Elementary celestial mechanics using Matlab

Abstract

The Trojan asteroids, discovered almost a century ago, are direct evidence for stability in the pure three-body problem. Two groups of asteroids share Jupiter's orbit, preceding or trailing it by an angle of 60 degrees. Because the Sun and Jupiter are by far the heaviest bodies, the restricted three body model is perfectly adequate to describe the relevant dynamics, and the perturbations due to the attraction of other planets do not significantly modify the orbit. For a hypothetical Lagrangian satellite bound to Earth and the Moon, we cannot discard the Sun's influence a priori. The problem of stability in this case is very hard; no analytic result is known, up to now. However, numerical analysis can give us a plausible answer. In spring 1999, I made this problem a classroom activity for my physics undergraduate students at the University of Parma. These students had an elementary background in classical mechanics but no computational-physics training. The choice of Matlab as a working environment was rather natural. With a little sacrifice in speed compared to Fortran or C, Matlab let us build a working program in a few days, including visualization and a friendly user interface. We easily found clear numerical evidence that the equilateral orbits L/sub 4/ and L/sub 5/ in the Earth-Moon system are unstable. I describe the program's setup, its Matlab implementation and the results.