We examine the effects that dynamical instability has on shaping the orbital properties of exoplanetary systems .
Using N-body simulations of non-EMS ( Equal Mutual Separation ) , multi-planet systems we find that the lower limit of the instability timescale t is determined by the minimal mutual separation K _ { min } in units of the mutual Hill radius .
Planetary systems showing instability generally include planet pairs with period ratio < 1.33 .
Our final period ratio distribution of all adjacent planet pairs shows dip-peak structures near first-order mean motion resonances similar to those observed in the Kepler planetary data .
Then we compare the probability density function ( PDF ) of the de-biased Kepler period ratios with those in our simulations and find a lack of planet pairs with period ratio > 2.1 in the observations—possibly caused either by inward migration before the dissipation of the disk or by planet pairs not forming with period ratios > 2.1 with the same frequency they do with smaller period ratios .
By comparing the PDF of the period ratio between simulation and observation , we obtain an upper limit of 0.03 on the scale parameter of the Rayleigh distributed eccentricities when the gas disk dissipated .
Finally , our results suggest that a viable definition for a ‘ ‘ packed ’ ’ or ‘ ‘ compact ’ ’ planetary system be one that has at least one planet pair with a period ratio less than 1.33 .
This criterion would imply that 4 % of the Kepler systems ( or 6 % of the systems with more than two planets ) are compact .