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Chaos in Navigation Satellite Orbits Caused by the Perturbed Motion Of Mon. Not. R. Astron. Soc. 000, 1–6 (2015) Printed 10 March 2015 (MN LATEX style file v2.2) Chaos in navigation satellite orbits caused by the perturbed motion of the Moon Aaron J. Rosengren,1⋆ Elisa Maria Alessi,1 Alessandro Rossi1 and Giovanni B. Valsecchi1,2 1IFAC-CNR, Via Madonna del Piano 10, 50019 Sesto Fiorentino (FI), Italy 2IAPS-INAF, Via Fosso del Cavaliere 100, 00133 Roma, Italy ABSTRACT Numerical simulations carried out over the past decade suggest that the orbits of the Global Navigation 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 a similar irregularity in the orbits of many celestial bodies in our Solar System. We find that secular 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 is threaded by a devious stochastic web. As in all cases in the Solar System, chaos ensues where resonances overlap. These results may be significant for the analysis of disposal strategies for the four constellations in this precarious region of space. Key words: celestial mechanics – chaos – methods: analytical – methods: numerical – planets and satellites: dynamical evolution and stability — planets and satellites: general. 1 INTRODUCTION behaviour. To identify the source of orbital instability, we inves- tigated the main resonant structures that organise and govern the Space debris—remnants of past missions, satellite explosions, and long-term orbital motion of navigation satellites. This paper clari- collisions—is a phenomenon that has existed since the beginning of fies the fundamental role played by secular resonances and reveals the space age; however, its significance for space activities, in par- the significance of the perturbed motion of the Moon. ticular the increasing impact risks posed to space systems, has been realised only in the past few decades (Kessler & Cour-Palais 1978; Rossi et al. 1999; Liou & Johnson 2006). The proliferation of space debris has motivated deeper and more fundamental analysis of 2 ANALYSIS arXiv:1503.02581v1 [astro-ph.EP] 9 Mar 2015 the long-term evolution of orbits about Earth (Breiter 2001b,a; Celletti & Gales¸2014). Orbital resonances are widespread within Orbital resonances are ubiquitous in the Solar System and this system as a whole (Hughes 1980), but particularly so amongst are harbingers for the onset of dynamical instability and the medium-Earth orbits (MEOs) of the navigation satellites in the chaos (Duncan et al. 1995; Lecar et al. 2001; Morbidelli 2002; region of semimajor axes between 4 and 5 Earth radii, and a clear Colwell et al. 2009; Tsiganis 2010; Asphaug 2014; Lithwick & Wu picture of their nature is of great importance in assessing debris mit- 2014). Our knowledge of such phenomena in celestial mechanics igation measures (Alessi et al. 2014). Indeed, the discovery that the comes mainly from studies on asteroid dynamics—attempts to de- recommended graveyard orbits of these satellites, located several scribe the stunningly complex dynamical structure of the asteroid hundred kilometres above the operational constellations, are poten- belt since the discovery of the Kirkwood gaps (Morbidelli 2002; tially unstable has led to a new paradigm in post-mission disposal— Tsiganis 2010). The orbital structure of the main belt is divided one that seeks to cleverly exploit these dynamical instabilities by unstable regions, narrow gaps in the distribution of the aster- and the associated eccentricity growth for re-entry and destruction oidal semimajor axes at 2.5 and 2.8 astronomical units (AU) from within the Earth’s atmosphere (Jenkin & Gick 2002; Chao & Gick the Sun, where the orbital periods (or mean motions) of the as- 2004; Rossi 2008; Deleflie et al. 2011). Previous studies have al- teroids are commensurable to that of Jupiter. The outer boundary ready noted the connection between the origin of the long-timescale is marked by the Jovian 2:1 mean-motion resonance at 3.3 AU. instabilities in the MEO region and a resonance phenomenon in- Secular resonances, which involve commensurabilities amongst the volving Earth oblateness and lunisolar perturbations, yet very little slow frequencies of orbital precession of an asteroid and the plan- attention has been given to a true physical explanation of the erratic ets, define the inner edge of the main belt near 2.1 AU and demar- cate the recognised subpopulations (asteroid families and groups). Secular resonances also occur embedded within the libration zones ⋆ E-mail: [email protected] of the prominent Jovian mean-motion resonances, and it is pre- 2 A. J. Rosengren et al. cisely the coupling of these two main types of resonance phenom- Rossi 2008; Celletti & Gales¸2014), and such resonance problems ena that generates widespread chaos (Lecar et al. 2001; Morbidelli have stood in the foremost rank of astrodynamical research work 2002; Tsiganis 2010). The creation of the Kirkwood gaps is thereby (Breiter 2001a). Undoubtedly, the most celebrated resonance phe- associated with a chaotic growth of the asteroids’ eccentricities nomenon in artificial satellite theory is the critical inclination prob- to planet-crossing orbits, which leads to the removal from these lem, which involves a commensurability between the two degrees resonant regions by collisions or close encounters with the ter- of freedom (apsidal and nodal motions) of the secular system restrial planets. The same physical mechanism that sculpted the (Hough 1981; Delhaise & Morbidelli 1993; Tremaine & Yavetz asteroid belt—the overlapping of resonances—is responsible for 2014). The extreme proximity to these critical inclinations, espe- the emergence of chaos in the Solar System on all astronomical cially in the case of the European Galileo and Russian GLONASS scales, from the dynamics of the trans-Neptunian Kuiper belt and constellations (Rossi 2008; Alessi et al. 2014), induced researchers, scattered disk populations of small bodies (Duncan et al. 1995) to very early, to associate the origin of the instabilities observed the apparent perfectly regulated clockwork motion of the planets in numerical surveys to these resonances (Jenkin & Gick 2002; (Lithwick & Wu 2014). Chao & Gick 2004). However, these inclination-dependent-only With the advent of artificial satellites and creation of space resonances are generally isolated (Cook 1962; Hughes 1980) and debris, similar dynamical situations, every bit as varied and rich thereby exhibit a well-behaved, pendulum-like motion, which, as those in the asteroid belt, can occur in closer proximity to our when considered alone, cannot explain the noted chaotic be- terrestrial abode (Upton et al. 1959; Hughes 1980; Breiter 2001a). haviours. The second class of geopotential resonances arises from The dynamical environment occupied by these artificial celestial the slight longitude-dependence of the geopotential, where the bodies is subject to motions that are widely separated in frequency: satellite’s mean motion is commensurable with the rotation of the the earthly day, the lunar month, the solar year, and various preces- Earth. Under such conditions, the longitudinal forces due to the sion frequencies ranging from a few years to nearly 26,000 years tesseral harmonics continually perturb the orbit in the same sense for the equinoxes. A vast and hardly surveyable profusion of per- and produce long-term changes in the semimajor axis and the mean turbations originates from the gravitational action of the Sun on the motion. For the inclined, slightly eccentric orbits of the naviga- Earth-Moon system. Primary among these irregularities is the re- tion satellites, where these commensurabilities abound, the tesseral gression the Moon’s line of nodes with a period of about 18.61 harmonics exhibit a multiplet structure, akin to mean-motion reso- years, and a progression of the lunar apsidal line with a period nances, whereby the interaction of resonant harmonics can overlap of roughly 8.85 years. The provision of frequencies in the Earth- and produce chaotic motions of the semimajor axis (Ely & Howell Moon-Sun system gives rise to a diverse range of complex reso- 1997; Ely 2002). These effects, however, are localised to a nar- nant phenomena associated with orbital motions (Hughes 1980). row range of semimajor axis (tens of kilometres) and are of Indeed, past efforts to identify and classify the resonances that gen- much shorter period than secular precession (Ely & Howell 1997; erate the orbital instability in the MEO region have been wholly ob- Celletti & Gales¸2014); consequently, tesseral resonances will not scured by the abundance of frequencies in the orbital and rotational significantly affect the long-term orbit evolution over timespans of motions of the Earth and of the Moon, and the great disparity of interest. timescales involved (Deleflie et al. 2011; Bordovitsyna et al. 2014; Celletti & Gales¸2014). While the orbits of most artificial satellites are too low to be 3 RESULTS AND DISCUSSION affected by mean-motion commensurabilities, even with the Earth’s moon (Dichmann et al. 2014), there exists a possibility of a some- These considerations have led us to investigate the role of lu- what more exotic resonance involving a commensurability of a sec- nar secular resonances in producing chaos and instability among ular precession frequency with the mean motion of the Sun or the the navigation satellites. We treat only the secular resonances of Moon (Cook 1962; Hough 1981; Breiter 2001b). Among the more the lunar origin, since, despite recent implications to the contrary compelling of these semi-secular commensurabilities is the evec- (Bordovitsyna et al. 2014), the solar counterparts will require much tion resonance with the Sun, where the satellite’s apsidal preces- longer timespans than what interests us here for their destabilising sion rate produced by Earth’s oblateness equals the Sun’s apparent effects to manifest themselves (Ely 2002)—a consequence of the mean motion, so that the line of apsides follows the Sun.
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