5 Conclusions and discussion

In this paper we were able to construct a complete linear secular theory for Trojan-type motion (i.e. based on expansions of the disturbing potential truncated at degree two in eccentricities and inclinations). This was achieved through the use of a Hamiltonian formulation which allowed us to generalise the theory presented in Paper I by including, in a rigorous way, the effect of an oblate central mass and the secular perturbations from additional massive bodies.

Using our theory we were able to locate secular resonances inside the co-orbital regions of the uranian satellites and the planets Mercury to Neptune. Comparison with numerical integrations showed that these locations are reasonably accurate and that secular resonances seem to play a major role in determining the stability of Trojan orbits.

We end by remarking that a next necessary step would be to develop a non-linear secular theory. In particular this would allow us to obtain even more accurate locations for these secular resonances (i.e. as functions of not only the proper semi-major axis, but also the proper eccentricity and the proper inclination). Moreover, a non-linear secular theory is also essential for application to the co-orbital structures with high eccentricity and/or inclination, studied by Namouni (1999).

Acknowledgements

I thank A. Morbidelli, D. Nesvorny and the reviewer, D. Hamilton, for useful comments. This work was supported by grants PRAXIS XXI/BD/5072/95 and SFRH/BPD/1586/2000 from the Fundação para a Ciência e a Tecnologia, Portugal.