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

Abstract

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.

accepted for publication in Icarus

References in corpus (12)

• Quantifying the Uncertainty in the Orbits of Extrasolar Planets
• Secular Evolution of Hierarchical Planetary Systems
• Secular dynamics of the three body problem: application to the upsilon Andromedae planetary system
• Evidence for Planet-Planet Scattering in Upsilon Andromedae
• Masses and orbital inclinations of planets in the PSR B1257+12 system
• A dynamical mechanism for establishing apsidal resonance
• The Dynamics of Two Massive Planets on Inclined Orbits
• The orbits of the extrasolar planets HD 82943 c,b
• Apsidal Alignment in Upsilon Andromedae
• Eccentricity Excitation and Apsidal Resonance Capture in the Planetary System Upsilon Andromedae
• Excitation of Orbital Eccentricities of Extrasolar Planets by Repeated Resonance Crossings
• Regular Motions in Extra-Solar Planetary Systems

Cited by in corpus (30)

• Secular Chaos and the Production of Hot Jupiters
• Hot Jupiters from Coplanar High-eccentricity Migration
• Survey of Kozai Dynamics Beyond Neptune
• Dynamical Evolution Induced by Planet Nine
• Compact planetary systems perturbed by an inclined companion: I. Vectorial representation of the secular model
• The 3-dimensional architecture of the Upsilon Andromedae planetary system
• 2/1 resonant periodic orbits in three dimensional planetary systems
• Secular Orbital Dynamics of Hierarchical Two Planet Systems
• Dynamical Stability of Terrestrial Planets in the Binary α Centauri System
• Planet Formation In Highly Inclined Binary Systems I. Planetesimals Jump Inwards And Pile Up
• A secular theory of coplanar, non-resonant planetary system
• Dynamics of planets in retrograde mean motion resonance
• Are retrograde resonances possible in multi-planet systems?
• Extrasolar planetary dynamics with a generalized planar Laplace-Lagrange secular theory
• A dynamical analysis of the 14 Her planetary system
• Secular dynamics of coplanar, non-resonant planetary system under the general relativity and quadrupole moment perturbations
• The Stationary Points of the Hierarchical Three Body Problem
• Secular Dynamics of S-type Planetary Orbits in Binary Star Systems: Applicability Domains of First- and Second-Order Theories
• Formation of '3D' multiplanet systems by dynamical disruption of multiple-resonance configurations
• The non-resonant, relativistic dynamics of circumbinary planets
• Trapping in three-planet resonances during gas-driven migration
• On the coplanar eccentric non restricted co-orbital dynamics
• Equilibria in the secular, non-coplanar two-planet problem
• On the 3D secular dynamics of radial-velocity-detected planetary systems
• Interesting dynamics at high mutual inclination in the framework of the Kozai problem with an eccentric perturber
• Librational KAM tori in the secular dynamics of the $\upsilon$ Andromedæ planetary system
• The phase-space architecture in extrasolar systems with two planets in orbits of high mutual inclination
• Secular resonance of inner test particles in hierarchical planetary systems
• Relativistic Lidov-Kozai resonance in binaries
• Dynamics of co-orbital exoplanets