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 & F814W) data and self-consistently cross-calibrate WFC3-IR JH (F110W & F120W) data using and absolute magnitude calibration of M_I = -4.05 mag as determined in the LMC 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 amongst these multi-wavelength calibrations. We go on to present a method for determining the reddening, extinction and the true modulus to the host galaxy using multi-wavelength 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.

中文翻译：

---

红巨星分支尖端多波长结构的数学基础

我们考虑了红巨星分支 (TRGB) 尖端在光学和近红外中的应用，以确定与附近星系的距离。我们分析了 ACS VI（F555W 和 F814W）数据，并使用 LMC 中使用分离的食双星几何视差确定的 M_I = -4.05 mag 的绝对星等校准和自洽交叉校准 WFC3-IR JH（F110W 和 F120W）数据. 我们演示了 TRGB 方法的光学和近红外校准如何在数学上自洽，并说明这些多波长校准之间的数学基础和关系。我们继续提出一种使用多波长数据确定宿主星系的变红、消光和真实模量的方法。该方法的强大之处在于，对于高精度的数据，