In popular usage the word librate has long passed out of style. It originally came from the Latin word librare, to balance. Librate means to vibrate slightly, as a balance or a scale does before it settles down. An object that librates is poised between two competing forces. Scientifically the word has had a longer lifetime, because the depths of outer space are sprinkled with what are called libration points: places where a satellite or a pebble or anything else that might get there would find itself perfectly balanced between competing gravitational forces. There are five of them shared by Earth and the sun, for instance, and another set of five shared by Earth and the moon. As the moon rotates around Earth, and Earth around the sun, these libration points rotate with them. Put a satellite at a libration point and it would appear motionless from Earth, hanging in space as though the laws of gravity had been suspended.
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In the past few years the study of libration points has gone from an academic exercise to a revelation. NASA now has four space missions in the works that will use the gravitational weirdness of libration points for everything from mapping the whisper of radiation left over from the Big Bang to photographing Earth 24 hours a day. Meanwhile, researchers at Caltech and Purdue University in Indiana have applied the mathematics of libration points to the solar system at large, creating a theory of how asteroids, comets, and dust move around and how spacecraft could follow the same invisible rivers of gravity to travel from planet to planet or moon to moon with little more fuel than it would take to drive a car from New York to Los Angeles. The study of libration points has become the pursuit of free rides. If mission planners do their math right, says Purdue astronautical engineer Kathleen Howell, once a spacecraft reaches the right velocity and position above Earth’s atmosphere, “you’ll never have to turn its engines on. It will just go where it has to go.”
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There is a tendency to think that the solar system is a simple place, to assume that the planets rotate easily around the sun, the moons around the planets, and that comets zing in and out in curvaceous orbits. The field of gravity that pervades this local space, we imagine, does so in a smooth, predictable manner. According to this view of the world, if you happen to drop into space near Earth, you’ll fall Earthward; if you’re closer to the sun, you’ll fall toward the sun. And somewhere in between, well, you’ll go one way or the other. But the solar system is not so mundane. With every planet and every moon continuously tugging away at one another, and the sun tugging away at all of them, a spacecraft that escapes Earth or the moon can find itself thrown into a complex and chaotic world of competing gravitational forces. What seems simple can quickly become the most complex and unpredictable of environments. (See “Unsolved Mystery,”)
Kathleen Howell (top) and Martin Lo map undulating manifolds that pull heavenly bodies through the solar system. They plan spacecraft trajectories that require astonishingly little fuel. |
That all began to change when mathematician Robert Farquhar got involved in the trajectory design business in the 1950s. He says his interest in trajectories was first stirred when Sputnik went up three weeks after he started a course in orbital mechanics at the University of Illinois. While studying at Stanford with John Breakwell, a legendary aeronautical engineer, Farquhar started working out the dynamics of libration points and “halo” orbits—three-dimensional loops around the points—so named because from Earth the orbit would look like a halo around the libration point. Halo orbits, however, were not so simple. For starters, they could be huge: a halo orbit around a libration point shared by Earth and the sun might be hundreds of thousands of miles around. And they had shapes unlike those of any orbits designers had ever encountered. “They look like a line drawn around the edge of a Pringle’s potato chip,” says Martin Lo, a mission designer at NASA’s Jet Propulsion Laboratory in Pasadena, California.
In 1966, Farquhar began arguing that halo orbits were ideal places from which to study Earth, the sun, and the depths of space. If you parked a spacecraft in a halo orbit, you could look down on the moon, back at Earth, or in toward the sun and stay there for years with minimal fuel to keep the spacecraft in orbit, a task mission designers call station-keeping. For many satellites, station-keeping eats up millions of dollars a year in fuel and labor costs.
Genesis will loop around libration points to collect bits of the sun’s atmosphere. |
After this success, Farquhar got out of the libration point business, but by that time Breakwell had enticed another graduate student to take over—Kathleen Howell. Howell set out to find a better way to plan mission trajectories than the trial-and-error methods used so far to discover halo orbits. Farquhar and his colleagues had calculated a single functioning halo orbit for the Explorer, as well as a “transfer orbit” to get it from an orbit around Earth—known as a parking orbit—out to the halo orbit. To find those orbits, Farquhar used a method called shooting. Howell describes it this way: “You guess what kind of conditions you need to launch from Earth”—for example, how strong a thrust to give the spacecraft, which way to point it, where to launch it. “Then you simulate the flight and just see where it goes. If you do enough simulations, you start to sort of see what might work.”

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Computing Manifolds of Halo Orbits











