Tides in the high-eccentricity migration of hot Jupiters: effects of nonlinear mode interaction

High eccentricity migration is a possible formation channel for hot Jupiters. However, in order for it to be consistent with the observed population of planets, tides must circularize the orbits in an efficient manner. A potential mechanism for such rapid circularization is the diffusive growth of the tidally driven planetary f-mode. Such growth occurs if the f-mode’s phase at pericenter varies chaotically from one pericenter passage to the next. Previous studies focused on the variation of the orbital period due to tidal back-reaction on the orbit 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 in fact dominates the shift induced by tidal back-reaction and significantly lowers the one-kick energy threshold for diffusive growth by about a factor of 5 compared to the linear theory’s prediction. Furthermore, nonlinear mode interaction plays a significant role in maintaining the subsequent orbital circularization. With linear theory alone, the diffusive evolution terminates when the eccentricity is still high (~ 0.95) because the orbital period decreases and the orbit is more tightly bound, both reducing the random phase shift necessary to maintain the diffusion. Nonlinear effects, however, helps to provide additional phase shift and drives the evolution to a lower eccentricity of around 0.8. Nonlinear interactions thus are crucial for both explaining the paucity of super-eccentric Jupiters and predicting the overall formation rate of hot Jupiters.