White dwarf binaries suggest a common envelope efficiency $\alpha \sim 1\mathrm{/}3$

Common envelope (CE) evolution, which is crucial in creating short period binaries and associated astrophysical events, can be constrained by reverse modeling of such binaries’ formation histories. Through analysis of a sample of well-constrained white dwarf (WD) binaries with low-mass primaries (7 eclipsing double WDs, 2 non-eclipsing double WDs, 1 WD-brown dwarf), we estimate the CE energy efficiency ${\alpha }_{\mathrm{C}\mathrm{E}}$ needed to unbind the hydrogen envelope. We use grids of He- and CO-core WD models to determine the masses and cooling ages that match each primary WD’s radius and temperature. Assuming gravitational wave-driven orbital decay, we then calculate the associated ranges in post-CE orbital period. By mapping WD models to a grid of red giant progenitor stars, we determine the total envelope binding energies and possible orbital periods at the point CE evolution is initiated, thereby constraining ${\alpha }_{\mathrm{C}\mathrm{E}}$. Assuming He-core WDs with progenitors of 0.9 - 2.0 $M_{\odot }$, we find ${\alpha }_{\mathrm{C}\mathrm{E}}\text{\hspace{0.17em}⁣}\sim \text{\hspace{0.17em}⁣}0.2-0.4$ is consistent with each system we model. Significantly higher values of ${\alpha }_{\mathrm{C}\mathrm{E}}$ are required for higher mass progenitors and for CO-core WDs, so these scenarios are deemed unlikely. Our values are mostly consistent with previous studies of post-CE WD binaries, and they suggest a nearly constant and low envelope ejection efficiency for CE events that produce He-core WDs.