Lecture 15: "The Watershed":
Tycho Brahe & Johannes Kepler

Key Ideas:

Tycho Brahe Johannes Kepler Kepler's Three Laws of Planetary Motion:

Tycho Brahe (1546-1601)

Tycho Brahe

Danish nobleman, brilliant astronomer and instrument builder.

The Man with the Golden Nose

Tycho was a brilliant and pugnacious man. As a student in Rostock in 1566, he lost part of his nose in a sword duel with another student (a third cousin, in fact). For the rest of his life, he wore a false nose he fashioned out of gold and silver (with a common copper one for everyday use), afixed to his face with an adhesive salve (click on the portrait above to see a larger version that makes this as plain as, well, the nose on his face).

The Nova of 1572:

In November of 1572, Tycho noticed a new bright star in the constellation of Cassiopeia. He proceeded to make detailed measurements of it over the next 16 months.

Seeing such change in the celestial realm was in serious contradiction to the Aristotelian ideal of celestial perfection. The event spurred Tycho's astronomical interests, and marked the beginning of his lifelong efforts to develop his own "Tychonic System", a hybrid of the Copernican and Ptolemaic systems. However, he knew that words were not enough, he needed hard observational evidence to prove that his ideas were correct.

The Tychonic System

Hybrid Geocentric/Heliocentric System:

The Tychonic System
[Click to view full size - 51k]

The Tychonic System uses the full machinery of epicycles, etc. to make it all work in details, but like Copernicus, without using equants. Tycho felt that his system explained planetary motions better, and preserved the ideas of an unmoving Earth and uniform circular motion.

The Lord of Uraniborg

Tycho knew that he needed the best astronomical data to prove that his system was correct. With the backing of the Danish crown, he built "Uraniborg" (Heavenly Castle) on Hven, a tiny island in the Øresund straits between Denmark and Sweden.

Uraniborg quickly became the premier center for astronomical research in Europe.

Tycho's 20 years of work at Uraniborg (from 1577 until 1597) produced the following results:

Astronomical Database:

Great Comet of 1577:

Intensive observation of Mars:

All of these observations attest not only to Tycho's considerable skills, but they also contain the seeds of the downfall of the Aristotelian view of the world. Comets and Novae (the latter now understood to be stellar explosions) were thought to be atmospheric phenomena, since change could not occur in the celestial realm. Further, the demonstration that the center of motion of the Great Comet of 1577 was the Sun and not the Earth was a blow to the strict geocentric point of view.

It was Tycho's careful observations of Mars and the planets, however, were to have an importance far beyond Tycho's imagining (or intentions).

"Let me not seem to have lived in vain."

Tycho had a falling out with the new Danish king, finally leaving Denmark in 1597.

1599: Appointed Imperial Mathematicus at Prague

1600: Hired Johannes Kepler as his assistant.

1601: Died in Prague, Kepler took his place.

Tycho made two essential contributions to science: he acquired the finest naked-eye data on the stars and planets, and he hired Johannes Kepler to anaylze these data. Contemporary accounts record that Tycho's dying words were "Let me not seem to have lived in vain". His wish was for Kepler to use his data to prove the truth of his Tychonic system. What he got was something quite different.

Johannes Kepler (1571-1630)

Johannes Kepler, anonymous oil painting c. 1620
[Oil Painting c.1620, Collegium Wilhelmitanum, Strasbourg]

Brilliant but tormented mathematician born in Germany.

Inherited Tycho's data along with his post of Imperial Mathematicus in 1601.

The Motions of Mars

Mars was the key to unlocking the secrets of planetary motion.

Kepler began analyzing the orbit of Mars in 1601. Task took 4 years.

The Watershed

Kepler listened to the data:

This led him to question his assumptions:

Published results of his work in 1609 (Astronomia Nova), which contained the first and second of his three laws of planetary motion.

The 1st Law of Planetary Motion

The orbits of the planets are ellipses with the Sun at one focus.

Click on the image to view full size (8Kb)

Ellipses are characterized by two numbers:

Example: The Orbit of Mars: a = 1.5237 AU, e = 0.0934

The 2nd Law of Planetary Motion

The line joining the Sun and the planet sweeps out equal areas in equal times.

Click on the image to view full size (10Kb)

The Second Law provides a geometric description of the change in speed that completely eliminates epicycles and all of the other baggage of the Ptolemaic and Copernican systems. It also signalled the final rejection of uniform circular motion.

The 3rd Law of Planetary Motion

The square of a planet's orbital period is proportional to the cube of the semimajor axis of the orbit.

Expressed Mathematically:


where P is the planet's orbital period measured in years, and a is the semimajor axis of the planet's orbit measured in AUs.

Kepler's Third Law applied to the Planets

Planet a (AU) P (yr) a3 P2 P2/a3
Mercury 0.3871 0.2408 0.0580 0.0580 1.0000
Venus 0.7233 0.6152 0.3785 0.3785 1.0000
Earth 1.0000 1.0000 1.0000 1.0000 1.0000
Mars 1.5237 1.8808 3.5373 3.5376 1.0001
Jupiter 5.2034 11.8626 140.8810 140.7216 0.9989
Saturn 9.5371 29.4475 867.4510 867.1551 0.9997

[The data used to create the table above are from the JPL Solar System Dynamics Group website, using their tables of Approximate Positions of the Major Planets and the table of Physical Characteristics of the Planets.]

The Third Law applies to all bodies orbiting the Sun: planets, comets, rocks, & spacecraft!

Something to try on your own: Look up Appendix 1 in the textbook, and compute P2/a3 for the 7 classical planets. There is a mistake in the table! Can you find it?

Kepler's Third Law did not appear until 1618 in his Harmonice Mundi (Harmony of the World). In this book Kepler made a much deeper exploration of his ideas of harmony in mathematics, music, and the heavens, even going so far as to set the motions of the planets to music, with rising pitch indicating faster orbital speed following his Second Law!

Despite the apparently mystical tone of this work, Kepler has deeply committed to the idea that physical causes underlied the motions of the planets, and he felt that the existance of a "harmonic" relation between the period and semi-major axis of an orbit was proof of this.

Empirical Laws

Kepler's Laws are Empirical:

To answer why, we need Physical Laws of planetary motion.

The correct physical explanation had to wait nearly 70 years until the work of Isaac Newton.

Further Reading:

I've only been able to give the barest outlines of this remarkable story. If you are interested in learning more, I highly recommend Tycho and Kepler, by Kitty Ferguson. Written in 2002, it is one of the finest accounts of the lives of the two men and their work that I have read. It fills in a lot of the details I have left out, and casts Tycho in a much better light than others have done.

The title of this lecture, The Watershed, is borrowed from the title of a biography of Kepler by Arthur Koestler (which itself is an excerpt from his longer book, The Sleepwalkers). Koestler's book is interesting if opinionated, and I do not agree with all of his ideas (in particular, I have issues with his rough treatment of Galileo, and his belief that Kepler's work proves his [Koestler's] ideas about of the essential role of the irrational in scientific discourse). Despite this, it is provocative and interesting reading.

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Updated: 2006 October 4
Copyright Richard W. Pogge, All Rights Reserved.