Dynamical Evolution of Multi-Resonant Systems: The Case of Gj 876

Abstract

The GJ 876 system was among the earliest multi-planetary detections outside of the Solar System, and has long been known to harbor a resonant pair of giant planets. Subsequent characterization of the system revealed the presence of an additional Neptune mass object on an external orbit, locked in a three body Laplace mean motion resonance with the previously known planets. While this system is currently the only known extrasolar example of a Laplace resonance, it differs from the Galilean satellites in that the orbital motion of the planets is known to be chaotic. In this work, we present a simple perturbative model that illuminates the origins of stochasticity inherent to this system and derive analytic estimates of the Lyapunov time as well as the chaotic diffusion coefficient. We then address the formation of the multi-resonant structure within a protoplanetary disk and show that modest turbulent forcing in addition to dissipative effects is required to reproduce the observed chaotic configuration. Accordingly, this work places important constraints on the typical formation environments of planetary systems and informs the attributes of representative orbital architectures that arise from extended disk-driven evolution.

An empirically derived three-dimensional Laplace resonance in the Gliese 876 planetary system

“…In Figure A3, we show the distribution of Lyapunov times for the 3-d set of solutions resulting from the integrations employing both timesteps, and the distribution of Lyapunov times for the coplanar set of solutions. These integrations all agree that the motion of these orbits is chaotic with a Lyapunov time of roughly 10 years, consistent with the analytic estimate of Batygin et al (2015). The small peak at 10 3 years in the distribution of Lyapunov times for the coplanar orbital solutions to the data corresponds to orbits which might be regular.…”

“…While the observed large libration amplitude and chaotic evolution of resonant angles is contrary to the predictions of the simplest migration models, Batygin et al (2015) recently proposed that that the observed libration of the resonant angles can be explained purely by a turbulent migration. We propose an alternative formation mechanism based on a phase of smooth disk migration which terminates abruptly (Appendix A).…”

“…P Batygin et al (2015) provide an elegant explanation of how such a chaotically evolving Laplace angle could have formed, and show that it can provide limits on the system's early formation: smooth migration can only form systems with relatively small libration amplitude, and regular non-chaotic evolution; stochastic migration is required to form chaotically evolving systems with large libration amplitudes.…”

A Dynamical Analysis of the Kepler-80 System of Five Transiting Planets

“…The most famous example of a three-body resonance is the Laplace resonance visible among the three inner Galilean moons. Such resonances are also important for chaos in the asteroid belt (e.g., Nesvorný & Morbidelli 1998) and have been seen in the Pluto system (Showalter & Hamilton 2015) and in the Gliese 876 exoplanetary system (e.g., Batygin et al 2015). Quillen & French (2014) find that three-body resonances can be comparably important as two-body resonances among the small inner moons of Uranus when the moons are near mean-motion resonances.…”

Planetary migration and the origin of the 2:1 and 3:2 (near)-resonant population of close-in exoplanets

“…The GJ876 system has been shown to be significantly chaotic (Martí et al 2013), although stable for time spans comparable to the age of the system. Batygin et al (2015) argued that the strong chaoticity is incompatible with a smooth migration in a laminar disk, proposing instead that resonance capture occurred in a turbulent disk characterized by low-amplitude stochastic perturbations. However, it is unclear why turbulence would have affected this system and not the other members of the resonant population.…”