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

by Adler, Charles L.

  It seems clear from these arguments that merpeople must be a species of hominids that left the land thousands or perhaps millions of years ago and adapted to the waters. This seems even clearer when we consider other…ahem… “interactions” that sailors have claimed to have had with these creatures. The best story detailing a liaison between a human and a mermaid is “The Professor and the Mermaid” by Giuseppe di Lampedusa, author of the novel The Leopard. In the story, the eponymous professor meets a mermaid while a graduate student studying the classics; the mermaid Lighea is described as a combination of goddess and animal, and the aging professor is implied to have returned to her by jumping off a ship at the end of the story. Other similar stories exist, the best known being “The Little Mermaid” by Hans Christian Andersen, in which a mermaid falls in love with a handsome sailor. This doesn’t seem too plausible if she were some species of fish. In the Disney version of the story, Ariel is just a girl with a tail, though a tail with clearly horizontal flukes.

  Interestingly enough, the mermaids in the Harry Potter stories appear to be fish. In the book Harry Potter and the Goblet of Fire, they are described as having gills; in the movie, although it is a little hard to tell, careful examination shows that they have vertical tails that move back and forth rather than up and down [203]. It makes one suspect there are two different unrelated species out there that, by chance or convergent evolution, look remarkably similar. The picture of the mermaid in the Prefect’s bathroom in the same film shows a much more hominid-looking mermaid, complete with horizontal flukes.

  4.3.2 Kleiber’s Law, Metabolic Rate, and Lung Capacity

  If merpeople are really seagoing mammals, how long can they stay underwater? All seagoing mammals, such as whales and dolphins, must come up for air from time to time. Let’s try to calculate this by making two assumptions:

  • The metabolic rate of any animal is proportional to the rate at which the animal can breathe in air (technically, the mass or [equivalently] volume flow rate.

  • The metabolic rate is proportional to M3/4.

  The first is easy to understand: metabolic processes need oxygen to proceed. The metabolic rate, or the rate at which energy is burned by the body, must be proportional to the amount of air taken in, divided by the time between breaths. The second assumption is known in biology as Kleiber’s law, after the biologist who first formulated the rule. There is a good deal of evidence to support this rule over a very large range of body masses, although there is some debate over the exact scaling exponent [243, pp. 62–63]. To the best of my knowledge the fundamental reason for Kleiber’s law is unknown.

  The metabolic rate is proportional to lung volume multiplied by the average rate at which the animal breathes:

  where M is the average metabolic rate, VL is lung volume, and R is the average number of breaths the animal takes per second. One tricky part: when whales breach the surface, after they blow water out their blowholes, they typically take a large number of breaths before diving again. This allows them to stay underwater longer.

  Studies show, unsurprisingly, that lung volume is proportional to body volume, that is, mass; assuming this, and Kleiber’s law, we find that, according to Walter Stahl [224],

  As the animal gets larger, the number of breaths per second it needs to take gets smaller, which explains why whales can stay underwater for long periods of time. Experimental data support this relationship. Stahl’s paper gives the formula [224]:

  for the average respiration rate as a function of mass (in kg). This formula gives a respiration rate of 17 per minute for an 80 kg adult male, which seems right: one breath every three seconds or so. Because of the water’s buoyancy, merpeople could be significantly more massive than people. If we assume a mass five times that of a human, or 300 kg, the respiration rate works out to be 12 per minute. That is, respiration rate scales only slowly with body mass. We therefore expect merpeople to stay under the surface of the water only about 40% longer than the average human (if my assumptions about mass are correct). Adaptations to the water may enable them to stay underwater for longer times, just as long as they hyperventilate (for want of a better word) when they surface. However, there seems to be no way they can stay underwater for the extended periods of time indicated in movies and fairy tales. An analogy: U-boats were German submarines of World War II. They were small craft with diesel engines and could remain submerged for only a few hours at a time. Today’s nuclear submarines are huge craft and can stay submerged for weeks at a time. Merpeople are to whales as U-boats are to today’s nuclear submarines.

  4.4 KLEIBER’S LAW, PART 2: OWLS, DRAGONS, HIPPOGRIFFS, AND OTHER FLYING BEASTS

  What is your name? What is your quest? What is the airspeed velocity of an unladen swallow?

  —OLD MAN, MONTY PYTHON AND THE HOLY GRAIL

  To quote J. B. S. Haldane:

  It is an elementary principle of aeronautics that the minimum speed needed to keep an aeroplane of a given shape in the air varies as the square root of its length. If its linear dimensions are increased four times, it must fly twice as fast. Now the power needed for the minimum speed increases more rapidly than the weight of the machine. [106, p.18]

  Figure 4.1. Airflow over a wing surface.

  To explore this point in detail, let’s consider air flowing over a wing with area A at speed v; this can represent any object flying at speed v through still air. Because of the shape of the wing, a lift force is generated upward on the wing (fig. 4.1). The lift is due to the fact that air is deflected downward by the wing, implying a force that pushes the air downward. By Newton’s third law, there must be an equal but oppositely directed force pushing up on the wing. The expression for this force is

  where ρ is the density of the fluid the wing is moving through (1 km/m3 for air), A is the area of the wing surface, and CL is a dimensionless lift coefficient that is usually about equal to 1. It varies with the shape of the wing and the speed, but not enormously. We can take it as 1 for our purposes here. In steady flight, neither rising nor descending, the lift must be equal to the weight. We can equate the two expressions and solve for the speed required:

  where M is the mass of the creature or plane and g is the acceleration of gravity. The speed therefore depends on the ratio of the mass to the surface area of the creature. If L represents some measure of the length of the beast, then . Bigger beasts must fly faster, proportional to the square root of body length. A Roc, a bird larger than an elephant, would have a flight speed of more than 100 mph just to keep it aloft.

  These formulas can be used for science fiction stories as well. For example, in Childhood’s End, the Overlords are a race of winged humanoids who live on an artificially low-gravity world [52]. They are somewhat larger than humans, about 7 feet tall, and presumably their mass follows the same scaling laws. A human visiting their world wasn’t too discomfited by the their world’s acceleration of gravity. Let’s assume that it’s about 80% of the acceleration of gravity on Earth. I estimate that their flight speed would need to be about 35 m/s (76 mph) to stay in the air, but could they fly at all? This is the question for the next section.

  4.4.1 Metabolic Rates and Flying

  The biggest problem with flight in large animals is that the metabolic rate for a large animal cannot be great enough to sustain the power needs for flying. This is indirectly related to the speed needed for flight: higher speeds are needed for larger animals, but higher speed means higher power requirements.

  The power requirement for flying can be found from the resistive forces acting on the animal as it flies. The drag force acting on the flying animal is

  This looks almost identical to the formula for lift force except that CD is a different coefficient, typically about 0.1 for streamlined fliers, and A′ is the effective projected forward area for the animal. This is a dissipative force, so power must be provided to keep the flier moving forward. This is an estimate; in particular, it may overestimate the power needs for larger fliers, which mostly
fly by soaring, using thermals for their lift. However, it is a useful rule of thumb.

  The power requirement can be found by using the formula P = F v. So the power requirement for flying is given by

  We are interested in how this scales as we increase the size or mass of the flier: in this formula, the two parameters of interest are the effective surface area (A′) and speed (v). As seen above, v ∝ L1/2; surface area scales as L2, so:

  A delightful book everyone should read is Bird Flight Performance: A Practical Calculation Manual by C. J. Pennycuick [191]. It is concerned with the computer modeling of avian flight. In it are formulas for estimating almost any quantity one would wish for when modeling flying creatures, on this world or any other. It is the only book I know that has a picture of a goose mounted in a wind tunnel [191, p. 51, fig. 5.6]. On p. 101 the author also mentions that metabolic rate scales as M3/4, while flying power scales, in principle, as M7/6, the derivation I gave above [191, p. 101]. However, his calculations are more detailed than mine, and he includes a graph of the ratio of flying power to (basal) metabolic rate derived from his computer simulations. If my very simple estimate were correct, the ratio should increase as mass increases:

  In fact, when plotted on a log-log graph, the slope is 0.35, reasonably close to our estimate. Whatever the true scaling, it is clear that the metabolic demands of flight increase more rapidly than metabolic rate as mass increases. This is the real reason why no massive biological fliers exist; Pennycuick estimates an upper mass limit between 15 and 20 kg for birds, which is close to the average mass of the California condor, the largest North American bird. The power required for flying is just too high for larger animals.

  This puts the kibosh on a large number of mythological beasts: flying dragons, hippogriffs, wyverns, and the like seem ruled out on the grounds that their metabolism simply can’t handle the power needed to fly. Perhaps this is why it is Mr. Weasley’s fondest desire, in Harry Potter and the Half-Blood Prince, to learn what keeps airplanes up [205]. This also seems to imply that flying humanoid aliens such as the Overlords are impossible. While human-powered flight in ultralight aircraft or hang gliders is possible, the wingspans required (of order 5 m or so) dwarf the size of the people being carried, and also require elaborate means of getting up in the air: being powered by bicycle train gears, or launched from hilltops or from behind airplanes or something similar.

  Poul Anderson thought up a good workaround to these problems in his stories concerning the Ythrians. These were flying intelligent aliens whose total mass was about 20–30 kg. To get around the mass restrictions, he postulated a very high metabolic rate for them: they were extreme carnivores with a special added digestive system to make their metabolism more efficient. With all of this, there were still issues with how they adapted to their environment: the population on their original home world was low because of the need for families to be separated by about 20 miles so that they could have enough hunting grounds to satisfy their enormous hunger. Before humans came to their world, they were limited to a Bronze Age culture because of their low population, which effectively forbade the growth of cities or even animal husbandry [23].1

  When I was a child, I had a coffee-table book called, I think, Dragons! The authors tried to provide a rational explanation for dragon “flight” by postulating that dragons were large blimps. Dragons, the book said, generated large amounts of … ahem … methane and other gases in their digestive tracts, giving them lift. They combusted the methane when breathing fire. This is ingenious, but I have no idea whether it is possible. There may be some reason that this mechanism is fundamentally impossible, as I can’t think of any animals that actually use it. I leave it as an exercise for the reader to work out the details.

  4.4.2 Owl Post

  One of the most pervasive aspects of life in the Harry Potter universe is the delivery of mail by owls, to the point that it is mocked by a character in Lev Grossman’s book The Magicians [103]. Ignoring the issue of whether owls can be trained to deliver mail, it’s worth considering what the upper mass limit for an owl parcel is (delivered by a single owl, that is).2

  Anyone who has read T. H. White’s The Once and Future King knows that owls eat mice, and that an owl can carry mice in its claws or beak [251]. So the mass of a mouse (call it 50 grams) is a lower bound. Estimating the greatest mass they can carry from first principles is probably an exercise in futility, so let’s assume that the mass of the largest prey they hunt represents an upper bound. Since Hedwig is a snowy owl, let’s consider them snowy owls. These large owls, with masses up to about 3 kg, will occasionally hunt hares, with mass up to perhaps 1 kg. So perhaps owl post isn’t a crazy idea.

  What about speed? In Harry Potter and the Goblet of Fire, Harry’s owl Hedwig delivers messages to the fugitive Sirius Black [203]. Black is implied to be hiding out on a tropical beach somewhere. I’m going to assume that his beach hideaway is at least 1,000 km away from chilly England. Owl flight speed is somewhere around 5 m/s, or 18 km/hr [191]. I’ll round up to 20 km/hr to make the math easy: to fly 1,000 km would take about 50 hours’ total flight time. Assuming that the owl flew eight hours each day, this implies a total of six days’ flying time to reach Black and six days to return. This is actually pretty consistent with times given in the books for exchanges of letters, so a point goes to Ms. Rowling for realism.

  4.4.3 God Makes Power, but Man Makes Engines

  The theme of this chapter is that animal behavior is limited by power generation. Humans as animals are capable of creating a few hundred watts of power, only a fraction of which is useful for work. Societies that run on human and animal power, common in fantasy settings, are extremely limited in scope compared to industrialized societies. Perhaps the great distinction between fantasy and science fiction is in the nature of the societies they portray. In fantasy, magic is a substitute for technology, a means of controlling “great powers” without the need for machines. In science fiction, men work through machines to control great powers.

  NOTES

  1. Anderson was a master at examining how biology and physics influence and limit science fiction. He was also very knowledgeable about history and sociology, all of which play a strong role in his stories. His essay “How to Build a Planet” was a direct inspiration for this book.

  2. As an aside, why do characters in the Harry Potter novels write with quill pens and ink on parchment? Try it; it’s not easy. Cutting a nib from a feather is highly skilled work, and writing with one is not much less so. Quill pens are three complete revolutions behind current technology, maybe more. And parchment is an expensive substitute for paper.

  PART II

  SPACE TRAVEL

  CHAPTER FIVE

  WHY COMPUTERS GET BETTER AND CARS CAN’T (MUCH)

  5.1 THE FUTURE OF TRANSPORTATION

  This second section of the book deals with spaceflight, mostly manned spaceflight. Manned exploration of other planets is perhaps the most common theme in all of science fiction, with the possible exception of alien contact; the parallels of this literature with novels of the Wild West have been thoroughly explored before, and I won’t go into them. Spaceflight is linked to the colonization of space, paralleling the colonization of the Americas and Africa by Europe in the eighteenth and nineteenth centuries. Space colonization has remained a persistent theme in science fiction to the present day; the website tvtropes.org refers to the 2009 movie Avatar as “[Disney’s] Pocahontas … in SPACE!” with good reason [15].

  However, on reading science fiction from the 1950s to the present, it is clear that reality hasn’t matched the prediction. Robert Heinlein in Expanded Universe wrote that the colonization of space was as inevitable as the Sun rising in the east [121]. He also wrote in the same work that by 2000, nudism would be common and casual, and that all of us would have flying cars and automated homes. Now, I’m not making fun of Heinlein here. Well, perhaps I am poking gentle fun, but with respect: I suspect the track record for most science fiction
writers in predicting the future is better than most academics, especially given that most science fiction authors, even authors of “hard” science fiction, aren’t working research scientists. Predicting the future is just hard.

  While science fiction writers aren’t usually research scientists, they tend to be highly intelligent amateurs of science, often with advanced degrees in a scientific field. For example, Isaac Asimov had a doctorate in biochemistry, though he never did any research in the field, and Larry Niven has an undergraduate degree in mathematics. Gregory Benford is a physics professor at the University of California–Irvine, but he is an exception to the general rule. I want to make a very strong statement here: many of these writers know a lot more about some fields of study than professional researchers do. They are amateurs of science, meaning they are “lovers” of the fields they study. However, they are also in the game of entertaining, meaning that for the sake of a good story, they incorporate things that go against modern scientific knowledge. For example, Robert Heinlein’s juvenile novel Red Planet is set on an inhabited Mars with thin but breathable air; even in 1949, the year the novel was published, a breathable atmosphere was known to be very unlikely. In fact, scientists had pointed this out as early as 1918 [25]. In the interests of a good story, however, it’s more interesting to discuss manned colonies on Mars than unmanned robots.