## Tuesday, January 17, 2012

### I get to re-use a graph from another problem

Last time I said it was interesting to calculate how effective the moon would be if it was a simple hunk of drywall, just a big round cut-out. Actually the interesting calculation is if we hang the drywall up in the sky, and then optimize it by tilting its angle so as it revolves around the earth, it always delivers the maximum amount of illumination.

It's  not hard to calculate that the effective illumination is maximized if the angle of tilt simply bisects the angle between the sun and the earth, as measured from the drywall. It's a consequence of Lambertian diffuse reflection and plain old geometry. So, for example, when the moon, sun, and earth form a right angle, the effective illumination is exactly 50% of what it was in "full moon" conditions: you tilt the drywall at 45 degrees so it intercepts 71% of the sunlight, and because of the viewing angle it's apparent size is only 71% of the full disk. Compound these two effects and you get 50% power. We can sketch the function of effective illumination power as the moon revolves around the earth. We imagine the sun is at the bottom of the picture, and the graph of radiance looks like this:

Look familiar? It's the same graph I plotted last month for the atom deposition pattern of a Stern Gerlach beam in a quadrupole field ! Interesting.

But that's nothing. The sheet of drywall does something very cool as it moves around the circle, always tilting itself so that it delivers the maximum power to the surface of the planet. It starts off at top dead center, square to the sun. As it moves around the ring clockwise, it tilts 45 degrees when it is at 3:00 position; then, in the 6:00 position (between the sun and the earth) it tilts at 90 degrees, parallel to the sun's rays, so it intercepts nothing; moving on, at 9:00 it is tilted 135 degrees to its original orientation, until finally when it returns to home, it is....flipped by 180 degrees! You have to make two full revolutions before it is restored to its original orientation. Now, where in all of physics has anyone ever heard of a situation where you need to make two full revolutions to get back to where you started? (HINT: That's a trick question!)

You better paint both sides of the sheet of drywall.