A Relativistic Motion Integrator: Numerical Accuracy and illustration with BepiColombo and Mars-Next

Today, the motion of spacecrafts is still described by the classical Newtonian equations of motion plus some relativistic corrections. This approach might become cumbersome due to the increase of tracking precision and clock stabilities requiring more relativistic corrections to be taken into account. In the appropriate framework of General Relativity, we use the Relativistic Motion Integrator (RMI) approach to numerically integrate the native relativistic equations of motion for a spacecraft. Considering a given metric, this approach includes all the (relativistic) gravitational effects up to the corresponding order of the metric. The principle of RMI is presented. We compare the results obtained with the RMI method with those from the usual Newton plus correction approach. In particular, the orbit determination of the Bepicolombo (around Mercury) and Mars-NEXT (around Mars) orbiters will be discussed. Finally, we present a numerical study of RMI and we show that the RMI approach is relevant to study the orbit of spacecrafts.