Vectors (24 page)

Against our requirement of a support length of 4,940 kms, how well do the substances that we have available today measure up?

Not too well. Now we see why no one has yet built a Beanstalk. Table 1 shows the strengths of currently available materials, their densities, and their support lengths. (The physical data that I am using here is drawn, wherever possible, from the "Handbook of Chemistry and Physics," 57th Edition. It is one of the most widely available reference texts and should be in any reasonable library.)

TABLE 1
STRENGTH OF MATERIALS

Not surprisingly, we won't be trying to make a Beanstalk support cable from lead. As we can see from the table, even the best steel wire that we can find has a support length only one hundredth of what we need. The last entry in the table, Fictionite, would be perfect but for one drawback: it doesn't exist yet. The strongest materials that we have today, graphite and silicon carbide whiskers, still fall badly short of our requirements (for Earth, that is. A Mars Beanstalk has a minimum support length of only 973 kms. We could make one of those nicely using graphite whiskers).

Does this mean that we have a hopeless situation? It depends what confidence you have in the advance of technology. Table 2 lists the strength of materials that have been available at different dates in human history. There is some inevitable arbitrariness in making a table like this, since no one really knows when the Hittites began to smelt iron, and there must have been poor control of times, temperatures and purity of raw materials in the Bronze Age and early Iron Age. All these factors have a big effect on the tensile strength of the products.

It is tempting to try and fit some kind of function to the values in the table, and see when we will have a material available with a support length of 5,000 kms. or better. It is also very dangerous to even think of such a thing. For example, consider a fit to the data of the form: Strength = B/(t - T), where B and T are to be determined by the data, and t is time in years before the year 2000 A.D. This fits the data fairly well if we choose B = 525,000 and T = 17.5 years. Unfortunately, such a form becomes infinite when t = T. If we were to believe such a fit, we would expect to have infinitely strong materials available to us some time in 1982!

TABLE 2
PROGRESS IN STRENGTH OF MATERIALS
 
AS A FUNCTION OF TIME

Note:
Years given indicate the dates when the materials could first be reliably produced in production quantities.

Not surprisingly, extrapolation of a trend without using physical models can lead us to ridiculous results. A much more plausible way of predicting the potential strength of materials is available to us, based on the known structure of the atom. In chemical reactions, only the outermost electrons of the atom participate, and it is the coupling of these outer electrons that decides the strength of chemical bonds. These bonds in turn set bounds on the possible strength of a material. Thus, so far as we are concerned the nucleus of the atom—which is where almost all the atomic mass resides—contributes nothing: strength of coupling, and hence material strength, comes only from those outer electrons.

In Table 3 we give the strengths of the chemical bonds for different pairs of atoms. These strengths, divided by the molecular weight of the appropriate element pair, decide the ultimate strength-to-weight ratio for a material entirely composed of that pair of elements. The final column of the table shows the support length that this strength-to-weight ratio implies, using the carbon-carbon bond of the graphite whisker as the reference case.

TABLE 3
POTENTIAL STRENGTH OF MATERIALS BASED
 
ON THE STRENGTH OF CHEMICAL BONDS

* Some of these element pairs do not exist as stable molecules, but can exist in a crystal lattice structure.
** We are using the support length of the graphite whisker as the standard of strength provided by the chemical bonds.

Examining the Table, we see that the hydrogen-hydrogen bond has by far the greatest potential strength. In this bond, every electron participates in the bonding process (each atom has only one!) and the hydrogen nucleus contains no neutrons, which offer added weight without adding anything to the possible strength. A substance that consisted of pure solid hydrogen could in principle have a support length of more than 9,000 kilometers—very similar to the Fictionite of Table 1.

Even this strength is very modest if we are willing to look at a rather more exotic composition for our cables. Positronium is an 'atom' consisting of an electron and a positron. The positron takes the place of the usual proton in the hydrogen atom, but it has a far smaller mass. Positronium has been made in the laboratory, but it is unstable with a very short lifetime. If, however, positronium could be stabilized against decay, perhaps by the application of intense electromagnetic fields, then the resulting positronium-positronium bond should have a strength comparable with that of the hydrogen-hydrogen bond, and a far smaller molecular weight. It will have a support length of 16,700,000 kilometers—the taper of a Beanstalk made from such a material would be unmeasurably small. This would be true even for a Beanstalk on Jupiter, where the strength requirement is higher than for any other planet of the Solar System.

The positronium cable is likely to remain unavailable to us for some time yet. Even the solid hydrogen cable offers us the practical problem that we don't know how to build it. Rather than insisting on any particular material for our Beanstalk, it is safer and more reasonable to make a less specific statement: the strength of materials available to us has been increasing steadily throughout history, with the most striking advance coming in this century. It seems plausible to look for at least an increase of another order of magnitude in strength in the next hundred years. Such an advance in materials technology would make the construction of a Beanstalk quite feasible by the middle of the next century, at least from the point of view of strength of materials. It could come far sooner.

Something with the properties of Fictionite would do very nicely. The taper ratio would be only 1.6, and a Beanstalk that was one meter in diameter at the lower end and of circular cross-section could support a load of nearly sixteen million tons.

WHERE TO BUILD THE BEANSTALK.

We have talked about what we will make the Beanstalk out of, but we have not discussed where we will find those materials. The answer to such a question is provided when we look at
how
we will build it.

For several reasons, the 'Tower of Babel' technique—start here on Earth and just build upwards—is not the way to go. The structure would be in
compression,
not tension, all the way up to beyond geostationary altitude, and we picked our material for its tensile strength. Worse still, structures in compression can buckle, which is a form of mechanical failure that does not apply to materials under tension.

Clearly, we will somehow begin
at the top,
with materials that we find up there. But
where
at the top? This is worth thinking about in more detail.

To a first approximation, the Earth is a sphere and its external gravity field is the same as that of a point mass. To a good second approximation, it is an oblate spheroid, with symmetry about the axis of rotation (the polar axis). The third order approximation gets much messier. Not only does the Earth "wobble" a bit about its axis of rotation, but there are fine inhomogeneities in the internal structure that show up as 'gravity anomalies' in the external gravitational field. These gravity anomalies are the deviations of the field from that which would be produced by a regular spheroid of revolution.

The anomalies are small—only a couple of milli-gals—but they are important. (In geodesy, a
gal
is not something that a male geodesist would like to snuggle up to; it is a unit of acceleration, equal to 1 centimeter per second per second. A
milligal
is a thousandth of that. Earth's surface gravity is about 980 gals. If the Earth's gravity field were to change by one milligal, you would weigh differently by about one four-hundredth of an ounce. Even a change of a full gal—a thousand milligals—would not be noticed.)

If we look at these small gravity anomalies in the region of the orbit of a geostationary satellite, we find that they give rise to local maxima and minima of the gravitational potential. Satellites in such orbits tend to 'drift' to where the potential has its nearest local maximum, and to oscillate about such a position. For this particular location (35,770 kilometers up, in the plane of the equator) these are the stable points of the gravitational field. At first sight, this looks like the best place to start to build your Beanstalk. You could put your source of materials there, and begin to extrude load-bearing cable up and down simultaneously, so as to keep a balance between the gravitational and centrifugal forces on the whole cable. Doing this, you might expect to be able to keep the cable Earth-stationary, always over the same fixed point of the surface.

Unfortunately, the gravitational potential is not so well-behaved. The positions of the stable points, the places where the potential has its local maxima, depend on the distance from the center of the Earth.

As you begin to extrude cable upwards and downwards, parts of the cable will move into regions where they are no longer at a local maximum of the potential. There will then be a strong tendency for the cable to "walk." It will begin to move steadily around the equator (and off the equator!), adjusting its position to the
average
of the gravity potential maxima encountered at all heights where a piece of the cable is present.

Such behavior is—at the very least—an annoyance. It means that you must allow for such motion in the design and construction, and you must tether the cable at the ground end when you have finished.

Such a tether is not a bad thing. We shall see later that it is an essential part of the Beanstalk design if we want a usable structure, one that can carry cargo and people up and down it. However, you can't tether the Stalk until you have
finished
building it. So we have still not answered the question, where do you do that construction? Remember, the geostationary location is full of other satellites—the communications satellites sit out there, and some of the weather satellites. It would be intolerable for the Beanstalk, half-built, to come drifting along through their
lebensraum
until it was finally long enough to tether.

What other options do we have? Well, there is the "bootstrap" method. In this, you fabricate a very thin Beanstalk, tether it, and use that to stop your main Beanstalk from wandering about during the construction.

My own favorite is more ambitious than a construction from geostationary orbit. You build
all
your Beanstalk well away from Earth, out at L-4 or L-5. When you have it all done, you fly it down. You arrange your timing so that the lower end arrives at a pre-prepared landing and tether site on the equator at the same time as the upper end makes a rendezvous with a ballast weight, way out beyond geostationary height. Once the Beanstalk has been tethered, the problem of a stable position for the orbit is not serious—it merely means that the Stalk doesn't follow the exact local vertical on the way up, because it tries to adapt to the mean gravity gradient all the way along its length.