How big is the Moon

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How Big is the Moon

INTRODUCTION. Pretend that you were living 500 or more years ago. With the technology available to you at that time - pretty much what you probably have around your present house other than books and computers and TV and radio - how could you determine how big the moon is? Answering this question requires you to do several projects - each one adding a piece to the puzzle.

  1. Puzzle Piece #1: What is the shape of the moon? You can see that the moon in the sky is at the very least a flat disk. If you use even a low power telescope, you will quite clearly see that the moon is a sphere - it is a disk with three-dimensional depth. So you conclude the moon is a sphere.

  2. Puzzle Piece #2: What is the shape of the earth? It would be nice if you could fly very, very high above the earth to look down to see what shape it is, but airplanes and rockets for doing so haven't been invented yet. It might be nice if we could use the sun's illumination to cast a shaddow of the earth on a wall somewhere out in space - at least that way you could see the shape of the earth's projection. But there is no such screen. Or is there? Yes, there is the moon every so often - during eclipses (a partially eclipsed moon is shown at the right), when a portion of the earth's edge is projected onto the lunar surface and we can see it from earth at night. And what shape is projected?

  3. Puzzle Piece #3: Is the moon larger or smaller than the earth? Here is a hint to help your thinking: can an elephant hide behind a twig? Can a twig hide behind an elephant? In other words, can a larger moon hide from the sun behind a smaller earth, or is it the other way around? Which of the two figures here (Fig A or Fig B) would be the case if the moon were larger than the earth?

  4. Puzzle Piece #4: Quantitatively, how much larger is the earth than the moon? Now that we crudely know that the moon is smaller than the earth, we can be more exact and calculate "relatively" how small it is. Is it 50% the diameter of the earth? 80%, or what? To figure this out you should have an enlargement of a partially eclipsed moon so that you can make comparisons of the two curves you see - the lunar edge and the shaddow of the earth's edge. Using some classic Euclidean geometry, you can determine the radii of both circles, and from those you can calculate the relative volumes of both the earth and the moon. You do this by drawing any two tangents (dictionary?), and then drawing perpendiculars at the midpoints of each tangent (white lines on the lefthand figure). Where the perpendiculars intersect is the center of the earth's circle. The radius is from the center to earth's circle - shown here by the arrow. Using the same magnification of the eclipsed moon, do the same using the moon's circular edge and determine the radius of the moon (red lines on the righthand figure). Now divide the earth's diameter by that of the moon's diameter to get a fraction that is how many times larger the earth is than the moon. The only thing to do is calculate the volumes of the two spheres, and then make a proportion. (The volumes will proportional to the cubes of their radii.)

  5. Puzzle Piece #5: How large is the earth? There is a way that you can do this using methods that were known more than 500 years ago (actually more than 2,000 years ago). This is easiest to do if you find a "correspondent" - a person with whom you can communicate, who lives exactly north or south of you. The further apart you are the more precise your answer will be. At exactly the same time both of you determine how high the sun is in the sky, or the lengths of the shaddows cast by equally high stakes set absolutely vertically in the ground. Using trigonometry, calculate angle α. You now have your latitude. Next you must determine as precisely as possible the distance between the two poles (that of your friend and your own pole).

a sequence leading up to totality

A Few Questions

  1. During total eclipse, how is it that you can still see the moon? Why is the moon rusty colored?
  2. When the moon is halfway to total eclipse, why is the earth's shaddow fuzzy rather than a sharp line? (Hint: see figure to the right, and label what parts of the diagram are umbra and penumbra.)

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