Semi-analytical Satellite Orbit Calculation as Possible Alternative for Numerical Integration

For space based geodesy (i.e. gravity field determination) relative orbits of satellites have to be tracked with very high precision. In the next years, the necessary experimental capabilities will increase by the use of laser interferometry and high precision space clocks. As a consequence, at the present technological level it has to be critically analyzed whether the usual post-Newtonian approximations are sufficient to fully exploit the instrumental capabilities. Today simulations of orbital arcs for satellites of gravity missions are often performed by making use of classical numerical integration techniques. Those results in essence only yield information on the very special case under consideration which prevents one from drawing universally valid conclusions. Numerous simulations had to be performed in order to gain insight into the various correlations between force field modeling, orbital configuration, and so on. Analytical orbit integration provides deeper insight into the physical causes of the orbit evolution than any special perturbation technique, because it operates directly in the spectral domain rather than in the time domain. Similar to a certain class of analytical perturbation techniques, Lie series can be used to numerically integrate satellite orbits. With that semi-analytical technique it is furthermore possible to investigate relativistic effects on orbits in cases, where analytical solutions are not available. Here we focus on the already existing approach for the Newtonian two-body problem, which is extended to the known Schwarzschild problem. The corresponding Hamiltonian for that static and spherically symmetric problem is given and a respective software package is implemented in Mathematica. Because of the complexity of the Lie series coefficients and the very time consuming calculation, the package is implemented using parallel computing. The semi-analytical results are compared with both numerical and analytical orbit integration to investigate the numerical stability and precision of the software. In a next step the software is extended to a more general solution in Kerr spacetime and with consideration of gravitational effects of multipoles. Limitations, possible extensions and further generalizations of the semi-analytical orbit calculation are discussed.