KAM Torus Frequency Generation from Two-Line Element Sets

The Kolmogorov Arnold and Moser (KAM) theorem states that a lightly perturbed Hamiltonian system will have solutions which lie on a torus. Earlier work by the second author has shown that most Earth satellite orbits perturbed by the geopotential lie on KAM tori. The problem then arises as to how to convert the current satellite tracking orbits to KAM tori. A KAM torus is characterized by three frequencies and three phase angles. The frequencies are essentially the rates of change of the mean anomaly, the longitude of the ascending node, and the argument of perigee. In this paper we explore the determination of these three rates from the fitting of SGP4 two line element sets (TLEs), and then constructing KAM tori with the specified frequencies. The success of this process, and an idea of the residual errors, can then be obtained by comparing the SGP4 predictions with the KAM torus predictions. Second order polynomials are fit to data from TLEs over 18 months using a least squares technique. The first order coefficients are used as the torus basis frequencies while the second order terms are used to account for perturbations to the satellite\u27s orbit such as air drag. Four cases are attempted using the Hubble Space Telescope and three rocket bodies as test subjects. A KAM torus with the desired basis frequencies is constructed and used to predict satellite position. For the final test case, this shows an oscillatory error with an amplitude of less than 80 km over a period of almost two years. The authors speculate that this is caused by periodic lunar and solar perturbations, masked in the SGP4 fits by frequent updates