Lunar Coincidence

Something fundamental has been on my mind (again) recently:
Why is the Moon almost exactly the same angular size as the Sun?????

To be clear: what I mean is the Moon and Sun appear to be the same size in the sky, which has the thrilling consequence of generating the occasional total solar eclipse, like this:

This is one of those big "whoa dude" factoids to me. How can the apparent size of the sun and moon be so close?!  Why should it be so?! Most people would say it's a coincidence. This is something I've wondered about for a long time.

Recently a very interesting paper by Steven Balbus discussed this phenomena, and the possible consequence it has on life. Consider: if the Moon was more dense but the same mass and distance, it would have nearly the same tidal effect on the Earth, yet wouldn't cause total eclipses. If the Moon was a bit further away it wouldn't raise the same tides (and possibly do a host of other interesting things), which might be fundamental for life as we know it...

So it's a handy fact that the Moon is the right mass and distance that helps create life, and a damned coincidence that it also happens to be the same angular size as the Sun in our sky! Consider the cultural implications our Sun/Moon being equal in size. The result is frequent appearance in myth and legend as opposing gods.

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So I started wondering... there are lots of moons in our solar system (we don't know of any moons in other planetary systems yet). Do any other moons exhibit this kind of coincidence, where the apparent diameter of the moon is the same as the Sun, as seen from the surface of the planet?

If we assume all moons are spheres, this is an easy enough calculation to do. You just need to gather the separations and sizes (radii) of the Sun, the planets, and all their moons throughout the solar system! A little geometry (see kids, not just useful for mini-golf) and you can figure out how large the Sun and each moon appears in the sky as seen from the "surface" of each planet...

Here's a graph to that effect:

To get total solar eclipses you need moons that land on the line of equality in this graph (dotted line).  Indeed, 3 other satellites (besides our bff, Luna) exhibit this coincidence! Of course, you can't stand on the "surface" of Saturn or Uranus, so this is all kind of silly... Let's take a look at the "winners":

Prometheus (orbiting Saturn) 

Pandora (orbiting Saturn)

Perdita (orbiting Uranus... maybe)

The first two are potato shaped rocks (each about 40miles across), not the grand sphere we're used to seeing in our night sky. Pandora isn't quite like James Cameron's imagined moon. The third may not even be a "moon"... It's the little fleck the yellow arrow points at. The discovery of Perdita was disputed for a while, and only recently been reconfirmed using HST.

That these moons (which could exactly cause total solar eclipses) are so small is really a statement about how far the giant planets truly are from the Sun. Out there the Sun is just a bright star in the sky!

There are lots of other moons that would appear very large in the sky as well. The famous Jovian moons are huge and close. Note how crazy big Charon appears compared to Pluto - this is really a "binary planet" configuration (yeah yeah yeah, I know Pluto's not a planet).
Aside: binary planets are something I've been muttering about for a couple years now... I've got $10 that says we find one in the next 5-10 years.

I'm tickled to imagine: what if beings lived in the clouds of Saturn, floating in the thin cold air, soaking up the faint sunlight. Very occasionally that somewhat brighter star would wink out completely, only to be re-lit by Prometheus, bringer of fire...