Numerical Non-Adiabatic Tidal Calculations with GYRE-tides: The WASP-12 Test Case

Abstract

We revisit the tidal evolution of the WASP-12 system using direct numerical calculations with the GYRE-tides code. WASP-12b is a hot Jupiter on a 1.1-day orbit around a slightly evolved F-type star. Its observed orbital decay rate, $|\dot{P}_{\rm orb}/P_{\rm orb}| \approx 3.2\,\mathrm{Myr}^{-1}$, provides a strong constraint on stellar tidal dissipation. We confirm that linear tides with radiative damping and convective damping, as currently implemented, are not sufficient to reproduce the observed inspiral timescale. Nevertheless, our calculations, based on fully non-adiabatic forced oscillations in MESA stellar models with convective envelopes, yield dissipation rates that are consistent with previous semi-analytic and adiabatic estimates, confirming the robustness of our numerical framework. As the only open-source, actively maintained tool capable of computing orbital evolution in exoplanet systems, GYRE-tides provides a benchmark calculation for WASP-12 and future applications. Our results validate GYRE-tides as a tool for analyzing combined radiative and convective damping, and indicate that the observed decay rate requires tidal dissipation operating in or near the fully damped regime, which may be achieved through nonlinear damping. These contributions could also be evaluated by computing the wave luminosity at the radiative-convective boundary using our tool. GYRE-tides offers an open-source framework for computing tidal dissipation in short-period exoplanet systems, including the many systems expected to show orbital decay in upcoming Roman surveys.

Figures (5)

• Figure 1: Synchronization timescale $t_{\rm sync}$ (top panel) and orbital decay timescale $t_a$ (bottom panel) as functions of orbital period, computed for the WASP-12 system considering only radiative damping of the dynamical tide. The results are based on linear tidal response functions calculated with GYRE-tides. Sharp decreases in the timescales arise from resonance with g-modes; the labels beneath the plot identify the resonant mode, and additionally indicate (in parentheses) the orbital harmonic $k$ involved in the resonance, and the propagation direction ("p": prograde; "r": retrograde) of the mode. The horizontal dashed lines in both panels denote the age of the Universe. Outside the resonances, the predicted timescales generally exceed the observed decay timescale by approximately three orders of magnitude, indicating that radiative damping alone cannot explain the rapid inspiral of WASP-12b. The red star marks the observed value of the orbital decay timescale, $|t_a| \sim 2.2\,\mathrm{Myr}$.
• Figure 2: As in Figure \ref{['fig:fig_secular_0.01pseudosync_rad_only']}, the same binary configuration and stellar structure are used, but here both with convective damping and radiative damping are included. The inclusion of convective damping enhances tidal dissipation across a broader range of orbital periods. The red star denotes the observed decay timescale.
• Figure 3: Characteristic frequencies, viscosity prescriptions, radial displacement amplitudes, and the gravitational potential perturbation response for a $1.3\,M_\odot$ main-sequence stellar model, as a function of pressure. Top: The plotted quantities include the Brunt–Väisälä frequency $N/2\pi$ (blue), the Lamb frequency $S_2/2\pi$ for $\ell=2$ (orange), the inverse thermal timescale $1/t_{\rm th}$ (orange dashed line), the inverse eddy turnover timescale $1/t_{\rm ed}$ (green dashed line), the dynamical frequency $1/t_{\rm dyn}$ (horizontal black dashed line), and the tidal forcing frequency $1/P_f$ (horizontal black dotted line). Horizontal royal blue and coral lines mark the propagation regions of multiple $g$-modes and $p$-modes, respectively. Bottom: Viscosity profiles standard mixing-length formulation (${\rm STD}$, blue solid line), eddy viscosity implemented in GYRE (orange solid line), and Zahn's reduced viscosity (${\rm Z}$, green solid line).
• Figure 4: Mode analysis of the $\ell=m=k=2$ tidal response in our fiducial 1.3 $M_{\odot}$ main-sequence stellar model. Top: Radial displacement amplitude $\xi_r$ (real and imaginary parts), including results from the full nonadiabatic forced oscillation calculation, the case with radiative diffusion damping only, and the analytical equilibrium tide (green solid); the dimensionless nonlinearity parameter $k_r\xi_r$ is also shown as red solid line. Middle: Horizontal displacement amplitude $\xi_h$ (real and imaginary parts), compared with the case with radiative diffusion damping only; the corresponding $k_h\xi_h$ is indicated as red solid line. Bottom: Eulerian gravitational potential perturbation $\Phi_{\ell m}$ (real and imaginary parts), with both full and radiative-only solutions. All quantities are plotted as a function of radius.
• Figure 5: Same as Figure \ref{['fig:mode_analysis_k_eq_2_ell_eq_2_MS']}, but for a $1.2\,M_{\odot}$ subgiant stellar model whose core is dominated by a radiative zone.