An Introduction to Thermonuclear Fusion. A Point-Form Summary. This section of the course notes, and the associated PowerPoint presentations, make the following critical points: in 1905, just a century ago, Einstein showed that there is a relationship between mass and energy (E = m c-squared). Matter is, in some sense, frozen energy. This explains the enormous energy released in atomic reactors and bombs, where a small amount of mass quite literally disappears. this seems to violate one of the great Conservation Laws that we met in Phys 015, where I said that mass cannot freely be created or destroyed. But modern physics adopts a modified Conservation Law in which the sum total of mass and energy is conserved, with the ability to convert mass to energy and vice-versa. Einstein's equation implies that if a tiny fraction of the mass of the sun could be continuously converted to a lot of energy, the sun could keep hot and essentially unchanging for billions of years while it slowly 'consumed itself' in fact the sun has enough mass to keep up its rate of energy output for trillions of years. It will not, however, last that long because it does not convert all of its mass to energy -- only about a tenth of a percent. this still gives it a potential life of ten billion years. The solar system was formed about 4.6 billion years ago. (See The Age of the Solar Systemin the Phys 015 course notes to remind yourself how we determine that.) So our solar system is just entering its middle age. sometimes, the conversion of mass to energy happens spontaneously, in a lump of undistinguished material. This is the source of the energy released in the radioactive breakdown of naturally-occuring elements like Uranium. by contrast, the conversion of mass to energy happens with great effectiveness in the sun simply because is so hot there, a circumstance that allows a series of thermonuclear reactions to take place. These reactions would not occur in a cool gas of the same composition: the ten-million-degree temperature is the key. in the sun, hydrogen atoms are being fused together to make helium. A lump of helium is less massive than the four hydrogens that were joined together, and the lost mass (a fraction of the total) appears as energy. this cannot happen unless it is very hot. The hydrogen nuclei, which are protons, repel each other because they have positive electical charges. (Remember that "opposite charges attract; like charges repel." ) The protons will only fuse together if they can be made to race about at high speed and slam into each other despite the electical repulsion. Once they get very close to each other, a new force takes over and binds them together. the small amounts of heavier elements present in the sun play no important role in the energy generation: hydrogen is the raw fuel, and helium is the product. We will see later that heavier elements can play a role in later stages of a star's life, especially in stars that are more massive than the sun. We will also learn that the heavy elements were themselves created in stars, and that we understand the varied abundances we see (which explains, for example, why uranium is rare but silicon is not). the sun is very stable: if you were to squeeze it, it would spring back to its original shape in a matter of minutes. Squeezing makes it hotter in the middle, which increases the sustaining pressure; it also increases the thermonuclear reactions, which releases more energy. For both reasons, is quickly 'pops' back to its original size, like a balloon which you briefly dimple and let go. these considerations all explain the remarkable longevity and stability of the sun (without which we would not be here to make these comments) I remind you all that astrophyicists make precise numerical calculations of stellar structure, the rates of the nuclear reactions, and so on. Astronomy is a mathematical subject, although I do not present that face of it to you. as noted, the material in the centre of the sun is a completely ionized plasma (a cloud of hot gas in which all the electrons have been stripped from the nuclei). This simplifies the behaviour (fortunately for physicists!). Thermonuclear reactions can take place between the nuclei. the binding energy curve tells us how much energy can be extracted by various nuclear processes, which come in two kinds: fission and fusion in fission, a big nucleus, like uranium, breaks into two or more smaller pieces. The sum of the masses of the pieces is less than the original, with the difference showing up in the form of energy. (Imagine cutting a kilogram of butter into two parts, and discovering that the two lumps weighed precisely 666 and 333 grams. Where did the missing gram go?) in fusion, by contrast, two or more small lumps are merged together to form a heavier nucleus. In this case, the new lump weighs less than the sum of the small pieces, and energy is released. (Imagine mixing 200 gm of sugar with 100 gm of beaten eggs, and finding that the mixture weighs only 295 grams. Where did the missing 5 grams go?) the binding energy curve allows the physicist to work out the energy released. The critical point is the shape of the curve. The steepness at the low-atomic-number end tells us that if you merge a few little pieces, like using Hydrogen to form Helium, you get out a LOT of energy. The curve is quite flat at the high-atomic-number end. This tells us that breaking a big lump, like Uranium, into smaller pieces yields considerably less energy -- but still rather a lot! (Think of an atomic bomb.) fission (the breaking apart of radioactive elements) can and does occur in nature, most often when a stray neutron hits the uranium nucleus (or whatever it is) and makes it unstable. Typically, the uranium breaks into two big nuclei (often Krypton and Barium), but it also releases two more neutrons, each of which can spark yet another fission event if they hit a new uranium nucleus. if uranium atoms are close together, this can give a chain reaction and release lots of energy. We do this in a controlled way in the Pickering reactors. (The energy is used to heat water to steam, which drives turbines to produce electricity.) if we have too many uranium nuclei in close proximity (more than the critical mass ) the reaction is uncontrollable and you have an atomic bomb. hydrogen bombs do not rely on chain reactions. Fusing one set of nuclei together will not automatically lead to another set doing so. But if you have a lot of ultra-hot material, you can get many such reactions happening here and there, with a colossal release of energy -- a hydrogen bomb the big difference is to create an atomic bomb you just need to accumulate the fuel into a lump that exceeds the critical mass, whereas the fuel in the hydrogen bomb needs to be heated to millions of degrees before the reactions will start. This is harder to do, but obviously not impossible. (In fact, the 'fuse' in a hydrogen bomb is an atomic bomb) fusion reactors could provide us with essentially limitless energy, and would not produce the 'dirty' waste products that fission reactors do. The problem is that it is hard to confine a gas which is at a temperature of millions of degrees, but we are developing the technology to do so in the sun, Hydrogen is fused to form Helium. Helium is about 4 times the mass of Hydrogen, so the process fuses four hydrogen nuclei together. The mechanism does not depend on a single collision of four protons at once -- extremely unlikely!! -- but instead relies on a steady buildup, piece by piece, through a chain of reactions known as the pp, or proton-proton, cycle . this cycle produces positrons ('anti-matter') and neutrinos the positrons carry off the positive charges of two of the original protons, leaving them as neutrons. (Helium has two neutrons and two protons.) But the positrons are short-lived. They promptly run into electrons and annihilate completely, forming gamma rays (radiant energy) as unlikely as it may sound, positrons have a usefulness in medicine, in an application called PET (Positron Emission Tomography). The positrons we use there are not formed by thermonuclear reactions, but are created in the spontaneous decay of trace amounts of radioactive elements introduced into the body the neutrinos have a great importance which we will return to later

Associated Readings from the Text.

Please look at: Chapter 15, pages 496-517. Chapter 4, pages 120-124, for a reminder of the nature of matter and the states we can find it in.

Energy From Mass.

As we saw in the previous set of notes, the continuing stable energy output of the sun, scarcely changed over the last few billion years, cannot be the result of slow gravitational contraction or conventional combustion. The correct answer stems from a new kind of physics, that first understood and made quantitative by perhaps the best-known scientist of all time: Albert Einstein. In his most famous formula E = m c**2, (E equals em cee-squared), Einstein encapsulated something which can be expressed qualitatively in the following simplified terms: Modern physics now recognizes that lumps of matter are, in a sense, frozen lumps of energy. The implication of the formula is that it is possible for a chunk of matter to be converted to pure energy (with this energy appearing in any of its possible forms -- the radiant energy we call light, the energy of motion, or in any other of its manifestations). Conversely, in a region of very high energy content (such as in the middle of an intense sea of radiation) it is possible for lumps of material to spontaneously appear, `condensing,' so to speak, out of the field of energy. We will encounter this later. Einstein's equation tells us that if we have a small amount of mass (m) we can, in principle at least, convert it to a determinable amount of energy (E), as given by the formula. In fact, an enormous amount of energy is created because, as you can see, the total is evaluated by a formula incorporating the speed of light (c) - a very large number itself which moreover appears as a `square' in the formula! So you can appreciate that even a small amount of matter can be converted to yield (for instance) the power of the hydrogen bomb, within which a little bit of the hydrogen fuel is converted to pure energy. And so too in the sun, which is in a sense a gigantic thermonuclear bomb. This revelation may worry you! Perhaps you remember our earlier discussion of the great conservation laws of physics, at which time we stated, apparently quite unequivocally, that energy is conserved - that is, it does not merely appear from nowhere or vanish without trace. Thus we noted that a piece of chalk held at arm's length has some potential energy. As it falls, this energy is converted to the kinetic energy of motion; and when it hits the floor the energy is dissipated into the random jiggling of the atoms of the floor (the floor is heated a little) and into the motion of air molecules (you hear the sound of the impact). At all stages, the total energy is the same. Given that, it makes sense to ask what effect Einstein's hypothetical conversion of mass to energy has on the Law of the Conservation of Energy. Well, in modern physics the law is slightly modified to a combined `Law of the Conservation of Mass and Energy.' The grand total of those two things together is still conserved. If mass vanishes, an exactly equivalent amount of energy must appear. Einstein's development of his famous equation predated, by about thirty years, even the first tentative development of an understanding of how stars undergo nuclear reactions. Yet there was no doubt in the minds of astrophysicists that this had to be the energy source of the stars, even if the detailed reactions were not yet understood. The discovery of the great age of the solar system (billions of years) completely ruled out other hypotheses, and it was inescapable that the conversion of mass to energy had to be the answer. Consider some numbers. The sun emits 4 thousand million trillion trillion ergs of energy every second. If this comes from the conversion of mass to energy, application of Einstein's formula implies that about four million metric tonnes of matter is being completely converted to energy every second in the sun. This sounds like an enormous amount, but must be considered in light of the fact that the sun is 2 billion trillion times as massive as this. Thus the fractional loss of mass is almost negligible, and the sun has enough mass that it could, in principle, keep up this rate of energy production for about ten trillion years - more than a thousand times the age of the solar system. (We will see, however, the sun does not last anywhere near as long as this, because other things happen which keep it from converting all its mass to energy. In the end, it converts less than one percent of its mass to energy, and lasts a correspondingly shorter time.)

The Importance of Temperature.

Nearly all the nuclear reactions take place very near the center of the stars, where the temperature is highest. Why is this? Indeed, why are there not nuclear reactions occurring in the materials around us every day? The answer is that the reactions in the sun are what are called fusion reactions: lumps of hydrogen (protons) are jammed (fused) together to build up helium nuclei, during which process a little bit of the mass is converted to energy. The important thing to recognize is that merging the protons is not a simple exercise. Why not? It is because each proton carries a positive charge, and will repel other protons strongly if any come drifting towards it. Simply put, it is very difficult to make two protons come into close adjacency in day-to-day life and conditions. On the other hand, if the protons can be made to come very close together, new forces (the so-called short-range forces ) can make them coalesce and release the energy you want. They will never do this, however, unless you can get them past the initial repulsion. This is done by slamming them together at high speed: that is, get the nuclei moving fast enough (i.e. by heating up the gas) and you will get nuclear reactions occurring. This is why we call these thermo nuclear reactions. In the final analysis, the energy released in the myriad fusion reactions in the sun exactly balances the total energy radiated away from the surface, and the equilibrium of the sun is maintained over billions of years. The details are by now quite well-understood, as we will see.

A Semantic Point.

I begin with an apology. Astronomers use the expression `nuclear burning' to refer to the nuclear reactions which lead to the energy release within stars. But these reactions are not the same as the kind of burning you are used to. In a fireplace, for instance, we have chemical reactions which leave the atoms intact but merely rearrange them into new chemical compounds (with a net release of energy). By contrast, in the nuclear reactions within stars, nuclei of one type, such as hydrogen, are fused (or merged) and thereby converted into other elements , like helium. A little of the mass is converted to energy in this process. I will try to use the expression `fusion reactions' as much as possible, but old habits die hard, and unfortunately I will probably succumb to the urge to speak of `burning.' Please try to remember the distinction!

The Composition of the Sun.

If we are to calculate the total energy generated by nuclear reactions in the sun, we obviously need to know the relative abundance of hydrogen atoms and other things. How do we get this? There are a few ways: We can examine the spectrum of the sun, and infer from the presence and strength of the absorption lines present (with due attention to the effects of temperature and the rather complex atomic physics involved) what the composition is, at least for the outer layers of the sun (which is where the light in our spectrum comes from). We can determine the composition of things like comets (from their spectra) and meteorites (which can actually be collected after they fall to Earth). These objects date from the time of formation of the solar system, and we know that the sun formed within the same cloud of gas as did these littler bits and pieces. We can determine the composition of the planets Jupiter and Saturn, since these are big enough and cool enough to have held onto all the elements in the raw material that once made up the primitive solar system. (By contrast, the Earth is too small and warm to have held onto any of the abundant original hydrogen gas in the nebula, for instance.) We can work out the relative abundances of at least the heavier elements from moon rocks and Earth rocks. (They are deficient in the very lightest elements.) We can study the spectra of other stars and gas clouds to work out their abundances, now that we understand the astrophysical factors which determine the spectral feature strengths. As I told you in an earlier lecture, this kind of analysis was first attempted by Henry Norris Russell, at Princeton, about 90 years ago. He concluded, wrongly, that the sun had a composition like that of the Earth, a finding which satisfied most everyone's expectations. Subsequent developments in quantum theory and atomic physics led to the realization that stars are all pretty much the same composition: nearly 3/4 hydrogen by mass and 1/4 helium by mass (with just traces of other things). We can now do even better! See page 564 of your text for a figure which shows the relative abundances of various elements in the cosmos (which is, roughly speaking, the same as the mix of materials in the outer parts of the sun). If you think about the proportions shown in that figure, you will see that about 90 of every 100 atoms in the universe is hydrogen, and about 9 of the remaining 10 are heliums. (Each of these weighs 4 times as a hydrogen, so for every 90 units of mass in the form of hydrogen, there are 36 heliums and bits of other heavier species. This justifies the fractions I quoted above.) For every hydrogen atom, there are about 10-to-the-minus-10 lithium atoms. Putting this the other way around, we can see that there are about ten-to-the-tenth hydrogen atoms (ten billion of them) for every lithium atom; thus lithium is not a particularly abundant element. We can make similar statements about the abundance of all known elements. But the figure shows a couple of very important other general features, which we will come back to later: The progressively heavier elements are progressively rarer, on average; and There is a pronounced `odd-even' effect. Carbon (with atomic number 6) is fairly abundant, as is oxygen (atomic number 8); but nitrogen, which is in between them at atomic number 7, is less abundant than either one. Continue the series: fluorine (9) is rare, neon (10) more common, sodium (11) is rare, magnesium (12) is common, aluminum (13) is rarer again, and silicon (14) is common. (Indeed, the most abundant elements in the crustal rocks of the Earth are silicon and oxygen.) We need to know the details of these compositional effects if we are to understand the structure and inner workings of the sun very well, so it is good to have worked out these abundances so very precisely. But there is something more to the story than just knowing these numbers. It is a remarkable and profound success of astrophysics that both of these general features - the reduced abundance of heavier elements and the `odd-even' effect - can be explained as a consequence of nuclear reactions in stars! It is the stars that produce all the heavier elements, and we even understand the relative abundances with which they do so! This gives us great confidence in at least the broad correctness of our theories.

How Structurally Stable is the Sun?

Imagine picking up a snowball and squeezing it so that it is compressed to a slightly smaller size. What happens when you release it? The answer is: nothing. The snowball stays in its new, more compact state. By contrast, if you pick up a balloon and squeeze it so that it is dimpled here and there (where your fingers are pressing), it quickly pops back out to its original size when you release it. This is because there is an internal restoring force (the air pressure) which restores the original shape. Of course, this internal pressure does not cause the balloon to keep swelling out to some even larger size because it is held in by the tension in the rubber material of which the balloon is made. The balloon is in equilibrium. Now, how do you think the sun would behave if you could, with god-like powers, reach your hands around it and squeeze it into something ten percent smaller than it is now? You might expect that it would stay compressed when you release it, since it has no structural rigidity, being only a ball of gas; but that's not right. What happens is this: When you squeeze the sun, you force the particles closer together. Those directed inward motions lead to an increase in the vigour with which the particles are colliding - in other words, the gas heats up. (This is a very common phenomenon: when you try to push atoms closer together, they collide more energetically and the gas temperature goes up. That is why a bicycle pump gets hot to the touch when you use it to compress air.) The faster motion of the particles means that they provide a greater pressure than they did before. (The speed of the particles determines how hard they hit against other things, which is what the pressure is.) In addition, the higher temperature means that the nuclear reactions themselves go faster. (The higher the temperature, the faster the particles are moving, which makes it easier to slam two protons together despite the repulsive force they exert because of their positive charges.) This liberates yet more energy, heats up the gas, and increases the pressure. For both these reasons, when you now release the sun from your squeeze hold, the extra internal pressure causes it to quickly expand back out to its original configuration. In this respect, the sun is like the balloon rather than the snowball. Now let us think about the opposite situation. Suppose you could `pull out' the outer parts of the sun, and make it a little bigger than before. The same arguments hold in reverse: the center would cool down, leading to a drop in the pressure (partly because of the slower motion of the particles and partly because of the reduced rate of nuclear reactions). Thus, when you let go of it, the sun would have too little pressure within it to hold it up against gravity, and it would quickly recontract to its original configuration. (The same sort of thing happens with the balloon: if you pinch the surface here and there and try to pull the skin of the balloon out to some larger size, it will quickly jump back to its original shape when released.) The really remarkable thing is how quickly this would all happen. When you squeeze a balloon briefly and then let go, it takes up its original shape in just a second or so - but it is only a few centimeters in diameter, quite a small object. The enormous sun, more than a million kilometers across, doesn't do much worse: it would only take fifteen minutes or so for its equilibrium to be restored! In other words, the shape and figure of the sun is not likely to change dramatically over time: it really is remarkably stable. When we say the sun is in equilibrium, we mean quite literally that it is in the same sort of configuration - shape, size, temperature - with no significant changes for some millions, or even billions, of years. Studying the structure of the sun is therefore a lot easier than studying something like evolving weather systems, which are very chaotic and unstable. By the way, we will see later that there are `pulsating' stars which are not stable to this degree: some of them expand and contract, like a beating heart, getting bigger and smaller by about ten percent in size on a timescale of days or weeks and continuing in this fashion for centuries. We will explore the special physics that makes this possible later.

A Mathematical Exercise.

Our objectives now are to develop a full and precise understanding of the interior of the sun, and to be able to calculate what is going on within it. How soon will the nuclear fuel in the sun be used up? Will the sun's properties, both interior and exterior, change dramatically as it ages? What is its eventual fate? The development of such an understanding is a challenging task, one which has benefitted enormously from the development of modern computers. Please remember that this is, in general principle at least, no different from the way in which you might work out how long your supply of firewood will last given the vigour with which your pot-bellied stove is consuming it. But in detail this is a supremely mathematical and complex physical problem, requiring a deep understanding of the interactions between the nuclei of atoms - and a lot of other physics. So although I will describe the findings in qualitative terms in what follows, you should always be aware that our understanding is more than just conversational! To make this point more strongly, I show, in the figure below, a cartoon representation of an astronomer at a computer, coding in some of the basic equations which describe the physical structure of the solar interior. The cartoon repeats the point I made much earlier: ours is an intensely mathematical science, although I do not present it to you in that form. As you can see from the figure, astrophysicists use the scientific equations which relate the temperature, pressure, chemical composition, density, nuclear reaction rates, and so on of the constituents of the sun. In the figure, you can also see some representative numbers. Let us consider a few of them. The left-hand column represents the radius of the sun, and if you compare that column with the third one (the density) you can see that within the central 10 percent of the sun's radius, the density is about 100 times that of water. In these inner regions, the temperature is almost 15 million degrees (the second column). Sixty-three percent of the mass of the sun (column four) is found within the innermost thirty percent of the radius (which encloses the innermost three percent of the total volume), and 91 percent of the energy generated by nuclear reactions (the last column) occurs within the innermost twenty percent of the sun's radius. Although we now use computers, a model of the interior of the sun can be worked out by hand from these equations. Indeed, exactly that sort of thing was done fifty or more years ago! It is more difficult to consider how the interior structure changes as time goes on: the nuclear reactions lead to a slowly changing composition, with attendant structural changes, but this means working out many different models. As noted, such work is moderately straightforward these days thanks to the advent of powerful computers. Qualitatively, then, our models describe stars which are densest and hottest at the centre, and it is in these dense central regions that the nuclear reactions occur and most of the energy is released.

The Sun in Cross-Section.

Our complete solutions, also known as `solar models,' reveal, among other things, that the sun looks in cross-section rather as shown on page 500 of your text. Energy from the innermost hot core diffuses out through the sun first by radiative transport (which means simply that the photons make their way through the material, without the material itself moving much), and then - in the outermost parts of the sun - by convective transport (which means that there are large churning motions in the material itself, just as in a pot of boiling soup). We encountered this kind of convective motion before, when studying how the heat flow in the Earth drives plate tectonics and continental drift. A couple of lectures from now, we will consider some more of the superficial features of the sun. For today, however, I will consider only what is going on deep in the hot central core.

The States of Matter.

The material in the interior of the sun is not like anything you encounter in day-to-day life. You are all familiar with the three common states of matter - solid, liquid, and vapour - but have not yet had to deal with plasmas, although these represent the most abundant form of matter in the visible universe! The figure on page 123 of your textbook introduces the subject by describing matter at a variety of different temperatures, as follows: Look first at the bottom panel on the page, which describes material at a very low temperature, barly above zero Kelvins (also known as absolute zero). Such a low temperature means that the constituent particles are not moving around vigorously, so the atoms can remain bonded together in a solid form, like a crystalline lattice. An example is a cold chunk of rock, such as an asteroid in the outer solar system. This cold lump emits only long-wavelength radiation (remember the Wien Displacement Law from the fall term). At a somewhat higher temperature, say 300 K (room temperature), the typical particles in a substance are jiggling about at speeds of about 1/2 km/sec, and such motions can disrupt the crystalline bonds to turn some materials into liquids at these temperatures, just as ice turns to water. (Obviously not all substances melt at the same temperature.) The emitted light is at infrared wavelengths. At 500 K, the particles move at about 1 km/sec, and molecules and atoms can escape liquids (water boils!) so that the particles are now moving freely as a gas or a vapour. At 5000 K, a temperature like that at the surface of the sun, typical speeds are about 10 km/sec, and collisions between atoms knock off electrons: many of the the atoms are ionised. Material at this temperature emits visible light. At 10 million K, the sort of temperature found in the core of a star, all atoms are completely ionised, which means that it consists of free-moving charged particles: we call this a plasma . Dense material at this temperature emits energetic X-rays and gamma rays. But now we encounter some new physics: in this hot material, thermonuclear reactions can take place, and a new source of energy can be tapped - the energy source of the stars. At still higher temperatures, off the scale shown in the figure, other things can happen. Such temperatures can be encountered near black holes and were dominant in the early universe; but we will leave this discussion until later.

A Brief Reminder of Atomic Structure.

On page 122, you will find a reminder of the basic structure of the two simplest elements, hydrogen and helium. The kind of atom you have - hydrogen, helium, carbon, etc - is determined by the number of protons in the nucleus, also known as the atomic number. Nuclei can also contain neutrons; typically, the number of neutrons in an atom is comparable to the number of protons. Remember, however, that these atoms can also appear in various isotopic forms in which a nucleus may contain one or more extra neutrons. For example, hydrogen with just one proton is simply called hydrogen. If it has an extra neutron, it still behaves chemically like hydrogen (since it still has just one proton and, in unionized matter, one electron), but this isotope is called deuterium, and the nuclei are called deuterons. ``Heavy water'' is merely water in which each hydrogen has been replaced by a deuteron. Helium has two protons, and can exist in two isotopic forms: one with a single neutron, and the other (everyday helium) with two neutrons. Within the core of the sun, all elements are completely ionized. We can ignore the electrons, which are flying about freely. What really matters is the interactions between the nuclei - the protons (hydrogen) and helium nuclei (plus other species, in the later stages of the evolution of a star).

The Binding Energy Curve.

As I have said, the sun slowly converts some of its mass to energy, but it does not do this with 100-percent efficiency. That is, when a bit of hydrogen fuel disappears from the center of the sun, it does not all simply vanish, having been completely coverted to radiant energy. Instead, it is turned into helium (which you might think of as the ashes of the reaction) and only a fraction of the material gets converted to energy - less than one percent of it, in fact. How, then, do we calculate how much energy we can get out of a lump of hydrogen? Einstein's famous formula says "E = m c ** 2" but this merely tells us how much we could get in principle, if all the mass (m) were to be converted. In reality, we will get less, but by how much? The energy which is actually available in nuclear reactions can be encapsulated in something called the binding energy curve, a simplified version of which is shown in the figure below. (A similar figure appears on page 563 of your text, but is expressed in a slightly different way, so it looks upside-down compared to mine. The critical point is the shape, however, with one steep side and one flatter part.) In the figure, look at the top panel first. You will see that I have plotted something called `binding energy' as a function of atomic number. In a sense, binding energy measures the different amounts of energy invested in the construction of nuclei of different kinds. If nuclei can be changed from one kind to another, some of the energy can be liberated. One important feature, as noted, is the shape of the curve - its relative flatness on the right-hand side, and its extreme steepness on the left. To understand what this means, consider the middle panel in the figure - the one which distinguishes the positions of uranium, barium, and krypton. What this really summarizes is the fact that uranium can break (or can be broken) into two smaller pieces (barium and krypton, in this case) with a net release of energy. The amount of energy released is measured by the difference in heights shown on the figure, and the splitting action is called fission. This is what happens in an atomic bomb. Now look at the bottom panel of the figure. You can see that hydrogen lies very far below helium on the diagram. What this means is that if we take several protons - that is, the nuclei of hydrogen - and jam, or fuse, them together to form a nucleus of helium, we get out a lot of energy. Once again, the energy available is measured by the difference of energies shown in the figure. The process of jamming nuclei together is fusion, and this is what happens in a hydrogen bomb. Notice that much more energy is liberated in hydrogen fusion than in the fission of uranium. In short, big heavy nuclei (like uranium or plutonium) will release energy if they break apart - which is something we can stimulate them to do. Light nuclei, like hydrogen, release energy if they can be encouraged to merge to form heavier things like helium - which we can also encourage, in a different way.

The Sun: Not Exactly a Bomb.

It is tempting, but misleading, to think of the sun as a gigantic hydrogen bomb. It is true that the basic physics is similar - in both locations, hydrogen is being converted to helium, with a release of energy - but in a bomb, there is an important complicating factor. The problem with igniting a bomb is that it explodes! That is, the fuel is blown outwards, scattered into the air. If the bomb is poorly designed, much of it will simply be thrown out, unused. The ideal, for those who work in the unpleasant area of armament research, is to have the ignition and burning happen so fast that all the fuel is consumed before the bomb has had much chance to expand. Interestingly, the study of this problem - the functioning of the hydrogen bomb - was one of the most important reasons for the development of large computers, shortly after the Second World War. No one then foresaw a widespread use of personal computers. In the sun and stars, there is no such difficulty, because the fuel is held in place by the intense gravitational field. The nuclear reactions continue at a steady pace until the fuel is (eventually) consumed. The sun keeps a trickle of energy-generation going, just enough to sustain itself against continuing gravitational contraction.

A Digression: The Atomic Bomb vs the Hydrogen Bomb.

What happens in an atomic bomb is as follows: In a lump of fuel (uranium, or plutonium, say), one of the nuclei spontaneously fissions (i.e. breaks into pieces) or is prompted to do so because of a collision with a randomly-moving neutron. If the nucleus in question is uranium, it usually fragments into barium and krypton, as I indicated above; but in addition to these lumps, there is also a large amount of pure energy released, and a couple of neutrons. These new neutrons are important because they can individually run into yet another uranium nucleus and cause it to break down. The original neutron produced one fission event and two more neutrons; those can lead to two fission events and a total of four neutrons; these can lead to four fission events and eight neutrons; and so on. This is a chain reaction. If you don't want the uranium to blow up in your face, all you have to do is keep the individual nuclei widely enough separated that most of the neutrons do not instantly run into another uranium atom. (There are other ways of controlling the reactions, such as using moderators that control the speed of the neutrons and make them more or less likely to interact with the uranium atoms). If you do want an explosion, all that you need to do is bring into a single lump an amount of fissionable material which exceeds the so-called critical mass. Fundamentally, in an atomic bomb two sub-critical masses of uranium are merely slapped together; the explosion follows directly. To help you visualise this, I described in class an old Walt Disney documentary film I saw as a child, several decades ago, and showed you a couple of videos of my recent attempts to reproduce the experiment demonstrated there. In it, a roomful of mousetraps was set, with a ping-pong ball on each one. Finally, a single ping-pong ball was thrown into the room. The trap onto which it landed snapped, throwing its ball, the original ball, and the trap itself high into the air. Where they came down, more traps snapped; and so on. Soon the air was a blur of flying ping-pong balls. This is not a bad model for the fission chain reaction. Finally, of course, I can tell you that nuclear reactors like the ones at the Pickering power plant are designed to permit controlled fission reactions, but enormous precautions are taken in the design so that no critical mass ever is established. There is really no way that they could every explode like an atomic bomb. What they can do, as at Chernobyl, is release so much energy that they get very hot. If there should then be a failure of the cooling mechanism, the heat can rupture the containment vessel and allow radioactive gases and the like to escape. (This is still quite a serious accident, of course, although not as apocalyptic as a bomb.) A hydrogen bomb is quite different. In it, we need to heat the hydrogen fuel prodigiously to force the hydrogen nuclei together. In practice, this is done by using an atomic bomb as a fuse! In both of these unpleasant applications, the objective is to release as much energy as possible as quickly a possible. As noted, this is not what is going on in the sun: there, the reaction rates are steady and indeed rather slow. By the way, controlled nuclear fusion is something which would be very useful for the human race. From abundant low-cost fuel (the hydrogen in sea water, say), we could produce more energy than in a fission reactor (reconsider the binding energy curve) and produce very little of the dirty waste which fission yields. The big problem is to get and keep the fuel very hot - if we use a container at millions of degrees, the container walls themselves melt. But there is hope in that we may be able to contain the hot gases using magnetic fields so that the gases need never touch the vessel itself. There is a lot of experimentation going on now, with real hopes of success in the near future.

Back to Fusion in the Sun: The Proton-Proton Chain.

Let us now return our attention to the interior of the sun. The figure on page 471 of your text will remind you of the basic problem: if two protons approach each other at moderate speed, their positive charges lead to mutual repulsion and they `veer off' without coming close enough to fuse together. In the sun, as we have noted, this resistance is overcome by the expedient of having a very hot gas. At millions of degrees, the particles are moving so rapidly (at average velocities of 500 km/sec, with occasional particles moving much faster still) that every so often a pair of them collide and combine. How exactly does the hydrogen get turned to helium? Well, helium is almost exactly four times as massive as hydrogen, so your first thought might be that four protons (hydrogen nuclei) would have to collide in a single spot and all stick together to form helium, with a consequent energy release. This simple picture has two basic problems: The first difficulty is that helium does not contain four protons! As you know, it has two protons and two neutrons (which are almost exactly the same mass as protons), so we would have to understand how two of the protons somehow turn into neutrons in the process of fusing hydrogens together. The second problem is considerably more serious. To explain the real difficulty, I offered you the following analogy in class: there are many two-car accidents, or accidents in which one car runs into a lamppost, say, because there are so many cars moving about (in ways which sometimes seem nearly random!). But it is rather rare to have a multiple collision in which three or four cars all come together at once. In other words, as a driver you are much more likely to have one other car run into you than a pair of them, or three of them, at the same time from different directions. The way we constrain traffic on busy highways actually makes multiple-car accidents distressingly common, but if you visualise the protons in the sun moving around independently and truly at random, I think you will agree that there are much better odds of collisions between two protons than collisions between four protons all at the same time. Moreover, in a four-particle collision, each proton would feel the repulsive effects of three other positive charges, so it would be hard to make them merge. How, then, can we understand the build-up of helium from protons (hydrogen)? The answer is that it is not done in a single step. The figure on page 502 of your text shows the set of reactions which constitute the so-called proton-proton chain. (NOTE: You do not need to commit this sequence of reactions to memory! I want you to appreciate the sense of the process, with a slow steady buildup.) The important steps are as follows:< BR> In step one, a pair of protons combine. In the process, one of the protons turns into a neutron. Its positive charge is carried off by a particle which has the mass of an electron, but a positive charge. (Such a particle is called a positron.) In addition, a special kind of particle called a neutrino is emitted; this will be the subject of a later lecture. The new lump has an atomic mass of 2, but has only one proton so is an isotope of hydrogen. It is a deuteron. The newly formed deuteron is floating about in a sea of protons, and after a bit it runs into another proton. They fuse together, forming a nucleus of `light' helium (that is, it has two protons, so it is helium; but it lacks a second neutron, so is not `regular' helium). Energy is released in the form of a high-energy photon (a gamma ray). Meanwhile, similar processes have created another chunk of `light' helium elsewhere in the sun, and in due course two such chunks meet and fuse. This liberates more energy - and two protons. Now let us do some number work. In the first step, two protons combine. In the second step, a third one comes in. A parallel sequence elsewhere consumes another three, so six protons altogether are processed. But in the third step, the lumps of light helium merge and liberate two protons, so the net effect is four protons ===> one helium, as advertised.

How Do We Know All This?

How can we speak so knowledgeably about what is happening in the interior of the sun? There are three answers: The species involved, hydrogen and helium, are the simplest nuclei, so the pure theory of their interactions can be handled remarkably well by mathematical physics. It is possible to carry out experiments, on Earth, in which individual pairs of particles are made to collide. In this way we augment our theoretical understanding so that we have a pretty good idea of how particles might fuse in the conditions in the core of the sun. Finally, we can use a computer to `build' a full solar model. We ask the question: is it possible to put together a self-consistent computer model in which a big blob of gas with the same total mass and original composition of the sun undergoes nuclear reactions for 4.5 billion years (the known age) and winds up looking much as the sun does now? That is, would it have the surface temperature, total brightness, radius, and other attributes of the present sun? The answer is that our models do exactly this, and there is real reason for confidence.

The Positron and Its Uses.

As noted, the first step of the proton-proton chain (also called the `the P-P Chain' or `the P-P Cycle') led to the release of a positron. This particle is an example what we call anti-matter. The name is unfortunate, because it sounds like something out of black magic. But it is no more mysterious and unnatural than ordinary matter. The positron is just, in some sense, the `mirror particle' or `anti-particle' of an electron: it is exactly like an electron in mass and so forth, but has the opposite charge. The interesting thing about the positron is that when it meets an electron the two simply annihilate each other, and their mass is converted to pure radiant energy (in the form of a pair of gamma rays). This behaviour, of course, explains why we don't see large quantities of anti-matter sitting around. It would simply vanish in a very explosive fashion when it came into contact with ordinary matter! But it can be, and is, produced for fleeting moments and in small quantities, in particle accelerators and so on. Other particles, like protons and neutrons, have anti-particles too, but the positron is the one we encounter in nuclear burning in stars. As unlikely as it sounds, modern medical science has been able to put positrons to work, in an application called Positron Emission Tomography (PET). Suppose you had a brain tumour or some sort of lesion on which a doctor was reluctant to do exploratory surgery. Well, it is possible to introduce into the blood stream a small amount of a radioactive `tracer' element which can become concentrated in the tumour. The radioactive tracer decays inside the tumour, and the decays can include the emission of positrons which travel almost no distance at all before encountering electrons and annihilating. This may sound pointless, but the gamma rays given off in the annihilation are so energetic that they come straight out through the skull (causing no damage, because there are very few of them) and can be detected. The directions they come from can be carefully measured, and it is possible to build up a detailed image of the tumour without any invasive surgery or other intervention. This technique has been an important area of research in the Physics Department at Queen's and at the Kingston General Hospital. Previous chapter:Next chapter


0: Physics 016: The Course Notes, spring 2005. 1: The Properties of the Sun: 2: What Is The Sun Doing? 3: An Introduction to Thermonuclear Fusion. 4: Probing the Deep Interior of the Sun. 5: The Sun in More Detail. 6: An Introduction to the Stars. 7: Stars and Their Distances: 8: The HR Diagram: 9: Questions Arising from the HR Diagram: 10: The Importance of Binary Stars: 11: Implications from Stellar Masses: 12: Late in the Life of the Sun: 13: The Importance of Star Clusters in Understanding Stellar Evolution: 14: The Chandrasekhar Limit: 15: Supernovae: The Deaths of Massive Stars, 16: Pulsars: 17: Novae: 18: An Introduction to Black Holes: 19: Gravity as Geometry: 20: Finishing Off Black Holes: 21: Star Formation: 22: Dust in the Interstellar Medium: 23: Gas in the ISM: 24: The Size and Shape of Our Galaxy: 25: The Discovery of External Galaxies: 26: Galaxies of All Kinds: 27: The Expanding Universe: 28: Quasars and Active Galaxies: 29: The Hot Big Bang: 30: The Geometry of the Universe: 31: Closing Thoughts:


Part 1:Part 2:Part 3:


Home.
Mystery destination!


(Thursday, 28 March, 2024.)

Barry's Place Speical Offer