About 250 years ago, mathematicians wrote the first equations describing where the James Webb Space Telescope, launched on Christmas Day 2021, now resides. Webb will stay put for around 20 years at its cosmic parking spot, surveying the universe’s galaxies. And we don’t have to worry about it wandering away: Its new home is a gravitationally balanced spot relative to Earth and the sun, called a Lagrange point.
Webb experiences the pull of gravity from both our own planet and the sun at Lagrange point 2 (L2), one of five such points in the sun-Earth system. Centripetal force—which makes objects move in a circle around an object with gravity—also accelerates the telescope into orbit with that system, causing it to revolve around, and get pulled toward, L2. Space explorers love Lagrange points because when viewed from Earth, the points appear to stay in fixed locations, making them convenient for communicating with spacecraft. In the 18th century, mathematicians pinpointed the five Lagrange points that rule the motion of satellites like Webb; it was an exercise in understanding the motions of a two-body system like Earth and the moon. But Lagrangian math must account for the motions of three bodies based on their gravitational attractions, initial positions, and velocities.
There’s an infinite number of solutions to this three-body problem, says astrophysicist Neil Cornish, who studies gravitational waves at Montana State University in Bozeman, Montana, and wrote an explanation of Lagrange points for NASA.
You have to look for the total force exerted on the smaller-mass body using Newton’s second law of motion, which states that the force acting on an object is equal to the mass of an object times its acceleration. You can feel this when you push an empty shopping cart and a full one; the full cart will move more slowly, and it takes more force to push it.
But you can’t ignore the movements of all three bodies. Earth is spinning on its axis, leading to the Coriolis effect, which causes objects to move in curved lines. (It’s the reason why hurricanes and projectiles trace a curved path.) Centripetal force also drives an object revolving around a central mass to be pulled toward the center of that mass.
Cornish compares Lagrange points to a marble on a hilly surface. “If I put a marble on the very top of a hill…the forces balance, but if I flick that marble a little tiny bit, it’s going to roll off the hill,” he says. “So we call that an ‘unstable’ point. Whereas, if it was right at the bottom of the valley, the forces balance. And if I was to knock it, it would actually just oscillate back and forth in the valley. So we call that a ‘stable’ equilibrium.”
To find these points of stability, the equations have to balance all the forces. To do so, you must solve a polynomial equation. (Polynomials help you find all the values of a variable that make an equation equal zero.) You can plow through many math operations to find Lagrange points, but that requires solving a messy 12th- or 15th-order polynomial equation. Unfortunately, it’s daunting to solve for high-order polynomials, Cornish says. In contrast, a fifth-order polynomial looks like this and is solvable:
p(x) = 2x5+x4–2x–1
“So I use physical intuition to kind of guide me to roughly where the points might be, because the math on its own can get really messy,” Cornish says. In the end, he only had to solve three fifth-order equations and one second-order equation. Cornish considered symmetry along the line formed by two points, with the sun and Earth on either side. This reasoning eliminates any Lagrange points outside of this plane of the ecliptic (the imaginary plane containing Earth’s orbit around the sun). Three Lagrange points are unstable and lie along that line (L1, L2, and L3), and two are stable (L4 and L5) and symmetric, above and below that line as points of an equilateral triangle.
“I was able to sort of eliminate an entire class of solutions just by thinking about it a bit, rather than just diving in and using brute force,” Cornish says.
We add calculus to the mix to describe the stability for each point, which is crucial for sending space missions to Lagrange points. Calculus gives the model a shake to see if the forces will hold an object in place or let it drift away over time.
If the Lagrange point is not fully stable, like Webb’s, spacecraft need regular course correction with a tiny fuel burn to nudge it back to the point’s center. In about 20 years, Webb’s fuel will run out, and it will drift from L2. From there, Cornish thinks it will wander out of our solar system and become an interstellar traveler.
Want to solve for Lagrange points yourself? An undergraduate student who’s taken an advanced mechanics class and vector algebra has all the tools they need to find those solutions.
Visualizing Lagrange Points
Let’s think of Lagrange points in simpler terms. Imagine a bowling ball (the sun) and a baseball (Earth) sitting on a horizontal plane, and each have their own gravitational pull. Because it’s much heavier, the bowling ball has an overall greater pull than the baseball. Next, toss a marble (a satellite) between the two. If it’s balanced perfectly between the two depressions, it’s akin to being on a “saddle point” sitting between two massive objects’ gravity wells on the plane. But if you push the marble in either direction too much, it will suffer from the pull of the larger object.