Kepler's Laws of Planetary Motion (1609-1619): Elliptical Orbits

Kepler’s laws of planetary motion are three rules describing how planets move around the Sun, discovered by the German astronomer Johannes Kepler between about 1602 and 1618. Published in Astronomia Nova (1609) and Harmonices Mundi (1619), they replaced the ancient assumption of perfectly circular orbits with ellipses and gave astronomy precise mathematical laws. Isaac Newton later showed that they follow from gravitation, making them a cornerstone of modern physics.

Background: circles and the Copernican system

Since antiquity astronomers had assumed that heavenly bodies move in circles at uniform speed. Ptolemy’s system, with the Earth at the centre, combined circles upon circles to reproduce the planets’ wandering paths. When Nicolaus Copernicus placed the Sun at the centre in 1543, he still used circles and small epicycles.

Neither model predicted planetary positions very accurately, partly because the observations they relied on were themselves poor. That changed with the Danish nobleman Tycho Brahe, who at his observatory on the island of Hven, described in the article on Uraniborg, built large instruments and measured the positions of stars and planets for about twenty years with an accuracy of around one to two minutes of arc.

Kepler and Tycho

Johannes Kepler was born on 27 December 1571 in Weil der Stadt in Württemberg. At the University of Tübingen his teacher Michael Maestlin introduced him to Copernican astronomy. While teaching mathematics in Graz he published Mysterium Cosmographicum (1596), which tried to explain the spacing of the six known planets by nesting the five regular Platonic solids between their spheres.

In 1600, under growing pressure in Graz from the Counter-Reformation because he was a Lutheran, Kepler joined Tycho Brahe, who had become imperial mathematician to Emperor Rudolf II in Prague. Their relationship was uneasy: Tycho guarded his data closely, and Kepler wanted full access. Tycho set him to work on the orbit of Mars, whose large eccentricity made it the hardest planet to model.

Tycho died on 24 October 1601. Two days later Kepler was appointed his successor as imperial mathematician, and after disputes with Tycho’s heirs he kept the use of the observations.

The war on Mars

Kepler called his struggle with the Mars orbit a war. He first built a sophisticated circular model, his “vicarious hypothesis”, which matched Tycho’s longitudes well. But when tested against other observations it was wrong by about eight minutes of arc. Earlier astronomers might have accepted such an error, but Kepler trusted Tycho’s precision and concluded that the model, not the data, must be wrong.

Along the way he found that the speed of a planet changes in a regular way. In 1602 he arrived at what is now called the second law: a line joining the planet to the Sun sweeps out equal areas in equal times. Only after this did he abandon the circle altogether. After testing various oval shapes, he found in 1605 that an ellipse, with the Sun at one focus, fitted the observations of Mars.

Kepler insisted that astronomy should be based on physical causes, not just geometry. He placed the source of planetary motion in the Sun, imagining a force, partly inspired by William Gilbert’s work on magnetism, that swept the planets around as the Sun rotated.

Astronomia Nova (1609)

Kepler’s account of his campaign was published in 1609 as Astronomia Nova (“New Astronomy”), with a subtitle stressing that it was based on causes, or “celestial physics”. Printing problems and disputes with Tycho’s family delayed it for years after the work was largely complete.

The book is unusual because Kepler describes his false starts and errors as well as his successes. It contains what are now called the first two laws, although Kepler did not number them, and he demonstrated them only for Mars. The order in which the laws are numbered today reflects their logical presentation in later textbooks, not the order of discovery: the area law came first.

The third law and Harmonices Mundi (1619)

Kepler moved to Linz in 1612 after the death of Rudolf II. His life there was hard: wars, religious tension and, from 1615, the long witchcraft trial of his mother, Katharina, whom he defended until her release in 1621.

He also pursued an old ambition: to find the harmonies that he believed God had built into the cosmos. On 15 May 1618 he confirmed the relationship now known as the third law: the square of a planet’s period of revolution is proportional to the cube of its mean distance from the Sun. It appeared in Harmonices Mundi (“The Harmony of the World”), published in 1619, amid long discussions of musical intervals linked to planetary speeds.

In his textbook Epitome Astronomiae Copernicanae (1617–1621) Kepler applied the laws to all the planets and to the Moon and the moons of Jupiter, which Galileo had discovered in 1610.

The three laws

First law (law of ellipses): each planet moves in an ellipse with the Sun at one of its two foci. The ellipses are mostly close to circles; that of Mars departs from a circle by an eccentricity of about 0.09.

Second law (law of equal areas): the line from the Sun to a planet sweeps out equal areas in equal intervals of time. A planet therefore moves fastest at perihelion, its closest point to the Sun, and slowest at aphelion.

Third law (harmonic law): the square of the orbital period is proportional to the cube of the semi-major axis of the orbit. Measuring periods in years and distances in astronomical units, the two numbers are equal for every planet. Jupiter, for example, lies about 5.2 times farther from the Sun than the Earth; 5.2 cubed is about 140, and the square root of 140 is about 11.9, which matches Jupiter’s orbital period of almost twelve years.

Taken together, the laws describe not only the shape of each orbit but also how fast a planet moves at each point and how the different orbits relate to one another. For the first time the whole planetary system could be described by a few simple rules rather than a collection of separate geometrical devices for each planet.

Key people

Johannes Kepler (1571–1630) was a Lutheran mathematician whose deep religious belief that God had created an orderly, mathematical universe drove his search for laws. He also made important contributions to optics, explaining how the eye forms images and how lenses work.

Tycho Brahe (1546–1601) was a Danish nobleman and the greatest observational astronomer before the telescope. He rejected Copernicus’s moving Earth and proposed his own system, in which the planets orbit the Sun while the Sun circles a stationary Earth.

Rudolf II (1552–1612), Holy Roman Emperor, was a patron of art, alchemy and astronomy who employed both Tycho and Kepler at his court in Prague, although his treasury often failed to pay Kepler’s salary.

Michael Maestlin (1550–1631), professor at Tübingen, taught Kepler the Copernican system and remained his correspondent and adviser.

Reception and consequences

The laws were not widely accepted at first. Galileo, Kepler’s contemporary and correspondent, never adopted elliptical orbits. Many astronomers found Kepler’s mixture of physics, mysticism and difficult mathematics hard to follow.

What changed minds was accuracy. Kepler’s Rudolphine Tables, completed with Tycho’s data and published in 1627, predicted planetary positions far better than earlier tables. In 1631 Pierre Gassendi in Paris observed a transit of Mercury close to the time Kepler had predicted, and in 1639 Jeremiah Horrocks in England observed a transit of Venus using corrected Keplerian tables.

Kepler died on 15 November 1630 in Regensburg. In 1687 Isaac Newton’s Principia showed that all three laws follow from a single inverse-square law of gravitation together with his laws of motion, and the laws became part of the foundation of classical mechanics.

Myths and facts

Myth: Kepler murdered Tycho Brahe to get his data. This theory, popularised in the early 2000s, has no supporting evidence. Analysis of Tycho’s remains after their exhumation in Prague in 2010 found no poisonous level of mercury.

Myth: Kepler simply guessed the ellipse. The surviving notes show years of calculation and the rejection of many models before the ellipse emerged as the shape that matched the observations.

Fact: Kepler’s laws are still used. They describe the orbits of planets, comets, moons and artificial satellites, and the third law is used to estimate the masses of stars and the orbits of planets around other stars.

Timeline

• 1571, 27 December: Johannes Kepler born in Weil der Stadt.
• 1596: Mysterium Cosmographicum published.
• 1600, February: Kepler begins working with Tycho Brahe near Prague.
• 1601, 24 October: Tycho Brahe dies; Kepler becomes imperial mathematician.
• 1605: Kepler finds that the orbit of Mars is an ellipse.
• 1609: Astronomia Nova published with the first two laws.
• 1618, 15 May: Kepler confirms the third law.
• 1619: Harmonices Mundi published.
• 1627: Rudolphine Tables published.
• 1630, 15 November: Kepler dies in Regensburg.

Why it matters

Kepler’s laws turned astronomy from a geometry of circles into a science of precise physical laws based on observation. They showed that careful data could overturn a two-thousand-year-old assumption, and they gave Newton the facts his theory of gravitation had to explain.