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Relative sizes

We previously determined that the diameter of the earth is 12742 km. Now we want to determine the size of the moon relative to the earth.

Image Credit: NASALet’s start with an even simpler question: Is the moon bigger or smaller than the earth? How could you prove this? Remember this question wouldn’t seem silly to someone standing on the Moon!

The world's first view of Earth taken by a spacecraft from the vicinity of the Moon. The photo was transmitted to Earth by the United States Lunar Orbiter I. This crescent of the Earth was photographed August 23, 1966. Image Credit: NASAThis question was tackled over 2300 years ago by Aristarchus of Samos (310-230 BC). His measurement begins with the observation of a total lunar eclipse. He correctly assumed that this was the result of the earth casting a shadow on the moon.

Total lunar eclipseFirst Aristarchus timed how long the moon took to travel through the earth’s shadow. As Eratosthenes did, he also made a simplifying assumption that the sun’s light reaches us in perfectly parallel lines. Which we know isn’t exactly true as two cone shaped shadows are produced: umbral and penumbral. Though it was close enough for a nice estimation.

Penumbral and umbral shadows. Image credit: Saregd

He was actually timing how long between the entry and exit of the darker (penumbral) shadow. Which to him appeared to be around 2.6 hours. Next, he compared this with the time it takes the moon to move a distance equal to it’s diameter. In the previous article we determined that this was about 1 hour. So he’s left with two numbers:

  • time it takes the moon to travel 1 moon diameter = 1 hour
  • time it takes the moon to travel 1 earth diameter  = 2.6 hours

How could we figure out the size of the moon from this?

If the times were the equal than it would imply that the moon is the same size as the earth. However the time of a lunar eclipse is much longer, which means the earth must be larger. How much larger? We need to setup a basic proportion:

time #1 / time #2 = moon diameter / earth diameter

1 / 2.6 = moon diameter / earth diameter

This led him to claim that the earth was about 8/3 the diameter of the moon. This is pretty close to the actual difference: the earth is about 3.7 times bigger than the moon. The main reason he underestimated was because the umbral shadow is narrower than the earth.

Actual size of the moon

We now know the relative size of the moon compared to earth. We also know the actual size of the earth from a previous calculation. Finally we can determine the approximate size of the moon!

moon diameter = earth Diameter / 3.7

moon diameter = 12742/3.7 = 3443 km

This is very close to the actual diameter of the moon: 3474.8 km

We determined the size of the earth and moon using nothing but shadows!