Domains of Practical Stability near $L_{4,5}$ in the 3D Restricted Three-Body Problem Carles Sim\'o$^{(1),*}$, Priscilla Sousa-Silva$^{(1)}$ and Maisa Terra$^{(2)}$} $^{(1)}$ Departament de Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona, Barcelona, Catalunya, Spain\\ $^{(2)}$ Departamento de Matem\'atica, Instituto Tecnol\'ogico de Aeron\'autica, S\~ao Jos\'e dos Campos, SP, Brazil} carles@maia.ub.es, priandss@maia.ub.es, maisa@ita.b Abstract It is well-known that the triangular relative equilibrium solutions (r.e.s.) of the 3D RTBP are linearly stable in a range of the mass parameter $\mu\in(0, \mu_1)$. Nonlinear stability is found also if $\mu$ is different from the exceptional values $\mu_2,\mu_3$, for a set of almost full measure on a small vicinity of the r.e.s. \cite{2}. Furthermore, as shown in \cite{1}, normal forms give rise to Nekhorosev-like estimates of diffusion: the possible escape is extremely slow, producing a \emph{practical stability}. It is natural to formulate the question: up to which distance escape is slow, and why crossing some \emph{quasi-boundaries} it becomes relatively fast. Some preliminary hints about the answer to the question can be found in \cite{3} and \cite{4}. The goal of this presentation is to describe some recent progress \cite{5} which clearly shows the role that several codimension-one manifolds play in the problem. Some of these manifolds could be expected: They are the stable and unstable manifolds of the centre manifold around the collinear libration point $L_3$. But other codimension-one manifolds have been found to play a key role, specially for orbits which reach large values of $z$. At least for small values of $\mu$ these manifolds are the stable and unstable manifolds of the centre manifolds associated to some families of periodic orbits of elliptic-hyperbolic type. The study is done by considering simultaneously all the levels of the Jacobi constant. The methodology, results and some indications about not too small values of $\mu$ will be reported. References {1} Giorgilli, A., Delshams, A., Fontich, E., Galgani, L. and Sim\'o, C., Effective stability for a hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem \emph{Journal of Differential Equations}, \textbf{77} (1989), 167--370. {2} Markeev, A.P., Stability of the triangular lagrangian solutions of the restricted three-body problem in the three-dimensional circular case \emph{Soviet Astronomy}, \textbf{15} (1972), 682--686. {3} Sim\'o, C., Effective computations in celestial mechanics and astrodynamics, in \emph{Modern Methods of Analytical Mechanics and their Applications}, Ed. V. V. Rumyantsev and A. V. Karapetyan, CISM Courses and Lectures \textbf{387}, 55--102, Springer, 1998. {4} Sim\'o, C., Boundaries of Stability, talk given at the U. of Barcelona on June 3, 2006, available at {\tt{http://www.maia.ub.es/dsg/2006/}}. {5} Sim\'o, C., Sousa-Silva, P. and Terra, M., Domains of Practical Stability and Hyperbolic Structures Near the Triangular Lagrangian Points, work in progress.