TwiddleLab / Physics / Gravity and Orbits

Ellipses and Kepler's Laws

Lesson 6 of 10 · 7 min · Free, no account needed

What you will learn

• Most orbits are ellipses with the Sun at one focus; a circle is the special case.
• A planet or comet sweeps equal areas in equal times, so it is fastest closest to the Sun.
• T² = a³ with T in years and a in AU: 4 times farther out, 8 times longer year.

Start the lesson

How the lesson goes

1. Halley's comet visits every 76 years, but rushes past the Sun in weeks. Why does it hurry there?

   Here is a smaller comet, far out and moving slowly. Draw the path you expect it to take.

2. Launch a probe sideways at Earth's distance. Find a circle, an ellipse and an escape.

3. Now the hurry. Play one orbit and watch what the comet sweeps in equal times.

4. Kepler's three laws

   1. Orbits are ellipses, with the Sun at one focus. a is half the long axis.

   2. The line from the Sun sweeps equal areas in equal times: fast near the Sun, slow far away.

   3. T² = a³ for anything orbiting the Sun, with T in years and a in AU. Simplified model: each planet feels only the Sun.

5. Test the third law. Drag each dot up to its T².

6. Game: Comet catcher. Launch a probe from Earth to meet the comet near the Sun.

   Drag the arrow: it is the probe's speed change. Less fuel, more stars.

7. Math: on log scales the planets line up. Which slope does that line have?

   log T = 1.5 × log a. On a graph with log scales on both axes that is a straight line through (1, 1) with slope 1.5.

8. A planet orbits at 4 AU. How long is its year?

   Kepler's third law in years and AU: T² = a³ = 64, so T = 8 years.

9. Your catch against the cheapest route. Fly both.

Previous lesson: Weightless Means Falling · Next lesson: Escape Velocity · Part of the course: Gravity and Orbits

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