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

TL;DR

This paper develops a unified formalism to analyze the phase space structure of two-planet extrasolar systems with high mutual inclination, exploring transitions from planar to Lidov-Kozai regimes and identifying nearly-integrable dynamics.

Contribution

It introduces a novel 'book-keeping' technique for the secular Hamiltonian and provides a semi-analytical approach to study the transition from near-integrability to Lidov-Kozai dynamics in inclined planetary systems.

Findings

Identifies the minimum truncation order for accurate dynamics representation.

Shows the phase space structure is similar in 3D and planar cases for certain regimes.

Estimates the inclination level up to which the system remains nearly-integrable.

Abstract

We revisit the secular 3D planetary three-body problem aiming to provide a unified formalism for studying the structure of the phase space for progressively higher values of the mutual inclination $i_{mut}$ between the two planets' orbits. We propose a `book-keeping' technique yielding (after Jacobi reduction) a clear decomposition of the secular Hamiltonian as $H_{sec}=H_{planar} +H_{space}$, where $H_{space}$ contains all terms depending on $i_{mut}$. We numerically compare several models obtained via expansion in the orbital eccentricities or via multipole expansion. We find the mimimum required truncation orders to accurately represent the dynamics. We explore the transition, as $i_{mut}$ increases, from a `planar-like' to a `Lidov-Kozai' regime. Using a typical (non-hierarchical) example, we show how the structure of the phase portraits of the integrable secular dynamics of the planar case is reproduced to a large extent also in the 3D case. We estimate semi-analytically the level of $i_{mut}$ up to which the dynamics remains nearly-integrable. In this regime, we propose a normal form method by which the basic periodic orbits of the nearly-integrable regime (apsidal corotation resonances) can be computed semi-analytically. On the other hand, as the energy increases the system gradually moves to the `Lidov-Kozai' regime. The latter is dominated by two different families of inclined periodic orbits ($C_1$ and $C_2$), of which $C_2$ becomes unstable via the usual Lidov-Kozai mechanism. We discuss the connection between the above families of periodic orbits. Finally, we study numerically the form of the phase portraits for different mass and semi-major axis ratios of the two planets, aiming to establish how generic are the phenomena reported above as the systems parameters are chosen close to one or more hierarchical limits.