Lesson Plan #30         http://www.phy6.org/stargaze/Lgravity.htm

# (20)  Newton's Theory of Universal Gravitation

This lesson re-derives Newton's result, that if the force of gravity decreases like the square of the distance from the center of attraction, the same force that causes objects on the Earth"s surface to fall can also be responsible for the motion of the Moon.

Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern, Code 695, Goddard Space Flight Center, Greenbelt, MD 20771
u5dps@lepvax.gsfc.nasa.gov or audavstern@erols.com

 This lesson plan supplements: "Newton's Theory of Universal Gravitation," section #20   http://www.phy6.org/stargaze/Sgravity.htm "From Stargazers to Starships" home page: ....stargaze/Sintro.htm Lesson plan home page and index:             ....stargaze/Lintro.htm

Lesson Plan #31         http://www.phy6.org/stargaze/Lkepl3rd.htm

# (21)  Kepler's 3rd Law

### (21a)  Application of Kepler's 3rd Law

In these closely linked units, Newton's calculation is applied to artificial Earth satellites. It is shown that at least for circular orbits, this calculation leads to Kepler's 3rd law. The velocity required for a low Earth orbit is derived, and a practical formula is obtained for the orbital period in a circular Earth orbit of any radius.

Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern, Code 695, Goddard Space Flight Center, Greenbelt, MD 20771
u5dps@lepvax.gsfc.nasa.gov or audavstern@erols.com

 This lesson plan supplements: "Kepler's 3rd Law," section #21   http://www.phy6.org/stargaze/Skepl3rd.htm "Application of Kepler's 3rd Law," section #21a   http://www.phy6.org/stargaze/Sappl3rd.htm "From Stargazers to Starships" home page: ....stargaze/Sintro.htm Lesson plan home page and index:             ....stargaze/Lintro.htm

Lesson Plan #32         http://www.phy6.org/stargaze/Lframes1.htm

# #22   Frames of Reference: The Basics

### #22b    The Theory of Relativity

Velocity and acceleration always need to be measured relative to some fixed benchmarks, which define a "frame of reference. " Do the laws of physics depend on which frame we use? This section shows that two frames of reference moving relative to each other with a constant velocity are completely equivalent, and the same laws hold in both.

Each such frame is consistent, but their observations may differ. Section 22a describes how the motion of the Earth modifies the apparent positions of stars and the arrival direction of the solar wind.

Optional Section 22b contains a brief non-mathematical discussion of what the theory of relativity is about.

Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern, Code 695, Goddard Space Flight Center, Greenbelt, MD 20771
u5dps@lepvax.gsfc.nasa.gov or audavstern@erols.com

 This lesson plan supplements:   "Frames of Reference: The Basics," section #22   http://www.phy6.org/stargaze/Sframes1.htm    "The Aberration of Starlight," section #22a        http://www.phy6.org/stargaze/Saberr.htm    "TheTheory of Relativity," section #22b http://www.phy6.org/stargaze/Srelativ.htm "From Stargazers to Starships" home page: ....stargaze/Sintro.htm Lesson plan home page and index:             ....stargaze/Lintro.htm

Goals: The student will learn

• The concept of frames of reference in physics.

• That two frames of reference, each moving with respect to the other with a constant velocity v (constant speed, constant direction), observe the same accelerations and therefore Newton's laws are the same in both.

• About the aberration of starlight and its explanation by James Bradley, including the story of the flag on a moving boat, which gave the essential clue.

• About the aberration of the solar wind and the behavior of comet tails.

• About the proposed "Solar Probe" space mission

• (optional sect. 22b) What Einstein's theory of relativity is about, in qualitative terms.

Terms: frame of reference, relative velocity, stellar aberration, solar wind, comet tails, principle of relativity, classical mechanics, relativistic mechanics, quantum mechanics.

Stories and extensions: The story of the aberration of starlight and of Bradley's observation on a boat in the river. About the solar wind, magnetosphere and comets.

Note to the teacher: This lesson is closely related to the one on vectors (section #14 of "Stargazers", lesson plan #23). Some ideas expanded here were already introduced in #14--for instance, the motion of an airplane flying with velocity v1 relative to the air, which itself (because of the blowing wind) has a velocity v2 relative to the ground. In that example, the air and the ground represent two frames of reference moving with respect to the other, and we have already shown that the velocity of the airplane with respect to the ground is the vector sum v1+v2 .

Here, however, two additional aspects come into play. One, we are also concerned with accelerations and forces. These are the simplest cases, where all velocities are constant in magnitude and direction, so that shifting from one frame to the other adds no new forces or accelerations (That will no longer hold when we come to discuss rotating frames). . And two, we study the changes created by the motion of the observer's own frame of reference.

Section (22a) is optional. It contains interesting stories, illustrating the lesson, but can be omitted (and perhaps assigned to some advanced students) if time runs short. It is also possible to teach only the first example, on the aberration of starlight and on its explanation by James Bradley.

Starting the lesson.

The starting paragraphs of Section #22 are quite appropriate for starting the lesson. After that, bring up the questions below, and continue with Section #22a.

Questions and tidbits:

--What is meant by a "frame of reference"?

A frame of reference is a set of reference points with respect to which motion is measured. These points move together and keep their relative distances and angles of view.

--Can you give examples of frames of reference?
--Interior of a house, a ship, airplane, car, railway carriage or spaceship.
--Surface of the Earth, the Moon or Mars.
--A moving elevator, merry-go-round, roller coaster car or other ride.
--The frame of the wind carrying a run-away balloon, or of a river carrying a swimmer.
--Also, in certain contexts, the frame of the distant stars.

We have two frames of reference: A is the inside an elevator rising with constant velocity u, B is the frame of the building in which the elevator is located. A rider drops a penny inside the elevator. Is the velocity of the penny the same as seen from A and B?

No, in A its velocity includes u, in B it does not. It may be falling with respect to the elevator cabin but actually rising with respect to the building.

In the preceding example, is the acceleration of the penny the same viewed inside the elevator and outside it?
--Yes, it is equal to g in both frames--as long as the elevator's velocity is constant.

You are the passenger in a car driving with velocity u on a rainy night. On the street outside, through the side window of the car, you see raindrops falling. They fall with a constant velocity v (because of air resistance, they no longer accelerate). As you watch them in the light of streetlights, how do they appear to move? What is their apparent velocity w? In what direction do they streak the windows?
--The outside of the car appears to be moving with velocity -u (like u but opposite direction) to the rear. Raindrops as viewed from the moving car seem to have a velocity -u in addition to their falling velocity v, causing them to slant backwards, and their streaks on the windows slant similarly. (Draw on the board)

Their velocity vector w has a vertical downwards component v (magnitude of v) and a horizontal component u (magnitude of -u) to the rear: in vector notation w = v-u = v+(-u). Since v and u are perpendicular to each other, by Pythagoras, w = SQRT(v2 + u2). Their streaks on the window are in the direction of w and the angle A between those streaks and the vertical satisfies sinA = u/w or tanA = u/v.

How are distances to stars measured by the parallax method?
The Earth's orbit around the Sun is approximately a circle whose radius is about 150,000,000 km ("one astronomical unit" or 1 AU). Therefore, on two dates 6 months apart, the Earth occupies positions (A,B) separated by 300,000,000 km (or 2 AU). Say the star is at point C, and assume the diameter AB of the Earth's orbit was chosen in such a way that AC is perpendicular to it (always possible!).

If the directions to C are slightly different when viewed from A and B, then the difference gives the "parallax" angle between AC and BC. Using that angle one can calculate all other properties of the triangle ABC, including the distances AC abd BC from Earth to the star.

What changes were observed around 1700 in the position of Polaris?
It seemed to move in a small ellipse, about 20" wide.

How did astronomers know that it was not Polaris that did the moving?
The motion around the ellipse took 1 year to complete, and it was highly unlikely that Polaris would match the Earth's orbital period. Also, other stars near Polaris displayed similar motions.

--How did James Bradley know that the shift of Polaris was not a parallax effect?
Because the greatest displacement of Polaris in any direction did not match the greatest displacement in the opposite direction by Earth in its orbit, but occured 3 months afterwards.

--In the end, how did Bradley explain the strange shift in the position of Polaris and other stars?
The velocity u of the Earth in its orbit made starlight observed from Earth appear to have an extra velocity (-u) added to its own. Since the added velocity had a sideways component, perpendicular to the direction from the star, it shifted the direction from which the light appeared to come.

--The aberration of starlight allows us to deduce that the Earth is indeed moving. Doesnŝt that contradict an earlier claim that absolute motion is undetectable?
The claim was that absolute motion with constant velocity in a straight line was undetectable. The motion of the Earth discussed here is around a circular orbit.

[Optional further discussion by the teacher:

Suppose Earth and the whole solar system did move with constant velocity u along a straight line. The positions of stars would then be shifted, too, but the shift would not change as time went on. Astronomers would see the positions of the stars fixed and not suspect their real direction was different.

Actually, we have other cues, and from them we know that the solar system is moving at about 20 km/s towards a point known as the solar apex, near the star Vega. But in principle, it could also be that we are at rest and all those stars are moving in our direction, away from the solar apex. The physical effects would be exactly the same. It is only our logic that tells us it is more likely that our sun is moving, rather that a large number of distant suns happen to move on parallel tracks.]

[Harder poser--perhaps to take home] How do you think would a star on the ecliptic appear to move? Hint: it's not a circle--not even close!

Why does the solar wind, on the average, appear to come not from the Sun but from a direction 4 degrees off the Sun?
Because of the orbital velocity u of the Earth. In the frame of the Earth the solar wind appears to move as if it has an extra added velocity -u, and that shifts its direction.

You are aboard a steamship traveling at velocity u while the wind blows at velocity v. From your point of view, at what velocity does the smoke seem to travel?
The velocity of the smoke alone relative to the ship is -u, combined with the wind velocity is is v-u, which can also be written v+(-u).

How is the plume of smoke from a steamship similar to the tail of a comet?
The comet releases tail material just as the steamer releases smoke, and like the steamer, it has its own velocity u.

The difference is that the comet releases two kinds of "smoke, " namely dust and plasma. Each of them responds to a different "wind". The dust responds to sunlight, whose velocity c is much greater than u, so in the frame of the comet, that light essentially arrives in its original direction, radially from the Sun. The dust tail is then stretched out radially too (even though its velocity, to be sure, is much smaller than c).

The solar wind also moves radially out, but its velocity v is only some 4-6 times larger than u. As a result, the plasma tail which it affects moves at v-u relative to the comet and makes an appreciable angle with the radial direction.

What do you know about the "Solar Probe" mission?

How would instruments aboard the "solar probe" detect solar wind particles, even though they are shielded from direct sunlight?
The solar probe near its closest approach to the Sun moves almost as fast as the solar wind, but in a direction perpendicular to that of the solar wind (which moves radially). In the probe's own frame of reference, therefore, solar wind ions move along slanting paths that brings them behind the heat shield.

### The Theory of Relativity

What is the principle of relativity?
When one frame of reference moves in a straight line and at a constant velocity relative to another, no physical process can distinguish which one is moving and which one is at rest.

How does the theory of relativity modify Newtonian mechanics?
The mass of any moving material (as seen from some other frame) increases as it approaches the speed of light, and it resists further acceleration more and more. As a result, the speed of light is a limit which no material velocity can cross.

What does relativity say about time in two moving frames of reference--especially if their relative velocity is close to the velocity of light??
Time does not flow at the same rate in the two frames. Two events which in one frame are a second apart, viewed from another frame may be two seconds apart.

In the late 1930s an unstable particle was discovered, named the muon (originally, "mu-meson"). Muons were fragments of collisions of very fast nuclei, and in the laboratory they decayed radioactively (into an electron and an unseen neutrino) in about 2 millionths of a second (microseconds). How far should muons traveling at the speed of light (300,000 km/s) be able to move, on the average, before decaying?

300,000 x 2/1000,000 = 0.6 kilometers

Muons moving close to the speed of light are produced in the atmosphere by collisions of fast atomic nuclei from space ("cosmic rays") at an altitude of about 12 kilometers. Yet a large fraction of them is still observed at sea level (they form the greater part of the cosmic radiation observed there). If they are so short-lived, how come they are not lost by decaying before reaching the ground?

The lifetime of these muons is 2 microseconds in its own frame of reference. Because of their speed, in the frame of the Earth is it much longer, allowing them to last long enough to reach the ground.

(In the frame of the muons, the lifetime remains 2 microseconds, but the distance from the top of the atmosphere to the ground, which is 12 km in the Earth's frame, may be only 0.6 km in the frame of the muon.)

 [Comment: After relativity was introduced, Newtonian mechanics also became known as "classical mechanics" to distinguish it from "relativistic mechanics." Later still different modifications to Newton's mechanics were found to be appropriate for atomic dimensions, and these became known as "quantum mechanics." (And in case you wonder: yes, there also exist "relativistic quantum mechanics")]

Lesson Plan #33         http://www.phy6.org/stargaze/Lflight.htm

# #22c   Airplane flight

In aviation it is usually more convenient to consider the motion of air over a wing or over a propeller blade in their own frames of reference. This lesson examines swept-back airliner wings (also at swept-forward and swiveled wings), and at the loss of propeller efficiency when the airplane gains forward speed.

Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern, Code 695, Goddard Space Flight Center, Greenbelt, MD 20771
u5dps@lepvax.gsfc.nasa.gov or audavstern@erols.com

 This lesson plan supplements:   "Airplane Flight," section #22c   http://www.phy6.org/stargaze/Sflight.htm "From Stargazers to Starships" home page: ....stargaze/Sintro.htm Lesson plan home page and index:             ....stargaze/Lintro.htm

Lesson Plan #34         http://www.phy6.org/stargaze/Lframes2.htm

# #23a   Frames of Reference: The Centrifugal Force

Uniformly moving frames of reference experience no new forces. Uniformly accelerating frames and rotating frames do so. If we want to express the equations of motion in their coordinates, we must always add "inertial forces" to represent the effects of their acceleration. The centrifugal force is one such force, described here and illustrated by examples.

Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern, Code 695, Goddard Space Flight Center, Greenbelt, MD 20771
u5dps@lepvax.gsfc.nasa.gov or audavstern@erols.com

 This lesson plan supplements:   "Accelerating Frames of Reference:   Inertial Forces," section #23   http://www.phy6.org/stargaze/Sframes2.htm   "Frames of Reference: The Centrifugal Force," section #23a   http://www.phy6.org/stargaze/Sframes3.htm "From Stargazers to Starships" home page: ....stargaze/Sintro.htm Lesson plan home page and index:             ....stargaze/Lintro.htm

Note: This lesson uses vectors, and some way of denoting them on the board and in the notebook must be agreed on by the class. In this lesson plan, all vector quantities will be underlined.

Lesson Plan #35         http://www.phy6.org/stargaze/Lrotfram.htm

# #24   Rotating Frames of Reference in Space and on Earth

This lesson continues to explore rotating frames of reference, focusing on the weightless environment in space and on a qualitative discussion of the Coriolis force.

Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern, Code 695, Goddard Space Flight Center, Greenbelt, MD 20771
u5dps@lepvax.gsfc.nasa.gov or audavstern@erols.com

 This lesson plan supplements:   "Rotating Frames of Reference in Space and on Earth," section #24   http://www.phy6.org/stargaze/Srotfram.htm "From Stargazers to Starships" home page: ....stargaze/Sintro.htm Lesson plan home page and index:             ....stargaze/Lintro.htm

Lesson Plan #36                                 http://www.phy6.org/stargaze/Lsun1lit.htm

# (S-1)   Sunlight and the Earth

A discussion of the solar heating of the Earth and atmosphere, its heat loss processes, and the way these relate to weather and climate.

Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern, Code 695, Goddard Space Flight Center, Greenbelt, MD 20771
u5dps@lepvax.gsfc.nasa.gov or audavstern@erols.com

 This lesson plan supplements: (S-1) Sunlight and the Earth http://www.phy6.org/stargaze/Sunlite1.htm "From Stargazers to Starships" home page: ....stargaze/Sintro.htm Lesson plan home page and index:             ....stargaze/Lintro.htm

 Note to the teacher: This first unit in a sequence on the Sun actually deals with solar heating of the Earth, and is also suitable for a course on weather and climate. It acquaints the student with concepts of heat radiation and convection. Originally the lesson plan included many additional aspects of the weather, later moved to a separate section, Weather and Climate. The teacher may include them if more extensive coverage of these topics is needed.

Goals: The student will learn

The student will learn

• How the Sun supplies almost all the energy we use.
• About the balance between incoming solar energy and outgoing heat, radiated by Earth back into space.
• About the "greenhouse effect" and gases that contribute to it.
• That convection of heat in the atmosphere is the cause of weather phenomena.
• That water vapor also carries solar heat and plays an important role in atmospheric convection.

Terms: radiation (visible, infra-red, ultra-violet), radiation balance, greenhouse effect, greenhouse gases, ozone, convection, thermal currents, buoyancy, stratosphere, troposphere, (humidity)

### Starting the lesson:

Today we start our discussions of the Sun, by first looking at the effect Sun has on our environment on Earth. Almost all the energy used by people on Earth comes from the Sun. Can anyone give an example?

• Food grown by crops exposed to sunlight, or obtained from animals which eat such crops.
• Solar cells that power satellites, calculators etc.
• Solar water heaters on rooftops.
• Cars use gasoline or other liquid fuels that originate from fossil plants.
• Electricity may be generated by coal--from fossil plants, too.
• Windmills are powered by winds, whose motion is caused by solar heat as discussed later. (Ocean wave energy also comes from winds.)
• Hydro-electric power comes from water descending from mountains: it was lifted into rain clouds by the Sun's heat.
• In addition, all the water we drink is distilled by sunlight. Were it not for the Sun, all water would be salty.
• Sunlight dries laundry strung on a line.
• In some countries, sunlight is used to produce salt from sea-water (in ponds) and to dry tomatoes, figs, fish etc.

Do you have an example of energy not from the Sun? (may be skipped if time is short)

• Nuclear power, from splitting up the nuclei of heavy atoms, as will be discussed in section S-8.

Nuclei carry a net positive electric charge, because of the positive protons which they contain. A heavy nucleus of uranium or plutonium can be made to split into two parts, each positively charged, and because the fragments strongly repel each other, each gains a lot of speed and energy. As the fragments collide with matter around them, their energy becomes heat, that heat is used to create high-pressure steam, which turns turbines turning electric generators. (Teacher draws diagram on the board: fission in reactor becomes heat in a boiler, which turns turbines to generate electricity.)

• Geothermal heat, e.g. the heat of volcanoes, geysers and hot springs, on land and on the ocean floor. In some places (e.g. Iceland) geothermal steam is used to produce electricity. This energy comes from radioactivity. The Earth contains a small amount of radioactive uranium and thorium (and radioactive substances created by them, "daughter products"), as well as radioactive potassium. All these emit fast particles which heat up the surrounding material. This is a slow process, governed by chance, and the probability of any atom of the above substances undergoing "radioactive decay" in a given year is less than one in a billion.

However, even a weak heating process can create high temperatures, if the heat has difficulty escaping.

You can think of an area of 1 meter2 in the ground on which you stand as the top of a deep wedge running all the way to the center of the Earth. On the average, the heat produced by radioactivity in that wedge must escape through that same area. So much volume, so little surface! That is why the Earth deep down has a temperature of thousands of degrees, able to melt lavas and produce volcanos.

• The tides, raised by the pull of the Sun and the Moon. In principle, tidal energy could be exploited, except that technically it is very difficult to run turbines on small differences of height.

• (Someone may mention starlight: yes, in principle...)

Then go into the lesson. Questions about the material:

How is the temperature of the Sun related to sunlight?

• The Sun radiates because it is hot. All hot objects radiate, though sometimes the radiation is outside the range of the human eye.

What does sunlight tell about the Sun's temperature?

• Judging by its color distribution, the emitting layer of the Sun has an absolute temperature of about 5780° (degrees centigrade from the "absolute zero" of -273.1° centigrade, at which all heat motions cease). The Sun's surface temperature will be again discussed in a later lesson.

What does this say about the surface of the Sun?

• The material on the surface of the Sun cannot be solid or liquid, because at 57800 all known substances are vaporized. In fact, some will be ionized, with at least one electrons ripped off and floating independently nearby.

The top layer of the Sun, from which sunlight comes, is known as the Sun's photosphere. All words that start with "photo" are related to light. Know any?

• Photography and words derived from it--photoelectric flash, photogenic, photostat (picture) etc.
Also photosynthesis, photo-electric cell (or "photo-electric eye"; also "phototube"), photon ("particle" of light--see later lesson; also "photon torpedo" in "Star Trek"), phot (unit of light, favorite in crossword puzzles), photolysis (chemical dissociation caused by light) etc.

If the Sun constantly heats the Earth, how come the Earth does not heat up?

• The Earth also radiates back into space, emitting infra-red light (IR) which our eyes cannot see. On the average, the energy the Earth receives is balanced by the energy it loses.

How can the Earth radiate back as much as it receives from the Sun, if its absolute temperature is only around 3000, while that of the Sun is 57800?

• Take an area A on the surface of the Earth. It receives "hot" sunlight only from a small patch of the sky, about 0.50 across, but it radiates back infra-red (IR) to half the sky. Clearly, if incoming and outgoing energy flows are to be equal, it does not have to shine anywhere as brightly as the Sun!

(optional discussion)

You probably know that with a magnifying glass you can use sunlight to create a lot of heat--enough even to start a fire. Why?

•   (Someone may say "because it concentrates the Sun's light.")

Correct. Is there a limit to the temperature you can get?

•   Someone may say "no" because all the light seems to converge to a point.

Actually, you cannot get a greater temperature than that of the surface of the Sun. Here is why.

Suppose you use a magnifying glass to set a piece of paper on fire. Viewed from the point on which the sunlight is focused, the Sun is magnified--no longer 0.50 across but maybe 30, 50 or 100. Its heating power is then magnified by the square of the ratio between the angles--36, 100 or 400 times, because the heat the paper receives goes like the area of the bright patch which shines on it.

There exist solar furnaces with arrays of mirrors that can magnify the Sun even more, so that viewed from the focus, it covers (for instance) 1/10 of the sky. The heat generated may be intense enough to melt iron. However, as the object at the focus heats up, it also radiates away more heat!

Suppose that by some clever arrangement of lenses and mirrors we have managed to illuminate the object at the focus from all directions. Whichever way it "looks" it sees the Sun. At equilibrium, the heat it gets equals the heat it radiates back. How hot does it get?

Suppose it is hotter than the Sun. If brightness depends only on temperature (very nearly true), then it is also brighter than the Sun, and in any direction it radiates back to the Sun more than it receives. That is obviously impossible, so we conclude that the sample can never exceed the temperature of the photosphere which provides its light in the first place.

(end of optional section)

So far we have assumed the Earth receives visible light radiated by the Sun and radiates it back into space as infra-red light, less concentrated but in all directions.

Actually this assumption is more appropriate for the Moon or for the planet Mercury. It is not quite accurate for Earth. Why?

• (someone may say "because the Earth has an atmosphere")

Correct. But why does the atmosphere make a difference?

• Two reasons:
1. Clouds reflect heat before it reaches the ground
2. Air absorbs infra-red light

Correct. One can understand the effect of clouds.
But with the atmosphere--isn't any energy absorbed by the atmosphere radiated out again--or else, the atmosphere would get hotter and hotter?     Yes, it is. Anything absorbed by the atmosphere is radiated away again. But some of it is radiated downwards and returns to Earth! Therefore the existence of the atmosphere impedes the outflow of heat.

Why is this process called "The Greenhouse Effect? "

• Because the same process keeps glass-covered greenhouses warm. The Sun heats the ground and greenery inside the greenhouse, but the glass absorbs the re-radiated infra-red and returns some of it to the inside.

What substances in the atmosphere contribute to the "greenhouse effect"?

• The ones that absorb IR light. Water vapor (H20) is important, so is carbon dioxide (CO2) produced by burning fuel, still another is methane (CH4) emitted by decaying vegetation and also by the digestion process in cows and related animals.

The concentration of CO2 in the atmosphere has increased significantly in the last century, because of the burning of coal and other fossil fuels. This might be responsible for the warming trend in the world's climate observed in recent decades.

What is ozone?

• Ozone is a form of oxygen, forming the molecule O3 instead of the usual O2. It forms in two places: high in the atmosphere, between 18 and 50 km (peak at 25) where it is beneficial in absorbing ultra-violet light, and near the ground, part of urban air polution. Ozone is very reactive chemically, so near the ground it erodes paint and is unhealthy to breathe.

Is ozone a greenhouse gas?

• Yes, it is one, too. It is true it also prevents solar UV from reaching the ground, and that reduces the heating of the ground: but it is a very slight effect, because UV does not carry much energy. Of greater concern are the harmful effects of increased UV on the skin and eyes.

However, ozone also absorbs infra-red light. As far as the heat balance is concerned, this effect is more important. Overall, therefore, the presence of ozone helps keep the Earth warm.

What other processes affect the heating and cooling of the Earth, besides absorption and emission of light and of radiations like IR and UV?

Someone may say "heat generated by human activities":

• Big cities are indeed slightly warmer, but overall the heat released by human activities is too small to have any great effect.

Someone may say "weather":

• Well--yes. But what are the two things that characterize weather besides the temperature of the air?
1. Wind
2. Rain
So let's talk about wind, and then about rain--or to use more technical terms
1. The flow of air
2. The evaporation and precipitation of water

Why does warm air rise?

• Warm air is less dense than cold air, and therefore floats up (the way oil that is less dense floats on top of water).

How does a hot-air balloon work?

• A burner heats the air that fills the balloon: it is lighter than the air around it and rises.

What does rising air have to do with the warming of the Earth by the Sun?

• It is one way the ground removes heat: it warms up the air near it, which rises. Later, higher up in the atmosphere, the air radiates its heat out to space, cools down and gets denser, then sinks down again, replaced by warm air that is still rising. This is known as the convection of heat.

How is heat convected near a cold window? How does air flow there?

• It flows downwards. Air touching the window cools, gets heavier and drops to the floor. It then spreads into the house where (if the house is heated) it is warmed up again. Meanwhile other air is replacing it near the window and is getting cooled in its turn.

(Note: Fiberglass is a good heat insulator, because it prevents the air inside it from flowing. As long as the air does not move, it does not carry away much heat. Wool blankets and sweaters work the same way.)

Convection in the atmosphere stops around 10-15 kilometers. What is the layer above that called, and what do you know about it?

• It is called the stratosphere and is pretty stable and cold (though its higher layers get heated by ultra-violet, absorbed by ozone.) The region below it--where weather is found--is the troposphere, and the boundary between the two is the tropopause.

(If you have seen an isolated thunderstorm from a distance, you might have noted that its top is flattened and spread out. It is flattened against the bottom of the stratosphere, which blocks the convection of the storm from rising any further.)

What role does water play in moving the Sun's heat outwards?

• Part of the energy of sunlight goes to heating the ground, but another part evaporates water, from the oceans, lakes, rivers and plants.
As warm humid air rises, it carries energy in two forms--some as heat, some as humidity.

What causes rain?

• As warm air rises, it expands and cools. Cold air cannot hold as much water and some is forced out, creating clouds and rain.

On a summer morning you get up and find the grass outside covered with dew--with drops of water. How did they get wet?

• The ground cooled down during the night, and cooled the air. The cooled-down air had to give up some water, and deposited it on the grass.

When humid air gives up water as rain, does it speed up its cooling or slow it down?

• Slow it down. When sunlight evaporated that water, it invested energy in the process. But energy in nature is always conserved! When the water is forced out, that energy is returned to the air, adding to its heat.

Lesson Plan #37               http://www.phy6.org/stargaze/Lsun2vue.htm

# (S-2) Our View of the Sun

An introduction to solar observations--by eye or telescope (with cautions) and during eclipse. Also, the distance of the Sun, its layers, the corona and the solar wind.

Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern, Code 695, Goddard Space Flight Center, Greenbelt, MD 20771
u5dps@lepvax.gsfc.nasa.gov or audavstern@erols.com

 This lesson plan supplements: (S-2) Our View of the Sun http://www.phy6.org/stargaze/Sun2view.htm "From Stargazers to Starships" home page: ....stargaze/Sintro.htm Lesson plan home page and index:             ....stargaze/Lintro.htm

 Goals: The student will learn Safety rules for observing the Sun How the distance of the Sun, about 150,000,000 km, must be measured indirectly, using Kepler's laws. About the visible layers of the Sun--photosphere, chromosphere, corona. About the high temperature of the corona, the evidence for it and what makes it so puzzling. About the solar wind and its connection to the Sun's corona. Terms: astronomical unit, plasma, photosphere, chromosphere, corona, solar wind.

### Start the lesson by discussing a solar eclipse:

 Who here has seen an eclipse of the Sun? Total? Partial? Actual observations, or over TV?     How does one observe an eclipse? What should be avoided? What may one do? --As long as any part of the Sun remains uncovered, do not look directly. Instead, project the Sun's image onto a flat surface from a pinhole or telescope, or view the Sun through a completely blackened B&W film. The Sun should appear comfortably dim. Looking directly can be harmful--and you will be too dazzled to see any details. Looking through binoculars or through a telescope is very dangerous--as if you focused a magnifying glass onto your eye! On the other hand, once the Sun is completely covered (in a total eclipse only), it is safe to look and to photograph.) What can one see during a total eclipse that is not visible otherwise? The chromosphere, a reddish ring around the Sun. --The corona with streamers extending above active sunspots, "Plumes" above the poles and arches above sunspot groups, both these suggesting magnetic fields.   Students who observe a total eclipse should be cautioned, however, not to expect to see all the fine details one finds in eclipse photographs. Pictures that appear in journals often use time exposures and superpositions to make the outer fringes of the corona appear brighter than they actually are. For instance, a picture shown on the web of the total eclipse of 11 August 1999 was actually obtained by blending more than 20 pictures.

### Additional points the teacher may raise:

 Total solar eclipses were once used to look for an additional planet orbiting very close to the Sun. Because of the glare of the Sun, such a planet would be invisible at any other time. Astronomers had even proposed a name for is--Vulcan. No such planet was ever seen, and astronomers today agree that none exists. In the early 1800s, the German amateur Heinrich Schwabe, using a telescope (with a projected or filtered image) watched the Sun day after day for years, trying to see Vulcan passing in front of the disk of the Sun, when it should be visible as a dark spot (click here for a picture of the planet Mercury in front of the Sun). To distinguish such spots from sunspots, Schwabe carefully noted down all the sunspots he saw. He never found a planet, but after about 17 years, in 1843, he discovered the 11-year cyclical rise and fall in the number of sunspots, which had eluded professional astronomers.
 A recent discovery (14 Nov'99): As discussed in connection with Kepler's first law, planets orbit not around their central star but around the common center of gravity of their planetary system (see here). The central star also orbits that point, and this causes its position in the sky to wobble slightly. Astronomers have used such wobbles--or more accurately, the changes of speed associated with them, which slightly shift spectral colors--to detect the existence of planets around more than 20 stars.     Most of those planets are as big as Jupiter or bigger, because obviously the heaviest planets shift the center of gravity by the greatest amount (the effect of an Earth-size planet is still too small to be measured). Many of them were detected by Paul Butler (Carnegie Inst.), Geoffrey Marcy (U. Cal Berkeley) and Steve Vogt (U. Cal Santa Cruz), who worked out a sensitive method for such observations.     The observations do not tell much about how the orbit is oriented in space, but for one recently discovered planet we now know a bit more. It orbits a star catalogued as HD 209458, some 153 light years from Earth, and happens to pass right between us and it. The researchers were hoping to find such a planet, and sure enough, Greg Henry of Tennesse State Univ, using an Arizona telescope, detected on November 7, 1999 a temporary drop in the light intensity by 1.7%. It is a big planet with about 2/3 the mass of Jupiter, and its orbital period is about 3.5 days. Because it orbits very close, and its size is expanded by the heat of its nearby star, it is considerably larger than Jupiter, able to block a measurable amount of starlight.
 A total solar eclipse was also used in 1919 by the British astronomer Arthur Eddington to check out the general theory of relativity of Albert Einstein. Einstein predicted that the Sun's gravity would bend starlight passing close to the Sun. If that happened, stars whose position in the sky during a total eclipse was close to that of the Sun should have been very slightly shifted from their usual positions. The effect was very small, but Eddington confirmed its existence. (This is related to the preceding point; illustrate by a sketch on the board) Today we know a handful of cases in which a very distant galaxy is obscured by a nearer one. You might think that the distant galaxy would be invisible, hiding (so to speak) behind the nearer one. However, the gravity of the nearer galaxy can bend light from the distant one, light which otherwise would have missed Earth, so that it does reach us. If the positioning is right, we see multiple images of the obscured galaxy. That phenomenon is known as gravitational lensing. (end of additional material) Chromosphere, corona--does the Sun have any other visible layers? -- The photosphere, the layer from which sunlight reaches us. It is below the chromosphere.

### Teacher's explanation of the heat outflow of the Sun:

The heat of the Sun is generated deep inside, in the Sun's core, by processes which require very high temperatures and great pressure (these will be studied in a later lesson).

That heat moves outwards from the core, towards the Sun's surface, by processes somewhat like the ones by which heat works its way through the atmosphere, from the surface of the Earth to space (section S-1).

No one of course can observe what happens deep inside the Sun, but a theory has been developed about the interior of stars, suggesting heat in the deepest layers travels by radiation. Atoms radiate light and neighboring atoms absorb it, but since the deep layers are compressed with atoms packed close to each other, the process is more like the conduction of heat. As heat moves outwards, the temperature keeps dropping, because net flow of heat can only take place from hot material to colder one.

Closer to the Sun's surface, the theory predicts, heat is carried by convection, as in the Earth's atmosphere--by gas flowing around closed paths. All these layers are still fairly dense, and any light emitted is quickly absorbed again. The photosphere is the final layer. Not enough material remains above it, allowing any light emitted there to spread out into space. It is a relatively thin layer, of the order of 100-200 kilometers

Because most sunlight comes from the photosphere, the much fainter chromosphere and corona are only seen during a total eclipse, when the light of the photosphere is completely blocked by the disk of the Moon.

However, astronomers can also view the Sun through special filters which reject all of the Sun's light except one or another narrow range of color ("spectral line," discussed in a later lesson). In some such colors, the chromosphere or the inner corona shine brightly enough to be seen even without a total eclipse.

### A calculation on the board

A proton's mass is 1.67 10-27 kg. Assuming the solar wind consists entirely of protons, how much mass does the Sun lose each second? Assume the mean Sun-Earth distance ("astronomical unit") is 150,000,000 km.

(This calculation tests ability to use scientific notation when working with very large and very small numbers. The teacher might call up a student to derive it on the board, with the class copying).

1 AU = 150,000,000 km = 1.5 108 km = 1.5 1011 meter = 1.5 1013 cm

The area of a sphere with radius r = 1 AU around the Sun is

4pr2 = (12.56) x (2.25 1026) cm2 = 2.826 1027 cm2

Imagine behind every square centimeter a column of solar wind 400 kilometers long, waiting to cross it during the next second! (Teacher could illustrate this with a sketch on the board).

The entire surface of the sphere is therefore crossed each second by the solar wind particles contained in a volume

(2.826 1027 cm2) x (4 107 cm) = 1.13 1035cm3

And with a density of 6 ions/cc, the number of ions is

(1.13 1035) x 6 = 6.78 1035

The total mass lost by the Sun each second is

(6.78 1035) x (1.67 10-27 kg) = 1.13 109 kg = 1.13 106 ton

The Sun therefore loses about a million tons each second. It sounds like a lot but really isn't--one cubic kilometer of the ocean contains about 1000 times more, and the Sun is not much diminished even if this loss continues for many billions of years.

(end of calculation)

What ultimately happens to the solar wind?

• It gets lost in interstellar space. Actually, a boundary is expected, the "heliopause" marking the outer limit of the solar wind domination, from where on the interstellar gas determines the ambient conditions. The space probe "Voyager 2" is going to reach it, but whether it will still be transmitting remains to be seen.

# (S-3) The Magnetic Sun

An overview of phenomena related to the magnetism of the Sun, in particular to sunspots and their 11-year cycle, solar flares and magnetic disturbances at Earth caused by "solar activity." Also reviews briefly the connection between electricity and magnetism

Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern, Code 695, Goddard Space Flight Center, Greenbelt, MD 20771
u5dps@lepvax.gsfc.nasa.gov or audavstern@erols.com

 This lesson plan supplements: "The Magnetic Sun," section #S-3           http://www.phy6.org/stargaze/Sun5wave.htm "From Stargazers to Starships" home page and index:          http://www.phy6.org/stargaze/Sintro.htm

Goals: The student will learn here

• That magnetic forces in nature rarely involve iron, but are actually forces between electric currents.

• About sunspots and their intense magnetism, with a strength of about 0.15 Tesla (1500 gauss).

• About the 11-year sunspot cycle and its discovery.

• About "solar activity" associated with sunspots and their cycles, e.g. the abrupt brightenings known as solar flares.

• That solar activity is probably associated with the release of magnetic energy, and that such releases can propel fast plasma flows towards Earth, causing there "magnetic storms."

Terms: Sunspot, magnetic field lines, magnetic fields, sunspot cycle, solar activity, solar flare, magnetic storm, magnetic energy

Stories: The discovery of electromagnetism by Oersted and Ampere, the discovery of the sunspot cycle by Heinrich Schwabe and the discovery of solar flares by Richard Carrington.

The teacher is advised to read those stories ahead the class, to be better able to present them to students, including the original articles, here and here. (Students can also be assigned to do so and to make the presentations.)

### Starting the lesson:

(As described here, the teacher would begin the lesson around the above stories of discovery)

No area of science draws as much interest as stories of discovery, and of the unusual people who made them. This class has already covered some interesting discoveries. Which of them were associated with the names of... Aristarchus? Erathostenes? Columbus (even if it wasn't a scientific discovery)? Copernicus? Kepler? Newton?

Louis Pasteur was a French biochemist in the 19th century, whose many discoveries included a way of preserving food by heating ("pasteurization") and a procedure for saving the lives of people bitten by animals infected with rabies, which up till then meant almost certain death. Commenting on scientific discoveries, Pasteur said "chance favors the prepared mind". Discoveries often depend on luck--but luck is not enough, the mind must be prepared to exploit its opportunity.

[a student might prepare a poster with that quote, to hang in class]

Today as we discuss magnetism and magnetic phenomena on the Sun, we will discuss three discoveries in which luck had a part--but luck wasn't the only reason. (List on the board--students copy.)

1. The discovery of electromagnetism in 1820 by Hans Christian Oersted, in Denmark, and its explanation by Andre-Marie Ampere, in France.

2. The discovery of the sunspot cycle (1843, widely accepted around 1851) by Heinrich Schwabe in Germany.

3. The discovery of solar flares (1859) by Richard Carrington in England

(possible comment: Each discovery occured in a different country!. Science is truly international.)

The discovery of the connection between electricity and magnetism

What do we know about magnetism of iron magnets?

• Like poles repel, unlike poles attract.
• The magnetic compass tends to point north-south.

[Why is plain iron attracted to a magnet? Because when iron is in the region of influence of a magnet--its "magnetic field"--it becomes temporarily magnetic itself, with the pole closest to a pole of the magnet having opposite polarity, causing it to be attracted.]

What magnetic phenomena do not involve iron, and why are they called "electromagnetic" phenomena?

• The attraction between parallel currents in the same direction, and the repulsion between them if the currents flow in opposite directions.
The forces between the coils, etc. used in electric machinery all follow from this basic property.

(optional)

The magnetic field which satellites observe in space is often different from what one would expect, based on the fields we observe on the ground.

The reason is that large electric currents often flow through the space surrounding the Earth, and they contribute their own magnetic fields as well. The currents can flow there, because of the presence of free electrons and ions (a "plasma").

What to you think--would such currents tend to spread out to cover as much space as possible, or would they narrow down to string-like filaments? You must give a reason.

[If no one answers:"It has to do with the forces between electric currents."]

• They might tend to narrow down. A current flowing along a wide tube may be viewed as composed of many narrower parallel currents, and we know that parallel currents attract each other.

(This tendency of currents to narrow down has also been observed in laboratory plasmas, where it is named the "pinch effect.")
(end of optional part)

Suppose you had a compass needle able to point in any direction in space, not just horizontally. How would the northward-pointing end of the needle point at

1. The northern magnetic pole
2. The southern magnetic pole
3. Halfway between
4. Elsewhere?

[Note: such needles are available--see end of section S-3, or click here: http://www.cochranes.co.uk/BNRVP30/edu5.htm]

• Straight down at the north magnetic pole, straight up at the southern pole, horizontally northward at the magnetic equator, northward slanting down in the northern hemisphere, northward slanting up in the southern hemisphere.
(By the way: the early Chinese who discovered the magnetic compass claimed it pointed south.)

How would such a needle point near a straight wire carrying an electric current? (Neglect the Earth's magnetism)

• Perpendicular to the current.

How was the connection between electric currents and magnetic force discovered? (Students tell about it, or else, the teacher.)

What are magnetic field lines? Base your definition on the magnetic needle described above.

• They are imaginary lines giving at each point the direction in which our 3-D needle would point, if placed there.

What are magnetic field lines used for?

• They were originally used to graphically describe magnetic fields. In the rarefied plasmas of space, however, they also guide the flow of particles and currents. This is why arching solar formations above magnetic sunspots sometimes resemble the field lines of bar magnets.

(The guiding property of field lines also makes possible the trapping around Earth of ions and electrons in the Earth's radiation belts. The motion of these particles stops and reverses before they hit the Earth, because they are also reflected from regions of stronger magnetic field, found closer to Earth.)

What is Andre-Marie Ampere remembered for?

• He explained the observations which baffled Oersted, and in doing so gave the first clear idea of what magnetism was. In his honor the unit of magnetic current is called the "Ampere."

[P.S.: He is also remembered as the man who was invited to dinner with Napoleon and forgot to go!]

The intensity of the Earth's magnetic field at the magnetic equator is about 31,000 nT (nanotesla) or 0.31 gauss. The field intensity goes down with distance r like 1/r3. If the intensity of the interplanetary magnetic field at the Earth's orbit is 5 nT (a typical value), at what distance--in Earth radii--is this matched by the Earth's field? (Needs calculator capable of extracting cube roots.)

• If at a distance of 1 Earth radius the field is 31,000 nT, at distance of r earth radii is is 31,000/r3 . If r is the distance where it drops to 5 nT, we get 31,000/r3 = 5 . Multiply both sides by r3, divide by 5 to get

r3 = 31000/5 = 6200 r = 18.4 Earth radii.

Then go on to the discovery of sunspots.

When and how were sunspots discovered?

• In 1609, when telescopes were first used in astronomy. Galileo, Fabricius and Scheiner all claim credit and might be independent discoverers.

What did the discoverers see?

• They saw dark spots on the Sun.

How do we know that the Sun rotates around its axis?

• Sunspots were observed to travel across the face of the Sun in a way that suggests they rotated with it.

What is unusual about the Sun's rotation?

• The rotation period depends on latitude: the equator rotates fastest, in about 27 days. Nearer to the pole it can be 2.5 days longer.

Optional: The teacher may draw a table with the observed latitude dependence of the rotation period (in days) and let students graph it:

 SolarLatitude ActualPeriod Period Viewedfrom Earth 0 25.03 26.87 10 25.19 27.06 20 25.65 27.59 30 26.39 28.45 40 27.37 29.65
 Question: Why is the period viewed from Earth longer by about 2 days? Say we track a sunspot at latitude 20°, which at a certain time faces Earth. After 25.65 days the spot again faces the same direction in space as before--in the direction of the same stars, for instance. However the Earth orbits the Sun, and has by then moved ahead in its orbit. It takes the Sun's rotation 2 more days to reach the position where the sunspot faces Earth again. (end of optional part) Here tell the story of Schwabe's discovery of the sunspot cycle. Why do we think that magnetism plays an important part in solar phenomena? Some clues: From the light of sunspots, it was found that they were intensely magnetic. During a total eclipse, the "plumes" of the corona above the poles and the "arches" above sunspot regions suggest the form of magnetic field lines near magnets. Eruptions on the Sun are linked to "magnetic storms" on Earth. What suggests that the 11-year sunspot cycle is a magnetic phenomenon? Sunspots are intensely magnetic In each sunspot cycle, the magnetic polarity order of the leading and trailing sunspots in sunspot pairs reverses. Each sunspot cycle the large-scale polar magnetic field of the Sun reverses. [Optional: Does the Earth's own magnetic field ever reverse? Yes, it apparently does, but at very infrequent intervals. That magnetic field is caused by electric currents in the Earth's molten core, and is observed to change slowly from year to year. On rare occasions, however, the dominant pattern of currents seems to reverse direction. How do we know? We know because the floor of the oceans contains fissures, caused when the sea-bottom is pulled away in opposite directions, away from them. The pulling-apart is caused by forces deep inside the Earth, and a typical speed of those "plates" is about one inch a year. The most famous fissure is the one running down the entire middle of the Atlantic ocean, in a roughly north-south direction but with bends and kinks. As the seafloor plates are pulled away to both sides, lava comes up from the fissure and solidifies into a black rock called basalt, which become part of those plates. However, basalt is weakly magnetic, and when it solidifes it becomes magnetized in the direction of the magnetic field existing at the time. Thus each part of the seafloor records the Earth's magnetism at the time it emerged from the fissure, just as the tape of a tape recorder records the magnetism of the recording head at the time it passes near it. In this way, the seafloor has recorded the magnetism of the Earth during the last 10-20 million years. It turns out that the seafloor is not magnetized in a uniform direction! Rather, it is magnetized in long parallel stripes, parallel to the central fissure, and each two neighboring stripes have opposite magnetic polarity. This suggests that between the times those two stripes emerged, the Earth's magnetic field reversed direction. The last time that has happened (assuming a rate of about an inch a year) was about 700,000 years ago.](End of optional section)

# (S-4) The Many Colors of Sunlight

An introduction to color and to both line spectra and continuous spectra, with applications to sunlight.

Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern, Code 695, Goddard Space Flight Center, Greenbelt, MD 20771
u5dps@lepvax.gsfc.nasa.gov or audavstern@erols.com

 This lesson plan supplements: "The Many Colors of Sunlight," section #S-4           http://www.phy6.org/stargaze/Sun4spec.htm "From Stargazers to Starships" home page and index:          http://www.phy6.org/stargaze/Sintro.htm

 (optional) If time allows, students can be shown how a diffraction grating works. The grating acts like many closely spaced slits: the light hitting between the slits is scattered irregularly, and only what hits the slits goes through. If light acts as a wave, one can show mathematically that each slit acts as a new source of the wave. (By the way--the "slits" are really the ridges between the grooves of the grating, which transmit light like a windowpane, not the grooves themselves) Suppose light arrives at the grating from a direction perpendicular to it. Waves have peaks and valleys, and when the arriving wave-front is at a peak, all slits also start their "local waves" with a peak. Suppose we continue in the same direction 1, 2, 3... wavelengths. The wave from each slit will then also have a peak at those locations. These peaks would be in the same distances if the wave passed intact through the grating, as if it wasn't there, suggesting that much of the wave goes straight through, with no modification. But wait! If each slit is the source of a wave spreading in all directions, what about the part spreading in slanted directions? Consider the part spreading at an angle q to the original direction of light. The wave front of a wave moving in that direction would be along the drawn slanted line. However, the distances of the points on that line from the various slits--each a separate "light source"--are all different! If at the slits the wave has a peak, at the wavefront, different parts of it are in different parts of the cycle. They are at a peak if the distance to a slit is an exact multiple Nl of the wavelength (N is some whole number), at a peak in the opposite direction if the distance is (N+0.5) l, and in other parts of the cycle for other distances. The sum total--say 2000 slits, 2000 different distances--is close to zero, so we get very little light scattered in the direction of q. (That cancellation of peaks is called "destructive interference between the waves.") Except... if the angle q is such that the distances of the wave-front from two neighboring slits differ by exactly one wavelength l. In that case, the distances from the next slits in line are 2l, 3l... and so forth, and the waves continue propagating "in step." That is "constructive interference" between the waves, and in those directions, you will see a fairly bright beam of light.
 As the second drawing shows, if D is the spacing between two neighboring slits, this requires D sinq = l Note that the angle q at which the light undergoes constructive interference depends on the wavelength l, that is, on the spectral color of the light. Each color is therefore bent by a different angle--just as it is in a prism. Because most of the beam goes straight through, the light may not be as bright as in a prism, but the separation of neighboring colors may be muchmore sensitive. The important thing to note is that such behavior is only expected from a wave. This therefore suggests that light is a wave, even though (at this stage of the discussion--like scientists for most of the 19th century) nothing tells us what exactly forms its peaks and valleys. In fact, the above formula allows the light's wavelength to be calculated. For example: you observe the yellow line of sodium with a grating having 1000 lines per centimeter, and find that light is brightest at an angle q=36°. What is the wavelength l? In fact, methods based on interference have measured wavelengths with such accuracy, that the international meter--originally defined by two scratches on a bar, kept in a vault in Paris--was at one time redefined in terms of the wavelength os a certain emission. Another interference effect are the colors seen when a thin layer of kerosene floats on a puddle of water--layers with a thickness of the order of a wavelength of light. Some light is reflected from the top of the kerosene layer, some from its bottom (which is the top of the water), and the two reflected waves interfere with each other. For some colors the interference is destructive, for others, constructive, leading to a shimmering of colors. (end of optional section) What does the spectrum of sunlight tell about the Sun? The continuous spectrum? ... It tells that the temperature of the photosphere is about 5780° Kelvin. The bright lines in the spectrum?... They tell us about the composition of the photosphere--mostly hydrogen, some helium, a bit of oxygen, carbon and heavier stuff (some lines, carefully recorded and analyzed, can also tell of the presence and strength of the local magnetic field). The dark lines of the spectrum?.... They identify cooler material in the higher levels of the photosphere. The spectrum of the corona, for instance, the presence of iron that has lost 13 electrons? ... It tells us the corona is very hot, and provides an estimate of its temperature. How was helium discovered? (The teacher can tell more about the discovery. The helium line was in the yellow part of the spectrum, and at first some astronomers credited it to sodium--but it had a different wavelength, and they gradually recognized that in no way could sodium produce it.)

# (S-5) Waves and Photons

This lesson introduces students to electromagnetic waves, at a qualitative high-school level. It then brings up the concept of photons, and the relation between photon wavelength and energy. This is tied to solar observations at various wavelengths.

Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern, Code 695, Goddard Space Flight Center, Greenbelt, MD 20771
u5dps@lepvax.gsfc.nasa.gov or audavstern@erols.com

 This lesson plan supplements: "Waves and Photons," section #S-5           http://www.phy6.org/stargaze/Sun5wave.htm "From Stargazers to Starships" home page and index:          http://www.phy6.org/stargaze/Sintro.htm

 Goals: The student will learn About the many types of electromagnetic waves, and about observing the Sun using different members of that family. Qualitatively, about the concept of an electromagnetic (EM) wave, as a linked oscillation of magnetic fields and electric currents, spreading through space. How James Clerk Maxwell proposed a slight modification of the equations of electricity, under which electromagnetic (EM) waves could exist, and how he identified light as such a wave, after which Heinrich Hertz created radio-frequency EM waves in his lab. That although light spreads like a wave, it only gives up its energy in well-defined amounts, known as photons. That the shorter the wavelength, the bigger the photon energy. Thus hot regions of the Sun, whose atoms move faster and therefore have more energy, are likely to emit shorter wavelengths. That light is also emitted in photons. When an individual atom emits light, it usually changes from some "excited state" of higher energy to one with lower energy. The energy (and hence, color) of the emitted photon is very precisely determined by the difference between those levels. Terms: wave, electromagnetic wave, wavelength, wave velocity, frequency, photon, Planck's constant, (atomic) energy level, excited atom, solar prominences. Stories: The discovery of electromagnetic waves. This lesson plan also includes (optional) the story of whistlers and a brief comment on laser action.

### The lesson may be started with a discussion of waves.

Lesson Plan #41                             http://www.phy6.org/stargaze/Lsun6new.htm

# (S-6)   Seeing the Sun in a New Light

A short section on features of the Sun's corona, observed from spacecraft in the extreme ultra violet (EUV) and in x-rays, including coronal holes and coronal mass ejections (CME). This section also discusses related phenomena in interplanetary space and on Earth.

## (S-6A)   Interplanetary Magnetic Field Lines

An optional class excercise in which students learn about "field line preservation" of flows in a highly conducting plasma, and use it to graphically obtain the shapes of interplanetary magnetic field lines.

Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern, Code 695, Goddard Space Flight Center, Greenbelt, MD 20771
u5dps@lepvax.gsfc.nasa.gov or audavstern@erols.com

 This lesson plan supplements: (S-6) Seeing the Sun in a New Light                http://www.phy6.org/stargaze/Sun6new.htm               (S-6A) Interplanetary Magnetic Field Lines                 http://www.phy6.org/stargaze/Simfproj.htm "From Stargazers to Starships" home page: ....stargaze/Sintro.htm Lesson plan home page and index:             ....stargaze/Lintro.htm

Goals

The student will learn

• About the observation of "coronal holes," by x-rays, also about related fast streams and moderate magnetic storms that recur at 27 day intervals

• About "coronal mass ejections" (CMEs), their effect near Earth and their monitoring from space.

• About high-energy ions and electrons accelerated by solar activity, probably from magnetic energy, and the hazard they pose to spacefarers.

• About NASA's "great observatories," expanding the coverage of the electromagnetic spectrum by astronomers.

Terms: Coronal holes, coronal mass ejections, magnetic storms, solar wind streams, interplanetary magnetic field

Stories, additions and features: A graphic excercise in which the expected shape of interplanetary magnetic field lines is obtained;   The "Chandra" X-ray observatory

### Starting the lesson

(This approach starts with a general discussion of X-rays before bringing up the Sun. Some student might look up beforehand information about the "Chandra" orbiting telescope and describe it to the rest of the class at the appropriate time; see here and here.
If time is short, the teacher can cut this part short and start right away with the solar corona)

Today we will talk largely about X-rays and the Sun. In the "Superman" comic books, Superman not only flies through the air and leaps over tall buildings, but also has the power of "X-ray vision" that allows him to see through walls.

X-rays can indeed penetrate walls, but even with X-ray vision, Superman would never see what is behind them--for at least two good reasons. Any guess?

1. X-rays are not too well reflected from objects the way light is--they usually go through until they are absorbed.
2. You only see objects around you when they are illuminated by sunlight or by lamps. In normal surroundings there are no X-rays. Unless he used an X-ray flashlight, Superman's X-ray vision would work no better than your own vision in total darkness.

Suppose you wanted to build an X-ray telescope. How can you focus X-rays? It turns out that they can be reflected if they hit at a very shallow angle--just as a flat stone will bounce off water if it hits at a shallow angle, but will surely go through and sink if dropped vertically.

The "Chandra" orbiting X-ray telescope, launched from the Space Shuttle on July 23, 1999, focuses X-rays in this way. Imagine you cut off the tread of a tire, to produce a ring with a curved cross section (drawing). The "Chandra" telescope has metal focusing surfaces shaped like the inside of this ring, and by shallow reflections they bring X-rays to a focus.

What part of the Sun do X-rays and related wavelengths see best?

•   The Solar Corona
Why?
•   Because the corona is very hot, about a million degrees centigrade. In a hot gas ions and electrons bounce around with greater speed, and when they collide, more energy is released. The photons they produce are also likely to have more energy--that is, rather than visible light, they may be in the region of X-rays or EUV.
What are coronal holes?
•   Dark areas of the corona, seen in EUV or X-rays. The first extensive observation of such "holes" was done from space station "Skylab" in 1973.

What is the connection between coronal holes and the solar wind?

•   The solar wind coming from coronal holes seems to move faster.

Any reason suggested?

•   Coronal holes probably occur in regions where the magnetic field does not greatly impede the flow of plasma.

(teacher may supplement)

Strong magnetic field lines can guide the motion of plasma--it moves easily along them, not so easily from one field line to its neighbor.

Weak magnetic fields, on the other hand, get instead pushed around by the plasma, which modifies their structure. (We'll come to that later.)

Sunspots are sources of strong magnetism. Usually, they come in pairs of opposite polarity--one is like the northward pointing end of a magnet, the other like the southward pointing end--and magnetic field lines seem to connect them, forming arches above the solar surface. Ions and electrons guided along such arches by their strong magnetic fields find it hard to escape.

In between sunspot groups, field lines stick out like blades of grass after a rain, and plasma can ride along them outwards.

So--where do you think sunspots are found--in coronal holes or outside them?

•   Outside them.

Over the poles, eclipse photographs show "plumes" sticking out (sketch on the board). Does this suggest "coronal holes" avoid the poles, or not?

•   They don't avoid the poles, on the contrary--each polar region seems to be a near-permanent "coronal hole." The space probe "Ulysses" passing over them observed fast solar wind streaming out of them, similar to the way it streams from coronal holes.

Spacecraft in the solar wind observe there a weak magnetic field. What is the origin of that field?

•   The field comes out from the Sun, and is dragged out by the solar wind. (Here, if time allows, the class may learn more about the interplanetary magnetic field and draw its configuration. That project is described here.)

What are "Coronal Mass Ejections" (CMEs)?

•   CMEs are big bubbles of plasma, threaded by field lines of the Sun, which get blown away by releases of magnetic energy.

Why are space researchers interested in CMEs?

1. The can cause magnetic storms if they reach Earth.
2. They are big and energetic, but we are still not sure how they are propelled.
3. Some move much faster than the solar wind and pile up shock fronts at their boundaries. Such shocks may generate high energy particles and might play an important role in magnetic storms

Note: an interesting shock-produced event occured on 24 March 1991. A student might read up on it and briefly tell the class about it. See http://www.phy6.org/Education/wbirthrb.html.

(Teacher explains) The main feature of a magnetic storm is that it greatly increases the amount of fast ions and electrons trapped in the Earth's magnetic field, typically at distances 2 to 8 Earth radii (RE). All one notes on the ground is a small change in the magnetic intensity--a drop of 1% at the equator is already a big storm--but at synchronous orbit (6.6 RE), so many fast ions and electrons are added that on a few occasions (in big storms) communication satellites have sustained damage.

The additional particles originate on the nightside of the Earth, in the long "magnetic tail" formed there. The same procedure also accelerates electrons along magnetic field that come down in the "auroral zone, " about 2500 km from the magnetic poles. When these electrons hit atoms of the upper atmosphere, around an altitude of 100 km., they produce light--the "northern lights" or "polar aurora. "

In the "auroral zone," aurora is no rarity: but during big magnetic storms, the added trapped plasma modifies the Earth's magnetic field in such a way that aurora forms on field lines closer to the equator. Therefore on such occasions people in the middle of Europe and the US also sometimes see auroras.

How can we be warned that an interplanetary shock is approaching?
(teacher supplements the answers of students)

•   Monitoring satellites are stationed at the L1 Lagrangian Point, an equilibrium position between the Earth and the Sun, at about 1% of the Sun's distance. Any ordinary feature in the solar wind arrives at that point about one hour before reaching Earth. Unfortunately, a shock moving at twice the solar wind speed gives only half that lead time.

--Special spacecraft cameras can actually see CMEs being ejected. If the spacecraft is near Earth and the CME is coming right at us, the detection of CMEs is difficult, because the Sun is on that same line of sight--although it has been done. In the future, however, spacecraft orbiting the Sun in the same orbit as Earth but located far from Earth (the "solar stereo" mission) should observe such CMEs from points far from their line of motion, and get a much better view.

What is "solar activity"?

•   Active solar phenomena associated with sunspots and their cycle, such as flares, CMEs, bursts of radio waves, bursts of X-rays, emission of high-energy particles from the Sun and related features. All these seem to be powered by magnetic energy.

(Teacher explains) When magnetic energy is released on the Sun, many electrons are accelerated (just as auroral electrons arise on Earth). Electrons are lightweight (they only form about 0.05% of the weight of matter to which they belong) and are therefore held much more tightly by magnetic forces. Because magnetic field lines tend to form arches with both ends on the Sun, they guide the electrons accelerated on them back to the Sun's denser atmosphere. When those electrons hit matter, X-rays are produced.

How are medical X-rays produced in a doctor's office?

•   Same way. Inside the X-ray machine is a glass tube empty of air, at one end of which electrons are emitted by a glowing wire. They are accelerated by a high voltage towards a metal target at the other end, and as they hit the target, X-rays are produced.

Do you think the polar aurora produces X-rays? Give your reason.

•   Yes, it does, because here too, fast electrons hit matter--in this case, atoms of the atmosphere.
(In fact, the "Polar" satellite carries an X-ray camera which takes auroral pictures in X-rays from above. This gives information about the energy of the electrons, which the visible auroral glow itself does not provide.)

Do you think that picture tubes of TVs and and computer monitors produce X-rays? Give your reason.

•   Yes, they do, because again, a beam of electrons is stopped by a screen. These X-rays are all inside the tube, however, and the thick glass does not let them get out. Good reasons exist for not watching too much TV, but X-rays are not among them.

Similar processes take place when electrons are accelerated near the Sun. (The teacher may discuss here the Yohkoh picture of X-rays in a magnetic arch and include some of the material below.) Monitoring satellites, such as the GOES satellites whose main task is weather observation from synchronous orbit, also record solar X-rays. They alert observers to unusual solar activity when the X-ray intensity suddenly rises.

CMEs and shocks are not the only factor in producing magnetic storms. The direction of the interplanetary magnetic field is also important--if it slants southward, arriving plasma is much more likely to produce magnetic storms than when it slants northward. Unfortunately, that direction cannot be sensed remotely. Only when a disturbance passes the L1 point can we tell what its magnetic field might be.

Why would the best time for a manned mission to Mars be the quiet part of the 11-year cycle, when solar activity is rare?

•   Because high energy ions associated with flares and CMEs can be hazardous to the health of astronauts.

(A student who has read Ben Bova's "Mars" may tell about the chapter in which the book tells about astronauts hiding in a "shelter" during a solar outburst.)

What is the idea behind NASA's "Great Observatories" such as "Hubble" and "Chandra"?

•   Each "great observatory" space mission is supposed to open new ranges of the electromagnetic spectrum, with a better resolution than that of earlier observations (i.e. distinguishing smaller and fainter objects). By using the new capabilities to survey the sky, new phenomena may be discovered.

Lesson Plan #42                             http://www.phy6.org/stargaze/Lsun7erg.htm

# (S-7)   The Energy of the Sun

This long section covers the generation of the Sun's energy through nuclear fusion, as well as some ideas about the evolution of stars like the Sun and their ultimate collapse, leading in some cases to supernova explosions. The section also acquaints the student with some fundamentals of nuclear physics and, if time allows, the class can proceed from here to section S-8 on nuclear power generation.

## (S-7A)   The Discovery of Atoms and Nuclei

A brief section on the emergence of our ideas on atoms, nuclei and their constituents. This section contains a historical introduction to the subject and may be omitted if time is short.

Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern, Code 695, Goddard Space Flight Center, Greenbelt, MD 20771
u5dps@lepvax.gsfc.nasa.gov or audavstern@erols.com

 This lesson plan supplements: (S-7) The Energy of the Sun                                     http://www.phy6.org/stargaze/Sun7erg.htm               (S-7A) The Discovery of Atoms and Nuclei                                     http://www.phy6.org/stargaze/Ls7adisc.htm "From Stargazers to Starships" home page: ....stargaze/Sintro.htm Lesson plan home page and index:             ....stargaze/Lintro.htm

 Goals The student will learn About the "solar constant," the average flow rate of solar energy to Earth. That stars contain appreciable gravitational energy, which they can release by shrinking. This is an important source of energy in the early and final stages of the evolution of a Sun-like star. That the composition of radioactive elements suggests the Earth is billions of years old. The only credible source that can power the Sun for so long is nuclear energy. About the structure of the atomic nucleus, an extremely compact object governed by 3 types of force: The electric repulsion of its protons, trying to blow it apart. The "strong nuclear force" which makes neutrons and/or protons stick together, provided they are first brought very close to each other, within the short range of that force. The "weak nuclear force" which tries to equalize the number of protons and neutrons in the nucleus and can, under suitable conditions, convert one kind to the other. It, too, has a short range. About the "curve of binding energy." That is the graph showing how the binding energy of a nucleus depends on its mass ("atomic weight"), telling us that the most stable nucleus is that of iron. Energy can be gained either by combining lighter nuclei ("nuclear fusion") to form nuclei up to iron, or by breaking up heavier ones. That the Sun gets its energy by fusion, combining hydrogen nuclei ("protons") to form helium. Because that energy comes from the strong nuclear force, fusion requires nuclei to come very close to each other. That usually happens only when atoms collide with great force. The high temperatures and pressures needed for such collisions appear to exist in the core of the Sun. About "controlled nuclear fusion" in which scientists try to release fusion power for commercial use, by magnetically confining and heating ions. About the way stars are believed to evolve. First a cloud of gas is pulled together by its own gravity, which supplies energy until nuclear fusion begins at the core of the new star. A long period of "nuclear burning" then follows. This ends when the star runs out of fuel and collapses, rapidly releasing a great amount of gravitational energy. About the remnants of such collapses--white dwarfs, neutron stars and black holes. About the gravitational collapse of larger stars in the ("type 2") supernova process, creating elements heavier than iron. The energy released by the collapse blows away the outer layers of the star. About the Crab Nebula, the remnanat of the 1054 supernova explosion. Terms: solar constant, gravitational energy, nucleus, proton, neutron, radioactivity, strong nuclear force, weak nuclear force, isotopes, binding energy of nuclei, mass defect, nuclear fusion, gravitational collapse, white dwarf, neutron stars, supernovas, Crab nebula, pulsars, black holes.

 (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. If time permits, an additional historical review is provided for this section, linked further below.)

### Starting the lesson

One 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):

• Matter is made up of atoms.

• Each atom has at its center a compact positive nucleus, surrounded by a cloud of negatively charged lightweight electrons.

• Nuclei in nature contain from 1 to 92 protons, positive particles which also form the nuclei of hydrogen atoms.

• Nuclei also contain neutrons, particles similar to protons but without any electric charge, in about equal number to that of protons.

All this we believe to be true, which is why I wrote it down and asked you to copy. But saying so isn't really physics. The physics is in the reasons we believe these statements hold.

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?

• That would be the power carried by sunlight falling perpendicular to the surface of the Earth, if the atmosphere did not scatter, absorb or reflect any of it.

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?

• Since only 5% of the solar energy is converted to electricity, the solar cells need to receive 20 times the power they deliver, or

20 x 3300 = 66,000 watt.

The power provided by sunlight is 0.5 of the solar constant or about 650 watt/m2 Therefore the required area is

66,000/650 ~ 100 m2 or about 1100 ft2

comparable to the area of the house itself.

Presumably all parts of the solar system--Earth, Sun, planets--came into existence together. How do scientists estimate the age of the Earth?
• They examine rocks containing long-lived radioactive elements and measure the accumulated percentage of decay products.

What age do such measurements suggest?
• Radioactive dating suggests the oldest rocks are several billions of years old. Perhaps the most reliable estimate is from Moon rocks brought back by Apollo astronauts, which remained relatively undisturbed from the time they were formed. They give about 4.7 billion years.

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?
• It was suggested that the Sun extracted gravitational energy by shrinking. All its mass was gradually falling down, inwards, and heating up from that fall.

What was the difficulty with this explanation?
• It did not provide enough energy. Even though the Sun has a much stronger gravity than Earth, it was estimated that its shrinkage could provide the Sun's energy for only about 20,000,000 years.

What source of energy is nowadays credited with the Sun's heat?
• Nuclear energy.

To understand nuclear energy, we need to know a few things about atomic nuclei. What are they made of?
• Atomic nuclei are made up of two kinds of particle, similar in mass and in the way they react to nuclear forces: protons and neutrons. The proton has a positive electric charge and the neutron is uncharged.

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).

What forces exist between protons and neutrons?
• Two protons of course repel each other electrically, both having the same kind of electric charge, a positive one.

• In addition, however, once they get very close to each other, nucleons attract very strongly--and it is this attraction, called the strong nuclear force, that holds them together in a nucleus.

There exists another nuclear force, much weaker. What does the weak nuclear force do?
• It tries to equalize the number of protons and neutrons inside the same nucleus.

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?
• It gains energy, that is, energy is released by the process.

On the other hand, to lift the stone from the floor against gravity, separating the two attracting objects, you must... ?

• ... you must invest energy.

When we add a neutron to a nucleus, do we gain or lose energy?
• Gain energy, since the neutron is attracted. (Think of a little magnet latching onto a refrigerator door!)

When we add a proton to a nucleus, what two kinds of force are involved--and do they give energy or absorb it?
This is more complicated:

• A nucleus is electrically charged, and so is the proton--both positively. So the two repel each other, and energy must be provided to let them approach each other. For instance the proton may be flung at the nucleus with great speed, and some of that speed (and of the associated energy) is lost as the two come closer, because of their mutual repulsion.

• Once the proton is very close to the nucleus, it is attracted by the nuclear force, which releases energy.

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?
• Up to iron, energy is gained by adding a proton to the nucleus. Energy must be invested in overcoming electric repulsion, but the energy gain from the nuclear attraction outweighs that. That is the fusion process, taking place inside the Sun and the source of its energy.

• For heavier nuclei, energy is lost. These atoms contain larger number of protons, their repulsion is stronger, and it outweighs the energy gain from the nuclear force.

By the same argument, though, if we could break up nuclei heavier than that of iron, we should gain energy. True or false?
• True. In fact, the heaviest atoms (uranium, for instance) do so spontaneously. They shoot out packets of two protons and two neutrons--"alpha particles" which are actually helium nuclei. That is one form of radioactivity.

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?
• The Sun gets its energy from converting hydrogen (the fuel) to helium (the product or "ashes").

This process is called...?
• "Nuclear fusion"

The teacher explains: 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)

1. First two hydrogen nuclei (protons) combine to form "heavy hydrogen," a proton plus a neutron. In this process, one proton converts into a neutron, and a "positron," the positive counterpart of an electron, is emitted.

(If a question is asked: isolated neutrons convert into protons, but inside the nucleus, with extra energy available, the conversion can also work the other way around.)

2. Then another proton is added, to create "light helium."

3. Finally a 4th proton is added, to create regular helium. Again a proton converts into a neutron, and another positron is emitted.

Where does the released energy appear? The nuclei emit gamma rays, while the positrons meet with electrons and both are "annihilated," also creating 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.

What is controlled nuclear fusion?
• Controlled nuclear fusion is the attempt by scientists to extract energy by fusion reaction in the laboratory.

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?
• They use magnetic fields.

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 larger than the Sun?
• It will probably collapse into a neutron star, as dense as the atomic nucleus. Here electrons are not just smeared out, but they combine with the protons to form neutrons, held together by the enormous gravity.

Why does the strength of the force of gravity and the energy released by it depend on the final size of the object?
• Because gravity in all these configurations acts as if mass were concentrated at the center, so that its force increases like 1/r2. The smaller the final value of r, the stronger is the force, and the more energy can be released by letting it move matter.

(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?

• Gravity is then so strong that the object collapses into a "black hole" of extremely small size. We cannot see its true size (or anything else about it) because the intense gravity does not allow any light to escape. However, the mass still exerts a gravitational pull on surrounding objects.

Calculation of Escape Velocities

(Teacher explains)

The escape velocity V from the surface of the Earth, at radial distance r, was calculated in an earlier lesson to satisfy

V2 = 2gr

where g is the acceleration due to gravity. Using SQRT to denote square root

V = SQRT (2gr)

By Newton's theory of gravitation, if m is the mass of the Earth

g = Gm/r2

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

V = SQRT [2Gm/r]

and for a star of mass M and radius R

V = SQRT [2GM/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

V1 = 12.4 + 30 = 42.4 km/sec.
So
42.4 km/s = SQRT [2GM/R1]

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.) What is the escape velocity V2 from the Sun's surface? The radius of the Sun is R2 = 700,000 km or about 1/200 AU. So V2 = SQRT [2GM/(R1/200)] =                 = SQRT[200 x (2GM/R1)] = SQRT[200] x SQRT[2GM/R1]             = 14.1 x 42.4 km/sec = 597.8 km/sec             In round numbers this is about 600 km/sec--twice the velocity of NASA's planned "solar probe" which at its closest approach will whiz by the Sun at 300 km/sec, at a distance of 4 solar radii. At 1/10,000 the velocity of light, that would be the fastest man-made object ever made, excluding accelerated atomic particles. Next, let the Sun shrink to the size of a white dwarf, say from a radius of R2=700,000 km to R3=7000 km, comparable to Earth's. What is the escape velocity V3 now? V3 = SQRT[2GM/(R2/100)] =     = SQRT[100 x (2GM/(R2)] = SQRT[100] x SQRT [2GM/(R2]             = 10 x 600 km/sec = 6000 km/sec A lot of energy can be released by material collapsing onto a white dwarf. If it were to come from far away, it would hit at 6000 km/sec, hundreds of times faster than any asteroid impact, fully 2% of the speed of light! Now let us go even further and let the Sun-size star collapse to a neutron star, of radius R4=7 km, 1000 times smaller. If the escape velocity is now V4, we get V4 = SQRT[2GM/(R3/1000)] = SQRT[1000] x SQRT(2GM/R3)]             = 31.6 x 6000 km/s = 189,600 km sec Say 190,000 km/sec (we neglect relativity here, which would change the result somewhat). That is close to 2/3 the velocity of light! Even a pebble hitting such a star will release a great amount of energy. And it is no wonder that with moderately larger masses, we get into black hole territory. (End of derivation)

 The final collapse of large stars creates supernova explosions. What happens there? So much energy is released by the collapse that the outer layers of the star are blown off with tremendous speed. And in addition? The final collapse creates a brief instant of tremendously rapid nuclear fusion, in which many heavy nuclei are created. All nuclei heavier than iron that are found on Earth are believed to have arisen in this way. 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 star became visible to the eye, and was studied extensively since that time (see picture). At the time of collapse, underground experiments in Japan and the US detected a simultaneous flow of more than a dozen neutrinos, simultaneously! This agreed with the theory of supernova collapse and with the brief but very intense spurt of nuclear fusion, creating new elements.

Author and curator: David P. Stern, u5dps@lepvax.gsfc.nasa.gov
Last updated 29 October 1999
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