A space elevator is a theoretical long tether that reaches all the way up into outer space and just hangs there all by itself. In theory, once you put one up, you can use it to climb up into outer space very cheaply compared to chemical rockets.
How could a long tether "just hang there?" Because it's in orbit around the Earth. A space elevator is a special type of satellite, specifically, one in a geosynchronous orbit.
The idea here is that it's possible to have a satellite always directly above the same spot on Earth. This isn't the normal state of things. Most satellites are in low-Earth orbit and whip around the planet in fast circles every few hours. But the higher up you put a satellite, the longer it will take to orbit. If you put it high enough, it will take 24 hours to orbit. That way, the Earth can spin underneath it at just the same pace it's orbiting, and by happy coincidence the satellite appears to stay in the same spot, to an observer on the surface of the Earth.
There are a lot of restrictions on where you can place such an orbit. First off, the shape and size of an orbit, along with the gravity of the planet and the laws of physics, determine how long an orbit takes. You can't just blast your rockets a little softer and expect to orbit slowly. If you do that, you won't be going fast enough to orbit any more, and gravity will pull you back to the ground.
If you want an orbit that takes 24 hours and is "smooth" (always has the same angular velocity), you're restricted to making it a circular orbit at a height that works out to be 36,000 km above the surface of the Earth. The radius of the Earth is only 6,000 km, so a space elevator is long enough to wrap around the Earth multiple times, and together they look roughly like a flying sparrow with a two-meter long hair trailing out its butt. Hold a hula hoop around a soccer ball. That is roughly the correct scale for how high up this circular orbit would be. By comparison, most orbits for satellites and space shuttle trips and things like that would be loops hovering about half a centimeter from the ball's surface.
Second, a geosynchronous orbit has to be over the equator. A satellite in a circular orbit doesn't have to be over the equator, of course. Take that soccer ball and hula hoop from before. Spin and twirl the hula hoop however you like. As long as the Earth is still at the center, you have a valid orbit. But it will only stay over the same spot if it's at the equator. Otherwise, the satellite wanders north and south again over the course of its day. If you try to cheat by simply sliding the hula hoop up some, so it's parallel to the equator but over say, the 45o line of latitude, it's no longer a possible physical orbit. The soccer ball isn't at the center any more, so its gravity would inevitably drag the orbit back down.
Imagine a satellite in geosynchronous orbit. Now imagine you drop a little fishing line from the bottom of the satellite, reaching down towards Earth. In response, the rest of the satellite will have to rise up a tiny little bit to keep the center of mass at the same height. Now drop that line lower and lower. The rest of the satellite continues creeping up a little bit higher, and the line keeps reaching lower, until eventually you have a tether that reaches all the way from Earth to geosynchronous orbit. That's a space elevator.
The first problem that creates is that the fishing line is under enormous tension. If it's made of any ordinary material, it will rip apart. To see why, imagine a spot on the line a few thousand kilometers above the Earth. If you cut the line there, the stuff below it would clearly not be in geosynchronous orbit - it isn't high enough up. So all that stuff would fall.
Since that material below the cut point isn't falling in an operational space elevator, something must be holding it up. That something is tension in the line. The higher up the space elevator you go, the more material there is to hold up, so the tension gets greater and greater. By the time you reach geosynchronous orbit, the line has to support the weight of tens of thousands of kilometers of line beneath it. That's a lot of weight, and any normal material would imply rip apart. So to build a space elevator you need something really strong. So far, there are no long ropes that are anywhere near strong enough, although some people speculate that it's possible to build one out of carbon nanotubes.
Geosynchronous Orbits II
The gravitational potential of two point masses is given by
is a constant, and are the objects' masses, and is the distance between them. We're giving the Earth (which in real life takes up lots of space) a single definite location at one point. Gauss' law allows us to do this for a body with spherical symmetry, but really it doesn't matter for the calculation I have in mind, because the distance for a space elevator will turn out to be much greater than the radius of the Earth.
Using a reference frame in which the center of mass of the Earth does not move, and assuming this is an inertial frame, the kinetic energy of the system is
Where is the magnitude of the satellite's velocity in this frame. We'll use ordinary spherical coordinates, so that the position of the satellite is given by . We're only interested in circular orbits, where is a constant, and . Then the velocity is
But we need a circular orbit, for which
The Lagrangian is
Then the Euler-Lagrange equations give
where is a constant depending on the gravitational constant and the mass of the Earth.
Setting and plugging through the numbers gives r=4.2*1024 km. That's the height of a geosynchronous orbit from the center of the Earth.
The force on a differential element of the cable, , is
where is the force of gravity on the segment and is the change in tension from below to above that point. is the mass of the length element, and is its acceleration. The positive direction is defined to be away from the center of the Earth. The element is accelerating in a circular orbit, with acceleration
If the mass of the element, , is written in terms of the cross-sectional area of the cable at a given length , and the density , we can then write the tension as
The difference between r and l is that l is a dummy variable for integrating over. It begins at re, the radius of the Earth, and goes to whatever radius we're considering.
The tension is zero at the surface of the Earth, then increases until the integrand becomes zero at geosynchronous orbit. Then the tension slowly slinks back down to zero at the far end of the elevator.
To determine whether or not the cable will rip, it's not the total tension that matters, but whether the tension is greater than the cable's tensile strength. And the cable's strength is proportional to it's cross section . This suggests a strategy for building the elevator - down near the ground where the tension is small, make the cable very thin. It doesn't need to be strong there. Up near geosynchronous orbit where the tension is very high, make the cable thick to deal with the high stress. This plan could be called "tapering" the cable.
The ideal taper for the cable is one in which the tension per unit area is constant all along the cable. To make the problem workable, assume the tension doesn't drop to zero at the surface of the Earth (in which case the "ideal" cable would be zero thickness there, and all the way up), but instead goes to some finite value .
Then we have:
this can be combined with the expression for the tension to give a differential equation for the cross-section length
which you can solve to find
where C is just a constant set so that the cable has the appropriate width at the surface of the Earth.