Satellite Motion

1
Satellite Motion
2
Low Orbit

• A container falls off the space station while in
  low earth orbit. It will move
• A) straight down toward Earth.
• B) curving slowly down toward Earth.
• C) in the same orbit as the space station.
• D) ever farther away due to lower mass.
• E) rapidly away into space.

3
Short Period

• An object in space would go in a straight line
  without another force.
• Gravity supplies a force to hold objects in
  circular orbits.

• In low orbit the period is related to the
  gravitational acceleration.

no gravity
gravity
Low Earth orbit period T lt 90 min.
4
Geosynchronous Orbit

• In higher orbits, the gravitational force is
  significantly less than on the surface.
• Use the force of universal gravitation.
• Fgrav G M m / r2

• The height for a satellite with a 24 hr period
  can be found.

radius r 4.22 x 107 m altitude is r - 6400 km
36,000 km
5
Testing Models

• Geocentric (or Ptolemaic) means the Earth is at
  the center and motionless.
• Heliocentric (or Copernican) means the Sun is at
  the center and motionless.
• Scholars wanted to differentiate models by
  comparing the predictions with precise
  observations.
• This originated the modern scientific method.

6
Keplers Work

• Tycho Brahe led a team which collected data on
  the position of the planets (1580-1600 with no
  telescopes).
• Mathematician Johannes Kepler was hired by Brahe
  to analyze the data.

• He took 20 years of data on position and relative
  distance.
• No calculus, no graph paper, no log tables.
• Both Ptolemy and Copernicus were wrong.
• He determined 3 laws of planetary motion
  (1600-1630).

7
Keplers First Law

• The orbit of a planet is an ellipse with the sun
  at one focus.

A path connecting the two foci to the ellipse
always has the same length.
8
Orbital Speed

• The centripetal force is due to gravity.
• GMm/r2 mv2/r
• v2 GM/r
• Larger radius orbit means slower speed.
• Within an ellipse larger distance also gives
  slower speed.

9
Keplers Second Law

• The line joining a planet and the sun sweeps
  equal areas in equal time.

Dt
The planet moves slowly here.
Dt
The planet moves quickly here.
10
Orbital Period

• An ellipse is described by two axes.
• Long semimajor (a)
• Short semiminor (b)
• The area is pab (becomes pr2 for a circle).

• The speed is related to the period in a circular
  orbit.
• v2 GM/r
• (2pr/T)2 GM/r
• T2 4p2r3/GM
• Larger radius orbit means longer period.
• Within an ellipse, a larger semimajor axis also
  gives a longer period.

11
Keplers Third Law

• The square of a planets period is proportional
  to the cube of the length of the orbits
  semimajor axis.
• T2/a3 constant
• The constant is the same for all objects orbiting
  the Sun

direction of orbit
semimajor axis a
The time for one orbit is one period T
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