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Orbit
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Model check
The science
One inverse-square force, three laws
A satellite feels only the pull of the central body, r'' = -μ r̂ / r². The delivered ring-track and Earth-satellite scenes are the special case where the launch speed equals the circular speed vc = √(μ/r) = ωr, so the path is a circle and ω = √(μ/r³). For the ring track the lab picks μ = ω²r³; for Earth it uses μ = 3.986×1014 m³/s² and r = 6371 km + altitude.
Change the launch speed and the circle becomes an ellipse. Launched sideways at r0 with v = f vc: semi-major axis a = r0/(2 - f²) and eccentricity e = |f² - 1|. For f above 1 the launch point is the closest point (periapsis), for f below 1 it is the farthest (apoapsis). The orbit is bound only for f below √2.
Kepler I: the path is an ellipse with the centre at one focus. Kepler II: the line from the centre to the satellite sweeps equal areas in equal times, dA/dt = ½ r²θ' = h/2: the coloured wedges are cut every twelfth of a period and have equal area. Kepler III: T = 2π√(a³/μ), so T² grows as a³. The speed follows vis-viva, v² = μ(2/r - 1/a).
Honest simplifications
This is two-body point-mass motion in a plane: no air drag, no flattening of the Earth (J2), no Moon or Sun, and the Earth is a point whose radius only sets r. The planet, hub and satellite are drawn at a size that fits the view; the orbit lines use the true ratio of orbit radius to planet radius for the chosen altitude. The 3D page imports the same physics module as the 2D page, so every number is identical.
Model checks (run live)
RK4 radius against Kepler's equation; energy conservation; the swept-area rate against h/2; the period against 2π√(a³/μ), all on a full independent run of one period; and the equal areas of the twelve wedges.