Nuclear Fusion in the Sun
One Ring to rule them all, One Ring to find them,
One Ring to bring them all and in the darkness bind them
(J.R.R.Tolkien Lord of the Rings)
150-154
To understand how the Sun is able to produce as much energy as it does we must look inward to the universe at the smallest scale currently detectable - the subnuclear scale of protons, neutrons and quarks.
Forces Within Nuclei
When you first learned about nuclei you may have wondered 'What holds the nucleus together?" Packed in a dense sphere only a few femtometers across are the protons and neutrons that make up ordinary nuclei. But the protons are all positively charged and you have known for a long time that "like charges repel". Prior to the 20th century there were only 2 known forces - the gravitational force and the electromagnetic force.
The electromagnetic force is the force that we see when magnets attract or repel or when clothing "clings together" when taken from a clothes dryer on a dry winter day. It is the electromagnetic force (often referred to as the Coulomb Force) that pushes protons apart or binds protons and electrons together. This force comes in two "flavours".
All particles exert attractive gravitational forces on each other. However, the gravitational force is extremely weak when compared to the electromagnetic force. In fact it is only about 1/1035 the strength of the electromagnetic force. The gravitational force is completely inadequate to explain why the protons of a nucleus can be packed together as they are.
So - there must be another force that comes into play in the nucleus. Early in the 20th century physicists discovered that there are actually two special forces that we encounter at the scale of the nucleus. They are the Weak Nuclear Force and the Strong Nuclear Force. The weak nuclear force is responsible for some forms of radioactive decay. It is the strong nuclear force that is responsible for holding the nucleus together.
The strong nuclear force is always attractive and two protons brought near each other would experience both the attractive power of the strong force and the repulsive effect of the Coulomb force. The strong force is about 1000 times stronger than the Coulomb force. But - there is a catch. The strong force only acts over a very small distance - only a few femtometers. So, unless the protons are very close together the affect of the strong force is negligible and they repel each other. The applet StrongForce (Figure 7.13) illustrates this. With the mouse move the right most proton toward the dotted vertical line. If you release it the proton will be repelled by the other proton. The dotted line represents the position at which the strong and Coulomb forces are equal in magnitude. If you put the proton very close together you will see that they attract and are bound together. (Note - because strong force changes so rapidly with distance the applet must use a much smaller time step in the region close to the nucleus - it will appear to run very slowly but this is necessary to maintain precision).
It is the strong nuclear force that is responsible for the energy produced in the core of the sun.
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Figure 7.13  StrongForce illustrates how the strong nuclear and Coulomb forces act on a pair of protons. |
The Power House
The center of the sun is hot! At about 14 000 000 K and with a density of about 160 g/cc (this is still a gas!!) atomic collisions are frequent and violent. Occasionally hydrogen nuclei fuse or stick together. This releases energy. Why energy is released was discovered in 1905 by Albert Einstein and can be expressed in the famous equation
The equation is called the mass-energy equivalence and tells you that "m" kilograms of matter (anything - uranium, coal, your old running shoe!) can, in principle be turned into pure energy. The "c" in the equation is the speed of light. In the case of the sun hydrogen atoms are fused together to produce helium in a process known as the proton-proton (or PP) cycle.
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| Figure 7.14 The proton-proton cycle |
Using the mass-energy equivalence equation we can precisely quantify how much energy is released during a fusion reaction. To do this we note that 4 hydrogen atoms have slightly more mass than 1 helium atom. We can summarize it this way:
- 4 H nuclei weigh 6.693 x 10-27 kg
- 1 He nucleus weighs 6.645 x 10-27 kg
- missing mass converted to energy is 0.048 x 10-27 kg
In symbols you can write the pp-reaction as:
In the pp-reaction you see a number of odd looking terms:
- e+ is a strange particle called the "anti-electron" or positron. This is a electron with a positive charge!
- γ is a gamma ray photon which is emitted and carries away some of the energy produced in the fusion process
- ν is a neutrino - an elusive particle that carries away a tiny amount of energy in the fusion process
- 2H and 3He are isotopes of hydrogen and helium respectively. Occasionally you see 2H written as 2D where D stand for deuterium which is an isotope of hydrogen.
Example 7.4 How much energy is released when 4 hydrogen atoms fuse to form a helium atom?
Solution: Use . In this case the mass "m" converted into energy is the missing 0.048 x 10-27 kg. In order to use consistent units you will need to express 'c' in m/s, so c = 3 x 108m/s. Inserting numbers then tells us that 1 fusion reaction liberates E = (0.048 x 10-27 kg)(3 x 108m/s)2 = 4.3 x 10-12 J of energy
The energy released in the fusion of 4 hydrogen atoms may not seem to be very much but there are a LOT of hydrogen atoms in the sun. Since we can measure the amount of energy we receive from the sun it is quite easy to estimate how many fusion reactions must occur each second to "power the Sun". Recall from Chapter 7.1 that the power output from the Sun is approximately 4 X 1026 J/s. To produce this energy s there must be:
Example 7.5 How many tonnes of hydrogen must be converted into helium each second in order for the Sun to shine as it does?
Solution: You have, in a sense already answered this when you concluded that there must be 1038 fusion reactions each second. Since there are 4 hydrogen atoms used in every fusion reaction this means that each second 4 x 1038hydrogen atoms get used up. The mass of 1 hydrogen atom is 1.67 x 10-12 kg so the total mass of hydrogen consumed each second is:
or about 700 million tonnes!
Fusion - A Closer Look
The proton-proton reaction starts when two protons (hydrogen nuclei) are able to overcome the repulsive force, or Coulomb barrier between them and get close enough for the strong nuclear force to bind them. Since the speed of the protons depends on temperature, fusion can only occur at extremely high temperatures. The applet ProtonCollide (Figure 7.15) allows you to investigate how proton-proton fusion depends on temperature. In this applet only the first step of the proton-proton cycle is shown.
| Figure 7.15 ProtonCollide allows you to investigate how fast protons must travel in order to overcome the Coulomb barrier. |
Example 7.6 Use the applet ProtonCollide to estimate the minimum temperature needed for the proton-proton reaction to proceed.
Solution: Proton-proton fusion will begin at approximately 10 million K. If this is to lead to sustained fusion reactions and the complete proton-proton cycle then the density of gas must be high enough for these reactions to occur in rapid succession. In the case of the sun the central temperature is about 14 million K and the density is quite high at 160 g/cm3.
Low mass stars, such as our Sun, use the proton-proton reaction to generate almost all of their energy. More massive stars can use "more efficient" reactions (which you learn about in the next unit). Regardless of reaction used, however, the net result is always the conversion of 4 hydrogen atoms into a helium atom. This turns out to be the most basic defining process for stars.
Where have all the neutrinos gone? Too few passing.
In the mid 1960's astronomers made an odd suggestion - "perhaps the Sun was going out!!" What prompted this rather ominous conclusion came as the first results of a new experiment began to emerge. If you look at the symbolic equation for the pp-reaction you will see the term 'n' which denotes the neutrino. Neutrinos are elusive to say the least. The consummate artful dodgers, they escape virtually unhindered from the sun and are expected to be produced in copious number. As elusive as they are they can be detected. Neutrino telescopes are located deep in the bowels of the earth - usually in unused mine shafts. Despite continued efforts of countless teams to observe solar neutrinos the observed production rate is only 1/3 of what would be expected given the p-p reaction and central temperature of the sun. This came as a shock to astronomers because if there is one thing we think we understand it is fusion reactions! This became known as the "Neutrino Problem" .
Resolution of the neutrino problem came through the efforts of Canadian and British physicists, using SNO (the Sudbury Neutrino Observatory). Beginning in 2000 they found evidence to support the idea that the neutrinos from the sun change their "type" from the one type produced in the solar core to one of the three types found in nature. This "neutrino" oscillation explains why we see 1/3 of the expected number. Prior to this, solar neutrino telescopes were detecting only one type of neutrino.
A Closer Look at the Sudbury Neutrino Observatory
One of Canada's most innovative observatories lies at the bottom of an abandoned Sudbury mine shaft! Poor planning? NO! This is the ideal location for a "telescope" whose job is to collect the elusive neutrinos that stream from the Sun. The neutrino is a bizarre particle. Typically a neutrino can pass through a light year of lead with only a 50% chance of being absorbed!! But they can be detected. The Sudbury telescope is similar to a number of other neutrino telescopes scattered around the world.
Figure 7.16 shows the entrance to the SNO. Located 2 km beneath the surface you will find the "detector" - a 12 m diameter acrylic plastic sphere filled with ultra-pure heavy water. The plastic sphere is shown in Figure 7.17 while under construction. Figure 7.18 shows the finished detector in place at the bottom of the mine. The outside of the plastic sphere is "encrusted" with nearly 10 000 photomultiplier tubes that detect tiny flashes of light produced when neutrinos are absorbed by deuterium nuclei. SNO detects about 10 neutrino absorptions per day. What's "Special" About SNO? Prior to late 1999 when SNO went into operation, neutrino telescopes employed a common dry-cleaning fluid to absorb neutrinos. This fluid, however, could absorb only 1 "flavour" of neutrino. By using ultra-pure heavy water (D2O) SNO was the first neutrino telescope to be able to detect all three forms of the neutrino and has beautiful confirmed that the neutrino production rate inferred from the SNO data is precisely what our theories of solar structure predict. Why the Bottom of a mine Shaft? All neutrino telescopes must be shielded from the steady "rain" of cosmic rays (mostly high energy protons" that strike the Earth. By placing the detector under several kilometers of rock this will effectively absorb any cosmic rays which would produce "false" flashes of light in the photomultiplier detectors. What Else Can SNO "Observe"? SNO and other neutrino telescopes can also provide us with extremely important information about supernovae - exploding stars. As you will see in Unit 4, it is ultimately the tremendous increase in neutrino flux from a collapsing massive star that triggers the explosion of the supernovae. The first supernova to be observed by neutrino telescopes was the 1987A supernova.
Example 7.7 If fusion in the Sun's core suddenly stopped how long would it be before astronomers would notice? Solution: Unlike what you saw in the previous chapter where you looked at the random walk of photons out off the sun (with time scales of hundreds of thousands to millions of years!) astronomers would detect the cessation of fusion within minutes! This is because neutrino telescopes worldwide would see a sudden drop in neutrino detections. Neutrinos have very little chance of being absorbed on their way out for the sun. Since the centre of the Sun is 149 million km from Earth it would only take about (149 000 000)km/(300 000 km/s) = 497 seconds or 8.3 minutes for the detection rate to shut off. |
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| Figure 7.16 Entrance to the Sudbury Neutrino Telescope at the Creighton #9 mine. (Image courtesy of SNO) | |
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| Figure 7.17 The acrylic "bubble" that holds the ultra-pure heavy water absorbing fluid. (Image courtesy of SNO) | |
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| Figure 7.18 The "SNO-ball" in place at the bottom of the Creighton #9 shaft. (Image courtesy of SNO) | |
Practice
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To understand how the sun produces its energy and how the energy is transported within the sun.
Chp 8.2
