Satellites in Orbit

As a satellite orbits the earth at any given altitude, it must travel at an exact speed. If it travels too slow, its orbit will decay and it will fall to the earth. If it travels too fast, it will achieve escape velocity (break out of orbit and travel off into space).

v = linear velocity of the satellite. The acceleration towards the Earth (aR ) is caused by the force of the Earth's gravity acting on the satellite. We'll take a look at a couple of different problems for keeping a satellite in orbit
based on what we want the satellite to do. 
The first type we'll take a look at is a satellite in "Near Earth Orbit" or "Low Earth Orbit". This type of satellite orbits the earth relatively close to the surface of the earth (within a few hundred kilometers of the Earth's surface, where the acceleration on the satellite due to Earth's gravity is approximately 80 - 90% that of the value on the surface of the Earth (9.8 m/s2). While the orbits for these vary, they are generally polar (pole to pole) and complete one period in about 90 minutes, give or take. In systems, to ensure world wide coverage, there are usually several satellites all dedicated to the same mission working together in a configuration known as a constellation. The GPS Satellite constellation is an example of a cluster of satellites in LEO. LEO satellites can have various missions, ranging from communications to weather to surveillance.

We can use Newton's Law of Universal Gravitation to find out where a satellite orbits or how fast it goes. From the above picture, we can see that the only force acting towards the center (our centripetal force) is gravity. So from Newton's Law of Universal Gravitation:   

This gravitational force is the centripetal force causing the centripetal acceleration.

The satellite's mass cancels from both sides and is not necessary

ar is defined as the linear velocity (v) squared divided by the radius of the orbit. This r is actually the radius of the Earth (6400 km) plus the altitude of the satellite. I'm going to do a very unmathematic-like thing and leave r in the denominator of both sides of the equation rather than cancel it out.

G is the Universal Gravitation constant (6.67 x 10-11 N-m2/kg2), v is the linear speed of the satellite, and mE is the mass of the Earth (5.97 x 1024 kg). Remember also from previous lessons that the speed of the satellite may be given by the equation

where r is the same distance described above and T is the period of the satellite. At this point I would provide a problem where you would have to solve for the period of rotation and the speed of the satellite given a hypothetical altitude (say, 300 km). Actually, this is a homework problem and you'll need to solve it for class. Better get started!

A second type of satellite is a Geosynchronous Satellite.  It may also be called Geostationary. A geosynchronous orbit is one where a satellite stays at a fixed spot above a point on the earth (only applicable to
points on the equator).  N
otice that as the planet rotates, the geosynchronous  satellite remains over the same longitude as the city. Geosynchronous  satellites are used to execute many different functions. Some of these functions include communications, satellite TV relay, weather examination,  military applications, etc. Geosynchronous satellites are especially great for  satellite TV because a dish can be pointed at a geosync satellite and never  moved unless the satellite’s orbit changes. A cluster of 3-4 of these  satellites working together can cover the entire Earth. These satellites are  typically at altitudes of about 37,000 km. Want to see the math? Let's pick up  where we left off above.

Given above, and

Also given above.

Substituting the second equation into the first for v

Simplify

 

Simplify some more

Solve for r

Substituting in the above values for G and mass of the Earth, and remembering that the period of the Earth is 24 hours, or 86,400 seconds, we find that r is 42,300 km. Subtracting the radius of the Earth, the altitude of the satellite must be 36,000 km. Give or take.           

More information on satellites and orbits can be found at:

http://octopus.gma.org/surfing/satellites/inorbit.html

http://www.noaa.gov/

For more information on geosynchronous satellites, try:

http://liftoff.msfc.nasa.gov/academy/rocket_sci/satellites/geosync.htm
http://www.wikipedia.org/wiki/Geosynchronous_satellite