  We study the long term orbital evolution of a terrestrial planet under the
gravitational perturbations of a giant planet. In particular, we are interested
in situations where the two planets are in the same plane and are relatively
close. We examine both possible configurations: the giant planet orbit being
either outside or inside the orbit of the smaller planet. The perturbing
potential is expanded to high orders and an analytical solution of the
terrestrial planetary orbit is derived. The analytical estimates are then
compared against results from the numerical integration of the full equations
of motion and we find that the analytical solution works reasonably well. An
interesting finding is that the new analytical estimates improve greatly the
predictions for the timescales of the orbital evolution of the terrestrial
planet compared to an octupole order expansion. Finally, we briefly discuss
possible applications of the analytical estimates in astrophysical problems.
