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

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.

Appendices

Appendix 1: Special functions in the lunisolar disturbing potential expansions

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

$$\begin{aligned} {\mathcal R}_\text {M}= \displaystyle \sum _{m = 0}^2 \sum _{s = 0}^2 \sum _{p = 0}^2 h_{2-2p, m, \pm s}^\text {M}\cos {\varPhi }_{2-2p, m, \pm s}^\text {M}, \end{aligned}$$

with harmonic angle and associated harmonic coefficient

$$\begin{aligned}&{\varPhi }_{2-2p, m, \pm s}^\text {M}= (2 - 2 p) \omega + m {\varOmega }\pm s ({\varOmega }_\text {M}- \pi /2) - y_s \pi , \\&h_{2-2p, m, \pm s}^\text {M}= \frac{\mu _\text {M}a^2}{a_\text {M}^3 (1 - e_\text {M}^2)^{3/2}} (-1)^{{\lfloor }{m/2}{\rfloor }}\\&\times \,\varepsilon _m \frac{(2 - s)!}{(2 + m)!} F_{2,m,p} (i) F_{2,s,1} (i_\text {M}) H_{2,p,2p-2} (e) (-1)^{m + s} U_2^{m, \mp s} (\epsilon ), \end{aligned}$$

so that a lunar resonance occurs when

$$\begin{aligned} \dot{\psi }_{2 - 2 p, m, \pm s} = (2 - 2 p) \dot{\omega }+ m \dot{\varOmega }\pm s \dot{\varOmega }_\text {M}\approx 0. \end{aligned}$$

The solar disturbing function can be written in the compact form

$$\begin{aligned} \displaystyle {\mathcal R}_\text {S}= \displaystyle \sum _{m = 0}^2 \sum _{p = 0}^2 h_{2-2p, m}^\text {S}\cos {\varPhi }_{2-2p, m}^\text {S}, \end{aligned}$$

with harmonic angle and coefficient

$$\begin{aligned} {\varPhi }_{2-2p, m}^\text {S}&= (2 - 2 p) \omega + m ({\varOmega }- {\varOmega }_\text {S}), \\ h_{2-2p, m}^\text {S}&= \frac{\mu _\text {S}a^2}{a_\text {S}^3 (1 - e_\text {S}^2)^{3/2}} \varepsilon _m \frac{(2 - m)!}{(2 + m)!} F_{2,m,p} (i) F_{2,m,1} (i_S) H_{2,p,2p-2} (e). \end{aligned}$$

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)

$$\begin{aligned} X_k^{l, m} (e) = \frac{1}{2 \pi } \int _0^{2 \pi } \left( \frac{r}{a} \right) ^l \cos (m f - k M)\, \mathrm {d} M. \end{aligned}$$

(39)

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

$$\begin{aligned} X_0^{l,m}&= \frac{1}{2 \pi } \int _0^{2 \pi } \left( \frac{r}{a} \right) ^l \cos m f\, \mathrm {d} M. \end{aligned}$$

(40)

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

$$\begin{aligned} X_0^{l,m}&= \left( -\frac{e}{2} \right) ^m \left( \begin{array}{c} l + m + 1 \\ m \end{array} \right) F \left( \frac{m - l - 1}{2}, \frac{m - l}{2}, m + 1; e^2\right) , \end{aligned}$$

(41)

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

$$\begin{aligned} X_0^{2,\pm 2} = \frac{5}{2} e^2, \quad X_0^{2,0} = 1 + \frac{3}{2} e^2. \end{aligned}$$

(42)

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

$$\begin{aligned} F_{2,m,p} (i)&= \sum _t \frac{(4 - 2t)!}{t! (2 -t)! (2 - m - 2t)! 2^{4 - 2t}} \sin ^{2 - m - 2t} i\nonumber \\&\quad \times \,\sum _{s = 0}^m \left( \begin{array}{c} m \\ s \end{array} \right) \cos ^s i \sum _c \left( \begin{array}{c} 2 - m - 2t + s \\ c \end{array} \right) \left( \begin{array}{c} m - s \\ p - t - c \end{array} \right) (-1)^{c - k}, \end{aligned}$$

(43)

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.

Table 2 Kaula’s inclination functions $F_{2,m,p} (i)$ from Kaula (1966)

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

Table 3 Giacaglia’s tesseral harmonics rotation functions $U_2^{m,\mp s}$ for the Moon, where $C = \cos (\epsilon /2)$ and $S = \sin (\epsilon /2)$

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.

Table 4 Lunar resonance conditions, harmonic angles, and associated coefficients needed for the resonant width computations of §2.2. Here we use the abbreviations $\text {s}= \sin$, $\text {c} = \cos$, $C = \cos (\epsilon /2)$, and $S = \sin (\epsilon /2)$

Table 5 Solar secular resonance conditions, harmonic angles, and associated harmonic coefficients needed for the resonant width computations of §2.2. Here we use the abbreviations $\text {s}= \sin$ and $\text {c} = \cos$

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

$$\begin{aligned} n_{1}\dot{\omega } + n_{2} \dot{{\varOmega }}&= \pm 2 \sqrt{\nu . \partial ^2 {\mathcal H}^\text {sec} / \partial X^2 \big \vert _{X = X_\star }} \end{aligned}$$

(44)

is equivalent to

$$\begin{aligned} \kappa P_{\mathbf{n}}(i)&= \pm 2 u(e) \sqrt{\nu . \partial ^2 {\mathcal H}^\text {sec} / \partial X^2 \big \vert _{X = X_\star }} \end{aligned}$$

(45)

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 $e \rightarrow 1$.

Fig. 14

Resonance centers and widths for the special class of inclination-dependent-only lunisolar resonances (q.v. Hughes 1980), plotted both with and without the solar contribution

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

(46)

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.

Fig. 15

Time-calibration for the FLI maps. Here we show the typical results obtained after 100, 200, and 530 years, respectively, for $a_{0}=25,500$ km, under force model 2

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.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.