CHAPTER NINETEEN

DYSON SPHERES AND RINGWORLDS

For all the richest and most powerful merchants life inevitably became rather dull and niggly, and they began to imagine that this was therefore the fault of the worlds they’d settled on. None of them was entirely satisfactory: either the climate wasn’t quite right in the later part of the afternoon, or the day was half an hour too long, or the sea was exactly the wrong shade of pink.

And thus was created the conditions for a staggering new form of specialist industry: custom-made luxury planet building.

—DOUGLAS ADAMS, THE HITCHHIKER’S GUIDE TO THE GALAXY

19.1  DYSON’S SPHERE

In June 1960 a paper titled “Search for Artificial Stellar Sources of Infrared Radiation” appeared in the journal Science [71]. Its author, Freeman Dyson, is a physicist who is currently a fellow of the Institute for Advanced Study at Princeton University (the same place where Einstein worked). We have encountered him before as one of the chief driving forces behind the Orion nuclear pulse propulsion drive. In the Science paper he presented an idea for finding alien civilizations that did not require searching for radio signals from them; instead, he suggested that astronomers look for sources of infrared radiation radiating at a temperature about right for life on a planet, but with the total power output of a star like our Sun.

The material factors which ultimately limit the expansion of a technologically advanced species are the supply of materials and the supply of energy. At present the material resources being exploited by the human species are roughly limited to the biosphere of the Earth…. The quantities of matter and energy which might conceivably become accessible to us within the solar system are … the mass of Jupiter … [and] the total energy output of the sun [71].

Let us consider the twin issues of energy and resource needs versus their supply.

19.1.1 Energy and Resource Needs

Figure 19.1 shows an estimate of world population since the year 14 CE. It is taken from a 1977 publication, supplemented by more recent data [69, table 5]:

As one can see, world population increased dramatically between around 1800 CE and now, presumably owing to the Industrial Revolution. Not only did population increase dramatically, so too did the growth rate. From about 1800 to the present, world population has increased at an average annualized rate of about 1.3%. This doesn’t sound like a lot, but I’ll reintroduce a rule of estimation from an earlier chapter: the time it takes to double any quantity increasing at an annual rate of q% is 70/q years. This means that at current rates of expansion, the world population doubles every 54 years. Let’s assume there are 100,000 habitable planets in our galaxy. If we could somehow shunt our excess population to these other (hypothetical) worlds, it would take only about 900 years until all of them had populations equal to that of present-day Earth.

If we believe that faster-than-light travel is impossible, then this is clearly impossible. It would take far longer than 900 years simply to get to the other planets. Also, the energy requirements of placing that many people off-planet are probably impossible to meet. Freeman Dyson considered the problem of exponentially increasing population and energy demands in a 1961 paper. His question was whether one could figure out a solution to the population problem without interstellar travel.

19.2  THE DYSON NET

Dyson considered building numerous habitable space stations in orbit around the Sun at roughly the same distance as Earth is from the Sun. His idea for this solution came directly from science fiction. In interviews and published work he has credited Olaf Stapledon’s book Star Maker, in which the members of an intergalactic civilization build large shells around their stars to capture the last remnants of energy as the stars die out [225]. Dyson’s stations would occupy a large fraction of the total surface area of a sphere around the Sun of radius 1 AU, so that they would intercept most of the light coming from the Sun. It should be note that this network of stations is not a Dyson sphere; the Dyson sphere would be a solid shell built around the Sun that would intercept all the power coming from it. I’ll refer to the multiple space stations design as a “Dyson net” to distinguish it from a Dyson sphere. The Dyson sphere has two major problems with it:

1. Newton’s shell theorem states there is no net force between a solid spherical shell such as a Dyson sphere and the sun it surrounds. This means there is no way to prevent the shell from drifting off-center and hitting the Sun. There are even worse problems involving Larry Niven’s Ringworld, which we look at in the next section.

2. Even though the net force is zero, there is still a stress on the sphere that tends to compress any given section of it. This stress is much higher than any known material can support. Again, we will consider this in detail for the Ringworld in the next section.

Large, individual space stations in orbit around the Sun don’t have this problem. Each of them is in orbit, meaning that centrifugal force balances out the force of gravity acting on them. The tricky part is preventing the hundreds of thousands of space stations in orbit around the Sun from colliding with each other.

The amount of matter needed to build this shell of space stations is enormous. As Dyson stated, if we limit ourselves to material in the Solar System, most of the mass (outside the Sun) is concentrated in the giant planet Jupiter. Jupiter’s mass is larger than all of the other planets’ masses combined. It has a mass of 1.9×10²⁷ kg, or roughly 318 times the mass of the Earth. Between Jupiter and Saturn we could probably scrape together about 400 M_E total, or about 2.5×10²⁷ kg of material. That is a lot, but if we spread it in a thin shell with radius 1 AU around the Sun, it would have a thickness of only about 9 meters if the density were the same as water. However, this is more than we need.

19.2.1 Large Structures in Space

In a later paper published in 1996, Dyson considered the engineering issues associated with building large space-based habitats for the Dyson net [74]. He examined three questions in the paper:

1. Is it possible to build large structures in space?

2. Is it possible to build light, rigid structures in space?

3. Is it possible to take planets apart for the material needed to build these structures?

The constraints on building large structures in space are different from those that apply to building large structures on the surface of a planet. The ultimate constraint on how large a space station can be made is tidal forces. In some sense we’ve encountered this constraint already in considering the space elevator, although that structure represented a rather extreme version of the issue.

If we imagine a space station in orbit around a planet or star, there is only one point, the center of gravity, where centrifugal force balances out the attraction of gravity. Any part of the station “below” the center of mass, closer to the primary body, will see gravity as slightly larger than centrifugal force, and anything “above” it will find centrifugal force stronger than gravity. This means that there will be a seeming force trying to pull the station apart. This is what is known as a tidal force, The first time I heard of it was in Larry Niven’s short story “Neutron Star,” in which the hero of the story is nearly torn apart by these tidal forces when his spacecraft tries to get too close to said star [176].

The net stress pulling apart a structure of size L in a circular orbit of radius r around a primary body of mass M is of order

where g = G M/r² is the acceleration of gravity owing to the primary and m is the mass of the space station. This is an approximation only; the stress will be a function of the shape of the craft and its distribution of mass. If the station has mean density ρ, then m ≈ ρL³. Working this out, we find that

where Y_max is the maximum possible stress the structure can take and h is the parameter defined in chapter 5 for the space elevator: the maximum height a material of a given stress/weight ratio can stand in a gravitational field. One big difference here is that the gravitational field is that due to the Sun rather than the Earth. Putting in numbers appropriate for steel, I get a maximum size of about 9×10⁸ m, or nearly 10⁶ km, which is what Dyson found in his paper as well.¹ This is a structure roughly 150 times larger than the Earth. Another way to look at it is that the structure is larger than the distance of the Moon from the Earth. Large as that is, carbon nanotube fibers, with higher strength/weight ratios by a factor of 1,000 or more, would allow structures more than thirty times larger to be built.

The distance from Earth to the Sun is 1.5×10¹¹ m. If we wanted enough of these structures to intercept all the light from the Sun, the approximate number is

19.2.2 Building Light Rigid Structures

The question Dyson addressed next was whether it was possible to build light, rigid structures using the least amount of material. For structures of constant density, the mass of a structure is related to its size by the relation

M ∼ L³.

Because of the cubic dependence on size, mass increases rapidly as we build very large structures. Dyson considered what we would call today a “fractal,” or self-similar, structure. This is built of subunits of octahedrons built into larger octahedral units, which are built into even larger octahedral units, and so on. Because of how the structure is built, and because it doesn’t have to support its own weight when in orbit, one can build very large, very light structures. The details are in Dyson’s paper [70]. Dyson estimated that building a structure 10⁶ km in size would require a mass of about 3×10¹⁴ kg, or about 5×10⁻¹¹M_E.

Because of its fractal construction, the mass scales as a lower power of the size:

M ∼ L^3/2

meaning that the overall density decreases as the structure gets larger. To build a Dyson shell one would need a total mass of about 10⁵ times the mass of one of these huge space stations, or about 10²⁰ kg.

This is still a lot of mass. The total mass of the Empire State Building is about 4×10⁸ kg, so the Dyson net represents about a trillion copies of it put into space. This is a mass much larger than that of all the buildings in all the cities in the world put together.

19.2.3 Taking Planets Apart

The total mass required to build a Dyson net, while less than the mass of the Earth, is still huge by any standard. Could we imagine taking the Earth or another planet apart to rebuild it into these structures?

The first question to ask is how much energy it takes to deconstruct a planet. The gravitational potential energy of a uniform sphere is

A planet isn’t a uniform sphere, but we’ll take this as a starting point. In the formula, M_p is the planet’s mass and R_p is its radius. Putting in values appropriate for Earth, we get U = −3.8×10³¹ J. In normalized units, we can write this (for any planet) as

in units where M is the mass of the planet relative to Earth’s mass and R is the radius relative to Earth’s. At least this much energy must be supplied to break up the planet. The luminosity of the sun is 3.86×10²⁶ J/s; the energy it would take to disassemble Earth represents the total output from the Sun for a day.

This leads to the question of how to do it. Dyson imagined an ingenious scheme to increase the Earth’s rotational speed until the planet flew apart. He proposed making an electrical motor out of the Earth. Laying large wires parallel to lines of latitude across it and running a current through them would give the Earth a sizable magnetic field. Then, running a current from pole to pole and out to large distances from the Earth and back again, one could generate a sizable torque either to speed it up or slow it down. Dyson envisioned doing this over the course of 40,000 years, which would require an average power of about 300 times the total power from the Sun intercepted by the Earth.

In Greg Bear’s Forge of God, hostile aliens destroy Earth much more rapidly by dropping two large, ultradense masses into the center of the Earth. One is matter, the other antimatter, both at neutronium densities. When they merged together, the resulting explosion destroyed the world [32]. Based on the calculation above and using E = Mc², it would require a mass of about 10¹⁴ kg to do this. The purpose of this was destructive rather than constructive, but the net effect was the same: a planet in pieces.

19.2.4 Detection of a Dyson Net

Dysons claimed that his motivation for writing these two papers was not to suggest that humanity should actually build such a structure. To quote Dyson,

When one discusses engineering projects on the grand scale, one can either think of what we, the human species, may do here in the future, or one can think of what extraterrestrial species, if they exist, may have already done elsewhere. To think about a grandiose future for humanity … is to pursue idle dreams…. But to think in a disciplined way about what we may be able to observe now astronomically … is a serious and legitimate form of science [74].

He wanted to see if astronomers could detect such structures if they existed. The issue is that the power output from a star like our Sun follows a blackbody curve with a characteristic temperature of about 6,000 K. This means that the peak in the spectrum is at a wavelength of about 0.5 µm, or 5×10⁻⁷ m. However, the Dyson net’s mean temperature would be a lot lower, around 300 K. The shell would absorb the sunlight at a relatively high temperature and reradiate it at a lower temperature. An astronomer would see a blackbody curve with a peak wavelength of about 10 µm, or 10⁻⁵ m. There are other astronomical sources of radiation at this wavelength. However, unlike other sources, it would look like a star with the same luminosity as our Sun but with a planet-like temperature. In fact, one can argue that if a civilization didn’t completely surround the star, astronomers would see two blackbody curves of similar strength: one resulting from the partly blocked star itself at a high temperature, and one from the shell at a much lower temperature. This seems a reasonable and unique signature for such a structure. I say “reasonable” because it is probably impossible to block the star entirely. The Dyson net isn’t a solid shell but a large number of relatively small satellites in orbit around it. To keep the satellites from colliding, there should be some room between them. To spot a Dyson net, one might look for stars like our Sun that radiate most of their energy in the visible region of the spectrum, but also a relatively large fraction (perhaps 10%–50%) in the far infrared. I can’t think of any naturally occurring astronomical objects that have a spectrum like that, but perhaps there are some.

19.3  NIVEN’S RINGWORLD

There are obvious issues with building a Dyson shell: the energy it requires, the need to destroy a world to create it, and so on. In 1970 Larry Niven came up with a smaller structure than a Dyson sphere for his novel Ringworld [177]. As the name implies, the Ringworld is a ring around a star, of radius 1 AU and width to be determined. Niven decided to spin it to provide gravity (in the form of centrifugal force) for its inhabitants.

What I’d like to do in this section is to play “Ringworld engineer”—that is, go through the process that (presumably) Larry Niven did when originally designing the thing. It’s a fun exercise, and the concept reflects the best of what hard science fiction has to offer; of course, our job here is much easier than Niven’s was: we have only to reconstruct the idea, not come up with the original notion. One other thing: I am deliberately not referring back to the book for the parameters but will estimate them based on what I know of physics and astronomy. At the end, we’ll compare them to the ones found in the book.

19.3.1 Ringworld Mass

First, let’s estimate its mass. It is considerably larger than a planet but smaller than a star. Unlike the Dyson net, it is a rigid structure. Oddly enough, its mass will be larger than what we calculated for the Dyson shell.

It’s larger than a planet but smaller than a star. Perhaps we can use the geometric mean of the mass of Earth and the Sun for its mass. Since the Earth’s mass is roughly 10⁻⁶ of the Sun’s mass, the geometric mean of the two masses is roughly 1/1,000 the Sun’s mass, or 1,000 times the mass of the Earth: about 6×10²⁷ kg. Another way to estimate the mass is to try to figure out where all this mass should come from. If we assume the designers used only the resources available in one stellar system, then we should estimate the Ringworld mass as the mass of everything in the Solar System but the Sun. As I stated above, most of the mass in the Solar System apart from the Sun is in the planet Jupiter. The combined mass of Jupiter and Saturn is roughly 390 M_E, which is not too far from my first estimate. Of course, astronomers have found “super-Jupiters” in other stellar systems with masses more than ten times Jupiter’s mass, so there are good reasons to think that I can be liberal by a factor of two or even more. I will use an estimate of 5×10²⁷ kg to make the calculations ahead come out nicer.

19.3.2 Ringworld Radius and Mean Temperature

One parameter of importance is the radius of this ring around the Sun. Because the Ringworld is inhabited by humans and humanoid aliens such as Kzinti, the distance from its star must be about the same as the distance from Earth to the Sun. If not, the radiant flux from the star will make the world too hot or too cold. On this assumption, R ≈1 AU = 1.5×10¹¹ m. There is one caveat: given this distance, one might naively expect the Ringworld temperature to about the same as Earth’s, but this is ignoring the fact that greenhouse warming by Earth’s atmosphere leads to a significant rise in Earth’s mean temperature (roughly 30°C). “So what?,” I hear the fan say: “Ringworld’s atmosphere, just about the same composition as Earth’s, leads to greenhouse warming of the Ringworld as well.” Yes, but Earth’s atmosphere extends completely around the planet, whereas Ringworld’s atmosphere is only on the inner side. One must apply the principle of detailed balance to calculate the temperature, using the fact that part of the heat flux escaping from the structure leaves through the back. Let’s calculate the mean temperature of the Ringworld using the principle of detailed balance.

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, T_RW 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 T_RW = 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).

or

Here, g_eff 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). However, 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 forces 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.

The stability issue is rather subtle. To begin with we need to look at why planetary orbits around the sun are stable. A planet, as it orbits its star, seems to feel two forces acting on it: the force of gravity pulling it in and a “centrifugal force” pushing it away. For a circular orbit, the two forces balance out at all times. For an elliptical orbit, as the planet approaches the star, the centrifugal force is stronger than the gravitational force, resulting in it being pushed away. As the planet gets farther from the star than its average distance, the force of gravity is stronger than the centrifugal force. This results in a net force toward the star. The final result is that the planetary orbit is stable: when the planet is too close, it is pushed away; when it is too far, it is pulled in.

This isn’t true for a Ringworld structure. Here the centrifugal force pushing outward is much stronger than the gravitational force pulling inward. The total force is balanced out by one that doesn’t exist for a planet: the tension in the ring structure. The Ringworld is a lot like a rapidly spinning gyroscope. If the structure slides off-center, it’ll start to wobble. The wobble will couple into the motion of the ring, sliding it further off center, and soon. This will eventually crash the structure into the star, but it will not be a smooth motion at all. The wobble may get so strong that it tears the structure apart before it hits the sun.

19.5.3 Deformation

The final issue is deformation of the structure. If the ring deforms a little bit from a perfect circle, does the structure tend to “bounce back” to a circle or does it tend to collapse itself out into a line? I don’t know the answer to this question. It probably depends on the material properties of scrith. Such stability problems tend to be highly nonlinear, and therefore difficult. My suspicion is that the extremely high centrifugal force will complicate the problem a lot.

19.5.4 Other Large Structures

In the recent novel Bowl of Heaven, Gregory Benford and Larry Niven write about another type of large structure. Essentially it is a half Dyson sphere. The structure looks like a parachute being dragged behind a racing car, with the Sun functioning as the car and the world-bowl as the parachute [37]. It is being used as some sort of very large starship. The structure was built for mysterious purposes; humans stumble across it during the course of the first interstellar exploration. One of the characters says of the structure:

The shell should fall into the star—it’s not orbiting. There’s some sort of force balance at play…. Just spinning it isn’t enough, either—the stress would vary with the curvature. You’d need internal support. [37, p. 33]

I have no idea whether Benford and Niven did a stability analysis for this structure. To give them credit, they mention in the book the misconception that Dyson’s original idea was a solid structure:

Only—the old texts reveal quite clearly that Dyson did not dream of a rigid structure at all. Rather, he imagined a spherical zone filled with orbiting habitats, enough of them to capture all of the radiant energy of a star. [37, p. 33]

In the 1970s Larry Niven wrote an essay “Bigger Than Worlds,” in which he discussed different structures from the size of large space stations to the galaxy [180]. I think they are all implausible for reasons that were brought up when Dyson published his original paper.

19.6  GETTING THERE FROM HERE—AND DO WE NEED TO?

Dyson originally proposed these structures as a cure for overpopulation. However, Dyson’s assumption was that the world population would continue to increase exponentially in a Malthusian fashion. Exponential increase is characteristic of populations that have plenty of resources. This can be seen in bacterial populations; given sufficient resources, they double every generation until the food available to them is exhausted. Human population has increased at an average rate of about 1% per year since around 1800 CE, that is, since the beginning of the Industrial Revolution [69]. However, this rate of increase was made possible by the enormous increase in resources available.

The growth of a bacterial or animal population doesn’t follow an exponential curve indefinitely. It follows the same sigmoidal curve that we investigated when looking at the Hubbert peak for oil use. It is characteristic of an expanding use of a finite resource. Dyson’s point was that the expansion of human population is limited by the resources, chiefly energy, available to it. The question is whether the human race will expand to this point if such resources are available to it. After Dyson’s original article was published, John Maddox, Eugene Sloane, and Poul Anderson published rebuttals to his thesis in Science magazine [156]. Anderson made the rather cogent point that unrestrained population growth would make the structure nearly impossible to build because, at the growth rates postulated by Dyson, the structure would take much longer to build than the time it would be needed in. To quote Anderson:

Even Dyson intimates that the project would take several thousand years to complete. In short, uncontrolled population growth will make the construction of artificial biospheres such as Dyson spheres impossible, and birth control will make them unnecessary. [156, p. 257]

It is a truism that First World civilizations have lower birth rates than Second or Third World ones. This is due to easy access to birth control and (probably) to other social factors, equality of rights for women being chief among them. If this trend holds for the developing world as in the developed, overall world birth rates would be expected to drop. (It’s interesting to note that one predicts the same result if one assumes resource depletion and the Four Horsemen.) In this optimistic scenario, world population should level off or possibly even decrease in the future. In any event, birth control seems easier in the long run. However, this still doesn’t rule out highly advanced civilizations.

Note

1. For the same weight/strength ratio, structures in Earth’s orbit are limited to about 100 km in size because of the higher value of g and lower value of r. This is another reason why it is difficult to build a space elevator.