Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Parada Torres, Javiera Fernanda
Doctor of Philosophy - PhD
University of British Columbia
This thesis introduces two new extragalactic distance determination methods; the first uses the median magnitude of carbon-rich asymptotic giant branch stars (CS), while the second uses the combined luminosity function of the red-giant branch (RGB) and oxygen-rich asymptotic giant branch stars (AGB). The sample of CS and RGB+AGB stars are selected from near-infrared JHKs-bands colour-magnitude diagrams (CMD), as in these filters RGs are bright and easy to identify. Both methods use the Magellanic Clouds (MC) as the fundamental calibrators and are tested in four Magellanic type galaxies: NGC 6822, IC 1613, WLM and NGC 3109, these target galaxies are all members of the Local Group. For the CS method, the CS J-band luminosity function is fitted using a Lorentzian distribution modified to allow the distribution to be asymmetric. The parameters of the best-fit distribution are then used to determine if the CS luminosity function of a given galaxy resembles that of the Large or Small MC (LMC or SMC). Based on this resemblance, either the LMC or SMC is used as the calibrator and the distance to the given galaxy is estimated using the median J-magnitude of the CS samples. The second method uses an un-binned maximum likelihood estimator to find the distance modulus that minimizes the difference between the luminosity function of the RGB+AGB stars in a target galaxy and the model distribution given by the luminosity function of the RGB+AGB stars in the MC. The model luminosity function can be given by the LMC and SMC individually or as a linear combination (LC) of both. The LC includes a "shape'' parameter that quantifies how "LMC-'' or "SMC-like'' a target galaxy is. Except for the NGC 3109 Ks luminosity function, the LC "shape'' results agree with the CS "LMC/SMC-like'' classification. Estimations of the distances through the tip of the RGB method are also included to test the performance and compare the results when the three different methods are applied to the same data set. The distance estimates for the target galaxies from the three different methods presented in this thesis are in good agreement within the error bars.
Red giant stars as standard candlesbyJaviera Fernanda Parada TorresBSc., Universidad de Concepcion, 2012MSc., University of British Columbia, 2016a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoral studies(Astronomy)The University of British Columbia(Vancouver)July 2022© Javiera Fernanda Parada Torres, 2022The following individuals certify that they have read, and recommend to the Facultyof Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Red giant stars as standard candlessubmitted by Javiera Fernanda Parada Torres in partial fulfillment of the re-quirements for the degree of Doctor of Philosophy in Astronomy.Examining Committee:Harvey Richer, Professor, Physics & Astronomy, University of British ColumbiaSupervisorJeremy Heyl, Professor, Physics & Astronomy, University of British ColumbiaCo-supervisorPaul Hickson, Professor, Physics & Astronomy, University of British ColumbiaUniversity ExaminerGrenfell Patey, Professor, Chemistry University of British ColumbiaUniversity ExaminerWendy Freedman, Professor, Astronomy & Astrophysics, University of ChicagoExternal ExaminerAdditional Supervisory Committee Members:Ingrid Stairs, Professor, Physics & Astronomy, University of British ColumbiaSupervisory Committee MemberChris Waltham, Professor, Physics & Astronomy, University of British ColumbiaSupervisory Committee MemberiiAbstractThis thesis introduces two new extragalactic distance determination methods; thefirst uses the median magnitude of carbon-rich asymptotic giant branch stars (CS),while the second uses the combined luminosity function of the red-giant branch(RGB) and oxygen-rich asymptotic giant branch stars (AGB). The sample of CS andRGB+AGB stars are selected from near-infrared 𝐽𝐻𝐾𝑠-bands colour-magnitudediagrams (CMD), as in these filters RGs are bright and easy to identify. Bothmethods use the Magellanic Clouds (MC) as the fundamental calibrators and aretested in fourMagellanic type galaxies: NGC 6822, IC 1613, WLM andNGC 3109,these target galaxies are all members of the Local Group.For the CS method, the CS 𝐽-band luminosity function is fitted using aLorentzian distribution modified to allow the distribution to be asymmetric. Theparameters of the best-fit distribution are then used to determine if the CS luminos-ity function of a given galaxy resembles that of the Large or Small MC (LMC orSMC). Based on this resemblance, either the LMC or SMC is used as the calibratorand the distance to the given galaxy is estimated using the median 𝐽-magnitude ofthe CS samples.The second method uses an un-binned maximum likelihood estimator to findthe distance modulus that minimizes the difference between the luminosity functionof the RGB+AGB stars in a target galaxy and the model distribution given by theluminosity function of the RGB+AGB stars in the MC. The model luminosityfunction can be given by the LMC and SMC individually or as a linear combination(LC) of both. The LC includes a “shape” parameter that quantifies how “LMC-” or“SMC-like” a target galaxy is. Except for the NGC 3109 𝐾𝑠 luminosity function,the LC “shape” results agree with the CS “LMC/SMC-like” classification.iiiEstimations of the distances through the tip of the RGBmethod are also includedto test the performance and compare the results when the three different methodsare applied to the same data set. The distance estimates for the target galaxies fromthe three different methods presented in this thesis are in good agreement withinthe error bars.ivLay SummaryMeasuring distances to objects outside our own Galaxy is a crucial and challengingfield of study in astronomy. When objects are too far for direct distancemeasurement(e.g. using a geometric technique), astronomers rely on standard candles. Standardcandles are objects of known brightness that serve as stellar yardsticks. If one ofthese is further away it will appear fainter and knowing that they should be the samebrightness allows you to work out the distance between them. Different types ofstars can be employed as standard candles, as long as their intrinsic brightness isknown. In this thesis two new distance determination methods are developed usingstars that are more evolved than the Sun as standard candles. These stars are brightand red, making them easy to identify and characterize in galaxies well beyond theconfines of our own Milky Way galaxy.vPrefaceThe research program followed to complete this thesis was developed by my super-visor Harvey Richer and myself, the author Javiera Parada (J.P.).In Chapter 2, the original photometric catalogues for the Magellanic Cloudswere developed by Paul Ripoche in [210] of which I am a co-author. For thisthesis these catalogues were updated to reflect the latest available data. The rawWIRCAm images for NGC 6822, IC 1613 and WLM were each pre-processed andstacked into a co-added image by Laurie Rousseau-Nepton. The raw images forNGC 3109 are publicly available through the ESO Science Archive Facility, allpre-processing steps were done by the author J.P. The photometry and calibrationfor the four data sets just mentioned were perform by the author J.P. The IC 1613Sibbons et al. photometric catalogue is publicly available through VizieR and onlycolour transformations had to be applied by the author J.P.The CS colour-selection range the robust median estimator (eq. 3.2) werederived in Ripoche et al. [210] of which I am a co-author.Parts of Chapters 3 and 6 are adapted from the published paper Parada et al.[186] [“Carbon stars as standard candles: II. The median 𝐽 magnitude as a distanceindicator.” Monthly Notices of the Royal Astronomical Society, 501, 933-947].The co-authors H. Richer, J. Heyl, P. Ripoche, and L. Rousseau-Nepton contributedto manuscript edits.Parts of Chapter 3 and 4 are adapted from the submitted manuscript [“Carbonstars as standard candles - III. Un-binned maximum likelihood fitting and compar-ison with TRGB estimations”]. The co-authors H. Richer, J. Heyl, P. Ripoche, andL. Rousseau-Nepton contributed to manuscript edits.The different carbon stars Milky Way catalogues used in Chapter 7 were con-vistructed in collaboration with P. Ripoche.Everything other than the above is an original, unpublished, independent workby the author J.P.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The local distance ladder . . . . . . . . . . . . . . . . . . . . . . 41.2 Colour, magnitudes and the distance modulus . . . . . . . . . . . 51.3 Why in the near-infrared? . . . . . . . . . . . . . . . . . . . . . . 71.4 How to make a red giant star: a brief summary of stellar evolution 91.5 The TRGB as a distance indicator . . . . . . . . . . . . . . . . . 111.6 Carbon stars as a distance indicator . . . . . . . . . . . . . . . . . 132 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17viii2.1 The Magellanic Clouds . . . . . . . . . . . . . . . . . . . . . . . 172.2 NGC 6822, IC 1613 and WLM . . . . . . . . . . . . . . . . . . . 202.2.1 Observations . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Photometry . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.4 Colour-magnitude diagram cleaning . . . . . . . . . . . . 272.3 NGC 3109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.1 Observations . . . . . . . . . . . . . . . . . . . . . . . . 302.3.2 Image reduction and photometry . . . . . . . . . . . . . . 312.3.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 322.4 IC 1613 (Sibbons et al. data) . . . . . . . . . . . . . . . . . . . . 333 Carbon stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.1 The Magellanic Clouds . . . . . . . . . . . . . . . . . . . 373.1.2 The target galaxies . . . . . . . . . . . . . . . . . . . . . 373.2 The median J magnitude and possible sources of uncertainties . . 393.3 The carbon star luminosity function . . . . . . . . . . . . . . . . 433.4 Metallicity and star formation history effects on the luminosityfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.4.1 Metallicity . . . . . . . . . . . . . . . . . . . . . . . . . 513.4.2 Star formation history . . . . . . . . . . . . . . . . . . . 543.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 563.5 Distance determination . . . . . . . . . . . . . . . . . . . . . . . 573.5.1 Determining the most suitable calibrator . . . . . . . . . . 573.5.2 The distance modulus . . . . . . . . . . . . . . . . . . . 583.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.6.1 NGC 6822 . . . . . . . . . . . . . . . . . . . . . . . . . 593.6.2 IC 1613 . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.6.3 WLM & NGC 3109 . . . . . . . . . . . . . . . . . . . . 623.7 The median vs the mean J magnitude . . . . . . . . . . . . . . . 633.8 Suitable versus less suitable calibrator . . . . . . . . . . . . . . . 66ix4 The tip of the red giant branch . . . . . . . . . . . . . . . . . . . . . 694.1 Red giant selection . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 TRGB detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 735 Red giants, a maximum likelihood method . . . . . . . . . . . . . . . 775.1 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2 Cumulative distribution luminosity function . . . . . . . . . . . . 805.3 The likelihood function . . . . . . . . . . . . . . . . . . . . . . . 815.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.5 A test of the techniques . . . . . . . . . . . . . . . . . . . . . . . 876 Three-dimensional structure of the Magellanic Clouds . . . . . . . . 896.1 The Large Magellanic Cloud . . . . . . . . . . . . . . . . . . . . 906.2 The Small Magellanic Cloud . . . . . . . . . . . . . . . . . . . . 1016.3 Crowding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 Reddening Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.1 Reddening maps and reddening factors . . . . . . . . . . . . . . . 1097.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.2.1 Carbon stars . . . . . . . . . . . . . . . . . . . . . . . . 1117.2.2 The tip of the red giant branch . . . . . . . . . . . . . . . 1137.2.3 LMC 3D-structure . . . . . . . . . . . . . . . . . . . . . 1167.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228 Carbon stars in the Milky Way . . . . . . . . . . . . . . . . . . . . . 1248.1 Catalogues of Galactic carbon stars . . . . . . . . . . . . . . . . . 1258.2 Standardising the catalogues . . . . . . . . . . . . . . . . . . . . 1268.3 Photometry and astrometry quality parameters . . . . . . . . . . . 1298.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 1348.4.1 Individual catalogues . . . . . . . . . . . . . . . . . . . . 1368.4.2 Master catalogue . . . . . . . . . . . . . . . . . . . . . . 1388.4.3 Grouped by spectral type . . . . . . . . . . . . . . . . . . 139x8.4.4 Lee et al. combined catalogue . . . . . . . . . . . . . . . 1408.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1449 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1469.1 Distance comparisons . . . . . . . . . . . . . . . . . . . . . . . . 1469.1.1 NGC 6822 . . . . . . . . . . . . . . . . . . . . . . . . . 1479.1.2 IC 1613 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1519.1.3 WLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1549.1.4 NGC 3109 . . . . . . . . . . . . . . . . . . . . . . . . . 1579.2 Metallicity effects on the distance estimations . . . . . . . . . . . 15810 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . 16210.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16210.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170xiList of TablesTable 2.1 WIRCam near-infrared observations . . . . . . . . . . . . . . 21Table 2.2 Zero-points for photometry calibration . . . . . . . . . . . . . 25Table 2.3 ISAAC near-infrared observations . . . . . . . . . . . . . . . . 32Table 3.1 Modified Lorentzian model best-fitting parameters and median𝐽-band magnitudes. . . . . . . . . . . . . . . . . . . . . . . . 47Table 4.1 TRGB absolute 𝑇 magnitudes for the Magellanic Clouds . . . . 72Table 4.2 TRGB apparent 𝑇 magnitudes for the target galaxies . . . . . . 72Table 4.3 Comparing distance moduli from the CS and TRGB methods . 76Table 5.1 Number of RG stars . . . . . . . . . . . . . . . . . . . . . . . 80Table 5.2 RG-MLE distance moduli using the LMC and SMC as singlecalibrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Table 5.3 RG-MLE distance moduli from the linear combination fittingtechnique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Table 6.1 LMC 3D structure analysis fields F1 to F9 parameters . . . . . 92Table 6.2 LMC 3D structure analysis fields F1 to F25 parameters . . . . 98Table 6.3 SMC 3D structure analysis fields F1 to F9 parameters . . . . . 102Table 7.1 Mean reddening values for the LMC and SMC . . . . . . . . . 111Table 7.2 Extinction coefficients for the LMC and SMC . . . . . . . . . 111Table 7.3 CS median 𝑀𝐽 magnitude for the LMC and SMC obtained withthe data corrected using different reddening maps . . . . . . . 114xiiTable 7.4 TRGB 𝐾𝑠 magnitude for the LMC and SMC obtained with thedata corrected using different reddening maps . . . . . . . . . 117Table 7.5 LMC 3D structure analysis fields F1 to F25 parameters . . . . 120Table 7.6 Target galaxies distance moduli when different reddening mapsare applied to the Magellanic Clouds . . . . . . . . . . . . . . 123Table 8.1 Number of stars in the Milky Way catalogues . . . . . . . . . . 127Table 8.2 Reddening factors in the Milky Way . . . . . . . . . . . . . . 129Table 8.3 Number of sources rejected with different Gaia astrometric so-lution goodness-of-fit selection criteria . . . . . . . . . . . . . 134Table 8.4 𝑀 𝐽 and 𝜎 for the individual Milky Way catalogues . . . . . . 138Table 8.5 𝑀 𝐽 and 𝜎 for the master Milky Way catalogue . . . . . . . . . 139Table 8.6 𝑀 𝐽 and 𝜎 for the CS of spectral type C-N . . . . . . . . . . . 140Table 8.7 𝑀 𝐽 and 𝜎 for the catalogue used by Lee et al. . . . . . . . . . 143Table 9.1 Distance moduli from this thesis . . . . . . . . . . . . . . . . 147Table 9.2 Independent distance estimations to NGC 6822 . . . . . . . . . 148Table 9.3 Independent distance estimations to IC 1613 . . . . . . . . . . 153Table 9.4 Independent distance estimations to WLM . . . . . . . . . . . 156Table 9.5 Independent distance estimations to NGC 3109 . . . . . . . . . 157xiiiList of FiguresFigure 1.1 Distance ladder . . . . . . . . . . . . . . . . . . . . . . . . . 5Figure 1.2 Stellar populations in the ((𝐽 −𝐾𝑠)0, 𝐽0) CMD. . . . . . . . . 7Figure 2.1 Cleaned and deredden ((𝐽−𝐾𝑠)0, 𝑀𝐽 ) CMDs for the LMC andSMC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Figure 2.2 SHARP and ROUND shape statistics from DAOPHOT routineFIND for WLM . . . . . . . . . . . . . . . . . . . . . . . . . 23Figure 2.3 𝐽𝐻𝐾𝑠 filters transmission profiles for 2MASS, WIRCam, WF-CAM and ISAAC instruments . . . . . . . . . . . . . . . . . 24Figure 2.4 Difference in magnitudes between the DAOPHOT photometryand their 2MASS counterparts for zero-point calibration . . . 26Figure 2.5 Calibrated and reddening corrected ((𝐽 − 𝐾𝑠)0, 𝐽0) colour-magnitude diagrams of NGC 6822, IC 1613 and WLM . . . . 27Figure 2.6 Using photometry statistics SHARP and CHI for CMD cleaning 29Figure 2.7 Calibrated, reddening corrected and cleaned ((𝐽 −𝐾𝑠)0, 𝐽0) ofNGC 6822, IC 1613 and WLM . . . . . . . . . . . . . . . . . 31Figure 2.8 Calibrated, reddening corrected and cleaned ((𝐽 −𝐾𝑠)0, 𝐽0)CMD of NGC 3109 and IC 1613 (Sibbons et al. data). . . . . 34Figure 3.1 LMC and SMC spectroscopic CS catalogues (𝐽 −𝐾𝑠)0, 𝑀𝐽CMDs plotted over the 2MASS CMDs. . . . . . . . . . . . . 36Figure 3.2 RA and DEC maps for the LMC and SMC . . . . . . . . . . . 37Figure 3.3 ((𝐽 −𝐾𝑠)0, 𝐽0) CMDs and CS distributions for NGC 6822, IC1613 (WIRCam), WLM, and NGC 3109 . . . . . . . . . . . . 38xivFigure 3.4 RA and DEC maps for NGC 6822 and WLM . . . . . . . . . 39Figure 3.5 RA and DEC map for IC 1613 and NGC 3109 . . . . . . . . . 40Figure 3.6 CMD with COLIBRI isochrones for TP-AGB . . . . . . . . . 42Figure 3.7 LMC and SMC CS luminosity functions; a comparison be-tween Gaussian and Lorentzian models. . . . . . . . . . . . . 44Figure 3.8 Best-fit modified Lorentzian function for the CS luminosityfunctions for the Magellanic Clouds . . . . . . . . . . . . . . 46Figure 3.9 Best-fit modified Lorentzian function for the CS luminosityfunctions for the target galaxies . . . . . . . . . . . . . . . . 49Figure 3.10 Correlations amongmodifiedLorentzianmodel parameters and𝑀 𝐽 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 3.11 CS luminosity functions of the target galaxies and their mostsuitable calibrator with distance estimations results. . . . . . . 61Figure 3.12 Faint magnitude selection effects on 𝑀 𝐽 and ⟨𝑀𝐽⟩ . . . . . . 64Figure 3.13 Magnitude selection effects on 𝑀 𝐽 and ⟨𝑀𝐽⟩ . . . . . . . . . 66Figure 3.14 Suitable versus less suitable distance calibrator. . . . . . . . . 68Figure 4.1 ((𝐽 −𝐾𝑠)0, 𝑇𝑀𝐾𝑠 ) CMD for the LMC and the SMC . . . . . . 70Figure 4.2 ((𝐽 −𝐾𝑠)0, 𝑇(𝐾𝑠)0) CMD for the target galaxies . . . . . . . . 71Figure 4.3 𝑇𝑀𝐾𝑠 red giant luminosity function for the Magellanic Clouds 73Figure 4.4 Red giants 𝑇(𝐾𝑠)0 luminosity functions for the target galaxies . 75Figure 5.1 ((𝐽 −𝐾𝑠)0, 𝑀𝐾𝑠 ) CMD for the Magellanic Clouds . . . . . . . 78Figure 5.2 ((𝐽 −𝐾𝑠)0, (𝐾𝑠)0) CMD for the target galaxies . . . . . . . . 79Figure 5.3 RGs 𝑀𝐾𝑠 luminosity functions for the Magellanic Clouds . . . 81Figure 5.4 RGs 𝐾𝑠 luminosity functions for the target galaxies . . . . . . 85Figure 5.5 Differential distance moduli for LMC RGs sub-samples . . . . 88Figure 6.1 RA and DEC map for the LMC and SMC with the location offields F1 to F9 . . . . . . . . . . . . . . . . . . . . . . . . . 91Figure 6.2 RA and DEC map for the LMC with the location of fields F1to F25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Figure 6.3 CS luminosity function for LMC fields F1 to F25 . . . . . . . 95xvFigure 6.4 RA and DEC map for the CS in the LMC with differentialdistance moduli . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 6.5 ((𝐽 −𝐾𝑠)0, 𝑇(𝐾𝑠)0) CMDs for the 25 different fields of the LMC 97Figure 6.6 Δ𝑀 𝐽 versus stellar density . . . . . . . . . . . . . . . . . . . 105Figure 7.1 Reddening map for the Magellanic Clouds . . . . . . . . . . . 110Figure 7.2 LMC ((𝐽 −𝐾𝑠)0, 𝑀𝐾𝑠 ) CMDs from data corrected using dif-ferent reddening maps. . . . . . . . . . . . . . . . . . . . . . 113Figure 7.3 CS luminosity functions from data corrected using differentreddening maps. . . . . . . . . . . . . . . . . . . . . . . . . 114Figure 7.4 LMC ((𝐽 −𝐾𝑠)0, 𝑇𝑀𝐾𝑠 ) CMDs from data corrected using dif-ferent reddening maps. . . . . . . . . . . . . . . . . . . . . . 115Figure 7.5 LMCRGs𝑇𝑀𝐾𝑠 luminosity functions fromdata corrected usingdifferent reddening maps. . . . . . . . . . . . . . . . . . . . . 116Figure 7.6 SMC RGs luminosity functions from data corrected using dif-ferent reddening maps. . . . . . . . . . . . . . . . . . . . . . 117Figure 7.7 TRGB bootstrap results for the LMCwhen using Skowron et al.[226] reddening maps . . . . . . . . . . . . . . . . . . . . . . 118Figure 8.1 CMD of the Milky Way catalogues . . . . . . . . . . . . . . . 130Figure 8.2 Milky Way CMDs with different 2MASS parameter selection 132Figure 8.3 Milky Way CMDs with different Gaia parameters selection . . 135Figure 8.4 Luminosity functions for the Milky Way CS catalogues . . . . 137Figure 8.5 CMD of the Milky Way CS of different spectral type . . . . . 141Figure 8.6 Luminosity function of the Milky Way CS of different spectraltypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142Figure 9.1 Independent distance estimations to NGC 6822 against year ofpublication. . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Figure 9.2 Independent distance estimations to IC 1613 against year ofpublication. . . . . . . . . . . . . . . . . . . . . . . . . . . . 154Figure 9.3 Independent distance estimations to WLM against year of pub-lication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155xviFigure 9.4 Independent distance estimations to NGC 3109 against year ofpublication. . . . . . . . . . . . . . . . . . . . . . . . . . . . 158Figure 10.1 Filters relative throughput for 2MASS and JWST/NIRCam . . 169xviiList of Symbols𝐻0 Hubble constant𝐽0 median apparent 𝐽-band magnitude, of the CS.⟨𝐽0⟩ mean apparent 𝐽-band magnitude, of the CS.𝑀 𝐽 median absolute 𝐽-band magnitude, of the CS.⟨𝑀𝐽⟩ mean absolute 𝐽-band magnitude, of the CS.M⊙ solar mass (1 M⊙ = 2×1030 kg)𝑀TRGB𝐾𝑠 absolute 𝐾𝑠-band magnitude of the TRGB𝜇0 distance modulus𝜇𝛼 proper motion in right ascension𝜇𝛿 proper motion in declination𝜛 parallaxxviiiList of AbbreviationsAGB asymptotic giant branchAMR age-metallicity relationCDF cumulative distribution functionCFHT Canada-France-Hawaii TelescopeCMD colour-magnitude diagramCS carbon starsCTIO Cerro Tololo International ObservatoryDEC declinationESA European Space AgencyESO European Southern ObservatoryFWHM full width at half maximumIRAS Infrared Astronomical SatelliteISAAC Infrared Spectrometer And Array CameraJWST James Webb Space TelescopeLMC Large Magellanic CloudMLE maximum likelihood estimatorMS Main SequenceNLLSQ non-linear least squaresPDF probability density functionxixPSF point-spread functionRA right ascensionRG red giantRGB red giant branchruwe renormalized unit weight errorSFH star formation historySFR star formation rateSMC Small Magellanic CloudSN Ia Type Ia supernovaeTP-AGB thermally pulsing AGBTRGB tip of the red giant branchUKIRT United Kingdom Infrared TelescopeVLT Very Large TelescopeWFCAM Wide Field CAMeraWIRCam Wide-field InfraRed CameraWLM Wolf-Lundmark-Melotte,one of the target galaxies.xxAcknowledgmentsI would like to express my deepest gratitude to my supervisors Prof. Harvey Richerand Prof. Jeremy Heyl for the help, guidance and patience throughout my time hereat UBC.I would also like to thank my family: my parents and my brothers, for theirunconditional love and motivation. My PhD would’ve not been successfully com-pleted without their support and encouragement.To my friends, old and new, you definitely make life more fun and grad schoolbearable. I really hope to see everyone again wherever this might be.Finally, I would like to thank my partner, Marta. Every time I was ready to quit,you did not let me and I am forever grateful. Despite (for some time both of us)going through grad school you managed to put aside the stress and make these past5 years memorable and amazing. This thesis is a testament to your unconditionallove and encouragement.xxiChapter 1IntroductionStandard candles are objects with a specific, known, or calibrated brightness usedto measure distances by comparing the observed apparent brightness of the objectto the intrinsic luminosity of the standard candle. This thesis explores the standardcandle aspect of red giant (RG) stars and develops a number of different methodsto estimate extragalactic distances using these stars.Measuring distances to objects beyond the boundaries of our Galaxy has beena crucial and challenging field of study. Even though galaxies were first cataloguednear the end of the 18th century by the French astronomer CharlesMessier, it was notuntil the 1920’s that the distance to these extragalactic objects was first measured.In fact, galaxies were first labeled by Messier as “spiral nebulae” and were thoughtto be clouds of gas within our own galaxy. With time, a controversial debate ofwhether these “spiral nebulae" were within or beyond the Milky Way developed.The debate raged on until 1923 when Edwin Hubble measured the distance to oneof these “fuzzy" objects (the Andromeda galaxy) placing it well beyond the outerlimits of the Milky Way [37].To measure the distance to the Andromeda galaxy, Hubble used Cepheid vari-ables – the first discovered standard candles. Cepheid variables are massive helium-burning stars that vary in luminosity periodically with time. The oscillations aredriven by the helium ionisation zone in the outer stellar envelope. When ionisedhelium (He+) is ionised into He++, the envelope becomesmore opaque thus trappingheat and increasing the pressure inside the envelope. To counteract the increase1in pressure the star expands and the envelope cools to a temperature where heliumcan recombine into He+ and the opacity decreases. A more transparent envelopelets heat escape to which the stars react by cooling its core and contracting theenvelope, releasing energy into the He+ envelope that in turn becomes He++ andthe cycle repeats [140, 38]. The standard candle aspect of Cepheid variables comesfrom this periodic pulse cycle that derives into a period-luminosity relation. Thisrelation was first discovered by astronomer Henrietta Leavitt in the early 1900’sby studying Cepheids in the Small Magellanic Cloud (SMC). In her study, Leavittfound that brighter Cepheids have longer periods [72]. If all the stars in the SMC areat the same distance this implies that brighter Cepheids are indeed more luminous.Leavitt’s relation at this time was only a law of proportionality, an independent dis-tance to a single Cepheid was still needed to calibrate this relation. In 1913, EjnarHertzsprung determined the distance to various Cepheids in the Milky Way thatallowed the first calibration of what it is now called “Leavitt’s law”[72]. Cepheidssoon proved the importance of standard candles to reach distances beyond thoseachieved through direct distance measurements such as parallax1. For example,in 1917 Harlow Shapley used variable stars to determined distances to galacticglobular clusters and estimate the diameter of the Milky Way and the position ofthe Sun within the Galaxy [37].Luminous standard candles are crucial to measuring distances beyond what canbe achieved through direct distance measurements. After Hubble measured thedistance to the Andromeda Galaxy using Cepheid variables, a variety of standardcandles were discovered and studied with the goal of obtaining the most accuratedistances to galaxies (e.g. the tip of the red giant branch (TRGB) [7, 148], Miravariables [100, 115], supernovae [8, 143]). This goal is primarily motivated by thesearch for a precise determination of the Hubble constant (𝐻0).Since defined by Edwin Hubble, 𝐻0 has become one of the most importantcosmological parameters. In his original work in 1929, Hubble [132] determinedthe distance to 24 galaxies (18 of them through Cepheids). Combining his results1Parallax is a triangulation technique, where the base of the triangle is the diameter of Earth’s orbitand the angle (𝑝) ismeasured based on the change in position of the objectwith respect to the stationarybackground of much more distant stars. Because distances to stars are much larger than the diameterof Earth’s orbit, the angle is always very small. Using the small-angle approximation and transformingradians to arcseconds the distance (𝑑) to the star can be written as 𝑑 [parsec] = 1/𝑝 [arcsec] [37].2with the radial velocities measured by V. M. Slipher, and additional data measuredby Milton Humason, Hubble realized there was a relation between the galaxy’srecessional velocity (𝜐) and its distance (𝑑): galaxies further away from us weremoving faster than their closer counterparts. With this discovery and evidence ofan expanding universe, he derived what is now known as the Hubble-Lamaître2 law(𝜐 = 𝐻0𝑑) and determined a value of 𝐻0 of 500 km s−1 Mpc−1 [72].Throughout the twentieth and early 21st century, the value of𝐻0 was determinedto lie within a range of 50 to 100 km s−1 Mpc−1 [37]. In the last decade early and lateuniverse measurements of 𝐻0 are coverging towards a number near 70 [67]. Oneof the most recent values is given by Riess et al. [208], who found using Cepheidsand supernovae that 𝐻0 = 73.3± 1.04 km s−1 Mpc−1. This result, obtained withobservations of the local (or late) universe, is currently in conflict with the valueof 𝐻0 based on observations of the early universe and the standard model ofcosmology. Observations of the cosmic microwave background temperature andpolarization anisotropies by the Planck Collaboration [195] yielded an 𝐻0 value of67.4±0.5 km s−1 Mpc−1, a 5𝜎 difference with Riess et al.’s result.The discrepancy between𝐻0 obtained from the early and late universe is referredto as “the 𝐻0 tension”. Attempts to reconcile this tension range from simplesystematic biases and uncertainties in standard candle calibration, to the need fornew physics (see Di Valentino et al. [67] and references therein). On the short-distance scales the determination of 𝐻0 depends sensitively on the accuracy of thedistances measured to galaxies in the local universe (see Section 1.1). A numberof standard candles, alternative to the Cepheid variables in the distance ladder,are being studied in order to further improve the local observational determinationof 𝐻0, e.g. the TRGB [84, 82], Mira variables [131] and carbon-rich asymptoticgiant stars, or carbon stars (CS)3 [154, 210, 253]. These attempts have resultedin a tension of their own. For example, two recent TRGB results have beenreported by Freedman [82] and Anand et al. [6]. The first author reports a value2Georges Lamaître used Slipher’s velocities and Hubble’s distances to published the first estimateof the expansion of the universe in 1927, two years before Hubble. Lamaître’s work was publishedin French in a Belgian journal resulting in limited exposure. It was only a few years ago that theIAU recommended that the previously known as Hubble’s Law be referred to as the Hubble-LamaîtreLaw [72]3Also defined as J-region stars by Nikolaev and Weinberg [181].3for 𝐻0 of 69.8± 0.6(𝑠𝑡𝑎𝑡) ± 1.6(𝑠𝑦𝑠) km s−1 Mpc−1, with a 1.8𝜎 difference withrespect to Riess et al. [208] but in agreement within the error bars with the PlanckCollaboration [195] result. The second study gives 𝐻0 = 71.5±1.8 km s−1 Mpc−1,in agreement with both Riess et al.’s and Freedman’s results but slightly over a 2𝜎tension with the Planck Collaboration.1.1 The local distance ladderThe distance ladder, as for any ladder, is constructed by “rungs” (See Figure 1.1). Inthis ladder, a rung is defined by a standard candle that can be observed over a rangeof distances so that it can be used to calibrate the following rung of the ladder. Agood standard candle, besides having a well-established absolute magnitude, mustthen be observable out to appreciably large distances.Riess et al. [208], for their first rung, standardized Cepheid variables usingdistances that were obtained through geometrical means, such as parallaxes ofCepheids in theMilkyWay, the distance to NGC 4258 frommasers Reid et al. [199],and the distance to the LargeMagellanic Cloud (LMC) obtained by Pietrzyński et al.[192] using detached eclipsing binaries. In the case of Freedman [82], the first rungis the TRGB which is calibrated again using distances from geometric techniques.Then Cepheid variables, or the TRGB, are used to calibrate the second rung ofthe ladder, most commonly Type Ia supernovae (SN Ia). SN Ia at their maximumbrightness achieve a visual absolute magnitude of ∼ −19, reaching distances anorder of magnitude further that Cepheids and well into the Hubble flow [37]. Forexample, Riess et al. [208] used 42 SN Ia in 37 host galaxies that had Cepheiddistances to standardize the luminosity of the SN Ia and use them in the next rung.Finally, the third rung of the distance ladder are the SN Ia at further distances,where both Cepheids and TRGB stars are too faint to be observed. Riess et al. [208]used 277 SN Ia at distances up to ∼ 600 Mpc to derive the above mentioned valueof 𝐻0 = 73.3±1.04 km s−1 Mpc−1.Each rung of the cosmological distance ladder depends on the calibration ofthe previous rung. Uncertainties in these calibrations will propagate up the laddermaking its way into the determination of 𝐻0.4Figure 1.1: The distance ladder in terms of distance modulus. The range ofthe rungs of the ladder are shown at the top of the figure. The first rungcalibrators whose distances are obtained through geometrical means are theMilky Way (MW) parallaxes (first bar) [43] and the LMC [192], the SMC[111] (DEB: detached eclipsing binaries), and NGC 4258 [199] (blue dots).The standard candles most commonly used as connection between the firstand second rung are the Cepheids [208] and the TRGB [82]. Connecting thesecond an third rung are the Type Ia supernovae (SN Ia) [208] with distancesthat reach the Hubble flow and are then used to estimate 𝐻0. NGC 3109,indicated with a red star, is the farthest galaxy studied in this thesis. TheCS red bar indicates the current limits of the CS method while the light redcontinuation shows the potential reach of this standard candle with the newJames Webb Space Telescope (JWST) [156].1.2 Colour, magnitudes and the distance modulusA colour-magnitude diagram (CMD) is a scatter plot showing, as the name suggest,the relation between the colour and either the apparent magnitude (brightness) orabsolute magnitude (intrinsic luminosity) of a star. The colour of a star is thedifference between a star’s apparent magnitude at a given wavelength to that at alonger wavelength. The colour is an indication of a star’s effective temperature and5it is independent of distance but it can be affected by reddening4. The apparentmagnitude (𝑚) of a star depends on the distance (𝑑) to the object, if the distanceis known then the absolute magnitude (𝑀) can be determined. The relationshipbetween magnitudes and distance is known as the distance modulus (𝜇0):(𝑚−𝑀)0 = 5 log(𝑑) −5 = 𝜇0, (1.1)where 𝑑 is the distance to the star in parsecs (1 pc= 3.086×1016 m) and the subscript0 indicates that the apparent magnitude has been corrected to account for the lineof sight extinction. From this formula it is possible to show that an uncertainty of0.1 magnitudes in 𝜇0 is equivalent to a 5% error in the distance.From Earth’s perspective, the stars in an external galaxy are all effectively at thesame distance5. Because of this, when plotting the CMD of a galaxy, it is possibleto observe a structured plot, and not just randomly scattered dots. Figure 1.2 showsa CMD of the LMC constructed from observations carried out in the near-infraredpart of the spectrum. Nikolaev and Weinberg [181] and Weinberg and Nikolaev[247] identified different stellar populations and matched them to the observablemorphological features on the CMD. Some of the features identified by Nikolaevand Weinberg [181] and Weinberg and Nikolaev [247] are included in the plot. Inthe original papers, each region was labeled with a single letter from A to L, inFigure 1.2, some of the features have been labeled with the name of the population.More specifically, in Figure 1.2, A, G and H follow the same nomenclature as theoriginal paper, while the RGB, AGB, CS, and X-CS correspond to E, F, J, and Kregions respectively. Region A, the bluest region at (𝐽 −𝐾𝑠)0 ≈ 0, is populated byblue supergiants and some of the hottest main-sequence stars. Region H is alsodominated by supergiants but of the cooler and redder type. To the right of region His region G, this area is occupied by young massive asymptotic giant branch (AGB)stars. Below is the normal oxygen-rich AGB (O-AGB) followed by the fainterpopulation of red giant branch (RGB) stars, the border between these two regions is4Reddening comes from the difference in the amount of extinction at two different wavelengths.Extinction is caused by dust in the line of sight which absorbs and scatters the light dimming themagnitude of the star. Because extinction has a greater effect on blue light it results in the reddeningof the color of the star.5For nearby galaxies three-dimensional effects such as the inclination and line of sight depth cansometimes be observed, see Chapter 6 and references therein.6Figure 1.2: Stellar populations in the different areas of the near-infrared ((𝐽−𝐾𝑠)0, 𝑀𝐽 ) CMD of the LMC. The LMC data were extracted from 2MASSas detailed in Section 2.1. The regions’ limits are based on the classificationfrom Nikolaev and Weinberg [181] and Weinberg and Nikolaev [247]. A, Gand H are blue supergiants, red supergiants and young AGB stars respectively.marked by the TRGB. On the red side of the AGB are the carbon-rich AGB stars (orCS). If these stars are embedded in thick, dusty circumstellar envelopes, their lightbecomes obscured and reddened, populating the red-most region with extreme CS(X-CS).1.3 Why in the near-infrared?A star is a rough approximation of an ideal emitter (i.e. a blackbody). Theseobjects emit energy at all wavelengths forming a continuous spectrum whose peakemission occurs at a wavelength 𝜆 = 𝜆𝑝𝑒𝑎𝑘 related to the object’s temperature (𝑇)7through Wien’s displacement law [105]:𝜆𝑝𝑒𝑎𝑘𝑇 = 2898 μmK, (1.2)where 𝑇 for a star is the surface temperature (equivalent to the effective temperatureof a blackbody with the same luminosity per surface area as the star). AccordingWein’s Law, objects with lower temperatures shine brighter at longer wavelengths.Infrared observations allow cooler objects that are faint or invisible in the opticalto stand out.The stars that have evolved to the TRGB and AGB stars have surface tempera-tures of ∼ 3000 K, with a slight dependence on the initial mass and metallicity ofthe star [191, 161, 222]. This temperature indicates that the peak of the spectrumfor these stars is located at ∼ 1𝜇m, at the beginning of the infrared region of theelectromagnetic spectrum.Beaton et al. [12] compared the brightness of the TRGB at different wavelengthsby using observations of the nearby galaxyNGC6822 obtainedwith different filters.The TRGB appears ∼ 2 magnitudes brighter in the 𝐼-band (0.8 μm) compared tothe 𝑉-band (0.55 μm), ∼ 1.5 magnitudes brighter in 𝐽-band (1.2 μm) compared tothe 𝐼-band, and a further 1 magnitude brighter in the 𝐾-band (2.2 μm) compared tothe 𝐽-band. In the even redder filters 𝐿 and 𝑀 at 3.8 and 4.5 μm respectively, theTRGB magnitude does not appear significantly brighter. The reason is that thesewavelengths are considerably longer compared to 𝜆𝑝𝑒𝑎𝑘 , and thus in a part of theRG’s spectrum where only a limited amount of energy is emitted from the star.CS are typically ∼ 1 magnitude brighter than the TRGB in the 𝐽-band ([147]and results from this thesis). CS have a carbon-to-oxygen ratio in their stellaratmosphere greater than unity. In an O-rich AGB star (C/O < 1) all the carbon istied up in carbon monoxide (CO) molecules while in CS (C/O > 1), the excesscarbon near the surface allows the formation of other carbon-containing moleculessuch as silicon carbide (SiC). SiC act as “dust” scattering blue light, making thestar appear redder than the bulk of the O-rich AGB population. Their red colourand high luminosity make CS a distinctive and easily identifiable feature in thenear-infrared CMD as can be seen in Figure 1.2.Among other advantages of observing in the infrared is the fact that light at8these wavelengths is less affected by dust and thus, less affected by extinction.Because extinction affects bluer light more than red light, it causes objects to lookredder than predicted by Wien’s law. Corrections for the reddening of light must beapplied when estimating distances, becoming a source of possible systematic error(more details are given in Section 9.1).1.4 How to make a red giant star: a brief summary ofstellar evolutionStar formation begins with the collapse and fragmentation of a molecular cloud.Once a star begins its life, generally defined as the onset of nuclear reactions inthe stellar core, it spends most of it converting hydrogen into helium in the stellarinterior. This evolutionary stage is called the Main Sequence (MS). The time a starspends on the MS and its subsequent evolution are determined by its mass. Thebrief description of stellar evolution in the following paragraphs focuses on low-mass stars (0.8𝑀⊙ ≲ 𝑀 ≲ 2.2𝑀⊙ [56, 140, 222]) which, as they evolve, populatethe RGB, AGB and, if certain conditions are met, can become CS.Once a low-mass star runs out of hydrogen in the core, the star leaves the MSand becomes a sub-giant. In this stage the helium core is inert and the star ispowered by a hydrogen-burning layer around the core. This shell keeps burninghydrogen feeding more inert helium into the core. To support the weight of the starthe core slowly contracts and the core temperature rises. To remain in equilibriumthe star responds by cooling the surface while increasing its luminosity. The star isnow going up the RGB. At the beginning of this phase, a convection zone developsnear the surface of the star. As the star evolves the inner limit of the convectionzone extends deeper into the interior of the star until it reaches the zone wherethe chemical composition has been contaminated by products of previous nuclearreactions [37]. This is called the first dredge-up event, which results in the mixingof the processed material with the rest of the star, changing even the compositionof the stellar surface.The end of the RGB is marked by the TRGB. In the optical and near-infraredCMDs, the TRGB can be interpreted as a boundary between the RGB and theoxygen-rich AGB (see Figure 1.2). The discontinuity observed in the CMDs is9a consequence of the sudden drop in luminosity caused by the onset of heliumburning in the core of low mass stars. The helium ignition occurs when thedegenerate helium core reaches high enough temperature (∼ 108 K) to fuse heliuminto carbon and oxygen via the triple-𝛼 process [37]. This explosive event changesthe structure of the star from a red giant to a horizontal branch star in a timemuch shorter than the dynamical time scale, making the star suddenly bluer andless luminous. Because the helium ignition occurs at a specific temperature, themagnitude of the TRGB can be predicted and used as a standard candle (see [12]and references therein).Once helium runs out in the core, the star is left with an inert carbon-oxygencore surrounded by a helium-burning shell and a hydrogen burning shell, these lasttwo separated by an almost pure helium intershell. The star is surrounded by aconvective H-He envelope and is now entering the AGB stage. During the earlyAGB phase the H-burning shell extinguishes while the He-burning shell remainsactive feeding more carbon and oxygen to the core. Once again the luminosityincreases and the star expands, while the convective envelope grows inwards. TheHe-burning layer advances, consuming and heating up the intershell. Becausehelium burns at much higher temperatures than hydrogen, this process reignites thehydrogen shell. After a short time the He-burning shell becomes unstable spendingmost of the time inactive with intermittent helium flashes (or thermal pulses) andthe star enters the thermally pulsing AGB (TP-AGB) phase. Analogous to the He-core flash, the helium shell flash is driven by the dumping of helium ashes from theH-burning layer. Once the temperature at the base of the shell is sufficient a pulseoccurs. The pulse pushes the H-burning shell outwards, which causes it to cool andeventually turn off. When the flash is finished, He-burning decreases allowing theH shell to recover and reignite, bringing the process to the starting point and thepulse cycle repeats [122, 37].During the thermal pulses, the intershell becomes convective, mixing the prod-ucts of helium burning bringing up carbon to the intershell. The convectiveenvelope also extends deeper into the star, reaching the region which has beenenriched with carbon and bringing this element up to the surface of the star. Thisconsequence of the thermal pulses is called third dredge-up6. If during this stage6Stars in the mass range described in this section do not go through a second dredge up event.10there are multiple third dredge-up events and the star experiences sufficient carbondredge-up to make the carbon-to-oxygen ratio in the stellar atmosphere greater thanunity, a CS is formed.1.5 The TRGB as a distance indicatorThe first indication that the TRGB was a promising distance indicator was foundby Baade [7] while studying the Andromeda galaxy and two of its satellite galaxiesMessier 32 and NGC 205. In his study, Baade found that the brightest red stars inall three galaxies had the same apparent magnitude and colour. Given that the threegalaxies form a system, and therefore are at the same distance, Baade concluded thatthese bright stars have the same absolute magnitude. Despite this early connectionit wasn’t until the early 1980’s that the TRGB began to be used to estimate distances(see Lee et al. [148] for a list of early TRGB distance estimates).Themain considerations when using the TRGB distance determinationmethodsare the detection and calibration of the TRGB. As detailed in Section 1.4, theevolution of low mass stars result in a discontinuity in the CMD at the locationof the TRGB, whose magnitude was initially estimated by eye. In the luminosityfunction of the RG stars (including stars below (RGB) and above (AGB) the TRGB),this discontinuity should be located at the point where the slope of the functionchanges the most. Therefore the first derivative of the luminosity function shouldhave a maximum at the magnitude of the TRGB.In 1993 Lee et al. [148] introduced a formal method to identify the TRGB,running an edge detection Sobel filter (as an approximation of the first derivative)in the binned luminosity function of the RGs. Different techniques have since beendeveloped to improve the detection. Sakai et al. [214] refined Lee et al.’s methodby using a continuous edge detection function and replacing the binned luminosityfunction with its smoothed version. This step reduced the error in the TRGBmagnitude determination by half. Méndez et al. [171] used Sakai et al.’s method inconjunction with a maximum likelihood analysis, fitting the RG luminosity functionwith a broken power law and locating the TRGB at the magnitude where the powerlaw is truncated (see also Makarov et al. [160], McQuinn et al. [170]). Referencesto additional methods can be found in Beaton et al. [12].11The TRGB has been extensively measured in the 𝐼-band where it has been foundto be nearly independent of intrinsic stellar properties (such as age or metallicity).Based on the empirical determination of the location of the RGB by Da Costa andArmandroff [61] in globular clusters and theoretical isochrones from Green et al.[112], Lee et al. determined that the 𝐼-band TRGB magnitude was insensitive tometallicity variations for stars with metallicities below ∼ −0.7 dex (and older than afew Gyr). Subsequent empirical and theoretical studies have confirmed the stabilityof the 𝐼-band TRGB magnitude and its modest dependence on metallicity (e.g.[38, 109, 222, 168] and reference therein). Metallicity effects can be observed inthe changing slope of the TRGB with colour. To account for these effects, differentcalibrations have been published with the 𝐼-band TRGB as a function of metallicity(e.g. [14]) or colour (e.g. [211]). Madore et al. [157] additionally developed acolour-dependent magnitude (𝑇-magnitude) system where the TRGB is flat. Thebattle to attain the most accurate determination of 𝐻0 has driven the search for thebest possible calibration of the 𝐼-band TRGB magnitude (see Table 1 of Freedman[82] for a list of recent calibrations) with values converging roughly at 𝑀𝐼 ∼ −4.Nevertheless, as mentioned in Section 1.3, the TRGB is brighter at wavelengthsredder than the 𝐼-band. During the early 2000’s, a number of studies found thatthe TRGB infrared magnitudes have a strong dependence on metallicity and starformation history (e.g. [78, 14, 215]). Even so, the development of the JamesWebb Space Telescope (JWST) (optimized for observations in the infrared), spikedthe interest and efforts for a precise calibration of the TRGB at these wavelengths.Valenti et al. [243] derivedmetallicity-based calibrations for 𝐽𝐻𝐾 based on globularclusters. Using these metallicity relationships, Górski et al. [109] derived a colour-based calibration for the 𝐽 and 𝐾𝑠 filters. Madore et al. [155] and Hoyt et al. [130]used the 𝑇-magnitude from Madore et al. [157] and calibrated the TRGB 𝐽, 𝐻 and𝐾𝑠 magnitudes using IC 1613 and the LMC respectively. Their results were ingood agreement with those found theoretically for 𝐽 and 𝐾 by Serenelli et al. [222].In the even redder 𝐿 and 𝑀 filters, the TRGB has been shown to be an unreliabledistance indicator as it exhibits variations of ∼ 0.15 mag in the TRGB even aftercorrections for metallicity effects [168].A recent analysis and multiwavelength (𝑉𝐼𝐽𝐻𝐾) calibration of the TRGB canbe found in Freedman et al. [85] and Hoyt [128].121.6 Carbon stars as a distance indicatorCS were first explored as distance indicators around the same time as the TRGB(in the early 1980’s), however, the quick development of the TRGB method drewthe attention away from this promising standard candle. In the following decades,very few studies focused on the standard candle aspect of CS, falling almost intooblivion from 2005 until 2020. The launch of JWST and the next generation oftelescopes optimized for infrared observations along with the search for alternativefirst rung standard candles have rekindled the interest in CS as a distance indicator[210, 154, 83, 186, 146, 253]The search for the absolute magnitude of CS was challenged by the insufficientavailable photometric data (see Crabtree et al. [58] and references therein). Thefirst attempts to define the absolute magnitude of CS were based on statisticalmethods (e.g. [217, 134]). Few efforts were made to measure individual absolutemagnitudes by estimating distances throughmethods such as trigonometric parallaxor using a non-CS companion star in a binary system (e.g Gordon [106], Richer[207], Olson and Richer [182]). However, even though at the time CS were mainlyobserved in the Galaxy, most were too distant for parallax measurements and wererarely members of binary systems [58].The lack of distance measurements to Galactic CS motivated the observationand search ofCS in stellar systemswith knowndistances such as galactic star clusters[189, 89] and the Magellanic Clouds [58, 204, 25, 88]. As more photometric databecame available, comparisons between the properties of CS in different galaxiesand stellar clusters were studied. For example Richer [202], using𝑉𝑅𝐼 photometry,found no differences in the colour of the CS in our Galaxy, the LMC and theSMC. Another paper by Frogel et al. [87] found similarities between the luminosityfunction of the CS (mainly their mean absolute bolometric magnitude) in the LMC,LMC clusters, the SMC, galactic star clusters, and the Sculptor and Fornax dwarfgalaxies. CS were also found to be approximately one magnitude brighter than theTRGB [202], the first hint of the advantage of CS over the TRGB.The first to use CS as a standard candle was Richer et al. [203] when theyestimated the distance to NGC 205. To distinguish CS from O-rich AGB stars,Richer et al. used narrow-band filters with central wavelengths at 710, 780, and 81013nm. CS have strong CN bands present at 810 nm, while O-rich AGB stars exhibit aTiO band at 780 nm and no CN band. When comparing the colours of the stars inthese narrow filters, CS will have large (81-78) and small (78-71) indices, whereasfor O-rich AGB stars it is the opposite. Once the CS were selected, assuming thatCS have the same mean luminosity in any galaxy, Richer et al. compared the meanapparent magnitude of the CS in NGC 205 in the 𝐼-band, to the mean absolute𝐼-band magnitude of the CS in the Magellanic Clouds. In the following years,the same method was applied to estimate the distances to NGC 300 [205] andNGC 55 [196]. Around the same time Cook et al. [57] independently developed asimilar method to differentiate CS but using only two intermediate-band (insteadof narrow-band) filters centred at 775 nm and 810 nm. Even though Cook et al.do not directly use CS to estimate distances, they noticed the similarities in the 𝐼mean magnitude and luminosity functions among the galaxies in their sample andsuggest the potential of these stars as distance indicators.During the 1990’s, the potential of CS as standard candles was nearly forgotten.A few studies in the early 2000’s did bring up the subject again. In 2001, Weinbergand Nikolaev [247] and van der Marel and Cioni [244] studied the structure ofthe LMC using near-infrared data. In both papers, the CS were used to identifiedspatial variations (such as inclination) by studying the CS luminosity function indifferent locations of the LMC and comparing the apparent magnitudes of CS in thedifferent fields to that of the entire galaxy. In the case of Weinberg and Nikolaev,they compared the mean apparent magnitudes by selecting the CS in a very narrow(∼ 0.1 mag) (𝐽 −𝐾𝑠) colour range as they concluded that such a small range wasnecessary to minimize the magnitude spread. On the other hand, van der Mareland Cioni [244] used 1.5 ≤ (𝐽 −𝐾𝑠) ≤ 2 (very close to the colour range we use inChapter 3) and compared the apparent mode of the CS distributions in the variousfields. In 2003, Demers et al. [64] found that for galaxies with one hundred CS ormore, the mean absolute 𝐼-band magnitude of the CS was nearly constant. Usingdata from the Local Group galaxies, Demers et al. [65] found a mean magnitude of−4.58±0.09. With this calibration, Demers et al. obtained the distance to IC 10.In 2005, the last review to date on the standard candle aspect of CS was writtenby Battinelli and Demers [11]. In their review they revisited recent results onthe subject and the possible advantages of CS over other distance indicators. The14authors highlight the fact that the mean absolute 𝐼 magnitude of the CS shows littledependence on metallicity, with a roughly constant mean magnitude for galaxieswith [Fe/H]> −1.5 . Battinelli and Demers also remark on the advantages ofobserving CS in the near-infrared, with a final note on future observations withJWST.Fifteen years after Battinelli and Demers’s review, the idea of CS as a distanceindicator has finally resurfaced. New studies such as Madore and Freedman [154],Freedman and Madore [83] and Lee et al. [146] adopt the same approach as Richeret al. [203] by defining an absolute mean 𝐽 magnitude for CS based on the absolutemean magnitudes of the CS in the Large and Small Magellanic clouds. Zgirskiet al. [253] also use the CS mean 𝐽 magnitude to determine distances with only theLMC as the fundamental calibrator. In their method, they first fit the luminosityfunction with a custom profile composed of the superposition of a Gaussian anda quadratic function. The mean 𝐽 magnitude is then defined as the mean of theGaussian component of the profile. Instead, in this thesis, the median 𝐽 magnitudeof the CS is used and either the LMC or SMC is use as the calibrator, determinedon a case by case basis.This thesis presents two independently developed methods to estimate distancesin the local universe using the CS (Chapter 3) and the RGs (Chapter 5). Estimationsof the distances through the TRGB method are also included (Chapter 4) to test theperformance and compare the results when the three different methods are appliedto the same data sets. Four target galaxies have been selected: NGC 6822, IC 1613,Wolf-Lundmark-Melotte (WLM), and NGC 3109, all Magellanic-type irregulargalaxies (which will be classified as “LMC-” or “SMC-like”). The four data setsare processed, calibrated and analyzed following the same procedure in order toachieve consistency (Chapter 2). The CS method is further explored by applyingit to study the three-dimensional structure of the Magellanic Clouds (Chapter 6),and to the CS in the Milky Way (Chapter 8). A test on how reddening may affectthe calibration of the absolute magnitudes of the Magellanic Clouds is presented inChapter 7. The results for the target galaxies from the three distance determinationmethods are compared with recent independent distance estimations (Chapter 9),including a discussion on the possible sources of disagreement (when present)15between the estimations. Finally, a brief conclusion and ideas for potential futurework are given in Chapter 10.16Chapter 2Data AnalysisIn this thesis, data from six different galaxies are used: the LMC and SMC as thecalibrator galaxies, and four target galaxies (NGC 6822, IC 1613, WLM and NGC3109). This Chapter details how the final photometric catalogues for each galaxywere obtained.2.1 The Magellanic CloudsWeobtained two separate catalogues for the LMC and SMC following the proceduredescribed in Ripoche et al. [210]. Photometric data are first retrieved from the TwoMicronAll Sky Survey (2MASS) 1 [227] point source catalogue. 2MASSuniformlyscanned the entire sky in three near-infrared bands; 𝐽 (1.235 𝜇𝑚), 𝐻 (1.662 𝜇𝑚),and𝐾𝑠 (2.159 𝜇𝑚) using two 1.3m telescopes located atMountHopkins inArizona,and Cerro Tololo International Observatory (CTIO) in Chile. Each telescope wasequipped with a three-channel camera, each channel consisting of a 256×256 arraydetector capable of observing the sky simultaneously in the 𝐽, 𝐻 and 𝐾𝑠 bands.The 2MASS catalogue includes a photometric quality flag (qflag) as a parameter foreach star measured. This three character string summarizes the overall photometricquality in each band (𝐽𝐻𝐾𝑠). Only stars with high quality valid measurements inall three filters (qflag=‘AAA’) are selected.1A joint project of the University of Massachusetts and the Infrared Processing and Analysis Cen-ter/California Institute of Technology, funded by the National Aeronautics and Space Administrationand the National Science Foundation, https://irsa.ipac.caltech.edu/Missions/2mass.html17In the direction of the LMC and SMC, there is high contamination from fore-groundMilkyWay stars. To remove these stars, it is necessary to utilize informationfrom a second source: Gaia 2 [96] Early Data Release 33 (EDR3, [94]). Gaia is aspace observatory of the European Space Agency (ESA) designed to observe eachof its targets about 70 times, with the goal of precisely measuring the positions,distances, movements, and changes in brightness of around one billion targets. Bymeasuring the displacement of the stars in the sky with time, Gaia measures theproper motions of its targets. Stars from the same object, such as a galaxy or starcluster, will all move at approximately the same rate and direction, having approxi-mately equal proper motions, while foreground and background sources will showrandom changes in position with time. Proper motions can then be used to removeforeground contamination by selecting stars that have nearly equal proper motionsand removing those that do not.To remove the foreground contamination in the LMC and SMC data, starsfrom 2MASS were cross-matched with stars found in Gaia EDR3. Not all starsin 2MASS have a Gaia counterpart, and not every counterpart found has a validproper motion measurement – only stars that have a Gaia counterpart with a validproper motion measurement are retained. A star is considered a member of theLMC or SMC when its total proper motion is smaller than 1 mas yr−1, with respectto the mean proper motion of the main body. The general proper motion selectioncriteria of a star can be expressed as:(𝜇𝛼∗− 𝜇𝛼∗)2 + (𝜇𝛿 − 𝜇𝛿)2 < 1masyr−1, (2.1)where 𝜇𝛼∗ = 𝜇𝛼𝑐𝑜𝑠𝛿 (Gaia parameter pmra) and 𝜇𝛿 (Gaia parameter pmdec) arethe proper motions of each individual star in right ascension (RA) and declination(DEC). 𝜇𝛼∗ and 𝜇𝛿 are the mean proper motions for the main object, in this casethe LMC or SMC. Taking the mean proper motions for the LMC and SMC from2https://www.cosmos.esa.int/web/gaia/home3Processed by the Gaia Data Processing and Analysis Consortium, https://www.cosmos.esa.int/web/gaia/dpac/consortium18Gaia Collaboration et al. [95], then the proper motion selection conditions are:(pmra−1.7608)2 + (pmdec−0.3038)2 < 1masyr−1(pmra−0.7321)2 + (pmdec+1.2256)2 < 1masyr−1(2.2)for the LMC and SMC respectively.The proper motion analysis removes most of the contaminating sources, never-theless some remain especially at the faint end. To ensure a clean sample of stars,a final cut is made based on the error in the parallaxes measured by Gaia. Themean parallax errors are 0.041 and 0.038 mas for the LMC and SMC respectively.Stars with parallax errors more than 2𝜎 away from the mean are removed from thesample. This cut affects mainly stars with 𝐽 fainter than 14 and (𝐽 −𝐾𝑠)0 redderthan 2.4, outside the range of the bulk of the CS.In the following chapters, the LMC and SMC will serve as fundamental dis-tance calibrators. Besides having a clean sample of stars, it is also important tohave a complete sample. The 2MASS 99% completeness magnitude limits are𝐽 =15.8, 𝐻 =15.1 and 𝐾𝑠 =14.3. The stars in both the LMC and SMC that will beused to calculate distances to other galaxies, are brighter than this limit, thereforeincompleteness is not an issue.The 2MASS apparent magnitudes need to be transformed to absolute magni-tudes. For the LMC,we adopt themean distancemodulus determined by Pietrzyńskiet al. [192]: 𝜇0,LMC = 18.477 ± 0.004 (statistical) ± 0.026 (systematic). For theSMC, we take the distance modulus found by Graczyk et al. [111]: 𝜇0,SMC = 18.977± 0.016 (statistical) ± 0.028 (systematic). To correct for reddening and extinctionwe use the reddening maps for both the Magellanic Clouds obtained by Górski et al.[110] and the extinction coefficients for the 2MASS bands fromGordon et al. [107].The resulting foreground cleaned ((𝐽 −𝐾𝑠)0, 𝑀𝐽 ) CMDs for the LMC andSMC are shown in Figure 2.1. A detailed description of how the LMC and SMCcatalogues were constructed can be found in Ripoche et al. [210].19Figure 2.1: ((𝐽 −𝐾𝑠)0, 𝑀𝐽 ) CMDs for the LMC (left) and SMC (right). Thedata shown are the resulting 2MASS catalogues after filtering foregroundcontamination and correcting for reddening.2.2 NGC 6822, IC 1613 and WLM2.2.1 ObservationsThe data for NGC 6822, IC 1613 andWLMwere obtained from observations madewith the Wide-field InfraRed Camera (WIRCam) at the Canada-France-HawaiiTelescope (CFHT). CFHT is located at the summit of Maunakea on the Island ofHawaii, at∼4200meters above sea level. The telescope is a Prime Focus/Cassegraincombination and is equipped with a 3.6 meters diameter mirror. WIRCam is a near-infrared camera located at the prime focus of CFHT. The instrument is composedof four 2048×2048 pixels detectors with a pixel scale of 0.3 arcseconds per pixeland a total field of view of ∼ 20 squared arcminutes [197].For NGC 6822 the images were taken in June 2016 and May 2019 (program ID17AD82, P.I. Laurie Rousseau-Nepton). For IC 1613 and WLM the observations20were made during September 2018 (program ID 18BD94, P.I. Laurie Rousseau-Nepton). For NGC 6822 and IC 1613 the WIRCam observations covered a totalfield of ∼ 25 arcmin2, for WLM the final field was slightly smaller: ∼ 20 arcmin2.Fields for the three galaxies are centred near the centre of the galaxy. Individualimages were taken with three different near-infrared filters: 𝐽 (1.253 𝜇m), 𝐻 (1.631𝜇m), and 𝐾𝑠 (2.146 𝜇m), with the exposures times of 60, 15 and 25 seconds for 𝐽,𝐻 and 𝐾𝑠 respectively. The number of exposures in each filter and total exposuretimes are presented in Table 2.1.Table 2.1: WIRCam near-infrared observations specifications. For all threegalaxies, individual frames exposure times are 60, 15 and 25 seconds for 𝐽, 𝐻and 𝐾𝑠 respectively.Galaxy N𝑜 of exposures Total exp. t (sec)𝐽 𝐻 𝐾𝑠 𝐽 𝐻 𝐾𝑠NGC 6822 40 160 120 2400 3000 3000IC 1613 20 84 60 1260 1260 1500WLM 19 80 60 1200 1140 15002.2.2 PhotometryAs the images obtained for NGC 6822, IC 1613 and WLM all come from thesame instrument, the same procedure for the photometry was followed for all threegalaxies.In every individual image, sources were initially identified using SExtrac-tor (Source Extractor, [20]). The SExtractor catalogues were then processed bySCAMP (Software for Calibrating AstroMetry and Photometry, [19]) to find theastrometric solution between the images in the same filter. This astrometic solutionwas used by SWarp [22, 21] to align the individual exposures and produce oneco-added image for each filter, for each galaxy.The photometry was performed on the co-added images using the digital stellarphotometry reduction programs DAOPHOT II [233] and ALLSTAR [232]. Firstthe DAOPHOT routine FIND is used to obtain a list of coordinates of sourcesbrighter than an specified star-detection threshold. For each source found, FINDgives two shape statistics: SHARP and ROUND, to identify whether a source is21distorted or is most likely a star. SHARP is the ratio of the heights of the best-fittingdelta function and the best-fitting Gaussian function of each source and it is usedto remove hot and cold pixels (pixels that are above or below the average of theneighbouring pixels). ROUND is obtained by fitting Gaussian functions in the 𝑥and 𝑦 direction and taking the ratio between the difference in height and the averageheight of the two Gaussians. Elongated objects will have ROUND values greaterthan or less than zero depending on which direction they stretch. Figure 2.2 shows,as an example, the value for the shape parameters for the 𝐽-band co-added imagefor WLM. The limits for the shape parameters were set to: 0.2 ≤SHARP≤ 1.0 and−1.0 ≤ROUND≤ 1.0. Any source with values outside these limits was discarded.Sources found in the three co-added images for the different filters are more likelyto be real sources. A comparison of the values of ROUND and SHARP for thesources found simultaneously in all three (𝐽𝐻𝐾𝑠) images against the sources foundonly in the 𝐽 image is shown in Figure 2.2.Using the task PHOT, aperture photometry is performed on all the remainingsources found in the previous step. PHOT allows the user to run aperture photometrywith multiple aperture sizes. For stellar sources, the signal-to-noise ratio is maxi-mized when the aperture size is equal to the full width at half maximum (FWHM)of the point-spread function (PSF) that describes the source. The PSF is associatedto the instrument in use but it also depends on environmental factors such as seeing.After analysing the PSF profiles of a number of stars of different magnitudes andat different locations in the detector, we set the aperture size at ∼4 pixels.From the aperture photometry file, stars are selected to derive the PSF that willbe used to perform PSF photometry using the DAPHOT routine PSF. On the firstcalculation, only stars found in all three filters are used. At the end of the firstrun, any star flagged as having its individual profile scatter greater than twice theaverage is removed from the list. After several iterations the PSF for each co-addedimage is set. Using the PSF just derived, ALLSTAR obtains PSF photometry of allthe stars in the aperture photometry list.Finally, the outputs from ALLSTAR are aligned and combined into a singlemaster catalogue (one for each galaxy) containing the pixel coordinates, instru-mental magnitudes, magnitude errors, and two statistical parameters describing theshape and goodness-of-fit of each star.22Figure 2.2: SHARP and ROUND shape statistics from DAOPHOT routineFIND for WLM. Sources with values outside the limits shown on the plots arediscarded. The orange markers represent all the sources found in the 𝐽-bandimage, blue markers instead are the sources that are found simultaneously inall three bands. Sources found in more than one image are more likely to bereal sources.2.2.3 CalibrationThe data presented in this thesis are all taken with 𝐽𝐻𝐾𝑠 set of filters. Nevertheless,the filters were manufactured specifically for each different instrument and as aconsequence are not identical. For example, the 𝐽𝐻𝐾𝑠 filters on 2MASS havedifferent transmission profiles and effective wavelengths than the 𝐽𝐻𝐾𝑠 filters usedby WIRCam. Figure 2.3 shows the relative 𝐽𝐻𝐾𝑠 filter responses for the differentinstruments used in this study.When comparing photometric data from different instruments, the differencesbetween the filters can introduce systematic effects in the observed stellar colours[35]. To compare photometry from different sources, it is important to place all23Figure 2.3: 𝐽𝐻𝐾𝑠 filters transmission profiles for 2MASS [55], WIRCam5,WFCAM6and ISAAC7systems. Transmission profiles for 2MASS are nor-malized to unity. When comparing data obtained with filters that have thesame name but are from different instruments, differences in the transmissionprofile need to be taken into consideration.data in the same photometric system. Because the data for the distance calibrators(the LMC and SMC) come from 2MASS, the 2MASS colours are adopted as thereference system for this study.Equations to transform WIRCam photometry data to the 2MASS system areprovided by the WIRwolf image stacking pipeline4:𝐽2MASS = 𝐽WIRCam +0.071× (𝐽 −𝐻)WIRCam𝐻2MASS = 𝐻WIRCam−0.034× (𝐽 −𝐻)WIRCam𝐾2MASS = 𝐾WIRCam−0.062× (𝐻 −𝐾)WIRCam.+0.002× (𝐽 −𝐻)WIRCam.(2.3)These equations were applied to the three WIRCam data sets.4A pipeline for calibrating and stacking CFHT WIRCam data - http://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/wirwolf/docs/filt.html24With the photometry now in the same colour system as 2MASS, 2MASS starspresent in the same area of the sky as the WIRCam data are used to calibrate theinstrumental magnitudes obtained with DAOPHOT. If the stars found in the 2MASSpoint source catalogue are in fact the same stars obtained from the WIRCam dataphotometry, then the difference in magnitude between the stars in the 2MASS cat-alogue and the DAOPHOT catalogue is expected to be constant. First, the stars arematched using the CDS cross-match service8. Second, for every match, the mag-nitude difference between the DAOPHOT photometry and 2MASS is calculated.Figure 2.4 shows the magnitude differences in 𝐽, 𝐻 and 𝐾𝑠 for IC 1613. Finally,the weighted average of the magnitude differences is taken as the zero-point andadded to the WIRCam photometry using only those stars below a separation andphotometry error threshold. This process is repeated for each filter for each galaxy.The zero-points with respect to 2MASS for all the galaxies are presented in Table2.2.Table 2.2: Zero-points for photometry calibration given by the difference inmagnitudes between the values obtained from the DAOPHOT photometry andthe stars found in the 2MASS catalogue. The DAOPHOT photometry from theWIRCam (NGC 6822, IC 1613 and WLM) and ISAAC (NGC 3109) data havebeen previously transformed to the 2MASS 𝐽𝐻𝐾𝑠 colour system.Galaxy mag(DAOPHOT) − mag(2MASS)Δ𝐽 Δ𝐻 Δ𝐾𝑠NGC 6822 -5.021 ± 0.010 -4.946 ± 0.015 -4.921 ± 0.010IC 1613 1.038 ± 0.006 1.048 ± 0.007 1.046 ± 0.008WLM 0.951 ± 0.009 1.024 ± 0.009 1.040 ± 0.017NGC 3109 3.324 ± 0.029 n/a 1.872 ± 0.031The final step to obtain the true apparent magnitudes is to correct the datafor reddening. After transforming the magnitudes to the 2MASS system usingequations 2.3, the final true apparent apparent magnitude (𝑋0) in any 𝑋-band (in5https://www.cfht.hawaii.edu/Instruments/Filters/wircam.html6https://about.ifa.hawaii.edu/ukirt/instruments/wfcam/description-of-wfcam/7http://www.eso.org/sci/facilities/paranal/decommissioned/isaac/inst/isaac_img.html8http://cdsxmatch.u-strasbg.fr/25Figure 2.4: Difference in magnitudes between the DAOPHOT photometry(already transform into the 2MASS colour system) and their 2MASS coun-terparts for IC 1613. The photometric band (𝐽𝐻𝐾𝑠) is indicated on the topleft corner of each panel. The blue markers are the stars selected (based onangular separation and photometry error cuts) for the zero-point calculation.The weighted average of the magnitude differences for the blue markers isrepresented by the black dotted line.this study 𝐽, 𝐻, or 𝐾𝑠) is written as:𝑋0 = 𝑋2𝑀𝐴𝑆𝑆 −Δ𝑋 − 𝐴𝑋, (2.4)where 𝐴𝑋 is the extinction in the 𝑋-band. 𝐴𝑋 is given by the extinction laws fromCardelli et al. [34] and Fitzpatrick [79] and using an extinction to reddening ratioin the 𝑉-band (𝐴𝑉/𝐸 (𝐵−𝑉)) of 3.1.NGC 6822 is located at a galactic latitude of about −18◦, which results ina considerable amount of foreground stellar contamination as well as significantreddening. According to Schlafly and Finkbeiner [218] the Milky Way reddening26value along the line-of-sight towards NGC 6822 is 𝐸 (𝐵−𝑉) = 0.21. Other studiesusing Cepheids [102, 90] and colour-magnitude diagrams [201] estimate the totalreddening in the line-of-sight to be about 𝐸 (𝐵 −𝑉) ∼ 0.35 mag. We adopt thelatter value, 𝐸 (𝐵−𝑉) = 0.35, to deredden our data as this value includes both theforeground reddening and an estimate of the internal reddening within the galaxy.For IC 1613 and WLMwe use the average line-of-sight extinction estimated bySchlafly and Finkbeiner [218]: 𝐸 (𝐵−𝑉) = 0.021 and 𝐸 (𝐵−𝑉) = 0.03 respectively.Internal reddening in IC 1613 has been found to be negligible [216, 81, 99].Therefore, no internal reddening is considered in the extinction corrections for thisgalaxy.The calibrated and reddening corrected CMDs for the WIRCam galaxies areshown in Figure 2.5.Figure 2.5: From left to right calibrated and reddening corrected ((𝐽 −𝐾𝑠)0,𝐽0) CMDs of NGC 6822, IC 1613 and WLM.2.2.4 Colour-magnitude diagram cleaningCMDs can often be contaminated by foreground stars and background galaxies.These will generally appear as an unexpected feature in the CMD or they can also27hide at the same magnitude and colour range as the stars from the object understudy. When studying the stars in the CMD, the presence of these sources can alterthe number of stars in a specific region of the diagram and hence the luminosityfunction of a group of stars.The most effective way to obtain a clean CMD is to use the displacement of thestars in the sky with time (proper motions). Stars from the same object such as agalaxy or star cluster will all move at approximately the same rate and in the samedirection, while foreground and background sources will show random changes inposition. To measure proper motions at least two observation of the same fieldat different times are needed. The more distant an object, the longer is the timebetween observations needed to detect changes in their positions on the sky.In the case of the galaxies in this study with a single epoch observation, propermotion measurements are not possible and other methods must be employed toaccount for CMD contamination.The left panel of Figure 2.6 displays the NGC 6822 CMD plotted over atheoretical model distribution of Milky Way stars obtained from the populationsynthesis code Trilegal9 [104]. Both the modeled foreground stars and the datafor NGC 6822 have been dereddened. Even though there are a large number offoreground stars, these are bluer than the CS blue limit ((𝐽−𝐾𝑠)0 = 1.4). Therefore,no corrections for the presence of foreground stars in the colour-magnitude diagramis necessary.The CMD of NGC 6822 also suffers from a considerable amount of contami-nation from background galaxies. Despite the fact that these sources are generallyfainter than the bulk of the CS, there is a large number of them located in thesame colour range (1.4 < (𝐽 −𝐾𝑠)0 < 2.0) [73]. Unlike stars, galaxies are extendedsources and can be distinguished from the rest of the point-like stellar sources. Inthe final step of the PSF photometry, ALLSTAR calculates a sharpness (SHARP10)and goodness-of-fit (CHI) index for each star. SHARP is obtained by comparing theshape of the observed sources to those of the PSF, while CHI is the ratio betweenthe observed and expected fitting errors. Sources with SHARP and CHI valueswell above the mean of the SHARP and CHI value distributions, are generally9http://stev.oapd.inaf.it/cgi-bin/trilegal10Different from the SHARP parameters output by the DAOPHOT routine FIND.28Figure 2.6: Left: Colour-magnitude diagram (CMD) of NGC 6822 show-ing in red the region of stars selected as the reference population to definethe typical values of SHARP and CHI. The orange markers represent thesynthetic population of foreground Milky Way stars generated with Trilegal[104]. Right: The top panels show the distributions of SHARP and CHI valuesfor the selected RG stars, the vertical continuous lines indicate the mean ofthe distribution and the other two vertical lines are placed at ±2.5𝜎 from themean. The centre and bottom panels show SHARP and CHI as a function of𝐽0 magnitude. The red dots are the red giant stars selected in the CMD. Theblack lines (now horizontal) are again located at the mean (continuous line)and at ±2.5𝜎 from the mean. The grey dots are all the stars from the CMD onthe left panel. Any star above or below the 2.5𝜎 limits is removed from theanalysis.background galaxies or blends and can be eliminated.To define the mean value of SHARP and CHI in NGC 6822 it is necessaryto determine a good reference population. A selection of stars in the red giantand asymptotic giant stage is made within a magnitude and colour range wherethere is little contamination from background and/or foreground sources. Theregion selected is shown in the left panel of Figure 2.6. These stars form a well-defined sequence in the CMD dominated by stars that are, most likely, members29of NGC 6822. This makes them a good reference population to define a typicalvalue of SHARP and CHI for stars from the galaxy. The distributions of SHARPand CHI values for the selected stars are shown in the top right panels of Figure2.6. SHARP and CHI cutoffs are set at the level at which these parameters reachvalues of 2.5 times the standard deviation from the mean of the distribution. A2.5𝜎 cut on SHARP and CHI removes most of the contamination to the CMD fromreal sources that are most likely background or foreground sources while keepingthe rejection of real members of the target galaxy to a minimum. These values areshown graphically in the right panels of Figure 2.6. Given the mean and dispersionvalues of the parameters for the red giants, the SHARP and CHI limits for a star tobe considered a well-defined point-like source are defined as:SHARPRG−2.5𝜎𝑅𝐺 < SHARP < SHARPRG +2.5𝜎𝑅𝐺CHI < CHIRG +2.5𝜎𝑅𝐺 .(2.5)The symbol𝜎𝑅𝐺 represents the standard deviation for SHARPandCHI respectively.As can be seen in the lower panel of Figure 2.6, nearly all the stars are above the2.5𝜎 limit below the median line, therefore it is not necessary to set a lower limitfor CHI.Figure 2.7 shows the CMDs for NGC 6822, IC 1613 and WLM after the datahave been through every step described in this section.2.3 NGC 31092.3.1 ObservationsThe data for NGC 3109 come from observations obtained by Soszyński et al. [228].These observations were made with the Infrared Spectrometer And Array Camera(ISAAC; decommission in 2013) mounted on the European Southern Observatory(ESO) Very Large Telescope (VLT) located at Cerro Paranal (2635 m above sealevel) in the Atacama desert in northern Chile. VLT is composed of four 8.2mseparate optical telescopes. ISAAC is mounted on the Nasmyth focus of unittelescope 3. The instrument’s short wavelength arm is composed of a single1024×1024 detector with a field of view of 2.5 squared arcminutes and a scale of30Figure 2.7: From left to right the calibrated, reddening corrected and cleanedcolour-magnitude diagrams of NGC 6822, IC 1613 and WLM.0.148 arcseconds per pixel.The observations consist of three fields (F1, F2 and F3) observed in 2004 inJanuary and February. Each field was observed on two different nights, except forF1 which was observed only once. The images were obtained using just two near-infrared filters: 𝐽 (1.25 microns) and 𝐾𝑠 (2.16 microns). Each individual frame hasan exposure time of 30 and 15 seconds in 𝐽 and 𝐾𝑠 respectively.2.3.2 Image reduction and photometryThe raw images were downloaded from the ESO Science Archive Facility. Someof the observed frames have large spatial offsets from the reference frames and/orare unfocused. When co-adding the images these frames were removed, thereforethe total exposure time for each field in the two filters varies. The total numberof frames and total exposure time used for this study for each field are detailed inTable 2.3.The images were processed with JITTER from the ECLIPSE package [66]developed by ESO for the reduction of near-infrared data. Dark frames and sky31Table 2.3: ISAAC near-infrared observations details for each field observedin NGC 3109. Individual frame exposure times are 30 and 15 seconds for 𝐽and 𝐾𝑠 respectively.Field N𝑜 of exposures Total exp. t (sec)𝐽 𝐾𝑠 𝐽 𝐾𝑠F1 12 38 360 570F2 17 65 510 975F3 19 68 570 1020(twilight) flat fields were taken for every observation night. Master dark andmaster flat frames were constructed using the ECLIPSE routines dark and twflatrespectively. The twflat routine also outputs a bad pixel mask that is used laterin the reduction process. For the sky estimation, sky frames away from the NGC3109 field were taken in between object frames.JITTER can perform flat fielding, dark subtraction, bad pixel corrections, skyestimation and subtraction, frame offset detection, and frame stacking. The re-duction process was done in two steps: first JITTER was run for each night up tothe sky subtraction process and in the second step JITTER was given the reducedimages for both observing nights for each field and filter. At the end of the processone co-added image per field and filter were obtained.PSF photometrywas performed independently in each co-added image using thesamemethod as for theWIRCam data. For each field we construct a photometry filewhich contains only the stars found in both the 𝐽 and 𝐾𝑠 images. These catalogueswere then cleaned using the SHARP and CHI parameters output by DAOPHOT IIas specified in Section 2.2.4.2.3.3 CalibrationTo calibrate the NGC 3109 photometry, it is first necessary to bring all three fieldsto a common instrumental zero-point. Both F1 and F3 partially overlap with F2allowing us to find common stars between F1 and F2, and F3 and F2. For eachcommon star the magnitude difference is calculated. The mean of the magnitudedifference is then used to adjust F1 and F3 zero-points relative to F2.The 𝐽𝐻𝐾𝑠 filters used by ISAAC closely match the 𝐽𝐻𝐾𝑠 filters defined in the32Las Campanas Observatory (LCO) (or Persson standards; [190]) system. Thereforecolour terms between the two systems are negligible [5, 152]. Comparison betweenthe LCO and 2MASS systems show that the colour terms do not differ from zero bymore than a few thousandths of a magnitude [35]. For the purposes of this study, thecolour terms between ISAAC and 2MASS 𝐽𝐻𝐾𝑠 system are considered negligibleand no transformations are applied.As described in Section 2.2.3 for the WIRCam data, to calibrate the NGC 3109data stars from 2MASS are matched to stars found within the three observed fields.Once the data has been zero-pointed to 2MASS, the reddening correction is added:𝐸 (𝐵−𝑉) = 0.056 [218]. The final CMD is shown in Figure 2.8.2.4 IC 1613 (Sibbons et al. data)A second catalogue for IC 1613 is used in this study. This second data set comesfrom Sibbons et al. [223] and and is publicly available through VizieR11. The dataare from observations made with the Wide Field CAMera (WFCAM) mounted onUnited Kingdom Infrared Telescope (UKIRT) in Hawaii. WFCAM consists of four2048×2048 pixels detectors that cover a field of 0.19 squared degrees with a scaleof 0.4 arcseconds per pixel. The catalogue presents calibrated 𝐽𝐻𝐾𝑠 magnitudes intheWFCAM system. As explained in Section 2.2.3, different instrument have filterswith slightly different properties. To transform the available data to the 2MASSsystem the transformation equations given by Hodgkin et al. [125] are used:𝐽2MASS = 𝐽WFCAM +0.075× (𝐽 −𝐻)WFCAM−0.002𝐻2MASS = 𝐻WFCAM−0.081× (𝐽 −𝐻)WFCAM +0.032𝐾2MASS = 𝐾WFCAM−0.010× (𝐽 −𝐾)WFCAM.+0.001× (𝐽 −𝐻)WFCAM.(2.6)The data in the catalogue have also already been corrected for foregroundreddening using the Schlegel et al. [219] extinction maps. Sibbons et al. [223]gives a range for the reddening of 𝐸 (𝐵−𝑉) = 0.02−0.03 but does not specify the11http://cdsarc.unistra.fr/viz-bin/cat/J/A+A/573/A8433exact value applied to the catalogue. Assuming an extinction to reddening ratio inthe V-band of 3.1, the difference between the minimum and maximum reddeningused in Sibbons et al. [223] would translate to a difference in the extinction in 𝐽 ofΔ𝐴𝐽 ≈ 0.01 magnitudes. The final CMD is shown in Figure 2.8.Figure 2.8: Calibrated, reddening corrected and cleaned ((𝐽−𝐾𝑠)0, 𝐽0) CMDof NGC 3109 and IC 1613 (Sibbons et al. data).34Chapter 3Carbon stars3.1 SelectionThe LMC and SMC catalogues obtained from 2MASS are compared against spec-troscopic catalogues of CS for both the LMC [142] and the SMC [198]. To obtainthe near-infrared photometry that go into the CS catalogues, these are cross-matchedwith the 2MASS stars using their RA andDEC coordinates. Once the 𝐽𝐻𝐾𝑠 coloursfor the spectroscopic catalogues are retrieved, these are plotted over the 2MASS((𝐽 −𝐾𝑠)0, 𝑀𝐽 ) CMD. In the plot in Figure 3.1, it can be seen that the majority ofthe stars from the spectroscopic catalogues fall in the same location of the CMD asthe horizontal feature visible at 𝑀𝐽 ∼ −6, identified as the CS region as explain inSection 1.2.The bluer stars in the spectroscopic catalogue fall on top of the O-rich AGB andRGB. Fortunately, there are few of these compared to the bulk of the CS population.To identify CS stars based solely on their (𝐽 −𝐾𝑠)0 colour, colour limits must beset so as to assure a sample compose exclusively of CS. With this in mind and forthe purpose of this work, a star will be catalogued as a CS if their (𝐽 −𝐾𝑠)0 colourfalls in the range:1.4 < (𝐽 −𝐾𝑠)0 < 2.0. (3.1)The blue limit ensures that the CS are well separated from the oxygen-rich AGBstars. The red limit is to avoid contamination from extreme CS (X-CS in Figure35Figure 3.1: (𝐽−𝐾𝑠)0, 𝑀𝐽 colour-magnitude diagrams for the Large (left) andSmall (right) Magellanic Clouds. Over-plotted in orange markers are thespectroscopic carbon stars catalogues. The vertical red lines represent thecolour limits for the carbon star region (1.4< (𝐽 −𝐾𝑠)0 < 2.0).1.2). Extreme (or obscured) CS are CS with larger (𝐽 −𝐾𝑠) colours due to dustycircumstellar envelopes [254, 181, 245, 28]. Contrary to the relatively constantbrightness in the J-band of the CS used in this paper, the J magnitude of extremeCS drops as they get redder. The values for the colour limits set in equation 3.1 arein agreement with the colour range found by Cioni et al. [53], Cioni and Habing[50] and Weinberg and Nikolaev [247].When considerable contamination of non-member stars in the CS colour rangeis encountered, a slight magnitude cut at the faint end may be necessary. Thiscut will be considered on a case-by-case basis. It is important to mention fornow, that the robustness of the distance determination method ensures that a slightmagnitude cut does not disrupt the estimation of the distance (see Sections 3.2and 3.7). However, an excess of faint stars in the CS region has an impact on theluminosity function model fitting procedure. A detailed analysis on the effects ofmagnitude selection will be presented in Section 3.7.363.1.1 The Magellanic CloudsFor the LMC and SMC it was possible to obtain a clean CMD by removing theforeground contamination (as detailed in Chapter 2). In addition, backgroundgalaxies are much fainter than the magnitude range observed by 2MASS. Therefore,for both Magellanic Clouds, it is not necessary to define a faint magnitude limitwithin the CS colour range. Figure 3.1 illustrates the CMD and CS region for theMagellanic Clouds. Within the colour limits defined in equation 3.1 there are 6,024LMC and 1082 SMC CS. The distribution on the sky of the CS within the LMCand SMC are shown in Figure 3.2.Figure 3.2: RA and DEC map for the LMC (left panel) and the SMC (rightpanel). The positions of the carbon stars in both galaxies are indicated by thered points.3.1.2 The target galaxiesFor all the target galaxies (except for the IC 1613 Sibbons et al. catalogue) mostof the background contamination in the CMD was removed using photometrystatistics (see Section 2.2.4). Nevertheless, some faint sources in the CS colourregion remain. These faint stars are not CS but most likely remaining backgroundcontamination.In NGC 6822 there is a significant number of faint sources that results in a smallbump in the luminosity function below magnitude ∼ 20. Examining the histogramin the left upper panel of Figure 3.3 a relatively flat distribution of stars can be seen37Figure 3.3: ((𝐽−𝐾𝑠)0, 𝐽0) CMDs for NGC 6822, IC 1613 (WIRCam), WLM,and NGC 3109. The red dashed lines represent the limits of the CS selectionin colour (1.4 < (𝐽 −𝐾)0 < 2.0). The panels to the right of each CMD showsthe distribution of stars within the colour range defined for the selection of theCS. The black horizontal lines in each plot mark the faint magnitude limit forthe selection of CS in each galaxy.between the two main features. A point roughly at the centre of the flat region isselected to define the faint magnitude limit at 𝐽0=19.15. A total of 484 stars withinthe limits of the CS region are found. The location of these stars in the galaxy isshown in Figure 3.4.For IC 1613 (WIRCam), WLM and NGC 3109, the remaining contamination38Figure 3.4: RA and DEC maps for NGC 6822 and WLMat the faint end of the CS region is minimal. Nevertheless, the few faint sources areremoved to avoid any interference with the luminosity function fitting procedure.The magnitude limits are set to 𝐽0=19.30, 𝐽0=20.20, and 𝐽0=20.60 for IC 1613(WIRCam), WLM and NGC 3109 respectively. Figure 3.3 shows the final limits ofthe CS regions defined on the CMD for these three galaxies. The total number ofCS found in each of these galaxies is 115, 68, and 183 stars and their location inthe galaxy can be seen in Figure 3.4 for WLM and 3.5 for IC 1613 and NGC 3109.In the case of IC 1613 (Sibbons et al.), the cleaning process using the photometryquality flags provided in the catalogue does not remove all faint sources in the CScolour range. The CMD in Figure 2.8 is highly contaminated at the faint end. Forconsistency, the faint magnitude cut for IC 1613 (Sibbons et al.) is set at the samelevel as for IC 1613 (WIRCam): 𝐽0=19.30 mag. Within this limits there are a totalof 125 CS whose locations within the galaxy are shown in Figure 3.5.3.2 The median J magnitude and possible sources ofuncertaintiesThe distance determination method to be presented in Section 3.5 relies on thevalues of the median apparent 𝐽-band magnitude (𝐽0) of the CS selected by their39Figure 3.5: RA and DEC maps for IC 1613 and NGC 3109. The red crosses(‘×’) signal the location of the CS. For IC 1613 the red crosses are specificallythe CS found in the WIRCam data set while blue plus-crosses (‘+’) are theCS found in the Sibbons et al. data. Most CS look as if indicated by anasterisk (‘*’) symbol. This is because most CS found are the same star in bothcatalogues.colour range. One of the possible sources of uncertainty in 𝐽0 are outliers: sourcesthat are not CS but fall in the colour window defined in equation 3.1. These outliersare likely massive oxygen rich stars (at the bright end) and background galaxies orforeground star contamination (at the faint end). In Chapter 2 it was described howmost of these outliers could be removed and also how the cleaning process is not100% effective. To minimize the effects of the remaining outliers, 𝐽0 is estimatedusing the robust median estimator derived in Ripoche et al. [210]. This robustmethod uses maximum likelihood statistics and the estimator 𝑄𝑛 [212] makingit less sensitive to outliers. 𝑄𝑛 (𝑥) is given by the 0.25 quantile of the distances{ 𝑥𝑖 − 𝑥 𝑗 ; 𝑖 < 𝑗}. 𝐽0 of the CS is obtained by minimizing the function given inequation 9 of Ripoche et al. [210]:𝐿 = 𝑁{𝑄2𝑛 (Δ) +𝑚𝑒𝑑2 (Δ)}+𝑁∑︁𝑖=112ln(𝜎2𝐽,𝑖 +𝜎2);Δ =𝐽0− 𝐽0[2(𝜎2𝐽 +𝜎2) ]2 , (3.2)where 𝑁 is the number of CS, the estimator 𝑚𝑒𝑑 returns the median, 𝜎𝐽 are theerrors in 𝐽 and 𝜎 is the intrinsic magnitude dispersion.The analysis to be carried out in Section 3.7 will show that the effect on 𝐽040of a magnitude selection (i.e. the inclusion or exclusion of outliers) is minimal.Nevertheless, there are other sources of uncertainty that can affect the median 𝐽0magnitude. The intrinsic properties of the stars such as metallicity and age mayhave an effect on the magnitude of each star. In order to asses the effect of theseproperties on 𝐽0, a comparison between the CS in the SMC and thermally pulsingAGB (TP-AGB) theoretical isochrones with different initial metallicities and atdifferent ages is carried out.Pastorelli et al. [188, 187] used detailed star-formation histories for the SMC(from [213]) and the LMC in order to characterize the TP-AGB population ofthese galaxies. Their work allowed them to improve the constraints on the physicalparameters that dominate the evolution of the stars at this stage of stellar evo-lution. These new constraints on the TP-AGB phase were incorporated into thePARSEC-COLIBRI stellar isochrones [30, 163]. The PARSEC code [30] evolves thestar from the pre-main sequence stage to the first thermal pulse in the AGB phase.The COLIBRI code [163] takes the star at the evolutionary step left by PARSEC andcontinues the evolution of the AGB star through the AGB thermal pulses until thestar loses its envelope.Theoretical isochrones were generated from the PARSEC-COLIBRI web inter-face1 with PARSEC v1.2S (latest code) and COLIBRI S_35 (which, according toPastorelli et al. [188], gives the best set of models that fit the observed SMC CMD)options. Figure 3.6 shows the ((𝐽−𝐾𝑠)0, 𝑀𝐽 ) CMD of the SMC with isochrones ofdifferent ages (left panel) and different metallicities (right panel). To simulate thebulk of the CS population in the SMC, isochrones of different ages are generatedwith 𝑍𝑖𝑛𝑖 = 0.004 and isochrones of different metallicities are all set to log(age/yr)= 9.1. The range of age and metallicity are selected in order to populate the areaof the CMD where SMC CS are observed. In the plot, all the stars labeled asTP-AGB in the isochrone models are included. Further inspection of the isochronedata, specifically to C/O ratio given for each star, shows that the area within thedefined colour limits for CS is populated predominantly by stars with C/O>1. Forall models, except for the youngest isochrone (log(age/yr) = 8.6), oxygen rich starsfall on the blue side of the CS colour range ((𝐽 −𝐾𝑠)0 <1.3) and all stars with C/Oratios greater than one have (𝐽 −𝐾𝑠)0 colour greater than 0.9.1Publicly available at http://stev.oapd.inaf.it/cmd41Figure 3.6: COLIBRI isochrones for TP-AGB plotted over the ((𝐽−𝐾𝑠)0, 𝑀𝐽 )CMD of the SMC. Left: The different marker types/colours representisochrones of different ages as specified in the inset. All the isochronesare generated using an initial metallicity value of 𝑍𝑖𝑛𝑖 = 0.004. Right: Thedifferent marker types/colours represent isochrones of log(age/yr)=9.1 gen-erated with different initial metallicities as specified in the inset. From theC/O ratio given for each star in the isochrones, for all isochrones (except forlog(age/yr)=8.6) there are no oxygen rich AGB stars in the color range de-marcated by the dashed black vertical lines. All the model stars with C/O<1have colour (𝐽 −𝐾𝑠)0 <1.3.As reflected in Figure 3.6, the theoretical isochrones are consistent with thelocations of the real stars. The bulk the CS from the SMC within the colour limitsspecified in Section 3.1 fall in the range of the isochrones with log(age/yr) between8.9 and 9.3. Older stars are bluer than the blue limit of the CS selection colourwindow. The younger isochrones cross through the CS area but, as seen in thefigure, there are not many SMC CS populating that region of the CMD. The effectsof metallicity on the magnitude of CS, as seen in the right panel of Figure 3.6, are42less pronounced than the change in magnitude with age. Pastorelli et al. [188] findthat the CS population of the SMC is largely populated by stars with 𝑍𝑖𝑛𝑖 = 0.004;nevertheless the presence of CS with different metallicities does not produce a largedispersion in magnitude within the CS colour range.Another possible source of uncertainty in the median 𝐽0 magnitude is stellarvariability. When the distribution of magnitudes of the CS are measured, starsthat might be inherently different and/or stars that could potentially be identicalbut are at different phases in their variation cycle are included. With single epochobservations it is impossible to determine the precise origin of the magnitudedispersion. However, unlike distance determination methods that depend on starby star measurement over a variation cycle (e.g. RR Lyrae and Cepheid variables),the CS method relies on the ensemble of all the CS within the given colour range.This ensemble is an average over the different types of stars but also the same star atdifferent parts of its cycle. Therefore, it is expected that the effects of variability onthe median magnitude average out. The fact that only a single epoch of observationsare needed is one of the great advantages of this method over the use of variablestars.3.3 The carbon star luminosity functionIn Ripoche et al. [210] an examination of the CS luminosity function for theMagellanic Clouds and the Milky Way was carried out. For each of these galaxiesthe median absolute 𝐽-band magnitude (𝑀 𝐽 ) and its dispersion were estimatedusing maximum likelihood statistics to avoid effects from outliers (see Sec. 2.3of [210] for details). In this initial study, it was assumed that (for any galaxy) the𝑀𝐽 magnitudes of the CS, in the given 1.4 < (𝐽 −𝐾𝑠)0) < 2.0 colour range, arenormally distributed about𝑀 𝐽 . However, a normal distribution does not thoroughlydescribe the shape of the CS luminosity functions. As can be seen in Figure 3.7, theGaussian model does not fit the extended tails and fails to describe the asymmetryof the luminosity functions, especially for the LMC.In the present study, the luminosity functions of the CS will be modelled usinga modified Lorentzian distribution function. The expression for the probability43Figure 3.7: LMC and SMC CS luminosity functions. In the left panel theluminosity functions are modelled with a Gaussian function centered at 𝑀 𝐽 .On the right, the same luminosity functions are now fitted using the modifiedLorentzian model specified in equation 3.4.density function (PDF) of a regular Lorentzian distribution can be written as:𝑓 (𝑥;𝑥0,𝑤) ≈ 11+ ( 𝑥−𝑥0𝑤 )2 , (3.3)where 𝑥0 is parameter that specifies the location of the peak (or mode) of thedistribution, and 𝑤 is the scale parameter that specifies the width of the distributionwith 2𝑤 defined as the FWHM.The modified Lorentzian function constructed to model the CS luminosityfunctions contains two extra parameters, one to allow for asymmetry (or skewness)and another to vary the weight of the tails (or kurtosis). The modified LorentzianPDF is defined as:𝑓 (𝐽;𝑚,𝑤, 𝑠, 𝑘) = 𝑎1+ ( 𝐽−𝑚𝑤 )2 + 𝑠 ( 𝐽−𝑚𝑤 )3 + 𝑘 ( 𝐽−𝑚𝑤 )4 , (3.4)where:𝑚 = mode of the distribution (we do not use 𝐽0 to avoid confusion with thecalibrated and deredden 𝐽 magnitude),𝑤 = scale parameter,44𝑠 = skewness parameter,𝑘 = kurtosis parameter,𝑎 = amplitude (height of the peak, used for plotting purposes).The skewness and kurtosis parameters are allowed to take positive or negativevalues, so the functional form may have minima far from the peak (in particularwhen 9𝑠2 > 32𝑘); consequently, Eq. 3.4 is applied only over the modest range ofmagnitudes that carbon stars exhibit.The model from equation 3.4 is fitted to the LMC, SMC, and all four targetgalaxies: NGC 6822, IC 1613 (both WIRCam and Sibbons et al. data sets), WLMand NGC 3109, using two different fitting methods. The first and most directapproach uses non-linear least squares (NLLSQ) fitting. The second method usesan un-binned maximum likelihood estimator (MLE). The main difference betweenthe application of the two fitting methods is how the data are used. For the NLLSQfitting the data are binned while the MLE fits the data directly without the need ofbinning.For the MLE method, the logarithm of likelihood function (𝐿) is given by:ln(𝐿) = 𝑛× ln(1𝑁)+∑︁𝑖ln( 𝑓𝑖), (3.5)where 𝑓 is given by equation 3.4, 𝑛 is the number of stars, and 𝑁 is given by:𝑁 =∫ 𝐽faint𝐽bright𝑓 (𝐽;𝑚,𝑤, 𝑠, 𝑘) 𝑑𝐽. (3.6)To estimate the best-fitting parameters, the model fitting procedure (for boththe binned and un-binned estimations) is repeated over ten-thousand bootstraprealizations for each galaxy. The values and statistical errors for the luminosityfunction parameters from the bootstrapping procedure are summarized in Table3.1. The results for the Lorentzian model parameters between the two methodsagree within the error bars (except for the amplitude which is not included in theun-binned MLE estimation). An important improvement introduced by the MLE isthe reduction in the error bars for the skewparameter. This parameter is fundamental45in determining the most suitable distance calibrator. The reduction in the error barsbetween binned and the MLE estimations is expected given that if the underlyingdistribution of the data is known, then the un-binned MLE is the optimal estimator[178]The parameters obtained for the LMC and SMC show a significant differencebetween the shapes of the luminosity functions of the CS in these two galaxies.TheLMC is skewed towards brighter magnitudes while the SMC is fairly symmetric.The difference in symmetry is also noticeable in the difference between the valueof the mode and 𝑀 𝐽 : in the LMC these two values differ by almost a tenth of amagnitude while in the SMC the values of the median and the mode are almostidentical. We see no significant difference between the values obtained from thebinned and un-binned techniques. The resulting model curves for the luminosityfunctions of the LMC and SMC are shown in Figure 3.8.Figure 3.8: Best-fit modified Lorentzian function for the CS luminosity func-tions for the Magellanic Clouds. In both panels, the straight orange linesrepresent the best-fit model using NLLSQ fitting to the binned data (B), whilethe dark dashed-line represents themodel obtained through an un-binnedMLE(L). The inset specifies the values for the parameters for the Lorentzian func-tion specified in Equation 3.4 for both fitting methods. The vertical lines atthe bottom of the plots indicate the location of 𝑀 𝐽 with its error bars.For NGC 6822, even though the difference in the skewness values obtainedfrom the binned and MLE methods falls slightly outside the error bars, both results46Table 3.1: Median 𝐽-band magnitude (𝑀 𝐽 ) for the Magellanic Clouds and 𝐽0 for the target galaxies) and best-fittingparameters for the CS luminosity function modified Lorentzian model specified in Eq. 3.4. Values and errors are obtainedfrom 10000 bootstrap realizations.Galaxy Median Method Mode Width Skew Kurtosis AmplitudeLMC -6.288+0.005−0.004NLLSQ -6.36+0.01−0.01 0.28+0.01−0.01 -0.43+0.05−0.05 0.12+0.03−0.02 1.34+0.03−0.03MLE -6.36+0.01−0.01 0.30+0.01−0.01 -0.46+0.04−0.04 0.16+0.02−0.02SMC -6.177+0.012−0.016NLLSQ -6.17+0.02−0.02 0.31+0.03−0.03 0.01+0.08−0.07 0.04+0.04−0.02 1.25+0.06−0.06MLE -6.17+0.02−0.02 0.36+0.03−0.03 0.00+0.002−0.003 0.17+0.08−0.06NGC 6822 17.26+0.03−0.02NLLSQ 17.16+0.03−0.03 0.22+0.04−0.03 -0.47+0.12−0.23 0.10+0.16−0.05 1.56+0.14−0.11MLE 17.20+0.02−0.02 0.22+0.02−0.02 -0.24+0.08−0.08 0.03+0.02−0.02IC 1613 18.27+0.06−0.03NLLSQ 18.22+0.09−0.09 0.27+0.10−0.12 -0.12+0.14−0.40 0.01+0.09−0.07 1.20+0.31−0.18(WIRCam) MLE 18.27+0.05−0.05 0.30+0.08−0.06 -0.03+0.04−0.06 -0.05+0.02−0.02IC 1613 18.31+0.04−0.03NLLSQ 18.24+0.20−0.13 0.27+0.11−0.14 -0.23+0.69−0.42 0.07+0.19−0.08 1.26+0.32−0.18(Sibbons et al.) MLE 18.28+0.05−0.05 0.28+0.06−0.04 -0.05+0.05−0.08 -0.03+0.03−0.01WLM 18.84+0.02−0.06NLLSQ 18.86+0.07−0.06 0.23+0.08−0.09 0.10+0.19−0.34 -0.01+0.07−0.05 1.49+0.53−0.28MLE 18.83+0.06−0.06 0.26+0.09−0.06 0.01+0.06−0.02 -0.04+0.03−0.04NGC 3109 19.40+0.03−0.02NLLSQ 19.39+0.05−0.06 0.30+0.08−0.06 -0.03+0.11−0.35 0.02+0.16−0.05 1.26+0.17−0.14MLE 19.40+0.03−0.04 0.29+0.07−0.05 -0.01+0.03−0.10 -0.02+0.07−0.0247indicate that the luminosity function is significantly skewed towards brighter mag-nitudes, similar to that for the LMC. For IC 1613 (both data sets), WLM, and NGC3109 the luminosity functions are more symmetric as for the SMC. As is the casefor the LMC and SMC, for all four of these galaxies, the difference between themode and 𝐽0 values reflects the value of the skewness for each distribution, withsymmetric luminosity functions having mode≈median. In the case of IC 1613, theresults for the WIRCam data set compared with the Sibbons et al. catalogue showno significant difference between the values of parameters obtained from the twodifferent catalogues. The median for the WIRCam data is slightly brighter than thatof the Sibbons et al. data, but it remains within the error bars. The resulting modelcurves for the luminosity functions of the target galaxies are shown in Figure 3.9.From the bootstrapping results we find that the median 𝐽0 magnitude correlatesonly slightly with the mode of the Lorentzian fitting model. The difference betweenthe median and the mode is an indication of the skewness of the distribution. Ifevery sample of stars has approximately the same skewness then the median mustchange as the the mode changes. Nevertheless the median is less affected by smallchanges in the sample compared to the mode and the skew parameter, that is whya correlation between the skewness and the mode is also found. The relationbetween these parameters is logical given the approximately constant behaviour ofthe median; in other words, if the mode moves away from the median, the skewnessincreases negatively or positively depending on the direction of the differencebetween the mode and the median.Among the rest of the model parameters, the width correlates with the kurtosisand has an anti-correlation with the amplitude. This is expected given that asample of CS with less dispersion in the magnitude should have a more pronouncedluminosity function peak. At the same time, given that the magnitude range remainsconstant, a wider distribution should also show more extended tails. For the LMC,a slight anti-correlation between the kurtosis and the skewness can be observed,however this relation is not seen for the rest of the galaxies included in this study.The corner plot between the luminosity function best-fit parameters for theLMC is shown in Figure 3.10.Using a Lorentzian for the fitting has the advantage that themode is less sensitiveto the magnitude selection compared to models that fit for the mean. An abrupt48Figure 3.9: Best-fit modified Lorentzian function for the CS luminosity functions for the target galaxies. All panels,except for the bottom right corner, follow the same description as Figure 3.8. The bottom right panel shows the luminosityfunctions obtained from the MLE method for both data sets superimposed over each other for visual comparison.49M J = − 6.288+0 .005− 0.004− 6.385− 6.370− 6.355− 6.340mm = − 6.36+0 .01− 0.010.2550.2700.2850.3000.315ww = 0.28+0 .01− 0.01− 0.56− 0.48− 0.40− 0.32− 0.24ss = − 0.43+0 .05− 0.050.090.120.150.18kk = 0.12+0 .03− 0.02− 6.295− 6.290− 6.285− 6.280− 6.275M J1.321.351.381.41a− 6.385− 6.370− 6.355− 6.340m0.2550.2700.2850.3000.315w− 0.56− 0.48− 0.40− 0.32− 0.24s0.090.120.150.18k1.321.351.381.41aa = 1.36+0 .03− 0.03− 6.37− 6.36− 6.35− 6.34m = − 6.36+0 .01− 0.01m0.2850.3000.315w− 0.55− 0.50− 0.45− 0.40− 0.35s0.120.150.180.21k− 6.295− 6.290− 6.285− 6.280− 6.275M J− 6.37− 6.36− 6.35− 6.34mw = 0.30+0 .01− 0.010.2850.3000.315ws = − 0.46+0 .04− 0.04− 0.55− 0.50− 0.45− 0.40− 0.35sk = 0.16+0 .02− 0.020.120.150.180.21kM J = − 6.288+0 .005− 0.004Un-binned MLENLLSQ (binned data)Figure 3.10: Corner plots for the LMC [80] showing the correlations amongthe parameters for the modified Lorentzian model and with 𝑀 𝐽 over 10,000bootstrap realizations of the CS sample. The reverse corner plots shows theresults when using the un-binned MLE fitting method, while the corner plotat the bottom shows the results for the NLLSQ fitting to the binned data.The values at the top of each distribution are the median and 16th and 84thpercentiles, about one-sigma for a normal distribution .50magnitude cut that prevents the luminosity function from smoothly declining willperturb the fitting of the tails of the distribution, affecting the kurtosis parameters.Even if the kurtosis parameter is affected by the location of the peak, the mode andskewness are not affected. As will be shown in Sec. 3.5, the mode and skewnessare the most important parameters in the distance determination method. Theseonly correlate with each other and are not significantly affected by the magnitudeselection.3.4 Metallicity and star formation history effects on theluminosity functionIt is evident from Figures 3.8 and 3.9 that the luminosity function of the CS isdifferent for different galaxies. From Table 3.1, the main differences betweenthe CS luminosity function of the LMC and SMC are the mode and the skewparameter. The LMC distribution of CS is skewed towards brighter magnitudeswhile for the SMC the distribution is more symmetric and it peaks at a faintermagnitude. Various studies have linked the differences in the characteristics of theCS luminosity functions to intrinsic global properties of the host galaxy such asmetallicity and star formation history (SFH). The determination of these propertiesfor galaxies in the local group (such as the galaxies in this study) has been the focusof multiple investigations. In the next subsections the possible effects on the CSluminosity function are discussed.3.4.1 MetallicityThere are a vast number of studies on the metallicity of the LMC and SMC. Moststudies agree the LMC has a metallicity gradient and on average a higher metallicitythan the SMC.Choudhury et al. [46, 45, 44] created high-resolutionmetallicitymapsof the LMC and SMC using three large photometric surveys: the Magellanic CloudPhotometry Survey, the Optical Gravitational Lensing Experiment (OGLE III) inthe optical bands, and the VISTA Survey of the Magellanic Clouds in the near-infrared. From the CMDs the authors selected red giant stars and used the slope ofthe RGB as an indicator of the mean metallicity of sub-regions of the galaxy. Theslopes were calibrated to metallicity using spectroscopic data of RGB stars. In their51study, they found shallow radial metallicity gradients for both the LMC and SMCwith average metallicities of [Fe/H]LMC = −0.38 and [Fe/H]SMC = −0.95 usingthe optical band data, and [Fe/H]LMC = −0.42 and [Fe/H]SMC = −0.97 using thenear-infrared survey.Another recent study by Nataf et al. [177] compiled RGB stellar spectral datafrom the Apache Point Observatory Galactic Evolution Experiment (APOGEE) andfound mean metallicities [Fe/H] = −0.64 and [Fe/H] = −0.88 for the LMC andSMC respectively.Gieren et al. [101], while studying the effects of metallicity on the Cepheidperiod-luminosity relations, compiledmetallicity values fromvariousCepheid stud-ies (see section 4 of [101] for references) arriving at mean metallicity values of[Fe/H] = −0.34 and [Fe/H] = −0.75 for the LMC and SMC respectively.AGB stars have also been used to estimate metallicity. Because the transitionfrom oxygen-rich to carbon-richAGB stars depends onmetallicity, the ratio betweenthe number of CS stars and spectral type M AGB stars (C/M ratio) can be used as atracer of metallicity in local group galaxies: e.g. [206, 196, 31, 50, 52, 49, 76, 116].Among the numerous publications, Cioni [49] found differences in the C/M ratioof the LMC and SMC (and M33). The authors updated the [Fe/H] vs. log(C/M)calibration from Battinelli and Demers [10] and used it to convert observed C/Mratios into metallicity values. Cioni results showed that the metallicity in the LMCdecreases radially with amaximum in themetallicity distribution at [Fe/H] ∼ −1.15while they found no clear trend for the SMC and a constant value of [Fe/H] ∼ −1.25.Variations between the estimates can be attributed to different tracers usedto measure metallicity and different samples of stars. The location of stars in theMagellanic Clouds can affect the metallicity estimates as there are natural variationsin chemical enrichment depending, for example, on the the SFH of the galaxy (seeSection 3.4.2).For the target galaxies, NGC6822, IC 1613,WLMandNGC3109, there are alsoa wide range of values in metallicity estimations. For NGC 6822 Venn et al. [246]found ⟨[Fe/H]⟩ = −0.49± 0.22 using the spectra of A-type super giants. Górskiet al. [108] used the colour of RGB stars to estimate a metallicity of ⟨[Fe/H]⟩ =−1.06± 0.09. Swan et al. [239] used Ca II triplet spectra of red giant stars andfound ⟨[Fe/H]⟩ = −0.84± 0.04. Recently, Hirschauer et al. [124] found a global52[Fe/H] = −1.286±0.096 using AGB stars. See Table 3 of Swan et al. [239] for asummary of other [Fe/H] values for NGC 6822.For IC 1613, Skillman et al. [225] used 𝐻𝑆𝑇 photometry to derive a broadmetallicity range −1.3 < [Fe/H] < −0.7 and evidence of constant star formation.Górski et al. [108] used the colour of RGB stars to estimate a mean metallicity of[Fe/H] = −1.50± 0.08. Using the spectra of red giants Kirby et al. [141] found[Fe/H] = −1.19± 0.01. Using the C/M ratio Sibbons et al. [223] and Chun et al.[47] obtained [Fe/H] = −1.26±0.07 and [Fe/H] = −1.23±0.06 respectively.For NGC 3109 and WLM there are less recent estimates of the metallicity.In the case of WLM Leaman et al. [145, 144] using RGB stars obtained a mean[Fe/H]= −1.28±0.02, while Urbaneja et al. [242] using blue super giant stars found[Fe/H]= −0.87±0.06. For NGC 3109, using RGB stars Hidalgo et al. [123] found⟨[Fe/H]⟩ = −1.84± 0.2. Using blue super giant stars Evans et al. [74] found andaverage metallicity of -0.93±0.07, while Hosek et al. [127], with a larger sample ofthe same type of stars, estimated the average metallicity to be -0.67±0.13.Theoretical studies have predicted a connection between the host galaxy’smetal-licity and the shape of its CS luminosity function. Marigo et al. [162] reproducedthe observed luminosity functions of the CS in the Magellanic Clouds using syn-thetic AGB star models. They concluded that the difference in the location ofthe peak of the luminosity function is due to the difference in metallicity betweenthe two galaxies, specifically how metallicity affects the third dredge-up efficiencyparameter. The authors found that the difference in the peak is reproduced bythe SMC having a higher efficiency. This is in agreement with other theoreticalstudies that indicate that dredge-up efficiency increases as metallicity decreases(e.g. [139, 59, 163, 27]). A lower metallicity means lower oxygen content in theatmosphere of the star. If there is less oxygen then less carbon is needed to bringthe C/O ratio above 1 during the third dredge-up events..Pastorelli et al. [188, 187] find that the current population of CS in the LMCis best represented as having a metallicity of 𝑍𝑖𝑛𝑖 = 0.008 while for the SMC𝑍𝑖𝑛𝑖 = 0.004, in agreement with the LMC having, on average, a higher observedmetallicity. Their best models also exhibit a lower efficiency in the third dredge-up parameter for the LMC. They explain that one of the consequences of a lowerefficiency is that the C/O > 1 transition takes place at brighter magnitudes. Through53its effect on the third dredge-up efficiency parameter, metallicity plays a role indefining the shape of the CS luminosity function with higher metallicities shiftingthe luminosity function peak to brighter magnitudes. However, this is a parameterfor which defining a global value for a galaxy is not straightforward. As we cansee from the various values quoted above, the average metallicity depends on themethod and stellar population used to measure it. As a result, the global metallicityof a galaxy may not necessarily be representative of the metallicity of the bulk ofthe CS population.3.4.2 Star formation historySFH may also play an important role in the shape of the CS luminosity function.Marigo et al. [162] claim that the drop at bright magnitudes in the CS luminosityfunction of the LMC can be attributed to a recent decrease in the star formationrate (SFR). A summary of the most important global aspects of the SFH for thegalaxies included in this study is presented below.The Magellanic Clouds have been the target of numerous SFH studies. In thecase of the LMC, Harris and Zaritsky [118] and Meschin et al. [173] find similarresults for the global SFH with the main events at slightly different ages. Bothstudies report an early burst followed by a quiescent period that lasted severalbillion years, placed at a time between approximately 12 to 5 Gyr ago by Harrisand Zaritsky and between ∼8 and 4 Gyr ago by Meschin et al. Both studies agreethat this period of low activity coincides with the so called age-gap. This periodreceives its name from the lack of star clusters with ages within the quiescentperiod. After this inactive period ∼ 5 Gyr ago, a second episode of star formationtook place characterized by bursts localized in specific regions of the galaxy. Mazziet al. [166] recently derived a spatially resolved SFH for the LMC finding the mostintense period of star formation between 4 and 0.5 Gyr ago with a SFR of 0.3 M⊙yr−1 . Harris and Zaritsky report an overall SFR of 0.2 M⊙ yr−1 with peaks at 2Gyr, 500 Myr, 100 Myr and 12 Myr ago. Meschin et al. finds regions where starformation ceased approximately ∼ 1 Gyr ago.Multiple studies arrive at the conclusion that the age-metallicity relation (AMR)of the LMC reflects its SFH. Harris and Zaritsky find the AMR history shows an54early epoch of rapid chemical enrichment, followed by constant metallicity valuesduring the quiescent period. Finally, a steady increase is observed following the re-ignition of star formation. The global AMRdescribed byHarris and Zaritsky agreeswith the AMR relation derived from the LMC’s star cluster population [183, 69]and other studies of field stars [36, 173].The SFH of the SMC is characterized by different aged populations havingdifferent spatial distributions [117, 48, 213]. Older populations seem to have asmooth spatial distribution, whereas recent star formation is limited to specificareas of the galaxy. Similar to the LMC, Harris and Zaritsky [117] and Rubeleet al. [213] find a quiescent period of star formation between ∼ 8 and ∼ 3 Gyr ago.Harris and Zaritsky [117] found that 50% of the stellar mass was formed earlierthan 8.4 Gyr ago with low star formation activity between 3 and 8.4 Gyr ago. Onthe other hand, Rubele et al. [213] found that 80% of the mass was formed between3.5 and 8 Gyr ago with 50% of the mass formed prior to 6.3 Gyr. Weisz et al. [248]find two pronounced peaks in the SFH of the SMC at 4.5 and 9 Gyr ago. The AMRderived by Rubele et al. [213] shows a considerable increase in metallicity between13 and 1.5 Gyr reaching Z∼0.003 followed a slow increase until 130 Myr ago whenit plateaus at present-day values ( Z∼0.004).Gallart et al. [98, 97] carried out one of the first attempts to construct an overallSFH for NGC 6822. The authors found a constant or slightly declining SFR overthe last few Gyr followed by an enhancement in the star formation activity between100-200 Myr ago. Wyder [252] found an overall constant or slightly increasingSFR with a possible period of low activity between 3 to 5 Gyr ago. Cannon et al.[33] found that over 50% of the stars in the galaxy were formed in the last ∼5 Gyrwith an increase in the SFR in the last 50 Myr. The results of Fusco et al. [91] aregenerally consistent with Cannon et al. [33], but they find a slow decrease in theSFR in some fields over the past Gyr. Additionally, Fusco et al. [91] are able toderive the age-metallicity relation for the galaxy finding an increase in metallicitywith time throughout the entire galaxy.For IC 1613, Skillman et al. [224] derived the mean SFH of the galaxy usingone HST/ACS field centered at the half-light radius of the galaxy. Their findingsindicate a constant SFH over the lifetime of the galaxy with variations of ∼30% orless in the SFR. The continuous star formation in this galaxy is accompanied by a55smooth build-up in metallicity over time.A similar continuous SFH is found for WLM. Dolphin [70] sets the beginningof the star formation at ∼12 Gyr ago with 50% of the stellar mass formed before9 Gyr ago. The SFR was found to decrease until 1-2.5 Gyr ago when it started toincrease again and has continued to rise until the present with an average SFR of∼0.006 solar mass (M⊙) yr−1. The rate found by Dolphin is in good agreementwith the average rate of ∼0.008 M⊙ yr−1 found recently by Mondal et al. [174].Albers et al. [3] also derived the SFH of WLM finding a constant rising SFR with50% of the stellar mass formed in the last 5 Gyr. In addition to the constant SFH,Albers et al. find constant chemical enrichment with the logarithm of the metallicityincreasing linearly with time.NGC 3109 does not seem to have a detailed study of its SFH. Bresolin et al.[29] and Greggio et al. [114] studied the star formation in this galaxy but only forthe last 1 Gyr. Both studies find that NGC 3109 is still actively forming stars.Additionally, Bresolin et al. [29] find no dependence of age on position. Recently astudy of the SFH of multiple dwarf galaxies gives a SFH for NGC 3109 composedof two star forming episodes at 10 and 1 Gyr ago with an average star forming rateof 0.0025 M⊙ yr−1.3.4.3 DiscussionPastorelli et al. [188] used the AMR derived from the SFH to find the initial metal-licity distribution for the TP-AGB stars in the SMC and LMC. They successfullymodeled the luminosity function of AGB stars in the SMC by calibrating syntheticTP-AGB evolution models. For the LMC, the best-fitting models calibrated for theSMC showed an excess of faint stars, a deficit of stars in the brighter magnitudebins, and an overestimation in the number of CS. Pastorelli et al. [187] resolved thediscrepancy between observations and synthetic population in the LMC by adjust-ing the third dredge-up parameters in the stellar evolution models. First, the authorsincreased the value of the core mass for which the efficiency parameter reaches itsmaximum value. This resulted in a decrease in the number of faint stars. Second,they decreased the maximum efficiency of the third dredge-up which, as mentionedpreviously, shifts the peak of the distribution to brighter magnitudes.56It is not possible to determine at this timewhether the shape of the CS luminosityfunction is driven mainly by metallicity or SFH. In Pastorelli et al. [188, 187], SFHis an essential part of their analysis to derive evolutionary models capable ofreproducing the observed TP-AGB populations of the SMC and LMC. However,the key parameters that defined the difference between the best-fitting models forthe LMC and SMC are the third dredge-up parameters, which as mentioned insection 3.4.1, are strongly related to metallicity.If the distance to a galaxy is measured against a calibrator with a significantlydifferent metallicity (or SFH) it may introduce a systematic error in the distancedetermination (see Section 3.8). Ripoche et al. [210] explored the possibility ofusing a composite magnitude to account for metallicity effects. This compositemagnitude includes a (𝐽 −𝐾𝑠)0 colour term and a metallicity correction term, bothmultiplied by the slope of the CS feature in the ((𝐽 −𝐾𝑠)0, 𝐽0) CMD within theCS colour limits (1.4< (𝐽 −𝐾𝑠)0 < 2.0). This proved useful to bring the values of𝑀 𝐽 for the LMC and SMC closer into agreement but it does not account for thedifference in the skew parameter.3.5 Distance determination3.5.1 Determining the most suitable calibratorFor the distance estimation method presented in this study, a fundamental step is thedetermination of the most suitable calibrator for each target galaxy; that is to say,whether a galaxy is "LMC-like" or "SMC-like". The first attempt was to comparethe global metallicities and SFHs of the target galaxies to those of the calibrators,as detailed in Section 3.4. These properties do not seem to be representative ofthe CS populations. Detailed analysis of TP-AGB stellar population, such as theones carried out by [188, 187] for the Magellanic Clouds, would be needed inthe target galaxies in order to use SFH as the basis for classification. Instead, it isplausible that it is the observed shape of the luminosity function that tells somethingfundamental about the properties of the galaxies. For example, the skewness couldbe an indication of the value of the third dredge-up efficiency parameter.To ascertain whether it is more appropriate to use the LMC or SMC as the57calibrator, the target galaxies are classified as "LMC-like" or "SMC-like" based onthe shape parameters of the CS luminosity function itself. In section 3.3 it wasmentioned that the main differences between the CS luminosity functions are themagnitude at which the distribution peaks (i.e. the mode of the distribution) andthe symmetry of the distribution. There are a few parameters from the Lorentzianmodel that can be used to asses the differences in the distributions. For example, thedifference between the values of the mode and the median changes; the smaller thedifference between the mode and the median, the more symmetric the distribution.Another indicator of the shape is the skew parameter; the deviation from zero ofthe skewness indicates how asymmetric the luminosity function is, while the signtells whether the luminosity function is inclined towards faint (positive skewness)or bright (negative skewness) magnitudes.In principle, the difference between the mean and median could also be used,as it is also an indicator of how asymmetric the distribution is. However, the meanis highly sensitive to outliers and variations in the limits defined for the CS region(see Sec. 3.7). Using the parameters from the Lorentzian model is preferable, asthis method is less sensitive to those variations and is able to identify the mode ofthe distribution in a robust manner.The mode can be well defined by the Lorentzian model but by itself doesnot give enough information. We will therefore employ the skew parameter todetermine whether the LMC or SMC is the better calibrator. The skew parameteris mostly defined by the structure of the luminosity function near the peak of thedistribution. This parameter has the advantage that it is not sensitive to variationsin the magnitude limits.3.5.2 The distance modulusTo find the true distance modulus (𝜇0) of a given galaxy (let it be called Galaxy X)the median 𝐽0 magnitude (𝐽0) of the CS in Galaxy X is compared to the medianof the absolute 𝐽 magnitude (𝑀 𝐽 ) of the CS in the most suitable calibrator galaxy.Then, given a Galaxy X with skewness 𝑠𝑋 and 𝐽0= 𝐽0,𝑋 the distance modulus of58Galaxy X is defined as:𝜇0,𝑋 =𝐽0,𝑋 −𝑀 𝐽LMC , if 𝑠𝑋 < −0.20𝐽0,𝑋 −𝑀 𝐽SMC , otherwise,(3.7)where the value −0.20 is approximately midway between the skewness of the hetwo calibrators (the LMC and SMC). The absolute magnitudes for the MagellanicClouds are set by adopting 𝜇0,LMC =18.477± 0.004 (statistical)± 0.026 (systematic)[192] for the LMC, and 𝜇0,SMC = 18.977 ± 0.016 (statistical) ± 0.028 (systematic)[111] for the SMC.To determine the distance modulus, the median magnitude is preferred over themode of the luminosity function. The median is a more robust estimator comparedto the mode which is obtained through the fitting of a function which includes other4 parameters (see Equation 3.4). As a result, for a given sample of CS, the statisticalerrors are generally larger on the mode than on the median. Even though the modeis not used for estimating the distance, the fitting procedure is still necessary toobtain the skew parameter and classify the target galaxy in order to determine themost suitable calibrator.3.6 ResultsEach target galaxy’s CS luminosity function was fitted using two different methods:a direct fit to the binned data using NLLSQ fitting and an un-binned MLE. Asdiscussed in section 3.3 the values obtained for the model parameters using bothmethods agree within the error bars. In this section the skew parameter is usedto classify the target galaxies into “LMC-like" or “SMC-like", determine the mostsuitable calibrator and estimate the target’s distance modulus.3.6.1 NGC 6822By examining the best-fit parameters in Table 3.1 it is possible to describe the shapeof the luminosity function. In the case of NGC 6822 the skew parameter indicatesthat the luminosity function is significantly skewed towards brighter magnitudeswhile its width seems narrower than the rest of the galaxies in the sample. Compared59to the LMC, the luminosity function of NGC 6822 is only slightly narrower withmoderately less weight on the tails. According to the values obtained for theskewness: 𝑠LSQ = −0.47 and 𝑠MLE = −0.24 both < −0.20, NGC 6822 is classifiedas an "LMC-like" galaxy. Using equation 3.7 and the LMC as the most suitablecalibrator we obtain a true distance modulus of 𝜇0 = 23.55±0.03 (statistical). TheCS luminosity function of NGC 6822 shifted by 𝜇0 is shown in Figure 3.11 plottedon top of the luminosity function of the CS in the LMC to compare the resemblancebetween the two distribution.Two other independent estimates of the distance to NGC 6822 using CS haverecently been published. Zgirski et al. [253] obtained a distance modulus of 𝜇0 =23.243±0.037 (stat), while Freedman and Madore [83] reports 𝜇0 = 23.44±0.02(stat) (not formally in the paper but as part of the appendix). All three resultsare in disagreement with each other. NGC 6822 is a challenging target due tothe considerable amount of reddening in the line-of-sight, in the case of the threeestimates mentioned here evenmodifying the different 𝐸 (𝐵−𝑉) values used in eachstudy does not bring the results into agreement. A more detailed discussion on theeffects of different extinction values and a comparison between these results andother independently measured distances to NGC 6822 is presented in Chapter 9.3.6.2 IC 1613The results for the luminosity function fitting parameters (mode, width, skewnessand kurtosis) are in very good agreement between the data sets for both the binnedNLLSQ fitting and the un-binned MLE methods for IC 1613. The one parameterthat stands out is the skewness obtained for the Sibbons data using the binnedmethod, which seems to be lower than expected. Nevertheless, we have to considerthat the error bars for this specific case are significantly larger than in all other skewparameter results.To classify IC 1613 we consider the skewness values obtained for both datasets, except for the binned estimation carried out on the Sibbons catalogue giventhe large uncertainty in the result. Considering the other three skewness valuesobtained for IC 1613, this galaxy is catalogued as an "SMC-like" galaxy. Using theSMC as calibrator we obtain 𝜇0 = 24.49± 0.04 (statistical) and 𝜇0 = 24.45± 0.0560Figure 3.11: CS luminosity functions of the target galaxies plotted over the luminosity function of the most suitablecalibrator. In the left panel the LMC (solid orange line) and NGC 6822 (dashed blue line). In the centre panel is theSMC (solid orange line) together with WLM (dashed light red line), and NGC 3109 (dashed-dotted blue line). In theright panel are shown the results for IC 1613 from the two different data sets available plotted over the SMC. Theapparent 𝐽0 magnitude CS distributions for the target galaxies from Figure 3.11 have been transformed to absolutemagnitudes using the distance modulus (𝜇0) specified in each plot. The errors shown for the distance moduli are thestatistical errors obtained from 10000 bootstrap realizations.61(statistical) for the Sibbons and WIRCam data respectively. The distance moduliobtained from the two different data sets are in good agreement within the errorbars.Only one other estimate of the distance modulus to IC 1613 using CS has beenperformed to date. Freedman and Madore [83] report 𝜇0 = 24.36± 0.03 (stat).This result is not consistent with the distance modulus found in this work, andthe statistical errors are not enough to bring them into agreement. The differencebetween the two estimates could be explained if different extinction corrections areapplied. The Freedman andMadore IC 1613 data come from the Sibbons cataloguebut the authors do not specify how the data were corrected for reddening. In theoriginal analysis of IC 1613 for this thesis, the data published by Sibbons et al.[223] was misinterpreted. The Sibbons et al. catalogue had already been correctedfor foreground reddening using the [219] extinction maps, because this detailedwas missed, the correction was applied again ending with an overly de-redden dataset. This over-correction of the reddening resulted in a distance modulus of 𝜇0 =24.34±0.05, in agreementwith Freedman andMadore. A similarmisunderstandingby Freedman and Madore would explain the difference with the value obtained inthis thesis.3.6.3 WLM & NGC 3109WLM and NGC 3109 show highly symmetric CS luminosity functions with skewvalues of 𝑠𝑏𝑖𝑛𝑛𝑒𝑑 = −0.10 and 𝑠𝑀𝐿𝐸 = 0.01 for WLM and 𝑠binned = −0.03 and𝑠MLE = −0.01 for NGC 3109. All these values are greater than the limit specified inequation 3.7, therefore, both galaxies are classified as "SMC-like" galaxies. Usingthe most suitable calibrator for these two galaxies we obtain distance moduli of𝜇0 = 25.02±0.04 (statistical) for WLM and 𝜇0 = 25.58±0.03 (statistical) for NGC3109.Two other studies using CS report a distance modulus for NGC 3109 and threefor WLM. Zgirski et al. [253] obtained a distance modulus of 𝜇0 = 24.954± 0.08(stat) for WLM and 𝜇0 = 25.517±0.046 (stat) for NGC 3109, while Freedman andMadore [83] reports 𝜇0 = 24.97±0.05 (stat) and 𝜇0 = 25.56±0.05 (stat) for WLMand NGC 3109 respectively. In the case of [146] they find a distance modulus to62WLM of 𝜇0 = 24.97±0.02 (stat). The results reported in the current study and thepapers cited agree within the error bars.Figure 3.11 summarizes the results presented above. The CS luminosity func-tion for each galaxy is shifted by its distance modulus and plotted over the CSluminosity function of its most suitable calibrator. The plots in Figure 3.11 allowthe reader to visually compare the shape of the distributions and to see how theyresemble the luminosity functions of the calibrators.3.7 The median vs the mean J magnitudeIn contrast with Madore and Freedman [154], in this work the median 𝐽 magnitudeis used instead of the mean. This selection is based on the fact that the medianis less affected by a 𝐽 magnitude selection as will be shown in this Section. Forthe LMC and SMC, magnitude limits are not necessary, and all the stars in the1.4 < (𝐽 −𝐾𝑠)0 < 2.0 colour range are used. By contrast, for the target galaxies, itwas necessary to establish a faint magnitude limit for the CS samples (see Section3.1).To test the effects of a faint magnitude cut on the mean (⟨𝑀𝐽⟩) and the median(𝑀 𝐽 ) of the CS, both statistics are calculated applying different faint magnitudecutoffs to the LMC and SMC CS samples. Figure 3.12 shows how the mean and themedian vary as the faint magnitude limit is pushed towards brighter magnitudes.For both the LMC and SMC the median varies less than the mean with respect tothe values of the full sample.Looking at the the luminosity functions in Figure 3.8 it is possible to see thatthe number of stars per bin starts to increase (at the faint end of the distribution) atmagnitude𝑀𝐽 =−5. The change between themedian for all the stars and themedianfor −∞ < 𝑀𝐽 ≤ −5 is 0.005 mag for the LMC and 0.001 mag for the SMC, whilethe mean varies 0.025 mag and 0.01 mag for the LMC and SMC respectively. Themedian is not significantly affected by a faint magnitude cut, in fact, the variationsdue to a magnitude cut are comparable to the error in the median. On the otherhand, the variations in the mean are of the order of the error bars in the distancemodulus obtained for NGC 6822.For the target galaxies, shifting the faint magnitude limit by 0.5 mag towards63fainter magnitudes does not significantly change 𝐽0 as most of the contamination atthe faint end of the CMD was removed. For all target galaxies, the differences in𝐽0 are ≲ 0.005 mag, except for the IC 1613-Sibbons catalogue where the differencereaches 0.04 mag (it is important to remind the reader that the Sibbons cataloguewas not processed in the same manner as the other data sets, see Chapter 2). Onthe other hand, shifting the faint magnitude limit by 0.5 mag significantly affects⟨𝐽0⟩, with Δ ⟨𝐽0⟩ ∼ 0.1. . On the other hand, shifting the faint magnitude selectiontowards brighter magnitudes by 0.5 mag for NGC 6822 changes 𝐽0 and ⟨𝐽0⟩ by∼ 0.01 and ∼ 0.02 mag respectively. In the same order the changes for IC 1613-WIRCam, WLM, NGC 3109 and IC 1613-Sibbons are: ∼ 0.05 and ∼ 0.1 mag,∼ 0.003 and ∼ 0.02 mag, ∼ 0.01 and ∼ 0.04 mag, and ∼ 0.04 and ∼ 0.09 mag. Inall cases the effects are greater on the mean than on the median.Figure 3.12: Values for the mean (⟨𝑀𝐽⟩; red crosses) and the median (𝑀 𝐽 ;green circles) 𝑀𝐽 magnitude for the CS in the LMC top panel and SMCbottom panel using different faint magnitude cutoffs. The magnitude rangeof the sample goes from −∞ to ≤ 𝑀𝐽,cutoff . The colour selection remainsalways at 1.4 < (𝐽 −𝐾𝑠)0 < 2.0. The dashed-red and solid-green lines markthe value of the mean and median for the entire sample of CS.64Testing for the changes in 𝑀 𝐽 and ⟨𝑀𝐽⟩ when a faint magnitude cut is applied,yielded results that indicate the median is a more reliable statistic compared tothe mean. Other CS magnitude selection criteria that can affect the values of 𝑀 𝐽and ⟨𝑀𝐽⟩ are the width in magnitude of the selection box or the location of thecentre of the box. The selection of CS used by [154] covers approximately ∼ 0.8magnitudes in 𝐽. To test the effects of different magnitude ranges a selection boxis centred at a fixed 𝑀𝐽 and the width Δ𝑀𝐽 is varied by 0.1 mag. If, for the LMC,a selection box is centred at 𝑀𝐽 = −6.2 with Δ𝑀𝐽 = ±0.4 mag (i.e. the box coversfrom magnitude −5.8 at the faint end to magnitude −6.6 at the bright end) then𝑀 𝐽 = −6.28 and ⟨𝑀𝐽⟩ = −6.26. Extending the box 0.1 mag on each end, so thatΔ𝑀𝐽 = ±0.5, changes the mean and the median by only ∼ 0.005 mag. If the boxkeeps expanding, at Δ𝑀𝐽 = ±0.6 mag the median reaches the median value of thecomplete sample of CS and it does not vary significantly as the magnitude limitsare pushed even farther apart. The mean only shows slightly larger variations thanthe median, with the tendency to approach the mean of the complete sample as thebox gets wider. The same process is repeated with the box centred at 𝑀𝐽 = −6.3finding similar results in the variations of 𝑀 𝐽 and ⟨𝑀𝐽⟩.On the other hand, if the width of the box is kept fixed at Δ𝑀𝐽 = ±0.4 mag andthe entire box is shifted up or down by 0.1 mag, the changes in 𝑀 𝐽 and ⟨𝑀𝐽⟩ aresignificantly larger with variations around ∼ 0.05 mag.The variations in 𝑀 𝐽 and ⟨𝑀𝐽⟩ when changing Δ𝑀𝐽 are of the order ofmagnitude of the errors in 𝑀 𝐽 ,LMC, as long as Δ𝑀𝐽 > ±0.4. However, changingthe location of the centre of the box produced variations an order of magnitudegreater than the errors in 𝑀 𝐽 ,LMC. Figure 3.13 shows the behaviour of the meanand the median when changing either the magnitude range with a fixed centre orchanging the location of the box with a fixed Δ𝑀𝐽 .The results of these tests indicate that making a tight magnitude selection has aconsiderable effect on the value of both ⟨𝑀𝐽⟩ and 𝑀 𝐽 . The impact of a magnitudeselection can be reduced by establishing a wide magnitude range. For example, ifa box is centred at 𝑀𝐽 = −6.2 but with Δ𝑀𝐽 = ± 1 mag, extending or reducing themagnitude range by ± 0.1 mag causes changes below ∼ 0.005 mag. Nevertheless,shifting the box up and down by ±0.1 mag while keeping a wide Δ𝑀𝐽 = ±1 mag,changes the mean by ∼ 0.01 while the median varies by less than ∼ 0.005.65Figure 3.13: Values for the mean (⟨𝑀𝐽⟩; red crosses) and the median (𝑀 𝐽 ;green circles) 𝑀𝐽 for sub-samples of CS from the LMC. On the top panel themagnitude width (Δ𝑀𝐽 ) is increased by 0.1 mag in every step while keepingthe centre of the selection box at 𝑀𝐽 = −6.2. On the bottom panel Δ𝑀𝐽is kept fixed at ±0.04 magnitudes while the centre of the box is shifted by0.1 mag at a time. The legend is the same as in Figure 3.12.3.8 Suitable versus less suitable calibratorTo test the systematic error introduced by using the less suitable calibrator galaxy104 sub-samples of 120 CS are randomly selected from the LMC and the distancemodulus is calculated using both the LMC and SMC as calibrators. Because thestars in the sub-sample come from the list of CS in the LMC each sub-sample isexpected to have the same distance as the LMC.When the distance to the sub-sampleis measured using the LMC as calibrator the peak of the distribution of distancemoduli is at the expected distance modulus. By contrast, when the distance isestimated using the SMC as calibrator, the distance modulus is off by 0.111 magfrom the expected value. If the same operation is repeated but now with CS drawn66from the SMC, the distance modulus obtained using the LMC as calibrator is againoff by 0.111 mag from the expected value. The top-left panel of Figure 3.14 showsthe distributions of distance moduli for the LMC sub-samples obtained using theLMC and SMC as calibrators.For WLM and IC 1613 (WIRCam) smaller samples of only 30 and 60 stars areused respectively. When IC 1613 is measured against the SMC, a difference of 0.02magnitudes is obtained with respect to the distance modulus for the full sample.This difference is well within the statistical errors. Instead, when using the LMCas calibrator, the distance modulus is off by 0.13 mag. For WLM the result for thedistance modulus using the SMC is only 0.01 mag from the value expected fromthis work. If the less suitable calibrator is used, in this case the LMC, the distancemodulus is off by 0.10 mag. The distance estimations to WLM and IC 1613 do notchange when using only ∼50% of the stars. The tests on WLM and IC 1613 allowsus to show that the distance determination method is robust enough that even for asmall sample of CS the median 𝐽 magnitude is not affected and remains a reliablestandard candle.For the remaining galaxies the subsamples were composed of 120 CS for NGC6822, 60 for IC 1613-Sibbons, and 90 for NGC 3109. The suitable versus lesssuitable calibrator tests return results that are at same order of magnitude as therest of the galaxies: 0.01 and 0.10 mag differences from the expected values for themost and less suitable calibrators respectively.The distributions of distance moduli for the CS sub-samples obtained fromNGC 6822, WLM and NGC 3109 are shown in Figure 3.14 as an example of theresults from this test.For all galaxies, measuring the distance modulus using the less suitable distancecalibrator introduces a systematic error of the order of a tenth of magnitude. A tenthof a magnitude is, for most cases, over twice the size of the statistical error bars onthe distance modulus. On the other hand, when using the most suitable calibrator,the results varied by only a hundredth of a magnitude, a difference below the valueof the average statistical errors.67Figure 3.14: Distributions of distance moduli for 104 random subsamples ofcarbon stars from the LMC, NGC 6822, WLM and NGC 3109. In eachplot the blue dashed line represents the distribution of distance moduli of thespecified galaxy using the LMC (full CS sample) as calibrator, and the orangesolid line is the distance moduli using the SMC (full CS sample) as calibrator.Each carbon star sub-sample is composed of 120 stars for the LMC and NGC6822, 90 for NGC 3109 and only 30 stars for WLM. The black dotted verticalline indicates the location of the expected distance modulus (𝜇0). For thetarget galaxies the expected 𝜇0 is given in Figure 3.11, while for the LMC𝜇0 = 18.477 [192]. Measuring the distance modulus to a galaxy using thewrong calibrator galaxy introduces a systematic error of the order of a tenthof magnitude.68Chapter 4The tip of the red giant branchIn this Chapter the distances to the target galaxies are estimated using the TRGBmethod. This will allow a comparison of the CS distance results with the resultsfrom a widely used method on the same data sets. It will also allow for a test of theprecision and accuracy of the CS method and compare its performance against theTRGB.4.1 Red giant selectionFor the TRGB detection method, instead of using the 𝐽0 or (𝐾𝑠)0 magnitudesdirectly, the compositemagnitude𝑇 introduced byMadore et al. [157] and calibratedfor the near-infrared 𝐽 and 𝐾𝑠 filters by Madore et al. [155] and Hoyt et al. [130]was used. The 𝑇 magnitude is meant to remove the effects of metallicity onthe magnitude of the TRGB. In the CMD, metallicity effects can be seen in thedeclining slope in the magnitudes of the TRGB with colour, in the red or bluedirection depending on the observing band. In the near-infrared filters, stars at thebluer end of the TRGB appear slightly fainter than at the redder end. The slope ofthe TRGB can be estimated and used to rectify it. The 𝑇 magnitude is constructedin an attempt to rectify the TRGB by correcting each star’s magnitude based onits colour and the slope of the TRGB, bringing the previously inclined TRGB to aleveled discontinuity between the RGB and the AGB.Based on the definition of𝑇 and the slopes found in [155] and [130], we assume69the composite (𝐾𝑠)0 magnitude to be:𝑇(𝐾𝑠)0 = (𝐾𝑠)0 +1.85[(𝐽 −𝐾𝑠)0−1.00] . (4.1)In the case of the LMC and SMC, (𝐾𝑠)0 become the absolute magnitude 𝑀𝐾𝑠 . Theresulting ((𝐽 −𝐾𝑠)0, 𝑇(𝐾𝑠)0) CMDs are presented in Figure 4.1 for the LMC andSMC, and Figure 4.2 for the target galaxies.Figure 4.1: ((𝐽 −𝐾𝑠)0, 𝑇𝑀𝐾𝑠 ) CMD for the LMC (left panel) and the SMC(right panel). The red-shaded area demarcates the region for the red giantsselection. The horizontal red line indicates the location of the TRGB deter-mined in Section 4.2.4.2 TRGB detectionTo detect the TRGB, the luminosity function of theMagellanic Clouds and the targetgalaxies are constructed by taking the normalized frequency of stars in magnitudebins of size 0.005 mag for the LMC, 0.01 mag for the SMC and NGC 6822, and0.015 mag for the IC 1613, WLM and NGC 3109. The luminosity function is thensmoothed using a 1-D Gaussian filter. Once the probability density function (i.e.binned luminosity function) has been smoothed, a Sobel filter is applied to detect70Figure 4.2: ((𝐽 −𝐾𝑠)0, 𝑇(𝐾𝑠)0) CMD for the target galaxies. The red-shadedarea demarcates the region for the red giants selection. The horizontal red lineindicates the location of the TRGB whose magnitude is detailed in the insetsof each panel.71the TRGB. This method of detection was first introduced by Lee et al. [148] andthe Sobel filter places a mask ([−1,0,+1]) along the smoothed luminosity functionwhich gives the value of the first derivative at each point along the curve. Followingthe procedure in Hoyt et al. [129], the Sobel filter output is then multiplied by aPoisson weighting scheme: (𝑁𝑖+1−𝑁𝑖−1) /√𝑁𝑖−1 +𝑁𝑖+1. This process works asan edge-response function as the maximum value of the Sobel filter marks wherethe slope of the luminosity function is the steepest. This maximum indicates thelocation of the TRGB. The binned and smoothed luminosity functions, and theSobel filter response are shown in Figure 4.3 for the LMC and SMC, and Figure4.4 for the target galaxies.For each galaxy, theTRGB is estimated over ten-thousand bootstrap realizations.The values and statistical errors for the TRGB from the bootstrapping procedureare summarized in Table 4.1 for the Magellanic Clouds and Table 4.2 for the targetgalaxies. The location of the TRGB on the CMDs are shown in Figure 4.1 for theLMC and SMC, and Figure 4.2 for the target galaxies.Table 4.1: TRGB absolute 𝑇 magnitudes (𝑇𝑇𝑅𝐺𝐵𝑀𝐾𝑠 ) for the Magellanic Clouds.The results and statistical errors come from ten-thousand bootstrap realizations.The TRGB magnitudes are also shown on the colour-magnitude diagrams inFigure 4.1.LMC SMC𝑇𝑇𝑅𝐺𝐵𝑀𝐾𝑠-6.290+0.015−0.010 -6.175+0.017−0.013Table 4.2: TRGB apparent 𝑇 magnitudes (𝑇𝑇𝑅𝐺𝐵(𝐾𝑠)0 ). The results and statisticalerrors come from ten-thousand bootstrap realizations. The TRGB magnitudeare also shown on the colour-magnitude diagrams in Figure 4.2.NGC 6822 IC 1613 WLM NGC 3109𝑇𝑇𝑅𝐺𝐵(𝐾𝑠)0 17.21+0.02−0.01 18.29+0.01−0.01 18.85+0.02−0.02 19.42+0.03−0.0272Figure 4.3: 𝑇𝑀𝐾𝑠 red giant luminosity function. Blue-dashed lines representthe LMC and orange lines the SMC. The most prominent (thicker) lines showthe smoothed luminosity functions plotted on top of the binned data (thinlines in the background). At the bottom of the plot we see the Sobel filteror edge-response functions where their maxima indicate the location of theTRGB and are marked by a vertical line of the same style and colour. Thevalues of the TRGB for the LMC and SMC are indicated in the inset.4.3 Results and discussionThe objective of using the 𝑇 magnitude is to remove possible effects of metallicityon the magnitude of the TRGB. When using this colour dependent magnitude, theabsolute TRGB magnitude in any filter is expected to be the same for the LMC andSMC. Nevertheless, the results show a difference of ∼0.11 magnitudes. Along withthe difference in the absolute magnitudes of the TRGB in the LMC and SMC, thereare differences between the RG luminosity functions in the two galaxies. Whenthe luminosity functions of the Magellanic Clouds are superimposed, differencesin the shapes of the 𝑇 magnitude luminosity functions can be observed. The most73prominent difference is how steep the drop is in the number of stars between theRGB and AGB, which seems more abrupt in the SMC than the LMC. Similarly,different luminosity function shapes are observed among the target galaxies. Thesedifferences in luminosity function shape and TRGB magnitude could indicate thatthe construction of a composite magnitude may not be enough to compensate formetallicity effects. It may also be that star formation history additionally plays animportant role in defining the absolute magnitude of the TRGB (see also Chapter7).As with the CS, for the TRGB method, both the LMC and SMC are used ascalibrators. To determine which calibrator is most suitable for each target galaxy itis necessary to determine whether these galaxies are more "LMC-like" or "SMC-like". For the CS method this was determined using their luminosity functions.Nevertheless, with only a smoothed RG luminosity function it is not possible toconstruct a procedure to quantify the differences between an "LMC-like" and an"SMC-like" like galaxy. The classification would need to be based entirely on avisual analysis of the luminosity functions.For the purposes of this thesis, in this Chapter the determination of the distancemoduli to the sample of galaxies will rely on the classification obtained fromthe CS analysis. From Section 3.6, NGC 6822 was classified as an "LMC-like"galaxy, while IC 1613, WLM and NGC 3109 were classified as "SMC-like". Withthis classification 𝜇0 = 23.50± 0.02 (stat) for NGC 6822, 𝜇0 = 24.46± 0.02 (stat)for IC 1613 (from the WIRCam data), 𝜇0 = 25.03 ± 0.03 (stat) for WLM and𝜇0 = 25.60±0.03 (stat) for NGC 3109. The results obtained from the CS median𝐽 magnitude and the TRGB are summarized in Table 4.3. The distance moduliobtained from both methods agree well within the statistical error bars.Themain purpose of estimating distance using the TRGBmethod, is to comparethe results from the methods developed in this thesis with a method that has beenextensively studied and tested both empirically and theoretically (see Section 1.5).As for the CS, the luminosity functions of the LMC and SMC are found to bedifferent (at least from visual inspection). Compared to the studies mentioned inSection 1.5, where the TRGB 𝐼 magnitude is found to be fairly constant across allobjects where the TRGB can be observed, in this thesis a statistically significant74Figure 4.4: Red giants𝑇(𝐾𝑠)0 luminosity functions for the target galaxies. Thestars selected to draw the luminosity function are shown in Figure 4.2. In eachpanel the red thin continuous line shows the binned luminosity function whilethe thicker red line on top is the smoothed luminosity function. The smoothedluminosity function of the LMC (blue dashed line) and the SMC (orangedashed-dotted line) are also plotted for comparison. The black line runningat the bottom of the plots shows the response of the edge-detection filter. Themaximum in the response function indicates the location (i.e. magnitude) ofthe tip of the red giant branch. The estimated distance modulus for each galaxyis specified in each plot along with the statistical error.75Table 4.3: Distance moduli values obtained from the median 𝐽 magnitude ofthe carbon stars compared with the distance moduli estimated using the TRGB.The errors presented are the statistical errors.Carbon Stars TRGBNGC 6822 23.55 ± 0.03 23.50 ± 0.02IC 1613 (WIRCam) 24.45 ± 0.05 24.46 ± 0.02WLM 25.02 ± 0.04 25.03 ± 0.03NGC 3109 25.58 ± 0.03 25.60 ± 0.03difference is found between the absolute 𝐾𝑠-band magnitude of the TRGB (𝑀TRGB𝐾𝑠 )of the LMC and SMC. If this difference is not taken into consideration, a systematicerror of ∼ 0.1 mag is introduced into the distance modulus estimation.The TRGB edge-detection method does not allow a quantifiable comparisonbetween the RG luminosity functions. In the next Chapter a method to measurethe differences in the RG luminosity functions is developed to confirm what in thisChapter could only be inferred from visual inspection.76Chapter 5Red giants, a maximum likelihoodmethodAs discussed in the previous chapter, the selection region to detect the TRGB covers∼ 2− 3 magnitudes, however, the rectification of the TRGB relies on a selectionof a few stars at the TRGB (see Figures 6 and 7 of [155] for an example) andthe follow up edge-detection procedure only effectively uses the stars at the TRGB.Nevertheless, the TRGB has proven that RG stars are an effective distance indicator.In this Chapter, the standard candle aspect of the RGs is explored further, ina way where more RG stars are used, and where no metallicity corrections arenecessary thus avoiding the introduction of an additional source of uncertainty.Instead, an un-binned maximum likelihood estimator (MLE) is used to find thedistance modulus based on the cumulative distribution functions of RG stars. Thismethod will be referred to as the RG-MLE method.5.1 SelectionIn the previous Chapter the RG stars were selected from the ((𝐽 −𝐾𝑠)0, 𝑇(𝐾𝑠)0)diagram, were 𝑇 is a rectified magnitude meant to remove the effects of metallicityon the magnitude of the TRGB [157]. In the case of the RG-MLE method, the RGsare selected directly from the ((𝐽 −𝐾𝑠)0, 𝐽0) and ((𝐽 −𝐾𝑠)0, (𝐾𝑠)0) CMDs.For the LMC and SMC the RG region magnitude range covers roughly three77magnitudes and the colour spans ∼ 0.16 mag (Figure 5.1). For the target galaxiesthe initial magnitude limits are selected based on the available data. NGC 6822 hasthe largest magnitude range covering approximately three magnitudes, for the restof the target galaxies the span is around 2 magnitudes. Because the photometricerrors are larger as stars get fainter, the colour range is widened to 0.20 mag forWLM and to 0.24 mag for NGC 3109. The ((𝐽 −𝐾𝑠)0, (𝐾𝑠)0) CMDs with theselection region of RGs for the target galaxies are shown in Figure 5.2. The numberof RG stars selected in each galaxy can be found in Table 5.1. When estimatingthe distance, the magnitudes considered in the calculation will be restricted so thatonly the stars that will fall within the magnitude limits of the calibrators are usedin the estimation. In other words, if the magnitude of a RG star in the target galaxy,once the distance modulus is added, is not within the range of the magnitude limitsof the calibrator’s RG region, this star is left out of the distance calculation.Figure 5.1: ((𝐽 − 𝐾𝑠)0, 𝑀𝐾𝑠 ) CMD for the Magellanic Clouds. The red-shaded area demarcates the region of the red giant selection.78Figure 5.2: (𝐽 −𝐾𝑠)0, (𝐾𝑠)0) CMDs for the target galaxies. The red-shadedarea demarcates the region of the RG stars selection.79Table 5.1: Number of RG stars in each galaxy selected from the ((𝐽 −𝐾𝑠)0,𝐽0) (first row), and ((𝐽 −𝐾𝑠)0, (𝐾𝑠)0) (second row) CMDs.LMC SMC NGC 6822 IC 1613 WLM NGC 3109𝐽 120913 22480 12087 2201 880 2191𝐾𝑠 135050 25446 13505 2283 1130 30285.2 Cumulative distribution luminosity functionFor the TRGBmethod, the luminosity function of theRGswas used in its probabilitydensity function form. For the RG-MLE method, the luminosity function of theRGs is used in its cumulative distribution function (CDF) form. As will be shownin the next section, the model for the RG stars’ luminosity function must be asmooth continuous function within the range of magnitudes of the data. To ensurethis condition is met, the luminosity functions of the LMC and SMC are modelledusing a smoothing spline fit. The luminosity functions of the target galaxies will bemodelled by a linear combination of the LMC and SMC spline models. The RGs𝑀𝐾𝑠 luminosity functions and spline fits for the Magellanic Clouds are shown inFigure 5.3.A visual inspection of the LMC and SMC RG luminosity functions reveals achange in the slope of the curve between magnitudes −6.5 and −6.0. The change inslope takes place in the magnitude range where the TRGB is found (𝑀TRGB𝐾𝑠 =−6.29for the LMC and−6.18 for the SMC). The value of the slope changes from galaxy togalaxy, thus indicating that the fraction of RGB to AGB stars is not constant amongall galaxies. As with the CS 𝐽-band and the RGs 𝑇-band luminosity functions,the different features in the LMC and SMC luminosity functions are an indicationof different intrinsic properties between the two galaxies reflected in the differentshapes of their luminosity functions. Similar dissimilarities can be observed in theluminosity functions of the target galaxies.With the RG-MLE method, as was the case with the CS, the construction ofa model to fit the luminosity functions will enable us to quantify the differencebetween an “LMC-like” and an “SMC-like” galaxy.80Figure 5.3: RGs 𝑀𝐾𝑠 luminosity functions for the LMC (cyan) and the SMC(orange). The smoothing spline fits are plotted over the luminosity functionswith a dashed-blue line for the LMC and red line for the SMC.5.3 The likelihood functionThe cumulative distributions of magnitudes (or the luminosity function) of theRGs in any Magellanic type galaxy is postulated to originate from an underlyingdistribution that can be described by a combination of the luminosity function ofthe RGs in the LMC and SMC. Assuming this hypothesis is true, an un-binnedmaximum likelihood estimator is used to fit the luminosity function of the RG starsfrom the target galaxies to a model distribution given by a linear combination ofthe luminosity functions of the RGs in the LMC and SMC. The distance modulus isdefined as the magnitude difference that minimizes the difference between the RGdistribution observed in the target galaxy and that of the Magellanic Clouds linearcombination.To construct the likelihood function consider, first, the simple case where themodel distribution (𝐹) is given by a single underlying population. In this context itwould be either the LMC or the SMC and not a linear combination. The case of asingle calibrator will allow us to obtain an estimation of the distance modulus foreach target galaxy from the direct comparison with the LMC and SMC, similarly81to the CS method. If a binned estimation was used, the fit would be done to thenumber of stars per bin in the cumulative distribution. Because stars are counted,the probability of having 𝑘 stars in each magnitude interval is given by a Poissondistribution:𝑃(𝑘;𝜆𝑖) = 𝑒−𝜆𝜆𝑘𝑖𝑘!. (5.1)where 𝜆 is a function ( 𝑓 ) of the bin properties (including the distance modulus)times the magnitude element (𝑑𝑀) of the bin.In the case of the un-binned estimation, the bins could be made as small asnecessary to the point where each bin will contain either 0 or 1 stars. Then, theprobability of finding 0 or 1 stars is given by:𝑃(𝑘;𝜆𝑖) ={𝑒−𝜆𝑖 𝑘 = 0,no stars,𝜆𝑖𝑒−𝜆𝑖 𝑘 = 1,one star.(5.2)The total likelihood of the sample is given by the product of the likelihoods of eachbin:𝐿 =>0∏𝑘=0𝑒−𝜆𝑖 +∏𝑘=1𝜆𝑖𝑒−𝜆𝑖 , (5.3)where the first term vanishes as it refers to the empty bins. Taking the logarithmand considering only the bins containing one star the likelihood function can bewritten as:ln(𝐿) =∑︁𝑖(ln(𝜆𝑖)) −∑︁𝑖(𝜆𝑖) , (5.4)where ∑︁𝑖ln (𝜆𝑖) =∑︁𝑖ln( 𝑓𝑖𝑑𝑀) =∑︁𝑖ln( 𝑓𝑖) +∑︁𝑖ln(𝑑𝑀), (5.5)and∑𝑖 (𝜆𝑖) is the sum over all the bins which gives the total number of starspredicted by the model (𝑁pred). Because the choice of bins (𝑑𝑀) does not playa role in determining the model that maximizes the likelihood of the observeddata, the last term in equation 5.5 can be dropped. The likelihood function is thenexpressed as:ln(𝐿) =∑︁𝑖(ln( 𝑓𝑖)) −𝑁pred. (5.6)82If 𝑓 is the derivative of the model 𝐹, the constructed model has to be a smoothcontinuous function in the range of magnitudes employed. A smoothing spline fit isused to make 𝐹 a smooth and continuous function. The predicted number of starswill be given by the limits of the cumulative distribution model, 𝐹: 𝑁pred = Δ𝐹.Because the cumulative distributions are normalized, 𝐹 is also normalized. Thisallows the inclusion of a numerical factor (henceforth called 𝐴) to go from thenormalized distribution to the real number and real distribution of the stars: 𝑁pred =𝐴Δ𝐹, and ln( 𝑓𝑖) =⇒ ln(𝐴 𝑓𝑖). The next step is to ensure that the likelihood functionhas only the distance modulus of the target galaxy as the unknown parameter. Inthis regard, it can be noted that 𝐿 will reach its maximum when the number ofpredicted stars is the same as the number of stars in the sample: 𝑁pred = 𝑁stars = 𝑁★.Putting all this together the likelihood function can be written as:ln(𝐿) =∑︁𝑖𝑁★ln(𝑓 [𝑚𝑖 − 𝜇0]Δ𝐹)−𝑁★. (5.7)In the above case the model distribution 𝐹 (and hence also 𝑓 ) is given by thedistribution of the RGs from a single calibrator, either the LMC or SMC. If a modelthat is a linear combination of the model for the LMC, 𝐹LMC, and the model for theSMC, 𝐹SMC, are considered, the likelihood function for the linear combination isdefined as:ln(𝐿) =∑︁𝑖𝑁★ln(𝑎 𝑓𝑖,LMC + (1− 𝑎) 𝑓𝑖,SMC𝑎Δ𝐹LMC + (1− 𝑎)Δ𝐹SMC)−𝑁★, (5.8)where the constant 𝑎 will indicate if the distribution of RGs in a given galaxy ismore “LMC-like” or “SMC-like”. The parameter 𝑎 can be considered a “shape”parameter where 𝑎 = 1 indicates a luminosity function identical to the LMC and𝑎 = 0 identical to the SMC, in other terms:𝑎> 0.5 “LMC-like”,< 0.5 “SMC-like”.(5.9)835.4 ResultsThe RG luminosity function of each target galaxy is compared with that of theMagellanic Clouds in both the 𝐽 and 𝐾𝑠-bands. The luminosity functions for thetarget galaxies in the 𝐾𝑠-band are shown in Figure 5.4The distance moduli are first estimated using the simple case where the RGluminosity function of each target galaxy is compared to the LMC and the SMCseparately. This is accomplished by minimizing the negative of the likelihoodfunction given in equation 5.7. The results for the distance moduli obtained againstthe two Magellanic Clouds are presented in Table 5.2. For every galaxy, the resultobtained against the LMC is significantly different to that obtained against the SMC(up to 0.2 magnitudes). As for the CS and the TRGB methods, the statistical errorsare obtained from 10000 bootstrap realizations.Table 5.2: RG-MLE distance moduli obtained from comparing the targetgalaxy’s RG luminosity function to both Magellanic Clouds separately. Theresults obtained against the LMC are significantly different to those obtainedagainst the SMC.vs LMC vs SMC𝜇0(𝐽) 𝜇0(𝐾𝑠) 𝜇0(𝐽) 𝜇0(𝐾𝑠)NGC 6822a 23.56+0.01−0.01 23.61+0.01−0.01 23.46+0.00−0.00 23.50+0.00−0.00IC 1613b 24.57+0.01−0.01 24.58+0.01−0.02 24.41+0.01−0.01 24.38+0.02−0.02WLMb 25.13+0.02−0.02 25.17+0.02−0.02 24.96+0.02−0.02 24.96+0.02−0.02NGC 3109b 25.70+0.02−0.02 25.60+0.01−0.01 25.53+0.01−0.02 25.42+0.03−0.03The most suitable calibrator as defined by the CS method:a = LMC ; b = SMCWhen comparing the distance moduli obtained with the methods from Chapters3 (CS) and 4 (TRGB) against those estimated using the RGs, the results determinedagainst the most suitable calibrator as defined by the CS method are, in general,in better agreement with the results from previous chapters. For NGC 6822 themost suitable calibrator was determined to be the LMC. In this case the resultobtained from the 𝐽-band luminosity function is in agreement with those fromthe CS and TRGB but the distance modulus from the 𝐾𝑠-band is slightly larger,84Figure 5.4: RGs 𝐾𝑠 luminosity functions for the target galaxies. In eachpanel, the luminosity function of the corresponding galaxy indicated in thetitle is represented by a red line. The luminosity functions of the LMC(dashed blue line) and the SMC (dash-dotted orange line) have been adjustedto the magnitudes covered by the target galaxy after the distance modulus isapplied. The original LMC and SMC luminosity functions are plotted in thebackground with thinner lines but following the same line style and colour.The distance modulus obtained from the linear combination fit is given in eachpanel along with the “shape” parameter 𝑎.85Table 5.3: RG-MLE distance moduli and shape parameter from the linearcombination fitting technique. For every galaxy, except for the shape parameterof NGC 3109, the results obtained from the 𝐽-band RG luminosity functionagree with those obtained using the 𝐾𝑠-band.𝜇0(𝐽) 𝑎(𝐽) 𝜇0(𝐾𝑠) 𝑎(𝐾𝑠)NGC 6822 23.51+0.01−0.01 0.64+0.05−0.05 23.55+0.02−0.01 0.69+0.07−0.06IC 1613 24.39+0.01−0.02 -0.15+0.08−0.08 24.35+0.02−0.02 -0.19+0.09−0.08WLM 24.97+0.03−0.03 0.06+0.21−0.17 24.96+0.02−0.02 -0.05+0.11−0.11NGC 3109 25.57+0.02−0.02 0.42+0.10−0.11 25.61+0.02−0.03 0.84+0.06−0.09placing the galaxy at a larger distance. For IC 1613 and WLM the most suitablecalibrator was deemed to be the SMC, in this case the results are slightly lower thanprevious results, placing the galaxies at shorter distances. For NGC 3109, whilethe appropriate calibrator according to the CS method was the SMC, for the RGsthe result using the LMC and the 𝐾𝑠-band, and the result using the SMC and the𝐽-band, are in agreement with that of the CS and TRGB.Secondly, the distance moduli are estimated by minimizing the negative of thelikelihood function given in equation 5.8. In this case, the luminosity functionof the target galaxy is compared against a linear combination of the luminosityfunctions of the LMC and SMC. Again the distances are estimated from both the𝐽 and 𝐾𝑠 bands luminosity functions. The results for this method are presented inTable 5.3 and plotted in Figure 5.4.The distance moduli obtained for NGC 6822, WLM and NGC 3109 are in goodagreement with those obtained from the CS and TRGB (within the error bars). Inthe case of IC 1613, compared to the results from the CS and TRGB, the distancemoduli are brighter (e.i. a shorter distance) with the 𝐾𝑠-band result falling slightlyoutside the statistical error bars.The results for the “shape” parameter 𝑎 indicate that the luminosity function ofthe RGs in NGC 6822 is “LMC-like” in agreement with the most suitable calibratordetermined using the CS luminosity function. For IC 1613 and WLM 𝑎 indicatesan “SMC-like” shape also in agreement with the CS method. For NGC 3109,the shape analysis on the different bands gives different results. For the 𝐽-band86the value of 𝑎(𝐽) = 0.42 is slightly inclined towards an “SMC-like” shape, while𝑎(𝐾𝑠) = 0.84 implies an “LMC-like” shape. It is not possible at the moment todetermined why, in the case of NGC 3109, the RG luminosity function resemblesthe luminosity function of different calibrators when using 𝐽 and 𝐾𝑠. Nevertheless,the difference between the values of 𝑎 from the different bands agrees with thedistances estimated against the Magellanic Clouds individually: 𝑎(𝐾𝑠) indicates“LMC-like”, going back to Table 5.2, when comparing NGC 3109 to the LMC,𝜇0(𝐾𝑠) agrees with the CS and TRGB, while 𝜇0(𝐽) does not. In the case of 𝑎(𝐽),the result reflects an “SMC-like” shape, in Table 5.2, when comparing NGC 3109to the SMC 𝜇0(𝐽) is the estimate that agrees with the previous methods.5.5 A test of the techniquesTo test the performance and possible systematics of the RG MLE method, thedistance estimation is performed on 104 random subsamples composed of 2000 RGstars taken from the LMC. The calculations are performed against each individualMagellanic Cloud, and also using a linear combination fitting. Because the starscome from the original selection of RGs in the LMC, each sub-sample is expectedto have the same distance modulus as the LMC (i.e. a differential distance modulus(Δ𝜇0) equal to zero), and their luminosity functions an “LMC-like” shape (i.e.𝑎 > 0.5).When the LMC RG sub-samples are compared against the entire LMC RGselection using the linear combination technique, the differential distance modulidistribution is centred at roughly the expected value of zero. Additionally, the linearcombination shape parameter, 𝑎 ∼ 1, indicates a luminosity function with an almostperfectly “LMC-like” shape. Instead, when comparing the LMC sub-samples to theSMC, Δ𝜇0 = −0.230 mag. The results for the LMC RG sub-samples are presentedin the top panels of Figure 5.5.The test is repeated using sub-samples of RG stars from the SMC. These sub-samples show a differential distance modulus of zero against the entire sample ofthe SMC RGs and -0.003 mag for the linear combination technique. The shape pa-rameter in this case is roughly zero indicating the sub-samples luminosity functionswere “SMC-like”. The SMC sub-samples show a Δ𝜇0 = 0.220 when compared to87Figure 5.5: Distribution of differential distance moduli (Δ𝜇0) for 104 randomsubsamples composed of 2000 RG stars from the LMC (top panels) and theSMC (bottom panels). The left panels show the distributions when comparingagainst the Magellanic Clouds individually, the centre panels depict Δ𝜇0 fromthe linear combination method. The right panels show the values for the“shape” parameter 𝑎. As expected, the sub-samples from the LMC show an“LMC-like” shape, likewise, the SMC samples show an “SMC-like” shape.the entire sample of LMC stars. This result is approximately equal in magnitudebut in the opposite direction to the result obtained for the LMC sub-samples whencompared to the SMC. The results for the SMC sub-samples are shown in thebottom panels of Figure 5.5.The results from the LMC and SMC sub-samples compared to each MagellanicCloud highlight the difference between the LMCandSMCRG luminosity functions.The estimations from the linear combination agree with the expected Δ𝜇0 = 0 andshape parameter results.88Chapter 6Three-dimensional structure ofthe Magellanic CloudsAn in-depth study of the three-dimensional (3D) structure of theMagellanic Cloudsgoes beyond the scope of this thesis. Instead, the following analysis is focused onexamining potential biases introduced by the the geometry of the the MagellanicClouds and its possible effects in the distance determination process, more exactly,in the calibration of 𝑀 𝐽 of CS and the 𝑀TRGB𝐾𝑠 .The Magellanic Clouds are 3D objects from which it is only possible to retrievea 3D projection. To examine the 3D projection effects, both the LMC and SMC aredivided into nine fields (a 3 by 3 binning, numbered F1 to F9). The RA and DEClimits of the fields are set by binning the CS data so that each field has an equalnumber of CS: 669-700 in the LMC, and 120-121 in the SMC. Because the fieldsare defined by the binning of equal numbers of CS, the number of RG stars variesin each bin. The location of the fields in the sky are shown in the RA and DECmaps in Figure 6.1. The distance to each field is estimated independently in thesame manner as for the whole sample of stars using the CS and the TRGB method.The CS and RG in each field are re-sampled ten-thousand times with replacement.For the CS, in each iteration, 𝑀 𝐽 is calculated and the CS luminosity functionis fitted with the modified Lorentzian model described in Section 3.3 using boththe NLLSQ and MLE methods. In the case of the RG, in each iteration the 𝑇𝐾𝑠luminosity function is smoothed and the edge-detection filter is used to estimate89𝑀TRGB𝐾𝑠 .The results for 𝑀 𝐽 and 𝑀TRGB𝐾𝑠 are presented in absolute magnitudes assuming𝜇0,LMC = 18.477 ± 0.004 (statistical) ± 0.026 (systematic) for the LMC [192] and𝜇0,SMC = 18.977 ± 0.016 (statistical) ± 0.028 (systematic) [111] for the SMC.The differences between 𝑀 𝐽 or 𝑀TRGB𝐾𝑠 of each field against the values for theentire samples of CS or RG stars are given by:Δ𝑀 𝐽 = 𝑀 𝐽 ,Fi−𝑀 𝐽 ,all,Δ𝑀TRGB𝐾𝑠 = 𝑀TRGB𝐾𝑠 ,Fi −𝑀TRGB𝐾𝑠 ,all ,(6.1)where 𝑎𝑙𝑙 refers to the LMC or SMC complete CS or RG samples.The differential distance modulus (Δ𝜇0) of each field with respect of the entireLMC or SMC samples is reflected in, and equal to, Δ𝑀 𝐽 and Δ𝑀TRGB𝐾𝑠 .6.1 The Large Magellanic CloudThe parameters obtained for each field from the bootstrapping analysis are shownin Table 6.1. The results for the shape parameters of CS luminosity function fitfor each field agree (within the error bars) with those reported for the complete CSsamples in Table 3.1. Only the skew parameter for F2 could be considered slightlyoutside of the range determined in Section 3.3.Examining the Δ𝑀 𝐽 values for the different fields of the LMC, we noticethat for the majority of the fields there are minimal variations with respect to theadopted distance modulus. The exceptions are fields 1 and 9 (F1 and F9), wherethe divergence of Δ𝑀 𝐽 from zero is significant.The median 𝑀𝐽 for F1 is 0.11 mag fainter than the complete sample whileF9 is 0.12 mag brighter. This difference in median 𝑀𝐽 is equal to the differencebetween the fields’ distance moduli and the distance modulus considered for theentire LMC. Our results show a total variation of 0.23 mag between the northeast(F9) and southwest (F1) field. This variation suggests that the LMC is tilted alongthe line of nodes with respect to the plane of the sky, with the north-east field closerto the observer and the south-west field more distant. This is in agreement withmultiple studies that have found evidence for and have been able to measure theinclination angle of the LMCusing different distance tracers: e.g. Cepheid variables90Figure 6.1: RA and DEC map for the LMC (top panel) and the SMC (bottompanel). The positions of the carbon stars in both galaxies are indicated bythe red markers. The black lines delimit the nine fields (F1 to F9), each fieldcontains an equal number of CS (120-121 for the SMC and 669-700 for theLMC). The large blue-dot in the top panel indicates the location of the starforming region 30 Doradus.91Table 6.1: Parameters for fields F1 to F9 from the LMC (as outlined in Figure 6.1). Columns 2 to 4 show the CSluminosity functions best-fit parameters. The next two columns show median absolute 𝐽 magnitude and differencebetween the 𝑀 𝐽 for each field outlined in Figure 6.1 and the entire sample of carbon stars (Δ𝑀 𝐽 = M𝐽,Fi−M𝐽,𝐿𝑀𝐶).The last two columns show the absolute 𝐾𝑠 magnitude difference between 𝑀TRGB𝐾𝑠 LMC and 𝑀TRGB𝐾𝑠 Fi. Δ𝑀 𝐽 and Δ𝑀TRGB𝐾𝑠can also be interpreted as the difference in distance modulus. Values and errors are obtained from 10000 bootstraprealizations.Field Mode Width Skew Kurtosis 𝑀 𝐽 Δ𝑀 𝐽 𝑀TRGB𝐾𝑠 Δ𝑀TRGB𝐾𝑠F1 -6.23+0.02−0.02 0.27+0.03−0.02 -0.35+0.11−0.11 0.12+0.06−0.05 -6.18+0.02−0.02 0.11±0.02 -6.20+0.01−0.01 0.09 ±0.02F2 -6.31+0.02−0.02 0.31+0.03−0.03 -0.19+0.11−0.12 0.12+0.06−0.04 -6.28+0.01−0.01 0.01±0.01 -6.22+0.02−0.01 0.07 ±0.02F3 -6.36+0.03−0.03 0.32+0.03−0.03 -0.25+0.15−0.16 0.17+0.06−0.06 -6.33+0.02−0.02 -0.04±0.02 -6.24+0.03−0.01 0.05 ±0.02F4 -6.33+0.03−0.02 0.29+0.03−0.03 -0.41+0.14−0.13 0.15+0.06−0.05 -6.27+0.01−0.02 0.02±0.02 -6.24+0.01−0.01 0.05 ±0.02F5 -6.38+0.03−0.03 0.32+0.04−0.03 -0.48+0.14−0.13 0.16+0.08−0.04 -6.30+0.01−0.02 -0.01±0.02 -6.30+0.04−0.02 -0.01 ±0.03F6 -6.41+0.03−0.02 0.30+0.03−0.03 -0.41+0.12−0.11 0.14+0.05−0.05 -6.33+0.02−0.02 -0.05±0.02 -6.27+0.02−0.01 0.02 ±0.02F7 -6.35+0.02−0.01 0.21+0.02−0.02 -0.28+0.07−0.06 0.05+0.02−0.01 -6.29+0.01−0.01 0.00±0.01 -6.30+0.01−0.01 0.01 ±0.02F8 -6.32+0.02−0.02 0.30+0.03−0.03 -0.37+0.08−0.08 0.09+0.04−0.02 -6.23+0.01−0.02 0.05±0.02 -6.22+0.03−0.04 0.07 ±0.04F9 -6.46+0.02−0.02 0.22+0.02−0.02 -0.31+0.08−0.09 0.07+0.04−0.02 -6.41+0.01−0.01 -0.12±0.01 -6.34+0.01−0.01 -0.05 ±0.0292[32], AGB and RGB star luminosity functions [247, 244], red clump stars [234],TRGB [130, 60], etc. Van derMarel and Cioni [244] used near-infrared data and themode of the luminosity function of AGB stars. They reported significant gradientsin both east-west and north-south directions with magnitude variations of up to∼ 0.25 mag in 𝐽, 𝐻, and 𝐾𝑠, in very good agreement with the magnitude differencefound between fields F1 and F9 reported in this thesis. Weinberg and Nikolaev[247] also used near-infrared data and AGB stars but with a smaller number offields and stars. They found a ∼ 0.3 mag gradient in the east-west direction, and novariations in the north-south direction. Comparing the magnitudes found betweenthe north and south fields, a small north-south gradient is seen with F1 0.15 magfainter than F3, F4 0.07 magnitudes fainter than F6, and F7 0.12 magnitudes fainterthan F9.The same analysis using the 𝑀TRGB𝐾𝑠 magnitudes yielded slightly different re-sults. The east-west gradient was found to be only 0.16 mag, with the north-eastfield (F9) only 0.05 magnitudes brighter compared to 𝑀TRGB𝐾𝑠 of the entire sam-ple, while 𝑀 𝐽 ,𝐹9 was found to be -0.11 mag away from 𝑀 𝐽 ,𝑎𝑙𝑙 . No significantdifferences between the north and south fields were found when using the TRGB.A nine-field division may not be sensitive enough to detect distance variationsusing RG stars. For the 3 by 3 binning, each field contains at least 10700 RG starswith an average of 14500. Because each bin is defined to have the same number ofCS, the number of RG stars varies from field to field.The 3D analysis is repeated dividing the LMC into 16 fields, in this case thenumber of CS is 376-377 per bin. Dividing the LMC into 16 fields produced similarresults to those obtained from the 9 fields analysis. For the CS, the south-west corneris 0.25 magnitudes fainter compared to the north-east corner, confirming a tilt inthe LMC along the direction perpendicular to the major axis of the bar. The fieldson the north edge are on average 0.15 mag fainter compared to their south edgecounterparts. Again, using the TRGB method, we find no significant differencesbetween north and south fields and only a 0.13 magnitude difference between thenorth-east and south-west corners. A 0.13 mag difference still suggests a slighttilt in the same direction as the CS, nevertheless the north-east corner is only 0.02magnitudes brighter than the 𝑀TRGB𝐾𝑠 value of the full sample of RG stars.Finally, the LMC is divided into 25 fields (Figure 6.2) with 241 CS per bin.93Figure 6.2: RA and DEC map for the LMC similar to Figure 6.1. The blacklines delimit the 25 fields described in Sec. 6.1, each field contains an equalnumber of CS (241). The blue star-marker points to the centre of the starforming region 30 Doradus.Figure 6.3 shows the best-fit modified Lorentzian function for each of the 25 fields,the value of the best-fit parameters are listed in Table 6.2. Except for field F8 beingslightly skewed towards fainter magnitudes, all the fields show a negative skewparameter. F5 and F10 have a skewness value closer to zero describing a moresymmetric distribution. Without considering the statistical error bars, 10 out ofthe 25 fields would have been catalogue as “SMC-like” galaxies with half of thesefields being significantly close (𝑠 ≤ −0.17) to the limit established in equation 3.7(𝑠 < −0.20).The 5 by 5 RA,DEC binning returned differential distance moduli that are inbetter agreement between the CS and TRGB methods. The location in the CMDof the TRGB in every field with respect to the TRGB for the full sample of LMCstars are shown in Figure 6.5. For the CS the results remain consistent with theprevious binning analysis. For the TRGB, the north fields (except F10) appear onaverage ∼0.10 magnitudes brighter than the south fields, comparable to the ∼0.1594Figure 6.3: CS luminosity function (dotted orange lines) for LMC fields F1to F25. In every panel the solid orange line shows the the best fit modifiedLorentzian model to the luminosity function through the NLLSQ techniquewhile the dark blue dashed-line depicts the best-fit from the MLE technique.95Figure 6.4: RA andDECmap for the CS in the LMCwith differential distancemoduli: Δ𝜇0 = 𝜇0,LMC − 𝜇0,Fi, obtained from the CS distance determinationmethod. The black lines delimit the same 25 fields as Figure 6.2. The colourbar shows the scale for the Δ𝜇0, with negative numbers indicating a smaller(or closer) distance modulus compared to the main body of the LMC.average mag difference between north and south fields using CS. In the case of thenorth-east versus south-west corner, the differences are 0.25 and 0.26 magnitudesfor the CS and TRGB methods respectively. The results for the 5 by 5 binning arepresented in Table 6.2 and in map form for the CS in Figure 6.4.96Figure 6.5: ((𝐽 −𝐾𝑠)0, 𝑇(𝐾𝑠)0) CMDs for the 25 different fields of the LMC.In each panel, the black horizontal line marks the location of the 𝑇TRGB𝑀𝐾𝑠magnitude for the entire sample of RG stars. The orange line indicates thelocation of the magnitude of the TRGB found for each specific field. F1 has aΔ𝜇0 = 0.12 while F25 Δ𝜇0 = −0.14 in agreement with the LMC north fieldsbeing closer to the Milky Way.97Table 6.2: Same as Table 6.1 but for the LMC divided into 25 fields. The location of the fields is outlined in Figure 6.2.Field Mode Width Skew Kurtosis 𝑀 𝐽 Δ𝑀 𝐽 𝑀TRGB𝐾𝑠 Δ𝑀TRGB𝐾𝑠F1 -6.21 +0.02−0.02 0.18+0.03−0.02 -0.20+0.10−0.10 0.03+0.04−0.02 -6.17+0.02−0.02 0.12 ±0.02 -6.17+0.01−0.02 0.12 ±0.02F2 -6.25 +0.05−0.05 0.29+0.05−0.04 -0.30+0.21−0.21 0.11+0.11−0.07 -6.18+0.02−0.04 0.11 ±0.03 -6.21+0.02−0.02 0.08 ±0.02F3 -6.28 +0.04−0.04 0.28+0.05−0.04 -0.18+0.14−0.18 0.07+0.10−0.05 -6.23+0.02−0.04 0.06 ±0.03 -6.24+0.08−0.08 0.05 ±0.08F4 -6.28 +0.03−0.03 0.27+0.05−0.04 -0.13+0.14−0.21 0.06+0.10−0.06 -6.27+0.02−0.02 0.02 ±0.02 -6.29+0.03−0.02 0.00 ±0.03F5 -6.35 +0.05−0.04 0.31+0.05−0.05 -0.02+0.20−0.25 0.14+0.14−0.12 -6.35+0.03−0.02 -0.06 ±0.03 -6.25+0.02−0.02 0.04 ±0.02F6 -6.27 +0.08−0.06 0.31+0.08−0.06 -0.48+0.46−0.32 0.21+0.34−0.09 -6.21+0.03−0.03 0.08 ±0.03 -6.24+0.01−0.02 0.05 ±0.02F7 -6.39 +0.03−0.03 0.24+0.04−0.04 -0.36+0.10−0.10 0.05+0.04−0.02 -6.29+0.03−0.03 -0.00 ±0.03 -6.28+0.02−0.02 0.01 ±0.02F8 -6.33 +0.04−0.04 0.33+0.06−0.05 0.11+0.17−0.14 0.06+0.12−0.07 -6.33+0.03−0.04 -0.04 ±0.03 -6.20+0.05−0.07 0.09 ±0.06F9 -6.38 +0.06−0.04 0.30+0.06−0.05 -0.37+0.25−0.18 0.10+0.07−0.06 -6.29+0.02−0.02 -0.00 ±0.02 -6.33+0.02−0.01 -0.04 ±0.02F10 -6.39 +0.05−0.05 0.34+0.06−0.05 -0.04+0.21−0.30 0.14+0.17−0.11 -6.36+0.01−0.03 -0.07 ±0.02 -6.27+0.04−0.08 0.02 ±0.03F11 -6.30 +0.05−0.04 0.27+0.05−0.04 -0.32+0.21−0.20 0.09+0.10−0.06 -6.25+0.04−0.02 0.04 ±0.03 -6.24+0.03−0.04 0.05 ±0.04F12 -6.38 +0.08−0.07 0.34+0.09−0.07 -0.56+0.37−0.31 0.19+0.31−0.07 -6.28+0.02−0.05 0.01 ±0.04 -6.31+0.12−0.02 -0.02 ±0.07F13 -6.37 +0.05−0.06 0.30+0.06−0.05 -0.44+0.25−0.27 0.16+0.14−0.08 -6.30+0.02−0.03 -0.01 ±0.02 -6.35+0.30−0.04 -0.06 ±0.17F14 -6.30 +0.05−0.05 0.32+0.06−0.05 -0.20+0.23−0.28 0.12+0.11−0.09 -6.27+0.03−0.02 0.02 ±0.03 -6.20+0.03−0.04 0.09 ±0.04F15 -6.43 +0.04−0.04 0.26+0.05−0.04 -0.24+0.25−0.23 0.11+0.10−0.08 -6.39+0.02−0.02 -0.10 ±0.02 -6.31+0.03−0.03 -0.02 ±0.0398Field Mode Width Skew Kurtosis 𝑀 𝐽 Δ𝑀 𝐽 𝑀TRGB𝐾𝑠 Δ𝑀TRGB𝐾𝑠F16 -6.30 +0.03−0.03 0.22+0.03−0.03 -0.19+0.13−0.14 0.03+0.04−0.03 -6.26+0.03−0.03 0.02 ±0.03 -6.24+0.03−0.01 0.05 ±0.02F17 -6.29 +0.03−0.03 0.20+0.03−0.02 -0.17+0.10−0.09 0.01+0.02−0.02 -6.24+0.03−0.01 0.05 ±0.02 -6.23+0.08−0.06 0.06 ±0.07F18 -6.36 +0.04−0.03 0.27+0.05−0.04 -0.39+0.12−0.13 0.06+0.05−0.03 -6.25+0.03−0.03 0.04 ±0.03 -6.25+0.13−0.04 0.04 ±0.08F19 -6.23 +0.05−0.05 0.38+0.07−0.06 -0.19+0.19−0.24 0.10+0.11−0.07 -6.21+0.05−0.01 0.08 ±0.03 -6.18+0.13−0.09 0.11 ±0.11F20 -6.46 +0.03−0.03 0.22+0.03−0.03 -0.31+0.13−0.13 0.05+0.05−0.03 -6.40+0.03−0.01 -0.11 ±0.02 -6.32+0.01−0.02 -0.03 ±0.02F21 -6.33 +0.02−0.02 0.18+0.03−0.02 -0.19+0.08−0.08 0.02+0.02−0.01 -6.29+0.01−0.02 -0.00 ±0.02 -6.29+0.02−0.01 0.00 ±0.02F22 -6.37 +0.03−0.03 0.23+0.04−0.03 -0.10+0.10−0.14 0.03+0.05−0.03 -6.35+0.03−0.02 -0.06 ±0.03 -6.32+0.17−0.02 -0.03 ±0.10F23 -6.36 +0.04−0.04 0.24+0.04−0.04 -0.27+0.16−0.17 0.06+0.07−0.04 -6.29+0.02−0.04 0.00 ±0.03 -6.19+0.04−0.08 0.10 ±0.06F24 -6.45 +0.03−0.03 0.19+0.03−0.03 -0.24+0.21−0.17 0.04+0.05−0.04 -6.41+0.02−0.01 -0.12 ±0.02 -6.34+0.05−0.04 -0.05 ±0.05F25 -6.44 +0.04−0.03 0.23+0.04−0.03 -0.20+0.17−0.18 0.07+0.08−0.05 -6.41+0.02−0.02 -0.12 ±0.02 -6.43+0.03−0.02 -0.14 ±0.0999For both distance determination methods, F19 yields a slightly larger distancemodulus compared to the main body of the LMC. The median 𝑀𝐽 magnitude ofthe CS in this field is 0.08 mag fainter, while for the TRGB F19 is 0.11 mag fainter.F19 overlaps with the star forming region 30 Doradus located at 𝛼 = 5ℎ38𝑚 and𝛿 = −69◦06′ [126]. 30 Doradus and its surrounding area have been cataloguedas the most extincted region in the LMC reaching 𝐸 (𝐵−𝑉) values of ∼ 0.5 mag[180, 119, 133]. . In this thesis the entire LMC is corrected using the mean𝐸 (𝐵−𝑉) = 0.127 reddening. The fainter magnitudes in F19 are consistent with aroughly higher extinction correction than applied in this study. F23 yields a fainterdistance modulus compared to the surrounding fields F22 and F24. The suddendrop in magnitude in F23 could also be explained by the proximity to 30 Doradusas the majority of the stars in F23 are close to the border with F19.Comparing the results obtained from the CS and TRGB methods field by field,most bins show similar variations from the average distance modulus, with Δ𝑀 𝐽values that agree with Δ𝑀TRGB𝐾𝑠 within the error bars. Significant differences arefound between fields F5, F8, F10, F15 and F20. Fields 5, 8 and 10 are also fieldsthat show significantly different skew values in their CS luminosity functions. Themost “controversial” field in this thesis is F8, which shows a positive skew value inthe CS luminosity function and it also has shows the largest difference between theCS and TRGB method, with the CS having a distance modulus 0.15 magnitudesbrighter than the one obtained with the TRGB. F8 overlaps with field 15 of Hoytet al. [130]. In their study the authors find a differential distance modulus of 0.15magnitudes between this particular field and their adopted LMC distance modulus.Recently Hoyt [128] indicates that this specific field does not show an appreciableTRGB in the apparent CMD and that reddening corrections for this region of theLMC does not improve the dispersion in the TRGB magnitudes. Although thedivision of the LMC into different fields in this thesis is not the same as in Hoyt’spapers, some fields can be compared directly. Our TRGB results agree with thefinding in Hoyt et al. [130]. The corner fields of Hoyt (fields 35, 38 and 39) overlapwith our field F1; all these fields show a differential distance modulus that placesthem further from the Milky Way than the other fields. The north west cornerfield 20 of Hoyt overlaps with F25. These corner fields appear to be closer to uscompared to the rest of the LMC.1006.2 The Small Magellanic CloudInspecting Table 6.3 it is possible to see that about half the fields (F1, F4, F7 andF9) show significant variations from 𝑀 𝐽 ,𝑎𝑙𝑙 placing the north-eastern corner (F9)field closer than the main body of the SMC and the south-west corner (F1) furtheraway. The most extreme case is F7 exhibiting a 𝑀 𝐽 0.15 magnitudes brighter than𝑀 𝐽 ,𝑎𝑙𝑙. For 𝑀TRGB𝐾𝑠 only the western fields (F1, F2 and F3) show a statisticallysignificant difference with respect to 𝑀TRGB𝐾𝑠 ,𝑎𝑙𝑙, placing these three fields slightlyfurther away from the observer compared to the main body of the SMC. None ofthe fields appear to be closer to the observer.The SMC structure is known to be more complex than the LMC. Studiesbased on RR Lyrae stars [138, 137, 120, 175], Cepheids [120, 209], red clump stars[236, 179, 235, 241] andRGB stars [136], have shown that the SMChas a significantline of sight depth with spatial variations and different dynamical structures. Thiscoupled with the low number statistics resulting from splitting the data into ninefields, could partially explain the disagreement between Δ𝑀 𝐽 and Δ𝑀TRGB𝐾𝑠 . Giventhe low number of stars, no further binning as for the LMC is performed on theSMC data.Kapakos and Hatzidimitriou [137] studied the RR Lyrae stars in the SMC andfound an average line of sight depth of ∼ 5 kiloparsecs (kpc), with the deepestregions in the north-east reaching ∼ 16 kpc. The authors also report evidenceof different dynamical structures within the SMC traced using the difference inmetallicity from the different populations. Haschke et al. [120] used both RR Lyraeand Cepheid stars to create a 3-D map of the SMC. The old population of RR Lyraestars present in the galaxy showed a low inclination angle of only a few degreeswith respect to the plane of the sky. In contrast, using the younger tracers (theCepheids) the authors found an inclination angle of 74◦±9◦.The recent studies based on red clump stars [179, 235, 241] have all identifieda foreground stellar substructure in the north-east regions of the SMC. This fore-ground population was inferred from the bi-modal distributions of the red clumppopulation. Recently, James et al. [136] found a bi-modality in the distribution ofradial velocities of RGB stars in the same region as the red clump studies. Bothforeground populations (the RGB and red clump stars) in the north-eastern region101Table 6.3: Same as Table 6.1 but for the SMCField Mode Width Skew Kurtosis 𝑀 𝐽 Δ𝑀 𝐽 𝑀TRGB𝐾𝑠 Δ𝑀TRGB𝐾𝑠F1 -6.05+0.04−0.04 0.26+0.10−0.06 0.00+0.01−0.14 0.03+0.22−0.05 -6.06+0.03−0.01 0.12 ±0.02 -6.02+0.00−0.00 0.15 ±0.01F2 -6.15+0.03−0.02 0.17+0.05−0.03 0.00+0.01−0.05 -0.02+0.01−0.01 -6.16+0.01−0.01 0.02 ±0.02 -6.11+0.03−0.01 0.07 ±0.03F3 -6.15+0.04−0.03 0.22+0.06−0.04 0.00+0.01−0.05 -0.03+0.02−0.01 -6.15+0.02−0.03 0.03 ±0.03 -6.08+0.03−0.01 0.09 ±0.03F4 -6.25+0.05−0.05 0.31+0.12−0.07 0.00+0.00−0.04 -0.02+0.15−0.02 -6.25+0.01−0.01 -0.07 ±0.02 -6.13+0.05−0.04 0.05 ±0.05F5 -6.22+0.04−0.04 0.27+0.08−0.06 0.00+0.01−0.00 -0.00+0.07−0.02 -6.24+0.03−0.02 -0.06 ±0.03 -6.18+0.03−0.02 -0.01 ±0.03F6 -6.14+0.05−0.05 0.30+0.10−0.07 0.00+0.02−0.02 -0.02+0.10−0.02 -6.12+0.04−0.05 0.05 ±0.05 -6.14+0.03−0.02 0.04 ±0.03F7 -6.28+0.07−0.07 0.58+1.29−0.18 0.00+0.01−0.22 0.22+59.82−0.24 -6.32+0.06−0.03 -0.15 ±0.05 -6.14+0.04−0.02 0.03 ±0.03F8 -6.15+0.04−0.03 0.24+0.11−0.06 0.00+0.01−0.01 -0.00+0.23−0.03 -6.17+0.02−0.03 0.01 ±0.03 -6.19+0.04−0.03 -0.01 ±0.04F9 -6.25+0.04−0.04 0.25+0.07−0.04 0.00+0.00−0.01 -0.02+0.02−0.01 -6.26+0.05−0.02 -0.08 ±0.04 -6.14+0.02−0.01 0.04 ±0.02102are found to be approximately ∼ 11 kpc in front of the main body of the SMC.All the studies mentioned above suggest that the formation of the foregroundsubstructure was induced by the latest tidal interaction between Magellanic Cloudsroughly 300 Myr ago, as predicted by the simulations from Besla et al. [23], Diazand Bekki [68]. If the foreground substructure was formed during an encounterbetween the LMC and SMC, then this substructure should be composed of starsolder than the time of the formation of the substructure. However, this feature is notobserved in RR Lyrae stars. Omkumar et al. [184] explain this discrepancy basedon the spatial distribution of RR Lyrae stars within the spheroidal component ofthe galaxy, while the stars that formed the foreground structure were stripped fromthe disc of the SMC.Even though no bi-modality is observed when analysing RR Lyrae stars andCepheid variables, studies based on these tracers still find a structural elongation ofthe SMC from east towards the west/southwest with the north-eastern region closestto us ([236, 221, 209, 175]).The distances found in this thesis for the south-west corner (F1) using CS andthe TRGB are in agreement with the above statement. Nevertheless, the north-eastcorner (F9) is found to be closer only when using the CS, while the TRGB detectionyields no significant deviation from the distance modulus found for the entire RGstars sample.Subramanian et al. [235] used the width of the distribution of magnitudes of redclump stars to estimate the depth of different regions of the SMC. In this study nosignificant variations in the width of the CS luminosity functions are found exceptfor F7 (south-east corner). The CS luminosity function is roughly twice as wide asthe luminosity function for the other fields. For F7, the kurtosis value is also verydifferent from the other eight fields. F7 could be dominated by a population of CSthat does not resemble the luminosity function of the entire sample. Nevertheless,given that kurtosis value is close to the value given for the MLE in Table 3.1(𝑘 = 0.17), it is reasonable to assume that the CS in this field dominate the weightof the tails of the entire sample.1036.3 CrowdingThe three-dimensional analysis also serves as a test of crowding effects in the results.In both the LMC and SMC, the fields have the same number of stars but cover areasof different dimensions. In Figure 6.6 we plot Δ𝑀 𝐽 against stellar density for eachfield. In the case of the 3 by 3 binning, the LMC fields F1 and F9 are outliers,Section 6.1 explains how these two fields appear to be at different distances due tothe direction in which the galaxy is tilted with respect to the observer. For the SMC,Section 6.2 discusses the complex intrinsic properties of the galaxy, evidence fromdetailed 3-D structure studies support the findings of F1 having a larger distancemodulus. For the SMCF7 is also an outlier, this field is inferred to have a populationof CS that show a much wider distribution compared to the main body of the SMC.Going through the rest of the fields we can see that the stellar density varies by overa factor of ∼10 in the various fields, but there is no obvious correlation with Δ𝑀 𝐽 .In the case of the 5 by 5 binning for the LMC, the Δ𝑀 𝐽 versus stellar densityplot reflects the results found in the 3 by 3 binning. For both binnings, F1 is locatedin the south-west corner, and F9 and F25 are both in the north-east corner andshow roughly equal stellar densities and Δ𝑀 𝐽 . With 25 fields the plot shows cleargroupings of fields; for the 9 central and most dense fields, they are all closer toΔ𝑀 𝐽 = 0 with the exception of F19 which is identified as an outlier due to thehigher reddening caused by the presence of the star forming region 30 Doradus.The corner fields all have lower densities and their Δ𝑀 𝐽 values reflect the tilt ofthe LMC along the line of nodes. In the middle of the density values, the southand western fields are all grouped towards fainter distance moduli values, whilethe north and eastern fields are concentrated towards negative values of Δ𝑀 𝐽 . F23seems to be located away from the rest of the eastern fields, but as pointed out inSection 6.1, part of this field also falls in the proximity of 30 Doradus. Again noclear trends dependent on density are observed.In all cases, besides the mentioned fields, there is no evidence of clear trends orlarger errors dependent on density. In fact, the central fields are the most crowdedregion in both the LMC and SMC and the distances are consistent with Δ𝑀 𝐽 = 0within the error bars.104Figure 6.6: Difference between the median absolute 𝐽 magnitude of a specificfield and the full CS sample (Δ𝑀 𝐽 ) versus stellar density. Δ𝑀 𝐽 can also beinterpreted as the difference in distance modulus (Δ𝜇0). In the top panel theresults for the 3 by 3 binning for both the LMC (black) and the SMC (grey)are shown. For the LMC, besides F1 and F9, that deviate from zero due to theinclination of the galaxy discussed in Section 6.1, there is no visible correlationbetween the plotted values. For the SMC, F7 has been flagged as an outlier(see Section 6.2). The bottom panel shows the results for the LMC whenbinned into 25 fields. The centre fields (black circles) are grouped closelyaround Δ𝑀 𝐽 = 0, with F19 slightly fainter than the rest of the group. F19 hasbeen flagged as a possible outlier due to a larger than average reddening inthat region of the LMC. As expected from the inclination of the galaxy withrespect to the plane of the sky, north (N) and east (E) fields (purple squares)are grouped at brighter magnitudes (Δ𝑀 𝐽 < 0) placed closer to the observerthan the centre fields, while the south (S) and west (W) fields (yellow crosses)show Δ𝑀 𝐽 > 0, further from the observer.1056.4 DiscussionFor both the LMC and SMC, the minimal variations in the luminosity functionmodel parameters tells that the contribution of CS stars from different locationsin the LMC and SMC does not have a significant impact on the overall estimatedluminosity function parameters. Based on the skew parameter, only one field outof nine in the LMC could have been paired with the less suitable calibrator. Furtherresearch is necessary to determine what produces the slightly different skew valuesin LMC-F2, but it may be a reflection of different star formation histories in differentparts of the galaxies. Nevertheless, the inclusion or exclusion of these fields doesnot alter the classification of the entire galaxy or the estimation of distances to othergalaxies.The variations in the distance modulus with respect to the main body of theLMC in the corner fields (F1 and F9, and F1 and F25 for the 3 by 3 and 5 by 5binnings respectively), and F19 and F23 are explained by the intrinsic 3-D structureand properties of the LMC. The corner fields follow the orientation of the LMCwhile F19 and F23 are affected by the high extinction in the star forming region 30Doradus. The variations in two out of the nine fields of the LMC do not affect thetotal median absolute 𝐽 magnitude and therefore have no effect in the determinationof the distance. The detection of these variations in the distance to different fieldsserves as a test of the sensitivity of the CS method.In Hoyt et al. [130] the authors also carried out a similar 3D structural analysisof the LMC using the TRGB technique in the 𝐾𝑠-band. Their results are in goodagreement with the findings in this thesis. Recently Hoyt [128] conducted anextensive analysis of the TRGB in the 𝐼-band in different fields of the LMC. Theauthor’s 3D-analysis of the LMC here also agrees with the findings in this thesis.In the SMC, evidence of variations in the distance modulus measured usingthe CS are consistent with studies that place the south-west corner (F1) furtherand the north-east field (F9) closer to the Sun. With the TRGB, F1 is placed ata similar distance but F9 shows no evidence of being further than the rest of theSMC. The inclusion or exclusion of any of these fields does not affect the luminosityfunction fitting procedure or the distance estimations. It is worth mentioning that inexternal galaxies beyond the Magellanic Clouds the line of sight depth is negligible106compared to the distance to the galaxy.When we sum up the stars over an entire galaxy, the signal will always bedominated by the population of stars most representative of the metallicity and starformation history of the galaxy. As we try to estimate distances to more distantgalaxies, the number of observable CS will decrease and it will be critical to useevery star available. For the most distant galaxies, the stars will be concentrated inone small area and excluding any will reduce the accuracy of the distance estimates.The purpose of this 3-D structure analysis is to test the potency of the CS as standardcandles. The results have shown that intrinsic properties of the galaxy do not greatlyaffect the final distance estimation.107Chapter 7Reddening EffectsHitherto, the Magellanic Clouds data were corrected using the mean 𝐸 (𝐵 −𝑉)reddening values from the Górski et al. [110] LMC and SMC reddening mapsalong with Gordon et al. [107] extinction factors.Since the start of the research carried out for this thesis, new reddening mapsand extinction factors for the Magellanic Clouds have been published. In thischapter we test the effects of reddening by repeating the CS and TRGB analysis onthe Magellanic Clouds data using the new available data.From the definition of distance modulus the main correction to this quantity isthe removal of reddening effects. To obtain the true distance modulus (𝜇0) to anyobject, it is necessary to account for the line of sight extinction. This is achievedby correcting the observed distance modulus using the appropriate extinction 𝐴𝑥:𝜇0 = (𝑚𝑥 −𝑀𝑥) − 𝐴𝑥 , (7.1)where 𝑥 is the wavelength at which the data is observed (in this thesis 𝑥 ∈ {𝐽,𝐻,𝐾𝑠})and 𝐴𝑥 is given by:𝐴𝑥 = 𝑅𝑥 𝐸 (𝐵−𝑉). (7.2)In the above equation 𝑅𝑥 is the extinction (or reddening) factor.Through out this thesis the Magellanic Clouds have been used as the primarycalibrators. In this case the distance to the LMC and SMC are known and it is used108to obtain the absolute magnitudes of the stars. Equation 7.1 can be re-written as:𝑀𝑥 = 𝑚𝑥 −5 log(𝑑) +5 − 𝐴𝑥 , (7.3)where 𝑑 is the distance to the star in parsecs.7.1 Reddening maps and reddening factorsThe Górski et al. [110] reddening maps of the Magellanic Clouds cover a consid-erably smaller area in the sky than that of the Magellanic Clouds photometric dataused in this thesis (see Figure 7.1). For this reason, it is not possible to obtain alocation dependent reddening for each individual star. To de-redden the LMC andSMC data, the magnitude of each stars was corrected using the mean 𝐸 (𝐵−𝑉)value from Górski et al.’s maps. The mean 𝐸 (𝐵−𝑉) reddening values for the LMCand SMC are presented in Table 7.1.Recently, Skowron et al. [226] developed new reddening maps that cover asignificantly larger area compared to Górski et al.’s maps. For the SMC, the extentof Skowron et al.’s map covers nearly the entire original SMC field. In the caseof the LMC, Skowron et al.’s map covers a slightly smaller area compared to theoriginal data, nevertheless over 95% of the CS selected in Chapter 3 fall within thelimits of Skowron et al.’s map.Figure 7.1 shows the reddening maps of the Magellanic Clouds obtained usingSkowron et al.’s data1. Over the reddening maps, the layout of Górski et al.’smaps are shown. It is possible to see that Górski et al.’s maps are centred in theareas of the LMC and SMC where the reddening is higher compared to other partsof the galaxy. The mean reddening values found from both maps are presentedin Table 7.1, Skowron et al.’s maps show a much lower mean but this numberis driven by the fact that the maps include the outskirts of the galaxies wherereddening is very low. In Skowron et al. the authors compared their maps to Górskiet al.’s maps using the intrinsic colour of red clump stars found in the two studies.In the overlapping areas Skowron et al. find red clump stars to have on averageredder intrinsic colours compared to those found by Górski et al., with differences1Publicly available at https://ogle.astrouw.edu.pl/cgi-ogle/get_ms_ext.py109Figure 7.1: Reddening map for the Magellanic Clouds. The colour excess𝐸 (𝑉 − 𝐼) for each star is retrieved from the Skowron et al. [226] reddeningmap. The black lines delimit the original fields used in this thesis. The dashedred lines show the boundaries of the reddening maps from Górski et al. [110]which are considerably smaller than the original fields.((𝑉 − 𝐼)0,Skowron − (𝑉 − 𝐼)0,Gorski) of 0.045 and 0.048 magnitudes in the LMC andSMC respectively. If the stars are found to be intrinsically redder, then the reddeningcorrection needed to be applied to the observed data is smaller.The maps give the reddening values in terms of the (𝐵−𝑉) and (𝑉 − 𝐼) colourexcess (𝐸). To correct the absolute magnitude of each stars it is necessary totransform 𝐸 (𝐵−𝑉) and 𝐸 (𝑉 − 𝐼) into 𝐴𝐽 and 𝐴𝐾 using the extinction factors givenin Equation 7.2. For the analysis in this thesis the 𝑅𝑥 values were taken fromGordon et al. [107]. Recently Chen et al. [40] published new 𝑅𝑥 values for theLMC and SMC. The extinction factors from the two studies are compared in Table7.2. The extinction coefficients for 𝐽 and 𝐾 are higher in Chen et al. for both theLMC and SMC. Most importantly the ratio between 𝑅𝐽 and 𝑅𝐾 changes drasticallywith 𝑅𝐽/𝑅𝐾 ∼ 8 for Gordon et al. and only 𝑅𝐽/𝑅𝐾 ∼ 2.5 for Chen et al. This willhave a impact on the analysis especially on the colour dependent magnitudes used110Table 7.1: Mean reddening values for the LMC and SMC from the Górskiet al. [110] and Skowron et al. [226] reddening maps. A lower mean valuefrom Skowron et al. is expected given their map covers also the outskirts ofthe galaxies where reddening is very low. The expression used in the fourthcolumn is taken from Skowron et al..Górski et al. [110] Skowron et al. [226] Skowron et al. [226]𝐸 (𝐵−𝑉) 𝐸 (𝑉 − 𝐼) 𝐸 (𝐵−𝑉) = (𝑉−𝐼)1.318LMC 0.127 0.100 0.076SMC 0.084 0.047 0.036Table 7.2: Extinction coefficients for the LMC and SMC from Gordon et al.[107] and Chen et al. [40].Gordon et al. [107] Chen et al. [40]𝑅𝐽 𝑅𝐻 𝑅𝐾 𝑅𝐽 𝑅𝐻 𝑅𝐾𝑠LMC 0.877 0.635 0.102 1.015 0.640 0.413SMC 0.378 0.487 0.046 0.742 0.468 0.302to obtained the TRGB magnitude.7.2 AnalysisIn this section the CSmedian𝑀𝐽 magnitude and TRGBmagnitude are re-calculatedfor the LMC and SMC using both Górski et al. [110] and Skowron et al. [226] mapswith the new extinction coefficients. Estimates without reddening corrections ofthe CS and TRGBmagnitudes are also included. In every calculation, only the starsthat fall within the limits of Skowron et al. [226] maps are used.7.2.1 Carbon starsFigure 7.2 shows the (𝐽−𝐾𝑠)0, 𝑀𝐾𝑠 LMCCMDs. In each panel the LMC data havebeen corrected using different reddening values. For Górski et al. [110] (indicated111as G20 in the plots) the absolute magnitudes are obtained with:𝑀𝑥 = 𝑚𝑥 −5 log(𝑑) +5 − 𝑅𝑥 𝐸 (𝐵−𝑉), (7.4)where 𝑑 is the distance to the LMC or SMC and 𝐸 (𝐵−𝑉) and 𝑅𝑥 are specified inTables 7.1 and 7.2 respectively. For Skowron et al. [226] (indicated as S21 in plots)the reddening term changes and 𝑀𝑥 is defined:𝑀𝑥 = 𝑚𝑥 −5 log(𝑑) +5 − 𝑅𝑥𝑅𝐼1.5𝐸 (𝑉 − 𝐼), (7.5)where 𝑅𝐼 = 2.2 is taken from Chen et al. [40] and 1.5 comes from the extinctionestimation given in Skowron et al.: 𝐴(𝐼) ∼ 1.5𝐸 (𝑉 − 𝐼). In the case of Skowronet al. maps, first the data are corrected using the location dependent reddening, i.e.each individual star is assign an 𝐸 (𝑉 − 𝐼) value based on their RA,DEC coordinates.Second, the data is corrected using the mean 𝐸 (𝑉 − 𝐼) value presented in Table 7.1.Only the first case is presented in Figure 7.2.For each different reddening correction applied, the CS are selected within the1.4 < (𝐽 −𝐾𝑠)0 < 2.0 colour limits. The luminosity function of the colour selectedCS is then fitted using the modified Lorentzian function:𝑓 (𝐽;𝑚,𝑤, 𝑠, 𝑘) = 𝑎1+ ( 𝐽−𝑚𝑤 )2 + 𝑠 ( 𝐽−𝑚𝑤 )3 + 𝑘 ( 𝐽−𝑚𝑤 )4 , (7.6)where:𝑚 = mode of the distribution,𝑤 = scale parameter,𝑠 = skewness parameter,𝑘 = kurtosis parameter,𝑎 = amplitude (height of the peak, used for plotting purposes),as specified in Section 3.3. The shape parameters 𝑤, 𝑠 and 𝑘 show no significantvariations when using different reddening corrections. Independent of the redden-ing correction applied, the LMC is always skewed towards brighter magnitudes112Figure 7.2: Comparison of LMC ((𝐽 −𝐾𝑠)0, 𝑀𝐾𝑠 ) CMDs obtained by cor-recting the data using different reddening maps. On the left data is correctedby using the mean 𝐸 (𝐵−𝑉) reddening from Górski et al. [110] (G20). Themiddle panel shows the data without any reddening corrections (NR). On theright the data has been corrected using the reddening map from Skowronet al. [226] (S21). The red vertical lines in each panel indicate the CS colourselection limits (1.4 < (𝐽 −𝐾𝑠)0 < 2.0).while the SMC exhibits a symmetric distribution. Figure 7.3 shows the resultingbest-fit and parameter values for the LMC and SMC when using Górski et al. [110](G20) and Skowron et al. [226] (S21) maps.Themedian𝑀𝐽 magnitude of theCS (𝑀 𝐽 ) for the different reddening correctioncases are shown in Table 7.3. The 𝑀 𝐽 values for the LMC are slightly fainter whenusing Skowron et al. [226] compared to the original value from this thesis. For theSMC the values remain in agreement within their error bars.7.2.2 The tip of the red giant branchThe absolutemagnitudes for the LMCand SMCare obtained as detailed in the abovesection. As in Chapter 4, the first step is transforming the absolute 𝐾𝑠 magnitudesfor the LMC and SMC data to the colour dependent 𝑇 magnitude defined for the113Figure 7.3: Best-fit modified Lorentzian model for the CS luminosity func-tions from the LMC (left panel) and SMC (right panel) from data correctedusing Górski et al. [110] (G20, dashed-blue line) and Skowron et al. [226](S21, orange line) maps. The shape of the of the luminosity function is notaffected by the change in reddening correction. The median 𝑀𝐽 magnitudefor these two cases are also shown at the bottom of each plot.Table 7.3: CS median 𝑀𝐽 magnitude (𝑀 𝐽 ) for the LMC and SMC obtainedwith the data corrected using different reddening maps. The first column showsthe original values obtained in Chapter 3. G20 means the data were correctedusing the mean 𝐸 (𝐵−𝑉) from Górski et al. [110]. NR indicates no reddeningcorrections. S21 are the data corrected using Skowron et al. [226] maps whilefor S21 (mean) the average 𝐸 (𝑉 − 𝐼) is applied. The last row shows thedifference |Δ| = |𝑀 𝐽,LMC −𝑀 𝐽,SMC |. When using Skowron et al. [226] mapsthe difference between 𝑀 𝐽 of the LMC and SMC is slightly lower compared tothe original estimations.Original G20 NR S21 S21 (mean)LMC -6.288+0.005−0.004 -6.303+0.004−0.004 -6.163+0.006−0.005 -6.256+0.005−0.005 -6.240+0.005−0.004SMC -6.177+0.012−0.016 -6.209+0.011−0.013 -6.146+0.012−0.012 -6.187+0.012−0.014 -6.179+0.012−0.016|Δ| 0.111 0.094 0.017 0.069 0.061114Figure 7.4: Comparison of LMC ((𝐽 −𝐾𝑠)0, 𝑇𝑀𝐾𝑠 ) CMDs obtained by cor-recting the data using different reddening maps. On the left data are correctedby using the mean 𝐸 (𝐵−𝑉) reddening from Górski et al. [110] (G20). Themiddle panel shows the data without any reddening corrections (NR). On theright the data have been corrected using the reddening map from Skowronet al. [226] (S21). The black horizontal line in each panel indicates the TRGBmagnitude found in each case (see Table 7.3). The shadowed regions delimitthe selection of red giants used to construct the luminosity functions shown inFigure 7.5.𝐾𝑠 filter [155, 130] as:𝑇(𝑀𝐾𝑠 )0 = (𝑀𝐾𝑠 )0 +1.85[(𝐽 −𝐾𝑠)0−1.00] . (7.7)The ((𝐽 −𝐾𝑠)0, 𝑇𝑀𝐾𝑠 ) CMDs for the LMC are shown in Figure 7.4, along with theRGs selection regions.Following the selection of the RGs, the TRGB detection procedure follows thesame steps as in Chapter 4. First the luminosity functions are smoothed using a1-D Gaussian filter. Second a Sobel filter is applied over the smoothed luminosityfunction acting as an edge-response function. The luminosity functions for theLMC and SMC are plotted in Figures 7.5 and 7.6 respectively. The 𝑀TRGB𝐾𝑠 forthe LMC and SMC when applying different reddening corrections are shown inTable 7.4. The values of 𝑀TRGB𝐾𝑠 and their statistical errors were obtained fromten-thousand bootstrap realizations. As with the CS, using Skowron et al. [226]115Figure 7.5: LMC RGs 𝑇𝑀𝐾𝑠 luminosity functions from data corrected usingdifferent reddening maps. The reddening used to correct the data is indicatedin the inset with G20 (blue-dashed lines) referring to Górski et al. [110], S21(orange lines) to Skowron et al. [226] and NR no reddening correction applied.The most prominent (thicker) lines show the smoothed luminosity functionsplotted on top of the binned data (thin line). At the bottom of the plot isthe Sobel filter or edge-response function where their maxima indicate thelocation of the TRGB marked by a vertical line and specified in the inset.maps gives significantly fainter 𝑀TRGB𝐾𝑠 values for the LMC, bringing 𝑀TRGB𝐾𝑠 ,LMC and𝑀TRGB𝐾𝑠 ,SMC into agreement within their error bars. However, when using the Skowronet al. reddening map the edge response shows two peaks very near in size with thesecond peak 0.06 magnitudes brighter. Figure 7.7 shows the bootstrapping resultsfor this case. The second brighter peak is detected ∼ 30% of the time.7.2.3 LMC 3D-structureWhen using the more recent Skowron et al. [226] reddening map we notice that theCS exhibit a slightly fainter magnitude while the TRGB seems significantly fainter.In this section we repeat the 3D analysis carried out in Chapter 6 for the LMC116Figure 7.6: Same as Figure 7.5 but for the SMC. In this case the edge-responsefunction have beenmoved downwards to avoid overlappingwith the luminosityfunctions.Table 7.4: Similar to Table 7.3 but for the TRGB 𝐾𝑠 magnitude (𝑀TRGB𝐾𝑠 ).As with the CS, using Skowron et al. [226] (S21) maps reduces the differencebetween |Δ| = |𝑀TRGB𝐾𝑠 ,LMC −𝑀TRGB𝐾𝑠 ,SMC |, bringing them to agree within the errorbars. Nevertheless, as shown in Figure 7.7, the bootstrapping results for theLMC find the TRGB to be brighter by 0.06 magnitudes ∼ 30% of the time.Original G20 NR S21 S21 (mean)LMC -6.290+0.015−0.010 -6.292+0.015−0.015 -6.102+0.020−0.011 -6.176+0.010−0.005 -6.202+0.030−0.015SMC -6.175+0.017−0.013 -6.204+0.030−0.020 -6.125+0.015−0.015 -6.161+0.030−0.020 -6.154+0.030−0.020|Δ| 0.115 0.088 0.023 0.015 0.048117Figure 7.7: TRGB bootstrap results histogram (grey-dotted line) for the LMCwhen using Skowron et al. [226] reddening map. The second peak at -6.236is detected ∼ 30% of the time.divided into 25 fields each containing the same number of CS. The focus of thissection is on the differential distance modulus of each field and not on the absolutemagnitude of the TRGB.Table 7.5 shows the results for the CS luminosity function best-fit parametersand differential distance moduli obtained from both CS and TRGB. Comparing tothe results fromChapter 6, in this case only 5 out of the 25 fieldswould be cataloguedas “SMC-like” galaxies (compared to 10 fields found in the original analysis). Ingeneral, the CS luminosity functions parameters do not vary significantly whenusing the two different reddening maps.For the CS no significant differences were found between the differential dis-tance moduli derived with the old Górski et al. [110] and new Skowron et al. [226]maps. In the case of the TRGB, fields F1 and F2 show a significant difference inthe differential distance modulus compared to the results using the old maps. Withthe new maps the TRGB of F1 and F2 are found to be consistent with the TRGBmagnitude found for the entire sample of stars (and not further away as with the oldmap). For both CS and TRGB, F19 shows a reduced differential distance modulus,most likely due to the use of the location dependent reddening which accounts forthis region having on average higher reddening.118The Hoyt [128] study of the TRGB in the LMC was conducted using theSkowron et al. [226] reddening map. Given that Hoyt [128]’s analysis was carriedout using data taken in a different band, the TRGBmagnitudes cannot be compareddirectly but, as mentioned in Chapter 6, the differential distance moduli results arein agreement with the results from this thesis.119Table 7.5: Same as Table 6.2 but with the LMC data corrected for reddening using the Skowron et al. [226] maps.Field Mode Width Skew Kurtosis 𝑀 𝐽 Δ𝑀 𝐽 𝑀TRGB𝐾𝑠 Δ𝑀TRGB𝐾𝑠F1 -6.19 +0.02−0.02 0.18+0.02−0.02 -0.19+0.09−0.11 0.03+0.03−0.02 -6.14+0.02−0.02 0.11 ±0.01 -6.12 +0.02−0.02 0.05 ±0.02F2 -6.25 +0.04−0.04 0.26+0.04−0.04 -0.36+0.16−0.15 0.08+0.07−0.04 -6.18+0.03−0.02 0.08 ±0.01 -6.16 +0.01−0.01 0.02 ±0.01F3 -6.25 +0.04−0.04 0.29+0.05−0.04 -0.22+0.15−0.19 0.09+0.09−0.06 -6.20+0.03−0.04 0.05 ±0.01 -6.19 +0.03−0.02 -0.01 ±0.03F4 -6.24 +0.04−0.04 0.29+0.05−0.04 -0.20+0.17−0.21 0.08+0.09−0.05 -6.22+0.02−0.02 0.04 ±0.01 -6.11 +0.02−0.08 0.07 ±0.04F5 -6.30 +0.04−0.03 0.28+0.05−0.04 -0.06+0.11−0.14 0.03+0.13−0.05 -6.29+0.04−0.02 -0.03 ±0.01 -6.12 +0.01−0.02 0.06 ±0.02F6 -6.24 +0.05−0.04 0.29+0.06−0.05 -0.37+0.25−0.23 0.14+0.11−0.08 -6.18+0.02−0.02 0.08 ±0.01 -6.17 +0.01−0.01 0.01 ±0.01F7 -6.34 +0.03−0.03 0.25+0.05−0.04 -0.37+0.10−0.10 0.06+0.04−0.02 -6.24+0.05−0.03 0.01 ±0.01 -6.19 +0.02−0.03 -0.01 ±0.03F8 -6.29 +0.05−0.05 0.35+0.07−0.06 0.08+0.18−0.16 0.07+0.14−0.09 -6.29+0.03−0.02 -0.04 ±0.01 -6.09 +0.03−0.04 0.09 ±0.04F9 -6.35 +0.05−0.04 0.30+0.07−0.05 -0.47+0.24−0.17 0.13+0.08−0.06 -6.26+0.03−0.03 -0.01 ±0.01 -6.14 +0.04−0.10 0.04 ±0.07F10 -6.32 +0.06−0.05 0.35+0.06−0.06 0.06+0.27−0.31 0.19+0.19−0.14 -6.31+0.03−0.03 -0.06 ±0.01 -6.17 +0.04−0.05 0.00 ±0.05F11 -6.28 +0.05−0.04 0.27+0.05−0.04 -0.42+0.20−0.16 0.11+0.09−0.06 -6.20+0.03−0.02 0.06 ±0.01 -6.15 +0.02−0.02 0.03 ±0.02F12 -6.35 +0.08−0.07 0.34+0.09−0.07 -0.62+0.37−0.28 0.20+0.31−0.06 -6.23+0.03−0.05 0.02 ±0.02 -6.20 +0.13−0.03 -0.02 ±0.08F13 -6.32 +0.05−0.06 0.32+0.06−0.05 -0.43+0.26−0.27 0.17+0.14−0.08 -6.24+0.02−0.03 0.02 ±0.01 -6.24 +0.02−0.02 -0.06 ±0.02F14 -6.28 +0.04−0.05 0.31+0.06−0.05 -0.22+0.20−0.27 0.09+0.11−0.07 -6.23+0.03−0.02 0.03 ±0.01 -6.16 +0.04−0.02 0.02 ±0.03F15 -6.37 +0.05−0.05 0.30+0.05−0.05 -0.27+0.28−0.27 0.18+0.16−0.11 -6.32+0.02−0.03 -0.07 ±0.01 -6.20 +0.02−0.01 -0.02 ±0.02120Field Mode Width Skew Kurtosis 𝑀 𝐽 Δ𝑀 𝐽 𝑀TRGB𝐾𝑠 Δ𝑀TRGB𝐾𝑠F16 -6.28 +0.03−0.03 0.23+0.04−0.03 -0.25+0.15−0.15 0.05+0.05−0.04 -6.23+0.04−0.02 0.02 ±0.01 -6.17 +0.02−0.01 0.01 ±0.02F17 -6.32 +0.03−0.03 0.22+0.03−0.03 -0.22+0.11−0.11 0.03+0.03−0.02 -6.25+0.01−0.01 0.01 ±0.01 -6.16 +0.05−0.03 0.01 ±0.04F18 -6.36 +0.04−0.03 0.28+0.05−0.04 -0.41+0.13−0.12 0.07+0.05−0.03 -6.27+0.03−0.01 -0.01 ±0.01 -6.21 +0.02−0.03 -0.03 ±0.03F19 -6.23 +0.04−0.04 0.36+0.07−0.06 -0.05+0.13−0.20 0.09+0.13−0.07 -6.22+0.02−0.02 0.04 ±0.01 -6.23 +0.20−0.05 -0.05 ±0.13F20 -6.42 +0.03−0.03 0.24+0.04−0.03 -0.28+0.16−0.14 0.06+0.06−0.04 -6.36+0.02−0.02 -0.11 ±0.01 -6.21 +0.01−0.01 -0.04 ±0.01F21 -6.32 +0.02−0.02 0.19+0.03−0.03 -0.25+0.08−0.08 0.03+0.03−0.02 -6.26+0.02−0.03 -0.00 ±0.01 -6.17 +0.07−0.02 0.00 ±0.05F22 -6.35 +0.04−0.03 0.25+0.04−0.03 -0.17+0.16−0.19 0.07+0.09−0.05 -6.32+0.02−0.02 -0.06 ±0.01 -6.24 +0.15−0.03 -0.06 ±0.09F23 -6.34 +0.04−0.03 0.24+0.04−0.04 -0.30+0.15−0.13 0.06+0.06−0.03 -6.29+0.03−0.02 -0.04 ±0.01 -6.19 +0.08−0.05 -0.02 ±0.07F24 -6.40 +0.03−0.02 0.19+0.04−0.03 -0.23+0.19−0.16 0.04+0.05−0.04 -6.36+0.01−0.02 -0.10 ±0.01 -6.25 +0.11−0.02 -0.08 ±0.07F25 -6.37 +0.04−0.04 0.26+0.04−0.04 -0.22+0.20−0.21 0.12+0.09−0.07 -6.34+0.02−0.02 -0.09 ±0.01 -6.31 +0.03−0.01 -0.13 ±0.091217.3 DiscussionFrom the results for 𝑀 𝐽 for the CS and 𝑀TRGB𝐾𝑠 in the previous section, it is possibleto see that correcting the data using Skowron et al. [226] reddening maps givesfainter magnitudes for the LMCwhile no significant effect is seen for the SMC. Thegreatest effects is seen in the TRGB magnitudes where using Skowron et al. [226]maps bring the values of 𝑀TRGB𝐾𝑠 of the LMC and SMC into agreement.The original work in this thesis uses the mean 𝐸 (𝐵 −𝑉) from Górski et al.[110]. This value was obtained by averaging the reddening in different parts of theMagellanic Clouds but from a much smaller area compared to the new Skowronet al. [226] maps. For the LMC, it is possible to see in Figure 7.1 that the Górskiet al. field is located in a high reddening area compared to the northern half ofthe galaxy. Averaging over the Górski et al. region gives a higher mean reddeningvalue compared to averaging over the Skowron et al. area. However, when the studypresented in this thesis was initiated, only Górski et al. maps were available andgiven the reduced size of their fields it was only viable to use the mean reddeningvalues given for the LMC and SMC.The use of different extinction coefficients (𝑅) seems to have a lesser impactcompared to the different reddening maps. The main difference between the coef-ficients used originally from Gordon et al. [107] and Chen et al. [40] is the valueof 𝑅 for the 𝐾 filter which is ∼ 4 times larger in the newer study. The differencein 𝑅𝐾 leads to a significant shift in the (𝐽 −𝐾𝑠)0 colour of the stars which can bemost clearly observed in the CMDs in Figure 7.4 where the selection box for thered giants had to be moved ∼ 0.05 magnitudes towards the red.Table 7.6 shows the distance moduli for the target galaxies using the calibrationsof 𝑀 𝐽 and 𝑀TRGB𝐾𝑠 in the different reddening cases. Except for the distance moduliobtained from the TRGB for NGC 6822, the results for each target galaxy are inagreement within the error bars with the original distance estimations of Chapters3 and 4.122Table 7.6: Distance moduli for the target galaxies when different reddeningmaps are used to calibrate the Magellanic Clouds. Except for the distancemoduli obtained from the TRGB for NGC 6822, the results for each galaxy arein agreement within the error bars.Original G20 S21 S21 (mean)Carbon starsNGC 6822 23.55 ± 0.03 23.56 ± 0.03 23.52 ± 0.03 23.50 ± 0.03IC 1613 24.45 ± 0.05 24.48 ± 0.05 24.46 ± 0.05 24.45 ± 0.05WLM 25.02 ± 0.04 25.05 ± 0.04 25.03 ± 0.04 25.02 ± 0.04NGC 3109 25.58 ± 0.03 25.61 ± 0.03 25.59 ± 0.03 25.58 ± 0.03TRGBNGC 6822 23.50 ± 0.02 23.50 ± 0.03 23.39 ± 0.02 23.41 ± 0.03IC 1613 24.46 ± 0.02 24.49 ± 0.03 24.45 ± 0.03 24.44 ± 0.03WLM 25.03 ± 0.03 25.05 ± 0.03 25.01 ± 0.03 25.00 ± 0.03NGC 3109 25.60 ± 0.03 25.62 ± 0.04 25.58 ± 0.04 25.57 ± 0.04123Chapter 8Carbon stars in the Milky WayChapter 3 explores CS and their standard candle capabilities based on the median 𝐽magnitude of the stars within 1.4 < (𝐽−𝐾𝑠)0 < 2.0 colour limits. The method usesthe LMC and SMC as primary calibrators to estimate distances to other Magellanictype galaxies. Ripoche et al. [210] explored the possibility of using the CS in theMilky Way as the calibrator instead of relying on the distance determination to theMagellanic Clouds. Ripoche et al. took the CS list from the galactic CS catalogueprovided by Alksnis et al. [4]. The distance to each individual CS was obtainedusing Gaia DR2 [93] parallaxes, while the near-infrared magnitudes (𝐽𝐻𝐾𝑠) wereacquired by cross-matching with the 2MASS point-source catalogue. They foundthat the Milky Way CS had a fainter 𝑀 𝐽 and a considerably wider luminosityfunction than the CS in the Magellanic Clouds. The substantial differences foundbetween the CS in the Milky Way and the Magellanic Clouds in this study wereattributed, in part, to the large uncertainty in the Gaia DR2 parallaxes.Gaia EDR3 includes updated and more precise parallaxes for many of theCS included in the Alksnis et al. [4] catalogue. In this Chapter, using updatedGaia EDR3 parallaxes, different galactic CS catalogues are analyzed to study thepotential usage of the CS in the Milky Way as the main distance calibrator.1248.1 Catalogues of Galactic carbon starsUp until the early 1990’s, a “General Catalogue of Cool Galactic CS” was main-tained by B.C. Stephenson [4]. Stephenson published two editions: in 1973 [230]and 1989 [231]. A third edition of this catalogue was published by Alksnis et al.[4] in 2001, containing a total of 6891 stars. The stars added to the third editioncome from a wide variety of sources (see references in Alksnis et al. for details)and are identified as CS using different spectral features (such as SiC emissionat 11.2 𝜇m, C2 molecular Swan bands in the visible spectra) or their location ininfrared colour-colour diagrams. The third edition of the catalogue widened thevariety of known CS, including R type or “warm” CS. For this reason the “Cool”part of the catalogue’s name was dropped.Recently, Suh [237] compiled a list of O-rich AGB stars and CS from differentsources available in the literature including Alksnis et al. [4] (see Suh [237] Tables1 and 2), to study different aspects of AGB stars such as their infrared variabilityand the properties of the dust envelopes around them. For a star to be a part of Suh’scatalogue it must have a counterpart identified by either the Infrared AstronomicalSatellite (IRAS) or by the Widefield Infrared Survey Explorer (WISE). The finalversion of Suh’s catalogue has a total of 7172 CS of which 5834 overlap withAlksnis et al., while most of the remaining stars come from Suh and Hong [238].Multiple Galactic CS catalogues (with new and/or known stars) have beenpublished with stellar selection criteria based on the research goals of the particularproject. For example, Abia et al. [2] selected 210 sources to study the propertiesof CS in the Solar neighbourhood through their luminosity function and kinematicproperties. A study by Pal and Worthey [185] found 12 CS in the Alksnis et al.catalogue that are likely members of different open clusters in the Galaxy. Li et al.[149] used a list of known CS to train a machine-learning algorithm to classifystars based on their Large Sky Area Multi-Object Fiber Spectroscopy Telescope(LAMOST) spectrum, successfully finding and classifying 2651 CS from a sampleof over seven million spectra. Lee et al. [147] combined two previously publishedcatalogues [250, 41] and used a colour-selected sample of stars to determine 𝑀 𝐽of the CS in the Milky Way. Because this study measures 𝑀 𝐽 in the Milky Wayin the same colour range as in this thesis, a closer look to these two catalogues is125warranted.Whitelock et al. [250] selected only 239 stars with near-infrared 𝐽𝐻𝐾𝐿 pho-tometry from different sources to specifically study carbon-rich Mira variables(C-Miras). The photometry for the CS in Whitelock et al. comes from the Aaron-son et al. [1] catalogue, the IRAS [13] point source catalogue andwas supplementedwith observations using the South African Astronomical Observatory (SAAO). Theobjective of Whitelock et al. was to construct a catalogue of stars with high prob-ability of being C-Miras rather than having a complete catalogue. The selectioncriteria for a star to be included in this catalogue were set depending on the originalsource catalogue. For stars coming from Aaronson et al. [1], the selection wasbased on their colour, (𝐻 −𝐾)0 < 0.8, whereas the stars from IRAS and SAAOwere picked based on their spectral type, selecting only stars that showed the SiCemission feature at 11.2 𝜇m. For IRAS sources, a flux ratio and total flux selectionlimits were also used, removing faint CS candidates. These selection criteria wererecognized by the authors as strict and likely to leave out real C-Miras but theyincrease the likelihood of having a clean sample.Chen andYang [41] compiled 974 stars frommultiple sources, with the objectiveof studying the CS evolutionary sequence (see Chen et al. [42] and referencestherein). The Chen and Yang catalogue includes CS that were selected usingdifferent spectral features (the SiC emission line and the HCN + C2H2 absorptionfeature at 3.1 𝜇m), their location in the NIR (𝐾 − 𝐿) vs IRAS ([12]-[25]) colour-colour diagram, or their IRAS flux ratios between 𝐹12/𝐹25 < 3 and 𝐹25/𝐹60 >2.75. For IRAS, the label numbers 12, 25 and 60 refer to the approximate centralwavelength of the filter. Chen and Yang removed from their catalogue any sourceclassified as a silicate CS, J-type CS, extreme CS or C-rich proto-planetary nebula.8.2 Standardising the cataloguesThe catalogues described in the previous section have all been constructed withdifferent purposes and from different sources. To be able to make comparisonsamong them or use all the stars as onemaster catalogue, it is necessary to standardisethe data.For this thesis six different catalogues [2, 4, 41, 149, 237, 250] are studied. As126a first step, each individual catalogue was retrieved independently from its corre-sponding reference. Then, the source identifications given in each catalogue wereused to cross-identify the stars with 2MASS using the SIMBAD1 [249] astronomicaldatabase. With the 2MASS identifier it is possible to retrieve the photometric data inthe 𝐽𝐻𝐾𝑠 bands from the 2MASS point-source catalogue. Table 8.1 gives the num-ber of stars in each catalogue after cross-matching with the 2MASS point-sourcecatalogue.Table 8.1: Number of stars in each Milky Way Catalogue after the specifiedcatalogue cross-match. Column 2 indicates the number of stars in the originalcatalogue. Column 3 gives the number of stars from the original catalogue thathave a 2MASS identifier. Column 4 is the number of stars that have a 2MASSidentifier and a distance measured by Bailer-Jones et al. [9]. Column 5 givesthe number of stars in column 4 that also have a reddening measured by Greenet al. [113].Catalogue Original 2MASS Bailer-Jones et al. Green et al.(distances) (reddening)(1) Alksnis et al. 6891 6688 6542 4397(2) Abia et al. 210 210 205 195(3) Chen and Yang 974 960 848 585(4) Li et al. 2651 2567 977 977(5) Suh 7172 6942 4507 3074(6) Whitelock et al. 239 239 228 107The photometric data from 2MASS is given in apparent magnitudes. To trans-form the available data into absolute magnitudes, the distance to the source andextinction in the observed band are required. Unlike the Magellanic Clouds, thestars in the Milky Way are in different directions and at different distances withrespect to the observer. Therefore, it is necessary to estimate the distance and theline-of-sight reddening for each star individually.1The SIMBAD database is operated at CDS, Strasbourg, France, and provides basic data, cross-identifications, bibliography and measurements for astronomical objects outside the solar system -http://simbad.u-strasbg.fr/simbad/127As mentioned in Chapter 2, Gaia has accurately measured positions, distances,motions, and changes in brightness for about one billion targets. The logicalnext step is to cross-match the stars with Gaia EDR3 in order to estimate thedistance to each source. With the 2MASS identifier it is possible to obtain theGaiacounterpart from the “most probable neighbour" catalogue developed by Marreseet al. [165, 164] provided in gaiaedr3.tmass_psc_xsc_best_neighbour. Thecross-matching method used to develop the “most probable neighbour" catalogueaccounts for epoch differences and propermotions between the two catalogues (Gaiaand 2MASS) on an object-by-object basis and assigns a score to each probablecounterpart based on the angular distance, position errors and the local surfacedensity. Once theGaia counterpart has been identified, the distance can be obtainedfrom the parallax (𝜛). However, the transformation between parallax and distance(𝑑) is non-linear (𝜛 ∼ 1/𝑑). Additionally, even small uncertainties in parallax cantranslate into large uncertainties in the distance (see Bailer-Jones et al. [9] andreferences therein). Therefore, simply using the inverse of the parallax is not areliable distance estimate, especially when the relative uncertainties are large (seeLuri et al. [153] for a recent review). Furthermore, Gaia parallaxes are known tohave systematic zero-point offsets (see Lindegren et al. [150] and references therein)that needs to be considered when transforming parallax to distance. Lindegrenet al. [150] found that the offset depends on the magnitude, colour, and eclipticlatitude of the source, and these dependencies are different for the five- and six-parameter solutions2. Correcting for the non-linear parallax-distance relation andthe systematics on the parallax measurements is a major enterprise. Fortunately,Bailer-Jones et al. [9] estimated the distances to 1.47 billionGaia sources accountingfor both the zero-point offset and the non-linear relation. Therefore, the distancesused in the analysis of the CS in the Milky Way are taken from Bailer-Jones et al.[9] catalogue.Since the stars are in the Milky Way, the extinction values will also depend onthe location to each star. The colour-excess of each source is obtained using the3-D dust-reddening map from Green et al. [113]. This map is derived considering2The Gaia EDR3 parallax determinations are of two kinds. Five-parameter solutions includeposition (RA,DEC), proper motion (𝜇𝛼∗,𝜇𝛿) and parallax. Six-parameter solutions additionallyinclude a pseudo-colour for sources where no reliable colour could be determined [94]128Gaia parallaxes, and photometric data from 2MASS and the Panoramic SurveyTelescope and Rapid Response System 1 (Pan-STARRS 13; [39]) surveys. Pan-STARRS covers the sky north of declination−30 deg, for this reason some of the CScatalogues lose a considerable number of sources (see Table 8.1). The extinction(𝐴) for each star is given by the dust-reddening (𝐸) from the 3-D map multipliedby the extinction factor (𝑅) derived for each band. Finally, the absolute magnitude(𝑀) is expressed as:𝑀𝑥 = 𝑚𝑥 −5 log(𝑑) +5−𝑅𝑥𝐸, (8.1)where 𝑥 ∈ {𝐽,𝐻,𝐾𝑠}, 𝑚 is the apparent magnitude in the 𝑥-band, 𝑑 is the estimateddistance by Bailer-Jones et al., and 𝑅𝑥 is listed in Table 8.2. The ((𝐽 −𝐾𝑠)0, 𝑀𝐽 )CMDs for the various Milky Way Catalogues are shown in Figure 8.1.Table 8.2: Reddening factors (𝑅𝑥 = 𝐴𝑥/𝐸 (𝐵 −𝑉)) in the Milky Way fromGreen et al. [113] in the 2MASS bands.𝑅𝐽 𝑅𝐻 𝑅𝐾𝑠0.7927 0.4690 0.30268.3 Photometry and astrometry quality parametersThe last column of Table 8.1 shows the number of stars in each catalogue for whichwe find 2MASS 𝐽𝐻𝐾𝑠 photometry, a distance derived from Gaia parallaxes byBailer-Jones et al. [9], and a reddening value obtained from the 3-D dust mapsdeveloped by Green et al. [113]. The process of standardizing the catalogues hadthe effect of reducing the number of stars available as not every star observed by2MASS has a parallax measured by Gaia or is in the part of the sky covered byGreen et al.’s reddening map.For the remaining stars it is important to examine the quality of the photometrygiven by 2MASS and of the astrometry given by Gaia. The 2MASS point-sourcecatalogue provides different “quality information” flags including the photometric3A 1.8 m optical and near-infrared telescope located on Mount Haleakala, Hawaii, designed forwide-field astronomical imaging in five broad-band filters (𝑔𝑟𝑖𝑧𝑦). https://panstarrs.stsci.edu/129Figure 8.1: CMD of the different Milky Way AGB stars catalogues used inthis thesis, except for Suh to avoid overcrowding ([237] overlaps almost in itsentirety with Alksnis et al. [4]). W + C&Y refers to the combined cataloguesof Whitelock et al. and Chen and Yang. The black-dotted lines delimit thecolour region defined for the CS: 1.4 < (𝐽 −𝐾𝑠)0 < 2.0.quality flag (ph_qual or qflg) and the read flag (rd_flg or rflg). Both flags arethree-character strings where each character gives information on a specific band(𝐽𝐻𝐾𝑠). For each source in the catalogue, the qflg gives an overall rating for thequality of the photometry, while the rflg indicates the photometry method usedto report the magnitude and uncertainty4. The photometry method used for eachsource is mainly determined by the brightness of the object. The default method isa PSF profile-fit that usually works well for faint sources (𝐾𝑠 > 13) and crowded42MASS point-source catalogue column descriptions: https://irsa.ipac.caltech.edu/data/2MASS/docs/releases/allsky/doc/sec2_2a.html130areas. However, the Milky Way stars in the catalogues described above are mostlybrighter than 𝐾𝑠 ∼ 13.2MASS covers a wide magnitude range (∼ 20 mag) thanks to the read out modeof the instrument. The 2MASS array has a reset timing that is identical to the read-out time. This allows 2MASS to execute a first read immediately after resetting thedetector and still obtain a 51 ms exposure. The second read is executed at 1.3 sto obtain the full exposure time per frame [227]. Objects brighter than magnitude𝐾𝑠 ∼ 8 saturate in the 1.3 s exposures. If the object does not saturate in the 51 msexposure its magnitude is then reported from aperture photometry and its rflg valueis set equal to “1”. For very bright sources that also saturate at 51 ms (𝐾𝑠 ≲ 4),the photometry is obtained from a 1-D radial profile fit to the unsaturated wingsof the PSF; in these cases rflg=“3”. Sources with magnitudes 𝐾𝑠 ≳ 8 are usuallyunsaturated in which case their photometry is reported from the PSF profile-fit; forthese stars their rflg=“2”. If a star has a rflg value different from “1”, “2” or “3” it isbecause there was either no detection of a source in a given band, or the photometryis of poor quality due to blending, contamination, artifacts, etc.The qflg has eight different quality ratings. “A”, “B”, “C” and “D” indicate avalid measurement with rflg=“1”, “2” or “3” with “A” having better signal-to-noiseratio and smaller photometric error. “E” indicates a poor PSF fit, “X” means nodetection, “U” only gives an upper limit of the magnitude and “F” is to flag sourceswhere no reliable error could be determined.Figure 8.2 shows where the stars with specific qflg and rflg values fall in theCMD. Because the AGB stars in theMilkyWay are on the bright end of the 2MASSmagnitude range, selecting stars with only valid PSF profile-fits biases the sampletowards fainter magnitudes. In fact, selecting stars with only rflg=“2” in the 𝐽-bandremoves ∼75% of the stars in the colour range defined for the CS (see middlepanel of Figure 8.2). If the selection is based on the qflg, and only stars rankedwith qflg=“A” were selected, as it was done for the Magellanic Clouds, then mostlybright stars would be removed (see left panel of Figure 8.2).For the purposes of this thesis, all rflg=“1”, “2” or “3” are considered validmeasures, except stars where no reliable error was measured (qflg=F). The useof rflg=“1”, “2” or “3” already removes the sources with qflg=“U”. This loosephotometry quality selection criteria brings along stars with high uncertainties131Figure 8.2: MilkyWay CMDs with different 2MASS parameter selection. Atthe bright magnitudes exhibited by the CS, 2MASS is close to its saturationlimit and a PSF profile fitting (rflg=“2”) sometimes is not possible. The leftpanel shows the location of the 2MASS sources classified with the highestphotometric quality results in orange (qflg=“A”), while the blue markers illus-trate how mostly bright stars have qflg≠“A”. The middle panel shows all thestars with a valid PSF photometry measurement. The right panel shows thelocation of stars with rflg=“1” (orange) and rflg=“3” (blue).(∼ 0.25 for stars with 𝐾𝑠 < 4), but it is the only way to avoid biasing the sample ofCS towards fainter magnitudes.Gaia’smain filtering criterion to exclude stars with poor astrometric solutions isthe renormalized unit weight error (ruwe) [151, 75]. The unit weight error (UWE)is a dimensionless factor that depends on the chi-squared of the astrometric fit, thenumber of good observations and the number of parameters in the solution (either5 or 6). A source with a good astrometric solution is expected to have a UWE valueclose to 1, but given Gaia’s calibration errors it tends instead to increase for bright,blue and very red sources. The renormalized UWE or ruwe includes an empiricalscale factor to compensate for the calibration errors. Lindegren et al. [150] studiedthe bias and zero-point offset of theGaia EDR3 parallaxes. In their analysis in orderto select a sample of LMC stars and another of galactic red clump stars, the authorssuggest setting ruwe< 1.4. Maíz Apellániz et al. [159] also studied biases in GaiaEDR3 parallaxes and included a short analysis on the degradation of the qualityof the parallaxes with increasing ruwe. They found that by increasing ruwe up to2, less than 1% of the stars fall outside of their original astrometry goodness-of-fit132selection criteria.Another parameter used to quantify the goodness-of-fit for the astrometricsolution is the parallax_over_error (𝜛/𝜎𝜛). This parameter allows one toselect sources whosemeasured parallax is larger than its uncertainty [75]. Althoughthe main astrometric solution Gaia paper [151] does not mention this parameter,other Gaia based studies have used 𝜛/𝜎𝜛 to select sources based on the parallaxprecision (e.g. [75, 92]). Fabricius et al. [75] give an illustrative example of a verytight selection at𝜛/𝜎𝜛 > 5, and suggest that the astrometric data processing noisegiving rise to negative parallaxes has the same probability of producing a spuriouspositive parallax. They compare the number of sources with 𝜛/𝜎𝜛 < −5 to thenumber of sources with 𝜛/𝜎𝜛 > 5. They find ∼ 3million sources in the negativeparallax sample suggesting that in their ∼ 190 million star sample with 𝜛/𝜎𝜛 > 5there are possibly 3 million spurious solutions (only ∼ 1.6%). Fabricius et al. alsofind that the fraction of spurious solutions strongly depends on magnitude and ismuch higher for sources with 6-parameter solutions than for 5-parameter ones. GaiaCollaboration et al. [92] analysed Milky Way stars in the direction of the Galacticanticentre. Their field covered 20 deg in Galactic longitude (𝑙), 100 deg in Galacticlatitude (𝑏) and was centred at (𝑙, 𝑏) = (180,0) deg. For their most crowded area(|𝑏 | < 10) the authors used 𝜛/𝜎𝜛 > 3, for the rest of the field they do not imposea 𝜛/𝜎𝜛 condition. Gaia Collaboration et al. [92] highlight the fact that findinga final sample that is large enough and not biased by the selection criteria (e.g.selections that might favour faint or bright stars), may be incompatible with havinga sample where all the stars have precise parallaxes.TheMilkyWayCS catalogues presented in this thesis have already been reducedin numbers mainly due to the lack of reddening measurements south of declination−30deg (see Table 8.1). If the six Milky Way catalogues are combined and dupli-cates removed, a list of 5607 unique sources is obtained. Of these 5607 sources,2500 are located within the 1.4 < (𝐽−𝐾𝑠)0 < 2.0 colour limits. Table 8.3 shows indetail the number of stars rejected when using different Gaia astrometric solutiongoodness-of-fit statistics selection criteria. A𝜛/𝜎𝜛 selection significantly reducesthe number of stars in the final sample, for example 𝜛/𝜎𝜛 > 3 and 𝜛/𝜎𝜛 > 5remove 16% and 35% of the stars found within 1.4 < (𝐽 −𝐾𝑠)0 < 2.0 respectively.On the other hand, the recommended ruwe< 1.4 cut only removes 2%. Figure 8.3133Table 8.3: Number of sources rejected with different Gaia astrometric solu-tion goodness-of-fit selection criteria. The first row indicates the number ofstars rejected from the entire 5607 stars sample. The second row shows thenumber of stars rejected in the CS 1.4 < (𝐽 −𝐾𝑠)0 < 2.0 colour range wherethe initial count is 2500 stars.ruwe 𝜛/𝜎𝜛< 1.4 < 2.0 > 5 > 4 > 3 > 2 > 1Total 467 184 1886 1472 1058 682 397CS 56 15 851 617 402 214 111illustrates the selection of stars based on ruwe< 1.4 and 𝜛/𝜎𝜛 > 3. The ruweselection discards mainly faint stars, while the 𝜛/𝜎𝜛 selection encompasses amuch larger number of sources but at the same time it removes outliers in the CS1.4 < (𝐽 −𝐾𝑠)0 < 2.0 colour range. The ruwe and 𝜛/𝜎𝜛 selections also removethe majority of sources with high parallax errors (independently of the parallaxmeasurement). The “clean” CMD for the Milky Way CS is shown in the rightpanel.The ruwe and𝜛/𝜎𝜛 selection criteria chosen can significantly impact the sizeof the sample and bias the selection towards fainter or brighter stars. At the momentit is not possible to determine the best selection criteria. For this reason and tostudy the effects of different ruwe and𝜛/𝜎𝜛 limits on the luminosity function and𝑀 𝐽 of the CS, the analysis will be performed and repeated using different limits.8.4 Results and discussionThe distance determination method using CS presented in this thesis relies on thecalibration of 𝑀 𝐽 for the CS in the LMC and SMC. The main objective of thisSection is to subject the CS in the Milky Way to the same analysis as the CS foundin the Magellanic Clouds. Therefore, the stars found in the catalogues mentionedabove are subjected to the same process detailed in Chapter 3.Most Milky Way catalogues have been constructed with specific research pur-poses and using specific selection criteria based on the properties of the stars,mainly their spectral type. Alksnis et al. [4] and Suh [237] main purpose was to134Figure 8.3: Milky Way CMDs with different Gaia parameters selection. Theleft panel highlights the sources with 𝜛/𝜎𝜛<3, and the centre panel thesources with ruwe> 1.4. The right panel shows the CMD after removing thestars highlighted in the previous two panels.make catalogues that were complete and hence are expected to be less biased to-wards a specific type of CS. Instead the other smaller catalogues have been designedfor particular research goals (see Section 8.1).To study the Milky Way CS distribution and to find their true 𝑀 𝐽 , four separateanalyses are performed. First, the six CS catalogues are analysed individually. Forthe second analysis the individual catalogues are combined into a master catalogue(with duplicate stars removed). Third, the analysis is performed separating the CSby spectral type (for those stars where a spectral type is available). Lastly, Chen andYang and Whitelock et al. catalogues are combined to replicate the catalogue usedby Lee et al.. The impact of differentGaia astrometric goodness-of-fit parameters isstudied by varying the ruwe and𝜛/𝜎𝜛 parameters limits. The criteria for 2MASSphotometry quality selection was determined in the previous section (any star witha valid photometric measurement either from PSF fitting or aperture photometry isused, except for those stars where an error in the 𝐽-band could not be determined).Even though the catalogues are listed as CS catalogues, it is important toremember that for the CS luminosity function and determination of the CS 𝑀 𝐽 ,only those stars that fall within 1.4 < (𝐽 −𝐾𝑠)0 < 2.0 colour limits are considered.1358.4.1 Individual cataloguesThe catalogues are first analysed individually using different Gaia astrometricgoodness-of-fit parameters selection criteria. For every catalogue the 𝑀 𝐽 and 𝜎are reported for two different ruwe values: <1.4 and <2. For each ruwe value twodifferent 𝜛/𝜎𝜛 limits are tested: > 5 and > 0. The CS luminosity function foreach catalogue is plotted in Figure 8.4, and the results for 𝑀 𝐽 and 𝜎 are shown inTable 8.4.For the individual catalogues it was not possible to fit the CS distribution withthe modified Lorentzian model used in Chapter 3. The catalogues of Alksnis et al.and Suh have a large intrinsicmagnitude dispersion and the luminosity function doesnot follow a Lorentzian shape. The data from Abia et al. shows two different peaksin the CS luminosity function, while the catalogues from Li et al. and Whitelocket al. have too few stars to fit any model. In Figure 8.4 the CS luminosity functions(except for Li et al. and Whitelock et al.) are accompanied by a Gaussian functioncentred at 𝑀 𝐽 and width equal to the intrinsic magnitude dispersion 𝜎.The 𝑀 𝐽 of the individual catalogues vary between −5.58 and −6.40 with 𝜎varying between 0.47 and 0.90magnitudes. The differentGaia parameters selectioncriteria do not showany significant effects on themedian 𝐽magnitude and dispersionvalues except for Li et al. and to a lesser extent for Suh. The number of CS is, inmost cases, significantly reduced for tighter criteria with a greater effect from thechange in 𝜛/𝜎𝜛 . The two catalogues which show changes in 𝑀 𝐽 for differentcriteria, are also the most affected by the tighter𝜛/𝜎𝜛 > 5 selection which reducesthe number of CS by ∼40%.The wide range of𝑀 𝐽 values obtained for the different catalogues is most likelydue to how the stars in each catalogue were selected, as mentioned in Section 8.1.Chen and Yang specifically removed CS which are classified as C-J type, these starswere found by Abia et al. to have fainter magnitudes compared to other CS spectraltypes. Whitelock et al. used a selection criteria based on flux values from IRASthat leaves out fainter CS, as their goal was to have a clean sample with sourceshaving a high probability of being C-rich Miras rather than to construct a completesample. Abia et al.’s sample of CS is dominated by C-N type stars, which they findto be on average brighter than the other spectral types. Suh only considered stars136Figure 8.4: CS luminosity functions for each catalogue listed in Table 8.1.The luminosity functions are shown for data selected with two different Gaiaastrometric goodness-of-fit parameters: ruwe< 2 and 𝜛/𝜎𝜛 > 0 (orange-straight line histograms), and ruwe< 1.4 and 𝜛/𝜎𝜛 > 5 (blue-dashed linehistograms). For Alksnis et al., Suh, Abia et al. and Chen and Yang theluminosity function is fitted with a Gaussian function centred at 𝑀 𝐽 . TheGaussian functions have been drawn using the same colour scheme as for thehistograms. In the insets the𝑀 𝐽 and intrinsic dispersion (𝜎) are specified. ForLi et al. and Whitelock et al. the reduced amount of data does not follow theGaussian shape, nevertheless 𝑀 𝐽 and 𝜎 can still be obtained using equation3.2.137Table 8.4: 𝑀 𝐽 and 𝜎 values obtained from 10000 bootstrap realizations. Oneach individual catalogue the calculations are perform using ruwe< 2 or < 1.4in combination with either 𝜛/𝜎𝜛 > 0 and > 5 .ruwe 𝜛/𝜎𝜛 𝑀 𝐽 𝜎 NCS 𝑀 𝐽 𝜎 NCSAlksnis et al. Suh<2 > 5 -5.78+0.02−0.02 0.67+0.01−0.01 1610 -5.60+0.03−0.03 0.67+0.02−0.02 941> 0 -5.79+0.02−0.02 0.70+0.01−0.01 2362 -5.65+0.02−0.03 0.68+0.02−0.02 1539<1.4 > 5 -5.78+0.02−0.03 0.67+0.01−0.01 1592 -5.60+0.03−0.03 0.67+0.02−0.02 924> 0 -5.80+0.02−0.02 0.69+0.01−0.01 2326 -5.65+0.02−0.03 0.67+0.02−0.02 1503Abia et al. Chen and Yang<2 > 5 -6.39+0.08−0.13 0.48+0.06−0.05 103 -6.37+0.09−0.07 0.53+0.05−0.05 141> 0 -6.39+0.08−0.13 0.48+0.06−0.05 103 -6.40+0.10−0.06 0.54+0.05−0.05 150<1.4 > 5 -6.40+0.08−0.15 0.47+0.06−0.06 99 -6.38+0.09−0.06 0.52+0.05−0.05 138> 0 -6.40+0.08−0.15 0.47+0.06−0.06 99 -6.40+0.10−0.05 0.53+0.05−0.05 147Li et al. Whitelock et al.<2 > 5 -5.74+0.25−0.13 0.89+0.10−0.15 36 -6.21+0.16−0.28 0.53+0.11−0.13 25> 0 -5.58+0.31−0.17 0.84+0.14−0.14 58 -6.21+0.16−0.26 0.53+0.11−0.13 25<1.4 > 5 -5.76+0.22−0.13 0.90+0.13−0.12 35 -6.20+0.15−0.26 0.53+0.11−0.13 25> 0 -5.66+0.27−0.12 0.87+0.15−0.15 55 -6.21+0.16−0.26 0.53+0.11−0.13 25with the highest photometric quality and this could bias the sample towards faintermagnitudes.8.4.2 Master catalogueThe individual catalogues discussed above were combined into a single mastercatalogue, with duplicate stars removed. Before any photometric or astrometricquality selection, the catalogue contains 5597 stars that have a 2MASS identifier,a measured distance and reddening information. Of the 5597 stars, 2498 are CSwithin the 1.4 < (𝐽 −𝐾𝑠)0 < 2.0 colour limits. Again, the catalogue is analysedusing differentGaia parameter selection criteria, this time including also results for138intermediate cuts at 𝜛/𝜎𝜛 > 3 and 𝜛/𝜎𝜛 > 1. The 𝑀 𝐽 and 𝜎 obtained for themaster catalogue are presented in Table 8.5.Themaster catalogue is dominated by the largest individual catalogue and hencethe results are roughly equal to those for the Alksnis et al. catalogue.Table 8.5: 𝑀 𝐽 and 𝜎 values obtained from 10000 bootstrap realizations onthe master catalogue data. A total of 10 different combinations of ruwe and𝜛/𝜎𝜛 selection criteria were used. Except for the number of CS, no significantdifferences are observed in the results when changing selection criteria.𝜛/𝜎𝜛ruwe< 2 ruwe< 1.4𝑀 𝐽 𝜎 NCS 𝑀 𝐽 𝜎 NCS> 5 -5.78+0.02−0.02 0.68+0.01−0.02 1641 -5.78+0.02−0.02 0.67+0.01−0.01 1621> 3 -5.77+0.02−0.02 0.68+0.01−0.01 2087 -5.77+0.02−0.03 0.68+0.01−0.01 2059> 1 -5.78+0.02−0.02 0.70+0.01−0.01 2372 -5.78+0.02−0.02 0.69+0.01−0.01 2336> 0 -5.78+0.02−0.02 0.70+0.01−0.01 2442 -5.79+0.02−0.02 0.69+0.01−0.01 24018.4.3 Grouped by spectral typeThe location of the CS of different spectral types in the CMD are shown in Figure8.5. It can be seen that in the (𝐽 −𝐾𝑠)0 colour range defined for the CS, the C-Nand C-J types dominate. The few SC type stars identified are found mainly in thiscolour gap as well.The luminosity functions for the CS in the 1.4 < (𝐽 −𝐾𝑠)0 < 2.0 colour range(before applying any Gaia selection cuts) are shown in Figure 8.6. The lumi-nosity function of the C-N type stars appears significantly wider compared to thedistributions of C-J and SC type stars.As for the individual and master catalogues, 𝑀 𝐽 and 𝜎 are obtained for samplesselected using different combinations of ruwe and𝜛/𝜎𝜛 criteria. For C-J only oneCS within 1.4 < (𝐽 −𝐾𝑠)0 < 2.0 is lost when going from ruwe< 2 to ruwe< 1.4.No SC type stars are lost when changing theGaia selection parameters. The valuesof 𝑀 𝐽 and 𝜎 for C-J and SC types are given below:139C− J𝑀 𝐽 = −5.83+0.08−0.12𝜎 = 0.44+0.06−0.09; SC𝑀 𝐽 = −6.14+0.25−0.40𝜎 = 0.46+0.14−0.17.Because so few stars are lost in the selection process, these results are not affectedby changes to ruwe and 𝜛/𝜎𝜛 .On the other hand, for C-N type stars, tightening the selection criteria has asignificant effect on the values of 𝑀 𝐽 and 𝜎 and on the number of CS. The resultsare shown in Table 8.6. For both ruwe< 1.4 and < 2 going from 𝜛/𝜎𝜛 > 0 to > 5removes ∼ 25% of the CS. Increasing the 𝜛/𝜎𝜛 selection limit removes mainlyfaint stars. It can be seen in Table 8.6 that as the 𝜛/𝜎𝜛 selection becomes tighter,𝑀 𝐽 becomes brighter and the intrinsic magnitude dispersion narrower.Table 8.6: Similar to Table 8.5 but for stars with spectral type C-N. In thiscase the use of different selection criteria has a significant effect on the valuesof 𝑀 𝐽 and 𝜎.𝜛/𝜎𝜛ruwe< 2 ruwe< 1.4𝑀 𝐽 𝜎 NCS 𝑀 𝐽 𝜎 NCS> 5 -6.31+0.12−0.11 0.77+0.10−0.10 75 -6.29+0.13−0.12 0.78+0.10−0.11 69> 3 -6.22+0.15−0.12 0.86+0.09−0.09 83 -6.20+0.16−0.11 0.86+0.11−0.10 77> 1 -6.02+0.18−0.16 0.99+0.09−0.11 97 -6.03+0.19−0.16 0.98+0.10−0.12 89> 0 -5.94+0.14−0.20 1.03+0.10−0.10 102 -5.99+0.17−0.17 1.00+0.10−0.12 928.4.4 Lee et al. combined catalogueAs mentioned in Section 8.1, Lee et al. [147] combined Chen and Yang andWhitelock et al. (W+C&Y) catalogues to estimate the median 𝐽-band magnitudeof the CS in the Milky Way. Lee et al. used these two catalogues in order to have asample composed purely of CS whose CS classification was confirmed by a methodother than just their (𝐽 −𝐾𝑠) 2MASS colours. Nevertheless, Lee et al. give 𝑀 𝐽based on a subsample of the combined catalogue selected from 2MASS colours(1.4 < (𝐽 −𝐾𝑠)0 < 2.0) and give 𝑀 𝐽 from 2MASS 𝐽-band photometric data. The140Figure 8.5: CMD of the different Milky Way AGB stars separated by spectraltype. The black-dotted lines delimit the colour region defined for the CS:1.4 < (𝐽 −𝐾𝑠)0 < 2.0.𝐽𝐻𝐾𝑠 data from Chen and Yang comes directly from the 2MASS point-sourcecatalogue while Whitelock et al. comes from SAAO and was transformed to the2MASS 𝐽𝐻𝐾𝑠 colour system. Chen and Yang point out that any data transformedto the 2MASS system might be different from that reported in the 2MASS point-source catalogue given that most CS are variable stars and the observations from2MASS and a different source were not likely taken at the same time. Both Chenand Yang and Whitelock et al. made their catalogues publicly available with thenear-infrared photometric data already corrected for extinction using Drimmel et al.[71] 3-D reddening maps. Chen and Yang do not apply an extinction correction tostars whose 2MASS magnitude uncertainty exceeds the reddening value estimatedfor a specific source.141Figure 8.6: Luminosity function of the Milky Way CS of different spectraltypes located within 1.4 < (𝐽 −𝐾𝑠)0 < 2.0 colour limits. The selection of theCS was made after a ruwe< 2 and 𝜛/𝜎𝜛 > 0 cut to the data. For the C-Nand C-J types, the luminosity function is accompanied by a Gaussian functioncentred at 𝑀 𝐽 .In this thesis, all near-infrared photometric information was obtained directlyfrom the 2MASS point-source catalogue with the extinction corrections fromGreenet al. [113]. To compare the results obtained by Lee et al. for the CS in the combinedW+C&Y catalogue, 𝑀 𝐽 and 𝜎 are estimated using, first, the magnitudes reportedby Lee et al. (see their Table 1), and then using the magnitudes from 2MASS withthe reddening values from Green et al. [113]. For the 2MASS magnitudes, thecalculations are initially done with reddening corrections applied to all the stars,and then a second time with reddening corrections applied only to those stars whosemagnitude uncertainty was less than its reddening value (simulating the Chen andYang approach). Finally, to compare the exact same stars from Lee et al. usingthe magnitudes reported in this thesis, 𝑀 𝐽 and 𝜎 are calculated using both sets ofmagnitudes for those stars selected by Lee et al. that have an available reddeningmeasurement by Green et al. [113]. The results are presented in Table 8.7. The142Table 8.7: Median 𝐽-band CS magnitude for the catalogue from Lee et al..Columns 2 and 3 show the 𝑀 𝐽 and 𝜎 obtained using 𝑄𝑛 statistics. Themagnitudes from Lee et al. do not include uncertainties therefore 𝑄𝑛 statisticscannot be used. Columns 4 and 5 show 𝑀 𝐽 and 𝜎 measured directly from thedata (given for all cases for comparison). For rows (i), (ii) and (iii) the CS areselected from the (𝐽 −𝐾𝑠)0 colour given in Lee et al.. For rows (iv), (v) and(vi) CS are selected from the (𝐽 −𝐾𝑠)0 derived for this thesis.𝑀 𝐽 𝜎 𝑀 𝐽 𝜎 NCSMagnitudes from Lee et al. [147](i) N/A N/A -6.15+0.05−0.04 0.60+0.05−0.03 153(ii) N/A N/A -6.09+0.08−0.05 0.66+0.05−0.07 99Magnitudes from 2MASS point-source catalogue(iii) -6.29+0.06−0.08 0.57+0.06−0.06 -6.29+0.02−0.09 0.71+0.08−0.10 99(iv) -6.28+0.06−0.05 0.57+0.05−0.06 -6.29+0.06−0.03 0.67+0.06−0.08 148(v) -6.11+0.07−0.09 0.60+0.05−0.06 -6.12+0.07−0.09 0.66+0.06−0.08 94(vi) -6.30+0.04−0.10 0.55+0.05−0.04 -6.30+0.03−0.10 0.61+0.07−0.05 155(i) All CS in Lee et al.. (ii) CS used in Lee et al. mag that havereddening values from [113]. (iii) Same CS as in (ii). (iv) CS selectedbased on 2MASS colours with reddening corrections applied usingChen and Yang approach. (v) No reddening correction applied.(vi) Reddening corrections applied to every source.magnitudes from Lee et al. do not include uncertainties therefore 𝑄𝑛 statisticscannot be used and only the normal median and dispersion for the CS 𝐽 magnitudesis presented. The normal median and dispersion is calculated for every case forcomparison.The values obtained for the 153 CS used by Lee et al. are the same as the onesreported in the original publication.The results obtained using the magnitudes and reddening derived for this thesisare brighter than those obtained by Lee et al. The comparison between rows (ii)and (iii) from Table 8.7 (where the same stars are used but with reddening valuesobtained from different sources) suggests that the main difference between the 𝑀 𝐽143reported by Lee et al. and the values obtained in this thesis come from the useof two different reddening maps. The 𝑀 𝐽 value obtain from 2MASS magnitudeswith no reddening corrections applied agrees with that from Lee et al. suggestingthe extinction factor applied from Drimmel et al. [71] might be underestimated. Ifreddening is underestimated the median 𝐽 magnitude will be fainter. Additionally,if the reddening is underestimated, it is likely that no corrections were applied giventhe condition used byChen andYang and the large 2MASSmagnitude uncertainties.The use of𝑄𝑛 statistics does not indicate significant differences from the normalmedian but it does exhibit smaller dispersion values.It is important to remember that the list of stars given by Lee et al. has alreadybeen "cleaned" usingGaia astrometric goodness-of-fit parameters criteria (ruwe< 2and𝜛/𝜎𝜛 > 5) although as shown in Section 8.4.1 that neither the Chen and Yangnor Whitelock et al. catalogues are affected by different Gaia selection criteria.8.5 ConclusionsAs seen in Table 8.4, the results obtained for𝑀 𝐽 and𝜎 show differences that exceedthe various error bars. The two catalogues, whose objective was to be complete(Alksnis et al. and Suh), alongwith the the catalogue fromLi et al. where all types ofCS were identified, show significantly fainter 𝑀 𝐽 and wider magnitude dispersioncompared to the Chen and Yang and Whitelock et al. catalogues where a detailedselection of stars was carried out. When separating the CS by spectral type, thereare also differences between the median magnitudes with C-J type stars showing afainter distribution and SC type a much brighter one compared to C-N type CS. Theluminosity function of C-N type CS shows a faint tail of stars that drives the mediantoward fainter magnitudes and to a much wider magnitude dispersion. However,this faint tail diminishes when applyingmore strictGaia astrometric goodness-of-fitselection criteria suggesting that these faint stars do not have reliable distances.The wide variety of CS in the Milky Way and the results for the more generalcatalogues suggest a significant difference between the CS in the Milky Way and inthe Magellanic Clouds. Selecting specific types of CS bring the differences closer,but such a selection is not generally possible outside the Milky Way. If the goalis to use CS in galaxies of similar properties as the Milky Way, a detailed study144of CS in grand design spirals is needed to determine the true CS 𝑀 𝐽 in this typeof galaxy. The increment in the accuracy of the parallaxes measured for each starin the next Gaia data release will help to determine the true intrinsic magnitudedispersion of the various CS spectral types. But if the ultimate goal is to use CS asdistance indicators, the apparent relative simplicity of Magellanic type galaxies (atleast as regards their CS component) compared to grand design spirals, may be acritical aspect in their usefulness as standard candles.145Chapter 9Discussion9.1 Distance comparisonsAs part of this thesis, the distance moduli for NGC 6822, IC 1613, WLM andNGC 3109 were estimated by calculating the median 𝐽-band magnitude of the CS,finding the TRGB magnitude in the 𝐾𝑠-band and using the luminosity function ofthe RGB plus the O-rich AGB stars in both the 𝐽 and 𝐾𝑠 bands. The results fromthis thesis are summarized in Table 9.1. In this Section, the results obtained fromthe three different methods are compared against each other and against previouslypublished, independently measured distances using different distance indicators. Itis important to acknowledge the fact that a direct comparison between the valuesfor 𝜇0 for a given galaxy is difficult as different studies use a variety of extinctionvalues and distance calibrations.For the four target galaxies, the error bars for all three methods are comparable,with the CS method exhibiting a slightly larger (∼ 0.01 mag) statistical error. Alarger error for the CS method is expected given that the number of CS in anyextra-galactic field is considerably smaller than the number of RGs. But, at thesame time, the uncertainties in the magnitudes of the CS are smaller as these starsare brighter than the RGs and thus have more accurate photometry.146Table 9.1: Distance moduli obtained in this thesis from the median 𝐽 mag-nitude of the CS, TRGB and RG-MLE methods. The errors presented are thestatistical errors. The last two rows show the results for the distance moduliwhen the Magellanic Clouds data are corrected using the new Skowron et al.[226] (S21) reddening maps.NGC 6822 IC 1613 WLM NGC 3109Carbon Stars 23.55 ± 0.03 24.45 ± 0.05 25.02 ± 0.04 25.58 ± 0.03TRGB 23.50 ± 0.02 24.46 ± 0.02 25.03 ± 0.03 25.60 ± 0.03RG-MLE (𝐽) 23.51 ± 0.01 24.39 ± 0.02 24.97 ± 0.03 25.57 ± 0.02RG-MLE (𝐾𝑠) 23.55 ± 0.02 24.35 ± 0.02 24.96 ± 0.02 25.61 ± 0.03CS (S21) 23.52 ± 0.03 24.46 ± 0.05 25.03 ± 0.04 25.59 ± 0.03TRGB (S21) 23.39 ± 0.02 24.45 ± 0.03 25.01 ± 0.03 25.58 ± 0.049.1.1 NGC 6822The four distance moduli obtained as part of this thesis for NGC 6822 are inagreement with each other within the statistical error bars. The TRGB and RG-MLE(𝐽) methods give nearly equal results (23.50 and 23.51) while the CS andRG-MLE (𝐾𝑠) give slightly larger distance moduli (23.55). When the LMC dataare corrected using the new Skowron et al. [226] reddening map (Chapter 7), thedistance modulus obtained through the CS method agrees with the four originalvalues obtained in this thesis. For the TRGB, changing the reddening map resultedin a significantly different TRGB magnitude for the LMC. In this case the distancemodulus does not agree with any of the other results.Table 9.2 displays some of the recent distance estimates to NGC 6822 usingvarious techniques and standard candles. The method, filter and extinction valuesfor each study are also given to allow for better comparison. The distance modulilisted in Table 9.2 are plotted in Figure 9.1As mentioned in Subsection 2.2.3, NGC 6822 is a challenging target due tothe considerable amount of reddening in the line-of-sight. The extinction valueused in this thesis, 𝐸 (𝐵 −𝑉) = 0.35 (equivalent to 𝐴𝐽 ∼ 0.29 and 𝐴𝑉 ∼ 1.05)[102, 90, 201], considers the total line-of-sight reddening (from the Milky Wayand NGC 6822 internal reddening). On the other hand, some studies find smaller147Table 9.2: Independent distance determinations to NGC 6822.Method Band E(𝐵−𝑉) 𝜇∗0 Reference YearCS 𝐽 0.35 23.55 ± 0.03 This thesis 2022TRGB 𝐾𝑠 0.35 23.50 ± 0.02 This thesis 2022RG-MLE 𝐽 0.35 23.51 ± 0.01 This thesis 2022RG-MLE 𝐾𝑠 0.35 23.55 ± 0.02 This thesis 2022Cepheids near-IR (3) 23.43±0.06 Bhardwaj et al. [24] 2016Cepheids (1) 0.35 23.38±0.02 Rich et al. [201] 2014Cepheids (2) (4) 23.40±0.05 Feast et al. [77] 2012Cepheids mid-IR 0.26 23.49±0.03 Madore et al. [158] 2009Cepheids 𝐽𝐾 0.356 23.29±0.02† Gieren et al. [102] 2006TRGB 𝐼 (5) 23.52±0.03‡ F&M 2020TRGB 𝐼 0.35 23.54±0.05 Fusco et al. [90] 2012TRGB 𝐼 0.231 23.32±0.10 McCall et al. [167] 2012TRGB 𝐽 0.23 23.31±0.05 Górski et al. [108] 2011TRGB 𝐾 0.23 23.26±0.07 Górski et al. [108] 2011TRGB 𝐼𝐽𝐾𝑠 0.24 23.34±0.12 C&H 2005RR Lyrae 𝑉 0.25 23.49±0.06 Feast et al. [77] 2012RR Lyrae 𝑉 0.25 23.36±0.17 Clementini et al. [54] 2003Mira 𝐽𝐻𝐾𝑠 (6) 23.56±0.03 Whitelock et al. [251] 2013CS 𝐽 0.356 23.24±0.04 Zgirski et al. [253] 2021CS 𝐽 (7) 23.44±0.02 F&M 2020Notes: ∗ statistical errors only, except for Clementini et al. and Cioni and Habing†Updated in Zgirski et al. [253]‡Average of multiple studies calibrated with 𝑀𝐼 = −4.05.(1) optical + 𝐽𝐻𝐾𝑠 + mid infrared.(2) 𝑉𝐼𝐽𝐻𝐾𝑠 and Infrared Array Camera 3.6𝜇𝑚, and 4.5𝜇𝑚; (3) Not specified.(4) A𝑉 = 0.667; (5) A𝐼 = 0.335; (6) A𝐽 = 0.20; (7) A𝐽 = 0.167C&H = Cioni and Habing [51]; F&M = Freedman and Madore [83]total extinction values or consider only the foreground reddening which can lead to𝐸 (𝐵−𝑉) values as low as ∼ 0.2 (e.g. [218, 90, 200]). The large range of extinctionvalues alone carries a systematic uncertainty of ∼ 0.15 mag in distance modulususing 𝐽-band and ∼ 0.45 mag in 𝑉-band.148Figure 9.1: Recent independent distance estimations to NGC 6822 from dif-ferent techniques. The different methods are plotted using different coloursand marker shapes. Others include Miras and RR Lyrae methods. The hor-izontal lines covering the full range of years indicate the distance modulusobtained in this thesis from the CS (straight purple line) and RG-MLE(𝐽)(dashed blue line) methods. The shaded areas depict the error bars for thevalues from this thesis.Except for Madore et al. [158], the Cepheid estimates do not agree with theresults from this thesis within the statistical error bars. Most estimates applieda lower extinction correction compared to this thesis, compensating with extrareddening corrections would only increase the difference between the Cepheiddistances and the estimates from this thesis. .Similar to the Cepheids, only one of the five TRGB values listed agrees withthe values from this thesis. The TRGB estimate by Fusco et al. [90] uses the sameextinction correction as in this thesis and agrees well with the values given in Table9.1. The other four TRGB estimates are considerably smaller and have smallerreddening corrections, any extra reddening would make the difference betweenthese values and the values from this thesis even larger. Freedman and Madore [83]give an average 𝐼-band 𝜇0(𝑇𝑅𝐺𝐵) = 23.52 from different studies including the 𝐼149TRGB estimates in Table 9.2. The average was done standardizing the individualvalues with 𝐴𝐼 = 0.355 and 𝑀𝑇𝑅𝐺𝐵𝐼 = −4.05. This average is in good agreementwith the results from this thesis.The estimate obtained from RR Lyrae stars by Feast et al. [77] is in agreementwith the values from this thesis. However, if the extinction is changed to 0.35, theresult no longer agrees. The Mira variables result from Whitelock et al. [251] isin good agreement with the values obtained in this thesis, even when the slightlydifferent extinction value used is considered.In the case of the CS, the distance moduli as presented in the two studies listeddo not agree with the values from Table 9.1. To calibrate the distances, this thesisuses the median 𝐽 magnitude of the CS in the LMC, 𝑀 𝐽 = −6.288. Zgirski et al.[253] instead use the mean 𝐽 magnitude of the CS in the LMC 𝑀𝐽=-6.212, whileFreedman and Madore [83] use the average of the means of the 𝐽 magnitudes ofthe CS in the LMC and SMC, 𝑀𝐽=-6.20. All three calibrations are based on thesame distance to the LMC. Standardizing the results from the independent studiesto 𝑀 𝐽 = −6.288 and 𝐸 (𝐵−𝑉) = 0.35, the distance moduli become 23.32 and 23.44for Zgirski et al. and Freedman and Madore respectively. The two values arestill not in agreement with the distance moduli determined in this thesis, nor witheach other. The alternatives left to explain the discrepancy between results are thedifference in the mean and median values of the CS 𝐽 magnitude, and possiblesystematics. The mean values for the apparent magnitudes of the CS in NGC 6822were found to be 17.031 by Zgirski et al. and 17.15 by Freedman and Madore. TheNGC 6822 CS selected in this thesis have a mean of 17.32, while the median usedto estimate the distance is 17.26. Using the mean magnitude would only increasethe difference. The difference between the two mean values from the independentstudies and the median value given in this thesis are enough to account for thediscrepancy between the results but does not explain why different distance moduliwere found. One possibility is in the different colour and magnitude limits usedto select the CS. For example Zgirski et al. use a colour selection in the LMC,SMC and WLM of 1.3 < (𝐽 −𝐾)0 < 2.0 mag while for two other galaxies theblue cut is set at 1.45. For every galaxy in their sample, their selection box has awidth Δ𝐽0 = 2.5 mag. No colour or magnitude limits are specified for NGC 6822.Freedman and Madore employ similar colour constrains but their magnitude range150seems significantly narrower compared to Zgirski et al. and this thesis. Anotherpossibility are systematic differences between the individual NGC 6822 data sets.For example, the near-infrared data employed by Zgirski et al. are the same data usedby Gieren et al. [102] to determined the distance to NGC 6822 using Cepheids.Madore et al. [158] found a systematic shift in the zero-point of this 𝐽 and 𝐾photometry and removed these data from their analysis. Madore et al. state thatinclusion of the 𝐽 and 𝐾 data in their analysis would have resulted in a 0.17 magbrighter (i.e. a shorter distance) distance modulus. The data set from Gieren et al.,is also used by Górski et al. [108], which explains the agreement between the resultsfrom these two studies and also with Zgirski et al..9.1.2 IC 1613From Table 9.1, the distance moduli obtained in this thesis for IC 1613 varied byup to 0.11 magnitudes. The result from the CS agrees with the result from theTRGB and the RG-MLE(𝐽) methods while the result from RG-MLE(𝐾𝑠) places IC1613 slightly closer. Because all distances were obtained from the same data set,the systematic errors in the photometry, reddening and calibration are expected tohave a similar effect on the three methods. Therefore, systematic errors originatingfrom the data analysis are unlikely responsible for large discrepancies between someof the IC 1613 distance moduli presented in Table 9.1. Examining the results inmore detail, the difference between the CS and RG-MLE(𝐾𝑠) estimations is small,after taking the error bars into consideration, the estimates differ by only 0.03 mag.The distance moduli obtained through the CS and TRGB methods using the newSkowron et al. [226] reddening maps on the Magellanic Clouds are in excellentagreement with the original CS and TRGB estimates.In the case of the TRGB versus RG-MLE(𝐾𝑠), the difference stretches to 0.07mag. A potential source of discrepancy is the contamination of the RG sampleclose to the TRGB with O-rich AGB stars. Another, is systematics in the methodsthemselves. In Chapter 4 it was explained that the TRGBmethod used in this thesisrelies on construction of a colour dependent magnitude that rectifies the TRGB.The construction of this magnitude itself depends on the calculation of the slopeof the TRGB. Additionally, to detect the TRGB, it is necessary to first smooth151the luminosity function followed by the use of a Sobel filter. Each of these stepscan add additional systematic uncertainties that could account for the difference.Additionally, to rectify the TRGB, predetermined slopes from Hoyt et al. [130]were used based on LMC data. In the three methods used in this thesis, differencesbetween the LMC and SMC luminosity functions have been seen for the RGs(including the rectified magnitude RGs luminosity function) and the CS. For theCS and RG-MLE method IC 1613 was classified as an “SMC-like” galaxy. It ispossible that the slope and calibration of the TRGB in the SMC differs from thatfound by Hoyt et al. in the LMC. However, the slopes found by Hoyt et al. areidentical to those from Madore et al. based on IC 1613 data.Table 9.3 presents a summary of recent independent distance estimates for IC1613. As for Table 9.2, the values are taken directly from their source withoutany correction for the use of different extinction values and/or zero-points for thecalibration of the distance. The results again cover a significant range of values withdifferences of up to ∼ 0.3 mag. To compare the different estimates, the distancemoduli listed in Table 9.3 are plotted in Figure 9.2 against the year of publication.Bernard et al. [17] provide three different Cepheid and two RR Lyrae distanceswhich have been averaged in Table 9.3. All five estimations are plotted. Theresults given in this thesis from the CS and RG-MLE(𝐾𝑠) methods are includedin Figure 9.2, represented by a horizontal line at the magnitude of the distancemodulus covering the full range of years. The error bars are depicted by the shadedareas around the corresponding line. Given the long list of available distancesmeasured to IC 1613, Figure 9.2 allows a faster comparison between individualmeasurements and the results from this thesis. Six of the 24 individual distanceestimates fall outside the range (including error bars) of the distances given in Table9.1 for IC 1613. Nine of the results shown agree within the error bars with all fourmeasurements presented in this thesis for IC 1613. No relation between the methodused and 𝜇0 is observed in the plot. From Table 9.3, it is also possible to discardany relation between distance measured and the band used to estimate it.Recently, Nagarajan et al. [176] used Gaia EDR3 data to estimate distances toseveral dwarf galaxies based on RR Lyrae variable stars. In their study, Nagarajanet al. report 𝜇0 = 24.42± 0.03 for IC 1613. That value is in agreement with theresults from this thesis within the statistical error bars (except for RG-MLE(𝐾𝑠)).152Table 9.3: Independent distance estimations to IC 1613.Method Band E(𝐵−𝑉) 𝜇∗0 Reference YearCS 𝐽 0.021 24.45 ± 0.05 This thesis 2022TRGB 𝐾𝑠 0.021 24.46 ± 0.02 This thesis 2022RG-MLE 𝐽 0.021 24.39 ± 0.02 This thesis 2022RG-MLE 𝐾𝑠 0.021 24.35 ± 0.02 This thesis 2022Cepheids near-IR (3) 24.26±0.07 Bhardwaj et al. [24] 2016Cepheids (1) 0.050 24.29±0.03 Scowcroft et al. [220] 2013Cepheids 𝐵𝑉𝐼 0.025 24.34±0.03 Tammann et al. [240] 2011Cepheids 𝑉𝐼 (3) 24.22±0.10 Bono et al. [26] 2010Cepheids 𝑉𝐼 0.025 24.47±0.12 Bernard et al. [17] 2010Cepheids (2) 0.08 24.27±0.02 Freedman et al. [86] 2009TRGB 𝐽𝐻𝐾 0.03 24.32±0.02 Madore et al. [155] 2018TRGB 𝐼 (4) 24.36±0.05† Hatt et al. [121] 2017TRGB 𝐼 (5) 24.39±0.03 McQuinn et al. [169] 2017TRGB 𝑉𝐼 0.021 24.44±0.09 Bernard et al. [18] 2013TRGB 𝑉𝐼 0.021 24.49±0.09 Bernard et al. [18] 2013TRGB 𝐽 0.02 24.36±0.08 Górski et al. [108] 2011TRGB 𝐾 0.02 24.24±0.08 Górski et al. [108] 2011TRGB 𝐼 0.025 24.38±0.06 Jacobs et al. [135] 2009TRGB 𝐼 0.025 24.37±0.05 Rizzi et al. [211] 2007RR Lyrae 𝑉𝐼 (3) 24.42±0.03 Nagarajan et al. [176] 2021RR Lyrae 𝑉 (6) 24.28±0.04 Hatt et al. [121] 2017RR Lyrae 𝑉 (7) 24.19±0.09 Dambis et al. [63] 2013RR Lyrae 𝑉𝐼 0.025 24.46±0.11 Bernard et al. [17] 2010Miras 𝐽𝐻𝐾𝑠 0.06 24.37±0.08 Menzies et al. [172] 2015FGLR‡ spec. (7) 24.39±0.11 Berger et al. [16] 2018CS 𝐽 (8) 24.36±0.03 F&M 2020Notes: ∗ statistical errors only, except for Bernard et al.‡flux-weighted gravity-luminosity relationship.†Updated in F&M = Freedman and Madore [83](1) 𝐽𝐻𝐾𝑠 and Infrared Array Camera 3.6𝜇𝑚, and 4.5𝜇𝑚(2) 𝐵𝑉𝑅𝐼𝐽𝐻𝐾 and Infrared Array Camera 3.6𝜇𝑚, and 4.5𝜇𝑚(3) Not specified; (4) A𝐼 = 0.038; (5) A𝑉 = 0.07; (6) No corrections applied(7) Estimated individually for each star; (8) A𝐽 = 0.018153Figure 9.2: Similar to Figure 9.1 but for IC 1613. Others include Miras,FGLR and CS methods.Nagarajan et al. also do an extensive comparison with other distance estimatesbased on RR-Lyrae and TRGB published over the last decade. They find thatpre-2014 estimates are closer to 𝜇 ∼ 24.45, compared to 𝜇 ∼ 24.3 for more recentestimates. Performing a similar analysis but considering also the distance estimatesfromCepheids and the three other methods included in Table 9.3, gives mean valuesfor before and after 2014 of 24.38 and 24.35 respectively.9.1.3 WLMThe various distance measurements for WLM are, in general, in good agreementwith each other. As for IC 1613, the estimate from the RG-MLE(𝐾𝑠) methodfalls slightly outside the reach of the statistical error bars when compared with theTRGB method. The distance moduli obtained using the new Skowron et al. [226]reddening maps on the SMC are in good agreement with all the original estimates.Table 9.4 presents a compilation of previous distance estimates for WLM basedon different standard candles. These values are plotted in Figure 9.3. Again, thereis a difference between results prior and after 2014 with distance modulus means154Figure 9.3: Similar to Figure 9.1 but for WLM.of 25.03 and 24.95 respectively. The estimates following 2014 have a smallerdispersion with a standard deviation of 0.02 mag compared to 0.09 mag for thevalues obtained before 2014. Except for the earliest Cepheid value given, all thevalues from Table 9.4 fall within the error bars of at least two of the estimates fromthis thesis. All the estimates published post 2020 are in good agreement with theresults from this thesis within the error bars, except for Lee et al. [146]’s TRGB𝐼distance. For this estimate, Lee et al. [146] report a systematic error of 0.06 mag,which would account for the small difference.155Table 9.4: Independent distance estimations to WLM.Method Band E(𝐵−𝑉) 𝜇∗0 Reference YearCS 𝐽 0.03 25.02 ± 0.04 This thesis 2022TRGB 𝐾𝑠 0.03 25.03 ± 0.03 This thesis 2022RG-MLE 𝐽 0.03 24.97 ± 0.03 This thesis 2022RG-MLE 𝐾𝑠 0.03 24.96 ± 0.02 This thesis 2022Cepheids 𝑉𝐼𝐽𝐻𝐾 0.05 24.98±0.03 Lee et al. [146] 2021Cepheids near-IR (1) 24.92±0.07 Bhardwaj et al. [24] 2016Cepheids 𝐵𝑉𝐼 0.037 24.95±0.03 Tammann et al. [240] 2011Cepheids 𝑉𝐼 (1) 25.09±0.07 Bono et al. [26] 2010Cepheids 𝑉𝐼𝐽𝐾 0.082 24.92±0.04 Gieren et al. [103] 2008Cepheids 𝑉𝐼 0.02 25.14±0.03 Pietrzyński et al. [193] 2007TRGB 𝐽𝐻𝐾 (2) 24.98±0.04 Lee et al. [146] 2021TRGB 𝐼 (3) 24.93±0.02 Lee et al. [146] 2021TRGB 𝐼 0.038 24.93±0.02 Albers et al. [3] 2019TRGB 𝐼 (4) 24.94±0.03 McQuinn et al. [169] 2017TRGB 𝐽 0.04 25.14±0.09 Górski et al. [108] 2011TRGB 𝐾 0.04 25.12±0.09 Górski et al. [108] 2011TRGB 𝐼 0.036 24.95±0.07 Jacobs et al. [135] 2009TRGB 𝐼 (5) 24.93±0.04 Rizzi et al. [211] 2007FGLR† spec. (6) 24.99±0.10 Urbaneja et al. [242] 2008CS 𝐽 0.082 24.95±0.08 Zgirski et al. [253] 2021CS 𝐽 (2) 24.97±0.02 Lee et al. [146] 2021CS 𝐽 (2) 24.97±0.05 F&M 2020Notes: ∗ statistical errors only; †flux-weighted gravity-luminosity relationship.(1) Not specified; (2) A𝐽 = 0.027, A𝐻 = 0.017, A𝐾 = 0.012; (3) A𝐼 = 0.057(4) A𝑉 = 0.1; (5)A𝐼 = 0.07; (6) determined individually for each star.F&M = Freedman and Madore [83]1569.1.4 NGC 3109For NGC 3109 the four original estimates from this thesis and the estimates withthe new reddening corrections to the SMC using Skowron et al. [226] reddeningmaps (see Table 9.1) are all in good agreement within the statistical error bars.Independent distance moduli determinations for NGC 3109 are presented inTable 9.5. Of the fourteen estimates found in the recent literature, ten are in verygood agreement with all four results from this thesis. The distancemoduli presentedin Table 9.5 are shown in plot form in Figure 9.4.Table 9.5: Independent distance estimations to NGC 3109.Method Band E(𝐵−𝑉) 𝜇∗0 Reference YearCS 𝐽 0.056 25.58 ± 0.03 This thesis 2022TRGB 𝐾𝑠 0.056 25.60 ± 0.03 This thesis 2022RG-MLE 𝐽 0.056 25.57 ± 0.02 This thesis 2022RG-MLE 𝐾𝑠 0.056 25.61 ± 0.03 This thesis 2022Cepheids near-IR (1) 25.49±0.06 Bhardwaj et al. [24] 2016Cepheids 𝑉𝐼 (1) 25.58±0.12 Bono et al. [26] 2010Cepheids 𝐽𝐾 0.087 25.57±0.02 Soszyński et al. [228] 2006Cepheids 𝑉𝐼 0.05 25.54±0.03 Pietrzyński et al. [194] 2006TRGB 𝐽 0.06 25.49±0.05 Górski et al. [108] 2011TRGB 𝐾 0.06 25.42±0.05 Górski et al. [108] 2011TRGB 𝑉 (2) 25.55±0.05 Dalcanton et al. [62] 2009TRGB 𝐼 (2) 25.50±0.04 Dalcanton et al. [62] 2009TRGB 𝐼 0.058 25.62±0.09 Jacobs et al. [135] 2009TRGB 𝑉𝐼 (3) 25.61±0.10 Hidalgo et al. [123] 2008TRGB 𝐼 (4) 25.56±0.05 Rizzi et al. [211] 2007FGLR† spec. (5) 25.55±0.09 Hosek et al. [127] 2014CS 𝐽 0.087 25.52±0.05 Zgirski et al. [253] 2021CS 𝐽 (6) 25.56±0.05 F&M 2020Notes: ∗ statistical errors only; †flux-weighted gravity-luminosity relationship.(1) Not specified; (2) A𝑉 = 0.201; (3) A𝑉 = 0.221, A𝐼 = 0.129(4) A𝐼 = 0.13 (5) determined individually for each star; (6) A𝐽 = 0.027F&M = Freedman and Madore [83]157Figure 9.4: Similar to Figure 9.1 but for NGC 3109.9.2 Metallicity effects on the distance estimationsThe CS and RG-MLE methods presented in this thesis are designed to accountfor metallicity effects without resorting to an extra calibration step. In the case ofthe CS, metallicity effects (reflected in the shape of the CS luminosity functions)are bypassed by finding the most suitable calibrator between the LMC and SMC.Ripoche et al. [210] tried to account formetallicity effects on the luminosity functionof the Magellanic Clouds by constructing a composite magnitude, similar to the 𝑇magnitude used in Chapter 4 developed byMadore et al. [157] to removemetallicityeffects on the magnitude of the TRGB. Ripoche et al. [210] find that the slope ofthe CS feature in the 𝐽 band is almost flat, and correcting the 𝐽 magnitudes to ametallicity independent magnitude increases the uncertainty in the median by anorder of magnitude. Freedman and Madore [83] attempted to investigate the effectsof metallicity on the mean 𝐽-band magnitude of the CS. For their sample of 12galaxies the authors find a linear relation between the mean 𝐽 magnitude of the CSand the host galaxy’s metallicity with a slope of 0.09±0.05 mag dex−1. Excludingthe galaxies with the lowest CS count, their results get closer to zero with a slope of−0.03±0.04 mag dex−1. Freedman andMadore [83] conclude that for their sample158of galaxies, the metallicity has no apparent effect on the mean magnitude of theCS. However, in the calibration of the of the mean 𝐽-band magnitude by the sameauthors in a companion paper, Madore and Freedman [154] find slightly differentmean magnitudes for the LMC (⟨𝑀𝐽⟩ = −6.22) and the SMC (⟨𝑀𝐽⟩ = −6.18). Asa calibration, they adopt the average of the two mean values.In Section 3.4, metallicity effects on the CS luminosity function were discussed,and how a higher metallicity drives the peak of the CS distribution towards brightermagnitudes. Section 3.4 also mentions how using different trace populations suchas RGB, AGB or super-giants, yield significantly different metallicity estimates forthe same galaxy. With this background information, it seems difficult to establisha relationship between mean 𝐽 magnitude and metallicity. On the other hand, thedifferences in the shape of the CS luminosity function for different galaxies arereflected in the parameters obtained for the best fit modified Lorentzian modelintroduced in Section 3.3.Additionally, Chapter 8 presents an extensive analysis of the CS in the MilkyWay. The MilkyWay is a grand design spiral with a higher metal content comparedto the Magellanic Clouds. Based on different catalogues of Galactic AGB stars, itwas found that the luminosity function of the CS in the Milky Way is substantiallydifferent from the CS luminosity functions of the LMC, SMC and the four targetgalaxies with a significantly fainter median 𝐽 magnitude. Lee et al. [147] presenta preliminary calibration of the CS method using the Milky Way CS. Their resultagrees with the calibrations based on the mean of the CS in the LMC. Nevertheless,in their study they employed a combination of only two of the catalogues presentedin Chapter 8, catalogues in which the CS were selected based first on their spectrafollowed by a (𝐽 −𝐾𝑠)0 colour selection and not solely based on their (𝐽 −𝐾𝑠)0colours. In Chapter 8 it is shown how highly selective samples can bias the valueof 𝑀 𝐽 towards fainter or brighter magnitudes depending on the selection. Usingthe largest available catalogue, the Milky Way median CS 𝐽 magnitude is half amagnitude fainter than 𝑀 𝐽 ,𝐿𝑀𝐶 . This means that using the CS in the Milky Wayto measure distances to Magellanic-type galaxies can introduce systematic errorsof ∼ 0.5 mag. Even if the selection of CS in the Galaxy based on stellar spectraimproves the agreement between the CS in the Milky Way and in the MagellanicClouds, as we explore more distant galaxies it becomes observationally expensive159and sometimes impossible to obtain this detailed information. Applying thismethodto other grand design spirals will require extensive studies of the CS in the MilkyWay and other nearby spirals to define a general calibration for this type of galaxy.In the RG-MLE method, the differences in the luminosity function of the RGstars is also noticeable from a visual inspection (see Figure 5.3). As for the CS, thisdifference is very likely due to metallicity effects (see Section 3.4. In this method,the luminosity function is modeled using a linear combination of the luminosityfunctions of the LMC and SMC. This process allowed us to quantify how “LMC-like” or “SMC-like” a galaxy is through the constant that multiplies each term ofthe linear combination (see Equation 5.8).The difference in the RGs luminosity function shape is analogous to the chang-ing slope of the TRGB, induced by metallicity effects. For the TRGB method, therectified 𝑇 magnitude defined by Madore et al. [157] is used to remove possiblemetallicity effects. The 𝑇 magnitude is defined to be independent and insensitiveto metallicity. However, differences in the RGs 𝑇-band luminosity functions aresill visible (see Figure 4.3). In this thesis, a difference in the absolute magnitude ofthe TRGB of the LMC and SMC of ∼ 0.1 mag was also found. The differences inthe luminosity function shape and TRGB magnitude between the LMC and SMCcould be an indication that metallicity effects are greater than what the 𝑇 magnitudeis able to correct for. To compensate for the unaccounted metallicity effects, thedistance moduli presented in Table 9.1 for the TRGB were estimated choosing themost suitable calibrator based on the categorization obtained from the CS method.Calibrations of the TRGB absolute magnitude in the 𝐼-band and 𝐽𝐻𝐾-bandshave also been based on metallicity [78, 15, 14, 243]. In these studies it was foundthat metallicity has a greater effect on the near-infrared TRGBmagnitude comparedto the optical 𝐼-band. But the near-infrared bands have the advantage of a brighterTRGB and less sensitivity to reddening. Studies (e.g. [61, 148]) have shown thatthe 𝐼-band TRGB magnitude for stars older than 4-5 Gyr and metallicities between−2.0 < [𝑀/𝐻] < −0.7 is almost constant with metallicity. Outside these ranges,assuming a single value for the TRGB can introduce large systematic errors in thedistance estimations (see Serenelli et al. [222] for a recent study).Recently McQuinn et al. [168] did an extensive theoretical analysis of thepotential of the TRGB as a standard candle in the near-infrared. Their study is160centred in determining how the magnitude of the TRGB changes as a function ofthe observing wavelength, and the stellar age and metallicity of the host galaxy.McQuinn et al. [168] find that in the 𝐽-band the TRGB magnitude varies by ∼ 0.1mag depending on the age, and ∼ 0.6 mag depending on the metallicity. McQuinnet al. [168] find that a slope and colour-based empirical calibration of the TRGB(similar to the observational calibration using the 𝑇 magnitude), can reduced thevariation to between 0.02 and 0.05 mag. This uncertainty in the TRGB magnitudewould be higher in real observation due to photometric uncertainties.The use of the RG stars to estimate distance through the MLE method avoidsthe need for a colour-based or metallicity-based calibration of the TRGB. The sameprocess that allows a metallicity independent distance estimation also limits themethod to galaxies with similar metallicities and star formation histories as theMagellanic Clouds. As for the CS method, the RG-MLE method has only beentested in Magellanic-type irregular galaxies.161Chapter 10Conclusions and future work10.1 SummaryIn this thesis two independent distance determination methods for nearby galaxieshave been developed. The first, and main focus of this thesis, uses the carbon-richAGB stars (the CS), while the second method combines the RGB and oxygen-richAGB stars. An estimate of the distances using the TRGBmethod was also includedfor comparison. For all three methods, the distance calibrator is either the LMCor the SMC, where the most suitable one for each target is chosen based on thecharacteristics of the luminosity function of the CS (or RGs). For the calibration,the adopted mean distance moduli for the Magellanic Clouds were 𝜇LMC = 18.477± 0.004 (statistical) ± 0.026 (systematic) [192] for the LMC, and 𝜇SMC = 18.977 ±0.016 (statistical) ± 0.028 (systematic) [111] for the SMC.The methods were tested in four members of the Local Group: NGC 6822,IC 1613, WLM and NGC 3109. All galaxies were corrected for extinction beforeselecting the CS or RGs. This is especially important for the CS as these stars areselected based solely on their (𝐽 −𝐾𝑠)0 colour. The use of extinction correctedcolours allows selecting the CS using the same colour limits for all the targetgalaxies.The CS method uses the median 𝐽 magnitude of the CS and compares it to theappropriate calibrator (LMC or SMC). To determine the most suitable calibratinggalaxy, the first step was to compare intrinsic properties such as metallicity and star162formation history. Most studies agree that, overall, the LMC is more metal rich thanthe SMC, but for the target galaxies, except IC 1613with consistently lowmetallicitymeasurements, the metallicity estimates cover a wide range of values. Metallicityestimates seem to depend greatly on the tracers used in the analysis and might notbe representative of the CS population. Given the wide spread of values in themetallicity estimates, it is preferable to compare the 𝐽−band luminosity functionsof the CS from the target galaxies to those from the LMC and SMC. The 𝐽-band CSluminosity functions of the LMC and SMC have different shapes. The differenceswere attributed to differences in the metallicities and star formation histories ofthe two galaxies. The main observable difference is that the LMC CS luminosityfunction is skewed towards brighter magnitudes. To quantify this difference, theCS distributions were modeled using a modified Lorentzian model that allows forasymmetry as well as different weights on the tails of the distributions. The resultsfor the skew parameter (𝑠) confirmed that for the LMC (𝑠 ∼ −0.4) and for the SMC(𝑠 ∼ 0.0) the luminosity functions have different shapes, with the LMC skewedtowards brighter magnitudes. The LMC also exhibits more extended tails in itsdistribution compared to the SMC, while the widths of the distributions are almostidentical.Fitting of the Lorentzian model was made with two different techniques. Thefirst uses a direct fit to the binned luminosity function using non-linear least squares.Given the large the number of stars in theMagellanic Clouds this fittingmethod gavereasonable-sized error bars for the parameters that define the Lorentzian model.This was not the case for the target galaxies as the number of CS available isconsiderably smaller. The second technique uses an un-binnedmaximum likelihoodmethod (MLE) to fit the modified Lorentzian model. The MLE methods reducedthe error bars in of the parameters of the Lorentzian model by at least a few percent,but for the skew parameter the error bars improved by an average of ∼ 75%. Theparameters obtained from the linear squares fit and the MLE method are consistentwithin the error bars.After fitting the Lorentzian model, the skew parameter was used to determinethat NGC 6822 is more "LMC-like" and IC 1613, WLM and NGC 3109 aremore "SMC-like". The limiting skew value (𝑠 = −0.2, see equation 3.7) for thedetermination of themost suitable calibrator was set to be themidway value between163the skewness of the LMC and SMC. Because for the targets in this thesis the skewparameters are roughly identical to the chosen calibrator and the sample of targetgalaxies is small, it is not possible to test if the limit defined in equation 3.7 is anappropriate limit.Onemajor difference between this thesis and the recent independent calibrationsof the CS [154, 253] as a distance indicator is the use of the median magnitude(𝑀 𝐽 ) as oppose to the mean (⟨𝑀𝐽⟩). Section 3.7 shows that tight constrains on themagnitude selection of the CS region considerably impacts the value of the medianand the mean of the distribution. Choosing a wider range of magnitudes reduces theeffects of magnitude selection. Different tests done on the CS samples varying themagnitude limits show that in every case the effects on ⟨𝑀𝐽⟩ (or ⟨𝐽0⟩) were greaterthan for 𝑀 𝐽 (or 𝐽0). For the LMC and SMC the CMDs were clean enough that nomagnitude selection was necessary for these data. However for the target galaxiesthe contamination to the CMDs at the faint end of the CS region required a faintmagnitude cut. Still, the magnitude ranges covered by the CS in these galaxies waslarge enough that changing the limit by up to half a magnitude only has an effecton the distance modulus of the same order as the error in the distance modulus.Additionally, the distances to the target galaxies were estimated using the TRGBedge-detection method. In this case the composite 𝑇 magnitude in 𝐾𝑠 is adoptedfollowing the equations in Madore et al. [155] and Hoyt et al. [130]. This color-based magnitude is constructed in an attempt to remove metallicity effects fromthe TRGB magnitude [157]. Nevertheless, a difference of ∼ 0.11 between TRGBabsolute magnitudes for the LMC and SMC was found.The discrepancy found between the TRGB values in the LMC and SMC couldindicate that the construction of the rectified𝑇 defined in equation 4.1 is not enoughto account for strong metallicity effects or other galaxy properties such as starformation history. McQuinn et al. [168] used simulated photometry to investigatethe dependency of the TRGB luminosity on stellar age and metallicity as a functionof wavelength. In the 𝐽 and 𝐾𝑠-band, using a transformed magnitude similar to the𝑇 magnitude, McQuinn et al. find that a Δ𝑚𝐾𝑠 ∼ 0.04 magnitude difference remainsafter correcting for metallicity effects. The difference found theoretically is smallerthan what was found in this thesis.The difference in the TRGB magnitudes found in this thesis is significant. Se-164lecting the most suitable calibrator for each target galaxy using only a smoothedluminosity function for the RGs is not enough. Because of the absence of aparametrized model, it is not possible to establish a method to quantify the differ-ence between the luminosity functions, instead themost suitable calibrator selectionrelied on the classification obtained from the CS luminosity function. Using the"LMC-like"-"SMC-like" classification obtained from the CS luminosity function,the distance moduli obtained from the CS median 𝐽 magnitude and TRGB magni-tude agree well within the statistical error bars.In Chapter 7, the calibration of the CS median 𝐽 magnitude and the TRGB 𝐾𝑠magnitude in the Magellanic Clouds is performed correcting the LMC and SMCdata using the more recent reddening maps developed by Skowron et al. [226]. Nosignificant differences were found for the SMC where 𝑀 𝐽 and 𝑀TRGB𝐾𝑠 are in goodagreement with the original values obtained in Chapters 3 and 4. For the LMCthe CS 𝑀 𝐽 value agrees with the original value, while a significant difference isfound for the TRGB with the new value turning out to be 0.11 magnitudes faintercompared to the original estimate in Chapter 4. This fainter estimate brings the𝑀TRGB𝐾𝑠 of the LMC and SMC into agreement within the error bars. No effectson the shape of the CS luminosity function are observed with the new reddeningcorrections, therefore the classification of the target galaxies remains unchanged.The new TRGB magnitude for the LMC affects only the distance modulus of NGC6822 as the other target galaxies are classified as “SMC-like” galaxies.The TRGB is a method that has been widely tested (see [85] and referencestherein), however metallicity effects require corrections to the RG magnitudes. Thedevelopment of a distance determination method that does not rely on metallicity(or metallicity-related) corrections and uses the majority of the stars in the RGBand AGB (instead of the few stars at the TRGB), will prove to be critical.The RG-MLEmethod developed in this thesis, selects the stars directly from thenear-infrared CMD and uses all the selected stars to find a model for the RG lumi-nosity function. In this case the luminosity functions were use in their cumulativedistribution form. The LMC and SMC functions were modelled using a smoothingspline, while for the target galaxies we used a linear combination of the LMC andSMC splines. The best model for the target galaxies is obtained using an un-binnedMLE that finds the distance modulus that minimizes the difference between the RG165distribution observed in the target galaxy and that of the Magellanic Clouds linearcombination. The linear combination includes a “shape” parameter that indicateswhether the luminosity function of a target galaxy most likely resembles that of theLMC or the SMC.The distance moduli obtained from the RG-MLE method are in general in goodagreement with the results from the CS and TRGB methods. For NGC 6822 the“shape” parameter indicated that the galaxy resembled the LMC, while IC 1613 andWLM were determined to be similar to the SMC, agreeing with the classificationfound from the CS. In the case of NGC 3109, using the luminosity function inthe 𝐽-band indicated a more “SMC-like” shape while using the 𝐾𝑠-band gave an“LMC-like” shape. NGC 3109 has the largest photometric errors of the four datasets, which could explain the difference.As mentioned, for the CS and TRGB methods the calibrator was either theLMC or SMC, as opposed to using both Magellanic Clouds in combination (e.g.[154]) or only one of the Clouds (e.g. [253]). For the RG-MLE method, the firsttechnique uses also either the LMC or SMC as single calibrator, while the secondtechnique uses a linear combination of the LMC and SMC. This decision was basedon the fact that the CS and RGs luminosity functions and the TRGB magnitudeof the LMC and SMC were found to be slightly different from one another. Thedifferences between the LMC and SMC are attributed to different metallicities andstar formation histories. In the case of the CS and TRGB the use of the less suitablecalibrator introduces an error on the distance modulus of ∼ 0.1 magnitudes. For theRG-MLE method comparing the RGs luminosity functions of the target galaxiesdirectly to the LMC or SMC (i.e. without using a linear combination, see Table5.2) gives differences in the distance moduli of up to ∼ 0.2 magnitudes, an order ofmagnitude larger than the typical statistical error bar in this method.Since the distance determination methods developed in this thesis rely on an"LMC/SMC-like" classification, they have thus only been used successfully onMagellanic-type irregular galaxies. As shown in Chapter 8, the CS luminosityfunction is notably different for the Milky Way, and this is expected to be the casefor all grand design spiral galaxies. In Chapter 8 some of the available CS cataloguesfor the Milky Way were analysed and significant differences were found in boththe median magnitude and intrinsic magnitude dispersion of the CS compared to166the Magellanic Cloud. A spectroscopic selection of CS in the Milky Way showedthat specific types of CS have more “Magellanic-type” properties and brings thedifferences closer. Nevertheless, such a selection might not be available in distantgalaxies where the ultimate plan is to use this method for a determination of theHubble Constant.10.2 Future workCS are potentially more efficient distance indicators than methods that requireobservations over variation cycles (such asCepheids, RRLyrae, andMira variables).For CS, no period determination is necessary as just a single observation of theirluminosity is used to determine the distance. Additionally, their 𝐽-band magnitudesshows to be independent of colour and has a small intrinsic dispersion. CS are alsobrighter and redder than TRGB stars which gives CS the potential to reach largerdistances in the universe. This advantages make CS an excellent alternative firstrung in the distance ladder that can provide an independent calibration of Type Iasupernovae (SN Ia). Additionally, CS stars can be found in all types all galaxies,while Cepheids for example, are only found in galaxies that had recent star formationlimiting the number of galaxies that can be used as calibrators of SN Ia, which canoccur in any type of galaxy.To use CS as SN Ia calibrators, the logical next step is to obtain distancesthrough the CS method to SN Ia host galaxies, unfortunately the currently availablesample of SN Ia host galaxies with good 𝐽𝐾𝑠 photometry is limited (if not zero).In the specific case of this thesis, one of the limitations of the low sample oftarget galaxies is the fact that it was not possible to see a wide range of skew valuesfor the CS luminosity function. Expanding the sample of target galaxies to includegalaxies with a wider range of characteristics will allow to determine whether theskew parameter is the best indicator from which to choose a calibrator.Currently the TRGB methods has been used in SN Ia host galaxies up to∼ 30 Mpc (𝜇0 ∼ 32.4) [82], while Cepheid variables have reach SN Ia host galaxiesup to ∼ 40 Mpc (𝜇0 ∼ 33) [208]. In this thesis the furthest galaxy in the sample wasestimated to be at ∼ 1.3 Mpc (𝜇0 ∼ 25.6). With the data available today, the CShave been used in galaxies up to ∼ 3.5 Mpc (𝜇0 ∼ 27.7) in the 𝐽-band [154, 253].167Recently, Madore et al. [156] reached a galaxy at ∼ 4 Mpc (𝜇0 ∼ 28) using the CSmean magnitude in the HST/WFC3 𝐹110𝑊 near-infrared filter (which the authorscategorized as a broad-band 𝐽 filter). Nevertheless, Madore et al. remark that theobservations available in 𝐹110𝑊 are not optimized for the observation of CS andclaim that with this instrument and filter it would be possible to reach distances upto ∼ 40 Mpc with under an hour of exposure time, but with a signal-to-noise ratioof only 5.The new JWST presents a unique opportunity to expand the CS distance deter-mination method to distances well beyond those achieved to date with Cepheids orTRGB stars. JWST is capable of reaching a signal-to-noise ratio large enough todistinguish single CS in galaxies far beyond the limits of HST at feasible exposuretimes. To reach at least 1 magnitude below the average 𝐽 magnitude for the CS ina particular galaxy, only a modest signal-to-noise ratio of ∼10 is required. For agalaxy at ∼ 50 Mpc this translate into an exposure time of ∼6 hours1.Moreover, this thesis and recent CS studies [154, 83, 210, 146, 253] base theselection of CS on their (𝐽 −𝐾𝑠) colour. JWST filter options do not include 𝐽 or𝐾𝑠 filters but both near infrared imagers on board JWST are equipped with filtersthat resemble 𝐽 and 𝐾𝑠, these are 𝐹115𝑊 and 𝐹200𝑊 . Figure 10.1 comparesthe 2MASS 𝐽 and 𝐾𝑠 band filters with 𝐹115𝑊 and 𝐹200𝑊 filters in JWST’sNear Infrared Camera (NIRCam). The JWST/NIRCam filters are bluer and widerthat their respective 2MASS counterparts. The larger bandwidth allows for aconsiderable amount of overlap among 𝐽 and 𝐹115𝑊 and between 𝐾𝑠 and 𝐹200𝑊 .Observations of galaxieswith JWSTwill allowdistances to bemeasured throughthe CS method out to 100 Mpc Madore et al. [156], well into the third rung of thedistance ladder. This will provide the possibility of bypassing the use of SN Iaentirely and measuring 𝐻0 directly. Precise distance measurements to more distantgalaxies will allow us to measure the Hubble Constant well out into the pure Hubbleflow and make a more definitive comparison with 𝐻0 measured from the CMB inthe early universe.1Exposure times were obtained with the JWST Exposure Time Calculator - https://jwst.etc.stsci.edu/168Figure 10.1: Filters relative throughput for 2MASS 𝐽𝐻𝐾𝑠 [55], and 𝐹115𝑊and 𝐹200𝑊 for JWST/NIRCam [229]. Throughput for both systems hasbeen normalized to unity. 2MASS filters are redder and narrower than JWSTfilters. Thanks to the greater bandwidth of the NIRCam filters, there is stilla ∼70% of overlap between NIRCam and 2MASS filters169Bibliography[1] Aaronson, Marc, Victor M. Blanco, Kem H. Cook, and Paul L. Schechter(1989), “Southern MilkyWay Carbon Stars: New Candidates, JHK Photom-etry, and Radial Velocities.” Astrophysical Journal Supplement, 70, 637.[2] Abia, C., P. de Laverny, S. Cristallo, et al. (2020), “Properties of carbon starsin the solar neighbourhood based on Gaia DR2 astrometry.” Astronomy &Astrophysics, 633, A135.[3] Albers, Saundra M., Daniel R. Weisz, Andrew A. Cole, et al. (2019), “Starformation at the edge of the Local Group: a rising star formation historyin the isolated galaxy WLM.” Monthly Notices of the Royal AstronomicalSociety, 490, 5538–5550.[4] Alksnis, A., A. Balklavs, U. Dzervitis, et al. (2001), “General Catalog ofGalacticCarbonStars byC.B. Stephenson. ThirdEdition.”Baltic Astronomy,10, 1–318.[5] Amico, P., J. G. Cuby, N. Devillard, et al. (2002), “ISAAC Data ReductionGuide 1.5.” ESO Very Large Telescope.[6] Anand, Gagandeep S., R. Brent Tully, Luca Rizzi, et al. (2021), “ComparingTip of the Red Giant Branch Distance Scales: An Independent Reductionof the Carnegie-Chicago Hubble Program and the Value of the HubbleConstant.” arXiv e-prints, arXiv:2108.00007.[7] Baade, W. (1944), “The Resolution of Messier 32, NGC 205, and the CentralRegion of the Andromeda Nebula.” Astrophysical Journal, 100, 137.[8] Baade, W. and F. Zwicky (1934), “On Super-novae.” Proceedings of theNational Academy of Science, 20, 254–259.[9] Bailer-Jones, C. A. L., J. Rybizki, M. Fouesneau, et al. (2021), “EstimatingDistances from Parallaxes. V. Geometric and Photogeometric Distances to1701.47 Billion Stars in Gaia Early Data Release 3.” Astronomical Journal, 161,147.[10] Battinelli, P. and S. Demers (2005), “The calibration of the metallicity versusC/M relation.” Astronomy & Astrophysics, 434, 657–663.[11] Battinelli, P. and S. Demers (2005), “The standard candle aspect of carbonstars.” Astronomy & Astrophysics, 442, 159–163.[12] Beaton, Rachael L., Giuseppe Bono, Vittorio Francesco Braga, et al. (2018),“Old-Aged Primary Distance Indicators.” Space Science Reviews, 214, 113.[13] Beichman, C. A., G. Neugebauer, H. J. Habing, et al., eds. (1988), InfraredAstronomical Satellite (IRAS) Catalogs and Atlases.Volume 1: ExplanatorySupplement., volume 1.[14] Bellazzini, M., F. R. Ferraro, A. Sollima, et al. (2004), “The calibration ofthe RGB Tip as a Standard Candle. Extension to Near Infrared colors andhigher metallicity.” Astronomy & Astrophysics, 424, 199–211.[15] Bellazzini, Michele, Francesco R. Ferraro, and Elena Pancino (2001), “AStep toward the Calibration of the Red Giant Branch Tip as a StandardCandle.” Astrophysical Journal, 556, 635–640.[16] Berger, Travis A., Rolf-Peter Kudritzki, Miguel A. Urbaneja, et al. (2018),“Quantitative Spectroscopy of Supergiants in the Local Group Dwarf GalaxyIC 1613: Metallicity and Distance.” Astrophysical Journal, 860, 130.[17] Bernard, Edouard J., Matteo Monelli, Carme Gallart, et al. (2010), “TheACS LCID Project. II. Faint Variable Stars in the Isolated Dwarf IrregularGalaxy IC 1613.” Astrophysical Journal, 712, 1259–1276.[18] Bernard, Edouard J., Matteo Monelli, Carme Gallart, et al. (2013), “TheACS LCID Project - VIII. The short-period Cepheids of Leo A.” MonthlyNotices of the Royal Astronomical Society, 432, 3047–3061.[19] Bertin, E. (2006), “Automatic Astrometric and Photometric Calibrationwith SCAMP.” In Astronomical Data Analysis Software and Systems XV(C. Gabriel, C. Arviset, D. Ponz, and S. Enrique, eds.), volume 351 ofAstronomical Society of the Pacific Conference Series, 112.[20] Bertin, E. and S. Arnouts (1996), “SExtractor: Software for source extrac-tion.” Astronomy & Astrophysics Supplement, 117, 393–404.171[21] Bertin, Emmanuel (2010), “SWarp: Resampling andCo-addingFITS ImagesTogether.” Astrophysics Source Code Library.[22] Bertin, Emmanuel, Yannick Mellier, Mario Radovich, et al. (2002), “TheTERAPIX Pipeline.” In Astronomical Data Analysis Software and SystemsXI (David A. Bohlender, Daniel Durand, and Thomas H. Handley, eds.),volume 281 of Astronomical Society of the Pacific Conference Series, 228.[23] Besla, Gurtina, Nitya Kallivayalil, Lars Hernquist, et al. (2012), “The roleof dwarf galaxy interactions in shaping the Magellanic System and implica-tions for Magellanic Irregulars.”Monthly Notices of the Royal AstronomicalSociety, 421, 2109–2138.[24] Bhardwaj, Anupam, ShashiM.Kanbur, LucasM.Macri, et al. (2016), “LargeMagellanic Cloud Near-infrared Synoptic Survey. II. The Wesenheit Rela-tions and Their Application to the Distance Scale.” Astronomical Journal,151, 88.[25] Blanco, V. M., M. F. McCarthy, and B. M. Blanco (1980), “Carbon andlate M-type stars in the Magellanic Clouds.” Astrophysical Journal, 242,938–964.[26] Bono, G., F. Caputo, M. Marconi, and I. Musella (2010), “Insights into theCepheid Distance Scale.” Astrophysical Journal, 715, 277–291.[27] Boyer, M. L., B. F.Williams, B. Aringer, et al. (2019), “ADramatic Decreasein Carbon Star Formation in M31.” Astrophysical Journal, 879, 109.[28] Boyer, Martha L., Sundar Srinivasan, Jacco Th. van Loon, et al. (2011),“Surveying the Agents of Galaxy Evolution in the Tidally Stripped, LowMetallicity Small Magellanic Cloud (SAGE-SMC). II. Cool Evolved Stars.”Astronomical Journal, 142, 103.[29] Bresolin, F., M. Capaccioli, and G. Piotto (1992), “Mapping the Star Forma-tion in Nearby Galaxies: the Magellanic Irregular NGC 3109.” In Morpho-logical and Physical Classification of Galaxies (G. Longo, M. Capaccioli,and G. Busarello, eds.), volume 178 of Astrophysics and Space ScienceLibrary, 447.[30] Bressan, Alessandro, Paola Marigo, Léo. Girardi, et al. (2012), “PARSEC:stellar tracks and isochrones with the PAdova and TRieste Stellar EvolutionCode.” Monthly Notices of the Royal Astronomical Society, 427, 127–145.172[31] Brewer, James P., Harvey B. Richer, and Dennis R. Crabtree (1995), “Late-Type Stars in M31. I. Photometric Study of AGB Stars and MetallicityGradients.” Astronomical Journal, 109, 2480.[32] Caldwell, J. A. R. and I. M. Coulson (1986), “The geometry and distance ofthe Magellanic clouds from cepheid variables.”Monthly Notices of the RoyalAstronomical Society, 218, 223–246.[33] Cannon, John M., Erin M. O’Leary, Daniel R. Weisz, et al. (2012), “On theOrigin of the Supergiant H I Shell and Putative Companion in NGC 6822.”Astrophysical Journal, 747, 122.[34] Cardelli, Jason A., Geoffrey C. Clayton, and John S. Mathis (1989), “TheRelationship between Infrared, Optical, and Ultraviolet Extinction.” Astro-physical Journal, 345, 245.[35] Carpenter, John M. (2001), “Color Transformations for the 2MASS SecondIncremental Data Release.” Astronomical Journal, 121, 2851–2871.[36] Carrera, R., C. Gallart, Eduardo Hardy, et al. (2008), “The Chemical Enrich-ment History of the Large Magellanic Cloud.” Astronomical Journal, 135,836–849.[37] Carroll, Bradley W. and Dale A. Ostlie (2006), An introduction to modernastrophysics. Addison-Wesley, San Francisco.[38] Cassisi, S. and M. Salaris (2013), Old Stellar Populations: How to Study theFossil Record of Galaxy Formation. Wiley, Weinheim.[39] Chambers, K. C., E. A. Magnier, N. Metcalfe, et al. (2016), “The Pan-STARRS1 Surveys.” arXiv e-prints, arXiv:1612.05560.[40] Chen, B. Q., H. L. Guo, J. Gao, et al. (2022), “Dust distributions in themagellanic clouds.” Monthly Notices of the Royal Astronomical Society,511, 1317–1329.[41] Chen, P. S. and X. H. Yang (2012), “A Catalog of Galactic Infrared CarbonStars.” Astronomical Journal, 143, 36.[42] Chen, Pei-Sheng, Xiao-HongYang, andHong-Guang Shan (2010), “Infraredspectral evolution of carbon stars.” Research in Astronomy and Astrophysics,10, 363–382.173[43] Chen, Seery, Harvey Richer, Ilaria Caiazzo, and Jeremy Heyl (2018), “Dis-tances to the Globular Clusters 47 Tucanae and NGC 362 Using Gaia DR2Parallaxes.” Astrophysical Journal, 867, 132.[44] Choudhury, Samyaday, Richard de Grijs, Kenji Bekki, et al. (2021), “TheVMC survey - XLIV: mapping metallicity trends in the large magellaniccloud using near-infrared passbands.” Monthly Notices of the Royal Astro-nomical Society, 507, 4752–4763.[45] Choudhury, Samyaday, Richard de Grijs, Stefano Rubele, et al. (2020), “TheVMC survey - XXXIX. Mapping metallicity trends in the Small Magel-lanic Cloud using near-infrared passbands.” Monthly Notices of the RoyalAstronomical Society, 497, 3746–3760.[46] Choudhury, Samyaday, Annapurni Subramaniam, and Andrew A. Cole(2016), “Photometric metallicity map of the Large Magellanic Cloud.”Monthly Notices of the Royal Astronomical Society, 455, 1855–1880.[47] Chun, Sang-Hyun, MiYoung Jung, Minhee Kang, et al. (2015), “Near-infrared photometric properties of asymptotic giant branch stars in the dwarfirregular galaxy IC 1613.” Astronomy & Astrophysics, 578, A51.[48] Cignoni, M., A. A. Cole, M. Tosi, et al. (2013), “Mean Age Gradient andAsymmetry in the Star Formation History of the Small Magellanic Cloud.”Astrophysical Journal, 775, 83.[49] Cioni, M. R. L. (2009), “The metallicity gradient as a tracer of historyand structure: the Magellanic Clouds and M33 galaxies.” Astronomy &Astrophysics, 506, 1137–1146.[50] Cioni, M. R. L. and H. J. Habing (2003), “AGB stars in the MagellanicClouds. I. The C/M ratio.” Astronomy & Astrophysics, 402, 133–140.[51] Cioni,M.R.L. andH. J.Habing (2005), “Near-IRobservations ofNGC6822:AGB stars, distance, metallicity and structure.” Astronomy & Astrophysics,429, 837–850.[52] Cioni, M. R. L., M. Irwin, A. M. N. Ferguson, et al. (2008), “AGB stars astracers of metallicity andmean age acrossM 33.”Astronomy&Astrophysics,487, 131–146.[53] Cioni, M. R. L., J. B. Marquette, C. Loup, et al. (2001), “Variability andspectral classification of LMC giants: Results from DENIS and EROS.”Astronomy & Astrophysics, 377, 945–954.174[54] Clementini, Gisella, Enrico V. Held, Lara Baldacci, and Luca Rizzi (2003),“RR Lyrae and Short-Period Variable Stars in the Dwarf Irregular GalaxyNGC 6822.” Astrophysical Journal Letters, 588, L85–L88.[55] Cohen, Martin, Wm. A. Wheaton, and S. T. Megeath (2003), “SpectralIrradiance Calibration in the Infrared. XIV. The Absolute Calibration of2MASS.” Astronomical Journal, 126, 1090–1096.[56] Collins, G. W. II (2003), The fundamentals of stellar astrophysics. WebEdition, URL http://bifrost.cwru.edu/personal/collins/AstroBook/.[57] Cook, K. H., M. Aaronson, and J. Norris (1986), “Carbon and M Stars inNearby Galaxies: A Preliminary Survey Using a Photometric Technique.”Astrophysical Journal, 305, 634.[58] Crabtree, D. R., H. B. Richer, and B. E. Westerlund (1976), “Carbon stars inthe LargeMagellanic Cloud.” Astrophysical Journal Letters, 203, L81–L85.[59] Cristallo, S., O. Straniero, L. Piersanti, and D. Gobrecht (2015), “Evolution,Nucleosynthesis, and Yields of AGB Stars at Different Metallicities. III.Intermediate-mass Models, Revised Low-mass Models, and the ph-FRUITYInterface.” Astrophysical Journal Supplement, 219, 40.[60] Cusano, F., M. I. Moretti, G. Clementini, et al. (2021), “The VMC Survey- XLII. Near-infrared period-luminosity relations for RR Lyrae stars andthe structure of the Large Magellanic Cloud.” Monthly Notices of the RoyalAstronomical Society, 504, 1–15.[61] Da Costa, G. S. and T. E. Armandroff (1990), “Standard Globular ClusterGiant Branches in the (M(I), (V - I)o) Plane.” Astronomical Journal, 100,162.[62] Dalcanton, Julianne J., Benjamin F. Williams, Anil C. Seth, et al. (2009),“The ACS Nearby Galaxy Survey Treasury.” Astrophysical Journal Supple-ment, 183, 67–108.[63] Dambis, A. K., L. N. Berdnikov, A. Y. Kniazev, et al. (2013), “RR Lyraevariables: visual and infrared luminosities, intrinsic colours and kinematics.”Monthly Notices of the Royal Astronomical Society, 435, 3206–3220.[64] Demers, S., P. Battinelli, and B. Letarte (2003), “Carbon star survey in theLocal Group. VII. NGC 3109 a galaxy without a stellar halo.” Astronomy &Astrophysics, 410, 795–801.175[65] Demers, S., P. Battinelli, and B. Letarte (2004), “A Carbon star approach toIC 10: Distance and correct size.”Astronomy&Astrophysics, 424, 125–132.[66] Devillard, N. (1997), “The eclipse software.” The Messenger, 87, 19–20.[67] Di Valentino, Eleonora, Olga Mena, Supriya Pan, et al. (2021), “In therealm of the Hubble tension-a review of solutions.” Classical and QuantumGravity, 38, 153001.[68] Diaz, Jonathan D. and Kenji Bekki (2012), “The Tidal Origin of the Mag-ellanic Stream and the Possibility of a Stellar Counterpart.” AstrophysicalJournal, 750, 36.[69] Dirsch, B., T. Richtler, W. P. Gieren, and M. Hilker (2000), “Age andmetallicity for six LMC clusters and their surrounding field population.”Astronomy & Astrophysics, 360, 133–160.[70] Dolphin, Andrew E. (2000), “Hubble Space Telescope Studies of the WLMGalaxy. II. The Star Formation History from Field Stars.” AstrophysicalJournal, 531, 804–812.[71] Drimmel, R., A. Cabrera-Lavers, andM. López-Corredoira (2003), “A three-dimensional Galactic extinction model.” Astronomy & Astrophysics, 409,205–215.[72] Elizalde, Emilio (2021), The True Story of Modern Cosmology: Origins,Main Actors and Breakthroughs. Springer International Publishing, Cham.[73] Ellis, R. S. and D. A. Allen (1983), “Infrared colours of a complete sampleof faint galaxies.” Monthly Notices of the Royal Astronomical Society, 203,685–693.[74] Evans, C. J., F. Bresolin, M. A. Urbaneja, et al. (2007), “The ARAUCARIAProject: VLT-FORS Spectroscopy of Blue Supergiants in NGC 3109-Classifications, First Abundances, and Kinematics.” Astrophysical Journal,659, 1198–1211.[75] Fabricius, C., X. Luri, F. Arenou, et al. (2021), “Gaia Early Data Release 3.Catalogue validation.” Astronomy & Astrophysics, 649, A5.[76] Feast, Michael W., Oyirwoth P. Abedigamba, and Patricia A. Whitelock(2010), “Is there a metallicity gradient in the Large Magellanic Cloud?”Monthly Notices of the Royal Astronomical Society, 408, L76–L79.176[77] Feast, Michael W., Patricia A. Whitelock, John W. Menzies, and NoriyukiMatsunaga (2012), “The Cepheid distance to the Local Group galaxy NGC6822.”Monthly Notices of the Royal Astronomical Society, 421, 2998–3003.[78] Ferraro, Francesco R., Paolo Montegriffo, Livia Origlia, and Flavio FusiPecci (2000), “A New Infrared Array Photometric Survey of Galactic Glob-ular Clusters: A Detailed Study of the Red Giant Branch Sequence as a Steptoward the Global Testing of Stellar Models.” Astronomical Journal, 119,1282–1295.[79] Fitzpatrick, Edward L. (1999), “Correcting for the Effects of InterstellarExtinction.” Publications of the Astron. Soc. of the Pac., 111, 63–75.[80] Foreman-Mackey, Daniel (2016), “corner.py: Scatterplot matrices inpython.” The Journal of Open Source Software, 24, URL http://dx.doi.org/10.5281/zenodo.45906.[81] Freedman, Wendy L. (1988), “Stellar Content of Nearby Galaxies. I. BVRICCD Photometry for IC 1613.” Astronomical Journal, 96, 1248.[82] Freedman, Wendy L. (2021), “Measurements of the Hubble Constant: Ten-sions in Perspective.” Astrophysical Journal, 919, 16.[83] Freedman, Wendy L. and Barry F. Madore (2020), “Astrophysical DistanceScale. II. Application of the JAGB Method: A Nearby Galaxy Sample.”Astrophysical Journal, 899, 67.[84] Freedman, Wendy L., Barry F. Madore, Dylan Hatt, et al. (2019), “TheCarnegie-Chicago Hubble Program. VIII. An Independent Determination ofthe Hubble Constant Based on the Tip of the Red Giant Branch.” Astrophys-ical Journal, 882, 34.[85] Freedman, Wendy L., Barry F. Madore, Taylor Hoyt, et al. (2020), “Cali-bration of the Tip of the Red Giant Branch.” Astrophysical Journal, 891,57.[86] Freedman, Wendy L., Jane Rigby, Barry F. Madore, et al. (2009), “TheCepheid Period-Luminosity Relation (The Leavitt Law) at Mid-InfraredWavelengths. IV. Cepheids in IC 1613.” Astrophysical Journal, 695, 996–998.[87] Frogel, J. A., J. H. Elias, J. G. Cohen, and S. E. Persson (1981), “Observedbolometric luminosities of carbon stars.” InPhysical Processes in RedGiants177(Jr. Iben, I. and A. Renzini, eds.), volume 88 of Astrophysics and SpaceScience Library, 159–164.[88] Frogel, J. A., S. E. Persson, and J. G. Cohen (1980), “Luminosities and tem-peratures of the reddest stars in three LMC clusters.” Astrophysical Journal,239, 495–501.[89] Frogel, J. A., S. E. Persson, and J. G. Cohen (1981), “Infrared photometry ofred giants in the globular cluster 47 Tucanae.” Astrophysical Journal, 246,842–865.[90] Fusco, F., R. Buonanno, G. Bono, et al. (2012), “Distance and reddening ofthe Local Group dwarf irregular galaxy NGC 6822.” Astronomy & Astro-physics, 548, A129.[91] Fusco, F., R. Buonanno, S. L. Hidalgo, et al. (2014), “A state-of-the-art anal-ysis of the dwarf irregular galaxy NGC 6822.” Astronomy & Astrophysics,572, A26.[92] Gaia Collaboration, T. Antoja, P. J. McMillan, et al. (2021), “Gaia EarlyData Release 3. The Galactic anticentre.” Astronomy & Astrophysics, 649,A8.[93] Gaia Collaboration, A. G. A. Brown, A. Vallenari, et al. (2018), “Gaia DataRelease 2. Summary of the contents and survey properties.” Astronomy &Astrophysics, 616, A1.[94] Gaia Collaboration, A. G. A. Brown, A. Vallenari, et al. (2021), “Gaia EarlyData Release 3. Summary of the contents and survey properties.” Astronomy& Astrophysics, 649, A1.[95] Gaia Collaboration, X. Luri, L. Chemin, et al. (2021), “Gaia Early DataRelease 3. Structure and properties of the Magellanic Clouds.” Astronomy& Astrophysics, 649, A7.[96] Gaia Collaboration, T. Prusti, J. H. J. de Bruijne, et al. (2016), “The Gaiamission.” Astronomy & Astrophysics, 595, A1.[97] Gallart, C., A. Aparicio, G. Bertelli, and C. Chiosi (1996), “The Local GroupDwarf Irregular Galaxy NGC 6822.III. The Recent Star Formation History.”Astronomical Journal, 112, 2596.[98] Gallart, C., A. Aparicio, G. Bertelli, and C. Chiosi (1996), “The Local GroupDwarf Irregular Galaxy NGC 6822.II.The Old and Intermediate -Age StarFormation History.” Astronomical Journal, 112, 1950.178[99] Georgiev, L., J. Borissova, M. Rosado, et al. (1999), “The Local Groupdwarf irregular galaxy IC 1613 . I. The young stellar content.” Astronomy &Astrophysics Supplement, 134, 21–37.[100] Gerasimovic, B. P. (1928), “The Absolute Magnitudes of Long Period Vari-able Stars.” Proceedings of the National Academy of Science, 14, 963–968.[101] Gieren, W., J. Storm, P. Konorski, et al. (2018), “The effect of metallicityon Cepheid period-luminosity relations from a Baade-Wesselink analysisof Cepheids in the Milky Way and Magellanic Clouds★.” Astronomy &Astrophysics, 620, A99.[102] Gieren, Wolfgang, Grzegorz Pietrzyński, Krzysztof Nalewajko, et al. (2006),“The Araucaria Project: An Accurate Distance to the Local Group GalaxyNGC 6822 from Near-Infrared Photometry of Cepheid Variables.” Astro-physical Journal, 647, 1056–1064.[103] Gieren, Wolfgang, Grzegorz Pietrzyński, Olaf Szewczyk, et al. (2008), “TheAraucaria Project: The Distance to the Local Group Galaxy WLM fromNear-Infrared Photometry of Cepheid Variables.” Astrophysical Journal,683, 611–619.[104] Girardi, L., M. A. T. Groenewegen, E. Hatziminaoglou, and L. da Costa(2005), “Star counts in theGalaxy. Simulating fromvery deep to very shallowphotometric surveys with the TRILEGAL code.” Astronomy& Astrophysics,436, 895–915.[105] Glass, I. S. (1999), Handbook of Infrared Astronomy. Cambridge UniversityPress, Cambridge.[106] Gordon, Courtney P. (1968), “The Absolute Magnitudes of Carbon Stars.”Publications of the Astron. Soc. of the Pac., 80, 597.[107] Gordon, Karl D., Geoffrey C. Clayton, K. A. Misselt, et al. (2003), “AQuantitative Comparison of the Small Magellanic Cloud, Large MagellanicCloud, and Milky Way Ultraviolet to Near-Infrared Extinction Curves.”Astrophysical Journal, 594, 279–293.[108] Górski, Marek, Grzegorz Pietrzyński, and Wolfgang Gieren (2011), “TheAraucaria Project. Infrared Tip of the Red Giant Branch Distances to FiveDwarf Galaxies in the Local Group.” Astronomical Journal, 141, 194.179[109] Górski, Marek, Grzegorz Pietrzyński, Wolfgang Gieren, et al. (2016), “TheAraucaria Project: On the Tip of the Red Giant Branch Distance Determi-nation to the Magellanic Clouds.” Astronomical Journal, 151, 167.[110] Górski, Marek, Bartłomiej Zgirski, Grzegorz Pietrzyński, et al. (2020), “Em-pirical Calibration of the Reddening Maps in the Magellanic Clouds.” As-trophysical Journal, 889, 179.[111] Graczyk, Dariusz, Grzegorz Pietrzyński, Ian B. Thompson, et al. (2020), “ADistance Determination to the Small Magellanic Cloud with an Accuracyof Better than Two Percent Based on Late-type Eclipsing Binary Stars.”Astrophysical Journal, 904, 13.[112] Green, Elizabeth M., Pierre Demarque, and Christopher R. King (1987), Therevised Yale isochrones and luminosity functions. Yale University Observa-tory, New Haven.[113] Green, Gregory M., Edward Schlafly, Catherine Zucker, et al. (2019), “A3D Dust Map Based on Gaia, Pan-STARRS 1, and 2MASS.” AstrophysicalJournal, 887, 93.[114] Greggio, L., G. Marconi, M. Tosi, and P. Focardi (1993), “Star Formation inDwarf Irregular Galaxies: DDO 210 andNGC 3109.”Astronomical Journal,105, 894.[115] Gyllenberg, W. (1929), “Studies on the proper motions of long-period vari-able stars.”Meddelanden fran Lunds Astronomiska Observatorium Serie II,53, 3–20.[116] Hamren, Katherine M., Constance M. Rockosi, Puragra Guhathakurta, et al.(2015), “A Spectroscopic and Photometric Exploration of the C/M Ratio inthe Disk of M31.” Astrophysical Journal, 810, 60.[117] Harris, Jason and Dennis Zaritsky (2004), “The Star Formation History ofthe Small Magellanic Cloud.” Astronomical Journal, 127, 1531–1544.[118] Harris, Jason and Dennis Zaritsky (2009), “The Star Formation History ofthe Large Magellanic Cloud.” Astronomical Journal, 138, 1243–1260.[119] Haschke, Raoul, Eva K. Grebel, and Sonia Duffau (2011), “New OpticalReddening Maps of the Large and Small Magellanic Clouds.” AstronomicalJournal, 141, 158.180[120] Haschke, Raoul, Eva K. Grebel, and Sonia Duffau (2012), “Three-dimensional Maps of the Magellanic Clouds using RR Lyrae Stars andCepheids. II. The Small Magellanic Cloud.” Astronomical Journal, 144,107.[121] Hatt, Dylan, Rachael L. Beaton, Wendy L. Freedman, et al. (2017), “TheCarnegie-Chicago Hubble Program. II. The Distance to IC 1613: The Tipof the Red Giant Branch and RR Lyrae Period-luminosity Relations.” Astro-physical Journal, 845, 146.[122] Herwig, Falk (2005), “Evolution of Asymptotic Giant Branch Stars.” AnnualReview of Astronomy and Astrophysics, 43, 435–479.[123] Hidalgo, Sebastian L., Antonio Aparicio, and Carme Gallart (2008), “TheDisc-Halo Structure of NGC3109.”Astronomical Journal, 136, 2332–2342.[124] Hirschauer, Alec S., Laurin Gray, Margaret Meixner, et al. (2020), “DustyStellar Birth and Death in the Metal-poor Galaxy NGC 6822.” AstrophysicalJournal, 892, 91.[125] Hodgkin, S. T., M. J. Irwin, P. C. Hewett, and S. J. Warren (2009), “TheUKIRT wide field camera ZYJHK photometric system: calibration from2MASS.” Monthly Notices of the Royal Astronomical Society, 394, 675–692.[126] Høg, E., C. Fabricius, V. V. Makarov, et al. (2000), “The Tycho-2 catalogueof the 2.5million brightest stars.”Astronomy&Astrophysics, 355, L27–L30.[127] Hosek, Jr., Matthew W., Rolf-Peter Kudritzki, Fabio Bresolin, et al. (2014),“Quantitative Spectroscopy of Blue Supergiants inMetal-poorDwarf GalaxyNGC 3109.” Astrophysical Journal, 785, 151.[128] Hoyt, Taylor J. (2021), “On Zero Point Calibration of the Red Giant BranchTip in the Magellanic Clouds.” arXiv e-prints, arXiv:2106.13337.[129] Hoyt, Taylor J., Rachael L. Beaton, Wendy L. Freedman, et al. (2021), “TheCarnegie ChicagoHubble ProgramX: Tip of the RedGiant BranchDistancesto NGC 5643 and NGC 1404.” Astrophysical Journal, 915, 34.[130] Hoyt, Taylor J., Wendy L. Freedman, Barry F. Madore, et al. (2018), “TheNear-infrared Tip of the Red Giant Branch. II. An Absolute Calibration inthe Large Magellanic Cloud.” Astrophysical Journal, 858, 12.181[131] Huang, Caroline D., Adam G. Riess, Wenlong Yuan, et al. (2020), “HubbleSpace Telescope Observations of Mira Variables in the SN Ia Host NGC1559: An Alternative Candle to Measure the Hubble Constant.” Astrophysi-cal Journal, 889, 5.[132] Hubble, E. (1929), “ARelation betweenDistance and Radial Velocity amongExtra-Galactic Nebulae.” Proceedings of the National Academy of Science,15, 168–173.[133] Inno, L., G. Bono, N. Matsunaga, et al. (2016), “The Panchromatic Viewof the Magellanic Clouds from Classical Cepheids. I. Distance, Reddening,and Geometry of the Large Magellanic Cloud Disk.” Astrophysical Journal,832, 176.[134] Ishida, K. (1960), “Statistical Study of Carbon Stars.” Publications of theAstronomical Society of Japan, 12, 214.[135] Jacobs, Bradley A., Luca Rizzi, R. Brent Tully, et al. (2009), “The Ex-tragalactic Distance Database: Color-Magnitude Diagrams.” AstronomicalJournal, 138, 332–337.[136] James, Dizna, Smitha Subramanian, Abinaya O. Omkumar, et al. (2021),“Presence of red giant population in the foreground stellar substructure ofthe Small Magellanic Cloud.” Monthly Notices of the Royal AstronomicalSociety, 508, 5854–5863.[137] Kapakos, E. and D. Hatzidimitriou (2012), “RR Lyrae variables in the SmallMagellanic Cloud - II. The extended area: chemical and structural analysis.”Monthly Notices of the Royal Astronomical Society, 426, 2063–2077.[138] Kapakos, E., D.Hatzidimitriou, and I. Soszyński (2011), “RRLyrae variablesin the Small Magellanic Cloud - I. The central region.” Monthly Notices ofthe Royal Astronomical Society, 415, 1366–1380.[139] Karakas, A. I., J. C. Lattanzio, and O. R. Pols (2002), “Parameterising theThird Dredge-up in Asymptotic Giant Branch Stars.” Publications of theAstronomical Society of Australia, 19, 515–526.[140] Kippenhahn, Rudolf, AlfredWeigert, andAchimWeiss (2012), Stellar Struc-ture and Evolution. Springer, Berlin, Heidelberg.[141] Kirby, Evan N., Judith G. Cohen, Puragra Guhathakurta, et al. (2013), “TheUniversal Stellar Mass-Stellar Metallicity Relation for Dwarf Galaxies.”Astrophysical Journal, 779, 102.182[142] Kontizas, E., A. Dapergolas, D. H. Morgan, and M. Kontizas (2001), “ACatalogue of carbon stars in the LMC.” Astronomy & Astrophysics, 369,932–938.[143] Kowal, C. T. (1968), “Absolute magnitudes of supernovae.” AstronomicalJournal, 73, 1021–1024.[144] Leaman, Ryan, Kim A. Venn, Alyson M. Brooks, et al. (2013), “The Com-parative Chemical Evolution of an Isolated Dwarf Galaxy: A VLT and KeckSpectroscopic Survey of WLM.” Astrophysical Journal, 767, 131.[145] Leaman, Ryan, KimA.Venn, AlysonM.Brooks, et al. (2012), “TheResolvedStructure and Dynamics of an Isolated Dwarf Galaxy: A VLT and KeckSpectroscopic Survey of WLM.” Astrophysical Journal, 750, 33.[146] Lee, Abigail J., Wendy L. Freedman, Barry F. Madore, et al. (2021), “TheAstrophysical Distance Scale. III. Distance to the Local Group GalaxyWLMUsing Multiwavelength Observations of the Tip of the Red Giant Branch,Cepheids, and JAGB Stars.” Astrophysical Journal, 907, 112.[147] Lee, Abigail J., Wendy L. Freedman, Barry F. Madore, et al. (2021), “A Pre-liminary Calibration of the JAGBMethod Using Gaia EDR3.” AstrophysicalJournal, 923, 157.[148] Lee, Myung Gyoon, Wendy L. Freedman, and Barry F. Madore (1993), “TheTip of the Red Giant Branch as a Distance Indicator for Resolved Galaxies.”Astrophysical Journal, 417, 553.[149] Li, Yin-Bi, A. Li Luo, Chang-De Du, et al. (2018), “Carbon Stars Identi-fied from LAMOST DR4 Using Machine Learning.” Astrophysical JournalSupplement, 234, 31.[150] Lindegren, L., U. Bastian, M. Biermann, et al. (2021), “Gaia Early DataRelease 3. Parallax bias versus magnitude, colour, and position.” Astronomy& Astrophysics, 649, A4.[151] Lindegren, L., S. A. Klioner, J. Hernández, et al. (2021), “Gaia Early DataRelease 3. The astrometric solution.” Astronomy & Astrophysics, 649, A2.[152] Lombardi, G., E. Mason, C. Lidman, et al. (2011), “A study of NIR atmo-spheric properties at Paranal Observatory.” Astronomy & Astrophysics, 528,A43.183[153] Luri, X., A. G. A. Brown, L. M. Sarro, et al. (2018), “Gaia Data Release 2.Using Gaia parallaxes.” Astronomy & Astrophysics, 616, A9.[154] Madore, Barry F. and Wendy L. Freedman (2020), “Astrophysical DistanceScale: The AGB J-band Method. I. Calibration and a First Application.”Astrophysical Journal, 899, 66.[155] Madore, Barry F., Wendy L. Freedman, Dylan Hatt, et al. (2018), “TheNear-infrared Tip of the Red Giant Branch. I. A Calibration in the IsolatedDwarf Galaxy IC 1613.” Astrophysical Journal, 858, 11.[156] Madore, Barry F., Wendy L. Freedman, and Abigail J. Lee (2022), “Astro-physical Distance Scale. IV. Preliminary Zero-point Calibration of the JAGBMethod in the HST/WFC3-IR Broad J-band (F110W) Filter.” AstrophysicalJournal, 926, 153.[157] Madore, Barry F., Violet Mager, andWendy L. Freedman (2009), “Sharpen-ing the Tip of the Red Giant Branch.” Astrophysical Journal, 690, 389–393.[158] Madore, Barry F., Jane Rigby, Wendy L. Freedman, et al. (2009), “TheCepheid Period-Luminosity Relation (The Leavitt Law) at Mid-InfraredWavelengths. III. Cepheids in NGC 6822.” Astrophysical Journal, 693, 936–939.[159] Maíz Apellániz, J., M. Pantaleoni González, and R. H. Barbá (2021), “Vali-dation of the accuracy and precision of Gaia EDR3 parallaxes with globularclusters.” Astronomy & Astrophysics, 649, A13.[160] Makarov, Dmitry, Lidia Makarova, Luca Rizzi, et al. (2006), “Tip of theRed Giant Branch Distances. I. Optimization of a Maximum LikelihoodAlgorithm.” Astronomical Journal, 132, 2729–2742.[161] Marigo, P. and L.Girardi (2007), “Evolution of asymptotic giant branch stars.I. Updated synthetic TP-AGBmodels and their basic calibration.” Astronomy& Astrophysics, 469, 239–263.[162] Marigo, Paola, Léo Girardi, and Alessandro Bressan (1999), “The thirddredge-up and the carbon star luminosity functions in the MagellanicClouds.” Astronomy & Astrophysics, 344, 123–142.[163] Marigo, Paola, Léo Girardi, Alessandro Bressan, et al. (2017), “A NewGeneration of PARSEC-COLIBRI Stellar Isochrones Including the TP-AGBPhase.” Astrophysical Journal, 835, 77.184[164] Marrese, P.M., S.Marinoni,M. Fabrizio, andG.Altavilla (2019), “GaiaDataRelease 2. Cross-match with external catalogues: algorithms and results.”Astronomy & Astrophysics, 621, A144.[165] Marrese, P. M., S. Marinoni, M. Fabrizio, and G. Giuffrida (2017), “GaiaData Release 1. Cross-match with external catalogues. Algorithm and re-sults.” Astronomy & Astrophysics, 607, A105.[166] Mazzi, Alessandro, Léo Girardi, Simone Zaggia, et al. (2021), “The VMCsurvey - XLIII. The spatially resolved star formation history across the LargeMagellanic Cloud.”Monthly Notices of the Royal Astronomical Society, 508,245–266.[167] McCall, M. L., O. Vaduvescu, F. Pozo Nunez, et al. (2012), “Fundamentalsof the dwarf fundamental plane.” Astronomy & Astrophysics, 540, A49.[168] McQuinn, Kristen. B. W., Martha Boyer, Evan D. Skillman, and Andrew E.Dolphin (2019), “Using the Tip of the Red Giant Branch As a DistanceIndicator in the Near Infrared.” Astrophysical Journal, 880, 63.[169] McQuinn, Kristen B.W., Martha L. Boyer, Mallory B.Mitchell, et al. (2017),“DUSTiNGS. III. Distribution of Intermediate-age and Old Stellar Popula-tions in Disks and Outer Extremities of Dwarf Galaxies.” AstrophysicalJournal, 834, 78.[170] McQuinn, Kristen. B. W., Evan D. Skillman, Andrew E. Dolphin, et al.(2016), “The Distance to M51.” Astrophysical Journal, 826, 21.[171] Méndez, Bryan, Marc Davis, John Moustakas, et al. (2002), “Deviationsfrom the Local Hubble Flow. I. The Tip of the Red Giant Branch as aDistance Indicator.” Astronomical Journal, 124, 213–233.[172] Menzies, JohnW., Patricia A.Whitelock, andMichaelW. Feast (2015), “TheLocal Group Galaxy IC 1613 and its asymptotic giant branch variables.”Monthly Notices of the Royal Astronomical Society, 452, 910–923.[173] Meschin, I., C. Gallart, A. Aparicio, et al. (2014), “Spatially resolved LMCstar formation history - I. Outside in evolution of the outer LMC disc.”Monthly Notices of the Royal Astronomical Society, 438, 1067–1080.[174] Mondal, Chayan, Annapurni Subramaniam, and Koshy George (2018),“UVIT Imaging of WLM: Demographics of Star-forming Regions in theNearby Dwarf Irregular Galaxy.” Astronomical Journal, 156, 109.185[175] Muraveva, T., S. Subramanian, G. Clementini, et al. (2018), “The VMCsurvey - XXVI. Structure of the Small Magellanic Cloud from RR Lyraestars.”Monthly Notices of the Royal Astronomical Society, 473, 3131–3146.[176] Nagarajan, Pranav, Daniel R. Weisz, and Kareem El-Badry (2021), “RRLyrae-based Distances for 39 Nearby Dwarf Galaxies Calibrated to GaiaeDR3.” arXiv e-prints, arXiv:2111.06899.[177] Nataf, David M., Santi Cassisi, Luca Casagrande, et al. (2021), “On theColor-Metallicity Relation of the Red Clump and the Reddening toward theMagellanic Clouds.” Astrophysical Journal, 910, 121.[178] Newey,W. andD.McFadden (1994), “Large Sample Estimation andHypoth-esis Testing.” In Handbook of Econommetrics (R. Engle and D. McFadden,eds.), volume 4 of Handbooks in economics.[179] Nidever, David L., Antonela Monachesi, Eric F. Bell, et al. (2013), “ATidally Stripped Stellar Component of theMagellanic Bridge.”AstrophysicalJournal, 779, 145.[180] Nikolaev, S., A. J. Drake, S. C. Keller, et al. (2004), “Geometry of the LargeMagellanic Cloud Disk: Results fromMACHO and the TwoMicron All SkySurvey.” Astrophysical Journal, 601, 260–276.[181] Nikolaev, Sergei and Martin D. Weinberg (2000), “Stellar Populations inthe Large Magellanic Cloud from 2MASS.” Astrophysical Journal, 542,804–818.[182] Olson, B. I. and H. B. Richer (1975), “The absolute magnitudes of carbonstars: carbon stars in binary systems.” Astrophysical Journal, 200, 88–94.[183] Olszewski, EdwardW., Nicholas B. Suntzeff, andMario Mateo (1996), “Oldand Intermediate-Age Stellar Populations in theMagellanic Clouds.” AnnualReview of Astronomy and Astrophysics, 34, 511–550.[184] Omkumar, Abinaya O., Smitha Subramanian, Florian Niederhofer, et al.(2021), “Gaia view of a stellar sub-structure in front of the Small MagellanicCloud.” Monthly Notices of the Royal Astronomical Society, 500, 2757–2776.[185] Pal, Tathagata and G. Worthey (2021), “The frequency by mass of Galacticcarbon stars inferred from Gaia measurements of star cluster members.”Monthly Notices of the Royal Astronomical Society, 506, 3669–3677.186[186] Parada, Javiera, Jeremy Heyl, Harvey Richer, et al. (2021), “Carbon starsas standard candles - II. The median J magnitude as a distance indicator.”Monthly Notices of the Royal Astronomical Society, 501, 933–947.[187] Pastorelli, Giada, Paola Marigo, Léo Girardi, et al. (2020), “Constrainingthe thermally pulsing asymptotic giant branch phase with resolved stellarpopulations in the Large Magellanic Cloud.” Monthly Notices of the RoyalAstronomical Society, 498, 3283–3301.[188] Pastorelli, Giada, Paola Marigo, Léo Girardi, et al. (2019), “Constrainingthe thermally pulsing asymptotic giant branch phase with resolved stellarpopulations in the Small Magellanic Cloud.” Monthly Notices of the RoyalAstronomical Society, 485, 5666–5692.[189] Persson, S. E., J. A. Frogel, J. G. Cohen, et al. (1980), “The spread inCO absorption and effective temperature among the giants in OME Cen.”Astrophysical Journal, 235, 452–469.[190] Persson, S. E., D. C. Murphy, W. Krzeminski, et al. (1998), “A New Systemof Faint Near-Infrared Standard Stars.” Astronomical Journal, 116, 2475–2488.[191] Pietrinferni, Adriano, Santi Cassisi, Maurizio Salaris, and Fiorella Castelli(2004), “A Large Stellar Evolution Database for Population Synthesis Stud-ies. I. Scaled Solar Models and Isochrones.” Astrophysical Journal, 612,168–190.[192] Pietrzyński, G., D. Graczyk, A. Gallenne, et al. (2019), “A distance tothe Large Magellanic Cloud that is precise to one per cent.” Nature, 567,200–203.[193] Pietrzyński, Grzegorz, Wolfgang Gieren, Andrzej Udalski, et al. (2007),“The Araucaria Project: The Distance to the Local Group Galaxy WLMfrom Cepheid Variables Discovered in a Wide-Field Imaging Survey.” As-tronomical Journal, 134, 594–603.[194] Pietrzyński, Grzegorz, Wolfgang Gieren, Andrzej Udalski, et al. (2006),“The Araucaria Project: A Wide-Field Photometric Survey for CepheidVariables in NGC 3109.” Astrophysical Journal, 648, 366–374.[195] Planck Collaboration, N. Aghanim, Y. Akrami, et al. (2020), “Planck 2018results. VI. Cosmological parameters.” Astronomy & Astrophysics, 641, A6.187[196] Pritchet, Christopher J., Harvey B. Richer, David Schade, et al. (1987), “TheLate-Type Stellar Content of NGC 55.” Astrophysical Journal, 323, 79.[197] Puget, Pascal, Eric Stadler, Rene Doyon, et al. (2004), “WIRCam: theinfrared wide-field camera for the Canada-France-Hawaii Telescope.” InGround-based Instrumentation for Astronomy (Alan F. M. Moorwood andMasanori Iye, eds.), volume 5492 of Society of Photo-Optical Instrumenta-tion Engineers (SPIE) Conference Series, 978–987.[198] Raimondo, G., M. R. L. Cioni, M. Rejkuba, and D. R. Silva (2005), “Pul-sation properties of C stars in the Small Magellanic Cloud.” Astronomy &Astrophysics, 438, 521–532.[199] Reid, M. J., D. W. Pesce, and A. G. Riess (2019), “An Improved Distanceto NGC 4258 and Its Implications for the Hubble Constant.” AstrophysicalJournal Letters, 886, L27.[200] Ren, Yi, Biwei Jiang, Ming Yang, et al. (2021), “The Sample of Red Super-giants in 12 Low-mass Galaxies of the Local Group.” Astrophysical Journal,923, 232.[201] Rich, Jeffrey A., S. E. Persson, Wendy L. Freedman, et al. (2014), “A NewCepheid Distance Measurement and Method for NGC 6822.” AstrophysicalJournal, 794, 107.[202] Richer, H. B. (1981), “Observations of a complete sample of C stars in theLarge Magellanic Cloud.” Astrophysical Journal, 243, 744.[203] Richer, H. B., D. R. Crabtree, and C. J. Pritchet (1984), “Luminous late-typestars in NGC 205.” Astrophysical Journal, 287, 138–147.[204] Richer, H. B., N. Olander, and B. E. Westerlund (1979), “Carbon stars inthe Large Magellanic Cloud: photometry, spectroscopy, and evolutionaryimplications.” Astrophysical Journal, 230, 724–735.[205] Richer, H. B., C. J. Pritchet, and D. R. Crabtree (1985), “Luminous late-typestars in NGC 300.” Astrophysical Journal, 298, 240–248.[206] Richer, H. B. and B. E. Westerlund (1983), “Carbon stars in Local Groupgalaxies.” Astrophysical Journal, 264, 114–125.[207] Richer, Harvey B. (1972), “Carbon Stars with Composite Spectra.” Astro-physical Journal Letters, 172, L63.188[208] Riess, Adam G., Wenlong Yuan, Lucas M. Macri, et al. (2021), “A Com-prehensive Measurement of the Local Value of the Hubble Constant with1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ESTeam.” arXiv e-prints, arXiv:2112.04510.[209] Ripepi, Vincenzo, Maria-Rosa L. Cioni, Maria Ida Moretti, et al. (2017),“The VMC survey - XXV. The 3D structure of the Small Magellanic Cloudfrom Classical Cepheids.” Monthly Notices of the Royal Astronomical Soci-ety, 472, 808–827.[210] Ripoche, Paul, Jeremy Heyl, Javiera Parada, and Harvey Richer (2020),“Carbon stars as standard candles: I. The luminosity function of carbonstars in the Magellanic Clouds.”Monthly Notices of the Royal AstronomicalSociety, 495, 2858–2866.[211] Rizzi, Luca, R. Brent Tully, Dmitry Makarov, et al. (2007), “Tip of the RedGiant Branch Distances. II. Zero-Point Calibration.” Astrophysical Journal,661, 815–829.[212] Rousseeuw, Peter J. and Christophe Croux (1993), “Alternatives to theMedian Absolute Deviation.” Journal of the American Statistical Associ-ation, 88, 1273–1283, URL https://www.tandfonline.com/doi/abs/10.1080/01621459.1993.10476408.[213] Rubele, Stefano, Giada Pastorelli, Léo Girardi, et al. (2018), “The VMCsurvey - XXXI: The spatially resolved star formation history of themain bodyof the Small Magellanic Cloud.”Monthly Notices of the Royal AstronomicalSociety, 478, 5017–5036.[214] Sakai, Shoko, Barry F. Madore, and Wendy L. Freedman (1996), “Tip of theRed Giant Branch Distances to Galaxies. III. The Dwarf Galaxy Sextans A.”Astrophysical Journal, 461, 713.[215] Salaris, Maurizio and Léo Girardi (2005), “Tip of the Red Giant Branchdistances to galaxies with composite stellar populations.” Monthly Noticesof the Royal Astronomical Society, 357, 669–678.[216] Sandage, Allan (1971), “The Distance of the Local-Group Galaxy IC 1613Obtained from Baade’s Work on its Stellar Content.” Astrophysical Journal,166, 13.[217] Sanford, Roscoe F. (1944), “Radial Velocities of 283 Stars of Spectral ClassesR and N.” Astrophysical Journal, 99, 145.189[218] Schlafly, Edward F. and Douglas P. Finkbeiner (2011), “Measuring Redden-ing with Sloan Digital Sky Survey Stellar Spectra and Recalibrating SFD.”Astrophysical Journal, 737, 103.[219] Schlegel, David J., Douglas P. Finkbeiner, and Marc Davis (1998), “Maps ofDust Infrared Emission for Use in Estimation of Reddening and Cosmic Mi-crowave Background Radiation Foregrounds.” Astrophysical Journal, 500,525–553.[220] Scowcroft, Victoria, Wendy L. Freedman, Barry F. Madore, et al. (2013),“The Carnegie Hubble Program: The Infrared Leavitt Law in IC 1613.”Astrophysical Journal, 773, 106.[221] Scowcroft, Victoria, Wendy L. Freedman, Barry F. Madore, et al. (2016),“The Carnegie Hubble Program: The Distance and Structure of the SMCas Revealed by Mid-infrared Observations of Cepheids.” Astrophys. J., 816,49.[222] Serenelli, A., A. Weiss, S. Cassisi, et al. (2017), “The brightness of thered giant branch tip. Theoretical framework, a set of reference models, andpredicted observables.” Astronomy & Astrophysics, 606, A33.[223] Sibbons, L. F., S. G. Ryan, M. Irwin, and R. Napiwotzki (2015), “The AGBpopulation in IC 1613 using JHK photometry.” Astronomy & Astrophysics,573, A84.[224] Skillman, EvanD., Sebastian L.Hidalgo, Daniel R.Weisz, et al. (2014), “TheACS LCID Project. X. The Star Formation History of IC 1613: Revisitingthe Over-cooling Problem.” Astrophysical Journal, 786, 44.[225] Skillman, Evan D., Eline Tolstoy, AndrewA. Cole, et al. (2003), “Deep Hub-ble Space Telescope Imaging of IC 1613. II. The Star Formation History.”Astrophysical Journal, 596, 253–272.[226] Skowron, D. M., J. Skowron, A. Udalski, et al. (2021), “OGLE-ing the Mag-ellanic System: Optical Reddening Maps of the Large and Small MagellanicClouds from Red Clump Stars.” Astrophysical Journal Supplement, 252, 23.[227] Skrutskie, M. F., R. M. Cutri, R. Stiening, et al. (2006), “The Two MicronAll Sky Survey (2MASS).” Astronomical Journal, 131, 1163–1183.[228] Soszyński, I., W. Gieren, G. Pietrzyński, et al. (2006), “The AraucariaProject: Distance to the Local Group Galaxy NGC 3109 from Near-InfraredPhotometry of Cepheids.” Astrophysical Journal, 648, 375–382.190[229] Space Telescope Science Institute, (STScI) (2017), “NIRCam Filters - JWSTUser Documentation.” https://jwst-docs.stsci.edu/near-infrared-camera/nircam-instrumentation/nircam-filters. [Published: 2017 November 29].[230] Stephenson, C. B. (1973), “A general catalogue of cool carbon stars.” Pub-lications of the Warner & Swasey Observatory.[231] Stephenson, C. B. (1989), “A general catalogue of cool carbon stars.” Pub-lications of the Warner & Swasey Observatory, 3, 53.[232] Stetson, P. (1994), “The Center of the Core-Cusp Globular Cluster M15:CFHT and HST Observations, ALLFRAME Reductions.” Publications ofthe Astronomical Society of the Pacific, 106, 250–280.[233] Stetson, P. B. (1987), “DAOPHOT - A computer program for crowded-fieldstellar photometry.” Publications of the Astronomical Society of the Pacific,99, 191–222.[234] Subramanian, S. and A. Subramaniam (2013), “Structure of the Large Mag-ellanic Cloud from near infrared magnitudes of red clump stars.” Astronomy& Astrophysics, 552, A144.[235] Subramanian, Smitha, Stefano Rubele, Ning-Chen Sun, et al. (2017), “TheVMC Survey - XXIV. Signatures of tidally stripped stellar populations fromthe inner Small Magellanic Cloud.”Monthly Notices of the Royal Astronom-ical Society, 467, 2980–2995.[236] Subramanian, Smitha and Annapurni Subramaniam (2012), “The Three-dimensional Structure of the Small Magellanic Cloud.” Astrophysical Jour-nal, 744, 128.[237] Suh, Kyung-Won (2021), “A New Catalog of Asymptotic Giant Branch Starsin Our Galaxy.” Astrophysical Journal Supplement, 256, 43.[238] Suh, Kyung-Won and Jinju Hong (2017), “A New Catalog of AGB StarsBased on Infrared Two-Color Diagrams.” Journal of Korean AstronomicalSociety, 50, 131–138.[239] Swan, J., A. A. Cole, E. Tolstoy, and M. J. Irwin (2016), “Ca II tripletspectroscopy of RGB stars in NGC 6822: kinematics and metallicities.”Monthly Notices of the Royal Astronomical Society, 456, 4315–4327.[240] Tammann, G. A., B. Reindl, andA. Sandage (2011), “New period-luminosityand period-color relations of classical Cepheids. IV. The low-metallicity191galaxies IC 1613,WLM, Pegasus, SextansA andB, and LeoA in comparisonto SMC.” Astronomy & Astrophysics, 531, A134.[241] Tatton, B. L., J. Th van Loon, M. R. L. Cioni, et al. (2021), “The VMCSurvey - XL. Three-dimensional structure of the Small Magellanic Cloud asderived from red clump stars.” Monthly Notices of the Royal AstronomicalSociety, 504, 2983–2997.[242] Urbaneja, Miguel A., Rolf-Peter Kudritzki, Fabio Bresolin, et al. (2008),“The Araucaria Project: The Local Group Galaxy WLM—Distance andMetallicity from Quantitative Spectroscopy of Blue Supergiants.” Astro-physical Journal, 684, 118–135.[243] Valenti, E., F. R. Ferraro, and L. Origlia (2004), “Red giant branch in near-infrared colour-magnitude diagrams - II. The luminosity of the bump andthe tip.”Monthly Notices of the Royal Astronomical Society, 354, 815–820.[244] van der Marel, Roeland P. and Maria-Rosa L. Cioni (2001), “MagellanicCloud Structure from Near-Infrared Surveys. I. The Viewing Angles of theLarge Magellanic Cloud.” Astronomical Journal, 122, 1807–1826.[245] van Loon, J. Th., J. R. Marshall, M. Cohen, et al. (2006), “Very LargeTelescope three micron spectra of dust-enshrouded red giants in the LargeMagellanic Cloud.” Astronomy & Astrophysics, 447, 971–989.[246] Venn, K. A., D. J. Lennon, A. Kaufer, et al. (2001), “First Stellar Abun-dances in NGC 6822 from VLT-UVES and Keck-HIRES Spectroscopy.”Astrophysical Journal, 547, 765–776.[247] Weinberg, Martin D. and Sergei Nikolaev (2001), “Structure of the LargeMagellanic Cloud from 2MASS.” Astrophysical Journal, 548, 712–726.[248] Weisz, Daniel R., Andrew E. Dolphin, Evan D. Skillman, et al. (2013),“Comparing the ancient star formation histories of the Magellanic Clouds.”Monthly Notices of the Royal Astronomical Society, 431, 364–371.[249] Wenger, M., F. Ochsenbein, D. Egret, et al. (2000), “The SIMBAD astro-nomical database. The CDS reference database for astronomical objects.”Astronomy & Astrophysics Supplement, 143, 9–22.[250] Whitelock, Patricia A., Michael W. Feast, Freddy Marang, and M. A. T.Groenewegen (2006), “Near-infrared photometry of carbon stars.” MonthlyNotices of the Royal Astronomical Society, 369, 751–782.192[251] Whitelock, Patricia A., John W. Menzies, Michael W. Feast, et al. (2013),“The Local Group galaxy NGC 6822 and its asymptotic giant branch stars.”Monthly Notices of the Royal Astronomical Society, 428, 2216–2231.[252] Wyder, Ted K. (2001), “The Star Formation History of NGC 6822.” Astro-nomical Journal, 122, 2490–2523.[253] Zgirski, Bartłomiej, Grzegorz Pietrzyński, Wolfgang Gieren, et al. (2021),“The Araucaria Project. Distances to Nine Galaxies Based on a StatisticalAnalysis of their Carbon Stars (JAGBMethod).” Astrophysical Journal, 916,19.[254] Zijlstra, Albert A., Cecile Loup, L. B. F. M. Waters, et al. (1996), “Obscuredasymptotic giant branch stars in the Magellanic Clouds - II. Near-infraredand mid-infrared counterparts.” Monthly Notices of the Royal AstronomicalSociety, 279, 32–62.193
Thesis/Dissertation
eng
Astronomy
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Richer, Harvey Brian, 1944-
Heyl, Jeremy
Red giant stars as standard candles
Text