Neglecting small fragments in space debris evolutionary models can lead to a significant underestimation of the collision risk for operational satellites. However, when scaling down to the millimetre range, the debris population grows to over a hundred million objects, making deterministic approaches too computationally expensive. On the contrary, probabilistic models provide a more efficient alternative, which however typically works under some simplifying assumptions on the dynamics, limiting their field of applicability. This work proposes an extension of the density-based collision risk models to any orbital dynamics and impact geometry. The impact rate with a target satellite is derived from a multi-dimensional phase space density function in orbital elements, which discretely varies over both phase space and time. The assumption of a bin-wise constant cloud density allows for the analytical transformation of the six-dimensional distribution in orbital elements into the three-dimensional spatial density function, guaranteeing an efficient and accurate evaluation of the fragments flux. The proposed method is applied to the assessment of the collision risk posed by occurred fragmentation events in different orbital regions on a high-risk target object. The effect on the impact rate of the modelling improvements, compared to previous probabilistic formulations, is discussed.

中文翻译：

---

基于密度的在轨碰撞风险模型适用于任何碰撞几何形状


忽略空间碎片演化模型中的小碎片可能会导致对运行卫星的碰撞风险的严重低估。然而，当缩小到毫米范围时，碎片数量会增长到超过一亿个物体，使得确定性方法的计算成本过于昂贵。相反，概率模型提供了一种更有效的替代方案，但通常在一些简化的动力学假设下工作，限制了其适用范围。这项工作提出将基于密度的碰撞风险模型扩展到任何轨道动力学和碰撞几何形状。与目标卫星的撞击率是从轨道元素中的多维相空间密度函数得出的，该函数在相空间和时间上离散变化。分箱恒定云密度的假设允许将轨道元素的六维分布解析转换为三维空间密度函数，保证碎片通量的高效和准确评估。该方法应用于评估高风险目标物体不同轨道区域发生的碎片事件所造成的碰撞风险。与之前的概率公式相比，讨论了建模改进对影响率的影响。