A Maximum Likelihood Calibration of the Tip of the Red Giant Branch
  Luminosity from High Latitude Field Giants using Gaia Early Data Release 3
  Parallaxes
  The calibration of the tip of the red giant branch (TRGB) in the I-band has a
direct role in determinations of the Hubble constant, a subject of recent
interest. We present a maximum likelihood (ML) method designed to obtain an
independent calibration of the brightness of TRGB using Gaia parallaxes from
the Early Data Release 3 (EDR3) of Milky Way field Giants at high Galactic
latitude. We adopt simple parameterizations for the Milky Way stellar
luminosity function and density law and and optimize the likelihood of the
observed sample as a function of those parameters. Using parameters to
partially constrain the luminosity function from other galaxies similar to the
Milky Way for which high quality TRGB data are available, we find values of the
TRGB magnitude of $ M_I^{TRGB} $ = $-3.978$ $\pm$ $0.031$ (stat) $\pm$ $0.045$
(sys) mag, where the systematic uncertainty covers the range of shape
parameters found in our reference galaxies. While current data are insufficient
to simultaneously characterize the shape of the Milky Way luminosity function
we estimate that the photometry from Gaia Data Release 3 (mid-2022) will allow
better constraints on the shape, and lower statistical uncertainties on the tip
by a factor of 3. With expected releases of improved parallax measurements from
Gaia, the method of calibrating the TRGB luminosity from field Giants is
expected to reach $\sim$ 0.01 mag uncertainty, which is an important step
toward a precise TRGB-based determination of the Hubble constant.
