Bifurcation of Lunisolar Secular Resonances for Space Debris Orbits

Abstract

Using bifurcation theory, we study the secular resonances induced by the Sun and Moon on space debris orbits around the Earth. In particular, we concentrate on a special class of secular resonances, which depend only on the debris' orbital inclination. This class is typically subdivided into three distinct types of secular resonances: those occurring at the critical inclination, those corresponding to polar orbits, and a third type resulting from a linear combination of the rates of variation of the argument of perigee and the longitude of the ascending node.The model describing the dynamics of space debris includes the effects of the geopotential, as well as the Sun's and Moon's attractions, and it is defined in terms of suitable action-angle variables. We consider the system averaged over both the mean anomaly of the debris and those of the Sun and Moon. Such multiply-averaged Hamiltonian is used to study the lunisolar resonances which depend just on the inclination.Borrowing the technique from the theory of bifurcations of Hamiltonian normal forms, we study the birth of periodic orbits and we determine the energy thresholds at which the bifurcations of lunisolar secular resonances take place. This approach gives us physically relevant information on the existence and location of the equilibria, which help us to identify stable and unstable regions in the phase space. Besides their physical interest, the study of inclination dependent resonances offers interesting insights from the dynamical point of view, since it sheds light on different phenomena related to bifurcation theory.2010 Mathematics Subject Classification. 37N05,37L10,70F15.

Dynamical lifetime survey of geostationary transfer orbits

“…Increasing A/m and adding drag to the model basically 'shift' the curves to higher q 0 . However, the dependence on A/m and drag is much smaller at high inclinations, where the main dynamical mechanism responsible for reentry is the overlapping of lunisolar resonances, as found in recent studies (see Celletti et al 2016;Gkolias et al 2016).…”

Drift and Its Mediation in Terrestrial Orbits

“…The canonically conjugated vector of angles is classically denoted (ℓ, g, h). Omitting details that might be found in Celletti et al [9,11], the disturbing function of the Sun's attraction, R ⊙ , reads as…”

“…This procedure involved the introduction of resonant coordinates through canonical transformations T k ∈ SL(3, Q) leading to an intuitive physical interpretation of Chirikov's overlap as a driver of chaos. However, since the work of Celletti et al [9], it has been observed that such a reduction does not always capture the features of the dynamics. In order to get a more refined and precise view of the extent of chaos, a superior way is instead to look at the destruction of KAM curves, e.g., using fast dynamical indicators.…”

“…The time evolution (over 25 lunar periods) of the eccentricity and inclination for the whole cluster of orbits (k = 200 orbits) is displayed in Figure 7. The spatial averaged orbit is displayed and superimposed with a bold red line 9 . Clearly, the eccentricities of the whole cluster evolve in an apparent regular fashion.…”

“…We are particularly interested by questions related to the stability of orbits. Based on the divergence of nearby trajectories, the existence of a mixed phase space where there is a cohabitation of stable and chaotic components has been recently pictured [7][8][9][10][11] and partially explained applying Chirikov's resonances overlap criterion [12]. The Hamiltonian flow obtained under the simplest assumptions for the disturbing effects of the perturbers (i.e., a development restricted to its lowest order and averaged over fast variables), Moon and Sun, encapsulates all the details of the dynamics in which we are interested [7].…”