Explore Johannes Kepler's Laws of Motion

orbits
The planets and comets of the solar system follow slightly elliptical orbits around the Sun. Moons and other satellites do the same around their planets. This diagram shows the orbits' shapes, although it is not to scale. NASA

Everything in the universe is in motion. Moons orbit planets, which in turn orbit stars. Galaxies have millions and millions of stars orbiting within them, and across very large scales, galaxies orbit in giant clusters. On a solar system scale, we notice that most orbits are largely elliptical (a sort of flattened circle). Objects closer to their stars and planets have faster orbits, while more distant ones have longer orbits.

orbits
A schematic of orbits of planets and comets of the solar system. They follow slightly elliptical paths around the Sun. Moons and other satellites do the same around their planets. Johannes Kepler devised the laws that explain these orbits. NASA

It took a long time for sky observers to figure these motions out, and we know about them thanks to the work of a Renaissance genius named Johannes Kepler (who lived from 1571 to 1630). He looked at the sky with great curiosity and a burning need to explain the motions of the planets as they seemed to wander across the sky.

Johannes Kepler
A portrait of Johannes Kepler by an unknown artist. Unknown artist/public domain

Kepler: The Man Behind the Laws

Kepler was a German astronomer and mathematician whose ideas fundamentally altered our understanding of planetary motion. His best-known work stems from his employment by Danish astronomer Tycho Brahe (1546-1601). He settled in Prague in 1599 (then the site of the court of the German emperor Rudolf) and became court astronomer. There, he hired Kepler, who was a mathematical genius, to carry out his calculations.

Kepler had studied astronomy long before he met Tycho; he favored the Copernican world-view that said the planets orbited the Sun. Kepler also corresponded with Galileo about his observations and conclusions.

Eventually, based on his work, Kepler wrote several works about astronomy, including Astronomia Nova, Harmonices Mundi, and Epitome of Copernican Astronomy. His observations and calculations inspired later generations of astronomers to build on his theories. He also worked on problems in optics, and in particular, invented a better version of the refracting telescope. Kepler was a deeply religious man and also believed in some tenets of astrology for a period during his life. 

Kepler's Laborious Task

Johannes Kepler
A portrait of Johannes Kepler by an unknown artist. Unknown artist/public domain

Kepler was assigned by Tycho Brahe the job of analyzing the observations that Tycho had made of the planet Mars. Those observations included some very accurate measurements of the position of the planet which did not agree with either Ptolemy's measurements or Copernicus's findings. Of all the planets, the predicted position of Mars had the largest errors and therefore posed the greatest problem. Tycho's data were the best available before the invention of the telescope. While paying Kepler for his assistance, Brahe guarded his data jealously and Kepler often struggled to get the figures he needed to do his job.

Accurate Data

Kepler's Third Law: The Hohmann Transfer Orbit
Kepler's Third Law: The Hohmann Transfer Orbit. NASA

When Tycho died, Kepler was able to obtain Brahe's observational data and attempted to puzzle out what they meant. In 1609, the same year that Galileo Galilei first turned his telescope towards the heavens, Kepler caught a glimpse of what he thought might be the answer. The accuracy of Tycho's observations was good enough for Kepler to show that Mars' orbit would precisely fit the shape of an ellipse (an elongated, almost egg-shaped, form of the circle).

Shape of the Path

Circular and Elliptical Orbits Having the Same Period and Focus
Circular and Elliptical Orbits Having the Same Period and Focus. NASA

His discovery made Johannes Kepler the first to understand that the planets in our solar system moved in ellipses, not circles. He continued his investigations, finally developing three principles of planetary motion. These became known as Kepler's Laws and they revolutionized planetary astronomy. Many years after Kepler, Sir Isaac Newton proved that all three of Kepler's Laws are a direct result of the laws of gravitation and physics which govern the forces at work between various massive bodies. So, what are Kepler's Laws? Here is a quick look at them, using the terminology that scientists use to describe orbital motions.

1. Planets move in ellipses with the Sun at one focus

Circular and Elliptical Orbits Having the Same Period and Focus
Circular and Elliptical Orbits Having the Same Period and Focus. NASA

Kepler's first law states "all planets move in elliptical orbits with the Sun at one focus and the other focus empty". This is also true of comets that orbit the Sun, too. Applied to Earth satellites, the center of Earth becomes one focus, with the other focus empty.

2. The radius vector describes equal areas in equal times

Illustrating Kepler's 2nd law: Segments AB and CD take equal times to cover.
Illustrating Kepler's 2nd law: Segments AB and CD take equal times to cover. Nick Greene

Kepler's 2nd law, the law of areas, states "the line joining the planet to the Sun sweeps over equal areas in equal time intervals". To understand this, think about when a satellite orbits. An imaginary line joining it to Earth sweeps over equal areas in equal periods of time. Segments AB and CD take equal times to cover. Therefore, the speed of the satellite changes, depending on its distance from the center of the Earth. Speed is greatest at the point in the orbit closest to the Earth, called perigee, and is slowest at the point farthest from the Earth, called apogee. It is important to note that the orbit followed by a satellite is not dependent on its mass.

3. Squares of periodic times are to each other as cubes of the mean distances

Kepler's Third Law: The Hohmann Transfer Orbit
Kepler's Third Law: The Hohmann Transfer Orbit. NASA

Kepler's 3rd law, the law of periods, relates time required for a planet to make 1 complete trip around the Sun to its mean distance from the Sun. "For any planet, the square of its period of revolution is directly proportional to the cube of its mean distance from the Sun." Applied to Earth satellites, Kepler's 3rd law explains that the farther a satellite is from Earth, the longer it will take to complete an orbit, the greater the distance it will travel to complete an orbit, and the slower its average speed will be. Another way to think of this is that the satellite moves fastest when it's closest to Earth and slower when it's farther away.