Modeling the 3--D Secular Planetary Three-Body Problem. Discussion on the outer ${\upsilon}$ Andromedae Planetary System

The three-dimensional secular behavior of a system composed of a central star and two massive planets is modeled semi-analytically in the frame of the general three-body problem. The main dynamical features of the system are presented in geometrical pictures allowing us to investigate a large domain of the phase space of this problem without time-expensive numerical integrations of the equations of motion and without any restriction on the magnitude of the planetary eccentricities, inclinations and mutual distance. Several regimes of motion of the system are observed. With respect to the secular angle $\Delta\varpi$, possible motions are circulations, oscillations (around 0 and $180^\circ$), and high eccentricity/inclination librations in secular resonances. With respect to the arguments of pericenter, $\omega_1$ and $\omega_2$, possible motions are direct circulation and high-inclination libration around $\pm 90^\circ$ in the Lidov-Kozai resonance. The regions of transition between domains of different regimes of motion are characterized by chaotic behavior. We apply the analysis to the case of the two outer planets of the $\upsilon$ Andromedae system, observed edge-on. The topology of the 3--D phase space of this system is investigated in detail by means of surfaces of section, periodic orbits and dynamical spectra, mapping techniques and numerical simulations. We obtain the general structure of the phase space, and the boundaries of the spatial secular stability. We find that this system is secularly stable in a large domain of eccentricities and inclinations.