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

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## Abstract

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.

## Keywords

Medium-Earth orbits Secular dynamics Orbital resonances Chaos Fast Lyapunov indicators (FLI) Stability maps Lunisolar resonances GNSS## Notes

### Acknowledgments

The present form of the manuscript owes much to the critical comments and helpful suggestions of many colleagues and friends. The authors are grateful to the two anonymous referees for their rapid, yet careful and incisive reviews. J.D. would like to thank M. Fouchard for discussions on the FLI computations, E. Bignon, P. Mercier, and R. Pinède for support with the Stela software, as well as the “Calcul Intensif” team from CNES, where numerical simulations were hosted. A.J.R. owes a special thanks to K. Tsiganis for hosting him at the Aristotle University of Thessaloniki in March, and for the numerous insightful conversations that ensued. A.J.R. would also like to thank N. Todorović, of the Belgrade Astronomical Observatory, and F. Gachet and I. Gkolias, of the University of Rome II, for discussions on the phase-angle dependencies of the FLI maps. Discussions with A. Bäcker, A. Celletti, R. de la Llave, G. Haller, and J.D. Meiss at the Global Dynamics in Hamiltonian Systems conference in Santuari de Núria, Girona, 28 June – 4 July 2015, have been instrumental in shaping the analytical component of this work. This research is partially funded by the European Commissions Framework Programme 7, through the Stardust Marie Curie Initial Training Network, FP7-PEOPLE-2012-ITN, Grant Agreement 317185. Part of this work was performed in the framework of the ESA Contract No. 4000107201/12/F/MOS “Disposal Strategies Analysis for MEO Orbits”.

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