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The science
Orbits around a gas giant
A moon of negligible mass on a circular orbit of radius a around a planet with gravitational parameter GM moves at v = √(GM/a) with period T = 2π√(a3/GM): Kepler's third law, T2/a3 = 4π2/GM, the same for every moon. The map integrates the equations of motion with a symplectic leapfrog scheme, which keeps energy E = ½v2 − GM/r and angular momentum L = x vy − y vx constant for an eccentric orbit. The bottom-right graph shows T2/a3 measured for each moon against the predicted level.
What the 3D page does
The 3D scene gives each moon a hand-picked constant angular speed (0.55, 0.35, 0.75 and 0.22 rad/s at radii 9, 12, 15.5 and 19). Those speeds are not Keplerian — the third moon is far too fast for its radius. Turn off Physical orbits to reproduce the page exactly; turn it on to see what gravity would actually do, with GM chosen so the inner moon keeps the same period.
Honest simplifications: the dynamics are in the orbital plane only; the inclinations of the 3D scene are drawn in the side view but do not change the forces. Moons attract each other point-to-point with a softening length of 0.2 units when the mass ratio slider is above zero; no tides, no planet oblateness, no resonance locking, no moon–ring interaction. Units are scene units, not kilometres.
What to try
- Start with As in 3D, then press Kepler (same inner period) and compare how the outer moons slow down.
- Look at the T²/a³ graph: flat when the orbits are physical, scattered when the speeds are the 3D page's hand-picked ones.
- Raise the moon mass slider to see moons tug each other and the periods drift.
- Choose Eccentric orbits and notice the moons speed up near the planet.