Equal Areas Demonstration
This activity helps visualize Kepler's Second Law: a line from the Sun to a planet sweeps out equal areas in equal time intervals.
Paper-Based Activity
- Print or draw a large ellipse on paper (about 20 cm across). Mark the Sun at one focus.
- Choose 8–12 points evenly spaced in time along the orbit. (Use the simulator or a table of mean anomaly vs. true anomaly to find these positions.)
- Draw a line from the Sun to each point, creating pie-slice-shaped sectors.
- Cut out the sectors and weigh them on a kitchen scale, or measure their area using a grid.
- You should find that all sectors have approximately the same area, even though their shapes are very different—short and fat near perihelion, long and thin near aphelion.
Why It Works
The equal-areas rule is a consequence of conservation of angular momentum. Because gravity pulls the planet straight toward the Sun (no sideways push), the planet's angular momentum never changes. The area swept per unit time is proportional to angular momentum, so it stays constant.
What to Notice
- Near perihelion, the planet is moving fast, so the "pie slice" is short (small r) but has a large angle. Near aphelion, the planet is slow, so the slice is long (large r) but has a small angle. Both slices have the same area.
- For a circular orbit (eccentricity = 0), all slices would be identical in shape too. The Second Law still holds; it just becomes trivial.