Speaking with one voice: simulations and observations discuss the common envelope α parameter

We present a comparative study between the results of most hydrodynamic simulations of the common envelope binary interaction to date and observations of post-common envelope binaries. The goal is to evaluate whether this data set indicates the existence of a formula that may predict final separations of post-common envelope systems as a function of precommon envelope parameters. Some of our conclusions are not surprising while others are more subtle. We find that: (i) Values of the final orbital separation derived from common envelope simulations must at this time be considered upper limits. Simulations that include recombination energy do not seem to have systematically different final separations; these and other simulations imply α_CE < 0.6-1.0. At least one simulation, applicable to double-degenerate systems, implies α_CE < 0.2. (ii) Despite large reconstruction errors, the post-RGB observations reconstructed parameters are in agreement with some of the simulations. The post-AGB observations behave instead as if they had a systematically lower value of α_CE. The lack of common envelope simulations with low-mass AGB stars leaves us with no insight as to why this is the case. (iii) The smallest mass companion that survives the common envelope with intermediate mass giants is 0.05-0.1 M☉. (iv) Observations of binaries with separations larger than ∼10 R☉, tend to have high M₂/M₁ mass ratios and may go through a relatively long phase of unstable Roche lobe mass transfer followed by a weakened common envelope (or with no common envelope at all). (v) The effect of the spatial resolution and of the softening length on simulation results remains poorly quantified.