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Tidal Inflation is Stronger for Misaligned Neptune-Sized Planets Than Aligned Ones

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper's statistical tests show misaligned Neptune-sized planets are more tidally inflated than aligned ones, at at least 90% confidence.

desk verdict The population comparison is a good idea and the case study is interesting, but the statistical evidence for 'at least 90% confidence' does not survive contact with the model-boundary and uncertainty problems. read the letter →

arxiv 2506.24100 v1 pith:7IWE5FTZ submitted 2025-06-30 astro-ph.EP

Ritika Sethi , Sarah Millholland This is my paper · ORCID
classification astro-ph.EP
keywords exoplanetstidalheatingradiusinflationspin-orbitmisalignmentpolarplanetsNeptune-sizedplanetaryevolutionWASP-107b
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

The paper tries to establish that the unusually large radii of many Neptune-sized exoplanets are tied to their orbital histories: planets whose orbits are misaligned with their host star's spin are more inflated by tidal heating than aligned planets. It compares 12 misaligned and 12 aligned planets, fitting each with and without tidal heating in a structural-evolution model, and measures the degree of inflation $R_{p,t}/R_{p,0}$, the factor by which tides enlarge the planet beyond the radius its envelope would have without tides. Four statistical tests separate the two populations at $\geq 90\%$ confidence, with average inflation of roughly 1.4 for misaligned planets versus 1.1 for aligned ones. If the result holds, a planet's observable size and internal energy budget carry information about its dynamical past, not just its composition.

What carries the argument

The central object is the degree of inflation, $R_{p,t}/R_{p,0}$: the ratio of the radius a tidal-heating model assigns to a planet (matching the observed radius) to the radius the same envelope mass fraction would have if tides were absent. It is extracted from a grid of about 15,000 two-layer thermal-evolution models by fitting each observed planet twice, once with and once without tidal heating, and comparing the inferred envelope fractions at a fixed reduced tidal quality factor $Q' = 10^4$. The tidal luminosity law used sets the heating from eccentricity tides with planetary obliquity set to zero, so the machinery isolates eccentricity-driven inflation rather than obliquity-driven tides.

What would settle it

A larger sample of Neptune/sub-Saturn planets with measured eccentricities, ages, and 3D obliquities could settle it: if the inflation excess among misaligned planets disappears once eccentricity and age are controlled, or if equally inflated planets appear at low obliquity, the central claim fails.

Watch

Extended reading notes

Core claim

For a sample of 24 Neptune-to-sub-Saturn planets with measured stellar obliquities, the paper finds that misaligned planets ($|\lambda| > 40^\circ$) have average tidally induced radius inflation $\langle R_{p,t}/R_{p,0}\rangle_m \approx 1.4$, compared with $\langle R_{p,t}/R_{p,0}\rangle_a \approx 1.1$ for aligned planets, evaluated at a fixed reduced tidal quality factor $Q' = 10^4$. The inferred envelope mass fraction drops by roughly 12% for misaligned planets when tides are included, versus about 5% for aligned planets. The Kolmogorov-Smirnov, Student's t, Welch's t, and Mann-Whitney U tests reject the null hypothesis that the two populations are drawn from the same inflation distribution at the 90%, 97%, 97%, and 95% confidence levels respectively. The paper interprets this as evidence that misaligned planets' dynamically excited histories, which place them in close-in, eccentric, highly inclined orbits, deposit more tidal heat into their envelopes and puff them up.

Load-bearing premise

The result stands on the assumption that the tidal-heating model with zero planetary obliquity, its simulation grid, and a linear extrapolation to a common tidal quality factor estimate inflation without a bias that tracks misalignment.

Editorial extensions

If this is right

  • Observed radii of misaligned Neptune/sub-Saturn planets should be interpreted with tidal inflation in mind; otherwise their envelope mass fractions are overestimated.
  • Low densities among misaligned planets can be read as a signature of past high-eccentricity migration or scattering, not necessarily of unusually massive envelopes.
  • Population studies of sub-Neptune and sub-Saturn composition should treat aligned and misaligned planets separately rather than combining them.
  • Dynamical modeling of close-in planets should couple radius evolution to orbital evolution; ignoring radius changes overestimates the tidal circularization rate.
  • The WASP-107 b case study favors $Q' \approx 10^4$-$10^6$, consistent with internal heat constraints from recent infrared observations.

Reading between the lines

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

  • A testable extension would be to measure internal temperatures across a larger sample with infrared emission spectroscopy; the model predicts misaligned planets should show systematically higher internal heat at fixed mass, radius, and incident flux.
  • If eccentricity tides are the driver, the inflation signal should weaken as orbits circularize, predicting a correlation between current eccentricity and inflation among misaligned planets and a deficit of inflated low-eccentricity misaligned planets at old ages.
  • The paper's exclusion of Jupiter-mass planets leaves open whether a similar alignment trend exists among hot Jupiters, where tidal inflation is also common but the dominant formation channels differ.
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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

5 major / 5 minor

Summary. The paper investigates whether misaligned (including polar) Neptune-sized and sub-Saturn-sized planets are more tidally inflated than aligned planets. Using 12 misaligned and 12 aligned planets with stellar obliquity measurements, the authors apply a MESA-based thermal evolution model with and without tidal heating, fix the reduced tidal quality factor at Q'=10^4, and infer envelope mass fractions (fenv,0 and fenv,t) and a dimensionless degree of inflation Rp,t/Rp,0. The reported mean inflation is about 1.4 for misaligned planets and 1.1 for aligned planets; four statistical tests (KS, Student t, Welch, Mann-Whitney) give p-values of 0.099, 0.022, 0.024, and 0.046, which the authors interpret as at least 90% confidence that misaligned planets are more inflated. The paper also presents a case study of WASP-107 b, coupling tidal orbital evolution with radius evolution, and derives Q' in the range 10^4-10^6, said to agree with JWST-based internal temperature constraints.

Significance. If the population-level claim is correct, it would establish a direct observational connection between a planet's dynamical history (spin-orbit misalignment and eccentricity excitation) and its present-day interior heat budget, with consequences for formation and migration theories. The assembled catalog of aligned and misaligned Neptune/sub-Saturn planets with obliquity measurements is a useful community resource, and the WASP-107 b coupled orbital-radius evolution model provides a concrete methodological advance by showing that radius feedback can slow tidal circularization. The paper also makes a falsifiable prediction: misaligned Neptunes should show systematically larger tidally-induced radius inflation under equilibrium-tide models. However, the statistical evidence as presented is not yet robust enough to support the stated confidence level, because the key tests ignore the quoted uncertainties and the sample treatment at the model boundary is inconsistent.

major comments (5)
  1. [Section 5.3 and Table 3] The four statistical tests are applied to single point estimates of Rp,t/Rp,0 with no propagation of the quoted uncertainties. The errors in Table 3 are standard deviations of Rp,t/Rp,0 over the log10 Q' bin [3.8,4.2] (Sections 4.1 and 4.2), not the full posterior or measurement uncertainties at fixed Q', so the effective noise is larger than reported. The p-values (especially the t-test p=0.022 and Welch p=0.024) would need to be recomputed using draws from the posterior distributions, or at least using Monte Carlo resampling from the individual standard deviations, before the claim of 'at least 90% confidence' is supported.
  2. [Section 4.1 and Table 3] The treatment of planets whose no-tide fits require fenv,0 > 50% is internally inconsistent: TOI-1842 b and HD 148193 b are excluded for this reason, while WASP-107 b, HAT-P-12 b, HAT-P-18 b, and Kepler-9 b are retained with fenv,0 > 50% in Table 3. Three of the four retained planets are misaligned and have some of the largest inflation ratios (Rp,t/Rp,0 between 1.68 and 1.89), so the exclusion rule directly affects the group means and the statistical comparison. The authors should specify a uniform, pre-selected boundary criterion and report results both with and without the fenv,0 > 50% planets.
  3. [Section 3.3 and Table 3] The assumption of e = 0.05 ± 0.02 for planets without measured eccentricities, combined with setting the planetary obliquity to ϵ = 0 for all planets, is not sufficiently stress-tested. The reported sensitivity analysis using only planets with measured eccentricities is reassuring in direction, but the subsample is small and no p-values are given. Because the inferred Rp,t/Rp,0 values depend strongly on the assumed eccentricity through equation (8), a quantitative sensitivity study (for example, comparing e = 0.05 vs e = 0.1, and including obliquity tides for polar planets) is needed to establish that the population difference is not an artifact of these choices.
  4. [Section 6.2.1] In the WASP-107 b case study, the authors fix fenv,t = 0.49 at Q' = 10^6 because the simulation grid ends at fenv = 50%, even though the MCMC fit at Q' = 10^4 gives fenv,t ≈ 24%. This clipping is a strong modeling assumption, and the resulting conclusion that Q' ≈ 10^4-10^6 for WASP-107 b should be labeled as conditional on that cap. A dedicated exploration of the sensitivity of the inferred Q' range to the fenv ceiling, or an extension of the simulation grid, would make the dynamical constraints more convincing.
  5. [Section 5.3 and Table 4] Multiple testing is not accounted for. Four separate tests are presented and the paper selects the 'at least 90% confidence' framing based on the ensemble. The KS p-value of 0.099 is the weakest, and the t-test p-value of 0.022 would not survive a strict 95% Bonferroni correction (0.022 × 4 = 0.088, though still below 0.1). The tests are also not independent because they are computed from the same point estimates. The authors should either pre-specify a primary test or report adjusted p-values; at present the evidence for the population difference is suggestive but not as strong as the abstract claims.
minor comments (5)
  1. [Figure 4 caption] The caption contains a doubled word: 'Impact of of tidal inflation'; please correct.
  2. [Throughout] The abbreviation for the Kolmogorov-Smirnov test appears as both 'K-S Test' and 'KS test'; please standardize to one form.
  3. [Table 3] The Ltide column lists values without uncertainties; if these are point estimates computed at Q' = 10^4, please clarify, or provide errors consistent with the other columns.
  4. [Section 6.2.1] The sentence 'we determine the value of fenv,t corresponding to this Q′ to be ≥ 50%' would be clearer as 'to be at least 50%', and the subsequent clipping at 0.49 should be described with an explicit justification.
  5. [Table 1] The reference list for the sample properties is long but not keyed to individual planets; a per-row citation column would aid reproducibility.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the population difference is emergent from MCMC fits to observed radii; self-cited tidal models are reused but are not load-bearing restatements of the conclusion.

full rationale

The central quantity, Rp,t/Rp,0, is not defined in terms of the alignment classification. Section 4.2 states that 'Rp,t equals the planet's observed radius by construction of the model' and that Rp,0 is the radius for the same envelope fraction without tidal heating. Both fenv,0 and fenv,t are obtained by fitting each planet's observed mass, flux, and radius independently with MCMC; the aligned-versus-misaligned difference emerges only after these fits are aggregated. Nothing in the model construction forces misaligned planets to have larger Rp,t/Rp,0, so the headline claim is not true by definition. The authors do reuse their own MESA simulation grid (Millholland 2019, 2020), but that grid is a public, physically motivated evolution code, and the transformation in Eq. 8 is an explicit luminosity-invariance mapping rather than an import of the paper's conclusion. The case-study Q' constraints for WASP-107 b are compared with independent JWST-based internal-temperature estimates, and the Louden & Millholland (2024) self-citation about long-term polar-orbit stability is peripheral, not the load-bearing step. The statistical caveats noted by critics, such as using point estimates without propagating the quoted uncertainties and retaining extrapolated boundary cases, are genuine robustness concerns but are not circularity: they affect whether the p-values are reliable, not whether the derived inflation ratios are equivalent to the inputs. Overall, the derivation chain is self-contained against external observations; the only circularity-adjacent feature is the repeated reliance on the authors' own previously published tidal models, which is not load-bearing in the logical sense.

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

The central claim depends on the tidal heating model and the simulation grid from prior work by the same group, plus several modeling choices such as epsilon=0, e=0.05 for unknown eccentricities, and fixed Q'=10^4 with extrapolation. No new entities are introduced. The WASP-107 b case study adds adopted timescales from the literature.

free parameters (5)
  • Q' (reduced tidal quality factor) fixed at 10^4 = 10^4
    Chosen as a fixed comparison value for all planets; the paper argues different choices do not affect the comparison, but the absolute Rp,t/Rp,0 values depend on it.
  • Assumed eccentricity for planets without measured e = 0.05 +/- 0.02
    Assigned in Section 3.3 to all planets lacking eccentricity measurements; a sensitivity test using only measured eccentricities yields 93% confidence, so the assumption is not fully load-bearing but affects the sample.
  • tau_inf (inflation timescale) = 5.8 Myr
    Adopted from Yu & Dai (2024) for the WASP-107 b radius evolution model; not fitted here.
  • tau_def (deflation timescale) = 0.5 Gyr
    Adopted from Thorngren et al. (2021) for the WASP-107 b radius evolution model; not fitted here.
  • fenv,t clipped at 0.49 for WASP-107 b = 0.49
    Set to the model grid maximum in the instantaneous radius evolution calculation (Section 6.2.1).
assumptions (6)
  • domain assumption Equilibrium tide theory with a constant time lag (viscous model) describes tidal dissipation in these planets.
    Adopted in Sections 3.1 and 6.1 from Leconte et al. (2010); the tidal luminosity and orbital evolution equations rely on it.
  • domain assumption The MESA simulation suite from Millholland (2019) and Millholland et al. (2020) accurately models the thermal and structural evolution of sub-Neptune planets with and without tidal heating.
    Used in Section 3.2; the paper's authors are among the creators of this suite, and the paper does not independently validate the grid beyond prior citations.
  • ad hoc to paper Planetary obliquity is zero for all planets, so only eccentricity tides contribute.
    Stated in Section 3.3; this isolates eccentricity tides but neglects obliquity tides, a potentially important heating source for misaligned planets.
  • domain assumption Ohmic dissipation is subdominant to tidal heating for the population as a whole.
    Estimated in Section 3.1 using Pu & Valencia (2017) relations; the paper notes it can be comparable for some systems, including WASP-107 b (Batygin 2025).
  • ad hoc to paper The linear extrapolation of log fenv,t and Rp,t/Rp,0 vs log Q' to Q'=10^4 is valid for planets whose posteriors do not reach that value.
    Used in Sections 4.1 and 4.2 for planets such as GJ 436 b and AU Mic b; justified by the appearance of the 2D posteriors but not rigorously tested.
  • domain assumption The angular momentum conservation relation (Eq. 13) links initial eccentricity and semi-major axis for WASP-107 b after ZLK suppression.
    Used in Section 6.1 to set initial conditions; assumes no other angular momentum loss or gain during tidal evolution.

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Pith. "Pith review of Tidal Inflation is Stronger for Misaligned Neptune-Sized Planets Than Aligned Ones." pith.science (2026). https://pith.science/paper/7IWE5FTZ

@misc{pith2026250624100,
  author       = {Pith},
  title        = {Pith review of: Tidal Inflation is Stronger for Misaligned Neptune-Sized Planets Than Aligned Ones},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7IWE5FTZ}},
  note         = {Machine review of arXiv:2506.24100}
}
abstract

Recent observations have revealed an intriguing abundance of polar-orbiting Neptune-sized planets, many of which exhibit unusually inflated radii. While such misaligned orbits point to a complex dynamical history, the connection between their orbital orientations and planetary structures remains poorly understood. In this study, we analyze a sample of 12 misaligned and 12 aligned planets using structure models that incorporate tidal heating. We use various statistical tests to demonstrate with at least $90\%$ confidence that misaligned planets experience more tidally-induced radius inflation compared to aligned planets. This inflation likely stems from their dynamically active histories, which often place them in close-in, eccentric, and highly inclined orbits. We further present a case study of WASP-107~b, an exceptionally inflated polar Neptune, and model its history using a simple coupled orbital and radius evolution approach. Our results place constraints on the planet's tidal quality factor that agree with recent JWST observations.

Figures

Figures reproduced from arXiv: 2506.24100 by the authors.

Figure 1
Figure 1. The 3D stellar obliquity, Ψ, of all planets with observed Rossiter-McLaughlin measurements, based on data from TEPCat (J. Southworth 2011). The distribution appears bimodal, consisting of planets with orbits nearly aligned or nearly perpendicular (Ψ ∈ [75◦ , 125◦ ]; highlighted by the blue background color) with respect to their host star’s equatorial planes. The pink squares represent Neptune-sized planets, which a… view at source ↗
Figure 2
Figure 2. Examples of fenv distributions from the MCMC analysis, shown for a handful of misaligned (top panel) and aligned planets (bottom panel). In each plot, the solid gray line represents the envelope mass fraction obtained for a planet using a tides-free model (fenv,0), with the gray-shaded area indicating the 1σ uncertainty. The other shaded regions – blue for misaligned planets and red for aligned planets – illustrate … view at source ↗
Figure 3
Figure 3. Examples of the posterior distributions of the degree of inflation, Rp,t/Rp,0, as a function of Q ′ . The top and bottom panels show examples for misaligned planets and aligned planets, respectively. the model neglecting tides would require fenv,0 > 50%, which lies outside the bounds of the simulation grid. As a result, we were unable to quantify the impact of tidal heating for these planets, and therefore excluded … view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Impact of of tidal inflation on the estimated envelope mass fraction. The top panel illustrates the reduction in the estimated envelope mass fraction, from fenv,0 (without tides) to fenv,t (including tides at Q ′ = 104 ). Aligned and misaligned planets are depicted wit…
Figure 5
Figure 5. Figure 5: Summary of results on the degree of inflation, Rp,t/Rp,0. The left panel shows Rp,t/Rp,0 as a function of log10(Ltide/Lirr), calculated for our entire sample at Q ′ = 104 . Aligned planets are represented by red dots and squares, while misaligned planets are represente…
Figure 6
Figure 6. Figure 6: shows the results of these calculations, in￾dicating the time required for the planet to reach its present day conditions, starting from an initial state (ei , Q′ i ) after the suppression of ZLK oscillations. We observe that as ei increases, there is a slight increase…
Figure 7
Figure 7. Figure 7: Tidal orbital evolution of WASP-107 b using the no radius evolution (dashed black curve), instantaneous ra￾dius evolution (blue curve), and radius evolution with time lag (red curve) approaches for Q ′ = 106 . We show the evolu￾tion of the eccentricity (top) and semi-m…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.