Numerical simulation of the post-Newtonian equations of motion for the near Earth satellite with an application to the LARES satellite

We study the post-Newtonian perturbations in the orbit of a near-Earth satellite by integrating them with a high-fidelity orbit propagation software KASIOP. The perturbations of the orbital elements are evaluated for various cases from a low-Earth orbit to a geostationary one, and from an equatorial to a polar orbit. In particular, the numerical simulation is applied to the LARES-like satellite under a realistic orbital configuration. The relativistic perturbations include the Schwarzschild term, the effects of Lense-Thirring precession, and the post-Newtonian term due to the quadrupole moment of the Earth as well as the post-Newtonian gravitoelectric and gravitomagnetic forces, which are produced by the tidal potential of the solar system bodies, are also modeled. The latter three terms are usually ignored in most orbit-propagation software. The secular variations of the orbital elements are evaluated from the orbital positions propagated for a half year. For a medium altitude orbit like that of the LARES mission, the magnitude of the relativistic perturbations ranges from the order of 10−7 m/s2 by the Schwarzschild effect to 10−15 m/s2 by the relativistic tidal effects. The orbital integration shows that the secular variations in three orbital elements - the ascending node, the argument of perigee, and the mean anomaly at epoch - are larger than the systematic error as results of the relativistic perturbations. The magnitudes of the secular variation are investigated in terms of the orbital altitude, inclination, and the size of each perturbation force. The numerical simulation rendered in this study shows that the secular post-Newtonian perturbations with the magnitude lying beyond the Schwarzschild and the Lense-Thirring effects need to be taken into account in current and upcoming space geodesy missions.