Analytical integration of KS element equations with J2 for short-term orbit predictions

A simple non-singular analytical theory for the short-term orbital motion of satellites in terms of KS elements, in closed form in eccentricity for the Earthʹs zonal harmonic J2, is developed. Owing to symmetry in the KS element equations, only one of the eight equations needs to be integrated analytically to obtain the state vector. Numerical experimentation with small (0.001) to very large (0.92) eccentricity orbits reveals that the orbital elements computed from the analytical expressions in a single step, during half a revolution, match quite well with numerically integrated values and accuracy of a few centimetres is observed in obtaining the most important orbital parameter (semi-major axis) with small analytical steps. Numerical results with other existing analytical solutions are found to be comparable. The present solution can be very effectively used for computation of mean elements with J2, and for trajectory computations during the coast phase of rocket trajectories. Also the theory can be used efficiently, owing to its low-memory requirements for on-board computation of the state vector in navigation and guidance software, where the modelling of J2 effect becomes necessary.