Quasi-secular evolution of mildly hierarchical triple systems: analytics and applications for GW-sources and hot Jupiters

In hierarchical triple systems, the inner binary is perturbed by a distant companion. For large mutual inclinations, the Lidov-Kozai mechanism secularly excites large eccentricity and inclination oscillations of the inner binary. The maximal eccentricity attained, $e_{\rm max}$ is simply derived and widely used. However, for mildly hierarchical systems (i.e. the companion is relatively close and massive), non-secular perturbations affect the evolution. Here we account for fast non-secular variations and find new analytic formula for $e_{{\rm max}}$, in terms of the system's hierarchy level, correcting previous work and reproducing the orbital flip criteria. We find that $e_{{\rm max}}$ is generally enhanced, allowing closer encounters between the inner binary components, thus significantly changing their interaction and its final outcome. We then extend our approach to include additional relativistic and tidal forces. Using our results, we show that the merger time of gravitational-wave (GW) sources orbiting massive black-holes in galactic nuclei is enhanced compared with previous analysis accounting only for the secular regime. Consequently, this affects the distribution and rates of such GW sources in the relevant mild-hierarchy regime. We test and confirm our predictions with direct N-body and 2.5-level Post-Newtonian codes. Finally, we calculate the formation and disruption rates of hot-Jupiters (HJ) in planetary systems using a statistical approach, which incorporates our novel results for $e_{{\rm max}}$. We find that more HJ migrate from further out, but they are also tidally disrupted more frequently. Remarkably, the overall formation rate of HJs remains similar to that found in previous studies. Nevertheless, the different rates could manifest in different underlying distribution of observed warm-Jupiters.