A Kinematic Analysis of S-Cluster Trajectories

This study investigates the spacecraft kinematics within the extreme gravitational environment of the Galactic Center, focusing on orbital dynamics near the supermassive black hole Sagittarius A* (Sgr A*). Because classical Newtonian mechanics fails in this strong-field regime, a high-fidelity numerical propagator based on the Schwarzschild metric was developed. By integrating the equations of motion of objects moving in strong gravitational fields using an explicit, adaptive-step Runge–Kutta DOP853 scheme, the relativistic dynamics are resolved in proper time. In this framework, the observed orbits of S-Cluster stars serve as natural dynamical benchmarks for evaluating spacecraft transfer trajectories targeting both the highly eccentric regime (S62 and S4714) and the wider cluster (S22 and S91). Our relativistic simulations show that there is a macroscopic discrepancy between the position predicted by classical mechanics and the actual location in curved spacetime. This targeting error is quantified by introducing a metric termed the “Relativistic Miss Distance” (RMD), which can exceed 1.4×107 km under classical open-loop Lambert guidance. From a mission-design perspective, the study characterizes the kinematic signatures of spacetime curvature: the attractive cubic term in the effective potential, the V-shaped excursion of the Newtonian osculating elements at pericenter, the step-wise Schwarzschild precession and the chronometric delay due to gravitational time dilation. Although well known and small in relative terms, these effects make a fully relativistic model necessary for precision targeting in the Milky Way’s core.