Orbiting in a figure-eight loop

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

Mathematicians Richard Montgomery of the University of California, Santa Cruz and Alain Chenciner of the Bureau des Longitudes in Paris describe their new solution of the equations of motion for three gravitationally interacting bodies in a report available at http://count.ucsc.edu/~rmont/Nbdy.html.

Newton’s laws provide a precise answer to the problem of determining the motion of two bodies. If the solar system consisted of the sun and a single planet, for example, the planet would orbit in an ellipse. When the system consists of more bodies, however, solving the equations is exceedingly difficult. Except in a few special cases, the motion turns out to be chaotic and unpredictable (SN: 2/22/92, p. 120).

The theoretical result obtained by Montgomery and Chenciner represents a new addition to the sparse list of exceptions. Given the well-known difficulty of the problem, “it certainly is exciting work,” says Gregory R. Buck of Saint Anselm College in Manchester, N.H. One can find approximate solutions that describe how a large number of masses can follow just about any given curve, Buck has shown (SN: 9/5/98, p. 149). In contrast, the new figure-eight orbit is an exact three-body solution.

Recent computer simulations by Carles Simá of the University of Barcelona have demonstrated that the figure-eight orbit persists even when the three masses aren’t precisely the same and can survive a small disturbance without serious disruption. Whether such a threebody system exists somewhere in the universe, however, isn’t known.

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