We can predict solar eclipses to the second. Here’s how.

On October 14, the Western Hemisphere will witness an annular solar eclipse. The moon will be too small and far away in our view to totally block out the sun’s disc. Instead, it will blot out its center, leaving a ring at the edges. The best locations to view that ring of fire in the sky will be along a path that cuts through Oregon, Texas, Central America, Colombia, and finally northern Brazil. You might decide to visit Albuquerque, New Mexico, where you’ll experience exactly 4 minutes and 48 seconds of an annular eclipse.

And if you’re seeking a true total eclipse, you only have to wait another six months. On April 8, 2024, at 2:10 p.m. Eastern (12:10 p.m. local time), Mazatlan, Mexico will become the first city in North America to see most of the sun vanish in shadow. The path of totality then arcs through Dallas and Indianapolis into Montréal, New Brunswick, and Newfoundland in Canada. We know all of these precise details—and more—thanks to our knowledge of where the moon and sun are situated in the sky at any given moment.

In fact, we can predict and map eclipses farther into the future, even centuries from now. Because they know the precise positions of the moon and the sun and how they shift over time, scientists can project the moon’s shadow onto Earth’s globe. And with cutting-edge computers, it’s possible to chart eclipse paths down to a range of a few feet.

A solar eclipse needs three things. It results when the moon blocks the sun’s light from our vantage point on Earth. So to predict an eclipse, you must know where and how the sun, moon, and Earth move in relation to each other. This isn’t quite as elementary as it may seem, because the solar system isn’t flat. The moon’s orbit slants about 5 degrees in relation to the sun’s path, which astronomers call the ecliptic. While our satellite passes between Earth and the sun around once a month—which we call a new moon—the two rarely seem to cross paths.

Solar eclipses can only occur when the moon is at one of the two points where the moon’s orbit crosses the ecliptic, known as a node. If the moon is new at this crossing, the result is a solar eclipse.

In centuries past, trying to predict eclipses meant predicting minute details of finicky orbits. But as astronomers learned more about how celestial objects moved, they began tabulating what they call ephemerides: predictions of where the moon, sun, and planets will be in the sky. Ephemerides are still the key to eclipse prediction.

[Related: Make a classic pinhole camera to watch the upcoming solar eclipse]

“All you need is the ephemeris data…you don’t have to actually track the orbit,” says C. Alex Young, a solar physicist at NASA’s Goddard Space Flight Center.

With ephemeris data, astronomers can pinpoint dates and times when the moon and sun cross paths. Once you know that date, mapping an eclipse is relatively straightforward. Ephemerides let scientists project the moon’s shadow onto Earth’s sphere; with 19th-century mathematics, they can calculate the shape and latitude of two features of that shadow, the umbra and penumbra. Then, by knowing what time it is and where Earth is angled in its rotation, it’s possible to determine the longitudes. Putting these together produces an eclipse map.

In the past, astronomers printed the ephemerides in almanacs, long tomes filled with page after page of coordinate tables. Just as all of astronomy has advanced into an era of computers, so have ephemerides. Scientists today mathematically model the paths of the moon, sun, planets, other moons, asteroids, and much more.

NASA’s Jet Propulsion Laboratory (JPL) regularly publishes a new compendium of celestial locations every few years. The most recent edition, 2021’s DE440, accounts for details like the moon’s core and mantle sloshing around and slowing its rotation. “Generally speaking, we know where the moon is from the Earth to about a meter, maybe a couple of meters,” says Ryan Park, an engineer at JPL. “We typically know where the sun is to maybe a couple hundred meters, maybe 300 meters.”

[Related: How to look at the eclipse without damaging your eyes]

Ephemerides serve other purposes, especially when planning spaceflight missions. But it’s largely due to more sophisticated ephemeris data that we can now reliably predict the motions of the moon for the centuries ahead. In fact, you can find detailed maps of solar eclipses nearly a millennium in the future. (If you’re lucky enough to be in Seattle on April 23, 2563 or in Amsterdam on September 7, 2974, prepare for total eclipse day.)

But these maps, like most eclipse maps, show the path of totality or annularity as a smooth line crossing Earth’s surface. That isn’t an accurate representation. “This was designed for pencil and paper calculation, so it makes a lot of simplifying assumptions that are just a tiny bit wrong,” says Ernie Wright, who makes eclipse maps for NASA Goddard, “for instance that the moon is a perfectly smooth sphere.”

Both the moon and Earth are jagged at the edge. Earth’s terrain can block some views of the sun, and the moon has its own patchwork of mountains and valleys. In fact, sunbeams passing through lunar vales create the Baily’s beads and “diamond ring” often seen at an eclipse’s edge. “We now have detailed terrain information of these mountains from the Lunar Reconnaissance Orbiter,” Young says.

Wright has helped devise a new way of mapmaking that swaps the Victorian-age mathematics out for modern computer graphics. His method turns Earth’s surface into a map of pixels, each one with different latitude, longitude, and elevation, with the sun and moon in the sky above. Then, the method calculates which pixels see which parts of the moon block which parts of the sun.

“You then make a whole sequence of maps at, say, one-second intervals for the duration of the eclipse,” Wright says. “You end up with a frame sequence that you can put together to make a movie of the shadow.” This new technique—only possible with modern computers and ultraprecise ephemerides—may allow us to make eclipse maps that clearly show whether you can see an eclipse from, say, your house.

“I think that’s going to provide a whole new set of maps in the future that are going to be much more accurate,” says Young. “It’s going to be pretty exciting.”