A Dynamical Systems Analysis of the Effects of the Launch Rate Distribution on the Stability of a Source-Sink Orbital Debris Model

Future launches are projected to significantly increase both the number of active satellites and aggregate collision risk in Low Earth Orbit (LEO). Ensuring the long-term sustainability of the space environment demands an accurate model to understand and predict the effect of launch rate distribution as a major driver of the evolution of the LEO orbital population. In this paper, a dynamical systems theory approach is used to analyze the effect of launch rate distribution on the stability of the LEO environment. A multi-shell, three-species source-sink model of the LEO environment referred to as MOCAT-3 for MIT Orbital Capacity Assessment Tool - 3 Species, is used to study the evolution of the species populations. The three species included in the model are active satellites, derelict satellites, and debris. Each shell is modeled by a system of three equations, representing each species, that are coupled through coefficients related to atmospheric drag, collision rate, mean satellite lifetime, post-mission disposal probability, and active debris removal rate. The major sink in the model is atmospheric drag, whereas the only source apart from collision fragments is the launch rate, making it the critical manageable factor impacting the orbital capacity. Numerical solutions of the system of differential equations are computed, and an analysis of the stability of the equilibrium points is conducted for numerous launch rate distributions. The stability of the equilibrium points is used to test the sensitivity of the environment to run-away debris growth, known as Kessler syndrome, that occurs at the instability threshold. Various bounding cases are studied from business-as-usual launch rates based on historic launch data, to high launch rates wherein a fraction of the satellite proposals filed with the International Telecommunication Union (ITU) are launched. An analysis of the environment's response to perturbations in launch rate and debris population is conducted. The maximum perturbation in the debris population from the equilibrium state, for which the system remains in a stable configuration, is calculated. Plots of the phase space about the equilibrium points are generated. The results will help to better understand the orbital capacity of LEO and the stability of the space environment, as well as provide improved guidelines on future launch plans to avoid detrimental congestion of LEO.