The Orbital Stability of Compact Four-Planet Systems

We investigate the orbital dynamics for four-planet systems consisting of Earth-mass planets on initially circular, coplanar orbits around a star of one solar mass. The initial semi-major axis of the first planet was designated at 1 A.U. with the remaining planets’ semi-major axes equally spaced by multiples of their mutual Hill radii. We explore the stability of the system until either a close encounter occurs (defined as when planets are within 0.01 AU of one another), the orbital eccentricity of a planet reaches or exceeds 1, or the simulated lifetime exceeds 10 billion years. Three sets of initial planetary longitudes were investigated: well-separated longitudes from Smith and Lissauer (Icarus, 2009), aligned longitudes, and random longitudes. We find that the simulated lifetime of four-planet systems follows a power-law with respect to planetary spacing that is intermediary to three and five planet systems. An unexpected result of our investigation is the appearance of a “phase shift” between the local minima (resonant dips in lifetime) and maxima of system lifetimes as a function of initial orbital separation between planets and resonance-induced local minima between systems with all planets starting at the same initial longitudes and those with the well-separated initial longitudes. On average, the locations in the resonant minima in lifetimes for the initially aligned systems are displaced inwards from those of the well-separated initial longitudes by an average distance of 0.04 of the mutual Hill sphere radius of neighboring planets.