Analytical Study of the Co-orbital Motion in the Circular Restricted Three-body Problem

In the restricted three-body problem(RTBP), if a small body and a planet stably orbit around a central star with almost exactly the same semimajor axis, and thus almost the same mean motion, this phenomenon is called the coorbital motion, or equivalently, the 1:1 mean motion resonance. The classical expansion of the disturbing function is divergent when the semimajor axis ratio of the small body to the planet is close to unity. Thus, most of the previous studies on the co-orbital dynamics were carried out through numerical integrations or semi-analytical approaches. In this work, we construct an analytical averaged model for the co-orbital motion in the framework of the circular RTBP. This model is valid in the entire coorbital region except in the vicinity of the collision singularity. The results of the analytical averaged model are in good agreement with the numerical averaged model even for moderate eccentricities and inclinations. The analytical model can reproduce the tadpole, horseshoe and quasi-satellite orbits common in the planar problem. Furthermore, the asymmetry of 1:1 resonance and the compound orbits(Icarus 137:293–314) in the general spatial problem can also be obtained from the analytical model.

A Data-driven Method for Realistic Covariance Prediction of Space Object with Sparse Tracking Data

Covariance of the orbital state of a resident space object(RSO)is a necessary requirement for various space situational awareness tasks,like the space collision warning.It describes an accuracy envelope of the RSO's location.However,in current space surveillance,the tracking data of an individual RSO is often found insufficiently accurate and sparsely distributed,making the predicted covariance(PC)derived from the tracking data and classical orbit dynamic system usually unrealistic in describing the error characterization of orbit predictions.Given the fact that the tracking data of an RSO from a single station or a fixed network share a similar temporal and spatial distribution,the evolution of PC could share a hidden relationship with that data distribution.This study proposes a novel method to generate accurate PC by combining the classical covariance propagation method and the data-driven approach.Two popular machine learning algorithms are applied to model the inconsistency between the orbit prediction error and the PC from historical observations,and then this inconsistency model is used for the future PC.Experimental results with the Swarm constellation satellites demonstrate that the trained Random Forest models can capture more than 95%of the underlying inconsistency in a tracking scenario of sparse observations.More importantly,the trained models show great generalization capability in correcting the PC of future epochs and other RSOs with similar orbit characteristics and observation conditions.Besides,a deep analysis of generalization performance is carried out to describe the temporal and spatial similarities of two data sets,in which the Jaccard similarity is used.It demonstrates that the higher the Jaccard similarity is,the better the generalization performance will be,which may be used as a guide to whether to apply the trained models of a satellite to other satellites.Further,the generalization performance is also evaluated by the classical Cramer von Misses test,which also shows that trained models have encouraging generalization performance.