The effects of positional uncertainty on the Gaussian probability computation for orbit conjunction are examined and an upper bound determined. Relative motion between two objects is assumed linear for a given encounter with time-invariant position covariance. A method is developed to assess the maximum probability for various satellite sizes, encounter geometries, and covariance sizes and shapes. The associated standard deviation then defines the boundary of probability dilution. The assertion is made that orbit positions should be sufficiently accurate to avoid this dilution region. This work shows how to calculate the upper bounds of probability by assuming worst-case covariance orientation and size. Power series approximations are developed for aspect ratios ranging from 1 to 50 to capture 99% of all conjunction possibilities. An analytical approximation is also given for an infinite aspect ratio to capture all possibilities. These expressions can be used as a simple pre-filter or to determine worst-case scenarios. Although desired, the actual covariances are not needed. What is needed is the ratio of major-to-minor axes of the projected combined covariance ellipse, the object sizes, and the relative distance at the point of closest approach.

中文翻译：

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将位置不确定性与最大连接概率相关

检查了位置不确定性对高斯概率计算的影响，并确定了上限。对于给定的时间不变位置协方差，两个对象之间的相对运动被假定为线性。开发了一种方法来评估各种卫星大小，遇到的几何形状以及协方差大小和形状的最大概率。然后，相关的标准差定义概率稀释的边界。有人断言，轨道位置应足够准确，以避开该稀释区。这项工作展示了如何通过假设最坏情况的协方差方向和大小来计算概率的上限。幂级数逼近针对1至50的宽高比而开发，以捕获所有连合可能性的99％。还给出了无限长宽比的解析近似值，以捕获所有可能性。这些表达式可以用作简单的预过滤器或确定最坏情况。尽管需要，但不需要实际的协方差。所需的是投影的组合协方差椭圆的长轴与短轴之比，对象大小以及最接近点的相对距离。