From Order to Chaos in Earth Satellite Orbits

Abstract

We consider Earth satellite orbits in the range of semi-major axes where the perturbing effects of Earth's oblateness and lunisolar gravity are of comparable order. This range covers the medium-Earth orbits (MEO) of the Global Navigation Satellite Systems and the geosynchronous orbits (GEO) of the communication satellites. We recall a secular and quadrupolar model, based on the Milankovitch vector formulation of perturbation theory, which governs the longterm orbital evolution subject to the predominant gravitational interactions. We study the global dynamics of this two-and-a-half degrees-of-freedom Hamiltonian system by means of the fast Lyapunov indicator (FLI), used in a statistical sense. Specifically, we characterize the degree of chaoticity of the action space using angle-averaged normalized FLI maps, thereby overcoming the angle dependencies of the conventional stability maps. Emphasis is placed upon the phase-space structures near secular resonances, which are of first importance to the space debris community. We confirm and quantify the transition from order to chaos in MEO, stemming from the critical inclinations, and find that highly inclined GEO orbits are particularly unstable. Despite their reputed normality, Earth satellite orbits can possess an extraordinarily rich spectrum of dynamical behaviors, and, from a mathematical perspective, have all the complications that make them very interesting candidates for testing the modern tools of chaos theory.

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

“…Anticipating a bit the next section, we are here interested in the dynamical mechanisms leading to transport; in particular we have not discriminated collisional orbits as we did in earlier work. As it has already been discussed several times and pointed out in several contributions [7,8,10], the inclination dependentonly resonances widen and develop chaos when ε is increasing, letting less and less room for invariant KAM tori. Eventually for a = 29, 600, there is a macroscopic chaotic component.…”

“…When invoking effective models, we always face the question of the relevance of the reduced model (how sound are the qualitative or quantitative informations derived from it). Gkolias et al [10] established the testimony of this doubly-averaged model against a singly-averaged approach. By simulating the two dynamics on different domains of the action-action, action-angle and angle-angle spaces 8 , we showed that dynamical features of interests were reproduced and in perfect agreement.…”

Dynamical cartography of Earth satellite orbits

“…The nonnegligible collision risks posed by these LEO-GEO transiting spacecraft has motivated both theoretical study and practical implementation (Armellin et al, 2015;Colombo et al, 2015;Merz et al, 2015). Even for the, seemingly more simple, inclined, nearly circular orbits of the navigation satellites, no official guidelines exist, and recent analyses have shown that the problem is far too complex to allow of an adoption of the basic geosynchronous graveyard strategy (Rosengren et al, 2015;Daquin et al, 2016;Celletti et al, 2016;Gkolias et al, 2016;Rosengren et al, 2017;Skoulidou et al, 2017) Fig. 1.…”

“…For near-Earth satellites the effects of these distant perturbing bodies are often negligible,comparison to that of the Earth's oblateness, but for satellite orbits that are very elongated or have semimajor axes of several Earth radii, these third-body gravitational perturbations can change the elements of the orbit to a measurable extent, albeit on longer timescales than the Earth's oblateness. Indeed, recent works suggest that the coupled resonant effects of the Earth's oblateness and lunisolar perturbations are the most important mechanisms for delivering high-altitude, Earth-orbiting satellites into regions where atmospheric drag can give lead to orbital decay (Upton et al, 1959;Breiter, 2001;Daquin et al, 2016;Gkolias et al, 2016). Hence, these perturbations are fully taken into account in our model (in an Nbody framework; i.e., without recourse to truncated Legendre expansions is done in averaged formulations).…”