TITLE
Consecutive quasi-collisions in the planar circular RTBP}
AUTHORS
Joaquim Font, Ana Nunes and Carles Sim\'o
J.F. Departament de Matem\`atica Aplicada i An\`alisi
Universitat de Barcelona
Gran Via 585, 08007 Barcelona
e-mail:quim@maia.ub.es
A.N. MAFUL/DFFCUL
Universidade de Lisboa
Av. Prof. Gama Pinto 2, 1619-003 Lisboa
e-mail:anunes@lmc.fc.ul.pt
C.S. Departament de Matem\`atica Aplicada i An\`alisi
Universitat de Barcelona
Gran Via 585, 08007 Barcelona
e-mail:carles@maia.ub.es
ABSTRACT:
In this paper we consider the planar
circular restricted three body problem and, in particular,
the existence of orbits which undergo consecutive close encounters with
the small primary. The number of revolutions of the small bodies around the
larger one between successive encounter can be chosen to be two arbitrary
sequences of natural number, with constraints depending on the Jacobi constant.
We prove that such orbits exist as a consequence of
the fact that, when the mass parameter $\mu $ is small, the first return map
defined on a region of phase space whose projection is a circle around the
small primary is a 'horseshoe' map. The proof is constructive, in the sense
that it is based on the computation of an approximate expression for this
return map. When $\mu $ is small, the approximate return map contains the
essential information about the dynamics from the quantitative as well as
from the qualitative point of view. Using this information, we have been
able to carry out a numerical study of this problem for $\mu $ up to $10^{-3}$.