Nonlinear effects on the diffusive tidal evolution

High eccentricity migration is a possible formation channel for hot Jupiters. However, in order for it to be consistent with the observed population of hot Jupiters, tides must circularize the orbits in less than ~ a Myr. A potential mechanism for such rapid circularization is the diffusive growth of the planetary f-mode. Such growth occurs if the f-mode's phase at pericenter vary chaotically from one pericenter passage to the next. Previous studies focused on the variation of the orbital period due to tidal back-reaction as the source of chaos. Here we show that nonlinear mode interactions can also be an important source. Specifically, we show that nonlinear interactions between a parent f-mode and daughter f-/p-modes induce an energy-dependent shift in the oscillation frequency of the parent. This frequency shift varies randomly from orbit to orbit because the parent's energy varies. As a result, the parent's phase at pericenter varies randomly, which we find can trigger it to grow diffusively. We show that the phase shift induced by nonlinear mode interactions can dominate the shift induced by tidal back-reaction and significantly lower the one-kick energy threshold for diffusive growth. Furthermore, since the daughter modes can be highly dissipative, the parent's nonlinear damping rate can be significantly higher than its own linear damping rate. Nonlinear damping plays a critical role in regulating the energy of a diffusively growing parent mode to always stay much below the binding energy of the planet. Combining the nonlinear frequency shift and damping, we find that nonlinear interactions may significantly speed up the rate of orbital circularization.