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.

Libration points are mathematical fictions, geometric fantasies, topological flights of fancy. There is nothing visible at a libration point. No signpost says HERE ALL FORCES CANCEL. But if you arrive at one, you may stay there with extraordinarily little effort, or you could orbit around one, as though the libration point were a planet rather than a spot of nothing. You could even surf from one libration point to another, from one side of Earth to the other, or one side of the solar system to the other, while barely exerting any effort.

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.”

**UNSOLVED MYSTERY**

While the solar system may seem like a relatively simple place, with moons orbiting planets, and planets orbiting the sun like clockwork, the mathematics that describes this system makes up one of the most famous unsolved problems in the field. Known as the n-body problem, it has stumped the world’s greatest mathematicians for 400 years. It goes like this: take empty space and sprinkle it with any number (hence the “n”) of planets, spaceships, suns, comets, and various celestial objects. Give them all some initial speeds and directions and let gravity go to work. Then calculate where those n bodies will be going, at what speed, and in what trajectories or orbits, from now until the end of time.

Mathematicians have made some progress on the problem, but only by simplifying it so much that it has little relevance to reality. Newton solved the two-body problem—the sun, for instance, and a single planet—and found that the two bodies, depending on their initial conditions, will always follow one of three possible trajectories, known as conics.

Add a third body, however—say, another moon or a spacecraft—and the problem gets very complicated. The eighteenth-century mathematician Joseph-Louis Lagrange spent much of his life obsessed with this three-body problem, and all he managed to come up with were five solutions—the libration points. (These five libration points are also known as Lagrange points, although Leonhard Euler, the Swiss mathematician, discovered three of them first.) A century later, Jules-Henri Poincaré developed two whole branches of mathematics—topology and dynamical systems—to get a handle on the n-body problem. Poincaré failed as well.

Between 1892, when Poincaré published his treatise on the problem, and the mid-1960s, a series of mathematicians followed Poincaré’s suggestion of painstakingly looking for weirdly shaped periodic orbits—the mathematical loops in the fabric of a three-body gravitational field. Computers made the task easier. Although Kathleen Howell, Martin Lo, and their colleagues haven’t solved the problem either, they’ve found approximations that can illuminate the structure of the solar system. —G. T.

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,”)

In its early days, the exploration of space moved forward ignoring these complexities. From Sputnik through the space shuttle, mission designers used a method that Jerry Marsden, a Caltech mathematician, calls Buck Rogering. They built huge rockets, such as the Saturn V, with enormous thrusters, lit the fuse, and they roared quickly out to the target and back, using enormous amounts of fuel. Huge rocket engines allowed the designers to ignore any gravitational effects more subtle than the tug of the planet they were leaving and that of the planet they were headed to.

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.

Farquhar was the first to employ halo orbits for a space mission, designing them into the trajectories for the International Sun Earth Explorer-3, launched in 1978. The spacecraft was sent to a halo orbit around a sun–Earth libration point, L1, which sits nearly a million miles into space on a line from Earth toward the sun. The mission of the Explorer was to study solar wind, and it needed to do that from a vantage point free from the influence of Earth’s magnetic field: the distant L1 halo orbit was a perfect spot. Because the Explorer was “very tenuously held by Earth,” says Farquhar, it was relatively easy to break that connection in 1982 and send the craft off on an unplanned mission—to fly through the tail of the comet Giacobini-Zinner. It did this in 1985.

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.”

To find a better method, Breakwell suggested that Howell study the mathematics of halo orbits. “He suggested we try to find out first what other halo orbits there might be. Is there only one? A whole bunch? Do they exist all around the solar system, or just near Earth? At all libration points, or just one?” Howell dedicated her graduate work as well as the next 15 years to that study, using techniques from a branch of mathematics called dynamical systems—or, more dramatically, chaos theory.

**MANIFOLD DESTINY**

To a mathematician, a manifold is nothing but a surface. Imagine a child making Christmas tree ornaments by painting various objects gold—a pinecone, a toy trumpet, a piece of coral. If each layer of paint were infinitely thin, those layers would be manifolds. The only catch is that these objects can exist in any number of dimensions. We’re used to thinking in three dimensions, of course, and even four isn’t much of a stretch, if you count time as the fourth. As you inflate a balloon and then let the air out, for example, it expands and contracts along all four axes: vertical, horizontal, depth, and time. If there are more than four dimensions, however, most people can’t quite visualize what’s going on—scientists have yet to invent n-D glasses for the mind’s eye. Nonetheless, mathematicians have worked out rules for how n-dimensional objects look and behave, and they trust the objects to follow the rules. For the manifolds that Kathleen Howell and Martin Lo study, each initial condition is another dimension. Varying one condition—a spacecraft’s starting point, say, or its initial velocity—will send the spacecraft onto a new path along the manifold or might even nudge it onto an entirely different manifold nearby.—Polly Shulman

Rather than try to calculate individual trajectories as though she were planning a space mission, Howell used computers to calculate tens of thousands of trajectories whose initial conditions differed only slightly. With one set of initial conditions, she would get a single trajectory. With, say, ten sets of slightly different initial conditions, she might get not just ten lines in space but ten lines that all lay on the same curvaceous surface, known in the lingo of mathematics as a manifold. These manifolds undulate with hills and dips, like a crumpled blanket or the surface of a walnut. (See “Manifold Destiny,” above.)

Because there are an infinite number of starting points for a spacecraft and an infinite number of starting velocities, there are also an infinite number of manifolds. “You can think of the solar system as foliated by these sheets and sheets, like an onion,” says Lo, “except that this onion is not just a sphere but some weird-shaped thing.”

Any group of three or more bodies will interact to create manifolds with bizarre, albeit subtle, gravitational effects. For instance, a spacecraft nudged off a libration point in one direction might float into space, following the curve of the manifold, while if it went off in another direction, it would float right back to the libration point. Similarly, there were manifolds that would drop a spacecraft onto a halo orbit, and other manifolds that would lift it off and drop it onto a different halo orbit. If you could find the right manifolds, Howell believed, you could place your spaceship anywhere you liked in the neighborhood of the Earth, sun, and moon.

Howell met Lo at a conference in the late 1980s. They shared a passionate conviction that the techniques of chaos theory had great potential for plotting mission trajectories. “We’d meet at conferences and really get into it,” says Howell, “trying to figure out how to convince people that we had a better way” to come up with spacecraft trajectories.

An opportunity came in 1995 with a mission known as Genesis. Proposed by Caltech geochemist Don Burnett, the idea was to put a spacecraft in orbit between the sun and Earth to collect particles of solar wind—electrically charged atoms from the sun’s atmosphere blown outward through the solar system. Planetary scientists believe the sun’s atmosphere, and hence solar wind, is probably the only piece of the solar system that has retained the system’s original chemical composition. Capturing some and studying it, Burnett believed, would help us understand what the primordial solar system was made of. The catch would be bringing it home. Genesis could collect solar particles on fragile, ultrapure wafers of silicon, sapphire, and germanium. Because a hard landing would shatter them, theGenesis payload would have to return to Earth in such a way that it could be snatched in midair by a helicopter—much gentler than splashing down. And because Genesis was not one of the more newsworthy missions, the whole thing had to be done with as little fuel and as small a spacecraft as possible.

Howell says Lo called her one Thursday in August 1996 and said they could get a chance to design the trajectory for the Genesis mission, but only if they could do it by Monday. So Howell and Brian Barden, her graduate student at Purdue, went to work using everything they had learned over the years about the local space of the Earth-sun system. By Sunday night, after what Howell calls a weekend from hell, she, Barden, and Lo had calculated the basic trajectory, a loopy path through space.

Genesis will be launched in January 2001, onto a manifold that will float it out onto a halo orbit around L1. After four orbits of six months each, it will float out of the halo orbit onto another manifold that will carry it past Earth. A gentle nudge will put it onto yet another manifold that will carry it out to L2, swing it around the libration point, and drop it back into Earth’s atmosphere, directly over Utah.

Genesis convinced NASA that libration points were essential for future missions. Dave Folta, a senior aerospace engineer with NASA’s Goddard Space Flight Center, says they are planning to use Howell’s techniques—which he admiringly calls “highfalutin math”—for at least four missions over the next decade, including the Next Generation Space Telescope, scheduled to replace the Hubble Space Telescope in eight years.

Meanwhile, Lo has moved his sights from local space missions to the solar system at large. He and his Caltech collaborators have taken to calculating libration points, halo orbits, and their attendant manifolds for all the planets in the solar system. What they’ve found has begun to confirm Lo’s suspicions that manifolds play crucial roles in determining the orbits and locations of all objects in the solar system smaller than planets and moons. For instance, Lo has shown that the manifolds surrounding the libration points of the outer planets all intersect. This suggests that any asteroids passing through such manifolds would most likely hitch a ride on the manifolds and drift right out of the solar system. This phenomenon might explain why there is an asteroid belt between Jupiter and Mars but none beyond Jupiter. Moreover, the orbits of some comets seem to trace the planes of manifolds with remarkable accuracy.

To Lo, the manifolds may provide a unified theory of the structure of the solar system, and the implications go well beyond space exploration. While it would theoretically be possible to use the manifolds to get from planet to planet, it would take far too long, he says. On the other hand, Jupiter or Saturn missions, for instance, could use local manifolds to explore those planets’ neighborhoods. Spacecraft could ride manifolds from one moon to another in weeks, with virtually no fuel. “You can sneak in on a stable manifold, be captured by one moon for a couple of periods and observe it, and then with greatly reduced energy go on to the next moon,” he says.

Closer to home, the manifolds in Earth’s vicinity might serve as inexpensive, low-fuel routes to and from the moon for commercial purposes. They might explain why Earth seems to be relatively free of asteroid and meteorite collisions compared with the other bodies in the solar system. Understanding the dynamical channels around the Earth-moon system, says Lo, might make it relatively easy to deflect a potentially Earth-bound asteroid, like the one that wiped out the dinosaurs, by gently nudging it onto a manifold that would take it away from Earth.

Meanwhile, Lo and Howell are working with Caltech’s Control and Dynamical Systems Department, a team of mathematicians and engineers, to plot the manifolds of the solar system in detail. For Lo, the experience has been a revelation. Back when he first started at JPL, he says, he felt frustrated as a mathematician reduced to doing mundane engineering. He was tempted to move to Wall Street, where he could do interesting math and make a nice living as well. Then he had a dream that showed him where his future lay. In his dream, the muddy waters of a river receded to reveal a river full of water buffaloes, wondrous animals “as happy as could be.” After that dream, he says, “I just knew I was not to leave here. I knew that there were great riches here to be discovered.”