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

Abstract

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 $\alpha_{\rm{CE}}$ 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 $\alpha_{\rm CE}$. Assuming He-core WDs with progenitors of 0.9 - 2.0 $M_\odot$, we find $\alpha_{\rm CE} \! \sim \! 0.2-0.4$ is consistent with each system we model. Significantly higher values of $\alpha_{\rm{CE}}$ 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.