ETD

• dynamical system
• low energy transfer
• lunar mission design
• phase-space structure
• poincarè map
• quasi-bicircular four body problem
• resonant orbits
• Sun-Earth-Moon system
• temporary lunar capture

This thesis investigates the dynamics of temporary lunar capture in the Sun–Earth–Moon system from the perspective of dynamical systems theory and low-energy trajectory design. The main objective is not the construction of a single optimized transfer, but rather the systematic exploration of families of trajectories that originate from the dynamical region associated with the Sun–Earth libration point and evolve naturally toward the lunar neighbourhood. In this framework, temporary capture is interpreted as a finite-time dynamical phenomenon: a trajectory enters the lunar regime, performs several revolutions around the Moon, and eventually escapes without requiring an imposed lunar insertion manoeuvre.

The study is developed within the Quasi-Bicircular Four-Body Problem, which is adopted as the main dynamical model. This formulation provides a coherent periodic approximation of the planar Sun–Earth–Moon system and allows the interaction between the Sun–Earth and Earth–Moon dynamics to be represented within a single framework. Unlike a purely patched sequence of restricted three-body problems, the QBCP preserves the continuous coupling between the solar and lunar gravitational regimes. This feature is essential for studying trajectories that depart from the Sun–Earth libration region and later interact with the Earth–Moon system. The Circular Restricted Three-Body Problem is also employed, mainly as a local model for the Earth–Moon phase and for the analysis of the temporary lunar capture itself.

A central role is played by the dynamical structures associated with the collinear libration region of the Sun–Earth system. In the periodically perturbed QBCP, the classical equilibrium points of the CR3BP are replaced by periodic solutions known as Dynamical Equivalents to the Libration Points. After computing the dynamical equivalent associated with the Sun–Earth L
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point, the local dynamics is analysed through linearization, Floquet theory, and a suitable change of variables that separates the central, stable, and unstable directions. The Parameterization Method is then used to construct high-order semi-analytical approximations of the center-unstable invariant manifold. This provides a structured set of initial conditions from which trajectories can be generated and propagated toward the Earth–Moon region.

The numerical exploration begins by constructing families of planar Lyapunov-type orbits on the central manifold and then perturbing them along the unstable direction. The resulting initial conditions are sampled with respect to both the phase of the QBCP and the position along the orbit. These conditions are propagated in the QBCP until they potentially reach the lunar sphere of influence. Once the lunar region is reached, the state is transformed into the Earth–Moon rotating frame and analysed in the CR3BP. A temporary capture criterion is introduced by requiring the trajectory to complete at least two revolutions around the Moon before escaping the lunar regime. Because close lunar encounters generate strong numerical sensitivity and possible collision singularities, Levi-Civita regularization is implemented to improve both accuracy and robustness in the propagation of near-collision trajectories.

The exploration reveals that temporary lunar captures are not uniformly distributed among the generated initial conditions. Instead, they appear in organized regions whose structure changes with the size of the initial Lyapunov orbit and with the phase of the four-body system. For smaller initial orbits, capture opportunities are concentrated in narrow phase intervals and are predominantly retrograde. As the initial orbit size increases, new branches of capture appear, the number of captured trajectories grows, and prograde cases become more visible, although they remain less frequent. The transfer phase from the Sun–Earth libration region to the lunar neighbourhood also becomes progressively richer, including both direct approaches and trajectories involving longer excursions within the Earth–Moon region.

To interpret the dynamical origin of these captures, Poincaré maps are constructed and used as the main diagnostic tool. These maps reduce the dimensionality of the problem while preserving the essential geometric organization of the phase space. In particular, the analysis focuses on the distribution of successive section crossings in terms of osculating orbital quantities, such as the semimajor axis and the argument of periapsis in the rotating frame. The maps reveal that many long temporary captures are not caused by random residence in the lunar region, but by the interaction of the trajectory with regular islands, chaotic layers, resonant structures, and sticky regions. Several representative cases show that captured trajectories often remain close to the boundary of stable regions before escaping, indicating that the phenomenon is governed by finite-time stickiness rather than by permanent stability.

The Poincaré analysis also highlights the presence of low-order resonant mechanisms. Some captured trajectories exhibit clear period-three or period-four patterns in the sequence of section crossings, showing that the temporary permanence around the Moon is organized by resonant structures located near the boundary of regular islands. In one case, an apparent anomalous crossing close to the interior of a stable region is interpreted as the effect of a phase shift generated near a transition layer, rather than as a true insertion into the stable island. This emphasizes the importance of interpreting the maps as projections of a higher-dimensional phase space, where apparent proximity in the plotted variables does not necessarily imply dynamical equivalence.

Finally, the thesis discusses the potential relevance of these structures for low-energy mission design. Since some natural trajectories approach the boundary of stable regions in the Poincaré map, small impulsive corrections can be used to move them from a sticky or resonant layer into the interior of a more regular domain. The required impulses are found to be small, because the natural dynamics has already transported the trajectory close to a favourable region of phase space. This suggests that temporary lunar captures may serve not only as objects of dynamical interest, but also as intermediate phases in broader low-energy mission architectures connecting the Sun–Earth libration region, the lunar neighbourhood, and potentially the vicinity of the Earth.

Overall, the work shows that temporary lunar capture in the Sun–Earth–Moon system is a structured dynamical phenomenon governed by invariant manifolds, resonances, chaotic transport, and sticky motion near regular regions. The combination of the QBCP, invariant-manifold generation, Levi-Civita regularization, and Poincaré map analysis provides a coherent methodology for identifying, classifying, and interpreting natural lunar capture opportunities. The results represent a first step toward the systematic use of temporary lunar capture as a functional element in low-energy trajectory design.