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 introduces a novel method for probing the interior structure of transiting exoplanets that reside in multi‑planet systems. The key idea is that tidal dissipation inside a short‑period transiting planet, when perturbed by an outer companion, can drive the system toward a “tidal fixed point.” At this fixed point the apsidal lines of the inner and outer planets are aligned, the eccentricities cease to evolve, and the rate of apsidal precession is the sum of three contributions: (1) secular planet‑planet interaction, (2) general‑relativistic precession, and (3) the quadrupole field generated by the inner planet’s tidal and rotational bulges. The last term depends directly on the planet’s tidal Love number k₂, which encodes how centrally concentrated the mass distribution is. Consequently, the equilibrium eccentricity e_b of the inner planet becomes a strong function of k₂ (and therefore of core mass, composition, and the tidal quality factor Q).

To make the connection quantitative, the authors employ an octupole‑order secular theory that accurately captures the long‑term gravitational coupling between the two planets even for moderately high eccentricities. They derive an analytic expression for the fixed‑point eccentricity:  e_b = (A_sec / (A_GR + A_sec + A_tide)) · e_c, where A_GR, A_sec, and A_tide are the precession rates due to general relativity, secular interaction, and tidal/rotational deformation, respectively. Since A_tide ∝ k₂, measuring e_b and the known outer‑planet eccentricity e_c allows a direct inference of k₂.

The method is applied to the HAT‑P‑13 system, which contains a 2.9‑day transiting gas giant (HAT‑P‑13b) and a massive, highly eccentric outer companion (HAT‑P‑13c) on a 428‑day orbit. The outer planet’s mass, semi‑major axis, and eccentricity are well constrained, providing the necessary input for the secular model. The authors construct a suite of interior structure models for HAT‑P‑13b, varying core mass from 0 to 120 M⊕ and the tidal quality factor Q from 10⁴ to 10⁶, while adjusting the internal heating rate to reproduce the observed radius (1.28 R_J). For each model they compute k₂ and the associated tidal precession term.

Using the observed inner‑planet eccentricity e_b = 0.021 ± 0.009, the fixed‑point relation yields a range 0.116 < k₂ < 0.425. This interval translates into a core‑mass constraint of 0 – 120 M⊕ (larger cores give smaller k₂) and an upper limit on the tidal quality factor Q < 3 × 10⁵, indicating that HAT‑P‑13b must dissipate tidal energy more efficiently than a typical Jupiter‑like planet. The authors emphasize that the current uncertainties are still large; however, a modest improvement in the eccentricity measurement (to the 10⁻³ level) would dramatically tighten the limits. For example, confirming e_b < 0.015 would force k₂ < 0.12, implying a substantial solid core (>30 M⊕), whereas e_b > 0.03 would point to k₂ > 0.4 and a core‑free interior.

The paper concludes that tidal fixed‑point analysis offers a powerful, largely model‑independent probe of exoplanet interiors, provided a suitable outer perturber exists. The HAT‑P‑13 case demonstrates that even with present‑day data one can place meaningful constraints on core mass, Love number, and tidal dissipation. Future high‑precision radial‑velocity monitoring and transit‑timing variations will refine e_b, enabling an unprecedented empirical test of planetary structure theories for extrasolar worlds.