  The minimum orbital separation of planets in long-stable planetary systems is
often modeled as a step function, parameterized with a single value
$\Delta_{min}$ (measured in mutual Hill radius of the two neighboring planets).
Systems with smaller separations are considered unstable, and planet pairs with
greater separations are considered stable. Here we report that a log-normal
distribution function for $\Delta_{min}$, rather than a single threshold value,
provides a more accurate model. From our suite of simulated planetary systems,
the parameters of the best-fit log-normal distribution are $\mu=1.97\pm0.02$
and $\sigma=0.40\pm0.02$, such that the mean, median, and mode of
$\Delta_{min}$ are 7.77, 7.17, and 6.11, respectively. This result is
consistent with previous estimates for $\Delta_{min}$ threshold values in the
range $\sim$5$-$8. We find a modest dependence of the distribution of
$\Delta_{min}$ on multiplicity within the system, as well as on planetary mass
ratios of the closest planet pair. The overall distribution of nearest-neighbor
planetary orbital spacings (measured in the mutual Hill radii and denoted
simply as $\Delta$) in long-term stable systems is also well fit with a
log-normal distribution, with parameters $\mu=3.14\pm0.03$ and
$\sigma=0.76\pm0.02$. In simulations of sets of many planets initially packed
very close together, we find that the orbital spacings of long-term stable
systems is statistically similar to that in the observed Kepler sample of
exo-planetary systems, indicating a strong role of sculpting of planetary
architectures by dynamical instabilities.

Earth and Planetary Astrophysics