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

by Adler, Charles L.

  Unfortunately, none of these options to reduce the fuel and energy costs reduces the most critical parameter for manned spaceflight, the time it takes. If we want to send people to the planets on relatively low-energy orbits, we are constrained by the laws of orbital dynamics to trips that take years, and we therefore must deal with the vexing problem of keeping people alive in space for years at a time. As we saw in chapter 4, this comes with a very high sticker price, such that to date, none of the space missions to Mars or farther out has been manned.

  9.7 COSTS

  Again the question arises, are the benefits worth the costs? Science fiction writers of the 1950s and 1960s blithely wrote about the exploration of Mars as if it were the exploration of some uncharted region on Earth, like the Conquistadors in Central and South America. Let’s consider the ground conditions on Mars to see whether this is reasonable.

  The average temperature on Mars is about the same as that of Antarctica. The atmosphere is almost too thin to be considered poisonous; at “sea level” on Mars the pressure is about the same as at the top of Mount Everest, and the atmosphere is an unbreathable mix mostly composed of carbon dioxide. There is very little surface water, even in the form of ice, even at the poles (which are mostly frozen carbon dioxide.) There is strong evidence that water once flowed on the surface of Mars billions of years ago, but it’s gone now. It is barely possible that life exists on Mars at the bacterial level. We have sent several unmanned probes there to see, and so far no evidence of life has been found.

  No one can stay on Mars without protection, meaning warmth and atmosphere. If astronauts were to stay there for many months, they would have two choices: either stay in a craft in orbit around Mars and explore the surface using robots, or descend to the surface and construct a habitable base. We already explore Mars using robots; the only advantage to controlling the robots from Mars orbit is that the delay time is considerably shorter (a second or so, compared to 4.3 minutes when Mars is at its closest to Earth). I’m not sure that this is enough of a gain over exploration of Mars using a robot probe to justify the factor of at least 100 in costs for such a mission.

  If they descended to the surface, the question becomes what they could do on the surface that a robot probe couldn’t. This is very relevant considering the added costs of creating a base they could live in. In previous chapters we’ve considered the costs of bringing an Earth-like environment into space; any base would most likely cost billions of dollars, and there would be no a priori way to test whether it would work for the long time which our hardy explorers would need to stay in it.

  The whole question of manned space exploration is a vexing one. Most science fiction writers of the 1940s through the 1980s and later foresaw the existence of colonies on Mars and on the moons of the giant planets. Robert Heinlein’s novel Farmer in the Sky, for example, posits that mankind has terraformed Jupiter’s moon Ganymede and used it for farming purposes to feed overpopulated Earth. On a more recent note, TV shows such as the Star Trek franchise, Babylon 5, and the old and new Battlestar Galactica series implicitly assume that mankind will spread out beyond planet Earth. This assumption, which is deeply ingrained in science fiction culture, comes from a time before the “rise of the robots”; no one thought that computing power, the basis of today’s unmanned robotic explorers, would increase as rapidly as it has, or be as cheap as it is. It isn’t clear there is anything that can be done by a human in space that can’t be done more cheaply and easily by a machine. At least one writer has called the notion of manned space travel “old-fashioned”. From a scientific standpoint, I think this is correct. Manned space travel is too costly and has too few benefits to justify its existence.

  Most readers might now expect a closing paragraph in which I extoll the nonscientific benefits of manned space exploration: the thrill of the exploration of the unknown; the idea that mankind needs new frontiers if it is not to stagnate; the worry that if mankind is stuck on one planet, a disaster could destroy us. These are appealing ideas. But manned space exploration clearly will not happen unless we find better ways of getting off-planet and creating homelike places elsewhere. I’d like to construct an analogy: we are in the same situation with regard to manned spaceflight today as Charles Babbage was with respect to computing in the 1860s. He invented the basic ideas for the modern computer and tried to implement them using the mechanical technology of his day. The technology was marginally not good enough to allow his analytical engine to be built. We seem to be in the same situation today: chemical rockets with exhaust speeds of a few thousand meters per second are marginally good enough to launch unmanned probes traveling slowly through the Solar System but are completely inadequate for manned missions. In the next chapter we examine solutions that have been proposed for getting out there more quickly.

  NOTES

  1. To quote the song “Shoehorn with Teeth” by the band They Might Be Giants: “Tour the world / In a Heavy Metal band / They run out of gas / The plane can never land.” Untrue for planes, true for spacecraft.

  2. For a full calculation, including escape velocities from Mars and Earth, see the paper “Journey to Mars” by Arthur Stinner and John Begoray [227]. One issue with this paper is that the authors do not discuss the fuel costs correctly. They assume fuel costs proportional to kinetic energy of the spacecraft instead of using the rocket equation.

  3. Technically speaking, this is a collision problem, even though the spacecraft doesn’t actually collide with the planet (at least, so we hope). The original idea for this may be due to the mathematician Stanislaw Ulam in his Los Alamos report LAMS-2219 [75, pp. 25–26] [240, chap. 9].

  4. This seems paradoxical and a violation of the conservation of energy. What is happening is that the spacecraft is losing gravitational potential energy by getting closer to the planet. By burning the fuel closer to the planet you are gaining not only the kinetic energy from the spent fuel but also the potential energy it has “lost” because the spent fuel will remain close to the planet.

  CHAPTER TEN

  ADVANCED PROPULSION SYSTEMS

  10.1 GETTING THERE QUICKLY

  In the last chapter we considered the Hohmann transfer orbit as a means of traveling from Earth to Mars or another planet. It was an energy-efficient method of getting there from here, but it had one big disadvantage: it took a long time. This was because the planets had to be in the right relative positions when the rocket was launched, and also because when the rocket was launched it became a planet in effect: a satellite of the Sun, acted on only by the force of gravity (except for short times when changing orbits). A trip to Mars took about 0.7 years, and trips to the outer planets took much longer. Everything is governed by Kepler’s third law.

  The reason it took so long is that the times when the spacecraft was accelerated by its engines were relatively short. For a minimum-energy, Hohmann-type orbit, the spacecraft is in free-fall orbit around the Sun except when making (brief) Δv maneuvers. However, if we want to travel to Mars over the weekend and be back in time to watch My Favorite Martian reruns on Monday night, we need spacecraft that are capable of acceleration for very large stretches of time, meaning they will need to expend a lot of energy. It is pretty easy to see that conventional rockets simply won’t work for this particular job.

  10.2 WHY CHEMICAL PROPULSION WON’T WORK

  Let’s take a hypothetical spacecraft that can accelerate continuously at 1 g so that we feel our normal weight along the voyage. The three questions we want to look at are:

  1. How long will it take?

  2. What is the greatest speed we reach along the voyage?

  3. How much fuel will we need for a 10,000 kg payload?

  We’ll use the final question as a criterion to evaluate the promise of different types of propulsion systems.

  The average distance of Mars from the Sun is 1.52 AU, that is, 1.52 times the average distance of Earth from the Sun. Therefore, at closest approach, Mars is 0.52 AU from Earth, or (about) 7.5×101
0 m, that is, 75 million km, or about 40 million miles. The acceleration of gravity is about 10 m/s2; I’ll assume that the ship accelerates halfway, then flips around and decelerates the other half. We don’t want to zoom by Mars at high speed, after all this effort.

  We start with a formula from freshman physics:

  That is, at constant acceleration the distance a body travels (d) is equal to one-half the acceleration (a) multiplied by the voyage time (t), squared. I’m assuming that the spaceship isn’t moving when the trip starts. Inverting this,

  One interesting thing about this equation: because the spacecraft is continuously accelerating on the way out (that is, traveling faster and faster all the time), going four times the distance takes only twice the time. From this, the time it takes to go halfway to Mars (remember, we are accelerating halfway and decelerating the other half) is

  The total transit time is t = 2t1/2 = 2 days. In more useful units, if we wish to go a distance of d AU at an acceleration of 1 g, the time it will take is

  So, if we can accelerate at 1 g, the Solar System is ours!

  The maximum velocity on our trip to Mars is at the midpoint and is given by

  As fast as this is, it is only about 0.2% of the speed of light. To achieve this velocity for a 10,000 kg payload with chemical rockets (i.e., a typical exhaust velocity of about 3,000 m/s), we need a mass ratio of approximately e287 ≈ 10124, which is clearly impossible. This is a physical impossibility, not merely a practical one, as there is not enough mass in the universe to achieve this.

  10.3 THE MOST FAMOUS FORMULA IN PHYSICS

  Chemical rockets rely on chemical potential energy for propulsion. Chemical energy is the energy released when you rearrange molecules to form other molecules—in other words, the energy released when atoms in some sort of compound change their relative positions with each other. This places an intrinsic limit on the amount of energy that can be released by a chemical reaction, as it is due to the electrical potential energy between different atoms.

  This is a limiting factor because atoms, although teensy on a human scale, are pretty far apart in the microworld. We can do a rough estimate of the electrostatic potential energy in a kilogram of matter by thinking about two adjacent atoms. Typically, atoms are separated by about 10−9 m in a solid or liquid. Let’s approximate this with the idea that the atoms are represented by two charges separated by a distance r about equal to the average interatomic spacing. This will underestimate the energy, but not by a whole lot:

  Here, k = 9×109 Jm/Coul2, e = 1.6×10−19 Coul, and r =10−9 m. This gives about 2×10−19 as the approximate potential energy between a pair of atoms. Since there are about 1026 atoms per kilogram of a solid, this implies about 2×107 J/kg (i.e., 20 MJ/kg) available in the form of chemical energy. This value is pretty good for a crude estimate: gasoline, for example, liberates about 80 MJ/kg when combusted.

  However, there is a lot more energy in matter; indeed, matter is a form of energy. This is what Einstein’s famous formula,

  states: there is an energy equivalent in matter to its mass (M) multiplied by the speed of light squared (c2). In other words, 1 kg is equivalent to 9×1016 J—90 million billion J, or about a billion times the energy present in chemical reactions. If we could somehow liberate even a small fraction of the energy present, we would not only have a working space drive but could also solve all of Earth’s energy problems, essentially forever (but more on that later.) But how?

  10.4 ADVANCED PROPULSION IDEAS

  10.4.1 Nuclear Energy Propulsion Systems

  The idea for nuclear propulsion of spacecraft dates back to at least 1945, as Richard Feynman documents in his book “Surely You’re Joking, Mr. Feynman!”:

  During the war, at Los Alamos, there was a very nice fella in charge of the patent office for the government, named Captain Smith. Smith sent around a notice to everybody that said something like, “We in the patent office would like to patent every idea you have for the United States government, for which you are working now [concerning nuclear energy].… Just come to my office and tell me the idea.”

  … I say to him, “That note you sent around: That’s kind of crazy to have us come in and tell you every idea.… There are so many ideas about nuclear energy that are so perfectly obvious, that I’d be here all day telling you stuff.”

  “LIKE WHAT?”

  “Nothin’ to it!” I say. “Example: nuclear reactor … under water … water goes in … steam goes out the other side … Pshshshsht—it’s a submarine. Or: nuclear reactor … air comes rushing in the front … heated up by nuclear reaction … out the back it goes … Boom! Through the air—it’s an airplane. Or: nuclear reactor … you have hydrogen go through the thing … Zoom!—It’s a rocket. Or: nuclear reactor … only instead of using ordinary uranium, you use enriched uranium with beryllium oxide at high temperature to make it more efficient.… It’s an electrical power plant. There’s a million ideas!” I said, as I went out the door. [84]

  Atomic energy is very attractive as an energy source for spacecraft propulsion because of the enormously high specific energy (energy/per kilogram of fuel) compared to chemical fuels. There are two types of nuclear reactions: fission and fusion. In fission, the capture of a neutron (n) causes large, unstable nuclei to fall apart, while in fusion reactions, energy is generated by the fusing together of lighter nuclei into heavier ones.

  I’m going to digress for a moment on the structure of the atom and the atomic nucleus. Atoms are mostly empty space. An atom is electrically neutral, but all of the negative charge is on the outside: the electron shells that make chemistry possible extend out to a distance of about 0.1 nm, or 10−10 m, from the center of the atom. All of the positive charge, in the form of protons, is at the center of the atom; for each electron in a neutral atom there is one proton as well. The protons are confined to a space that is about 10−15 m in radius, or 1/100,000 of the extent of the electron shells. Because like charges repel each other, something must act to “glue” the nucleus together. This glue is the neutron. For every proton, there is at least one neutron in the nucleus that provides a force (called the “strong nuclear force”) that keeps the protons from exploding outward. Neutrons are neutral (i.e., they carry no charge) and are about the same mass as the proton, which is about 1,800 times heavier than the electron.

  The total electrostatic potential energy of the nucleus is very much larger than the electrostatic potential energy between a pair of atoms—about 100,000 to 1,000,000 times bigger because of the size factor involved. The energy liberated from the nucleus is still pretty small compared to the ultimate amount of energy stored in matter, but is large compared to the chemical energy.

  Currently, all commercial nuclear reactors work through the fission of heavier elements into lighter ones. As elements become heavier and heavier (i.e., having more and more protons and neutrons in the nucleus), they become unstable and the largest ones can fall apart spontaneously. This process liberates energy: by fissioning into two nuclei, one large nucleus moves about half the protons far away from the other half.

  10.4.2 Fission Reactions

  In each process, the energy released can be calculated by the difference in mass between the end products and the initial nuclei. A typical fission reaction used in reactors is [246, pp. 1198–1199]:

  (keep in mind that 1 MeV = 1.6 × 10−13 J).

  Because an extra neutron is generated by the fission process, the process can be self-sustaining if enough 235U is present; this amount is called a critical mass. Nuclear reactors use an amount very slightly over criticality to generate energy in a controlled way, whereas nuclear bombs suddenly throw together two or more very slightly subcritical masses so that the chain reaction is fast, resulting in a huge, short burst of energy. Both types of reactions have been proposed for propulsion systems.

  The energy density of the fission of 1 kg of 235U can be calculated from the molar mass (235 g/mol) of this isotope:

  However, in a reactor, only ab
out 3.5% of the uranium is enriched into 235U; most is in the form of 238U, which is less reactive, meaning that the energy density drops to about 3×1012 J/kg. This is still about 105 times greater than the specific energy available from chemical reactions.

  Most work on nuclear propulsion systems for spacecraft are in drives that use a nuclear reactor to heat hydrogen gas as an exhaust fuel. Hydrogen is used because, as the smallest atom, it achieves the highest exhaust velocity for a given amount of energy dumped into it. For reasons discussed below, the exhaust velocity is limited to about 8,500 m/s, or roughly twice the maximum achievable by chemical fuels.

  Work on nuclear spacecraft engines was stopped by the Nuclear Test Ban treaties of the 1970s. Up until 1972, NASA was working on a series of nuclear rocket engines dubbed the NERVA, or Nuclear Engine for Rocket Vehicle Applications series. Interest in them has revived because of a series of initiatives started during the second Bush administration and carried on by the Obama administration to have manned missions to the Moon and Mars. Nuclear rockets could potentially shorten the time to Mars to under 100 days by allowing a Δv for the initial orbital maneuver greater than 34 km/s.

  Both NERVA and the Orion concept have several advantages over chemical propellants. One is that they tend to work better for larger payloads (meaning they can be designed to give higher thrust and impulse). There is a minimum size at which nuclear reactors can be built, but scaling them up is comparably less difficult. There are, of course, obvious disadvantages of using nuclear fuel, but they tend to be overstated.