Kepler’s Laws of Planetary Motion
The Dance of the Planets
The most used scientific theory of all time must be Newton’s laws of motion (see our previous episode). Nearly every bit of mechanical engineering ever done depends on those equations. And, combining those with Newton’s law of universal gravitation, we put satellites and people in orbit and send probes to other worlds. But not everyone knows that Newton’s laws were based on a previously revolutionary theory—Kepler’s laws of planetary motion.
In the early 17th century, Johannes Kepler, a German mathematician, greatly contributed to our understanding of planetary motion. Facing religious and political difficulties in Graz, Austria, he moved his family to Prague, Czech Republic, to work as an assistant for Tycho Brahe, the leading European astronomer of that time.
Portrait of Johannes Kepler by unknown artist
Tycho Brahe was famous for accurately observing stars and planets. He spent twenty years studying the sky at his observatory Uraniborg, located on the Danish island of Hven. He had met Kepler before and was impressed by the work Kepler was doing in his studies. However, Brahe held suspicions toward Kepler, fearing that the talented young intern might outshine him as the leading astronomer. Consequently, Brahe selectively revealed only a portion of his vast planetary data to Kepler.
Kepler was assigned the challenging task of understanding the orbit of Mars, which posed difficulties within the existing Aristotelian and Ptolemaic descriptions of the universe. Brahe likely gave this task to occupy Kepler while he focused on perfecting his geocentric model, where the Earth was in the center of the solar system. According to Brahe’s model, the planets Mercury, Venus, Mars, Jupiter, and Saturn orbited the Sun, which, in turn, orbited the Earth. However, Kepler firmly believed in the heliocentric Copernican model, which correctly positioned the Sun at the center. The problem with Mars’ orbit stemmed from the Copernican system’s assumption that the planet’s orbit, like others, was circular.
After considerable struggle, Kepler eventually came to the realization that the planets did not orbit in perfect circles but instead followed elongated or flattened circles known as ellipses. Mars, in particular, posed challenges due to its highly elliptical orbit. Ironically, it was this crucial part of Brahe’s data that enabled Kepler to formulate the correct theory of planetary motion, thereby disproving Brahe’s own geocentric model.
With the understanding that planets move in elliptical orbits, Johannes Kepler formulated three laws of planetary motion. These laws not only accurately described the motion of planets but also proved applicable to the movement of comets.
Kepler’s first law states that the orbit of each planet around the Sun is shaped like an ellipse. An ellipse has two foci (singular: focus) on the longer axis (see picture below). Within this elliptical orbit, the Sun occupies one of the foci, while the planet moves along the path. As a result, the distance between the planet and the Sun continuously varies throughout its orbit.
The Sun at the focus of an elliptical orbit
As a planet orbits the Sun, Kepler’s second law states that the line connecting the planet and the Sun sweeps out equal areas in equal time intervals. This means that planets do not travel at a constant speed along their orbits. Instead, their speed varies, causing the line joining the Sun and the planet to cover equal areas in the same amount of time. According to this law, a planet moves at its highest speed when it is nearest to the Sun and its slowest speed when it is farthest from the Sun.
The same (blue) area is swept out in a fixed time period. The green arrow is velocity. The purple arrow directed toward the Sun is the acceleration.
Kepler’s third law, the law of harmonies, states that the period of a planet’s orbit, squared, is proportional to its average distance from the sun, cubed. This means that as the radius of a planet’s orbit increases, its orbital period also increases. For example, Mercury, being the closest planet to the Sun, completes an orbit in just 88 days. On the other hand, the Earth takes 365 days, and Saturn requires a whopping 10,759 days to complete a full orbit.
While Kepler formulated his laws without knowledge of gravity, they played a crucial role in Isaac Newton’s development of the law of universal gravitation. Newton’s theory explains the mysterious force responsible for Kepler’s Third Law.
Tycho Brahe and Johannes Kepler were scientists who had different approaches to their work. Brahe relied on direct observation, while Kepler focused on calculations and testing ideas. Brahe’s precise measurements of celestial objects would have lacked meaning without Kepler’s interpretation and analysis. Similarly, Kepler’s efforts to comprehend planetary motion would have been mere speculation without Brahe’s crucial data to substantiate and test his theories. Their contributions to science were mutually dependent and significant. This is how scientific progress unfolds—building upon prior insights and occasional fortuitous discoveries.
Quotes
“I much prefer the sharpest criticism of a single intelligent man to the thoughtless approval of the masses.” —Johannes Kepler
“Truth is the daughter of time, and I feel no shame in being her midwife.” —Johannes Kepler
“We ought not to ask why the human mind troubles to fathom the secrets of the universe. The diversity of the phenomena of nature is so great, and the treasures hidden in the skies so rich, precisely in order that the human mind shall never be lacking in fresh nourishment.” —Johannes Kepler
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References
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