Vertical stability of co-orbital planetary systems: from Marchal’s family to P12?

Although many theoretical works suggest that co-orbital exoplanets may also exist, such a pair of planets has yet to be observed. In particular, out of the hundreds of multi-planetary systems discovered by the Kepler mission, none were found in co-orbital configuration thus far. Although this result points to a relative rarity of the configuration, part of this absence might be due to observational biases (Robutel, Leleu 2024). We are interested here in one of these: the fact that the planets do not orbit in the same plane, which can prevent the transit of, at least, one of the bodies. In this talk, we explore the stability domain in the spatial case, paying a particular attention to a family of periodic orbits, in a rotating frame, emanating from the Lagrange’s equilateral configuration in the vertical direction.  Theses solutions have been first discovered by Marchal (2009) in the special case of the restricted secular problem. We propose a numerical study to follow this family in a more general case. We show that the family discovered by Marchal exists in the full three-body problem (neither restricted nor secular) for a wide range of masses, and that the stability of its orbits evolves along the family, sometimes leading to stable systems for masses exceeding the Gascheau’s value (also known as Routh’s critical value). We try to continue these families increasing the planetary masses the two extreme cases, that is between the Marchal’s and P12 families.  we try to extend these families by increasing the planetary masses until they reach that of the central body, in the hope of discovering links between these orbits of the planetary problem and the P12 family, containing the famous “Figure 8” when the 3 masses are equal (Chenciner, Montgomery, 2000).

 

Bibliographie

Chenciner A. and Montgomery R. “A remarkable periodic solution of the three-body problem in the case of equal masse”. Annals of Mathematics, 152 (2000), 881–901

Marchal C. “Long term evolution of quasi-circular Trojan orbits”. Celestial Mechanics and Dynamical Astronomy, 105(2009), 53-67

Robutel P. and Leleu A. “Trojan Exoplanets” (2024) arXiv:2402.15168. To be published in: Handbook of Exoplanets, 2nd Edition, Hans Deeg and Juan Antonio       Belmonte (Eds. in Chief), Springer International Publishing AG, part of Springer Nature.