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In this work, we present a symplectic integration scheme to numericallycompute space debris motion. Such an integrator is particularly suitable toobtain reliable trajectories of objects lying on high orbits, especiallygeostationary ones. Indeed, it has already been demonstrated that such objectscould stay there for hundreds of years. Our model takes into account theEarth's gravitational potential, luni-solar and planetary gravitationalperturbations and direct solar radiation pressure. Based on the analysis of theenergy conservation and on a comparison with a high order non-symplecticintegrator, we show that our algorithm allows us to use large time steps andkeep accurate results. We also propose an innovative method to model Earth'sshadow crossings by means of a smooth shadow function. In the particularframework of symplectic integration, such a function needs to be includedanalytically in the equations of motion in order to prevent numerical drifts ofthe energy. For the sake of completeness, both cylindrical shadows and penumbratransitions models are considered. We show that both models are not equivalentand that big discrepancies actually appear between associated orbits,especially for high area-to-mass ratios.

Symplectic integration of space debris motion considering several Earth's shadowing models