  We investigate the common envelope binary interaction, that leads to the
formation of compact binaries, such as the progenitor of Type Ia supernovae or
of mergers that emit detectable gravitational waves. In this work we diverge
from the classic numerical approach that models the dynamic in-spiral. We focus
instead on the asymptotic behaviour of the common envelope expansion after the
dynamic in-spiral terminates. We use the SPH code {\sc phantom} to simulate one
of the setups from Passy et al., with a 0.88~\ms, 83~\rs \ RGB primary and a
0.6~\ms \ companion, then we follow the ejecta expansion for $\simeq 50$~yr.
Additionally, we utilise a tabulated equation of state including the envelope
recombination energy in the simulation (Reichardt et al.), achieving a full
unbinding. We show that, as time passes, the envelope's radial velocities
dominate over the tangential ones, hence allowing us to apply an homologous
expansion kinematic model to the ejecta. The external layers of the envelope
become homologous as soon as they are ejected, but it takes $\simeq 5000$~days
($\simeq 14$~yr) for the bulk of the unbound gas to achieve an homologous
expanding regime. We observe that the complex distribution generated by the
dynamic in-spiral evolves into a more ordered, ring-like shaped one in the
asymptotic regime. We show that the thermodynamics of the expanding envelope
are in very good agreement with those expected for an adiabatically expanding
sphere under the homologous condition and give a prediction for the location
and temperature of the photosphere assuming dust to be the main source of
opacity.
