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On the Binding Energy Parameter $λ$ of Common Envelope Evolution

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