A matter of gravity

Natural History, Feb, 2003

In his "Universe" column ["Going Ballistic" November 2002], Nell deGrasse Tyson eloquently covered many different and interesting aspects of "free fall." Of particular interest to me was his discussion of the chaotic motion of the planetary orbits and of the slingshot effect that can give spacecraft a planetary boost. (The motion of an object is chaotic if at each moment it can move in infinitely many possible directions, resulting in an erratic path. Think of the motion of a leaf blown about by the wind, or a drunk trying to walk a straight line.)

The chaos of planetary orbits is extremely subtle, and it takes careful measurement to notice it. But if a space probe passes a body such as the Moon in such a way that the probe is almost captured by the body's gravitational pull, the probe will usually linger for a few hours in the vicinity of the larger body, moving in a complicated, chaotic fashion before leaving abruptly on another path that is difficult to predict. The effect is called weak ballistic capture.

In 1990, I designed a new kind of route to the Moon for the Japanese spacecraft Hiten, requiring three months instead of the usual three days for the spacecraft to make the trip and leading to weak ballastic capture. The resultant chaotic motion offered a lot of flexibility to mission planning.

Permanent capture could be achieved using almost no fuel, an attractive option because Hiten carried so little. A more complicated plan was successfully followed, however: the craft stayed in weak capture for a few hours, then moved away from the Moon for six months to explore the Earth-Moon system, and finally returned to the Moon for placement in permanent capture. It turns out that weak capture is a slow version of the slingshot effect; recently weak capture was mathematically proved to be truly chaotic.

Edward Belbruno Princeton University Princeton, New Jersey

The figure-eight orbit that Nell Tyson mentions, as well as other newly discovered solutions to the old three-body problem, are so elegant that a name has been coined for them: choreographies. The figure eight is only the simplest example; other orbits are wildly more complex, with shapes that look more like fluttering butterflies. And those three-body orbits have been quickly generalized to even more fascinating dances for four or more bodies. Animated examples can be viewed on the Web at www.ams. org/new-in-math/cover/ orbits1.html.

As a theoretical astrophysicist working in stellar dynamics, I am sobered by the fact that great mathematicians and physicists worked on the three-body problem over the past three centuries without having any idea that orbits of this kind were awaiting discovery. And who knows what else there is to be found. It is not only with telescopes that new astronomical objects can be discovered. With even a small personal computer, a lucky guess, and enough persistence, anybody is now in a position to find new solutions to age-old problems of a kind that were completely beyond what Newton and the Le's and La's of celestial mechanics (Leverrier, Legendre, Lagrange, and Laplace, to name a few) could handle.

Piet Hut Institute for Advanced Study Princeton, New Jersey

Neil Tyson's article brought to mind my own attempt to dig a hole from Oklahoma to China. I was seven or eight at the time and asked my dad if I could dig down behind the house. "Sure, honey," he replied, barely looking up from the paper. The project was called off several weeks later when his tractor nearly fell into the hole. I was terribly disappointed, but now I have learned that a major catastrophe was averted.

First of all I would have been vaporized by the fierce heat of the iron core. At best I would have popped out in the southern Indian Ocean, which would have been dangerous because I couldn't swim. At least it is a relief to know the ocean water pouring into the tunnel would have surged back and forth, rather than flooding the Midwest.

Gloria Jones- Wolf Elk Falls, Kansas