# Nekhoroshev Estimates for the Survival Time of Tightly Packed Planetary Systems

## TL;DR

This paper derives a Nekhoroshev-based estimate for the survival time of tightly packed planetary systems, explaining the exponential growth of stability time with interplanetary spacing and planetary mass through high-order resonances.

## Contribution

It introduces a new heuristic formula for planetary system survival time based on Nekhoroshev's theorem, improving understanding of long-term stability beyond previous power-law models.

## Key findings

- The proposed formula matches N-body simulation results well.
- Survival time increases exponentially with interplanetary spacing and planetary mass.
- High-order resonances drive large deviations from initial conditions.

## 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 large deviations of the system from its initial conditions is due to 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: $T/P=c_1 \frac{a}{\Delta a} \exp \left(c_2 \frac{\Delta a}{a} /\mu^{1/4}\right)$, where $P$ is the average Keplerian period, $a$ is the average semi major axis, $\Delta a\ll a$ is the difference between the semi major axes of neighbouring planets, $\mu$ 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 $c_1=5 \cdot 10^{-4}$ and $c_2=8$.