TITLE:
On the dynamics of the Trojan asteroids.

AUTHOR:
Frederic Gabern

Departament de Matematica Aplicada i Analisi.
Universitat de Barcelona.
Gran Via 585, 08007, Barcelona, Spain.
E-Mail: gabern@mat.ub.es

ABSTRACT:
The Trojans are a group of asteroids that move in a neighbourhood of
the triangular points of the Sun-Jupiter system. The goal of this
thesis is to go deeply into the knowledge of the dynamical properties
of their motion.

Some of the known results on this subject use, as the model for their
dynamics, the well-celebrated Restricted Three Body Problem (RTBP). In
this model, it is assumed that the Sun and Jupiter are punctual masses
that revolve in circular orbits around their center of gravitation
(following the Kepler laws), and that the motion of the asteroid is
given by the gravitational attraction of these masses.

The first purpose of this work is to develop some other models for the
motion of the asteroid. In these models, we take into account the main
perturbations into the RTBP, that mainly come from the eccentricity of
Jupiter's orbit and from the perturbation that other planets (such as
Saturn and Uranus) causes to Jupiter's motion.

The second purpose is to make a semi-analytic study of the models. By
means of standard (for simpler cases) techniques, such as normal forms
or approximated first integrals, we describe the local non-linear
dynamics around the triangular points. The main novel aspect of the
application of these techniques to the developed models is the
implementation of the symplectic reducibility of the (periodic and
quasi-periodic) time-dependent equations. Another important
application of these techniques is the computation of a zone of
effective stability, that is a zone of the phase space where a
particle remains at least the expected lifetime of the Solar System.

The third purpose is to redo a numerical study of the dynamical
properties of the Trojan orbits in a more ``realistic'' model and to
compare it with the semi-analytical models. We use, as most of
astronomers do when they study this problem, the Outer Solar System
(OSS). That is, the N-body problem formed by the Sun, Jupiter, Saturn,
Uranus, Neptune and the massless particle. This study is based on the
frequency analysis of long (about 5 Millions of years) integrations of
the Trojan orbits. In order to produce such long integrations with a
good accuracy, a symplectic integrator has been used.

The thesis has been organized as follows: in the first chapter, we
give an extensive introduction to the problem. In chapter two, a
periodic perturbation of the RTBP (that comes from introducing the
effect of Saturn into the problem) is developed and studied in
detail. In the third chapter, the eccentricity of Jupiter is taken
into account giving rise to a model that is a periodic perturbation of
the RTBP and that contains an intrinsic resonance, showing up new
dynamical features. In chapter four, two quasi-periodic perturbations
of the RTBP are presented. The first one is constructed in order to
include both the effect of Saturn and the eccentricity of Jupiter's
orbit. In the second one, the effect of Saturn and Uranus are taken
into account. In the fifth chapter, we show the results of the
frequency analysis of the orbits generated by a symplectic integration
of the actual positions and velocities of 420 Trojans in the OSS, we
also compute the proper frequencies of the asteroids in the
semi-analytical models and we compare them with the results obtained
in the former OSS study.