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Astro-Marathon Problem 5 (Special)

“Aristarchus’ Method of Determining the Distance to the Moon”

(This is  astandard BDOAA/IOAA style problem you will has on Geometry section. If you can solve these types of problem it will be very beneficial for you to grab more marks!)

Use Chis Cook's composite photograph of a lunar eclipse to determine the radius of the moon and its distance from the earth (in units of the radius of the earth). The sketch below illustrates the appropriate geometry to use. Make use of small angle approximations

a) Assume that only the darkest part of the Earth’s shadow (umbra) corresponds to total eclipse. Draw a circle (with a compass if you have one) that best represents the umbral shadow.
b) Note that the center of the shadow does not lie on the line connecting the path of the center of the moon. Explain why not.

c) Measure the diameter of your circle as well as the diameter of one of the lunar images.

d) Estimate the percentage error by which you might have under– or overestimated the size of the earth’s shadow.

e) Compute the radius of the moon compared to that of the earth. Be sure to take into account the proper geometry of the umbral shadow at the distance of the moon (For this purpose you can take the angular size of both the sun and the moon to be 1/2◦.) Estimate the uncertainty in your answer. Compare to the known value of 6378/1738 = 3.67.

f) If the angular size of the Moon is 1/2◦, calculate the Earth–Moon distance, D, in terms of the radius of the earth, R⊕.

 

Note that there are two parallel vectors, displaced by one Earth’s diameter, both pointing toward the center of the Sun. This is a reasonable approximation since the diameter of the Sun is ∼ 100 times that of the Earth.

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