Priyank Behera, Clemson University; Aditya Gopalan, Purdue University; Harsha Honnappa, Purdue University
https://doi.org/10.64861/SYRV9892
Keywords: orbital capacity, space debris, stochastic modeling, carrying capacity, debris removal, fast-slow dynamics
Abstract:
The capacity of Low Earth Orbit (LEO) to support sustained space operations is under increasing pressure.
The number of tracked objects larger than 10 cm has grown sharply over the past two decades, driven by the rapid deployment of commercial megaconstellations, legacy debris from historical fragmentation events, and the emerging prospect of orbital data centers and other non-traditional payloads. The concentration of activity in preferential altitude bands — particularly the 500–600 km regime — raises urgent questions about how much operational capacity remains and how quickly it could be lost.
Collision avoidance maneuvers for Starlink alone have grown exponentially, with the annual maneuver rate scaling super-linearly with constellation size.
This trajectory suggests a potential tragedy of the commons, in which the aggregate behavior of many operators degrades the orbital environment for all.
Existing models for assessing orbital capacity and debris evolution are predominantly deterministic. More precisely, they track mean population counts of intact satellites and fragments using ordinary differential equations (ODEs). While these models provide valuable baseline forecasts, they cannot capture the inherent randomness of the LEO environment. Individual collision events are probabilistic, the number of fragments produced in any given breakup varies substantially, and launch schedules introduce further variability.
This paper develops a stochastic extension of the two-compartment Lotka-Volterra model introduced by Bradley and Wein [1], which tracks two interacting species — intact satellites and debris fragments — and investigates how randomness affects predictions of orbital capacity, critical density thresholds, and the onset of collisional cascading. The Bradley and Wein model captures the essential dynamics through a generalized Lotka-Volterra system. Intact satellites are injected at a prescribed launch rate, lost through atmospheric drag, and destroyed through collisions with other intacts or with fragments. Fragments are created by those same collisions and removed through drag.
The ODE system has a unique equilibrium representing the carrying capacity of the orbit, viz., the maximum number of intact satellites the shell can support without triggering runaway debris growth. This equilibrium is governed by a criticality parameter, Δ, defined as the difference between the fragment drag rate and the product of the intact-fragment collision efficiency and the equilibrium intact count; this is also defined as the carrying capacity of the orbit. When Δ is positive, drag exceeds fragment creation and the orbit is stable. When Δ reaches zero, the system is at critical carrying capacity. When Δ becomes negative, collisional cascading begins.
If LEO is operated near its capacity, we expect that the numbers of intacts and fragments differ by many orders of magnitude. In this paper, we study different deterministic and stochastic limits of a certain density-dependent Markov chain which is in the spirit of the Bradley and Wein [1] model. Due to the difference in order of magnitude of the intact and fragment counts, limits occur on different time-scales: depending on the type of randomness one wishes to study, one needs to examine different time horizons. There are several types of limits (depending on the time-scale), and thus several distinct pathways through which a collisional cascade could begin. In this paper, we specifically study the situation when there are many more fragments than intacts. Here, intact satellite population fluctuates rapidly on a fast timescale, while fragments evolve slowly, undergoing sustained high-amplitude excursions. Fragments can drift upward over long periods while the intact count appears roughly stable. Deterministic models are structurally unable to capture this phenomenon.
The multiple time-scale behavior also yields a third time-scale in which a collisional cascade can begin purely from the fluctuation behavior.
This is in stark contrast to results from fluid limits, wherein the collisional cascades are purely a result of the drift exceeding a threshold; functional central limit theorems around these fluid limits also cannot address the fact that fluctuations can cause collisional cascades.
Our results are as follows: on a fast time-scale for the intacts, we obtain both a fluid limit and a functional central limit theorem. We also obtain a fluid limit and a functional central limit theorem on an intermediate time-scale for the fragments. On this time-scale, we obtain a drift threshold above which the fluid limit exhibits collisional cascades. On a third, slow time-scale, we show that the limiting process is a Feller diffusion, whose collisional cascade behavior is stochastic, and determined purely by the fluctuation behavior, instead of by the drift. This final result is analogous to heavy traffic limits in queueing theory, although the result is a Feller diffuion.
The practical implications are significant: if debris runaway can occur substantially sooner and with higher probability than deterministic models predict, current capacity assessments may be systematically overoptimistic. The fast-slow structure means warning signs may be subtle. For instance, the intact population can appear well-behaved while fragments quietly accumulate risk. Constellation deployment, debris removal investment, and orbital slot allocation should all account for these stochastic effects. In addition, the slowest time-scale result indicates that planning for debris runaway cannot be a function only of the mean: the variance also plays a key role in determining pathways for debris runaway.
Date of Conference: September 15-18, 2026
Track: Space Debris