4. Gravity & 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

Mars Motion: 2005/06 & 2011/12

2006

2005

2012

2011

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

Deferents & Epicycles

An Ancient Heliocentric Model• Aristarchus 3rd 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 0.317 Earth years

– Venus’s synodic year is 1.599 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 1.092 Earth years

– Saturn’s synodic year is 1.035 Earth years

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 Galilei

http://ircamera.as.arizona.edu/NatSci102/NatSci102/images/galileo.jpg

Galileo Galilei’s Telescope Lenses

http://galileotelescope.org/new&original-galileo-telescope3b-sml.jpg

http://galileotelescope.org/2-pc-glass-hand-sml.jpg

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 totally contrary to geocentric models– Precisely opposite in sequence to the Moon’s phases

• Angular diameter varies, 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

Galileo’s Notes on Jupiter’s Moons

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

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 µ (Semimajor axis)3

• Very small discrepancies near massive Jupiter & SaturnKepler did describe but did not explain

Every Ellipse Has Two Foci

Kepler’s First & Second Laws

Area 1 = Area 2

Area 2

Area 1

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 FG

– Every pair of objects attract each other– FG is directly proportional to product of masses– FG is inversely proportional to square of distance

FG = force of gravity between 2 objectsm1 = mass of object number 1m2 = mass of object number 2r = distance between objects 1 & 2G = universal gravitational constant

= 6.67 . 10-11 newton . m2 / kg2

= 6.67 . 10-11 newton . m2 . kg–2

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 surfaceIt 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 12h 25m 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

Tides: A Simple Model

Tides: The Earth & the Moon

Geometry of Spring & Neap Tides

~ 55%

~ 55%

~ 45%

~ 45%

Tidal Geometry• Lunar & solar gravitational force alignment

– Along same line New & full moon• Tidal forces are greatestof the month Spring

tides– At right anglesFirst 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

Tides Affect Merging Galaxies

• 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– P2 µ A3

• 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: 1

• 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

Important Concepts: 2