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.
The formula
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
Worked example
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
- T² = k · a³
- T² = 1 · 4³ = 64
T = 8 years
Test yourself
A different planet orbits at a semi-major axis of 9 astronomical units. What is its orbital period?
- 9 years
- 81 years
- Correct answer: 27 years
Correct! T² = a³ = 9³ = 729, and the square root of 729 is 27 years.
Along its elliptical orbit, where does a planet move the fastest?
- Correct answer: Closest to the sun
- Farthest from the sun
- Speed never changes
Right! Equal areas in equal times means the planet must speed up when it's closest to the sun.
Where you see this
Every planet in the solar system, every moon around a planet, and every satellite around Earth traces an ellipse obeying these three laws — they were discovered from naked-eye observations of Mars decades before Newton explained why gravity produces exactly this behavior. Comets on very stretched-out ellipses show the second law most dramatically, whipping around the Sun at close approach and crawling near their farthest point.
Common mistakes
It's easy to picture planetary orbits as circles with the Sun at the center — real orbits are ellipses with the Sun at one focus, not the middle, which is why a planet's distance from the Sun actually changes over its year. Students also sometimes assume a planet moves at constant speed — the second law says the opposite: a planet speeds up near the Sun and slows down far from it, sweeping out equal areas in equal times either way.
How it connects
Kepler discovered these three patterns by fitting Mars's observed positions to the data, decades before Newton showed they all fall out of one force law: universal gravitation combined with centripetal motion. Together, gravitation and Kepler's laws close out the mechanics sequence that began with free fall — the same handful of ideas (force, acceleration, energy, momentum) that described a dropped ball also describe an entire solar system.