Gravitation

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CHAPTER 2 Gravitation and the Waltz of the
Planets
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Historical Background

• Ancient Greeks
• Ptolemy
• Copernicus
• Galileo
• Brahe
• Kepler
• Newton

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Two Views of the Universe

• Geocentric model (earth-centered)
• Heliocentric model (sun-centered)

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Ancient Greeks
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Geocentric Model of the Universe

• Stationary Earth
• All other objects circle the Earth.
• Universe consisted of stars, six planets (none
  beyond Saturn), the Sun and Moon.
• Stars were fixed onto a crystal sphere that
  surrounded the Earth.
• Since the sphere rotated around the Earth, so did
  the stars.

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Geocentric Model of the Universe

• Planets shift slowly eastward relative to the
  fixed stars.
• Planets were also observed to occasionally move
  backwards (east to west or westward with
  respect to the background stars)
• This was called retrograde motion.
• This was a problem to explain.

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Looking South
As the Earth rotates during the night (diurnal
motion) the planet appears to move from east to
west.
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The ancient Greeks knew that the planets slowly
shifted relative to the fixed stars in the
constellations.
Looking South
From night to night, the planets appear to move
from west to east relative to the background
stars (direct motion).
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Looking South
But sometimes, the planet would slow down, stop
and reverse its motion from east to west relative
to the background stars (retrograde motion).
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Geocentric Model
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Early models of the universe attempted to explain
the motion of the five visible planets against
the background of fixed stars. The main
problem was that the planets do not move
uniformly against the background of stars, but
instead appear to stop, move backward, then move
forward again. This backward motion is referred
to as retrograde motion.
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Ptolemys Role in the Geocentric Model 1st Century
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Geocentric Model of the Universe

• Hipparcus was first to use epicycles to address
  the problem of retrograde motion of the planets.
• Ptolemy expanded upon this.
• Ptolemy determined that retrograde motion of
  planets could be described by using deferents
  (orbital path around Earth) and epicycles
  (planets orbital path around the deferent).
• All orbits, deferents and epicycles were in
  perfect circles.

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Epicycles and Deferents
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Ptolemy explained this motion using a geocentric
(Earth-centered) model of the solar system in
which the planets orbited the Earth indirectly,
by moving on epicycles which in turn orbited the
Earth.
Ptolemy expanded upon Hipparcuss work
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Epicycles and Deferents
Occams Razor the best explanation is the
simplest
Stop the madness!
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Copernicus and the Heliocentric Model 1500s
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Copernicuss Heliocentric Model

• Sun placed at the center of the solar system.
• Planets placed in proper order away from the Sun.
• Synodic and sidereal periods for each planets
  orbit were measured.
• Copernicus still assumed perfect circular orbits
  for planets.
• This was a problem.

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Nicolaus Copernicus in the 16th century developed
the first heliocentric (sun-centered) model of
the solar system. In this model, the retrograde
motion of Mars is seen when the Earth passes Mars
in its orbit around the Sun. Lack of an
explanation for this motion was a significant
failing of Copernicus model.
FYI Aristarchus in 6th century
B.C. first suggested that the Sun was fixed and
the Earth moved around the Sun in a circle.
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Heliocentric vs. Geocentric Model
heliocentric
geocentric
Ptolemys geocentric model failed to reproduce
observed planetary motions.
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Tycho Brahe and KeplerTrue Planetary Motion Late
1500s to 1600
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Tycho Brahes observations

• Set up an observatory called Uraniborg in 1576
  (later, he built another observatory called
  Stjerneborg).
• He accurately measured the position and angle of
  stars and planets without a telescope (it had not
  been invented).
• Huge collection of data.
• Brahe invited Johannes Kepler to join him in at
  Uraniborg in 1600.

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After Tycho Brahes death, Johannes Kepler
(pictured here with Tycho in the background) used
Tychos observations to deduce the three laws of
planetary motion.
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Keplers Breakthrough

• Brahe asked Johannes Kepler to join him in
  Denmark in 1600.
• Kepler used Brahes data to make highly precise
  calculations of planetary orbits.
• Accuracy of orbits matched only if orbits were
  considered less than perfect circles (an ellipse).

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Keplers Breakthrough

• Kepler developed three laws that could be used to
  describe planetary motion.
• Laws are based upon the understanding of the
  ellipse.

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LAW 1. The orbit of a planet around the Sun is
an ellipse with the Sun at one focus.
Semi-major axis is half the major axis
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LAW 2 A line joining the planet and the Sun
sweeps out equal areas in equal intervals of time.
Planet moves slower in its orbit when farther
away from the Sun.
Planet moves faster in its orbit when closer to
the Sun.
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LAW 3 The square of a planets sidereal period
around the Sun is directly proportional to the
cube of its semi-major axis.
This law relates the amount of time for the
planet to complete one orbit around the Sun to
the planets average distance from the Sun. If
we measure the orbital periods (P) in years and
distances (a) in astronomical units, then the law
mathematically can be written as P2 a3.
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The amount of elongation in a planets orbit is
defined as its orbital eccentricity. An orbital
eccentricity of 0 is a perfect circle while an
eccentricity close to 1.0 is nearly a straight
line.
In an elliptical orbit, the distance from a
planet to the Sun varies. The point in a
planets orbit closest to the Sun is called
perihelion, and the point farthest from the Sun
is called aphelion.
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Galileo and the Telescope 1609 -1610
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Galileo was the first to use a telescope to
examine celestial objects. His discoveries
supported a heliocentric model of the solar
system.
1. Galileo discovered that Venus, like the Moon,
undergoes a series of phases as seen from Earth.
In the Ptolemaic (geocentric) model, Venus would
be seen in only new or crescent phases. However,
as Galileo observed, Venus is seen in all phases,
which agrees with the Copernican model as shown.
Phases of Venus
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2. Galileo also discovered moons in orbit around
the planet Jupiter. He concluded that they are
orbiting Jupiter because they move across from
one side of the planet to the other This was
further evidence that the Earth was not the
center of the universe.
Moons of Jupiter
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Galileos Contributions

• 3.      Imperfections on the Moons surface The
  Moons surface was irregular and crater-filled
• 4.       Dark spots on the Sun The Sun was
  observed to have dark spots

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Galileos Contributions

• Galileos observations began to erode the notion
  of celestial perfection and provided support for
  the heliocentric view of the universe.

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Isaac Newton formulated three laws to describe
the fundamental properties of physical reality.
NEWTONS THREE LAWS OF MOTION LAW 1 A body
remains at rest or moves in a straight line at
constant speed unless acted upon by a net outside
force. LAW 2 The acceleration of an object is
proportional to the force acting on it. LAW 3
Whenever one body exerts a force on a second
body, the second body exerts an equal and
opposite force on the first body.
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Newton also discovered that gravity, the force
that causes objects to fall to the ground on
Earth, is the same force that keeps the Moon in
its orbit around the Earth.
NEWTONS LAW OF UNIVERSAL GRAVITATION Two objects
attract each other with a force that is directly
proportional to the product of their masses and
inversely proportional to the square of the
distance between them.
With his laws, Newton was able to derive Keplers
three laws, as well as predict other possible
orbits.
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Newtons laws were applied to other objects in
our solar system.
Using Newtons methods, Edmund Halley worked out
the details of a comets orbit and predicted its
return.
Deviations from Newtons Laws in the orbit of
the planet Uranus led to the discovery of the
eighth planet, Neptune.
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We define special positions of the planets in
their orbits depending where they appear in our
sky. For example, while at a conjunction, a
planet will appear in the same part of the sky as
the Sun, while at opposition, a planet will
appear opposite the Sun in our sky.
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However, the cycle of these positions (a synodic
period) is different from the actual orbital
period of the planet around the Sun (a sidereal
period) because both the Earth and the planet
orbit around the Sun.
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Measuring Distances

• Parallax view
• http//www-astronomy-mps.ohio-state.edu/pogge/Ast
  162/Movies/parallax.html
• http//instruct1.cit.cornell.edu/courses/astro101/
  java/parallax/parallax.html

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When a new star appeared in the sky during the
16th century, a Danish astronomer named Tycho
Brahe reasoned that the distance of the object
may be determined by measuring the amount of
parallax.
The apparent change in the location of an object
due to the difference in location of the observer
is called parallax.
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Because the parallax of the star was too small
to measure, Tycho knew that it had to be among
the other stars, thus disproving the ancient
belief that the heavens were fixed and
unchangeable.
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Heliocentric Model

• Who is responsible?
• Evidence?
• Lacking evidence?
• What did Galileo discover?
• What are planetary configurations?

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WHAT DID YOU THINK?

• What makes a theory scientific?
• If it makes predictions that can be objectively
  tested and potentially disproved.
• What is the shape of the Earths orbit around the
  Sun?
• Elliptical
• Do the planets orbit the Sun at constant speeds?
• The closer a planet is to the Sun in its orbit,
  the faster it is moving. It moves fastest at
  perihelion and slowest at aphelion.

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WHAT DID YOU THINK?

• Do all the planets orbit the Sun at the same
  speed?
• No. A planets speed depends on its average
  distance from the Sun.
• How much force does it take to keep an object
  moving in a straight line at a constant speed?
• Unless an object is subject to an outside force,
  it takes no force at all to keep it moving in a
  straight line at a constant speed.
• How does an objects mass differ when measured on
  the Earth and on the Moon?
• Its mass remains constant.

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Key Terms
acceleration angular momentum aphelion astronomica
l unit configuration (of a planet) conjunction con
servation of angular momentum cosmology ellips
e elongation focus (of an ellipse) force Galilean
moons (satellites) gravity heliocentric cosmology
hyperbola inferior conjunction Keplers
laws kinetic energy law of equal areas law of
inertia light-year mass model momentum Newtons
laws of motion Occams razor opposition parabola p
arallax parsec
perihelion physics potential energy retrograde
motion scientific method scientific
theory semimajor axis (of an ellipse) sidereal
period superior conjunction synodic
period universal constant of gravitation universal
law of gravitation velocity weight work
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WHAT DO YOU THINK?

• What is the shape of the Earths orbit around the
  Sun?
• Do the planets orbit the Sun at constant speeds?
• Do all the planets orbit the Sun at the same
  speed?
• How much force does it take to keep an object
  moving in a straight line at a constant speed?
• How does an objects mass differ when measured on
  the Earth and on the Moon?

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• .Compare and contrast the Ptolemaic and
  Copernican cosmologies by explaining a variety of
  naked-eye observations, using both models.
• 2. State Keplers three laws of planetary motion
  describe the geometric content and observational
  consequences of each.
• 3. List Galileos telescopic observations and
  explain the success or failure of Ptolemaic and
  Copernican models in accounting for them.

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• 4. State and identify examples of Newtons three
  laws of motion.
• 5. State Newtons law of universal gravitation
  identify the characteristics of this law that
  explain Keplers laws in terms of Newtons laws.

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• 1.      How do we interpret the history of the
  geocentric and heliocentric views of the
  universe?
• 2.      How do Keplers laws describe planetary
  motion?
• 3.      How are Newtons laws of motion, Newtons
  law of gravitation and Keplers laws related?
• How do the planets move with respect to the Earth
  and the Sun

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1. What is the shape of the Earths orbit around the
   Sun?
2. Do planets orbit the Sun at constant speeds?
3. Do planets orbit the Sun at the same speed?
4. How much force does it take to keep an object
   moving in a straight line at a constant speed?
5. What is the difference between heliocentric and
   geocentric cosmology?

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• What are the contributions to astronomy made by
  the following individuals Ancient greeks,
  Ptolemy, Copernicus, Galileo, Brahe, Newton,
  Kepler?
• How do Keplers laws describe planetary motion?
• What is an ellipse?
• Does Venus show phases just like the Moon?
• How to the planets orbit the Sun?

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You will discover

• clues suggesting that Earth is not the center of
  the universe
• the scientific revolution that dethroned Earth
  from its location at the center of the universe
• Copernicuss argument that the planets orbit the
  Sun
• why the direction of motion of the planets on the
  celestial sphere sometimes appears to change
• that Keplers determination of the shapes of
  planetary orbits depended on the careful
  observations of his mentor, Tycho Brahe
• how Isaac Newton formulated an equation to
  describe the force of gravity
• how Isaac Newton explained why the planets and
  moons remain in orbit