L13. Kepler’s Rules: How Planets Really Move

Unit 3 · Gravity and Motion

R-report

L13. Kepler’s Rules: How Planets Really Move

For 1,500 years astronomers assumed planets must move in perfect circles, because circles seemed more elegant. Then one man spent years wrestling with the observations of Mars, and discovered the universe had chosen something else.

Orbits are ellipses

Johannes Kepler had access to decades of unusually precise measurements of planetary positions, recorded by Tycho Brahe before telescopes existed. When Kepler tried to fit circles to the path of Mars, the predictions were off by an amount too small to ignore and too large to explain away. Rather than dismiss the mismatch, he followed it — and found that the orbit is an ellipse. An ellipse is a stretched circle with two special interior points called foci. Kepler’s first rule says the Sun sits at one focus, and the other focus is empty space. How stretched an ellipse is gets described by its eccentricity, from 0 (a perfect circle) upward. In practice most planets are barely stretched: Earth’s eccentricity is about 0.017, so our orbit would look circular to the eye. Mercury is more noticeably elliptical at 0.21, and comets can exceed 0.9, swinging from deep space to close by the Sun. The lesson in method is as important as the result: a small, stubborn discrepancy in the data overturned a 1,500-year-old assumption.

Planets speed up when they are closer

Kepler’s second rule sounds abstract but describes something intuitive: a planet moves fastest when it is nearest the Sun and slowest when it is farthest away. The formal statement is that a line drawn from the Sun to the planet sweeps out equal areas in equal times. Near the Sun the planet must move a long way along its path to sweep the same area that a short, wide sweep covers when it is distant. You can see this in Earth’s own year. We are closest to the Sun in early January and farthest in early July, and Earth moves slightly faster in January. That is part of why the northern hemisphere’s winter is a few days shorter than its summer. The physical reason came later, from Newton. Closer to the Sun, gravity is stronger, so the planet is accelerated more; as it swings away, gravity gradually slows it down again — much like a ball thrown upward, speeding up as it falls back.

Farther out means a longer year

Kepler’s third rule links the size of an orbit to the time it takes to complete. The farther a planet is from the Sun, the longer its year — and not just proportionally, but considerably more. - Mercury: 0.39 times Earth’s distance, year of 88 days. - Earth: 1 unit, year of 365 days. - Jupiter: 5.2 units, year of about 12 Earth years. - Neptune: 30 units, year of about 165 Earth years. Neptune is 30 times farther out than Earth but takes 165 times as long, because it has both a far longer path to travel and a slower speed. The precise relationship is that the square of the orbital period is proportional to the cube of the average distance. This rule is genuinely useful. Measure how long something takes to go around, and you can calculate how far out it must be — a technique astronomers still apply to moons, binary stars, and planets around other stars.

Putting the investigation together

Kepler’s three rules describe planetary motion completely: orbits are ellipses with the Sun at one focus, planets sweep equal areas in equal times so they speed up when close, and orbits farther out take disproportionately longer to complete.

What makes this story matter is how Kepler got there. He trusted careful measurement over an elegant assumption, and a discrepancy most people would have rounded away led to a better description of reality. Newton later showed all three rules follow from gravity — a good example of one theory explaining another.