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

Long-term instability in multiplanet exosystems is a crucial consideration when confirming putative candidates, analysing exoplanet populations, constraining the age of exosystems and identifying the sources of white dwarf pollution. Two planets that 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 log10 x ~ 5.2[μ/(MJupiter/M☉)]-0.18 and μ is the planet-star mass ratio. 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 per cent 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.