In contrast to a widespread cliché, the satellite problems still require the research on the level more fundamental than just tracing the microscopic influence of yet another tesseral harmonic. (Breiter, op. cit., p. 90)

(Sławomir Breiter, 2001)

## 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.

This is a preview of subscription content,

to check access.## Notes

Celletti et al. (2015) presents a detailed proof of Lane’s mixed-reference-frame expansion of the lunar disturbing function (Lane 1989), correcting a subtle error that appears therein. Note that here, the disturbing function is cast in a particularly compact and transparent form, so that the additional quantity \(\varepsilon _s\) (analogous to \(\varepsilon _m\)) and the factor 1 / 2, appearing in Lane (1989), disappear from the expression for the harmonic coefficient.

Note that this refinement was not taken into account in Ely and Howell (1997) and is more suitable for the calculation of the widths for the inclination-dependent-only resonances, where solar perturbations play a non-negligible role (see Appendix 2).

Recall that we need the resonance width in order to determine when neighboring resonances overlap and hence when a system is unstable, à la Chirikov.

This is essentially the same conclusion reached by Todorović and Novaković (2015), who treat the mean-motion resonances in a particular region of the main asteroid belt (i.e., the location of the Pallas asteroid family): “\(\ldots \) the FLI maps depend on the choice of initial angles. \(\ldots \) However, we underline that this does not change the global dynamical picture of the region \(\ldots \)”

We note that since the FLI maps have an angle dependency, it would have been more appropriate here to compare the resonance geometries, computed analytically, with FLI maps obtained using random initial phase angles (Todorović and Novaković 2015). While this procedure would have yielded a better estimate of the real width of the chaotic zones associated to each resonance, it would have compromised the sharpness of the pictures.

Just before the integration stops, the FLI value of a re-entry orbit is still small but the slope of the FLI curve increases exponentially all of the sudden. Thus, there is a risk to consider a chaotic orbit as stable.

## References

Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics, 3rd edn. Springer-Verlag, Berlin (2006)

Barrio, R., Borczyk, W., Breiter, S.: Spurious structures in chaos indicators maps. Chaos Solitons Fractals

**40**, 1697–1714 (2009)Batygin, K., Morbidelli, A., Holman, M.J.: Chaotic disintegration of the inner solar system. Astrophys. J.

**799**, 120–135 (2015)Breiter, S.: Lunisolar apsidal resonances at low satellite orbits. Celest. Mech. Dyn. Astron.

**74**, 253–274 (1999)Breiter, S.: Lunisolar resonances revisited. Celest. Mech. Dyn. Astron.

**81**, 81–91 (2001)Breiter, S.: Fundamental models of resonance. Monografías de la Real Academia de Ciencias de Zaragoza

**22**, 83–92 (2003)Celletti, A., Galeş, C.: On the dynamics of space debris: 1:1 and 2:1 resonances. J. Nonlinear Sci.

**24**, 1231–1262 (2014)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)Cook, G.E.: Luni-solar perturbations of the orbit of an Earth satellite. Geophys. J.

**6**, 271–291 (1962)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)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)Delhaise, F., Morbidelli, A.: Luni-solar effects of geosynchronous orbits at the critical inclination. Celest. Mech. Dyn. Astron.

**57**, 155–173 (1993)Ely, T.A.: Eccentricity impact on east-west stationkeeping for global position system class orbits. J. Guid. Control Dyn.

**25**, 352–357 (2002)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)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)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)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)Garfinkel, B.: Formal solution in the problem of small divisors. Astron. J.

**71**, 657–669 (1966)Giacaglia, G.E.O.: Lunar perturbations of artificial satellites of the Earth. Celest. Mech.

**9**, 239–267 (1974)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)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)Haller, G.: Chaos Near Resonance. Springer-Verlag, New York (1999)

Hughes, S.: Earth satellite orbits with resonant lunisolar perturbations. I. Resonances dependent only on inclination. Proc. R. Soc. Lond. A

**372**, 243–264 (1980)Jupp, A.H.: The critical inclination problem: 30 years of progress. Celest. Mech.

**43**, 127–138 (1988)Kaula, W.M.: Theory of Satellite Geodesy. Blaisdell, Waltham (1966)

Lane, M.T.: An analytical treatment of resonance effects on satellite orbits. Celest. Mech.

**42**, 3–38 (1988)Lane, M.T.: An analytical modeling of lunar perturbations of artificial satellites of the Earth. Celest. Mech. Dyn. Astron.

**46**, 287–305 (1989)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)Lithwick, Y., Wu, Y.: Theory of secular chaos and Mercury’s orbit. Astrophys. J.

**739**, 31–47 (2011)Mardling, R.A.: Resonances, chaos and stability: the three-body problem in astrophysics. Lect. Notes Phys.

**760**, 59–96 (2008)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)

Milani, A., Nobili, A.M.: An example of stable chaos in the solar system. Nature

**6379**, 569–571 (1992)Morbidelli, A.: Modern Celestial Mechanics: Aspects of Solar System Dynamics. Taylor & Francis, London (2002)

Morbidelli, A., Froeschlé, C.: On the relationship between Lyapunov times and macroscopic instability times. Celest. Mech. Dyn. Astron.

**63**, 227–239 (1996)Morbidelli, A., Giorgilli, A.: On a connection between KAM and Nekhoroshev’s theorems. Phys. D

**86**, 514–516 (1995)Morbidelli, A., Guzzo, M.: The Nekhoroshev theorem and the asteroid belt dynamical system. Celest. Mech. Dyn. Astron.

**65**, 107–136 (1996)Murray, N., Holman, M.: Diffusive chaos in the outer asteroid belt. Astron. J.

**114**, 1246–1259 (1997)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)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)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)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)Skokos, Ch.: The Lyapunov characteristic exponents and their computation. Lect. Notes Phys.

**790**, 63–135 (2010)Todorović, N., Novaković, B.: Testing the FLI in the region of the Pallas asteroid family. Mon. Not. R. Astron. Soc.

**451**, 1637–1648 (2015)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)Upton, E., Bailie, A., Musen, P.: Lunar and solar perturbations on satellite orbits. Science

**130**, 1710–1711 (1959)Varvoglis, H.: Diffusion in the asteroid belt. Proc. Int. Astron. Union

**IAUC197**, 157–170 (2004)

## 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”.

## Author information

### Authors and Affiliations

### Corresponding author

## Appendices

### Appendix 1: Special functions in the lunisolar disturbing potential expansions

Recall that the lunar disturbing function can be written in the compact form

with harmonic angle and associated harmonic coefficient

so that a lunar resonance occurs when

The solar disturbing function can be written in the compact form

with harmonic angle and coefficient

A solar commensurability occurs when \((2 - 2p) \dot{\omega }+ m \dot{\varOmega }\approx 0\), or equivalently when \(\dot{\psi }_{2 - 2 p, m, 0} \approx 0\). We give here the the explicit formula needed to calculate the widths of the 29 distinct curves of secular resonances, six of which are locations of lunisolar resonance (i.e., the inclination-dependent-only cases), appearing in Fig. 1.

The general Hansen coefficient \(X_k^{l,m} (e)\), which permits the full disturbing function (prior to averaging) to be developed in terms of the mean anomalies, is a function of the orbit eccentricity and is given by the integral (Hughes 1980)

For \(k = 0\), exact analytical expressions exist for the zero-order Hansen coefficients \(X_0^{l,m}\) for all values of *l* and *m*. The integral for \(X_0^{l, m}\) can be written as

Note that since cosine is an even function, it is only necessary to obtain expressions for \(m > 0\) as \(X_0^{l,m} = X_0^{l,-m}\). For \(l \ge 1\) and \(0 \le m \le l\), the integrals can be evaluated as

where \(F (\alpha , \beta , \gamma ; x)\) is a hypergeometric function in \(e^2\). Several formulae of recurrence have been derived that greatly facilitate the calculation of these coefficients, however, for our purposes, \(H_{2,p,2p-2} (e) = X_0^{2,2-2p} (e)\), for \(p = 0, 1, 2\), can be easily evaluated as

The Kaula inclination functions \(F_{2,m,p} (i)\) and \(F_{2,s,1} (i_\text {M})\) are given by (Kaula 1966)

where *k* is the integer part of \((2 - m)/2\), *t* is summed from 0 to the lesser of *p* or *k*, and *c* is summed over all values making the binomial coefficients nonzero. Expressions for \(F_{2,m,p} (i)\) up to \(m = 2\), \(p = 2\), are given in Table 2.

The expression for the Giacaglia functions, computed from Eq. 4, are listed in Table 3.

The commensurability condition, harmonic angle, and associated harmonic coefficient for all resonance center curves are given in Tables 4 and 5, color coded to match that of Fig. 1. Note that when the angular arguments are the same (excepting a constant phase), as in each of the five resonance curves stemming from the critical inclination (\(63.4^\circ \)) at \(e = 1\) or the lunisolar inclination-dependent-only resonances, the associated harmonic coefficients must be combined into one single term for the calculation of the resonant width, as detailed in Sect. 2.2.

### Appendix 2: Inclination-dependent-only lunisolar resonances

Figure 14 shows the lunisolar inclination-dependent-only resonance centers (solid lines) and widths (transparent lobes) for increasing values of the satellites’s semi-major axis, computed with the solar contribution and without (shaded gray and without an edge color). The solar harmonics, neglected in previous works (Ely and Howell 1997), play an increasingly important role as the semi-major axis is increased, either by widening or shrinking the resonance domain. The lobe shape of the lunisolar apsidal resonances (q.v. Breiter 2001) and the rocket shape of the nodal resonance are easily explained by the dependence of the widths on the orbit eccentricity. The widths depend on *e* through their associated harmonic coefficients (Appendix 1), which comes in through the zero-order Hansen coefficients, and through Eq. 33. From Eqs. 3, 7, 26, and 42, \(\nu \rightarrow 0\) as \(e \rightarrow 0\) for \(p = 0\) or \(p=2\), but not for \(p = 1\).

To extend the result to the case where \(e \rightarrow 1\), let us note that the equation

is equivalent to

where \(\kappa \) a constant (independent of *e* and *i*), \(u(e)=(1-e^{2})^{2}\) and \(P_{\mathbf{n}}(i)=n_1 (5 \cos ^{2} i -1)/2 - n_2 \cos i\). When \(e \rightarrow 1\), the right-hand side of Eq. (45) goes to zero, and, as a result, *i* must be a root of \(P_{\mathbf{n}}\) in the limit \(e \rightarrow 1\).

### Appendix 3: Numerical setups

The dynamical model we used in the propagation of the orbits accounts for the perturbations stemming from the Earth, the Moon, and the Sun. To simplify the mathematical problem, all of these gravitational perturbations have been analytically averaged over the mean anomaly *M* of the satellite (the fast variable), and propagated using numerical integrations, which is much faster thanks to the absence of short-periodic variations. This is a well known and efficient semi-analytical approach to study the qualitative evolution of orbits over very long timespans. The averaging procedure has been done for the Earth’s geopotential up to \(J_5\) (including the \(J_2^2\) term). For tesseral resonances (located for certain semi-major axes), where there exists a commensurability between the frequency of the satellite’s mean motion and the sidereal time, a partial averaging method was applied to retain only the long-periodic perturbations. This is equivalent to retaining in the Earth’s geopotential only the slowly varying quantities associated with the critical resonant angle, as it is technically detailed in Morand (2013). Tesseral resonances are analytically well modeled by a 1.5-DOF Hamiltonian and they primary affect the semi-major axis of the satellite, leading to a thin chaotic response of the system (the 1.5-DOF bounded chaos is physically equivalent to an intermittency phenomenon on the semi-major axis). Despite the fact that chaotic response of the system is very confined in phase-space (on the order of kilometers), we have included these perturbations in the full system to check if such effects may couple with the lunisolar effects on longer timescales (Ely 2002). The coefficients of the Earth’s gravity field come from the GRIM5-S1 model. Our semi-analytical propagator is configurable and the third-body perturbations from the Moon and the Sun have been developed up to degree 4 and 3, respectively. The positions of the Moon and the Sun were computed from accurate analytical ephemerides (Simon et al. 1994). All of the equations of motion have been formulated through the equinoctial elements, related to the Keplerian elements by

which are suitable for all considered dynamical configurations. The variational system (i.e., the equations of motion for the state and tangent vector) are then propagated using a fixed step size (set to 1 day) Runge-Kutta, 6th-order integration algorithm.

To produce the different maps of the atlas, only the initial eccentricity and inclination were varied, distributed in a regular grid of \(320 \times 115\) initial conditions for the four semi-major axes of Fig. 3. The computation of the FLIs on a grid of initial conditions allows a clear distinction, in short CPU time, of invariant tori and resonances (Lega et al. 2010; Guzzo et al. 2002). The number of initial conditions chosen was a good balance between CPU time and the final pictures offered by the resolution. Concerning the simulations, after a calibration procedure, we decided to present the results of the FLI analysis after 530 years of propagation. By increasing the iteration time, the basic features of the FLI maps are not changed, but small higher-order resonances can be detected in a proper resolution (Todorović and Novaković 2015). Our chosen timescale may seem prohibitive and might also not be the best trade-off between clear distinction of the separation of nearby orbits (if ever) and CPU time. However, as showed by Fig. 15, a 250 years propagation may leads to partially erroneous conclusions concerning the stability in MEO. Moreover, for weakly hyperbolic orbits (usually for moderately eccentric orbits), the time necessary to detect the divergence is longer. Consequently, there is no risk about the conclusions by presenting the results after this propagation time, at the cost of somewhat more CPU time (Todorović et al. 2008). For the zoomed-in portion, the resolution and propagation duration have been increased to get finer details, structures, and more contrast in the maps. Unless explicitly stated, all others parameters (initial phases, initial epoch, initial tangent vector) have been fixed for each map. Concerning the initial tangent vector, we used the same for the whole map, a fixed normed vector orthogonal to the flow, *f*, in Eq. 36. The robustness of our results with respect to the choice of \(w_{0}\) was tested by computing the maximal Lyapunov exponent. The Lyapunov exponent, which is first of all a time average, is a property of the orbit, independent of the initial point of that orbit. By drawing the mLCE maps, we observe a nice agreement with the FLI maps in terms of the detected structures (chaotic or stable). The only (obvious) disagreement was the contrast in the maps, easily explained: Lyapunov exponents are slow to converge. This argument is strong enough to attest the relaxation that we can have with respect of \(w_{0}\). Finally, the final values of the FLIs are coded, and projected into the plane of *i*–*e* initial conditions, using a color palette ranging from black to yellow. Darker points correspond to stable orbits, while lighter colors correspond to chaotic orbits. The FLIs of orbits whose propagation stop before reaching the final time, \(t_{f}\), have not been considered in the maps: they appear as a white to avoid the introduction of spurious structures.^{Footnote 7} This is the reason why some maps seems to be perforated. Re-entry orbits were in practice declared if there exist a time \(t < t_{f}\) such that \(a\big (1-e\big ) < 120\) km.

## Rights and permissions

## About this article

### Cite this article

Daquin, J., Rosengren, A.J., Alessi, E.M. *et al.* The dynamical structure of the MEO region: long-term stability, chaos, and transport.
*Celest Mech Dyn Astr* **124**, 335–366 (2016). https://doi.org/10.1007/s10569-015-9665-9

Received:

Revised:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s10569-015-9665-9