• methods: numerical;
  • planets and satellites: formation;
  • planet;
  • disc interactions


Assuming that giant planets are formed in thin protoplanetary discs, a ‘3D’ system – i.e. a planetary system composed of two (or more) planets, whose orbital planes have large values of mutual inclination – can form, provided that the mutual inclination is excited by some dynamical mechanism. Resonant interactions and close planetary encounters are thought to be the primary inclination-excitation mechanisms, resulting in a resonant and non-resonant system, respectively. If by the end of planet formation, the system is dynamically ‘hot’, then a phase of planet–planet scattering can be expected; however, this need not be the case in every system. Here we propose an alternative formation scenario, starting from a system composed of three giant planets in a nearly coplanar configuration. As was recently shown for the case of the Solar system, planetary migration in the gas disc (Type II migration) can force the planets to become trapped in a multiply resonant state (similar to the Laplace resonance in the Galilean satellites). We simulate this process, assuming different values for the planetary masses and mass ratios. We show that such a triple resonance generally becomes unstable as the resonance excites the eccentricities of all planets and planet–planet scattering sets in. One of the three planets is typically ejected from the system, leaving behind a dynamically ‘hot’ (but stable) two-planet configuration. The resulting two-planet systems typically have large values of semimajor axial ratios (α=a1/a2 < 0.3), while the mutual inclination can be as high as 70°, with a median of ∼30°. These values are quite close to the ones recently obtained for the υ-Andromedae system. A small fraction of our two-planet systems (∼5 per cent) ends up in the stability zone of the Kozai resonance. In a few cases, the triple resonance can remain stable for long times and a ‘3D’ system can form by resonant excitation of the orbital inclinations; such a three-planet system could be stable if enough eccentricity damping is exerted on the planets. Finally, in the single-planet resulting systems, which are formed when two planets are ejected from the system, the inclination of the planet’s orbital plane with respect to the initial invariant plane – presumably the plane perpendicular to the star’s spin axis – can be as large as ∼40°.