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REVIEW 1 major objections 6 minor 53 references

A Radon-Transform Perspective on Exoplanet Transits

T0 review · 1 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read In a clean limit, an exoplanet's ingress and egress are two Radon projections of its shadow, so a single transit fixes the attenuation map on only two radial Fourier slices.

desk verdict A clean and honest Radon-transform reformulation of transit shape inference; the core identity is correct and the paper deserves refereeing and publication. read the letter →

arxiv 2608.13163 v1 pith:VDSBKEVR submitted 2026-08-13 astro-ph.EP astro-ph.IM

Shotaro Tada This is my paper · ORCID
classification astro-ph.EPastro-ph.IM
keywords exoplanetsexoplanetstructuretransitphotometryRadontransformFourier-slicetheoreminverseproblemsnon-circularsilhouettesplanetaryoblateness
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

In a clean limiting case, this paper establishes exactly what a single transit light curve can tell us about a non-circular planet. When the stellar limb is treated as a locally straight, uniformly bright knife edge, the time derivative of the light curve during ingress or egress becomes literally one Radon projection, a line integral, of the planet's two-dimensional attenuation map, with the projection direction set by the local normal to the stellar limb. Since ingress and egress supply at most two projection angles, the Fourier-slice theorem says the data constrain the map's two-dimensional Fourier transform only along two radial slices, leaving a large null space of invisible map features. The paper then shows that limb darkening and finite exposure only blur or reweight those slices rather than adding new angles, while limb curvature adds only weak sensitivity to transverse Fourier derivatives, so the non-uniqueness is structural. If correct, this reframes silhouette and atmosphere-asymmetry claims from transits: the inferred shape is always a prior-dependent choice among a continuum of maps that fit the same light curve.

What carries the argument

The load-bearing machinery is the two-dimensional Radon transform, $R[f](s,\,\theta)=\int_{\mathbb{R}^2} f(x)\,\delta(n(\theta)\cdot x - s)\,dx\,dy$, combined with the Fourier-slice theorem, which identifies the one-dimensional Fourier transform of a projection with a slice of the map's two-dimensional Fourier transform. In the transit context, the orbital velocity maps time into the Radon offset $s_j(t)$ and supplies the prefactor $n_j\cdot v(t)$, so each limb crossing mechanically scans out one projection of $A_p$. The inverse-problem consequence is the two-slice null-space condition of Eq. (33): any map whose transform vanishes on the two radial slices is silent. Departures from the ideal limit are handled by the same machinery: limb darkening convolves along the offset direction, finite exposure multiplies the slice by a sinc, and limb curvature adds terms built from tangential moments $M_1$ and $M_2$, which correspond to transverse derivatives of $\widehat{A}_p$ about the ideal slices.

What would settle it

Take two physically allowed silhouettes that the Radon limit declares identical, such as an ellipse and a more complicated shape from the paper's Figure 5, compute noiseless light curves with a full limb-darkened, curved-limb transit model, and check whether the ingress and egress derivatives differ by more than the paper's curvature-level residuals (roughly tens of ppm for $R_p/R_s \sim 0.1$); if they differ measurably, the practical null space is strictly smaller than the two-slice claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that a transit ingress or egress is an exact Radon measurement in the knife-edge, uniform-brightness limit. The time derivative of the flux satisfies $\dot{F}_j(t) = -I_0\,(n_j\cdot v(t))\,R[A_p](s_j(t),\,\theta_j)$, so each event gives one projection of the attenuation map $A_p$; the temporal Fourier transform of that segment then samples $\widehat{A}_p(-\nu\cos\theta_j/\alpha_j,\,-\nu\sin\theta_j/\alpha_j)$, a single radial line through the origin of the map's Fourier plane. Ingress and egress therefore constrain at most two radial slices, and the null space of the forward map contains every attenuation map whose Fourier transform vanishes on those two slices. Consequently, no unique non-circular silhouette can be recovered from the light curve alone; physically allowed maps that differ only off those slices are exactly indistinguishable in this limit. The same structure carries over to secondary eclipses with the planetary brightness map in place of the attenuation map.

Load-bearing premise

The result depends on treating the stellar limb near ingress and egress as a straight, uniformly bright knife edge with a static planet shadow; if the limb is curved or limb-darkened beyond a tiny correction, the light-curve derivative is no longer a pure Radon projection and the two-slice null-space statement is only approximate.

Editorial extensions

If this is right

  • A single transit can never pin down a unique non-circular silhouette: many physically allowed attenuation maps produce exactly the same ingress and egress light-curve derivatives in the Radon limit.
  • Oblateness inference is inseparable from the orbit: changing projected oblateness can be compensated by changing impact parameter and scaled semi-major axis, so external stellar and orbital constraints are required to break the degeneracy.
  • Limb darkening adds no new information; it only reweights the same two Fourier slices, so an unknown limb-darkening profile can bias the recovered attenuation map.
  • Finite exposure time and discrete sampling set a resolution floor along each slice; structures narrower than roughly twice the corresponding offset spacing cannot be recovered.
  • Stellar-limb curvature slightly softens the strict null space by making the data weakly sensitive to transverse Fourier derivatives around the ideal slices, with the most complementary coverage at impact parameter $b\sim 1/\sqrt{2}$, which explains why oblateness constraints tend to be strongest at high impact parameters.

Reading between the lines

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

  • Beyond the paper, the same two-slice picture gives a quantitative explanation for why eclipse-mapping analyses need phase curves or rotation information: the eclipse analog constrains the brightness map on only two radial slices, so any claimed longitudinal map is largely prior-driven.
  • If the curvature correction scales as the paper estimates, then near-grazing transits of large planets around small stars, where curvature residuals reach tens to hundreds of ppm, may be the practical place to recover some off-slice Fourier structure from high-precision photometry alone.
  • The framework suggests a testable prediction: a flexible non-parametric inversion of a high-signal-to-noise transit should return a continuous family of maps aligned with the Radon-invariant directions, and any point estimate that looks unique is a property of the prior, not the data.
  • One could extend the analysis by treating the two slices as weak measurements of the map's second moments; this would make quantitative the intuition that ingress and egress constrain a planet's projected area and its projections onto two directions, and could be turned into a model-independent moment estimator.
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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

1 major / 6 minor

Summary. Transit light curves are usually modeled with a circular sky-projected silhouette. This paper asks what a single transit can reveal when the planetary attenuation map A_p(x) is non-circular. Under a local knife-edge approximation of the stellar limb, with uniform stellar brightness (Eq. 12), the paper derives that the time derivative of the flux during ingress or egress equals the Radon transform of A_p at the projection angle set by the limb normal, multiplied by the limb-normal velocity component (Eq. 20). Section 3 uses the Fourier-slice theorem, together with a constant-velocity approximation, to show that the temporal Fourier transform of the ingress/egress derivative samples the 2D Fourier transform of A_p along at most two radial lines (Eqs. 30-31); hence the null space of a single transit consists of maps whose Fourier transforms vanish on those two slices (Eq. 33). Section 4 examines departures: limb darkening acts as a convolution along the same projection direction without introducing new projection angles; finite exposure time and sampling smooth and limit the accessible spatial frequencies along the slices; stellar-limb curvature adds weak sensitivity to transverse Fourier derivatives about the ideal slices; time-dependent planetary maps make the problem even more underconstrained. Section 5 applies the framework to oblateness inference (with a comparison to the Kepler-51d posterior of Liu et al.

Significance. If the result holds, this paper provides a compact, parameter-free explanation of several known degeneracies in transit shape inference and sharpens them into a single statement: a single transit constrains only two radial slices through the Fourier transform of the attenuation map. The central identity (Eq. 20) is derived from stated assumptions with no fitted constants, and I verified the key steps, including the Fourier-transform prefactor in Eq. (31), the limb-darkening convolution and slice invariance (Eqs. B16-B17), and the curvature expansion (Eqs. C15-C20); the algebra is clean throughout. The paper is also commendably honest about its frame: the two-slice null-space statement is explicitly conditional on the idealized half-plane model, and the extensions in Sections 4.1-4.4 quantify how realistic effects lift the degeneracy only weakly. The numerical checks against three independent codes (Figure 3) and the alignment of the analytic degeneracy curves with the published Kepler-51d posterior (Figure 7) strengthen confidence in the leading-order picture.

major comments (1)
  1. [§2.3, Figure 3] The text quotes the agreement between the Radon approximation and the model codes at the level of the light-curve residuals (about 100 ppm and 20 ppm for the two adopted area ratios), but the quantity that carries the Radon information in Eq. (20) is the time derivative of the light curve. The derivative residuals are displayed in the lower rows of Figure 3 but are neither quantified nor interpreted. From the curvature expansion in Eq. (C20), the leading curvature correction to the derivative is of order s/(2R_s) relative to the leading Radon term, i.e., up to about R_p/(2R_s) of the local derivative and R_p/(4R_s) of the peak derivative, which for the R_p/R_s = 0.1 case is a few percent, much larger in relative terms than the quoted ppm-level flux residuals. Please report the derivative residuals for the three comparisons and state explicitly that Eq. (20) holds only at the percent level for the largest area ratio considered, so that readers can gauge the accuracy of the derivative-level identification.
minor comments (6)
  1. [§5.1, Figure 7] The paper uses posterior samples from Q. Liu et al. (2024) with their permission, but it does not state whether the samples are publicly available or describe how the 'selected reference samples' in Figure 7 were chosen; please add a data-availability statement (or deposit the samples) and specify the selection procedure so that the comparison is reproducible.
  2. [Sections 2.3, 3.2, 5.1] The paper lists the software versions used (jaxoplanet, squishyplanet, catwoman, JAX, etc.) but provides no code for the Radon-approximation light curves, the non-uniqueness examples of Figure 5, or the analytic degeneracy curves of Figures 6 and 7; a short repository or even a pseudocode listing would make the numerical claims independently checkable.
  3. [Abstract and Section 1] The abstract motivates non-circular silhouettes by listing 'planetary rotation, tides, rings, or atmospheric inhomogeneities,' but the body does not discuss ringed silhouettes except for a passing reference in Section 5.1; either add a brief treatment of rings or remove them from the motivating list in the abstract.
  4. [Equation (13)] The Heaviside function is defined only for x>0 and x<0, leaving H(0) unspecified; stating H(0) (or noting that values on a measure-zero set are irrelevant for the distributional argument) would close a small analytic gap.
  5. [Equation (33)] The null-space condition is written with 'θ ∈ {θ_i, θ_e}' inside the universal quantifier; writing the two slice conditions separately (θ = θ_i and θ = θ_e) would be easier to read.
  6. [Throughout] There are small typesetting artifacts (e.g., 'Radon-T ransform' in the title line and the spacing in 'Keywords:Exoplanets') that should be cleaned up in the final version.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the Radon-limit identity is derived from an explicit forward model, and the paper's own extensions show the main corrections preserve or weaken the two-slice picture.

full rationale

The central identity Eq. (20) is a distributional consequence of the stated knife-edge uniform-half-plane approximation Eq. (12), with no fitted constants and no parameter that encodes the target result. The Fourier-slice and null-space statements (Eqs. 26, 31, 33) are standard Radon theory applied to that identity; I checked Eq. (31) algebraically and the prefactors and arguments are self-consistent. The paper explicitly conditions its main claim on Eq. (12) and then independently derives how limb darkening (Eqs. 39, 42, B16), finite exposure (Eqs. 44-49), curvature (Eqs. C20, C24), and time-dependent maps (Eqs. 56-57) modify the ideal picture, so the claimed scope is not inflated. The Kepler-51d comparison in Section 5.1 anchors degeneracy curves at selected posterior samples rather than fitting them ('Each curve is anchored at one selected sample but is not fitted to the posterior'), which is an external benchmark, not a circular validation. The only self-citation is S. Tada et al. (2025) in Section 5.2, used to note consistency with an existing ingress/egress asymmetry method; it is illustrative and not load-bearing for the Radon-limit derivation. Thus no step reduces by construction to its input.

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

No free parameters are introduced; the theory is a parameter-free identity under stated geometric approximations. The main axioms are the local half-plane stellar approximation, static maps, constant projected velocity across each limb crossing, the distributional Fourier-slice machinery, and full silhouette overlap. These are all stated in Sections 2 and 3 and are relaxed or discussed in Sections 4.1-4.4.

assumptions (5)
  • domain assumption Stellar limb is locally a straight edge and stellar surface brightness is uniform near ingress and egress (knife-edge approximation).
    Invoked in Eq. (12) and Section 2.2; it is the premise that makes the light-curve derivative an exact Radon projection and therefore yields the two-slice null space. Curvature and limb darkening relax it (Sections 4.1 and 4.3).
  • domain assumption The sky-projected planetary attenuation and emission maps are static over the transit.
    Assumed in Section 2.1; needed for ingress and egress to be two projections of the same map. Relaxed for rotating planets in Section 4.4, which only makes the inversion more underconstrained.
  • domain assumption Projected velocity is approximately constant over each ingress and egress.
    Used in Eq. (27) and Section 3.1 to map temporal frequency to spatial frequency along the Fourier slice; without it, the correspondence is a time-rescaled version with the same angular coverage.
  • standard math Fourier-slice theorem and distributional identities for the Heaviside and delta functions.
    Used in Eq. (26) and Eq. (14) to connect Radon projections to Fourier slices.
  • domain assumption The whole silhouette is fully superimposed on the stellar disk at some point of the transit.
    Stated in Section 2.2; ensures both ingress and egress scan the full map rather than grazing only part of it.

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Cite this review

Pith. "Pith review of A Radon-Transform Perspective on Exoplanet Transits." pith.science (2026). https://pith.science/paper/VDSBKEVR

@misc{pith2026260813163,
  author       = {Pith},
  title        = {Pith review of: A Radon-Transform Perspective on Exoplanet Transits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VDSBKEVR}},
  note         = {Machine review of arXiv:2608.13163}
}
read the original abstract

Transit light curves are usually analyzed under the assumption that the transiting planet has a circular sky-projected silhouette. However, planetary rotation, tides, rings, or atmospheric inhomogeneities can produce non-circular silhouettes. This raises the question of what information transit light curves can provide about the underlying two-dimensional attenuation map. In this paper, we show that, in a simple and transparent limit, the time derivative of the transit light curve during ingress or egress can be interpreted as a Radon-transform measurement of the planetary attenuation map, with the projection direction set by the local normal to the stellar limb. This viewpoint makes the information content of a single transit clear. Ingress and egress provide at most two projection angles, so the data constrain the Fourier transform of the attenuation map only along at most two radial slices, leaving a large null space. Physical constraints on the attenuation values and shape priors can reduce the range of viable solutions, but non-uniqueness generally remains. We further examine how realistic effects modify this picture. In particular, small stellar-limb curvature introduces weak sensitivity to transverse Fourier-space structure around the ideal slices, a sensitivity that is absent in the strict Radon-transform limit. These results provide a framework for understanding what transit light curves can and cannot reveal about non-circular planetary silhouettes.

Figures

Figures reproduced from arXiv: 2608.13163 by the authors.

Figure 1
Figure 1. Geometry and coordinate conventions. The sky– plane coordinates X ≡ (X, Y ) T are centered on the star, while x ≡ (x, y) T are centered on the planet. The origins are denoted by Os (stellar center) and Op (planet center), respectively, with Op located at (Xp(t), Yp(t)) in the (X, Y ) frame. The large circle represents the projected stellar disk, and the black-filled shape represents the projected planetary silhouett… view at source ↗
Figure 2
Figure 2. (a) Schematic illustration of the Radon trans￾form. (b,c) Close-up views near the stellar limb during (b) ingress and (c) egress. In the knife-edge approximation, the curved stellar limb is locally approximated by a straight edge; the dashed lines indicate the locally linearized limb at differ￾ent times. The blue arrows indicate the local inward normals to the stellar limb. In the Radon-transform limit, these nor￾ma… view at source ↗
Figure 3
Figure 3. Comparison of light curves from several transit model codes with those predicted by the Radon-transform approxi￾mation. Only the portions of the light curves around ingress and egress are shown. We show cases with planet-to-star area ratios of 0.01 (upper panels) and 0.003 (lower panels). From left to right, the columns show results for a circular silhouette computed with jaxoplanet (S. Hattori et al. 2024), an elli… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Radon projections and Fourier-domain representations of two synthetic spatial patterns of Ap(x, y). Each row shows, from left to right, the spatial map, its Radon sinogram, the log-scaled Fourier amplitude, and the Fourier phase. The white and black lines indicate the …
Figure 5
Figure 5. Figure 5: Examples of non-uniqueness for opaque silhou￾ettes in the Radon-transform limit. The four panels in the top row show silhouettes that produce the same light curve for (θi, θe) = (30◦ , 150◦ ): a circular silhouette, an elliptical silhouette, and two more complicated sh…
Figure 6
Figure 6. Figure 6: Ingress (or egress) duration as a function of the signed oblateness parameter ˜f and the impact parameter b for circular orbits in the Radon-transform limit. Over each of ingress and egress, the projected velocity is assumed to be constant (Equation (27)). The projecte…
Figure 7
Figure 7. Figure 7: Comparison between the posterior distribu￾tion obtained by Q. Liu et al. (2024) for an elliptical transit model of Kepler-51d and the shape–orbit degeneracy pre￾dicted by the Radon-transform approximation. The upper panels show the posterior in b, f cos(2ϕ), f sin(2ϕ),…
Figure 8
Figure 8. Figure 8: Radon projections and Fourier-domain representations of two synthetic spatial patterns of Ip(x, y). As in [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

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