The Web Between the Worlds

Beanstalk basics

The scientific literature about beanstalks, in all its different versions (we’ll get to those later), has grown steadily over the past twenty years. Now there exist many varieties of proposed forms, for use in a variety of places, ranging from Earth to Mars to the Lagrange points of the Earth-Moon system. This book uses what I will term the “standard beanstalk,” a structure which extends from the surface of the Earth up into space, and stands in static equilibrium.

To understand how any beanstalk is possible, even in principle, we begin with a few facts of orbital mechanics. A spacecraft that circles the Earth around the equator, just high enough to avoid the main effects of atmospheric drag, makes a complete revolution in about an hour and a half. A spacecraft in a higher orbit takes longer, so for example if the spacecraft is 1,000 kilometers above the surface, it will take about 106 minutes for a complete revolution about the Earth.

If a spacecraft circles at a height of 35,770 kilometers above the Earth’s equator, its period of revolution will be 24 hours. Since the Earth takes 24 hours to rotate on its axis (I am ignoring the difference between sidereal and solar days), the spacecraft will seem always to hover over the same point on the equator. Such an orbit is said to be geostationary. A satellite in such an orbit does not seem to move relative to the Earth. It is clearly a splendid place for a communications satellite, since a ground antenna can point always to the same place in the sky; most of our communications satellites in fact inhabit such geostationary orbits.

A 24-hour circular orbit does not have to be geostationary. If the plane of its orbit is at an angle to the equator, it will be geosynchronous, with a 24-hour orbital period, but it will move up and down in latitude and oscillate in longitude during the course of one day. The class of geosynchronous orbits includes all geostationary orbits.

All geostationary orbits share the property that the gravitational and centrifugal forces on an orbiting object there are exactly equal. If by some means we could erect a long, thin pole vertically on the equator, stretching all the way to geostationary orbit and beyond, then every part of the pole below the height of 35,770 kilometers would feel a net downward force because it would be moving too slowly for centrifugal acceleration to balance gravitational acceleration. Similarly, every element of the pole higher than 35,770 kilometers would feel a net upward force, since these parts of the pole are traveling fast enough that centrifugal force exceeds gravitational pull.

The higher that a section of the pole is above geostationary height, the greater the total upward pull on it. So if we make the pole just the right length, the total downward pull from all parts of the pole below geostationary height will exactly balance the total upward pull from the parts above that height. The pole will then hang free in space, touching the Earth at the equator but not exerting any downward push on it.

How long does such a pole have to be? If we were to make it of uniform material along its length, and of uniform cross section, it would have to extend upward for 143,700 kilometers, in order for the upward and downward forces to balance exactly. This result does not depend on the cross-sectional area of the pole, nor on the material of which the pole is made. However, it is clear that in practice we should not make the pole of uniform cross section. The downward pull the pole must withstand is far greater up near geosynchronous height than it is near the surface of the Earth. At the higher points, the pole must support the weight of more than 35,000 kilometers of itself, whereas near Earth it supports only the weight hanging below it. Thus the logical design will be tapered, with the thickest part at geostationary altitude where the pull is greatest, and the thinnest part down at the surface of the Earth.

We now see that “pole” is a poor choice of word. The structure is being pulled, everywhere along its length, and all the forces at work on it are tensions. We ought to think of the structure as a cable, not a pole. It will be of the order of 144,000 kilometers long, and it will form the load-bearing cable of a giant elevator which we will use to send materials to orbit and back.

The structure will hang in static equilibrium, rotating with the Earth. It will be tethered at a point on the equator, and it will form a bridge to space that replaces the old ferry-boat rockets. It will revolutionize traffic between its end points, just as the Golden Gate Bridge and the Brooklyn Bridge have made travel between their end points a daily routine for hundreds of thousands of people.

That is the main concept. Now we have to worry about a number of “engineering details.”

Designing a beanstalk

A number of important questions need to be answered in the process of beanstalk design:

* What is the shape of the load-bearing cable?

* What materials should be used to make it?

* Where will we obtain those materials?

* How will we use the main cable to move loads up and down from Earth?

* Will a beanstalk be stable, against the gravitational pull of the Sun and Moon, against weather, and against natural events here on Earth?

* And finally, when might we be able to build a beanstalk?

The first question is the easiest to answer. The most efficient design is one in which the stress on the material, per unit area, is the same all the way along the beanstalk’s length. With such an assumption, it is a simple exercise in statics to derive an equation for the cross-sectional area of the cable as a function of distance from the center of the Earth.

The equation is: A(r) = A(R).exp(K.f(r/R).d/T.R)

where A(r) is the cross-sectional area of the cable at distance r from the center of the Earth, R is the radius of a geostationary orbit, K is Earth’s gravitational constant, d is the density of the material from which the cable is made, T is its tensile strength per unit area, and f is a function given by:

f(x) = 3/2 — l/x — x²/2

The equation for A(r) shows that the important parameter in beanstalk cable design is not simple tensile strength, but rather T/d, the strength-to-density ratio of the material. The substance from which we will build the beanstalk must be strong, but more than that it must be strong and light.

Second, the equation shows that the tapering shape of the cable is tremendously sensitive to the strength-to-density ratio of the material, and in fact depends exponentially upon it. To see the importance of this, let us define the taper factor as the cross-sectional area of the cable at geostationary height, divided by the cross-sectional area at the surface of the Earth. Suppose that we have some material with a taper factor of 10,000. Then a cable one square meter in area at the bottom would have to be 10,000 square meters in area at geostationary height.

Now suppose that we could double the strength-to-density ratio of the material. Then the taper factor would drop from 10,000 to 100. If we could somehow double the strength-to-density ratio again, the taper factor would reduce from 100 to 10. An infinitely strong material would need no taper at all.

It is clear that we must make the beanstalk of the strongest, lightest material that we can find. What is not obvious is whether any material will allow us to build a beanstalk with a reasonable taper factor. Before we can address that question, we need to know how strong the cable has to be.

The cable must be able to support the downward weight of 35,770 kilometers of itself, since that length hangs down below geostationary height. However, that weight is less than the weight of a similar length of cable down here on Earth, for two reasons. First, the downward gravitational force decreases as the square of the distance from the center of the Earth; and second, the upward centrifugal force increases linearly with that distance. Both these effects tend to decrease the tension that the cable must support. A straightforward calculation shows that the tension in a cable of constant cross-section will be equal to the weight of 4,940 kilometers of such a cable, here on Earth. This is in a sense a “worst case” calculation, since we know that the cable will not be of constant cross-section; rather, it will be designed to taper. However, the figure of 4,940 kilometers gives us a useful standard, in terms of which we can calibrate the strength of available materials. Also, we want to hang a transportation system onto the central load-bearing cable, so we need an added margin of strength for that.

Now let us compare our needs with what is available. Let us define the “support length” of a material as the length of itself that a cable of such a material will support, under one Earth gravity, before it breaks under its own weight.

The required support length is 4,940 kms. What are the support lengths of available materials?

TABLE 1 lists the support lengths for a number of different substances. It offers one good reason why

TABLE 1

Strength of materials

Material Density (gm/cc) | Tensile Strength (kgm/cm²) | Support Length (km)

Lead 11.4 | 200 | 0.18

Gold 19.3 | 1,400 | 0.73

Cast iron 7.8 | 3,500 | 4.5

Manganese steel 7.8 | 16,000 | 21.

Drawn steel wire 7.8 | 42,000 | 54.

KEVLAR^TM 1.4 | 28,000 | 200.

Silicon whisker 3.2 | 210,000 | 660.

Graphite whisker 2.0 | 210,000 | 1,050.

no one has yet built a beanstalk. The strongest steel wire is a hundred times too weak. The best candidate materials that we have today, silicon and graphite dislocation-free whiskers, fall short by a factor of five.

This is no cause for despair. The strength of available materials has increased throughout history, and we can almost certainly look for strength increases in the future. A new class of carbon compounds, the fullerenes, are highly stable and seem to offer the potential of enormous tensile strength.

We would like to know how much strength is reasonable, or even possible. We can set bounds on this by noting that the strength of any material ultimately depends on the bonding between the outer electrons of its atoms. The inner electrons, and the nucleus, where almost all the mass of the atom resides, contribute nothing. In particular, neutrons in the nucleus add mass, and they do nothing for bonding strength. We should therefore expect that materials with the best strength-to-density ratios will be made of the lightest elements.

TABLE 2

Potential strength of materials

Element pairs* | Molecular weight (kcal/mole) | Bond strength (kms) | Support length

Silicon-carbon 40 | 104 | 455

Carbon-carbon 24 | 145 | 1,050

Fluorine-hydrogen 20 | 136 | 1,190

Boron-hydrogen 11 | 81 | 1,278

Carbon-oxygen 28 | 257 | 1,610

Hydrogen-hydrogen 2 | 104 | 9,118

Muonium 2.22 | 21,528 | 1,700,000

Positronium 1/919 | 104 | 16,750,000

*Not all element pairs exist as stable molecules.

TABLE 2 makes it clear that this argument is correct. The strongest material by far would use a hydrogen-hydrogen bond. In such a case, each electron (there is only one in each hydrogen atom) contributes to the bond, and there are no neutrons to add wasted mass.

A solid hydrogen cable would do us quite nicely in beanstalk construction, with a support length about twice what we need. However, solid crystalline hydrogen is not available as a working material. Metallic hydrogen has been made, as a dense, crystalline solid at room temperature — but at half a million atmospheres of pressure.

It is tempting to introduce a little science fiction here, and speculate on a few materials that do not yet exist in stable, useful form. The last two items in TABLE 2 both fall into the category of Fictionite (also known as Unobtainium), materials we would love to have available but do not.

A muonium cable would be made of hydrogen in which the electrons in each atom have been replaced by muons. The muon is like an electron, but 207 times as massive, and the resulting atom will be 207 times as small, with correspondingly higher bonding strength. Unfortunately the muonium cable is not without its problems, quite apart from the difficulty of making it in solid form. The muon has a lifetime of only a millionth of a second; and because muons spend a good part of the time close to the proton of the muonium atom, there is a good probability of spontaneous proton-proton fusion.

Time to give up? Not necessarily. It is worth remembering that a free neutron, not forming part of an atom, decays to a proton and an electron with an average lifetime of twelve minutes. Within an atom, however, the neutron is stable for an indefinite period. We look to future science to provide means of stabilizing the muon, perhaps by binding it, as the neutron is bound, within some other structure or material.

Positronium takes the logical final step in getting rid of the wasted mass of the atomic nucleus completely. It replaces the proton of the hydrogen atom with a positron. Positronium has been made in the lab, but it too is highly unstable. It comes in two varieties, depending on spin alignments. Para-positronium decays in a tenth of a nanosecond. Ortho-positronium lasts a thousand times as long — a full tenth of a microsecond.

We are unlikely to have these materials available for some time. Fortunately, we don’t need them. A solid hydrogen cable will suffice to build a beanstalk. Its taper factor is 1.6, from geostationary height to the ground. A cable one centimeter in diameter at its lower end is still only 1.3 centimeters across at geosynchronous altitude. To give an idea just how long this thin cable must be, note that our one-centimeter wire will mass 30,000 tons. And it’s strong. Slender as it is, it will be able to lift payloads of 1,600 tons to orbit.

TABLE 3

Beanstalks around the solar system

Body | Radius of stationary satellite orbit* (kms) | Taper factor (hydrogen cable)

Mercury 239,731 | 1.09

Venus 1,540,746 | 1.72

Earth 42,145 | 1.64

Mars 20,435 | 1.10

Jupiter 159,058 | 842.00

Saturn 109,166 | 5.11

Uranus 60,415 | 2.90

Neptune 82,222 | 6.24

Pluto** 20,024 | 1.01

Luna 88,412 | 1.03

Callisto 63,679 | 1.02

Titan 72,540 | 1.03

* Orbit radius is planetary equatorial radius plus height of a stationary satellite.

** Pluto’s satellite, Charon, is in synchronous orbit. If so, a beanstalk directly connecting the two bodies is possible.

Beanstalks are much easier to build for some other planets. TABLE 3 shows what beanstalks look like around the solar system, assuming we use solid hydrogen as the construction material. As Regulo said, Mars is a snap and we could make a beanstalk there with materials available today. Kim Stanley Robinson included a Mars beanstalk in his Mars Trilogy, Red Mars, Green Mars, Blue Mars. My only objection is that he destroyed the stalk cataclysmically in Red Mars, and in so doing obliterated the town of Sheffield that stood at its tether point.

Building the beanstalk

We cannot build a beanstalk from the ground up. The structure would be in compression, rather than tension, and it would buckle under its own weight long before it reached geostationary height.

We build the beanstalk from the top down. In that way, by extruding cable simultaneously up and down from a production factory in geostationary orbit, we can preserve at all times the balance between outward and inward forces. We also make sure that all the forces we must deal with are tensions, not compressions.

The choice of location for production answers another question raised earlier: Where will we obtain the materials from which to make the beanstalk?

Clearly, it will be more economical to use materials that are already in space, rather than fly them up from Earth’s deep gravity well. There are two main alternatives for their source: the Moon, or an asteroid. My own preference by far is to use an asteroid. Every test shows the Moon to be almost devoid of water or any other ready source of hydrogen. Two of the common forms of asteroid are the carbonaceous and silicaceous types, and coincidentally carbon and silicon fibers are today’s strongest known materials. A small asteroid (a couple of kilometers across) contains enough of these elements to make a substantial beanstalk.

If the solid hydrogen cable proves to be the only acceptable answer, then we need to seek farther afield for construction materials. Hydrogen is readily available in the solar system, but not on small asteroids whose orbits bring them anywhere near the Earth. Their volatile materials have long since boiled off due to solar heating. However, if we look farther out, hydrogen as components of water and methane becomes plentiful. A comet, which is little more than a huge dirty snowball, would serve us very well to make a beanstalk; and quite a small comet, with a head a few kilometers across, is big enough.

We must tether the lower end of the beanstalk cable at the equator. As a fringe benefit of the system, if we send mass all the way to the end of the beanstalk, far beyond geostationary orbit, then we will also have a free launch system. A mass released from 100,000 kilometers out can be thrown to any part of the solar system. The energy for this is, incidentally, free. It is provided by the rotational energy of the Earth itself.

Using the beanstalk

A load-bearing cable is not a transportation system, any more than an isolated elevator cable is an elevator. To make the transportation system, several additional steps are needed. First, we strengthen the tether, down on Earth’s equator, so that it can support a pull of many thousands of tons without coming loose. Next we go out to the far end of the cable and hang a big ballast weight there. The ballast pulls outward, so the whole cable is under an added tension, balancing the pull of the ballast against the tether.

We are going to attach a superconducting drive train to the cable. This will employ linear synchronous motors to move payloads up and down the length of the beanstalk. These motors are well-established in both principles and practice, so we can use off-the-shelf fixtures — except that we will want about 100,000 kilometers of drive ladder, and will need appropriate construction facilities and abundant materials. Here we will find a use for an asteroid of different composition, one high in metallic ores.

The motors will drive cargo cars up and down the beanstalk. Passengers, too, if the traveler is willing to put up with a rather long journey. At a uniform travel speed of 300 kms an hour, a journey to synchronous orbit will take five days. Much slower than a rocket but a lot more restful, and with spectacular scenery, this trip may resemble a leisurely transatlantic crossing on one of the great ocean liners.

The added tension provided by the ballast is very important. Each time a payload is attached to the drive train, the upward force on the tether is reduced by the weight of the payload. However, provided that the payload weighs less than the outward pull of the ballast weight, the whole system is stable. If the payload weighed more than the ballast’s pull, we would be in trouble. The whole beanstalk would be dragged down towards the Earth.

There is another advantage to using a really massive ballast. It allows use of a shorter cable. If we hang a big ballast weight out at, say 80,000 kilometers, there is no need to extend cable beyond that point. Another modest-sized asteroid, say a kilometer across, will do nicely for ballast. It will mass up to a billion tons.

We have not mentioned the source of energy to power the whole system. That could be provided by a solar power satellite, but will more likely be a fusion plant, sitting on the beanstalk at a geostationary orbit location. Superconducting cables run the length of the beanstalk, and can if appropriate provide power to the ground as well as running the motors on the space elevator itself. Since any energy used in the drive train to take mass up the beanstalk can be recovered by making the same mass do work as it comes down, a remarkably efficient system is possible. And by using the beanstalk as a slingshot, we have the energy-free launch system for payloads going to destinations anywhere in the solar system.

Any engineering structure has vulnerabilities, and the beanstalk is no exception. It easily withstands the buffeting of winds, since its cross-sectional area is minute compared with its strength; and the perturbing forces introduced by the attraction of the Sun and Moon are not enough to cause trouble, provided that resonance effects on the structure are avoided in its design. Accidental severing of the cable by impact with an incoming meteorite would certainly be catastrophic, but again the small cross-section of the cable makes that a most unlikely event.

In fact, by far the most likely cause of danger is a man-made problem: sabotage. A bomb exploding halfway up the beanstalk would create unimaginable havoc in both the upper and lower sections of the structure. All security measures will be designed to prevent this.

Alternative forms of beanstalk

There are two pacing items that decide when we can we build a beanstalk: the availability of strong enough materials, and a substantial off-Earth manufacturing capability. However, the first of these applies only to the “basic beanstalk” used in this novel. We now consider some interesting alternatives which remove the need for super-strong materials. We will term these alternatives the rotating beanstalk and the dynamic beanstalk.

The rotating, or non-synchronous, beanstalk was suggested in 1977, by Hans Moravec. It is a shorter stalk, non-tethered, that moves around the Earth in low orbit and dips its ends into the Earth’s atmosphere and back a few times a revolution. The easiest way to visualize this rotating structure is to imagine that it rolls around the Earth’s equator, touching down like the spoke of a wheel, vertically, with no movement relative to the surface.

Payloads are attached to the ends of the stalk at the moment of closest approach to the ground. But you have to be quick. The end of the stalk comes in at about 1.4 gees, then whips up and away again at the same acceleration.

The great virtue of the rotating beanstalk is that it can be made with less strong materials, and it would be possible to construct one today using graphite whiskers in the main cable. The taper factor is about ten. There is, of course, no need to have such a rotating stalk in orbit around the Earth. It could be sitting in free space, and as such it would serve as a “momentum bank.” It can provide momentum to spacecraft and thus forms a handy method for transferring materials around the solar system.

The dynamic beanstalk, which I think of an “Indian Rope Trick” for reasons I will give later, is an even nicer concept than the rotating beanstalk. It is not clear who first had the idea. Marvin Minsky, Robert Forward, and John McCarthy all seem to have had a hand in it, and I think I did the first analysis of its stability.

The dynamic beanstalk works as follows.

Consider a continuous stream of objects, such as steel bullets, launched up the center of a long, evacuated vertical tube. Suppose that the initial speed of these bullets is very high, faster than Earth’s escape velocity. This could be arranged using an electromagnetic accelerator at and below ground level. Suppose also that the tube is surrounded by the coils of a linear induction motor, so that there is electromagnetic coupling between the motor’s coils and moving objects within the tube.

Now, as the bullets ascend they are slowed by gravity; however, they can be given additional slowing by electromagnetic coupling. When this is done, the rising bullets transfer upward momentum to the surrounding coils.

At the top of the long tube (it can be any length, but let us say that it runs to geosynchronous altitude) the bullets are slowed and brought to a halt. Then they are moved over to another evacuated tube, parallel to the first one, and allowed to drop down toward the surface. As they fall they are accelerated downward by another set of coils surrounding the tube. Again, the result is an upward transfer of momentum to the coils. At the bottom, the bullets are slowed, caught, given a large upward velocity, and moved back into the original tube to be fired upward again. We thus have a continuous stream of bullets, ascending and descending in a closed loop.

If we arrange the initial velocity and the rate of slowing of the bullets correctly, the upward force contributed by the bullets at any height can be made to match the total downward gravitational force at that height. The whole structure stands in dynamic equilibrium, and it has no need for any super-strong materials.

Note the word “dynamic.” This type of beanstalk requires a continuous stream of bullets, with no time out for repair or maintenance. This is in contrast to our basic “static beanstalk,” which can stand on its own in stable equilibrium, without requiring dynamic elements, or the rotating beanstalk, which will also continue to operate without requiring an engine.

One advantage of a dynamic beanstalk is that it can be made of any length. A prototype could stretch upwards a few hundred kilometers, or even just a few hundred meters. In any case, seen from the outside there is no indication as to what is holding the structure up; hence the “Indian Rope Trick” label. Such a beanstalk would still be most useful if it went all the way to geosynchronous orbit, since at that height materials raised with the beanstalk can be left in position without requiring an additional boost to hold a stable orbit; but it doesn’t have to be made that way.

It is tempting to rule out the dynamic beanstalk on “environmental” grounds. What if the drive were to fail, and the whole thing come crashing down from space?

And yet we are quite used to systems that must keep working successfully, or suffer catastrophic failure. Two hundred years ago, our ancestors would have been appalled at the idea of hundreds of tons of metal hanging above their towns, operated by an engine that had to operate perfectly or the whole thing would fall. Given the technology of the day, they would have been right to be afraid.

Yet we live with such a situation every day. We have aircraft flying over us all the time, but we seldom think about the possibility that one will come crashing down on top of us. We have faith in today’s technology. Our grandchildren will have faith in a much greater technology, whose failure rates will be unimaginably lower than today. Machines and structures that are seldom inspected now will be under continuous computer supervision, including smart sensors in all their key components.

In that future environment, static beanstalks, rotating beanstalks, and dynamic beanstalks, or some later invention that supersedes all of them, will be both technologically feasible and socially acceptable. I think we are closer to a dynamic beanstalk, today, than we were to successful space flight in 1900.