(SC)RMI: A (S)emi-(C)lassical (R)elativistic (M)otion (I)integrator, to model the orbits of space probes around the Earth and other planets

TL;DR

This paper introduces a semi-classical relativistic motion integrator for accurately modeling spacecraft orbits within General Relativity, improving precision over classical methods by directly integrating relativistic equations.

Contribution

It presents a novel numerical integrator based on relativistic equations of motion, incorporating perturbations and ensuring proper velocity norm conservation.

Findings

Accurately models spacecraft orbits using relativistic equations.

Maintains proper velocity norm with symplectic integrator.

Bridges classical and relativistic orbital modeling.

Abstract

Today, the motion of spacecrafts is still described according to the classical Newtonian equations plus the so-called "relativistic corrections", computed with the required precision using the Post-(Post-)Newtonian formalism. The current approach, with the increase of tracking precision (Ka-Band Doppler, interplanetary lasers) and clock stabilities (atomic fountains) is reaching its limits in terms of complexity, and is furthermore error prone. In the appropriate framework of General Relativity, we study a method to numerically integrate the native relativistic equations of motion for a weak gravitational field, also taking into account small non-gravitational forces. The latter are treated as perturbations, in the sense that we assume that both the local structure of space-time is not modified by these forces, and that the unperturbed satellite motion follows the geodesics of the local space-time. The use of a symplectic integrator to compute the unperturbed geodesic motion insures the constancy of the norm of the proper velocity quadrivector. We further show how this general relativistic framework relates to the classical one.