Orbital Motion

How the model works

The satellite feels only the planet's gravity, an inverse-square pull towards the centre:

a = −GM · r / |r|³

Units are scaled so that GM = 1 and the launch distance is r₀ = 1. Two speeds matter:

• Circular speed v_c = √(GM/r₀) = 1.
• Escape speed v_esc = √(2GM/r₀) ≈ 1.414. At or above it, the total energy E = ½v² − GM/r is no longer negative and the satellite never returns.

Below escape speed the path is an ellipse with the planet at one focus (Kepler's first law). The satellite sweeps equal areas in equal times (second law), which the simulator checks using angular momentum L = r × v. Period squared is proportional to the semi-major axis cubed (third law): T = 2π√(a³/GM).

Integration uses velocity Verlet with adaptive sub-stepping near the planet, so energy and angular momentum drift stay tiny.

Two-body point-mass model: no drag, no other planets, no relativity. The planet is drawn to scale (radius 0.3).

Guided experiments

1. Circular orbit. Launch at exactly v = 1 straight sideways. Prediction: the distance stays at 1 and the period is 2π ≈ 6.28 time units. v = 1.00
2. Slow launch makes an ellipse. Launch at v = 0.8. Prediction: the launch point becomes the farthest point (apoapsis). The satellite speeds up as it falls closer, which is equal areas in action. v = 0.80
3. Escape. Launch at v = 1.45. Prediction: total energy is positive, so the path opens into a hyperbola and the satellite does not return. v = 1.45

Challenge

At v = 1.2 (sideways), what is the semi-major axis, and how long does one orbit take? Predict, then compare with the readout.

References

• OpenStax, University Physics Volume 1, §13.5 “Kepler's Laws of Planetary Motion”.
• NASA Science, “Orbits and Kepler's Laws”.