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REVIEW 4 major objections 5 minor 118 references

On the Ordering of Exoplanet Systems

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Kepler multi-planet systems show a persistent tendency for inner planets to be smaller than outer planets, even after accounting for detection biases.

desk verdict Useful catalog of size-ordering trends in Kepler multi-planet systems, but the paper overreaches when it claims the inner-smaller trend is not primarily a selection artifact; the bias controls don't actually test the selection function. read the letter →

arxiv 2508.13274 v1 pith:LTOCMT6D submitted 2025-08-18 astro-ph.EP

Michael Lozovsky , Hagai B. Perets This is my paper · ORCID
classification astro-ph.EP
keywords exoplanetsystemsplanetsizeorderingKeplermulti-planetformationorbitalmigrationplanetaryresonancesstellarmetallicityobservationalbiases
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 argues that within multi-planet exoplanet systems, planet sizes are not arranged by chance: inner planets tend to be smaller than their outer neighbours, and this 'smaller-inner' ordering is present even after restricting the sample to a region where transit detection biases are weak. The evidence comes from Kepler systems with two to four planets in the public exoplanet catalog, classified by the relative-size sequence of their planets and by the radius ratios of every planet pair. If the trend is real, the ordering of planet sizes inside a system becomes a new observational handle on planet formation and evolution, favouring models in which larger planets assemble farther out and migrate inward while photoevaporation shrinks close-in planets. The paper also finds that the ordering depends on host-star metallicity and that resonant pairs do not have a distinctive size-ratio distribution, which it reads as possible evidence that resonant chains formed early and were later destabilized.

What carries the argument

The argument runs on ordinal size configurations combined with pair radius-ratio distributions. Each system is assigned a sequence such as '12' for a smaller inner planet or '321' for a largest innermost planet, and the frequencies of these configurations are compared across the full and de-biased samples, stellar types, metallicities, and planet multiplicities. For each pair, the radius ratio $R_{\mathrm{in}}/R_{\mathrm{out}}$ and period ratio $P_{\mathrm{in}}/P_{\mathrm{out}}$ carry the quantitative signal, tested with Anderson–Darling comparisons of distributions and Fisher exact tests on small configuration counts. The de-biased sample is defined by hand-chosen cuts of $R > 2\,R_\oplus$ and $P < 50$ days, anchored to Kepler's roughly 90 percent detection-completeness region. Synthetic samples—random shuffles within multi-planet systems and random pairings of single-planet hosts matched in stellar mass and Hill-stability—serve as null models for what chance or bias alone would produce.

What would settle it

Forward-model Kepler's detection efficiency: inject synthetic multi-planet populations with no intrinsic radial size gradient, run them through the selection function, and check whether the observed '12' versus '21' counts and the $R_{\mathrm{in}}/R_{\mathrm{out}}$ distributions can be reproduced by bias alone. If they can, the claim that the ordering is intrinsic collapses; if the inner-smaller trend disappears when the same ordering analysis is applied to a sample with an independent selection function, the de-biasing is insufficient to establish it.

Watch

Extended reading notes

Core claim

The central claim is that inner planets in Kepler multi-planet systems are systematically smaller than outer planets, and that this ordering survives the authors' de-biasing cut (radii above two Earth radii and periods below fifty days), so it is not merely a product of transit detection biases. For two-planet systems, the configuration with a smaller inner planet outnumbers the reverse by 273 to 93 in the full sample and 153 to 68 after de-biasing. In three-planet systems the effect is strongest for the innermost pair and weakens for the outermost pair, which instead shows similar-sized planets in the 'peas in a pod' style. The observed radius-ratio distributions differ from a synthetic homogeneous sample and from synthetic pairs built from single-planet systems, supporting an intrinsic origin. The paper also reports a metallicity dependence of the inner-to-outer radius-ratio distribution, and no significant difference between resonant and non-resonant pairs, a null result it argues is in tension with simple resonant-capture expectations.

Load-bearing premise

The trend's robustness rests on the de-biased sample, a hand-picked window of radii above two Earth radii and periods under fifty days that the paper concedes may still have minor selection effects; if that window preferentially hides small inner planets or large outer planets, the apparent inner-smaller ordering could be manufactured by the cut itself.

Editorial extensions

If this is right

  • The inner-smaller ordering becomes a testable constraint for planet formation models, which the paper notes have not yet produced predictions for ordering.
  • Because the ordering's strength varies with pair location and multiplicity, formation and evolution codes will need to reproduce not just individual planet sizes but their relative arrangement within a system.
  • The metallicity-dependent radius-ratio distribution ties final system architecture to protoplanetary disk composition, giving observers a way to connect initial conditions to outcomes.
  • The null result for resonant pairs implies that if resonance capture shaped these systems, later destabilization must be common enough to erase any expected size-ratio signature.
  • Transit-selected multi-planet samples are biased toward nearly coplanar, dynamically quiet systems, so the trend may describe that subset rather than the full planetary population.

Reading between the lines

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

  • If the ordering is genuinely physical, mass ordering from transit-timing or radial-velocity measurements could be tested as a cleaner surrogate, because radii can be inflated by atmospheres and blur the formation signal.
  • A natural extension is to split resonant pairs by resonance order (2:1 versus 3:2) or by direct libration confirmation; the paper's null result may conceal a signal specific to certain resonances.
  • One discriminating prediction: if photoevaporation is a main driver, the inner-smaller trend should weaken or strengthen with stellar age and irradiation in a way the paper does not test, since envelope loss accumulates over time.
  • The synthetic single-planet comparison implies that blindly pairing single-planet hosts yields the opposite ordering; explaining why single-planet hosts differ from multi-planet hosts may itself be a clue about divergent formation pathways.
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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 / 5 minor

Summary. The paper analyzes the relative radii ordering of planets within multi-planet systems, using NASA Exoplanet Archive data, and focuses on Kepler systems with two to four planets. Systems are classified by the sequence of planet sizes from the innermost to outermost orbit (e.g., "12" vs "21" for pairs), and the analysis is repeated on a hand-defined "de-biased" sample with R > 2 R_Earth and P < 50 days. The paper reports central counts of 273 vs 93 for "12" vs "21" in the full two-planet sample and 153 vs 68 in the de-biased sample, finds that the trend is strongest for inner pairs in three-planet systems, claims a metallicity dependence using a split at [Fe/H] = -0.2, finds no significant difference between resonant and non-resonant pairs, and interprets the results as evidence that larger planets form farther out and migrate inward. The central conclusion is that the inner-smaller ordering is intrinsic and not solely a product of observational biases.

Significance. If the intrinsic-ordering claim were established, the paper would introduce a useful and underused observable: the relative size ordering within a system, which can constrain formation, migration, and subsequent dynamical evolution. The paper has clear strengths: transparent contingency tables with counts, use of public data, simple and reproducible statistical tests, and a falsifiable synthetic null. The observed trend in the Kepler sample is genuinely interesting and worth reporting. However, the load-bearing robustness claim rests on incomplete bias controls: the de-biased box does not remove the within-pair detectability gradient, the synthetic single-planet pairing is not a Kepler completeness forward model, and the metallicity split is chosen post hoc. The paper is therefore best viewed as a solid observed-trend study whose interpretation as an intrinsic property needs substantially more support.

major comments (4)
  1. [§3.1, §5.1, §6] The de-biased sample does not establish that the trend is unbiased. The cuts R > 2 R_Earth and P < 50 days remove small planets but do not correct the within-pair detection gradient: for two transiting planets, the outer planet has a longer period, fewer transits, and a lower geometric transit probability, so a population with random or even mildly inner-larger radii will preferentially yield detected pairs with larger outer planets. The Fisher exact tests in §5.1 (Table 8) only show that applying the cut does not significantly change the configuration mix compared with the full sample (p-values 0.2219 to 0.8918), not that the mix equals the intrinsic population. Since the central claim in §6 that the trend is "not solely a product of observational biases" rests on this control, that claim is currently unsupported. The paper itself concedes in §3.1 that the region "may still experience some minor selection effects," but the relevant point is that the selection effect is not minor for the ordering statistic.
  2. [§6.3, Table 13, Figure 15] The synthetic single-planet pairing is not a Kepler completeness forward model. The synthetic pairs are drawn from single-planet detections, preserving the marginal period and radius distributions, but the pairs are never passed through a detection pipeline, and single-planet and multi-planet samples have different selection functions (multiplicity-dependent completeness, mutual inclination, pipeline efficiency). Demonstrating that the observed ordering differs from this synthetic null only shows non-random pairing relative to that particular null; it does not establish that an unbiased population would produce the observed ordering. The text in §6.3 states that the synthetic sample "was constructed to replicate these biases," but the construction replicates only the marginal distributions of single-planet detections, not the joint detection probability for pairs. This is load-bearing for the "intrinsic" part of the central claim.
  3. [§4.3, §6.1, Figures 13-14] The metallicity split at [Fe/H] = -0.2 is chosen post hoc from the same data used to test metallicity dependence. The text in §4.3 states that the value "divides the two-planet sample into two roughly equal parts," and this same split is then used in §6.1 to claim a significant metallicity dependence of the radius-ratio distribution (Table 12). This is circular for the metallicity claim. The analysis should either use a pre-specified threshold, demonstrate robustness across a range of thresholds, or otherwise treat the split as a discovery that requires independent confirmation. In addition, the many pairwise Anderson-Darling and Fisher tests in Tables 9-12 are reported without any multiple-testing correction, so some of the "significant" results, including the metallicity contrast, may be chance findings.
  4. [§3.1, Tables 1-3, Figures 3-4] The analysis compares planetary radii without propagating their uncertainties. For typical Kepler radius uncertainties of several percent to ten percent, many adjacent planets in a system may be consistent with equal sizes, and the configuration labels "12" versus "21" as well as the ratios R_in/R_out can flip under plausible radius errors. The central count statistics, such as the 273 vs 93 in Table 1, therefore mix real ordering signal with measurement noise. The robustness of the trend should be checked by Monte Carlo resampling of radii within published uncertainties, or by restricting the ordering analysis to pairs with a radius difference that is significant at, say, the 2-sigma level. Without such a test, the quantitative strength of the trend is uncertain.
minor comments (5)
  1. [§3.1 and abstract] Please clarify whether the "full" sample is restricted to Kepler detections or includes all missions in the NASA Exoplanet Archive. Section 3.1 describes a "wide and heterogeneous sample" while the abstract and Section 6 refer specifically to "Kepler multi-planet systems." If the full sample contains non-Kepler planets, the comparison between full and de-biased samples mixes different selection functions.
  2. [§5.2] The text says that shuffling the two-planet sample 100 times results in 34,600 synthetic pairs, but 366 pairs repeated 100 times gives 36,600 pairs; please correct the arithmetic.
  3. [Table 11 and surrounding text] The description of the bold p-values is inconsistent: the text says bold values indicate failure to reject the null hypothesis, while the table caption says bold values indicate rejection of the null hypothesis. Please align the notation with the intended meaning.
  4. [Figure 1 and §3.1] The figure caption and the text disagree about which region is the de-biased sample: Figure 1 shows a shaded rectangle R > 2 R_Earth and P < 50 days, but the following paragraph says "the planet above the dashed green line is part of the de-biased sample," which describes a different selection. Please correct the wording.
  5. [§3.1] There are several typographical errors, including "de-baised" for "de-biased" and "refereed" for "referred," and the phrase "we select planes in a region" should be "we select planets in a region."

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central ordering trend is a direct statistical comparison, the de-biased cut and metallicity split are not fitted to the outcome, and self-citations are not load-bearing.

full rationale

The central claim of inner-smaller ordering is derived from direct radius and period measurements from the NASA Exoplanet Archive, classified into ordinal configurations (Section 3.2) and tested against a shuffled 'homogeneous' null in Section 5.2. The Anderson-Darling comparisons in Table 11 show observed two-planet and three-planet-pair radius-ratio distributions differ from the permutation null with p = 0.001-0.004, so the ordering result does not reduce to an input parameter; it is an empirical comparison. The 'de-biased' sample is an a priori box (R > 2 R_Earth, P < 50 d) chosen from a published Kepler completeness region, not tuned to maximize the inner-smaller configuration, and the Fisher tests in Section 5.1 only compare full vs. de-biased configuration counts. The paper also concedes that this box 'may still experience some minor selection effects' (Section 3.1) and discusses multiplicity bias in Section 6.2; these caveats weaken the bias-free interpretation but are not circularity. The metallicity split at [Fe/H] = -0.2 is chosen 'to divide the two-planet sample into two roughly equal parts' (Section 4.3), i.e., by sample size, not by the Rin/Rout distributions that are later compared; the High Z / Low Z categories are defined by stellar metallicity, not by the outcome statistic, so the metallicity-dependence test is not equivalent to its input. The synthetic single-planet pairing in Section 6.3 preserves marginal period/radius distributions of single-planet detections and applies Gladman stability, serving as an external null rather than a fitted prediction; whether it adequately replicates all Kepler biases is a validity concern, not a circular one. Self-citations (Lozovsky et al. 2018, 2021; Helled et al. 2016) are used for secondary inputs such as the 1.6 R_Earth rocky/non-rocky threshold (alongside Rogers 2015), stellar-type radius scaling, and radius-period correlations; none of these supplies the ordering result, and no uniqueness theorem or load-bearing premise is imported from the authors' prior work. No step in the derivation defines a quantity in terms of the target result, and no fitted parameter is renamed as a prediction. The paper is therefore self-contained with respect to its central ordering claim, and any remaining concerns are about selection-effect adequacy rather than circularity.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The analysis relies on public archive data, but the central claims depend on several hand-chosen thresholds (metallicity split, de-biased region, resonance tolerance) and on implicit assumptions about the completeness and accuracy of the catalog. No new physical entities are introduced. The pair-level tests also assume independence of pairs from the same system, which is not strictly valid.

free parameters (4)
  • metallicity split [Fe/H]_thresh = -0.2 dex
    Chosen in Section 4.3 to divide the two-planet sample into two roughly equal 'High Z' and 'Low Z' subsamples; the subsequent Anderson-Darling test showing metallicity dependence uses this data-informed threshold.
  • de-biased sample radius lower limit = 2 R_Earth
    Chosen in Section 3.1 to define a region presumed to suffer less from Kepler selection effects; not derived from a completeness model.
  • de-biased sample period upper limit = 50 days
    Same as the radius lower limit, chosen by hand to define the de-biased region in Section 3.1.
  • resonance proximity tolerance = 0.02 in period ratio
    Set in Section 4.1 based on Steffen & Hwang (2015) to classify pairs as near a first-order mean-motion resonance; a fixed threshold that affects the null result for resonant pairs.
assumptions (6)
  • domain assumption Observed planet multiplicity reflects the true number of planets in the system.
    Systems are classified by the number of planets with measured radii and periods in the archive; undetected or uncharacterized planets (e.g., non-transiting or missing parameters) could change the ordering counts. Invoked throughout Section 3.2.
  • domain assumption Planet radii are accurate enough to determine relative ordering within a system.
    The analysis assigns ordinal positions using point estimates of radii without error bars; if uncertainties are comparable to intra-system radius differences, configuration counts are partly noise. Assumed throughout Sections 3-5.
  • domain assumption The Petigura et al. (2013) 90% detection threshold applies to the Kepler multi-planet sample used here.
    Used in Section 3.1 to motivate the de-biased region; if this completeness boundary does not hold for the heterogeneous archive sample, the de-biased sample is not unbiased.
  • domain assumption Near-resonance classification by period ratio within 0.02 is a valid proxy for actual resonance.
    Adopted in Section 4.1 from Steffen & Hwang (2015); direct dynamical confirmation via libration is not performed, so the null result for resonant pairs depends on this proxy.
  • ad hoc to paper The synthetic pairing of single-planet systems with similar stellar mass and Gladman stability replicates observational biases.
    Used in Section 6.3 to argue the observed ordering is not due to biases; if the synthetic construction misses multiplicity bias or other detection effects, the conclusion is unsupported.
  • standard math Pairs extracted from the same three-planet system are treated as independent samples in the statistical tests.
    In the ab/ac/bc pair analysis, each three-planet system contributes three pairs, and the Anderson-Darling and KS tests treat them as independent, ignoring within-system correlations. This affects the significance levels reported in Tables 11 and 12.

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Pith. "Pith review of On the Ordering of Exoplanet Systems." pith.science (2026). https://pith.science/paper/LTOCMT6D

@misc{pith2026250813274,
  author       = {Pith},
  title        = {Pith review of: On the Ordering of Exoplanet Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LTOCMT6D}},
  note         = {Machine review of arXiv:2508.13274}
}
read the original abstract

We present a comprehensive analysis of planetary radii ordering within multi-planet systems, namely their ordinal position with respect to their size in a given system, utilizing data from the NASA Exoplanet Archive. In addition, we consider not only the ordinal positions but also the specific period ratios and radius ratios of planetary pairs in multi-planet systems. We explore various dependencies on stellar host type and metallicity, as well as planetary types, and explore the differences between planetary systems with different planet multiplicities and different planetary pairs in the same system. Focusing on Kepler systems with two to four planets, we account for observational biases and uncover a robust trend of smaller inner planets. This trend is particularly pronounced in inner pairs of three-planet systems and exhibits variations in stellar metallicity and planet multiplicity. Notably, we find that the distribution of inner-to-outer planet radii ratios depends on the system's metallicity, suggesting a link between initial conditions and the resulting system architecture. Interestingly, planet pairs in resonance do not exhibit significantly different size ratios compared to non-resonant pairs, challenging current theoretical expectations, again, possibly suggesting that initially resonant systems could have been later destabilized. Our findings align with planet formation and migration models where larger planets form farther out and migrate inward. Importantly, we emphasize the significance of planet ordering as a novel and crucial observable for constraining planet formation and evolution models. The observed patterns offer unique insights into the complex interplay of formation, migration, and dynamical interactions shaping planetary systems.

Figures

Figures reproduced from arXiv: 2508.13274 by the authors.

Figure 1
Figure 1. Sample of two- and three-planet systems used in this study. The dashed green line is a 90% detection threshold of Kepler, based on Petigura et al. (2013),the shaded area is area that is considered to suffer less from Kepler selection effect, and related as ”debiased” later. See sections 3.1 and 3.2. approach eliminates planets within the Period-Radius region that may be affected by data incompleteness. However, taki… view at source ↗
Figure 2
Figure 2. A planetary system is classified by the order of its planets’ relative sizes, counting from the innermost one outwards. Here we schematically present three types of systems (out of the possible six configurations): 123; 213 and 231. this way, each three-planet system is studied as three two-planet subsystem configurations( ab/3, ac/3, bc/3). The counts of pairs out of three-planet systems are shown in Figure 4and [… view at source ↗
Figure 3
Figure 3. The distribution of planetary ordering for two- (upper left), three- (upper-right) and four-planet(lower) systems. The histogram shows the full sample versus de-biased samples.The corresponding contingency tables for two-planet and three-planet systems’ sample are listed in Tables 1 and 2. Four-planet systems were not studied, due to low numbers of planets in each configuration [PITH_FULL_IMAGE:figures/full_fig_p00… view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Two planet systems compared to pair out of three planets systems.Corresponded contingency table is [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: The ordering distribution of two-planet systems. The full sample and the de-biased ones are divided into stellar types. Corresponding contingency table is [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: The ordering distribution of three-planet systems. The full sample and the de-biased ones are divided into stellar types. Corresponding contingency table is [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Two-planet systems divided into sub-groups by the planetary sizes: all the planets are above 1.6 R⊕, all the planets are below R⊕, or mixed.The corresponding contingency table is [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Same as [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Top: Cumulative distribution function (CDF) of the ratio of inner to outer planets’ radii for two-planet systems. Middle: same for two-planet pairs in three-planet systems. Bottom: Probability density function (PDF) of the ”debiased” samples as in the middle. System cl…
Figure 10
Figure 10. Figure 10: Histograms of inner-to-outer planet ratios Rin/Rout, for two-planet systems, and pairs out of three-planet systems. The dashed lines show integer rations. It is reasonable to assume that the properties of a planetary system are correlated to stellar properties, or mor…
Figure 11
Figure 11. Figure 11: Ratios of planetary radii and periods, for two-planet systems and pairs out of three-planet systems. with the highest observed for G-type stars and the lowest for M-type stars. These results suggest that the de-biasing process did not significantly alter the distribut…
Figure 12
Figure 12. Figure 12: Ratios of planetary radii and periods, for two planet systems and for sub-samples including small planets only, large planets only, or systems with one large planet and one small planet. Stellar Type Fisher Exact Test p-value Reject Null Hypothesis (True/False) M-plan…
Figure 13
Figure 13. Figure 13: Normalized cumulative distribution function (CDF) of radius ratios Rin/Rout. Planets with period ratios close to resonances (tolerance of 0.02) are colored blue, those not close to resonances are colored red, and the combined data is colored black. drawn from statisti…
Figure 14
Figure 14. Figure 14: Ratios of planetary radii and orbital periods for two-planet systems with subdivisions. ”High Z” and ”Low Z” refer to planets orbiting stars with [Fe/H] above and below -0.2, respectively. Planets with period ratios close to resonances (tolerance of 0.02) are colored …
Figure 15
Figure 15. Figure 15: Normalized distributions of synthetic two-planet systems made of one-planet systems (”Synthetic 1-planet”) and two-planet systems (”Synthetic 2-planet”) and real ”full” two-planet sample. To test the influence of observational biases on the ordering of planets in two-…

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