Daniel Jang, Massachusetts Institute of Technology; Andrea D’Ambrosio, Massachusetts Institute of Technology; Miles Lifson, Massachusetts Institute of Technology; Celina Pasiecznik, Massachusetts Institute of Technology; Richard Linares, Massachusetts Institute of Technology
Keywords: Stability analysis, source-sink, LEO environment, spacecraft collision and fragmentation, space debris population
Abstract:
The new space age has brought with it a congested LEO environment, where new commercial and government mega constellations are being proposed at a faster rate that require strict management of the orbital architecture. Effective policies and technologies to enable deorbiting measures for post mission disposal (PMD) and active debris removal (ADR) are needed to remove payloads that are past their lifetimes and other debris objects. Anthropogenic fragmentation events such as destructive anti-satellite tests produce thousands of debris pieces. Natural conjunctions happen as well, with around 300 on-orbit fragmentation events having occurred to date. As the number of satellites and debris continue to grow, it is imperative to calculate the relative and absolute risk of collisions in LEO.
There are largely two methods to model the LEO environment and collision risk: statistical sampling methods such as Monte Carlo methods, and evolutionary models. Statistical sampling methods propagate the best known object states with high fidelity propagators to estimate the future space environment. This method allows for accurate near-future predictions of potential collisions and is used operationally today for conjunction avoidance. Existing models include NASA’s Orbital Debris Engineering Model (ORDEM) and LEGEND, European Space Agency’s Orbital Debris Evolutionary Model (ODEM), Chinese Academy of Sciences’ SOLEM (Space Objects Long-term Evolution Model), University of Southampton’s Debris Analysis and Monitoring Architecture for the Geosynchronous Environment (DAMAGE) and more. For each of these models, the debris population and densities are outputted for some initial condition and assumptions. Computing a debris environment with different sets of assumptions, however, requires high computational cost as each object must be propagated. Sampling over a distribution of uncertainties on the states and parameters would require an exponential number of propagations. The high cost is due to the small time steps required to accurately model a collision and semi-analytical propagators requiring high compute cost to propagate far into the future. For each collision or fragmentation event, some breakup model such as NASA Standard Breakup Model is used to model the debris cloud generation. Though the outputted debris distribution for some assumed initial condition and future traffic model exist, all of these models are closed-source and inputting arbitrary assumptions is difficult if not impossible.
Evolutionary models describe the interactions between populations of objects with ordinary differential equations. For example, if all space objects of interest are categorized as payloads, derelict satellites or debris, three ordinary differential equations can describe the interaction between these populations, much like the Lotka–Volterra system of equations. Average values are often used to set the parameters. This method removes the need for computationally expensive propagation of object states to estimate a future debris environment. Gross populations are propagated forward according to the governing differential equations, which allows for fast solutions even far into the future. Exploration of a wide set of initial conditions and parameters is much more approachable. Kessler and Cour-Palais first described the feedback runaway phenomena and identified the risk of an exponential increase in the number of space debris, and since then a few evolutionary models have been proposed in the literature. Talent introduces the particles-in-box (PIB) model where a population within an orbital shell is assumed to have some average characteristic and interactions. University of Southampton’s Fast Debris Evolution (FADE) used simplified first order differential equations to describe the population interaction. JASON describes a three-population model for one shell and a given launch cadence. While other models have expanded the evolutionary model to analyze multiple shells, optimal control schemes and economic equilibrium for maximum policy intake, none have analyzed the multi-shell stability using evolutionary models, inclusion of non-circular orbits and periodic atmospheric density fluctuations. A higher fidelity evolutionary model will require these attributes, and that is described in this work.
We develop an evolutionary dynamical system model for the LEO population and identify the stable equilibrium points for a debris environment with non-circular debris population and periodic atmospheric density. The LEO environment is modeled as a multi-shelled system with a feedback control loop, where the control variables are the launch rate and PMD rate. Time-varying atmospheric model is used and its effect on the periodic equilibrium manifold is analyzed. Fragmentation model from NASA’s standard breakup model and the empirical data from Gabbard diagram formation theory are compared and levereged in this model. A range of launch profiles and PMD rates are identified to create a stable state of LEO population in the dynamical system.
Furthermore, various equilibrium points are defined and discussed, such as partial stability and global stability. Partial stability refers to the case where a particular population is in equilibrium without the others necessarily being in equilibrium. Global stability refers to the equilibrium point where there is no change in any of the populations, and the rate of change for all populations is 0. The significance and implication of these definitions are discussed.
Date of Conference: September 27-20, 2022
Track: Space Debris