Vertical instability and inclination excitation during planetary migration

TL;DR

This paper investigates how planetary migration can lead to inclination excitation through vertical instability of periodic orbits, revealing the conditions under which planets develop non-coplanar orbits during resonance capture.

Contribution

It identifies the role of vertical critical orbits in driving inclination excitation and computes these orbits for key resonances across different planetary mass ratios.

Findings

Inclination excitation occurs at vertically unstable periodic orbit points.

Vertical critical orbits lead to stable 3D periodic orbit families.

Limits of eccentricity for inclination resonance are determined.

Abstract

We consider a two-planet system, which migrates under the influence of dissipative forces that mimic the effects of gas-driven (Type II) migration. It has been shown that, in the planar case, migration leads to resonant capture after an evolution that forces the system to follow families of periodic orbits. Starting with planets that differ slightly from a coplanar configuration, capture can, also, occur and, additionally, excitation of planetary inclinations has been observed in some cases. We show that excitation of inclinations occurs, when the planar families of periodic orbits, which are followed during the initial stages of planetary migration, become vertically unstable. At these points, {\em vertical critical orbits} may give rise to generating stable families of $3D$ periodic orbits, which drive the evolution of the migrating planets to non-coplanar motion. We have computed and present here the vertical critical orbits of the $2/1$ and $3/1$ resonances, for various values of the planetary mass ratio. Moreover, we determine the limiting values of eccentricity for which the "inclination resonance" occurs.