The California Legacy Survey. IV. Lonely, Poor, and Eccentric: A Comparison between Solitary and Neighborly Gas Giants

This work measures the occurrence of single and multiple giant planets as a function of multiple orbital properties. Many studies of RV or transit surveys use the intuitive occurrence measurement method known as "inverse detection efficiency" (Howard et al. 2012; Petigura et al. 2013). According to this procedure, one estimates occurrence in a region of parameter space by counting up the planets found in that region, with each planet weighted by the search completeness in that region. One can measure the search completeness map of a survey by injecting many synthetic signals into each data set, and computing the fraction of signals in a given region that are recovered by the search algorithm in use. Inverse detection efficiency is actually a specific case of a Poisson likelihood method, in which one models an observed planet catalog as the product of an underlying Poisson process and empirical completeness map (Foreman-Mackey et al. 2014).

In this work, as in previous papers in the CLS (Fulton et al. 2021; Rosenthal et al. 2022), we used the Poisson likelihood method to model the occurrence of giant planets, taking measurement uncertainties into account. Given an observed population of observed planets with orbital and $M\sin i$ posteriors { ω }, and associated survey completeness map Q( ω ), and assuming that our observed planet catalog is generated by a set of independent Poisson process draws, we can evaluate a Poisson likelihood for a given occurrence model Γ( ω ∣ θ ), where θ is a vector of model parameters. The observed occurrence $\hat{{\rm{\Gamma }}}({\boldsymbol{\omega }}| {\boldsymbol{\theta }})$ of planets in our survey can be modeled as the product of the measured survey completeness and some underlying occurrence model,

$$\begin{eqnarray}&&\hat{{\rm{\Gamma }}}({\boldsymbol{\omega }}| {\boldsymbol{\theta }})=Q({\boldsymbol{\omega }}){\rm{\Gamma }}({\boldsymbol{\omega }}| {\boldsymbol{\theta }}).\end{eqnarray} \tag{ 1 }$$

The Poisson likelihood for an observed population of objects is

$$\begin{eqnarray}&&{ \mathcal L }={e}^{-\int \hat{{\rm{\Gamma }}}({\boldsymbol{\omega }}| {\boldsymbol{\theta }})\,d{\boldsymbol{\omega }}}\displaystyle \prod _{k=1}^{K}\hat{{\rm{\Gamma }}}({{\boldsymbol{\omega }}}_{k}| {\boldsymbol{\theta }}),\end{eqnarray} \tag{ 2 }$$

where K is the number of observed objects and ω _k is the kth planet's orbital parameter vector. The Poisson likelihood can be understood as the product of the probability of detecting an observed set of objects (the product term in Equation (2)), and the probability of observing no additional objects in the considered parameter space (the integral over parameter space). Another way to understand this model is that it represents a Poisson process, with some probability of generating planets with a given mass and semimajor axis rate density.

Equations (1) and (2) serve as the foundation for our occurrence model, but do not take into account the uncertainties in our measurements of planetary orbits and minimum masses. In order to do this, we use Markov Chain Monte Carlo methods to sample the orbital posteriors of each system empirically (Rosenthal et al. 2021). We can hierarchically model the orbital posteriors of each planet in our catalog by summing our occurrence model over many posterior samples for each planet. The hierarchical Poisson likelihood is therefore approximated as

$$\begin{eqnarray}&&{ \mathcal L }\approx {e}^{-\int \hat{{\rm{\Gamma }}}({\boldsymbol{\omega }}| {\boldsymbol{\theta }})\,d{\boldsymbol{\omega }}}\displaystyle \prod _{k=1}^{K}\displaystyle \frac{1}{{N}_{k}}\displaystyle \sum _{n=1}^{{N}_{k}}\displaystyle \frac{\hat{{\rm{\Gamma }}}({{\boldsymbol{\omega }}}_{k}^{n}| {\boldsymbol{\theta }})}{p({{\boldsymbol{\omega }}}_{k}^{n}| {\boldsymbol{\alpha }})},\end{eqnarray} \tag{ 3 }$$

where N_k is the number of posterior samples for the kth planet in our survey and ${{\boldsymbol{\omega }}}_{k}^{n}$ is the nth sample of the kth planet's posterior. p( ω ∣ α ) is our prior on the individual planet posteriors. We placed uniform priors on ln(Msini) and ln(a). We used emcee (Foreman-Mackey et al. 2013) to sample our hierarchical Poisson likelihood.

As done in prior eccentricity studies (Kipping 2013; Van Eylen et al. 2019; Bowler et al. 2020), we model the eccentricity distribution of a population of exoplanets with a beta distribution, because it is [0, 1] bound, flexible in its shape, and has only two model parameters. The beta distribution follows the probability density function

$$\begin{eqnarray}&&p(x|\alpha ,\beta )=\displaystyle \frac{\left.{\rm{\Gamma }}(\alpha +\beta \right)}{\left.{\rm{\Gamma }}(\alpha ){\rm{\Gamma }}(\beta \right)}{x}^{\alpha -1}{\left(1-x\right)}^{\beta -1},\end{eqnarray} \tag{ 4 }$$

where Γ(x) is the gamma function, rather than the occurrence rate in Equation (3). We only consider planets with $M\sin i$ ≥ 0.1 M_J. In other words, we are only considering the eccentricity distributions of planets with masses greater than one-third of Saturn's mass.

When we apply our hierarchical modeling framework to the eccentricity posteriors of our giant planets, we produce the likelihood

$$\begin{eqnarray}&&{ \mathcal L }({\boldsymbol{e}}| {\boldsymbol{\theta }})\propto \displaystyle \prod _{k=1}^{K}\displaystyle \frac{1}{N}\displaystyle \sum _{n=1}^{N}p({e}_{k}^{n}| {\boldsymbol{\theta }}),\end{eqnarray} \tag{ 5 }$$

where θ is the set of model parameters, K is the number of planets, N is the number of samples drawn from each planet's posterior, and e_k ⁿ is the eccentricity of the nth sample drawn from the kth planet's posterior. The normalization term in Equation (3) disappears because Equation (4) is a normalized probability density function.

In order to compare the host-star metallicities and other properties of distinct groups of planetary systems, we use simple statistical tests to compare the cumulative distribution functions, otherwise known as empirical distributions, of these groups. Specifically, we use the Anderson–Darling test, a statistical method for computing the probability that two empirical distributions were drawn from the same underlying distribution.

We use the hierarchical model described in Section 3 to characterize the distinct eccentricity distributions of single and multiple giant planets. Figures 3 and 4 show our results when we fit a beta distribution to these populations. For single giants, our model returns beta parameters $\alpha ={0.60}_{-0.12}^{+0.14}$ and $\beta ={1.45}_{-0.30}^{+0.34}$. For multiple giants, our model returns $\alpha ={1.06}_{-0.18}^{+0.21}$ and $\beta ={3.68}_{-0.70}^{+0.81}$. While we define multiplicity as the number of giant planets at any semimajor axis, our joint fits only include planets beyond 0.3 au, since giants within that distance have short tidal circularization timescales (Ogilvie 2014). For reference, our population fit to all giant planets beyond 0.3 au returns $\alpha ={0.85}_{-0.11}^{+0.13}$ and $\beta ={2.27}_{-0.32}^{+0.36}$.

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Both the beta distribution models and the raw posteriors show that single giant planets have a pileup of circular orbits and a long tail that extends to e ≤ 1, while the multiple giant planets have a more extended range of moderate eccentricity and a sharp cutoff around e ∼ 0.7. Hints of this behavior are visible in Figure 13 of Wright et al. (2009), which compared the eccentricity distributions of all single and multiplanet systems discovered to date and found a long single-planet tail not shared by the multiples. Bryan et al. (2016) reported a similar result with a larger planet sample.

One possible explanation is that multiple giant systems have some probability of experiencing secular interactions or scattering events, whereas truly single giants need strong disk interactions or Kozai–Lidov interactions with a stellar binary to reach high eccentricities. This would imply that the observed population of more circular single giants have not experienced companion-driven excitation, while the moderately eccentric multiples and highly eccentric singles represent different outcomes from planet–planet interactions. Eccentric giants that appear to be lonely now may have been born with nearby companions and scattered them into ejection or tidal engulfment, while systems with two observed, moderately eccentric giants experienced dynamical interactions that were strong enough to excite but not eject or cause engulfment. The alternative of Kozai–Lidov excitation by binary stars is entirely possible, particularly since a Gaia query reveals that most of the 12 CLS singles with e > 0.5 have wide stellar companions.

We compare the host-star metallicity distributions of systems with only one observed giant planet and multiple observed giant planets to determine whether these two groups are different. Figure 5 shows that hosts of multiple giant planets are distinctly more metal rich than hosts of only one detected giant planet, with 82.2% significance, that being the fraction of Anderson–Darling tests performed on many samples from our distributions that fall within a p-value p ≤ 0.01. This finding is in agreement with the trend first observed in Fischer & Valenti (2005). We split our giant host sample along its median metallicity and found that the search completeness contours for the two subsamples are nearly identical, as seen in Figure 6. This provides confidence that the difference in these two distributions is not caused by more extensive RV observations of our most metal-rich host stars.

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We can interpret this phenomenon in several ways. One possible explanation is that the probability of more than one giant planet forming around a star increases with increasing host metallicity. This is consistent with expectations from the core accretion theory of giant formation (Lissauer et al. 2009). Another possible explanation is that metal-richer stars are more likely to form multiple giant planets closer in, and that our sample of single giants contains additional undetected giant companions beyond 30 au. This is disfavored by the results from Fulton et al. (2021), which found evidence that giant planets are less common beyond 8 au than within 1–8 au.

In light of our single–multiple metallicity result, we explore whether the eccentricity distribution or semimajor axis distribution of the giant planet population is also metallicity dependent. The hosts of circular (e < 0.2) giant planets and hosts of eccentric (e ≥ 0.2) giant planets have median metallicities separated by less than 1σ in their standard errors, and Anderson–Darling tests produce p-values almost entirely above 0.01. The same is true of hot (a < 0.1) giant hosts with respect to colder (a ≥ 0.1) giant hosts. This rules out the possibility that our single–multiple metallicity result is driven by a or e as confounding factors, rather than multiplicity itself.

Splitting single giants along e ≥ 0.5, further along the long tail of eccentric singles, leads to inconclusive results. Figure 7 implies that there is not a conclusive difference in host metallicity between single e ≥ 0.5 hosts and hosts of less eccentric singles, and Anderson–Darling tests also show no conclusive difference. The medians of the two groups are separated by just 1.1σ. However, the same is all true of the difference between single e ≥ 0.5 hosts and all multiple giant hosts; Anderson–Darling tests are inconclusive, and the medians of the two groups are separated by 1σ. We therefore conclude that our sample of single eccentric planets is not large enough to differentiate its metallicity distribution from that of either the population of single planets on circular orbits or the multigiant population. This is not surprising, as our sample only contains 12 single planets with eccentricities higher than 0.5. The two comparison samples are proportionally larger, with 52 systems with single giants on low eccentricity orbits (e ≤ 0.5) and 31 systems with multiple giant planets.

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We use our Poisson likelihood model to compute occurrence as a function of orbital distance for the distinct single giant and multiple giant populations. Figure 8 shows our results. There is a hot Jupiter pileup within 0.6 au for lonely giants, but no such pileup for neighborly giants. We further investigate this phenomenon by generating occurrence posteriors for three classes of giants: warm Jupiters (0.1–0.4 au), moderately hot Jupiters (0.06–0.1 au), and very hot Jupiters (0.03–0.06 au). The single giant and multiple giant distributions of warm and hot Jupiters are indistinguishable, but the very hot Jupiter distributions are separated with 97.6% confidence. We measured this confidence as the fraction of random draws from the difference between distributions that is greater than zero. When we define multiplicity to include all massive companions, including brown dwarfs and stellar binaries, one very hot Jupiter changes in multiplicity status, and our separation confidence decreases to 94.4% significance.

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While very hot Jupiters may be lonely, warm Jupiters (≥0.1 au) may tell another story. Jackson et al. (2021) predict that warm Jupiters have outer giant companions with a significant nonzero frequency, which could imply that these companions interact secularly to bring the warm Jupiters to their final states. Alternatively, Huang et al. (2016) found that warm Jupiters are more likely to have small planet companions than hot Jupiters. This would imply that warmer giants are less likely to have migrated inward, since migration would disrupt the orbits of small companions. We test these hypotheses with our sample of warm Jupiters, shown in Figure 9. In a sample of 21 systems, 10 show detected outer giant companions, while 11 do not. We satisfy the prediction of Jackson et al. (2021), and therefore cannot rule out secular migration. Further resolving this question requires a sample that is uniformly sensitive to both cold gas giants and small rocky planets, which is not the case for the CLS.

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By investigating occurrence as a function of mass for single and multiple giant planets separately, we may be able to infer whether the two groups form via different pathways. Figure 10 shows occurrence rates with respect to $M\sin i$ of both single and multiple giants within 0.1–2 au (multiples can include planets outside this range). These two distributions are ≤2σ separated in all four considered ranges of $M\sin i$, but the multiples distribution tends toward higher masses than the singles distribution. Leaving aside our occurrence model, the median $M\sin i$ of our single giant planets is 0.92 M_J, while the median $M\sin i$ of giant multiples is 1.71 M_J. While we cannot assume that the giant masses are normally distributed, we can report that the standard error in the mean $M\sin i$ for singles is 0.33 M_J, and that the standard error in the mean $M\sin i$ for multiples is 0.42 M_J.

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Stepping back from our occurrence model, we can compare the cumulative distributions of the two groups and make assumptions that the average completeness contours for the single hosts and multiple hosts are nearly identical, similar to the metallicity-split contours in Figure 6. Figure 11 shows these cumulative distributions. Anderson–Darling tests show that there is a >99.7% chance of a p-value below 0.01 for matching single and multiple giants within 3 au. This can be interpreted as the degree of confidence in the claim that giants in multiples are consistently more massive than single giants. Additionally, while eccentric singles appear to be more massive than circular singles, this could be a selection effect of more eccentric detections with greater mass.

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Comparative work requires taking a broader view of giant multiplicity, beyond hot or warm Jupiters. Figure 12 shows the observed giant multiplicity distribution of the CLS. Approximately two-thirds of all stars that host a giant planet host only one observed giant planet, while one-third of giant hosts show two or more observed giants. This distribution is not corrected for search completeness; it only shows the multiplicity of detected giant planets. This means that at least one-third of stars that host one giant planet also host two or more giant planets, unless we are somehow less sensitive to single giants than to multiple giants.

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By looking at systems of multiple giant planets, we can also explore how important global disk properties are for setting the final planet masses compared to the time line and location of each individual planet's formation. Focusing on multiple giant systems, we investigate whether paired giant planets have correlated masses. Specifically, we compare the $M\sin i$ values of neighboring giant planets, for each neighboring giant pair in a system. We measure a Pearson correlation coefficient of R = 0.476 ± 0.010 (the error comes from bootstrapping the measured $M\sin i$ values with uncertainties). We use two independent methods to investigate whether sampling a population of uncorrelated giant pairs may produce this value of R.

In one case, we use bootstrapping to approximate uncorrelated populations using the observed CLS catalog. First, we repeatedly draw two giant planets from the CLS catalog until we have 31 pairs, the same size as the number of observed multigiant systems. We sample without replacement to ensure that we are shuffling our entire giant catalog, rather than drawing the same planets multiple times. Then, we measure the Pearson correlation coefficient of this shuffled population. We also measure the standard deviation in the log ratio of planet pair $M\sin i$ values, and assume coplanarity to simplify this to ${\sigma }_{\mathrm{ln}({M}_{2}/{M}_{1})}$. It must be noted that coplanarity between giant planets is not a given, and that this limits the significance of our results.

We repeat this process 10⁵ times, until we build up bootstrapped distributions of R and ${\sigma }_{\mathrm{ln}({M}_{2}/{M}_{1})}$. Our measured R falls just outside the 99.7% confidence interval of the bootstrapped distribution, at the 99.71% confidence interval. This means that there is a 0.29% probability that an uncorrelated giant planet population could produce a Pearson correlation coefficient of 0.45. Our measured ${\sigma }_{\mathrm{ln}({M}_{2}/{M}_{1})}$ falls on the 1.4% confidence interval of the bootstrapped distribution.

We also use a synthetic population experiment to measure the significance of our observed correlation independently. We drew giant planet pairs from a uniform ln($M\sin i$ ) distribution and the semimajor axis distribution reported in Fulton et al. (2021), with a minimum period semimajor axis ratio ln(a₂/a₁) > 0.5. We impose this limit on the semimajor axis ratio in order to prevent unlikely or unphysical draws of pairs within 3:2 resonance. We took our search completeness contours into account to only select simulated planet pairs for which both planets are detected. We detail our synthetic population algorithm in the list below.

• 1.  
  Draw two values of ln($M\sin i$) from a uniform distribution between ln(30 M_⊕) and ln(6000 M_⊕), and two values of ln(a) from the nonparametric giant planet model reported in Fulton et al. (2021), bounded between 0.1 and 10 au. We approximate this distribution as piecewise uniform in ln(a), with a break at 1 au, integrated probability of 2/11 between 0.1 and 1 au, and integrated probability of 9/11 between 1 and 10 au.
• 2.  
  Measure the search completeness and draw a random number from a uniform 0–1 distribution for each set of ln(a) and ln($M\sin i$). For each set, if the random number drawn is less than the detection probability at those parameters, report the planet corresponding to that set as detected.
• 3.  
  Keep the planet pair if we detect both planets and ln(a₂/a₁) > 0.5, otherwise reject it.
• 4.  
  Repeat until we have drawn the desired number of simulated planet pairs for one synthetic population.
• 5.  
  Repeat until we have drawn the desired number of synthetic populations.

With 10⁴ synthetic populations of 31 observed planet pairs each, this test finds that our observed correlation falls just within the 99% confidence interval of the distribution that our uncorrelated simulations produces.

Dawson & Murray-Clay (2013) found that giants with metal-rich hosts show signatures of planet–planet scattering. In particular, giant planets between 0.1 and 1 au are likely to be more eccentric when they orbit stars with [Fe/H] > 0. This higher eccentricity could be caused by planet–planet scattering events, which would imply that metal-rich stars are more likely to host multiple giant planets. Our results support this hypothesis, as we find that systems of multiple giant planets are more common around metal-rich stars, and that systems with multiple giant planets have higher average eccentricities than those in single giant planet systems. Table 1 shows the above properties side by side for the two populations.

Table 1. Summary of Population Comparison between Single Giant Systems and Multigiant Systems

Property	Characteristic	Single Giant Planets	Multiple Giant Planets
Eccentricity
	10th percentile, e10%	0.03	0.03
	Median, e50%	0.20	0.23
	90th percentile, e90%	0.77	0.47
	Beta distribution α	${0.60}_{-0.12}^{+0.14}$	${1.06}_{-0.18}^{+0.21}$
	Beta distribution β	${1.45}_{-0.30}^{+0.34}$	${3.68}_{-0.70}^{+0.81}$
Semimajor Axis
	Comparison	Hot Jupiter pileup	Hot Jupiter deficit
Mass
	Median	0.92 MJ	1.71 MJ
	Mean	2.07 ± 0.33 MJ	2.88 ± 0.42 MJ
	Feature	Less massive	Intrasystem uniformity
Iron Abundance
	〈[ Fe/H ]〉	0.129 ± 0.019 dex	0.228 ± 0.027 dex
	Comparison	Enriched over solar	More enriched
Stellar Binarity
	Fraction Bound (Lit.)	26.2% ± 6.3%	25.8% ± 9.1%

Note. Fraction of bound binaries comes from a literature estimate with Simbad and the Washington Double-star Catalog. Uncertainties come from Poisson counting errors.

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Although this story is clear, our data are less definitive on the question of how the subpopulation of single giants on highly eccentric orbits acquired their eccentricities. There are few other mechanisms that could explain the metal-poor, high-eccentricity tail of lonely giants above e ∼ 0.6. The most likely alternative to scattering is the Kozai–Lidov mechanism, by which a massive outer companion secularly perturbs the orbit of an inner companion, potentially exciting giants to e > 0.6, but only given a high mutual inclination and absence of other dynamical factors (Naoz 2016). Even if this could account for all of our single, highly eccentric giants, it would require the presence of stellar binary companions or undetected giant companions in these systems. We went back and examined the RV data sets for the eccentric single planets in our sample, and found that four out of 65 single giant systems have ≥3σ significant parabolic or linear trends, and two of these (HD 34445 and HD 156668) show strong a linear correlation between their RV trends and S-values, which we can treat as a proxy for magnetic activity. Therefore, only two out of 65 single giant systems show RV evidence for a stellar binary companion (HD 195019 and HD 145934), and neither of these systems host giants with e ≥ 0.2. HD 195019 has a visually resolved wide-orbit binary companion (Lu et al. 1987), while HD 145934's status is unclear.

Alternatively, it is possible that most of the single planets in our sample do in fact have additional sub-Jovian giant companions located outside 10 au. Our survey may be missing giant planets beyond this distance, and therefore misidentifying multigiant systems as single giant systems. As seen in Figure 13, the majority of CLS-detected giant planet pairs have a semimajor axis ratio less than 10, with a peak ratio frequency around 3. This means that our coldest single giants around 10–20 au could have companions out around 30–60 au, where they would be undetectable if they were small enough and on circular orbits. Again, this is disfavored by the giant planet falloff beyond 8 au reported in Fulton et al. (2021). Additionally, there could be highly inclined giant planets in our sample that evade RV detection and enhance the apparent population of highly eccentric singles.

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From where do our lonely and highly eccentric giants come? Out of 50 CLS single giant planets beyond 0.3 au, 11 have median eccentricities greater than 0.5. CLS contains 30 multigiant systems with at least one giant beyond 0.3 au. If the eccentric singles all used to be scattered multiples, that would imply that 11 out of 41 multiple systems, or roughly a quarter, scattered themselves into solitude. Carrera et al. (2019) found that the observed eccentricity distribution of giants within 5 au is compatible with scattering accounting for all giants with e ≥ 0.3. This supports the claim that many highly eccentric singles used to be multiples that experienced scattering events. However, our poorly constrained metallicity distribution for high-eccentricity giant hosts limits our ability to make claims about their original multiplicity.

We leave this discussion with more questions than when we began, but are hopeful that future work will elucidate these mysteries. Longer RV baselines for eccentric single systems and imaging campaigns to look for distant stellar or substellar companions could clarify the statuses of our single and eccentric giants, while synthetic population models might have something interesting to say about the origins of our observed eccentricity and metallicity distributions.

Knutson et al. (2014) performed an RV-trend search for massive companions to hot Jupiters, and found that 50% ± 10% of giants within 0.7–11 days host an outer companion between 1–13 M_J and 1–20 au. More than two-thirds of their hot Jupiters with companions have orbital periods within 6 days. In the CLS sample, three out of 15 hot Jupiters within 10 days host an outer companion in the same range (HD 187123, HD 217107, and HD 9826). That gives a lower bound on the companion rate of 20% ± 12%.

These two rates are compatible within 2.14σ, and therefore consistent with the companion occurrence rate reported in Knutson et al. (2014). In order to make a more direct comparison we would need to calculate the completeness-corrected companion rate for the hot Jupiters in the Legacy sample, but this is outside the scope of the current study. It is also worth noting that many of the companions reported in Knutson et al. (2014) were detected as linear trends with 5–8 yr baselines. These systems would benefit from additional RV follow up in order to explore potential false positive scenarios due to stellar activity more rigorously and better constrain the masses and orbital semimajor axes of the companions.

Previous studies of sub-Neptune-sized transiting planets have found that planets located in the same system have correlated masses and radii (Millholland et al. 2017; Weiss et al. 2018). Our study reveals that giant planets forming in the same system also appear to have correlated masses. This could imply that the global properties of protoplanetary disks strongly influence the final masses of their nascent giant planets, and that these properties are more important than solid surface density and other properties that vary radially or over time. That condition could be the initial protoplanetary disk mass, since disk mass determines how much material is available during giant planet formation, solid mass for large core formation, or disk lifetime, since quickly dissipating disks have less time to make giants.

Weiss et al. (2018) measured a Pearson correlation of R = 0.65 for the paired planet radii of a larger sample than the CLS. This means that the magnitude of the "small peas in a pod" effect is significantly greater than that of the "giant peas in a pod" effect. The final masses of giant planets are very sensitive to the relative timing of the onset of runaway accretion. Giant planets can also cut off the flow of gas to the inner disk, which could affect the ability of an inner companion to accrete a massive gas envelope. This might explain the weaker correlation between giant masses.