After what I ran into the summer of last year, I practically lost all heart in writing. Did post some manuscripts I’d already written to a critique group, but nothing really new. Then I ran into this and thought some would find it interesting.

Some weeks ago I noticed what looked like a water tower at a distant tree line. Normally wouldn’t have made much of it, but there is nothing but swamp in that direction. Making a best guess of the bearing, I checked that direction on Google Earth. Lo and behold, there was a town in that direction, except it was over 8.5 miles away (13.7 km) Sure enough, it had a water tower, and the terrain and drop due to the curvature of the earth was such that it was theoretically possible that the top would be visible. But near as I could tell from Google Earth, the water tower was only about 40 feet (12.2m) wide. Should I be able to see it’s a water tower from that distance?

I decided to calculate what the water tower’s angular width. I divided half the width in feet by the distance in feet, took the anti-tangent, and multiplied it by 2. That gave 0.051 degrees, about 3 minutes of an arc. That’s small.

Or is it? It turns out that the average human eye can see down to 28 arc seconds. An arc minute is 1/60 of a degree and an arc second is 1/60 of an arc minute, so an arc second is 1/3600 of a degree. Twenty-eight arc seconds is about 0.008 degrees, smaller than the apparent size of the water tank. But wouldn’t it still be hard to tell that it was a water tower?

Posing this question elsewhere, some suggested it could be a type of mirage, and, after observing the water tower more and noticing it seemed to be larger at times and once a darker color, and once not visible, I agreed. But observing it on different days, more often than not it appeared consistent in size, color, and location.

It was then I recalled that the moon, though it seems large in the sky, has an angular width of only about half a degree. Brilliant Venus appears as a tiny disk without the aid of telescope or binoculars, and at times as an angular width almost as large as an arc minute. In 1945, the USS New York noticed Venus high above and, thinking it was a Japanese bomb balloon, opened fire. After a few minutes the navigator realized they were shooting at Venus. Yes, Venus looked that large in the sky.

That came to mind one day last week when I saw a brilliant reflection from a road sign two miles (3.2 km) away. The width would have been about 42.43 inches (1.08 m). I could have estimated the angular width as I did with the water tank, but decided to pretend it was an arc and use radians. A radian is 1/2π of a circle. Since the circumference of a circle is 2πR, where R is the radius, then we can calculate the distance of an arc of circle by multiplying the radius times the angle in radians. If we call that distance S, then S = Rθ, where θ is the angle in radians. Working backwards, we can can calculate the angle in radians by θ = S/R, and that gives us 0.000 335 radians as the arc width of the sign. We can convert radians to degrees by degrees = radians * 180/π, so that works out to 0.019 degrees, or 1.14 arc minutes. If we do this “properly,” using the anti-tangent of half the sign width divided by the distance, then multiplying by 2, that gives us practically the same value. Rounding it to the same amount, we have 0.019 degrees, or 1.14 arc minutes. With angles that small, we can get by pretending the sign is part of the curve with a radius of 2 miles (3.2 km), and really, a curve that big is nearly flat for no more than the width of the sign.

This has not been lost on the world’s militaries. If you’ve seen a typical lensatic compass sold in the US, likely as not the dial was marked both in degrees and mils. Mils, short for milliradians, as used by the US and other NATO members, are 1/6400 of a circle. Since a radian is 1/2π of a circle, that works out to about 1/6.283 of a circle. A milliradian would then be 1/6283 of a circle. But 1/6283 is unwieldy, and it’s more convenient to use 1/6400 in its place. Not precise, but close enough. Using mils, if you have an idea of how tall or wide an object is and can measure its angular width, you can then estimate the range. Since θ=S/R and a milliradian is 1/1000 of a radian, then mils = Sx1000/R, which means R = Sx1000/mils. So, if you have an object 1m wide and it has an angular width of 1 mil, then you know it’s 1,000m away.

In the case of the water tank, its angular width comes to 0.891 mils, but 28 arc seconds is 0.136 mils. That water tank is over six times the smallest angular width the average human eye can resolve. Things with surprisingly small angular width can look larger than we think, just as the moon covers only about half a degree of the sky.

This is only about discernible width. We see stars other than our sun as points of light because they’re so far away we can’t see their angular width. If there’s enough photons from an object and it’s significantly brighter than the background, we can see it even if it just looks like a point. The only reason the Andromeda Galaxy looks like a dim fuzzy blob instead of a cloud of sparkling lights is that it’s so far away that not enough photons reach our eyes to see it clearly without a telescope. If that road sign had been over five miles (8 km) away, I might have seen a brilliant point of light but wouldn’t have been able to see the width.

After being satisfied I was really seeing the water tank in a town about 8.5 miles away, I happened to be driving along a different road, topped at hill, and clearly saw the water tower in the next town. Later I checked and found that hill is about 6 miles (9.7 km) away. At that distance, the tank has an angular width of a little over 4 arc minutes. That seems small, yet the tank didn’t look all that tiny. It’s something I’d seen many times before but had forgotten about it. Had I remembered, I probably would have accepted that the other water tank was at the town across the swamp and never would have delved into things like angular width.

Of course, if you’ve read this far, you might be thinking that would have been a good idea.