Journal of The British Interplanetary Society, Vol
52,
pp. 175-180, 1999.
GEOFFREY A. LANDIS
Ohio Aerospace Institute, NASA Glenn
Research Center 302-1, 21000 Brookpark Road, Cleveland, OH 44135, USA.
E-mail: geoffrey.landis@lerc.nasa.gov
CRAIG CAFARELLI
Worcester Polytechnic Institute, Worchester, MA 01549, USA.
* Paper IAF-95-V.4.07, presented at the 46th International Astronautics Federation Congress, Oslo Norway, 2-6 October 1995.
In many ways, a tower is more practical than a tether system. A tower can be built up incrementally from the surface of the Earth, allowing materials to be fabricated on the ground and emplaced without launch costs. A tower is also less susceptible to damage from space debris in critical areas. This paper re-examines the feasibility of a Tsiolkovski tower in view of modern composite materials.
A difficulty of a tower system, of course, is that for towers which are many times longer than their thickness, failure is by buckling and not by simple compressive failure. It is assumed that an active stabilisation will be employed for any such system and that this failure mode can be eliminated since active stabilisation is now a well-developed technology.
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(1)
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The physical meaning of Lc is the length at which a member of constant cross-section will fail under its own load, under a uniform force of one gravity. The critical length for a material can be equally well defined for tension as well as for compression. The compression critical length will be different from the tension critical length, since the ultimate strength is different. In general, compression crjti cal strengths tend to be lower than tension critical strength.
The same basic equation applies for analysis of a compression ("tower") structure or a tension ("tether") system. If the tower is tapered so that constant stress is maintained throughout, the area taper ratio at point L is:
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(2)
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where Re is the radius of the Earth and w the angular rotation rate of the Earth. The total mass can be found by integrating the taper ratio. This is done using the computer program Mathematica.
Figure 1 shows a sample case, showing the mass as a function of length for a tether (extending down from geosynchronous orbit) and for a tower (extending upward from the surface of the Earth). Units on the vertical scale are the log10 of the mass. The mass of the tower grows very rapidly with length for the first few thousand kilometres, since it is deep in the Earth's gravitational field. The growth then flattens out, as the remainder of the length is at a lower gravitational field. For a tether extending from GEO, the opposite is true. The tether mass grows only slowly with length for lengths of tens of thousands of kilometres. Only at the last few thousand kilometres, when the tether extends very close to the surface of the Earth, does the mass start rising extremely rapidly. At the full length from surface to GEO, the tether mass is actually higher than the tower mass. Most of the mass increase is in the last thousand kilometres.
Fig. 1 Shows the relationship between tower (topmost curve) and tether (lower curve) mass as a function of length. Lc is assumed to be 150 km and payload mass is assumed to be 20 tons.
By integrating eq. (2). the total mass of a
tether (down ward deployed from GEO in tension) or a tower (upward constructed
in compression) system can be calculated if the mass to be carried is known.
The tower or tether mass is proportional to the required payload, hence
the mass is best expressed as a mass ratio, tether (or tower) mass to payload
mass. This is a very strong function of the characteristic length of the
material used to construct the tower. This is shown in fig.
2, for a system extending from the surface to geosynchronous orbit,
where the vertical axis is the log (base 10) of the payload ratio.
Fig. 2 Tether or tower mass/payload mass as a function of Lc, for an Earth to GEO tower ("Tsiolkovski tower"), tower curve; and for an Earth to GEO tether ("synchronous skyhook"), upper curve.
Surprisingly, for the same value of Lc, the Tsiolkovski tower to geosynchronous orbit has a lower mass than the comparable tensile skyhook. This is because both the tower and the tether taper most rapidly near the Earth's surface. Thus, the greatest length of the tether is at its highest thickness, while the greatest length of a tower is at its lowest thickness. Unfortunately, this advantage is mostly washed out by the fact that compression strengths are not, in general, as high as tensile strengths.
While these whiskers have only been made on a small scale, they could be theoretically woven together to make one long fibre. The capability to weave these thin whiskers together is not a near-future process. The materials from which the whiskers are made are widely available but, unfortunately, have very weak characteristics on their own. For example, while graphite whiskers have a Lc of 1239 km. commercially available Graphite has a Lc of only 16.8 km.
Another material type with promise for far future tether materials is
the glass family. While the best available chemically-treated glass has
a Lc of 58.5 km, the theoretical
maximum values for glass are much higher. Theoretical values show that
glass fibres have a potential ultimate tensile strength of 27.6 GPa. This
would give glass a characteristic length of 1170.2 km.
It is possible to do this by making the tower in the form of a hollow shell and then pressurising the interior of the shell. (It is interesting to note that G. David Brin introduced the concept of a pressurised tower in the book Sundiver, albeit with considerably lower heights proposed than the heights of towers considered here [9]. Presumably his reasoning was similar to the reasoning here.)
As noted in the Errata, the mass of pressurizing gas required in this case makes the approach infeasible for towers of the heights being considered here.
A 2240 km tall tower is analysed as a sample case, sized to support a payload at the top of the tower of 22.8 tons, i.e., equal to the space shuttle payload.
The first case to consider is the case of a tower built of compression material. Here it is assumed that the material is boron/epoxy, the best available material. If the tower were a single solid rod, a diameter of only 1 centimetre is required to support the weight at the top of the tower. The required taper ratio from the surface to the top is 700,000. The diameter of the base of the tower would then be 9 meters. (In actual design. the tower would probably not be solid but a hollow shell, and the actual diameter would thus be larger.) The mass of the tower is 17 million tons. For comparison, the mass of the Empire State Building is 356,000 tons. The volume of material required is 105,000 m3.
The second case examined is a pressurised compression tower constructed of PBO fibre with an internal pressure of 1 atmosphere.
ERRATUM: This calculation, as presented in the original version of the paper, was in error, and is deleted in this version.
The utility of towers of considerably lower height (15 km) as elements of space transportation systems are considered elsewhere [10].
An ultimate fuel-free access to orbit would be a skyhook connecting the surface to geosynchronous orbit. For comparable values of materials strength, it is actually easier to do this with a tower than with a tether. However, by far the most efficient way to do this is to use both. A tower of height 3000 km, connecting to a tether extending the rest of the way to geosynchronous orbit, could be significantly less massive than a tensile skyhook alone, built with existing fibres.
The mass of a skyhook is often cited as making impossible. It is instructive to compare this to the mass of other engineering artifacts. The mass of the skyhook system consisting of a tower connected to the GEO tether is an order of magnitude greater than that of the Empire State Building, 365,000 tons. The mass is somewhat more than that of a recent Norwegian North-Sea oil drilling platform, 1 million tons, but somewhat less than the 5 million tons of the Great Pyramid of Giza. On the other hand, other engineering artifacts of the transportation infrastructure considerably dwarf the size and the mass of such a skyhook. For example, the American Interstate highway system extends over a total length on the order 100,000 km (considerably more than the distance from the surface to GEO), with a mass of several thousand million tons, and was initially put into place in roughly a decade. A skyhook transport system could be constructed with a similar effort.
Acknowledgement: I would like to acknowledge David M. Palmer, who pointed out the error in the analysis of gas mass for the pressurized tower.