On the energy equation and efficiency parameter of the common envelope evolution

We have investigated the structure of evolved giant stars with masses 3 - 10 M_sun in order to evaluate the binding energy of the envelope to the core prior to mass transfer in close binary systems. This binding energy is expressed by a parameter λ which is crucial for determining the outcome of binaries evolving through a common envelope (CE) and spiral-in phase. We discuss the λ-parameter and the efficiency of envelope ejection in the CE-phase, and show that λ depends strongly on the evolutionary stage (i.e. stellar radius) of the donor star at the onset of the mass transfer. The existence of this relation enables us to introduce a new approach for solving the energy equation. For a given observed binary system we can derive a unique solution for the original mass and age of the donor star, as well as the pre-CE orbital period. We find that the value of λ is typically between 0.2 and 0.8. But in some cases, particularly on the asymptotic giant branch of lower-mass stars, it is possible that λ > 5. A high value of λ (rather than assuming a high efficiency parameter, η_CE > 1) is sufficient to explain the long final orbital periods observed among those binary millisecond pulsars which are believed to have evolved through a CE-phase. We also present a tabulation of λ as a function of stellar radius and mass, which is useful for a quick estimation of the orbital decay during a common envelope and spiral-in phase.