Kepler's Third Law: Orbital Period vs. Radius
1 · Predict
A satellite circles a central mass in a stable circular orbit. When you move it to a larger orbit radius, how does the time for one full revolution change?
- A larger orbit takes longer per revolution.
- The period stays the same at any radius.
- A larger orbit takes less time per revolution.
2 · Set Up
- Open the grav-kepler-law preset. A satellite orbits the central mass; the orbit timer and period probe are already attached.
- Set the satellite's orbit radius r, then launch it sideways at the circular-orbit speed shown on the v = √(GM/r) formula card so the orbit stays circular.
- Let the satellite complete at least one full loop and read the orbital period T from the timer once the orbit is steady.
3 · Collect Data
| Orbit radius r (m) | Circular speed v (m/s) | Measured period T (s) |
|---|---|---|
| 2.5 | ||
| 3.5 | ||
| 4.5 |
Plot T² (vertical axis) against r³ (horizontal axis) for your three orbits. Draw the best-fit straight line through the origin and find its slope.
4 · Analyze
- For each row compute T²/r³. Are the three values roughly equal? Compare their average to 4π²/GM using GM = 40 (so 4π²/GM ≈ 0.99 s²/m³).
- Your T²-vs-r³ line should pass through the origin. Explain why that straight-line, through-origin shape is exactly what T² = (4π²/GM)·r³ predicts, and what the slope represents.
5 · Extend
- Real planets obey the same law: Mercury (small orbit) races around the Sun in 88 days while Neptune (large orbit) takes 165 years. Use T²/r³ = constant to explain this huge difference.
- Kepler's constant here is 4π²/GM, so it depends on the central mass, not on the satellite. Predict what happens to every orbit's period if GM of the central body were doubled.
The Physics Behind This Experiment
Kepler's Third Law
For any orbit around a given central mass, the period squared is proportional to the mean radius cubed: T²/r³ = 4π²/GM. The ratio is the same for every orbit, so timing one orbit at a known radius reveals GM.
Circular Orbital Speed
A body in a circular orbit of radius r must travel at v = √(GM/r): the speed that makes gravity supply exactly the centripetal force needed. Launching at this speed keeps the orbit circular rather than elliptical.