The Common Envelope Evolution Outcome. III. the Improvement of Stellar Binding Energy with the Envelope Residual

Abstract

Common-envelope evolution (CEE) is a key process in the evolution of close binary systems. Many important astrophysical objects and evolutionary stages are closely related to CEE, including white dwarf binaries, hot subdwarfs, and gravitational wave mergers. In the standard energy formalism of CEE, the binding energy of the donor envelope plays a crucial role, as it directly affects the final orbital period after CEE and serves as a key physical parameter in binary population synthesis studies. However, the currently adopted binding energy suffers from large uncertainties, mainly because the envelope binding energy of giant-branch stars varies strongly near the helium-core boundary. In addition, the expansion of the star during CEE can also affect the binding energy. To address these issues, we introduce an improved binding energy for the envelope mass residual. Based on adiabatic mass loss models, we recalculate the distribution of the CEE binding-energy parameter lambda for stars with different masses and at different evolutionary stages, and we analyse the effects of envelope mass residual and adiabatic expansion. Due to the envelope mass residual, the lambdas of some donors can increase by one to two orders of magnitude at the late red giant branch and asymptotic giant branch stages. Furthermore, we provide interpolation grids and fitting formulae for these results, which can be readily applied to various binary population synthesis codes.