Old-Aged Stellar Population Distance Indicators

Old-Aged Stellar Population Distance Indicators

Rachael L. Beaton Rachael L. Beaton Hubble Fellow Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, The Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena CA 91101, 44email: rbeaton@princeton.eduGiuseppe Bono Department of Physics, University of Rome Tor Vergata INAF-Osservatorio Astronomico di Roma, Vittorio Francesco Braga Department of Physics, University of Rome Tor Vergata ASDC Massimo Dall’Ora INAF-Osservatorio Astronomico di Capdoimonte, Giuliana Fiorentino INAF—OAS Osservatorio di Astrofisica & Scienza dello Spazio di Bologna, In Sung Jang Leibniz-Institut für Astrophysic Potsdam, D-14482 Potsdam, Germany, Clara E. Martínez-Vázquez Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile, Noriyuki Matsunaga Department of Astronomy, School of Science, The University of Tokyo, Japan, Matteo Monelli IAC- Instituto de Astrofísica de Canarias, Calle Vía Lactea s/n, E-38205 La Laguna, Tenerife, Spain Departmento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain Jillian R. Neeley Department of Physics, Florida Atlantic University, 777 Glades Rd, Boca Raton, FL 33431 Maurizio Salaris Astrophysics Research Institute, Liverpool John Moores University 146 Brownlow Hill, L3 5RF Liverpool, UK    Giuseppe Bono Rachael L. Beaton Hubble Fellow Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, The Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena CA 91101, 44email: rbeaton@princeton.eduGiuseppe Bono Department of Physics, University of Rome Tor Vergata INAF-Osservatorio Astronomico di Roma, Vittorio Francesco Braga Department of Physics, University of Rome Tor Vergata ASDC Massimo Dall’Ora INAF-Osservatorio Astronomico di Capdoimonte, Giuliana Fiorentino INAF—OAS Osservatorio di Astrofisica & Scienza dello Spazio di Bologna, In Sung Jang Leibniz-Institut für Astrophysic Potsdam, D-14482 Potsdam, Germany, Clara E. Martínez-Vázquez Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile, Noriyuki Matsunaga Department of Astronomy, School of Science, The University of Tokyo, Japan, Matteo Monelli IAC- Instituto de Astrofísica de Canarias, Calle Vía Lactea s/n, E-38205 La Laguna, Tenerife, Spain Departmento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain Jillian R. Neeley Department of Physics, Florida Atlantic University, 777 Glades Rd, Boca Raton, FL 33431 Maurizio Salaris Astrophysics Research Institute, Liverpool John Moores University 146 Brownlow Hill, L3 5RF Liverpool, UK    Vittorio Francesco Braga Rachael L. Beaton Hubble Fellow Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, The Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena CA 91101, 44email: rbeaton@princeton.eduGiuseppe Bono Department of Physics, University of Rome Tor Vergata INAF-Osservatorio Astronomico di Roma, Vittorio Francesco Braga Department of Physics, University of Rome Tor Vergata ASDC Massimo Dall’Ora INAF-Osservatorio Astronomico di Capdoimonte, Giuliana Fiorentino INAF—OAS Osservatorio di Astrofisica & Scienza dello Spazio di Bologna, In Sung Jang Leibniz-Institut für Astrophysic Potsdam, D-14482 Potsdam, Germany, Clara E. Martínez-Vázquez Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile, Noriyuki Matsunaga Department of Astronomy, School of Science, The University of Tokyo, Japan, Matteo Monelli IAC- Instituto de Astrofísica de Canarias, Calle Vía Lactea s/n, E-38205 La Laguna, Tenerife, Spain Departmento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain Jillian R. Neeley Department of Physics, Florida Atlantic University, 777 Glades Rd, Boca Raton, FL 33431 Maurizio Salaris Astrophysics Research Institute, Liverpool John Moores University 146 Brownlow Hill, L3 5RF Liverpool, UK    Massimo Dall’Ora Rachael L. Beaton Hubble Fellow Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, The Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena CA 91101, 44email: rbeaton@princeton.eduGiuseppe Bono Department of Physics, University of Rome Tor Vergata INAF-Osservatorio Astronomico di Roma, Vittorio Francesco Braga Department of Physics, University of Rome Tor Vergata ASDC Massimo Dall’Ora INAF-Osservatorio Astronomico di Capdoimonte, Giuliana Fiorentino INAF—OAS Osservatorio di Astrofisica & Scienza dello Spazio di Bologna, In Sung Jang Leibniz-Institut für Astrophysic Potsdam, D-14482 Potsdam, Germany, Clara E. Martínez-Vázquez Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile, Noriyuki Matsunaga Department of Astronomy, School of Science, The University of Tokyo, Japan, Matteo Monelli IAC- Instituto de Astrofísica de Canarias, Calle Vía Lactea s/n, E-38205 La Laguna, Tenerife, Spain Departmento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain Jillian R. Neeley Department of Physics, Florida Atlantic University, 777 Glades Rd, Boca Raton, FL 33431 Maurizio Salaris Astrophysics Research Institute, Liverpool John Moores University 146 Brownlow Hill, L3 5RF Liverpool, UK    Giuliana Fiorentino Rachael L. Beaton Hubble Fellow Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, The Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena CA 91101, 44email: rbeaton@princeton.eduGiuseppe Bono Department of Physics, University of Rome Tor Vergata INAF-Osservatorio Astronomico di Roma, Vittorio Francesco Braga Department of Physics, University of Rome Tor Vergata ASDC Massimo Dall’Ora INAF-Osservatorio Astronomico di Capdoimonte, Giuliana Fiorentino INAF—OAS Osservatorio di Astrofisica & Scienza dello Spazio di Bologna, In Sung Jang Leibniz-Institut für Astrophysic Potsdam, D-14482 Potsdam, Germany, Clara E. Martínez-Vázquez Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile, Noriyuki Matsunaga Department of Astronomy, School of Science, The University of Tokyo, Japan, Matteo Monelli IAC- Instituto de Astrofísica de Canarias, Calle Vía Lactea s/n, E-38205 La Laguna, Tenerife, Spain Departmento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain Jillian R. Neeley Department of Physics, Florida Atlantic University, 777 Glades Rd, Boca Raton, FL 33431 Maurizio Salaris Astrophysics Research Institute, Liverpool John Moores University 146 Brownlow Hill, L3 5RF Liverpool, UK    In Sung Jang Rachael L. Beaton Hubble Fellow Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, The Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena CA 91101, 44email: rbeaton@princeton.eduGiuseppe Bono Department of Physics, University of Rome Tor Vergata INAF-Osservatorio Astronomico di Roma, Vittorio Francesco Braga Department of Physics, University of Rome Tor Vergata ASDC Massimo Dall’Ora INAF-Osservatorio Astronomico di Capdoimonte, Giuliana Fiorentino INAF—OAS Osservatorio di Astrofisica & Scienza dello Spazio di Bologna, In Sung Jang Leibniz-Institut für Astrophysic Potsdam, D-14482 Potsdam, Germany, Clara E. Martínez-Vázquez Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile, Noriyuki Matsunaga Department of Astronomy, School of Science, The University of Tokyo, Japan, Matteo Monelli IAC- Instituto de Astrofísica de Canarias, Calle Vía Lactea s/n, E-38205 La Laguna, Tenerife, Spain Departmento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain Jillian R. Neeley Department of Physics, Florida Atlantic University, 777 Glades Rd, Boca Raton, FL 33431 Maurizio Salaris Astrophysics Research Institute, Liverpool John Moores University 146 Brownlow Hill, L3 5RF Liverpool, UK    Clara E. Martínez-Vázquez Rachael L. Beaton Hubble Fellow Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, The Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena CA 91101, 44email: rbeaton@princeton.eduGiuseppe Bono Department of Physics, University of Rome Tor Vergata INAF-Osservatorio Astronomico di Roma, Vittorio Francesco Braga Department of Physics, University of Rome Tor Vergata ASDC Massimo Dall’Ora INAF-Osservatorio Astronomico di Capdoimonte, Giuliana Fiorentino INAF—OAS Osservatorio di Astrofisica & Scienza dello Spazio di Bologna, In Sung Jang Leibniz-Institut für Astrophysic Potsdam, D-14482 Potsdam, Germany, Clara E. Martínez-Vázquez Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile, Noriyuki Matsunaga Department of Astronomy, School of Science, The University of Tokyo, Japan, Matteo Monelli IAC- Instituto de Astrofísica de Canarias, Calle Vía Lactea s/n, E-38205 La Laguna, Tenerife, Spain Departmento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain Jillian R. Neeley Department of Physics, Florida Atlantic University, 777 Glades Rd, Boca Raton, FL 33431 Maurizio Salaris Astrophysics Research Institute, Liverpool John Moores University 146 Brownlow Hill, L3 5RF Liverpool, UK    Noriyuki Matsunaga Rachael L. Beaton Hubble Fellow Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, The Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena CA 91101, 44email: rbeaton@princeton.eduGiuseppe Bono Department of Physics, University of Rome Tor Vergata INAF-Osservatorio Astronomico di Roma, Vittorio Francesco Braga Department of Physics, University of Rome Tor Vergata ASDC Massimo Dall’Ora INAF-Osservatorio Astronomico di Capdoimonte, Giuliana Fiorentino INAF—OAS Osservatorio di Astrofisica & Scienza dello Spazio di Bologna, In Sung Jang Leibniz-Institut für Astrophysic Potsdam, D-14482 Potsdam, Germany, Clara E. Martínez-Vázquez Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile, Noriyuki Matsunaga Department of Astronomy, School of Science, The University of Tokyo, Japan, Matteo Monelli IAC- Instituto de Astrofísica de Canarias, Calle Vía Lactea s/n, E-38205 La Laguna, Tenerife, Spain Departmento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain Jillian R. Neeley Department of Physics, Florida Atlantic University, 777 Glades Rd, Boca Raton, FL 33431 Maurizio Salaris Astrophysics Research Institute, Liverpool John Moores University 146 Brownlow Hill, L3 5RF Liverpool, UK    Matteo Monelli Rachael L. Beaton Hubble Fellow Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, The Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena CA 91101, 44email: rbeaton@princeton.eduGiuseppe Bono Department of Physics, University of Rome Tor Vergata INAF-Osservatorio Astronomico di Roma, Vittorio Francesco Braga Department of Physics, University of Rome Tor Vergata ASDC Massimo Dall’Ora INAF-Osservatorio Astronomico di Capdoimonte, Giuliana Fiorentino INAF—OAS Osservatorio di Astrofisica & Scienza dello Spazio di Bologna, In Sung Jang Leibniz-Institut für Astrophysic Potsdam, D-14482 Potsdam, Germany, Clara E. Martínez-Vázquez Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile, Noriyuki Matsunaga Department of Astronomy, School of Science, The University of Tokyo, Japan, Matteo Monelli IAC- Instituto de Astrofísica de Canarias, Calle Vía Lactea s/n, E-38205 La Laguna, Tenerife, Spain Departmento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain Jillian R. Neeley Department of Physics, Florida Atlantic University, 777 Glades Rd, Boca Raton, FL 33431 Maurizio Salaris Astrophysics Research Institute, Liverpool John Moores University 146 Brownlow Hill, L3 5RF Liverpool, UK    Jillian R. Neeley Rachael L. Beaton Hubble Fellow Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, The Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena CA 91101, 44email: rbeaton@princeton.eduGiuseppe Bono Department of Physics, University of Rome Tor Vergata INAF-Osservatorio Astronomico di Roma, Vittorio Francesco Braga Department of Physics, University of Rome Tor Vergata ASDC Massimo Dall’Ora INAF-Osservatorio Astronomico di Capdoimonte, Giuliana Fiorentino INAF—OAS Osservatorio di Astrofisica & Scienza dello Spazio di Bologna, In Sung Jang Leibniz-Institut für Astrophysic Potsdam, D-14482 Potsdam, Germany, Clara E. Martínez-Vázquez Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile, Noriyuki Matsunaga Department of Astronomy, School of Science, The University of Tokyo, Japan, Matteo Monelli IAC- Instituto de Astrofísica de Canarias, Calle Vía Lactea s/n, E-38205 La Laguna, Tenerife, Spain Departmento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain Jillian R. Neeley Department of Physics, Florida Atlantic University, 777 Glades Rd, Boca Raton, FL 33431 Maurizio Salaris Astrophysics Research Institute, Liverpool John Moores University 146 Brownlow Hill, L3 5RF Liverpool, UK    and Maurizio Salaris Rachael L. Beaton Hubble Fellow Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, The Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena CA 91101, 44email: rbeaton@princeton.eduGiuseppe Bono Department of Physics, University of Rome Tor Vergata INAF-Osservatorio Astronomico di Roma, Vittorio Francesco Braga Department of Physics, University of Rome Tor Vergata ASDC Massimo Dall’Ora INAF-Osservatorio Astronomico di Capdoimonte, Giuliana Fiorentino INAF—OAS Osservatorio di Astrofisica & Scienza dello Spazio di Bologna, In Sung Jang Leibniz-Institut für Astrophysic Potsdam, D-14482 Potsdam, Germany, Clara E. Martínez-Vázquez Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile, Noriyuki Matsunaga Department of Astronomy, School of Science, The University of Tokyo, Japan, Matteo Monelli IAC- Instituto de Astrofísica de Canarias, Calle Vía Lactea s/n, E-38205 La Laguna, Tenerife, Spain Departmento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain Jillian R. Neeley Department of Physics, Florida Atlantic University, 777 Glades Rd, Boca Raton, FL 33431 Maurizio Salaris Astrophysics Research Institute, Liverpool John Moores University 146 Brownlow Hill, L3 5RF Liverpool, UK
Abstract

Old-aged stellar distance indicators are present in all Galactic structures (halo, bulge, disk) and in galaxies of all Hubble types and, thus, are immensely powerful tools for understanding our Universe. Here we present a comprehensive review for three primary standard candles from Population II: (i) RR Lyrae type variables (RRL), (ii) type II Cepheid variables (T2C), and (iii) the tip of the red giant branch (TRGB). The discovery and use of these distance indicators is placed in historical context before describing their theoretical foundations and demonstrating their observational applications across multiple wavelengths. The methods used to establish the absolute scale for each standard candle is described with a discussion of the observational systematics. We conclude by looking forward to the suite of new observational facilities anticipated over the next decade; these have both a broader wavelength coverage and larger apertures than current facilities. We anticipate future advancements in our theoretical understanding and observational application of these stellar populations as they apply to the Galactic and extragalactic distance scale.

Contents

1 Introduction

Standard candles drawn from old stellar populations have a significant advantage for distance scale as, with the exception of young Galactic star clusters, old stellar populations are found in every galactic structural component (disk, halo, bulge), galaxies of all Hubble types, and galaxies of all luminosities (from ultra-faint to ultra-luminous). Most importantly, from these distance indicators it is possible to map both Galactic and extra-galactic objects using tracers pulled from the same underlying stellar population, if not the same class of star. Moreover, due to the presence of old stars in most structural components of galaxies, it is possible to study nearby galaxies in three dimensions (e.g., measuring depths, orientations, etc.) and then to evaluate if there systematics in mean distances due to these complex structures (see e.g., Kunder et al, 2018). In turn, this helps us to better understand how projection effects and line-of-sight depth could bias mean distances. Thus, by virtue of being “old” standard candles have immense potential.

The goal of this chapter is to provide a comprehensive review for the primary standard candles drawn from Population II stars. The term Population II (Pop II) is an old one that originated from Baade (1944) in which the nebulous central regions of the Andromeda, M 32, and NGC 205 were first resolved into individual stars. Baade (1944) realized that these stars more closely resembled those in Galactic globular clusters (GGCs) than the “slow moving” stars in the solar neighborhood (e.g., disk stars). Later work would frame the differences as a function of age and metallicity, as Baade (1958a) concisely summarized at the Vatican conference. Interestingly, the nature of the variable stars and their proper classification into Pop I or Pop II was intimately entwined (Baade, 1958b). We focus our attention in this chapter to relatively luminous tracers that can be used for a broad range of distances and can be considered “primary”, in that there exist some absolute calibrations using parallaxes for these standard candles from before the onset of progressive Gaia data releases (DR1 and DR2 at the time of writing). These considerations result in three distance indicators: (i) the RR Lyrae (RRL), (ii) the Type II Cepheids (T2C), and (iii) the tip of the red giant branch (TRGB) stars. The basic properties of these standard candles are given in Table 1.

Both the RRL and the T2C are pulsational variables that occur when specific Pop II sequences cross the classical instability strip and these stars adhere to specific period–luminosity (PL) relationships from which distances can be determined to individual stars. In contrast, the stars that comprise TRGB are non-variable in nature111These stars likely have some intrinsic variability – as most stars do, but it is on a much smaller scale than the pulsational variables that have amplitudes 0.3 to >1 mag. and as a result, the distance measurement is performed using a population of stars, which makes it statistical in nature.

Figure 1: Optical color magnitude diagrams for the (a) LMC and (b) SMC from the Magellanic Clouds Photometric Survey (black; MCPS Zaritsky et al, 2002, 2004) with variable star identifications from OGLE-III as follows RRLs (green pluses), T2Cs (red five-pointed stars), and the Classical Cepheids (blue open circles) over plotted (Soszyński et al, 2016b, 2008, 2015, respectively). The approximate location of the TRGB is indicated with a thick orange arrow. The axes for both panels are identical for ease of comparison. We note that no corrections for extinction (Galactic or internal to the LMC) have been applied and, thus, the color widths of the populations may be broader, to the red, than their intrinsic range.

Figures 1a and 1b show the relative positions of the Classical Cepheids (blue), RRLs (green), T2Cs (red), and TRGB (orange arrow) for the Large and Small Magellanic Clouds (LMC, SMC) using variable star identifications from OGLE-III222The full variable star catalog can be queried here: http://ogledb.astrouw.edu.pl/~ogle/OCVS/ (Soszyński et al, 2008, 2015, 2016b) overplotted on a color magnitude density diagram from Magellanic Clouds Photometric Survey333Data is available here: http://djuma.as.arizona.edu/~dennis/mcsurvey/ (greyscale Zaritsky et al, 2002, 2004). Figures 1a and 1b keenly demonstrate how the T2Cs could have been confused with the Classical Cepheids, as the two populations overlap in magnitude. The three sub-classifications for the T2Cs are also visible by the distinct clumps in the LMC (Figure 1a) with these distinctions being less clear in the lower-metallicity SMC (Figure 1b). The difference in the population size for the three variable star classes is also apparent, with the T2Cs being less abundant than either the Classical Cepheids or the RRLs. Figures 1a and 1b reinforce the mean magnitude differences given in Table 1, with the TRGB being more luminous than the bulk of the T2Cs and the T2Cs being brighter than the RRLs.

As a class, the RRL have a long history in the optical, having been discovered in the 19th century in cluster diagrams, but recognition of their great potential in the infrared has come only recently (most notably, Longmore et al, 1986). The short period of RRLs make identifying them relatively simple with observations spanning only a few nights and with numerous Globular Clusters being sufficiently nearby for them to have been readily discovered by early photometric monitoring campaigns (a detailed history is given in Smith, 1995). In contrast, the T2Cs were only separated from the classical Cepheids in 1956 by Baade and, despite being a solution to a difficulty in reconciling from the distance ladder and cosmological theory, have received little attention until the long-term monitoring from the OGLE project unveiled them en masse in the LMC (Figure 1a, Soszyński et al, 2008). The recognition of the TRGB as a luminosity indicator came later still when the work of Da Costa and Armandroff (1990) provided high quality homogeneous photometry for a number of Galactic globular clusters (GGCs) transformed into absolute units by their RRL distances. The TRGB was first used to determine distances for galaxies by Lee et al (1993), who developed the analysis techniques necessary to detect the tip from color-magnitude diagrams.

Each of these distance indicators, thus, has a different volume of literature accompanying them and, as a result, have different depths of both theoretical understanding and observational applications. Often these vary not only by distance indicator, but also by wavelength regimes and objects (field stars, star clusters, galaxies) in which the techniques have been employed. Thus, the depth and breath of information provided in this review varies for each distance indicator.

Star Sub-Type [mag] [mag] [days]
RR Lyrae (RRL)
Fundamental Mode (RRab)  +0.6  -0.6 0.3 <  < 1.0
First Overtone Mode (RRc)  +0.6  -0.4 0.2 <  < 0.5
Type II Cepheids (T2C)
BL Herulis (BL Her)  -0.5  -1.0 1 < < 4
W Virginis (W Vir)  -1.0  -4.0 4 < < 20
RV Tauri (RV Tau)  -2.5  -5.0 20 < < 80
Tip of the Red Giant Branch Stars (TRGB)
Metal-Poor  -4.0  -5.5
Metal-Rich  -3.9  -6.5
Table 1: Basic Properties of Population II Distance Indicators.

Generally, a single book chapter cannot fully describe any one of these standard candles. Thus, we refer the reader to more detailed discussions that are in the literature. Of particular note are the following books: Smith (1995) on RRLs and Catelan and Smith (2015) on pulsational variables of all kinds, including both RRLs and T2Cs. Additionally, McWilliam (2011) is a set of online conference articles that present reviews of many aspects of RRL beyond those that will be discussed here, as well as other discussions relating to metal-poor, old stellar populations. Salaris and Cassisi (2005) is an excellent resource on stellar evolution and stellar populations that provides insight into all three of our distance indicators, but most especially the TRGB. Lastly, Beaton et al (2016) provides comparison of RRL and TRGB methods with Cepheids in terms of the extragalactic distance scale that may help the reader understand the recent resurgence of interest Pop II standard candles.

Our goal in this chapter is to place these Population II standard candles into the context of the distance scale by providing a sense of the current theoretical understanding and observational application of these tools. Where possible, we take a multi-wavelength approach discussing optical, near-infrared, and mid-infrared characteristics and applications. In the sections that follow we discuss each of the distance indicators in turn, with parallel discussions of theory and practice for the RRL in Section 2, a description and homogeneous PL relations for T2C in Section 3, and both a physical description and application of the TRGB is given in Section 4. The absolute scale for each distance indicator and inter-comparisons are described in Section 5. We conclude in Section 6 with an outlook for the future, in particular improvements to our physical understanding from Gaia and the observational application with future large-aperture facilities.

2 The RR Lyrae variables

Figure 2: (a) Example multi-wavelength light curves for an RRab RRL, WY Ant and (b)Example multi-wavelength light curves for an RRc RRL, RZ Cep (data from Monson et al, 2017). In each panel, the light curves are shown in ten photometric bands, which are from bottom to top , , , , , , , , , and . The optical bands are the Johnson-Cousins system, the near-infrared bands are in the 2MASS system, and the mid-infrared bands are in the Spitzer-IRAC system; a detailed discussion of the photometry homogenization is given in Monson et al (2017). In addition to the raw data points, a smoothed light curve is shown in black and a template light curve is shown in gray, if no data is present. The horizontal line indicates the mean magnitude and the dashed vertical line indicates maximum light in the optical ().

The first variable in a GGC was discovered by Williamina Fleming and reported in Pickering (1889). In 1893, Solon Bailey initiated a large scale program for imaging GGCs from the Harvard College Observatory in Arequipa, Peru with Williamina Fleming serving as his assistant. Very quickly several variables were discovered in the brightest GGCs in the Southern sky, in particular for  Centauri ( Cen). By the end of this project, Bailey had discovered over 500 variables in GGCs, which was equal in number to that found over the entire remainder of the sky! A more detailed story of the discovery of RRLs is given in the introduction to Smith (1995).444We also refer the reader to Sobel (2016) for a description of the role of Williamina Fleming in these discoveries. More specifically, Williamina Fleming was first E. C. Pickering’s maid, became the first of the Harvard ‘computers,’ and was involved in a large number of projects at the Harvard Observatory. Her accomplishments were honored when she became the first American woman elected to the Royal Astronomical Society. RR Lyrae, itself, was discovered by W. Flemming prior to 1899 and reported in Pickering et al (1901).

The observations reported in Bailey (1902) define the nomenclature for RRLs that is used to this day. RRLs come in two primary types; more specifically, those that pulsate in the fundamental mode (FU) known as RRab (a combination of the original Bailey types of a and b) and those that pulsate in the first overtone (FO) known as RRc (the Bailey type c stars). The connection between the modes of pulsation and their Bailey types was first inferred by Schwarzschild (1940) using high quality photometric-plate light curves in the cluster M 3; more specifically, Schwarzschild (1940) concluded that the the ‘c’ stars needed to pulsate in a different mode to account for the strong deviation of this class from the period-density relation for the ‘a’ and ‘b’ type variables. Schwarzschild (1940) further defined the color-edges of the instability strip concluding that no non-pulsating stars could exist in this temperature-luminosity range. Additionally, there are the RRd type stars that show two pulsation modes and an ensemble of stars discovered from the OGLE program have more complex or atypical pulsation behavior (e.g., Soszyński et al, 2016a, and references therein). The mean magnitudes of the RRL place them on the horizontal branch (HB) of GGCs, whereas at any given epoch, they will scatter above or below based on their amplitudes.

Some RRLs show an amplitude-modulation phenomenon known as the Blazhko effect (first identified in Blažko, 1907) that currently has no consensus for its physical origin, though its impacts on the light curve are well documented. A particularly interesting demonstration of the complexity of the Blazhko phenomenon can be found in Chadid et al (2014, see also references therein) that presents data at very high cadence tracing a single star during Antarctic winter for 150 days. The periods of the Blazhko effect can be from tens to hundreds of pulsation cycles for a given star (see for instance, the study of Skarka, 2014). The effect is well known and studied in the optical, but has recently been shown to persist into the band (Jurcsik et al, 2018). The impact of the Blazhko effect on distances is that these stars have more uncertainty with respect to both their periods and mean magnitudes, with the impact being proportional to the level of amplitude-modulation. Thus, these stars, when identified, are not always used for PL fitting.

Bailey et al (1919) paved the road for use of RRLs as both tracers for old stellar population (Pop. II) and as distance indicators. Subsequent investigations demonstrated that the RRL visual mean magnitude, within a given cluster, was nearly constant. The first strong evidence that RRL visual mean magnitude and the metallicity were correlated came in Baade (1958c), in which the populations of variables were compared between disk and halo star clusters and the Galactic Bulge. Additional evidence accumulated over the next few years; in particular, (i) additional probes of the HB magnitude for GGCs with metallicity estimates by Sandage and Wallerstein (1960) and (ii) comparisons between variables found in more environments, in particular the Draco dwarf Spheroidal that showed both similarities and differences to stellar sequences in the GGCs (Baade and Swope, 1961). Since those realizations, RRLs have been commonly used to estimate distances by means of a calibrated -band magnitude versus metallicity relation (e.g., Sandage, 1982, and references therein), subsequently rediscussed and calibrated several times, that is still widely used. The mean magnitude of RRLs in the -band is nearly constant ( mag), with a dependence on their chemical composition, thus making RRLs solid standard candles.

Observations of RRLs in a number of GGCs supported the evidence that the topology of the Instability Strip (IS) changes with metallicity. In particular, the first overtone blue edge (FOBE) is virtually independent of the metallicity as originally suggested by linear radiative models (Baker and Kippenhahn, 1965; Cox, 1963; Iben, 1971). This finding and observations brought forward the opportunity to use the FOBE vs period relation as both a reddening and a distance indicator. The theoretical scenario concerning the red edge of the IS was more complex, because it is caused by the increased efficiency of the convective transport when moving toward cooler effective temperatures. Pioneering nonlinear models that included a time dependent treatment of the convective transport (Deupree, 1977b; Stellingwerf, 1982), suggested that the red edge becomes cooler (redder) as the metallicity increases.

The use of RRLs as distance indicators has had a quantum jump thanks to the empirical discovery by Longmore et al (1986) that RRLs obey a linear PL relation at near–infrared wavelengths. This discovery was later soundly supported by nonlinear convective models Bono et al (1994, 2001). More recently, the advent of mid–infrared facilities on board of space telescopes, like WISE and Spitzer, led to the derivation of empirical PL relations at longer wavelengths (e.g., than 3.6m; Section 2.2). Another popular diagnostic used to derive distances is the Wesenheit function (van den Bergh, 1975; Madore, 1982), which is a reddening-free formulation of the less common PL–color relations (or PLC), to be discussed in Section 2.2.5.

RRLs are readily recognizable from their light variation. Figure 2 presents an example light curve for each of the two dominant sub-classes of RRL, an RRab (Figure 2a) and an RRc (Figure 2b) in ten photometric broadband filters from Monson et al (2017). Figure 2 illustrates the large amplitudes and unique shapes in the optical for both types of stars, while also demonstrating how the amplitudes decrease strongly as a function of wavelength. Indeed, the amplitude for RZ Cep is 0.1 mag in the IR compared to 0.5 mag in the optical. The RRab stars, in particular, have a “sawtooth” shape in the optical, but become more sinusoidal at longer wavelengths as the impact of the temperature changes become less important. The RRc stars (right), in contrast, have a shape that changes comparatively little as a function of wavelength.

RRLs are nearly ubiquitously present in Local Group galaxies. Indeed, they have been identified in all the stellar systems that host an old ( Gyr) stellar component. This evidence makes them excellent probes to investigate the structure and the old stellar populations at the early epochs of galaxy formation. They can be used to trace the components of our Galaxy (bulge, halo, thick disk) and to determine the distance and characterize the old population in Local Group (LG, distances within 1 Mpc) galaxies. Thus, RRLs provide a crucial first step to the extragalactic distance scale for Pop II stellar systems, letting us control possible systematics affecting the commonly used distance scale based on classical Cepheids.

In the following sections, we focus our attention on the theoretical and semi–empirical background for the use of RRLs as distance indicators, with a special attention to some recent developments. Despite great progress in the determination of the RRL PL over the past decade, theoretical PL relations are often used for distance determination due to the lingering uncertainties associated with the absolute zero-point and, in particular, the effect of metallicity. Thus, the sections to follow describing theoretical efforts are quite detailed to motivate the strengths and weaknesses of the theoretical PLs.

2.1 A physical description of RR Lyrae

RRLs are radially pulsating low-mass () stars in their central helium-burning phase. The radial oscillations are an envelope phenomenon that takes place in a well–defined range of effective temperatures, therefore, they populate a relatively narrow region in the Hertzsprung-Russell diagram, which is the intersection between the so called Cepheid IS and the HB. We will shortly describe the physical mechanisms driving the radial pulsation and we will analyze with some detail the different approaches to estimate distances using RRLs, highlighting their advantages and disadvantages.

The idea of a pulsating gaseous sphere was developed for the first time by Ritter (1879)555Several authors also cite Ritter’s work from a series of papers published between 1878 and 1883 in Wiedemann’s Annalen 5-20 that the authors of the present manuscript have been unable to find. Such references begin as early as Shapley (1914) and are cited as recently as Smeyers and van Hoolst (2010)., who found a simple relation between pulsation period and mean density, but it was only in Shapley (1914) that the radial pulsation hypothesis was advocated for Cepheid-like stars. The dispute between binarity and radial oscillations was settled in favor of the latter by the so-called Baade-Wesselink method. Radial pulsation is a phenomenon that involves the stellar envelope for certain values of the surface effective temperature (T) and defines in the color magnitude diagram (CMD) a region in which the stars are unstable to pulsation, the IS. It is worth mentioning that a star radially pulsates only during its crossing(s) of the IS. As a result, the most numerous pulsators are those that have long evolutionary lifetimes within the IS, such as Scuti (central hydrogen burning) or RR Lyrae and Cepheids (central helium burning).

The physical mechanisms underlying this phenomenon are related to the cyclic variations of the opacity and of the equation of state for regions in which H and He are partially ionized, which are referred to as the and mechanisms. The physics of radial oscillations was introduced by Eddington (1926) and in the early models by Cox (1958) and Zhevakin (1959). Contrary to what happens in the rest of the envelope, these regions trap energy during the contraction and loose energy during the expansion, thus acting similar to a mechanical valve. The initial perturbation may originate from a stochastic fluctuation in the external layers of the envelope. The mechanical work can either be driven, if the envelope mass located on top of the ionization regions is large enough (below a critical effective temperature that defines the blue edge of the IS) and it can be quenched by the efficiency of convective transport that penetrates deep into the stellar envelope, toward effective temperatures lower than the red edge of the IS.

Simple linear adiabatic models, such as the one developed by Eddington (1918), can predict the pulsation period and the pulsation mode, but they cannot predict the other observables such as the mean magnitudes, amplitudes, shapes of the light curve, and edges of the IS. This implies that non-adiabatic effects have to be considered to properly model the growth of the pulsational instability and the blue edge of the IS (Baker and Kippenhahn, 1965; Cox, 1963; Iben, 1971). The other fundamental pulsation observables, such as the pulsation amplitudes, the morphology of the light curves, and the topology of the instability strip (modal stability), can only be predicted with accuracy with the inclusion of non-linear terms in the hydrodynamical equations (Christy, 1966; Stellingwerf, 1974) and by taking into account the coupling between pulsation and convection (Stellingwerf, 1982, 1983; Feuchtinger et al, 1993).

Current state-of-the-art models adopt a non-linear, time dependent formalism, which also takes into account the effects of convection and its non-linear coupling with the pulsation (e.g., Bono and Stellingwerf, 1994; Bono et al, 1997b; Marconi et al, 2003; Di Criscienzo et al, 2004; Szabó et al, 2004; Marconi et al, 2015). These models assume a spherical envelope (with no rotation or magnetic fields) and then solve the hydrodynamical equations; these are the conservation of mass and momentum plus a treatment for the radiative transfer that includes convection. These equations are solved as a function of time until they approach the limit cycle stability. Although this is one of the most comprehensive approaches, we need to highlight that there is still room for improvement. In particular, the treatment of the convective transfer is non-local and time dependent. Despite this limitation, these models can account for all the observables, including the red boundary of the IS. Moreover, they appear capable of explaining more complicated observables including the double–mode pulsators and the Blazhko effect (Szabó et al, 2014).

For the sake of the subsequent discussion, we note that van Albada and Baker (1971) formalized the relation between the pulsation periods and the structural parameters of mass (), luminosity (), effective temperature (), to which Marconi et al (2015) adds composition (). The relations take the following form:

(1)

The most recent physical pulsation relations are given in Marconi et al (2015) and are as follows:

(2)
(3)

for FU (Equation 2) and FO pulsators (Equation 3), respectively. These physical relations are the basis for all the theoretically-determined pulsation relations (e.g., those projected into observable quantities), which are discussed in the following subsections.

We conclude here highlighting that recent theoretical investigations (e.g., Marconi et al, 2011, 2016) have shown that the pulsation properties of RRLs are also affected by the helium content. The Helium abundance impacts RRL evolutionary properties, like the total bolometric luminosity and the evolutionary timescales, which in turn impact the pulsation properties and the the interpretation of observed quantities. Figure  3 shows theoretical HB models in color-magnitude space from “a Bag of Stellar Tracks and Isochrones” (BaSTI) (most recently, Pietrinferni et al, 2004, with earlier works referenced therein)666The models are publicly available: http://basti.oa-teramo.inaf.it/ for a fixed total metal content (), but with different Helium contents of (solid hashing), (dotted hashing), and (dashed hashing). The First Overtone Blue Edge (FOBE) is shown for reference. From Figure 3, the impact on the mean absolute magnitude () is demonstrated, with those stars having greater Helium enrichment being systematically brighter and having a slightly different temperature distribution. Beyond this, at fixed metal abundance, the pulsation period is expected to increase and the pulsation amplitude to decrease as the helium abundance increases. The width of the IS, however, is minimally affected.

Figure 3: Theoretical HB models from in the color-magnitude diagram. Solid, dotted and dashed regions indicate different assumptions of the helium content for a fixed overall metallicity abundance. The vertical, blue line shows the color of the First Overtone Blue Edge (FOBE) that is almost constant for the metallicity range of the RRLs. All the theoretical models used here are taken from BaSTI (Pietrinferni et al, 2004).

2.2 RR Lyrae as Standard Candles

This subsection focuses on the origin of various relationships that can be used to determine distances using RRL. First, optical relationships are explored, these being (i) the -Metallicity relationship is explained in Section 2.2.1 and (ii) the FOBE method in Section 2.2.2. Then the, relatively new, NIR and MIR relationships are explained in Section 2.2.3. The multi-wavelength slopes are explored in Section 2.2.4. Lastly, reddening-free relationships constructed from multi-band observations are explored in Section 2.2.5.

2.2.1 The visual magnitude–metallicity relation

The original idea of a relation between the mean luminosity of RRLs and their metallicity dates back to Baade and Swope (1955) and Sandage (1958), who discovered a correlation between the mean periods and iron abundances for cluster variables (see also Arp, 1955). These studies were driven by the evidence of a well defined dichotomy among the RRLs belonging to GGCs (Oosterhoff, 1939). Oosterhoff (1939) found two groupings of fundamental mode RRLs, one with mean periods  days and [Fe/H] and one with  days and [Fe/H]-1.5. These two groups were later christened the Oosterhoff I (Oo I) and II (Oo II) groupings, respectively. Subsequently, this effect was also confirmed for field stars by Preston (1959). Figure 4 demonstrates the Oosterhoff dichotomy for GGCs using data from Catelan (2009). Figure 4 demonstrates two groupings of clusters in P-[Fe/H] space that separated by a region known as as the “Oosterhoff gap.” The dichotomy is explained as the intrinsic luminosity for the RRLs in two Oo groups being different, with the higher metallicity Oo I clusters being fainter (Sandage, 2006). This physical understanding is largely supported by stellar evolutionary models (e.g., Cassisi and Salaris, 2013), where higher metallicity stars have both a lower helium-core mass (main parameter that setting the HB luminosity) and a higher opacity in the envelope.

Figure 4: Demonstration of the Oosterhoff classification for GGCs with more than ten RRab using the compilation of Catelan (2009). Both “young” and “old” halo clusters have been included in this visualization. Left panel: The distribution of the mean period (P) against the metallicity ([Fe/H]) of the cluster. Right panel: Histogram of of the mean periods that illustrates the distinct separation between the Oo-I and Oo-II type clusters. The lack of clusters with P 0.60 days is known as the Oosterhoff gap.

We demonstrate the metallicity-trend in Figure 5 where a population of synthetic HB stars are shown for different chemical compositions () from the RRL portion of the IS (drawn from the same models shown in Figure 3). For these stars we have computed a period using the pulsation relations given in Marconi et al (2015). We show that for metal contents ranging from ([Fe/H] dex) to ([Fe/H] dex) and for a fixed -enhancement ([/Fe] dex), synthetic periods become shorter by and luminosities become fainter by a factor of  mag (from bottom to top in Figure 3). However, the offset in luminosity cannot be the only cause of the difference in the mean period of fundamental RRLs between Oo I and Oo II clusters. Bono et al (1997a, and references therein) show that the dichotomy is largely due to the different pulsation behavior in the “OR region,” which is the intersection between the FU and FO IS, as a result of a hysteresis mechanism originally suggested by van Albada and Baker (1973).

Figure 5: Synthetic period-magnitude diagrams for a population of RRLs as a function of metallicity from (bottom panel) to Solar (top panel). A range of mean masses have been adopted for each simulation as indicated by color-coding and symbols shapes to highlight the population effects on both the mean magnitude and period.

Given this relation between magnitude and metallicity, it is common to assume a linear relation between the RRL -band absolute magnitude, , and the stellar metallicity, typically expressed as [Fe/H] in the following form:

(4)

Many different calibrations have been suggested in the literature for the slope () and the zero point (; e.g., Liu and Janes, 1990; Carney et al, 1992; McNamara, 1997; Clementini et al, 2003; Bono et al, 2003, among others)

Despite the simplicity of Equation 4, the -[Fe/H] relation as a distance diagnostic has a number of drawbacks. First of all, the reddening value and the extinction law have an impact on the absolute magnitude. The extinction coefficient is quite large in optical band, = 3.1 (Cardelli et al, 1989), but it decreases by one order of magnitude when approaching longer wavelengths (see also Section 2.2.3). The reddening law, itself, may not be universal (Kudritzki and Urbaneja, 2012). Moreover, several complicated astrophysical effects have a strong impact on the stability of the relationship including:

  1. intrinsic deviations from the linear form of Equation 4 (e.g., a form non-linear with iron abundance; Bono et al, 2003, 2007),

  2. evolutionary effects within RRL populations,

  3. measurement uncertainties for the metallicity, which include both systematic differences between metallicity scales (e.g., calibrations) and methodologies (e.g., what abundance is actually being measured),

  4. measurement uncertainties for the -enhancement, and

  5. deviation from static magnitudes (see also Caputo et al, 2000).

Some additional information on these effects are given in the list below:

  1. Linearity:   The assumption of the linear –[Fe/H] relation (Equation 4) has been challenged many times. There is strong empirical evidence that the –[Fe/H] relation is not linear over the whole metallicity range covered by the observed GGCs (e.g., Rey et al, 2000; Caputo et al, 2000). This is also supported by both pulsational (e.g., Bono, 2003; Caputo et al, 2000; Di Criscienzo et al, 2004) and evolutionary (e.g., Catelan et al, 2004) models. The slope is observed to get steeper at approximately [Fe/H] and, as a result, a quadratic form of this relation has been proposed (see e.g., Caputo et al, 2000; Catelan et al, 2004; Di Criscienzo et al, 2004; Sandage and Tammann, 2006; Sandage, 2006; Bono et al, 2007, among others). To provide the reader with a quantitative idea of the dependence on the metallicity and of the uncertainties, we report here the calibration by Clementini et al (2003) that is widely used:

    (5)

    where the zero-point reflects the distance to LMC and the quoted uncertainties are only those on the coefficient of the linear regression. In the literature, it has been pointed out that theoretical calibrations of the –[Fe/H] relation consider only the Zero Age HB (ZAHB) models, and hence they do not fully take into account evolutionary effects (e.g., Catelan et al, 2004; Sandage and Tammann, 2006). However, empirical calibrations such as Caputo et al (2000) are built on observed data, with RRL variables at different evolutionary stages, and therefore they represent a “mean” evolutionary state/status for the HB. The quadratic form of the calibration, suitable for -2.4 <[Fe/H] <0.0, determined in Bono et al (2007) and applied to the Galactic Bulge and Sagittarius dwarf in Kunder and Chaboyer (2009) is as follows:

    (6)

    However, it is worth noting that no deviation from linearity has been observed within the LMC over a broad metallicity range (Clementini et al, 2003). Thus, the interplay between evolutionary effects (discussed below) and metallicity is difficult to decouple in observational work. A canonical example of this difficulty is  Centauri, which is home multiple stellar populations, and has been deemed a “red herring” in the review of Smith (1995) due to the complexity of disentangling these effects (a more recent study is Braga et al, 2016, 2018, with additional references therein).

  2. Evolutionary Effects:  The evolutionary effects on the –[Fe/H] relation are shown in Figures 3 and  5. We have plotted for each selected metallicity several realizations of the HB morphology coming from different assumptions on the mean mass distribution on the HB. This is mimicking different possible values of the mass loss efficiency during the Red Giant Branch (RGB) phase, which has the result of changing the HB morphology. A comprehensive discussion of the role of mass loss on the HB morphology can be found in Salaris et al (2002, and references therein). By an inspection of Figure 5, it is clear that the HB width is not narrow due to the overlap in color-space of stars on the ZAHB and stars evolving off the ZAHB on their path to the Asymptotic Giant Branch (AGB) phase. The evolved-RRL have higher luminosities. The presence of both ZAHB-RRL and evolved-RRL on the HB defines the “evolutionary effect” As a result of the time spent on the HB, we observe RRLs until they have burned most of the He in their cores (up to 90 %). The region where a RRL spend most of its life time is also shown in Figure 3 for different He values and a given mean metallicity. This effect implies a broadening of the optical magnitude distribution across the HB that is also correlated with the metal content, which is supported by both theory (Bono et al, 1995) and observations (Sandage and Katem, 1982).

  3. Size of the He Core:  There are other “ingredients” that affect the theoretical calibration of the –[Fe/H] relation, which are related to the size of the He core reached before the He ignition. These include:

    1. The initial He value: This can be summarized as the primordial He content plus a Y/Z, which is commonly measured from H II regions. There is evidence that Y/Z may depend on the environment (e.g., Peimbert and Peimbert, 2010). An increase in total He abundance of  dex implies  mag as is demonstrated in Figure 3.

    2. Details of Red Giant Evolution: More specifically, the processes occurring along the RGB phase that delay the ignition of the 3- reaction in the electron degenerate core. These include atomic diffusion, electron conductive opacity, loss of energy via neutrinos, and core rotation. Their impact on the HB luminosity can be up to  dex or  mag (Salaris et al, 2002; Serenelli et al, 2017).

2.2.2 The FOBE method

The location in the period–magnitude diagram of the FOBE can be used as a distance indicator for a population of RRL independent of the -[Fe/H] relationship. The FOBE method was extensively described for the first time by Caputo (1997). This is a graphical, or topological, method that produces accurate distances for those stellar systems with sizable samples of FO RRLs (Fiorentino et al, 2010). However, it can be reliably used only when the blue part of the IS is well populated (Stetson et al, 2014b; Martínez-Vázquez et al, 2017). Once the metallicity is known, preferably from spectroscopic measurements, and a reliable mass for the RRLs can be estimated (typically in the range 0.5 ; Bono et al, 2003), the following theoretical relation from Caputo et al (2000) can be used to fit the blue edge defined by FO RRLs to predict the absolute magnitude:

(7)

where is the period of the bluest FO pulsators and is the magnitude of the bluest FO pulsators for a given composition and mass. Although, an assumption on the RRL mass has to be made (Bono et al, 2003), the possibility to use the FOBE method relies on the well understood negligible dependence of the color of the FOBE on the metallicity (Bono et al, 1997a), which is demonstrated in Figure 3. For the bluest FO RRLs the period is most strongly dependent on luminosity and mass.

The sharp, blue edge of the IS occurs because for a given mass and luminosity, as the surface temperature increases there is less mass above the ionization zones, and in turn, their contribution to the work integral () decreases. Then, the FOBE color is essentially fixed by the minimum difference between the temperature of the stellar surface and that of the Hydrogen ionization region, within which pulsation is efficient. The minimum difference is almost constant and does not depend on metallicity, at least for the metallicity range typical of RRLs. On the contrary, the red edge of the IS strongly depends on metallicity, because the quenching mechanism for the radial pulsations is related to convection that increases with the stellar opacity, and thus by increasing the metal content (Deupree, 1977a; Stellingwerf, 1982). The above theoretical prediction has been observed in GGCs; more specifically, the FOBE color is always 0.2 mag. Thus, the FOBE method can also be used to estimate the mean reddening of a stellar system (see Walker, 1998, for an example).

2.2.3 The NIR period–luminosity relations

Unlike in the optical, the observations in the IR show a true PL relationship (first identified by Longmore et al, 1986). In Figure 6, the mean magnitudes for RRL in the star cluster, Reticulum, are shown for eight photometric bands () against the fundamentalized period, which is defined as follows (Bono et al, 2001):

(8)

Reticulum is an ideal system to visualize the behavior of the PL slope with wavelength, because it has no appreciable metallicity spread on the RGB (Grocholski et al, 2006) and, thus, scatter due to metallicity differences are minimized. In Figure 6, preliminary PL fits are shown for each band as solid lines with the scatter about the PL shown as the shaded regions (Catelan and Smith, 2015, Monson et al., in prep.). As previously discussed, the band shows no PL slope, whereas a slope appears and becomes progressively steeper at the longest wavelengths. In the next sections, we discuss empirical and theoretical results on the RRL PL.

Figure 6: Mean apparent magnitudes for RRL in the Reticulum star cluster in the optical (), NIR (), and MIR ([3.6],[4.5]) versus the fundamentalized period (Equation 8). PL relationships are overplotted (solid lines) for each of the bands (adapted from Catelan and Smith, 2015, Monson et al., in prep.). The change in the slope from the optical into the mid-infrared is noticeable, with being a nearly flat slope to the mid-infrared being quite steep.

The -band period–luminosity relationship

Period–luminosity–color (PLC) relations have been predicted for RRL by even the earliest theoretical models (e.g., van Albada and Baker, 1971), but their use has always been hampered by several uncertainties on the structural parameters of the stars, in particular mass, effective temperature and metallicity. Evolutionary effects also play a role, because during their evolution (the He burning phase lasts at most for 100 Myr), RRLs change their luminosities and effective temperatures. Moreover, observational uncertainties, such as those associated with reddening and extinction, have to be decoupled from true astrophysical differences. Thus, empirical confirmation of the PLC has only come of late.

Figure 7: PL distribution from optical (bottom panel) to mid-infrared (top panel). We have used synthetic HB models as in Figures 3 and 5 for fixed abundance () and a canonical He abundance () with different mean mass for the population distribution.

Longmore et al (1986) demonstrated, on pure empirical basis, that RRLs follow a tight linear PL relation in the -band (PL). They also demonstrated that the PL relation can be derived from the general equation of the pulsation, since the period dependence on the -band can be derived from the proxy of the effective temperature (as a direct consequence of the fact that RRLs follow PLC relations). In other words, a PL arises because of the increasing bolometric correction as a function of the effective temperature, with the redder RRLs having -band magnitudes brighter than the bluer RRL (see Figure 7, for the K band). Moreover, since for a given cluster, the effective temperature distribution inside the IS is related to the period distribution, such that the stars with longer periods are also cooler, the magnitude-effective temperature relation becomes a magnitude-period relation. A more general empirical calibration of the PL was given by Janes and Liu (1992) and these authors also considered the dependence on the metallicity, which was shown to have little impact on the slopes.

A sound and complete theoretical derivation of the -band period–luminosity–metallicity (PLZ) relation was subsequently explored by Bono et al (2001) and Bono et al (2003), finding the following:

  1. the uncertainties on the mass and luminosity (e.g., on the evolutionary effects) have only a mild impact on the PLZ.

  2. the overall dependence on the metallicity is lower, at  mag dex in [Fe/H], than that in the optical bands ( >0.2 mag dex, see Equation 5).

On the latter aspect, we remark that Sollima et al (2006) found an even lower metallicity term (=0.08 mag dex in [Fe/H]), on the basis of an empirical calibration. Muraveva et al (2015) combined low-dispersion spectroscopy of 70 RRLs in the LMC, with  dex, with NIR photometry collected with VISTA finding a very low dependence on the metallicity ( mag dex in [Fe/H]), and a quite steep dependence on the period (). All these aspects reflect on a very low dispersion of the PLZ relation, being  mag rms or <2% in distance (Bono et al, 2001).

PL and PLZ observational relations have been successfully tested on several stellar systems, among the others we mention the LMC cluster Reticulum (Dall’Ora et al, 2004) and the Galactic GGCs IC 4499 (Storm, 2004), M 92 (Del Principe et al, 2005), Omega Centauri (Del Principe et al, 2006), M 5 (Coppola et al, 2011), and M 4 (Braga et al, 2015). The observed scatter about the PL ranges from 0.03 mag (Reticulum) to 0.09 mag (IC 4499), which suggests a single-star distance uncertainty ranging from 1.5% to 4.5% is feasible. Additional studies have occurred in dwarf galaxies such as the LMC (Ripepi et al, 2012a; Moretti et al, 2014), SMC (Szewczyk et al, 2009; Muraveva et al, 2018c), Fornax (Karczmarek et al, 2017a), Sculptor (Pietrzyński et al, 2008), Carina (Karczmarek et al, 2015), and IC 1613 (Hatt et al, 2017).

Compared to the –[Fe/H] relationship, the key advantages distance determination at longer wavelengths are:

  1. the impact of extinction is less by a factor of ten or more and, as a result, the impact of differential extinction is also greatly reduced;

  2. the impact of the uncertainties the reddening measurement, itself, are minimized (this being what is determined to estimate extinction);

  3. the pulsation amplitude is lower than in the optical bands (Figure 2) and the light curves are more symmetrical, accurate mean magnitudes can be measured even with a few data points;

  4. the availability of light curve templates (e.g., Jones et al, 1996; Monson et al, 2017; Hajdu et al, 2018), permits the estimation of the mean magnitudes even with a single data point, if the ephemerides of the variable are known from the optical photometry (see applications in Beaton et al, 2016, Rich et al. submitted).

On the other hand, finding the variables, measuring their periods, and determining their pulsation mode (FU or FO) is more difficult due to smaller amplitudes, sinusoidal shape, and the more uniform light curve shapes between the sub-classes. Moreover, templates is primarily only available for for the RRab type stars.

Mid-IR period–luminosity relations

The advantages of using PL relations at mid-infrared wavelengths (MIR, from 3.4 to 22 m) with Cepheids have been well-recognized in the literature since McGonegal et al (1982), but only very recently it has been possible to explore this wavelength regime with RRLs. Indisputable advantages to the MIR PLs is that at 3.4 m the impact of interstellar absorption is more than one order of magnitude lower than in the optical bands, which permits study of heavily obscured and differentially obscured regions like the Galactic bulge that contain a large population of RRL (see discussion in Kunder et al, 2018).

Indeed, on the basis of the Wide Field Infrared Survey Explorer (WISE) measurements of field RRL variables, Klein et al (2011) were able to produce the first calibration of the PL relations at the W1 (3.4 m), W2 (4.6 m), and W3 (12 m) bands. They subsequently refined their calibrations in Klein et al (2012) and Klein et al (2014), with no metallicity term. Another absolute calibration of the MIR PL relations was published by Madore et al (2013), using four field RRLs for which individual trigonometric parallaxes, via the HST Fine Guidance Sensors (HST-FGS), were available (Benedict et al, 2011). Even if with only four objects, the observed dispersion obtained by Madore et al (2013) is a very promising 0.1 mag (5% in distance). Madore et al (2013) also found, as expected, only a marginal dependence on the metallicity, because at the MIR wavelengths stars with the temperature of RRLs have few molecular bands and metallic lines. Based on a much larger sample of 129 RRLs with Hipparcos-based statistical-parallaxes distances, Klein et al (2014) obtained very nice results, with observed spreads ranging from 0.01 mag to 0.05 mag, depending on the filter, on the sample (FU or FO pulsators), and assumptions on the intrinsic width of the PL. Neeley et al (2017) expanded the Spitzer-analysis to a larger sample of RRL with parallaxes from the the TGAS catalog in Gaia-DR1 (e.g., Michalik et al, 2015), finding consistent solutions within the observational uncertainties. A complementary analysis from Sesar et al (2017) used WISE photometry with the TGAS parallaxes employing a Bayesian algorithm, which Muraveva et al (2018a) has expanded upon recently using Gaia-DR2.

The first calibration of the MIR PL using cluster variables was published by Dambis et al (2014), which presented an analysis of WISE data for 15 GGCs. Neeley et al (2015, 2017) published data of the GGC M 4, collected with the Infrared Array Camera (IRAC) on-board the Spitzer Space Telescope, obtaining PL relations in the IRAC and bands with a scatter of 0.05 mag (2.5% in distance). Muraveva et al (2018b) has recently published a calibration in the LMC star cluster Reticulum.

2.2.4 The RR Lyrae period–luminosity slopes

Because the bolometric correction sensitivity to effective temperature is already at work starting from the -band (see Cassisi et al, 2004; Catelan et al, 2004; Marconi et al, 2015), PL relations have also now been derived theoretically for the photometric bands , , and , which are shown in Figure 7. At shorter wavelengths, the effects related to the finite width of the IS and evolution of RRLs are larger and produce PL relations with larger intrinsic scatter (compare the panels of Figure 7). The relationships in , , and are still of great interest, because these wavelengths can be observed easily from the ground.

Figure 8: Following Madore et al (2013, their figure 4), the PL slopes are plotted a function of wavelength from both theoretical (circles; Catelan et al, 2004; Marconi et al, 2005a, 2015; Neeley et al, 2017) and empirical (triangles; Klein et al, 2011; Madore et al, 2013; Braga et al, 2015; Dambis et al, 2014; Neeley et al, 2015, 2017) studies. We note that studies in the same photometric band have been offset slightly in their effective wavelength for visualization purposes. The shaded region is drawn to guide the eye following Madore et al (2013). The slopes for the long-wavelength filters asymptotically approach that predicted by period-radius relationship for both theory (dashed line Marconi et al, 2015) and empirical determinations (dotted line Burki and Meylan, 1986).

In Figure 8, the PL slopes determined from a subset of both empirical (triangles) and theoretical (circles) investigations are plotted versus wavelength following the example of Madore et al (2013). These slopes are for the “global” population, e.g., the first-overtone periods have been fundamentalized following Equation 8. The shaded region in Figure 8 is meant to guide the eye to the behavior with wavelength and this is adapted from the visualization in Madore et al (2013, their figure 4). Generally, the slopes between the different studies, both theoretical and empirical, agree well. Still, there are comparatively few studies on the PL for wavelengths longer than and further investigations, in particular empirical studies in star clusters with differing chemical compositions and evolutionary states, are highly relevant to determining a “consensus” PL slope for a given filter.

In Figure 8, the “predicted” slopes from the period-radius (PR) or period-radius-metallicity (PRZ) relationships are also shown following Madore et al (2013). NIR and MIR wavelengths sample the Rayleigh-Jeans tail for RRLs and, as a result, the magnitudes are much less sensitive to temperature than to the radial variations. This behavior is readily seen in the light curves shown and, indeed, the light curves beyond are nearly indistinguishable. Thus, predicting the PL slope from the period-radius relationship places an asymptotic limit on the PL slopes.

Starting from and converting into a logarithmic form yields the following,

(9)

which can be converted into magnitudes as follows,

(10)

From here, we can substitute either the period-radius (PR) or period-radius-metallicity (PRZ) relationship to remove the radius term. The PR has been determined in several ways, (i) empirically by Burki and Meylan (1986) using the Baade-Wesselink technique (BW), (ii) stellar evolutionary models by Marconi et al (2005b) (evol), and (iii) pulsation models by Marconi et al (2015) (puls). The PR/PRZ takes the form, , and the three studies find period slopes of , , and , respectively. Substituting, these into Equation 10, we estimate determine period slopes of , , and .

If we assume that the temperature term contributes very little at long wavelengths then we can ignore that term to obtain an estimate for the slope of the PL relationship. The estimated slopes from the theoretical PR and PRZ (dashed lines; Marconi et al, 2015) and the empirical PR (dotted lines Burki and Meylan, 1986) are shown in Figure 8. The overall agreement between these asymptotic values and the slopes in Figure 8 serves as a good cross-check on our theoretical and empirical measurements.

2.2.5 Reddening-free relations

Even if PLZ relations have advantages for estimating individual distances, they are still affected by the uncertainties in the reddening corrections, and the problem can be severe in regions affected by strong differential reddening (e.g., as in M4 Braga et al, 2015, or in the Galactic Bulge).

To circumvent this problem, van den Bergh (1975) and Madore (1982) developed the Wesenheit pseudo-magnitudes. They are, by construction, “reddening free” functions that redefine the observed magnitude. The fiducial form of the transformation is defined as:

(11)

where is the selective-to-total extinction ratio. In the common practice, “Cardelli’s law” is adopted (Cardelli et al, 1989).

The Wesenheit period–luminosity (PW) and period–luminosity–metallicity (PWZ) relations have the following form:

(12)

where each combination of filters will have its own set of PWZ parameters. The advantages and disadvantages of PW and PWZ relations have been discussed several times in the literature, in particular for classical Cepheids (e.g., Inno et al, 2013; Fiorentino et al, 2007; Ngeow et al, 2005; Storm et al, 2011; Ripepi et al, 2012b) and, more recently, with RRLs (e.g., Braga et al, 2015; Marconi et al, 2015, and references therein).

Briefly, the main features can be summarized as follows:

  1. the individual distances are independent of the uncertainties in the reddening or in differential reddening within an RRL population;

  2. a universal reddening law is assumed, but there are indications of deviations from “Cardelli’s law” in regions of high obscuration (Cardelli et al, 1989);

  3. PW and PWZ relations mimic PLC relations, since they host the color term: this means that the effect of the width of the IS is reduced, and individual distances are much more precise than those obtained with simple PL relations.

Thus, in regions where the assumption of universal reddening law is reasonable, the PW/PWZ is a powerful tool.

On the practical side, however, use of the PW/PWZ are more observationally expensive than common PL and PLZ relations, because they need accurate mean magnitudes determined in two photometric bands. According to the models by Marconi et al (2015), the dispersion around the PW and PWZ relations range from  mag (in the NIR bands) to 0.07–0.08 mag (in the optical bands). Braga et al (2015) provide complementary empirical PW/PWZ relations determined from M 4 in the optical and NIR. The PW relation is almost independent of metallicity and are particularly useful.

We also note that, on empirical basis, Riess et al (2011) proposed to use three-band PW relations, where the magnitude is reconstructed from the following form,

(13)

where is the ratio between the selective absorption in the band, and the color excess for the color . The advantages of triple band PW/PWZ relations are a smaller dispersion respect to the two band relations and lower systematics from correlated errors having used the same magnitude in the color term. The drawback is that they need accurate mean magnitudes determined for three bands. The most advanced theoretical study on this topic to date is that of Marconi et al (2015), which provides theoretical three-band relations for RRL in the optical and NIR.

2.3 Case studies using RR Lyrae

In the previous section, a general physical understanding of RRLs as distance indicators was developed using both theoretical and empirical investigations. In this section, a series of case studies are presented on the practice of using RRLs for the different wavelength regimes discussed previously. In Section 2.3.1, the –[Fe/H] relationship and the optical PW relationships are applied to the Sculptor dwarf galaxy In Sections 2.3.2 and 2.3.3, the optical, near-infrared and mid-infrared PL relations are determined and applied to the nearby GGC M 4 that suffers from strong differential reddening in the optical.

2.3.1 An optical study of RR Lyrae in Sculptor

Figure 9: Period-amplitude diagrams for the RR Lyrae stars in Sculptor in three different optical bands: (left), (middle) and (right). RRab and RRc are shown as black circles. For the sake of clarity, RRd stars are not plotted as their periods are less certain. The RRc and RRab occupy specific regions in the Bailey diagram, with the RRc having shorter periods and smaller amplitudes (i.e., in the bottom left) than the RRab’s (right). The RRab’s also have a broader range of amplitudes. These primary groups in the Bailey diagram can be further refined into specific sequences for the Oo groups, with two clear sequences visible for the RRab’s.

Here we present a brief summary of a comprehensive study of the RRLs in the Sculptor dwarf galaxy (Martínez-Vázquez et al, 2015, 2016a, 2016b). Sculptor is a Milky Way dwarf spheroidal (dSph) satellite with a complex chemical enrichment history (Smith and Dopita, 1983; Da Costa, 1984; Majewski et al, 1999; Hurley-Keller et al, 1999; Tolstoy et al, 2004; de Boer et al, 2012; Starkenburg et al, 2013). Tolstoy et al (2004) demonstrated that the red HB stars (RHB) are more centrally concentrated than the blue HB stars (BHB) that mirrors gradients from detailed spectrocopic studies on the RGB Battaglia et al (2008); Walker et al (2007, 2009); Kirby et al (2009); Leaman et al (2013); Ho et al (2015), which suggests a that range of metallicites in also present the RRL population.

Time series photometry was produced for stars Sculptor from a set of 5149 individual images collected over 24 years. Variables were identified using the Welch-Stetson variability index (Welch and Stetson, 1993) and periods obtained from a simple string-length algorithm (Stetson et al, 1998) from which 536 RRLs were identified using the the shapes of the light curve. The final dataset contains pulsational properties and mean magnitudes for 536 RRLs in the , , and bands that is complete over 70% of Sculptors area.

Figure 10: The distribution for Sculptor RRL. Left panel: Zoom-in on the HB stars in the color-magnitude diagram of Sculptor. The horizontal line (at  mag) displays the split between bright (gray) and faint (black) RRL subsamples. Open diamonds and circles represent the RRab and RRc stars, respectively. Right panel: Luminosity distribution function in band for Sculptor RRLs. The arrow marks the magnitude adopted to split into bright and faint RRL subsamples. We note that the RRd and other more complicated RRL sub-types have been removed from this sample.

Figure 9 shows the Bailey Diagrams for the , , and bands. The RRab and RRc stars separate into long- and short-period groups, that have their own amplitude behavior. The overall reduction of the amplitudes from the to the band is apparent. In the and the sub-division of the primary RRab and RRc sequences into Oo groups can also be seen, which is anticipated from the large metallicity spread.

Figure 10 shows the magnitude distribution for the Sculptor RRLs. A zoom-in on the CMD is shown in the left panel and the corresponding luminosity function for the RRLs is shown in the right panel. The RRL stars of Sculptor show a spread in magnitude of 0.35 mag, which is significantly larger than the typical uncertainties in the mean magnitude ( mag), larger than anticipated for a mono-metallicity population showing evolutionary effects, and larger than the intrinsic spread for a mono-metallicity population.

Figure 11: Left panel Metallicity distribution obtained from the PL in band for the RRLs of Sculptor (dark grey histogram). The black solid line show the Gaussian fit performed. The parameters of the fit are labeled in the panel. Right panel Metallicity distribution for the bright (light grey histogram) and faint RRL subsample. Both distributions have been fitted to a Gaussian (dashed curves). The parameters of each fit are labeled in the panel.

The -band PLZ (PLZ) can used to obtain the metallicity distribution for the RRLs following Martínez-Vázquez et al (2016a):

  1. A cross comparison of the photometric metallicities to the spectroscopic measurements from Clementini et al (for several dozen sources; 2005).

  2. Application of the technique to RRL in Reticulum, which has a well-defined spectroscopic metallicity (e.g., Mackey and Gilmore, 2004) and well-sampled light curves (Kuehn et al, 2013).

Both tests validated the photometric metallicities obtained using the PL and the results of this analysis for the Sculptor RRLs are given in Figure 11. In Figure 11, the left panel shows the full distribution for Sculptor and the right panel emphasizes the difference in the population split into the Bt and Ft groups identified from the -magnitudes.

The distribution of RRLs are centered at  dex, with 90% of the sample between and  dex (left panel of Figure 11). The Bt sample is on average more metal-poor, with mean metallicty of  dex, than the Ft sample that has a mean metallicity of  dex (right panel of Figure 11).

The distance to Sculptor was determined using the metal-independent PW relations for the , and , filter-color combinations (e.g., Equation 11 and 13) with parameters from Marconi et al (2015, their table 9) and Martínez-Vázquez et al (2015, their table 2), respectively. Marconi et al (2015) demonstrated that these PW relations are relatively insensitive to metallicity, which is important for the large metallicity spread in Sculptor. These PW relations were applied to each of three subsamples within the RRL population: (i) RRab, (ii) RRc, and (ii) the global sample (both RRab and fundamentalized RRc).

The zero-points of the PW relationships are not well constrained from observations as there are only good (<10%) trigonometric parallaxes for five RRL, even after the release of the TGAS parallaxes from Gaia DR1 (Lindegren et al, 2016, and references therein) Thus, three different approaches were used to quantify the effect of both the zero-point () and the slope () of the relation on the distances. These are:

  1. Theoretical: Using the zero-points and slopes of the predicted PW relations

  2. Semi-empirical: Using the same theoretical zero-points and the empirical slopes, which were obtained from the fit performed in the plane of both observational Wesenheit magnitudes versus (P) the RRLs.

  3. Empirical: Adopting observed slopes used in the previous method, but using the zero-points determined from HST trigonometric parallaxes (Benedict et al, 2011). More specifically, RR Lyr, itself, to calibrate both the RRab and the global samples and RZ Cep to calibrate the RRc sample.

The full details of these distance determinations can be found in Martínez-Vázquez et al (2015, in particular, section 4 and their table 3).

The zero-points are determined for each of the four RRab, global, and RRc PW relations. For the RRab and global relations, the zero-points agree quite well with theoretical ones. However, in the case of the RRc, the new zero-points are 30% smaller than predicted, which is in agreement with the mean RRc magnitude determined from Kollmeier et al (2013) from statistical parallax and highlights the need to use a large sample to determine robust zero points.

The final result is  mag with  mag, which is in good agreement with previous determinations (e.g., Rizzi, 2002; Pietrzyński et al, 2008, among others). The multiple methods required to set the zero-points and slopes of the PW relationships also emphasizes the role that high-quality trigonometric parallaxes for field RRLs, and later cluster RRLs, will play in improving the accuracy of the distances determined via the RRL PL, PLZ, and PW relationships.

2.3.2 A near-IR study of the RRLs in M4

We present a case study using the GGC M 4. M 4 is the closest GGC to the Sun and it hosts a sizable sample of RRLs (Stetson et al, 2014b), but the cluster suffers from strong differential extinction. The mean reddening is  mag with a range of  mag – or a mean () of 1.81 mag (1.37 mag), with variations at the 0.4 mag level across the cluster. In contrast, the IR reddening is  mag with a variation of only  mag – or a mean  mag with variation at the 0.1 mag level.

Figure 12: Optical and NIR CMDs for M 4. (a): Optical , CMD for the HB in M 4 (data from Stetson et al, 2014a). The RRab (circles), RRc (squares), and Blazhko candidate RRLs are indicated. (b): magnitude histogram. The dispersion of the distribution is 0.18 mag, which, despite the differential reddening, is much narrower than that of Sculptor (Figure 10) due to the lack of strong metallicity gradients. (c): NIR , CMD for the HB in M 4 (data from Stetson et al, 2014a). Unlike the optical (a), the NIR HB has a noticeable slant that gives rise to the PL. (d): magnitude histogram where the width is due to the PL. The y-axis for all panels is 2.0 mag for ease of comparison.

The data for M 4 is similar in its volume and time-baseline to that for the Sculptor dwarf from Section 2.3.1 and is described in full in Stetson et al (2014b). The optical data was obtained in 18 runs over 16 years for a total of over 5000 images in , , , , and filters. The NIR data was obtained in 18 runs over 10 years in , , and . The final RRL census in M 4 contains 32 RRab and 12 RRc for a total of 45 RRL.

Figure 12a shows the optical CMD for M 4 and Figure 12b is a histogram of the mean magnitudes in for the RRLs, both of which can be compared directly to the corresponding panels of Figure 10 showing the same for Sculptor. M 4 has a dramatically smaller metallicity spread than Sculptor, but it suffers from strong differential extinction in the optical, which creates a broadened distribution of RRL magnitudes. Figure 12c is the NIR CMD for M 4 and Figure 12d is a histogram of the mean magnitudes in for the RRLs.

Figure 13: NIR PL relations for RRL in M 4. Top: Empirical -band PL relations of the RRLs of M 4. Light circles represent RRcs, while dark squares mark the position of RRabs. Slopes () and dispersions () of the relations are labeled. Middle: same as top, but for -band PL relations. Bottom: same as top, but for -band PL relations

Period–luminosity relations

Figure 13 shows the empirical PL relations in the (top), (middle), and (bottom) bands for the RRab and RRc in M 4. The RRab and RRc RRL are shown independently in Figure 13 with their native periods. The resulting PL relations are given below. First, for the RRab stars:

(14)

For the RRc,

(15)

Lastly, the global PL uses fundamentalized periods for the RRc variables (e.g, Equation 8), which is not shown explicitly in Figure 13:

(16)

These slopes and their dispersions are also labeled on the top of each panel in Figure 13. As expected, the slope increases with wavelength as PL approaches the PR relation (e.g., Figure 8). The dispersion in (0.06 mag) is half that of the for this cluster (Braga et al, 2015), further emphasizing a strength of the NIR PL.

The distance to M 4 can be determined once the zero-point of the PL is calibrated. Similar to the methods for Sculptor, this can be done either on an empirical or on a theoretical basis, as we describe below:

  1. Empirical: Zero points are determined by adopting the HST parallaxes (Benedict et al, 2011) and the mean magnitudes (Sollima et al, 2008) for RR Lyr (an RRab). Note that the HST parallaxes have smaller uncertainties than those from TGAS in Gaia DR1 (Lindegren et al, 2016).

  2. Theoretical: Zero points are adopted from the predictions of the theoretical PLZ relations (Marconi et al, 2015). The metallicity term is taken into account by adopting a fixed [Fe/H] dex, which is the average iron abundance of M 4 (Marino et al, 2008; Malavolta et al, 2014).

The two methods provide distances of (standard error) (standard deviation) mag and (standard error)  mag (standard deviation), respectively (see Braga et al, 2015, for additional details). Both are in agreement within 1  with estimates of the distance modulus from the literature, in particular that derived from eclipsing binaries of  mag (1- agreement; Kaluzny et al, 2013).

Figure 14 shows two PW relationships in , and , for M 4 from Braga et al (2015) that combine both optical and NIR magnitudes. Using these relationships, calibrated in a similar fashion, provide  mag, in perfect agreement with the distance moduli determined from the PLs. For the two relationships shown in Figure 14, the dispersions are between 0.04 and 0.05 mag. The small dispersions and the agreement between these distance moduli and those from the NIR PL provide empirical validation that the PW relationships are indeed insensitive to reddening and that the assumptions of the PW relationship (e.g., “Cardelli’s Law”) work well for M 4 in the NIR.

Figure 14: Three-band PW relationships for RRL in M4. Top: Empirical PW(,) relations of the RRLs of M4. Light circles represent RRcs, while dark squares mark the position of RRabs. The dispersions () of the relations are labeled. Bottom: same as top, but for PW(,) relations

2.3.3 A MIR study of RR Lyrae in M4

Figure 15: MIR photometry for the HB in M 4. (a) Joint opt-MIR CMD for the HB in M 4 (data from Stetson et al, 2014a; Neeley et al, 2015) with the RRL common to both datasets indicated. (b) Distribution of [3.6]-mag for the RRLs in M 4. (c) MIR CMD for the HB in M 4 (data from Neeley et al, 2015) that demonstrates the HB is not readily identifiable in MIR CMDs. Identification and classification of RRLs is much more readily completed in the opt or NIR, but the RRc variables are noticeably fainter. (d) Histogram of [4.5] magnitudes for the RRLs in M 4.

The MIR case study also uses the closest GGC, M 4. Due to the smaller amplitudes at longer wavelengths, it is not practical to identify and derive the pulsation characteristics using only MIR data (e.g., Figure 2). Therefore we rely on ground-based optical and NIR data described previously to provide the coordinates and periods of our target RRL, which then sets the optimal strategy to derive well sampled light curves in the MIR (see also Hendel et al, 2018; Muraveva et al, 2018b).

Intensity mean magnitudes are computed either from Fourier fitting techniques (as in Dambis et al, 2014) or the GLOESS method (as in Monson et al, 2017). Using the exquisite multi-band data available for M 4 and described in this chapter, the extinction and distance to M 4 can be measured using a semi-empirical method, as described in Neeley et al (2017). The photometry was simultaneously fit in each band to theoretical PLZ relations, finding an extinction of (assuming measured directly in the foreground of M 4 by Hendricks et al, 2012) and a distance modulus of  mag. Both this value and the fully empirical results above are in good agreement with other distance methods, (e.g. eclipsing binaries from Kaluzny et al, 2013) as well as from RRL in the NIR (Braga et al, 2015).

The empirical PL relations using both Spitzer and WISE data are given in Figure 16. Periods of the RRab stars (shown as open circles) have been fundamentalized following Equation 8. The empirical MIR PL relations for M 4 for the “global sample” are as follows:

(17)

The dispersion on the IRAC relations is 0.05 mag and is approaching the expected intrinsic scatter of the PL relation (0.03 mag). The difficulty of deriving accurate photometry in clusters using WISE data (due to the larger PSF) results in a larger dispersion of 0.09 mag. The observed slopes are in good agreement with theoretical PL relations derived from pulsation models (see discussion in Neeley et al, 2017).

Figure 16: IRAC (left) and WISE (right) PL relations for RRL in M 4 using data from Neeley et al (2015) and Dambis et al (2014). Periods of the first overtone pulsators (open circles) have been fundamentalized (e.g., Equation 8).

To use the PL to determine a distance, the absolute zero point must be calibrated with RRL of known distances. Similar to the NIR study of M 4, there are two methods:

  1. Empirical : Adopting observed slopes from M 4 RRLs and zero points are determined using trigonometric parallaxes for Galactic RRLs (either from HST-FGS or Gaia).

  2. Semi-empirical: The distance and extinction are measured simultaneously by fitting multi-wavelength data to the theoretical PLZ relations (Marconi et al, 2015; Neeley et al, 2017).

The distance modulus of M 4 can be derived by comparing the empirical and calibrated PL relations, as was done in Neeley et al (2017). This results in  mag and  mag, which agrees with the Kaluzny et al (2013) and Braga et al (2015) measurements. Using the exquisite multi-band data available for M 4 and described in this chapter, the extinction to M 4 can be measured by fitting the distance and extinction law directly to each star as was described in Neeley et al (2015, 2017), finding  mag using measured directly in the foreground of M 4 by Hendricks et al (2012, taking into account M 4 is directly behind the  Oph cloud).

2.4 Summary

RRLs have been so far excellent standard candles (e.g., Tammann et al, 2008). RRL can be used in individual Galactic structures, extra-galactic systems, and are likely the only relatively-luminous standard candle that can be used to trace ultra-faint galaxies and tidal streams) that populate the Galactic halo (e.g., Fiorentino et al, 2015, 2017; Andrievsky et al, 2018). Although they are typically discovered and classified in the optical, photometry of RRL can be used to determine distances in a number of passbands, from the optical to the mid-infrared using theoretical and/or empirical calibrations of the RRL PLZ or the FOBE method.

On the theoretical side, current uncertainties in the PL relations come from our lack of perfect knowledge of the evolutionary effects (mixing length, turbulent convection, mass loss, helium abundance), of the input physics (atomic diffusion, extra-mixing, neutrino losses), and of the color-temperature transformations that are adopted to transform bolometric magnitudes and effective temperatures, into the observed magnitudes and colors. Given that theoretical relations are often preferred over empirical ones, these improvements are important.

As demonstrated by the case studies, the distances from the RRL are limited by the incomplete knowledge of the RRL PLZ (in all bands), in particular by the lack of a strong trigonometric parallax calibration. As such, single studies will often apply several calibrations to determine the distance. Studies using RRL are also limited by the imprecise knowledge of the metallicity for individual RRLs, although there are some forms of the RRL PLZ that have only a small dependence on metallicity.

The next generation of large telescopes will permit use of RRLs to distances of 5–6 Mpc (see e.g., Deep et al, 2011), which have previously only been accessible via Cepheids. The proposed adaptive optics systems for 30-meter class telescopes will be optimized for the NIR wavelengths, which necessitates sound calibration of the NIR PLZ and PWZ relations.

3 The type II Cepheids

3.1 Overview

Type II Cepheids (T2Cs) are evolved, low-mass stars (typically < 1 M) and have periods between 1 and 100 days, which roughly spans the range between RRLs and Miras (for the former, see the previous section and for the latter see Subramanian et al, 2017). The discovery of T2Cs played a critical role in the establishment of the modern extragalactic distance scale in the 1950s (formalized in the reviews of Baade, 1958b, c, at the seminal Vatican Conference). Before its revolutionizing discovery, the stars adopted to build the PL relation of Cepheids were a mixture of both (i) young, intermediate-mass stars characterized by a radial distribution typical of thin disk stars and (ii) old, low-mass stars with a radial distribution typical of the Galactic halo or bulge. The former are the classical Cepheids of Population-I, while the latter are known as Type-II Cepheids (T2Cs), to further emphasize the difference in age and stellar mass of their progenitors. By discovering the presence of two distinct classes of variables Baade effectively doubled the size (and the age) of the Universe (Baade, 1956; Fernie, 1969) by identifying and resolving a major issue in the determination of the Hubble constant at that time.

The discovery of T2Cs can be considered as one of three seminal discoveries that formed the modern distance ladder, joining (i) the measurement of stellar parallax by Bessell (1871) and (ii) the discovery of the Leavitt Law777The consensus of the scientific community present for the Thanks to Henrietta Leavitt Symposium held on Nov 8, 2008 was to officially adopt the term ‘Leavitt Law’ for the Cepheid PL relationship. The Council of the American Astronomical Society provided similar advice shortly thereafter. The reader is refferred to https://www.cfa.harvard.edu/events/2008/leavitt/ and https://aas.org/archives/Newsletter/Newsletter_146_2009_05_May_June.pdf, respectively. or PL, first noted by Leavitt (1908) and then quantified by Leavitt and Pickering (1912). For a review of the Leavitt Law, readers are referred to Subramanian et al (2017). After the separation of classical Cepheids and T2Cs, the latter have received far less attention than the former. Nevertheless, there are several good reviews on T2Cs and related objects which are recommended to interested readers: Harris (1985); Wallerstein and Cox (1984); Wallerstein (2002); Sandage and Tammann (2006); Welch (2012); Feast (2010, 2013); Catelan and Smith (2015).

Dating back to almost half a century ago (Wallerstein, 1970), T2Cs have been associated with GGCs characterized by a blue (hot) HB morphology. This means that the HB for these clusters is well populated on the blue tail, which is comprised of hot and extreme HB stars. Roughly one hundred of these variables have been detected in GGCs (Clement et al, 2001a). They have also been observed in stellar systems with complex star formation histories like the Magellanic Clouds (\al@Soszynski_2008,Soszynski_2010; \al@Soszynski_2008,Soszynski_2010, Groenewegen and Jurkovic, 2017a, b) and there are a preliminary identifications in the extragalactic stellar systems with examples being: IC 1613, M 31, M 33, M 106 and NGC 4603 (Majaess et al, 2009). On the other hand, Fornax is the only nearby dSph galaxy for which even a handful of T2Cs have been detected (Bersier and Wood, 2002); if these stars were a signpost for old metal-poor populations, then dSphs, which are home to many RRLs, would be expected to be hosts as well. The paucity of these objects in dSphs has been explained by their lack of blue HB morphologies. However, an observational bias cannot be excluded, since we still lack long-term photometric monitoring for the bulk of distant and diffuse stellar systems, although recent efforts have dramatically improved our variable star census (e.g., Stetson et al, 2014b, among others).

3.2 Observed pulsational properties

Example light curves for T2Cs for a wide span of periods are given in Figure 17 using data from OGLE (S08). The T2Cs show a striking degree of variation in comparison to the RRLs (Figure 2), because T2Cs are often divided into subpopulations. The period distribution from S08 is given in Figure 18a and shows three groupings by period. Considering this period distribution, S08 presented a classification scheme for the T2Cs in the LMC using the pulsation properties of the stars, including their amplitudes, light curve shapes, and periods. These pulsation properties are both distance and reddening independent and, thus, are good means for typing the stars. The three groups are divided into the classical sub-types with the following criteria:

  1. BL Herculis (or BL Her) with 1P4 days,

  2. W Virginis (or W Vir) with 4P20 days,

  3. RV Tauri (or RV Tau) with 20 days.

The period-amplitude diagram in the -band () for the LMC T2Cs is given in Figure 18b with the amplitudes ranging from 0.1 to 1 mag. Figure 18b further demonstrates that there is a set of structure in the period-amplitude diagram similar to what is seen in the Bailey diagram for RRLs (see e.g., Figure 9), albeit the T2C groups are less well defined than the RRLs. In addition to these classical groupings, S08 have also proposed a new group of T2Cs, the peculiar W Vir (or pW) stars known as pW variables, that display peculiar light curves and are brighter than typical T2Cs at a fixed period; these stars are indicated by open circles in Figure 18b. This classification scheme, however, has not been well settled, because the two minima of the period distribution that drive the definition of the sub-groups depends on the metal abundance and, likely, also on the environment. This means that the quoted period ranges will not universally define the three sub-groups and, thus, further investigations in a more diverse set of objects may yield further sophistication to our sub-classifications.

It is also important to note that different classification schemes have been adopted in different studies throughout the years. For example, General Catalog of Variable Stars888Available: http://www.sai.msu.su/gcvs/gcvs/ (GCVS, Samus et al, 2017) uses a boundary of 8 days to separate the T2Cs, ‘CWA’ for  days and ‘CWB’ for  days. In addition, GCVS uses the label ‘RV’ for RV Tau stars for radially pulsating supergiants characterized by the presence of alternating primary and secondary minima. Different schemes were proposed by Joy (1949) and later by Diethelm (1990) that take into account factors beyond the period, such as the light curve shape and spectroscopic features (see also Sandage and Tammann, 2006). Several authors have further suggested that the RV Tau stars are also heterogeneous as a class. While six objects in GGCs have been claimed to be RV Tau stars, some authors have doubted this classification from both the photometric (Zsoldos, 1998) and the spectroscopic point of view (Russell, 1998). The reader should be cautious, when exploring this topic, due to the complexities of these classifications.

Figure 17: Example optical, -band, light curves for T2C in the LMC from the OGLE survey (S08). From top to bottom there are two representative light curves for stars of each of the BL Her, W Vir, and RV Tau classes. The light curves have been normalized to amplitudes of 1 mag to allow comparison of the shapes and structures, with the minor-tick marks being 0.1 mag. The light curve shapes are easily distinguished from those of RRL with similar periods (e.g., Figure 2).
Figure 18: Basic properties of the T2Cs as defined in the LMC from the OGLE survey (S08). (a) Period distribution of the T2Cs as divided into the BL Her, W Vir, and RV Tau objects following the S08 classification scheme. (b) -band amplitude () versus period diagram (known as the Bailey diagram for the RRLs). The pentagons correspond to BL Her stars, the squares to W Vir stars, the open circles to peculiar W Vir objects, and triangles to RV Tau stars.
Figure 19: Predicted topology of the instability strip in the luminosity and effective temperature range typical of RRLs and T2Cs (Marconi et al, 2015). The the hot and cool edges of the strip are indicated for the FU mode (solid edges), FO mode (dashed edges), and their overlap region (“OR”; double hatched). Stars more luminous than the transition point (“TP”; see the text for definition) do not pulsate in the FO. Only in the region demarcated by the dashed edges can FO pulsators (RRc) attain a stable limit cycle. In the region labeled “OR”, variables can pulsate simultaneously in the FO and in the FU mode (RRd, or mixed-mode pulsators). The region bounded by the solid edges represent the regions where FU pulsators can attain a stable limit cycle. The luminosity for the bulk of the T2Cs excludes them from the FO region.

3.3 Theoretical pulsation predictions

There is a general consensus that T2Cs are in a post-HB phase with a C+O degenerate core with H-shell burning as the main energy source. As is well known, Zero-Age HB stars have a wide range of effective temperatures depending on the total mass (or the envelope mass). After the He exhaustion in the core, they start to climb up in the HR diagram evolving mainly into AGB, but this climb can begin at a range of effective temperatures on the HB. The evolutionary paths of the post-HB phase are predicted to be highly dependent on the ZAHB position. Stars coming from the blue HB tail that go through the IS at luminosities brighter than those of RRL, are found to be pulsating stars and commonly called T2Cs. Calculated evolutionary tracks were pioneered by Gingold (1976, 1977) with Smolec (2016) and Bono et al (2016) providing up-to-date theoretical calculations. However, we still lack a firm understanding of the production routes for T2Cs, in particular specific observational evidence for or against the postulated paths (though, we note that detailed studies such as Groenewegen and Jurkovic, 2017a, b, provide a solid starting point). An interesting question is, for example, the possibility of excursions, first suggested by Schwarzschild and Härm (1970), from the AGB to the hotter side reaching the instability strip. The early investigations Gingold (1976, 1977) suggested such routes (Gingold nose) play an important role on explaining a fraction of T2Cs, but such excursions are not expected to take place commonly if at all according to more recent calculations (see Smolec, 2016; Bono et al, 2016).

In contrast to the RRLs, theoretical investigations of the pulsation properties of T2Cs are quite limited. Early studies based on linear models were discussed by Wallerstein and Cox (1984) together with their period distribution, light- and velocity curves, and their location in the color magnitude diagram. Subsequently, a detailed analysis of T2Cs pulsation properties was performed by Fadeev and Fokin (1985) using non linear radiative models. In particular, Fadeev and Fokin (1985) investigated the modal stability and provided theoretical constraints on the PL relation for T2Cs, but their approach neglected the convective transport and, in turn, the physical mechanism that causes the quenching of radial oscillations. This means that this approach cannot predict the location of the cool (red) boundary of the IS. Using a similar approach, a thorough analysis of the limit cycle stability of T2Cs was presented by Kovacs and Buchler (1988). They have shown that the pulsation behavior changes from single periodic, to period doubling, and eventually to chaotic when increasing the pulsation period.

A more comprehensive theoretical scenario including a time-dependent convective transport was presented for BL Her stars by Bono et al (1997c); Marconi and Di Criscienzo (2007); Di Criscienzo et al (2007). In Bono et al (1997c), a systematic theoretical investigation of T2Cs has shown that fundamental pulsators are primarily expected among this class of variable stars. Moreover, they derived a period–luminosity–amplitude relation and found that it is independent of metallicity for the metal poor regime of that was explored in the study. These theoretical predictions match quite well the pulsation properties observed for GGCs.

Marconi and Di Criscienzo (2007) presented the full morphology of the theoretical IS for both fundamental and first overtone pulsators and a detailed atlas of both the light and radial velocity curves. Their models confirm that the FO IS is both narrow and limited to faint luminosities, providing new constraints on the transition point. The topology of the IS for the luminosity and effective temperature range of the RRLs and T2Cs is given in Figure 19. T2Cs populate the IS at luminosities brighter than the so-called transition point (Stellingwerf, 1979). The transition point corresponds to the luminosity at which the cool (red) edge of the FO IS matches the luminosity and the effective temperature of the hot (blue) edge of the FU. This means that FO pulsators do not attain a stable limit cycle for luminosities that are brighter and effective temperatures that are cooler than the transition point. In consequence, most T2Cs are fundamental pulsators. The first overtones, if any, are only permitted for the low luminosity or short period T2Cs. The empirical evidence, originally brought forward by McNamara (1995) and more recently by S08 and S10 for Galactic and Magellanic T2Cs, confirms that they primarily pulsate in the fundamental mode. There is theoretical evidence that some stability islands for the FO can appear at luminosities brighter than the transition point, but their true nature needs to be further clarified (Marconi et al, 2015). Furthermore, the FO IS tends to vanish in the metal-rich regime (e.g., [Fe/H]). Finally, using these same pulsation models and the evolutionary tracks from Pietrinferni et al (2004), Di Criscienzo et al (2007) derived analytical relations for the boundaries of the IS as a function of the adopted stellar parameters, as well as the PL and PW relations for these objects. They derived the T2C-based distances to GGCs and concluded that there is good agreement with the distance scale set by the RRLs.

Similarly to what happens for classical Cepheids, the PL relation for T2Cs can be derived using the fundamental pulsation relation given in Equation 1 (van Albada and Baker, 1971) valid for radial pulsators, together with the mass–luminosity relation typical of low-mass AGB evolutionary models (e.g., for BL Her stars, Di Criscienzo et al, 2007). Indeed, these AGB pulsators follow an inverse relation between stellar mass and luminosity, since lower mass pulsators are crossing the IS at a luminosity brighter than those with larger masses (Bono et al, 2016). The pulsation relation can then be averaged in temperature because the width of the IS is quite narrow (1500 K), originating a PL relation (see also Fadeev and Fokin, 1985; Matsunaga et al, 2006). Interestingly enough, the slope of the overall PL for T2Cs is less steep than that of classical Cepheids, which is also predicted by theoretical models (e.g., Bono et al, 1999; Di Criscienzo et al, 2007).

3.4 Optical and near-IR period–luminosity and period–Wesenheit relations

It has long been recognized that T2Cs follow, at least, a loose PL relation. From previous studies on T2Cs in GGCs, it is known that the T2Cs do follow a PL in . However, the relation has not been well established until recent work. While Harris (1985) and McNamara (1995) claimed that the optical PL slopes become steeper at around , Pritzl et al (2003) did not find such a feature for the T2Cs in NGC 6388 and NGC 6441 based on HST photometric data. As Pritzl et al (2003) claimed, many earlier studies for the T2Cs were based on magnitudes determined from photographic plates and, from these earlier works, it had been unclear if this class were useful as a distance indicator.

Breakthroughs were brought by modern photometric surveys in both the optical and the infrared. In particular, large-scale microlensing surveys that started in the late 1990s have systematically discovered T2Cs in the Magellanic Clouds. In the MACHO microlensing survey, Alcock et al (1998) determined that W Vir and RV Tau sub-types follow a PLC relation in the form of

(18)

with a relatively small scatter of 0.15 mag, which is identical (c.f. Bono and Marconi 1999) to a PW relation of the form,

(19)

The photometric analysis performed by S08 using optical data from the OGLE survey provided solid, empirical optical PL relations that support the theoretical predictions. The LMC sample is larger than the samples in GGCs and, because they are in the same system, can be studied as an ensemble without uncertainties from distances to the individual clusters. Figures 20a and 20b show the period-luminosity distribution for the T2Cs from S08 for the and bands, respectively; the BL Her stars are shown as filled hexagons, the W Vir as filled squares, and the RV Tau stars as filled triangles, while the pW stars are removed (following the classification by S08). The PL relations can be fit in three ways:

  1. First, a linear regression can be performed for the entire sample, which results in the following relations,

    (20)

    A linear regression over the entire period range shows a decrease of the dispersion from 0.6 to 0.4 mag when moving from the to the band, indicating that the scatter decreases for longer wavelengths.

  2. As is evident in Figures 20, the PL relations for T2Cs in these photometric bands are far from being linear and they could be better represented by a higher order fit, more specifically a polynomial quadratic in . Fitting this form of the PL to the S08 data results in the following fits,

    (21)

    This reduces the scatter by 0.1 mag, corresponding to improvements in distance precision by 5 %.

  3. As has been done in previous works, we also consider excluding RV Tau stars from the sample, because they show a larger spread in luminosity at fixed period for all wavelengths. Effectively, this is a cut in period for the S08 sample to those stars with  days. Fitting a linear relation over this period range results in the following,

    (22)

    However, it is noteworthy that the scatter of these relations and those of the quadratic fits for all three sub-types are similar. Exploration of additional systems with different metallicities and in different environments may help distinguish which of the relations is more representative for a larger range of properties.

Figure 20: The PL relations for T2Cs in the LMC: (a) -band and (b) -band PL relations. The S08 T2C classification is maintained, but the peculiar W Vir stars are removed. The three PLs discussed in the text are shown for both panels: (i) a linear global fit for all three sub-types (solid line; Equation 20), (ii) a quadratic global fit that better accounts for the structure of the RV Tau types (dashed line; Equation 21), and (iii) a linear fit just to the BL Her and W Vir type stars (solid line; Equation 22).

One has to keep in mind the presence of peculiar T2Cs that are scattered above the PL relation for BL Her and W Vir type T2Cs. They have some distinctive characteristics compared to W Vir stars (e.g., light curve shapes) but otherwise may be difficult to distinguish without long term monitoring. Soszyński et al (2017) further investigated the light curves of this pW group. A significant fraction of the pW stars identified by S08 and S10 were found to be in eclipsing binary systems. These objects are interesting in terms of evolutionary paths to produce T2Cs (and related objects), but to obtain accurate distances to stellar systems with T2Cs these atypical objects need to be identified and then excluded.

There is also evidence that T2Cs display well defined NIR PL relations that are linear over the entire period range. Matsunaga et al (2006) reported relations with a scatter of 0.15 mag in for BL Her to RV Tau stars in GGCs. Because they used RRL-based distances to individual GGCs to combine T2Cs onto the single relation, the well-defined relations suggest some consistency with the RRL distance scale. Ripepi et al (2015) found smaller dispersions in both the (0.13 mag) and (0.09 mag) for T2Cs in the LMC, further supporting the evidence that they are good distance indicators. Recent updates on the NIR relations based on a larger photometric sample can be found in Bhardwaj et al (2017a, b).

The slopes of these NIR PL relations do not change dramatically over the metallicity range spanned by these observations, as shown by Figure 21 that displays the values for PL relations obtained from GGCs (Matsunaga et al, 2006) and the LMC (Ripepi et al, 2015). Spectroscopic measurements are still too limited to draw firm conclusions (only a few dozen have measurements); those stars with metallicity measurements, however, span a range of metallicities similar to that of RRLs. The agreement becomes more stringent if we account for the theoretical NIR PL relations provided by Di Criscienzo et al (2007), which are shown in Figure 21 as the open circles. Note that a similar agreement is also found in a detailed comparison of the zero-points. Theoretical work, however, suggested a dependence of the zero-point on the metal content at the level of 0.04 to 0.06 mag dex. Observations, on the other hand, suggest a minimal, if any, dependence (Matsunaga et al, 2006). A different result is found when considering the T2Cs in the SMC (Ciechanowska et al, 2010); more specifically, the slopes are shallower with in the -band ( mag) and in the -band ( mag). However, the difference with the slopes shown in Figure 21 is within the associated 1  errors, and it is probably due to the high scatter of the mean NIR magnitudes derived from single epoch observations in Ciechanowska et al (2010). In addition, the NIR sample of T2Cs in Ciechanowska et al (2010) includes only 50% of the total optical sample for the SMC (40 objects, S10, ) and thus a more complete sample with more well-sampled light curves may resolve the discrepancy.

Figure 21: Observed slopes of NIR () PL relations for T2Cs in GGCs (xes, Matsunaga et al, 2006) and in the LMC (triangles, S08). The slopes predicted by (Di Criscienzo et al, 2007) are plotted as open circle. The vertical bars display the uncertainties on the slopes.

The use of empirical relations to determine the LMC distance employed by Ripepi et al (2015) returned a distance modulus in excellent agreement with RRLs, but with a difference of 0.1 mag with respect to the distance modulus based on classical Cepheids (most likely due to parallax sample used, see Section 5.2). Even though this is not a statistical significant difference (e.g., within the formal 1  uncertainty), it suggests a possible population bias in the LMC distances (Inno et al, 2016). Theoretical models for the BL Her period range are in extremely good agreement with the above evidence, since they predict dispersions in these passbands equal to  mag and  mag (Di Criscienzo et al, 2007).

Matsunaga et al (2006) also pointed out that the T2Cs and RRLs seem to follow the same, or continuous, PL relations in the infrared. The fact that RRLs and T2Cs follow very similar NIR PL relations (see also Feast, 2011) seems to be supported by the smooth and natural transition of the evolutionary properties of evolved RRLs and BL Her type stars. Although it was also discussed that BL Her and W Vir stars could have distinct NIR PL relations, recent findings based on accurate NIR mean magnitudes for LMC T2Cs suggest they are, within the errors, very similar (Ripepi et al, 2015, and references therein). This does not apply to RV Tau variables, because they typically display a larger spread compared to W Vir and BL Her stars. The increase in the spread might also mask a possible change in the slopes of the NIR PL relations (Ripepi et al, 2015). This occurrence appears to be associated with the properties of the circumstellar envelopes typical of these variables, which may also contribute to the larger scatter for this sub-type observed in the optical bands (see Figure 20).

PW relations are also commonly used to reduce the impact of extinction in distance determination and here we derive these for T2Cs based on the most conspicuous samples available in literature (see also Bhardwaj et al, 2017a).

  1. The most common Wesenheit magnitude in the optical takes the form of . Performing a linear regression to the BL Her and W Vir stars we find:

    (23)

    for the LMC (S08) and SMC (S10) data, respectively. To compare the relations on the absolute scale, we assume the consensus distance moduli,  mag (de Grijs et al, 2014, and references therein) and  mag (de Grijs and Bono, 2015, and references therein), respectively (see also de Grijs et al, 2017, in this series). Figure 22a compares these relations. It is worth noticing that when decreasing the host galaxy metallicity (from that of the LMC to that of the SMC), the slopes get steeper and the dispersion around the mean relation increases, although the difference of slopes is still within 1 .

  2. The most common NIR Wesenheit magnitude takes the form of . Performing a linear regression to the BL Her and W Vir type stars for the LMC (as given by Ripepi et al, 2015), SMC (IR magnitudes and periods from Ciechanowska et al, 2010) and GGCs (data from Matsunaga et al, 2006), we find the following PW relations:

    (24)

    Figure 22b compares NIR PW relations; the SMC relation is not included because the mean magnitudes have a large dispersion about the relation due to sparse sampling of the NIR light curve. As shown in Figure 22b, the slopes for LMC and GGCs are very similar, suggestive of only a mild dependence on the metal content. However, for the SMC we note that the slope decreases down to  mag dex, which is opposite to what happens in the optical regime. There are no doubts that a more detailed investigation of T2Cs in the SMC will be crucial to understand the universality of the NIR PW relations.

Figure 22: (a) Optical (,) PW relations for T2Cs in the LMC and SMC derived as described in the text using data are taken from (S08) and (S10), respectively. (b) NIR (,) PW relations for T2Cs. The LMC data is adopted from Ripepi et al (2015), whereas the for GGCs it has been derived using NIR magnitudes from Matsunaga et al (2006). The precise form of the relations is given in Equations 23 and 24 and described in the text.

3.5 T2Cs in context with RRLs and CCs

In this section we discuss the pulsation and evolutionary properties of T2Cs as compared to the most popular distance indicators, namely RRLs and classical Cepheids. The aim is to highlight their possible new role for these stars in the Gaia era that is expected to refine the extragalactic distance scale. In the following, we list their properties and assess their advantages and disadvantages compared to classical Cepheids and RRLs:

  1. NIR PL relation:  As discussed in the previous section, T2Cs follow well defined optical and NIR PL relations (Matsunaga et al, 2006, 2013). Moreover, there is evidence that their pulsation properties are either independent of, or minimally affected by, the metal abundance (Bono et al, 1997c; Di Criscienzo et al, 2007; Lemasle et al, 2015a). RRLs also follow linear NIR PL relations, however their luminosity (and luminosity range) is much smaller, which largely limits their application to within the Local Group (e.g., distances up to 1 Mpc). T2Cs are roughly 1.5 mag fainter than classical Cepheids at fixed period with the precise difference being period dependent. Moreover, the long period tail of classical Cepheids exceeds 100 days, while the long period tail of T2Cs approaches 70 days.

  2. Linearity:  The current empirical evidence indicates that BL Her and W Vir stars follow NIR PL relations that are linear over the entire period range. With current data, however, it is unclear if RV Tau objects follow a long period extension of the same PL relations, or whether their PL relation is distinct. Thus, RV Tau subgroup is typically neglected when fitting PL relations and in determining distances.

  3. Brightness:  T2Cs are brighter than RRLs and cover a similar magnitude range as Classical Cepheids. The difference with RRLs ranges from at least half magnitude ( mag) to roughly three magnitudes ( mag). This means that they can be employed to measure distances and trace old (blue HB) stellar populations beyond the Local Group.

  4. (Short) Evolutionary Lifetime:  T2Cs are typically approaching the AGB phase. This means that their evolutionary time scale is roughly two orders of magnitude faster than RRLs (Marconi et al, 2015; Bono et al, 2016). RRLs are low-mass stars during their core He burning phase, they have been identified in all stellar systems hosting stellar populations older than 10 Gyr, and from a theoretical point of view stellar mass and chemical abundance ranges that can give origin to RRLs are well known (Bono et al, 1996, 1997a, 1997b). The same is true for classical Cepheids, that are young core helium burning stars. The ratio of evolutionary times of classical Cepheids to the entire lifetime of their progenitors is longer than for T2Cs (Marconi et al, 2005a). This has a strong impact on the occurrence of T2Cs in a given system such that T2Cs are more rare than both RRLs and classical Cepheids. As shown in Figure 1a, the LMC contains 3,000 classical Cepheids, 20,000 RRLs, but only 200 T2Cs (Fiorentino and Monelli, 2012).

  5. Binary or Evolutionary Channel:  The fraction of T2Cs in binary systems is not well constrained. Very recently Soszyński et al (2017) and Soszyński et al (2018) presented additional monitoring of T2Cs in the LMC and in the Bulge that find new irregularities in the light curves consistent with binary companions. Maas et al (2002) present evidence that several RV Tau stars may be in binary systems. More importantly, there is photometric evidence that the pW stars may also be binaries (most recently Pilecki et al, 2017). This working hypothesis is also supported by spectroscopic evidence of binary companions for some T2Cs (Maas et al, 2007; Soszyński et al, 2010; Jurkovic et al, 2016) and, in turn, casts some doubts on the proposed small initial mass of their progenitor stars. These kind of objects have been dubbed Binary Evolution Pulsators (Karczmarek et al, 2016, 2017b). Long term monitoring of T2Cs can help identify their evolutionary path and determine if binary formation is the dominant formation path. Period changes can tell us the directions of evolution within the instability strip and this, in turn, provides additional evidence for the evolutionary paths of T2Cs (e.g, Diethelm, 1996; Wallerstein, 2002; Rabidoux et al, 2010).

  6. Shape of the light curve:  The coupling between shape of the light curve, e.g., the Fourier parameters and luminosity amplitudes, and the pulsation period is less distinctive for T2Cs than for RRLs (e.g., compare Figure 17 to Figure 2 for light curves and Figure 18 and Figure 9 for the amplitude-magnitude diagrams). They partially overlap with classical Cepheids (S08).

  7. Host stellar systems:  T2Cs have been identified in a variety of stellar systems hosting an old ( Gyr) stellar population, e.g., both early and late type stellar systems. Although this appears as a solid empirical evidence, we still lack firm identification of T2Cs in dSphs. The lack of T2Cs in dSphs can hardly be explained as an observational bias, since they are systematically brighter than RRLs and some of the classical dSphs have been surveyed for very long time scales (e.g., Stetson et al, 2014a; Coppola et al, 2015; Martínez-Vázquez et al, 2016b, among others). The working hypothesis suggested by Bono et al (2011, 2016) to explain the desert of T2Cs in dSphs is that they typically lack hot and extreme HB stars (Salaris et al, 2013). This means that the fraction of HB stars evolving from the blue (hot) to the red (cool) side of the CMD is small. This is relevant, given that the stellar populations of dSphs do cover the same metallicity range of GGCs that do have well extended blue tails on the HB. The lack of hot/extreme HB stars, and in turn of T2Cs, may be due to an environmental effect between dSphs and GGCs, that otherwise are composed of similar stellar populations.

On the whole, T2Cs have many enticing properties and show potential as distance indicators, but also still have some more shadowy areas left to be resolved before their application en masse.

3.6 Summary and final remarks

In this section, the observational and theoretical properties of T2Cs have been described. Although there is no doubt that they are radially pulsating stars crossing the IS, their evolutionary origin has not yet been completely addressed (e.g., the Binary Evolution Pulsator scenario, see Karczmarek et al, 2016, 2017a). Both theoretical and observational studies demonstrate that T2Cs mainly pulsate in the fundamental mode, given that they reach luminosities brighter than the transition point. The possible occurrence of short-period, fainter T2Cs evolving along the so-called blue hanger (Bono et al, 2011) cannot be excluded. However, they are expected to add little contamination to the fundamental mode NIR PL and PW relations.

T2Cs are largo sensu solid distance indicators, for they follow well defined optical and NIR PL relations, and both theoretical and empirical results suggest a minimal dependence on metal-abundance. Furthermore, they also follow similar optical and NIR PW relations. The key advantages of these diagnostics is to be independent of reddening uncertainties, but rely on the assumption that the reddening law is universal. This further suggests that T2Cs can be considered stricto sensu ideal distance indicators, given that they can provide very accurate relative distances, independent of uncertainties in the zero-point of the PL relations. The small metallicity dependence of the slopes of the NIR PW relations further supports the use of these variables as distance indicators for stellar systems affected by differential reddening (e.g., as with M 4 in Braga et al, 2015). To obtain the tighter PL relations, only BL Her and W Vir stars (i.e., those T2Cs with periods less than 20 days and fainter than  mag) are considered, excluding the brighter RV Tau objects. This means that one can detect and employ T2Cs as distance indicators out to 10 Mpc with HST (limiting magnitude  mag). However, because their periods have a wide distribution ranging from a few days to 20 days, complete photometric surveys with long temporal baseline have to be conducted in order to collect complete T2C samples.

The absolute distance scale of T2Cs, i.e., the zero point of the PL relation, is calibrated using the populations in the Magellanic Cloud and GGCs. There are only a handful of trigonometric parallax measurements for T2Cs and the uncertainties for the measurements that do exist make it difficult to establish a distance scale directly tied to parallax. The uncertainties affecting the zero-points of these relationships, and in turn, the absolute distances based on T2Cs, are going to be largely reduced during the next few years. This is thanks to the very accurate trigonometric parallaxes that will be provided by Gaia, not only for the nearest T2Cs, but for more than 100 field T2Cs and at least a dozen in nearby GGCs using end-of-mission predictions (Harris, 1985; Matsunaga et al, 2006).

The observational outlook concerning T2Cs appears even more promising if we take account of the fact that JWST will be capable of producing a complete census of T2Cs for Local Group and Local Volume galaxies. The halos of large galaxies are, indeed, marginally affected by crowding problems. Moreover, 30 meter-class telescopes will provide a unique opportunity to trace old stellar populations in the bulges and the innermost regions of galaxies in the Local Volume due to their unprecedented spatial resolution (Bono et al, 2017).

4 The Tip of the Red Giant Branch

The ultimate origin of the tip of the red giant branch (TRGB) as a standard candle is the canonical work of Baade (1944), who used red-sensitive plates to first resolve the stellar populations in early-type stellar systems, M 32, NGC 205 and the central region of M 31. From these observations developed the concept of the stellar classes of Pop I and Pop II. Baade also concluded that the brightest stars in these objects had similar magnitudes and colors. Sandage (1971) later pointed out that galaxies in the Local Group invariably contained a background sheet of red stars. Comparing the Cepheid distances to three Local Group galaxies, M 31, M 32, and IC 1613, he determined a mean absolute magnitude of the brightest red stars to be =-3.0 0.2 mag. This work led to the early applications using RGB stars for distance determination in the early eighties (see a summary in Lee et al, 1993).

A formal method to detect the TRGB in color-magnitude-diagrams (CMDs) of galaxies, a calibration and a thorough analysis of its uncertainties were first published by Lee et al (1993) and Madore and Freedman (1995), respectively. More specifically, Lee et al (1993) compared empirical RGB loci for 6 GGCs (Da Costa and Armandroff, 1990) with theoretical stellar evolutionary tracks (Yale Models, Green et al, 1987), and concluded that the -band absolute magnitude of the TRGB is accurate to  mag for resolved stellar systems of low metallicity (e.g., [Fe/H]<). Lee et al (1993) introduced Sobel “edge detection” kernels for detecting the discontinuity caused by the TRGB in -band luminosity functions and this methodology was rigorously tested by Madore and Freedman (1995). Since those works, the TRGB has been used extensively in Pop II dominated systems. With current observational capabilities the TRGB can be used to determine distances out to 20 Mpc (see e.g., Jang and Lee, 2015, 2017a; Jang et al, 2018, and references therein). Indeed, several authors have discussed the TRGB as an alternative to CCs for the basis of the extragalactic distance scale (e.g., Mould and Sakai, 2008; Beaton et al, 2016). So far, more than 400 galaxies have been measured their distances with the TRGB (Tully et al, 2016).

The TRGB as a precise distance indicator for resolved stellar systems has held against several observational tests (Lee et al, 1993; Ferrarese et al, 2000; Bellazzini et al, 2001; Sakai et al, 2004). Several studies have shown that the TRGB distance estimates to nearby galaxies agree well with those based on other precise distance indicators, such as CCs, RRLs, and Megamasers (Freedman, 1988; Lee et al, 1993; Sakai et al, 2004; Ferrarese et al, 2000; Tammann et al, 2008; Mager et al, 2008; Freedman and Madore, 2010; Jang and Lee, 2017b). It indicates that the precision of the TRGB is comparable to those of other primary distance indicators.

In this section, physical understanding of the TRGB is given in Section 4.1. Section 4.2 discusses empirical methods to measure the TRGB. A case study of using TRGB in the optical is described in Section 4.3. In Section 4.4, the IR properties of the TRGB are discussed. We conclude looking to future improvements of the method in Section 4.5.

4.1 Physical Description

The TRGB corresponds to the termination of the RGB evolution in old stellar populations, due to the onset of the He-flash in the electron degenerate cores of low-mass stars. For a given initial chemical composition, the bolometric luminosity of the TRGB is determined by the He-core mass at the He-flash. This mass is basically constant, e.g., within a few 0.001, for stellar masses that reach He-ignition at ages from 1.5-3 Gyr to the age of the Universe (the starting age is a function of the initial metallicity).

The use of the TRGB for distance estimates has been traditionally based on -band photometry, in the implicit assumption that the observed RGB of a galaxy has an age comparable to the age of GGCs, or in any case larger than 4-5 Gyr and metallicity [M/H] below 0.7 (see, e.g., Da Costa and Armandroff, 1990; Lee et al, 1993; Salaris et al, 2002). In this age range is roughly constant for a given [M/H], but becomes brighter with increasing [M/H]. Two effects influence in more metal rich stars: (i) increased efficiency of the hydrogen burning shell, which makes the stars brighter, and (ii) decreasing He-core mass, which makes the stars fainter; overall, is more luminous in metal-rich stars from these two effects. At the same time the effective temperature of the TRGB decreases with increasing [M/H] and the bolometric corrections, , decrease with decreasing . The resulting rate of decrease of with increasing metallicity very nearly matches the rate of increase of with [M/H]. Given that =, this implies that stays almost constant with [M/H] for the range between 2.0 and 0.7 and for ages older than 4-5 Gyr.

Figure 23: distribution for the TRGB from theoretical models from BaSTI (Pietrinferni et al, 2004). (a) The full range of ages (1.5< t < 14 Gyr) and metallicity (2.2 < [M/H]<0.3 dex) for the models and (b) only models with ages older than 4 Gyr. In both panels, the blue open circles are for [M/H] < 0.65 dex and the red filled circles are for [M/H] > 0.65 dex. The solid orange line represents the empirical calibration of =-4.05 mag (Lee et al, 1993; Salaris and Cassisi, 1997; Tammann and Reindl, 2013), with the shading giving the typical uncertainty of 0.10 mag. The ‘classical’ calibration is a reasonable approximation to the theory for stars in this age and metallicity range (corresponding to V-I from 1.0 to 2.5 mag).

From the variation of the absolute magnitude from theoretical models, it follows that applying a single-value for the absolute magnitude of the TRGB to galaxies with an extended star formation history can cause large systematic errors. Adopting a single value in a system assumes that the RGB stars in these galaxies are all older than 4-5 Gyr with [M/H]0.7, which can only be assumed safely in specific conditions. This effect has been studied theoretically by Barker et al (2004), Salaris and Girardi (2005), Cassisi and Salaris (2013), and also verified empirically by Górski et al (2016). A recent, detailed study is given in Serenelli et al (2017).

Figure 23 displays a theoretical calibration in Johnson-Cousins filters using the Pietrinferni et al (2004, BaSTI) stellar evolution models after correcting the TRGB bolometric magnitudes for the effect of including the updated electron conduction opacities by Cassisi et al (2007) and using the empirical bolometric corrections by Worthey and Lee (2011). In Figure 23 [M/H] lower(larger) than 0.65, are displayed with open(filled) circles, respectively. The BaSTI theoretical models cover a range of metallicity [M/H] between 2.2 and 0.3 and ages between 1.5 and 14 Gyr. The full age and metallicity range of the models is shown in Figure 23a, which demonstrates that decreasing age at constant [M/H] pushes to bluer colors, whereas increasing [M/H] at fixed age has the opposite effect. The smooth relation displays a maximum around 1.6, with the TRGB magnitude increasing steadily for redder colors.

Figure 23b shows TRGB magnitudes and colors for ages above 4 Gyr with [M/H]0.65 and [M/H]0.65 using the same symbols as Figure 23a compared with a ‘classical’ calibration (orange). This ‘classical’ calibration provides a median =4.05, with typical observational uncertainties of 0.10 mag over the range of from 1.0 to 2.5 mag (shown with the shading; Lee et al, 1993; Salaris and Cassisi, 1997; Tammann and Reindl, 2013). The theoretical behavior of vs in this age and metallicity range is approximately quadratic, but using a constant average value of is a decent approximation as is shown by the shading in Figure 23b.

Figure 24: Theoretical distribution at the TRGB for the full age range of the BaSTI models, but limited to sub-solar [M/H]. Two different sets of bolometric corrections are employed, more specifically those of Worthey and Lee (2011, blue circles) blue circles and of Gustafsson et al (2008, black squares). We also show the empirical estimate (Bellazzini et al, 2001) for  Centauri (orange triangle; Bellazzini et al, 2001) and the empirical relationship by Rizzi et al (2007) based on a sample of galaxies (red line). The range of the ‘classical’ calibration is shown in gray. The the zero point uncertainties are shown by the shading. See also Serenelli et al (2017).

It is evident that the relationship is very smooth for all [M/H], and irrespective of any age sub-selection, and can be used as distance indicator for any stellar population with an RGB. Figure 24 compares the theoretical relationship of Figure 23 (open circles) to one obtained using the same models, but applying the bolometric corrections by Gustafsson et al (2008, open squares). For these bolometric corrections we include the TRGB only up to [M/H] solar, because at super-solar metallicities TRGB models have negative log() (surface gravity) that are not covered by these bolometric corrections (additional discussion is given in Serenelli et al, 2017). We compare the theoretical relations to the empirical relationship determined by Rizzi et al (2007) on a sample of galaxies (solid red line) and the determination of the TRGB absolute magnitude for  Centauri, which has a broad range of metallicities and ages (Bellazzini et al, 2001) and the ‘classical’ calibration from Figure 23.

Rizzi et al (2007) calibration has roughly the same slope of the theoretical ones for above 1.6-1.7, but with a magnitude offset of 0.1 mag. The uncertainty for the  Centauri datapoint is large and does not put very strong constraints on the TRGB absolute magnitude. The two sets of bolometric corrections applied to the theoretical calculations provide similar results as long as is below 2.0. Increasingly larger discrepancies appear for redder colors.

Figure 25: Theoretical distribution at the TRGB for the full age and [M/H] range of the models, more specifically the panels are: (a) and (b) . The color coding matches that of Figure 24, more specifically open circles denote results obtained with the Worthey and Lee (2011) bolometric corrections, whilst open squares display results obtained with the Gustafsson et al (2008) bolometric corrections. Overall, the NIR-TRGB magnitude shows an approximately linear relationship with color and, compared to the optical (Figures 23 and 24), it more than 1 magnitude brighter, spans a range of 1 mag over this range of [M/H] and ages, but spans a much more narrow color-range. See also Serenelli et al (2017).

The panels of Figure 25 shows theoretical TRGB absolute magnitude-color calibrations for the infrared filters and (Figure  25a and  25b, respectively). Notice an almost linear behavior and again a smooth and tight correlation over the whole age and metallicity range covered by the models, with the added bonus of a lower sensitivity to reddening. Notice also how the dynamical range of the TRGB magnitude in is reduced compared to the case of the and bands.

The panels of Figure 25 also compare the results from the different bolometric corrections, as in Figure 24. The uncertainty on the bolometric corrections is at the moment the major drawback for theoretical calibrations in these filters. Notice the different overall slope of the two sets of results, that cause systematic differences of the TRGB absolute magnitudes at fixed color up to 0.2 mag.

Wu et al (2014) have also derived empirically relationships in the corresponding infrared filters of the WFC3-HST system, namely and following the same methods as Rizzi et al (2007). Figure 26 compares their TRGB relationship (red boxes) with the theoretical one based on Gustafsson et al (2008) bolometric corrections (filled circles). The overall shapes are different in the sense that the empirical calibration has a clear change of slope at 0.95 that is not seen in the theoretical counterpart. Overall differences of the TRGB absolute magnitude at fixed color are however typically within 0.05 mag.

Figure 26: Comparison of theoretical and empirical calibrations of the NIR-TRGB color-magnitude distribution for the HST+WFC3/IR filters F110W and F160W, which are similar to the 2MASS and . The filled black circles are the theoretical predictions from BaSTI using the Gustafsson et al (2008) bolometric corrections and the open squares are the empirical measurements from Wu et al (2014). The error bar on the zero point of the empirical calibration is also displayed.

4.2 Detecting the TRGB

In this subsection, we discuss the different techniques employed to determine the apparent magnitude of the TRGB, which is a sharp discontinuity along the observed luminosity function (LF). In the following, we discuss the major components of the TRGB detection process: edge-detection algorithms in Section 4.2.1, strategies used to account for the TRGB shape in Section 4.2.2, and brief caution regarding the overlap between the TRGB and other stellar sequences in Section 4.2.3.

Figure 27: Following Jang et al (2018, their figure 9), F814W luminosity functions (blue histograms) and edge-detection responses (red lines) applied to NGC 5011 C. Each panel applies a specific TRGB algorithm to the same underlying data, as follows: (a) Lee et al (1993), (b) Madore and Freedman (1995), (c) Sakai et al (1996), (d) Méndez et al (2002), (e) Mager et al (2008), (f) Madore et al (2009), (g) Jang and Lee (2017a), and (h) Hatt et al (2017). The histograms show modifications for smoothing as employed in each of the studies with y-axis labels of for binned starcounts and for smoothing. The TRGB is detected in each plot as the maximum of the response function, which is indicated by the dashed vertical line. Generally the algorithms agree for this dataset, but differences can occur as in Jang et al (2018).
Reference Derivative Approximation
Lee et al (1993)
Madore and Freedman (1995)
Sakai et al (1996)
Méndez et al (2002)
Mager et al (2008)
Madore et al (2009)
Jang and Lee (2017a) [–1, –2, –1, 0, +1, +2, +1]
Hatt et al (2017) [–1, 0, +1]
Table 2: Discrete Approximations to the First Derivative Used for TRGB Detection

4.2.1 Edge-Detection Techniques

Practically, the LF is constructed from binning the magnitudes of RGB stars from selection over an appropriate color range. The TRGB is detected by finding the point of greatest change in the LF, either using tools that approximate the first derivative or tools that model the components of the LF itself (e.g., individual stellar sequences). Poisson noise occurs in the LF, especially in smaller samples (like in GGCs), which in turn create spurious spikes in the edge-detection response. Thus, smoothing is generally used to mitigate this noise, which can either be (i) incorporated into the functional form of the edge-detection kernel, (ii) applied to the LF itself, or (iii) folded into the model. Edge detection techniques for the TRGB come in the following forms:

  1. discrete approximations to the derivative (e.g., the Sobel kernal),

  2. discrete approximations to the derivative that incorporate smoothing (e.g., a Gaussian formulation of the Sobel kernel),

  3. maximum-likelihood fitting techniques (e.g., fitting a functional form to the LF with the TRGB as a parameter).

These methods can applied LF that are either discrete (e.g., with large bins) or a “continuous” (e.g., with bins over very small intervals). Each method has advantages and disadvantages that should be weighed for the science application in question. As an example, simulations by Madore and Freedman (1995) demonstrated that at least 100 stars within one magnitude below the TRGB to detect unambiguously the TRGB for their algorithm and similar tests should be performed for any algorithmic approach to provide guidance for its application. Popular forms of the discrete derivative approximations are given in Table 2. Alternative parametric and non-parametric methods have been also applied to determine the TRGB magnitude in a number of stellar systems; the studies of Cioni et al (2000); Conn et al (2011); Makarov et al (2006) are representative examples.999Additional non-parametric studies are listed in Jang et al (2018).

Figure 27 compares applications of eight different forms for TRGB detection from the literature similar to the comparison in NGC 1365 made by Jang et al (2018, their figure 9). The same photometry for NGC 5011 C,a dwarf galaxy in the Centaurus A group (Tully, 2011), has been used in each panel. The histograms are slightly different owing to the prescriptions of the individual method; those that use direct binned LFs have labels and those that smooth and/or resample the LF have are labeled . Comparison of the LF histograms, show that they look relatively similar irregardless of the smoothing, thus differences in the edge-response (red) can be largely attributed to the formulation of edge-detection algorithm.

The edge-detection response function is shown in red in each panel of Figure 27. The form of the algorithm is given in Table 2. The Lee et al (1993), Madore and Freedman (1995), Jang and Lee (2017a), and Madore et al (2009) kernels are similar and are applied to binned LFs, except that additional elements are progressively added into the kernel that act to “smooth” the response; this is seen by comparing Figures 27a, 27b, 27g, and 27f that show a progressive smoothing of the edge-detection response. The algorithms for Sakai et al (1996), Méndez et al (2002), and Mager et al (2008) implement smoothing directly into the algorithm using various means of suppressing the noise; in particular, Sakai et al (1996) uses adaptive binning and both Méndez et al (2002) and Mager et al (2008) use a poison-noise weight applied to a logarithmic form. The impact of these forms can be seen as Figure 27c largely shows the same features as the previous discrete forms, but slightly broader and smoother and both Figure 27d and 27e show a slight amplification of the smaller scale features due both their weighting and logarithmic forms. Figure 27h that was proposed in Hatt et al (2017) and used subsequently by Jang et al (2018) and Hatt et al (2018) uses a Guassian smoothing function on the LF but applies the simple Sobel kernel from Lee et al (1993), which results in an unambiguous single peaked edge-response.

The magnitude bin of maximum response for each panel are not significantly different from each other (especially considering the discrete forms use magnitude bins of 0.05 mag): = 24.18, 24.03, 24.04, 24.02, 24.03, 24.03, 24.03, and 24.04 mag for panels (a) to (h), respectively. The total range of th detection is 0.16 mag or 8% in distance. The edge-response peak is asymmetric in many of the panels, but only in Lee et al (1993) (Figure 27a) is the maximum response in the secondary peak; excluding this result, the range is 0.02 mag or 1% variation in distance. This is a more consistent result than for NGC 1365, which was the example shown in Jang et al (2018). This LF showed a much larger degree of variation in the peak response for these same algorithms, albeit the authors ultimately concluded the results were not dissimilar within the statistical uncertainties. Thus, it is important to choose the algorithm carefully for a given application.

The uncertainties on the TRGB detection are particularly difficult to determine and there is a similar degree of variation in how this measurement is made as in the TRGB detection algorithm. The strategies incorporate one or more of the following elements: boot-strap or jack-knife resampling of the LF, the bin-size, the mean photometric uncertainty at the tip, the width of the response function, the width of the kernel, and the signal-to-noise of the detection. Certainly each of these elements play a role in the underlying uncertainty, but each contributes differently in terms of random or systematic elements. One particular uncertainty that is often overlooked is the start point of the binning itself; even with bins of 0.05 mag, the TRGB result will change as the bins are shifted small amounts.

To counter this, Hatt et al (2017) presents a detailed discussion of how the uncertainties, from the magnitudes and colors to the binning, are coupled in complex ways. As a result of this investigation, Hatt et al (2017) developed end-to-end simulations that insert an artificial TRGB into their images, photometer it, and measure the TRGB in statistical fashion. This technique combines many of the strategies listed, but adds an ability to measure both random and systematic uncertainty as the TRGB inserted into the images was known a priori. Using these simulations, they not only produce rigorous estimates of their uncertainties, but use the statistical distributions to select the best smoothing parameters for their LF. The Hatt et al methodology naturally incorporates the effects of incompleteness and crowding. This method has been applied thus far to five galaxies out to 20 Mpc distant with good results (Hatt et al, 2017; Jang et al, 2018; Hatt et al, 2018).

Figure 28: Comparison of optical TRGB calibrations. (a) Calibrations of M in terms of . The uncertainties on the zero points are 0.10 mag the span of which is shown by the gray shading, with the exception of With the exception of the QT calibration at 0.06 mag using multiple calibration paths (for details see Jang and Lee, 2017b). (b) Calibrations of M in terms of [Fe/H] or [M/H], the uncertainty on the zeropoint of B04 and B01 is 0.12 mag and for SC97, which is from theoretical models, see Section 4.1. The labels are as follows, in panel a: L93 – Lee et al (1993); R07 – Rizzi et al (2007), B08 – Bellazzini (2008), T: Madore et al (2009), and QT: Jang and Lee (2017b); and in panel b: SC97 – Salaris and Cassisi (1997); B01 – Bellazzini et al (2001), and B04 Bellazzini et al (2004).
Reference Calibration of
Color Based Calibrations:
Lee et al (1993) -4.0 0.1
Rizzi et al (2007) -4.05 + 0.217[() - 1.6] none given
Bellazzini (2008) -3.939 - 0.194( + 0.080() none given
Madore et al (2009) -4.05 + 0.2[() - 1.5] none given
Jang and Lee (2017a) -4.015 - 0.007[() - 1.5] +0.091[() - 1.5] 0.058
Metallicity Based Calibrations:
Salaris and Cassisi (1997) -3.732 + 0.588[M/H] +0.193[M/H] Theoretical
Bellazzini et al (2001) -3.66 + 0.48[Fe/H] + 0.14[Fe/H] 0.12
Bellazzini et al (2004) -3.629 + 0.679[M/H] + 0.258[M/H] 0.12
Table 3: Calibrations of the Shape of the OPT-TRGB

4.2.2 Techniques to “Sharpen” the Tip Edge

The absolute magnitude of the TRGB is often approximated to a single value: 4.05 0.10 mag (see, e.g., Lee et al, 1993; Salaris and Cassisi, 1997; Tammann and Reindl, 2013; Bellazzini et al, 2001, and references therein). However, as shown in Figures 23 and Figures 24, outside of the old, metal-poor regime, is not, in fact, constant. Fortunately, the age or metallicity effect is projected into the color of the star, e.g., more metal rich and/or younger stars are fainter and redder in , CMDs than old their metal-poor counterparts. To both boost the signal at the TRGB and to avoid spurious detections, many authors take this effect into consideration.

The two primary ways to counteract the shape of the TRGB are as follows:

  1. Magnitude-Color Calibration:  Instead of determining a single-valued calibration, the calibration can be measured as a function of the mean color known as the relation. Here, the TRGB color basically accounts for the star formation history of the observed population. Typically, users of this technique will use the mean color of the RGB sequence to to estimate for their system. An alternate formulation relates directly to the metallicity. Given that the metallicity has to be measured in a independently, preferably from spectroscopy, but often in reference to theoretical stellar tracks (isochrone fitting), this is not necessarily an advantage.

  2. T magnitude: Madore et al (2009) suggested that, instead of fitting a more complex zero-point, to calibrate the slope out of the color-magnitude data. Madore et al used theoretical tracks to produce an alternate magnitude system, known as the “T” for TRGB-magnitude system. This technique transforms the observed photometry into a system where the TRGB is flat – effectively torquing the CMD (a useful schematic can be found in figure 1 of Madore et al, 2009). Then, a single calibration to the metal-poor portion of the TRGB can be applied to the data over the full range. This can be done independent of priors on metallicity. The Madore et al (2009) formulation is as follows:

    (25)

    where is a fiducial color (preferably where the absolute zero point is determined) and was determined relative to theoretical models to be = 0.20 from Bellazzini et al (2001), but values ranging from 0.15 (Mager et al, 2008) to 0.22 (Rizzi et al, 2007) have been used.

  3. QT magnitude: More recently, Jang and Lee (2017b) expanded upon this initial formulation proposing the magnitude system, where the stands for “quadratic” because the functional form is a quadratic in color. The QT magnitude takes on the following form:

    (26)

    where = 0.159 0.010, = -0.047 0.020, and = 1.1 for the , magnitude-color combination (similar to the ground based , ). Jang and Lee (2017b) provide calibrations in several common HST filter combinations.

Table 3 summaries several of the color-magnitude and metallicity-magnitude calibrations of the optical TRGB and their uncertainties, if available. The calibrations are compared in Figure 28, where Figure 28a shows the - calibrations and Figure 28b shows the - calibrations. The common single-value from Lee et al (1993) and its approximate color-range is also shown for comparison and its uncertainty largely encompasses the span of the more complex forms for the blue-side of the RGB. Moreover, Hatt et al (2017) and Jang et al (2018) both demonstrated that there was no formal different between the TRGB detections for in native, T or QT magnitudes for metal-poor systems. As with the edge-detectors, the application of these different techniques depends largely on the situation; in particular, only when there are well measured colors, there are significant red populations, and the uncertainties due to the transform be estimated, is it advisable to adopt more complex forms of the absolute calibration.

4.2.3 Contamination to the LF

Unfortunately, over its color-magnitude span, the RGB is not the only stellar sequence. A particularly vexing containment are the asymptotic giant branch stars (AGB), that cover a magnitude range overlapping with the brightest portion of the RGB, and eventually, depending on the galaxy star formation history, can extend to much brighter magnitudes than the TRGB (for more description of the AGB see Habing and Olofsson, 2003; Rosenfield et al, 2014, 2016, and references therein). The AGB are particularly daunting because the RGB is their progenitor population and so, where there are strong RGB sequences, there is likely an AGB sequence, albeit at lower overall numbers.

The start of the AGB sequence for stars more luminous than the RGB can be confused for the TRGB. This can be seen in Figure 27, especially for the Méndez et al (2002) and Mager et al (2008) algorithms, there are small increases in the LF and edge-response at 1 magnitude brighter than the TRGB. While in the NGC 5011 C case the effect is small, it is not so in galaxies with more recent star formation. An example being attempts to detect the TRGB in the Antennae galaxy (NGC 4038/39) by Saviane et al (2004, 2008), who found (random) (systematic) mag. A later study, Schweizer et al (2008), took into account the AGB population and used fields outside of the main star forming disk to find (random) (systematic) mag, which has been independently confirmed by Jang and Lee (2015). The Saviane et al physical distance was 13.3 Mpc compared to Schweizer et al measuring 20 Mpc, which demonstrates the impact neglecting AGB stars can have.

Mitigation strategies for the AGB include: fitting the full form of the LF, including and AGB component, as in Méndez et al (2002, among others) or working in regions of galaxies where the liklihood of significant contamination is small like in stellar halos (see discussion in Beaton et al, 2016). It is worth mentioning, that in regions that are dominated by younger stellar populations (like in disks), other populations like red super giants or highly extincted yellow super giants (Cepheids), can also cause difficulty measuring the TRGB confidently (see, e.g., in the LMC and SMC in Figure 1). These latter populations show much more intrinsic variations as well as being embedded in dust (e.g. local changes in extinction) such that they are much more difficult to model. For the highest precision and accuracy, the TRGB is best applied to old population regimes.

4.3 Case Study for the Optical TRGB

Figure 29: Demonstration of the TRGB measurement process for the Local Group galaxy NGC 185 using HST+ACS data. (a) F814W - (F606W-F814W) CMD of resolved stars in its outer region ( > 4.4). The TRGB is marked by a dashed line. (b) F814W-band luminosity function of resolved stars (histogram) and corresponding edge-detection response (solid line). A strong edge-detection response is seen at the TRGB ( mag).

Here we present an example of the TRGB distance measurement for the Local Group dwarf elliptical galaxy, NGC 185 (E3 pec) using the dataset described in Geha et al (2015, Proposal ID:GO11724). Figure 29a shows the F814W(F606W-F814W) CMD (similar to ground based , ) for a halo field in NGC 185 and it is evident that the bulk of the bright stars are Pop II. The CMD, itself, shows multiple stellar populations, including the main sequence, blue stragglers, HB, AGB, and RGB, but few unambiguously Pop I sequences (e.g., compare to the LMC and SMC in Figures 1a and 1b). The TRGB is the brightest part of RGB sequence and is marked by a dashed line.

The apparent magnitude of the TRGB is determined from the I () luminosity function, which is given in Figure 29b. Because the TRGB is, effectively, the ending point of the RGB sequence, the LF should show a strong discontinuity at the TRGB. Here, we applied the edge-detection algorithm introduced in Mager et al (2008) and plotted the response in Figure 29b as a solid line. The edge-detection response shows a strong peak at  mag. Using bootstrap resampling of the LF and subsequent TRGB detection, the uncertainty on the TRGB measurement is  mag or a 1.5% statistical uncertainty for the TRGB measurement.

The distance modulus is then determined using a calibration of the TRGB absolute magnitude in the F814W filters and we adopt the Rizzi et al (2007) calibration given in Table 3. In this formulation, the age-metallicity behavior of the TRGB is modeled using the color of the RGB star, effectively fitting a tilted line to the TRGB absolute magnitude (shown in Figure 24). The median TRGB color in Figure 29a is  = 1.430.02 mag. Foreground extinction is estimated to be E(B-V)=0.162 mag, corresponding to and  mag (Schlafly and Finkbeiner, 2011). Combining the absolute magnitude of the TRGB () with the apparent magnitude of the TRGB (), we obtained a value for the distance modulus, mag.

This value is in good agreement with those based on red clump stars, mag (Pietrzyński et al, 2010), and RRLs,  mag (Salaris and Cassisi, 1997, 1998; Tammann et al, 2008). The distance summary for NGC 185 provided by de Grijs and Bono (2014) quotes the mean distance of 24.0270.333 mag from 26 studies for the TRGB, 23.9930.128 mag for the RR Lyrae from 8 studies, and 23.9970.119 mag for the full sample of distances.

Figure 30: The TRGB morphology as a function of wavelength for the Local Group dwarf galaxy NGC 6822. From left to right, the magnitudes in the , , , , , , , and bands are plotted against the