Title:

Natural and perturbed dynamics about Trojan bodies

This thesis analyses the dynamics of a massless spacecraft (or point mass) around an inhomogeneous Trojan body in a system composed of three primaries lying on a plane at the vertexes of an equilateral triangle, with their mutual positions fixed over the course of the motion. This configuration will here be referred to as "Lagrangian" or "equilateral triangle", implicitly meaning that the three primaries lie on a common plane. To this end two suitable models are identified to represent the system, depending on the distance from the primary. The first model, adopted for use close to the asteroid, where the dynamics is dominated by this sole body, is the Restricted Two Body Problem. In this model the inhomogeneities of the asteroid are taken into account as they have a dominant effect on the dynamics of the spacecraft. This model will therefore be referred to as the Inhomogeneous R2BP. The second model is the Lagrangian Circular Restricted Four Body Problem (CR4BP), with the primaries lying on the same plane, which is adopted far from the asteroid1, where the gravitational perturbations of the Sun and Jupiter are dominant while the inhomogeneities of the asteroid are negligible. Lowthrust propulsion perturbations are incorporated into this model. The possibility to determine the range of validity of each model using an application of a Weak Stability Boundary (WSB) theory, a relatively novel approach to dynamics for designing lowenergy transfers, is investigated and applied. A completely new, analytical definition of the Weak Stability Boundary, coherent with the previous algorithmic definitions, is thus developed in this work for the first time. The existing (algorithmic) WSB theory, previously always treated numerically and mainly applied to Circular Restricted Three Body Problems (CR3BP), is here rebuilt from an analytical point of view and extended to the Lagrangian CR4BP. Moreover some topological properties of the WSB are introduced and to analytical Moreover some topological properties of the WSB are introduced and applied, leading to analytical estimations of the set of stable orbits around the small primary. An estimation of the range of validity of the models is thus derived, which is based on the region of stable orbits. The dynamics in a Restricted Two Body Problem incorporating the shape/density inhomogeneities of the body, is analysed, suitable for modelling the spacecraft dynamics inside the estimated reference region. The irregular gravitational potential is formulated using spherical harmonics, the coefficients describing the physical properties of the body. An analytical, arbitrary degree, perturbation theory, assuming the spherical harmonics of the body as known, is derived. This result generalizes to arbitrary degree the previous closed form (i.e. valid for every eccentricity) perturbation theories which are usually limited to second degree (namely to the inclusion of two spherical harmonics coefficients). The theory here developed, double averaging the system by means of two canonic Lie transformations, leads to an integrable, arbitrarily accurate approximation of the system whose explicit second order Hamiltonian formulation, derived in closed form, is thus stated. From this theory an analytic method for determining initial conditions for frozen orbits around any irregular body is derived for the first time. Such Frozen orbits are orbits with no secular perturbations in the inclination, argument of pericentre, and eccentricity. Results are shown for a major Jupiter Trojan: 624Hektor. As the spherical harmonics of this Trojan are unknown and not present in any previous literature, a method is here applied, which deduces these coefficients from a three dimensional polyhedric model of the body, assuming a constant density. Finally the dynamics of the Lagrangian CR4PB is studied, for modelling the system outside the estimated reference boundary. The natural equilibria and Lyapunov stability of the linearized system are analysed. A study of the changes in the topology of the linearly stable zone for different conceivable masses of the Trojan is shown in this work for the first time. Lowthrust propulsion perturbations, in all previous literature confined to two and three body problems, are here incorporated into the four bodies system examined, enabling the generation of surfaces of artificial equilibria. Applications are shown for the main example of Lagrangian configuration in the Solar system, the SunJupiterTrojanspacecraft system. Numerical simulations for 624Hektor confirm the validity of the model once its real tadpole orbit around the triangular point is taken into account.
