Satellite Motion

1
Chapter 14

• Satellite Motion

2
Earth Satellites

• An Earth satellite is a satellite that falls
  around the Earth not into it.
• Gravity keeps a satellite in orbit
• Without gravity, the satellite would keep moving
  in a straight line
• How fast must a stone be thrown to horizontally
  orbit the Earth?
• 8 km/s (29,000 km/h 18,000 mi/h)
• The geometric curvature of the Earth is 5 m
  vertical for every 8 km horizontal
• An object in free fall falls 5 m in 1 s
• In order to match the curve of the Earth, the
  object must travel 8 km horizontally in 1 s

3
Earth Satellites

• The orbital speed for a close orbit Earth
  satellite is 8 km/s
• A close orbit satellite must be at least 150 km
  above the Earths surface
• If it wasnt it would burn up against the
  friction of the atmosphere (like a falling star)

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Circular Orbits

• Circular orbit speed is not affected by gravity
• Gravity pulls the satellite straight downward
  (perpendicular to the orbit speed)
• In order for speed to be affected, work must be
  done on the satellite
• Since force and distance are perpendicular, there
  is no work done and no change in KE the speed is
  not affected
• KEY whenever anything moves in a circular
  paththe velocity (tangential) and the force are
  perpendicular

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Circular Orbits
6
Circular Orbits

• It takes about 90 minutes for a satellite close
  to the Earth to complete it orbit
• For higher altitudes the speed is less and the
  time of orbit (period) is longer
• The moon is a satellite of Earth and its period
  is 27.3 days
• Communications satellites have a period of 24
  hours (so they are always above the same point on
  the Earth)
• In order to have a period of 24 hours, the
  satellites are 6.5 Earth radii away from the
  center of the Earth

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Circular Orbits

• Would a cannonball fired upward at 8 km/s go into
  Earth orbit?
• Satellites in close circular orbit fall about 5 m
  each second. How can this be if they are not
  getting any closer to the Earth?
• True or False. Satellites remain in orbit instead
  of falling into the Earth because they are beyond
  the pull of gravity.

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Elliptical Orbits

• What happens if a close orbit projectile is
  launched from the Earth with a tangential
  velocity greater than 8 km/s?
• The satellite will overshoot a circular path
• The orbital path will be an ellipse (oval)
• Ellipsea closed path in which the sum of its
  distances from 2 fixed points (foci) is constant
• For satellite motion the center of the planet is
  a focus of the elliptical path
• The other focus may lie somewhere within the
  planet or outside the planet (it depends on the
  orbit speed)
• The closer the focithe closer the ellipse is to
  a circle
• Satellite speed in an elliptical orbit varies
• Initially the satellite is at its launch speed
  as it moves away from the planet the satellite
  loses speed until it is no longer moving away
  from the planet as it moves back towards the
  planet it gains speed until it is finally back to
  its launch speed (at the same point where it
  started)
• This pattern continues and the satellite traces
  an ellipse in each cycle

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Elliptical Orbits
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Elliptical Orbits

• If a cannonball is fired horizontally from
  Newtons mountain at a tangential velocity less
  than 8 km/s, it soon strikes the ground below.
  Will its speed of impact be greater than, the
  same as, or less than its initial speed?
• If it is fired at 9 or 10 km/s, how does the
  speed change?

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Energy Conservation Satellite Motion

• Moving objects have KINETIC energy
• Objects above the Earths surface have POTENTIAL
  energy
• A satellite has both KE and PE
• The sum of the KE and PE everywhere is constant
• Energy is conserved

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Energy Conservation Satellite Motion

• Circular Orbitthe distance from the Earth does
  not change
• PE is constant
• Energy is conservedKE is constant
• If KE is constantvelocity is constant
• Elliptical Orbitthe distance from the Earth and
  the speed of the satellite changes
• PE is greatest where the satellite is farthest
  away (at the apogee)
• KE is smallest where the satellite is farthest
  away (apogee)the satellite is moving at its
  slowest speed
• PE is smallest where the satellite is closest
  (perigee)
• KE is greatest where the satellite is closest
  (perigee)the satellite is moving at its fastest
  speed

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Energy Conservation Satellite Motion
Where does the satellite have a. Maximum
speed b. Maximum velocity c. Maximum
gravitational attraction to the Earth d.
Maximum kinetic energy e. Maximum gravitational
potential energy f. Maximum total energy g.
Maximum acceleration h. Minimum speed i.
Increasing speed j. Decreasing speed k. Force
perpendicular to its velocity
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Escape Speed

• What goes up must come downsometimes
• If you launch a rocket with a vertical speed of 8
  km/s (and no horizontal speed), what will happen?
• The rocket will come crashing back with a speed
  of 8 km/s
• Isnt there some vertical speed that will allow
  the rocket to escape and not come down?
• Yes!
• Anything fired vertically (neglecting air
  resistance) with a speed greater than 11.2 km/s
  from the Earth will leave Earth (deceleratingbut
  never stopping)

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Why 11.2 km/s?
where G 6.67 x 10 -11 Nm2/kg2 M mass of
attracting body and d distance between
attracting bodys center and the object
Earth has a mass of 5.98 x 1024 kg and a radius
of 6.38 x 106 m. What is the escape speed of a
rocket launched on Earth?
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Escape Speed

• The escape speed for various bodies in the solar
  system are all different
• The escape speed of the sun is 620 km/s
• 11.2 km/s is the speed an object needs to escape
  the Earthbut not necessarily the moon or the sun
• An escaped object from the Earth will take up
  orbit around the sun
• Escape speeds for different bodies refer to the
  initial speed given by a brief force (not a
  constant speed)
• One could escape Earth at any constant speed
  greater than zero if it had enough time
• A rocket could go to the moon without ever
  attaining 11.2 km/s as long as the engines are
  sustained for a long period of time
• Most satellites are launched from the equator,
  towards the east
• This is done to take advantage of the Earths spin

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Keplers Laws

• Tycho Brahe (1546-1601) followed and charted the
  paths of planets (Mars) for years.
• Before telescopes were even invented!
• Johannes Kepler joined Brahe in 1600 and
  continued his observations after Brahes death
• Kelper derived three laws of orbital motion from
  the data
• Kepler could not explain why planets obeyed his
  laws
• This phenomenon could finally be explained by
  Newtons Law of Universal Gravitation

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Keplers 1st Law

• Planets follow elliptical orbits, with the sun at
  one focus of the ellipse.
• Remember a circle is a special case of an
  ellipsethe foci are together at the same point
  (the center of the circle)
• Newton was able to show this because the force of
  gravity decreases with the distance (inverse
  square law 1/r2)
• Closed orbits must have the form of ellipses or
  circles due to this
• Not all orbits are closed
• A comet that passes by the sun once and leaves
  the solar system has a hyperbolic or parabolic
  orbit

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Keplers 2nd Law

• As a planet moves in its orbit, it sweeps out an
  equal amount of area in an equal amount of time.
• The force of gravity on a planet is directly
  toward the sun (inward). Therefore, gravity
  exerts no torque and angular momentum must be
  conserved (L mvr).
• http//home.cvc.org/science/kepler.gif
• The Earths orbit is slightly elliptical. In
  fact, the Earth is closer to the sun during the
  northern hemisphere winter than it is during the
  summer. Is the speed of the Earth during the
  winter (a) greater than, (b) less than, or (c)
  the same as its speed during the summer (northern
  hemisphere)?

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Keplers 3rd Law

• The period, T, of a planet increases as its mean
  distance from the sun, r, raised to the 3/2
  power.
• This is directly derived from Newtons Law of
  Universal Gravitation and centripetal force
• Keplers Third Law is also written as
• r3 kT2
• In our solar system k 1.

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Keplers Law Problems

• The Earth revolves around the sun once a year at
  an average distance of 1.50 x 1011 m.
• Use this information to calculate the mass of the
  sun.
• Find the period of revolution for the planet
  Mercury, whose average distance from the sun is
  5.79 x 1010 m.

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Keplers 3rd Law Problems

• Mercury is 0.387 AUs from the sun. What is
  Mercurys orbital period in years? In days?
• Jupiter has an orbital period of 11.86 years.
  How far away from the sun is Jupiter (in AUs)?