Relativistic Dynamical Stability Criterion of Multi-Planet Systems with a Distant Companion

TL;DR

This paper develops an analytical stability criterion for multi-planet systems with a distant inclined companion, incorporating relativistic precession effects, and validates it with numerical simulations and observational data.

Contribution

It extends previous stability criteria by including relativistic effects and provides a practical analytical tool for assessing system stability with distant perturbers.

Findings

The criterion is consistent with numerical simulations for 2- and 3-planet systems.

It constrains the orbital parameters of potential unseen companions.

The criterion broadens the stable parameter space for inclined distant companions.

Abstract

Multi-planetary systems are prevalent in our Galaxy. The long-term stability of such systems may be disrupted if a distant inclined companion excites the eccentricity and inclination of the inner planets via the eccentric Kozai-Lidov mechanism. However, the star-planet and the planet-planet interactions can help stabilize the system. In this work, we extend the previous stability criterion that only considered the companion-planet and planet-planet interactions by also accounting for short-range forces or effects, specifically, relativistic precession induced by the host star. A general analytical stability criterion is developed for planetary systems with $N$ inner planets and a relatively distant inclined perturber by comparing precession rates of relevant dynamical effects. Furthermore, we demonstrate as examples that in systems with $2$ and $3$ inner planets, the analytical criterion is consistent with numerical simulations using a combination of Gauss's averaging method and direct N-body integration. Finally, the criterion is applied to observed systems, constraining the orbital parameter space of a possible undiscovered companion. This new stability criterion extends the parameter space in which an inclined companion of multi-planet systems can inhabit.