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

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) timescale, ITDs approach remarkably similar lognormal distributions with standard deviations ≈ 0.43±0.16 dex, despite the instability times varying across our sample from 10 4 −10 8 orbits. We find excellent agreement between these predictions, derived from ≈ 450 closely packed configurations of three planets, and a much wider validation set of ≈ 10, 000 integrations, as well as on ≈ 20, 000 previously published integrations of tightly packed five-planet systems, and a seven-planet resonant chain based on TRAPPIST-1, despite their instability timescales extending beyond our analyzed timescale. 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.

The path to instability in compact multi-planetary systems

“…Thus the standard deviation does not depend on the order of magnitude of the instability time. This is a remarkable result, as it has been shown in numerical simulations (Hussain & Tamayo 2020) that the standard deviation of the survival time of extremely close initial conditions has the same properties. Besides, Hussain & Tamayo (2020) measured the standard deviation to be 0.43 ± 0.16 dex which is consistent with the value we obtain for initial conditions not too close initially to two-planets MMR, as can be seen in Fig.…”

“…Top panel: probability distribution function of log 10 T surv /T 0 for various values of u 0 , the normalised distance of the initial condition of the system to the two neighbouring first-order two-planet MMRs. We see that for u 0 = 0.5, the time distribution is log-normal as observed in Hussain & Tamayo (2020). Bottom panels: mean and standard deviation of the distribution as a function of u 0 .…”

“…Beyond the trend already observed by Chambers et al (1996), Obertas et al (2017) showed that the survival time is reduced in the vicinity of low-order two-planet MMR. Hussain & Tamayo (2020) show that the spreading around the linear trend for log T surv /P is roughly constant and follows a normal distribution with a standard deviation of 0.43 ± 0.16 dex, indicating that the instability emerges from a chaotic diffusion process. Beyond the EMS initial conditions, Pu & Wu (2015) explored the impact of small variations of the initial conditions by drawing the spacings, eccentricities, and inclinations from distributions and showed that the exact spacing can be replaced by the minimal separation between the orbits.…”

“…2a system can experience almost no AMD evolution during most of its lifetime before a rapid increase shortly before instability. d. The survival time distribution suggests that the evolution is driven by a diffusion process (Hussain & Tamayo 2020). The dips close to first-order two-planet MMRs indicate that these latter play a fundamental role in enabling the orbit crossing.…”

Breaking Resonant Chains: Destabilization of Resonant Planets Due to Long-term Mass Evolution

“…To account for the chaotic nature of the orbital evolution of multiple planet systems, we perform two additional simulations in 21 cases and four additional simulations in five cases. While previous studies suggested that the standard deviation of the logarithm orbital crossing time of planets that are not in resonances is 0.2 dex (Rice et al 2018;Hussain & Tamayo 2020), standard deviations of 85 % of our cases with additional simulations are less than 0.2 dex. Their median and average values are 0.084 dex and 0.13 dex, respectively.…”