The numerical detection of resonances using dynamical indicators is an efficient tool for eval- uating the long-term stability and for discriminating between order and chaos. In this regard, methods based on the analysis of vectors deviating from an initial reference orbit, stemming from the pioneering work of Lyapunov, have been widely employed in the past decades. Especially High Earth Orbits (HEO), governed by the interplay of multiple perturbations accumulated over long time spans, show an orbital evolution that can result in a strong sensitivity to the initial condi- tions. Using an orbital model defined in terms of Milankovitch vectorial elements, stability maps are developed to target natural disposal trajectories in the osculating space. However, these maps have to be initialized to account the short-periodic terms that may have relevant effects on the re-entry solutions. Using the direct approach of Kozai, we derive the analytical short-period cor- rection for J2 and design a numerical algorithm based on the Fast Fourier Transform for validation purposes. Indeed, for HEO orbits, the lunisolar effect becomes relevant and the short-period correc- tions must be included for an accurate satellite dynamics. Therefore, we extend both the analytical and numerical developments to include the lunisolar perturbations as well, and to obtain a first- order, singularity-free, full-perturbative transformation. The proposed methodology has important applications for satellite lifetime prediction and for passive space debris removal.
Evolution of Chaos in Long-term Propagation of High Earth Orbits
The numerical detection of resonances using dynamical indicators is an efficient tool for eval- uating the long-term stability and for discriminating between order and chaos. In this regard, methods based on the analysis of vectors deviating from an initial reference orbit, stemming from the pioneering work of Lyapunov, have been widely employed in the past decades. Especially High Earth Orbits (HEO), governed by the interplay of multiple perturbations accumulated over long time spans, show an orbital evolution that can result in a strong sensitivity to the initial condi- tions. Using an orbital model defined in terms of Milankovitch vectorial elements, stability maps are developed to target natural disposal trajectories in the osculating space. However, these maps have to be initialized to account the short-periodic terms that may have relevant effects on the re-entry solutions. Using the direct approach of Kozai, we derive the analytical short-period cor- rection for J2 and design a numerical algorithm based on the Fast Fourier Transform for validation purposes. Indeed, for HEO orbits, the lunisolar effect becomes relevant and the short-period correc- tions must be included for an accurate satellite dynamics. Therefore, we extend both the analytical and numerical developments to include the lunisolar perturbations as well, and to obtain a first- order, singularity-free, full-perturbative transformation. The proposed methodology has important applications for satellite lifetime prediction and for passive space debris removal.
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