Lesson Plan #16     http://www.phy6.org/Stargaze/Larist.htm

(9a)     May Earth be
Revolving around the Sun?

(9b) The Earth's Shadow    

  These two sections belong together (9b is an elaboration which may be omitted). This lesson traces the beginning of the heliocentric theory of the solar system--the idea that the solar system revolves around the Sun--to an observation by the Greek astronomer Aristarchus, which convinced him that the Sun was much bigger than the Earth.

Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern

This lesson plan supplements: "Aristarchus: Is the Earth revolving around the Sun?," section #9a : on disk Sarist.htm, on the web

"From Stargazers to Starships" home page and index: on disk Sintro.htm, on the web

Goals: The student will

  • How Aristarchus used the position of the half-full Moon to estimate the distance to the Sun, and the results he obtained.

  • Another application of "pre-trigonometry," to a 3° triangle.

  • The strange ways of scientific progress: Aristarchus made a great error--yet his final conclusion, that the Sun is much larger than Earth, still held true.

  • [If (9b) is included: The fact the Sun covers a 0.5° disk in the sky means that its rays cover a cone of 0.5° opening angle, and therefore the shadow of the Earth is also such a cone.
    At points more distant than the tip of the cone, no shadow exists, for the Earth is not big enough to cover all of the Sun.]

Terms: (none new)

Stories and extras: The entire section is a story, of how Aristarchus was (probably) led to his heliocentric theory.

    Start the class by discussing what is a scientific discovery? Historians of science often argue about "who was first"--but what does it mean?

   The Greek philosopher Democritus argued that all matter consisted of "atoms", a Greek word meaning "undividable. " He pointed out that a collection of very small particles--e.g. sand or poppyseeds--can be poured like a continuous fluid, so maybe water, too, consists of many tiny "atoms" of water. Does this qualify as a prediction of the atomic theory of matter?

   In the early 1700s, the Irish writer Jonathan Swift wrote "Gulliver's Travels, " a satire of the politics and society of his times, in the form of voyages to distant fantastic countries (today it might have been called "science fiction.") In his third voyage he visits an island floating in the air, which is ruled by an academy of scientists (a spoof on the "Royal Society", an association of Britain's top scientists which still exists). He reports that by using improved telescopes, members of the academy had discovered that two small moons orbiting Mars at a close range.

   A century and a half later, an astronomer discovered that Mars indeed had two such satellites, quite similar to what Swift had described. Does it mean that Swift had predicted those moons?

   By our standards, these are just lucky guesses. To qualify as a prediction, a claim needs not only to be stated, but also justified, it needs a logical reason. In this lesson we discuss a proposal by Aristarchus, around 270 BC, that the Earth went around the Sun, rather than vice versa. It took 1800 years before this claim was made again, and another century before it was generally accepted.

   However, this was not guesswork. Aristarchus, who also estimated the distance of the Moon, had a serious reason for his claim: the Sun, he showed, was much larger than the Earth, making it likely that the Sun, not Earth, was at the center.

   Let us go through his arguments.

Give the material of section 9a of "Stargazers. Start by assuming that the shadow of the Earth had the same width as the Earth, and that the Earth had twice the width of the Moon. Later, if time and the level of the class allow it, the teacher may continue with a discussion of the actual shadow of the Earth, which is cone-shaped [Section 9b].)

Guiding questions and additional tidbits

-- Who was Aristarchus of Samos?

    Aristarchus was an early Greek Astronomer, living between 310-230 BC. Samos is a Greek island.
        [The teacher may point out that dates BC seem to proceed in the opposite direction to what we are used to--e.g. born -310, died -230.]

-- What did Aristarchus establish about the Moon?
    He was the first to estimate its distance, about 60 Earth radii, 380,000 km or 240,000 miles.

-- What was the revolutionary proposal Aristarchus made about the Sun?
    Two correct answers exist here:
            That the Sun was much bigger than the Earth
            That the Earth went around the Sun, not vice versa

-- On what observation did Aristarchus base his claims about the Earth?
    Aristarchus tried to see where the Moon was, relative to the Sun, when it appeared to be exactly half-full.

-- What is the Moon's relation to the Earth and Sun, when it is half-full?
    When the Moon is half full, the angle Sun-Moon-Earth (corner at the Moon) must be exactly 90°.

-- What does the Sun-Earth-Moon angle (corner at Earth) at such times tell about the Sun's distance?
    You can measure that angle, for instance, if the half-moon is visible in the daytime, as often happens. It allows one to construct the full Earth-Sun-Moon triangle.

    [Draw diagram of the triangle on the blackboard.]
    If the Sun is very, very far away, the Sun-Earth-Moon angle is also be very close to 90°. In fact, that is the case: the amount by which that angle differs from 90° is too small to be reliably measured. The only thing one could conclude from it is that the Sun was very distant.

    As it happened, the measurement made by Aristarchus was inaccurate. It is hard to tell when the Moon is exactly half full! He believed the Sun-Earth-Moon angle was 87°, short of 90° by 3°. The Earth-Sun-Moon triangle then has a sharp corner of 3 degrees, and its proportions were such, that the Sun was about 20 times further than the Moon.

-- If the Sun is 20 times more distant than the Moon, what does it say about the Sun's size?
    Since the Sun's size in the sky is about the same as that of the Moon, it must also be 20 times bigger in diameter.

-- What did Aristarchus believe about the relative size of the Earth, compared to the Moon and Sun?
    From observation of the Earth's shadow during an eclipse of the Moon, he concluded that the Moon had half the diameter of the Earth (Actually, it is less than 1/3 that diameter). By his estimate, therefore, the Sun's diameter was 10 times that of Earth (in reality, it is more than 100 times larger).

-- How did Aristarchus view the Sun-Earth system?
    He guessed the Earth went around the Sun--probably, because the Sun was very much bgger. Others at the time claimed the Sun went around the Earth. After all, astronomers (including Aristarchus) had shown that the Moon went around Earth, so why not the Sun, too?

--Did other astronomers agree with Aristarchus?
    No, they continued to disagree.

--What was their argument? (teacher may help fill details)
    Aristarchus determined that the Sun's distance was 20 times that of the Moon, or about 1200 Earth radii. If the Earth went in a circle around the Sun, at that distance, its positions half a year apart would be a full diameter apart, 2400 Earth radii, about 10 million miles.

        To the ancient Greeks, that was an enormous distance. The stars were clearly more distant than the Sun, but it was hard to imagine that from two positions so far apart, there should not exist some difference in the apparent positions of stars in the constellations of the sky. Yet none could be seen.

        That was the argument of Hipparchus and Ptolemy, leading Greek astronomers. Their view prevailed for about 1800 years.

--Was there a flaw in that argument?
    The argument was valid. It would have been even stronger, had it been realized that the diameter of the Earth's orbit was about 20 times larger than in the estimate by Aristarchus.

        However, the stars were much more distant than anyone had held possible. Any shift in their positions was too small to be observed by the eye.

        (Such a shift was first observed in 1838, using some of the best telescopes of the era, and even then, only for the stars nearest to us. See the section on parallax.)


(9b) The Earth's Shadow [optional]

       This is a detail that may be skipped in the classroom, only perhaps assigned as a project to advanced students.
       One should start it by making clear that the Sun covers a 0.5° disk of the sky. If we select some point P on Earth and trace all the sun's rays that reach it from that disk, those rays form a narrow cone.
       That cone contains all the directions in which the Sun's rays arrive at the Earth's vicinity, and the full shadow of the Earth only extends over the region where all those directions are blocked by the Earth.
       It will only extend a certain distance behind Earth. At greater distances, the Earth will cover less than 0.5° of the sky and will appear smaller than the Sun. At those distances, one can never be in the full shadow of the Earth.

        The Lagrangian L2 point, 236 Earth radii from Earth on the side opposite from the Sun, is a good location for a distant observatory. Being more distant from the Sun, it should orbit it more slowly than Earth, but because of the added pull of the Earth, it can move a little faster and thus keep up with Earth (more about Lagrangian points is in the last part of "From Stargazers to Starships").

        NASA plans to place its next large infrared telescope at this position. It would be just outside the shadow cone. The Earth would be a little too small to cover all the Sun, which would shine in a bright ring around the dark Earth. In full shadow, the telescope would get very cold--a desirable feature for detecting infra-red light. As it is, it will need a light shield to protect it from the remaining sunlight.

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Author and Curator:   Dr. David P. Stern
     Mail to Dr.Stern:   stargaze("at" symbol)phy6.org .

Last updated: 12 September 2004