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

by Adler, Charles L.

  The laws of mechanics as observed by a traveler on a small rotating spaceship or space station are so bizarre as to be cartoonlike. In the novel Orphans of the Sky, Robert Heinlein writes of a generation ship traveling between the stars, on a journey so long that its inhabitants forget that an outside universe exists [117]. One wonders what laws of nature that world’s Galileo or Newton would invent to describe it.

  Figure 7.2. The “Over-the-Shoulder” Trajectory.

  Figure 7.3. The “Wiley Coyote” Trajectory.

  Figure 7.4. The “Etch-a-Sketch” Trajectory.

  7.5 THE LAGRANGE POINTS

  In previous chapters we discussed satellites in low Earth or geosynchronous orbit. These two orbits are common settings for space stations in science fiction novels, but there is a third setting that is at least as common as the other two: the L4 and L5 Lagrange points of the Earth-Moon system.

  So far we’ve discussed satellite orbits that are solutions to the two-body problem: orbits in which only the gravitational attraction of the satellite to the Earth is important. The Lagrange points are solutions to the far more difficult three-body problem, or problems involving the gravitational interactions of three separate masses. While such problems can be solved on the computer, there are almost no solutions that lead to periodic orbits as in two-body problems.

  The major exceptions are the Lagrange points, first discovered by the physicist Joseph-Louis Lagrange. These are solutions to the three-body problem under the restricted assumption that one of the three is much less massive than the other two, that is, that the orbit of the other two points can be described only by considering of their own masses and motions. The five Lagrange points are shown in figure 7.5.

  The points L1 through L3 are easily enough explained by considering the balance of forces acting on them. Essentially, the force of gravity due to the Earth balances out the gravitational attraction of the Moon and the “centrifugal” force resulting from the rotation of the satellite about the Earth (or, more precisely, the center of gravity of the Earth-Moon system). However, these points are unstable: objects placed in these points will tend to drift away from them over long periods of time. They are still useful, as the time periods can be large compared to the duration of any missions. For example, the proposed orbit of the James Webb Space Telescope (the successor to the Hubble Space Telescope) is at the Earth-Moon L2 point. We can approximately calculate the position of the L1 point by balancing the force of gravity from the Earth acting on the station with that of the Moon acting on the station:

  Figure 7.5. Lagrange Points of the Earth-Moon System.

  In this equation,

  • r is the distance from the center of the Earth to the station;

  • R is the distance from the center of the Earth to the center of the Moon (= 3.84 × 108 m);

  • M is the mass of the Earth (= 5.97 × 1024 kg);

  • m is the mass of the Moon (= 7.35 × 1022 kg); and

  • G is the universal gravitational constant (= 6.67 × 10−11 Nm2/kg2).

  One can solve this, to find

  Because the mass of the Earth is much bigger than the mass of the Moon, the L1 point will be located much closer to the Moon than the Earth—by my calculations, nine-tenths of the distance between Earth and Moon. This is not quite all, however: there is also a centrifugal force, owing to the rotation of the space station around the center of the Earth, that seems to push the station away from the center of the Earth. (It’s the same pseudo force that we use to generate “gravity” on the space station.) Because it tends to offset the force of Earth’s gravity (for this point) its effect is to move the position of the L1 point about 5% closer to Earth, at least by my calculations. The effect of the centrifugal force is relatively small for the L1 point but must be taken into account to get the position of the L2 point. The L2 point is located beyond the Moon, so the centrifugal force must balance out the gravitational attraction of both Earth and Moon. The L3 point is located on the opposite side of the Moon orbiting the Earth: again, one must balance the “centrifugal force” against the attraction of both Earth and Moon. There are a number of science fiction stories involving “counter-Earths” circling the Sun exactly opposite Earth (essentially at the Earth-Sun L3 point); we wouldn’t see it because the Sun would block our view of it all the time. I think that this is the position of the Bizarro world of the Superman comics but cannot find a reference for this.

  The L4 and L5 points are respectively 60 degrees ahead of and behind the Moon in its orbit around the Earth. They are equidistant from Earth and Moon; O’Neill proposed them as useful places for building these colonies [189]. The reason why an object that is placed there will continue to orbit is subtle; the force vector due to the resultant forces acting on it from the Moon and the Earth points toward the center of mass of the system. If the satellite’s rotational speed matches the rotational speed of the Moon around the center of mass, the satellite will stay in the orbit [232].

  Orbits at the L4 and L5 points are quasi-stable: if the station is pushed out of the orbit, it tends to orbit around the Lagrange point rather than be pushed away from it, as with L1 through L3 [232]. However, these points are far from both Earth and Moon, and hence energetically expensive to reach and to place satellites in. Why choose them?

  His original papers show that O’Neill chose these points because he was thinking big. Remember that he projected that by the middle of the twenty-first century, more people would be living off-Earth than on it. He therefore needed room for extremely large structures, further envisioning that a network of these structures could one day be transformed into a Dyson shell, a truly huge structure surrounding the Sun that a very advanced civilization could use to harvest all the Sun’s available power [188]. O’Neill also considered that mining the Moon for raw materials would be needed to build these colonies; the Lagrange points are in some sense a compromise position between building a colony in orbit around the Earth or building one in orbit around the Moon. He pointed out, sensibly, that because escape velocity on the Moon is only about one-third the escape velocity from the Earth, the energy costs of lifting objects to the Lagrange points from the Moon is much less expensive than from the Earth. However, this requires the ability to mine the moon for raw materials, something he assumed would be possible relatively quickly post-Apollo but has not yet happened (if it ever will).

  7.6 OFF-EARTH ECOLOGY AND ENERGY ISSUES

  Now that we have supplied our colony with gravity and found a place in the heavens for it, what other needs do its inhabitants have? Food, air, an ambient temperature between about 0°C and 30°C: these can serve as basic human needs. However, we can’t go about supplying them in the same way that we supply these needs for the Space Shuttle astronauts; this is supposed to be a self-sustaining colony, ready for longterm human habitation, not a trip lasting a few days to a week. This implies that the station needs some sort of ecology: the people on board need to grow their own food and recycle waste products so that they can live as independently of Earth as possible. Otherwise there is no way that the station can become economically viable, given the cost of transporting food and other goods into space. Building such an artificial environment will not be easy, especially when one considers the size and interdependence of ecosystems on spaceship Earth.

  7.6.1 Food

  An adult male with a mass of about 80 kg needs approximately 2,000 kcal of food energy to sustain him for one day; an adult female with a mass of about 70 kg needs about 1,800 kcal. (The kilocalorie, or food calorie, is a unit of energy equal to 4,200 J.) Other nutritional needs aside, this can be supplied in a variety of ways: meat is the most energy-dense food, with a caloric content of about 3,000 kcal/kg, while fruits and vegetables are less energy dense (about 600 kcal/kg). Meat, however, is very expensive in energy terms to produce, as one must raise the food animals on other food, usually some type of grain. It is about ten times more expensive to grow the grain to feed the animal, which is then slaughtered to feed th
e people, than it is to feed the people directly on the grain. Agricultural land on a space station will be some of the most expensive real estate anywhere, leading me to believe that very little of it will be used to raise beef, pork, or chicken. Assuming that the inhabitants are essentially vegetarian, they will need about 3 kg of grain and vegetables to eat per person per day, or about 30,000 kg/day for a station with a population of 10,000. This works out to about 107 kg of food per year.

  At this point the choices available to the science fiction writer branch out exponentially. Will the food be grown traditionally, meaning that somehow the station must have room for acres of soil spread out and access to sunlight, or will it be grown hydroponically? Will plants be fertilized naturally or artificially? If artificially, how is the fertilizer to be produced? As an aside, the creation of ammonia via the Haber-Bosch process is very expensive, energetically speaking: modern-day agriculture depends strongly on this process, which is estimated to use some 1%–2% of all the energy used in the world [105]. Heinlein’s novel Farmer in the Sky explores some of these issues. The novel is set on a colony on Ganymede rather than a space station. It goes into the ecology of farming in detail. The farming is done in a very low-tech manner, with topsoil being generated from Ganymedan rocks fertilized with bacteria and other organisms brought from Earth, mostly by hand and stoop labor [111]. This is an odd decision by Heinlein given the level of technology exhibited in the book, which includes limitless energy generation through the conversion of matter to antimatter.

  One way to clear out these complications is by considering the conservation of energy and the idea that in growing and eating food, we are transforming energy from one form to another. At every step of the way, some fraction of the energy is lost. The prospective author must work through all the details, but let’s make some assumptions and see what we get.

  • First, let’s assume that the space station is in orbit around the Earth or the Moon, or is in one of the Lagrange points. This is probably the most common assumption in science fiction, but it lets us assume that the flux of sunlight reaching the station is about the same as that reaching Earth from the Sun.

  • Second, let’s assume we have a large station to feed, and one that is somehow self-sustaining. Whether it is perfectly self-sustaining is another issue, but let’s assume that most of the food consumed on the station is somehow produced on the station. Further, we’ll assume that the station has some 10,000 people on board, again about the size of the station in Babylon 5.

  • Third, let’s assume that the energy to grow the crops is coming from the Sun. Again, we need not assume this, but this is the energy source of agriculture on Earth, and is free. We need not assume that we are shining the Sun directly on the growing plants; we could use solar power to generate electricity to run lights for hydroponic tanks. The solar constant above Earth’s atmosphere is 1,360 W/m2, meaning that every square meter of the (projected area) of the station (turned toward the Sun) is illuminated by 1,360 J of energy every second.

  • Finally, let’s assume a food consumption of 2,500 kcal/day per person. This is an energy usage rate; in metric units, it works out to about 120 W, or a relatively bright light bulb, per person. This means that 120 W per person × 104 people = 1.2 × 106 W must be available in the form of food energy.

  Inefficiencies crop up in at least two places in this chain:

  • The conversion of solar power into energy stored in a plant; and

  • The fact that only a small portion of a plant is edible.

  Plants are very roughly 1% efficient in the conversion of solar power into energy stored as biomass via photosynthesis. There are many reasons for this overall efficiency; the maximum possible efficiency is around 13% conversion from basic energetic grounds, but there are larger overall inefficiencies [245]. The second point is that only a small portion of the plant is edible; corn kernels or wheat endosperm make up only a small fraction of the overall mass of the plant, say another 1%. The overall efficiency is therefore only about (10−2)2 = 10−4. Therefore, the total power needed to supply the station for its agricultural needs is 1.2 × 106 W × 104 ≈ 1010 W. Using the solar constant, the total illuminated area therefore is

  It is relatively easy to show that the other power needs for the people on board the station are small compared to the energy required for crop growth (assuming that the energy needs are comparable to that of the average American today). We can account for this by adjusting the area of the station slightly upward, say, to 107 m2. If the station is in the shape of a cylinder 2 km in diameter, then the length of the station must therefore be about 5 km to supply these energy needs.2 This also works out to about 1,000 m2 per person to supply the energy needed to grow the food for their survival. A useful reference for doing these sorts of estimates in detail is the nifty little book The Fire of Life, by Max Kleiber. The subtitle of the book, An Introduction to Animal Energetics, says it all. Table 19.5 of the book is the aptly titled “Area Yielding Food Energy for One Man Per Year”; ignoring the entry on algae (presumably Chlorella), we see that we need between 600 m2 to 1,500 m2 per person, and more if we want meat and eggs. This is right in line with what I estimated [140, p. 341].

  Of course, this is only an order-of-magnitude estimate, and there are other options. For example, instead of using natural lighting, one might illuminate the plants by means of white LEDs powered by large solar panels plating the outside the station or floating nearby. Even though the conversion of sunlight into electricity into the light emitted by the LEDs involves energy losses at each step of the way, one might overall do better than by using sunlight alone because much of the solar energy spectrum is useless for photosynthesis, being in the infrared spectrum. By concentrating light in the spectral region where plants can use it, the space station’s inhabitants might be able to boost overall energy efficiency. This is the approach taken by Eric Yam in his design for the space station Asten. He takes a very detailed bottom-up approach, whereas the analysis in this chapter is top down, but his design has a total solar cell area of about 3 × 106 m2, which is pretty close to what I calculate here [11]3.

  7.6.2 Atmosphere

  Have you heard about the new restaurant on the Moon? Great food, but no atmosphere.

  —ANONYMOUS

  Earth’s atmosphere is composed of 74% N2, 24% O2, and roughly 2% trace gases. Earth’s mean temperature is about 288 K, or 15°C, and atmospheric pressure at sea level on Earth is about 1 × 105 N/m2. I’m going to assume that we need this mix of atmospheric components on our space station, although this isn’t necessarily true; scuba divers use a different mix of gases (namely, oxygen and helium) when executing deep-sea dives. Oxygen is a highly reactive gas; it is not present in the atmosphere of any other planet in the solar system because it reacts with other gases or solids. On Mars, much of the oxygen is bound up in the soil or in the form of CO2 in the atmosphere. The reason why Earth has so much oxygen in its atmosphere is life. The respiration cycle in which plants take up CO2 and liberate oxygen, which animals then breathe and exhale CO2, is taught in every kindergarten class in the nation. Plant life thus serves a dual use on our station, providing not only food but also oxygen for the inhabitants to breathe. This has long been realized by science fiction writers. For example, in George Smith’s story “Venus Equilateral,” the engineers aboard the station must circumvent disaster when the new station head misguidedly clears out the “weeds” in the station’s air recycling plant, not knowing that they are the air recycling plant(s) [222].

  We must imagine that there are enough crops or other vegetation to supply the station with its atmospheric needs. Let’s do a quick, back-of-the-envelope calculation to see if we have enough plants already from agriculture to supply these needs. I’ll assume the following:

  • The average adult breathes about 20 times per minute.

  • The volume taken in each breath is about 1 liter (= 10−3 m3).

  • Oxygen is generated via photosynthesis powered by
the Sun at an efficiency of about 1% (as in the previous section).

  • The atmosphere on the station is at the same pressure and has the same content (about 74% N2, 24% O2, and 2% trace gases) as Earth’s atmosphere.

  • There are 10,000 people on the station.

  By my calculations the plants on the station will need to generate about 0.5 kg of oxygen per second to supply the needs of the people. (This is an overestimate, because not all the oxygen in each breath is taken up by the body.) The energy needs to generate this amount of oxygen can be found by considering the chemical reaction for photosynthesis [125, p. 515]:

  It’s an endothermic reaction, meaning that 2,870 kJ of energy in the form of visible light must be supplied to create one mole of glucose from 6 moles of CO2 and water. In the process, 6 moles of O2 are formed, which can be used for human and animal respiration; since each mole of O2 has a mass of 32 grams, oxygen can be supplied by photosynthesis at an energy cost of 1.5×107 J/kg. If again we assume that the overall efficiency of the process is about 1%, it will take about 7.5 × 108 W from sunlight to supply the respiration needs for the colony. This is about 6% of the energy needed for agricultural purposes, so we should be well covered by the agricultural energy budget. Another way to put it is that we need about 50 m2 of plant area per person to provide sufficient oxygen in the atmosphere, but nearly 1,000 m2 to provide food.4 Interestingly enough, in the novel Space Cadet, Robert Heinlein uses what is clearly too low a figure for the area of plants needed to provide oxygen for a human being (10 ft2, or about 1 m2); the figure is off by about two orders of magnitude, as our calculations and other data show [110].

  7.7 THE STICKER PRICE