be used to expel the common envelope to infinity. We can also find the position that a final binary would have if it kept the same orbital separation, for any valid post-CE mass of the donor $m_\mathrm{c, min}<m_\mathrm{d,c}<m_\mathrm{d}$. Clearly, a widened orbit would violate energy conservation.

We also plot the energy “expense”, the initial energy value plus the energy needed to disperse the envelope above a given core mass, as the dashed lines in Fig.\[alpha\] (assuming $\alpha=1$ — the most efficient case possible, a smaller value would shift the curve higher). The different points on these tracks are determined by removing the outer mesh point, computing the amount of energy needed to expel that mass shell to infinity, and removing that energy from the binary orbit. The angular momentum for these states is computed using the remaining masses and orbital energy to satisfy Kepler’s law. As this energy expense is the smallest possible, the final binary state should lie anywhere *above* the dashed lines for a given core mass. For comparison, we show a solution where $\lambda$ is calculated from the stellar structure, and where $\lambda=1$ is assumed.

Energy considerations limit the range of possible post-CE binary separations, as there 