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 lambda which is crucial for determining the outcome of binaries evolving through a common envelope (CE) and spiral-in phase. We discuss the lambda-parameter and the efficiency of envelope ejection in the CE-phase, and show that lambda 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 lambda 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 lambda > 5. A high value of lambda (rather than assuming a high efficiency parameter, eta_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 lambda 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