Jaewon Choi, Myongji University; Souvik Dhara, Georgia Institute of Technology; Harsha Honnappa, Purdue University
https://doi.org/10.64861/TCAS5266
Keywords: LEO environment modeling, source–sink model, stochastic diffusion, orbital shell capacity
Abstract:
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Low Earth orbit (LEO) has entered a regime in which anthropogenic space objects—active spacecraft, failed payloads, and fragments—interact nonlinearly through launch, decay, and collision processes. Source–sink models, often called “particle-in-a-box” frameworks, have become a mainstay for long-horizon, system-level assessments of the orbital environment. These models compress the environment into altitude shells and propagate populations with coupled ordinary differential equations (ODEs), enabling fast policy exploration compared with object-by-object propagation. Recent exemplars include multi-species, multi-shell frameworks that estimate capacity or stability under varying launch and post-mission disposal (PMD) assumptions. However, most existing risk-based source–sink models use relatively coarse altitude bins (often 10–50 km) and treat the evolution of satellite, derelict, and debris populations as purely deterministic. A growing line of work has highlighted that such deterministic treatments can obscure variability that matters for safety and sustainability: stochastic formulations reveal tails—debris run-ups that may be policy-relevant even when mean trajectories look benign. At the same time, intrinsic geometric capacity studies show that coordination choices and shell geometry strongly influence how densely shells can be packed without self-conjunctions, introducing constraints that interact with, but are distinct from, debris-evolution risk._x000D_
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Current and planned constellations increasingly operate in tightly spaced orbital shells (often 5 km or less in vertical separation), where intra-shell and inter-shell fluxes, discrete collision events, and PMD compliance variability can create genuinely stochastic dynamics at the shell level. Capacity ranking exercises that aggregate risk over broad altitude bins underscore the value of shell-wise assessment but also hint that bin size itself can alter the apparent severity ordering across LEO, reinforcing the need to resolve narrow shells when local demand and coordination compress vertical spacing._x000D_
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This paper argues that narrow-shell stochasticity is a first-order feature of the contemporary LEO environment. We formalize a Markov jump process (MJP) for launches, collisions (including derelict–derelict and satellite–satellite interactions), debris generation, and drag-driven cross-shell fluxes. Each shell tracks three species—active satellites, derelicts, and debris—and all transitions are modeled as Markov jump events with intensities per unit volume. For any pair of species, the raw encounter intensity scales with a geometric collision kernel combining a squared impact parameter and the mean relative speed. Each encounter is resolved into lethal, disabling, or no-effect outcomes using physically interpretable coefficients. Debris yields follow the NASA Standard Breakup Model (SBM). Atmospheric drag induces a Markovian downward flux from shell to shell at per-object rates derived from a standard analytic drag mapping using local atmospheric density and ballistic coefficients. Launches enter as independent Poisson processes, and post-mission disposal acts on active satellites with a configurable success probability, where failure diverts a satellite to the derelict population._x000D_
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From this MJP we derive two complementary continuum limits. First, by the Kurtz–Ethier law of large numbers, we obtain a deterministic ODE limit for mean behavior in which each jump intensity is replaced by its expectation, recovering the familiar coupled ODEs of prior source–sink models (e.g., MOCAT-3) while retaining our explicit jump accounting for bookkeeping. Second, by a functional central limit theorem, we derive a diffusion (SDE) approximation that captures O(V^{-1/2}) fluctuations in shell populations, where V denotes shell volume. The finite-volume fluctuations scale as V^{-1/2}, so the density approximation including noise converges to a Gaussian limit and yields an Itô-form SDE with independent Brownian drivers for each collision channel, drag flux, launch process, and PMD event. The framework is parameterized in a manner fully compatible with prior source–sink models—kinetic-theory collision rates, NASA breakup scalings, and exponential-atmosphere drag—so it can be directly compared against established ODE-based analyses while explicitly representing randomness at the shell scale._x000D_
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Numerical experiments employ both an adaptive embedded Runge–Kutta ODE solver (Dormand–Prince) with automatic stiffness detection and fallback to an implicit BDF method, and Euler–Maruyama time-stepping for the SDE, supplemented by a Gillespie-style discrete-event sampler for cross-validation. We enforce nonnegativity via projection, apply conservative shell-coupling updates to maintain mass balance across adjacent shells, and verify convergence by confirming that halving the time step shifts ensemble means by less than one percent over 100-year horizons. Results aggregate many independent trajectories per scenario, reporting means and 95% pointwise bands._x000D_
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Using this formulation, we show three principal findings. First, deterministic equilibria may mask sizable dispersion in long-run outcomes at fine shell widths: where an ODE predicts a single steady state, the stochastic ensemble reveals a distribution of outcomes whose spread grows as shells narrow and effective volumes shrink. Second, stochastic trajectories can produce debris cascades that depart markedly from mean ODE paths—even when average flows remain bounded, individual realizations can undergo runaway debris growth. Third, these differences are most pronounced in the narrow, tightly spaced shells that characterize modern mega-constellation altitude bands, where the effective volume per shell is smallest._x000D_
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These results bear on ongoing work on shell design and slotting, where tighter, geopotential-aware packing raises intrinsic capacity. Our findings suggest such packing may also heighten sensitivity to rare events, a trade-off between geometric capacity and stochastic risk that warrants joint study._x000D_
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In sum, our contribution is to integrate narrow-shell stochasticity into a source–sink compatible framework and to quantify how randomness at the shell level can reshape long-term risk assessments and policy trade-offs for PMD and launch pacing. Our framework links operator decisions—launch scheduling, avoidance efficacy—to environment-wide risk, offering actionable guidance for safe, high-density use of LEO._x000D_
Date of Conference: September 15-18, 2026
Track: Space Debris