$Gaia$ white dwarfs within 40 pc II: the volume-limited northern hemisphere sample.

We present an overview of the sample of northern hemisphere white dwarfs within 40 pc of the Sun detected from $Gaia$ Data Release 2 (DR2). We find that 521 sources are spectroscopically confirmed degenerate stars, 111 of which were first identified as white dwarf candidates from $Gaia$ DR2 and followed-up recently with the William Herschel Telescope and Gran Telescopio Canarias. Three additional white dwarf candidates remain spectroscopically unobserved and six unresolved binaries are known to include a white dwarf but were not in our initial selection of white dwarfs in the $Gaia$ DR2 Hertzsprung-Russell diagram (HRD). Atmospheric parameters are calculated from $Gaia$ and Pan-STARRS photometry for all objects in the sample, confirming most of the trends previously observed in the much smaller 20 pc sample. Local white dwarfs are overwhelmingly consistent with Galactic disc kinematics, with only four halo candidates. We find that DAZ white dwarfs are significantly less massive than the overall DA population ($\overline{M}_\mathrm{DAZ} = 0.59\,\mathrm{M}_\odot$, $\overline{M}_\mathrm{DA} = 0.66\,\mathrm{M}_\odot$). It may suggest that planet formation is less efficient at higher mass stars, producing more massive white dwarfs. We detect a sequence of crystallised white dwarfs in the mass range from $0.6\ \lesssim M/\mathrm{M}_\odot \lesssim\ 1.0\,$ and find that the vast majority of objects on the sequence have standard kinematic properties that correspond to the average of the sample, suggesting that their nature can be explained by crystallisation alone. We also detect 56 wide binaries including a white dwarf and 26 double degenerates.