Measuring the Moon: Unlocking Ancient Geometry During a Lunar Eclipse
The total solar eclipse back in April was marvelous and hopefully many of you were able to make your way into the path of totality to experience it. Total solar eclipses are pretty tough to beat! We still have two eclipses remaining in 2024. While neither of them is going to be spectacular, we can still have some fun with at least one of them.
On the night of Tuesday, Sept. 17th, a partial lunar eclipse will be visible from pretty much the entire western hemisphere. Parts of Africa and Europe will see the eclipse during the pre-dawn hours.
You can use Starry Night or SkySafari to tell you the exact time for your location. Select the Moon and then go to the Info button. Look in the Visibility section and you will see the date and time for the next lunar eclipse visible from your location. Hit the little clock icon on the far right and Sky Safari will automatically jump straight to the exact second of maximum eclipse.
This will be a very partial lunar eclipse. Even at maximum eclipse, Earth’s shadow will cover only 8% of the Moon’s disk. That is probably not enough coverage to see even a slight change of color. On the positive side, the partial eclipse will last just over an hour.
The other big positive is that we can do some fun things with partial lunar eclipses. Partial lunar eclipses are how we first measured the diameter of the Moon. Aristarchus succeeded at this way back in 240 B.C. The Greeks saw geometry as the pinnacle of human understanding, so it is no surprise that they were the ones to figure out the geometry behind how to measure the Moon’s diameter.
During the partial phase of a lunar eclipse is when we can see the Earth’s curved shadow in the sky. The Earth’s shadow is larger than the Moon, so we will only ever see a small part of the Earth’s shadow cast onto the Moon. However, if you take a photo or make a careful sketch, you can extend the curve from the small part of the Earth’s shadow that we do see and get a circle which represents Earth’s entire shadow (see the image below).
Fair warning: finding the circle which represents the exact size of Earth’s shadow can be tricky because the Earth’s shadow is fuzzy rather than a clear, distinct line. That is the key piece to the entire process, though. If you can find the right fit, all of the following measurements and results should come out well.
Aristarchus had to sketch the Moon and Earth’s shadow, figure out a circle similar to the one shown above and then take measurements from his sketch. Thankfully, photography makes this process a lot easier for us.
So how do you get from this sketch/photo to the diameter of the Moon?
Begin by making two measurements:
- The diameter of the Moon’s disk in your image.
- The diameter of Earth’s shadow from your image (the yellow circle in the image above).
Use whatever units you like - millimeters if you work from a printed image or you can use software to measure the two diameters in pixels.
Once you have these measurements, calculate the ratio of the Earth to the Moon:
The next thing to consider is that the Earth’s shadow is not the same size as Earth itself because the shadow tapers off (narrows) the further from Earth you go. How do we adjust for this before continuing our calculations? Remember that during a total solar eclipse, the Moon’s shadow tapers down to almost a single point. Earth’s shadow tapers by a similar amount during a lunar eclipse. Without going too deep into the geometry, we correct for this by simply adding 1.0 to our ratio (the equation below includes this adjustment).
At this point, we only have one calculation left to go:
Aristarchus knew the diameter of the Earth (12,742 km) because Erastosthenes measured it about a century earlier. The real life diameter of the Moon is what we are trying to measure, so that is our variable. Simply plug in the numbers we know and do a little algebra to solve the equation and calculate the real-life diameter of the Moon!
The accepted value for the diameter of the Moon is 3,475 km. If you drew your circle and measured it carefully, you should be able to get within a few percent of that accepted value.
Merely looking at the size of Earth’s shadow compared to the size of the Moon itself shows us that Earth is physically much larger than the Moon. But just how much larger?
Instead of just comparing diameters, let's talk about surface area. Once you know the Moon’s diameter, you can calculate the surface area of the entire Moon. It isn’t much. The entire surface area of the Moon is just a little bit greater than the surface area of Africa. Yes, if continents were wrapping paper, you could wrap the entire Moon with Africa and a few small islands!
Another way to make the comparison is using volumes. Knowing the diameter of the Earth and Moon, respectively, allows us to calculate the volume of each. Run the numbers and how many Moons would fit inside the Earth? Just over 49 Moons!
How incredible it is that we can see such things simply by paying close attention to shadows.
P.S. At the beginning of the article, I mentioned two eclipses. The second one is an annular solar eclipse. Don’t get too excited about it unless you’ve already bought your plane tickets. The path of that eclipse is almost entirely over open water in the southern Pacific Ocean (here’s a more detailed preview video for both of this fall’s eclipses).
