Modeling Medium-Term Debris Cloud of Satellite Breakup via Probabilistic Method

Accurately modeling the evolution of debris clouds from satellite breakups is necessary for assessing the collision risk to space assets. Generally, the evolution of debris clouds can be divided into short-, medium-, and long-term phases. Compared to the short-and long-term phases, the study on the medium-term phase was insufficient. This paper proposes a probabilistic method to model the medium-term evolution of a debris cloud. First, a satellite breakup event can be characterized by a probabilistic density distribution of the initial velocity vectors of all fragments. Then, the distribution in the initial velocity can be transformed to that in three classic orbital elements, i.e., semimajor axis, eccentricity, and inclination. Given that the driving force of medium-term evolution is the J₂ perturbation and that J₂ has no secular contribution to these three classic orbital elements, the distribution of classic orbital elements remains stationary during the evolution. This feature can be exploited to express the probabilistic density distribution of any given target position in terms of the density distribution of the classic orbital elements by changing variables. Finally, 16 one-variable equations with analytical first derivatives were formulated to find the proper inverse mapping from the given position to the classic orbital elements, and the Jacobian matrix of the inverse mapping was derived. Medium-term debris clouds resulting from two typical breakup velocity distributions were tested using the proposed method. Results show that the proposed method can accurately simulate the evolution of the clouds’ geometrical extent and density flow.