On the Energy Equation and Efficiency Parameter of the Common Envelope Evolution

TL;DR

This paper investigates the structure of evolved giant stars to evaluate the envelope binding energy parameter lambda, revealing its dependence on stellar evolution stage and introducing a new method to determine binary system parameters post-common envelope phase.

Contribution

It provides a detailed analysis of the lambda parameter's dependence on stellar radius and introduces a novel approach for solving the energy equation in binary evolution.

Findings

Lambda ranges from 0.2 to 0.8 for most stars.

High lambda values (>5) can explain long orbital periods in some binary pulsars.

A tabulation of lambda as a function of stellar radius and mass is provided.

Abstract

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.