A Correlation Between the Final Separation and Mass Ratio from Common Envelope Simulations

Analytical models for common envelope evolution (CEE), particularly the energy formalism, are used in binary population synthesis to predict post-CEE configurations. This formalism is based on an efficiency parameter alpha, which relates the orbital energy released during CEE to that required to unbind the envelope of the giant. However, one of the main challenges is that CEE is a multiscale, multiphysics process. As a result, there may not be a universal value for alpha, or even a general expression. Using 13 3D simulations of CEE with RGBs (1 and 2 M$_\odot$ primary; four mass ratios; with and without corotation), we present an empirical linear correlation between the post-plunge-in separation and the mass ratio, normalized by the giant radius. This trend for the plunge-in phase of CEE persists across RGB, AGB, and supergiant simulations in the literature, even for partially bound envelopes. Therefore, alpha from simulations should not be used to predict the final separation, but rather as a diagnostic of whether sufficient orbital energy has been liberated to completely eject the envelope immediately after the radial plunge. If this condition is not met, further in-spiral is expected in later stages of CEE, which may explain why the final separation of post-CEE observations is generally smaller than those predicted by the linear fit. Our results reinforce the idea that a better description could emerge if CEE is treated as a sequence of distinct phases, rather than treating it as a single event governed by alpha.