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Kepler’s Laws of Planetary Motion Tyco Brahe (1546-1601) • Danish astronomer • Had an island observatory and the best measurements of the positions for all know planets and the moon • Mercury, Venus, Mars, Jupiter, Saturn Johannes Kepler (1571-1630) • Austrian mathematician • Interested in how the planets move around the sun • When to Tyco’s island to get these accurate measurements • At that time, many astronomers believed that planets orbited around the sun in perfect circles • Tyco’s accurate measurements for Mars didn’t fit a circle • Kepler found that the orbit of Mars fit an ellipse the best First Law of Planetary Motion: A planet’s orbit is an ellipse with the sun at one focus and nothing at the other focus What is an ellipse? 2 foci • An ellipse is a geometric shape with 2 foci • A circle has 1 central focus An ellipse also has… …a major axis Perihelion …and a minor axis Aphelion Semi-major axis • Perihelion: When Mars or any other planet is closest to the sun • Aphelion: When Mars or any other planet is farthest from the sun Second Law of Planetary Motion: the line joining the planets to the Sun sweeps out “equal areas in equal times” as the planet travels around the ellipse • Kepler also found that Mars changed speed as it orbited around the sun – Faster when closer to the sun – Slower when farther from the sun • Areas A and B swept out by a line from the sun to A B Mars, were equal over the same amount of time • Kepler found a relationship between the time it took a planet to go completely around the sun (T, year) and the average distance from the sun (R, semi-major axis) T1 2 T2 2 = T1 R1 T2 R1 3 R2 3 R2 Third Law of Planetary Motion: the ratio of the square of the revolution time for two planets is equal to the ratio of the cubes of their semi-major axes T2 R2 • Earth’s year (T) is 1 year • Earth’s semi-major axis (R), the distance from the Earth to the Sun, is 1 astronomical unit (AU) T1 2 R1 3 or T1 2 = R1 3 or T 2 = R 3 = 1 1 2 3 1 1 T2 R2 WHAT DOES THIS MEAN?!? • If you know the distance from the sun and year of planet A, and you know the distance from the sun for planet B you can find the year of planet B. OR • If you know the distance from the sun and year of planet A, and you know the year of planet B you can find the distance from the sun for planet B When we compare the orbits of the planets… Planet T(yrs) R(au) T2 R3 Venus 0.62 0.72 Earth 1.00 1.00 1.00 1.00 Mars 1.88 1.52 3.53 3.51 Jupiter 11.86 0.38 0.37 5.20 141 141 We find that T2 and R3 are essentially equal. Kepler’s Laws apply to any celestial body orbiting any other celestial body • • • • • Any planet around a sun The moon around the Earth Any satellite around the Earth The International Space Station Any rings around any planet Earth’s Movement Barycenter • The point between two objects where they balance each other – The center of mass where two or more celestial bodies orbit each other. – When a moon orbits a planet, or a planet orbits a star, both bodies are actually orbiting around a point that lies outside the center of the larger body. (1,710 km) • The Sun is not stationary in our solar system. • It actually moves as the planets tug on it, causing it to orbit the solar system’s barycenter. • The sun never strays too far from the solar system barycenter. • http://www.youtube.com/watch?v=1iSR3Yw6 FXo Precession • Change in the direction of the axis of the Earth, but without any change in tilt • This changes the stars near (or not near) the Pole, but does not affect the seasons • The Earth’s axis currently points at Polaris as the “Northern Star”, but it wont always be.. Each gyration around the cone takes 26,000 years Nutation • Wobbling around the precessional axis • Change in the angle: ½ degree one way or the other • Occurs over an 18 year period and is due to the Moon • Very slightly increases or decreases seasonal effects