NUMERICAL INTEGRATION OF A SATELLITE ORBIT WITH KS TRANSFORMATION

A satellite orbit is mainly influenced by central body gravitational forces. For a satellite in LEO (Low Earth Orbit), MEO (Medium Earth Orbit) or GEO (Geosynchronous Earth Orbit) the Earth´s gravity distribution and other perturbations determine the position and velocity changes in function of time. If the motion is around a spherical body with homogenous mass distribution and without perturbative forces, the orbit must be cyclic like the Two Body Problem (TBP) or Keplerian Orbit. Different numerical methods can be applied for solving the Ordinary Differential Equations (ODE´s). In this work a fourth-order fixed step-size Runge-Kutta numerical integrator (RK4) was implemented. With satellite´s position and velocity in inertial reference frame at zero time (orbit initial conditions) and solving the ODE´s with RK4 it is possible to know the satellite position and velocity at any time, with a certain level of accuracy. When the integration time is equal to the orbit period time, in a Keplerian orbit, the initial and final orbit data are compared to obtain the integration error in position and velocity. To better accuracy it is recommended to change the ODE´s from a Newtonian system by a time transformation to a stable Liapunov system and finally to Kunstaanheimo-Stiefel (KS) transformed system. In this paper the results obtained by applying the KS transformation to the orbit ODE´s, its accuracy and error analysis for different step-sizes of integration in a satellite orbit propagation are presented. Additionally, the orbits propagated are compared in terms of performance, CPU time and degradation of accuracy. Finally conclusions are drawn showing the beneficial aspects of using the KS transformation as an efficient technique for precise orbit integration of Earth satellites.  Keywords: Satellite Orbit, Numerical Integration, Time Transformation, KS transformation.