Limitations of the cube method for assessing large constellations

Due to the significant computational demands involved in the long-term projection of large debris populations, evolutionary models make use of fast and efficient algorithms for collision risk assessment. A commonly used algorithm for collision risk assessment is the cube method, introduced by Liou et al. in 2003. Relatively little research has been undertaken to understand the collision probability estimation errors that arise from the use of the cube method. These errors are especially important when assessing the collision risk associated with large constellations because of the structure and the relatively high number of satellites involved. This paper investigates the cube method, with the aim of determining how the choice of the cube size and time-step affects collision rate estimates. The work used the University of Southampton's Debris Analysis and Monitoring Architecture to the Geosynchronous Environment (DAMAGE) tool to build hypotheses about the influence of the model parameters on the collision rate estimates for orbiting satellites in idealized scenarios, including a large constellation. The results showed that cube sizes between 4 km and 16 km were required to deliver collision rate estimates for discrete object pairs, with smaller cube sizes requiring time-steps of less than 0.1 days. For long-term projections involving many objects in the Low Earth Orbit (LEO) environment, such small time-steps are impractical, as simulation run-times for a single Monte Carlo run would increase from hours to days, or even weeks.