1b. The Earth Moves. Planetary Motion (Part 2) C. Heliocentric Models. C. Heliocentric Models. 1) The Revolution. 2) Tycho and Kepler

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1 Planetary Motion (Part ) C. Heliocentric Models I CAN easily conceive, most Holy Father, that as soon as some people learn that in this book which I have written concerning the revolutions of the heavenly bodies, I ascribe certain motions to the Earth, they will cry out at once that I and my theory should be rejected. Therefore, when I considered this carefully, the contempt which I had to fear because of the novelty and apparent absurdity of my view, nearly induced me to abandon utterly the work I had begun. [excerpt from Dedication of the evolutions of the Heavenly Bodies to Pope Paul III, by Nicolaus Copernicus 1543] 9 C. Heliocentric Models 1) The evolution ) Tycho and Kepler 3) Newton and Halley The evolution 31 1b. The Earth Moves 3 a) Aristarchus of Samos ( BC) had calculated that the sun was much bigger than the earth. He proposed that therefore it was more important and should be the center of the solar system. However, parallax of stars was not observed, and it seemed silly to say the earth moved. Copernicus (1543 AD) shows that the complex motion of the planets can be more simply explained if the sun is at rest, and the earth revolves around it ( heliocentric theory ). However, he doesn t really address the issue of why falling bodies don t fall sideways if the earth is moving. Nicolaus Copernicus ( AD) Geocentric Theory of the universe 1b. Copernican System The only thing that orbits the earth is the moon. Much simpler than geocentric model (which needs 5 deferents and 5 epicycles) 33 1c. etrograde Motion The Earth travels around the Sun in a smaller orbit than Mars, and moves more rapidly than Mars. Consequently, as the Earth overtakes and passes this slower-moving planet, Mars appears for a few months to fall behind and move backward with respect to the background stars. 34 Fig 1-15, p.34 1

2 One of the most important books ever Nicolaus Copernicus On the evolution of Heavenly Spheres (1543) 35 The 1000 Zlotych bill features Copernicus. Due to inflation, it 36 was worth about 10 cents USD when I was last in Poland 1d. Synodic vs Sidereal The synodic period of a planet is the time that elapses between two successive configurations, as seen from Earth - from one opposition to the next, for instance, or from one conjunction to the next. Because the Earth moves while the planet is moving, a planet's synodic period is not equal to its sidereal period (the time to complete one orbit around the Sun). 37 a. Galileo Galilei ( ) A contemporary of Kepler Galileo was one of the very first scientists to do experiments to understand Nature 38 He was the first astronomer to use a telescope (in 1610) to study the sky. a.1 Inventor of Telescope? 39 a. Mountains of the Moon Hans Lippershey invented the telescope 1609 Galileo made his own, and let everyone think that he invented it (won an award!) Galileo s drawings of the Moon with his telescope, 1610 He measured the heights of mountains from the shadows cast and found them to be about same size as mountains on the earth!

3 a.3 Milky Way made of stars 41 a.4 Jupiter has Moons 4 The Milky Way appears as a faint milky white across the sky. With his telescope, Galileo could magnify it and show it was made of stars! Galileo saw four little stars moving with Jupiter across the sky, and changing their positions every night. He quickly and correctly concluded that they are in orbit around Jupiter. This confirmed the Copernican view that the Earth is NOT the center of all motion in the Universe. 43 Saw spots on the sun that moved, interpreted that the sun rotates in 8 days a.6 Venus Has Phases 44 Galileo saw that Venus was bigger at crescent and smaller when full. a.6 Venus Phases Prove Heliocentric 45 Galileo saw that the phases were inconsistent with the geocentric model, proved Venus goes around the sun! b. Aristotle s Physics Natural state is rest continuation of motion depends on continued action of a force [e.g. the prime mover (god) must continue to push the moon] 46 Antiperistasis: To explain how a ball rolls when no-one is pushing it: as ball rolls, it leaves empty space behind it. Nature abhors a vacuum, so air rushes in to fill the space vacated by the ball, hence continues to push the ball along, but it will eventually stop. Aristotle: 384 BC-3 BC 3

4 b. Natural Motion is Straight 47 b. Galileo s Law of Inertia 48 Giambattista Benedetti Bodies at rest tend to stay at rest Natural motion is straight, not circular When projectile released from a sling, it goes straight. Bodies in uniform motion will tend to stay in uniform motion His work influences Galileo Unless acted upon by an outside force If string breaks, ball goes straight natural state IS motion Galileo Galilei Demo: Play 11:35-14:5 of MU #4: b. Galileo: no absolute motion 49 Galileo: (163) Motion is relative. A ball dropped from the crow s nest will hit at the base of the ship s mast, even if the ship is moving. Hence, there is no absolute measure of motion (or rest). b Aristotle and Falling Motion the rate of falling is proportional to the weight and inversely proportional to the density of the medium 50 the peripatetics (followers of Aristotle) postulated that the speed was proportional to the distance fallen. x v x t Aristotle: 384 BC-3 BC Galileo shows that this must be wrong, for an object starting at zero speed would never acquire any speed! Demo: Play 11:35-0:3 of MU #4: a.8 Galileo s Experiment at Pisa 1590 Galileo s Principle: All bodies fall at the same rate, regardless of mass 51 a.8 Law of Falling 5 "Men are like wine flasks," he once said to a group of students. "...look at...bottles with the handsome labels. When you taste them, they are full of air or perfume or rouge. These are bottles fit only to pee into!" -Galileo expressing what he thought of the other faculty members of the University of Pisa. Needless to say, his teaching contract was not renewed! Aristotle: Heavier balls will fall faster Galileo: They fall at the same rate! 4

5 b. Tycho 53 b. Tycho Brahe s Uraniborg Observatory 54 Tycho Brahe measuring star positions (without a telescope) Measurements of position of Mars showed deviations from Copernican model! He built a big observatory with gigantic protractors (no telescopes yet!) b.3 Tycho Brahe ( ) 55 Because he could not measure any parallax of stars, he concludes the earth is at rest. He suggested a weird hybrid model where planets go around sun, but sun goes around earth c. Johannes Kepler ( ) Tycho at first invited Kepler to help in analysis of his data, but then jealously wouldn t let him have the information. On his deathbed he gave Kepler the data. 56 Kepler used it (particular data on Mars), to develop three laws of planetary motion. c.1 Kepler s 1 st Law: Orbits are Ellipses 1605: Kepler realized that the motion of Mars could not be explained with a circular orbit, or the multiple circles proposed by Ptolemy. He accepted Copernicus view that Mars was in orbit around the Sun, rather than around the Earth. He experimented (mathematically) with orbits of various shapes, and found that Mars orbit best fits an ellipse. 57 Ellipses, circles (parabolas and hyperbolas) are conic sections, studied first by the greeks. But it would NEVE occur to the greeks that an orbit is an ellipse. (why?) 58 Fig -3, p.45 5

6 The Ellipse Highly eccentric Focus Focus Focus Focus An ellipse has two foci. Using a fixed length of string around the foci, one draws an ellipse Fig -4, p.45 Not very eccentric c.1 Kepler s 1 st Law (1605) 61 6 Planet orbits tend to have low eccentricity (nearly circular). Comet orbits tend to be highly eccentric. Law No. 1. Each planet moves around the Sun in an orbit that is an ellipse, with the Sun at one focus (and nothing at the other!) Fig -10, p c. Kepler s nd Law (1609) 64 Kepler also noticed that when Mars is closest to the Sun in its elliptical orbit, it moves faster than when it is farther away. This led him to formulate his Second Law of Planetary Motion. 6

7 c. Kepler s nd Law (Equal areas in Equal Times) c. Kepler s nd Law 66 According to his second law, a planet moves fastest when closest to the Sun (at perihelion) and slowest when farthest from the Sun (at aphelion). As the planet moves, an imaginary line joining the planet and the Sun sweeps out equal amounts of area (shown as colored wedges in the animation) in equal intervals of time. c.3 Kepler s 3 rd Law: Harmonic Law 67 c.3 Kepler s 3 rd Law (1618) 68 Planets closer to the sun move faster. This is consistent with his nd law, that showed a planet will move faster at perihelion. He searched for a relationship between orbital period and distance to the sun. The square of the orbital period (P) is directly proportional to the cube of the semimajor axis of the orbit (a). P = a 3 This law explains the proportions of the sizes of the orbits of the planets and the time that it takes them to make one complete circuit around the Sun. [Note: in physics, the symbol a is also used to represent acceleration. Confused?] Why is it called the harmonic law? Kepler thought the spacing between planets was related to musical intervals. 69 An example of Kepler s third law: The orbit of Mars (ecall: P = a 3 ) Mars orbit period (P) is 1.88 years. P = Kepler s law says that P = a 3, so 3.53 = a 3. So then a = (3.53) 1/3 (the cube root of 3.53), or 1.5. Thus, the semimajor axis (average distance of Mars from the Sun) is 1.5 Astronomical Units. 7

8 c. How large is the AU=Astronomical Unit? The average distance from the Earth to the Sun 400 further than moon 4,000 radius of earth 150,000,000 kilometers, or 93,000,000 miles About 500 light-seconds 71 Measuring the AU Transits of Venus (1761 & 1769) observed from different places on the earth were used by Lalande (1771) to get the first accurate measurement of the AU. 7 But how was this measured? Venus transits happen twice 8 years apart, then over a 100 years till the next! Transit of Venus June 01 If you missed it, there is not another one until 117! Isaac Newton ( ) Epitaph: "Nature and Nature's laws lay hid in night: God said, 'Let Newton be!' and all was light." 73 Isaac Newton ( ) 74 Famous book the Principia, published 1687 Based on work done in 1666 Edmund Halley had to twist his arm to get him to write the book and paid for the publication! 1. Newton s First Law 75. Second law: Force is the cause 76 1 st Law: (adopted from Galileo s laws of inertia), Every body preserves in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed. In other words: bodies at rest tend to stay at rest, bodies in motion tend to stay in constant motion, unless compelled to change by an outside force. (a) Statement of Law (from Principia) A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed. In other words: Force = mass acceleration Force: F is the cause of acceleration (unit: Newtons) Acceleration a is the response to force (unit: meter/sec ) Mass: m is the inertia or resistance to force (unit: kg) 8

9 3. Newton s 3 rd Law of eciprocity (1687) Orbits and Gravity Orbital velocity 78 To any action there is always an opposite and equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts This means, as much as the earth is pulling on you You are pulling back on the earth the same amount Without this law, energy and momentum would not be conserved in the universe. (a) Galileo proposed that throwing a ball at different speeds causes it to travel farther before it falls to Earth. Throw it fast enough ( orbital velocity ), and as it falls the earth s curve falls underneath it, and it falls forever ( free fall ) By matching the centripetal acceleration of a circular path to the acceleration of gravity he derived the orbital speed (near surface of earth) For Earth (=6000 km), orbital velocity v is 17,500 miles/hr, or 8 km/sec v g 9.8 m s 4b Inverse Square Law Edmond Halley ( ) deduced that Kepler s 3 rd law implies that gravity (of sun) must decrease with square of distance. v P v ac 4 P 3 P 1 a Moon is 60 earth radii away, hence earth s pull on moon is 3600 weaker, orbital speed only 1 km/s 79 4c. Gravity: Newton s 4 th Law (i) The apple tree story "After dinner, the weather being warm, we went into the garden and drank tea, under the shade of some apple trees," wrote Stukeley, in the papers published in 175 and previously available only to academics. "He told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. It was occasion'd by the fall of an apple, as he sat in contemplative mood. Why should that apple always descend perpendicularly to the ground, thought he to himself." 80 (ii) The Law of Gravity (1680) Newton s 4 th law: The mutual force between two bodies is proportional to their masses, and inversely proportional to square of distance. Gravitation Constant G measured 100 years later by Cavendish. 81 iii. Newton s Orbit GmM ma F Combining Newton s nd and v GM 4 th laws, we see that the mass a of the orbiting body cancels GM out! v Orbital speed depends only on size of orbit (and mass of planet) 8 9

10 (iv) The Equivalence Principle 83 v. The Newton-Kepler Law 84 Weight is the Force of Gravity on an object Force of field g on mass m is: F=mg In the Principia Newton also deduced Kepler's third law, but in an important new form Equating this force with Newton s second law, we find that the mass cancels ma=mg Hence, (as Galileo said), all bodies fall at same rate! GM g Mass of central body: M = a 3 /P Orbital adius a (in astronomical units) Period P (in years) Mass M in units of solar masses To measure mass of Earth, use moon s orbit Jupiter, use Galilean moons Sun, use orbits of planets Galaxy, use orbits of stars around galaxy Escape velocity 85 Escape Velocity for the planets 86 The velocity of escape, V, from a planet is given by Throw the ball hard enough and it will escape the gravity pull of the Earth V GM How fast do you have to throw it? It depends on which planet you live on G is the universal gravitational constant m is the mass of the planet r is the radius of the planet The escape velocity of Earth is 11 km/sec, or About 5,000 miles per hour. Escape Velocities for the planets, the Sun, and the Moon 87 (e1). Special elativity 88 Object Mass (Earth = 1) adius (Earth = 1) Escape Velocity Sun 330, km/sec Jupiter Saturn Neptune Uranus Venus Mars Mercury Moon Einstein (6 years old) publishes theory of special relativity Speed of light is the same for all observers Motion is relative (Galileo) there is no experiment one can do to determine absolute motion relative to space. Laws of physics must hold in all reference frames which differ only by a constant velocity 10

11 e. Galileo s Experiment at Pisa 1590 Galileo s Principle: all bodies fall at the same rate, regardless of mass 89 e3. The Equivalence Principle eference at rest with Gravity is indistinguishable to a reference frame which is accelerating upward in gravity free environment Strong EEP (Einstein Equivalence Principle) same result, but argued from a different way. He proposed that falling bodies in gravity are equivalent to being in an accelerated frame (e.g. in an accelerating elevator) The apple accelerating downward due to gravity looks the same as an apple at rest in space, with the floor accelerating upward towards it. eferences: Updated 9/4/ Good Java Demo: Galileo interview: Galileo: x 11