Eclipses and a bit of History

Recently, I've been doing some research for an upcoming talk (a sequel to my Science of Anime talk).

One of the topics I'll be hitting is that of the duration of solar eclipses and how, in popular media, they occur, from the moment the first bit of the moon's limb covers the sun's disk, to the moment it leaves it, in the span of about 1 minute.

In reality, totality can last several minutes (the maximum theoretical length of totality is about 7 3/4 minutes). The entire eclipse lasts closer to two hours. What I wanted to know is just how much we'd have to change physical parameters to make the eclipse happen as it does in TV world.

I won't give away the answer, but wow is it a pain to figure out! The reason is there's a ton of different variables that go into it. First off, consider the shadow of the moon holding still as it would appear to in this static picture. In that, the earth would turn through the shadow, making the eclipse occur.

But in reality, the moon is moving too. So you have to add that motion in. Que moving coordinate frames with spherical trig.

Trying to quantify all this has not been particularly easy. When I first thought this question up, I figured it would take 15 minutes of derivation. I've been tinkering with it for 3 months here and there, and still haven't solved the full system, although I think I've accounted for all the variables to an order of magnitude approximation which is all I really needed in the first place.

I was hoping to find a nice little equation (and by little, I don't literally mean little) that was behind one of these eclipse calculators, but surprisingly, I haven't been able to find it with all the powers of the internet behind me! I've been using Google books, and going through quite a few texts available and still haven't found the full equation (only approximations).

Most of the more recent ones (starting around 1970) all have their solutions written as computer languages (and I don't care enough to sit there and reverse engineer their Fortran or C++). So interestingly, the most useful books are ones that date back to the turn of last century!

I've always thought of astronomical history as very interesting so I find myself reading more than strictly necessary. One of the most interesting things I've come across in this venture was a note on "Flame-like proturberances" during solar eclipses:

Immediately after the commencement of the total obscuration, red protuberances, resembling flames, appear to issue from the edge of the moon's disk. These appearances, which were first noticed by Vassenius, on the occasion of the total solar eclipse which was visible at Gottenberg on the 3rd of May, 1733, have been re-observed on the occurrence of every total solar eclipse which has taken place since that time, and constitute one of the most curious and interseting effects attending this class of phenomena.

- From Handbook of Astronomy, 1875.

Obviously today we call them solar prominences and know they're a result of matter captured by the Sun's magnetic fields or blown out through other stellar activity. But it's interesting seeing just how far Astronomy has come in the last 134 years.

For those that are curious, there's a review article on the history of solar prominences that's quite interesting as well. In the 1840's some thought these prominences were "mountains on the sun". It wasn't until the 1850's that it was realized these were more likely clouds of some sort, and it wasn't the 1900's that a full interpretation was realized.

Think of what we'll learn in the next 100 years....