January 8 – First Quarter, First Heretic

Today’s Factismal: The Sun is 389 times farther away than the Moon.

One of the first astronomers was also one of the first heretics. Back around 260 BCE, a Greek astronomer by the name of Aristarchus discovered a way to measure the distance to the Sun and Moon and was nearly exiled for his innovation. Today, you can honor Aristarchus by making his measurement for yourself.

As his contemporaries did, Aristarchus realized that the Moon orbited the Earth and that its phases were related to its position relative to the Sun and Earth. To understand this, think about a little kid running around his mother while she walks around a campfire. When the kid is between the fire and the mom, all the mom can see is the kid’s shadow; this is the New Moon. When the kid is behind the mom, she can see his smiling face; this is the Full Moon. And when the kid is halfway around, she can see half of the kid and the other half is in shadow; this is either the First Quarter or the Last Quarter, depending on how far around the kid has gotten.

The Greeks knew that the phases of the Moon depended on where it was in relation to the Sun

The Greeks knew that the phases of the Moon depended on where it was in relation to the Sun

But Aristarchus took this one step further. He realized that the Sun, Earth, and Moon formed an ever-changing celestial triangle and that the angles of that triangle could tell us something very interesting about our universe. Aristarchus knew that when the Moon appears to be half-way illuminated, then it must make a right angle with the Sun. He then reasoned that the ratio of the distances between the Earth and the Sun (S on the diagram) and the Earth and the Moon (L on the diagram) could be found from the angle between the Sun and the Moon as seen from Earth (θ on the diagram). And, by constructing a similar but smaller triangle here on Earth, he could measure that ratio. And that is exactly what he did.

When the Moon is in the First or Last Quarter, it makes a right angle to the Sun. Apply a little geometry and you know how far away the Sun is!

When the Moon is in the First or Last Quarter, it makes a right angle to the Sun. Apply a little geometry and you know how far away the Sun is!

On a day much like today, Aristarchus went out and looked up at the Moon. He carefully measured the angle that it made with the Sun (he measured it to be 87°; modern measurements make it closer to 89°, 50’), and then built a triangle to match the three angles (87°, 3°, and 90°). When he was done, he found that the Sun had to be between 18 and 20 times farther away than the Moon. If you do that using the more accurate modern values, you’ll discover that the Sun is really 389 times farther away!

Aristarchus went on to use eclipses to determine the actual distance to the Sun and Moon. From that, he was able to infer their sizes, and found that the Sun had to be much, much larger than the Earth, which was bigger than the Moon. Since it didn’t seem reasonable to him that something so big would circle something so small (relatively speaking), he decided that the Earth actually orbited the Sun and went around telling people of his idea. The local philosophers didn’t think much of it, and tried to get Aristarchus banished for impiety. But his logic was unassailable and so they decided that he had just measured something wrong and let the matter lie there. It would take another 1,700 years before the heliocentric model of the universe became popular – but that’s a factismal for another day!

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