近地轨道接近满容量时的随机动力学
Stochastic Dynamics of Low Earth Orbit Near Full Capacity
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中文总结 AI 辅助
该研究针对近地轨道容量评估的确定性模型缺陷，构建两物种Lotka–Volterra模型的随机扩展，揭示不同时间尺度下的Kessler级联特性，指出碎片失控概率高于确定性预测，需考虑碰撞动力学方差。
中文摘要 AI 辅助
近地轨道（LEO）维持空间活动的容量正受到巨型星座、遗留碎片和新型有效载荷类别的日益增长的压力。现有对轨道容量和碎片演化的评估大多是确定性的，通过常微分方程跟踪完好卫星和碎片的平均数量；它们无法捕捉碰撞、碎裂尺寸和发射计划的固有随机性。我们开发了Bradley和Wein提出的两物种Lotka–Volterra模型的随机扩展，将其构建为依赖于密度的马尔可夫链，并研究其确定性和随机尺度极限。由于完好物体和碎片的数量级差异很大，这些极限出现在不同的时间尺度上，只有在相应的时间范围内，碰撞性Kessler级联的不同路径才会显现。在快速的完好物体时间尺度上，我们得到了一个常微分方程近似和一个高斯随机微分方程近似；在中等的碎片时间尺度上，我们得到了一个常微分方程近似、一个高斯随机微分方程近似，以及临界Kessler阈值，超过该阈值后，常微分方程近似会在Kessler综合征中失控。至关重要的是，在临界阈值的第三个慢时间尺度上，碎片数量收敛到Feller扩散，其中失控纯粹由波动而非漂移触发——这是常微分方程近似及其高斯随机微分方程近似无法观测到的效应。碎片失控可能比确定性模型预测的更早发生，且概率更高：完好物体种群可能表现良好，而碎片却在悄悄积累风险。星座部署、碎片清除投资和轨道位置分配应考虑这些随机效应，对失控的规划必须依赖于碰撞动力学的方差，而非仅依赖于均值。
英文摘要
The capacity of Low Earth Orbit (LEO) to sustain space operations is under mounting pressure from megaconstellations, legacy fragmentation debris, and new payload classes. Existing assessments of orbital capacity and debris evolution are largely deterministic, tracking mean populations of intact satellites and fragments with ordinary differential equations; they cannot capture the inherent randomness of collisions, breakup sizes, and launch schedules. We develop a stochastic extension of the two-species Lotka--Volterra model of Bradley and Wein, formulated as a density-dependent Markov chain, and study its deterministic and stochastic scaling limits. Because intacts and fragments differ by many orders of magnitude, these limits emerge on distinct time-scales, and different pathways to a collisional Kessler cascade become visible only on the appropriate time horizon. On a fast intact time-scale we obtain an ODE approximation and a Gaussian SDE approximation; on an intermediate fragment time-scale we obtain an ODE approximation, a Gaussian SDE approximation, and the critical Kessler threshold, above which the ODE approximation runs away in Kessler syndrome. Crucially, on a third, slow time-scale at the critical threshold, the fragment count converges to a Feller diffusion, in which runaway is triggered purely by fluctuations rather than by the drift---an effect the ODE approximations and their Gaussian SDE approximations cannot see. Debris runaway may occur sooner, and with higher probability, than deterministic models predict: the intact population can appear well-behaved while fragments quietly accumulate risk. Constellation deployment, debris-removal investment, and slot allocation should account for these stochastic effects, and planning for runaway must depend on the variance of the collision dynamics, not on the mean alone.
