The fraction of polar aligned circumbinary disks

Circumbinary gas disks that are misaligned to the binary orbital plane evolve toward either a coplanar or a polar-aligned configuration with respect to the binary host. The preferred alignment depends on the dynamics of the disk: whether it undergoes librating or circulating nodal precession, with librating disks evolving to polar inclinations and circulating disks evolving to coplanar. We quantify the fraction of binary star systems whose disks are expected to have polar orbits $f_\text{polar}$, extending previous work to include disks with non-zero mass. Our results suggest that, for low mass disks, the polar fraction is highly sensitive to the distribution of binary eccentricity with a higher fraction expected for higher binary eccentricities, $f_{\rm polar}\sim e_{\rm b}$. However, for massive discs, the fraction is independent of the binary eccentricity and $f_{\rm polar}\approx 0.37$. The value of $f_\text{polar}$ is always reduced in a population with a greater preference for low initial mutual inclination. We also explore the consequences of the finite lifetime and non-zero radial extent of a real disk, both of which affect a disk's ability to complete its evolution to a stationary configuration. Our findings can be used to make predictions given populations with well-understood distributions of binary eccentricity, initial mutual inclination, and disk angular momentum.