MCR-TRGB: A Multiwavelength-Covariant, Robust Tip of the Red Giant Branch Measurement Method
M. J. Durbin, R. L. Beaton, J. J. Dalcanton, B. F. Williams, M. L. Boyer
DDraft version June 16, 2020
Typeset using L A TEX twocolumn style in AASTeX63
MCR-TRGB: A Multiwavelength-Covariant, Robust Tip of the Red Giant Branch MeasurementMethod ∗ M. J. Durbin, R. L. Beaton, J. J. Dalcanton, B. F. Williams, and M. L. Boyer Department of Astronomy, University of Washington, Box 351580, U.W., Seattle, WA 98195-1580, USA Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA (Received February 3, 2020; Revised June 12, 2020; Accepted June 12, 2020)
Submitted to ApJABSTRACTWe present a new method to measure colors and magnitudes of the tip of the red giant branch inmultiple bandpasses simultaneously by fitting an n -dimensional Gaussian to photometry of candidatetip stars. We demonstrate that this method has several advantages over traditional edge detection,particularly in regimes where the TRGB magnitude is strongly color-dependent, as is the case inthe near-infrared. We apply this method to a re-reduction of a set of optical and near-IR HST dataoriginally presented in Dalcanton et al. (2012a, D12). The re-reduction takes advantage of the increaseddepth and accuracy in the NIR photometry enabled by simultaneous reduction with higher resolutionoptical data in crowded fields (Williams et al. 2014). We compare three possible absolute calibrationsof the resulting apparent TRGB measurements, one adopting the same distance moduli as in D12,and two based on predicted TRGB absolute magnitudes from two widely-used, modern sets of modelisochrones. We find systematic offsets among the model absolute calibrations at the ∼ . Keywords: distance scale; galaxies: distances and redshifts; galaxies: dwarf; galaxies: halos; galaxies:irregular; galaxies: stellar content; infrared: stars; stars: Population II INTRODUCTIONThe tip of the red giant branch (TRGB) is defined asthe truncation of the RGB phase of stellar evolution.The TRGB is reached when the helium flash ignites,terminating the expansion and cooling of the outer lay-ers (Salaris & Cassisi 2006). Helium ignition occurs ata more or less fixed temperature, and thus the maxi-mum bolometric luminosity ( L bol ) produced by the coreis well-constrained (see e.g., Sweigart & Gross 1978;VandenBerg et al. 2000; Salaris & Cassisi 2006; Serenelli Corresponding author: M. [email protected] ∗ Based on observations made with the NASA/ESA Hubble SpaceTelescope, obtained at the Space Telescope Science Institute,which is operated by the Association of Universities for Researchin Astronomy, Inc., under NASA contract NAS 5-26555. et al. 2017). However, both the bolometric luminosityand the observed luminosity in a given bandpass willvary from star to star depending on the effective tem-perature, atmospheric chemistry, and on which elementsand molecules selectively absorb and emit flux. Whilethe TRGB can be used as a “standardizable candle”,care must be taken to understand the wavelength depen-dence of the observed TRGB luminosity (see Serenelliet al. 2017, for a discussion of additional physical de-tails).Baade (1944), when first resolving M 31 into stars,noticed a field of red-stars of roughly equal brightness,which we now associate with the TRGB of “old” stellarpopulations. However, the optical TRGB (OPT-TRGB)was not used as a distance indicator until Lee et al.(1993), who leveraged precise color-magnitude diagrams(CMDs) of globular clusters from Da Costa & Arman-droff (1990) to demonstrate an effective technique to a r X i v : . [ a s t r o - ph . GA ] J un Durbin et al. “detect” the truncation of the RGB sequence observa-tionally, and thereby determine a distance to the hostsystem.The Lee et al. (1993) methods are conceptually sim-ple; to detect the truncation of the RGB sequence, oneidentifies the magnitude at which there is a sharp jumpin star counts, as expected for the edge of the RGB se-quence. Lee et al. (1993) applied an edge-detection algo-rithm that approximates the first-derivative of a discretefunction (a Sobel filter; Sobel & Feldman 1968) to mea-sure the point of greatest change in the RGB luminosityfunction, which they identified as the apparent magni-tude of the TRGB. Since Lee et al. (1993), algorithmsto detect the TRGB and calibrations of the absoluteTRGB have evolved (a review and comparison is givenin Beaton et al. 2018), but the core of the techniquehas stayed the same. In general, the OPT-TRGB em-ployed in the I filter is thought to have a near-constantmagnitude M I ∼ − t > < − . ∼
31 Mpc (Jang & Lee 2017a).While the OPT-TRGB is a powerful tool with sev-eral key science drivers (for example, Tully et al. 2013;Trujillo et al. 2019; Anand et al. 2019b; Freedman et al.2019, among others), extending this method to the near-infrared (IR-TRGB, hereafter) has several advantages:(i) the stars themselves are ∼ − . P ∼
10 day Cepheids (seeFig. 30 in Beaton et al. 2018); (ii) the impact of extinc-tion is reduced by up to a factor of 6 (Indebetouw et al.2005) (Casagrande & VandenBerg 2014), permitting ex-ploration of galaxies behind high extinction (see e.g.,Anand et al. 2019a) and reducing any dust-based sys-tematics; and (iii) the next generation of astronomicalfacilities, whether 30-m class telescopes on the ground,wide-field telescopes in space, or large-aperture tele-scopes in space, are likely to realize their highest effi-ciency in the near- to mid-infrared. Thus, there is enor-mous potential for the IR-TRGB, although there remainchallenges to its implementation at high precision.The first detailed characterization of the IR-TRGBin the Hubble Space Telescope (HST)’s WFC3/IR fil-ters was presented in Dalcanton et al. (2012a, D12 here-after) in which 23 galaxies with optical imaging from theACS Nearby Galaxy Survey Treasury (ANGST, Dal-canton et al. 2009, hereafter D09) were supplementedwith WFC3/IR imaging in the F110W and F160W fil-ters (Dalcanton 2009, GO-11719). D12 detected the IR-TRGB applying a Sobel filter to F110W–F160Wcolor-magnitude diagrams, and then converted the dust-corrected apparent magnitudes to an absolute scale viadistances derived in D09, using the OPT-TRGB cali-brated to models described in Girardi et al. (2008). D12found a strong correlation between the absolute F160Wmagnitude of the IR-TRGB and the F110W-F160Wcolor, such that redder TRGB stars had a brighter ab-solute magnitude. The correlation was expected due tometallicity variations among the sample, such that moremetal rich stars had redder colors, pushing a larger frac-tion of their bolometric flux into the NIR. However, theD12 IR-TRGB was brighter than contemporaneous the-oretical models by 0.05 to 0.10 mag and, generally, wasnotably different from globular cluster observations thathad been converted from 2MASS into the WFC3/IRsystem. The general conclusion from this paper wasthat while the IR-TRGB was promising, there were sig-nificant unresolved issues. A subsequent and similaranalysis by Wu et al. (2014), however, essentially foundthe same underlying mag-color relationship for the IR-TRGB, albeit these authors argued for a break in theslope at F110W–F160W = 0.95 mag.More recent, ground-based work in the 2MASS filtersystem by Hoyt et al. (2018), Madore et al. (2018), andG´orski et al. (2018) produced empirical color-magnituderelations for the IR-TRGB. These, however, are signifi-cantly different from those determined for WFC3/IR onHST. In their review, Beaton et al. (2018) compared theWFC3/IR and 2MASS IR-TRGB slopes to demonstratethat these independent WFC3/IR and 2MASS calibra-tions largely agree when considered within a given fil-ter system and that the apparent differences are morelylikely due to inherent differences between the filter sys-tems. As a result, calibrations from the ground-based2MASS systems are likely inapplicable to the space-based WFC3/IR system.In addition to advancing empirical measurements ofthe IR-TRGB, recent papers have also updated theo-retical relationships derived from stellar models. A keywork is that of Serenelli et al. (2017), which directlycompared the theoretical IR-TRGB for a range of metal-licities and ages in the BaSTI (Pietrinferni et al. 2013)model suite. Serenelli et al. (2017) report both physi-cal and color-magnitude relationships for the IR-TRGB,but note that uncertainties in the bolometric correctionsand stellar T eff scale make direct use of these relation-ships challenging (as discussed further in Beaton et al.2018). McQuinn et al. (2019) studied the variation inthe TRGB with age and metallicity from the opticalto the mid-IR using simulated photometry based onthe PARSEC (Bressan et al. 2012; Marigo et al. 2017) CR-TRGB: A Multiwavelength TRGB Measurement Method h m s s s s ◦ D ec ( J ) DDO712 h m s m s s s ◦ DDO782 h m s s m s s s ◦ DDO822 h m s s s s s − ◦ D ec ( J ) ESO540-0302 h m s m s s s ◦ HS1172 h m s s m s s s ◦ IC2574-SGS2 h m s s s m s ◦ D ec ( J ) KDG732 h m m m ◦ KKH372 h m s m s s s s ◦ M81-DEEP2 h m s s s s − ◦ RA (J2000) D ec ( J ) NGC03002 h m s s s s ◦ RA (J2000)NGC2403-HALO-62 h m s m s s s ◦ RA (J2000)NGC2976-DEEP2 h m s s m s s ◦ D ec ( J ) NGC3077-PHOENIX2 h m s s s ◦ NGC37412 h m s s s s s ◦ NGC41632 h m s s s s s − ◦ D ec ( J ) NGC7793-HALO-62 h m s s m s s s − ◦ SCL-DE12 h m s s s m s ◦ SN-NGC2403-PR2 h m s m s s s m s ◦ D ec ( J ) UGC43052 h m s s s m s ◦ UGC44592 h m s m s s s s ◦ UGC51392 h m s s m s s s ◦ RA (J2000) D ec ( J ) UGC85082 h m s s s s ◦ RA (J2000)UGCA2922 Figure 1.
Footprints of the HST observations originally presented in D09 and D12, which we reanalyze in this work. ACS/WFCfootprints are shown by thin white lines, and WFC3/IR footprints are in thick orange. Background images are PanSTARRs z + g for all targets except NGC 300 & NGC 7793-HALO-6, which use DSS2. All background images were retrieved through theHiPS thumbnail service provided by the Universit´e de Strasbourg. model suite, and found that rectifying the photometryto a fiducial tip reduced the range of variations in themeasured F160W TRGB to 0.04 mag. Thus, while thepotential for the IR-TRGB is well-recognized (see e.g.,Beaton et al. 2018, among others), the empirical evi-dence for its reliability remains unclear.D12 presented a number of concerns regarding theiranalysis that ranged from the relatively new data pro-cessing and calibration of WFC3/IR data, to crowdingin the images (for which the higher-resolution optical im-ages are clearly deeper and more complete). However,since D12, major large-scale projects like the Panchro-matic Hubble Andromeda Treasury (PHAT; Dalcantonet al. 2012b; Williams et al. 2014), the Cosmic AssemblyNear Infrared Deep Extragalactic Legacy Survey (CAN-DELS; Grogin et al. 2011; Koekemoer et al. 2011), andthe Ultra Deep Field (Koekemoer et al. 2013; Borlaffet al. 2019), have led to substantial improvement bothin our technical knowledge of the WFC3/IR camera andin the development of multiwavelength data-processingtechniques that significantly improve the WFC3/IRphotometric quality. Additionally, there have been mul- tiple internal efforts to improve WFC3/IR calibrationand data products (for a comprehensive overview seeMack 2018). It is the purpose of this work to apply thesetechniques to the D12 dataset and revisit the discrepan-cies identified in D12 regarding the IR-TRGB (Durbin2017). We also take advantage of and expand upon re-cent works (e.g. Hoyt et al. 2018; Madore et al. 2018;Freedman et al. 2020) that have demonstrated the effec-tiveness of calibrating the TRGB in multiple bandpassesby selecting a set of fiducial “tip stars” and fitting theirmultiwavelength behavior; we present a generalized ver-sion of this method here.The outline of the paper is as follows. Section 2 de-scribes the observations, image processing, photometry,and artificial star tests. Section 3 presents techniquesto isolate the RGB, identify candidate TRGB stars, andtrace their multiwavelength behavior. Section 4 presentsthe final measured TRGB apparent magnitudes and col-ors, and compares the absolute magnitudes and dis-tance moduli obtained from previously published dis-tances and then from calibration to two sets of theoret-ical isochrones. Section 5 presents a discussion of our Durbin et al. results, attempts to resolve concerns from D12, and dis-cusses lingering concerns regarding the full realization ofthe IR-TRGB. Section 6 presents a summary of our workand discusses directions of future research. Throughoutthe main text, we limit visualizations to a representativeset of galaxies; figures for the full sample are given asfigure sets. DATA2.1.
Observations
We re-reduced the optical and near-infrared
HST imaging data described in D09 and D12. The D12 ob-servations were a WFC3/IR imaging follow-up (SNAP-11719) to the optical ACS/WFC data presented inD09. The F110W+F160W observations cover 26 point-ings in 22 Local Volume galaxies with a range of star-formation histories. The majority of the galaxies arelow-metallicity dwarfs, with the exception of M81. Ta-ble 1, reproduced from D12, presents summary informa-tion about the galaxies in our sample, including coordi-nates, angular diameter, apparent B magnitude, fore-ground reddening, T-type, H I line widths, and groupmembership. We note that not all of these galaxies havethe purely old stellar populations that are consideredoptimal for measuring the TRGB.We analyzed 24 of the 26 datasets that were includedin D12. To maintain uniformity in the final datasetand analyses, we excluded two targets (NGC404 andNGC2403-DEEP) because their optical data were takenby WFPC2 rather than ACS. Additionally, we com-bined the two pointings of Holmberg II (UGC4305-1 andUGC4305-2 in D12) into a single target UGC4305 here,as they have slight overlap in the NIR and substantialoverlap in the optical. All targets have ACS imaging inF814W (comparable to Johnson-Cousins I ) and at leastone of the F475W, F555W, and F606W filters (compara-ble to SDSS g , Johnson-Cousins V , and broad Johnson-Cousins V respectively). Figure 1 shows the footprintsof the ACS/WFC (white) and WFC3/IR (orange) oneither PanSTARRS or DSS2 imaging for each of the 23distinct targets used here.Table 2 describes the ACS/WFC and WFC3/IR ob-servations used for this work including references to theoriginal proposals, total F814W exposure time, and off-sets of the observation from the galaxy center.We retrieved all data in the form of calibrated indi-vidual exposures ( *flt files for WFC3/IR and CTE-corrected *flc files for ACS/WFC) from the MikulskiArchive for Space Telescopes (MAST) with Astroquery(Ginsburg et al. 2017, 2019) on January 28, 2019, andobtained up-to-date reference files with the HST CRDS bestref tool (Swam et al. 2004). 2.2. Alignment & Photometry
Alignment RMS (mas) N ( i m ag e s ) F814WF110WF160W
Figure 2.
Distributions of the RMS scatter of alignmentresiduals for F814W, F110W, and F160W. Both near-IR fil-ters have a residual scatter on the order of 0.025 . (cid:48)(cid:48) , or ∼ . (cid:48)(cid:48) , but there is a long tail of images with higherscatter, likely due to variations in exposure depth and sourcedensities. We aligned all exposures using
TweakReg and
Drizzlepac
TweakReg aligns images by calculating an affine trans-form (shifts, rotation, and scale) that best describes thetransformation between astrometric catalogs from twoimages, one of which is treated as the fiducial “reference”image. It then calculates an updated WCS solution forthe non-reference image using the affine transform.By default,
TweakReg extracts astrometric source cat-alogs from input images with a point source extractionroutine based on DAOFIND (Stetson 1987), which isoptimized for point source detection. However, manyof our exposures are too sparsely populated with brightstars to produce a reliable cross-filter alignment solu-tion from point sources alone, requiring the additionof background galaxies to the astrometric source cat-alogs. We therefore followed the procedure describedin Lucas (2015) to align images on Source Extractor(Bertin & Arnouts 1996) catalogs rather than
TweakReg -produced catalogs. Source Extractor’s detection algo-rithm is largely morphology-agnostic, which enables therobust detection of both point and extended sources. Weused SEP (Barbary 2016), a Python and C reimplemen-tation of core Source Extractor algorithms, to derive allcatalogs used in alignment. Although it is true that extended sources are less optimal foralignment, as their morphologies may vary across filters affect-ing their calculated centroids, they are nonetheless useful in theabsence of sufficient point sources.
CR-TRGB: A Multiwavelength TRGB Measurement Method Table 1.
Sample galaxiesGalaxy Alt. RA Dec Diam. B T A V m − M T W GroupNames (J2000) (J2000) ( (cid:48) ) ( km s − )DDO53 U4459 08:34:06.5 66:10:45 1.6 14.55 0.104 27.79 10.0 25 M81DDO78 10:26:27.9 67:39:24 2.0 15.80 0.058 28.18 -3.0 M81DDO82 U5692 10:30:35.0 70:37:10 3.4 13.57 0.112 27.90 9.0 M81HoI U5139,DDO63 09:40:28.2 71:11:11 3.6 13.64 0.137 27.95 10.0 29 M81HoII U4305 08:19:05.9 70:42:51 7.9 11.09 0.087 27.65 10.0 66 M81HS117 10:21:25.2 71:06:58 1.5 16.50 0.316 27.91 10.0 13 M81I2574 U5666,DDO81 10:28:22.4 68:24:58 13.2 10.84 0.100 27.90 9.0 115 M81KDG2 E540-030,KK9 00:49:21.1 -18:04:28 1.2 16.37 0.064 27.61 -1.0 SclKDG63 U5428,DDO71 10:05:07.3 66:33:18 1.7 16.01 0.270 27.74 -3.0 19 M81KDG73 10:52:55.3 69:32:45 0.6 17.09 0.052 28.03 10.0 18 M81KKH37 06:47:45.8 80:07:26 1.2 16.40 0.204 27.56 10.0 20M81 N3031,U5318 09:55:33.5 69:04:00 26.9 7.69 0.232 27.77 3.0 422 M81N300 00:54:53.5 -37:40:57 21.9 8.95 0.034 26.50 7.0 149 14+13N2403 U3918 07:36:54.4 65:35:58 21.9 8.82 0.110 27.50 6.0 231 M81N2976 U5221 09:47:15.6 67:54:49 5.9 11.01 0.241 27.76 5.0 97 M81N3077 U5398 10:03:21.0 68:44:02 5.4 10.46 0.188 27.92 10.0 65 M81N3741 U6572 11:36:06.4 45:17:07 2.0 14.38 0.066 27.55 10.0 81 14+07N4163 U7199 12:12:08.9 36:10:10 1.9 13.63 0.055 27.29 10.0 18 14+07N7793 23:57:49.4 -32:35:24 9.3 9.70 0.054 27.96 7.0 174 SclSc22 Sc-dE1 00:23:51.7 -24:42:18 0.9 17.73 0.042 28.11 10.0 SclU8508 IZw60 13:30:44.4 54:54:36 1.7 14.12 0.042 27.06 10.0 49 14+07UA292 CVnI-dwA 12:38:40.0 32:46:00 1.0 16.10 0.043 27.79 10.0 27 Note —Reproduced from D12, with updates to A V from Schlafly & Finkbeiner (2011). Name, position, diameter, B T , W ,and T-type taken from Karachentsev et al. (2004). m − M from D09 and Karachentsev et al. (2003) for NGC 7793; Groupmembership from Karachentsev (2005) or Tully et al. (2006). We chose ACS/WFC F814W as our “reference” filterfor all targets, as it is the only optical filter commonto all targets, and in most cases it is the deepest andmost likely to contain sources that are detected acrossmultiple filters. We aligned all frames for each targetwith the following steps:1. Extract initial source catalogs from all F814W ex-posures with SEP and align these with
TweakReg ;2. Combine all aligned F814W exposures into asingle distortion-corrected reference image with
AstroDrizzle , and extract a deep reference cata-log from the drizzled image;3. Realign all F814W exposures to the reference im-age using catalogs from the cosmic ray cleaned( *crclean ) images produced by
AstroDrizzle ;4. Align all other exposures to the reference imagewith
TweakReg .We did not attempt to derive an absolute astromet-ric solution for any of our targets, as the majority are severely limited by the ∼ (cid:48) × (cid:48) WFC3/IR field of viewand do not have enough bright sources to reliably matchagainst external astrometric catalogs such as
Gaia . Forthe purposes of this work, internally consistent align-ment on a per-target basis is sufficient.Figure 2 compares the RMS scatter of the alignmentresiduals for the common filters of F814W, F110W, andF160W. The residuals for the two WFC3/IR filters arevery similar, with a residual scatter of ∼ . (cid:48)(cid:48)
025 or 0.2WFC3/IR pixels. The residuals for F814W are morescattered, with a peak at 0 . (cid:48)(cid:48)
01 (0.2 ACS/WFC pixels)and a long tail, likely due to differences in the underly-ing image datasets themselves (e.g., different exposuredepths and source densities).We carried out photometry on the aligned images withthe pipeline described in Williams et al. (2014), whichwraps the
HST photometry package DOLPHOT (Dol-phin 2000). Briefly, DOLPHOT uses a set of fiducialPSF models that are empirically scaled for each frame toaccount for frame-to-frame differences, such as those in-duced by “breathing” (Hasan & Bely 1994). The cross-
Durbin et al.
Table 2.
Observations
Galaxy Target name Date obs. Offset( (cid:48) ) Exptime(s) Σ min Σ max N (cid:63) Opt.propid Opt. filtersKDG63 DDO71 2010-04-21 16:33:04 0.97 9000 0.00 0.86 68477 GO-9884 F606W, F814WDDO78 DDO78 2010-04-20 15:13:25 0.34 2292 0.02 0.54 56458 GO-10915 F475W, F814WDDO82 DDO82 2010-05-07 07:27:41 0.38 2442 0.01 2.99 187699 GO-10915 F475W, F606W,F814WKDG2 ESO540-030 2009-12-17 12:32:10 0.15 7840 0.00 0.60 28087 GO-10503 F606W, F814WHS117 HS117 2010-02-24 02:35:38 0.13 900 0.00 0.46 13011 GO-9771 F606W, F814WI2574 IC2574-SGS 2010-02-25 03:34:37 3.28 6400 0.10 1.40 286852 GO-9755 F555W, F814WKDG73 KDG73 2010-06-09 18:17:42 0.43 2274 0.00 0.22 12721 GO-10915 F475W, F814WKKH37 KKH37 2009-09-29 11:12:38 0.09 3441 0.00 1.36 30966 GO-10915,GO-9771 F475W, F606W,F814WM81 M81-DEEP 2010-06-13 01:26:19 13.88 29953 0.02 0.23 63093 GO-10915 F475W, F606W,F814WN300 NGC0300 2010-04-19 18:17:32 6.26 2982 0.05 0.59 197750 GO-10915,GO-9492 F475W, F555W,F606W, F814WN2403 NGC2403-HALO-6 2010-04-25 04:57:54 5.58 720 0.01 0.46 20441 GO-10523 F606W, F814WN2976 NGC2976-DEEP 2010-02-25 02:34:59 3.03 27191 0.02 1.50 96662 GO-10915 F475W, F606W,F814WN3077 NGC3077-PHOENIX 2010-02-21 23:20:39 3.89 19200 0.02 0.38 70482 GO-9381 F555W, F814WN3741 NGC3741 2009-11-07 02:03:02 0.51 2331 0.00 2.14 48369 GO-10915 F475W, F814WN4163 NGC4163 2010-03-23 18:11:32 0.23 3150 0.01 3.89 153523 GO-10915,GO-9771 F475W, F606W,F814WN7793 NGC7793-HALO-6 2010-06-14 19:43:15 6.02 740 0.01 0.45 20079 GO-10523 F606W, F814WSc22 SCL-DE1 2009-09-08 01:16:49 0.18 17920 0.00 0.24 18967 GO-10503 F606W, F814WN2403 SN-NGC2403-PR 2010-04-22 08:27:47 0.90 1450 0.37 7.59 433196 GO-10182,GO-10402 F475W, F606W,F814WHoII UGC4305 2010-02-26 10:10:22 0.54 9920 0.02 1.65 327523 GO-10605,GO-10522 F555W, F814WDDO53 UGC4459 2010-04-23 11:46:34 0.25 4768 0.02 0.47 63451 GO-10605 F555W, F814WHoI UGC5139 2009-08-21 23:26:49 0.35 5936 0.04 0.52 105305 GO-10605 F555W, F814WU8508 UGC8508 2009-10-14 20:11:32 0.18 2349 0.00 1.57 73755 GO-10915 F475W, F814WUA292 UGCA292 2010-05-18 13:08:17 0.35 2274 0.00 0.21 17668 GO-10915,GO-10905 F475W, F606W,F814W
Note —Here the date observed is the date of the last IR exposure; the offset is the distance between the center of the IRfootprint and the galaxy coordinates as given in Table 1; the exposure time is the total F814W exposure time (all IRobservations have uniform exposure times of 600s in F110W and 900s in F160W); and N (cid:63) is the number of stars that weredetected in all of F814W, F110W, and F160W. camera wrapper utilizes a single underlying source listsuch that DOLPHOT can iteratively measure each in-dividual source simultaneously across frames employingtechniques optimized for crowded fields. As described inWilliams et al. (2014), the output photometry for eachsource requires additional characterization to have real-istic uncertainties incorporating all concerns; these arederived via artificial star tests that are described in thefollowing subsection.The key difference in the procedure adopted here com-pared to that of D12 is that we perform simultane-ous cross-camera photometry rather than reducing thedatasets independently and then matching cataloguedsources. Due to the differences in the native angular res-olution between ACS/WFC and WFC3/IR (0 . (cid:48)(cid:48) . (cid:48)(cid:48) . (cid:48)(cid:48)
75 (15 WFC3/IR pixels) and extracting sourcesfrom the convolved images with SEP. We used the ellipseparameters a , b , and θ of the sources to mask potentiallycontaminated pixels, with a and b multiplied by 5 to en-sure that a sufficient fraction of the contaminating fluxwas masked. 2.3. Artificial Star Tests
The primary sources of photometric uncertainty inthese data are total exposure depth, which determines
CR-TRGB: A Multiwavelength TRGB Measurement Method . . . . . F110W − F160W − − − − − F W AST input IR photometry . . . . . . . . . l og ( N ? ) Figure 3.
Hess diagram of input AST photometry in thenear-IR. The densest portions (orange to yellow) are fromthe CMDs, whereas the uniform sampling is purple.
As the AST input locations were assigned at random,they do not necessarily reflect the true distributions ofdensity and depth for any single target. We therefore resampled the full set of AST results to match the dis-tribution of these quantities for each target as closely aspossible, as follows.We evaluated stellar surface densities using kernel den-sity estimation (Rosenblatt 1956; Parzen 1962) with thePython package
KDEpy (Odland 2018). We selected thephotometry to be used for density estimation using thesame near-IR selection box as in the ASTs, with theadditional criteria of having a mean near-IR signal-to-noise greater than 3. We then constructed stellar sur-face density maps by convolving source coordinates witha Gaussian kernel with a width of 5 (cid:48)(cid:48) , and tagged allphotometry with their local densities. Density maps forthree example targets are shown in Figure 4. In theanalysis presented in Section 3 we used only photome-try with local densities less than 1.5 stars/ (cid:3) (cid:48)(cid:48) , exceptfor the high-density target SN-NGC2403-PR, where weused a maximum local density of 3 stars/ (cid:3) (cid:48)(cid:48) . Fig. Set 4. Surface density maps
While all near-IR exposures were taken with identicalexposure times and are therefore of comparable depth,there is considerable variation in the optical exposuredepths, which in turn may affect DOLPHOT’s sourcedetection and subsequent deblending of near-IR sources.To characterize exposure depth, we use the weight mapsgenerated by
Astrodrizzle for the combined F814Wreference images to assign fiducial total exposure timesto the locations of each source.For each target we separated to the photometry into10 bins according to density vs. depth using K-meansclustering (Arthur & Vassilvitskii 2007; Sculley 2010),and resampled the full set of ASTs to match the ob-served distributions of densities and depths.We then use the resampled ASTs to assign fiducialphotometric uncertainties, biases, and completenesses toall of our photometry. We define the photometric biasto be the median of the differences between observedand input AST magnitudes, the photometric error to bethe interquartile range of the same, and the complete-ness to be the fraction of stars with non-null observedmagnitudes. We calculate these quantities as a functionof AST input magnitudes in each filter.We subtract filter-appropriate foreground extinctionsfrom all photometry, with values obtained from Schlafly& Finkbeiner (2011); the corresponding V -band extinc-tions are listed in Table 1. We assume negligible in-ternal extinction for all targets, as the majority of ourtargets are either low-metallicity dwarfs or halos of spi-ral galaxies. Target SN-NGC2403-PR is an exception,but in that case we find that the photometric uncertain-ties due to crowding are large enough that an attempt Durbin et al.
F110W − F160W F W HS117 h m s s s s ◦ RA D ec . . . . . . . S t e ll a r s u r f a ce d e n s i t y F110W − F160W F W NGC0300 h m s s s − ◦ RA D ec . . . . . . . S t e ll a r s u r f a ce d e n s i t y F110W − F160W F W NGC4163 h m s s s s ◦ RA D ec . . . . . . . S t e ll a r s u r f a ce d e n s i t y Figure 4.
Left: NIR Hess diagrams of three galaxies in oursample, with the selections of stars included in our surfacedensity calculations highlighted. Right: corresponding stel-lar surface density maps for each target. All density mapsare scaled to the same limits (0 to 1.5 RGB stars per squarearcsecond) to highlight the range of stellar densities in oursample. Gaps in the density images show where contaminat-ing sources such as foreground stars and background galaxieswere rejected. The complete figure set (23 images) is avail-able in the online journal. to analyze or correct for internal extinction would likelynot be productive.2.4.
Comparison to D12 photometry
Here we directly compare this generation of photom-etry to that of D12 by crossmatching individual stars.We first convert the IR pixel coordinates of the originalphotometry to the WCS defined by our realigned im-ages. We select an initial sample of stars within 1 magof the D12 TRGB values and maximum per-filter old-to-new magnitude differences of 0.5 mag, and match on RAand Dec using a kd-tree (Bentley 1975) with a maximumdistance of 2 (cid:48)(cid:48) . We then find the robust coordinate trans-formation parameters between the new and old photom- etry using RANSAC regression (Fischler & Bolles 1987)on the matched initial sample with a maximum resid-ual value of 0 . (cid:48)(cid:48)
1. We apply this transformation to thefull set of old photometry coordinates and match thetransformed coordinates again with a kd-tree, this timewith a maximum distance of 0 . (cid:48)(cid:48)
1. Figure 5 shows thechanges in magnitude as a function of the original D12magnitudes in F160W, with the D12 TRGB ± . Fig. Set 5. Photometry comparisons
Interestingly, we find that near the tip, the medianmagnitude difference is typically very small (on the or-der of 0.01 mag) but negative for uncrowded stars, in-dicating that this generation of photometry is slightlybrighter than the previous. However, even the sparsestfields show a population of high-crowding stars that areseveral tenths of a magnitude dimmer than their D12counterparts. TRGB MEASUREMENTIn this section we describe the steps we use to measurethe apparent magnitudes and colors of the IR-TRGB.We adopt a multiwavelength approach, which we call“MCR-TRGB”, that we summarize for the reader inadvance of detailed descriptions. First, we isolate theRGB sequence from the other stellar populations. Fromthe RGB sample, we do a tip detection to select starsin the vicinity of the TRGB. This initial sample is thenseparated into potential sub-populations to isolate thosethat have colors and magnitudes consistent with beingTRGB stars. The color and magnitude distributions ofthe candidate tip stars are then fitted jointly for all ap-plicable color-magnitude spaces to build the final color-magnitude calibrations. This approach has several ad-vantages over traditional Sobel edge-detection for thepurpose of this work, which we discuss in detail in Sec-tion 5.Throughout this section, the methods are demon-strated using galaxies that span a range in metallicityand RGB shape. Identical figures for each of the 23galaxies in the sample are provided as figure sets.3.1.
Initial RGB Star Selection
Fig. Set 6. RGB selection CMDs
A maximally complete and minimally contaminatedsample of RGB stars is essential for characterizing theTRGB and the RGB luminosity function near the tip.Unfortunately there are many stars that have colors andmagnitudes similar to RGB stars, such as red helium-burning (RHeB) and asymptotic giant branch (AGB)stars. These “contaminant” populations can blur theTRGB edge or distort its measured magnitude (see dis-cussion in D12).
CR-TRGB: A Multiwavelength TRGB Measurement Method . . . . . . m F160W (D12) − . − . . . . ∆ m F W (t h i s w o r k − D ) HS117, F160W
Median ∆ m Mean ∆ mm TRGB (D12) ± . . . . . . . C r o w d i n g ( m ag ) , t h i s w o r k . . . . . . m F160W (D12) − . − . . . . ∆ m F W (t h i s w o r k − D ) NGC0300, F160W
Median ∆ m Mean ∆ mm TRGB (D12) ± . . . . . . . C r o w d i n g ( m ag ) , t h i s w o r k . . . . . . m F160W (D12) − . − . . . . ∆ m F W (t h i s w o r k − D ) NGC4163, F160W
Median ∆ m Mean ∆ mm TRGB (D12) ± . . . . . . . C r o w d i n g ( m ag ) , t h i s w o r k Figure 5.
Changes in photometry between D12 and thiswork for HS117 (top), NGC 300 (middle), and NGC 4163(bottom), with the D12 magnitude on the x-axis and ∆ m on the y-axis. The color-coding indicates the DOLPHOTcrowding parameter of the new photometry, which is thenumber of magnitudes subtracted from the initial measure-ment due to neighboring sources. The rolling mean and me-dian are shown by the solid and dashed lines respectively.The complete figure set for both F110W and F160W (46images) is available in the online journal. Typically, RGB stars are selected using strict binarycolor-magnitude cuts; we describe two particular exam-ples. D12 initially select stars with colors in the range0 . < F110W − F160W < . red vs. blue–red CMDs using F814W, F110W, and F160W asthe red filters, and using all available optical filters otherthan F814W as the blue. We also construct CMDs inF814W vs. F814W–F160W, F110W vs. F814W–F110W,and F110W vs. F110W–F160W for all targets. Thenumber of unique color-magnitude combinations variesfrom 6 to 12 depending on the number of available op-tical filters for each target. We apply broad initial colorand magnitude cuts based on the D12 TRGB measure-ments. Figure 6 provides example CMDs after cuts forNGC 4163 in the color-magnitude combinations used forthis analysis.Next, we define an RGB locus in each filter combi-nation by fitting a predicted RGB color-magnitude se-quence to the photometry in each color-magnitude com-bination independently. We minimize the median dis-tance between the observed photometry and a grid ofsynthetic photometry derived from PARSEC (Marigoet al. 2017) isochrones of ages 4 to 14 Gyr and [Fe/H] − . − . M F110W = − Durbin et al.
F475W − F814W . . . . . . F W F475W − F110W . . . . . . . F W F475W − F160W F W . . . F606W − F814W . . . . . . F W F606W − F110W . . . . . . . F W F606W − F160W F W .
25 0 .
50 0 .
75 1 . F814W − F110W . . . . . . . F W . . . . F814W − F160W F W .
50 0 .
75 1 .
00 1 . F110W − F160W F W . . . . . . R G B + p r o b a b ili t y NGC4163 RGB+ selection
Figure 6.
A demonstration of filter-by-filter RGB selections for NGC4163. Each of the 9 panels contains a color-magnitudecombination used to determine P (RGB)+ for stars in the color-magnitude range. The points in each panel are color-coded bythe final probability as indicated on the color-bar. The best-fit PARSEC synthetic RGB in each combination is shown. Thecomplete figure set (23 images) is available in the online journal. extrapolate the fitted RGB sequences out to at least 1.5mag brighter than the measured D12 TRGB apparentmagnitudes in all filters. As a result, stars brighter thanthe TRGB that fall along the predicted color-magnitudeloci will be assigned high RGB-sequence probabilities.These probabilities should be understood as estimates ofa star’s proximity to the color-magnitude relations char-acteristic of each target’s RGB sequence, rather than asidentifications of only the stars that are truly on theRGB.The individual RGB-sequence probabilities are thenaveraged across all color-magnitude combinations toproduce global RGB-sequence probabilities, which wecall P (RGB)+. 3.2. Edge Detection
We make an initial selection of candidate tip stars byapplying a Sobel edge detection to the RGB-weightedluminosity function (LF). For each target we choose thefilter with the sharpest LF; that is, the filter in whichthe tip magnitude is least dependent on color. This isF814W for most targets, and F110W for targets withF110W − F160W > .
95 mag, as measured in D12).We first construct a luminosity function (shown in themiddle column of Figure 7) by marginalizing P (RGB)+over color as a function of magnitude. We use a bin sizeof 0.01 mag, which is a factor of ∼ CR-TRGB: A Multiwavelength TRGB Measurement Method
KDEpy implementation of the Improved Sheather-Jones algorithm (Botev et al. 2010), which chooses anoptimal kernel width based on the overall density of thedata. We then multiply this fiducial width by the squareof the photometric uncertainties scaled by their medianvalue as a function of magnitude, which de-emphasizesLF variation fainter than the TRGB, where photometricuncertainties are higher. The final smoothed LF, shownoverlaid in black on the raw LFs in Figure 7, is thenused for the initial TRGB detection.To detect the TRGB, we begin by applying a Sobel fil-ter, which is one of the most widely used means of find-ing the tip (see summary and comparisons in Beatonet al. 2018). The Sobel filter approximates the firstderivative of a discrete dataset via convolution with akernel. In its simplest form, this kernel is [ − , , i − i + 1 bin to determine the edge-response, η , for bin i . This kernel is applied to the smoothed LF, and theresponse is shown for each galaxy in the right panels ofFigure 7. In Figure 7, the magnitude of maximum So-bel response, m ( η max ), is indicated by the dashed lineacross all panels.We then select candidate tip stars near m ( η max )within a range we call ∆ η . The value of ∆ η is deter-mined using two quantities: i) the median photomet-ric error within ± . m ( η max ), σ η max phot , and ii), aminimum number of tip candidate stars N min (cid:63) . We de-fine N min (cid:63) as the square root of the number of stars 1magnitude below m ( η max ), with a hard minimum of 30stars. For each target, we make an initial selection ofstars within ± σ TRGBphot , and then iteratively expand theselection range by 0 . σ TRGBphot on each side until either N min (cid:63) is reached or ∆ η is over 0.2 mag. For the majorityof our targets, the initial selection window of ± σ TRGBphot is enough to meet N min (cid:63) . Our final ∆ η is shown by theblue band in the panels of Figure 7 for our examplepointings. Out of the stars that fall within the fiducial tip mag-nitude range, we first select likely RGB stars as thosewith P (RGB)+ > .
6, which roughly corresponds tostars that were identified as RGB+ sequence candidateswith over 90% probability in at least two-thirds of thefilter combinations we used to assign RGB probabilities.We then reject stars with anomalous magnitudes in atleast one filter with Local Outlier Factor outlier detec-tion (Breunig et al. 2000), which evaluates the relativeisolation of points using k -nearest neighbors. We takethis trimmed sample of stars to be our final set of tipstar candidates, which we then use to measure tip mag-nitudes and colors as described in the following section. Fig. Set 7. Initial TRGB star selection
We note that this selection of likely RGB tip stars isperformed based on the resuts of applying the Sobel fil-ter to the filter where the tip is “flat” with color. TheSobel filter, by design, looks for an sharp edge in a one-dimensional distribution. Two dimensional implemen-tations of the Sobel Filter exist, but still require conver-sion of our CMDs into a binned form. Thus, applica-tion of the one-dimensional Sobel filter to a distributionthat has magnitude-color behavior may not fully detectthe true edge in the distribution. Lastly, where thereis strong magnitude-color trend, because our colors aremore imprecise than our magnitudes, the intrinsic slopecan be distorted by the color-spread in our data. Thus,in the next section, we develop a method to utilize thetip stars we have just identified to trace the intrinsicTRGB slope across our set of color-magnitude combina-tions. 3.3.
Multiwavelength tip fitting
We characterize the color and magnitude distributionsof our candidate tip stars using Extreme Deconvolution(XDGMM, Bovy et al. 2011), a modification of Gaussianmixture modeling that accounts for uncertainties in theinput data. Specifically, we use XDGMM to fit a singlesix-dimensional Gaussian to the F814W, F110W, andF160W magnitudes and the F814W–F160W, F814W–F110W, and F110W–F160W colors of the tip star candi-dates. Although the underlying distribution of tip starsin this parameter space is not intrinsically Gaussian, wefind that a single Gaussian is a reasonable approxima-tion for the majority of our tip star samples. Addition-ally, for the faintest and sparsest of our targets, low starcounts and photometric uncertainties on the same orderas the width of the tip star selection windows do not al-low us to place reasonable constraints on more complexmodels, such as multi-component Gaussian mixtures.We discuss potential alternative modeling approachesin subsection 5.2.2
Durbin et al. .
00 1 .
25 1 .
50 1 .
75 2 . F814W − F160W . . . . . . . F W HS117 Normalized RGBluminosity function
RawSmoothed
Edge response(arbitrary units) .
50 0 .
75 1 .
00 1 . F814W − F110W . . . . . . . . F W NGC0300 Normalized RGBluminosity function
RawSmoothed
Edge response(arbitrary units) . . . F814W − F160W . . . . . . . F W NGC4163 Normalized RGBluminosity function
RawSmoothed
Edge response(arbitrary units)
Figure 7.
Tip star selection with edge detection for threedemonstrative galaxies in our sample. From top to bot-tom, HS117, NGC 300, and NGC 4163. For each galaxy,three panels are shown, from left to right, the CMD of high-probability RGB stars, the raw (gray) and smoothed (black)luminosity function, and the Sobel edge response ( η ). Theinitial magnitude of the TRGB is identified as the magni-tude at η max , which is identified as the dashed line in eachpanel. TRGB candidate stars are selected within the blueband, the width of which is deteremined by the photometricuncertainty at the tip and by the number of stars on the up-per RGB as described in the text. The complete figure set(23 images) is available in the online journal. For the uncertainties we use as inputs to XDGMM,we divide each star’s individual photometric uncertain-ties by P (RGB+), effectively weighting the input pointsby P (RGB+). We emphasize that XDGMM, as a tool,allows us to take into account these uncertainties andweights on the RGB+ likelihood to trace the tip in fil-ters where the Sobel edge is less effective due to color-magnitude slopes. We take the means of the fitted distributions to beour final apparent tip magnitudes and colors. Resultsof these fits are shown for our sample galaxies in Fig-ure 8, where we plot ellipses showing the 95% confidenceregions of the XDGMM fits in three color-magnitudecombinations. The width, height, and position angle ofeach ellipse are derived from two-dimensional slices ofthe full six-dimensional covariance matrix.Potential systematic and statistical biases of thismethod are discussed in Appendix A; overall, we findthat the results are comparable to those of edge detec-tion in most cases. Fig. Set 8. TRGB fitting results RESULTS4.1.
Apparent TRGB magnitudes and colors
In this section we compare the TRGB apparent magni-tudes and colors we have measured using the techniquesdeveloped in this paper to those used in D12. All revisedapparent magnitudes and errors are reported in Table 3.First, Figure 9 compares the change in apparentF160W magnitude and F110W–F160W color betweenthis work (blue points) and D12 (orange points) foreach target in our sample. The 68% confidence inter-vals are shown for our measurements and demonstratethat the difference between this work and D12 is almostalways larger than our measurement uncertainties, al-beit, as shown in the lower right, most are within thecolor-magnitude photometric error circle for an individ-ual source at the tip.The origin of these offsets can be determined by com-paring the individual differences between the photom-etry. Figure 10 compares the relative change betweenthe measurements of this work and that of D12 forthe F110W (x-axis) and F160W (y-axis). For both∆ m F160W and ∆ m F110W (defined as this work minusD12), the median difference is approximately +0.05mag; histograms are shown in Figure 10 on each axis.Interestingly, the offsets are highly correlated; in the up-per panel, a one-to-on line is shown in black-dashed linewith a fit to the results given in blue solid, and the 95%confidence interval (shown in the shaded region) encom-passes the one-to-one line.Figure 11 displays the F814W–F110W (left) andF814W–F160W (right) to F110W–F160W color-colordiagrams for the results of this work (blue) comparedto that of D12 (orange). Relative to D12, the measure-ments from this work move the color-color relations tothe left in this diagram – bluer in F814W–F110W andF814W–F160W and slightly redder in F110W–F160W.In Figure 11 we provide reference lines to highlightthe color behavior, using a linear function for the D12
CR-TRGB: A Multiwavelength TRGB Measurement Method . . . F814W − F160W . . . . . . . F W HS117 . . . . F814W − F110W . . . . . . F W . . . F110W − F160W . . . . . . . F W . . . . F814W − F160W . . . . . . . F W NGC0300 .
50 0 .
75 1 .
00 1 . F814W − F110W . . . . . . F W . . . . F110W − F160W . . . . . . . F W . . . F814W − F160W . . . . . . F W NGC4163 . . . . F814W − F110W . . . . . . F W . . . F110W − F160W . . . . . . . F W Figure 8.
Results of XDGMM fits to candidate TRGB stars for HS117 (top), NGC 300 (middle), and NGC 4163 (bottom). Foreach galaxy the fits are shown for the following color-magnitude combinations: F814W, F814W–F160W (left), F110W, F814W–F110W (center), and F160W, F110W–F160W (right). The solid horizontal lines in each panel identify the mean magnitudes ofthe tip from XDGMM (which are typically very close to the Sobel-detected edge), and the overplotted ellipses show the 95%color-magnitude confidence regions of the two-dimensional fitted tip. We note that because this method uses a set of candidatetip stars selected based on their magnitudes in a single band (either F814W or F110W), the 2-D ellipses may not follow thevisual impression of the tip in other bandpasses. This is especially apparent when the color-width of the RGB is of order thecolor uncertainties, which is typical in the NIR. The complete figure set (23 images) is available in the online journal. photometry and a logistic function for our new photom-etry. (We caution that these fitting relations should notbe taken as physically meaningful.)4.2.
The TRGB color-absolute magnitude relation
To derive the color dependence of the NIR TRGB ab-solute magnitude, we must adjust the apparent magni-tudes in Figure 9 by the appropriate distance modulusfor each galaxy.We first present a revised NIR color-absolute magni-tude relation adopting the same distances as in D12, and then explore the use of the most up-to-date stellarmodels to derive revised distance moduli and absolutemagnitudes. 4.2.1.
Adopting D12 distances
The distance moduli used in D12 were deteminedusing the F814W TRGB, which enables a fully self-consistent study of the TRGB across bandpasses. Withthe exception of NGC 7793, these distances were orig-inally published in Dalcanton et al. (2009), whereasthe distance for NGC 7793 is from Karachentsev et al.4
Durbin et al.
Table 3.
Apparent TRGB magnitudes
F814W F110W F160WTarget m σ fit σ phot m σ fit σ phot m σ fit σ phot N (cid:63) DDO71 23.742 0.002 0.015 22.990 0.022 0.047 22.134 0.040 0.048 1478DDO78 23.730 0.005 0.032 22.966 0.050 0.042 22.056 0.066 0.042 1948DDO82 23.864 0.005 0.044 23.040 0.041 0.051 22.123 0.060 0.051 4525ESO540-030 23.617 0.007 0.036 22.940 0.023 0.039 22.092 0.031 0.043 631HS117 23.845 0.011 0.036 23.154 0.009 0.045 22.318 0.019 0.040 592IC2574-SGS 23.875 0.005 0.040 23.091 0.031 0.062 22.211 0.050 0.057 5227KDG73 23.887 0.021 0.031 23.258 0.023 0.046 22.483 0.024 0.052 312KKH37 23.542 0.004 0.032 22.819 0.020 0.044 21.959 0.030 0.048 902M81-DEEP 24.074 0.103 0.013 22.892 0.021 0.039 21.892 0.024 0.035 551NGC0300 22.493 0.043 0.017 21.565 0.009 0.022 20.602 0.022 0.022 1350NGC2403-HALO-6 23.340 0.020 0.027 22.497 0.036 0.027 21.593 0.036 0.028 378NGC2976-DEEP 23.734 0.055 0.027 22.857 0.016 0.044 21.910 0.028 0.040 1771NGC3077-PHOENIX 23.972 0.080 0.016 22.990 0.015 0.044 22.010 0.034 0.041 1136NGC3741 23.488 0.004 0.034 22.795 0.005 0.043 21.981 0.006 0.043 798NGC4163 23.241 0.007 0.030 22.508 0.039 0.038 21.623 0.059 0.039 2429NGC7793-HALO-6 23.868 0.007 0.051 23.029 0.046 0.032 22.101 0.042 0.038 866SCL-DE1 24.007 0.028 0.015 23.348 0.034 0.049 22.554 0.030 0.052 454SN-NGC2403-PR 23.416 0.076 0.067 22.457 0.030 0.100 21.489 0.032 0.079 1641UGC4305 23.569 0.004 0.034 22.803 0.034 0.050 21.954 0.049 0.049 4883UGC4459 23.708 0.004 0.034 22.985 0.028 0.044 22.162 0.031 0.048 1292UGC5139 23.893 0.003 0.021 23.133 0.031 0.047 22.296 0.048 0.050 2015UGC8508 23.018 0.005 0.027 22.315 0.021 0.032 21.503 0.035 0.034 1402UGCA292 23.750 0.021 0.028 23.168 0.023 0.044 22.411 0.035 0.048 177 (2003), which also uses the F814W TRGB. The absolutecalibration of the F814W (ground-based I -band) TRGBat <
5% precision is unclear (Jang & Lee 2017b; Beatonet al. 2018; Freedman et al. 2019; Yuan et al. 2019; Reidet al. 2019; Freedman et al. 2020). Historically, it hasbeen assumed to be a constant value of approximately M I TRGB ∼ − .
05 mag (Lee et al. 1993; Salaris & Cassisi1997). However, this magnitude is only anticipated tobe roughly constant for uniformly old and metal-poorpopulations (Beaton et al. 2018; Serenelli et al. 2017;Salaris & Cassisi 2006); more specifically, [M/H] < − . > M F814WTRGB , Dalcanton et al. (2009) used the mean opti-cal colors of stars within 0.2 mag of the apparent F814WTRGB to choose fiducial Girardi et al. (2008) isochroneswith corresponding colors. Dalcanton et al. (2009) thendetermined the predicted F814W TRGB absolute mag-nitude for each galaxy from the isochrone sets, sub-tracted that from their measured F814W TRGB appar- ent magnitudes, and corrected for foreground extinctionto obtain their distance moduli.We calculate NIR TRGB absolute magnitudes by sub-tracting the D12 distance moduli from our apparent tipmagnitudes, as reported in Table 4. The resulting NIRabsolute magnitudes and color-magnitude relation arecompared to that of D12 in Figure 12.We fit a linear relation to the absolute F160W mag-nitudes and F110W–F160W colors determined in thiswork using orthogonal distance regression (ODR, Boggset al. 1987), and find: M F160WTRGB = − . − F160W) − .
475 (1)The uncertainties on our slope and zeropoint are 0.057mag color − and 0.050 mag, respectively. Compared tothe equivalent fit from D12 (their eq. 1), M F160WTRGB = − . − F160W) − . , (2)we find an 0.02 mag fainter zero-point ( <
1% in dis-tance) and a change in the slope of less than 0.04 magcolor − , both of which are well within our uncertainties.As expected, the difference in the zeropoint is roughlyequivalent to the differences in measured TRGB pho-tometry observed in Figure 9. CR-TRGB: A Multiwavelength TRGB Measurement Method Table 4.
Absolute TRGB magnitudes from D12 distances
Target µ (D12) M F814W σ F814W M F110W σ F110W M F160W σ F160W
DDO71 27 . − .
998 0 . − .
750 0 . − .
606 0 . . − .
090 0 . − .
854 0 . − .
764 0 . . − .
036 0 . − .
860 0 . − .
777 0 . . − .
993 0 . − .
670 0 . − .
518 0 . . − .
065 0 . − .
756 0 . − .
592 0 . . − .
025 0 . − .
809 0 . − .
689 0 . . − .
143 0 . − .
772 0 . − .
547 0 . . − .
018 0 . − .
741 0 . − .
601 0 . . − .
696 0 . − .
878 0 . − .
878 0 . . − .
007 0 . − .
935 0 . − .
898 0 . . − .
160 0 . − .
003 0 . − .
907 0 . . − .
026 0 . − .
903 0 . − .
850 0 . . − .
948 0 . − .
930 0 . − .
910 0 . . − .
062 0 . − .
755 0 . − .
569 0 . . − .
049 0 . − .
782 0 . − .
667 0 . . − .
092 0 . − .
931 0 . − .
859 0 . . − .
103 0 . − .
762 0 . − .
556 0 . . − .
084 0 . − .
043 0 . − .
011 0 . . − .
081 0 . − .
847 0 . − .
696 0 . . − .
082 0 . − .
805 0 . − .
628 0 . . − .
057 0 . − .
817 0 . − .
654 0 . . − .
042 0 . − .
745 0 . − .
557 0 . . − .
040 0 . − .
622 0 . − .
379 0 . Recalibrating Distances to Recent Models
Both the physical isochrones and the filter transforma-tions described in Girardi et al. (2008) have undergonemany revisions in the intervening years (Bressan et al.2012; Marigo et al. 2017), and thus the F814W TRGBzeropoints adopted in D09 and D12 may no longer be ap-propriate. Here we apply a similar distance estimationmethod as in D09 to our revised measurements, usingsynthetic photometry from the model suites PARSECv. 1.2S (Bressan et al. 2012; Marigo et al. 2017) andMIST v. 1.2 (Choi et al. 2016), both of which are usedroutinely for stellar populations work. We retrieved thesynthetic photometry directly from the cited web ser-vices. For both sets, we use isochrones with ages span-ning 8 to 14 Gyr with log(age) spacing of 0.05 dex. ThePARSEC metallicities span − . ≤ [Fe / H] ≤ http://stev.oapd.inaf.it/cgi-bin/cmd 3.3 http://waps.cfa.harvard.edu/MIST/index.html span − . ≤ [Fe / H] ≤ Z (cid:12) = 0 . Y (cid:12) = 0 . Z (cid:12) = 0 . Y (cid:12) = 0 . Durbin et al. .
70 0 .
75 0 .
80 0 .
85 0 .
90 0 .
95 1 .
00 1 . F110W − F160W . . . . . . . . . m T R G B ( F W ) Median photometric error
Dalcanton 2012 → this work m TRGB (F160W) + 1 (NGC 300)68% confidence
Figure 9.
Comparison of the revised tip F160W apparentmagnitudes and F110W – F160W colors from this work (bluepoints) to those of D12 (orange points). Arrows indicate per-target correspondence between D12 and the new measure-ments. The blue ellipses show the 68% confidence regions onthe measurements of this paper from XDGMM fitting, andthe bottom right errorbars indicate the median photometricuncertainties in color and magnitude for an individual star.For NGC 300 (unfilled points) we plot M F160W + 1 ratherthan M F160W , as it is ∼ The results of this procedure are shown in Fig-ure 13 where the left panel shows the adopted values of M TRGB (F814W)–(F814W-F160W) and the right panelshows the inferred values of M TRGB (F160W)–(F110W-F160W). In each panel, sets of synthetic photometryat single ages (8-13 Gyr), with metallicities spanning − . < [ F e/H ] < − .
25 dex, are plotted as transparentlines, with PARSEC models plotted in green and MISTmodels in orange. Overall, the mag-color behavior of thetwo sets is qualitatively similar, but the absolute magni-tudes differ by ∆ M F814W ∼ .
15 mag, with PARSECbeing brighter than MIST for the same color ( ∼
8% indistance).The fits of our data to the predicted M TRGB (F814W)–(F814W-F160W) distributions are shown as the pointsin the panels of Figure 13. From this, we determine adistance modulus to each galaxy, which we denote as µ F814W . In the right panel, we use µ F814W to trans-late m TRGB (F160W) to M TRGB (F160W), and comparethese values to the same isochrone sets used to derive µ F814W . The PARSEC-based distances place the ob-served NIR TRGB ∼ .
05 mag fainter than predicted,but they do trace the same underlying variations with . . . ∆ m F160W (this work − D12) − . . . . . . ∆ m F W (t h i s w o r k − D ) Medianphotometricerror
Linear fit (95% confidence)∆ m F110W = ∆ m F160W . . . . F110W − F160W (this work) − . . . . . . ∆ m F W (t h i s w o r k − D ) Medianphotometricerror
Linear fit (68% confidence)
Figure 10.
Top: changes in the M F160W tip magnitudescompared to changes in the M F110W
TRGB magnitudes be-tween this work and D12. Bottom: changes in the M F160W tip magnitudes between this work and D12 against the NIRcolor measured in this work. Both panels show histogramswith overlaid kernel density estimates of the marginal distri-butions of the quantities on each axis, linear fits with shadedconfidence intervals, and scale bars with typical photometricuncertainties. color. In contrast, the MIST-derived values are less off-set overall in magnitude, but show shape deviations thatbecome particularly pronounced at red colors.Figure 14 repeats this process in reverse by de-termining a distance modulus, µ F160W , based off ofthe M TRGB (F160W)–(F110W-F160W) model predic-tions (left panel) and then comparing the absolute
CR-TRGB: A Multiwavelength TRGB Measurement Method . . . . . F814W − F110W . . . . . . . . F W − F W Medianphotometricerror
Dalcanton 2012 → this work 1 . . . . . . F814W − F160W
Medianphotometricerror
68% confidenceLinear fitLogistic fit
Figure 11.
Comparison of D12 color-color relation to the TRGB color-color relation determined in this work, with F110W– F160W against F814W–F110W (left) and F814W–F160W (right). As in Figure 9, orange points are from D12, blue pointsare from this work, arrows connect the corresponding results, and the blue shading indicates our two-dimensional uncertaintiesfrom XDGMM. We show a linear fit to the D12 values (as done in that work) and a generalized logistic fit to values from thiswork to highlight the changes in morphology in our new color-color relations. . . . . F110W − F160W − . − . − . − . − . − . M T R G B ( F W ) Median photometricerror
Dalcanton 2012(95% confidence)This work(95% confidence)Dalcanton 2012(95% confidence)This work(95% confidence)
Figure 12.
Comparison of revised NIR color-absolute mag-nitude relations to D12 with revised absolute magnitudesderived using the same distances as in D12. Blue points arevalues from the current work and orange points are fromD12. The corresponding color-coded lines show linear fits toeach dataset and the shaded regions show 95% confidenceintervals. Again, we see that our results are slightly red-der and slightly fainter than D12 using their distances. Thiscolor-color relation is distance-independent. M TRGB (F814W)–(F814W-F160W) empirical relation-ship to the models in the right panel. In this case,different behavior is observed: for both model sets, M TRGB (F814W) from our data are too bright by ∼ ∼ New Distance Moduli Compared to D12
Figure 15 compares the distance moduli determinedin the previous subsection to those from D12. The toppanel compares the distances calibrated to F814W (thesame filter in either case) and the bottom panel com-pares the distances from F160W. No difference (∆ µ = 0mag) is indicated by the vertical dashed line. In bothcases, the PARSEC-based calibration is systematicallylarger than in D12 by median values of 0 . ± . . ± . ± ± − . ± . − . ± . ± ± DISCUSSION8
Durbin et al. .
25 1 .
50 1 .
75 2 .
00 2 .
25 2 . F814W − F160W − . − . − . − . − . − . − . M T R G B ( F W ) Adopted µ (F814W) PARSECMIST 0 . . . . F110W − F160W − . − . − . − . − . − . M T R G B ( F W ) Inferred
Figure 13.
Revised absolute tip magnitudes using distance moduli calibrated to M F814W versus F814W–F160W derivedfrom synthetic photometry from the MIST (orange) and PARSEC (green) model suites. Each solid line represents a set oftheoretical tip star colors and absolute magnitudes at a single age. In the left panel, we tie the observed colors to an absolutemagnitude in either isochrone set to determine µ F814W . On the right, we use µ F814W to determine M F160WTRGB and plot againstour F110W–F160W color; we find that these measurements are systematically offset from the corresponding isochrone models. . . . . F110W − F160W − . − . − . − . − . − . M T R G B ( F W ) Adopted µ (F160W) PARSECMIST 1 .
25 1 .
50 1 .
75 2 .
00 2 .
25 2 . F814W − F160W − . − . − . − . − . − . − . M T R G B ( F W ) Inferred
Figure 14.
We invert the demonstration of Figure 13. In the left panel, we tie the observed colors to an absolute magnitude ineither isochrone set to determine µ F160W . On the right, we use µ F160W to compute M F814WTRGB and plot against our F814W–F160Wcolor; we again find that our measurements are systematically offset from the corresponding isochrone models. Taken withFigure 13, this suggests that distance moduli calibrated to models in one band will systematically mispredict the correspondingtip behavior in other bands.
In this section, we discuss the advantages and limita-tions of our adopted methods to trace the TRGB acrossmultiple wavelengths. Once established, we then dis-cuss more fundamental limitations to our investigation,which include knowledge of the absolute magnitude ofthe TRGB, details of the physical models underlying the isochrone suites, and possible systematics that aredifficult to disentangle with the data at hand.5.1.
Advantages of the MCR-TRGB method
MCR-TRGB simultaneously measures the distribu-tions of a pre-selected group of stars across an arbi-trary number of color-magnitude combinations. As a
CR-TRGB: A Multiwavelength TRGB Measurement Method Table 5.
New distance moduli
PARSEC MISTF814W F160W F814W F160WTarget µ σ µ σ µ σ µ σ
DDO71 27.842 0.065 27.899 0.093 27.710 0.065 27.747 0.093DDO78 27.822 0.093 27.939 0.100 27.700 0.093 27.785 0.100DDO82 27.948 0.101 28.026 0.109 27.835 0.101 27.864 0.109ESO540-030 27.719 0.072 27.839 0.080 27.566 0.072 27.694 0.080HS117 27.948 0.072 28.042 0.075 27.796 0.072 27.885 0.075IC2574-SGS 27.969 0.094 28.027 0.115 27.847 0.094 27.866 0.115KDG73 27.978 0.072 28.077 0.090 27.761 0.072 27.898 0.090KKH37 27.643 0.072 27.733 0.087 27.506 0.072 27.585 0.087M81-DEEP 28.043 0.162 28.068 0.070 27.938 0.162 27.788 0.070NGC0300 26.551 0.079 26.639 0.047 26.441 0.079 26.435 0.047NGC2403-HALO-6 27.423 0.061 27.456 0.060 27.311 0.061 27.306 0.060NGC2976-DEEP 27.804 0.098 27.893 0.079 27.696 0.098 27.699 0.079NGC3077-PHOENIX 28.015 0.124 28.118 0.083 27.909 0.124 27.877 0.083NGC3741 27.590 0.065 27.659 0.075 27.429 0.065 27.501 0.075NGC4163 27.340 0.079 27.450 0.091 27.209 0.079 27.290 0.091NGC7793-HALO-6 27.948 0.091 28.026 0.077 27.838 0.091 27.858 0.077SCL-DE1 28.104 0.064 28.188 0.094 27.926 0.064 28.020 0.094SN-NGC2403-PR 27.467 0.158 27.546 0.155 27.358 0.158 27.334 0.155UGC4305 27.668 0.084 27.705 0.099 27.537 0.084 27.557 0.099UGC4459 27.810 0.074 27.857 0.087 27.664 0.074 27.695 0.087UGC5139 27.993 0.075 28.022 0.099 27.858 0.075 27.864 0.099UGC8508 27.119 0.064 27.178 0.069 26.964 0.064 27.015 0.069UGCA292 27.811 0.071 27.961 0.091 27.586 0.071 27.787 0.091
Note —Quoted errors are the quadrature sum of the photometric and fitting errors. result, the method ensures self-consistency in the mea-sured color-magnitude behavior as it is determined fromthe same underlying set of stars. In contrast, tra-ditional edge detection is done on a per-filter basis,and it is not guaranteed to detect the tip using pre-cisely the same stars across color-magnitude combina-tions. This limitation is particularly important in thecases of steeply-sloped tips where the correspondingcolor-baseline changes significantly relative to the color-uncertainty.Fitting the full color-magnitude covariance has thefurther benefit of characterizing spread of colors andmagnitudes both intrinsically, e.g., within a galaxy, andexperimentally, e.g., with respect to our photometricprecision. This differs from Sobel-based edge detectionwhere it is difficult to properly account for measure-ment uncertainties in both magnitude and color. More-over, Sobel-based edge detection does not automaticallyaccount for intrinsic color-magnitude variation across a given TRGB and, as a result, the Sobel-edges are gen-eralized to a mean color on a per-filter basis.Unlike the T -magnitude system of Madore et al.(2009), which rectifies the photometry to an assumedTRGB slope, MCR-TRGB relies only on the assump-tion that there is one filter in which the tip magnitudehas a weak enough color dependence enough to make aninitial selection of tip star candidates. This permits usto use the well-established Sobel method to find a “flat”edge to define candidate tip stars and then utilize thosestars in regimes where the tip is more difficult to detectusing Sobel-based methods. This provides a fundamen-tal advantage toward revealing the underlying intrin-sic behavior of TRGB stars to construct self-consistentcolor-magnitude calibrations.We posit that MCR-TRGB is a more effective toolto define the underlying color-magnitude calibrationsfor local, well-studied, and well-observed galaxies thanrelying on techniques more suited for distant galaxies.0 Durbin et al. − . − . − . . . . . ∆ µ , this work − D12 N ( ga l a x i e s ) ∆ µ (F814W) PARSECMIST − . − . − . . . . . ∆ µ , this work − D12 N ( ga l a x i e s ) ∆ µ (F160W) PARSECMIST
Figure 15.
Histograms and overlaid biweight kernel densityestimates of ∆ µ , where ∆ µ is the difference between distancemoduli derived in this work and the distance moduli used inD12. The top panel shows ∆ µ using distance moduli cali-brated to M F814W and the bottom shows ∆ µ using distancemoduli calibrated to M F160W . These differences are muchlarger than those measured using the same distances (Fig-ure 12) and thus can be interpreted as differences betweenmodel predictions of tip magnitudes.
Stated differently, if your goal is to provide the best char-acterization of the behavior of tip stars across multiplebands, MCR-TRGB will perform better than standardSobel-edge detection. It also provides a fully empiri-cal basis to explore the ultimate precision of the TRGBas distance-measurement tool, from which we can gainunderstanding of both systematic and statistical biasesfor more distance measurements where there is a lowerability to probe these terms with available data.5.2.
Limitations of MCR-TRGB
Our method requires multi-wavelength data and rel-atively good data quality, which makes it less gener-ally applicable to all distance measurement applications.More specifically, MCR-TRGB requires fairly stringentinitial rejection of potential contaminants, which maynot be feasible for all datasets due to photometric un-certainties or the complexity of the underlying stellarpopulations. Moreover, the multiwavelength tracing ofindividual TRGB stars may also be infeasible for manycontexts where the acquisition of multiband imaging is too expensive. Thus, as just discussed, we considerMCR-TRGB’s most significant role is as a tool to de-fine the underlying systematics affiliated with TRGB-based distances as the community explores differentcolor-magnitude regimes.The XDGMM algorithm used in MCR-TRGB losesits ability to resolve the shape of an intrinsic distribu-tion when the typical uncertainties on the input datapoints are comparable to the full range of the inputdata. This method loses its advantages for low signal-to-noise photometry, filter combinations with very shortcolor baselines, and simple stellar populations with lit-tle color spread near the TRGB. (See Appendix A forfurther discussion.)Another limitation of XDGMM is its assumption ofGaussianity. In practice, the intrinsic distribution ofTRGB stars within a given magnitude range is farfrom Gaussian. More complex models of this distri-bution should be investigated, ideally on high-precisionphotometry of systems with well-populated RGB se-quences. For example, a multi-component Gaussianmixture could be of use in distinguishing remaining con-taminants, such as a low-density AGB “background”,from the TRGB population. Alternate fitting methods,such as Gaussian process regression, should also be con-sidered. We reserve tests of this nature for future ex-ploration using local galaxies with properties and obser-vations that are better matched to the requirements ofdrawing conclusions from such tests.5.3.
Absolute Calibration
In section 4.2.2, Figures 13 and 14 introduce a puzzlewith regards to using synthetic photometry as the ab-solute calibration for the TRGB. Inspection of Figures13 and 14 reveal two concerns for the isochrone sets:(i) a systematic offset between PARSEC and MIST ispresent for the absolute magnitudes regardless of filter,such that PARSEC is consistently brighter than MIST,and (ii) there is a relative filter-to-filter offset betweenthe models’ predictions and our measurements, suchthat the absolute NIR magnitudes of our measurementsderived by adopting the models’ optical predictions donot correspond to the models’ NIR predictions, and viceversa. Understanding these differences requires consid-ering how stellar interior models, which predict stellarstructure, are mapped to stellar atmospheres used toconstruct the synthetic photometry shown in Figures 13and 14. An excellent discussion of this process is givenby Casagrande & VandenBerg (2014).Before examining the models more closely, we notethat such offsets should not be surprising when viewedin the context of the larger literature. Even in the
CR-TRGB: A Multiwavelength TRGB Measurement Method I -band, there is a current debatein the value of the absolute magnitude of the TRGB,which is central to determining H (e.g., see Freedmanet al. 2019; Yuan et al. 2019; Reid et al. 2019; Freedmanet al. 2020, and references therein). Even the most de-tailed and careful calibrations have total uncertaintiesat the 0.05 mag level due to various systematic terms.The range of recently used F814W absolute values spans ∼ .
10 mag (as reviewed by Beaton et al. 2018). Thesediscrepancies in the absolute magnitude of the tip canpropagate into stellar models depending on exactly howthe isochrone sets cross-check their own absolute scales.As reviewed by Beaton et al. (2018), such discrepan-cies also affect the RR Lyrae and horizontal branches,which imparts uncertainty on the absolute scale for glob-ular clusters (as are explored in detail by Casagrande& VandenBerg 2014). Therefore, no theoretical predic-tions can be expected immune to the downstream effectsof systematics in the empirical distance scale.In this subsection, we first present a preliminary com-parison of our results to the empirical TRGB relation de-rived for the (F606W–F814W)- M F814W color-magnitudeplane by Jang & Lee (2017b), and discuss its implica-tions with regard to assessing differences in the models’behavior. We then consider two aspects of the syntheticphotometry that might contribute to the discrepanciesin Figures 13 and 14: (i) differences in adopted stellar at-mospheres, which affect the conversion from bolometricluminosity to observed fluxes in specific filters, discussedin section 5.3.2, and (ii) differences in the underlyingstellar evolution physics, discussed in section 5.3.3.5.3.1.
Comparison to empirical optical results
The discrepancies between our measurements andboth sets of synthetic photometry in Figures 13 and 14make it unclear which model should be preferred, if ei-ther. As an alternative, we turn to the empirical F814WTRGB calibration presented by Jang & Lee (2017b).Like the majority of existing empirical F814W/ I cali-brations, it is based on a specific optical color baseline(F606W–F814W), which precludes us from adopting itfor our entire sample, as F814W is the only optical fil-ter common to all our targets. However, the subset ofour sample with F606W observations (14 out of 23 tar-gets) allows us to make a preliminary comparison as abenchmark against theoretical calibrations. Jang & Lee (2017b) employ a quadratic functionalform (the QT system) for the M F814W vs. (F606W– A more detailed analysis that incorporates recently revised dis-tances to the two absolute-scale zeropoint anchors used by Jang& Lee (2017b) is currently in preparation.
F814W) relation, which they calibrate over an extensivecolor range of 0 . < F606W − F814W < QT relation for the galaxiesin our sample with F606W coverage in Figure 16.Figure 16 shows that the Jang & Lee calibration fallssquarely between the PARSEC and MIST predictions inF814W. Adopting the distances from this calibration, wethen plot the inferred absolute magnitude in the F160Wband in the center panel. The resulting F160W TRGBabsolute magnitudes agree quite well with the MIST pre-dictions in the NIR. We explore this apparent color mis-match further in the right panel, which compares the op-tical F606W–F814W colors to the NIR F110W–F160Wcolors for the 14 galaxies with F606W data. Althoughthe center panel of Figure 16 appears to favor MISTpredictions in the NIR at all but the reddest colors, theoptical-IR color-color behavior strongly favors PARSEC.5.3.2. Bolometric corrections
We make a direct comparison of the models’ phys-ical predictions in the left panel of Figure 17, whichcompares the MIST and PARSEC model TRGB intemperature-luminosity space for the same range ofages and metallicities as in Figures 13 and 14. Wesee that the models are offset from each other in T eff and log( L/L (cid:12) ), indicating differences in the underlyingstellar structure, with PARSEC running approximately ∼
10% more luminous and 50-150 K warmer than MISTat the same age and metallicity. (For comparison, Choiet al. (2018) find uncertainties on the absolute T eff scaleof ±
100 K due to boundary conditions.) The PARSECpredictions also show a slightly larger spread in T eff thanMIST over the same range of age and metallicity.We attempt to isolate filter-to-filter differences be-tween the models’ predictions by examining their color-color behavior. The middle panel of Figure 17 showsthe F814W–F110W to F110W–F160W color-color be-havior of the two model sets . There is a divergence onthe order of 0.1 mag for stars with F814W–F110W > Durbin et al. . . . . F606W − F814W − . − . − . − . − . − . − . − . M T R G B ( F W ) Adopted
Jang & Lee 17PARSECMIST . . . . F110W − F160W − . − . − . − . − . M T R G B ( F W ) Inferred . . . F606W − F814W . . . . . . F W − F W Medianphotometricerror
Figure 16.
As in Figure 13, we infer absolute F160W magnitudes via adopted F814W absolute magnitudes. Here theF814W absolute magnitudes are calibrated to the F814W vs. F606W–F814W QT relation presented by Jang & Lee (2017b)for the subset of our sample with F606W coverage. We overplot MIST (orange) and PARSEC (green) synthetic photometryfor comparison. Left: The QT relation with adopted M F814W values. Center: Inferred M F160W vs. F110W–F160W. Right:Distance-independent F110W–F160W vs. F606W–F814W color-color plot. T eff . . . . . . l og ( L / L (cid:12) ) PARSECMIST 1 . . F814W − F110W . . . . . F W − F W . . . . . F814W − F110W . . . . . . F W − F W This work
Figure 17.
Left: Bolometric luminosity vs. effective temperature for the MIST (orange) and PARSEC (green) model tip starsused in this work (ages 8 - 14 Gyr and − ≤ [Fe / H] ≤ − .
25 dex). PARSEC luminosities are found to be consistently ∼ F814W–F110W (which appears to effectively rule outunaccounted-for extinction as a source of disagreement).Comparison to the right panel of Figure 16, which showsgood agreement between observations and PARSEC pre-dictions for F606W–F814W and F110W–F160W colors,suggests that PARSEC’s predicted TRGB colors areoverall accurate in the optical and IR independently, butthat there may be offsets in the relative cross-calibrationbetween the two wavelength regimes in either the stel-lar atmosphere models or in our data. More specifi-cally, if PARSEC’s predicted F110W and F160W abso-lute magnitudes were shifted to be ∼ . T eff , and both [Fe/H] and [ α /H].While the differences in predicted bolometric luminosi-ties shown in the left panel of Figure 17 suggest thatPARSEC and MIST would still predict different abso-lute magnitudes for a tip star of a given age and metal-licity, bolometric corrections and the stellar T eff scaleare likely to be a source of some of the filter-to-filteroffsets we observe.As of this writing, the PARSEC web service (CMDv. 3.3) uses PHOENIX (Allard et al. 2012) bolometric cor-rections for stars with T eff < ∼ ATLAS12 (Kurucz 2014). Chen et al. (2019) have explored howthe
PHOENIX (Allard et al. 2012) and
ATLAS9 (Kurucz
CR-TRGB: A Multiwavelength TRGB Measurement Method
PHOENIX bolometric corrections produce RGB colors that are bi-ased red by up to 0.1 mag in V − I , which translatesto an artificially bright TRGB in the NIR consistentwith what we see in Figures 13 and 14. Although Chenet al. (2019) claim that the PHOENIX bolometric correc-tions are preferable for giants because they are com-puted with spherical geometry, Fu et al. (2018, sec-tion 3.2.3) find that the
PHOENIX bolometric correctionscannot reproduce the observed RGB colors in 47 Tuc,which they term the “RGB-too-red” problem. While wehave been unable to locate any similar such studies ofMIST’s predictions for the RGB, Fu et al. (2018) cau-tion that
ATLAS12 atmospheres may be unreliable for T eff < > I -band TRGB from the BaSTImodels (Pietrinferni et al. 2013) using four sets of bolo-metric corrections (see their fig. 8). They see differencesat the ∼
10% level when applying different sets of bolo-metric corrections to models using the same underlyingphysics, comparable to the amplitude of discrepancieswe see here. Although this investigation focused on theoptical, their finding of ∼ . Physical properties of TRGB stars
The left-most panel of Figure 17 suggests that thereare currently real differences in the predicted stellarstructure at the TRGB due to the different physical as-sumptions between the models, even before differencesin atmospheres or bolometric corrections are included.Our comparison of PARSEC and MIST broadly agreeswith the conclusions from a more detailed model-focusedstudy by Serenelli et al. (2017), who investigated boththe physical and computational factors contributing todifferences between the predicted TRGB luminositiesof two sets of stellar models (BaSTI and GARSTEC).Serenelli et al. (2017) were able to produce identicalpredictions of tip stars’ physical properties from two different model suites only when certain physical pro-cesses, such as neutrino energy loss and electron screen-ing, as well as some numerical criteria such as integra-tion timestep, were implemented consistently betweenstellar evolution codes (the full set of which they term“concordance physics”). While a comparable investi-gation of such sources of difference between MIST andPARSEC is well outside the scope of this work, we find itreasonable to conclude that some aspects of their phys-ical differences are likely due to limitations in our cur-rent understanding of certain “cutting edge” topics instellar astrophysics, and may also be in part due to dif-fering computational approaches. Tip stars, in additionto being both cool and luminous, are at an evolutionarytransition point, and so may be especially sensitive tothese details.Another possible source of disagreement between themodels’ predictions and our data is elemental abun-dances, including the helium fraction Y and α enhance-ment, the latter of which is of particular concern at[Fe/H] (cid:46) − (cid:46) . α -enhanced models, although they areslated to be included in future releases of both (Choiet al. 2016; Fu et al. 2018). Serenelli et al. (2017) reportthat, at least after the adoption of their concordancephysics, α enhancement produces what they consider tobe negligible effects on the TRGB bolometric luminosity( <
1% difference between [ α /Fe] = 0.4 and [ α /Fe] = 0at fixed [M/H]), T eff ( <
2% difference), and predicted
VIJK magnitudes ( < .
01 mag difference at constantcolor). Similarly, they find that a change in the heliumfraction Y of 0.01 (approximately the range over whichestimates of the primordial helium mass fraction vary)has no more than a 1% effect on TRGB temperaturesand luminosities. Nonetheless, as forthcoming versionsof PARSEC will offer options for variable α enhance-ment and helium abundance (Fu et al. 2018) as well asupdated bolometric corrections , we anticipate that adirect analysis of these quantities’ impacts on predic-tions of TRGB behavior as they pertain to this workwill be both easily achievable and informative.5.4. TP-AGB contamination The
PHOENIX bolometric corrections currently employed in thePARSEC web service only take total Z into account in their colortransformations (Fu et al. 2018), which is problematic for un-derstanding potential photometric effects of varying abundanceratios. ATLAS12 model atmospheres for α -enhanced PARSECisochrones down to T eff = 4000 K are currently in development(Fu et al. 2018; Chen et al. 2019). Durbin et al.
While we believe that our method of determining P (RGB)+ is overall effective at rejecting contaminatingpopulations, such as red supergiants and the bulk of theAGB, there may be some amount of remaining contam-ination, particularly from thermally-pulsing AGB (TP-AGB) stars, which we briefly discuss here.Although TP-AGB stars are generally intrinsicallybrighter than the TRGB, extinction from circumstellardust can substantially impact their observed magnitudesand bring them closer to luminosities typical of the up-per RGB. However, as they are also heavily reddened,their colors are inconsistent with RGB stars (Boyer et al.2017, see their fig. 8), so it is likely that our method ofTRGB candidate selection successfully rejected most, ifnot all, of these stars.TiO absorption is another factor that may bring cer-tain TP-AGB stars, particularly M-type stars at highmetallicity, closer in luminosity to the TRGB (Boyeret al. 2019). While the bulk of our targets are low-metallicity dwarf galaxies, this may be an issue for someof the larger galaxies in our sample, such as M81.Finally, TP-AGB stars may cross the TRGB whenthey reach the minimum point in their pulsation cycle.In this case, the fact that our NIR data were taken sev-eral years later than our optical data is an advantage;TP-AGB stars at their minimum in our optical obser-vations are unlikely to be at their minimum in the NIR,and vice versa.5.5. Limited Empirical Constraints on TRGBMagnitude Stability
We briefly discuss other physical concerns that mayaffect the colors and magnitudes of TRGB stars, includ-ing binarity, low amplitude pulsational variability, massloss, and planetary engulfment.First, the presence of a companion star may affect theintrinsic photometric properties of a TRGB star throughmass transfer in a binary interaction. Preliminary in-vestigations suggest that the photometric effects of theformer phenomenon are overall secondary to the vari-ation of TRGB magnitude with metallicity (Eldridge2019, private communication). Thus, we qualitativelyconclude that binarity is likely to be a source of someamount of residual scatter in our measurements ratherthan a primary driver of TRGB variation across popu-lations.Second, as recent high-precision, high-cadence pho-tometry has demonstrated, low-amplitude variability ex-ists for many to most stellar types. Pulsational variabil-ity was first proposed for stars on the upper RGB byIta et al. (2002) and has since been observationally con-firmed (Ita et al. 2004; Lebzelter & Wood 2005; Wood 2015). Variability may contribute some amount of un-certainty to TRGB measurements by effectively blurringthe TRGB edge. However, there are few establishedconstraints on relevant characteristics, such as typicalperiods, amplitudes, fractions of stars that exhibit vari-ability, and dependence on stellar properties such as ageand metallicity. We have thus disregarded TRGB vari-ability as a potential systematic in this work due to lackof empirical constraints. We expect that any overalleffects are small compared to our dominant sources ofuncertainty.RGB stars are known to experience mass loss drivenby chromospheric activity (Origlia et al. 2007; Groe-newegen 2012; Pasquato et al. 2014), which may be am-plified by either of the first two properties. Jimenez et al.(2020) predict that variations in the mass loss param-eter η at the TRGB may affect individual stars’ lumi-nosities by over 5%, although they estimate that the neteffect on measured TRGB distances does not exceed 2%,and that it is strongly metallicity-dependent. Addition-ally, although mass loss has been correlated with theblueshifting of optical and near-IR spectral lines suchas H α and the calcium triplet (McDonald & van Loon2007; Wood 2015), the impact of this blueshifting onbroadband photometry has not been quantified.Jimenez et al. (2020) also consider planetary engulf-ment, wherein an RGB star consumes one or more plan-ets in close orbit as it expands. They predict that theincreased turbulence in the star’s convective envelope,corresponding to an increase in mixing length, may re-sult in a net decrease in TRGB luminosity by up to 5%for a single star that has consumed a giant planet. How-ever, they conclude that both detailed hydrodynamicalsimulations and further studies of planetary system for-mation are required to accurately constrain potentialimpacts of this phenomenon on the TRGB as a distanceindicator.Again, we expect that these effects are overall wellwithin the uncertainties of this work, but may need tobe taken under consideration in future high-precisionTRGB studies. CONCLUSIONS AND FUTURE WORK6.1.
Conclusions
We have developed a method to measure TRGB mag-nitudes and colors in multiple filters simultaneously.This method, MCR-TRGB, was designed to use a setof likely RGB stars, which were defined where tradi-tional TRGB-detection methods using edge-detectioncan be employed reliably, to study the multi-wavelengthbehavior of the TRGB using those same stars. Weapplied MCR-TRGB to a re-reduction of optical+NIR
CR-TRGB: A Multiwavelength TRGB Measurement Method HST data originally presented in D12; these new reduc-tions use the optical observations, which have higherspatial resolution and are generally more complete atthe TRGB, to produce more complete and precise pho-tometry in the infrared-bands. When using the samedistances as D12, we find only minor adjustments tothe color-magnitude behavior of the IR-TRGB. How-ever, the D12 absolute magnitudes were determined rel-ative to color-magnitude predictions from stellar models.Thus, we compared three different absolute-magnitudecalibrations of the measured TRGB magnitudes, one us-ing the same distance moduli as in D12, and two usingdistance moduli derived from the predicted TRGB abso-lute magnitudes from two commonly used isochrone sets(PARSEC and MIST). We find that the isochrone-basedabsolute calibrations are inconsistent with each other atthe ∼ . T eff scale arelikely to explain a large part of the inconsistences wehave found. 6.2. Future work
An empirical absolute TRGB calibration in WFC3/IRbandpasses remains elusive. A fully model-independentcalibration, as in Jang & Lee (2017b), is clearly nec-essary. However, there are a limited number of sys-tems that are distant enough that their apparent TRGBmagnitudes are easily observable with
HST , but nearbyenough to have distances that are well-constrained byother means.Another limitation of our study is the lack of preciseindependent distances to the galaxies in this work. In-deed, the majority of these systems only have distancesdetermined from the TRGB itself. Some of these galax-ies are within a volume for variable-star based distanceswith
HST , though we are cautious about their precisiongiven the metallicity dependence of such relations and the difficulty of inferring stellar metallicities for galaxiesat these distances (for RR Lyrae see Beaton et al. 2018).Our initial goal in this work was to relate empiricalresults on the multi-wavelength TRGB to the physi-cal characteristics of the underlying stellar populations,such as age and metallicity. We found that goal chal-lenging due to the internal mismatches we observe inthe isochrone sets, given that the aforementioned phys-ical parameters are ultimately inferred via comparisonto those from isochrone sets once a distance, also oftenisochrone-dependent, is assumed.ACKNOWLEDGMENTSWe gratefully acknowledge the anonymous referee,Emily Levesque, ˇZeljko Ivezi´c, Anil Seth, and Evan Skill-man for helpful feedback on this manuscript, as well asJJ Eldridge, Olivia Jones, and several members of theCarnegie-Chicago Hubble Program team for illuminat-ing discussions. We also thank the GalRead group atPrinceton University, in particular Andy Goulding, fordirecting us to Extreme Deconvolution.We acknowledge the people of the Dkhw’Duw’Absh,the Duwamish Tribe, the Muckleshoot Tribe, theLenape, and other tribes on whose traditional lands wehave performed this work.Support for this work was provided by NASA throughgrant
Facilities:
HST(ACS/WFC), HST(WFC3/IR)
Software:
AstroML (VanderPlas et al. 2012, 2014),Astropy (Astropy Collaboration et al. 2013, 2018), As-troquery (Ginsburg et al. 2017, 2019), Dask (Rock-lin 2015; Dask Development Team 2016), DOLPHOT(Dolphin 2000, 2016), Drizzlepac (STSCI DevelopmentTeam 2012; Hack et al. 2013; Avila et al. 2015), KDEpy(Odland 2018), Matplotlib (Hunter 2007), NumPy (vander Walt et al. 2011), Pandas (McKinney 2010, 2011),Seaborn (Waskom et al. 2018), SciPy (Jones et al. 2001),Scikit-learn (Pedregosa et al. 2011), SEP (Barbary 2016,2018), Vaex (Breddels & Veljanoski 2018a,b)6
Durbin et al.
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CR-TRGB: A Multiwavelength TRGB Measurement Method Durbin et al.
CR-TRGB: A Multiwavelength TRGB Measurement Method A. TESTS ON ARTIFICIAL DATAIn this section, we diagnose potential biases and sys-tematics induced by our technique by applying theabove methods to simulated CMDs with known theo-retical TRGB magnitudes. We use the results of thisanalysis to determine bias corrections to our final tipmagnitudes and to refine the uncertainties on our mea-surements.We first generate a set of idealized (i.e., error-free)photometry of artificial RGB sequences with MATCHbased on the PARSEC model suite. Serenelli et al.(2017) demonstrated that metallicity is the primarydriver of variation in TRGB colors and magnitudes forold ages, so we hold all parameters except metallicityconstant. We use a Chabrier IMF with a slope of 1.3,a binary fraction of 0.3, and a constant star formationrate with an age range of 100 Myr to 14 Gyr. We varymetallicity between − . < [Fe/H] < − . ∼ ∼ − . < F814W < − . − . < F110W < − . − . < F160W < − . N T +1 (cid:63) ) in the filter used to measure the Sobel edge. Wealso adjust all magnitudes by a value randomly gener- ated from a Gaussian with a mean of 0 mag and standarddeviation set to the star’s photometric uncertainty.A.1. Luminosity function sampling
Here we test our method against N T +1 (cid:63) over a rangeof 200 ≤ N T +1 (cid:63) ≤ N T +1 (cid:63) values for the majority of the galaxies in our sample.At each N T +1 (cid:63) we run 20 end-to-end XDGMM tip fit-ting iterations and calculate the offsets ∆ M TRGB ≡ M TRGB (measured) – M TRGB (true) for each filter. Werepeat these tests at four different metallicities ([Fe/H = {− . , − . , − . , − . } dex) to check for possible color-dependent effects.Figure 18 shows the median per-filter differences inthe measured versus theoretical TRGB values against N T +1 (cid:63) for each of the four metallicities. The error barson the points in Figure 18 show the interquartile range ofthe results. The dashed horizontal lines are color-codedto match the filter and show the range of the per-filtermean uncertainty on the tip-fitting results; we definethe uncertainty as the quadrature sum of the XDGMMfitting uncertainty and the median photometric error ofthe tip stars.For all but [Fe/H] = − . M TRGB start out around 0.04mag in the most undersampled case, increase approxi-mately with N T +1 (cid:63) for N T +1 (cid:63) (cid:46) M TRGB ∼ .
01 mag.These results are broadly similar to what is seen for tra-ditional edge detection methods. Madore et al. (2009)found that Sobel edge detection is prone to bias whenthe RGB luminosity function is undersampled, and thata sample of N T +1 (cid:63) (cid:38)
500 stars is required for edge de-tection to function accurately.For [Fe/H] = − . Metallicity
For each metallicity in our artificial dataset, we run20 end-to-end tip fitting iterations with N T +1 (cid:63) = 4000stars, in the regime where sampling effects are minimal.Figure 19 shows the median per-filter differences in the0 Durbin et al. − . . . . . . ∆ M T R G B ( m e a s u r e d − p r e d i c t e d ) [Fe/H] = − . − . F814WF110WF160W ± σ ± σ ± σ
200 500 1000 2000 5000 N ? < M TRGB − . . . . . . ∆ M T R G B ( m e a s u r e d − p r e d i c t e d ) [Fe/H] = − .
200 500 1000 2000 5000 N ? < M TRGB [Fe/H] = − . Figure 18.
Differences between measured and theoretical TRGB magnitude versus N T +1 (cid:63) (the number of stars within 1 magfainter of the TRGB), for [Fe/H] = − . − . − . − . σ TRGB uncertainty, with the exception of poorly populated CMDs ( N T +1 (cid:63) < ∼
500 stars). measured versus the theoretical TRGB values againstthe measured IR-TRGB color; the IR-TRGB color in-creases approximately monotonically with metallicity inthe artificial data.The jump in offset values at F110W–F160W > Photometric uncertainties
To isolate the impact of photometric uncertainties onour TRGB-measurements, we use the artifical datasetwith N T +1 (cid:63) = 2000 stars and [Fe/H] = − . XDGMM vs. Sobel edge detection
Here we investigate the behavior of XDGMM tip fit-ting relative to the standard method of Sobel edge de-tection using the same set of trials as in subsection A.1.As our method for XDGMM tip fitting itself uses So-bel edge detection to set the color-magnitude center of
CR-TRGB: A Multiwavelength TRGB Measurement Method .
70 0 .
75 0 .
80 0 .
85 0 .
90 0 .
95 1 . F110W − F160W (measured) − . − . . . . . ∆ M T R G B ( m e a s u r e d - p r e d i c t e d ) F814WF110WF160W ± σ ± σ ± σ Figure 19.
The difference between measured and theoreti-cal TRGB values against the median measured NIR tip colorfor the ensemble of simulated datasets. Results for F814W,F110W, and F160W are shown in blue, orange, and green,respectively. The horizontal lines show ± σ , where σ is themean quadrature sum of the photometric and fitting errorsfor each filter This bias is within the TRGB detection uncer-tainty in almost all cases. − . − .
01 0 .
00 0 .
01 0 .
02 0 . ∆ M TRGB (measured - predicted) N (tr i a l s ) F814WF110WF160W
Figure 20.
Histograms of the difference between the mea-sured and predicted M TRGB for the with N T +1 (cid:63) = 2000 starsand [Fe/H] = − . the initial tip star selection window, we can make afully self-consistent comparison of the Sobel edge magni-tudes to the XDGMM mean magnitudes for detectionsin F814W and F110W. (We do not perform edge detec-tion on F160W in our method and so do not make thecomparison.)The top panel of Figure 21 shows the relation betweenthe median difference between the XDGMM-fitted mean .
00 0 .
02 0 . ∆ M TRGB , Sobel − . . . . . . . ∆ M T R G B , X D G MM Linear fit(68% confidence) y = x .
05 0 .
10 0 .
15 0 . Width of tip star selection region ∆ η − . . . . . ∆ M T R G B , X D G MM − S o b e l Linear fit(95% confidence)
Figure 21.
Top: The median difference between XDGMM-fitted and theoretical tip magnitudes versus the median dif-ference between the Sobel edge magnitude and theoreticaltip magnitude in either F814W or F110W. Each point is themedian result for one set of trials with fixed metallicity and N T +1 (cid:63) . The solid line and flanking filled region show a linearfit to the data and its 68% confidence interval, whereas thedashed line shows a one-to-one relation. Bottom: Differencebetween the XDGMM-fitted mean and Sobel edge magni-tude versus ∆ η , which is the width in magnitudes of the tipstar selection region in the luminosity function. Each pointrepresents the offset for a single trial with fixed metallicityand N T +1 (cid:63) . The solid line and flanking filled region showa linear fit to the data and its a 95% confidence interval,respectively. and the theoretical tip versus the median difference be-tween the Sobel edge and theoretical tip. The linear fitto the data is consistent within 1 σ with a one-to-onerelation, indicating that the methods produce overallconsistent tip mangitudes.The bottom panel of Figure 21 shows the relation be-tween the width of the tip star selection region, ∆ η ,and the difference between the XDGMM- and Sobel-2 Durbin et al. derived tip magnitudes, ∆ M TRGB . The quantities arecorrelated, albeit with some scatter on the order of 0.01mag, and are fit by the linear relation ∆ M X − STRGB =0 . η ) − .
01 mag.A.5.
Adjustments to measurements
In the previous subsections, a number of tests wereperformed to quantify the statistical and systematic un-certainties of the MCR method using artificial photom-etry. Here we match our observed galaxies to their ar-tificial tests to determine both systematic terms thatare applied in the form of bias corrections and statisti-cal terms that are applied in the form of inflating thealgorithmic uncertainties. The corrections are parame-terized by two key observables: (i) how well populatedthe RGB is as a proxy for the total mass, and (ii) theF814W–F160W color as a proxy for the underlying stel-lar population properties. All such adjustments are sum-marized in Table 6 and if a given target does not appearin the table, then it did not require a modification.
Table 6.
Applied bias corrections
Value( ± error) subtracted from m TRGB
Target name F814W F110W F160WKDG73 0 . ± .
02 0 . ± .
02 0 . ± . . ± .
02 0 . ± .
02 0 . ± . . ± .
02 0 . ± .
02 0 . ± . . ± .
02 0 . ± .
02 0 . ± . . ± . . ± . . ± . . ± . . ± . − . ± . − . ± . For each target, we determine the most appropriatesets of tests to use to determine the bias based on N T +1 (cid:63) and F814W–F160W color. The quoted adjustment val-ues are adopted from the relevant set of trials, with un-certainties determined as the median and interquartilerange of the offsets (measured – predicted value) in eachfilter. We subtract the offsets from the measured TRGBapparent magnitude and add the associated uncertaintyin quadrature to the fitting uncertainty. We also modifyall relevant colors based on these adjustments.The first four targets in Table 6 (KDG73, NGC2403-HALO-6, SCL-DE1, and UGCA292) all have N T +1 (cid:63) <
500 stars. (Although these targets do not all have thesame colors, we found that differences between offsetswere negligible at the relevant colors.) For these wetake the median and interquartile range of offsets for alltrials with N T +1 (cid:63) <
500 stars and [Fe / H] ≤ − . − F160W ∼ − F160W ∼ .
25 mag.We match these colors by adopting the median andinterquartile range of offsets for trials with − . ≤ [Fe / H] ≤ − . − . ≤ [Fe / H] ≤ − ..