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Retrieving interior properties of hot Jupiters with Love numbers and atmospheric measurements

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read WASP-19Ab's measured tidal response, combined with its atmospheric metallicity, points to a core of about 79 Earth masses.

desk verdict Solid retrieval framework and useful precision thresholds, but the WASP-19Ab core detection leans on a linear-tidal calculation that the authors themselves flag as potentially biased by more than the measurement error. read the letter →

arxiv 2505.10304 v1 pith:7YKPLW3T submitted 2025-05-15 astro-ph.EP

Esther van Dijk , Yamila Miguel This is my paper · ORCID
classification astro-ph.EP
keywords hotJupitersLovenumbersinteriorstructurecoremassatmosphericmetallicityBayesianretrievalWASP-19Abexoplanetinteriors
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 two observable constraints together—the tidal Love number $k_2$ and the atmospheric metallicity—can lift the degeneracies that usually leave hot Jupiter interior models with upper limits instead of detections. The authors build a Bayesian retrieval on a standard planetary structure code, using both a simple two-layer homogeneous model and a more complex dilute-core inhomogeneous model. Applied to the five hot Jupiters with published Love numbers, only WASP-19Ab has data precise enough to pin down the interior, and both models confirm the presence of a core. Under the homogeneous model the core mass fraction is $0.21^{+0.05}_{-0.04}$, about $79^{+21}_{-18}$ Earth masses. This matters because it shows how upcoming JWST-quality phase curves and atmospheric spectra can turn tides into a direct probe of where heavy elements sit inside giant planets.

What carries the argument

The load-bearing observable is the second-order Love number $k_2$, the surface value of the Love function $K_n(r)$ that measures how strongly a planet's gravitational potential responds to an external tide. It is computed from the interior density profile by integrating a first-order differential equation for $\eta_2(r)$ under the assumption of hydrostatic equilibrium and a linear tidal response. The retrieval uses the planetary structure code CEPAM to generate density profiles for a homogeneous two-layer model (compact heavy-element core plus uniform H/He envelope) and an inhomogeneous model with a dilute core whose heavy-element fraction falls off through an error-function gradient; a nested-sampling Bayesian engine then fits predicted radius and $k_2$ to observations, with atmospheric metallicity entering as a Gaussian prior on the envelope metal fraction. The $k_2$ measurement carries the core-mass information because a more centrally concentrated mass distribution (larger core) lowers the Love number.

What would settle it

A non-linear tidal calculation for WASP-19Ab (for example with the concentric MacLaurin spheroid method used for WASP-12b) that changes the predicted $k_2$ by the ~19% bias would move the observed value relative to the model grid; if the resulting posterior no longer excludes a coreless interior, the detection claim collapses. A direct JWST phase-curve measurement of $k_2$ for this planet that differs from the Bernabò et al. (2024) value by more than the combined uncertainties would likewise settle the question.

Watch

Extended reading notes

Core claim

The central claim is that accurate interior constraints for hot Jupiters require a Love number measurement with precision better than about 40% for a homogeneous model and 15% for a dilute-core model, and that this must be paired with an atmospheric metallicity measurement to break the remaining degeneracy. For the only planet in the current sample that meets both criteria, WASP-19Ab, the retrieved posteriors rule out a coreless interior: the probability that both the core mass fraction and the dilute-core extent are below 0.1 is only 0.02%. The homogeneous model gives a core mass fraction of $0.21^{+0.05}_{-0.04}$ ($79^{+21}_{-18}$ $M_\oplus$), while the inhomogeneous model gives a compact core of $55^{+25}_{-29}$ $M_\oplus$ that extends to zero but is almost never zero. The paper does not claim to distinguish a compact from a diluted core, only that some form of dense core is present.

Load-bearing premise

The core detection for WASP-19Ab rests on interpreting its measured Love number with a linear, hydrostatic first-order tidal model, even though the planet's rotation parameter ($q_0 \approx 0.06$) is large enough that non-linear effects could shift the Love number by about 19%, larger than the 12% observational uncertainty.

Editorial extensions

If this is right

  • Love number measurements with uncertainties above roughly 40% provide little interior information, so future observation campaigns should target phase-curve precisions below that threshold.
  • Only two current planets, HAT-P-13b and WASP-19Ab, meet the precision threshold; for the other three the retrievals return upper limits rather than detections.
  • WASP-19Ab becomes a benchmark for formation models: any successful giant-planet formation theory must reproduce a core of roughly 80 Earth masses inside a 1.15 Jupiter-mass planet.
  • Applying the same retrieval to predicted JWST phase-curve detections (~17% precision for WASP-12b-like planets) would move several planets into the regime where a core is detected rather than merely bounded.
  • The Love number also sharpens the constraint on the heating efficiency parameter $\gamma$, linking tidal response to the inflation mechanism.

Reading between the lines

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

  • If the non-linear tidal bias for WASP-19Ab is as large as the paper's cited estimate for the similar planet WASP-12b (~19% in $k_2$), the retrieved core mass fraction could shift noticeably; the authors themselves recommend modelling non-linear effects before trusting the exact value.
  • A natural extension is to apply the retrieval to the upcoming JWST Love-number and atmospheric-metallicity measurements for a larger planet sample, which would test whether the mass–metallicity trend seen in the current five planets persists.
  • The large spread between atmospheric and bulk metallicity found in Figure 10 suggests atmospheric metallicity alone is a weak proxy for a planet's total heavy-element content, so future atmospheric surveys should not be read as direct interior constraints.
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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

3 major / 5 minor

Summary. The paper presents a Bayesian retrieval framework that combines mass, radius, equilibrium temperature, atmospheric metallicity, and Love number measurements to infer the interior structure of hot Jupiters, using CEPAM-based interior models in two configurations: a homogeneous two-layer model and an inhomogeneous dilute-core model. The framework is validated on a synthetic test planet with known interior properties, and the authors quantify the Love-number precision required to recover bulk metallicity and core mass, finding thresholds of approximately 40% for the homogeneous model and 15% for the dilute-core model. The framework is then applied to five hot Jupiters with measured Love numbers, of which WASP-19Ab is the only planet with both a precise Love number and an atmospheric metallicity constraint; for this planet the paper reports a homogeneous-model core mass fraction of 0.21+0.05/-0.04, corresponding to about 79 Earth masses, and a claimed 0.02% probability of a coreless interior in the inhomogeneous model.

Significance. If the WASP-19Ab result holds, the paper would provide the first exoplanet core detection based on a Love number measurement combined with an atmospheric metallicity constraint, which is a notable advance. The test-planet validation with injected values, the explicit comparison of two interior models, and the quantitative precision-threshold analysis are strengths that make the retrieval methodology useful for planning JWST-era observations. However, the flagship core-detection claim relies on a linear hydrostatic Love number calculation that the authors themselves identify as potentially biased for WASP-19Ab; because the potential systematic is larger than the observational uncertainty, the core detection should be tested against this systematic before the claim is treated as settled.

major comments (3)
  1. [Section 5.1, Table 3, Section 4.2.2, Abstract] The WASP-19Ab core-detection claim is not robust to the non-linear tidal systematic that the authors themselves quantify. Section 5.1 states that WASP-19Ab has rotational parameter q0 approximately 0.06, which exceeds the q0 << 0.01 regime where the linear approximation holds, and that Wahl et al. (2021) estimate that neglecting non-linear effects underestimates k2 by about 19% for WASP-12b, a planet with similar q0. This is larger than the 12% observational uncertainty adopted in the retrieval, and k2 is the main constraint on core mass. The abstract and Section 4.2.2 present the core mass of 0.21+0.05/-0.04 and the coreless-probability statement without this caveat. I request that the authors add a retrieval with k2 shifted upward by approximately 19% (or with a non-linearly corrected k2) and report whether the core detection and the 0.02% probability survive; if they do not, the abstract and conclusions should be revised to present the core detection as conditional on the linear-tidal interpretation.
  2. [Section 4.2.2] The claim that a coreless interior is "effectively ruled out" relies on a probability threshold that is not physically defined. The 0.02% figure is computed as the probability that both m_core and m_dilute are below 0.1, but a diluted-core model with these parameters can still contain substantial heavy elements in the envelope (the retrieved Z_atm for WASP-19Ab is about 0.07), so this is not a direct posterior probability for the absence of a compact core. The authors should either justify the threshold as a physically meaningful definition of coreless or replace the statement with a model-comparison metric that directly contrasts interiors with and without a compact core.
  3. [Section 4.1] The systematic tendency for the retrieved Love numbers to fall at the lower end of, or below, the observed values for most planets in the sample is discussed by the authors as a possible observational bias or missing physics. Since WASP-19Ab is the only planet with a precise enough measurement to drive a core detection, the absence of a quantitative test of how this systematic discrepancy would affect the WASP-19Ab inference is a gap. The authors should state explicitly whether the WASP-19Ab agreement is expected to be immune to the same bias, or add a sensitivity test in which the observed k2 is shifted within the plausible systematic range.
minor comments (5)
  1. [Equation (1)] The notation Z_atm appears in Equation (1) while the text and Table 1 use Z_env for the envelope metal mass fraction; please unify the notation.
  2. [Figure 5] The middle panel label reads "Love number k22" but should read "Love number k2".
  3. [Table 1] The prior notation "LU(0, l_age_min)/U(1, L_age_min)" for L_grav is confusing, particularly because the text in Section 2.3 says the lower limit is set to 1 L_J; please clarify the intended ranges in the table.
  4. [Appendix B] The sentence "In Figures B2, B3, B4, B5 and B6 we the results obtained" is missing the verb "show"; please correct it.
  5. [Abstract] The phrase "interiors structures" should be "interior structures" for grammatical correctness.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the interior retrieval is a forward-model fit to independent Love number and atmospheric metallicity constraints, not a self-referential prediction.

full rationale

The derivation chain is self-contained. The forward interior model (CEPAM) computes radius and k2 from structure parameters via the hydrostatic equations and the first-order Love number relation (Eqs. 4-5), and the likelihood (Eq. 6) compares those model outputs to observed Rp and k2. The core mass fraction mcore and envelope metallicity Zenv are free parameters with physically chosen priors (Table 1); neither is defined in terms of the quantities the paper claims to infer. The atmospheric metallicity enters as an independent Gaussian prior and the Love number as an independent likelihood term, and the WASP-19Ab core detection is a posterior inference driven by the measured k2 = 0.20(+0.02/-0.03), not a restatement of the prior (mcore is uniform on [0,1]). The test-planet validation in Section 3 uses injected true values to check the retrieval, providing an internal consistency check rather than a tuned input for the science claim. The self-citation to Bloot et al. (2023) for the dilute-core gradient and Lgrav prior is methodological and not load-bearing; no conclusion depends on accepting that citation as proof. The non-linear tidal caveat in Section 5.1 is a physical model-dependence concern that the authors flag, not a circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central WASP-19Ab core detection rests on the model family (homogeneous or dilute core) and the linear tidal response assumption; the largest unquantified systematic is the non-linear rotational effect discussed in Section 5.1. The structural parameters (mcore, Zenv, mdilute, Zdilute) are retrieval targets, not ad hoc inputs; the two genuinely free nuisance parameters are gamma and Lgrav. No new physical entities are postulated.

free parameters (2)
  • gamma (heating efficiency) = WASP-19Ab: 0.02+0.02/-0.01 (homogeneous, log-uniform prior, Table C4)
    Introduced to model inflation; log-uniform prior between 1e-5 and 0.1; constrained by radius and Love number; affects radius and therefore core mass inference.
  • Lgrav (intrinsic luminosity)
    Prior from evolutionary models and stellar age, lower limit 1 L_J; log-uniform or uniform prior; this prior choice changes results for WASP-12b, WASP-103b, and HAT-P-13b (Appendix B).
assumptions (6)
  • domain assumption Planets are in hydrostatic equilibrium; the first-order Love number k2 is computed from the density profile with the Sterne ODE (Eqs. 4-5).
    Section 2.1.3; this is the basis for all k2 calculations. The authors note in Section 5.1 that non-linear rotational effects can bias k2 by about 19% for WASP-19Ab, making this the key fragile premise.
  • domain assumption Envelope composition uses a proto-solar H/He ratio (Lodders 2021) with heavy elements represented by water; EOS: MH13-H, SCH95-He, Mazevet et al. 2019 water, Hubbard and Marley 1989 core.
    Section 2.1; choice of EOS and heavy-element proxy affects radius and k2. The authors argue EOS differences are smaller than observational uncertainties (citing Howard et al. 2025), but this is not demonstrated in this paper.
  • ad hoc to paper Hot Jupiter inflation is modeled as L_int = L_grav + gamma times L_irr, with gamma free between 1e-5 and 0.1 and heat deposited uniformly.
    Section 2.1.2; no physical mechanism is specified. The retrieved heating efficiencies depend on the log-uniform prior choice (Appendix B), and for weakly constrained planets the prior changes bulk metallicities substantially.
  • domain assumption Observed atmospheric metallicity [M/H] equals the envelope metal mass fraction Zenv and is converted to mass fraction assuming solar abundances and proto-solar H/He (Appendix A).
    Section 2.3, Table 1; used to set Gaussian priors on Zenv. Atmospheric enrichment or depletion processes could break this equivalence.
  • domain assumption Dilute core heavy-element gradient is parameterized by an error function with fixed width dm = 0.075 (Eq. 1).
    Section 2.1, taken from Bloot et al. 2023; the width is fixed, not retrieved, and could influence how easily k2 constrains mdilute and Zdilute.
  • standard math Nested sampling (PyMultinest) with 1000 live points and sampling efficiency 0.1 gives converged posterior estimates.
    Section 2.2; standard Bayesian machinery; no convergence diagnostics are shown.

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Cite this review

Pith. "Pith review of Retrieving interior properties of hot Jupiters with Love numbers and atmospheric measurements." pith.science (2026). https://pith.science/paper/7YKPLW3T

@misc{pith2026250510304,
  author       = {Pith},
  title        = {Pith review of: Retrieving interior properties of hot Jupiters with Love numbers and atmospheric measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7YKPLW3T}},
  note         = {Machine review of arXiv:2505.10304}
}
abstract

Understanding exoplanet interiors is crucial for interpreting atmospheric observations and constraining their evolution and formation. However, due to limited observational constraints, interiors structures remain poorly understood. In this work, we investigate how new observational constraints, such as the Love number and atmospheric metallicity, improve our ability to characterize the interiors of hot Jupiters, planets for which Love number measurements are most feasible. We assess the precision required in Love number measurements to derive interior properties using both a simple two-layer homogeneous model and a more complex dilute core model. To account for observational uncertainties, we implement a retrieval framework. Our results show that accurately constraining core mass and bulk metallicity requires a high-precision Love number measurement, better than 40% for a homogeneous model and 15% for a dilute core model, along with an atmospheric metallicity measurement. We apply our retrieval framework to five planets with observed Love numbers, of which only WASP-19Ab has both an atmospheric metallicity constraint and a highly precise Love number measurement, with a precision of 12%. For this flagship planet, both models confirm the presence of a core, although we cannot yet distinguish between a compact core or diluted core. With the homogeneous model, we find a core mass fraction of $0.21^{+0.05}_{-0.04}$, corresponding to $79^{+21}_{-18}$ $M_\mathrm{earth}$. Upcoming JWST observations are expected to provide high-precision Love number measurements and precise atmospheric data, offering new insights into the structure and composition of gas giant interiors.

Figures

Figures reproduced from arXiv: 2505.10304 by the authors.

Figure 1
Figure 1. The two different interior structure models used in this work. Both models have an isothermal rocky core, with a heavy element fraction of 1. In the homogeneous model (left) there is a sharp transition at 𝑚core to the envelope, consisting out of hydrogen (X), helium (Y) and heavy elements (𝑍env). In the inhomogeneous model (right) we add a dilute core region, where the heavy element fraction reduces gradually from 𝑍… view at source ↗
Figure 2
Figure 2. Posterior distributions (upper panels) and summary statistics (lower panels) of the bulk metal fraction (𝑍planet), core mass fraction (𝑚core), dilute core parameters (𝑚dilute, 𝑍dilute) and heating efficiency (𝛾) for the test planet with properties in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. The predicted posteriors of the interior parameters resulting using different uncertainties in the Love number as input for the test planet. The posteriors are represented by the median and 1-sigma confidence intervals for the metal mass and dilute core parameters and by 2-sigma upper limits for the core mass and heating efficiency. The dashed black lines indicate the true values. Results for the homogeneous (blue) … view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: The posterior distributions (upper panels) and corresponding confi￾dence intervals (lower panels) of the bulk metallicity resulting from retrievals with Love number uncertainties of 1 and 10%. Results of the homogeneous (shaded) and inhomogeneous (not shaded) model are…
Figure 5
Figure 5. Figure 5: The planetary radius (left), Love number 𝑘22 (middle) and planetary mass (right) that the model predicts and the observed values for the five planets. The results for both the homogeneous and inhomogeneous model are shown. 0.0 0.2 0.4 0.6 0.8 1.0 mcore 10 2 10 1 10 0 Z…
Figure 6
Figure 6. Figure 6: The interior structure parameters of the planets in the sample. In the left panel we show the core mass fraction and atmospheric metallicity. The background lines represent equal bulk metallicity (𝑍planet) for the homogeneous model. The right panel shows the dilute cor…
Figure 8
Figure 8. Figure 8: The bulk metal mass as a function of planet mass for the five planets in the sample. The background lines represent lines of equal bulk metallicity (𝑍planet). We compare our results to the mass-metallicity trends derived by Thorngren et al. (2016) and Müller & Helled (…
Figure 11
Figure 11. Figure 11: The heating efficiency fraction and the equilibrium temperatures of the planets with Love number measurements. We compare our results to the Heating Efficiency Equilibrium Temperature (HEET) distributions derived by Thorngren & Fortney (2018) and Sarkis et al. (2021) …
Figure 10
Figure 10. Figure 10: The bulk metallicity and atmospheric metallicity of the five plan￾ets in the sample. Jupiter and Saturn are shown for comparison with bulk metallicities from Sur et al. (2025) and atmospheric metallicities calculated from the O/H (for Jupiter) and C/H (for Saturn) atm…

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    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.stat...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.