Hill stability in the AMD framework

In a two-planet system, due to Sundman (1912) inequality, a topological boundary can forbid close encounters between the two planets for infinite time. A system is said Hill stable if it verifies this topological condition. Hill stability is widely used in the study of extra solar planets dynamics. However people often use the coplanar and circular orbits approximation. In this paper, we explain how the Hill stability can be understood in the framework of Angular Momentum Deficit (AMD). In the secular approximation, the AMD allows to discriminate between a priori stable systems and systems for which a more in depth dynamical analysis is required. We show that the general Hill stability criterion can be expressed as a function of only the semi major axes, the masses and the total AMD of the system. The proposed criterion is only expanded in the planets-to-star mass ratio $\epsilon$ and not in the semi-major axis ratio, in eccentricities nor in the mutual inclination. Moreover the expansion in $\epsilon$ remains excellent up to values of about $10^{-3}$ even for two planets with very different mass values. We performed numerical simulations in order to highlight the sharp change of behaviour between Hill stable and Hill unstable systems. We show that Hill stable systems tend to be very regular whereas Hill unstable ones often lead to rapid planet collisions. We also remind that Hill stability does not protect from the ejection of the outer planet.