They stay up there thanks to the magic of orbital mechanics. Not as complicated as they might seem, orbital mechanics are in play all around us. When you throw a ball, you are really putting it into orbit -- only a very shallow one doomed to end in a matter of a few yards. Now imagine shooting a cannon from the surface of the moon, at its highest point. Since the moon has no atmosphere, and no air resistance, if you launched the cannon ball at the appropriate speed it would orbit the entire moon without dropping to the ground -- fast enough to trace the curve of the moon and remain aloft. It's similar for a satellite, which needs to be about 200 miles (320 kilometers) above Earth's atmosphere to be free of it. Once there, satellite operators set the speed so that it essentially "falls" to Earth at the same speed that the curve of the Earth falls away from the satellite -- thus the satellite never comes down; it just keeps falling its way around Earth.
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