Of Particular Significance

Why a Partial Solar Eclipse is Totally Awesome!

POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON 03/14/2024

On April 8th, 2024, a small strip of North America will witness a total solar eclipse. Total solar eclipses are amazing, life-changing experiences; I hope you have a chance to experience one, as I did.

Everyone else from Central America to northern Canada will see a partial solar eclipse. What good is a partial solar eclipse?

Astonishingly, the best thing about a partial solar eclipse seems to have essentially disappeared from public knowledge. We’ve got three weeks to change that, and I hope you’ll help me — especially the science journalists and science teachers among you.

The best thing about a partial solar eclipse is that you can measure the Moon’s true diameter, armed with only a map, a straight-edge, and a leafy tree or a card with a pinhole in it. No telescope, no eclipse glasses, no equipment that costs real money or is hard to obtain. You’ll get an answer that’s good within 10-20 percent, with no effort at all. And in the process you can teach kids about shadows, about simple geometry (no trigonometry needed), about the Earth, Moon, and Sun, and about how scientists figure things out.

About a week later, you can measure the Moon’s distance from Earth, too. All you need for that is a coin, a ruler, and drawings of similar triangles.

In this post, I’ll explain

  • First, the two easy steps necessary to measure the size of the Moon;
  • Second, evidence that this actually works;
  • Third, the reasons why it works;
  • Fourth, a brief comment (and a link to an older, longer post about it) showing how to measure the Moon’s distance once you know its size

The Amazingly Easy Way to Measure the Size of the Moon

  1. Before the eclipse, find a map that shows your location and the “totality strip” where the eclipse is total. On that map, measure the smallest linear distance (not the driving distance!) between you and the edge of the strip. Call that distance L.
  2. Next, right at the maximum of the eclipse, when the Moon’s silhouette blocks as much of the Sun as it can from your location, determine the fraction of the Sun’s diameter that remains visible, measured at the thickest part of the Sun’s shape. (You can see the shape of the eclipsed Sun where it is projected onto the ground or a piece of paper, after the sunlight has passed through leaves, or through a colander, or through a grid of your own fingers, or through any other little hole through which light can pass.) Call that fraction F.
  • D \approx L/F

That’s all there is to it!

This estimate should be accurate to within 10-20% (as long as you’re not too close to the Earth’s poles or too close to the totality strip; we can talk about why later.) Any child over ten can carry this out with a little help.

Click here to see an example of how this works

From Boston, here’s how I can measure L, using Google Maps and an eclipse map showing the location of the totality strip:

From Boston to the totality strip is about 145 miles.

Meanwhile timeanddate.com predicts that in Boston at mid-eclipse the Sun will appear as shown below: a fraction of around 6 to 8% of the Sun will be unblocked. (We’re so close to the totality strip that measuring F is quite difficult to do accurately, so our estimate of the Moon’s size will be more uncertain than for people further away.)

The fraction of the Sun’s diameter that will be visible at mid-eclipse in Boston is less than 10%; the full diameter of the Sun is shown in orange. Prediction from timeanddate.com .

That will give us (if we take F=7% as a best guess) an estimate of the Moon’s size

  • D = L / F = 143 miles / 0.07 = 2050 miles .

(If we let F range from 6% to 8%, this estimate really ranges between 1900 and 2200 miles.) The Moon’s true diameter is about 2160 miles, so this is wonderfully successful, given its ease.

Is this too good to be true? Let’s see.

Some Basic Evidence

Here’s a map from NASA (via Wikipedia) showing where the eclipse is total and partial. Let’s call the region of totality the “totality strip”, and the regions where the eclipse is partial the “northern partiality zone” and the “southern partiality zone.”

Figure 1: As with all total eclipses, the one on April 8th will have a narrow totality strip (dark blue) where the eclipse is total, surrounded by two large “partiality zones” (light blue) in which the eclipse will be partial. Eclipse Predictions by Fred Espenak, NASA’s GSFC

Now here’s the same map with the Moon, correctly sized, superposed on the two partiality zones. You see that indeed the width of each partiality zone is roughly the size of the Moon — slightly larger, but much less than twice as large. [Because the eclipse is north of the equator, the northern partiality zone is closer to the north pole, and the Earth’s curvature causes it to be somewhat larger than the southern partiality zone. This is discussed in an aside later in this post.]

Figure 2: Each partiality strip is a little wider than the diameter of the Moon, for reasons explained later.

Let’s say you’re in one of the partiality zones, and let’s call its width W. If you’re right near the outer edge of that zone, then L is approximately W. That’s also where the eclipse barely happens — the Moon just clips the edge of the Sun — and so, since the Sun’s diameter is hardly blocked at all, F is close to 1. Your estimate will then be

  • D = L / F = W / 1 = W,

which is in accord with Figure 2.

Suppose instead that you are halfway between the totality strip and the outer edge of your zone (measured on a line perpendicular to the totality strip.) Then L = W/2. But also the Moon will block half the Sun’s diameter, leaving half of it unblocked: F = 1/2. That means that you will estimate, again,

  • D = L/F = (W/2) / (1/2) = W.

More generally, if your distance from the totality strip is L, and L is a fraction P of the width W of the partiality zone, i.e.

  • L = P W ,

then the fraction F of the Sun’s diameter that will be unblocked at mid-eclipse will also be approximately P. Therefore, no matter where you are in the zone,

  • D = L / F = (P W) / P = W ,

so your estimate of the Moon’s diameter will always come out more or less right.

(That said, if you’re very close to the totality strip, F will be hard to measure precisely, so your estimate may be very uncertain; and if you’re close to the poles, W will be significantly larger than D, so your estimate will be poor. Fortunately, that won’t be true for most of us.)

What’s behind this clever trick? Here’s the reasoning.

Why Does This Work?

What’s great about this trick is that it’s not hard to understand, though it does take a few steps. [I’m not sure I yet have the best pedagogical strategy for laying out those steps; suggestions welcome.]

Why the Partiality Zones are as Wide as the Moon

First, we need to understand why the width of each partiality zone is roughly the same as the width of the Moon — why D and W are almost the same, as long as we are far from the Earth’s poles. It all has to do with shadows — moon shadows, of both types.

You may have noticed (kids often do) that there are often two types of shadows visible when you’re at home and lit by a single central light. If a thumbtack lit by a small light bulb is close to a wall, it casts a shadow that is crisp and about the same size as the tack. But as you move the tack away from the wall, the shadow becomes fuzzier. If you look closely, you’ll see that the inner dark part of the shadow (the “umbra”) is shrinking, while there’s an outer part (the “penumbra”), quite hard to see, that is growing.

Eventually, when the tack is far enough away, the inner dark part will become almost a dot. At that point, the outer dim shadow — you may only barely see it — has a diameter about twice the diameter of the tack. See Figure 3.

Figure 3: How a shadow of a tack changes as it moves away from the wall. (Left) The shadow is crisp and the same size as the tack. (Center) The dark umbra is narrower than the tack, while the dim penumbra is wider than the tack. (Right) As the dark umbra becomes very narrow, the penumbra becomes twice the width of the tack (as indicated by the two tack-sized disks). Credit: the author.

For the solar eclipse, it’s the same idea, except that instead of bulb, tack and wall, we have Sun, Moon and Earth.

There’s simple geometry behind this shadow-play; I’ve drawn it vertically in the figure below, so that you can see it no matter how narrow your screen. Let’s assume that the Sun is much further away from Earth than the Moon is (a fact that you can also verify during daytime, either a week after the eclipse or a week before.) I’ve drawn four lines, two of them red, two of them black; watch where they go.

Figure 4: The Sun is totally eclipsed in the little gap between the black lines; it is partially eclipsed between the red and black lines. Not to scale.

Inside the black lines, the Moon totally blocks the Sun; both the left edge and right edge of the Sun in the figure are blocked. Inside the red lines, the Moon partially blocks the Sun. And so, at the moment shown, the little space between the two black lines is where the eclipse is total; that’s a location within the totality strip. Meanwhile, the distance between each red line and the nearest black line is the width W of one of the partiality zones.

For maximum simplicity, I’ve drawn this where the Sun, Moon and Earth are perfectly lined up, so that the total eclipse is occuring where the Earth’s surface is nearest the Sun and Moon. That makes both partiality zones the same size. In the aside below, I’ll show you what happens if this isn’t the case. But let’s not get distracted by that yet.

The important thing is that because the Sun is so much further than the Moon, the red and black lines from the left edge of the Sun are almost parallel. Where they meet the Moon, they are separated by the Moon’s diameter — that is, by the distance D. But the distance between two parallel lines is constant, so the distance between two nearly parallel lines changes very slowly. This means they are still a distance D apart when they reach the Earth.

On top of this, the two black lines almost meet; the totality strip is very narrow. Taken together, these facts imply that W, the width of the region on Earth’s surface between a red line and the closest black line, is roughly the same size as D!

As noted in an aside below, the Earth’s curvature tends to make the partiality zones a bit larger than the Moon’s diameter, while for an eclipse nearing a pole of the Earth, the partiality zone closest to the pole will be larger than the other partiality zone. But these are details; they don’t change the basic story.

Click here for an brief discussion of why W and D aren’t quite the same

As noted, W isn’t quite D. Even with the eclipse centered on the Earth as in Figure 5, W is larger than D because the Earth’s surface is a sphere. (This is somewhat compensated for by measuring L to the edge of the totality zone rather than its center or opposite edge.)

The width W of the partiality zone is wider than the diameter D of the Moon because of the Earth’s shape.

Second, if the eclipse is far from the equator, the partiality zone nearer to a pole of the Earth will be larger than the other.

If the location of totality is offset from the line connecting Moon and Sun, then the totality zone on the side of the offset will be larger than the other, and than D, due to the Earth’s curvature.

In short, this is not a method designed to get a precise or accurate measurement of the Moon’s diameter. But it’s perfectly fine if one’s aim is merely to get a rough idea of how nature works, which is often more than enough for scientists, as well as for everyone else. There is an opportunity here to talk to students about the nature of approximations, when and why it’s okay to use them, and how to improve upon them.

Why the Unblocked Fraction of the Sun’s Diameter Plays the Key Role

Second, we have to understand why the unblocked fraction F of the Sun’s diameter is roughly the same as what we called P (the distance L to the totality strip divided by the width of the partiality zone W.) This is illustrated in Figure 5, where again I’ve drawn four lines on each image, but now all four begin from the location at which we are observing the eclipse, and they indicate where the Moon and Sun appear to us.

Figure 5: What one sees in the partiality zone between the red and black dots; the orange and grey lines show the location and width on the sky of the Sun and Moon.(Far Left) In the totality zone, F=P=0. (Far Right) Near the edge of the partiality zone, F and P are close to 1. (Near Left and Right) Throughout the partiality zone, F=P, as shown for P=1/6 and 1/2. Not to scale.

In each image in Figure 5, the totality strip is indicated by the black dot, and the edge of a partiality zone by the red dot. The Sun’s disk in the sky is spanned by the green lines, and its apparent width in the sky is shown by the orange line; the Moon’s disk is spanned by the blue lines, and its apparent width is shown by the grey line.

In the far left image, the observer is in the totality strip, so P=0; the Moon blocks the Sun completely, so the grey line aligns with the orange line and F=0. At far right, the observer is at the outer edge of the partiality zone, so P is almost 1; the grey line almost misses the orange line completely, so almost all the Sun’s diameter remains unblocked and F is almost 1. In between are shown intermediate situations where F and P are both 1/6 or both 1/2. The closer the observer gets to the totality strip, making P smaller, the less of the Sun is unblocked, making F smaller by the same amount.

Bonus: Measure the Distance to the Moon in Daylight

I’ve explained how this works in this older post. For today, here’s a quick summary.

Within a week after the eclipse, the Moon will reach first quarter, which means that the Moon will rise around noon. In the afternoon, then, you can see it in the East. Then you can have one person hold a penny, and another person move until the penny perfectly eclipses the Moon; a third person can measure the distance s between the person’s eyes and the penny. Then, measuring the diameter d of the penny, we can use similar triangles to convince ourselves that the distance S to the Moon, divided by the diameter of the Moon D, is the same as the distance from observer to penny divided by the diameter of the penny:

  • S/D = s/d

and so

  • S = (s/d) D

Since we measured D during the eclipse, we now know S also.

By the way, in another old post, I showed that we can also confirm that our original assumption, that the Sun is much, much further than the Moon, is correct. At first quarter — i.e. one quarter of the way through the Moon’s monthly cycle — one can verify two things at once, by eye:

  • the Moon is half lit
  • the Moon is 90 degrees away from the Sun in the sky

These two things can only both be true if the Sun is much further than the Moon.

Spread the Word

So you see, it’s not only easy to measure the Moon’s size during an eclipse, it’s relatively easy to explain how and why the method works. To do so requires a range of logical reasoning tools, some drawings of lines and triangles, and a little experimentation with shadows, but no actual math. (I’m sure it can be done better than I did it here.) On top of that, the measurement can be carried out in daylight, while most children are in school. I think it’s a great opportunity for science education — a chance for a meaningful fraction of the 600 million people in North America to experience scientific reasoning for themselves, and to observe how it leads to consistent, reliable knowledge.

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