Kepler's Laws

Where the laws came from

The astronomer Tycho Brahe kept a continuous record of the positions of the Sun, Moon and planets for almost 20 years. He was the last and greatest of Europe’s observers before the telescope.^[1] After Tycho’s death in 1601, his assistant Johannes Kepler spent more than 20 years analysing them.^[1] He published his first two laws in 1609, in The New Astronomy.^[1]

The first law: ellipses

Every planet travels around the Sun along an ellipse, and the Sun sits at one of that ellipse’s two foci.^[1]

An ellipse is a stretched circle with two special points inside it, its foci. How stretched it is, is called its eccentricity. Half its longest width is the semimajor axis, which sets the size of the orbit.^[1]

The second law: equal areas

An imaginary line from the Sun to a planet covers the same area in the same amount of time, wherever the planet is in its orbit.^[1]

Imagine a string from the Sun to the planet. In one month, the string sweeps out a slice of the ellipse. Kepler found that every month’s slice has the same area. Near the Sun the slice is short and wide; far from the Sun it’s long and thin. That means a planet has to move faster when it’s closer to the Sun.^[1]

The third law: bigger orbits, longer years

Square a planet’s orbital period (in years) and you get the cube of its semimajor axis (in astronomical units, AU): P² = a³.^[1]

One AU is Earth’s average distance from the Sun. For Earth, P = 1 and a = 1, and the law checks out.

Worked example (optional). Mars has a semimajor axis of 1.52 AU.^[1] Then a³ ≈ 1.52 × 1.52 × 1.52 ≈ 3.51, so P ≈ √3.51 ≈ 1.88 years: a little less than two Earth years.^[1]

Why the laws work

Kepler described how planets move, but not why. The answer came from Isaac Newton. He showed that a gravitational force weakening with the square of distance produces exactly the orbits described by Kepler’s laws.^[2] The second law turns out to be a consequence of the conservation of angular momentum.^[3] Newton also refined the third law: the full version includes the masses of both bodies.^[2]