Tidally driven inertial waves enhance eccentricity damping and spin evolution in planets and stars

TL;DR

The paper analyzes tidally driven inertial waves in convective envelopes as a mechanism to boost eccentricity damping and alter spin evolution beyond standard equilibrium tides. By deriving energy and angular momentum transfer via inertial modes and performing targeted numerical experiments with rotating polytropes and incompressible shells (including magnetic braking and frequency-dependent eddy viscosity), the authors show that inertial waves can drive eccentricity damping by orders of magnitude and produce observable spin-orbit signatures. A key finding is the emergence of discrete spin equilibria and a broader cool core of nearly circular binaries, extending circularization to longer orbital periods than classical tides would allow. These results offer a potential reconciliation between theory and observations from Kepler, TESS, and Gaia, while highlighting sensitivities to interior structure and viscosity prescriptions.

Abstract

Tidal interactions influence the orbital motions of binary star systems and extrasolar planets alike. Tides also affect stellar and planetary rotation rates. We demonstrate that in addition to altering spin synchronization and pseudosynchronization, tidally driven inertial waves in the convective envelopes of low-mass stars and gas giant planets can enhance tidal eccentricity damping. Analytically, we find that eccentricity damping by inertial waves can be orders of magnitude faster than equilibrium tides, independent of any eddy viscosity prescription. We use simplified numerical experiments to demonstrate this enhancement, and to explore the effects of different mixing length treatments of convective turbulence, as well as a spin-down torque from magnetic braking. These calculations demonstrate that tidally driven inertial waves can produce an extended cool core of nearly circular binaries, helping to reconcile a longstanding discrepancy between observed and predicted main-sequence binary circularization. Our calculations additionally suggest that tidally driven inertial waves may leave identifiable signatures in the ratios of orbital to rotation periods for stellar binaries, including synchronous and sub-synchronous rotation periods reminiscent of populations identified in Kepler, TESS, and Gaia data.

Figures (14)

• Figure 1: Inverse eccentricity damping timescales $\dot{e}/e,$ normalized by dimensionless amplitude $\epsilon_T=q(R/a)^5$, squared mean motion $n,$ and effective constant time lag $\tau$. The thin lines show the predictions of Equations \ref{['eq:psedot']} and \ref{['eq:tbedot']} for evolution due to equilibrium tides (black), and torque balances involving inertial modes (purple to orange colorscale). For the latter, each line is truncated at the point where Equation \ref{['eq:tbreak']} predicts that the torque balance should break. The thick curves show results from directly integrating Equations \ref{['eq:adot']}-\ref{['eq:Jdot']} for a polytropic model of a $1M_\odot$ star with a reference radius $R_0=2R_\odot$, a constant kinematic viscosity of $\nu_0=10^{-5}R_0^2\omega_{\rm dyn,0}$, and the initial parameters $P_{\rm orb}=10$ days, $\Omega/\omega_{\rm dyn}=0.1,$$e=0.5$ at $t=0$ (solutions computed from other initial conditions follow very similar trajectories in this parameter space). Tidally driven inertial modes can enhance eccentricity damping by orders of magnitude, relative to equilibrium tides.
• Figure 2: Left: imaginary parts of tidal Love numbers, scaled by $(\Omega/\omega_{\rm dyn})^2\omega_t/\Omega$ and plotted as a function of tidal frequency for isentropic polytropes (top) and incompressible shells (bottom) with a constant Ekman number $E_k=\nu/(R^2\Omega)=10^{-6}.$ The different curves show different rotation rates for the tidally perturbed object. The dashed lines in the top panel indicate "f-mode tide" calculations that exclude the inertial oscillations of the polytropic model. Right: meridional slices illustrating the radial velocity perturbation (in units with $G=M=R=1$, and normalized by the amplitude of the tidal potential) corresponding to the tidal frequency indicated by the black dots in the lefthand panels. The presence of a central core produces a much more complicated spectrum of resonances in the incompressible shell.
• Figure 3: Time evolution of the ratio between rotation rate and mean motion (top) and eccentricity (bottom) for the integrations depicted by thick lines in \ref{['fig:edot_cpr']}. In calculations including the inertial modes of the model (orange lines), a progression through discrete spin ratios $\Omega/n\simeq j/2.7$ (for integer $j$; gray lines) leads to much more rapid eccentricity damping than in an equilibrium tidal model with the same underlying eddy viscosity (taken here to have a constant value of $\nu=10^{-5}R_0^2\omega_{\rm dyn,0},$).
• Figure 4: Evolution of semimajor axis divided by equatorial radius (top), period ratio $\Omega/n=P_{\rm orb}/P_{\rm rot}$ (middle), and eccentricity (bottom), for polytropic models with $M=1M_\odot$, $R=2R_\odot$, constant kinematic viscosity $\nu=10^{-5}R_0^2\omega_{\rm dyn,0},$ mass ratio $q=1,$ initial orbital periods and rotation rates of $P_{\rm orb}=10$ days and $\Omega/\omega_{\rm dyn}=0.05$ (respectively), and initial eccentricities increasing from 0 (light yellow) to $0.65$ (dark blue). The dots at $t\simeq13$Myr indicate the simultaneous endpoints for the parameterized curves shown in \ref{['fig:n15_eaO']} (left). Larger initial eccentricities lead to more rapid circularization, regardless of the tidal model. For any given initial condition, tidally driven inertial modes (solid lines) damp eccentricities much more rapidly than quasi-equilibrium tides (dashed lines).
• Figure 5: Parameterized curves showing the same simulations as \ref{['fig:n15_aoe_vs_t']}, but with eccentricity plotted as a function of $a/R$ (left) and $\Omega/n$ (middle, right). The middle and righthand panels demonstrate that the evolution progresses along the surface of vanishing tidal torque (indicated by the colormap, which transitions from log to linear scale near $\mathcal{T}=0$). The red line in the middle panel shows the zero-torque surface predicted for a constant time lag. The torque is qualitatively modified by the inclusion (right) or omission (middle) of inertial waves in the tidal model. The lefthand panel shows that inertial modes' modification of tidal energy dissipation produces circularization out to larger $a/R$ in an equivalent amount of time ($t\approx13$Myr).