The Distribution of Planet Radius in Kepler Multiplanet Systems Depends on Gap Complexity

We find that the gap complexity of a system does not correlate with the host star’s properties. The sample Pearson correlation coefficients, $r$, between $C$ and stellar temperature, mass, radius, log(g), and metallicity are all below 0.1 with slopes consistent with zero. The average radius uncertainty and the average log-flux of planets in a system has no correlation with gap complexity. Additionally, we find sensitivity-related parameters such as the Multiple Event Statistic (MES) have no correlation with gap complexity both when all values for the planets are used and when values are averaged in a system. We also find the distributions of MES in low and high complexity systems are consistent with each other.

However, the gap complexity is sensitive to other period-related measures. While not correlated with inner period, the outer period has a correlation coefficient of 0.37 with $C$. From a slope, $b$, of a linear regression ($y=a+bx$), the outer planet period around a $C$=1.0 system is expected to be 155 days longer than a $C$=0 system.

Correlations also exist between gap complexity and the range of periods in log-space ($b=0.72$ days, $r=0.47$) and the normalized standard deviation of period ($\sigma_{P}/\mu_{P}$, $b=0.53$, $r=0.49$). The strongest correlation, $r=0.64$, is shown in Fig. 3 left axis between gap complexity and the average $\Delta P/P_{i}$ in a system. Where the average $\Delta P/P_{i}$ is equal to

Since the ratio of the outer planet to inner planet period in each nearest pair is equal to $1+\Delta P/P_{i}$, the figure shows a majority of systems with C$<$0.1 have average period ratios less than 3:1. While systems at large gap complexities may have low period ratios, they have a wider spread in period ratios with some over 10:1.

We find that systems with larger gap complexities tend to be more widely spread out in planet periods. This has not been considered in previous works and for our purposes is considered as an underlying variable in the subsequent sections.

Additionally, other works use different measures to describe how planets are spaced in a system. Following upon the work of Goyal & Wang (2022) and Goyal et al. (2023), Goyal & Wang (2024) describes the spacing uniformity in systems with the adjusted Gini index (Deltas, 2003; Gini, 1912) of a system’s period ratios ($\mathcal{G}_{P_{ratio}}$). The Gini index is an economic measure of inequality and is used here to describe the uniformity of period ratios in a system. In Fig. 3 right axis, we show the correlation between gap complexity and the Gini index of period ratios. The Gini index is also normalized to be between zero and one, but the maximum in our systems is 0.88. While the correlation is not 1:1 the two parameters to measure the uniformity of spacing in systems are highly correlated.

In this section, we aim to examine the radius distribution in systems split into samples of low- and high-$C$. First, we determine at what gap complexity to divide the systems. We perform a two-sample Kolmogorov–Smirnov (K-S) test between samples of planet radius divided into low-/high-$C$ with various maximum/minimum values of $C$ determining the samples. A two-sample K-S test gives the probability that both samples were drawn from the same underlying population without assumption of that underlying distribution. We emphasize that the K-S test is sensitive to sample size and cannot be used to tell how different or in what ways two distributions are different. For example, a p-value less than 0.05 can be achieved by drawing 30 values from $\mathcal{N}(0,1)$ and from $\mathcal{N}(1,1)$ or similarly by drawing 3000 values from $\mathcal{N}(0,1)$ and from $\mathcal{N}(0,0.9)$.

In Fig. 6, we show the p-value of the K-S test with different divisions. When using all planet radii ($<$6 R_⊕), the most significance occurs when the sample is divided into systems with $C$$<$0.105 and systems of $C$$>$0.11. However, our primarily interest is in in the region between one and three Earth-radii which features the radius valley. Thus, we also perform K-S tests only considering planets in this region. The significance comes when low-$C$ is defined as $C$$<$0.165 and high-$C$ is defined as $C$$>$0.35.

Moving forward we will divide out samples into low- and high-$C$ based upon the minimum p-value of the K-S tests since no division exists a priori. However, Fig. 6 shows these points are not exceptional with significance varying slowly and smoothly in most directions from the chosen division. When only considering the planets with 1$<$R$<$3 R_⊕, there is significance across all cuts of low-$C$ for any high-$C$ cut above 0.3.

In Fig. 7, we show the radius distribution for systems below and above the determined gap complexities. For each of the low- and high-$C$ systems, we show 100 distributions where each planet’s radius is randomly chosen from a Gaussian with observed median value and a standard deviation equal to the average of the plus/minus observed uncertainties. We calculate the p-value of a 2-sample K-S test between each sampled radius distribution for the 100 distributions.

When considering all planets, there are 118 systems with $C$$<$0.105 and 117 systems with $C$$>$0.11 and 3 systems between. The median p-value of a 2-sample K-S test between these two distributions is 0.0044. We see when considering all planets, Fig. 7 left, the abundance of sub-Earths at low-$C$ dominate the significance.

In Fig. 7 right, we show the entire radius distribution for the highest significance in the 1-3 R_⊕ region. There are 138 systems with $C$$<$0.165 and 62 systems with $C$$>$0.35. The median p-value of a 2-sample K-S test between the planets in the analysis region of 1-3 R_⊕ is 0.0008. We see the frequency of sub-Neptunes in the high-$C$ systems has been suppressed to below the frequency of planets in the radius valley. The frequency of super-Earths increases from the low-$C$ to the high-$C$.

For the region of 1-3 R_⊕, we also perform a test of modality on the low- and high-$C$ planets. Using a Hartigan dip-test of unimodality at a p-value of 0.01 and 1000 trials we find the low-$C$ planets to have two peaks while the high-$C$ only has one peak. The test find the two peaks to be from 1.08 to 1.9 R_⊕ and from 1.90 to 2.40 R_⊕. While for the high-gap complexity planets, the test returns one broad peak across the analysis range from 1.01 to 2.98 R_⊕.

In our entire sample, 47 systems have 95 planets with median planet radius less than the Earth. Systems with sub-Earths span gap complexities from 0-1. However, we see in Fig. 5 sub-Earths have a 1.5$\sigma$ lower gap complexity than expected. In Fig. 4, we see a bump in the sub-Earth frequency at low-$C$ which decreases at gap complexities over 0.2 to the frequency consistent with a tail of a Gaussian which is centered in the super-Earth region.

When dividing based on the K-S test in Fig. 6, 27 of the 47 sub-Earth hosting systems have $C$$<$0.105 and host 67 sub-Earths. While, 19 sub-Earth systems have $C$$>$0.11 and host 27 sub-Earths. At this division there are approximately the same amount of total systems and planets in the low- and high-$C$ samples (Fig. 7). However, low-$C$ systems host approximately 2.5 times more sub-Earths than high-$C$ systems.

The minimum frequency of planet radius in the region of 1-3 R_⊕, the bottom of the radius valley, occurs at approximately 1.9 R_⊕ in our total planet sample. We will define super-Earths as planets larger than Earth and smaller than 1.9 R_⊕. Sub-Neptunes will be shorthand for those above 1.9 R_⊕ and below 3 R_⊕. The planets are colored by categories in Fig. 2. We see the frequency of super-Earths increasing and sub-Neptunes decreasing with increasing gap complexity in Fig. 4. In Fig. 5, super-Earths are more likely to have a high-$C$ while sub-Neptunes are more likely to have a low-$C$.

We calculate the percentage here of each type of planet considering only planets between 1-3 R_⊕ in our systems divided into $C$$<$0.165 and $C$$>$0.35. In low-$C$ systems, 47% of planets (163 planets) have their median planet radius in the super-Earth regime while 53% (182) are in the sub-Neptune regime. In high-$C$, 63% (103) are super-Earths while only 37% (60) are sub-Neptune. Super-Earths are 1.3 times more prevalent while sub-Neptunes are similarly 1.4 times less prevalent in high-$C$ systems compared to low-$C$ systems.

While the gap complexities of systems with sub-Earths for the most part follow the frequency of all of the systems, shown in Fig. 3, there are eight systems with gap complexities between 0.07 and 0.105. Although this is a small number, it is approximately twice the expected amount and results in the sharp increase in the significance of the K-S tests in Fig. 6.

Furthermore, three of the eight systems have exactly three sub-Earths and three have four or more sub-Earths. Included in these three high-multiplicity sub-Earth systems are KOI-2859 (Kepler-1371) and KOI-4032 (Kepler-1542)—two systems of all 5 planets having median planet radius below that of Earth (see Fig. 2). The 6 systems with 3+ sub-Earths that are all in this narrow region of gap complexity (0.07$<$C$<$0.105) have 23 sub-Earths. Removing these 23 planets accounts for about half of the over-abundance of sub-Earths in low-$C$ systems.

While KOI-2859 and KOI-4032 have non-negligible gap complexities, their average $\Delta P/P_{i}$ are 0.29 and 0.26 respectively. In Fig. 1, we see the sub-Earths are primarily under 20 days. To find multiple sub-Earths under 20 days the spacing between planets must be small keeping the planets within 20 days. A tight spacing is correlated with a low gap complexity as shown in Fig. 3.

This brief analysis of specific systems indicates that the increase in sub-Earths at low-$C$ may be an effect of observational bias toward shorter periods for sub-Earth observations. The Kepler telescope can only find sub-Earths with a high number of transits thus favoring shorter orbital periods. The Solar System is an example of a system of sub-Earths with a higher $C$ than the sub-Earth systems found by Kepler. With an incomplete survey of small planets, we may be missing systems of sub-Earths with high-$C$.

In addition to missing irregularly spaced systems of sub-Earths, we could also be missing sub-Earths in our already observed high-$C$ systems. Gilbert & Fabrycky (2020) presents several lines of evidence that systems of $C$$>$0.33 have unobserved planets. The value of $C$ for an intrinsically 4-planet system that is evenly spaced in period ratio where one interior planet is unobserved is 0.336. These missing interior planets may not be transiting because of higher mutual inclinations (see next paragraph), but Zhu et al. (2018) find that higher intrinsic multiplicity systems tend to have lower mutual inclinations. Our missing sub-Earths in high-$C$ systems could support that these missing interior planets are small and below the detection threshold. However, the lack of sub-Earths begins at a much lower value of $C$ than suggested in Gilbert & Fabrycky (2020) for missing interior planets with the deficit most prominent at $C$$>$0.11.

If Kepler high-$C$ systems do host unobserved sub-Earths but they are not transiting, this could point to outer giants disrupting smaller planets in the inner system and raising their mutual inclinations. Raising the inclination and ejecting planetesimals in the region of Mars through the migration of Jupiter and Saturn or an instability amongst the giant planets is a common solution to the “small Mars problem” in the Solar System (Morbidelli & Raymond, 2016; Clement et al., 2018, 2019). As in the early Solar System (Morbidelli & Crida, 2007), cold giant planets in exoplanet systems should similarly form in resonant chains driven by gravitation torques of the disk. The giant planets then scatter small objects causing the resonant chains to break and further scattering materials as the planets migrate (Malhotra, 1993; Gomes et al., 2005). High-$C$ systems may represent those with multiple giant planets which have migrated and ejected smaller sub-Earth bodies from the system or raised their mutual inclinations to the point where they are unobserved.

Our discussion in this section would be aided by a future work detailing how $C$ changes with changing the inner planet’s period, changing the outer planet’s period, and discovering a new planet interior or exterior to the current planets.

Unlike the sub-Earths, we find that the change in the abundance of super-Earths and Sub-Neptunes is not easily explained by the periods sampled in low- and high-$C$ systems. In Fig. 1, we see there is a small number of hot, near Earth-radii planets which for the most part have high gap complexities. However, if we remove all systems which contain either a shorter than 2 day orbit or longer than 365 day orbit our results remain the same. If this restriction is made there are only 41 systems with $C$$>$0.35, making the uncertainties in the frequency much larger, but the over-abundance of super-Earths and under-abundance of sub-Neptunes remains in high-$C$ systems with approximately the same K-S significance when compared to low-$C$.

The division between low- and high-$C$ does correspond with the value of a system of 3-observed planets and one interior unobserved planet (discussed in the Sect. 5.1. However, it is less clear why the unobserved interior planet would be preferentially a sub-Neptune over a super-Earth. Nearly two-thirds of high-$C$ systems would need a missing sub-Neptune to equalize the amount of super-Earths and sub-Neptunes. High-$C$ systems do represent a larger spread in periods and sub-Neptunes tend to have longer periods than super-Earths, so there may be a slight bias to missing sub-Neptunes but not enough to match the proportions in low-$C$ systems. An injection-recovery test with varied initial distributions or planet radius and intrinsic gap complexities is reserved for a future work.

Our results agree with those of Goyal & Wang (2024). They split Kepler multis into ones of only planets less than 1.6 R_⊕ and systems with only planets between 1.6 and 4.0 R_⊕. They find the systems with only smaller planets have a higher uniformity in period ratios measured by a small ($\leq$0.1) Gini index. The Gini index of period ratios has a much larger spread and higher maximum value in the systems of larger planets. Goyal & Wang (2024) also finds a greater mass uniformity amongst the larger planets and discusses that the lower spacing uniformity does not fit in with the low-entropy dynamical pathways which are preferred by their other results.

Thus, we too turn our discussion to astrophysical formation mechanisms. If the high-$C$ systems initially had equal amounts of super-Earths and sub-Neptunes, it is possible that the mechanism that raised their gap complexities also changed approximately 10-20% of sub-Neptunes to super-Earths. High-$C$ systems may represent high entropy formation environments which is further supported by He & Weiss (2023) finding cold giant planets in high-$C$ systems. In contrast, Kong et al. (2024) find that the dynamical stirring from giant planets preferentially collide the most different planetesimals in mass and spacing leaving the final system more compact and uniform than in a low-entropy formation environment (also supported by Lammers et al. (2023)).

If planets in high-$C$ systems experience more collisions, this lack of sub-Neptunes could be evidence of impact-driven atmosphere loss (Schlichting et al., 2015; Lozovsky et al., 2023). The formation simulations of Dawson et al. (2016) show that systems with low solid surface density and less gas during inner system formation tend to produce rockier planets with wider spacings. Chance et al. (2022) analyzes the simulations of Dawson et al. (2016) for the effect of impact-driven atmosphere erosion and finds it is able to reproduce the radius valley although a less empty radius valley than predicted by photoevaporation. Similarly, the formation models of Izidoro et al. (2022) invoke a period of instability amongst the inner planets in resonant configurations which lead to giant impacts. They argue that the giant impacts are efficient at stripping all primordial atmospheres and the radius valley is between water-rich larger planets and rocky smaller planets.

Alternatively, the difference could form earlier with systems that evolve to high-$C$ initially having less volatile-rich planets. The mechanism proposed for changing the radius distribution of S-type planets in binary star systems in Sullivan et al. (2023) and Sullivan et al. (2024) is the binary decreasing the timescale and mass of the disk which then inhibits core formation. Similarly giant planets, proposed to exist in high-$C$ systems, may decrease the inward pebble drift (Lambrechts & Johansen, 2014) and decrease the mass available in the protoplanetary disk Thommes (2003); Haghighipour & Scott (2012). The decrease of inward pebble drift especially that of volatile-rich material may lead to the overabundance of super-Earths. The shortening of the lifespan of the protoplanetary disk may decrease core masses and leave the planets in more excited orbits which may be unobserved and/or lead to atmosphere erosion by collisions.

The over abundance of super-Earths and under abundance of sub-Neptunes in high-$C$ systems is an effect of relatively small numbers. An additional 10 systems of three sub-Neptunes would change the result. Thus future work should continue to grow the sample of multis and searches should look for missing planets in high-$C$ systems.