Astronomy 161: An Introduction to Solar System Astronomy Prof. Richard Pogge, MTWThF 2:30

Lecture 15: The Watershed:
Tycho Brahe & Johannes Kepler

Key Ideas:

• Superb observer who amassed 20 years of precise planetary data.

• Brilliant theorist who analyzed Tycho's data

• 1st Law: Orbits are Ellipses with Sun at one focus.
• 2nd Law: Equal Areas Law
• 3rd Law: (Period)²=(Semi-major Axis)³

Tycho Brahe (1546-1601)

Danish nobleman, brilliant astronomer and instrument builder.

• Admired Copernicus as a mathematician.
• Did not like the idea of a moving Earth.

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.

• Failed to measure a parallax, concluded it was in the celestial realm beyond the Moon!
• Saw this as a blow to the Ptolemaic system.

The Tychonic System

[Click to view full size - 51k]

• The Earth is at the center.
• The Moon & Sun orbit the Earth.
• but, the planets orbit the Sun!

The Lord of Uraniborg

• Equipped it with the best instruments he could design.
• Achieved an unprecedented 1-2 arcminute measurement precision.

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:

• Positions of 777 stars with 1-2 arcminute precision.
• 20 years of precise planetary data.

Great Comet of 1577:

• Showed it was farther away than the Moon and orbiting the Sun!

Intensive observation of Mars:

• Tried and failed to observe the parallax of Mars
• Amassed data on Mars oppositions.

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."

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)

Brilliant but tormented mathematician born in Germany.

• Staunch Copernican
• Convinced the Universe was governed by physical laws.
• Obsessed with finding harmony in the heavens.
• Had a genius for analyzing data.

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

The Motions of Mars

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

• Started by determining the orbit of the Earth using successive oppositions of Mars.
• Fit an off-center circle (i.e., not centered on the Sun) to the first 4 data points.
• A 5th test data point did not fit by 8 arcminutes...

The Watershed

• Knew that Tycho's data were accurate to 1-2 arcminutes.
• But Tycho was never off as much as 8 arcminutes.

This led him to question his assumptions:

• Forced to abandon uniform circular motion.
• Concluded Mars' orbit was not a circle.

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

The 1^st 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:

• Semimajor Axis: (a) size of the longest axis
• Eccentricity: (e) shape of the ellipse.

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

The 2^nd 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)

• Planets move fastest at Perihelion (closest to the Sun)
• Planets move slowest at Aphelion (farthest from the Sun)

The Second Law (sometimes called the "Equal Areas 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 3^rd 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:

P²=a³

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!

Kepler's Third Law did not appear until 1618 in his Harmonice Mundi (The 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

• They describe how the planets move.
• They don't explain why they move that way.

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

• Kepler made a start, but he had incorrect ideas about how forces work.
• Kepler was strongly motivated by notions about universal harmonies and convinced that the motions could be understood physically.

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

Further Reading:

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.