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 Δ 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 Δ 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 μ = 1.97 ± 0.02 and σ = 0.40 ± 0.02, such that the mean, median, and mode of Δ min are 7.77, 7.17, and 6.11, respectively. This result is consistent with previous estimates for Δ min threshold values in the range ∼5–8. We find a modest dependence of the distribution of Δ 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 Δ) in long-term stable systems is also well fit with a log-normal distribution, with parameters μ = 3.14 ± 0.03 and σ = 0.76 ± 0.02. 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 exoplanetary systems, indicating a strong role of sculpting of planetary architectures by dynamical instabilities.