Parametric approximations of fast close encounters of the planar three-body problem as arcs of a focus-focus dynamics (Q6608794)

A gravitational close encounter of a small body with a planet maybe produces a substantial change of its orbital parameters which is studied with the circular restricted three-body problem (CRTBP). The author of this paper provides parametric representations of the fast close encounters with the secondary body of the planar CRTBP, as arcs of non-linear focus-focus dynamics. The paper concerns the representation of the arcs of solutions which intersect a small neighbourhood of P2, obtained from computations of series. As a result, in the consequence of a remarkable factorization of the Birkhoff normal forms (Bnf) of the Hamiltonian, the problem is represented with the Levi-Civita regularization. The parameterizations are computed using two different sequences of Birkhoff normalizations of given order N, using a computer algebra system. In this paper, the properties of the first sequence of saddle-saddle Bnf are described and all the details needed to construct the second sequence of focus-focus Bnf, and the solution of its Hamilton equations are provided. Some numerical demonstrations of the method for values of the mass parameter representative of the Sun-Earth and the Sun-Jupiter cases are given. The author also presents the technical details of the construction of the Bnf for 2-degree of freedom Hamiltonian systems, and he provides, also, the generating functions which define the Bnf of order N = 6. An error analysis based on the Gronwall lemma is also given, and conclusions and perspectives are provided.