Studying network of symmetric periodic orbit families of the Hill problem via symplectic invariants, Extension of the creep tide theory to exoplanet systems with high stellar obliquity

Studying network of symmetric periodic orbit families of the Hill problem via symplectic invariants In the framework of the spatial circular Hill three-body problem we illustrate the application of symplectic invariants to analyze the network structure of symmetric periodic orbits families. The extensive collection of families within this problem constitutes a complex network, fundamentally comprising the so-called basic families of periodic solutions, including the orbits of the satellite g, f, the libration (Lyapunov) a,c, and collision ℬ_0 families. Since the Conley–Zehnder index leads to a grading on the local Floer homology and its Euler characteristics, a bifurcation invariant, the computation of those indices facilitates the construction of well-organized bifurcation graphs depicting the interconnectedness among families of periodic solutions. The critical importance of the symmetries of periodic solutions in comprehending the interaction among these families is demonstrated. Extension of the creep tide theory to exoplanet systems with high stellar obliquity The fundamental basis of the rheophysical tide theory – a.k.a. creep tide theory or hydrodynamical tide – was established by the seminal work of Ferraz-Mello (2013). Later, Folonier et al. (2018) presented their so-called parametric equations of the creep, the most elaborated version of such a theory. Folonier et al. (2025) have recently extended this latest version to tridimensional configurations, including the orbital and axial inclinations within the dynamics. Not only are the shapes of the bodies and their rotations simultaneously determined, but other motions are also considered, such as the (equatorial) nodal precession, the nutation, and the libration of the spin axes. In this talk, the main concepts of the 3D version of the creep theory are summarized, as well as its first application to the rotational dynamics of the CoRot-3b’s exoplanetary system. The method’s advantages will be highlighted, as well as the open concerns still to be treated in future works.