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A gravitational close encounter of a small body with a planet may produce asubstantial change of its orbital parameters which can be studied using thecircular restricted three-body problem. In this paper we provide parametricrepresentations of the fast close encounters with the secondary body of theplanar CRTBP as arcs of non-linear focus-focus dynamics. The result is theconsequence of a remarkable factorization of the Birkhoff normal forms of theHamiltonian of the problem represented with the Levi-Civita regularization. Theparametrizations are computed using two different sequences of Birkhoffnormalizations of given order N. For each value of N, the Birkhoffnormalizations and the parameters of the focus-focus dynamics are representedby polynomials whose coefficients can be computed iteratively with a computeralgebra system; no quadratures, such as those needed to compute action-anglevariables of resonant normal forms, are needed. We also provide some numericaldemonstrations of the method for values of the mass parameter representative ofthe Sun-Earth and the Sun-Jupiter cases.

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