A $Gaia$ Data Release 2 catalogue of white dwarfs and a comparison with SDSS

We present a catalogue of white dwarf candidates selected from the second data release of $Gaia$ (DR2). We used a sample of spectroscopically confirmed white dwarfs from the Sloan Digital Sky Survey (SDSS) to map the entire space spanned by these objects in the $Gaia$ Hertzsprung-Russell diagram. We then defined a set of cuts in absolute magnitude, colour, and a number of $Gaia$ quality flags to remove the majority of contaminating objects. Finally, we adopt a method analogous to the one presented in our earlier SDSS photometric catalogues to calculate a probability of being a white dwarf ($P_{\mathrm{WD}}$) for all $Gaia$ sources which passed the initial selection. The final catalogue is composed of $486\,641$ stars with calculated $P_{\mathrm{WD}}$ from which it is possible to select a sample of $\simeq 260\,000$ high-confidence white dwarf candidates in the magnitude range $8<G<21$. By comparing this catalogue with a sample of SDSS white dwarf candidates we estimate an upper limit in completeness of $85$ per cent for white dwarfs with $G \leq 20$ mag and $T_{\mathrm{eff}} > 7000$ K, at high Galactic latitudes ($|b|>20^{\circ}$). However, the completeness drops at low Galactic latitudes, and the magnitude limit of the catalogue varies significantly across the sky as a function of $Gaia$'s scanning law. We also provide the list of objects within our sample with available SDSS spectroscopy. We use this spectroscopic sample to characterise the observed structure of the white dwarf distribution in the H-R diagram.