# UTSA astrophysicist reveals the math behind eclipse predictions

## You may want to dig out the protractors, calculators

SAN ANTONIO – I’ve taken plenty of math classes in my day, but when it comes to figuring out when and where a total eclipse will occur, that’s next level math.

Thankfully, there are people far smarter than myself, including UTSA’s Dr. Angela Speck, who heads up the physics and astronomy department, who can explain such things.

When I arrived at her office, I found her armed with a grease board and a marker. Below are the Cliff Notes of our conversation.

My first question to Speck, who was donning a dress covered in constellations, was simply “how difficult is it to determine the path of an eclipse?”

That question prompted a long pause. That says a lot. I mean, Dr. Speck is an astrophysicist for goodness sakes! But, as she’ll tell you, there’s a lot and I mean A LOT to consider. We can simplify the players: the sun, the moon, and the earth. That’s where the simplicity ends.

”The moon isn’t always between us and the sun, and so you get kind of a weird mismatch when they do intersect. ‘Where is going to hit on the earth?’” she said.

She began identifying angles and distances between the three major players. I had flashbacks to high school geometry. Sine, cosine, tangent... remember those?

“Yeah, this is all geometry,” Speck said.

So, to figure out this problem, you should probably know that the average distance from the earth to the sun is one astronomical unit, or AU. For the math enthusiasts out there this is A³=GMD²/k² where A= unit distance, G= gravitational constant , M=solar mass, k=numerical value of Gaussian gravitational constant (of course! 😜). Long story short, it’s roughly 93 million miles.

”This is about one four hundredth of an AU,” Speck explained, drawing the distance between the moon and the earth. “So you can start to figure out, ‘OK what is this?’ So you end up with ‘How big is this and where does it land on the earth?’”

Trust me when I tell you there’s much more math than that, but we’ll skip forward for the sake of time.

”Once you worked out one [eclipse], you can start to predict when they will be in the future and when they will be in the past,” Speck said.

But this is also when it gets even more complicated, if you can believe that. Keep in mind that our spheres (moon, earth, and sun) are not perfect spheres. They are also all spinning, wobbling, and orbiting each other.

”You’re going to get all the cycles of the moon going around the earth and the earth going around the sun,” Speck said. “They all come together every 18 years, 11 days, and 8 hours.”

That seems very specific. Turns out, it’s got a name: the saros cycle. It’s defined as “a period of exactly 223 synodic months, approximately 6585.321 days, or 18 years, 10, 11, or 12 days (depending on the number of leap years), and 8 hours, that can be used to predict eclipses of the Sun and Moon. Weirdly enough, out of all of that, it’s the 8 hours that throws everything off.

”So, we just had an annular eclipse and if you’ve seen the path of that, you’ve got an annular eclipse, you’ve got a total eclipse, you’ve got this lovely cross,” said a demonstrative Speck, showing how the two paths cross over San Antonio. “There was a cross like that 18 years, 11 days ago, but because of the eight hours its not over San Antonio, it’s a third of the way around the planet, because that’s a third of a day. So it was in North Africa... Libya.”

In other words, we’re lucky to be living in San Antonio at this moment. Very lucky. Additionally, we’re also lucky to be living on Earth. For as complicated as the math seems to be, the equations all work out in favor of us Earthlings. Take that Martians! 👽

”We have a moon that is exactly the right size and distance from us, so that it looks the same size as the sun in the sky, it’s much closer so it looks much bigger,” Speck explained.

I also learned that the adage “what a time to be alive” has never been more true. In the past, the moon was too close and the eclipses blacked out everything and...

”In the future, the moon will be too far away, and you’ll only get annular eclipses,” Speck said.

This is really just skimming the surface of what brilliant minds over eons have been able to figure out to determine the path and length of an eclipse. Incredibly, that prediction can now be pinpointed all the way down to street level. With the edge of the path passing over San Antonio, it’s possible that one side of a street could witness 100% totality, while the other side of the street is technically not in totality (it’ll still be somewhere around 99%). It’s incredible to think that math and science can predict this years in advance!