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The New Generation Planetary Population Synthesis (NGPPS) VIII. Impact of host star metallicity on planet occurrence rates, orbital periods, eccentricities, and radius valley morphology

T0 review · 4 major / 9 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A single, untuned planet-formation model reproduces the observed relationships between host star metallicity and planetary properties, including the shape of the radius valley.

desk verdict Solid, honest NGPPS paper with real new predictions, but the f_D/G–[Fe/H] calibration is the load-bearing assumption; deserves peer review with a demand for sensitivity tests. read the letter →

arxiv 2507.09874 v1 pith:2E2NF7IT submitted 2025-07-14 astro-ph.EP astro-ph.SR

Di-Chang Chen , Christoph Mordasini , Alexandre Emsenhuber , Remo Burn , Ji-Wei Xie , Ji-Lin Zhou This is my paper · ORCID
classification astro-ph.EPastro-ph.SR
keywords planetformationplanetarypopulationsynthesisstellarmetallicityradiusvalleyexoplanetoccurrenceratescoreaccretionorbitalmigrationphotoevaporation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether a single, physically motivated model of planet formation and evolution can explain how the host star's iron abundance shapes the kinds of planets that form. Using the Generation III Bern model, which simulates core accretion, orbital migration, N-body interactions, disk evolution, and atmospheric escape, the authors generate 1000 synthetic planetary systems at different metallicities and compare them with observed exoplanet populations after applying the same detection biases. They find that the model reproduces, without any metallicity-specific tuning, the observed correlations between [Fe/H] and planet occurrence rates, orbital periods, eccentricities, and the morphology of the radius valley—the dip in planet radii around 1.9 Earth radii. The most striking result is quantitative: all five metrics that describe the radius valley's shape match the LAMOST-Gaia-Kepler observations within about 1–2 sigma. If the model is right, the broad demographics of exoplanets can be understood as consequences of standard planet-formation physics acting on disks whose solid content scales with stellar metallicity.

What carries the argument

The load-bearing object is the Generation III Bern model, a global planet formation and evolution code that simultaneously tracks a viscous protoplanetary disk, the growth of planetary cores by planetesimal and gas accretion, Type I and Type II migration, N-body interactions among embryos and planets, giant impacts, and—after 100 million years—the long-term cooling, contraction, and photoevaporative mass loss of each planet. The quantity that carries the metallicity dependence is the disk's dust-to-gas ratio, which the model sets from the stellar iron abundance through a single mapping, $f_{D/G}/f_{D/G,\odot} = 10^{[\mathrm{Fe/H}]}$. This mapping converts every metallicity bin into an initial solid content, and all of the paper's metallicity trends—occurrence rates, periods, eccentricities, and radius-valley morphology—are produced by propagating this one scaling through the full formation and evolution calculation, after applying Kepler and radial-velocity detection biases to the synthetic systems.

What would settle it

Measure the dust-to-gas ratio in protoplanetary disks around stars spanning $-0.5 < [\mathrm{Fe/H}] < +0.5$ and test whether it follows the assumed scaling $10^{[\mathrm{Fe/H}]}$; a clear deviation would miscalibrate the synthetic population's metallicity axis and require re-examining the quantitative agreement with Chen et al. (2022).

Watch

Extended reading notes

Core claim

The paper's central claim is that the nominal Generation III Bern model—a global end-to-end simulation of planet formation and evolution that was not adjusted to reproduce any metallicity-dependent observations—produces a synthetic planetary population whose statistical properties depend on host star metallicity in the same way as the observed one. In the synthetic population, the occurrence rates of giant planets and Neptune-size planets rise with [Fe/H] (with slopes β ≈ 1.3 and β ≈ 0.5–0.8 respectively), small planets of 1–3.5 Earth radii first become more common and then less common as [Fe/H] increases past about 0.1 dex, and sub-Earths become rarer around metal-rich stars. The radius valley deepens with increasing [Fe/H]: the contrast between valley and non-valley planets grows, the ratio of super-Earths to sub-Neptunes falls, and the average radius of planets above the valley increases, while the average radius below the valley stays constant. For all five radius-valley morphology metrics defined in Chen et al. (2022), the trends in the synthetic population are quantitatively consistent with the observed trends within roughly 1–2 sigma error bars. The model also predicts that planets inside 10-day orbits are preferentially hosted by metal-rich stars and that eccentric planets are more common around metal-rich stars, though both of these correlations are significantly weaker in the model than in observations; the authors attribute the discrepancy to processes omitted from the model, such as long-term dynamical interactions and the influence of binary companions.

Load-bearing premise

The entire metallicity axis of the synthetic population rests on the assumption that a star's iron abundance maps exactly and linearly into the disk's dust-to-gas ratio through $10^{[\mathrm{Fe/H}]}$, with no dependence on disk evolution, radial drift, grain growth, or other elements.

Editorial extensions

If this is right

  • The observed diversity of exoplanet demographics as a function of stellar metallicity can be explained by standard core-accretion physics operating on disks with different solid content; no separate, metallicity-dependent formation mode is required.
  • The radius valley's deepening with [Fe/H] emerges naturally from the formation of two distinct populations—rocky super-Earths formed in situ and water-rich sub-Neptunes that migrated inward—rather than from post-formation processes alone.
  • The predicted occurrence-rate scaling laws (giants with β ≈ 1.3, Neptunes with β ≈ 0.5–0.8, sub-Earths anti-correlated) give quantitative targets for future surveys such as PLATO and Roman.
  • The model's failure to reproduce the full strength of the observed period and eccentricity correlations pinpoints missing physics: long-term dynamical instabilities and the effects of stellar companions.
  • Because the same synthetic population is compared with both transit (Kepler) and radial-velocity surveys, the work demonstrates a multi-method consistency check on planet formation theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the assumed dust-to-gas scaling $f_{D/G} \propto 10^{[\mathrm{Fe/H}]}$ is replaced by a relation that accounts for radial drift or grain growth, the predicted metallicity trends would shift; testing the model against carbon-to-oxygen ratio or other abundance diagnostics could reveal whether iron alone is the right tracer of disk solids.
  • The predicted anti-correlation between sub-Earth occurrence and [Fe/H] is a falsifiable prediction that current Kepler samples are too small to test; a dedicated search for sub-Earths around metal-poor and metal-rich stars would provide a sharp test of the model.
  • The inflection point in small-planet occurrence at [Fe/H] ≈ 0.1 dex might be a signature of the onset of giant-planet perturbation; checking whether the multiplicity of small-planet systems drops at super-solar metallicity could distinguish this from alternative explanations.
  • The discrepancy between the model's weak period/eccentricity-metallicity correlations and the stronger observed ones suggests that late dynamical evolution, rather than the initial formation environment, dominates the hot and eccentric populations; including binaries and secular chaos in a next-generation model would directly test this interpretation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 9 minor

Summary. This paper presents the eighth NGPPS paper, using the Generation III Bern model population NG76Longshot (1000 systems) to predict how planet occurrence rates, orbital periods, eccentricities, and radius-valley morphology depend on host-star [Fe/H]. After applying Kepler (KOBE) and RV detection biases, the synthetic population is compared with observational samples from Chen et al. (2022, 2023), Zhu (2019), Buchhave et al. (2014), and An et al. (2023). The main results are positive occurrence-rate slopes of β≈1.3 for giant planets and β≈0.5–0.8 for Neptune-size planets, an inflection near [Fe/H]≈0.1 dex for small planets, an anti-correlation for sub-Earths, a deepening radius valley with increasing [Fe/H], and weak but statistically significant period–metallicity and eccentricity–metallicity correlations. The authors acknowledge that the synthetic eccentricity and period trends are weaker than observed and attribute this to the neglect of long-term dynamical evolution and stellar/binary environment effects.

Significance. If accepted at face value, the claimed quantitative consistency of the radius-valley metrics with Chen et al. (2022) is a notable success for a forward population-synthesis model that was not re-fit to the metallicity trends. The paper's strengths include the use of a previously published population without parameter tuning, the application of realistic detection biases, the transparent reporting of uncertainties via bootstrap and Bayesian methods, and the explicit discussion of model discrepancies. Its main limitation is that all metallicity dependence is injected through the assumed mapping in Eq. (1); if that mapping is miscalibrated, the quantitative agreement would be a coincidence rather than a validation of the formation physics. The paper nonetheless provides a useful benchmark for the Bern model and a clear set of falsifiable predictions.

major comments (4)
  1. [Sect. 3, Eq. (1)] The assumed mapping f_D/G = 0.0149 × 10^[Fe/H] is the only channel through which stellar metallicity enters the synthetic initial conditions, so every quantitative comparison in Sections 4–7 is contingent on this calibration. The paper states the relation but does not test it against protoplanetary disk observations, nor does it quantify the impact of scatter in disk dust-to-gas ratios, of [M/H] versus [Fe/H] differences, or of refractory/volatile fractionation. The authors should add a sensitivity study that varies the exponent of Eq. (1) or adds a dispersion, and use it to bound the systematic error on the claimed 1–2σ agreements.
  2. [Sect. 4.1, Fig. 2] The hot-Jupiter analysis is based on only 12 planets, giving β = 1.3+0.9−0.6, a probability for β > 0 of 96.76% (about 2σ), and an AIC difference of 6.6 relative to a constant model. This is too weak to support the statement that the synthetic hot-Jupiter slope is quantitatively consistent with the observed β ≈ 1.6 ± 0.3; the text should either soften this claim or provide a formal assessment of how large a slope difference the 12-planet sample could actually detect.
  3. [Sect. 5, Fig. 10] The central claim that all five radius-valley metrics are consistent with Chen et al. (2022) 'within ~1–2σ' is supported only by visual inspection of Fig. 10. No formal statistic (per-metric chi-square, p-value, or overlap probability) is reported, so the reader cannot distinguish genuine agreement from agreement driven by large error bars. Please provide a quantitative comparison for each of the five metrics, including the no-dependence case R−valley.
  4. [Sect. 4.3, Fig. 4] The inflection at [Fe/H] ≈ 0.1 dex for 1–3.5 R⊕ planets is not determined by a statistical search: the data are split at 0.1 dex and monotonic fits are performed on each side. Such a procedure makes a maximum near the chosen split almost inevitable. A change-point or piecewise regression over a grid of breakpoints should be used to test whether 0.1 dex is preferred over neighboring values before this value is quoted as a quantitative result in the abstract.
minor comments (9)
  1. [Figs. 2 and 3 captions] The figure captions refer to 'the best-fits of Equation (1)', but the exponential occurrence-rate fit is Eq. (2); Eq. (1) is the dust-to-gas ratio mapping.
  2. [Table 1] The rows for Cvalley and Avalley both cite Eq. (11) and state 'positively-correlated'; the Avalley row should cite Eq. (12) and state that the slope is negative.
  3. [Sect. 7.1] The acronym 'HIRES/KICK' appears twice and should be 'HIRES/Keck'.
  4. [Sect. 6] 'Form observations' should read 'From observations'.
  5. [Sect. 4.4] The sentence beginning 'We initialize our planetary sample...' is grammatically incomplete; it should be split into two sentences.
  6. [Eq. (17)] The Gaussian kernel is missing the 1/σ normalization factor; the factor cancels in the weighted mean of Eq. (16), but the formula as written is not the stated log-normal kernel.
  7. [Sect. 4.1, references] The in-text citation 'Chen et al. 2025, submitted' has no corresponding entry in the reference list.
  8. [Sect. 5] The phrase 'p−value < 0.003 from maximizes Hartigan's dip statistic' is ungrammatical; it should read 'from a maximized Hartigan's dip statistic' or similar.
  9. [Sect. 5 vs. other sections] The paper evaluates the synthetic population at 2 Gyr for the radius-valley analysis but at 5 Gyr for all other analyses; please justify this choice and cite the age determination for the Chen et al. (2022) sample.

Circularity Check

0 steps flagged · score 1.0 of 10

No definitional circularity found: the metallicity-dependent predictions emerge from the forward simulation; the untested f_D/G–[Fe/H] mapping (Eq. 1) is an input assumption, not a circular reduction.

full rationale

The paper's central claims are that the nominal Bern model, without re-fitting to metallicity-dependent observations, reproduces observed occurrence-rate slopes, radius-valley morphology trends, and period/eccentricity correlations as functions of [Fe/H]. Walking the derivation chain, the only direct input coupling [Fe/H] to the model is Eq. (1), f_D/G / f_D/G,sun = 10^[Fe/H]. This is an explicit initial-condition mapping, not a fitted output: the synthetic population's [Fe/H] distribution is an input, and the occurrence-rate slopes (e.g. beta ~ 1.3 for giants, ~0.5–0.8 for Neptunes) and the five radius-valley metrics are computed forward from the Bern model. No parameter in this paper is fit to the observed metallicity trends and then renamed as a prediction. The comparisons in Sect. 4–7 use the five metrics defined in Chen et al. (2022) and observed slopes from Chen et al. (2023) and An et al. (2023), whose author lists overlap with the present paper. However, these are external observational analyses based on LAMOST-Kepler/CKS data, and the same trends are also benchmarked against independent works such as Johnson et al. (2010), Zhu (2019), Buchhave et al. (2014), Mulders et al. (2016), and Mills et al. (2019). The model description is cited from prior NGPPS papers by the same group, but that is normal method citation rather than a load-bearing self-citation chain; the model's physics is not justified solely by a prior paper. The paper also explicitly reports where the model disagrees with observations (e.g., weaker period and eccentricity dependences), which further indicates the comparisons are not forced by construction. Eq. (1) is a potentially fragile assumption about star–disk metallicity coupling, and the realism of this mapping is a legitimate correctness concern, but it is not circular: the predicted quantities are not defined in terms of the observed quantities they are compared with. Therefore, no circular step can be quoted, and the appropriate circularity score is low.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. It inherits a large set of fixed parameters from the Bern model (alpha, planetesimal size, embryo mass, initial conditions), none of which are re-fitted to the metallicity trends. The main ad hoc element is the visually chosen inflection point at 0.1 dex, plus the descriptive fits (beta, C, valley-metric slopes) applied to the simulation output. The load-bearing domain assumptions are the [Fe/H]-to-dust ratio mapping and the representativeness of the initial conditions.

free parameters (6)
  • turbulent viscosity parameter alpha = 2e-3
    Fixed across the population, adopted from prior NGPPS calibration; controls disk evolution and migration, thereby shaping all metallicity trends.
  • planetesimal size R_plts = 300 m
    Fixed model parameter from prior calibration; affects growth efficiency and the resulting radius distribution.
  • initial embryo mass = 0.01 M_Earth
    Fixed initial protoplanet mass inserted in every disk; influences the outcome of oligarchic growth.
  • inflection point [Fe/H]_inf = 0.1 dex (and -0.1 dex for the <2 R_Earth subsample)
    The split point between increasing and decreasing occurrence-rate regimes is chosen by visual inspection of Fig. 4 rather than fitted, then reported as a result.
  • exponential fit coefficients beta and C = e.g., beta ~ 1.3 for giants, ~0.5-0.8 for Neptunes; C varies by class
    These are fitted to the synthetic occurrence-rate data in each [Fe/H] bin; they are descriptive of the simulation output rather than inputs to the model.
  • linear slopes of radius-valley metrics = C_valley slope 15.9; A_valley slope -1.0; R+ slope 0.06; f_NP slope 0.15
    Fit to synthetic data to quantify the metallicity dependence; descriptive statistics of the simulation output.
assumptions (5)
  • domain assumption Stellar [Fe/H] maps directly to disk dust-to-gas ratio via f_D/G / f_D/G_sun = 10^[Fe/H] (Eq. 1).
    All metallicity trends in the paper scale through this mapping; if the real dust-to-gas ratio does not track stellar iron abundance, the synthetic metallicity axis is miscalibrated.
  • domain assumption Initial disk properties (gas mass, metallicity, size, inner edge, lifetime) sampled from observed young disk distributions are representative of the disk population.
    Stated in Sect. 3; the population's range and median [Fe/H] are compared to Kepler host stars via a KS test, but the disk parameter sampling itself is assumed representative.
  • domain assumption The Generation III Bern model's physical prescriptions (core accretion, type I/II migration, photoevaporation, N-body for 100 Myr) are correct as described in Emsenhuber et al. (2021a).
    The paper relies entirely on the model as an external benchmark; it does not re-derive or test these prescriptions.
  • domain assumption The KOBE program accurately simulates Kepler detection biases and completeness.
    Used for the radius-valley and small-planet eccentricity analyses in Sects. 5 and 7.2; any incompleteness in KOBE propagates into the synthetic samples.
  • domain assumption Observational sample selection (LAMOST-Gaia-Kepler, RV surveys) is comparable to the synthetic selection after applying detection biases.
    Comparisons in Sects. 4.5-7 assume the synthetic and observed samples probe the same population; the paper checks the [Fe/H] distributions but not other selection dimensions.

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Cite this review

Pith. "Pith review of The New Generation Planetary Population Synthesis (NGPPS) VIII. Impact of host star metallicity on planet occurrence rates, orbital periods, eccentricities, and radius valley morphology." pith.science (2026). https://pith.science/paper/2E2NF7IT

@misc{pith2026250709874,
  author       = {Pith},
  title        = {Pith review of: The New Generation Planetary Population Synthesis (NGPPS) VIII. Impact of host star metallicity on planet occurrence rates, orbital periods, eccentricities, and radius valley morphology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2E2NF7IT}},
  note         = {Machine review of arXiv:2507.09874}
}
read the original abstract

The dust-to-gas ratio in the protoplanetary disk, which is likely imprinted into the host star metallicity, is a property that plays a crucial role during planet formation. We aim at constraining planet formation and evolution processes by statistically analysing planetary systems generated by the Generation III Bern model, comparing with the correlations derived from observational samples. Using synthetic planets biased to observational completeness, we find that (1) the occurrence rates of large giant planets and Neptune-size planets are positively correlated with [Fe/H], while small sub-Earths exhibit an anti-correlation. In between, for sub-Neptune and super-Earth, the occurrence rate first increases and then decreases with increasing [Fe/H] with an inflection point at 0.1 dex. (2) Planets with orbital periods shorter than ten days are more likely to be found around stars with higher metallicity, and this tendency weakens with increasing planet radius. (3) Both giant planets and small planets exhibit a positive correlation between the eccentricity and [Fe/H], which could be explained by the self-excitation and perturbation of outer giant planets. (4) The radius valley deepens and becomes more prominent with increasing [Fe/H], accompanied by a lower super-Earth-to-sub-Neptune ratio. Furthermore, the average radius of the planets above the valley increases with [Fe/H]. Our nominal model successfully reproduces many observed correlations with stellar metallicity, supporting the description of physical processes and parameters included in the Bern model. However, the dependences of orbital eccentricity and period on [Fe/H] predicted by the synthetic population is however significantly weaker than observed. This discrepancy suggests that long-term dynamical interactions between planets, along with the impact of binaries/companions, can drive the system towards a dynamically hotter state.

Figures

Figures reproduced from arXiv: 2507.09874 by the authors.

Figure 1
Figure 1. Probability density functions of the stellar metallicity ([Fe/H]) for the synthetic population (black) and Kepler Sun-like planet-host stars (cyan). In the top-left corner, we also print the KS p−value and the median values (1-σ interval) of the two samples. tion, for planets whose mass is dominated by accreted solids, is how accreted volatile ices (mainly H2O, CO2, CH4, CH3OH, NH3, but the model also allows for acc… view at source ↗
Figure 2
Figure 2. Left panels: The occurrence rate of hot Jupiters (Top), warm Jupiters (Middle) and cold Jupiters (Bottom) as a function of stellar metallicity [Fe/H] from the synthetic NGPPS population and evolution model at 5 Gyr. The solid lines denote the best-fits of Equation (1). For the occurrence rate, we are more concerned with the increasing trend rather than the absolute magnitude. Thus, to compare the synthetic results w… view at source ↗
Figure 3
Figure 3. Left panels: The occurrence rate of hot Neptunes (Top), warm Neptunes (Middle) and cold Neptunes (Bottom) as a function of stellar metallicity [Fe/H] from the synthetic NGPPS population by Bern planet formation and evolution model at 5 Gyr. The solid lines denote the best-fits of Equation (1). Right panels: Marginalized posterior probability density distributions of the parameters (C, β) for the occurrence rates of … view at source ↗
Figures from the paper (18 more)
Figure 4
Figure 4. Figure 4: The occurrence rate F, the fraction of stars hosting planetary system ηsystem and the average multiplicity k as a function of stellar metallicity [Fe/H] for Kepler-like planets (Rp : 1 − 4R⊕, P < 400 d, Left panels), planets with radii between 2 − 4R⊕ (Middle panels) a…
Figure 5
Figure 5. Figure 5: Similar to [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 7
Figure 7. Figure 7: The period-radius diagram of planets selected from the synthetic population at 2 Gyr after applying the detection selection similar to PAST III (Chen et al. 2022). Planets with different compositions are plotted in different colours: red for H/He-rich envelopes, blue f…
Figure 6
Figure 6. Figure 6: The occurrence rate F (Top), the fraction of stars hosting plan￾etary system ηsystem (Middle) and the average multiplicity k (Bottom) as a function of stellar metallicity [Fe/H] for the sub-Earths from the synthetic population by Bern planet formation and evolution mod…
Figure 8
Figure 8. Figure 8: Orbital period-radius diagram of planets with different metallicity [Fe/H] from the synthetic population at 2 Gyr after applying the detection bias similar to PAST III. 1 2 4 6 0 0.05 0.10 0.15 1 2 4 6 1 2 4 6 1 2 4 6 1 2 4 6 1 2 4 6 1 2 4 6 1 2 4 6 0 0.05 0.10 0.15 […
Figure 9
Figure 9. Figure 9: Radius distribution of planets with different metallicity [Fe/H] from the synthetic population after applying a detection selection similar to PAST III. 3. For R + valley, the linear increasing model is preferred over the constant model with ∆AIC = 10.5. The best-fit i…
Figure 10
Figure 10. Figure 10: The five metrics to characterize the radius valley morphology (i.e. Cvalley, Avalley, R + valley, R − valley, and fNP) as functions of [Fe/H] for synthetic population at 2 Gyr after applying the detection bias. The solid lines denote the best fits derived from synthet…
Figure 11
Figure 11. Figure 11: The mass-radius distribution of planets with different metallicity [Fe/H] from the synthetic population after applying the detection selection similar to PAST III. distribution. Here we adopt the same log-normal kernel with σ = 0.29 as Mulders et al. (2016): K(log P, …
Figure 12
Figure 12. Figure 12: Host star metallicities vs. planet orbital period (grey dots) from the synthetic sample. The red line denotes the kernel regression of the mean metallicity of the planet population (From Equation 16). The shaded red area shows the 68% (1-σ) confidence interval on the …
Figure 15
Figure 15. Figure 15: Planetary occurrence rate as a function of orbital period for super-solar and sub-solar metallicity stars derived from the synthetic population. F(P<10days) F(P≥10days ). To obtain the uncertainties, we assume that the num￾bers of planets obey the Poisson distribution…
Figure 14
Figure 14. Figure 14: Difference in metallicity between stars hosting planets with orbital period P < 10 and > 10 days. The solid red points and error-bars show the results derived from the synthetic sample generated by the Bern model. The grey and cyan circles denote the results from Buch…
Figure 16
Figure 16. Figure 16: The semi-major axis (a) vs. effective mass (Mp sin i) for giant planets in the biased RV synthetic population at 5 Gyr. Solid points and open circles denote planets above and below detection limit, respec￾tively. The dashed line represents the typical detection limit …
Figure 17
Figure 17. Figure 17: The planetary orbital eccentricity (e) vs. stellar metallicity ([Fe/H]) for giant planets in the synthetic population at 5 Gyr after se￾lection according to the RV detection bias. We also print the Pearson coefficient and corresponding p−value. With similar criteria a…
Figure 18
Figure 18. Figure 18: The cumulative distributions of stellar metallicities [Fe/H] for high-eccentricity (red) and low-eccentricity giant planets (blue). The two sample KS test p−value is printed at the corner [PITH_FULL_IMAGE:figures/full_fig_p015_18.png]
Figure 19
Figure 19. Figure 19: The cumulative distributions of stellar metallicities [Fe/H] (Top panel) and planetary eccentricity (Bottom panel) for single giant plane￾tary systems (blue) and multiple giant planetary systems. The two sam￾ple KS test p−values are printed at the lower-right corner o…
Figure 21
Figure 21. Figure 21: The orbital eccentricity vs. stellar metallicity ([Fe/H]) for Kepler-detectable small planets in single (blue) and multiple (orange) systems in the synthetic population. We also plot the mean planetary eccentricity and their 1-σ interval (50 ± 34.1% percentiles) from …
Figure 23
Figure 23. Figure 23: The average masses of small planets (regardless whether could be detected or not) in single transiting systems without OGPs vs. stellar metallicity (grey circles). The diamonds and errorbars denote the mean and 1-σ (50 ± 34.1) interval of the average masses in differe…
Figure 22
Figure 22. Figure 22: Top: The fraction of single transiting small planets with outer giant planets (OGP) vs. stellar metallicities derived from the Kepler￾biased synthetic sample. Bottom: The orbital eccentricity vs. stel￾lar metallicity for small transiting planets with/without OGPs (pur…

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