Comparative Analysis of TRGBs (CATs) from Unsupervised, Multi-halo-field Measurements: Contrast is Key

Jang et al. (2021) and Anand et al. (2022) show that younger stellar populations can contaminate the tip, primarily through the presence of intermediate-age AGB stars and also from supergiants blueward of the RGB. One way to eliminate the contamination is using the spatial distribution of MS stars as a proxy for regions with younger stellar populations. Alternate routes would be a cut on the distance from the disk of the galaxy, which depends significantly on the shape of the galaxy, or a cut on the local sky brightness (Anand et al. 2018). Here we follow the conceptual approach of Anand et al. (2022), i.e., use MS stars to identify the position of young stellar populations, and develop a specific procedure for this step.

We illustrate our algorithm in Figure 4. First, we correct the CMD for Milky Way (MW) reddening (Schlafly & Finkbeiner 2011). As shown in Figure 2, there is a vertical red line, which equals F606W−F814W = 0.3 mag after MW extinction correction, as well as a horizontal green line, which is the magnitude where S/N reaches 10. Then we utilize the MS stars bluer than the vertical red line and brighter than the horizontal green line as a proxy for Population I stars, as marked blue in the bottom-left panel of Figure 4, and create a density map showing the number of blue stars per 16 arcsec² area. The green line is necessary here to prevent counting in parts of the CMD where the MS and RGB merge into each other. We also calculate the number of other stars (black dots in the bottom-left panel of Figure 4), as well as the ratio of these two densities. Density plots of the ratio and the blue star population are shown in the first two panels of Figure 4. We smooth the density map of MS stars using a two-dimensional Gaussian-windowed, Locally Weighted Scatterplot Smoothing (GLOESS) algorithm, which uses a two-dimensional Gaussian weighting function as

$$\begin{eqnarray}&&w([i,j]\to [m,n])={e}^{-\tfrac{{\left(x(i)-x(m)\right)}^{2}+{\left(y(j)-y(n)\right)}^{2}}{2{\sigma }^{2}}},\end{eqnarray} \tag{ 1 }$$

where functions x and y are the bin center in units of pixels. The smoothing scale is set by the σ parameter in Equation (1). For this analysis, we fix the smoothing scale for all fields, independent of distance. For this algorithm, we must determine the number for the smoothing scale as well as the spatial quality cut level and manner. Using the metrics discussed in Section 4.1, we find the optimal manner to cut is using the number density of blue MS stars rather than the ratio of blue-to-red stars. We use a cut level that is relative to the peak number density, and we determine from the metrics that this value should be ∼10%, such that all areas with a number density of blue stars >10% of the peak number density for that field are removed. Furthermore, we find a smoothing scale σ of 160 pixels is best; we show the results of smoothing in the top right of Figure 4. We present the success of this algorithm in the bottom row of Figure 4. The bottom-left and right panels show the original and clipped CMDs, respectively, while the bottom middle panel shows how the field is carved out. The figure shows that the younger population as seen from the CMD (seen in the bottom-left panel) is removed to a much greater extent than the RGB (as seen in the bottom-right panel).

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To further reduce contamination of the TRGB it is common to apply a slanted band in color–magnitude space that follows the RGB branch (Hatt et al. 2018; Jang et al. 2018), so the measurement is limited to the bluer, metal-poor region that is found to have a nearly constant absolute magnitude with color. An example of such a band is shown in Figure 3. A challenge of defining a proper band is that the mean color or slope of this branch may vary across fields or galaxies and that there is a balance to be struck between using a narrow range of color for population uniformity and retaining more stars for better statistics. In the literature, the type of color band has varied substantially. For example, Hatt et al. (2017, 2018) used a band to include the largest number of stars along the RGB branch, typically of width 0.5–2.0 mag. In Freedman et al. (2019), it is said that no band is used for the distance ladder measurement. In some analyses, when measured in F814W and F606W, the slope of the RGB branch is set to a fixed parameter (e.g., k = −6 mag · color⁻¹ in Jang et al. 2018). In other analyses like that of Hoyt et al. (2019, 2021), there is no color band applied (which we call vertical as it is equivalent to a band slope of ∞).

We parameterize a general color band with two boundaries, a blue and red limit with

$$\begin{eqnarray}\begin{array}{rcl}{\rm{F}}814{{\rm{W}}}_{\mathrm{blue}} & & =s\ast ({\rm{F}}606{\rm{W}}-{\rm{F}}814{\rm{W}})+b\\ {\rm{F}}814{{\rm{W}}}_{\mathrm{red}} & = & s\ast ({\rm{F}}606{\rm{W}}-{\rm{F}}814{\rm{W}}-w)+b,\end{array}\end{eqnarray} \tag{ 2 }$$

where the slope is s (Δ mag/Δ color) and the width of the band is w and intercept b. For an unsupervised algorithm, we fix the bandwidth w, and provide a range of the slope s (−7 ∼ −5 for F606W/F814W CMDs, and −3 ∼ −1 for F555W/F814W CMDs), and the values for s and b are found to maximize the number of stars in the band. This optimization can be done iteratively with the more global optimization of the width and slope range (see the Appendix) and redetermining the offset.

We utilize the ACS F606W and F814W photometry from the GHOSTS data set to construct CMDs. To create the LFs, we bin the stars by their ACS F814W magnitude in 0.01 mag intervals, and use the GLOESS method to smooth the LF (Persson et al. 2004). GLOESS distinguishes itself from other local regression smoothing techniques by applying weight from all bins with an emphasis locally such that the ith bin when smoothing locally at the jth bin as

$$\begin{eqnarray}&&w(i\to j)={e}^{-\tfrac{{\left(\mathrm{mag}(i)-\mathrm{mag}(j)\right)}^{2}}{2{\sigma }^{2}}},\end{eqnarray} \tag{ 3 }$$

where mag(i) is defined at the bin center and σ is the smoothing scale. Hatt et al. (2018) offer an algorithmic approach to the optimal smoothing method through the use of artificial star tests and find that the smoothing scale should roughly equal the level of the photometric uncertainty near the tip magnitude. However, this approach does not appear to be adopted for other TRGB measurements (e.g., Freedman et al. 2019). We vary the smoothing scale as an option, as shown in Table 1 and discussed in the Appendix, to minimize the field-to-field dispersion of the tip measurement around the mean for each galaxy.

The second step is generating the edge detection response (EDR) curve by applying a Sobel detection. The Sobel kernel [−1,0,+1] was first introduced into the context of TRGB by Lee et al. (1993) and is equivalent to a centered first derivative of the function

$$\begin{eqnarray}&&\mathrm{EDR}(i)={\rm{LF}}(i+1)-{\rm{LF}}(i-1),\end{eqnarray} \tag{ 4 }$$

so it gives the maximum response at the position where LF changes fastest in star number density, that is, the magnitude of TRGB. This corresponds to the physical interpretation of TRGB, due to a termination of the RGB branch, thus resulting in a sudden drop of the stellar population. In Table 1, we denote this first-derivative Sobel detection as simple. One modification of this method considers the S/N by applying a Poisson weighting

$$\begin{eqnarray}&&\mathrm{EDR}(i)=\sqrt{{\rm{LF}}(i+1)}-\sqrt{{\rm{LF}}(i-1)},\end{eqnarray} \tag{ 5 }$$

which we denote as Poisson. Hatt et al. (2017) employed another form of Poisson weighting

$$\begin{eqnarray}&&\mathrm{EDR}(i)=\displaystyle \frac{\mathrm{LF}(i+1)-\mathrm{LF}(i-1)}{\sqrt{\mathrm{LF}(i+1)+\mathrm{LF}(i-1)}},\end{eqnarray} \tag{ 6 }$$

which we denote as Hatt. Since the model of LF predicts an exponential growth of star population below the tip (Méndez et al. 2002), the simple Sobel filter might result in false detection in the faint end. The Poisson weighting can suppress faint, false peaks. For our analysis, we use Equation (6) for the Sobel detection weighting, and the optimization process is shown in the Appendix.

After generating the EDR curve under a specific smoothing and weighting technique, we introduce a tip detection method that ultimately identifies the strongest tip as well as tips of comparable size. We first find all local maximum points that satisfy

$$\begin{eqnarray}&&\mathrm{EDR}(i-1)\lt \mathrm{EDR}(i)\gt \mathrm{EDR}(i+1)\end{eqnarray} \tag{ 7 }$$

and denote the set of these points as [EDR_tot].

We define two properties of candidate peaks measured after applying a spatial and color cut: the contrast and the number of stars below the tip. These can be used as both quality indicators for tip measurements and used for further correlation studies.

• 1.  
  Contrast, R: For every F814W magnitude in the observable range, we define a contrast value that is the ratio of the number of stars below versus above that magnitude, given bounds of 0.5 mag. This can be defined as

  $$\begin{eqnarray}&&R={N}_{+}/{N}_{-},\end{eqnarray} \tag{ 8 }$$

  where N₊ and N₋ are the number of stars between the given magnitude and + or −0.5 mag respectively. This ratio resembles that used by Hoyt et al. (2021) (there ±1 mag) to pass a set threshold as a quality indicator of the CMD, though is a smaller range to reduce the impact of photometric noise.
• 2.  
  # Stars below the tip, N_+,1.0: The number of stars below the tip (applicability of Sobel filter). Following past studies, we calculate this as the total number of stars 1 magnitude below a tip such that

  $$\begin{eqnarray}&&{N}_{+,1.0}=\mathrm{number}\,\mathrm{of}\,\mathrm{stars}\,1\,\mathrm{mag}\,\mathrm{below}\,\mathrm{tip}.\end{eqnarray} \tag{ 9 }$$

  Madore et al. (2008) suggest that an effective TRGB measurement requires around 400 ∼ 500 stars within 1 mag below the tip, whereas Makarov et al. (2006) state that 50–100 is sufficient for a steady detection, though this is for the maximum-likelihood method. We include N_+,1.0 = 50, 100, 200, and 400 as lower range limits for this optimization.

When we apply quality cuts on [EDR_tot], we create a new array of possible peaks, which we call [EDR_qual].

In order to further remove spurious peaks, we find the maximum peak in EDR space

$$\begin{eqnarray}&&{\mathrm{EDR}}_{\mathrm{Max}}=\max ([{\mathrm{EDR}}_{\mathrm{qual}}]).\end{eqnarray} \tag{ 10 }$$

After that, we qualify every peak in [EDR_tot] (no matter its value of R) if it has a peak value greater than $0.6\times {\mathrm{EDR}}_{\mathrm{Max}}$, and we denote the set as [EDR_Peak]. A threshold scale of 0.6 is a conservative value chosen to keep the real tip included, since it may not be located at the maximum derivative location (i.e., maximum EDR). The intention of this algorithm is to find all the possible peaks, retain most of them by applying fairly loose initial selection criteria, and plan to further refine the selection in the rest of the paper.

We correct the tip magnitude for both MW extinction (Schlafly & Finkbeiner 2011) and predicted host-galaxy extinction based on the distance to the center of the galaxy and the galaxy shape (Ménard et al. 2010). These corrections are on the small (on the ∼0.01 mag level) and discussed further in Section 5.1.

Ultimately, we are attempting to minimize the dispersion of tip measurements between fields, while maintaining as many fields with a detection as possible. To evaluate the impact of different selection criteria, including spatial clipping, color band selection, smoothing scale, and R and N_+,1.0 filter, we introduce three different metrics:

• 1.  
  σ_tot. The dispersion of magnitudes of all valid peaks that pass certain selection criteria. For a global σ_tot value for multiple galaxies, we weight the σ_tot for each galaxy by the number of tip measurements within that galaxy.
• 2.  
  P_valid. The percentage of fields with a peak that passes the criteria. The ideal scenario is P_valid = 100%.
• 3.  
  N_TPF. The average number of tips detected per field (TPF). This is calculated by dividing the number of valid peaks that pass selection criteria by the number of fields with at least one valid peak. The ideal scenario is N_TPF = 1 as a value >1 implies some ambiguity in the identification of the peak.

In the Appendix, we discuss our optimization process of the values to be used from Table 1, and only the optimized parameters are used in the analysis unless otherwise stated. The optimization uses data from all the fields in galaxies with multiple fields.

We focus on NGC 3031 as a training set to establish a robust algorithm and then we will apply it to other galaxies. In Figure 5, we show the EDR for a nominal analysis case with Hatt style weighting and both spatial and color clipping applied as previously described in Sections 3.1 and 3.2. We show the responses for two different smoothing values (σ = 0.05, 0.10) for the LF. We can see the dissimilarity of the shape and wiggles of the edge responses, even with the high S/N data, revealing true stochasticity likely from astrophysical variations in the field CMDs. As can be seen from the bottom panels in Figure 5, σ = 0.10 merges most adjacent peaks found for σ = 0.05 into a single peak. Further, we find the R selection can filter out false detections at the faint end, and the N_+,1.0 selection can filter out the extremely large R tips at the bright end (likely due to AGB stars). This can be seen from the shape of LF in Figure 3, whose value is smaller at the bright end. Consequently, N_+,1.0 selection will filter out detections at the bright end where star density is small. The full detection files and how R and N_+,1.0 make selections can be found on our website. ⁸

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A table that summarizes the results of the analysis steps discussed in Section 3 is shown in Table 2. The steps include spatial clipping, color band filtering, ratio R cut, N_+,1.0 cut, and smoothing scale modification. We apply these methods sequentially and cumulatively to show the improvement in the recovered dispersion. We find for an initial (Step 1 in Table 2) assessment of tip detection without any of the optimization choices described in the previous discussion, the total dispersion is σ_tot = 1.34 and there is a mean of approximately nine detected tips per field indicating the wiggly nature of the EDR. The different steps have different impacts on either σ_tot, P_valid, or N_TPF defined in Section 4.1 as seen in Table 2. For instance, R and N_+,1.0 cuts have better efficiency in decreasing σ_tot and N_TPF than spatial clipping and color band. As can be seen in Figure 5, even after the spatial removal of young regions and the selection by color band, there is high dispersion due to the presence of multiple peaks in the Sobel response for each field. Spatial clipping and color band can only suppress the noise introduced by non-RGB stars, but cannot change the wiggling pattern of EDR much, thus slightly decreasing σ_tot and N_TPF. On the contrary, R and N_+,1.0 provide a quality selection for the peaks of wiggling EDR, where R blocks peaks at the faint end, and N_+,1.0 denies peaks at the bright end. As shown in Figure 8, sufficiently large R can decrease σ_tot and N_TPF to extremely ideal values, regardless of the threshold value of N_+,1.0. Therefore, we find that the selection of tips with greater values of R is the most powerful method for eliminating false measurements while reducing the intra-host dispersion. Since there are still outliers in Figure 5 after a moderate R > 3.5 selection, N_+,1.0 is also significant to further confine the tip detections, especially the false detections around the edge of AGB. We also find that the N_+,1.0 cut reduces the dispersion but also reduces P_valid substantially, from 100% to below 50%. Moreover, we find a relatively larger smoothing scale is beneficial, as shown in Figure 5 and the Appendix because it can merge nearby peaks and suppress the wiggly EDR to give smaller σ_tot and N_TPF.

Ultimately, we find we can reach σ_tot = 0.088 mag for this host while ∼45% of the fields pass the entirety of selection requirements, as shown in Step 6 of Table 2. For this case, we measure an N_TPF = 1.2, still not quite one tip for each field, and the deviation from 1 is due to a single field with two tip detections. We show a graphical representation of the reduction in dispersion for NGC 3031 in Figure 6. We also show how the residuals are improved if we change the step order. As seen, the R cut is the most powerful one, since it decreases the dispersion from 1.32 to 0.38 and the N_TPF goes down from 7.82 to 1.73 for the first application sequence. Besides, we find that R pulls down the initial dispersion from 1.34 to 0.80 if applied directly to raw data. As a comparison, we calculate the impact of applying only N_+,1.0 cut to the initial data set of σ_tot = 1.34, and we find N_+,1.0 can only decrease the total dispersion to 1.15. Therefore, R is the most powerful selection rule, though N_+,1.0 is necessary to further refine the validity of tip detections.

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While we focus on the total dispersion of the recovered tip values, a related question is whether the location of the measured peaks changes due to the various analysis steps. In practice, this is difficult to quantify because there are often multiple tips detected for a specific field. However, if we attempt to trace the displacement of tip measurements due to the analysis steps, we find two general trends. The first is that the non-smoothing steps (Steps 2–5 in Table 2) have the impact of producing a brighter detected tip compared to that recovered in Step 1, typically on the order of ∼0.035 mag. We find the two steps that cause this shift are the spatial and color cuts. The smoothing step has more of a stochastic effect on the recovered tip magnitude, with impacts on the order of ±0.05 mag for a change in smoothing from 0.05 to 0.1.

When considering which fields will yield the best TRGB measurements, it is interesting to consider the example of the spatial positions of the fields in NGC 3031 and Figure 1. Top-quality measures (R ≥ 7 and N_+,1.0 ≥ 200) are available only in three of the 11 Halo fields 01, 04, and 08. Fields 02 and 10 have only fair contrast (R ∼ 4–6 with field 02 having two possible tips separated by ∼0.2 mag) and 05 and 11 have fair contrast and low N_+,1.0 (<200). The rest, fields 03, 06, 07, and 09 are of low quality and do not yield a reliable tip measurement. We find the best detection comes from the region just outside the disk (in this case just beyond the 25th mag arcsec⁻² isophot and at a projected radius of 15–20 kpc). However, further fields are poorly constrained due to the small stellar population, which is hard for the Sobel filter to work well and the detections will be excluded by N_+,1.0 cut.

With a processing algorithm shown to be successful with NGC 3031, we apply it to the nine other GHOSTS galaxies with multiple fields. We show the improvement in the peak measurements for all the galaxies in Table 2 and find results similar to what we found for NGC 3031. We find that σ_tot drops from 1.35 to 0.058 when including all the steps of the algorithm. The dispersion of tip brightnesses measured per galaxy can be seen in Figure 7. We find, similar to NGC 3031, that ∼50% of the fields pass these quality cuts. We also find that N_TPF = 1.08 for the full set, showing that we measure close to one tip per field, which is the ideal scenario. There is one notable galaxy, i.e., NGC 4736 in Figure 7, which has one outlier that is 0.3 mag away from the other two detections. This is because, in one of the two fields of NGC 4736, there are two detections that pass through the selection criteria of N ≥ 4, N_+,1.0 ≥ 200.

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We show the key trends of our three metrics in Figure 8. For each of these metrics, we vary the minimum R (R_min) value and calculate these metrics (left panel). We also show these trends as a function of the mean R using a rolling bin size of 3. We separate the measurements for fields of different minimum N_+,1.0 values at N_+,1.0 = 50, 100, 200, and 400. We find a significant trend in the reduction of σ_tot with R_min; this reduction is sharpest at ${R}_{\min }\sim 2.5$, but continues to the highest R_min values.

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For high values of R_mean ∼ 8 and N_+,1.0 ≥ 1000, the dispersion reaches a very low ∼0.03 mag indicating the successful approach of the algorithm. For lower N_+,1.0 values and lower R values (<4), the dispersion may be as high as 0.8–0.9 mag owing to the frequent occurrence of multiple tips per field. We also show that P_valid steadily declines as R_min increases. For R values higher than 6, we are unable to recover a tip value for more than 50% of the fields. We also show that as we increase the quality requirements, N_TPF becomes closer to 1.

We can use the multi-field dispersion to approximate the uncertainty for a tip measurement as a function of its individual value of R and N_+,1.0 (instead of a group minimum threshold). We find that the multi-field dispersion (and hence, the TRGB uncertainty) can be roughly modeled by

$$\begin{eqnarray}&&\sigma =\sqrt{{\left[\left(\displaystyle \frac{2{e}^{1.5(3-R)}}{{e}^{1.5(3-R)}+1}\right){\left(\displaystyle \frac{1}{{N}_{+,1.0}-100}\right)}^{0.1}\right]}^{2}+{0.04}^{2}}\ \mathrm{mag},\end{eqnarray} \tag{ 11 }$$

where R and N_+,1.0 are the specific values for the field. This formula is roughly modeled from an ad hoc Fermi distribution, and parameters are set to best fit the numerical results. Also, note that this formula is only valid for N_+,1.0 > 100. As the formula indicates, an ideal (but rarely observed) sharp tip with R ≥ 10 and N_+,1.0 ≥ 1000 can yield an uncertainty of σ ∼ 0.04 mag. The more typical case of a less sharp tip measurement with R ∼ 4 and a well-populated tip N_+,1.0 ∼ 500 yields σ ∼ 0.15 mag. A very fuzzy tip with poor contrast, R = 3, produces much larger uncertainties with σ ranging from 0.3–1 mag depending on whether the tip is well-populated (N_+,1.0 = 1000) or not (N_+,1.0 = 100). Considering tips with R < 3 invariably allows multiple values and more information must be used to select among the tips.

Our analysis thus far has shown that R is especially powerful for predicting the recovered dispersion of peak magnitudes. In Figure 9, we compare this contrast value to the residuals of tip measurements relative to the median of all tips in the same host. We notice a clear trend such that the higher the R value, the brighter the apparent tip. Assuming that uncertainties follow Equation (11), we fit a line to the trend shown in Figure 9. We find a best-fit slope of −0.023 ± 0.0046 mag, which is highly significant at ∼5σ. If we remove one outlier at R ∼ 5 with a residual ∼ −0.2 mag, which is ∼3.5σ away, the fit has a slope of −0.025 ± 0.0047. An unweighted fit gives a slope of −0.033 ± 0.0072 mag (4.5σ). If we further restrict N_+,1.0 ≥ 400, a weighted fit gives a slope of −0.029 ± 0.0054 mag, and an unweighted fit results in a slope of −0.037 ± 0.0071 mag (both with a 5σ significance). The host extinction, as discussed in Section 5.1, has little effect on this relation: if we increase the host extinction derived from Ménard et al. (2010) by an order of magnitude, the relation of the weighted fitting is little changed to −0.020 ± 0.0046 mag for N_+,1.0 ≥ 200, and −0.025 ± 0.0054 mag for N_+,1.0 ≥ 400. It also matches the scale and sense seen in other multi-field studies such as in NGC 4258 (Jang et al. 2021, see Figure 12) and in LMC fields (Hoyt et al. 2021).

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Using this relation to standardize a set of tips of mixed contrast ratios (e.g., R ≥ 3) lowers their dispersion by ∼10%–30% (depending on the number of stars). The standardization can be achieved by setting a fiducial point (we use R = 4) and correcting the tip magnitude counting in the tip-contrast relation (TCR) with a slope of −0.023 mag/unit ratio. For greater numbers of stars N_+,1.0 ≥ 400 (and a broad range of contrasts, R ≥ 3) we see the greatest improvement with the dispersion declining from 0.095 to 0.074 mag.

There are two sources of dust extinction that impact our results. The first is extinction due to the MW. Many of the fields are separated by angular separations that are comparable to the resolution of the MW dust map (Schlafly & Finkbeiner 2011) of ∼5'. We find on average variations of ∼3 mmag between fields, significantly smaller than the dispersion shown in Figure 6. A second source of dust extinction is internal to the host galaxy. However, because the fields are far from the center of the host, internal extinctions will be small. To quantify and account for these, we follow (Ménard et al. 2010, Equation (30)) to estimate the host extinction defined by the projected radii of the field from the core of a galaxy. The median field extinction estimate is 0.012 mag (standard deviation = 0.006 mag) and while included, has little impact on the TCR.

In Figure 9, we presented a 5σ trend between tip-brightness residuals and measured R values. To better understand the origin of this relation, we use stellar population models to find the relation between stellar population age and contrast, as shown in Figure 10. These calculations were carried out for single-age populations with ages ranging from 1.5–10 Gyr at a metallicity of [M/H] = −2 and [M/H] = −1.5 using the Padova CMD3.7 stellar models from Girardi et al. (2010). As can be seen, there is a strong relation between age and R. This finding is consistent with a result in Girardi et al. (2010), which studies old metal-poor galaxies and found R values are all >20, significantly above any found in this work. From model predictions of McQuinn et al. (2019), it is expected that the F814W absolute luminosity of the TRGB should have a small dependence on stellar age of roughly 0.02 mag across an age range of 5 Gyr and 0.04 mag across 0.5 dex of metallicity. It is thus possible that the variation in R is related to population differences which cause an apparent R-luminosity relation at the level seen here.

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It is also possible that the value of R correlates with patchy dust (i.e., anisotropic around a host as opposed to the mean, isotropic dust quantified by Ménard et al. (2010) and removed from the tips) with that dust as an underlying cause of the TCR. For the purpose of standardizing TRGB distance measurements, the cause of an apparent TCR is less material. For any of these causes, standardization and calibration using R would remove the present variation of the tip with R along the distance ladder and could also remove related biases due to differences in field demographics.

We also note that we do not find any significant evidence (<1σ) for a correlation between TRGB residuals and the mean color of the TRGB over the range of mean colors sampled by our tip measurements, 1.15 <F606W−F814W < 1.65 mag. Such a correlation may be present at our sensitivity level of ≤0.1 mag per magnitude change in color, but our use of a color band reduces our sensitivity to color differences.

An important use of TRGB is in measurements of the Hubble constant (Freedman et al. 2019; Anand et al. 2022). In this role, the brightness of the TRGB in one or more anchor galaxies with geometric distance measurements is compared to that in SNe host galaxies. To quantify whether the inferences found for the GHOSTS set may be applied to the TRGB measurements in SN hosts, we compare the main diagnostics of these two sets, using the properties CMDs/TRGB catalog (Anand et al. 2021b) on the Extragalactic Distance Database ⁹ (Tully et al. 2009), which provides TRGB distances and the underlying HST photometry for of order 500 galaxies. We calculate the R and N_+,1.0 values for these fields before applying a spatial cut or a color band. As shown in Figure 11, we find an overall good overlap of the R parameter, our key diagnostic, indicating good applicability. There are some SN hosts with R ≤ 4 which according to Figure 8 may not yield a reliable TRGB (though spatial clipping may be used to increase R). We do find that the number of stars in the SN fields is somewhat higher than that found for the GHOSTS fields, though this is likely due to the fact that the SN fields are more distant on average. Externally, we also examine other properties like RGB width and slope, and do not find any obvious inconsistencies.

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While differences in tip magnitude across different fields in the same galaxy have been seen in many previous analyses (e.g., Hoyt et al. 2021; Jang et al. 2021; Anand et al. 2022), some of these analyses have chosen to favor brighter tips measured in some fields while discarding fainter tips in others. The value of this approach would depend on the goal of the analysis. If the goal is to measure the brightest apparent magnitude attained by the TRGB and nothing else, this approach is reasonable. If, however, the goal is to produce a comparison or calibration of populations with similar luminosity, our results suggest it is necessary to standardize the luminosity of the fields by accounting for their differing contrasts. Indeed, differences in TRGB contrast may play a role in the ∼0.1 mag range of recent calibrations of the absolute luminosity of the TRGB (see Blakeslee et al. 2021, Table 3 and Li et al. 2022, see Table 4 and Figure 6). We hope to explore such standardization in future work.

The goals of this analysis were to establish unsupervised measurements of the TRGB and quantify the field-to-field dispersion for a single galaxy. We find a dispersion as small as ∼0.03 mag, however, less than 50% of GHOSTS fields have a high enough contrast to produce some precise measure. We measure significant trends of the dispersion with the R contrast parameter and motivate the use of quality cuts on this value in the future. Furthermore, we find a trend between tip magnitude and the R value at ∼5σ. The slope of this relation is a trend of up to 0.3 mag over the range of usable R values seen for the GHOSTS fields (from R = 3–12). We do not see any obvious differences between the GHOST fields analyzed here and the fields analyzed for distance ladder measurements. In follow-up work, we will apply the standardization methods and insights found here to refine measurements of H₀ from TRGB.