Statistical Distribution Function of Orbital Spacings in Planetary Systems

Abstract

The minimum orbital separation of planets in long-stable planetary systems is often modeled as a step function, parameterized with a single value ${{\rm{\Delta }}}_{\min }$ (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 ${{\rm{\Delta }}}_{\min }$ , 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 μ = 1.97 ± 0.02 and σ = 0.40 ± 0.02, such that the mean, median, and mode of ${{\rm{\Delta }}}_{\min }$ are 7.77, 7.17, and 6.11, respectively. This result is consistent with previous estimates for ${{\rm{\Delta }}}_{\min }$ threshold values in the range ∼5–8. We find a modest dependence of the distribution of ${{\rm{\Delta }}}_{\min }$ 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 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.

Keywords

• Exoplanet systems
• Exoplanet dynamics