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 envelopes 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.
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