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The maximum eccentricity method (MEM, (Dvorak et al. in Astron Astrophys 426(2):L37–L40, 2004)) is a standard tool for the analysis of planetary systems and their stability. The method amounts to estimating the maximal stretch of orbits over sampled domains of initial conditions. The present paper leverages on the MEM to introduce a sharp detector of separatrices and chaotic seas. After introducing the MEM analogue for nearly-integrable action-angle Hamiltonians, i.e., diameters, we use low-dimensional dynamical systems with multi-resonant modes and junctions, supporting chaotic motions, to recognise the drivers of the diameter metric. Once this is appreciated, we present a second-derivative-based index measuring the regularity of this application. This quantity turns to be a sensitive and robust indicator to detect separatrices, resonant webs and chaotic seas. We discuss practical applications of this framework in the context of N-body simulations for the planetary case affected by mean-motion resonances, and demonstrate the ability of the index to distinguish minute structures of the phase space, otherwise undetected with the original MEM.

Near-earth space continues to be the focus of critical services and capabilities provided to the society. With the steady increase of space traffic, the number of Resident Space Objects (RSOs) has recently boomed in the context of growing concern due to space debris. The need of a holistic and unified approach for addressing orbital collisions, assess the global in-orbit risk, and define sustainable practices for space traffic management has emerged as a major societal challenge. Here, we introduce and discuss a versatile framework rooted on the use of the complex network paradigm to introduce a novel risk index for space sustainability criteria. With an entirely data-driven, but flexible, formulation, we introduce the Resident Space Object Network (RSONet) by connecting RSOs that experience near-collisions events over a finite-time window. The structural collisional properties of RSOs are thus encoded into the RSONet and analysed with the tools of network science. We formulate a geometrical index highlighting the key role of specific RSOs in building up the risk of collisions with respect to the rest of the population. Practical applications based on Two-Line Elements and Conjunction Data Message databases are presented.

We describe the phase space structures related to the semi-major axis of Molniya-like satellites subject to tesseral and lunisolar resonances. In particular, the questions answered in this contribution are: (1) we study the indirect interplay of the critical inclination resonance on the semi-geosynchronous resonance using a hierarchy of more realistic dynamical systems, thus discussing the dynamics beyond the integrable approximation. By introducing ad hoc tractable models averaged over fast angles, (2) we numerically demarcate the hyperbolic structures organising the long-term dynamics via fast Lyapunov indicators cartography. Based on the publicly available two-line elements space orbital data, (3) we identify two satellites, namely Molniya 1-69 and Molniya 1-87, displaying fingerprints consistent with the dynamics associated to the hyperbolic set. Finally, (4) the computations of their associated dynamical maps highlight that the spacecraft are trapped within the hyperbolic tangle. This research therefore reports evidence of actual artificial satellites in the near-Earth environment whose dynamics are ruled by hyperbolic manifolds and resonant mechanisms. The tools, formalism and methodologies we present are exportable to other region of space subject to similar commensurabilities as the geosynchronous region.

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, iL. Since iL is comparable to the resonance’s separatrix width, the parametrization of all resonance bifurcations has to be done in terms of the proper inclination ip, instead of the mean one. (iii) The subset of circular orbits constitutes an invariant subspace embedded in the full phase space, the center manifold C, where actual navigation satellites lie. Using ip as a label, we compute its range of values for which C becomes a normally hyperbolic invariant manifold (NHIM). The structure of invariant tori in 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.

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.

We present our recent contributions to the theory of Lagrangian descriptors for discriminating ordered and deterministic chaotic trajectories. The class of Lagrangian descriptors we are dealing with is based on the Euclidean length of the orbit over a finite time window. The framework is free of tangent vector dynamics and is valid for both discrete and continuous dynamical systems. We review its last advancements and touch on how it illuminated recently Dvorak’s quantities based on maximal extent of trajectories’ observables, as traditionally computed in planetary dynamics.

Lagrangian Descriptors (LDs) are scalar quantities able to reveal separatrices, manifolds of hyperbolic saddles, and chaotic seas of dynamical systems. A popular version of the LDs consists in computing the arc-length of trajectories over a calibrated time-window. Herein we introduce and exploit an intrinsic geometrical parametrisation of LDs, free of the time variable, for 1 degree-of-freedom Hamiltonian systems. The parametrisation depends solely on the energy of the system and on the geometry of the associated level curve. We discuss applications of this framework on classical problems on the plane and cylinder, including the cat's eye, 8-shaped and fish-tail separatrices. The developed apparatus allows to characterise semi-analytically the rate at which the derivatives of the geometrical LDs become singular when approaching the separatrix. For the problems considered, the same power laws of divergence are found irrespective from the dynamical system. Some of our results are connected with existing estimates obtained with the temporal LDs under approximations. The geometrical formalism provides alternative insights of the mechanisms driving this dynamical indicator.

Lagrangian descriptors (LDs) based on the arc length of orbits previously demonstrated their utility in delineating structures governing the dynamics. Recently, a chaos indicator based on the second derivatives of the LDs, referred to as \\(\\vert\\vert \\Delta \\rm{LD} \\vert\\vert\\), has been introduced to distinguish regular and chaotic trajectories. Thus far, the derivatives are numerically approximated using finite differences on fine meshes of initial conditions. In this paper, we instead use the differential algebra (DA) framework as a form of automatic differentiation to introduce and compute \\(\\vert\\vert \\Delta \\rm{LD} \\vert\\vert\\) up to machine precision. We discuss and exemplify benefits of this framework, such as the determination of reliable thresholds to distinguish ordered from chaotic trajectories. Our extensive parametric study quantitatively assesses the accuracy and sensitivity of both the finite differences and differential arithmetic approaches by focusing on paradigmatic discrete models of Hamiltonian chaos, namely the Chirikov's standard map and coupled \\(4\\)-dimensional variants. Our results demonstrate that finite difference techniques for \\(\\vert\\vert \\Delta \\rm{LD} \\vert\\vert\\) might lead to significant misclassification rate, up to \\(20\\%\\) when the phase space supports thin resonant webs, due to the difficulty to determine appropriate thresholds. On the contrary, \\(\\vert\\vert \\Delta \\rm{LD} \\vert\\vert\\) computed through DA arithmetic leads to clear bimodal distributions which in turn lead to robust thresholds. As a consequence, the DA framework reveals as sensitive as established first order tangent map based indicators, independently of the underlying dynamical regime. Finally, the benefits of the DA framework are also highlighted for non-uniform depleted meshes of initial conditions.

Lagrangian descriptors (LDs) based on the arc length of orbits previously demonstrated their utility in delineating structures governing the dynamics. Recently, a chaos indicator based on the second derivatives of the LDs, referred to as \\(\\vert\\vert \\Delta \\rm{LD} \\vert\\vert\\), has been introduced to distinguish regular and chaotic trajectories. Thus far, the derivatives are numerically approximated using finite differences on fine meshes of initial conditions. In this paper, we instead use the differential algebra (DA) framework as a form of automatic differentiation to introduce and compute \\(\\vert\\vert \\Delta \\rm{LD} \\vert\\vert\\) up to machine precision. We discuss and exemplify benefits of this framework, such as the determination of reliable thresholds to distinguish ordered from chaotic trajectories. Our extensive parametric study quantitatively assesses the accuracy and sensitivity of both the finite differences and differential arithmetic approaches by focusing on paradigmatic discrete models of Hamiltonian chaos, namely the Chirikov's standard map and coupled \\(4\\)-dimensional variants. Our results demonstrate that finite difference techniques for \\(\\vert\\vert \\Delta \\rm{LD} \\vert\\vert\\) might lead to significant misclassification rate, up to \\(20\\%\\) when the phase space supports thin resonant webs, due to the difficulty to determine appropriate thresholds. On the contrary, \\(\\vert\\vert \\Delta \\rm{LD} \\vert\\vert\\) computed through DA arithmetic leads to clear bimodal distributions which in turn lead to robust thresholds. As a consequence, the DA framework reveals as sensitive as established first order tangent map based indicators, independently of the underlying dynamical regime. Finally, the benefits of the DA framework are also highlighted for non-uniform depleted meshes of initial conditions.

Near-Earth space continues to be the focus of critical services and capabilities provided to the society. With the steady increase of space traffic, the number of Resident Space Objects (RSOs) has recently boomed in the context of growing concern due to space debris. The need of a holistic and unified approach for addressing orbital collisions, assess the global in-orbit risk, and define sustainable practices for space traffic management has emerged as a major societal challenge. Here, we introduce and discuss a versatile framework rooted on the use of the complex network paradigm to introduce a novel risk index for space sustainability criteria. With an entirely data-driven, but flexible, formulation, we introduce the Resident Space Object Network (RSONet) by connecting RSOs that experience near-collisions events over a finite-time window. The structural collisional properties of RSOs are thus encoded into the RSONet and analysed with the tools of network science. We formulate a geometrical index highlighting the key role of specific RSOs in building up the risk of collisions with respect to the rest of the population. Practical applications based on Two-Line Elements and Conjunction Data Message databases are presented.