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Planetary Edge Trends (PET). I. The Inner Edge-Stellar Mass Correlation

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The innermost orbits of small planets are set by the dust sublimation radius of the protoplanetary disk, not by stellar magnetism.

desk verdict A careful measurement of the inner-edge–mass correlation in Kepler multis, with a steeper slope after metallicity correction; the slope is robust in direction, but the exact value and the dust-sublimation preference rest on an ad hoc functional form and a weak model comparison. read the letter →

arxiv 2501.02215 v2 pith:KNNITLVN submitted 2025-01-04 astro-ph.EP astro-ph.SR

Meng-Fei Sun , Ji-Wei Xie , Ji-Lin Zhou , Beibei Liu , Nikolaos Nikolaou , Sarah C. Millholland This is my paper · ORCID
classification astro-ph.EPastro-ph.SR
keywords exoplanetsinneredgestellarmassrelationmetallicitycorrectiondustsublimationradiusKeplermulti-planetsystemssuper-Earthssub-Neptunes
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 tries to establish what determines the innermost orbit of small planets in compact multi-planet systems. Using Kepler multi-planet systems with Gaia-derived stellar masses and LAMOST metallicities, the authors fit the inner edge as a power law in stellar mass and, after correcting for the metallicity projection, find $\gamma_1 \approx 0.6$–$1.1$. Comparing that slope with theoretical predictions, the pre-main-sequence dust sublimation radius of the protoplanetary disk matches best, whereas hot Jupiters are thought to be halted by the magnetospheric truncation of the gas disk. If correct, the result means small planets and giant planets stop at different physical boundaries in their natal disks.

What carries the argument

The load-bearing object is the metallicity-corrected power-law fit $a_{\mathrm{in}} = \gamma_0\,(M_\star/M_\odot)^{\gamma_1}\,10^{\gamma_2\,[\mathrm{Fe/H}]}$, fitted in log space by multiple linear regression. The argument also rests on defining the inner edge as the semimajor axis of the innermost planet in coplanar multi-transiting systems rather than inferring it from occurrence rates; an appendix quantifies how occurrence rates dilute the slope, giving $\Delta\gamma \approx -0.64$ for a $1\,M_\odot$ versus $0.5\,M_\odot$ comparison. Theoretical slopes for dust sublimation, tidal, and co-rotation mechanisms are tabulated and compared with the measured $\gamma_1$, with the active $\alpha=1$ dust-sublimation model matching the all-multiple-systems slope within $1\sigma$.

What would settle it

Split the multi-planet sample into metal-poor and metal-rich bins at fixed stellar mass and refit the inner-edge slope in each bin: if the separable power-law form is right, the mass slope should be the same in both bins, while a divergent slope would falsify the correction. A complementary test would resolve the dust sublimation radius in pre-main-sequence disks around stars of known mass and compare it directly with the inner edges of the resulting planetary systems.

Watch

Extended reading notes

Core claim

The central claim is that the inner edge of systems of small planets scales with stellar mass more steeply than previously reported once stellar metallicity is controlled: for the four population samples the metallicity-corrected power-law index is $\gamma_1 = 0.6$–$1.1$, with all multiple systems at $0.81^{+0.09}_{-0.08}$, mixed systems at $1.12^{+0.08}_{-0.07}$, super-Earth systems at $0.57^{+0.10}_{-0.11}$, and sub-Neptune systems at $0.67^{+0.17}_{-0.18}$. The authors argue that this slope agrees with the pre-main-sequence dust sublimation radius, so the innermost orbits of small planets are likely limited by the dust-destruction region of the protoplanetary disk. They further show that earlier occurrence-rate based estimates near $\gamma_1 \approx 1/3$ underestimated the correlation because outer planets dilute the inner-edge signal, and that transit selection bias does not produce the observed trend.

Load-bearing premise

The central result assumes that metallicity enters the inner-edge relation as a single power law in [Fe/H] that multiplies a mass power law, with no cross-term or break; the authors state this functional form has no strong theoretical or astrophysical basis, so if the true metallicity dependence differs, the reported slope is an artifact of the assumed model.

Editorial extensions

If this is right

  • For all multi-planet systems the metallicity-corrected slope is $\gamma_1 = 0.81^{+0.09}_{-0.08}$, notably steeper than the $\sim 1/3$ slope obtained from occurrence rates in earlier work.
  • Mixed systems containing both super-Earths and sub-Neptunes show the strongest mass dependence ($\gamma_1 = 1.12^{+0.08}_{-0.07}$), suggesting that samples containing both populations are the most sensitive to stellar mass.
  • The measured $\gamma_1 = 0.6$–$1.1$ range is consistent with active dust sublimation and, at the lower end, with stellar tides, while co-rotation and planetary tides are disfavored as the dominant controls.
  • Extrapolating the all-multiple-systems slope to A-type stars places the inner edge near $0.21$–$0.27$ AU, consistent with the observed rarity of close-in planets around A-type stars.
  • The comparison with hot-Jupiter results implies that different planet populations have different inner-edge regulators: dust sublimation for small planets versus magnetospheric truncation for giant planets.

Reading between the lines

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

  • If dust sublimation sets the inner edge, then the inner edge should also respond to disk luminosity and grain size; systems with larger grains or dimmer disks should permit planets closer in, a prediction that could be tested with spatially resolved disk surveys.
  • The innermost-planet method used here could be applied to TESS multi-planet systems to check whether the $\gamma_1 \approx 0.6$–$1.1$ slope holds outside the Kepler field and is not a survey-specific artifact.
  • By the authors' logic, single-transiting systems, once corrected for inclination and geometric bias, should show the same intrinsic slope; measuring it would separate detection geometry from the physical truncation mechanism.
  • If the boundary is set in the pre-main-sequence phase, the inner edge should correlate more tightly with stellar properties than with planet mass or system age; age-dated samples could test this ordering.
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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 / 4 minor

Summary. The paper measures the correlation between the inner edge of multi-planet systems and stellar mass using Kepler DR25, Gaia (Berger et al. 2020), and LAMOST metallicities (PAST II). Restricting to small planets in multi-transiting systems, it fits a single power law a_in ∝ M^γ1, then a multiple linear regression that adds a metallicity term 10^{γ2 [Fe/H]}. The reported metallicity-corrected exponent is γ1 = 0.6–1.1 across all multiple systems, super-Earths, sub-Neptunes, and mixed systems. The paper argues that transit selection bias does not induce the correlation, that the occurrence-rate method used by earlier work underestimates γ1, and that comparison with theoretical models favors the pre-main-sequence dust sublimation radius as the physical mechanism setting the inner edge of small planets, in contrast to hot Jupiters.

Significance. If the measurement is robust, it sharpens an important observational constraint on the inner-edge–stellar-mass relation and reconciles the discrepancy with occurrence-rate-based estimates (γ1 ≈ 0.3) by attributing it to methodological bias. The paper is careful in several respects: uncertainties are propagated by resampling stellar parameters, p-values are computed with permutation tests, the transit-selection effect is examined in dedicated simulations, and the authors explicitly acknowledge the ad hoc nature of the metallicity-correction form. The final physical conclusion, however, rests on a model comparison that is only weakly discriminating, and the headline exponent depends on a functional form whose robustness is asserted rather than demonstrated.

major comments (4)
  1. [Sec. 3.3, Eq. (8)] The headline result γ1 = 0.6–1.1 depends on the assumed log-separable metallicity term 10^{γ2 [Fe/H]} in Eq. (7)/(8). The text itself states that this form 'does not have a strong theoretical or astrophysical basis.' Since [Fe/H] correlates with log M (γ3 = 0.09–0.17 in Fig. 5) and γ2 is fitted negative, any misspecification of the metallicity dependence (e.g., a nonlinear or non-separable [Fe/H] effect) will be projected directly into the corrected γ1. The paper says alternative forms were tested but gives no quantitative results. Please report the alternative fits (e.g., quadratic [Fe/H], interaction terms, or binned analyses) and show whether γ1 remains in the quoted range; otherwise the 0.6–1.1 interval is a property of the assumed model rather than a measured range.
  2. [Sec. 4.1 and Fig. 7] The theoretical comparison fixes all model intercepts to the observed intercept and compares only slopes. With the reported 1σ uncertainties, active dust sublimation (α = 1, slope 0.78) and stellar tides (0.69) are both within 1σ of the all-multiple MLR slope 0.81 ± 0.09, and passive sublimation (k = 2, 1.0) is within 2σ. The statement that the pre-main-sequence dust sublimation radius 'best matches' the data is therefore not a quantitatively supported discrimination. Please provide a formal model comparison (e.g., likelihood or information-criterion based, with the intercept as a free parameter, or with the observed scatter included) and state which mechanisms are rejected at what confidence.
  3. [Sec. 2.1 and Table 1] The analysis uses only the 166 LAMOST-matched multi-planet systems from the larger Kepler multis sample. Requiring a metallicity measurement can introduce selection effects in stellar mass and metallicity, and the paper does not test whether the LAMOST subsample is representative of all Kepler multis. Please compare the mass, radius, and period distributions and the uncorrected γ1 of the LAMOST subsample versus the full Kepler multis sample, or apply a weighting/selection correction. This is needed to support the claim that the measured correlation is intrinsic rather than a property of the metallicity-matched subset.
  4. [Appendix A, Eqs. (A.1)–(A.10)] The appendix's quantitative claim that occurrence rates bias γ1 down to ≈0.36 from an intrinsic value of 1.0 relies on the identification f_occ ∝ a_occ = mean(a_n) in Eq. (A.3) and on adopting N_L = 3, N_H = 2 from Yang et al. (2020). The step f_occ ∝ a_occ is not derived from the occurrence-rate equations of Mulders et al. (2015) that are quoted in Eqs. (A.1)–(A.2); in particular, the star-count N* and the planet-multiplicity N are conflated. Please either derive this relation carefully from the occurrence-rate definition or reframe the appendix as an illustrative toy estimate rather than the explanation for the discrepancy with previous work.
minor comments (4)
  1. [Sec. 3.1, Fig. 3] The sub-Neptune panel reports p = 0.2177, so the statement in the text that 'a correlation ... can be observed' for sub-Neptunes should be explicitly qualified as not statistically significant at the usual threshold; the subsequent power-simulation discussion is helpful and should be connected directly to this p-value.
  2. [Sec. 3.4, Fig. 6] The normalization in Eq. (11), P_max = R*(max)/a_in(min), uses the maximum of one quantity and the minimum of another; clarify that this is indeed the intended maximum over the sample and that the resulting P_transit is bounded by unity in the simulation.
  3. [Appendix A, Fig. A.2] The caption says the systems are ordered 'from left to right' by increasing stellar mass, but the horizontal axis labels and tick marks are unclear in the figure as printed; please make the ordering explicable without reference to the caption alone.
  4. [Sec. 3.3] In the paragraph after Eq. (8), the statement that alternative functional forms 'do not have a strong theoretical or astrophysical basis' is used to justify the chosen form; this is fine, but the same paragraph should contain the quantitative robustness check requested in the major comments, not only a caveat.

Circularity Check

1 steps flagged · score 2.0 of 10

Central gamma1 measurement is an empirical MLR fit to independent Kepler/LAMOST/Gaia data; only a minor self-citation in Appendix A supports a secondary explanation, not the headline result.

  1. self citation load bearing [Appendix A, Eqs. (A.9)-(A.10); invoked in Sec. 4.2]
    "Since the number of planets in a system decreases with increasing stellar mass (Yang et al. 2020)... According to the results of Yang et al. (2020), systems with 1.0 M⊙ host stars have an average of two planets (NH = 2), while systems with 0.5 M⊙ host stars have an average of three planets (NL = 3)."

    Yang et al. (2020) is authored by J.-Y. Yang, J.-W. Xie, and J.-L. Zhou, with Xie and Zhou also co-authors of the present paper. The appendix uses this internal multiplicity-mass relation to compute the predicted attenuation Delta_gamma ~ -0.64 of the occurrence-rate method, and then inserts the paper's own conclusion (gamma1 ~ 1.0) to recover gamma_occ ~ 0.36, which it says aligns with Mulders et al. (2015). This makes the paper's explanation of the discrepancy with Mulders et al. depend on a chain beginning and ending in the authors' own results. However, this is not the central claim: the headline gamma1 = 0.6-1.1 is fitted directly from the sample via Eq. (8), and the theoretical slopes in Table 2/Fig. 7 come from external published models.

full rationale

The quantitative claim gamma1 = 0.6-1.1 is obtained by a least-squares / multiple linear regression fit of log(ain) against log(M*) and [Fe/H] (Eq. 8), using public Kepler DR25, Berger20, and PAST II data. It is an empirical fit, not a derived prediction, and no parameter is fit to a subset and then renamed as a prediction for the same subset. The metallicity correction in Eq. (7) is admittedly an adopted functional form 'without a strong theoretical or astrophysical basis'; that is a model-dependence and robustness caveat, not circularity, because the paper does not claim to derive that form from first principles and explicitly notes that alternative forms were tested. The theoretical comparison in Fig. 7 uses independent published model scalings (Liu et al. 2019; Dullemond et al. 2001; Jackson et al. 2009; Mulders et al. 2015), so it is not importing a uniqueness theorem or fitting a model onto the data. The only flagged issue is the Appendix A self-citation to Yang et al. (2020), which supports the secondary explanation for why occurrence-rate studies found a shallower slope. Because that appendix also assumes the paper's own gamma1 ~ 1.0 to reproduce Mulders' 0.3, it is more a consistency check than an independent derivation. The central measurement stands on its own regression, so the circularity score is low (2).

Assumptions & free parameters 10 free parameters · 7 assumptions · 0 invented entities

The measurement itself is a multi-variable regression; the reported gamma1 values are fitted, not predicted. The physical conclusion additionally assumes the present-day inner edge preserves the primordial dust sublimation scaling and that the model comparison can be done with the intercept fixed to the observed value. No new physical entities are introduced.

free parameters (10)
  • gamma1, all multiple systems (MLR) = 0.81 (+0.09/-0.08)
    Least-squares slope of log a_in vs log M_star in Eq. (8); central quantitative result.
  • gamma1, mixed multiple systems (MLR) = 1.12 (+0.08/-0.07)
    Subsample slope; contributes the upper bound of the 0.6 to 1.1 range.
  • gamma1, super-Earths (MLR) = 0.57 (+0.10/-0.11)
    Subsample slope after metallicity correction.
  • gamma1, sub-Neptunes (MLR) = 0.67 (+0.17/-0.18)
    Subsample slope; the underlying one-dimensional correlation is not significant (p=0.2177).
  • gamma2, metallicity coefficient, all systems (MLR) = -0.28 (+0.03/-0.03)
    Fitted coefficient for [Fe/H] in Eq. (8), applied as the metallicity correction.
  • log gamma0_amf, MLR intercept, all systems = -1.195
    Reported in Sec. 4.1 and used to set the intercept of theoretical model lines in Fig. 7.
  • alpha, active dust sublimation accretion-rate exponent = 1 and 2
    Chosen from the literature (Alcala et al. 2014); sets model slope 7/9 or 11/9.
  • k, passive dust sublimation luminosity-mass exponent = 1 and 2
    Chosen from pre-main-sequence luminosity evolution; sets model slope 0.5 or 1.0.
  • c, planet semimajor-axis ratio in occurrence-rate toy model = ~2
    Estimated from average period ratios of the sample in Fig. A.2; used to derive Delta gamma.
  • N_L, N_H, multiplicities at 0.5 and 1.0 solar mass = 3 and 2
    Taken from Yang et al. (2020); used in Eq. (A.10) to estimate the occurrence-rate bias.
assumptions (7)
  • domain assumption Kepler DR25 period and radius measurements, and Berger20 Gaia-Kepler stellar masses, are accurate enough for the fitted slopes.
    Used throughout; stellar mass errors are resampled, but systematic accuracy is not propagated.
  • domain assumption The radius valley equation of Zhu and Dong (2021) applies to classify super-Earths versus sub-Neptunes.
    Used in Eq. (1) to split samples; misclassification could change population slopes, though the authors' simulation suggests a moderate effect.
  • ad hoc to paper The metallicity correction model a_in proportional to M_star^gamma1 times 10^(gamma2 [Fe/H]) is the correct separable functional form.
    Authors state it has no strong theoretical or astrophysical basis; it is inherited from occurrence-rate papers (Johnson et al. 2010; Zhu and Dong 2021).
  • domain assumption Kepler multis are coplanar enough that the innermost transiting planet is the true inner edge.
    Used to exclude singles and define the sample in Sec. 2.2; if false, single systems would bias the edge measurement.
  • domain assumption The theoretical dust sublimation and tidal scaling laws from Liu et al. (2019), Dullemond et al. (2001), and Jackson et al. (2009) are applicable to pre-main-sequence disks.
    Used in Sec. 4.1 to generate the model slopes compared with the observation.
  • ad hoc to paper In Appendix A, f_occ proportional to a_occ equals the mean of the semimajor axes of planets in a system.
    This re-derives the occurrence-rate inner edge differently from Mulders et al. (2015) and is a toy assumption.
  • domain assumption The multiplicity-stellar mass relation from Yang et al. (2020), used to set N_L=3 and N_H=2, applies to this sample.
    Used in Eq. (A.10) to compute Delta gamma; not verified on the current sample.

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Pith. "Pith review of Planetary Edge Trends (PET). I. The Inner Edge-Stellar Mass Correlation." pith.science (2026). https://pith.science/paper/KNNITLVN

@misc{pith2026250102215,
  author       = {Pith},
  title        = {Pith review of: Planetary Edge Trends (PET). I. The Inner Edge-Stellar Mass Correlation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KNNITLVN}},
  note         = {Machine review of arXiv:2501.02215}
}
read the original abstract

The position of the innermost planet (i.e., the inner edge) in a planetary system provides important information about the relationship of the entire system to its host star properties, offering potentially valuable insights into planetary formation and evolution processes. In this work, based on the Kepler Data Release 25 (DR25) catalog combined with LAMOST and Gaia data, we investigate the correlation between stellar mass and the inner edge position across different populations of small planets in multi-planetary systems, such as super-Earths and sub-Neptunes. By correcting for the influence of stellar metallicity and analyzing the impact of observational selection effects, we confirm the trend that as stellar mass increases, the position of the inner edge shifts outward. Our results reveal a stronger correlation between the inner edge and stellar mass with a power-law index of 0.6-1.1, which is larger compared to previous studies. The stronger correlation in our findings is primarily attributed to two factors: first, the metallicity correction applied in this work enhances the correlation; second, the previous use of occurrence rates to trace the inner edge weakens the observed correlation. Through comparison between observed statistical results and current theoretical models, we find that the pre-main-sequence (PMS) dust sublimation radius of the protoplanetary disk best matches the observed inner edge stellar mass. Therefore, we conclude that the inner dust disk likely limits the innermost orbits of small planets, contrasting with the inner edges of hot Jupiters, which are associated with the magnetospheres of gas disks, as suggested by previous studies. This highlights that the inner edges of different planetary populations are likely regulated by distinct mechanisms.

Figures

Figures reproduced from arXiv: 2501.02215 by the authors.

Figure 1
Figure 1. Distribution of stars and planets for all multis studied in this paper. The left panel illustrates the distribution of stars, while the right panel presents the distribution of planets. Solid points denote the inner edge of the system, while hollow points represent other planets within the system. Red points indicate super-Earths and blue points represent sub-Neptunes, classified according to the radius valley defin… view at source ↗
Figure 2
Figure 2. Architecture of the three population samples of all multis studied in this paper. The left panel represents super-Earths, the middle panel represents sub-Neptunes, and the right panel represents mixed systems containing both super-Earths and sub-Neptunes within the same system. The point colors and styles are the same as those in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. The inner edge of planetary systems as a function of stellar mass for the datasets of all multiple systems, super-Earth systems, sub-Neptune systems, and mixed multiple systems. The four panels illustrate the correlation between the inner edge and stellar mass for different populations. Different colors are used to represent each population: green for all multiple systems, red for super-Earth systems, blue for sub-N… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Effect of stellar metallicity on the stellar mass–inner edge correlation. The metallicity dependence of the inner edge and stellar mass, as well as their projection (top two panels) on the inner edge and stellar mass diagram (bottom-left panel). Green solid points repr…
Figure 5
Figure 5. Figure 5: Dependence of the inner edge on stellar mass and metallicity as well as the corresponding dependence between mass and metallicity obtained through the MLR model. The left and middle panels show the power-law indices, which correspond to γ1 and γ2 in Eqs. (7) and (8), r…
Figure 6
Figure 6. Figure 6: Simulated analysis of observational selection bias. The panels in the top row represent Ntest (with a larger number of sample sets set to 10,000), while those in the bottom row represent Nobs (the same number of sample sets as the observational data). The left panels s…
Figure 8
Figure 8. Figure 8: Schematic comparison of the conclusions of this work and Mendigutía et al. (2024), revealing that the inner edges of different plan￾etary systems correspond to different mechanisms. The results obtained in this work for small planets align more closely with the dust su…
Figure 7
Figure 7. Figure 7: Correlation γ1 comparison between different theoretical mod￾els and the observational result. The green band represents the result of the dataset’s all multiple systems population obtained through the MLR model (same as the top panel in [PITH_FULL_IMAGE:figures/full_f…

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    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.