Lec06-Feb04[1]

Lec06-Feb04[1] - 2/4/11 Quiz Ques>on #1 Johannes...

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Unformatted text preview: 2/4/11 Quiz Ques>on #1 Johannes Kepler Astro 109 Lecture 6: Kepler's Laws of Planetary Mo>on Feb. 4 A. made the first accurate measurements of Earth's size. B. was the first astronomer to use a telescope to observe the phases of Venus. C. was famous for highly accurate observa>ons of planet and star posi>ons, made without a telescope. D. was the first to discover that the orbits of the planets about the Sun are ellip>cal. E. was the first to successfully find physical laws that explain both terrestrial and celes>al mo>on. Feb. 4 Quiz Ques>on #2 Which of the following is not an aspect of Kepler's laws? A. The orbit of each planet around the Sun is an ellipse with the Sun at one focus. B. A given planet moves faster when it is closer to the Sun and slower than it is farther away. C. Each planet spins (rotates) in the same direc>on as its mo>on around the Sun. D. A planet with a larger orbit around the Sun has a slower average speed than a planet with a smaller orbit. Feb. 4 Feb. 4 Quiz Ques>on #3 Earth is about 1 AU from the Sun, and its orbital period is 1 yr. Venus is about 0.7 AU from the Sun, and its orbital period is 0.6 yr. How long is the orbital period of a spacecra] in a circular orbit 0.8 AU from the Sun? A. B. C. D. E. less than 0.6 yr 0.6 yr between 0.6 yr and 1 yr 1 yr more than 1 yr Key Concepts orbits are conic sec>ons (circles, ellipses, hyperbolas) equal areas in equal >mes / angular momentum P2 a3 universality Thought Ques>on If the Moon were 42,200 km from the center of Earth, in the equatorial plane, how would it appear to an observer on Earth? A. B. C. D. E. We would see all the phases in a year. We would see all the phases in a month. We would see all the phases in a day. We would not see any phases. I don't know how to answer this ques>on. Feb. 4 Feb. 4 1 2/4/11 Kepler's Laws 1. The orbit of a planet about the Sun is an ellipse with the Sun at one focus. Conic sec>ons Circle Ellipse Wikipedia focus (plural foci) Feb. 4 Feb. 4 Ellip>cal orbit semimajor axis major axis minor axis Can't fit the data? Change your hypothesis! "If I had believed that we could ignore these eight minutes [of arc], I would have patched up my hypothesis accordingly. But, since it was not permissible to ignore, those eight minutes pointed the road to a complete reforma>on in astronomy." -- Kepler Note: we measure semimajor axis from the center, but remember the Sun is at a focus. Feb. 4 Feb. 4 Specifying a circle or ellipse Circle: radius r Example Halley's comet is about 0.6 AU from the Sun at perihelion, and 35 AU from the Sun at aphelion. What is its semimajor axis? Ellipse: semimajor axis a, semiminor axis b - OR - semimajor axis a, eccentricity e major axis = 35.6 AU semimajor axis = 17.8 AU 0.6 AU 35 AU Not to scale! Feb. 4 Feb. 4 2 2/4/11 Kepler's Laws 1. The orbit of a planet about the Sun is an ellipse with the Sun at one focus. 2. A line joining a planet and the Sun sweeps out equal areas in equal intervals of >me. Feb. 4 Feb. 4 Why? Casual ice skater effect Physics conserva>on of angular momentum Feb. 4 Feb. 4 Discussion Ques>on Suppose we plot the shape of planetary orbits and the corresponding speed versus >me. Which of the following combina>ons is correct? A B Kepler's Laws 1. The orbit of a planet about the Sun is an ellipse with the Sun at one focus. 2. A line joining a planet and the Sun sweeps out equal areas in equal intervals of >me. C D 3. The square of the sidereal period of a planet is directly propor>onal to the cube of the semimajor axis of the orbit. Feb. 4 Feb. 4 3 2/4/11 Kepler's original formula>on Express the semimajor axis, a, in AU Express the period, P, in yr Then Planet Planets in the Solar System a (AU) 0.387 0.723 1 1.52 5.20 9.55 a2 0.150 0.522 1 2.32 27.1 91.3 a3 0.058 0.378 1 3.54 141 872 P (yr) 0.241 0.615 1 1.88 11.86 29.46 P2 0.058 0.378 1 3.53 141 868 P3 0.014 0.233 1 6.64 1668 25568 Mercury Venus P 2 = a3 Earth Mars Note: depends on semimajor axis not eccentricity not perihelion or aphelion distance Feb. 4 Feb. 4 Jupiter Saturn Example Halley's comet is about 0.6 AU from the Sun at perihelion, and its sidereal period is 75.3 yr. How far is it from the Sun at aphelion? A generalized version For orbits around objects other than the Sun... a3 = P 2 = 5670 a = 17.8 AU perihelion + aphelion = 2a = 35.6 AU --> aphelion = 35 AU [P (yr)]2 = [a (AU)]3 M (M ) Feb. 4 Feb. 4 Discussion Ques>on Suppose the Sun suddenly turned into a black hole with the same mass. What would happen to the Earth? A. B. C. D. E. It would fall into the black hole. It would stay in the same orbit but move faster. It would stay in the same orbit at the same speed. It would stay in the same orbit but move slower. It would move to a smaller orbit. A modified version More useful for orbits around planets... Express the semimajor axis, a, in km Express the period, P, in s Then [P (s)]2 Feb. 4 Feb. 4 = 10-4 [a (km)]3 M (M ) 4 2/4/11 Jupiter's Moons Moon Io Europa a (km) 4.22105 6.71105 a3 7.51016 3.01017 1.21018 6.71018 3.110-7 a3 2.31010 9.41010 3.81011 2.11012 P (d) 1.769 3.551 7.155 16.689 P (s) 1.5105 3.1105 6.2105 1.4106 P2 2.31010 9.41010 3.81011 2.11012 Universality Kepler's laws apply to: planets orbi>ng the Sun moons orbi>ng planets planets orbi>ng other stars stars orbi>ng black holes ... Ganymede 1.07106 Callisto 1.88106 [P (s)]2 = 3.1 10-7 [a (km)]3 Feb. 4 Feb. 4 Discussion Ques>on We can measure how fast exoplanets move in their orbits. What can we learn about those planets by applying Kepler's laws? A. B. C. D. E. The size of exoplanet orbits. The shape of exoplanet orbits. The size of exoplanets. Both A and B. All three A, B, and C. hop://exoplanets.org/massradiiframe.html Feb. 4 Feb. 4 The Center of the Milky Way http://www.astro.ucla.edu/~ghezgroup/gc/pictures/orbitsMovie.shtml Feb. 4 http://www.mpe.mpg.de/ir/GC/index.php Feb. 4 5 2/4/11 Example There is a black hole at the center of the Milky Way. A star orbits it with a semimajor axis a = 920 AU and period P = 14.5 yr. What is the mass of the black hole (in units of Msun)? Recap orbits are conic sec>ons (circles, ellipses, hyperbolas) equal areas in equal >mes / angular momentum P2 a3 universality [P (yr)]2 = M (M ) = = = = Feb. 4 [a (AU)]3 M (M ) [a (AU)]3 [P (yr)]2 9203 14.532 7.8 108 2.1 102 3.7 106 Feb. 4 Discussion Ques>on If the Moon were 42,200 km from the center of Earth, in the equatorial plane, how would it appear to an observer on Earth? A. B. C. D. E. We would see all the phases in a year. We would see all the phases in a month. We would see all the phases in a day. We would not see any phases. I don't know how to answer this ques>on. Example A satellite orbits Earth in a circular orbit with a radius of 42,200 km. What is its period? [P (s)]2 = 10-4 P = 7.4 109 = 24 hr [a (km)]3 M (M ) = 8.6 104 s Feb. 4 Feb. 4 6 ...
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This note was uploaded on 03/24/2011 for the course PHYS 123 taught by Professor Wenkstern during the Spring '08 term at Rutgers.

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