  The complex interplay between the Solar Wind and the lunar surface serves as
a quintessential example of space weathering. However, uncertainties persist
regarding the influence of plasma originating from Earth's ionosphere,
necessitating a comprehensive understanding of its quantitative impact.
Hitherto, the dearth of reliable models has impeded accurate computation of ion
flux from Earth to the Moon under varying solar wind conditions.The objective
of this study is to adapt a kinetic model for the challenging conditions of
having both the Earth and the Moon in a single simulation box. IAPIC, the
Particle-In-Cell Electromagnetic Relativistic Global Model was modified to
handle the Sun-Earth-Moon system. It employs kinetic simulation techniques that
have proven invaluable tools for exploring the intricate dynamics of physical
systems across various scales while minimizing the loss of crucial physics
information such as backscattering. The modeling allowed to derive the shape
and size of the Earth's magnetosphere and allowed tracking the O$^+$ and H$^+$
ions escaping from the ionosphere to the Moon: $\mathrm{O^+}$ tends to escape
towards the dayside magnetopause, while $\mathrm{H^+}$ travels deeper into the
magnetotail, extending up to the Lunar surface. In addition, plasma temperature
anisotropy and backstreaming ions were simulated, allowing for future
comparison with the experiment. This study shows how a kinetic model can
successfully be applied to study the transport of ions in the Earth-Moon
environment. A second paper will detail the effect on the Lunar environment and
the impact on the Lunar water.
