A simple scaling for the minimum instability time-scale of two widely spaced planets

TL;DR

This paper presents an empirical scaling law for the minimum time-scale before two widely spaced, Hill stable planets become Lagrange unstable, aiding long-term stability predictions in planetary systems.

Contribution

It introduces a simple, empirically derived relation linking planetary mass ratio and initial orbit count to the onset of instability, applicable to various planetary masses and configurations.

Findings

Instability occurs after about 10^{5.2 mu^{-0.18}} initial orbits.

Low-eccentricity, Hill stable terrestrial planets remain stable throughout white dwarf lifetimes.

Giant planets may become unstable even if Hill stable, especially beyond the critical Hill separation.

Abstract

Long-term instability in multi-planet exosystems is a crucial consideration when confirming putative candidates, analyzing exoplanet populations, constraining the age of exosystems, and identifying the sources of white dwarf pollution. Two planets which are Hill stable are separated by a wide-enough distance to ensure that they will never collide. However, Hill stable planetary systems may eventually manifest Lagrange instability when the outer planet escapes or the inner planet collides with the star. We show empirically that for two nearly coplanar Hill stable planets with eccentricities less than about 0.3, instability can manifest itself only after a time corresponding to X initial orbits of the inner planet, where log_{10}(X) is of the order of 5.2 mu^{-0.18} and mu is the planet-star mass ratio measured in (Jupiter mass/Solar mass). This relation applies to any type of equal-mass secondaries, and suggests that two low-eccentricity Hill stable terrestrial-mass or smaller-mass planets should be Lagrange stable throughout the main sequence lifetime of any white dwarf progenitor. However, Hill stable giant planets are not guaranteed to be Lagrange stable, particularly within a few tens of percent beyond the critical Hill separation. Our scaling represents a useful "rule of thumb" for planetary population syntheses or individual systems for which performing detailed long-term integrations is unfeasible.