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

TL;DR

This paper introduces a symplectic integration method for accurately simulating space debris trajectories, incorporating Earth's shadowing effects with smooth functions, and demonstrates its advantages over non-symplectic methods for high orbits.

Contribution

The paper presents a novel symplectic integrator that models Earth's shadow crossings analytically, improving long-term accuracy for space debris orbit simulations.

Findings

The symplectic integrator maintains energy conservation with large time steps.

Smooth shadow functions effectively model Earth's shadow crossings.

Discrepancies between cylindrical and penumbra shadow models impact high area-to-mass ratio debris trajectories.

Abstract

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