If you know my work at all you know I am pretty interested in light and shadow. Not to say ‘obsessed’ or anything. A few days ago we had a show of light and shadow on a planetary scale – a total solar eclipse. I didn’t see much of it because we were clouded over here in Minneapolis but I followed it on TV. As I watched I found myself wondering how Flat Earth believers would explain it. I don’t know why I thought of this, I just did. So I went online to find out.
Most of the Flat Earth questions and explanations are – as you might expect – pretty silly, but one point in particular was not. Several people pointed out that although the moon is claimed to be 3474 km in diameter, the shadow that falls on the earth during a total eclipse is only 112 km in diameter. Assuming the rays of the sun that strike the moon and the earth are parallel rays (as it is often said), given parallel or slightly divergent rays the shadow would have to be the same size or larger than the Moon itself. How is it possible that a shadow could be smaller than the occluding object?
That’s actually not a stupid question, and while it may be uninformed, few are informed enough to know the answer. I wasn’t either. That’s why it intrigued me. The answer – like the answer to nearly every question about light and shadow – is ‘it’s all relative’.
- The light source is large enough (relative to the occluder)
- The occluder (the object throwing the shadow) is small enough yet in the right (relative) placement to completely or nearly completely block the light source from the point of view of the receiver
- The receiver (where the shadow falls) is in the right (relative) placement to the occluder
then the umbra and/or the antumbra will be smaller than the occluder. The penumbra will be the same size or larger but the umbra is actually a cone pointing towards (or even beyond) the plane of the receiver.
One of the things I find most interesting about this when visualized is that it clearly shows that a shadow is not 2 dimensional and not simply an ‘absence’ of light, but a 3 dimensional projection of darkness. You can also see in this illustration that whether we see a total (umbral) eclipse or an annular (antumbral) eclipse depends on the relative distance between the earth and the moon.
I don’t have to contend with this particular formula, although it would be fun. The conditions required to create this effect are an intensely bright and very large light source – like the sun – and very large distances, not something that can be recreated in a studio. But this kind of thing is familiar to me. I wrestle with similar relationships every day I work on my photographs. Light is very weird.
And to get back to that question of ‘parallel rays’. For all practical purposes we can treat rays of light from the Sun as ‘parallel’ when they get to Earth. In the practice of imitating natural light effects they can (must) be treated as such, but in truth they aren’t. The divergence, however, is so slight that at our human scale, we don’t perceive it.
The reason they seem to be parallel is that only a small fraction of the sun’s rays strike the earth and those that do are all going in (roughly) the same direction. Although this specific effect can be approximated/faked in a studio, some things the sun does cannot, like the actual focal depth of shadows thrown by the sun. Interestingly, direct sunlight – in the absence of lens interference like a heavy atmosphere – does not enlarge shadows because the rays are more or less parallel. That’s one of the ways you can tell the difference between reflected light, especially if reflected from a convex surface, and direct sunlight. Like I said, light is weird.
The reason that household lamps do enlarge shadows is that the rays from a lamp are wildly divergent.
I spend a ridiculous amount of time on things like this.