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

by Adler, Charles L.

  We ran out. Well, not exactly; we’re not out of oil yet, but we have run low. It’s a simple idea: you can’t pour 11 gallons out of a 10-gallon jug. The year 1972 was the first year in which the United States produced less oil than it consumed, making it vulnerable to the OPEC oil embargo. Everyone, including the OPEC countries, was pretty surprised at how well it worked, or at least how much confusion and panic it caused. And no one saw it coming.

  Well, not quite no one. M. King Hubbert, a geophysicist working for Shell, saw the oil deficit coming a long time before it happened. In 1956 he published a visionary paper, “Nuclear Energy and the Fossil Fuels,” in which he compared the search for oil during the late 1800s and early 1900s to the age of exploration in the late Middle Ages, when Western civilization put together a map of the world. His point was that during the early age of petroleum exploration, just as in the earlier age of exploration, one couldn’t be sure what one would find. No one knew where the oil was or how much there was of it. But as of 1956, Hubbert felt that enough of the Earth’s great oil deposits (particularly in the United States) had been located to allow a reasonable estimate of the world’s oil supply. He wrote, “What we seem to have achieved is an abundance of detailed charts of local areas with only an occasional attempt to construct a map of the whole world which, despite its inherent imperfections, is still neccessary if we are to have an approximate idea of where we are now and where we are going” [126].

  The two key ideas Hubbert drew on in his paper were, (1) there is a finite amount of the fossil fuels (coal, oil, and natural gas) and (2) world civilization uses these resources at an ever-increasing rate. Indeed, their usage is at an exponentially increasing rate. Historically, fossil fuels were very little used before about 1800—that is, at the beginning of the Industrial Revolution. However, as the world became more industrialized, this use rate increased, at one point in the 1950s being something like an increase of 9% per year. This worked as long as we had enough oil in the ground to sustain such increases. However, as Hubbert pointed out, the total amount of oil in the ground could be estimated with relatively good accuracy: the United States had some 200 billion barrels, while the total world supply was roughly two trillion barrels [127].2

  Current world use rate is roughly 80 million barrels per day, or about 30 billion barrels per year. Of this, the United States consumes about one-fourth of the total. Currently, the rate at which oil is produced in the world matches the demand. However, an oil industry rule of thumb states that the maximum amount of oil that can be extracted from a well per year is roughly 10% of the remaining oil left in it. Therefore, as oil wells become depleted, not only is there less left, we can take out less every year. So we are faced with exponential growth in oil use until the day comes when the demand for it becomes greater than the supply, at which point the rate at which we use oil must begin to decline. Again, the idea is simple: if a resource is finite, at some point you have to start using less of it. Exponential growth rates are unsustainable.

  One way to describe this is via a logistic function. This is the function satisfied by the differential equation

  where γ represents the (yearly) growth rate characterizing the curve. Here, x(t) represents the total fraction of the resource that has been used up by time t; it goes from 0 to 1, 1 meaning that all of it has been used up. We are actually more interested in the rate dx/dt, as this is more relevant to the economics of oil production. Without even solving the equation, we can see that the usage rate will reach a maximum when x = 1/2, with a value of γ/4.

  This is the simplest possible model describing such a situation because the factor γ is tied to demand for oil or whatever natural resource is being consumed. Thus, γ was essentially zero before about the year 1900, when the world switched from coal to oil as its primary fuel source. Because of this, it will generally be a function of time. In particular, as the oil runs out, if the world switches to another type of energy source, the demand rate will decrease significantly again, maybe to zero, more likely to a lower but non-zero value. For now I’ll assume that γ is a constant value and see what results we get.

  Current world oil use is roughly 30 billion barrels per year and the world used roughly one trillion barrels between the year 1900 and 2010. Therefore, at the present time x is approximately .43 (i.e., we’ve gone through slightly less than half the oil in the ground), and we can work out that γ ≈ 0.12 from this data. We can back-project from this figure to see whether this figure represents a realistic model for world oil usage. Please note that these values are very approximate, for reasons I discuss below.

  The solution to the logistical equation is

  and the fractional use rate from this model is

  Note that on this scale we have a maximum value of the use rate when t = 0. Also, in principle, the assumption is that the world has been using oil indefinitely into the past, which isn’t true. Industrial society only began using it in large quantities around the year 1900 CE. However, in practice the model has such a small use rate for times more than 100 years ago that it doesn’t matter. We can then fit this to the total world use rate by multiplying by the total amount of oil in the world, Q:

  I assume a value Q ≈ 2.3 trillion barrels, or 2,300 billion barrels.

  The predictions of the model are interesting. Figure 17.1 shows the results of this calculation. Using these numbers, and looking at our current world usage rate of 30 billion barrels per year, we are approximately sixteen years away from the peak, at which point the world will be using about 70 billion barrels per year. Thereafter, there is a rapid decline. Sixteen years after the peak, world usage is back down to 30 billion barrels per year and falling exponentially. However, this curve doesn’t fit the actual data from the past very well, which is not too surprising. The exponential rise part of the curve indicates a 12% growth rate per year, which is almost impossible for any society. Two other curves are shown: one made using γ = 0.09 and a second using γ = 0.06. Interestingly, all curves pass the 30 billion barrel per year mark at around 10–15 years before the peak, but the peak production rate is proportional to γ, and therefore is only about 35 billion barrels per year for the lowest value. To compensate, however, the oil lasts longer.

  Figure 17.1. Logistic graph showing a mathematical model of world oil use. The graphs had been scaled so that oil usage is approximately 30 billion barrels per year in 2010.

  One point about all models: they are extremely dependent on the assumptions one makes going in. This is one major problem with economic models, which drives physicists like me to despair. Most economic models are based on shaky assumptions regarding supply and demand. Most of the parameters are fit by hand, as in the example given above rather, than derived from first principles. Also, there is nothing particularly special about a logistic curve. Many different mathematical models predict similar things. Hubbert, to his credit, understood this. If you go back to his original 1956 paper and his 1982 survey article in The American Journal of Physics, you are struck by how bright he was. His assumptions are based on rather general mathematical principles. Many different curves look globally similar to a logistic curve, the difference lying in the fine details. Other examples of such sigmoidal curves are the error function and the Gompertz curve.

  So oil will, perhaps, begin running out in about ten years, maybe sooner. A recent survey by the Kuwaiti government indicated that the peak might occur as soon as 2014; other estimates place it nearer 2020. What happens after that? According to whom you believe, either not much or Doomsday.

  Lets look at Doomsday scenarios first. Imagine that I could wave a wand and utter the invocation “Petrolevaporatus,” magically destroying all the world’s oil supply, drying up all the gas in every gas tank in every car, emptying every oil well on the planet. If that happened, about 99% of the U.S. population would die. Period. Think about it: you don’t grow your own food—at least, not enough to feed yourself. Only 2% of Americans nowadays are farmers, and the bulk of our food is grown on
large industrial farms. Our major food crop is corn, mainly used to feed the livestock that end upon as protein our tables. A good discussion of this can be found in the first section of Michael Pollan’s The Omnivore’s Dilemma [194]. Modern high-yield corn cannot be grown by hand. It requires intensive cultivation using industrial fertilizers and industrial harvesting. Also, most of it has to be transported several hundred miles to the feedlots. Most food in the United States travels a long way to reach our tables: it’s been estimated that for every Kcal of food energy we consume 7–10 Kcal were spent bringing it to us. This means lots of energy spent on food transportation. If cheap transportation goes away suddenly, our food supply disappears. Everybody starves.

  However, the oil supply won’t evaporate overnight. Instead, the demand for light, sweet crude will exceed its supply sometime in the future. What happens then? Presumably the price of oil goes up, maybe rapidly, such as the $4-plus per gallon prices people in the United States saw a few years ago. What happens after that is unclear. We haven’t developed any good substitutes for gasoline for transportation. This isn’t to say there aren’t any but that we haven’t developed them yet. Supply-side economists predict that once the price of oil begins to rise, people will rapidly develop substitutes and the world will move away from crude oil as an energy source. This is possible: for example, synthetic gasoline can be produced from coal using the Fisher-Tropsch process, which kept the Germans fighting for at least one year after they were surrounded by the Allies at the end World War II. America has vast coal reserves. It isn’t a great backup plan, as the process pollutes like crazy. Other sources possibly include biofuels. Energy conservation also helps, although it’s a lot harder than most people think. The issue isn’t so much overall energy resources but the fact that there is no really good substitute for oil to use for energy to power transportation.

  Perhaps the best short story every written about the oil crisis is a fantasy story called “Not Long before the End” by Larry Niven. It was published in 1969, before the oil crisis. It can be read as an allegory of the energy crisis of the 1970s, although calling it this demeans the story somewhat. The story is set several thousand years ago in a fairly standard sword-and-sorcery setting. However, the main character, a wizard known as the Warlock, discovers that mana, the source of magic, is a nonrenewable resource. Use of magic by thousands of wizards across the world is draining away the source of their powers, and magic is becoming scarcer and harder to perform. A speech from the Warlock has stuck with me: “The swordsmen, the damned stupid swordsmen, will win in the end until mankind found another way to bend nature to his will.” Several other stories were set in the same world, including the novella The Magic Goes Away, the short story collection The Magic May Return, and the novels The Burning City and The Burning Tower. The best-known movies concerning a dystopian future after the oil runs out are the Mad Max series, Mad Max, The Road Warrior, and Mad Max: Beyond Thunderdome, which took place after a war brought on by fighting over dwindling energy resources.

  Whether or not the oil runs out 30 years from now, it will run out eventually. One other thing: the amount of time doesn’t depend very much on how much oil there is, for the rate at which the world uses fossil fuels is increasing roughly exponentially. If the demand for a quantity doubles every 10 years, say, then in 20 years you will need to supply four times as much; in 30 years, eight times as much; in 100 years, 1,000 times as much. In 200 years the demand for the quantity has increased to 1 million times the initial demand, and it just keeps on going. Put differently, let’s say we initially think we have enough oil to last the world for the next 30 years. Then we discover we actually have twice the amount we initially thought. If the doubling time is 10 years, then this is only an extra 10 years, supply, not an extra 30 years’ supply. If there is four times the amount we thought, this buys an extra 20 years, not an extra 120 years. And so on. The time it will take to run out increases logarithmically (i.e., very slowly) with the increase in the initial quantity.

  Of course, we don’t have to restrict ourselves to only one option for ending the world. Global warming combined with rising fuel costs due to Hubbert’s peak can lead to a synergistic situation in which the stresses from the combination of the two are greater than the sum of the parts. We live in a world where a lot of the smaller, poorer nations have nuclear weapons; in the last five years, India and Pakistan have come scarily close to war. Rising food and fuel costs resulting from the two factors listed above could certainly lead to war, and the ubiquity of nuclear arms means that a country’s destructive capacity doesn’t depend primarily on its size or wealth. Also, the world’s great oil reserves are in areas of great political instability.

  17.2.4 Longer-Term Survival

  Beyond this? If we last through our own turmoil, Mother Earth is very capable of doing away with us. The Earth goes through climate cycles, possibly caused by variations in its axial tilt. These variations are small, but enough to cause 100,000-year cycles of glaciation followed by briefer interglacial periods. We live in an interglacial period right now. If our world civilization can survive for several thousand more years, past the heating brought on by the man-made greenhouse effect, then we will see a cooling period as the next glaciers appear. All of modern human civilization—the development of agriculture, domestication of animals, writing, towns and cities—has emerged since the last glaciers retreated. Can human civilization survive the next glacial period?

  The last 50 years have demonstrated that humankind can affect global climate. Perhaps we can do it in a safe way as climate cools. The temperature difference between the glacial periods and the interglacials isn’t much: maybe 6°C. This is about how much the IPCC predicts the Earth will warm up due to anthropogenic greenhouse gases under the worst-case scenario. We have observational evidence that humanity can warm the Earth; the unanswered question is whether we can do it safely. It is an interesting moral question whether we really should tinker with the Earth’s climate and ecology on such a large scale.

  Beyond this, a bevy of disasters wait for us. Beyond the glaciation cycle is the prospect of Earth’s climate becoming completely chaotic. The Moon stabilizes Earth’s axial tilt, but in 100 million years the Moon will have receded far enough away that the stabilizing effect will vanish. Earth’s axis will wander drunkenly, and the long-term variation in temperature may destroy all life. But worse is to come: the luminosity of the Sun is gradually increasing. In a billion years the Sun’s output will increase by 10%, leading to a runaway greenhouse effect that will make Earth a worse hell than Venus. If we cannot somehow temper this, our long-term descendants will have to find a new home, create one, or simply move the Earth.

  NOTES

  1. A very readable guide to the IPCC findings is Michael Mann and Lee Kump’s book Dire Predictions: Understanding Global Warming [157].

  2. To be very specific, these estimates apply to the “light, sweet crude” oil which is the oil of choice for the transportation industry: it is the type most easily refined into gasoline. Other types, such as the oil found in the Canadian oil sands, require more refinement before they can be used as a transportation fuel and are costlier to process.

  CHAPTER EIGHTEEN

  WORLD-BUILDING

  18.1 TERRAFORMING

  The noted physicist Kip Thorne first defined the Sagan problem as one that tested what the laws of physics would allow on a fundamental level [236]. The question Thorne was considering was whether physics allowed time travel and faster-than-light travel. This is a question that clearly probes the ultimate laws of nature. In these final chapters we’re going to consider a series of Sagan problems, each progressively more fundamental than the last, to see what might be possible for a sufficiently advanced civilization. These discussions are based on the laws of physics as they’re known now. However, all of them are practical impossibilities (to say the least) for human civilization as it is now, and possibly will be forever.

  This chapter is titled “World-B
uilding” because that’s exactly what we’re going to consider. A large fraction of science fiction deals with the transformation, or even construction, of habitable worlds for settlement by humanity. Varied reasons are given for doing this, but the underlying reasons seem to be variants on “because they’re there.” There’s already a lot of literary or historical critique of the assumptions behind this sort of colonization. The fundamental one is that it is an inappropriate projection into the future of the history of European and American colonial expansion of the nineteenth century. Perhaps a better critique of such ventures is to look at them from an economics point of view, which relatively few people have done so far.

  We’ve already discussed the idea of building space stations to house humanity in low Earth orbit and space travel to other planets. This is distinct from the idea which we are considering in this chapter, actually “building” a habitable planet. In this context, “building” is a misnomer: in reality, we are considering the process of terraforming a world, that is, turning a planet such as Mars or Venus, on which people cannot survive in the open, into one in which people can survive in the open. Hence the name: we are forming other planets into ones like Earth.

  Let’s consider the preconditions needed for a planet to be terraformed. First, it must have a solid surface, which rules out the Jovian planets Jupiter, Saturn, Uranus, and Neptune. Second, it must be neither too hot nor too cold to support life. As we’ve seen in chapter 2, this is determined by two things: the distance of the planet from the Sun and the composition and density of the planetary atmosphere. The first condition is something we don’t have any control over right now. However, we can change the composition of a planetary atmosphere. Since the Industrial Revolution we have been running an uncontrolled experiment on Earth’s atmosphere to see what the results of changing the net atmospheric composition of CO2 by a factor of two. So there are very good indications that if we can set up an industrial project of large enough scale on another world, we can change the climate of that planet significantly.