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

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.