Analysis of the Representation of Orbital Errors and Improvement of their Modelling

Sammanfattning: In Space Situational Awareness (SSA), it is crucial to assess the uncertainty related to thestate vector of resident space objects (RSO). This uncertainty plays a fundamental role in, forexample, collision risk assessment and re-entry predictions. A realistic characterization of thisuncertainty is, therefore, necessary.The most common representation of orbital uncertainty is through a Gaussian (or normal)distribution. However, in the absence of new observations, the uncertainty grows over timeand the Gaussian representation is no longer valid under nonlinear dynamics like spacemechanics. This study evaluates the time when the uncertainty starts becoming non-Gaussianin nature. Different algorithms for evaluating the normality of a distribution were implemented andMonte Carlo tests were performed on them to assess their performance. Also, the distancesbetween distributions when they are propagated under linear and nonlinear algorithms werecomputed and compared to the results from the Monte Carlo statistics tests in order to predictthe time when the Gaussianity of the distribution breaks. Uncertainty propagation using StateTransition Tensors and Unscented Transform methods were also studied. Among theimplemented algorithms for evaluating the normality of a distribution, it was found thatRoyston’s method gives the best performance. It was also found that if the Normalized L 2distance between the linear and non-linear propagated distributions is greater than 95%, thenuncertainty starts to become non-Gaussian. In the best case scenario of unperturbed two-bodymotion, it is observed that the Gaussianity is preserved for at least three orbital periods in thecase of Low-Earth and Geostationary orbits when initial uncertainty corresponds to the meanprecision of the space debris catalog. If the initial variances are reduced, then Gaussianity ispreserved for a longer period of time. Time for which Gaussian assumption is valid on orbitaluncertainty is also dependent on the initial mean anomaly. Effect of coordinatestransformation on Gaussianity validity time is also analyzed by considering uncertainty inCartesian, Keplerian and Poincaré coordinate systems. This study can therefore be used to improve space debris cataloguing.