Nekhoroshev Estimates for the Survival Time of Tightly Packed Planetary Systems

Abstract

N -body simulations of non-resonant tightly-packed planetary systems have found that their survival time (i.e. time to first close encounter) grows exponentially with their interplanetary spacing and planetary masses. Although this result has important consequences for the assembly of planetary systems by giants collisions and their long-term evolution, this underlying exponential dependence is not understood from first principles, and previous attempts based on orbital diffusion have only yielded power-law scalings. We propose a different picture, where the deviations of the system from its initial conditions is not limited by orbital diffusion, but by the lifetime of a series of confining barriers in phase-space-invariant KAM tori-that are slowly destroyed with time. Thus, we show that survival time of the system T can be estimated using the Nekhoroshev's stability limit and obtain a heuristic formula for systems away from overlapping two-body mean-motion resonances as: T /P = c 1 a ∆a exp c 2 ∆a a /µ 1/4 , where P is the average Keplerian period, a is the average semi major axis, ∆a a is the difference between the semi major axes of neighbouring planets, µ is the planet to star mass ratio, and c 1 and c 2 are dimensionless constants. We show that this formula is in good agreement with numerical N-body experiments for c 1 = 5 · 10 −4 and c 2 = 8, supporting our proposal that the lifetime of non-resonant planetary is primarily determined by the lifetime of KAM tori, which are likely destroyed by three-body resonances.

Predicting the long-term stability of compact multiplanet systems

“…This is an important empirical test since, if true, it implies a robustness of our stability classifications to distant unseen planets. This is crucial for reliable stability constrained characterization of exoplanet systems, and is consistent with previous numerical experiments with equal separation planets showing an insensitivity to additional bodies beyond somewhat larger multiplicities of five [Chambers et al, 1996], as well as theoretical arguments showing that the Fourier amplitudes of the perturbation potential due to an additional planet fall off exponentially with separation [Quillen, 2011, Yalinewich andPetrovich, 2019].…”

“…Several previous studies have fit functional forms to instability times recorded in large suites of N-body integrations [e.g., Chambers et al, 1996, Marzari and Weidenschilling, 2002, Faber and Quillen, 2007, Smith and Lissauer, 2009, Obertas et al, 2017. They found that instability times rise steeply with increasing interplanetary separation measured in mutual Hill radii, i.e., the characteristic radius around the planets in which their gravity dominates that of the star [see also Quillen, 2011, Yalinewich andPetrovich, 2019],…”

“…where ai is the semimajor axis of the inner planet in the pair, mi and mi+1 are the respective planet masses, and M is the stellar mass † . While this provides insight into the underlying dynamics [Quillen, 2011, Yalinewich andPetrovich, 2019], other orbital parameters also strongly influence stability. Follow-up studies have considered the effects of finite eccentricities and inclinations [e.g., Yoshinaga et al, 1999, Zhou et al, 2007, Funk et al, 2010, Wu et al, 2019, but make various simplifying assumptions (e.g., equal interplanetary separations and eccentricities).…”

“…† We note that the Hill sphere scales as the planet-star mass ratio µ to the one third power. Other authors [e.g.,Quillen, 2011, Hadden and Lithwick, 2018, Yalinewich and Petrovich, 2019 argue that a µ 1/4 scaling is better motivated. These scalings are close to one another, and given the poor performance of such models (Sec.…”

The path to instability in compact multi-planetary systems

“…(3) and ( 4) is easily explained by the fact that most studies only considered a limited mass range and very small difference between the exponents. Zhou et al (2007) estimated T surv as a power-law in the spacing and using Nekhoroshev estimates, Yalinewich & Petrovich (2020) proposed a scaling similar to Eq. (4).…”