Completeness of the Gaia-verse IV: The Astrometry Spread Function of
  Gaia DR2
  Gaia DR2 published positions, parallaxes and proper motions for an
unprecedented 1,331,909,727 sources, revolutionising the field of Galactic
dynamics. We complement this data with the Astrometry Spread Function (ASF),
the expected uncertainty in the measured positions, proper motions and parallax
for a non-accelerating point source. The ASF is a Gaussian function for which
we construct the 5D astrometric covariance matrix as a function of position on
the sky and apparent magnitude using the Gaia DR2 scanning law and demonstrate
excellent agreement with the observed data. This can be used to answer the
question `What astrometric covariance would Gaia have published if my star was
a non-accelerating point source?'.
  The ASF will enable characterisation of binary systems, exoplanet orbits,
astrometric microlensing events and extended sources which add an excess
astrometric noise to the expected astrometry uncertainty. By using the ASF to
estimate the unit weight error (UWE) of Gaia DR2 sources, we demonstrate that
the ASF indeed provides a direct probe of the excess source noise.
  We use the ASF to estimate the contribution to the selection function of the
Gaia astrometric sample from a cut on astrometric_sigma5d_max showing high
completeness for $G<20$ dropping to $<1\%$ in underscanned regions of the sky
for $G=21$.
  We have added an ASF module to the Python package SCANNINGLAW
(https://github.com/gaiaverse/scanninglaw) through which users can access the
ASF.
