They were not all quotes, but many were. Here, for the curious, are their origins.
Chapter 1. “Praise, my soul, the King of Heaven, to His feet thy tribute bring.” — from the hymn beginning with these words, by Reginald Heber.
Chapter 3. “Go and catch a falling star…” — from the poem beginning with these words, by John Donne.
Chapter 4. “Busy old fool, unruly Sun…” — from the poem beginning with these words, by John Donne.
Chapter 5. “The light of other days…” — from the poem beginning, “Oft, in the still night, ere slumber’s chain has bound me…” by Thomas Moore.
Chapter 8. “To meet with Caliban…” — from William Shakespeare’s The Tempest.
Chapter 9. “Pluck from the memory a rooted sorrow, raze out the written troubles of the brain…” — from William Shakespeare’s Macbeth.
Chapter 11. “What seest thou else, in the dark backward and abysm of time?” — from William Shakespeare’s The Tempest.
Chapter 12. “…at the quiet limit of the world, a white-haired shadow roaming like a dream…” — from Tennyson’s “Tithonus.”
Chapter 15. “I do begin to have bloody thoughts…” — from William Shakespeare’s The Tempest.
Chapter 16. “Then I saw that there was a way to Hell, even from the gates of Heaven…” — from John Bunyan’s A Pilgrim’s Progress.
Chapter 18. “Cor contritum quasi cinis, gere curam mei finis.” — from the Dies Irae in the Latin Mass for the Dead; these lines are often translated as, “See like ashes my contrition, help me in my last condition.”
APPENDIX 2: Beanstalks in Fact and Fiction
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.
KEVLARTM 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 aroun
d 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.
The Web Between the Worlds Page 28
