Fundamental limits from chaos on instability time predictions in compact planetary systems

Abstract

ABSTRACT Instabilities in compact planetary systems are generically driven by chaotic dynamics. This implies that an instability time measured through direct N-body integration is not exact, but rather represents a single draw from a distribution of equally valid chaotic trajectories. In order to characterize the ‘errors’ on reported instability times from direct N-body integrations, we investigate the shape and parameters of the instability time distributions (ITDs) for ensembles of shadow trajectories that are initially perturbed from one another near machine precision. We find that in the limit where instability times are long compared to the Lyapunov (chaotic) time-scale, ITDs approach remarkably similar lognormal distributions with standard deviations ≈0.43 ± 0.16 dex, despite the instability times varying across our sample from 104 to 108 orbits. We find excellent agreement between these predictions, derived from ≈450 closely packed configurations of three planets, and a much wider validation set of $\approx 10\, 000$ integrations, as well as on $\approx 20\, 000$ previously published integrations of tightly packed five-planet systems, and a seven-planet resonant chain based on TRAPPIST-1, despite their instability time-scales extending beyond our analysed time-scale. We also test the boundary of applicability of our results on dynamically excited versions of our Solar system. These distributions define the fundamental limit imposed by chaos on the predictability of instability times in such planetary systems. It provides a quantitative estimate of the instrinsic error on an N-body instability time imprinted by chaos, approximately a factor of 3 in either direction.