Monday, January 16, 2012

The Polished Steel Moon

Not everyone knows this, but twenty years ago I used to be the math guy on community access TV here Winnipeg. “Math With Marty” started in 1989 and ran for three years; it quickly attracted what everyone calls a cult following. Just recently I started putting up old video clips on YouTube. I guess it was one of those old shows that got me started on the topic of the moon.
I originally posed the problem in terms of billiard ball collisions: If the moon is a giant billiard ball, and you shower it with regular billiard balls at high speed, how would you describe the distribution of billiard balls bouncing off the moon? You can watch me solve this problem here Actually, in the opening clip, it’s not me solving the problem, it’s my friend Neil, who was co-host and lead guitar player on the show with me. I pick it up toward the end of the clip, and…well, you can see for yourself.

Although I solve the problem in terms of billiard ball collision, it’s obviously exactly the same problem if you consider it as light reflecting off a polished sphere. And the answer is the same: the scattered light is uniformly distributed through space. All angles are equally illuminated. If you can’t see the source beam, you have no way of knowing even what direction it came from based on the scattered light, because the scattered light goes equally in all directions.
Despite its deceptive simplicity, this is not a trivial result. In fact, it is only true in three dimensions, as we see from Neil’s attempt to solve the two-dimensional equivalent. In my way of thinking, when things come out this way it seems to illustrate some kind of cosmic property of the universe. In other words, I don’t know what it means, but it must mean something.

What would the moon look like if it were a polished steel ball? Evidently it would appear equally bright from whatever angle we viewed it, regardless of the relative angle of illumination. In fact, all we would se would be a bright glint of the sun, and we wouldn’t even know whereabouts on the face of the moon it came from. (except that the silhouette off the moon was blocking whatever stars would have been behind it.)

The interesting question is: would this polished ball be a better or worse source of illumination, on average, than our ordinary every-day moon? Bu ordinary and everyday, I mean of course the theoretical moon which ought to be a Lambertian scatterer, an ideal bright-white sheet of paper. In other words, all the light that comes in must go out, just like the polished ball, except now the scattering is diffuse. Cosine-law and all that.

Which one would provide more illumination to the earth, on average? It’s a funny question, and for the longest time I was drawn to the tantalizaing prospect that either moon would be equally good. Actually, in practical terms, the real moon is more useful because it scatters preferentially in the backwards direction, so it is a better night-light than a daytime light. This is in contrast to the polished ball, which scatters in all directions equally, so much of its utility is wasted in brightening the day by an infinitesimal amount. The point is, what goes in must go out, and since over the course of a whole month the moons are on average located at all possible angles with repect to the sun, the earth must receive the same total illumination from either one of them.

Except it’s not quite right. The moon does a circular orbit in the place of the sun, but it is never found, for example, above the north pole. This screws up everything. It’s actually a case where the two-dimensional case has a much tidier solution. For cylindrical earth-moon systems, the average illumination of polished versus difffuse is of course equal. Not for the three dimensional case.

Because of the cosine-law for the scattering angle, the diffuse scatterer keeps more of its illumination in the equatorial plane, which is of course where the earth is. The polished sphere wastes more of its scattered light outside the plane. So if the purpose of the moon is to illuminate the night, the actual moon is actually a better moon than the polished sphere after all. Partly because it keeps more of its light in the plane of the solar system, and more importantly because it is a more effective back-scatterer, so it wastes less power on the daylit skies. Did God maybe know what he was doing when he put it up there?

The unfortunate thing about the “real” moon, the diffuse scatterer, is that I haven’t found any neat and tidy way to do the calculation. I wanted to use its equivalence “on average” with the polished moon to draw some nifty conclusions, but I still don’t know how. The polished moon, at worst case, reduces to a Grade 11 science problem in focal lengths, so I think I can do it. I just don’t know exactly what  I’ll do with it.

A couple of funny things about the real moon. First, in terms of total reflected radiance as an illuminator of the nighttime sky, it ought to have an equivalent polished version. Not the polished sphere: that is clearly different. I’m saying that for some distorted, squashed-down version of the sphere, we should be able to generate a polished surface that has the same illuminating effect as the real, diffuse moon. That would be an interesting thing to calculate. I’m going to guess that it might be a cycloid of revolution, but that’s just a wild guess.

The other point I still wanted to come back to was the one I talked about last time, the departure from Lambertian diffusion. As Wikipedia points out, if the moon were a Lambertian scatterer, it should look darker around the edges and brighter in the center: in other words, it should looks more spherical. One of the most obvious facts of the moon is that it just doesn’t: it looks more like a flat dish than a round ball. Wikipedia explains this as a departure from Lambertian scattering: since the outer edges are just as bright as the inside, the scattered power must be greater at lower scattering angles.

I said last time that I didn’t buy it, and now I have a good reason to back this up. The Wikipedia theory would seem to explain the flat-dish appearance of the full moon, but then it’s a total contradiction with the half-moon, which also appears uniformly bright. If the flatness of the full moon is indeed due to enhanced low-angle scattering around the outside, the the half moon should show even more drastic darkening toward the diametral line, which is the zone of very oblique illumination. In other words, if the scattering is enhanced at the low angles, it must be depleted at the high angles. But the half-moon looks just as uniform as the full moon. You can’t have it both ways.

I still say it’s a psycho-visual effect having to do with the saturation of the eye receptors, the rods or the cones or whatever they are. Some of those photo-shots of the full moon look pretty dramatically spherical. I guess the effect shows up when you balance your light levels properly.

There was one more type of moon configuration which turns out to interesting to analyze, and that’s the big flat drywall cutout moon. After all, when we look at the moon, it looks flat…so why now analyze how it would behave if it were just a big round hunk of drywall stuck up there in the sky. I figured out some cool stuff about it, but I think we’ll leave that for out next post.

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