Numerical Investigations of Non-equal Mass and Non-equal Spacing Packing of Planetary Bodies

We study the optimal packing of non-equally massed and non-equally spaced multi-planet systems through numerical N-body simulations. Previous studies have generally assumed that a system of equal mass planets will be optimally packed if they are also equally spaced, i.e., if the semi-major axis ratios between planet pairs is a constant. We explicitly test this assumption by obtaining the stability timescales of 5-planet systems around a Sun-like star (with masses varying from 3 Earth masses to 3 Jupiter masses) with increasing degrees of non-uniform-spacing represented by the parameter k. Such systems are simulated using N-body integrations until they reach the point of gravitationally unstable close encounters. For planets with equal masses, a value of k = 1 corresponds to equal spacing, whereas a value of k < 1 leads to the inner planets being more widely spaced than outer planets. We study the optimal value of k for optimal planet packing (i.e., longest stability time) under both equal mass and non-equal mass scenarios and find evidence that k = 1 is optimal under most (but not all) initial conditions; we discuss the scenarios where k < 1 may be preferable. We also study the role that distance to mean-motion resonances (MMRs) play in determining the configurations of optimal planet packing.