Rung 2: The distance to the Moon

The distance to the Moon was determined by first finding the size of the Moon relative to the size of the Earth. This determination of the relative sizes of the Earth and Moon predated the estimate of the absolute size of the Earth due to Eratosthenes and was first carried out by Aristarchus of Samos (310-230 BC). Once again a model is required to make the determination of the relative sizes of the Earth and Moon, in particular a good model for what is taking place during a lunar eclipse. It was surmised that during a lunar eclipse the full Moon is passing through the shadow of the Earth. Aristarchus timed how long the Moon took to travel through Earth's shadow and compared this with the time required for the Moon to move a distance equal to its diameter (this could be done by timing how long a bright star in obscured by the Moon). He found that the shadow was about 8/3 the diameter of the Moon. The model is extended from the above because we need to add the Moon:

2. The Sun is very far away.

3. The Moon orbits the Earth in such a way that eclipses can occur.

Question 6: How is the second statement important to the measurement of the relative size of the Moon to the Earth? (Hint: Consider what would happen to the shadow of the Earth if the Sun was placed very close to the Earth.)

Question 7: What evidence did the ancient Greek Astronomers have that the Moon orbits the Earth? (Give as many as you can think of.)

Question 8: How much time does it take the Moon to move 0.5 degrees in the sky? This is the time that the Moon requires to sweep out its own diameter on the sky. To determine this remember that it takes the Moon 28 days to sweep out 360 degrees (once around the Earth). Use the following ratio to determine this:
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Time for the Moon to move its own diameter on the sky       0.5 degrees
-----------------------------------------------------   =   -----------
Time for the Moon to go 360 degrees (in hours)            360 degrees

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• Let's put this timing method to work.

Question 9: How much time did it take the Moon to sweep out Earth's shadow during the lunar eclipse (take the difference of the two times you recorded)? Divide this time by the time it takes the Moon to sweep out its own diameter, which you determined above. This number is approximately how much bigger the Earth is than the Moon.

 Although Aristarchus used a timing method, we can get a crude estimate of the relative size of the Moon to the Earth by looking at the curvature of the Earth's shadow during a lunar eclipse. All of the above model statements are still important to this conceptually simpler method. Caveat Corner: Umbra and Penumbra during a Lunar Eclipse

• Now use the curvature of the Earth's shadow during a lunar eclipse to determine the relative sizes of the Earth and Moon.

Question 10: Why would this method have been difficult for the ancient Greek Astronomers to perform, and why was the timing method more favoured?

Question 11: How does your answer using the timing method compare to what you got by looking at the curvature of Earth's shadow during the lunar eclipse?

Question 12: Use the radius of the Earth that you determined from rung 1 of our distance ladder along with the relative size of the Moon determined to find the radius of the Moon in kilometers.

Radius of the Moon: _____________________ km.

Once we have the absolute diameter of the Moon we can easily determine its distance from the Earth by measuring its angular diameter on the sky. The angle subtended by the Moon on the sky is about 0.5 degrees.

Question 13: Calculate the distance to the Moon using its absolute diameter and its angular diameter. For help with the geometry of this problem and with the trigonometry involved with this calculation, take a peek at this hint.

Distance to the Moon: _____________________ km.