The Math That Helps the James Webb Space Telescope Sit Steady in Space

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.

-  MANASEE WAGH

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.