Self-consistent Dynamical and Chaotic Tides in the REBOUNDx framework

TL;DR

This paper presents a self-consistent method to couple dynamical tides with N-body orbital dynamics by implementing a drag-force description within the REBOUNDx framework. By modeling planets as $\gamma=2$ polytropes and using the $l=m=2$ f-mode with an energy-exchange map, the authors translate tide–mode interactions into a drag term that updates at apoapsis and influences subsequent periapsis passages. The approach reproduces known chaotic-tides behavior and demonstrates rapid high-$e$ migration in both isolated-planet and vZLK-driven scenarios, confirming the method’s accuracy and utility for fast N-body studies across exoplanetary and broader astrophysical contexts. The work enables efficient exploration of extreme eccentricity systems and provides a flexible tool for investigating tidal processes in a range of stellar and planetary settings.

Abstract

At high eccentricities, tidal forcing excites vibrational modes within orbiting bodies known as dynamical tides. In this paper, we implement the coupled evolution of these modes with the body's orbit in the \texttt{REBOUNDx} framework, an extension to the popular $N$-body integrator \texttt{REBOUND}. We provide a variety of test cases relevant to exoplanet dynamics and demonstrate overall agreement with prior studies of dynamical tides in the secular regime. Our implementation is readily applied to various high-eccentricity scenarios and allows for fast and accurate $N$-body investigations of astrophysical systems for which dynamical tides are relevant.

Figures (7)

• Figure 1: Mode evolution in the iterative map defined by \ref{['eq:dE']} evolved over $10^3$ orbits for a variety of orbital parameters. The black lines correspond to various values of $|\Delta \hat{P}_\text{crit}|$.
• Figure 2: Chaotic migration outcomes for planets with ${M_p=M_J}$ and ${R_p=1.6\,R_J}$ orbiting a star of mass ${M_\star=M_{\odot}}$ with various initial eccentricities and pericenter distances. Bright yellow regions of the heatmap correspond to planets that migrated to a small fraction of their initial semi-major axis. Dark purple regions correspond to planets whose initial and final semi-major axes are similar.
• Figure 3: Two examples of high-eccentricity migration as a result of chaotic tides. Each system consists of a Jupiter-sized planet of mass ${M=M_J}$ and ${R_p=1.6\,R_J}$ orbiting a star of mass $M_{\odot}$ with initial semi-major axis ${a_0=1.5}$ AU. The orange curves correspond to initial eccentricity $e_0=0.985$ and the blue curves correspond to $e_0=0.98501$. The trajectories quickly diverge due to the chaos of the system. In this case, $E_\text{bind} = 3.8 E_{B,0}$.
• Figure 4: Example of low-amplitude oscillations as a result of dynamical tides. The system is the same as that of \ref{['fig:high-e-migration']}, except $e_0 = 0.982$. The trajectories are quasi-periodic.
• Figure 5: Evolution of f-mode over time for the same system as \ref{['fig:high-e-migration']} over 5 kyr, with $e_0=0.985$ (top panel) and $e_0=0.98$ (bottom panel). Points that are lighter correspond to later times in the simulation.