Dynamical tide modified Roche limit in eccentric, asynchronous binaries

TL;DR

This work develops a comprehensive nonlinear hydrodynamics framework for the Roche limit in eccentric, asynchronous binaries by combining exact affine-model equations with a VanHoolst-style mode expansion to capture dynamical tides. It shows that dynamical tides accumulate across pericenter passages, elevating the instability threshold relative to the static limit (up to about 30% for high eccentricity and low damping), while single-pass analyses can yield smaller thresholds. The study reveals that nonlinear three- and four-wave couplings drive runaway growth of quadrupolar f-modes, connecting to the diffusive (chaotic) tide and explaining mass transfer phenomena in diverse systems, including high-eccentricity exoplanet migration, rpTDEs, and QPEs. The three-wave truncation reproduces Chandrasekhar’s Roche limit for homogeneous ellipsoids with good accuracy and provides a pathway to apply the theory to realistic stars/planets via the VanHoolst expansion, enabling broader astrophysical applications and population-level insights.

Abstract

The Roche limit, or the threshold separation within which a celestial object (the donor) M cannot remain in a stable configuration due to a companion's tidal field, has been well established when M is 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 of M, 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. When M moves 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 from M can 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 chaotic fluid evolution inside M even with moderate eccentricities, complementing previous studies of chaotic 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.

Figures (8)

• Figure 1: Equilibrium sequences under the static limit. Curves in gray correspond to synchronized donors (with $p=1$), while those in yellow are for the non-rotating case ($p=0$). The solid and dashed lines are evaluated at the three-wave and four-wave orders, respectively. Vertical lines mark the locations where $\epsilon$ reaches the maxima. The classic results of Chandrasekhar:63 are shown in the red horizontal lines.
• Figure 2: The threshold pericenter separation at which hydrodynamical instability occurs as a function of the orbital eccentricity. The plot is for the equilibrium tide model and ignores the dynamical tide. Different colors represent different levels of spin synchronization (gray for pseudo-synchronized rotation; yellow for zero spin).
• Figure 3: Numerical evolution of the $q_2$ mode for a non-rotating, homogeneous donor star in an eccentric orbit with $e=0.3$. The left panel uses the exact nonlinear equations, while the right one uses the expanded version to the three-wave order. Runaway of the mode amplitude happens when $D_p/R_t \lesssim 2.72$ (left) or $D_p /R_t \lesssim 2.66$. This matches Fig. \ref{['fig:Dp_th_eq_tide']} where the threshold separation is predicted to be $D_p^{\rm (th)}\simeq 2.66 R_t$.
• Figure 4: Numerical evolution of the $q_2$ mode for a homogeneous donor in an eccentric orbit with $e=0.9$. The donor is assumed to be non-rotating in the left panel and pseudo-synchronized in the right. Over multiple cycles' accumulation, runways in mode amplitude may happen at pericenter separation much greater than the prediction of Fig. \ref{['fig:Dp_th_eq_tide']}.
• Figure 5: Numerical evolution of the $q_2$ mode for a non-rotating donor with $(n, \Gamma)=(1,2)$ in an eccentric orbit with $e=0.9$. Both the exact (left) and three-wave (right) nonlinear equations are considered. Runaways in mode amplitudes are observed when $D_p/R_t\lesssim 2.7$, matching nicely the predictions of Guillochon:11. The threshold $D_p/R_t$ can be even greater if the system is integrated for more orbits.