Ivar's Peterson's Math Trek-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 surprising pattern of motion arises from the force of gravity acting on three bodies of equal mass. The movements are timed so that each body in turn passes between the other two. Moreover, such coordinated movements aren't the only intricate maneuvers possible when several bodies attract one another gravitationally.

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.

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

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's 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.

In 1993, Cris Moore, now at the University of New Mexico, added to the sparse list of exceptions. He discovered, via computer calculations, that three equal masses can chase each other around the same figure-eight curve in the plane.

In 2000, mathematicians Richard Montgomery of the University of California, Santa Cruz and Alain Chenciner of the Université Paris VII-Denis Diderot rediscovered the figure-eight orbit found by Moore, working out an exact solution to the equations of motion for three gravitationally interacting bodies. Montgomery described the discovery in the May 2001 Notices of the American Mathematical Society.

Three equal masses (red, blue, and green), shown at five equal time intervals, chase each other around a figure-eight-shaped curve. Initially, the three masses are in a straight line.
Courtesy of Michael Nauenberg

Computer simulations by Carlès Simò of the University of Barcelona 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 noted.

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. Nonetheless, advanced imaging techniques and the recent discovery of unusual extrasolar planetary systems are providing new space-time venues in which such motions could occur.

The existence of the three-body, figure-eight orbit 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, Simò 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 noted. Nonetheless, "they make beautiful patterns: flowers, chains, and so on." For examples, see http://www.soe.ucsc.edu/~charlie/3body/.

In the latest development, Moore and Michael Nauenberg of the University of California, Santa Cruz have unveiled a slew of new, periodic n-body orbits. Whereas previously discovered orbits were confined to the plane, the new orbits are three-dimensional. For a glimpse of these orbits, see http://www.santafe.edu/~moore/gallery.html.

One striking example (below) has 12 equal masses following four, roughly circular, interlocked orbits. Topologically, these orbits form the edges of a cuboctahedron.

Nauenberg gives additional examples of these three-dimensional orbits at http://physics.ucsc.edu/~michael/ (see section on recent publications). Even in the case of three bodies, it's possible to have intriguing nonplanar, figure-eight orbits.

Somewhere in the universe, triple stars and weird planetary systems may be doing crazy eights—and more! It would certainly be heavenly choreography.

Chenciner, A., and R. Montgomery. 2000. A remarkable periodic solution of the three body problem in the case of equal masses. Annals of Mathematics 152:881-901. Preprint available at http://www.bdl.fr/Equipes/ASD/preprints/prep.2000/huit.pdf.

Chenciner, A., J. Gerver, R. Montogmery, and C. Simò. In press. Simple choreographies of n bodies: A preliminary study. Geometry, Mechanics and Dynamics. Preprint available at http://www.bdl.fr/Equipes/ASD/preprints/prep.2001/CGMS.pdf.

Montgomery, R. 2001. A new solution to the three body problem. Notices of the American Mathematical Society 48(May):471-481.

Nauenberg, M. Preprint. Continuity and Stability of families of figure eight orbits with finite angular momentum. Available at http://physics.ucsc.edu/~michael/marc9.pdf.

______. 2001. Periodic orbits for three equal mass particles with finite angular momentum. Physics Letters A 292:93-99.

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

______. 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.

**********
A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the Mathematical Association of America (MAA) book Mathematical Treks: From Surreal Numbers to Magic Circles. See http://www.maa.org/pubs/books/mtr.html.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.

Ivars Peterson's MathTrek

Strange Orbits