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