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

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 $\bullet$ .

Figures (11)

• Figure 1: Coordinate system linked to the invariant plane (perpendicular to the total angular momentum and containing the most massive body). The angular positions of the bodies are indicated by the angles $w_j$ (draconic true anomalies) originating at the ascending node ($N_j$) of the corresponding orbit.
• Figure 2: The two families of fixed points derived from Lagrange equilibria in the average problem. The two branches of ${\cal VF}_L$ are identical up to permutation of the two small bodies.
• Figure 3: (a): Behavior of eigenvalues and precession frequencies of nodes along families ${\cal VF}_E$ (purple) and ${\cal VF}_L$ (red). (b): Precession of the ascending node.
• Figure 4: Stability map of the co-orbital region in the plane ($\lambda_1 - \lambda_2, J_0)$ for $m_1 = m_2 = 10^{-3}m_0$. The stability index, from blue to red, corresponds to the maximum eccentricity values reached by the two planets during their integration in the full problem, while the white is associated with orbits that either left co-orbital resonance before the end of integration or collided. The green curve is the projection of ${\cal VF}_L$ (computed in the average problem) onto the stability map.
• Figure 5: ${\cal VF}_L$ and ${\cal VF}_E$ for ${\varepsilon} = 10^{-3}$, in the $( w_1 - w_2, J)$ plane.