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REVIEW 3 major objections 5 minor 46 references

Prospects of detecting rotational flatness of exoplanets from space-based photometry

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Space-based photometry can detect a Saturn-like exoplanet's rotational flattening in a single transit, provided the host star's density is known to about one percent.

desk verdict Useful detectability map for single-transit oblateness, but the central 3-sigma claim rests on an unresolved prior-width inconsistency (0.24 vs 0.04) in the a/R* constraint; needs a sensitivity run before the numbers can be trusted. read the letter →

arxiv 2507.15359 v1 pith:HBR4MI3K submitted 2025-07-21 astro-ph.EP

Sz. Kálmán , Sz. Csizmadia , L. M. Bernabó , R. Szabó , Gy. M. Szabó This is my paper · ORCID
classification astro-ph.EP
keywords oblatenessexoplanetstransitphotometryrotationalflatteningGauss-Legendrequadratureasteroseismologylight-curveretrievalspace-based
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 asks whether the rotational flattening of an exoplanet, the slight difference between its equatorial and polar radii, leaves a measurable imprint in a space-based transit light curve. The authors build a numerical transit model that uses Gauss-Legendre quadrature to integrate the overlap between a limb-darkened star and a sky-projected ellipse, verify it against an established spherical-transit model, and run 800 injection-and-retrieval experiments with realistic white and time-correlated noise. They find that a single transit of a Saturn-like planet (oblateness $f\approx 0.09$) around a bright, quiet star yields a $3\sigma$ detection of sky-projected oblateness in about 59% of tested configurations, provided the stellar mean density is known to about 1% from asteroseismology. They also establish a noise floor: at point-to-point scatter of 256 ppm per 60-second exposure, oblateness retrieval is not reliable, and oblateness values below Jupiter's ($f\approx 0.06$) are not recoverable at any tested noise level.

What carries the argument

The carrying mechanism is a numerical transit model in which the sky-projected planetary disk is an ellipse and the blocked stellar flux is computed by two-dimensional Gauss-Legendre quadrature (a numerical integration rule sampling the integrand at Legendre-polynomial roots), evaluating the limb-darkened stellar surface brightness at each quadrature point and summing only points inside the stellar disk. The load-bearing identity is Kepler's third law written as $(a/R_\star)^3 = P^2 G(1+q)/(3\pi\rho_\star)$, which converts an external stellar-density measurement into a Gaussian prior on the scaled semi-major axis and thereby breaks the $b$\textendash$f$\textendash$\vartheta$ degeneracy that otherwise hides the oblateness signal. In the spherical limit the model matches the standard analytical transit light curve to within a few ppm at 96 quadrature points, while running about 25% faster, and its discrepancy with the analytical model shrinks as $n^{-2}$ with only a mild rise in runtime.

What would settle it

Re-run the same 800-case injection grid with eccentric orbits (for example $e=0.05$, all other parameters and priors unchanged): if the $3\sigma$ detection rate for Saturn-like oblateness falls substantially below 59%, or the recovered $f$ is biased by more than 0.02, the circular-orbit assumption is the part of the claim that fails.

Watch

Extended reading notes

Core claim

The central claim is that a rotating exoplanet imprints its non-spherical shape on the ingress and egress of the transit light curve, and that this imprint can be recovered once the degeneracy with the impact parameter is broken. The paper shows that a precise stellar mean density, obtained for example from asteroseismology, supplies that break: Kepler's third law ties $a/R_\star$ to $\rho_\star^{1/3}$, so a 1% density measurement becomes a Gaussian prior of $\sigma = 0.24$ on $a/R_\star$, which in turn pins down the transit geometry and separates oblateness $f$ from impact parameter $b$ and sky-projected obliquity $\vartheta$. In 800 synthetic single-transit light curves spanning $f = 0.03$\textendash$0.30$, $\vartheta = 0^\circ$\textendash$90^\circ$, and noise levels $\sigma_w = 1$\textendash$256$ ppm, a $3\sigma$ detection of $f$ is achieved in 559 of 700 cases below 256 ppm; Saturn-like oblateness ($f \approx 0.09$) is detected in about 59% of configurations, with retrieved $f$ and $\vartheta$ in $1\sigma$ agreement with the truth in roughly two-thirds of those detections. No tested configuration yields a reliable detection at $\sigma_w = 256$ ppm, and values below $f \approx 0.06$ remain below the method's reach.

Load-bearing premise

The retrieval depends on an externally known stellar mean density (about 1% precision) and on a strictly circular orbit, since the density-based prior on $a/R_\star$ is what separates oblateness from the impact-parameter degeneracy and the prior is only strictly valid for circular orbits.

Editorial extensions

If this is right

  • A single transit of a Saturn-like planet ($f\approx0.09$) around a bright, asteroseismically characterized star can deliver a $3\sigma$ oblateness measurement in about 59% of tested geometries, making dedicated follow-up observations of individual transits worthwhile.
  • Oblateness below about $f=0.06$, the value of Jupiter, is not recoverable at any tested noise level, so single-transit non-detections cannot yet constrain the flattening of most known planets.
  • At $\sigma_w=256$ ppm per 60 s exposure, only 54 of 100 configurations reach $3\sigma$ and the recovered values are frequently inaccurate, establishing a noise floor for planning transit-oblateness programs.
  • The method relies on prior knowledge of the host star's mean density to about 1% and of limb-darkening coefficients to 0.01, so the target list is restricted to bright, quiet, asteroseismically characterized stars.
  • Because the model is about 25% faster than the analytical spherical model at full precision, large retrieval grids over many planets are computationally feasible.

Reading between the lines

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

  • By extension, co-adding several transits of the same planet should reduce the effective white noise by roughly $\sqrt{N}$, so targets that fail at 256 ppm in a single transit may become detectable in multi-transit campaigns; the paper's single-transit setting is conservative.
  • An untested extension is to apply the same density-anchor retrieval to eccentric orbits; since the paper's prior is only strictly valid for circular orbits, such systems would need independent eccentricity and periastron constraints from radial velocities.
  • The measured sky-projected obliquity $\vartheta$, combined with independent spin-orbit measurements, could begin to constrain the three-dimensional spin geometry of hot planets; the paper does not pursue that combination.
  • Existing space-photometry archives of bright stars that already meet the sub-256 ppm noise requirement could be re-analyzed for oblateness without new observations.
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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 new numerical transit model for oblate (biaxial ellipsoid) exoplanets based on Gauss-Legendre quadrature, implemented in the TLCM framework. The model is benchmarked against the Mandel-Agol analytical model in the spherical limit and against the squishyplanet model for one oblate configuration. The authors then perform 800 injection-and-retrieval tests with different oblateness values f, sky-projected obliquities ϑ, white-noise levels, and injected red noise, and they report 3σ oblateness detection rates as a function of f, ϑ, and noise. Their headline claims are that Saturn-like oblateness (f≈0.09) is detectable in about 59% of tested configurations around bright stars when the stellar density is known from asteroseismology, and that noise levels of 256 ppm or higher make oblateness detection unreliable. The paper also introduces the Δ metric for quantifying light-curve model differences and reports a modest speed advantage over the analytical model.

Significance. If the detectability results are robust, the paper provides a practical roadmap for constraining sky-projected oblateness with CHEOPS, JWST, PLATO, or ARIEL photometry, and the publicly archived input/output light curves support reproducibility. The paper is also transparent about a key limitation: the stellar-density prior that breaks the b–f–ϑ degeneracy is stated to be valid only for circular orbits. However, the central 3σ detectability claim is not yet pinned down because of a quantitative inconsistency in the adopted prior width, and the external validation of the oblate model is thin. The work is potentially valuable but currently requires revision before the headline numbers can be taken at face value.

major comments (3)
  1. [Section 4.2 and Table 7] The text in Section 4.2 states that a Gaussian prior on a/R⋆ is applied with mean 73.26 and standard deviation 0.24, justified by a 1% stellar-density precision via Eq. (30), while Table 7 lists the prior as N(73.26, 0.04), which is six times narrower. This is not cosmetic: the narrower width corresponds to Δρ⋆/ρ⋆ ≈ 0.16%, far better than the 0.5–2.6% range cited from Silva Aguirre et al. (2017), and Fig. 17 shows a very strong a/R⋆–b correlation (Pearson r = 0.988). Because the a/R⋆ prior is the main lever that breaks the b–f–ϑ degeneracy described in Section 4.2, the reported 3σ detection rates (e.g., 59% for f≈0.09) and the claim that f≥0.15 is 'guaranteed' at low noise depend directly on which width was used. The manuscript must state which value was actually used in the 800 retrievals, reconcile the text and table, and provide a sensitivity run at the wider prior.
  2. [Section 3.3] The only external benchmark for the oblate model is a single configuration in which TLCM and squishyplanet differ by roughly 13 ppm in amplitude, while the estimated oblateness signal is roughly 45 ppm. The systematic model difference is therefore about 29% of the signal amplitude, and it is comparable to the lowest tested white-noise levels (σw = 2–16 ppm). Since the injection and retrieval steps both use the same TLCM Gauss-Legendre integrator, systematic integration errors partly cancel in the recovery statistics, making the 3σ detection claim model-conditioned. I recommend adding comparisons over at least a small grid of f and ϑ values, or otherwise quantifying the bias that the 13 ppm-level model difference would induce in the retrieved f and in the detection rates.
  3. [Abstract and Section 4.2] The paper explicitly notes that the stellar-density prior trick is valid strictly for circular orbits, and all 800 injection tests use circular orbits. The abstract's unqualified statement that a 3σ oblateness detection is possible for a planet orbiting a bright enough star therefore overstates the scope: eccentric orbits with unknown eccentricity and argument of periastron are not covered by the reported detection rates. Since the limitation is already flagged in Section 4.2, this is a scope-and-presentation issue rather than an internal inconsistency, but the abstract should carry the circular-orbit qualifier or otherwise clearly state the restricted applicability of the method.
minor comments (5)
  1. [Throughout] There are several typographical errors that should be corrected in a revision, including 'oblanteness' in Section 4.1, 'charaterized' in Section 4.3, 'lowe' in Section 3.3, and 'the the model' in Section 4.1.
  2. [Abstract, Section 3.2, Section 5] The reported speed advantage of the numerical model over the analytical model is inconsistent: the abstract says 'about 25% faster,' Section 3.2 says the analytical model takes about 28% more time, and the conclusion says 'about 20% faster.' These numbers should be harmonized.
  3. [Eq. (26) and Fig. 5] The quadratic coefficient in the piecewise fit for log Δ versus log(a/R⋆) is given as −0.314(5) in Eq. (26) but as −0.134(5) in the caption of Fig. 5. Please check which value is correct and make the two consistent.
  4. [Table 7] The table uses the notation N(73.26, 0.04) for the a/R⋆ prior, while the text in Section 4.2 uses a standard deviation of 0.24; beyond resolving this discrepancy, the table would benefit from a footnote stating the corresponding assumed stellar-density precision.
  5. [Section 4.2] The limb-darkening priors are described as 'strict' with standard deviation 0.01, but the text in Section 4.1 refers to limb-darkening coefficients 'A = B = 1.3' while Table 7 uses uA and uB; please unify the notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central claim is an injection-retrieval sensitivity study with external benchmarks (Mandel-Agol and squishyplanet); the text/Table-7 prior discrepancy is a correctness issue, not a circular reduction.

full rationale

The paper's central derivation chain is a Monte Carlo power study: it injects transit light curves generated with a numerical quadrature model at known oblateness values, then retrieves parameters with the same model and reports 3-sigma detection rates. This is a self-consistency test, not a derivation in which the output is equivalent to the input by construction. The forward model is independently benchmarked in the spherical limit against the analytic Mandel-Agol model and in the oblate case against squishyplanet, so the validation is not purely self-referential. The detection claim is explicitly conditional on stated priors: a 1% stellar-density prior on a/R* (Eq. 30) and 1% limb-darkening priors. These priors are adopted from external asteroseismic precision estimates (Silva Aguirre et al. 2017), not fitted from the target data, so they are not a fitted input renamed as a prediction. The manuscript itself flags the circular-orbit restriction of the a/R* prior trick, which is a limitation but not circularity. The internal inconsistency between the text value sigma(a/R*) = 0.24 and Table 7's N(73.26, 0.04) affects the realism or reproducibility of the assumed prior and the resulting detection rates, but it does not make the derivation circular. Self-citations to TLCM and Kálmán et al. are software and noise-model references, not load-bearing uniqueness arguments. No equation defines the predicted quantity in terms of the input, and no parameter is fitted to a subset and then reported as a prediction. Therefore no significant circularity is present.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The retrieval strategy rests on three hand-chosen inputs: the 1% stellar density prior that sets the a/R* width, the 1% limb darkening priors, and the scaled ARIMA red-noise model. The outer validation against Mandel-Agol and squishyplanet gives independent grounding, but the detectability numbers are conditional on these assumed priors. No new physical entities are introduced.

free parameters (3)
  • a/R* Gaussian prior width = 0.24 (text) / 0.04 (Table 7)
    Chosen to encode 1% stellar density precision (Eq. 30) and to break the b-a/R* degeneracy; the text/table inconsistency is a concrete uncertainty in the central retrieval setup.
  • Limb darkening Gaussian prior widths = 0.01
    Adopted for uA and uB in Table 7 based on assumed knowledge of the star; detection rates depend on these tight priors.
  • Red noise amplitude scaling = 1/10 of ARIMA clone from Kalman et al. 2024
    Hand-chosen scaling for the time-correlated noise model; the detectability results are conditional on this noise representation.
assumptions (5)
  • standard math Gauss-Legendre quadrature with sum of weights = 2 (Eq. 8)
    Used for the 2D integration in Eq. (9) and (18); the paper admits the implemented point placement in Sect. 2.2 does not strictly follow this premise.
  • domain assumption Planet shape is a biaxial ellipsoid (spheroid) with rigid-body rotation; potential V = -GMp/rPol = -GMp/rEq - (1/2) omega^2 rEq^2 (Eq. 3)
    Neglects tidal forces and quadrupole moment; gives maximal oblateness fcrit = 1/3 and differs from the critical value cited from Berardo & de Wit.
  • domain assumption Star is spherical with static limb darkening; the 1.1 factor in the delta^2 < (1.1(1+rEq))^2 transit-phase cut is arbitrary (Sect. 2.2)
    The 1.1 factor is explicitly called arbitrarily selected; any number >= 1 would work, and it does not affect the light curve values.
  • domain assumption ARIMA clone of HST photometry, scaled by 1/10, represents space-based telescope red noise (Sect. 4.1)
    Adopted from Kalman et al. 2024; the detectability conclusions are conditional on this noise model.
  • domain assumption Stellar density known to 1% via asteroseismology, implying Delta(a/R*)/(a/R*) = 0.0033 (Eq. 30)
    Basis for the Gaussian prior on a/R*; the paper notes the trick is valid strictly for circular orbits.

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Pith. "Pith review of Prospects of detecting rotational flatness of exoplanets from space-based photometry." pith.science (2026). https://pith.science/paper/HBR4MI3K

@misc{pith2026250715359,
  author       = {Pith},
  title        = {Pith review of: Prospects of detecting rotational flatness of exoplanets from space-based photometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HBR4MI3K}},
  note         = {Machine review of arXiv:2507.15359}
}
abstract

In the era of photometry with space-based telescopes, such as CHEOPS (CHaracterizing ExOPlanets Satellite), JWST (James Webb Space Telescope), PLATO (PLAnetary Transits and Oscillations of stars), and ARIEL (Atmospheric Remote-sensing Infrared Exoplanet Large-survey), the road has opened for detecting subtle distortions in exoplanet transit light curves -- resulting from their non-spherical shape. We investigate the prospects of retrieval of rotational flatness (oblateness) of exoplanets at various noise levels. We present a novel method for calculating the transit light curves based on the Gauss-Legendre quadrature. We compare it in the non-rotating limit to the available analytical models. We conduct injection-and-retrieval tests to assess the precision and accuracy of the retrievable oblateness values. We find that the light curve calculation technique is about $25$\% faster than a well-known analytical counterpart, while still being precise enough. We show that a $3 \sigma$ oblateness detection is possible for a planet orbiting bright enough stars, by exploiting a precise estimate on the stellar density obtained e.g. from asteroseismology. We also show that for noise levels $\geq 256$ ppm (expressed as point-to-point scatter with a $60$~s exposure time) detection of planetary oblateness is not reliable.

Figures

Figures reproduced from arXiv: 2507.15359 by the authors.

Figure 1
Figure 1. Gauss-Legendre quadrature points used for the transit light curve calculations of an oblate planet (red ellipsoidal contours) in front of a star (yellow disk). When µ ∈ R, the integration point overlaps with the star, thus it is included in the transit computations (red points). On the other hand, when µ /∈ R, the (blue) point does not block light from the stellar disk. More quadrature points yield more precise ligh… view at source ↗
Figure 2
Figure 2. Dynamic distribution of the Gauss-Legendre quadrature points used for the transit light curve calculations of an oblate planet, from the improved model. The overlapping (transiting) area is highlighted in purple. -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -1.0 -0.5 0.0 0.5 1.0 log RP/R* log Δ[ ppm h] log Δ 1 ppmh = 4.914(52) + 2.955(52)log RP R ∗ [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Discrepancy between the analytical and numerical models for a given set of transit parameters, with changing RP/R⋆. of a runtime increase by two orders of magnitude (on the same computers). Given that the light curve modelings that are described below last for ≈ 1 day, this is not feasible yet. A possible solution might be the utilization of GPU-based calculations, however, that is beyond the scope of this work [PI… view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Discrepancy between the analytical and numerical models for a given set of transit parameters, with changing b. 0.8 1.2 1.6 2.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 log a/R* log Δ[ ppm h] log Δ 1 ppmh = − 1.27(12) + 1.37(11) log a R ∗ log Δ 1 ppmh = − 0.582(1) + 0.321(5) lo…
Figure 5
Figure 5. Figure 5: Discrepancy between the analytical and numerical models for a given set of transit parameters, with changing a/R⋆ (and corresponding P) [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Difference between the analytical model (ϕMA) and the numerical model presented here (ϕMA) for a transit generated with parameters taken from case I ( [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Discrepancy between the analytical and numerical models for a given set of transit parameters, with changing n number of Gauss-Legendre quadrature points (top panel). The light curve calculation time is shown on the bottom panel. We note that the intrinsic limitation o…
Figure 8
Figure 8. Figure 8: Difference between a transit light curve of an oblate planet simulated with TLCM and squishyplanet. Relative flux 0.994 0.996 0.998 1.000 σw = 2 ppm -0.005 0.000 0.005 Orbital phase [ppm] Residuals -300 0 300 Relative flux 0.994 0.996 0.998 1.000 σw = 4 ppm -0.005 0.00…
Figure 9
Figure 9. Figure 9: Example light curves with red noise (bigger panels, blue dots) for ϑ = 63◦ and f = 0.21 at all 8 σw noise levels. The best-fit transit models are shown with solid red lines. The residuals are shown on the smaller panels for every noise level. Orange dots show the resid…
Figure 10
Figure 10. Figure 10: Difference between the transit light curve of an oblate planet and a spherical planet, for a particular choice of f and ϑ. The shaded regions highlight the ingress and egress phases of the transit. f = (REq-RPol)/REq ϑ [degrees] 0 9 18 27 36 45 54 63 72 81 0.03 0.06 0…
Figure 11
Figure 11. Figure 11: Distortion signal in the transit light curve cause by the oblateness at every f–ϑ grid point as compared to a circular planet. The distortion ∆ is expressed as the area under the curve of the residuals of an oblate planet and a circular one with the same effective rad…
Figure 12
Figure 12. Figure 12: Accuracy of the retrieved oblateness parameter in the light curve which included red noise, for all eight σw noise levels. Squares in the input f–ϑ grid are coloured based on the accuracy of the fitted f: purple if the retrieved parameter is within 0.02 of the injecte…
Figure 13
Figure 13. Figure 13: Precision of the retrieved oblateness parameter in the light curve which included red noise, for all eight σw noise levels. Squares in the input f–ϑ grid are coloured based on the precision of the fitted f: purple if the retrieved parameter is within 1σ of the injecte…
Figure 14
Figure 14. Figure 14: Accuracy of the retrieved obliquity parameter in the light curve which included red noise, for all eight σw noise levels. Squares in the input f–ϑ grid are coloured based on the accuracy of the fitted ϑ: purple if the retrieved parameter is within 5◦ of the truth, blu…
Figure 15
Figure 15. Figure 15: Precision of the retrieved obliquity parameter in the light curve which included red noise, for all eight σw noise levels. Squares in the input f–ϑ grid are coloured based on the precision of the fitted θ: purple if the retrieved parameter is within 1σ of the injected…
Figure 16
Figure 16. Figure 16: Distribution of the fitted semi-major axes (left), impact parameters (middle) and equatorial radii (right). The top row (red) shows the absolute deviations from the injected values, the bottom row (blue) shown the relative deviations from the injected values. We compa…
Figure 17
Figure 17. Figure 17: Posterior distribution in the b – a/R⋆ space from the f = 0.06, ϑ = 9◦ case. 5. DISCUSSION & CONCLUSION We present a novel numerical approach for modelling the transit light curves of exoplanets whose shape is described by a biaxial ellipsoid due to their rapid rotati…

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