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

Lagrange

Euler

Poincaré

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,”)

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.

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.

Genesis will loop around libration points to collect bits of the sun’s atmosphere.

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.”

 



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Computing Manifolds of Halo Orbits