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.
When asked to help the Japanese mission, Belbruno looked up and silently said “thank you.” Then, in a flash of insight, he saw how chemical rockets and fuzzy boundaries might be joined.
Belbruno approached the problem with the eye of an explorer. Since so little was known about the fuzzy boundary, he began studying it to see if he could find some properties that could be exploited. Using a computer, he began sampling lunar orbits, searching for the place where the instability and chaos began. What he quickly found was that nature could be kind as well as complex. “Even though the structure inside the fuzzy boundary was impossibly complicated,” Belbruno recalls, “I found that the chaotic region itself began at a very well defined place.” That clearly demarcated edge was the last stable lunar orbit, a gravitational path that was barely attached to the moon and yet didn’t disappear into the fuzzy boundary. There was a similar weakly bound terrestrial orbit leading from Earth to the sharp outer edge of the fuzzy boundary. Finding those two orbits was the breakthrough Belbruno needed. “An almost unbound lunar orbit can be easily connected with an almost unbound Earth orbit,” he says. “The two orbits have nearly the same velocity, so almost no rocket power is required to switch from one to the other.”
It’s possible to imagine the fuzzy boundary as a kind of raised lip encircling the gravity well of the moon. The chaos of orbits in the boundary resides along the top of the lip. Belbruno’s two weakly bound orbits are just on the safe side of that chaos. A spacecraft could ride the first weakly bound orbit from Earth right up to the outer edge of the lip and then, with a slight nudge, quickly cross through it to the weakly bound lunar orbit on the other side. As in the Hohmann transfer, the crossing of the unstable region would occur fast enough to keep the spacecraft untouched by the chaos there.
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Belbruno knew his work would become practical only if fuzzy boundaries could be exploited by the faster, more powerful chemical rockets. But he soon found that the high velocities that make chemical rockets so attractive also made them useless for his new trajectories. Fast-moving rockets could be connected with the gentle fuzzy-boundary trajectory only through maneuvers that cost so much in time and rocket fuel as to make the whole exercise pointless. And slow-moving rockets that could make the transfer simply took too long.
By the spring of 1990 Belbruno, like the JPL managers, had almost given up on fuzzy boundaries. He had already accepted a position as an assistant professor of mathematics at Pomona College in California; he was also spending more time on his serious avocation—painting. Then, in one of those twists that make life worth imitating in art, a stranger appeared at Belbruno’s office doorway.
The stranger was James Miller, a JPL engineer who was trying to find a way to help the Japanese reconfigure their lunar mission, called Muses. In January of that year two probes had been launched. One, Muses-B, had been sent to the moon. The larger probe, Muses-A, was left in Earth orbit to perform its own experiments and, at the same time, to act as a relay station for the smaller craft, which would beam its radio signals back to Earth. After reaching the moon, however, Muses-B stopped transmitting. Still hoping to fulfill all the mission’s objectives, the Japanese asked JPL to look into the possibility of sending Muses-A to the moon to take over the duties of its lost companion. Miller told Belbruno that he had tried to set up a standard Hohmann transfer but that the fuel remaining on Muses-A was a fraction of what was needed. When Miller asked if it would be possible to use a fuzzy-boundary trajectory, two things happened. Belbruno looked up and silently said “thank you,” and then, in a flash of insight, he saw how chemical rockets and fuzzy boundaries might be joined. The answer was really as clear as day.
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
It was the sun that solved the problem for both Belbruno and Miller. Belbruno had thought that chemical rockets were too fast to be easily switched onto a gentle Earth-moon fuzzy-boundary trajectory. But there are other fuzzy boundaries. In fact there is always a fuzzy boundary between any pair of gravitationally interacting bodies. If chemical rockets were too powerful to connect with the Earth-moon fuzzy boundary, perhaps they could connect with the more distant one that surrounds the Earth and separates its bound orbits from those of the sun.
The rocket’s speed would allow the craft to ride quickly out and up to the lip of the Earth-sun fuzzy boundary. Then, without crossing into the region of chaos, it could roll back down Earth’s gravitational well to connect with an Earth-moon fuzzy-boundary trajectory, ending, finally, in a stable lunar orbit. This would all happen so gently and gradually that it wouldn’t need powerful retro-rockets, and the craft could coast all the way to its goal, like a marble rolling around the inside of a bowl at such an oblique trajectory that it lands in the center with no residual velocity at all. You might think a spacecraft with sufficient velocity to get out of Earth’s gravity well all the way to the lip of the sun’s well would come flying back with so much velocity that it would sail right by the Earth-moon ridge. But the three-body problem makes the real situation much more complex: the simultaneous relative motions of the sun, Earth, moon, and spacecraft weave a tangle of orbits and velocities far beyond the power of an analogy that relies on a simple one-dimensional curve. Given the added complexity of the real situation, says Belbruno, “it’s amazing that this simple bowl analogy can take us so far. But you can’t expect the bowl to get the mechanics right for you.”
In all the work of the previous four years, Belbruno had simply missed the possibility of bringing the sun into the equation. “You get so conditioned by working with the Hohmann transfer that there is a subconscious tendency to always stay within the Earth-moon system,” he says. “You never think about what else is out there.” But by traveling first to the Earth-sun fuzzy boundary 930,000 miles away and then connecting with the Earth-moon fuzzy boundary, Belbruno was able to piece together a five-month trajectory that would put Muses-A into a lunar orbit despite its meager fuel resources.
In April 1991 Muses-A, renamed Hiten after a Buddhist angel of music, fired its rockets and started onto its new trajectory. On October 2, in a stunning confirmation of Belbruno’s ideas, the spacecraft reached the moon with enough fuel left over to allow it a kind of victory lap, flying out into space again to perform extra experiments before settling into a lunar orbit. The Japanese Muse had become Belbruno’s redeeming angel.
Today the potential of fuzzy boundaries is just beginning to be felt. Belbruno and Miller have shown, for instance, that NASA’s now-defunct Lunar Observer mission would have required 25 percent less fuel with a fuzzy-boundary trajectory. Recently, Rex Ridenoure of JPL, working with Belbruno, demonstrated that the fuzzy-boundary method will even allow small, missile-size rockets dropped from jets to deliver significant payloads to the moon. The potential fuel savings of fuzzy-boundary trajectories fit in nicely with demands by Congress that NASA learn to do more with less. According to Belbruno, fuzzy-boundary trajectories can, in general, produce savings of 30 percent, allowing spacecraft to double their payloads. If so, they might be used to exploit some abundant but now inaccessible natural resources. Imagine, for example, mining the moon, and the costs needed to ship raw material back to Earth orbit. Using fuzzy-boundary trajectories, a low-cost pipeline could be established, with cargo ships continuously “running the ridge” with minimum fuel consumption.
Dick McGehee, the acting director of the Geometry Center, thinks it is just a matter of time before fuzzy boundaries become a tool for interplanetary navigation. “Many of these chaotic regions have been exploited by mathematicians for years,” he says. “But this is the first time they’ve been explored as a way of getting from here to there—perhaps even to completely unexplored regions of space.”
