Mathematical Underpinnings of the Multiwavelength Structure of the Tip of the Red Giant Branch

We consider the application of the tip of the red giant branch (TRGB) in the optical and in the near-infrared for the determination of distances to nearby galaxies. We analyze ACS VI (F555W and F814W) data and self-consistently cross-calibrate WFC3-IR JH (F110W and F120W) data using an absolute magnitude calibration ofM(I) = -4.05 mag as determined in the Large Magellanic Cloud using detached eclipsing binary star geometric parallaxes. We demonstrate how the optical and near-infrared calibrations of the TRGB method are mathematically self-consistent, and illustrate the mathematical basis and relations among these multiwavelength calibrations. We go on to present a method for determining the reddening, extinction, and the true modulus to the host galaxy using multiwavelength data. The power of the method is that with high-precision data, the reddening can be determined using the TRGB stars themselves, and decreases the systematic (albeit generally small) uncertainty in distance due to reddening for these halo stars.