Unlocking Galactic Wolf-Rayet stars with $\textit{Gaia}$ DR2 I: Distances and absolute magnitudes

We obtain distances to 383 Galactic Wolf-Rayet (WR) stars from $\textit{Gaia}$ DR2 parallaxes and Bayesian methods, with a prior based on H$\mathrm{\scriptsize{II}}$ regions and dust extinction. Distances agree with those from Bailer-Jones et al. for stars up to 2 kpc from the Sun, though deviate thereafter due to differing priors, leading to modest reductions in luminosities for recent WR spectroscopic results. We calculate visual and K-band absolute magnitudes, accounting for dust extinction contributions and binarity, and identify 187 stars with reliable absolute magnitudes. For WR and O stars within 2 kpc, we find a WR/O ratio of 0.09. The distances are used to generate absolute magnitude calibrations and obtain the $\mathrm{\textit{Gaia}}$ colour magnitude diagram for WR stars. Average $v^{\rm WR}$-band absolute magnitudes for WN stars range from -3.6 mag (WN3-4) to -7.0 mag (WN8-9ha), and -3.1 (WO2-4) to -4.6 mag (WC9), with standard deviations of $\sim$0.6 mag. Using H$\mathrm{\scriptsize{II}}$ region scale heights, we identify 31 WR stars at large (3$\sigma$, |z|$\geq$156 pc) distances from the mid-plane as potential runaways accounting for the Galactic warp, of which only 4 involve WN8-9 stars, contrary to previous claims.