Elisa Maria Alessi

Chaos in navigation satellite orbits caused by the perturbed motion of the Moon

ABSTRACT Numerical simulations carried out over the past decade suggest that the orbits of the GlobalNavigation Satellite Systems are unstable, resulting in an apparent chaotic growth of the ec-centricity. Here we show that the irregular and haphazard character of these orbits reflects asimilar irregularity in the orbits of many celestial bodies in our Solar System. We find thatsecular resonances, involving linear combinations of the frequencies of nodal and apsidal pre-cession and the rate of regression of lunar nodes, occur in profusion so that the phase space isthreaded by a devious stochastic web. As in all cases in the Solar System, chaos ensues whereresonances overlap. These results may be significant for the analysis of disposal strategies forthe four constellations in this precarious region of space.Keywords: celestial mechanics– chaos– methods:analytical–methods :numerical– planetsand satellites: dynamical evolution and stability — planet s and satellites: general. 1 INTRODUCTIONSpace debris—remnants of past missions, satellite explosi ons, andcollisions—is a phenomenon that has existed since the begin ning ofthe space age; however, its significance for space activitie s, in par-ticular the increasing impact risks posed to space systems, has beenrealised only in the past few decades (Kessler & Cour-Palais1978;Rossi et al. 1999; Liou & Johnson 2006). Theproliferation of spacedebris has motivated deeper and more fundamental analysis ofthe long-term evolution of orbits about Earth (Breiter 2001b,a;Celletti & Gales¸ 2014). Orbital resonances are widespread withinthis system as a whole (Hughes 1980), but particularly so amongstthe medium-Earth orbits (MEOs) of the navigation satellites in theregion of semimajor axes between 4 and 5 Earth radii, and a clearpicture of their nature isof great importance inassessing debris mit-igation measures (Alessi et al. 2014). Indeed, the discovery that therecommended graveyard orbits of these satellites, located severalhundred kilometres above the operational constellations, are poten-tiallyunstable has led toa new paradigm in post-mission disposal—one that seeks to cleverly exploit these dynamical instabilitiesand the associated eccentricity growth for re-entry and destructionwithin the Earth’s atmosphere (Jenkin & Gick 2002; Chao & Gick2004; Rossi 2008; Deleflie et al. 2011). Previous studies hav e al-ready noted theconnection between theorigin of the long-timescaleinstabilities in the MEO region and a resonance phenomenon in-volving Earth oblateness and lunisolar perturbations, yet very littleattention has been given to a true physical explanation of the erratic

The dynamical structure of the MEO region: long-term stability, chaos, and transport

It has long been suspected that the Global Navigation Satellite Systems exist in a background of complex resonances and chaotic motion; yet, the precise dynamical character of these phenomena remains elusive. Recent studies have shown that the occurrence and nature of the resonances driving these dynamics depend chiefly on the frequencies of nodal and apsidal precession and the rate of regression of the Moon’s nodes. Woven throughout the inclination and eccentricity phase space is an exceedingly complicated web-like structure of lunisolar secular resonances, which become particularly dense near the inclinations of the navigation satellite orbits. A clear picture of the physical significance of these resonances is of considerable practical interest for the design of disposal strategies for the four constellations. Here we present analytical and semi-analytical models that accurately reflect the true nature of the resonant interactions, and trace the topological organization of the manifolds on which the chaotic motions take place. We present an atlas of FLI stability maps, showing the extent of the chaotic regions of the phase space, computed through a hierarchy of more realistic, and more complicated, models, and compare the chaotic zones in these charts with the analytical estimation of the width of the chaotic layers from the heuristic Chirikov resonance-overlap criterion. As the semi-major axis of the satellite is receding, we observe a transition from stable Nekhoroshev-like structures at three Earth radii, where regular orbits dominate, to a Chirikov regime where resonances overlap at five Earth radii. From a numerical estimation of the Lyapunov times, we find that many of the inclined, nearly circular orbits of the navigation satellites are strongly chaotic and that their dynamics are unpredictable on decadal timescales.

Desaturation manoeuvres and precise orbit determination for the BepiColombo mission

This work analyses the consequences that the desaturation manoeuvres can have on the precise orbit determination corresponding to the Mercury Orbiter Radioscience Experiment (MORE) of the BepiColombo mission to Mercury. This is an ESA/JAXAjoint project with challenging objectives regarding geodesy, geophysics and fundamental physics. We will show how these manoeuvres affect the orbit of the s/c and the radio science measurements and how to include them in the orbit determination and parameter estimation procedure. The non-linear least-squares fit is applied on a set of observational arcs separated by intervals of time where the probe is not visible. With the current baseline of two ground stations, two manoeuvres are performed per day, one during the observing session and the other in the dark. To reach the scientific goals of the mission, they have to be treated as ‘solve for quantities’. We developed a specific methodology based on the deterministic propagation of the orbit, which is able to deal with these variables, by connecting subsequent observational arcs in a smooth way. The numerical simulations demonstrate that this constrained multi-arc strategy is able to determine all the manoeuvres together with the other parameters of interest at a high level of accuracy.

Galileo disposal strategy: stability, chaos and predictability

Recent studies have shown that the medium-Earth orbit (MEO) region of the Global Navigation Satellite Systems is permeated by a devious network of lunisolar secular resonances, which can interact to produce chaotic and diffusive motions. The precarious state of the four navigation constellations, perched on the threshold of instability, makes it understandable why all past efforts to define stable graveyard orbits, especially in the case of Galileo, were bound to fail; the region is far too complex to allow of an adoption of the simple geosynchronous disposal strategy. We retrace one such recent attempt, funded by ESAs General Studies Programme in the frame of the GreenOPS initiative, that uses a systematic parametric approach and the straightforward maximum-eccentricity method to identify long-term stable regions, suitable for graveyards, as well as large-scale excursions in eccentricity, which can be used for post-mission deorbiting of constellation satellites. We then apply our new results on the stunningly rich dynamical structure of the MEO region toward the analysis of these disposal strategies for Galileo, and discuss the practical implications of resonances and chaos in this regime. We outline how the identification of the hyperbolic and elliptic fixed points of the resonances near Galileo can lead to explicit criteria for defining optimal disposal strategies.

Solar radiation pressure resonances in Low Earth Orbits

The aim of this work is to highlight the crucial role that orbital resonances associated with solar radiation pressure can have in Low Earth Orbit. We review the corresponding literature, and provide an analytical tool to estimate the maximum eccentricity which can be achieved for well-defined initial conditions. We then compare the results obtained with the simplified model with the results obtained with a more comprehensive dynamical model. The analysis has important implications both from a theoretical point of view, because it shows that the role of some resonances was underestimated in the past, but also from a practical point of view in the perspective of passive deorbiting solutions for satellites at the end-of-life.

Natural highways for end-of-life solutions in the LEO region

We present the main findings of a dynamical mapping performed in the Low Earth Orbit region. The results were obtained by propagating an extended grid of initial conditions, considering two different epochs and area-to-mass ratios, by means of a singly averaged numerical propagator. It turns out that dynamical resonances associated with high-degree geopotential harmonics, lunisolar perturbations and Solar radiation pressure can open natural deorbiting highways. For area-to-mass ratios typical of the orbiting intact objects, these corridors can be exploited only in combination with the action exerted by the atmospheric drag. For satellites equipped with an area augmentation device, we show the boundary of application of the drag, and where the Solar radiation pressure can be exploited.

Ranking in-orbit fragmentations and space objects

The future space debris environment will be dominated by the production of fragments coming from massive fragmentations. In order to identify the most relevant parameters influencing the long term evolution of the environment and to assess the criticality of selected space objects in different regions of the circumterrestrial space, a large parametric study was performed. In this framework some indicators were produced to quantify and rank the relevance of selected fragmentations on the long term evolution of the space debris population. Based on the results of the fragmentation studies, a novel analytic index, the Criticality of Spacecraft Index, aimed at ranking the environmental criticality of abandoned objects in LEO, has been devised and tested on a sample population of orbiting objects.