Ph.D Thesis

Ph.D StudentIzzo Paolo
SubjectEvolution of Chaos in Long-Term Propagation of High
Earth Orbits
DepartmentDepartment of Aerospace Engineering
Supervisors PROF. Pinchas Gurfil
ASSOCIATE PROF. Christian Circi
DR. Aaron Rosengren
Full Thesis textFull thesis text - English Version


The study of reciprocity and complementarity between order and chaos in deterministic dynamical systems is a crucial aspect in many scientific fields, and particularly in the domains of astrodynamics and celestial mechanics. The identification of the unstable portions of phase space will shed light on the behaviour of such systems, giving a portrait of the physical significance of the resonance spectrum. In this context, the numerical detection of resonances using dynamical indicators is an efficient tool for evaluating the long-term stability and for discriminating between order and chaos. In the past decades, methods based on the analysis of vectors deviating from an initial reference orbit have been widely employed. They mostly belong to the large family of methods stemming from the pioneering work of Lyapunov. However, thus far, chaos indicators for modeling, analysis and long-term propagation, have not been systematically applied to the study of High Earth-Orbits (HEO). These orbits include, among others, geostationary orbits and geostationary-transfer orbits, hosting both active satellites and space debris, subjected to the simultaneous action of various concurrent forces. In fact, HEO are governed by the delicate interplay of multiple perturbations accumulated over long time spans, resulting in local and global instabilities, and in a strong sensitivity to initial conditions and model parameters.

The main goal of the proposed research is to create a complete cartography of the HEO dynamical environment, while utilizing singularity-free, high-fidelity orbital modelling. We propose to formulate the orbital dynamics using the Milankovitch formalism, resulting in a vectorial orbital propagator, able to characterize, through averaging processes, the long-term orbital dynamics of HEO under natural perturbations. Thus, cartographic stability maps, uncovering orbital resonances, will be obtained through averaging methods; this is essential from the computational efficiency standpoint. At the same time, from an operational perspective,  it must be specified where in the osculating space one should actually target to place the satellite. Therefore, these maps will be then upgraded to account for the short-periodic terms by the use of a proper mean-to-osculating transformation.

To investigate these considerations, we present herein an analytical short-period correction for the dominant Earth-oblateness perturbation and compare our results to the classical Brouwer-Lyddane transformation. We further develop the short-period corrections for the lunisolar perturbations. Finally, for validation purposes, we will compare these analytical solutions to a numerical averaging approach based on the fast-Fourier transform. In the larger context of passive debris removal using resonances and instabilities, detailed and extensive maps have been produced based on semi-analytical methods. Yet, with an appropriate mean-to-osculating transformation, we have generated a more reliable detection of the phase-space portions suitable for satellite disposal.

The proposed methodology results in noticeable applications, ranging from the design of spacecraft disposal strategies to the prediction of space debris distribution and decay.