giant star at the onset of the CE and $\varepsilon$ is the specific internal energy. $E_\mathrm{bind}$ therefore consists of the potential energy of the envelope and its internal energy, and can be found directly from the stellar structure for any choice of core mass. In our calculations, we adopt to include in $\varepsilon$ only the thermal energy of the gas and the radiation energy, but not the recombination energy. There are also alternative definitions for $E_\mathrm{bind}$, where ionization energy [e.g. @Han02] or enhanced winds [e.g. @Soker04] are taken into account.

Another parameter, $\alpha_\mathrm{CE}$, is introduced as a measure of the efficiency with which energy is transferred from the orbit into envelope expansion. Invoking energy conservation, one can then find the final orbital separation from

$$\alpha_\mathrm{CE} {\lambda} 
    \left ( \frac{ G m_\mathrm{d,c} m_\mathrm{a}} {2 a_\mathrm{f}} -\frac{ G m_\mathrm{d} m_\mathrm{a}} {2 a_\mathrm{i}} \right ) 
        =
  \frac {G m_\mathrm{d} m_\mathrm{d,e}} {r_\mathrm{d}}.
\label{allam}$$

The obvious bounds on this parameter are $0<\alpha_\mathrm{CE} \leq 1$, since energy cannot be generated. The parameter $\lambda$ can be calculated from detailed stellar-structure models. For many low-mass giant stars of interest $\lambda\approx 1$, and therefore many authors assume $\alpha \lambda =1$ or $0.5$. @vdsluys2010 find $0.027 < \lambda < 1.73$ 