How many fixed steps per orbit does a twin need before an eccentric orbit stops drifting?

What it found

Explicit Euler, semi-implicit Euler, velocity Verlet and RK4 against the exact Kepler solution (Newton on Kepler's equation), eccentricity 0 to 0.9. Explicit Euler never works. The symplectic pair has bounded but large energy excursions at 100 steps per orbit (semi-implicit 12% of |E₀| at e = 0.5, 71% at 0.7) and a periapsis direction that wanders (−0.62 rad after 50 orbits at e = 0.5), while RK4 creeps 0.004 rad. For position error under 0.01 a over 10 orbits RK4 needs 64, 128, 512, 4096 steps per orbit at e = 0, 0.3, 0.7, 0.9 against 512, 1024, 8192, >8192 for the symplectic pair, so RK4 is cheaper even at four force evaluations per step. At e = 0.9 every fixed step fails at 100 per orbit. Unit-scale two-body problem, ten-orbit horizon, textbook schemes, not MuJoCo or PyBullet. Every input is exact or swept; nothing is assumed.

A figure from its own output

Paper, code and results

• README and method
• Experiments and results
• Tests

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Produced with AI assistance (Claude). Results are from the code in this repository and are simulated or stylised unless the abstract says otherwise; nothing here implies affiliation with or endorsement by any lab, researcher or company cited. MIT licensed.