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14 декабря, 2021
Fissionable uranium (U235) cannot break up into two iron atoms because iron has 56 nucleons, and uranium has 235, which is a lot more than two times 56. To break it up into three or four big pieces would be very unlikely. So uranium fissions into two atoms larger than iron: typically, into krypton (Kr89) and barium (Ba144), whose atomic numbers add up to 233. There are then two neutrons left over, and it is these that carry off the generated energy and keep the chain reaction going. The energy released is not maximal since uranium moves only about halfway down the right-hand slope.
Now look at fusion at the extreme left of Fig. 4.2. When heavy hydrogen in the form of deuterium (H2) and tritium (H3) combine to form helium (He4), with one
Box 4.1 What is Binding Energy?
Suppose we have two pitchers, one 30 cm (1 feet) tall and the other 60 cm (2 feet) tall. We then drop a ripe tomato into the short pitcher. The tomato releases some energy by making a plunk! sound. It is bound to the pitcher because it takes energy to lift it out. Now we drop another ripe tomato into the tall pitcher. It releases more energy by, perhaps, going splat! It is more tightly bound to its pitcher because it takes twice as much energy to lift it out. When each tomato drops down, it loses gravitational potential energy and gains binding energy. Therefore, binding energy is the negative of potential energy. That is why Fig. 4.2 makes more sense than Fig. 4.1. Since the sum of potential energy and kinetic (motion) energy remains the same, kinetic energy is increased when potential energy is decreased; or, equivalently, binding energy is increased. In nuclear reactions, the increase in kinetic energy goes mainly to the lightest resultant particles, usually the neutrons. In both fission and D-T fusion, the neutrons are captured and their kinetic energies turned into heat.
Number of nucleons in nucleus
Fig. 4.2 An inverted binding energy diagram showing that going downhill from either side will release energy
neutron left over, there is a very sharp increase in binding energy. The curve is so steep that a lot more energy is released than in fission. However, this is energy per nucleon, and uranium has many more nucleons than hydrogen. After this is accounted for, the total energy gained per reaction is larger in fission than in fusion. This is not important. The end result is that both processes create large amounts of energy by forming elements closer to the middle of the periodic table.
The materials involved are, however, very different. In fission, uranium has to be mined and transported to huge isotope separation plants. Raw uranium is mostly U238. Only 0.7% of it is U235, the part that is fissionable. The separation plants enrich the mix so that there is a higher percentage of the good stuff. The products of fission are highly radioactive, some for thousands or millions of years. This is a well — known problem with fission.
By contrast, fusion uses only hydrogen, which occurs in three forms. Normal hydrogen, labeled as H1 in Fig. 4.1, contains only a single proton. Deuterium (H2) contains one proton and one neutron; it is “heavy hydrogen.” Tritium (H3) is heavier, containing one proton and two neutrons. The sun produces its energy by converting H1 hydrogen into helium through a sequence of reactions which we cannot duplicate on earth. Here, we cannot do as well and must be content with converting heavy hydrogen, H2 or H3, into helium, but the energy gain is still very large. The reaction product is helium, whose nucleus, also called an alpha particle, consists of two protons and two neutrons. It is very tightly bound, so helium is very stable. This stability causes it to be the harmless gas used to fill birthday balloons. Deuterium, which we will call D, occurs naturally in water. In heavy water, D replaces the H in H2O. There is one part of D2O for every 6,400 parts of H2O, and it is easy to separate it out. No mining or large separation plants. However, the other fuel, tritium (H3 or T for short), does not occur naturally. It is also radioactive and decays in 12.3 years. It has to be bred from lithium in a fusion reactor. You may have noticed that deuterium contains one proton and one neutron, while helium contains two protons and two neutrons. Why not fuse two D’s together to get helium? Well, this is hard and will come only in the second generation of fusion reactors. Right now, we are trying to fuse D with T to get helium plus an extra neutron. That neutron carries away most of the energy generated, but it also causes some radioactivity, but much less than in fission. For the future (Chap. 10), there are other advanced reactions involving helium-3 (He3), lithium, or boron which are completely free of radioactivity. Note that lithium and boron are abundant and safe elements on earth.
Since tokamaks depend on an internal plasma current to produce the required twist of the magnetic field lines, the current has to be produced even if it is not needed for ohmic heating. Fortunately, the plasma automatically generates such a current,
figuratively “pulling itself up by its own bootstraps.” This comes about as follows. Since the plasma is not perfectly confined but gradually diffuses to the wall, there will be a density gradient, with the density high in the center and low near the walls, where the plasma can leave quickly. Think of a packed football or soccer stadium where, at the end of a game, the crowds storm onto the field in spite of the guards. The density of people is high at the top, but the crowd disperses on the field, where the density is low, forming a density gradient. It is this density gradient in a tokamak that causes the bootstrap current. Technically, it is the pressure gradient, where the pressure is density times temperature. Consider a tokamak with its helical magnetic lines, as shown in Fig. 7.17. The twist in the lines of the magnetic field is created by a toroidal current J, which generates the poloidal component, Bp of the field. It is this poloidal part of the field which is important here.
Figure 7.18 is a closer look at the minor cross section of the plasma showing the same tokamak current J seen in Fig. 7.17. The black arrows show the force on the electrons exerted by the plasma pressure pushing outwards. We can neglect the ions because they move so slowly that they cannot carry much current. The electrons gyrate in small circles, so we need only to consider the drift of their guiding centers. In Chap. 5, we showed the gyroscopic effect on guiding centers, in which a force moves the guiding center in a direction perpendicular to that force. The relevant part of Fig. 5.7 is reproduced in Fig. 7.18a, showing that the B-field, pressure force, and electron velocity are mutually perpendicular. Note that the current is opposite to the electron velocity because of their negative charge. In Fig. 7.18b, the force is in the radial direction, pushing outwards, while the poloidal field Bp is in the azimuthal direction, going around in the circular direction. The electrons therefore drift in the direction perpendicular to both, which is the toroidal direction, the same as that of J. This toroidal electron drift constitutes the bootstrap current. It turns out that this current is always in the same direction as J, so that it adds to the total current. Once a seed current is induced in the torus so that the field lines twist enough to confine the plasma, the bootstrap current can then take over most of the work. There is, of course, also a pressure-caused drift perpendicular to the main toroidal component of the B-field, but this drift is in circles in the poloidal direction and does not contribute to the main tokamak current J.
Mother Nature made it hard for us to confine a plasma in a torus by requiring that the field lines be helical, but she then provided the benefit of bootstrap current so that this helicity can be mostly self-generated. It does not matter which direction the toroidal field is in, or which direction the toroidal current is in; the bootstrap current will always add to the toroidal current. In present experiments, the bootstrap current has been observed to contribute more than half the total current. In planned experiments, the bootstrap fraction will be more than 70%, and in fusion reactors more than 90%.
Detailed calculations3 of bootstrap current can be made using the neoclassical banana theory described in Chap. 6. Although collisions between passing particles and those in banana orbits cause the major part of bootstrap current, the final answer does not depend on knowing the collision rate. Collisions cause the pressure gradient, but it is only the resulting pressure gradient that matters. Going back to the stadium full of fans, we see that a density gradient of fans will occur regardless of whether they bump and shove one another or whether they do not touch.
Fig. 7.17 A tokamak with a toroidal current J, which generates a poloidal field Bp, giving a twist to the magnetic field |
Fig. 7.18 Generation of the bootstrap current perpendicular to both the pressure gradient and the poloidal field |
In designing tokamaks, the shape of the bootstrap current depends on the shape of the magnetic field, which itself depends on the bootstrap current, so a delicate optimization problem has to be solved. The so-called Advanced Tokamak designs with high bootstrap fraction have hollow current profiles with larger current at the edge than in the center.
Bringing the plasma up to fusion temperatures is done with the injection of neutral atoms and the excitation of different types of plasma waves. In addition, waves are also used to drive the plasma current without using transformers — so-called noninductive current drive. There are many physics problems involved in these processes. Neutral beams also fuel the plasma and give it rotational velocity. Waves not only heat the plasma and drive its current but are also used to change local conditions inside the plasma and shape the current profile. In this chapter, we are concerned with technology and therefore concentrate on the hardware and discuss only the main types of waves that can be used.
Originally, glass microballoons containing DT gas were used as targets. They were like the glass beads used to coat projector screens but had to be perfectly round and smooth.
One is shown in Fig. 10.45a, and a number of them are shown on a coin in Fig. 10.45b. When hit with lasers, the glass exploded, half going out and half going in, compressing the gas. This is how the first fusion neutrons were observed.
Later targets used low-Z ablators to have a more controlled compression (Z is the atomic number). Examples of target designs are shown in Fig. 10.46. All of them have a shell of frozen DT as the fuel. In panel (a), there is also a bit of DT at the center, confined by a heavy pusher. This is supposed to ignite first, giving energy to help ignite the main fuel. In panel (b), the ablator is polystyrene foam, which allows DT gas to be permeated into the capsule without using a fine tube, as in Fig. 10.45a. The DT is frozen at cryogenic temperatures, and is melted and smoothed by the little bit of heat from the decay of the tritium. In panel (c), a beryllium
Fig. 10.45 Glass microballoons used as laser fusion targets, (a) magnified, and (b) in real size. (Photo from the author’s archives; original from a national laboratory: Livermore, Los Alamos, or Sandia.) |
Fig. 10.46 Examples of capsule designs: (a) with central ignition [43]; (b) with plastic foam [42]; (c) with beryllium ablator [44] |
ablator is used in a design to optimize shock heating. To improve compression, multiple shocks can be created by shaping the laser pulse into increasingly strong steps. Since strong shocks travel faster than weak ones, multiple shocks can be timed to catch up with one another just when they reach the center.
Target design is very computation-intensive, since the progression of the implosion has to be predicted. Designs differ depending on their purpose and the driver. Making just one of these targets takes great skill and cost. In a reactor, each pellet can cost no more than $0.50. Surprisingly, it is predicted that in mass production, these targets can be made for only $0.16 each [45]. Tens of thousands can be made at once in fluidized beds, and the infusion of DT into the spheres and the freezing of a layer at 18 K can be done to a whole batch at once since injection of DT through individual micron-size tubes is no longer necessary.
Normal batteries like the AA — and AAA-size ones we use everyday are sandwiches of three materials made into long sheets, as shown in Fig. 3.55a. The anode and cathode materials are separated by a thin insulating sheet, and all three are made as thin as possible and rolled up tightly to fit the largest area into the smallest space. The anode and cathode materials have a chemical potential between them such that the anode is negative and the cathode is positive. They are connected to the contacts at the bottom and top of the battery, respectively. When a light bulb is connected to the contacts, an electric current flows, lighting the bulb, and discharging the built-up charges between the sheets. The chemical potential sets the voltage of the battery, typically 1.5 V, and the area of the sheets determines how much charge they can hold, and therefore the “life” of the battery. Most batteries are not rechargeable.
Lithium-ion batteries are rechargeable. How they work is illustrated in Fig. 3.55b, where the anode and cathode layers are represented by shelves holding Li ions. The anode material is usually graphite (loosely packed carbon) holding some positive lithium ions. The cathode can be made of any of a number of materials, including proprietary ones, which largely determine the performance of the battery. Before the two electrodes are connected together, the chemical potential between them
Fig. 3.55 (a) Construction of a battery; (b) Layers of a lithium-ion battery [33] draws the lithium ions from the anode to the cathode until the extra positive charge added to the cathode cancels out the chemical potential. The ions travel through an electrolyte, which is a conducting liquid like salt water, only thicker. It is the gooey stuff that leaks out of an old battery. A thin plastic sheet, the separator, prevents the electrodes from touching each other. The separator is thin enough to allow the ions to pass through. A short circuit develops if there is a hole in the separator. Now if the battery is connected to a load, electrons which are attracted by the extra positive charge on the cathode can flow through the load to do useful work. As shown, the electric current is in the opposite direction to the electron motion because the electrons carry negative charge. To recharge, a negative voltage is applied to the anode to draw the lithium ions back. This is what takes hours. A large battery pack could consist of 100 cells, each 5 cm in diameter and 20 cm long (4 x 8 in.), divided up into modules so that overheating in one module does not spread to others.
As for cathode materials, cobalt-containing compounds such as cobalt dioxide have high-energy density and are commonly used for small Li-I cells, but they are not suitable for cars because of a tendency toward thermal runaway. The best found for cars so far is iron phosphate, which is more stable and less likely to overheat. It gives lower voltage, so that chains of batteries have to be longer to provide a high output voltage. Higher power and longer life are claimed if the cathode is made with nano-sized divots to increase surface area [33]. More on this will come in the next section. The race to make the best iron phosphate battery has already led to patent fights among battery companies.
The long charging times for Li-I batteries have been overcome by Ceder et al. [34] working with LiFePO4 (lithium-iron-phosphate) cathode material.
A123 Systems, a company started in Boston, has expanded into a $91M business in Asia using this material in small batteries for power tools and hobbyists.59 Employing techniques from ultracapacitors (next section), Ceder et al. form the cathode in such a way that it has large surface area with channels aligned so that Li ions can get in and out of the cathode rapidly. In small samples, discharge times of the order of seconds were observed, more than ten times faster than normal. Critics, including J. Goodenough, an inventor of LiFePO4 cathodes, doubted that charging times could be as short as discharging times.60 However, Ceder claims that the rates are for both charging and discharging. If we accept that, there is still a problem with charging a car, even a hybrid, in 10 minutes. It requires a lot of power. A plug-in hybrid using 0.24 kWh/mile can go 40 miles (64 km) on about 10 kWh of electricity. To put that much energy into a battery in 10 minutes would require 60 kW of power, enough to run an office building. Charging at home would have to be scheduled so that not everyone on a grid line plugs in at once. However, there is no need to charge that fast at home; overnight will do. Where fast charging is needed is in filling stations en route. To charge nine cars at once would require half a megawatt of power. Probably high-voltage lines and a small substation would be required at each “gas” station. Some people suggest that such stations should have large battery banks to store the energy slowly and continuously so that not so much instantaneous power is needed. In any case, building the infrastructure to support electric cars is worthwhile for saving oil and cleaning up the environment. Ultimately, when oil runs out and fission and fusion plants generate most of the energy for transportation, the electric grid will have to handle the power for all vehicles.
So far, we have encountered no insuperable problems. We can build a torus with a helical magnetic field and nested magnetic surfaces which should contain a plasma. We know how to make coils that will generate the 1-T magnetic field to hold the plasma pressure. Even if the plasma pressure is higher than 3-4 atm, a mere doubling of the field strength to 2 T will hold four times as much plasma because the field pressure increases as the square of the field strength. That toroidal fields which hold single particles for millions of traverses around a torus was shown very early in the game [3]. As we shall see, the Lawson criterion on the nt product would be easy to attain if a plasma behaved like a normal gas. The problem is that plasma is a special kind of gas, and an ornery one at that.
We said before that “plasma” is a misnomer because a plasma is not easily shaped or formed. Nature abhors a vacuum. A magnetic field is a vacuum. Plasma will try to cross the field and expand to fill the material container. Although the magnetic field keeps each ion and electron spiraling in a Larmor orbit so that each particle by itself cannot cross the field lines, the ensemble of particles can form ways to escape. This is because the particles are charged and can clump together to create electric fields, and these electric fields can take plasma across field lines. The plasma behaves more like a fluid (air or water, for instance) than like a collection of particles, each acting by itself. Since the particles are charged, a plasma can pull Houdini tricks that air or water cannot. Like an ant colony, the community can accomplish more than the individuals. Metaphors aside, these escape mechanisms are called instabilities, which are responsible for the slow progress in fusion up to now, and which are the subject of most of the technical literature on plasma physics. Before we can describe instabilities, we have to tell more about how a plasma behaves.
In fusion, ELMs are not trees but edge-localized modes. The name itself suggests that they are not understood, not unlike the term assigned to the Irritable Bowel Syndrome. The name has even spawned an adjective, ELMy, and a participle, ELMing, which should give philologists conniptions. ELMs occur at the pedestal in H-mode plasmas (Chap. 7). Recall that in this high-confinement mode, a transport barrier, shown earlier in Fig. 7.25, is formed at the edge of the plasma.
This thin layer holds back the plasma because it quenches all instabilities with strong electric field shear. But it can’t do that forever. If the plasma escaped at the classical diffusion rate due to collisions alone, the plasma pressure in the interior would rise so high that the barrier would break down. This breakdown occurs in short bursts, called ELMs, so that there is a steady release of plasma to the outside. Actually, this is a good thing because the “ash” of the DT reaction has to be taken out. This ash is the cleanest ash ever — pure helium — but it has to be removed because otherwise the expensive magnetic field would be used up in confining the ash rather than the fuel.
The H-mode occurs only when the heating power exceeds a certain threshold value. ELMs occur when the power is just above this threshold and are really localized near the plasma edge. Recall that the “edge” of the plasma is defined by the divertor, like the one at the bottom of Fig. 8.13. The plasma edge is defined by the last closed magnetic surface, the one at the X made by the field lines just above the divertor. Plasma venturing beyond that is led into the divertor, where it strikes high- temperature materials with heroic cooling to dissipate the heat. Also shown in the figure is the layer where the H-mode barrier exists and, inside that, the core plasma. The problem with ELMs is that the heat comes in short bursts — less than 1 ms — occurring a few times a second, and divertors cannot handle a heat flow that is not steady. A single ELM, while it lasts, can carry 20 GW of power, an energy flow
Blanket and
first wall
Region I
Core plasma
Region ll
Plasma edge and
H-mode
confinement barrier
Region ill
Scrape-off layer
Divertor plasma
Divertor chamber
Fig. 8.13 Cross-section of a tokamak with a single-null divertor, showing the scrape-off layer [16]
comparable to that of the Three Gorges Dam in China [17]. There are thus three tasks: measuring what ELMs do, explaining what causes them, and devising a way to suppress them.
It’s hard to measure what goes on inside the thin barrier layer during the unpredictable time when a burst occurs, but there is a large data base on the different types of ELMs and the conditions before and after they occur [18]. Three types of ELMs have been observed. As the heating power is increased past the H-mode threshold, Type 3 ELMs first occur. These occur rapidly, each with a small energy release. They come after a magnetic precursor signal can be detected. As the power is raised, the ELM frequency decreases until there are no ELMs at all. Then Type 2 ELMs, called “grassy” ELMs, occur; they are very small, rapid bursts whose time traces resemble grass. Further increase in power produces Type 1 ELMs. These occur in most H-mode tokamaks and release energy in rather regular bursts. Each pulse occurs when the density and temperature at the top of the pedestal reach critical values, and these drop when an ELM occurs. Density and temperature then recover slowly until the next burst is triggered. Although ELM-free discharges can be produced, they cause the temperature and density at the top of the pedestal to be rather low, and these control the quality of the fusion plasma in the main volume. It is found that the best fusion conditions can be produced by ELMy H-mode plasmas, in which the plasma is allowed to escape in regular Type 1 ELMs.
Many theorists [19] have worked on the ELM problem, and the consensus is that ELMs are a magnetic instability called a “peeling-ballooning” instability. Computations can predict the temperature and density values in the pedestal that can trigger an ELM, but they are far from explaining all the features that have been observed. And, as usual, there is no guarantee that another theory can’t also explain the ELM threshold. There is, however, good news. The DIII-D team at General Atomics have figured out a way to suppress ELMs without degrading the quality of the core plasma [20]. They apply “resonant magnetic perturbations” with an array of small coils just outside the plasma edge. These produce small magnetic islands in the edge region which work some kind of magic. Experimental results are promising enough that such coils are being considered and designed to be added to ITER.5
The engineering of a fusion reactor will require solution of a number of serious technological problems, as we have seen above. ITER will take decades to build and operate, and it is not designed to solve many of these problems. It is therefore prudent to build smaller machines specially designed for technology development so that this work can proceed in parallel with ITER. Many proposals have been made for a fusion development facility (FDF). A few of these will be described here.
IFMIF: International Fusion Materials Irradiation Facility
A favorite proposal of the European Union, together with Japan, is the IFMIF, a large linear accelerator that has been in the planning stage for 16 years. A diagram of it is shown in Fig. 9.30. As you can see, this is a large installation. The accelerator occupies a building of several hundred meters in length. It is designed to produce neutrons with energies matching those that would enter a tokamak blanket. This is done by accelerating to 40 MeV a beam of deuterons onto a target of liquid lithium. Reactions like the reverse of that in Fig. 9.10 would occur: a deuteron on lithium-6 would produce beryllium and a neutron, and a deuteron on lithium-7 would produce beryllium and two neutrons. The neutrons would then be used to bombard different materials to see how they stand up.
The key parameters for assessing radiation damage are neutron flux, neutron fluence, and dpa. Flux is how many neutrons per second go through each square meter. Fluence is how many have gone through the area during the whole life of the material. Dpa measures the damage, either per year or for the whole life. The flux produced by IFMIF is comparable to that expected in ITER, and about four times less than that in DEMO. The dpa per year in IFMIF is comparable to that in DEMO (about 30) and much larger than at in ITER.5 The fluence cannot compare with that in DEMO, but could duplicate that in the limited life of ITER.
The IFMIF will cost about $700M [22]. It has been severely criticized because only small samples, a few square centimeters in size, can be tested. This is entirely inadequate to test the large components of ITER and DEMO, especially the blanket modules.
Ion Source
High Energy Beam Transport
Fig. 9.30 Diagram of the International Fusion Materials Irradiation Facility [A. Moslang (Karlsruhe), Strategy of Fusion Materials Development and the Intense Neutron Source IFMIF]
Proposals to build small but powerful tokamaks to test burning plasmas were made well before ITER. In the late 1980s, a Compact Ignition Tokamak was initiated in the USA, but was soon canceled. In 1999, Dale Meade at Princeton designed a 10-T, 2-m diameter tokamak call Fusion Ignition Research Experiment (FIRE), but this was never funded. These early ideas were based on the hope that very high magnetic fields produced without superconductivity could be used to achieve ignition on a small scale. This philosophy, promulgated by Bruno Coppi at the Massachusetts Institute of Technology, resulted in the Alcator tokamaks at M. I.T. and the Ignitor in Italy. In 2010, Italy and Russia signed an agreement to build a 13-T Ignitor-type tokamak to study burning plasma physics before ITER is finished. These small, pulsed machines cannot expose the steady-state problems that ITER will face. Engineering problems such as tritium breeding and plasma exhaust can be studied only with sufficient neutron flux. There are several proposals for large machines designed specifically for problems not tackled by ITER which will run simultaneously with ITER. None of these has been funded so far.
Research on closed magnetic bottles started with stellarators such as the figure-8 stellarator shown in Fig. 4.18. In 1969, the Model C Stellarator in Princeton, the largest at the time, was converted to a tokamak because of the good results coming from Russia with their configuration. Soon almost all new machines were tokamaks. This was because of the self-healing properties of tokamaks, as described in Chap. 7. When the temperature profile in the plasma became too peaked, sawtooth oscillations would arise and smooth it out to maintain stability. All this is now changed, and stellarators have come back as a hope for the future.
The difference between these two toroidal devices, tokamaks and stellarators, is the way the poloidal magnetic field (the component that gives the field lines their helical twists) is generated. In tokamaks, a large current in the plasma generates that field. In stellarators, external coils generate that field, and no large plasma current is necessary. But these external poloidal-field coils are hard to make. Present — day Advanced Tokamaks no longer rely on the self-healing features that were initially useful. We have learned how to shape the plasma current with radiofrequency and microwave power to keep the plasma stable. In fact, sawtooth oscillations are now deliberately eliminated. Since self-organization is no longer necessary, we can reconsider stellarators. Stellarators are less subject to effects such as disruptions that are connected with the large plasma current. In effect, they eliminate a source of energy that allows a plasma to self-organize destructively in its efforts to escape from confinement. Furthermore, since transformer action to drive the plasma current is not necessary, stellarators are more suitable for steady-state, continuous operation.
• Developing fusion power will cost less than putting a man on the moon. The Manhattan and Apollo programs have shown that the scientific and engineering communities have the ingenuity to achieve almost unimaginable goals once it is driven by national priorities, a sense of urgency, personal challenge, and a sense of national pride. With seven nations having banded together to push forward on fusion, the USA has lost its chance to do this alone. However, we are still far
from the goal because the most difficult problems of materials engineering have yet to be solved. The USA can regain its former leadership in fusion research by building one or more large FDFs to solve these problems simultaneously with ITER to shorten the time to a working reactor.
• The development of wind and solar power in private industry has stimulated the economy. Fusion machines are big and must be funded by the government, but the economic stimulus can also be generated by the subcontracts awarded to small companies. For instance, such components as superconducting strands, silicon carbide tiles, blanket modules, RF antennas, and even 3D computations can all be parceled out to start-up companies. New jobs will be created, and new financing will be secured.
• A high-priority Apollo-like program to put fusion on a fast track will cost less than Apollo did and will solve the CO2 problem, the fossil-fuel shortage problem, and the oil dependence problem all at once.