Drift and its mediation in terrestrial orbits

The slow deformation of terrestrial orbits in the medium range, subject to lunisolar resonances, is well approximated by a family of Hamiltonian flow with $2.5$ degree-of-freedom. The action variables of the system may experience chaotic variations and large drift that we may quantify. Using variational chaos indicators, we compute high-resolution portraits of the action space. Such refined meshes allow to reveal the existence of tori and structures filling chaotic regions. Our elaborate computations allow us to isolate precise initial conditions near specific zones of interest and study their asymptotic behaviour in time. Borrowing classical techniques of phase- space visualisation, we highlight how the drift is mediated by the complement of the numerically detected KAM tori.Comment: 22 pages, 11 figures, 1 table, 52 references. Comments and feedbacks greatly appreciated. This article is part of the Research Topic `The Earth-Moon System as a Dynamical Laboratory', confer https://www.frontiersin.org/research-topics/5819/the-earth-moon-system-as-a-dynamical-laborator

Analytical development of the lunisolar disturbing function and the critical inclination secular resonance

We provide a detailed derivation of the analytical expansion of the lunar and solar disturbing functions. Although there exist several papers on this topic, many derivations contain mistakes in the final expansion or rather (just) in the proof, thereby necessitating a recasting and correction of the original derivation. In this work, we provide a self-consistent and definite form of the lunisolar expansion. We start with Kaula's expansion of the disturbing function in terms of the equatorial elements of both the perturbed and perturbing bodies. Then we give a detailed proof of Lane's expansion, in which the elements of the Moon are referred to the ecliptic plane. Using this approach the inclination of the Moon becomes nearly constant, while the argument of perihelion, the longitude of the ascending node, and the mean anomaly vary linearly with time. We make a comparison between the different expansions and we profit from such discussion to point out some mistakes in the existing literature, which might compromise the correctness of the results. As an application, we analyze the long--term motion of the highly elliptical and critically--inclined Molniya orbits subject to quadrupolar gravitational interactions. The analytical expansions presented herein are very powerful with respect to dynamical studies based on Cartesian equations, because they quickly allow for a more holistic and intuitively understandable picture of the dynamics.Comment: 30 pages, 4 figure

From order to chaos in Earth satellite orbits

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 long-term 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.Comment: 30 pages, 9 figures. Accepted for publication in the Astronomical Journa

Medium Earth Orbit dynamical survey and its use in passive debris removal

The Medium Earth Orbit (MEO) region hosts satellites for navigation, communication, and geodetic/space environmental science, among which are the Global Navigation Satellites Systems (GNSS). Safe and efficient removal of debris from MEO is problematic due to the high cost for maneuvers needed to directly reach the Earth (reentry orbits) and the relatively crowded GNSS neighborhood (graveyard orbits). Recent studies have highlighted the complicated secular dynamics in the MEO region, but also the possibility of exploiting these dynamics, for designing removal strategies. In this paper, we present our numerical exploration of the long-term dynamics in MEO, performed with the purpose of unveiling the set of reentry and graveyard solutions that could be reached with maneuvers of reasonable DV cost. We simulated the dynamics over 120-200 years for an extended grid of millions of fictitious MEO satellites that covered all inclinations from 0 to 90deg, using non-averaged equations of motion and a suitable dynamical model that accounted for the principal geopotential terms, 3rd-body perturbations and solar radiation pressure (SRP). We found a sizeable set of usable solutions with reentry times that exceed ~40years, mainly around three specific inclination values: 46deg, 56deg, and 68deg; a result compatible with our understanding of MEO secular dynamics. For DV <= 300 m/s (i.e., achieved if you start from a typical GNSS orbit and target a disposal orbit with e<0.3), reentry times from GNSS altitudes exceed ~70 years, while low-cost (DV ~= 5-35 m/s) graveyard orbits, stable for at lest 200 years, are found for eccentricities up to e~0.018. This investigation was carried out in the framework of the EC-funded "ReDSHIFT" project.Comment: 39 pages, 23 figure

Non-averaged regularized formulations as an alternative to semi-analytical orbit propagation methods

This paper is concerned with the comparison of semi-analytical and non-averaged propagation methods for Earth satellite orbits. We analyse the total integration error for semi-analytical methods and propose a novel decomposition into dynamical, model truncation, short-periodic, and numerical error components. The first three are attributable to distinct approximations required by the method of averaging, which fundamentally limit the attainable accuracy. In contrast, numerical error, the only component present in non-averaged methods, can be significantly mitigated by employing adaptive numerical algorithms and regularized formulations of the equations of motion. We present a collection of non-averaged methods based on the integration of existing regularized formulations of the equations of motion through an adaptive solver. We implemented the collection in the orbit propagation code THALASSA, which we make publicly available, and we compared the non-averaged methods to the semi-analytical method implemented in the orbit propagation tool STELA through numerical tests involving long-term propagations (on the order of decades) of LEO, GTO, and high-altitude HEO orbits. For the test cases considered, regularized non-averaged methods were found to be up to two times slower than semi-analytical for the LEO orbit, to have comparable speed for the GTO, and to be ten times as fast for the HEO (for the same accuracy). We show for the first time that efficient implementations of non-averaged regularized formulations of the equations of motion, and especially of non-singular element methods, are attractive candidates for the long-term study of high-altitude and highly elliptical Earth satellite orbits.Comment: 33 pages, 10 figures, 7 tables. Part of the CMDA Topical Collection on "50 years of Celestial Mechanics and Dynamical Astronomy". Comments and feedback are encourage

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

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.Comment: Submitted to Celestial Mechanics and Dynamical Astronomy. Comments are greatly appreciated. 28 pages, 15 figure

Assessing and Minimizing Collisions in Satellite Mega-Constellations

We aim to provide satellite operators and researchers with an efficient means for evaluating and mitigating collision risk during the design process of mega-constellations. We first establish a baseline for evaluating various techniques for close-encounter prediction and collision-probability calculation (Hoots et al. 1984, Gronchi 2005, JeongAhn and Malhotra 2015) by carrying out brute-force numerical simulations and using a sequence of filters to greatly reduce the computational expense of the algorithm. Next, we estimate conjunction events in the orbital environment following the anticipated deployments of the OneWeb LEO and SpaceX Starlink mega-constellations. As a final step, we investigate Minimum Space Occupancy (MiSO) orbits (Bombardelli et al. 2018), a generalization of the well-known frozen orbits that account for the perturbed-Keplerian dynamics of the Earth-Moon-Sun-satellite system. We evaluate the ability of MiSO configurations of the proposed mega-constellations, as suggested by Bombardelli et al. 2018, to reduce the risk of endogenous (intra-constellation) collisions. The results indicate that the adoption of the MiSO orbital configuration can significantly reduce risk with nearly indistinguishable adjustments to the nominal orbital elements of the constellation satellites.Comment: 32 pages, 19 figures; submitted for publication in Adv. Space Res.; comments welcom