First Order Analytical Solution for Distant Retrograde Orbits in the Circular Restricted Three-Body Problem

The work carried out in this thesis uses the Circular Restricted Three Body Problem(CR3BP model). The CR3BP attracted the attention of many researchers and mathematicianssince it was considered for the first time three centuries ago. The mainreason for this continuous interest is that this mathematical model represents a goodfirst approximation in a series of real scenarios in orbital mechanics. In the frameworkof the planar CR3BP this thesis focuses on the study of distant retrograde orbits andin particular aims at finding an approximate analytical solution for this type of orbit.The periodic distant retrograde orbit derives from the work of H´enon who developed asystematic theory for the study of periodic orbits in the CR3BP. The work of H´enon islimited to the problem of Hill, which is a special case of the CR3BP in which the massparameter of the system tends to zero. H´enon calculated periodic and non-periodicorbits; in the present work, we limit our attention only to periodic orbits. In particular,only the family f is taken into consideration and analysed. Starting from validinitial conditions given by H´enon for the model of Hill, we reproduced the evolutionof these families in the CR3BP (that is, where the gravitational parameter is a finiteand constant value). A differential correction method, together with a continuationmethod, has been exploited to find the initial conditions for each orbit in the CR3BPobtaining the complete continuation of these periodic orbits. The most interestingfeature of the DROs is the great distance they reach from the secondary attractor,remaining, however, to orbit around it. The problem was written with a Lagrangianapproach and then, using the variational principles, in the Hamiltonian form. This latterform is convenient because it allows to reduce the problem to a series of differentialequations of the first order. Before applying the perturbation theory, a Taylor seriesexpansion, respect mass ratio, on CR3BP was done in order to obtain a model that isa intermediate step between that of Hill and the CR3BP that we will call CR3BP-1.Since this latter model is still complex, the less relevant terms, in mass, have been neglected,this model will be called CR3BP-113. Subsequently, the classical perturbationtheory was applied, applying a canonical transformation to the non-perturbed part ofHamiltonian (quadratic form of the Hamiltonian) and subsequently applying the Lietransform. This leads to a series of differential equations of more immediate analyticalsolution. This procedure was created for the Hill model and then for the CR3BP-113.These solution was then compared with the numerical solution for DROs, both interms of maximum error and in terms of computational speed, resulting very reliableand computationally fast compared to the numerical model. However, the solutionfound does not have a noticeable improvement over that of Hill unless we take verylow mass relations into consideration. This is due to the fact that the potential of the primary is considered, in part, as a disturbance and the closer we get to the primary,the less this assumption is valid.