Summer 1997

New Computer Algorithm Puts Asteroids in the Cross Hairs

by Teresa Brumback

The Chicken Littles of the world can relax. A JMU physics professor's new celestial mechanics program now gives scientists a way to put sideswiping asteroids in Earth's cross hairs long before the sky falls.

And fall it might - loosely speaking, says physics professor Joseph Rudmin, who has developed the celestial mechanics program for calculating the paths of asteroids and comets. It was Luis and Walter Avarez' 1980 discovery that an asteroid impact 60 million years ago exterminated the dinosaurs that prompted Rudmin and former student Tim MacDevitt to try to improve the techniques for calculating asteroid trajectories.

Although those early efforts using polynomial approximations proved futile, Rudmin eventually crossed paths with JMU math professors James Sochacki and G. Edgar Parker. Their own mathematical breakthrough provided Rudmin with a computer algorithm that permits orbits to be calculated more accurately and quickly than ever before.

That is good news for Earth, because, Rudmin says, catastrophic impacts are not solely a prehistoric phenomenon. Rudmin points to sulfur-dioxide deposits in the Greenland ice cap 5,200 years ago and evidence of massive flooding in Mesopotamia (known to Westerners as the story of Noah's Ark and the Great Flood) at about the same time. Both, he says, could have been caused by an asteroid or comet hitting the Persian Gulf. The earliest story of the flood, the 2600 B.C. Epic of Gilgamesh, Rudmin says, quite clearly points to a celestial catastrophe. He believes the traditional interpretation of this catastrophe wrongly suggests a hurricane instead of asteroid or comet impacts. The latter would account for craters formed at the time in Saudia Arabia, Argentina, Estonia and possibly Canada.

Rudmin, though, is no doomsayer. Instead, he has drawn a bead on celestial objects in hopes of providing ample warning for those who can deflect an impending object away from Earth. His celestial mechanics program, which he has shared with the Naval Observatory in Washington, D.C., turns on the computer algorithm developed by Parker and Sochacki.

The new method for calculating the trajectories of asteroids and comets - and the routes for all celestial bodies in space - "allows it to be done much better than it was before, with far more precision," Rudmin says, unequivocally.

No wonder the Naval Observatory is interested. Its Astronomical Almanac publishes 200-day projections of the solar system with planetary positions accurate to about 200 miles. "With the Parker-Sochacki method," Rudmin says, "I can easily calculate the orbits to within 15 miles or even 15 feet, if I so choose."

It's a happy outcome for theorist Parker and applied mathematician Sochacki, who say they have not reinvented the wheel, but just used an algebraic tool found in all high school algebra textbooks - the Binomial Theorem - to prove a theorem which applies to Rudmin's work.

Sochacki, in a lecture to students in 1989, showed a phenomenon that caught the interest of Parker, who was in the audience. "I went to work on his problem using a process I knew how to estimate and I noticed a pattern of coefficients in my approximations," says Parker.

Adds Parker, "We wondered if the pattern had to be there, or if it was just coincidental to this particular problem [Henon's equation]. We were first able to prove it for problems with polynomial generators, then later for many problems with analytic generators. Our most recent work has extended the method to partial differential equations."

"I presented the Henon problem," Sochacki says. "Ed played with it and saw a phenomenon on the computer and asked me, 'Is this true?'" Just down the hall, Carter Lyons proved that what Parker and Sochacki observed was indeed true. Sochacki later made the proof in its full generality, showing that it wasn't just luck on the computer.

Since differential equations model so many phenomena in so many disciplines, the ability to solve a broad category of problems in one fell swoop with a single computable method is nothing short of a coup.

"We wanted to prove that the phenomenon we saw for the Henon differential equation is true for all differential equations of a very general class," Sochacki says. To do that, they took a classical technique from the 19th century, the Picard iteration (iteration, roughly speaking, means repeating a process over and over) and made a link with a theorem from the 18th century, Taylor's Theorem. To their satisfaction, they proved a body of theorems that have historical connections, wide applicability and are directly transportable to computers. "We have a method whose proof is elementary, so elementary that undergraduates can understand this," he adds.

They have, in short, discovered a theorem of classical origins that has striking contemporary applicability.

While their method has also been used on campus in economics and biology, Rudmin's celestial mechanics program was the first application to fully incorporate Parker's and Sochacki's findings. And it just so happens that this particular application has, well, worldwide impact.

"We ought to be worried about asteroids," says Rudmin. "If we had enough warning when the next would hit, we could deflect them."

Rudmin is working on a paper for the American Journal of Physics showing the Parker-Sochacki Method's astronomical applications for Comet Swift-Tuttle, which is expected to return toward Earth on a close trajectory between the years 2126 and 2261, and other celestial objects. He also presented a well-received paper on the technique at the American Physical Society Meeting in Washington, D.C., in April.

He is among the two math professors' biggest cheerleaders. "Their achievement is great enough so that they should become famous for it," Rudmin concludes. Parker and Sochacki are hopeful simply that their method, which they are already teaching to their students, will eventually wind up in textbooks. "Fame would be nice," says Parker, "but doing the mathematics has been the real excitement. We are particularly pleased that communications across the disciplines of physics and mathematics has made our work useful."

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