Orbital Stability of Closely-Spaced Four-planet Systems

Abstract

We investigate the orbital dynamics of four-planet systems consisting of Earth-mass planets on initially-circular, coplanar orbits around a star of one solar mass. In our simulations, the innermost planet's semimajor axis is set at 1 AU, with subsequent semimajor axes spaced equally in terms of planets' mutual Hill radii. Several sets of initial planetary longitudes are investigated, with integrations continuing for up to $10^{10}$ orbits of the innermost planet, stopping if a pair of planets pass within 0.01 AU of each other or if a planet is ejected from the system. We find that the simulated lifetimes of four-planet systems follow the general trend of increasing exponentially with planetary spacing, as seen by previous studies of closely-spaced planets. Four-planet system lifetimes are intermediate between those of three- and five-planet systems and more similar to the latter. Moreover, as with five-planet systems, but in marked contrast to the three-planet case, no initial semimajor axes spacings are found to yield systems that survive several orders of magnitude longer than other similar spacings. First- and second-order mean-motion resonances (MMRs) between planets correlate with reductions in system lifetimes. Additionally, we find that third-order MMRs between planets on neighboring orbits also have a substantial, though smaller, destabilizing effect on systems very near resonance that otherwise would be very long-lived. Local extrema of system lifetimes as a function of planetary spacing occur at slightly smaller initial orbital separation for systems with planets initially at conjunction relative to those in which the planets begin on widely-separated longitudes. This shift is produced by the asymmetric mutual planetary perturbations as the planets separate in longitude from the initial aligned configuration that cause orbits to spread out in semimajor axis.