Ivars Peterson's MathTrek

April 9, 2001

# Strange Orbits

Like toy cars chasing each other on a looped racetrack, three stars can, in principle, trace out a figure-eight orbit in space. This newly discovered, mathematically surprising pattern of motion arises from the force of gravity acting on three bodies of equal mass. Their movements are timed so that each body in turn passes between the other two.

Newton's laws provide a precise answer to the problem of determining the motion of two bodies under the influence of gravity. If the solar system consisted of the sun and a single planet, for example, the planet would follow an elliptical orbit. When the system consists of more than two bodies, solving the relevant equations of motion gets very tricky.

For three interacting bodies (described as the three-body problem), mathematicians have found a small number of special cases in which the orbits of the three masses are periodic. In 1765, Leonhard Euler (1707-1783) discovered an example in which three masses start in a line and rotate so that they stay in line. Such a set of orbits is unstable, however, and it would not be found anywhere in the solar system.

In 1772, J.L. Lagrange (1736-1813) identified a periodic orbit in which three masses are at the corners of an equilateral triangle. In this case, each mass moves in an ellipse in such a way that the triangle formed by the three masses always remains equilateral. A so-called Trojan asteroid, which forms a triangle with Jupiter and the sun, moves according to such a scheme.

Three gravitationally interacting bodies of equal mass: Euler configuration (left) and Lagrange configuration (right).

Subsequent work by Henri Poincaré (1854-1912) and others demonstrated that, in general, it is impossible to obtain a general solution, expressed as an explicit formula, to the three-body problem. In other words, given three bodies in a random configuration, the resulting motion nearly always turns out to be chaotic. No one can predict precisely what paths those bodies would follow.

Now, mathematicians Richard Montgomery of the University of California, Santa Cruz and Alain Chenciner of the Université Paris VII-Denis Diderot have added to the sparse list of exceptions. They found a new, exact solution to the equations of motion for three gravitationally interacting bodies. "The three equal masses chase each other around the same figure-eight curve in the plane," Montgomery reports in the May Notices of the American Mathematical Society.

Three equal masses (red, blue, and green) chasing each other around a figure-eight-shaped curve. Initially, the three masses were in a straight line, with red at the midpoint between blue and green. Courtesy of R. Montgomery.

Computer simulations by Carles Simó of the University of Barcelona have demonstrated that the figure-eight orbit is stable. The orbit persists even when the three masses aren't precisely the same, and it can survive a tiny disturbance without serious disruption.

"What stability means physically is that there is some chance that the [figure-eight orbit] might actually be seen in some stellar system," Montgomery says. The chance that such a three-body system exists somewhere in the universe, however, is very small. Numerical experiments suggest that the probability is somewhere between one per galaxy and one per universe.

The existence of the three-body, figure-eight orbit has prompted mathematicians to look for similar orbits involving four or more masses. Joseph Gerver of Rutgers University, for instance, found one set in which four bodies stay at the corners of a parallelogram at every instant, while each body follows a curve that looks like a figure-eight with an extra twist.

Using computers, Simó has found hundreds of exact solutions for the case of N equal masses traveling a fixed planar curve. "They are not stable, except for the original figure-eight case," Montgomery notes. Nonetheless, "they make beautiful patterns: flowers, chains, and so on."

References:

Casselman, B. 2001. A new solution to the three body problem\$#151;and more. Available at http://www.ams.org/new-in-math/cover/orbits1.html.

Chenciner, A., and R. Montgomery. In press. A remarkable periodic solution of the three-body problem in the case of equal masses. Annals of Mathematics. Available at http://orca.ucsc.edu/~rmont/index.html.

Chenciner, A., J. Gerver, R. Montgomery, and C. Simó. Preprint. Simple choreographic motions of N bodies: A preliminary study. Available at http://www.maia.ub.es/dsg/.

Montgomery, R. 2001. A new solution to the three body problem. Notices of the American Mathematical Society 48(May):471. Available at http://www.ams.org/notices/200105/fea-montgomery.pdf.

Moore, C. 1993. Braids in classical dynamics. Physical Review Letters 70(June 14):3675. Abstract available at http://prola.aps.org/abstract/PRL/v70/i24/p3675_1.

Peterson, I. 2000. Orbiting in a figure-eight loop. Science News 157(April 1):219.

______. 1999. Prophet of chaos. MAA Online (Nov. 15).

______. 1998. Following gravity's loops and knots. Science News 154(Sept. 5):149.

______. 1993. Newton's Clock: Chaos in the Solar System. New York: W.H. Freeman.

______. 1992. Chaos in the clockwork. Science News 141(Feb. 22):120.

Richard Montgomery has a Web site at http://orca.ucsc.edu/~rmont/index.html.

Computer simulations by Carles Simó of the University of Barcelona of novel N-body orbits can be found at http://www.maia.ub.es/dsg/nbody.html.

Ivars Peterson is the mathematics/computer writer and online editor at Science News (http://www.sciencenews.org). He is the author of The Mathematical Tourist, Islands of Truth, Newton's Clock, Fatal Defect, and The Jungles of Randomness. He also writes for the children's magazine Muse (http://www.musemag.com) and is working on a book about math and art.

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Math Trek 2: A Mathematical Space Odyssey by Ivars Peterson and Nancy Henderson. For children ages 10 and up. New York: Wiley, 2001. ISBN 0-471-31571-0. \$12.95 USA (paper).