Anyone who’s seen an eclipse like the one coming to the U.S. on Monday is likely to tell you that they’re special because of their strangeness, ethereal beauty and awesome power. But they’re also special because they require such a specific alignment of objects and events. That we can even see a total solar eclipse from Earth in the first place is a glorious mathematical accident, one that has led to a lot of unique scientific discoveries.

A total eclipse is only possible because the moon is positioned juuuust right relative to the Earth and the sun. This doesn’t happen on every planet with a moon and won’t always be possible for Earth. In about 600 million years, we won’t have total eclipses anymore, because by then, tidal forces will have flung the moon to a spot too distant to completely block the sun’s disk. Even today, total eclipses are only visible across a narrow path on Earth.

That leaves only several hundred thousand millennia for us to do some fun science. This year, math buffs are using the eclipse to measure the sun’s diameter and even to bask, very briefly, in the levels of illumination one would experience on the other planets.

The task of determining the sun’s size is trickier than it might seem because the sun is a roiling ball of plasma with no surface. It’s also constantly spewing gas and radiation and magnetism, so the diameter of its “disk” is constantly fluctuating. But it’s easier to measure during an eclipse. In Kansas City, St. Louis, and Minden, Nebraska, citizen-scientists will fan out to measure the sun. Working in pairs, the researchers will stand at 175-foot intervals across the very edges of the so-called path of totality, the 70-mile-wide swath darkened by the shadow of the moon. The path’s location is well-known, but the precise location of its edges is a little more uncertain, because of the fluctuating sun and the movement of the moon. One person will shoot smartphone video while another sketches naked-eye observations of exactly when the shadow appears. Assuming that the phone is synchronized to global positioning satellites and can take a time-stamped video, this system should allow for a hyper-accurate measurement of where and when the shadow arrives. By making precise measurements, scientists hope to figure out the exact edge of totality and how much it might vary from predictions. This will help scientists understand how precisely they can measure changes in the size of the sun.

While eclipses provide a unique view of Earth and the sun, they can also give us a glimpse of what being on other planets may be like. Michael Zeiler, a cartographer and eclipse chaser who runs the website GreatAmericanEclipse.com, determined when the sunlight during the eclipse will be of the same intensity as it is elsewhere in the solar system. If you were standing on the center of the moon’s shadow in the path of totality at 28 minutes, 40 seconds before totality, you’d experience the intensity of sunlight on Mars. A breath before totality, just 7.7 seconds before the sun is swallowed, you’d experience the sunlight on Pluto. At 59 seconds before totality, you would experience the sunlight on Saturn, which — as it turns out — is special for more than just its rings.

In the excitement surrounding this year’s eclipse, some astronomers have speculated that Earth is the only place where conditions for a total solar eclipse, in which only the sun’s corona is visible, exist in the solar system. Is there anywhere else we could find an alignment of planet, moon and sun that could produce total, corona-only eclipses as well as an alignment that could produce ring-of-fire eclipses, in which the sun’s outer circumference is visible behind the moon?

Eclipse chaser Bill Kramer, who first saw a total eclipse as a 13-year-old standing on a cruise ship in the North Atlantic, aimed to find out. Kramer, who is a retired computer engineer and runs what you might call an online eclipse museum, first gathered NASA data on all of the solar system’s planets and their moons. He then wrote a simple code that analyzed their mutual orbits, the moons’ apparent sizes from the planets’ surfaces — or, in the case of gas giants, their cloud tops — and other statistics. He threw out some moons because of their weird shapes. All told, he ended up with 141 moons with the potential to cause eclipses.

Kramer didn’t expect to find any that could do what Earth’s moon does. “When you think of Carl Sagan’s ‘billions and billions’ — 131 ain’t a big number,” he said, referencing the renowned astronomer’s imagined number of stars and planets in the universe.

In the end, he found two moons with eclipse potential — both orbiting Saturn. The moon Pandora, which looks like a dented potato, can produce a total eclipse as viewed from Saturn’s cloud tops, but its weird shape makes it less likely to produce a perfect black hole surrounded by a corona. There’s just one moon with that distinction: Epimetheus, a dinky thing just 84 miles across at its longest point that hangs out in a gap amid Saturn’s rings and zips around the planet every 17 hours.

We won’t get images of an Epimetheus eclipse any time soon. The Cassini spacecraft, designed to study Saturn and its rings and moons, will end its 20-year mission in a few weeks by plunging into Saturn’s atmosphere, and its orbit was never aligned to see the sun eclipsed by this moon. Even if the spacecraft’s orbit had hit the right spot, an Epimetheus eclipse would be breathtakingly short: just 0.6 seconds of totality, according to Kramer’s calculations.

What all this means is that Kramer got what he was after — sort of. “I just wanted to be able to state, unequivocally, that, yes, we have the only total eclipse of this style in the solar system,” he said. Now he knows that’s not the case. But, he said, “I can now state unequivocally that we have the best solar eclipses in the solar system. I believe that strongly.”