Rotating Station: Spin Gravity (2D)

Spin gravity

A ring of radius R turning at angular speed omega pushes everything on the inside of its rim with the centripetal acceleration a = omega2 R, which residents feel as weight. One Earth gravity needs omega = sqrt(g / R): R = 60 m gives 3.9 rpm, R = 450 m only 1.4 rpm. The floor moves at omega R (about 23 m/s for R = 60 m at 1 g).

Why it is not quite like Earth

Weight falls off linearly toward the axis, g(z) = omega2 (R - z), so the head of a 1.8 m person weighs 1.8 / R less than the feet (3 % at R = 60 m). In the rotating frame every moving object also feels the Coriolis acceleration aC = -2 omega x v, perpendicular to its velocity: walking spinward makes you heavier, anti-spinward lighter, and moving your head quickly (when omega is large) causes the dizziness that limits useful spin rates to about 2 rpm, with 4-6 rpm the upper end of what trained people tolerate.

Equations of motion

In the rotating frame, with the origin on the axis, x'' = 2 omega y' + omega2 x and y'' = -2 omega x' + omega2 y (Coriolis plus centrifugal terms). The page integrates them with fourth-order Runge-Kutta, steps of 1/240 s, and finds the landing by linear interpolation across the floor circle |r| = R. A drop from rest at height h falls for t = sqrt(2h / (omega2 R)) and drifts backwards by about (1/3) omega g t3.

The inertial-frame view

Seen from outside, a released ball simply moves in a straight line at constant velocity, the sum of its throw velocity and the floor velocity omega x r at the release point; the floor rotates under it. The ball lands where that straight line meets the circle |p| = R, which is exactly what the page uses as an independent reference.

Independent checks run in the page

1) The rotating-frame RK4 landing point and time must match the inertial straight-line solution rotated back by omega t. 2) The Jacobi constant C = v2/2 - omega2 r2/2 is conserved in the rotating frame (the Coriolis force does no work), so its drift measures the integrator error. 3) For a plain drop from h < 8 % R the simulated back-drift must agree with (1/3) omega g t3 within 6 %.

Honest simplifications

The model is two-dimensional (the plane of rotation) and ignores air: in a real habitat the air co-rotates with the ring, so light objects feel drag toward the floor velocity and a thrown ball is slowed relative to the air. The ring is rigid and spins at a constant rate, the ball is a point, floor collisions are not modelled and the comfort bands are rule-of-thumb values from the literature, not medical limits. Playback speed is slowed so that the motion can be followed.

Things to try

• Drop a ball at R = 20 m and at R = 450 m and compare the sideways drift.
• Throw spinward and anti-spinward at the same speed: the flight times differ because the ball's inertial speed differs.
• Switch to the inertial frame: the ball flies straight while the ring turns underneath.

Educational model of an idealised rotating habitat.