The planetary exospheres are poorly known in their outer parts, since the neutral densities are low compared
with the instruments detection capabilities. The exospheric models are thus often the main source
of information at such high altitudes. We present a new way to take into account analytically the additional
effect of the radiation pressure on planetary exospheres. In a series of papers, we present with an
Hamiltonian approach the effect of the radiation pressure on dynamical trajectories, density profiles and
escaping thermal flux. Our work is a generalisation of the study by Bishop and Chamberlain (Bishop, J.,
Chamberlian, J.W. [1989]. Icarus 81, 145–163). In this first paper, we present the complete solutions of
particles trajectories, which are not conics, under the influence of the solar radiation pressure with some
assumptions. This problem is similar to the classical Stark problem (Stark, J. [1914]. Ann. Phys. 348,
965–982). This problem was largely tackled in the literature and more specifically, recently by Lantoine
and Russell (Lantoine, G., Russell, R.P. [2011]. Celest. Mech. Dynam. Astron. 109, 333–366) and by
Biscani and Izzo (Biscani, F., Izzo, D. [2014]. Mon. Not. R. Astron. Soc. 439, 810–822) as we will discuss
in this paper. We give here the full set of solutions for the motion of a particle (in our case for an atom
or a molecule), i.e. the space coordinates and the time solution for bounded and unbounded trajectories
in terms of Jacobi elliptic functions. We thus provide here the complete set of solutions for this so-call
Stark effect (Stark, J. [1914]. Ann. Phys. 348, 965–982) in terms of Jacobi elliptic functions (Jacobi, C.G.
J. [1829]. Fundamenta nova theoriae functionum ellipticarum. Sumtibus fratrum), which may be used
to model the trajectories of particles in planetary exospheres.