Noah Frere

# Lunar Eclipse Trigonometry

The shadow across the Moon during a lunar eclipse is caused by the Earth blocking the Sun’s light — it is Earth’s shadow. While looking at my photos of the January 20th, 2019 eclipse, I got curious about the size of the shadow. I drew a circle, using the terminator (line that separates the light from the shadow) as a guide, as my best guess of Earth’s shadow. This picture eventually ended up on the back of my business card.

As you can see, my shadow was 2 ½ times larger than the Moon. I wanted to know how large the shadow really was, so I drew a schematic showing the trigonometry and distance relationships between the Sun/Earth/Moon system, and tried to derive it. If I did it correctly, then by using some rough figures, the shadow should instead be 2.7 times as large as the Moon. Not bad, with a 7% error!

I can’t remember three years ago very well, but apparently I must have recalculated the shadow size using more accurate numbers, because I found this picture in my files:

The main point with this last picture is contained in the yellow text: hence the difficulty in drawing the correct sized circle.

So this week I tried to draw the correct sized circle for May 16th, 2022 lunar eclipse.

I thought I did a good job, but then I did the math, and the shadow was only a little more than twice the size of the moon. That didn’t seem right. So I drew another schematic, but this time solved for the distance between the Earth and Moon using this shadow size of of 2.157. This time I was more precise.

I ended up with an Earth-Moon distance that was impossibly too large: 568,662 km, when in fact the real distance is only 358,440 — that’s 1.6 times over-estimated! I thought maybe my shadow terminator was too fuzzy to for me to draw a proper circle encompassing the shadow, so I tried a different photo:

But I got similar results. I’ve increased the width of the circles here so you can se them easier.

So I tried a different photo, but again similar results:

My ratio of shadow/moon size is way too small. So I looked up the real shadow size for guidance, and found this graphic:

Overlaying my yellow and blue size circles atop this picture, you can see how vastly under-estimated my yellow shadow circle is. I kept thinking “how could my shadow size be so incorrect?”

Then I had a moment of serendipity. I happened to be looking at image 3748 zoomed out, and when seen from that point of view, it was clear that my shadow circle was too small! I had been looking at the micro view, trying to get the my yellow circle to match the contours of the terminator in my photo. However, it’s like missing the forest for the trees. You have to consider both the micro and the macro for a healthy perspective. So I redrew the circle from a zoomed out position.

I overshot this time, drawing a circle a bit too large, because I was tired of under-doing it for so long :). Well, it was pretty close, so I went into my Mac Numbers table and played with the size until I found the correct circle diameter: 2.91”, which gives a ratio of 2.72. In other words, the shadow of earth was 2.72 times larger than the moon at the time of this photo. Running through the math, this gives the correct Earth-Moon distance within 1,346 km, which is a 0.4% error. That’s the best I can do while drawing these circles to two decimal places of inches.

I’ll take it! I believe I’ll be able to draw a more accurate circle next time without having to look it up first.