Statistical Distribution Function of Orbital Spacings in Planetary Systems

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.