Why Does E=mc²? (And Why Should We Care?) - Brian Cox, Jeffrey R. Forshaw (2009)
Chapter 6. And Why Should We Care? Of Atoms, Mousetraps, and the Power of the Stars
We have seen how Einstein’s famous equation forces us to reconsider the way we think about mass. We have come to appreciate that rather than being simply a measure of how much stuff something contains, mass is also a measure of the latent energy stored up within matter. We have also seen that if we could unlock it, then we would have a phenomenal source of energy at our disposal. In this chapter we will spend some time exploring the ways in which mass energy can actually be liberated. But before we turn to such useful practicalities, we would like to spend a little more time exploring our newfound equation, E = mc2+mυ2, a little more carefully.
Remember, this version of E = γmc2 is only an approximation, although a pretty good one for speeds even as high as 20 percent of the speed of light. Writing it like this makes the separation into mass energy and kinetic energy most apparent, and we won’t bother to remind you that it is just an approximation. Recall also that we can construct a vector in spacetime whose length in the space direction represents a conserved quantity, which reduces to the old-fashioned law of conservation of momentum for velocities that are small compared with the speed of light. Just as the length of the new spacetime momentum vector in the space direction is conserved, so too must its length in the time direction be a conserved quantity, and this length is mc2 +mυ2. We recognized thatmυ2 is the formula for a quantity long familiar to scientists, the kinetic energy, and so we identified the conserved quantity as energy. Very important, we didn’t start off looking for the conservation of energy. It emerged quite unexpectedly when we were trying to find a spacetime version of the law of conservation of momentum.
Imagine a bucket of armed mousetraps, all storing energy in the springs. We know that wound-up springs store energy because when the trap is triggered there is a loud bang (which is energy being released as sound) and the trap might jump up in the air (energy being turned into kinetic energy). Now imagine that one trap goes off and triggers the rest. There is a huge clatter as the energy stored in the springs is liberated and the mousetraps snap shut. The conservation of energy says that the energy before the mousetraps snap shut must equal the energy afterward. Moreover, since the traps were initially all sitting at rest, the total energy must equal mc2, where m is the total mass of the bucket of primed traps. Afterward, we have a bunch of spent traps plus the energy that was liberated. To balance the energy before with that afterward, it therefore follows that the bucket of armed mousetraps is actually more massive than the bucket of triggered traps. Let’s think of another example, this time involving a contribution to mass arising from kinetic energy. A box full of hot gas has more mass than an identical box containing the same gas at a lower temperature. The temperature measures how fast the molecules are whizzing around inside the box—the hotter the gas, the faster the molecules move around. Because they are moving faster, they have more kinetic energy (i.e., the result of adding together the values ofmv2 for each molecule is bigger for the hot gas) and hence the box has more mass. The logic extends to everything that stores energy. A new battery is more massive than a used battery, a hot flask of coffee is more massive than a cold one, and a steaming-hot meat and potato pie bought at halftime on a wet Saturday afternoon at Oldham Athletic’s football ground is more massive than the same uneaten pie at the end of the game.
The conversion of mass to energy is therefore not such an exotic process. It is happening all the time. As you relax by a crackling fire you are absorbing heat from the burning coals, and that heat takes energy away from the coal. In the morning, when the fire has died away, you could very carefully sweep up every last piece of ash and weigh it with scales of unfeasible accuracy. Even if you miraculously managed to get every atom of ash, you would find that it weighed less than the original coals weighed. The difference would be equal to the amount of energy liberated divided by the speed of light squared, as predicted by E= mc2, i.e., according to m = E/c2. We can quickly figure out how tiny the change in mass would be for the kind of fire that might warm your house as the night draws near. If the fire generates 1,000 watts of power for 8 hours, then the total energy output is equal to 1,000 x (8 x 60 x 60) joules (because we have to work in seconds, not hours, in order to get an answer in joules), which is just less than 30 million joules. The corresponding loss of mass must therefore be equal to 30 million joules divided by the speed of light squared, and that is equal to less than one-millionth of a gram. The explanation for the tiny reduction in mass is a direct consequence of the conservation of energy. Before igniting the fire, the total energy of the coals is equal to the total mass of coal multiplied by the speed of light squared. As the fire burns, energy leaves the fire. Eventually, the fire dies and we are left with ash. According to the law of conservation of energy, the total energy of the ash must be less than the total energy of the coal by an amount equal to the energy that went into warming the room. The energy of the ash is equal to its mass multiplied by the speed of light squared, which must be lighter than the original coal by the amount we just calculated.
The process of converting mass into energy and energy into mass is therefore absolutely fundamental to the workings of nature; it really is an everyday occurrence. For anything to happen at all in the universe, energy and mass must be continually sloshing back and forth. How on earth did anyone manage to explain anything involving energy before we knew this seemingly most basic of facts about the workings of nature? It’s worth remembering that Einstein first wrote down E = mc2 in 1905 in a world that was far from primitive. The first intercity passenger railway, powered by coal-burning steam locomotives, was opened in 1830 between Liverpool and Manchester. Coal-burning ocean liners had been crossing the Atlantic for almost seventy years, and the golden age of steam was in full swing with advanced steam-turbine-powered liners, such as the Mauretania and Titanic , about to enter service. The Victorians certainly knew how to burn coal efficiently and to spectacular effect, but how did the scientists of the day think of the physics behind a burning fire before Einstein? A nineteenth-century engineer would have said the coal has latent energy stored within it (rather like the energy stored in lots of miniature mousetraps) and the chemical reactions that burn the coal spring the traps and liberate that energy. This picture works, and allows calculations to be made with the accuracy required to design a beautiful machine like an ocean liner or an express steam locomotive. The post-Einstein view does not disagree with this picture but rather it adds to it. That is to say, we now understand that latent energy is irrevocably intertwined with the concept of mass. The more latent energy something has, the more massive it is. It would not have occurred to scientists before Einstein that there was a link between mass and energy, because they had not been forced to think in that way. Their view of nature was accurate enough to explain the world they observed and to solve the problems they encountered, because the changes in mass were so tiny that they never needed to know them.
Here lies another insight into science. With each new level of understanding, a more accurate worldview emerges. The current worldview is never claimed to be correct, in the very important sense that there are no absolute truths in science. The body of scientific knowledge at any point in history, including now, is simply the collection of theories and views of the world that have not yet been shown to be wrong.
All of the examples we just looked at lead to very tiny fractional changes in mass, but of course the release of the corresponding energy can be very significant. A fire keeps us warm and a hot pie is much tastier than a cold one. In the case of burning coal, the stored energy is chemical in origin. The molecules that make up the coal get rearranged and turn to ash as a result of a chemical chain reaction initiated by a lighted match. As the bonds between the molecules snap and reform and atoms recombine with atoms to make new molecules, energy is released and the mass reduces. Chemical energy has its origins in the structure of atoms. The simplest example is a single hydrogen atom, which is a single electron in orbit around a single proton. It is simple enough that physicists can use the quantum theory to calculate how the mass of the atom should change as the electron moves around. There is a smallest value for the mass of a hydrogen atom. It is an utterly miniscule 0.00000000000000000000000000000000002 kilograms less than the combined mass of an electron and a proton that are far apart. Nevertheless, that difference, when converted into energy, is a very big deal. Ask any chemist or experience its effect yourself sitting in front of that nice coal fire.
Because particle physicists are as lazy as the next guy, they don’t like writing very small numbers down with lots of zeros and decimal places, so they don’t usually use kilograms to measure mass. Instead they use a unit called the electron volt, which is actually a measurement of energy. An electron volt is the amount of energy an electron gets when it is accelerated through a potential difference of 1 volt. This is a mouthful, and we are again in danger of covering ourselves in chalk dust. In more normal-sounding language, if you get a 9 volt battery and build a little particle accelerator out of it, you would be able to give an electron 9 electron volts of energy. The electron volt is turned into a mass by dividing it by c2 (remember E = mc2). In this rather more convenient language, the hydrogen atom has a smallest mass, which is 13.6 eV/c2 less than the masses of the proton (938,272,013 eV/c2) and electron (510,998 eV/c2) combined (1 eV is the abbreviation for an energy of 1 electron volt). Notice that by keeping a factor of c2 “in the units,” it is easy to figure out how much energy is stored within a proton at rest. Since the energy is obtained by multiplying the mass by c2, the c2 factors cancel out and the energy is just 938,272,013 eV.
Notice also that the mass of a hydrogen atom is smaller, not bigger, than the sum of its component parts. It is as if the atom has some negative energy stored within it. There is nothing mystical about negative energy in this context: “Negative stored energy” just means that it takes effort to dismantle the atom, and it often goes by the name “binding energy.” The next smallest mass of a hydrogen atom is 10.2 eV/c2smaller than the sum of its parts.7 The mystical-sounding and oft-misunderstood quantum theory actually derives its name from the fact that masses like these come in discrete (“quantized”) values. For example, there is no hydrogen atom with a mass 2 eV/c2 bigger than the smallest mass. This is really all there is to the word “quantum.” The different masses actually correspond to the electrons being in different orbits around the atomic nucleus, which in the case of hydrogen is a single proton.
That said, one has to be very careful in picturing electron orbits, because they are not really like the orbits of planets around the sun. Loosely speaking, the atom with the smallest mass has the electron closer to the proton than the atom with the next smallest mass, and so on. The hydrogen atom with the electron as close as it can be to the proton is said to be in its “ground state” and it is as light as it can be. Add just the right amount of energy and the electron will jump up to the next available orbit and the atom will become a bit heavier, simply because a bit of energy has been added. In that sense, adding energy to an atom is like winding up the spring in a mousetrap.
All of this does beg the question of how we know such fine detail about hydrogen atoms. Surely we don’t go around measuring these tiny mass differences using weighing scales? At the heart of the quantum theory is an equation called the Schrödinger wave equation, and we can use it to predict what the masses should be. Legend has it that Schrödinger discovered the equation, one of the most important in modern physics, while on a winter sojourn with his mistress in the Alps over Christmas and New Year’s of 1925-1926. Quite how he explained this to his wife is rarely discussed in physics textbooks. We can only hope his mistress enjoyed the fruits of his labors as much as the generations of physics students who know the eponymous equation by heart. The calculation is not too difficult for an atom as simple as hydrogen, and it has graced many an undergraduate examination paper. But mathematical tractability means little without the corroborating evidence provided by experiments. Fortunately, the results of the quantum nature of atomic structure are pretty easy to observe. In fact, we all observe them every day. There is a general rule in quantum theory that roughly goes like this: Left alone, a heavier thing will turn into a lighter thing if at all possible. It is not a hard concept to understand. If the thing is left alone it cannot possibly go to a heavier thing because there is no energy being added, whereas there is always the chance it can shed some energy and become lighter. Of course, the third option is that it does nothing and stays the same, and sometimes that is the case. For the hydrogen atom this means that the heavier version will eventually shed some of its mass. It does so by emitting a single particle of light, the photon we met earlier. For example, a next-to-lightest hydrogen atom will at some point spontaneously convert into a lightest hydrogen atom as a consequence of a change in the orbit of the electron. The excess energy is carried away by a photon.8 The reverse process can occur too. A photon, if one just happens to be around, can be absorbed by the atom, which then jumps to a higher mass because the energy absorbed promotes the electron to a higher orbit.
Perhaps the most everyday way of getting energy into atoms is to heat them up. This causes the electrons to jump up into the higher orbits and subsequently drop back down again, emitting photons as they go (this is the physics behind a sodium vapor street lamp). These photons carry an energy that is exactly equal to the energy difference between the orbits, and if we could detect them, we would have a direct window into the structure of matter. Fortunately, we are detecting them all the time because our eyes are nothing more (or less) than photon detectors, and the energy of the photons is registered directly as color. The azure blue of an island-pitted tropical ocean, the jagged diamond yellow of Van Gogh’s stars, and the iron-red of your blood are a direct measurement by your eyes of the quantized structure of matter. The origin of the colors emitted by hot gases was one of the driving forces behind the discovery of quantum theory at the turn of the twentieth century. The years of careful observation of the light emitted from anything and everything by legions of diligent scientists are commemorated in our language by the name of the gas that fills party balloons. “Helium” is derived from the Greek word “helios,” which means “sun,” because the signature of this atom was first discovered by French astronomer Pierre Janssen in the light from a solar eclipse in 1868. In this way we discovered helium on our star before we found it on Earth. Today, astronomers search for signs of life on distant worlds by looking for the characteristic fingerprint of oxygen in the starlight shining through the atmospheres of planets as they pass across the face of their parent stars. Spectroscopy, as this branch of science is known, is a powerful tool for exploring the universe without and within.
All of the atoms in nature come in a tower of energies (or masses), depending on where the electrons are, and since there is more than a single electron in every atom except hydrogen, the light emitted from them spans all the colors of the rainbow and beyond, which is ultimately the reason why the world is so colorful. Chemistry is, very crudely, the area of science that is concerned with what happens when bunches of atoms come close together (but not too close). As two hydrogen atoms approach each other, the protons repel because they both carry positive electric charge, but that repulsion is overcome because the electron in one atom attracts the proton in the other. The result is that there is an optimal configuration where the two atoms are bound together to make a hydrogen molecule. The atoms are bound in the same sense that the electron is bound into orbit around a single hydrogen nucleus. Being bound means simply that it takes some effort to pull them apart and “it takes some effort” is a sloppy way of saying that we need to supply some energy. If we need to add energy just to break the molecule apart, then it follows that the molecule is less massive than the sum of the original two hydrogen atoms, just as the hydrogen atom is less massive than the sum of the masses of its constituents. In both cases, the binding energy comes about because of the force of electromagnetism that we met at the beginning of the book.
As everyone who has spent time in a school chemistry lab with a box of matches and an inattentive teacher knows, chemical reactions can sometimes lead to the production of energy. A coal fire is a perfect, nicely controlled example; a little nudge from a lighted match and energy is released steadily for hours. More dramatic, an exploding stick of dynamite releases similar amounts of energy to a coal fire, albeit rather more quickly. The energy doesn’t come from the match that lit the fire or the fuse, but from the energy stored within. The bottom line is always that the combined mass of the products of the reaction must be less than the mass we started with if some energy has been lost.
A final example may serve to further illustrate the idea of energy release through chemical reactions. Imagine sitting in a room full of hydrogen and oxygen molecules. We would be able to breathe perfectly well, and at first sight it would appear quite safe and comfortable since it takes energy to pull apart two hydrogen atoms bound together in a molecule. This would seem to suggest that molecular hydrogen should be a stable substance. It can, however, be broken up via a chemical reaction that generates an impressive amount of energy; so impressive in fact that hydrogen gas is very dangerous stuff. It is highly flammable in air, needing only a tiny spark to trigger disaster. In our newfound language, we can analyze the process in a little more detail. Suppose we mix together a gas of hydrogen molecules (two hydrogen atoms bound together) and a gas of oxygen molecules (two oxygen atoms bound together). Now, you might well become very nervous sitting in your room when you discover that the combined mass of two hydrogen molecules and one oxygen molecule is bigger than the combined mass of two water molecules, each of which is made of two hydrogen atoms and an oxygen atom. In other words, the four hydrogen atoms and two oxygen atoms that started as molecules are more massive than two lots of H2O. The excess mass is approximately 6 eV/c2. The hydrogen and oxygen molecules would therefore quite like to be rearranged into two water molecules. All that will be different is the configuration of the atoms (and their associated electrons). At first glance the energy release per molecule is tiny, but a roomful of gas contains in the region of 1026 molecules,9 and that translates into around 10 million joules of energy, which is plenty enough to rearrange your own personal molecules as a side effect. Fortunately, if we are careful, then we are not destined to be incinerated because although the final products have a mass that is smaller than the initial products, it takes a bit of effort to put them, and their electrons, into the right configuration. It is a bit like pushing a bus over a cliff edge—it takes some effort to get it started but when it goes, there is no stopping it. That said, it would be very unwise to light a match, which would supply plenty enough energy to trigger the molecular rearrangement process and get the water production under way.
Liberating chemical energy by shuffling atoms around or gravitational energy by shuffling heavy things around (like huge volumes of water in hydroelectric plants) provides our civilization with a means to generate and harness energy. We are also becoming increasingly adept at harvesting the abundant resources of kinetic energy found in nature. As the wind blows, molecules of air rush along, and we can convert that wild kinetic energy into useful energy by putting a wind turbine in the way. The molecules bang into the blades of the turbine and as a result the molecules slow down, delivering their kinetic energy to the turbine, which starts to rotate (incidentally, that is another example of the conservation of momentum). In this way, the kinetic energy of the wind gets transformed into rotational energy of the turbine, and that in turn can be used to power a generator. Harnessing the power of the sea works in much the same way, except that it is the kinetic energy of water molecules that gets converted into useful energy. From a relativistic perspective, all forms of energy contribute to mass. Imagine a giant box filled with flying birds. You could put the box on a set of measuring scales and weigh it, thereby inferring the mass of the birds plus the box. Since the birds are flying around, they have some kinetic energy, and as a result the box will weigh a tiny bit more than it would if the birds were all asleep.
The energy released in chemical reactions has been the primary source of power for our civilization since prehistoric times. The amount of energy that can be liberated for a given amount of coal, oil, or hydrogen is at the most fundamental level determined by the strength of the electromagnetic force, since it is this force that determines the strength of the bonds between atoms and molecules that are broken and reformed in chemical reactions. However, there is another force of nature that offers the potential to deliver vastly more energy for a given amount of fuel, simply because it is much stronger.
Deep inside the atom lies the nucleus—a bunch of protons and neutrons stuck together by the glue of the strong nuclear force. Being glued together, it takes effort to pull a nucleus apart, just as it does for atoms and molecules, and its mass is therefore less than the sum of the masses of its individual proton and neutron parts. Entirely analogous to the goings-on in chemical reactions, we might wonder whether it is possible to make nuclei interact with each other in such a way that allows this mass difference to be emitted as useful energy. Breaking chemical bonds and releasing the stored energy in the atoms can be as easy to achieve as lighting a match, but releasing the energy bound up in a nucleus is an entirely different matter. It is often hard to access and usually requires some clever apparatus. Not always, though; there are occasions where nuclear energy is liberated naturally and spontaneously, with extremely important and unexpected consequences for planet Earth.
The heavy element uranium has 92 protons and, in its most stable naturally occurring form, 146 neutrons. In this guise, it has a half-life of around 4.5 billion years, which simply means that in 4.5 billion years, half of the atoms in a lump of uranium will have spontaneously split up into lighter things, the heaviest of these being the element lead, and liberated energy as a result. In the language of E = mc2, the uranium nucleus splits into two smaller nuclei, whose combined mass is a little less than the mass of the original nucleus. It is that loss of mass that manifests itself as nuclear energy. The process whereby a heavy nucleus splits up into two lighter nuclei is called nuclear fission. Along with the 146-neutron form of uranium, there also exists a less-stable naturally occurring form with 143 neutrons that splits into a different form of lead with a half-life of 704 million years. These elements can be used to accurately date rocks almost as old as the earth itself, which is around 4.5 billion years old.
The technique is beautifully simple. There exists a mineral known as zircon that naturally incorporates uranium into its crystalline structure, but not lead. It can therefore be assumed that any lead present in the mineral comes from the radioactive decay of uranium, which allows the date of formation of the zircon to be measured with high precision simply by counting the number of lead nuclei present and knowing the rate of decay of the uranium. The heat generated when uranium splits up also plays a crucial role in keeping the earth warm, and that heat helps provide the power that drives plate tectonics and pushes up new mountains. Without this impetus, fueled by nuclear energy, the land would crumble into the sea as a result of natural erosion. We shall say no more about nuclear fission. It is now time to zoom in on the atomic nucleus and learn a little more about its stored energy and the other important process that can occur to facilitate its release: nuclear fusion.
Take two protons (no electrons are around this time, so we have no chance to make them stick together in a hydrogen molecule). Left alone, they would fly apart in opposite directions because they both carry positive electric charge. So it seems pretty pointless to try to push them closer together. Even so, let us imagine pushing the protons closer together and investigate what happens. One way to do this would be to hurl them at each other with increasing speed. The force of repulsion between the protons gets larger and larger as the protons get closer and closer together. In fact, it doubles in strength for every halving of the distance. It therefore seems that our protons are always destined to be flung apart. If the electrical repulsion were the only force in nature, this would certainly be the case. There are, however, the strong and weak nuclear forces to contend with. When the protons get so close together that they are almost touching each other (protons are not solid balls, so we can even think of them as overlapping) something very remarkable happens. Not always, but some of the time, when we bring two protons together like this, one of the protons will spontaneously turn itself into a neutron and the excess positive electric charge (the neutron being electrically neutral, hence its name) is shed as a particle called a positron. Positrons are identical to the electron except that they carry positive charge. Also emitted is a particle called a neutrino. Compared to the proton and neutron, which have very similar masses, the electron and neutrino are very light and they whiz off into the sunset, leaving the proton and neutron behind. The details of this transmutation process are very well understood using the theory of weak interactions developed by particle physicists in the second half of the twentieth century. We will show how it works in the next chapter. All we need to know here is that the process can and does occur. Free from the electric repulsion, the proton and neutron can snuggle together under the influence of the strong nuclear force. A proton and neutron bound up like that is called a deuteron, and the process of a proton turning into a neutron with the emission of a positron (or vice versa, with the emission of an electron, which can also happen) is called radioactive beta decay.
How does all of that fit with our understanding of energy? Well, the two original protons each have a mass of 938.3 MeV/c2. 1 MeV is equal to 1 million eV (the “M” stands for “mega” or “million”). The conversion between MeV/c2 and kilograms is easy enough: 938.3 MeV/c2 corresponds to a mass of 1.673 x 10-27 kilograms.10 The two original protons have a total mass of 1876.6 MeV/c2. The deuteron has a mass of 1875.6 MeV/c2, and the energy associated with the 1 MeV remainder is carried away by the positron and neutrino, of which approximately half is used up to manufacture the positron since it has a mass of around ½ MeV/c2 (neutrinos have almost no mass at all). So when two protons convert into a deuteron, a relatively tiny fraction (around 1/40 of 1 percent) of the total mass is destroyed and converted into the kinetic energy of the positron and the neutrino.
Squeezing two protons together to make a deuteron is one way to liberate the energy bound up in the strong force, and it is an example of nuclear fusion. The term “fusion” is used to describe any process that releases energy as a result of fusing together two or more nuclei. In contrast to the energy released in a chemical reaction, which is a result of the electromagnetic force, the strong nuclear force generates a huge binding energy. For example, compare the ½ MeV released when a deuteron is formed to the 6 eV released in our hydrogen-oxygen explosion. This is in keeping: The energy released in a nuclear reaction is typically a million times the energy released in a chemical reaction. The reason that fusion doesn’t happen all the time in our everyday experience here on Earth is that, because the strong force operates only over short distances, it only kicks in when the constituents are very close together and declines very rapidly at distances much greater than a femtometer (which is roughly equal to the size of one proton). But it is not easy to push protons together to that distance because of their electromagnetic repulsion. One way to do it requires the protons to be moving extremely fast, and this in turn means a very high temperature indeed because temperature is essentially nothing more than a measure of the average speed of things; the molecules of water in a hot cup of tea are jiggling around more than the molecules in a cold pint of beer. At the very least a temperature of around 10 million degrees is necessary for fusion to begin, and preferably significantly more. Fortunately for us, there are places in the universe where temperatures meet and exceed those necessary for nuclear fusion—deep in the hearts of stars.
Let us journey back in time to the cosmic dark age, less than half a billion years after the big bang when the universe is filled with only hydrogen, helium, and a sprinkling of the lighter chemical elements. Slowly, as the universe continues to expand and cool, the primordial gases begin to fall in on themselves in clumps under the influence of gravity, picking up speed as they rush toward each other, just as this book will speed up toward the ground if you drop it. Faster-moving hydrogen and helium means hotter hydrogen and helium, so the big balls of gas become increasingly hot and increasingly dense. At a temperature of 10,000 degrees, the electrons are ripped from their orbits around the nuclei, leaving behind a gas of protons and electrons known as a plasma. Together the individual electrons and protons continue to fall inexorably inward, faster and faster in a relentlessly quickening collapse. The plasma is rescued from a seemingly irretrievable fall when the temperature approaches 10 million degrees, when something very important happens, something that transforms the hot ball of protons and electrons into the life and light of the universe; a magnificent source of nuclear energy; a star. Individual protons fuse together to make a deuteron, which itself can fuse with another proton to produce helium, and all the while precious binding energy is released. In this way the new star slowly converts a small fraction of the original mass into energy, which heats up the core of the star and allows it to halt and resist any further gravitational collapse, at least for a few billion years—time enough for cold, rocky planets to be warmed, liquid water to flow, animals to evolve, and civilizations to rise.
Our sun is a star that is currently in just such a comfortable midlife phase: It is burning hydrogen to make helium. In the process, it loses 4 million tons of mass every second of every day of every millennium as it converts 600 million tons per second of hydrogen into helium. This profligacy, the foundation of our existence, cannot continue forever, even for our local ball of plasma, large enough to contain a million earths. So what happens when a star runs out of hydrogen fuel in its core? Without the nuclear source of outward pressure, the star will once again start to collapse, getting hotter and hotter as it does so. Eventually, at a temperature of around 100 million degrees, helium begins to burn and once again the star’s collapse is arrested. We are using the word “burn,” but that isn’t really very precise. What we really mean is that nuclear fusion is taking place and the net mass of the final products is less than the mass of the original fusing material—the loss of mass leading to the production of energy in accord with E = mc2.
The process of burning helium is really worth a closer look. When two helium nuclei fuse, they make a particular form of beryllium, made up of four protons and four neutrons. This form, called beryllium-8, lives for only one ten-millionth-of-a-billionth of a second before it falls apart into two helium nuclei again. The brief life of beryllium-8 is so fleeting that it is very unlikely it will hang around long enough to fuse with anything else. In fact, without a helping hand, that is pretty much what would always happen, and the pathway to synthesizing heavier elements inside stars would be blocked. In 1953, when the understanding of the nuclear physics of stars was still in its infancy, astronomer Fred Hoyle realized that carbon had to be manufactured inside stars, irrespective of what the nuclear physicists told him, because he strongly believed that there is nowhere else in the universe to make it. Coupled with his astute observation that astronomers exist, he theorized that this could happen only if a slightly heavier type of carbon nucleus exists such that it can be formed very efficiently as the result of fusion between the short-lived beryllium-8 and a third helium nucleus. For the theory to work out, Hoyle figured out that the heavy carbon should be 7.7 MeV/c2 heavier than ordinary carbon. Once this new form of carbon has been made in the star, the pathway to heavier elements opens up. At the time, no such form of carbon was known but, spurred on by Hoyle’s prediction, scientists wasted no time in hunting for it. It was a matter of days after Hoyle made his prediction that nuclear physicists working in the Kellogg Laboratory at Caltech confirmed his prediction without any shadow of doubt. This is a remarkable story, not least because of the way it helps us build confidence in our understanding of how stars work: There is no better vindication of a beautiful theory than the verification in an experiment of a prior prediction.
Today we have a great deal more evidence that supports the theory of stellar evolution. One striking example comes from the study of the neutrinos produced every time a proton turns into a neutron in the fusion process. Neutrinos are ghostly particles that hardly ever interact with anything, and as such, most of them stream out from the sun as soon as they are produced without hindrance. The neutrino flux is so great, in fact, that around 100 billion of them pass through each square centimeter of the earth every second. This is an easy fact to read but an astonishing thing to imagine. Hold your hand up in front of you and look at your thumbnail. Each second, 100 billion subatomic particles from the core of our star will pass through it. Fortunately for us, the neutrinos nearly always pass through our hands, and in fact the entire earth, as if they did not exist. However, on rare occasions, a neutrino will interact, and the trick is to build experiments that are able to catch these extremely rare events. The Super-Kamiokande experiment, located deep in the Mozumi mine near the city of Hida in Japan, is up to the challenge. Super-Kamiokande is a huge cylinder 40 meters across and 40 meters tall, containing 50,000 tons of pure water, surrounded by over 10,000 photomultiplier tubes that are capable of detecting the very faint flashes of light that are produced when a neutrino collides with an electron in the water. As a result, the experiment is able to “see” the neutrinos streaming from the sun, and the number arriving turns out to agree with expectations based upon the theory that they are produced as a result of fusion processes inside the sun.
Eventually, the star will exhaust its supply of helium and begin to collapse even further. As the core temperature rises past 500 million degrees, it becomes possible for the carbon to burn, producing a variety of heavier elements all the way up to iron. Your blood is red because it contains iron, the end point of fusion in the core of stars. Elements heavier than iron cannot be manufactured through fusion in the core because there is a law of diminishing returns, and for nuclei heavier than iron there is no more energy to be released from fusing with extra nuclei. In other words, adding protons or neutrons to an iron nucleus can only make it heavier (not lighter, as would be necessary for fusion to act as a source of energy). Nuclei heavier than iron prefer instead to shed protons or neutrons, as we saw earlier in the case of uranium. In these cases, the sum total of the masses of the products is less than the mass of the initial nucleus, and so energy is released when a heavy nucleus divides. Iron is the special case; it is the Goldilocks nucleus and that means that iron is exceptionally stable.
With no other source of energy available to prevent the inevitable, a star that has an iron-rich core is really at the point of no return, and gravity resumes its relentless work. There is now only one last chance for the star to prevent total collapse. It becomes so dense that the electrons that have been hanging around ever since they were ripped off the hydrogen atoms during its birth resist further collapse as a result of the Pauli exclusion principle. The principle is an important one in quantum theory and it is crucial for the stability and structure of atoms. Crudely put, it says that there is a limit to how closely you can pack electrons together. In a dense star, the electrons exert an outward pressure that increases as the star collapses until it is eventually so large that it can prevent any further gravitational collapse. Once that happens, the star is trapped in an enfeebled but incredibly long-lived state. It has no fuel to burn (that is why it was collapsing in the first place) and it cannot collapse any further because of the electron pressure. Such a star is called a white dwarf—a slowly fading memorial to a majesty irredeemably diminished—the once-bright creator of the elements of life compressed into a remnant the size of a small planet. In a time far longer than the age of the universe today, the white dwarfs will have cooled so much that they fade from view. We are reminded of the beautiful sentiments of the father of the big bang theory, Georges Lemaitre, when reflecting on the inevitable universal journey from light into darkness from which even stars cannot escape: “The evolution of the universe can be likened to a display of fireworks that has just ended: some few wisps, ashes and smoke. Standing on a well-cooled cinder, we see the fading of the suns, and try to recall the vanished brilliance of the origins of the worlds.”
It has been our goal throughout this book to be careful to explain why things are as they are and to provide arguments and evidence as we progress. The description we presented here of how a star works might seem fanciful, and we have certainly deviated from our careful, explanatory style. You might even object that since it is not possible to do laboratory experiments directly on stars, we cannot possibly be certain how they work. But that isn’t why we were brief. We have been brief because it would take us too far from the point to go into more detail. The remarkable work of Hoyle and the success of experiments like Super-Kamiokande will have to suffice by way of supporting evidence, along with one last beautiful prediction made by Indian physicist Subrahmanyan Chandrasekhar. In the early 1930s, armed only with already well-established physics, he predicted that there should be a largest possible mass for any (nonrotating) white dwarf star. Chandrasekhar originally estimated the largest mass to be around 1 solar mass (i.e., the mass of the sun), and more refined calculations later led to a value of 1.4 solar masses. At the time of Chandrasekhar’s work, only a handful of white dwarf stars had been observed. Today, around 10,000 white dwarf stars have been observed, and they typically have a mass close to that of the sun. Not a single one has a mass that exceeds Chandrasekhar’s maximum value. It is one of the true joys of physics that laws discovered in tabletop experiments in a darkened laboratory on earth pertain throughout the universe, and Chandrasekhar exploited that universality to make his prediction. For that work he received the 1983 Nobel Prize. The validation of his prediction is one of the pieces of evidence that allows physicists to be very confident that they really know how stars work.
Are all stars fated to end their lives as white dwarf stars? The narrative in the previous paragraph suggests so, but it is not the whole story and there was a clue. If there can never be a white dwarf star with a mass larger than 1.4 solar masses, what happens to stars that are bigger than that? Putting aside the possibility that big stars can shed material so that they sneak in under Chandrasekhar’s limit, two alternative fates await. In both cases, the large initial mass means that the electrons eventually start to move around at close to the speed of light as the collapse continues. Once that happens, there really is nowhere else to go; their pressure will never be sufficient to resist the force of gravity. For these massive stars, the next stop is a neutron star, in which nuclear fusion steps in for a final time. The protons and electrons move so fast that they reach a point where they have sufficient energy to initiate proton-electron fusion, producing a neutron. The reaction is the reverse of the radioactive beta decay process, whereby a neutron spontaneously decays into a proton and an electron with the emission of a neutrino. In this way, all of the protons and electrons gradually convert into neutrons and the star is nothing but a ball of neutrons. The density of a neutron star is phenomenal: A single teaspoon of neutron star matter weighs more than a mountain. Neutron stars are stars that are more massive than our sun yet are compressed to the size of a city.11Many of the known neutron stars spin at phenomenal rates and blast beams of radiation out into space like cosmic lighthouses. These stars are known as pulsars, and they are truly wonders of the universe. Some known pulsars are approaching twice the mass of our sun, measure only 20 kilometers in diameter, and spin more than five hundred times every second. Imagine the violence of the forces on such an object. We have discovered wonders beyond imagination.
Beyond neutron stars, a final fate awaits the biggest stars. Just as the electrons can approach the speed of light in white dwarfs, the neutrons in a neutron star can bump up against the limit Einstein imposed on them. When this happens, no known force will prevent complete collapse, and the star is destined to form a black hole. Today our knowledge of the physics of space and time inside black holes is incomplete. As we shall see in the final chapter, the presence of mass causes spacetime to warp away from the Minkowski spacetime that we have become so familiar with, and in the case of a black hole, that warping is so extreme that not even light can escape its clutches. In such extreme environments, the laws of physics as we currently know them break down, and figuring out the way forward is one of the great challenges for twenty-first-century science, for only then will we be able to complete the story of the stars.