20 May/June 2022
Deep Math
5PHOTO^ ILLUSTRATIONBYALYSE^ MARKEL
Visualizing Lagrange PointsLet’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 amarble (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.venient for communicating with spacecraft.
In the 18th century, mathematicians pinpointed
the five Lagrange points that rule the motion of sat-
ellites 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 Cor-
nish, 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
empt y 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 f lick 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 bal-
ance. 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 poly-
nomial 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) = 2x^5 +x^4 –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-or-
der 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 eclip-
tic (the imaginary plane containing Earth’s orbit
around the sun). Three Lagrange points are unsta-
ble 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 sta-
bility 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 correc-
tion 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, Cor-
nish thinks it will wander out of our solar system
and become an interstellar traveler.saddle
point