Death by Black Hole: And Other Cosmic Quandaries - Neil deGrasse Tyson (2014)
SECTION 2. THE KNOWLEDGE OF NATURE
Chapter 9. THE FIVE POINTS OF LAGRANGE
The first manned spacecraft ever to leave Earth’s 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 thrusting the command module and its three occupants up to a speed of nearly seven miles per second. Half the energy to reach the Moon had been expended just to reach Earth’s orbit.
The engines were no longer necessary after the third stage fired, except for any midcourse tuning the trajectory might require to ensure the astronauts did not miss the Moon entirely. For 90 percent of its nearly quarter-million-mile journey, the command module gradually slowed as Earth’s gravity continued to tug, but ever more weakly, in the opposite direction. Meanwhile, as the astronauts neared the Moon, the Moon’s 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 each other out. But Earth and the Moon also orbit a common center of gravity, which resides about a thousand miles beneath Earth’s surface, along an imaginary line connecting the centers of the Moon and Earth. When objects 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 contraption spins up, rotating faster and faster, you feel a stronger and stronger force pinning you against the wall. At top speeds, you can barely move against the force. That’s just when they drop the floor from beneath your feet and twist 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 wonder if 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 (1736-1813), 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 directly toward or away from Earth, ever so slightly, it will irreversibly fall either toward Earth or the Moon, like a barely balanced marble atop a steep hill, a hair’s-width away from rolling 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, all forces balance. But unlike the other 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’s 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 space colonies. All you need do is ship raw construction materials to the area (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-gravity environment, you could build an enormous space station—tens of miles across—with very little stress on the construction materials. And 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 “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. 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 appear to “hover” over Earth’s surface and serve as an ideal relay station for radio communications from one nation to another. Today, hundreds of communication satellites do just that.
Where is this magical place? It’s not low Earth orbit. Occupants there, such as the Hubble Space Telescope and the International Space Station, take about 90 minutes to circle Earth. Meanwhile, 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 happens to lie 22,300 miles above Earth’s 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 darling of astrophysics 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 earns 24 hours of continuous view of the entire night sky because Earth has shrunk to insignificance. Conversely, from low Earth orbit, the location of the Hubble telescope, Earth is so close and so big in the sky, that it blocks nearly half the total field of view. The Wilkinson Microwave Anisotropy Probe (named for the late Princeton physicist David Wilkinson, a collaborator on the project) reached L2 for the Sun-Earth system in 2002, and has been busily taking data for several years on the cosmic microwave background—the omnipresent signature of the big bang itself. The hilltop for the Sun-Earth L2 region in space is even broader and flatter than that for the Earth-Moon L2. By saving only 10 percent of its total fuel, the space probe has enough to hang around this point of unstable equilibrium for nearly a century.
The James Webb Telescope, named for a former head of NASA from the 1960s, is now being planned by NASA as the follow-on to the Hubble. It too will live and work at the Sun-Earth L2 point. Even after it arrives, plenty of room will remain—tens of thousands of square miles—for more satellites to come.
Another Lagrangian-loving NASA satellite, known as Genesis, librates around the Sun-Earth L1 point. In this case, L1 lies a million miles toward the Sun. For two and a half years, Genesis faced the Sun and collected pristine solar matter, including atomic and molecular particles from the solar wind. The material was then returned to Earth via a midair recovery over Utah and studied for its composition, just like the sample return of the Stardustmission, which had collected comet dust. Genesis will provide a window to the contents of the original solar nebula from which the Sun and planets formed. After leaving L1, the returned sample did a loop-the-loop around L2 and positioned its trajectory before it returned 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, in fact, predicted 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 was 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 that of Jupiter’s. Behaving for all the world as though they were responding to tractor beams, these asteroids are eternally tethered by the gravitational and centrifugal forces of the Sun-Jupiter system. 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 a ship leaving dry dock, gently cast 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 halfway 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 it reads, is not entirely far-fetched. 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.