2. 9/13/2005PHY 101 Week 31 The solar system Sun Planets Asteroids Comets Pluto Neptune Uranus Saturn Jupiter Mars Earth Venus Mercury Sun

3. 9/13/2005PHY 101 Week 32 Historical figures in the Copernican Revolution Ptolemy – the geocentric model, that the Earth is at rest at the center of the Universe. Copernicus – published the heliocentric model. Galileo – his observations by telescope verified the heliocentric model. Kepler – deduced empirical laws of planetary motion from Tycho’s observations of planetary positions. Newton – developed the full theory of planetary orbits.

4. 9/13/2005PHY 101 Week 33 The Copernican Revolution

5. 9/13/2005PHY 101 Week 34

6. 9/13/2005PHY 101 Week 35 Nicolaus Copernicus The Earth moves, in two ways. It rotates on an axis (period = 1 day). It revolves around the sun (period = 1 year).

7. 9/13/2005PHY 101 Week 36

8. 9/13/2005PHY 101 Week 37 Johannes Kepler (1571 – 1630) … discovered three empirical laws of planetary motion in the heliocentric solar system 1.Each planet moves on an elliptical orbit. 2.The radial vector sweeps out equal areas in equal times. 3.The square of the period is proportional to the cube of the radius. (needed for the CAPA)

9. 9/13/2005PHY 101 Week 38 How did Kepler determine the planetary orbits? Compare the heliocentric model to naked-eye astronomy The most complete data had been collected over a period of many years by Kepler’s predecessor, Tycho Brahe of Denmark. The inner planet is Earth; the outer one is Mars. Plot their positions every month. Mars lags behind the Earth so its appearance with respect to the Zodiac is shifting. Earth Mars

10. 9/13/2005PHY 101 Week 39 Ellipse Geometry To draw an ellipse: Take a string. Tack down the two ends. Put a pencil in the string and pull the string taut. Move the pencil around keeping the string taut. An ellipse is the locus of points for which the sum of the distances to two fixed points is fixed. The two fixed points are called the focal points of the ellipse.

11. 9/13/2005PHY 101 Week 310 Parameters of an elliptical orbit (a,e) ► Semi-major axis = a = one half the largest diameter ► Eccentricity = e = ratio of the distance between the focal points to the major diameter For example, this ellipse has a = 1 and e = 0.5. Perihelion = r2 = 0.5 Aphelion = r1 = 1.5 ► Perihelion and aphelion

12. 9/13/2005PHY 101 Week 311 Isaac Newton

13. 9/13/2005PHY 101 Week 312 The observed solar system at the time of Newton Sun Mercury Venus Earth Mars Jupiter Saturn (all except Earth are named after Roman gods, because astrology was practiced in ancient Rome) Three outer planets discovered later… Uranus (1781, Wm Herschel) Neptune (1846 Adams; LeVerrier) Pluto (1930, Tombaugh)

14. 9/13/2005PHY 101 Week 313 To explain the motion of the planets, Newton developed three ideas: 1.The laws of motion 2.The theory of universal gravitation 3.Calculus, a new branch of mathematics Newton solved the premier scientific problem of his time --- to explain the motion of the planets. Isaac Newton “If I have been able to see farther than others it is because I stood on the shoulders of giants.” --- Newton’s letter to Robert Hooke, probably referring to Galileo and Kepler m F a  2 21 r mGm F 

15. 9/13/2005PHY 101 Week 314 Newton’s Theory of Universal Gravitation Newton and the Apple (The apple never fell on his head, but sometimes stupid people say that, trying to be funny.) Newton asked good questions  the key to his success. Observing Earth’s gravity acting on an apple, and seeing the moon, Newton asked whether the Earth’s gravity extends as far as the moon.

16. 9/13/2005PHY 101 Week 315 In fact there are nine planets. The center of mass of the solar system is fixed (  ). To a first approximation the center of mass is at the Sun. The motion of the planets Diagram of a planet revolving around the sun. The eccentricity e is grossly exaggerated ― real orbits are very close to circular. (  ) actually it moves around the center of the galaxy

17. 9/13/2005PHY 101 Week 316 Centripetal acceleration Answer : 790 N Example. Determine the string tension if a mass of 5 kg is whirled around your head on the end of a string of length 1 m with period of revolution 0.5 s. For an object in circular motion, the centripetal acceleration is a = v 2 /r. (Christian Huygens)

18. 9/13/2005PHY 101 Week 317 Three concepts --– Centripetal acceleration Centripetal force Centrifugal force

19. 9/13/2005PHY 101 Week 318 Circular Orbits (a pretty good approximation for all the planets because the eccentricities are much less than 1.) Centripetal acceleration Newton’s second law There is a subtle approximation here: we are approximating the center of mass position by the position of the sun. This is a good approximation. (velocity) (acceleration)

20. 9/13/2005PHY 101 Week 319 Circular Orbits The planetary mass m cancels out. The speed is then Time = distance / speed i.e., Period = circumference / speed  Kepler’s third law: T 2  r 3 Period of revolution

21. 9/13/2005PHY 101 Week 320 Generalization to elliptical orbits (and the true center of mass!) where a is the semi-major axis of the ellipse The calculation of elliptical orbits is difficult mathematics. The story of Newton and Halley Many applications...

22. 9/13/2005PHY 101 Week 321 FIN

23. 9/13/2005PHY 101 Week 322 Newton’s figure to explain planetary orbits Newton’s theory has stood the test of time. We use the same theory today for planets, moons, satellites, etc.

24. 9/13/2005PHY 101 Week 323 The Law of Equal Areas Newton: Angular momentum is conserved—in fact that’s true for any central force—so the areal rate is constant. Perihelion : fastest Aphelion : slowest

25. 9/13/2005PHY 101 Week 324 Newton said of himself… “ I know not what I appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier sea-shell, whilst the great ocean of truth lay all undiscovered before me.”

26. 9/13/2005PHY 101 Week 325 Applications Earth and Sun Other planets Moon and Earth Artificial satellites Exploration of the solar system Some of these are covered in the CAPA problems. Here is the general formula. We can always neglect m (mass of the satellite) compared to M (mass of the center).