The stability and fates of hierarchical two-planet systems

We study the dynamical stability and fates of hierarchical (in semi-major axis) two-planet systems with arbitrary eccentricities and mutual inclinations. We run a large number of long-term numerical integrations and use the Support Vector Machine algorithm to search for an empirical boundary that best separates stable systems from systems experiencing either ejections or collisions with the star. We propose the following new criterion for dynamical stability: $a_{\rm out}(1-e_{\rm out})/[a_{\rm in}(1+e_{\rm in})]>2.4\left[\max(\mu_{\rm in},\mu_{\rm out})\right]^{1/3}(a_{\rm out}/a_{\rm in})^{1/2}+1.15$,whichshouldbeapplicabletoplanet-starmassratios$\mu_{\rm in},\mu_{\rm out}=10^{-4}-10^{-2}$,integrationtimesupto$10^8$orbitsoftheinnerplanet,andmutualinclinations$\lesssim40^\circ$. Systems that do not satisfy this condition by a margin of $\gtrsim 0.5$ are expected to be unstable, mostly leading to planet ejections if $\mu_{\rm in}>\mu_{\rm out}$,whileslightlyfavoringcollisionswiththestarfor$\mu_{\rm in}<\mu_{\rm out}$. We use our numerical integrations to test other stability criteria that have been proposed in the literature and show that our stability criterion performs significantly better for the range of system parameters that we have explored.