Perturbative methods combined with machine learning techniques to cluster space debris

We develop a new procedure, based on perturbation theory for dynamical systems and machine learning, to approach the problem of clustering groups of space debris surrounding the Earth. Precisely, we study the simulations of the explosion of multiple satellites, each one generating a group of fragments with different numerosity. For each fragment, we determine the orbital elements at the initial time and then we follow their evolution for some years. The objective is to distribute the fragments into a number of clusters that corresponds to the initial count of satellites, so that each cluster is associated to its parent body. We adopt a model described by a Hamiltonian that encompasses the Keplerian component, the geopotential (restricted to the most significant harmonic coefficients), the lunar and solar gravitational influences, as well as the solar radiation pressure. The mean elements are computed by implementing an averaging process of the Hamiltonian over the fast variables. Employing perturbation theory, we also calculate the so-called proper elements, which prove to be quasi-integrals for the Hamiltonian system. Taking advantage of the information provided by mean and proper elements, we proceed to implement clusterization methods. The efficacy of these methods requires a careful analysis of different factors, that we highlight and analyze in detail. In particular, we study the impact in the variation of the orbital elements of the fragments over time, how much they overlap at the initial time, the dependence on the times of observation, the choice of the distance between fragments, the phase space region where the break-up events occur. The analysis of such factors, together with the implementation of clustering methods using mean and proper elements, leads to the definition of a well-defined procedure to cluster fragments after multiple break-up events.