Space debris families: from perturbative methods to machine learning techniques

We study the secular dynamics of a space debris around the Earth through different analytical and computational tools, ranging from Hamiltonian perturbation theory to machine-learning techniques. Precisely, we consider the gravitational attraction of the Earth, including up to second degree terms (J2 and J3), the influence of Sun and Moon, as third-body perturbations, and the effects of Solar radiation pressure. This model depends on several variables that vary with different rates. This motivates the implementation of a hierarchical perturbation theory to compute the so-called proper elements, which are constants of motion for the normalized Hamiltonian without the remainder function. It should be mentioned that proper elements have been introduced by Hirayama in 1918 in the context of asteroid families.

The novelty of the work presented here is that we compute proper elements for space debris, namely fragments generated by a break-up event, and use the results to classify the fragments into families. To this end, we construct in JAVA a simulator of catastrophic events (a collision or an explosion) based on the NASA model EVOLVE 4.0. We propagate the fragments at different epochs to obtain the osculating or mean elements by integrating Newton’s equations or the averaged Hamiltonian, respectively, and we compute the proper elements at each epoch. Finally, we compare the osculating (or mean) and proper elements using statistical data analysis and we cluster the fragments using machine-learning techniques. The results clearly show that proper elements are a unique characteristic signature of the fragments, that allows us to group them into families.

The impact of the method is clearly shown by tests on real cases: Ariane 44lp, Atlas V Centaur, CZ-3, Titan IIIc Transtage. Using the elements in the database “Stuff in Space”, we propagate backward in time to successfully reconnect the space debris to their parent body (even not knowing the break-up time). The potentiality of the procedure is further witnessed by a sample where the proper elements allow to distinguish between fragments generated by nearby break-up events.