Long-Term Analytical Boundedness Conditions for Relative Orbits Under Third-Body Perturbations

What is it about?

The singly-averaged relative distances and rates induced by the third-body perturbation, where the third body is in an elliptic inclined orbit, can be used for predicting the long-term evolution of the relative motion under differential third-body effects. The averaged relative distance changes nonlinearly with respect to time. In addition, the distances in the radial and out-of-plane directions vary nearly periodically, but with large oscillations, and a significant secular drift exists in the along-track direction. According to the analysis, the differences in $\bar e$, $\bar i$, $\bar {\Omega}$ are the main factors responsible for the large periodic variations of the relative distance. By initializing these differential elements to zero, a new boundedness condition yielding small periodic variations is proposed to mitigate differential third-body effects. The relative distance based on the new analytical boundedness condition yields smaller periodic variations, as expected, and drifts slower than scenarios designed based on $J_2$-only invariance conditions.

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