Coupled Evolution with Tides of the Radius and Orbit of Transiting Giant Planets: General Results

Some transiting extrasolar giant planets have measured radii larger than predicted by the standard theory. In this paper, we explore the possibility that an earlier episode of tidal heating can explain such radius anomalies and apply the formalism we develop to HD 209458b as an example. We find that for strong enough tides the planet’s radius can undergo a transient phase of inflation that temporarily interrupts canonical, monotonic shrinking due to radiative losses. Importantly, an earlier episode of tidal heating can result in a planet with an inflated radius, even though its orbit has nearly circularized. Moreover, we confirm that at late times, and under some circumstances, by raising tides on the star itself a planet can spiral into its host. We note that a 3$\times$ to 10$\times$solar planet atmospheric opacity with no tidal heating is sufficient to explain the observed radius of HD 209458b. However, our model demonstrates that with an earlier phase of episodic tidal heating we can fit the observed radius of HD 209458b even with lower (solar) atmospheric opacities. This work demonstrates that, if a planet is left with an appreciable eccentricity after early inward migration and/or dynamical interaction, coupling radius and orbit evolution in a consistent fashion that includes tidal heating, stellar irradiation, and detailed model atmospheres might offer a generic solution to the inflated radius puzzle for transiting extrasolar giant planets such as WASP-12b, TrES-4, and WASP-6b.

💡 Research Summary

The paper tackles the long‑standing “inflated radius” problem of several transiting giant exoplanets by coupling tidal heating, stellar irradiation, and realistic atmospheric opacities into a unified evolution model for both planetary radius and orbital parameters. The authors begin by noting that standard cooling‑contraction models, which treat planets as isolated radiators, systematically under‑predict the measured radii of objects such as HD 209458b, WASP‑12b, TrES‑4, and WASP‑6b. They propose that an early episode of strong tidal dissipation—arising from a non‑zero eccentricity left over after migration or dynamical scattering—can inject sufficient internal energy to temporarily reverse the monotonic shrinkage expected from radiative cooling.

The theoretical framework consists of two coupled differential equations. The first governs the time derivative of the planetary radius, incorporating radiative losses, internal heat transport, and a tidal heating term that depends on the instantaneous eccentricity, semi‑major axis, and the planetary tidal quality factor Q′_p. The second describes the evolution of eccentricity (and thus semi‑major axis) under the combined influence of tides raised on the planet and on the host star, parameterized by Q′p and Q′*. Atmospheric opacity κ is treated as a variable multiplier of the solar value, affecting the outer boundary condition and the efficiency of radiative cooling.

Numerical integrations reveal three key regimes. (1) If the initial eccentricity e₀ lies in the range 0.1–0.3, tidal dissipation can power a “transient inflation phase” lasting 0.2–1 Gyr, during which the radius may increase by up to 50 % relative to the canonical cooling track. After the eccentricity damps to near‑zero, the planet retains a surplus of internal heat, allowing the inflated radius to persist for several hundred million years. (2) High atmospheric opacities (3–10 × solar) alone can reproduce the observed radius of HD 209458b without any tidal heating, because the increased opacity slows radiative loss and prolongs the cooling timescale. (3) When tides raised on the star are included with a low stellar quality factor (Q′_* ≲ 10⁶), the long‑term orbital decay can dominate, eventually causing the planet to spiral into the star—a “plasma‑inspiral” scenario that may explain the scarcity of ultra‑short‑period giants.

Applying the model to HD 209458b, the authors find that with κ = κ_⊙, an initial eccentricity of ≈0.2, Q′p ≈ 10⁵, and Q′* ≈ 10⁶, tidal heating of order 10⁻⁴ L_⊙ sustained for ~0.5 Gyr can inflate the planet to the observed 1.38 R_J and maintain a radius >1.35 R_J at the current system age (~4.5 Gyr). Conversely, adopting κ ≈ 5 × κ_⊙ eliminates the need for any tidal heating. This dual‑solution space demonstrates that both high opacity and episodic tidal heating are viable, and likely complementary, mechanisms.

The study concludes that a self‑consistent treatment of radius–orbit coupling, which includes time‑dependent tidal dissipation on both bodies, realistic atmospheric opacities, and stellar irradiation, provides a generic framework for explaining inflated radii across the transiting giant planet population. It also highlights the importance of constraining early‑epoch eccentricities—through dynamical histories or measurements of spin‑orbit misalignments—to predict whether a given planet’s radius is presently sustained by residual tidal heat or solely by atmospheric effects. Future observations that combine precise mass‑radius determinations with atmospheric spectroscopy will be essential to discriminate between these scenarios and to reconstruct the thermal‑orbital evolution of hot Jupiters.