  $N$-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 $c_2=8$.
