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I know there are many different satellites, but why would they put a satalite in a geosynchronous orbit vs a low earth orbit? What are the benefits or disadvantages of each?

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It depends on the mission your satellite needs to perform. The orbit is something like "where" your satellite is, after all. So what it needs to do dictates its orbit.

A geosynchronous or geostationary orbit will keep the satellite in the same general area. This is useful for communications and weather satellites.

Sun-synchronous orbits have the satellite pass over parts of the ground at repetitive solar times, helping with e.g. ecological observations.

Lunar orbits are useful if you want to go to the moon.

Low-earth orbits are quick, and useful for things that essentially just need to be in orbit but not much else (since they require less energy to get to).

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  • $\begingroup$ Navigation satellites like GPS need a very precise orbit and a good coverage of the Earth surface. Using the almanac data, the position of the satellite should be predictable very precisely. $\endgroup$ – Uwe Feb 24 '18 at 9:39
  • $\begingroup$ Low-earth orbits don't require "much" less energy to get to because orbital velocity still needs to be reached. Less energy, but not much less. Low-Earth are also partly protected by Earth's magnetic field, which is beneficial for people on the space shuttle and it reduces (but doesn't eliminate) bombardment from coronal mass ejections. $\endgroup$ – userLTK Feb 24 '18 at 11:55
  • $\begingroup$ @Uwe should we add that on to the answer proper? Maybe make it a Community Wiki answer? $\endgroup$ – Erin Anne Feb 24 '18 at 17:40

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