Spectroscopic classification of a complete sample of
  astrometrically-selected quasar candidates using Gaia DR2
  Here we explore the efficiency and fidelity of a purely astrometric selection
of quasars as point sources with zero proper motions in the {\it Gaia} data
release 2 (DR2). We have built a complete candidate sample including 104
Gaia-DR2 point sources brighter than $G<20$ mag within one degree of the north
Galactic pole (NGP), all with proper motions consistent with zero within
2$\sigma$ uncertainty. In addition to pre-existing spectra, we have secured
long-slit spectroscopy of all the remaining candidates and find that all 104
stationary point sources in the field can be classified as either quasars (63)
or stars (41). The selection efficiency of the zero-proper-motion criterion at
high Galactic latitudes is thus $\approx 60\%$. Based on this complete quasar
sample we examine the basic properties of the underlying quasar population
within the imposed limiting magnitude. We find that the surface density of
quasars is 20 deg$^{-2}$, the redshift distribution peaks at $z\sim1.5$, and
that only eight systems ($13^{+5}_{-3}\%$) show significant dust reddening. We
then explore the selection efficiency of commonly used optical, near- and
mid-infrared quasar identification techniques and find that they are all
complete at the $85-90\%$ level compared to the astrometric selection. Finally,
we discuss how the astrometric selection can be improved to an efficiency of
$\approx70\%$ by including an additional cut requiring parallaxes of the
candidates to be consistent with zero within 2$\sigma$. The selection
efficiency will further increase with the release of future, more sensitive
astrometric measurement from the Gaia mission. This type of selection, purely
based on the astrometry of the quasar candidates, is unbiased in terms of
colours and emission mechanisms of the quasars and thus provides the most
complete census of the quasar population within the limiting magnitude of Gaia.
