> Can anyone supply me with any information on earth to orbit
> elevators? In particular, do they require (as in Arthur C Clarke's
> book) exotic materials, or *could* one be constructed with 20th
> century technology? Secondly, what are the limits on placing
> the 'bottom' end? How close to the equator does it need to be?
>
> TIA -- Richard Tibbetts

Copyright © 1996 by Joshua W. Burton ( burton AT het DOT brown DOT edu).  All Rights Reserved.
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I did a lot of calculations about this a few years back; here are some results that might interest you. Here's the apparent strength of gravity as you go up the elevator, allowing for both the earth's rotation and the 1/r² field:

Weightlessness comes at the Clarke point, of course, 35950 km up. Above that, there is a centrifugal effect, and the earth appears to be 'above' you---but you would have to be nearly 200,000 km up before the apparent gravity reaches -1.0 m/s². In practice, no one would build it out that far; you just want to go far enough to keep the center of gravity at the Clarke point, plus a bit more to put the lower end of the elevator in tension. A big mass just slightly above synchronous orbit is probably the way to go.

Midway Station, the lowest point where you go into an elliptical orbit instead of hitting the ground if you jump off, is 23450 km up, and has a tiny apparent gravity of 0.29 m/s². The total energy cost from ground to the Clarke point is just over 13 kW-hr per kg lifted, which means $100 a ticket at today's energy prices, minus savings for energy generated by the 'down' cars, plus (rather large) financing charges on the capital investment.

Next come strength-of-materials considerations. We need a material with the highest possible (breaking strength)/(density), which is a tough sell, because Kevlar, good piano wire, and nearly everything else has essentially the same optimum value for this parameter. They all have breaking strengths of a 'few' billion Pa, and a density of a 'few' thousand kg/m³, where 'few' is the same number in both cases. The strongest high-tensile materials are the heaviest, by and large. Exotic materials like spun sapphire or diamond do better on the micron scale, and buckytubes get close to the theoretical limit (the strength of the chemical bonds themselves). In principle, such materials should be anywhere from 40 to 120 times stronger than the optimal value above, which I shall call '1x piano wire'. But Griffith theory teaches us that the length of the 'critical' crack (one that releases enough energy to drive its own spontaneous propagation) goes down as 1/(stress)². So even if exotic materials can be machined in gigaton lots, we may find that they are unusable at the huge stresses we need. The first woodpecker that comes along may bring the whole thing down if the critical crack is a few microns long.

But let's assume we can cope with this issue, if necessary with nanobot inspectors checking for micro-cracks, or simply a sheath of unstressed material around the structural members. The tension is essentially zero at the bottom: if we wanted we could leave the cable hanging loose a foot from the ground. (We want some tension there, of course, when we build an actual elevator, or the dynamic oscillations will kill us.) At the Clarke point, where the stress is largest, the stress depends on the weight of the tower below, which depends on the strength of the material. It's like rocketry, ironically enough: the 'fuel' for the upper stages is 'payload' cost for the lower ones. In this case, of course, it's upside-down: we have to keep the lower part of the tower as light as we dare, so that the upper part doesn't have to be exponentially heavy. And a high-tensile steel tower, like a rocket powered by Wisconsin butter (happy now, Senator Proxmire?), just doesn't have enough juice.

Assuming each wire has to take a thousand tonnes of tension at the bottom (add wires as needed, depending on what you want to send up the tower...), we get a minimum thickness profile like this:

These have no margin for error, so in reality you would need a material 20 or 30 times stronger than the best steel wire before you could even think about the 6x scenario. And the figure I used for 'ordinary' piano wire is itself way out at the bleeding edge of 1990s metallurgy; we can't produce such wire in suspension-bridge lots yet. The 6x version is already at the point of exponentially diminishing returns: we need a 5-foot-thick steel cable at the top to run a two-inch cable to ground. For 1x piano wire, a cable fifty feet in radius will carry a thousand tonnes only down to about 11000 km, and will have to taper to dental floss at 0 km.

In short, the prospects look pretty bleak: we need a smaller planet, or one with a faster rotation rate, in order to do this with any known engineering materials. The mass of my kT-load cable is 1.4 MT for 15x, 2.9 MT for 10x, 8 MT for 8x, 25 MT for 7x, 95 MT for 6x, 580 MT for 5x, and over 500 GT (!) for 3x---that's right, lowering the breaking strength from 5x to 3x multiplies the weight of cable a thousandfold. I don't even have the heart to calculate 1x; the cable would weigh as much as a moon. For real fantasy, at 100x, the cable only weighs 6 kT, a few times its 'payload', and we are finally in the regime of conventional engineering, where supporting the structure's own weight isn't the sole concern.

If we want the upper end of the elevator to sling payloads into deep space, we need to run it up about 11000 km beyond the Clarke point, but when you consider the cost of construction, it might be simpler to just take a conventional rocket engine up the elevator in pieces, and launch it from geosynchronous orbit. Other arrangements are possible: Robert Forward has done some interesting calculations involving 'skyhooks' that don't quite touch the ground. You get a considerable advantage, because it's only the bottom thousand km that are under anything like one gee. Also, you avoid weather. You can land on the bottom platform of such a skyhook with a very low-tech suborbital rocket---a V2 or a Scud-B gets that high. Even more interesting is the idea of lowering the platform from a non-geosync orbit, thus greatly reducing the weight of cable, and spinning the cable end-over-end at a speed that leaves the lower end almost stationary each time it swings into the earth's atmosphere. If the rotation period and the orbital period are commensurate with each other and with 24 hours, you can have regularly scheduled pickup points at fixed (ocean!) locations. Many such solutions are within reach of 2x or even 1x cable, but again you must allow a (5-fold?) safety factor before you apply for a permit.