On the Binding Energy Parameter $\lambda$ of Common Envelope Evolution

The binding energy parameter $\lambda$ plays an important role in common envelope (CE) evolution. Previous works have already pointed out that $\lambda$ varies throughout the stellar evolution, though it has been adopted as a constant in most of the population synthesis calculations. We have systematically calculated the binding energy parameter $\lambda$ for both Population I and Population II stars of masses $1-20M_{\text{sun}}$, taking into account the contribution from the internal energy of stellar matter. We present fitting formulae for $\lambda$ that can be incorporated into future population synthesis investigations. We also briefly discuss the possible applications of the results in binary evolutions.