The Perfect Eclipse*

*warning: lots of math in this posting.

As you may have learned from all the eclipse hype this month, the Sun and Moon are almost the exact apparent size (their sizes as observed from Earth). This is a function of four factors: the Sun’s diameter, the Moon’s diameter, the distance of the Earth to the Sun, and the distance of the Earth to the Moon. Their apparent size (measured as the apparent angle) is so close, when the Moon and Sun are each at their average distances to Earth, the difference between their respective apparent angles is only 0.014 degrees (I would be glad to share my math, if you are so inclined). Suffice it to say, it’s a very, very small difference.

To highlight just how small this difference is, let’s talk about Bailey’s beads. Francis Bailey was a 19th century British astronomer. He was the first to observe that during the totality of a solar eclipse (when the Sun slips completely behind the Moon), some particles of sunlight still squeeze through and are observable around the Moon’s limb (outer edge). He correctly identified this phenomenon as due to the light passing through the Moon’s low valleys, and the spots became known as Bailey’s beads. So even when the Sun is completely behind the Moon, the fit is so tight, that the light that has traveled 93 million miles from the Sun can be  seen through a few low spots through the Moon’s rugged landscape. That demonstrates just how closely the apparent size of the Sun and Moon are to each other—namely, that just a difference of a few thousand feet on the Moon’s surface is enough to create a light gap in the eclipse. Pretty amazing when comparing that few thousand feet with the Sun’s diameter of over 865,000 miles. You begin to understand just how close the apparent size of each celestial object is to each other. As an astronomical note, the Sun and Moon’s distance from Earth vary throughout each of their respective orbits; so much so, that during some solar eclipses, the Sun never quite goes behind the Moon, and a small sliver of light surrounds the Moon. This is called an annular eclipse. But I digress.

I read this week that someone had calculated the odds of all these sizes and distances being in just the right proportions to each other and to Earth. In the October 2000 edition of Astronomy magazine, they explored this same idea, and said that the odds of the occurrence were enormous and pretty close to zero. The article I read this week said the probability was 1 million to the sixth power. Now, I can’t claim that the methodology for this figure is correct, but there are lots of potential sizes for suns (some are so large they would swallow every planet in our solar system), and sizes and shapes for moons, and for the distances of each to a specific planet, so it’s easy to see there are an almost countless number of possible combinations. Given our lack of precise knowledge of planets and moons outside of our little solar system, it’s probably impossible to fully quantify this probability, but I believe it’s easy to understand that, given the number of size and distance combinations, the probability is extraordinarily low. So, lacking a more precise figure, we will use this probability figure (1 million to the sixth power) for the remainder of our discussion.

Before moving forward, it would be helpful to name this phenomenon—of the Moon and Sun almost perfecting aligning in size during a solar eclipse. For lack of a better term, let’s call it a Perfect Eclipse. Now, it’s worth noting that you can have a solar eclipse that completely hides the sun (totality), without needing a Perfect Eclipse. If our Sun was just a bit smaller or further away, or the Moon a bit larger or closer, you would still witness an eclipse with the Sun disappearing behind the Moon. So, what we are talking about is the rarity of our Sun and our Moon being an almost perfect fit in apparent sizes. It’s also worth noting that there’s no known utility for having a Perfect Eclipse. I like to think it was the whimsy of our Creator. But however you want to interpret this phenomenon, let’s examine just how rare a Perfect Eclipse is. Here comes the math.

To begin, 1 million to the sixth power, is 10e+36, or 1 with 36 zeroes behind it (in the short scale naming system used in the U.S. and U.K., it’s called an undecillion). It’s a very big number. And when working with really big numbers, it’s very easy to lose sight of their relative size. So here’s a few examples that might make it easier to understand just how improbable it is that our Earth, Sun, and Moon line up the way they do.

The odds of you winning the Mega Millions lottery is 1 in 302,575,350. Not very good (but one can hope!). As small as your odds are of winning, they are still 3.3e+27 times greater than the odds that a Perfect Eclipse would occur (that’s 3.3 octillion, and it is also a very, very large number). So maybe we need another comparison to gain a better understanding of the rarity of a Perfect Eclipse.

There are an estimated 2 trillion galaxies in the universe, with each galaxy having an average of 100 million stars. Multiplying these two figures to get to the total estimated number of stars in the universe produces the value of 2e+23 (this is many times more than the total number of individual grains of sand on Earth). So let’s try a thought experiment. Let’s say you are stranded somewhere out in the cosmos and you want to get back to our solar system. You are given a chance to do so by drawing one, and only one, slip of paper out of a giant cosmic hat, into which the name of every star has been placed (now I know that most stars aren’t named, but this is just a thought experiment, so stay with me). The odds of you selecting our solar system is 1 in 2e+23. Not good. However, your odds are still 5 trillion times better than the odds of a Perfect Eclipse occurring. Starting to get the idea just how rare this phenomenon is?

One more example. Let’s say that you had the ability to create random, multiple combinations of sizes of the Moon and Sun, and their respective distances from Earth. And that you could create a new combination every second starting from the universe’s beginning (the Big Bang, about 13.7 billion years ago) to this day—that’s about 4.323e+17 seconds. Now, what are the odds that you have hit upon just the right combination to create a Perfect Eclipse? Only about 1 in 2,312,999,110,513,060,000 (if you are sounding out this number, it is 2 quintrillion, 312 quadrillion, 999 trillion, 110 billion, 513 million, and 60 thousand)! In other words, you are still a long, long way away from finding the combination for a Perfect Eclipse.

The Encyclopedia Britannica says that the almost perfect alignment of apparent sizes of the Moon and Sun during an eclipse is just a “mere coincidence.” I don’t agree, and I think we have shown just how rare this phenomenon really is. In fact, while unprovable, it’s quite likely, from a purely statistical standpoint, to be the only instance of a Perfect Eclipse in the universe!