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1 Reminders! Website: Lectures 15 are available for download as study aids. Reading: You should have Chapters 14 read, Chapter 5 by the end of today, and Chapters 6 and 7 by the end of this weekend. Homework: You should have homework #1 turned in already. Homework #2 is due next Wednesday. Discussion: Next week, we ll review for the midterm. Remember you can attend ANY discussion section. The schedule is Monday 12, Tuesday 12, Wednesday 910, all in Evans 264.
2 Observing Project Remember that for one of your observing projects you can go to a star party (stargazing). This is available at the Lawrence Hall of Science every 1st and 3rd Saturday of the month. For more information (be sure to call ahead) see: Also: The Chabot Space & Science Center has stargazing every Friday and Saturday 7:3010:30. This can also count for your star party observing project. This Saturday, there will be a special lecture (James Allen Telescope Array  ATA) and star party, starting at 7:30 pm. Chabot Space & Science Center: East Bay Astronomical Society:
3 Early Astronomy and Gravity Today s Lecture: Review from yesterday (chapter 5, pages ) Early astronomy, Copernicus, Galileo Kepler and Newton (chapter 5, pages ) Kepler s 3 Laws Newton s Laws of Motion
4 The Solar System Early history of Solar System Studies (Greece) Planets wander slowly (over many weeks) among the fixed stars. (This is NOT to be confused with the daily easttowest movement due to the Earth s rotation.) Earth Usually the planets drift from west to east. It is natural (but incorrect) to think of the Earth as the center of the Universe Spheres for each planet, the Moon, the Sun, and the stars seem to rotate around it. S Stars embedded in the celestial sphere
5 Retrograde Motion Retrograde motion is a problem for the simplest geocentric models. Each year, planets drift from east to west for a while! E W Ptolemy (~140 AD) devised a new geocentric theory to explain this. E W
6 From Ptolemy to Copernicus Ptolemy s geocentric system was very complicated, but also very accurate. It lasted for nearly 1500 years! But most people still thought that the perfect reality was a bunch of nested spheres as Aristotle originally suggested. Copernicus wrote about heliocentric theory, first published in 1543 (after death) Galileo s used a telescope to see that Venus goes through an entire set of phases (like the Moon) confirming the heliocentric hypothesis (1610)
7 The Phases of Venus Venus according to Copernicus: all phases Venus according to Ptolemy: only crescent and new Since Venus always appear close to the Sun in the sky, it could not go through a complete set of phases in Ptolemy s system, but it should in Copernicus system.
8 Tycho Brahe He saw a partial solar eclipse at age 14 and was impressed by the prediction. He decided to measure very accurate positions of planets (especially Mars) for 20 years. Hired Johannes Kepler, a superb mathematician, to analyze the data.
9 Johannes Kepler He analyzed Tycho s data. Revised the Copernican system with 3 laws: 1. Planetary orbits are ellipses (and NOT circles), with the Sun at one focus (nothing at the other focus) 2. A line between the Sun and a planet sweeps out equal areas in equal times 3. The square of a planet s orbital period is proportional to the cube of its semimajor axis P 2 R 3 or P 2 = kr 3, where k = some constant These laws were purely empirical; Kepler had no physical explanation for them.
10 Information on ellipses Eccentricity: e = (distance between foci) / (major axis) If both foci are at the same point, then it s a circle and e = 0. The maximum value is e=1, which makes a long, skinny ellipse. e = 0 e = 0.5 e = 0.9
11 Ellipse: set of points P such that a + b = constant a P b Kepler s 1st Law: Move on ellipses with on the Sun at one focus. Kepler s 2nd Law: Equal areas are swept by the planets in equal time (planets go faster when closer to the Sun). Sun t 1 t 2 Each sector in this ellipse has the same area, thus t 1 = t 2
12 Kepler s 3rd Law P 2 R 3 or P 2 = kr 3 where k is a constant It s convenient to use units based on the Earth s orbit! Example: or P earth = 1 year and R earth = 1 Astronomical Unit (A. U.) If P mars = 1.88 years, then what is R mars? R mars = (1.88) 2/3 = 1.52 A.U.
13 Isaac Newton ( ) 1661 Goes to Cambridge University 1665 Flees Cambridge due to the plague. Invents calculus, discovers universal law of gravitation, develops laws of motion, begins studies of optics Becomes Lucasian Professor of Mathematics 1671 Invents the reflecting telescope Publishes The Principia Goes temporarily insane 1696 Becomes Warden of the Mint Publishes Opticks 1705 Knighted (for service to the government)
14 Isaac Newton The Principia (1687) Three Laws of motion: (1) If no forces act on a body, its speed and direction of motion stay constant (2) Force = mass x acceleration or F = ma (Note: acceleration is the change in speed OR direction of motion) (3) When two bodies interact, they each exert equal and opposite forces on each other.
15 Newton s Law of Universal Gravitation d m 2 m 1 and m 2 = masses of objects 1 and 2 d = distance between objects G = the gravitational constant Note that the force is symmetric. The force on m 1 is the same as on m 2! m 1
16 Cannon ball thought experiment
17 How does gravity produce orbits? I. No gravity: Straight line motion at constant speed. II. Gravity: Body starts from rest and accelerates due gravity to velocity v = at in a time t III. I + II Constant tangent speed plus gravity gives an orbit! Imagine many small steps v
18 An orbit = fall and miss The Moon is constantly falling toward the Earth it just keeps missing! (that s good, right?) m 2 m 1 No m 2! This shows that the acceleration of an object acted on by gravity is independent of its mass!
19 Newton derives Kepler s Laws! From his laws of motion and universal gravitation, Newton derived and generalized Kepler s three empirical laws. (1) Orbits in general are conic sections which depend on the transverse velocity. If bound, it will be an ellipse or circle, but it can easily be a hyperbola or parabola. (2) A planet sweeps out equal areas in equal times: property of gravity being strongest when the distance is smallest (difficult to derive simply).
20 Conic sections and orbits
21 Newton derives Kepler s Laws! (cont.) (3) Combining Newton s 2nd law of motion with gravitation and assuming a circular orbit: Before Kepler derived his law for things orbiting the Sun. This new law applies for everything (for example, the moon of Jupiter). is nearly constant if m 1 >> m 2
22 Newton calculates the Sun s mass! IMPORTANT: If m 1 >> m 2, then we can ignore m 2 so If we can measure P and R, then we can derive m 1! Example: What is the Sun s mass? Plug in numbers > M sun = 2 x grams! We can also estimate the mass of the Earth from the Moon!
23 Orbital Speeds of Planets For any planet: If the orbit is roughly circular, then 2πR = VP (distance = speed x time) This shows that distant planets should move more slowly (also related to Kepler s 2nd law)
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