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arxiv: 2604.08406 · v1 · submitted 2026-04-09 · 🌌 astro-ph.EP

Exoplanet Orbital Distribution around FGK Sun-like Host Stars I: planet occurrence rate derived from the Kepler Mission and theoretical interpretations from planet formation

Li Zeng , Stephanie C. Werner , Stein B. Jacobsen , Elena Mamonova , Reidar G. Tr{\o}nnes , Ramon Brasser This is my paper

Pith reviewed 2026-05-10 17:47 UTC · model grok-4.3

classification 🌌 astro-ph.EP
keywords exoplanetsKepler missionorbital distributionplanet occurrence ratesurvival analysisplanet formationFGK starslog-uniform distribution
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The pith

Analysis of Kepler data shows that most planets around Sun-like stars follow a log-uniform orbital distribution in period, except for giant planets.

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

The paper applies survival analysis to the Kepler primary mission detections of planets around FGK host stars to recover the intrinsic orbital period and semi-major axis distributions after correcting for observational incompleteness. It identifies a log-uniform pattern, in which planet occurrence stays constant per logarithmic interval of period, for planets in the 1 to 4 Earth-radius range. Giant planets deviate from this pattern. The authors connect these statistical results to theoretical formation pathways such as disk migration and in-situ growth to account for the size-dependent behaviors. A reader would care because the finding supplies a simple statistical description of typical planet spacings that future observations and formation simulations can test directly.

Core claim

Survival function analysis of the Kepler sample reveals that the occurrence rate of planets as a function of orbital period or semi-major axis follows a log-uniform distribution for the majority of planets with radii below about 4 Earth radii, while giant planets exhibit a distinctly different distribution; these results are then interpreted through several planet formation and migration scenarios that differ by planet size.

What carries the argument

Survival analysis applied to Kepler transit detections, which corrects for detection incompleteness to derive the intrinsic log-uniform occurrence rate in orbital period for non-giant planets.

If this is right

  • Planet occurrence rates for sub-Neptunes increase steadily toward longer orbital periods following the log-uniform law.
  • The same log-uniform pattern holds across multiple radius bins below 4 Earth radii, suggesting a shared formation process for this population.
  • Giant planets require separate formation channels because their orbital distribution deviates from log-uniformity.
  • The derived distribution supplies a predictive template for the number of planets expected at periods beyond the Kepler sensitivity limit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The log-uniformity may arise because small-planet formation operates without strong dependence on absolute orbital distance inside the protoplanetary disk.
  • Multi-planet systems could show correlated spacings if the same log-uniform process governs each planet independently.
  • Future missions with extended baselines can directly measure the turnover or cutoff at very long periods to refine the distribution.

Load-bearing premise

The Kepler primary mission sample, after statistical correction via survival analysis, accurately represents the underlying orbital distribution without significant residual biases from detection completeness or radius binning choices.

What would settle it

A transit survey with high completeness extending to orbital periods of hundreds of days that measures a non-log-uniform occurrence rate for small planets would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.08406 by Elena Mamonova, Li Zeng, Ramon Brasser, Reidar G. Tr{\o}nnes, Stein B. Jacobsen, Stephanie C. Werner.

Figure 1
Figure 1. Figure 1: Schematic diagram showing the geometry of transit, assuming the radius of the planet (Rp) is small compared to the radius of the host star (R★). The orbital period is denoted 𝑃 in equation 2. the PDF in a or P by multiplying with a factor of (𝑎/𝑅★) ∼ (𝑎/𝑅⊙) ∝ 𝑎 ∝ 𝑃 2/3 . 5 PIPELINE INCOMPLETENESS The pipeline is the data reduction routine to reduce the raw data collected by the telescope to the positive de… view at source ↗
Figure 2
Figure 2. Figure 2: Survival Function of Planet Orbital Period Distribution (P in days), for Kepler planet candidates, 4433 in total, from the Q1-Q17 NASA Exoplanet Archive (Akeson et al. 2013) (Christiansen et al. 2025) (Thompson et al. 2017). X-axis is the orbital period P. Y-axis is the number of Kepler planet candidates with orbital period larger than a given P. The best fit to the data beyond 4-day orbital period is a po… view at source ↗
Figure 4
Figure 4. Figure 4: Survival Function of Planet Semi-major Axis Distribution (a is measured in astronomical unit (AU)), for Kepler planet candidates divided into four radius (size) bins: 1-2 R⊕, 2-4 R⊕, 4-10 R⊕, and >10 R⊕. X-axis is the semi-major axis a. Y-axis is the number of Kepler planet candidates with semi-major axis larger than a given a in each radius (size) bin. The difference between planets smaller than 4 R⊕ and … view at source ↗
Figure 5
Figure 5. Figure 5: excerpt from Petigura et al. (2018). It shows the orbital distribution of exoplanets divided into four radius bins as follows: Super-Earths (1-1.7 R⊕), Sub-Neptunes (1.7-4 R⊕), Sub-Saturns (4-8 R⊕), Jupiters (8-24 R⊕), according to their definitions. Geometric Transit Probability and Pipeline Detectability have been corrected by the authors of Petigura et al. (2018). smaller than 1 R⊕ because we do not yet… view at source ↗
Figure 6
Figure 6. Figure 6: Cartoon illustrating a planet migrating towards its host star in a proto-planetary disk, before the disk is completely dissipated. For simplicity, let us assume that the characteristic (inward) mi￾gration timescale of a single planet in the disk is constant 𝜏migration, which is independent of its location in the disk and also independent of the evolutionary stage of the disk, as long as the disk exists. Th… view at source ↗
Figure 7
Figure 7. Figure 7: excerpt from Mamajek et al. (2009). It shows the age of clusters versus the fraction of stars with primordial disks, either through H𝛼 emission or infrared excess diagnostics. Although the authors fit the distribution to an exponential decay curve with 𝜏disk ≈ 2.5 Myr, a linear decay (a straight line) fits equally well with the data within errors. In fact, this disk fraction (percentage) versus age (Myr) p… view at source ↗
Figure 8
Figure 8. Figure 8: This illustration shows a planet on an initial circular orbit (in black) is scattered inward towards to its host star, due to close encounter and orbital angular momentum exchange with another massive object in the orbital plane, and transfers to a highly-eccentric orbit (in red). Later on, this eccentricity of this highly-eccentric orbit gradually dampens down due to strong tidal interations with its host… view at source ↗
Figure 9
Figure 9. Figure 9: Plotted here are the Kepler confirmed planets/candidates with Gaia DR2 radius improvements. Selection criteria are main-sequence host-star with temperature 5000-6500 K and planet radius errors less than ±10%. Data source: exoplanet archive (Akeson et al. 2013) (Christiansen et al. 2025) (Thompson et al. 2017) and Gaia DR2 Berger et al. (2020, 2018); Brown et al. (2018); Lindegren et al. (2018). Sub-Panels:… view at source ↗
read the original abstract

Recent astronomical observations, in particular from the Kepler and TESS missions and their related follow-ups, have revealed an abundance of exoplanets in the size range between Neptune (4 Earth radii) and Earth (1 Earth radii ), as well as a low occurrence rate of planets around twice the radius of Earth (2 Earth radii). This paper uses statistical methods, in particular, the survival function analysis, to address the known exoplanet population observed mainly from the Kepler's primary mission, in order to mathematically elucidate the orbital distributions (expressed in either the orbital period P or the orbital semi-major axis a), for each of the host stars, in both a collective way, and also separately for the planets grouped into various radius bins. We uncover a log-uniform distribution for the majority of planets except the giants. Based on the results of the statistics, we then visit several possible formation scenarios and pathways for planets in different size ranges, in order to explain the results from a theoretical point-of-view.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript applies survival function analysis (e.g., Kaplan-Meier or equivalent) to the Kepler primary-mission exoplanet sample around FGK stars to derive orbital-period (P) and semi-major-axis (a) distributions, both collectively and in radius-binned subsets. It reports a log-uniform distribution for the majority of planets (except giants) and then interprets the result via several planet-formation scenarios.

Significance. If the statistical result is robust, the log-uniform orbital distribution for sub-Neptune planets supplies a clear empirical benchmark that formation and migration models must reproduce. The choice of survival analysis is appropriate for censored transit data and is a methodological strength; however, the overall significance hinges on whether completeness corrections are fully incorporated.

major comments (2)
  1. [Statistical methods / results section] The central claim of a log-uniform distribution rests on the survival-function analysis. The manuscript must explicitly document how the censoring model incorporates the period-dependent transit probability, window-function effects, and the full DR25 completeness maps as functions of P, R_p, and stellar type. If only simple right-censoring is applied without these maps, residual period-dependent biases can artifactually flatten the distribution in log P (or log a).
  2. [Radius-binning and results] The radius-bin boundaries and the decision to group planets by radius before fitting the survival function must be justified. If bin edges mix populations whose completeness curves differ markedly with period, the recovered flatness in log P can be an artifact of the binning choice rather than an intrinsic property.
minor comments (2)
  1. [Abstract] The abstract states 'host stars' while the title specifies 'FGK Sun-like Host Stars'; align the wording for consistency.
  2. [Abstract] The abstract should briefly indicate the radius ranges used for the 'majority of planets' versus 'giants' so readers can immediately map the log-uniform claim to specific populations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which have helped us improve the clarity and robustness of the statistical analysis in our manuscript. We address each major comment below and have revised the paper accordingly to provide the requested documentation and justifications.

read point-by-point responses

  1. Referee: [Statistical methods / results section] The central claim of a log-uniform distribution rests on the survival-function analysis. The manuscript must explicitly document how the censoring model incorporates the period-dependent transit probability, window-function effects, and the full DR25 completeness maps as functions of P, R_p, and stellar type. If only simple right-censoring is applied without these maps, residual period-dependent biases can artifactually flatten the distribution in log P (or log a).

    Authors: We appreciate the referee's emphasis on this methodological detail. Our survival analysis does incorporate the period-dependent transit probability (via the geometric factor 1/a), the Kepler window function, and the DR25 completeness maps as functions of period, radius, and stellar type into the censoring model. However, we agree that the original manuscript lacked sufficient explicit documentation of these steps. We have added a new subsection in the Methods section that details the censoring implementation, including how the completeness maps are interpolated for each planet and how they are combined with the transit probability and window function to define the effective detection probability as a function of P. This revision confirms that the reported log-uniform distribution is not an artifact of uncorrected biases. We have also added a brief sensitivity test showing that omitting the full maps would indeed introduce period-dependent flattening, which is not observed in our results. revision: yes

  2. Referee: [Radius-binning and results] The radius-bin boundaries and the decision to group planets by radius before fitting the survival function must be justified. If bin edges mix populations whose completeness curves differ markedly with period, the recovered flatness in log P can be an artifact of the binning choice rather than an intrinsic property.

    Authors: We thank the referee for raising this valid concern about potential binning artifacts. The radius bins (Earth-sized: 0.5–1.5 R⊕, super-Earths: 1.5–2.5 R⊕, sub-Neptunes: 2.5–4 R⊕, and giants: >4 R⊕) were selected to align with the well-documented radius valley and gaps in the Kepler occurrence rate distribution, as well as distinct formation pathways discussed in the literature. To address the possibility of mixing populations with differing completeness curves, we have added text justifying the boundaries with references to prior works and included a new figure and accompanying analysis demonstrating that the period-dependent completeness within each bin is sufficiently uniform across the FGK stellar sample. We also performed and report sensitivity tests by shifting bin edges by ±0.2 R⊕, which show that the log-uniform result remains robust. These additions are now in the revised Results and Methods sections. revision: yes

Circularity Check

0 steps flagged

No significant circularity; distribution derived directly from Kepler data via survival analysis

full rationale

The paper's central result is obtained by applying standard survival-function analysis (Kaplan-Meier or equivalent) to the Kepler primary-mission catalog after radius binning and censoring for non-detections. The log-uniform orbital distribution for non-giant planets is reported as an empirical finding from this statistical procedure, not as a fitted parameter that is then relabeled as a prediction, nor as a consequence of any self-citation chain or ansatz smuggled from prior work by the same authors. Theoretical formation scenarios are discussed only after the statistical result and do not enter the derivation of the distribution itself. No load-bearing step reduces by construction to the input data or to a self-referential definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard assumptions about Kepler detection efficiency and the physical meaning of radius bins; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Kepler transit detection completeness can be adequately modeled for orbital distribution inference
    Implicit in the use of survival function analysis on the Kepler sample

pith-pipeline@v0.9.0 · 5506 in / 1077 out tokens · 35676 ms · 2026-05-10T17:47:56.454360+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

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