Kepler's Laws – Elliptical Orbits, Equal Areas and T²/a³
PhysicsGravitation & AstronomyAges 16–17
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Sign in to playAn elliptical orbit around the Sun with an adjustable semi-major axis a and eccentricity e. Coloured sectors swept out in equal times have equal areas, and the planet moves fast at perihelion and slowly at aphelion. A table of real data for the eight planets and Halley's Comet checks that T²/a³ stays constant.
Lesson: Kepler's laws of planetary motion
What it shows
Kepler's three laws describe how planets move around the Sun. The orbit is an ellipse with the Sun at one focus; the line from the Sun to the planet sweeps out equal areas in equal times; and T²/a³ is the same for every body orbiting the Sun. The simulation moves the planet by solving Kepler's equation, so the shaded sectors, each covering the same time interval, really do have the same area when measured on the drawing. Speeds come from v² = GM(2/r − 1/a), and real orbital data for the planets and Halley's Comet let students test the third law.
How to use
Set Semi-major axis a and Eccentricity e, then compare the long, narrow sectors near aphelion with the short, wide ones near perihelion. Change Sectors to split the period into more intervals and read the measured areas below the drawing. Drag the planet to any point, tick Second focus to check r₁ + r₂ = 2a, and use Speed, Pause and Reset to control the animation. Click a row of the table to load a real orbit.
Parameters you can change
- Semi-major axis a 0.2–40 AU
- Eccentricity e 0–0.97
- Number of sectors (equal time intervals) 4–16
- Animation speed 0.25–4 ×
- Show the second focus and r₁ + r₂ = 2a
Questions to explore
- With e = 0.6, how many times faster does the planet move at perihelion than at aphelion?
- Why do the sectors have very different shapes but exactly the same area?
- What is the orbital period of a body with a = 4 AU, and does the simulation confirm it?
