4. Gravitation & Planetary Motion. Mars Motion: 2005 to 2006

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1 4. Gravitation & Planetary Motion Geocentric models of ancient times Heliocentric model of Copernicus Telescopic observations of Galileo Galilei Systematic observations of Tycho Brahe Three planetary laws of Johannes Kepler Three motion laws of Isaac Newton Newton explains Kepler s laws Differential gravity causes tides Ancient Geocentric Planetary Models Basic observations The sky Everything in the heavens moves constantly The Earth Absolutely no sensation of Earth motion Fundamental conclusions Earth is perfectly stationary in space Heavens constantly revolve around the Earth Stars do so with extreme regularity & circularity Arbitrarily imposed concept of uniform circular motion Fundamental problems Sun, Moon & planets known to move irregularly All vary in speed during direct [West to East] motion Planets at times exhibit retrograde [East to West] motion 2006 Mars Motion: 2005 to One Solution to Retrograde Motion Ptolemy s variation on uniform circular motion Deferent: A circle centered on the Earth Epicycle: A circle centered on the deferent The Sun [or Moon or planet] is attached to the epicycle Epicycle center moves uniformly along the deferent Epicycle itself rotates uniformly following the deferent Ptolemy s problems Determining relative sizes of deferent & epicycle Determining relative speeds of deferent & epicycle Explaining why this should occur No simple mathematical relationship between planets Occam s razor 1852 Simplest explanation is probably the correct one Cut to the heart of the matter The Greek Geocentric Cosmogony Deferents & Epicycles

2 An Ancient Heliocentric Model Aristarchus 3 rd century B.C. The Earth is a planet like all the others He showed that the Sun is much larger than the Earth It is reasonable that smaller objects orbit larger ones All planetary orbits lie in nearly the same plane Simplicity takes precedence over complexity The idea eventually called Occam s razor Aristarchus s critics prevail We must be the center of the Universe Humans are the apex of creation Earth must therefore be unique We have no sensation the Earth is moving Physical senses prevail over rational thought Complexity is one price for preserving a preconception! Planet Categories & Configurations Planetary categories Inferior Closer to Sun than Earth Never seen very far from the Sun & never near midnight Mercury & Venus Superior Farther from Sun than Earth Often seen very far from the Sun & often near midnight Mars, Jupiter, Saturn, Uranus, Neptune & Pluto Planetary configurations Conjunctions Aligned with Sun Inferior conjunction Planet between Earth & Sun Superior conjunction Sun between Earth & planet Elongations Away from the Sun Maximum eastern As far E of the Sun as possible Maximum western As far W of the Sun as possible Copernicus s Heliocentric Model Cultural context ~ 1500 Almost 2,000 years of unnecessary ambiguity Circle sizes & speeds determined by pure guesswork No logical sequence to the planets Scientific context Simplicity replaces complexity & ambiguity Mercury & Venus closer to the Sun than Earth They do exhibit maximum eastern & western elongation They never exhibit opposition Mars, Jupiter & Saturn farther from the Sun than Earth They never exhibit maximum eastern & western elongation They do exhibit opposition Uranus, Neptune & Pluto visible only through telescopes They never exhibit maximum eastern & western elongation They do exhibit opposition Heliocentric Retrograde Motion Heliocentric Planetary Configurations Synodic Years Inferior planets These planets orbit the Sun faster than the Earth The closer to the Sun, the shorter the synodic year Mercury s synodic year is Earth years Venus s synodic year is Earth years Superior planets These planets orbit the Sun slower than the Earth The farther from the Sun, the shorter the synodic year Jupiter s synodic year is Earth years Saturn s synodic year is Earth years

3 Inferior Planet Synodic Periods Origins of the Telescope Invention of the telescope Giambattista della Porta (Naples) 1589 Wrote about combining convex & concave lenses Objects appear enlarged & upright (rather than inverted) Hans Lippershey (Holland) 1608 Petitioned the Belgian government to get a patent Made the design principles known Galileo s telescope Heard a rumor about Lippershey s invention Built a telescope himself in just 24 hours July 1609 Credited a Dutchman for the original invention Claimed his improvements made the telescope famous Published observations of celestial phenomena Galileo s Telescopic Observations Several celestial surprises 1610 The Milky Way is a mass of innumerable stars The Moon has craters & mountains Saturn has handles rings The Sun has spots Two celestial phenomena 1610 Venus Exhibits phases completely contrary to geocentric models Precisely opposite in sequence to the Moon s phases Varies in angular diameter synchronized with its phases Largest when new & smallest when full Jupiter Four aligned yet constantly moving points of light Same relationships as shown by the planets Venusian Phases During 2001 Venus: The Heliocentric Model Tycho s Systematic Observations Perfectly unchanging heavens challenged Seemingly changeable objects Nova of 1572 Exhibited no measurable parallax Comet of 1577 Exhibited no measurable parallax Ptolemy s view cannot be correct King of Denmark rewards Tycho Brahe Two observatories Biggest & best measuring instruments ever made Basic approach Search diligently for stellar parallax Test relative merits of geocentric & heliocentric models Multiple observations with multiple instruments Successful attempt to identify instrument errors Concluded Earth is at rest; supported hybrid model First stellar parallax measured in 1838

4 Parallax: Apparent Shift In Position Tycho s Real Success Planetary data Unprecedented accuracy measuring time Clock invented by Galileo Unprecedented number of position measurements Unprecedented accuracy of position measurements Within one minute of arc Strategic hiring 1600 Tycho hired Johannes Kepler to analyze data Tycho died in 1601, possibly of alcohol poisoning Kepler worked on Tycho s data for 9 years Kepler s Mathematical Calculations An expert & imaginative geometer Believed in a true heliocentric planetary system Tried various possibilities Ovals Worked better than circles but not perfectly Ellipses Worked to the limit of measurement accuracy Kepler s three laws of planetary motion All planets orbit the Sun on an elliptical path The Sun is at one focus & nothing is at the other focus Perihelion Orbital point closest to the Sun Aphelion Orbital point farthest from the Sun All planets sweep out equal areas in equal time Measured by a line connecting the planet & the Sun (Sidereal period) 2 (Semimajor axis) 3 Very small discrepancies near massive Jupiter & Saturn Kepler did describe but did not explain Kepler s First & Second Laws Elliptical path 2 1 Area 1 = Area 2 Newton s Three Laws of Motion Bodies remain undisturbed unless acted upon Commonly called the principle of inertia Acceleration is proportional to applied force F = m. A [mass. acceleration] Every action has an equal & opposite reaction Without friction, the ground could not push back Newton s Law of Universal Gravitation Newton postulated a force called gravity F G Every pair of objects attract each other F G is directly proportional to product of masses F G is inversely proportional to square of distance " F G = G # m 1 m 2 r 2 \$ % F G = force of gravity between 2 objects m 1 = mass of object number 1 m 2 = mass of object number 2 r = distance between objects 1 & 2 G = universal gravitational constant = newton. m 2 / kg 2 = newton. m 2. kg 2

5 Newton s Insight Legend An apple fell from a tree & hit Newton on the head He discovers the force of gravity Reality Falling apples & orbiting Moons have same cause The apple has no sideways motion & falls straight down The Moon has sideways motion & stays in orbit Devise a thought experiment Assume that There is no atmospheric friction There is a very high mountain Imagine three balls Drop one & it falls straight down Throw one slowly sideways & it falls nearby Throw one fast sideways & it follows Earth s curved surface It orbits the Earth!!! Newton s Model of Orbiting Objects Newton Explains Kepler s Laws Newton s great contribution Consider his three laws of motion Consider his law of universal gravitation Newton s form of Kepler s third law Newton s great discovery All orbits are conic sections Circles Ellipses with both foci at the same location Orbiting objects remain the same distance away Ellipses Elongated closed curves with 2 foci Orbiting objects have constantly changing distance Parabolas Elongated open curves with 1 focus Orbiting objects will return infinitely far into the future Hyperbolas Elongated open curve pairs Orbiting objects will never return All Orbits Are Conic Sections Tidal Effects Basic phenomena Periodic rise & fall of the ocean surface About 1.0 meter in the middle of the ocean Periodic rise & fall of the land surface About 0.5 meter in the middle of the continents Typical timing About 12 h 25 m between successive high or low tides Typical patterns Daily About 2 high & 2 low tides Successive high or low tides are usually not equal height Highest daily high tide, lowest daily low tide, etc. Monthly Spring tides: Highest high & lowest low monthly tides Neap tides: Lowest high & highest low monthly tides Differential Gravity Causes Tides Basic phenomena Gravity inversely proportional to distance squared Close celestial bodies exert relatively strong gravity Nearest side has stronger gravity than farthest side Nearest side gets pulled most Farthest side gets pulled least Objects causing Earth tides The Moon ~55% of tidal height on average The Moon is quite small but also quite close The Sun ~45% of tidal height on average The Sun is quite large but also quite far

6 Tides: A Simple Model Tides: The Earth & the Moon Geometry of Spring & Neap Tides ~ 55% ~ 45% ~ 55% ~ 45% Tidal Geometry Lunar & solar gravitational force alignment Along same line New & full moon Tidal forces are greatest of the month Spring tides At right angles First quarter & third quarter moon Tidal forces are least of the month Neap tides Some variations on a theme Summer Sun is far North of the equator New moon Highest daily high tide in N hemisphere Full moon Both daily high tides about equal Winter Sun is far South of the equator New moon Highest daily high tide in S hemisphere Full moon Both daily high tides about equal Important Concepts: 1 Ancient geocentric planetary models The sky moves constantly, but There is no hint that Earth moves Ptolemy s approach Evidence Only the stars move uniformly Planets exhibit retrograde motion Explanation Uniform circular motion A system of deferents & epicycles Occam s razor: Choose simplicity An ancient heliocentric alternative Aristarchus: The Earth is also a planet Copernicus s heliocentric model Maximum elongation explained Mercury & Venus are inferior planets Retrograde motion explained Earth overtakes superior planets Sidereal & synodic years Galileo s telescopic studies Lunar craters, sunspots etc. Venus has phases & changes diameter Jupiter has four moons Tycho Brahe s measurements More & more accurate than ever Johannes Kepler s calculations Elliptical orbits w/sun at one focus Equal areas in equal times P 2 A 3 Isaac Newton s physical laws Three laws of force & motion Law of universal gravitation Together, these explain Kepler s laws Rigorous description of gravity Keeps objects on the Earth surface Keeps objects in conic-section orbits Important Concepts: 2 Tidal effects Periodic rise & fall of Earth s surface Caused by differential gravity Nearest side has strongest gravity Farthest side has weakest gravity Tidal geometry Moon & Sun aligned: Spring tides Moon & Sun orthogonal: Neap tides

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