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A&A
Volume 687, July 2024
Article Number A25
Number of page(s) 13
Section Planets and planetary systems
DOI https://doi.org/10.1051/0004-6361/202449371
Published online 27 June 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.

1 Introduction

Thanks to the improvement of technology and long-term surveys, the number of known planets has exceeded 5000, and thousands of candidates are yet to be confirmed (NASA Exo-planet Archive; Akeson et al. 2013). Among these surveys, the Kepler mission has played an important role (e.g. nearly half of identified planets have been found by Kepler) and provided an unprecedented high-precision sample for exoplanet science (Brown et al. 2011; Huber et al. 2014; Mathur et al. 2017). The majority of planets (candidates) detected by Kepler are compact super-Earths (≲1.7R⊙) and sub-Neptunes (≳2.l R⊙) with lower bulk densities, which are separated by a ‘radius valley’ (∽1.9 ± 0.2 R⊙; e.g. Fulton et al. 2017; Owen & Murray-Clay 2018; Van Eylen et al. 2018; Weiss et al. 2023). Super-Earths are generally assumed to be ‘water-poor’ rocky planets (iron and silicates with no or a thin gas envelope) and are probably formed in situ (e.g. Santerne et al. 2018; Bonomo et al. 2019; Astudillo-Defru et al. 2020).

However, the composition of sub-Neptunes (as well as the origin of the radius valley) remains debated. Plenty of theoretical models have been proposed, and they can generally be classified into two categories. Some studies suggest that sub-Neptunes are born as ‘gas dwarfs’ consisting of rocky cores plus an H/He-dominated gas envelope massive enough (i.e. larger than a few percent in mass) to significantly increase the radius. During their subsequent evolution, some of these planets lose their gas envelope and evolve into super-Earths, resulting in the appearance of a radius valley. The energy source that drives the gas-loss process could either be from the radiation of the host star (the photoevaporation; Owen & Wu 2013, 2017; Jin et al. 2014; Lopez & Fortney 2016; Jin & Mordasini 2018) or from the cooling power of the planet core (the core-powered model; Ginzburg et al. 2016, 2018; Gupta & Schlichting 2019, 2020). Alternatively, some other studies have proposed that sub-Neptunes are ‘water worlds’ containing significant amounts (several tens of percent of their total masses) of H2O-dominated fluid and/or ice (Zeng et al. 2019; Mousis et al. 2020; Aguichine et al. 2021). To obtain such large water contents, these water-rich sub-Neptunes are expected to have formed beyond the ice line and migrated inwards to their current orbits due to mechanisms such as planet– disc interactions (Ward 1997; Ida et al. 2013; Bitsch et al. 2019; Venturini et al. 2020a,b; Izidoro et al. 2022; Burn et al. 2024)1.

To reveal the origin of the radius valley and the composition of sub-Neptunes, numerous of studies have investigated the planetary mass-radius curves and the dependence of the radius valley on planetary and stellar properties (e.g. orbital period, stellar mass, metallicity, age; Van Eylen et al. 2018; Owen & Murray-Clay 2018; Berger et al. 2020; Chen et al. 2022). However, the current observational data provides no conclusive result to distinguish these theoretical hypotheses, as none of them could explain all the observational evidence. For example, Chen et al. (2022) shows that the average radii of sub-Neptunes shrink with age using the LAMOST-Gaia-Kepler kinematic catalogue (Chen et al. 2021a,b), supporting that some sub-Neptunes are gas dwarfs containing a significant gas envelope. In contrast, some other studies have provided supporting observational evidence that many sub-Neptunes are water worlds from the planetary mass-radius curves (Zeng et al. 2019) and the planetary radius distribution around M-type stars (Luque & Pallé 2022).

Besides atmospheric spectroscopy (e.g. Kempton et al. 2023), more clues on the formation and composition of sub-Neptunes can be derived from their orbital spacing (the period ratio of adjacent planets) in multiple planetary systems. Specifically, if sub-Neptunes are water rich, they are thought to have formed beyond ice lines and then migrated inwards to current orbits. Convergent migration in a gas-damped disc can lead to capture in mean motion resonances (MMRs) such that water-rich sub-Neptunes would more likely exhibit close spacing and binding in MMRs (e.g. Snellgrove et al. 2001; Lee & Peale 2002; Pierens & Nelson 2008; Emsenhuber et al. 2021a). Thus, if sub-Neptunes are formed ex situ, sub-Neptune pairs would have a larger fraction in MMR compared to a random distribution. On the contrary, if sub-Neptunes and super-Earths have the same formation pathway and only differ by their subsequent evolution (rocky cores that have kept or lost their thick gas envelopes), they would probably both be formed more or less in situ via solid accretion and/or giant impacts of embryos due to gravitational interactions (e.g. Chambers 2001; Yin et al. 2002; Touboul et al. 2007; Kokubo & Ida 2007; Emsenhuber et al. 2021a). In this case, we would not expect distinct frequencies of MMRs from a random distribution. The subsequent long-term gravitational interactions would lead to the widening of the orbital spacing and the breaking of some unstable MMRs (e.g. Zhou et al. 2007; Lithwick & Wu 2012; Pu & Wu 2015; Millholland & Laughlin 2019). Some recent studies have also suggested that atmospheric escape would contribute to the breaking of MMRs (Matsumoto & Ogihara 2020; Wang & Lin 2023). Thus, generally, planets formed via an in situ pathway are less likely to be in MMRs because of these above mechanisms. Therefore, by exploring the distributions of the orbital period ratio of sub-Neptune pairs versus super-Earth pairs, one can put constraints on their origin and the compositions, therefore clarifying what the predominant compositions of sub-Neptunes (rocky+H/He versus water rich) are and how much these two different compositions and thus formation pathways contribute.

In this paper, by using the Kepler data and a synthetic population generated by the Bern model of planet formation and evolution, we investigate the distribution of period ratios of adjacent planet pairs. The rest of this paper is organised as follows. In Sect. 2, we describe how we constructed our observed and synthetic planetary samples. In Sect. 3, by analysing the observational sample selected from Kepler DR25, we present the observational evidence that sub-Neptunes show a preference to be captured in near-MMR. In Sect. 4, we obtain the theoretical predictions of period ratio distributions for the planet pairs with different compositions by using the synthetic planet population. In Sect. 5, we put constraint on the formation and composition of planets by comparing the observational evidence with the theoretical results. Finally, we summarise the paper in Sect. 6.

2 Sample

In this section, we describe how we constructed the observational sample from the Kepler data in Sect. 2.1. In Sect. 2.2, we show how to construct the synthetic sample from the planet population generated by the Bern model.

2.1 Observational sample

For the observational sample, we initialised our data selection based on the Kepler data release (DR) 25, which contains 8054 Kepler objects of interest (KOIs, Mathur et al. 2017). We excluded KOIs flagged as false positives (FAPs), leaving 4034 planets (candidates). Since the synthetic systems are generated around single 1 M⊙ stars, we only kept systems around Sun-like stars (i.e. effective temperature in the range of 4700–6500 K and surface gravity log g > 4). We also removed potential binaries by excluding planet host stars with a Gaia DR2 re-normalised unit-weight error (RUWE) greater than 1.2. For the planetary sample, we adopted the following criteria: (i) excluding the KOIs identified as false positives; (ii) removing KOIs with radii greater than 4 R⊕; (iii) removing KOIs with a period greater than 400 days to ensure detection efficiency; (iv) excluding KOIs with a period less than five days to avoid the influence of tide (e.g. Jackson et al. 2008; Lai 2012); and (v) keeping planetary systems with multiplicity Np ≥ 2. Planets (candidates) with radii ≥2.1 R⊕, ≤1.7 R⊕, and 1.7–2.1 R⊕ are classified as sub-Neptunes, super-Earths, and valley planets, respectively. The final observational sample contains 697 planets (candidates), 373 sub-Neptunes, 239 super-Earths, and 85 valley planets, in 293 multiple systems.

2.2 Synthetic sample

We adopted a combined planetary formation and evolution model (the Generation III Bern model) to synthesise a population of model planetary systems. The Bern model (Alibert et al. 2004, 2005, 2013; Mordasini et al. 2009a, 2012; Emsenhuber et al. 2021a) simulates the initial formation and subsequent long-term evolution phases of planetary systems in a self-consistently coupled way. A detailed description of the Generation III model used here can be found in Emsenhuber et al. (2021a); therefore, we only give a short overview in this work. Starting from planetary embryos and background planetesimals embedded in a gaseous disc, the formation stage includes the following main physical processes: the viscous evolution of the protoplanetary gas disc, including disc photoevaporation; the evolution of the surface density and dynamical state of planetesimals; the simultaneous accretion of planetesimals and gas by the embryos; the orbital migration and eccentricity and inclination damping of the protoplanets by the gas disc; and the gravitational N-body interactions among the multiple protoplanets forming in each system, including impacts. The following evolution stage follows the long-term evolution of each planet individually. It includes the following processes: the thermodynamic evolution (cooling and contraction of the interior), mass loss via atmospheric escape, and tidal migration due to stellar tides. The individual sub-modules included in the model can be of considerable complexity, and whenever possible, underlying differential equations are solved rather than using semi-empirical closed-form expressions. However, in order to still be able to numerically simulate many systems and planets over gigayears, a low-dimensional approach was used (except for the N-body integrator, which is 3D). The gas disc was assumed to be rotationally symmetric, the planet interiors were assumed to be spherically symmetric, and the planetesimals were described on a 1D grid (radial) by a surface density rather than by individual particles.

For the analysis, we used an improved version of the nominal synthetic population for solar-mass stars (NG76) that was presented in Emsenhuber et al. (2021b). The improvements in the population studied here (called NG76longshot) regard two aspects (see Weiss et al. 2022, Emsenhuber et al. 2023 and Burn et al. 2024, where this improved data set was also already analysed). Firstly, the formation stage, which includes N-body interactions, was prolonged from 20 to 100 Myr to capture late dynamical events (Izidoro et al. 2017). Secondly, during the evolutionary phase, the originally employed simple energy-limited XUV-driven escape model (Jin & Mordasini 2018) was replaced with a full hydrodynamic model (Kubyshkina et al. 2018; Affolter et al. 2023), and the equation of state for water (AQUA, Haldemann et al. 2020) included the different physical phases. Furthermore, it was assumed that the water mixes with H/He (where present), instead of an onion-like structure with separated layers. This has important implications for the radii of water-rich close-in sub-Neptunes because of the runaway greenhouse radius inflation effect (Turbet et al. 2020).

Despite its high content in physical processes, it should be noted that the model is based on classical concepts. More recent developments such as a description of growth starting from dust over pebbles and planetesimals to protoplanets (Voelkel et al. 2020, 2022); hybrid pebble and planetesimal accretion (Alibert et al. 2018; Kessler & Alibert 2023); MHD-wind driven disc evolution (Weder et al. 2023); the impact of structured discs (Lau et al. 2022; Jiang & Ormel 2023); or a torque densities approach for orbital migration (Schib et al. 2022) are not yet included. These limitations should be kept in mind when assessing the results of this work.

For the population synthesis studied here (New Generation Planetary Population Synthesis; NGPPS), the 1000 systems were obtained by varying key initial disc conditions in a Monte-Carlo way (Ida & Lin 2004; Mordasini et al. 2009b). The probability distributions were derived from observations of protoplanetary discs (Tychoniec et al. 2018) and are described in detail in Emsenhuber et al. (2021b). Each disc was initially seeded with 100 lunar-mass embryos at t = 0. The variation of the disc properties over the observed range led to a large diversity of resulting planetary systems, from systems with only low-mass super-Earths to multiple massive giants.

From this synthetic population, we selected Kepler-like planets with the same criteria as the observational sample (i.e. planets with radii ≤4 R⊕ and periods within 5–400 days) at 5 Gyr. In order to make comparisons with Kepler observations, we then applied the synthetic detection bias using the KOBE programme (for more details, see Appendix C of Mishra et al. 2021). Specifically, we first used the KOBE-Shadow module to test whether the generated planets could transit. Then, for these transiting planets, we used the KOBE-transit module to calculate their transit parameters and kept those with numbers of transit greater than or equal to three and S/N ≥ 7.1. Finally, we used the KOBE-Vetter module to simulate the detection completeness and reliability. After applying the detection biased, only systems of two or more detected synthetic planets were retained. The final synthetic sample consisted of 1851 planets (740 sub-Neptunes, 920 super-Earths, and 191 valley planets) in 757 multiple systems. Figure 1 shows the radius-period diagram of the selected observational and synthetic samples.

thumbnail Fig. 1

Radii as a function of orbital period of planets selected from Kepler DR 25 data (blue) and the synthetic population after applying the bias of the Kepler survey (pink). We restricted the sample to planets with radii between 1–4 R⊕ and orbital periods between 5 and 400 days in multiple systems.

3 Analysis of the selected Kepler sample

In this section, using the selected observational sample, we obtain the period ratios (PR) of adjacent planet pairs in multiple systems. Then we investigate whether they have a larger fraction in near-first-order MMRs compared to random distributions.

3.1 Constructing a control sample

To evaluate whether the observational planets are randomly paired or not, we constructed a control sample from the selected observational sample. Specifically, for each planet (sub-Neptune/super-Earth/valley planet) in a given system, we randomly assigned an orbital period from the observed distribution of planets in the select planetary sample. We then calculated its signal-to-noise ratio and transit detection efficiency. If the resulting S/N ≥ 7.1 and detection efficiency was ≥10%, we considered the planet with the newly assigned orbital period observable by Kepler and kept it. Otherwise, we reassigned the orbital period to it and repeated the above process until the above criteria were met. Finally, we calculated the period ratios for the adjacent planet pairs in the control sample and compared them with the observational results from Kepler data.

3.2 Ensuring the stability of multiple systems

To evaluate the stability of multiple systems in the observational control sample, we calculated the Hill stability criterion from the orbital separation, H≡aout−ainRH,$H \equiv {{{a_{{\rm{out}}}} - {a_{{\rm{in}}}}} \over {{R_{\rm{H}}}}},$(1)

where the two planets are indexed as ‘in’ and ‘out’ and a denotes the semi-major axis. We selected planet pairs with H > 7.1 as Hill stable when the distribution of the period ratio for the control sample was most similar (with the largest KS p-value) to that of the observed sample (see Appendix A for details). Actually, the theoretical limit for Hill stability K is dependent on various parameters, such as multiplicity Np (Funk et al. 2010), planetary mass ratio over stellar mass µ (Chambers et al. 1996; Zhou et al. 2007), the orbital eccentricities and inclinations (Yoshinaga et al. 1999; Zhou et al. 2007), and stable timescales (Smith & Lissauer 2009). The value of K is ~2 3$\~2\sqrt 3 $ to 12 from simulation and observation (Gladman 1993; Chambers et al. 1996; Malhotra 2015; Pu & Wu 2015; He et al. 2019; Dietrich et al. 2024). In Appendix A, we present detailed discussions on the effect of the variation of K. We evaluated the Hill stability by assuming adjacent planets on circular orbits (i.e. e = 0) since most of the Kepler planets have no (accurate) eccentricity measurements, and on average, the Kepler planets in multiple systems are on nearly circular orbits (e ~ 0.04; e.g. Xie et al. 2016). The term RH is the mutual Hill radius relevant for dynamical interactions (Marchal & Bozis 1982) given by RH=(Min+MoutM*)1/3(ain+ain)2, ${R_{\rm{H}}} = {\left( {{{{M_{{\rm{in}}}} + {M_{{\rm{out}}}}} \over {{M_*}}}} \right)^{1/3}}{{\left( {{a_{{\rm{in}}}} + {a_{{\rm{in}}}}} \right)} \over 2}{\rm{, }}$(2)

where M is the mass of the planets (in and out) and star (*). Since most planets in our observational control sample have no (accurate) mass measurements, we estimated their masses by converting their measured radii according to an empirical broken power-law mass-radius relationship (Lissauer et al. 2011b; Wolfgang & Laughlin 2012; Otegi et al. 2020; Ramos et al. 2017; Müller et al. 2023; Huang & Ormel 2023): MpM⊕=C (RpR⊕)α,${{{M_{\rm{p}}}} \over {{M_ \oplus }}} = C{\left( {{{{R_{\rm{p}}}} \over {{R_ \oplus }}}} \right)^\alpha },$(3)

where C and α are set to 3.98 and 0.92 for Rp ≤ 3.80 R⊕ and 0.74 and 2.18 for R > 3.80 R⊕, respectively (Ramos et al. 2017). It is clear that this power-law only reflects a mean mass-radius relation. On the other hand, the dependency of the Hill sphere on mass is weak (proportional to only M1/3).

For systems of three or more planets, the long-term interaction between planets could potentially induce additional instabilities (Chambers et al. 1996; Smith & Lissauer 2009). To further ensure the stability of planet pairs in systems with Np > 3, we adopted a conservative heuristic criterion suggested by Fabrycky et al. (2014): Hin +Hout >18,${H_{{\rm{in }}}} + {H_{{\rm{out }}}} > 18,$(4)

where ‘in’ and ‘out’ denote the inner pair and the outer pair of three adjacent planets.

thumbnail Fig. 2

Probability density functions of period ratios of adjacent planet pairs derived from the Kepler DR 25 data (blue) and from a control sample with the assumption that planets are randomly paired (grey). The grey regions and lines represent the control sample by combining the 1000 times of randomly pairing and one example with similar fractions in near-first-order MMRs in the 1000 sets, respectively. The dotted lines and purple band represent the centres and regions of the first-order resonances.

3.3 Comparing the period ratio distributions of the observational sample with the control sample

Figure 2 displays the histogram distribution function of the period ratio derived from the observational sample and control sample for all planet pairs (top panel), sub-Neptune pairs (i.e. pairs of two sub-Neptunes; middle panel), and super-Earth pairs (i.e. pairs of two super-Earths; bottom panel). As can be seen, the actual Kepler planet pairs seem to prefer to be near first-order MMRs (i.e. 5:4, 4:3, 3:2, and 2:1) compared to the random-paired control sample. Furthermore, sub-Neptune pairs show an obvious preference in near-MMRs when compared to the corresponding control sample, which in contrast is not seen in super-Earth pairs. To describe the above feature mathematically, according to previous studies (e.g. Jiang et al. 2020; Huang & Ormel 2023), we selected planet pairs with PRs between [ 3j+43j+1,3j+23j−1 ]$\left[ {{{3j + 4} \over {3j + 1}},{{3j + 2} \over {3j - 1}}} \right]$ (the closest third-order resonances), and |Δ| < 0.03 were selected as near-MMRs, where Δ is a dimensionless parameter to measure the offset of a given PR to j + 1 : j MMR, Δ≡jj+1 PR −1.$\Delta \equiv {j \over {j + 1}}{\rm{ PR }} - 1.$(5)

We then calculated the fractions of planet pairs that are near-MMRs for both the actual observational sample FMMR obs $F_{{\rm{MMR }}}^{{\rm{obs }}}$ and the randomly paired control sample FMMRcon$F_{{\rm{MMR}}}^{{\rm{con}}}$. We adopted 0.03 as a typical value for the boundary of near-MMRs. Actually, the expected value of the boundary depends on the planetary masses and eccentricities and is approximately several of 10−2 both from numerical simulations (e.g. Silburt & Rein 2015; Xie 2014; Migaszewski 2015) and observations (e.g. Baluev & Beaugé 2014; Nelson et al. 2016; Ramos et al. 2017). In Appendix B, we provide a detailed discussion on the influence of the variation of the resonance offset boundary.

To quantify the differences between the observational sample and the randomly paired control sample, we defined a metric, the normalised fraction of planet pairs being near-MMRs, F^MMR${\hat F_{{\rm{MMR}}}}$, which is mathematically expressed as F^MMR=FMMRobsFMMRcon.${\hat F_{{\rm{MMR}}}} = {{F_{{\rm{MMR}}}^{{\rm{obs}}}} \over {F_{{\rm{MMR}}}^{{\rm{con}}}}}.$(6)

This metric is thus a measure of how much more frequently actual planets are near-MMRs compared to a random-paired situation. To obtain the uncertainty, we resampled the counted numbers from the Poisson distribution for 10000 times and derived the corresponding F^MMR${\hat F_{{\rm{MMR}}}}$. The uncertainty (1σ interval) was set as the range of 16–84% percentiles of the 10 000 calculations.

Figure 3 shows the normalised fractions in near-MMRs F^MMR${\hat F_{{\rm{MMR}}}}$ as well as their 1σ intervals for all the actual planet pairs, sub-Neptune pairs, and super-Earth pairs. The plot also compares the observed results to the results obtained for the synthetic population. These results are discussed in the next section.

As can be seen, as a whole, the Kepler planet pairs show a statistically weak preference to be near-MMRs (F^MMR>1)$\left( {{{\hat F}_{{\rm{MMR}}}} > 1} \right)$ by a factor of 1.3−0.2+0.2$1.3_{ - 0.2}^{ + 0.2}$ (with a confidence level of 91.21%) compared to the random-paired control sample, which is consistent with previous studies (Lissauer et al. 2011a; Fabrycky et al. 2014). However, when splitting the actual Kepler pairs into the two sub-samples (sub-Neptunes and super-Earths), a more interesting picture arises: Compared to the control sample, sub-Neptunes pairs show a significant preference of near-MMRs, with a F^MMR${\hat F_{{\rm{MMR}}}}$ of 1.7−0.3+0.3$1.7_{ - 0.3}^{ + 0.3}$. Out of the 10 000 sets of resampled data, the F^MMR${\hat F_{{\rm{MMR}}}}$ of sub-Neptune pairs is larger than one for 9843 times, corresponding to a confidence level of 98.43%. On the contrary, the super-Earth pairs have a F^MMR${\hat F_{{\rm{MMR}}}}$ of 1.3−0.3+0.2$1.3_{ - 0.3}^{ + 0.2}$, which is consistent with that of the randomly paired control sample within 1σ error bars. Furthermore, for the sub-Neptune pairs, there exists an obvious asymmetry in the distribution of period ratios near first-order resonances, with an excess of planet pairs lying wide of resonance by a factor of 9.5−4.2+13.9$9.5_{ - 4.2}^{ + 13.9}$, compared to planets lying narrow of resonance with a confidence level of 99.95%. for super-Earth pairs, the distribution of period ratios near first-order resonances is nearly symmetric, and the number ratio of planet pairs lying wide of resonance over those lying narrow of resonance is 1.1−0.4+0.4$1.1_{ - 0.4}^{ + 0.4}$, which is similar to a random distribution.

Based on the above analyses, we conclude that sub-Neptunes in the actual Kepler sample have a significant preference to be captured in a near-MMR configuration. In contrast, super-Earths are statistically randomly paired. This could suggest that some sub-Neptunes have experienced a different formation process than super-Earth planets, where orbital migration and resonant capture play a more important role.

thumbnail Fig. 3

Normalised fraction of near-MMRs pairs, F^MMR${\hat F_{{\rm{MMR}}}}$ (solid points) with 1σ uncertainties (vertical bars) for all actual planet pairs, sub-Neptune pairs, and super-Earth planet pairs in Kepler multiple transiting systems. The two horizontal solid lines and shaded regions represent the theoretical predictions and 1σ uncertainties of water-rich (blue) and water-poor planet pairs (red) derived from the synthetic sample. The two horizontal dashed lines denote the theoretical predictions of simulated sub-Neptune (brown) and super-Earth (green) pairs derived from the synthetic sample. The result of the randomly paired control sample is plotted as a dotted black line and by definition is equal to unity.

4 Theoretical predictions for planet pairs for different compositions

In this section, we analyse the period ratio distribution of adjacent planet pairs in the synthetic population. The model provides not only the orbital parameters and radius of the planets but also their mass, composition, and dynamic and formation histories.

To illustrate how synthetic planetary systems and their system architecture emerge, we first describe the formation and evolution of one specific system out of the 1000 in the synthetic population (Fig. 4). The physical processes and mass scales leading to the emergence of the four different planetary system architectures have been described in Emsenhuber et al. (2023). For the analysis here, the Class I and Class II systems identified there are the most relevant. They both contain (close-in) low-mass planets but originate from different formation pathways.

Class I systems contain in the inner part (inside of about 1 AU) super-Earth planets that have formed approximately in situ. Thus, they have rocky (silicate-iron) cores without much water. Their formation pathway is characterised by growth initially via planetesimal accretion and then giant impacts (embryo-embryo collisions) and only limited orbital migration (only inside of the ice line). After the dissipation of the gaseous disc, atmospheric escape removes in most cases their H/He envelopes. The resulting evaporated, bare cores have radii ≤1.7 R⊕ and populate the super-Earth peak of the observed radius distribution.

Class II systems, in contrast, contain larger ex situ ice-rich sub-Neptunian planets that have primarily accreted beyond the ice line. They first accrete icy planetesimals outside of the water ice line at approximately constant orbital separation and then migrate inwards at approximately constant mass in a ‘horizontal branch’ (Mordasini et al. 2009b). They contain about 50% water in mass. Because of the runaway greenhouse radius inflation effect (Turbet et al. 2020), they have radii ≥2.1 R⊕ and populate the observed sub-Neptune peak. In the model, the radius gap is thus caused by the distinct formation and evolution pathways of the planets above and below it. Their formation (orbital migration) leads to the presence of ex situ ice-rich sub-Neptunes, and their evolution (atmospheric escape) leads to the presence of approximately in situ rocky super-Earths.

The governing mass scale in the Class I systems is the Goldreich mass. It corresponds to the mass resulting from the final giant impact phase where mutual scattering increases the eccentricities, which in turn defines the width of an effective feeding zone. In the Class II systems, the governing mass scale is the equality mass (when migration and planetesimal accretion timescales become equal) or the saturation mass (when the co-rotation torques saturate). When these masses are reached, the planets start to migrate inwards. In mass (instead of radius) space, the two classes overlap much more, but below (above) about 5 M⊕, super-Earths (sub-Neptunes) dominate (see also Venturini et al. 2020a,b).

thumbnail Fig. 4

Example of the formation of a synthetic planetary system from initially 100 lunar-mass embryos in a typical protoplanetary disc with solar metallicity ([Fe/H] = 0.0). The epochs in time (in years) are shown in the top-left of the six panels. Solid points show (proto)planets with the semi-transparent part scaling with the radius. Grey crosses (fading in time) show the last position of protoplanets that were accreted by other more massive bodies. The colours of points represent the mass fraction of ice in the core. Horizontal black bars go from the periastron to the apoastron (i.e. represent orbital eccentricity). Lines show the growth tracks in the semi-major axis-mass plane.

4.1 Formation of one synthetic system

The specific example shown in Fig. 4 displays the temporal evolution of a planetary system in the semi-major axis-mass plane for a typical proptoplanetary disc with solar metallicity ([Fe/H] = 0.0). The initial masses of gas and dust are 0.027 M⊕ and 105 M⊕, respectively. Both of these values are close to the mean values of the probability distribution of the disc initial conditions (Emsenhuber et al. 2021b). At t = 0, 100 lunar-mass embryos are uniformly seeded inside of 40 AU in the logarithm of the semi-major axis.

As shown in the top-left panel of Fig. 4, shortly after the beginning (105 yr), the dominating process is the quasi in situ accretion of planetesimals in the feeding zone of the embryos. Some dynamical interactions also start among some embryos, leading to some collisions and mergers, which are indicated by grey crosses. Orbital migration is not yet important since the timescales are quite long for very low masses, whereas planetes-imal accretion is fast, inducing nearly vertical upward tracks in the semi-major axis-mass plane. The specific pattern of the maximum mass as a function of distance can be understood in the following way. From 0.1 AU to the largest masses at 0.4-0.8 AU, the masses are increasing because planets are (except for giant impacts) stuck at the local planetesimal isolation mass, which is an increasing function of distance for the minimum-mass solar nebula (MMSN)-like planetesimal surface profile assumed here (Lissauer 1993). Outside of this, the masses are initially decreasing again. This is the consequence of the increase of the growth timescale with orbital distance for oligarchic growth (Thommes et al. 2003). Slightly inside of 3 AU, there is again a sudden increase in the masses. This is caused by the water ice line, which increases the planetesimal surface density by approximately a factor of two. Even further out, the masses decrease, which is again caused by the slower planetesimal accretion at larger distances.

At 1 Myr (the top-right panel of Fig. 4), (just) beyond the ice line, a handful of protoplanets with masses between about 1 and 4 M⊕ have formed and some inward migration has started, leading to the tracks bending inwards. This occurs at the saturation mass where the co-rotation torque saturates (Msat or at the equality mass where the migration timescale becomes shorter than the growth timescale; see Emsenhuber et al. 2023). For protoplanets inside the ice line, the growth via giant impacts (embryo-embryo collisions) plays a more important role and allows some proto-planets in the inner disc to grow beyond the local isolation mass. We also observed that some orbital migration has started inside the ice line.

At 5 Myr (the middle-left panel of Fig. 4), in the outer disc, the massive protoplanets that initially formed beyond the ice lines have reached masses of several Earth masses and migrated further inwards. In this configuration (Type I migration), outer and more massive planets generally migrate faster, and the pro-toplanets capture each other in resonant convoys and migrate together (e.g. Pierens & Nelson 2008; Alibert et al. 2013). During their migration inwards, the icy planets kept almost constant masses, leading to tracks in the ‘horizontal branch’ (Mordasini et al. 2009b). This is due to the fact that the planetesimals inside their position were already accreted by the embryos growing there. The outer ice planets pushed three inner rocky protoplan-ets to the inner edge of the disc at about 0.02 AU. In this inner disc, the growth via giant impacts became the dominant effect among the rocky planets. In particular, tens of protoplanets with masses of a few percent to tenths of M⊕ that existed at 1 Myr formed these three super-Earths via numerous giant impacts.

After the disc disappears (~5.6 Myr), we continued to make N-body dynamic interaction to 100 Myr. As can be seen in the last three panels of Fig. 4, during 10–100 Myr, the dynamic interaction between (proto)planets induced further giant impacts and scattering, reducing the number of planets and destroying MMRs, especially for the super-Earths at very close orbits.

This system illustrates how in the synthetic population, the larger sub-Neptune planets initially formed beyond the ice line with large mass fractions of water and/or ice in their interiors are expected to have a larger fraction of MMRs compared to the smaller, volatile, and poor rocky super-Earths formed inside the ice line. While the formation pathway dominated by inwards migration characterising this system corresponds to Class II systems, in this specific system, one inner rocky planet has remained. This is in contrast to most Class II systems, where only ice-rich sub-Neptunes remain. The compositional ordering with rocky planets inside and ice ones outside is rather characteristic of Class I systems with little orbital migration. Thus, this system has a partially mixed character. It is an interesting example with planets spanning the valley. A number of such systems are observationally known (Carter et al. 2012; Owen & Campos Estrada 2020). Furthermore, this system contains seven planets (five sub-Neptunes and two super-Earths) with periods between 1 and 400 days. After applying the synthetic detection bias using the KOBE programme, three of the planets (two sub-Neptunes and one super-Earth) can be observed. Interestingly, the two sub-Neptunes were both born beyond the ice line and contain a significant amount (more than 40%) of water in their envelopes (Class II), while the super-Earth was born at ~0.8 AU (inside the ice line), and the mass fraction of water in its envelope is only ~1%. That is to say, water-rich planets and water-poor planets locate on different sides of the radius valley.

4.2 Statistical analysis of the synthetic systems

We next study whether the different formation pathways of planets in the synthetic system are reflected in the frequency of pairs in MMRs. We evaluated the Z value at the moment when the parent protoplanetary disc disappears. We divided the planets into two different categories. The first is Z ≥ 0.1 water-rich planets2 corresponding to bodies that have accreted ice-rich material, either directly by accreting water-rich planetesimals or indirectly by colliding with a water-rich protoplanet. The ice content serves here as a proxy for formation pathways where orbital migration was of importance (Class II). The second category is water-poor (Z < 0.1) planets that mainly formed in the region inside of the ice line and for which migration was less important (Class I architectures).

Figure 5 displays the cumulative distributions of the starting positions for planets of different radii and compositions. As can be seen, and as expected, water-rich planets are mainly formed beyond the ice line, while water-poor planets are formed inside the ice line. In the selected synthetic sample, a large majority of the sub-Neptunes are water rich (706 of 740; i.e. 95%). The super-Earths consist of 437 water-rich planets and 483 water-poor planets. The former are planets that originally had an envelope with a Z ≥ 0.1 that was subsequently lost. The selected synthetic sample contains 796 stable adjacent planet pairs, which are composed of 596 pairs of two water-rich planets (hereafter ‘water-rich pairs’), 135 pairs of two water-poor planets (hereafter ‘water-poor pairs’), and 65 pairs of one water-rich planet and one water-poor planet (hereafter ‘mixed pairs’). That is to say, most (over 90%) of the planets are adjacent to planets with the same composition category.

We then investigated the period ratios for adjacent planet pairs in the synthetic sample with the same method as described in Sect. 3 for the actual observed planets. Figure 6 compares the probability density distributions of period ratios derived from the synthetic sample and the control sample for planet pairs of different compositions. To quantify the preference in near-MMRs, we also calculated their normalised fractions in near-MMR, F^MMR${\hat F_{{\rm{MMR}}}}$ relative to the random-paired sample. As shown in Fig. 3 by the blue line and region, water-rich planet pairs exhibit a significant preference to be near-MMRs compared to the randomly paired control sample by a factor of 2.3−0.2+0.2$2.3_{ - 0.2}^{ + 0.2}$ and with a confidence level ≳5σ. On the contrary, water-poor planet pairs (red line and region) have a normalised fraction in near-MMRs (0.8−0.1+0.2)$\left( {0.8_{ - 0.1}^{ + 0.2}} \right)$ that is a bit lower than (but statistically indistinguishable from) that of the random-paired control sample.

Since the presence of water cannot be directly observed, we also calculated F^MMR${\hat F_{{\rm{MMR}}}}$ of some subgroups distinguishing (synthetic) super-Earths and sub-Neptunes. The derived results are 2.2−0.2+0.3,2.1−0.3+0.3,0.8−0.4+0.6, and 0.7−0.2+0.2$2.2_{ - 0.2}^{ + 0.3},2.1_{ - 0.3}^{ + 0.3},0.8_{ - 0.4}^{ + 0.6}{\rm{, and }}0.7_{ - 0.2}^{ + 0.2}$ for water-rich sub-Neptune pairs, water-rich super-Earth planet pairs, water-poor sub-Neptune pairs, and water-poor super-Earth pairs, respectively. These values are consistent with the results from the whole water-rich and water-poor planet pairs within 1σ uncertainties.

For direct comparison with the observational results, we also divided the synthetic sample into sub-Neptunes and super-Earths only according to radius, yielding 154 adjacent sub-Neptune pairs (146 water rich, six water poor, two mixed) and 273 super-Earth pairs (121 water rich, 137 water poor, 15 mixed). We then calculated their F^MMR${\hat F_{{\rm{MMR}}}}$, which are 2.1−0.2+0.2 and 1.4−0.1+0.2$2.1_{ - 0.2}^{ + 0.2}{\rm{ and }}1.4_{ - 0.1}^{ + 0.2}$, respectively. As shown in Fig. 3, the synthetic results for sub-Neptune (brown) and super-Earth (green) pairs are higher than the observational results, which implies that the planet population generated by the Bern model contains larger fractions of water-rich planets compared to the Kepler sample (see Sect. 5 for detailed discussions).

thumbnail Fig. 5

Cumulative distributions of the starting position of the embryos that eventually became planets with different compositions and radii. The dashed line represents the typical snow line location.
thumbnail Fig. 6

Probability density functions of period ratios of adjacent water-rich (top panel) and water-poor (bottom panel) planet pairs derived from the synthetic sample and the corresponding control sample with the assumption that planets are randomly paired (grey). The grey regions and lines represent the control sample by combining the 1000 times of randomly pairing and one example with similar fractions in near-first-order MMRs in the 1000 sets, respectively.

5 Implication on the formation and composition of planets

In this subsection, we compare the observational results (Sect. 3) with the theoretical predictions (Sect. 4) of the period ratio distributions. From the comparison of the model and observed F^MMR${\hat F_{{\rm{MMR}}}}$, we can also put a nominal prediction on the proportion that formed as water-rich (or more exactly ex situ) planets ƒrich with the following formula: frich =F^MMR−F^MMRpoorF^MMRrich −F^MMRpoor,${f_{{\rm{rich }}}} = {{{{\hat F}_{{\rm{MMR}}}} - \hat F_{{\rm{MMR}}}^{{\rm{poor}}}} \over {\hat F_{{\rm{MMR}}}^{{\rm{rich }}} - \hat F_{{\rm{MMR}}}^{{\rm{poor}}}}},$(7)

where the ‘rich’ and ‘poor’ superscripts denote results derived respectively from water-rich and water-poor planets in the synthetic sample. The uncertainty of ƒrich is calculated from the uncertainties of F^MMR,F^MMR rich , and F^MMR poor ${\hat F_{{\rm{MMR}}}},\hat F_{{\rm{MMR }}}^{{\rm{rich }}}{\rm{, and }}\hat F_{{\rm{MMR }}}^{{\rm{poor }}}$ by means of error propagation. Here, we ignore the contributions of mixed pairs, as these are rare (≲5%) in simulated samples. To verify Eq. (7), we calculated ƒrich for the sub-Neptune and super-Earth pairs in the synthetic sample generated by the Bern models from their period ratio distributions. The resulting ƒrich are 87−20+13%$87_{ - 20}^{ + 13}\% $ and 40−17+22%$40_{ - 17}^{ + 22}\% $, which are close to the fractions of sub-Neptunes (706/740 = 95%) and super-Earths (437/920 = 48%) denoted as water rich in the synthetic sample.

The derived constraints on the formation and composition of Kepler planets are as follows:

  • 1.

    For the sub-Neptune pairs, the actual Kepler sample shows that they exhibit a preference to be near-MMRs. Quantitatively, the derived observational F^MMR ,1.7−0.3+0.3${\hat F_{{\rm{MMR }}}},1.7_{ - 0.3}^{ + 0.3}$ is smaller than the F^MMR${\hat F_{{\rm{MMR}}}}$ derived for synthetic ex situ water-rich planet pairs, 2.3−0.2+0.2$2.3_{ - 0.2}^{ + 0.2}$ but larger than synthetic in situ water-poor planet pairs, 0.8−0.1+0.2$0.8_{ - 0.1}^{ + 0.2}$. In the 10 000 sets of resampled data, the derived F^MMR${\hat F_{{\rm{MMR}}}}$ from the observational sample is smaller than those of water-rich planet pairs for 9460 times and lager than water-poor planet pairs for 9878 times, corresponding to confidence levels of 94.60% and 98.78%, respectively. That is to say, the hypothesis that observed sub-Neptunes are all water rich or water poor could be confidently (≳2σ) rejected.

    We also calculated the ƒrich using Eq. (7), and the resulting ƒrich is 60−28+20%$60_{ - 28}^{ + 20}\% $ for sub-Neptunes in Kepler multiple transiting systems. Taken at face value, this would mean that, formation-wise, the actual sub-Neptunes are approximately a half-half combination of two populations that formed via in situ and ex situ ways.

  • 2.

    For the super-Earth pairs in actual Kepler multiple systems, we find that they are statistically randomly paired and show no preference to be captured in near-MMRs. Quantitatively, F^MMR${\hat F_{{\rm{MMR}}}}$ derived from the Kepler data, 1.3−0.3+0.2$1.3_{ - 0.3}^{ + 0.2}$, is a bit higher than the theoretical prediction of water-poor planet pairs, 0.80−0.1+0.2$0.80_{ - 0.1}^{ + 0.2}$, but significantly smaller than that of water-rich synthetic super-Earth planet pairs, 2.3−0.2+0.2$2.3_{ - 0.2}^{ + 0.2}$, with a confidence level of 99.82%. The derived ƒrich from Eq. (7) is 28−17+21%$28_{ - 17}^{ + 21}\% $ for super-Earth pairs, demonstrating that the observed super-Earth planets are mainly (~50% – 90%) born as water-poor (i.e. iron plus silicate) cores inside the ice line.

    Matsumoto & Ogihara (2020) and Wang & Lin (2023) have proposed a hypothesis that could also contribute to the lower fraction in near-MMRs for super-Earths. If a planetary system formed around a star with strong XUV emissions, planets born with a significant gas envelope may lose their envelope via photoevaporation and evolve to bare super-Earth planets (e.g. Owen & Wu 2013; Jin et al. 2014). Their semi-major axes could be impulsively changed if they quickly lost ≳10% of their total mass within a timescale of ≲104–4105 yr. In this way, some pairs may escape from MMR configurations, further reducing the fraction of super-Earth near-MMRs. Therefore, considering the effect of mass loss on the planetary orbits3, the theoretical expectation of water-poor super-Earth pairs in near-MMRs should be lower. According to Eq. (7), the fraction of super-Earth planets that formed as water rich in Kepler transiting multiple systems would be higher; thus, the above ƒrich (28−17+21%)$\left( {28_{ - 17}^{ + 21}\% } \right)$ is a somewhat lower limit.

In Appendix B, we also calculate the F^MMR${\hat F_{{\rm{MMR}}}}$ and ƒrich for the Kepler by taking different boundaries of the resonance offset as near-MMRs. The derived results (Fig. B.1) are consistent with the above results (Fig. 3), demonstrating that the variation of boundary of resonance offset has little influence on our main conclusions.

6 Summary

Based on the Kepler DR 25 catalogue (Mathur et al. 2017), we have investigated the distributions of orbital period ratios of adjacent planet pairs in the multiple transiting Kepler systems, separating the sample into super-Earth and sub-Neptune pairs. We find that sub-Neptune pairs show a significant preference (98.43%) to be captured in near-MMR configurations by a factor of 1.7−0.3+0.3$1.7_{ - 0.3}^{ + 0.3}$ compared to a random-paired control sample, while the super-Earth planet pairs show no significant difference from a random distribution (Sect. 3, Fig. 2).

Using a synthetic planetary population generated by the Generation III Bern model (Emsenhuber et al. 2021a,b), we studied the frequency of MMRs in the two classes of close-in planets identified in Emsenhuber et al. (2023) that differ by their formation pathways: Class I approximately are in situ water-poor planets that have formed within the ice line and Class II are ex situ water-rich planets that were born beyond the ice line and have migrated inwards (see Sect. 4 and Figs. 4–5). For Class I, a final giant impact phase governs the resulting architecture (orbits and masses), whereas for Class II, the effects of Type I orbital migration are dominant. The two classes correspond mainly to super-Earth (Class I) and sub-Neptunes (Class II). As for the actual Kepler planets, we derived their period ratio distributions and normalised fractions that are near-MMRs (Fig. 6). The prevalence of near-MMRs relative to the random-paired sample are 2.3−0.2+0.2 and 0.8−0.1+0.2$2.3_{ - 0.2}^{ + 0.2}{\rm{ and }}0.8_{ - 0.1}^{ + 0.2}$ for the synthetic sub-Neptunes and super-Earths, respectively.

Based on comparing the observational results with the theoretical predictions (Sect. 5 and Fig. 3), we rejected the hypothesis that the actual sub-Neptunes in Kepler multiple transiting systems are only water-rich ex situ or only water-poor in situ planets with a confidence level of ~2σ. Instead, our result suggest that they should be made from a mixture of both the two populations and the two formation pathways. Our nominal estimation is that 60−28+20%$60_{ - 28}^{ + 20}\% $ of the actual sub-Neptunes formed via a formation pathways where orbital migration was important, potentially implying an origin beyond the ice line and thus an ice-rich composition.

In contrast, actual super-Earth planet pairs have a normalised fraction of near-MMR that is significantly (99.82%) smaller than that of synthetic water-rich planet pairs but statistically consistent with that of synthetic water-poor planet pairs within about 1σ error bars. This suggests that the majority of actual super-Earth planets correspond to water-poor planets born within the ice line with no or limited importance of orbital migration.

Our results on observed MMRs thus suggest that close-in sub-Neptunes and super-Earths have partially different formation pathways. This supports the view (Burn et al. 2024) where the radius valley is both a consequence of the formation (orbital migration; larger water-rich sub-Neptunes born ex situ versus smaller rocky super-Earth born in situ; see also Venturini et al. 2020a,b) as well as evolution (atmospheric escape) that populates the super-Earth peak in the radius distribution.

Future studies, both from observational analyses (e.g. larger sample of planets in multiple systems, studies of atmospheric composition; Gardner et al. 2006; Rauer et al. 2014) and from numerical simulations of theoretical formation and evolution models will test our results and provide more clues on the formation, evolution, and compositions of close-in low-mass exoplanets, the most common type of known planets.

Acknowledgements

This work is supported by the National Key R&D Program of China (nos. 2019YFA0405100, 2019YFA0706601) and the National Natural Science Foundation of China (NSFC; grant nos. 11933001, 12273011). We also acknowledge the science research grants from the China Manned Space Project with NO.CMS-CSST-2021-B12 and CMS-CSST-2021-B09. J.-W.X. also acknowledges the support from the National Youth Talent Support Program. D.-C.C. also acknowledges the Cultivation project for LAMOST Scientific Payoff, Research Achievement of CAMS-CAS and the fellowship of Chinese postdoctoral science foundation (2022M711566). This research was supported by the Excellence Cluster ORIGINS which is funded by the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation) under Germany’s Excelence Strategy – EXC-2094-390783311. C.M. acknowledges the support from the Swiss National Science Foundation under grant 200021_204847 “PlanetslnTime”. Parts of 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.

Appendix A Influence of the limit for Hill stability

thumbnail Fig. A.1

Cumulative distributions for all the adjacent planet pairs (top, cyan), sub-Neptune pairs (middle, orange), and super-Earth pairs (bottom, green) selected as Hill stability with different lower limits of orbital spacing in the observed sample. The distributions of the corresponding randomly paired control samples are plotted in grey. We also show all the KS p-values of the observed planet pairs compared to those in the control sample.

In the main text, we adopted the Hill stable criterion to ensure the generated planet pairs in the control sample is stable. However, the expected value for the limit of Hill stability K is still uncertain. Specifically, the minimum separation required to be Hill stable is 2 3$2\sqrt 3 $ when considering two planets in circular and co-planar orbits (Gladman 1993). Pu & Wu (2015) found that K ~ 10 – 12 is required for long-term stability for the adjacent planets in Kepler systems showing four or more transiting planets. He et al. (2019) generated sets of simulated ‘Kepler-like’ planetary systems and reported a best fit of K ~ 8, compared to the actual observational data. Dietrich et al. (2024) carried out numerical simulations and provided a median value of K = 7.17 from simulated planet pairs in long-stable planetary systems. Thus, generally, the previous estimates for K are in the range of ~2 3−12$\~2\sqrt 3 - 12$.

To evaluate the influence of the variation of the limit for Hill stability for the control sample, we re-selected planet pairs as Hill stable (H > K & Hin + Hout > 18) in the observed and synthetic samples with different values of K. Figure A.1 shows the cumulative distributions of the period ratio of planet pairs selected as Hill stable with five typical values of K. We also performed the two-sample Kolmogorov–Smirnov (K–S) test between the distributions of period ratios derived from the observed and control samples. Figure A.2 displays the resulting p-values as a function of K. As can be seen, when K is small (≲5), the control sample has more planet pairs with period ratios less than 1.5. When K exceeds approximately ten, most of the planet pairs with period ratios less than 1.5 become ‘unstable’ in the observed sample, indicating that the theoretical limit of approximately more than ten is too strict for our selected Kepler sample. Besides, the control sample generally has a larger distribution in period ratios compared to the observed sample. Thus, the resulting p-values first increase and then decrease, and we therefore took K with the maximum p-value of 7.1 as the best value for the control sample. This result is consistent with previous estimates in the range of roughly five to eigth (Malhotra 2015; He et al. 2019; Dietrich et al. 2024) but is smaller than that of Pu & Wu (2015) (~ 10 – 12). This was not expected because our selected observed sample mainly consists of systems with two or three transiting planets, and the theoretical limit K would be smaller than that of Pu & Wu (2015) due to the decrease in planet multiplicity.

thumbnail Fig. A.2

KS p-values between the distributions of the period ratio of all the adjacent planet pairs for the observed and control samples when taking different values for the Hill stability K limit.

To further quantify the influence of the variation of K, we re-selected planet pairs with H>2 3$H > 2\sqrt 3 $, 5, 10, and 12 as Hill stable from the observed and control samples. For systems with three or more planets, we also required Hin + Hout > 18. Then we calculated the normalised fraction in MMRs F^MMR${\hat F_{{\rm{MMR}}}}$ for the observed sample and compared F^MMR${\hat F_{{\rm{MMR}}}}$ with those of water-rich and water-poor pairs in the synthetic sample.

Figure A.3 shows the normalised fraction in MMRs for the observed and synthetic samples of different K. As can be seen, all the cases show similar results: The sub-Neptune pairs in Kepler multiple systems have an F^MMR${\hat F_{{\rm{MMR}}}}$ smaller than that of synthetic water-rich planet pairs but larger than that of the synthetic water-poor planet pairs, while for the super-Earth pairs, Kepler data show an F^MMR${\hat F_{{\rm{MMR}}}}$ that is statistically indistinguishable from that of the synthetic water-poor planet pairs but smaller than that of the synthetic water-rich planet pairs. Furthermore, the proportions of sub-Neptunes and super-Earths to be water rich are (48−26+24%,27−22+12%),(65−32+20%,30−25+20%),(63−43+23%,23−20+28%)$\left( {48_{ - 26}^{ + 24}\% ,27_{ - 22}^{ + 21}\% } \right),\left( {65_{ - 32}^{ + 20}\% ,30_{ - 25}^{ + 20}\% } \right),\left( {63_{ - 43}^{ + 23}\% ,23_{ - 20}^{ + 28}\% } \right)$, and (64−46+25%,2926+32%)$\left( {64_{ - 46}^{ + 25}\% ,29_{26}^{ + 32}\% } \right)$ when K is set to 2 3$2\sqrt 3 $, 5, 10, and 12, respectively, which are all consistent with the results of K as 7.1 within uncertainties of 1σ. Based on the above analyses, we concluded that the variation of the limit of Hill stability has no significant effect on our main conclusions (Fig. 3, Sect. 5).

Appendix B Influence of the boundaries of the resonance offset as near-MMRs

In this paper, we adopted the resonance offset Δ≡jj+1PR−1$\Delta \equiv {j \over {j + 1}}{\rm{PR}} - 1$ to indicate the distance of a planet pair with a given period ratio (PR) from a j : k resonance. In the aforementioned nvestigation in Sect. 3–4, we set 0.03 as the boundary of Δ to select near-MMRs, following previous studies (e.g. Fabrycky et al. 2014; Jiang et al. 2020) that selected 0.03 as the boundary since the overabundance of planet pairs just outside resonances are mainly within this interval.

However, the expected value of the boundary depends on the planetary masses and eccentricities and is not certain. From theory, N-body simulations of resonance captures generally provide a prediction of Δ ~ 10−1 (e.g. Silburt & Rein 2015), and the offset could further increase when considering the effect of more complex disc models on the migration of planetary pairs (e.g, photoevaporation, opacity laws; Migaszewski 2015, 2016) or some dissipative evolutions, such as tidal dissipation (Lithwick & Wu 2012; Petrovich et al. 2013; Xie 2014). From observations, previous studies have found that Δ ~ several of 10−2 (e.g. Baluev & Beaugé 2014; Nelson et al. 2016). Ramos et al. (2017) investigated the Kepler systems and suggested a possible smooth trend in which Δ decreases with P and the boundaries of Δ are approximately 0.02 – 0.04 for P in the range of 5 to 400 days (see Fig.1 of their paper).

thumbnail Fig. A.3

Similar to Fig. 3 in the main text but with a different limit of Hill stability K.

To evaluate the effect of the variation of the resonance offset boundary on our results, we re-selected planet pairs with Δ less than 0.02 and 0.04 and a period ratio within the closest third-order resonances as near-MMRs. With these criteria, by adopting the same procedure as in Sect. 3 and 4, we recalculated the normalised fractions in MMRs for the observational sample and compared the results with the theoretical predictions derived from the synthetic sample.

Figure B.1 displays the results when the boundaries of Δ are set to 0.02 (top panel) and 0.04 (bottom panel). As can be seen, the derived F^MMR${\hat F_{{\rm{MMR}}}}$ generally decreases with an increasing Δ boundary for the observed sub-Neptune pairs and synthetic water-rich planet pairs, but it changes little for the observed super-Earth pairs and synthetic water-poor planet pairs. This result is as expected because sub-Neptune (water-rich planet) pairs in the observational (synthetic) sample exhibit an overabundance just outside the exact resonances, which, however, could not be seen in the random-paired control sample (see Fig. 2, 6). Thus, when the Δ boundary expands, a larger proportion of planet pairs in the control samples are selected as near-MMRs compared to the sub-Neptunes (water-rich planets) in the observational (synthetic) sample, resulting in a smaller F^MMR${\hat F_{{\rm{MMR}}}}$· Nevertheless, the main conclusions of our work (see Sect. 5) hold. Specifically, for the sub-Neptune pairs in Kepler multiple systems, when the boundary is set to 0.02 (0.04), the derived F^MMR${\hat F_{{\rm{MMR}}}}$ is smaller than that of the synthetic water-rich planet pairs with a confidence level of 92.15% (95.77%) but larger than that of the synthetic water-poor planet pairs with a confidence level of 97.24% (98.37%). For the super-Earth pairs, Kepler data show an F^MMR${\hat F_{{\rm{MMR}}}}$ that statistically indistinguishable (≲ 1σ) from that of the synthetic water-poor planet pairs but significantly smaller than that of the synthetic water-rich planet pairs with a confidence level of 99.91% (99.22%) when the boundary is set to 0.02 (0.04). Furthermore, the proportions of water-rich sub-Neptunes and super-Earths are (54−27+25%,7−7+23%)$\left( {54_{ - 27}^{ + 25}\% ,7_{ - 7}^{ + 23}\% } \right)$ and (60−24+23%,25−12+18%)$\left( {60_{ - 24}^{ + 23}\% ,25_{ - 12}^{ + 18}\% } \right)$ when the boundary is set to 0.02 and 0.04, respectively, which is consistent with the results of the boundary of 0.03 within 1σ uncertainties. Based on the above analysis, we therefore concluded that our conclusions (Fig. 3, Sect. 5) are not (significantly) affected by the variation of the boundary of resonance offset.

thumbnail Fig. B.1

Similar to Fig. 3 in the main text but the boundaries of resonance offset Δ are set to 0.02 (top panel) and 0.04 (bottom panel).

Appendix C Comparison with the results derived from the synthetic in situ population

We analysed the distribution of the period ratios of the planet population generated by the Bern model involving the disc migration in the Sect. 4. We found that ex situ water-rich sub-Neptunes (mainly Class II) have a larger fraction of planet pairs to be near-MMRs. For close planets that approximately formed in situ (Class I), their frequency in MMRs is a bit lower but statistically indistinguishable from that of a random distribution. These results are in agreement with the proposal of the previous studies that convergent disc migrations lead to more capture in resonance (e.g. Snellgrove et al. 2001; Lee & Peale 2002; Pierens & Nelson 2008; Emsenhuber et al. 2021a).

To further verify the above proposal, we also generated a synthetic in situ planet population using the Bern model without considering disc-planet interactions (i.e. without orbital migration and damping of eccentricities and inclinations; see Appendix A of Emsenhuber et al. 2021b). With the same criteria described in Sect. 2 of the main text, we selected Kepler-like planets (Rp ≤ 4R⊕) from the in situ population at 5 Gyr and only retained systems of two or more planets. The selected in situ sample contained 1031 planets in 419 systems, and their radius-period diagram is shown in Fig. C.1. In this case, we did not adopt the KOBE programme to apply the Kepler bias because very few systems would be identified as multiple systems after applying the bias. The main reasons are as follows: (1) The in situ population is farther away from the host stars and less planets (especially super-Earths) are detectable. (2) More importantly, their eccentricities and inclinations have not been damped during the disc phase because all disc-planet interactions were turned off (see Fig. C.2). Multiple planets in such systems are very difficult to detect simultaneously via transits.

thumbnail Fig. C.1

Radii as a function of orbital period of planets selected from the synthetic in situ population without applying the bias of the Kepler survey. We restricted the sample to planets with radii between 1 – 4R⊕ and orbital periods < 400 days in multiple systems.
thumbnail Fig. C.2

Cumulative distributions of the orbital eccentricity (top) and inclination dispersion (bottom) for the select synthetic in situ sample.

We then calculated the period ratio of adjacent planet pairs in the select in situ sample. We also constructed a randomly paired control sample using the same method described in Sect. 3.1. Since the eccentricities of the planets of the in situ population are relatively large, we calculated the Hill stability metric with the following equation: H≡aout (1−eout)−ain(1+ein)RH.$H \equiv {{{a_{{\rm{out }}}}\left( {1 - {e_{{\rm{out}}}}} \right) - {a_{{\rm{in}}}}\left( {1 + {e_{{\rm{in}}}}} \right)} \over {{R_{\rm{H}}}}}.$(C.1)

Then we kept planet pairs as Hill stable with the same criteria described in Sect. 3.2 of the main text.

Figure C.3 shows the cumulative distribution function (top) and probability density function (bottom) of the orbital period ratios of planet pairs. As can be seen, the period ratios of planet pairs in the selected in situ sample are statistically indistinguishable from the randomly paired control sample with a K–S p-value of greater than 0.05 and show no preference in near-MMR. We also calculated the normalised fraction in near-MMR and the derived F^MMR=0.9−0.3+0.4${\hat F_{{\rm{MMR}}}} = 0.9_{ - 0.3}^{ + 0.4}$, which is consistent with that of the random distribution. The above analyses demonstrate that, as expected, the synthetic in situ population behaves in a similar way as a random distribution (but still originates from the same formation model including the N-body interactions but excluding migration and damping).

thumbnail Fig. C.3

Cumulative distribution function (top) and probability density function (bottom) of the period ratio of adjacent planet pairs derived from the selected in situ sample (purple) and the corresponding control sample (grey). In the top-right corner of the top panel, we show the two-sample K-S p-value.

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1

It is worth noting that the water-rich and water-poor planets are classified based on their formation characteristics (e.g. composition, location, etc.), whereas super-Earth and sub-Neptune planets are classified based on their current physical radius (i.e. after formation and any evolution that occurred, such as migration or atmospheric loss).

2

When the envelope mass is small compared to the total mass of the planet (or even zero in the case that the envelope has fully evaporated during the long-term evolution), the absolute amount of ice these planets contain is low or even zero at 5 Gyr. However, we still refer to these planets for simplicity as ‘ice rich’, understanding this as a sign that orbital migration was important.

3

Our model currently does not include evaporation and N-body interactions at the same time; therefore, it does not include the effect that atmospheric escape may break MMRs.

All Figures

thumbnail Fig. 1

Radii as a function of orbital period of planets selected from Kepler DR 25 data (blue) and the synthetic population after applying the bias of the Kepler survey (pink). We restricted the sample to planets with radii between 1–4 R⊕ and orbital periods between 5 and 400 days in multiple systems.
In the text
thumbnail Fig. 2

Probability density functions of period ratios of adjacent planet pairs derived from the Kepler DR 25 data (blue) and from a control sample with the assumption that planets are randomly paired (grey). The grey regions and lines represent the control sample by combining the 1000 times of randomly pairing and one example with similar fractions in near-first-order MMRs in the 1000 sets, respectively. The dotted lines and purple band represent the centres and regions of the first-order resonances.
In the text
thumbnail Fig. 3

Normalised fraction of near-MMRs pairs, F^MMR${\hat F_{{\rm{MMR}}}}$ (solid points) with 1σ uncertainties (vertical bars) for all actual planet pairs, sub-Neptune pairs, and super-Earth planet pairs in Kepler multiple transiting systems. The two horizontal solid lines and shaded regions represent the theoretical predictions and 1σ uncertainties of water-rich (blue) and water-poor planet pairs (red) derived from the synthetic sample. The two horizontal dashed lines denote the theoretical predictions of simulated sub-Neptune (brown) and super-Earth (green) pairs derived from the synthetic sample. The result of the randomly paired control sample is plotted as a dotted black line and by definition is equal to unity.
In the text
thumbnail Fig. 4

Example of the formation of a synthetic planetary system from initially 100 lunar-mass embryos in a typical protoplanetary disc with solar metallicity ([Fe/H] = 0.0). The epochs in time (in years) are shown in the top-left of the six panels. Solid points show (proto)planets with the semi-transparent part scaling with the radius. Grey crosses (fading in time) show the last position of protoplanets that were accreted by other more massive bodies. The colours of points represent the mass fraction of ice in the core. Horizontal black bars go from the periastron to the apoastron (i.e. represent orbital eccentricity). Lines show the growth tracks in the semi-major axis-mass plane.
In the text
thumbnail Fig. 5

Cumulative distributions of the starting position of the embryos that eventually became planets with different compositions and radii. The dashed line represents the typical snow line location.
In the text
thumbnail Fig. 6

Probability density functions of period ratios of adjacent water-rich (top panel) and water-poor (bottom panel) planet pairs derived from the synthetic sample and the corresponding control sample with the assumption that planets are randomly paired (grey). The grey regions and lines represent the control sample by combining the 1000 times of randomly pairing and one example with similar fractions in near-first-order MMRs in the 1000 sets, respectively.
In the text
thumbnail Fig. A.1

Cumulative distributions for all the adjacent planet pairs (top, cyan), sub-Neptune pairs (middle, orange), and super-Earth pairs (bottom, green) selected as Hill stability with different lower limits of orbital spacing in the observed sample. The distributions of the corresponding randomly paired control samples are plotted in grey. We also show all the KS p-values of the observed planet pairs compared to those in the control sample.
In the text
thumbnail Fig. A.2

KS p-values between the distributions of the period ratio of all the adjacent planet pairs for the observed and control samples when taking different values for the Hill stability K limit.
In the text
thumbnail Fig. A.3

Similar to Fig. 3 in the main text but with a different limit of Hill stability K.
In the text
thumbnail Fig. B.1

Similar to Fig. 3 in the main text but the boundaries of resonance offset Δ are set to 0.02 (top panel) and 0.04 (bottom panel).
In the text
thumbnail Fig. C.1

Radii as a function of orbital period of planets selected from the synthetic in situ population without applying the bias of the Kepler survey. We restricted the sample to planets with radii between 1 – 4R⊕ and orbital periods < 400 days in multiple systems.
In the text
thumbnail Fig. C.2

Cumulative distributions of the orbital eccentricity (top) and inclination dispersion (bottom) for the select synthetic in situ sample.
In the text
thumbnail Fig. C.3

Cumulative distribution function (top) and probability density function (bottom) of the period ratio of adjacent planet pairs derived from the selected in situ sample (purple) and the corresponding control sample (grey). In the top-right corner of the top panel, we show the two-sample K-S p-value.
In the text

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