Weekly Review Session

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Weekly Review Session

• Monday or Wednesday after 430

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Geocentric Geostatic Models
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Platos model of the cosmos (400 - 300 BCE)

• Stars revolve once a day on one axis
• Sun, moon and planets have same motion
• - but second circular motion which moves them
  relative to stars
• As stars revolve, sun, moon and planets will be
  in different position

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Eudoxus model of the cosmos

• 27 spheres
• 3 for sun
• 3 for moon
• 4 for each of 5 planets
• 1 for fixed stars

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Hippòpedon

• additional spheres combined with suitable speeds
  and choice of axes
• Gives planetary deviation from ecliptic and
  retrograde motion

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Aristotles Universe
55 spheres
real made of aether
- no space between them
Celestial bodies attached
Rotate at different velocities
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Dominance of Aristotelian world system
Why geocentric-geostatic model?
Earthly
- corruptible
- earth absolutely heavy
- reaches proper place and stops
Celestial

• - Heavens perfect

- Objects in Heavens unchanging
- Only motion in Heavens is circular motion
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Why cant the Earth be moving?

• Rotation
• Earth 24,000 miles in circumference at equator
• 1 rotation in 24 hours
• Speed at equator 1,000 miles per hour (1467 ft/s)
• If 1 sec for object to fall floor moves
  almost 1500 ft

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Why cant the Earth be moving?

• Revolution
• Earths period (assume circular 2pr) miles
  5.84 x 108 miles
• Period 365 days x 24 hr/day 8760 hr
• Velocity 66,705 mi/hr

Parallax
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Parallax Effect
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Parallax Effect
Large Parallax Angle
Small Parallax Angle
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Problems

• observations - not possible with simple
  circular models
• apparent size / brightness of planets vary
• apparent speed of planets vary
• planets have different shapes of retrograde
  motion

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Ancient Modelsgeocentric geostatic

• Eudoxus nested homocentric spheres
  (non-physical, mathematical entities)
• Aristotle solid crystalline spheres

Cant account for Retrograde Motion or Varying
brightness of planets
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Retrograde Motion
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Ancient Modelsgeocentric geostatic

• Apollonius
• - introduced the epicycle to explain
  planetary and retrograde motion
• Hippararchus
• - database of around 850 stars
• - suns motion represented by a path around the
  Earth that was an off-centered circle (eccentric)
• Both reject idea of crystalline spheres

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Eccentric Circle Epicycle
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Epicycle on Deferent
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Explains retrograde motion varying brightness
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Dominance of Aristotelian Thought

• Heavens perfect

• Objects in Heavens unchanging

? Only motion in Heavens is circular motion
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Ptolemy (85-165) Egyptian-born Greek astronomer
- synthesized Hipparchus' observations with his
own and extended to Apollonius' model of the
universe by introducing more epicycles
produced a model that was to become accepted as
the standard model until the 1600s
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Ptolemaic Model

• spherical, unmoving Earth in central region of
  Universe ? its natural place
• not strictly geocentric - model used eccentrics
  10,000 Earth diameters from the center) but
  geostatic
• stars fixed on celestial sphere that rotated once
  every 24 hours.
• Sun and the five planets had their motions
  explained by combinations of epicycles, deferents
  and eccentrics. In total some seventy circles and
  spheres were required

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Ptolemaic Model

• inspired by Plato - model should fit observations
  despite appearances introduced equant into his
  model
• Each heavenly body had its own equant which did
  not coincide with the center of a planet's
  deferent
• concept of equant broke with the concept that
  motion of spheres should be uniform but effective
  in accounting for the variations noted in the
  retrograde motion of planets

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Ptolemys model complex but standard for 1,500
years

• It worked - could predict the position of a
  planet to within 2
• It accounted for observed planetary motions,
  retrograde motion and variations in brightness
• It placed the Earth in its natural place at the
  center of things, satisfying Aristotelian
  philosophy
• It matched with common sense. We do not feel the
  Earth move and Ptolemy's model had a static Earth
  

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Strictly Uniform Circular Motion
the speed of the yellow object along the circle
is uniform
the circle shares a center with the Earth
the object sweeps out equal angles in equal times
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Basic Construction - Epicycle
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Epicycle on Deferent
Deferent produces eastward motion seen most of
the time
Epicycle produces retrograde motion
explains varying size brightness of planets
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Basic Construction - Eccentric
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Eccentric Circle
Green object sweeps out equal angles in equal
times, maintaining a constant speed in its
perfectly circular path
But from the Earth, the object appears to move
faster in the bottom half of its motion than in
the top half
The motion of the green object is perfectly
uniform - it will appear nonuniform to observer
on Earth because the Earth does not coincide with
the center of the motion 
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Basic Construction - Equant
Basic Construction - Equant
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Equant Circle
Body on circular path that does not share a
center with Earth
Object's speed varies in its orbit around that
circle--moves faster in the bottom half, slower
on the top half
Motion is uniform in that red object sweeps out
equal angles in equal times from a point inside
the circle
This equant point, not at center of circle nor at
center of Earth
Speed of object is nonuniform from perspective of
equant point, center of equant circle, and center
of Earth 
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Combinations of three constructions
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All together
Deferent only yellow
Deferent eccentric green
Deferent eccentric equant red
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Ptolemys Model
Centers of the inferior planet between Earth and
sun epicycles move on an imaginary line
connecting the Earth and the Sun
Lines connecting superior planets outside of
Earth-sun pair to the centers of their epicycles
parallel to the line connecting Earth and Sun
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Why would ancient astronomers believe in a
geocentric model?
1. All motion in the heavens is uniform circular
motion
2. Objects in the heavens are made from perfect
material and cannot change their properties
(e.g., brightness)
3. Earth is a the center of the Universe
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Medieval Aristotelian Astronomy

• Philosophy of Aristotle wedded to Medieval
  theology

• Prime mover God

• Outermost sphere heaven

• Earth in center concern God had for mankind

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Ptolemaic Model
Successfully accounted for naked eye observations
made by the ancient Greeks

• Epicycles within the epicycles existed (as many
  as 28 for some planets)

Sun-linked epicyles of the inferior planets
explain why they always remain close to the Sun
in the sky
The epicycles of the superior planets account for
their occasional bouts of retrograde motion
Displacement of Earth from center of deferants,
introduction of equant as center of uniform
rotation explain why the planets speed up
slightly when they are close to the Earth (and,
hence, appear brighter in the night sky), and
slow down when they are further away