The phase-space architecture in extrasolar systems with two planets in orbits of high mutual inclination , Bounds on energy and angular momentum loss in the full n-body problem

Abstract

Authors: Rita Mastroianni, Christos Efthymiopoulos 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 $\mathcal{H}_{sec}=\mathcal{H}_{planar}+\mathcal{H}_{space}$, with $\mathcal{H}_{space}$ collecting all terms depending on the planets mutual inclination $i_{mut}$. We numerically compare several models obtained by multipole (Legendre) or Laplace-Lagrange expansions of $\mathcal{H}_{sec}$, aiming to define suitable truncation orders for these models. We explore the transition, as $i_{mut}$ 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 $i_{mut}$ 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. Authors: D. J. Scheeres, G. M. Brown A new bound on the amended potential in an n-body system is derived and applied to the partitioning of energy and angular momentum in a disrupting gravitational aggregate. This result provides analytic insight into how energy and angular momentum can be lost or partitioned between different collections of bodies as they escape from each other.