Wizards, Aliens, and Starships: Physics and Math in Fantasy and Science Fiction
Page 8
The maximum travel distance R occurs for an angle of 45 degrees. Table shows the minimum speed for our ballistic flying car needed for various travel distances.
The third column is the total flight time in minutes (based on the speed and distance traveled). It isn’t bad: we can travel a thousand kilometers in under ten minutes. One issue is that the formula can’t be trusted for distances much greater than 1,000 km because at that point the curvature of the Earth must be taken into account. For example, the estimate of velocity for a 10,000 km hop is 10,000 m/s, which is significantly higher than the speed it would take to put the car into orbit. At this point we are into orbital flight, which is covered in the next chapter. The kinetic energy of the flying car is derived under the assumption that the car has a mass of 1,000 kg, which is probably too low but ballpark correct.
Table 5.1
Projectile Distance as a Function of Launch Speed
The fuel costs kill this idea, at least at the longest distances. Under some reasonable assumptions, the 1,000 km hop is going to run at least a few thousand dollars in fuel. This is ignoring the practical engineering stuff about how to fling someone around in a can like this and still have him or her survive the fall. That step will involve a lot of fuel as well. I don’t think we’ll see the flying car anytime soon.
NOTES
1. The subject of reversible computing is too complex to go into here. For a good discussion of why, in theory, computation can take no energy, see The Feynman Lectures on Computation for a very readable introduction to the subject [82, chap. 5, “Reversible Computation and the Thermodynamics of Computation”].
CHAPTER SIX
VACATIONS IN SPACE
6.1 THE FUTURE IN SCIENCE FICTION: CHEAP, EASY SPACE TRAVEL?
The last conjectures may seem absurd to those who look still on manned space flight in terms of todays multimillion dollar productions. But … the cost and difficulty of space travel will be reduced by orders of magnitude in the decades to come.
—ARTHUR C. CLARKE, THE PROMISE OF SPACE
In an early scene in the movie 2001: A Space Odyssey, Dr. Heywood Floyd travels via a commercial (Pan Am) shuttle to a space station in orbit around the Earth. The humdrum way in which this is shown in the movie makes it clear that in the future world of 2001, such “flights” are common. This scene is almost unique among science fiction movies for actually getting the dynamics of space flight correct. Clarke’s observations from his popular 1968 book on space exploration reinforced this idea in the movie: in the future, space travel will be as easy and as common as traveling on an airplane.
This is the great theme of science fiction, one that has permeated the genre since it became a separate branch of literature. Essentially every science fiction writer between 1900 and 1980 published stories involving interplanetary travel, even those writers who weren’t primarily known for writing hard science fiction. A sampling of some well-known stories from this time period makes the point clear:
• From the Earth to the Moon, by Jules Verne. This is not the first story of space travel (Verne was anticipated by Cyrano de Bergerac and Johann Kepler), nor is it scientifically accurate: his astronauts build a cannon to launch the spacecraft, which in reality would smash them to jelly on launch [241].
• First Men in the Moon, by H. G. Wells. Again, this isn’t scientifically accurate: Prof. Cavour invents a metal that shields anything placed above it from gravity. Even at the time, this was known to be impossible because it violated the first law of thermodynamics: one could use such a metal to build a perpetual motion machine. Regardless, the intrepid heroes build a spacecraft made from Cavourite and travel to the Moon where they meet the insect-like Selenites living in caves beneath the surface [249].
• Ralph 124C41+, by Hugo Gernsback. This represents the prehistory of American hard science fiction. Space travel isn’t central to the theme, but the eponymous hero travels into space at the end of the novel [91].
• Rocket Ship Galileo, by Robert Heinlein. Heinlein is one of the best of the golden age space writers, and this book represents a wonderful blend of good writing and good science. In the book three Boy Scouts and an adult build a nuclear-powered spacecraft, anticipating the NERVA program of the 1970s. It’s still worth reading, and is one of the books that drew me into science fiction in the first place. It is also worth looking at because it exemplifies what the science fiction writers paid attention to (the science) and ignored (the enormous infrastructure costs associated with projects like this). But more on this below [109].
• The Martian Chronicles, by Ray Bradbury. The science is unimportant to the story, but it does, of course, center on the exploration of Mars by humanity [40].
• Ubik, by Philip K. Dick. This is a very weird novel mostly taking place in an artificially created afterlife, but the first scenes of the novel involve a trip to the Moon. The world in which the story takes place, as in most of Dick’s stories, is a near-future Earth where travel to outer space is as common as airplane travel is today, although such details are seldom important to the story. One should note that the “near future” date of Ubik was sometime in the 1980s, now thirty years past [67].
• Neuromancer, by William Gibson. Usually thought of as the first cyberpunk novel, Neuromancer is firmly set in a world consistent with earlier science fiction expectations: the main characters travel to inhabited space stations in orbit around the Earth toward the end of the novel. Gibson is an interesting case because his earlier novels are firmly set in this milieu but his later ones essentially abandon these science fiction trappings [95].
I could cite hundreds of other examples. The entire genre was essentially considered the literature of space travel from about 1930 to 1980 and is still strongly linked to it in the popular imagination, especially in movies, TV, and other expressions of popular culture. One important point is that space travel is commonplace even in stories in which the plot doesn’t depend on it at all. Of the novels mentioned above, both Ubik and Neuromancer could easily have been written without space travel. So pervasive was the idea of space travel as a commonplace of the future that both Dick and Gibson included it in their novels as a matter of course.
Accurate depictions of the science of space travel and rocketry began to appear in science fiction stories in the 1930s and 1940s; this is probably associated with the publication of Robert Goddard’s massive paper, “A Method of Achieving Extreme Altitudes” in 1919 [98]. He was not the first to do serious scientific work on the subject—Tsiolkovsky in Russia and Oberth in Germany anticipated his work—but his work represents the first English-language publication on the applications of rocketry to space travel. In the article he derives a number of results, including the “rocket equation” I show below. The founding of the British Interplanetary Society in the early 1920s also increased popular interest in the subject and spurred further research on these ideas. Again, the thought was that in the near future, interplanetary travel, or at least travel into orbit, would be relatively common. This didn’t happen, although in the early 2000s we began to see the beginnings of space tourism, albeit for the very rich. So, what happened? Why don’t we fly to the Moon on gossamer wings today?
6.2 ORBITAL MECHANICS
We’ll begin by considering the cheapest trip into space: a flight into orbit around the Earth. There have been seven people who have booked a vacation in orbit through Space Adventures, a company devoted to space tourism; Dennis Tito in 2001 supposedly paid $20 million for a ride to the International Space Station. This is clearly far beyond the reach of the ordinary Joe or Jane, but the flight was supposed to inaugurate the era of space tourism, with prices falling and larger numbers of people heading up and out. However, there has been little progress on this front in the last decade.
At first glance it would seem that the biggest problem with traveling into space is that of energy. It takes a lot of energy to get into orbit. The cost of this energy is expensive in today’s society. So we need to ask two quest
ions:
1. How much energy does it take to go into space? What are the costs of this energy usage?
2. Is this the limiting factor that makes space travel expensive? If not, what other factors are important?
6.3 HALFWAY TO ANYWHERE: THE ENERGETICS OF SPACEFLIGHT
Once you’re in orbit, you’re halfway to anywhere.
—ATTRIBUTED TO ROBERT A. HEINLEIN
There seems to be a common misconception that the late, great Space Shuttle could fly to the Moon if it had needed to. This idea was fostered by the heritage of the Apollo space program and an unfamiliarity with the scale of space. The shuttle orbit was typically 350 km (about 200 miles) above the surface of the planet, or less than the distance between Washington, D.C., and New York City. The distance from the center of the Earth to the Moon is some 380,000 km (about 238,000 miles), or about a thousand times greater than the highest shuttle orbit. If Earth were shrunk to the size of a basketball, the orbit of the shuttle would be about half an inch above the surface of the ball and the Moon would be about the size of a tennis ball, 30 feet away.
If the orbit is so close, why is it so hard to put the shuttle into orbit? You don’t need huge launchers to go from Washington to New York, after all. Maybe it’s because you have to go 200 miles up, instead of horizontally. This is part of the answer, but not all of it. The big reason has to do with that word “orbit.” To simply launch the shuttle 200 miles straight up would take less energy than to put it into orbit (by my calculations, about one-fourth as much). However, such a rocket would rapidly fall back to Earth because it wouldn’t have the speed to stay in orbit once there.
6.3.1 What Goes Up
People on the surface of the Earth don’t fall through the Earth because of the solidity of the ground. The ground exerts an upward force on our feet that balances the downward force of gravity. Airplanes don’t fall out of the sky because of the lift generated by the flow of air around their wings. But what keeps the shuttle up? It does have wings, but they’re only for landing at the very end of a mission. In space, as we’re told, no one can hear you scream because there’s no air, and without air there is no lift to keep the shuttle up.
What is very interesting is that nothing keeps the shuttle up. In fact, it is continually falling toward the Earth but always missing it. The orbit of the shuttle is nearly circular: we can think of the orbit as something like a stone swing on a rope by a child at play. If the rope broke, the rock would fly off at a high speed at a right angle to the rope, following Newton’s first law: if no net force acts on an object, it moves in a straight line at a constant speed. The stone flying free moves on a straight line in the direction it was just moving in before the rope broke (fig. 6.1 has a picture of the stone’s path). However, the rope constrains it to move in a circle. The shuttle is just like that stone, with the rope being the attractive force of gravity between it and Earth.
If any object (the shuttle, a stone on a rope, a car turning around a curve) moves in a circle at some speed v, it must be acted on by a force directed toward the center of the circle. The size of the force is given by
Figure 6.1. Rock whirled on a string with speed v and radius r.
where M is the mass of the object and r is the radius of the circle. Again, the force can be anything so long as it is centrally directed (“centripetal,” in physics jargon). The force on the shuttle is the force of gravity and is given by
Here, G is a universal constant (which has the value 6.67 × 10−11 Nm/kg2 in metric units), M is the mass of the Earth (5.98 × 1024 kg), and r is the distance from the shuttle to the center of the planet, which is equal to the radius of the Earth plus the height of the shuttle orbit (about 6,800 km, or 6.8 × 106 m). We can combine the two equations to solve for the speed to the keep shuttle in orbit:
One should note that the speed doesn’t depend on the mass of the shuttle. Putting the numbers in, vorbit = 7,600 m/s (7.6 km/s, or a whopping 17,000 mph). If it were any slower it would crash into the ground. Bigger speeds lead first to noncircular orbits, and eventually to the spacecraft never coming back.
The difficulty of putting the shuttle into orbit isn’t mainly how high its orbit is but how fast the shuttle is moving. The payload mass of the shuttle is about 100,000 kg (roughly 100 tons). The kinetic energy of any object moving at a speed v is given by the formula
Using this formula, the kinetic energy of the shuttle is about 3 × 1012 J, or about 3 million MJ (megajoules). The energy density of a gallon of gasoline is roughly 100 MJ per gallon, meaning that about 30,000 gallons of gasoline contain the equivalent energy. At an average U.S. gasoline cost of $3.50 per gallon, we get a cost of about $105,000 for the amount of gasoline containing the equivalent kinetic energy. However, that isn’t the entire story.
6.3.2 The Rocket Equation
The tricky part is that a rocket burns fuel to reach such a high speed. A lot of fuel. Typically, the mass of the fuel is much more than the mass of the payload for any rocket; for example, the total liftoff mass of the shuttle orbiter plus solid rockets plus tanks is about 2,000,000 kg, or twenty times the payload mass. The reason for this is that when you burn fuel to move a spaceship, you’re moving not only the spaceship itself but also the mass of the fuel you are carrying. How rockets work is pretty straightforward and has to do with Newton’s third law that for every action there is an equal but oppositely directed reaction. To give an analogy, let’s say you are standing on a patch of frictionless ice with your little sister. You have a mass of 80 kg, she has a mass of 40 kg. She’s bugging you, so you give her a shove so that she goes flying off at a speed of 4 m/s to the right. You will find yourself moving at a speed of 2 m/s to your left. Because you exerted a force on her to push her away, she automatically exerted a force on you of exactly the same size but in the opposite direction. Because you have a great mass than she does, you end up moving more slowly than she does, but you still move.
This is how a rocket works: the engine of a rocket burns fuel. The combustion of the fuel supplies the energy to move the combustion products at high speed; the engine is designed to channel the rapidly moving gases “backward”; action and reaction dictate that the ship experiences a force that moves it forward. The fuel is characterized by an ejection speed, u, which is typically a few thousand meters per second. The ejection speed is the speed at which the burned fuel is thrown from the back of the rocket. The thrust (net force pushing the rocket forward) is then given by the product of the exhaust velocity and the rate at which fuel is being burned, which we label dm/dt. We can use Newton’s second law to write down a differential equation for the speed, v, of the rocket:
where m is the mass of the fuel + payload of the rocket at any given time and dm/dt is the rate at which the fuel is being consumed. Two points need to be made here:
1. The acceleration of the rocket isn’t constant because the mass is continually changing.
2. We are ignoring the effects of gravity on the speed of the rocket. In fact, getting to some final velocity will require more fuel than we will calculate using this equation, but the difference is relatively small in the case of the shuttle.
We can now solve the equation using elementary calculus. Assuming that the rocket started with zero speed and ended up with speed vf, we find
Here, mf is the total fuel mass and mr is the mass of the rocket minus its fuel. This is the famous “rocket equation,” which should be memorized by any serious science fiction writer (or fan, for that matter). The shuttle booster rockets have an exhaust speed of about 2,600 m/s. Table 6.1 shows final speeds given various initial fuel mass to rocket mass ratios.
Table 6.1
Final Velocity of a Rocket as a Function of Mass Ratio
mf/mr
Final Velocity (m/s)∗
0.1
248
0.3
682
1
1,802
3
3,604
10
6,234
20
7.915†
30
8,958
50
10,222
100
12,000
∗ The fuel ejection speed u = 2,600 m/s.
† Approximate orbital velocity of Earth.
Note that for large fuel ratios, the rocket can travel faster than its exhaust speed. It turns out that the shuttle needs a mass ratio of 20:1 to achieve orbital speed, according to the rocket equation. This estimate is pretty accurate: the total liftoff mass is about 2 million kg, or about twenty times the mass of the orbiter. We’re off by a bit because we’ve assumed only one stage; that is, we’ve assumed that our rocket burned all of its fuel in one continuous burn, whereas the shuttle has three stages (three separate fuel burns) as it goes into orbit.1.
Two other concepts are important for rocketry:
• Specific impulse (Isp): This is just the ejection speed, u, in disguise: Isp = u/g, where g is the acceleration of gravity. I’m not sure why rocket scientists use this instead of u, but there you are.
• Thrust (T): The net force that pushes the rocket forward, which is given by the formula
This is also essentially a materials characteristic, determined by the type of fuel you are using. Rockets are characterized by these two concepts: some rockets have high thrust but low specific impulse, like chemical propellants; low thrust but high specific impulse, like ion rockets; or high impulse, high thrust, such as the Orion propulsion system. Low-impulse, low-thrust rockets are useless for spaceflight. I mostly discuss high-thrust propulsion systems since they are the only ones capable of lifting materials from Earth into orbit.