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

by Adler, Charles L.

  The basic principle of calculating the mean temperature of this structure is the same as calculating the mean temperature of a planet: there is a certain flux absorbed on average from the Sun by the structure, which depends on the total flux from the star and the mean albedo. It is radiated away in the form of infrared radiation; however, some of the infrared radiation is trapped by the atmosphere, leading to an increase in the mean temperature. There are two complications: first, as mentioned above, half of the flux is radiated away by the back of the structure where there is no atmosphere. All other things being equal, this will lead to a net cooling effect. Second, day/night alternation and latitude variations in mean insolation for a planetary surface lead to a reduction in the effective flux by a factor of four. Control of solar flux heating the Ringworld is by means of “shadow squares” in orbit around the Sun that periodically interrupt the sunlight to effectively give variations between day and night. The godlike engineers who construct the structure will choose the spacing and width of the shadow squares to maintain the proper temperature, which we will assume to be 288 K, the same as for Earth. (Another means of controlling temperature is via atmospheric composition, that is, by changing the mix of greenhouse gases in the atmosphere.) A little algebra gives us

  In this expression, TRW is the mean temperature of the structure, F is the mean “solar” flux illuminating the surface (assumed to be 1000 W/m2), σ is the Stefan-Boltzmann constant (5.67×10−8 W/m2K4), f is the fraction of emitted infrared radiation reabsorbed by the atmosphere (0.77, as for our model of Earth’s atmosphere), and η is the fraction of time the Ringworld is sunlit; it is our free parameter. The Ringworld is made up of a particular brand of unobtainium called scrith. I am assuming that scrith perfectly absorbs all light incident on it; that is, it has an albedo of zero. Playing around with the model leads to η = 0.65 for a temperature TRW = 288 K. This means that the shadow squares should be set up in such a way that 65% of the time, the Ringworld is in daylight and 35% of the time it is night.

  19.3.3 “Gravity” and Rotational Velocity

  Because of the small thickness of the structure (discussed below), its gravitational attraction will be completely insufficient to hold anything to it. Because of this, Niven posited the structure was orbiting with such a velocity that the centrifugal force effectively served the purpose of gravity, exactly as in a rotating space station. From this, and the assumption that the acceleration of gravity is effectively the same as on Earth (g ≈ 10 m/s2), we can calculate the rotational speed of the structure:

  This is forty times the orbital velocity of Earth; it also means that the kinetic energy of the structure will be humungous:

  Pay close attention to this number.

  Moving on: Let’s say, as a wild guess, that scrith’s density is typical for solid matter: about 5,000 kg/m3. Then the total volume of the Ringworld is 1024 m3. We can view the Ringworld as a thin ribbon with radius R, thickness T, and width W. The volume is given by the formula

  Only R is specified initially; for Earthlike life to flourish (assuming the star is similar to the Sun), R ≈ 1 AU = 1.5×1011 m. Since the Ringworld is nominally created to handle overpopulation, the inner surface area should be as large as possible, meaning we want to make the width as large as possible, or the thickness as small as possible. As a wild guess, I’m going to assume (similar to what we did for the total mass) that the width is the geometrical mean between the radius and the thickness:

  From this guess, and using equation 19.8, we arrive at an equation for the thickness:

  This leads to a width of 5,400 km, which is roughly half the Earth’s diameter, and a surface area given by

  or roughly 105 times the surface area of the Earth. Looking up what Niven wrote, he assumed a diameter of about 1 million miles, or 1.6×106 km, meaning a surface area of about 1.5×1021 m2. This is about three million times the surface area of Earth [177]. However, it means the structure is now only about 700 m thick.

  19.3.4 Ringworld Structural Strength

  Larry Niven once wrote that the Ringworld could be understood as a suspension bridge with no endpoints [180]. This description is worth examining in some detail. First, like the cables in a suspension bridge, the structure is entirely in tension. This is because the structure is in such rapid rotation that the centrifugal force pushing out on it is much greater than the attraction of the star’s gravity on it. Planets are different: a planet’s orbit is essentially defined by the balancing point of centrifugal force and gravity, so (apart from tidal forces) one doesn’t need to worry about these considerations. In the rotating coordinate frame of the structure, the centrifugal force pushing outward on the ring is balanced by the net component of the tension in the ring pointing inward. Figure 19.2 illustrates this: the net force per unit circumference acting on the element of the ring shown is twice the tension multiplied by the ring curvature (= 1/2R).

  Figure 19.2. Ringworld Structural Tension.

  or

  Here, geff is the effective acceleration of gravity on the structure (= v2/R), which is assumed to be 10 m/s2, M is the total mass, and ρ is the density of scrith. This has a nice interpretation: the net tension in the structure is the total centrifugal force on the structure divided by pi. The stress in the structure is the tension per unit cross-sectional area,

  which is again easy to work out:

  The tension in the structure is some five orders of magnitude higher than the tension in the space elevator, and there are no materials that currently exist that we could build that structure with. In the “small favors” department, the stress doesn’t depend on the structure’s thickness, meaning that effectively we can build as thin as we like. One point, however, is that the bulk modulus of the structure should be at least an order of magnitude larger than the tension, or else the structure will begin to deform significantly: this implies a bulk modulus of about Y = 1017 N/m2 at a bare minimum. However, the speed of sound (i.e., the speed of compressional waves) in a structure is given by the formula

  or a whopping 1.5% of the speed of light!

  Scrith almost certainly can’t exist. Ordinary matter is held together by electrostatic forces; the maximum possible value for the bulk modulus is going to be of the order of the stored energy per unit volume for bulk matter, which is about 1012 N/m2. This makes the space elevator a marginally possible concern but the Ringworld an almost certain impossibility. Scrith is not quite as extreme a material as the gleipnirsmalmi needed for faster-than-light travel, but it’s approaching it. The h parameter (assuming a density of 5,000 kg/m3) is about 2×1012 m, or ∼ 10−3 light-years. Scrith isn’t quite gleipnirsmalmi, but it isn’t trivial to find either.

  19.3.5 Energetics

  Assuming that it is possible to build, how long would it take to build such a structure? This isn’t an easy question to answer: the structure is so far beyond what humanity can do now, and there are so many assumptions one would have to make about the society that could build it, that, well, words fail me. However, maybe we can make a lower bound based on energy.

  We calculated the kinetic energy of the structure: 3.65×1039 J. Somehow the civilization has to generate the energy to give it this rotational kinetic energy. The handiest source of energy is the star the Ringworld circles. If the star is similar to our Sun, its luminosity is about 3.6×1026 W. If we assume that this super-civilization can harness all the energy of the star, it will take a total time of about 1013 s, or 300,000 years, to generate all this energy from the star. However, it’s probably unreasonable to assume the super-civilization can use all the energy from the star. If it harnesses only 10% of the energy, the figure goes up to three million years.

  Is there another way to generate this energy? Well, Einstein tells us that the total energy content of matter is E = Mc2; if we could extract the total energy content of matter to rotate this structure, the mass we would need to convert to energy is 3.6×1039 J/3 × 108 m/s = 4×1022 kg. This is roughly the mass of Earth’s Moon (7.35×1022 kg). H
owever, you need some means of extracting this energy; rumbling about antimatter won’t cut it, as it takes energy to make antimatter, as we saw in an earlier chapter. If the Ringworld engineers don’t have a handy moon made of antimatter (which they might—one of Niven’s Known Space stories has Beowulf Schaeffer investigating just such an object), they will need to extract the energy another way. The only reasonable means is to toss the object into a rapidly rotating black hole; one can extract up to 50% of the mass-equivalent energy by doing so (more on this later). Again, story continuity helps us here: another of the Beowulf Schaeffer stories involves him dealing with a “space pirate” who is using a miniature black hole to swallow starships. Of course, the black hole must be large enough that Hawking radiation hasn’t caused it to evaporate, but again, more on this later.

  19.4 THE RINGWORLD, GPS, AND EHRENFEST’S PARADOX

  One neat thing about the Ringworld is that it is a living embodiment of one of the most puzzling features of Einsteinian relativity, as the physicist Paul Ehrenfest pointed out in 1909. The idea is simple: imagine that Louis Wu, the protagonist of the first three Ringworld novels, is in a spacecraft that is hovering directly over Ringworld’s star. We’ll approximate the Ringworld as a perfect circular ring with radius of 1 AU, centered on the star. If the Ringworld were unmoving, then its circumference would be equal to 2π AU; however, since it is rotating, the theory of relativity predicts that it is foreshortened in the direction of its motion. Therefore, by that argument, its circumference should be less than 2π times its radius.

  This is not a big effect. Even with the Ringworld’s high rotational speed of 1,200 km/s, the effect is only about one part in 105. However, the Ringworld is huge: the amount “missing” from the circumference is more than half the size of the Earth! Another way to put this is that this relativistic effect is more than 100,000 times larger than relativistic effects induced by Earth’s rotation, and those effects are easily measurable by atomic clocks. Indeed, it might be difficult to design a GPS-equivalent system for the Ringworld because of this huge correction.

  Ehrenfest’s paradox is notoriously difficult to handle. One can argue that the circumference should, in fact, really be larger than 2π AU—at least for some observers. Some would argue that Wu, because he is observing the ring from an inertial reference frame, should measure the circumference as 2π times its radius. However, let’s look at what happens when someone on the Ringworld tries to measure the circumference. Let us say that Chmeee, a Kzinti and a friend of Louis Wu, has a large supply of measuring sticks exactly 1 m long. He is going to use them to measure the circumference of the Ringworld by laying them in place end to end. From Louis Wu’s perspective, each measuring stick is foreshortened in the direction of motion, so it takes more of them to measure the circumference than if the Ringworld weren’t rotating. Chmeee is going to measure the structure as being longer than 2π R!

  Most textbooks on relativity give short shrift to Ehrenfest’s paradox, mostly because it doesn’t have a clear-cut solution. One that has a decent discussion of the issue is Relativistic Kinematics, by Henri Arzeliès [28, pp. 204–243]. It devotes an entire chapter to the problem. To summarize a long and complicated argument, Arzeliès concludes that the problem is incomplete without recourse to the general theory of relativity. This is understandable: general relativity is a theory of curved space-time that allows for non-Euclidean geometries where the ratio of the circumference of a ring to its radius is not equal to 2π. His conclusion is that one must know the material properties of the ring, which then must be fed into Einstein’s field equations to discuss its subsequent motion and shape.

  Most writers don’t agree with Arzeliès, although it is more a matter of interpretation than of fact. The consensus seems to be that initial conditions matter a lot for this problem. How you start the structure rotating is important in determining its geometry. The question is a fascinating one, however, and far from settled. A relativistically rotating Ringworld would be the ideal object to settle the debate once and for all.

  19.5 THE RINGWORLD IS UNSTABLE!

  Oh, the Ringworld is unstable,

  The Ringworld is unstable

  Did the best that he was able

  And that’s good enough for me!

  —SCIENCE FICTION CONVENTION FILKSONG

  Shortly after the publication of the original novel Ringworld, Larry Niven was greeted by chants from students at a lecture at MIT, “The Ringworld is unstable!” This means that, left to its own devices, the Ringworld has a tendency to slide into its sun in a fairly short time. And by “fairly short time,” I mean a fairly short time on human timescales, not astronomical ones: at most, a few years. This instability drove the plot of the second novel, The Ringworld Engineers, as Niven was compelled to find a solution to it.

  Interestingly enough, while the instability is pretty famous in science fiction circles, there aren’t any complete descriptions of its causes. Larry Niven wrote in an essay that his friend Dan Alderson, a scientist at the Jet Propulsion Laboratory in Pasadena, spent two years working out the exact mechanism of the instability; unfortunately, as far as I can discover, he never published it. There have been at least two scientific papers written by others on the nature of the instability, but they are unconvincing; while they are correct as far as they go, they are fairly simplistic and ignore one or more of the complexities imposed by the nature of the structure. There are three issues at stake: a static instability, a dynamic instability, and a tendency for the structure to tear itself apart.

  Before I begin my analysis, I want to mention three things: first, my assumption throughout this is that both Larry Niven and the late Dan Alderson are (were) very bright people, at least as bright as I am. Also, I don’t have two years of leisure time to try to repeat his entire analysis. However, what one person can do, another can imitate, and I am bright enough (I think) to be able to repeat the key features. Second, I am standing on the shoulders of giants: what follows is based on a paper written by the third greatest physicist of all time, James Clerk Maxwell, on the stability of Saturn’s rings [161]. Maxwell’s paper is a tour-de-force analysis showing that Saturn’s rings can’t be solid or liquid in nature but must consist of small fragments, and forms the basis of the mathematical field of stability analysis as it exists today. The paper is key to understanding the Ringworld instability in its fullness, although the techniques must be adapted to the problem at hand, for reasons discussed below. Finally, the mathematics of such an analysis go beyond the level of mathematics used in this book; I will discuss it in only a general way. Also, the instability discussed relates only to in-plane motions of the structure, that is, motions of the ring in the plane that contains its sun. Out-of-plane motions aren’t considered here at all.

  19.5.1 Static Instability

  This is the issue that has been written about. If the Sun is at the exact center of the ring, then the net force on the ring is zero because the gravitational attraction of the Sun on one part of the ring is exactly cancelled out by the force on a part exactly opposite it. However, if the ring slides a little off-center or is displaced slightly upward or downwards, what then? If it tends to come back to the original position, then the system is stable. Small perturbations of the structure tend to restore themselves. If, however, it tends to slide further off-center, then it is unstable, and the ring will be destroyed when it hits the Sun.

  Colin McInnes did a stability analysis focused on this question [162]. He found that the Ringworld was unstable to motions in the plane. That is, if the Ringworld slides off to one side, gravitational forces tend to force it more to that side instead of back to the center. The analysis is straightforward, but the mathematics involves hypergeometric functions, so we will not reproduce it here. Out-of-plane motions are stable, however. If the structure is displaced upward, gravity tends to force it back downward, and vice versa.

  This is not surprising. No static configuration of masses can be in equilibrium owing to gravitational f
orces alone. This is a consequence of what is called Earnshaw’s theorem. A good paper on Earnshaw’s theorem for the advanced reader is W. Jones’s 1980 paper [133]. A planet orbiting the sun is in dynamic equilibrium: gravitational forces are balanced by “centrifugal” ones. This is why I think that the Ringworld instability is much more subtle than a mere static problem. Planetary motion is stable over timescales of billions of years, so why couldn’t the Ringworld be stabilized the same way?

  19.5.2 Dynamic Instability

  The issue of the stability of solid ring systems goes back about 150 years. The issue did not concern man-made structures. Instead, the question was whether Saturn’s rings were solid, liquid, or composed of many smaller bodies rotating around Saturn.

  Before Voyager 2 flew by Saturn we had no detailed pictures of the ring system, but it had long been understood that they couldn’t be solid because of an analysis done by James Clerk Maxwell, one of the greatest physicists who ever lived. Maxwell is best known today for his discovery of the full set of equations governing the electromagnetic field, but he made contributions to all areas of physics. In 1856 he published an essay on the stability of Saturn’s rings that included their motion [161]. He showed that if Saturn’s rings were solid, they couldn’t be stabilized by their rotation in the same way that a planet orbiting the Sun could be. They would inevitably slide off-center and hit the planet. Because they didn’t do this, he concluded that the rings were made from small chunks of material orbiting the planet independently. He also considered whether they could be liquid; the answer was also no.