The Watershed : Tycho & 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 2nd Law: Equal Areas Law 3rd Law: (Period) 2 = (Semi-major Axis) 3 Tycho Brahe (1546-1601) Danish nobleman, brilliant astronomer and instrument builder. Staunch Aristotelian and anti-copernican. Developed his own Tychonic System. Built Uranibourg (Heavenly Castle) on the island of Hven with royal support. The best instruments he could design. Achieved an unprecedented 1-2 arcminute measurement precision. The Man with the Golden Nose Lost tip of his nose in a student duel at Wittenberg in 1566. Wore a metal insert, including one of gold for special occasions. Uranibourg Castle Tools of Tycho Brahe Sextant Mural Quadrant 1
Tycho Brahe (1546-1601) The Last (and Best) of the Old Fashioned Astronomers Astronomical Database: Positions of 777 stars to 1-2 arcmin precision. 20 years of precise planetary data measured regularly rather than only at special times (conjunctions etc) Nova of 1572: Failed to measure a parallax, concluded it was in the celestial realm beyond the moon! Great Comet of 1577: Showed it was beyond the moon and orbiting the Sun! The (Lack of) Parallax A large part of Ptolemaic thinking was that the distant heavens, the Celestial sphere, were perfect and immutable. By failing to measure a parallax for the supernova of 1572 or a comet in 1577, Tycho proved that these objects were not atmospheric phenomenon (as was the belief) but astronomical. He was using the size of the Earth as his baseline, not the size of the Earth s orbit assuming he had 1 arcminute accuracy, this means his limit was that they were more than 20 million km away. R <1arcmin D>R / Lunar Parallax Comet of 1577 Lima, Peru 20 o Rochester, NY Tycho s (Type Ia) Supernova Today ESA/XMM Tychonic System A Hybrid System: Earth at Center. Moon & Sun Orbit Earth. Planets Orbit Sun! Full mechanism of epicycles, equants, etc. This is an X-ray image of the hot gas left after the supernova. It s about 20 light years across after 430 years, and about 8000 light years away. 2
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 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. Asked to work with Tycho because he wanted access to his data started in 1600 Inherited Tycho s data along with his post of Imperial Mathematicus in 1601 Johannes Kepler (1571-1630) Theories based on a deferent, an epicycle and eccentric can match the positions of most planets to 0.1 o and of the Sun to 0.01 o the biggest problem is Mars, where the errors are 0.5 o Until Tycho, almost all this could be hidden as problems in measurement (Copernicus was happy to get fits that matched to 1/6 o ) Tycho first assigned Kepler the problem of working out the motions of Mars 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. He could not obtain a good fit using uniform circular motions. Kepler's calculations from the Astronomia Nova (1609) Kepler listened to the data, so he questioned the assumptions: Forced to abandon uniform circular motion. Concluded Mars orbit was not a circle, but instead an ellipse with the Sun at one focus. Published results in 1609 (Astronomia Nova). The Watershed 3
The 1 st 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. Orbit of Mars: a = 1.5237 AU, e = 0.0934 Semimajor Axis F 1 Center F 2 Ellipses I Ellipses are a type of conic section the curves you get by cutting a cone with a plane circle, ellipse, parabola, hyperbola Closed orbits must be ellipses (a circle is just an ellipse with no eccentricity) Ellipses II Geometric construction tie a string (yellow) to two points called the foci (F 1 and F 2 ) and draw the curve found by stretching the string tightly (the distance from one focus to the curve and over to the other focus is the same for all points on the curve). The length of the string is twice the semimajor axis. Open orbits are parabolas and hyperbolas. The Circle An ellipse with no ellipticity Semimajor axis = a = radius of the circle Eccentricity = e = 0 Ellipses III We can write everything else in terms of the semimajor axis a and ellipticity e Pericenter = closest approach = a(1 e) Apocenter = most distant point = a(1+e) pericenter Semimajor axis=a Semiminor Axis=b ae apocenter Put the two focii at the same point and you get a circle. Sun Planet 4
Planetary Ellipticities The range of planetary eccentricities is Venus smallest with e=0.007 Pluto largest with e=0.25 Earth relatively small with e=0.017 Mars relatively big with e=0.093 one reason why its orbit created the biggest problems for models based on circles Comets can have eccentricities approach unity Halley s comet has semi-major axis =a = 17.9 AU eccentricity =e = 0.967 pericenter = a(1 e) = 0.6 AU apocenter = a(1+e) = 35 AU The 2 nd Law of Planetary Motion The line joining the Sun and the planet sweeps out equal areas in equal times. Planets move fastest when closest to the Sun (perihelion) Planets move slowest when farthest from the Sun (aphelion). Geometric description of the change in speed. Completely eliminates epicycles & other stuff. T=0 d T=10 d Equal Areas in Equal Times T=10 d T=0 d 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: P P= P 2 a 3 =a a a P = Period in years, a = Semimajor axis in AU Planet a (AU) P (yr) a 3 P 2 Mercury 0.387 0.241 0.058 0.058 Venus 0.723 0.615 0.378 0.378 Earth 1.000 1.000 1.000 1.000 Mars 1.524 1.881 3.537 3.537 Jupiter 5.203 11.862 140.8 140.7 Saturn 9.534 29.456 867.9 867.7 Empirical Laws Kepler s Laws are of 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. The Third Law applies to all bodies orbiting the Sun: planets, comets, rocks, & spacecraft! This was to wait for Isaac Newton. 5