The Carnegie-Chicago Hubble Program. IX. Calibration of the Tip of the
  Red Giant Branch Method in the Mega-Maser Host Galaxy, NGC4258 (M106)
  In the nearby galaxy NGC 4258, the well-modeled orbital motion of H$_2$O
masers about its supermassive black hole provides the means to measure a
precise geometric distance. As a result, NGC 4258 is one of a few "geometric
anchors" available to calibrate the true luminosities of stellar distance
indicators such as the Tip of the Red Giant Branch (TRGB) or the Cepheid
Leavitt law. In this paper, we present a detailed study of the apparent
magnitude of the TRGB within NGC 4258 using publicly-available HST observations
optimally situated in the gas- and dust-free halo along the minor axis,
spanning distances ranging from 8 to 22 kpc in projected galactocentric radius.
We undertake a systematic evaluation of the uncertainties associated with
measuring the TRGB in this galaxy, based on an analysis of 54 arcmin$^2$ of
HST/ACS imaging. After quantifying these uncertainties, we measure the TRGB in
NGC 4258 to be F814W$_0$ = 25.347 $\pm$ 0.014(stat) $\pm$ 0.042(sys) mag.
Combined with a recent 1.5% megamaser distance to NGC 4258, we determine the
absolute luminosity of the TRGB to be $M_{F814W}^{TRGB}$ = -4.050 $\pm$
0.028(stat) $\pm$ 0.048(sys) mag. This new calibration agrees to better than 1%
with an independent calibration presented in Freedman et al. (2019, 2020) that
was based on detached eclipsing binaries (DEBs) located in the LMC.
