The Degree of Alignment Between Circumbinary Disks and Their Binary Hosts

All four circumbinary (CB) protoplanetary disks orbiting short-period ($P < 20$ day) double-lined spectroscopic binaries (SB2s)---a group that includes UZ Tau E, for which we present new ALMA data---exhibit sky-plane inclinations $i_{\rm disk}$ which match, to within a few degrees, the sky-plane inclinations $i_\star$ of their stellar hosts. Although for these systems the true mutual inclinations $\theta$ between disk and binary cannot be directly measured because relative nodal angles are unknown, the near-coincidence of $i_{\rm disk}$ and $i_\star$ suggests that $\theta$ is small for these most compact of systems. We confirm this hypothesis using a hierarchical Bayesian analysis, showing that 68% of CB disks around short-period SB2s have $\theta < 3.0^\circ$. Near co-planarity of CB disks implies near co-planarity of CB planets discovered by Kepler, which in turn implies that the occurrence rate of close-in CB planets is similar to that around single stars. By contrast, at longer periods ranging from $30-10^5$ days (where the nodal degeneracy can be broken via, e.g., binary astrometry), CB disks exhibit a wide range of mutual inclinations, from co-planar to polar. Many of these long-period binaries are eccentric, as their component stars are too far separated to be tidally circularized. We discuss how theories of binary formation and disk-binary gravitational interactions can accommodate all these observations.