As we were watching last Monday's eclipse, from the moment the Moon's disk was first noticeable over the Sun, we were trying to figure out how fast the Sun's light was diminishing. First it starts at 0 (0%), of course, and at totality it's 1 (100%).
But how fast does it proceed between the beginning and totality? It's a calculation of how much one disk (the Moon) obscures the other (the Sun).
I had actually tried to calculate this before the eclipse, and while it's just geometry it's a bit tricky, especially for someone who rarely calculates anything anymore. My brother-in-law is a laser physicist, and he said he once had to calculate this once regarding two laser spots (or some such), and only found it after a few hours of work.
Eventually I started hunting around the Web, and found this nice derivation from Adrian Jannetta, a math instructor in England. He also built this interactive calculator (at the page's top) to calculate the Sun's obscuration for different radii of the Sun and Moon.
Eclipse astronomers classify how far along the eclipse is by the magnitude M, which is how much of the Sun's diameter is covered up, at any given time, by the Moon. Also, the algebra simplifies considerably if you take the Sun and Moon to have the same radii -- remember, this is the radii as seen in the sky, not in actuality, and the very reason a solar eclipse happens is because the angular diameter of the two is very nearly equal.
(Actually for Monday's eclipse I read the Moon, at totality, obscured 103% of the Sun's area, so its angular radii was just a bit bigger than the Sun's, but I'm going to ignore that here.)
So setting the Sun and Moon radii equal in Janetta's equations gives
where again, M is the magnitude, and α is just an intermediate parameter to make the math look simple.
M is going to be proportional to time, assuming the Moon glides uniformly across the Sun's disk. (Probably not exactly true, but close enough for this example. At our location it took 1 hr 11 m 55 s from beginning to totality.)
Then I get the following for the obscuration as a function of magnitude:
|The eclipse banana|
I guess I thought there might have been a quickening, nonlinear obscuration closer to totality -- when the "banana" shows up, then continually shrinks. But we were all too excited to really judge it objectively.
Anyway, I spent some spare hours working on this because I couldn't get it out of my head. Now's it's gone.