Transit light curve and inner structure of close-in planets

Planets orbiting very close to their host stars have been found, some of them on the verge of tidal disruption. The ellipsoidal shape of these planets can significantly differ from a sphere, which modifies the transit light curves. Here we present an easy method for taking the effect of the tidal bulge into account in the transit photometric observations. We show that the differences in the light curve are greater than previously thought. When detectable, these differences provide us an estimation of the fluid Love number, which is invaluable information on the internal structure of close-in planets. We also derive a simple analytical expression to correct the bulk density of these bodies, that can be 20% smaller than current estimates obtained assuming a spherical radius.

💡 Research Summary

The paper investigates how the strong tidal forces acting on planets that orbit very close to their host stars deform these bodies from a sphere into a triaxial ellipsoid, and how this deformation imprints itself on transit light curves. The authors develop a simple analytical framework that links the planet’s shape to its fluid second Love number (h_f), which quantifies the response of a fluid body to external potentials.

Starting from the standard ellipsoid equation X²/a² + Y²/b² + Z²/c² = 1, they express the semi‑axes a (the long axis pointing toward the star), b (the equatorial axis orthogonal to the star‑planet line), and c (the polar axis) in terms of a dimensionless deformation parameter q = h_f (m_/m)(b/r₀)³, where m_ and m are the stellar and planetary masses, r₀ is the orbital distance, and b is the mean equatorial radius. The parameter q scales as r₀⁻³, reaching a maximum near the Roche limit (q_max ≈ h_f/30). Consequently, a ≈ b(1+3q) and c ≈ b(1−q), indicating that the planet can be up to ~25 % longer along the star‑direction and ~8 % flatter at the poles when close to disruption.

To compute the transit signal, the authors transform the ellipsoid into the observer’s frame using the planet’s orbital phase θ (θ = n(t−t₀) for a synchronous circular orbit) and inclination i. The projection onto the sky plane yields an ellipse described by A(x−x₀)² + B(x−x₀)(y−y₀) + C(y−y₀)² = 1, where the coefficients A, B, and C depend on a, b, c, θ, and i. The overlap between this projected ellipse and the stellar disk (assumed spherical with radius R_*) gives the instantaneous loss of stellar flux. The authors show that, for the level of deformation considered, the choice between a uniform stellar surface brightness and a realistic quadratic limb‑darkening law has negligible impact on the differential signal.

Two characteristic signatures appear in the light curve. The first, already known, is a small oscillation at ingress and egress caused by the polar flattening (c < b). The second, newly highlighted, is a “bump” that persists throughout the entire transit. This bump arises because the long axis a rotates with the planet’s orbital phase, causing the projected area to vary continuously. The effect is strongest for edge‑on transits (i ≈ 90°) and for planets with large q (i.e., those very close to their star). Simulated light curves for real systems such as Kepler‑78b, WASP‑12b, and WASP‑19b show differences up to 10⁻⁴ relative flux—well above the typical stellar variability floor (~10⁻⁵) on transit timescales. For more distant planets the signal drops below 10⁻⁵, making detection more challenging.

Because the projected ellipse depends only on the single additional parameter q (all other quantities are observable), fitting the analytical model to high‑precision photometry directly yields q, and thus h_f, via the relation q = h_f (m_*/m)(b/r₀)³. The fluid Love number h_f encodes the planet’s internal mass distribution; through the Darwin‑Radau relation it connects to the normalized moment of inertia I/(mR²). Therefore, a measurement of h_f provides a rare observational constraint on the degree of internal differentiation (e.g., core size, envelope rigidity).

The authors also address the impact of the ellipsoidal shape on bulk density estimates. The true volume of a triaxial ellipsoid is V = 4πabc/3. Substituting the expressions for a and c in terms of b and q yields V ≈ V_s (1 + q), where V_s = 4πb³/3 is the volume of a sphere with radius b. For q ≈ 0.1, the volume is ~10 % larger, implying that densities derived under the spherical assumption can be overestimated by up to ~20 %. This correction is particularly important for planets near the Roche limit, where the deformation is strongest.

In summary, the paper provides a tractable, analytic method to incorporate tidal deformation into transit modeling, enabling the extraction of the fluid Love number and a corrected bulk density from existing photometric data. The approach avoids the need for complex numerical interior models while still capturing the essential physics of tidal bulges. It is readily applicable to current and upcoming high‑precision transit surveys (e.g., TESS, PLATO, JWST), opening a pathway to probe the interior structure of close‑in exoplanets directly from their light curves.