by Sam Dinkin
Monday, July 23, 2007

In 1975, Jerome Pearson wrote a great paper on space elevators that became the basis for excellent work by Brad Edwards including his 2002 book with Eric Westling, The Space Elevator: A Revolutionary Earth-to-Space Transportation System. For decades elevator enthusiasts have had their hopes pinned on carbon nanotubes or “graphite whiskers” as they were called in Pearson’s era.

Pearson had a chart (Figure 2 on p. 789) indicating that diamonds have a “characteristic height” of over 3,000 kilometers for space elevator construction. This suggests that a 3,000-kilometer untapered diamond cable can support its own weight hanging at 1g. Pearson calculated that integrating the lower gravity along the cable to geosynchronous orbit results in a cable that is strong enough to hold its own weight from only “4900 km high in a uniform one-g field”. That suggests that a taper ratio of three would be conservative with the maximum cross section at synchronous altitude.

Modern techniques of growing diamond crystals allow the orientation of the crystal to be stretched along the direction of the highest tensile strength, which can be seven times as large as along other directions. While diamonds are not as light as carbon nanotubes so they do not have as large a characteristic height at their theoretical maximum strength, that’s not a problem because diamonds can handle the load in theory.

Diamonds have a big advantage over carbon nanotubes. They are currently being produced in commercial quantities. Chemical vapor deposition (CVD) is being used to flood the jewelry market with extremely high quality diamonds at 10-30 percent below wholesale prices of natural diamonds. Substrates as wide as 15 centimeters have been reported as a base for diamond growth. That’s hundreds of times the area that’s needed for a space elevator cable.

For now, the retail market price for CVD diamonds is about $25,000 per gram. The United States Geological Survey last year predicted that the wholesale price of 2-gram (10-carat) diamonds can be reduced to $25 per gram. This is credible. The total world market for diamonds is about 125,000 kilograms. The average cost of industrial diamonds in the US, which produces half of the world’s industrial diamonds by weight, is $5,000 per kilogram for 50,000 kilograms or about 20 percent of anticipated wholesale CVD prices. By growing the CVD installed base, prices of CVD diamonds can indeed be driven down to cost, especially with some assurance that they won’t be sold to jewelry customers.

At $25,000 per kilogram, an 18-metric-ton starter space elevator would cost $450 million in materials cost to manufacture and would double the money demand for industrial diamonds from the US that year.

At $25,000 per kilogram, it does not make sense to build an elevator that stretches 150,000 kilometers like Pearson’s or 100,000 kilometers like Edwards’. The elevator should be 50,000 kilometers long or even less. An economic equation for how long the cable should be versus how heavy and close-in the ballast should be can be derived as follows. If your ballast is hanging at 1 g, suppose you need ballast that weighs 1,000 kilograms. If you reduce the length of the cable above geosynchronous orbit so that the ballast is being pulled out by centripetal force at 0.5 g, you would need about twice as much ballast plus the weight of the cable that would have been strung between the 1 g point and the 0.5 g point. For the 1/3 g point, you would need roughly three tmes the ballast and save the cable distance to drop 1/6 g. For the 1/4 g point, you would need roughly four times the ballast and save the distance to drop 1/12 g. The savings in cable length falls at roughly approaches 1/n(n–1) where n is the multiple of ballast from what would be needed if the ballast hung at 1 g. So if cable costs $25,500 per kilogram delivered and ballast costs $500 per kilogram to go up the elevator, then it makes sense to have a very short distance above geosynchronous orbit with a large ballast.

As a back of the envelope calculation, consider the economics of adding a second cable of similar lift ability to an existing starter cable with cable mass averaging 0.25 kg/km up to geosynchronous orbit (about nine metric tons), then extend the second cable out to about where the tip experiences lunar gravity, then have a ballast that weighs about six times what it would on the end of a longer cable where the ballast would experience 1 g. So if the nine metric tons of cable exerts about the same as three tons entirely concentrated at the 1g point (assuming a taper of 3-to-1 with no force at geosynchronous orbit), we would need about 18 tons of ballast, but it would cost $9 million delivered versus $153 million to build and deliver an extra six tons of cable. The extra delivery cost is well worth the savings in manufacturing cost. Going to the 1/12 g point, we could save another $37.5 million in manufacturing cost and would spend another $9 million in delivery cost. Cost minimization is the point above geosynchronous orbit experiencing around six percent of Earth’s gravity.

Cost minimization is even closer to geosynchronous orbit if there is excess demand for additional mass to be anchored at the top of the cable. If a space hotel wants to locate there and is willing to pay the $500 per kilogram freight and wants to be one million kg ($500 million delivery cost) then the ballast is essentially free during the early stages of elevator development and the cable should stretch out to less than the 1/20 g point. If the cost per kilogram of delivery is only $10 per kilogram instead of $500 per kilogram, then again, it’s a lot cheaper to have the ballast closer in.

Having a shorter cable would reduce the maximum velocity of payload launched from the end of the elevator, but most of what is going to go up the elevator at first is more elevator and ballast, geosynchronous orbit payloads, and other payloads that don’t require the extra length. With the ballast so close to geosynchronous orbit, a case can be made to have a constant taper and simply weld new cable on to the end of the elevator and take up slack at the bottom of the elevator. No welding would need to be done in transit, only at the terminal. Old cable at the Earth end of the elevator could be retired if damaged or recycled by taking it back up to the top where it could become one-third of the big end of the elevator. In this fashion, quality assurance of manufacturing could be carefully controlled and occur mostly on Earth with only welding of new cable lengths occurring slightly above geosynchronous orbit.

Cutting the price of cargo delivered to geosynchronous orbit is worth billions of dollars a year. A space elevator is indeed worth its weight in diamonds.