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Abstract The Roche limit, or the threshold separation within which a celestial object (the donor)Mcannot remain in a stable configuration due to a companion’s tidal field, has been well established whenMis in hydrostatic equilibrium and has synchronous rotation in a circular orbit. However, limited analyses exist considering corrections to the Roche limit due to hydrodynamical effects. We fill in the gap by providing a general theoretical framework involving nonlinear hydrodynamics. We consider both exact nonlinear equations derived from an affine model describing incompressible ellipsoids and series-expanded ones that can be calculated for realistic stars and planets. Our formulation addresses the Roche problem in generic orbits and synchronization levels ofM, and fully accounts for the history-dependent hydrodynamical effects. We show that, as the orbital eccentricity increases, fluid instability is more likely to develop at the pericenter due to the increased dynamical tide that accumulates over multiple orbits. WhenMmoves in a highly eccentric orbit (with eccentricity around 0.9) and the damping of the fluid is small, the threshold pericenter separation at which mass loss fromMcan occur can be at least 30% higher than the value predicted for a circular orbit with hydrostatic equilibrium. If only a single passage is considered, however, the threshold separation is 20% smaller than the static limit. The nonlinear interaction at each pericenter passage can also trigger a diffusive fluid evolution insideMeven with moderate eccentricities, complementing previous studies of diffusive tides caused by random propagation phases. Our work has broad implications for interacting binaries in eccentric orbits, including migrating gaseous exoplanets, repeated partial tidal disruption events, and more. 

This paper investigates nonparametric quantile regression using recurrent neural networks (RNNs) and sparse recurrent neural networks (SRNNs) to approximate the conditional quantile function, which is assumed to follow a compositional hierarchical interaction model. We show that RNN- and SRNN-based estimators with rectified linear unit (ReLU) activation and appropriately designed architectures achieve the optimal nonparametric convergence rate, up to a logarithmic factor, under stationary, exponentially β-mixing processes. To establish this result, we derive sharp approximation error bounds for functions in the hierarchical interaction model using RNNs and SRNNs, exploiting their close connection to sparse feedforward neural networks (SFNNs). Numerical experiments and an empirical study on the Dow Jones Industrial Average (DJIA) further support our theoretical findings. 

WASP-107 b seems to be a poster child of the long-suspected high-eccentricity migration scenario. It is on a 5.7 day, polar orbit. The planet is Jupiter-like in radius but Neptune-like in mass with exceptionally low density. WASP-107 c is on a 1100 day,e= 0.28 orbit with at least Saturn mass. Planet b may still have a residual eccentricity of 0.06 ± 0.04: the ongoing tidal dissipation leads to the observed internally heated atmosphere and hydrodynamic atmospheric erosion. We present a population synthesis study coupling octupole Lidov–Kozai oscillations with various short-range forces, while simultaneously accounting for the radius inflation and tidal disruption of the planet. We find that a high-eccentricity migration scenario can successfully explain nearly all observed system properties. Our simulations further suggest that the initial location of WASP-107 b at the onset of migration is likely within the snowline (<0.5 au). More distant initial orbits usually lead to tidal disruption or orbit crossing. WASP-107 b most likely lost no more than 20% of its mass during the high-eccentricity migration, i.e., it did not form as a Jupiter-mass object. More vigorous tidally induced mass loss leads to disruption of the planet during migration. We predict that the current-day mutual inclination between the planets b and c is substantial: at least 25°–55°, which may be tested with future Gaia astrometric observations. Knowing the current-day mutual inclination may further constrain the initial orbit of planet b. We suggest that the proposed high-eccentricity migration scenario of WASP-107 may be applicable to HAT-P-11, GJ-3470, HAT-P-18, and GJ-436, which have similar orbital architectures.