Web edition: April 6, 2001
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.
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.
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."