Hamiltonian secular theory and KAM stability in exoplanetary systems with 3D orbital architecture

In the present thesis, we first revisit the secular 3D planetary three-body problem aiming to provide a unified formalism for studying the structure of the phase space for progressively higher values of the mutual inclination i_mut between the two planets’ orbits. We propose a “book-keeping” technique yielding (after Jacobi reduction) a clear decomposition of the secular Hamiltonian as H_sec = H_planar + H_space , where H space contains all terms depending on i_mut . We explore the transition from a “planar-like” to the Lidov-Kozai regime. We show how the structure of the phase portraits of the integrable secular dynamics of the planar case is reproduced to a large extent also in the 3D case. We estimate semi-analytically the level of i_mut up to which the dynamics remains nearly-integrable, and propose a normal form method to compute the basic periodic orbits (apsidal corotation resonances) and quasi-periodic orbits in this regime. We explore the families of periodic orbits dominant in the other limit, of the Lidov-Kozai regime, as well as the connection between all the above families of periodic orbits. We study numerically the form of the phase portraits for different mass and semi-major axis ratios of the two planets, for systems’ parameters close to one or more hierarchical limits (in the planets’ mass or distance ratio). Secondly, we introduce a quasi-periodic restricted Hamiltonian to describe the secular motion of a small-mass planet in a multi-planetary system. As an example, we refer to the motion of υ-And b (the innermost planet in the extrasolar υ-Andromedæ system). We reconstruct the orbits of υ- And c and υ-And d in a stable configuration through Frequency Analysis of their secular motions. These orbits are then injected in the equations describing the orbital dynamics of υ-And b, ending up with a Hamiltonian model having 2+3/2 degrees of freedom validated through a comparison of numerical integrations with the complete 4-body problem. We also add relativistic corrections to the above model. We study the stability of υ-And b as a function of the initial values of the inclination and of the longitude of the node, which are subject to observational uncertainties. Studying the evolution of the eccentricity, we show how to exclude orbital configurations with long-time high probability of (quasi)collision with the central star. We introduce a normal form approach, based on averaging of the angles associated with the secular motions of the major exoplanets, leading to a further reduced model with 2 dof. This allows to quickly preselect, by a numerical criterion, the domains of stability for υ-And b. After the above steps, we implement the Kolmogorov normalization algorithm in order to provide a computer-assisted proof of existence of KAM (Kolmogorov-Arnold-Moser) tori in the framework of the above secular models for planet υ-And b. We compute the Kolmogorov normal form and provide a computer-assisted proof of existence of KAM tori for the inner planet’s secular motion for pre-selected initial conditions in i) the secular Hamiltonian model found after the elimination of all the fast angles of the problem (SQPR 2+2/2 model), and ii) in the reduced model after normalizing the secular motions of the outer planets (SQPR 2 model). We show how the KAM tori persist when general-relativistic corrections are added to the models. Finally, we present a Kolmogorov-like normal form algorithm in the neighborhood of an invariant torus in Hamiltonian systems H = H_0 + εH_1 , where H_0 is the Hamiltonian of N linear oscillators, and H_1 is a polynomial series. This yields a normal form analogue of a corresponding Lindstedt method for coupled oscillators. We comment on the possible use of the Lindstedt method itself under two distinct schemes, i.e., one analogous to the Birkhoff and another analogous to the Kolmogorov (torus fixing) normal form.