[Paper Review] Stability of the co-orbital resonance under dissipation: Application to its evolution in protoplanetary discs

A. Leleu, Gavin A. L. Coleman+1 more|arXiv (Cornell University)|Jan 22, 2019

Astrophysics and Star Formation Studies3 references4 citations

TL;DR

This paper develops a stability criterion for co-orbital exoplanets in protoplanetary discs under generic dissipation and mass evolution, showing that migration and disc-induced forces can either stabilize or destabilize Lagrangian equilibria. Key results reveal that leading massive planets and resonant chains stabilize co-orbital configurations, while inward migration and mass asymmetry increase libration amplitudes, potentially ejecting planets from resonance or increasing eccentricities and inclinations, reducing transit detectability.

ABSTRACT

Despite the existence of co-orbital bodies in the solar system, and the prediction of the formation of co-orbital planets by planetary system formation models, no co-orbital exoplanets (also called trojans) have been detected thus far. In this paper we investigate how a pair of co-orbital exoplanets would fare during their migration in a protoplanetary disc. To this end, we computed a stability criterion of the Lagrangian equilibria L4 and L5 under generic dissipation and slow mass evolution. Depending on the strength and shape of these perturbations, the system can either evolve towards the Lagrangian equilibrium, or tend to increase its amplitude of libration, possibly all the way to horseshoe orbits or even exiting the resonance. We estimated the various terms of our criterion using a set of hydrodynamical simulations, and show that the dynamical coupling between the disc perturbations and both planets have a significant impact on the stability: the structures induced by each planet in the disc perturb the dissipative forces applied on the other planets over each libration cycle. Amongst our results on the stability of co-orbitals, several are of interest to constrain the observability of such configurations: long-distance inward migration and smaller leading planets tend to increase the libration amplitude around the Lagrangian equilibria, while leading massive planets and belonging to a resonant chain tend to stabilise it. We also show that, depending on the strength of the dissipative forces, both the inclination and the eccentricity of the smaller of the two co-orbitals can be significantly increased during the inward migration of the co-orbital pair, which can have a significant impact on the detectability by transit of such configurations.

Motivation & Objective

• To understand the stability of co-orbital exoplanets during type I and type II migration in protoplanetary discs.
• To identify conditions under which co-orbital configurations (L4/L5) are stabilized or destabilized by disc-induced dissipation and mass accretion.
• To assess how migration, mass asymmetry, and resonant chains affect the evolution of libration amplitudes, eccentricities, and inclinations.
• To evaluate the detectability of co-orbital exoplanets via transit and radial velocity methods under evolving disc conditions.
• To quantify the impact of disc structure and planet-disk interactions on the long-term survival of co-orbital systems.

Proposed method

• Developed an integrable analytical model for co-orbital resonance under generic dissipation and slow mass evolution in the coplanar-circular case.
• Derived a stability criterion for Lagrangian equilibria (L4/L5) based on the balance between destabilizing migration and stabilizing mass accretion.
• Calibrated the model using hydrodynamical simulations of planet-disc interactions to estimate torque and dissipation terms.
• Simulated long-term evolution in evolving protoplanetary discs with varying disc profiles and planet masses.
• Analyzed the influence of planet mass ratio, migration direction, and resonant chains on co-orbital stability and orbital element evolution.
• Investigated inclination and eccentricity evolution using analytical and simulation-based estimates of damping and excitation forces.

Experimental results

Research questions

• RQ1 Under what conditions does inward migration stabilize or destabilize co-orbital configurations in protoplanetary discs?
• RQ2 How do disc-induced torques and mass accretion affect the libration amplitude of co-orbital planets?
• RQ3 What role do planet mass ratios and resonant chains play in stabilizing or destabilizing L4/L5 Lagrangian equilibria?
• RQ4 How do eccentricity and inclination evolve during co-orbital migration, and what is their impact on transit detectability?
• RQ5 Can co-orbital systems remain stable long enough to be detectable via transit timing variations or radial velocity measurements?

Key findings

• Inward migration and mass asymmetry tend to increase libration amplitudes, potentially driving co-orbitals from tadpole to horseshoe or resonance-exiting states.
• Leading massive planets significantly stabilize the co-orbital configuration, reducing the risk of ejection or resonance escape.
• Resonant chains with a third planet can stabilize otherwise unstable co-orbital systems, even if perturbations from spiral arms are neglected.
• Mass accretion stabilizes co-orbitals, but this effect is strongest in early migration phases and diminishes over time.
• Eccentricities and inclinations can be significantly excited during inward migration, especially when disc damping is weak, reducing transit probabilities.
• Mutually inclined co-orbital systems may form along the m1I1 = m2I2, Ω1 = Ω2 + π direction, particularly when the more massive planet creates a deep gap, reducing detectability.