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
Mars Motion: 2005 to 2006
2006
2005
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
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
• Superior conjunction
– Elongations
• Maximum eastern
• Maximum western
Planet between Earth & Sun
Sun
between Earth & planet
Away from the Sun
As far E of the Sun as possible
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’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
• Comet of 1577
Exhibited no measurable parallax
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
– Tycho hired Johannes Kepler to analyze data
• Tycho died in 1601, possibly of alcohol poisoning
• Kepler worked on Tycho’s data for 9 years
1600
Kepler’s Mathematical Calculations
• An expert & imaginative geometer
– Believed in a true heliocentric planetary system
– Tried various possibilities
• Ovals
• Ellipses
Worked better than circles but not perfectly
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 FG
– Every pair of objects attract each other
– FG is directly proportional to product of masses
– FG is inversely proportional to square of distance
æ m1 × m2 ö
FG = G × è
2
r ø
FG
m1
m2
r
G
=
=
=
=
=
=
=
force of gravity between 2 objects
mass of object number 1
mass of object number 2
distance between objects 1 & 2
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 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 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%
~ 45%
~ 55%
~ 45%
Tidal Geometry
• Lunar & solar gravitational force alignment
– Along same line New & full moon
• Tidal forces are greatest of the month
– At right angles
Spring tides
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
• Full moon
– Winter
Highest daily high tide in N hemisphere
Both daily high tides about equal
Sun is far South of the equator
• New moon
• Full moon
Highest daily high tide in S hemisphere
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
•
•
•
– Lunar craters, sunspots etc.
– Venus has phases & changes diameter
– Jupiter has four moons
Ptolemy’s approach
– Evidence
• Only the stars move uniformly
• Planets exhibit retrograde motion
– Explanation
• Uniform circular motion
• A system of deferents & epicycles
•
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
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  A3
– Occam’s razor: Choose simplicity
•
Sidereal & synodic years
Galileo’s telescopic studies
•
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