Satellite collisions or fragmentations generate a huge number of space debris; over time, the fragments might get dispersed, making it difficult to associate them to the configuration at break-up. In this work, we present a procedure to back-trace the debris, reconnecting them to their original configuration. To this end, we compute the proper elements, namely dynamical quantities which stay nearly constant over time. While the osculating elements might spread and lose connection with the values at break-up, the proper elements, which have been already successfully used to identify asteroid families, retain the dynamical features of the original configuration. We show the efficacy of the procedure, based on a hierarchical implementation of perturbation theory, by analyzing the following four different case studies associated to satellites that underwent a catastrophic event: Ariane 44lp, Atlas V Centaur, CZ-3, Titan IIIc Transtage. The link between (initial and final) osculating and proper elements is evaluated through tools of statistical data analysis. The results show that proper elements allow one to reconnect the fragments to their parent body.
Proper elements are quasi-invariants of a Hamiltonian system, obtained through a normalization procedure. Proper elements have been successfully used to identify families of asteroids, sharing the same dynamical properties. We show that proper elements can also be used within space debris dynamics to identify groups of fragments associated to the same break-up event. The proposed method allows to reconstruct the evolutionary history and possibly to associate the fragments to a parent body. The procedure relies on different steps: (i) the development of a model for an approximate, though accurate, description of the dynamics of the space debris; (ii) the construction of a normalization procedure to determine the proper elements; (iii) the production of fragments through a simulated break-up event. We consider a model that includes the Keplerian part, an approximation of the geopotential, and the gravitational influence of Sun and Moon. We also evaluate the contribution of Solar radiation pressure and the effect of noise on the orbital elements. We implement a Lie series normalization procedure to compute the proper elements associated to semi-major axis, eccentricity and inclination. Based upon a wide range of samples, we conclude that the distribution of the proper elements in simulated break-up events (either collisions and explosions) shows an impressive connection with the dynamics observed immediately after the catastrophic event. The results are corroborated by a statistical data analysis based on the check of the Kolmogorov-Smirnov test and the computation of the Pearson correlation coefficient.
Perturbative methods have been developed and widely used in the XVIII and XIX century to study the behavior of N-body problems in Celestial Mechanics. Such methods apply to nearly-integrable Hamiltonian systems and they have the remarkable property to be constructive. A well-known application of perturbative techniques is represented by the construction of the so-called proper elements, which are quasi-invariants of the dynamics, obtained by removing the perturbing function to higher orders. They have been used to identify families of asteroids; more recently, they have been used in the context of space debris, which is the main core of this work. We describe the dynamics of space debris, considering a model including the Earth’s gravitational attraction, the influence of Sun and Moon, and the Solar radiation pressure. We construct a Lie series normalization procedure and we compute the proper elements associated to the orbital elements. To provide a concrete example, we analyze three different break-up events with nearby initial orbital elements. We use the information coming from proper elements to successfully group the fragments; the clusterization is supported by statistical data analysis and by machine learning methods. These results show that perturbative methods still play an important role in the study of the dynamics of space objects.
Normal form methods allow one to compute quasi-invariants of a Hamiltonian system, which are referred to as proper elements. The computation of the proper elements turns out to be useful to associate dynamical properties that lead to identify families of space debris, as it was done in the past for families of asteroids. In particular, through proper elements we are able to group fragments generated by the same break-up event and we possibly associate them to a parent body. A qualitative analysis of the results is given by the computation of the Pearson correlation coefficient and the probability of the Kolmogorov-Smirnov statistical test.
The computation of the proper elements of a Hamiltonian system can be done by using perturbation theory techniques. For the space debris problem, the normalization procedure is an iterative method of reducing the initial Hamiltonian system to a simplified form, by removing the fast and semi-fast angles of the model which describes the dynamics of the system. It is worth mentioning that in the space debris problem, the perturbations to be included in the model depend on the altitude of the space debris. The new elements obtained after the normalization process define the proper elements, which are quasi-invariants of motion, hence quantities nearly constant in time. We show that these elements are particularly useful in the classification and back-tracing of space debris, as well as in the possible recognition of their origin. The results are supported by appropriate simulations and data analysis. Work in collaboration with A. Celletti and G.Pucacco
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