The phase-space architecture in extrasolar systems with two planets in orbits of high mutual inclination

Abstract

We revisit the secular 3D planetary three-body problem, aiming to provide a unified formalism representing all basic phenomena in the phase space as the mutual inclination between the planetary orbits increases. We propose a ‘book-keeping’ technique allowing to decompose the Hamiltonian as mathcal {H}_textrm{sec}=mathcal {H}_textrm{planar}+mathcal {H}_textrm{space}, with mathcal {H}_textrm{space} collecting all terms depending on the planets mutual inclination i_textrm{mut}. We numerically compare several models obtained by multipole (Legendre) or Laplace–Lagrange expansions of mathcal {H}_textrm{sec}, aiming to define suitable truncation orders for these models. We explore the transition, as i_textrm{mut} increases, from a ‘planar-like’ to a ‘Lidov–Kozai’ regime. Using a numerical example far from hierarchical limits, we find that the structure of the phase portraits of the (integrable) planar case is reproduced to a large extent also in the 3D case. A semi-analytical criterion allows to estimate the level of i_textrm{mut} up to which the dynamics remains nearly integrable. We propose a normal form method to compute the basic periodic orbits (apsidal corotation orbits A and B) in this regime. We explore the sequence of saddle-node and pitchfork bifurcations by which the A and B families are connected to the highly inclined periodic orbits of the Lidov–Kozai regime. Finally, we perform a numerical study of phase portraits for different planetary mass and distance ratios and qualitatively describe the approach to the corresponding hierarchical limits.