Marchal's family of periodic orbits. I: Stability of inclined co-orbital planetary systems

TL;DR

This paper studies the stability of inclined co-orbital planetary systems, extending classical results to a broader range of masses and inclinations through analytical and numerical methods.

Contribution

It demonstrates the persistence of a family of periodic orbits in the three-body problem for various masses and inclinations, including in the full, non-averaged system.

Findings

The family of orbits exists in the full three-body problem for a wide range of masses.

Inclined systems remain stable beyond Routh's critical value.

High mutual inclinations (>60°) lead to high instability.

Abstract

At the Lagrange relative equilibrium of the three-body problem, for all values of the masses, the elliptic eigenvalues associated with vertical eigenvectors give rise to spatial quasi-periodic orbits, which become periodic in a rotating frame. In 2009, by averaging out the fast frequencies, Christian Marchal showed that these orbits, which are fixed points in the restricted average problem, form a oneparameter family connecting L 4 to L 5 . Using perturbation methods, we show the persistence of this family in the average three-body problem for nonzero masses in the limit where one mass is dominant over the other two (known as the planetary problem). We also give an analytical approximation valid for mutual inclinations less than 60 $\bullet$ . Then, using purely numerical methods, we show that this family exists in the full three-body problem (neither restricted nor average) for a wide range of masses, beyond the planetary case. We also show that the stability of its orbits evolves along the family, with inclined systems remaining stable for masses exceeding the Gascheau's value (also known as Routh's critical value). Finally, we show the impact of this family's stability on the global dynamics of the co-orbital region as well as its high instability for mutual inclinations exceeding 60 $\bullet$ .