The Neptunian Final Parsec Problem: How Inertial Waves Enable Neptunes and Gas Giants to Survive Tidal Migration

Abstract: Observations indicate that some hot Neptunes and Jupiters underwent high eccentricity migration (HEM). A longstanding issue facing HEM theory remains, however: in order to shrink their orbits by ${\sim}1{-}2$ orders of magnitude, planets must dissipate ${\gtrsim}10{\times}$ more orbital energy than is contained in their structural binding energy. Previous work suggests that excitation of planets' fundamental modes (''$f$-modes'') at very high eccentricities, $e{\sim}$0.99, can shrink semi-major axes by ${\sim}$an order of magnitude (${\sim}5{\rightarrow}0.5$ au) while safely bleeding tidal energy to space. We examine how planets traverse the remaining order of magnitude in orbital shrinkage, $a{\sim}$0.5${\rightarrow}$0.05 au, which still poses a danger for Neptunian planets due to their lower binding energies (their ''final parsec'' problem). To assess whether tidally-migrating Neptunes balance their energy budgets, we perform coupled orbit/structure calculations tracking their frequency-dependent tidal response under inertial waves (IWs). We show that tidal radius inflation at moderate eccentricities sweeps Neptunes' fundamental modes back into resonance with tidal forcing; only a small amount of tidal heat relative to a Neptune's binding energy is needed to drive this $f$-mode excitation. This excitation of Neptunes' $f$-modes may offer a route to safe circularization, provided the $f$-mode dissipates without harming the planet. In contrast, eccentric gas giants ($e{\sim}0.9$) circularize without drama over ${\sim}$Gyr. Our theory predicts that warm Neptunes on a HEM track ($a{\gtrsim}0.1 \ \mathrm{au}, \ e{\gtrsim}0.8$) should be rare, in opposition to warm gas giants and lower eccentricity Neptunes. Observational confirmation of our predictions would constrain the interior structures of Neptunian planets via tidal theory.