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The minimum orbital separation of planets in long-stable planetary systems isoften 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 withgreater separations are considered stable. Here we report that a log-normaldistribution 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 isconsistent with previous estimates for $\Delta_{min}$ threshold values in therange $\sim$5$-$8. We find a modest dependence of the distribution of$\Delta_{min}$ on multiplicity within the system, as well as on planetary massratios of the closest planet pair. The overall distribution of nearest-neighborplanetary orbital spacings (measured in the mutual Hill radii and denotedsimply as $\Delta$) in long-term stable systems is also well fit with alog-normal distribution, with parameters $\mu=3.14\pm0.03$ and$\sigma=0.76\pm0.02$. In simulations of sets of many planets initially packedvery close together, we find that the orbital spacings of long-term stablesystems is statistically similar to that in the observed Kepler sample ofexo-planetary systems, indicating a strong role of sculpting of planetaryarchitectures by dynamical instabilities.

Statistical Distribution Function of Orbital Spacings in Planetary Systems