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Masses and envelope binding energies of primary stars at the onset of a common envelope

We present basic properties of primary stars that initiate a common envelope (CE) in a binary, while on the giant branch. We use the population-synthesis code described in Politano et al. (2010) and follow the evolution of a population of binary stars up to the point where the primary fills its Roche lobe and initiates a CE. We then collect the properties of each system, in particular the donor mass and the binding energy of the donor's envelope, which are important for the treatment of a CE. We find that for most CEs, the donor mass is sufficiently low to define the core-envelope boundary reasonably well. We compute the envelope-structure parameter {λ_\mathrm{env}} from the binding energy and compare its distribution to typical assumptions that are made in population-synthesis codes. We conclude that {λ_\mathrm{env}} varies appreciably and that the assumption of a constant value for this parameter results in typical errors of 20--50%. In addition, such an assumption may well result in the implicit assumption of unintended and/or unphysical values for the CE parameter {α_\mathrm{CE}}. Finally, we discuss accurate existing analytic fits for the envelope binding energy, which make these oversimplified assumptions for {λ_\mathrm{env}}, and the use of {λ_\mathrm{env}} in general, unnecessary.