AI 中文总结
该研究针对LEO源汇模型，将其建模为马尔可夫跳跃过程，推导得到随壳体积变窄而与确定性ODE偏离的SDE近似，可用于更准确评估窄壳下的LEO碰撞风险与可持续性。
AI 中文摘要
确定性源汇模型被广泛用于评估低地球轨道（LEO）环境的长期演化、容量及可持续性。这些模型通过常微分方程（ODE）传播壳平均种群，依赖单个碰撞、处置和衰变事件在足够大的高度壳内实现平均化。随着星座流量日益组织为千米级和亚千米级壳，这一平均假设不再成立。我们将多壳、多物种的LEO环境建模为马尔可夫跳跃过程，从该建模中恢复出传统源汇ODE作为大体积极限，并推导得到随机微分方程（SDE）近似，其波动按每壳体积V的V^{-1/2}缩放。该扩散近似通过对基础跳跃过程的精确离散事件模拟得到验证。在450-800 km频段上以固定空间密度扫描V，我们发现两种描述在与现有源汇模型所用壳体积相当的情况下一致，但在壳变窄时出现分歧。在所考虑的最细壳体积下，平均碎片种群达到ODE预测值的约4.5倍，且多个实现经历ODE轨迹中不存在的失控增长。这种偏离由非线性碰撞项驱动，通过该类项，种群方差和协方差提高了预期碰撞率，进而产生更多碎片并强化碰撞-碎片反馈。因此，窄壳随机性既拓宽了结果分布，又改变了预期轨迹。由于该随机模型与确定性模型共享参数化，它为现有源汇模型提供了尺度一致的扩展，用于评估壳配置、碰撞风险和长期LEO可持续性。
英文摘要
Deterministic source-sink models are widely used to assess the long-term evolution, capacity, and sustainability of the low Earth orbit (LEO) environment. These models propagate shell-averaged populations through ordinary differential equations (ODEs), relying on individual collision, disposal, and decay events to average out within sufficiently large altitude shells. As constellation traffic is increasingly organized into kilometer- and sub-kilometer-scale shells, this averaging assumption becomes strained. We formulate the multi-shell, multi-species LEO environment as a Markov jump process and, from this formulation, recover the conventional source-sink ODE as a large-volume limit and derive a stochastic differential equation (SDE) approximation whose fluctuations scale as V^{-1/2} in the per-shell volume V. The diffusion approximation is validated against an exact discrete-event simulation of the underlying jump process. Sweeping V at fixed spatial density over the 450-800 km band, we find that the two descriptions agree at shell volumes comparable to those used in established source-sink models but diverge as shells narrow. At the finest shell volume considered, the mean debris population reaches roughly 4.5 times the ODE prediction, with several realizations undergoing runaway growth absent from the ODE trajectory. This departure is driven by nonlinear collision terms, through which population variance and covariance raise expected collision rates, generating further debris and reinforcing the collision-debris feedback. Narrow-shell stochasticity thus both widens the outcome distribution and shifts the expected trajectory. Because the stochastic model shares its parameterization with the deterministic one, it provides a scale-consistent extension of existing source-sink models for evaluating shell configuration, collision risk, and long-term LEO sustainability.
