KAM Tori from Two-Line Element Sets: a Comparison to SGP4

Kenneth J. Stuart

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Simplified General Perturbations 4 (SGP4), the current analytical model for daily tracking of objects in Earth orbit, provides at best kilometer-level accuracy with an average error growth of 1 to 3 kilometers per day. An improved analytical model with increased accuracy is necessary to reliably track the ever-growing number of objects in Earth orbit. This research examines if a Kolmogorov- Arnold-Moser (KAM) torus constructed from SGP4 two-line element (TLE) sets provides a more consistent orbit prediction than SGP4. One year of TLEs are processed as pseudo observations to identify the time rate of change of orbital elements for eight objects in different Earth orbits. The rates are then used to calculate torus basis frequencies, and an attempt is made to construct a torus for each test case. A least squares algorithm is implemented to fit SGP4 position vectors to the surface of the derived torus. The orbit path along the surface of the torus is compared to the orbit produced by SGP4 via a root-mean-square analysis. The results show that KAM tori basis frequencies can be extracted from TLEs, but SGP4 position vectors are not valid sources of pseudo observation data for the KAM torus analytical model with the current methodology.