Analytical expressions for the envelope binding energy of giants as a function of basic stellar parameters

The common-envelope (CE) phase is an important stage in the evolution of binary stellar populations. The most common way to compute the change in orbital period during a CE is to relate the binding energy of the envelope of the Roche-lobe filling giant to the change in orbital energy. Especially in population-synthesis codes, where the evolution of millions of stars must be computed and detailed evolutionary models are too expensive computationally, simple approximations are made for the envelope binding energy. In this study, we present accurate analytic prescriptions based on detailed stellar-evolution models that provide the envelope binding energy for giants with metallicities between Z = 10^-4 and Z = 0.03 and masses between 0.8 M _⊙ and 100 M _⊙, as a function of the metallicity, mass, radius, and evolutionary phase of the star. Our results are also presented in the form of electronic data tables and Fortran routines that use them. We find that the accuracy of our fits is better than 15% for 90% of our model data points in all cases, and better than 10% for 90% of our data points in all cases except the asymptotic giant branches for three of the six metallicities we consider. For very massive stars (M ≳ 50 M _⊙), when stars lose more than 20% of their initial mass due to stellar winds, our fits do not describe the models as accurately. Our results are more widely applicable - covering wider ranges of metallicity and mass - and are of higher accuracy than those of previous studies.