radius-based transition at 1.5 R⊕. Wolfgang et al.
(2016) provide a best fit “average” relation for the sample of RV-measured
transiting Sub-Neptunes (R $<$ 4 R⊕), and quantify the astrophysical scatter
of this relation. Otegi et al. (2020) modified the perspective on these
relationships by introducing a discontinuity and segregating rocky planets
from more volatile-rich ones using the equation of state for pure water. This
introduced a physically grounded criterion for distinguishing between the two
populations. However, with the updated catalog, the discovery of planets
spanning the gap between more massive rocky planets and less massive volatile-
rich ones has made the transition between these populations less clear. It is
plausible that the transition between rocky and volatile-rich populations
depends on other parameters, such as insolation or stellar characteristics.
Still, recently, Edmondson et al. (2023) provided evidence supporting the
consistency and validity of using the pure water composition line for this
transition. Nevertheless, this composition line lacks physical validity, as a
planet with such a composition would be unable to form and sustain its
existence. In a different way, Kanodia et al. (2019) and Parviainen et al.
(2024) present Python modules that can be used to fit a M-R relationship
without assuming an underlying power law. Kanodia et al. (2019), however, do
not consider any separation between the different populations, whereas
Parviainen et al. (2024) use composition lines to represent 3 populations,
”rocky planets”, ”water worlds” and ”Sub-Neptunes”, constituting the three-
component mixture model.

Regarding the transition to giant planets, Weiss et al. (2013) identified a
transition at 150 M⊕ based on visual inspection of mass-radius and mass-
density diagrams. Hatzes & Rauer (2015) conducted a similar analysis,
examining the change in slope in mass-density relationships, and identified
the transition at 95 M⊕. Chen & Kipping (2017) utilized a detailed forecasting
model employing a probabilistic M-R relation, placing the transition at
130$\pm$22 M⊕. In contrast, Bashi et al. (2017) identified two empirical
regimes on the M-R relation with a transition at 124$\pm$7 M⊕ and 12.1$\pm$0.5
R⊕. Edmondson et al. (2023) employed piecewise regression and located the
transition at 115 M⊕. However, when considering the equilibrium temperature of
the gas giants, it was found that, for giant planets, the radius only depends
on the temperature. Finally, Müller et al. (2023) have leveraged the updated
PlanetS catalog to investigate M-R relationships in a purely statistical
manner, finding the transitions to be at 7.80 M⊕ and 125 M⊕ to separate the
three populations.

In this study, we decide to use the 20% water line at 650 K from Luo et al.
(2024) to distinguish between rocky planets and those with higher volatility.
This model is described in more details in section 3.3 and allows us to
separate the two groups in a way that is consistent with what we observe
visually. Finally, this 20% is consistent from a formation point of view, with
the water fraction corresponding to a body that has to have formed beyond the
ice line of the planetary disk. We also limit ourselves to planets below 10 M⊕
for the rocky population as can be seen in the observations (see next sections
and Fig. 2). For the transition between intermediate-mass and giant planets,
we employ the piecewise regression Python package from
DataDog777https://github.com/DataDog/piecewise, which applies piecewise
regression with automated breakpoints detection, coupled with a MCMC to take
into account the uncertainties of the mass and radius measurements. We find
the transition at 138$\pm^{+21}_{-42}$ M⊕. This is in line with the values
found in the literature, and is consistent with planet formation theory
(Helled, 2023). Indeed, Helled (2023) suggested that the transition to gas
giant planets occurs at about Saturn’s mass, with an uncertainty of $\sim 20$
M⊕, this means that above this mass, the planetary composition is expected to
be H-He dominated. Below this mass, however, the composition is expected to be
very diverse, and while some H-He is expected, the ratio between H-He and
heavy elements can vary significantly, depending on the exact formation
conditions. As a result, for intermediate-mass planets there should not be a
distinct density/radius as in the case of small and giant planets. Instead,
for this planetary type the planetary mass is the physical property that is
more natural to use when separating the different populations.

We fit the three populations using a Orthogonal Distance Regression (ODR)
method, in which observational errors on both dependent and independent
variables are considered, and the results are the following:

$R=\begin{cases}(1.02\pm 0.01)M^{(0.28\pm 0.01)}&,\text{ if
}\rho>\rho_{20\%\mathrm{H}_{2}\mathrm{O}}\text{ and }M<10M_{\oplus}\\\
(0.61\pm 0.04)M^{(0.67\pm 0.02)}&,\text{ if
}\rho<\rho_{20\%\mathrm{H}_{2}\mathrm{O}}\text{ and }M<138M_{\oplus}\\\
(11.9\pm 0.7)M^{(0.01\pm 0.01)}&,\text{ if }M>138M_{\oplus}\end{cases}$

or

$M=\begin{cases}(0.98\pm 0.01)R^{(3.57\pm 0.13)}&,\text{ if
}\rho>\rho_{20\%\mathrm{H}_{2}\mathrm{O}}\text{ and }M<10M_{\oplus}\\\
(1.64\pm 0.11)R^{(1.49\pm 0.04)}&,\text{ if
}\rho<\rho_{20\%\mathrm{H}_{2}\mathrm{O}}\text{ and }M<138M_{\oplus}\\\
(0.08\pm 0.01)R^{(100\pm 100)}&,\text{ if }M>138M_{\oplus}\end{cases}$

Despite the change in separation between the two groups, our two relations for
small and intermediate-mass planets align with Otegi et al. (2020). The
increased number of planets in the ”rocky” category suggests a slight
elevation in the exponent of the power law, but this remains within the error
bars of the equation from Otegi et al. (2020). They also exhibit slight
deviations from Edmondson et al. (2023)’s relations. This discrepancy can be
attributed to differences in object selection between our two samples and the
different separation criteria. In the gas giant regime, our relation is
relatively close to the one presented in Bashi et al. (2017). The observed
population in this regime displays significant dispersion, with a notable
dependence on incoming irradiation. Planets subjected to higher irradiation
tend to exhibit hotter and more expanded atmospheres, contributing to the
observed variability (Batygin et al., 2011; Sestovic et al., 2018; Thorngren &
Fortney, 2018).

However, it’s worth remembering that no physical interpretations about
composition, internal structure or formation should be drawn from these M-R
relationships and purely arbitrary category separations. For example, the
composition line of 20% water depends on the planet’s temperature (see Sect.
3.3) and so a more precise examination of all conditions must be made in order
to assert the rocky or more volatile nature of an individual planet. A good
use of these relationships is to estimate the mass/semi-amplitude of the
radial velocity signal during radial velocity follow-up campaigns of planets
detected by TESS or other photometric spacecrafts/instruments. For this
purpose, we recommend employing the relationship for rocky planets up to 1.75
R⊕. From 1.75 to 2.1 R⊕, it’s advisable to use both the rocky and intermediate
relationships. Beyond that, the intermediate relationship serves as a good
estimate. These values are derived from the minimum and maximum radius values
of the two currently observed populations, with an added safety margin. It’s
important to note that the dispersion of the observed population remains
extensive around these relations, indicating that the estimated mass may vary
significantly for a given radius. Using the two relations ”rocky” and
”intermediate” with their uncertainties, for R = 1.75 R⊕, the mass varies from
3.45 to 7.85 M⊕ and for R = 2.1 R⊕, it varies from 4.49 to 14.31 M⊕.

Figure 2: M-R diagram of small planets around M-dwarfs, K-dwarfs and FG-dwarfs
from the PlanetS catalog. The planets are color-coded by their equilibrium
temperature calculate within the catalog with a bond albedo of 0 and a total
heat redistribution. The composition lines of pure-silicates (brown) from Zeng
et al. (2016), Earth-like planets (yellow), and 50% water (blue) from Zeng et
al. (2019) are displayed. The vertical dotted lines correspond to the minimum
mass of the Sub-Neptunes across spectral types (at 1.9, 3.4 and 4.3 M⊕ for M-,
K- and FG-dwarfs) discussed in Sect. 4.1 and 4.4.

##  3 M-R diagram of small exoplanets around M-dwarfs

###  3.1 Sample selection

We use our robust and comprehensive PlanetS catalog to select the sample of
small planets around M-dwarfs. We apply selection criteria similar to those of
Luque & Pallé (2022) (hereafter LP22) in order to compare our results. We
choose planets with a radius of less than 4 R⊕, orbiting stars with an
effective temperature of less than 4000 K. We obtain 46 planets with precise
mass and radius measurements orbiting stars with an effective temperature
ranging from 2566 to 3997 K corresponding to stellar masses from 0.09
(TRAPPIST-1) to 0.63 M⊙ (TOI-1235). This sample contains planets all robustly
confirmed with photometric and spectroscopic instruments, including new
discoveries up to December 2023. By way of comparison, the sample of LP22
study contains 34 planets. A complete comparison between the two samples are
detailed in the appendix B.

The resulting M-R diagram for our sample is shown in the first panel of Fig.
2, with the planets colored by their equilibrium temperature computed in the
PlanetS catalog (appendix A.2, using an albedo of 0). For comparison the two
theoretical composition lines used by LP22 (Earth-like composition and 50%
water/50% silicates from Zeng et al. (2019)) are drawn in Fig. 2, but are
discussed in more details in section 3.3. We also draw the pure-silicates
composition line (Zeng et al., 2016) to represent the lowest density limit of
a rocky planet.

A first examination of this M-R diagram shows us that there are exoplanets
lying between the two composition lines of Earth-like and 50% water shown in
the figure. These are both new planets characterized after LP22’s paper
catalog (July 2021), but also some of the exoplanets whose properties they
have re-analyzed (see section B of the appendix). The substantial 30% increase
in the number of planets within the sample has considerable implications for
the conclusions drawn regarding planetary populations. It clearly demonstrates
the importance of acquiring more data in a parameter space that is inherently
incomplete and biased, such as the M-R diagram.

A subset of exoplanets closely follows the Earth’s composition line, featuring
masses ranging from TRAPPIST-1 h (0.33 M⊕) to 10 M⊕ and associated radii of
approximately 0.8-1.8 R⊕. Within this cluster, a prominent group of planets,
known as Super-Earths, exhibits masses around 2 M⊕ and radii around 1.2 R⊕.
Following this, there are planets with larger radii (1.5-2.8 R⊕) and masses
between 2 and 10 M⊕, identified as Sub-Neptunes positioned above the pure
silicate line. These Sub-Neptunes are presumed to contain more volatile
elements to account for their observed density. Subsequently, there are only
two planets with radii around 3.5 R⊕ (K2-25 b : Rp = 3.43 $\pm$ 0.12 R⊕
(Stefansson et al., 2020) ; and TOI-1231 b : Rp =
3.43${}^{\text{+0.16}}_{\text{-0.15}}$ R⊕ (Burt et al., 2021)) and very few
objects surpassing 2.5 R⊕. Another noteworthy feature in this diagram is the
ultra-dense sub-Earth GJ 367 b with a mass of 0.633 M⊕ and a radius of 0.699
R⊕ (Goffo et al., 2023). This planet boasts an exceptionally short orbital
period (P = 0.32 days) and experiences intense irradiation (T${}_{\text{eq}}$
= 1365 K). The phase curve from JWST revealed that it appears to be a hot,
dark barren rock (Zhang et al., 2024). Positioned just below a theoretical
100% Iron composition line, GJ 367 b stands out as the densest planet in our
sample.

###  3.2 Density and radius distributions

####  3.2.1 Histograms

To evaluate the probability distribution of a given variable, the first idea
that naturally comes to mind is to use a histogram - it’s well-known, simple
to understand and works reasonably well. But histograms have clear
limitations: the positioning and quantity of bins are arbitrarily determined
and the estimated distribution lacks smoothness, whereas the true distribution
has a continuous, smooth nature (Boels et al. 2019). Moreover, as they stand,
they do not take into account the uncertainties of the chosen variable. The
limits are even greater when a histogram is applied to a very small sample
(see appendix C). So the first objective is to consider the overall sample to
plot the histogram.

In Figure 3, we plot a comparison of the histograms of the normalized density
of the LP22 sample and our filtered PlanetS catalog (section 3.1). We
normalize the planetary bulk density by the Earth-like composition: i.e. we
divide by the density that a planet of the same mass would have with
terrestrial composition (32.5% iron and 67.5% silicates, represented as
$\rho_{\oplus\text{,s}}$ in this study). To make the figure easier to read,
we’ve separated the colors of the two samples inside the bins. We draw the
line at 0.65 $\rho_{\oplus\text{,s}}$ found to be the gap in composition by
LP22.

As already seen in the M-R diagram (Fig. 2), some planets populate
intermediate compositions in a continuous way between rocky and volatile-rich,
making the existence of the 0.65 $\rho_{\oplus\text{,s}}$ gap between the two
populations less obvious. This conclusion can also be drawn from the
histograms of the two samples compared in figure 3. Indeed, by taking the
entire population of LP22 planets instead of dividing them into the three
categories, the gap becomes very faint, whereas for our sample it is filled by
several planets. This implies that there is a continuous transition between
the two population. While this representation is biased by the choice of the
number of bins, the histogram shows two main components: one centered around
0.9 $\rho_{\oplus\text{,s}}$ and the other around 0.4
$\rho_{\oplus\text{,s}}$, with a similar width of about 0.4
$\rho_{\oplus\text{,s}}$. Once again, as noted in the previous section, we
find the very dense outlier GJ 367 b.

Figure 3: Histogram of the normalized density distribution of small planets
orbiting M-dwarfs. In blue, for the LP22 sample. In pink, for this study. For
reasons of clarity, the bins are separated in two with the two colors
representative of the two samples. A dotted line is drawn at 0.65
$\rho_{\oplus\text{,s}}$ to represent the compositional gap suggested by LP22.
Figure 4: Kernel Density Estimates of the density (left) and radius (right)
distributions of the small planets orbiting M-dwarfs in the PlanetS catalog.
The 10,000 KDEs realizations are plotted in the background (light red), along
with the resulting mean KDE (dark red solid line) and its standard deviation
(dark red filled area). The black dashed lines are the two Gaussians that
provide the best representation of each distribution (fitted to the mean KDE).

####  3.2.2 Kernel Density Estimation

In this study, we go beyond the simple histogram and look at the Kernel
Density Estimation (KDE) of our sample. A KDE can be seen as an extension of
the histogram. Like the latter, its aim is to estimate an unknown probability
density function from a sample of data. However, so far, the uncertainties are
not taken into account in the construction of the histogram. Despite our
selections on mass and radius, our sample has a median relative error in
density of 16% and can go up to 31%, which can shift a planet from one
composition to another but also change the appearance of the total
distribution. To take these uncertainties into account, we perform a re-
sampling technique: a large number of new data sets (N = 10 000 samples) are
created, which lie within the normal distribution of the density measurements
with the maximum uncertainty taken between the upper and lower uncertainty as
standard deviation. For each of these new data sets, the KDE is calculated. We
then calculate the mean and standard deviation of all the KDEs obtained.

The choice of kernel and bandwidth are the two adjustable KDE parameters, and
it is widely accepted in the literature that the choice of bandwidth is more
important than the choice of kernel (Silverman, 1981; Wand & Jones, 1995;
Silverman, 2018). A larger bandwidth makes the estimate smoother, while a
smaller bandwidth captures more detail. The KDE is calculated using the
statsmodels Python package888https://github.com/statsmodels/statsmodels/
(Seabold & Perktold 2010) that allows for continuous data and for an automatic
search of the optimal bandwidth. This optimal bandwidth, crucial for capturing
data distribution accurately, is automatically determined through a
combination of cross-validation and maximum likelihood operator (Heidenreich
et al. 2013). Cross-validation assesses KDE’s performance by partitioning data
into validation sets, while the maximum likelihood operator identifies the
bandwidth that maximizes the likelihood of observing the data. This integrated
approach ensures an effective selection of bandwidth, enhancing the fidelity
of density estimation. However, as this approach is computationally expensive,
we search for the optimum bandwidth for 1,000 samples randomly selected from
the 10,000 and use the resulting median bandwidth for all our resampled data.
A test with 1 000 vs. 10 000 samples shows that the median bandwidth agrees in
both cases.

We apply this method to the distribution of bulk density of the planets in our
sample but also to the distribution in radius. Figure 4 shows the result: we
have plotted the 10,000 realizations of the KDE calculation for the density
and radius distribution of our small planets orbiting M-dwarfs, and we have
indicated the mean KDE as well as the standard deviation of all KDEs.

Firstly, concerning the density distribution, again normalized here by the
Earth’s composition as in the histogram, we can see the effect of measurement
uncertainty allowing a wide dispersion in the possible shapes of the
distribution. A bimodality in composition seems to appear when looking at this
distribution: we fit to the mean KDE two Gaussians to model this bimodality
and find a population centered around 0.35 $\rho_{\oplus\text{,s}}$with a
sigma of 0.17 and another centered around 0.91 $\rho_{\oplus\text{,s}}$with a
sigma of 0.23 (dashed black lines Fig. 4). However, the two distributions
overlap considerably between 0.25 and 0.75 $\rho_{\oplus\text{,s}}$.

Looking at the right tail of the distribution, we see that there are very few
planets denser than Earth’s composition. Consequently, there is no clear
evidence of the existence of a population of Super-Mercuries in this sample.
For planets with a radius and mass greater than Earth, there is no bias in
detecting such objects below the Earth’s composition line. On the
distribution, they should lie between 1.3 and 1.45 in normalized density.
Concerning the left tail of the distribution, we’re biased by our cut-off at 4
R⊕, but we also suffer from the lack of planets with a radius $\geq$ 2.5 R⊕,
which we should be detecting and which should have a normalized density
between 0.15 and 0.5 $\rho_{\oplus\text{,s}}$. This lack of Sub-Neptunes
relative to the maximum of our distribution, which is around the Earth’s
compositional density, is consistent with occurrence rate studies of
transiting planets around M-dwarfs showing that Sub-Neptunes are less abundant
than Super-Earths around especially mid-to-late type M-dwarfs (Ment &
Charbonneau, 2023). However, the maximum of the distribution is highly biased
by the incompleteness of the sample. The TRAPPIST-1 system and the few planets
with mass less than 1 M⊕ bias this maximum around the terrestrial composition.
Indeed, due to our detection limits in radial velocity and in transit
(preventing us from going to longer periods), this part of the parameter space
is highly incomplete. The resulting density and radius distributions of the
planets around M-dwarfs without TRAPPIST-1 and the few planets with mass less
than 1 M⊕ can be found as a dashed red line in Fig. 7.

Secondly, by examining the radius distribution produced, we can see that there
is a maximum around 1.3 R⊕. This is again in agreement with Ment & Charbonneau
(2023) that Super-Earths appear to be more abundant around M-dwarfs than Sub-
Neptunes. Examining the left-hand tail of the KDE, we are constrained by the
limits of mass detection before those in radius. Indeed, we have 30 sub-Earth
size planets around M-dwarfs validated by TESS and Kepler/K2 (from NASA
Exoplanet Archive). We are therefore not biased towards the right-hand side of
the distribution. The abrupt decrease in the distribution around 2.5 R⊕, or
”radius cliff”, is a feature not biased by detection limits. The latter is
still largely unexplored, and studies such as Kite et al. (2019) try to
explain it by equilibrium processes between the atmosphere and a magmatic
surface. It is also worth noting the bias regarding Earth-size planets around
M-dwarfs detected by TESS. In fact, it appears that given current sensitivity
limits, TESS may miss dozens of planets in transit around nearby late
M-dwarfs, and that the planet occurrence rate is unlikely to increase and may
even decrease for the latest M-dwarfs (Brady & Bean, 2022; Dietrich et al.,
2023).

An important characteristic of this distribution is that it does not
distinctly show the radius gap of Kepler planets between Super-Earths and Sub-
Neptunes: the radius valley, or Fulton gap (Fulton et al. 2017). Here, the
radius valley fades, but there is still a break in the slope of the
distribution around its value of 1.5-2 R⊕, a small plateau before another
decrease in the distribution. This feature could then represent the transition
between two populations of planets. Considering the median of the stellar
masses of our sample ($\sim$0.33 M⊙), one would anticipate the radius valley
to be more around 1.3-1.5 R⊕ (Fulton & Petigura, 2018; Berger et al., 2020; Ho
& Van Eylen, 2023). Interestingly, in our analysis, this range aligns more
with the maximum of our distribution. We can once again fit two Gaussians to
the mean KDE to model this bimodality and we find a population centered around
1.21 R⊕ with a sigma of 0.30 and another centered around 2.07 R⊕ with a sigma
of 0.44 (dashed black lines Fig. 4). Here again, the two distributions overlap
($\sim$1.2-2 R⊕). Another small bump in the distribution around 3.5 R⊕ is due
to the 2 planets around this value and reflects the lack of planets in this
region.

Finally, the KDE allowed us to explore the properties of density and radius
distributions, while not preventing us from analyzing the biases and
incompleteness of our sample. While the KDE can enhance the portrayal of a
distribution by considering its continuous characteristics, as the histogram,
its effectiveness is constrained by the sample size of planets actually
observed (see appendix C). This limitation becomes even more pronounced when
we consider the subtleties of planet formation and the evolutionary mechanisms
that give rise to the real and complex distribution we seek to derive,
compounded by observational biases that result in an incomplete sample. Only
by taking these aspects into account can we now discuss the implications for
the potential internal structure and conditions of formation and evolution of
these planets.

###  3.3 Impact on internal structure and composition

Figure 5: Numerical radius-density probability model applied to small planets
orbiting M-, K- and FG-dwarfs from the PlanetS catalog (Top, middle and bottom
panel, respectively) using spright and the water-rich planet density model
proposed by Aguichine et al. (2021). Grey data points represent measured
values with associated uncertainties for planets in our sample. The blue map
depicts the logarithm of the posterior probability, and the black lines
indicate posterior means for each of the three planet populations (”rocky”,
”water-worlds” and ”Sub-Neptunes”). Solid lines denote radius regimes where
the component has a weight of unity, meaning all planets in this range belong
to this specific component, while dashed lines mark transition regimes between
populations. This clearly shows that the ”water-worlds” population is only
considered as a transition between the other two populations, and not as a
distinct population. Figure 6: Mass-Radius diagrams of small planets orbiting
M-dwarfs, focusing on those larger and more massive than Earth, up to 3 R⊕.
The three rows correspond to three equilibrium temperature intervals, with
planets with T${}_{\text{eq}}\leq$ 450 K at the top, 450 K ¡ T${}_{\text{eq}}$
¡ 700 K in the middle, and at the bottom T${}_{\text{eq}}\geq$ 700 K. The two
columns correspond to the two internal structure models, from Aguichine et al.
(2021) (A21) on the left and from Luo et al. (2024) (LDD24) on the right. The
composition lines are drawn from these models at the average temperatures of
planets in the intervals, 400 K, 550 K and 800 K for cold, warm and hot
respectively. Each panels feature Zeng et al. (2019)’s terrestrial and 50%
water composition lines represented by black dash-dotted lines and dotted
respectively.

The exact positioning of a planet within the M-R diagram, and thus its bulk
density, prompts a deeper exploration of its internal composition and
structure in terms of the interplay between rocky and volatile elements. At
any specific point within the diagram, numerous potential compositions for the
planet come into consideration. Additionally, a crucial factor to consider is
the irradiation received by the planet from its host star, which we quantify
by the equilibrium temperature. This temperature estimation provides insights
into the planet’s thermal state and allows us to tailor our understanding of
the pressure/temperature profiles of the planet’s constituent elements. Hence,
it transcends the traditional two-dimensional M-R problem and introduces
equilibrium temperature as a pivotal parameter. Consequently, there emerges a
unique combination of mass, radius, and equilibrium temperature to juxtapose
against the composition lines that can be delineated on the M-R diagram.

It is at this juncture that LP22’s study exhibits a shortcoming: when
categorizing ”rocky-worlds” from ”water-worlds”, while the rocky planets seem
to have a terrestrial composition (63% silicates, 27% iron), they rely on a
single composition line for the water-rich planets (comprising 50% water and
50% silicates). However, in this model, the water is in condensed form, which
is not justified for equilibrium temperature above 400 K. Out of the 6 planets
on LP22’s water line, only one has an equilibrium temperature below this
threshold: TOI-270 d. This exoplanet is no longer even situated on this
composition line in our sample (see appendix B). Another limitation that could
be seen in this composition is the absence of an iron core in the planet’s
structure.

Moreover, as highlighted earlier, the determination of a planet’s internal
composition and structure based solely on the Mp/Rp/T${}_{\text{eq}}$
parameters is profoundly challenging due to the high degree of degeneracy.
Consequently, it is not feasible to definitively pinpoint a single composition
and structure for the planet using this trio of parameters alone. LP22 also
explores alternative compositions, such as terrestrial or aqueous compositions
with and without an atmospheric layer of H/He using compositions from Zeng et
al. (2019). They conclude from the observations that a slight change in mass
fraction of H/He envelopes or in their temperatures would lead to substantial
discrepancies in the planets’ radii and thus that these aligned planets are
more likely to be composed primarily of water rather than being predominantly
gas-rich in their composition. Furthermore, it’s noteworthy that the
temperature of these lines actually corresponds to the corresponding entropy
temperature at 100 bar, not the equilibrium temperature. Zeng et al. (2019)
assume a uniform specific entropy across exoplanets of different masses and
this assumption is not justified particularly in case of exoplanets that have
undergone thermodynamic processes such as cooling and mass loss as well as
giant impact. These processes naturally lead to a decrease in a planet’s
specific entropy, a phenomenon influenced by multiple factors, including
planetary mass and equilibrium temperature.

They also conducted a comparative analysis with the model proposed by Turbet
et al. (2020), which accounts for runaway greenhouse radius inflation by the
potential presence of a hydrosphere expanded due to the intense irradiation
experienced by these planets. Nevertheless, it is essential to note that the
data presented in this study primarily encompass terrestrial-like planets with
a mass ranging from 0 to 2 Earth masses, leading to substantial extrapolation
in LP22.

In this study, we want to highlight other models that could explain the
composition and internal structure of these planets. Rogers et al. (2023)
shows that these ”water-worlds” could well correspond to rocky planets with an
H/He atmosphere. The comparison between their models and the sample of planets
from LP22 can be found in Figure 1 of their study. The differences in our
sample are consistent with this comparison. However, they use a pure
evolutionary model (mass-loss, cooling) and theoretical initial quantities of
H/He not taken from global formation models (that include core formation,
planet migration, collisions…).

Mousis et al. (2020), Aguichine et al. (2021), Vivien et al. (2022) and
Pierrehumbert (2023) extend the work of Turbet et al. (2020) to more massive
planets by considering the runaway greenhouse effect of water-rich planets,
considering in particular a layer of water in supercritical form in the
planetary structure. These models allow for a much larger planetary radius
with a smaller amount of water than Zeng et al. (2019) model. This is because
the latter considers an atmosphere with an isothermal profile (and a condensed
form for water), whereas the others consider a radiative-convective profile.
The recent study by Selsis et al. (2023) shows that the truth lies somewhere
in between, isothermal near the surface and convective in the upper
atmosphere.

More recently, Parviainen et al. (2024), employing Bayesian model comparison,
uses an updated sample from LP22 and shows evidence against the presence of a
”water-world” population around M-dwarfs when considering the mass-density-
radius relationships for small planets derived from the ”water-worlds” models
of Zeng et al. (2019) and Aguichine et al. (2021). Consequently, our study
aligns with Parviainen et al. (2024) findings, supporting the notion that
”water-worlds” do not constitute a distinct and discernible population.
Indeed, we use the python module spright presented in Parviainen et al. (2024)
to fit our sample and obtain the water world population strength $\omega$
consistent with 0. The result is shown Fig. 5.

However, all these models are based on the assumption that water is solely
present on the surface. Recently, following Dorn & Lichtenberg (2021), Luo et
al. (2024) revisited this assumption by allowing water to be part of the
planet’s inner layers: mantle and core. This is following recent studies for
the Earth, that indicate that the largest portion of Earth’s water may be
stored deep within its core (although not in molecular form). This discovery
has substantially expanded the estimated total water inventory to 1.7% by
weight, signifying a considerable increase compared to previous assessments
(Li et al., 2020; Tagawa et al., 2021). New ab initio calculations from Luo et
al. (2024) now show that the majority of a planet’s water budget (even more
than 95%) can be stored deep in the iron cores and the rocky mantles, not at
their surfaces. As a consequence, the limited amount of steam present at the
surface results in significantly reduced planetary radii for a given mass.
This revelation underscores the critical role of internal water distribution
in shaping planetary characteristics and prompts a reevaluation of our
assumptions regarding the interplay between a planet’s composition and its
observable features.

In Figure 6, we have depicted the composition lines from the models of
Aguichine et al. (2021) (left-column) and Luo et al. (2024) (right-column) in
conjunction with our small planets around M-dwarfs and the Earth-like and 50%
water composition lines from Zeng et al. (2019). In comparison with Fig. 2,
we’ve zoomed in on the area of interest in the ”water-worlds” discussion, so
we are ignoring planets larger than 3 R⊕ and smaller than Earth. This
representation allows us to estimate the composition bounds of the M-dwarf
planets based on these theoretical models. To enhance the clarity of this
comparison, we have categorized and color-coded the equilibrium temperature of
the planets into distinct intervals: $\leq$ 450 K for the top panels, [450 K,
700 K] for the middle panels and $\geq$ 700 K for the bottom panels. The
contrast in shapes between Aguichine and Luo’s models, which incorporate steam
water and solubility, and Zeng’s model, which assumes condensed water, is
immediately noticeable.

On one hand, Aguichine’s models offer insights into the potential composition
of planets with larger radii, expanded by the presence of water in vapor form
in the atmosphere and the greenhouse effect induced by the planets’
temperatures. With these models, it becomes apparent that planets
corresponding to the 50% water line of Zeng’s model (and the water-world
population of LP22) can be interpreted as having a composition of
approximately 10% water in the atmosphere, with a terrestrial composition for
the core. These planets are associated with equilibrium temperature ranges of
300 to 700 K. As we move from these 10% composition lines toward terrestrial
composition, the planets are slightly more irradiated, resulting in
substantially lower quantities of water. On the contrary, larger planets may
contain more water - 20, 40 or even 50% - in their atmosphere.

Conversely, Luo et al. (2024) model excels in depicting the diversity of
mass/radius/temperature combinations between Zeng’s water and terrestrial
compositions. The radius in this model is less influenced by the irradiation
due to the presence of most of the water in the planet’s mantle and core. For
the Sub-Neptunes, even the most water-rich scenario with 50% cannot represent
those planets with radii above 2.2 R⊕, as these planets require some H/He in
their atmosphere to explain their observed radius. Also, LP22’s water-world
population would have variable water mass fractions between 10% and 50% with
water distributed between the surface, mantle and core for equilibrium
temperatures between 350 and 850 K.

Finally, Aguichine’s model, while incorporating irradiation effects, has
limitations stemming from the lack of interaction between the atmospheric
layers and the planet’s interior. Notably, it does not consider solubility and
solely focuses on surface water in form of the steam atmosphere. The model by
Luo et al. (2024) seems more coherent in this respect. But both models are
static in time, and don’t take into account the cooling and possible
compression of the planet as it evolves. In this context, all the Sub-Neptunes
around M-dwarfs cannot be solely explained by the presence of water in their
atmosphere and core; for some of them, an atmosphere containing H/He is needed
to account for their observed bulk density.

In conclusion, despite being expected to represent the same population of
planets with identical composition, these two models yield markedly different
outcomes in terms of mass or final radius. Despite the inherent degeneracy in
this parameter space, achieving greater consensus among theories of internal
structure, formation, and evolution is imperative to accurately interpret
observables and predict the likely atmospheric compositions to be observed.

In the end, discriminating between these diverse planetary models proves
challenging when relying solely on observable parameters such as mass, radius,
and equilibrium temperature. In addition, this latter is contingent upon
numerous assumptions about the atmospheric and surface characteristics of the
planets in question. A comprehensive understanding of the atmospheric
compositions becomes imperative for achieving a meaningful distinction among
these varied compositions.

###  3.4 M-R trends in the context of formation and evolution models

Examining the M-R diagram alongside density and radius distributions offers
valuable demographic insights into the formation and evolution of planets.
Conversely, observational data serves as a crucial constraint, refining and
enhancing the precision of these theoretical frameworks.

As in LP22, we find that the radius valley fades for M-dwarfs (Fig.2), i.e, no
distinct value in planet radius separates planets crossing the Earth-like
composition from the 50% water line. Venturini et al. (2024) expand the pebble
accretion model for Sun-like stars of Venturini et al. (2020) to a range of
stellar masses spanning 0.1 to 1.5 $M_{\oplus}$. They predict a filling of the
radius valley for M-dwarfs, in line with our findings. The reason behind the
fading radius valley is type-I planetary migration, which happens for lower
planetary masses when the stellar mass diminishes (Paardekooper et al., 2010).
Consequently, smaller icy planets (formed beyond the iceline) reach the inner
planetary system compared to the Sun-like case, producing a larger overlap in
mass and radius with the rocky planets formed within the iceline. This effect
of planet migration acting on smaller planets for M-dwarfs was reported as
well in Burn et al. (2021). Ho et al. (2024) also identifies a shallower
radius valley around lower-mass stars when revisiting planetary parameters
from Kepler 1-minute short cadence light curves. They conclude that
photoevaporation models alone are insufficient to explain this phenomenon.
Instead, they propose three alternative scenarios: variations in the stars’
XUV output, dispersion in planetary core composition, and collision events.

LP22 explicates the dual compositional nature of their findings as a
consequence of planetary formation through pebble accretion (Venturini et al.,
2020). Indeed, the formation and evolution simulations of Venturini et al.
(2020) for Sun-like stars revealed that the formed planets exhibit a stark
bimodal distribution in Water Mass Fraction (WMF), manifesting typically as 0
or 0.5, with very few planets with intermediate values. However, for
M-planets, Venturini et al. (2024) show that the number of planets with
intermediate compositions (in between pure rock and 50%water), increases
compared to planets formed around Sun-like stars. The bimodality in water mass
fractions still persist, but the gap occurs at 0 ¡ WMF ¡ 20%. When considering
that most of the real exoplanets are hot enough to host steam atmospheres, the
radius of planets with $\sim 20\%$ WMF is similar to the ones having condensed
water at 50% by mass as shown by LP22. Overall, in the mentioned pebble-based
model, two groups of planets in M-R can be easily identified due to the
mentioned dichotomy in composition. This is not really observed in our PlanetS
Catalog for M-planets. On the other side of the theoretical possibilities,
planetesimal-based models yield more mixed, intermediate compositions compared
to pebble-based models (Burn et al., 2021; Brügger et al., 2020). Thus, our
results could give support to a planetesimal or hybrid pebble-planetesimal
(Alibert et al., 2018; Guilera et al., 2020) accretion scenario.
Alternatively, when N-body interactions are taken into account among hundreds
of planetary embryos growing in the same disc, giant impacts can yield as well
more intermediate compositions that blurs the radius and density valley (e.g
Emsenhuber et al., 2021)

##  4 M-R diagram of small exoplanets around FGK-dwarfs

###  4.1 Sample selection

Similar to our approach with M-dwarfs, we are leveraging the PlanetS catalog
to broaden our investigation to FGK-dwarfs for comparative analysis. To
categorize planets based on their spectral type, we focus on stars with
T${}_{\text{eff}}\geq$ 4000 K, effectively excluding the previously examined
M-dwarfs. Subsequently, we utilize the ”Spectral Type” column in the catalog,
determined through the table from Pecaut & Mamajek (2013), to specifically
target K, G, and F-dwarfs for our study. Similarly, we select planets with a
radius of less than 4 R⊕.

In the final dataset, we have a total of 61 planets orbiting K-dwarfs, 63
planets orbiting G-dwarfs, and 9 planets orbiting late F-dwarfs. To conduct a
robust analysis, we combine the G- and F-dwarfs into a single sub-sample,
considering that the F-dwarfs alone are not numerous enough to support a
statistically reliable study. The sample sizes of planets around K- and
F/G-type stars are comparable to those around M-dwarfs, facilitating a
statistically fair comparison.

A test was carried out to reduce the relative uncertainties on mass and radius
in the PlanetS catalog from 25% to 15% on mass and from 8% to 5% on radius.
This reduced the samples to 26, 32 and 31 planets for M-, K- and FG-dwarfs
respectively (i.e. 40%, 48% and 57% respectively). In view of the results in
Appendix C, we chose to keep the ”full” samples, so as to be able to interpret
the resulting distributions more confidently.

The effective temperature range of our stars goes from 4065 to 5320 K for the
K-dwarfs and from 5325 to 6321 K for the FG-dwarfs. Our separation into
spectral types obviously induces a classification by host star mass, useful
for the discussions in the following sections. The median stellar masses are
0.33 M⊙, 0.8 M⊙ and 0.96 M⊙, for M-dwarfs, K-dwarfs and FG-dwarfs
respectively. We’d expect a higher stellar mass value for FG-dwarfs, but since
we’re focusing on the PlanetS catalog’s sample of planets of precise mass
(Sect 2.1), we’re biased towards the lower stellar masses where it’s easier to
detect and characterize a small planet.

The resulting M-R diagram for the two sub-samples is shown on the center and
left panels of the Fig. 2, with the planets colored by their equilibrium
temperature. Upon comparing our two new samples with the panel on the left,
which depicts small planets around M-dwarfs, a noticeable detection bias is
evident around FGKs. This bias is characterized by an increase in the minimum
mass and minimum radius of the well-characterized planets in our catalog.
Additionally, there is a notable prevalence of larger, more massive planets
around FGKs compared to M-dwarfs. Remarkably, for all three subsamples,
planets with an Earth-like composition seem to exhibit a consistent maximum
mass of 10 M⊕. Another noteworthy mass limit to consider in the comparison of
our three samples is the minimum mass of Sub-Neptunes. It appears that as
stellar mass increases, so does the minimum mass of these planets. We observe
values of around 1.9 M⊕, 3.4 M⊕ and 4.3 M⊕, for M-dwarfs, K-dwarfs, and FG-
dwarfs, respectively. These observations are major elements in the comparison
of our observed population with synthetic populations derived from theoretical
models of formation and evolution (see 4.4).

Figure 7: Kernel Density Estimates of the density (left) and radius (right)
distributions of the small planets orbiting M- (red), K- (orange) and FG-
dwarfs (yellow) from the PlanetS catalog. For visibility, only the resulting
mean KDE (solid line) of the 10,000 realizations and its standard deviation
(filled area) are plotted. The red dashed lines are the density/radius
distributions of the sample orbiting M-dwarfs without the TRAPPIST-1 system
and planets below 1 M⊕.

###  4.2 Comparing the density and radius distributions

We employ the same methodology as detailed in section 3.2.2 to derive the KDEs
of our two new planet samples around FG- and K-dwarfs. In Figure 7, the
outcomes of the KDE calculations are depicted, illustrating distinct density
and radius distributions for M, K, and FG-stars. For greater visibility, the
10,000 samples of each test are not shown, and only the mean and standard
deviation of the result are displayed.

Starting with the analysis of density distributions, a striking contrast
emerges between the distribution for planets orbiting M-dwarfs and those
around FG- and K-dwarfs. The distributions for FG- and K-dwarfs exhibit a
notable similarity, while the distribution for M-dwarfs is nearly inverted
concerning the occurrence of densities between rocky planets and those
composed of more volatile materials. Examining the three distributions around
0.75-1 in normalized density, a noticeable discrepancy is observed; although
they are considerably lower for FGKs, there exists a detection bias
disfavoring Super-Earths around FGKs compared to M-dwarfs. In addition, we
recall that the significant peak around Earth density is driven by the
TRAPPIST-1 planets whose uncertainties are very low compared to the rest of
our sample. Upon their removal, the density distribution assumes a shape more
akin to that of FGK-stars (see dashed line Fig. 7), revealing a relative
decrease in the number of rocky planets with an increase in stellar mass,
attributed to detection limits. In the same way as for M-dwarfs, there appears
to be no discernible population of Super-Mercuries around FGK stars, although
this is subject to the limits of transit detection as it involves small planet
sizes around bigger stars. Nevertheless, at lower densities (around 0.25-0.4
$\rho_{\oplus\text{,s}}$) corresponding to larger and more massive planets,
the impact of the detection limit diminishes. Consequently, a conspicuous
shortage of volatile planets around M-dwarfs becomes evident when compared to
their counterparts around FGK-stars.

Regarding the radius distributions, distinctions between M-dwarfs and FGK-
dwarfs are once more apparent. This contrast is primarily influenced by the
left tail of the distributions, reaffirming the detection bias favoring Super-
Earths toward the latest spectral types. Additionally, the less clear gap
between the two populations of Super-Earths and Sub-Neptunes in the radius
distribution around M-dwarfs contributes to this divergence. Indeed, our
analysis reveals a bimodal distribution between Super-Earths and Sub-Neptunes
for planets around FGK-type stars, consistently demonstrating a radius valley
around 1.8 Earth radii (Fulton et al., 2017; Van Eylen et al., 2018). However,
it’s important to note that the KDEs do not provide the granularity needed to
discern variations in this value based on changes in stellar mass (Fulton &
Petigura, 2018; Berger et al., 2020; Ho & Van Eylen, 2023). The plateau
observed in the M-dwarfs distribution corresponds to the Fulton gap apparent
in the FGK-stars distributions. Once more, the absence of Sub-Neptunes (with
radii between 2 and 4 R⊕) around M-dwarfs is clearly evident in the data. The
oscillations in the radius distribution of the FG-dwarfs on the right-hand
side are due to the bandwidth of the KDE, revealing both the radius valley and
the subtleties of the larger-radius distribution. Additionally, it prompts
inquiries regarding the evolution of the ”radius cliff” concerning spectral
type.

The augmented sample sizes of small planets around FGK-type stars provide a
more robust foundation for precise estimation of density and radius
distributions through KDEs. We were then able to carry out a comparative study
of the evolutionary trajectories of these distributions across spectral types.
This analysis is closely intertwined with considerations of detection limits
and sample incompleteness, offering valuable insights into the potential
implications for the internal structure, composition, formation, and evolution
of planets orbiting diverse stellar types as discussed in the following sub-
sections.

###  4.3 Internal structure and composition across spectral type

After an extensive discussion of various models representing different
compositions of small planets around M-dwarfs in section 3.3, we extend our
analysis to planets orbiting FGK-dwarfs. In the supplementary material of
LP22, an attempt is made to identify the same composition gap observed for
planets around M-dwarfs in the case of planets around FG- and K-dwarfs.
However, their sample size is considerably smaller compared to ours (82 vs.
133 planets). Nevertheless, akin to our findings, they also report scant
evidence of a clear gap between a population of ”rocky” planets and a
population of ”water-rich” planets. Notably, Figure 2 in this work vividly
illustrates the presence of planets falling between the two composition lines.
As in section 3.3, we use the spright module from Parviainen et al. (2024) to
fit our samples and find that it refutes the hypothesis of the existence of a
distinct population of ”water-worlds” around K- and FG-dwarfs (middle and
bottom panels of Fig.5).

Once more, this prompts a reconsideration of the validity of drawing a single
equilibrium temperature line, especially given the broader range of
temperatures exhibited by planets around FGK-stars, which are notably more
irradiated than those around M-dwarfs. The degeneracy between the
compositions, as discussed in section 3.3, persists unchanged for these two
new samples.

A noteworthy observation is the greater prevalence of Sub-Neptunes with larger
radii (R${}_{\text{p}}$¿2.5 R⊕) around FGK-stars compared to M-dwarfs, a trend
previously highlighted in the M-R diagrams and in the density and radius
distributions in the preceding sections. This trend could be attributed to the
intense XUV flux during the early stages of M-dwarf development, potentially
leading to the atmospheric escape of Sub-Neptunes. Indeed, the XUV luminosity
of M-dwarfs is enhanced by factor 10-50 relative to solar-type stars during
their prolonged premain-sequence evolution (Shkolnik & Barman, 2014; Luger &
Barnes, 2015; Ribas et al., 2016; Peacock et al., 2020) and an increase in
this XUV flux implies an increase in atmospheric escape (Lammer et al., 2003;
Vidal-Madjar et al., 2004). Furthermore, Kubyshkina & Vidotto (2021) modeled
the evolution of a wide range of sub-Neptune-like planets orbiting stars of
different masses and evolutionary histories and found that atmospheric escape
of planets with the same equilibrium temperature ranges occurs more
efficiently around lower mass stars. This might also suggest richer H/He
envelope compositions or atmospheres containing water, potentially inflated by
the heightened irradiation received by planets around FGK-type stars.

Despite evidence suggesting that M-dwarfs tend to form more rocky planets,
another noteworthy observation pertains to the density distinction of the Sub-
Neptunes around M-dwarfs and FGK-dwarfs. Specifically, when focusing on the
smaller Sub-Neptunes (1.8 R⊕ ¡ R${}_{\text{p}}$ ¡ 2.8 R⊕), indications emerge
suggesting that those orbiting FGK-stars demonstrate a marginally elevated
bulk density compared to those in orbit around M-dwarfs. The selection of this
interval was made to exclude planets on the terrestrial composition line with
a mass of 10 M⊕ and a corresponding radius of 1.79 ⊕. Additionally, it aimed
to include all planets around M-dwarfs while avoiding the lack of objects
beyond 2.8 R⊕. This difference can be seen on the M-R diagrams (Fig. 2) but
also by plotting the density distributions of these samples using the same
method as in Sect. 3.2.2. The mean KDEs and their corresponding uncertainties
are plotted in Fig. 8. These distributions do indeed show a shift towards the
lowest densities for planets around M-dwarfs compared with those around FGKs.
To test whether this difference in densities is statistically significant, we
utilize a Mann–Whitney U test (Wilcoxon, 1945; Mann & Whitney, 1947), which
studies whether the two data samples come from the same distribution. With a
p-value of 0.013, we find that the densities of small Sub-Neptunes orbiting M-
and FGK-dwarfs belong to different distributions with over 95% confidence.
This can be because these planets are ice-rich, and hence likely migrated
objects that accreted most of their solids beyond the ice line (Alibert &
Benz, 2017; Venturini et al., 2020; Burn et al., 2021).

Furthermore, by looking at the M-R diagrams, a notable increase in the
dispersion of rocky planets around the terrestrial composition line is evident
around FGK-dwarfs compared to M-dwarfs. This variability could be indicative
of greater structural diversity in terms of core-to-mantle ratios or may
reflect heightened uncertainties in the masses and radii of planets orbiting
stars of earlier spectral types.

Figure 8: Kernel Density Estimates of the density distributions of the small
Sub-Neptunes (1.8 R⊕ ¡ Rp ¡ 2.8 R⊕) orbiting M- (red) and FGK-dwarfs (yellow)
from the PlanetS catalog.

###  4.4 Formation and evolution: trends from M-dwarfs to FGK-types

Examining the formation and evolution of systems gains added complexity and
interest when we incorporate a new dimension into the comparison between
observations and theory: the mass of the host star. The variation in stellar
mass introduces distinctive effects and adjustments within the models, like
changes in the mass of the disk or in migration processes. Venturini et al.
(2024) extends the results of Venturini et al. (2020) by studying the
influence of stellar mass. Similarly, Burn et al. (2021) perform this exercise
for planetesimal accretion formation.

The following demographic trends are worth to understand in the context of
formation and evolution models:

  1. 1.

The minimum mass of planets falling near the 50% condensed water composition
seems to increase with stellar mass (see black dotted lines Fig.2).

  2. 2.

The maximum mass of rocky planets seems to be approx. 10 M⊕ for all spectral
types.

  3. 3.

The radius valley gets filled when the stellar mass decreases.

  4. 4.

There is a depletion of Sub-Neptunes with sizes $2.5\leq\text{R}_{\rm{p}}\leq
4\,\text{R}_{\oplus}$ and $\text{M}_{\rm{p}}\geq 10\,\text{M}_{\oplus}$ around
M-dwarfs compared to FGK-dwarfs (see Fig.2).

Regarding the first point, in our PlanetS calatog, planets falling on the 50%
condensed water line have a minimum mass of approximately 1.8 M⊕ for M-planets
(M${}_{\star}\sim 0.33\leavevmode\nobreak\ \text{M}_{\odot}$), of 3.4 M⊕ for
K-planets (M${}_{\star}\sim 0.8\leavevmode\nobreak\ \text{M}_{\odot}$), and of
4.3 M⊕ for FG-planets (M${}_{\star}\sim 0.96\leavevmode\nobreak\
\text{M}_{\odot}$) as noticed in Sect. 4.1. In planetesimal-based models, Burn
et al. (2021) form water-rich planets with a minimum mass of 1 and 2 M⊕ for
M${}_{\star}=0.5\,\text{and}\,1.0\,\text{M}_{\odot}$, respectively. In a
similar way, but for pebble-based models, Venturini et al. (2024) find a
minimum mass of water-worlds of 1.4, 2.8, and 4.5 M⊕ for
M${}_{\star}=0.4,0.7\,\text{and}\,1.0\,\text{M}_{\odot}$, respectively. We
note that particularly the minimum masses of the pebble-based models are in
very good agreement with our PlanetS catalog. As we already mentioned in
Sect.3.4, this trend of an increasing planet mass of water-worlds with stellar
mass in the theoretical models may be a consequence of type-I migration, which
acts more efficiently for low mass stars. This also explains point 3, the
filling of the radius valley when moving towards M-dwarfs (Venturini et al.,
2024). However, these minimum values may be influenced by observational biases
that render this parameter space incomplete. Despite finding planets smaller
and less massive than these typical limits across spectral types, we are still
approaching the current detection limits, which are around $\sim$1 m/s in
radial velocity precision. Nonetheless, the increasing trend appears reliable
as it persists in our sample test with increased precision in mass and radius
(as mentioned in Sect. 4.1).

Regarding the maximum mass of Super-Earths as a function of host star mass,
our findings remain consistent with a limit of 10 M⊕ for all the spectral
types. Instead, theoretical studies of planet formation and evolution show a
dependence on this maximum mass of Super-Earths with stellar mass. Indeed,
Burn et al. (2021) find values of approximately 10 and 20 M⊕ for
M${}_{\star}=0.5\,\text{and}\,1.0\,\text{M}_{\odot}$, respectively for
planetesimal-based models. Venturini et al. (2024) find lower values for
pebble-based models, and a flatter dependence, of 5, 6, and 7 M⊕ for
M${}_{\star}=0.4,0.7\,\text{and}\,1.0\,\text{M}_{\odot}$, respectively. It is
worth to mention that the differences between the two models might not be due
to the assumption of planetesimals vs. pebbles alone. The simulations of Burn
et al. (2021) consider 50 planetary seeds per disc and N-body interactions
which result in planet merging, while the simulations of Venturini et al.
(2024) consider 7 seeds per disc, and N-body interactions are not directly
modeled. They are included ”a posteriori” but assuming that each planet
suffered from only one major giant impact. In any case, a constant maximum
mass of rocky planets for stellar masses ranging from 0.3 to 1.0 M⊙ as we find
in the current study, seems to be difficult to reproduce with the current
formation models. An interesting possibility to explore in the future is that
such maximum mass of rocky planets is linked to evolutionary processes. The
Super-Earths with masses close to 10 Earth masses in our sample have
equilibrium temperatures above 1000 K (see Fig.2). At such high levels of
irradiation, even a steam atmosphere could escape (see Venturini et al.
(2024), their App.B.2), rendering the planet as a remnant rocky core.

Concerning the final point (4), the results from Venturini et al. (2024) show
the same qualitative trend. In that work, the deficit of Sub-Neptunes with
masses larger than 10 M⊕ around M-dwarfs is a result of the icy cores reaching
smaller masses than around FGK, due to the formation in less massive discs.

##  5 Conclusions

We have updated the PlanetS catalog, incorporating its extension and update.
This has allowed us to re-assess the Mass-Radius (M-R) relationships, which
are pivotal tools for transit follow-up radial velocity missions, to estimate
the masses of objects detected through transit methods by space telescopes
like Kepler/K2, TESS, or the upcoming PLATO mission (Rauer et al., 2014).

Moving beyond the histogram, we delved into the study of density and radius
distributions of small planets around M-dwarfs from our PlanetS catalog. Our
approach involved utilizing Kernel Density Estimation coupled with a
resampling technique allowing us to address the uncertainties associated with
planetary parameters and to acknowledge the continuous nature of the
distribution. The examination of the M-R diagram and KDEs revealed several
trends:

  1. 1.

There is a continuous transition of composition of the planets in the
transition between Super-Earths and Sub-Neptunes around M-dwarfs, ranging from
terrestrial to more volatile compositions.

  2. 2.

There is a dearth of Sub-Neptunes around M-dwarfs relative to Super-Earths.

  3. 3.

The radius valley of M-planets is fading.

  4. 4.

A Super-Mercuries population is absent (at least for planets bigger than Earth
due to observational bias).

Subsequently, these observations were correlated with theories of internal
structure, formation, and evolution, aiming to gain a deeper understanding.
Our investigation extended to questioning various composition line models in
the M-R diagram and their implications for the potential water composition of
Sub-Neptunes orbiting M-dwarfs, introducing equilibrium temperature as a third
crucial parameter. Regarding the first point (1), the Sub-Neptunes in question
exhibit a diverse range of compositions contingent on their level of
insolation, and so do not constitute a distinct population of ”water-worlds”
but rather represent a continuum within the broader spectrum of small planets
around M-dwarfs. Advanced models of interior and atmosphere suggest that these
planets can possess a water-rich interior, but some of their observed radii
can only be accounted for by the presence of an H/He-rich atmosphere. Ongoing
theoretical and experimental studies are continually enhancing our
comprehension of the complexity of planet interiors with possible
compositional coupling between atmospheres, mantle and iron cores. The
composition lines resulting from the P-T profiles, linked to an advanced
internal structure model, will be a subject of keen interest in the upcoming
months. Recent investigations into planetary formation through pebble
accretion around M-dwarfs reveal a more diverse range of water compositions
compared to Solar-type stars. However, the absence of a compositional gap in
our observations suggests that these small planets around M-dwarfs may be
primarily formed through planetesimal accretion or a hybrid combination of
pebble and planetesimal accretion processes.

Expanding our investigation, we included FGK-dwarfs to compare the M-R
diagrams and KDEs across different spectral types and highlighted tendencies:

  1. 5.

There is a scarcity of Sub-Neptunes around M-dwarfs in contrast to FGK-dwarfs.

  2. 6.

Small Sub-Neptunes (1.8 R⊕ ¡ R${}_{\text{p}}$ ¡ 2.8 R⊕) observed around
M-dwarfs exhibit significantly lower density compared to their counterparts
around FGK-dwarfs.

  3. 7.

The minimum mass of volatile-rich Sub-Neptunes increases as stellar masses
increase.

  4. 8.

The maximum mass of a planet with a terrestrial composition appears consistent
across spectral types, remaining close to 10 M⊕.

These demographic trends were then also interpreted in the context of
formation and evolution models. The discrepancy of the point (5) might be
attributed to the intense XUV flux emanating from M-dwarfs in their youth,
potentially leading to heightened atmospheric evaporation from planets in
these systems. The point (6) may be explained by the fact that M-planets are
richer in ice than FGK-planets, and migrated more efficiently after accreting
most of their solids beyond the ice line. The point (7) may be an effect of
type-I migration, but the exact values of minimum mass could still be affected
by observational biases. This migration effect is also responsible for the
fading of the radius valley around M-dwarfs compared with FGKs. However,
pinpointing this value precisely of 10 M⊕ for the point (8) through formation
and evolution models poses challenges.

Ultimately, the constrained sample sizes pose limitations, hindering a
comprehensive interpretation and differentiation of various theoretical
phenomena underlying these distributions and diagrams. We saw that the
significant 30% increase in the number of M-planets within the sample has
notable implications for the conclusions drawn regarding planetary
populations. This underscores the critical need for more data in a parameter
space that inherently lacks completeness and exhibits biases, such as the M-R
diagram. We need to enlarge this parameter space to ensure the robustness of
further statistical studies. On the one hand, instruments such as NIRPS
(Bouchy et al. 2017), the new infrared spectrograph installed on ESO’s 3.6m
telescope at La Silla and dedicated to the study of M-dwarfs and their
planetary systems, can help us achieve this. We also need to explore planets
with longer orbital periods, as the median for this sample is 3.4 days. The
NIRPS Guaranteed Time of Observation (GTO), and in particular Work Package 2
(WP2) devoted to radial velocity follow-up of transiting planets, is focused
to determining the mass of planets around M-dwarfs in this parameter space. On
the other hand, the James Webb Space Telescope (JWST, Gardner et al. 2023)
emerges as a pivotal player in helping to lift the degeneracy of the planetary
compositions. It is poised to significantly contribute to this endeavor
through its capabilities in unveiling the atmospheric complexities of these
Sub-Neptunes. We calculate the transmission spectroscopy metric (TSM) proposed
by Kempton et al. (2018) for the Sub-Neptunes of our samples
(R${}_{\text{p}}\geq$ 1.5 R⊕), and find out that with a TSM ranging from 20 to
457, with a median of 71, the Sub-Neptunes around M-dwarfs therefore stands
out to be excellent candidates for atmospheric characterization via
transmission spectroscopy. For the Sub-Neptunes of the planets orbiting FGK-
stars (R${}_{\text{p}}\geq$ 2 R⊕), they are less good candidates for
atmospheric characterization via transmission spectroscopy with a TSM ranging
from 1 to 240, with a median of 34. High-resolution atmospheric spectroscopic
observations by instruments such as ESPRESSO and NIRPS complement space
telescopes, including the potential detection of crucial components such as
water, helium, carbon dioxide, carbon monoxide or methane. Finally, PLATO will
play a crucial role in advancing our understanding of host stars. For
instance, it will contribute to determining the age of the stars: this
information will be crucial for unraveling the temporal evolution of the stars
and their associated planetary systems.

###### Acknowledgements.

We thank the anonymous referee for valuable comments that helped improve the
manuscript. This work has been carried out within the framework of the NCCR
PlanetS supported by the Swiss National Science Foundation under grants
51NF40_182901 and 51NF40_205606. J.V. acknowledges support from the Swiss
National Science Foundation (SNSF) under grant PZ00P2$\\_$208945\. C.D.
acknowledges support from the Swiss National Science Foundation under grant
TMSGI2_211313. This publication makes use of The Data & Analysis Center for
Exoplanets (DACE), which is a facility based at the University of Geneva (CH)
dedicated to extrasolar planets data visualization, exchange and analysis.
DACE is a platform of the Swiss National Center of Competence in Research
(NCCR) PlanetS, federating the Swiss expertise in Exoplanet research. The DACE
platform is available at https://dace.unige.ch. This research has made use of
the NASA Exoplanet Archive, which is operated by the California Institute of
Technology, under contract with the National Aeronautics and Space
Administration under the Exoplanet Exploration Program. This work presents
results from the European Space Agency (ESA) space mission Gaia. Gaia data are
being processed by the Gaia Data Processing and Analysis Consortium (DPAC).
Funding for the DPAC is provided by national institutions, in particular the
institutions participating in the Gaia MultiLateral Agreement (MLA). The Gaia
mission website is https://www.cosmos.esa.int/gaia. The Gaia archive website
is https://archives.esac.esa.int/gaia.

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##  Appendix A Classical calculations in the PlanetS catalog

###  A.1 Insolation Flux

To calculate the flux received by the planet in a homogeneous way, we use the
following formula (Bolmont et al. 2016):

$F=\frac{L_{\star}}{4\pi a^{2}\sqrt{1-e^{2}}}$ (1)

where L⋆ is the star’s luminosity, $a$ is the semi-major axis, and $e$ is the
eccentricity of the orbit. The luminosity used is that taken from Gaia DR3 if
it exists, otherwise the value from the reference article is used. If neither
exists, the value is not calculated.

###  A.2 Equilibrium Temperature

We use the insolation flux to calculate the equilibrium temperature with the
formula from Méndez & Rivera-Valentín (2017):

$T_{\text{eq}}=\left[\frac{(1-A)F}{4\sigma}\right]^{\frac{1}{4}}$ (2)

where $F$ is the insolation flux received calculated in A.1, $A$ the planet
albedo or Bond Albedo (we use A = 0) and $\sigma$ the Stefan-Boltzmann
constant. We consider a full heat redistribution.

###  A.3 Expected Radial Velocity Semi-Amplitude

In the PlanetS catalog, we also compute the expected radial-velocity semi-
amplitude considering the inclination, the eccentricity, the semi-major axis
and the masses of the system’s bodys with the formula from Lovis & Fischer
(2010) :

$K=\frac{28.4329\mathrm{\leavevmode\nobreak\ m}\mathrm{\leavevmode\nobreak\
s}^{-1}}{\sqrt{1-e^{2}}}\frac{M_{p}\sin
i}{M_{\mathrm{Jup}}}\left(\frac{M_{\star}+M_{p}}{M_{\odot}}\right)^{-1/2}\left(\frac{a}{1\mathrm{AU}}\right)^{-1/2}$
(3)

where $M_{p}$ and $M_{\star}$ are the masses of the planet and star
respectively, $a$ is the semi-major axis, $i$ is the inclination and $e$ is
the eccentricity of the orbit.

##  Appendix B Comparison of the samples : LP22 vs this work

Figure 9: Mass-Radius diagram comparing the samples from LP22 and this work.
The light green circles represent the planets having the same reference and
parameters in both studies. The blue triangles correspond to planets only in
the PlanetS catalog (new characterizations). Red crosses are planets only in
the sample of LP22. Then, the planets with different references/parameters in
both samples are in different colors with a triangle for the parameters from
the PlanetS catalog and a cross for the parameters from LP22 sample. The
composition lines of Earth-like planets (yellow) and 50% water (blue) from
Zeng et al. (2019) are displayed.

To complete the section 3.1, this appendix offers a comparison between the
sample from LP22 and from this work, from the PlanetS catalog. Fig. 9 shows
the differences in the planets and parameters chosen in a Mass-Radius diagram.
The sample of LP22 study contains 34 planets and the sample from this work,
from the PlanetS catalog, contains 46 planets. 24 planets have the same
reference and parameters between the two samples (light green circles). Our
sample has 12 new planets corresponding to characterizations dating from after
the LP22 catalog creation (July 2021) (blue triangles). 1 planet is only in
the LP22 catalog (red cross): it is K2-18 b because there is no reference with
the required precision, or rather, those that do are flagged controversial by
NASA Exoplanet Archive due to the controversial nature of planet c in the
system, which in turn influences the analysis of planet b’s mass in radial
velocities. 9 planets are in both samples, but with different parameters as a
result of a choice of a different reference. LTT 3780, TOI-1235, L 98-59 and
GJ 1214 systems was re-analyzed by LP22 but the uncertainties of the derived
parameters are larger than those the references used in the PlanetS catalog :
Nowak et al. (2020); Bluhm et al. (2020); Demangeon et al. (2021); Cloutier et
al. (2021) respectively. A special case is the TOI-270 system and its 3
planets. The reference paper for both samples is Van Eylen et al. (2021), but
the radii of the planets don’t match999Since this publication dates from
October 2021 and the LP22 catalog from July 2021, we suspect that these are
preliminary study values that have been taken.. In our sample, the radii are
$R_{b}=1.21\pm 0.03$, $R_{c}=2.35\pm 0.07$, $R_{d}=2.13\pm 0.06$, whereas LP22
considers $R_{b}=1.15\pm 0.05$, $R_{c}=2.33\pm 0.07$, $R_{d}=2.00\pm 0.07$.
This difference between the reference paper and the values taken by LP22 leads
to significant errors in their results, given that planet d in this system is
part of their ”water-world” population. Kaye et al. (2022) re-analyzed this
system but the masses were determined using TTVs and the analysis do not
respect our robustness criteria (described Sect. 2.1). All the planets with
different parameters between LP22 and this study are shown in different colors
in the figure, with a triangle for PlanetS catalog parameters and a cross for
LP22 catalog parameters. All the parameters used for this work are online via
the PlanetS catalog101010https://dace.unige.ch/exoplanets/ using the selection
criteria set out in Sect. 3.1.

##  Appendix C The limitations of histograms and KDEs with small sample

Figure 10: Comparison of histogram and KDE performance in representing a
random draw in a Gaussian distribution. Each panel corresponds to a specific
number of draws in the Gaussian distribution (n=34, 46, 70, 220). The actual
distribution is depicted in red, with red dashed lines indicating randomly
drawn points from this real distribution (using a fixed seed for
repeatability). The histogram is presented with blue outlines, and the KDE is
represented by a blue line. The legend provides information about the bins and
bandwidth, determined by Sturges’ rule and cross-validation, respectively, for
each case.

Despite an increase of 10 planets in our sample of small planets around
M-dwarfs compared to LP22, our sample size is a limiting factor for robust
statistical analysis. The purpose of this appendix is to show the limitations
of histograms and KDEs in relation to this number.

To do this, we drew 34, 46, 70 and 220 bins from a Gaussian distribution with
mean 0.53 and standard deviation 0.3 (representing the density distribution
found in Fig.4). We then plotted histograms using a Sturges rule (Sturges
1926) to define the number of bins. We also calculated the KDEs in the same
way as explained in section 3.2.2. The results are shown in Fig. 10.

In the first two cases, with sample sizes of 34 and 46, a ”false” bimodality
emerges in the histogram. However, for the Kernel Density Estimation (KDE),
bimodality is only apparent in the first case. The continuous nature of KDE
aids in representing the actual distribution more accurately, and as the
number of draws in the Gaussian distribution increases, the observed
distribution converges to the real one.

To quantify this convergence, we use the Kullback-Leibler (K-L) divergence
formula between two probability distributions P and Q, here P is the actual
distribution and Q the calculated KDE :

$D_{KL}(P\,||\,Q)=\sum_{i}\left(p_{i}\cdot\log\left(\frac{p_{i}}{q_{i}}\right)\right)$
(4)

where pi and qi represent the respective probabilities of events in the P and
Q distributions.

We have then plotted in Fig. 11 the calculation of this divergence between the
actual pdf of the distribution and the KDE as a function of the number of
samples drawn from the distribution. This can be modeled as a power-law
evolution :

$KL=310.67\times n^{-0.81}$ (5)

with KL the resulting K-L divergence and n the number of samples drawn.

Figure 11: K-L divergence as a function of the number of samples drawn from
the Gaussian distribution. The K-L divergence is calculated between this
latter and the KDE calculated from the drawn samples. The red line correspond
to the fit by the power law and the grey filled area the uncertainty of the
fit.

The two distributions can be considered close when the value of the divergence
of K-L is less than 10. This is the case here for a number of samples drawn
greater than 68$\pm$8.

In conclusion, the KDEs proves to be more effective in representing a
continuous distribution with a low number of elements compared to the
histogram. A larger sample size is essential for a more accurate
quantification of the resulting distribution. Moreover, in the real case,
uncertainties in the measurements of the parameter under consideration further
emphasize the need for a larger sample size.