Probabilistic multi-dimensional debris cloud propagation subject to non-linear dynamics

The permanent power loss and the deviation of the trajectory of satellites impacted by centimetre and sub-centimetre sized debris have highlighted the need of taking into account such small fragments in the evolutionary models of the debris population and in the assessment of the in-orbit collision risk. When scaling down to the centimetre-millimetre range, deterministic models for propagating the fragments’ orbit suffer from the massive computational cost required. The continuum approach for modelling the debris clouds is a well-established alternative to the piece-by-piece propagation. A density function is formulated to describe the distribution of fragments over a suitable phase space. Accurate and efficient continuum formulations have been developed to propagate clouds of fragments under atmospheric drag and J2 perturbations, but a general model able to work under any dynamical regime has still to be found. This paper proposes a continuum approach that combines the method of characteristics with the discretisation of the domain in Keplerian elements and area-to-mass ratio into bins. The problem of using a binning approach with such a multi-dimensional phase space is addressed bounding and partitioning the domain, through probabilistic models on the way the fragments distribute over the phase space, as consequence of a fragmentation event. The proposed approach is applied to the modelling and propagation of a space debris cloud under the full set of orbital perturbations, and compared against a Monte Carlo simulation in terms of objects’ number and distribution. The method proves to be accurate on the medium scale, in both space and time, and guarantees statistical validity with a reduced computational effort, leveraging its probabilistic nature.