The Carnegie–Chicago Hubble Program. III. The Distance to NGC 1365 via the Tip of the Red Giant Branch^∗

We obtained optical imaging over 16 orbits on 2014 September 17, 21, and 25 using the ACS/WFC instrument aboard HST (PID:GO13691, PI: Freedman; Freedman 2014). Six and ten orbits were used for the F606W and F814W filters, respectively. Pointings were centered on ${\rm{R}}.{\rm{A}}.\,=\,{3}^{{\rm{h}}}{33}^{{\rm{m}}}52\buildrel{\rm{s}}\over{.} 4$ and $\mathrm{decl}.\,=\,-{36}^{{\rm{h}}}{12}^{{\rm{m}}}05\buildrel{\rm{s}}\over{.} 0$, which is 50 southeast of the center of NGC 1365. The field was selected to be safely in the stellar halo of NGC 1365, and care was taken to place the pointing sufficiently far from the spiral arm by inspection of WISE and GALEX imaging.

Individual exposures were approximately 1200 s each, yielding total exposure times of 14,676 and 24,396 s for the F606W and F814W filters, respectively. The exposure times were designed to obtain a signal-to-noise ratio (S/N) of 10 in F814W at the anticipated apparent magnitude of the TRGB, as predicted using previous distance estimates to NGC 1365 (Freedman et al. 2001; Riess et al. 2016). The F606W S/N is lower (typically by a factor of 3), but since the color is intended to serve only as a means of isolating the TRGB by removing RGB contaminants, the lower precision in the color does not directly affect the I-band TRGB measurement itself. This observation strategy was designed to provide reliable photometry reaching at least one magnitude below the anticipated TRGB, thereby meeting the sampling requirements for a robust TRGB identification as defined in Madore et al. (2009). A summary of the observations, split into the three HST visits, is given in Table 1.

Table 1.  Summary of ACS/WFC Observations for NGC 1365

Dates	Filter	No. obs	R.A. (2000)	Decl. (2000)	Field Size	Exposure Time (s)
2014 Sep 17	F606W	12	${03}^{{\rm{h}}}{33}^{{\rm{m}}}51\buildrel{\rm{s}}\over{.} 4$	−36°12'050	337 × 337	∼1200
2014 Sep 21	F814W	10	${03}^{{\rm{h}}}{33}^{{\rm{m}}}51\buildrel{\rm{s}}\over{.} 4$	−36°12'050	337 × 337	∼1200
2014 Sep 25	F814W	10	${03}^{{\rm{h}}}{33}^{{\rm{m}}}51\buildrel{\rm{s}}\over{.} 4$	−36°12'050	337 × 337	∼1200

Note. See also Figure 1 for the coverage of the imaging.

Download table as:  ASCIITypeset image

Figure 1(a) shows the ACS/WFC pointing relative to the galaxy using a wide-area (10' × 12') J-band image from the FourStar near-infrared imager on the Magellan–Baade telescope (for a description of the instrument see Persson et al. 2013) taken as part of the CSP follow-up campaign for SN 2012fr (C. Contreras et al. 2017, in preparation). Figure 1(b) is a gray-scale map of the ACS/WFC observations based on a "drizzled" co-add, and Figure 1(c) is a 10'' × 10'' region of the ACS/WFC image. Here we circle point sources that have magnitudes and colors of RGB stars. This clearly illustrates that the RGB stars in our halo pointing are well isolated from neighboring sources.

Zoom In Zoom Out Reset image size

Individual ACS/WFC images used were of the FLC data type, which are calibrated, flat-fielded, and CTE-corrected in the Space Telescope Science Institute (STScI) CALACS pipeline. The non-uniform pixel area due to ACS/WFC geometric distortions was corrected using the STScI-provided Pixel Area Maps.⁸ All further analysis is conducted on these pixel-area-corrected FLC frames.

Instrumental magnitudes were derived for individual FLC images via point-spread function (PSF) fitting in the software DAOPHOT (Stetson 1987). We generated stellar grids on the ACS footprint using the synthetic HST PSF Tiny Tim (Krist et al. 2011) and then constructed a PSF model for each filter with DAOPHOT. A direct test of the Tiny Tim PSFs against direct frame-by-frame PSF modeling with isolated, bright stars is described in Paper II and was found to agree within the photometric uncertainties. Images were aligned using the routines DAOMATCH and DAOMASTER (Stetson 1987). A master star list was created from a co-add of all images to perform photometry simultaneously using ALLFRAME (Stetson 1994).

We transformed the instrumental magnitudes to the ACS Vega magnitudes following Equations (2) and (4) of Sirianni et al. (2005). A correction from the PSF magnitudes to the 05 aperture magnitudes for each CCD chip was determined by comparing the curve of growth generated from aperture magnitudes to the PSF magnitude (also measured at a 05 radius). We find aperture corrections of −0.065 mag (chip 1) and −0.063 mag (chip 2) for F814W and −0.023 mag (chip 1) and −0.018 mag (chip 2) for F606W. The statistical uncertainty (the error on the mean) for the aperture corrections is <0.01 mag. However, we found that independent efforts in the selection of bright, isolated stars resulted in differences at the level of ∼0.01 mag. We therefore adopt a systematic uncertainty of 0.01 mag in the photometric calibration due to the aperture correction at 05.

The values of 05 to infinite aperture correction are 0.095 mag for F606W and 0.098 mag for F814W (Bohlin 2016). Bohlin (2016) describes a ∼4% uncertainty in encircled energy (EE) for cool, late-type stars that are used to compute the 05 to infinite aperture correction. Table 6 of Sirianni et al. (2005), however, shows that the total variation in EE due to changes in effective wavelength via spectral type is <0.01 mag for WFC at 05, implying that the EEs for these cooler stars are consistent with hotter stars that are constrained at the level of 1%. We therefore adopt half of the estimate of Bohlin (2016), or 2% error in flux (0.02 mag) as another systematic uncertainty due to the scatter in measured EE for the RGB stars that are the focus of this work.

We used photometric zero-point values of 26.412 mag for F606W and 25.524 mag for F814W, which were provided for a given observation date by the online STScI ACS Zero-points Calculator.⁹ There is a systematic uncertainty in the observed flux of Vega, which forms the basis of the ACS zero-point calculation. As per Sirianni et al. (2005), we adopt a conservative systematic uncertainty of 0.02 mag for the zero-points.

Mean magnitudes for each filter were computed from the individual frame intensities with a median-based σ-clip algorithm, setting the clip at 3σ. We also use the image-quality parameter "sharpness" to remove non-stellar sources. The selection criteria we used are sharpness_F814W < 0.1 exp(F814W – 23.5) and sharpnesss_F814W ≥ −0.4, which, from visual inspection of the objects in the images, adequately removes obvious extended sources and cosmic rays.

The calibrated CMD is shown in Figure 2. A dramatic change in the source density in the range 27.3 ≲ F814W ≲ 27.4 mag corresponds to the TRGB, which is marked by an arrow but is also visible by eye. Stars brighter than the TRGB are likely either thermally pulsing asymptotic giant branch (TP-AGB) stars or blended RGB stars. The blue shaded region highlights the approximate color–magnitude space of the RGB stars. The slope (−6 mag/color) and the colors of edges of the shaded region (=1.0 and 1.5 at the anticipated TRGB level) have been determined from the investigation of stellar isochrones for old (12 Gyr age) and metal-poor ([Fe/H] ≤ −0.9) RGB stars provided by the Dartmouth group (Dotter et al. 2008).

Zoom In Zoom Out Reset image size

To quantify the completeness of our photometry, we performed extensive artificial star tests spanning a wide range in the color–magnitude space. We input 2000 stars with pixel coordinates randomly sampled from a uniform distribution. Their input F814W magnitudes are randomly sampled from the range 20 ≤ F814W ≤ 29 mag, and colors were sampled from 1.0 ≤ F606W – F814W ≤ 1.5 in order to approximate the range of likely TRGB stars as seen in the blue shaded region of Figure 2. This process was repeated 100 times, producing 200,000 artificial stars.

We performed photometry on these new images identically to the original frames. The results of the tests are shown in Figure 3. Figure 3(a) shows the completeness of the artificial stars as a function of F814W magnitude, split into two color regions:

$$\begin{eqnarray*}&&1.00\leqslant F606W-F814W\lt 1.25,\\ &&1.25\leqslant F606W-F814W\leqslant 1.50.\end{eqnarray*}$$

We find that the completeness of the photometry remains constant up to the anticipated TRGB magnitude. The completeness curves in Figure 3 are an average over all physical coordinates, and the only deviation from the average is in the upper 10% of CCD 1 (closest to NGC 1365), where the completeness falls to ∼75% from ≳90%. In general 100% completeness is unobtainable due to, for example, source blending with background galaxies and bright foreground stars, and the incompleteness of observations at the edges of the imaging dither pattern.

Zoom In Zoom Out Reset image size

One concern for the TRGB analysis is the potential blending of stars fainter than the anticipated TRGB, which could obscure the location of the tip. By sub-sampling the artificial star luminosity function (ASLF) from the uniform distribution to one that follows the commonly adopted slope of the RGB luminosity function (LF), e.g., 0.3 dex mag⁻¹ (Méndez et al. 2002), and by limiting the number of stars found to that within a magnitude of the anticipated TRGB (∼10,000), we find that <1% of measured stars fainter than the anticipated TRGB magnitude exceed their input magnitudes by greater than 3σ, suggesting that negligibly few stars should systematically obscure the TRGB.

In Figures 3(b) and (c), the recovered photometry is compared to the input photometry for the entire sample of simulated stars for F814W and F606W – F814W, respectively. We find that the recovered photometry in F814W is in agreement to within the uncertainties to roughly ∼1 magnitude fainter than the anticipated TRGB magnitude. This result suggests that crowding (which would alter sky values and hence the final measured magnitudes) is not a significant concern.

We observe that the uncertainty in color grows much more quickly due to the lower S/N of the F606W imaging, though, as is discussed later, the color–magnitude behavior of the TRGB has little to no impact on its measurement for our data set because we preferentially target metal-poor stars in the halo of NGC 1365. In the next section, we make our measurement of the TRGB and estimate its associated uncertainties.

In order to make an optimal measurement of the TRGB, we seek the level of smoothing in the LF that most reduces the associated statistical and systematic errors. Sections 3.1.1 and 3.1.2 below describe the creation of an ASLF as well as simulations of TRGB measurements in order to model the properties of our GLOESS smoothing function and the [−1, 0, +1] edge detection kernel.

3.1.1. Artificial Star LFs

We conservatively assumed that the RGB LF has a relative count of stars four times greater than the AGB LF within 0.1 mag of the TRGB, which is the upper range that has recently been observed in nearby galaxies (Rosenfield et al. 2014). We assumed that the LF for the RGB has a slope of 0.3 dex mag⁻¹, which has been confirmed empirically in several TRGB studies (see, e.g., Méndez et al. 2002; Makarov et al. 2006; Conn et al. 2012). To account for the possible contamination of the TRGB by TP-AGB stars, we further assumed an AGB component to the ASLF with a slope of 0.1 dex mag⁻¹. This slope is an intermediate value between commonly adopted flat LFs (Durrell et al. 2002; McConnachie et al. 2004) and steeper (though uncertain) direct estimates (see, e.g., Makarov et al. 2006). Since we have found little evidence of blending/crowding from the artificial star tests described in Section 2.4, the small magnitude range that is relevant for the TRGB analysis (ΔF814W ∼ 0.1 mag) suggests that these slopes are largely symbolic to produce more realistic simulations.

The RGB component of our ASLF begins at the estimated tip magnitude F814W = 27.37 in the ACS Vega system (determined retroactively from an initial assessment of the TRGB), and it extends to F814W = 28.37 mag. The AGB component begins 1 mag brighter than the tip magnitude and extends to the same depth as the RGB. As discussed above, we normalize the AGB LF relative to the RGB LF such that it comprises one-fifth of the total stars within ±0.1 mag of the TRGB. We assign a central color of F814W − F606W = 1.25 for each component of the ASLF and sample uniformly in color for individual stars such that the RGB spans 1.0 ≤ F606W – F814W ≤ 1.5.

Two thousand stars were sampled at random from this ASLF distribution and placed into each individual FLC frame at pixel coordinates uniformly distributed in X and Y. These stars were manually added to the "master list" of sources and the ALLFRAME photometry was performed as previously described. The artificial star process was repeated 250 times for CCD 1, which comprises ∼80% of stars belonging to NGC 1365. In a separate test, we observed that there was no difference in the quality of artificial star photometry between CCDs, and we therefore focused on the more populated CCD 1.

These simulations produced a total of 500,000 artificial RGB stars, for which ∼445,000 stars were successfully measured for ∼88% completeness over the RGB magnitude range. Figure 4(a) shows the input and output ASLFs as blue and yellow histograms, respectively. While the input ASLF has a hard bright edge to represent the TRGB, the output ASLF demonstrates the broadening of the TRGB edge due to measurement uncertainties and the increasing incompleteness as a function of magnitude. In black, we display a scaled LF of the observed data set to show the good agreement in the relative number of AGB and RGB stars around the TRGB.

Zoom In Zoom Out Reset image size

3.1.2. Simulating TRGB Edge Detections

Figure 5(a) presents the final CMD used to determine the distance to NGC 1365. We first consider a color–magnitude restriction, described in the previous sections and displayed as the shaded region, that isolates the RGB. We note that these limits coincide with the color range over which the absolute magnitude of the TRGB is known to be flat with color (Lee et al. 1993; Jang & Lee 2017b), within the present uncertainties on that determination. Moreover, we note that our highlighted data are also consistent with no trend in color to within our measurement uncertainties. We therefore do not modify the CMD data in any way for the TRGB measurement (see Section 4.1.2 below for a discussion of this point). Figure 5(b) is the resulting LF for stars in the blue shaded region after smoothing using the GLOESS algorithm and our optimized scaling parameter, ${\sigma }_{s}=0.11$ mag, as determined in the previous section. Figure 5(c) is the result of applying the [−1, 0, +1] Sobel kernel to the LF, which shows a strong peak at 27.37 mag (indicated by the dashed lines in Figures 5(a) and (b)).

Zoom In Zoom Out Reset image size

We next consider the TRGB measurement without the color–magnitude restriction. As with the study of IC 1613 in Paper II, we find that the inclusion of all stars within the CMD displaces the TRGB by at most a negligible ∼0.01 mag from the case of a restricted color region. Since the restricted color–magnitude region has stars (with better constrained average colors), it is no surprise that the measured TRGB magnitude is roughly unchanged because most stars fall within the blue shading. In terms of simulating the TRGB detection for the full CMD sample, the systematic error remains unchanged while the statistical error decreases by ∼0.01 mag given the larger sample of stars used to define the TRGB.

Based on the simulations in the previous subsection, we assign a statistical uncertainty of 0.03 mag and a systematic uncertainty of 0.04 mag. Incorporating the estimated systematic uncertainties from the HST photometric calibration, our TRGB determination is F814W = 27.37 ± 0.03_stat ± 0.04_sys mag before correcting for line-of-sight reddening.

The Milky Way foreground extinction is estimated to be small: 0.051 mag for F606W and 0.031 mag for F814W, or E(F606W – F814W) = 0.020 mag (Schlafly & Finkbeiner 2011, retrieved from NED, the NASA/IPAC Extragalactic Database). Applying the foreground estimates to our TRGB measurement from the previous subsection, we find an extinction-corrected TRGB magnitude of F814W = 27.34 mag. Since the uncertainty in the color excess (${\sigma }_{{E}_{B-V}}\approx 0.03$ via Schlegel et al. 1998) is comparable to the estimated level of reddening, we adopt half of the reddening as an additional systematic uncertainty.

Reddening internal to NGC 1365 itself is an unknown, though the halo observations ensure that it is minimized relative to the disk of the galaxy. To test for the presence of halo dust within our observations, we looked for differential reddening in the measurement of the TRGB in four regions of equal star count that represent concentric annuli from NGC 1365. The measured TRGB values from each of these distinct regions are F814W = 27.44, 27.33, 27.39, and 27.40 mag in order from the inner to outer regions. The average of these TRGB measurements is 27.39 ± 0.04 mag, where the uncertainty is taken to be the standard error on the mean. The TRGB magnitudes do not show a clear gradient as a function of galactocentric radius, and the average TRGB measurement is in good agreement with the TRGB value from the full CMD sample.

Because the four regions are reduced to ∼25% of the population count from the observed data, we performed an additional test in which we divided the total imaging into only two regions to make use of a larger sample of stars. In these two regions, we measured TRGB values of F814W = 27.42 mag and 27.39 mag for the regions closest to and furthest from NGC 1365, respectively. The difference between these two measured TRGB values is consistent with the uncertainty in the measurement of the TRGB itself using the full data set. From the results above, we conclude that there is insufficient evidence for systematic halo dust within the uncertainties of the current set of observations. Nonetheless, we do note that the faintest TRGB measurement is in the region closest to NGC 1365 for both experiments. Any individual galaxy within the CCHP sample may not have adequate population statistics to assess the level of internal reddening, though the results across several galaxies will be enlightening in determining the existence of halo dust.

Currently, the absolute magnitude of the TRGB in the optical has no direct calibration from trigonometric parallaxes, although Gaia will be providing these measurements in the near future. In the interim, we have chosen to adopt an absolute magnitude for the TRGB for the CCHP. Justification for this adopted value is presented in a forthcoming analysis of the main body of the Large Magellanic Cloud (LMC) and is anchored to direct distance determination to detached eclipsing binaries in the bar of the LMC. Our adopted zero-point is ${M}_{I}^{\mathrm{TRGB}}=-3.95\pm {0.03}_{\mathrm{stat}}\pm {0.05}_{\mathrm{sys}}$ mag (T. J. Hoyt et al. 2017, in preparation). The adoption of a provisional zero-point is a strategy similar to that employed in the mid-stages of the Key Project for the purpose of internal consistency. This provisional calibration is also consistent to within ±1σ with the canonical TRGB calibration based on globular clusters, ${M}_{I}^{\mathrm{TRGB}}\approx -4$ mag, and is consistent with other calibration efforts (e.g., Rizzi et al. 2007; Jang & Lee 2017b, among others). We note that Jang & Lee (2017b) find ${M}_{I}^{\mathrm{TRGB}}\,=-3.968\pm 0.106$ mag in the LMC using a similar TRGB measurement process to that adopted in this paper (e.g., spanning a similar color range).

Applying this provisional zero-point to our TRGB apparent magnitude, we find a true distance modulus to NGC 1365 of μ₀ = 31.29 ± 0.04_stat ± 0.06_sys mag, or a distance of D = 18.1 ± 0.3_stat ± 0.5_sys Mpc. Table 2 summarizes the values for the TRGB magnitude, the distance modulus, its uncertainties, and the adopted reddening.

Table 2.  TRGB Distance and Error Budget

Parameter	Value	${\sigma }_{\mathrm{ran}}$	${\sigma }_{\mathrm{sys}}$
TRGB F814W magnitude	27.37	0.03	0.04
${A}_{F814W}$	0.03	⋯	0.02a
Provisional ${M}_{I}^{\mathrm{TRGB}}$	−3.95	0.03	0.05
True distance modulus (mag)	31.29	0.04	0.06
Distance (Mpc)	18.1	0.3	0.5

Note.

^aTaken to be half of ${A}_{F814W}$.

Download table as:  ASCIITypeset image

The objective of the CCHP is to obtain a high-fidelity measurement of the Hubble constant, minimizing systematics by observing and applying a homogeneous analysis of the TRGB in galaxies spanning 10 magnitudes in distance modulus (see Paper I, Table 5 for a summary of the TRGB targets). We have developed a data-reduction strategy that can be applied to galaxies spanning this wide range in distance. As a result, the data processing, treatment of the sloped TRGB, and edge detection strategies differ from similar studies using the TRGB at these distances. In the subsections to follow, we compare our methods to those used in other studies.

4.1.1. Data Processing

4.1.2. Rectification of the TRGB Slope

4.1.3. Edge Detectors

Previously published estimates of the distance modulus to NGC 1365, based on Cepheids, a Type II supernova (SN 2001du), and the Tully–Fisher relation (NED-D), range from ${\mu }_{0}=29.52$ to 32.09 mag, with mean and median values of 31.20 and 31.26 mag, respectively. Cepheids are the only fully independent measure of the distance to NGC 1365, and we therefore focus our distance comparison on them.

There are roughly 30 distance estimates for NGC 1365 based on Cepheids tabulated in NED up to late 2017. However, nearly all of these estimates are based on the same imaging data set that was obtained by the Hubble Key Project: 12 epochs of F555W and four epochs of F814W taken with the WFPC2 instrument (Silbermann et al. 1999; Freedman et al. 2001), with later works simply updating the calibration of the original study. These updated distance moduli show a large spread, ranging from ${\mu }_{0}=31.18$ to 32.09 mag, resulting primarily from uncertainties in the color and metallicity dependence of the Cepheid period–luminosity (PL) relation. Because of this uncertainty and the systematic offsets introduced by comparing the results of consecutive publications differing only in zero-points, we have chosen the result of Freedman et al. (2001) to represent the results from this ensemble of publications. We have further considered a recent analysis by Riess et al. (2016), who analyze new near-IR photometry for a subset of the Cepheids originally discovered within the Key Project (Freedman et al. 2001).

Updating the final, metallicity-corrected Key Project Cepheid distance to reflect the updated LMC anchor distance of ${\mu }_{0,\mathrm{LMC}}=18.49$ gives ${\mu }_{0}=31.26\pm 0.05\pm 0.14$ mag. Riess et al. (2016) find ${\mu }_{0}=31.307\pm 0.057$ mag using near-IR Cepheids and anchoring the zero-point of the PL relation to a number of different techniques in the Galaxy, the LMC, and NGC 4258.¹⁰ Figure 6 illustrates the consistency of the two independent Cepheid distances with the value derived in this study. The sample error on the mean is only 0.04 mag, and gives no indication of a significant difference in the distances derived from stars of Pop I and Pop II for NGC 1365 at this time. The weighted average of these results suggests a true distance modulus $\langle {\mu }_{0}\rangle =31.30\pm 0.04$ mag, which is statistically indistinguishable from the TRGB measurement presented here based on the provisional TRGB luminosity in the LMC.

Zoom In Zoom Out Reset image size

4.3.1. Comparing the Distance Scales for Population I and Population II

4.3.2. The TRGB Error Budget

The RGB stars measured in this study are as faint as F814W ≈ 29 mag and F606W ≈ 30 mag. In individual frames (20 for F814W and 12 for F606W), these stars are measured at low S/N. There are two independent approaches to producing photometry for these sources:

• 1.  
  Generate a master source list from a high-S/N median image and use it as an input to force-photometer individual frames (as is done in the ALLFRAME software). The photometry is completed on the flc image products and we will refer to this technique as flc.
• 2.  
  Directly photometer co-added images, defining an empirical PSF based on high-S/N sources in the median image. The photometry is completed on a drc image product and we will refer to this technique as drc.

The former (flc) is the technique described in the main text, for which we utilize the theoretical Tiny Tim PSFs (Krist et al. 2011). It has the disadvantage of the stellar FWHM being undersampled (though we note that because stellar crowding is low, our stellar profile fitting is not limited to the stellar FWHM). The latter technique (drc) has been used more broadly in the literature for TRGB-based analyses at these distances (e.g., Caldwell 2006; Aloisi et al. 2007; Bird et al. 2010; Jang & Lee 2017a, and references therein) and comes with the advantage of producing stellar profiles that are Nyquist-sampled within the stellar FWHM. In this Section, we provide quantitative comparisons between the flc and drc methods.

We adopt the flc photometry from the main text and the drc photometry is produced as follows. Drizzled image stacks are constructed using DrizzlePac (Fruchter & Hook 2002). We carefully selected ∼100 relatively bright sources in each CCD chip and used them to refine image alignment with the Tweakreg task; the mean residual rms for the X and Y shifts determined with Tweakreg were smaller than 0.1 pixel. We then used Astrodrizzle to make a combined drizzled image for each filter with final_pixfrac = 0.8 and final_scale = 0.03 arcsec pixel⁻¹. The output drizzled images have stellar FWHMs of ∼3 pixels, corresponding to ∼009. PSF photometry on the drizzled images and standard calibration were performed following the method described in the main text with the exception of the PSF modeling. We generated empirical PSFs with DAOPHOT that were constructed from ∼25 bright isolated stars in each of the F606W and F814W images. The F814W source catalog is used as the "master catalog" and the two frames are simultaneously photometered using ALLFRAME (Stetson 1994). The magnitudes are calibrated identically as described in the main text (Section 2.2).

Figures 7(a) and (b) provide star-by-star comparisons of the flc and drc photometry in the F606W and F814W filters, respectively. Bright stars with F606W ≲ 24 mag and F814W ≲ 23 mag are in excellent agreement with median offsets smaller than 0.01 mag for both filters. However, we measure small systematic offsets for the fainter stars. At the TRGB magnitude (F814W ≈ 27.4 mag and F606W ≈ 28.7 mag), median offsets are measured to be 0.04 mag in F606W and 0.03 mag in F814W. The precise origin of the offsets for fainter sources remains unclear, but could be due to (i) the relatively small number of sources used to determine the empirical PSF¹¹ or (ii) documented differences between the Tiny Tim and empirical magnitudes for faint sources that were described by Krist et al. (2011). For sources in the range 27.34 < F814W < 27.40 mag the median magnitude uncertainty is 0.07 mag for F814W and 0.13 mag for F606W (the latter measurement is for the same stars in the F814W range). Thus, the differences identified in Figures 7(a) and (b) for the fainter sources are within the magnitude uncertainties.

Zoom In Zoom Out Reset image size

While some star-to-star differences are demonstrated in Figures 7(a) and (b), a more relevant question is the results from the TRGB detection. Thus, we apply the same techniques described in the main text to the drc photometry. The result is given in Figure 7(c). The CMD shows a well-defined RGB with a visible discontinuity due to the TRGB at F814W ∼ 27.4 mag. A visual comparison to Figure 2 reveals that the drizzle-based CMD looks more well populated (i.e., more complete) and this is consistent with having performed PSF photometry on a higher-S/N image. We select stars in the shaded region (identical to that of Figure 2 in the main text), construct an LF, apply the GLOESS smoothing, and lastly apply the [−1, 0, 1] Sobel filter. The edge detection response is shown in red in Figure 7(c) and the maximal response occurs at F814W = 27.36 ± 0.03 mag. The derived TRGB magnitude is statistically consistent with the value from the individual frame photometry, F814W = 27.37 ± 0.03 mag (Table 2).

From the comparisons given in the panels of Figure 7, we conclude that the two reduction procedures are statistically identical both for their output photometry and in their TRGB measurements. Thus, the choice to use individual frame photometry for the CCHP project, motivated by the need for a homogeneous image processing strategy for both nearby and distant SNe Ia hosts, is entirely consistent with the body of work derived from drizzled photometry (e.g., Jang & Lee 2017a).

A great benefit of the TRGB as a distance indicator is that the metallicity sensitivity of the absolute magnitude is projected into the color of the star. Furthermore, for metal-poor stars that populate the "blue" edge of the TRGB, ${(V-I)}_{0}\lesssim 2$, the I magnitude of the TRGB is relatively insensitive to metallicity (i.e., it is flat with color; see Lee et al. 1993; Rizzi et al. 2007; Madore et al. 2009; Jang & Lee 2017b, among others). Thus, with only an optical color-cut (as is done in Figure 2), the I, and by proxy the TRGB in F814W, requires no correction for metallicity to convert to an absolute magnitude system.

The color dependence of the TRGB in I, however, is not negligible everywhere; in particular, for the color range ${(V-I)}_{0}\gtrsim 2.0$ the I magnitude becomes noticeably fainter. Madore et al. (2009) presented an empirical technique to rectify or transform the TRGB magnitudes for metal-rich sources to the metal-poor (flat) portion of the TRGB. The general form of a transformation into the ${T}_{{\lambda }_{1},{\lambda }_{2}}$ (or TRGB) magnitude system is defined as

$$\begin{eqnarray}&&{T}_{{\lambda }_{1},{\lambda }_{2}}={m}_{0,{\lambda }_{1}}-{\beta }_{{\lambda }_{1},{\lambda }_{2}}[{({m}_{{\lambda }_{1}}-{m}_{{\lambda }_{2}})}_{0}-{\gamma }_{{\lambda }_{1},{\lambda }_{2}}]\end{eqnarray} \tag{ 1 }$$

where ${m}_{0,{\lambda }_{1}}$ is the initial magnitude corrected for the Milky Way extinction and ${\beta }_{{\lambda }_{1},{\lambda }_{2}}$ is the slope of the TRGB to a fiducial color, ${\gamma }_{{\lambda }_{1},{\lambda }_{2}}$. In the standard Johnson–Cousins system used in Madore et al. (2009), the slope is ${\beta }_{I,V}=0.2$ and the fiducial color was ${\gamma }_{I,V}=1.5$ to produce ${T}_{I,V}$ magnitudes from I photometry. The parameter values were determined from a linear approximation to the TRGB predicted by theoretical models described by Bellazzini et al. (2001, 2004). A standard edge detection algorithm can be applied to the rectified CMD and the distance modulus is computed as ${(m-M)}_{0}=T-{M}_{\mathrm{TRGB}}$, where M_TRGB is defined at the fiducial color, ${\gamma }_{{\lambda }_{1},{\lambda }_{2}}$.

For use in this work, we convert the T magnitude system of Madore et al. (2009) into the ACS/WFC system for ${\lambda }_{1}\,=F814W$ with ${\lambda }_{2}=F606W$ and ${\lambda }_{2}=F555W$, utilizing the photometric transformations from the flight magnitude system to the Johnson–Cousins system given in Sirianni et al. (2005, their Section 8.3 and Table 22). We find ${\beta }_{F814W,F606W}\,=0.27$ and the fiducial color is ${\gamma }_{F814W,F606W}=1.18$, and ${\beta }_{F814W,F555W}=0.19$ and the fiducial color is ${\gamma }_{F814W,F555W}\,=1.59$. The uncertainties associated with this transformation are not well constrained. Sirianni et al. (2005) mention differences at the level of 3%–4% for stars with V – I > 1.0 between the calibrating photometry. More specifically, the authors write that the transformations "can be applied to main-sequence and giant stars without introducing errors of more than a few percent. However, it should not be expected that the 1%–2% accuracy of ACS photometry will be preserved in the transformed data." Because we have transformed the ${T}_{I,V}$ terms into ${T}_{F814W,F606W}$ terms, the full impact of these systematics is difficult to constrain, but it must be larger than 4% (e.g., double the accuracy quoted by Sirianni et al. (2005) for their ACS absolute calibration).

Jang & Lee (2017b) investigated the color dependence of the TRGB from the HST/ACS photometry of eight nearby galaxies and find that the run of the I TRGB with the V – I color can be described with two components: a flat one for the blue color range ($V-I\lesssim 1.9$) and a steep one for the red color range ($V-I\gtrsim 1.9$). From this, they introduced the QT magnitude, a quadratic form of the TRGB magnitude corrected for the color dependence of the TRGB. ${{QT}}_{{\lambda }_{1},{\lambda }_{2}}$ is given by

$$\begin{eqnarray}{{QT}}_{{\lambda }_{1},{\lambda }_{2}} & = & {m}_{0,{\lambda }_{1}}-{\beta }_{{\lambda }_{1},{\lambda }_{2}}[{({m}_{{\lambda }_{1}}-{m}_{{\lambda }_{2}})}_{0}-{\gamma }_{{\lambda }_{1},{\lambda }_{2}}]\\ & & -{\alpha }_{{\lambda }_{1},{\lambda }_{2}}{[{({m}_{{\lambda }_{1}}-{m}_{{\lambda }_{2}})}_{0}-{\gamma }_{{\lambda }_{1},{\lambda }_{2}}]}^{2}\end{eqnarray} \tag{ 2 }$$

where ${\alpha }_{F814W,F606W}=0.159\pm 0.010$, ${\beta }_{F814W,F606W}=-0.047\,\pm 0.020$, and ${\gamma }_{F814W,F606W}=1.1$. Unlike the T conversion described previously, the QT transformation was measured in the ACS native photometry system. We estimate that the systematic uncertainty from the QT transformation is approximately 0.02 mag from propagation of the slope uncertainties through the constant terms in Equation (2).

We applied the ${T}_{F814W,F606W}$ and ${{QT}}_{F814W,F606W}$ magnitude transformations to the flc photometry of NGC 1365, and the resulting CMDs are shown in Figures 8(a) and (b), respectively. To construct an LF, we apply the color–magnitude restriction indicated by the blue shading in Figures 8(a) and (b), which is identical to that applied in Figures 2 and 5(a) in the main text. We use GLOESS smoothing with our idealized ${\sigma }_{s}$ and use the Sobel filter, [−1, 0, +1]. The edge detection response function is shown in Figures 8(a) and (b) and has strong peaks at $T\simeq {QT}\simeq 27.4$ mag. The TRGB magnitude is derived by adopting the maximum edge detection response and the corresponding uncertainty is determined following the procedure outlined in Jang & Lee (2017a), which uses the results of bootstrap resampling. We obtain TRGB magnitudes ${T}_{I,V}=27.32\pm 0.03$ mag and ${{QT}}_{F814W,F606W}=27.33\pm 0.03$ mag, which agree within their mutual uncertainties.

Zoom In Zoom Out Reset image size

Comparing the ${T}_{F814W,F606W}$ and ${{QT}}_{F814W,F606W}$ results to an extinction-corrected TRGB magnitude from the main text, F814W₀ = 27.34 ± 0.03 mag, we find a good agreement within the quoted uncertainties. We note that our color–magnitude restrictions largely avoid the regions of F606W – F814W color where the ${T}_{F814W,F606W}$ and ${{QT}}_{F814W,F606W}$ magnitudes are expected to provide the most benefit by bringing these fainter TRGB sources to the same magnitude as the bluer TRGB. Thus, we conclude that our results from the raw F814W magnitudes are fully consistent with those determined with the rectified ${T}_{F814W,F606W}$ and ${{QT}}_{F814W,F606W}$ systems.

The detection of the apparent magnitude of the TRGB is one of the most critical steps in estimating its distance. Broadly, two independent approaches have been developed for identifying the TRGB:

• 1.  
  Direct edge detection algorithms as in Lee et al. (1993), Madore & Freedman (1995), Sakai et al. (1996), Méndez et al. (2002), Mager et al. (2008), Madore et al. (2009), and Jang & Lee 2017a that typically make use of a form of the Sobel filter that is an approximation to the first derivative of a discrete function. These can take on discrete (N) and continuous forms (Φ) based on the smoothing that is applied to the LF before application of the edge detection algorithm.
• 2.  
  Template fitting as in Cioni et al. (2000), Méndez et al. (2002), Frayn & Gilmore (2003), Mouhcine et al. (2005), Makarov et al. (2006), and Conn et al. (2011).

Based on a review of the literature, we have selected seven forms of TRGB edge detection in addition to that adopted by the CCHP. These are similar to those applied in Appendix B of Paper II.

We have applied these eight Sobel filters to the LFs of the selected stars in NGC 1365 and plot them in Figure 9. We used a 0.06 mag bin to construct the LF for those edge detectors that operate directly on the histogram (e.g., Figures 9(a), (b), (e)–(g)). The bin width chosen here is larger than with other methods because no additional smoothing is applied to reduce statistical noise. In the case of the continuous forms of Sobel filters, we used a bin width of 0.01 mag for deriving the Gaussian-smoothed LFs (e.g., Figures 9(c) and (d)). Figure 9(h) is a re-visualization of the algorithm applied in the main text. The magnitude of the TRGB is determined by choosing the maximum edge detection response. Qualitatively, all eight edge detection responses in the panels of Figure 9 have peaks at F814W ∼ 27.4 mag.

Zoom In Zoom Out Reset image size

The results in the quantitative tip detection in the panels of Figure 9 are as follows: F814W = 27.46 mag from Lee et al. (1993) algorithm (a), F814W = 27.46 mag from Madore & Freedman (1995) algorithm (b), F814W = 27.50 mag from Sakai et al. (1996) algorithm (c), F814W = 27.40 mag from Méndez et al. (2002) algorithm (d), F814W = 27.46 mag from Mager et al. (2008) algorithm (e), F814W = 27.40 mag from Madore et al. (2009) algorithm (f), F814W = 27.40 mag from Jang & Lee (2017a) algorithm (g), and F814W = 27.37 mag from the algorithm adopted in the main text (h). Our estimate in the main text, F814W = 27.37 mag, is 0.07 mag brighter than the mean of all other edge detectors tested here.

In each panel of Figure 9, the TRGB is indicated by the vertical dashed line in each panel. While there is broad agreement, the techniques produce results that vary over a range of 0.13 mag, which is four times larger than the quoted uncertainty on our measurement of 0.03 mag. For a more quantitative analysis, we estimated random (statistical) and systematic uncertainties of each TRGB detection via simulations of ASLFs as described in the main text. These results are summarized in Table 3. Derived random uncertainties range from 0.01 mag to 0.07 mag with a mean of 0.04 mag. We noted that the systematic uncertainties, which are mostly not quantitatively addressed in previous studies, are comparable with or sometimes even larger than the random uncertainties. Allowing for these additional uncertainties, all of these values are consistent with our measurement (Figure 9(h)).

Table 3.  Comparison of TRGB Measurements from Different Edge Detectors

Method	Figure	Bin	TRGB	${\sigma }_{\mathrm{ran}}$	${\sigma }_{\mathrm{sys}}$
Reference	Panel	Width	(mag)	(mag)	(mag)
Lee et al. (1993)	a	0.06	27.46	0.07a	0.05
Madore & Freedman (1995)	b	0.06	27.46	0.06a	0.04
Sakai et al. (1996)	c	0.01	27.50	0.02	0.08
Méndez et al. (2002)	d	0.01	27.40	0.01	0.01
Mager et al. (2008)	e	0.06	27.46	0.04a	0.03
Madore et al. (2009)	f	0.06	27.40	0.04a	0.02
Jang & Lee (2017a)	g	0.06	27.40	0.04a	0.05
This study	h	0.01	27.37	0.03	0.02

Note.

^aFor larger bin size, 0.03 mag is added to the uncertainties.

Download table as:  ASCIITypeset image

In Paper II, a thorough discussion of the advantages and disadvantages of the wide range of edge detectors was presented, and for brevity we will discuss only the implications that can be interpreted from the panels of Figure 9. First, the use of large bins is sometimes problematic since the size of the bin has an effect on the quantitative Sobel response. Thus, in our binned LFs, an additional uncertainty of ∼0.03 (50% of a bin) should be added to the algorithmic measurement uncertainty. This additional uncertainty arises due to the inability to distinguish the location of the peak within the set binning strategy and this must be applied to the results in Figures 9(a), (b), (e)–(g). We note that some previous studies (Dalcanton et al. 2009; Jang & Lee 2017b) have undertaken Monte Carlo simulations to find a statistically more reliable TRGB magnitude and to minimize uncertainties associated with the bin size.

Second, many of the edge detection algorithms themselves employ smoothing directly in the algorithm itself. If applied to a "raw" LF, then this is not problematic, but many of the algorithms are applied to LFs that have already been smoothed. This is clearly evident in Figure 9(c), which has not only a heavily smoothed LF, but also a heavily smoothed algorithm. This "double smoothing" results in the most deviant of the TRGB values (27.50 mag) and the response of the edge detection is not a peak, but a plateau that reduces the precision of the tool. The bias in the algorithm of Sakai et al. (1996) is evident by comparing Figures 9(c) and (d), which, while having nearly identical LFs, have very different edge responses with a large systematic offset (${\sigma }_{\mathrm{sys}}=0.08$ mag).

Lastly, there are algorithms that attempt to model the uncertainties in the data (both magnitude uncertainties and completeness), but these rely critically on the ability to assess these values well for a data set. After being modeled, these uncertainties are folded into the detection algorithm itself, instead of applying modifications to the LF directly. The difficulty with this approach is that it is not fully reproducible by an independent team. As has been shown in previous sections of this Appendix, there are quantitative differences at the level of 0.04 mag between photometry derived from the same underlying images due to subtle choices in the data processing. We have demonstrated that our algorithms for the LF and for the edge detection are robust to these differences, but algorithmic approaches that use the photometry characterizations directly from one's own photometry would not be reproducible by an independent process. This is particularly concerning for the template-fitting strategies that (i) rely on the input idealized model of the LF to be well matched to the actual intrinsic LF for the field of interest and (ii) are highly sensitive to the completeness in both bands of the photometry (not just the band used for the LF).

In conclusion, we see quantitative differences between our adopted strategies for smoothing the LF and for applying an edge detection algorithm (Figure 9). As we have shown, these differences can be understood within the true uncertainties of the various techniques. As discussed in depth in Paper II, our modeling of uncertainties is explicitly designed to minimize the total uncertainty from the photometric data, to address true uncertainties including the systematics, and to take into account the full scale of photometric uncertainties. Also, our edge detection algorithm combined with the LF binning is reproducible by others and independent from the uncertainty associated with the bin size. Indeed, our approach measures the TRGB with a total uncertainty of ∼0.05 mag, small enough to determine the value of the Hubble constant accurate to 2.5% (Section 4.3.2).