  When dealing with satellites orbiting a central body on a highly elliptical
orbit, it is necessary to consider the effect of gravitational perturbations
due to external bodies. Indeed, these perturbations can become very important
as soon as the altitude of the satellite becomes high, which is the case around
the apocentre of this type of orbit. For several reasons, the traditional tools
of celestial mechanics are not well adapted to the particular dynamic of highly
elliptical orbits. On the one hand, analytical solutions are quite generally
expanded into power series of the eccentricity and therefore limited to
quasi-circular orbits [17, 25]. On the other hand, the time-dependency due to
the motion of the third-body is often neglected. We propose several tools to
overcome these limitations. Firstly, we have expanded the disturbing function
into a finite polynomial using Fourier expansions of elliptic motion functions
in multiple of the satellite's eccentric anomaly (instead of the mean anomaly)
and involving Hansen-like coefficients. Next, we show how to perform a
normalization of the expanded Hamiltonian by means of a time-dependent Lie
transformation which aims to eliminate periodic terms. The difficulty lies in
the fact that the generator of the transformation must be computed by solving a
partial differential equation involving variables which are linear with time
and the eccentric anomaly which is not time linear. We propose to solve this
equation by means of an iterative process.
