  A gravitational close encounter of a small body with a planet may produce a
substantial change of its orbital parameters which can be studied using the
circular restricted three-body problem. In this paper we provide parametric
representations of the fast close encounters with the secondary body of the
planar CRTBP as arcs of non-linear focus-focus dynamics. The result is the
consequence of a remarkable factorization of the Birkhoff normal forms of the
Hamiltonian of the problem represented with the Levi-Civita regularization. The
parametrizations are computed using two different sequences of Birkhoff
normalizations of given order N. For each value of N, the Birkhoff
normalizations and the parameters of the focus-focus dynamics are represented
by polynomials whose coefficients can be computed iteratively with a computer
algebra system; no quadratures, such as those needed to compute action-angle
variables of resonant normal forms, are needed. We also provide some numerical
demonstrations of the method for values of the mass parameter representative of
the Sun-Earth and the Sun-Jupiter cases.
