Tides in the High-eccentricity Migration of Hot Jupiters: Triggering Diffusive Growth by Nonlinear Mode Interactions

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 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 parents' energy varies. As a result, the parents' 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 lowers the one-kick energy threshold for diffusive growth by about a factor of 5 compared to the prediction from the linear theory. For a given pericenter distance, this reduces the semimajor axis needed to trigger diffusive growth from ≳2 au to ≃1 au. Nonlinear interactions could thus enhance the formation rate of hot Jupiters through the high-eccentricity migration channel and potentially mitigate the discrepancy between the observed and predicted occurrence rates for close-in gas giants as compared to those farther from the star.