PhysicsLAB Lab
Determining the Distance to the Sun

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The word "moon" derives from a Germanic word meaning "month" or "measure of time." Early man relied on the consistency of the Moon’s motion and phases to predict such things as the changing of seasons, the movement of animals, and the changing of weather. The phases of the Moon are created by the relative positions of the Earth, Sun, and Moon. We now know that the moon's orbital period exactly matches its rate of rotation; both being, 27 days, 7 hours, and 43.1 minutes. This results the same face of the Moon always facing the Earth. It wasn't until the Apollo missions that man first saw the "far side" of the Moon.
Among the first scholars in the Western world to offer a scientific explanation for the Moon was the Greek philosopher Anaxagoras (d. 428 BC), who reasoned that the Sun and Moon were both giant spherical rocks, and that the latter reflected the light of the former. His atheistic view of the heavens was later cause for his imprisonment and eventual exile. He was later honored by having a crater on the Moon named after him, Crater Anaxagoras.
The Greek mathematician Aristarchus (310-230 BC) proposed that during a half-full moon the Sun’s light would be perpendicular to the Moon.

image courtesy of NASA
By measuring the shadow cast toward the half-full moon at the Sun’s zenith you can depict the triangle formed by the Earth-Moon-Sun system as shown below.
Aristarchus measured the angle to be 87º and then used simple geometry to determine the distance from the Earth to the Sun, provided the distance from the Earth to the Moon was known.
Refer to the following information for the next five questions.

Begin by constructing a right triangle with one of its angles being 87º. You may NOT use the above picture as it is not to scale. Call the 90º angle, C; the 87º angle, B; and the 3º angle, A. Now measure the lengths of each of its sides.
side a 

side b 

hypotenuse c 

Using right-triangle trigonometry and the fact that the actual distance from the Earth to the Moon is 3.844 x 102 Mm, determine Aristarchus’ distance to the Sun. 

The actual angle of the sunlight towards the half-moon at the Sun’s zenith is 89.85º not 90º degrees. Describe the significance of this "missed" angle measurement.

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