Astronomy Specials:

The jump Aristarchus made from terrestrial measurements of scale to the measurement of distances in the heavens was courageous!

    No laser ranging experiments. He (possibly) knew about the Earth (its size was known from the measurements by Eratosthenes). He also used a bit of clever geometry. Then by watching the Moon travel through the Earth's shadow (a lunar eclipse), he determined its size and distance.

For our purposes, let's assume that the sun is REALLY far away. Aristarchus actually used a clever trick to show that the sun is many times further then the moon. In reality it is ~ 400 times further:


In the picture above you can use a trigonometry relationship to find the relative distances: Sun distance / Moon distance:

sin (1/7)                    sin (89.86)
----------        =         --------------
Dist. moon            Dist to the sun

From which:

Dist. to the sun        sin (89.86)
------------------    =  -------------    = 400
Dist to the moon      sin (1/7)

Note: Aristarchus used a slightly more complicated method, because trig was not invented yet!

Part 1: If the sun is far away, the shadow of the earth at the moon's orbit is similar to the size of the earth:

Imagine that in the picture above the sun is ~100 times further to the left. What happens to earth's shadow near the moon? The size of the shadow WILL BECOME SIMILAR TO THE SIZE OF THE EARTH.

Aristarchus knew that the moon moves eastwards when compared with the stars. He correctly hypothesized that this was due to the motion of the moon around the earth roughly once a month. He also knew that on average twice a year, the surface of the full Moon became dark for a period ranging up to about 3.5 hours.

He correctly reasoned that during these eclipses of the Moon, the Moon was passing through the shadow cast by the Earth. This is called a lunar eclipse. The eclipse would only occur during the full Moon because that's the time when the Moon is on the opposite side of the Earth from the Sun. It didn't occur every time because the Moon's orbit and the Sun's orbit were tilted slightly relative to one another, so sometimes the Moon would pass above or below the Earth's shadow rather than right through it. Aristarchus realized that the longest lunar eclipses must occur when the Moon passes right through the center of the Earth's shadow.

So, in 3.5 hours the moon moves a distance equal to the diameter of the earth. In a month it moves (2 hours x 12)  x 27 days (Lunar month) = 192 earth diameters.

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The relationship between the lunar orbit circumfrance and the distance to the moon is simply:

2 x pi  x Distance = Circumfrance

Where pi=3.1412

Since the circumfrance is 192 earth's diameter, the distance is:

Distance = Circumfrance / (2 x pi)
             = 192 earth diameters / (2 x 3.1412)
             =  30 earth diameters

Using the result derived at by Eratosthenes for the diameter of the earth of 1 earth diameter ~8000 miles (or 12,000 km), we get that the distance to the moon is 240,000 miles. That's very close to the modern value of 238,906 miles!!!

Now, using the first ratio in this page, the solar distance  over the lunar distance = 400, we get the distance to the sun as:

400 x 240,000 = 96,000,000 miles

This is incredibly close to the 93 million miles accepted value!

Also, note that we can easily find the sizes of the moon and sun now: