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

Goals: The student will learn

  • How Newton tied together the gravity observed on Earth and the motion of the Moon.

  • To apply an earlier lesson about centripetal forces.

  • To appreciate the philosophical implication of Newton's calculation: all parts of the universe seem to obey the same laws of nature.

    Terms: Gravitation (or gravity), inverse-square law, solar constant.

    Stories: The Story of Newton's apple is traced to its source. Also, mention is made of Cavendish and Eötvös (and in this lesson plan, of Coulomb).

    Starting the lesson:

    Today we will learn about Newton's theory of universal gravitation. The story is often told that Newton was inspired to this by a falling apple--with some going so far as to suggest it bonked him on the head as he was sitting under a tree. We will come back to that story.

    Of course, Newton did not "discover" gravity. People in his time were well aware that the Earth attracted all objects near its surface, and Galileo, as you learned some time ago, experimented with falling objects. But all these observations had to do with what happened close to Earth, in a region accessible to experimenters. The objects in the sky--planets orbiting in space, the Moon circling around the Earth--seemed to belong to a different class of phenomena, with separate rules, such as Kepler's laws. Astronomers were only beginning to understand those phenomena.

    Newton's achievement was in making the connection, in proposing that the same force that pulled apples down also pulled the Moon.

    Suppose, he said, that the force decreased like the square of the distance from the center of Earth. The Moon is 60 times more distant than the surface of the Earth, so the force of gravity should be 60 times 60, or 3600 times weaker. Would that equal the centripetal force mv2/R or 4p2N2R needed to hold the Moon in its orbit?

    Newton could calculate that--he knew R from Aristarchus and Hipparchus, and he knew N = 1/T, where T was the orbital period of the Moon. As you will see, the result fit well, and Newton concluded that it was indeed the gravity of the Earth that held the Moon to its orbit. He also guessed (correctly) that the Sun had a force of gravity much like the Earth's, and that was what kept the planets going around it. That is why he named his idea the theory of universal gravitation.

    Was the inverse squares law just a lucky guess? It was more. Imagine a source of light--say, the Sun--shining at equal intensity in all directions, sending out each second sunlight carrying an energy of I joules--that is, radiating a power of I watts. What is the density at which solar power flows at a distance of R meters, across an area of, say, 1 square meter (1 meter2)?

    A sphere of radius R centered on the Sun has an area 4pR2 meters2. Since all parts of the sphere receive equal illuminaton, each square meter receives a power I/4pR2 watts, and obviously, that intensity decreases like 1/R2. At Earth's orbit it equals 1360 watt/m2, a quantity known as the solar constant. At the orbit of Jupiter, R is about 5 times larger, and the light intensity is only 1/ 52 = 1/25 of the solar constant, which is why Pioneers 10 and 11, and Voyagers 1 and 2, preferred a power source other than solar cells. And although a lightbulb does not shine with equal intensity in all directions, your reading illumination, too, will decrease approximately like the square of the distance from it.

      [By the way: for historical reasons, phenomena which spread energy and obey the 1/R2 law are often classified as radiation. Microwaves used to heat food in the kitchen are called radiation, fast ions and electrons emitted by radioactive substances are also called radiation--even though the two phenomena are different and unrelated. Microwaves are similar to light and radio, ions and electrons are fast-flying pieces of matter.

      Students should be made aware that the damaging kinds of radiation are the ones also known as ionizing radiation, "ionizing" meaning they deliver enough energy to atoms and molecules to tear off electrons. These can be fast ions and electrons from radioactivity, or high-energy "electromagnetic" radiation similar to microwaves, radio waves and light, but of the types known as x-rays and gamma rays.]

    Newton was aware of such results, and the 1/R2 law was a natural assumption for him. Today that calculation is easy to repeat--but remember, one needed at least some idea of the laws of motion and of the centripetal force which they prescribe. Galileo and Kepler lacked that knowledge, and could not have tested the 1/R2 law without it.

    Newton's "universal" gravitation led to a principle which today is all too easily taken for granted: that the same physical laws hold anywhere in the universe. All the astronomical evidence since Newton's time seems to confirm it: not just the laws of gravity, but the chemical elements, the speed of light, the way light is produced and other physical processes seem to be the same on Earth and in distant stars, or even in other galaxies.

    Now let us go back to that apple. (Continue with the original story of Newton's apple as given in "Stargazers").

    Questions to ask
    Since the entire lesson focuses on a single calculation, not much can be asked beyond its details.

    What idea did the falling apple supposedly inspire in Newton?

    --That the force of gravity which causes objects to fall was also the force that kept the Moon in its orbit around Earth.

    What did Newton have to assume, to test his guess?

    --He had to assume something about the rate at which gravity decreased with distance. His assumption was an "inverse square law"--that gravity decreased like 1/R2.

    Concluding comments (Optional)

    The French military engineer Augustine Coulomb lived a century after Newton. In 1777 he won a prize for a new method to measure the Earth's magnetism--by suspending a horizontal bar magnet by its middle, from a long string, with a pointer attached--or better, a small mirror (draw on the board). When a beam of light is reflected from the mirror, even small magnetic variations can be measured, even those that twist the magnet by just a tiny amount from its quiet position.

    Coulomb also adapted this "torsion balance" instrument to measure the force between two magnetic poles--either repelling or attracting. You bring the pole end of a second bar magnet near one of the poles of the suspended magnet. The force between them can then be measured by the twisting of the suspension string, which exerts an elastic force. Coulomb found that magnetic poles, like gravity, attracted and repelled with a force that decreased like the inverse of the square of the distance.

    Coulomb's experiment was so sensitive that static electricity, which generated its own forces, sometimes interfered with its observations. That gave Coulomb the idea of measuring electric forces the same way. He replaced the magnet with a little dumb-bell shaped stick, with small spheres at the end. He then charged one sphere with static electricity, brought close to it another charged sphere, and measured the force. Guess what? Electric forces also decreased like 1/R2.

    A reclusive English gentleman named Henry Cavendish set out to use the same method to measure the force of gravity directly. Instead of two lightweight spheres at the end of the stick, he used two heavy ones, and attracted one of them by a third sphere, a big one. The force was tiny and because of the big masses, the instrument reacted agonizingly slowly. But in 1796, he finally succeeded--showing that not just the sphere of the Earth, but even spheres of lead in the lab, exerted a measurable force of gravity.

  • 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

    Goals: The student will learn

  • To derive the velocity in a circular orbit of any radius.

  • To derive the Earth escape velocity.

  • To prove Kepler's 3rd law for circular orbits

  • A simple formula for the orbital period in a circular Earth orbit of given radius.

    Terms: Orbital velocity, escape velocity, synchronous orbit.

    Starting the lesson:

    Today we will apply Newton's calculation to circular Earth orbits. As already noted, all three of Kepler's laws can be derived from an inverse-square gravitational force, assuming only that it acts primarily between only two bodies--e.g. Sun and planet, Earth and Moon--while the pull of all others can be neglected.

    With our limited math we can handle here only circular orbits, not elliptical ones. Circular orbits are a somewhat special and simple case. The first law for instance is automatically satisfied, because yes, the circle is an ellipse--of zero eccentricity and with both its foci combined at the center.

    The second law alsodoes not add any information about the motion. The speed of a satellite or planet in a circular orbit is constant, and so is its distance from the center. The area swept by the radius per unit time is then automatically constant.

    The third law, however, is meaningful even for circular orbits. Newton's calculation allows it to be checked, and today we will show that the inverse-squares law indeed gives the expected result.

    In this day and age we are interested not just in planetary orbits but also in the orbits of artificial satellites. We will therefore use such orbits to illustrate the calculation, and some useful results will also be derived.

    Let us start by calculating the velocity v1 required by a spacecraft in a circular low Earth orbit--what is called in Russia "the first cosmic velocity."

    [Go over lesson 21, mentioning that what we call "escape velocity" is called in Russia the "second cosmic velocity."]

    Questions and worked examples

    This section is again mostly concerned with one calculation, and the questions asked are all related to it.

    Express the orbital velocity v1 at r = 1 RE, given the acceleration of gravity at that distance as g = 9.81 m/s2 and the radius of the Earth RE = 6,371,000 meters.

      This derivation is found in lesson 21)

    If you give the above velocity v1 to a rocket launched vertically, straight up, how far from Earth will it get before falling back?
    Hint: the semi-major axis of an orbit depends only on the energy.

      The rocket will rise and then come down along the same radial line, following what is essentially a very narrow and elongated ellipse (eccentricity close to 1).

      Since the energy is the same as the one required for low Earth orbit, the semi-major axis a will also be the same. For the low Earth orbit, a = 1 RE, hence the same value holds here, too, making the ellipse 2RE long. The highest point of the trajectory--the apogee of the orbit--will therefore be at the height of 1 RE above ground.

    Given the orbital velocity for a circular orbit at 1 RE as 7.9 km/sec, what is the escape velocity at the surface of the Earth (r = 1 RE)?
    What sort of "escape" does it provide, and what sort doesn't it?

      The escape velocity is v2 = 7.9*SQRT(2) = 7.9*1.4142 = 11.2 km/s. A velocity greater than v2 makes it possible for the launched object to escape Earth and not fall back. However, it will still be bound by the Sun's gravity, and will stay in an orbit around the Sun, similar to the Earth's.

    If an object above the atmosphere has a velocity greater than the escape velocity v2, in what upward direction must it move to escape the Earth's gravity?

      It does not matter. An escape orbit is the limiting case of an orbit with very, very large semi-major axis, and therefore a very distant apogee. With a starting speed close to the escape velocity, in whichever upward direction the object is moving, its starting point is always close to the perigee, and it is always headed for the apogee. In the limit (=at escape velocity), this means it is heading for infinity.

    The Earth orbits the Sun in a near circular orbit of radius 1 AU (Astronomical Unit) at 30 km/sec. How much extra velocity, above and beyond 11.2 km/s, does a spacecraft escaping the Earth's gravity need, to escape the Sun's gravity as well?

      The calculation is similar to the preceding one. Escape velocity from the Sun at 1 AU is
          v2s = 30*SQRT(2) = 30*1.4142 = 42.4 km/s.
      Thus an additional velocity of (42.4 - 30) = 12.4 km/s is required.

    In some weird alternate universe (already met in the lesson on Newton's 2nd law) weight and mass are not proportional. Two materials, astrite and barite, have the same weight per unit volume, but a volume of astrite has twice the mass of a similar volume of barite.

    Both are strong and light metals, and are a natural choice for spacecraft construction; we can assume that barite behaves the way aluminum does on Earth. Which of the two would be a better choice?

      Say an Earth-like planet in the alternate universe has a radius RE and surface gravitational acceleration g. In a circular orbit having a radius of R planetary radii, Newton's equation for the orbital velocity of a barite satellite is the same as for terrestrial aluminum in an Earth orbit, i.e.

              mg/R2 = mv2/(RE R) and hence v2 = g RE/R

      An astrite satellite has the same weight but twice the mass, so Newton's equation becomes

              mg/R2 = 2mv2/(RE R) and hence v2 = g RE/2R

      The orbital velocity required by the astrite satellite at distance R is therefore smaller by a factor SQRT(2) = 1.4142 than that of a barite satellite in the same orbit, making astrite the obvious choice.

    --The planet Mars has a radius of RM=3390 km and its satellite Deimos orbits it in a near-circular orbit with orbital period TD = 1.26244 days and a mean distance of RD = 23,436 kilometers. What is (1) The acceleration gM due to gravity at the surface of Mars and (2) The escape velocity there?

      If the acceleration on the surface is gM, then for a circular orbit at distance RD

              gM (RM/RD)2 = v2/ RD

      where v is the velocity of Deimos in its orbit. From a previous calculation, if N is the number of orbits per second and TD is also given in seconds

              v = 2pRDN = 2pRD/TD

              TD = 1.26244*86400 sec = 109,075 sec.

              v = 6.2832*(23,436,000 m)/109,075sec = 1350 m/sec

      In calculating a ratio we can use kilometers, so RD/RM = 23,436/3390 = 6.9133

      We then have

              gM = v2RD/RM2 = (v2/RD) (RD/RM)2

              gM = ((1350)2/23,436,000)*(6.9133)2 =

              = 0.077764*47.793 = 3.717 m/s2

      i.e. slightly more than 1/3 the acceleration of free fall on Earth. The orbital velocity V1m at the surface of Mars, in analogy with V1 derived for Earth, is found from

              V1m2 = gM RM = 3.717*3,390,000 = 12,600,000

              V1m = 3549.6 m/s = 3.5496 km/s

      The escape velocity is SQRT(2)=1.4142 times that, or 5.02 km/sec.

    According to Kepler's 3rd law, T is proportional to a, where T is the orbital period in seconds and a the semi-major axis in meters. That implies T2 = ka3 . In a circular orbit around Earth, a=r where r is the orbital radius. Can you derive k for such orbits?

    The result is derived in "Stargazers" and is     k = g RE/4p2

    Here the teacher may continue with lesson 21a (which is optional). Some questions related to that lesson:

    From Kepler's 3rd law, the orbital period T around Earth in a circular orbit at distance R is

                    T = Q *R*SQRT(R)
    If R is given in Earth radii, what is the value of Q?

      The calculation in "Stargazers" gives 5063.5 seconds.

    --At what distance R is the orbital period around Earth 24 hours? Why is that orbit important?

      We have 86400 sec = 5063.5 R*SQRT(R) = 5063.5 R3/2

      R3/2 = 17.0633         R = (17.0633)2/3 = 6.628 RE

      The orbit is important because a satellite in an equatorial orbit at this distance stays all the time above the same point on the Earth's equator, and rotates with the Earth as if it were attached to it by a rigid rod. That makes such a"synchronous orbit" desirable for communication satellite, because they can then be tracked from the ground by a fixed antenna.

    --Navigational satellites of the GPS (Global Positioning System) move in 12 hour orbits. What is their distance?

    --The calculation is very similar to the preceding one.

    We have 43200 sec = 5063.5 R*SQRT(R) = 5063.5 R3/2

    R3/2 = 8.53164         R = (8.53164)2/3 = 4.175 RE

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

    #22   Frames of Reference: The Basics   

    #22a    The Aberration of Starlight

    #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

    "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

    Note: This section is optional. Also, if time is short, the discussion of forces on a propeller (the more difficult part) may be omitted.


    The student will learn:

    • Basic concepts about airplane flight.

    • The reason jetliner wings are swept back.

    • Why jet engines have replaced propellers in high-speed flight

    Terms: frame of reference, relative velocity, lift, drag, thrust, angle of attack, wind tunnel, sweepback of wings.

    Stories and side excursions: The wind tunnel of the Wright brothers, the swept-back wing, the X-29 airplane with swept-forward wings and the swivel-wing idea.

    Starting the lesson:

    Today we will discuss the application of frames of reference to an airplane flying with a constant velocity v through the air. Viewed in the frame of reference of the ground, in the absence of wind, the airplane is moving through still air.

    If on the other hand we prefer to use the airplane as our frame of reference, then the air is what moves, blowing in the opposite direction to the flight and flowing around the wings and the aircraft.

    We can look at it either way, but the second way is usually more convenient. In any case, it is the flow of air over the wings of an airplane that supports the airplane in the air, creating an upward force known as lift. Lift increases rapidly with velocity--as a matter of fact, it grows like the velocity squared--so with enough speed, even an airliner weighing 100-200 tons can be supported

    We don't have the time here to discuss how lift is generated. Let it just be said, that the cross-section of the wing--flat on the bottom, curved on top--makes air flow faster over the top than over the bottom. This only happens when the front of the wing faces the wind (directly or lifted slightly, at a small "angle of attack").

    When the airplane stands on the ground, not moving, air presses on the top and bottom of the wing with equal force. In flight, the faster flow on top of the wing creates lower pressure there, and the extra pressure from below is then what produces the lift.

    Another force on the wing and on the airplanes is the drag--that is the name given to the air resistance, and it also grows like the square of the velocity. The drag is overcome by the thrust, the forward pull of the propeller or the push of the jet engine. And finally, the lift is opposed by the weight of the airplane and its cargo.

    [As part of this discussion, it may help to draw on the board a side view of an airplane, and each time a force is mentioned, illustrate it by an appropriately directed and labeled arrow.]

    Then continue from the text of section #22b, starting at the subhead "Frames of Reference."

    Questions and side excursions

    What are the 4 forces acting on an airplane in flight, and what are their directions?

      Thrust of the engine--pulls the aircraft forward
      Drag--the resistance of the air, opposes the thrust
      Lift--the upward force on the wing
      Weight--the downward pull of gravity.

    What creates the lift on an airplane wing?

      The pattern of air flow over the top and bottom of the wing reduces the air pressure on the wing's top surface.

    [Optional discussion:
    Can the same lift force be applied to transportation by water?

      A.: Yes! Such wing-like surfaces are known as hydrofoils, a name which is sometimes also applied to boats and ships using them. The hydrofoils extend below the hull and across its width--e.g., one in front and one in the rear. The boat starts moving like an ordinary boat, floating on the water. Then, when an appropriate speed is reached, the hydrofoils lift the hull out of the water, leaving only the propellers and hydrofoils submerged. Such boats are capable of much greater speeds than ordinary motor boats, e.g. 70 mph and more.]

    Why do jetliners avoid flying above the speed of sound?

      Because at supersonic speeds, air piles up in front of the aircraft and its wing, forming a compressed layer ("shock"), rather than smoothly flowing around the airplane. Compression heats up the air, and since heat is a form of energy, the process robs energy from the motion and greatly increases the drag force. It also reduces the lift of a wing, which depends on the orderly flow of air around it.

    Why do jetliners avoid flying faster than even 85% of the speed of sound?

      Because as part of the lift-generating process, air flows faster over the top of the wing. When the airliner's speed only approaches the speed of sound, the flow over the top of the wing may exceed that speed and form shocks.

    Why do swept-back wings allow an airliner to fly closer to the speed of sound?

      Because in a crude approximation, the flow of air over a swept-back wing can be resolved into a component flowing along the wing, not strongly involved in lift and drag, and a component flowing across the wing, perpendicular to it, which acts like the ordinary flow across a wing that is not swept back.

      However, a component of a velocity is always less than the full velocity. Therefore, in a swept-back wing the speed of the perpendicular flow will be smaller than that of the airplane. This allows the airplane to get closer to the speed of sound before shocks form on top of its wings.

    Will wings swept forward have the same effect?

      Yes, they will, as demonstrated on the X-29 airplane. However, the flexing of wings makes this design less stable.

    Can the same advantage be obtained from a wing that turns around a swivel after the airplane has attained cruising speed--one wing is swept back, the other forward?

      Yes it can, and such designs have been tested with models. The swivel-wing airplane can fly along a straight path, but cannot be safely steered.

    Optional peripheral discussion--or riddle to ponder at home:

    In the lobby of the Air and Space Museum in Washington hangs the "Voyager" airplane which flew non-stop around the world, taking more than a week. It took off at 138 miles-per-hour, using two engines, but it came back at only 78 miles per hour, with one engine turned off. Why the difference?

      Flying nonstop around the world took a lot of fuel--in fact, fuel weighing many times more than the rest of the airplane. On take-off and in early stages of its flight, Voyager was heavily loaded, and to let its wings create enough lift to hold it aloft, it had to fly fast. Coming home, most of the fuel had been consumed, there was no need for such speed, and one engine could supply all the thrust that was needed.]

    How does an aircraft propeller work?

      It could be viewed as a small rotating wing, whose lift pulls the airplane forward.
      (The blade ends, which move fastest, produce most of the lift)

    Before take-off, when the airplane stands on the end of the runway and the pilot "revs up" the propeller, how does air flow in the frame of a propeller blade?

      The air flows in a way similar to its flow over an airplane wing.

    How does the above flow change when the airplane has appreciable forward speed?

      In addition to the flow experienced by the propeller on the ground before take-off, it now also feels a flow of air from the front, due to the motion of the airplane.

      If the propeller blade is viewed like the wing of a flying airplane, the added flow is like an added wind, blowing vertically downward. The combined flow appears to the blade like a head wind slanting downwards.

    What is done to remedy this?

      The blades of the propeller can be twisted (even in flight), in a way making them face the slanting flow of air.

    What limits the usefulness of this remedy?

      The lift force on the propeller is now at an angle. Only part of it pulls the airplane forward, the rest contributes a greater air resistance to the motion of the propeller. The faster the airplane flies, the smaller is the part that pulls and the greater the part that resists and must be overcome.

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

    #23   Accelerating Frames of Reference:   Inertial Forces   

    #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.

    Goals: The student will learn

    • That when applying the laws of motion in an accelerating frame to a body with mass m, using coordinates defined in such frame, one must always add an "inertial force" caused by the frame's acceleration..

    • In a uniformly accelerating frame, the inertial force opposes the acceleration. For instance, in a braking car (acceleration directed backward, opposed to velocity), the inertial force tends to push forward.

    • In a uniformly rotating frame, to the forces on a mass m must be added a "centrifugal force" mv2/r, directed away from the axis of rotation.

    • To clearly distinguish between the centripetal and centrifugal forces.
      • The centripetal force is used when we calculate motion, using coordinates, velocities and accelerations of a non-rotating frame of reference.

      • The centrifugal force is needed to describe the same motion, using coordinates, velocities and accelerations of a rotating frame of reference.

    • Examples of the inertial forces:
      1. The force on objects inside a braking bus,
      2. The centrifugal force on objects on the rotating Earth
      3. The centrifugal force in a roller-coaster car going around a "loop the loop".

    Terms: inertial force, accelerated frame of reference, rotating frame of reference, centrifugal force, bulge of the Earth.

    Stories: Variation of g and the bulge of the Earth. (optional, at the end of this lesson plan: pumping a swing)

    Starting the lesson:
    Up to now we have dealt with uniformly moving frames of reference. That was easy: moving from one such frame to another, it was only necessary to re-define velocities and coordinates in terms appropriate to the new frame. The forces were the same, because forces, as Newton showed, depend only on acceleration.

    Tody we go a step further, to accelerating frames of reference--in particular, two examples:

    • a bus whose driver suddenly hit the brakes, and
    • a rotating frame, say a bus turning a sharp corner.

    From experience everyone knows what to expect. In the first case, the passengers are flung forward, in the second, they are pushed to the outside of the curve, by what is known as a centrifugal force.

    Both these are a rather special type of force, called inertial forces: they only appear in the accelerating frame.

    • If we treat the motion in the coordinates of the accelerating frame--e.g. if your coordinates are those of the inside of the bus--they must be included.

    • If you solve the problem using the coordinates of an outer frame of reference, which does not accelerate or rotate--e.g. if your coordinates refer to position in the outside world--they do not exist.

    For example: the bus which suddenly stops.

    • If we calculate position relative to the outside world, the situation is very simple. The bus is slowed down by its brakes, but the passengers keep going, until something--the seat belt, the seat in front--stops them.
    • If we calculate position relative to the bus, the passengers are pushed forward, to the seat in front, to the limits of the seat belt.

    This can be quite confusing. Some teachers and textbooks avoid inertial forces altogether or call them "fictional forces". As we will see, however, they can be quite useful. One thing you should remember: be sure you know to which frame of reference your coordinates belong. With accelerated frames, all frames are not all equivalent.


    The best known example, of course, is a rotating frame of reference, where the centripetal force pushes inwards, and the centrifugal force pushes outwards. Which to use? Here is the rule: [on the board--all please copy]

      The centripetal force is only used in a nonrotating frame of reference. It is the force holding the rotating mass to its curved path, and is directed towards the axis of rotation.

      The centrifugal force is only used in a rotating frame. It must be added to other forces in that frame, in order to take into account the frame's rotation, and is directed away from the axis of rotation.

    As you will see, both describe the same physics, and either can be used to study motion. However, the centrifugal force seems more intuitive: this is what you feel when you sit in a car going around a sharp curve. You feel yourself pushed to the outside of the curve. Your mind is keyed to the frame of reference of the car that surrounds you, and in that frame you feel pushed outwards.

    As a result, almost anyone knows intuitively what a centrifugal force is, while "centripetal force" is only familiar if you learned about it in school.

    (Then continue with the material of section #23 in "Stargazers.")


    An astronaut lies horizontally on a couch, flat on the back, inside a space shuttle as it takes off. In which direction is the astronaut accelerating?

      --The acceleration is directed forward (=upwards, at least at first).

    Does the astronaut feel pushed--and if so, which way?

      The astronaut feels pushed down, towards the couch, opposite to the acceleration

    We agree the astronaut's body is pressed towards the couch. Viewed from the outside world, why that pressure?
      Because the shuttle is accelerating, the couch is accelerating with it, and it in turn accelerates the astronaut's mass, through the pressure the couch exerts on it.

    Viewed from inside the cabin, why that pressure?
      The cabin is an accelerating frame of reference with acceleration a, and it exerts an inertial force -ma on anything inside it.

    If you have cartesian coordinates (x,y), what are the unit vectors xu and yu?
      Vectors of unit length in the directions of the coordinate axes

    What are unit vectors useful for?
      Vectors combine two properties--direction and magnitude. Sometimes it is convenient so look at each of these separately. A vector V in the x-direction can be written V xu: the factor V contains all the information about the magnitude, the unit vector tells everything about the direction.

    Say you fire a gun at an angle upwards, so that
    • the initial upwards velocity is u, and
    • the initial horizontal velocity is w.
    [draw it all on the board].
    Let the direction of u be the y axis, and of w the x axis After t seconds, the upward velocity is (u-gt) while the horizontal one remains w. Express the velocity vector V after time t.
      V = (u-gt) xu + w yu

    Is motion around a circle with constant speed ("uniform rotation") an accelerated motion?
      Yes, it is.

    Why is it accelerated, if the speed does not change?
      The speed doesn't, but the direction of the motion does. By Newton's laws, only if a body moves with constant speed along a straight path are no forces needed to maintain that motion.

    So, for a body with mass m to move with constant speed V around a circle of radius R, a force is needed. What is that force called, what is its magnitude and what is its direction?
    • It is called the centripetal force
    • Its magnitude is m V2/R
    • Its direction is towards the center of the circle.

    You are standing on the floor, and your body is subject to two forces:
    • Your weight of F1 newtons,
    • the reaction F2 newtons of the floor, not letting you sink any further.
    Your body is in equilibrium--it does not move. What can you say about those two forces?
      They must cancel, that is, add up to zero: F1 + F2 = 0

    [The discussion that follow may be helped by a crude drawing on the board].
    You are sit on a whirling turntable, part of a carnival ride, at a distance r from the axis, going around at a velocity V. As before, F1 is your weight and F2 is the force exerted on you by the ride.
        You hold tight and keep your position. If ru is the unit vector pointing radially outwards from where you sit, which of these equations correctly describes your force balance?

    F1 + F2 = (mV2/R) ru        

    F1 + F2 = -(mV2/R) ru      ?

    The total force F1 + F2 on you--in the first case it is outwards, in the second inwards. Which is it?
      Inwards: F1 + F2 = - mV2/R Ru

      Because... as your body goes around the curve, in the absence of forces it would continue straight--but holding on to the turntable, you are instead pulled into the circle, towards the center. You need an inwards-directed centripetal force to keep you going around the circle.

    Another way of writing the same equation is F1 + F2 + (mV2/R) ru = 0 How is this interpreted in the rotating frame?
      You keep your position in the rotating frame only if all forces on your body add up to zero--including the centrifugal force (mV2/R) ru

    You live on a rotating Earth. How does the centrifugal force affect you on the equator?
      The force on your body is the sum of gravity and the centrifugal force. On the equator, the two are opposed, and therefore the effective value of g is slightly smaller there.

    How does the centrifugal force affect you away from the equator?
      It reduces the pull of gravity by a lesser amount than at the equator. In addition, it also slightly shifts the direction in which bodies fall, because while gravity is radial, the direction of the centrifugal force is parallel to the equatorial plane.

    In which direction is the shift?
      [sketch on the board the two forces and their head-to-tail vector addition: the gravity arrow goes down towards the center of the Earth, the centrifugal arrow continues from its end parallel to the equator.]

      Falling bodies are slightly displaced towards the equator.

    Does the centrifugal force of the Earth's rotation affect the Earth itself?
      Yes, it, makes the earth oval instead of a perfect sphere. The equatorial radius of Earth is 6357 km, while the polar radius is 6378km (average radius, 6371 km)

    Originally the meter was defined in terms of the distance from the equator to the pole, which was to be equal to 10,000,000 meters. If that were still the definition,would you expect the length of the equator to be 40,000 km? A little more? A little less?
      A little more. Note: the actual meter was defined as the distance between two fine scratches on a "standard meter" metal bar kept in Paris, supposedly based on the above distance. Later it was redefined in terms of wavelengths of light, which can be very precisely measured.

    Jupiter and Saturn both rotate around their axes in about 10 hours. Would you expect them to be more oval or more spherical than the Earth?
      More oval; in fact, they look oval in the telescope. For Earth the difference between polar and equatorial radii is 1/297 of the radius; for Jupiter 1/15, for Saturn 1/9.5.

    You are sitting on a rotating platform, right on the middle. You want one of the people sitting nearer to the edge of the platform to join you at the middle, so you extend your hand and pull that person in, against the centrifugal force.

    What do you think: do you need energy to overcome the centrifugal force? Does your arm perform work?

      If you guessed that you did need energy, you are right.

    There exists an interesting application. You are probably aware children can "pump up" a swing and keep it going; you may even have done so yourself. How is it done?

    The process is best seen in swings on which you stand rather than sit. The centrifugal force on the swing varies: it is greatest at the bottom of the swing, where motion is fastest, but zero when the swing briefly stops at the end points. What you do is, near the end points you lower your body, then near the bottom of the swing you stand up straight again.

    By standing up, you overcome the centrifugal force, which is directed towards your legs, so you invest energy. But energy is conserved and must go to somewhere else in the motion! It actually goes to the energy of the swinging motion, making the swing move more vigorously, or at least making it overcome the energy loss to friction, which would otherwise gradually slow it down.

    The "pumping" of a swing by moving legs and body while sitting down is somewhat similar.

    [Incidentally: the "Exploratorium" science museum in San Francisco had--and maybe still has--a swing which can be "pumped" by a rope which controls a weight on it. By pulling up the weight near the bottom of the swing and letting go near the end points, you can increase its swings and keep them going.]

    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

    Goals: The student will learn

    • Understand the "weightlessness" or "zero g" environment aboard spacecraft

    • Realize that weightlessness will occur in any frame of reference which moves freely, subject only to gravity (e.g. "freely falling").

    • Learn about the Coriolis force, another inertial force not covered previously. Like other inertial forces, it too appears only in a rotating (or otherwise accelerating) frame. While the centrifugal force acts on any object in the rotating frame, the Coriolis force only acts on objects that move in that frame, in a way that affects their distance from the rotation axis.

    Terms: weightlessness (or zero-g), artificial gravity, Coriolis force.

    Stories and extensions: The "Vomit Comet" airplane used by NASA to simulate weightlessness; "artificial gravity" in a rotating space station; the swirl of draining sinks and of storms, north and south of the equator


    Starting the lesson:

    This lesson continues the exploration of rotating frames of reference, in particular two of their features: weightlessness and the Coriolis force.

    Every one here must have already heard of "weightlessness" in space, and has probably seen on TV astronauts floating weightlessly in the "zero g" environment of the space shuttle. It is not that the Earth's gravity does not extend to the shuttle's altitude--without gravity, how could it stay in its orbit?

    So what is happening to gravity on the shuttle? To get a better understanding, let us look at a few examples of gravity in action:

    Example 1: When you jump down from a truck or a wall, gravity pulls your body down and gives it an acceleration g--until your feet are stopped by the ground.

    Example 2: When you stand on the floor, gravity also pulls your body down. But you do not move and do not accelerate, because the floor won't let you. It produces an opposing force which prevents any motion, and your body is now in an equilibrium.

    In example 1, gravity is the only force present, and you accelerate. In example 2, two opposing forces cancel each other, and nothing moves.

    Now, Example 3: You watch on TV an astronaut inside the orbiting shuttle, standing in front of the camera. The way gravity acts on that astronaut's body--which of the preceding two examples does it resemble? The person jumping from a truck, or the one standing on the floor?

    [The class may discuss the question briefly.]

    It sure looks like example 2, the person standing on the floor. But actually it is much closer to example 1, the jump from the truck.

    Once again: In example 2, two opposing forces cancel each other, and nothing moves.

    In example 1, gravity is the only force present, and your body is accelerating.

    • The astronaut in front of the camera is moving, at the orbital velocity of 8 km/sec.

    • The astronaut is accelerating, because motion in a circular orbit (or in any non-straight orbit!) is accelerated. And

    • Gravity is the only force present.

    The only reason motion is not evident is that the shuttle and camera are also moving and accelerating, just as fast. Gravity is already fully employed, it is producing all the acceleration it can, and there remains no additional force that would push the astronaut against the floor. So to the person in space, this feels like weightlessness.

    If the orbit is an exact circle, gravity supplies the centripetal force needed to maintain it, so we have....
    [continue with the first equation in section #24, up to "The Coriolis Force" which is introduced below.]

    The second half of this unit is concerned with the Coriolis force.

    The centrifugal force, you will recall, is an inertial force--one which only enters the calculation if the motion is described in a rotating frame of reference. In that frame, if the centrifugal force on some body is balanced by some other force, that body is at rest.

    If however the body is is moving in the rotating frame, and we want to use its equations of motion in the rotating coordinates, we not only need to add the centrifugal force, but may also need another inertial force--the Coriolis force. Unlike the centrifugal force, it only comes into play when an object moves in the rotating frame, in such a way that it changes its distance from the rotation axis.

    [The discussion that follows can use a sketch on the board.]

    At any point in a rotating system, you move around the axis with a certain velocity. The greater your distance from the axis, the greater the velocity. And when you move in the rotating frame to a greater and smaller distance from the axis, your velocity should also increase or decrease. Wouldn't you think that inertia would resist such changes? It does--and that is the essence of the Coriolis force.

    A good example is given by a rotating space station, the kind that was first proposed in the early 1950s. It was then suggested that the centrifugal force might provide an "artificial gravity" in space, inside a wheel-shaped rotating space station...

    [Continue in section #24, from the heading "The Coriolis Force"]


    Are astronauts weightless in space because they have gone beyond the reach of Earth's gravity?

      No. Even as far as the Moon, Earth's gravity is still felt, since that is what keeps the Moon in its orbit.

    If astronauts are not free from the Earth's gravity, why do they feel weightless in orbit?

      The way we feel gravity is by the way we interact with our surroundings. When an astronaut is inside an orbiting spacecraft, both of them move with the same acceleration g (or g(rE/r)2 at distance r). No other force acts, and therefore no force exists that pushes the astronaut towards the floor--or makes standing up require an effort.

    In a high dive into a pool of water, in the time between leaving the diving board and entering the water, is the diver weightless?

      Essentially, yes (air resistance interferes--but more so with skydivers). That is one reason why divers can perform intricate twists and somersaults. .

    One of the rides in the "Great America" amusement park in Santa Clara, California, is the "Drop Zone" (other parks have similar rides), centered on a 220-foot tower with vertical rails running down its side. Riding on each rail is a frame holding several seats. At the foot of the tower, riders are strapped into those seats and and then the frame is pulled up to the top of the tower.

    At the appropriate moment, the frame is released and plunges down, in what is essentially a free fall. At 1/3 of the tower's height, brakes become active and break the fall, so that by the time the ground is reached, all the falling speed is again lost. How many gravities do the riders feel (1) in the free-fall phase, (2) in the braking phase (assuming a constant rate (-a) of deceleration)?

      (1) Zero gravities. Free fall is a weightless state.

      (2) Suppose the tower's height is 3h: the riders fall a distance 2h with acceleration g, then stop within distance h with acceleration -a (here a negative number). You would guess the riders would have to decelerate twice as fast

      a = 2g

      producing on them an additional inertial force 2mg. That, however, is in addition to their own weight mg, making the total force 3mg, or 3 gravities.

      [ Optional] That guess is correct and here is a formal calculation: Suppose V is the velocity achieved at the end of the free fall. From conservation of energy

      2hmg = mV2/2     hence     V2 = 4hg

      The formulas for accelerated motion are like the ones for free fall, but with the acceleration (-a) in place of g. Suppose the braking motion takes a time t. Then

      vfinal = vstarting - at     or     0 = V - at

      From which:     at = V
      For the distance, using V = at
                  h = vstartingt - (1/2)at2 = Vt - (1/2)Vt
      or         h = 1/2 Vt
      Replace t = V/a     to get     h = V2/2a     or     V2 = 2ah
      and since also     V2 = 4hg,     a = 2g. .

    A space payload must achieve 8 km/sec to go into orbit. Can NASA get a "head start" on that motion by launching in the direction of the Earth's rotation? Is this equivalent to starting out with a velocity equal to the velocity of rotation at the Earth's surface, about 400 meter/sec near the equator? If not--why not?

      Launching in the direction of the Earth's rotation (namely, eastward) does provide an initial velocity equal to the surface velocity of rotation. If the Earth did not rotate, NASA would have to furnish all the 8 km/s, but because it does, rockets launched eastward get a head start. That is one reason why Cape Canaveral was chosen as the prime launching site, as was the European site at Kourou, French Guiana. Satellites launched into a polar orbit (e.g. from Vandenberg, California) lack this advantage. Israel's satellites have to be launched westward over the Mediterranean sea and require a boost larger than 8 km/sec. .

    A large low pressure area in the northern hemisphere has air flowing into it and swirling (as was seen in the lesson) counterclockwise. Suppose instead we have a high pressure area, with air flowing out. Will it swirl? And if so, in what direction? Give reasons.

      Arguments similar to the ones used for low pressure areas show that here swirling also does occur and is (north of the equator) clockwise. In meteorology, low pressure areas are sometimes called cyclones and their circulation is "cyclonic," while high pressure areas are anticyclones and their circulation is "anti-cyclonic" in the opposite sense.

    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 radiation emitted by hot objects ("black body radiation").
    • 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:

    1. 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.

    How thick is the chromosphere?
    • About 5000 kilometers.

    How does this thickness compare to the radius of the Sun?
    • It is less than 1% of the Sun's radius, which is about 700,000 kilometers.

    How does the radius of the Sun compare to that of Earth?
    • It is more than 100 times bigger.

    What are the approximate temperatures of the chromosphere and Corona, in degrees centigrade?
    • About 50,000° and 1,000,000°.

    Why is that so puzzling?

    • Because the only credible source of the heat is the Sun itself. However, heat always flows from a hot temperature to a lower one--never the other way around. The photosphere below these layers is cooler: how then can it heat the chromosphere and corona?

    How can we be sure that the corona is so hot?

    • The light from the corona is characteristic of atoms which have lost quite a few electrons (more about that, in a later lesson). At the density of the corona, this requires collisions between fast atoms, and the speed of the atoms of a gas is directly related to its temperatures.

    So, what can be the explanation of the heat of the corona?

    • The energy must arrive at the corona in a form different from heat--perhaps as sound-like waves, or as a population of fast ions produced by such waves.

    Any older explanations?

    • At one time it was proposed that a steady flow of small meteorites was hitting the Sun. However, nothing like them was ever observed passing near Earth.

    What is the solar wind?

    • A fast flow of ions, streaming out from the Sun in all directions.

    What produces the solar wind?

    • Its energy comes from the heat of the corona. Its top layers are so hot that they manage to escape the Sun's gravity.

      [If the corona were like the Earth's atmosphere, its temperature would gradually drop with height. However, the corona is hot enough to be a plasma, a mixture of free-floating positive ions and negative electrons. Plasmas conduct heat well, and this does not allow the high layers to stay cool.]

    How fast does the solar wind move, what is its density at the Earth's orbit, and what is it made up of?

    • Its velocity is about 400 km/sec, density at the Earth's orbit about 6 ions/cm3 , both these varying widely (at a distance of r astronomical units, the average density becomes 6/r2). Its composition resembles that of the Sun--mostly ionized hydrogen (protons), some helium and a small amount of heavier elements.

    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.

    Lesson Plan #38     http://www.phy6.org/Stargaze/Lsun3mag.htm

    (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

    "From Stargazers to Starships" home page and index:

    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.


    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:

    Period Viewed
    from 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)

    Lesson Plan #39         http://www.phy6.org/Stargaze/Lsun4spe.htm

    (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

    "From Stargazers to Starships" home page and index:

    Goals: The student will learn

    • How a glass prism resolves light into its rainbow components, which can be recombined to create white light.

    • The difference between spectral colors and the colors perceived by the eye.

    • That hot solids (or dense gases) radiate a continuous spectrum, related to their temperature.

    • That the colors of a glowing rarefied gas are characteristic of the atoms emitting them, with examples.

    • That cold atoms in a gas absorb the same colors as the ones they emit when hot, and that such absorbtion causes dark lines in the Sun's spectrum.

    • That light is a wave spreading through space, with a very short wavelength, or the order of a few microns (0.001 millimeter).

    Note: at this level, the student is told that the wave nature and wavelength are deduced from the way parts of a beam of light resonate with each other. The question of why these waves are called "electromagnetic" is left for the next lesson.)

    Terms: [Refraction of light, dispersion of light], spectrum, spectral color, perceived color, 3-color theory, black body spectrum, spectral lines, diffraction grating, wave, wavelength, absorption spectrum.

    Stories: Discovery of helium.

    Starting the lesson:

      (One way of introducing the subject of color is through the rainbow. Accordingly, unless time is short, the teacher may devote the beginning of the class to the phenomenon of the rainbow. The discussion here also extends into optics--the study of light.)
    Today we will discuss the colors of sunlight, but before we can talk about the Sun, we need to get an idea of what color is. In earlier times, this question confused many people--not just scientists, but also artists, who studied color for their own reasons.

    One striking phenomenon involving color is the rainbow. People generally divide the rainbow into 7 colors. Anyone knows what they are, in the order in which they appear?

    (Students may answer--put on board, make sure class copies). Red, orange, yellow, green, blue, indigo, violet

    (All of you know "indigo"? It is the name of a dye. What color are blue jeans?

    • dark ones are indigo, light ones are blue)

    We obtain the same sequence when light passes a prism--red is bent least, violet the most (draw on board). Anyone knows why? (students may answer)

    Because (draw on board, students copy) light hitting a flat surface of glass gets bent by an angle. (That is known as refraction--"The beam of light is refracted.")
      However, different colors are bent by different amounts. (That feature is known as the dispersion of light).

    When a beam of light hits a glass windowpane (draw on the board) with two parallel sides, the beam is refracted one way as it enters the glass, then the opposite way as it leaves. The net result is that it emerges with the same direction as before. So, inside the glass diffect colors may be separated, but when they emerge again they all move again in the same direction, and no separation remains.

    In a prism (refer to drawing already on the board), the entry side and the exit side are inclined, not parallel, and therefore the direction is changed--by a different amount for each color.


          How about rainbows? When do you see rainbows?

    • We see rainbows when the Sun is shining on falling rain. Typically, a rain cloud has passed over you and you are now in the sunlight again. (Students may also mention rainbows in sprays from large waterfalls or from garden sprinklers.)

    Has anyone noticed where in the sky the rainbow appears?

    • Always on the side opposite the Sun. In fact, it is part of a circle whose center is exactly in the opposite direction from the Sun. And the outermost color is always red.

    Now, the big question: why the colors?

      Let me give you a hint. When a beam of light goes through a windowpane (refer the drawing on the board), not all the light goes through. Some of it, instead of being refracted the second time, is reflected back, as if the back of the glass was a mirror. (add the reflected beam to the drawing on the board, a broken line.)

    Given this hint--does anyone know how the colors arise, or can you guess?

      Raindrops are tiny spheres (draw on board) and act very much like prisms. The light that is reflected inside is separated into colors, and when it comes out--at a point where the angle of the surface is different, just as it is in a prism--the different colors are separated.

    Sometimes you see a second, outer rainbow--fainter, larger, with red now the innermost color. Any guess how it gets produced?

    • By light reflected twice inside the raindrop
    . (end of optional section)

    How did Newton show that white light was a mixture of all colors?

    • He used a second prism to bring all the colors together again, and the result was white light, like the one which entered the first prism.

    In 1800 William Herschel, the astronomer who discovered the planet Uranus, made the following experiment. He split a beam of sunlight using a prism, and let the spectrum coming out of it--the spread of colors--fall on a table. (Draw on the board, adding details as described below.)

    He then placed a thermometer on the place where the spectrum fell, and noted that the temperature rose.

    He then moved the thermometer past the red edge of the spectrum, and noted that even there a higher temperature was recorded. What did this mean?

    • The heating of the thermometer in the middle of the spectrum meant that light carried heat (or in our terms, carried energy).

    • The heating of the thermometer past the edge of the spectrum suggested that light was shining there as well--a new invisible color which the eye could not see, but the prism was able to separate. Today we would call it infra-red ("infra" means "below"--related to "inferior")

    So we have now 7 colors in the rainbow, plus perhaps others which we cannot see (but instruments can). Now let us look at a TV screen.

    In a TV with a black-and-white picture tube, the screen contains many small dots which glow when a beam of electrons passes them. The beam sweeps over the picture many times each second, and by controlling its strength, each dot it passes over can be made to shine brightly, dimly or not at all. This way a picture is painted on the screen. The newer flat screens have no electron beams, but the glowing dots are directly connected electrically--otherwise it is very similar.

    Anyone knows how a color TV gets its color?

    • The screen of a color TV has 3 types of dots, each glowing in a different color, each connected separately to a source of electricity. At any time only one color is connected, and only it responds to the beam. Then the next color is switched on, then the third, and then the cycle repeats. The electrons therefore always paint, very quickly, a red picture, a yellow one and a blue one, and the eye sees the three combined.

    But then the eye can only see three possible colors! How can we see all colors of the rainbow?

    • Our eye's ability to distinguish color is limited. It contains three types of cell, each sensitive to a different range of colors. Any color--of the rainbow or otherwise-- is therefore seen as equal to some combination of the 3 "primary colors. " Color printers, too, need only use inks in three colors.

    Does combining yellow and blue produce green?

    • To the eye, yes. To the prism spectrograph, no.

    Does combining yellow and red produce orange?

    • (see preceding answer)

    Who discovered the "three color theory"?

    • Tell about James Clerk Maxwell.

    How did he show that 3 colors were sufficient?

    • A combination of the 3 colors produced white.

    What is a spectrum?

    • A distribution of colors, as obtained (for instance) by a prism.

    From now on we only discuss "spectral color" as seen by a prism instrument ("spectrometer").

    You are given a closed box with a hole, from which a beam of light emerges. You are given a spectrometer and asked to tell whether the light comes from a filament lightbulb or a fluorescent lamp. How can you tell?

    • The light of a glowing filament has a smooth continuous spectrum ("black body spectrum"), most intense at a color which depends on temperature.

    • The light of glowing gas, like that in a fluorescent tube, has a "line spectrum" concentrated in "spectral lines" of well-defined colors, characteristic of the gas. If you see them in the spectrometer, the light source in the box is a fluorescent tube; if they are absent, the light comes from a hot filament.

    (The fluorescent coating inside the tube absorbs light and re-emits it in a broad continuous spectrum, so even if you see spectral lines, you will also see a continuous spread.)

    The star Antares (in the constellation of Scorpio) is reddish, Sirius is white and our Sun is yellow by comparison. How would the surface temperature of the Sun compare to the ones of the two stars?

    • The Sun is hotter than Antares, not as hot as Sirius.

    When the battery in your flashlight runs down, the light it produces looks more orange than yellow. Why?

    • Because the battery cannot supply as much electric current as before, the filament in the lightbulb isn't as hot as it should be.

    Why are most flames yellow?

    • Heat easily makes sodium atoms emit a distinct yellow "spectral line" (actually two lines, very close in color). Firewood, paper, candle wax and most other combustible materials contain at least traces of table salt (sodium chloride), which colors the flame.
    (Here is the place to discuss neon lights, mercury and sodium streetlights, etc.)

    The polar aurora or "northern lights" comes from beams of fast electrons from space that slam into the upper atmosphere, at a typical height of 100 km. These electrons are guided by the Earth's magnetic field lines and therefore are generally observed only in an "auroral zone" around the magnetic pole, typically 2500 km from the pole. At locations like Fairbanks, Alaska, aurora is not all that rare. (See here for more about the aurora, and here about the auroral zone.)

    Light is emitted when those electrons collide with atoms, a bit like the way light is emitted when a beam of fast electrons inside a TV picture tube hits the screen. Different spectral lines are observed, but the brightest and most common one is a green line at 5.577 micron. A red line at 6.300 micron is also sometimes emitted.

    These emissions used to puzzle scientists, because they failed to match spectra in the laboratory. Did the upper atmosphere contain any new unknown substance?

    Around 1925, the answer was found--both colors came from oxygen. When an oxygen atom was activated ("excited") by a collision in a certain way, it took about half a second before it emitted light, an unusually long time. In laboratory experiments, the gas used was so dense that other atoms usually collided with the excited atom and carried away its extra energy. Only high in the atmosphere could the emission process proceed undisturbed.
    (end of the optional section)

    A spectrometer can use a prism to separate light into its other components. Do any other ways exist?

    • Yes, for instance the so-called diffraction grating. (Teacher may explain what it is and perhaps demonstrate.)

    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.)

    Lesson Plan #40     http://www.phy6.org/Stargaze/Lsun5wav.htm

    (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

    "From Stargazers to Starships" home page and index:

    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.

    What is a wave?
      The precise definition of a wave involves too much math (involving the so-called "the wave equation," which requires calculus), so instead let us take as a working definition of a wave "a disturbance spreading through space, usually carrying energy." It isn't completely accurate, but it covers the waves studied here.

    What kinds of waves to you know--other than light and its relatives? (List on the board the types proposed by students, then fill in gaps)

    • --Ripples forming on the surface of water, after a stone is thrown in. (Strictly speaking, those are surface waves, almost 2-dimensional.)

    • --Ocean waves. That also includes tides and tsunamis, a Japanese word for waves generated in the ocean by earthquakes (widely misnamed "tidal waves").

    • Sound waves, in air, water and solid materials.
          "Ultrasound" is sound too high-pitched for the human ear to hear (though dogs can hear some of its range).
          "Infrasound" are sound-like waves too low in frequency for the ear--for instance, the shaking of the ground when a heavy truck or train passes nearby.

    • Earthquakes propagate as waves in the solid earth.

    • Waves on a string such as on a guitar string, or a "slinky" toy, or a rope whose end is waved (and even a flag waving in the wind). These can be viewed as one-dimensional waves, spreading along a given direction only.

    • (Someone may name shock waves. Shocks are propagating disturbances and carry energy, but the equation they satisfy is not the wave equation, and they lose energy as they spread, so that scientists usually omit the "wave" designation. We never claimed our working definition was perfect!)

    Light belongs to the family of "Electromagnetic Waves," described more accurately later. Anyone knows other waves in this family? (List on the board, then fill gaps)

    • Infra red (IR) waves (also known as infra red light).
    • Ultra violet (UV) waves or UV light (also "extreme ultra violet").
    • Radio Waves.( AM, FM, TV are all ways in which radio waves are used to carry signals, not different kinds of waves.)
    • Microwaves
    • X-rays
    • Gamma rays

        (But not cosmic rays--these are particles. So are electrons and ions.
          Unless raised by a student, the following is better skipped, to save time and potential confusion. Strictly speaking, any moving particles are also associated with so-called "matter waves." Such waves, describing the particle's "wave function" in quantum mechanics, are of a different kind, giving the probability of the particle being observed at various locations.)

      (If someone mentions Alfven waves or "whistlers" these are among known types of plasma waves, many of which are electromagnetic waves modified by the presence of plasma, for instance, in space around Earth.)

      (If someone mentions "standing waves," these are waves of any kind confined between boundaries and therefore not traveling outside them. For instance, a wave on a guitar string, confined to the section between the bridge and the fret.)

    What properties of a wave can we measure?

    • Amplitude, the height of the crests, a measure of how strong it is; e.g. loudness of sound.

    • Wavelength l (Greek "lambda") the distance between two crests.

    • Wave velocity v, the velocity with which crests advance in space. The "velocity of light" at which electromagnetic waves spread in vacuum is usually denoted by the letter c.

    • Frequency f or n (Greek "nu"), the number of crests passing a fixed point each second

    How do we find the relation between the wavelength, wave velocity and frequency? (Teacher starts the explanation)

    The length v of the wave passing in one second through a given point in space contains many up-and-down swings, each of length l. Their total number in that second is.... (a student may try complete on the board, or else, teacher continues)

          (length of wave train)
          (length of one wave)
                              = v/l
          So....   f = v/l

    What difference does it make whether light is a wave or an array of "rays," of streams of "light particles"?

          --In principle, a big difference:

    • A ray propagates along a single line only.
    • A wave fills all of space,

    (Teacher explains) For many years it was believed that light consisted of rays. Then, however, scientists discovered interference--when two or more rays from the same source overlapped, they could reinforce or weaken each other--depending whether their crests overlapped (reinforced) or crest matched dip (=crest in opposite direction), in which case they were weakened.

    The diffraction grating is an example. It was then realized that light was a wave, but with a very small wavelength. Beams of light are well defined on distance scales much larger than a wavelength, but they "fuzz out" when one tries to define their boundaries within a wavelength or less.

    What is a magnetic field?

      (This should be familiar from section S-3 "The Magnetic Sun").
          A magnetic field is the region in which magnetic forces can be observed--forces on an iron magnet or on an electric current.

    What causes magnetic fields?

      Iron magnets and electric currents are also the sources of magnetic fields.

    (Teacher explains) An electric field is similarly the region where electric forces can be observed. You comb your hair on a dry winter day and find the comb is electrified--it can attract little bits of paper, the way a magnet attracts pins. In that case, an electric field exists around the comb.

      Similarly, friction among dry garments creates an electric field, which makes them cling to each other. In a laser printer or a xerox-type copier, a roller is electrified and finely powdered carbon sticks to it; later, under the influence of light, electric charge is removed preferentially, leaving it (and the carbon sticking to it) only where it contributes to the printed picture.

    All these electric fields persist, because the comb, fabric, roller and surrounding air are all insultating materials, which do not allow electric charges to move freely. On the other hand, an electric field in a metal wire that conducts electricity (=allows charges to move) will move charges and will create an electric current. That is how home electricity works.

      Because such currents expend energy, electric fields in a conductor quickly die out, unless they are maintained by an electric battery or generator, which provide a steady input of energy.

    Now the big question: can electric currents ever drive electric currents in a vacuum or in an insulator--currents that creat magnetic fields? Usually--no. However... (students answer, teacher supplements)

      James Clerk Maxwell in 1861 found that if one added to the basic equations of electricity (now known as "Maxwell's equations") a certain term that permitted such a current, those equations had, as one solution, a wave which advanced in vacuum at the speed of light.

      The additional term was only significant if the electric field changed very rapidly. When the electric field stayed steady, the extra term was absent. Maxwell showed that such a wave had properties actually observed in light, and proposed that the added term was indeed necessary, and that light was an electromagnetic (EM) wave.

    Light is usually produced by heat, or by glowing gases. If Maxwell was right, however, it should be possible to produce EM waves by purely electrical means. Can it be done?

    •   Yes, and that is how radio waves are produced--for instance, by a current bouncing back and forth, very rapidly, in a conductor we call antenna. Heinrich Hertz in Germany did something like this in 1886.

    What differences do you know between sound waves and electromagnetic (EM) waves?

    • EM waves in air travel nearly a million times faster than sound waves
    • EM waves are carried by rapidly varying magnetic and electric fields. Sound waves in air are carried by rapidly varying pressure changes.
    • EM waves are transmitted through electrical insulators but are stopped by conductors. Sound propagation through a material depends on its elastic properties, not on electrical ones.

    [Do not raise the following point unless some student does. In electromagnetic waves, the disturbances--the electric and magnetic forces which mark the waves--are transverse to the direction in which the wave travels--see animated figure from section (S-5). The shaking is directed sideways--like a wave on a rope. Sound waves in air are longitudinal, with the oscillating pressure force along the direction in which the wave spreads. However, sound-like waves in solids--e.g. earthquake waves--may also be transverse.]

    As an illustration of the 3rd difference above--how are windows on microwave ovens constructed--and why?

      --They are of glass, but with a metal mesh in them. The glass allows one to see the cooking food, but the microwaves are reflected back by the mesh and cannot escape.

    Most electromagnetic waves have a much higher frequency than sound waves--but not always. Ultrasound used in medicine and in cleaning small objects has a much higher frequency than the kind of sound we hear. And whistlers are electromagnetic waves in the same frequency range as regular sound, modified by the plasmas surrounding Earth and guided by the Earth's magnetic field lines--sometimes even bouncing back and forth along them, from one hemisphere to the other.

    Whistlers can be picked up by long wire antennas. They were occasionally heard along the end of the 19th century, on long-distant telephone lines, sounding like long whistles dropping in pitch. In World War I (1914-8) they were heard on telephone lines employed by the armies in the field, and in 1919 the German scientist Heinrich Barkhausen proposed they were coming from outside Earth.

    In 1953 Owen Story showed that they were broadcast from lightning strokes. Lightning emits a wide range of radio frequencies, as first shown in 1895 by Aleksandr Stepanovich Popov in Russia--indeed, its crackling is easily picked up by ordinary radios during thunderstorms. Storey showed that whistlers often started from lightning in the opposite hemisphere, near the other end of the observer's magnetic field line, whose attached plasma guided them. Whistlers have since then provided unique information about the density of ions and electrons in space around Earth.
    (end of optional section)

    Light and other EM radiations spread like waves, all over space. However, the way they give up their energy is distinctly not wavelike. Can you describe it?

    • Absorption of energy always occurs in well-defined chunks of energy, known as photons.

    What quality of light determines the energy of its photons?

      Its frequency n (nu)--or in other, equivalent terms, its wavelength l (lambda), or for light, its color (all three statements are equivalent). When the frequency gets higher, the wavelength shorter or the color moves from red towards violet, the energy of each photon also gets larger.

      Photons of blue light have more energy then those of red, which in turn have more than those of infra-red, and all these are less energetic than photons of ultra-violet. The energies of X-rays are even larger, and gamma-ray photons have energies larger still.

    What formula gives the connection between photon energy E and the light's frequency n?

    E = h n

      where h stands for a measurable quantity, a constant of nature, called "Planck's constant" because it was discovered by Max Planck in Germany in 1900. Today the chief scientific institution in Germany is the "Max Planck Institute."

    You have an electric circuit triggered by an "electric eye," a tube empty of air with metal contacts inside it. One is a plate of suitable metal, so that when light falls on it, it knocks out electrons which complete the circuit, causing an alarm to ring.
      You find that such a tube is triggered by blue light but not by red light. Can you guess the reason?

    • The red photons do not have enough energy to knock out electrons, whereas the blue photons do, because each of them has a higher energy. Note that it is the quality of the light that makes the difference, not the quantity. An intense red beam will not trigger the tube, but a faint blue one will!

    When developing film in a darkroom, no light is allowed, or else it would darken the exposed parts of the film. However, with old-type orthochromatic film it was permissible to use a deep-red "safelight" in the darkroom. Why was this safe?

    • Same reason as before. The red photons had too little energy to affect the light-sensitive material on the film.

    Why do glowing gases (e.g. neon lights) emit well-defined spectral colors?

    • They do because each of their atoms emits light independently, and can do so only after it is elevated (by collision or by absorbing light) to one of its higher energy levels. These levels (different for each atom) are well defined, and so is the final "ground level" of each type of atom in its usual (unexcited) state.

      When the atom changes from its high level to a lower one (or to its ground level), the amount of energy given to the resulting photon is always fixed, equal to the difference between the energy levels. Therefore the emitted color is also fixed.

    Optional: Lasers
        Atoms in a gas emit photons individually, and the peaks of the waves they produce are randomly distributed. On the other hand, in a laser, between tuned mirrors (and for appropriate energy levels) the atoms can match their emissions, the way crickets or frogs at night sometimes coordinate their sounds into a single chorus. These ordered ("coherent") light waves can be modified to carry signals through "optical cables" of super-transparent glass fibers, much more efficiently than metal cables can do so
    (end of optional section)

    Why do pictures of the Sun taken in X-rays show mainly the corona?

    • Because the speed of atoms in a hot gas, and therefore their kinetic energy, increases with temperature. Atoms in the photosphere and chromosphere move too slowly for their collisions to provide enough energy to create an X-ray photon--for instance, by exciting one of the high energy levels of an atom (or ion).

    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
                  (S-6A) Interplanetary Magnetic Field Lines

    "From Stargazers to Starships" home page: ....stargaze/Sintro.htm
    Lesson plan home page and index:             ....stargaze/Lintro.htm


    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
    •   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

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

    "From Stargazers to Starships" home page: ....stargaze/Sintro.htm
    Lesson plan home page and index:             ....stargaze/Lintro.htm


    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:

      1. The electric repulsion of its protons, trying to blow it apart.

      2. 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.

      3. 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.
    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.)

    1. 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.

    2. 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!

    3. 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.

    Back to IndexNASA Home PageGoddard Space Flight Center

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