A deep dive into the $2g+h$ resonance: separatrices, manifolds and phase space structure of navigation satellites

Abstract

Despite extended past studies, several questions regarding the resonant structure of the medium-Earth orbit (MEO) region remain hitherto unanswered. This work describes in depth the effects of the $2g+h$ lunisolar resonance. In particular, (i) we compute the correct forms of the separatrices of the resonance in the inclination-eccentricity (i, e) space for fixed semi-major axis a. This allows to compute the change in the width of the $2g+h$ resonance as the altitude increases. (ii) We discuss the crucial role played by the value of the inclination of the Laplace plane, $i_{L}$. Since $i_L$ is comparable to the resonance’s separatrix width, the parametrization of all resonance bifurcations has to be done in terms of the proper inclination $i_{p}$, instead of the mean one. (iii) The subset of circular orbits constitutes an invariant subspace embedded in the full phase space, the center manifold ${\mathcal {C}}$, where actual navigation satellites lie. Using $i_p$ as a label, we compute its range of values for which ${\mathcal {C}}$ becomes a normally hyperbolic invariant manifold (NHIM). The structure of invariant tori in ${\mathcal {C}}$ allows to explain the role of the initial phase h noticed in several works. (iv) Through Fast Lyapunov Indicator (FLI) cartography, we portray the stable and unstable manifolds of the NHIM as the altitude increases. Manifold oscillations dominate in phase space between $a =$ 24,000 km and $a=$ 30,000 km as a result of the sweeping of the $2g+h$ resonance by the and resonances. The noticeable effects of the latter are explained as a consequence of the relative inclination of the Moon’s orbit with respect to the ecliptic. The role of the phases in the structures observed in the FLI maps is also clarified. Finally, (v) we discuss how the understanding of the manifold dynamics could inspire end-of-life disposal strategies.

Leading terms coefficients

The formal coefficients of the leading terms appearing in the resonant Hamiltonian of Eq. (33), the center-manifold Hamiltonian of Eq. (53) and the coupling terms of Eq. (56) are summarized within Tables 2, 3 and 4, respectively. The physical units adopted in this study read $\mu _{\odot }=1.32712 \times 10^{11} \text {km}^{3}/\text {s}^{2}$, , $\mu _{\text {E}}=398\,600 \, \text {km}^{3}/\text {s}^{2}$, $J_{2}=1.082 \times 10^{-3}$, $R_{\text {E}}=6\,387.1$km, $r_{\odot }=1.49579 \times 10^{8}$ km, km. The shortcuts and denote the cosine and sinus of the inclination of the Moon to the ecliptic . In the same way, $c_{\varepsilon }$ and $s_{\varepsilon }$ denote the cosine and sinus of the obliquity of the ecliptic $\varepsilon =23^{\circ }44$, $i_{\star }=56^{\circ }06$ and n denotes the mean motion of the test-particle, $n=\sqrt{\mu _{\text {E}}/a^{3}}$.

Table 2 Coefficients of the resonant Hamiltonian ${\mathcal {H}}_{\text {R}}$

Table 3 Coefficients of the centre-manifold Hamiltonian ${\mathcal {H}}_{\text {CM}}$

Table 4 Coefficients of the coupling terms Hamiltonian ${\mathcal {H}}_{\text {C}}$