FIRST-ORDER RESONANCE OVERLAP AND THE STABILITY OF CLOSE TWO-PLANET SYSTEMS

Abstract

Motivated by the population of observed multi-planet systems with orbital period ratios 1 < P 2/P 1 ≲ 2, we study the long-term stability of packed two-planet systems. The Hamiltonian for two massive planets on nearly circular and nearly coplanar orbits near a first-order mean motion resonance can be reduced to a one-degree-of-freedom problem. Using this analytically tractable Hamiltonian, we apply the resonance overlap criterion to predict the onset of large-scale chaotic motion in close two-planet systems. The reduced Hamiltonian has only a weak dependence on the planetary mass ratio m 1/m 2, and hence the overlap criterion is independent of the planetary mass ratio at lowest order. Numerical integrations confirm that the planetary mass ratio has little effect on the structure of the chaotic phase space for close orbits in the low-eccentricity (e ≲ 0.1) regime. We show numerically that orbits in the chaotic web produced primarily by first-order resonance overlap eventually experience large-scale erratic variation in semimajor axes and are therefore Lagrange unstable. This is also true of the orbits in this overlap region which satisfy the Hill criterion. As a result, we can use the first-order resonance overlap criterion as an effective stability criterion for pairs of observed planets. We show that for low-mass (≲ 10 M ⊕) planetary systems with initially circular orbits the period ratio at which complete overlap occurs and widespread chaos results lies in a region of parameter space which is Hill stable. Our work indicates that a resonance overlap criterion which would apply for initially eccentric orbits likely needs to take into account second-order resonances. Finally, we address the connection found in previous work between the Hill stability criterion and numerically determined Lagrange instability boundaries in the context of resonance overlap.