The Chaos of Celestial Physics and Astrodynamics

For mathematician Edward Belbruno, by embracing “chaos” he was better able to understand the three-body problem of celestial physics. His notion of chaos describes motion that defies precise long-term predictions.

Published January 1, 2005

By William Tucker
Academy Contributor

In 1990, Edward Belbruno was packing his belongings, getting ready to leave the Jet Propulsion Laboratories in Pasadena. His five-year effort to interest NASA in low-energy trajectories for spaceflight had failed.

A graduate of the Courant Institute of Mathematics in New York, Belbruno had long been playing with the idea of charting very precise flight paths through the sky or into space. He wanted to allow space probes to slip into orbit around a moon or planet without the use of powerful, fuel-consuming retrorockets. His task was made immensely complicated – if not impossible – by the three-body problem of celestial physics.

When first formulating the laws of gravity, Isaac Newton had calculated the interaction of two bodies. They could be a stone falling to Earth, a spacecraft in orbit, or the Earth itself on its trajectory about the Sun. In each case, the two bodies both revolve around the center of mass – a point somewhere between their two centers, like the balancing point of a see-saw.

The interaction of three bodies, however, is immensely more difficult. In fact, in the late 1950s, V. Arnold, a Russian mathematician, and J. Moser, a German, independently proved that the three-body problem could not be solved at all. The proof came from solving the more general problem of chaos in nearly periodic motion, as outlined by Arnold’s teacher, A. N. Kolmogorov, in the 1920s. It is now known as the Kolmogorov-Arnold-Moser (KAM) theorem.

Order in Chaos

The obstacle to finding a solution is that the three-body problem leads, literally, to chaos. To a mathematician, that does not mean a dark abyss or a mad frenzy. Rather, chaos describes motion that defies precise long-term predictions.

However, mathematics offers tools even for dealing with the unknowable. Using the mathematics of chaos, Belbruno felt that he could fudge the three-body problem enough to create a proper trajectory. The difficulty was that his slow dance to the Moon would take two years, whereas conventional rockets can make the trip in three days. NASA lost interest, and Belbruno was shown the door.

Then a miracle happened. The Japanese had launched a two-part Moon probe, Muses A, the size of a desk, and Muses B, the size of a grapefruit. The two had separated while in Earth orbit and the grapefruit headed for the Moon. Upon arrival, however, Muses B’s radio failed, and the probe was lost. Now Muses A was circling the Earth with very little fuel and nothing to do. A JPL engineer remembered Belbruno’s work. Suddenly Belbruno had an audience. Could he help? Belbruno said he could.

“In the same instant, I realized that I could add the Sun’s gravitational field to the equation,” Belbruno says. Ten months later, Muses A – now rechristened Hiten, after a Buddhist angel – fired half its remaining fuel and, guided by Belbruno’s equations, glided into a 2-million-mile itinerary beyond the Moon and back again. It was like flicking a paper airplane into space, hoping it would eventually settle into a trajectory where its momentum perfectly matches the Moon’s gravity.

The Angel of Chaos

Belbruno’s formulas worked, and the mission was saved. “They used it again for the Genesis probe of the Sun and the European Space Agency mission SMARTONE,” says Belbruno. “NASA now takes my work a lot more seriously.”

So seriously that Belbruno was commissioned to call a conference at the University of Maryland in 2003 to investigate astrodynamics and chaos. Also under study were formation flying, navigation and control of unmanned spacecraft, orbital dynamics, mission proposals, and possible propulsion methods for pushing probes deep into the solar system. The results have been collected as Astrodynamics, Space Mission, and Chaos, Volume 1017 in Annals of the New York Academy of Sciences.

Although Belbruno and his fellow authors could not know it, space probes were about to be brought back front and center by President George W. Bush’s announcement of a mission to Mars, somewhere around 2020. “The cost for delivering cargo to the Moon is now $1 million per pound,” says Belbruno. “Every pound of fuel we can save is another pound of payload that can be delivered.

“I don’t agree with everything the president does, but I think he has shown great vision on this initiative,” he adds. “The idea of going step by step to the Moon, building a base, and then moving on to Mars and back is very practical. I think there’s a good possibility we’ll succeed.”

Also read: Exploring the Ethics of Human Settlement in Space