Kepler's Second Law: Equal Areas & Angular Momentum
1 · Predict
A satellite on an elliptical orbit is launched sideways at perihelion (closest approach). As it coasts out to aphelion and back, does the line joining it to the central mass sweep out area faster near perihelion, faster near aphelion, or at the same rate everywhere?
- The radius line sweeps area fastest near perihelion, where the satellite moves quickest.
- The radius line sweeps equal areas in equal times — the same rate everywhere on the orbit.
- The radius line sweeps area fastest near aphelion, where the satellite has the most room to move.
2 · Set Up
- Open the equal-areas simulation and press Reset. The central mass sits at the middle of the stage with gravitational parameter GM = 40 m³/s², and the shaded sector wedges trace the area the satellite's radius line sweeps out.
- Enable the specific-angular-momentum probe (h) and the eccentricity probe (e) on the satellite so both invariants are shown live.
- For each trial below, set the perihelion radius r_p and aphelion radius r_a, launch the satellite from perihelion with a purely sideways (tangential) velocity, press Play, and let it complete at least one full orbit before recording the settled h reading.
3 · Collect Data
| Perihelion radius r_p (m) | Aphelion radius r_a (m) | Escape speed at perihelion v_esc (m/s) | Specific angular momentum h (m²/s) |
|---|---|---|---|
| 2.5 | 5 | ||
| 2 | 6 | ||
| 3 | 4 |
Plot the measured specific angular momentum h (y-axis) against the perihelion radius r_p (x-axis) for your three orbits. Then, watching a single orbit run, sketch how the satellite's speed changes from perihelion to aphelion — where is it fastest, and where slowest?
4 · Analyze
- For one orbit, note the h reading at perihelion and again at aphelion. Within reading error they should match. Using h = r·v, explain why the satellite must move slower when r is large (aphelion) and faster when r is small (perihelion) to keep h constant.
- The area swept per unit time is dA/dt = h/2. Since h is constant, equal areas are swept in equal times. Use your measured h for the first orbit to compute dA/dt, and confirm h = √(2·GM·r_p·r_a/(r_p + r_a)) reproduces the value you read off the sim.
- Compare the satellite's perihelion speed (v_p = h/r_p) to the escape speed v_esc you computed for each row. Is v_p above or below v_esc, and what does that tell you about whether the orbit stays bound?
5 · Extend
- Halley's Comet whips past the Sun in months but takes decades to crawl back from the outer solar system. Using the equal-areas law, explain why a comet spends so little of its orbit near the Sun and so much of it far away.
- Specific angular momentum h = r·v carries no mass term. If you doubled the satellite's mass but launched it identically, would the measured h change? Would its full angular momentum L = m·r·v change? Explain the difference.
The Physics Behind This Experiment
Escape Speed
The minimum speed at radius r that would let the satellite break free of the central mass. The satellite's real perihelion speed stays below this value, so the orbit remains bound — a closed ellipse that sweeps equal areas in equal times.
Circular Orbital Speed
The tangential speed that produces a perfect circle at radius r. Launching faster than this at perihelion (as every elliptical trial here does) sends the satellite outward to a distant aphelion before gravity pulls it back.