Stability Limits in Resonant Planetary Systems

The relationship between the boundaries for Hill and Lagrange stability in orbital element space is modified in the case of resonantly interacting planets. Hill stability requires the ordering of the planets to remain constant while Lagrange stability also requires all planets to remain bound to the central star. The Hill stability boundary is defined analytically, but no equations exist to define the Lagrange boundary, so we perform numerical experiments to estimate the location of this boundary. To explore the effect of resonances, we consider orbital element space near the conditions in the HD 82943 and 55 Cnc systems. Previous studies have shown that, for non-resonant systems, the two stability boundaries are nearly coincident. However the Hill stability formula are not applicable to resonant systems, and our investigation shows how the two boundaries diverge in the presence of a mean-motion resonance, while confirming that the Hill and Lagrange boundaries are similar otherwise. In resonance the region of stability is larger than the domain defined by the analytic formula for Hill stability. We find that nearly all known resonant interactions currently lie in this extra stable region, i.e. where the orbits would be unstable according to the non-resonant Hill stability formula. This result bears on the dynamical packing of planetary systems, showing how quantifying planetary systems' dynamical interactions (such as proximity to the Hill-stability boundary) provides new constraints on planet formation models.