A NUMERICAL EXPLORATION OF THE EVOLUTION OF TROJAN-TYPE ASTEROIDAL ORBITS

Numerical N-body integrations have been conducted to study the stability of orbits of asteroids, which are co-orbital with the major planets. In Jupiter's case such celestial bodies are the well-known Trojan asteroids, while the newly discovered asteroid 1990MB is the first known "Martian Trojan." In the outer solar system our 20 million year integrations indicate the possible existence of stable Trojan-type tadpole orbits for all the other giant planets. Horseshoe orbits were found to be unstable on the timescale of several million years. Since the evolution of the unstable orbits is likely to be descriptive of the general evolutionary patterns of asteroidal orbits we describe briefly the observed behavior of these orbits, too. The unstable orbits have typically two ways of possible evolution: either to move up in the solar system to greater semimajor axis beyond Neptune's orbit, sometimes to orbits resembling that of Pluto (over millions of years), or to be ejected out of the solar system, most often by Jupiter. In many cases the asteroids are trapped into resonant orbits (resonance with Neptune) for times up to millions of years. These and many other evolved orbits have perihelion distances near the distance of Neptune. For the stable Trojan-type orbits the probability density of the planet-Trojan (libration) angle is discussed to show where observational searches should be directed. In the inner solar system our two million year calculations indicate the existence of long-term stability for both tadpole and horseshoe type orbits of asteroids co-orbital with the planets Venus, Earth, and Mars. Comparison of the numerical integrations with a simple analytic theory shows good agreement. This suggests that analytical theories can provide good approximations for the Trojan-type orbits in the inner solar system. The Liapunov exponents were calculated for some of the orbits. These, however seem not to provide any clear answer to the question of very long term persistence of the types of motion.