Dynamics of a satellite orbiting a planet with an inhomogeneous gravitational field |
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Authors: | J F Palacián |
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Institution: | (1) Departamento de Ingeniería Matemática e Informática, Universidad Pública de Navarra, 31006 Pamplona, Spain |
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Abstract: | We study the dynamics of a satellite (artificial or natural) orbiting an Earth-like planet at low altitude from an analytical
point of view. The perturbation considered takes into account the gravity attraction of the planet and in particular it is
caused by its inhomogeneous potential. We begin by truncating the equations of motion at second order, that is, incorporating
the zonal and the tesseral harmonics up to order two. The system is formulated as an autonomous Hamiltonian and has three
degrees of freedom. After three successive Lie transformations, the system is normalised with respect to two angular co-ordinates
up to order five in a suitable small parameter given by the quotient between the angular velocity of the planet and the mean
motion of the satellite. Our treatment is free of power expansions of the eccentricity and of truncated Fourier series in
the anomalies. Once these transformations are performed, the truncated Hamiltonian defines a system of one degree of freedom
which is rewritten as a function of two variables which generate a phase space which takes into account all of the symmetries
of the problem. Next an analysis of the system is achieved obtaining up to six relative equilibria and three types of bifurcations.
The connection with the original system is established concluding the existence of various families of invariant 3-tori of
it, as well as quasiperiodic and periodic trajectories. This is achieved by using KAM theory techniques. |
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Keywords: | Satellite dynamics Zonal and tesseral harmonics Delaunay normalisation Reduction and invariant theories Bifurcation lines Non-linear stability KAM theory Invariant tori Quasiperiodic and periodic orbits |
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