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

"From Stargazers to Starships" home page and index: on disk Sintro.htm, on the web
          http://www-spof.gsfc.nasa.gov/stargaze/Sintro.htm

• 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.

   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 we might have called it "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].)

-- 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.]

He was the first to estimate its distance, about 60 Earth radii, 380,000 km or 240,000 miles.

Two correct answers exist here:
        That the Sun was much bigger than the Earth
        That the Earth went around the Sun, not vice versa

Aristarchus tried to see where the Moon was, relative to the Sun, when it appeared to be exactly half-full.

When the Moon is half full, the angle Sun-Moon-Earth (corner at the Moon) must be exactly 90°.

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.

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.

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).

He guessed the Earth went around the Sun, not the Sun around the Earth, as others claimed at the time, and continued to do for the next 1800 years.

   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 distance, one can never be in the full shadow of the Earth.