Seminar | Series

Speakers

• Anargyros Dogkas
• Alexandre Prieur

Abstract

In celestial mechanics, the computation of local quasi-integrals of the motion has been a central subject of study. Proper elements have been historically used for the classification of objects (like asteroids and Earth orbiters) as well as for the construction of local analytical solutions for the evolution of orbital elements. However, the domain where these quasi-integrals are well defined is naturally restricted by the existence of resonances. In this article, we use analytical methods to define quasi-integrals of motion in the case of resonant systems. This is done using the quasiperiodic properties of libration regions in isolated resonances, as well as in the multi-resonant case, allowing us to apply perturbation theory effectively even though the initial system was resonant. Finally, we introduce the problem of Earth orbiters, like satellites and space debris, and we illustrate the results of these methods in a variety of examples in mean motion and secular resonances, including thinner resonances
and multi-resonant systems.

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 one-parameter family connecting
to . 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 . 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 .