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 due to the discrepancy between direct and indirect estimates of its value. We present a maximum likelihood 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 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 ${M}_{I}^{\mathrm{TRGB}}=-3.91\pm 0.05$ (stat) ± 0.09 (sys) mag, where the systematic uncertainty covers the range of shape parameters found in our Milky Way sample and in reference galaxies. While APASS Data Release 9 all-sky photometry is insufficient to provide a reliable constraint on 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 the expected releases of improved parallax measurements from Gaia, the method of calibrating the TRGB luminosity from field giants is expected to reach ~0.01 mag uncertainty, which is an important step toward a precise TRGB-based determination of the Hubble constant.