Following gravity's loops and knots.

The force of gravity governs the motion of planets, asteroids, and other bodies in the solar system. Predicting their orbits requires solving equations representing the gravitational attraction between interacting masses.

Now, a mathematician has discovered a new set of approximate solutions of those equations. Each solution corresponds to a string of equally spaced masses--like the beads of a necklace--chasing each other around a closed loop at just the right speed.

Although the solar system is unlikely to harbor such behavior, it may occur among filaments and vortices in a plasma of huge numbers of charged particles, suggests Gregory R. Buck of Saint Anselm College in Manchester, N.H. He reports his findings in the Sept. 3 Nature.

Newton's laws of movement provide a precise answer to the problem of determining the movement of two gravitationally interacting bodies. For example, if the solar system consisted of the sun and a single planet, the planet would follow an elliptical orbit.

When the system consists of three or more bodies, however, solving the equations proves immensely difficult. The motion turns out to be chaotic and unpredictable (SN: 2/22/92, p. 120), except in a few special cases. For example, the sun is so much more massive than the planets that the solar system behaves roughly as if each planet were influenced only by the sun.

Another special case occurs for three bodies at the corners of an equilateral triangle. Such a configuration rotates as if the masses were fixed to a turntable. The sun, Jupiter, and the so-called Trojan asteroids, for instance, form such a triangle, which rotates about the system's center of mass.

"With such a configuration, appropriate initial conditions can be supplied so that the motion keeps the same configuration for all time," says Donald G. Saari of Northwestern University in Evanston, Ill.

Instead of considering specific geometric configurations, in which forces partially cancel out to produce a simple type of motion, Buck looked at what sorts of cancellations would occur among an enormous number of bodies.

He found that an infinite number of masses following a looped path satisfies Newton's equations. In effect, gravity pulls the bodies one after the other around each bend of the loop. Such loops can have any shape, no matter how tangled or knotted, as long as they don't intersect anywhere and there are no loose ends.

The equations may not be precisely satisfied, however, for a finite number of masses traveling in a loop, Saari notes. In that case, gravitational forces would tend to move masses away from a loop configuration. What isn't clear yet is how long a given loop configuration involving a finite number of masses would last, Buck says.