Orbital stability of compact three-planet systems, I: Dependence of system lifetimes on initial orbital separations and longitudes

TL;DR

This study investigates the long-term stability of three-planet systems with Earth-mass planets, analyzing how initial orbital spacing and configurations influence system lifetimes over billions of years.

Contribution

It provides a detailed quantitative analysis of the factors affecting the stability and lifetime scatter of three-planet systems, extending previous work beyond rough exponential trends.

Findings

System lifetimes increase exponentially with orbital separation.

A significant fraction of three-planet systems survive much longer than similar systems.

Initial longitude shifts cause about a factor of two variation in system lifetime.

Abstract

We explore the orbital dynamics of systems consisting of three planets, each as massive as the Earth, on coplanar, initially circular, orbits about a star of one solar mass. The initial semimajor axes of the planets are equally spaced in terms of their mutual Hill radius, which is equivalent to a geometric progression of orbital periods for small planets of equal mass. Our simulations explore a wide range of spacings of the planets, and were integrated for virtual times of up to 10 billion years or until the orbits of any pair of planets crossed. We find the same general trend of system lifetimes increasing exponentially with separation between orbits seen by previous studies of systems of three or more planets. One focus of this paper is to go beyond the rough trends found by previous numerical studies and quantitatively explore the nature of the scatter in lifetimes and the destabilizing effects of mean motion resonances. In contrast to previous results for five-planet systems, a nontrivial fraction of three-planet systems survive at least several orders of magnitude longer than most other systems with similar initial separation between orbits, with some surviving $10^{10}$ years at much smaller orbital separations than any found for five-planet systems. Substantial shifts in the initial planetary longitudes cause a scatter of roughly a factor of two in system lifetime, whereas the shift of one planet's initial position by 100 meters along its orbit results in smaller changes in the logarithm of the time to orbit crossing, especially for systems with short lifetimes.