The Geometry of Planetary Orbits

The basis for everything we understand about the orbits of the planets comes from the work of early-17th-century German astronomer Johannes Kepler.

Kepler, without knowing about the force of gravity, was able to mathematically describe the motions of the planets around the sun in his three laws of planetary motion — an amazing achievement! His work was also key in providing evidence for Copernicus' theory that the sun, not the earth, was at the center of our solar system. Newton too, used Kepler's laws as a foundation for developing his own laws of motion and universal gravitation. So, what are Kepler's three laws?

Kepler's First Law  — The Law of Orbits or Law of Ellipses

Planets revolve around the sun in elliptical orbits with the sun at one focus. (An ellipse is like a circle that has been "squashed" or stretched, where the sum of the distances from any point on its curve to two fixed points (called foci) is always constant.) In the diagram below, A and B are the two foci, and the sum d_{1 +} d₂ is always constant.

Credit: mathmonks

Kepler's Second Law — The Law of Areas or Law of Equal Areas

The [imaginary] line joining a planet to the sun sweeps out equal areas in equal intervals of time. This means a planet moves faster when it's closer to the sun and slower when it's farther away.

Kepler's Third Law — The Law of Periods or Law of Harmonies

There is a mathematical relationship between the orbital period of a planet and the size of its orbit: the square of the orbital period is directly proportional to the cube of the semi-major axis of the orbit.

Credit: Spacenasa via Wikimedia Commons

Kepler's laws are followed not only by the planets in our solar system, but also by the satellites that orbit the earth, and by any moons that orbit a planet (with the earth or the "host" planet as one of the foci instead of the sun). Comets also follow Kepler's laws, with Halley's Comet being a well-known example.

Elliptical orbit of Halley's comet. Credit: nagualdesign via Wikimedia Commons

In case you are wondering, the asteroids in our solar system, and even Uranus, which orbits the sun lying on its side, also follow Kepler's laws! These laws are so universal that they are followed by planets and moons in many solar systems that are similar to ours!