Nekhoroshev Estimates for the Survival Time of Tightly Packed Planetary Systems

N-body simulations of nonresonant, 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 giant 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 large deviations of the system from its initial conditions is due to a few slowly developing high-order resonances. Thus, we show that the survival time of the system T can be estimated using a heuristic motivated by Nekhoroshev's theorem, and obtain a formula for systems away from overlapping two-body mean-motion resonances as $T/P={c}_{1}\tfrac{a}{{\rm{\Delta }}a}\exp \left({c}_{2}\tfrac{{\rm{\Delta }}a}{a}/{\mu }^{1/4}\right)$, where P is the average Keplerian period, a is the average semimajor axis, Δa ≪ a is the difference between the semimajor axes of neighboring planets, μ is the planet-to-star mass ratio, and c1 and c2 are dimensionless constants. We show that this formula is in good agreement with numerical N-body experiments for c1 = 5 × 10−4 and c2 = 8.