Calibration of Extragalactic Distances on Differentmetallicity Environments

Calibration of extragalactic distances on diﬀerent metallicity environments

Juanita Cardona Rocha

A thesis submitted in partial fulﬁllment for the degree of Bachelor of Physics

Director: Ph.D. Alejandro Garc´ıa

Universidad de los Andes Science Faculty, Physics Department Bogot´a,Colombia July 27, 2020 For my grandmothers Clara and Beatriz, who never doubted of me

And for William, for being my rock. Acknowledgments

First, I would like to thank my director, Alejandro Garc´ıa,for his help and patience with this thesis. I would have never been able to ﬁnish this work alone without your useful tips and advice. I would also like to thank my parents and my brother, for all the support and for always encouraging me during this year. You are my anchor, my biggest support and everything I am I owe it to you. Thank you for your patience and unconditional love. I would like to thank the friends I’ve made in these 4 years, especially Ana Mar´ıa,Nathalia, Laura, Juan David, Santiago & Camilo. The journey of studying Physics has been full of laughter, all-nighters and support thanks to all of you. I would also like to thank my choir friends, especially the ones in “Combo Cedritos” along with Santi, Anita and Natis. You’ve seen me change majors and supported me through thick and thin and for that I will always carry you in my heart. I thank Willy wholeheartedly for being my best friend, lifting me up and never letting me give up. You’ve made my life brighter and happier and I am never going to be able to thank you enough for everything you’ve given me.

Last but not least, even if they can’t read this, I would like to thank my cats for cheering me up whenever I am down and for their constant company. They kept me sane while writing this thesis. Abstract

In this work the distances to NGC 247, M33, IC 1613 and NGC 55 were found using Henrietta Leavitt’s Period-Luminosity Relation (PLR) in the VIJK filters in order to see if the galaxies’ metallicity affected their distance calculus. This has been a long debated topic and the results will determine if the PLR is universal, which is relevant for the Cepheid extragalactic distance scale. The relations were calculated using an MM -estimator, since this kind of robust regression deals with the heteroscedasticity problems present in astronomical data and allows the inclu- sion of influential points. The PLRs were also calculated using the slope of the Large Magellanic Cloud’s PLR, since it is considered the zero point of this relation. Two distance moduli were used for the LMC: (18.5 ± 0.1) mag found by Freedman et al. (2001) and (18.477 ± 0.004) mag calculated by Pietrzyńskiet al. (2019). Hence, for each galaxy two distances and an excess color were found. Comparing these results with the ones reported by the Araucaria project for the same galaxies, it was concluded that the excess colors found in this thesis are evidence of the non universality of the PLR, since the values with more discrepancy correspond to the galaxies with a significantly different metallicity than the LMC. This shows that the metallicity affects the slope of the PLR and that this distance calculus method only works cor- rectly for galaxies with a similar metallicity to the LMC.

Keywords: Period-Luminosity Relation - Cepheids - distance scale - Large Magellanic Cloud - metallicity Resumen

En este trabajo las distancias a NGC 247, M33, IC 1613 y NGC 55 fueron en- contradas usando la RelaciónPeriodo-Luminosidad (RPL) de Henrietta Leavitt en los filtros VIJK para ver si la metalicidad de las galaxias afecta el cálculode distancia. Este ha sido un tema muy debatido y los resultados van a determinar si la RPL es universal, lo cual es importante para la escala extragalactica de Cefei- das. Las relaciones fueron calculadas usando un estimador MM, ya que este tipo de regresiónrobusta lidia con los problemas de heterocedasticidad presentes en datos astronómicosy permite la inclusiónde puntos influyentes. Las RPL tambiénfueron calculadas usando la pendiente de la RPL de la Nube Mayor de Magallanes (LMC), debido a que es considerada el punto cero de esta relación.Dos módulosde distancias fueron utilizadas para la LMC: (18.5±0.1) mag, encontrada por Freedman et al. (2001), y (18.477 ± 0.004) mag calculado por Pietrzyńskiet al. (2019). Por lo tanto, dos distancias y un exceso de color fueron encontrados para cada galaxia. Al com- parar estos resultados con los reportados por el proyecto Araucaria para las mismas galaxias, se concluyóque los excesos de color encontrados en esta tesis son evidencia de la no universalidad de la RPL, debido a que los valores con mayor discrepancia corresponden a las galaxias con una metalicidad significativamente distinta a la de la LMC. Esto muestra que la metalicidad afecta la pendiente de la RPL y que este método de cálculode distancia solo funciona correctamente para galaxias con una metalicidad similar a la LMC.

Palabras clave: Relaci´onPeriodo-Luminosidad - Cefeidas - escala de distancias - Nube Mayor de Magallanes - metalicidad Contents

Chapter 1

The story behind variable stars and the facts of some distant galax- ies1 1.1 NGC 247...... 4 1.2 M33...... 6 1.3 IC 1613...... 7 1.4 NGC 55...... 9 1.5 Data obtention...... 10

Chapter 2

Theoretical framework 13 2.1 Photometric concepts...... 13 2.2 Extinction laws...... 16 2.3 Variable stars...... 17 2.4 Mechanism of stellar pulsation...... 21 2.4.1 Period-density relation for radial pulsations...... 21 2.4.2 Radial modes of pulsation...... 23 2.4.3 Eddington’s Valve Mechanism...... 24 2.4.4 The κ and γ Mechanisms...... 26 2.5 Period-Luminosity Relations...... 27 2.6 Distance determination...... 29 2.7 Robust regression...... 32

Chapter 3

Results and discussion 36 3.1 Large Magellanic Cloud...... 36 3.2 NGC 247...... 36 3.3 M33...... 41 3.4 IC 1613...... 46 3.5 NGC 55...... 50 3.6 Discussion...... 56 Chapter 4

Conclusions and future work 62

Appendix 1 64

Appendix 2 65

Appendix 3 66

Appendix 4 67

Appendix 5 68

Appendix 6 69

Appendix 7 70

Appendix 8 71

References 72

7 Chapter 1

The story behind variable stars and the facts of some distant galaxies.

Stars have been captivating mankind’s imagination since the beginning of time. They have been the source of myths, the guide of sailors and the basis for the creation of calendars. For centuries, those who watched the night sky thought that the stars’ brightness remained constant. So when David Fabricius, a German Lutheran pastor and astronomer, noticed in 1595 that the brightness of the o Ceti star changed with time, so he renamed it to Mira which means wonderful. Fabricius noticed that its luminosity faded slowly and eventually returned to its original brilliance in an 11 month period. This astonishing discovery was ﬁrst attributed to dark blotches on the star’s surface, so when this spot on the rotating star faced Earth it seemed that the star had changed its brightness (Carroll & Ostlie, 2007). In reality, Fabricius had discovered the ﬁrst pulsating star.

Almost two centuries passed before another pulsating star was discovered. John Goodricke discovered that the brightness of the star δ Cephei varied with a period of 5 days, 8 hours and 48 minutes. This revelation costed Goodricke’s short life, since he developed pneumonia while discovering this star through the night and died afterwards. Nevertheless, this English astronomer found the ﬁrst Cepheid, a type of pulsating star (Carroll & Ostlie, 2007), and this discovery changed astronomy forever.

Cepheids became highly important for astronomers, thanks to the discoveries made by Henrietta Swan Leavitt while she was working as a human computer for Edward Charles Pickering at Harvard College Observatory at the beginning of the 20th century (Carroll & Ostlie, 2007). When Leavitt was searching for variable stars in the Small Magellanic Cloud (SMC), she noticed that the brighter Cepheids had a longer pulsation period, as seen in Figure A1a of Appendix 1. When she plotted these stars’ apparent magnitude as a function of their period’s logarithm, a linear relationship was found as can be seen in Figure A1b of Appendix 1. Pickering and Leavitt proposed that the deviations from the straight line in Figure A1b could be corrected by using an absolute scale of magnitudes (Leavitt & Pickering, 1912). Currently it is known that the absolute magnitude of a Cepheid can be obtained from the period- luminosity relation (PLR), which in turn can be used to calculate the star’s distance.

1 Hence, knowing the absolute magnitude of the galaxy, with its PLR, its distance can be calculated. This has been a tool used for astronomers for decades and will be used in this project. Here the distances to the galaxies NGC 247, M33, IC 1613 and NGC 55 will be calculated using the PLR of each galaxy’s Cepheids, which will be found using robust regressions. This type of regression was chosen to deal with the heteroscedasticity problems related to this kind of astronomical data and to minimize the influence of outliers. The MM -estimator is the most adequate of the robust regression estimators, since it combines the M -estimator’s high asymptotic relative efficiency along with the high breakdown of the S-estimator (Montgomery et al., 2012). This means that the MM -estimator can describe efficiently large samples of data and is resistant to data contamination.

The PLR will be obtained ﬁrst for the Large Magellanic Cloud (LMC) and then for each galaxy in the VIJK ﬁlters, resulting in a distance modulus for each case. By combining these results along with the ratio of total to selective extinction obtained from Schlegel’s law (Schlegel et al. 1998), an impressively precise distance modulus to the galaxy can be found, which is denominated the true distance modulus (Gieren et al., 2009).

The galaxies chosen for this thesis were selected mainly because their photometric data are public, which does not happen every time. Also, the distances and metallicities between the galaxies vary signiﬁcantly, which can be seen in Figures1 and2, respectively. This is of great advantage since the discussion of the results will revolve around the eﬀect of the metallicity on the universality of the PLR.

2 α(º)

NGC 247

M33 LMC NGC 55 IC1613

Distance (Mpc)

Figure 1: Distances and coordinates to LMC, NGC 247, M33, IC 1613 and NGC 55 (Freedman et al. 2001; Gieren et al. 2009; Gieren et al. 2013; Pietrzy´nskyet al. 2006 and Gieren et al. 2008, repectively).

Figure 2: Metallicities of M33, IC 1613, SMC, NGC 55, NGC 247 and the LMC (Ibata et al. 2007; Pietrzy´nskyet al. 2006a; Choudhury et al. 2018; Pritchet et al. 1987; Kacharov et al. 2018 and Choudhury et al. 2016, respectively).

3 Now, a brief overview of the studied galaxies and their data obtention will be made. Then the theoretical framework will be presented followed by the results of the robust PLR for each galaxy and of its distance calculus. The results’ analysis will follow and ﬁnally the conclusions will be given.

1.1 NGC 247

Figure 3: NGC 247 galaxy as seen from the ESO’s WFI at La Silla Observatory in Chile.1 In this 33.770 × 21.000 ﬁeld of view the spiral form of the galaxy is clearly seen. Here the North is 90◦ right from the vertical.

NGC 247 is a spiral galaxy (Figure3) belonging to the Sculptor group, which is a “ﬁlament-shaped group of galaxies extending over ∼ 5 Mpc along the line of sight” (Patrick et al. 2017). Speciﬁcally, NGC 247 is located in the Cetus constellation, is a type SAB(s) galaxy (as reported in the NED/IPAC Extragalactic Database2) h m s ◦ 0 00 and has coordinates of α = 0 47 8. 55 and δ = −20 45 37. 4 (Cook et al. 2014), which means that this galaxy is near the South Galactic Pole and far away from the Galactic Plane (Garc´ıa-Varela et al. 2008). For this reason, the probability of contamination on the galaxy’s data by blue galactic stars is negligible (Rodr´ıguezet al. 2019).

1Image taken from https://www.eso.org/public/images/eso1107a/ 2Available at http://ned.ipac.caltech.edu/

4 8 NGC 247 has a HI disk of limited extension with a mass of (8.0 ± 0.3) × 10 M (Carignan & Puche, 1990), where the Hα emission happens mainly in the galaxy’s spiral arms (Rodr´ıguezet al. 2019). Hence, this galaxy has a mass-luminosity ratio 10 of MHI /LB = 0.34 and a total mass of 7.5 × 10 M (Burlak, 1996). It also has a −5 −3 −1 derived mean density of 1.6 × 10 stars pc , a star formation rate of 0.1M yr for the last 16 Myr (Rodr´ıguezet al. 2019) and a metallicity of [F e/H] = −0.5 dex (Kacharov et al. 2018). Because of the inclination of the galaxy (i = 74◦; Rodr´ıguez et al. 2019) a variable internal extinction is expected since the line of sight can pass through dust clouds several times.

The distance modulus of NGC 247 was calculated for the first time by Carignan (1985), who found the value of 27.01 mag by photographic plates. The most recent calculus of this distance modulus using Cepheids was made by Gieren et al. (2009) and Madore et al. (2009) The former combined the results found on the VI filters by Garc´ıa-Varela et al. (2008) along with observations in the JK filters of 23 Cepheids to determine a distance of (3.38 ± 0.06) Mpc. The latter used 9 Cepheids to calculate a distance modulus equal to (3.65 ± 0.08) Mpc for the VRI band. This distance has also been estimated using the tip of the red giant branch (TRGB) method by Monachesi et al. (2016), who found a value of (3.66±0.1) Mpc. This method consists on calculating the distance of a star when it reaches the Red Giant Branch phase and starts burning triple-α helium at T = 107 K. In this moment, a star can not maintain the equilibrium between energy production and expansion, resulting in a thermonuclear runaway. This ignition happens at MI ≈ 4 mag, which can be used as standard candle in order to find the distance to the galaxy (Rizzi et al., 2007).

5 1.2 M33

Figure 4: The Triangulum galaxy taken with the ESO’s VST at Paranal Observatory in Chile.3 This image shows a ﬁeld of view of 68.010 × 57.040 which captured the spiral form of M33 and its high brightness. Here North is located 90◦ left of the vertical.

The M33 galaxy, otherwise known as the Triangulum Galaxy, is a spiral galaxy of type SA(s)cd younger than the Milky Way since its gas fraction is higher and its stellar colors are bluer (Gratier et al. 2012), as seen in Figure4. It is part of the Local group, being its third brightest galaxy with an absolute magnitude of MV = −18.9 (Ibata et al. 2007). It is located in the Triangulum constellation and has coordi- h m s ◦ 0 00 nates of α = 1 33 50. 89 and δ = 30 39 36. 8 (Cook et al. 2014). This galaxy has a metallicity of [F e/H] = −1.5 dex, an inclination of i = 53.8◦. It also has an elliptical radius of s = 0.75◦ (Ibata et al. 2007), an interstellar reddening of E(B −V ) = 0.042 10 mag (McConnachie et al. 2004) and a mass around 5−25×10 M . M33 is gas-rich, 9 having a gas disc with a total HI mass of 1.4 × 10 M which stretches 22 kpc from its center of mass (Patel et al. 2017). The eﬀects of the galaxy’s orientation on

3Image taken from https://www.eso.org/public/images/eso1424a/

6 the distance modulus will be minimal due to the high inclination, making it a great candidate to study its distance (Scowcroft et al. 2009).

This galaxy’s distance modulus was calculated using near-infrared photometry for the ﬁrst time by Madore et al. (1985), who used 15 Cepheids and found a value of (24.5±0.2) mag. Using data from single-epoch I -band of 32 Cepheids, Gyoon-Lee et al. (2002) calculated two distance moduli to the galaxy: (24.52±0.13sys ±0.14ran) mag using the PLR and (24.52 ± 0.11sys ± 0.15ran) mag using the Wesenheit func- tions. For both of these distance moduli the ﬁrst error is the systematic one and the second is the random error. Kim et al. (2002) also calculated M33’s distance using the TRGB and Red Clump Stars, resulting in distance moduli of (24.81±0.04) mag and of (24.76 ± 0.05sys ± 0.04ran) mag, respectively. More recently, Gieren et al. (2013), who also used near-infrared photometry of 26 Cepheids, determined an extinction-corrected distance modulus of (24.62 ± 0.03) mag.

1.3 IC 1613

Figure 5: IC 1613 galaxy as seen by the OmegaCAM camera on ESO’s VLT Survey Telescope at Paranal Observatory in Chile.4 This clear image shows the irregular shape of the galaxy, as well as its small size in the 25.340 × 17.230 ﬁeld of view. In this case the North aligns with the vertical.

7 IC 1613 is an isolated irregular dwarf galaxy (see Figure5) of type IB(s)m discovered by Max Wolf in 1906 in the Cetus constellation. This galaxy is part of the Local group and has a distance modulus of (24.291 ± 0.035) mag, which was calculated by Pietrzyńskiet al. (2006a) using the data from 39 Cepheids in the JK filters. This distance was also found with 44 Cepheids by Bernard et al. (2010), who found a distance modulus of (24.50 ± 0.11) mag. These authors also used 90 RR Lyrae stars to find the distance to this galaxy, which resulted in a value of (24.39 ± 0.12) mag.

IC 1613 has been widely studied since it is located far from the Galactic plane, h m s ◦ 0 00 having the coordinates of α = 1 4 47. 8 and δ = 2 7 4 (Ross et al. 2015). For this reason, this galaxy has small interstellar reddening of 0.025 mag. It has an inclina- ◦ 8 tion of i = 38 , stellar mass of ≈ 2 × 10 M and a star formation rate in its center −3 −1 −2 of (1.6 ± 0.8) × 10 M yr kpc . This galaxy has several HI supershells, including 7 a HI hole of 1 kpc in diameter with a mass of 2.8 × 10 M . The total HI mass is of 7 6.0 × 10 M (Azim-Hashemi et al. 2019).

The metallicity of this galaxy is around [F e/H] = −1.0 dex, a value that is increasing because of the generations of stars that synthesize metals and return them to the galaxy’s interstellar medium. But thanks to the overall low metallicity, IC 1613 is a very important object of investigation since it can provide information about the eﬀects of metallicity in the PLR and on other stellar distance indicators (Pietrzy´nskiet al. 2006a).

4Image taken from https://www.eso.org/public/images/eso1603a/

8 1.4 NGC 55

Figure 6: Image of NGC 55 galaxy taken with the ESO’s WFI on the 2.2-metre MPG/ESO telescope at La Silla Observatory in Chile.5 This image is based on data obtained in the BV and Hα ﬁlters and shows the irregular shape of the galaxy and its inclination in the 34.880 × 31.800 ﬁeld of view. Here the North aligns with the vertical.

The highly inclined galaxy NGC 55 is a late-type galaxy in the Sculptor group of h m s ◦ 0 00 type SB(s)m (Figure6) and has coordinates of α = 00 14 53. 6 and δ = −39 11 47. 9 (Westmeier et al. 2013). Because of its inclination (i ≈ 80◦; Patrick et al. 2017) with respect to the line of sight, the reddening of the galaxy may be strong (Gieren et al. 2008). This galaxy has been compared to the LMC due to its barred spiral where the bar is seen along the line of sight (Patrick et al., 2017). NGC 55 has two large HII complexes, supergiant ﬁlaments and shells, all of which are responsible of the transportation of the ionizing radiation to the halo. This galaxy has a star formation

5Image taken from https://www.eso.org/public/images/eso0914a/

9 −1 rate of 0.22M yr (Patrick et al., 2017) and a metallicity of [F e/H] ≈ −0.6 dex (Pritchet et al. 1987).

NGC 55 has an abundant young stellar population which is a sign of blue supergiant and Cepheid stars within the galaxy, both objects being useful for distance determination. Pietrzyńskiet al. (2006b) found 100 blue supergiants and 143 Cepheids, the latter having a mean reddening of E(B − V ) = (0.127 ± 0.0102) mag. These authors found a distance modulus of (26.40 ± 0.09sys ± 0.05int) mag, where the first error corresponds to the systematic error and the second to the intrinsic one, using the data from Cepheids in the VI filters. Using these results and analyzing the data of these same Cepheids in the JK filters, Gieren et al. (2008) determined that the true distance based on the VIJK filters is (1.94 ± 0.03) Mpc. This distance was also found by the TRGB by Tanaka et al. (2011) in the VI bands, resulting in the distance of (2.1 ± 0.1) Mpc. So the distance to NGC 55 is ≈ 2 Mpc, meaning that its proximity allows the obtention of accurate photometry and spectroscopy (Gieren et al, 2008).

1.5 Data obtention The observational data used for this thesis were taken mainly from OGLE-II and the Araucaria project. The OGLE-II project was the second phase of the Optical Gravitational Lensing Experiment (OGLE) project. The main goal of OGLE is the search of dark matter using the microlensing phenomenon. The first phase started in 1992, with the first observations beginning in 1995 using the 1 m Swope telescope and a 2048 × 2048 Ford/Loral CCD camera at Las Campanas Observatory in Chile. The need of more observation time and better equipment led to the creation of the second phase of the project in 1997. This phase motivated the construction of the 1.3 m Warsaw telescope and the use of a 2048 × 2048 Scientific Imaging Technolo- gies CCD chip, which gave a scale of 0.417 arcsec/pixel at the focus of the telescope (Udalski et al. 1997). With this equipment, the optical data from both the Large Magellanic Cloud (as seen in Figure7) and IC 1613 was collected by Udalski (2000) and Udalski et al. (2001), respectively. The specific details of these observations can be found in Appendix 2.

10 Figure 7: OGLE-II Cepheids in the LMC. It can be seen the way the team took the data by moving the CCD chip along the galaxy (Udalski et al. 1999). The drift-scan technique was used to take images of large areas, as seen in the red rectangles.

At this moment, OGLE is in its fourth phase, using the Warsaw telescope with a 256 Megapixel mosaic camera composed by 32 CCD chips. Between 2010 to 2016, OGLE-IV made observations of the Magellanic Clouds in the I band in order to complete its variable star catalog. Although the team covered a total area of 670 square degrees encompassing the whole Magellanic System, they focused on variable stars in the outskirts of the clouds, since their centers had been surveyed by OGLE-II and OGLE-III already. This project led to the discovery of 115 Classical Cepheids, 994 RR Lyrae stars and 12 anomalous Cepheids that were not included in the previous OGLE Collection of Variable Stars (OCVS). The updated collection contains a total of 9649 Classical Cepheids (composed of 5229, 3568 and 117 stars pulsating in the Fundamental mode, First Overtone and Second Overtone, respectively), 262 anomalous Cepheids and 46433 RR Lyr variables in the Magellanic System. Since this is the third time that the system has been observed by the project, it can be said that the OGLE Collection of Classical Cepheids in the Magellanic System is prac- tically complete, thus concluding Henrietta Leavitt’s work (Soszy´nskiet al. 2017). The reason that all of these 9649 Cepheids will not be used for the Large Magellanic Cloud’s distance calculation is that only the Cepheids which pulsate in the fundamental mode are needed, and the OGLE-IV catalog contains Cepheid that pulsate in the ﬁrst and second overtone. Moreover, OGLE-II has reddening corrected data,

11 while the data from OGLE-IV must be processed using an extinction map. Since this process is long and does not have a general procedure, the corrected data from OGLE-II were chosen for the distance calculus.

The Araucaria Project was created around 2009 “with the main goal of improving the distance scale calibrations based on observations of major distance indicators in several nearby galaxies” (Pietrzyński& Gieren, 2009). Based on these observations, this project tries to solve the three main problems in the calibration of primary stellar distance indicators: the zero point, environmental dependencies and the proper treatment of internal extinction. Hence, the distances obtained by the project have been calculated using several standard candles (such as Cepheids, TRGB, RR Lyare and red clump stars) along with observations in optical and infrared bands collected in various telescopes in order to achieve a high precision in the results (Pietrzyński& Gieren, 2009). The data for NGC 247 and NGC 55 in the VIJK filters (Pietrzyński et al. 2006b; Gieren et al. 2008; Garc´ıa-Varela et al. 2008; Gieren et al. 2009) along with the data for M33 and IC 1613 in the JK filters (Pietrzyńskiet al. 2006a; Gieren et al. 2013) were taken from various papers of the Araucaria project.

Apart from these two projects, the data on the near infrared bands for the Large Magellanic Cloud and on the optic bands for M33 were taken from Persson et al. (2004) and Macri et al. (2001b), respectively. The speciﬁc details of these observations along with the ones used from the Araucaria project can be found in Appendix 2.

12 Chapter 2

Theoretical framework

2.1 Photometric concepts

As explained by Karttunen et al. (2017), the term of Iν, which is the speciﬁc intensity of radiation at a frequency ν, refers to the energy that passes through a surface element dA within a solid angle dω, which makes a θ angle with the surface’s normal as can be seen in Figure8.

Figure 8: The speciﬁc intensity of light from a sources that passes through a surface depends on the area’s surface and the angle it forms with the normal.

The amount of energy within a frequency of [ν, ν + dν] entering through ω in a time dt is

dEν = Iν cos θdAdνdωdt, (1) −2 −1 where Iν is the speciﬁc intensity of the radiation and has units of W m Hzsterad (Karttunen et al. 2017). This intensity is determined for a frequency ν and a solid angle ω, so to obtain the total intensity I for all frequencies we must integrate Iν over all frequencies: Z ∞ I = Iνdν. (2) 0

13 With the specific and total intensity, the flux density can be found. This concept gives the power radiation per unit area and has units of W m−2. This can be measured in a detector that collects energy from a radiation source, which is equal to the flux density integrated over the instrument’s area in a time interval. So the flux density Fν for a frequency ν is 1 Z Z Fν = dEν = Iν cos θdω. (3) dAdνdt S S Where S is the surface of the detector (Karttunen et al. 2017). The total flux density F is Z F = I cos θdω, (4) S and if the radiation is isotropic, the equation (4) becomes Z F = I cos θdω. (5) S The flux is important since it is related to a star’s luminosity. The luminosity is defined as the flux density emitted by a star into a solid angle by the following equation:

L = ωr2F, (6) where F is the flux density observed at a distance r. From equation (6) it is evident that the flux density and the distance to the source are inverse, meaning that F ∝ r−2 (Karttunen et al. 2017). Therefore, the flux density is given by the inverse square law.

Another use of the flux density is in the determination of stars’ magnitude, which can be divided in apparent and absolute magnitude. The apparent magnitude m is the apparent brightness of a star as seen from the Earth (Karttunen et al. 2017). If a star of magnitude of 0 has a flux density of F0, then all the other magnitudes can be found using the following equation: F m = −2.5 log . (7) F0 The absolute magnitude measures the brightness of a star by calculating its apparent magnitude at a distance of 10 pc. The difference between both magnitudes results in the distance modulus which is defined as

14 F (r) r m − M = −2.5 log = 5 log , (8) F (10) 10 where F (r) is the ﬂux density at a distance r and F (10) is the ﬂux density at a distance of 10 parsecs (pc). Equation (8) is only valid as long as the distance r is expressed in parsecs6 (Karttunen et al. 2017).

All the previous equations were formulated assuming that there was no interstellar medium between the observer and the radiation source. This of course is far from reality since there are particles in space that absorb, re-emit and scatter incident waves away from the line of sight. These radiation loses are deﬁned as extinction and can be measured in magnitudes. Hence including this extinction A due to the entire interstellar medium between the star and the observer results in r m − M = 5 log + A, (9) 10pc where A = (2.5 log e)τ and A ≥ 0. Here τ is a dimensionless quantity called the optical thickness (Karttunen et al. 2017).

Another eﬀect of the interstellar medium is the reddening of light. This happens when the particles (with typical sizes in the order of microns) absorb and scatter light, which mainly aﬀects blue light. If we have that the visual and blue magnitude of a star are r V = M + 5 log + A , (10) V 10 V r B = M + 5 log + A , (11) B 10 B then the observed color index will be

B − V = MB − MV + AB − AV = (B − V )0 + E(B − V ), (12)

where (B − V )0 = MB − MV is the intrinsic color of the star and E(B − V ) = AB − AV is its color excess. The ratio between AV and E(B − V ) is almost constant for all stars and is equal to A R = V ≈ 3.0. (13) V E(B − V )

61 pc ≈ 3.0857 × 1016 m

15 This makes the calculation of AV possible which in turn can be used in equation (9) to ﬁnd the distance to the star (Karttunen et al. 2017).

2.2 Extinction laws Galactic extinction is an important topic in the calculation of distances since, as seen in equation (9), as the extinction increases the computed distance will decrease. For this reason, astronomers have attempted to create extinction laws in order to accurately calculate distances. Cardelli et al. (1989) used the extinction data of several sources for ultraviolet, optical a near infrared in order to ﬁnd an extinction law A(λ)/A(V ) for 0.125µm ≤ λ ≤ 3.5µm. This extinction law can be applied for both diﬀuse and dense regions of interstellar medium and only depends on RV = A(V )/E(B − V ). The authors formulated this extinction law as

hA(λ)/A(V )i = a(x) + b(x)/RV , (14) where a(x) and b(x) take the coeﬃcients according to Table1

Filter x(µm−1) a(x) b(x) A(λ)/A(V ) U 2.78 0.9530 1.9090 1.569 B 2.27 0.9982 1.0495 1.337 V 1.82 1.0000 0.0000 1.000 R 1.43 0.8686 -0.3660 0.751 I 1.11 0.6800 -0.6239 0.497 J 0.80 0.4008 -0.3679 0.282 H 0.63 0.2693 -0.2473 0.190 K 0.46 0.1615 -0.1483 0.114 L 0.29 0.0800 -0.0734 0.056

Table 1: Coeﬃcients and extinctions at standard optical/near-infrared wavelengths for RV = 3.1 (Cardelli et al. 1989)

More recently, Schlegel et al. (1998) made a full-sky 100 µm map composed of several reprocessed and composite sky maps. The authors removed the zodiacal light7 contamination as well as possible cosmic infrared background from their dust

7Zodiacal light is a glow that appears in the night sky due to “background radiation associated with solar light scattered by the tenuous ﬂattened interplanetary cloud of dust particles surrounding the Sun and the planets” (Lasue et al. 2020).

16 Filter λeff A/A(V ) A/E(B − V ) (A˚) Landolt U 3372 1.664 5.434 Landolt B 4404 1.321 4.315 Landolt V 5428 1.015 3.315 Landolt R 6509 0.819 2.673 Landolt I 8090 0.594 1.940 CTIO U 3683 1.521 4.968 CTIO B 4393 1.324 4.325 CTIO V 5519 0.992 3.240 CTIO R 6602 0.807 2.634 CTIO I 8046 0.601 1.962 UKIRT J 12660 0.276 0.902 UKIRT H 16732 0.176 0.576 UKIRT K 22152 0.112 0.367 UKIRT L 38079 0.047 0.153

Table 2: Schlegel relative extinction for selected bandpasses (Schlegel et al. 1998) maps in order to use them as new Galactic extinction estimators. To calibrate their maps, the authors assumed a standard reddening law and used a sample of 106 cluster elliptic galaxies as well as 384 elliptic galaxies with B-V and Mg line strength measurements. They used the colors of this galaxies to measure the reddening per unit ﬂux density of 100 µm emissions. With this, they derived the relative extinction for selected bandpasses, which is shown in Table2.

2.3 Variable stars As noted in the introduction, stars with changing magnitudes or luminosities are called variable stars. In reality, all stars are variable since their luminosity changes as they evolve, which is a very long process. The presence of stellar spots can also change the star’s luminosity phase. Nevertheless, if the star is variable, its magnitude can be plotted as function of time (also known as the light curve) and the shape of the light curve along with the star’s spectral class and radial pulsation mode determine its classiﬁcation. There are diﬀerent kind of stars that pulsate, as shown in Table 3. Also, these stars are part of a transient phenomenon, which means that their position in the Hertzprung-Russell (HR) diagram is not along the main sequence as can be seen in Figure9 (Carroll & Ostlie, 2007). It is worth noting that pulsating variables (such as Classical Cepheids, RR Lyrae and δ Scuti) occupy a narrow

17 band in the diagram called the instability strip. This band is about 600-1100 K wide and as stars evolve and enter this strip, they begin to pulsate (Carroll & Ostlie, 2007).

More recently, the ESA space mission Gaia has made observations with nearly si- multaneous measurements in astrometry, photometry and spectroscopy. Its data are homogeneous throughout the entire sky since the mission only uses one instrument which is not affected by Earth’s atmosphere or seasons. With this, the project has released accurate astrometric measurements for an unprecedented number of objects. The project has made its own HR diagram called the colour-absolute magnitude diagram (CaMD) (Figure 10a) using the second Gaia data release. They located the variable stars in the CaMD diagram, resulting in Figure 10b (Gaia Collaboration et al. 2019). There it can be seen that the instability strip is not as clear as the one in the traditional HR diagram, but nevertheless it is clear that the Cepheids occupy a specified region of the CaMD. So it is evident that Cepheids’ pulsation is a transient episode in their evolution since in this diagram they are located in a different place than the main sequence.

Type of pulsating variable Number Pulsation period Spectral Type ∆m Classical Cepheids (δ Cep, W Vir) 800 1 - 135 F-K-I . 2 RR Lyrae 6100 < 1 A-F8 . 2 Dwarf cepheids (δ Scuti) 200 0.05 - 7 A-F . 1 β Cephei 90 0.1 - 0.6 B1-B3 III & 0.3 Mira variables 5800 80-1000 M-C & 2.5 RV Tauri 120 30-150 G-M & 4 Semiregular 3400 30-1000 K-C & 4.5 Irregular 2300 - K-M & 2 Table 3: Properties of pulsating stars. Here the “Number” refers to the number of stars of a given type according to Kukarkin’s catalogue and “∆m” to the pulsation amplitude in magnitudes (Karttunen et al. 2017)

18 Figure 9: Hertzprung-Russel diagram with the position of several types of variable stars (Carroll & Ostlie, 2007). Here the Cepheid instability strip is evident (between 600-1100 K wide), which denotes the evolution phase in which stars enter and pulsate. It can be seen that the star’s mass dictates the kind of variable star it will turn into. Hence the pulsation period is also related to the star’s density.

19 (a) Gaia’s CaMD with its must striking and basic astronomical features. The gray objects have a parallax greater than 1 mas. It is worth noting that these objects are not corrected for interstellar extinction, stellar multiplicity, rotation, inclination of the rotation axis, and chemical composition.

(b) Gaia’s known pulsating variable stars placed in the CaMD. It can be seen that there are still clear groups of variable stars in the diagram and that the instability strip (consisting of Cepheids, RR Lyrae and δ Scuti) can still be identiﬁed and marked with a black ellipse.

Figure 10: Gaia’s Collaboration results of stellar variability in the CaMD (Gaia Collaboration et al. (2019).

20 2.4 Mechanism of stellar pulsation After the discovery of variable stars, the reason of their pulsation remained a mystery for centuries. As mentioned before, it was thought that the pulsation was caused by dark spots that rotated in the star’s surface, changing its brightness. Another theory claimed that the changes in the luminosity were the effect of tidal waves in the atmospheres of binary stars. But in 1914 astronomer Harlow Shapley contradicted this theory, by affirming that the size of the variable star would then exceed its orbit. He proposed that for Cepheids the pulsations were radial, meaning that they changed size homogeneously and in the process they became brighter and dimmer. This theory was further developed in 1918 by Sir Arthur Stanley Eddington, who provided a firm theoretical framework for this kind of pulsation (Carroll & Ostlie, 2007). Both the radial pulsation method and its theoretical framework are the must accepted theories of why Cepheids pulsate, and this will all be explained in the next subsections.

2.4.1 Period-density relation for radial pulsations There can be two kinds of pulsation: radial and non-radial. The former refers to the case where a star pulsates while maintaining its spherical shape, while in the latter this spherical shape is lost. There are several types of stars that pulsate radially, including Classical Cepheids as seen in Table4. For radial pulsations, the change of size (and therefore of luminosity) is caused by resonating sound waves inside the star. Knowing this, an estimate of the pulsation period Π can be obtained by considering the time it would take for the wave to cross the diameter of a star of radius R and density ρ. This pulsation period is roughly

Type of pulsating variable Population type Pulsation type Long-Period Variables I,II R Classical Cepheids I R W Virginis II R RR Lyrae II R δ Scuti I R, NR β Cephei I R, NR ZZ Cet I NR

Table 4: Population and pulsation type of several pulsating stars. “R” refers to radial pulsations and “NR” to non-radal pulsations (Carroll & Ostlie, 2007)

21 Z R dr Π ≈ 2 , (15) 0 vs

where vs is the adiabatic sound speed and is given by s γP v = , (16) s ρ where P is the pressure and γ the adiabatic index (Caroll et al. 2007). The pressure can be found using the hydrostatic equilibrium equation as

dP GM ρ G( 4 πr3ρ)ρ 4 = − r = − 3 = − πGρ2r, (17) dr r2 r2 3 by assuming that the pressure of the star is constant and it has the shape of a sphere. Integrating equation (17) and using the boundary condition that P (r = R) = 0, the pressure as a function of the radius is 2 P (r) = πGρ2(R2 − r2). (18) 3 Replacing this equation in (16) gives s γ 2 πGρ2(R2 − r2) r2 v = 3 = γπGρ(R2 − r2), (19) s ρ 3 and then replacing this in (22), the period will be

Z R dr Π ≈ 2 q . (20) 0 2 2 2 3 γπGρ(R − r ) Replacing r = R sin x and dr = R cos xdx, it follows that

2 Z R R cos xdx 2 Z R Π ≈ ≈ dx. (21) q p 2 q 2 0 R2(1 − sin x) 2 0 3 γπGρ 3 γπGρ √ 2x r Integrating will result in Π ≈ 2 and since x = arcsin( R ) then 3 γπGρ R 2 r 2 π r 3π

Π ≈ q arcsin ≈ Π ≈ q ≈ . (22) 2 R 2 2 2γGρ 3 γπGρ 0 3 γπGρ

22 This means that the pulsation period and the star’s mean density are inverse. This period-mean density relation explains why the pulsation period is greater for supergiants and smaller for dwarfs, as seen in Table3. The result given by equation (22) somewhat agrees to the period of Cepheids. For example, if M = 5M and R = 50R (such as in a typical Cepheid), then Π ≈ 10 days, which falls within the range of periods measured for Classical Cepheids (Carroll & Ostlie, 2007). Using this last equation and referring to Figure9 it can be seen that the Cepheids with small densities have large pulsation periods and therefore are the brightest. This also happens to RR Lyrae and δ Scuti, as seen on Figure9.

2.4.2 Radial modes of pulsation The resonating waves inside the star are standing waves, which means that there can be some modes of oscillation inside the star. For all modes, there is a node where the gases do not move at the star’s center and an antinode at the star’s surface. In the fundamental mode, the waves move in the same direction at every point of the star. In the ﬁrst overtone there is a node between the center and the surface of the star while in the second overtone there are two nodes. All of these cases along with the gas movement’s direction are shown on Figure 11. Almost all classical Cepheids and W Virginis stars pulsate in the fundamental mode, while the RR Lyrae and long-period variables oscillate either on the fundamental mode or the ﬁrst overtone (Carroll & Ostlie, 2007).

23 Figure 11: Analogy of the modes of oscillation of standing waves between an organ pipe (top) and a star (bottom). Here (a) corresponds to the fundamental mode, (b) to the ﬁrst overtone and (c) to the second overtone (Carroll & Ostlie, 2007).

2.4.3 Eddington’s Valve Mechanism In order to explain the mechanism that powers the sound waves, Eddington modelled pulsating stars as thermodynamic heat engines. He proposed that the gases in a layer of the star compress and expand, doing work in the process. When heat enters this layer, the net work done by the gases will be positive and this will drive the oscillations. Moreover when the heat leaves the layer, the oscillations will be dampened. In this mechanism, the opacity plays a huge role in the production of the oscillations. According to Kramer’s law, the opacity κ depends on the density and temperature of the stellar material by κ ∝ ρT −3.5. It is evident that κ is more sensitive to changes of temperature than of pressure, which means that if the temperature increases, the opacity will increase too and vice versa (Carroll & Ostlie, 2007). This is the basis of Eddington’s valve mechanism, which is shown in Figure 12. The steps shown in there are:

1. A layer of stellar material will fall inwards due to the eﬀect of gravity.

2. The compressing layer will have an increase of temperature and density, making it more opaque.

24 3. Heat (Q) builds up inside the layer since its opacity will act as a barrier to the energy ﬂowing toward the surface.

4. The pressure will build up inside the layer, which will push the layer outwards.

5. The expanding layer will have a decrease of temperature and density, making it more transparent.

6. The trapped heat can now escape the layer, which will drop the pressure inside of it and make it collapse, repeating the cycle

11 4

2 5

3 Q 6 Q

Figure 12: Steps of the Eddington valve mechanism

25 2.4.4 The κ and γ Mechanisms In Eddington’s valve mechanism, it takes special circumstances to overcome the damping effect of most stellar layers. Thus there must be some conditions that excite and maintain the pulsations once they begin. These conditions where identified by S.A. Zhevakin and then verified by Rudolph Kippenhahn, Norman Baker and John P. Cox (Carroll & Ostlie, 2007).

These authors found that the regions where the valve mechanism can work are the partially ionization zones of a star. In these layers the gases are partially ionized and when a layer is compressed, it will produce further ionization due to the increase of temperature. Therefore part of the heat will be used in the ionization of the gas and the temperature of the layer will not increase as much as expected, making the opacity more sensitive to the density. It follows that during the expansion of the layer, the gases’ ions will recombine with electrons and release energy, which will not decrease the temperature of the layer as much as expected. This means that Kramer’s law will depend mainly on the layer’s density, which implicates that the layer will absorb heat as it shrinks and release heat when it expands. This is known as the κ mechanism. This mechanism is reinforced by the fact that heat tends to enter the layer during its compression due to the fact that this zone is colder that the adjacent layers of stellar mass and vice versa. This is known as the γ mechanism (Carroll & Ostlie, 2007).

Moreover the partial ionization zones regulate the ﬂow of energy in the star’s layers, causing their pulsation. In most stars, there are two of these layers: the Hy- drogen partial ionization zone and the Helium II partial ionization zone. The ﬁrst one is a broad zone of temperature between 1 to 1.5×104 K where neutral Hydrogen and Helium are ionized (H I → H II & He I → He II). These ionizations require energies of 13.6 eV and 24.53 eV, respectively. In the second zone, which is found deeper in the star and has a temperature of 4×104 K, Helium is ionized again (He II → He III). This requires 54.41 eV, so the radiation emitted in either ionization zone is going to have values of ultraviolet wavelengths.

The location of these zones determine whether the star can pulsate or not. If the star is too hot, the layers will be near the surface (as seen in Figure 13) and there would not be enough density and mass to drive the oscillations. If the star is cooler, the zones will be located deeper in the star which may excite the ﬁrst overtone since there is more mass available. Furthermore, in cold stars the partial ionization regions will be found deep enough for the fundamental mode to happen. However the surface

26 temperature of the star can not be too cold, since it could dampen the oscillations. It has been found that the He II partial ionization zone is mainly responsible for the pulsation of stars within the instability strip. Besides, the star is least brightest when the least mass lies between the Hydrogen ionization zone and the surface. This means that after the minimum radius is reached, the star will attain its greatest luminosity (Caroll et al. 2007).

Figure 13: Location of the partial ionization zones in stars of diﬀerent temperatures (Carroll & Ostlie, 2007).

2.5 Period-Luminosity Relations As mentioned in Chapter 1, Henrietta Swan Leavitt was the first person to no- tice the connection between the period of a variable star and its luminosity. Her job was to compare photographs of the same field taken at different times in order to identify variable stars. She discovered 2400 Cepheids, mainly in the SMC (with periods between 1 and 50 days) and determined in PLR in Figure A1b. Moreover, since all the stars in the SMC are roughly at the same distance from Earth, then the differences of their apparent magnitudes must be equal to the ones in their absolute magnitudes. Consequently the observed differences in the star’s brightness must reflect the difference of the star’s apparent luminosity. In addition to Leavitt’s

27 discoveries, it was found that the absolute magnitude of a pulsating star could be calculated by measuring its period and luminosity thanks to its PLR. Therefore the distance to the star’s galaxy could be found using equation (8) (Carroll & Ostlie, 2007). Usually the PLR is found by ﬁtting a straight line to the data of the star’s mean magnitude as function of the logarithm of its period.

Observations in infrared bands result in more accurate PLR since the effect of the interstellar extinction is decreased in these filters. Hence, the scatter of the data from the fitted line is decreased dramatically. Adding a color term to the fit can further re- duce this scatter, resulting in accurate distances to the stars (Carroll & Ostlie, 2007).

After the PLR was discovered, astronomers established diﬀerent zero points to the relation, which is crucial for the determination of both Galactic and extragalactic distance scales. Nowadays, the most widely used zero point is the distance to the LMC, because the galaxy’s plane faces Earth and therefore the data are not affected by the galaxy’s depth. Another reason is that the distances derived from this galaxy’s PLR are independent of any error related to the reddening zero-point (Feast & Catchpole, 1997). The distance to this galaxy has been a debate for decades since it is so important to the distance scale to galaxies. In 2000, Udalski used the data from the OGLE-II microlensing experiment to calculate the distance to this galaxy using Cepheids, RR Lyrae and Red Clump Stars. He used 3300 Fundamental Mode Cepheids (with periods between 1 - 31 days) to ﬁnd a corrected PLR for VI equal to

V0 = (−2.775 ± 0.031) log P + (17.066 ± 0.021), (23)

I0 = (−2.977 ± 0.021) log P + (16.593 ± 0.014). (24) Furthermore, Persson et al. (2004) calculated the infrared PLR for the LMC. They used 92 Cepheids with periods between 3 to 100 days in the J, H, K and Ks bands. After correcting the extinction for the stars, they obtained the following relations:

J0 = (−3.154 ± 0.051) log P + (16.336 ± 0.064), (25)

H0 = (−3.234 ± 0.042) log P + (16.079 ± 0.053), (26)

K0 = (−3.261 ± 0.042) log P + (16.036 ± 0.053), (27)

28 KS0 = (−3.281 ± 0.108) log P + (16.051 ± 0.05). (28)

2.6 Distance determination In order to calculate the distance to any galaxy, the PLR of the LMC must be taken into account, since it is considered the zero point of the relation. For this reason, the ﬁrst step is to calculate the robust PLR of the LMC for each ﬁlter. Suppose that the following robust PLR is obtained for the V band

V0 = β log P + ZLMC , (29)

where β is the slope and ZLMC is the intercept of the linear ﬁt for the LMC. Equation (8) can be rewritten as

V0 = 5 log rLMC − 5 + MV , (30) and replacing (29) in the previous equation it follows that

β log P + ZLMC = 5 log rLMC − 5 + MV , (31)

where rLMC is the distance to the LMC in parsecs. As mentioned before, the PLR of the LMC is considered to be universal, meaning that any slope β of any PLR of any galaxy must be equal to the one of the LMC. For this reason, the only parameter that changes is the intercept Z. Supposing that the PLR of a galaxy A is calculated and the same procedure as before is applied, the following result must be expected:

β log P + ZA = 5 log rA − 5 + MV , (32)

where rA is the distance to the galaxy in parsecs. It can be said that if the Cepheids of both galaxies have the same periods, then their absolute magnitudes are equal. So subtracting (31) from (32) yields

ZA − ZLMC = 5 log rA − 5 log rLMC . (33) The only way to solve this equation is to use the distance modulus to the LMC. This topic has been extensively debated since this value can vary from 18.3 mag to 18.7 mag. Fitzpatrick et al. (2002) obtained a distance modulus to the LMC bar center of (18.52 ± 0.06) mag by using the HV 982 eclipsing binary. Alves et al. (2002) used Red Clump stars in the K ﬁlter and got a distance of (18.493 ± 0.003)

29 mag. Furthermore, Freedman et al. (2001) used a distance modulus of (18.50±0.10) mag for their calculation of the Hubble constant arguing that this distance modulus “agrees with the mean and median of the distribution for other methods at the 2.5% level”, as seen in Figure 14. Since this distance modulus is dereddened, using equation (9) the distance to the LMC is (50.119 ± 2.308) kpc. More recently, Pietrzyński et al. (2019) calculated the distance to the LMC that is precise to 1% using 20 eclipsing binaries and a calibration of the surface brightness-colour relation in order to determine the geometrical distance to the galaxy. Their final distance modulus is 18.477 ± 0.004stat ± 0.026sys mag, where the first error corresponds to the statistical one and the second to the systematic one. Using again equation (9), this distance to the LMC is (49.591 ± 0.091) kpc. Based on all of this, both the distance modulus of Freedman of 18.50 ± 0.01 mag and of Pietrzyńskiof 18.477 ± 0.004 mag will be used in this thesis. For the case of Freedman, the distance modulus of the LMC can be written as

18.5 = (m − M)LMC = 5 log rLMC − 5. (34)

Clearing 5 log rLMC from this equation and replacing it in equation (33) gives

ZA − ZLMC = 5 log rA − 18.5 − 5, (35) from which the distance modulus of the galaxy A will be

(m − M) = 5 log rA − 5 = 18.5 + ZA − ZLMC . (36) Using equation (36), the distance modulus to any galaxy can be calculated thanks to the PLR. This same procedure can be done using the distance to the LMC found by Pietrzy´nskiby changing 18.5 mag to 18.477 mag in equation (34).

30 Figure 14: Image taken from Freedman et al. (2001) showing the “distribution of LMC distance moduli as compiled by Gibson (2000) plotted as a continuous probability density distribution”. This clearly shows that the distance modulus to the LMC of 18.5 mag agrees with the median and mean of this distribution.

In order to calculate the distance modulus without the eﬀects of reddening, the following equation can be used:

(m − M)0 = (m − M)λ − Aλ = (m − M)λ − E(B − V )Rλ, (37)

where (m − M)λ is the reddened distance modulus to a galaxy obtained in the λ ﬁlter and Rλ is the ratio between Aλ and E(B − V ), which is obtained from Schlegel’s law in Table2. So by plotting ( m − M)λ as a function of Rλ for a galaxy and making a linear regression, the slope will be the color excess and the intercept will be (m − M)0, which is known as the true distance modulus (Gieren et al. 2009). So the distance in parsecs to galaxy A will be:

0.2((m−M)0+5) rA = 10 . (38) In order to ﬁnd the error of the distance, last equation will be rewritten as

(m − M) + 5 ln r log r = 0 = A . (39) 10 A 5 ln 10

Deriving with respect to rA and changing the diﬀerentials for deltas the result is

31 1 ∆(m − M) 1 1 0 = , (40) 5 ∆rA ln 10 rA

So by clearing ∆rA, the error of the distance will be: ∆(m − M) ∆r = r · ln 10 · 0 , (41) A A 5

where ∆(m−M)0 is the error of the distance modulus obtained from the intercept of the linear regression made for equation (37).

2.7 Robust regression When a least squares regression is applied to a set of data using n observations, for a p-parameter model Y = Xβ + ε, it is assumed that the error vectors ε are distributed as N(0, σ2I) (Draper & Smith, 1998). Also, the least squares method assumes that the error term ε has a constant variance σ2I, the errors are uncorrelated and that all observations have the same weight (Garc´ıa-Varela et al. 2016). In reality there are many situations where the distribution of ε is not normal and presents a heavy-tailed distribution. This tends to generate influential points that affect the least squares regression since they pull the fit too much in a certain direction (Mont- gomery et al. 2012). This means that a robust regression must be made, since the robust method weighs observations unequally, down-weighing observations with large residuals so that they don’t affect the result of the regression (Draper& Smith, 1998).

There are several kind of robust estimators. One is the “maximum likelihood” estimator, one type of M-estimator. This method assumes that the errors are inde- pendently distributed by the same distribution f(ε). Then the maximum likelihood of β is given by βˆ, which maximizes the quantity

n Y 0 f(Yi − xiβ), (42) i=1 0 where xi is the ith row of X and i = 1, 2, ... n, in the least squares model (Draper & Smith, 1998). This is not necessarily scale-invariant, so to solve this problem equation (42) can be rewritten as

n n 0 X ei X nYi − x β o ρ = ρ i . (43) s s i=1 i=1

32 Here ρ(u) is a deﬁned function of u (Draper & Smith, 1998) and s is a robust estimate of scale and is usually (Montgomery et al. 2012):

median|e − median(e )| s = i i . (44) 0.6745

To minimize equation (43) with respect to βj, j = 1, 2, ..., k, the ﬁrst partial derivative of ρ(ui) must be calculated as

0 ∂ρ nYi − xiβ o = ψ(ui) = ψ . (45) ∂βj s Then the next equation can be obtained

n 0 X nYi − x β o x ψ i = 0, j = 1, 2, .., k, (46) ij s i=1 where k is the number of parameters in the model (Garc´ıa-Varela et al. 2016), 0 ψ is the partial derivative ∂ρ/∂u and xij is the j th entry of xi = (1, xi1, xi2, ..., xik) (Draper & Smith, 1998). Since ψ is nonlinear then equation (46) must be solved by iterative methods. The iteratively reweighted least squares (IRLS) is the most ˆ used method. It assumes that an initial estimate β0 is available in order to write the p = k + 1 equations of equation (46) as

n 0 n 0 0 0 X nYi − x β o X xij{ψ[(Yi − x β)/s]/(Yi − x β)/s}(Yi − x β) x ψ i = i i i , (47) ij s s i=1 i=1 which is equal to

n X 0 xijwi0(Yi − xiβ) = 0, (48) i=1

where wi,0 are the weights and are deﬁned as

( ψ[(Y −x0 βˆ )/s] i i 0 if y 6= x0 βˆ , (Y −x0 βˆ )/s i i 0 wi0 = i i 0 (49) 0 ˆ 1 if yi = xiβ0, In matrix notation equation (48) can be written as

0 0 X W0Xβ = X W0Y, (50)

33 where W0 is an n × n diagonal matrix of weights given by equation (49). Then ˆ β1 can be found as

ˆ 0 −1 0 β1 = (X W0X) X W0Y. (51) ˆ Now the weights must be recalculated using β1 in equation (49) and so on until convergence is achieved (Montgomery et al. 2012). It is worth noting that the weight function will depend on the ρ(u) that is chosen.

It is important to check that there are no bad influential points in the data (points that are outliers and influential at the same time) before doing the M -estimator procedure. Otherwise, the MM -estimator should be used since it has a high breakdown point (BDP). The BDP is “the smallest fraction of contaminated data to cause a divergence of the estimator from the value that it would take if the data were not contaminated” (Garc´ıa-Varela et al. 2016). This estimator combines the asymptotic efficiency of M -estimators with high BDP estimators. In order to find the MM -estimator the following three steps must be followed:

1. Use a high BDP parameter estimator to compute the residuals of the model.

2. Based on these residuals, calculate an M-estimate with high BDP.

3. Calculate the model parameter’s using M-estimators and the scale estimator computed in the previous step.

On both the M and MM -estimator methods, it is assumed that the errors are uncorrelated. Also these methods are based on asymptotic results, which means that they could fail for small data sets (Garc´ıa-Varela et al. 2016).

This process can be done computationally in the RStudio software through either the robust or robustbase packages. Both packages use the S-estimator in the first step since it has a 50% BDP. Also, both take ψ as the bisquare function and both use an asymptotic efficiency of 95% for normal errors. After downloading the data of each galaxy in RStudio, either the lmRob (for the robust package) or the lmrob (for the robustbase package) command can be used. In either case, the independent and dependent variables of the data set must be specified to the program, after which it will calculate the slope and intercept of the robust fit between both variables. The results will also include the standard error of the calculated parameters, the residual standard error and the R-squared. On one hand, the robust package tests for bias between the calculated MM -estimation with the M-estimate and the Least Squares

34 estimate. On the other hand, the robustbase package’s results include the value of the weights used and points the outliers in the data set. Hence, since the PLR will be calculated using the MM -regression, when a robust regression is mentioned further on this refers to this type of regression.

35 Chapter 3

Results and discussion

3.1 Large Magellanic Cloud As mentioned in Section 2.6, the first step in order to find the distance to a galaxy is to calculate the PLR for the LMC. First the NICMOS photometric system used by Persson et al. (2004) was converted to the UKIRT system, since this was the system used by the Araucaria project (from where the IR data of all galaxies were obtained). According to Hawarden et al. (2001), there are only zero-point offsets between both systems in the JHK filters. This means that a value of 0.034 and 0.015 mag must be subtracted from the LMC data in the J and K filters respectively. This way the data are corrected from the radiation absorption made by the interstellar medium and be converted to the UKIRT system, which is used in the calculus of the PLR. After doing this process, the data in IR have to be dereddened, since the data in the optical band had already been corrected by the OGLE project. This way, although the true distance modulus for a galaxy is calculated using its reddened data, the systematic error of the extinction of the LMC will not affect the results. A robust regression was made for each filter, which can be seen in the Figure A2 in Appendix 3. Hence, the following PLRs were obtained for the LMC:

V0 = −2.743 log P + 17.054, (52)

I0 = −2.966 log P + 16.593, (53)

J0 = −3.085 log P + 16.241, (54)

K0 = −3.247 log P + 16.007. (55) Having this, the slope and intercepts of these PLRs will be used in the distance calculus for the rest of the galaxies.

3.2 NGC 247 Using the PLRs from the LMC, a robust regression was made for each ﬁlter of NGC 247 following the procedure described in Section 2.6. With this, the plots in Figure A3 in Appendix 4 were obtained, resulting in the following PLRs for NGC 247:

36 V = −2.743 log P + 26.689, (56) I = −2.966 log P + 26.095, (57) J = −3.085 log P + 25.469, (58) K = −3.247 log P + 24.221. (59) From equations (56)-(59) the intercepts were used to calculate the distance modulus to the galaxy from equation (36). Then using the LMC distance modulus found by Freedman et al. (2001) the results from Table5 were calculated. Recall that the ratio of total to selective extinction Rλ was obtained from Schlegel’s Law (Table2). The extinction in a ﬁlter for any galaxy can be found at the NED/IPAC Extragalactic Database, which uses the extinction maps of Schlegel et al. (1998). Plotting (m − M)λ as a function of Rλ results in Figure 15.

Filter (λ) (m − M)λ (mag) Rλ V 28.136 ± 0.111 3.24 I 28.002 ± 0.114 1.962 J 27.728 ± 0.131 0.902 K 27.714 ± 0.119 0.367

Table 5: Distance modulus for NGC 247 in the VIJK ﬁlters using the distance modulus for the LMC found by Freedman et al. (2001)

37 Figure 15: True distance modulus of NGC 247 using the distance modulus for the LMC found by Freedman et al. (2001)

A weighted least squares regression was made in RStudio in order to find the relation between the reddened distance moduli and their respective Rλ, as described by equation (37). This type of regression was used because the error bars of the data points are taken into account, such that points with little errors affect more the regression than the ones with big errors. After fitting the weighted line, the following values were obtained:

E(B − V ) = 0.159 ± 0.024 mag, (60)

(m − M)0 = 27.638 ± 0.052 mag, (61) where the errors were calculated by RStudio when the regression was applied to the data. This regression has a value of R2 = 0.9553. Using equation (38), this corresponds to a distance of NGC 247 of (3.34 ± 0.08) Mpc. It is evident that this value agrees with the distance modulus found in the VIJK ﬁlters reported in Table5. This value is also very close to the distance of (3.38 ± 0.06) Mpc found by the Araucaria project (Gieren et al. 2009), having only a 1.2% discrepancy. The diﬀerence between

38 these distances can be seen in the intercepts of the calculated true distance modulus with the one used by the Araucaria Project, as seen in Figure 17.

This same procedure was repeated using the LMC distance modulus found by Pietrzy´nskiet al. (2019). Table6 has the distance moduli found with this distance in the VIJK ﬁlters, which were plotted against their corresponding Rλ in Figure 16.

Filter (λ) (m − M)λ (mag) Rλ V 28.113 ± 0.049 3.24 I 27.979 ± 0.054 1.962 J 27.705 ± 0.085 0.902 K 27.690 ± 0.064 0.064

Table 6: Distance modulus for NGC 247 in the VIJK ﬁlters using the distance modulus for the LMC found by Pietrzy´nskiet al. (2019)

Figure 16: True distance modulus of NGC 247 using the distance modulus for the LMC found by Pietrz´nskiet al. (2019)

39 Fitting another weighted line in the data from Figure 16, the values for the excess color and the second true distance modulus were found:

E(B − V ) = 0.159 ± 0.024 mag, (62)

(m − M)0 = 27.615 ± 0.052 mag, (63) and this regression has to a value of R2 = 0.9553. This corresponds to a distance of (3.33 ± 0.08) Mpc for NGC 247. As with the first true distance modulus in equation (61), the second value is in excellent agreement with the reddened distance moduli of Table6. It is also closer to the distance modulus found by the Araucaria project since it only has a 0.3% discrepancy compared to it. It can be seen that the two calculated distances are pretty similar, having only a difference of 10 kpc. The reason for this variation comes from the two LMC distances used, since between the values of Freedman et al. (2001) and Pietrzyńskiet al. (2019) there is a 528 pc difference. For the excess color, the value obtained from the two distance modulus has a 10.2% discrepancy compared to the 0.177 mag value found by Gieren et al. 2009. These differences can be seen in the slopes of the lines in Figure 17. Using the values of extinction for NGC 247 reported by NED/IPAC, the excess color has a value of 0.018 mag. Hence by subtracting the obtained excess color with the one found with the NED/IPAC data, the galaxy’s internal excess color is 0.141 mag. This high value is caused by the galaxy’s inclination (74◦), which shows that NGC 247 is almost “edge on” to the sight of view. Thus, all the accumulated dust in the spiral’s arms will cause this internal extinction, resulting in the significant excess color found in equation (62).

40 Figure 17: Comparison between the obtained distance moduli and excess colors (blue line and green line) with the ones found by Gieren et al. 2009 (red line). The main diﬀerence is found in the excess color, as seen by the slopes of the obtained distance moduli with the one used by the Araucaria project.

3.3 M33 The same procedure for NGC 247 was repeated with the data from M33. The robust PLRs can be found in Figure A4 in Appendix 5, resulting in these PLRs for M33:

V = −2.743 log P + 23.515, (64) I = −2.966 log P + 22.825, (65) J = −3.085 log P + 22.450, (66) K = −3.247 log P + 22.205. (67) From these, the distance modulus was found for each ﬁlter using the distance modulus to the LMC of Freedman et al. (2001), which resulted in the values pre-

41 sented in Table7. These values were then plotted against their respective ratio between total to selective extinction, as seen in Figure 18.

Filter (λ) (m − M)λ (mag) Rλ V 24.962 ± 0.107 3.24 I 24.732 ± 0.106 1.962 J 24.709 ± 0.144 0.902 K 24.698 ± 0.118 0.367

Table 7: Distance modulus for M33 in the VIJK ﬁlters using the distance modulus for the LMC found by Freedman et al. (2001)

Figure 18: True distance modulus of M33 using the distance modulus for the LMC found by Freedman et al. (2001)

From the weighted ﬁt applied to the data in Figure7, the value of R2 is equal to 0.7676 and the following results were obtained:

E(B − V ) = 0.074 ± 0.029 mag, (68)

42 (m − M)0 = 24.653 ± 0.040 mag. (69) This corresponds to a distance for M33 of (852.315 ± 17.700) kpc. The value for (m − M)0 found is very similar to the reddened distance moduli in Table7 and this distance is close to the one found by the Araucaria project (Gieren et al. 2013) of (839.460 ± 11.598) kpc. Although both distances have a diﬀerence of around 13 kpc, this only constitutes a discrepancy of 1.5%, so it can be said that the true distance modulus found is precise. The comparison between the true distance modulus found by Gieren et al. 2013 and the values found by the calculation using the robust regression can be seen in Figure 20.

This whole procedure was repeated using the LMC distance modulus of Pietrzy´nski et al. (20019). The reddened distance modulus for each ﬁlter is presented in Table 8 and these values were plotted against their respective Rλ in Figure 19.

Filter (λ) (m − M)λ (mag) Rλ V 24.938 ± 0.038 3.24 I 24.710 ± 0.036 1.962 J 24.686 ± 0.103 0.902 K 24.675 ± 0.062 0.062

Table 8: Distance modulus for M33 in the VIJK ﬁlters using the distance modulus for the LMC found by Pietrzy´nskiet al. (2019)

43 Figure 19: True distance modulus of M33 using the distance modulus for the LMC found by Pietrz´nskiet al. (2019)

After ﬁtting a weighted least squares regression in the data presented in Figure 19, the results for the excess color and the second true distance modulus were found to be:

E(B − V ) = 0.074 ± 0.029 mag, (70)

(m − M)0 = 24.630 ± 0.040 mag, (71) meaning that this regression also has an R2 = 0.7676. This results in a distance of (843.335 ± 15.535) kpc for M33. This second true distance modulus is very close to the reddened values of Table8 and it is similar to the distance found by the Araucaria project for this galaxy. The value for the second distance only has a 0.5% discrepancy with respect to the value calculated by Gieren et al. (2013). Also, the two found distances have a diﬀerence of around 9 kpc, which is due to the diﬀerent LMC distance modulus used in the calculations. The excess color, which is the same as the one obtained using the distance modulus of the LMC of Freedman, has a 61.0% discrepancy compared to the value found by Gieren et al. 2013 of 0.19 mag.

44 This distance along with the found excess color for both regressions can be seen in analogy with the values found by the Araucaria project in Figure 20. The diﬀerences can be seen in the slopes of the lines and their intercepts. Using the extinctions for M33 found in the NED/IPAC database, the color excess has a value of 0.040 mag. This means that there is a 0.034 mag excess color within the galaxy, which may be caused by its inclination (53.8◦) and because it is located outside the Galactic ◦ 0 00 plane, since its declination is of δ = 30 39 36. 8. Hence the accumulated dust withing the galactic plane is not present in the line of sight, which in turn results in a low extinction for galaxies outside the galactic plane.

Figure 20: Comparison between the obtained distances and excess colors (blue line and green line) with the ones found by Gieren et al. 2013 (red line). Here the contrast between the excess colors is evident, since the value found by the Araucaria project diﬀers extremely from the calculated results.

45 3.4 IC 1613 This process was repeated for IC 1613. The robust regressions of this galaxy which resulted in the following PLRs for the ﬁlters VIJK :

V = −2.743 log P + 22.940, (72) I = −2.966 log P + 22.546, (73) J = −3.085 log P + 22.077, (74) K = −3.247 log P + 21.684. (75) These PLRs can be found ﬁtted to the data for each ﬁlter in Figure A5 in the Appendix 6. After using the equations in Section 2.6, the results in Table9 were found using the LMC distance modulus of 18.5 ± 0.1 mag of Freedman et al. (2001). These results were plotted, as seen in Figure 21.

Filter (λ) (m − M)λ (mag) Rλ V 24.387 ± 0.110 3.24 I 24.453 ± 0.104 1.962 J 24.336 ± 0.148 0.902 K 24.176 ± 0.152 0.367

Table 9: Distance modulus for IC 1613 in the VIJK ﬁlters using the distance modulus for the LMC found by Freedman et al. (2001)

46 Figure 21: True distance modulus of IC 1613 using the distance modulus for the LMC found by Freedman et al. (2001)

A weighted least squares regression was applied to the data, which resulted in the values of the excess color and the true distance modulus according to equation (37). The found values are:

E(B − V ) = 0.066 ± 0.046 mag, (76)

(m − M)0 = 24.231 ± 0.090 mag. (77) This results in a distance of IC 1613 of (701.778 ± 29.086) kpc. This value of the distance is close to the distance found by the Araucaria project of (721.440 ± 29.070) kpc (Pietrzyńskiet al. 2006a), since it only has a 2.7% discrepancy compared to the value of the Araucaria project, as seen in the difference of intercepts of the calculated line and the Araucaria line in Figure 23. This value also agrees to the values of Table9, which shows the reddened distance moduli for the galaxy in the VIJK filters.

Repeating this procedure with the distance modulus to the LMC found by Pietrzy´nski et al. (2019), the results from Table 10 were found. Then these distance moduli were

47 plotted against their respective ratio of total to selective extinction, as seen in Figure 22.

Filter (λ) (m − M)λ (mag) Rλ V 24.364 ± 0.046 3.24 I 24.430 ± 0.029 1.962 J 24.313 ± 0.110 0.902 K 24.154 ± 0.114 0.367

Table 10: Distance modulus for IC 1613 in the VIJK ﬁlters using the distance modulus for the LMC found by Pietrzy´nskiet al. (2019)

Figure 22: True distance modulus of IC 1613 using the distance modulus for the LMC found by Pietrz´nskiet al. (2019)

After applying the weighted least squares regression to the data in Figure 22, the following results were obtained:

E(B − V ) = 0.066 ± 0.046 mag, (78)

48 (m − M)0 = 24.208 ± 0.090 mag. (79) Both weighted regressions have a value of R2 = 0.5128. This second true distance modulus corresponds to a distance to IC 1613 of (694.384 ± 28.780) kpc. The value of (m − M)0 is very similar to the distance modulus for each filter at table 10. This value is also in agreement to the distance modulus to the galaxy calculated by Pietrzyńskiet al. (2006a) in the Araucaria project, since it only has a 3.8% disrep- ancy compared to it. Also, the difference between the two calculated distances is of approximately 7 kpc, which shows that both values are pretty similar. Further- more, the excess color has a 26.7% discrepancy to the value found by Pietrzyński et al. 2006 of E(B − V ) = 0.090 mag. Figure 23 shows an analogy between both calculated results with the one obtained by the Araucaria project, which is evident in the slopes and in the intercepts of the lines. Comparing the obtained result with the one found using NED/IPAC’s extinction values for IC 1613 (E(B − V ) = 0.024 mag), it can be seen that the internal extinction of this galaxy is of 0.042. This low extinction is caused by the fact that IC 1613 is far from the Galactic plane and has a low inclination (38◦). Also since this galaxy has an irregular shape, the dust isn’t as concentrated in the line of sight as in the case of an spiral galaxy.

49 Figure 23: Comparison between the obtained distances and excess colors (blue line and green line) with the ones found by Pietrzy´nskiet al. 20016a (red line). Here the distance and the excess color obtained by the Araucaria project diﬀer considerably from the ones calculated.

3.5 NGC 55 Finally for NGC 55, the robust PLRs found by repeating once more this procedure are on Figure A6 in Appendix 7 and resulting in the following PLRs for NGC 55:

V = −2.743 log P + 25.342, (80) I = −2.966 log P + 24.771, (81) J = −3.085 log P + 24.253, (82) K = −3.247 log P + 23.976. (83) The results of the reddened distance modulus for the VIJK ﬁlters using the LMC distance modulus of Freedman et al. (2001) can be found on Table 11. These values

50 were plotted against their respective ratio of total to selective extinction in Figure 24.

Filter (λ) (m − M)λ (mag) Rλ V 26.788 ± 0.109 3.24 I 26.678 ± 0.105 1.962 J 26.512 ± 0.143 0.902 K 26.469 ± 0.129 0.367

Table 11: Distance modulus for NGC 55 in the VIJK ﬁlters using the distance modulus for the LMC found by Freedman et al. (2001)

Figure 24: True distance modulus of NGC 55 using the distance modulus for the LMC found by Freedman et al. (2001)

The values from the plot in Figure 24 were also modelled using a weighted least squares regression. With that and considering equation (37), the following results were obtained:

E(B − V ) = 0.118 ± 0.011 mag, (84)

51 (m − M)0 = 26.421 ± 0.017 mag. (85) From this, the distance of NGC 55 is of (1.92 ± 0.02) Mpc. The true distance modulus of equation (85) is very similar to the distance moduli in Table5. This value also agrees to the distance found by the Araucaria project of (1.94 ± 0.03) Mpc (Gieren et al. 2008) since it only has a 1.0% discrepancy with respect to this distance, as seen in the intercepts of the found true distance modulus (blue) with the one of the Aracuaria project (green) in Figure 26.

Now with the distance modulus to the LMC of Pietrzy´nski,the results of Table 12 were found.

Filter (λ) (m − M)λ (mag) Rλ V 26.765 ± 0.044 3.24 I 26.655 ± 0.0311 1.962 J 26.489 ± 0.102 0.902 K 26.446 ± 0.081 0.367

Table 12: Distance modulus for NGC 55 in the VIJK ﬁlters using the distance modulus for the LMC found by Pietrzy´nskiet al. (2019)

52 Figure 25: True distance modulus of NGC 55 using the distance modulus for the LMC found by Pietrz´nskiet al. (2019)

The weighted least squares regression was applied to the data in Figure 25 in order to ﬁnd the excess color and true distance modulus. Both weighted regressions have the same value of R2 = 0.982. From this, the following results were obtained:

E(B − V ) = 0.118 ± 0.011 mag, (86)

(m − M)0 = 26.398 ± 0.017 mag. (87) This second true distance modulus results in a distance of (1.90 ± 0.01) Mpc. This distance compared to the one found by Gieren et al. (2008) with the Araucaria project has a discrepancy of 2.1%, which makes it a really precise value. Also, this true distance modulus agrees with the results of Table 12, which indicates that the true distance modulus is in agreement with the values in the VIJK ﬁlters. Further- more, this value is very close to the one calculated using the LMC distance modulus of Freedman et al. (2001), since there are only 20 kpc between both values. The excess color, which turned out to be the same for both calculated true distance moduli, has a 7.1% discrepancy compared to the value of 0.127 mag obtained by Gieren

53 et al. (2008). This is evident in Figure 26, in which the red line corresponds to the true distance modulus of the Araucaria project and the blue and green lines are the moduli obtained with the robust PLRs. It is evident that the found values of the excess color and of the distances are really close to the ones of Gieren et al. (2008), since the slopes and intercepts of the lines are really similar. By calculating the excess color using NED/IPAC’s Database, a very diﬀerent value of 0.012 mag is found. This means that the internal extinction is of 0.106 mag for this galaxy, which is a signiﬁcant value. It is caused by NGC 55’s inclination (80◦), this galaxy is almost “edge on” from the observer. As with NGC 247, this causes the accumula- tion of dust in its spiral arms which causes an increased extinction in the line of sight.

Figure 26: Comparison between the obtained distances and excess colors (blue line and green line) with the ones found by Gieren et al. 2008 (red line). Here the distance and the excess color obtained by the Araucaria project are really similar to the ones obtained using both LMC distance moduli.

It is worth noting that the values of the intercepts of all PLRs and the value of the slopes of the LMC’s PLR were conﬁrmed using the fast bootstrapping method in

54 RStudio. This method consists of resampling the data points and reﬁt the statistical model in order to obtain the sample variance. With this, the values of all PLRs were calculated again and they were almost equal to the values found with the robust regression. These comparisons can be seen in Table A2 of Appendix 8. The summary of the results is found in Table 13 and the plotted distances to each galaxy can be seen in the 3D Figure 27.

Distance in Mpc using Distance is Mpc using Galaxy Excess color (mag) (m − M)LMC from (m − M)LMC from Freedman et al. (2001) Pietrzy´nskiet al. (2019) NGC 247 0.159 3.34 3.33 M33 0.074 0.85 0.84 IC 1613 0.066 0.70 0.69 NGC 55 0.118 1.92 1.90

Table 13: Results of the excess color and distances for each galaxy

Figure 27: Distances for the four galaxies as seen in a 3D plot

55 3.6 Discussion First, the distance moduli obtained with the robust PLRs for each galaxy will be compared with the ones reported by the Araucaria project. This is seen in Figure 28, in which these distances were plotted all at once in order to see the contrast between the results. It is obvious that the obtained results are not very diﬀerent than the ones found by the Araucaria project since the discrepancies between both values are so low, as seen in Figure 29.

Figure 28: Obtained distance moduli compared to the ones obtained by the Araucaria project. It can be seen that the results were pretty precise, since they aoverlap with the Aracuaria distance moduli.

56 IC 1613

NGC 55

M33 NGC 247

Figure 29: Discrepancies between the two found distances for each galaxy compared to their corresponding distance reported by the Araucaria project

The relationship between the metallicity and the obtained excess colors for each galaxy are shown in Figure 30. Here it can be seen that generally as the excess color increases, the metallicity decreases. The exception to this is M33, but it is worth mentioning that this is the galaxy with the metallicity furthest away from the LMC’s metallicity, which is represented by the straight line in Figure 30. But the inverse relation between the metallicity and the excess color is evident with the rest of the data points.

57 M33

IC 1613

NGC 55 NGC 247

LMC

Figure 30: Relation between the excess colors of the galaxies and their metallicities. Here the straight line is the LMC’s metallicity, which is used as a reference point since there is evidence that the PLR only works well for galaxies with metallicities close to the one of the LMC (Garc´ıa-Varela et al. 2016).

The obtained excess colors were compared with the ones found by the Araucaria project in Figure 31. Here it can be seen that the calculated excess colors of NGC 247 and NGC 55 are closest to the values found by the Araucaria project, since they have the lowest discrepancies compared to the reported values as seen in Figure 32. It is also evident that these are the galaxies with the metallicities most similar to the one of LMC within the studied galaxies. Since their results are in agreement to the reported results by the Araucaria project, it can be said that the metallicity does have an effect on the calculated excess color. This is supported by the fact that M33 has the most different metallicity from the LMC, and its excess color differs greatly from the the result of the Araucaria project. With this it can be concluded that the PLR is not universal, because it depends on the metallicites of the galaxies and how similar are they to the one of the LMC. This is clearly seen in the case of M33, since it has a 1.37 dex difference from the metallicity of the LMC, and its excess color’s

58 error bars do not even overlap the value obtained by the Araucaria project. This is more evident in Figure 32, since this galaxy has over a 60% discrepancy with the reported value by the Araucaria project. Since the discrepancies for the excess color are greater than the ones of the distance for galaxies with a very diﬀerent metallicity than the LMC, it can be said that the metallicity aﬀects the results of the excess color and not of the distance calculus. Because of this, forcing the slope of the PLR of the LMC to be the same of the PLR’s of other galaxies is incorrect and only works if the galaxy has a metallicity similar to the LMC.

Figure 31: Comparison between the obtained results using the robust regression and the ones published by the Araucaria project. It is evident that a big discrepancy occurs between the values in M33. This galaxy is also the one with the most diﬀerent metallicity from the LMC.

59 M33

IC 1613

NGC 247 NGC 55

Figure 32: Discrepancies between the found excess color for each galaxy compared to their corresponding value reported by the Araucaria project

The dependence of the PLR on external factosr has been debated for years. Proofs of the non universality of the LMC’s PLR has been shown by several authors. Romaniello et al. (2009) researched the dependency of the PLR with the chemical components of the Cepehids, and concluded that “Our data show a clear increasing trend of the PL residuals in the V band with the iron content (i.e. metallicity). This result is in disagreement with an independence of the PLR on iron abundance and with the linearly decreasing trend found by other observational studies in the literature”. This supports a link between the PLR with the galaxies’ metallicities, which complies with conclusion that in order for the PLR to work, the galaxy must have a metallicity similar to the one of the LMC. Tammann et al. (2003) also studied PLRs and the P-C (perior-color) relations in the Cepehids of the Milky Way, LMC and SMC. They concluded that the PLR depends on the metallicity of the galaxy and that this affects the slope of the relation. They found that the ridge line of the instability strip of the Magellanic Cloud shifted bluewards, and argumented that “these shifts are caused by a metallicity-dependent blanketing effect as well as by intrinsic temperature differences that depend on luminosity. These effects explain at the same time the different slopes of the P-C relations of the three galaxies”. This

60 supports the argument that forcing the slope of the LMC on other PLRs is incorrect since, as the authors said, “The deﬁnitive conclusion is that the notion of a universal slope of the P-L relation cannot be maintained in the presence of metallicity.” All of this supports the eﬀect of the metallicity on the distance calculus using the PLR, which is shown in the resultant excess colors as mentioned before, and hence proves the non universality of the PLR.

One possible solution to the metallicity problem is to anchor the Cepheid distance scale to a diﬀerent galaxy. This galaxy must have a low metallicity and be near the Milky Way in order to be observed with telescopes with moderate apertures. One candidate for this galaxy is NGC 4258, which was proposed by Macri et al. (2006), since it has masers and thus its distance can be calculated precisely using radio astronomy and possesses 281 Cepheids and has similar metallicity to the LMC (there is only a 0.47 dex diﬀerence between both galaxies). With this geometrical distance calculus, Macri et al. (2006) found that using the maser distance to this galaxy, the distance modulus to the LMC was calculated resulting in (18.41 ± 0.13sys ± 0.10ran) mag. This shows that with this new anchor the distance modulus to the LMC changes, which in turn would change all the results found with this zero point. Macri et al. (2006) also found a relation between the metallicity and the distance modulus −1 of γ = (−0.29 ± 0.05sys ± 0.09ran) mag dex . Thus, not only does NGC 4258 have a geometrically calculated distance, which reduces the uncertainty of the extragalactic distance scale, but it also provides a metallicity correction to the distance, making it the perfect anchor for the Cepheid distance scale.

61 Chapter 4

Conclusions and future work

In this project the distance to the galaxies NGC 247, M33, IC 1613 and NGC 55 was calculated using the PLR. This relation was found for each galaxy in the VIJK filters by applying an MM -regression to their photometric data. The greatest benefit of this regression is that it does not require the rejection of data (such as outliers or influential points) since it has a 50% breakdown point. This results in a highly precise fit which gives PLRs similar to the ones obtained by the OGLE-II and the Araucaria projects. After applying the procedure described in Section 2.6 and using the LMC distance modulus of (18.5 ± 0.10) mag (Freedman et al. 2001) and of (18.477 ± 0.004) mag (Pietrzyńskiet al. 2019), two distances and an excess color were calculated for each galaxy.

Comparing the obtained distances to the ones reported in the Araucaria project, it was found that these values are really similar since all calculated distances have a low discrepancy, as seen in Figure 28. Nevertheless, the found color excesses varied in a signiﬁcant way from the values obtained by the Araucaria project, as seen in Figure 31. By plotting the galaxy’s metallicity against its calculated excess color (Figure 30) it is obvious that there is a relation between these variables. This is evident in the disagreement of the calculated values with the reported ones, since the galaxy with the most diﬀerent metallicity from the LMC (M33) had a greater error than the galaxies (NGC 247, IC 1613 and NGC 55) with a similar metallicity to the one of the LMC.

Hence this thesis presents more evidence that the PLR is not universal since it depends on the galaxies’ metallicity. As long as this metallicity is similar to the one of the LMC, the distance calculus using the PLR will give precise results. If not, the galaxies’ color excess will differ significantly from their true value. As mentioned in subsection 3.6, the metallicity mostly affects the PLR’s slope. For this reason it can be said that forcing this slope to be the same as the one of the LMC’s PLR is wrong for galaxies that have a substantially different metallicity from the LMC; causing significant errors in their color excess.

Future work must be done in order to reinforce these results. Although the results of NGC 247 and NGC 55 are precise because their metallicities are similar to the one

62 of the LMC, there is not a clear range of metallicities in which forcing the LMC’s PLR is correct. So this range must be decided in order to use the correct zero point of the PLR and obtain correct results. For the values outside this range of metallicities, a correction must be found in order to precisely calculate the distance modulus using the LMC’s PLR. Finally the possibility of using galaxies containing masers and Cepheids (such as NGC 4258) must be studied, since the geometrical distance to these galaxies can be calculated with little uncertainty using radio astronomy. This opens the opportunity of anchoring the Cepheid distance scale to one of these galaxies, allowing the continuation of these stars as standard candles and the use of Leavitt’s PLR for the calculation of extragalactic distances with high precision.

63 Appendix 1 64

(b) Leavitt’s period-luminosity relation for the star’s maximum (a) Leavitt’s plot of the apparent magnitude (ordinate)of 25 and minimum brightness. The abscissa is equal to the logarithm stars in the SMC against their periods (abcissa).The uppermost of the period and the ordinate corresponds to the stars apparent curve corresponds to the Cepheids’maximum period and the magnitude. The uppermost curve corresponds to the Cepheids’ bottom curve corresponds to its minimum period. For both maximum period logarithm and the bottom curve corresponds cases it can be seen that there is a relation between the star’s to its minimum period logarithm. It is evident that in either magnitude and its period. case a straight line can be made for each case,which shows the relation between the magnitude and the period’s logarithm

Figure A1: Henrietta Leavitt’s results for the 25 studied stars in the SMC (Leavitt & Pickering, 1912). Appendix 2

Galaxy Filters Telescope Exposition time (s) Observation Date

VI Warsaw telescope 174 (V ), 125 (I ) 8 1997-2000 LMC JK Swope & duPont telescopes Not reported 9 1993-1997 900 (VI ), 700 (V )& VI Warsaw, MPG/ESO & Blanco telescopes 300,722 (V ), 300-360 2002-2005 NGC 247 (I ), respectively JK ESO VLT 1020 (J ), 3420 (I ) 2005 Fred L. Whipple & 10 VI 900 (V ), 600 (I ) 1996-1997 M33 Michigan-Darthmout-MIT telescopes JK ESO VLT 720 (J ), 360 (K ) 2008-2009 VI Warsaw telescope 900 (VI ) 2000 IC 1613

65 JK ESO NTT 1020 (J ) ,3900 (K ) 2004 VI Warsaw telescope 900 (VI ) 2002 NGC 55 JK ESO VLT 720 (J ), 2460 (K ) 2004

Table A1: Observation log for the galaxies used in this thesis

8Taken from Udalski et al. (1999) 9This value was searched in the paper where the data of the LMC in the JK bands was taken from (Persson et al. 2004) and in the paper where the authors explained the observation procedure and data reduction used for infrared stars (Persson et al. 1998). In neither of these papers was the exposition time used reported 10Taken from Macri et al. (2001a) Appendix 3

Robust PLR for LMC

(a) Robust PLR for the LMC in the V band (b) Robust PLR for the LMC in the I band 66

Figure A2: Robust PLR for the LMC Appendix 4

Robust PLR for NGC 247

(a) Robust PLR for NGC 247 in the V band (b) Robust PLR for NGC 247 in the I band 67

Figure A3: Robust PLR for NGC 247 Appendix 5

Robust PLR for M33

(a) Robust PLR for M33 in the V band (b) Robust PLR for M33 in the I band 68

Figure A4: Robust PLR for M33 Appendix 6

Robust PLR for IC 1613

(a) Robust PLR for IC 1613 in the V band (b) Robust PLR for IC 1613 in the I band 69

Figure A5: Robust PLR for IC 1613 Appendix 7

Robust PLR for NGC 55

(a) Robust PLR for NGC 55 in the V band (b) Robust PLR for NGC 55 in the I band 70

Figure A6: Robust PLR for NGC 55 Appendix 8

Galaxy Filter β Z βFBT ZFBT V -2.74308 17.05350 -2.7430785 17.5483795 I -2.96622 16.59284 -2.9662165 16.5928407 LMC J -3.08475 16.24091 -3.0847514 16.2409122 K -3.2472 16.0074 -3.2472446 16.0073781 V 26.68945 26.689450 I 26.0949 26.094860 NGC 247 - - J 25.46869 25.469 K 25.22096 25.2209576 V 23.515 23.515 I 22.82538 22.825 M33 - - J 22.44996 22.4499603 K 22.20534 22.205 V 22.94031 22.940 I 22.54589 22.546 IC 1613 - - J 22.07679 22.076788 K 21.6839 21.683932 V 25.34195 25.342 I 24.77132 24.771 NGC 55 - - J 24.2528 24.2527904 K 23.97618 23.976178

Table A2: Comparison between the values found for the slope β and the intercept Z by the robust regression and by fast bootstrapping (FBT)

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