Ejection-collision orbits in the restricted three-body problem

• Autores: Óscar Rodríguez del Río
• Directores de la Tesis: Mercè Ollé Torner (dir. tes.) , Jaume Soler Villanueva (codir. tes.)
• Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2021
• Idioma: español
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• Resumen

  • The main objective of this dissertation is the study of the ejection-collision (EC) orbits in the circular and planar Restricted Three Body Problem (RTBP from now on). In particular, we will focus on the analytical and numerical study of a very specific type of EC orbits, that we denote as n-EC orbits. An n-EC orbit is an orbit such that the particle ejects from one primary and reaches n times a relative maximum in the distance with respect to the primary from which it ejected before colliding with it. In this way, we will study numerically in depth this kind of orbits and we will show analytically that for a sufficiently large value of the Jacobi constant (for which we will give an expression in terms of the mass parameter and the value of n) there exist exactly four n-EC orbits with well-defined characteristics. These results generalize and improve the previous results for the particular case of n=1, and we will see that they can be easily extrapolated to the Hill problem. Besides, we will observe numerically that the evolution of these four original families of n-EC orbits present a very rich dynamics.It is well-known that the system that defines the motion of the particle is not well defined at the points where the primaries are located. For this reason, we have used two different techniques to regularize the collision, the McGehee regularization and the Levi-Civita regularization. Thus, in this dissertation we have analyzed the advantages and disadvantages of each regularization and the different methods that can be used to detect collisions. Since this dissertation will be mainly focused on values of the Jacobi constant greater than those associated to the equilibrium point L1, these two local regularizations will be enough. For less restrictive values of the Jacobi constant we will see that there exist other global regularizations or alternatively, we can simply work with local regularizations in a neighbourhood of each primary.On the other hand, from the numerical point of view we have analyzed the global behaviour of the ejection orbits in the RTBP. We have studied the relation between the family of Lyapunov periodic orbits around the equilibrium point L1 and the ejection orbits for values of the Jacobi constant such that the associated Hill regions only allow a bounded motion for these orbits. In particular, we have seen that a chaotic infinity of heteroclinic connections between one primary and the Lyapunov periodic orbits around the equilibrium point L1 are obtained. As a consequence a chaotic infinity of ejection-collision orbits is also derived. Besides, we will see that we can construct colour diagrams that allow to describe the global dynamics of the ejection orbits given a range of time. These colour diagrams provide a very precise understanding of the dynamics of these orbits.Finally, we have made a first exploration of the spatial case of the circular restricted three body problem (RTBP 3D). In this first approach we have not used the classical Kustaanheimo–Stiefel regularization, instead we have decided to use a 3D version of the McGehee regularization. This presents some problems that we have analyzed and addressed,