Aristarchus' method of estimating the size and distance of the Moon

Aristarchus, all the way back in ~300 BCE, measured the angular separation of the Sun and the Moon when seen during its first or third quarter phases.

*NOT TO SCALE

He measured the angular distance between the Moon and the Sun and determined it was "one-thirteen of a quadrant" (~3 degrees) less than vertical 90 degrees, or  ~87 degrees.  From this he determined that the length to the Sun (S) was about ~19-times longer than the length to the Moon (L).

*The true value of this angle is closer to 89.85(0) degrees, and so the Sun's distance is actually about 389-times the Moon's distance to the Earth. As you can see, only a small 2.85 degree (3.17%) error in the angle estimate resulted in a ~20-fold error in the distance. Yikes! This probably is a result of the fact that 1) it's hard to know exactly when the Moon is half-full and thus creating the exactly 90 degree angle, and 2) he had to estimate the relative depths to the center of the Earth, the Moon and the Sun and, as we know, Aristarchus could've have possibly imagined how huge the Sun was prior to determining its distance.

Never-the-less, Aristarchus did show that the Sun was very, very far away (or much farther away than the Moon).

Aristarchus knew that occasionally, the Moon would pass entirely through the shadow cast by the Earth: a total lunar eclipse.  He saw this as an excellent opportunity to estimate the radius of the Moon.



First he measured the time it took from the moment when the edge of the Moon first entered the full shadow (the umbra) and the moment when the Moon was first totally eclipsed. At the same time, Aristarchus also measured the total duration of eclipse.  He found that both times were the same, so the Moon must be only half as wide as Earth's shadow.  In other words, he concluded that the diameter of the Moon was ~0.5 times diameter of the Earth.  

Aristarchus also knew that during a total solar eclipse, the Moon's body nearly perfectly covered the Sun.  He cleverly reasoned that since he had earlier shown that the Sun was 19.1-times farther way from Earth than the Moon, that since they are nearly the same size in the sky, the Sun's radius must be ~19.1-times larger than than the Moon's to keep their apparent sizes seen from Earth correct.

Drawing what he felt was the correct conformation of a total lunar eclipse, he reasoned that:

Since (s / ℓ) = 19.1, and (d / ℓ) = 2:

(1 + (ℓ / s)) / (1 + (d / ℓ)) = (ℓ / t)
(1 + (1 / 19.1)) / (1 + 2) = (ℓ / t) = 1 / 2.85

In other words, the Earth's diameter was 2.85 times the diameter of the Moon. The actual ratio is about 3.7.

Again, we see that Aristarchus' mistake measurement error for the correct angle on the quarter-moons angle brings the 19.1 /1 (S / L) ratio (versus the actual 389 /1 ratio) and applies the same errant ratio to (s / ℓ), which would also be ~389/1.  Small errors, especially ones applied trigonometry, lead to big errors.

He also determined that:

(1 + (s / ℓ) / (1 + (d / ℓ)) = (s / t)
(1 + 19.1) / (1 + 2) = (s / t)  = 6.70

In other words, Aristarchus estimated that the Sun's radius was 6.7 times larger than the Earth's radius.  The real ratio is 109.


[List of Aristarchus' Propositions]

[Aristarchus' On the Sizes and Distances]

A solid effort, none-the-less. Later Hipparchus did a bit better.  Giovanni Cassini later complete the job by using the observed parallax of Mars to determine the distance of all the planets in the Solar System.

 Below are the actual distances:

The Moon is 384,400 km (238,900 miles), on average, from the Earth. This varies because the Moon is in a elliptic orbit around the Earth. [image]

The Moon-Earth distance is actually increasing by ~38 mm per year because of a phenomena known as tidal-locking.

[The Earth Moon distance drawn to scale]

The Earth is about 149,597,870.700 km (92,955,807.273 miles) from the Sun