(S-7) The Energy of the Sun
(S-7A)   The Discovery of Atoms and Nuclei
Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern
This lesson plan supplements: (S-7) The Energy of the Sun : on disk Sun7enrg.htm, on the web
The student will learn
To the teacher:
A problem faced in covering modern science at the high school level is that so much must be accepted on faith, because too much time and effort would be needed to explain the reasons why basic ideas are held to be true.
This is a delicate subject. Most students will accept taught facts without questioning them--for instance, accept that Mt. Everest exists, even if none of them ever saw it. Yet sometimes so much is accepted on faith that the entire structure becomes suspect.
Students need to realize that all the abstract concepts of science--atoms, nuclei, protons and neutrons, none of them visible to us--evolved gradually, that scientists questioned them at every step, and in the end accepted them only because no other interpretation seemed possible.
One is reminded of a story (possibly even true) about a 19th century meeting in which British teachers discussed the math curriculum. The question arose whether to teach Euclid's classical geometry, with its interlocking sequence of theorems and proofs. One teacher rose up and said something like the following: "If a duly accredited teacher tells the student that the sum of the angles in a triangle is 180 degrees, the student should accept it and not be required to prove it as well."
Today we smile at this argument, the very opposite of the scientific approach we try to impart to students. Yet in physics and astronomy, just as much as in mathematics, the student must learn to appreciate the reasons, not just memorize the contents.
An additional historical review is provided for this section, linked further below. If time permits, cover its contents as well.
Starting the lessonOne problem in high school physics is that so much must be covered! Physics has advanced tremendously in the 20th century, but many of its recent advances involve complicated theory and intricate observations--so much that even university professors find it hard to explain everything.
Today we discuss energy generation by the Sun, which involves atoms and nuclei. Most high school teachers (and most texts) simply tell students (teacher writes on the blackboard, and students copy):
No one has ever seen an atom, nucleus, electron, proton or neutron.The fact is, it took well over a century to reach these conclusions. And as in the rest of physics, the existence of these objects was accepted only after the evidence of observations and experiments left us with no alternative.
(Click here for a brief history, S-7A The Discovery of Atoms and Nuclei.)
Questions in Class.Some of these questions are not easy, and depending on the class, the teacher might prefer to provide their answers and use them as part of the teaching process.
You read that "the solar constant is 1.3 kilowatt/m2." What does this mean?
You air-condition your house on a hot summer day, and the air conditioner draws a current of 30 Amperes at 110 volts, consuming 30 x 110 = 3300 watt. Suppose you live in the age of solar power, and obtain your energy from an array of solar cells which converts 5% of the energy of sunlight into electricity. Also, because of the atmosphere and other limitations, these cells only receive an energy flow of half the "solar constant". What area of solar cells do you need to run the air conditioner?
Presumably all parts of the solar system--Earth, Sun, planets--came into existence together. How do scientists estimate the age of the Earth?
What age do such measurements suggest?
Presumably, the Sun has been shining at least for as long as the age of the oldest rocks. What energy source for the Sun, based on physical laws, was the first to be proposed?
What was the difficulty with this explanation?
What source of energy is nowadays credited with the Sun's heat?
To understand nuclear energy, we need to know a few things about atomic nuclei. What are they made of?
The teacher may supplement: Since protons and neutrons create very similar nuclear forces, they are sometimes given a common name "nucleons. "
Neutrons are slightly heavier, and if free neutrons are produced, they convert spontaneously into (proton + electron) in about 10 minutes (a 3rd particle, a very light "neutrino," is also produced). One may say that the free neutron is "radioactive."
What forces exist between protons and neutrons?
(The teacher can demonstrate one sort of "short range force" by a cluster of small "button magnets" as are used to post notes on (say) refrigerator doors. The magnets will stick together, but if you pull one away from the rest, you only need remove it a short distance before the attraction is reduced to practically zero.
This is also a "short range force," although mathematically it behaves differently from the nuclear force.)
There exists another nuclear force, much weaker. What does the weak nuclear force do?
If two particles are attracted to each other, and we let their attraction move them--is energy released or absorbed?
(if students are not sure) Suppose I hold a stone in my hand--here. The Earth attracts it downwards. If I let it fall in the direction it is attracted--does it gain energy or does energy have to be invested?
On the other hand, to lift the stone from the floor against gravity, separating the two attracting objects, you must... ?
When we add a neutron to a nucleus, do we gain or lose energy?
When we add a proton to a nucleus, what two kinds of force are involved--and do they give energy or absorb it?
In the above process, then, electric forces absorb energy and nuclear forces release it.
Taking both into account--is net energy lost or gained? The answer depends on how big the target nucleus is. Can anyone explain?
By the same argument, though, if we could break up nuclei heavier than that of iron, we should gain energy. True or false?
Teacher supplements: Practically all the helium atoms we use to fill balloons and blimps started out as alpha-particles from radioactivity!
How do we know? From the light emitted by helium on the Sun, we know that it contains a small percentage of "light helium" whose nuclei have two protons but only one neutron. The light of stars suggests that they, too, contain a little of this variety.
But on Earth, this kind of helium is very rare! Its rarity suggests that almost all of the helium present when the Earth first formed was lost to space. Meanwhile, however, new "ordinary" helium was produced in rocks, in the form of alpha particles from uranium and similar elements. Some of it diffused, over millions of years, into natural gas, and that is where we get most of our helium today.
What is the particular process believed to be responsible for the Sun's energy? What is the fuel, and what is the final product?
This process is called...?
The teacher may explain: nuclear fusion does not happen in one step. That would require 4 protons colliding at the very same instant, something that is not too likely.
Instead, the reaction occurs in stages (outline on the board)
Where does the released energy appear? The nuclei emit gamma rays, while the positrons meet with electrons and both are "annihilated,"in the process, also leaving behind gamma rays. All this gamma radiation is absorbed in matter and heats it up.
Interestingly, the helium nucleus is lighter, it has less mass than the combined mass of the 4 protons that the Sun started with. If m is the difference in mass (the term is "mass defect"), then the energy E released is given by E=mc2, Einstein's famous formula.
The calculation below helps drive home the energy-mass equivalence.
The Sun generates energy constantly by converting hydrogen to helium in its deep core. As noted above, the helium produced has slightly lower mass than the sum of the masses of protons (= hydrogen nuclei) being fused together, and by Einstein's famous equation
(Let the class vote, and write on the blackboards the number voting for each guess--which of the 5 choices comes closest:
less than 1000 tons--1000 ton--1,000,000--1,000,000,000 --more ? )
Let's start by estimating the energy output of the Sun. The numbers are BIG, so we need use scientific notation, explained (if necessary; the unit linked here assumes the user is familiar with powers of numbers, especially of 10) here. We must work in consistent units, so all distances will be in meters, time in seconds, mass in kilograms, and energy then is naturally expressed in joules.
The area of a sphere of radius R is
The energy flowing through each square meter is the solar constant S, about 1300 watt or 1300 joule per second. So the total energy output of the Sun is
E = A S = 2.83 1023 × 1300 joule = 3.676 1026 joule/sec
The velocity of light is (approx)
of about 4 million tons per second. Who voted for 1,000,000 ?
The Sun also loses mass by emitting the solar wind, a fast stream of protons and other ions, evaporated at high speed from the top of the outermost layer of the Sun, the million-degree corona.
Which mass loss is greater?
The density of the solar wind at the Earth's orbit is of the order 107 protons per cubic meter (or a little less), moving typically at a velocity of 400 km/sec or 4 105 meter/second.
That means that each second, the flow of protons through a meter-squared area of the sphere we drew earlier, involves all protons in a stack 1 m2 wide and 4 105 meter long. They number
Now the mass of the proton is 1.673 10–27 kg. To derive this we need to know the number of atoms in a gram of hydrogen ("Avogadro's number"), a hard problem which took many years and efforts to solve, so we won't even try. The mass of the sun's total loss through the solar wind is therefore
It is remarkable how close these two numbers are--one dictated by processes in the innermost core of the Sun, the other by processes in its outermost layer.
Coincidence, you say?
What is controlled nuclear fusion?
Lacking the enormous pressure of the Sun"s core, how do laboratory experiments in controlled nuclear fusion manage to hold the very hot hydrogen together?
The teacher may explain further: no magnetic field produced in the lab can match the enormous pressure at the center of the Sun. However, fusion is also possible at lower pressures and temperatures, with fuels that "fuse" more easily--for instance, heavy forms ("isotopes") of hydrogen, which besides a proton contain one or two neutrons. Even with them, however, no commercially useful fusion power has as yet been released.
If stars get their heat by the fusion of hydrogen to form helium--what happens when all the hydrogen is used up and converted to helium?
(Teacher might explain) For a while the star may gain energy from the fusion of nuclei larger than hydrogen, but that energy source does not last long. When the star is no longer able to generate heat, gravity takes over and heat is released by shrinkage--the process originally proposed for the Sun.
A Sun-sized star has a complicated final evolution, including a "red giant" stage when it "puffs up" with a radius greater than that of the Earth's orbit, relatively cool and rarefied. In the end, what is left probably becomes a "white dwarf," a star in which gravity has crushed all atoms and smeared out their electrons. This is an extremely dense star, no bigger than Earth, but with a mass that is still an appreciable fraction of the mass of the Sun. After energy generation dies out, it becomes a dark dwarf, and it is anybody's guess how many of those are hidden in space, because we have no way of observing them.
What is the fate of a star 4 times heavier than the Sun?
Why does the strength of the force of gravity and the energy released by it depend on the final size of the object?
(Teacher: in a while we will try to calculate that force and energy)
What does the general theory of relativity suggest about the final fate of a star 50 times more massive than the Sun?
Calculation of Escape Velocities (Optional)
The escape velocity V from the surface of the Earth, at radial distance r, was calculated in an earlier lesson to satisfy
where g is the acceleration due to gravity. Using SQRT to denote square root
By Newton's theory of gravitation, if m is the mass of the Earth
Here G is the number that measures the strength of the gravitational pull, the one which the delicate experiments by Cavendish and Eötvös determined. Then
and for a star of mass M and radius R
That is the velocity needed for an object to fly off the star to infinity, starting with distance r. But by the conservation of energy, it is also the final velocity of an object coming from far away and hitting the surface. It is therefore a measure of the energy that a star releases by collapsing to radius R.
Let us go through some very approximate calculations, just to get orders of magnitude. We start from a result derived in section #21 of "From Stargazers to Starships," by which a space vehicle launched from the Earth's orbit needs 12.4 km/sec to escape the solar system altogether.
This is above and beyond the 30 km/sec which it already has from the Earth's motion around the Sun, making the total "escape velocity" from a distance 1 AU from the Sun
with M the mass of the Sun and R1 the Earth-Sun distance, the "astronomical unit" (AU).
(The teacher may continue, or may call students to do the next 3 stages of the calculation.) |
(End of optional derivation)
Note: We now know that at the center of our galaxy is a huge black hole, whose mass has been astimated at 3.7 million times the mass of the Sun. Other galaxies may well have similar concentrated centers. See #7A. The Black Hole at the Center of Our Galaxy.
The final collapse of large stars creates supernova explosions. What happens there?
The teacher may explain further: One product of nuclear reactions are neutrinos, particles with no electric charge and (probably) a very small mass, which respond neither to electric forces (they carry no electric charge) nor to nuclear forces. As a result they can easily go through the thickness of the Earth or even the Sun without hitting anything. Only very rarely do they interact with matter, through the weak nuclear force.
Experiments in large tanks of water or special fluids, buried deep underground (to shield out the effects of "cosmic rays," fast ions from space) can detect occasional neutrinos, usually from the Sun's core. On 24 February 1987, a supernova became visible (mainly from the southern hemisphere) in the Large Magellanic Cloud, a small galaxy attached to ours. Although some 150,000 light years distant, that object became visible to the eye, and was studied extensively since that time.
[Mainly for astronomy buffs--however, this phenomenon is also discussed at the end of the 2nd section on the Doppler effect, added to this web-site collection after this lesson plan was written.]
Supernovas created in the above process are known as "Type II" supernovas. Those of "Type I" are believed to be neutron stars which belong to a "double star" pair, orbiting around a common center of gravity. The neutron star keeps attracting gas from its companion, ultimately passing the limit for the creation of a black hole, and bang! A supernova!
The point here is, that this always happens at about the same mass, and the maximum brightness, as well as the rate of its decay, should always be the same. Actually, variations exist because of differences in the composition, but they can be recognized and corrected for. Thus a type I supernova is like a precalibrated light source. By observing such events in distant galaxies, and noting how bright they appear to us, we can estimate their distance.
The spectral lines of distant galaxies are also moved towards the red end of the spectrum by a "Doppler shift" because they all recede from us. That is part of the expansion of the universe which started at the "big bang", between (it is estimated) 13 and 14 billion years ago. Comparing the recession to the distance derived from supernovas, astronomers have concluded that the expansion of the universe it actually accelerating.
That observation remains a great mystery. One could explain it if the expansion of the universe were slowing down, by proposing that the attraction of galaxies was working against it. Instead, there seems to exist an unknown force driving the universe apart more and more. Astronomers are still trying to understand it.
Author and Curator: Dr. David P. Stern