We study the evolution of asteroid orbits in a restricted three-body problem formulation in two dimensions, consisting of the Sun, the planet Jupiter and an unspecified asteroid of negligible mass. It was discovered by Kirkwood [l] that the distribution of asteroid orbits contains gaps for orbits whose period is commensurate with that of Jupiter. Detailed computations in three-dimensional, many-body formulations found that test bodies initially placed in a forbidden orbit did not develop large eccentricities or leave the gap even after the passage of 10 5 years [2]. While previous two-dimensional, three-body simulations, an extension of earlier work [3], showed significant departure of asteroids placed in forbidden orbits in fewer than 10 revolutions, our present work shows such orbits to be stable for at least 25,000 years. The results suggest two things: One, the two-dimensional, three-body, reduced problem (modeled in figure 3) is consistent with more detailed three-dimensional, many-body models, contrary to previous work. Secondly, the numerical integrations of the coupled equations of motion for Jupiter and the asteroid are highly sensitive to the precision in the method of computation. The demand for precise calculations for accurate predictions brings forth a problem of time complexity. With the resources available, simulations longer than 25,000 years were not practically possible
