The world as we know it

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The world as we know it

• Everything up to now leads to a beautiful
  consistent cosmogony model of the Universe.
• Celestial sphere with stars fixed to it rotates
  East to West daily.
• Sun (annually) and Moon (monthly) move West to
  East on ecliptic.

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• Ptolemys model included sizes and rotation
  speeds for all cycles, and predicted planetary
  positions accurately successful scientific
  model!
• E.g. Venus always found near Sun, so its
  deferent turns annually (aligned with Sun) while
  epicycle accounts for deviations.

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• Copernicus (1500) revives heliocentric model of
  Aristarchus.
• Planets (and Earth) orbit Sun. Apparent motion
  of planets in sky comes from relative motion.
• Here inferior planets always near Sun because
  their orbits smaller than Earths.
• Copernicus computes periods and (relative)
  distances from Sun for all planets.
• Farther planets move slower.

• Copernician model is much simpler (Occams razor)
  as explanation of observations.
• Small deviations lead him to add epicycles.

No smoking gun yet rejected on cultural grounds
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Calculating Periods

• Planets motion in sky results from combination
  of true motion and Earths motion. Planet orbits
  Sun each sidereal period.
• What we see recurs every synodic period (relative
  configuration of Earth, Sun, planet).
• For an inferior planet, a synodic period has
  elapsed when the planet has lapped Earth.
• For a superior planet when Earth has lapped it.
  
• If sidereal period is P days, planet moves 360/P
  degrees a day. Earth moves 360/E degrees a day
  where E365 is sidereal period of Earth.
• Planet laps Earth (completing a synodic period of
  S days) when S(360/P) S(360/E) 360. This is
  same as 1/P 1/E 1/S.
• For a superior planet find similarly 1/P 1/E
  1/S.

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Brahe and Kepler

• First steps to deeper insight were careful
  observations by Brahe (1580) of planetary motion
  to great precision.
• Using these, Kepler (1609) finds three laws of
  planetary motion.

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Keplers Laws

• 1. Orbit of a planet is an ellipse with Sun at
  one focus.
• Ellipse is shape of all points such that sum of
  their distances from two points (foci) is
  constant.
• Eccentricity (e) measures how far the foci are
  relative to size. e0 is a circle.

• Other focus is nothing (not even same for all
  planets).
• Typically e small, .017 for Earth, .2 for Mercury.

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• 2. Line connecting planet to Sun sweeps out
  equal areas in equal time intervals as planet
  orbits.
• Planet moves faster at perihelion, slower at
  aphelion.
• This causes slight change in rate of Suns motion
  discussed earlier.

• This effect much more dramatic for comets which
  follow highly eccentric orbits.

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• 3. Square of sidereal period is proportional to
  cube of semimajor axis.
• This relates the orbital motions of different
  planets orbiting same Sun.
• Write this as P2 a3. This is valid if P is
  measured in years and a in AU.

• Recall 1AU 1.496 108 km 93 million miles is
  average Earth-Sun distance.

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• Keplers laws are amazing progress. They give
  planetary motion with unprecedented accuracy.
  Whats more, they are universal they apply to
  any orbital system, from an atom through Saturns
  moons to Galaxy clusters.
• In physics such universality means there are
  fundamental laws at work here.
• These were found by Newton (1670) who at first
  was not thinking at all about Astronomy.

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Galileos Smoking Scope

• Smoking gun evidence for Copernician model
  required new technology.
• Galileo (1610) turns new telescope up and finds
  phases of Mercury.
• The correlation between phase and position in sky
  agrees with heliocentric model, not with
  Ptolemaic model.

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Cultural Issues

• Galileo also discovers that Jupiter is itself
  accompanied by moons that orbit the planet, much
  like our Moon orbits Earth. Nature repeats on
  different scales.
• Motion of Jupiters moons studied closely, forms
  first Nautical clock for longitude measurement.
• Despite all this, Galileo tried for heresy and
  sentenced to house arrest.

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More on Galileo

• Galileo made many other discoveries of
  importance. With his telescope, he discovered
  mountains on the moon, size and shape of planets,
  nature of Milky Way, moving spots on Sun, among
  others.
• He also studied mechanics, properties of motion
  in general. Formulated principle of inertia
  object tends to remain in its state of motion
  unless disturbed externally. We know we need to
  work to move things. Galileos insight reminds us
  we also work to stop or turn them.
• Galileo almost got mechanics right. What stopped
  him was the fact that the mathematics needed to
  formulate the theory was not known. To make
  progress, Newton had to invent Calculus.

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Newtons Laws of Motion

• 1. An object upon which no forces act will move
  in a straight line with constant velocity.
  (inertia)
• Familiar when velocity is zero object at rest
  will stay there.
• Need to remember in our world two forces (at
  least) always get in the way gravity pulls us
  down friction slows all motion. To see Newton 1
  need to minimize these or imagine them removed.
• Velocity is a vector has direction as well as
  magnitude. So constant velocity means no change
  in direction or speed.

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• 2. When a force acts on an object, it will
  change its velocity. The acceleration will be
  proportional to the force (and pointed in the
  same direction). The proportionality constant is
  called mass.
• F ma
• Acceleration a is rate of change of velocity v.
  So measured in (m/sec)/sec or m/sec2.
• Like v it is a vector and has direction. Note
  that changing direction of v requires
  acceleration, just as does changing magnitude of
  v.
• m is mass. Measured in kg total amount of
  stuff.
• F is force. Measured in kg (m/sec2) N(ewton)
  
• Acceleration of gravity here is g 9.8 m/sec2,
  so force of gravity on 1kg. is 9.8 N.

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Uniform Circular Motion

• Planet moves at uniform speed v around circle of
  radius R. Period is P2pR/v
• Is velocity constant?

• NO. Direction changes.

• Guess acceleration
• Points inwards
• Grows with larger v (m/sec).
• Smaller with larger R (m).
• Measured in m/sec2.

• a v2/R.
• So F ma m v2/R

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Numbers for Earth

• As Earth spins, we move at
• Vspin (2pR)/P 4107 m/24 hr
• 463 m/sec 1036 mph
• As Earth orbits, we move at
• Vorbit (2pR)/P
• 6.28 1.5 1011 m/36586400 sec.
• 29871 m/sec 100,595 mph

• a v2/R
• 4632/6.38106
• 0.034 m/sec2

• F ma
• 100kg. 0.034m/sec2
• 3.4 N

• a v2/R
• 298712/1.51011
• 0.0059 m/sec2

• F ma
• 100kg. 0.0059m/sec2
• 0.59 N

• When rocks in space hit Earth the relative
  velocities are about 100,000 mph. That is why
  they burn in atmosphere as meteors!

• Lets compute the forces required to keep in
  these circular motions a person of mass m100 kg.
  He weighs 9.8 m/sec2 100 kg 980 N

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• 3. When one object applies a force to another,
  the latter applies a force to the former, equal
  in magnitude and opposite in direction. (action
  and reaction).
• This explains how we walk. I push Earth back, it
  pushes me forward!
• As Moon orbits Earth, Earth orbits Moon both in
  fact orbit common center of mass

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Inertial Forces

• When a car stops, you are hurled at the
  windshield. What force acted to make you move?
• None really. Absent any force, your body would
  continue to move at its former velocity.
• To keep you stationary in a car with
  acceleration a requires a force Fma
• From the point of view of someone in the car,
  there is an inertial force F -ma that must be
  balanced.
• Inertial forces are not truly forces, they are
  the result of measuring motion with respect to
  accelerating frames of reference. Like gravity,
  the inertial force is proportional to mass.