Stretching the time to orbital catastrophe.

The solar system has apparently survived in roughly its present configuration for more than 4 million years. Yet computer simulations of the evolution of planetary orbits have revealed evidence of chaos.

Such uncertain behavior sharply restricts to a few tens of millions of years how far into the future one can predict the precise shape and orientation of a typical orbit, whether that of a plant or an asteroid (SN: 2/22/92, p.120). It leaves open the possibility that Mars, for example, could someday swing out of its present orbit and smash into Earth, or that Pluto may evantually escape the solar system.

Astronomers have now taken a step toward resolving the question of why chaos and unpredictability seem compatible with the solar system's apparent equanimity over billions of years. Using computer simulations of certain types of orbits, Myron Lecar, Fred Franklin and Marc Murison of the Harvard-Smithsonian Center for Astrophysics in Cambridge, Mass., derived a numerical relationship0 linking the characteristic time over which an orbit remains predictable and the much longer time after which an orbit is likely to drastically change its shape or orientation in space.

For the solar system, the results imply that no catastrophe would likely occur for at least a trillion years. This represents "a long time, even by astronomical standards, but not infinitely long," Lecar says.

Is there a chance that the solar system might get into trouble much sooner? "Yes," Lecar says, "but it's very small."

The researchers studied more than 1,000 examples of three types of orbits. In one case, they computed sample orbits traced out by an asteroid gravitationally influenced only by the sun and Jupiter. In another scenario, they looked at orbits of hypothetical asteroids stationed between Jupiter and Saturn. The third situation involved tracking the orbit of a tiny body initially circling the smaller of two stars in a binary system.

For each situation, they compounded a quantity known as the Lyapunov time, commonly used in the study of chaotic systems to characterize how rapidly two orbits starting at minutely different positions separate and go their own way. This number provides an estimate of how far into the future one can predict chaotic behavior.

The researchers then compared the computed Lyapunov time with the time it takes orbits to change sufficiently for an orbiting body to cross a planet's path or escape the system. They discovered that despite the great differences between the three types of orbits considered, all led to approximately the same numerical relationship between the Lyapunov time and the time for an orbit to make a sudden transition.

According to this relationship, the transition time is proportional to the Lyapunov time to the 1.8 power. "You always get this exponent, 1.8," Lecar says.

However, the researchers looked at only three special cases, involving gravitational interactions much less complicated and among fewer bodies than those in the solar system. Whether this analysis applies to real planets and other chaotic systems isn't clear.

"The correction is interesting," says Scott Tremaine of the Canadian Institute for Theoretical Astrophysics at the University of Toronto. "But there's unlikely to be a single universal law that describes exactly what's going on. My general experience is that the details of what happens depend fairly strongly on the configuration."

Also missing is a convincing explanation of why the exponent in the cases studied happens to be 1.8. "I don't think anyone really understands the exact nature of the chaos [present]," says Martin Duncan of Queen's University in Kingston, Ontario, who has been computing the dynamical behavior of hypothetical asteroid orbits beyond Neptune. "This apparent stability in the face f unpredictability remains a puzzle."