# Gravity and the Motion of the Planets

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1 Gravity and the Motion of the Planets

2 Foundations of Modern Science More than 2500 years ago Pythagoras put forth the idea that nature can be described with mathematics. Aristotle asserted that the Universe is governed by physical laws.

3 Ancient Greek astronomers knew of Sun, Moon, Mercury, Venus, Mars, Jupiter, Saturn, and stars. Aristotle vs. Aristarchus (3 rd century B.C.): Aristotle: Sun, Moon, Planets and stars rotate around fixed Earth. Aristarchus: Used eclipses and phases to show Sun bigger than Earth (and Moon smaller), so guessed Earth orbits Sun. Also guessed Earth spins on axis once a day => apparent motion of stars. Aristotle: But there's no wind or parallax. Early Greek Models

4 Greek models were based on: Geocentric Universe: Earth at the Center of the Universe. Perfect Heavens : Motions of all celestial bodies described by motions involving objects of perfect shape, i.e., spheres or circles. Difficulty in explaining periodic changes in the speed and direction of the planets. Greek Astronomy

5 Retrograde Motion Most of the time planets move from east to west on the celestial sphere, just like the Sun and Moon Direct Motion But, sometimes they reverse direction and move west instead Retrograde Motion

6 Ptolemaic System Ptolemy (AD ) Almagest Geostatic Model Planets move in epicycles Centers of epicycles move in larger circle called deferents Deferents are not centered on the Earth for all objects Eccentric

7

8 Ptolemaic System Used for 1500 years Despite being wrong why did this model survive for so long? It worked, that is it could predict the position of a planet to within 2 It accounted for the observed planetary motions, retrograde motion and variations in brightness. Unlike Aristarchus' model it did not predict the unobserved stellar parallax. It placed the Earth in its natural place at the centre of things, agreeing with the philosophy of the time. It matched with common sense. We do not feel the Earth move and Ptolemy's model had a static Earth.

9 Occam s Razor States that among competing hypotheses, the one that makes the fewest assumptions should be selected. This is not absolute rule, but a good way to dismiss theories that are unnecessarily complicated. Einstein's Constraint: "Everything should be kept as simple as possible, but no simpler."

10 The Copernican Revolution Nicolaus Copernicus ( ): Heliocentric Solar System Earth rotates daily on it axis The other planets each orbit the Sun The period of the planets' orbits increases with increasing distance from the Sun.

11 The Copernican Revolution Why was Copernicus model better than Ptolemy s? It could predict planetary positions to within 2, the same as that of Ptolemy. Retrograde motion of planets was explained by the relative motion between them and the Earth.

12

13 The Copernican Revolution Distances between planets and the Sun could be accurately determined in units of the Earth-Sun distance (ie Astronomical Units). Orbital periods could be accurately determined. It explained the difference between the inferior planets (Mercury and Venus) that were always observed close to the Sun and the superior ones (Mars, Jupiter and Saturn).

14 The Copernican Revolution It preserved the concept of uniform circular motion without the need for equants. It preserved Aristotle's concept of real spheres nestled inside one another. Unlike Ptolemy's model it did not require the Moon to change in size.

15 Arrangement of the Planets

16 Some Geometry Why do Mercury and Venus go through cycles?

17 Phases of Inferior Planets

18 Planetary Periods and Orbit Sizes Synodic Period: Time between to identical configurations as seen from Earth. Sidereal Period: True orbital period of planet measured with respect to the stars. Planet Synodic Sidereal Copernican Distance Modern Distance Mercury 116 days 88 days 0.38 AU 0.39 AU Venus 584 days 225 days 0.72 AU 0.72 AU Earth year 1.00 AU 1.00 AU Mars 780 days 1.9 years 1.52 AU 1.52 AU Jupiter 399 days 11.9 years 5.22 AU 5.20 AU Saturn 378 days 29.5 years 9.07 AU 9.55 AU Uranus 370 days 84.1 years AU Neptune 368 days years AU

19 Why Others opposed the Copernican Model No annual stellar parallax could be detected. Copernicus explained this as due to the fact that the stars were a vast distance hence any parallax would be very small and difficult to detect. It required a moving Earth By removing the Earth from its natural place it was philosophically and theologically unacceptable to many scholars. It was no more accurate than Ptolemy's in predicting planetary positions. It was actually more complicated then Ptolemy's model. In his efforts to avoid the equant but retain uniform circular motion he had to introduce more devices to fit his observations.

20 Tycho Brahe ( ) Dismissed the idea of the unchanging heavens Tycho s supernova, 11/11/1572 Measured precisely the position of planets and stars without using a telescope. Ultimately concluded (wrongly) that the Copernican model was incorrect, based on his failure to measure parallax.

21 Parallax

22 Johannes Kepler ( ) German mathematician Tycho s assistant Developed modern theory of orbital motion based on Tycho Brahe s measurements 3 Laws Works for ALL orbiting celestial objects, i.e. moons, satellites, stars, etc.

23 Kepler s Laws First Law: Planets move in ellipses with the Sun at one focus.

24 Ellipses

25 Second Law: A line joining a planet and the Sun sweeps out equal areas in equal intervals of time. Kepler s Laws

26 Kepler s Laws Third Law (1618): The square of the sidereal period of a planet is directly proportional to the cube of the semimajor axis of the orbit P 2 a 3 P = planet s sidereal period, in years a = planet s semimajor axis, in AU

27 Kepler s Laws are empirical, that is accurately described planetary motion, but could not explain why. These same laws describe any orbiting body

28 Galileo ( ) Telescope invented by Dutch opticians Galileo was the first to observe the sky and publish his findings. Surface features on the Moon Phases of Venus Milky Way consists of stars Moons of Jupiter Sunspots

29 Phases of Inferior Planets

30 Lunar Surface

31 Jupiter s Moons The moons of Jupiter as drawn by Galileo over successive nights. How they look through a small telescope

32 Stars in the Milky Way

33 Sunspots

34 Isaac Newton ( ) English physicist, mathematician, astronomer, natural philosopher, alchemist and theologian Considered by many to be the greatest and most influential scientist who ever lived. Newton s Laws of Motion

35 Some of Newton s Discoveries and Inventions Laws of Motion Basis for Physics Calculus Theory of Gravity Reflecting Telescope

36 Newton s Laws First Law: An object remains at rest, or moves in a straight line at a constant speed, unless acted upon by a net outside force. Second Law: Force = Mass * Acceleration (F=ma) Third Law: Whenever one object exerts a force on a second object, the second object exerts and equal and opposite force on the first object. Collection is now called Newtonian Mechanics

37 Some Terminology Speed(velocity): How fast an object is moving, measured in units of length/time, i.e. km/s, mph, ft/s Acceleration: How fast the velocity changes, measured in units of length/time/time or length/time 2, i.e. m/s 2

38 Newtonian Gravity F = gravitational force between two objects m 1 = mass of first objects m 2 = mass of second object r = distance between objects G = universal constant of gravitation

39 Newtonian Gravity Explains well the motions of planets, stars, galaxies, satellites, etc. Orbits don t have to be ellipses, but can be any conic section. Fails to explain some observed phenomena, but more on this later

40 Explanation of Orbits, Conic Sections

41 Tidal Forces Differences in the gravitational pull at different points in an object Mostly a lunar effect Sun can also affect tides Spring tides ( New moon or Full Moon) Neap Tides (Quarter Moon)

42 Tidal Forces

43 Tidal Forces

44 Tidal Forces

45 The Interplanetary Transport Network How can we efficiently navigate through the solar system? The ITN is a collection of pathways through the solar system, governed by gravity that require little energy for an object to traverse them.

46 Recall Conic sections and orbits: Some orbits require very little energy or no energy at all, i.e. parabolic orbits, hyperbolic orbits Gravitational Slingshots

47

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