Dynamical and Secular Stability of Mutually Inclined Planetary Systems

Multiple analytical and empirical stability criteria have been derived in the literature for two planet systems. But, the dependence of the stability limit on the initial mutual inclination between the inner and outer orbits is not well modeled by previous stability criteria. Here, we derive a semi-analytical stability criteria for two planet systems, at arbitrary inclinations, in which the inner planet is a test particle. Using perturbation theory we calculate the characteristic fractional change in the semi-major axis of the inner binary $\beta=\delta a_1/a_1$ caused by perturbations from the companion. A stability criteria can be derived by setting a threshold on $\beta$ Focusing initially on circular orbits, we derive an analytical expression for $\beta$ for co-planar prograde and retrograde orbits. For non-coplanar configurations, we evaluate a semi-analytical expression. We then generalize to orbits with arbitrary eccentricities and account for the secular effects. Our analytical and semi-analytical results are in excellent agreement with direct N-body simulations. In addition, we show that contours of $\beta\sim0.01$ can serve as criteria for stability. More specifically, we show that (1) retrograde orbits are generally more stable than prograde ones; (2) systems with intermediate mutual inclination are less stable due to vZLK dynamics; and (3) mean-motion resonances (MMRs) can stabilize intermediate inclination secularly unstable regions in phase space, by quenching vZLK secular processes (4) MMRs can destabilize some of the dynamically stable regions. We also point out that these stability criteria can be used to constrain the orbital properties of observed systems and their age. We also point out that these stability criteria can be used to constrain the orbital properties of observed systems (in particular inclination) and their age.