Kepler's Laws of Planetary Motion

Planets don't move in perfect circles — they trace ellipses (stretched-out circles) with the sun sitting at one focus, not the center. As a planet swings closer to the sun it speeds up, and as it drifts farther away it slows down, sweeping out equal areas in equal times. The farther out a planet's orbit is, the longer its year takes, because period squared grows with the orbit's size cubed.

T² = k · a³

• T — period (yr): the time the planet takes to complete one full orbit
• a — semi-major axis (AU): the planet's average distance from the sun
• k — proportionality constant (yr²/AU³): equals 1 for anything orbiting our sun in these units

A planet orbits the sun at a semi-major axis of 4 astronomical units. How many years does it take to complete one orbit?

• a = 4 AU
• k = 1 yr²/AU³

1. T² = k · a³
2. T² = 1 · 4³ = 64

T = 8 years

A different planet orbits at a semi-major axis of 9 astronomical units. What is its orbital period?

• 9 years
• 81 years
• 27 years

Along its elliptical orbit, where does a planet move the fastest?

• Closest to the sun
• Farthest from the sun
• Speed never changes