Dynamical instability and its implications for planetary system architecture

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 $K_{\rm min}$ 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.