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.
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.”