Simulation of the space debris environment in LEO using an analytical approach

Several numerical approaches exist to simulate the evolution of the space debris environment. These simulations usually rely on the propagation of a complete population of objects in order to determine the collision probability for each object. Using a Monte Carlo (MC) approach the chances for events, such as explosions and collisions, are triggered based on an assumed probability distribution. So in many different scenarios different objects are fragmented and contribute to a different version of the space debris environment. Finally, the results of the different scenarios are averaged to get a statistically significant estimation. This method is computationaly very expensive due to the propagation of the objects and the application of the MC method. At the Institute of Aerospace Systems (ILR) an analytical model capable of describing the evolution of the space debris environment has been developed and implemented. The model is based on source and sink mechanisms, where yearly launches as well as collisions and explosions are considered as sources. The natural decay and post mission disposal measures are the only sink mechanisms. This method reduces the computational costs tremendously. In order to achieve this benefit a few simplifications have been applied. The approach of the model partitions the LEO into altitude shells. Only two kinds of objects are considered, intact bodies and fragments, which are also divided into diameter bins. As an extension to the previously presented model the eccentricity has additionally been taken into account with 67 eccentricity bins. While a set of differential equations has been implemented in a generic manner, the Euler method has been chosen to integrate the equations for a given time span. For this paper parameters have been chosen so that the model is able to reflect the results of the numerical MC-based simulation LUCA, which is also being developed at the ILR. The evolution of the population in LEO for a 200 years time span is shown and compared for both approaches using step sizes of 1 year. For selected objects in LEO the flux and environmental criticality values are shown. In conclusion the field of application for such a fast analytical model is shown.