Wizards, Aliens, and Starships: Physics and Math in Fantasy and Science Fiction

by Adler, Charles L.

  Governmental interest in high-orbital manufacturing stems in part from calculations on its economics. These suggest that a community in space could supply large amounts of energy to the Earth, and that a private, perhaps multinational investment in a first space habitat could be returned several times over in profits.

  —GERARD K. O’NEILL, THE HIGH FRONTIER

  O’Neill’s space stations were always meant to be economically feasible enterprises. When counting costs he assumed, as did many others at the time, that the dominant expense of putting people into space would be the fuel costs. He therefore studied a number of alternative propulsion systems such as railguns for launching materials from the Earth or Moon. However, fuel costs are not the dominant costs for space travel today. The dominant costs currently are infrastructure costs, which most science fiction writers tended to grossly underestimate when writing their books. I think that O’Neill probably fell into this trap as well; his $30 billion investment to construct the station seems low, even when measured in 1977 currency. Let’s estimate the cost of putting a station into space using numbers appropriate for today’s space program.

  Robert Heinlein’s novel Space Cadet featured Terra Station, a large, autonomous space colony in geosynchronous orbit around Earth. In the book, it mentions that the mass of the station is 600,000 tons, or roughly 6×108 kg [110, p. 25]. From chapter 2, the current cost of putting things into space is about $9,000/lb, or nearly $20,000/kg. Let’s assume we can lower this cost by an order of magnitude, or $2,000/kg. Then, the cost of putting the material to make such a station would be about $1.2 trillion.

  This seems about right. Eric Yam’s proposal for Asten estimated that the total cost of building it would be $500,000,000, or half a trillion dollars, or roughly a twentieth of the GDP of the United States [11]. This figure strikes me as a low estimate. If you look at Yam’s space station budget, several items are persistently underestimated, such as the construction processes (including mining the Moon for raw materials), which are estimated at only $22 billion. Be that as it may, the proposal is incredibly detailed, and I would certainly like to live on space station Asten if it ever gets built. But let’s play a hard-nosed bean counter who is asked to evaluate whether the station is worth investing in. An investment of $500 billion over 20 years at 5% interest means that ultimately $790 billion will need to be paid back to investors. Since there are 10,000 people aboard the station, each of them is responsible for generating $79 million in revenue these 20 years, or about $3.2 million per year per person. This isn’t impossible. A handful of the top tech companies have similar per capita revenues. Let’s look at Yam’s proposals for the uses of the station Asten, which are similar to ones that other space station designers have proposed:

  • Mining the moon for raw materials. This is the biggest proposed use for the station. The minimum energy required to transport 1 kg of materials from geosynchronous orbit to the Moon is 4 × 106 J, as compared to 6 × 107 J from Earth to the Moon. We therefore achieve a ten times reduction in energy costs by building this satellite—more, in fact, once the rocket equation comes into the mix. One question: do we need to mine the Moon? The reason given is to build more space stations and larger spaceships, which sounds suspiciously like a circular argument to me.

  • Crystal growing. This has been a mantra of space enthusiasts for the past twenty years: one can grow very large, very defect-free crystals for the computer industry in space. However, there are two problems with this proposal. First, because of improvements in crystal-growing technology on Earth, one can already grow very large, perfect crystals for the computer industry. Second, getting them from geosynchronous orbit down to Earth would cost a lot. Even if we assume something like the space elevator, the costs would still exceed $100/kg, and perhaps exceed $1,000/kg. Crystal growing is estimated as 9% of the space station’s GDP; this implies that it must return a profit of at least $90 billion over the 20-year investment period we assumed, or about $4.5 billion per year. This is comparable to the yearly profits of the entire U.S. semiconductor industry, which seems a stretch for one small space station.

  • Building large space telescopes and other space construction projects. Again, we can build and launch them from Earth without the enormous initial price tag.

  • Other stuff. Here the Asten proposal gets plain silly. There is a discussion of creating homogeneous mixtures, “perfect spheres,” biomedical research, and so forth, all of which can be done on Earth for a tiny fraction of the price tag. The author is repeating the NASA line that proponents of space station Freedom spouted in the early 1990s. They didn’t make scientific sense then, they don’t make scientific sense now.

  The only way to build such a habitat economically is to reduce the costs of transport into space by about two orders of magnitude, to a few hundred dollars per kilogram. Let’s take a look at one suggestion for doing this in the next chapter: the space elevator.

  NOTES

  1. The idea that water in the toilet spins the way it does because of this effect is an urban myth.

  2. I’m assuming that the effective cross-sectional area for light absorption for the station is a rectangle whose length is the length of the station and whose width is equal to its diameter. This takes into account the fact that light will be absorbed obliquely because of its overall shape.

  3. Asten won the 2009 Grand Prize for the NASA Space Settlement Design contest. This is a yearly contest open to high school and middle school students to design workable models for space stations. Although I am leery of whether Asten makes any financial sense, the basic physics and engineering underlying it appear to be sound.

  4. Another way to do this calculation is to use the fact that in the presence of enough sunlight and a sufficiently high CO2 concentration in the atmosphere, plants can produce a maximum of about 20 × 10−6 moles/m2/s of leaf area, and work from there [245, p. 139, fig. 6.4]. Using this, I estimate 178 m2 of leaf area per person needed to provide the oxygen, which is about three times larger than my back-of-the-envelope calculation. However, the moral is still the same: atmospheric oxygen needs will be provided amply by crops grown for food.

  CHAPTER EIGHT

  THE SPACE ELEVATOR

  “… Come, let us build a city and a tower whose top may reach to heaven …”

  —GENESIS 11:4

  8.1 ASCENDING INTO ORBIT

  It’s time for your summer vacation, but you’re bored with Aruba, Tokyo is so blasé, and even the wonders of Antarctica have begun to pall. So you get online and decide to book a trip to the Moon. The first leg is simple: a plane ride to Ecuador to the new Space Hub. Touching down in the airport near the Hub, you look in vain for rockets, but see only a large, circular building with what seems to be a thin ribbon reaching up as far as you can see into the sky. You go past immigration, get passports stamped to go to the R.I.L. (República Independiente de la Luna), and head into a large set of rooms the size of a train car in the shape of a ring. It has small sleeping compartments, a tiny cafeteria at the center, and an observation booth on the outside. You head over to the observation room and realize that while you were wandering around, you’re now a thousand feet above the ground and climbing steadily! Three days later, you step out into the spaceship that will take you to the Moon—in geosynchronous orbit 36,000 km above the surface of the Earth.

  The idea of a tower reaching into the heavens is as old as the Bible, and stands as a symbol of man’s ambition (and hubris) to this day. Could one build a tower high enough to reach the heavens? Konstantin Tsiolkovsky was perhaps the first person to investigate the physics of such a structure. In his work, Dreams of Earth and Sky, he writes,

  On the tower, as one ascends it, gravity decreases gradually … due to the removal from the planet’s center [and] also to the proportionally increasing centrifugal force … On climbing such a tower to a height of 34,000 versts [= 36,000 km] gravity will disappear, while still higher there will again appear a force that will increase proportionally
to removal from the critical point, but acting in the opposite direction [26].

  The idea is that if you could build a structure tall enough, by climbing up it you could reach a point at which the centrifugal force resulting from the rotation of the Earth itself perfectly balanced the force of gravity; any object released from that point would not fall to the ground but would remain in orbit over the surface of the Earth, orbiting once every 24 hours. Tsiolkovsky probably was illustrating the principles of orbital mechanics rather than proposing a real structure, but real engineering studies of such a tower began in the 1960s. Yuri Artutsanov in the USSR and John Issacs, Hugh Bradner, and George Backus in the United States came up with independent but similar elaborations on this basic idea and started to look at the means of engineering such a structure [27] [128]. An elevator to take you into space may seem crazy, but there are a lot of attractive features about it. To understand them, we need to delve into the idea of a geosynchronous orbit.

  8.2 THE PHYSICS OF GEOSYNCHRONOUS ORBITS

  If you have satellite TV, you may wonder where the satellites that beam down your programs are: how high are they above the Earth, and exactly where do they circle? These aren’t dumb questions; a lot of satellites circle Earth in a variety of orbits. In the last chapter we found that the shuttle goes into a low Earth orbit with an orbital speed of 7 km/s. At that speed, it will circle the Earth every 90 minutes.

  Any satellite in this same orbit will circle the Earth with exactly the same period, which makes it a very bad orbit for satellite TV. Such a satellite would move so fast that anyone watching it from the surface of the Earth would see it flash across the sky and vanish in a matter of minutes. The satellite receiver would be able to track it for only a minute or two every hour and a half, meaning rather intermittent coverage. We want a satellite that stays over the same spot of ground forever, meaning one that takes exactly 24 hours to rotate around the Earth.

  In chapter 6 we discussed the dynamics of the Space Shuttle orbit. A geosynchronous satellite behaves in exactly the same way as the shuttle does except that it has a higher orbit. As a reminder, the speed of the satellite in orbit around the Earth is given by the formula

  Note that the mass of the satellite doesn’t enter into this. What we really want is a relation between the period of the satellite and r; this relationship, which is known as Kepler’s third law, is one of the most useful in all of astronomy. If T is the period of the satellite, then

  Here, ME is the mass of the Earth. One must use the correct units in this formula, which (in the metric system) for T is seconds and for ME is kilograms. This gives an answer for r in meters. Using T = 24 hours = 86,400 s, ME = 5.9 × 1024 kg and G = 6.67 × 10−11 N/m2 kg2, we find r = 4.2 × 107 m = 42,000 km. This is from the center of the planet: it is approximately 35,600 km (22,000 miles) above the surface of the Earth. A satellite placed in an orbit at this distance will always stay in the same spot above the surface of the Earth as long as the orbit is around Earth’s equator.

  We can make the formula more straightforward if we work in nonstandard units. This is something I will do often in this book. It is sometimes awkward to always have to refer to very large numbers, like the mass of the Earth, or very small ones, like G, when doing these calculations. It is instead more useful to work in units where by the mass of the planet is expressed as a multiple of the mass of the Earth and time is measured in units of days, as these are pretty typical values for planetary masses and satellite periods. In our Solar System, the mass of the planets ranges from about one-tenth the mass of the Earth to about 300 times it. So we can rewrite the formula given above as

  That is, express the mass in units of the Earth, and the orbital period in units of days; the distance is then 42,000 km times the cube root of the mass times the cube root of the square of the period.

  8.3 WHAT IS A SPACE ELEVATOR, AND WHY WOULD WE WANT ONE?

  Imagine taking a satellite in geosynchronous orbit and streeeetching it out, being careful to balance the upper section and lower sections so that the center always stayed in geosynchronous orbit. If we were careful, we might be able to extend one section to the ground, where we would anchor it, while the top section extended far out into space. In essence, this is the space elevator, also called a “geosynchronous skyhook” by some writers, and a “beanstalk” by others. Tsiolkovsky called it a tower, but it’s really a cable hanging down from space. It doesn’t crash into Earth because the center of gravity of the structure is in geosynchronous orbit and is moving at exactly the right speed to stay there. It is a truly audacious idea, but people have taken it seriously enough to fund studies on its construction [76][77].

  It is a popular theme in science fiction. The first I read of such a structure was in the novel The Fountains of Paradise, by Arthur C. Clarke, in which the engineer Vannevar Morgan is engaged in building one [56]. The science is up-to-date: the structure is built from the carbon nanotubes discussed below. Morgan loses a fingertip to monomolecular filament built from these materials. It is a gripping story: in the climax, Morgan sacrifices himself to save several people stranded on the elevator. Charles Sheffield’s novel The Web between the Worlds also revolves around the concept, and David Gerrold’s novel Jumping Off the Planet involves a trip to a geosynchronous space station made using a space elevator [93][216]. The cartoonist Randall Munroe of the webcomic xkcd is fond of it as well.

  The reason for its existence can be stated in a nutshell—energy. In principle, it costs significantly less energy to bring a person or satellite into orbit by the space elevator than by rocket. To put a satellite into orbit using rockets, you need to get the rocket moving fast on launch—7 km/s to get it into low Earth orbit, and nearly 11 km/s for a geosynchronous orbit. This takes a lot of fuel and energy. Indeed, because of the rocket equation, there is a lot of energy wasted because of the need to carry the fuel along with the rocket. These needs disappear with the space elevator. In principle, you can ascend as slowly as you want because the Earth’s rotation supplies the kinetic energy needed. In fact, if the elevator extended upward above 46,000 km above the ground (52,000 km from the center of the Earth), a satellite launched from the top would escape Earth’s gravitational pull completely. We’ll calculate the energy needs for ascending the elevator later; they’re small compared to rocketry. But how do we build such a tall structure?

  8.4 WHY BUILDINGS STAND UP—OR FALL DOWN

  Buildings, bridges, and other structures come in two main types: ones that are supported in compression, that is, by being pushed on, and ones that are supported partly or mostly (but not entirely) in tension — being pulled on. Building a structure depends on knowing exactly how materials respond to tension and compression: some materials are much stronger in compression than in tension, while others, such as rope or rubber bands, can take a lot of tension but can’t be used at all in compression. Stone is a good example of a material that can stand a lot of compression but can’t take much tension; it tends to break when pulled on strongly. European cathedrals and the Egyptian and Mesoamerican pyramids are good examples of structures that are supported almost entirely by compression. Such buildings tend to be much wider than they are high so that they won’t bend when loaded from the sides by high wind or ground movements.

  Figure 8.1. Bending of a shelf loaded with books.

  In general, structures will be in both tension and compression simultaneously. Figure 8.1 shows how this works. Consider a bookshelf piled with heavy books. The shelf bows under the weight of the books. Because of the bending, the surface immediately under the books is compressed, while the opposite side of the shelf is stretched out. Along the center of the bookshelf runs a neutral plane in which the material is being neither stretched nor compressed. From this example, you can see that bending places a structure under both tension and compression. Wood is a good example of a material that is strong in both compression and tension, which is why it is such a popular building material for small structures.

  Even t
hough there are a lot of elements in tension in a skyscraper or a suspension bridge, ultimately they are supported by the ground, or in other words, in compression. All structures that rest on the ground must be supported at least partly by compression. The space elevator is different: it is basically a very long cable or rope. Since ropes have no strength in compression, it must be supported entirely by tension. Exactly how it supports itself is subtle, however.

  Imagine a section of the elevator that is below geosynchronous orbit—say, 200 km above the ground, the same as the shuttle orbit. As was explained in the last section, the lower the orbit, the faster a satellite must move to stay in orbit; the shuttle takes only 90 minutes to go around the Earth. However, this section of the elevator is moving more slowly than the orbital speed appropriate for that height. Left to itself, it “wants” In tension to fall to Earth. What prevents it from doing so is the section of the space elevator immediately above it pulling it up. Newton’s third law mandates that there must be a reaction force on that section pulling it down. Similarly, a section of the cable above geosynchronous orbit is moving faster than orbital speed at that height; it wants to fly away, and is held back by the section below it. Because of this, below geosynchronous orbit there is a force pulling down on the structure, while above it, there is a force pulling up. This is tension.

  The tension in this structure is enormous. At the end of this chapter I include a section on calculating parameters of the structure exactly, but we can do a back-of-the-envelope calculation to estimate how strong this structure has to be. If we imagined a skyhook holding a cable dangling down above the Earth’s surface, the force acting on the hook (which we will use as an approximation for the tension in the cable) is its “effective weight,” which is given by