Calibration of the Tip of the Red Giant Branch (TRGB)

The Tip of the Red Giant (TRGB) method provides one of the most accurate and precise means of measuring the distances to nearby galaxies. Here we present a VIJHK absolute calibration of the TRGB based on observations of TRGB stars in the Large Magellanic Cloud (LMC),grounded on detached eclipsing binaries (DEBs). This paper presents a more detailed description of the method first presented in Freedman et al. (2019) for measuring corrections for the total line-of-sight extinction and reddening to the LMC. In this method, we use a differential comparison of the red giant population in the LMC, first with red giants in the Local Group galaxy, IC 1613, and then with those in the Small Magellanic Cloud. As a consistency check, we derive an independent calibration of the TRGB sequence using the SMC alone, invoking its geometric distance also calibrated by DEBs. An additional consistency check comes from near-infrared observations of Galactic globular clusters covering a wide range of metallicities. In all cases we find excellent agreement in the zero-point calibration. We then examine the recent claims by Yuan et al. (2019), demonstrating that, in the case of the SMC, they corrected for extinction alone while neglecting the essential correction for reddening as well. In the case of IC 1613, we show that their analysis contains an incorrect treatment of (over-correction for) metallicity. Using our revised (and direct) measurement of the LMC TRGB extinction, we find a value of Ho = 69.6 +/-0.8 (+/-1.1% stat) +/- 1.7 (+/-2.4% sys) km/s/Mpc.


INTRODUCTION
The Tip of the Red Giant Branch (TRGB) sequence marks the well-understood, abrupt evolutionary transition of low-mass RGB stars onto the lower-luminosity horizontal branch. These RGB stars are powered by a hydrogen-burning shell surrounding an isothermal helium core that is supported by electron degeneracy. Their transition away from the TRGB is initiated by a brief helium flash, a period of thermal run-away whereupon the degeneracy of the core is lifted, and the triple-alpha (helium-burning) process can proceed at a lower luminosity in a gas-pressure-supported core. At this transition, the tip stars rapidly fade in luminosity, settling onto the horizontal branch, burning helium in their core and hydrogen in a surrounding shell. The basic underlying physical explanation for a universal upper limit to the luminosity of an RGB star is theoretically well understood (e.g., Salaris et al. 1997, Bildsten et al. 2012Serenelli et al. 2017); and empirically, this distinctive feature in the observed RGB luminosity function is well-known to be an excellent means of measuring the distances in the halos of nearby, resolved galaxies: in the I band the TRGB is a 'standard candle' (e.g., Rizzi et al. 2007;Serenelli et al. 2017).
Over the past 30 years the TRGB sequence has been measured and successfully used by many independent groups to determine high-precision distances to large numbers of nearby galaxies (e.g., Sakai et al. 2004;Makarov et al. 2006;Rizzi et al. 2007;Dalcanton et al. 2009Dalcanton et al. , 2012Jang & Lee 2017a,b;McQuinn et al, 2019;Freedman et al, 2019 and references therein). The unique imaging capabilities of the Hubble Space Telescope (HST ), combined with the ease with which the I-band tip in particular (see below) can be measured and used to determine distances, have made this method readily accessible and widely applied by the general astronomical community. Indeed, a recent search of the literature (NED August 2019) shows that there are currently some 900 applications of the TRGB method published for over 300 individual galaxies. A program aimed at optimally determining TRGB distances simply requires targeted observations of the halos of any nearby galaxy, made to sufficient depth to resolve and measure the brightest old Population II red giant stars, which are known to inhabit all stellar halos. Although the method itself is not new, it is only recently that the TRGB method has been applied directly to the determination of H o (Karachentseva et al. 2003, Mould et al. 2008Jang & Lee 2017;Freedman et al. 2019).
One of the distinct advantages of the TRGB method is the fact that the TRGB stars populate the halos of the host galaxy (in contrast to the Cepheids, for example, which are found only in the higher-surface-brightness disk regions). Additionally, the halos of undisturbed galaxies are demonstrably lacking in gas and devoid of dust, reddening and extinction. While this latter advantage holds true in the application of the TRGB method, the situation is somewhat more complicated when it comes to calibrating the TRGB. We consider this aspect of the TRGB calibration in what follows.
It is anticipated that the zero-point calibration of the TRGB will clearly be strengthened when absolute trigonometric parallaxes from Gaia become available for significant samples of nearby TRGB stars in the halo of the Milky Way. A preliminary calibration based on the Gaia Data Release 2 (DR2) has been published by Mould et al. (2019), which is in broad agreement with earlier calibrations. However, the early-release parallax measurements are still preliminary; and, they are subject to known systematic uncertainties (as discussed in Arenou et al. 2018). Anticipating the upcoming DR3 results from Gaia, which are still some years away from publication, we have, in the meantime, chosen to use the Large Magellanic Cloud (LMC) to set the absolute calibration for the TRGB. We have then compared this TRGB zero point with additional, parallel calibrations, each grounded in geometric distances (as is the LMC) derived from the analysis of detached eclipsing binaries (DEBs) in the SMC (Graczyk et al. 2014) and 47 Tucanae (Thompson et al. 2019). We have used the da Costa & Armandroff (1990; hereafter DCA90) study of RGB stars in Galactic globular clusters, extending the analysis into the near-infrared using 2MASS data and using the new geometric distance to 47 Tucanae based on detached eclipsing binaries for the zero-point calibration.
For well-separated (detached) double-line spectroscopic, binary star systems, the stellar radii can be measured from their photometric and radial-velocity curves. As articulated by Paczyski (1997), the distance to the DEB can then be determined through an empirical calibration of the surface brightness (angular diameter) and color (temperature). The DEB method has been recently and extensively developed by Pietrzyski and collaborators (see Pietrzyski et al. 2019 and references therein) concentrating on late-type giants. These authors find an extremely tight relation between the optical/near-IR (V − K) color and surface brightness, with an rms scatter of only 0.018 mag, from which they have determined a 1% distance to the LMC.
Using the Hubble Space Telescope Advanced Camera for Surveys (HST/ACS ), Freedman et al. (2019) applied a calibration of the TRGB, which was subsequently applied to galaxies hosting Type Ia supernovae (SNe Ia), and used to determine a new value of the Hubble constant, anchored to the DEB distance to the LMC. In that work, we briefly described a new method for measuring the extinction for TRGB stars in the LMC, and we provided a preliminary calibration at V IJHK wavelengths for the TRGB. Here we provide a more detailed description and update of the method, which makes use of a consistent set of TRGB stars with multi-wavelength data. A clear advantage of this method is that it provides a direct measurement of the extinction, derived from, and applicable to the TRGB stars themselves.
We first begin by illustrating the expected behavior of the TRGB, based on published isochrones spanning a range in metallicity, color and age. Based on VI and JHK photometry for TRGB stars in the LMC we then detail our method for measuring the extinction and reddening, and determine a calibration of the TRGB zero point. We discuss our measurement of the LMC extinction in the context of other recent studies, provide two external consistency checks on our calibration, and present a value for H 0 based on our adopted calibration.
In Appendix A, we address some serious issues arising in the recent paper by Yuan et al. (2019;hereafter Y19). We demonstrate that the adoption of incorrect assumptions has led to an erroneous result; in specific, their underestimate of the line-of-sight extinction to the TRGB stars in the LMC.

TRGB THEORETICAL ISOCHRONES
To provide some overall context for the method that we have developed, we first show the expected behavior of the TRGB as a function of metallicity, age and bandpass. For illustrative purposes, we use the PARSEC (Padova and Trieste Stellar Evolutionary Code: http://stev.oapd.inaf.it/cgi-bin/cmd) isochrones (CMD Version 3.3; Bressan et al. 2012;Marigo et al. 2017). We are not aiming here to provide a comprehensive discussion of different TRGB models, but simply to convey, in broad outline, the general behavior of the TRGB as a function of wavelength, age and metallicity. We note that here and throughout this paper, all magnitudes are on the Vega System.
In Figure 1 (upper panel), we examine the behavior with wavelength of red giant branch stars with a range of metallicities at V , I, and K-band wavelengths. Shown are giant-branch isochrones with a large range of metallicities spanning -2.0 < [M/H] < −0.1 dex. As is well-known, with increasing metallicity the V -band isochrones become fainter with wavelength, the I-band isochrones have a nearlyconstant magnitude, and the near-infrared isochrones become brighter.
In Figure 1 (lower panel) we also examine the behavior with wavelength of red giant branch stars with a range of ages. Shown are giant-branch isochrones at V , I, and K-band wavelengths, with an age spread covering 4 × 10 9 to 10 10 years, at a fixed metallicity of [M/H] = -1.8 dex. This plot illustrates the overall insensitivity of Figure 1. PARSEC isochrones for red giant branch stars with a constant age and a metallicity spread (upper panel) and a constant metallicity, but with an age spread (lower panel), are shown for V (blue), I (black) and K (red) bandpasses, to the same scale, for comparison. In the upper panel, the isochrones have a constant age of 10 Gyr and a metallicity range from -2.0 < [M/H] < -0.1 dex; in the lower panel, the isochrones have a fixed metallicity of [M/H] = -1.8, and an age spread of 4 < t < 10 Gyr. The x-axis for the V and I isochrones is the (V − I) color, while for K it is (J − K); however, for clarity the V -band isochrones have been shifted by -3.5 mag in (V − I), and the K-band isochrones have been shifted by +4.0 mag in (J − K). As can be seen, for older stellar populations (>4 Gyr), the RGB colors are very insensitive to age, while the colors track differences in metallicity very clearly. the giant branch luminosity, at a given wavelength, to the age of older RGB stars. Most of the color width of a composite old (>4 Gyr) stellar population results from a spread in metallicity, not a spread in age. At fixed metallicity an age spread of 4×10 9 to 10 10 years introduces an effective (color-correlated) magnitude scatter of ± 0.040 (at V), 0.005 (I), 0.012 (J), 0.023 (H) and 0.026 (K) mag, respectively. In Figure  2 we show the isochrones varying in metallicity for a single 10 Gyr age at V IJHK wavelengths. These wavelengths correspond to those for which we have data in the LMC, SMC and IC 1613, which we turn to in §3 below. We note here that the I-band magnitude is remarkably constant over this illustrated range of metallicities (and corresponding colors), and gives rise to a single (standard candle) I-band tip magnitude, while at other wavelengths there is instead a slanting distribution, or a (decreasing and/or increasing)"run" of -absolute magnitudes with increasing metallicities/colors. There is therefore no unique "tip" magnitude in these other bands, but there is nevertheless a very well-defined (and theoretically predicted) correspondence of absolute magnitude with intrinsic color. We refer to these distributions as "TRGB sequences". The same stars defining the tip of the red giant branch at a given metallicity will also define the tip in different band color-magnitude diagrams, not at arbitrary magnitudes, but in the logical order that is predicted by the models. Moreover, the fact that higher metallicity manifests itself by making stars observed in the optical fainter while simultaneously resulting in brighter stars in the near-infrared, allows for the effects of metallicity and reddening to be decoupled and individually solved for. This direct dependence of the TRGB color with metallicity, and the negligible dependence on the age for older populations, provides a powerful means of accounting for differences in metallicity, while simultaneously correcting for extinction.
To summarize this section, there is little dependence on age for all TRGB stars older than about 4 Gyr (e.g., Serenelli et al. 2017). Conversely, there is an extremely tight relation between the color of the stars defining the TRGB and the metallicity of the TRGB population, which provides a basis to empirically measure the extinction.
In what follows, we do not use the theoretical isochrones in our analysis to determine reddenings or distances: we undertake an entirely empirical analysis. However, we confirm empirically that the behavior of the RGB with color and wavelength follows the general predictions from theory, thereby lending confidence to the overall method and its application.

THE METHODOLOGY: A MULTI-WAVELENGTH APPROACH TO THE TRGB CALIBRATION
In this paper, we use V I photometric data from the third phase of the Optical Gravitational Lensing Experiment (OGLE-III; Udalski et al. 2008) for both the LMC and the SMC. For the LMC, we use the "Shallow" survey data published by Ulacyk, et al. (2012); both the LMC and SMC data are available at the following website http://www.astrouw.edu.pl/ogle/ogle3/maps/. Yuan et al. (2019) have pointed out that the coarser pixel scale in the optical (U BV I) survey of the two Magellanic Clouds undertaken by Zaritsky et al. (2002Zaritsky et al. ( , 2004 probably suffer from blending issues. While the Zaritsky data for the SMC (but not the LMC) were used in the analysis by Freedman et al. (2019), here we now self-consistently use only the OGLE-III data for both the LMC and the SMC. In the near-infrared, JHK observations of both Clouds were obtained in the course of the 2MASS all-sky survey. We have cross-identified the LMC and SMC stars from the OGLE-III surveys with those in 2MASS. This merged catalog forms the basis for the following study and the LMC calibration of the TRGB. The IC 1613 data analyzed in this paper are from Hatt et al. (2017).
The method that we are using is conceptually very simple. As noted above, the individual stars that define the tip in the I band (at 8000Å) are precisely the same stars that must also define the TRGB at both shorter and longer wavelengths. Using a set of TRGB stars defined in the I band, we then make use of those same stars at a range of wavelengths from the optical to the near-infrared, where the extinction decreases with increasing wavelength. The overall methodology is similar in concept (but not identical) to that developed in the past for determining extinctions to Cepheid variables (e.g., Freedman 1988, Freedman, Wilson & Madore 1991. Before proceeding further we note that in this study we confine ourselves to the color range 1.4 < (V-I) < 2.2 mag (-1.4 > [Fe/H] > -0.6), where the I-band magnitude of the TRGB is observed to be approximately constant (see F19 and references therein). At colors beyond this range the theoretical dependence of the Iband TRGB is non-linear, as illustrated in Figure 3 of Mager, Madore & Freedman (2008) (which, in turn, was derived from Bellazzini et al. 2001Bellazzini et al. , 2008; however, the color range specified above encompasses virtually all of the low-metallicity halo stars used in extragalactic distance determinations, and it symmetrically straddles the peak of the Bellazzini relation in that color range. For all practical applications the TRGB I-band magnitude (in that restricted color range) is effectively constant; the formal scatter of less than ±0.01 mag (deviating by only ±0.015 mag, peak-to-peak).

Schematic Illustration of the Method
In Figure 3, we show a schematic representation of the steps involved in measuring both extinction and reddening. The upper dashed lines represent the run of the TRGB absolute magnitude as a function of intrinsic color. The I-band relation is flat with color. The brightest RGB stars decline in luminosity as a function of color for wavelengths shorter than the I band, while these same stars increase in luminosity with color for wavelengths longer than the I band. The solid lines (marked 'True') midway down in each of the three diagrams are the intrinsic TRGB sequences displaced to fainter magnitudes by identical (true distance modulus) offsets labeled µ o . Further downward displacements of the TRGB lines are tagged by the wavelengthdependent extinctions (A V , A I & A K ) which systematically decrease in amplitude with wavelength, followed by wavelength-independent reddenings, each of the same amount, shown as E(V − I) in each diagram. The resulting extincted, reddened and distance-modulus displaced traces of the three example TRGBs are labeled 'Apparent'. The upward arrows separating the 'Apparent' and 'True' moduli in V , I and K have magnitudes of ∆ V , ∆ I and ∆ K as marked. It should be noted that these vertical displacements are not to be confused or directly equated with the extinction values appropriate to each of those wavelengths, except for the I-band, where, coincidentally, the slope of the TRGB is flat with color. For all downward sloping ("blue") TRGB relations ∆ λ underestimates A λ ; and for all upward sloping and ("infrared") TRGB relations ∆ λ overestimates A λ . The magnitude of that error (the difference between A λ and ∆ λ ) is simply s λ × E(V − I), where s λ is the slope of the TRGB in the selected color-magnitude diagram. For clarity, Using simulated error-free data, Figure 4 illustrates the method for using TRGB data to simultaneously determine the true distance modulus and simultaneously, the total line-of-sight extinction and reddening from multi-wavelength data. The steep solid line (labeled True) is the run of extinction as a function of wavelength fit to the observations. Scatter around this particular fit is, by definition, zero for this simulated set of error-free data. The apparent magnitude off-sets ∆ λ between the intrinsic TRGB relations and the apparent TRGB relations (as described in Figure  Figure 3. A schematic illustration of multi-wavelength TRGB seen in three representative bands labeled V, I and K from left to right, illustrating the effects of distance, extinction and reddening. See text for details. 3) are shown as squares with inset filled circles. They show a shallow run of extinction with wavelength that is significantly less than the input value, and they show measurable scatter, resulting from not having applied the shifts required by the reddening terms. By iteratively compensating for the difference between ∆ λ and A λ , re-plotting, re-fitting and each time re-calculating the sum of the squares of the residuals, a minimum is approached and then exceeded (when the applied reddening exceeds the input value), as illustrated by the two surrounding solutions in Figure  4 (which have measurable scatter clearly in excess of zero) shown as dashed lines through open circles. Figure 4. A schematic illustration of the process of using TRGB data to simultaneously determine the true distance modulus and reddening from multi-wavelength data. Dots enclosed in squares are the result of the first iteration on the reddening simply differencing the apparent magnitude of the tip stars with the intrinsic relations. The scatter is nonnegligible. Open circles show alternate reddening solutions on both sides of the input value. These solutions also show increasing (non-zero) scatter, most easily seen in the dashed-line solutions passing above and below the I-band data point. The input reddening is shown by the circled dots and fit by the steep solid line which was recovered by simply mimimizing the dispersion about the fit. See text for details.

Application of the Method
We begin with the I vs (V − I) CMD where the slope of the TRGB is minimally impacted by color/metallicity variations, as noted above. The full set of CMDs for the OGLE-III LMC data is shown in Figure 5. For the LMC we spatially restricted the CMD sample to those stars outside of a one-degree radius circle centered on the bar of the LMC, where crowding within the high-surface-brightness (high-crowding) portions of the bar could bias the magnitudes of the TRGB stars therein.
The I vs (V − I) CMD is shown in the upper right panel. We selected a small subset of 306 tracer stars in the CMD that fall symmetrically about the calibrated fit to the tip, selected within the color range 1.4 < (V-I) < 2.2 mag. These stars have OGLE I-band photometry (as noted above, from the "Shallow" survey data published by Ulacyk, et al. 2012), with quoted errors of less than ±0.02 mag at I ∼ 14.5 mag. They sample the tip with a scatter of ±0.04 mag, giving a scatter on the mean of ±0.002 mag. In the optical, the slope in the I-band has been set to zero (thereby resulting in a V-band slope of unity), while in the near-infrared the slopes are taken from Madore et al. (2018). We give these here for completeness, along with their uncertainties: at J, H and K, the slopes are -0.85 ± 0.09, -1.62 ± 0.16, and -1.85 ± 0.19, respectively. Our specific goal here is the difference in zero points, not a full solution of all possible parameters. As we shall see, these slopes are consistent with the observed CMDs shown in Figures 5 and 6 discussed below. Thus we solve here for two parameters only: the total line-of-sight reddening and the true distance modulus for our selected sample of LMC TRGB stars.
Given the set of tracer stars described above (taken from Freedman et al. 2019 and derived from their Figure 4), we mapped each of the I-band tracers into the CMDs at shorter (V ) and longer (JHK) wavelengths. Using the adopted slopes of the TRGB in the flanking CMDs shown in Figure 5, we then determined their individual zero points by minimizing the scatter between the intrinsic line of predetermined slope and the tracer stars. Errors on the mean for each of the zero points were then calculated from the measured scatter of the 306 tracer stars in each color.
For our TRGB zero-point calibration via the LMC, we use two fiducial galaxies, IC 1613 and the SMC, each with very low line-of-sight reddenings. Before undertaking the multi-wavelength, differential reddening analysis of the TRGB population of the LMC with respect to these galaxies, we have first subtracted the small foreground reddenings of E(B − V ) = 0.022 (IC 1613) mag and 0.033 mag (SMC), respectively, based on the Galactic foreground reddening maps of Schlafly & Finkbeiner (2011), which is a re-calibration of Schlegel, Finkbeiner & Davis 1998), as tabulated in NED. To be clear, these maps do not include any correction for extinction in the main bodies of galaxies along the extended line of sight, but the host-galaxy TRGB stars are expected to have negligible extinction, given that they reside in the gas-and dustfree halos of these galaxies. In Figure 6, we show a similar set of the five V IJHK CMDs for IC 1613 and the SMC. The fits to their TRGB sequences were obtained using the identical procedure as discussed above for the LMC stars. Subtracting the respective fits at each of the wavelengths for both galaxies, we determined differential apparent distance moduli to our subset of TRGB stars in the LMC for each of the five (V IJHK) bandpasses. This step yields a first estimate of the differential distance moduli, each of which contains a fixed pillar, which is the difference in their true moduli. This common (geometric) offset is, of course, independent of wavelength. We emphasize, however, that these simple differential distance moduli at this stage do not yet account for the reddening of the TRGB stars, and they are therefore only a first approximation to the extinction. As we describe below, to measure both extinction and total reddening we iterate on the solution, solving simultaneously for distance, extinction and reddening by minimizing the sum of the squares of the residuals around the fit of extinction versus inverse wavelength. In what follows, we have adopted the reddening curve used in NED which is taken from Schlafly & Finkbeiner (2011), which in turn is derived from Fitzpatrick (1999) using R v = 3.1. We have used a source spectrum appropriate to a K-type giant, closely matching a typical elliptical galaxy, or as in this case, TRGB stars.  Hatt et al. (2017) and the SMC optical data are from OGLE-III. The slanting lines mark the apparent magnitude levels of the TRGB at each of the wavelengths, using predetermined slopes, and fit as described in the text. The arrows are as described in Figure 5.

Extinction Curves
In Figure 7 (lower panel) we show the iterated extinction plot derived from a comparison of the apparent magnitudes of the TRGB stars in the LMC with respect to IC 1613. The fit, as shown by the solid black line, gives a difference in their true distance moduli of ∆µ = +5.899 mag, giving a true distance modulus for IC 1613 of µ IC1613 = 24.376 mag. This distance agrees reasonably well with the recentlypublished Cepheid distance modulus of 24.29 ± 0.03 mag given by . (The mean and  It should be noted that the two unknowns (true distance modulus and reddening) cannot (mathematically) be determined from one pair of colors. At least three bands (two independent colors) are required. If this minimal number is adopted in future studies then, as large a wavelength baseline as possible should be chosen. With current facilities V , I and K bands would be a competitive choice.

Zero Points
We adopt the DEB distance modulus to the LMC to be 18.477 mag (Pietrzyski et al. 2019). Taking the average of the two LMC reddenings derived above then gives E(B − V ) LM C = 0.097 mag, which converts to A V = 0.300, A I = 0.165, A J = 0.078, A H = 0.049 and A K = 0.032 mag. Applying these corrections to the observed magnitudes of the TRGB stars in the LMC at their respective wavelengths gives the following absolute multi-wavelength calibrations of the TRGB method normalized to an intrinsic color of (V − I) o = 1.80 mag and applicable over the color range 1.4 < (V −I) o < 2.2 mag for the optical (V I), and normalized to (J −K) o = 1.00 mag for the near-infrared (JHK) calibration. In updating our calibration with respect to earlier ones we have here rationalized the pivot-point colors in the optical and infrared so that the fits in both cases are set to colors that correspond to the same metallicity, i.e., [Fe/H] = -1.0 dex. This simply required moving the (V-I) normalization from 1.60 to 1.80 mag. This shift does not impact the I-band calibration, which is taken to be flat over the above color range. For the I-band, M T RGB I = 14.595 -18.477 -0.165 = -4.047 ± 0.022 (stat) ± 0.039 (sys) mag, where the uncertainties are summarized in Table 2  Adopting ∆µ(SM C − LM C) = 0.488 mag and the DEB distance modulus for the SMC of 18.965 ± 0.025 (stat) ± 0.048 (sys) mag, our detection of the I-band TRGB at 14.93 mag and an extinction of A I =0.056 mag (NED) gives M I (SM C) = -4.09 ± 0.03 (stat) ± 0.05 (sys) mag, in systematic agreement, at the one-sigma level, with our LMC measured value of -4.05 ± 0.03 (stat) ± 0.05 (sys) mag. The SMC tip detection was made using a standard Sobel filter applied to a GLOESS-smoothed I-band luminosity function based on stars within the color range 1.4 < (V − I) < 1.8 mag, using software and methods identical to those used in Freedman et al. (2019). Within that same color range 138 tip tracer stars, used to measure the tip magnitudes in VJH & K, yielded an error on the mean of σ I(T RGB) = ± 0.007 mag.  Table 2, converted to extinctions adopting a Fitzpatrick (1999) reddening law (solid black line). These extinction values are determined for a variety of stellar populations. The mean of the distribution is shown by an open white circle. In blue is the extinction value suggested by Yuan et al. (2019), based on a crowding correction, applied to the SMC photometry of Zaritsky et al. (2002). See Appendix A for a detailed description of why this post-processing of the results presented in Freedman et al. (2019) is problematic. In red and orange are the values determined here for TRGB stars, using IC 1613 or the SMC, respectively, to set the reddening floor. These values agree at the millimag level with the equivalent measurements made in Freedman et al. (2019). For reference, the thick white arrow plotted is the Schlafly and Finkbeiner (2011) estimate of the foreground reddening to the LMC. We note that the total extinction adopted by Yuan et al. is smaller than the foreground extinction. The two LMC extinctions derived in this paper, using the SMC and IC 1613, are shown as orange and red stars, respectively, and highlighted by two lines pointing to the label "This Study".

DEB-Based Galactic Globular Cluster Calibration of the TRGB
We now present a second independent check on the calibration of the TRGB. The zero point of this calibration of the near-infrared TRGB is based upon the recently measured detached eclipsing binary (DEB) based geometric distance to 47 Tucanae (Thompson et al. 2019). 1 Following da Costa & Armandroff (1990; DCA90) we use a selection of Galactic globular clusters, covering a wide range of metallicities ([Fe/H] = -2.2 to -0.7) 2 to produce multi-wavelength (JHK) composite color-magnitude diagrams. The composite CMD can then serve as the basis for an independent check on the zero-point calibration of the TRGB sequences, for comparison with the mixed-metallicity populations seen in the halos of nearby galaxies. Our sample includes M2, M4, M5, M15, M55, NGC 0362, NGC 1851, NGC 6362, NGC 6397, NGC 6752, and 47 Tuc. We use the DCA90 compilation because it provides a set of consistent relative distances to the various clusters. We set the absolute zero point using the recently determined DEB measurements. In a more detailed, multi-wavelength study of the TRGB calibration in preparation, we will be applying proper-motion selection using high-resolution Gaia DR2 data to study a significantly larger sample of 45 Galactic globular clusters (observed in 10 band-passes from the optical to the near-infrared).
At present, we have revisited the TRGB calibration of the slopes and zero points primarily in the near-infrared for two main reasons: (1) There is homogeneous, high precision and high accuracy JHK photometry for all of these targets, obtained in the course of the 2MASS Survey all-sky (Cutri et al. 2006). 3 (2) The impact of line-ofsight reddening on the zero-point calibration of the TRGB is greatly diminished by working in the infrared. The globular cluster sample studied by DCA90 is a relatively low line-of-sight reddening subset of Galactic globular clusters, with reddenings ranging from E(B − V ) =0.02 to 0.10 mag, which translate to extinction corrections in the near infrared of A J = 0.016 to 0.080 mag, A H = 0.01 to 0.05 mag and A K = 0.007 to 0.034 mag (after dropping NGC 6397 because of its large foreground extinction of A V = 0.56 mag.) Reddening corrections have been adopted from DCA90. We note also the JHK TRGB study for 24 globular clusters belonging both to the bulge and the halo of the Galaxy, undertaken by Valenti et al. (2004a,b). In general the reddenings for this sample are larger (with E(B − V ) values up to 1.3 mag), so we have not (at this time) used them for our purposes.  Thompson et al. (2019) have determined direct distances to two DEBs in the metal-rich Galactic globular cluster 47 Tuc. These two stars have geometrically determined distances of 4.41 ± 0.06 and 4.60 ± 0.09 kpc, respectively, giving an average true distance modulus of µ o = 13.27± 0.07 mag. As noted in Thompson et al. the analysis by Chen et al. (2018) of Gaia parallaxes to 47 Tuc yield a distance modulus of 13.24 ± 0.005 (stat) ± 0.058 mag (sys). The Gaia parallax corresponds to a distance of 4.45 kpc, which agrees with the DEB distance to within 1%. Adopting the DEB geometric calibration, we re-zero the true distance moduli to each of the DCA90 globular clusters, with the required offset being -0.12 mag.

Composite Globular Cluster Color-Magnitude Diagrams and Luminosity Functions
In Figure 9 we show a composite JHK CMD which combines each of the individual datasets for the least-reddened clusters. The upward-slanting lines in each of the three panels trace the TRGB calibration, previously published in Madore et al. (2018). The correspondence between the two provides an independent check and confirmation of the result derived from the LMC TRGB stars. The two vertical lines mark the intrinsic color range (0.65 < (J −K) < 1.25 mag) over which a linear calibration can be observationally defined. The horizontal broken line shows the magnitude of the TRGB increasing in brightness towards longer wavelengths as measured at the fiducial color of (J − K) = 1.00 mag. Figure 10. A composite CMD for the clusters shown in Figure 9 , where in this case, the magnitudes have been rectified by flattening out the color dependence seen in the raw CMDs. Using the measured slopes we flatten or "rectify" the color-magnitude diagrams such that their marginalized luminosity functions show the greatest contrast in the TRGB discontinuity. The rectified CMDs are shown in Figure 10. Figure 11 shows the marginalized luminosity functions of the rectified composite CMDs for the Galactic globular clusters shown above. 117 RGB stars contribute to the ±0.3 mag interval within which the detection and measurement of the tip was made in this figure. The statistical uncertainty on the detections is found to be ±0.051 mag in J and K and ±0.057 mag in H. Averaged over the three wavebands, the derived zero point in the I-band is calculated to be M I = -4.056 ± 0.053 (stat) ± 0.080 (sys) mag, where the systematic error is that carried over directly from the DEB systematic error on the distance to 47 Tuc.

Geometry of the LMC
A further test that we have undertaken is to determine the effect of geometry on the measurement of the TRGB in the LMC. Using the OGLE III photometric data from Ulaczyk et al. (2012), we adopted a position angle of 132 deg and an inclination angle of 25 deg for the LMC disk, as determined by Pietrzyski et al. (2019) (see also Figure 3 of Jacyszyn-Dobrzeniecka et al. 2016, based on Cepheids). As these and earlier studies of the LMC have shown, the northeast side of the LMC is closer to us.
We calculated the displacements of individual stars along the line of sight direction between the tilted (observed) and non-tilted (inclination corrected) disks. The displacement is measured to be up to ∼1400 pc (∼3% in distance and ∼0.06 mag in distance modulus). The median/mean of the displacements is very small, amounting to only ∼10 pc. This is naturally expected, as the spatial coverage of the OGLE data is quite symmetric centered on the LMC. We then applied the edge detection algorithm to the original and the inclination-corrected OGLE catalogs. As can be seen in Figure 12, the two measured tip values are entirely consistent, with values of 14.606 mag and 14.604 mag, respectively, with uncertainties of 0.02 mag.

Summary of Consistency Checks on the Zero Point
As summarized in Table 1, the independent determinations of the I-band magnitude for the TRGB based on the 1) DEB measurements for the SMC and 2) JHK 2MASS observations for a sample of Galactic globular clusters, calibrated by DEB distance measurements in 47 Tucanae, agree very well with our determination for the LMC (see §3.4). We adopt the latter measurement, M T RGB I = -4.047 ± 0.022 (stat) ± 0.039 (sys) for our calibration of the TRGB. For clarity, we note that the calibration for the F814W filter, differs from the I-band calibration by -0.0068 mag (see Freedman et al. 2019, §3.4), giving M T RGB F 814W = -4.054 ± 0.022 (stat) ± 0.039 (sys).

IMPLICATIONS FOR THE HUBBLE CONSTANT
We adopt the average of our measured values from §3.4 of A LM C I = 0.165 ± 0.02 mag. Based on the calibration for HST/ACS F814W from Freedman et al. (2019), applied to a sample of nearby galaxies with TRGB distances, and tied to the Carnegie Supernova Project distant sample of SNe Ia, we determine a slightly revised value of H 0 = 69.6 ± 0.8 (±1.1% stat) ± 1.7 (2.4% sys) km s −1 Mpc −1 , (a difference of 0.23%). We note that despite the criticisms leveled by Y19, our value of H 0 agrees to within 1-σ with their quoted value of H 0 = 72.4 ± 1.9. In Figure 13, we show our value of H 0 compared to other recently published determinations in the literature. Our purpose in this paper is not to undertake a detailed assessment of current H 0 values. At this time, we note simply that the Hubble 'tension' issue remains.

SUMMARY AND CONCLUSIONS
In this paper we have introduced and described in detail a new methodology for the simultaneous determination of true distance moduli and total line-of-sight reddenings for TRGB stars, self-consistently using multi-wavelength data for the TRGB stars themselves. We have used a set of stars in the outer main body of the LMC, combined with the recently published high-accuracy and high-precision distance to the LMC, to calibrate the TRGB method at five independent wavelengths ranging from the optical (V I) to the near infrared (JHK).
The multi-wavelength calibration resulting from the LMC analysis alone is given in §3.4. We find an absolute I-band magnitude for the TRGB of M T RGB I = -4.047 ± 0.022 (stat) ± 0.039 (sys); and for F814W, M T RGB F 814W = -4.054 ± 0.022 (stat) ± 0.039 (sys). Independent consistency checks on this calibration have also been presented. The first is a calibration through TRGB stars in the SMC, using a geometric distance to the SMC based on five DEBs. This calibration gives an I-band zero point of M I = -4.09 ± 0.03 (stat) ± 0.05 (sys) mag. The second is a near-infrared calibration of using a composite set of Galactic globular clusters with a range of (spectroscopicallydetermined) metallicities from [Fe/H] = -2.2 to -0.7 dex. Without any additional free parameters, we set the zero-point calibration adopting the new DEB distance modulus to 47 Tuc, tied to the relative distances from DCA90. This gives an I-band zero point of M I = -4.06 ± 0.05 (stat) ± 0.08 (sys) mag. Each of these independent checks result in zero points consistent with our adopted value of our primary calibration using the LMC alone.
We find a value for the LMC TRGB extinction of A I = 0.160 ± 0.024 mag using the nearby galaxy, IC 1613, for the differential reddening comparison. This value agrees well with the independent value of A I = 0.169 ± 0.014 mag, using OGLE-III photometry for the SMC. This photometry suffers less from crowding/blending effects than that of Zaritsky et al. (2002), and leads to an improved estimate of the LMC TRGB extinction than that obtained by F19. Our measurement technique has the advantage of self-consistently being determined by and for the TRGB stars themselves. Our (combined) value of A I = 0.165 ± 0.02 mag lies within one sigma of the mean for other recent estimates of the LMC extinction. It differs by 0.005 mag from that adopted in Freedman et al. (2019).
Adopting the HST/ACS calibration from Freedman et al. (2019), applied to a sample of nearby galaxies with TRGB distances, and tied to the Carnegie Supernova Project distant sample of SNe Ia, we determine a value of H 0 = 69.6 ± 0.8 (±1.1% stat) ± 1.7 (2.4% sys) km s −1 Mpc −1 .
We note that the measurement of the extinction can continue to be improved with future study. We have already obtained new high-resolution optical data using the 6.5-meter Magellan Baade telescope for several nearby galaxies, in order to further improve the accuracy of our LMC reddening estimate.
Results published recently by Y19 differ from ours for several reasons. The first point of divergence is that we use the OGLE-III photometry of the SMC to derive the LMC extinction, rather than attempting, as Y19 do, to correct the problematic photometry of Zaritsky et al. (2002). Second, Y19 adjust the F19 result for singleband extinction alone, neglecting the concomitant, iterative corrections for reddening, which must be applied (as discussed in detail in §3.2). Third, Y19 apply a theoretical correction for metallicity effects to the Freedman et al. (2019) result for the IC 1613 calibration, neglecting the fact that the method outlined in Freedman et al., and again in more detail in this paper, already explicitly accounts for metallicity. The methodology employed in Y19 differs significantly from the analysis process that we have explicated here and applied in Freedman et al.; and in major respects their interpretation of the method is simply in error. We conclude that our direct measurement of the LMC TRGB extinction based on the TRGB stars themselves is to be preferred over a(n arbitrary) value determined for a different population of stars, and which is systematically low compared to other recent studies in the published literature (see Figure 8). Despite these differences, we note that the Freedman et al. (2019) and Y19 H o values agree to within 1-σ.
An accurate measure of the extinction to TRGB stars in the LMC will remain an important (but not the only) component of the calibration and application of the TRGB method to the determination of H o . Future improvements will come from higher precision multi-wavelength photometry for LMC TRGB stars, and from new multi-wavelength photometry and Gaia DR3 parallaxes to their counterparts in the halo of the Milky Way. Future Gaia parallaxes for Milky Way globular clusters will also strengthen the Galactic calibration of the TRGB. We note that the TRGB method, as applied to the determination of H 0 , is relatively new compared to the application of the Cepheid distance scale or cosmic microwave background measurements. Additional calibrators for the TRGB method will be forthcoming from HST and eventually with the James Webb Space Telescope. The high precision measured for the TRGB, and the ability to work in the isolated, low-reddening halos of galaxies of all morphological types, will be critical for breaking the impasse in current measurements of H 0 . hereafter Y19) concluded that the results of our paper were incorrect. Here, we show that there are a number of misunderstandings and incorrect assumptions made in the Y19 paper. For example, in their critique of our results based on a differential analysis of the LMC reddening with respect to the SMC and IC 1613, they (back)-apply corrections to the F19 result, leading them to erroneous conclusions. We now list and discuss these issues in detail below.
1. The SMC: Use of the Zaritsky et al. (2002) data Y19 noted that the OGLE-III photometry for the SMC is less subject to blending issues than that available to Freedman et al. (2019), which was based on the survey of Zaritsky et al. (2002;hereafter Z02). As seen earlier in §3.2, we have now analyzed the OGLE-III photometry, and emphasize that the Y19 criticism of the SMC reddening (as being due to blending issues) is no longer applicable to the results in this current paper. Indeed, as shown in their Figure 5, there is no apparent trend of the photometric offsets with local number density. We note that the difference in reddening, resulting from adopting the OGLE-III data over the Z02 data for the SMC, amounts to only +0.005 mag, and falls within the one-sigma uncertainty quoted by Freedman et al. (2019).

Y19 "correction" to the Z02 SMC data
We note that the re-analysis by Y19 for the SMC data is based on the Z02 photometry, which they have demonstrated are less accurate. Rather than use the better OGLE-III photometry, Y19 correct the Z02 results for blending effects, and then back-correct the Freedman et al. (2019) results. We point out that in doing this 'back-correction', Y19 correct for a magnitude effect (the extinction), but do not iteratively and simultaneously correct for the color (reddening) term that must also be accounted for (see the discussion in §3.2 and Figures 3, and 4; without taking these effects into account, the extinction and reddening terms will be underestimated.) We further note their blending correction may or may not be applicable to the actual stars used in the Freedman et al. analysis. We conclude that the Y19 'back-corrected' result for the SMC extinction value based on the Z02 data is incorrect.
3. IC 1613: Y19 statement about the "V" data Y19 conclude that they cannot make use of the data for IC 1613, used here and in Freedman et al. (2019), for a photometry comparison. They state that "the IC 1613 detections are presented in Hatt et al. (2017) where ground-based V photometry is calibrated to a redder ACS filter, F606W." We wish to be clear here, that as outlined in §3.2, we do not make use of such a transformation for our derived fit to the extinction law. The optical data that enter into the derivation of the extinction fit are zeroed to the V I photometry from Hatt et al. (2017), based on the well-calibrated set of Stetson photometry (http://www.cadc-ccda.hia-iha.nrccnrc.gc.ca/en/community/STETSON/standards/). In Freedman et al. and Hatt et al. we compare these ground-based data to the F814W data for IC 1613, and find agreement to ±0.02 mag (1%). To briefly summarize, we fit an extinction law to a multi-wavelength (V IJHK) set of data to determine A I for the LMC, which we then apply to the I-band TRGB magnitude. The transformation to the HST/ACS F814W system is done only in the last step, when we calibrate the TRGB magnitude. The data referred to in Hatt et al. are not used and are not relevant for the analysis for determining the reddening described here.
4. IC 1613: Y19 double correction for metallicity Y19 cite McQuinn et al. (2019; hereafter M19) stating that "the use of a single slope to 'rectify' the TRGB for its color dependence at different metallicities produces differences in the TRGB of ∼0.04 mag for the difference in metallicity between -2 to -1 dex. Quoting directly, Y19 state that they "naively use the comparison in McQuinn et al. (2019) for the same SFH and ∆[F e/H] (from -2 to -1 dex)" to illustrate the impact of rectifying the TRGB colors on the extinction estimate. These corrections amount to 0.026, 0.033 and 0.038 mag, making the TRGB stars fainter in J, H and K, respectively; and amounting to -0.019 and -0.014 mag offsets in V and I, respectively, making stars at those wavelengths brighter. After making these corrections, Y19 find a value for A LM C I = 0.10 mag "in agreement with earlier determinations." We have discussed the (broad) range of earlier published reddening determinations in §3.5. However, we point out that the correction applied by Y19 for metallicity is simply incorrect, as illustrated by both the theoretical isochrones shown in Figures 1 and 2, and by the empirical color-magnitude diagrams shown in Figures 5 and 6. The wellknown metallicity-tracking trends with color for RGB stars are directly exhibited in the CMDs. Y19 have thus apparently applied corrections for metallicity effects twice.
We summarize the results of Figure 2 once again here: The TRGB stars at I, for example, map uniquely into their respective positions in the CMDs at other colors, and they do so in accordance with their metallicities as reflected and calibrated by their colors. As expected, the most metal-rich stars are fainter at V , and brighter at K. These other wavebands do not have a flat distribution with magnitude; there is a well-defined run or sequence of magnitude with color that is not arbitrary. Using a self-consistent sample of precisely the same RGB stars defined in the I-band CMD, one can make use of the (very fortunate) fact that the optical (V and I), and the near-infared (JHK) bandpasses exhibit distinctly different behaviors with metallicity. The opposite signs of the optical/near-IR behavior resulting from metallicity differences is in contrast to that of extinction, which is a steadily decreasing function of inverse wavelength. The different dependencies of metallicity and extinction with wavelength thus allows the two effects to be separated, and the extinction can be independently measured, as demonstrated in Figure 3, and detailed in §3.1.

LMC Reddening
Y19 argue that the extinction estimates from Haschke et al. (2011) are the best available estimate of the LMC reddening. The recent results of Gorski et al. (2020) dispute that claim, arguing for reddenings that are 0.06 mag higher in E(B − V ) than the Haschke et al. calibration, which is based on red clump (RC) stars and RR Lyrae stars. Unlike the case for TRGB stars, the evolutionary tracks for RC stars are well-known to be strong functions of both age and metallicity (see, for example, Girardi & Salaris (2001) and Williams et al. (2009)). Finally, as discussed by Joshi & Panchal (2019), and in agreement with Gorski et al., it is found that the Haschke extinction estimates are systematically lower than all other estimates reviewed in their summary. For completeness, we note also that a higher value for the RC star extinction is also found in the earlier analysis by Pawlak (2016). At the time of their study, Jang & Lee (2017) had no direct estimate of the TRGB extinction and adopted the Haschke et al. extinction value, with its originally published uncertainty of 0.07 mag.