An explicit integrator for the 3D Kepler problem with exact time–true anomaly relation and exact preservation of integrals of motion

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ID: 332393
2026
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Abstract
Abstract We develop and present a numerical scheme for the Kepler problem that preserves exactly all integrals of motion: angular momentum, total energy, and the Laplace–Runge–Lenz vector. This property ensures that orbital trajectories retain their precise shape and orientation over long times, avoiding the spurious precession typical of many standard methods. The scheme uses an adaptive time step derived from a constant angular increment that allows determining the true anomaly as a function of time up to round-off errors. It means that the induced discretization of the Kepler problem is exact. Analytical considerations and numerical experiments demonstrate that the algorithm combines high accuracy with long-term stability, even for orbits with extreme eccentricity, where other methods fail.
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Authors Jan L. Cieśliński, Maciej Jurgielewicz
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Journal monthly notices of the royal astronomical society
Year 2026
DOI
10.1093/mnras/stag1928
URL
https://doi.org/10.1093/mnras/stag1928
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