NavList:

Dear Frank,

I like how we increasingly move towards more and more refined tricks in every iteration. I was not originally thinking about using the central slope to estimate the width of the penumbra. My goal was to just get another estimate on the size, which would actually be a lower bound, as the sigmoid will always be wider than the linear extension of its middle part.

However, as you note, we can infer the true width of the sigmoid from the slope, rather than just making bounds estimates. Clever! With a tiny bit of calculus, I obtain that the linear extension of the central slope will underestimate the true width of the penumbra by the factor of pi/4. When I divide the original 10m penumbra-based estimate of the crater diameter by that factor, I obtain 12.7m, which is conveniently close to the estimate based on the apparent size of the Earth.

The factor pi/4 is different from yours (pi/2), since the sigmoids in question are different. While a half-(co)sinewave is indeed pi/2-times wider than its central slope would suggest, the penumbra sigmoid is different. Its slope is f(z) = 2 · sqrt(1-z²) / pi, where z ranges from -1 (entire Sun just about obscured) to +1 (entire Sun just about visible), and I choose F(z) = (z · sqrt(1-z²) + asin(z)) / pi + 1/2 as its integral. Then the values of F represent illumination from 0 to 1. For that curve, the relevant factor is pi/4.

Matus

File: