A Radon-Transform Perspective on Exoplanet Transits

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.

• The paper shows that, in the knife-edge uniform-brightness limit, the time derivative of ingress or egress directly measures a Radon projection of the planet’s attenuation map, with validation residuals of about 100 ppm for a 1% planet-to-star area ratio and 20 ppm for 0.3%.
• A single transit provides at most two projection angles, leaving an infinite-dimensional null space in silhouette reconstruction and creating persistent degeneracies with orbital parameters, even when physical priors or multi-wavelength observations are added.
• Limb darkening and finite exposure blur existing projections, while stellar-limb curvature adds weak transverse information and complementary ingress–egress constraints that are strongest near an impact parameter of approximately 0.7.

Overview

Transit photometry has traditionally relied on the assumption that the transiting planet presents a circular sky-projected silhouette. As photometric precision and cadence improve—particularly with JWST—this assumption is increasingly tested by physical effects that can distort the projected shape: rapid rotation, tidal deformation, rings, and azimuthally asymmetric atmospheric opacity. In this paper, Tada develops an analytical framework that clarifies the fundamental information content of transit light curves for non-circular silhouettes. The central result is that, under a set of transparent approximations, the time derivative of the flux during ingress or egress constitutes a Radon-transform measurement of the planetary attenuation map, with the projection direction fixed by the local normal to the stellar limb (2608.13163). This identification makes the degeneracies of transit shape inference mathematically explicit.

Forward model and the Radon-transform limit

The author defines a planet-centered attenuation map $$A_{\mathrm p}(\bm{x}) \equiv 1 - \mathcal{T}_{\mathrm p}(\bm{x})$$, where $\mathcal{T}_{\mathrm p}$ is the transmittance, and allows arbitrary non-circular silhouettes. The key derivation proceeds by differentiating the observed flux under the assumption that the stellar intensity map is static in the star-centered frame, so that the time derivative of the flux reduces to an integral of the attenuation map weighted by the time derivative of the stellar intensity.

The critical approximation is the "knife-edge, uniform-brightness" limit: the stellar limb is locally linearized as a straight edge and limb darkening is neglected. Under this approximation, the gradient of the stellar intensity is a Dirac delta on the tangent line, and the flux derivative becomes

$$\dot{F}_j(t) = -I_0\,\bm{n}_j\cdot\bm{v}(t)\, \mathcal{R}[A_{\mathrm p}]\!\left(s_j(t),\theta_j\right),$$

where $\mathcal{R}$ is the Radon transform, $\bm{n}_j$ is the inward unit normal to the stellar limb at ingress ($j=\mathrm i$) or egress ($j=\mathrm e$), and $s_j(t)$ is the Radon offset determined by the orbital motion. Each ingress or egress thus traces a single projection of $A_{\mathrm p}$ at a single angle.

The author validates this approximation against established forward models (jaxoplanet, squishyplanet, catwoman) for circular, elliptical, and two-lobed silhouettes. For a planet-to-star area ratio of 0.01 (roughly a transiting hot Jupiter), light-curve residuals remain within about 100 ppm; for a ratio of 0.003, within about 20 ppm. These residuals are quantitatively consistent with the derived curvature error bound, $$\lesssim 106\,\mathrm{ppm}\,(R_{\mathrm p}/R_{\mathrm s}/0.1)^3$$ for a circular star, confirming that the dominant discrepancy is the neglected stellar-limb curvature rather than any failure of the Radon identification.

Information content and the null space

The Fourier-slice theorem provides the sharpest statement of the information content: the one-dimensional Fourier transform of a Radon projection at angle $\mathcal{T}_{\mathrm p}$0 equals the two-dimensional Fourier transform of the map evaluated along a single radial line. Since a single transit supplies at most two projection angles (one at ingress, one at egress), the data constrain $\mathcal{T}_{\mathrm p}$1 only along at most two radial slices in the Fourier plane. The null space of the inverse problem comprises all maps whose Fourier transforms vanish on those slices, and this space is infinite-dimensional.

The author demonstrates this non-uniqueness constructively, showing explicit examples of distinct opaque silhouettes—circles, ellipses with different position angles, and more complicated shapes—that produce identical light curves for given ingress/egress angles. Physical priors (bounded attenuation $\mathcal{T}_{\mathrm p}$2, binary opacity, compact support) and multi-wavelength data reduce the feasible set but do not eliminate the geometric null space, since all wavelengths sample the same two projection angles.

A crucial qualification is that the mapping from observation time to Radon offset depends on the orbital geometry. The ingress–egress interval constrains only a combination of period, $\mathcal{T}_{\mathrm p}$3, and impact parameter $\mathcal{T}_{\mathrm p}$4, so silhouette inference is degenerate with the orbital solution unless external constraints (radial velocities, stellar density, secondary eclipses) are supplied. Any inferred silhouette is therefore a product of the data plus the adopted prior structure, not a unique reconstruction.

Departures from the idealized limit

The paper examines four realistic modifications, each with a specific consequence for the accessible information.

Limb darkening: under a local one-dimensional brightness profile, limb darkening replaces the ideal Radon measurement by a one-dimensional convolution along the offset direction. In the Fourier domain, it multiplies the accessible slices by a transfer function but does not open new projection angles. Even a perfectly known limb-darkening profile therefore adds no information about $\mathcal{T}_{\mathrm p}$5 beyond the uniform-brightness limit; if the profile is fitted simultaneously, it becomes degenerate with the measurable projections and can bias the inferred silhouette.

Finite exposure time: exposure averaging smooths the signal along the Radon-offset direction with width $\mathcal{T}_{\mathrm p}$6, and discrete sampling imposes a Nyquist limit $\mathcal{T}_{\mathrm p}$7 on recoverable spatial frequencies. For hot-Jupiter parameters with JWST-like cadence, the native sampling resolves spatial scales of a few percent to a few tenths of a planetary radius, though photon noise and systematics will usually dominate in practice.

Stellar-limb curvature: to first order in the limb curvature $\mathcal{T}_{\mathrm p}$8, the flux derivative acquires terms involving the first and second tangential moments $\mathcal{T}_{\mathrm p}$9 and $$\dot{F}_j(t) = -I_0\,\bm{n}_j\cdot\bm{v}(t)\, \mathcal{R}[A_{\mathrm p}]\!\left(s_j(t),\theta_j\right),$$0 of the map. In the Fourier plane, these correspond to the first and second transverse derivatives of $$\dot{F}_j(t) = -I_0\,\bm{n}_j\cdot\bm{v}(t)\, \mathcal{R}[A_{\mathrm p}]\!\left(s_j(t),\theta_j\right),$$1 about the ideal slices. Curvature thus does not provide new projection angles, but it weakly constrains a neighborhood of the slices—a sensitivity entirely absent in the strict Radon limit. The complementarity between ingress and egress information is maximized when the two slices are orthogonal, at $$\dot{F}_j(t) = -I_0\,\bm{n}_j\cdot\bm{v}(t)\, \mathcal{R}[A_{\mathrm p}]\!\left(s_j(t),\theta_j\right),$$2, which explains why prior oblateness studies (Barnes & Fortney 2003; Dholakia et al. 2025) find optimal detectability near $$\dot{F}_j(t) = -I_0\,\bm{n}_j\cdot\bm{v}(t)\, \mathcal{R}[A_{\mathrm p}]\!\left(s_j(t),\theta_j\right),$$3–0.8.

Time-dependent maps: for tidally locked short-period planets such as WASP-121b, WASP-18b, and WASP-43b, a multi-hour transit corresponds to tens of degrees of rotational phase change. If ingress and egress sample different maps, each event provides one projection of a different map, making the inverse problem strictly more underconstrained. Relating the two maps requires additional assumptions, such as a common three-dimensional structure anchored by a general circulation model.

Implications for shape inference

The framework yields concrete results for practical inference. For elliptical silhouettes, each Radon slice of a centered uniform ellipse is determined by the projected area and the support function $$\dot{F}_j(t) = -I_0\,\bm{n}_j\cdot\bm{v}(t)\, \mathcal{R}[A_{\mathrm p}]\!\left(s_j(t),\theta_j\right),$$4; ingress and egress constrain $$\dot{F}_j(t) = -I_0\,\bm{n}_j\cdot\bm{v}(t)\, \mathcal{R}[A_{\mathrm p}]\!\left(s_j(t),\theta_j\right),$$5 only up to the uncertain orbital scaling factors, producing extended families of degenerate solutions in $$\dot{F}_j(t) = -I_0\,\bm{n}_j\cdot\bm{v}(t)\, \mathcal{R}[A_{\mathrm p}]\!\left(s_j(t),\theta_j\right),$$6 space. The author demonstrates this quantitatively by overlaying analytically derived degeneracy curves on the JWST/NIRSpec posterior for Kepler-51d from Liu et al. (2024): the curves, anchored at single posterior samples without fitting, closely track the posterior contours, confirming that the Radon-transform picture captures the dominant shape–orbit degeneracy in a real observational analysis.

For atmospheric inhomogeneity, the framework supports limb-resolved analyses that treat ingress and egress separately, consistent with recent JWST detections of morning–evening terminator asymmetries. It also flags two caveats: limb-darkening errors can bias separately fitted ingress and egress offsets (though symmetric profiles cause partial cancellation for morning–evening asymmetry), and the centroid of the effective silhouette can differ from the center of mass, complicating the separation of orbital geometry from atmospheric asymmetry—a problem the author notes is not yet fully resolved in current analyses.

Limitations and open questions

The framework is explicitly a leading-order construction. The knife-edge approximation requires $$\dot{F}_j(t) = -I_0\,\bm{n}_j\cdot\bm{v}(t)\, \mathcal{R}[A_{\mathrm p}]\!\left(s_j(t),\theta_j\right),$$7, and the quantitative validation covers only area ratios down to 0.003; for Earth–Sun analogs (area ratio $$\dot{F}_j(t) = -I_0\,\bm{n}_j\cdot\bm{v}(t)\, \mathcal{R}[A_{\mathrm p}]\!\left(s_j(t),\theta_j\right),$$8), the signal is far smaller, and the applicability of the framework at those scales is not demonstrated. The curvature corrections are derived only to first order in $$\dot{F}_j(t) = -I_0\,\bm{n}_j\cdot\bm{v}(t)\, \mathcal{R}[A_{\mathrm p}]\!\left(s_j(t),\theta_j\right),$$9, so the extent to which they genuinely break the exact degeneracies of the strict Radon limit is characterized perturbatively rather than exhaustively. The static-map assumption fails for rapidly rotating planets, and the proposed remedy—treating ingress and egress as projections of two different maps—leaves the relationship between those maps dependent on adopted dynamical priors whose influence on the recovered solution is not quantified. Finally, the paper establishes the null-space structure analytically but does not develop a practical inversion algorithm that optimally exploits the weak curvature-induced sensitivity; how much of the ideal degeneracy can be broken with realistic noise remains an open quantitative question.

Conclusion

This paper recasts transit ingress and egress as Radon-transform measurements of the planetary attenuation map, providing a compact analytical account of why a single transit cannot uniquely determine a non-circular silhouette: the data constrain the Fourier transform of the map along at most two radial slices, leaving a large null space, and the interpretation is further entangled with orbital parameters. The framework clarifies the roles of limb darkening (a convolution, not new angles), finite sampling (a spatial-frequency ceiling), stellar-limb curvature (weak transverse sensitivity peaking in complementarity at $\mathcal{R}$0), and rotation (two projections of two different maps). Its principal value is diagnostic: it delineates precisely what transit light curves can and cannot reveal about planetary shape and atmospheric asymmetry, and it explains quantitatively the degeneracy structure seen in existing JWST oblateness analyses.