Libre à moi

Free lifting for a space elevator

January 7, 2009 – 12:20 am

As I was reading an article about a new kind of lifter for the space elevator, I started thinking about the energy you would need to bring anything to the geostationary orbit. I also remember thinking that over the geostationary orbit, moving still upward gives you energy. If you imagine the space elevator as a tube and you put a rock in it, under the geostationary orbit it falls downward, but over it it falls upward. So there is energy to be salvaged if you let the object go above the orbit.

The potential of this rock looks like that

[this curb is shamefully stolen from this page I hope the author will forgive me but it looks a lot better than the one I made]

It comes from the gravitational potential [from wikipedia but can be easily calculated as the integral of the force]

With G the gravitational constant, Mt the mass of the earth and r the distance from the center of the earth.

And the centrifugal one which is V = - w^2 * r^2 / 2 [as the integral of the force |F| = w^2 * r].

With w the angular speed of the earth and r the distance from the center of the earth.

I bet you could notice that the top of the curb is the geostationary point where the potential is maximum, it produces an instable equilibrium. Something else that I would like to point out is that if you suppose that the space elevator has no counterweight and its diameter does not change, then this is similar to the equation of the integral of all the forces on the cable ! Which means that the cable will fall on the earth unless it is longer than DE (the point where the equation reach 0).

DE ~ 144000 km

So DE is 4 times higher than the geostationary orbit which is at 36000 km, and about a third of the distance to the moon.

So the center of mass of the cable is at half DE which is at about twice Dgeo (the distance to the geostationary orbit). So the center of mass is usually quite different from the geostationary point ! This has already been noticed and explained here but wikipedia still has an image that states the opposite on the space elevator article, let’s hope this will be fixed soon.

Now let’s think of this curb in another way: it represents the minimum energy needed to get an object to a certain height on the space elevator. So it means that you don’t need any energy to get an object to DE, the energy that you consume fighting gravity to get the object to the geostationary point can be regained by the centrifugal force. It will absorb the kinetic energy of the rotating earth and slow down the earth rotation on itself.

And what if we make the elevator longer than DE ? Then you can actually gain energy by moving matter all the way up the elevator ! This can also be thought of as a siphon: if you attach a giant drinking straw at the bottom of the ocean that goes all the way above DE and start to siphon water off the straw, then once you initiate the move, water will continue flowing into space, emptying the oceans until the earth stand still or the oceans are empty (I leave it to you to determine which one happens first), if water doesn’t freeze in the straw of course.

This is all nice and theoretical but what about practice ?

Let’s put a pulley on the earth and possibly one in space and have a cable turn around them, so that the cable is moving upward and downward. Now attach some rocks regularly along the cable on the ground (not too much the total mass of the rocks must be a lot smaller than the mass of the cable) and let go of them at the top of the cable. If the cable is longer than DE then the cable will be accelerated by that ! And it could be used to lift some passenger or material to geostationary orbit instead of rocks, which might possibly be more useful.

This could also be used as a clean energy source. We could use the rotation of the wheel inside the pulley on the ground to produce electricity. It is also interesting to notice that it is possible to slow down the rotation of the earth on itself without consuming any energy and without any interaction with another body such as the moon.

This may not be an original idea but I haven’t found any reference to it while googling.

Compared to a “standard” elevator with a counterweight it has various good and bad points.

Good:

- lifting doesn’t consume any energy, in fact it could produce some

- no lifting problem: you just have to attach yourself to the moving cable

- you can go up and down at the same time

- the tension on the cable isn’t necessarily higher than on a “conventional” elevator except for the fact that the thickness of the cable must be constant

Bad:

- need a ~6 times longer cable than an elevator with a counterweight

- the thickness of the cable must be constant

- must be careful that the cable going up won’t touch the cable going down

[edit a nicer schema contributed by a friend]