Kepler's Laws of Planetary Motion, Explained

Before Kepler, the dominant model of planetary motion was Copernicus’s: the sun at the center, the planets circling it along perfectly round orbits. But Tycho Brahe’s observations — the astronomer who had gathered the most precise data of the pre-telescope era — didn’t line up with that model. It was Johannes Kepler whom Brahe assigned the task of figuring out how Mars really moved, and that work produced the three laws that carry his name.

Kepler’s first law: orbits are ellipses

Kepler’s first law says that every planet’s orbit around the sun is an ellipse, with the sun occupying one of the two foci of the shape — not the center. The ellipse was already a familiar figure in Euclidean geometry long before Kepler, but no one had yet applied it to the motion of real planets.

How far an ellipse departs from a circle is measured by its eccentricity, a value between zero and one: at zero the ellipse is a circle, and the closer it gets to one, the flatter the shape becomes, until it’s almost a straight line. Every planet has its own eccentricity, but no orbit in the solar system is an exact circle.

Kepler’s second law: the law of areas

The second law concerns how fast a planet moves along its orbit, and the technical statement translates into a simple picture: the imaginary line joining the planet and the sun sweeps out equal areas in equal times. When the planet is close to the sun, that line is short, so to cover the same area in the same time the planet has to travel a longer stretch of orbit: it speeds up. When it’s far from the sun, the line is long and a short arc is enough to cover the same area: the planet slows down.

The point of the orbit closest to the sun is called perihelion, the farthest one aphelion. No planet moves at a constant speed; the second law says exactly why.

Kepler’s third law: the relationship between period and distance

The third law connects two quantities that look independent at first glance: the time a planet takes to complete one orbit (the period) and its average distance from the sun. The law states that the square of the orbital period is proportional to the cube of the average distance: planets farther out take much longer to circle the sun, and not in direct proportion.

Worked example: Earth, by definition, has a period of 1 year and an average distance of 1 astronomical unit (AU); so its period squared is 1 and its distance cubed is 1. Mars has a period of 1.88 years and a distance of 1.524 AU: the period squared gives 3.5344, the distance cubed gives 3.5396 — almost the same number. Jupiter has a period of 11.9 years and a distance of 5.203 AU: the period squared gives 141.61, the distance cubed gives 140.85, again nearly identical. The small gaps come only from rounding in the data, not from an exception to the law.

The table below compares five planets, from the closest to the sun to the farthest among those the source reports:

Planet	Period T (years)	Average distance a (AU)	T²	a³
Mercury	0.241	0.387	0.05808	0.05796
Earth	1	1	1	1
Mars	1.88	1.524	3.5344	3.5396
Jupiter	11.9	5.203	141.61	140.85
Saturn	29.5	9.539	870.25	867.98

In every row, T² and a³ are practically the same number: the numerical confirmation of the third law, valid for planets whose mass is negligible next to that of the star they orbit.

Solar system orbits, seen through Kepler’s laws

The three laws don’t describe some abstract case: they hold for every planet in the solar system. Every orbit is an ellipse with the sun at one focus, every planet speeds up as it nears the sun and slows down as it moves away, and the period-distance relationship checks out for Mercury just as it does for Saturn, as the table above shows.

From Tycho Brahe to Newton

Kepler published the first two laws in 1609, in the work Astronomia Nova. The third law arrived ten years later, in 1619, in Harmonices Mundi. In between came a long stretch of calculation on Tycho Brahe’s data, particularly on the orbit of Mars, the planet Brahe had assigned to Kepler.

Kepler didn’t know about gravity: his laws describe how the planets move, and it would be Isaac Newton, decades later, who explained why, with his theory of universal gravitation. Kepler’s laws were instrumental to that work.