# 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”.

## References

- Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics, 3rd edn. Springer-Verlag, Berlin (2006)MATHGoogle Scholar
- Barrio, R., Borczyk, W., Breiter, S.: Spurious structures in chaos indicators maps. Chaos Solitons Fractals
**40**, 1697–1714 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar - Batygin, K., Morbidelli, A., Holman, M.J.: Chaotic disintegration of the inner solar system. Astrophys. J.
**799**, 120–135 (2015)ADSCrossRefGoogle Scholar - Breiter, S.: Lunisolar apsidal resonances at low satellite orbits. Celest. Mech. Dyn. Astron.
**74**, 253–274 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar - Breiter, S.: Lunisolar resonances revisited. Celest. Mech. Dyn. Astron.
**81**, 81–91 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar - Breiter, S.: Fundamental models of resonance. Monografías de la Real Academia de Ciencias de Zaragoza
**22**, 83–92 (2003)MathSciNetMATHGoogle Scholar - Celletti, A., Galeş, C.: On the dynamics of space debris: 1:1 and 2:1 resonances. J. Nonlinear Sci.
**24**, 1231–1262 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar - Celletti, A., Galeş, C., Pucacco, G., Rosengren, A.J.: On the analytical development of the lunar and solar disturbing functions. arXiv:1511.03567 (2015)
- Chirikov, B.V.: A universal instability of many-dimensional oscillator systems. Phys. Rep.
**52**, 263–379 (1979)ADSMathSciNetCrossRefGoogle Scholar - Cook, G.E.: Luni-solar perturbations of the orbit of an Earth satellite. Geophys. J.
**6**, 271–291 (1962)ADSCrossRefMATHGoogle Scholar - Daquin, J., Deleflie, F., Pérez, J.: Comparison of mean and osculating stability in the vicinity of the (2: 1) tesseral resonant surface. Acta Astronaut.
**111**, 170–177 (2015)ADSCrossRefGoogle Scholar - Deleflie, F., Rossi, A., Portmann, C., Métris, G., Barlier, F.: Semi-analytical investigations of the long term evolution of the eccentricity of Galileo and GPS-like orbits. Adv. Space Res.
**47**, 811–821 (2011)ADSCrossRefGoogle Scholar - Delhaise, F., Morbidelli, A.: Luni-solar effects of geosynchronous orbits at the critical inclination. Celest. Mech. Dyn. Astron.
**57**, 155–173 (1993)ADSCrossRefMATHGoogle Scholar - Ely, T.A.: Eccentricity impact on east-west stationkeeping for global position system class orbits. J. Guid. Control Dyn.
**25**, 352–357 (2002)ADSCrossRefGoogle Scholar - Ely, T.A., Howell, K.C.: Dynamics of artificial satellite orbits with tesseral resonances including the effects of luni-solar perturbations. Int. J. Dyn. Stab. Syst.
**12**, 243–269 (1997)MathSciNetCrossRefMATHGoogle Scholar - Froeschlé, C., Gonczi, R., Lega, E.: The fast Lyapunov indicator: a simple tool to detect weak chaos. Application to the structure of the main asteroidal belt. Planet. Space Sci.
**45**, 881–886 (1997)ADSCrossRefGoogle Scholar - Froeschlé, C., Lega, E.: On the structure of symplectic mappings. The fast Lyapunov indicator: a very sensitive tool. Celest. Mech. Dyn. Astron.
**78**, 167–195 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar - Galeş, C.: A cartographic study of the phase space of the elliptic restricted three body problem. Application to the Sun–Jupiter–Asteroid system. Commun. Nonlinear Sci. Numer. Simul.
**17**, 4721–4730 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar - Garfinkel, B.: Formal solution in the problem of small divisors. Astron. J.
**71**, 657–669 (1966)ADSMathSciNetCrossRefGoogle Scholar - Giacaglia, G.E.O.: Lunar perturbations of artificial satellites of the Earth. Celest. Mech.
**9**, 239–267 (1974)ADSMathSciNetCrossRefMATHGoogle Scholar - Guzzo, M., Lega, E., Froeschlé, C.: On the numerical detection of the effective stability of chaotic motions in quasi-integrable systems. Phys. D: Nonlinear Phenom.
**163**, 1–25 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar - Hadjidemetriou, J.D.: A symplectic mapping model as a tool to understand the dynamics of 2/1 resonant asteroid motion. Celest. Mech. Dyn. Astron.
**73**, 65–76 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar - Haller, G.: Chaos Near Resonance. Springer-Verlag, New York (1999)CrossRefMATHGoogle Scholar
- Hughes, S.: Earth satellite orbits with resonant lunisolar perturbations. I. Resonances dependent only on inclination. Proc. R. Soc. Lond. A
**372**, 243–264 (1980)ADSMathSciNetCrossRefGoogle Scholar - Jupp, A.H.: The critical inclination problem: 30 years of progress. Celest. Mech.
**43**, 127–138 (1988)ADSCrossRefMATHGoogle Scholar - Kaula, W.M.: Theory of Satellite Geodesy. Blaisdell, Waltham (1966)MATHGoogle Scholar
- Lane, M.T.: An analytical treatment of resonance effects on satellite orbits. Celest. Mech.
**42**, 3–38 (1988)ADSCrossRefGoogle Scholar - Lane, M.T.: An analytical modeling of lunar perturbations of artificial satellites of the Earth. Celest. Mech. Dyn. Astron.
**46**, 287–305 (1989)ADSCrossRefMATHGoogle Scholar - Lega, E., Guzzo, M., Froeschlé, C.: A numerical study of the hyperbolic manifolds in a priori unstable systems. A comparison with Melnikov approximations. Celest. Mech. Dyn. Astron.
**107**, 115–127 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar - Lithwick, Y., Wu, Y.: Theory of secular chaos and Mercury’s orbit. Astrophys. J.
**739**, 31–47 (2011)ADSCrossRefGoogle Scholar - Mardling, R.A.: Resonances, chaos and stability: the three-body problem in astrophysics. Lect. Notes Phys.
**760**, 59–96 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar - Morand, V.: Semi analytical implementation of tesseral harmonics perturbations for high eccentricity orbits. In: Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, Hilton Head, South Carolina, Paper AAS pp. 13–749 (2013)Google Scholar
- Milani, A., Nobili, A.M.: An example of stable chaos in the solar system. Nature
**6379**, 569–571 (1992)ADSCrossRefGoogle Scholar - Morbidelli, A.: Modern Celestial Mechanics: Aspects of Solar System Dynamics. Taylor & Francis, London (2002)Google Scholar
- Morbidelli, A., Froeschlé, C.: On the relationship between Lyapunov times and macroscopic instability times. Celest. Mech. Dyn. Astron.
**63**, 227–239 (1996)ADSCrossRefMATHGoogle Scholar - Morbidelli, A., Giorgilli, A.: On a connection between KAM and Nekhoroshev’s theorems. Phys. D
**86**, 514–516 (1995)MathSciNetCrossRefMATHGoogle Scholar - Morbidelli, A., Guzzo, M.: The Nekhoroshev theorem and the asteroid belt dynamical system. Celest. Mech. Dyn. Astron.
**65**, 107–136 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar - Murray, N., Holman, M.: Diffusive chaos in the outer asteroid belt. Astron. J.
**114**, 1246–1259 (1997)ADSCrossRefGoogle Scholar - Richter, M., Lange, S., Bäcker, A., Ketzmerick, R.: Visualization and comparison of classical structures and quantum states of four-dimensional maps. Phys. Rev. E
**89**, 022902 (2014)ADSCrossRefGoogle Scholar - Rosengren, A.J., Alessi, E.M., Rossi, A., Valsecchi, G.B.: Chaos in navigation satellite orbits caused by the perturbed motion of the Moon. Mon. Not. R. Astron. Soc.
**449**, 3522–3526 (2015)ADSCrossRefGoogle Scholar - Robutel, P., Laskar, J.: Frequency map and global dynamics in the solar system I: short period dynamics of massless particles. Icarus
**152**, 4–28 (2001)ADSCrossRefGoogle Scholar - Simon, J., Bretagnon, P., Chapront, J., Chapront-Touzé, M., Francou, G., Laskar, J.: Numerical expressions for precession formulae and mean elements for the Moon and the planets. Astron. Astrophys.
**282**, 663–683 (1994)ADSGoogle Scholar - Skokos, Ch.: The Lyapunov characteristic exponents and their computation. Lect. Notes Phys.
**790**, 63–135 (2010)ADSCrossRefGoogle Scholar - Todorović, N., Novaković, B.: Testing the FLI in the region of the Pallas asteroid family. Mon. Not. R. Astron. Soc.
**451**, 1637–1648 (2015)ADSCrossRefGoogle Scholar - Todorović, N., Lega, E., Froeschlé, C.: Local and global diffusion in the Arnold web of a priori unstable systems. Celest. Mech. Dyn. Astron.
**102**, 13–27 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar - Upton, E., Bailie, A., Musen, P.: Lunar and solar perturbations on satellite orbits. Science
**130**, 1710–1711 (1959)ADSMathSciNetCrossRefMATHGoogle Scholar - Varvoglis, H.: Diffusion in the asteroid belt. Proc. Int. Astron. Union
**IAUC197**, 157–170 (2004)Google Scholar