Lec06-Feb05

Lec06-Feb05 - 2/5/10 Discussion Ques>on If the...

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Unformatted text preview: 2/5/10 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? Astro 109 Lecture 6: Kepler’s Laws of Planetary Mo>on A.  B.  C.  D.  E.  Feb. 5 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. 5 Quiz Ques>on #1 Quiz Ques>on #2 Kepler’s laws of planetary mo>on A planet in an ellip>cal orbit around the Sun moves most rapidly when it is A.  were ini>ally derived from fundamental physics laws. B.  were ini>ally found from observa>on and are empirical rules that describe how the planets move. C.  were handed down from the ancient Greeks. D.  were discovered by Kepler in ancient Babylonian texts. E.  are now known to be grossly incorrect. A.  B.  C.  D.  E.  Feb. 5 at aphelion. at perihelion. halfway between perihelion and aphelion. closer to aphelion than perihelion. None of the above; it has a constant speed. Feb. 5 Quiz Ques>on #3 Two planets are in iden>cal orbits around iden>cal stars. One planet is Jupiter’s size, whereas the other is Earth’s size. Which of the following is true? A.  The Jupiter ­size planet has a longer period than the Earth ­size planet. B.  The Earth ­size planet has a longer period than the Jupiter ­size planet. C.  Both planets have the same period. D.  The Earth ­size planet has a period exactly twice that of the Jupiter ­size planet. E.  Not enough informa>on is given to determine the correct answer. Feb. 5 Key Concepts •  orbits are conic sec>ons (circles, ellipses, hyperbolas) •  equal areas in equal >mes / angular momentum •  P2 ∝ a3 •  universality Feb. 5 1 2/5/10 Conic Sec>ons Kepler’s Laws 1.  The orbit of a planet about the Sun is an ellipse with the Sun at one focus. Circle Ellipse focus (plural foci) Feb. 5 Feb. 5 ★ semimajor axis major axis minor axis Can’t fit the data? Change your hypothesis! Feb. 5 Note: we measure semimajor axis from the center, but remember the Sun is at a focus. Feb. 5 Specifying a circle or ellipse •  Circle: –  radius r •  Ellipse: –  semimajor axis a, semiminor axis b  ­ OR  ­ 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? –  semimajor axis a, eccentricity e major axis = 35.6 AU semimajor axis = 17.8 AU 0.6 AU Feb. 5 35 AU Feb. 5 2 2/5/10 ★ 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. 5 Feb. 5 ★ Discussion Ques>on Why? Casual •  ice skater effect Suppose we plot the shape of planetary orbits and the corresponding speed versus >me. Which of the following combina>ons is correct? A B C D Physics •  conserva>on of angular momentum Feb. 5 Feb. 5 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. 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. 5 Kepler’s original formula>on •  Express the semimajor axis, a, in AU •  Express the period, P, in yr •  Then P 2 = a3 •  Note: depends on semimajor axis –  not eccentricity –  not perihelion or aphelion distance Feb. 5 3 2/5/10 Example Planets in the Solar System Planet a (AU) a2 a3 P (yr) P2 Mercury 0.387 0.150 0.058 0.241 0.058 0.014 Venus 0.723 0.522 0.378 0.615 0.378 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? P3 0.233 Earth 1 1 1 1 1 1.52 2.32 3.54 1.88 3.53 6.64 Jupiter 5.20 27.1 141 11.86 141 1668 Saturn 9.55 91.3 872 29.46 868 = P 2 = 5670 a = 17.8 AU 1 Mars a3 25568 Feb. 5 •  perihelion + aphelion = 2a = 35.6 AU •   ­ ­> aphelion = 35 AU Feb. 5 Discussion Ques>on A generalized version Suppose the Sun suddenly turned into a black hole with the same mass. What would happen to the Earth? For orbits around objects other than the Sun… [P (yr)]2 = A.  B.  C.  D.  E.  [a (AU)] M (M⊙ ) 3 Feb. 5 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. Feb. 5 A modified version More useful for orbits around planets… Jupiter’s Moons Moon a (km) a3 3.1×10 ­7 ×a3 P (d) P (s) P2 Io 4.22×105 7.5×1016 2.3×1010 1.769 1.5×105 2.3×1010 Europa 6.71×105 3.0×1017 9.4×1010 3.551 3.1×105 9.4×1010 Ganymede 1.07×106 1.2×1018 3.8×1011 7.155 6.2×105 3.8×1011 Callisto 1.88×106 6.7×1018 2.1×1012 16.689 1.4×106 2.1×1012 •  Express the semimajor axis, a, in km •  Express the period, P, in s •  Then [P (s)]2 Feb. 5 = 10−4 × [a (km)]3 M (M⊕ ) [P (s)]2 = 3.1 × 10−7 × [a (km)]3 Feb. 5 4 2/5/10 Discussion Ques>on Universality We can measure how fast exoplanets move in their orbits. What can we learn about the planets by applying Kepler’s laws? Kepler’s laws apply to: •  •  •  •  •  planets orbi>ng the Sun moons orbi>ng planets planets orbi>ng other stars stars orbi>ng black holes … A.  B.  C.  D.  E.  Feb. 5 The size of exoplanet orbits. The shape of exoplanet orbits. The size of exoplanets. Both A and B. All three A, B, and C. Feb. 5 exoplanets.org Feb. 5 The Center of the Milky Way Feb. 5 http://www.mpe.mpg.de/ir/GC/index.php 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)? [P (yr)]2 = M (M⊙ ) = = = http://www.astro.ucla.edu/~ghezgroup/gc/pictures/orbitsMovie.shtml = Feb. 5 Feb. 5 [a (AU)]3 M (M⊙ ) [a (AU)]3 [P (yr)]2 9203 14.532 7.8 × 108 2.1 × 102 3.7 × 106 5 2/5/10 Example A satellite orbits Earth in a circular orbit with a radius of 42,200 km. What is its period? [P (s)]2 P = 10−4 × [a (km)]3 M (M⊕ ) = 7.4 × 109 = 8.6 × 104 s = 24 hr Feb. 5 6 ...
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This note was uploaded on 09/15/2011 for the course PHYS 109 taught by Professor Pryor during the Spring '09 term at Rutgers.

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