Quantifying Uncertainties on the Tip of the Red Giant Branch Method

We present an extensive grid of numerical simulations quantifying the uncertainties in measurements of the Tip of the Red Giant Branch (TRGB). These simulations incorporate a luminosity function composed of 2 magnitudes of red giant branch (RGB) stars leading up to the tip, with asymptotic giant branch (AGB) stars contributing exclusively to the luminosity function for at least a magnitude above the RGB tip. We quantify the sensitivity of the TRGB detection and measurement to three important error sources: (1) the sample size of stars near the tip, (2) the photometric measurement uncertainties at the tip, and (3) the degree of self-crowding of the RGB population. The self-crowding creates a population of supra-TRGB stars due to the blending of one or more RGB stars just below the tip. This last population is ultimately difficult, though still possible, to disentangle from true AGB stars. In the analysis given here, the precepts and general methodology as used in the Chicago-Carnegie Hubble Program (CCHP) has been followed. However, in the Appendix, we introduce and test a set of new tip detection kernels which internally incorporate self-consistent smoothing. These are generalizations of the two-step model used by the CCHP (smoothing followed by Sobel-filter tip detection), where the new kernels are based on successive binomial-coefficient approximations to the Derivative-of-a-Gaussian (DoG) edge detector, as is commonly used in modern digital image processing.