Lagrange Points and Polynomial Equations: Part 1

I recently read a terrific article in the American Mathematical Monthly about Lagrange points, which are (from Wikipedia) “points of equilibrium for small-mass objects under the gravitational influence of two massive orbiting bodies.” There are five such points in the Sun-Earth system, called , , , , and .

To describe these Lagrange points, I can do no better than the estimable Isaac Asimov. I quote from his essay “Colonizing the Heavens” from his book The Beginning and the End, which was published in 1977. I read the book over and over again as a boy in the mid-1980s. (Asimov’s essay originally concerned the Earth-Moon system; in the quote below, I changed the words to apply to the Sun-Earth system.)

Imagine the Sun at zenith, exactly overhead. Trace a line due eastward from the Sun down to the horizon. Two-thirds of the way along that line, one-third of the way up from the horizon, is one of those places. Trace another line westward away from the Sun down to the horizon. Two-thirds of the way along that line, one-third of the way up from the horizon, is another of those places.

Put an object in either place and it will form an equilateral triangle with the Sun and Earth…

What is so special about those places? Back in 1772, the astronomer Joseph Louis Lagrange showed that in those places any object remained stationary with respect to the Sun. As the Earth moved about the Sun, any object in either of those places would also move about the Sun in such a way as to keep perfect step with the Earth. The competing gravities of the Sun and Earth would keep it where it was. If anything happened to push it out of place it would promptly move back, wobbling back and forth a bit (“librating”) as it did so. The two places are called “Lagrangian points” or “libration points.”

Lagrange discovered five such places altogether, but three of them are of no importance since they don’t represent stable conditions. An object in those three places, once pushed out of place, would continue to drift outward and would never return.

The last paragraph of the above quote represents a rare failure of imagination by Asimov, who wrote prolifically about the future of spaceflight. Points and are indeed stable equilibria, and untold science fiction stories have placed spacecraft or colonies at these locations. (The rest of Asimov’s essay speculates about using these points in the Earth-Moon system for space colonization.) However, while the points and are unstable equilibria, they do have practical applications for spacecraft that can perform minor course corrections to stay in position. (The point is especially unstable to outside gravitational influences and thus seems unsuitable for spacecraft.) Again from Wikipedia,

Sun–Earth L₁ is suited for making observations of the Sun–Earth system. Objects here are never shadowed by Earth or the Moon and, if observing Earth, always view the sunlit hemisphere… Solar and heliospheric missions currently located around L₁ include the Solar and Heliospheric Observatory, Wind, Aditya-L1 Mission and the Advanced Composition Explorer. Planned missions include the Interstellar Mapping and Acceleration Probe(IMAP) and the NEO Surveyor.

Sun–Earth L₂ is a good spot for space-based observatories. Because an object around L₂ will maintain the same relative position with respect to the Sun and Earth, shielding and calibration are much simpler… The James Webb Space Telescope was positioned in a halo orbit about L₂ on January 24, 2022.

Earth–Moon L₁ allows comparatively easy access to Lunar and Earth orbits with minimal change in velocity and this has as an advantage to position a habitable space station intended to help transport cargo and personnel to the Moon and back. The SMART-1 Mission passed through the L₁ Lagrangian Point on 11 November 2004 and passed into the area dominated by the Moon’s gravitational influence.

Earth–Moon L₂ has been used for a communications satellite covering the Moon’s far side, for example, Queqiao, launched in 2018, and would be “an ideal location” for a propellant depot as part of the proposed depot-based space transportation architecture.

While the locations and are easy to describe, the precise locations of and are found by numerically solving a fifth-order polynomial equation. This was news to me when I read that article from the American Mathematical Monthly. While I had read years ago that finding the positions of the other three Lagrange points wasn’t simple, I did not realize that it was no more complicated that numerically finding the roots of a polynomial.

The above article from the American Mathematical Monthly concludes…

[t]he mathematical tools that Lagrange uses to arrive at a solution to this three-body problem lie entirely within the scope of modern courses in algebra, trigonometry, and first-semester calculus. But surely no ordinary person could have pursued the many extraordinarily complicated threads in his work to their ends, let alone woven them together into a magnificent solution to the problem as he has done. Lagrange noted in the introduction to his essay, “This research is really no more than for pure curiosity …” If only he could have watched on Christmas Day as the James Webb Space Telescope began its journey to the Lagrange point !

In this short series, we discuss the polynomial equations for finding and .