Astronomy 171
Solar System Astronomy
Prof. Paul Martini

Lecture 14: Brahe and Kepler

Key Ideas

Tycho Brahe: Amassed 20 years of precise planetary data
Johannes Kepler: Brilliant theorist who analyzed Tycho's data
Kepler's Three Laws of Planetary Motion: 1st Law: Orbits are Ellipses with Sun at 1 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 greatest naked-eye astronomer of the pre-telescope age.

Nova of 1572

Bright "New Star" (Nova Stella) in Cassiopaeia: Failed to measure a parallax for the Nova; Must be in the celestial realm beyond the Moon!; Saw this as a blow to the Ptolemaic system
Tycho developed his own "Tychonic" system to replace the Ptolmaic

Tychonic System

Hybrid System: Earth at Center; Moon and Sun orbit the Earth; Planets orbit the Sun!
Full machinery of epicycles, etc., but no equant, like the Copernican system

Uraniborg

Tycho needed the best astronomical data to prove that his system was correct.
Built "Uraniborg" (Heavenly Castle) on the island of Hven with royal support.: Equipped with the best instruments; Achieved an unprecendented 1-2 arcminute measurement precision
Became an important center for astronomical research in Europe.

The Lord of Uraniborg

Astronomical Database: Positions of 777 stars to 1-2 arcmin precision; 20 years of precise planetary data
Great Comet of 1577:: Showed it was farther away than the Moon and orbiting the Sun!
Intensive Observations of Mars: Tried and failed to observe the parallax of Mars; Amassed data on Mars oppositions

"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

Johannes Kepler (1571 - 1630)

Brilliant German Mathematician: Staunch Copernican; Convinced the Universe was governed by physical laws; Obsessed with finding harmony in the heavens; Had a genius for data analysis
Inherited Tycho's data and post as 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. It took him 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 fifth test data point did not fit by eight arcminutes!

The Watershed

Kepler listened to the data: Knew Tycho's data were accurate to 1-2 arcminutes
Kepler questioned his assumptions:: Forced to abandon uniform circular motion; Concluded Mars' orbit was not a circle
Published the results in 1609 (Astronomia Nova)

1st Law of Planetary Motion

The orbits of the planets are ellipses with the Sun at one focus
Ellipses are characterized by two numbers:: Semimajor Axis (a): size of the longest axis; Eccentricity (e): shape of the ellipse (deviation from a perfect circle)
Orbit of Mars:: a = 1.5237 AU, e = 0.0934

2nd Law of Planetary Motion

The line joining the Sun and the planet sweeps out equal areas in equal times: Planets move fastest at Perihelion; Planets move slowest at Aphelion
Kepler's Second Law provides a geometric description of the change in speed.
It completely eliminates epicycles

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:: P² = a³
P = Period in years, a = semimajor axis in AU

Empirical Laws

Kepler's Laws are Empirical Laws: They describe how the planets move; They do not explain why they move that way
Not yet Physical Laws: Kepler made a start, but he had incorrect ideas about forces; Kepler's thinking was strongly motivated by his notions of universal harmony.
Correct explanation had to wait until the work of Isaac Newton

See A Note about Graphics to learn why some of the graphics shown in the lectures are not reproduced with these notes.