Weighing the Galactic disk using phase-space spirals III. Probing
  distant regions of the disk using the Gaia EDR3 proper motion sample
  We have applied our method for weighing the Galactic disk using phase-space
spirals to the Gaia EDR3 proper motion sample. For stars in distant regions of
the Galactic disk, the latitudinal proper motion has a close projection with
vertical velocity, such that the phase-space spiral in the plane of vertical
position and vertical velocity can be observed without requiring that all stars
have available radial velocity information. We divided the Galactic plane into
360 separate data samples, each corresponding to an area cell in the Galactic
plane in the distance range of 1.4-3.4 kpc, with an approximate cell length of
200-400 pc. Roughly half of our data samples were disqualified altogether due
to severe selection effects, especially in the direction of the Galactic
centre. In the remainder, we were able to infer the vertical gravitational
potential by fitting an analytic model of the phase-space spiral to the data.
This work is the first of its kind, in the sense that we are weighing distant
regions of the Galactic disk with a high spatial resolution, without relying on
the strong assumptions of axisymmetry. Post-inference, we fit a thin disk scale
length of $2.2\pm 0.1$ kpc, although this value is sensitive to the considered
spatial region. We see surface density variations as a function of azimuth on
the order of 10-20 %, which is roughly the size of our estimated sum of
potential systematic biases. With this work, we have demonstrated that our
method can be used to weigh distant regions of the Galactic disk despite strong
selection effects. We expect to reach even greater distances and improve our
accuracy with future Gaia data releases and further improvements to our method.
