which the first phase of mass loss appears to have been accompanied by orbital expansion.

Preliminaries
=============

We now outline the two canonical prescriptions for treating common-envelope evolution, and develop the standard formalism for treating stable mass transfer in the case of Roche-lobe overflow.

Energy balance versus angular-momentum balance {#sec:2.1}
----------------------------------------------

In the standard treatment of CE outcomes, the final separation of the binary is determined via the “energy formalism” [@Webbink84], in which the binding energy of the (expelled) envelope is equated to the decrease in the orbital energy $E_\mathrm{orb}$:

$$E_\mathrm{bind} = E_\mathrm{orb,i} - E_\mathrm{orb,f} 
        = -\frac{ G m_\mathrm{d} m_\mathrm{a}} {2 a_\mathrm{i}} 
            + \frac{ G m_\mathrm{d,c} m_\mathrm{a}} {2 a_\mathrm{f}}.$$

Here $a_\mathrm{i}$ and $ a_\mathrm{f}$ are the initial and final binary separations, $m_\mathrm{d}$ and $m_\mathrm{a}$ are the initial star masses (donor and accretor, respectively) and $ m_\mathrm{d,c}$ is the final mass of the donor, after losing its envelope.

A parameter $\lambda$ is introduced to characterise the central concentration of the donor envelope:

$$E_\mathrm{\lambda, bind} = \int_\mathrm{core}^\mathrm{surface} 
                \left (  \frac{Gm}{r(m)} - \varepsilon (m) \right ) dm 
            = \frac {G m_\mathrm{d} m_\mathrm{d, e}} {\lambda r_\mathrm{d}}.
\label{lambda}$$

Here $m_\mathrm{d, e}$ is the mass of the removed (giant’s) envelope, $r_\mathrm{d}$ is the radius of the 