Determination of the Interior Structure of Transiting Planets in Multiple-Planet Systems

Tidal dissipation within a short-period transiting extrasolar planet perturbed by a companion object can drive orbital evolution of the system to a so-called tidal fixed point, in which the apsidal lines of the transiting planet and its perturber are aligned, and for which variations in the orbital eccentricities of both planet and perturber are damped out. Significant contributions to the apsidal precession rate are made by the secular planet-planet interaction, by general relativity, and by the gravitational quadropole fields created by the transiting planet’s tidal and rotational distortions. The fixed-point orbital eccentricity of the inner planet is therefore a strong function of the planet’s interior structure. We illustrate these ideas in the specific context of the recently discovered HAT-P-13 exo-planetary system, and show that one can already glean important insights into the physical properties of the inner transiting planet. We present structural models of the planet, which indicate that its observed radius can be maintained for a one-parameter sequence of models that properly vary core mass and tidal energy dissipation in the interior. We use an octopole-order secular theory of the orbital dynamics to derive the dependence of the inner planet’s eccentricity, on its tidal Love number. We find that the currently measured eccentricity, implies 0.116 < k2_{b} < 0.425, 0 M_{Earth}<M_{core}<120 M_{Earth}$, and Q_{b} < 300,000. Improved measurement of the eccentricity will soon allow for far tighter limits to be placed on all three of these quantities, and will provide an unprecedented probe into the interior structure of an extrasolar planet.

💡 Research Summary

The paper presents a novel method for probing the interior structure of short‑period transiting exoplanets that reside in multi‑planet systems, using the HAT‑P‑13 system as a concrete example. The authors focus on the concept of a “tidal fixed point,” a dynamical configuration in which the inner transiting planet (b) and its outer companion (c) have their apsidal lines aligned (or anti‑aligned) and precess at the same rate. In this state, the inner planet’s orbital eccentricity (e_b) is no longer a free parameter but is set by a balance between several contributions to its apsidal precession: secular planet‑planet interactions, general relativistic (GR) precession, and the quadrupole precession arising from the planet’s tidal and rotational bulges. Crucially, the quadrupole term depends on the planet’s tidal Love number (k₂_b), which encodes the degree of central condensation and therefore directly reflects the internal density distribution, including the mass of any solid core.

The authors first develop the dynamical framework using an octupole‑order secular theory (Mardling 2007) that is valid for systems far from mean‑motion resonance and where the inner planet’s eccentricity is much smaller than that of the outer companion. They write down the time‑derivatives of the eccentricities and longitudes of pericenter, incorporating the tidal dissipation term characterized by the quality factor Q_b. The total precession rate of the inner planet is expressed as the sum of four terms: secular, tidal‑induced, GR, and rotational. The tidal precession term is proportional to k₂_b (R_b/a_b)^5 n_b f₂(e_b), where f₂(e_b) is a known eccentricity function. The Love number itself is computed from the internal density profile ρ(r) via the differential equation originally derived by Sterne (1939).

Having established the theoretical link between k₂_b, Q_b, and e_b, the authors apply the model to the observed parameters of HAT‑P‑13b: mass ≈0.85 M_J, radius ≈1.28 R_J, orbital period ≈2.916 days, and measured eccentricity e_b = 0.021 ± 0.009. The outer companion, HAT‑P‑13c, has M_c sin i ≈15 M_J, period ≈428 days, and eccentricity ≈0.691. Assuming the system has already relaxed to the tidal fixed point (reasonable given the system’s age and the rapid tidal circularization timescale for the inner planet), the condition ˙ω_b,total = ˙ω_c,sec yields a relationship between e_b and k₂_b.

To explore the interior structure, the authors generate a grid of planetary evolution models using a modified Berkeley stellar evolution code that solves the standard equations of planetary structure, including contributions from gravitational contraction, radiative cooling, and tidal heating. The two free parameters in this grid are the solid core mass (M_core) and the internal tidal luminosity (L_int). For each (M_core, L_int) pair they compute the planetary radius, Love number k₂_b, and the implied quality factor Q_b (via the tidal heating formula). By demanding that the model reproduce the observed mass, radius, and effective temperature (≈1650 K), they isolate a set of viable interior models.

The resulting models span a range of core masses from 0 to 120 M_⊕ and Love numbers from 0.116 to 0.425. Correspondingly, the inferred Q_b values are all below 3 × 10⁵, indicating relatively efficient tidal dissipation. The authors also calculate the equilibrium eccentricity e_eq(b) that would satisfy the fixed‑point condition for each k₂_b. They find that ignoring the planet’s quadrupole precession would predict e_eq ≈0.0336, whereas including it (with a representative k₂_b≈0.3) reduces e_eq to ≈0.0161—a difference that is observationally detectable. A fourth‑order polynomial fit (Equation 16) relates e_eq to k₂_b, providing a practical tool for converting a measured eccentricity into a Love number and thus into a core mass estimate.

The paper emphasizes that the current uncertainty in e_b (±0.009) translates into a broad range of possible interior structures, but that modest improvements in eccentricity measurement—via secondary eclipse timing, refined radial‑velocity data, and long‑baseline transit‑timing variations—could shrink the error bars dramatically. A precision of Δe_b ≈0.001 would allow discrimination between core‑less models and those with cores of order 40–80 M_⊕. Moreover, the authors discuss potential complications: stellar quadrupole precession (if the host star rotates rapidly) could mimic the effect of a larger planetary k₂_b, and non‑coplanarity between the two orbital planes would alter the secular precession rates. Both effects can be constrained with future measurements of stellar rotation period, spin‑orbit alignment (via the Rossiter‑McLaughlin effect), and precise TTV analyses.

In conclusion, the study demonstrates that a transiting planet in a multi‑planet system, when driven to a tidal fixed point, offers a powerful indirect probe of its interior. The HAT‑P‑13b system already yields meaningful constraints on core mass, tidal quality factor, and internal heating, and the method is readily applicable to other systems as more multi‑planet transiting configurations are discovered. Continued high‑precision photometry and spectroscopy will sharpen these constraints, opening a new window onto the composition and thermal evolution of extrasolar giant planets.