Global dynamics of the L4 and L5 Trojans

The first asteroid, librating on a stable orbit close to Jupiter`s equilateral equilibrium point preceding this planet in its orbit was observed in 1906 by Max Wolf in Heidelberg and was named after the hero of the Trojan war (588) Achilles. Nowadays (April 2002) 696 asteroids - all of which were given the names of warriors of the Trojan war - are known to be moving close to 60 ahead of Jupiter (L4) and 519 objects are trailing Jupiter at about 60 (L5). The reason for the difference of the number of Trojans around both points is still unclear. To find an answer to this question is one goal of the project we plan to do. On the other hand it was found that some of the Trojans, namely the ones on the edge of the Trojan cloud (around L4 or L5) have only lifetimes of several hundreds of million years and thus we do not know why they are still here. Consequently the second aim is to find out the time scales of diffusion of asteroids from the kernel of the Trojans (small eccentricities and librations) to the edge of the Trojan belt. To reach these goals we plan to do extensive numerical experiments in the framework of a realistic dynamical model (sun + masses of the inner planets, the 4 outer giant planets and massless fictitious asteroids) for timescales up to 108 years. Three different types of orbits will be investigated; the real Trojans, orbits close to the real Trojans to explore the phase space in their vicinity and orbits for a grid of initial conditions around L4 and L5. The results will be analysed by different methods (eg. Lyapunov Exponent, Analysis of the frequencies) to be able to determine the transport mechanism in the region of motion around L4 and L5. At the same time we will be able to determine stability times for theTrojans and the differences of the transport around L4 and around L5.

The aim of this project was to have a global view of the motion of asteroids close to the Lagrangian points 60 degrees ahead (L4) and 60 degrees behind (L5) Jupiter with the same semi-major axis as this gas giant. These two areas are populated with a large number of small bodies with approximately 1100 respectively 900 Trojan asteroids. The primary goal of the study was the determination of the largeness of this region around this stable equilibrium points, known as the Lagrange equilateral equilibrium points. With the aid of new methods we could determine stability or instability of an orbit already before the escape from the stable region. The results of our long-term integrations have shown that the L5 Trojans were unstable with smaller initial orbital inclinations and eccentricities than the L5 Trojans. This is a possible explanation for the difference in the number of asteroids close to the two equilibrium points of Jupiter. In the continuation of our work we also studied these stability regions around the equilibrium points of the other gas giants, namely Saturn (which turned out to be an unstable region in time scales of several ten million of years), Uranus and also Neptune. As important new subject we then extended our investigations to the stability of terrestrial planets in extra-solar planetary systems. During the last 15 years a great number of stars hosting large planets were discovered in the solar neighbourhood. All of these more than 200 planets are of a size comparable to Jupiter, but no small body, an earth-like planet has been discovered up to now. Depending on the distance of such a big planet to the host star additional terrestrial planet may move on stable orbits in the so-called habitable zone (where water could exist in liquid form) around a sun-like star. In our project, as an extension of the stability study of the Lagrange points we investigated these equilibrium points with giant planets in the EPSs, which move themselves in this habitable zone. With our work we opened another possibility for the research in Astrobiology, a new scientific branch, which is looking for the possibility of life outside our Earth, namely for Earth-like planets moving in stable orbits in extra-solar planetary systems in the same orbit like a very large planet like Jupiter.