The Five Points of Lagrange
by Neil deGrasse Tyson
From Natural History Magazine, April 2002
The first manned spacecraft ever to leave Earth orbit was Apollo 8. This achievement remains one of the most remarkable, yet unheralded firsts of the twentieth century. When that moment arrived, the astronauts fired the third and final stage of their mighty Saturn V rocket, rapidly reaching nearly seven miles per second for the spacecraft and its three occupants. Half the energy to reach the Moon had been expended just to achieve Earth orbit. At about this time, a well-known television news anchor declared that the astronauts had just left Earth's gravity. But the astronauts were on their way to the Moon. And last anybody had checked, the Moon was in orbit around Earth by the action of mutual gravitational forces. So Earth's gravity must extend at least as far as the Moon. Fact is, the force of gravity for any object extends to the infinite reaches of space, even as it grows exponentially weaker.
After the third stage fired, engines were unnecessary, except for any mid-course tuning the trajectory might require to ensure the astronauts did not miss the Moon entirely. For ninety percent of its nearly quarter-million-mile journey, the spacecraft gradually slowed as Earth's gravity continued to tug in the opposite direction. Meanwhile, as the astronauts neared the Moon, its force of gravity grew stronger and stronger. A spot must therefore exist, en route, where the Moon's and Earth's opposing forces of gravity balance precisely. When the command module drifted across that point in space, its speed increased once again as it accelerated toward the Moon.
If gravity were the only force to be reckoned, then this spot would be the only place in the Earth-Moon system where the opposing forces canceled. But Earth and the Moon also orbit a common center of gravity, which lives about a thousand miles beneath Earth's surface along an imaginary line connecting Earth's center and the Moon's center. When things move in circles of any size and at any speed, they create a new force that pushes outward, away from the center of rotation. Your body feels this
centrifugal force when you make a sharp turn in your car or when you survive amusement park attractions that turn in circles. In a classic example of these nausea-inducing rides, you stand along the edge of a large circular platter, with your back against a perimeter wall. As the thing spins up, rotating faster and faster, you feel a stronger and stronger force pinning you against the wall. You can't now move. That's when they drop the floor from below your feet and turn the thing sideways and upside down. When I rode one of these as a kid, the force was so great that I could barely move my fingers, they, being stuck to the wall along with the rest of me.
If you actually got sick on such a ride, and turned your head to the side, the vomit would fly off at a tangent. Or it might get stuck to the wall. Worse yet, if you didn't turn your head, it might not make it out of your mouth due to the extreme centrifugal forces acting in the opposite direction. (Come to think of it, I haven't seen this particular ride anywhere lately. I bet they've been outlawed.)
Centrifugal forces arise as the simple consequence of an object's tendency to travel in a straight line after being set in motion, and so are not true forces at all. But you can calculate with them as though they are. When you do, as did the brilliant eighteenth-century French mathematician Joseph Louis Lagrange, you discover spots in the rotating Earth-Moon system where the gravity of Earth, the gravity of the Moon and the centrifugal forces of the rotating system balance. These special locations are known as the points of Lagrange. And there are five of them.
The first point of Lagrange (affectionately called L1) falls between Earth and the Moon, slightly closer to Earth than the point of pure gravitational balance. Any object placed there can orbit the Earth-Moon center of gravity with the same monthly period as the Moon and will appear to be locked in place along the Earth-Moon line. Although all forces cancel there, this first Lagrangian point is a precarious equilibrium. If the object drifts sideways in any direction, the combined effect of the three forces will return it to its former position. But if the object drifts toward or away from Earth ever so slightly, it will irreversibly fall either toward Earth or the Moon, like a barely balanced cart atop a steep hill, a hair's-width away from falling down one side or the other.
The second and third Lagrangian points (L2 and L3) also lie on the Earth-Moon line, but this time L2 lies far beyond the far side of the Moon, while L3 lies far beyond Earth in the opposite direction. Once again, the three forces—Earth's gravity, the Moon's gravity, and the centrifugal force of the rotating system—cancel in concert. And once again, an object placed in either spot can orbit the Earth-Moon center of gravity with the same monthly period as the Moon.
The gravitational hilltops represented by L2 and L3 are much broader than the one represented at L1. So if you find yourself drifting down to Earth or the Moon, only a tiny investment in fuel will bring you right back to where you were.
While L1, L2, and L3 are respectable space places, the award for best Lagrangian points must go to L4 and L5. One of them lives far off to the left of the Earth-Moon centerline while the other is far off to the right, each representing a vertex of an equilateral triangle, with Earth and Moon serving as the other vertices. At L4 and L5, as with their first three siblings, forces are in equilibrium. But unlike the first three Lagrangian points, which enjoy only unstable equilibrium, the equilibria at L4 and L5 are stable; no matter which direction you lean, no matter which direction you drift, the forces prevent you from leaning farther, as though you were in a valley surrounded by hills. For each of the Lagrangian points, if your object is not located exactly where all forces cancel, then its position will oscillate around the point of balance in paths called librations. (Not to be confused with the particular spots on Earth surface where one's mind oscillates from ingested libations.) These librations are equivalent to the back-and-forth rocking a ball would undergo after rolling down a hill and overshooting the bottom.
More than just orbital curiosities, L4 and L5 represent special places where one might build and establish colonies. All you need do is ship to the area raw construction materials (mined not only from Earth, but perhaps from the Moon or an asteroid), leave them there with no risk of drifting away, and return later with more supplies. After all the raw materials were collected in this zero-G environment, you could build an enormous space station—tens of miles across—with very little stress on the construction materials. By rotating the station, the induced centrifugal forces could simulate gravity for its hundreds (or thousands) of residents. The space enthusiasts Keith and Carolyn Henson founded the
L5 Society in August 1975 for just that purpose, although the society is best remembered for its resonance with the ideas of Princeton physics professor and space visionary Gerard K. O'Neill, who promoted space habitation in his writings such as the 1976 classic The High Frontier: Human Colonies in Space. The L5 Society was founded on one guiding principle:
to disband the Society in a mass meeting at L5, presumably inside a space habitat, thereby declaring their mission accomplished. In April 1987, the L5 Society merged with the National Space Institute to become the National Space Society, which continues today.
The idea of locating a large structure at libration points appeared as early as 1961 in Arthur C. Clarke's novel A Fall of Moondust. Indeed, Clarke was no stranger to special orbits. In 1945, he was the first to calculate, in a four-page, hand-typed memorandum, the location above Earth's surface where a satellite's period exactly matches the 24-hour rotation period of Earth. A satellite with that orbit would
hover over Earth's surface and serve as an ideal relay station for radio communications from one part of Earth to another. Today, hundreds of communication satellites do just that. Where is this magical place? Objects in low Earth orbit, such as the Hubble Space Telescope and the International Space Station, take about ninety minutes to circle Earth. Objects at the distance of the Moon take about a month. Logically, an intermediate distance must exist where an orbit of 24-hours can be sustained. That distance lies about 22,000 miles above Earth surface.
Actually, there is nothing unique about the rotating Earth-Moon system. Another set of five Lagrangian points exist for the rotating Sun-Earth system. The Sun-Earth L2 point in particular has become the destination of choice for many scientific satellites. The Sun-Earth Lagrangian points all orbit the Sun-Earth center of gravity once per Earth year. At a million miles from Earth, in the direction opposite that of the Sun, a telescope at L2 will have 24-hours of continuous view of the night sky because Earth has shrunk to the size of the Moon in Earth's sky. For observers in low orbits, such as that of the Hubble Telescope, Earth blocks a significant field of view. The recently launched Microwave Anisotropy Probe (MAP for short), reached L2 for the Sun-Earth system in a couple of months and is now librating there, busily taking data on the cosmic microwave background—the omnipresent signature of the big bang itself. The real estate for the Sun-Earth L2 is even wider than that for the Earth-Moon L2. By saving only ten percent of its total fuel, the MAP satellite has enough to hang around this point of unstable equilibrium for nearly a century.
The next generation space telescope, now being planned by NASA as the follow-on to the Hubble, is also being designed to work at the Sun-Earth L2 point. And there is plenty of room—tens of thousands of square miles—for more satellites to come.
Another Lagrangian-loving NASA satellite, known as Genesis, will librate around the Sun-Earth L1 point. In this case, L1 lies a million miles toward the Sun. For two and a half years, Genesis will face the Sun and collect pristine solar matter, including atomic and molecular particles from the solar wind. The material would then be returned to Earth via a mid-air recovery over Utah and be studied for its composition, which provides a window to the contents of the original solar nebula from which the Sun and planets formed. After leaving L1, the returned sample will do a loop-the-loop around L2 and position its trajectory before it returns to Earth.
Given that L4 and L5 are stable points of equilibrium, one might suppose that space junk would accumulate near them, making it quite hazardous to conduct business there. Lagrange made a prediction that space debris would be found at L4 and L5 for the gravitationally powerful Sun-Jupiter system. A century later, in 1905, the first of the Trojan family of asteroids were discovered. We now know that for L4 and L5 of the Sun-Jupiter system, thousands of asteroids lead and follow Jupiter around the Sun, with periods that equal Jupiter's period around the Sun. As though they were responding to tractor beams, these asteroids are forever tethered in place by the gravitational and centrifugal forces of the Sun-Jupiter system. (These asteroids pose no risk to life on Earth, they, being stuck in the outer solar system, and out of harm's way.) Of course, we expect space junk to accumulate at L4 and L5 of the Sun-Earth system as well as the Earth-Moon system. It does. But not nearly to the extent of the Sun-Jupiter encounter.
As an important side benefit, interplanetary trajectories that begin at Lagrangian points require very little fuel to reach other Lagrangian points or even other planets. Unlike a launch from a planet's surface, where most of your fuel goes to lift you off the ground, launching from a Lagrangian point would resemble the a ship leaving dry-dock—becoming adrift into the ocean with only a minimal investment of fuel. In modern times, instead of thinking about self-sustained Lagrangian colonies of people and farms, we can think of Lagrangian points as gateways to the rest of solar system. From the Sun-Earth Lagrangian points you are half way to Mars; not in distance or time but in the all-important category of fuel consumption.
In one version of our space-faring future, imagine fuel stations at every Lagrangian point in the solar system, where travelers fill up their rocket gas tanks en route to visit friends and relatives elsewhere among the planets. This travel model, however futuristic is reads, is not entirely farfetched. Note that without fueling stations scattered liberally across the United States, your automobile would require the proportions of the Saturn V rocket to drive coast to coast: most of your vehicle's size and mass would be fuel, used primarily to transport the yet-to-be-consumed fuel during your cross-country trip. We don't travel this way on Earth. Perhaps the time is overdue when we no longer travel that way through space.
Neil deGrasse Tyson, an astrophysicist, is the Frederick P. Rose Director of New York City’s Hayden Planetarium and a visiting research scientist at Princeton University.
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