April 13, 2013

I was watching an old episode of Cheers the other day in which Sam and Woody attempt to set up a new satellite TV system in the bar.  When Woody asks about how satellite TV works, Sam describes the satellites, and mentions that they’re like 87 million miles away.  This is of course a gross exaggeration; the sun is only 93 million miles away.  It would be essentially impossible for something to orbit the earth from that distance.  Geosynchronous satellites (those that orbit at the same rate as the earth’s rotation) typically orbit at around 22,000 miles.  The exact figure is actually a little higher (22,236), but satellites in an elliptical orbit will be closer than this along their minor axis (the shorter “radius” of an ellipse).  There are actually two types of geosynchronous satellites: regular geosynchronous satellites which are synchronized with the rotation of the earth and appear in the same general area throughout the day, and geostationary satellites which orbit around the equator and appear at the exact same spot throughout the day.  Satellites can only be in geostationary orbit around the equator, because physics.

Okay, let’s dig a little deeper.  One object orbits the center of another object, so we can’t have a satellite in geostationary orbit around New York City because the satellite would have to orbit a point in the earth above its center (if the Arctic is “up” and the Antarctic is “down”).  This means that a satellite in geosynchronous orbit at 30 degrees latitude above the equator sits 22,000 miles above a swath of earth 60 degrees wide.  You may be familiar with this pattern if you’ve watched any space-themed movies: when tracking a space shuttle or other space vehicle it’s path on the ground looks somewhat like a sine wave.  Why does the satellite have to be at exactly 22,236 miles?  Because math.

Okay, okay, we’ll look at an actual explanation.  For one object to orbit another, its centripetal acceleration has to equal its gravitational attraction, i.e., it has to be traveling forward as fast as it’s falling.  Things can orbit the earth at much lower altitudes: the space shuttles often hung out around 200 miles up.  But if we want the satellite to sync up with the rotation of the earth, we need it to be higher, because otherwise the centripetal acceleration required to keep it in orbit will be too fast for the earth to keep up.  Note that 22,236 miles is actually a minimum.  We can have a satellite much further out and traveling much faster and have it still sync up with the earth’s rotation.  Of course, there’s no reason we would ever want to do that because it would cost more money and make communications more difficult.