At first glance, because the ramjet picks up its fuel en route, it appears that we can keep accelerating forever, not subject to the limitations imposed by the rocket equation. Let’s make some assumptions:
1. The ramjet, of total mass M, is under way and traveling at some speed v relative to the Earth.
2. It has a funnel of some sort that scoops matter into the fusion engines. The funnel has area A.
3. Fusion uses a fraction f of the hydrogen for fuel, which yields energy with an efficiency α and ejects the rest to the rear at speed u (which depends on both f and α).
4. The interstellar hydrogen has a mass density ρ.
The thrust of the spacecraft is proportional to two things, the fuel speed, u, and the rate at which mass is being fed to the engine, dm/dt, which can be shown to be equal to ρAv. This makes sense: the faster a ship moves forward, the more hydrogen it scoops up; the large the funnel area, the more hydrogen it scoops up; and the higher the density, the more it scoops up:
The thrust increases with increasing speed, all other things being equal [44]. If we want the acceleration to be constant (meaning that the thrust is constant), we need to control the fraction f of the fuel undergoing fusion. If we intake a certain amount of hydrogen equal to m into the ramjet, the exhaust speed of the fuel can be found (in the nonrelativistic limit):
or
This means that we can write the equation for the acceleration (=T/M) in the form
If we want the acceleration to be constant (say, 1 g for the entire flight) then we must make f, the fraction of hydrogen used as fuel used by the ramjet, depend on v. To calculate f we must solve the quadratic equation
If we want a certain value of acceleration, there will be a minimum speed below which this is impossible to do. The reason is simple: at low speeds, the amount of mass going into the ramscoop will not provide enough thrust.
There is another force acting on the ship. By analogy with motion through fluids on Earth, there will be a “drag force” exerted on the ship. This drag force will have the general form
where β is a dimensionless coefficient of order 1; its exact value will depend on the design of the ramscoop. This implies a maximum velocity for the ship: as the speed gets higher, the drag force increases until it exactly balances the thrust. At this point,
The maximum value of f (1 − f) occurs at f = 1/2, so the maximum speed of the ship is
At best, α = 7×10−3 for the proton-proton cycle, so the maximum speed is going to be around 6% of the speed of light. This depends on β, of course, so there might be some way of “streamlining” the ramscoop to minimize this problem.
There are a number of problems with the idea, which probably make it untenable, apart from the speed limitation mentioned above:
• Unknown in 1960, when Bussard first published his paper, is that the Solar System is in the middle of a high-temperature “bubble” of interstellar gas about 650 light-years across whose density is relatively low, about .006 molecules per cm3, or about 6,000 /m3 [130, p. 435]. This density is about two orders of magnitude lower than Bussard’s estimates [44].
• Most of the matter in interstellar space is hydrogen; however, as noted above, fusing hydrogen is difficult because of the slowness of the reaction.
• Fusion of deuterium is easier, but deuterium is many thousands of times less common in interstellar space than hydrogen.
• Because the ramjet will work only if the ratio of the funnel area to the mass is very large, most papers concentrate on using magnetic fields as funnels.
These work only with charged particles; some have suggested using either lasers to ionize the interstellar medium or using strong magnetic fields. Materials limitations on our ability to produce large magnetic fields reduce the acceleration of the ramjet significantly once it reaches relativistic speeds, even if we ignore drag [158]. The scoop diameter also needs to be about 107 km at low speeds for it to work [159].
• Losses due to radiation from the conversion of matter into energy will limit the ship to speeds significantly less than the speed of light [215].
• When charged particles are accelerated they lose energy. These Bremsstrahlung losses may exceed energy gained by fusion of the interstellar medium by a factor of 109 [123].2
• Finally, a point I don’t think anyone else has ever brought up: an influx of charged particles at high speeds entering the ramscoop magnetic field will distort its shape in the same way that the solar wind distorts the Earth’s and Jupiter’s magnetic fields [130, p. 275]. It will compress the field in the direction of the ship’s motion and drag it behind the ship, making it less effective as a funnel.
Figure 11.1. Particles funneled into the ramscoop.
More problems with the idea are discussed in detail in John Mauldin’s book, Prospects for Interstellar Travel [160, pp. 110–116, 326]. If the ramship idea can be made to work, the issues associated with it will probably limit its speed to a small fraction, probably less than 10%, of the speed of light.
Most science fiction writers ignore this and assume that the ramjet can accelerate indefinitely. However, this speed limit puts the kibosh on novels like Tau Zero and A World out of Time. While this speed is pretty good, if we really want to get to the stars within a human lifetime, we need to go faster—much faster. How do we do that?
11.1.3 Matter-Antimatter Drives
Popular culture once again comes to our aid. As everyone knows, the Starship Enterprise, serial number NCC-1701 (or 1701A, or 1701D, etc., depending on which movie or show you watch), is powered by the reaction of matter and antimatter. It is in fact unclear from the show whether the ship is powered by these reactions or whether the ship is propelled by them when using its impulse engines. (I’m typing this in the realization that some diehard fan is going to email me, quoting chapter and verse, exactly how it is used.) Maybe we can use antimatter for spacecraft propulsion, just as the Federation of Planets does.
What is antimatter, anyway? It’s a long story. Back in the 1930s, one of the major challenges facing physics was how to marry the two newly discovered theories of special relativity and quantum mechanics. An English physicist, Paul Dirac, proposed an equation that described the relativistic quantum behavior of the electron. When the equation was solved it predicted there should be a companion particle to the electron, one with identical mass and other properties but with opposite electrical charge. Because the electron has negative charge, this new particle was dubbed the positron. The Caltech physicist Carl David Anderson detected the positron in cloud chamber experiments in 1932. It quickly became apparent to physicists that all elementary particles must have antiparticles associated with them, and the antiproton and antineutron were discovered within the next few decades.
There’s a tendency for nonphysicists to somehow think that antimatter is exotic, or a theory that hasn’t been proved yet. All these notions are untrue: antimatter particles by the trillions are produced in accelerator experiments all over the world. This is not to say that much has been produced: in 2001 CERN, a European consortium running the largest particle accelerators in the world, estimated that antimatter production is less than one-billionth of a gram per year at an equivalent cost of several trillion dollars per gram [2].
Why so expensive? Because antimatter doesn’t exist in nature, at least to any appreciable amount. Most matter is “ordinary”; antiparticles are produced naturally only in radioactive decays or in processes that have temperatures equivalent to the center of the Sun or higher. It all hinges on E = Mc2. To produce a kilogram of antimatter, you have to supply the energy equivalent to it, or roughly 9×1016 J. Put differently all of the energy used by the United States in the course of one year (1.4 × 1020 J) is equivalent to the energy produced by the annihilation of about 750 kg of matter with 750 kg of antimatter. If antimatter existed in large quantities, it could be used to fuel civilization; because it doesn’t, the tables are turned. It takes enormous amounts of energy to produce it.
This
doesn’t mean that it has to be as exquisitely expensive to produce as it currently is. If the energy costs were the only issue involved in producing it, the cost would be less than one-millionth of the current costs: gasoline, for example, has an energy density of about 108 J per gallon, meaning that we’d have to burn 900 million gallons of gas to give us the energy equivalent to 1 kg of antimatter. This would cost about $2.25 billion at 2009 U.S. prices of $2.50 per gallon. There have been several serious proposals for large antimatter creation facilities, mostly authored by the late Robert L. Forward, which would bring the cost of antimatter production low enough to make it a viable starship fuel [87].
On to the question of how to use antimatter in propulsion systems. One issue is that we have to get rid of the notion that the “annihilation” of a particle with its antiparticle somehow produces “pure” energy. Such a statement is meaningless. What really happens is that the interaction of a particle with its antiparticle produces different reactant products, similar to what happens in a chemical reaction. The amount of energy available for use in a propulsion system is determined by what the reactants are. The simplest thing to consider is an electron-positron reaction occurring in which the two particles are at rest with respect to each other. The reaction here is
This is shorthand notation for the reaction of an electron and a positron to create 2 gamma rays, that is, two very high-energy photons. If the electron and positron are at rest, the total energy of the gammas will be equal to the equivalent rest-mass energy of the electron and positron, that is, about 1 MeV or 1.6×10−13 J. “MeV” is the abbrevation for the unit mega-electron volt, the electron volt being a unit of energy equal to 1.6×10−19 J. Most particle masses are quoted in units of MeV or GeV (= 1,000 MeV) for the sake of convenience. Physicists learn to quickly convert from these units into more standard units of energy when needed. Because the electron is one of the lightest particles that have mass (only the neutrinos have less mass), an electron-antielectron annihilating at rest can produce only photons. This is good, because in principle, 100% of the rest mass of the electrons can be used as energy for propulsion.
One method of driving the spacecraft forward is to use the momentum of the photons created to propel it. A light beam shining on your hand carries a force, though a very weak one under normal circumstances. If we shine a light beam on a mirror, there will be a force acting to push the mirror forward equal to
where P is the power of the light beam (i.e., the energy per unit time hitting the mirror) and c is the speed of light. Under most circumstances, this force is weak: for example, if we shine a 1 W flashlight on the mirror, the net force is only 3×10−9 N. However, if we took 1 kg of matter and reacted it with 1 kg of antimatter to produce a beam of gamma rays directed against the mirror, and did it in a time of 1 second, the net force would be 6×108 N. Even if the mirror had a mass of 10,000 kg (10 tons), it would end up moving at a speed of 60,000 m/s!
11.2 MASS RATIOS FOR MATTER-ANTIMATTER PROPULSION SYSTEMS
Proponents of such a technology, including Forward and Robert Frisbee at the Jet Propulsion Laboratory (JPL), have put together elaborate studies of propulsion systems based on matter-antimatter reactions. There are a lot of nitty-gritty details that we won’t go into here; in particular, Frisbee’s studies seem to indicate that proton-antiproton reactions are better than electron-positron reactions for spaceship drives, even though less of the initial “mass-energy” of the reactants is available for propulsion [88]. For the sake of simplicity I will consider only cases in which the reactants are transformed into high-energy gamma rays and bounced against some sort of mirror to propel the spacecraft.
Using equation (11.5) with u = c, we find
where c is the speed of light, R is the mass ratio (mi/mf), and v is the final speed reached by the spaceship (assuming that it starts from rest). For v ≪ c, this is approximately v/c ≈ R − 1. Since c = 3×105 km/s, to get a 1,000 kg payload to Earth’s escape speed of 11 km/s takes a mass of 18 grams of normal matter reacted with 18 grams of antimatter.
Table 11.1
Final Velocity of an Antimatter Rocket as a Function of Mass Ratio
Ratio (R)
v/c
v(m/s)
1.01
0.01
3.00×106
1.05
0.05
1.50×107
1.1
0.1
3.00×107
1.2
0.18
5.40×107
1.3
0.26
7.80×107
1.5
0.38
1.14×108
2
0.6
1.80×108
5
0.92
2.76×108
10
0.98
2.94×108
20
0.995
About 3×108
100
0.9998
About 3×108
Table 11.1 shows the real advantages of using a matter-antimatter propulsion system. A mass ratio of 10 gets you to 98% of the speed of light. Unfortunately, this means that for every kilogram of payload mass you need 5 kg of antimatter to react with 5 kg of normal matter, but 5 kg of antimatter exceeds the world’s supply by a factor of about a trillion.
There are no naturally occurring sources of antimatter handy. As far as we can tell, the universe is overwhelmingly made of normal matter. This means that any antimatter will have to be created. This is possible; antimatter is created in particle accelerators all over the world, but in very small quantities. Typical generation rates for antimatter are of the order 1010–1012 particles per second, or (assuming the particles are antiprotons) 10−17–10−15 kg/s. Generously, generating 1 kg of antimatter using current accelerators will take about 1015 s, or 30,000,000 years.
However, we’re not trying to produce antimatter for spacecraft propulsion right now. Without going into details, essentially the question is one of energy. If we have access to enough energy, we can create antimatter. The questions are how much can we create, and how much will it cost?
11.2.1 The Cost and Time of Producing Antimatter
From the relation E = Mc2, we can’t generate more mass than E / c2 given energy E. So let’s make the following assumptions:
1. We have a power plant that supplies energy at a rate of P J/s to supply us with energy.
2. The energy costs us s cents per kilowatt-hour (= 3.6×106 J).
3. We can convert the energy into mass with efficiency f < 1.
4. We want to generate 1 kg of antimatter (which is equivalent to 9×1016 J).
The twin questions we want to ask are: How long will it take as a function of P and f. How much will it cost?
In time t, the power plant will generate a total energy Pt joules, or a mass of P ft/c2 kg of antimatter. Therefore, the time to generate 1 kg is
using the fact that a year is 3.16 × 107 s. Figure 11.2 shows the time it takes given power plants with energy generation rates running from 107 W to 1017 W. Two points to note: 1013 W is about the total power the world currently produces, and 1017 W is about the total power the Earth receives from the Sun.
The total cost (in dollars) can be worked out as well in terms of the energy cost s:
where M stands for “money.” Figure 21.8 shows the total cost for three different values of s: 10−3 cents/kW-hr, 10−1, and 10 (which is about the current cost of electricity production in the United States). It should be noted that both figures are on logarithmic scales, as I want to show this as a function of widely differing values of efficiencies, costs, and total power allocated to the task. Also, f ≤ 0.5 because for every particle of antimatter created, one particle of normal matter is created because of the conservation laws of physics.
Figure 11.2. Time and cost to produce 1 kg of antimatter.
What are realistic values for f given today’s technology? Robert Forward looked into this problem in detail. In a 1984 Air Force technical
report he estimated that by using specially designed linear accelerators, one might achieve an overall energy efficiency of f = 2.5×10−4 [87, p. 3].3 To the best of my knowledge, such specially dedicated linear accelerators have never been built. However, a recent paper in Physical Review Letters demonstrated anti-electron production using high-energy-density laser pulses with an overall efficiency of 1011 electrons per kilojoule, which corresponds to an efficiency of f = 8×10−6 [104]. This was a tabletop experiment, so it’s not that difficult to imagine that if it were properly scaled up, one could gain a few orders of magnitude in efficiency, to maybe f = 10−4. I don’t believe any specific physics limits overall efficiency, merely engineering details, but I could be very wrong. This is ignoring the other vexing issue of how we contain all of the antimatter we produce.
The figures make it clear: under almost any reasonable conditions, this is an expensive and lengthy task. I’ve marked a point on each using dashed lines: assuming that we put all of the power of a 1 GW power plant toward this task (which is just barely possible today) and figured out a way to do this at 10% efficiency (i.e., f = 0.1) (which is probably impossible), it would take about 30 years and cost $25 billion.
Using a more reasonable efficiency of 10−4 we could produce 1 gram of antimatter in the same time at the same cost, which is more or less the goal Forward was hoping to achieve in his report. Forward’s estimates of costs are similar to mine: he calculated that with f = 10−4, the cost would be 107 $/mg [87, p. 150].
Wizards, Aliens, and Starships: Physics and Math in Fantasy and Science Fiction Page 17