The Sciences

Gravity's Rim

Between the gravitational basin of one celestial body and another lies a fuzzy, chaotic boundary. Now one mathematician has found a way to ride the edge of chaos to the moon.

By Adam FrankSep 1, 1994 5:00 AM


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For more than 30 years we’ve been navigating the solar system with our fists. Our spacecraft, perched atop thundering rockets the size of skyscrapers, are blasted off the face of Earth. Arriving at their destination in a headlong rush, they must pivot, then blast their retro-rockets in a desperate effort to slow down enough so that they will be lassoed by the gravity of the target planet or moon and pulled into orbit.

This brute-force approach to celestial mechanics has served us well in the past, but it is a dangerous and expensive game. For example, the freight costs to the moon, our closest neighbor, are a whopping $1 million per pound. And the retro-rockets must fire at exactly the right moment to put the spacecraft into lunar orbit. If they fail, all those millions of dollars of hardware become instant space junk, sailing right past the moon into infamy and the infinite. This fuel-hungry strategy may have been appropriate when the world’s space agencies had money to burn. But now that NASA is under intense pressure to do more with less, our future capability to explore and even exploit space may well hinge on our ability to find a gentler way through the solar system.

Ed Belbruno, a former NASA mathematician now working at the Geometry Center in Minneapolis, may well have found that way. In effect, Belbruno has learned to use the mathematics of chaos theory to chart safe and efficient paths through the turbulent topology of space-time. His own path to discovery, however, was neither safe nor cheap.

Just over four years ago, in fact, Belbruno was packing his bags after NASA’s project managers at the Jet Propulsion Laboratory (JPL) in Pasadena gently pushed him out the door. Belbruno had been immersed in celestial mechanics—the study of gravity and motion—ever since his doctoral studies at the prestigious Courant Institute of Mathematics in New York City. He continued his work at Boston University, where he focused on one of the most vexing problems in the field—the notorious three-body problem. While figuring out the gravitational interactions of two objects is relatively simple (Kepler and Newton did it more than 300 years ago for the sun and orbiting planets), adding a third plunges the problem into chaos. “The gravity of each object is constantly affecting the movement of the other two,” explains Belbruno, “so the system is always in a complex state of flux.”

From the perspective of a celestial mechanic, an object’s gravity creates a deep well or basin in space-time, with more massive bodies creating deeper wells. Imagine a bowling ball sitting on a water bed; the ball’s weight creates a valley in the flat landscape of the bed. Larger, heavier balls create still deeper valleys. Since every object makes its own gravity well, the depressions overlap, giving space a kind of reverse alpine topology. The moon makes a depression in the broader basin made by Earth. Earth does the same in the vast gravity well of the sun.

The three-body problem is frustrating because of the complex nature of the ridges between the wells. Strictly speaking, these regions should be null points where the competing gravitational forces cancel out; and in fact such points were first located by French mathematician Joseph-Louis Lagrange in the eighteenth century. They are like the point where a pencil standing on its sharpened end would balance, or where a marble perched at the top of a steep hill would stand exactly still. In these situations, gravity pulls with equal force in all directions, making the net force null. But in reality, these points are extremely unstable. The most infinitesimal nudge, like the bump of an electron, can knock the pencil off its point and the marble off the hill. With an almost infinite number of infinitesimal nudges occurring in every direction at every second, it is impossible to predict when or how something will be knocked off an unstable point. In the same way, the instability at, say, the gravitational ridge between Earth and the moon makes the mathematical description of orbits there disappear into a kind of haze nearly impossible for mathematicians and physicists to peer through.

NASA scientists plotting orbits to the moon and planets haven’t worried much about the effects of these chaotic ridges. Their rockets simply travel too fast. They ride over these bumps the way a 747 might ride the top of a thunderhead with barely a perceptible jiggle. They do not labor at the top long enough to get pulled about by the chaotic forces. A slower plane, however, might not have sufficient forward thrust to escape.

Normally, to maintain a simple circular orbit around an object, a rocket needs to find an exact balance between its tendency to fly outward into space and its inward attraction toward the moon or planet it circles. It needs enough velocity so that it doesn’t fall into the gravity well, but not so much velocity that it switches to a higher orbit or breaks the gravitational bonds entirely and zooms off, free. In a sense the stable orbits are like tracks carved into the topology of space itself. The moon, planets, and spacecraft all “ride” these orbits like railroad cars ride on their tracks. The terrain they ride—the hills and valleys and mountains—is sculpted by the planets and moons and stars. For an orbiting spacecraft, the trick is to get—and stay—on the right track. If its velocity should change, it would immediately get off one orbit and onto another.

In 1985 Belbruno headed for JPL, eager to escape academia and enter the applied world of celestial mechanics. When he got there, he found NASA had a purely practical engineering perspective on the three-body problem: they made it into two two-body problems. First the spacecraft blasted off Earth’s surface (out of the deepest part of the gravity well) and eased into a “parking orbit” around Earth. From there another rocket blast switched the craft onto the tracks of a high-velocity, highly elliptical orbit. Though still gravitationally bound to Earth, this high-speed orbit carried the spacecraft so quickly toward the moon that it barely noticed the ridge it crossed. When it passed the moon, the craft fired powerful retro-rockets and abruptly slowed down, switching onto a new orbit, one gravitationally bound to the moon.

This dance of gravity and chemical fuel is called the Hohmann transfer, named for Walter Hohmann, who worked it out in the 1920s. While the Hohmann transfer is very fast, it requires huge amounts of fuel. “You have to lose 600 miles per second of velocity to get off the Earth orbit and allow the spacecraft to be captured by the moon’s gravity,” says Belbruno. “Slowing down that much takes a lot of energy.”

Belbruno began questioning the efficiency of this approach. He saw that the Hohmann transfer ignored the subtleties of the gravitational topology of space. Perhaps, he reasoned, you could use the instability at the ridge to your advantage and cause a spacecraft to be captured by the moon without using retro-rockets. If you used just enough energy to make it to the top of the ridge from the Earth side, like a roller coaster cresting a hill, then with just the right nudge you could roll down the other side and be caught in the moon’s gravity well. “I thought one might find a single orbit that connected Earth and the moon,” says Belbruno. “With no orbits to transfer between, no retro-rockets would be needed. It would be cheap and safe.”

Massive bodies distort the fabric of space, with more massive objects creating bigger gravity wells. The huge gravity well created by the sun is itself warped by the smaller gravity well of Earth; Earth’s well, in turn, is dented by the mass of the moon.

Any such single orbit, however, would be highly unstable because it would sit on top of the ridge separating the gravity well of the moon from the well of Earth. A rocket riding this trajectory would have to behave like a marble that shot up from the bottom of a bowl and landed on the lip with just enough velocity to roll around on the edge without falling back in, or over. The idea was to get the spacecraft to stop fully poised at the top of the ridge, with zero residual energy. Then a rocket would nudge the craft onto a highly oblique new orbit that would send it rolling around the inside of the well. It would be like stopping a roller-coaster car at the top of the hill, then shifting it onto a new track with a far gentler slope. Engineers knew that actually getting a spacecraft onto one of those teetering orbits would require a near miracle. It’s about the same degree of difficulty as getting a pencil to stand on its point by throwing it across the room.

Belbruno got his chance to attempt the impossible when NASA decided to explore the feasibility of using electric propulsion to get a small probe to the moon. Electric propulsion works by driving a stream of plasma, or electrically charged gas, out the back of a spacecraft. It’s cheap but very, very slow. Compared with the roar of chemical rockets, electric propulsion produces only a gentle purr. Belbruno was asked to find a trajectory that could get the electrically driven probe into lunar orbit. “You actually can get to the moon with electric propulsion by spiraling slowly away from Earth for a year and building up speed,” says Belbruno. “What you can’t do with it is slow down fast enough to transfer onto a lunar orbit.”

As a mathematician, Belbruno knew that finding the right trajectory meant tackling the most difficult part of the three-body problem: chaos. The signature of chaos is the sensitive dependence on initial conditions. Small changes in where you start lead to huge changes in where you end. The ridge between Earth’s well and the moon’s is rife with it. On one side the orbits are gravitationally bound to Earth; on the other side they are bound to the moon. In between, in Belbruno’s description, lies a “fuzzy boundary” where the orbits attached to Earth and those attached to the moon mix in an impossible tangle—a tightly woven tapestry of dynamics whose threads are an infinity of different possible orbits. Orbits with velocity low enough to get caught in the boundary become lost in the tangle of paths there. Even if you could find a single orbit that emerged from the boundary and led to the moon, another orbit infinitesimally close by might lead a million miles away in the opposite direction. If Belbruno was to find a single path to the moon, he would first have to navigate the chaos in the fuzzy boundary.

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