The H$\alpha$ sky in three dimensions

We combine parallax distances to nearby O stars with parsec-scale resolution three-dimensional dust maps of the local region of the Milky Way (within 1.25 kpc of the Sun) to simulate the transfer of Lyman continuum photons through the interstellar medium. Assuming a fixed gas-to-dust ratio, we determine the density of ionized gas, electron temperature, and H$\alpha$ emissivity throughout the local Milky Way. There is good morphological agreement between the predicted and observed H$\alpha$ all-sky map of the Wisconsin H$\alpha$ Mapper. We find that our simulation underproduces the observed H$\alpha$ emission while overestimating the sizes of HII regions, and we discuss ways in which agreement between simulations and observations may be improved. Of the total ionizing luminosity of $5.84 \times 10^{50}~{\rm photons~s^{-1}}$, 15% is absorbed by dust, 64% ionizes "classical'' HII regions, 11% ionizes the diffuse warm ionized medium, and 10% escapes the simulation volume. We find that 18% of the high altitude ($|b| > 30^{\circ}$) H$\alpha$ arises from dust-scattered rather than direct emission. These initial results provide an impressive validation of the three-dimensional dust maps and O-star parallaxes, opening a new frontier for studying the ionized ISM's structure and energetics in three dimensions.